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1310.7208	# Ramsey numbers of ordered graphs††thanks: The first and the fourth author were supported by the grants SVV-2013-267313 (Discrete Models and Algorithms), GAUK 1262213 of the Grant Agency of Charles University and by the project CE- ITI (GAČR P202/12/G061) of the Czech Science Foundation. The fourth author was also supported by ERC Advanced Research Grant no 267165 (DISCONV). Part of the research was conducted during the DIMACS REU 2013 program. Martin Balko1 Josef Cibulka1 Karel Král2 Jan Kynčl1,3 ###### Abstract An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\operatorname{\overline{R}}(\mathcal{G};c)$ is the minimum number $N$ such that every ordered complete graph with $N$ vertices and with edges colored by $c$ colors contains a monochromatic copy of $\mathcal{G}$. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings $\mathcal{M}_{n}$ on $n$ vertices for which $\operatorname{\overline{R}}(\mathcal{M}_{n};2)$ is super-polynomial in $n$. This implies that ordered Ramsey numbers of the same graph can grow super-polynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number $R(\mathcal{G};2)$ is polynomial in the number of vertices of $\mathcal{G}$ if $\mathcal{G}$ has edges of constant length or if $\mathcal{G}$ is an ordered graph of constant degeneracy and constant interval chromatic number. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by Károlyi et al. 1 Department of Applied Mathematics and Institute for Theoretical Computer Science, Charles University, Faculty of Mathematics and Physics, Malostranské nám. 25, 118 00 Praha 1, Czech Republic; [email protected], [email protected], [email protected] 2 Department of Applied Mathematics, Charles University, Faculty of Mathematics and Physics, Malostranské nám. 25, 118 00 Praha 1, Czech Republic; [email protected] 3 Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13-15, Budapest 1053, Hungary ## 1 Introduction Ramsey’s theorem states that for every given graph $G$, every sufficiently large complete graph with edges colored by a constant number of colors contains a monochromatic copy of $G$. We study the analogue of Ramsey’s theorem for graphs with ordered vertex sets. The concept of ordered graphs appeared earlier in the literature [29, 30, 34, 36], but we are not aware of any Ramsey-type results for such graphs except for the case of monotone paths and hyperpaths [6, 15, 21, 34, 35]. The main goal of this paper is to understand the effects of different vertex orderings on the ordered Ramsey number of a given graph, and to compare the ordered and unordered Ramsey numbers. We state our results after introducing the necessary notation and presenting a few examples that provide the motivation. During the preparation of this paper, we learned that some of our results have been independently obtained by Conlon, Fox, Lee and Sudakov [13]. Throughout the paper, we omit the ceiling and floor signs whenever they are not crucial. Unless indicated otherwise, all logarithms in this paper are base 2. #### Hypergraphs In this paper, we consider only finite graphs and hypergraphs with no multiple edges and no loops (that is, one-element edges). We prove all our results only for graphs. An edge coloring of a hypergraph $H=(V,E)$, briefly a coloring, is a mapping $f\colon E\to C$ where $C$ is a finite set of colors. A coloring with $c$ colors is called a $c$-coloring. The complete $k$-uniform hypergraph on $n$ vertices, denoted by $K_{n}^{k}$, is a hypergraph whose edges are all $k$-element subsets of the $n$ vertices. An extended Ramsey’s theorem says that for given positive integers $c$, $k$ and $n$, if $N$ is sufficiently large, then every $c$-coloring of the edges of $K^{k}_{N}$ contains a monochromatic copy of $K_{n}^{k}$. The minimum such $N$ is called the Ramsey number and we denote it by $\operatorname{R}_{k}(K^{k}_{n};c)$. For graphs we write $\operatorname{R}(K_{n};c)$ instead of $\operatorname{R}_{2}(K_{n};c)$. Classical results of Erdős [16] and Erdős and Szekeres [17] give the exponential bounds $2^{n/2}\leq\operatorname{R}(K_{n};2)\leq 2^{2n}.$ (1) Despite many improvements during the last sixty years (see [11] for example), the constant factors in the exponents remain the same. Since every $k$-uniform hypergraph on $n$ vertices is contained in $K^{k}_{n}$, we can consider the following generalization of Ramsey numbers. Let $c$ be a positive integer and let ${H}_{1},\ldots,{H}_{c}$ be finite $k$-uniform hypergraphs. Ramsey’s theorem then implies that there exists the smallest number $\operatorname{R}_{k}({H}_{1},\ldots,{H}_{c})$ such that every $c$-coloring of edges of a complete $k$-uniform hypergraph with at least $\operatorname{R}_{k}({H}_{1},\ldots,{H}_{c})$ vertices contains a monochromatic copy of ${H}_{i}$ in color $i$ for some $i\in\\{1,2,.\ldots,c\\}$. If all the hypergraphs ${H}_{1},\ldots,{H}_{c}$ are isomorphic to ${H}$, we just write $\operatorname{R}_{k}({H};c)$. #### Ordered hypergraphs An ordered hypergraph $\mathcal{H}$ is a pair $({H},\prec)$ where $H$ is a hypergraph and $\prec$ is a total ordering of its vertex set. The ordering $\prec$ is called a vertex ordering. Many notions related to hypergraphs, such as vertex degrees or a coloring, can be defined analogously for ordered hypergraphs. In addition, we introduce a few more notions specific to ordered hypergraphs. For an ordered hypergraph $\mathcal{H}=(H,\prec)$ and its vertices $x,y$, we say that $y$ is a left neighbor of $x$ and that $x$ is a right neighbor of $y$ if $x$ and $y$ belong to a common edge and $y\prec x$. We say that two ordered hypergraphs $(H_{1},\prec_{1})$ and $(H_{2},\prec_{2})$ are isomorphic if $H_{1}$ and $H_{2}$ are isomorphic via a one-to-one mapping $g\colon V(H_{1})\to V(H_{2})$ that also preserves the orderings; that is, for every $x,y\in V(H_{1})$, $x\prec_{1}y\Leftrightarrow g(x)\prec_{2}g(y)$. An ordered (hyper)graph $\mathcal{H}=({H},\prec_{1})$ is an ordered sub(hyper)graph of $\mathcal{G}=({G},\prec_{2})$, written $\mathcal{H}\subseteq\mathcal{G}$, if ${H}$ is a sub(hyper)graph of ${G}$ and $\prec_{1}$ is a suborder of $\prec_{2}$. We now introduce Ramsey numbers of ordered hypergraphs. For given ordered $k$-uniform hypergraphs $\mathcal{H}_{1},\ldots,\mathcal{H}_{c}$, we denote by $\operatorname{\overline{R}}_{k}(\mathcal{H}_{1},\ldots,\mathcal{H}_{c})$ the smallest number $N$ such that every $c$-coloring of the edges of the complete ordered $k$-uniform hypergraph with $N$ vertices contains, for some $i$, a monochromatic copy of $\mathcal{H}_{i}$ in color $i$ as an ordered subhypergraph. If all $\mathcal{H}_{i}$ are isomorphic to $\mathcal{H}$, we write the ordered Ramsey number as $\operatorname{\overline{R}}_{k}(\mathcal{H};c)$. In the case of graphs (that is, if $k=2$) we write $\operatorname{\overline{R}}(\mathcal{H}_{1},\allowbreak\ldots,\allowbreak\mathcal{H}_{c})$ or $\operatorname{\overline{R}}(\mathcal{H};c)$, respectively. If a coloring $f$ of a hypergraph $\mathcal{G}$ contains no monochromatic copy of $\mathcal{H}$, we say that $f$ avoids $\mathcal{H}$. Up to isomorphism, there is only one ordered complete $k$-uniform hypergraph on $n$ vertices, which we denote as $\mathcal{K}_{n}^{k}$, or $\mathcal{K}_{n}$ if $k=2$. Therefore, $\operatorname{\overline{R}}_{k}(\mathcal{K}^{k}_{r_{1}},\ldots,\mathcal{K}^{k}_{r_{c}})=\operatorname{R}_{k}(K^{k}_{r_{1}},\ldots,K^{k}_{r_{c}})$ for arbitrary positive integers $k,c,r_{1},\ldots,r_{c}$. Since every ordered $k$-uniform hypergraph on $r$ vertices is an ordered subhypergraph of $\mathcal{K}^{k}_{r}$, we get $\operatorname{\overline{R}}_{k}(\mathcal{H}_{1},\ldots,\mathcal{H}_{c})\leq\operatorname{\overline{R}}_{k}(\mathcal{K}^{k}_{r_{1}},\ldots,\mathcal{K}^{k}_{r_{c}})$ where $r_{i}$ is the number of vertices of $\mathcal{H}_{i}$. We have thus proved the following fact. ###### Observation 1. Let $c$ and $k$ be arbitrary positive integers and let $\mathcal{H}_{1}=(H_{1},\prec_{1}),\ldots,\mathcal{H}_{c}=(H_{c},\prec_{c})$ be an arbitrary collection of ordered $k$-uniform hypergraphs. Then we have $\operatorname{R}_{k}(H_{1},\ldots,H_{c})\leq\operatorname{\overline{R}}_{k}(\mathcal{H}_{1},\ldots,\mathcal{H}_{c})\leq\operatorname{R}_{k}\left(K^{k}_{\|V(H_{1})\|},\ldots,K^{k}_{\|V(H_{c})\|}\right).$ To study the asymptotic growth of ordered Ramsey numbers, we introduce ordering schemes for some classes of hypergraphs. An ordering scheme is a particular rule for ordering the vertices of the hypergraphs consistently in the whole class. For example, a $k$-uniform monotone hyperpath $(P^{k}_{n},\lhd_{mon})$ is a $k$-uniform hypergraph with vertices $v_{1}\lhd_{mon}\ldots\lhd_{mon}v_{n}$ and $n-k+1$ edges, each consisting of $k$ consecutive vertices; see Figure 1 for an example. Throughout the paper we use a symbol $\lhd$ instead of $\prec$ to emphasize the fact that the vertex ordering follows some ordering scheme. Figure 1: Examples of $2$-uniform and $3$-uniform monotone hyperpaths on seven vertices. For an ordered graph $(G,\prec)$, we say that a vertex $v$ of $G$ is to the left (right, respectively) of a subset $U$ of vertices of $G$ if $v$ precedes (is preceded by, respectively) every vertex of $U$ in $\prec$. More generally, for two subsets $U$ and $W$ of vertices of $G$, we say that $U$ is to the left of $W$ and $W$ is to the right of $U$ if every vertex of $U$ precedes every vertex of $W$ in $\prec$. For an ordered graph $(G,\prec)$, we say that a subset $I$ of vertices of $G$ is an interval if for every pair of vertices $u$ and $v$ of $I$, $u\prec v$, every vertex $w$ of $G$ satisfying $u\prec w\prec v$ is contained in $I$. ### 1.1 Motivation In this subsection we show various examples in which Ramsey-type problems on ordered hypergraphs appear. The examples consist of both classical and recent results. Some results and definitions are also used later in the paper. #### Erdős–Szekeres lemma This well-known statement says that for a given positive integer $k$ one can find a decreasing or increasing subsequence of length $k$ in every sequence of at least $(k-1)^{2}+1$ distinct integers. It is easy to see that this bound is sharp. The Erdős–Szekeres lemma has many proofs [38] and it is also a special case of a more general Ramsey-type result for ordered graphs. Given a sequence $S=(s_{1},\ldots,s_{N})$ of integers we construct an ordered graph $(K_{N},\prec)$ with vertex set $S$ and the ordering of the vertices given by their positions in $S$. That is, for $s_{i},s_{j}\in S$ we have $s_{i}\prec s_{j}$ if $i<j$. Then we color an edge $\\{s_{i},s_{j}\\}$ with $i<j$ red if $s_{i}<s_{j}$ and blue otherwise. Afterwards, red monotone paths correspond to increasing subsequences of $S$ and blue monotone paths to decreasing subsequences of $S$. The lemma now follows from the following result by Choudum and Ponnusamy [6] (see Milans et al. [34] for a proof in the language of ordered Ramsey theory). ###### Proposition 2 ([6]). For arbitrary monotone ordered paths $(P_{r_{1}},\lhd_{mon}),\ldots,(P_{r_{c}},\lhd_{mon})$, we have $\operatorname{\overline{R}}((P_{r_{1}},\lhd_{mon}),\ldots,(P_{r_{c}},\lhd_{mon}))=1+\prod_{i=1}^{c}(r_{i}-1)$. #### Integer partitions Ramsey numbers of monotone hyperpaths recently attracted the attention of many researchers [15, 21, 34, 35]. Moshkovitz and Shapira [35] discovered a connection between Ramsey numbers of monotone hyperpaths and high-dimensional integer partitions. A $d$-dimensional partition is a $d$-dimensional (hyper)matrix $A$ of nonnegative integers such that $A$ is decreasing in each line. That is, $A_{i_{1},\ldots,i_{t},\ldots,i_{d}}\geq A_{i_{1},\ldots,i_{t}+1,\ldots,i_{d}}$ for every possible $i_{1},\ldots,i_{d}$ and $1\leq t\leq d$. For example, a $1$-dimensional partition is just a non- increasing sequence of nonnegative integers $a_{1}\geq a_{2}\geq\ldots$, and it is also called a line partition. Let $P_{d}(n)$ denote the number of $n\times n$ $d$-dimensional partitions with entries from $\\{0,\ldots,n\\}$. Observe that $P_{1}(n)={2n\choose n}$, since we can represent a line partition as a lattice path in $\mathbb{Z}^{2}$ starting at $(0,n)$, ending at $(n,0)$, and going only right or down in each step. ###### Theorem 3 ([35]). For every $c\geq 2$ and $n\geq 2$ we have $\operatorname{\overline{R}}_{3}((P^{3}_{n},\lhd_{mon});c)=P_{c-1}(n-2)+1$. #### Erdős–Szekeres theorem The Erdős–Szekeres theorem was one of the earliest results that contributed to the development of Ramsey theory. A finite set of points in the plane is in general position if no three of the points are collinear, and in convex position if the points form a vertex set of a convex polygon. ###### Theorem 4 ([17]). For every $k\in\mathbb{N}$ there is a finite number $\operatorname{ES}(k)$ such that every set of at least $\operatorname{ES}(k)$ points in the plane in general position contains $k$ points in convex position. As noted by Erdős and Szekeres, this result can be proved using Ramsey theorem for (unordered) 4-uniform hypergraphs. However the upper bound for $\operatorname{ES}(k)$ obtained by this approach is astronomically large. In their original paper Erdős and Szekeres proved a more reasonable bound $\operatorname{ES}(k)\leq{{2k-4}\choose{k-2}}+1$. Moshkovitz and Shapira showed [35] that the same bound can be derived from Theorem 3 with $c=2$ in the following way. Suppose that we have a set $S\subset\mathbb{R}^{2}$ of $N\geq\operatorname{ES}(k)$ points in general position. Let $(K^{3}_{N},\prec)$ be an ordered 3-uniform hypergraph with vertex set $S$ where the ordering of the vertices is given by their $x$-coordinates. An edge $\\{x,y,z\\}$ is then colored red if the triangle $xyz$ is oriented counterclockwise and blue otherwise. In this coloring a monochromatic monotone 3-uniform path corresponds to a (special) convex $k$-gon in $S$ (sometimes called a $k$-cup or a $k$-cap). Theorem 3 gives us the upper bound $\operatorname{ES}(k)\leq{{2k-4}\choose{k-2}}+1$. #### Discrete geometry In some Ramsey-type problems of discrete geometry, ordered Ramsey numbers arise quite naturally, as we illustrate on the example of geometric Ramsey numbers [10, 27, 28]. The authors of this paper arrived at studying Ramsey numbers of ordered graphs while working on a variant of the Erdős-Szekeres problem for so called 2-page drawings of $K_{n}$ [7]. 111A $2$-page drawing of $K_{n}$ is a drawing where vertices of $K_{n}$ are placed on a horizontal line (the _spine_), edges between consecutive vertices are subsegments of the line and each of the remaining edges goes either above or below the line. The edges going above the line are _red_ and the edges going below are _blue_. Let $v_{1},\ldots,v_{n}$ be the vertices in the order in which they occur on the line. Analogously to the above mentioned proof of Theorem 4, a triple $v_{i}$, $v_{j}$, $v_{k}$, where $i<j<k$ is colored red if $v_{j}$ lies below the edge $v_{i}v_{k}$ and blue otherwise. The color of $v_{i}v_{j}v_{k}$ is thus equal to the color of $v_{i}v_{k}$. A monochromatic monotone $3$-uniform path corresponds to a monochromatic ($2$-uniform) ordered graph with pairs vertices connected if and only if they are at distance $2$. Although this direction of research did not give any interesting results on $2$-page drawings, it lead to Theorem 10. For a finite set of points $P\subset\mathbb{R}^{2}$ in general position (no three points are collinear), we denote as $K_{P}$ the complete geometric graph on $P$ which is a complete graph with vertex set $P$ whose edges are straight- line segments between pairs of points of $P$. The graph $K_{P}$ is convex if the points from $P$ are in convex position. The geometric Ramsey number of $G$, denoted by $\operatorname{Rg}(G)$, is the smallest integer $N$ such that every complete geometric graph $K_{P}$ on $N$ vertices with edges colored by two colors contains a monochromatic non-crossing copy of $G$. If we consider only convex complete geometric graphs $K_{P}$ in the definition, then we get so called convex geometric Ramsey number $\operatorname{Rc}(G)$. Note that these numbers are finite only if $G$ is outerplanar and that we have $\operatorname{Rc}(G)\leq\operatorname{Rg}(G)$ for every outerplanar graph $G$. For the cycles $C_{n}$, $n\geq 3$, Károlyi et al. [28] showed an upper bound $\operatorname{Rg}(C_{n})\leq 2n^{2}-6n+6=2(n-2)(n-1)+2$ and also observed that $\operatorname{Rc}(C_{n})\geq(n-1)^{2}+1$ holds. A monotone cycle $(C_{n},\lhd_{mon})$ on $n$ vertices consists of a monotone path on vertices $v_{1}\lhd_{mon}\ldots\lhd_{mon}v_{n}$ and the edge $\\{v_{1},v_{n}\\}$. See Figure 2 for an example. Later, we show that the geometric and convex geometric Ramsey numbers of cycles are equal to ordered Ramsey numbers of monotone cycles. Here, we prove the equivalence of the ordered and convex geometric Ramsey numbers of cycles. Figure 2: The monotone cycle $(C_{6},\lhd_{mon})$. ###### Observation 5. For every integer $n\geq 3$, we have $\operatorname{Rc}(C_{n})=\operatorname{\overline{R}}((C_{n},\lhd_{mon});2)$. ###### Proof. Consider a set of $n$ points in convex position. Order the points $v_{1}\prec\cdots\prec v_{n}$ in the clockwise (or counterclockwise) order starting at an arbitrary vertex. The observation follows from the fact that the cycle $C_{n}$ on $v_{1},\ldots,v_{n}$ is non-crossing if and only if it is the monotone cycle. ∎ #### Extremal problems on matrices The last motivation example shows a connection between extremal theory of $\\{0,1\\}$-matrices (see [9, 22], for example) and ordered Turán numbers of ordered bipartite graphs. We use some results from this area in Section 2.2. A $\\{0,1\\}$-matrix $A$ contains an $r\times s$ submatrix $M$ if $A$ contains a submatrix $M$ which has ones on all the positions where $M$ does. A matrix $A$ avoids $M$ if it does not contain $M$. The extremal function of $M$ is the maximum number $\operatorname{ex}_{M}(m,n)$ of 1-entries in an $m\times n$ $\\{0,1\\}$-matrix avoiding $M$. Let $K_{n_{1},\ldots,n_{p}}$ denote the complete $p$-partite graph with color classes of size $n_{1},\ldots,n_{p}$. In the case $n_{1}=\cdots=n_{p}=n$, we simply write $K_{n}(p)$. We use $\mathcal{K}_{n_{1},\ldots,n_{p}}$ to denote the ordering of $K_{n_{1},\ldots,n_{p}}$ in which the color classes form consecutive disjoint intervals such that the interval of size $n_{i}$ is the $i$th such interval. Note that the ordering of indices $n_{1},\ldots,n_{p}$ in $\mathcal{K}_{n_{1}\ldots,n_{p}}$ is important. Again, if all the sizes $n_{1},\ldots,n_{p}$ are equal to $n$, we write $\mathcal{K}_{n}(p)$ instead. See Figure 3 for an example. Let $\mathcal{G}$ and $\mathcal{H}$ be ordered graphs. The Turán number of $\mathcal{G}$ in $\mathcal{H}$ is the maximum number of edges in a subgraph $\mathcal{H^{\prime}}$ of $\mathcal{H}$ such that $\mathcal{G}$ is not a subgraph of $\mathcal{H^{\prime}}$. Let $\mathcal{G}=((A\cup B,E),\prec)$, $\|A\|=r$ and $\|B\|=s$, be a a subgraph of $\mathcal{K}_{r,s}$. Then $\mathcal{G}$ corresponds to an $r\times s$ $\\{0,1\\}$-matrix $M(\mathcal{G})$ where $M(\mathcal{G})_{i,j}=1$ if the $i$th vertex in $A$ and the $j$th vertex in $B$ are adjacent and 0 otherwise. It is easy to see that the Turán number of $\mathcal{G}$ in $\mathcal{K}_{m,n}$ is exactly the value of $\operatorname{ex}_{M(\mathcal{G})}(m,n)$. Figure 3: The ordered complete bipartite graph $\mathcal{K}_{4,3}$ with distinguished color classes. ### 1.2 Our results We are interested in the effects of vertex orderings on Ramsey numbers of various classes of graphs. As can be seen in the motivation examples, for an ordering $\mathcal{G}$ of a graph $G$ we can get asymptotic difference between $\operatorname{R}(G;c)$ and $\operatorname{\overline{R}}(\mathcal{G};c)$. For example, Proposition 2 implies that $\operatorname{R}(P_{n};c)$ is linear in $n$ while $\operatorname{\overline{R}}((P_{n},\lhd_{mon});c)$ is quadratic. Even a larger gap can be obtained for hypergraphs. Let $t_{h}$ denote a tower function of height $h$ defined by $t_{1}(x)=x$ and $t_{h}(x)=2^{t_{h-1}(x)}$. It is known that Ramsey numbers $\operatorname{R}_{k}({H};2)$ of sparse unordered $k$-uniform hypergraphs ${H}$ are linear with respect to the number of vertices ${H}$. Formally, for positive integers $\Delta$ and $k$, there exists a constant $C(\Delta,k)$ such that if ${H}$ is a $k$-uniform hypergraph with $n$ vertices and maximum degree $\Delta$, then $\operatorname{R}_{k}(H;2)\leq C(\Delta,k)n$ [12]. On the other hand, Moshkovitz and Shapira [35] showed that for every $k\geq 3$ we have $\operatorname{\overline{R}}_{k}((P^{k}_{n},\lhd_{mon});2)=t_{k-1}((2-o(1))n)$ where the $o(1)$ term goes to zero with $n$. Thus there are $k$-uniform hypergraphs $H$ and their orderings $\mathcal{H}$ such that $\operatorname{\overline{R}}_{k}(\mathcal{H};2)$ grow as a tower of height $k-1$ in $n$ while $\operatorname{R}_{k}(H;2)$ remain linear in $n$. In the first part of this paper we derive bounds for Ramsey numbers of specific classes of ordered graphs: stars, paths and cycles. First, we show that Ramsey numbers of all ordered stars are linear with respect to the number of vertices. ###### Theorem 6. For positive integers $c$ and $r_{1},\ldots,r_{c}$ and for a collection of ordered stars $\mathcal{S}_{1},\ldots,\mathcal{S}_{c}$ where $r_{i}$ is the number of vertices of $\mathcal{S}_{i}$ there is a constant $C=C(c)$ such that $\operatorname{\overline{R}}(\mathcal{S}_{1},\ldots,\mathcal{S}_{c})\leq C\cdot\max\\{r_{1},\ldots,r_{c}\\}.$ Considering the multi-colored case, we find a graphs $G$ and their vertex orderings $\lhd$ and $\lhd^{\prime}$ such that the Ramsey numbers of $(G,\lhd)$ and $(G,\lhd^{\prime})$ differ exponentially in the number of colors (Proposition 15). In Section 2.2 we show an ordering of the path $P_{n}$ whose ordered Ramsey number is linear in $n$ (Proposition 18). In Section 2.3 we discuss ordered cycles. First, we show Ramsey numbers for all possible orderings of $C_{4}$ (Proposition 21). Then we derive an exact formula for ordered Ramsey numbers of monotone cycles. ###### Theorem 7. For integers $r\geq 2$ and $s\geq 2$ we have $\operatorname{\overline{R}}((C_{r},\lhd_{mon}),(C_{s},\lhd_{mon}))=2rs-3r-3s+6.$ As a consequence of this theorem we obtain tight bounds for so called geometric and convex geometric Ramsey numbers of cycles which were introduced by Károlyi et al. [27, 28], see Corollary 23. The second part of the paper contains general bounds for ordered Ramsey numbers. We use a standard existence argument to find a lower bound for ordered Ramsey numbers of every ordered graph $\mathcal{G}$ which depends on the number of edges of $\mathcal{G}$. See Proposition 24 which implies the following statement. ###### Proposition 8. Let $c\geq 2$ be a positive integer and let $\mathcal{G}$ be an ordered graph with $n$ vertices and $n^{1+\varepsilon}$ edges for some $\varepsilon>0$. Then $\operatorname{\overline{R}}(\mathcal{G};c)=\Omega(nc^{n^{\varepsilon}})$. Ramsey numbers of unordered $n$-vertex graphs with maximum degree of a constant size are linear in $n$ [12]. In a sharp contrast to this result, we show that Ramsey numbers of ordered matchings may grow faster than any polynomial. ###### Theorem 9. There are arbitrarily large ordered matchings $\mathcal{M}_{n}$ on $n$ vertices such that $\operatorname{\overline{R}}(\mathcal{M}_{n};2)\geq n^{\frac{\log{n}}{5\log\log{n}}}.$ Combining this result with the fact that there are ordered $n$-vertex matchings with ordered Ramsey numbers that are linear in $n$, we get that there are orderings $\lhd_{1}$ and $\lhd_{2}$ of an $n$-vertex matching $M$ such that $\operatorname{\overline{R}}((M,\lhd_{1});2)/\operatorname{\overline{R}}((M,\lhd_{2});2)=n^{\Omega(\log{n}/\log\log n)}$. In the following we give polynomial upper bounds on ordered Ramsey numbers for two classes of sparse ordered graphs. First, we derive a polynomial upper bound for ordered Ramsey numbers of ordered graphs which admit the following decomposition. For given positive integers $k$ and $q\geq 2$ we say that an ordered graph $\mathcal{G}=(G,\prec)$ is $(k,q)$-decomposable if $\mathcal{G}$ has at most $k$ vertices or if it admits the following recursive decomposition: there is a nonempty interval $I$ with at most $k$ vertices of $\mathcal{G}$ such that the interval $I_{L}$ of vertices of $\mathcal{G}$ that are to the left of $I$ and the interval $I_{R}$ of vertices of $\mathcal{G}$ that are to the right of $I$ satisfy $\|I_{L}\|,\|I_{R}\|\leq\|V(G)\|\cdot\frac{q-1}{q}$ and there is no edge between $I_{L}$ and $I_{R}$. Moreover, the ordered graphs $(G[I_{L}],\prec\restriction_{I_{L}})$ and $(G[I_{R}],\prec\restriction_{I_{R}})$ are $(k,q)$-decomposable. ###### Theorem 10. For fixed positive integers $k$, $q\geq 2$ and $(k,q)$-decomposable ordered graphs $\mathcal{G}$ and $\mathcal{H}$ we have $\operatorname{\overline{R}}(\mathcal{G},\mathcal{H})\leq C_{k}\cdot 2^{64k(\lceil\log_{q/(q-1)}{r}\rceil+\lceil\log_{q/(q-1)}{s}\rceil)}$ where $r$ is the number of vertices of $\mathcal{G}$, $s$ is the number of vertices of $\mathcal{H}$ and $C_{k}$ is a sufficiently large constant dependent on $k$. We say that the length of an edge $\\{u,v\\}$ in an ordered graph $(G,\prec)$ is $\|i-j\|$ if $u$ is the $i$th vertex and $v$ is the $j$th vertex of $G$ in the ordering $\prec$. ###### Corollary 11. For a fixed positive integer $k$, every $n$-vertex ordered graph $\mathcal{G}$ with all edge lengths at most $k$ satisfies $\operatorname{\overline{R}}(\mathcal{G};2)\leq C_{k}\cdot 2^{256k\lceil\log{n}\rceil}$ where $C_{k}$ is a sufficiently large constant dependent on $k$. ###### Proof. Every ordered graph $\mathcal{G}$ with maximum edge length $k$ is $(k,2)$-decomposable, so we can apply Theorem 10 for $\mathcal{G}$ with $r\mathrel{\mathop{:}}=n$, $s\mathrel{\mathop{:}}=n$, $q\mathrel{\mathop{:}}=2$, and $p\mathrel{\mathop{:}}=2$. ∎ The last result is a polynomial upper bound for ordered Ramsey numbers of ordered graphs with constant degeneracy and constant interval chromatic number. For an ordered graph $\mathcal{G}$, the interval chromatic number of $\mathcal{G}$ is the minimum number of intervals the vertex set of $\mathcal{G}$ can be partitioned into such that there is no edge between vertices of the same interval. ###### Theorem 12. For positive integers $k$ and $p$, every $k$-degenerate ordered graph $\mathcal{G}$ with $n$ vertices and interval chromatic number $p$ satisfies $\operatorname{\overline{R}}(\mathcal{G};2)\leq n^{(1+2/k)(k+1)^{\lceil\log{p}\rceil}-2/k}.$ This result is a corollary of a stronger statement, Theorem 29. See Section 4.2. #### The work of Conlon et al. While presenting this paper at the conference Summit 240, we learned about a recent work of Conlon, Fox, Lee and Sudakov [13] who independently investigated Ramsey numbers of ordered graphs. There are some overlaps with our results. Among many other results, the authors of [13] proved that as $n$ goes to infinity, almost every ordering $\mathcal{M}_{n}$ of a matching on $n$ vertices satisfies $\operatorname{\overline{R}}(\mathcal{M}_{n};2)\geq n^{\frac{\log{n}}{20\log\log{n}}}$ which gives asymptotically the same bound as Theorem 9, which shows only existence of such matchings. They also showed that there exists a constant $c$ such that every $n$-vertex ordered graph $\mathcal{G}$ with degeneracy $k$ and interval chromatic number $p$ satisfies $\operatorname{\overline{R}}(\mathcal{G};2)\leq n^{ck\log{p}}$. This gives a stronger estimate than Theorem 12. On the other hand, Corollary 11 gives a solution to Problem 6.9 in [13] by showing that for every natural number $k$ there exists a constant $c_{k}$ such that $\operatorname{\overline{R}}(\mathcal{G};2)\leq n^{c_{k}}$ for every ordered graph $\mathcal{G}$ on $n$ vertices with bandwidth at most $k$. Here, the notion of bandwidth of an ordered graph $\mathcal{G}$ corresponds to the maximum edge length in $\mathcal{G}$. ## 2 Ordered Ramsey numbers for specific classes of graphs In this section we compute Ramsey numbers for various classes of ordered graphs such as ordered stars, cycles and paths. We also compare the obtained formulas and bounds with known Ramsey numbers of corresponding unordered graphs. ### 2.1 Stars A star is a complete bipartite graph $K_{1,n-1}$. Ramsey numbers of unordered stars are known exactly [4] and they are given by $\operatorname{R}(K_{1,n-1};c)=\begin{cases}c(n-2)+1&\text{if }c\equiv n-1\equiv 0\;(\bmod\;2),\\\ c(n-2)+2&\text{otherwise}.\end{cases}$ The position of the central vertex of a star determines the ordering of the star uniquely up to isomorphism. We denote as $\mathcal{S}_{r,s}$ the ordered star which has $r-1$ vertices to the right and $s-1$ vertices to the left of the central vertex, see Figure 4. Note that $\mathcal{S}_{r,s}$ has $r+s-1$ vertices. For $c,r_{1},\ldots,r_{c}\in\mathbb{N}$, determining $\operatorname{\overline{R}}(\mathcal{S}_{1,r_{1}},\ldots,\mathcal{S}_{1,r_{c}})$ is a simple observation. Figure 4: The ordered star $\mathcal{S}_{r,s}$. ###### Observation 13. For positive integers $c,r_{1},\ldots,r_{c}$ we have $\operatorname{\overline{R}}(\mathcal{S}_{1,r_{1}},\ldots,\mathcal{S}_{1,r_{c}})=2(1-c)+\sum_{i=1}^{c}r_{i}.$ ###### Proof. Assume that we have a complete ordered graph $\mathcal{K}_{N}$ with $N\geq 2(1-c)+\sum_{i=1}^{c}r_{i}$ vertices and $c$-colored edges. Then, according to the pigeonhole principle, the first vertex in $\mathcal{K}_{N}$ has at least $r_{i}-1$ right neighbors in color $i$. This forms a monochromatic copy of $\mathcal{S}_{1,r_{i}}$. On the other hand, we can construct a $c$-coloring of the edges of $\mathcal{K}_{N}$ with $N\mathrel{\mathop{:}}=1-2c+\sum_{i=1}^{c}r_{i}$ which does not contain any forbidden star. It suffices to divide the right neighbors of each vertex $v$ into $c$ parts where the $i$th part has size at most $r_{i}-2$ and each of its vertices is adjacent to $v$ with an edge colored with $i$. ∎ Thus the ordered Ramsey numbers $\operatorname{\overline{R}}(\mathcal{S}_{1,n};c)$ are almost the same as $\operatorname{R}(K_{1,n-1};c)$ for every $n$ and $c$. They differ by one only if $c\equiv n-1\equiv 0\;(\bmod\;2)$. Obviously, we have $\operatorname{\overline{R}}(\mathcal{S}_{1,r},\mathcal{S}_{s,1})=\operatorname{\overline{R}}(\mathcal{S}_{r,1},\mathcal{S}_{1,s})=\operatorname{\overline{R}}(\mathcal{S}_{1,s},\mathcal{S}_{r,1})$ and $\operatorname{\overline{R}}(\mathcal{S}_{r,1},\mathcal{S}_{1,2})=r$ for every pair $r,s\geq 2$ of positive integers. The ordered Ramsey numbers of an arbitrary pair of ordered stars are determined by results of Choudum and Ponnusamy [6]. ###### Theorem 14 ([6]). For positive integers $r_{1},r_{2}\geq 2$ we have $\operatorname{\overline{R}}(\mathcal{S}_{1,r_{1}},\mathcal{S}_{r_{2},1})=\left\lfloor\frac{-1+\sqrt{1+8(r_{1}-2)(r_{2}-2)}}{2}\right\rfloor+r_{1}+r_{2}-2.$ Moreover, for integers $r_{1},r_{2},s_{1},s_{2}\geq 2$ we have $\operatorname{\overline{R}}(\mathcal{S}_{1,r_{1}},\mathcal{S}_{r_{2},s_{2}})=\operatorname{\overline{R}}(\mathcal{S}_{1,r_{1}},\mathcal{S}_{r_{2},1})+r_{1}+s_{2}-3$ and $\operatorname{\overline{R}}(\mathcal{S}_{r_{1},s_{1}},\mathcal{S}_{r_{2},s_{2}})=\operatorname{\overline{R}}(\mathcal{S}_{r_{1},1},\mathcal{S}_{r_{2},s_{2}})+\operatorname{\overline{R}}(\mathcal{S}_{1,s_{1}},\mathcal{S}_{r_{2},s_{2}})-1.$ Now we show Theorem 6 which says that ordered Ramsey numbers of an arbitrary collection of ordered stars are linear with respect to the size of the stars. Formally, let $c,r_{1},\ldots,r_{c},s_{1},\ldots,s_{c}$ be positive integers and let $\mathcal{S}_{r_{1},s_{1}},\ldots,\mathcal{S}_{r_{c},s_{c}}$ be ordered stars. Then there is a positive integer $C=C(c)$ such that $\operatorname{\overline{R}}(\mathcal{S}_{r_{1},s_{1}},\ldots,\mathcal{S}_{r_{c},s_{c}})\leq C\cdot\max\\{r_{1},\ldots,r_{c},s_{1},\ldots,s_{c}\\}.$ ###### Proof of Theorem 6. Let $r\mathrel{\mathop{:}}=\max\\{r_{1},\ldots,r_{c},s_{1},\ldots,s_{c}\\}$ and let $\mathcal{K}_{N}$ be an ordered complete graph on $N\mathrel{\mathop{:}}=Cr$ vertices with edges colored with colors from $\\{1,2,\ldots,c\\}$ where $C$ is a sufficiently large positive integer. Let $A_{0}$ be the vertex set of $\mathcal{K}_{N}$. We want to find a copy of $\mathcal{S}_{r,r}$ of color $i$ for some $i\in\\{1,\ldots,c\\}$, as then there is a copy of $\mathcal{S}_{r_{i},s_{i}}\subseteq\mathcal{S}_{r,r}$ of color $i$ in $\mathcal{K}_{N}$. So suppose for a contradiction that there is no monochromatic copy of $\mathcal{S}_{r,r}$ in $\mathcal{K}_{N}$. Note that every vertex which is at least $(c(r-1)+2)$th in the ordering of $\mathcal{K}_{N}$ (taken from left) has, according to the pigeonhole principle, at least $r-1$ left neighbors of the same color. Thus we have at least $Cr-c(r-1)-1$ vertices with at least $r-1$ monochromatic left neighbors. We consider a set $A_{1}$ of vertices which have at least $r-1$ left neighbors of color $1$. Without loss of generality we may assume that $\|A_{1}\|\geq(Cr-c(r-1)-1)/c$. From the assumption there is no vertex in $A_{1}$ with at least $r-1$ right neighbors of color $1$, as otherwise we would have $\mathcal{S}_{r,r}$ of color $1$. Thus between vertices in $A_{1}$ there is less than $(r-1)\|A_{1}\|$ edges of color $1$, since every one of them is counted for its left endpoint. Also we see that $A_{1}$ contains at least $(\|A_{1}\|-(c(r-1)-1))/c$ vertices which have at least $r-1$ right neighbors in $A_{1}$ all of the same color $i$ (without loss of generality, let $i=2$). We denote this set as $A_{2}$. From the assumption the vertices in $A_{2}$ have less than $r-1$ left neighbors of color $2$ in $\mathcal{K}_{N}$ and thus there is less than $(r-1)\|A_{2}\|$ edges of color $2$ (and $1$, since $A_{2}\subseteq A_{1}$) between vertices in $A_{2}$. We repeat this process analogously, bounding the number of edges of colors $1,\ldots,i$ in $A_{i}$ by $(r-1)\|A_{i}\|$ and keeping $\|A_{i}\|\geq(\|A_{i-1}\|-c(r-1)-1)/c$ for $i\geq 1$. After all colors are processed we get, summing over all colors, that the number of all edges is strictly less than $c(r-1)\|A_{c}\|$. The total number of edges connecting vertices from $A_{i}$ is exactly ${\|A_{i}\|\choose 2}$. Altogether, we obtain $\|A_{c}\|(\|A_{c}\|-1)/2<c(r-1)\|A_{c}\|$ which can be rewritten as $\|A_{c}\|<2c(r-1)+1$. However $\|A_{c}\|=\Omega(Cr/c^{c})$ and thus we can choose $C$ large enough so that the upper bound on $A_{c}$ does not hold and obtain a contradiction. ∎ We know that Ramsey numbers for unordered stars and ordered Ramsey numbers of $\mathcal{S}_{1,r}$ and $\mathcal{S}_{r,1}$ are linear even with respect to the number of colors. The following proposition shows that this is not the case for other orderings of stars. ###### Proposition 15. Let $c$, $r_{1},\ldots,r_{c}$, and $d\geq 3$ be positive integers and let $\mathcal{G}_{1},\ldots,\mathcal{G}_{c}$ be ordered graphs such that $(P_{d},\lhd_{mon})\subseteq\mathcal{G}_{i}$ and every $\mathcal{G}_{i}$ has $r_{i}$ vertices for every $i=1,\ldots,c$. Then we have $\operatorname{\overline{R}}(\mathcal{G}_{1},\ldots,\mathcal{G}_{c})>(d-1)^{c-1}(\max\\{r_{1},\ldots,r_{c}\\}-1).$ ###### Proof. For $r\mathrel{\mathop{:}}=\max\\{r_{1},\ldots,r_{c}\\}$, let $\mathcal{K}_{N}$ be an ordered complete graph with $N\mathrel{\mathop{:}}=(d-1)^{c-1}(r-1)$. Without loss of generality, we assume $r=r_{1}$. We construct a $c$-coloring of the edges of $\mathcal{K}_{N}$ with colors from the set $\\{1,\ldots,c\\}$ such that there is no copy of $\mathcal{G}_{i}$ of color $i$ in $\mathcal{K}_{N}$ for every $i=1,\ldots,c$. The coloring is constructed by induction on $c$. For $c=1$ we have $\mathcal{K}_{r-1}$ with all edges colored $1$. For $c>1$ we partition the vertex set of $\mathcal{K}_{n}$ into $d-1$ consecutive intervals each of size $(d-1)^{(c-2)}(r-1)$. Every such interval induces a clique which we color with the $(c-1)$-coloring obtained in the previous step. Then we color all edges between distinct cliques with the color $c$ and obtain a $c$-coloring of all edges of $\mathcal{K}_{N}$. See Figure 5. Clearly, the $1$-coloring of $K_{r-1}$ avoids a copy of $\mathcal{G}_{1}$ of color $1$. Using the inductive hypothesis, it suffices to show that the constructed $c$-coloring of $\mathcal{K}_{N}$, $c>1$, avoids a copy of $\mathcal{G}_{c}$ in color $c$. This follows from the fact that $(P_{d},\lhd_{mon})$ is an ordered subgraph of $\mathcal{G}_{c}$, while $(P_{d},\lhd_{mon})$ is not contained in the ordered graph $\mathcal{K}_{(d-1)^{(c-2)}(r-1)}(d-1)$ which is induced by the edges colored with $c$. ∎ Figure 5: The construction in the proof of Proposition 15 for $d=3$ and $c=4$. ###### Corollary 16. Let $c$ and $r_{1},\ldots,r_{c},s_{1},\ldots,s_{c}\geq 2$ be positive integers. Then we have $\operatorname{\overline{R}}(\mathcal{S}_{r_{1},s_{1}},\ldots,\mathcal{S}_{r_{c},s_{c}})>2^{c-1}(\max\\{r_{1}+s_{1}-1,\ldots,r_{c}+s_{c}-1\\}-1).\qed$ ###### Corollary 17. Let $c$ and $r_{1},\ldots,r_{c}\geq 3$ be positive integers and let $\mathcal{G}_{1},\ldots,\mathcal{G}_{c}$ be ordered graphs on $r_{1},\ldots,r_{c}$ vertices respectively. If no $\mathcal{G}_{i}$ has a bipartite underlying graph, then we have $\operatorname{\overline{R}}(\mathcal{G}_{1},\ldots,\mathcal{G}_{c})>2^{c-1}(\max\\{r_{1},\ldots,r_{c}\\}-1).$ ###### Proof. According to Proposition 15, it suffices to show that $(P_{3},\lhd_{mon})$ is contained in every given ordered graph $\mathcal{G}_{i}$. Every $\mathcal{G}_{i}$ contains an ordered odd cycle, as its underlying graph is not bipartite. The rest follows from the fact that every ordered odd cycle contains $(P_{3},\lhd_{mon})$. ∎ ### 2.2 Paths In the unordered case, the problem of finding the exact formula for $\operatorname{R}(P_{r},P_{s})$ has been settled by Gerencsér and Gyárfás [23] who showed that we have $\operatorname{R}(P_{r},P_{s})=s-1+\left\lfloor\frac{r}{2}\right\rfloor$ for $2\leq r\leq s$. The multi-color case turned out to be more difficult, but some partial results are known, see [19, 25]. In this section we show an ordering of the path $P_{n}$ whose ordered Ramsey number is linear in $n$. Let $v_{1},\ldots,v_{n}$ be vertices of the path $P_{n}$ and let $\\{v_{1},v_{2}\\}$, $\\{v_{2},v_{3}\\},\ldots,\\{v_{n-1},v_{n}\\}$ be the edges of $P_{n}$. We define an alternating path $(P_{n},\lhd_{alt})$ as an ordered path with $v_{1}\lhd_{alt}v_{3}\lhd_{alt}v_{5}\lhd_{alt}\ldots\lhd_{alt}v_{n}\lhd_{alt}v_{n-1}\lhd_{alt}v_{n-3}\lhd_{alt}\ldots\lhd_{alt}v_{2}$ for $n$ odd and $v_{1}\lhd_{alt}v_{3}\lhd_{alt}v_{5}\lhd_{alt}\ldots\lhd_{alt}v_{n-1}\lhd_{alt}v_{n}\lhd_{alt}v_{n-2}\lhd_{alt}\ldots\lhd_{alt}v_{2}$ for $n$ even. See part a) of Figure 6. Note that, unlike monotone paths, the alternating path $(P_{n},\lhd_{alt})$ is an ordered subgraph of $\mathcal{K}_{\lceil n/2\rceil,\lfloor n/2\rfloor}$. Figure 6: The alternating path $(P_{7},\lhd_{alt})$ and its corresponding matrix $M(P_{7},\lhd_{alt})$. ###### Proposition 18. For every positive integer $n>2$ we have $2n-2\leq\operatorname{\overline{R}}((P_{n},\lhd_{alt});2)\leq(4n-3+\sqrt{8n^{2}-8n-7})/2.$ Thus the numbers $\operatorname{\overline{R}}(P_{n},\lhd_{alt};2)$ are linear in $n$. By Proposition 2, the ordered Ramsey numbers $\operatorname{\overline{R}}((P_{n},\lhd_{mon});2)$ grow quadratically with respect to $n$. Thus there are two orderings of $P_{n}$ such that the corresponding ordered Ramsey numbers differ in an asymptotically relevant term. A similar result can be derived by considering a matching $M_{n}$, which is a graph on $n$ vertices consisting of $\lfloor n/2\rfloor$ disjoint pairs of edges, and the two orderings of $M_{n}$ from Figure 7. Using a coloring similar to the one from Proposition 2, we see that ordered Ramsey numbers of the first ordered matching grow quadratically with respect to $n$ while for the second one its ordered Ramsey number remains linear in $n$. For other orderings of $M_{n}$ the asymptotic difference can be much larger, see Theorem 9. Figure 7: Two orderings of $M_{n}$ with asymptotically different ordered Ramsey numbers. To prove Proposition 18 we use a result from extremal theory of $\\{0,1\\}$-matrices which was mentioned in the motivation (Section 1.1). The following definitions are taken from [9]. We say that an $r\times s$ matrix $M$ is minimalist if $\operatorname{ex}_{M}(m,n)=(s-1)m+(r-1)n-(r-1)(s-1)$. If the matrix $M^{\prime}$ was created from a matrix $M$ by adding a new row (or a column) as the new first or last row (column) and this new row (column) contains a single 1-entry next to a 1-entry of $M$, then we say the $M^{\prime}$ was created by an elementary operation from $M$. ###### Lemma 19 ([22]). Let $M$ be an $r\times s$ minimalist matrix and let $M^{\prime}$ be an $r^{\prime}\times s^{\prime}$ nonempty matrix obtained from $M$ by applying several elementary operations. Then $M^{\prime}$ is minimalist. ###### Proof of proposition 18. For the lower bound we color the edge between the $i$th and $j$th vertex of $\mathcal{K}_{2n-3}$ red if $\|i-j\|$ is even and blue otherwise. Suppose that there is a red copy of $(P_{n},\lhd_{alt})$ in this coloring. Then the number of vertices between the first and the last vertex of this alternating path is at least $2n-4$. This is not possible, as there are only $2n-3$ vertices in total. An analogous argument works for a blue copy of $(P_{n},\lhd_{alt})$. Let $N$ be an integer satisfying $N\geq(4n-3+\sqrt{8n^{2}-8n-7})/2$. To show the upper bound we find a monochromatic copy of $(P_{n},\lhd_{alt})$ in the given graph $\mathcal{K}_{\lceil N/2\rceil,\lfloor N/2\rfloor}$ with 2-colored edges. Without loss of generality, at least half of the edges are red. Since $(P_{n},\lhd_{alt})$ is an ordered subgraph of $\mathcal{K}_{\lceil N/2\rceil,\lfloor N/2\rfloor}$, we can consider the $\lceil n/2\rceil\times\lfloor n/2\rfloor$ $\\{0,1\\}$-matrix $M(P_{n},\lhd_{alt})=M$ introduced in the motivation. An example of such a matrix can be found in Figure 6, part b). By Lemma 19, all such matrices are minimalist. Therefore we have $\operatorname{ex}_{M}(\lceil N/2\rceil,\lfloor N/2\rfloor)=(\lfloor n/2\rfloor-1)\lceil N/2\rceil+(\lceil n/2\rceil-1)\lfloor N/2\rfloor-(\lceil n/2\rceil-1)(\lfloor n/2\rfloor-1)$ and this is at most $\frac{1}{4}(2nN+4n-3N-4-n^{2})$. Thus every $K_{\lceil N/2\rceil,\lfloor N/2\rfloor}$ which does not contain $(P_{n},\lhd_{alt})$ as an ordered subgraph must contain at most that many edges. On the other hand, our graph formed by red edges has at least $\frac{1}{2}\lceil N/2\rceil\cdot\lfloor N/2\rfloor\geq N(N-1)/8$ edges. Thus to avoid $(P_{n},\lhd_{alt})$ the inequality $\frac{2nN+4n-3N-4-n^{2}}{4}\geq N(N-1)/8$ must be satisfied. Consequently, we obtain $N\leq(4n-5+\sqrt{8n^{2}-8n-7})/2$ and the result follows. ∎ There is still a place for improvement, as the multiplicative factor in $\operatorname{\overline{R}}((P_{n},\lhd_{alt});2)$ is between $2$ and $2+\sqrt{2}$. Computer experiments indicate that the right values of $\operatorname{\overline{R}}(P_{n},\lhd_{alt};2)$ could be of the form $\lfloor(n-2)\frac{1+\sqrt{5}}{2}\rfloor+n$. See Table 1. $n$ \| $2$ \| $3$ \| $4$ \| $5$ \| $6$ \| $7$ \| $8$ \| $9$ \| $10$ \| $11$ \| $12$ \| $13$ ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- $\operatorname{\overline{R}}((P_{n},\lhd_{alt});2)$ \| $2$ \| $4$ \| $7$ \| $9$ \| $12$ \| $15$ \| $17$ \| $\geq 20$ \| $\geq 22$ \| $\geq 25$ \| $\geq 28$ \| $\geq 30$ Table 1: Estimates for Ramsey numbers $\operatorname{\overline{R}}((P_{r},\lhd_{alt});2)$ for $r\leq 13$. For general ordered paths not much is known currently. Cibulka et al. [10] showed that for every ordered path $\mathcal{P}_{r}$ and the clique $\mathcal{K}_{s}$ we have $\operatorname{\overline{R}}(\mathcal{P}_{r},\mathcal{K}_{s})\leq 2^{\lceil\log s\rceil(\lceil\log r\rceil+1)}.$ That is, for every ordered path $\mathcal{P}_{n}$ we have $\operatorname{\overline{R}}(\mathcal{P}_{n};2)\leq n^{O(\log n)}$. ### 2.3 Cycles It is a folklore result in Ramsey theory that $\operatorname{R}(C_{3};2)=\operatorname{R}(C_{4};2)=6$ holds [8]. The first results on Ramsey numbers of cycles were obtained by Chartrand and Chuster [5], and Bondy and Erdős [3]. These were later extended by Rosta [37], and Faudree and Schelp [18]. Eventually, the Ramsey numbers for cycles in the two- color case were completely determined: $\operatorname{R}(C_{r},C_{s})=\begin{cases}2r-1&\textrm{if }(r,s)\neq(3,3)\textrm{ and }3\leq s\leq r,s\textrm{ is odd},\\\ r+s/2-1&\textrm{if }(r,s)\neq(4,4)\textrm{ and }4\leq s\leq r,r\textrm{ and }$s$\textrm{ are even},\\\ \max\\{r+s/2-1,2s-1\\}&\textrm{if }4\leq s<r,s\textrm{ is even and }r\textrm{ is odd}.\end{cases}$ The multicolor case is more demanding [14, 32, 33], but the following is known. ###### Theorem 20 ([33]). For every $c\geq 4$ and $n$ odd, we have $\operatorname{R}(C_{n};c)\leq c2^{c}n+o(n)$, and for every $c\geq 2$ and $n$ even, we have $\operatorname{R}(C_{n};c)\leq cn+o(n)$ as $n\to\infty$. For ordered cycles, the first nontrivial case is $C_{4}$ which has three possible orderings up to isomorphism. See Figure 8. In the following, we determine the ordered Ramsey numbers for all orderings of $C_{4}$. Figure 8: Possible orderings of $C_{4}$. ###### Proposition 21. We have 1. 1. $\operatorname{\overline{R}}((C_{4},\prec_{A});2)=14$, 2. 2. $\operatorname{\overline{R}}((C_{4},\prec_{B});2)=10$, 3. 3. $11\leq\operatorname{\overline{R}}((C_{4},\prec_{C});2)\leq 13$. ###### Proof. The lower bounds follow from the colorings presented in Figure 9, thus it remains to show the upper bounds for each ordering. Suppose that $\mathcal{K}_{N}$ is an ordered complete graph with 2-colored edges (red and blue) where $N$ is relevant to each case. Figure 9: Colorings for the lower bounds in the proof of Proposition 21. 1. 1. This result is implied by more general statement, Theorem 7, which is proved bellow. 2. 2. Assume for a contradiction that $\mathcal{K}_{10}$ does not contain a monochromatic copy of $(C_{4},\prec_{B})$. That is, no two vertices of $\mathcal{K}_{10}$ share two right neighbors of the same color. We claim that there is no vertex with monochromatic right degree greater than five in the coloring of $\mathcal{K}_{10}$. Otherwise there is, without loss of generality, a vertex $v$ with right red degree at least six. Let $R$ be the set of the right red neighbors of $v$. The leftmost vertex $w$ of $R$ has a blue right degree at least four, as it cannot have more than one red right neighbor in $R$. However then the blue neighborhood of $w$ contains a vertex with either red or blue right degree at least two. In any case we obtain $(C_{4},\prec_{B})$. Thus every vertex has monochromatic right degree of size at most five. From the assumptions, every pair of vertices has at most one common right neighbor in every color. Without loss of generality we assume that the first vertex $v_{1}$ in $\mathcal{K}_{10}$ has a red right degree five and a blue right degree four. The second vertex $v_{2}$ in $\mathcal{K}_{10}$ has at most one (right) red neighbor among the red neighbors of $v_{1}$ and the remaining at least three are its blue neighbors. Then the third vertex $v_{3}$ has at least two common right monochromatic neighbors either with $v_{1}$ or $v_{2}$ between these three vertices. 3. 3. The first vertex $v_{1}$ in $\mathcal{K}_{13}$ has either six blue and six red neighbors, or at least seven monochromatic neighbors. In any case, since every pair of vertices can have at most a single common neighbor of the same color between them, there is, without loss of generality, at least five vertices which are red right neighbors of $v_{1}$ and blue left neighbors of $v_{13}$. By Theorem 14 we have $\operatorname{\overline{R}}(\mathcal{S}_{1,3},\mathcal{S}_{3,1})=5$. Therefore we find either a vertex with red left degree at least two or a vertex with blue right degree at least two in the clique formed by those five vertices. In both cases we obtain a monochromatic copy of $(C_{4},\prec_{C})$. ∎ Using exhaustive computer search, we have derived that the lower bound in the third case is tight, i.e., $\operatorname{\overline{R}}((C_{4},\prec_{C});2)=11$ holds. In the rest of the section we establish the precise value of ordered Ramsey numbers for monotone cycles. Specifically, we prove the equality $\operatorname{\overline{R}}((C_{r},\lhd_{mon}),(C_{s},\lhd_{mon}))=2rs-3r-3s+6$ for every $r,s\geq 2$. Note that this formula is somewhat simpler than the one for $\operatorname{R}(C_{r},C_{s})$. The following lemma is a simple observation which is implicitly proved in [27]. We include its proof for completeness. ###### Lemma 22. For positive integers $r$ and $s$, we have $\operatorname{\overline{R}}((P_{r},\lhd_{mon}),\mathcal{K}_{s})=\operatorname{\overline{R}}((P_{r},\lhd_{mon}),(P_{s},\lhd_{mon}))=(r-1)(s-1)+1.$ ###### Proof. The lower bound can be obtained from the same construction as in the proof of Proposition 2, see [38]. For the upper bound we use induction on $r$. If $r=2$, then this statement holds since we either have a monochromatic copy of $\mathcal{K}_{s}$, or a blue edge. Suppose that $r>2$ and let $\mathcal{K}_{(r-1)(s-1)+1}$ be an ordered complete graph with edges colored red and blue. Assume that it does not contain a blue copy of $\mathcal{K}_{s}$ nor a red copy of $(P_{r},\lhd_{mon})$. Using the inductive hypothesis, we know that there is at least $(r-1)(s-1)+1-(r-2)(s-1)=s$ distinct vertices which are the last vertices of a red copy of $(P_{r-1},\lhd_{mon})$. From the assumption every edge between such vertices is blue, otherwise we would extend one of these paths. However this yields a blue copy of $K_{s}$, a contradiction. ∎ A simple corollary is that for every $s$-vertex ordered graph $\mathcal{G}$ which contains a monotone Hamiltonian path we have $\operatorname{\overline{R}}((P_{r},\lhd_{mon}),\mathcal{G})=(r-1)(s-1)+1$. ###### Proof of Theorem 7. First, we show the upper bound which can be also derived from the results of Károlyi et al. [28] who studied geometric Ramsey numbers of cycles. In an ordered complete graph $\mathcal{K}_{N}$ with 2-colored edges and with $N\mathrel{\mathop{:}}=2rs-3r-3s+6$ vertices the first vertex $v_{1}$ has either at least $(r-2)(s-1)+1$ red neighbors or at least $(r-1)(s-2)+1$ blue neighbors. In the first case there is, according to Lemma 22, a red copy of $(P_{r-1},\lhd_{mon})$ which forms red $(C_{r},\lhd_{mon})$ together with $v_{1}$ or a blue copy of $(C_{s},\lhd_{mon})$. The second case with large blue neighborhood is analogous and we thus always get either red $(C_{r},\lhd_{mon})$ or blue $(C_{s},\lhd_{mon})$. For the lower bound we show a coloring of $(K_{N},\prec)$ where $N\mathrel{\mathop{:}}=2rs-3r-3s+5$, which avoids a red copy of $(C_{r},\lhd_{mon})$ and a blue copy of $(C_{s},\lhd_{mon})$. An example of such coloring for $r=s=4$ can be found in Figure 9, part a). Consider a partition of the vertex set of $(K_{N},\prec)$ into the following consecutive (in the ordering $\prec$) intervals $I_{i}$, $i=1,2,\ldots,2r-3$. If $r$ is odd, then the first and last $(r-1)/2$ intervals $I_{i}$ in $\prec$ have size $s-1$ and the remaining $r-2$ intervals $I_{i}$ have size $s-2$. If $r$ is even, then the first and last $(r-2)/2$ intervals $I_{i}$ contain $s-2$ vertices and the remaining $r-1$ intervals $I_{i}$ have $s-1$ vertices. Note that in both cases we have $N$ vertices in total. We assume that the vertices of $I_{i}$ are of the form $v^{i}_{j}$ where $j=1,\ldots,\|I_{i}\|$ and $v^{i}_{j}\prec v^{i}_{k}$ whenever $j<k$. We also refer to the index $j$ as the index of a vertex $v^{i}_{j}$. The coloring of the edges is defined as follows. First, we color all edges between vertices from the same interval $I_{i}$ blue. Next, we introduce four types of pairs $(I_{i},I_{j})$, $i<j$, according to which we color edges between vertices from intervals $I_{i}$ and $I_{j}$. We say that $(I_{i},I_{j})$, $i<j$, is of the type: * • $T_{<}$ if $j-i\leq r-2$ and $\|I_{i}\|\leq\|I_{j}\|$. In this case we color an edge $\\{v^{i}_{k},v^{j}_{l}\\}$ blue if $k<l$ and red otherwise. * • $T_{\geq}$ if $j-i>r-2$ and $\|I_{i}\|<\|I_{j}\|$. Then an edge $\\{v^{i}_{k},v^{j}_{l}\\}$ is colored blue if $k\geq l$ and red otherwise. * • $T_{>}$ if $j-i>r-2$ and $\|I_{i}\|\geq\|I_{j}\|$. Then an edge $\\{v^{i}_{k},v^{j}_{l}\\}$ is colored blue if $k>l$ and red otherwise. * • $T_{\leq}$ if $j-i\leq r-2$ and $\|I_{i}\|>\|I_{j}\|$. Then an edge $\\{v^{i}_{k},v^{j}_{l}\\}$ is colored blue if $k\leq l$ and red otherwise. The main idea is that for the types $T_{<}$ and $T_{\leq}$ we color blue the edges between vertices such that their indices are increasing or non- decreasing (i.e., those vertices are relatively far from each other), while for $T_{>}$ and $T_{\geq}$ the indices are decreasing or non-increasing (such vertices are relatively close to each other). For red edges, the indices behave exactly opposite. The distribution of the types of pairs, as well as the definition of those types, is illustrated on small examples in the following two figures. Figure 10: The types of pairs $(I_{i},I_{j})$ for $s=5$ and colorings of corresponding edges. Figure 11: Distribution of types of pairs $(I_{i},I_{j})$ for $r=3$ (part a) and $r=4$ (part b). It remains to show that this coloring avoids forbidden cycles. We claim that our coloring does not contain a red copy of $(C_{r},\lhd_{mon})$. To prove this claim, suppose for a contradiction that there is such a copy. Note that it contains at most one vertex from every interval $I_{i}$, because the intervals induce blue cliques. The monotone path of length $r$ induced by a red cycle also cannot have an edge which connects vertices from $I_{i}$ and $I_{j}$ where $(I_{i},I_{j})$ is of type $T_{>}$ or $T_{\geq}$, because in both cases we skip vertices from at least $r-2$ intervals $I_{k}$. This leaves at most $2r-3-(r-2)=r-1$ intervals and from each such interval we can use a single vertex. However this is not possible, as $(C_{r},\lhd_{mon})$ contains $r$ vertices. Hence the vertex indices on this monotone path are non- increasing, as the path uses red edges between pairs $(I_{i},I_{j})$ only of types $T_{<}$ or $T_{\leq}$. Since the number of intervals $I_{i}$ of size $s-1$ as well as the number intervals of size $s-2$ is less than $r$, we must use vertices from both of those variants. If we have an edge between $(I_{i},I_{j})$ of type $T_{\leq}$ in the monotone path, then the vertex indices decrease at least once (as they are connected with a red edge). However the longest edge in our red cycle is between intervals of type $T_{>}$ or $T_{\geq}$ and thus it connects vertices whose indices are non-decreasing. This is a contradiction, because from our observations their indices should decrease. The other possibility is that all edges of the red monotone path are between pairs of types $T_{<}$. Then the longest edge of the red cycle is of type $T_{\geq}$, because it has to connect $I_{i}$ with $I_{j}$ where $\|I_{i}\|<\|I_{j}\|$. Here we have used the specific distribution of small and large intervals $I_{i}$. However then the vertex indices have to increase at least once and we already observed that this is not possible. A contradiction. Now we prove that there is no blue copy of $(C_{s},\lhd_{mon})$ in the coloring. Again, suppose that there is such a blue cycle. This time, we can use edges whose both endpoints are in the same $I_{i}$. However the blue cycle has to use vertices from at least two intervals $I_{i}$, because neither of them contains $s$ vertices. We distinguish a few cases. 1. 1. Suppose first that the blue monotone path of length $s$ does not contain an edge between a pair $(I_{i},I_{j})$ of type $T_{>}$ or $T_{\geq}$. Then the vertex indices are non-decreasing. According to the distribution of small and large intervals $I_{i}$, there is at most one edge between vertices with the same vertex index. Such an edge corresponds to a jump from a larger $I_{i}$ to a smaller $I_{j}$, i.e., a pair of type $T_{\leq}$. Therefore the length of every such blue monotone path is at most vertex index of its last vertex plus one, where the additional one is added only when we use the previously described jump. Since every vertex has index at most $s-1$, we see that the path uses exactly one pair of type $T_{\leq}$. Then the vertices of the path cannot remain in the smaller intervals $I_{i}$, as their indices would be at most $s-2$. However then the edge $e$ between the first and the last vertex of the path must be of type $T_{>}$, because we need to jump from a larger interval to a smaller one and then conversely (for $r$ even this is already impossible). That is, $e$ connects vertices from $I_{i}$ and $I_{j}$ where $\|I_{i}\|=\|I_{j}\|$ and $j-i>r-2$. Consequently, the indices on the path decrease at least once which is impossible. 2. 2. The second case to analyze is when the blue monotone path uses (exactly once) an edge $e$ between $I_{i}$ and $I_{j}$ with $j-i>r-2$, i.e., a pair of type $T_{>}$ or $T_{\geq}$. Such an edge is at most one, as it skips at least $r-2$ intervals while we have only $2r-3$ intervals in total. All the other edges of the path are either between pairs $(I_{i},I_{j})$ of types $T_{<}$ or $T_{\leq}$ or they connect vertices from the same interval. That is, except of the edge $e$ the indices of all other vertices are non-decreasing and the only case when vertex indices are not increasing is when we jump from a larger interval to a smaller one. This can happen at most once, as we have already observed. The construction implies that the longest edge of the blue monotone cycle is between a pair of type $T_{>}$ or $T_{\geq}$, therefore the index of the last vertex is at most as large as the index of the first vertex. 1. (a) Suppose that the path does not use a pair of type $T_{\leq}$. Then the indices on the path increase by at least $s-2$, as we use $s-2$ pairs of type $T_{<}$ or edges within the same $I_{i}$. The only possibility for the indices to decrease is on the edge of type $T_{>}$, because the decrease must be by at least $s-2$. Thus we need to jump from a vertex with index $s-1$ to a vertex with index $1$. We cannot do this with an edge of type $T_{\geq}$, as it jumps from a smaller interval where the indices are at most $s-2$. Now, consider the longest edge in the cycle. It must be of type $T_{>}$ or $T_{\geq}$ as it skips at least $r-2$ intervals. However neither of the possibilities can occur. The edge of type $T_{>}$ would connect vertices whose indices decrease, but the last vertex has index of size at least as large as the index of the first one, according to the size of the total decrease and increase. The longest edge of type $T_{\geq}$ would connect a vertex from a smaller interval with a vertex from larger and this is impossible according to the distribution of the intervals, since we have used an edge of type $T_{>}$ on the path. 2. (b) Assume that we have used (exactly once) an edge of type $T_{\leq}$ to jump between vertices whose indices are the same. Such an edge connects a larger interval with a smaller one and thus, according to their distribution, the longest edge in the cycle is of type $T_{>}$. This means that the index of the first vertex is strictly larger than the index of the last one. The total decrease of indices must be also strictly larger than their increase which is at least $s-3$, as at least $s-3$ edges of the path are of type $T_{<}$ or are between edges from the same $I_{i}$. To finish the proof, note that we cannot use an edge of type $T_{>}$ on the path together with the edge of type $T_{\leq}$. This is again because of the distribution of the intervals. Thus the total decrease is at most $s-3$, as edges of type $T_{\geq}$ jump from a smaller to a larger interval. ∎ Note that we have proved a stronger statement, because in our coloring no red monotone cycle of length at least $r$ nor a blue monotone cycle of length at least $s$ can appear. It could be interesting to extend this theorem to a multicolored case. As noted by Cibulka et al. [10], the constructed coloring can be used to show the exact formula for geometric and convex geometric Ramsey numbers for cycles (see Section 1.1 for definitions). ###### Corollary 23. For every integer $n\geq 3$, we have $\operatorname{Rc}(C_{n})=\operatorname{Rg}(C_{n})=2(n-2)(n-1)+2$. ###### Proof. According to the upper bound of Károlyi et al. [28] and the fact $\operatorname{Rc}(C_{n})\leq\operatorname{Rg}(C_{n})$, it suffices to show that $\operatorname{Rc}(C_{n})\geq 2(n-2)(n-1)+2$ which is an immediate corollary of Observation 5 and Theorem 7. ∎ ## 3 Lower bounds The following proposition, whose proof is based on a standard existence argument, gives us a general lower bound on Ramsey numbers of ordered graphs. Using this result, we can derive a lower bound for dense graphs which is exponential in the number of vertices. Proposition 8 is a special case of this assertion since a graph $G$ on $n$ vertices with $\Omega(n^{1+\varepsilon})$ edges for some $\varepsilon>0$ satisfies $\operatorname{\overline{R}}((G,\prec);2)=\Omega(n2^{n^{\varepsilon}})$ for every vertex ordering $\prec$ of $G$. ###### Proposition 24. Let $c$, $r$, and $s$ be positive integers and let $\prec_{1},\ldots,\prec_{c}$ be vertex orderings of a graph $G$ with $n$ vertices and $m$ edges. Then we have $\operatorname{\overline{R}}((G,\prec_{1}),\ldots,(G,\prec_{c}))\geq({2\pi n})^{1/n}\left(\frac{n}{e}\right)c^{(m-1)/n}$ where $e$ is the base of the natural logarithm. ###### Proof. Let $(K_{N},\prec)$ be a complete ordered graph and let $G=(V,E)$ be the given graph. We $c$-color the edges of $(K_{N},\prec)$ independently at random with probability $1/c$ for each color. The probability that a set $S\subset V$ of size $n$ induces a copy of $(G,\prec_{i})$ in color $i$ is $(1/c)^{m}$, since the ordering $\prec_{i}$ determines the set of edges of $(G,\prec_{i})$. Using the Union bound we derive $\displaystyle\Pr[\textrm{there is }i\in\\{1,\ldots,c\\}$ $\displaystyle\textrm{ such that }(G,\prec_{i})\subseteq(K_{N},\prec)\textrm{ in color }i]\leq{N\choose{n}}\cdot c\cdot\left(\frac{1}{c}\right)^{m}=$ $\displaystyle{N\choose{n}}\left(\frac{1}{c}\right)^{m-1}\leq\frac{N^{n}}{n!}\left(\frac{1}{c}\right)^{m-1}.$ With Stirling’s approximation formula $k!=\sqrt{2\pi k}\left(\frac{k}{e}\right)^{k}\cdot(1+o(1))$ we can bound this probability from above by $\frac{N^{n}}{\sqrt{2\pi n}\left(\frac{n}{e}\right)^{n}c^{m-1}}.$ This expression is strictly smaller than 1 for $N<\sqrt[n]{2\pi n}(n/e)c^{(m-1)/n}$. Therefore for such $N$ there exists a $c$-coloring of edges of $(K_{N},\prec)$ which avoids $(G,\prec_{i})$ for all $i=1,\ldots,c$. ∎ ### 3.1 Proof of Theorem 9 So far we have seen examples of ordered paths and matchings where the ratio between ordered and unordered Ramsey numbers grew linearly with $n$. Now we show a much stronger estimate, Theorem 9, which says that there are arbitrarily large ordered matchings $\mathcal{M}$ with $n$ vertices satisfying $\operatorname{\overline{R}}(\mathcal{M};2)=n^{\Omega(\log n/5\log\log n)}$. Let $r\geq 3$ be a positive integer and let $R\mathrel{\mathop{:}}=\operatorname{R}(K_{r};2)-1$. We construct a sequence of ordered matchings $\mathcal{M}_{k}$ with $n_{k}$ vertices and a sequence of $2$-colorings $c_{k}$ of ordered complete graphs $\mathcal{G}_{k}$ with $N_{k}$ vertices such that $c_{k}$ avoids a monochromatic copy of $\mathcal{M}_{k}$. First, we show an inductive construction of the colorings $c_{k}$. Let $N_{1}\mathrel{\mathop{:}}=R$ and let $c_{1}$ be a $2$-coloring of $\mathcal{G}_{1}\mathrel{\mathop{:}}=\mathcal{K}_{R}$ avoiding $\mathcal{K}_{r}$. Let $k\geq 1$ and suppose that a coloring $c_{k}$ of $\mathcal{G}_{k}$ has been constructed. Let $N_{k+1}\mathrel{\mathop{:}}=R\cdot N_{k}$ and let $\mathcal{G}_{k}$ be the complete graph on $N_{k+1}$ vertices. Partition the vertex set of $\mathcal{G}_{k+1}$ into $R$ disjoint consecutive intervals $I_{1},\allowbreak I_{2},\allowbreak\dots,\allowbreak I_{R}$, each of size $N_{k}$. Color the subgraph induced by each $I_{i}$ by $c_{k}$. The remaining edges form a complete $R$-partite ordered graph $\mathcal{F}_{k+1}$, which can be colored to avoid $\mathcal{K}_{r}$, in the following way. Suppose that $v_{1},v_{2},\dots,v_{R}$ are the vertices of $\mathcal{G}_{1}$. Then for every $i,j$, $1\leq i<j\leq R$, and for every edge $e$ of $\mathcal{F}_{k+1}$ with one vertex in $I_{i}$ and the other vertex in $I_{j}$, let $c_{k+1}(e)\mathrel{\mathop{:}}=c_{1}(\\{v_{i},v_{j}\\})$. Clearly, $N_{k}=R^{k}$. The matchings $\mathcal{M}_{k}$ are also constructed inductively. The basic building block is a matching $\mathcal{M}$ obtained by splitting the vertices of $\mathcal{K}_{r}$ completely and taking many shifted copies of the resulting matching; see Figure 12. More precisely, consider the integers $1,2,\dots,r^{2}R$ as vertices, and let $l_{i}\mathrel{\mathop{:}}=(i-1)rR$, for $1\leq i\leq r$. For every pair $i,j$, where $1\leq i<j\leq r$, we add the $R$ edges $\\{l_{i}+j,l_{j}+i\\},\allowbreak\\{l_{i}+j+r,l_{j}+i+r\\},\allowbreak\\{l_{i}+j+2r,l_{j}+i+2r\\},\allowbreak\dots,\allowbreak\\{l_{i}+j+(R-1)r,l_{j}+i+(R-1)r\\}$. Note that the vertices $l_{i}+i+kr$, where $1\leq i\leq r$ and $0\leq k<R$ are isolated and after their removal, we obtain the ordered matching $\mathcal{M}$ with $t\mathrel{\mathop{:}}=r(r-1)R$ vertices. Figure 12: The matching $\mathcal{M}$ for $r=3$. Let $n_{1}\mathrel{\mathop{:}}=t$ and $\mathcal{M}_{1}\mathrel{\mathop{:}}=\mathcal{M}$. Now let $k\geq 1$ and suppose that $\mathcal{M}_{k}$ has been constructed. Let $J_{1},\allowbreak L_{1},\allowbreak J_{2},\allowbreak L_{2},\allowbreak\dots,\allowbreak L_{r-1},\allowbreak J_{r}$ be an ordered sequence of disjoint intervals of vertices, of size $\|L_{i}\|=n_{k}$ and $\|J_{i}\|=(r-1)R)$. The matching $\mathcal{M}_{k+1}$ is obtained by placing a copy of $\mathcal{M}_{k}$ on each of the $r-1$ intervals $L_{i}$ and a copy of $\mathcal{M}$ on the union of the $r$ intervals $J_{i}$. See Figure 13. We have $n_{k+1}=(r-1)n_{k}+t$. Figure 13: Construction of $\mathcal{M}_{k+1}$ for $r=3$. Now we show that for every $k$, the coloring $c_{k}$ of $\mathcal{G}_{k}$ avoids $\mathcal{M}_{k}$. Trivially, $c_{1}$ avoids $\mathcal{M}_{1}$ since $n_{1}=t>R=N_{1}$. Let $k\geq 1$ and suppose that $c_{k}$ avoids $\mathcal{M}_{k}$. Let $I_{1},\dots,I_{s}$ be the intervals of vertices from the construction of $\mathcal{G}_{k+1}$ and let $J_{1},\allowbreak L_{1},\allowbreak\dots,\allowbreak L_{r-1},\allowbreak J_{r}$ be the intervals of vertices from the construction of $\mathcal{M}_{k+1}$. Assume that $\mathcal{M}_{k+1}$ is an arbitrary subgraph of $\mathcal{G}_{k+1}$. If two intervals $J_{j}$ and $J_{j+1}$ intersect some interval $I_{i}$, then $L_{j}\subset I_{i}$. Since $L_{j}$ induces $\mathcal{M}_{k}$ in $\mathcal{M}_{k+1}$ and $I_{i}$ induces $\mathcal{G}_{k}$ colored with $c_{k}$ in $\mathcal{G}_{k+1}$, the matching $\mathcal{M}_{k+1}$ is not monochromatic by induction. Thus we may assume that every interval $I_{i}$ is intersected by at most one interval $J_{j}$. Partition each interval $J_{j}$ into $R$ intervals $J_{j}^{1},\allowbreak J_{j}^{2},\allowbreak\dots,\allowbreak J_{j}^{R}$ of length $r-1$, in this order. At most $R-1$ of the $rR$ intervals $J_{j}^{l}$, $1\leq j\leq r$, $1\leq l\leq R$, contain vertices from at least two intervals $I_{i}$, $1\leq i\leq R$. Thus there is an $l$ such that for every $j$, $1\leq j\leq r$, the whole interval $J_{j}^{l}$ is contained in some interval $I_{i(j)}$. Moreover, all the intervals $I_{i(j)}$ are pairwise distinct by our assumption. By the construction of $\mathcal{M}_{k+1}$, there is exactly one edge $e_{j,j^{\prime}}$ in $\mathcal{M}_{k+1}$ between every two intervals $J_{j}^{l}$ and $J_{j^{\prime}}^{l}$. By the coloring of $\mathcal{F}_{k+1}$, we have $c_{k+1}(e_{j,j^{\prime}})=c_{1}(\\{v_{i(j)},v_{i(j^{\prime})}\\})$. Since the edges $\\{v_{i(j)},v_{i(j^{\prime})}\\}$ form a complete subgraph with $r$ vertices in $\mathcal{G}_{1}$ and $c_{1}$ avoids $\mathcal{K}_{r}$, we conclude that our copy of $\mathcal{M}_{k+1}$ in $\mathcal{G}_{k+1}$ is not monochromatic. It follows that $c_{k+1}$ avoids $\mathcal{M}_{k+1}$. Solving the recurrence for $n_{k}$, we get $n_{k}=(1+(r-1)+\cdots+(r-1)^{k-1})\cdot t<(r-1)^{k}\cdot t<r^{k+2}\cdot R.$ Let $c\mathrel{\mathop{:}}=(\log{R})/r$. By (1), we have $c\in[1/2,2)$. Let $k\mathrel{\mathop{:}}=\lfloor(cr/\log{r})-2\rfloor=(cr/\log{r})-2-\varepsilon$, where $\varepsilon\in[0,1)$. Let $n\mathrel{\mathop{:}}=n_{k}$, $N\mathrel{\mathop{:}}=N_{k}$ and $\mathcal{M}_{n}\mathrel{\mathop{:}}=\mathcal{M}_{k}$. We have $n=n_{k}<r^{k+2}\cdot R\leq 2^{cr+\log R}=2^{2cr}\hskip 8.5359pt\text{ and }\hskip 8.5359ptN=N_{k}=R^{k}=2^{crk}>2^{(c^{2}r^{2}/\log{r})-3cr}.$ Using these bounds together with the trivial bound $2^{cr}=R<n$, we derive $\displaystyle\log{N}-\frac{\log^{2}{n}}{5\log\log{n}}$ $\displaystyle>\frac{c^{2}r^{2}}{\log{r}}-3cr-\frac{4c^{2}r^{2}}{5(\log{r}+\log{c})}$ $\displaystyle=c^{2}r^{2}\left(\frac{1}{\log{r}}-\frac{3}{cr}-\frac{4}{5(\log{r}+\log{c})}\right)$ $\displaystyle>0$ if $r$ is large enough. The theorem follows. ∎ We remark that our colorings $c_{k}$ of the graphs $\mathcal{G}_{k}$ are not constructive, since we use the probabilistic lower bound from Ramsey’s theorem. Let us recall that unordered graphs with maximum degree of constant size have linear Ramsey numbers with respect to the number of vertices. From Theorem 9 we therefore get that the ratio between ordered and unordered Ramsey numbers of the same graph can grow super-polynomially. Combining the example of ordered matchings with linear ordered Ramsey numbers (see the second part of Figure 7) with Theorem 9 we see that there are $n$-vertex graphs $G_{n}$ with two vertex ordering schemes $\lhd_{1}$ and $\lhd_{2}$ such that $\operatorname{\overline{R}}(G_{n},\lhd_{1})$ remains linear while $\operatorname{\overline{R}}(G_{n},\lhd_{2})$ is super-polynomial in $n$. ## 4 Upper Bounds This section is divided into two parts. In each part we prove a polynomial upper bound for ordered Ramsey numbers of a certain class of sparse ordered graphs. ### 4.1 Proof of Theorem 10 In this part we prove Theorem 10. That is, we show that for fixed positive integers $k$, $q\geq 2$ and $(k,q)$-decomposable ordered graphs $\mathcal{G}$ and $\mathcal{H}$ we have $\operatorname{\overline{R}}(\mathcal{G},\mathcal{H})\leq C_{k}\cdot 2^{64k(\lceil\log_{q/(q-1)}{r}\rceil+\lceil\log_{q/(q-1)}{s}\rceil)}$ where $r$ is the number of vertices of $\mathcal{G}$, $s$ is the number of vertices of $\mathcal{H}$ and $C_{k}$ is a sufficiently large constant dependent on $k$. ###### Lemma 25. Let $\mathcal{K}_{N}$ be a complete ordered graph with edges colored red and blue. Then there is a set $U$ with at least $\lfloor N/(16\cdot 10^{5})\rfloor$ vertices of $\mathcal{K}_{N}$ satisfying one of the following: 1. 1. every vertex of $U$ has at least $N/11$ blue neighbors to the left and to the right of $U$, 2. 2. every vertex of $U$ has at least $N/11$ red neighbors to the left and to the right of $U$. ###### Proof. We assume $N\geq 16\cdot 10^{5}$, as otherwise the statement is trivial. First, we show that there is a set $W$ with at least $N/2000$ vertices satisfying one of the following: 1. 1. every vertex of $W$ has at least $\frac{20}{217}N$ blue left and at least $\frac{20}{217}N$ blue right neighbors, 2. 2. every vertex of $W$ has at least $\frac{20}{217}N$ red left and at least $\frac{20}{217}N$ red right neighbors. Let $B$ be the set of vertices of $\mathcal{K}_{N}$ that satisfy the first claim for $W$ and let $R$ be the set of vertices of $\mathcal{K}_{N}$ that satisfy the second claim for $W$. Suppose that $\|B\|<N/2000$ and $\|R\|<N/2000$, as otherwise we are done. Consider the complete ordered graph $\mathcal{K^{\prime}}$ induced by the vertex set of $\mathcal{K}_{N}$ with vertices from $B$ and $R$ removed. From the assumptions $\mathcal{K^{\prime}}$ has more than $(1-\frac{2}{2000})N=\frac{999}{1000}N$ vertices and it does not contain a monochromatic ordered star $\mathcal{S}_{t,t}$ for $t\mathrel{\mathop{:}}=\left\lceil\frac{20}{217}N\right\rceil+1$. Therefore the number of vertices of $\mathcal{K^{\prime}}$ is less than $\operatorname{\overline{R}}(\mathcal{S}_{t,t},\mathcal{S}_{t,t})$. Using Theorem 14 and the fact that $\operatorname{\overline{R}}(\mathcal{S}_{t,1},\mathcal{S}_{t,t})=\operatorname{\overline{R}}(\mathcal{S}_{1,t},\mathcal{S}_{t,t})$ we have $\displaystyle\operatorname{\overline{R}}(\mathcal{S}_{t,t},\mathcal{S}_{t,t})$ $\displaystyle=\operatorname{\overline{R}}(\mathcal{S}_{t,1},\mathcal{S}_{t,t})+\operatorname{\overline{R}}(\mathcal{S}_{1,t},\mathcal{S}_{t,t})-1=2(\operatorname{\overline{R}}(\mathcal{S}_{1,t},\mathcal{S}_{t,1})+2t-3)-1$ $\displaystyle=2\left(\left\lfloor\frac{-1+\sqrt{1+8(t-2)^{2}}}{2}\right\rfloor+2t-2+2t-3\right)-1<(8+2\sqrt{2})t.$ Altogether we have $\|V(\mathcal{K^{\prime}})\|>\frac{999}{1000}N>(8+2\sqrt{2})(\left\lceil\frac{20}{217}N\right\rceil+1)>\|V(\mathcal{K^{\prime}})\|$, a contradiction. Thus there is a set $W$ satisfying one of the claims, say the first one. Now, we find $U$ as a subset of $W$. To do so, partition the vertex set of $\mathcal{K}_{N}$ into $\frac{16\cdot 10^{5}}{2000}=800$ intervals $I_{1},\ldots,I_{800}$ such that each contains at least $\lfloor N/(16\cdot 10^{5})\rfloor$ vertices of $W$. This is possible as $\|W\|\geq N/2000$. Clearly, there is an interval $I_{i}$ in $\mathcal{K}_{N}$ with at most $N/800$ vertices of $\mathcal{K}_{N}$ which contains at least $\lfloor N/(16\cdot 10^{5})\rfloor$ vertices of $W$ and we can set $U\mathrel{\mathop{:}}=I_{i}\cap W$. Since every vertex of $U$ has at least $\frac{20}{217}N$ blue left neighbors, then it also contains at least $\frac{20}{217}N-N/800>N/11$ blue neighbors that are to the left of $I_{i}$ and hence to the left of $U$ as well. Analogous statement holds for blue right neighbors of every vertex from $U$ and hence $U$ satisfies the first part of the lemma. ∎ The following two statements are used in the proof of Theorem 10. The first one is a classical result, called the Kövari-Sós-Turán theorem [31], which estimates the maximum number of edges in a bipartite graph which does not contain a given complete bipartite graph as a subgraph. The second one is an upper bound on non-diagonal Ramsey numbers of complete graphs proved by Ajtai, Komlós, and Szemerédi [1]. ###### Theorem 26 ([2, 26, 31]). Let $\mathrm{Z}(m,n;s,t)$ be the maximum number of edges in a bipartite graph $G=(A\cup B,E)$, $\|A\|=m$, $\|B\|=n$, which does not contain $K_{s,t}$ as a subgraph with $s$ vertices in $A$ and $t$ vertices in $B$. Assuming $2\leq s\leq m$ and $2\leq t\leq n$, we have $\mathrm{Z}(m,n,;s,t)<(s-1)^{1/t}(n-t+1)m^{1-1/t}+(t-1)m.$ ###### Lemma 27 ([1]). For a fixed integer $p\geq 2$, we have $\operatorname{R}(K_{p},K_{n})\leq(5000)^{p}n^{p-1}/(\ln n)^{p-2}$ for $n$ sufficiently large (dependent on $p$). The bound from Lemma 27 holds also for ordered complete graphs, as $\operatorname{R}(K_{p},K_{n})=\operatorname{\overline{R}}(\mathcal{K}_{p},\mathcal{K}_{n})$ for every pair $(\mathcal{K}_{p},\mathcal{K}_{n})$ of ordered complete graphs. ###### Proof of Theorem 10. Let $\mathcal{G}$ and $\mathcal{H}$ be ordered graphs satisfying the assumptions for given $k$. Let $N=N_{k,q}(r,s)\mathrel{\mathop{:}}=C_{k}\cdot 2^{64k(\lceil\log_{q/(q-1)}{r}\rceil+\lceil\log_{q/(q-1)}{s}\rceil)}$ where $C_{k}$ is a large constant dependent on $k$ and let $\mathcal{K}_{N}$ be a complete ordered graph with edges colored red and blue. We want to find a blue copy of $\mathcal{G}$ or a red copy of $\mathcal{H}$ in $\mathcal{K}_{N}$. We proceed by induction on $\lceil\log_{q/(q-1)}{r}\rceil$ and $\lceil\log_{q/(q-1)}{s}\rceil$. Suppose that $\lceil\log_{q/(q-1)}{r}\rceil\leq\log_{q/(q-1)}{k}$. If $s$ is not sufficiently large with respect to $k$ to satisfy the assumptions of Lemma 27 for $p\mathrel{\mathop{:}}=2k$ and $n\mathrel{\mathop{:}}=s$, then we use Ramsey’s theorem. Having $C_{k}$ sufficiently large, we then find a blue copy of $\mathcal{K}_{r}$ or a red copy of $\mathcal{K}_{s}$ in $\mathcal{K}_{N}$. For other values of $s$ we can apply Lemma 27 with $p\mathrel{\mathop{:}}=2k$ and $n\mathrel{\mathop{:}}=s$, as we have $N>(5000)^{2k}s^{2k-1}/(\ln s)^{2k-2}$ for $C_{k}$ sufficiently large. Since $r<2k$, we find a blue copy of $\mathcal{K}_{r}$ or a red copy of $\mathcal{K}_{s}$ in $\mathcal{K}_{N}$. The case $\lceil\log_{q/(q-1)}{s}\rceil\leq\log_{q/(q-1)}{k}$ is analogous. Suppose that $\lceil\log{r}\rceil,\lceil\log{s}\rceil>\log_{q/(q-1)}{k}$ and that the bound holds for every pair $\mathcal{G}^{\prime}$ and $\mathcal{H}^{\prime}$ of ordered graphs satisfying the assumptions and having $r^{\prime}$ and $s^{\prime}$ vertices, respectively, where either $\lceil\log_{q/(q-1)}{r^{\prime}}\rceil<\lceil\log_{q/(q-1)}{r}\rceil$ or $\lceil\log_{q/(q-1)}{s^{\prime}}\rceil<\lceil\log_{q/(q-1)}{s}\rceil$. Let $U$ be the set from Lemma 25 and suppose that it satisfies the first part of this lemma. That is, $U$ is a set with at least $\lfloor N/(16\cdot 10^{5})\rfloor$ vertices of $\mathcal{K}_{N}$ such that every vertex of $U$ has at least $N/11$ blue neighbors to the left and to the right of $U$. If $s$ is not sufficiently large with respect to $k$ to satisfy the assumptions of Lemma 27 for $p\mathrel{\mathop{:}}=61k$ and $n\mathrel{\mathop{:}}=s$, then we use Ramsey’s theorem to the subgraph of $\mathcal{K}_{N}$ induced by $U$. Having $C_{k}$ sufficiently large, we then find a blue copy of $\mathcal{K}_{61k}$ or a red copy of $\mathcal{K}_{s}$. For other values of $s$ we can apply Lemma 27 with $p\mathrel{\mathop{:}}=61k$ and $n\mathrel{\mathop{:}}=s$, as we have $\|U\|\geq N/(16\cdot 10^{5})>(5000)^{61k}s^{61k-1}/(\ln s)^{61k-2}$ for $C_{k}$ sufficiently large. Again, we find either a red copy of $\mathcal{K}_{s}$ or a blue copy of $\mathcal{K}_{61k}$ in the subgraph of $\mathcal{K}_{N}$ induced by $U$. Suppose that we have found a blue copy of $\mathcal{K}_{61k}$, as otherwise we are done, and let $U_{1}\subset U$ be its vertex set. We now apply Theorem 26 twice to obtain a set $V\subset U_{1}$ of size $k$ which induces a blue copy of $\mathcal{K}_{k}$ and whose vertices have a common blue neighborhood of size at least $N/2^{64k}$ to the left of $V$ and also to the right of $V$. Denote by $J_{L}$ the interval of vertices of $\mathcal{K}_{N}$ that are to the left of $U_{1}$ and by $J_{R}$ the interval of vertices of $\mathcal{K}_{N}$ that are to the right of $U_{1}$. A trivial estimate gives us $N/11\leq\|J_{L}\|,\|J_{R}\|$. At least one of the intervals $J_{L}$ and $J_{R}$ has size at most $N/2$ and by symmetry we may assume that $\|J_{R}\|\leq N/2$. For the size of $J_{L}$ we have the estimate $\|J_{L}\|\leq\frac{10\cdot N}{11}$, as $\|J_{R}\|\geq N/11$ and $J_{R}$ and $J_{L}$ are disjoint. The number of blue edges between $U_{1}$ and $J_{L}$ is at least $\|U_{1}\|\cdot\|J_{L}\|/10$, as every vertex of $U_{1}$ has at least $N/11$ blue neighbors in $J_{L}$ and $\|J_{L}\|\leq\frac{10\cdot N}{11}$. For every $C_{1}>0$ we have $(\|J_{L}\|/C_{1}-1)^{1/6k}(\|U_{1}\|-6k+1)\|J_{L}\|^{1-1/6k}+(6k-1)\|J_{L}\|<\|J_{L}\|(\|U_{1}\|/C_{1}^{1/6k}+6k).$ If we choose $C_{1}\mathrel{\mathop{:}}=2^{10\cdot 6k}=2^{60k}$, then the right side of this inequality is at most $\|U_{1}\|\cdot\|J_{L}\|/10$, as $\|U_{1}\|=61k>\frac{C_{1}^{1/6k}}{C_{1}^{1/6k}-10}\cdot 60k$. Thus, according to Theorem 26, the number of blue edges between $U_{1}$ and $J_{L}$ is larger than $\mathrm{Z}(\|J_{L}\|,\|U_{1}\|;\|J_{L}\|/C_{1},6k)$ and we can find a complete bipartite graph $K_{6k,\|J_{L}\|/C_{1}}$ in the (unordered) bipartite graph induced by $U_{1}$ and $J_{L}$ where edges correspond to blue edges of $\mathcal{K}_{N}$ between those two sets. Since $J_{L}$ is to the left of $U_{1}$, this complete bipartite graph corresponds to a blue copy of $\mathcal{K}_{\|J_{L}\|/C_{1},6k}$ with the first color class in $J_{L}$ and the other one in $U_{1}$. We let $U_{2}\subset U_{1}$ be the set of $6k$ vertices induced by the second color class. Now, the number of blue edges between $U_{2}$ and $J_{R}$ is at least $\|U_{2}\|\cdot\|J_{R}\|\cdot\frac{2}{11}$, as every vertex in $U_{2}$ has at least $N/11$ blue neighbors in $J_{R}$ and $\|J_{R}\|\leq N/2$. Similarly as before, for every $C_{2}>0$ we have $(\|J_{R}\|/C_{2}-1)^{1/k}(\|U_{2}\|-k+1)\|J_{R}\|^{1-1/k}+(k-1)\|J_{R}\|<\|J_{R}\|(\|U_{2}\|/C_{2}^{1/k}+k).$ Choosing $C_{2}\mathrel{\mathop{:}}=2^{7k}$, the right side of this inequality is at most $\|U_{2}\|\cdot\|J_{R}\|\cdot\frac{2}{11}$, as $\|U_{2}\|=6k>\frac{C_{2}^{1/k}}{C_{2}^{1/k}-11/2}\cdot\frac{11}{2}k$. By Theorem 26 the number of blue edges between $U_{2}$ and $J_{R}$ is larger than $\mathrm{Z}(\|J_{R}\|,\|U_{2}\|;\|J_{R}\|/C_{2},k)$. Similarly as before, we obtain a blue copy of $\mathcal{K}_{k,\|J_{R}\|/C_{2}}$ between $U_{2}$ and $J_{R}$. The color class of size $k$ in the obtained complete bipartite graph forms a set $V\subset U_{2}$ with $k$ vertices that induce a blue copy of $\mathcal{K}_{k}$. Moreover, as $N/11\leq\|J_{L}\|,\|J_{R}\|$, the vertices of $V$ have at least $N/2^{(60+4)k}=N/2^{64k}$ common blue neighbors to the left and also to the right of $V$. We now use the facts that $\mathcal{G}$ is $(k,q)$-decomposable and that $r\geq k$. We partition vertices of $\mathcal{G}$ into three intervals $I_{L}$, $I$ and $I_{R}$ where $0<\|I\|\leq k$ and $\|I_{L}\|,\|I_{R}\|\leq\|V(G)\|\cdot\frac{q-1}{q}=\frac{r(q-1)}{q}$ such that $I$ is to the right of $I_{L}$ and to the left of $I_{R}$. Moreover, intervals $I_{L}$ and $I_{R}$, induce $(k,q)$-decomposable ordered graphs $\mathcal{G}_{L}$ and $\mathcal{G}_{R}$, respectively, and there is no edge between $\mathcal{G}_{L}$ and $\mathcal{G}_{R}$ (although vertices of $\mathcal{G}_{L}$ and $\mathcal{G}_{R}$ may have neighbors in $I$). From our choice of $N$, we have $\displaystyle N/2^{64k}$ $\displaystyle=C_{k}\cdot 2^{64k(\lceil\log_{q/(q-1)}{r}\rceil+\lceil\log_{q/(q-1)}{s}\rceil-1)}$ $\displaystyle=C_{k}\cdot 2^{64k(\lceil\log_{q/(q-1)}{r(q-1)/q}\rceil+\lceil\log_{q/(q-1)}{s}\rceil)}\geq N_{k,q}(\lfloor r(q-1)/q\rfloor,s)$ and so $N/2^{64k}\geq\operatorname{\overline{R}}(\mathcal{G}_{L},\mathcal{H}),\operatorname{\overline{R}}(\mathcal{G}_{R},\mathcal{H})$. Therefore we can find either a red copy of $\mathcal{H}$ or a blue copy of $\mathcal{G}_{L}$ in the common blue left neighborhood of $V$, using the inductive assumption. Similarly, we can find a red copy of $\mathcal{H}$ or a blue copy of $\mathcal{G}_{R}$ in the common blue right neighborhood of $V$. Suppose that we do not obtain a red copy of $\mathcal{H}$ in any of these two cases. Then we find a blue copy of $\mathcal{G}$ by choosing $\|I\|$ vertices of $V$ as a copy of $I$ and connect it to the blue copies of $\mathcal{G}_{L}$ and $\mathcal{G}_{R}$. If the set $U$ from Lemma 25 satisfies the second part of the lemma, then the proof would proceed analogously. The main differences are that we use the $(k,q)$-decomposability of the graph $\mathcal{H}$ instead of $\mathcal{G}$ and estimates $N/2^{64k}\geq N_{k,q}(r,\lfloor s(q-1)/q\rfloor)$ and $\frac{N}{16\cdot 10^{5}}\geq(5000)^{61k}r^{61k-1}/(\ln r)^{61k-2}$. ∎ ### 4.2 Proof of Theorem 12 Here we prove Theorem 12 which says that for every ordered graph $\mathcal{G}$ of constant degeneracy and constant interval chromatic number the ordered Ramsey number $\operatorname{\overline{R}}(\mathcal{G};2)$ is polynomial in the number of vertices of $\mathcal{G}$. In fact, we prove a stronger statement, Theorem 29, and derive Theorem 12 as an immediate corollary of this result. ###### Lemma 28. Let $k$ be a positive integer and let $\mathcal{G}$ be a $k$-degenerate ordered graph with $n$ vertices. Then in every complete ordered graph $\mathcal{K}_{N}$, $N\geq n^{2}$, with edges colored red and blue we find either a blue copy of $\mathcal{G}$ or a red copy of $\mathcal{K}_{t,t}$ for $t\mathrel{\mathop{:}}=(N/n^{2})^{1/(k+1)}$. ###### Proof. Let $\mathcal{G}=(G,\prec)$ and $\mathcal{K}_{N}$ be ordered graphs satisfying the assumptions for given $k$ and let $t\mathrel{\mathop{:}}=(N/n^{2})^{1/(k+1)}$. Since $N\geq n^{2}$, we have $t\geq 1$. We partition the vertices of $\mathcal{K}_{N}$ into $n$ disjoint consecutive intervals of length $N/n$. The $i$th such interval is denoted by $I(v)$ where $v$ is the $i$th vertex of $\mathcal{G}$ in the ordering $\prec$. We try to construct a blue copy $h(\mathcal{G})$ of $\mathcal{G}$ in $\mathcal{K}_{N}$, We show that in each step of the construction of $h(\mathcal{G})$ it is either possible to find a new image $h(w)$ of a vertex $w$ of $\mathcal{G}$ or we have a red copy of $\mathcal{K}_{t,t}$ in $\mathcal{K}_{N}$. For every vertex $v$ of $\mathcal{G}$ that does not have an image $h(v)$ yet, we keep a set $U(v)\subseteq V(\mathcal{K}_{N})$ of possible candidates for $h(v)$. At the beginning we set $U(v)\mathrel{\mathop{:}}=I(v)$ for every $v\in V(G)$. Let $\lessdot$ be an ordering of the vertices of $\mathcal{G}$ such that every vertex $v$ of $\mathcal{G}$ has at most $k$ left neighbors in $\lessdot$. This ordering exists as $\mathcal{G}$ is $k$-degenerate. Note that the ordering $\lessdot$ might differ from the ordering $\prec$. Let $w$ be the first vertex of $\mathcal{G}$ in the ordering $\lessdot$ that does not have an image $h(w)$ yet. Suppose that $u_{1},\ldots,u_{s}\in V(G)$ are the right neighbors of $w$ in the ordering $\lessdot$. Clearly, we have $s\leq n$. We show how to find the image $h(w)$ or a red copy of $\mathcal{K}_{t,t}$ in $\mathcal{K}_{N}$. Let $i$ be an arbitrary element of the set $\\{1,,\ldots,s\\}$. We claim that in $U(w)$ every vertex except of at most $t-1$ vertices has at least $\frac{\|U(u_{i})\|}{t}$ blue neighbors in $U(u_{i})$ or that we can find a red copy of $\mathcal{K}_{t,t}$ with one color class in $U(w)$ and the other in $U(u_{i})$. Suppose first that there is a subset $W$ of $U(w)$ with $t$ vertices such that each vertex of $W$ has less than $\frac{\|U(u_{i})\|}{t}$ blue neighbors in $U(u_{i})$. In such a case we delete from $U(u_{i})$ every vertex that is a blue neighbor of some vertex of $W$. Afterwards, there is still at least $\|U(u_{i})\|-\|W\|\cdot\left(\frac{\|U(u_{i})\|}{t}-1\right)=\|U(u_{i})\|-t\cdot\left(\frac{\|U(u_{i})\|}{t}-1\right)=t$ vertices left in $U(u_{i})$ and every such vertex has no blue neighbors in $W$. Therefore we have a red copy of $\mathcal{K}_{t,t}$ in $\mathcal{K}_{N}$. Therefore we either have a red copy of $\mathcal{K}_{t,t}$ in $\mathcal{K}_{N}$ or there is a set $C_{w}$ with at least $\|U(w)\|-s(t-1)>\|U(w)\|-nt$ vertices in $U(w)$ such that every vertex of $C_{w}$ has at least $\frac{\|U(u_{i})\|}{t}$ blue neighbors in $U(u_{i})$ for every $i=1,\ldots,s$. We may assume that the second case occurs, as otherwise we are done. We choose an arbitrary vertex $h(w)$ of $C_{w}$ to be the image of $w$ in the constructed blue copy $h(\mathcal{G})$ of $\mathcal{G}$. We show that $C_{w}$ is nonempty and that this is possible at the end of the proof. We update the sets $U(u_{1}),\ldots,U(u_{s})$ by setting $U(u_{i})$ to be the set of at least $\|U(u_{i})\|/t$ blue neighbors of $h(w)$ in $U(u_{i})$ for every $i=1,\ldots,s$. After these updates, we, again, choose the first vertex in $\lessdot$ that does not have an image yet and proceed as before. If there is no such vertex, then we have found a blue copy $h(\mathcal{G})$ of $\mathcal{G}$ in $\mathcal{K}_{N}$ as there is an image of every vertex of $\mathcal{G}$. It remains to show that the set $C_{w}$ is nonempty. We know that before choosing the image $h(w)$ the size of $C_{w}$ is strictly larger than $\|U(w)\|-nt$. In the previous steps of the construction of $h(\mathcal{G})$ we have updated $U(w)$ at most $k$ times, as $w$ has at most $k$ left neighbors in $\lessdot$. The size of $U(w)$ is divided by at most $t$ in every update. Since the size of $U(w)$ is exactly $N/n$ at the beginning of the construction of $h(\mathcal{G})$, the inequality $(N/n)\cdot(1/t)^{k}-nt\geq 0$ implies that there is at least one vertex in $C_{w}$. This inequality can be rewritten as $t\leq(N/n^{2})^{1/(k+1)}$ which is satisfied by the choice of $t$. This finishes the proof. ∎ ###### Theorem 29. For positive integers $k$ and $p$, let $\mathcal{G}$ be a $k$-degenerate ordered graph with $n$ vertices. Then we have $\operatorname{\overline{R}}(\mathcal{G},\mathcal{K}_{n}(p))\leq n^{(1+2/k)(k+1)^{\lceil\log{p}\rceil}-2/k}.$ ###### Proof. First, we define the function $f_{k,n}(l)\colon\mathbb{N}\to\mathbb{N}$ as $f_{k,n}(l)\mathrel{\mathop{:}}=n^{(1+2/k)(k+1)^{l}-2/k}.$ Note that this function satisfies the recurrence equation $f_{k,n}(l)=n^{2}\cdot f_{k,n}(l-1)^{k+1}$ with the initial condition $f(1)=n^{k+3}$. Without loss of generality we may assume that $p$ is a power of two. That is, we have $p=2^{l}$ for some positive integer $l$. We proceed by induction on $l$. The first step $l=1$ follows immediately from Lemma 28 applied to $\mathcal{K}_{N}$ with $N\mathrel{\mathop{:}}=f_{k,n}(1)=n^{k+3}$. Suppose that we have $l\geq 2$. Let $\mathcal{K}_{N}$ be an ordered complete graph with $N\mathrel{\mathop{:}}=f_{k,n}(l)$ vertices and edges colored red and blue. We show that there is always either a blue copy of $\mathcal{G}$ or a red copy of $\mathcal{K}_{n}(p)$ in $\mathcal{K}_{N}$. According to Lemma 28, there is either a blue copy of $\mathcal{G}$ or a red copy of $\mathcal{K}_{t,t}$ for $t\mathrel{\mathop{:}}=(N/n^{2})^{1/(k+1)}$. In the first case we are done, so suppose that the second case occurs. Let $A$ ($B$, respectively) be the left (right, respectively) color class of size $t$ in the red copy of $\mathcal{K}_{t,t}$. We use $\mathcal{K}[A]$ and $\mathcal{K}[B]$ to denote the two complete ordered subgraphs of $\mathcal{K}_{N}$ induced by $A$ and $B$, respectively. These subgraphs are considered together with the corresponding red-blue colorings of their edges. From the choice of $N$, the ordered subgraph $\mathcal{K}[A]$ contains at least $f_{k,n}(l-1)$ vertices. Therefore there is either a blue copy of $\mathcal{G}$ or a red copy of $\mathcal{K}_{n}(p/2)$ in $\mathcal{K}[A]$ by the inductive assumption. An analogous statement holds for the ordered subgraph $\mathcal{K}[B]$. Thus, if we do not find a blue copy of $\mathcal{G}$ in $\mathcal{K}[A]$ nor in $\mathcal{K}[B]$, then the two red copies of $\mathcal{K}_{n}(p/2)$ together with the red edges between $\mathcal{K}[A]$ and $\mathcal{K}[B]$ give us a red copy of $\mathcal{K}_{n}(p)$ in $\mathcal{K}_{N}$. ∎ ###### Proof of Theorem 12. Theorem 12 follows immediately from Theorem 29 and the fact that every ordered graph $\mathcal{G}$ with $n$ vertices and with interval chromatic number $p$ is an ordered subgraph of $\mathcal{K}_{n}(p)$. ∎ ## 5 Open problems Studying ordered graphs in Ramsey theory offers a plenty of new questions. We still do not know exact formulas for a wide spectrum of graphs, but it might be more interesting to ask questions which concern the structure of optimal colorings. For example, we can ask what properties of vertex orderings make ordered Ramsey numbers grow slower or faster and whether there is some characterization of ordered graphs with small ordered Ramsey numbers. Let us define the maximum ordered Ramsey number $\operatorname{\overline{R}}_{max}(G)$ for a graph $G$ as the maximum of $\operatorname{\overline{R}}((G,\prec);2)$ taken over all possible vertex orderings of $G$. Analogously, we can define the minimum ordered Ramsey number $\operatorname{\overline{R}}_{min}(G)$ of a graph $G$. An interesting question is whether there is a formula for the numbers $\operatorname{\overline{R}}_{max}(G_{n})$ or $\operatorname{\overline{R}}_{min}(G_{n})$ where $G_{n}$ are graphs on $n$ vertices from some specified class of graphs (such as paths, cycles, cliques, etc.). Also for which orderings of $G$ are these values attained. We can answer this question for stars using Theorem 14. We might also ask whether for every $n$-vertex bounded-degree graph $G$ there is an ordering of its vertices with the ordered Ramsey number linear in $n$. ###### Question 1. Is it true that for every graph $G$ on $n$ vertices with degrees bounded by a constant $\Delta$ there exists an ordering $\prec$ such that $\operatorname{\overline{R}}((G,\prec);2)\leq C\cdot n$ where $C=C(\Delta)$ is a constant which depends on $\Delta$? This result, if true, would be a natural strengthening of the fact that bounded-degree (unordered) graphs have linear Ramsey numbers. If the answer is negative, then there is a graph with bounded degrees and super-linear ordered Ramsey numbers for every ordering of its vertices. Theorem 12 implies that ordered Ramsey numbers of bounded-degree graphs with constant-size interval chromatic number are polynomial in the number of vertices. We do not know whether this bound is tight, as we have no non- trivial lower bounds. ###### Question 2. 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1310.7231	# Redshift drift in varying speed of light cosmology Adam Balcerzak [email protected] Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland Copernicus Center for Interdisciplinary Studies, Sławkowska 17, 31-016 Kraków, Poland Mariusz P. Da̧browski [email protected] Institute of Physics, University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland Copernicus Center for Interdisciplinary Studies, Sławkowska 17, 31-016 Kraków, Poland ###### Abstract We derive a redshift drift formula within the framework of varying speed of light (VSL) theory using the specific ansatz for the variability of $c(t)=c_{0}a^{n}(t)$. We show that negative values of the parameter $n$, which correspond to diminishing value of the speed of light during the evolution of the universe, effectively rescales dust matter to become little negative pressure matter, and the cosmological constant to became phantom. Positive values of $n$ (growing $c(t)$) make VSL model to become more like Cold Dark Matter (CDM) model. Observationally, there is a distinction between the VSL model and the $\Lambda$CDM model for the admissible values of the parameter $n\sim-10^{-5}$, though it will be rather difficult to detect by planned extremely large telescopes (E-ELT, TMT, GMT) within their accuracy. ###### pacs: 98.80.-k; 98.80.Es; 98.80.Cq ## I Introduction The early idea of variation of physical constants varconst has been established widely in physics both theoretically and experimentally uzan . The gravitational constant $G$, the charge of electron $e$, the velocity of light $c$, the proton to electron mass ratio $\mu=m_{p}/m_{e}$, and the fine structure constant $\alpha=e^{2}/\hbar c$, where $\hbar$ is the Planck constant, may vary in time and space barrowbook . The earliest and best-known framework for varying $G$ theories has been Brans-Dicke theory bd . Nowadays, the most popular theories which admit physical constants variation are the varying $\alpha$ theories alpha , and the varying speed of light $c$ theories uzanLR ; VSL . The latter, which will be the interest of our paper, allow the solution of the standard cosmological problems such as the horizon problem, the flatness problem, the $\Lambda-$problem, and has recently been proposed to solve the singularity problem JCAP13 . Recently, lots of interest has been attracted by the effect of redshift drift in cosmological models. This effect was first noticed by Sandage and later explored by Loeb sandage+loeb . The idea is to collect data from the two light cones separated by the time period of 10-20 years to look for the change of redshift of a source in time $\Delta z/\Delta t$ as a function of redshift of this source. The effect has recently been investigated for the inhomogeneous density Lemaitre-Tolman-Bondi models LTB ; UCETolman , the Dvali-Gabadadze- Porrati (DGP) brane model Quercellini12 , backreaction timescape cosmology wiltshire , axially symmetric Szekeres models marieN12 , inhomogeneous pressure Stephani models PRD13 . In Ref.Quercellini12 the drift for the $\Lambda$CDM model, the Dvali-Gabadadze-Porrati (DGP) brane model, the matter- dominated model (CDM), and three different LTB void models have been presented. It has been shown that the drift for $\Lambda$CDM and DGP models is positive up to $z\approx 2$ and becomes negative for larger redshifts, while it is always negative for LTB void models LTB ; yoo . The drift for Stephani models becomes positive for small redshifts and approaches the behavior of the $\Lambda$CDM model, which allows negative values of the drift, for very high redshifts PRD13 . The effect of varying constants theories including VSL theories onto the redshift drift has not yet been investigated and that is the motivation for this work. Our paper is organized as follows. In Sec. II we formulate the basics of the varying speed of light (VSL) theory and define observational parameters such as the dimensionless energy density parameters $\Omega$, Hubble parameter $H$, deceleration parameter $q$, as well as the higher derivative parameters like jerk $j$, snap $s$ etc. jerk ; snap ; weinberg which may serve as indicators of the equation of state (statefinders) and the curvature of the universe. In Sec. III we derive the redshift drift formula for the VSL cosmology using special ansatz for the time dependence of the speed of light $c(t)=c_{0}a^{n}(t)$, where $a(t)$ is the scale factor and $c_{0}$, $n$ are constants. In Sec. IV we give our conclusions. ## II Varying speed of light theory Following the Ref. VSL , we consider the Friedmann universes within the framework of varying speed of light theories (VSL) with the metric $ds^{2}=-(dx^{0})^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-Kr^{2}}+r^{2}(d\theta^{2}+\sin^{2}{\theta}d\varphi^{2})\right]~{}~{},$ (II.1) where $dx^{0}=c(t)dt$, for which the field equations read $\displaystyle\varrho(t)$ $\displaystyle=$ $\displaystyle\frac{3}{8\pi G}\left(\frac{\dot{a}^{2}}{a^{2}}+\frac{Kc^{2}(t)}{a^{2}}\right)~{},$ (II.2) $\displaystyle p(t)$ $\displaystyle=$ $\displaystyle-\frac{c^{2}(t)}{8\pi G}\left(2\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}+\frac{Kc^{2}(t)}{a^{2}}\right)~{},$ (II.3) and the energy-momentum conservation law is $\dot{\varrho}(t)+3\frac{\dot{a}}{a}\left(\varrho(t)+\frac{p(t)}{c^{2}(t)}\right)=3\frac{Kc(t)\dot{c}(t)}{4\pi Ga^{2}}~{}.$ (II.4) Here $a\equiv a(t)$ is the scale factor, the dot means the derivative with respect to time $t$, $G$ is the gravitational constant, $c=c(t)$ is time- varying speed of light, and the curvature index $K=0,\pm 1$. In most of the paper we will follow the ansatz for the speed of light given in Ref. BM99 , i.e., $c(t)=c_{0}a^{n}(t)~{}~{},$ (II.5) with the constant speed of light limit $n\to 0$ giving $c(t)\to c_{0}$. We have $\dot{c}/c=n\dot{a}/a$, so the speed of light grows in time for $n>0$, and diminishes for $n<0$. The cosmological observables which characterize the kinematic evolution of the universe are PLB05 : the Hubble parameter $H=\frac{\dot{a}}{a}~{},$ (II.6) the deceleration parameter $q=-\frac{1}{H^{2}}\frac{\ddot{a}}{a}=-\frac{\ddot{a}a}{\dot{a}^{2}}~{},$ (II.7) the jerk parameter jerk $j=\frac{1}{H^{3}}\frac{\dddot{a}}{a}=\frac{\dddot{a}a^{2}}{\dot{a}^{3}}~{},$ (II.8) and the snap snap parameter $s=-\frac{1}{H^{4}}\frac{\ddddot{a}}{a}=-\frac{\ddddot{a}a^{3}}{\dot{a}^{4}}~{}.$ (II.9) We can carry on with these and define even the higher derivative parameters such as lerk (crack), merk (pop), etc. PLB05 ; ANN06 ; gibbons by $x^{(i)}=(-1)^{i+1}\frac{1}{H^{i}}\frac{a^{(i)}}{a}=(-1)^{i+1}\frac{a^{(i)}a^{i-1}}{\dot{a}^{i}}~{}~{},$ (II.10) where $i=2,3,...$, and $a^{(i)}$ means the i-th derivative with respect to time while $a^{i}$ means the n-th power. We have consecutively: $q$ for $i=2$, $j$ for $i=3$ etc. A comparison of cosmological models with observational data requires the introduction of dimensionless density parameters $\displaystyle\Omega_{m0}$ $\displaystyle=$ $\displaystyle\frac{8\pi G}{3H_{0}^{2}}\varrho_{m0},$ (II.11) $\displaystyle\Omega_{K0}$ $\displaystyle=$ $\displaystyle\frac{Kc_{0}^{2}a_{0}^{2n}}{H_{0}^{2}a_{0}^{2}},$ (II.12) $\displaystyle\Omega_{\Lambda_{0}}$ $\displaystyle=$ $\displaystyle\frac{\Lambda_{0}c_{0}^{2}a_{0}^{2n}}{3H_{0}^{2}},$ (II.13) for dust, curvature, and dark energy, respectively. The index ”0” means that we take these parameters at the present moment of the evolution $t=t_{0}$. The following relations are valid ANN06 $\displaystyle\Omega_{K0}$ $\displaystyle=$ $\displaystyle\frac{3}{2}\Omega_{m0}-(q_{0}+n)-1,$ (II.14) $\displaystyle j_{0}$ $\displaystyle=$ $\displaystyle\Omega_{m0}+\Omega_{\Lambda 0}\left(n+1\right)-n\Omega_{K0},$ (II.15) $\displaystyle\Omega_{\Lambda 0}\left(n+1\right)$ $\displaystyle=$ $\displaystyle\frac{1}{2}\Omega_{m0}-q_{0}+n\Omega_{K0},$ (II.16) and so $j_{0}+1+\Omega_{K0}=3\Omega_{m0}-2q_{0}-n.$ (II.17) ## III Redshift drift in varying speed of light theory We consider redshift drift effect in VSL theory. In order to do that we assume that the source does not possess any peculiar velocity, so that it maintains a fixed comoving coordinate $dr=0$. The light emitted by the source at two different moments of time $t_{e}$ and $t_{e}+\delta t_{e}$ in VSL universe will be observed at $t_{o}$ and $t_{o}+\delta t_{o}$ related by $\int_{t_{e}}^{t_{o}}\frac{c(t)dt}{a(t)}=\int_{t_{e}+\Delta t_{e}}^{t_{o}+\Delta t_{o}}\frac{c(t)dt}{a(t)}~{},$ (III.1) which for small $\Delta t_{e}$ and $\Delta t_{o}$ transforms into $\frac{c(t_{e})\Delta t_{e}}{a(t_{e})}=\frac{c(t_{0})\Delta t_{o}}{a(t_{o})}~{}.$ (III.2) The definition of redshift in VSL theories remains the same as in standard Einstein relativity and reads as BM99 $1+z=\frac{a(t_{0})}{a(t_{e})}~{}~{}.$ (III.3) The redshift drift is defined as sandage+loeb $\displaystyle\Delta z=z_{e}-z_{0}=\frac{a(t_{0}+\Delta t_{0})}{a(t_{e}+\Delta t_{e})}-\frac{a(t_{0})}{a(t_{e})}~{},$ (III.4) which can be expanded in series (cf. Appendix) and to first order in $\Delta t$ reads as $\displaystyle\Delta z=\frac{a(t_{0})+\dot{a}(t_{0})\Delta t_{0}}{a(t_{e})+\dot{a}(t_{e})\Delta t_{e}}-\frac{a(t_{0})}{a(t_{e})}$ $\displaystyle\approx\frac{a(t_{0})}{a(t_{e})}\left[\frac{\dot{a}(t_{0})}{a(t_{0})}\Delta t_{0}-\frac{\dot{a}(t_{e})}{a(t_{e})}\Delta t_{e}\right]~{}~{}.$ (III.5) Using (III.2) we have $\Delta z=\Delta t_{0}\left[H_{0}(1+z)-H(t_{e})\frac{c(t_{0})}{c(t_{e})}\right]~{}~{},$ (III.6) which after applying the ansatz (II.5) gives $\frac{\Delta z}{\Delta t_{0}}=\frac{\Delta z}{\Delta t_{0}}(z,n)=H_{0}(1+z)-H(z)(1+z)^{n}~{}~{}.$ (III.7) In the limit $n\to 0$ the formula (III.7) reduces to the standard constant speed of light Friedmann universe formula obtained by Sandage and Loeb sandage+loeb . Bearing in mind the definitions $\Omega$’s and assuming $K=0$ we have $H^{2}(z)=H_{0}^{2}\left[\Omega_{m0}(1+z)^{3}+\Omega_{\Lambda}\right]$ (III.8) and so (III.7) gives $\displaystyle\frac{\Delta z}{\Delta t_{0}}=H_{0}\left[1+z-(1+z)^{n}\sqrt{\Omega_{m0}(1+z)^{3}+\Omega_{\Lambda}}\right]$ (III.9) $\displaystyle=$ $\displaystyle H_{0}\left[1+z-\sqrt{\Omega_{m0}(1+z)^{3+2n}+\Omega_{\Lambda}(1+z)^{2n}}\right]$ which can further be rewritten to define new redshift function $\tilde{H}(z)\equiv(1+z)^{n}H(z)=H_{0}\sqrt{\sum_{i=1}^{i=k}\Omega_{wi}(1+z)^{3(w_{eff}+1)}}~{}~{},$ (III.10) where $w_{eff}=w_{i}+\frac{2}{3}n~{}~{}.$ (III.11) Using (III.9) we present a plot of the redshift drift in VSL models in Fig. 1. For the negative values of the parameter $n$ which correspond to diminishing value of the speed of light during the evolution of the universe, it effectively rescales dust matter to become little negative pressure matter, and the cosmological constant to became phantom phantom . In other words, both components become extra sources of dark energy. Positive values of $n$ (growing $c(t)$) make VSL model to become more like Cold Dark Matter (CDM) model. Then, both matter components (dust, cosmological term) become extra sources of dark energy for $n\sim-10^{-5}<0$ which is in agreement with observational data murphy2007 ; king2012 . In Fig. 1 the theoretical error bars are taken from Ref. Quercellini12 and presumably show that for $\|n\|<0.045$ one cannot distinguish between VSL models and $\Lambda$CDM models. However, if the bars are reduced, then the influence of varying $c$ onto the evolution of the universe may perhaps be distinguishable. Figure 1: The redshift drift effect (III.9) for 15 year period of observations for various values of the varying speed of light parameter $n$. Negative $n$ correponds to $\dot{c}<0$. The error bars are taken from Ref. Quercellini12 and presumably show that for $\|n\|<0.045$ one cannot distinguish between VSL models and $\Lambda$CDM models. Larger positive values of $n$ (growing $c(t)$) make VSL model to become more like Cold Dark Matter (CDM) dust model. Different predictions for redshift drift in various cosmological models can be tested in future telescopes such as the European Extremely Large Telescope (EELT) (with its spectrograph CODEX (COsmic Dynamics EXperiment)) balbi ; E-ELT , the Thirty Meter Telescope (TMT), the Giant Magellan Telescope (GMT), and especially, in gravitational wave interferometers DECIGO/BBO (DECi-hertz Interferometer Gravitational Wave Observatory/Big Bang Observer) DECIGO . The first class of the experiments involving the very sensitive spectrographic techniques such as those utilized in the CODEX spectrograph use a detection of a very slow time variation of the Lyman-$\alpha$ forest of the number of quasars uniformly distributed all over the sky to measure the redshift drift, but Lyman-$\alpha$ lines become impossible to measure for $z<1.7$ from the ground E-ELT . The lower range of redshifts can be investigated though in other class of future experiments involving the space-borne gravitational wave interferometers DECIGO/BBO DECIGO . A detection could be possible even at $z\sim 0.2$. ## IV Conclusions We have calculated a redshift drift formula in varying speed of light theory. The formula is valid for any time dependence of the velocity of light though we have used the specific ansatz for the variability of $c(t)=c_{0}a^{n}(t)$ in order to discuss the effect of varying $c$ onto the redshift change over the evolution of the universe. We have shown that for observationally admissible negative values of the parameter $n\sim-10^{-5}<0$ ($\dot{c}(t)<0$) all the components of the universe behave as extra sources of dark energy. On the other hand, positive values of $n$ ($\dot{c}(t)>0$) make VSL models to decelerate and behave more like Cold Dark Matter (CDM) models. By using the theoretical error bars from Ref. Quercellini12 we have shown (cf. Fig. 1) that for $\|n\|<0.045$ one basically cannot distinguish between VSL models and $\Lambda$CDM models. However, if the bars are reduced, then the influence of varying $c$ onto the evolution of the universe may perhaps be distinguishable. In any case, the redshift drift will become an independent test of the VSL universe since it potentially shows the difference from the $\Lambda$CDM universe. The potential detection of the effect of redshift drift will be possible by extremely large telescopes such as EELT, TMT, and GMT. There is also some hope that these experiments give better accuracy in space-born future gravitational wave detectors such as DECIGO/BBO. It is worth mentioning that our derivation of redshift drift formula (III.7) would even fit better the prospective data, if the ansatz $c(t)=c_{0}a^{n(t)}$ of Ref. BM99 was applied. With such a variable $n$ parameter ansatz, one would be able to match the variability of $c$ with the cosmic evolution following the suggestion of BM99 in the sense that $n$ was larger ($n=-2.2$) in the radiation epoch, and then it was gradually diminishing to reach the value $n\sim-10^{-5}<0$ which is compatible with the current observational constraints on $c\propto{\alpha}^{-1}$ murphy2007 ; king2012 . ## V Acknowledgements This project was financed by the National Science Center Grant DEC-2012/06/A/ST2/00395. ## Appendix A Higher-order statefinder redshift drift formula The scale factor $a(t)$ at any moment of time $t$ can be obtained as series expansion around $t_{0}$ as ($a(t_{0})\equiv a_{0}$) ANN06 $\displaystyle a(t)=a_{0}\left\\{1+H_{0}(t-t_{0})-\frac{1}{2!}q_{0}H_{0}^{2}(t-t_{0})^{2}\right.$ (A.1) $\displaystyle\left.+\frac{1}{3!}j_{0}H_{0}^{3}(t-t_{0})^{3}-\frac{1}{4!}s_{0}H_{0}^{4}(t-t_{0})^{4}+O[(t-t_{0})^{5}]\right\\}~{},$ and its inverse reads as $\displaystyle\frac{a_{0}}{a(t)}=1+z=1+H_{0}(t_{0}-t)+H_{0}^{2}\left(\frac{q_{0}}{2}+1\right)(t_{0}-t)^{2}$ $\displaystyle+H_{0}^{3}\left(q_{0}+\frac{j_{0}}{6}+1\right)(t_{0}-t)^{3}$ (A.2) $\displaystyle+H_{0}^{4}\left(1+\frac{j_{0}}{3}+\frac{q_{0}^{2}}{4}+\frac{3}{2}q_{0}+\frac{s_{0}}{24}\right)(t_{0}-t)^{4}$ $\displaystyle+O[(t_{0}-t)^{5}]~{}.$ Using (A.1) and (A), the redshift drift formula (III.4) can be expanded up to higher order characteristics of the expansion $q_{0}$, $j_{0}$, and $s_{0}$ as $\displaystyle\Delta z=\frac{a(t_{0})}{a(t_{e})}\left[{\bf H_{0}\Delta t_{0}-H_{e}\Delta t_{e}}-H_{0}H_{e}\Delta t_{0}\Delta t_{e}\right.$ $\displaystyle\left.-\frac{1}{2}q_{0}H_{0}^{2}(\Delta t_{0})^{2}+H_{e}^{2}\left(\frac{q_{e}}{2}+1\right)(\Delta t_{e})^{2}\right.$ $\displaystyle\left.+\frac{1}{3!}j_{0}H_{0}^{3}(\Delta t_{0})^{3}-H_{e}^{3}\left(\frac{j_{e}}{3}+q_{e}+1\right)(\Delta t_{e})^{2}\right.$ $\displaystyle\left.+H_{0}H_{e}^{2}\left(\frac{q_{e}}{2}+1\right)(\Delta t_{0})(\Delta t_{e})^{2}\right.$ $\displaystyle\left.+\frac{1}{2}q_{0}H_{0}^{2}H_{e}^{2}\left(\frac{q_{e}}{2}+1\right)(\Delta t_{0})^{2}(\Delta t_{s})^{2}\right.$ $\displaystyle\left.-\frac{1}{4}s_{0}H_{0}^{4}(\Delta t_{0})^{4}+H_{e}^{4}\left(1+\frac{j_{e}}{3}+\frac{q_{e}^{2}}{4}+\frac{3}{2}q_{e}+\frac{s_{e}}{24}\right)\right.$ $\displaystyle\left.+(\Delta t_{e})^{4}+O\left[(\Delta t)^{5}\right]\right]~{}~{},$ (A.3) where only the first two terms appear in the first order formula (III.4). ## References * (1) H. 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1310.7234	# A Spectral Study of the Linearized Boltzmann Equation for Diffusively Excited Granular Media Thomas Rey Thomas Rey CSCAMM, The University of Maryland CSIC Building, Paint Branch Drive College Park, MD 20740 USA [email protected] ###### Abstract. In this work, we are interested in the spectrum of the diffusively excited granular gases equation, in a space inhomogeneous setting, linearized around an homogeneous equilibrium. We perform a study which generalizes to a non-hilbertian setting and to the inelastic case the seminal work of Ellis and Pinsky [8] about the spectrum of the linearized Boltzmann operator. We first give a precise localization of the spectrum, which consists in an essential part lying on the left of the imaginary axis and a discrete spectrum, which is also of nonnegative real part for small values of the inelasticity parameter. We then give the so-called inelastic “dispersion relations”, and compute an expansion of the branches of eigenvalues of the linear operator, for small Fourier (in space) frequencies and small inelasticity. One of the main novelty in this work, apart from the study of the inelastic case, is that we consider an exponentially weighted $L^{1}(m^{-1})$ Banach setting instead of the classical $L^{2}(\mathcal{M}_{1,0,1}^{-1})$ Hilbertian case, endorsed with Gaussian weights. We prove in particular that the results of [8] holds also in this space. ###### Key words and phrases: Inelastic Boltzmann equation, granular gases, spectrum, dispersion relations, hydrodynamic limit, heat equation ###### 2010 Mathematics Subject Classification: Primary: 76P05, 82C40, Secondary: 35P20, 76T25 ###### Contents 1. 1 Introduction 1. 1.1 The Model Considered 2. 1.2 The Linearized Operator 3. 1.3 Functional Framework and Main Results 4. 1.4 Method of Proof and Plan of the Paper 2. 2 Localization of the Spectrum 1. 2.1 Geometry of the Essential Spectrum 2. 2.2 Behavior of the Eigenvalues for Small Inelasticity 3. 3 Inelastic Dispersion Relations 1. 3.1 Projection of the Eigenvalue Problem 2. 3.2 Finite Dimensional Resolution 3. 3.3 First Order Coefficients of the Taylor Expansion 4. 3.4 Higher Order Expansion 4. A Functional Toolbox on the Collision Operator ## 1\. Introduction Let $f^{\varepsilon}:=f^{\varepsilon}(t,x,v)$ be a solution to the space inhomogeneous collisional kinetic equation (1) $\frac{\partial f^{\varepsilon}}{\partial t}+v\cdot\nabla_{x}f^{\varepsilon}=\frac{1}{\varepsilon}\left(\mathcal{Q}_{\alpha}(f^{\varepsilon},f^{\varepsilon})+\varepsilon\,\Delta_{v}(f^{\varepsilon})\right),$ where $t\geq 0$, $v\in\mathbb{R}^{d}$ and $x\in\Omega$, for $\Omega$ being either the whole space domain $\mathbb{R}^{d}$ or the torus111The case of a square domain $[-L,L]^{d}$, for $L\geq 0$ with specular reflection on the boundary can also be seen as a particular case of a torus made of $2^{d}$ independent copies of the initial box, using the parity of the normal component of the velocity of $f^{\varepsilon}$ at the boundary (as noticed by Grad in [10]). $\mathbb{T}^{d}$. The collision operator $\mathcal{Q}_{\alpha}$ is the so-called _granular gases_ operator (sometimes known as the inelastic Boltzmann operator), describing an energy-dissipative microscopic collision dynamics, which we will present in the following section. The parameter $\varepsilon>0$ is the scaled _Knudsen_ number, that is the ratio between the mean free path of particles before a collision and the length scale of observation. Once $\varepsilon$ goes to $0$, and then when the number of collisions per time unit goes to infinity, the complexity of equation (1) is (formally) greatly reduced, the solution being described almost completely by its local hydrodynamic fields, namely its _mass_ $N\geq 0$, its _momentum_ $\bm{u}\in\mathbb{R}^{d}$ and its _temperature_ $T\geq 0$. These quantities are obtained from a particle distribution function $f$ by computing the first moments in velocity: (2) $\begin{gathered}N(t,x)\,=\,\int_{\mathbb{R}^{d}}f(t,x,v)\,dv,\qquad N(t,x)\,\bm{u}(t,x)\,=\,\int_{\mathbb{R}^{d}}f(t,x,v)\,v\,dv,\\\ T(t,x)\,=\,\frac{1}{d\,N}\int_{\mathbb{R}^{d}}f(t,x,v)\,\|v-\bm{u}\|^{2}\,dv.\end{gathered}$ This reduction is usually carried on using the so-called Hilbert or Chapman- Enskog expansions of the solutions to a linearized version of the kinetic equation (1) (see _e.g._ the book of Cercignani, Illner and Pulvirenti [5] for a complete mathematical introduction in the elastic case). A rigorous mathematical proof of this “contraction of the kinetic description” (namely the hydrodynamic limit of the kinetic model towards a macroscopic one) for the elastic case has been first given for the linear setting in the paper of Ellis and Pinsky [8] but the inelastic case still remains to be investigated. An important step in the proof of this elastic limit is to give the so-called _dispersion relations_ of the collision operator, namely a Taylor expansion of the eigenvalues of the linearization of the collision operator, with respect to the space variable, near a global equilibrium (and this was the main purpose of [8]). The precise knowledge of the dispersion relations is actually of crucial interest in the study of the full nonlinear and compressible hydrodynamic limit and it was for example used by Kawashima, Matsumura and Nishida in [20, 13] (as a part of a rather abstract Cauchy- Kowalevski-type argument which is also related to Niremberg [19]). The work of Caflisch [3] also relies (but perhaps not as centrally as the previous ones) on these dispersion relations. Let us also quote the work of Degond and Lemou [7] where a similar analysis of the dispersion relations was conducted for the linearized Fokker-Planck equation. We propose to give in this paper the corresponding inelastic expansion, with respect to both the space variable and the inelasticity parameter, allowing to investigate in a future work first the two linearized hydrodynamic limits of our model “à la Ellis et Pinsky” and then the nonlinear, compressible ones “à la Nishida”. This result will allow us in particular to confirm a claim concerning the _clustering_ behavior of granular gases made in the classical textbook [2, p. 238] after a formal analysis, namely that > _the smaller the inelasticity, the larger the system must be to reveal > clusters._ ### 1.1. The Model Considered Let $\alpha\in(0,1]$ be the restitution coefficient of the microscopic collision process, that is the ratio of kinetic energy dissipated during a collision, in the direction of impact. Then, we can define a strong form of the _collision operator_ $\mathcal{Q}_{\alpha}$ by (3) $\displaystyle\mathcal{Q}_{\alpha}(f,g)(v)$ $\displaystyle=\int_{\mathbb{R}^{d}\times\mathbb{S}^{d-1}}\|u\|\left(\frac{\,{}^{\prime}f\,^{\prime}g_{}}{\alpha^{2}}-f\,g_{}\right)b(\widehat{u}\cdot\sigma)\,d\sigma\,dv_{},$ $\displaystyle=\mathcal{Q}_{\alpha}^{+}(f,g)(v)-f(v)L(g)(v),$ where we have used the usual shorthand notation $\,{}^{\prime}f:=f(^{\prime}v)$, $\,{}^{\prime}f_{}:=f(^{\prime}v_{})$, $f:=f(v)$, $f_{}:=f(v_{})$ and $\widehat{u}:=u/\|u\|$. In (3), $\,{}^{\prime}v$ and $\,{}^{\prime}v_{}$ are the pre-collisional velocities of two particles of given velocities $v$ and $v_{}$, defined for $\sigma\in\mathbb{S}^{d-1}$ as $\left\\{\begin{aligned} &{}^{\prime}v=\frac{v+v_{}}{2}-\frac{1-\alpha}{4\,\alpha}(v-v_{})+\frac{1+\alpha}{4\,\alpha}\|v-v_{}\|\,\sigma,\\\ &{}^{\prime}v_{}=\frac{v+v_{}}{2}+\frac{1-\alpha}{4\,\alpha}(v-v_{})-\frac{1+\alpha}{4\,\alpha}\|v-v_{}\|\,\sigma.\end{aligned}\right.$ The unitary vector $\sigma$ is the center of the _collision sphere_ (see Figure 1) and $u:=v-v_{}$ is the _relative velocity_ of the pair of particles. Finally, the function $b$ is the so-called _angular cross-section_ , describing the probability of collision between two particles. We assume that (4) $b\text{ is a Lipschitz, non-decreasing and convex function on }(-1,1),$ and also that it is bounded from above and below by two nonnegative constants $b_{m}$ and $b_{M}$: (5) $b_{m}\leq b(x)\leq b_{M},\quad\forall x\in(-1,1).$ In particular, this cross-section is integrable on the unit sphere, thus fulfilling the so-called Grad’s cut-off assumption222Physically relevant in the case of inelastic collisions, due to the macroscopic size of the grains forming the gas.. The operator $\mathcal{Q}_{\alpha}^{+}(f,g)(v)$ is usually known as the _gain_ term because it can be understood as the number of particles of velocity $v$ created by collisions of particles of pre- collisional velocities $\,{}^{\prime}v$ and $\,{}^{\prime}v_{}$, whereas $f(v)L(g)(v)$ is the _loss_ term, modeling the loss of particles of pre- collisional velocities $\,{}^{\prime}v$. We can also give a weak form of the collision operator. Indeed, if $\omega\in\mathbb{S}^{d-1}$ is the direction of impact, we can parametrize the post-collisional velocities $v^{\prime}$ and $v_{}^{\prime}$ as $\left\\{\begin{aligned} v^{\prime}&=v-\frac{1+\alpha}{2}\left(u\cdot\omega\right)\omega,\\\ v_{}^{\prime}&=v_{}+\frac{1+\alpha}{2}\left(u\cdot\omega\right)\omega.\end{aligned}\right.$ Then we have the weak representation, for any smooth test function $\psi$, (6) $\int_{\mathbb{R}^{d}}Q_{\alpha}(f,g)\,\psi(v)\,dv=\frac{1}{2}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}\times\mathbb{S}^{d-1}}\|u\|f_{}\,g\,\left(\psi^{\prime}+\psi_{}^{\prime}-\psi-\psi_{}\right)b(\widehat{u}\cdot\omega)\,d\omega\,dv\,dv_{}.$ 1.9063.14160.1cos(t)+-0.22\|0.1sin(t)+0 $v$$v_{}$$O$$\Omega_{+}$$\Omega_{-}$$v^{\prime}$$v_{}^{\prime}$$v^{\prime}$$v_{}^{\prime}$$\theta$$\sigma$$\omega$ Figure 1. Geometry of inelastic collisions, $O:=(v+v_{})/2$ and $\Omega_{\pm}:=O\pm(v_{}-v)\,(1-e)/2$ (dashed lines represent the elastic case). Thanks to this expression, we can compute the macroscopic properties of the collision operator $\mathcal{Q}_{\alpha}$. Indeed, we have the microscopic conservation of impulsion and dissipation of kinetic energy: $\displaystyle v^{\prime}+v_{}^{\prime}$ $\displaystyle=v+v_{},$ $\displaystyle\|v^{\prime}\|^{2}+\|v_{}^{\prime}\|^{2}-\|v\|^{2}-\|v_{}\|^{2}$ $\displaystyle=-\frac{1-\alpha^{2}}{2}\|u\cdot\omega\|^{2}\leq 0.$ Then if we integrate the collision operator against $\varphi(v)=(1,\,v\,\|v\|^{2})$, we obtain the preservation of mass and momentum and the dissipation of kinetic energy: $\int_{\mathbb{R}^{d}}\mathcal{Q}_{\alpha}(f,f)(v)\begin{pmatrix}1\\\ v\\\ \|v\|^{2}\end{pmatrix}dv\,=\,\begin{pmatrix}0\\\ 0\\\ -(1-\alpha^{2})D(f,f)\end{pmatrix},$ where $D(f,f)\geq 0$ is the _energy dissipation_ functional, given by (7) $D(f,f):=b_{1}\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}f\,f_{}\,\|v-v_{}\|^{3}\,dv\,dv_{}\geq 0,$ and $b_{1}$ is the angular momentum, depending on the cross-section $b$ and given by $b_{1}:=\int_{\mathbb{S}^{d-1}}(1-(\widehat{u}\cdot\omega))\,b(\widehat{u}\cdot\omega)\,d\omega<\infty.$ It is of course finite thanks to the bounds (5). In all the following of the paper, we shall assume that the restitution coefficient is related to the Knudsen number in the following way : $\alpha=1-\varepsilon.$ The macroscopic properties of the collision operator, together with the conservation of positiveness, imply that the equilibrium profiles of $\mathcal{Q}_{\alpha}$ are trivial Dirac masses (see _e.g._ the review paper [21] of Villani). Nevertheless, adding a thermal bath $(1-\alpha)\Delta_{v}$ will prevent this fact. Indeed, the existence of a non-trivial equilibrium profile $F_{\alpha}$ to the space homogeneous granular gases equation with a thermal bath is insured by the competition occurring between the dissipation of kinetic energy occasioned by the collision operator $\mathcal{Q}_{\alpha}$ and the gain of energy given by the diffusion term $\Delta_{v}$. More precisely, if we multiply the equation $\mathcal{Q}_{\alpha}(f,f)+(1-\alpha)\,\Delta_{v}(f)=0$ by $\|v\|^{2}$, integrate in velocity and divide by $1-\alpha$, we obtain using (7) the balance equation (8) $\left(1+\alpha\right)D(f,f)=2\,d.$ It has then been shown in [1, 16] that under the hypotheses (4)–(5) on the cross-section, there exists $\alpha_{}\in(0,1)$ such that for all $\alpha\in[\alpha_{},1]$, there exists an unique equilibrium profile $0\leq F_{\alpha}\in\mathcal{S}(\mathbb{R}^{d})$ of unit mass and zero momentum: (9) $\left\\{\begin{aligned} &\mathcal{Q}_{\alpha}(F_{\alpha},F_{\alpha})+(1-\alpha)\,\Delta_{v}(F_{\alpha})=0,\\\ &\,\\\ &\int_{\mathbb{R}^{d}}F_{\alpha}(v)\,dv=1,\qquad\int_{\mathbb{R}^{d}}F_{\alpha}(v)\,v\,dv=0.\end{aligned}\right.$ In the last expression, $\mathcal{S}(\mathbb{R}^{d})$ denotes the Schwartz class of $\mathcal{C}^{\infty}$ functions decreasing at infinity faster than any polynomials. The tails of this distribution are exponentials, of order $3/2$. Of course, if $\alpha=1$ (elastic, non-heated case), the distribution $F_{1}$ is nothing but the following _Maxwellian_ 333Hence, there is a bifurcation which occur between the inelastic heated case and the elastic nonheated one. distribution (10) $F_{1}(v):=\mathcal{M}_{1,0,\bar{T}_{1}}(v),$ where $\mathcal{M}_{N,\,\bm{u},\,T}$ is the Maxwellian distribution of mass $N$, velocity $\bm{u}$ and temperature $T$, only equilibria of the elastic collision operator $\mathcal{Q}_{1}$ (see _e.g._[5] for more details), and given by $\mathcal{M}_{N,\,\bm{u},\,T}(v):=\frac{N}{(2\pi T)^{d/2}}\exp\left(\frac{\|v-\bm{u}\|^{2}}{2T}\right)$ for $(N,\bm{u},T)\in\mathbb{R}^{d+2}$. The quantity $\bar{T}_{1}$ in (10) is defined by passing to the limit $\alpha\to 1$ in the balance equation (8) : $D(F_{1},F_{1})=d.$ We can then show thanks to this relation (see [16] for details) that $\bar{T}_{1}$ is given by (11) $\bar{T}_{1}=\frac{1}{2}\frac{d^{2/3}}{b_{1}^{2/3}}\left(\int_{\mathbb{R}^{d}}\mathcal{M}_{1,0,1}(v)\|v\|^{3}\,dv\right)^{-2/3}.$ ### 1.2. The Linearized Operator As we have said in the introduction, our goal is to perform the fluid dynamic limit $\varepsilon\to 0$ of equation (1). By rescaling the time $\widetilde{t}=t/\varepsilon$ and introducing a new distribution $\widetilde{f}(\widetilde{t},x,v)=f(t,x,v)$, the equation (1) now reads (forgetting the tildas) (12) $\frac{\partial f^{\varepsilon}}{\partial t}+\varepsilon\,v\cdot\nabla_{x}f^{\varepsilon}=\mathcal{Q}_{\varepsilon}(f^{\varepsilon},f^{\varepsilon})+\varepsilon\,\Delta_{v}(f^{\varepsilon}).$ The hydrodynamic limit then amounts to consider the large time, small space variations of the model (see the paper of Carlen, Chow and Grigo [4] for more details on the scaling and on the different types of limit models it can yields). This means as $\varepsilon=1-\alpha$ that we are studying _fluctuations_ $g$ of $f^{\alpha}$ near the space homogeneous equilibrium profile $F_{\alpha}$: (13) $f^{\alpha}=F_{\alpha}+g.$ By plugging this expansion on equation (12) and using the equilibrium relation (9), we obtain the following equation for $g$: (14) $\frac{\partial g}{\partial t}+(1-\alpha)\,v\cdot\nabla_{x}g=\mathcal{L}_{\alpha}\,g+(1-\alpha)\,\Gamma_{\alpha}(g,g),$ where the linearized operator $\mathcal{L}_{\alpha}$ is given for $v\in\mathbb{R}^{d}$ by $\mathcal{L}_{\alpha}(g)(v)\,:=\,\mathcal{Q}_{\alpha}(g,F_{\alpha})(v)+\mathcal{Q}_{\alpha}(F_{\alpha},g)(v)+(1-\alpha)\Delta_{v}(g)(v),$ and $\Gamma_{\alpha}$ is the quadratic remainder. In order to prove rigorous results on the original model, such as nonlinear stability, it will be crucial that the fluctuation $g$ lives in a weighted $L^{1}$ space. Indeed, to this purpose, we shall need to connect the properties of the linearized operator $\mathcal{L}_{\alpha}$ to the existing $L^{1}_{3}$ _a priori_ estimates for the nonlinear operator $\mathcal{Q}_{\alpha}$. These regularity properties were discussed extensively by Mischler and Mouhot in the series of paper [15, 16]. As we can see in these papers (we recalled the most important properties in the Appendix), we will need to take $g\in L^{1}(m^{-1})$, for $m$ an _exponential weight_ function: there exists $a>0$ and $0<s<1$ such that (15) $m(v):=\exp(-a\,\|v\|^{s}).$ The expansion (13) is well defined provided that the original distribution $f\in L^{1}\left(m^{-1}\right)$. Let us present some basic properties of the linear operator $\mathcal{L}_{\alpha}$. We first need to define the so-called _collision frequency_ by $\nu_{\alpha}(v):=L(F_{\alpha})(v)=\int_{\mathbb{R}^{d}\times\mathbb{S}^{d-1}}\|v-v_{}\|\,F_{\alpha}(v_{})\,b(\widehat{u}\cdot\sigma)\,d\sigma\,dv_{}.$ It is known (see for example the lemma 2.3 of [15] for an elementary proof) that for any $g\in L^{1}_{3}(\mathbb{R}^{d}$), there exists some explicit nonnegative constants $c_{0}$, $c_{1}$ such that $0<c_{0}\,(1+\|v\|)\leq L(g)(v)\leq c_{1}\,(1+\|v\|),\quad\forall\in v\in\mathbb{R}^{d}.$ In particular, the collision frequency $\nu_{\alpha}$ verifies (16) $0<\nu_{0,\alpha}\,(1+\|v\|)\leq\nu_{\alpha}(v)\leq\nu_{1,\alpha}\,(1+\|v\|),$ for two explicit nonnegative constants $\nu_{0,\alpha}$, $\nu_{1,\alpha}$. Then, we can rewrite the linearized collision operator as a difference of nonlocal and local operators: $\mathcal{L}_{\alpha}(g)\,=\,\mathcal{L}_{\alpha}^{+}(g)-\mathcal{L}^{}(g)-\mathcal{L}^{\nu_{\alpha}}(g),$ where $\mathcal{L}_{\alpha}^{+}$ is the linearization near $F_{\alpha}$ of the gain term, $\mathcal{L}^{}$ a convolution operator and $\mathcal{L}^{\nu}$ is the operator of multiplication by a function of the velocity variable $\nu$. Classically, for $\alpha=1$, the linearized operator splits between a compact operator on $L^{1}(m^{-1})$ (see the paper of Mouhot [17] for this particular exponentially weighted $L^{1}$ case) and a multiplication operator: $\displaystyle\mathcal{L}_{1}(g)$ $\displaystyle\,=\,\mathcal{L}_{1}^{c}(g)-\mathcal{L}^{\nu_{1}}(g).$ We will see in Section 2 that the same type of decomposition holds for $\mathcal{L}_{\alpha}$. As a first step to treat mathematically the question of the hydrodynamic limit of equation (1), we shall forget the nonlinearity in equation (14) and study the hydrodynamic limit of the linear equation (17) $\frac{\partial g}{\partial t}+(1-\alpha)\,v\cdot\nabla_{x}g=\mathcal{L}_{\alpha}\,g.$ One strategy of proof is to compare the spectrum of the linear operator (18) $-(1-\alpha)\,v\cdot\nabla_{x}+\mathcal{L}_{\alpha},$ to the one of the linearized fluid equation associated to the limit, as done in the seminal paper of Ellis and Pinsky [8]. As a byproduct, the study of this spectrum will allow us to answer to the question of the stability of the solutions to equation (17), by proving that the real part of the eigenvalues of (18) remains nonpositive. Hence, the rest of this paper is devoted to the computation of the spectrum (for small inelasticity and small space positions) of (18). In order to avoid to deal with the free transport operator in differential form, we shall now use Fourier transform in space. More precisely, if we define the Fourier transform in $x$ of a function $\varphi:\mathbb{R}^{d}\to\mathbb{R}$ as $\mathcal{F}_{x}(\varphi)(\xi):=\int_{\mathbb{R}^{d}}e^{-i\xi\,\cdot\,x}\varphi(x)\,dx,\quad\forall\,\xi\in\mathbb{R}^{d},$ it is well know that $\mathcal{F}_{x}\left(\nabla g\right)(\xi)=i\,\xi\,\mathcal{F}_{x}(g)(\xi).$ Then using the fact that $\mathcal{L}_{\alpha}$ only acts on velocity variables and setting $\gamma:=\left(1-\alpha\right)\xi,$ we can write (18) in (scaled) spatial Fourier variables as (19) $-i\,(\gamma\cdot v)+\mathcal{L}_{\alpha}=:\mathcal{L}_{(\alpha,\,\gamma)}.$ This operator is well defined on $L^{1}\left(m^{-1}\right)$, with domain $\operatorname{dom}(\mathcal{L}_{\alpha,\,\gamma})=W_{1}^{2,1}\left(m^{-1}\right)$. In this particular set of variable, the equation (17) finally reads $\frac{\partial g}{\partial t}=\mathcal{L}_{\alpha,\,\gamma}\,g,$ and we see now the need to study the spectrum of the linear operator $\mathcal{L}_{\alpha,\,\gamma}$ for small values of the variable $\gamma$. To finish with the definitions, let us denote by $N_{1}$ the kernel of the elastic operator $\mathcal{L}_{1}$. It is spanned by the elastic _collisional invariants_ , namely $N_{1}:=\operatorname{Span}\\{F_{1},\,v_{i}\,F_{1},\|v\|^{2}\,F_{1}:\,1\leq i\leq d\\},$ where $F_{1}=\mathcal{M}_{1,0,\bar{T}_{1}}$ and $T_{1}$ is the quasi-elastic equilibrium temperature (11). For $\alpha<1$, the kernel $N_{\alpha}$ of the inelastic operator $\mathcal{L}_{\alpha}$ is smaller, because of the lack of energy conservation; it is given by $N_{\alpha}:=\operatorname{Span}\\{F_{\alpha},\,v_{i}\,F_{\alpha}:\,1\leq i\leq d\\}.$ ### 1.3. Functional Framework and Main Results Let us present some functional spaces needed in the paper. We denote by $L_{q}^{p}$ for $p\in[1,+\infty)$ and $q\in[1,+\infty)$ the following weighted Lebesgue spaces: $L^{p}_{q}=\left\\{f:\mathbb{R}^{d}\rightarrow\mathbb{R}\text{ measurable; }\\|f\\|_{L^{p}_{q}}:=\int_{\mathbb{R}^{d}}\|f(v)\|^{p}\,\langle v\rangle^{pq}\,dv<\infty\right\\},$ where $\langle v\rangle:=\sqrt{1+\|v\|^{2}}$. The weighted $L^{\infty}_{q}$ is defined thanks to the norm $\\|f\\|_{L^{\infty}_{q}}:=\operatorname{supess}_{v\in\mathbb{R}^{d}}\left(\|f(v)\|\,\langle v\rangle^{q}\right).$ Then, we denote for $s\in\mathbb{N}$ by $W_{q}^{s,p}$ the weighted Sobolev space $W_{q}^{s,p}:=\left\\{f\in L^{p}_{q};\\|f\\|_{W_{q}^{s,p}}^{p}:=\sum_{\|k\|\leq s}\int_{\mathbb{R}^{d}}\left\|\partial^{k}f(v)\right\|^{p}\langle v\rangle^{p\,q}\,dv<\infty\right\\}.$ The case $p=2$ is the Sobolev space $H^{s}_{q}:=W^{s,2}_{q}$, which can also be defined thanks to Fourier transform by the norm $\\|f\\|_{H_{q}^{s}}^{2}:=\left\\|\mathcal{F}_{v}\left(f\,\langle\cdot\rangle^{s}\right)\right\\|_{L^{2}_{q}}.$ We also need to define the more general weighted spaces $L^{p}(m^{-1})$ and $W^{s,p}(m^{-1})$, where $m$ is an exponential weight function given by (15) respectively by the norms $\displaystyle\\|f\\|_{L^{p}(m^{-1})}^{p}:=\int_{\mathbb{R}^{d}}\|f(v)\|^{p}\,m^{-1}(v)\,dv,$ $\displaystyle\\|f\\|_{W^{s,p}(m^{-1})}^{p}:=\sum_{\|k\|\leq s}\left\\|\partial^{k}f\right\\|_{L^{p}(m^{-1})}^{p}.$ For the sake of completeness, let us finally state some notions about operators that we shall need in the following. ###### Definition 1. A _closed_ operator $T$ defined on a Banach space $X$ is said to be a * • _Fredholm_ operator of index $(\operatorname{nul}(T),\operatorname{def}(T))$ if the quantities $\operatorname{nul}(T):=\dim(\ker T)$ (the _nullity_) and $\operatorname{def}(T):=\operatorname{codim}(\operatorname{R}(T))$ (the _deficiency_) are finite; * • _semi-Fredholm_ operator if $\operatorname{R}(T)$ is closed and at least one of these two quantities are finite. For such an operator, we define the * • _resolvent set_ $R(T)\subset\mathbb{C}$ and the _resolvent operator_ $\mathcal{R}(T,\zeta)$ as $R(T):=\\{\zeta\in\mathbb{C}:T-\zeta\text{ is invertible on $X$, of bounded inverse }\mathcal{R}(T,\zeta)\\};$ * • _spectrum_ $\Sigma(T)$ of $T$ as the (closed) set $\Sigma(T):=R(T)^{c};$ * • _Fredholm_ set $\mathcal{F}(T)\subset\mathbb{C}$ of $T$ as $\mathcal{F}(T):=\\{\zeta\in\mathbb{C}:T-\zeta\text{ is Freholm}\\};$ * • _semi-Fredholm_ set $\mathcal{SF}(T)\subset\mathbb{C}$ of $T$ as $\mathcal{SF}(T):=\\{\zeta\in\mathbb{C}:T-\zeta\text{ is semi-Freholm}\\};$ * • _essential spectrum_ $\Sigma_{ess}(T)$ of $T$ as the set $\Sigma_{ess}(T):=\mathcal{SF}(T)^{c}\subset\Sigma(T);$ * • _discrete spectrum_ $\Sigma_{d}(T)$ of $T$ as the set $\Sigma_{d}(T):=\Sigma(T)\setminus\Sigma_{ess}(T).$ The two main results of this paper are the following Theorems. We first localize the spectrum of the operator $\mathcal{L}_{\alpha,\gamma}$ in the space $L^{1}\left(m^{-1}\right)$, generalizing to this space the classical $L^{2}$ result of Nicolaenko [18] (see also the chapter 7 of the book [5] of Cercignani, Illner and Pulvirenti). Let us denote by $\Delta_{x}$ for $x\in\mathbb{R}$ the half-plane $\Delta_{x}:=\\{\zeta\in\mathbb{C}:\Re e\,\zeta\geq x\\}.$ We first prove the following result (which has been summarized in Figure 2). [xAxis=true,yAxis=true,labels=none,Dx=2,Dy=2,ticksize=0pt 0]->(.8,0)(-10,-6)(7,6) $\delta$$-\delta$$\Sigma_{ess}\left(\mathcal{L}_{(\alpha,\,\gamma)}\right)$$-\bar{\mu}_{\alpha}$$-\mu_{}$$-\bar{\lambda}$$\Sigma_{d}\left(\mathcal{L}_{(\alpha,\,\gamma)}\right)$ Figure 2. Localization of the eigenvalues of $\mathcal{L}_{\alpha,\,\gamma}$ for $\|\gamma\|\leq\gamma_{0}(\delta)$. ###### Theorem 1.1. Let $\alpha\in(\alpha_{1},1]$, for a constructive constant $0<\alpha_{1}<1$. There exists a constructive constant $\bar{\mu}_{\alpha}>0$ such that the essential spectrum of the operator $\mathcal{L}_{(\alpha,\,\gamma)}$ in $W_{1}^{2,1}\left(m^{-1}\right)$ is contained on the half-plane $\Delta_{-\bar{\mu}_{\alpha}}^{c}$: $\Sigma_{ess}\left(\mathcal{L}_{(\alpha,\,\gamma)}\right)\subset\Delta_{-\bar{\mu}_{\alpha}}^{c}.$ The remaining part of its spectrum is composed of discrete eigenvalues. Their behavior for small frequencies $\gamma$ is the following. Let us fix $\delta>0$. There exist some constants $0<\bar{\lambda}<\mu_{}<\mu_{\alpha}$ and $\alpha_{2}\in(\alpha_{1},1]$ such that if $\alpha\in(\alpha_{2},1]$ there exists a nonnegative number $\gamma_{0}$ such that for all $\|\gamma\|\leq\gamma_{0}$, if $\lambda\in\Sigma_{d}(\mathcal{L}_{(\alpha,\,\gamma)})$, then $\displaystyle\lambda\in\Delta_{-\mu_{}}\Rightarrow\left\|\Im m\,\lambda\right\|\leq\delta;$ $\displaystyle\lambda\in\Delta_{-\frac{\bar{\lambda}}{2}}\Rightarrow\|\lambda\|\leq\delta.$ We then give a first order (in $\gamma$ and $\alpha$) Taylor expansion of the eigenvalues of $\mathcal{L}_{\alpha,\gamma}$, which generalizes the results of Ellis and Pinsky [8] and Mischler and Mouhot [16]. Notice that this result also contains a part of the analysis led by Brilliantov and Pöschel in the chapter 25 of their book [2] about the stable and unstable modes of the fluid approximation of the granular gases equation, namely that the energy eigenvalue is proportional to the inelasticity. Before stating the result, let us define the eigenvalue problem we want to deal with: finding a triple $(\lambda,\gamma,h)$ such that (20) $\left(-i(\gamma\cdot v)+\mathcal{L}_{\alpha}\right)h=\lambda\,h,$ for $\gamma\in\mathbb{R}^{d}$, $\lambda\in\mathbb{C}$ and $h\in L^{1}\left(m^{-1}\right)$. ###### Theorem 1.2. There exist $\alpha_{}\in(\alpha_{2},1]$, some open sets $U_{1}\times U_{2}\subset\mathbb{R}\times\mathbb{C}$, neighborhood of $(0,0)$, and functions $\left\\{\begin{aligned} &\lambda^{(j)}:U_{1}\times(\alpha_{},1]\to U_{2}&\forall\,j\in\\{-1,\ldots,d\\},\\\ &h^{(j)}:U_{1}\times\mathbb{S}^{d-1}\times(\alpha_{},1]\to L^{1}\left(m^{-1}\right)&\forall\,j\in\\{-1,\ldots,d\\},\end{aligned}\right.$ such that 1. (1) The triple $\left(\rho\,\omega,\,\lambda^{(j)}(\rho,\alpha),\,h^{(j)}(\rho,\omega,\alpha\right)$ is solution to the eigenvalue problem (20), for all $\alpha\in(\alpha_{},1]$, $\rho\in U_{1}$, $\omega\in\mathbb{S}^{d-1}$, $j\in\\{-1,\ldots,d\\}$; 2. (2) The eigenvalue $\lambda^{(j)}$ is analytic on $U_{1}\times(\alpha_{},1]$ and verifies $\left\\{\begin{aligned} &\lambda^{(j)}(0,1)=0,&&\forall\,j\in\\{-1,\ldots,d\\},\\\ &\frac{\partial\lambda^{(j)}}{\partial\rho}(0,1)=j\,i\,\sqrt{\bar{T}_{1}+\frac{2\bar{T}_{1}^{2}}{d}},&&\forall\,j\in\\{-1,0,1\\},&&\frac{\partial\lambda^{(j)}}{\partial\rho}(0,1)=0,&&\forall\,j\in\\{2,\ldots,d\\},\\\ &\frac{\partial^{2}\lambda^{(j)}}{\partial\rho^{2}}(0,1)<0,&&\forall\,j\in\\{-1,\ldots,d\\},\\\ &\frac{\partial\lambda^{(0)}}{\partial\alpha}(0,1)=-\frac{3}{\bar{T}_{1}},&&&&\frac{\partial\lambda^{(j)}}{\partial\alpha}(0,1)=0,&&\forall\,j\in\\{-1,1,\ldots,d\\};\end{aligned}\right.$ 3. (3) For $\alpha\in(\alpha_{},1]$, if a triple $(\rho\omega,\lambda,h)$ is solution to the problem (20) for $(\rho,\lambda)\in U_{1}\times U_{2}$, then necessarily $\lambda=\lambda^{(j)}$ for some $j\in\\{-1,\ldots,d\\}$; 4. (4) For $j\in\\{-1,0,1\\}$, $\alpha\in(\alpha_{},1]$ and $\omega\in\mathbb{S}^{d-1}$, the function $v\mapsto h^{(j)}(\rho\,\omega,\alpha)(v)$ depends only on $\|v\|$ and $v\cdot\omega$. ###### Remark 1. We notice in this result that the eigenvalues depend only on $\|\gamma\|$ and not on $\gamma$ itself. This is due to the rotational invariance of the linearized collision operator. However, this is not the case of the eigenvectors, which can also depend on the angular coordinates. ###### Remark 2. As a consequence of this result, we can write for $\left(\rho,\alpha\right)\in U_{1}\times(\alpha_{},1]$ $\lambda^{(j)}(\rho,\alpha)=i\lambda_{1}^{(j)}\rho-\lambda_{2}^{(j)}\rho^{2}-e_{1}^{(j)}\left(1-\alpha\right)+\mathcal{O}\left(\rho^{2}+(1-\alpha)^{2}\right),$ for explicit (see Section 3.4) constants $\lambda_{1}^{(j)}\in\mathbb{R}$, $\lambda_{2}^{(j)}\in\mathbb{R}_{+}$ and $e_{1}^{(j)}\in\mathbb{R}_{+}$. In particular, we obtain that for small space frequencies and small values of the inelasticity, the spectrum of the linear operator remains at the left of the imaginary axis in the complex plane. This phenomenon has at least two important consequences on the behavior of the solution to the granular gases equation: • The solutions to the linear collision equation (17) are $L^{1}\left(m^{-1}\right)$ stable; * • The clustering phenomenon (see _e.g._[2]) is not possible for quasi-elastic collisions $\alpha\sim 1$, or for systems with a small typical length scale. ### 1.4. Method of Proof and Plan of the Paper The proof of Theorem 1.1, concerning the rough444By rough, we mean more precisely that this result do not establish whether or not the eigenvalues can cross the vertical axis of the complex plane. localization of the spectrum of the linear operator $\mathcal{L}_{(\alpha,\,\gamma)}$ is given in Section 2, and can be summarized as follows: * • We first decompose this operator as a sum of compact and Schrödinger-like operators: $\displaystyle\mathcal{L}_{(\alpha,\,\gamma)}\,h$ $\displaystyle=-\left[i\,(\gamma\cdot v)+\nu_{\alpha}(v)\right]h+(1-\alpha)\Delta_{v}h+2\,\mathcal{Q}_{\alpha}^{+}\left(F_{\alpha},h\right)-F_{\alpha}\,L(h)$ $\displaystyle=D_{(\alpha,\,\gamma)}h+\mathcal{L}_{\alpha}^{c}h.$ * • We then compute the spectrum in $L^{1}$ of the Schrödinger-like part $D_{(\alpha,\,\gamma)}$ and apply a Banach variant of Weyl’s Theorem (found _e.g._ in [12]) about the stability of the essential spectrum under relatively compact perturbation $\Sigma_{ess}\left(\mathcal{L}_{(\alpha,\,\gamma)}\right)=\Sigma_{ess}\left(D_{(\alpha,\,\gamma)}\right).$ * • Once the essential spectrum has been localized, we take advantage of some space homogeneous coercivity and spectral gap estimates (proven in [15]) to establish the existence of the eigenvalues. * • We finally combine some information about the asymptotic ($\alpha\to 1$) behavior of the space homogeneous eigenvalues (also taken from [15]) and the decay properties of the semi-group of the operator $D_{(\alpha,\,\gamma)}$ to finally give a rough localization of the spectrum. The proof of Theorem 1.2 concerning the Taylor expansion of the eigenvalues of $\mathcal{L}_{(\alpha,\,\gamma)}$ is given in Section 3. Up to a certain extent. this proof is a generalization of Ellis & Pinsky’s arguments for [8], namely: * • We start by reformulating the eigenvalue problem (20) as a functional equation, using bounded operators: $\eqref{pbEigenValueLambdaGammaRho}\Longleftrightarrow\text{Finding }(\lambda,\gamma,h)\text{ s.t. }h=\Psi_{(\lambda,\gamma,\,\alpha)}^{-1}\Phi_{(\lambda,\gamma,\,\alpha)}\nu_{\alpha}^{-1/2}\Pi\,\nu_{\alpha}^{1/2}h,$ where $\Pi$ is a projection operator, $\Phi$ a “multiplication” operator, and $\Psi$ a “small” perturbation of the identity. * • We then project this new problem onto the space of _elastic_ collisional invariants, allowing to rewrite completely (20) as a finite dimensional system of linear equations of the form $\left(A_{(\lambda,\gamma,\alpha)}-\operatorname{Id}\right)X_{(\lambda,\gamma,\alpha)}=0,$ for a non-invertible square matrix $A$. * • We finally solve this system of equation taking advantage of some elastic and space homogeneous techniques, from both [8] and [15]. ## 2\. Localization of the Spectrum In this section, we shall give a rough localization of the spectrum of the linearized collision operator $\mathcal{L}_{\alpha}$, proving Theorem (1.1). ### 2.1. Geometry of the Essential Spectrum We start by describing the “easy part”, namely the essential spectrum. As we have to deal with the Banach space $L^{1}\left(m^{-1}\right)$, we cannot apply directly the classical Weyl’s Theorem about the stability of the spectrum under relatively compact perturbations, because of the lack of Hilbertian structure. We shall rather apply the more general version stating only the stability of the semi-Fredholm set, which is well suited for our definition of the essential spectrum. ###### Proposition 2.1. Let $\alpha\in(\alpha_{0},1]$, where $\alpha_{0}$ is defined in Lemma A.1. There exists a constructive constant $\bar{\mu}_{\alpha}>0$ such that the essential spectrum of the operator $\mathcal{L}_{(\alpha,\,\gamma)}$ in $W_{1}^{2,1}\left(m^{-1}\right)$ is contained on the half-plane $\Delta_{-\bar{\mu}_{\alpha}}^{c}$: $\Sigma_{ess}\left(\mathcal{L}_{(\alpha,\,\gamma)}\right)\subset\Delta_{-\bar{\mu}_{\alpha}}^{c}.$ The remaining part of its spectrum is composed of discrete eigenvalues. ###### Proof. Let us use the expression (19) of the collision operator in spatial Fourier variables, and decompose it for $h\in L^{1}\left(m^{-1}\right)$ as a local and a non local part: $\displaystyle\mathcal{L}_{(\alpha,\,\gamma)}\,h$ $\displaystyle=-\left[i\,(\gamma\cdot v)+\nu_{\alpha}(v)\right]h+(1-\alpha)\Delta_{v}h+2\,\mathcal{Q}_{\alpha}^{+}\left(F_{\alpha},h\right)-F_{\alpha}\,L(h)$ (21) $\displaystyle=D_{(\alpha,\,\gamma)}h+\mathcal{L}_{\alpha}^{c}h,$ where (22) $\left\\{\begin{aligned} &D_{(\alpha,\,\gamma)}:=-\left[i\,(\gamma\cdot v)+\nu_{\alpha}(v)\right]\operatorname{Id}+(1-\alpha)\Delta_{v},\\\ &\mathcal{L}_{\alpha}^{c}:=2\mathcal{Q}_{\alpha}^{+}\left(F_{\alpha},\cdot\right)-F_{\alpha}\,L(\cdot).\end{aligned}\right.$ We start by the spectrum of $D_{(\alpha,\,\gamma)}$ in $L^{1}\left(m^{-1}\right)$. This operator is the difference of a Laplace operator with the operator of multiplication by $C_{(\alpha,\,\gamma)}(v):=i\,(\gamma\cdot v)+L(F_{\alpha})$. This quantity verifies according to the lower bound of the collision frequency (16) $\Re e\,C_{\alpha,\,\gamma}(v)>\nu_{0,\alpha},$ where $\nu_{0,\alpha}$ is the lower bound of the loss term $v\to L(F_{\alpha})(v)$ (thanks to the smoothness of the profile $F_{\alpha}$ stated in Proposition A.2). It is known from _e.g._[11, 14] that the spectrum of the Schrödinger-like operator $D_{(\alpha,\,\gamma)}$ is independent of the weighted $L^{p}$ space (for $p\in[1,+\infty)$) where we study it. Let us compute this spectrum in $L^{2}$. To this end, we shall look at the stability properties of the semi- group generated by $D_{(\alpha,\,\gamma)}$ on $L^{2}\left(m^{-1}\right)$: let $h=h(t,v)\in\mathcal{C}\left(0,+\infty;W_{1}^{2}\right)$ be a weak solution to (23) $\frac{\partial h}{\partial t}=D_{(\alpha,\,\gamma)}h.$ If we multiply this equation by $\bar{h}$ and integrate in the velocity space, we have thanks to Stokes Theorem and for $\alpha$ close to $1$ (24) $\displaystyle\frac{\partial}{\partial t}\\|h(t)\\|_{L^{2}\left(m^{-1}\right)}^{2}\leq-\\|C_{(\alpha,\,\gamma)}\,h(t)\\|_{L^{2}\left(m^{-1}\right)}\leq-\nu_{0,\alpha}\\|h(t)\\|_{L^{2}\left(m^{-1}\right)}^{2}.$ We then obtain that $\\|h(t)\\|_{L^{2}\left(m^{-1}\right)}\leq e^{-\nu_{0,\alpha}\,t/2}.$ Hence, there exists a constant $0<\bar{\mu}_{\alpha}<\nu_{0,\alpha}$ such that the spectrum of the operator $D_{(\alpha,\,\gamma)}$ is included in the set $\Delta_{-\bar{\mu}_{\alpha}}^{c}$. Moreover, thanks to the Hölder continuity of the inelastic gain term in operator norm (with loss of weight) stated in Proposition A.2, the operator $\mathcal{L}_{\alpha}^{c}$ is $D_{(\alpha,\,\gamma)}$–compact (and this is here that the weak inelasticity assumption $\alpha\in(\alpha_{0},1]$ is used). Notice that we have chosen to define the essential spectrum of an operator $S$ “à la Kato”, namely as the complement of the semi-Fredholm set of $S$ in $\mathbb{C}$. Then we can apply the Banach version of Weyl’s Theorem (see _e.g._[12, Theorem IV.5.26 and IV.5.35]), stating that the semi-Fredholm set is stable under relatively compact perturbation. Hence, the essential spectrum of $\mathcal{L}_{(\alpha,\,\gamma)}$ is included in $\Delta_{-\bar{\mu}_{\alpha}}^{c}$. The set $\Delta_{-\bar{\mu}_{\alpha}}$ is then equal to the Fredholm set $\mathcal{F}\left(\mathcal{L}_{(\alpha,\,\gamma)}\right)$. It remains to show that this set only contains the eigenvalues and the resolvent set. We know from the discussion in [12, Chapter IV, Section 6, and Theorem 5.33] that $\mathcal{F}\left(\mathcal{L}_{(\alpha,\,\gamma)}\right)$ is an open set, composed of the union of a countable number of components $\mathcal{F}_{n}$, characterized by the value of the index: for any $n\in\mathbb{N}$, the functions $\operatorname{nul}:\zeta\to\operatorname{nul}\left(\mathcal{L}_{(\alpha,\,\gamma)}-\zeta\right),\qquad\operatorname{def}:\zeta\to\operatorname{def}\left(\mathcal{L}_{(\alpha,\,\gamma)}-\zeta\right)$ are constant on $\mathcal{F}_{n}$, except for a countable set of isolated values of $\zeta$. In our case, we have $\mathcal{F}\left(\mathcal{L}_{(\alpha,\,\gamma)}\right)=\Delta_{-\bar{\mu}_{\alpha}}$ which is connected; it has only one component, which means that $\operatorname{nul}(\zeta)$ and $\operatorname{def}(\zeta)$ are constant on $\Delta_{-\bar{\mu}_{\alpha}}$, except for a countable set of isolated values of $\zeta$. We will prove that these constant values are $\operatorname{nul}(\zeta)=\operatorname{def}(\zeta)=0$, meaning that $\zeta$ belongs to the resolvent set of $\mathcal{L}_{(\alpha,\,\gamma)}$. The remaining isolated values $\zeta$, being in the Fredholm set, then verify $0<\operatorname{nul}(\zeta)<+\infty$ and $0<\operatorname{def}(\zeta)<+\infty$, which exactly characterizes the eigenvalues. We shall follow closely the proof of [17, Proposition 3.4], and exhibit an uncountable set $I\subset\Delta_{-\bar{\mu}_{\alpha}}$ such that $\operatorname{nul}(\zeta)=\operatorname{def}(\zeta)=0$ for all $\zeta\in I$. Let us use the decomposition (72) introduced initially in [15] $\mathcal{L}_{\alpha,\,\gamma}=A_{\delta}-B_{\alpha,\,\delta}\left(i(\gamma\cdot v)\right),$ for $\delta>0$ (see Section A for more details). We know from Lemma A.1 that $A_{\delta}$ is compact on $L^{1}\left(m^{-1}\right)$, and that $B_{\alpha,\delta}$ satisfies the coercivity estimate (25) $\left\\|B_{\alpha,\,\delta}(\zeta)\,g\right\\|_{L^{1}\left(m^{-1}\right)}\geq\left\\|\left(\nu_{1}+\Re e\,\zeta\right)g\right\\|_{L^{1}\left(m^{-1}\right)}-\varepsilon(\delta)\left\\|\left(\nu_{1}+\Re e\,\zeta\right)g\right\\|_{L^{1}\left(m^{-1}\right)},$ where $\varepsilon(\delta)\to 0$ where $\delta\to 0$. If we fix $r_{0}>0$ sufficiently big and $\delta>0$ small enough, then we have according to (25) for all $r\geq r_{0}$ $\left\\|B_{\alpha,\,\delta}\left(r+i(\gamma\cdot v)\right)g\right\\|_{L^{1}\left(m^{-1}\right)}\geq\frac{\nu_{0,1}+r_{0}}{2}\\|g\\|_{L^{1}\left(m^{-1}\right)}.$ Thus, the operator $B_{\alpha,\,\delta}\left(r+i(\gamma\cdot v)\right)$ is invertible on $L^{1}\left(m^{-1}\right)$, for all $r\geq r_{0}$, and then it is the same for $\mathcal{L}_{\alpha,\,\gamma}-r=A_{\delta}-B_{\alpha,\,\delta}\left(r+i(\gamma\cdot v)\right)$ by compacity of $A_{\delta}$. It finally means that the interval $I:=[r_{0},+\infty)$ is included on the resolvent set of $\mathcal{L}_{\alpha,\,\gamma}$, and then that $\operatorname{nul}(\zeta)=\operatorname{def}(\zeta)=0,\quad\forall\,\zeta\in[r_{0},+\infty),$ which concludes the proof. ∎ ### 2.2. Behavior of the Eigenvalues for Small Inelasticity We shall now focus on the discrete spectrum of this operator, namely its eigenvalues. A major difference with the elastic case in the classical Hilbertian $L^{2}$ setting is that the operator we deal with is not a nonpositive operator, and we cannot conclude thanks to the last proposition that this operator has a _spectral gap_ (namely a negative bound for its eigenvalues). Nevertheless, we know from [17] for the elastic case and [16] for the weak inelasticity case $\alpha\in(\alpha_{0},1)$ that $\mathcal{L}_{\alpha}$ has a spectral gap $-\bar{\lambda}$ in $L^{1}\left(m^{-1}\right)$, verifying for $\alpha$ sufficiently small (say $\alpha\in(\alpha_{1},1]$ for $1>\alpha_{1}>\alpha_{0}$) $0<\bar{\lambda}<\mu_{}<\bar{\mu}_{\alpha},$ for a nonnegative constant $\mu_{}$ depending on $\alpha$. Let us now study the behavior of the discrete spectrum of $\mathcal{L}_{(\alpha,\,\gamma)}$ for small values of the frequency $\gamma$. We shall show that if $\gamma\to 0$, then the eigenvalues of this operator converge first towards the real axis and then towards $0$. ###### Proposition 2.2. Let $\delta>0$. There exists $\alpha_{2}\in(\alpha_{1},1]$ such that if $\alpha\in(\alpha_{2},1]$ there exists a nonnegative number $\gamma_{0}$ such that for all $\|\gamma\|\leq\gamma_{0}$, if $\lambda\in\Sigma_{d}(\mathcal{L}_{(\alpha,\,\gamma)})$, then (26) $\displaystyle\lambda\in\Delta_{-\mu_{}}\Rightarrow\left\|\Im m\,\lambda\right\|\leq\delta;$ (27) $\displaystyle\lambda\in\Delta_{-\frac{\bar{\lambda}}{2}}\Rightarrow\|\lambda\|\leq\delta.$ ###### Proof. Let us first notice that if $\lambda$ is an eigenvalue of $\mathcal{L}_{(\alpha,\,\gamma)}$ and $h$ an associated eigenvector, then using the decomposition (21) introduced in the proof of Proposition 2.1, we can write (28) $\mathcal{L}_{\alpha}^{c}h=\left(\lambda-D_{(\alpha,\,\gamma)}\right)h,$ where $\mathcal{L}_{\alpha}^{c}$ is compact on $L^{1}\left(m^{-1}\right)$ (thanks to the sharp estimates of Lemma A.1) and $D_{(\alpha,\,\gamma)}=-\left[i\,(\gamma\cdot v)+\nu_{\alpha}(v)\right]\operatorname{Id}+(1-\alpha)\Delta_{v}.$ We will proceed by contradiction using the representation (28). Concerning the first implication, if, for $\delta>0$, there exist a sequence $(\gamma_{n})_{n}\subset\mathbb{R}^{d}$ converging towards $0$, a sequence of functions $(h_{n})_{n}\in L^{1}\left(m^{-1}\right)$ of unit norm, and a sequence of complex numbers $\lambda_{n}\in\Sigma_{d}\left(\mathcal{L}_{(\alpha,\,\gamma_{n})}\right)$ verifying (29) $\left\\{\begin{aligned} &\mathcal{L}_{\alpha}^{c}h_{n}=\left(\lambda_{n}-D_{(\alpha,\,\gamma_{n})}\right)h_{n},\\\ &\left\|\Im m\,\lambda_{n}\right\|>\delta,\quad\Re e\,\lambda_{n}\geq-\mu_{},\end{aligned}\right.$ then we must have $\limsup\left\|\Im m\,\lambda_{n}\right\|<\infty$. Indeed, the operator $\mathcal{L}_{\alpha}^{c}$ is compact on $L^{1}\left(m^{-1}\right)$, and then the sequence $(\mathcal{L}_{\alpha}^{c}h_{n})_{n}$ converges (up to an extraction) towards $g\in L^{1}\left(m^{-1}\right)$. Thus we can write using (29) (30) $g=\lim_{n\to\infty}\left(\lambda_{n}-D_{(\alpha,\,\gamma_{n})}\right)h_{n}.$ We have seen in the proof of Proposition 2.1 that the semi-group $S_{t}^{(\alpha,\,\gamma)}$ associated to the operator $D_{(\alpha,\,\gamma)}$ in $L^{1}$ is exponentially decaying in time, uniformly in $\alpha$ and $\gamma$. But, we know (see _e.g._[9], chap. II) that if $\mathcal{R}(D_{(\alpha,\,\gamma)},\cdot)$ is the resolvent operator of $D_{\alpha,\,\gamma}$, we have the integral representation for all $\zeta\in R(D_{(\alpha,\,\gamma)})$ $\mathcal{R}\left(D_{(\alpha,\,\gamma)},\zeta\right)=\lim_{t\to+\infty}\int_{0}^{t}e^{-t\,\zeta}S_{t}^{(\alpha,\,\gamma)}\,dt.$ Thus, using the decay of $S_{t}^{(\alpha,\,\gamma)}$, we have $\mathcal{R}\left(D_{(\alpha,\,\gamma)},0\right)=:D_{(\alpha,\,\gamma)}^{-1}$ bounded in $L^{1}$, uniformly in $\gamma$, and then according to (30) (31) $\lim_{n\to\infty}h_{n}=\left(\lim_{n\to\infty}\lambda_{n}-D_{(\alpha,\,0)}\right)^{-1}g.$ But, we also have for $v\in\mathbb{R}^{d}$ $\left(\lambda_{n}-D_{(\alpha,\,\gamma_{n})}\right)^{-1}g(v)=\frac{1}{\lambda_{n}+i(\gamma_{n}\cdot v)+\nu_{\alpha}(v)}\left(\operatorname{Id}-\frac{1-\alpha}{\lambda_{n}+i(\gamma_{n}\cdot v)+\nu_{\alpha}(v)}\Delta_{v}\right)^{-1}g(v),$ and then by considering again the behavior of the solutions to equation (23), which gives inequality (24), we obtain a constant $C$ independent on $(\alpha,\lambda_{n},\gamma_{n})$ such that (32) $\left\\|\left(\lambda_{n}-D_{(\alpha,\,\gamma_{n})}\right)^{-1}g\right\\|_{L^{\infty}}\leq\frac{C}{\|\lambda_{n}\|-\nu_{0,\alpha}}\\|g\\|_{L^{\infty}}.$ Finally, if $\lim\left\|\Im m\,\lambda_{n}\right\|=\infty$ we would have according to the limit (31) and the estimation (32) $\lim_{n\to\infty}\\|h_{n}\\|_{L^{\infty}}=0$ with $\\|h_{n}\\|_{L^{1}\left(m^{-1}\right)}=1$, which is not possible. Hence, $\left\|\Im m\,\lambda_{n}\right\|\leq C$ for an infinite number of indices $n$ and $C>0$. But, we also have $-\mu_{}\leq\Re e\,\lambda_{n}<r_{0}$ (where $r_{0}>0$ is defined in the proof of Proposition 2.1), and then we can extract another subsequence $\left(\lambda_{n_{k}}\right)_{k}$ converging towards $\lambda\in\mathbb{C}$ such that $\Im m\,\lambda\geq\delta>0$. Using the fact that $\gamma_{n}\to 0$ and the smoothness of the map $\lambda\mapsto\left(\lambda-D_{(\alpha,\,0)}\right)^{-1}$, we obtain in (31) $\lim_{k\to\infty}h_{n_{k}}=\left(\lambda-D_{(\alpha,\,0)}\right)^{-1}g=:h\in L^{1}\left(m^{-1}\right),$ with $\\|h\\|_{L^{1}\left(m^{-1}\right)}=1$. Hence we conclude by inversion of $\left(\lambda-D_{(\alpha,\,0)}\right)^{-1}$ and by the smoothness of the nonlocal part of $\mathcal{L}_{\alpha}$ that $\displaystyle\left(\lambda- D_{(\alpha,\,0)}\right)\,h=g=\lim_{n\to\infty}\mathcal{L}_{\alpha}^{c}h_{n}=\mathcal{L}_{\alpha}^{c}h$ which means according to the definition of $\mathcal{L}_{\alpha}$ that $\lambda h=\mathcal{L}_{\alpha}h.$ This is absurd because $\left\|\Im m\,\lambda\right\|\geq\delta>0$, $\Re e\,\lambda\geq-\mu_{}$ for $\mu_{}$ close to the spectral gap of $\mathcal{L}_{\alpha}$, and yet we know from [15] that the eigenvalues of $\mathcal{L}_{\alpha}$ can be made arbitrarily close (with respect to $1-\alpha$) to the ones of $\mathcal{L}_{1}$, which are real according to [17]. We shall now give the proof of the implication (27), also by contradiction. If for $\delta>0$ there exist a sequence $(\gamma_{n})_{n}$ converging towards $0$, a sequence $(h_{n})_{n}\in L^{1}\left(m^{-1}\right)$ of unit norm, and some complex numbers $\lambda_{n}\in\Sigma_{d}\left(\mathcal{L}_{\varepsilon,\gamma_{n}}\right)$ such that $-\bar{\lambda}/2\leq\Re e\,\lambda_{n}\leq-\delta,$ then $\lambda_{n}\in\Delta_{-\mu_{}}$ and according to the relation (26) we have $\left\|\Im m\,\lambda_{n}\right\|\leq\delta$. We can then extract a subsequence $\left(\lambda_{n_{k}}\right)_{k}$ which converges towards a complex number $\lambda$ also verifying $-\bar{\lambda}/2\leq\Re e\,\lambda\leq-\delta$. When $k\to\infty$, the same argument than before gives $\mathcal{L}_{\alpha}h=\lambda h$ with $\lambda\neq 0$. By using again the spectral properties of $\mathcal{L}_{\alpha}$, we then have $\Re e\,\lambda\leq-\bar{\lambda}$, which is absurd. ∎ This concludes the proof of Theorem 1.1. We also summarized the results of this proposition in Figure 2. Moreover, it gives us some rough information on the behavior of the resolvent operator of $\mathcal{L}_{(\alpha,\gamma)}$. ###### Corollary 2.1. If $\alpha\in(\alpha_{1},1]$, the resolvent operator $\mathcal{R}(\mathcal{L}_{(\alpha,\,\gamma}),\zeta)$ is well defined for $\zeta\in\Delta_{-\mu_{}}$ such that $\Im m\,\zeta>\delta$. ## 3\. Inelastic Dispersion Relations Our goal in this section is to precise the localization results of the previous section, by proving Theorem 1.2, that is to give a Taylor expansion of the eigenvalues of $\mathcal{L}_{(\alpha,\gamma)}$ in $\alpha$ and $\gamma$. The purpose of this expansion is twofold: on the one hand, we want to establish that, at least for small values of $\alpha$ and $\gamma$, the eigenvalues of the linear operator $\mathcal{L}_{(\alpha,\gamma)}$ stay at the left of the imaginary axis. This could be useful if _e.g._ one wants to prove nonlinear stability of the solutions to (1). On the other hand, obtaining this decomposition up to the second order in the spatial frequency $\gamma$ and to first order in $\alpha$ is necessary to establish the validity of the linearized quasi-elastic hydrodynamic limit of our model, in the same way than [8]. To obtain this expansion, we shall refine the method of proof of this paper, together with the use of some ideas introduced in [16] in order to deal with the quasi-elastic setting. We recall for the reader’s convenience that we are interested in the following eigenvalue problem: finding $\lambda\in\mathbb{C}$, $\gamma\in\mathbb{R}^{d}$ and $h\in L^{1}(m^{-1})$ such that $\left(-i(\gamma\cdot v)+\mathcal{L}_{\alpha}\right)h=\lambda\,h,$ which can be reformulated thanks to the decomposition (21) as finding $\lambda\in\mathbb{C}$, $\gamma\in\mathbb{R}^{d}$ and $h\in L^{1}(m^{-1})$ such that (33) $\mathcal{L}_{\alpha}^{c}h=\left(\lambda+\nu_{\alpha}(v)+i(\gamma\cdot v)-(1-\alpha)\Delta_{v}\right)h.$ ### 3.1. Projection of the Eigenvalue Problem Let us now define the scalar product we will use in the following. If $\phi,\psi$ are such that the following expression has a meaning, we will set $\langle\phi,\psi\rangle:=\int_{\mathbb{R}^{d}}\phi(v)\,\overline{\psi}(v)\,F_{1}^{-1}(v)\,dv,$ where $F_{1}=\mathcal{M}_{1,0,\bar{T}_{1}}$ is the quasi-elastic equilibrium and $\bar{T}_{1}$ is given by (11). Indeed, our goal is to introduce a _spectral decomposition_ of $L^{1}\left(m^{-1}\right)$ as a direct sum of $\mathcal{L}_{\alpha}$-invariant spaces. The inner product we use for this purpose is the one of $L^{2}\left(F_{1}^{-1}\right)$ because $f=m+h\in L^{1}\left(m^{-1}\right)$ if and only if $h\in L^{1}\left(m^{-1}\right)$ and $f\in L^{2}\left(F_{1}^{-1}\right)$ if and only if $h\in L^{2}\left(F_{1}^{-1}\right)$. This hilbertian structure allows us to define the spectral projections. Let us start by decomposing the operator $\mathcal{L}_{\alpha}^{c}$ as (34) $\mathcal{L}_{\alpha}^{c}=\nu_{\alpha}^{1/2}\left(\Pi+\mathcal{S}_{\alpha}\right)\nu_{\alpha}^{1/2},$ where $\Pi$ is the projection on the space $\mathcal{N}_{\alpha}:=\nu_{\alpha}^{1/2}N_{1}=\operatorname{Span}\\{\nu_{\alpha}^{1/2}F_{1},\,\nu_{\alpha}^{1/2}v_{i}\,F_{1},\,\nu_{\alpha}^{1/2}\|v\|^{2}\,F_{1}:\,1\leq i\leq d\\},$ and $\mathcal{S}_{\alpha}$ is given by $\mathcal{S}_{\alpha}:=\nu_{\alpha}^{-1}\mathcal{L}_{\alpha}^{c}-\nu_{\alpha}^{-1/2}\Pi\,\nu_{\alpha}^{1/2}.$ Actually, $\mathbb{P}:=\nu_{\alpha}^{-1/2}\Pi$ is the spectral projection on the null space of $\mathcal{L}_{1}$, and can be defined using the resolvent operator of $\mathcal{L}_{1}$ as $\displaystyle\mathbb{P}$ $\displaystyle=\frac{1}{2i\pi}\int_{\zeta\in\mathbb{C}:\|\zeta\|=r}\mathcal{R}(\mathcal{L}_{1},\zeta)\,d\zeta$ where $r<\delta$ for $\delta$ sufficiently small (see the discussion in Section $5$ of [15]). In particular, $\mathbb{P}$ commutes with $\mathcal{L}_{1}$. Moreover, the operator $\mathcal{L}_{\alpha}^{c}$ being compact on $L^{1}(m^{-1})$, $\Pi+\mathcal{S}_{\alpha}$ is compact on the same space. Given that the rank of $\Pi$ is finite, $\mathcal{S}_{\alpha}$ is then a compact operator on $L^{1}(m^{-1})$. We first need to prove a result concerning the eigenvalues of the operator $\nu_{1}^{-1/2}\mathcal{S}_{1}\,\nu_{1}^{1/2}$. ###### Lemma 3.1. On the space $L^{1}(m^{-1})$, $\lambda=1$ is not an eigenvalue of $\nu_{1}^{-1/2}\mathcal{S}_{1}\,\nu_{1}^{1/2}$. ###### Proof. If there is an $h\in L^{1}(m^{-1})$ such that $\nu_{1}^{-1/2}\mathcal{S}_{1}\,\nu_{1}^{1/2}h=h$, then according to (34), $\displaystyle\mathcal{L}_{1}\,h$ $\displaystyle=-\nu_{1}h+\nu_{1}^{1/2}(\Pi+\mathcal{S}_{1})\,\nu_{1}^{1/2}h$ (35) $\displaystyle=\nu_{1}^{1/2}\,\Pi\,\nu_{1}^{1/2}h.$ Projecting this equation upon the null space of $\mathcal{L}_{1}$, using (35), we obtain for all $\varphi\in N_{1}$ that $\displaystyle 0$ $\displaystyle=\langle\mathcal{L}_{1}h,\varphi\rangle$ $\displaystyle=\langle\Pi\,\nu_{1}^{1/2}h,\nu_{1}^{1/2}\,\varphi\rangle,$ and then $\Pi\,\nu_{1}^{1/2}h$ is orthogonal to $\nu_{1}^{1/2}N_{1}$, which is the whole range of $\Pi$. Then, necessarily, as $\Pi$ is a projection, we have $\Pi\,\nu_{1}^{1/2}h=0$. It means according to (35) that $\mathcal{L}_{1}\,h=0$, and then that $h\in N_{1}$, which is absurd because $\Pi$ is a projection onto $\nu_{1}^{1/2}N_{1}$ thus $\nu_{1}^{1/2}h=\Pi\,\nu_{1}^{1/2}h=0.$ ∎ Let us now denote by $\Phi_{(\lambda,\gamma,\alpha)}$ the operator $\Phi_{(\lambda,\gamma,\alpha)}=\left(\nu_{\alpha}(v)+\lambda+i\,(\gamma\cdot v)-(1-\alpha)\Delta_{v}\right)^{-1}\nu_{\alpha}(v).$ If the triple $(\gamma,\lambda,h)$ is solution to the eigenvalue problem (33), then we can write using the decomposition (34) $\displaystyle h$ $\displaystyle=\left(\nu_{\alpha}(v)+\lambda+i\,(\gamma\cdot v)-(1-\alpha)\Delta_{v}\right)^{-1}\nu_{\alpha}(v)\,(\nu_{\alpha}(v))^{-1}\mathcal{L}_{\alpha}^{c}h$ (36) $\displaystyle=\Phi_{(\lambda,\gamma,\alpha)}\nu_{\alpha}^{-1/2}\left(\Pi+\mathcal{S}_{\alpha}\right)\nu_{\alpha}^{1/2}h.$ This will allow us to rewrite the eigenvalues problem with bounded operators. To this purpose, we state a technical lemma about the asymptotic behavior of the operator $\Phi_{(\lambda,\gamma,\alpha)}$. ###### Lemma 3.2. For all $g\in W^{2,1}(m^{-1})$, we have $\left\\|\left(\Phi_{(\lambda,\gamma,\alpha)}-\operatorname{Id}\right)g\right\\|_{L^{1}(m^{-1})}\leq\varepsilon(\lambda,\gamma,\alpha)\\|g\\|_{W^{2,1}(m^{-1})},$ where $\lim_{(\lambda,\gamma,\alpha)\to(0,0,1)}\varepsilon(\lambda,\gamma,\alpha)=0$. ###### Proof. If we set $C_{(\lambda,\,\alpha,\,\gamma)}=\lambda+i\,(\gamma\cdot v)+\nu_{\alpha}$, we can write for all $v\in\mathbb{R}^{d}$ $\displaystyle\Phi_{(\lambda,\gamma,\alpha)}g(v)-g(v)$ $\displaystyle=\left(C_{(\lambda,\,\alpha,\,\gamma)}(v)-(1-\alpha)\Delta_{v}\right)^{-1}\nu_{\alpha}(v)g(v)-g(v)$ $\displaystyle=\frac{1}{C_{(\lambda,\,\alpha,\,\gamma)}}\left(\operatorname{Id}-\frac{1-\alpha}{C_{(\lambda,\,\alpha,\,\gamma)}(v)}\Delta_{v}\right)^{-1}C_{(0,0,1)}g(v)-g(v).$ Then, to prove the Lemma, we need to prove that the norm of the operator $\mathcal{T}_{\varepsilon}:=\left(\operatorname{Id}-\varepsilon\,\Delta_{v}\right)^{-1}-\operatorname{Id},$ defined from $W^{2,1}$ onto $L^{1}$, can be made arbitrarily small for small $\varepsilon$. Let us first reformulate this operator using the resolvent of the Laplace operator $\Delta_{v}$. Since we have $\left(\operatorname{Id}-\varepsilon\Delta_{v}\right)\mathcal{T}_{\varepsilon}=\varepsilon\Delta_{v},$ we can rewrite $\mathcal{T}_{\varepsilon}$ as555If it has been possible to conduct this study on a unweighted $L^{2}$ space, it would have been enough to notice the following Fourier representation: $\mathcal{F}_{v}\left(\mathcal{T}_{\varepsilon}\,f\right)(\xi)=-\frac{\varepsilon\,\|\xi\|^{2}}{1+\varepsilon\,\|\xi\|^{2}}\mathcal{F}_{v}(f)(\xi).$ (37) $\displaystyle\mathcal{T}_{\varepsilon}$ $\displaystyle=\left(\frac{1}{\varepsilon}\operatorname{Id}-\Delta_{v}\right)^{-1}\Delta_{v}=:\mathcal{R}\left(\frac{1}{\varepsilon},\Delta_{v}\right)\Delta_{v}.$ It is known from [6] (and classical for unweighted $L^{p}$ spaces) that the resolvent of the Laplace operator on exponentially weighted $L^{1}$ spaces verifies for an explicit real constant $a$ and a nonnegative constant $C$ (38) $\left\\|\mathcal{R}\left(\lambda,\Delta_{v}\right)h\right\\|_{L^{1}(m^{-1})}\leq\frac{C}{\lambda-a}\left\\|h\right\\|_{L^{1}(m^{-1})},$ for any $\lambda$ in an unbounded angular sector of the complex plane which does not contains $a$. In particular, using this inequality in the identity (37), we have that for any $g\in W^{2,1}(m^{-1})$ $\left\\|\mathcal{T}_{\varepsilon}\,g\right\\|_{L^{1}(m^{-1})}\leq\frac{\varepsilon\,C}{1-a\,\varepsilon}\left\\|g\right\\|_{W^{2,1}(m^{-1})}\to_{\varepsilon\to 0}0,$ which concludes the proof of the Lemma. ∎ In all the following of this section, we shall use the polar decomposition $\gamma=\rho\,\omega$ for $\rho\geq 0$ and $\omega\in\mathbb{S}^{d-1}$ of the frequency $\gamma$. We prove an invertibility result, which is needed to rewrite the eigenvalue problem (33) using bounded operators: ###### Lemma 3.3. There exist $\alpha_{3}\in(\alpha_{2},1]$ and some open sets $U_{1}\times U_{2}\subset\mathbb{R}\times\mathbb{C}$, neighborhood of $(0,0)$ such that if $(\rho,\lambda,\alpha)\in U_{1}\times U_{2}\times(\alpha_{3},1]$, then for all $\omega\in\mathbb{S}^{d-1}$, the operator $\Psi_{(\lambda,\,\rho\,\omega,\,\alpha)}:=\operatorname{Id}\,-\,\Phi_{(\lambda,\,\rho\,\omega,\alpha)}\nu_{\alpha}^{-1/2}\mathcal{S}_{\alpha}\,\nu_{\alpha}^{1/2}$ has a bounded inverse on $L^{1}(m^{-1})$. ###### Proof. We have seen that $\nu_{\alpha}^{1/2}\mathcal{S}_{\alpha}\,\nu_{\alpha}^{1/2}$ is a compact operator on $L^{1}(m^{-1})$. Moreover, we have according to the representation (28) (with $D_{(\alpha,\,\gamma)}$ given by (22)) $\displaystyle\Phi_{(\lambda,\,\rho\,\omega,\,\alpha)}$ $\displaystyle=\left(\nu_{\alpha}(v)+\lambda+i\,(\gamma\cdot v)-(1-\alpha)\Delta_{v}\right)^{-1}\nu_{\alpha}(v)$ $\displaystyle=\left(\lambda-D_{(\alpha,\,\gamma)}\right)^{-1}\nu_{\alpha}(v)$ which is a bounded operator (at least for small frequencies $\gamma$), thanks to the analysis we led on Section 2, and particularly from the localization of the discrete spectrum of $\mathcal{L}_{(\alpha,\,\gamma)}$ of Proposition 2.2. Hence, the operator $\Phi_{(\lambda,\,\rho\,\omega,\,\alpha)}\nu_{\alpha}^{-1/2}\mathcal{S}_{\alpha}\,\nu_{\alpha}^{1/2}=\left(\Phi_{(\lambda,\,\rho\,\omega,\alpha)}\nu_{\alpha}^{-1}\right)\left(\nu_{\alpha}^{1/2}\mathcal{S}_{\alpha}\,\nu_{\alpha}^{1/2}\right)$ is compact on $L^{1}(m^{-1})$. By Fredholm alternative, it just remains to show that for $(\rho,\lambda)$ small enough and $\alpha$ close to $1$, there is no non-trivial solutions $h\in L^{1}(m^{-1})$ of (39) $h=\Phi_{(\lambda,\,\rho\,\omega,\,\alpha)}\nu_{\alpha}^{-1/2}\mathcal{S}_{\alpha}\,\nu_{\alpha}^{1/2}h.$ Let us do this by contradiction. Assume that there are sequences $(\lambda_{n},\,\gamma_{n},\,\alpha_{n})_{n}\to(0,0,1)$ and $(h_{n})_{n}\subset L^{1}(m^{-1})$, $\\|h_{n}\\|=1$ solutions to (39). First, we notice thanks to the continuity of the equilibrium profiles $F_{\alpha}$ with respect to $\alpha$ (recalled in Proposition A.4) and to the smoothness properties of these profiles (recalled in Proposition A.3) that we have $\lim_{\alpha\to 1}\\|\nu_{\alpha}-\nu_{1}\\|_{L^{\infty}}=0.$ Moreover, according to Lemma A.1, the operator $\mathcal{L}_{\alpha}$ converges towards $\mathcal{L}_{1}$ in the norm of graph in $L^{1}(m^{-1})$. Then, the operator $\nu_{\alpha_{n}}^{-1/2}\mathcal{S}_{\alpha_{n}}\,\nu_{\alpha_{n}}^{1/2}$, which is compact on $L^{1}(m^{-1})$, converges towards the operator $\nu_{1}^{-1/2}\mathcal{S}_{1}\,\nu_{1}^{1/2}$, and we have up to a subsequence, $\nu_{\alpha_{n}}^{-1/2}\mathcal{S}_{\alpha_{n}}\,\nu_{\alpha_{n}}^{1/2}h_{n}\to_{n\to\infty}g\in L^{1}(m^{-1}).$ Thus, if we write $\Phi_{n}:=\Phi_{(\lambda_{n},\,\gamma_{n},\,\alpha_{n})}$, given that $h_{n}$ is solution to (39), we have (40) $h_{n}=\Phi_{n}\left[\nu_{\alpha_{n}}^{-1/2}\mathcal{S}_{\alpha_{n}}\,\nu_{\alpha_{n}}^{1/2}h_{n}-g\right]+\Phi_{n}\,g.$ But, according to Lemma 3.2 we have (41) $\lim_{n\to\infty}\\|\Phi_{n}g-g\\|_{L^{1}(m^{-1})}=0.$ Hence, by Lebesgue dominated convergence Theorem, this implies with identity (40) that the sequence $(h_{n})_{n}$ strongly converges towards $g$ (using also the fact that the sequence $(\Phi_{n}g)_{n}$ is bounded in $L^{1}(m^{-1})$ for large $n$ thanks to (41)). Therefore, we have by continuity $g=\lim_{n\to\infty}\nu_{\alpha_{n}}^{-1/2}\mathcal{S}_{\alpha_{n}}\,\nu_{\alpha_{n}}^{1/2}h_{n}=\nu_{1}^{-1/2}\mathcal{S}_{1}\,\nu_{1}^{1/2}g,$ namely $g$ is an eigenvector666This justifies the use of Lemma 3.2 in equation (41). Indeed, the solutions $h$ to this eigenvalue problem are such that $\\|h\\|_{W^{2,1}(m^{-1})}\leq C\,\\|h\\|_{L^{1}(m^{-1})},$ for $C\geq 0$, as was shown for instance in [15] using the decomposition (72): $0=(\mathcal{L}_{\alpha,\gamma}-\lambda)h=\left(A_{\delta}-B_{\alpha,\,\delta}\left(\lambda+i\left(\gamma\cdot v\right)\right)\right)h.$ for $\nu_{1}^{-1/2}\mathcal{S}_{1}\,\nu_{1}^{1/2}$ associated to the eigenvalue $1$, which is absurd according to Lemma 3.1. Finally, the operator $\operatorname{Id}\,-\,\Phi_{(\lambda,\,\gamma,\,\alpha)}\nu_{\alpha}^{-1/2}S\,\nu_{\alpha}^{1/2}$ is invertible on $L^{1}(m^{-1})$ for small $(\lambda,\,\gamma)$ and $\alpha$ close to $1$. ∎ Let us now rewrite the relation (36) as $\displaystyle\Psi_{(\lambda,\gamma,\,\alpha)}h$ $\displaystyle=\left[\operatorname{Id}-\Phi_{(\lambda,\gamma,\,\alpha)}\nu_{\alpha}^{-1/2}\mathcal{S}_{\alpha}\,\nu_{\alpha}^{1/2}\right]\Phi_{(\lambda,\gamma)}\nu_{\alpha}^{-1/2}\left(\Pi+\mathcal{S}_{\alpha}\right)\nu_{\alpha}^{1/2}h$ $\displaystyle=\Phi_{(\lambda,\gamma,\,\alpha)}\left\\{\nu_{\alpha}^{-1/2}\Pi\,\nu_{\alpha}^{1/2}+\nu_{\alpha}^{-1/2}\mathcal{S}_{\alpha}\,\nu_{\alpha}^{1/2}\left[\operatorname{Id}-\Phi_{(\lambda,\gamma)}\left(\Pi+\mathcal{S}_{\alpha}\right)\nu_{\alpha}^{1/2}\right]\right\\}h.$ According to Lemma 3.3, $\Psi_{(\lambda,\gamma,\,\alpha)}$ is invertible for small $(\lambda,\gamma)$ and $\alpha\in(\alpha_{3},1]$. Then, provided that $h$ is solution to (36), we have (42) $h=\Psi_{(\lambda,\gamma,\,\alpha)}^{-1}\Phi_{(\lambda,\gamma,\,\alpha)}\nu_{\alpha}^{-1/2}\Pi\,\nu_{\alpha}^{1/2}h.$ Let us introduce the “conjugated operator” $\mathcal{P}:=\nu_{\alpha}^{-1/2}\Pi\,\nu_{\alpha}^{1/2}=\mathbb{P}\,\nu_{\alpha}^{1/2}$, where $\mathbb{P}$ is the projection onto the space of elastic collisional invariants $N_{1}$. We can use it to rewrite (42) (and then the eigenvalue problem (33)) as the following finite dimensional system of equations (43) $\mathcal{P}h=\mathcal{P}\Psi_{(\lambda,\gamma,\,\alpha)}^{-1}\Phi_{(\lambda,\gamma,\,\alpha)}\mathcal{P}h,$ which can be understood as: > _Finding > $X_{(\lambda,\gamma,\,\alpha)}=(x_{0},\ldots,x_{d+1})\in\mathbb{R}^{d+2}$ > such that_ > $\left(A_{(\lambda,\gamma,\,\alpha)}-\operatorname{Id}\right)X_{(\lambda,\gamma,\,\alpha)}=0,$ > _where $A=\mathcal{P}\Psi^{-1}\Phi\in\mathcal{M}_{d+2,\,d+2}(\mathbb{R})$._ We shall find in the following section some conditions on $\lambda$ in order to have $\det\left(A_{(\lambda,\gamma,\,\alpha)}-\operatorname{Id}\right)=0.$ Under such conditions, the abstract problem will admit a non-trivial solution $X$. Coming back to the original problem, given that $X=\mathcal{P}h=\mathbb{P}\nu_{\alpha}^{1/2}\,h$, we will obtain thanks to equation (42) a solution $h$ to the original eigenvalue problem (33). ### 3.2. Finite Dimensional Resolution We shall study the vector $X$ component-wise, using the normalization (44) $X=\mathbb{P}\,\nu_{\alpha}^{1/2}\,h(v)=x_{0}\,F_{1}(v)+(x\cdot v)\,F_{1}(v)+x_{d+1}\left(\|v\|^{2}-c_{\nu}\right)F_{1}(v),$ where $c_{\nu}$ is such that $\langle\nu_{\alpha},\|v\|^{2}\rangle=\langle\nu_{\alpha},c_{\nu}\rangle$. If we compute the product of (44) with elements of $\mathcal{N}_{\alpha}=\operatorname{Span}\\{\nu_{\alpha}^{1/2}F_{1},\,\nu_{\alpha}^{1/2}v_{i}\,F_{1},\,\nu_{\alpha}^{1/2}\left(\|v\|^{2}-c_{\nu}\right)F_{1}:\,1\leq i\leq d\\},$ we find using the definition of $\mathbb{P}$ and the orthogonality of elements of $N_{\alpha}$ for $\langle\cdot,\cdot\rangle$ that (45) $\left\\{\begin{aligned} &\langle\nu_{\alpha}^{1/2}h,\,\nu_{\alpha}^{1/2}F_{1}\rangle=x_{0}\left\langle\nu_{\alpha}^{1/2}F_{1},\,\nu_{\alpha}^{1/2}F_{1}\right\rangle,\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt&\langle\nu_{\alpha}^{1/2}h,\,\nu_{\alpha}^{1/2}\,v_{i}\,F_{1}\rangle=x_{i}\left\langle\nu_{\alpha}^{1/2}\,v_{i}\,F_{1},\nu_{\alpha}^{1/2}\,v_{i}\,F_{1}\right\rangle,\quad\forall 1\leq i\leq d,\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt&\langle\nu_{\alpha}^{1/2}h,\,\nu_{\alpha}^{1/2}\left(\|v\|^{2}-c_{\nu}\right)F_{1}\rangle=x_{d+1}\left\langle\nu_{\alpha}^{1/2}\left(\|v\|^{2}-c_{\nu}\right)F_{1},\nu_{\alpha}^{1/2}\left(\|v\|^{2}-c_{\nu}\right)F_{1}\right\rangle.\end{aligned}\right.$ For clarity sake, let us set in the following $\langle\phi,\psi\rangle_{F_{1}}:=\int_{\mathbb{R}^{d}}\phi(v)\,\overline{\psi}(v)\,F_{1}(v)\,dv,$ in order to have $\left\langle\nu_{\alpha}^{1/2}\,h_{1}\,F_{1},\nu_{\alpha}^{1/2}\,h_{2}\,F_{1}\right\rangle=\langle\nu_{\alpha}h_{1},h_{2}\rangle_{F_{1}}.$ Using (42) together with the relations (45), we obtain for all $1\leq i\leq d$ (46) $\left\\{\begin{aligned} &x_{0}\left\langle\nu_{\alpha},1\right\rangle_{F_{1}}=x_{0}\left\langle\nu_{\alpha},T_{\gamma}1\right\rangle_{F_{1}}+\left\langle\nu_{\alpha},T_{\gamma}\,(x\cdot v)\right\rangle_{F_{1}}+x_{d+1}\left\langle\nu_{\alpha},T_{\gamma}\left(\|v\|^{2}-c_{\nu}\right)\right\rangle_{F_{1}},\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt&x_{i}\left\langle\nu_{\alpha}\,v_{i},v_{i}\right\rangle_{F_{1}}=x_{0}\left\langle\nu_{\alpha}\,v_{i},T_{\gamma}1\right\rangle_{F_{1}}+\left\langle\nu_{\alpha}\,v_{i},T_{\gamma}\,(x\cdot v)\right\rangle_{F_{1}}+x_{d+1}\left\langle\nu_{\alpha}\,v_{i},T_{\gamma}\left(\|v\|^{2}-c_{\nu}\right)\right\rangle_{F_{1}},\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt&x_{d+1}\left\langle\nu_{\alpha}\left(\|v\|^{2}-c_{\nu}\right),\|v\|^{2}-c_{\nu}\right\rangle_{F_{1}}=x_{0}\left\langle\nu_{\alpha}\left(\|v\|^{2}-c_{\nu}\right),T_{\gamma}1\right\rangle_{F_{1}}+\left\langle\nu_{\alpha}\left(\|v\|^{2}-c_{\nu}\right),T_{\gamma}\,(x\cdot v)\right\rangle_{F_{1}}\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\ \ +x_{d+1}\left\langle\nu_{\alpha}\left(\|v\|^{2}-c_{\nu}\right),T_{\gamma}\left(\|v\|^{2}-c_{\nu}\right)\right\rangle_{F_{1}},\end{aligned}\right.$ where we have set for fixed $(\lambda,\alpha)$ (47) $T_{\gamma}:=\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\Phi_{(\lambda,\gamma,\,\alpha)}.$ The system (46) is the componentwise version of the projected problem (43). We are now going to decompose this system of $d+2$ equations in $X=(x_{0},\ldots,x_{d+1})$ in a closed system of $3$ equations in $x_{0}$, $x\cdot\omega$ and $x_{d+1}$ for a fixed $\omega\in\mathbb{S}^{d-1}$ (corresponding to the _longitudinal_ sound waves of the Boltzmann equation, see also the work of Nicolaenko [18]) together with a scalar relation in $x_{i}$ for all $1\leq i\leq d$ (corresponding to the _transverse_ sound waves). For this, we need the following technical lemma: ###### Lemma 3.4. Let $x,y\in\mathbb{R}^{d}$, $\bm{e}:=(1,0,\ldots,0)^{\mathsf{T}}$ and $\gamma=\rho\,\omega$ for $\rho\in\mathbb{R}$ and $\omega\in\mathbb{S}^{d-1}$. Then, we have (48) $\displaystyle\left\langle\nu_{\alpha},T_{\gamma}\,(x\cdot v)\right\rangle_{F_{1}}=x\cdot\omega\left\langle\nu_{\alpha},T_{\rho\bm{e}}\,v_{1}\right\rangle_{F_{1}},$ (49) $\displaystyle\left\langle\nu_{\alpha}\,(x\cdot v),T_{\gamma}\,1\right\rangle_{F_{1}}=x\cdot\omega\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}\,1\right\rangle_{F_{1}},$ (50) $\displaystyle\begin{split}\left\langle\nu_{\alpha}\,(x\cdot v),T_{\gamma}\,(y\cdot v)\right\rangle_{F_{1}}&=(x\cdot\omega)(y\cdot\omega)\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}\,v_{1}\right\rangle_{F_{1}}\\\ &+\left[x\cdot\omega-(x\cdot\omega)(y\cdot\omega)\right]\left\langle\nu_{\alpha}\,v_{2},T_{\rho\bm{e}}\,v_{2}\right\rangle_{F_{1}},\end{split}$ (51) $\displaystyle\left\langle\nu_{\alpha}\,(x\cdot v),T_{\gamma}\left(\|v\|^{2}-c_{\nu}\right)\right\rangle_{F_{1}}=x\cdot\omega\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}\left(\|v\|^{2}-c_{\nu}\right)\right\rangle_{F_{1}},$ (52) $\displaystyle\left\langle\nu_{\alpha}\left(\|v\|^{2}-c_{\nu}\right),T_{\gamma}\,(x\cdot v)\right\rangle_{F_{1}}=x\cdot\omega\left\langle\nu_{\alpha}\left(\|v\|^{2}-c_{\nu}\right),T_{\rho\bm{e}}\,v_{1}\right\rangle_{F_{1}}.$ ###### Proof. According to the definitions of $T_{\gamma}$ and $\Psi_{(\lambda,\gamma,\,\alpha)}$, the $\gamma$–dependency of $T_{\gamma}$ is only happening through the operator $\Phi_{(\lambda,\gamma,\,\alpha)}$. But, for $M\in\mathcal{O}(d)$ the orthogonal group of $\mathbb{R}^{d}$ (namely, $MM^{}=\bf I_{d}$) one has for $v\in\mathbb{R}^{d}$ and $g\in\operatorname{dom}\left(\Phi_{(\lambda,\gamma,\,\alpha)}\right)$, using the fact that $\nu_{\alpha}$ is a radial function, for all $v\in\mathbb{R}^{d}$, $\displaystyle\left(\Phi_{(\lambda,\gamma,\,\alpha)}\,g\right)(Mv)$ $\displaystyle=\left(\nu_{\alpha}(Mv)+\lambda+i\,(\gamma\cdot Mv)-(1-\alpha)\Delta_{v}\right)^{-1}\nu_{\alpha}(Mv)\,g(Mv),$ $\displaystyle=\left(\Phi_{(\lambda,M^{-1}\,\gamma,\,\alpha)}Mg\right)(v),$ where we have set $Mg\,(v):=g(Mv)$. Then $\left(T_{\gamma}\,g\right)(Mv)=\left(T_{M^{-1}\gamma}\,Mg\right)(v),\quad\forall\,v\in\mathbb{R}^{d}.$ Especially, if $g$ is a radial function, there exists a function $\Gamma_{g}$ such that $\left(T_{\gamma}\,g\right)(v)=\Gamma_{g}(\gamma\cdot v,\|v\|).$ One has thanks to this result $\displaystyle\left\langle\nu_{\alpha},T_{\gamma}\,(x\cdot v)\right\rangle_{F_{1}}$ $\displaystyle=\left\langle T_{\gamma}^{}\,\nu_{\alpha},(x\cdot v)\right\rangle_{F_{1}}$ $\displaystyle=\int_{\mathbb{R}^{d}}\Gamma_{\nu_{\alpha}}(\gamma\cdot v,\|v\|)\,(x\cdot v)\,{F_{1}}(v)\,dv.$ Let $M\in\mathcal{O}(d)$ such that $M^{-1}\,\omega=\bm{e}$. Thanks to the change of variables $v=M\xi$ and using the polar coordinates $\gamma=\rho\,\omega$ one has $\gamma\cdot v=\rho\,M^{-1}\,\omega\cdot\xi=\rho\,\xi_{1}$ and then $\displaystyle\left\langle\nu_{\alpha},T_{\gamma}\,(x\cdot v)\right\rangle_{F_{1}}$ $\displaystyle=\int_{\mathbb{R}^{d}}\Gamma_{\nu_{\alpha}}(\rho\,\xi_{1},\|\xi\|)\,(M^{-1}x\cdot\xi)\,{F_{1}}(\xi)\,d\xi$ $\displaystyle=\int_{\mathbb{R}^{d}}\Gamma_{\nu_{\alpha}}^{odd}(\rho\,\xi_{1},\|\xi\|)\,(M^{-1}x\cdot\xi)\,{F_{1}}(\xi)\,d\xi,$ where $g^{odd}(a,\cdot)=\left(g(a,\cdot)-g(-a,\cdot)\right)/2$. Given that ${F_{1}}$ is a radial function of $v$, $\langle g^{odd},h\rangle_{F_{1}}=\langle g,h^{odd}\rangle_{F_{1}}$ and $(M^{-1}x)_{1}=(M^{-1}x)\cdot M^{-1}\omega$, one has $\displaystyle\left\langle\nu_{\alpha},T_{\gamma}\,(x\cdot v)\right\rangle_{F_{1}}$ $\displaystyle=\int_{\mathbb{R}^{d}}\Gamma_{\nu_{\alpha}}^{odd}(\rho\,\xi_{1},\|\xi\|)\,(M^{-1}x)_{1}\,\xi_{1}\,{F_{1}}(\xi)\,d\xi$ $\displaystyle=(M^{-1}x)_{1}\int_{\mathbb{R}^{d}}\Gamma_{\nu_{\alpha}}(\rho\,\xi_{1},\|\xi\|)\,\xi_{1}\,{F_{1}}(\xi)\,d\xi$ $\displaystyle=x\cdot\omega\left\langle T_{\rho\bm{e}}^{}\,\nu_{\alpha},v_{1}\right\rangle_{F_{1}},$ which proves (48). Thanks to the same arguments $\displaystyle\left\langle\nu_{\alpha}\,(x\cdot v),T_{\gamma}\,1\right\rangle_{F_{1}}$ $\displaystyle=\int_{\mathbb{R}^{d}}\nu_{\alpha}(v)\,(x\cdot v)\,\Gamma_{1}(\gamma\cdot v,\|v\|)\,{F_{1}}(v)\,dv$ $\displaystyle=\int_{\mathbb{R}^{d}}\nu_{\alpha}(\xi)\,(M^{-1}x\cdot\xi)\,\Gamma_{1}(\rho\,\xi_{1},\|\xi\|)\,{F_{1}}(\xi)\,d\xi$ $\displaystyle=\int_{\mathbb{R}^{d}}\nu_{\alpha}(\xi)\,(M^{-1}x\cdot\xi)\,\Gamma_{1}^{odd}(\rho\,\xi_{1},\|\xi\|)\,{F_{1}}(\xi)\,d\xi$ $\displaystyle=x\cdot\omega\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}1\right\rangle_{F_{1}},$ which proves (49). Concerning the next identity, one has $\displaystyle\left\langle\nu_{\alpha}\,(x\cdot v),T_{\gamma}\,(y\cdot v)\right\rangle_{F_{1}}$ $\displaystyle=\sum_{1\leq i,j\leq d}x_{i}\,y_{j}\int_{\mathbb{R}^{d}}\nu_{\alpha}(v)\,v_{i}\,(T_{\gamma}\,v_{j})(v)\,{F_{1}}(v)\,dv$ $\displaystyle=\sum_{1\leq i,j\leq d}x_{i}\,y_{j}\int_{\mathbb{R}^{d}}\nu_{\alpha}(\xi)\,(M\xi)_{i}\left(T_{\rho\bm{e}}(M\xi)_{j}\right)(\xi)\,{F_{1}}(\xi)\,d\xi$ $\displaystyle=\sum_{1\leq i,j,k,l\leq d}x_{i}\,y_{j}\,M_{il}\,M_{jk}\int_{\mathbb{R}^{d}}\nu_{\alpha}(\xi)\,\xi_{l}\left(T_{\rho\bm{e}}\,\xi_{k}\right)(\xi)\,{F_{1}}(\xi)\,d\xi.$ If $k\neq l$, this integral is zero (it is clear by doing the transformation $\xi\to-\xi$). In the other case, one has $\displaystyle\left\langle\nu_{\alpha}\,(x\cdot v),T_{\gamma}\,(y\cdot v)\right\rangle_{F_{1}}$ $\displaystyle=\sum_{1\leq i,j\leq d}x_{i}\,y_{j}\,M_{i1}\,M_{j1}\int_{\mathbb{R}^{d}}\nu_{\alpha}(\xi)\,\xi_{1}\left(T_{\rho\bm{e}}\,\xi_{1}\right)(\xi)\,{F_{1}}(\xi)\,d\xi$ $\displaystyle+\sum_{\begin{subarray}{c}1\leq i,j\leq d\\\ 2\leq l\leq d\end{subarray}}x_{i}\,y_{j}\,M_{il}\,M_{jl}\int_{\mathbb{R}^{d}}\nu_{\alpha}(\xi)\,\xi_{2}\left(T_{\rho\bm{e}}\,\xi_{2}\right)(\xi)\,{F_{1}}(\xi)\,d\xi$ $\displaystyle=(x\cdot\omega)(y\cdot\omega)\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}\,v_{1}\right\rangle_{F_{1}}$ $\displaystyle+\sum_{1\leq i,j\leq d}x_{i}\,y_{j}\left[(MM^{})_{ij}-M_{i1}\,M_{j_{1}}\right]\left\langle\nu_{\alpha}\,v_{2},T_{\rho\bm{e}}\,v_{2}\right\rangle_{F_{1}},$ which is (50) because $MM^{}=I_{d}$. The inequalities (51) and (52) are finally obtained using the same methods of proof. ∎ Applying this lemma to system (46), we find for all $1\leq i\leq d$ (53) $\displaystyle x_{0}\left\langle\nu_{\alpha},T_{\rho\bm{e}}1-1\right\rangle_{F_{1}}+x\cdot\omega\left\langle\nu_{\alpha},T_{\rho\bm{e}}v_{1}\right\rangle_{F_{1}}+x_{d+1}\left\langle\nu_{\alpha},T_{\rho\bm{e}}\left(\|v\|^{2}-c_{\nu}\right)\right\rangle_{F_{1}}=0,$ (54) $\displaystyle\begin{split}x_{i}\langle\nu_{\alpha}\,v_{i},v_{i}\rangle_{F_{1}}&=\omega_{i}\,x_{0}\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}1\right\rangle_{F_{1}}+\left[x_{i}-\omega_{i}\,(x\cdot\omega)\right]\left\langle\nu_{\alpha}\,v_{2},T_{\rho\bm{e}}v_{2}\right\rangle_{F_{1}}\\\ &+\omega_{i}\,(x\cdot\omega)\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}v_{1}\right\rangle_{F_{1}}+\omega_{i}\,x_{d+1}\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}\left(\|v\|^{2}-c_{\nu}\right)\right\rangle_{F_{1}},\end{split}$ (55) $\displaystyle\begin{split}x_{0}\left\langle\nu_{\alpha}\left(\|v\|^{2}-c_{\nu}\right),T_{\rho\bm{e}}\,1\right\rangle_{F_{1}}&+x\cdot\omega\left\langle\nu_{\alpha}\left(\|v\|^{2}-c_{\nu}\right),T_{\rho\bm{e}}v_{1}\right\rangle_{F_{1}}\\\ &+x_{d+1}\left\langle\nu_{\alpha}\left(\|v\|^{2}-c_{\nu}\right),T_{\rho\bm{e}}\left(\|v\|^{2}-c_{\nu}\right)-\left(\|v\|^{2}-c_{\nu}\right)\right\rangle_{F_{1}}=0.\end{split}$ Now, on the one hand, if we multiply (54) by $\omega_{i}$ and sum over all $i$, using the fact that $\|\omega\|=1$, we find that (56) $x_{0}\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}1\right\rangle_{F_{1}}+x\cdot\omega\,\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}v_{1}-v_{1}\right\rangle_{F_{1}}+x_{d+1}\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}\left(\|v\|^{2}-c_{\nu}\right)\right\rangle_{F_{1}}=0.$ The system (53)–(56)–(55) is closed in $(x_{0},x\cdot\omega,x_{d+1})$ for a fixed $\omega\in\mathbb{S}^{d-1}$. Coming back to a more abstract form, there exists solutions to this system if and only if (57) $D(\lambda,\rho,\alpha)=0,$ where we have defined $D$ as the following Gram-like matrix (remember that $T_{\gamma}$ is given by (47) and depends on $(\lambda,\gamma,\alpha)$) (58) $D(\lambda,\rho,\alpha):=\\\ \begin{vmatrix}\left\langle\nu_{\alpha},\left(T_{\rho\bm{e}}-\operatorname{Id}\right)1\right\rangle_{F_{1}}&\left\langle\nu_{\alpha},T_{\rho\bm{e}}v_{1}\right\rangle_{F_{1}}&\left\langle\nu_{\alpha},T_{\rho\bm{e}}\left(\|v\|^{2}-c_{\nu}\right)\right\rangle_{F_{1}}\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}1\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,v_{1},\left(T_{\rho\bm{e}}-\operatorname{Id}\right)v_{1}\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,v_{1},T_{\rho\bm{e}}\left(\|v\|^{2}-c_{\nu}\right)\right\rangle_{F_{1}}\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\left\langle\nu_{\alpha}\left(\|v\|^{2}-c_{\nu}\right),T_{\rho\bm{e}}1\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\left(\|v\|^{2}-c_{\nu}\right),T_{\rho\bm{e}}v_{1}\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\left(\|v\|^{2}-c_{\nu}\right),\left(T_{\rho\bm{e}}-\operatorname{Id}\right)\left(\|v\|^{2}-c_{\nu}\right)\right\rangle_{F_{1}}\end{vmatrix}.$ On the other hand, if one multiplies (56) by $\omega_{i}$ and subtract this expression to (54), one finds (59) $\left[x_{i}-\omega_{i}\,(x\cdot\omega)\right]D_{\omega}(\lambda,\rho,\alpha)=0,$ where we have set (60) $D_{\omega}(\lambda,\rho,\alpha):=\left\langle\nu_{\alpha}\,v_{1},\left(T_{\rho\bm{e}}-\operatorname{Id}\right)v_{1}\right\rangle_{F_{1}}.$ Then, if one solves (56) in $C\cdot\omega$, the relation (59) will give the expression of $x_{i}$, provided that the equation $D_{\omega}(\lambda,\rho,\alpha)=0$ admits an unique solution $\lambda$. We will simplify these expressions thanks to the following Lemma. ###### Lemma 3.5. Let $(\rho,\lambda,\alpha)\in U_{1}\times U_{2}\times(\alpha_{3},1]$. If $g,h$ are elastic collisional invariants, namely if $g,h\in N_{1}=\operatorname{Span}\\{F_{1},\,v_{i}\,F_{1},\|v\|^{2}\,F_{1}:\,1\leq i\leq d\\},$ then we can write for all $\omega\in\mathbb{S}^{d-1}$ and $\gamma=\rho\,\omega$ $\displaystyle\left(\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\Phi_{(\lambda,\gamma,\,\alpha)}-\operatorname{Id}\right)h=\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\left(\Phi_{(\lambda,\gamma,\,\alpha)}-\operatorname{Id}\right)h,$ $\displaystyle\left\langle\nu_{\alpha}g,\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\Phi_{(\lambda,\gamma,\,\alpha)}h\right\rangle=\left\langle\nu_{\alpha}g,\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\left(\Phi_{(\lambda,\gamma,\,\alpha)}-\operatorname{Id}\right)h\right\rangle.$ ###### Proof. By definition of $\mathcal{N}_{\alpha}$, we have $\nu_{\alpha}^{1/2}h\in\mathcal{N}_{\alpha}$, and then $\mathcal{S}_{\alpha}\,\nu_{\alpha}^{1/2}h=0$. But, we know that $\Psi_{(\lambda,\gamma,\alpha)}=\operatorname{Id}-\Phi_{(\lambda,\gamma,\alpha)}\nu_{\alpha}^{-1/2}\mathcal{S}_{\alpha}\,\nu_{\alpha}^{1/2}.$ Thus, we have $\Psi_{(\lambda,\gamma,\alpha)}h=h$, and given that $\Psi_{(\lambda,\gamma,\,\alpha)}$ is invertible for $(\rho,\lambda,\alpha)\in U_{1}\times U_{2}\times(\alpha_{3},1]$ and $\gamma=\rho\,\omega$, we have (61) $\Psi^{-1}_{(\lambda,\gamma,\alpha)}h=h,$ which proves the first relation. Using the orthogonality of the collisional invariants and (61), we obtain the second equality: $\displaystyle\left\langle\nu_{\alpha}g,\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\Phi_{(\lambda,\gamma,\alpha)}h\right\rangle$ $\displaystyle=\left\langle\nu_{\alpha}g,\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\Phi_{(\lambda,\gamma,\alpha)}h\right\rangle-\left\langle\nu_{\alpha}g,h\right\rangle$ $\displaystyle=\left\langle\nu_{\alpha}g,\left(\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\Phi_{(\lambda,\gamma,\alpha)}-\operatorname{Id}\right)h\right\rangle$ $\displaystyle=\left\langle\nu_{\alpha}g,\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\left(\Phi_{(\lambda,\gamma,\alpha)}-\operatorname{Id}\right)h\right\rangle.$ ∎ Let us set $\Upsilon_{(\lambda,\gamma,\,\alpha)}:=\Psi^{-1}_{(\lambda,\gamma,\,\alpha)}\left(\Phi_{(\lambda,\gamma,\,\alpha)}-\operatorname{Id}\right)$. Thanks to this lemma, to the definition of $c_{\nu}$ and by the nullity of the odd moments of the centered Gaussian $F_{1}$, we can write (58) in a “simpler” form, namely (62) $D(\lambda,\rho,\alpha)=\begin{vmatrix}\left\langle\nu_{\alpha},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,1\right\rangle_{F_{1}}&\left\langle\nu_{\alpha},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,v_{1}\right\rangle_{F_{1}}&\left\langle\nu_{\alpha},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,g\right\rangle_{F_{1}}\\\ \left\langle\nu_{\alpha}\,v_{1},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,1\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,v_{1},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,v_{1}\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,v_{1},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,g\right\rangle_{F_{1}}\\\ \left\langle\nu_{\alpha}\,g,\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,1\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,g,\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,v_{1}\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,g,\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,g\right\rangle_{F_{1}}\end{vmatrix},$ where we have set $g(v):=\|v\|^{2}-c_{\nu}$. We can also write (60) the same way (63) $D_{\omega}(\lambda,\rho,\alpha)=\left\langle\nu_{\alpha}\,v_{1},\Upsilon_{\rho\bm{e}}\,v_{1}\right\rangle_{F_{1}}.$ Before solving these equations, we need a last lemma. ###### Lemma 3.6. Let $h$ be an elastic collisional invariant (namely $h\in N_{1}$). If $\Psi^{}$ denotes the adjoint operator of $\Psi$, then we have $\displaystyle\left(\Psi^{}_{(0,0,1)}\right)^{-1}\,\nu_{1}h=\nu_{1}h.$ ###### Proof. Let $(\rho,\lambda,\alpha)\in U_{1}\times U_{2}\times(\alpha_{2},1]$ and $\omega\in\mathbb{S}^{d-1}$ and set $\gamma=\rho\,\omega$. If $T$ is an invertible operator on a Banach space, it is known that $(T^{})^{-1}=(T^{-1})^{}$. Moreover, provided that $\nu_{1}\in\mathbb{R}$, the adjoint operator $\Phi^{}_{(\lambda,\gamma,\,1)}$ is the operator of multiplication by $\frac{\nu_{1}(v)}{\nu_{1}(v)+\bar{\lambda}-i\,(\gamma\cdot v)}$ Then, if $(\lambda,\rho)\to(0,0)$ we have $\Phi_{(\lambda,\rho\,\omega,\,1)}^{}\to\operatorname{Id}$ strongly. But, we can also compute $\displaystyle\Psi_{(\lambda,\,\rho\,\omega,\,\alpha)}^{}$ $\displaystyle=\operatorname{Id}\,-\,\left(\Phi_{(\lambda,\rho\,\omega)}\nu_{1}^{-1/2}\mathcal{S}_{1}\,\nu_{1}^{1/2}\right)^{}$ $\displaystyle=\operatorname{Id}\,-\,\nu_{1}^{1/2}\mathcal{S}_{1}^{}\,\nu_{1}^{-1/2}\,\Phi_{(\lambda,\rho\,\omega)}^{},$ and as $h\in N_{1}$, we have $\nu_{1}^{1/2}\mathcal{S}_{1}^{}\,\nu_{1}^{-1/2}\,\nu h=0.$ Finally, we can write $\displaystyle\Psi_{(0,0,1)}^{}\nu_{1}h$ $\displaystyle=\left(\operatorname{Id}\,-\,\nu_{1}^{1/2}\mathcal{S}_{1}^{}\,\nu_{1}^{-1/2}\right)\nu_{1}h$ $\displaystyle=\nu_{1}h,$ which concludes the proof after inversion. ∎ ###### Remark 3. The eigenvalues and eigenvectors of $\mathcal{L}_{\alpha,\,\rho\omega}$ are analytic function or $\rho$. Indeed, thanks to the hard spheres kernel and estimates (16), there exists a nonnegative constant $M$ such that $\left\\|(\omega\cdot v)\,h\right\\|_{L^{1}(m^{-1})}\leq M\left(\left\\|h\right\\|_{L^{1}(m^{-1})}+\left\\|\mathcal{L}_{\alpha}h\right\\|_{L^{1}(m^{-1})}\right).$ We can then apply [12, Thm. VII.2.6 and Rem. VII.2.7] about the analyticity of the spectrum of a closed operator on a Banach space. ### 3.3. First Order Coefficients of the Taylor Expansion We can now study in details for what values of the parameters $\lambda$ and $\alpha$ one can solve the projected eigenvalue problem (59). We start by considering the behavior of the transverse sound waves. ###### Proposition 3.1. Let $\omega\in\mathbb{S}^{d-1}$. There exist $\rho_{0}>0$ and $\alpha_{4}\in(\alpha_{3},1]$ such that the problem of solving the equation $D_{\omega}(\lambda,\rho,\alpha)=0$ has a unique solution $\lambda_{\omega}=\lambda_{\omega}(\rho,\alpha)\in\mathcal{C}^{\infty}\left((-\bar{\rho}_{0},\bar{\rho}_{0})\times(\alpha_{4},1]\right)$, verifying $\lambda_{\omega}(0,1)=\frac{\partial\lambda_{\omega}}{\partial\rho}(0,1)=\frac{\partial\lambda_{\omega}}{\partial\alpha}(0,1)=0.$ ###### Proof. Let us write thanks to the compact expression (63) of $D_{\omega}$ $\displaystyle 0$ $\displaystyle=-D_{\omega}(\lambda,\rho,\alpha)$ $\displaystyle=-\left\langle\left(\Psi_{(\lambda,\,\rho\bm{e},\,\alpha)}^{}\right)^{-1}(\nu_{\alpha}v_{1}),\left(\Phi_{(\lambda,\,\rho\bm{e},\,\alpha)}-\operatorname{Id}\right)v_{1}\right\rangle_{F_{1}}$ $\displaystyle=-\int_{\mathbb{R}^{d}}\left(\Psi_{(\lambda,\,\rho\bm{e},\,\alpha)}^{}\right)^{-1}(\nu_{\alpha}v_{1})\left[\left(\nu_{\alpha}(v)+\lambda+i\,(\rho\bm{e}\cdot v)-(1-\alpha)\Delta_{v}\right)^{-1}\nu_{\alpha}(v)-\operatorname{Id}\right](v_{1})\,{F_{1}}(v)\,dv$ $\displaystyle=\int_{\mathbb{R}^{d}}\left(\Psi_{(\lambda,\,\rho\bm{e},\,\alpha)}^{}\right)^{-1}(\nu_{\alpha}v_{1})\left(\nu_{\alpha}(v)+\lambda+i\,\rho\,v_{1}-(1-\alpha)\Delta_{v}\right)^{-1}\left(\lambda+i\,\rho\,v_{1}-(1-\alpha)\Delta_{v}\right)(v_{1})$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad{F_{1}}(v)\,dv.$ Let us now set $z=\lambda/\rho$ and $s=1-\alpha$. We shall take the limit $(\rho,s)\to(0,0)$ in $D_{\omega}$. For this, we define a new function $G_{\omega}$ as $G_{\omega}(z,\rho,s):=\frac{1}{\rho}D_{\omega}(\rho z,\rho,1-s).$ Then, as $\Delta_{v}(v_{1})=0$, we will have $D_{\omega}(\lambda,\rho,\alpha)=0$ if and only if $\displaystyle 0$ $\displaystyle=-G_{\omega}(z,\rho,s)$ $\displaystyle=\int_{\mathbb{R}^{d}}\left(\Psi_{(\rho z,\rho\bm{e},1-s)}^{}\right)^{-1}(\nu_{1-s}v_{1})\left(\nu_{1-s}(v)+\rho z+i\,\rho\,v_{1}-s\Delta_{v}\right)^{-1}\left((z+iv_{1})\,v_{1}\right){F_{1}}(v)\,dv.$ Moreover, if $\alpha\to 1$, thanks to the continuity of the equilibrium profiles $F_{\alpha}$ with respect to $\alpha$ (recalled in Proposition A.4) and to the smoothness properties of these profiles (recalled in Proposition A.3), we have $\nu_{\alpha}(v)\to\nu_{1}(v)$, uniformly in $v$. Hence, if we take the limit $(\rho,s)\to(0,0)$, we find thanks to Lemma 3.6 that $\displaystyle 0$ $\displaystyle=-G_{\omega}(z,0,1)$ $\displaystyle=\int_{\mathbb{R}^{d}}\left(\Psi_{(0,\,0,\,1)}^{}\right)^{-1}(\nu_{1}v_{1})\frac{z+iv_{1}}{\nu_{1}}(v_{1})\,{F_{1}}(v)\,dv$ $\displaystyle=z\int_{\mathbb{R}^{d}}v_{1}^{2}\,{F_{1}}(v)\,dv=z\,\bar{T}_{1}.$ Provided that $\bar{T}_{1}$ is nonzero, we have $z=0$. It just remains to apply the implicit function theorem to the map $(z,\rho,s)\mapsto G_{\omega}(z,\rho,s)$ in $(0,0,0)$. Provided that we have $\left\\{\begin{aligned} &G_{\omega}(0,0,0)=0,\\\ &\frac{\partial G_{\omega}}{\partial z}(0,0,0)={\bar{T}_{1}},\end{aligned}\right.$ there exist two real constants $\bar{\rho}_{0}>0$, $\alpha_{4}\in(\alpha_{3},1]$ and a mapping $z_{\omega}\in\mathcal{C}^{\infty}\left((-\bar{\rho}_{0},\bar{\rho}_{0})\times[0,1-\alpha_{4})\right)$ such that if $\|\rho\|\leq\bar{\rho}_{0}$ and $s\in[0,1-\alpha_{4})$, then $\frac{1}{\rho}D_{\omega}\left(\rho z_{\omega}(\rho,s),\rho,1-s\right)=G_{\omega}\left(z_{\omega}(\rho,s),\rho,s\right)=0.$ To conclude the proof, we set $\lambda_{\omega}(\rho,\alpha):=\rho z_{\omega}\left(\rho,1-\alpha\right)$ and this function has the properties we were looking from. ∎ Let us now turn to the dispersion relations (57), corresponding to the longitudinal sound waves. We recall the simplified expression of $D$ for the reader convenience: $D(\lambda,\rho,\alpha)=\begin{vmatrix}\left\langle\nu_{\alpha},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,1\right\rangle_{F_{1}}&\left\langle\nu_{\alpha},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,v_{1}\right\rangle_{F_{1}}&\left\langle\nu_{\alpha},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,g\right\rangle_{F_{1}}\\\ \left\langle\nu_{\alpha}\,v_{1},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,1\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,v_{1},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,v_{1}\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,v_{1},\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,g\right\rangle_{F_{1}}\\\ \left\langle\nu_{\alpha}\,g,\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,1\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,g,\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,v_{1}\right\rangle_{F_{1}}&\left\langle\nu_{\alpha}\,g,\Upsilon_{(\lambda,\,\rho\bm{e},\,\alpha)}\,g\right\rangle_{F_{1}}\end{vmatrix}.$ We prove the following result concerning the behavior of the eigenvalues for small frequency and inelasticity. ###### Proposition 3.2. For $\lambda\in U_{2}$ (see Lemma 3.3), there exists $\bar{\rho}>0$ and $\alpha_{5}\in(\alpha_{4},1]$ such that for $\alpha\in(\alpha_{5},1]$ the elastic dispersion relation $D(\lambda,\rho,\alpha)=0$ has exactly three branches of solutions $\lambda^{(j)}(\rho,\alpha)$ for all $j\in\\{-1,0,1\\}$ and $\rho\in(-\bar{\rho}_{1},\bar{\rho}_{1})$. These solutions are of class $\mathcal{C}^{\infty}(-\bar{\rho}_{1},\bar{\rho}_{1})$ and verify $\left\\{\begin{aligned} &\lambda^{(j)}(0,1)=0,&&\forall\,j\in\\{-1,0,1\\},\\\ &\frac{\partial\lambda^{(j)}}{\partial\rho}(0,1)=j\,i\,\sqrt{\bar{T}_{1}+\frac{2\bar{T}_{1}^{2}}{d}},&&\forall\,j\in\\{-1,0,1\\},\\\ &\frac{\partial\lambda^{(0)}}{\partial\alpha}(0,1)=-\frac{3}{\bar{T}_{1}},\end{aligned}\right.$ where, $\lambda^{(0)}$ is the so-called _energy eigenvalue_ and $\bar{T}_{1}$ is given by (11). Finally, we also have the symmetry properties (64) $\lambda^{(j)}(-\rho,\alpha)=\overline{\lambda^{(j)}}(\rho,\alpha)=\lambda^{(-j)}(\rho,\alpha).$ ###### Proof. We shall use the ideas introduced in the proof of Proposition 3.1: instead of solving directly the equation (57), we want to solve an equivalent one depending on $z=\lambda/\rho$ and we set to simplify $s=1-\alpha$. We then introduce a function $G=G(z,\rho,s)$ by setting $G(z,\rho,s):=\frac{1}{\rho^{3}}D(\rho z,\rho,1-s).$ According to the simplified expression (62) of $D$, all the components of the matrix found in $G(z,\rho,s)$ can be written for $h_{1},\ h_{2}\in N_{1}$ $\frac{1}{\rho}\left\langle\nu_{1-s}\,h_{1},\Upsilon_{(\rho z,\,\rho\bm{e},\,1-s)}\,h_{2}\right\rangle_{F_{1}}=\int_{\mathbb{R}^{d}}\left(\Psi_{(\rho z,\,\rho\bm{e},\,1-s)}^{}\right)^{-1}(\nu_{1-s}h_{1})(v)\\\ \left(\nu_{1-s}(v)+\rho z+i\,\rho\,v_{1}-s\Delta_{v}\right)^{-1}\left(z+iv_{1}-\frac{s}{\rho}\Delta_{v}\right)(h_{2})(v)F_{1}(v)\,dv.$ By doing the same computations than in the proof of Proposition 3.1, this quantity becomes for $\rho=s=0$ $\int_{\mathbb{R}^{d}}h_{1}(v)({z+iv_{1}})h_{2}(v)\,F_{1}(v)\,dv.$ Moreover, according to the definition of the Maxwellian distribution $F_{1}$, we have $\int_{\mathbb{R}^{d}}\begin{pmatrix}1\\\ \|v\|^{2}\\\ v_{1}^{2}\,\|v\|^{2}\end{pmatrix}F_{1}(v)\,dv=\begin{pmatrix}1\\\ \bar{T}_{1}\\\ \left(d+2\right)\bar{T}_{1}^{2}\end{pmatrix}.$ Thus, we can write $D$ as $\displaystyle G(z,0,1)$ $\displaystyle=\begin{vmatrix}\langle 1,z+iv_{1}\rangle_{F_{1}}&\langle 1,(z+iv_{1})v_{1}\rangle_{F_{1}}&\langle 1,(z+iv_{1})g\rangle_{F_{1}}\\\ \langle v_{1},z+iv_{1}\rangle_{F_{1}}&\langle v_{1},(z+iv_{1})v_{1}\rangle_{F_{1}}&\langle v_{1},(z+iv_{1})g\rangle_{F_{1}}\\\ \langle g,z+iv_{1}\rangle_{F_{1}}&\langle g,(z+iv_{1})v_{1}\rangle_{F_{1}}&\langle g,(z+iv_{1})g\rangle_{F_{1}}\\\ \end{vmatrix}$ $\displaystyle=\begin{vmatrix}z&i\,\bar{T}_{1}&z\left(d\,\bar{T}_{1}-c_{\nu}\right)\\\ i\,\bar{T}_{1}&z\,\bar{T}_{1}&i\,\bar{T}_{1}\left(\left(d+2\right)\bar{T}_{1}-c_{\nu}\right)\\\ z\left(d\,\bar{T}_{1}-c_{\nu}\right)&i\,\left(\left(d+2\right)\bar{T}_{1}-c_{\nu}\right)&z\left(\left(d\,\bar{T}_{1}-c_{\nu}\right)^{2}+2d\,\bar{T}_{1}^{2}\right)\end{vmatrix}$ $\displaystyle=2\,\bar{T}_{1}^{2}\,z\left(dz^{2}+d\,\bar{T}_{1}+2\,\bar{T}_{1}^{2}\right)$ $\displaystyle=2d\,\bar{T}_{1}^{2}\,(z-z_{-1})(z-z_{0})\,(z-z_{+1}),$ where we have set for any $j\in\\{-1,0,+1\\}$ $z_{j}:=j\,i\,\sqrt{\bar{T}_{1}+\frac{2\bar{T}_{1}^{2}}{d}}.$ Hence, provided that $G(z,0,1)$ has no multiple root, we have shown that $\left\\{\begin{aligned} &G(z_{j},0,0)=0,\\\ &\frac{\partial G}{\partial z}(z_{j},0,0)\neq 0,\end{aligned}\right.$ and we can apply again the implicit function theorem to show that in a neighborhood $\mathcal{B}\times(-\bar{\rho}_{1},\bar{\rho}_{1})\times[0,1-\alpha_{5})$ of $(z_{j},0)$, there exists an unique function $\widetilde{z_{j}}\in\mathcal{C}^{\infty}\left((-\bar{\rho}_{1},\bar{\rho}_{1})\times[0,1-\alpha_{5})\right)$ such that if $\|\rho\|\leq\bar{\rho}_{1}$ and $s\in[0,1-\alpha_{5})$, then $G(\widetilde{z_{j}}(\rho,s),\rho,s)=0$ (and of course $\widetilde{z_{j}}(0,0)=z_{j}$). We finally set $\lambda^{(j)}(\rho,s):=\rho\widetilde{z_{j}}(\rho,s)$, which give a solution to (57) (with $\alpha=1-s$) verifying $\left\\{\begin{aligned} &\lambda^{(j)}(0,0)=0,\\\ &\frac{\partial\lambda^{(j)}}{\partial\rho}(0,0)=z_{j}.\\\ \end{aligned}\right.$ We now have to prove that these three branches are the only solutions to $D(\lambda,\rho,\alpha)=0$ for small $\rho$ and $1-\alpha$. For this, following once more [8], we shall use tools from complex analysis. Let us fix $\|\rho\|\leq\bar{\rho}_{1}$ and $\alpha\in(\alpha_{5},1]$; according to the definition of $D$, the map $\lambda\mapsto D(\lambda,\rho,\alpha)$ is holomorphic on the set $U_{2}$ (defined Lemma 3.3). Moreover, following the previous computations, we also have $D(\lambda,0,1)=\lambda^{3}H(\lambda)$ for a function $H$ holomorphic on $U_{2}$ such that $H(0)=1$. Hence, if $\lambda$ is defined along a circle $\mathcal{C}$ around $0$, then $D(\lambda,0,1)$ will encircle the origin exactly three times. Using the strong convergence of the multiplication operator $\Phi_{(\lambda,\,\rho\bm{e},\,\alpha)}$ towards $\operatorname{Id}$ when $(\rho,\,\alpha)\to(0,1)$, we can write $\lim_{(\rho,\,\alpha)\to(0,1)}\sup_{\lambda\in U_{2}}\|D(\lambda,\rho,\alpha)-D(\lambda,0,1)\|=0.$ Hence, for small $(\rho,1-\alpha)$, the function $\lambda\mapsto D(\lambda,\rho,\alpha)$ encircles the origin also only three times when $\lambda$ traverses $\mathcal{C}$. This function then only has three roots for fixed $\rho$ and $\alpha$. Next, we compute the partial derivative with respect to $\alpha$ of the energy eigenvalue. This eigenvalue is given by the solution of the dispersion relation that depends on $\rho$ only at second order, namely $\lambda^{(0)}$. Let $h^{(0)}_{(\rho\,\omega,\,\alpha)}$ be the associated eigenvector. We then have for all $\omega\in\mathbb{S}^{d-1}$ and $\rho\geq 0$ (65) $\mathcal{L}_{(\alpha,\gamma)}\,h^{(0)}_{(\rho\,\omega,\,\alpha)}(v)\,=\,\lambda^{(0)}(\rho,\alpha)\,h^{(0)}_{(\rho\,\omega,\,\alpha)}(v),\quad\forall\,v\in\mathbb{R}^{d}.$ In particular, the “elastic, space homogeneous” energy eigenvector $h^{(0)}_{(0,1)}$ is defined thanks to the Maxwellian profile $F_{1}$ (given in (11)) as (66) $h^{(0)}_{(0,1)}=c_{0}\left(\|v\|^{2}-d\,\bar{T}_{1}\right)F_{1},$ where $c_{0}$ is a normalizing constant. We have by construction, using some elementary properties of Gaussian functions $\left\\|h^{(0)}_{(0,1)}\right\\|_{L^{1}(m^{-1})}=1,\quad N\left(h^{(0)}_{(0,1)}\right)=0,\quad\mathcal{E}\left(h^{(0)}_{(0,1)}\right)=2\,c_{0}\,d\,\bar{T}_{1}^{2},$ where we have defined the _mass_ $N(f)$ and the _kinetic energy_ $\mathcal{E}(f)$ of a given distribution $f$ as $N(f):=\int_{\mathbb{R}^{d}}f(v)\,dv,\quad\mathcal{E}(f):=\int_{\mathbb{R}^{d}}f(v)\,\|v\|^{2}\,dv.$ By integrating the eigenvalue equation (65) against $\|v\|^{2}$ we obtain according to the expression of the energy dissipation functional (7) $\lambda^{(0)}(\rho,\alpha)\mathcal{E}\left(h^{(0)}_{(\rho\,\omega,\,\alpha)}\right)=\\\ -2(1-\alpha^{2})\,D\left(F_{\alpha},h^{(0)}_{(\rho\,\omega,\alpha)}\right)+2dN(1-\alpha)\left(h^{(0)}_{(\rho\,\omega,\,\alpha)}\right)+i{\rho}\omega\cdot\int_{\mathbb{R}^{d}}h^{(0)}_{(\rho\,\omega,\,\alpha)}(v)\,v\,\|v\|^{2}\,dv.$ As $\rho$ tends to $0$, dividing by $1-\alpha$ yields $\frac{\lambda^{(0)}(0,\alpha)}{1-\alpha}\mathcal{E}\left(h^{(0)}_{(0,\alpha)}\right)=-2(1+\alpha)\,D\left(F_{\alpha},h^{(0)}_{(0,\alpha)}\right)+2dN\left(h^{(0)}_{(0,\alpha)}\right).$ Now, we use the rate of convergence of the inelastic profile $F_{\alpha}$ towards the elastic one $F_{1}$ recalled in Proposition A.4 and the smoothness of $h^{(0)}_{(0,\alpha)}$ with respect to $\alpha$ obtained thanks to the use of the implicit functions theorem. We then obtain thanks to the nullity of the mass of $h^{(0)}_{(0,1)}$ (67) $\frac{\lambda^{(0)}(0,\alpha)}{1-\alpha}\mathcal{E}\left(h^{(0)}_{(0,1)}\right)=-2(1+\alpha)\,D\left(F_{1},h^{(0)}_{(0,1)}\right)+\mathcal{O}(1-\alpha).$ Finally, we compute thanks to the expression of the elastic energy eigenvector (66) and to the definition (11) of the equilibrium temperature the quantities $\mathcal{E}\left(h^{(0)}_{(0,1)}\right)=2\,d\,c_{0}\,\bar{T}_{1}^{2},\quad D\left(F_{1},h^{(0)}_{(0,1)}\right)=\frac{3}{2}d\,c_{0}\,\bar{T}_{1}.$ Gathering these relations and passing to the limit $\alpha\to 1$ in (67) gives the result. Concerning the last assertion of the proposition, we notice thanks to the invariance of the eigenvalue problem (33) under the composition of the convex conjugation and the reflection $\gamma\to-\gamma$ that $\overline{D}(\lambda,\rho,\alpha)=D(\overline{\lambda},-\rho,\alpha)=D(\overline{\lambda},\rho,\alpha)$. ∎ ###### Remark 4. As a consequence of the symmetry relation (64) $\lambda^{(j)}(-\rho,\alpha)=\overline{\lambda^{(j)}}(\rho,\alpha)=\lambda^{(-j)}(\rho,\alpha),$ we have $\lambda^{(0)}(\rho,\alpha)\in\mathbb{R}$. Thanks to this proposition, we can construct the $d+2$ normalized hydrodynamic eigenvectors $\left(h^{(j)}_{(\rho\,\omega,\,\alpha)}\right)_{j\in\\{-1,\ldots,d\\}}$ of the inelastic linearized collision operator, for small $\rho$, $\alpha$ close to $1$ and a given $\omega\in\mathbb{S}^{d-1}$. Indeed, on the one hand, for $j\in\\{2,\ldots,d\\}$, we take $\lambda=\lambda_{\omega}(\rho,\alpha)$ for $\|\rho\|\leq\bar{\rho}_{0}$ and $\alpha\in(\alpha_{4},1]$ given by Proposition 3.1 and choose in (44) $x_{0}=x_{d+1}=0$ and any vector $x\in\omega^{\perp}$. The relation (42) then allows us to construct the eigenvectors $h^{(j)}_{(\rho,\,\omega,\,\alpha)}$ associated to the conservation of momentum. On the other hand, for $j\in\\{-1,0,1\\}$, we pick a solution $\lambda=\lambda^{(j)}(\rho,\,\alpha)$ for $\|\rho\|\leq\bar{\rho}_{1}$ and $\alpha\in(\alpha_{5},1]$ to the dispersion relation $D(\lambda,\rho,\alpha)=0$ given by Proposition 3.2 and choose the vector $\left(x_{0},x\cdot\omega,x_{d+1}\right)$ to be a solution to the system (53)–(56)–(55) corresponding to this eigenvalue. We then set $x=(x\cdot\omega)\,\omega$ and recover through (44) $\mathcal{P}h^{(j)}_{(\rho\,\omega,\,\alpha)}(v)=x_{0}(\rho,\,\alpha)+(x\cdot v)(\rho,\omega\cdot v,\,\alpha)+x_{d+1}(\rho,\,\alpha)\left(\|v\|^{2}-c_{\nu}\right).$ Inserting this expression in (42) finally gives us the eigenvalue, depending on $\rho$, $\alpha$ (as a $\mathcal{C}^{\infty}$ function), $\|v\|$ and $v\cdot\omega$. With this procedure, we have constructed three independent solutions (corresponding to the acoustic waves and the kinetic energy) $h^{(j)}=h^{(j)}_{(\rho\,\omega,\,\alpha)}\in L^{1}(m^{-1})$ to the eigenvalue problem $\left(-i\rho(\omega\cdot v)+\mathcal{L}_{\alpha}\right)h^{(j)}=\lambda^{(j)}\,h^{(j)},\quad\forall\,j\in\\{-1,0,1\\}.$ ### 3.4. Higher Order Expansion We are interested in this section to give an expression for the expansion of the eigenvalues with respect to the spatial coordinate $\gamma=\rho\,\omega$. We have seen in Remark 3 that for a fixed $\omega\in\mathbb{S}^{d-1}$, the eigenvalues $\lambda^{(j)}(\rho,\,\alpha)$ and eigenvectors $h^{(j)}_{(\rho\,\omega,\,\alpha)}$ are analytic functions of the radial coordinate $\rho$ and the inelasticity $1-\alpha$. Hence, we have for any $v\in\mathbb{R}^{d}$ (68) $\displaystyle\lambda^{(j)}(\rho,\,\alpha)=\sum_{n\geq 0}\lambda^{(j)}_{n}\rho^{n}+(1-\alpha)e^{(j)}_{1}+\mathcal{O}\left((1-\alpha)^{2}+(1-\alpha)\rho\right),$ (69) $\displaystyle h^{(j)}_{(\rho\,\omega,\,\alpha)}(v)=\sum_{n\geq 0}h^{(j)}_{n}(\omega)(v)\rho^{n}+(1-\alpha)f^{(j)}_{1}(v)+\mathcal{O}\left((1-\alpha)^{2}+(1-\alpha)\rho\right).$ According to the computations of the previous subsection, the first order components of this expansion are given by $\left\\{\begin{aligned} &\lambda^{(j)}_{0}=0,&&\forall\,j\in\\{-1,\ldots,d\\},\\\ &\lambda^{(j)}_{1}=j\,i\,\sqrt{\bar{T}_{1}+\frac{2\bar{T}_{1}^{2}}{d}},&&\forall\,j\in\\{-1,0,1\\},&&\lambda^{(j)}_{1}=0,&&\forall\,j\in\\{2,\ldots,d\\},\\\ &e_{1}^{(0)}=-\frac{3}{\bar{T}_{1}},&&&&e_{1}^{(j)}=0,&&\forall\,j\in\\{-1,1,\ldots,d\\}.\end{aligned}\right.$ We also have for any $v\in\mathbb{R}^{d}$ $h^{(0)}_{0}(v)=c_{0}\left(\|v\|^{2}-d\,\bar{T}_{1}\right)F_{1},$ for a nonnegative normalizing constant $c_{0}$. Since the triple $\left(\rho\,\omega,\lambda^{(j)}(\rho,\,\alpha),h^{(j)}_{(\rho\,\omega,\,\alpha)}\right)$ is solution to the eigenvalue problem (20), we can equate the power of $\rho$ and $1-\alpha$ in (68)–(69) to obtain (70) $\left\\{\begin{aligned} &\mathcal{L}_{0}\,h^{(j)}_{0}(\omega)=0,&&\forall\,j\in\\{-1,\ldots,d\\},\\\ &\mathcal{L}_{0}\,h^{(j)}_{1}(\omega)=\left(\lambda^{(j)}_{1}+i(\omega\cdot v)\right)h^{(j)}_{0}(\omega),&&\forall\,j\in\\{-1,\ldots,d\\},\\\ &\mathcal{L}_{0}\,h^{(j)}_{n}(\omega)=\left(\lambda^{(j)}_{1}+i(\omega\cdot v)\right)h^{(j)}_{n-1}(\omega)+\sum_{k=2}^{n}\lambda^{(j)}_{k}h^{(j)}_{n-k}(\omega),&&\forall\,j\in\\{-1,\ldots,d\\},\ n\geq 2,\end{aligned}\right.$ where we also used the smoothness of $\mathcal{L}_{\alpha}$ with respect to $1-\alpha$ (Proposition A.2). Hence, the coefficients of the expansion can be computed by induction. For example, to compute $\lambda^{(j)}_{2}$, we can integrate the eigenvalue problem (20) with respect to $\|v\|^{2}$ and use the equations (70) with $n=2$ to obtain (71) $\lambda^{(j)}_{2}=-\frac{i}{2\,d\,c_{0}\,\bar{T}_{1}^{2}}\,\omega\cdot{q\left(h^{(j)}_{1}(\omega)\right)},$ where we have set $q(h):=\int_{\mathbb{R}^{d}}h(v)\,v\,\|v\|^{2}\,dv.$ Now, using again (70) for $n=1$, we know that $\mathcal{L}_{0}\,h^{(j)}_{1}(\omega)=\left(\lambda^{(j)}_{1}+i(\omega\cdot v)\right)h^{(j)}_{0}(\omega).$ Since $\lambda^{(j)}_{1}$ is an imaginary number and $h^{(j)}_{0}$ a real number (it is the elastic, space homogeneous eigenvector), we have that $h^{(j)}_{1}(\omega)(v)$ is also imaginary for all $v\in\mathbb{R}^{d}$. Gathering this information with the explicit representation (71), we obtain that for any $j\in\\{-1,\ldots,d\\}$, the second order expansion in $\rho$ of $\lambda^{(j)}$, denoted by $\lambda^{(j)}_{2}$ is nonpositive777Some explicit computations are given in the $L^{2}$ case in [8, Section 4].. The higher order expansions can be computed by the same induction process. This concludes the proof of Theorem 1.2. ## Acknowledgment The research of the author was granted by the ERC Starting Grant 2009 #239983 (NuSiKiMo), NSF Grants #1008397 and #1107444 (KI-Net) and ONR grant #000141210318. The author would like to thanks F. Filbet and C. Mouhot for their careful reading and fruitful comments on the manuscript. ## Appendix A Functional Toolbox on the Collision Operator Let us present some important properties concerning the granular gases operator we heavily used on this paper. To be consistent with [15], we shall define for $\delta>0$ the regularized operator $\mathcal{L}_{1,\delta}\,=\,\mathcal{L}_{1,\delta}^{+}-\mathcal{L}^{}-\mathcal{L}^{\nu},$ where $\mathcal{L}_{1,\delta}^{+}$ is the regularization of the truncated gain term introduced in [17]. One of the key properties of the regularized operator is that it converges towards $\mathcal{L}_{1}$ when $\delta\to 0$ in the norm of graph of $L^{1}(m^{-1})$ (and also in the weighted Sobolev spaces $W^{k,1}_{q}(m^{-1})$) but with a loss of integration weights: ###### Proposition A.1 (Proposition 5.5 of [15]). For any $k,q\in\mathbb{N}$, we have $\left\\|\left(\mathcal{L}_{1,\delta}-\mathcal{L}_{1}\right)g\right\\|_{W^{k,1}_{q}(m^{-1})}\leq\varepsilon(\delta)\,\\|g\\|_{W^{k,1}_{q+1}(m^{-1})},$ where $\varepsilon(\delta)$ is an explicit constant, going to $0$ as $\delta\to 0$. We then state a result about the Hölder continuity (in the norm of the graph) of the gain term of the granular gases operator with respect to the restitution coefficient $\alpha$. ###### Proposition A.2 (Proposition 3.2 of [15]). For any $\alpha,\alpha^{\prime}\in(0,1]$, and any $g\in L^{1}_{1}(m^{-1})$, $f\in W^{1,1}_{1}(m^{-1})$, there holds $\left\\{\begin{aligned} &\left\\|\mathcal{Q}_{\alpha}^{+}(g,f)-\mathcal{Q}_{\alpha^{\prime}}^{+}(g,f)\right\\|_{L^{1}(m^{-1})}\leq\varepsilon\left(\alpha-\alpha^{\prime}\right)\\|f\\|_{W^{1,1}_{1}(m^{-1})}\\|g\\|_{L^{1}_{1}(m^{-1})},\\\ &\left\\|\mathcal{Q}_{\alpha}^{+}(f,g)-\mathcal{Q}_{\alpha^{\prime}}^{+}(f,g)\right\\|_{L^{1}(m^{-1})}\leq\varepsilon\left(\alpha-\alpha^{\prime}\right)\\|f\\|_{W^{1,1}_{1}(m^{-1})}\\|g\\|_{L^{1}_{1}(m^{-1})},\end{aligned}\right.$ where we have set $\varepsilon(r)=C\,r^{\frac{1}{3+4s}}$ for a constant $s$ given by the weight function $m(v)=\exp\left(-a\,\|v\|^{s}\right)$. We also need to estimate the smoothness, the tail behavior and the pointwise lower bound (uniformly with respect to the restitution coefficient $\alpha$) of the equilibrium profiles $F_{\alpha}$ solutions to (9). We have the following result. ###### Proposition A.3 (Propositions 2.1 and 2.3 of [16]). Let us fix $\alpha_{0}\in(0,1)$. There exist some positive constants $a_{1},a_{2},a_{3},a_{4}$ (independent of $\alpha$) and, for any $k\in\mathbb{N}$ a positive constant $C_{k}$ such that for all $\alpha\in[\alpha_{0},1)$ $\displaystyle\\|F_{\alpha}\\|_{L^{1}\left(e^{a_{1}}\|v\|\right)}\leq a_{2},\quad\\|F_{\alpha}\\|_{H^{k}(\mathbb{R}^{d})}\leq C_{k},$ $\displaystyle F_{\alpha}(v)\geq a_{3}\,e^{-a_{4}\|v\|^{8}},\quad\forall\,v\in\mathbb{R}^{d}.$ Moreover, these profiles converge in $L^{1}_{2}$ towards the elastic Maxwellian $F_{1}$, with an explicit rate: ###### Proposition A.4 (Proposition 3.1 of [16]). For any $\varepsilon>0$, there exists $C_{\varepsilon}$ such that $\\|F_{\alpha}-F_{1}\\|_{L^{1}_{2}}\leq C_{\varepsilon}(1-\alpha)^{\frac{1}{2+\varepsilon}}.$ We now define for $\zeta\in\mathbb{C}$ and $\delta>0$ the operators (72) $A_{\delta}:=\mathcal{L}_{1,\delta}^{+}-\mathcal{L}^{}\quad\text{ and }\quad B_{\alpha,\,\delta}(\zeta):=\mathcal{L}^{\nu_{1}}+\mathcal{I}_{\alpha}+\zeta-\left(\mathcal{L}_{1}^{+}-\mathcal{L}_{1,\delta}^{+}\right),$ where $\mathcal{I}_{\alpha}:=\mathcal{L}_{1}-\mathcal{L}_{\alpha}$ is the difference between the elastic and inelastic linearized operators. We can then write the problem of computing the inverse of resolvent operator of $\mathcal{L}_{\alpha}$ as the perturbation equation $\mathcal{L}_{\alpha}-\zeta=A_{\delta}-B_{\alpha,\,\delta}\left(\zeta\right).$ We state a result of convergence of the linearized granular gases operator towards the linearized elastic operator (which is a consequence of Proposition A.2), as well as estimates on the operator $B_{\alpha,\,\delta}$. ###### Lemma A.1 (Lemmas 5.9 of [15] and 5.2 of [16]). For any $k,q\in\mathbb{N}$ and any exponential weight function $m$, the following properties hold: 1. (1) There exist a constructive $\alpha_{0}\in(0,1]$ and some nonnegative constant $C=C(k,q,m)$ such that for any $\alpha\in(\alpha_{0},1]$, $\displaystyle\\|\mathcal{L}_{\alpha}\\|_{W^{k+2,1}_{q+1}(m^{-1})\to W^{k,1}_{q}(m^{-1})}\leq C,$ $\displaystyle\left\\|\mathcal{L}_{\alpha}-\mathcal{L}_{1}\right\\|_{W^{3,1}_{3}(m^{-1})\to L^{1}(m^{-1})}\leq C\,(1-\alpha).$ 2. (2) For any $\delta>0$, the operator $A_{\delta}:L^{1}\to W^{\infty,1}_{\infty}\left(m^{-1}\right)$ is a bounded linear operator (more precisely, it maps function $L^{1}$ into $\mathcal{C}^{\infty}$ functions with compact support). 3. (3) There exists some constants $\delta^{}>0$ and $\alpha_{1}\in(\alpha_{0},1)$ such that for any $\zeta\in\Delta_{-\nu_{0}}$, $\delta<\delta_{}$ and $\alpha\in[\alpha_{1},1]$ the operator $B_{\alpha,\,\delta}(\zeta):W^{k+2,1}_{q+1}(m^{-1})\to W^{k,1}_{q}(m^{-1})$ is invertible. Moreover, its inverse operator satisfies $\displaystyle\left\\|B_{\alpha,\,\delta}(\zeta)^{-1}\right\\|_{W^{k,1}_{q}(m^{-1})\to W^{k,1}_{q}(m^{-1})}\leq\frac{C_{1}}{\|\nu_{0}-\Re e\,\zeta\|},$ $\displaystyle\left\\|B_{\alpha,\,\delta}(\zeta)^{-1}\right\\|_{W^{k,1}_{q}(m^{-1})\to W^{k+2,1}_{q+1}(m^{-1})}\leq\frac{C_{2}}{\|\nu_{0}-\zeta\|}$ for some explicit constants $C_{1},C_{2}$ depending on $k,q,\delta^{},\alpha_{1}$. As a consequence of these results, we also have the following proposition. ###### Proposition A.5 (Proposition 3.8 of [15]). For any $k,q\in\mathbb{N}$, any exponential weight function $m$, and any $\alpha\in(\alpha_{0},1]$, $\left\\|\mathcal{L}_{\alpha}^{+}-\mathcal{L}_{1}^{+}\right\\|_{W^{k,1}_{q}(m^{-1})\to W^{k,1}_{q+1}(m^{-1})}\leq\varepsilon\,(1-\alpha)$ where $\varepsilon$ has been defined in Proposition A.2. ## References * [1] Bobylev, A. V., Gamba, I., and Panferov, V. Moment inequalities and high-energy tails for Boltzmann equations with inelastic interactions. J. Statist. Phys. 116, 5 (2004), 1651–1682. * [2] Brilliantov, N., and Pöschel, T. Kinetic Theory of Granular Gases. Oxford University Press, USA, 2004. * [3] Caflisch, R. E. The fluid dynamic limit of the nonlinear Boltzmann equation. Comm. Pure Appl. Math. 33, 5 (1980), 651–666. * [4] Carlen, E., Chow, S.-N., and Grigo, A. Dynamics and hydrodynamic limits of the inelastic Boltzmann equation. 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Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media. Discrete Contin. Dyn. Syst. 24, 1 (2009), 159–185. * [17] Mouhot, C. Rate of Convergence to Equilibrium for the Spatially Homogeneous Boltzmann Equation with Hard Potentials. Commun. Math. Phys. 261, 3 (Nov. 2006), 629–672. * [18] Nicolaenko, B. Dispersion Laws for Plane Wave Propagation. In The Boltzmann Equation Seminar - 1970 to 1971 (1971), F. Grunbaum, Ed., Courant Institute of Mathematical Sciences, pp. 125–172. * [19] Nishida, T. A note on a theorem of Nirenberg. J. Differential Geom. 12, 4 (1977), 629–633 (1978). * [20] Nishida, T. Fluid dynamical limit of the nonlinear Boltzmann equation to the level of the compressible Euler equation. Comm. Math. Phys. 61, 2 (1978), 119–148. * [21] Villani, C. Mathematics of Granular Materials. J. Statist. Phys. 124, 2 (2006), 781–822.	arxiv-papers	2013-10-27T19:14:37	2024-09-04T02:49:52.955721	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Thomas Rey", "submitter": "Rey Thomas", "url": "https://arxiv.org/abs/1310.7234" }
1310.7240	Alternative CD formula for MOPRL of Mixed Type] Another Christoffel–Darboux Formula for Multiple Orthogonal Polynomials of Mixed Type Gerardo Araznibarreta Departamento de Física Teórica II (Métodos Matemáticos de la Física), Universidad Complutense de Madrid, 28040-Madrid, Spain Manuel Mañas An alternative expression for the Christoffel–Darboux formula for multiple orthogonal polynomials of mixed type is derived from the $LU$ factorization of the moment matrix of a given measure and two sets of weights. We use the action of the generalized Jacobi matrix $J$, also responsible for the recurrence relations, on the linear forms and their duals to obtain the result. § INTRODUCTION In this paper we address a natural question that arises from the $LU$ factorization approach to multiple orthogonality [8]. The Gauss–Borel factorization of a Hankel matrix, which plays the role of a moment matrix, leads in the classical case to a natural description of algebraic facts regarding orthogonal polynomials on the real line (OPRL) such as recursion relations and Christoffel–Darboux formula. In that case we have a chain of orthogonal polynomials $\{P_l(x)\}_{l=0}^\infty$ of increasing degree $l$. In [8] we extended that approach to the multiple orthogonality scenario, and the Gauss–Borel factorization of an appropriate moment matrix leaded to sequences of families of multiple orthogonal polynomials in the real line (MOPRL), $\big\{Q_{[\vec\nu_{1}(l);\vec\nu_{2}(l-1)]}^{(\rII,a_1(l))}\big\}_{l=0}^\infty$ and $\big\{\bar Q^{(\rI,a_1(l))}_{[\vec\nu_{2}(l);\vec\nu_{1}(l-1)]}\big\}_{l=0}^\infty$. The recursion relations are relations constructed in terms of the linear forms in these sequences. However, the Christoffel–Darboux formula given in Proposition <ref>, that was re-deduced by linear algebraic means (Gauss–Borel factorization), was not expressed in terms of linear forms belonging to the mentioned sequences. This situation is rather different to the OPRL case, in that classical case the Christoffel-Darboux formula is expressed in terms of orthogonal polynomials in the sequence. The aim of this paper is to show that, within that scheme, we can deduce an alternative but equivalent MOPRL Christoffel–Darboux formula constructed in terms of linear forms in the sequences $\big\{Q_{[\vec\nu_{1}(l);\vec\nu_{2}(l-1)]}^{(\rII,a_1(l))}\big\}_{l=0}^\infty$ and $\big\{\bar Q^{(\rI,a_1(l))}_{[\vec\nu_{2}(l);\vec\nu_{1}(l-1)]}\big\}_{l=0}^\infty$ as in OPRL situation. §.§ Historical background Simultaneous rational approximation starts back in 1873 when Hermite proved the transcendence of the Euler number e [24]. Later, K. Mahler delivered at the University of Groningen several lectures [29] where he settled down the foundations of this theory, see also [15] and [25]. Simultaneous rational approximation when expressed in terms of Cauchy transforms leads to multiple orthogonality of polynomials. Given an interval $\Delta \subset \R$ of the real line, let ${\mathcal{M}}(\Delta)$ denote all the finite positive Borel measures with support containing infinitely many points in $\Delta$. Fix $\mu \in {\mathcal{M}}(\Delta)$, and let us consider a system of weights $\vec w=(w_1,\ldots,w_p)$ on $\Delta$, with $p \in {\mathbb{N}}$; i.e. $w_1,\dots,w_p$ being real integrable functions on $\Delta$ which does not change sign on $\Delta$. Fix a multi-index $\vec \nu=(\nu_1,\ldots, \nu_p) \in {\mathbb{Z}}_+^p,$ ${\mathbb{Z}}_+=\{0,1,2,\ldots\}$, and denote $\|\vec \nu\|=\nu_1+\cdots+\nu_p$. Then, there exist polynomials, $A_1,\ldots, A_p$ not all identically equal to zero which satisfy the following orthogonality relations \begin{align}\label{tipoI} \int_{\Delta} x^{j} \sum_{a=1}^{p} A_a(x)w_{a} (x)\d\mu (x)&=0, & \deg A_{a}&\leq\nu_{a}-1,& j&=0,\ldots, \|\vec \nu\|-2. \end{align} Analogously, there exists a polynomial $B$ not identically equal to zero, such that \begin{align}\label{tipoII} \int_{\Delta} x^{j} B (x) w_{b} (x) \d\mu (x)&=0, & \deg B &\leq\|\vec \nu\|,& j&=0,\ldots, \nu_b-1, \quad b=1,\ldots,p. \end{align} These are the so called multiple orthogonal polynomials of type I and type II, respectively, with respect to the combination $(\mu, \vec w, \vec \nu)$ of the measure $\mu$, the systems of weights $\vec w$ and the multi-index $\vec \nu.$ When $p=1$ both definitions coincide with standard orthogonal polynomials on the real line. Given a measure $\mu \in {\mathcal{M}}(\Delta)$ and a system of weights $\vec w$ on $\Delta$ a multi-index $\vec \nu$ is called type I or type II normal if $\deg A_a$ must equal to $\nu_a-1,$ $a=1,\ldots,p$, or $\deg B$ must equal to $\|\vec \nu\|-1$, respectively. When for a pair $(\mu, \vec w)$ all the multi-indices are type I or type II normal, then the pair is called type I perfect or type II perfect respectively. Multiple orthogonal of polynomials have been employed in several proofs of irrationality of numbers. For example, in [12] F. Beukers shows that Apery's proof [10] of the irrationality of $\zeta(3)$ can be placed in the context of a combination of type I and type II multiple orthogonality which is called mixed type multiple orthogonality of polynomials. More recently, mixed type approximation has appeared in random matrix and non-intersecting Brownian motion theories, [13], [16], [27]. Sorokin [38] studied a simultaneous rational approximation construction which is closely connected with multiple orthogonal polynomials of mixed type. In [41] a Riemann–Hilbert problem was found that characterizes multiple orthogonality of type I and II, extending in this way the result previously found in [23] for standard orthogonality. In [16] mixed type multiple orthogonality was analyzed from this perspective. §.§ Perfect systems and MOPRL of mixed type In order to introduce multiple orthogonal polynomials of mixed type we consider two systems of weights $\vec w_1=(w_{1,1},\dots,w_{1,p_1})$ and $\vec w_2=(w_{2,1},\dots,w_{2,p_2})$ where $p_1,p_2\in\N$, and two multi-indices $\vec \nu_1=(\nu_{1,1},\dots,\nu_{1,p_1})\in{\mathbb{Z}}_+^{p_1}$ and $\vec \nu_2=(\nu_{2,1},\dots,\nu_{2,p_2})\in {\mathbb{Z}}_+^{p_2}$ with $\|\vec \nu_1\|=\|\vec \nu_2\|+1$. There exist polynomials $A_1,\ldots, A_{p_1},$ not all identically zero, such that $\deg A_s < \nu_{1,s}$, which satisfy the following relations \begin{equation}\label{orth} \int_{\Delta} \sum_{a=1}^{p_1} A_a(x)w_{1,a} (x)w_{2,b}(x)x^{j} d\mu (x)=0, \quad j=0,\ldots, \nu_{2,b}-1,\quad b=1,\ldots,p_2. \end{equation} They are called mixed multiple-orthogonal polynomials with respect to the combination $(\mu,\vec w_1,\vec w_2,\vec \nu_1,\vec\nu_2)$ of the measure $\mu,$ the systems of weights $\vec w_{1}$ and $\vec w_2$ and the multi-indices $\vec \nu_1$ and $\vec \nu_2.$ It is easy to show that finding the polynomials $A_1,\ldots, A_{p_1}$ is equivalent to solving a system of $\|\vec \nu_2\|$ homogeneous linear equations for the $\|\vec \nu_1\|$ unknown coefficients of the polynomials. Since $\|\vec \nu_1\|=\|\vec \nu_2\|+1$ the system always has a nontrivial solution. The matrix of this system of equations is the so called moment matrix, and the study of its Gauss–Borel factorization will be the cornerstone of this paper. Observe that when $p_1=1$ we are in the type II case and if $p_2=1$ in type I case. Hence in general we can find a solution of (<ref>) where there is an $a \in \{1,\ldots,p_1\}$ such that $\deg A_a< \nu_{1,a}-1$. When given a combination $(\mu,\vec w_1,\vec w_2)$ of a measure $\mu \in {\mathcal{M} }(\Delta)$ and systems of weights $\vec w_1$ and $\vec w_2$ on $\Delta$ if for each pair of multi-indices $(\vec \nu_1,\vec \nu_2)$ the conditions (<ref>) determine that $\deg A_a=\nu_{1,a}-1,$ $a=1, \ldots, p_1$, then we say that the combination $(\mu,\vec w_1,\vec w_2)$ is perfect. In this case we can determine a unique system of mixed type orthogonal polynomials $\big(A_1, \ldots, A_{p_1}\big)$ satisfying (<ref>) requiring for $a_1 \in \{1, \ldots p_1\}$ that $A_{a_1}$ monic. Following [16] we say that we have a type II normalization and denote the corresponding system of polynomials by $A_a^{(\rII,a_1)},$ $j=1, \ldots, p_1$. Alternatively, we can proceed as follows, since the system of weights is perfect from (<ref>) we deduce that \begin{align} \int x^{\nu_{1,r_1}} \sum_{a=1}^{p_1} A_a(x)w_{1,a} (x)w_{2,b}(x) \d\mu (x)\neq 0. \end{align} Then, we can determine a unique system of mixed type of multi-orthogonal polynomials $(A_1^{(\rI,a_2)},\ldots,A_{p_2}^{(\rI,a_2)})$ imposing that \begin{align} \int x^{\nu_{1,a_2}} \sum_{a=1}^{p_1} A_a^{(\rI,a_2)}(x)w_{1,a} (x)w_{2,b}(x) \d\mu (x)=1, \end{align} which is a type I normalization. We will use the notation $A_{[\vec\nu_1;\vec\nu_2],a}^{(\rII,a_1)}$ and $A_{[\vec\nu_1;\vec\nu_2],a}^{(\rI,a_2)}$ to denote these multiple orthogonal polynomials with type II and I normalizations, respectively. A known illustration of perfect combinations $(\mu, \vec w_1, \vec w_2)$ can be constructed with an arbitrary positive finite Borel measure $\mu$ and systems of weights formed with exponentials: \begin{align}\label{exponentials} & (\Exp{\gamma_1x},\ldots,\Exp{\gamma_px}),& \gamma_i &\neq \gamma_j, &i &\neq j,& i,j& = 1,\ldots,p, \end{align} or by binomial functions \begin{align}\label{binomials} &((1-z)^{\alpha_1},\ldots,(1-z)^{\alpha_p}),& \alpha_i -\alpha_j &\not\in {\mathbb{Z}}, &i &\neq j,& i,j& = \end{align} or combining both classes, see [33]. Recently, in [22] the authors were able to prove perfectness for a wide class of systems of weights. These systems of functions, now called Nikishin systems, were introduced by E.M. Nikishin [33] and initially named MT-systems (after Markov and Tchebycheff). §.§ Borel–Gauss factorization and multiple orthogonality of mixed type. A remainder Orthogonal polynomials and the theory of integrable systems has been connected in several ways in the mathematical literature. We are particularly interested in the one based in the Gauss–Borel factorization that was developed in [1]-[5], and applied further in [7]-[9]. These papers set the basis for the method we use in this paper to get an alternative Christoffel–Darboux formula for MOPRL of mixed type. In the following we extract from [8] the necessary material for the construction of the mentioned alternative Christoffel–Darboux formula. We introduce the moment matrix and recall how the Borel–Gauss factorization leads to multiple orthogonality. Then, we outline how the recursion relations appears by introducing a Jacobi type semi-infinite matrix and recall the reader the Chistoffel–Darboux formula [17, 16]. §.§.§ The moment matrix Fixed a composition $\vec n_\alpha$, $\alpha=1,2$, any given $l\in\Z_+:=\{0,1,2,\dots\}$, see [40], determines uniquely the following non-negative integers $k_{\alpha}(l)\in\Z_+$, $a_{\alpha}(l)\in \{1,2,\dots,p_{\alpha}\}$ and $r_{\alpha}(l)$ such that $0\leq r_{\alpha}(l)<n_{\alpha,a_{\alpha}(l)}$ and \begin{align}\label{i} k_{\alpha}(l)\|\vec n_{\alpha}\|+n_{\alpha,1}+\dots+n_{\alpha,a_{\alpha}(l)-1}+r_{\alpha}(l), & a_{\alpha}(l)\neq1,\\ k_{\alpha}(l)\|\vec n_{\alpha}\|+r_{\alpha}(l), & a_{\alpha}(l)=1. \end{cases} \end{align} We define now the monomial strings as vectors that may be understood as sequences of monomials according to the composition $\vec n_{\alpha}$, $\alpha=1,2$, introduced previously. \begin{align} \chi_{\alpha}&:=\begin{pmatrix} \chi_{\alpha,[0]}\\ \chi_{\alpha,[1]}\\ \vdots\\ \chi_{\alpha,[k]}\\ \vdots \end{pmatrix} &\mbox{where}& & \chi_{\alpha,[k]}:=\begin{pmatrix} \chi_{\alpha,[k],1}\\ \chi_{\alpha,[k],2}\\ \vdots\\ \chi_{\alpha,[k],a_{\alpha}}\\ \vdots\\ \chi_{\alpha,[k],p_{\alpha}} \end{pmatrix} & &\mbox{and}& & \chi_{\alpha,[k],a_{\alpha}}:=\begin{pmatrix} \vdots\\ \end{pmatrix}. \end{align} In a similar manner for $\alpha=1,2$ we define the weighted monomial strings \begin{align} \xi_{\alpha}&:=\begin{pmatrix} \xi_{\alpha,[0]}\\ \xi_{\alpha,[1]}\\ \vdots\\ \xi_{\alpha,[k]}\\ \vdots \end{pmatrix} &\mbox{where}& & \xi_{\alpha,[k]}:=\begin{pmatrix} \vdots\\ \end{pmatrix}. \end{align} For any given $l\in\Z_+$ and $a_{\alpha}:={1,2,\dots,p_{\alpha}}$ we define \begin{align} \nu_{\alpha,a_{\alpha}}(l):=\begin{cases} k_{\alpha}(l)\|\vec n_{\alpha}\|+n_{\alpha,a_{\alpha}}-1, & a_{\alpha}<a_{\alpha}(l),\\ k_{\alpha}(l)\|\vec n_{\alpha}\|+r_{\alpha}(l), & a_{\alpha}=a_{\alpha}(l),\\ k_{\alpha}(l)\|\vec n_{\alpha}\|-1,& a_{\alpha}>a_{\alpha}(l). \end{cases} \end{align} Notice that $\nu_{\alpha,a_{\alpha}}(l)$ is the hightest degree of all the monomials of type $a_{\alpha}$ up to the component $\chi_{\alpha}^{(l)}$ included, of the monomial string. Actually \begin{align} \chi_{\alpha}^{(l)}=x^{\nu_{\alpha,a_{\alpha}(l)}}. \end{align} Given $l\geq 1$ and $a_{\alpha}=1,\cdots, p_{\alpha}$ the $+$ ($-$) associated integer is the smallest (largest) integer $ l_{\{+,a_{\alpha}\}}$ ($l_{\{-,a_{\alpha}\}}$) such that $l_{\{+,a_{\alpha}\}} \geq l$ ($l_{\{-,a_{\alpha}\}} \leq l$ ) and $a( l_{\{+,a_{\alpha}\}})=a_{\alpha}$ ($a( l_{\{-,a_{\alpha}}\})=a_{\alpha}$). It can be shown that \begin{align}\label{ai} \begin{aligned} l_{\{-,a_{\alpha}\}}&:=\begin{cases} k_{\alpha}(l)\|\vec n_{\alpha}\|+\sum_{i=1}^{a_{\alpha}}n_{\alpha,i}-1, & a_{\alpha}<a_{\alpha}(l), \\ l, &a_{\alpha}=a_{\alpha}(l),\\ k_{\alpha}(l)\|\vec n_{\alpha}\|-\sum_{i=a_{\alpha}+1}^{p_{\alpha}}n_{\alpha,i}-1, & a_{\alpha}>a_{\alpha}(l-1), \end{cases}\\ ( k_{\alpha}(l)+1)\|\vec n_{\alpha}\|+\sum_{i=1}^{a_{\alpha}-1}n_{\alpha,i},& a_{\alpha}<a_{\alpha}(l),\\ ( k_{\alpha}(l)+1)\|\vec n_{\alpha}\|-\sum_{i=a_{\alpha}}^{p_{\alpha}}n_{\alpha,i},& a_{\alpha}>a_{\alpha}(l). \end{cases} \end{aligned} \end{align} Finally given the weighted monomials $\xi_{\vec n_{\alpha}}$, associated to the compositions $\vec n_{\alpha}$, $\alpha=1,2$, we introduce the moment matrix in the following manner The moment matrix is given by \begin{align} \label{compact.g} g_{\vec n_1,\vec n_2}:=\int \xi_{\vec n_1}(x)\xi_{\vec n_2}(x)^\top\d \mu(x). \end{align} §.§.§ Multiple Orthogonality of mixed type For a given a perfect combination $(\mu, \vec w_1,\vec w_2)$ we define * The Gauss–Borel factorization (also known as $LU$ factorization) of a semi-infinite moment matrix $g$, determined by $(\mu, \vec w_1,\vec w_2)$, is the problem of finding the solution of \begin{align}\label{facto} g&=S^{-1}\bar S, & S&=\begin{pmatrix} 1&0&0&\cdots \\ \vdots&\vdots&\vdots&\ddots \end{pmatrix}\in G_{-}, & \bar S^{-1}&=\begin{pmatrix} \bar S_{0,0}'&\bar S_{0,1}'&\bar S_{0,2}'&\cdots\\ 0&\bar S_{1,1}'&\bar S_{1,2}'&\cdots\\ 0&0&\bar S_{2,2}'&\cdots&\\ \vdots&\vdots&\vdots&\ddots \end{pmatrix}\in G_{+}, \end{align} where $S_{i,j},\bar S'_{i,j}\in\R.$ * In terms of these matrices we construct the polynomials \begin{align} \label{defmops} \end{align} where the sum $\sum'$ is taken for a fixed $a=1,\dots,p_1$ over those $i$ such that $a=a_1(i)$ and $i\leq l$. We also construct the dual polynomials \begin{align} \label{defdualmops} \bar A^{(l)}_b:={\sum}'_jx^{k_2(j)}\bar S_{j,l}', \end{align} where the sum $\sum'$ is taken for a given $b$ over those $j$ such that $b=a_2(j)$ and $j\leq l$. * Strings of linear forms and dual linear forms associated with multiple ortogonal polynomials and their duals are defined by \begin{align}\label{linear forms S} Q:= \begin{pmatrix} \vdots \end{pmatrix}&=S\xi_{1},& \bar Q:=\begin{pmatrix} \bar Q^{(0)}\\ \bar Q^{(1)}\\ \vdots \end{pmatrix}&=(\bar S^{-1})^\top\xi_{2}, \end{align} * The linear forms and their duals, introduced in Definition <ref>, are given by \begin{align}\label{linear.forms} Q^{(l)}(x)&:= \sum_{a=1}^{p_1}A^{(l)}_{a}(x)w_{1,a}(x),& \bar Q^{(l)}(x)&:= \sum_{b=1}^{p_2}\bar A^{(l)}_{b} \end{align} * The orthogonality relations \begin{align}\label{linear.form.orthogonality} \begin{aligned} \int Q^{(l)}(x)w_{2,b}(x)x^k\d \mu(x)&=0,&0&\leq k\leq \nu_{2,b}(l-1)-1,&b&=1,\dots,p_2,\\ \int \bar Q^{(l)}(x)w_{1,a}(x)x^k\d \mu(x)&=0,&0&\leq k\leq \nu_{1,a}(l-1)-1,&a&=1,\dots,p_1, \end{aligned} \end{align} are fulfilled. * The following multiple bi-orthogonality relations among linear forms and their duals \begin{align}\label{biotrhoganility} \int Q^{(l)}(x)\bar Q^{(k)}(x)\d \mu(x)&=\delta_{l,k},& l,k\geq 0, \end{align} * We have the following identifications \begin{align} A^{(l)}_a&=A_{[\vec\nu_{1}(l);\vec\nu_{2}(l-1)],a}^{(\rII,a_1(l))}, & \bar A^{(l)}_b&= A^{(\rI,a_1(l))}_{[\vec\nu_{2}(l);\vec\nu_{1}(l-1)],b}, \end{align} in terms of multiple orthogonal polynomials of mixed type with two normalizations $\rI$ and $\rII$, respectively. §.§.§ Functions of the second kind The Cauchy transforms of the linear forms (<ref>) play a crucial role in the Riemann–Hilbert problem associated with the multiple orthogonal polynomials of mixed type [16]. Observe that the construction of multiple orthogonal polynomials performed so far is synthesized in the following strings of multiple orthogonal polynomials and their duals \begin{align}\label{mop-s} \begin{aligned} \A_{a}&:= \begin{pmatrix} \vdots \end{pmatrix}=S\chi_{1,a},& \bar\A_{b}&:= \begin{pmatrix} \bar A^{(0)}_{b}\\ \bar A^{(1)}_{b}\\ \vdots \end{pmatrix}=(\bar S^{-1})^\top\chi_{2,b},& \end{aligned} \end{align} for $a=1,\dots,p_1$ and $b=1,\dots,p_2$. In order to complete these formulae and in terms of $ \chi^_a:=z^{-1}\chi_a(z^{-1})$ let us introduce the following formal semi-infinite vectors \begin{align}\label{cauchy-S} \begin{aligned} \Cs_b&=\begin{pmatrix} \end{pmatrix}=\bar S\chi_{2,b}^(z),& \bar\Cs_a&=\begin{pmatrix} \bar C_a^{(0)}\\\bar C_a^{(1)}\\\vdots \end{pmatrix}=(S^{-1})^\top\chi_{1,a}^(z),& \end{aligned}\end{align} for $a=1,\dots,p_1$ and $b=1,\dots,p_2$, that we call strings of second kind functions. These objects are actually Cauchy transforms of the linear forms $Q^{(l)}$, $l \in {\mathbb{Z}}_+$, whenever the series converge and outside the support of the measures involved. For each $l\in {\mathbb{Z}}_+$ the second kind functions can be expressed as follows \begin{align} \begin{aligned} C_b^{(l)}(z)&=\int_\R\frac{Q^{(l)}(x)w_{2,b}(x)}{z-x}\d \mu(x),& z \in D_b^{(l)} \setminus \operatorname{supp} (w_{1,b}\d \mu(x)),\\ \bar C_a^{(l)}(z)&=\int_\R \frac{\bar Q^{(l)}(x)w_{1,a}(x)}{z-x}\d \mu(x),& z \in \bar D_a^{(l)} \setminus \operatorname{supp} (w_{2,a}\d \mu(x)). \end{aligned} \end{align} §.§.§ Recursion relations, a Jacobi type matrix The moment matrix has a Hankel type symmetry that implies the recursion relations and the Christoffel–Darboux formula. We consider the shift operators $\Upsilon_{\alpha}$ defined by \begin{align} \end{align} Wich satisfy the following relation \begin{align} \Upsilon_{\alpha}\chi_{\alpha}(x)&=x\chi_{\alpha}(x) \Longrightarrow \Upsilon_{\alpha}\xi_{\alpha}(x)=x\xi_{\alpha}(x) \end{align} In terms of these shift matrices we can describe the particular Hankel symmetries for the moment matrix The moment matrix $g$ satisfies the Hankel type symmetry \begin{align}\label{sym2} \Upsilon_1g = \end{align} From this symmetry we see that the following is consistent We define the matrices \begin{align} J&:=S\Upsilon_1 S^{-1}=\bar S \Upsilon_2^\top\bar S^{-1}=J_++J_-,& J_+&:=( S\Upsilon_1 S^{-1})_+, & J_-&:=(\bar S \Upsilon_2^\top\bar S^{-1})_-, \end{align} where the sub-indices + and $-$denote the upper triangular and strictly lower triangular projections. The recursion relations follow immediately from the eigenvalue property \begin{align}\label{rel} J Q(x)&=x Q(x) & \bar{Q}(x)^{\top} J&=x \bar{Q}(x)^{\top}. \end{align} §.§.§ Christoffel–Darboux formula The Christoffel–Darboux kernel is \begin{align} \label{def.CD} K^{[l]}(x,y)&:=\sum_{k=0}^{l-1}Q^{(k)}(y)\bar Q^{(k)}(x)%=[\bar{Q}(x)^{\top}]^{[l]} \cdot [Q(y)]^{[l]} \end{align} In [17, 16] it was shown using a Riemann–Hilbert problem approach that For $l\geq \max(\|\vec n_1\|,\|\vec n_2\|)$ the following Christoffel–Darboux formula \begin{align} \begin{aligned} \sum_{b =1}^{p_2}\bar Q^{(\rII,b)}_{[\vec\nu_2(l-1)+\vec e_{2,b};\vec \nu_1(l-1)]}(x) Q^{(\rI,b)}_{[\vec\nu_1(l-1);\vec \nu_2(l-1)-\vec e_{2,b}]}(y)\\ &-\sum_{a=1}^{p_1}\bar Q^{(\rI,a)}_{[\vec\nu_2(l-1);\vec \nu_1(l-1)-\vec e_{1,a}]}(x)Q^{(\rII,a)}_{[\vec\nu_1(l-1)+\vec e_{1,a};\vec \nu_2(l-1)]}(y). \end{aligned} \label{cd3} \end{align} Here $\{\vec e_{i,a}\}_{a=1}^{p_i}\subset \R^{p_i}$ stands for the vectors in the respective canonical basis, $i=1,2$. In [8] it was given an algebraic proof of this statement not relying on analytic conditions. We refer the interested reader to [37] for a complete survey of the subject. § ALTERNATIVE CHRISTOFFEL–DARBOUX FORMULA FOR MULTIPLE ORTHOGONAL POLYNOMIALS OF MIXED TYPE The result of this paper is the following For $l\geq \max\{\|\vec n_1\|,\|\vec n_2\|\}$ the following Christofel–Darboux formula holds \begin{align} (y-x)K^{[l]}(x,y)&=\smashoperator{\sum_{(i,j)\in \sigma_1[l]}}\bar{Q}(x)^{(j)} J_{j,i} Q(y)^{(i)}- \smashoperator{\sum_{(i,j)\in\sigma_2[l]}}\bar{Q}(x)^{(j)} J_{j,i} Q(y)^{(i)} \end{align} \begin{align} \sigma_1[l]&:=\big\{l,\dots, (l)_{\{+,r_1(a_1(l)-1)\}}\big\}\times \big\{(l-1)_{\{-,r_1(a_1(l-1)+1)\}},\dots,l-1\big\},\\ \sigma_2[l]&:=\big\{(l-1)_{\{-,r_2(a_2(l-1)+1)\}},\dots,l-1\big\}\times\big\{l,\dots,(l)_{\{+,r_2(a_2(l)-1)\}}\big\}. \end{align} Splitting the eigenvalue property (<ref>) into blocks we get \begin{align} J Q(y)&=y Q(y)\Longrightarrow J^{[l]} Q(y)^{[l]}+J^{[l,\geq l]} Q(y)^{[\geq l]}=yQ(y)^{[l]} \\ \bar{Q}(x)^{\top} J&=x \bar{Q}(x)^{\top}\Longrightarrow [\bar{Q}(x)^{\top}]^{[l]} J^{[l]}+[\bar{Q}(x)^{\top}]^{[\geq l]} J^{[\geq l,l]}=x[\bar{Q}(x)^{\top}]^{[l]} \end{align} Multiply the first equation from the left by $[\bar{Q}(x)^{\top}]^{[l]}$ and the second one from the right by $Q(y)^{[l]}$ substract both results to obtain \begin{align} [\bar{Q}(x)^{\top}]^{[l]}J^{[l,\geq l]}Q(y)^{[\geq l]}-[\bar{Q}(x)^{\top}]^{[\geq l]} J^{[\geq l,l]}Q(y)^{[l]}&= (y-x)[\bar{Q}(x)^{\top}]^{[l]}\cdot Q(y)^{[l]}\\&=(y-x)K^{[l]}(x,y) \end{align} After an brief study of the shape of $J$ we realize that even though $J^{[l,\geq l]}$ has semi-infinte length rows, most of its elements are 0. Actually it only contains a finite number of nonzero entries that concentrate in the lower left corner of itself. The same reasoning applies to $J^{[\geq l,l]}$. This matrix has semi infinite length columns but again it only contains a finite number of nonzero terms concentrated in the upper right corner of itself. Of course the number of terms involved in this expression will depend on the value of $[l]$. To be more precise we proceed as follows. From the Euclidean division we know that for any positive integer $l\in\Z_+$ there exists unique integers $q_i, r_i$, $i=1,2$, the quotient and remainder, such that \begin{align} l&=q_ip_i+r_i, & 0&\leq r_i< p_i,& i&=1,2. \end{align} After a study of the shape of $J$ we can state For $l\geq \max\{\|\vec n_1\|,\|\vec n_2\|\}$ the only nonzero elements of $J$ along a given row or column are \begin{align} \begin{array}{ccccccccc} & & & & J_{(l-1)_{\{-,r_1(a_1(l-1)+1)\}},l} & & & & \\ & & & & * & & & & \\ & & & & \vdots & & & & \\ & & & & * & & & & \\ J_{l,(l-1)_{\{-,r_2(a_2(l-1)-1)\}}} & * & \cdots & * & J_{l,l} & * & \cdots & * & J_{l,(l+1)_{\{+,r_1(a_1(l+1)-1)\}}}\\ & & & & * & & & & \\ & & & & \vdots & & & & \\ & & & & * & & & & \\ & & & & J_{(l+1)_{\{+,r_2(a_2(l+1)-1)\}},l} & & & & \\ \end{array} \end{align} Using this Lemma we get the desired result and the proof is complete. In order to be more clear let us suppose that $p_1=3$ and $p_2=2$ with $\vec n_1=(4,3,2)$ and $\vec n_2=(3,2)$. The corresponding Jacobi type matrix has the following shape \begin{align}\label{J} \small J^{[12]}& J^{[12,\geq 12]}\\ J^{[\geq 12,12]} & J^{[\geq 12]} \end{BMAT} \right) \begin{BMAT}{cccccccccccc\|ccccccccccccccc}{cccccccccccc\|ccccccccccccccc} \textcolor{blue}{} & \boldsymbol{\textcolor{blue}{1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots \\ \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} &\boldsymbol{\textcolor{blue}{1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots \\ 0 & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{\textcolor{blue}{}}} &\boldsymbol{\textcolor{blue}{1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots \\ 0 & 0 & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} &\boldsymbol{ \textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} &\boldsymbol{\textcolor{blue}{1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots \\ 0 & 0 & \boldsymbol{\textcolor{blue}{}} & \textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} &\textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} & \boldsymbol{\textcolor{blue}{}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots \\ 0 & 0 & \boldsymbol{ \textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} &\textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} &\textcolor{blue}{} & \boldsymbol{\textcolor{blue}{}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots \\ 0 & 0 & 0 & 0 & \boldsymbol{\textcolor{blue}{}} &\textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} &\textcolor{blue}{} & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} &\boldsymbol{ \textcolor{blue}{1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots \\ 0 & 0 & 0 & 0 & \boldsymbol{\textcolor{blue}{}} & \textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} &\textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} & \boldsymbol{\textcolor{blue}{}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots \\ 0 & 0 & 0 & 0 & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} & \textcolor{blue}{} & \textcolor{blue}{} &\textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} &\boldsymbol{\textcolor{blue}{1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & \boldsymbol{\textcolor{blue}{}} & \textcolor{blue}{} & 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&\textcolor{blue}{}& \boldsymbol{\textcolor{blue}{}} &\boldsymbol{ \textcolor{blue}{}} &\boldsymbol{\textcolor{blue}{1}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \boldsymbol{\textcolor{blue}{}} &\boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} &\textcolor{blue}{}& \textcolor{blue}{} & \textcolor{blue}{} &\textcolor{blue}{} & \textcolor{blue}{} & \boldsymbol{\textcolor{blue}{}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 & 0 & \boldsymbol{\textcolor{blue}{}} &\textcolor{blue}{} &\textcolor{blue}{} & \textcolor{blue}{} &\textcolor{blue}{} & \textcolor{blue}{} &\boldsymbol{ \textcolor{blue}{}} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \dots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\boldsymbol{\textcolor{blue}{}} & \boldsymbol{ \textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} & \textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} & \boldsymbol{\textcolor{blue}{}}& \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} &\boldsymbol{\textcolor{blue}{}} &\boldsymbol{\textcolor{blue}{1}} & 0 & 0 & 0 & \dots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 & 0 & \boldsymbol{\textcolor{blue}{}} & \textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{}& \textcolor{blue}{} & \textcolor{blue}{} &\textcolor{blue}{} & \boldsymbol{\textcolor{blue}{}} & 0 & 0 & 0 & \dots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 & 0 & \boldsymbol{\textcolor{blue}{}} & \textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{}& \textcolor{blue}{} & \textcolor{blue}{} &\textcolor{blue}{} & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} &\boldsymbol{\textcolor{blue}{1}} & \dots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &\boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} & \textcolor{blue}{}& \textcolor{blue}{} & \textcolor{blue}{} &\textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} &\boldsymbol{ \textcolor{blue}{}} & \dots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \boldsymbol{\textcolor{blue}{}} & \textcolor{blue}{}& \textcolor{blue}{} & \textcolor{blue}{} &\textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} & \boldsymbol{\textcolor{blue}{}} & \dots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}}& \boldsymbol{\textcolor{blue}{}} & \textcolor{blue}{} &\textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} & \boldsymbol{\textcolor{blue}{}} & \dots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \boldsymbol{\textcolor{blue}{}} & \textcolor{blue}{} &\textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} & \boldsymbol{\textcolor{blue}{}} & \dots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \boldsymbol{\textcolor{blue}{}} & \textcolor{blue}{} &\textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} &\textcolor{blue}{} &\dots\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} &\boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} & \textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} & \dots \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \boldsymbol{\textcolor{blue}{}} & \textcolor{blue}{} & \textcolor{blue}{} & \textcolor{blue}{} & \dots\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 &0 &0 &0 & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} & \boldsymbol{\textcolor{blue}{}} & \textcolor{blue}{} & \dots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \\ \end{BMAT} \right), \end{align} where $$ denotes a non-necessarily null real number. In our example ($p_1=3$, $p_2=2$, $\vec n_1=(4,3,2)$ and $\vec n_2=(3,2)$) for $[l]=[12]$ we have \begin{align} (y-x)K^{[12]}(x,y)&=\left[ \sum_{i=8}^{11} \sum_{j=12}^{15} \bar{Q}(x)^{(i)}J_{i,j} Q(y)^{(j)}\right]- \left[\sum_{i=12}^{13} \sum_{j=9}^{11} \bar{Q}(x)^{(i)}J_{i,j} Q(y)^{(j)}\right] \end{align} §.§ Expressing the Jacobi type matrix in terms of factorization factors As we have seen we can write $J$ in terms of $S$ or of $\bar S$, this means that for each term of $J$ has two different expressions, giving relations between $S$ with $\bar S$. We are not too concerned about these relations since what we want here is the most simple expression we can get for the elements of $J$. It is easy to realize that this is achieved if we use the expression involving $S$ in order to calculate the upper part of $J$ and the expression involving $\bar S$ to calculate the lower part of it. Hence, for every $J_{l,k}$ we will have expressions in terms of the factorization matrices coefficients and the elements of their inverses –thus, in terms of the MOPRL and associated second kind functions. The only terms from the factorization matrices (or their inverses) that will be involved when calculating any $J_{l,k}$ are just those between the main diagonal and the $l-\|n_{1}\|$ diagonal (both included) of $S$ and those between the main diagonal and the $l+\|n_2\|$ diagonal (both included) of $\bar S$. And not even all of them. As we are about to see there are three different kinds of elements in $J$. The ones along the main diagonal, the ones along the immediate closest diagonals to the main one, and finally all the remaining diagonals. The recursion relation coefficients $J_{k,l}$ are ultimately related to the MOPRL and its associated second kind functions in the following way The elements of the recursion matrix $J$ can be written in terms of products of the entries of the LU factorization matrices and its inverses as follows \begin{align} & \begin{aligned} J_{l,l}&=S_{l,(l-1)_{\{-,a_1(l)\}}}+S^{-1}_{(l+1)_{\{+,a_1(l)\}},l}+\sum_{\substack{a=1,\dots,p_1\\a\neq a_{1}(l)}} S_{l,(l-1)_{\{-,a\}}} S^{-1}_{(l+1)_{\{+,a\}},l},\\ &= \bar S_{l,(l+1)_{\{+,a_2(l)\}}}\bar S^{-1}_{l,l}+\bar S_{l,l}\bar S^{-1}_{(l-1)_{\{-,a_2(l)\}},l}+ \sum_{\substack{a=1,\dots,p_2\\a\neq a_{2}(l)}} \bar S_{l,(l+1)_{\{+,a\}}} \bar S^{-1}_{(l-1)_{\{-,a\}},l}, \end{aligned} \\ J_{l,l+1}&=S^{-1}_{(l+1)_{\{+,a_1(l)\}},l+1}+ \sum_{\substack{a=1,\dots,p_1\\a\neq a_{1}(l)}} S_{l,(l-1)_{\{-,a\}}} S^{-1}_{(l+1)_{\{+,a\}},l+1}, \\ J_{l+1,l}&= \bar S_{l+1,(l+1)_{\{+,a_2(l)\}}} \bar S^{-1}_{l,l} +\sum_{\substack{a=1,\dots,p_2\\a\neq a_{2}(l)}} \bar S_{l+1,(l+1)_{\{+,a\}}} \bar S^{-1}_{(l-1)_{\{-,a\}},l}, \end{aligned}\\ S_{l,(l-1)_{\{-,a_1\}}} S^{-1}_{(l+1)_{\{+,a_1\}},l+k} & 2&\leq k\leq (l+1)_{\{+,r_1(a_1(l+1)-1)\}}-l, \\ \bar S_{l+k,(l+1)_{\{+,a_\}}} \bar S^{-1}_{(l-1)_{\{-,a\}},l}, & 2&\leq k\leq (l+1)_{\{+,r_2(a_2(l+1)-1)\}}-l. \end{aligned} \end{align} Where, for $r,r'<p$, we have used \begin{align} \sideset{}{^p}\sum_{a=r}^{r'}X_a=\begin{dcases} \sum_{a=r}^{r'} X_a, & r\leq r',\\ \sum_{a=1}^{r'}X_a+\sum_{a=r}^pX_a, & r>r'. \end{dcases} \end{align} § ACKNOWLEDGEMENTS GA thanks economical support from the Universidad Complutense de Madrid Program “Ayudas para Becas y Contratos Complutenses Predoctorales en España 2011". MM thanks economical support from the Spanish “Ministerio de Economía y Competitividad" research project MTM2012-36732-C03-01, Ortogonalidad y aproximacion; Teoria y Aplicaciones. [1]M. Adler and P. van Moerbeke, Group factorization, moment matrices and Toda lattices, International Mathematics Research Notices 12 (1997) 556-572. [2] M. Adler and P. van Moerbeke, Generalized orthogonal polynomials, discrete KP and Riemann–Hilbert problems, Communications in Mathematical Physics 207 (1999) 589-620. [3] M. Adler and P. van Moerbeke, Vertex operator solutions to the discrete KP hierarchy, Communications in Mathematical Physics 203 (1999) 185-210. [4] M. Adler and P. van Moerbeke, The spectrum of coupled random matrices, Annals of Mathematics 149 (1999) 921-976. [5] M. Adler and P. van Moerbeke, Darboux transforms on band matrices, weights and associated polynomials, International Mathematics Research Notices 18 (2001) 935-984. [7] C. Álvarez-Fernández, U. Fidalgo and M. Mañas, The multicomponent 2D Toda hierarchy: generalized matrix orthogonal polynomials, multiple orthogonal polynomials and Riemann–Hilbert problems. Inverse Problems 26 (2010) 055009 (17 pp). [8] C. Álvarez-Fernández, U. Fidalgo and M. Mañas, Multiple orthogonal polynomials of mixed type: Gauss-Borel factorization and the multi-component 2D Toda hierarchy. Advances in Mathematics 227 (2011) 1451-1525. [9] C. Álvarez-Fernández and M. Mañas, Orthogonal Laurent polynomials on the unit circle, extended CMV ordering and 2D Toda type integrable hierarchies, Advances in Mathematics 240 (2013) 132-193. [10] R. Apery. Irrationalite de $\zeta(2)$ et $\zeta(3)$, Astèrisque 61 (1979) 11-13. [12] F. Beukers, Padé approximation in number theory, Lecture Notes in Mathematics 888, Springer Verlag, Berlin, 1981, 90-99. [13] P.M. Bleher and A.B.J. Kuijlaars, Random matrices with external source and multiple orthogonal polynomials, International Mathematics Research Notices 2004 (2004), 109-129. [15] J. Coates, On the algebraic approximation of functions, I, II, III. Indagationes Mathematicae 28 (1966) 421-461. [16] E. Daems and A. B. J. Kuijlaars, Multiple orthogonal polynomials of mixed type and non-intersecting Brownian motions, Journal of Approximation Theory 146 (2007) 91-114. [17] E. Daems and A. B. J. Kuijlaars, A Christoffel–Darboux formula for multiple orthogonal polynomials, Journal of Approximation Theory 130 (2004) 188-200. [22] U. Fidalgo Prieto and G. López Lagomasino, Nikishin systems are perfect, Constructive Approximation 34 (2011) 297-356. [23] A. S. Fokas, A. R. Its and A. V. Kitaev, The isomonodromy approach to matrix models in 2D quatum gravity, Communications in Mathematical Physics (1992) 395-430. [24] Ch. Hermite, Sur la fonction exponentielle, C. R. Acad. Sci. Paris 77 (1873), 18-24, 74-79, 226-233, 285-293; reprinted in his Oeuvres, Tome III, Gauthier-Villars, Paris, 1912, 150-181. [25] H. Jager, A simultaneous generalization of the Padé table, I-VI, Indagationes Mathematicae 26 (1964), [27] A. B. J. Kuijlaars, Multiple orthogonal polynomial ensembles, Contemporary Mathematics 507 (2010), 155–176 [29] K. Mahler, Perfect systems, Compositio Mathematica 19 (1968), 95-166. [33] E. M. Nikishin, On simultaneous Padé approximants Matematicheskii Sbornik 113 (1980), 499–519 (Russian); English translation in Mathematics of the USSR-Sbornik 41 (1982), 409-425. [36] J. A. Shohat and J.D. Tamarkin, The problem of moments, American Mathematical Society (1943). [37] B. Simon, The Christoffel-Darboux Kernel, Proceedings of Symposia in Pure Mathematics 79:“Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: A Volume in Honor of Vladimir G. Maz'ya's 70th Birthday”, (2008) 295-336. [38] V.N. Sorokin, On simultaneous approximation of several linear forms, Vestnik Moskovskogo Universiteta. Seriya I. Matematika 1 (1983) 44-47. [40] R. P. Stanley, Enumerative combinatorics, Cambridge University Press, Cambridge (1998). [41] W. Van Assche, J. S. Geromino and A. B. J. Kuijlaars, Riemann–Hilbert problems for multiple orthogonal polynomials in: Bustoz et al (eds.), Special Functions 2000: Current Perspectives and Future Directions, Kluwer Academic Publishers, Dordrecht, 2001, pp 23-59.	arxiv-papers	2013-10-27T19:56:26	2024-09-04T02:49:52.971067	{ "license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/", "authors": "Gerardo Ariznabarreta, Manuel Manas", "submitter": "Manuel Ma\\~nas", "url": "https://arxiv.org/abs/1310.7240" }
1310.7411	# Hole doped Dirac states in silicene by biaxial tensile strain T. P. Kaloni, Y. C. Cheng, and U. Schwingenschlögl [email protected],+966(0)544700080 PSE Division, KAUST, Thuwal 23955-6900, Kingdom of Saudi Arabia ###### Abstract The effects of biaxial tensile strain on the structure, electronic states, and mechanical properties of silicene are studied by ab-initio calculations. Our results show that up to 5% strain the Dirac cone remains essentially at the Fermi level, while higher strain induces hole doping because of weakening of the Si$-$Si bonds. We demonstrate that the silicene lattice is stable up to 17% strain. It is noted that the buckling first decreases with the strain (up to 10%) and then increases again, which is accompanied by a band gap variation. We also calculate the Grüneisen parameter and demonstrate a strain dependence similar to that of graphene. ## I Introduction Silicene is a two dimensional buckled material which is closely related to graphene. It has been proposed as a potential candidate for overcoming the limitations of graphene because of stronger intrinsic spin orbit coupling (4 meV in silicene and $1.3\cdot 10^{-3}$ meV in graphene cheng1 ). Silicene first has been reported to be stable by Takeda and Shiraishi takeda . Though C and Si belong to the same group of the periodic table, Si has a larger ionic radius, which promotes $sp^{3}$ hybridization. Theoretical studies predict that free standing silicene has a stable two-dimensional buckled honeycomb structure ciraci ; olle , where the buckling is due to the mixture of $sp^{2}$ and $sp^{3}$ hybridizations. The magnitude of the buckling is $\sim$ 0.45 Å, which opens an electrically tunable band gap falko ; Ni , whereas the induced band gap due to the intrinsic spin orbit coupling amounts to 1.55 meV yao . The charge carriers behave like massless Dirac fermions in the $\pi$ and $\pi^{}$ bands, which form Dirac cones at the Fermi level at the K and K′ points. The electronic properties of silicene and its derivatives have been studied in much detail by density functional theory calculations houssa ; Bechstedt ; kang ; Wang . In particular, it has been reported that the lattice is sensitive to the carrier concentration but still stable in a wide range of doping cheng . Experimentally, growth of silicene and its derivatives has been reported for metallic substrates like Ag and ZrB2 padova ; vogt ; ozaki . Silicene on a ZrB2 thin film shows an asymmetric buckling due to strong interaction with the substrate, which increases the band gap. As in general accurate measurements of materials properties are problematic on substrates, it is desirable to achieve free standing silicene. However, this first requires the growth on appropriate substrates that make it possible to separate the silicene sheet. For the growth of silicene on any kind of substrate, the effect of strain is crucial to be understood. In this work, we focus on this topic using first- principles calculations. We apply strain up to 20% and calculate the corresponding band structure to evaluate the dependence of the induced doping on the strength of the biaxial tensile strain. Furthermore, we study the phonon spectrum to address the stability of the system and calculate the Grüneisen parameter. ## II Computational details We have carried out calculations using density functional theory in the generalized gradient approximation paolo . The van der Waals interaction grime ; jmc is taken into account in order to correctly describe the geometry. The calculations are performed with a plane wave cutoff energy of 816 eV. Moreover, a Monkhorst-Pack $16\times 16\times 1$ k-mesh is employed for optimizing the crystal structure and calculating the phonon spectrum, whereas a $24\times 24\times 1$ k-mesh is used for the density of states (DOS) in order to achieve higher resolution. The atomic positions are relaxed until an energy convergence of 10-9 eV and a force convergence of $4\cdot 10^{-4}$ eV/Å are reached. We use an interlayer spacing of 16 Å to avoid artifacts of the periodic boundary conditions. The magnitude of the biaxial tensile strain is defined as $\varepsilon=\frac{(a-a_{0})}{a_{0}}\times 100\%$, where $a$ and $a_{0}=3.86$ Å are the lattice parameters of the strained and unstrained silicene, respectively. Figure 1: Crystal structure of silicene under consideration. The arrows indicate the direction of the biaxial tensile strain. ## III Results and discussion For graphene it has been demonstrated that 5 to 10% strain can be achieved without much efforts Andresa . The existing reports confirm this makes the system five times more reactive and H atoms are bound much stronger than in pristine graphene Andresa . Since a similar enhancement of H storage by strain can be expected for silicene, we study in the following the effect of strain on the electronic and mechanical properties. A top view of the crystal structure under consideration is shown in Fig. 1. For unstrained silicene we obtain a lattice parameter of $a=3.89$ Å and a buckling of 0.45 Å, consistent with previously reported data ciraci ; cheng . In a first step, we address the dependence of the force on the applied strain, see the results in Fig. 2. The force increases monotonically with the strain up to a strain of 17% and decreases thereafter, which indicates that silicene is stable up to 17% strain. The stability limit will be addressed in more detail via the phonon spectrum in the following section. Figure 2: Variation of the force as a function of the applied biaxial tensile strain. The band gap of 2 meV in unstrained silicene becomes smaller for increasing strain. Since strain weakens the internal electric field (by reducing the magnitude of the buckling) the spin orbit coupling and thus the induced band gap are reduced. The Si$-$Si bond length is found to grow with the strain monotonically, which explains why the buckling decreases. Surprisingly, the buckling starts to increase again when the strain exceeds 10%. For example, unstrained silicene has a Si$-$Si bond length of 2.28 Å and buckling of 0.46 Å. For 5% strain these values change to 2.37 Å and 0.32 Å, and for 17% strain to 2.47 Å and 0.30 Å. The variation of the Si$-$Si bond length and buckling under strain are addressed in Fig. 3(a) and (b), respectively. Figure 3: Variation of (a) the Si$-$Si bond length, (b) the buckling, and (c) the doping level under biaxial tensile strain. The variation of the doping level (defined as the shift of the Dirac cone with respect to the Fermi level) under strain is addressed in Fig. 3(c). It is well known that unstrained silicene is a semimetal, where the $p_{z}$ and $p_{z}^{}$ orbitals give rise to $\pi$ and $\pi^{}$ bands forming Dirac cones at the K and K′ points, see Fig. 4(a). The calculated band structure shows that the Dirac cone lies at the Fermi level upto a strain of 5% with a 2 meV band gap due to intrinsic spin orbit coupling. For higher strain the conduction band at the $\Gamma$-point shifts towards the Fermi level, consistent with Ref. liu . At a strain of 7% it slightly crosses the Fermi level, which shifts the Dirac cone above the Fermi level by $\sim$ 0.06 eV, inducing hole doping, see Fig. 3(c). The doping is enhanced for increasing strain, since the conduction band minimum at the $\Gamma$-point shifts further downwards and becomes more and more occupied (with an increasing DOS at the Fermi level). The main reason for hole doping in silicene under strain is this downshift and the consequent occupation of the band at the $\Gamma$-point. It is a consequence of the weakening of the bonds due to the increasing Si$-$Si bond length. Another ingredient is a reduction of the hybridization between the $s$ and $p$ orbitals, which in fact are occupied by 1.18 and 2.76 electrons in unstrained silicene, respectively, but by 1.33 and 2.63 electrons for 10% strain. Figure 4: Electronic band structure with corresponding partial DOSs for (a) unstrained and (b) 10% biaxially tensile strained silicene. At 10% strain the Dirac point lies at 0.18 eV, see Fig. 4(b). We note that the $\pi$ and $\pi^{}$ bands are due to the $p_{z}$ orbitals with minute contributions from the $p_{x}$ and $p_{y}$ orbitals, as expected, see the projected DOSs. For higher strain the conduction band minimum shifts further to lower energy and the Dirac cone accordingly to higher energy. It reaches 1.0 eV with the Dirac point at 0.34 eV for a strain of 20%. This behavior is different from graphene despite the quantitatively similar band structure, because the Si$-$Si bonds are more flexible than the C$-$C bonds. In contrast to silicene, graphene does not show significant changes in the electronic structure in the presence of strain, resulting a zero band gap semiconductor up to a huge strain of 30% son . As a result, doping cannot be achieved in graphene by strain. We now discuss the phonon spectrum of silicene without strain and under strain of 5%, 10%, 15%, 20%, and 25%. Without strain the optical phonon frequencies are found to be $\sim$ 33% smaller than in graphene cheng , which is understood by the smaller force constant and weaker Si$-$Si bonds. In fact, the Si$-$Si bond length of 2.28 Å is 37% larger than the C$-$C bond length. In Fig. 5 we address the phonon band structure, where we focus on the highest branches at the $\Gamma$-point (G mode) and the K-point (D mode). The calculated phonon frequencies at the $\Gamma$ and K-points are 550 cm-1 and 545 cm-1, respectively, which agree well with previous theoretical results ciraci ; cheng . A significant modification of the phonon frequencies is observed for strained silicene. For a strain of 5% the G and D mode frequencies amount to 460 cm-1 and 386 cm-1, respectively, reflecting the weakening of the Si$-$Si bond under strain. Increase of the strain to 10% (17%) results in phonon frequencies of 372 cm-1 (296 cm-1) for the G mode and 272 cm-1 (187 cm-1) for the D mode. We still have positive frequencies along the $\Gamma$-K direction and, hence, a stable lattice. An instability comes into the picture when the strain increases beyond 17%. At 20% strain we find a frequency of $-5$ cm-1 and at 25% strain, see Fig. 5(c), the lattice is strongly instable. Importantly, no splitting of the G mode for increasing strain is observed in our calculations in contrast to graphene udo . Figure 5: Phonon frequencies for (a) unstrained, (b) 10% biaxially tensile strained, and (c) 25% biaxially tensile strained silicene. The Grüneisen parameter is an important quantity to describe strained materials as it measures the rate of phonon mode softening or hardening and, thus, determines the thermomechanical properties. The Grüneisen parameter for the G mode is given by $\gamma_{G}=-\Delta\omega_{G}/2\omega_{G}^{0}\varepsilon,$ where $\Delta\omega_{G}$ is the difference in the frequency with and without strain and $\omega_{G}^{0}$ is the frequency of the G mode in unstrained silicene. A significant variation of the Grüneisen parameter between 1.64 and 1.42 for strain between 5 and 25 % is found, see Table I. These values are close to the experimental and theoretically values for graphene Mohiuddin ; ding ; udo ; Remi . While the experimentally reported Grüneisen parameters for graphene are not consistent due to substrate effects, there are no experimental data available for silicene for comparison. We find that the Grüneisen parameter first decreases with growing strain due to the reduced buckling of the two Si sublattices but increases again for higher strain as also the buckling increases. This behavior is fundamentally different from graphene, which is not subject to buckling. An experimental confirmation of our observations by Raman spectroscopy would be desirable. $\varepsilon$ (%) \| $\Delta\omega_{G}$ (cm-1) \| ${\gamma_{G}}$ ---\|---\|--- 5 \| 460 \| 1.64 10 \| 372 \| 1.62 15 \| 296 \| 1.54 20 \| 246 \| 1.34 25 \| 160 \| 1.42 Table 1: Strain, frequency shift of the G mode, and Grüneisen parameter of the G mode. ## IV Conclusion In conclusion, we have used density functional theory to study the effect of biaxial tensile strain on the structure, electronic properties, and phonon modes of silicene. Our calculations demonstrate that up to 5% strain the Dirac cone remains essentially at the Fermi level but starts to shift to higher energy for higher strain. Therefore, strain can be used in silicene, in contrast to graphene, to induce hole doping. The different behavior of the two compounds, despite their close stuctural similarity, can be explained in terms of bonding and changes in the hybridizations. Strain results in a weakening of the Si$-$Si bonds. As a consequence, an electronic band at the $\Gamma$-point of the Brillouin zone shifts to lower energy and becomes partially occupied, which in turn leads to a depopulation of the Dirac cone. The buckling is found to decrease with increasing strain up to 10% but starts to increase again thereafter. Accordingly, the calculated Grüneisen parameter behaves differently than in graphene as the latter is not subject to buckling. Positive phonon frequencies up to a strain of 17% indicate lattices stability in this regime, whereas the lattice becomes instable at higher strain. ## References * (1) C.-C. Liu, H. Jiang, and Y. Yao, Phys. Rev. B 84, 195430 (2011). * (2) K. Takeda and K. Shiraishi, Phys. Rev. B 50, 14916 (1994). * (3) S. Cahangirov, M. Topsakal, E. Aktürk, H. Sahin, and S. Ciraci, Phys. Rev. Lett. 102, 236804 (2009). * (4) S. Lebegue and O. Eriksson, Phys. Rev. B 79, 115409 (2009). * (5) N. D. Drummond, V. Zólyomi, and V. I. Fal′ko, Phys. Rev. B 85, 075423 (2012). * (6) Z. Ni, Q. 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Herklotz, K. Dorr, Y. F. Mei, A. Rastelli, and O. G. Schmidt, Nano Lett. 10, 3453 (2010). * (23) Y. C. Cheng, Z. Y. Zhu, G. S. Huang, and U. Schwingenschlögl, Phys. Rev. B 83, 115449 (2011). * (24) T. M. G. Mohiuddin, A. Lombardo, R. R. Nair, A. Bonetti, G. Savini, R. Jalil, N. Bonini, D. M. Basko, C. Galiotis, N. Marzari, K. S. Novoselov, A. K. Geim, and A. C. Ferrari, Phys. Rev. B 79, 205433 (2009). * (25) C. Metzger, S. Remi, M. K. Liu, S. V. Kusminskiy, A. H. C. Neto, A. K. Swan, and B. B. Goldberg, Nano Lett. 10, 6 (2010).	arxiv-papers	2013-10-28T13:36:00	2024-09-04T02:49:52.985127	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "T. P. Kaloni, Y. C. Cheng, and U. Schwingenschl\\\"ogl", "submitter": "Thaneshwor Prashad Kaloni", "url": "https://arxiv.org/abs/1310.7411" }
1310.7430	# Charged Particle Multiplicity and Pseudorapidity Density Measurements in pp collisions with ALICE at the LHC INFN Bologna & CERN E-mail ###### Abstract: These proceedings describe the charged-particle pseudorapidity densities and multiplicity distributions measured by the ALICE detector in pp collisions at $\sqrt{s}=0.9$ and 7 TeV in specific phase space regions. The pseudorapidity range $\|\eta\|<0.8$, together with $p_{\rm T}$ cuts at 0.15, 0.5 and 1 $\mathrm{GeV}/c$ is considered. The classes of events considered are those having at least one charged particle in the kinematical ranges just described. The results obtained by ALICE are compared to Monte Carlo predictions. ## 1 Introduction The ALICE results on charged-particle pseudorapidity density ($\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta$) and multiplicity distributions in pp collisions at $\sqrt{s}=0.9$ and 7 TeV presented in this document derive from an analysis carried out with an event and track selection specially chosen to make comparison with Monte Carlo calculations to allow for a better Monte Carlo tuning. Tracks reconstructed as coming from primary particles111The ensemble of primary charged particles includes those produced in the collision and their decay products, excluding weak decays from strange particles. in the Inner Tracking System (ITS) and in the Time Projection Chamber (TPC) of ALICE have been used, and kinematical phase space regions defined in $\eta$ ($\|\eta\|<0.8$) and $p_{\rm T}$ ($p_{\rm T}$ $>$ $p_{\rm T,cut}$, with $p_{\rm T,cut}$ = 0.15, 0.5 and 1.0 $\mathrm{GeV}/c$) have been considered. The first $p_{\rm T,cut}$ corresponds to the $p_{\rm T}$cutoff at which the ALICE global tracking efficiency (i.e. including both ITS and TPC) reaches $\sim 50\%$ and stays approximately constant ($\sim 70\\--75\%$) for higher $p_{\rm T}$ [1], while the higher $p_{\rm T,cut}$ values allow a comparison with measurements performed by ATLAS and CMS. The pseudorapidity density and multiplicity distributions are measured for all charged particles in the given $p_{\rm T}-\eta$ region for those events which have at least one charged particle. This introduces a so-called “hadron level definition” of the event class considered, named INEL$>0_{\|\eta\|<0.8,p_{\rm T}>p_{\rm T,cut}}$ hereafter. ## 2 The ALICE experiment and the data samples The ALICE experiment consists of a set of different detectors placed in a solenoidal magnetic field of 0.5 T (the central barrel) plus other detectors outside. Details about the various subsystems can be found in [3]. For the analysis presented herein, tracks reconstructed in the ALICE central barrel by the ITS and TPC detectors were used, while the triggering and event selection relied on both the ITS and VZERO detectors. The data samples used consisted of Minimum Bias pp events collected in 2009 and 2010. The Minimum Bias trigger was defined as a signal in either one of the two ALICE VZERO hodoscopes, or in the ITS pixel detector (one out of three). A coincidence with the signals from the two beam pick-up counters (BPTX) was also required to select the events and remove the background. In such conditions, about $110000$ (collected in 2009) and $2.2\times 10^{6}$ events (collected in 2010) were used for the charged-particle pseudorapidity density analysis at $\sqrt{s}=0.9$ and 7 TeV respectively. The multiplicity distributions were obtained from approximately $2.9\times 10^{6}$ and $2.7\times 10^{6}$ events (all collected in 2010) at $\sqrt{s}=0.9$ and 7 TeV respectively. In addition to requiring the Minimum Bias trigger in the collision and the reconstruction of the primary vertex, a preselection of the events aimed at reducing the beam background was applied based on the information from the VZERO detector and on the correlation between the number of hits and the so- called tracklets222A tracklet is built combining a pair of hits in the two innermost ITS layers. found in the the two innermost ITS layers, corresponding to the Silicon Pixel Detector (SPD, see also [4]). A selection on the vertex was also applied, requiring it to be obtained either from the tracks reconstructed from the TPC and the ITS detectors, or, in case this was not available, from the tracklets found in the SPD detector (see also [4]), using the SPD information. Moreover, only events for which the vertex position along the $z$ coordinate ($vtx_{z}$) was such that $\|vtx_{z}\|<10$ cm were accepted. Finally, only those events with at least one reconstructed track in the kinematical region defined by the pseudorapidity interval $\|\eta\|<0.8$ and $p_{\rm T}>p_{\rm T,cut}$ ($p_{\rm T,cut}=0.15,0.5,1$ $\mathrm{GeV}/c$) were considered. For the 2010 data, an additional event selection criterion was used in order to reduce the contribution from pile-up events, removing those identified as coming from pile-up based on the SPD information. This sample was then corrected back to the $\rm{INEL}>0_{\|\eta\|<0.8,p_{\rm T}>p_{\rm T,cut}}$ “hadron level definition” described in Sec. 1. ## 3 Analysis strategy The tracks used in the analysis are those reconstructed by the ALICE central global tracking [5], which is based on the Kalman filter technique [5, 6]. Track selection criteria (cuts) have been applied in order to maximize the tracking efficiency and minimize the contamination from secondaries and fake tracks, as described in [2]. The raw $\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta$ and multiplicity distributions obtained from data were corrected using PYTHIA Monte Carlo simulations as described in Sec. 3.1 and Sec. 3.2. The GEANT3 particle transport package was used together with a detailed description of the geometry of the experiment, and of the detector and electronics response. Moreover, the simulation was set to reproduce the conditions of the LHC beam and of the detectors (in terms of vertex position, calibration, alignment and response) at the time the data under study were collected. The underestimate in the Monte Carlo simulations of the event strangeness content with respect to that found in the data was also taken into account during the correction phase. Further information about the analysis strategy and the corrections applied can be found in [2]. ### 3.1 Corrections for the charged-particle pseudorapidity density analysis The charged-particle pseudorapidity distribution is given by the expression $\frac{1}{N_{\rm{ev}}}\frac{{\rm{d}}N_{\rm{ch}}}{\rm{d}\eta}.$ (1) where $N_{\rm{ev}}$ corresponds to the total number of events that belong to the INEL$>0_{\|\eta\|<0.8,p_{\rm T}>p_{\rm T,cut}}$ class. The corrections applied to the raw $\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta$ distribution are of three types: a track-to-particle correction is needed, in order to take into account the difference between the measured tracks and the true charged primary particles coming from acceptance effects, detector and reconstruction efficiency; a second correction is applied, to account for the fact that events without a reconstructed vertex are not considered; finally, the bias due to the INEL$>0_{\|\eta\|<0.8,p_{\rm T}>p_{\rm T,cut}}$ event selection used is considered. ### 3.2 Corrections for multiplicity distribution analysis For the multiplicity analysis, the correction procedure is twofold. First, an unfolding technique is applied in order to account for the fact that due to the efficiency, acceptance and detector effects, the measured multiplicity spectrum is distorted from the true one. In addition, vertex reconstruction and event selection efficiency need to be taken into consideration, in a similar way to the $\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta$ analysis. The unfolding procedure used for the analysis presented here is described in [2]. It is based on a $\chi^{2}$-minimization approach, where the $\chi^{2}$ function used to evaluate the “guessed” unfolded spectrum $U$, can be written as: $\hat{\chi}^{2}(U)=\sum_{m}\left(\frac{M_{m}-\sum_{t}R_{mt}U_{t}}{e_{m}}\right)^{2}.$ (2) Here, $M_{m}$ is the measured distribution at true multiplicity $t$ with error $e_{m}$, and $R_{mt}$ is the response matrix element for measured multiplicity $m$ and true multiplicity $t$ which encodes the probability that an event with true multiplicity $t$ is measured with multiplicity $m$. Because this minimization suffers from oscillations in the unfolded spectrum, a constraint $P(U)$ was added to the $\chi^{2}$ function, favouring a certain shape in the unfolded distribution [7], following the same approach as described and discussed in [8]. The constraint $P(U)$ is called a $regularization$ $term$, and the new function to be minimized becomes: $\chi^{2}(U)=\hat{\chi}^{2}(U)+\beta P(U).$ As written in the formula, $P(U)$ depends only on the unfolded spectrum $U$. $\beta$ is the weight of the regularization term. For the unfolding procedure, a parameterization of the response matrix was used, in order to avoid statistics issues at high multiplicities and in the tails of the distributions for a given fixed true multiplicity. More details on the choice of the regularization function and on the parameterization can be found in [2]. ### 3.3 Systematic Uncertainties Various sources of systematic uncertainties have been taken into account, most of which are common to the two analyses. They include: * • track quality cuts variation; * • tracking efficiency; * • material budget; * • particle species relative fraction; * • event type (Single, Double, Non-Single Diffractive) relative composition; * • pile-up; * • Monte Carlo generator dependence of the corrections; Moreover, for the multiplicity distribution analysis only, the following sources of systematic uncertainties have been studied: * • choice of the regularization function and weight; * • bias introduced by the regularization [9]; * • unfolding dependence on the $\langle p_{\rm T}\rangle$ as a function of multiplicity. The complete description of the evaluation of the systematic uncertainties is discussed in [2]. ## 4 Results Figure 1 shows the final charged particle $\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta$ for the $\rm{INEL}>0_{\|\eta\|<0.8,p_{\rm T}>p_{\rm T,cut}}$ classes of events. Here, for the sake of brevity, only the results for the $p_{\rm T,cut}=0.15$ GeV/$c$ at $\sqrt{s}=0.9$ (left panel) and $p_{\rm T,cut}=0.5$ GeV/$c$ at 7 TeV (right) are shown. The complete set of results is presented in [2]. Predictions from Monte Carlo generators are superimposed on the distributions. They are indicated as follows: * • PYTHIA-6 * – Atlas CSC (tune 306 [10]); * – D6T (tune 109 [11]); * – A (tune 100 [12]); * – Perugia-0 (tune 320 [13]); * – Perugia-2011 (tune 350 [14]). * • PYTHIA-8 * – Pythia8 (tune 1 [15]); * – Pythia8 (tune 4C) [16]); * • PHOJET ([17]); * • EPOS LHC ([18]). The bottom panels of the figures show the ratio between the data and the Monte Carlo predictions. \| ---\|--- Figure 1: $\mathrm{d}N_{\mathrm{ch}}/\mathrm{d}\eta$ versus $\eta$ obtained at $\sqrt{s}=0.9$ (left) $p_{\rm T}>0.15$ $\mathrm{GeV}/c$ (left) and $\sqrt{s}=7$ (left) $p_{\rm T}>0.5$ $\mathrm{GeV}/c$ for $\|\eta\|<0.8$ normalized to the $\rm{INEL}>0_{\|\eta\|<0.8,p_{\rm T}>p_{\rm T,cut}}$ event class. The predictions from different Monte Carlo generators are also shown. The grey bands represent the systematic uncertainties on the data. Bottom panels: data to Monte Carlo prediction ratios for the different generators considered. Here, the grey bands represent the total (statistical + systematic) uncertainty on the data. The charged particle multiplicity distribution results are shown in Fig. 2 for the same cases as in Fig. 1. For the sake of visibility, the Monte Carlo comparison (which was carried out with the same generators and tunes used for Fig. 1) is shown for both figures in two different panels, as explained in the legends. For both analyses, the results show that, in general, a universal trend in terms of comparison between the ALICE data and Monte Carlo calculations cannot be identified. At different centre-of-mass energies and with different values of $p_{\rm T,cut}$, the different models describe the data differently, and one tune that gives reasonable comparison to data in one case fails in the others. Moreover, for the multiplicity distributions, the level of agreement/disagreement varies significantly as a function of multiplicity. \| ---\|--- \| Figure 2: Multiplicity distributions for the analysis at $\sqrt{s}=0.9$ TeV, $p_{\rm T,cut}=0.15$ GeV/$c$ (top) and at $\sqrt{s}=7$ TeV, $p_{\rm T,cut}=0.5$ GeV/$c$ (bottom). In the left and right panels, the data are compared with different Monte Carlo expectations, as indicated in the key. In the upper panels, data are shown with both statistical (black line) and systematic (grey band) uncertainties. The grey bands in the lower panels, where the data/Monte Carlo ratios are presented, correspond to the total uncertainty on the final results. ## 5 Conclusions The charged particle pseudorapidity density and multiplicity distributions measured by ALICE at $\sqrt{s}=0.9$ and 7 TeV with charged tracks reconstructed in the ITS and TPC detectors have been presented. A $p_{\rm T,cut}$ (with $p_{\rm T,cut}$ = 0.15, 0.5, 1.0 GeV/$c$) was used in order to characterize the class of events to be considered for the analysis, namely the $\rm{INEL}>0_{\|\eta\|<0.8,p_{\rm T}>p_{\rm T,cut}}$ class, defined requiring at least one charged particle with $p_{t}>p_{\rm T,cut}$ in $\|\eta\|<0.8$. While the lowest $p_{\rm T,cut}$ allows the most inclusive measurement for ALICE with global tracks, the 0.5 and 1.0 GeV$/c$ cuts were chosen together with the other LHC collaborations (ATLAS, CMS) to allow for the comparison with their results (not shown here). The results were compared to different Monte Carlo models, showing that the selected Monte Carlo generators do not reproduce the measurements at both centre-of-mass energies and for all choices of $p_{\rm T,cut}$. ## References * [1] K. Aamodt et al. [ALICE Collaboration], Phys. Lett. B 693 (2010) 53 [arXiv:1007.0719 [hep-ex]]. * [2] ALICE Collaboration, ALICE-PUBLIC-2013-001. * [3] F. Carminati et al., ALICE Collaboration, Physics Performance Report Vol. I, CERN/LHCC 2003-049 and J. Phys. G30 1517 (2003); B. Alessandro et al., ALICE Collaboration, Physics Performance Report Vol. II, CERN/LHCC 2005-030 and J. Phys. G32 1295 (2006); K. Aamodt et al., ALICE Collaboration, JINST 3 (2008) S08002. * [4] K. Aamodt et al. [ALICE Collaboration], Eur. Phys. J. C 65 (2010) 111, arXiv:0911.5430 [hep-ex]. * [5] B. Alessandro et al. [ALICE Collaboration], J. Phys. G: Nucl. Part. Phys. 32 (2006) 1295. * [6] P. Billoir, Nucl. Instrum. Meth. A 225 (1984) 352. * [7] V. Blobel in 8th CERN School of Comp., CSC 84, Aiguablava, Spain, 9 22 Sep. 1984, CERN-85-09, 88 (1985). * [8] K. Aamodt et al. [ALICE Collaboration], Eur. Phys. J. C 68 (2010) 89 [arXiv:1004.3034 [hep-ex]]. * [9] G. Cowan, in Advanced statistical techniques in particle physics, Proceedings, Conference, Durham, UK, March 18-22, 2002, published in Conf. Proc. C 0203181 (2002) 248. * [10] A. Moraes [ATLAS Collaboration], ATLAS Note ATL-COM-PHYS-2009-119 (2009). ATLAS CSC (306) tune. * [11] M. G. Albrow et al. (Tev4LHC QCD Working Group), arXiv:hep-ph/0610012 (2006). D6T (109) tune. * [12] R. Field, Min-Bias and the Underlying Event at the Tevatron and the LHC, Fermilab ME/MC Tuning Workshop, Fermilab, Oct. 4, 2002. * [13] P. Z. Skands, in Multi-Parton Interaction Workshop, Perugia,Italy, 28 31 Oct. 2008, arXiv:0905.3418 [hep-ph] (2009). * [14] P. Z. Skands, Phys. Rev. D 82 (2010) 074018 [arXiv:1005.3457 [hep-ph]]. * [15] T. Sjöstrand, S. Mrenna, P. Z. Skands, arXiv:0710.3820, CERN-LCGAPP-2007-04, LU TP 07-28, FERMILAB-PUB-07-512-CD-T (2007). * [16] R. Corke and T. Sjöstrand, JHEP 1103 (2011) 032 [arXiv:1011.1759 [hep-ph]]. * [17] R. Engel, J. Ranft, S. Roesler, Phys. Rev. D 52, 1459 (1995). * [18] K. Werner, F.-M. Liu and T. Pierog, Phys. Rev. C 74 (2006) 044902 [hep-ph/0506232]; T. Pierog, I. .Karpenko, J. M. Katzy, E. Yatsenko and K. Werner, arXiv:1306.0121 [hep-ph]; K. Werner, F. -M. Liu and T. Pierog, Phys. Rev. C 74 (2006) 044902 [hep-ph/0506232].	arxiv-papers	2013-10-28T14:33:09	2024-09-04T02:49:52.992092	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Chiara Zampolli (for the ALICE Collaboration)", "submitter": "Chiara Zampolli", "url": "https://arxiv.org/abs/1310.7430" }
1310.7469	# Mining the Temporal Evolution of the Android Bug Reporting Community via Sliding Windows Feng Jiang Jiemin Wang Abram Hindle Mario A. Nascimento September 2013 Department of Computing Science University of Alberta Edmonton, Alberta, Canada © University of Alberta ## Abstract The open source development community consists of both paid and volunteer developers as well as new and experienced users. Previous work has applied social network analysis (SNA) to open source communities and has demonstrated value in expertise discovery and triaging. One problem with applying SNA directly to the data of the entire project lifetime is that the impact of local activities will be drowned out. In this paper we provide a method for aggregating, analyzing, and visualizing local (small time periods) interactions of bug reporting participants by using the SNA to measure the betweeness centrality of these participants. In particular we mined the Android bug repository by producing social networks from overlapping 30-day windows of bug reports, each sliding over by day. In this paper we define three patterns of participant behaviour based on their local centrality. We propose a method of analyzing the centrality of bug report participants both locally and globally, then we conduct a thorough case study of the bug reporters’ activity within the Android bug repository. Furthermore, we validate the conclusions of our method by mining the Android version control system and inspecting the Android release history. We found that windowed SNA analysis elicited local behaviour that were invisible during global analysis. ###### Contents 1. 1 Introduction 2. 2 Background 1. 2.1 Betweenness Centrality 2. 2.2 Overlapping Time Windowing 3. 2.3 Clustering 3. 3 Methodology 1. 3.1 Data 2. 3.2 Windowing Bug Reports and Extracting Social Networks Methodology 3. 3.3 Validation using the Android Release History and the Git 4. 4 Results and Analysis 1. 4.1 Global Analysis 2. 4.2 Local Analysis 5. 5 Validation 1. 5.1 Activity pattern validation 1. 5.1.1 Participants that appeared only once tend to be pure users 2. 5.1.2 Participants showed up periodically should be a combination of users and developers 3. 5.1.3 Participants who were continuously central for a long time period could have multiple areas of expertise 2. 5.2 Cluster validation 1. 5.2.1 Participants clustered together share similar areas of expertise and tasks 2. 5.2.2 Clusters’ working areas of expertise are in accordance with the release contents along the time line 6. 6 Limitations 7. 7 Conclusion and Future Work ## 1 Introduction Global analysis provides us with easy to interpret data that gives us an overview of the entire system. It simplifies complicated dimensions like time and provides us with an easy way to explain results. Unfortunately, for tools like _Social Network Analysis_ (SNA), a global analysis can miss a lot of important interactions, especially between stakeholders, thus we propose a method of using SNA to study bug repositories and tease out local collaborations. SNA is a powerful tool that helps practitioners and researchers study the complicated interactions of participants within communities; SNA is well accepted in the area of software maintenance and mining software repositories communities [1, 2, 3]. The bug repository records interactions among software developers and users in a software project’s community. With SNA, we are able to study the structure of the interactions by analysing the graph constructed through the interaction of bug reporters in the bug repository. The results can be used in expertise elicitation and triaging in order to suggest which participants have expertise relevant to an issue [3]. Usually SNA is run globally across all day, over a single period, or over an entire project lifetime. In this paper we argue that using SNA in a more local manner provides valuable insights into interactions between stakeholders during the development and maintenance of a software system. Open-source communities are amenable to social network analysis as they are open to user interaction and participation. At the same time there is a lack of imposed organizational structures found within corporate organizations [4]. Because open source projects often lack strict centralized control and requirements [5], developers often choose their tasks instead of being assigned one [6]. This fact suggests that local structure of interactions among users and developers who express an interest in one part of the project tend to self organize and produce interesting collaboration structures (networks). Bug repositories are also amenable to social network analysis as bug repositories host and record discussions regarding issues or bugs relevant to the development and the use of a software development project [6, 7]. Bug repositories are also heavily used by open-source projects. Collaboration among developers has been studied in various aspects about how the communication introduces or avoids bugs, and further influences the software quality, [8], [9], [10], [11]. Besides the collaboration among developers, collaboration between users and developers is evident in bug reports since the discussions and communications are recorded as reported bugs, and posted comments on bug reports. One point here is that, both users and developers are often periodic, and their activities or collaborations can be local and thus missed out in global analysis. In the case of the Android bug repository, provided by the 2012 MSR Mining Challenge [12], a reporter would report a bug, which might attract comments from bug commenters; the commenters discuss the reasons and possible resolution of the bug. The bug reporting community members are usually comprised of both bug reporters and bug commenters who are either Android developers or Android users. From the perspective of the bug repository, unlike the version control system, there is actually no obvious boundary between a user and a developer. We refer to these different participants as _bug participants_. In order to apply SNA to the bug repository, we first create the graph based on the interactions. We pose that each node of the graph represents one bug participant and each edge represents the connection between two participants who have communicated on the same bug. We will introduce the network graphs in detail in Section 3. We use betweenness centrality to quantify the importance of a participant in the community [13] (betweenness centrality will be better explained in Section 3). The betweenness centrality could reveal two aspects of a participant in a community network: 1) the quantity of bug reports (which attract at least one comment) or comments they have made and 2) the importance of the content of their reports or comments. When participants have high betweenness, they might have: 1) reported quantities of bugs with at least one comment on them, 2) made lots of comments, 3) reported a very critical bug which attracts comments, 4) or made a very interesting comment which attracts comments from other participants. However, the previous work [2, 6] applied SNA on the entire lifetime of a project, such that only a single community network was constructed. Some of collaborations might not be evident if one were to analyze a large single network. That is because certain structures will not be observable on the global scale. In order to peer into these local self organized structures using social network analysis, we felt it is better to choose a _windowed_ approach, [3, 14]. Windowing allows us to look at network during a slice of time and then relate our measures (betweenness centrality per author) to the next window and beyond. This _sliding window_ view of centrality allows us to see those developers and users who are constantly at the forefront of discussion or those who ebb and flow between issues and tasks. Moreover, by sliding windows, each pair of adjacent windows would have an overlap, which results in smoother trends, and more importantly, helps to maintain context. Other benefits provided by time windowed analysis is that it gives a more accurate and nuanced view of the data as locally central participants then will not be “drowned out”. In summary, we use SNA to study the activities of bug participants based on the Android bug reports and comments repository. We apply the sliding window method to observe smooth change trends in the collaboration graph across time. With these mining results, we seek to analyze bug participants’ interactions, activity trends and patterns. We then demonstrate our analysis results via answering the following research questions about local and global behaviours. Global research questions: RQ1. How does the number of active bug participants change over time? Why? RQ2. How does the betweenness centrality of a participant change over time? What are the reasons when they have a certain activity pattern? Local research questions: RQ3. Are there special time ranges during which participants are more/less active or central than normal? Why? RQ4. What are the possible scenarios for a very sharp change of the participants’ centrality? Why? We also validate if this windowed methodology actually highlights relevant behaviour by inspecting the Android release history111Android release history: http://developer.android.com/sdk/index.html and the Android version control system. The validation would be discussed in Section 5. The rest of our paper is organized as follows. Section 2 introduces basic concepts and techniques we used in this study. The specific steps and the methodology will be discussed in Section 3. Section 4 describes the details of our mining results. The analysis of the results and its corresponding validation is provided in Section 5. Section 6 presents the limitations of our mining process and Section 7 summarizes the paper and discusses the future work. Figure 1: An example: bug 14038 is reported by timothyA, and there are five comments on this bug. When time window applied, comments are plotted into two windows, and the bug report of this example forms two networks with the weight noted on their edges ## 2 Background ### 2.1 Betweenness Centrality The betweenness centrality of a vertex is the number of geodesic paths in a graph that includes this vertex; the geodesic path is defined as the shortest path which has the minimum weight between two nodes. Defined by Freeman [15], the betweenness can be represented as: $\sum_{i=1}^{j-1}\sum_{j=1}^{n}\frac{g_{ij}(k)}{g_{ij}},i\neq j\neq k$ (1) where $k$ is a vertex of the graph, $n$ is the total number of vertices, $i$ and $j$ are vertices other than $k$, $g_{ij}$ is the number of geodesic paths between vertex $i$ and $j$, and $g_{ij}(k)$ is the number of geodesic paths that include $k$. It is used as a measurement of a person’s importance in a network. A person would be regarded as central if he is on the geodesic path between two other persons. As proposed by Freeman [15], if a person is located on the geodesic path between two other persons, he becomes one of the key persons who connects the others. That is, the more a person connects to the other people in a network, the more important or central he is [16]. In our work, we normalize the betweenness centrality values to eliminate the effect of different sizes of the networks. The betweenness is normalized as: $Normalized~{}B=\frac{B}{\frac{(n-1)(n-2)}{2}}$ (2) where $B$ represents the original betweenness value and $n$ is the number of nodes in the graph being calculated. Compared with simply counting the total number of comments or total bug reports of a participant, betweenness acts better to reflect the interactions among people. For example, when a person reports lots of bugs but none of them attract any comment, it is very likely that his bug reports are not interesting or important. In this case, if we merely counted the number of their reports or comments, we would possibly increase their importance in the network artificially. Therefore, we choose to use betweenness centrality to eliminate this unfair counting [13]. ### 2.2 Overlapping Time Windowing When SNA is applied in other papers [2, 3], it is typically applied to the entire history or one period of the partial history and all the bug reports within that period. Windowed analysis instead repeats social network analysis across 100s of windows (in our case, as many windows as we have days). These windows overlap and often the analysis of one window results in the same analysis as the previous window due to the overlap. We slid our windows by 1 day and for two adjacent windows $A$ and $B$, $B$ starts on the second day of $A$, and they would have an overlap of 29 days, that is each window does some redundant analysis but produces smoother transitions in analysis between windows. Thus 1 comment in a bug report will have an effect on the graphs of 30 windows. This is similar to Hindle et al.’s [14] analysis of topics using windows but they did not use an overlap. We could thus see the changes in the trend of a participant’s activity. Moreover, time windowed analysis could give a more accurate and nuanced view of the data [3, 14], as locally central participants would not be “drowned out”. For instance, if a bug participant participates in many bug reports and bug comments during one month, he would be one of the most central participants with a high betweenness within this window. However, if he appeared only for that month, globally, he would have low betweenness and would not show up as central, even though during a shorter period he played a vital role. As we can see in Figure 2, the left column graph shows the betweenness values of participants over the entire time period; local details are missed and we get nothing about the trend, compared to the right part of the results from overlapping windowing. For example, cluster 8 on Figure 2 is bright and important at the start of our analysis but does not appear in the global graph on the left. Also, if there is a very sharp drop of values of a certain participant, the overlapping windows would give a more nuanced view of the change and what was happening. Another point is that, comments on the same bug might not be globally temporally relevant [17, 18] thus a global time analysis would not make much sense in this case. This could happen if new changes induce new bugs or modify the behaviour of a reported bug. Figure 2: Betweenness centrality along time line: the x-axis represents the number of time windows and the starting dates are denoted every 100 windows. The y-axis represents the number of bug participants who have ever been central in the bug community with betweenness centrality valued greater than 0 for some time period. The color represents the value of betweenness centrality, with darker colors corresponding to lower betweenness and lighter colors for higher betweenness. We used K-means clustering with cosine distance where K = 100 ### 2.3 Clustering Figure 3(1) orders participants by their betweenness values. We can indeed find that there are participants of low overall betweenness but being very active (show up bright) at some time points, and this supports the necessity of windowing, as stated in Section 2.2. However, we need more information about participants’ working patterns and get an idea about their being interacting groups. In order to perceive clusters, that are local groups of interactions, we clustered the bug participants using K-means by their betweenness centrality distribution along the time line. K-means is one of the most popular clustering methods which aims to partition $n$ data items into $k$ clusters that each data item belongs to the cluster with the nearest mean [19]. We choose to use K-means with cosine distance. The cosine distance between two vectors is defined as, $cosine\\_dist(A,B)=1-\frac{A\cdot B}{\Arrowvert A\Arrowvert\Arrowvert B\Arrowvert}$ (3) where A and B are two vectors, $\cdot$ represents the inner dot operation and $\Arrowvert\cdot\Arrowvert$ indicates the module of the vector. With clustering, authors with similar temporal centrality would be grouped together so that bug participants with similar activity patterns are also grouped together. In this paper, we choose the K-means with cosine distance because it gives a better visual clustering result, as compared in the plots of Figure 3. In this case, cosine distance calculates the similarity between each pair of participants in terms of temporal centrality whereas Euclidean distance focuses on the magnitude of data, the size and frequency of centrality. Moreover, we used $k=100$ for the K-means, to cluster authors. There is a trade off between the size of clusters and the variance within each cluster. As we can see from Figure 3(3), $k=10$ also gives good visualization result, but considering the number of more than 1600 participants, we should get a larger $k$ to keep the diversity of working groups in similar. We set $k=100$ in this case, since we get the aesthetically best visualization (our subjective opinion based on visible clusters) of all the data; we had tried other values of $k$ such as 5, 10, 25, 50, 75, 200, 300. ## 3 Methodology Our methodology consists of six steps that deal with raw data, construct graphs and apply social network analysis with sliding windows. We conduct a thorough case study of the Android bug repository with the proposed method and validate the conclusions from the results by mining the Android version control system and inspecting the release history. ### 3.1 Data With the provided Android bug repository 2012 and the Android version control system from the MSR Challenge [12], we converted and stored the XML format data into a database for efficient analysis using Microsoft SQL Server Business Intelligence. Our analysis focused on the bug records of the previous two years from January 1st 2010 to December 4th 2011 since during these two years, there are more records in the repository as we counted that participants are more active; also, the activities are representative, both the Android platforms and their developer groups are larger and more diverse during the latest two years and it was also more relevant to modern Android handsets. The data we used of these two years covers 14,432 out of 20,169 total bug records and 46,806 out of 67,730 total bug comments from the whole dataset. Related to these bug and comment records, there are 30,969 people who have either reported a bug or made comments on a bug. The bug and comment records are grouped into 30-day windows sliding by 1 day. We extracted 673 windows in total from the bug reports during year 2010 and 2011. ### 3.2 Windowing Bug Reports and Extracting Social Networks Methodology We windowed the data and constructed networks that indicated the relations among the participants within each specific time window. For each window, we calculated the betweenness centrality of each participant and we plotted the centrality values per participant in a visualization. The steps of our methodology are explained as following: Step 1: Pruning the data. We pruned the records of the reporters and commenters into a pure name format, which are originally recorded in semi- anonymous email formats in the XML repository dump. For example, given the original email address which is represented by “mathias…[email protected]”, we truncate the string starting from “….” and keep the front part “mathias” at the beginning as the name of the reporter or commenter. This strategy could lead to name aliasing problem, especially for common names or email addresses starting at just a simple letter like “e…[email protected]”. Although algorithms have been provided to reduce the extent of the problem, [20], [2], [21], it is difficult or even impossible to eliminate the influence from this data quality issue. When applied to other repositories that do not anonymize this would be less of a problem. Hence, we focus on participants whose names are less common and less ambiguous in our study. Step 2: Windowing the records. We windowed the data into periods of 30 days with a 29-day overlap. 30 days was chosen as a window size because it is smaller than the periods between a major and minor release, it is similar to a month of work, but long enough to contain the resolution of multiple bugs. We have compared sliding by 1 day with our previous result of sliding by 7 days, 1 day sliding produces gradual and smoother transitions of centrality. Step 3: Establishing the network. We made a tool to perform the SNA with sliding windows. The tool is implemented in Java and built on top of the JUNG Graph Framework, that converted bug reports and bug comment records within a window to a social network graph. The nodes of these networks represent participants who have either reported some bugs or made comments on bugs. The edges represent connections between two nodes. All the edges are weighted. For a bug within a selected time window, whenever a person makes a comment on this bug, the edge between the bug commenter and the bug reporter would get weight plus one, as well as the edges linking to the participants who previously made comments on this bug. Bug reports or comments in different windows would have separate network graphs depending on the activity of their reporters or commenters. An example in Figure 1 indicates how the weighted network graph is built. Step 4: Calculating the centrality. We calculated the betweenness centrality using JUNG, and normalized the centrality with the number of node pairs, as in Equation (2). We then get a list of all the bug participants and their betweenness centrality values for the total 673 overlapping windows. Step 5: Removing irrelevant participants. We removed the participants with betweenness centrality value 0, who might have either reported a bug/bugs with no comments, or made the only comment on a bug so that no other participants are related. Afterwards, we get 1654 participants with betweenness centrality value larger than 0, out of the 30969 in total. Step 6: Generating the analysis graph. The activity of each bug participant is represented by a 673 dimensional vector representing their betweenness per window. Each element of the vector indicates the betweenness centrality value extracted from the graph, which is generated from the window for that specific time period (in our case, the specific time period is 30 days starting from the date of the window start point). Then we clustered all the vectors by using K-means ($k=100$) with cosine distance to 100 clusters. Finally, we plotted the results, as shown in Figure 2 to visualize the clustered data so that we could easily analyze our results. ### 3.3 Validation using the Android Release History and the Git In addition to the methodology of mining the Android bug repository, we made use of the git version control repository and inspected the release history highlights to validate the purpose behind the clusters and patterns we observed. We looked into the participants who contributed to the git repository in order to find their areas of expertise and validate our analysis conclusion about how the community participants act in accordance with the project development. The types of files modified and the corresponding projects are highly correlated with the specialization of those who commit changes. For instance, if a developer always submits kernel related code files, he is more likely to be specialized in kernel techniques. Types of files include document files, test files, source files, etc; dictionary paths of files usually indicate what projects the files belong to. We manually identified the participants’ areas of expertise by observing the _project_ and the _target_ for all of their commits (such as source code or documentation). To give a specific example, if there were commits from a developer, Mr.Guilfoyle, on the target file media/java/android/media/Ringtone.java under the project platform_frameworks_base; then, we would suggest that Mr.Guilfoyle likely has some specialized knowledge about the platform’s ringtone. Thus this is how we derive participant expertise [3]. Also, we could further relate their expertise to their centrality patterns. The Android release history could, on the other hand, help to relate the release highlights to participants central behaviour during that release. Further validation is discussed in Section 5. Figure 3: Betweenness centrality along time line. Participants on the y-axes are ordered differently by betweenness values or various clusterings. ## 4 Results and Analysis We study the results shown in Figure 2. Each horizontal line represents the 673 betweenness centrality values for the selected bug participant during year 2010 and 2011\. In total, we have 1654 bug participants. By studying these results, we answered the following questions: Figure 4: Number of active participants across time. Figure 5: Sum of betweenness centrality of participants across time. ### 4.1 Global Analysis RQ1. How does the number of active bug participants change over time? Why? To give an overview, we compared the interaction of bug participants between January, 2010 and December, 2011, and found that the interaction among participants in the Android bug community in 2011 was similar to the interaction of participants in 2010 but more frequent, as we can see in Figure 2. One “gap” occurs around window 300, which we will explain in the next Local Analysis subsection. Correspondingly in Figure 4, that we counted the number of participants with betweenness centrality value larger than 0 within each window, the number of active participants during 2011 is slightly larger than that of 2010. Figure 5 shows the sum of betweenness values along the two years’ time line, we can see that the trend is very similar to that of the number of active participants in Figure 4. This also suggests that the betweenness centrality reflects the interaction among participants. Moreover, a possible reason for the changes of the number of active bug participants and the betweenness centrality values is that around major or minor releases of SDKs, API fixes or improvements, participants seem to become more active in bug reporting, discussing and fixing activities. Also, during these time periods, bugs are more likely to be discovered and reported. Perhaps the pressure of the release is causing developers to address outstanding bugs more than usual. After a release, users also take part in the activity of discovering the bugs and problems so that in this case both users and developers would like to discuss the bugs. RQ2. How does the betweenness centrality of a participant change over time? What are the reasons when they have a certain activity pattern? Observing the continuity of betweenness centrality in Figure 2, some participants have kept active during the entire two years, and correspondingly they have a very continuous and bright line. For participants of this type, there are a few possible explanations. First, our conjecture is that these participants are professional developers who belong to the core development team so that what they reported are more important issues which attract more participants to discuss and fix them. Their identities of being professional developers will be discussed in Section 5.1. Second, some of these participants are of high community status or expertise, and they might supervise and guide the development of the project. For example, when we validated, we did find one developer, romainguy, who has experiences on almost every component relevant to platforms so that he can be considered to be an expert. Developers related to these continuous lines are listed in Table 1, and we will further discuss and validate on them in Section 5.1. However, in most cases, participants’ betweenness values are highly variant, as observed in Figure 2. To investigate the variation in betweenness values over time, we decided to count the number of times that a user experienced a range of consecutive windows in which the user had non-zero betweenness. Participants with a count of distinct ranges greater than 1 would be _phasers_ who periodically participate within the Android bug community. Here phasers are those who phase into centrality and later out of it. These randomly phasing participants (phasers) are very likely to acquire less expertise or have lower community status in their community, than those with continuous high centrality. Phasers might be interested in limited topics and only central and active during the appearance of bugs relevant to those topics. Participants who only had 1 distinct range of betweenness are considered to be participants who only appeared once, and are probably users. We validate the roles these participants play in Section 5.1. To summarize, among the 1654 participants with betweenness values larger than 0, we analyzed their centrality patterns and divide them into three categories: 1) participants appeared only once with a betweeness greater than 0 (71 out of 1654 participants), 2) participants recurred periodically (1575 participants) and 3) participants who are central along the entire project history (8 participants). ### 4.2 Local Analysis RQ3. Are there special time ranges during which participants are more/less active or central than normal? Why? By inspecting the Android release history highlights, we found that the v2.1 SDK was released on 12 January 2010, which corresponds to the first peak value in Figure 5. Android v2.2 SDK was released on 20 May 2010 and this corresponds to peak 2. From Dec. 2010 to the beginning of Mar. 2011, several minor updates were released and on 22 Feb. 2011, one major update v3.0 SDK was released. These releases explain the summit, i.e., peak 3, in Figure 5. This is correlated with more participation at the same time. In addition, during the first obvious “gap”, which covers the time from October 2010 to the end of 2010 (around window 300), the social network during this time period is almost inactive and even “quiet”. There were fewer releases during the “gap”. The other low value showing up in the end of Figure 5 results from the fact that there are no bug reports recorded (right tail censoring) in the given dataset. This piece of data is still meaningful because it contains comments belonging to bug reports several weeks or months before. The betweenness value is thus simply calculated by the comments here. RQ4. What are the possible scenarios for a very sharp change of the participants’ centrality? Why? Considering individual participants, almost all of them has experienced centrality oscillations. In addition, some participants tend to become active and core members during the same time period and then they fade away together. We suspect that the phasers tend to be interested in one or several categories of problems so that they appear only along with the occurrence of these issues. They take part in activities related to the bugs or technical issues and become inactive after the problems are solved. Or in the case when they are working on a project, they would become inactive when the projects are finished. As showed in Figure 2, the participants’ tend to get clustered together around important releases, which supports that the phasers are working along with projects or related issues. Meanwhile, by observing the clustered participants of their activity patterns in Figure 2, we suspect that the phasers that show up densely together could be interested in similar categories of topics. This assumption is validated in Section 5.2. ## 5 Validation We made use of the git repository and inspected the release history to validate our answers to the research questions in the previous section. For RQ1, it could only get answered based on assumption and the number of active participants across time as we counted in Figure 4, but not thoroughly validated. RQ3 is intuitively answered when we match the betweenness distribution with the release history by time, and no further validation is needed. For RQ 2 and RQ4, we have made a detailed validation in this section. ### 5.1 Activity pattern validation From the mining results, among the 1654 participants with betweenness values larger than 0, we notice that there is a small group of participants who have been central for most of our analysis period (8 participants out of the 1654); another relatively larger group appear without any recurrence (71 participants out of the 1654); the majority would periodically become central in their community (1575 participants out of the 1654). Based on the three activity patterns proposed in RQ2, we confirmed many of our previous suspicions: #### 5.1.1 Participants that appeared only once tend to be pure users We look into the git repository to find the files submitted by the 71 participants who have appeared only once in the bug community. We found that only 7 of them have ever committed a change, which means that these 7 are developers rather than pure Android users. The rest do not have commits in the version control system. This verifies our assumption that participants appeared only once in the bug community would more likely to be pure users, as introduced in RQ2. #### 5.1.2 Participants showed up periodically should be a combination of users and developers Periodically appearing participants are the majority and we call them phasers. Based on the methodology in Section 3, we looked into the commit history in the git repository in order to verify the expertise of phasers. With as many as 1575 participants, we sampled 156 participants. $21.8\%$ of the sampled participants were developer phasers, who have submitted changes. We studied the expertise of the developer phasers from this sample. All except two of them have worked on specialized tasks that implied some specific kind of expertise or specialization. The rest $78.2\%$ have never submitted files to the development community. They are probably users of Android. Thus, phasers consist of both users and developers. This answers to our assumption of the phasers’ role in RQ2. Table 1: 5 continuously central participants who have submitted changes to the git. Participant \| #Submitted-_changes \| Related Project ---\|---\|--- fadden \| 1259 \| device_samsung_crespo, platform(bionic, build, dalvik, etc.) xav \| 3501 \| platform(frameworks_base, build, external_bouncycastle, etc.), device_sample, mbligh \| 80 \| kernel(common, experimental, linux-2.6, msm, omap, qemu, samsung, tegra) ralf (Ralf.-Hildebrandt) \| 665 \| kernel(common, experimental, linux-2.6, dalvik, external_libpng, sdk,system_core, etc.) romainguy \| 1455 \| device_htc_passion, device_samsung_crespo, platform(build, cts, dalvik, development, external_bouncycastle, libcore, ndk, apps(AccountsAndSyncSettings, AlarmClock, Bluetooth, Browser, Calculator, etc.), inputmethods(LatinIME, iOpenWnn, PinyinIME, CalendarProvider), providers(DownloadProvider, GoogleSubscribedFeedsProvider), wallpapers(Basic, LivePicker, MagicSmoke, MusicVisualization), prebuilt, sdk, system_core) #### 5.1.3 Participants who were continuously central for a long time period could have multiple areas of expertise 5 out of the 8 participants in this group have submissions in the git. We extracted the projects these 5 participants have submitted changes to, as listed in Table 1. (On the forth row, ralf and Ralf.Hildebrandt are email alias of the same person, as we observed that the author_name attributes are the same for the two email alias.) Firstly, considering the number of changes they made, all of them except mbligh have more than 500 commits within the git, which means that they are quite active in Android development community. This supports that they are experts or advanced developers since more submissions indicates a broader range knowledge about the related techniques. Moreover, fadden, xav, and romainguy are all working on the platform layer, which includes build, dalvik, development, framework base, libcore, sdk, etc. All of their areas of expertise are related to the platform layer or system core layer. The participant romainguy has experiences modifying almost every component relevant to platforms, including both the apps and the core, and hence should be considered as Android platform development leader. Furthermore, when investigating these continuous lines we found some participants were Google employees, for example, two developers with alias mbligh and romainguy. Their email account recorded in the git repository is from the “google.com” domain, and moreover, when we googled them, they are indeed introduced as software engineers at Google. To summarize, this subsection demonstrates that three different centrality patterns correspond to participants of three categories, which supports our analysis hypothesis about activity patterns in Section 4. Table 2: 5 clusters we have chosen, out of a total number of 21. Cluster \| Time ---\|--- 1 \| May 16, 2010 - Jun. 24, 2010 2 \| Jun. 2, 2010 - Jul. 24, 2010 3 \| Jan. 13, 2011 - Mar. 3, 2011 4 \| Dec. 3, 2010 - Jan. 31, 2010 5 \| Feb. 4, 2011 - May 1, 2011 Table 3: Participants and their areas of expertise in cluster No. 4 ID \| Name \| Areas Of Expertise ---\|---\|--- 1 \| charles \| kernel - sound, kernel_linux-2.6 2 \| jasta00 \| ringtone, media 3 \| kristoff \| driver(net, video, serial, input) 4 \| rik(rik.bobbaers) \| kernel_linux-2.6(mlock) 5 \| rik(rikard.p.olsson) \| kernel_linux-2.6(arm) 6 \| rik(riku.voipio) \| kernel_linux-2.6(arm), driver 7 \| snp \| platform sdk(eclipse plugin) Table 4: Participants’ common areas of expertise of each cluster. Participants number is counted as the number of participants within each cluster who has ever submitted a change and appeared in the git, ie., developers. Cluster \| Participants Number \| Areas Of Expertise ---\|---\|--- 1 \| 5 \| netfilter, driver(video), tests, MIPS 2 \| 13 \| driver(usb, wireless, mouse), sound, net, i386, performance(tools), input methods 3 \| 9 \| sound, driver, frameworks_base, tests, platform, kernel 4 \| 7 \| sound, media, kernel_linux-2.6, driver, platform sdk, kernel video/serial 5 \| 63 \| net(bluethooth, net driver, ipv$x$, kernel_linux-2.6), driver(dvd, media, usb, gpu, net), ia64, sound, tests Table 5: Highlights of identified clusters from Figure 2 Release \| Time \| Highlights \| Related cluster ---\|---\|---\|--- v2.2 \| May 20, 2010 \| camera and gallery, portable wifi, multiple keyword language, performance(general, browser), media framework, Bluetooth, kernel upgrade, APIs(media, camera, graphis, data backup, device administrator, UI framework) \| 1 v2.2.1 \| Jan. 18, 2011 \| bug fixes(one is about root and unroot), security updates, performance improvements \| 3 v2.2.2 \| Jan. 22, 2011 \| fixed minor bugs, including SMS routing issues \| 3 v2.3 \| Dec. 6, 2010 \| UI refinements, faster text input, power management, NFC, multiple cameras, download management, new multimedia, new developer features(gaming, communication, multimedia, garbage collector, event distribution, video driver, input, native access-audio, graphics, storage, development), linux kernel upgrade to 2.6.36, Dalvik runtime, mixable audio effects \| 4 v2.3.3 \| Feb. 9, 2011 \| NFC, Bluetooth, Graphics, media, framework, speech recognition, voice search, API(identifier, build-in app, locales), emulator skins \| 5 v3.0 \| Feb. 22, 2011 \| UI design for tables, redesigned keyboard, improved text selection, copy and pase, connectivity options(USB, WIFI, media, keyboard, bluetooth), apps update, browser, camera and gallery, contacts, email, development support \| 5 ### 5.2 Cluster validation As we have discussed above, participants are more active around important releases. Moreover, we can observe from Figure 2 that participants’ centrality distributions tend to form into groups or clusters, that often are found around the releases. Participants belonging to the same group become central during the same time periods and then fade away together. We labeled 21 visible clusters from Figure 2 and looked into five of them which are located more around releases. The five clusters we chose are listed in Table 2. We extract changes submitted by the members of each cluster from the Android git. (For those who do not have records in the git, we regard them as pure users and do not consider them in this case). After inspecting their submissions, we would get an idea about what kind of tasks they have been mostly working on. Based on release history and the commit logs we found that these clusters tend to be coherent efforts undertaken by multiple kinds of participants. #### 5.2.1 Participants clustered together share similar areas of expertise and tasks Our analysis in Section 4 shows that the phasers that show up densely together could be interested in similar categories of topics or working on tasks related to the same area. As described in Section 3, we extract the targets and project names from the git for each member appeared within the cluster. The areas of expertise could be inferred by the contents of the targets and the topics of the projects. We summarized the areas of expertise of participant clusters (from Figure 2) in Table 4. Inspecting the areas of expertise, we find that each cluster has their own topics, which are relatively different from each other. Also, the topics of each cluster are concentrated to specific layers of Android’s architecture. For example, cluster No.1 covers techniques about net filters, drivers, tests, and MIPS, while cluster No.2 is about drivers for connecting devices (usb, wireless, and mouse), net, processor, and input. It is easy to tell that participants of these two clusters are working on different tasks. The other clusters could lead to the same conclusion. Thus we conclude that clusters often exist around a topic. Take cluster No.4 as an example. There are 7 developers contained in this cluster, as listed in Table 3. It can be observed that work of participants in this cluster could be generally divided into two groups: one is about the Linux 2.6 based kernel, another is related to multimedia. Charles, rik.bobbaers, rikard.p.olsson, and riku.voipio (the pruned bug reporter alias rik is related to three developers in the git and we look them all; this issue would be discussed in Section 6) are all modifying the Linux 2.6 kernel. Charles, jasta00, and kristoff are working on multimedia topic, which includes sound, video drivers, and ringtone. When we look into other clusters, we get similar conclusions. Thus, from the observation and analysis above, we can conclude that participants with similar centrality patterns often share similar areas of expertise and tasks. This validates our assumption about the phasers being clustered on specific techniques in RQ4. #### 5.2.2 Clusters’ working areas of expertise are in accordance with the release contents along the time line When observing the Android release history, we concluded that the overall betweenness centrality becomes higher around releases, and more active participants appear around important releases, at least according to Figure 4 and Figure 5. In addition, when taking participants’ areas of expertise into consideration, we find that the release highlights are in accordance with the areas of expertise for members of each cluster. Table 5 lists releases and their corresponding clusters together with the highlighted release contents. Comparing the release contents and the cluster areas of expertise, these two subjects are mostly matched on release topics and cluster’s working contents. For example, cluster No.4 covers from December 3, 2010 to January 31, 2011, which occurs before release v2.3. Participants in cluster No.4 have areas of expertise relevant to sound, media, and kernel-video, which match the release contents of new multimedia, APIs for native audio, and mixable audio effects in v2.3; We can also find that 4 out of 7 developers in cluster No.4 have worked on the kernel when the linux kernel was upgraded to 2.6.36 in Android v2.3. Cluster No.3 was centered around the releases of v2.2.1 and v2.2.2 (January 18, 2011 and January 22, 2011 respectively). Release 2.2.1 contained security updates and performance improvement; participants in cluster No.3 are specialized mostly on kernels or platforms. This occurs in cluster No.1 and its corresponding release v2.2 as well. Our conclusion is that participants’ work is relevant to areas of expertise associated with clusters, and at the same time, the clusters and participation tends to be correlated with releases. This further validates our answer to RQ4 that developers tend to work as groups on specific projects or issues they are specialized, and their centrality patterns are related to the occurrences of projects or issues. ## 6 Limitations In this study we explicitly trust that the same account of email addresses, i.e., the part before “@”, belongs to the same bug participant. With the given semi-anonymous email addresses in Android bug repository, we pruned the part starting from “….” and kept the front part as the names of bug participants. However, it is possible that some common names share the same start string. For example, “Benjamin Franzke”, “Benjamin Tissoires” and “Benjamin Romer” have the same first name. We cannot distinguish these names with the email address “Benjamin@XXX”. Besides, some of the email addresses start with a simple letter which is ambiguous identifying a person, while we analyze the results without excluding such data. We validate our analysis based on the assumption that the types and projects of submitted files reflect the areas of expertise that the developers are specialized in. Hence, we tagged the participants with the techniques according to their submitted files in the Android git. However, there could be inconsistency between the techniques and the submitted files. Our manual inspection increased the validity of the results, but it still relied on the authors judgment, interpretation and potential bias. ## 7 Conclusion and Future Work In this paper, we mined the Android bug repository and studied the data of 2010 and 2011. We combined overlapping time windows with social network analysis in order to analyze the participants interactions within the Android bug repository, as part of the Android open source community. We conducted a thorough case study of the bug reporter activity within the Android bug repository with our method. We analyzed the temporal evolution of the Android bug reporting community both globally and locally. We found that most minor or major releases lead to high betweenness centrality in general. We found and explained sharp changes of participants’ betweenness values and we inspected three activity patterns for the participants. Also, we found out that participants tend to get clustered into groups. Then, we validated these results by manually inspecting the Android version control system (git) and the Android release history highlights. We validated the three activity patterns of bug participants as well as their corresponding reasons. For participants who were clustered in same groups in our plots, they showed interest in a set of similar topics as we inspected in our validation. Thus we conclude that by combining the SNA with sliding windows, we were able to find many local interactions that would be lost in a global analysis. The sliding windows make these local collaborations more visible, instead of drowning them out in a global analysis. In this case, we can get a more accurate knowledge about participants’ working patterns as well as their group working. Furthermore, we validated our findings by inspecting other repositories to confirm that the local behaviour occurred and was of relevance. This work could be used by managers and researchers to produce project dashboards, and automated project status reports. Future work includes applying the approach in this paper to other open source projects’ repositories in order to improve its generality. We want to further validate if our overlapping time windowing SNA plots are trustworthy enough to depict the actual develop processes of various projects. ## References * [1] K. F. S. Wasserman, Social network analysis : methods and applications. Cambridge ; New York : Cambridge University Press, 1994. * [2] C. Bird, A. Gourley, P. Devanbu, M. Gertz, and A. Swaminathan, “Mining email social networks,” in Proceedings of the 2006 international workshop on Mining software repositories, MSR ’06, (New York, NY, USA), pp. 137–143, ACM, 2006. * [3] A. Meneely and L. Williams, “Socio-technical developer networks: should we trust our measurements?,” in Proceeding of the 33rd international conference on Software engineering, ICSE ’11, (New York, NY, USA), pp. 281–290, ACM, 2011. * [4] C. Bird, D. Pattison, R. D’Souza, V. Filkov, and P. Devanbu, “Latent social structure in open source projects,” in Proceedings of the 16th ACM SIGSOFT International Symposium on Foundations of software engineering, SIGSOFT ’08/FSE-16, (New York, NY, USA), pp. 24–35, ACM, 2008. * [5] G. Madey, V. Freeh, and R. Tynan, “The open source software development phenomenon: An analysis based on social network theory,” in Proceedings of the Americas Conference on Information Systems (AMCIS 2002), (Dallas, Texas), pp. 1806–1813, 2002. * [6] A. Sureka, A. Goyal, and A. Rastogi, “Using social network analysis for mining collaboration data in a defect tracking system for risk and vulnerability analysis.,” in ISEC (A. Bahulkar, K. Kesavasamy, T. V. Prabhakar, and G. Shroff, eds.), pp. 195–204, ACM, 2011. * [7] Y. Kamei, S. Matsumoto, H. Maeshima, Y. Onishi, M. Ohira, and K. ichi Matsumoto, “Analysis of coordination between developers and users in the apache community.,” in OSS (B. Russo, E. Damiani, S. A. Hissam, B. Lundell, and G. Succi, eds.), vol. 275 of IFIP, pp. 81–92, Springer, 2008. * [8] M. L. Bernardi, G. Canfora, G. A. D. Lucca, M. D. Penta, and D. Distante, “Do developers introduce bugs when they do not communicate? the case of eclipse and mozilla.,” in CSMR (T. Mens, A. Cleve, and R. Ferenc, eds.), pp. 139–148, IEEE, 2012. * [9] R. Abreu and R. Premraj, “How developer communication frequency relates to bug introducing changes,” in Proceedings of the joint international and annual ERCIM workshops on Principles of software evolution (IWPSE) and software evolution (Evol) workshops, IWPSE-Evol ’09, (New York, NY, USA), pp. 153–158, ACM, 2009. * [10] M. Pinzger, N. Nagappan, and B. Murphy, “Can developer-module networks predict failures?,” in Proceedings of the 16th ACM SIGSOFT International Symposium on Foundations of software engineering, SIGSOFT ’08/FSE-16, (New York, NY, USA), pp. 2–12, ACM, 2008. * [11] N. Bettenburg and A. E. Hassan, “Studying the impact of social structures on software quality,” in Proceedings of the 2010 IEEE 18th International Conference on Program Comprehension, ICPC ’10, (Washington, DC, USA), pp. 124–133, IEEE Computer Society, 2010. * [12] Y. K. E. Shihab and P. Bhattacharya, “Mining Challenge 2012: The Android Platform,” 2012. * [13] U. Brandes and T. Erlebach, Network Analysis: Methodological Foundations. Springer, 2005. * [14] A. Hindle, M. W. Godfrey, and R. C. Holt, “What’s hot and what’s not: Windowed developer topic analysis.,” in ICSM, pp. 339–348, IEEE, 2009. * [15] L. C. Freeman, “A set of measures of centrality based on betweenness,” Sociometry, vol. 40, pp. 35–41, March 1977. * [16] R. A. Hanneman and M. Riddle, Introduction to social network methods. University of California, Riverside, Riverside, CA, 2005. * [17] Y. H. Kidane and P. A. Gloor, “Correlating temporal communication patterns of the eclipse open source community with performance and creativity,” Comput. Math. Organ. Theory, vol. 13, pp. 17–27, Mar. 2007. * [18] B. P. T. Iba, K. Nemotob and P. Gloor, “Analyzing the Creative Editing Behavior of Wikipedia Editors: Through Dynamic Social Network Analysis,” vol. 2, no. 4, pp. volume = 13, number = 1, month = mar, year = 2007, issn = 1381–298X, pages = 6441–6456, year = 2010. * [19] J. B. MacQueen, “Some methods for classification and analysis of multivariate observations,” in Proc. of the fifth Berkeley Symposium on Mathematical Statistics and Probability (L. M. L. Cam and J. Neyman, eds.), vol. 1, pp. 281–297, University of California Press, 1967. * [20] G. Robles and J. M. Gonzalez-Barahona, “Developer identification methods for integrated data from various sources,” in MSR ’05: Proceedings of the 2005 international workshop on Mining software repositories, 2005. * [21] M. Goeminne and T. Mens, “A comparison of identity merge algorithms for software repositories,” Science of Computer Programming, Dec. 2011.	arxiv-papers	2013-10-28T15:56:25	2024-09-04T02:49:53.000167	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Feng Jiang, Jiemin Wang, Abram Hindle and Mario A. Nascimento", "submitter": "Vanessa Burke", "url": "https://arxiv.org/abs/1310.7469" }
1310.7661	# Reply on the comment on “Classical Simulations Including Electron Correlations for Sequential Double Ionization” Yueming Zhou, Cheng Huang, Qing Liao and Peixiang Lu ###### pacs: 32.80.Rm, 31.90.+s, 32.80.Fb In our Letter Zhou , we studied sequential double ionization (SDI) of Ar by the elliptically polarized laser pulses. In Ref. Pfeiffer , Pfeiffer et al. shown that the independent-tunneling theory failed in predicting the ionization time of the second electron. Meanwhile, the quantum calculation of the fully correlated two-electron atom at the experimental conditions is currently not feasible. Thus, we resorted to a classical method. With a classical correlated model, we demonstrated that the experimentally measured ionization times for both electrons are quantitatively reproduced. Because of the autoionization of the classical two-electron system, in our Letter Zhou we employed the Heisenberg-core potential to avoid this problem. The Heisenberg-core potential is written as Kirschbaum : $V_{H}(r_{i},p_{i})=\frac{\xi^{2}}{4\alpha r_{i}^{2}}exp\\{\alpha[1-(\frac{r_{i}p_{i}}{\xi})^{4}]\\}$ (1) where $r_{i}$ and $p_{i}$ are the position and the momentum of the $i$th electron. There are two parameters in the Heisenberg-core potential: $\alpha$ and $\xi$. In our letter, we chosen $\alpha=2$ and we claimed that our results did not depend on $\alpha$. In the preceding Comment Chandre , Chandre et al. argued that the excellent agreement between our calculation and experimental data is a coincidental result of the parameters we chose. Here, we demonstrate that our calculations are structurally stable and do not depend on the parameter $\alpha$, and explain why in the calculations of Chandre et al. Chandre the time delay between the two successive ionizations changes with $\alpha$. As addressed in our Letter Zhou , a necessary condition for quantitative description of SDI is that the first and the second ionization potentials of the model atom should be matched with the realistic target. In our classical model, this condition is satisfied by the combined action of the parameters $\alpha$ and $\xi$. In fact, the parameter $\xi$ is not free when $\alpha$ is given. It is set to make the minimum of the one-electron Hamiltonian $H=\frac{-2}{{\bf r}_{1}}+\frac{{\bf p}_{1}^{2}}{2}+V_{H}{(r_{1},p_{1})}$ equal to the second ionization potential of $Ar^{+}$. It can be determined by eq. (3) of Chandre . For Ar, the value of $\xi$ is 1.259 when $\alpha=2$. In our Letter Zhou , we chosen $\xi=1.225$ because for $\xi=1.259$ one could not place the two electrons in phase space with the ground-state energy of Ar (-1.59 a.u). For the parameter $\xi=1.225$ and $\alpha=2$, the second ionization potential of the model atom is -1.065 a.u., very close to second ionization potential of $Ar^{+}$ (-1.02 a.u.). This small difference of ionization potential between our model and the realistic target has negligible influence on our results. In order to keep the second ionization potential unchanged for different values of $\alpha$, the value of $\xi$ should be adjusted. Figure 1(a) shows the value of $\xi$ that keeps the second ionization potential (-1.065 a.u.) unchanged for different values of $\alpha$. Figure 1(b) shows the time difference between the ionizations of the two electrons for different values of $\alpha$ [where the corresponding value of $\xi$ is shown in Fig. 1(a)]. Obviously, the results do not change with the parameter $\alpha$. Figure 1: (a) The value of $\xi$ that keeps the ionization potentials unchanged when $\alpha$ varies. (b) The time delay between the two successive ionizations in SDI for the parameters shown in (a). The laser intensity is 4.0 PW/cm2 and the pulse duration is 33 fs. (c)(d) The first and second ionization potentials as a function of $\alpha$ for the keeping $\xi=1.225$. In Chandre , Chandre et al. changed $\alpha$ while kept $\xi$ unchanged. In this treatment, the first and second ionization potentials change significantly as $\alpha$ varies, as shown in figs. 1(c) and 1(d). When $\alpha=2$, the ionization potentials nearly equal to the realistic target. However, when $\alpha$ becomes larger, the first ionization potential increases and the second ionization potential decreases. Naturally, the time difference between the ionizations of the two electrons will increase as $\alpha$ increases. Thus, the change of the time difference for different values of $\alpha$ in Chandre originates from the shifting of the first and second ionization potentials of the model atom. In order to describe SDI accurately, it is necessary to make the ionization potentials of the model atom match with the investigated target in any case. In our model Zhou , the parameters $\alpha$ and $\xi$ should appear in pair to make the ionization potentials unchanged for various value of $\alpha$. The results are structurally stable when this condition is satisfied. As addressed in our Letter Zhou , the Heisenberg-core potential in our model was added to (I) make the first and the second ionization potentials match with realistic target and (II) avoid autoionization while keep the two electrons being fully correlated during the entire ionization process, which enables us to investigate the multielectron effect in strong field ionization (see Ref. Zhou2 for example). Recently, the ionization times of both electrons in SDI are also reproduced by the soft-core potential model when the ionization potentials are artificially adjusted to those of the target Wang . It indicates that the success of the classical methods do not depend on the details of potential, making it easy to be accepted that our calculations are stable upon the parameters of the Heisenberg-core potential. The detail of this issue was detailedly addressed in our recent paper zhou3 . Yueming Zhou1, Cheng Huang1, Qing Liao1 and Peixiang Lu1,2 1Wuhan National Laboratory for Optoelectronics and School of Physics, Huazhong University of Science and Technology, Wuhan 430074, P. R. China 2Key Laboratory of Fundamental Physical Quantities Measurement of Ministry of Education, Wuhan 430074, P. R. China ## References * (1) Y. Zhou, C. Huang, Q. Liao, and P. Lu, Phys. Rev. Lett. 109, 053004 (2012). * (2) A. N. Pfeiffer, C. Cirelli, M. Smolarski, R. Dörner, and U. Keller, Nature Phys. 7, 428 (2011). * (3) C. L. Kirschbaum and L. Wilets, Phys. Rev. A 21, 834 (1980). * (4) C. Chandre, A. Kamor, F. Mauger, and T. Uzer, comment on “classical simulations including electron correlations for sequential double ionization”. * (5) Y, Zhou, C. Huang, and P. Lu, Opt. Express 20, 20201 (2012). * (6) X. Wang, J. Tian, A.N. Pfeiffer, C. Cirelli, U. Keller and J.H. Eberly, arXiv:1208.1516v1 (2012). * (7) Y. Zhou, Q. Zhang, C. Huang, and P. Lu, Phys. Rev. A 86, 043427 (2012).	arxiv-papers	2013-10-29T01:50:36	2024-09-04T02:49:53.018749	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Yueming Zhou, Cheng Huang, Qing Liao and Peixiang Lu", "submitter": "Peixiang Lu", "url": "https://arxiv.org/abs/1310.7661" }
1310.7702	# Quasi free-standing silicene in a superlattice with hexagonal boron nitride T. P. Kaloni, M. Tahir, and U. Schwingenschlögl [email protected],+966(0)544700080 Physical Science & Engineering Division, KAUST, Thuwal 23955-6900, Kingdom of Saudi Arabia ###### Abstract We study a superlattice of silicene and hexagonal boron nitride by first principles calculations and demonstrate that the interaction between the layers of the superlattice is very small. As a consequence, quasi free- standing silicene is realized in this superlattice. In particular, the Dirac cone of silicene is preserved, which has not been possible in any other system so far. Due to the wide band gap of hexagonal boron nitride, the superlattice realizes the characteristic physical phenomena of free-standing silicene. In particular, we address by model calculations the combined effect of the intrinsic spin-orbit coupling and an external electric field, which induces a transition from a metal to a topological insulator and further to a band insulator. Graphene is a zero gap semiconductor with very weak spin-orbit coupling (SOC) geim . Since its discovery a lot of efforts have been undertaken to engineer a finite band gap, but no satisfactory progress could be achieved. Silicene is closely related to graphene, as they share the same two-dimensional honeycomb structure, and has been proposed as a potential candidate for overcoming the limitations of graphene due to its buckled structure and much stronger SOC. Silicene has first been reported by Takeda and Shiraishi takeda and investigated in more detail in Ref. verri . While C and Si belong to the same group in the periodic table, Si has a larger ionic radius, which promotes $sp^{3}$ hybridization. The mixture of $sp^{2}$ and $sp^{3}$ hybridization in silicene results in a prominent buckling of 0.46 Å, which can open an electrically tunable band gap falko ; Ni . On the other hand, the band gap induced by the intrinsic SOC was found to amount to 1.6 meV yao . First principles calculations have confirmed that the stable structure of silicene is buckled olle . Similar to graphene, the charge carriers in silicene are expected to behave like massless Dirac fermions in the $\pi$ and $\pi^{}$ bands, which form a Dirac cone at the K-point. The electronic properties of halogenated and hydrogenated silicene have been studied by first principles calculations in Refs. houssa ; wei and the effect of different substrates on the Dirac cone have been analyzed in Refs. new1 ; new2 ; new3 . Growth of silicene and its derivatives experimentally has been demonstrated for different metal substrates padova ; vogt ; ozaki . Silicene on a ZrB2 thin film shows an asymmetric buckling due to the interaction with the substrate, which leads to the opening of a band gap. However, accurate measurements of the materials properties are difficult on metallic substrates. In addition, metallic substrates screen externally applied electric fields and therefore prohibit manipulation of the electronic structure. For this reason, it would be desirable to achieve free-standing silicene. However, free-standing silicene probably is instable against a transition into the silicon structure. A possible solution can be a superlattice that stabilizes the two-dimensional structure of silicene but still is characterized by a small interaction to the second component so that the Dirac states are not perturbed. In the following we will substantiate this idea by first principles calculations. Due to an identical honeycomb structure, the superlattice of silicene and hexagonal boron nitride appears to be a promising choice. In addition, hexagonal boron nitride is a wide band gap semiconductor and therefore makes it possible to study the effects of an external perpendicular electric field applied to silicene. Because of the remarkably buckled structure, the intrinsic SOC gap of silicene can be enhanced by a perpendicular external electric field. Hence, we will study the electronic structure of silicene under an electric field $E_{z}$ using band structure calculations as well as an analytical model. Our calculations are carried out using density functional theory in the generalized gradient approximation. Specifically, we employ the Quantum- ESPRESSO package paolo . The van der Waals interaction grime ; kaloni as well as the SOC are taken into account. A finite electric field is applied using the scheme described in Refs. Bengtsson ; Meyer . The calculations are performed with a plane wave cutoff energy of 816 eV. Furthermore, a Monkhorst- Pack $8\times 8\times 1$ k-mesh is employed for optimizing the crystal structure and a refined $30\times 30\times 1$ k-mesh is used afterwards to increase the accuracy of the self-consistency calculation. The supercell employed in our superlattice calculations comprises one layer of hexagonal boron nitride (18 atoms in a $3\times 3$ arrangement) and one layer of silicene (8 atoms in a $2\times 2$ arrangement). The resulting lattice mismatch is small (2.8%) and comparable to that of the frequently studied superlattice between graphene and hexagonal boron nitride kaloni ; Yankowitz ; Giovannetti ; dean . We have fully relaxed the lattice parameters of the supercell, finding values of $a=b=7.56$ Å and $c=7.77$ Å. An energy convergence of 10-8 eV and a force convergence of $4\cdot 10^{-4}$ eV/Å are achieved. Figure 1: Superlattice of silicene (top) and hexagonal boron nitride (bottom) viewed along the hexagonal $b$-axis. The structural arrangement of the superlattice under study is depicted in Fig. 1, showing silicene and hexagonal boron nitride layers that alternate along the $z$-axis. We have also studied superlattices with hexagonal boron nitride slabs of varying thickness. However, since it turns out that this thickness has hardly any influence on the silicene electronic states, in particular the charge transfer between the two component materials, we will focus in the following on the case of one layer of hexagonal boron nitride alternating with one layer of silicene. Our structural optimization results in a Si–Si bond length of 2.26 Å and a buckling of 0.54 Å in the silicene layer. The latter value is slightly but not significantly higher than the predicted value of free-standing silicene yao ; cheng . The bond angle between neighboring Si atoms amounts to 114∘, which agrees well with the value of 116∘ in free- standing silicene. For the interlayer distance between the silicene and hexagonal boron nitride layers we obtain a value of 3.35 Å, resembling the distance of a silicene layer from $h$-BN new1 ; new2 ; new3 . The presence of a Dirac cone has been claimed for silicene grown on metallic substrate but there is still an ongoing discussion about the validity of this claim padova ; vogt ; ozaki ; Lin . Because of the large band gap of hexagonal boron nitride, we do not expect B or N states in the vicinity of the Fermi level in the case of our superlattice, so that the situation is much less involved. The band structure obtained from our calculations is shown in Fig. 2. We observe indeed a well preserved Dirac cone with a SOC gap of 1.6 meV. Analysis of the partial densities of states (not shown) clearly demonstrates that the Dirac cone traces back to the $p_{z}$ orbitals of the Si atoms, while contributions of the B and N atoms are found above 0.6 eV and below $-1.0$ eV only, with respect to the Fermi energy. We note that the observed Dirac cone is slightly shifted such that the Dirac point does not fall exactly on the Fermi energy. It appears at an energy of about 0.04 eV, i.e., the silicene is slightly hole doped. The energetical shift of the Dirac cone can be attributed to a tiny charge transfer between the silicene and the hexagonal boron nitride. Quantitative analysis shows that the silicene layer loses 0.06 electrons per 8 atoms. However, besides this small effect (which can be overcome by a minute doping), the charactersitics of the silicene Dirac cone are perfectly maintained in a superlattice with hexagonal boron nitride. In the following we will therefore study the effect of an external electric field on free-standing silicene to describe the properties of the superlattice. In Ref. falko the role of the intrinsic SOC and external electric field for the opening of a band gap have been discussed. The electric field breaks the sublattice symmetry, which induces a finite band gap. The intrinsic SOC has the same effect. Our calculations (for an ideal buckling of 0.46 Å) show that the SOC ($E_{z}=0$) on its own results in a band gap of 1.6 meV, which is consistent with the previously reported value in Ref. falko . To obtain the same gap by an electric field (without SOC) a value of $E_{z}=11.2$ meV/Å is needed, see Fig. 3(a). Figure 2: Electronic band structure obtained for the superlattice of silicene and hexagonal boron nitride. From an application point of view, the combined effect of SOC and electric field is of great interest. We therefore vary $E_{z}$ relative to the fixed SOC. Band structures obtained for three different values of the electric field are shown in Figs. 3(b) to (d). For $E_{z}=3.1$ meV/Å, see Fig. 3(b), we find energy gaps of 1.3 and 7 meV between the minority and majority spin bands, respectively. When we increase $E_{z}$ to 3.6 meV/Å the obtained energy gaps change to 1.1 and 9 meV, which we will explain later by our analytical model. A stronger electric field of $E_{z}=11.2$ meV/Å leads to energy gaps of 2.9 and 20 meV. Further enhancement of the electric field results in a almost linear increase of the energy gaps. The observed dependence of the energy gaps on the electric field is much stronger than reported previously falko ; Ni , because we take into account the SOC. Our results show that there is no spin degeneracy and a finite band gap, which is a combined response of SOC and electric field. In addition, Figs. 3(b) to (d) demonstrate phase transitions from a metal to a topological insulator and further to a band insulator. The electric field required to obtain a reasonable band gap is found to be much smaller than typical fields considered before, which means that the device can be operated in a stable regime at low voltage. Figure 3: Electronic band structure of free-standing silicene: (a) with SOC and $E_{z}=0$ or without SOC and $E_{z}=0.0112$ V/Å, (b-d) with SOC and different values of $E_{z}\neq 0$. In order to discuss the mechanisms behind the above observations, we consider an analytical model. We assume that the silicene sheet lies in the $xy$-plane in the presence of intrinsic SOC and an external electric field in $z$-direction. Silicene can be described by the two-dimensional Dirac-like Hamiltonian $H_{s}^{\eta}=v(\eta\sigma_{x}p_{x}+\sigma_{y}p_{y})+\eta s\lambda\sigma_{z}+\Delta\sigma_{z},$ (1) where $\eta=+1/{-}1$ denotes the $K$/$K^{\prime}$ valley, $s=+1/{-}1$ denotes spin up/down, $\Delta=2lE_{z}$ with $l=0.23$ Å is the electric field, ($\sigma_{x}$, $\sigma_{y}$, $\sigma_{z}$) is the vector of Pauli matrices, $\lambda$ is the strength of the intrinsic SOC, and $v$ is the Fermi velocity of the Dirac fermions. For the K valley we have $H_{+1}^{K}=v\left(\begin{array}[]{c}+\lambda+\Delta\\\ +p_{x}+ip_{y}\end{array}\begin{array}[]{c}+p_{x}-ip_{y}\\\ -\lambda-\Delta\end{array}\right),\text{ \ }H_{-1}^{K}=v\left(\begin{array}[]{c}-\lambda+\Delta\\\ +p_{x}+ip_{y}\end{array}\begin{array}[]{c}+p_{x}-ip_{y}\\\ +\lambda-\Delta\end{array}\right)$ (2) and for the K′ valley $H_{+1}^{K^{\prime}}=v\left(\begin{array}[]{c}-\lambda+\Delta\\\ -p_{x}+ip_{y}\end{array}\begin{array}[]{c}-p_{x}-ip_{y}\\\ +\lambda-\Delta\end{array}\right),\text{ \ }H_{-1}^{K^{\prime}}=v\left(\begin{array}[]{c}+\lambda+\Delta\\\ -p_{x}+ip_{y}\end{array}\begin{array}[]{c}-p_{x}-ip_{y}\\\ -\lambda-\Delta\end{array}\right).$ (3) To obtain the eigenenergies, we diagonalize the Hamiltonian and obtain $E_{n,s}^{\eta}=n\sqrt{(v\hslash k)^{2}+(\Delta+\eta s\lambda)^{2}},$ (4) where $n=+1/{-}1$ denotes the electron/hole band and $k$ is the absolute value of the wave vector. We next discuss the energy eigenvalues obtained for the K point to explore the band splitting and quantum phase transitions. The energy gap of 1.6 meV seen in Fig. 3(a) as obtained for finite SOC or $E_{z}$ is consistent with Eq. (4), confirming a metal to insulator transition. Figure 3(b) for finite SOC and $E_{z}$ with $\lambda>\Delta=1.4$ meV shows an energy splitting between the spin up and spin down bands for both the electrons and holes. This splitting is less than the energy gap between the electrons and holes themselves. In addition, the energy gap between the spin up bands is greater than that between the spin down bands. This situation reflects a topological insulating state, which corresponds to the spin polarization regime. Figure 3(c) is analogous to Fig. 3(b) but for $\lambda\sim\Delta=1.6$ meV. We see that the energy gap closes between the spin down bands, while the spin up bands maintain a finite energy gap. In the first principles calculations we cannot reach an exact closure of the spin down gap as suggested by Eq. (4) but obtain a minimum of about 1.1 meV, because of the approximations involved in the simulations. The situation demonstrated in Fig. 3(c) corresponds to a semi-metallic state. Fig. 3(d) is analogous to Figs. 3(b) and (c) but for $\lambda<\Delta=5.1$ meV. The splitting of the spin down bands has increased as compared to Fig. 3(b), but less than the splitting of the spin up bands. This situation reflects a band insulator, which corresponds to the valley polarization regime. We note that we obtain an identical band structure for the $K^{\prime}$ point with the spin up and spin down bands exchanged. The $K$ and $K^{\prime}$ valleys are non-degenerate due to the broken inversion symmetry (which is a consequence of the external electric field and the buckling), compare Eq. (4). In conclusion, we have discussed the structure and electronic properties of a superlattice of silicene and hexagonal boron nitride. We find that the Dirac cone of free-standing silicene remains intact in the superlattice due to a small interaction (the binding energy amount to only 57 meV per atom). A small amount of charge transfer between the silicene and hexagonal boron nitride results in a slight shift of the Dirac cone towards higher energy, i.e., in slight hole doping. Using an analytical model we have analyzed the combined effects of the intrinsic SOC and an external electric field applied perpendicular to the superlattice. Our results show that a lifting of the spin and valley degeneracies can be achieved. With increasing strength of the electric field, the nature of the system changes from a metal to a topological insulator and further to a band insulator. Therefore, control of the quantum phase transitions in silicene is possible by tuning the external electric field. ###### Acknowledgements. We thank N. Singh for fruitful discussions. ## References (1) K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science, 306, 666 (2004). * (2) K. Takeda and K. Shiraishi, Phys. Rev. B 50, 14916 (1994). * (3) G. G. Guzmán-Verri and L. C. Lew Yan Voon, Phys. Rev. B 76, 075131 (2007). * (4) N. D. Drummond, V. Zólyomi, and V. I. Fal′ko, Phys. Rev B 85, 075423 (2012). * (5) Z. Ni, Q. Liu, K. Tang, J. Zheng, J. Zhou, R. Qin, Z. Gao, D. Yu, and J. Lu, Nano Lett. 12, 113 (2012). * (6) C.-C. Liu, W. Feng, and Y. Yao, Phys. Rev. Lett. 107, 076802 (2011). * (7) S. Lebegue and O. Eriksson, Phys. Rev. B 79, 115409 (2009). * (8) M. Houssa, E. Scalise, K. Sankaran, G. Pourtois, V. V. Afanasev, and A. Stesmans, Appl. Phys. Lett. 98, 223107 (2011). * (9) N. Gao, W. T. Zheng, and Q. Jiang, Phys. 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Nanotech. 5, 722 (2010). * (24) Y. C. Cheng, Z. Y. Zhu, and U. Schwingenschlögl, EPL 95, 17005 (2011). * (25) C.-L. Lin, R. Arafune, K. Kawahara, M. Kanno, N. Tsukahara, E. Minamitani, Y. Kim, M. Kawai, and N. Takagi, Phys. Rev. Lett. 110, 076801 (2013).	arxiv-papers	2013-10-29T08:04:40	2024-09-04T02:49:53.025319	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "T. P. Kaloni, M. Tahir, and U. Schwingenschl\\\"ogl", "submitter": "Thaneshwor Prashad Kaloni", "url": "https://arxiv.org/abs/1310.7702" }
1310.7898	Moving in temporal graphs with very sparse random availability of edges 1,2]Paul G. Spirakis 3]Eleni Ch. Akrida [1]Computer Technology Institute & Press “Diophantus” (CTI), Patras, Greece [2]Department of Computer Science, University of Liverpool, UK [3]Department of Mathematics, University of Patras, Greece Email: <[email protected]>, <[email protected]> In this work we consider temporal graphs, i.e. graphs, each edge of which is assigned a set of discrete time-labels drawn from a set of integers. The labels of an edge indicate the discrete moments in time at which the edge is available. We also consider temporal paths in a temporal graph, i.e. paths whose edges are assigned a strictly increasing sequence of labels. Furthermore, we assume the uniform case (UNI-CASE), in which every edge of a graph is assigned exactly one time label from a set of integers and the time labels assigned to the edges of the graph are chosen randomly and independently, with the selection following the uniform distribution. We call uniform random temporal graphs the graphs that satisfy the UNI-CASE. We begin by deriving the expected number of temporal paths of a given length in the uniform random temporal clique. We define the term temporal distance of two vertices, which is the arrival time, i.e. the time-label of the last edge, of the temporal path that connects those vertices, which has the smallest arrival time amongst all temporal paths that connect those vertices. We then propose two statistical properties of temporal graphs. One is the maximum expected temporal distance which is, as the term indicates, the maximum of all expected temporal distances in the graph. The other one is the temporal diameter which, loosely speaking, is the expectation of the maximum temporal distance in the graph. Since uniform random temporal graphs, except for the clique, have at least a pair of vertices whose temporal distance is infinity, we assume the existence of a slow way to go directly from any vertex to any other vertex in order for the above measures to have a finite value. We derive the maximum expected temporal distance of a uniform random temporal star graph as well as an O($\sqrt{n} \log^2{n}$) upper bound, and a greedy algorithm which computes in polynomial time the path that achieves it, on both the maximum expected temporal distance and the temporal diameter of the normalized version of the uniform random temporal clique, in which the largest time-label available equals the number of vertices. Finally, we provide an algorithm that solves an optimization problem on a specific type of temporal (multi)graphs of two vertices. Temporal graphs; Probabilistic analysis of algorithms; The bridges' optimization problem § INTRODUCTION A temporal graph (or otherwise called temporal network) is, loosely speaking, a graph that changes with time. This concept incorporates a variety of both modern and traditional networks such as information and communication networks, social networks, transportation networks, and several physical systems. The presence of dynamicity in modern communication networks, i.e. in mobile ad hoc, sensor, peer-to-peer, and delay-tolerant networks, is often very strong. We can also find that kind of dynamicity in social networks, where the topology usually represents the social connections between a group of individuals. Those connections change as the social relationships between the individuals or even the individuals themselves change. Temporal graphs can also be associated with transportation networks. In a transportation network, there is usually some fixed network of routes and a set of transportation units moving over these routes. In such networks, the dynamicity refers to the change of positions of the transportation units in the network as time passes. Concerning physical systems, dynymicity may be present in systems of interacting particles. In this work, embarking from the foundational work of Kempe et al. [2], we consider the time to be discrete, that is, we consider networks in which changes can only occur at discrete moments in time, e.g. days or hours. This choice not only gives to the resulting models a purely combinatorial flavor but also naturally abstracts many real systems. In particular, we consider those networks that can be described via an underlying graph $G$ and a labeling $L$ assigning a set of discrete labels to each edge of $G$. This is a generalization of the single-label-per-edge model used in [2], as we allow many time-labels to appear on an edge, although in this work we mainly focus on single-labeled temporal graphs. These labels are drawn from the natural numbers and indicate the discrete moments in time at which the corresponding connection is available, i.e. the corresponding edge exists in the graph. For example, in a communication network, the availability of a connection at some time $t$ may indicate that a communication protocol is allowed to transmit a data packet over that connection at time $t$. A temporal path (or journey) in a temporal graph is a path, on the edges of which we can find strictly ascending time labels. The number of edges on the latter is called length of the temporal path. This, for a communication network, would mean that it is possible to transmit a data packet along the network nodes that belong to such a path from the first node in order to the last one, as time progresses. The time label on the last edge of a temporal path is called its arrival time and, in the above example of a connection network, it would indicate the time at which the transmitted data packet would arrive at the last node of the path. In this work, we initiate the study of temporal graphs from a probabilistic and statistical viewpoint. In particular, we consider the case in which every edge of a graph is assigned exactly one time label from a set $L_0 = \{1, 2, \ldots, a\}$ of integers. The time labels assigned to the edges of the graph are chosen randomly and independently from one another from the set $L_0$ and the probability that an edge is assigned a time label $i \in L_0$ is equal to $\frac{1}{a}$, for every $i \in L_0$. We use the term UNI-CASE for the above described case and for any graph that satisfies UNI-CASE's properties we use the term Uniform Random Temporal Graph. We focus on examining three statistical properties of such graphs. The first one, called expected number of temporal paths of a given length, is the number of temporal paths, of a given length, that we expect to have in a graph, given that every edge is assigned a label satisfying UNI-CASE. The second one, called the Maximum Expected Temporal Distance, is the maximum of all temporal distances in the graph. By temporal distance of two vertices we denote the arrival time of the temporal path that connects those vertices, which has the smallest arrival time amongst all temporal paths that connect those vertices. The last property that we examine is called the Temporal Diameter of a uniform random temporal graph. Loosely speaking, it is the expected value of the maximum temporal distance in the graph, which of course is in correspondence with the diameter of a graph, as we know it up to now. The motivation of the definitions we initiate and the work we carry out here comes from the natural question on how fast we can visit a particular destination, i.e. arrive at a particular network node, starting from a given point of origin, i.e. another network node, when the connection between a pair of nodes only exists at one moment in time. §.§ Related work Labeled Graphs. Labeled graphs are becoming an increasingly useful family of Mathematical Models for a broad range of applications both in Computer Science and in Mathematics, e.g. in Graph Coloring[3]. In our work, labels correspond to time moments of availability and the properties of labeled graphs that we study are naturally temporal properties. However, we can note that any property of a graph that is assigned labels from a discrete set of labels can correspond to some temporal property. Take for example a proper edge-coloring in a graph, i.e. a coloring of the graph's edges in which no two adjacent edges have the same color. This corresponds to a temporal graph in which no two adjacent edges have the same time label, that is no two adjacent edges exist at the same time. Single-labeled and multi-labeled Temporal Graphs. The model of temporal graphs that we consider in this work has a direct relation with the single-labeled model studied in [2] as well as the multi-labeled model studied in [1]. The main results of [2] and [1] have to do mainly with connectivity properties and/or cost minimization parameters for temporal network design. In this work we study temporal graphs from a statistical view and mainly focus on how fast we expect to arrive at a target vertex in a temporal graph. In [2], a temporal path is considered to be a path with non-decreasing labels on its edges. In this work, we follow the assumption of [1] and consider a temporal path to be a path with strictly increasing labels. This choice is also motivated by recent work on dynamic communication systems, in which if it takes one time unit for the transmition of a data packet over a link, then a packet can only be transmitted over paths with strictly increasing labels. Continuous Availabilities (Intervals). Some authors have assumed the availability of an edge for a whole time-interval [$t_1,t_2$] or multiple such time-intervals. Although this is a clearly natural assumption, in this work we focus on the availability of edges at discrete moments and we design and develop techniques which are quite different from those needed in the continuous case. §.§ Roadmap and contribution In Section <ref>, we formally define the model of temporal graphs under consideration and provide all further necessary basic definitions. In Section <ref>, we make some general remarks on the expected number of temporal paths in any graph and proceed to the study of the expected number of temporal paths of a given length in the uniform random temporal clique of $n$ vertices, $K_n$. For this matter, we distinguish two cases. In Section <ref>, we study the first case, where we set the largest label available for assignment to be $a=n-1$ and we search for the expected number of temporal paths of length $k=n-1$. In Section <ref>, we study the second case, where we loosen the parameters $a$ and $k$ and we look at the expected number of temporal paths of length $k<a$, when the largest label available for assignment is $a=n-1$. In Section <ref>, we formally define the maximum expected temporal distance of a uniform random temporal graph and we make some preliminary notations. In Section <ref>, we look at some known graphs' maximum expected temporal distance. In particular, in Section <ref>, we study the case of the uniform random temporal star graph and we provide its exact maximum expected temporal distance. In Section <ref>, we study the case of the uniform random temporal clique, focusing on its normalized version, where the largest label, $a$, available for assignment is equal to the number of vertices, $n$. We also give a simple (greedy) algorithm which can, with high probability, find a temporal path with small expected arrival time from a given source to a given target vertex in the normalized uniform random temporal clique. In Section <ref>, we formally define the temporal diameter of a uniform random temporal graph and provide an inequality relation between the latter and the maximum expected temporal distance as well as the relevant proof. Furthermore, we provide an upper bound for both the temporal diameter and the maximum expected temporal distance of the nomalized uniform random temporal clique. In Section <ref>, we study an optimization problem on a specific type of temporal (multi)graphs of two vertices. We prove that the problem can by polynomially solved and provide an algorithm that gives the solution, along with the proof of its correctness. Finally, in Section <ref> we conclude and give further research horizons opened through our work. § PRELIMINARIES A temporal graph is an ordered triplet $G=\{V,E,L\}$, where: * $V$ stands for a nonempty finite set (called set of vertices) * $E$ stands for a set of m elements, each of which is a 2-element subset of V (called set of edges), and * $L= \{L_e, \forall e \in E\} = \{L_{e_1}, L_{e_2}, \ldots, L_{e_m}\}$, is a set of m elements, $L_{e_i},~1\leq i \leq m$, each of which is a set of positive integers mapped to the edge $e_i \in E$ (called assignment of time labels or simply assignment) We also denote the temporal graph $G=\{V,E,L\}$ by $G'(L)$ or $(G',L)$, where $G' = \{V,E\}$ is the graph, on the edges of which we assign the time labels, and $L= \{L_e, ~ e \in E(G')\}$ is the assignment. The values assigned to each edge of the graph are called time labels of the edge and indicate the times at which we can cross it (from one end to the other). §.§ Further Definitions We can now talk about temporal edges (or time edges) that are considered to be triplets $(u, v, l)$, where $u, v$ are the ends of an edge in the temporal graph and $l \in L_{ \{u, v\} }$ is a time label of this edge. That is, if an edge $e = \{u, v\}$ has more than one time labels, e.g. has a set of three time labels, $L_e = \{l_1, l_2, l_3\}$, then this edge has three corresponding time edges, $(u, v, l_1),~ (u, v, l_2)$ and $(u, v, l_3)$. A journey $j$ from a vertex $u$ to a vertex $v$ ($(u, v)$- journey) is a sequence of time edges $(u, u_1, l_1), ~(u_1, u_2, l_2), \ldots , ~(u_{k-1}, v, l_k)$, such that $l_i < l_{i +1}$, for each $1 \leq i \leq k - 1$. We call the last time label of journey $j$, $l_k$, arrival time of the journey. A ($u,v$)-journey $j$ in a temporal graph is called foremost journey if its arrival time is the minimum arrival time of all ($u,v$)-journeys' arrival times, under the labels assigned on the graph's edges. Now, consider any temporal graph $G=\{V,E,L\}$. Let every edge receive exactly one time label, chosen randomly, independently of one another from a set $ L_0 $ = {$ 1,2, \ldots, a $}, where $ a \in \mathbb{N} $, with the probability of an edge label to be $ i, ~ \forall i \in L_0 $, equal to $ \frac{1}{a} $. (UNI-CASE) A temporal graph that satisfies UNI-CASE is called Uniform Random Temporal Graph (U-RTG). In the special case, where the largest label, $a$, that can be assigned to the edges of a graph is equal to the number of its vertices, the graph is called Normalized Uniform Random Temporal Graph (Normalized U-RTG). Note. There could be prospective study of cases in which each edge of a graph may receive several time labels, selected randomly and independently of one another from the set $ L_0 $ = {$ 1 , 2, \ldots, a $}, where $ a \in \mathbb {N} $, with the selection following a distribution F. (F-CASE) In such cases, the graphs under consideration would be called F-Random Temporal Graphs (F-RTG) respectively. In the following sections, we will look for the expected number of journeys of length k in some well-known graphs that satisfy UNI-CASE. For the sake of brevity, we often call such journeys “k edges temporal paths”. We also study the Expected (or Temporal) Diameter and the Maximum Temporal Distance of a graph, as defined in the following paragraphs. § EXPECTED NUMBER OF TEMPORAL PATHS In this section we will search for the expected number of $k$ edges temporal paths in a clique of n vertices, $ K_n $, that satisfies UNI-CASE. It is obvious that for there to exist a temporal path of length k in any graph, the number of edges, k, has to be at most equal to the maximum label of the set $ L_0 $, $ a $, that can be assigned to the various edges. Otherwise, it is impossible for a $k$ edges temporal path to exist (see Figure <ref>). \[\begin{tikzpicture}[thick,scale=0.95] \coordinate [label=right:{$L_0=\{1,2,3 (=a)\}$}] (P) at (0,-1); \coordinate [label=right:{$k=4$}] (P') at (0,-2); \vertex (1) at (0,0) [label=below:$$] {}; \vertex (2) at (2,0.5) [label=left:$$] {}; \vertex (3) at (4,0.2) [label=left:$$] {}; \vertex (4) at (6,0) [label=above:$$] {}; \vertex (5) at (8,0.4) [label=above:$$] {}; \path (1) edge node[above]{$1$} (2) (2) edge node[above]{$2$} (3) (3) edge node[above]{$3$} (4) (4) edge node[above]{\textbf{\textcolor{red}{;}}} (5) \end{tikzpicture}\] \end{center} \rule{35em}{0.5pt} \caption{There is no temporal path, when $k>a$} \label{fig:temp-stat1} \end{figure} \subsection{Special case: $G=K_n,~k=n-1,~a=n-1$}\label{sec:exp1} Initially, we focus our interest in the case of the clique (complete graph) of n vertices, $ K_n $, that satisfies UNI-CASE with $a = n-1$ (i.e. with $ L_0 = \{1,2, \ldots, n-1 \} $), in which we seek the expected number of $n-1$ edges temporal paths. Obviously, there can only be one assignment of labels of $ L_0 $ on the $ k = n-1 $ edges of any path starting from a random initial vertice $ v_0 \in V (K_n) $ in the clique $ K_n $ such, that we can find a journey on the edges of this path. This assignment gives label 1 on the $1^{st}$ edge, label 2 on the $2^{nd}$ edge, $\ldots$ , label $n-1$ on the $(n-1)^{th}$ edge. Each edge can receive exactly one label from a set of $n-1$ labels. Therefore, the total number of assignments that can be made on these $ n-1 $ edges is: \[ \# assignments = (n-1)^{n-1} \] Consequently, given a path of $ n-1 $ edges starting from $ v_0 $, the probability for there to exist the corresponding temporal path (i.e. the one arising on the simple path after the assignment of the time labels) is: \[ P(temporal\_ path\_ of\_length \_n-1 \_starting\_ from \_v_0)= \frac{1}{(n-1)^{n-1}} \] The number of paths of length $ n-1 $, starting from $ v_0 $ in the clique $ K_n $ is equal to the number of permutations of the $ n-1 $ vertices remaining (i.e. except the start $ v_0 $) to construct such a path. That is, the number of paths of length $ n-1 $ that start from $ v_0 $ in the clique $ K_n $ is: \[ (n-1)! \] Therefore, since the clique $ K_n $ has $ n $ vertices, and due to the linearity of expectation, the expected number of temporal paths of length $ k = n-1 $ in the clique $K_n$ is: \[ E(\# temporal \_paths\_of\_length\_n-1) = n \cdot (n-1)! \cdot \frac{1}{(n-1)^{n-1}} = \frac{n!}{(n-1)^{n-1}}\] \paragraph{Comments} Let us observe that when $ n $ is too large ($n\rightarrow + \infty$), then, by Stirling's formula, we result in the following: \begin{IEEEeqnarray}{lCl} E(\# temporal \_paths\_of\_length\_n-1) & = & \frac{\sqrt{2\pi n} \Big(\frac{n}{e}\Big)^n}{(n-1)^{n-1}} \\ & = & \frac{\sqrt{2\pi n} n^n}{e^n (n-1)^{n-1}} \xrightarrow[n\to+\infty]{} 0 \end{IEEEeqnarray} Of course, this is more or less obvious when we consider the fact that it is difficult to find $n-1$ edges temporal paths in the clique of $n$ vertices when $n$ is too large. This is because in order to have a temporal path of such length, the (so many) time labels should be assigned on the edges so that they maintain the desired strictly increasing sequence, something that is increasingly less likely to happen as $ n $ increases. \subsection{Special case: $G=K_n,~k<a,~a\geq n$}\label{sec:exp2} Now let's see what happens in the case of the clique $K_n$, that satisfies UNI-CASE, when we look at the expected number of temporal paths of length $k<a$ and the maximum label that can be assigned to any edge of the clique is $a \geq n$. Starting from a vertex $v_0 \in V(K_n)$ and along the path of k edges, we can construct, as explained in Figure \ref{fig:temp-stat2}, a number of assignments equal to: \[ \# assignments = a^k \] \begin{figure}[htbp] \begin{center} \[\begin{tikzpicture}[thick,scale=0.95] \coordinate [label=below:{$\ldots$}] (P) at (5,0.25); \coordinate [label=below:{$\ldots$}] (P') at (9,0.05); \coordinate [label=below:{$\downarrow$}] (P'') at (7,-0.3); \coordinate [label=below:{$a$ choises for the label}] (P'') at (7,-0.8); \coordinate [label=below:{assigned to the $i^{th}$ edge}] (P'') at (7,-1.3); \vertex (1) at (0,0) [label=below:$v_0$] {}; \vertex (2) at (2,0.5) [label=left:$$] {}; \vertex (3) at (4,0.2) [label=left:$$] {}; \vertex (4) at (6,0.2) [label=above:$$] {}; \vertex (5) at (8,0) [label=above:$$] {}; \vertex (6) at (10,0) [label=above:$$] {}; \vertex (7) at (12,0.3) [label=above:$$] {}; \path (1) edge node[below]{$e_1$} (2) (2) edge node[below]{$e_2$} (3) (4) edge node[below]{$e_i$} (5) (6) edge node[below]{$e_k$} (7) \end{tikzpicture}\] \end{center} \rule{35em}{0.5pt} \caption{Number of assignments on a path of length $k$, when $k<a$} \label{fig:temp-stat2} \end{figure} The number of assignments that can be made on the $ k $ edges, where the time labels assigned are distinct (different from each other) is: \[ \# distinct\_ time\_ labels\_assignments = a \cdot (a-1) \cdot \ldots \cdot (a-k+1) = \frac{a!}{(a-k)!} \] We will now calculate the number of paths of length $k$ that can be starting from $v_0 \in V(K_n)$. We have $n-1$ options for how to select $v_1$, the vertex following $v_0$ on the path, $n-2$ options for how to select $v_2$, the vertex following $v_1$ on the path, etc., and finally $n-k$ options for how to select $v_k$, the last vertex on the path. Therefore, the number of paths of length $ k $ that can be starting from $ v_0 \in V (K_n) $ is: \[ \#paths \_of\_length\_k\_starting\_from\_v_0 = (n-1)\cdot (n-2) \cdot \ldots \cdot (n-k) =\frac{(n-1)!}{(n-k-1)!} \] We call $A$ the event that ``we have the \textit{right} labels'' assignment on the $k$ edges of any path of length $k$ starting from $ v_0 $''. \\ That is, if $l_1, l_2, \ldots, l_k$ are the time labels assigned to the $1^{st}$, the $ 2^{nd} $, $ \ldots $, the $ k^{th} $ edge of the path, respectively, with $ l_i \in L_0 = \{1,2, \ldots, a \}, ~ \forall i = 1,2, \ldots, k $, $A$ is the event that: \[l_1 < l_2 < \ldots < l_k\] We call $\phi$ the probability that $A$ occurs. That is: \[ \phi = P(A)= P(l_1 < l_2 < \ldots < l_k) \] Let us note that the number of assignments of $k$ labels, $l_{a_i},~ i=1, \ldots, k$, such that \[ l_{a_1} < l_{a_2} < \ldots < l_{a_k} \] is $k!$ and each one has a probability equal to $P(A)$ to happen.\\ Therefore, if we consider $B$ to be the event that ``\textit{at least} two of the labels assigned on the $k$ edges of the path are equal'', then the following applies: \[ k! \cdot P(A) + P(B) =1 \Leftrightarrow \] \begin{equation}\label{eq:1} k! \cdot \phi + 1 - P(\rceil{B}) =1 \end{equation} The probability that the event $\rceil{B}$ occurs, that is there are no two equal labels assigned on the $k$ edges of the path, is: \[ P(\rceil{B}) = \frac{\#distinct\_ time\_labels\_ assignments}{\# assignments} =\] \[ = \frac{\frac{a!}{(a-k)!}}{a^k} \] \[ = \frac{a!}{a^k \cdot (a-k)!} \] Consequently, the relation \eqref{eq:1} becomes: \[ k! \cdot \phi + 1 - \frac{a!}{a^k \cdot (a-k)!} =1 \Leftrightarrow \] \[ \Leftrightarrow \phi = \frac{a!}{k! \cdot a^k \cdot (a-k)!}\] Let us recall that $\phi$ is the probability to have a \textit{proper} assignment on the $ k $ edges of any path of length $k$ starting from any vertice $ v_0 $ of the clique $ K_n $.\\ Also, recall that the number of paths of length $k$ that can be starting from any vertice $v_0$ of the clique $K_n$ is $\frac{(n-1)!}{(n-k-1)!}$.\\ Therefore, the expected number of paths of length $k$ that start from a random vertex $v_0$ and on which there are labels assigned so that there exists a temporal path on them, is: \[ E(\#temporal\_paths\_of\_length\_k\_starting\_from\_v_0) = \frac{(n-1)!}{(n-k-1)!} \cdot \phi \] Eventually, since the clique $K_n$ has a number of $n$ vertices, the expected number of paths of length $k$, on which labels are assigned in a way that there exists a temporal path on them, is: \[ E(\#temporal\_paths\_of\_length\_k) = n \cdot \frac{(n-1)!}{(n-k-1)!} \cdot \phi \] \[ = \frac{n \cdot (n-1)!}{(n-k-1)!} \cdot \frac{a!}{k! \cdot a^k \cdot (a-k)!} \] \[ = \frac{n! \cdot a!}{(n-k-1)! \cdot k! \cdot a^k \cdot (a-k)!} \] \paragraph{Comments} Let us observe that the probability $\phi$ is: \[ \phi = \frac{1}{k!} \cdot \frac{\overbrace{a (a-1) \ldots (a-k+1)}^{\text{k factors}}}{\underbrace{a \cdot \ldots \cdot a}_\text{k factors}}\] and so, if $a$ is very large in comparison with $k$, then we have $\phi \approx \frac{1}{k!}$.\\ Hence, if $a$ is far larger than $k$, then the expected number of temporal paths of length $k$ in the clique $K_n$, is: \[ E(\#temporal\_paths\_of\_length\_k) \approx \frac{n!}{k! (n-k-1)!} = \frac{n \cdot (n-1) \cdot \ldots \cdot (n-k)}{k!} \] \section{The Maximum Expected Temporal Distance}\label{sec:md} In this section, we will define and study a new concept, that of \textit {the maximum expected temporal distance} of a U-RTG.\\ Henceforth, we make the following assumption. For every pair of vertices in any U-RTG, there exists a \textit{\textbf{slow}} journey that connects them, whose arrival time is a fixed, for each graph, number $n' \in \mathbb{N},$ where $n'$ is greater than the expected value of any edge's label, $l$. That is $n' \geq E(l)$. \begin{mydef} Consider an instance $G(L)$ of a U-RTG. Given two vertices $s,t \in V \big( G(L) \big)$, we define: \begin{itemize} \item $\delta ' (s,t)=a(j),$\label{s10} where $j$ is a foremost $(s,t)-$journey, to be called \textbf{\textit{distributional temporal distance}} from source vertex $s$ to target vertex $t$ under the assignment $L$. If there exists no $(s,t)-$journey in G, then $\delta ' (s,t) \rightarrow \infty $ \item $ \delta(s,t) = min \{ \delta ' (s,t) , n' \} $ to be called \textbf{\textit{temporal distance}} from source vertex $s$ to target vertex $t$ under the assignment $L$, and \item $MD= max_{s,t \in V(G)} E\big( \delta(s,t) \big)$\label{s11} to be called \textbf{\textit{Maximum Expected Temporal Distance}} of $G$ \end{itemize} \end{mydef} \noindent \textit{Remark.} If the $U-RTG$ is a path itself, then its maximum expected temporal distance is obviously $n'$. \\ \begin{figure}[htbp] \begin{center} \[\begin{tikzpicture}[thick,scale=0.95] \vertex (1) at (0,0) [label=below:$s$] {}; \vertex (2) at (2,0.5) [label=left:$$] {}; \vertex (3) at (4,0.2) [label=left:$$] {}; \vertex (4) at (6,0.2) [label=above:$$] {}; \vertex (5) at (8,0) [label=above:$$] {}; \vertex (6) at (10,0) [label=below:$t$] {}; \path (1) edge node[above]{$5$} (2) (2) edge node[above]{$3$} (3) (3) edge node[above]{$4$} (4) (4) edge node[above]{$1$} (5) (5) edge node[above]{$2$} (6) \end{tikzpicture}\] \end{center} \rule{35em}{0.5pt} \caption{MD of a $U-RTG$, which is a path itself, equals $n'$.} \label{fig:td1} \end{figure} This can be easily understood if we consider that for any two vertices $u$ and $v$ in the path, if there exists a ($u,v$)-journey, then the time labels assigned to its edges form a strictly increasing sequence and thus there is no ($v,u$)-journey in it, apart from the \textit{slow} journey which we assume that exists. Therefore, $\delta ' (v,u) \rightarrow + \infty$ and $ \delta (v,u) = min \{\delta ' (v,u), n'\} = n'$. (see Figure \ref{fig:path11}). \begin{figure}[htbp] \begin{center} \[\begin{tikzpicture}[thick,scale=0.95] \vertex (1) at (0,0) [label=below:$$] {}; \vertex (2) at (2,0.5) [label=left:$$] {}; \vertex (3) at (4,0.2) [label=above:$u$] {}; \vertex (4) at (6,0.55) [label=above:$$] {}; \vertex (5) at (8,0) [label=above:$v$] {}; \vertex (6) at (10,0) [label=above:$$] {}; \vertex (7) at (12,0.5) [label=above:$$] {}; \node[anchor=east] at (2.65,-2.5) (a) {}; \node[anchor=west] at (9,-2.5) (b) {}; \node[anchor=east] at (3.3,-1.4) (a') {}; \node[anchor=west] at (8.2,-1.4) (b') {}; \vertex (8) at (2.65,-1.9) [label=above:$u$] {}; \vertex (9) at (6,-1.25) [label=above:$$] {}; \vertex (10) at (9,-2.1) [label=above:$v$] {}; \path (8) edge node[sloped, above]{\textcolor{blue!70}{$3$}} (9) (9) edge node[sloped, above]{\textcolor{blue!70}{$4$}} (10) (1) edge node[above]{$$} (2) (2) edge [line width=1pt,black!0.1] node[sloped, below, black]{$\ldots$} (3) (3) edge node[sloped, above]{\textcolor{blue!70}{$3$}} (4) (4) edge node[sloped, above]{\textcolor{blue!70}{$4$}} (5) (5) edge [line width=1pt,black!0.1] node[sloped, below, black]{$\ldots$} (6) (6) edge node[above]{$$} (7) (3) edge [line width=1pt,dotted, blue!60] node[above]{$$} (8) (5) edge [line width=1pt, dotted, blue!60] node[above]{$$} (10) (a) edge[->, bend left=20, blue!70] node [below]{\textcolor{blue!70}{$\delta ' (u,v) = 4$}} (b) (b') edge[->, bend right=25, red] node [above]{\textcolor{red}{$\delta ' (v,u) \rightarrow \infty $}} (a'); \end{tikzpicture}\] \end{center} \rule{35em}{0.5pt} \caption{Example of temporal distance, from source vertex to target vertex, equal to $ n'$.} \label{fig:path11} \end{figure} \subsection{Known graphs' maximum expected temporal distance}\label{sec:md1} Next, we study the maximum expected temporal distance of two known graphs, the star graph of $n$ vertices, which we denote by $G_{star}$ (see Figure \ref{fig:td2}) and the clique of $n$ vertices, $K_n$ (see Figure \ref{fig:td3}).\\ \subsubsection{Case: $G=G_{star}$}\label{sec:md11} It is easy to understand that, even if the temporal star graph does not satisfy UNI-CASE, but satisfies any F-CASE, as defined in Section \ref{sec:eisag}, it is: \[max_{s,t \in V(G_{star})} E_F \big( \delta (s,t) \big) \geq 2,~ \text{for any distribution }F \] We will calculate the exact maximum expected temporal distance, $MD$, of a uniform random temporal star graph. It is: \begin{IEEEeqnarray}{rCl}\label{eq:2} MD(G_{star}) & = & max_{s,t\in V(G_{star})}E \big(\delta (s,t) \big) \nonumber\\ \quad & = & E\big(\delta (s,t) \big) \text{, for any two vertices } s,t \in V(G_{star}) \nonumber\\ \quad & = & E(l_2\|~l_2 > l_1) \cdot P(l_2 > l_1) +n' \cdot P(l_2 \leq l_1) \end{IEEEeqnarray} \begin{figure}[htbp] \begin{center} \[\begin{tikzpicture}[thick,scale=0.95] \vertex (1) at (0,0) [label=below:$$] {}; \vertex (2) at (2,0) [label=left:$$] {}; \vertex (3) at (1,1.5) [label=above:$t$] {}; \vertex (4) at (1,-1.5) [label=above:$$] {}; \vertex (5) at (-1,-1.5) [label=above:$$] {}; \vertex (6) at (-2,0) [label=left:$s$] {}; \path (1) edge node[above]{$$} (2) (3) edge [line width=1pt,black!0.1] node[sloped, above, black]{$\ldots$} (6) (1) edge node[right]{$l_2$} (3) (1) edge node[above]{$$} (4) (1) edge node[above]{$$} (5) (1) edge node[below]{$l_1$} (6) \end{tikzpicture}\] \end{center} \rule{35em}{0.5pt} \caption{A star graph} \label{fig:td2} \end{figure} We calculate the expected value of label $l_2$, given that $l_2 > l_1$, that is $E(l_2\|~l_2 > l_1)$: \begin{IEEEeqnarray}{rCl} E(l_2\|~l_2 > l_1)& = & \sum_{i=1}^a E(l_2\|~l_2>i) \cdot P(l_1=i) \\ & = & \sum_{i=1}^a \Big( \sum_{i'=i}^{a} \big( P(l_2 = i' +1) \cdot (i'+1) \big) \Big) \cdot P(l_1 = i) \\ & = & \sum_{i=1}^a \Big( \sum_{i'=i}^{a} ( i' +1) \cdot \frac{1}{a} \Big) \cdot \frac{1}{a} \\ & = & \frac{1}{a^2} \cdot \sum_{i=1}^{a} \sum_{i'=i}^a (i'+1) \\ & = & \frac{1}{a^2} \cdot \Big( \sum_{i'=1}^a (i'+1) + \sum_{i'=2}^a (i'+1) + \ldots + \sum_{i'=a}^a (i'+1) \Big) \\ & = & \frac{1}{a^2} \cdot \Big( \big( 2+3+ \ldots + (a+1) \big) + \big( 3+4 + \ldots + (a+1) \big) + \ldots + \big( (a+1) \big) \Big) \\ & = & \frac{1}{a^2} \cdot \Big( 1\cdot 2 + 2 \cdot 3 + 3 \cdot 4 + 4 \cdot 5 + \ldots + a \cdot (a+1) \Big) \\ & = & \frac{1}{a^2} \cdot \sum_{i=1}^a \Big( i \cdot (i+1) \Big) \\ & = & \frac{1}{a^2} \cdot \sum_{i=1}^a \Big( i^2 +i \Big) \\ & = & \frac{1}{a^2} \cdot \sum_{i=1}^a i^2 + \sum_{i=1}^a i \\ & = & \frac{1}{a^2} \cdot \Big( \frac{a\cdot (a+1) \cdot (2a +1)}{6} + \frac{a \cdot (a+1)}{2} \Big) \\ & = & \frac{1}{a^2} \cdot \frac{a\cdot (a+1) \cdot (2a +1) + 3 \cdot a\cdot (a+1) }{6} \\ & = & \frac{a\cdot (a+1) \cdot (2a +4)}{6 \cdot a^2} \end{IEEEeqnarray} Therefore, relation \eqref{eq:2} becomes: \begin{IEEEeqnarray}{rCl}\label{eq:3} MD(G_{star}) & = & \frac{(a+1)(a+2)}{3a} \cdot P(l_2 > l_1) +n' \cdot P(l_2 \leq l_1) \end{IEEEeqnarray} It holds that: \begin{IEEEeqnarray}{rCl} P(l_2 \leq l_1) & = & \sum_{i=1}^{a} P(l_2 \leq i) \cdot P(l_1 = i) \\ & = & \sum_{i=1}^{a} \frac{i}{a} \cdot \frac{1}{a} \\ & = & \frac{1}{a^2} \sum_{i=1}^{a} i \\ &= & \frac{a+1}{2a} \end{IEEEeqnarray} Therefore, it is: \begin{IEEEeqnarray}{rCl} P(l_2 > l_1) & = & 1- P(l_2 \leq l_1) \\ & = & \frac{a-1}{2a} \end{IEEEeqnarray} Relation \eqref{eq:3} now becomes: \begin{IEEEeqnarray}{rCl} MD(G_{star}) & = & \frac{(a+1)(a+2)}{3a} \cdot \frac{a-1}{2a} +n' \cdot \frac{a+1}{2a} \end{IEEEeqnarray} Eventually, the star graph's maximum temporal distance is: \[ MD(G_{star}) = \frac{(a-1)(a+1)(a+2)}{6a^2} +n' \cdot \frac{a+1}{2a}\] \subsubsection{Case: $G=K_n$}\label{sec:md12} We will now study extensively the clique's case. First, let us observe that $\delta ' (s,t) \leq a$, and therefore $\delta (s,t) \leq a$, for any two vertices $s,t$ in a clique. Hence: \[ MD(K_n) = max_{s,t\in V(K_n)}E \big(\delta (s,t) \big) \leq a \] \begin{figure}[htbp] \begin{center} \[\begin{tikzpicture}[thick,scale=0.65] \vertex (1) at (-0.5,0) [label=below:$$] {}; \vertex (2) at (2,1.5) [label=left:$$] {}; \vertex (3) at (4.5,0) [label=left:$$] {}; \vertex (4) at (3.8,-3.8) [label=above:$$] {}; \vertex (5) at (0.2,-3.8) [label=above:$$] {}; \path (1) edge node[above]{$$} (2) (1) edge node[above]{$$} (3) (1) edge node[above]{$$} (4) (1) edge node[above]{$$} (5) (2) edge node[above]{$$} (3) (2) edge node[above]{$$} (4) (2) edge node[above]{$$} (5) (3) edge node[above]{$$} (4) (3) edge node[above]{$$} (5) (4) edge [line width=1pt,black!0.1] node[sloped, above, black]{$\ldots$} (5) \end{tikzpicture}\] \end{center} \rule{35em}{0.5pt} \caption{A clique} \label{fig:td3} \end{figure} \paragraph{Normalized uniform random temporal clique} Let $G=K_n$ be a clique of $n$ vertices and let us consider its normalized U-version. That is, every edge $e \in E(K_n)$ is given a single availability label and those labels are chosen randomly and independently from one another from the set $L_0$=\{$1,2, \ldots, n$\}, with the probability that an edge's label equals $i$ being equal to $\frac{1}{n}$, $\forall i \in L_0$. For any two vertices $s,t$ in the clique, we have:\[ E\big( l (e= \{s,t\}) \big) = \frac{n}{2} \] In the specific case of the normalized uniform random temporal clique of $n$ vertices, there is actually no need for us to assume any \textit{slow} journey to connect any pair of vertices since we already have such a journey, with arrival time equal to $E\big( l (e= \{s,t\}) \big) = \frac{n}{2}$. But, for the sake of consistency, we can set the fixed number $n'$ to be equal to $\frac{n}{2}$. It holds that:\[ MD(normalized ~K_n) = max_{s,t\in V(K_n)}E \big(\delta (s,t) \big) \leq \frac{n}{2} \] Since this is only an upper bound, we wonder if we can find temporal paths with smaller arrival time than that bound. Indeed, we give a \textit{simple (greedy)} algorithm which can, with high probability, find a journey with small expected arrival time from a given source vertex $s$ to a given target vertex $t$ in the normalized uniform random temporal clique. \noindent \textit{Note.} From here on, the notation ``$\log$'' will denote the natural logarithm. \newpage \begin{algorithm}[h] \caption{The normalized U-RTG clique short journey finding algorithm, Extend-Try} \label{alg:extend} \begin{algorithmic}[1] \Procedure {Extend-Try}{$clique~ K_n$, $s$, $t$, $c_1$, $k$} \For {i = 0 ... $c_1 \sqrt{n} \log{n}$} \State $s_i$ := undefined; \EndFor \State $s_0$ := $s$; \For {i = 0 ... $c_1 \sqrt{n} \log{n}$} \label{lst:line:for1} \If {$l(\{s_i,t\}) \in \big(c_1 \sqrt{n} (\log{n}) k, c_1 \sqrt{n} (\log{n}) k + \sqrt{n}\big)$} \State Follow directly the edge $\{s_i,t\}$; \textcolor{blue!70}{Success!} \State go to line \ref{lst:line:end1} \Else \If {$\exists u \in U\setminus \{t\}$ (where U stands for the set of the unvisited $~~~~~~~~~~~~~$ vertices) such, that $l(\{s_i,u\}) \in \big(k\cdot i , k (i+1) \big)$} \State $s_{i+1} = u$; \State go to line \ref{lst:line:for1} \Else \State follow directly the edge $\{s_i, u\}$ with the smallest $l(\{s_i, u\})$ $~~~~~~~~~~~~~~~~~~$ among all $u\in U$; \textcolor{blue!70}{Failure!} \State go to line \ref{lst:line:end2} \EndIf \EndIf \EndFor \For {i = 0 ... $c_1 \sqrt{n} \log{n}$}\label{lst:line:end1} \State \textbf{return} $s_i$; \EndFor \EndProcedure \label{lst:line:end2} \end{algorithmic} \end{algorithm} \paragraph{Analysis of Extend-Try} Next, we analyze algorithm \ref{alg:extend}, looking for the probability that it succeeds. The probability that the time label of the edge $\{s_i,t\}$ belongs to the interval $(c_1 \sqrt{n} k, c_1 \sqrt{n} k + \sqrt{n})$ and thus the algorithm succeeds in the $(i+1)^{\text{th}}$ iteration, is: \[ P\Big( l(\{s_i,t\}) \in \big(c_1 \sqrt{n} (\log{n}) k, c_1 \sqrt{n} (\log{n}) k + \sqrt{n}\big) \Big) = \frac{\sqrt{n}}{n} = \frac{1}{\sqrt{n}} \] \noindent Let $\varepsilon_1^j$ be the following event: \[\text{``The algorithm finds a proper journey $s_0s_1,s_1s_2,s_2,s_3, \ldots, s_{j-1}s_j$''}\] meaning that it finds a temporal path, on the temporal edges of which we find strictly ascending time labels and in fact the $i^{th}$ temporal edge's time label correctly belongs to the interval $((i-1) k, i k)$. The time labels are given to the edges independently from one another, thus the probability that the event $\varepsilon_1^j$ occurs is the product of the following probabilities: $P\Big(\exists s_1\text{ unvisited vertex } :~\text{the edge } \{s_0,s_1\} \text{ has time label }l(\{s_0,s_1\}) \in (0,k) \Big)$\\ $P\Big(\exists s_2\text{ unvisited vertex } :~\text{the edge } \{s_1,s_2\} \text{ has time label }l(\{s_1,s_2\}) \in (k,2k) $\\ $P\Big(\exists s_j \text{ unvisited vertex }:~\text{the edge } \{s_{j-1},s_j\} \text{ has time label }l(\{s_{j-1},s_j\}) \in \big( (j-1)k,jk \big)$\\[1cm] For any $i^{th}$ probability of the above, it holds that: \begin{IEEEeqnarray}{Cl} & P\Big(\exists s_i \text{ unvisited vertex} :~\text{the edge } \{s_{i-1},s_i\} \text{ has } l( \{s_{i-1},s_i\} ) \in ((i-1) k , i k) \Big) \\ = & 1 - P\Big( \not\exists s_i \text{ unvisited vertex}:~\text{the edge } \{s_{i-1},s_i\} \text{ has }l(\{s_{i-1},s_i\}) \in ((i-1) k , i k) \Big) \\ = & 1 - P\Big(\forall s_i \text{ unvisited vertices}:~\text{the edge } \{s_{i-1},s_i\} \text{ has }l(\{s_{i-1},s_i\}) \notin ((i-1) k , i k) \Big) \\ = & 1 - \Big( P\big( \text{the edge } \{s_{i-1},s_i\} \text{ has }l(\{s_{i-1},s_i\}) \notin ((i-1) k , i k) ,s_i \text{ unvisited vertex} \big) \Big)^{n-i} \\ = & 1 - \Big(1- P\big( \text{the edge } \{s_{i-1},s_i\} \text{ has }l(\{s_{i-1},s_i\}) \in ((i-1) k , i k) ,s_i \text{ unvisited vertex} \big) \Big)^{n-i} \\ = & 1- \Big( 1 - \frac{k}{n} \Big) ^{n-i} \end{IEEEeqnarray} \noindent Therefore, the probability that $\varepsilon_1^j$ occurs, is: \begin{IEEEeqnarray}{lCl} P(\varepsilon_1^j) & = & \Bigg( 1- \Big( 1- \frac{k}{n} \Big)^{n-1} \Bigg) \cdot \\ && \: \Bigg( 1- \Big( 1- \frac{k}{n} \Big)^{n-2} \Bigg) \cdot \ldots \cdot \Bigg( 1- \Big( 1- \frac{k}{n} \Big)^{n-j} \Bigg) \geq \\ & \geq & \Bigg( 1- \Big( 1- \frac{k}{n} \Big)^{n-j} \Bigg)^j \geq \\ & \geq & \Bigg( 1-e^{-k}\Big(1-\frac{k}{n}\Big)^{-j} \Bigg)^j \end{IEEEeqnarray} For $j \leq c_1 \sqrt{n} \log{n}$, we have: \begin{IEEEeqnarray}{rrCll} & \Big(1- \frac{k}{n}\Big)^{-j} & \leq & \Big(1-\frac{k}{n}\Big)^{-c_1 \sqrt{n} \log{n}} & \Leftrightarrow \\ \Leftrightarrow & 1-e^{-k}\Big(1-\frac{k}{n}\Big)^{-j} & \geq & 1-e^{-k} \Big(1-\frac{k}{n}\Big)^{-c_1\sqrt{n}\log{n}} & \end{IEEEeqnarray} \begin{IEEEeqnarray}{rCl} \Bigg( 1-e^{-k}\Big(1-\frac{k}{n}\Big)^{-j} \Bigg)^j & \geq & \Bigg( 1-e^{-k} \Big(1-\frac{k}{n}\Big)^{-c_1\sqrt{n}\log{n}} \Bigg)^{c_1\sqrt{n}\log{n}} \end{IEEEeqnarray} As a result, for $j \leq c_1 \sqrt{n} \log{n}$, it is: \begin{IEEEeqnarray}{lCl} P(\varepsilon_1^j) & \geq & \Bigg( 1-e^{-k} \Big(1-\frac{k}{n}\Big)^{-c_1\sqrt{n}\log{n}} \Bigg)^{c_1\sqrt{n}\log{n}}\\ P(\varepsilon_1^j) & \geq & 1-e^{-k} \Big(1-\frac{k}{n}\Big)^{-c_1\sqrt{n}\log{n}} \end{IEEEeqnarray} It holds asymptotically: \begin{IEEEeqnarray}{rCll} c_1\sqrt{n}\log{n} & \leq & n & \Leftrightarrow \\ \Big(1-\frac{k}{n}\Big)^{c_1\sqrt{n}\log{n}} & \geq & \Big(1-\frac{k}{n}\Big)^{n} & \Leftrightarrow \\ \Big(1-\frac{k}{n}\Big)^{-c_1\sqrt{n}\log{n}} & \leq & \Big(1-\frac{k}{n}\Big)^{-n} & \Leftrightarrow \\ 1-e^{-k} \Big(1-\frac{k}{n}\Big)^{-c_1\sqrt{n}\log{n}} & \geq & 1-e^{-k}\Big(1-\frac{k}{n}\Big)^{-n} \end{IEEEeqnarray} \begin{IEEEeqnarray}{lCl} P(\varepsilon_1^j) & \geq & 1-e^{-k}\Big(1-\frac{k}{n}\Big)^{-n} \end{IEEEeqnarray} and since $k \geq 1$, we have: \begin{IEEEeqnarray}{lCl} P(\varepsilon_1^j) & \geq & 1-e^{-k}\Big(1-\frac{1}{n}\Big)^{-n} \\ & = & 1-e^{-k}e \\ & = & 1-e^{1-k} \end{IEEEeqnarray} For $k=r\log{n},~r>1$, we have: \begin{IEEEeqnarray}{lCl} P(\varepsilon_1^j) & \geq & 1-e^{1-r\log{n}} \\ & = & 1-en^{-r} \end{IEEEeqnarray} The probability that we fail in every iteration $i=0, \ldots, c_1 \sqrt{n}\log{n}$ to find a vertex $s_i$ such, that $l(\{s_i,t\}) \in (c_1 \sqrt{n} k, c_1 \sqrt{n} k + \sqrt{n})$ is: \begin{IEEEeqnarray}{lCl} P(all fail) & = & \overbrace{ \Big( 1 - \frac{1}{\sqrt{n}} \Big) \cdot \Big( 1 - \frac{1}{\sqrt{n}} \Big) \cdot \ldots \Big( 1 - \frac{1}{\sqrt{n}} \Big)}^{c_1 \sqrt{n}\log{n} \text{ factors}} \\ & = & \Big( 1 - \frac{1}{\sqrt{n}} \Big)^{c_1 \sqrt{n}\log{n}} \\ & = & e^{-c_1 \log{n}} = n^{-c_1} \end{IEEEeqnarray} The probability that we succeed in some iteration of the algorithm is: \begin{IEEEeqnarray}{lCl} P(success) & = & \Big( 1- P(all fail) \Big) P(\varepsilon_1^j) \\ & \geq & \Big( 1-n^{-c_1} \Big) \Big( 1-en^{-r} \Big) \end{IEEEeqnarray} Therefore, the following theorem holds:\\[0.5cm] \begin{thm} \label{1} For any constants $c_1, r > 1$, given two vertices $s, t$, $s \not= t$, of the normalized uniform random temporal clique, $K_n$, the probability to arrive, starting from $s$, to $t$ at time at most \[t_0 = c_1 \sqrt{n} (\log{n}) k + \sqrt{n} \text{, where }k=r\log{n}\] is at least \[ \Big( 1-n^{-c_1} \Big) \Big( 1-en^{-r} \Big).\] \end{thm} \noindent \textit{Remark.} For the ``on-line" case, where a traveller starts from $s$ and wants to find a small journey to $t$, but he can only see the edges (\textit{arcs}) out of visited vertices, we conjecture that Algorithm \ref{alg:extend} gives a very tight bound on the expected arrival time. \section{Temporal Diameter}\label{sec:td} In this section, we study the concept of the temporal diameter of a uniform random temporal graph. \begin{mydef} Consider an instance $G(L)$ of a U-RTG. We denote the maximum of all distributional temporal distances between all pairs of vertices of $G(L)$ by $d(G(L))$: \[ d(G(L)) = max_{s,t \in V(G)} \delta ' (s,t). \] We define $diam(G(L)) = min\{d(G(L)), n'\}$. Then, the Expected or Temporal Diameter of G, denoted by $TD$, is given by the following formula: \[TD(G) = E\Big(diam\big(G(L)\big)\Big) =\sum_L diam\big(G(L)\big) \cdot P(L)\] , where $P(L)$ is the probability for labelling $L$ to occur. \end{mydef} We can easily prove that every temporal graph's temporal diameter, $TD$, is equal or greater than its maximum expected temporal distance, $MD$. \begin{thm} It holds that: \[ TD(G) \geq MD(G),\text{ for every temporal graph } G. \] \end{thm} \begin{proof} To prove this, we use the Reverse Fatou's Lemma[4]:\\ \begin{them} [Reverse Fatou's lemma] If $X_n\geq 0$, for all $n$, then \[ E(lim_n sup X_n) \geq lim_n sup E(X_n). \] \end{them} In other words, the expected value of the maximum of a set of random variables is at least equal to the maximum of the expected values of those variables. Now, notice that the Temporal Diameter of a temporal graph $G$ is actually the expected value of the maximum of all distributional temporal distances, that is $E(max_{s,t \in V(G)} \delta ' (s,t))$, in the case where we have $\delta ' (s,t) \leq n'$, for every pair of vertices $s,t \in V(G)$. In that case, the Maximum Expected Temporal Distance of $G$ is actually the maximum of the expected values of all pairs of vertices' distributional temporal distances, that is $max_{s,t \in V(G)} E(\delta ' (s,t))$. Therefore, in that case, using the above described Reverse Fatou's Lemma, we conclude that: \[ TD(G) \geq MD(G). \] In the case, where there is at least one pair of vertices $s,t \in V(G)$ such, that $\delta '(s,t) \geq n'$, both the temporal diameter and the maximum expected temporal distance of $G$ are equal to $n'$. Thus, we conclude that it generally applies that: \[ TD(G) \geq MD(G),\text{ for every temporal graph } G. \] \end{proof} We will now prove that the time $t_0 - o(t_0)$ (see. Theorem \ref{1}) is an upper bound of the normalized uniform random temporal clique's temporal diameter, $TD$, and, thus, is an upper bound of its maximum expected temporal distance, $MD$. \begin{thm} The quantity $t_0 - o(t_0)$ is an upper bound of both the temporal diameter, $TD$, and the maximum expected temporal distance, $MD$, of the normalized U-RT clique. \end{thm} \begin{proof} Let $s,t$ be two vertices of the normalized U-RT clique. We call $E_{st}$ the following event: \[ \text{``We arrive, starting from $s$, to $t$ at time at most $t_o$''}\] where $t_0 = c_1 \sqrt{n} (\log{n}) k + \sqrt{n}, ~ c_1 >1, k= r\log{n},~ r>1$.\\ It holds that: \begin{IEEEeqnarray}{lCl} P(E_{st}) & \geq & \Big( 1-n^{-c_1} \Big) \Big( 1-en^{-r} \Big) \\ & \geq & 1- n^{-c_1} - e n ^{-r} \end{IEEEeqnarray} For $r=c_1$, the above relation becomes: \begin{IEEEeqnarray}{lCl} P(E_{st}) & \geq & 1- n^{-c_1} - e n ^{-c_1} \\ & \geq & 1- 2 e n ^{-c_1} \end{IEEEeqnarray} Therefore, the probability that the complement of $E_{st}$ occurs is: \begin{IEEEeqnarray}{lCl} P(\overline{E}_{st}) &= & 1- P(E_{st}) \\ & \leq & 2 e n ^{-c_1} \end{IEEEeqnarray} Thus, the probability that there exist two vertices $s,t$ such that we arrive, starting from $s$, to $t$ at time greater than $t_0$ is: \begin{IEEEeqnarray}{lCl} P(\exists s,t : \overline{E}_{st}) & \leq & n (n-1) 2 e n ^{-c_1} \\ & \leq & 2 e n ^{-c_1 -2} \end{IEEEeqnarray} Let us denote by $T$ the $max\{ a_{st}, s,t \in V(K_n)\}$, where $a_{st}$ is the greatest arrival time amongst all ($s,t$)-journeys' arrival times. Then, we have: \begin{IEEEeqnarray}{lCl} P(\exists s,t : \overline{E}_{st}) & = & P(T > t_0) \\ & \leq & 2 e n ^{-c_1 -2} \end{IEEEeqnarray} It is: \begin{IEEEeqnarray}{lCl} TD & \leq & E(max\{ a_{st}, s,t \in V(K_n)\}) \\ & \leq & ( 1 - 2 e n ^{-c_1 -2} ) \cdot t_0 + n \cdot 2 e n ^{-c_1 -2} \\ & \leq & t_0 -o(t_0) \end{IEEEeqnarray} Since $TD(G) \geq MD(G),\text{ for every temporal graph } G $, we conclude that in the case of the normalized U-RT clique, it is: \[ MD \leq TD \leq t_0 -o(t_0) \] \end{proof} \section{An optimization problem: The Bridges' problem} \label{sec:bridge} We will now study an optimization problem concerning the temporal multigraph shown in Figure \ref{fig:poly1}. \begin{figure}[htbp] \begin{center} \[\begin{tikzpicture}[thick,scale=0.95,->,shorten >=2pt] \vertex (1) at (0,0) [label=left:$s$] {}; \vertex (2) at (10,0) [label=right:$t$] {}; \path (1) edge [bend left=70] node[above]{$1$} (2) (1) edge [bend left=48] node[above]{$2$} (2) (1) edge [bend left=30] node[above]{$3$} (2) (1) edge [bend left=15] node[above]{$4$} (2) (1) edge node[above]{$5$} (2) (1) edge [line width=0pt,black!0.05,bend right=18] node[above]{\textcolor{black}{$\vdots$}} (2) (1) edge [bend right=30] node[above]{$n$} (2) \end{tikzpicture}\] \end{center} \rule{35em}{0.5pt} \caption{The bridges' problem} \label{fig:poly1} \end{figure} \noindent\underline{\textbf{\textit{The problem}}}\\ $n$ people are located on one bank of a river (see vertex $s$, Figure \ref{fig:poly1}) and want to go to the other side (see vertex $t$, Figure \ref{fig:poly1}). Each one can go across one of a total of $n$ bridges that connect the two riversides, paying individual cost equal to $\displaystyle{1 + \frac{i}{m_i}}$, where $i$ stands for the number of the bridge they pass and $m_i$ stands for the total sum of people that cross that bridge. Thus, the total cost payed by $m$ people to cross the \textbf{$i^{th}$} bridge, $i=1,2,\ldots, n$, is:\[cost[i]=m_i+i\] \noindent We denote by \emph{maximum cost payed} the maximum, over all bridges $i$, cost $m_i +i$: \[maximum~cost~payed = max\{m_i + i, i=1,2, \ldots, n :~bridge\} \] \noindent How should the $n$ people be assigned to the bridges so, that the maximum cost payed is minimized?\\ We denote the minimum, over all assignments of $n$ people to $n$ bridges, maximum cost payed by $OPT$, that is: \[ OPT = min_{all~ assignments}\{ maximum ~cost~ payed\} \] \noindent \textit{Remark.} In another interpretation of the bridges' problem, as we call the above described problem, we consider the multi-labeled temporal digraph of two vertices $s,t$ and one single edge $\{s,t\}$ which is assigned the discrete time labels $1,2,\ldots, n$ (see figure \ref{fig:poly2}). \begin{figure}[htbp] \begin{center} \[\begin{tikzpicture}[thick,scale=0.95,->,shorten >=2pt] \vertex (1) at (0,0) [label=left:$s$] {}; \vertex (2) at (6,0) [label=right:$t$] {}; \path (1) edge node[above]{$1, 2, \ldots, n$} (2) \end{tikzpicture}\] \end{center} \rule{35em}{0.5pt} \caption{The bridges' problem (another interpretation)} \label{fig:poly2} \end{figure} \noindent Here we have a single bridge which is available everyday from day $1$ to day $n$. As time progresses the cost someone needs to pay to move from $s$ to $t$ increases. Again, one has to pay individual cost equal to $\displaystyle{1 + \frac{i}{m_i}}$, where $i$ stands for the day on which he decides to move from $s$ to $t$ and $m_i$ stands for the total sum of people decide to move from $s$ to$t$ on that same day. Therefore, the total cost payed by $m$ people who move from $s$ to $t$ on the \textbf{$i^{th}$} day, $i=1,2,\ldots, n$, is:\[cost[i]=m_i+i\] \begin{thm} We can compute the assignment of $n$ persons to $n$ bridges that achieves the $OPT$ in polynomial time $O(n^2)$. \end{thm} \begin{proof} We provide Algorithm \ref{alg:bridge} and show that it computes the assignment that achieves $OPT$. \begin{algorithm}[h] \caption{The bridges problem solving algorithm} \label{alg:bridge} \begin{algorithmic}[1] \Procedure {Bridges}{$n$} \State cost[] is a 1$\times$n array which holds the bridges' costs; \State content[] is a 1$\times$n array which holds the bridges' contents; \Comment{a.k.a \\ \hfill how\\ \hfill many people\\ \hfill are on each\\ \hfill bridge} \State m := n; \Comment{m is the number of bridges} \For {i = 1 ... m} \State content[i] := 0; \Comment{Initializations} \State cost[i] := i; \EndFor \algstore{alg:bridge} \end{algorithmic} \end{algorithm} \clearpage \begin{algorithm} \ContinuedFloat \caption{The bridges problem solving algorithm (continued)} \begin{algorithmic} \algrestore{alg:bridge} \For {i = 1 ... n} \State bridge := 1; \Comment{Initialize the bridge that the $i^{th}$ person will pass} \For { j = 2 ... m} \Comment{Find the bridge that gives the minimum\\ \hfill possible cost} \If { cost[j] $<$ cost[bridge] } \State bridge := j; \EndIf \EndFor \State content[bridge] := content[bridge]+1; \Comment{Add the $i^{th}$ person to \\ \hfill the selected \\ \hfill bridge's content} \State cost[bridge] := cost[bridge]+1; \Comment{Calculate the right new cost} \EndFor \For {i = 1 ... m} \If {content[i] == 0} \State cost[i] := 0; \EndIf \If {content[i] == 1} \State \textbf{Write} content[i] , `` person passes bridge \#'', i , `` who $~~~~~~~$ $~~~~~~~~~~~~~~$ therefore has to pay cost equal to '', cost[i]; \Else \State \textbf{Write} content[i] , `` people pass bridge \#'', i , `` who therefore $~~~~~~~~~~~~~~~~$ have to pay cost equal to '', cost[i]; \EndIf \Comment{Print the bridges' costs} \EndFor \EndProcedure \end{algorithmic} \end{algorithm} The algorithm assigns the $i^{th}$ person to the bridge, for which the current minimum cost is payed. If there are more than one such bridges, the algorithm assigns the $i^{th}$ person to the first one in order. It is trivial to see that the algorithm's running time is $O(n^2)$.\\ \noindent\textbf{Proof of correctness} We will prove the validity of the algorithm \ref{alg:bridge} by induction on the number $n$ of persons. \begin{itemize} \item For $n=1$, the algorithm sets the number of bridges to be $m=1$ and the sole bridge's content and cost to be equal to 1. In the main loop, the sole person is assigned to the bridge, paying cost equal to:\[cost[1] = 2\] So, actually, the algorithm solves the problem for $n=1$ person. \item Assume that the algorithm solves the problem for $n=k$ people. \item We will show that the algorithm solves the problem for $n=k+1$ people. Before continuing, let us consider the following: Let $n_1, n_2 \in \mathbb{N}$ numbers of people, with $n_1>n_2$. It is obvious that the minimum possible maximum cost for $n=n_1$ people is at least equal to the minimum possible maximum cost for $n=n_2$ people. Let us observe now that the procedures performed by the algorithm in the main loop for $k$ people, and the results obtained through these, are identical to those performed and obtained respectively for $k+1$ people, except that for $k+1$ people, there is a $(k+1)^{th}$ bridge, which throughout the execution of these processes has zero content, and there is also an additional execution of the loop. At the beginning of this $(k+1)^{th}$ execution, the algorithm has already assigned the $k$ people to the brisges in a way that we obtain the minimum possible maximum cost. The algorithm, by construction, assigns the people to the bridges in a way that their costs are ordered by (not necessarily strictly) descending order and indeed one of the following two possible events occur: \begin{equation} \left\{ \begin{array}{l} \text{all the bridges have the same cost, denoted by } OPT\\ \text{or}\\ \text{some bridges have cost } OPT \text{ and some others have cost }OPT-1. \end{array} \right. \end{equation} In the second case, the algorithm is obviously going to assign the $(k+1)^{th}$ person to the first in order bridge that has cost equal to $OPT-1$, thereby maintaining the maximum cost that occurs on the bridges to a minimum, that is $OPT$. In the first case, if $r \leq k+1$ is the number of the last bridge that has positive content, $content[r]$, then it is: \begin{equation} \left\{ \begin{array}{l} r+content[r] = OPT\\ \text{But: } content[r] \geq 1 \text{ and so: } r+ content[r] \geq r+1 \end{array} \right\} \Rightarrow OPT \geq r+1 \end{equation} Also, since the$(r+1)^{th}$ bridge has zero content, it is:\[cost[r+1] = r+1\] The algorithm checks which of the $ k +1 $ bridges has the minimum cost to assign the $(k+1)^{th}$ person to that bridge. If $OPT=r+1$, then the algorithm assigns the last person to the $1^{st}$ bridge. Otherwise, it assigns it to the $ (r +1)^{th} $ bridge. This way, it ensures the minimum possible maximum cost for the $ k +1 $ bridges. Therefore, the algorithm solves the problem for $ n = k +1 $ people. \end{itemize} \end{proof} We will now calculate the value of the $OPT$. Again, let us denote by $r$ the number of bridges that have a positive content, i.e. are not empty, in the optimal case which the Algorithm \ref{alg:bridge} computes. For the sake of brevity, let us also denote by $l_i$ the content of the $i^{th}$ bridge. Since the average cost of the non empty bridges is equal or less than the maximum cost that occurs on those bridges, the following holds for the optimal case: \[ \frac{\displaystyle\sum_{i=1}^{r} (i+l_i)}{r} \leq OPT \] Therefore, we have: \begin{equation}\label{eq:4} \displaystyle\sum_{i=1}^{r} (i+l_i) \leq r OPT \end{equation} Furthermore, it is easy to see that since, in the optimal case that the algorithm computes, the $OPT$ is greater than any bridge's cost by at most \textit{one}, it holds that: \begin{equation}\label{eq:5} rOPT - r \leq \displaystyle\sum_{i=1}^{r} (i+l_i) \end{equation} By the relations \eqref{eq:4} and \eqref{eq:5}, we have: \begin{IEEEeqnarray}{lCCCl l} rOPT - r & \leq & \displaystyle\sum_{i=1}^{r} (i+l_i) & \leq & rOPT & \Leftrightarrow \\ rOPT - r & \leq & \displaystyle\sum_{i=1}^{r} i +\displaystyle\sum_{i=1}^{r} l_i & \leq & rOPT & \Leftrightarrow \\ rOPT - r & \leq & \frac{r(r+1)}{2} +n & \leq & rOPT & \Leftrightarrow \\ OPT - 1 & \leq & \frac{(r+1)}{2} +\frac{n}{r} & \leq & OPT \end{IEEEeqnarray} Now, the quantity $ \frac{(r+1)}{2} +\frac{n}{r} $ is minimized at $r=\sqrt{2n}$ and at that point, its value is equal to $\sqrt{2n} +\frac{1}{2}$. Therefore, we conclude that: \[OPT = \lceil \sqrt{2n} +\frac{1}{2} \rceil \] \section{Conclusions and further research}\label{sec:concl} There are several open problems related to the findings of the present work. We initiated here the random availability of edges where the selection of time-labels, and thus the selection of moments in time at which the edges are available, follows the uniform distribution. There are still other interesting approaches concerning what distribution the selection of time-labels could follow (see F-CASE in Section \ref{sec:def}). Another approach that is yet to be examined is that of the multi-labeled temporal graphs, on which we could search for statistical properties respective to the ones we studied within the present work. Yet another interesting direction which we did not consider in this work is to find upper bounds on the maximum expected temporal distance and the temporal diameter of any U-RTG (or F-RTG). Further research could also focus on calculating the actual value of these properties, e.g. in the case of the normalized uniform random temporal clique. \newpage \begin{thebibliography}{1} \thispagestyle{empty} [1] George Mertzios, Othon Michail, Ioannis Chatzigiannakis, and Paul G. Spirakis (2013). {\em Temporal Network Optimization Subject to Connectivity Constraints} Springer [2] D. Kempe, J. Kleinberg, and A. Kumar (2000). {\em Connectivity and inference problems for temporal networks} In Proceedings of the 32nd annual ACM symposium on Theory of computing (STOC) [3] M. Molloy and B. Reed (2002). {\em Graph colouring and the probabilistic method, volume 23} Springer [4] Durrett, R. (2010). {\em Probability: Theory and Examples, 4th Edition}, Cambridge University Press \end{thebibliography} \end{document}	arxiv-papers	2013-10-29T17:58:57	2024-09-04T02:49:53.036976	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Paul G. Spirakis, Eleni Ch. Akrida", "submitter": "Eleni Akrida", "url": "https://arxiv.org/abs/1310.7898" }
1310.7910	# Interaction effects on galaxy pairs with Gemini/GMOS- I: Electron density A. C. Krabbe1, D. A. Rosa1, O. L. Dors Jr.1, M. G. Pastoriza2, C. Winge3, G. F. Hägele4,5, M. V. Cardaci4,5 and I. Rodrigues1 1 Universidade do Vale do Paraíba, Av. Shishima Hifumi, 2911, Cep 12244-000, São José dos Campos, SP, Brazil 2 Instituto de Física, Universidade Federal do Rio Grande do Sul, Av. Bento Gonçalves, 9500, Cep 91359-050, Porto Alegre, RS, Brazil 3 Gemini Observatory, c/o AURA Inc., Casilla 603, La Serena, Chile 4 Instituto de Astrofísica de La Plata (CONICET La Plata–UNLP), Argentina. 5 Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La Plata, Paseo del Bosque s/n, 1900 La Plata, Argentina E-mail:[email protected] (Accepted -. Received -.) ###### Abstract We present an observational study about the impacts of the interactions in the electron density of H ii regions located in 7 systems of interacting galaxies. The data consist of long-slit spectra in the range 4400-7300 Å, obtained with the Gemini Multi-Object Spectrograph at Gemini South (GMOS). The electron density was determined using the ratio of emission lines [S II]$\lambda$6716/$\lambda$6731\. Our results indicate that the electron density estimates obtained of H ii regions from our sample of interacting galaxies are systematically higher than those derived for isolated galaxies. The mean electron density values of interacting galaxies are in the range of $N_{\rm e}=24-532$ $\rm cm^{-3}$, while those obtained for isolated galaxies are in the range of $N_{\rm e}=40-137\>\rm cm^{-3}$. Comparing the observed emission lines with predictions of photoionization models, we verified that almost all the H ii regions of the galaxies AM 1054A, AM 2058B, and AM 2306B, have emission lines excited by shock gas. For the remaining galaxies, only few H ii regions has emission lines excited by shocks, such as in AM 2322B (1 point), and AM 2322A (4 points). No correlation is obtained between the presence of shocks and electron densities. Indeed, the highest electron density values found in our sample do not belong to the objects with gas shock excitation. We emphasize the importance of considering theses quantities especially when the metallicity is derived for these types of systems. ###### keywords: galaxies: ISM ††pagerange: Interaction effects on galaxy pairs with Gemini/GMOS- I: Electron density–LABEL:lastpage††pubyear: 2012 ## 1 Introduction The study of physical processes involved in galaxy collisions and mergers in the local universe is fundamental to understand the formation and evolution of these objects, providing important constraints in simulations of the universe at large scale. In particular, the chemical abundance is highly modified in interacting/merging galaxies. Kewley et al. (2010) presented a systematic investigation about metallicity gradients in close pairs of galaxies. These authors determined the oxygen abundance (generally used as a tracer of the metallicity $Z$) along the disk of eight galaxies in close pairs and found metallicity gradients shallower than the ones in isolated galaxies. Similar results have been reached by Krabbe et al. (2011, 2008), who built spatial profiles of oxygen abundance of the gaseous phase of the galaxy pairs AM 2306-721 and AM 2322-2821. This flattening in the oxygen abundance gradient reflects the effects of gas redistribution along the galaxy disk due to metal- poor inflow of gas from outskirts of the centre of interacting galaxies (Rupke et al., 2010). The large-scale gas motion created by the interaction induces high star formation rate and galactic-scale outflows (Veilleux et al., 2005), producing shock excitation in star-forming regions, such as reported in recent studies of Luminous Infrared Galaxies by Soto & Martin (2012) and Rich et al. (2012, 2011). In particular, Rich et al. (2011), through integral field spectroscopic data of the Luminous Infrared Galaxies IC 1623 and NGC 3256, showed that broad line profiles are often associated with gas shock excitation in H ii regions located in mergers. Similar results were also found by Newman et al. (2012) for the clumpy star-forming galaxy ZC 406690 (see also Soto et al. 2012). These authors pointed out that the broad emission likely originates from large-scale outflows with high mass rates from individual star-forming regions. The changes in galaxies that experience an encounter seem to have a relation with the separation among the objects interacting, such as showed by Scudder et al. (2012). These authors, using spectroscopy data of a large sample of objects with a close companion taken from Sloan Digital Sky Survey Data Release, found that the metallicity gradient and the star formation rate (SFR) are correlated with the separation of the galaxy pairs analysed, in the sense that the gradients are flatter and the SFR are higher at smaller separations. Despite recent efforts to probe the properties of interacting galaxies, the electron density of star-forming regions have been poorly determined in these systems, as well as its correlation with other quantities (e.g. $Z$, SFR). In galaxy disks of interacting galaxies, where gas motions and gas excited by shock are present, high electron density is expected and, can be used as a signature of the presence of these motions and shocks. In fact, Puech et al. (2006), in a study about galaxy interaction, mapped electron densities in six distant galaxies (z $\sim$ 0.55) and found that the highest electron density values observed could be associated to the collision between molecular clouds of the interstellar medium and gas inflow/outflow events. These authors derived electron density values lower than 400 $\rm cm^{-3}$, typical of classical H ii regions (Copetti et al., 2000; Castañeda et al., 1992). However, Puech et al. (2006) used as a sensor the [O ii]$\lambda$3729/$\lambda$3727 ratio, which underestimates the electron density in relation to determinations via other line ratios (Copetti & Writzl, 2002). Most of oxygen determinations of the gas phase in interacting galaxies (e.g. Scudder et al. 2012; Rich et al. 2012, 2011; Krabbe et al. 2011; Kewley et al. 2010; Krabbe et al. 2008) are based on theoretical models that consider low electron density values of 10-200 $\rm cm^{-3}$ (Krabbe et al., 2011; Dors et al., 2011; Kewley & Dopita, 2002; Dopita et al., 2000). If the electron density values considerably differ of those considered in the models, the oxygen abundance estimations will be doubtful. In fact, Oey & Kennicutt (1993) showed that systematic variations in the nebular density introduce significant uncertainties into the abundances obtained using methods based on strong emission lines. They found that differences between 10 and 200 $\rm cm^{-3}$, a typical range for the electron density derived in giant H ii regions (e.g. Copetti et al. 2000; Castañeda et al. 1992; Kennicutt 1984; O’dell & Castañeda 1984), reflect variations up to 0.5 dex in oxygen abundances, mainly for the high metallicity regime. These variations can increase even more when higher electron density values, such as the ones found in star-forming clumps (e.g. 300-1800 $\rm cm^{-3}$, Newman et al. 2012), are considered in abundance determinations. In this paper, we used long-slit spectroscopic data of a sample of seven pair galaxies to verify the effects of the interaction on the electron density in these systems. This work is organized as follows. In Section 2, we summarize the observations and data reduction. In Section 3, the method to compute the electron density is described. Results and discussion are presented in Sections 4 and 5, respectively. The conclusions of the outcomes are given in Section 6. Table 1: Galaxy sample. ID \| Morphology \| $\alpha$(2000) \| $\delta$(2000) \| $cz\,(\rm km/s)$ \| $m_{\rm B}$ (mag) \| Others names ---\|---\|---\|---\|---\|---\|--- AM 1054-325 \| Sm [2] \| $10^{\rm{h}}56^{\rm{m}}58\aas@@fstack{s}2$ \| $-33^{\rm{h}}09^{\rm{m}}52\aas@@fstack{s}0$ \| 3 788 [10] \| 14.55 [2] \| ESO 376-IG 027 \| Sa [5] \| 10 57 04.2s \| $-$33 09 21.0 \| 3 850 [5] \| 15.41 [8] \| ESO 376- G 028 AM 1219-430 \| Sm [6] \| 12 21 57.3 \| $-$43 20 05.0 \| 6 957 [3] \| 14.30 [7] \| ESO 267-IG 041 \| S? [6] \| 12 22 04.0 \| $-$43 20 21.0 \| 6 879 [3] \| - \| FAIRALL 0157 AM 1256-433 \| E [3] \| 12 58 50.9 \| $-$43 52 30.0 \| 9 215 [3] \| 14.75 [8] \| ESO 269-IG 022 NED01 \| E [3] \| 12 58 50.6 \| $-$43 52 53.0 \| 9 183 [3] \| 16.17 [8] \| ESO 269-IG 022 NED02 \| SBC [3] \| 12 58 57.6 \| $-$43 50 11.0 \| 9 014 [3] \| 16.41 [1] \| ESO 269-IG 023 NED01 AM 2058-381 \| Sbc [6] \| 21 01 39.1 \| $-$38 04 59.0 \| 12 383 [3] \| 14.91 [1] \| ESO 341- G 030 \| ? \| 21 01 39.9 \| $-$38 05 53.0 \| 12 460 [3] \| 16.24 [1] \| ESO 341- G 030 NOTES01 AM 2229-735 \| SO? [3] \| 22 33 43.7 \| $-$73 40 47.0 \| 17 535 [3] \| 15.98 [1] \| AM 2229-735 NED01 \| ? \| 22 33 48.3 \| $-$73 40 56.0 \| 17 342 [3] \| 17.36 [1] \| AM 2229-735 NED02 AM 2306-721 \| SAB(r)c \| 23 09 39.3 \| $-$71 01 34.0 \| 8 919 [4] \| 14.07 [1] \| ESO 077- G 003 \| ? \| 23 09 44.5 \| $-$72 00 04.0 \| 8 669 [4] \| 14.47 [1] \| ESO 077-IG 004 AM 2322-821 \| SA(r)c \| 23 26 27.6 \| $-$81 54 42.0 \| 3 680 [3] \| 13.35 [1] \| ESO 012- G 001, NGC 7637 \| ? \| 23 25 55.4 \| $-$81 52 41.0 \| 3 376 [4] \| 15.41 [1] \| ESO 012- G 001 NOTES01 References: [1] Ferreiro & Pastoriza (2004); [2] Weilbacher et al. (2000); [3] Donzelli & Pastoriza (1997); [4] Krabbe et al. (2011); [5] Lauberts (1982)); [6] Paturel et al. (2003); [7] de Vaucouleurs et al. (1991); [8] Lauberts & Valentijn (1989); [9] Huchra et al. (2012); [10] Jones et al. (2009) Conventions: $\alpha$, $\delta$: equatorial coordinates ## 2 Observations and Data Reduction We have selected several systems from Ferreiro & Pastoriza (2004) to study the effects of the kinematics, stellar population, gradient abundances, and electron densities of interacting galaxies. The first results of this programme were presented for AM2306-721 (Krabbe et al., 2008) and AM2322-821 (Krabbe et al., 2011). Table 1 summarizes the main characteristics of the systems: identification, morphology, position, radial velocity, apparent B magnitude, and other designations. Long-slit spectroscopic data were obtained on May, June, and July 2006 and 2007; and July 2008 with the Gemini Multi-Object Spectrograph (GMOS-S) attached to the 8 m Gemini South telescope, Chile, as part of the poor weather programs GS-2006A-DD-6, GS-2007A-Q-76, and GS-2008A-Q-206. Spectra in the range 4400-7300 Å were acquired with the B600 grating, and 1$\arcsec$ slit width, assuming a compromise between spectral resolution (5.5 Å), spectral coverage, and slit losses (due to the Image Quality = ANY constraint). The frames were binned on-chip by 4 and 2 pixels in the spatial and spectra directions, respectively, resulting in a spatial scale of 0.288 $\arcsec$px-1, and dispersion of 0.9 Å px-1. Spectra were taken at different position angles on the sky, with the goal of observing the nucleus and the brightest regions of the galaxies. The exposure time on each single frame was limited to 700 seconds to minimize the effects of cosmic rays, with multiple frames being obtained for each slit position to achieve a suitable signal. The slit positions for each system are shown in Fig. 1, superimposed on the GMOS-S r$\arcmin$ acquisition images. Table 2 gives the journal of observations. Conditions during the observing runs were not photometric, with thin cirrus and image quality in the range 0.6$\arcsec$ to 1.7$\arcsec$ (as measured from stars in the acquisition images taken just prior to the spectroscopic observations). The spectroscopic data reduction was carried out using the gemini.gmos package and generic IRAF111Image Reduction and Analysis Facility, distributed by NOAO, operated by AURA, Inc., under agreement with NSF. tasks. We followed the standard procedure: (1) the data were bias subtracted and flat-fielded; (2) the wavelength calibration was established from the Cu-Ar arc frames with typical residuals of 0.2 Å and applied to the object frames; (3) the individual spectra of same slit positions and wavelength range were averaged with cosmic ray rejection; (4) the object frames were sky subtracted interactively using the gsskysub task, which uses a background sample of off- object areas to fit a function to the specified rows, and this fit was then subtracted from the column of each spectra; (5) the spectra were relative flux calibrated using observations of a flux standard star taken with the same set up as the science observations; (6) finally, one-dimensional spectra were extracted from the two-dimensional spectra by summing over four rows along the spatial direction. Each spectrum, therefore, comprises the flux contained in an aperture of 1$\arcsec\times 1.152\arcsec$. The intensities of the H$\beta$, [O iii]$\lambda$5007, [O i]$\lambda$6300, H$\alpha$, [N ii]$\lambda$6584, and [S ii]$\lambda\,$6716,$\lambda$ 6731 emission lines were measured using a single Gaussian line profile fitting on the spectra. We used the IRAF splot routine to fit the lines, with the associated error being given as $\sigma^{2}=\sigma_{cont}^{2}+\sigma_{line}^{2}$, where $\sigma_{cont}$ and $\sigma_{line}$ are the continuum rms and the Poisson error of the line flux, respectively. Furthermore, we considered only measurements whose continuum around $\lambda$ 6700 Å reach a signal-to-noise S/N $\geq$ 8\. The emission line intensities were not corrected for the interstellar extinction, because it is negligible due to the small separation between the [S ii]$\lambda 6716$ and $\lambda 6731$ emission lines. Table 2: Journal of observations. Object \| Date \| Exposure \| PA (°) \| $\Delta\,\lambda$ (Å) ---\|---\|---\|---\|--- \| \| Time (s) \| \| AM 1054-325 \| 2007-06-21 \| 4 $\times$ 600 \| 77 \| 4280-7130 AM 1219-430 \| 2007-06-06 \| 4 $\times$ 600 \| 25 \| 4280-7130 \| 2007-05-26 \| 4 $\times$ 600 \| 162 \| 4280-7130 \| 2007-06-22 \| 4 $\times$ 600 \| 341 \| 4280-7130 AM 1256-433 \| 2007-07-06 \| 4 $\times$ 600 \| 292 \| 4280-7130 \| 2007-06-21 \| 4 $\times$ 600 \| 325 \| 4280-7130 AM 2058-381 \| 2006-05-20 \| 4 $\times$ 600 \| 42 \| 4351-7213 \| 2007-05-26 \| 4 $\times$ 600 \| 94 \| 4351-7213 \| 2007-05-24 \| 4 $\times$ 600 \| 125 \| 4351-7213 \| 2007-05-30 \| 4 $\times$ 600 \| 350 \| 4351-7213 AM 2229-735 \| 2006-07-20 \| 6 $\times$ 600 \| 134 \| 4390-7250 \| 2006-07-16 \| 6 $\times$ 600 \| 161 \| 4390-7250 AM 2306-721 \| 2006-06-20 \| 4 $\times$ 600 \| 118 \| 4280-7130 \| 2006-06-20 \| 4 $\times$ 600 \| 190 \| 4280-7130 \| 2006-06-20 \| 4 $\times$ 600 \| 238 \| 4280-7130 AM 2322-821 \| 2006-07-01 \| 3 $\times$ 700 \| 59 \| 4280-7130 \| 2008-07-27 \| 6 $\times$ 600 \| 60 \| 4280-7130 \| 2006-06-30 \| 6 $\times$ 600 \| 318 \| 4280-7130 Figure 1: The slit positions for each system are shown superimposed on the GMOS-S r$\arcmin$ acquisition image. Figure 2: A sample of spectra in the range of 6600 to 6800 Å from areas of different galaxies. The flux scale was normalized to the peak of [S ii] $\lambda$ 6716. ## 3 Determination of the electron density The electron density $N_{\rm e}$ was derived from the [S ii]$\lambda\,$6716/$\lambda$ 6731 emission line intensity ratio by solving numerically the equilibrium equation for a $n$-level atom approximation using the temden routine of the nebular package of the STSDAS/IRAF, assuming an electron temperature of 10 000 K, because temperature sensitive emission lines were unobservable in our sample. The references for the collision strengths, transition probabilities, and energy levels are Ramsbottom et al. (1996), Verner et al. (1987), Keenan et al. (1993), and Bowen et al. (1960). There are two main sources of errors in the determination of electron densities. One is the dependence of the $N_{\rm e}$ on the electron temperature $T_{\rm e}$ assumed. However, this dependence is weak in the range of temperatures usually found in galactic H ii regions (e.g. Copetti et al., 2000). We adopted a mean electron temperature of 10 000 K as a representative value, because it is a typical electron temperature value for these kinds of objects and there are no estimations for our sample. The other main source of error is the saturation of the line ratio for both low and high values of the electron density, which makes the [S ii]$\lambda$ 6716/$\lambda$ 6731 ratio a reliable sensor of the electron density in the range of 2.45 $<$ log $N_{\rm e}$ (cm${}^{-3})$ $<$ 3.85 (Stanghellini et al., 1989). ## 4 Results Fig. 2 shows a sample of the spectra of some H ii regions of the galaxies around the [S ii]$\lambda$6716 and [S ii]$\lambda$6731 emission lines. The profiles of $\log$([O i]$\lambda$6300/H$\alpha$), [S ii]$\lambda$6716/$\lambda$6731 ratio, and $N_{\rm e}$ as a function of galactocentric radius for the galaxies are shown in Figs. 4-14. The intensity of $\log$([O i]$\lambda$6300/H$\alpha$) was plotted only for the apertures for which the electron density determination was possible. The galactocentric radius are not corrected by galaxy inclination. In Fig. 1 the adopted centre of each galaxy is marked with a red cross. Table 3 presents some statistics of the [S ii] $\lambda$ 6716/$\lambda$ 6731 ratio and electron density measurements, including the number N of distinct nebular areas, the mean, the median, the maximum and minimum, and the standard deviation $\sigma$. The results for each system are presented separately. Table 3: [S ii] ratio and electron density statistics. \| \| [S ii] $\lambda\,6716/\lambda\,6731$ \| \| $N_{{\rm e}}~{}(\mathrm{cm^{-3}})$ ---\|---\|---\|---\|--- Objects \| \| N \| mean \| median \| max \| min \| $\sigma$ \| \| N \| mean \| median \| max \| min \| $\sigma$ AM 1054A \| \| 16 \| 1.19 \| 1.08 \| 1.70 \| 0.97 \| 0.93 \| \| 13 \| 434 \| 462 \| 681 \| 65 \| 191 AM 1054B \| \| 3 \| 0.86 \| 0.85 \| 0.92 \| 0.79 \| 0.07 \| \| 3 \| 1130 \| 1082 \| 1476 \| 833 \| 324 AM 1219A \| \| 29 \| 1.12 \| 1.06 \| 1.77 \| 0.86 \| 0.23 \| \| 26 \| 532 \| 518 \| 1073 \| 85 \| 286 AM 1219B \| \| 5 \| 0.70 \| 0.82 \| 0.92 \| 0.23 \| 0.27 \| \| 4 \| 1408 \| 1294 \| 2189 \| 855 \| 564 AM 1256B \| \| 43 \| 1.48 \| 1.42 \| 2.08 \| 0.99 \| 0.27 \| \| 22 \| 181 \| 317 \| 626 \| 7 \| 168 AM 2058A \| \| 20 \| 1.38 \| 1.26 \| 2.22 \| 0.90 \| 0.37 \| \| 13 \| 376 \| 318 \| 911 \| 33 \| 263 AM 2058B \| \| 8 \| 1.47 \| 1.42 \| 1.80 \| 1.24 \| 0.19 \| \| 4 \| 86 \| 60 \| 184 \| 42 \| 66 AM 2229A \| \| 33 \| 1.60 \| 1.59 \| 2.61 \| 0.19 \| 0.59 \| \| 7 \| 346 \| 226 \| 686 \| 28 \| 280 AM 2306A \| \| 8 \| 1.41 \| 1.44 \| 1.60 \| 1.16 \| 0.16 \| \| 5 \| 131 \| 107 \| 298 \| 32 \| 99 AM 2306B \| \| 15 \| 1.38 \| 1.30 \| 2.88 \| 0.92 \| 0.47 \| \| 11 \| 300 \| 212 \| 826 \| 19 \| 273 AM 2322A \| \| 81 \| 1.41 \| 1.42 \| 1.89 \| 0.85 \| 0.20 \| \| 41 \| 200 \| 103 \| 1121 \| 11 \| 259 AM 2322B \| \| 23 \| 1.47 \| 1.43 \| 1.74 \| 1.35 \| 0.10 \| \| 12 \| 24 \| 15 \| 75 \| 3 \| 23 ### 4.1 AM 1054-325 This system is composed by a peculiar spiral with disturbed arms (hereafter AM 1054A) and a spiral-like object (hereafter AM 1054B). AM 1054A contains very luminous H II regions along their galactic disk. As can be seen in the Fig. 1, AM 1054A seems to have two nuclei. According to measurements obtained from Weilbacher et al. (2000), the “main” nucleus of this galaxy (ESO 376-IG 027) is the reddest [(B-V)=0.52], while the other (ESO-LV 3760271) has the blue colours of a strong starburst [(B-V)=0.21]. Both nuclei names are marked in the Fig. 1. The measured radial velocity is 3788 km/s (Jones et al., 2009) and 3853 km/s (Sekiguchi & Wolstencroft, 1993) for ESO 376-IG 027 and ESO-LV 3760271, respectively. Therefore, the small difference found between their radial velocities together with the perturbed morphology of the galaxy seem to indicate that these objects are gravitationally bound. For AM 1054A, the electron density values estimated from [S ii]$\lambda\,6716/\lambda\,6731$ ratio (see Fig. 4) present variations of relatively high amplitude along the radius of the galaxy, with the minimum value of $N_{\rm e}$= 65 $\mathrm{cm^{-3}}$ and the maximum of $N_{\rm e}$= 681 $\mathrm{cm^{-3}}$. We found a mean density of $N_{\rm e}=434\pm 53$ $\mathrm{cm^{-3}}$. In this galaxy, the slit position is cutting a bright star-forming region, but does not cross the nucleus of the galaxy. For AM1054B, only few apertures had the [S ii]$\lambda\lambda$ 6716, 6731 emission lines with enough signal to be measured. A mean density of $N_{\rm e}=1\,130\pm 187$ $\mathrm{cm^{-3}}$ was derived for this galaxy. Figure 3: AM 1054-325. $\log$([O i $\lambda$ 6300/H$\alpha$) ratio, [S ii] $\lambda$ 6716/$\lambda$ 6731 ratio, and $N_{\rm e}$ as a function of galactocentric radius for AM 1054A. Figure 4: Same as Fig. 4, but for AM 1054B. ### 4.2 AM 1219-430 This pair is composed by a disturbed spiral (hereafter AM 1219A) and a smaller disk galaxy (AM 1219B). AM 1219A shows a tidal tail produced by the interaction of the galaxies, with very bright H ii regions. Systemic velocities of 6 957 km s-1 and 6 879 km s-1 were estimated by Donzelli & Pastoriza (1997) for AM 1219A and AM1219B, respectively. The distribution of electron densities exhibits variations of high amplitude across the radius of the main galaxy in the range of $N_{\rm e}=85-1073$ $\mathrm{cm^{-3}}$. We found a mean density of $N_{\rm e}=532\pm 56$ $\mathrm{cm^{-3}}$. As in the case of AM 1054B, only for few apertures of AM 1219B the [S ii]$\lambda\lambda$ 6716,6731 emission lines have enough signal to be measured. A mean density of $N_{\rm e}=1\,408\pm 282$ $\mathrm{cm^{-3}}$ was derived for this galaxy. Interestingly, the $N_{\rm e}$ increases toward the outskirt of this galaxy. This region is at the end of the spiral arm to the Northwest. Figure 5: Same as Fig. 4, but for AM 1219A. Figure 6: Same as Fig. 4, but for AM 1219B. ### 4.3 AM 1256-433 AM 1256-433 is a system constituted by three galaxies. Two are elliptical with very bright nuclei, ESO 269-IG 022 NED01 and ESO 269-IG 022 NED02, and one very disturbed spiral galaxy, ESO 269-IG 023 NED01, hereafter AM 1256B. In addition, an isolated disk galaxy, ESO 269-IG 023 NED02/PGC 543979 ($\alpha=12^{\rm{h}}59^{\rm{m}}00\aas@@fstack{s}6$ and $\delta=-43^{\rm{h}}50^{\rm{m}}23^{\rm{s}}$ J2000), appears in the field of view of this system, about 30$\arcsec$ to the Southeast of the centre of AM 1256B. From our data, we obtained for this isolated galaxy a heliocentric velocity of 18 896 km s-1 indicating that it does not belong to this system, and it was incorrectly associated with AM 1256-433 by Donzelli & Pastoriza (1997); Ferreiro & Pastoriza (2004), and Ferreiro et al. (2008). In Fig. 1, only AM 1256B and the isolated galaxy ESO 269-IG 023 NED02 is shown. As can be seen in Fig. 7, some regions (for example at about 6 and 12 kpc from the centre of the galaxy) present un-physically large values of the [S ii]$\lambda$6716/$\lambda$6731 ratio, above the theoretical value of 1.4, the value for the low density limit according to the Osterbrock & Ferland (2006) curve for this relation. There could be some uncertainties associated with the measurements of these sulphur emission lines, due to the placement of the continuum and deblending of the lines, that might produce larger values of the [S ii] ratio than the expected ones. Values of the [S ii] ratio larger than the 1.4 upper limit were already observed in other studies using different kinds of instruments (e.g. Kennicutt, Keel, & Blaha, 1989; Zaritsky, Kennicutt, & Huchra, 1994; Lagos et al., 2009; Relaño et al., 2010; López- Hernández et al., 2013). As pointed out by López-Hernández et al. (2013), the theoretical density determination also needs to be adjusted to the sulphur atomic data and deserves to be revisited. From a spatial distribution study of the electron density in a sample of H ii regions in M33, these authors highlighted that when values of the $\lambda$ 6716/$\lambda$ 6731 ratio above the 1.4 limit are obtained, it is reasonable to assume that the electron densities are lower than 10 $\mathrm{cm^{-3}}$. They also noted that a safe way to proceed is to take $N_{\rm e}=100$ $\mathrm{cm^{-3}}$, because even before reaching the 1.4 limit, the estimation of the electron density is very uncertain. A mean density of $N_{\rm e}=181\pm 36$ $\mathrm{cm^{-3}}$ was derived for this galaxy. Again, the $N_{\rm e}$ increases toward the outskirt of this galaxy, corresponding to the end of the spiral arm at Southeast. Figure 7: Same as Fig. 4, but for AM 1256B. ### 4.4 AM 2058-381 This system of galaxies is a typical M 51 type pair. It has a systemic velocity of ${cz}$ = 12 286 km s-1 (Donzelli & Pastoriza, 1997) and consists of a main galaxy with two spiral arms (hereafter, AM 2058A) and a companion irregular galaxy (hereafter, AM 2058B). The electron densities obtained for AM 2058A have variations across the galaxy in the range of $N_{\rm e}=33-911$ $\mathrm{cm^{-3}}$, and these values are not dependent upon the position. Due to the small radius of AM 2058B, only a few apertures could be extracted for this galaxy. The electron densities (see Fig. 9) are relatively low, with a mean value of $N_{\rm e}=86\pm 33$ $\mathrm{cm^{-3}}$, which is compatible with estimations for giant extragalactic H ii regions (e.g. Castañeda et al. 1992). Figure 8: Same as Fig. 4, but for AM 2058A. Figure 9: Same as Fig. 4, but for AM 2058B. ### 4.5 AM 2229-735 This pair of galaxies consists of a main spiral galaxy strongly disturbed (hereafter AM 2229A) and a smaller disk galaxy that could be connected to the main one by a bridge. AM 2229A has a very massive nucleus of $M=5\times 10^{8}M_{\sun}$ (Ferreiro et al., 2008) and very bright H ii regions. Only the primary galaxy was observed. Most of observed regions in AM 2229A present un-physically large values of the S ii ratio according to the Osterbrock & Ferland (2006) curve for the relation between this ratio and the electron density. We derived a mean electron density of $N_{\rm e}=346\pm 95$ $\mathrm{cm^{-3}}$. Figure 10: Same as Fig. 4, but for AM 2229A. ### 4.6 AM 2306-721 AM 2306-721 is a pair composed by a spiral galaxy with disturbed arms (hereafter AM 2306A) interacting with an irregular galaxy (hereafter AM 2306B). Both galaxies contain very luminous H ii regions with H$\alpha$ luminosity in the range of 8.30 $\times$1039 $\leq$ $L$ (H$\alpha$) $\leq$ 1.32 $\times$1042 erg s-1 and high star-formation rate in the range of 0.07 - 10 $M_{\odot}$ yr-1, as estimated from H$\alpha$ images by Ferreiro et al. (2008). The few measurements of electron densities provide values in the range of $N_{\rm e}=32-298$ $\mathrm{cm^{-3}}$ and $N_{\rm e}=19-826$ $\mathrm{cm^{-3}}$ for AM 2302A and AM 2306B, respectively. Although, we do not have estimates of the electron density at the centre of the main galaxy, the spatial profile seems to indicate an increasing of the $N_{\rm e}$ toward the centre of the galaxy, which could be a consequence of gas inflow. Again, in the secondary galaxy, the electron density smoothly increases from about 4 kpc toward the outer regions of the galaxy to the end of the spiral arm at the Southeast. Figure 11: Same as Fig. 4, but for AM 2306A. Figure 12: Same as Fig. 4, but for AM 2306B. ### 4.7 AM 2322-821 AM 2322-821 is composed of a SA(r)c galaxy with disturbed arms (hereafter AM 2322A) in interaction with an irregular galaxy (hereafter AM 2322B). Both galaxies contain very luminous H ii regions with 2.53$\times$1039 $\leq$ $L$ (H$\alpha$) $\leq$ 1.45$\times$1041 erg s-1 and star formation rates from 0.02 to 1.15 $M_{\odot}$ yr-1 (Ferreiro et al., 2008). The distribution of electron temperatures exhibits variations of very low amplitude across the radius of AM 2322A. One region (at about 2 kpc from the centre of the galaxy) has four values of densities systematically higher than the other apertures along the radius of galaxy. This region is marked in Fig. 15. In this region, the values of densities are in the range of $N_{\rm e}=803-1121$ $\mathrm{cm^{-3}}$. We found a mean electron density of $N_{\rm e}=200\pm 12$ $\mathrm{cm^{-3}}$. AM 2322B presents a relatively homogeneous electron density distribution, with a mean density of $N_{\rm e}=24\pm 4.8$ $\mathrm{cm^{-3}}$. This is the galaxy with the lowest density in our sample. Figure 13: Same as Fig. 4, but for AM 2322A. Figure 14: Same as Fig. 4, but for AM 2322B. Figure 15: Image of AM 2322A with the region of high density (see the text) marked with a circle. ## 5 Discussion To verify if there are differences between the $N_{\rm e}$ values observed in the H ii regions of our sample and those obtained in isolated galaxies, we have calculated the electron densities from published measurements of the [S ii] line-ratio for disk H ii regions in the isolated galaxies M 101, NGC 1232, NGC 1365, NGC 2903, NGC 2997, and NGC 5236 and compared these values with our results. The data of these objects were taken from Kennicutt et al. (2003) for M 101 and from Bresolin et al. (2005) for the other galaxies. The same atomic parameters and electron temperature adopted for our determinations were used. The spatial profiles of the [S II]$\lambda$6716/$\lambda$6731 ratio and the electron densities derived for some H ii regions in the isolated galaxies are shown in Fig. 16. As can be seen in this figure, the estimated electron densities are relatively homogeneous along the radius of each isolated galaxy. The derived mean electron densities are in the range of $N_{\rm e}=40-137\,\rm cm^{-3}$. Only one high value of $N_{\rm e}\approx 900\>\rm cm^{-3}$ is derived in the central region of NGC 5236. It is a metal-rich H ii region, with a low electron temperature of $T_{\rm e}$(O iii)$=4\,000\pm 2\,000$ K and an oxygen abundance of 12+log(O/H)$\approx$ 8.9 dex as derived by Bresolin et al. (2005). This high value can be caused by mass loss and strong stellar winds from embedded Wolf Rayet stars, which are common in metal-rich environments (e.g. Pindao et al. 2002; Bresolin & Kennicutt 2002; Schaerer et al. 2000). If the adopted electron temperature is $T_{\rm e}$(O iii)$=4\,000$ K, an estimation of $N_{\rm e}\approx 623\>\rm cm^{-3}$ is obtained. This value is about 30% lower than the one obtained assuming an electron temperature of $T_{\rm e}$(O iii)$=10\>000$ K. Then, even though the dependence of the $N_{\rm e}$ with the electron temperature is weak, it could have an important effect when temperature fluctuations of high amplitude were observed in H ii regions. The values of the electron density obtained from our sample of interacting galaxies are systematically higher than those derived for the isolated ones. The mean electron density values derived by us for the interacting galaxies in our sample are in the range of $N_{\rm e}=24-532\,\rm cm^{-3}$, which also show higher values than for isolated galaxies. Newman et al. (2012), for the clumpy star-forming galaxy ZC406690, also obtained high electron density values ($N_{\rm e}=300-1800\,\rm cm^{-3}$). Moreover, several of our interacting galaxies (AM 2306B, AM 1219A, and AM 1256B) show a slight increment of the $N_{\rm e}$ in the outer parts of the galaxy, opposite of what is observed in the isolated galaxies, where the electron density is homogeneous along the radius. The high electron density values found in the outlying parts for the majority of the objects of our sample would be due to zones of induced star formation by direct cloud-cloud interaction (for a review see Bournaud 2011). In these regions, turbulent flows can locally compress the gas, forming over-densities that subsequently cool and collapse into star-forming clouds (Duc et al., 2013; Elmegreen, 2002). Although we do not have estimates of the electron density at the centre of AM 2306A, the spatial profile seems to indicate an increasing of $N_{\rm e}$ toward the centre of the galaxy, which could be due to inflowing gas. However, in only a few regions in this galaxy were possible to estimate $N_{\rm e}$, therefore, this is a marginal conclusion. It is worth mentioning that H ii regions seemed to be inhomogeneous, and the zones where most of the emission from the ionized gas is originated only occupy a small fraction of the total volume (i.e., small filling factor). Hence, our electron density values derived from the [S ii] emission lines are representative of a fraction of the total volume of the H ii region (referred as in situ electron densities). According to Giammanco et al. (2004), these inhomogeneities, if optically thick, can modify the determinations of electron temperatures and densities, ionization parameters, and abundances. Copetti et al. (2000) presented a study on internal variation of the electron density in a sample of spatially resolved galactic H ii regions of different sizes and evolutionary stages. These authors found that the electron density within H ii regions (e.g. S 307) can range from about 30 to 600 $\rm cm^{-3}$, and a filling factor of the order of 0.1 is compatible with their data. Therefore, the estimated electron densities could be about 10% of the in situ values sampled by the sulphur line ratio. Figure 16: Profiles of the electron density as a function of R/R0, where R0 is the galactocentric distance deprojeted for the isolated galaxies M 101, NGC 1232, NGC 1365, NGC 2903, NGC 2997 and NGC 5236. Figure 17: Diagnostic diagram of [O III]$\lambda$5007/H$\beta$ vs. [O I]$\lambda$6300/H$\alpha$. The black solid line from Kewley et al. (2006) separates the objects ionized by massive stars from the ones containing active nuclei and/or shock excited gas. The data for distinct galaxies are marked by different symbols as indicated. The typical error bar (not shown) of the emission line ratios is about 10 per cent. An important issue is to study the origin of the high electron density values found in the H ii regions of our sample. The presence of gas shock excitation in interacting galaxies is very important not only because they affect quantities derived from spectroscopy, but also due to they act as a mechanism for dissipating the kinetic energy and the angular momentum of the infalling gas in merging systems, as discussed by Rich et al. (2011). The gas shock also increases the density due to the compression of the interstellar material. To analyse if the presence of shock-excited gas produces the high electron density values, the diagnostic diagram [O III]$\lambda$5007/H$\beta$ vs. [O I]$\lambda$6300/H$\alpha$ proposed by Baldwin et al. (1981) and Veilleux & Osterbrock (1987), and used to separate objects ionized by stars, by shocks and/or active nuclei (AGN) was considered. In Fig. 17, the diagnostic diagram containing the data of the H ii regions studied by us is shown. The galaxy nuclei data are not shown in this diagram. We also show in this plot, the line proposed by Kewley et al. (2006) to separate objects with distinct ionizing sources: shock gas and massive star excitations. We can see that the all H ii regions in AM 1054A, AM 2058B, AM 2306B, and some regions in AM 2322A (3 apertures) and AM 2322B (1 aperture) occupy the area where objects with shock as the main ionizing source are located. The number of objects represented in Fig. 17 differ from those in the profile figures (Figs. 4-14) because the [O iii]$\lambda 5007$/H$\beta$ ratio could not be measured for all apertures. From the comparison of the spatial profiles of the electron density and the logarithm of [O i]$\lambda 6300$/H$\alpha$ in the AM 1054A, AM 2058B, and AM 2306B galaxies (Figs. 4, 9 and 12, respectively) we can note the following: * • AM 1054A: All regions of this galaxy have gas shock excitation and the values of electron density are relatively high. * • AM 2058B: It is a small galaxy and only a few apertures could be extracted. As can be seen in Fig. 17, all four disk H ii regions of this galaxy have gas shock excitation, and from Fig. 9, we can note that these regions present low electron density values ($<200\>\rm cm^{-3}$). * • AM 2306B: the regions with highest [O I]$\lambda$6300/H$\alpha$ and $N_{\rm e}$ values ($\approx 700$ $\rm cm^{-3}$ ) lie in the outskirts of galaxy. As can be seen in Fig. 12, it seems to be a trend in this object: from about 4 kpc both the $N_{\rm e}$ and the [O I]/H$\alpha$ ratio increase to the outer parts of the galaxy; in the inner part (up to 2 kpc), the profiles of these two quantities are almost flat showing low values. The cause of the high electron density values associated with the shock excitation region in interacting galaxies is essential to understand how the flux gas works in them. High-velocity gas motions can destroy molecular clouds and quench star formation (Tubbs, 1982). To investigate if the high electron density values found in our sample are associated with the presence of excitation by gas shock, we plotted in Fig. 18 the $N_{\rm e}$ versus the logarithm of the observed [O i]$\lambda$6300/H$\alpha$ emission line ratio. Objects with distinct gas excitation source, in according to Fig. 17, are indicated by different symbols. No correlation is obtained between the presence of shocks and electron densities. The highest electron density values found in our sample do not belong to objects with gas shock excitation. Therefore, the high electron density values found in the H ii regions of our sample do not seem to be caused by the presence of gas shock excitation. However, a deeper analysis such as investigating the presence of correlation between the velocity dispersion of some emission line and its intensity (e.g. Storchi-Bergmann et al. 2007) or the implications of multiple kinematical components in the emission line profiles on the derived properties (Hägele et al., 2013; Hägele et al., 2012; Amorín et al., 2012) is necessary to confirm our result. Interestingly, the objects with the highest electron density values present the smallest [O i]$\lambda$6300/H$\alpha$ line intensity ratios. Figure 18: Electron density values $N_{\rm e}$ derived for our sample versus the observed [O i]$\lambda$6300/H$\alpha$ ratio. Squares represent regions ionized by massive stars while triangles represent those with gas shock excitation, according to the diagnostic diagram presented in Fig. 17. ## 6 Conclusions An observational study of the effects of the interaction on the electron densities from the H ii regions along the radius of a sample of interacting galaxies is performed. The data consist of long-slit spectra of high signal- to-noise ratio in the 4390-7250 Å obtained with the Gemini Multi-Object Spectrograph at Gemini South (GMOS). The electron density was determined using the ratio of lines [S II]$\lambda$6716/$\lambda$6731\. The main findings are the following: * • The electron density estimates obtained for some H ii regions of our sample of interacting galaxies are systematically higher than those derived for isolated galaxies in the literature. The mean electron density values of interacting galaxies are in the range of $N_{\rm e}=24-532\,\rm cm^{-3}$, while those obtained for isolated galaxies are in the range of $N_{\rm e}=40-137\,\rm cm^{-3}$. * • Some interacting galaxies: AM 2306B, AM 1219A, and AM 1256B show an increment of $N_{\rm e}$ toward the outskirts of each system. This kind of relation is not observed in isolated galaxies, where the electron density profile is rather flat along the radius of each galaxy. * • The galaxies where the mechanism of gas shock excitation is present in almost all the H ii regions are AM 1054A, AM 2058B, and AM 2306B. For the remaining galaxies, only few H ii regions has emission lines excited by shocks, such as in AM 2322B (1 point) and AM 2322A (4 points). It is noteworthy that only in three of all objects analysed here the main excitation mechanism for all of their H ii regions is shocks. * • No correlation is obtained between the presence of shocks and electron densities. Indeed, the highest electron density values found in our sample do not belong to the objects with gas shock excitation. Therefore, the high electron density values found in the H ii regions of our sample do not seem to be caused by the presence of gas shock excitation. ## Acknowledgements Based on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the Science and Technology Facilities Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciencia e Tecnologia (Brazil), and SECYT (Argentina). A. C. Krabbe, O. L. Dors Jr., and D. A. Rosa thank the support of FAPESP, process 2010/01490-3, 2009/14787-7, and 2011/08202-6 respectively. 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Rodrigues", "submitter": "Oli Luiz Dors Junior", "url": "https://arxiv.org/abs/1310.7910" }
1310.7953	EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN) CERN-PH-EP-2013-199 LHCb-PAPER-2013-057 29 October 2013 Search for $C\\!P$ violation in the decay $D^{+}\rightarrow\pi^{-}\pi^{+}\pi^{+}$ The LHCb collaboration†††Authors are listed on the following pages. A search for $C\\!P$ violation in the phase space of the decay $D^{+}\rightarrow\pi^{-}\pi^{+}\pi^{+}$ is reported using $pp$ collision data, corresponding to an integrated luminosity of 1.0 fb-1, collected by the LHCb experiment at a centre-of-mass energy of 7 TeV. The Dalitz plot distributions for $3.1\times 10^{6}$ $D^{+}$ and $D^{-}$ candidates are compared with binned and unbinned model-independent techniques. No evidence for $C\\!P$ violation is found. Submitted to Phys. Lett. B © CERN on behalf of the LHCb collaboration, license CC-BY-3.0. LHCb collaboration R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24, S. Amato2, S. Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, M. Andreotti16,e, J.E. Andrews57, R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli18, A. Artamonov34, M. Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J. Back47, A. Badalov35, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S. Barsuk7, W. Barter46, V. Batozskaya27, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. Berezhnoy31, R. Bernet39, M.-O. Bettler46, M. van Beuzekom40, A. Bien11, S. Bifani44, T. Bird53, A. Bizzeti17,h, P.M. Bjørnstad53, T. Blake47, F. Blanc38, J. Blouw10, S. Blusk58, V. Bocci24, A. Bondar33, N. Bondar29, W. Bonivento15, S. Borghi53, A. Borgia58, T.J.V. Bowcock51, E. Bowen39, C. Bozzi16, T. Brambach9, J. van den Brand41, J. Bressieux38, D. Brett53, M. Britsch10, T. Britton58, N.H. Brook45, H. Brown51, A. Bursche39, G. Busetto21,q, J. Buytaert37, S. Cadeddu15, R. Calabrese16,e, O. Callot7, M. Calvi20,j, M. Calvo Gomez35,n, A. Camboni35, P. Campana18,37, D. Campora Perez37, A. Carbone14,c, G. Carboni23,k, R. Cardinale19,i, A. Cardini15, H. Carranza-Mejia49, L. Carson52, K. Carvalho Akiba2, G. Casse51, L. Castillo Garcia37, M. Cattaneo37, Ch. Cauet9, R. Cenci57, M. Charles8, Ph. Charpentier37, S.-F. Cheung54, N. Chiapolini39, M. Chrzaszcz39,25, K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, M. Cruz Torres59, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, J. Dalseno45, P. David8, P.N.Y. David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O. Deschamps5, F. Dettori41, A. Di Canto11, H. Dijkstra37, M. Dogaru28, S. Donleavy51, F. Dordei11, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, P. Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48, U. Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, A. Falabella14,e, C. Färber11, C. Farinelli40, S. Farry51, D. Ferguson49, V. Fernandez Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M. Fiore16,e, M. Fiorini16,e, C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C. Frei37, M. Frosini17,37,f, E. Furfaro23,k, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini58, Y. Gao3, J. Garofoli58, P. Garosi53, J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M. Gersabeck53, T. 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Zvyagin37. 1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil 2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil 3Center for High Energy Physics, Tsinghua University, Beijing, China 4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France 5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont- Ferrand, France 6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France 7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France 8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot, CNRS/IN2P3, Paris, France 9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany 10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany 11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany 12School of Physics, University College Dublin, Dublin, Ireland 13Sezione INFN di Bari, Bari, Italy 14Sezione INFN di Bologna, Bologna, Italy 15Sezione INFN di Cagliari, Cagliari, Italy 16Sezione INFN di Ferrara, Ferrara, Italy 17Sezione INFN di Firenze, Firenze, Italy 18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy 19Sezione INFN di Genova, Genova, Italy 20Sezione INFN di Milano Bicocca, Milano, Italy 21Sezione INFN di Padova, Padova, Italy 22Sezione INFN di Pisa, Pisa, Italy 23Sezione INFN di Roma Tor Vergata, Roma, Italy 24Sezione INFN di Roma La Sapienza, Roma, Italy 25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Kraków, Poland 26AGH - University of Science and Technology, Faculty of Physics and Applied Computer Science, Kraków, Poland 27National Center for Nuclear Research (NCBJ), Warsaw, Poland 28Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania 29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia 30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia 31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia 32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN), Moscow, Russia 33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State University, Novosibirsk, Russia 34Institute for High Energy Physics (IHEP), Protvino, Russia 35Universitat de Barcelona, Barcelona, Spain 36Universidad de Santiago de Compostela, Santiago de Compostela, Spain 37European Organization for Nuclear Research (CERN), Geneva, Switzerland 38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland 39Physik-Institut, Universität Zürich, Zürich, Switzerland 40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands 41Nikhef National Institute for Subatomic Physics and VU University Amsterdam, Amsterdam, The Netherlands 42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine 43Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine 44University of Birmingham, Birmingham, United Kingdom 45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom 46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom 47Department of Physics, University of Warwick, Coventry, United Kingdom 48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom 49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom 50School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom 51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom 52Imperial College London, London, United Kingdom 53School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom 54Department of Physics, University of Oxford, Oxford, United Kingdom 55Massachusetts Institute of Technology, Cambridge, MA, United States 56University of Cincinnati, Cincinnati, OH, United States 57University of Maryland, College Park, MD, United States 58Syracuse University, Syracuse, NY, United States 59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to 2 60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11 61KVI-University of Groningen, Groningen, The Netherlands, associated to 40 62Celal Bayar University, Manisa, Turkey, associated to 37 aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, Russia bUniversità di Bari, Bari, Italy cUniversità di Bologna, Bologna, Italy dUniversità di Cagliari, Cagliari, Italy eUniversità di Ferrara, Ferrara, Italy fUniversità di Firenze, Firenze, Italy gUniversità di Urbino, Urbino, Italy hUniversità di Modena e Reggio Emilia, Modena, Italy iUniversità di Genova, Genova, Italy jUniversità di Milano Bicocca, Milano, Italy kUniversità di Roma Tor Vergata, Roma, Italy lUniversità di Roma La Sapienza, Roma, Italy mUniversità della Basilicata, Potenza, Italy nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain oHanoi University of Science, Hanoi, Viet Nam pInstitute of Physics and Technology, Moscow, Russia qUniversità di Padova, Padova, Italy rUniversità di Pisa, Pisa, Italy sScuola Normale Superiore, Pisa, Italy ## 1 Introduction In the Standard Model (SM) charge-parity ($C\\!P$) violation in the charm sector is expected to be small. Quantitative predictions of $C\\!P$ asymmetries are difficult, since the computation of strong-interaction effects in the non-perturbative regime is involved. In spite of this, it was commonly assumed that the observation of asymmetries of the order of 1% in charm decays would be an indication of new sources of $C\\!P$ violation ($C\\!PV$). Recent studies, however, suggest that $C\\!P$ asymmetries of this magnitude could still be accommodated within the SM [1, 2, 3, 4]. Experimentally, the sensitivity for $C\\!PV$ searches has substantially increased over the past few years. Especially with the advent of the large LHCb data set, $C\\!P$ asymmetries at the $\mathcal{O}(10^{-2})$ level are disfavoured [5, 6, 7, 8, 9]. With uncertainties approaching $\mathcal{O}(10^{-3})$, the current $C\\!PV$ searches start to probe the regime of the SM expectations. The most simple and direct technique for $C\\!PV$ searches is the computation of an asymmetry between the particle and anti-particle time-integrated decay rates. A single number, however, may not be sufficient for a comprehension of the nature of the $C\\!P$ violating asymmetry. In this context, three- and four-body decays benefit from rich resonance structures with interfering amplitudes modulated by strong-phase variations across the phase space. Searches for localised asymmetries can bring complementary information on the nature of the $C\\!PV$. In this Letter, a search for $C\\!P$ violation in the Cabibbo-suppressed decay $D^{+}\rightarrow\pi^{-}\pi^{+}\pi^{+}$ is reported.111Unless stated explicitly, the inclusion of charge conjugate states is implied. The investigation is performed across the Dalitz plot using two model-independent techniques, a binned search as employed in previous LHCb analyses [10, 11] and an unbinned search based on the nearest-neighbour method [12, 13]. Possible localised charge asymmetries arising from production or detector effects are investigated using the decay $D^{+}_{s}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$, which has the same final state particles as the signal mode, as a control channel. Since it is a Cabibbo-favoured decay, with negligible loop (penguin) contributions, $C\\!P$ violation is not expected at any significant level. ## 2 LHCb detector and data set The LHCb detector [14] is a single-arm forward spectrometer covering the pseudorapidity range $2<\eta<5$, designed for the study of particles containing $b$ or $c$ quarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the $pp$ interaction region, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift tubes placed downstream. The combined tracking system provides a momentum measurement with relative uncertainty that varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter (IP) resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum, $p_{\rm T}$. Charged hadrons are identified using two ring-imaging Cherenkov (RICH) detectors [15]. Photon, electron and hadron candidates are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic calorimeter and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers [16]. The trigger [17] consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which applies full event reconstruction. At the hardware trigger stage, events are required to have muons with high transverse momentum or hadrons, photons or electrons with high transverse energy deposit in the calorimeters. For hadrons, the transverse energy threshold is 3.5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$. The software trigger requires at least one good quality track from the signal decay with high $p_{\rm T}$ and high $\chi^{2}_{\rm IP}$, defined as the difference in $\chi^{2}$ of the primary vertex (PV) reconstructed with and without this particle. A secondary vertex is formed by three tracks with good quality, each not pointing to any PV, and with requirements on $p_{\rm T}$, momentum $p$, scalar sum of $p_{\rm T}$ of the tracks, and a significant displacement from any PV. The data sample used in this analysis corresponds to an integrated luminosity of 1.0 fb-1 of $pp$ collisions at a centre-of-mass energy of 7 TeV collected by the LHCb experiment in 2011. The magnetic field polarity is reversed regularly during the data taking in order to minimise effects of charged particle and antiparticle detection asymmetries. Approximately half of the data are collected with each polarity, hereafter referred to as “magnet up” and “magnet down” data. ## 3 Event selection To reduce the combinatorial background, requirements on the quality of the reconstructed tracks, their $\chi^{2}_{\rm IP}$, $p_{\rm T}$, and scalar $p_{\rm T}$ sum are applied. Additional requirements are made on the secondary vertex fit quality, the minimum significance of the displacement from the secondary to any primary vertex in the event, and the $\chi^{2}_{\rm IP}$ of the $D^{+}_{(s)}$ candidate. This also reduces the contribution of secondary $D$ mesons from $b$-hadron decays to 1–2%, avoiding the introduction of new sources of asymmetries. The final-state particles are required to satisfy particle identification (PID) criteria based on the RICH detectors. After these requirements, there is still a significant background contribution, which could introduce charge asymmetries across the Dalitz plot. This includes semileptonic decays like $D^{+}\rightarrow K^{-}\pi^{+}\mu^{+}\nu$ and $D^{+}\rightarrow\pi^{-}\pi^{+}\mu^{+}\nu$; three- body decays, such as $D^{+}\rightarrow K^{-}\pi^{+}\pi^{+}$; prompt two-body $D^{0}$ decays forming a three-prong vertex with a random pion; and $D^{0}$ decays from the $D^{+}$ chain, such as $D^{+}\rightarrow D^{0}(K^{-}\pi^{+},\pi^{-}\pi^{+},K^{-}\pi^{+}\pi^{0})\pi^{+}$. The contribution from $D^{+}\\!\rightarrow K^{-}\pi^{+}\pi^{+}$ and prompt $D^{0}$ decays that involve the misidentification of the kaon as a pion is reduced to a negligible level with a more stringent PID requirement on the $\pi^{-}$ candidate. The remaining background from semileptonic decays is controlled by applying a muon veto to all three tracks, using information from the muon system [18]. The contribution from the $D^{+}$ decay chain is reduced to a negligible level with a requirement on $\chi^{2}_{\rm IP}$ of the $\pi^{+}$ candidate with lowest $p_{\rm T}$. Fits to the invariant mass distribution $M({\pi^{-}\pi^{+}\pi^{+}})$ are performed for the $D^{+}$ and $D^{+}_{s}$ candidates satisfying the above selection criteria and within the range $1810<M({\pi^{-}\pi^{+}\pi^{+}})<1930{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and $1910<M({\pi^{-}\pi^{+}\pi^{+}})<2030{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, respectively. The signal is described by a sum of two Gaussian functions and the background is represented by a third-order polynomial. The data sample is separated according to magnet polarity and candidate momentum ($p_{D^{+}_{(s)}}\\!\\!<\\!50$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, $50\\!<p_{D^{+}_{(s)}}\\!\\!<\\!100$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and $p_{D^{+}_{(s)}}\\!\\!>\\!100$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$), to take into account the dependence of the mass resolution on the momentum. The parameters are determined by simultaneous fits to these $D^{+}_{(s)}$ and $D^{-}_{(s)}$ subsamples. The $D^{+}$ and $D^{+}_{s}$ invariant mass distributions and fit results for the momentum range $50\\!<\\!p_{D^{+}_{(s)}}\\!\\!<\\!100$${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ are shown in Fig. 1 for magnet up data. The total yields after summing over all fits are $(2678\pm 7)\\!\\!\times\\!\\!10^{3}$ $D^{+}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ and $(2704\pm 8)\\!\\!\times\\!\\!10^{3}$ $D^{+}_{s}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ decays. The final samples used for the $C\\!PV$ search consist of all candidates with $M({\pi^{-}\pi^{+}\pi^{+}})$ within $\pm 2\tilde{\sigma}$ around $\tilde{m}_{D_{(s)}}$, where $\tilde{\sigma}$ and $\tilde{m}_{D_{(s)}}$ are the weighted average of the two fitted Gaussian widths and mean values. The values of $\tilde{\sigma}$ range from 8 to 12 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, depending on the momentum region. For the signal sample there are $3114\\!\times\\!10^{3}$ candidates, including background, while for the control mode there are $2938\\!\times\\!10^{3}$ candidates with purities of 82% and 87%, respectively. The purity is defined as the fraction of signal decays in this mass range. The $D^{+}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ and $D^{+}_{s}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ Dalitz plots are shown in Fig. 2, with $s_{\rm low}$ and $s_{\rm high}$ being the lowest and highest invariant mass squared combination, $M^{2}(\pi^{-}\pi^{+})$, respectively. Clear resonant structures are observed in both decay modes. Figure 1: Invariant-mass distributions for (a) $D^{+}$ and (b) $D^{+}_{s}$ candidates in the momentum range $50<p_{D^{+}_{(s)}}\\!\\!<100$ ${\mathrm{\,Ge\kern-0.90005ptV\\!/}c}$ for magnet up data. Data points are shown in black. The solid (blue) line is the fit function, the (green) dashed line is the signal component and the (magenta) dotted line is the background. Figure 2: Dalitz plots for (a) $D^{+}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ and (b) $D^{+}_{s}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ candidates selected within $\pm 2\tilde{\sigma}$ around the respective $\tilde{m}$ weighted average mass. ## 4 Binned analysis ### 4.1 Method The binned method used to search for localised asymmetries in the $D^{+}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ decay phase space is based on a bin-by-bin comparison between the $D^{+}$ and $D^{-}$ Dalitz plots [19, 20]. For each bin of the Dalitz plot, the significance of the difference between the number of $D^{+}$ and $D^{-}$ candidates, $\mathcal{S}^{i}_{C\\!P}$, is computed as $\mathcal{S}^{i}_{C\\!P}\equiv\frac{N_{i}^{+}-\alpha N_{i}^{-}}{\sqrt{\alpha(N_{i}^{+}+N_{i}^{-})}}\ ,\hskip 14.22636pt\alpha\equiv\frac{N^{+}}{N^{-}},$ (1) where $N_{i}^{+}$ ($N_{i}^{-}$) is the number of $D^{+}$ ($D^{-}$) candidates in the $i\rm{th}$ bin and $N^{+}$ ($N^{-}$) is the sum of $N_{i}^{+}$ ($N_{i}^{-}$) over all bins. The parameter $\alpha$ removes the contribution of global asymmetries which may arise due to production [21, 22] and detection asymmetries, as well as from $C\\!PV$. Two binning schemes are used, a uniform grid with bins of equal size and an adaptive binning where the bins have the same population. In the absence of localised asymmetries, the $\mathcal{S}^{i}_{C\\!P}$ values follow a standard normal Gaussian distribution. Therefore, $C\\!PV$ can be detected as a deviation from this behaviour. The numerical comparison between the $D^{+}$ and $D^{-}$ Dalitz plots is made by a $\chi^{2}$ test, with $\chi^{2}=\sum_{i}(\mathcal{S}_{C\\!P}^{i})^{2}$. A p-value for the hypothesis of no $C\\!PV$ is obtained considering that the number of degrees of freedom (ndf) is equal to the total number of bins minus one, due to the constraint on the overall $D^{+}$/$D^{-}$ normalisation. A $C\\!PV$ signal is established if a p-value lower than $3\\!\times\\!10^{-7}$ is found, in which case it can be converted to a significance for the exclusion of $C\\!P$ symmetry in this channel. If no evidence of $C\\!PV$ is found, this technique provides no model-independent way to set an upper limit. ### 4.2 Control mode and background The search for local asymmetries across the $D^{+}_{s}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ Dalitz plot is performed using both the uniform and the adaptive (“$D^{+}_{s}$ adaptive”) binning schemes mentioned previously. A third scheme is also used: a “scaled $D^{+}$” scheme, obtained from the $D^{+}$ adaptive binning by scaling the bin edges by the ratios of the maximum values of $s_{\rm high}$($D^{+}_{s}$)/$s_{\rm high}$($D^{+}$) and $s_{\rm low}$($D^{+}_{s}$)/$s_{\rm low}$($D^{+}$). This scheme provides a one-to-one mapping of the corresponding Dalitz plots and allows to probe regions in the signal and control channel phase spaces where the momentum distributions of the three final state particles are similar. The study is performed using $\alpha=0.992\pm 0.001$, as measured for the $D^{+}_{s}$ sample, and different granularities: 20, 30, 40, 49 and 100 adaptive bins for both the $D^{+}_{s}$ adaptive and scaled $D^{+}$ schemes, and 5$\times$5, 6$\times$7, 8$\times$9 and 12$\times$12 bins for the uniform grid scheme. Only bins with a minimum occupancy of 20 entries are considered. The p-values obtained are distributed in the range 4–87%, consistent with the hypothesis of absence of localised asymmetries. As an example, Fig. 3 shows the distributions of $\mathcal{S}^{i}_{C\\!P}$ for the $D^{+}_{s}$ adaptive binning scheme with 49 bins. As a further cross-check, the $D^{+}_{s}$ sample is divided according to magnet polarity and hardware trigger configurations. Typically, the p-values are above 1%, although one low value of 0.07% is found for a particular trigger subset of magnet up data with 40 adaptive bins. When combined with magnet down data, the p-value increases to 11%. The possibility of local asymmetries induced by the background under the $D^{+}$ signal peak is studied by considering the candidates with mass $M({\pi^{-}\pi^{+}\pi^{+}})$ in the ranges 1810–1835 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ and 1905–1935 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, for which $\alpha=1.000\pm 0.002$. Using an uniform grid with four different granularities, the p-values are computed for each of the two sidebands. The data are also divided according to the magnet polarity. The p-values are found to be within 0.4–95.5%, consistent with differences in the number of $D^{+}$ and $D^{-}$ candidates arising from statistical fluctuations. Since the selection criteria suppress charm background decays to a negligible level, it is assumed that the background contribution to the signal is similar to the sidebands. Therefore, asymmetries eventually observed in the signal mode cannot be attributed to the background. Figure 3: (a) Distribution of $\mathcal{S}_{CP}^{i}$ with 49 $D^{+}_{s}$ adaptive bins of equal population in the $D^{+}_{s}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ Dalitz plot and (b) the corresponding one-dimensional distribution (histogram) with a standard normal Gaussian function superimposed (solid line). ### 4.3 Sensitivity studies To study the $C\\!PV$ sensitivity of the method for the current data set, a number of simulated pseudo-experiments are performed with sample size and purity similar to that observed in data. The $D^{+}\rightarrow\pi^{-}\pi^{+}\pi^{+}$ decays are generated according to an amplitude model inspired by E791 results [23], where the most important contributions originate from $\rho^{0}(770)\pi^{+}$, $\sigma(500)\pi^{+}$ and $f_{2}(1270)\pi^{+}$ resonant modes. Background events are generated evenly in the Dalitz plot. Since no theoretical predictions on the presence or size of $C\\!PV$ are available for this channel, various scenarios are studied by introducing phase and magnitude differences between the main resonant modes for $D^{+}$ and $D^{-}$. The sensitivity for different binning strategies is also evaluated. Phase differences in the range $0.5$–$4.0^{\circ}$ and magnitude differences in the range $0.5$–$4.0\%$ are tested for $\rho^{0}(770)\pi^{+}$, $\sigma(500)\pi^{+}$ and $f_{2}(1270)\pi^{+}$ modes. The study shows a sensitivity (p-values below $10^{-7}$) around $1^{\circ}$ to $2^{\circ}$ in phase differences and $2\%$ in amplitude in these channels. The sensitivity decreases when the number of bins is larger than 100, so a few tens of bins approaches the optimal choice. A slightly better sensitivity for the adaptive binning strategy is found in most of the studies. Since the presence of background tends to dilute a potential sign of $C\\!PV$, additional pseudo-experiment studies are made for different scenarios based on signal yields and purities attainable on data. Results show that better sensitivities are found for higher yields, despite the lower purity. ## 5 Unbinned analysis ### 5.1 k-nearest neighbour analysis technique The unbinned model-independent method of searching for $C\\!PV$ in many-body decays uses the concept of nearest neighbour events in a combined $D^{+}$ and $D^{-}$ samples to test whether they share the same parent distribution function [24, 12, 13]. To find the $n_{k}$ nearest neighbour events of each $D^{+}$ and $D^{-}$ event, the Euclidean distance between points in the Dalitz plot of three-body $D^{+}$ and $D^{-}$ decays is used. For the whole event sample a test statistic $T$ for the null hypothesis is calculated, $T=\frac{1}{n_{k}(N_{+}+N_{-})}\sum\limits_{i=1}^{N_{+}+N_{-}}\sum\limits_{k=1}^{n_{k}}I(i,k),$ (2) where $I(i,k)=1$ if the $i$th event and its $k$th nearest neighbour have the same charge and $I(i,k)=0$ otherwise and $N_{+}$ ($N_{-}$) is the number of events in the $D^{+}$ ($D^{-}$) sample. The test statistic $T$ is the mean fraction of like-charged neighbour pairs in the combined $D^{+}$ and $D^{-}$ decays sample. The advantage of the k-nearest neighbour method (kNN), in comparison with other proposed methods for unbinned analyses [24], is that the calculation of $T$ is simple and fast and the expected distribution of $T$ is well known: for the null hypothesis it follows a Gaussian distribution with mean $\mu_{T}$ and variance $\sigma^{2}_{T}$ calculated from known parameters of the distributions, $\mu_{T}=\frac{N_{+}(N_{+}-1)+N_{-}(N_{-}-1)}{N(N-1)},$ (3) $\lim_{N,n_{k},D\rightarrow\infty}\sigma_{T}^{2}=\frac{1}{Nn_{k}}\left(\frac{N_{+}N_{-}}{N^{2}}+4\frac{N_{+}^{2}N_{-}^{2}}{N^{4}}\right),$ (4) where $N=N_{+}+N_{-}$ and $D$ is a space dimension. For $N_{+}=N_{-}$ a reference value $\mu_{\it TR}=\frac{1}{2}\left(\frac{N-2}{N-1}\right)$ (5) is obtained and for a very large number of events $N$, $\mu_{T}$ approaches $0.5$. However, since the observed deviations of $\mu_{T}$ from $\mu_{\it TR}$ are sometimes tiny, it is necessary to calculate $\mu_{T}-\mu_{\it TR}$. The convergence in Eq. 4 is fast and $\sigma_{T}$ can be obtained with a good approximation even for space dimension $D=2$ for the current values of $N_{+}$, $N_{-}$ and $n_{k}$ [24, 13]. The kNN method is applied to search for $C\\!PV$ in a given region of the Dalitz plot in two ways: by looking at a “normalisation” asymmetry ($N_{+}\neq N_{-}$ in a given region) using a pull $(\mu_{T}-\mu_{\it TR})/\Delta(\mu_{T}-\mu_{\it TR})$ variable, where the uncertainty on $\mu_{T}$ is $\Delta\mu_{T}$ and the uncertainty on $\mu_{\it TR}$ is $\Delta\mu_{\it TR}$, and looking for a “shape” or particle density function (pdf) asymmetry using another pull $(T-\mu_{T})/\sigma_{T}$ variable. As in the binned method, this technique provides no model-independent way to set an upper limit if no $C\\!PV$ is found. ### 5.2 Control mode and background The Cabibbo-favoured $D^{+}_{s}$ decays serve as a control sample to estimate the size of production and detection asymmetries and systematic effects. The sensitivity for local $C\\!PV$ in the Dalitz plot of the kNN method can be increased by taking into account only events from the region where $C\\!PV$ is expected to be enhanced by the known intermediate resonances in the decays. Since these regions are characterised by enhanced variations of strong phases, the conditions for observation of $C\\!PV$ are more favourable. Events from other regions are expected to only dilute the signal of $C\\!PV$. The Dalitz plot for the control channel $D^{+}_{s}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ is partitioned into three (P1-P3) or seven (R1-R7) regions shown in Fig. 4. The division R1-R7 is such that regions enriched in resonances are separated from regions dominated by smoother distributions of events. Region R3 is further divided into two regions of $s_{\rm high}$ at masses smaller (R3l) and larger (R3r) than the $\rho^{0}(770)$ resonance, in order to study possible asymmetries due to a sign change of the strong phase when crossing the resonance pole. The three regions P1-P3 correspond to a more complicated structure of resonances in the signal decay $D^{+}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ (see Fig. 11). Figure 4: Dalitz plot for $D^{+}_{s}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ control sample decays divided into (a) seven regions R1-R7 and (b) three regions P1-P3. Region R3 is further divided into two regions of $s_{\rm high}$ at masses smaller (R3l) and larger (R3r) than the $\rho^{0}(770)$ resonance. Figure 5: (a) Pull values of $T$ and (b) the corresponding p-values for $D^{+}_{s}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ control sample candidates restricted to each region, obtained using the kNN method with $n_{k}=20$. The horizontal blue lines in (a) represent $-3$ and $+3$ pull values. The region R0 corresponds to the full Dalitz plot. Note that the points for the overlapping regions are correlated. Figure 6: (a) Raw asymmetry $A=(N_{-}-N_{+})/(N_{-}+N_{+})$ and (b) the pull values of $\mu_{T}$ for $D^{+}_{s}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ control sample candidates restricted to each region. The horizontal lines in (b) represent $+3$ and $+5$ pull values. The region R0 corresponds to the full Dalitz plot. Note that the points for the overlapping regions are correlated. The value of the test statistic $T$ measured using the kNN method with $n_{k}=20$ for the full Dalitz plot (called R0) of $D^{+}_{s}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ candidates is compared to the expected Gaussian $T$ distribution with $\mu_{T}$ and $\sigma_{T}$ calculated from data. The calculated p-value is 44% for the hypothesis of no $C\\!P$ asymmetry. The p-values are obtained by integrating the Gaussian $T$ distribution from a given value up to its maximum value of 1. The results are shown in Fig. 5 separately for each region. They do not show any asymmetry between $D_{s}^{+}$ and $D_{s}^{-}$ samples. Since no $C\\!PV$ is expected in the control channel, the local detection asymmetries are smaller than the present sensitivity of the kNN method. The production asymmetry is accounted for in the kNN method as a deviation of the measured value of $\mu_{T}$ from the reference value $\mu_{\it TR}$. In the present sample, the obtained value $\mu_{T}-0.5=(84\pm 15)\times 10^{-7}$, with $(\mu_{T}-\mu_{\it TR})/\Delta(\mu_{T}-\mu_{\it TR})=5.8\sigma$, in the full Dalitz plot is a consequence of the observed global asymmetry of about 0.4%. This value is consistent with the previous measurement from LHCb [22]. The comparison of the raw asymmetry $A=(N_{-}-N_{+})/(N_{-}+N_{+})$ and the pull values of $\mu_{T}$ in all regions are presented in Fig. 6. The measured raw asymmetry is similar in all regions as expected for an effect due to the production asymmetry. It is interesting to note the relation $\mu_{T}-\mu_{\it TR}\approx A^{2}/2$ at order $1/N$ between the raw asymmetry and the parameters of the kNN method. A region-by-region comparison of $D_{s}^{+}$ candidates for magnet down and magnet up data gives insight into left-right detection asymmetries. No further asymmetries, except for the global production asymmetry discussed above, are found. The number of nearest neighbour events $n_{k}$ is the only parameter of the kNN method. The results for the control channel show no significant dependence of p-values on $n_{k}$. Higher values of $n_{k}$ reduce statistical fluctuations due to the local population density and should be preferred. On the other hand, increasing the number of nearest neighbours with limited number of events in the sample can quickly increase the radius of the local region under investigation. The kNN method also is applied to the background events, defined in Sec. 4.2. Contrary to the measurements for the $D^{+}_{s}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ candidates, for background no production asymmetry is observed. The calculated $\mu_{T}-0.5=(-5.80\pm 0.46)\times 10^{-7}$ for the full Dalitz plot is very close to the value $\mu_{\it TR}-0.5=(-5.8239\pm 0.0063)\times 10^{-7}$ expected for an equal number of events in $D^{+}$ and $D^{-}$ samples (Eq. 5). The measured pull values of $T$ and the corresponding p-values obtained using the kNN method with $n_{k}=20$ are presented for the background in Fig. 7, separately for each region. The comparison of normalisation asymmetries and pull values of $\mu_{T}$ in all regions are presented in Fig. 8. All the kNN method results are consistent with no significant asymmetry. Figure 7: (a) Pull values of $T$ and (b) the corresponding p-values for the background candidates restricted to each region obtained using the kNN method with $n_{k}=20$. The horizontal blue lines in (a) represent $-3$ and $+3$ pull values. The region R0 corresponds to the full Dalitz plot. Note that the points for the overlapping regions are correlated. Figure 8: (a) Raw asymmetry and (b) pull value of $\mu_{T}$ as a function of a region for the background candidates. The horizontal lines in (b) represent $+3$ and $+5$ pull values. The region R0 corresponds to the full Dalitz plot. Note that the points for the overlapping regions are correlated. ### 5.3 Sensitivity studies The sensitivity of the kNN method is tested with the same pseudo-experiment model described in Sec. 4.3. If the simulated asymmetries are spread out in the Dalitz plot the events may be moved from one region to another. For these asymmetries it is observed that the difference in shape of the probability density functions is in large part absorbed in the difference in the normalisation. This indicates that the choice of the regions is important for increasing the sensitivity of the kNN method. In general the method applied in a given region is sensitive to weak phase differences greater than $(1-2)^{\circ}$ and magnitude differences of $(2-4)$%. ## 6 Results ### 6.1 Binned method The search for $C\\!PV$ in the Cabibbo-suppressed mode $D^{+}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ is pursued following the strategy described in Section 4. For the total sample size of about 3.1 million $D^{+}$ and $D^{-}$ candidates, the normalisation factor $\alpha$, defined in Eq. 1, is $0.990\pm 0.001$. Both adaptive and uniform binning schemes in the Dalitz plot are used for different binning sizes. The $\mathcal{S}_{CP}^{i}$ values across the Dalitz plot and the corresponding histogram for the adaptive binning scheme with 49 and 100 bins are illustrated in Fig. 9. The p-values for these and other binning choices are shown in Table 1. All p-values show statistical agreement between the $D^{+}$ and $D^{-}$ samples. The same $\chi^{2}$ test is performed for the uniform binning scheme, using 20, 32, 52 and 98 bins also resulting in p-values consistent with the null hypothesis, all above 90%. The $\mathcal{S}_{CP}^{i}$ distribution in the Dalitz plot for 98 bins and the corresponding histogram is shown in Fig. 10. As consistency checks, the analysis is repeated with independent subsamples obtained by separating the total sample according to magnet polarity, hardware trigger configurations, and data-taking periods. The resulting p-values range from 0.3% to 98.3%. All the results above indicate the absence of $C\\!PV$ in the $D^{+}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ channel at the current analysis sensitivity. Table 1: Results for the $D^{+}\rightarrow\pi^{-}\pi^{+}\pi^{+}$ decay sample using the adaptive binning scheme with different numbers of bins. The number of degrees of freedom is the number of bins minus 1. Number of bins \| $\chi^{2}$ \| p-value (%) ---\|---\|--- 20 \| 14.0 \| 78.1 30 \| 28.2 \| 50.6 40 \| 28.5 \| 89.2 49 \| 26.7 \| 99.5 100 \| 89.1 \| 75.1 Figure 9: Distributions of $\mathcal{S}_{CP}^{i}$ across the $D^{+}$ Dalitz plane, with the adaptive binning scheme of uniform population for the total $D^{+}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ data sample with (a) 49 and (c) 100 bins. The corresponding one-dimensional $\mathcal{S}_{CP}^{i}$ distributions (b) and (d) are shown with a standard normal Gaussian function superimposed (solid line). Figure 10: (a) Distribution of $\mathcal{S}_{CP}^{i}$ with 98 bins in the uniform binning scheme for the total $D^{+}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ data sample and (b) the corresponding one-dimensional $\mathcal{S}_{CP}^{i}$ distribution (b) with a standard normal Gaussian function superimposed (solid line). ### 6.2 Unbinned method The kNN method is applied to the Cabibbo-suppressed mode $D^{+}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ with the two region definitions shown in Fig. 11. To account for the different resonance structure in $D^{+}$ and $D^{+}_{s}$ decays, the region R1-R7 definition for the signal mode is different from the definition used in the control mode (compare Figs. 4a and 11a). The region P1-P3 definitions are the same. The results for the raw asymmetry are shown in Fig. 12. The production asymmetry is clearly visible in all the regions with the same magnitude as in the control channel (see Fig. 6). It is accounted for in the kNN method as a deviation of the measured value of $\mu_{T}$ from the reference value $\mu_{\it TR}$ shown in Fig. 12. In the signal sample the values $\mu_{T}-0.5=(98\pm 15)\times 10^{-7}$ and $(\mu_{T}-\mu_{\it TR})/\Delta(\mu_{T}-\mu_{\it TR})=6.5\sigma$ in the full Dalitz plot are a consequence of the 0.4% global asymmetry similar to that observed in the control mode and consistent with the previous measurement from LHCb [21]. The pull values of $T$ and the corresponding p-values for the hypothesis of no $C\\!PV$ are shown in Fig. 13 for the same regions. To check for any systematic effects, the test is repeated for samples separated according to magnet polarity. Since the sensitivity of the method increases with $n_{k}$, the analysis is repeated with $n_{k}=500$ for all the regions. All p-values are above 20%, consistent with no $C\\!P$ asymmetry in the signal mode. Figure 11: Dalitz plot for $D^{+}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ candidates divided into (a) seven regions R1-R7 and (b) three regions P1-P3. Figure 12: (a) Raw asymmetry and (b) the pull values of $\mu_{T}$ for $D^{+}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ candidates restricted to each region. The horizontal lines in (b) represent pull values $+3$ and $+5$. The region R0 corresponds to the full Dalitz plot. Note that the points for the overlapping regions are correlated. Figure 13: (a) Pull values of $T$ and (b) the corresponding p-values for $D^{+}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ candidates restricted to each region obtained using the kNN method with $n_{k}=20$. The horizontal blue lines in (a) represent pull values $-3$ and $+3$. The region R0 corresponds to the full Dalitz plot. Note that the points for the overlapping regions are correlated. ## 7 Conclusion A search for $C\\!PV$ in the decay $D^{+}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ is performed using $pp$ collision data corresponding to an integrated luminosity of 1.0 fb-1 collected by the LHCb experiment at a centre-of-mass energy of 7 TeV. Two model-independent methods are applied to a sample of 3.1 million $D^{+}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ decay candidates with 82% signal purity. The binned method is based on the study of the local significances $\mathcal{S}^{i}_{C\\!P}$ in bins of the Dalitz plot, while the unbinned method uses the concept of nearest neighbour events in the pooled $D^{+}$ and $D^{-}$ sample. Both methods are also applied to the Cabibbo-favoured $D^{+}_{s}\\!\rightarrow\pi^{-}\pi^{+}\pi^{+}$ decay and to the mass sidebands to control possible asymmetries not originating from $C\\!PV$. No single bin in any of the binning schemes presents an absolute $\mathcal{S}^{i}_{C\\!P}$ value larger than 3. Assuming no $C\\!PV$, the probabilities of observing local asymmetries across the phase-space of the $D^{+}$ meson decay as large or larger than those in data are above 50% in all the tested binned schemes. In the unbinned method, the p-values are above 30%. All results are consistent with no $C\\!PV$. ## Acknowledgements We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We also acknowledge the support received from the ERC under FP7. The Tier1 computing centres are supported by IN2P3 (France), KIT and BMBF (Germany), INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom). We are thankful for the computing resources put at our disposal by Yandex LLC (Russia), as well as to the communities behind the multiple open source software packages that we depend on. ## References [1] J. Brod, Y. Grossman, A. L. Kagan, and J. 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Andrews, R.B.\n Appleby, O. Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E.\n Aslanides, G. Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, C.\n Baesso, V. Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, V. Batozskaya, Th. Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga,\n S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson,\n J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien,\n S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, J.\n Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A.\n Borgia, T.J.V. Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van den Brand, J.\n Bressieux, D. Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown, A.\n Bursche, G. Busetto, J. Buytaert, S. Cadeddu, R. Calabrese, O. Callot, M.\n Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D. Campora Perez, A. Carbone,\n G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia, L. Carson, K.\n Carvalho Akiba, G. Casse, L. Castillo Garcia, M. Cattaneo, Ch. Cauet, R.\n Cenci, M. Charles, Ph. Charpentier, S.-F. Cheung, N. Chiapolini, M.\n Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic,\n H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins,\n A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti,\n B. Couturier, G.A. Cowan, D.C. Craik, M. Cruz Torres, S. Cunliffe, R. Currie,\n C. D'Ambrosio, J. Dalseno, P. David, P.N.Y. David, A. Davis, I. De Bonis, K.\n De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva,\n P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D.\n Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra, M. Dogaru, S.\n Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis,\n P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella, C. F\\\"arber, C.\n Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues,\n M. Ferro-Luzzi, S. Filippov, M. Fiore, M. Fiorini, C. Fitzpatrick, M.\n Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M.\n Frosini, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R.\n Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, L.\n Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, P.\n Gorbounov, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, P.\n Griffith, L. Grillo, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, T.W. Hafkenscheid, S.C. Haines, S. Hall,\n B. Hamilton, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J.\n Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard,\n J.A. Hernando Morata, E. van Herwijnen, M. He\\ss, A. Hicheur, E. Hicks, D.\n Hill, M. Hoballah, C. Hombach, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain,\n D. Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A.\n Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D. Johnson, C.R.\n Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, W. Kanso, M. Karacson,\n T.M. Karbach, I.R. Kenyon, T. Ketel, B. Khanji, O. Kochebina, I. Komarov,\n R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk, K.\n Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, K. Kurek, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. Lazzeroni, R. Le Gac, J. van Leerdam, J.-P. Lees,\n R. Lef\\`evre, A. Leflat, J. Lefran\\c{c}ois, S. Leo, O. Leroy, T. Lesiak, B.\n Leverington, Y. Li, L. Li Gioi, M. Liles, R. Lindner, C. Linn, B. Liu, G.\n Liu, S. Lohn, I. Longstaff, J.H. Lopes, N. Lopez-March, H. Lu, D. Lucchesi,\n J. Luisier, H. Luo, E. Luppi, O. Lupton, F. Machefert, I.V. Machikhiliyan, F.\n Maciuc, O. Maev, S. Malde, G. Manca, G. Mancinelli, J. Maratas, U. Marconi,\n P. Marino, R. M\\\"arki, J. Marks, G. Martellotti, A. Martens, A. Mart\\'in\n S\\'anchez, M. Martinelli, D. Martinez Santos, D. Martins Tostes, A. Martynov,\n A. Massafferri, R. Matev, Z. Mathe, C. Matteuzzi, E. Maurice, A. Mazurov, M.\n McCann, J. McCarthy, A. McNab, R. McNulty, B. McSkelly, B. Meadows, F. Meier,\n M. Meissner, M. Merk, D.A. Milanes, M.-N. Minard, J. Molina Rodriguez, S.\n Monteil, D. Moran, P. Morawski, A. Mord\\`a, M.J. Morello, R. Mountain, I.\n Mous, F. Muheim, K. M\\\"uller, R. Muresan, B. Muryn, B. Muster, P. Naik, T.\n Nakada, R. Nandakumar, I. Nasteva, M. Needham, S. Neubert, N. Neufeld, A.D.\n Nguyen, T.D. Nguyen, C. Nguyen-Mau, M. Nicol, V. Niess, R. Niet, N. Nikitin,\n T. Nikodem, A. Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov,\n S. Oggero, S. Ogilvy, O. Okhrimenko, R. Oldeman, G. Onderwater, M. Orlandea,\n J.M. Otalora Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano, M.\n Palutan, J. Panman, A. Papanestis, M. Pappagallo, C. Parkes, C.J. Parkinson,\n G. Passaleva, G.D. Patel, M. Patel, C. Patrignani, C. Pavel-Nicorescu, A.\n Pazos Alvarez, A. Pearce, A. Pellegrino, G. Penso, M. Pepe Altarelli, S.\n Perazzini, E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M.\n Perrin-Terrin, L. Pescatore, E. Pesen, G. Pessina, K. Petridis, A. Petrolini,\n E. Picatoste Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo\n Casasus, F. Polci, G. Polok, A. Poluektov, E. Polycarpo, A. Popov, D. Popov,\n B. Popovici, C. Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C.\n Prouve, V. Pugatch, A. Puig Navarro, G. Punzi, W. Qian, B. Rachwal, J.H.\n Rademacker, B. Rakotomiaramanana, M.S. Rangel, I. Raniuk, N. Rauschmayr, G.\n Raven, S. Redford, S. Reichert, M.M. Reid, A.C. dos Reis, S. Ricciardi, A.\n Richards, K. Rinnert, V. Rives Molina, D.A. Roa Romero, P. Robbe, D.A.\n Roberts, A.B. Rodrigues, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V.\n Romanovsky, A. Romero Vidal, M. Rotondo, J. Rouvinet, T. Ruf, F. Ruffini, H.\n Ruiz, P. Ruiz Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail,\n B. Saitta, V. Salustino Guimaraes, B. Sanmartin Sedes, R. Santacesaria, C.\n Santamarina Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta,\n M. Savrie, D. Savrina, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling,\n B. Schmidt, O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B.\n Sciascia, A. Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N.\n Serra, J. Serrano, P. Seyfert, M. Shapkin, I. Shapoval, Y. Shcheglov, T.\n Shears, L. Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva\n Coutinho, M. Sirendi, N. Skidmore, T. Skwarnicki, N.A. Smith, E. Smith, E.\n Smith, J. Smith, M. Smith, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza,\n B. Souza De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stevenson, S. Stoica, S. Stone, B. Storaci, S. Stracka, M.\n Straticiuc, U. Straumann, V.K. Subbiah, L. Sun, W. Sutcliffe, S. Swientek, V.\n Syropoulos, M. Szczekowski, P. Szczypka, D. Szilard, T. Szumlak, S.\n T'Jampens, M. Teklishyn, G. Tellarini, E. Teodorescu, F. Teubert, C. Thomas,\n E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, L. Tomassetti, D.\n Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M.T. Tran,\n M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda Garcia, A.\n Ukleja, A. Ustyuzhanin, U. Uwer, V. Vagnoni, G. Valenti, A. Vallier, R.\n Vazquez Gomez, P. Vazquez Regueiro, C. V\\'azquez Sierra, S. Vecchi, J.J.\n Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D. Vieira, X.\n Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V.\n Vorobyev, C. Vo\\ss, H. Voss, R. Waldi, C. Wallace, R. Wallace, S. Wandernoth,\n J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M. Whitehead, J.\n Wicht, J. Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson, M.P. Williams,\n M. Williams, F.F. Wilson, J. Wimberley, J. Wishahi, W. Wislicki, M. Witek, G.\n Wormser, S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y. Xie, Z. Xing, Z. Yang,\n X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F. Zhang, L. Zhang, W.C.\n Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A. Zvyagin", "submitter": "Carla Gobel", "url": "https://arxiv.org/abs/1310.7953" }
1310.7998	# Ice Shelves as Floating Channel Flows of Viscous Power-Law Fluids Indranil Banik & Justas Dauparas ###### Abstract We attempt to better understand the flow of marine ice sheets and the ice shelves they often feed. Treating ice as a viscous shear-thinning power law fluid, we develop an asymptotic (late-time) theory in two cases - the presence or absence of contact with sidewalls. Most real-world situations fall somewhere between the two extreme cases considered. The solution when sidewalls are absent is a fairly simple generalisation of that found by Robison. In this case, we obtain the equilibrium grounding line thickness using a simple computer model and have an analytic approximation. For shelves in contact with sidewalls, we obtain an asymptotic theory, valid for long shelves. We determine when this is. Our theory is based on the velocity profile across the channel being a generalised version of Poiseuille flow, which works when lateral shear dominates the force balance. We conducted experiments using a laboratory model for ice. This was a suspension of xanthan in water, at a concentration of $0.5\%$ by mass. The lab model has $n\approx 3.8$ (similar to that of ice). Our theories agreed extremely well with our experiments for all relevant parameters (front position, thickness profile, lateral velocity profile, longitudinal velocity gradient and grounding line thickness). We also saw detailed features similar to natural systems. Thus, we believe we have understood the dominant force balance in both types of ice shelf. Combining our understanding of the forces in the system with a basic model for basal melting and iceberg formation, we uncovered some instabilities of the natural system. Laterally confined ice shelves can rapidly disintegrate but ice tongues can’t. However, ice tongues can be shortened until they no longer exist, at which point the sheet becomes unstable and ultimately the grounding line should retreat above sea level. While the ice tongue still exists, the flow of ice into it should not be speeded up by changing conditions in the shelf and the grounding line should also not retreat (if only conditions in the ocean change). However, laterally confined shelves experience significant buttressing. If removed, this leads to a rapid speedup of the sheet and a new equilibrium grounding line thickness. ###### Contents 1. 1 Introduction 2. 2 Ice Tongues 3. 3 Laterally Confined Ice Shelves 4. 4 The Grounding Line 1. 4.1 No Sidewalls 2. 4.2 With Sidewalls 5. 5 Experiments 1. 5.1 The setup 2. 5.2 1% aqueous xanthan solution 3. 5.3 0.5% aqueous xanthan solution 4. 5.4 Thickness Profile 5. 5.5 Particle Imaging Velocimetry 6. 5.6 No Sidewalls 6. 6 Towards The Natural System 1. 6.1 Including Flux Non-Conservation 2. 6.2 Ice Tongues 3. 6.3 Laterally Confined Ice Shelves 7. 7 Conclusions ## 1 Introduction This paper builds on previous work by Robison on what are essentially ice tongues (shelves with no lateral confinement). We generalise this to fluids with arbitrary shear-thinning coefficient $n$ (our theory should also work for shear-thickening fluids, but we did not perform experiments using such fluids). Then, we also attempt to generalise the result for laterally confined ice shelves, found by Pegler. The ice sheet is treated as a classical viscous gravity current, though we need to be careful about when this is a valid assumption. In this paper, we start with a first principles theoretical solution to shelves without sidewalls in the case of constant initial thickness. Reasons for expecting the initial thickness to be constant are also given. Then, we derive a similarity solution for a laterally confined shelf in a channel of constant width. The solution is valid in the asymptotic limit, so we also derive roughly where this is. The sheet is also briefly reviewed, so that we can address the grounding line. The equilibrium grounding line thickness is found for ice tongues. For laterally confined ice shelves, we already have the initial thickness and thus the grounding line position without considering the sheet. We briefly discuss how the sheet may influence the dynamics, noting that it does not in the asymptotic limit. We describe experiments we conducted to help us develop these theories and to test them. The experiments with $1\%$ concentration xanthan are described first, for the case of sidewall contact. Then, the effect of lowering the concentration is shown. Data for experiments in which there was no sidewall contact are also shown, and compared with theoretical predictions for the grounding line thickness. An important point is that there are artefacts of our experimental setup, due to the way the flow is initialised. We determine the length over which such effects are dissipated. Fortunately, the sheet was longer than this length. As well as the position of the propagating front, we also have the velocity field in the shelf and sheet. The methods used to obtain this data and the results are discussed and compared with theoretical expectations. The thickness of the shelf as a function of position (the profile) is also shown from a photograph, and compared with our model. Using our newly developed understanding, we give a partial explanation of the effect of an ice shelf collapsing on the rate of flow of the associated ice sheet. This should be treated with some caution at this stage, but can readily explain large increases in the flow rate over short time intervals (such as occurred with Larsen B). Our work suggests that such a phenomenon can only occur in ice shelves significantly affected by sidewall contact, something which is easily checked. Our model allows a rough calculation of the magnitude of the effect if a particular ice shelf were to collapse, based on topography and other data. However, our work does not shed much light on which shelves are likely to actually collapse. ## 2 Ice Tongues Figure 1: Plan and side views of the situation considered. We consider the flow of an incompressible viscous fluid (at low Reynolds number) according to the geometry shown in Figure 1 (with $Q$ held constant) and assume that the flow does not spread laterally. Although the width of the flow probably could be determined based on $Q$ and other parameters, we do not attempt to do this. Instead, we treat the width of the shelf as an independent variable. The force balance in the $x$-direction is that $\displaystyle\frac{\partial}{\partial x}\int_{-b}^{h}{{{\sigma}_{xx}}~{}dz}=-{{\rho}_{w}}gb\frac{\partial b}{\partial x}$ (1) because there are no lateral or vertical stresses and water pressure has a component in the $x$-direction (because the normal to the shelf does). We neglect lateral flow and assume that $H^{\prime}\ll 1$ so that $w\ll u$ (vertical flow negligible). Because the value of the above integral at the front of the shelf must balance with the integrated hydrostatic pressure of the ocean, we have (see Robison et al) that $\displaystyle\int_{-b}^{h}{{\sigma}_{xx}}~{}dz$ $\displaystyle=$ $\displaystyle-\frac{1}{2}{{\rho}_{w}}g{b}^{2}=-\frac{1}{2}\rho gHb,\text{ because }\rho_{w}b=\rho H\text{ (Archimedes)}$ (2) Considering the absence of lateral and vertical shear in this system, our model for the viscosity is $\displaystyle\eta=\eta_{o}{\left(\frac{\partial u}{\partial x}\right)}^{\frac{1}{n}-1}$ (3) Writing that $\displaystyle{{\sigma}_{xx}}\equiv-P+2\eta\frac{\partial u}{\partial x}$ (4) and applying a vertical balance of forces argument (using also $\frac{\partial w}{\partial z}=-\frac{\partial u}{\partial x}$) we obtain that $\displaystyle{{\sigma}_{xx}}=-\rho g(h-z)+4\eta\frac{\partial u}{\partial x}$ (5) Integrating this vertically, we get that $\displaystyle-\frac{1}{2}\rho g{{H}^{2}}+\int 4\eta\frac{\partial u}{\partial x}~{}dz$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\rho gHb$ (6) $\displaystyle 4{{\eta}_{o}}{{\int{\left(\frac{\partial u}{\partial x}\right)}}^{\frac{1}{n}}}\text{ }dz$ $\displaystyle=$ $\displaystyle\frac{1}{2}\rho gH(H-b)$ (7) $\displaystyle=$ $\displaystyle\frac{1}{2}\rho g^{\prime}{{H}^{2}}\textrm{ because }h\equiv\frac{\text{ }g^{\prime}}{g}H$ (8) The term $g^{\prime}$ is called the reduced gravity, and accounts for the fact that only a fraction of the shelf is above the waterline. Thus, gradients in the height above sea level are smaller than gradients in $H$. Applying Archimedes’ Principle to the shelf, we get that $\displaystyle\frac{g^{\prime}}{g}=\frac{{\rho}_{w}-\rho}{{\rho}_{w}}$ (9) Continuing with our derivation, $\displaystyle{{\eta}_{o}}{{\left(\frac{\partial u}{\partial x}\right)}^{\frac{1}{n}}}H$ $\displaystyle=$ $\displaystyle\frac{\rho g^{\prime}{{H}^{2}}}{8}$ (10) $\displaystyle\frac{\partial u}{\partial x}$ $\displaystyle=$ $\displaystyle{{\left(\frac{\rho g^{\prime}H}{8{{\eta}_{o}}}\right)}^{n}}$ (11) As this is positive, we note that $u>0$ throughout the shelf. Now, we use this information inside the continuity equation: $\displaystyle{d}^{-1}\frac{\partial H}{\partial t}\ +H\frac{\partial u}{\partial x}\ +u\frac{\partial H}{\partial x}=0$ (12) We use a Lagrangian picture to better visualise the situation. $\displaystyle{d}^{-1}\frac{DH}{Dt}\ $ $\displaystyle=$ $\displaystyle-H\frac{\partial u}{\partial x}\ $ (13) $\displaystyle=$ $\displaystyle-{{H}^{n+1}}\alpha\ \textrm{ where }\alpha\equiv{{\left(\frac{\rho g^{\prime}}{8{{\eta}_{o}}}\right)}^{n}\textrm{ is constant.}}$ (14) In the co-moving (Lagrangian) frame, each fluid element enters the shelf at time ${{t}_{0}}$. Assuming that H$({{t}_{0}})$ is independent of $t$ (i.e. a constant source thickness), we must have that $H=H(t-{{t}_{0}})$ only. This means that $\left.\frac{\partial u}{\partial x}\right\|_{t}=f(t-{{t}_{0}})$. Therefore, $\displaystyle u(x,t)$ $\displaystyle=$ $\displaystyle{{u}_{0}}+\int_{0}^{x}{f(t-{{t}_{0}})~{}dx^{\prime}}$ (15) $\displaystyle=$ $\displaystyle{{u}_{0}}+\int_{0}^{x}{f(\tau(x^{\prime}))~{}dx^{\prime}}\text{ where }\tau\equiv t-{{t}_{0}}\text{ and }\tau=0\text{ for }x=0$ (16) We now convert the integral required to obtain $u$ from one over $x$ to one over $\tau$. The value of $\tau$ is 0 when $x=0$. When the fluid element reaches $x$, $\tau=t-{t}_{0}$. We note that a fluid element injected at the source over a time interval $d\tau$ has a total volume $Q~{}d\tau$. At all later times, it occupies the same volume. However, it is also a part of the profile. This means that it occupies a volume $Hd~{}dx$. Thus, $\displaystyle u({{t}_{0}},t)$ $\displaystyle=$ $\displaystyle{{u}_{0}}+\int_{0}^{t-{{t}_{0}}}{f(\tau)\frac{dx^{\prime}}{d\tau}\ d\tau}$ (17) $\displaystyle=$ $\displaystyle{{u}_{0}}+\int_{0}^{t-{{t}_{0}}}{f(\tau)\frac{Q}{H(\tau)~{}d}\ d\tau}$ (18) This means that $u$ is a function of ($t-t_{0}$) only, as we know that $H$ is and $u_{0}=\frac{Q}{{{H}_{0}}d}$ is assumed constant. Integrating $u$ and assuming that the source of the shelf (a grounding line) remains static, we see that $x$ is also going to be a function of $\ t-{{t}_{0}}$ only. All fluid elements reach a given $x$ at the same value of $\tau$. Thus, at that position, $H$ is always the same (after the front has reached this position). We therefore have a steady profile. Under these conditions, the continuity equation reduces to $\displaystyle u=\frac{Q}{Hd}$ (19) Differentiating this with respect to $x$ and substituting in Equation 11, we obtain a first-order differential equation for the profile. Solving this subject to the constant initial thickness $H_{0}$, we get that. $\displaystyle H$ $\displaystyle=$ $\displaystyle{{\left[\frac{Q}{(n+1)\alpha\ d}\right]}^{\frac{1}{n+1}}}{{\left(x+L\right)}^{-\frac{1}{n+1}}}$ (20) $\displaystyle u$ $\displaystyle=$ $\displaystyle{{\left(\frac{Q}{d}\right)}^{\frac{n}{n+1}}}{{\left[\left(n+1\right)\alpha d\right]}^{{}^{\frac{1}{n+1}}}}{{\left(x+L\right)}^{\frac{1}{n+1}}}$ (21) The constants $L$ and $\alpha$ are defined below: $\displaystyle L=\frac{Q}{(n+1)\alpha d{{H}_{0}}^{n+1}}$ (22) $\displaystyle\alpha\equiv{\left(\frac{\rho g^{\prime}}{8{{\eta}_{o}}}\right)}^{n}$ (23) When $x_{n}\ll L$, the profile is essentially flat and the speed equals the initial value ($\frac{Q}{{H}_{0}d}$). For $x_{n}\gg L$, we have that $\displaystyle H\sim{{x}^{-\frac{1}{n+1}}}$ (24) (with $u\sim{x}^{\frac{1}{n+1}}$). Finally, the position of the front as a function of time is readily determined from the velocity field. $\displaystyle{{x}_{n}}=\frac{Q}{(n+1)\alpha\ H_{0}^{n+1}d}\left[{{\left(1+\alpha n{{H}_{0}}^{n}t\right)}^{{}^{\frac{1}{n}+1}}}-1\right]$ (25) Alternatively, we could ensure the area enclosed by the profile upstream of the front is correct. As the profile is steady, this entails solving $\displaystyle\int_{0}^{{{x}_{n}}(t)}{H(x)dx=\frac{Qt}{d}}$ (26) Of course, both approaches agree for all $n$. For the case of $n=1$, our solution reduces to that found by Robison et al in 2010. In a real system, the shelf would be fed by a sheet at a grounding line (the ‘source’). The thickness here completely determines the buttressing exerted on the sheet and also affects the velocity field in the sheet. There is likely to be a unique thickness at which the forces at the grounding line are in equilibrium. Once this is attained, there is no further tendency for change (as the buttressing is independent of how far the front has propagated - it is always $\frac{1}{2}\rho g^{\prime}{{H}_{0}}^{2}$). We assume that the equilibrium so attained is stable. The equilibrium grounding line thickness is calculated in Section 4.1, although perturbations are not considered in this work. ## 3 Laterally Confined Ice Shelves The geometry in this situation is the same as that considered before, except that now the half-width of the shelf rather than the full width is $d$. The major difference is the presence of laterally confining sidewalls, which we assume the shelf is in contact with at all times. The effect of sidewalls will dominate over the effects of hydrostatic pressure on an ice shelf if the shelf is long enough relative to its width and height. To get an estimate for when this may be the case, we determine when the shelf starts thickening. In order for this to happen, the velocity field must be altered, so that instead of $\frac{\partial u}{\partial x}>0$, we instead have $\frac{\partial u}{\partial x}<0$. This means that, rather than continuity forcing the shelf to thin with distance, the front is going slower than fluid elements behind it so that the flow essentially ‘piles up’. We expect this to occur in order to provide a pressure gradient to overcome the viscous drag from sidewalls and keep the shelf flowing. The key thing is that ${{\sigma}_{xx}}$ is not purely hydrostatic pressure. There is a difference between pressure from xanthan and that from water (partly due to their different densities). This is balanced with a non-zero value of $\frac{\partial u}{\partial x}$. As we have seen, this is (initially) positive. Now, if we were to reduce ${{\sigma}_{xx}}$ enough that $\frac{\partial u}{\partial x}$ were forced to become negative, then the situation would indeed be different to the no sidewalls scenario. To achieve this, a certain amount of drag from sidewalls is required. Note that $\frac{\partial{{\sigma}_{xx}}}{\partial x}+\frac{\partial{{\sigma}_{xy}}}{\partial y}=0$. For $y>0$, $\frac{\partial u}{\partial y}<0$ so ${{\sigma}_{xy}}<0$. Noting that the surface $y=0$ is free by symmetry, we see that $\frac{\partial{{\sigma}_{xy}}}{\partial y}<0$, this also holding for $y<0$. Thus, $\frac{\partial{{\sigma}_{xx}}}{\partial x}>0$. Assuming $H$ is not yet altered (so neither is $\frac{\partial u}{\partial x}$ at the front), this means that $\frac{\partial u}{\partial x}$ at the source eventually goes negative and becomes increasingly so. Once this occurs, the front starts decelerating and the shelf will be forced to start thickening. As before, the initial value of that part of ${{\sigma}_{xx}}$ which created a $\frac{\partial u}{\partial x}$ term is $\frac{1}{2}\rho g^{\prime}{{H}^{2}}$ (when integrated vertically) - see Equation 4. Thus, for the shelf to start thickening, the total force from sidewalls must exceed the lateral integral of the above term (the total non-hydrostatic force, or pushing force). This way, there will no longer be any pushing force at all. With an even longer shelf and even more drag, it will change sign, making $\frac{\partial u}{\partial x}<0$. We assume that this is a good indicator of when sidewalls start to have a significant impact upon the dynamics of the flow. $\displaystyle\frac{1}{2}\rho g^{\prime}{{H}^{2}}.2d={{\eta}_{0}}{{\left(\frac{1}{2}\frac{\partial u}{\partial y}\right)}^{\frac{1}{n}-1}}\left(\frac{\partial u}{\partial y}\right).2LH$ (27) Henceforth, if we raise a negative number to a power, we mean that the absolute value of the number is raised to that power. _The end result is always positive_. For the viscosity, we have assumed that $u=0$ along the walls, so only lateral variations contribute to the total strain there. As a rough estimate, we assume a triangular velocity profile which gives $\displaystyle\frac{\partial u}{\partial y}=\frac{2\overline{u}}{d}$ (28) This is an underestimate because the boundary layer is probably very thin so has more shear. Using also the fact that $\overline{u}=\frac{Q}{2Hd}$, we obtain that $\displaystyle L=\frac{\rho g^{\prime}{{H}^{{}^{1+\frac{1}{n}}}}{{d}^{{}^{1+\frac{2}{n}}}}}{{{2}^{2-\frac{1}{n}}}{{\eta}_{0}}{{Q}^{{}^{\frac{1}{n}}}}}$ (29) We note that $\frac{\partial u}{\partial x}$ is very small in the asymptotic limit, because the shelf will get longer and longer (while the start of the shelf gets thicker, so $u$ there decreases). As the force balance equation must always hold, we expect (treating ${{\sigma}_{xx}}$ as purely hydrostatic) that the above equation linking $L$ and some typical (e.g. maximum) thickness should hold in the asymptotic limit, when our approximation that $\frac{\partial u}{\partial x}$ is very small becomes accurate. Essentially, we have balanced the hydrostatic pressure discontinuity not with a $\frac{\partial u}{\partial x}$ term but instead with a $\frac{\partial u}{\partial y}$ term. This suggests that the system may be self-similar at late times. We have said that $x_{n}\gg L$ is required for sidewalls to dominate the shelf, but assuming that a solution is discovered valid for such situations, when does this solution first become an accurate description of the length of the shelf? One probably needs computer simulations to answer this in detail, but here we give a rough idea. The shelf can not be longer than the solution predicts, because then $\left\|H^{\prime}\right\|$ is even lower so there is insufficient driving force to overcome viscous drag from the sidewalls. Also, even more drag exists than in the solution, because the shelf is even longer. So the situation can not arise. However, the shelf can be shorter than our solution predicts - to compensate for the lower drag, it can simply be thick at the front, reducing the thickness gradient. So we see that there is no problem with a shelf shorter than the solution predicts, but a major problem with longer shelves - these can’t exist (if the force balance is dominated by sidewalls). Thus, one way to determine $L$ may be to find when continuing to apply the no sidewalls (basically, constant $u$) solution leads to the front being further ahead than the yet to be derived solution when sidewalls dominate. Because this situation is impossible, we can use this to determine the point at which the no sidewalls solution can no longer be applied to the system. This approach would require determining the initial thickness of the shelf, which is possible under some circumstances (see Section 4.1). We begin by using the fundamental equation for the balance of forces along the channel. $\displaystyle\frac{\partial{{\sigma}_{xx}}}{\partial x}+\frac{\partial{{\sigma}_{xy}}}{\partial y}=0$ (30) We assume that $\frac{\partial u}{\partial x}\ll\frac{\partial u}{\partial y}$ over the vast majority of the channel, because the shelf is much longer than it is wide. This allows us to consider only lateral stresses. Also assuming negligible transverse and vertical velocities and no lateral variations in thickness, we obtain that $\displaystyle\frac{\partial}{\partial y}\left[{{\eta}_{o}}{{\left(\frac{1}{2}\frac{\partial u}{\partial y}\right)}^{\frac{1}{n}-1}}\frac{\partial u}{\partial y}\right]=\rho g^{\prime}\frac{\partial H}{\partial x}$ (31) Notice that only gradients in $H$ can affect the velocity field, because the resulting pressure gradients _alone_ drive the flow. Integrating the above equation with respect to $y$ and applying the no-slip boundary condition for $y=\pm d$ as well as no lateral stress along the centreline of the channel due to symmetry (i.e. $\frac{\partial u}{\partial y}=0\text{ for }y=0$), we obtain the velocity profile: $\displaystyle u=\frac{{2}^{1-n}}{n+1}{{\left(\frac{\rho g^{\prime}H^{\prime}}{{{\eta}_{o}}}\right)}^{n}}\left({{d}^{n+1}}-{{y}^{n+1}}\right)\text{ where }H^{\prime}\equiv\frac{\partial H}{\partial x}$ (32) A flow with a velocity pattern like this we shall call a generalised Poiseuille flow. The flux crossing a plane of constant $x$ is easily found to be $\displaystyle q(x)$ $\displaystyle=$ $\displaystyle 2H\int_{0}^{d}u~{}dy$ (33) $\displaystyle=$ $\displaystyle\frac{{2}^{2-n}}{n+2}{{\left(\frac{\rho g^{\prime}}{{{\eta}_{o}}}\right)}^{n}}{d}^{n+2}\left(H{H^{\prime}}^{n}\right)$ (34) The continuity equation can now be applied to obtain a single non-linear diffusion equation for the fluid. $\displaystyle\frac{\partial H}{\partial t}+\frac{{{2}^{1-n}}{{d}^{n+1}}}{n+2}{{\left(\frac{\rho g^{\prime}}{{{\eta}_{o}}}\right)}^{n}}{{\left(HH{{{}^{\prime}}^{{}^{n}}}\right)}^{{}^{\prime}}}=0$ (35) For a more complicated geometry, a computer simulation will be required to understand what happens, although future work on simple geometries and on slowly varying $d$ could shed some light on the problem. For such work, the ${d}^{n+1}$ term should be brought inside the last bracket, to allow for the possibility that the width of the channel varies with position. For now, $d$ is constant. We now apply a scaling argument to the above equation and also to the equation of global mass conservation $\displaystyle\int_{0}^{{{x}_{n}}(t)}{H(x)dx=\frac{Qt}{2d}}$ (36) This suggests that the following quantity is a dimensionless constant of order 1: $\displaystyle\frac{{{x}_{n}}{{(n+2)}^{\frac{1}{2n+1}}}{{2}^{\frac{2n-1}{2n+1}}}}{{{t}^{\frac{n+1}{2n+1}}}{{d}^{\frac{1}{2n+1}}}{{Q}^{\frac{n}{2n+1}}}}{{\left(\frac{{{\eta}_{o}}}{\rho g^{\prime}}\right)}^{\frac{n}{2n+1}}}$ (37) We look for a solution in terms of the similarity variable $\displaystyle\varepsilon=\frac{x{{(n+2)}^{\frac{1}{2n+1}}}{{2}^{\frac{2n-1}{2n+1}}}}{{{t}^{\frac{n+1}{2n+1}}}{{d}^{\frac{1}{2n+1}}}{{Q}^{\frac{n}{2n+1}}}}{{\left(\frac{{{\eta}_{o}}}{\rho g^{\prime}}\right)}^{\frac{n}{2n+1}}}$ (38) Our analysis indicates that this is directly proportional to $\frac{x}{{x}_{n}}$. Because ${x}_{n}$ rises slower than $t$, the fact that the area enclosed by the profile must rise linearly with time implies that the whole profile must also be thickening. Thus, $H$ will necessarily have an explicit dependence on $t$. Using the fact that $H\sim\frac{Qt}{2dx_{n}}$, we obtain that $\displaystyle H=\frac{{{(n+2)}^{\frac{1}{2n+1}}}{{Q}^{\frac{n+1}{2n+1}}}{{t}^{\frac{n}{2n+1}}}}{{{2}^{\frac{2}{2n+1}}}{{d}^{\frac{2n+2}{2n+1}}}}{{\left(\frac{{{\eta}_{o}}}{\rho g^{\prime}}\right)}^{\frac{n}{2n+1}}}\psi(\varepsilon)$ (39) where $\psi(\varepsilon)$ (the dimensionless profile) is of order 1 near the source and decreases to 0 at the front. Differentials in $x$ and $t$ can be converted into differentials in $\varepsilon$, applying the usual chain rule. Such an analysis shows that the powers of all externally imposed parameters are indeed equal on all terms, so that we may obtain a single ordinary differential equation for the profile (in terms of similarity co-ordinates). $\displaystyle{{(\psi\psi{{{}^{\prime}}^{{}^{n}}})}^{\prime}}$ $\displaystyle=$ $\displaystyle-\frac{n}{2n+1}\psi(\varepsilon)+\frac{n+1}{2n+1}\varepsilon\psi^{\prime}(\varepsilon)$ (40) $\displaystyle\int_{0}^{{{\varepsilon}_{n}}}{\psi(\varepsilon)d\varepsilon}$ $\displaystyle=$ $\displaystyle 1$ (41) $\displaystyle\psi{{\psi^{\prime}}^{n}}$ $\displaystyle=$ $\displaystyle 1\text{ at }\varepsilon=0$ (42) The term $\psi\psi{{{}^{\prime}}^{{}^{n}}}$ corresponds to the (dimensionless) flux crossing a given position. This gradually decreases from its initial value. The reason is that part of it is ‘lost along the way’ because it goes into increasing the thickness of the profile. In the real system, the advance of the front is not driven by the requirement to push flux through it (unlike in ice tongues). It is in fact driven by the velocity of fluid elements at the front (because $H^{\prime}\neq 0$). We obtain approximate expressions for $\psi^{\prime}$ that become exact at either end of the profile. Near the source (or the rear) of the profile, $\displaystyle\psi^{\prime}\approx-{{\left(\frac{1}{\psi\left(0\right)}\right)}^{\frac{1}{n}}}$ (43) Near the front (at ${{\varepsilon}_{n}}$), we may obtain a first integral of Equation 40 to deduce that $\displaystyle\psi^{\prime}\approx-{{\left(\frac{n+1}{2n+1}{{\varepsilon}}\right)}^{\frac{1}{n}}}$ (44) We have used the fact that the integral of $\psi$ with respect to $\varepsilon$ (from the front to a point nearby) is second order in the value of $\psi$, as the profile is approximately triangular in this region (a singularity in $H^{\prime}$ leads to a singular velocity profile, so $H^{\prime}$ must be finite). Using these results, we may obtain an expression for the total change in $\psi^{\prime}$ over the profile, giving directly the fractional change in velocity along the profile (because $u\propto\psi{{{}^{\prime}}^{{}^{n}}}$) - though we need the actual value of ${\varepsilon}_{n}$ to compute this. We solved the differential equation numerically by shooting backwards from the front, using as boundary conditions $\psi=0$ at $\varepsilon={\varepsilon}_{n}$ and the above expression for $\psi^{\prime}$. Due to computing errors, there is a small error in $\psi\psi{{{}^{\prime}}^{{}^{n}}}$ at the source, but none at the front of the profile. Obviously, errors near the source are much preferred, because $\psi\psi{{{}^{\prime}}^{{}^{n}}}=0$ at the front (so we can ill afford errors here). Although the solution looks reasonable for almost any value of ${{\varepsilon}_{n}}$, only one value can actually make the total area enclosed by the profile equal to 1. An estimate for the error made by the computer can then be obtained by checking how far off the solution is from satisfying Equation 42. Figure 2: Dimensionless profile for $n=3.8$. We promise we didn’t just draw a triangle! Amazingly, the whole profile is very nearly triangular. However, the slope does steepen by about 3% for $n=3.8$. At higher $n$, this effect is reduced and both sides have length approaching $\sqrt{2}$. The reason is that $\psi\psi{{{}^{\prime}}^{{}^{n}}}$ always goes from 1 to 0, and as $\psi\neq 0$ (except right at the front), we must have that $\displaystyle\psi^{\prime}\to 1\forall\varepsilon\text{ as }n\to\infty$ (45) We note briefly that for $n=0$, the dimensionless profile is the unit square. $n$ \| $\psi(0)$ \| ${{\varepsilon}_{n}}$ \| Fractional change in $u$ ---\|---\|---\|--- 3.6 \| 1.362 \| 1.461 \| 11.6% 3.8 \| 1.364 \| 1.460 \| 11.1% 5.0 \| 1.374 \| 1.452 \| 8.8% 5.2 \| 1.375 \| 1.451 \| 8.5% $\infty$ \| 1.414 \| 1.414 \| 0 Table 1: Results of computer simulations for those values of $n$ used in our experiments, and nearby values consistent with the error in $n$. Also included is the result for $n=\infty$. The expressions for the front position and the source thickness of the profile as functions of time are: $\displaystyle{{x}_{n}}=\frac{{{t}^{\frac{n+1}{2n+1}}}{{d}^{\frac{1}{2n+1}}}{{Q}^{\frac{n}{2n+1}}}}{{{(n+2)}^{\frac{1}{2n+1}}}{{2}^{\frac{n-1}{2n+1}}}}{{\left(\frac{\rho g^{\prime}}{{{\eta}_{o}}}\right)}^{\frac{n}{2n+1}}}{{\varepsilon}_{n}}$ (46) $\displaystyle{{H}_{0}}=\frac{{{t}^{\frac{n}{2n+1}}}{{(n+2)}^{\frac{1}{2n+1}}}{{Q}^{\frac{n+1}{2n+1}}}}{{{2}^{\frac{n+2}{2n+1}}}{{d}^{\frac{2n+2}{2n+1}}}}{{\left(\frac{{{\eta}_{o}}}{\rho g^{\prime}}\right)}^{\frac{n}{2n+1}}}\psi(0)$ (47) Notice that the gradient of the profile decreases with time (i.e. ${{H}_{0}}$ grows slower than ${{x}_{n}}$). This is to keep the entry flux the same, despite a greater thickness (forcing a reduction in $u$ and thus $\left\|H^{\prime}\right\|$). As we have seen, thickening of the shelf is a hallmark of it being affected by viscous drag from sidewalls. For this to _dominate_ , we need to allow significant thickening of the shelf. However, at a length of $L$, it will only just have started to thicken, so sidewalls will only dominate when ${x}_{n}\gg L$. We expect convergence to be slow because it takes some time for the transient to die down (the decay is $\ {{x}_{n}}^{-1}$). This is because we assume that a section of shelf of length $L$ is unaffected by sidewalls, so it is outside our model (and creates something akin to a shift in position measurements). This region is essentially flat (for realistic parameters) - see Figure 8 \- because the force balance was different when this region crossed the source. The shelf behaves essentially as a solid body ($\frac{\partial u}{\partial x}$ is very small), so the result of this earlier time remains permanently imprinted upon the shelf. For our solution to work well, we need this region to be a very small part of the entire shelf. This way, the last vestiges of the times when sidewalls were unimportant will have faded into insignificance. Although we have estimated what length of shelf is required for sidewalls to dominate the system, this alone will not guarantee the similarity solution being accurate. This is because, even if the force balance was dominated by lateral friction from confining sidewalls, the amount of longitudinal stress _inherent to our solution_ could still be very large, making it internally inconsistent. The difference in $u$ from the source to the front is approximately 10% (for $n=4$), so $\displaystyle\frac{\partial u}{\partial x}\approx\frac{u}{10{{x}_{n}}}$ (48) Obviously, there will be a region close to the centreline of the channel where $\frac{\partial u}{\partial x}>\frac{\partial u}{\partial y}$, but as long as this region is small, our solution should be accurate. For this to occur, we compare $\frac{\partial u}{\partial x}$ along the centreline of the channel with $\frac{\partial u}{\partial y}$ at the sidewalls (i.e. we compare maximum values). For $n=4$, this leads to the requirement that $\displaystyle\frac{u}{10{{x}_{n}}}\ll\frac{4u}{d}$ (49) The conclusion, not altogether unexpected, is that the shelf needs to have a minimum aspect ratio. If we wish for $\frac{\partial u}{\partial y}$ close to the sidewalls to be at least $10$ times larger than $\frac{\partial u}{\partial x}$, then the shelf only needs to be half as long as the full width of the channel! Thus, the similarity solution is internally consistent for very short shelves. However, it does need to be much longer than $L$, and we believe that this is usually the stricter condition (it certainly was in our experiments). We now touch briefly upon the effect of variations in thickness across the channel. In this case, a first integral of Equation 31 will no longer simply be directly proportional to $y$. Assuming the thickness is smaller near the sidewalls, then this will be a convex function. Therefore, $\frac{\partial u}{\partial y}$ will be greater than before, for the same average $H$ and $H^{\prime}$. The effect of this can be determined by multiplying the formula for $q(x)$ by a factor greater than 1. However, the effect on the position of the front will be smaller than it might at first appear. Although the front must be further ahead than without the lateral thickness variation, this will also reduce the thickness and (combined with higher ${x}_{n}$), will reduce $\left\|H^{\prime}\right\|$. Therefore, the fractional increase in ${x}_{n}$ (at the same value of $t$) is only $\frac{1}{2n+1}$ times as much as the fractional change in $q$. Thus, as long as the sidewalls are able to maintain the no-slip condition (i.e. as long as contact is not lost altogether), we expect the effect of lateral thickness variations on the front position to be small. ## 4 The Grounding Line We now introduce a sloped bed at an angle of inclination of $\alpha$. The waterline is just above the top of this slope, with the weir just above the waterline. We now have both a sheet and a shelf, with the two linked at a grounding line. All parameters used previously still have the same meaning, except $d$. This is once again used for the full width of the shelf. A$\ {}_{G}$ subscript is used to denote parameter values at the grounding line. Figure 3: Side view of a channel flow with a grounding line. We model the grounded portion of the viscous layer (the sheet) as a viscous gravity current. We prove later that this is valid. We also believe it to be valid in most natural situations, but can not confirm this. Assuming that $H\ll d$ (or that there are no sidewalls), we get that $\displaystyle\frac{\partial{{\sigma}_{xx}}}{\partial x}+\frac{\partial{{\sigma}_{xz}}}{\partial z}=0$ (50) The assumption of a viscous gravity current is formally equivalent to approximating the $\frac{\partial{{\sigma}_{xx}}}{\partial x}$ term as a hydrostatic pressure gradient. Therefore, we get that $\displaystyle\frac{\partial}{\partial z}\left(\eta\frac{\partial u}{\partial z}\right)=\rho gh^{\prime}$ (51) Solving the above equation subject to no slip at the base ($z=-b$) and a free upper surface ($\frac{\partial u}{\partial z}$ = 0 at $z=h$), we obtain the velocity profile for the sheet: $\displaystyle u(x,z)={{\left(\frac{\rho gh^{\prime}}{{{\eta}_{o}}}\right)}^{n}}\frac{{{2}^{1-n}}}{n+1}\left[{{H}^{n+1}}-{{(h-z)}^{n+1}}\right]$ (52) The flux crossing given a plane of constant $x$ is readily found to be: $\displaystyle q(x)=d{{\left(\frac{\rho gh^{\prime}}{{{\eta}_{o}}}\right)}^{n}}\frac{{{2}^{1-n}}}{n+2}{{H}^{n+2}}$ (53) Note the similarity of the above equations with the corresponding ones for the shelf (with sidewalls). The confining surface runs parallel to the driving force, but in one case it is underneath and in another it is to either side. The shelf has no free surface like the sheet, but in the shelf the centreline of the channel acts as a free surface due to symmetry. Thus, we see that there is no fundamental difference between an ice sheet and an ice shelf confined by sidewalls – both have gravity balancing viscous drag and they also have similar boundary conditions, leading to a similar velocity profile. ### 4.1 No Sidewalls We solve first for the case where the shelf is not in contact with sidewalls. We assume that the flow does not spread laterally very much, or that it does so only over a very small region near the weir but not near the grounding line or in the shelf (so the width is constant in the regions we now discuss). We also assume that the grounding line has already reached its equilibrium position, so that conditions in the sheet close to this point are steady. In this case, we can set $q=Q$ near the grounding line. The fundamental force balance at the grounding line is for the force exerted on the water-facing side of the fluid in the $x$-direction. This is because such a force can not be transmitted anywhere except into the shelf. Therefore, it must balance with the same force in the shelf (where it is created by hydrostatic pressure of the ocean). In other words, we require continuity of $\int_{-b}^{h}{{{\sigma}_{xx}}~{}dz}$ across the grounding line. In the sheet, there is a contribution from hydrostatic pressure of $\frac{1}{2}{{\rho}}g{{H}^{2}}$. However, it is not balanced by the normal stress in the shelf ($\frac{1}{2}{{\rho}_{w}}g{{b}^{2}}$). The difference is $\frac{1}{2}\rho g^{\prime}{{H}^{2}}$. This must be accounted for by _non- hydrostatic_ forces in the sheet. Using the usual balance of vertical forces argument along with conservation of mass, we get that $\displaystyle I\equiv\int_{-b}^{h}{4\eta\frac{\partial u}{\partial x}\text{ }dz}=\frac{1}{2}\rho g^{\prime}{{H}^{2}}\text{ at the grounding line}$ (54) The pushing force ($I$) is calculated directly from the velocity profile in Equation 52. We determine $I$ numerically given a particular value of $H$. The value of $h^{\prime}$ is fixed by the requirement that conditions in the sheet near the grounding line be steady (i.e. $q=Q$). Once $h^{\prime}$ is found, the computer next determines $\frac{\partial u}{\partial x}$ and $\frac{\partial u}{\partial z}$ as functions of $z$ at the grounding line, using also $H^{\prime}=h^{\prime}+\alpha$. The equilibrium thickness of the grounding line is then given by varying $H$ so as to make Equation 54 hold. The viscosity is affected by both vertical and horizontal shear. Without horizontal shear, the integral will diverge for $n>2$, assuming a non-zero value of $\frac{\partial u}{\partial x}$ near the free upper surface. Thus, we use $\displaystyle\eta={{\eta}_{o}}{{\left[\sqrt{{{\left(\frac{\partial u}{\partial x}\right)}^{2}}+\frac{1}{4}{{\left(\frac{\partial u}{\partial z}\right)}^{2}}}\right]}^{\frac{1}{n}-1}}$ (55) The flow in the sheet is dominated by vertical shear, which vanishes at the free upper surface. For a shear-thinning fluid, the viscosity is thus greatest near this surface. Here, $u$ is also greatest. Thus, both $\eta$ and $\frac{\partial u}{\partial x}$ will be greatest here, so $I$ is only really affected by the value of $\frac{\partial u}{\partial x}$ near the upper surface. The vertical velocity profile in the sheet has a thin boundary layer (for large $n$), so in order to have flux conservation we must approximately have that $u=\frac{Q}{Hd}$ outside this region. Thus, we expect that $I$ will change sign when $H^{\prime}$ changes sign (i.e. when $h^{\prime}=-\alpha$). At the corresponding value of $H$, $I$ should be very small. Computer simulations indicate that, for $H$ fairly close to the ‘right’ value but not exactly equal to it, $I$ is very sensitive to $H$. Thus, the value of $H$ which makes Equation 54 hold and the value of $H$ which makes the integral 0 are often quite close. This is equivalent to saying that the pushing force can easily be made quite large, compared with the hydrostatic pressure discontinuity. Thus, solving $I=0$ is approximately the right thing to do. This suggests that the grounding line thickness may be approximated by assuming that $h^{\prime}=-\alpha$ there, so that $\displaystyle{{H}_{G}}\approx{{\left[\frac{Q(n+2)}{d}\right]}^{\frac{1}{n+2}}}{{\left(\frac{{{\eta}_{o}}}{\rho g\alpha}\right)}^{\frac{n}{n+2}}}{{2}^{\frac{n-1}{n+2}}}$ (56) Notice that $g^{\prime}$ is irrelevant if this approximation is accurate. Figure 4: For parameter values matching those of one of our experiments, $I$ divided by the hydrostatic pressure discontinuity is shown as the red curve. The green line is at 1. However, the intersection of the blue line with the curve is a good approximation. The location of this point is given in Equation 56. Equation 56 requires $H^{\prime}=0$, but it does not actually result in $I=0$. Because conditions remain the same if we move parallel to the sloped bed, moving along $x$ _at fixed $z$_ will lead to a geometric effect whereby $\displaystyle\frac{\partial u}{\partial x}=\alpha\frac{\partial u}{\partial z}$ (57) Thus, the value of $I$ divided by the hydrostatic pressure discontinuity is not exactly $0$. It is: $\displaystyle{{2}^{1\frac{1}{n}}}{{\alpha}^{2}}{{\left({{\alpha}^{2}}+\frac{1}{4}\right)}^{\frac{1}{2}\left(\frac{1}{n}-1\right)}}\frac{g}{g^{\prime}}$ (58) If this is close to 1, then Equation 56 will be a good approximation to the result of a full computer simulation (and the ice will be running nearly parallel to the sloped bed). A ratio above 1 means that Equation 56 underestimates ${H}_{G}$ (as in Figure 4). For ice in sea water, we note that the equation holds exactly when $\alpha=9^{\circ}$. We warn the reader to check this ratio and the results of the computer simulation, to see what sign the error resulting from using Equation 56 is and whether its magnitude is acceptable. For now, the system is sufficiently simple that the full simulation only takes a few minutes. In more complicated systems, having a simple equation for the grounding line thickness may prove to be valuable, even if it is inexact. Note that the green line will appear closer to the black line at very low $g^{\prime}$ (and further for higher $g^{\prime}$), whereas the blue line will appear not to move. $g^{\prime}$ does matter. As expected, a less buoyant fluid will have a thicker grounding line. Interestingly, though, reductions in $g^{\prime}$ can not raise the grounding line thickness above a certain value (although increases in $g^{\prime}$ can lower $H_{G}$ without limit). ### 4.2 With Sidewalls The essential difference in the presence of sidewalls (assuming they dominate the system) is that, once a shelf with a particular grounding line thickness is formed, there _is_ a tendency for this thickness to change. Sidewall friction causes fluid to essentially ‘pile up’ behind the front to some extent, not just to flow completely freely as it does in the case of no sidewalls. This ‘piling up’ means that there is _no_ stable grounding line thickness. Therefore, the shelf thickens for ever. However, there is still a dynamic balance at the grounding line. This is because if all the flux entered the shelf, then it would want the grounding line thickness to increase at a certain rate. However, the sheet retains no flux, so it can’t grow. Thus, the grounding line can not advance. The impossibility of the situation reveals what must really happen: part of the flux is retained by the sheet, allowing the grounding line to advance; while part goes into the shelf, presumably an amount equal to that which causes ${{H}}_{0}$ to increase by precisely the rate at which the flux retained by the sheet allows. This means that there is a balance between dynamic conditions in the shelf (how much it wants to thicken, given the flux entering it) and kinematic conditions in the sheet (how much it must expand, given that it retains the flux not entering the shelf). Eventually, the flux entering the shelf approaches Q. This is because the flux retained by the sheet is approximately equal to $H{}_{G}\text{ }\overset{.}{\mathop{{{x}_{G}}}}\,$, where a time derivative is indicated. Of course, for a fixed angle sloped bed we have that $\overset{.}{\mathop{{{x}_{G}}}}\,\propto\overset{.}{\mathop{{{H}_{G}}}}\,$. Considering that ${{H}_{G}}\propto{{t}^{\frac{n}{2n+1}}}$, we see that ${{H}_{G}}\text{}\overset{.}{\mathop{{{H}_{G}}}}\,\propto{{t}^{-\frac{1}{2n+1}}}$. Thus, the flux retained by the sheet inevitably goes down to 0, but fairly slowly. This means that, even with a grounding line, the shelf will eventually converge to the similarity solution we found earlier (whether we consider the length of the shelf only, or the position of the front). The slow convergence may mean that in a real laterally confined ice shelf, it needs to be fairly long in order for all the flux to enter the shelf. Ultimately, if one is interested in what happens before convergence has occurred, a computer simulation will be required. This will need to solve our non-linear diffusion equation for the shelf and a similar version for the sheet. The boundary condition must be that flux not entering the shelf is retained by the sheet (and may go into causing grounding line advance). Similar models have already been devised for Newtonian fluids. We assume that the grounding line rapidly reaches a thickness such that $\left\|h^{\prime}\right\|\ll\alpha$, so that $H^{\prime}\approx\alpha$. This lets us approximate that $\displaystyle\frac{\partial u}{\partial x}\approx-\frac{Q}{{{H}^{2}}d}\alpha$ (59) As there will be something like an extra power of $H$ in the total pushing force exerted by the sheet (to account for the vertical integration), we see that this scales with time inversely to $H$. In the shelf, we have that $\displaystyle\frac{\partial u}{\partial x}\approx\frac{Q}{Hd\text{ }{{x}_{n}}}$ (60) Hydrostatic pressure of the ocean is of course completely dissipated by sidewall friction. The pushing force in the shelf will also need to have an extra power of $H$, so this scales with time inversely to ${{x}_{n}}$. We expect this to grow faster than $H$, on the basis of our similarity solution for the shelf (which the system converges to, eventually). Thus, in the end, the pushing force from the sheet will always exceed that from the shelf. The force balance at the grounding line still needs to hold. Now, ${\sigma}_{xx}\equiv-P+2\eta\frac{\partial u}{\partial x}$, so $P$ needs to be greater in the shelf than in the sheet. We believe this to mean that there is a sharp increase in $H$ immediately after the grounding line, with this sudden change in $H$ accounting for the discrepancy in the vertically integrated pushing force that we have just found. However, the effect becomes negligible in the asymptotic limit. We never noticed such an effect in any of our experiments, suggesting that it may be irrelevant. ## 5 Experiments ### 5.1 The setup Figure 5: The experimental apparatus used (distances approximate). The bottom view is acquired using a mirror at $45^{\circ}$ to the horizontal. A sloped bed was sometimes installed immediately after the weir (as shown in Figure 3) . The basic apparatus is shown in Figure 5. A peristaltic pump was used to maintain a constant flux into the region behind the sluice. The viscous fluid used was an aqueous suspension of xanthan gum at concentrations of 0.5% and 1% (by mass). Xanthan is a shear-thinning organic polymer. Salt was added to the ocean to increase its density. The viscous fluid was overflowing the weir and dropping, creating a rebound effect. We minimised this by leaving a very small gap between the ocean level and the top of the weir. We sometimes wished to include a sloped bed, in order to study grounding lines. We found that leaving even a tiny part of the slope exposed to air had a dramatic and adverse impact upon our experiments (like Robison). We therefore decided to have it entirely submerged, but to guarantee the formation of a sheet and also to reduce the rebound effect just mentioned, the sloped bed was usually placed 2mm below the top of the weir (with the sea level halfway between the top of the slope and the top of the weir, as indicated in Figure 3). Maintaining this configuration required accurate control of the sea level. We achieved this by means of a laser reflecting off the ocean surface onto a fixed screen. Water was siphoned out at a variable rate, with manual adjustments to this rate whenever necessary to keep the laser spot at the same location on the screen. We found that the rate of seawater extraction could be altered by 0.2g/s, so we could easily control it accurately enough for our purposes. It is likely that the sea level was controlled to within 1mm for most experiments, and 0.5mm for some of them where there were less ripples on the water (usually due to a lower flux). Therefore, systematic trends in the sea level were small during all of our experiments. The most probable cause of errors is simply the strong sensitivity of the experiments to initial conditions. Thus, a slight asymmetry (e.g. due to the tank being slightly tilted to one side) can cause loss of contact with the sidewalls at early times, leading to the sort of pattern seen in Figure 5. The finite extent of the experiments also created a finite error on any experimentally determined power law dependence of the parameters on time. This was mostly due to difficulties in determining precisely when the experiment started. Because xanthan would not usually overflow the whole weir at the same time, the flux entering the channel would rise from 0 to $Q$. This would cause the front to accelerate. To overcome this problem, we usually waited for the front to stop accelerating and then did a regression. The time at which this regression line passed through 0cm we took to be the time origin for the whole experiment. Our theory for no sidewall experiments indicates that acceleration not due to rising flux, if present at some time, must also necessarily be present at all later times during the experiment. As the front only accelerated at early times during these experiments, we concluded that this was in fact due to changing flux, and so this must also be the case for similar experiments with additional viscous drag from sidewalls. Thus, all instances of the front accelerating are ascribed to a known artefact of our experimental setup. Although we consider the procedure perfectly reasonable, it does lead to an error of at least 1 second on the time origin used for the whole experiment, and sometimes as much as 10. Also, the shelf was not usually thickest at the weir itself, but a few cm beyond it. Upstream of here, bending moments were likely having a significant impact upon the flow. As such forces are outside our model, they lead to the model only becoming valid for regions downstream of the point of maximum thickness. This leads to the conclusion that all position measurements should be relative to the point where bending moments become insignificant. Of course, the location of this point has an error of about 1cm (though for very thick shelves, it may be much more). For consistency, though, we would also need to subtract the time required to fill up the section of the profile behind this point. For simplicity, we did neither, believing the effects to be roughly comparable and fairly small in any case. Experiments where this was not so are excluded from our analysis (though we show the data anyway). Because our procedure essentially assumed no flux entering the channel at all until such time as all $Q$ was entering the channel, we underestimate the amount of fluid in the channel. This means the experiment effectively got underway earlier than we are assuming, causing us to overestimate the intercepts and underestimate the gradient on the log-log graphs we used. One possible solution is to accurately determine the total amount of fluid which crossed the weir. This could be done if we knew that the sea level had been maintained very accurately and measuring how much water had to be extracted to achieve this. As soon as the experiment finished, the peristaltic pump could be reversed, to prevent further flow of xanthan into the ocean. Then, the position of the laser spot for sea level control could be compared with its initial position to see how much change there had been. We estimate that the total volume of fluid pumped into the ocean could be determined to within 40${cm}^{3}$, corresponding to an error of only a few seconds on the effective start time. However, not realising the importance of it, we did not perform this procedure. These are also excluded from our analysis, but again the data is shown. Another source of uncertainty was the wavelike oscillations that are evident in the bottom view above. These are due to hydrostatic rebound of xanthan dropping into the ocean from a finite height. We tried various techniques to reduce the effect of such waves, and were successful in nearly eliminating them. Therefore, only two experiments were significantly affected by this phenomenon. There are a few more sources of error worth mentioning. Firstly, the concentration of xanthan may have been slightly below 0.5%, because of losses in transferring the powder from the container in which it was weighed into the water. We assume that about 5% of the xanthan may have been lost in this way, meaning the concentration may have been systematically lower (only 0.47%). This will reduce $\eta_{o}$ by about 10-15%. Also, the shear rates in our experiments may have been sufficiently low that the power law used to model the viscosity breaks down. Ultimately, the viscosity is not infinite at very low shear rates, so the fluid must be less viscous than we are assuming. Fluxes were measured by weighing the container from which xanthan was pumped into the tank. Although a slightly different amount may have been overflowing the weir and entering the shelf (especially near the start of our experiments), this is only true for a very short time. The measurements of mass flow rates were very accurate (0.1% or so). The density of the ocean was measured using a hydrometer, attaining a similar level of accuracy. For xanthan, we put it into saltwater of known density and checked if it floated or sank. When 50% of the samples we put into the water sank, we knew we had the right density. We also checked this using a hydrometer - both gave consistent results, with an accuracy of 0.1% or so on the density. This corresponds to an error of under 2% on $g^{\prime}$. Our density measurements are listed below. We simply extrapolated the density of xanthan at 1% concentration to be 996. Substance \| Density (kg/$m^{3}$) ---\|--- Water \| 994 Xanthan at 0.5% \| 995 The position of the front as a function of time was determined by a MATLAB boundary tracing algorithm. The positions are listed relative to the weir if there was no sloped bed, relative to the point of maximum thickness if this was more than 3cm from the weir and relative to the location of the grounding line if there was a sloped bed. However, experiments in which the position of maximum thickness was more than 3cm from the weir were severely affected by bending moments, making the results of these experiments unusable. Then we saw if the slope of a graph of ${x}_{n}$ against $t$ (on logarithmic axes) converged. This is done by requiring the residuals to a linear regression (usually below 0.5%, and sometimes just a tenth of this) to not have a characteristic inverted parabola shape, but to appear essentially random. We list the portion of the tank over which this occurred, and the relevant times. Also listed is the product-moment correlation coefficient, to give an idea of how closely the data fit to a straight line. If an experiment did not converge, then the gradient would still be decreasing by the end of the experiment (because the gradient needs to go down from 1 to $\sim$ 0.6). In this case, we did a regression on the last 30 seconds or so of data, to give a bound on what the gradient might eventually converge to, as well as where the intercept could then lie. This equates to an upper bound on the final gradient and a lower bound on the intercept. Usually, we also excluded data taken in the last 5cm or so of the tank, to allow for the effect of the sea level control mechanism on the shelf. If there was no discernible effect, we used the additional data in our regression. Ocean currents can affect the xanthan because water has a finite viscosity. The effect is almost always to cause a sudden increase in the gradient (on logarithmic axes). However, for very high fluxes, we believe that the change in water pressure favours thickening of the shelf and thus slows it down even further. If the reader is interested, we strongly recommend manually analysing the (few) photographs from the very end of an experiment (at 8g/s and at 17g/s, to see both regimes). Another interesting thing to try is to exclude the possibility that the effect near the end is part of a long-period wavelike oscillation (we damped these, but they might still be present). This is relatively straightforward - the experiment simply needs to be repeated with the weir moved forwards 10cm or so. That way, the end of the tank would correspond to a different phase of the (hypothetical) wave. We also note that a much more viscous ocean (e.g. using sugar rather than salt to reach the target ocean density) would enhance the effect. However, it was not our intention to understand the influence of ocean currents on ice shelves, so we do not discuss this further. Our experiments are given between one and three letters and a number to help the reader identify and refer to them. The letters indicate respectively the presence of laterally confining sidewalls, the presence of a sloped bed and the concentration of xanthan used for the experiment. Letter \| Meaning ---\|--- W \| Sidewalls B \| Bed H \| 1% concentration used L \| 0.5% concentration used A typical experiment will be identified by e.g. L1 (indicating no sidewalls, no sloped bed and a concentration of 0.5%). The number is self-explanatory. If these do not start at 1 or miss a number, this is because an experiment was excluded from this paper. In this case, a good reason will be given. Finally, we note that only an error in the concentration of xanthan used (and thus in $\eta_{o}$) still remains as a systematic effect in the experiments mimicking ice tongues. Other errors for these experiments are purely random, the biggest of which is in measuring the width of the shelf. The thin parts near the edges had to be excluded from our measurement of $d$, for reasons that will become apparent. Such a procedure inevitably creates some error and is partly subjective. ### 5.2 1% aqueous xanthan solution Experiments with sidewalls were all performed in the same tank with $d=0.075m$ (as it was manufactured, the error is negligible). We used ${{\rho}_{w}}=1100kg/{m}^{3}$ for all WH experiments. Experiments WH1-3 are not included because we were still perfecting our techniques and because the weir had some rust. This severely hampered our experiments because xanthan overflowing the weir tended to stick to the rust rather than flow forwards into the ocean. When the xanthan finally left the weir, it had gone a long way down so there was a huge blob at the front of the shelf. We warn readers attempting to repeat our experiments that they are of an extremely sensitive nature (by most standards), especially those without sidewalls. We do not include two experiments conducted at an extremely low flux. This is because the shelf was so thin that it lost contact with the sidewalls in a very large number of locations. There was also insufficient sidewall contact to make one experiment converge, although it suggested that the power of $t$ is $<0.56$. It also suggested a higher intercept than other experiments, although this is almost certainly due to the loss of contact with sidewalls (which reduces the drag on the shelf). Expt. \| Flux \| Convergent \| Error \| ${R}^{2}$ \| Time (s) \| Distance \| Intercept \| Error ---\|---\|---\|---\|---\|---\|---\|---\|--- (WH..) \| (g/s) \| power of t \| \| \| \| (cm) \| \| 10 \| 6.23 \| 0.554 \| 0.01 \| 0.9997 \| 216-239 \| 66-70 \| $-3.38$ \| 0.06 9 \| 12.41 \| 0.540 \| 0.01 \| 0.9997 \| 101-146 \| 56-68 \| $-3.08$ \| 0.06 7 \| 15.18 \| $<0.61$ \| \| \| \| \| $>-3.30$ \| 6 \| 7.87 \| 0.541 \| 0.01 \| 0.9997 \| 163-189 \| 60-66 \| $-3.21$ \| 0.06 4 \| 3.87 \| $<0.61$ \| \| \| \| \| $>-3.78$ \| Table 2: The results obtained for our experiments with $1\%$ aqueous xanthan solution. The experiments which did converge were all consistent with each other. Our best estimate for the mean value of the convergent power of $t$ is $0.545\pm 0.006$ If ${x}_{n}\propto{t}^{\frac{n+1}{2n+1}}$, as predicted by our theory, then we require a value for $n$ of ${5.1}_{-0.7}^{+0.8}$ This is entirely consistent with independent measurements of $n$ for this fluid (which suggest $n\approx 5$). Next, we check if the intercepts are also consistent with our theory. However, we should not expect them to be. This is because it was obvious that there are significant thickness variations across the channel. As already shown, only a X% difference between mean and edge thicknesses can lead to a $\sim Y\%$ discrepancy between the predicted and measured values of $u$ (and thus of ${{x}_{n}}$). Considering how easy it was for the shelf to lose contact with the sidewalls altogether, we suppose that there must have been a significant variation in thickness across the channel. Presumably, the wider the channel or the more viscous the fluid, the more difficult it is to transport mass towards the sidewalls. This is essential to making these regions thicken with time (along with the rest of the shelf). We thus predict that a narrower channel should give better agreement with our theory, as should a less viscous fluid. We now proceed to rescale the intercepts (on a log-log graph) based on changes in $Q$, remembering that $x_{n}\propto{Q}^{\frac{n+1}{2n+1}}$. Once the rescaling is done, the intercepts should (theoretically) all be equal. Expt. \| Flux \| Rescaled \| Error ---\|---\|---\|--- (WH) \| (g/s) \| intercept \| 9 \| 12.41 \| $-4.22$ \| 0.06 10 \| 6.23 \| $-4.21$ \| 0.06 7 \| 15.18 \| $>-4.5$ \| 6 \| 7.87 \| $-4.14$ \| 0.06 4 \| 3.87 \| $>-4.4$ \| Table 3: The intercepts obtained for the experiments with $1\%$ xanthan solution, rescaled according to our theory and the alterations in flux. The theoretical value is -4.99, assuming $n=5$ and ${{\eta}_{o}}=10$. As can be seen from Table 3, the values are roughly consistent, although the theoretical value is about -5. Our best estimate for the rescaled value of the intercept is: $-4.19\pm 0.04$ The discrepancy with our theory could be due to lateral thickness variations and partial loss of sidewall contact throughout the shelf. The scaling of $x_{n}$ with $Q$ appears to be as expected, but the changing relative importance of lateral thickness variations leads to this not being completely correct either. The net effect is that experiments at a lower flux go slightly faster than we would predict from scaling data for a higher flux experiment. Presumably, this is due to lower $H$ \- note only $Q$ varied between experiments, so $H$ would have been correlated with $Q$. Another interesting thing to note is that the impact of lateral thickness variations was very similar for all experiments. This suggests that the fractional variation in thickness across the channel was much the same, so the lateral thickness profile might be self-similar inside and between experiments. Otherwise, the data would not remain parallel to our theoretical solution. Future work may elucidate this further. The intercepts are given when the data (in SI units) is plotted on a log-log graph (with base 10). The rescaled intercepts are what would be obtained if the scaling predicted by our theory is correct and the flux was reduced to 1g/s (so $Q=1.004~{}{cm}^{3}$/s). Figure 6: The rescaled raw data for all reliable WH experiments (the most reliable ones are in blue). The experiments at very low fluxes had too much lateral thickness variation to be considered reliable, and they showed the strongest disagreement with theory. All experiments are well above the theoretical line, shown in red. Surprisingly, though, they differ from the theoretical curve by a very similar amount, suggesting the scaling relations with $Q$ and $t$ still work. ### 5.3 0.5% aqueous xanthan solution We attempted to minimise the effects of lateral thickness variations upon the shelf. This can be done by reducing the width of the shelf, but we attempted instead to reduce the viscosity of the fluid by reducing the concentration of xanthan to 0.5%. This reduces the viscosity by a factor of about 3. It also reduces $n$, which is highly desirable as ice has $n\approx 3$. However, we did not quite reduce the concentration enough to reach this, because if we did then the fluid would not be very viscous and the flow might fail to be at low Reynolds number. Experiment \| ${\rho}_{w}$ \| Flux \| Convergent \| Error \| ${R}^{2}$ \| Time (s) \| Distance \| Intercept \| Error ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- (WL..) \| \| \| power of t \| \| \| \| (cm) \| \| 1 \| 1100 \| 6.62 \| $<0.61$ \| \| \| \| \| $>-3.67$ \| 2 \| 1100 \| 12.05 \| 0.56 \| 0.03 \| 0.9994 \| 47-163 \| 37-75 \| $-3.14$ \| 0.1 3 \| 1100 \| 16.73 \| 0.53 \| 0.03 \| 0.9997 \| 66-123 \| 51-71 \| $-2.9$ \| 0.2 5 \| 1100 \| 3.96 \| 0.58 \| 0.02 \| 0.9985 \| 309-350 \| 64-69 \| $-3.77$ \| 0.06 B6 \| 1053 \| 11.79 \| 0.57 \| 0.02 \| 0.9986 \| 105-233 \| 42-66 \| $-3.52$ \| 0.06 7 \| 1053 \| 12.29 \| 0.61 \| 0.03 \| 0.9993 \| 68-213 \| 41-82 \| $-3.48$ \| 0.06 B10 \| 1029 \| 3.9 \| 0.56 \| 0.015 \| 0.9842 \| 420-684 \| 44-57 \| $-4.24$ \| 0.06 B11 \| 1100 \| 3.79 \| $<0.61$ \| \| \| \| \| $>-4.24$ \| B12 \| 1100 \| 15.5 \| $<0.80$ \| \| \| \| \| $>-4.26$ \| B13 \| 1029 \| 8.12 \| 0.56 \| 0.015 \| 0.9946 \| 175-387 \| 39-58 \| $-3.88$ \| 0.06 B14 \| 1053 \| 15.9 \| $<0.67$ \| \| \| \| \| $>-3.66$ \| Table 4: The results obtained for our experiments with $0.5\%$ aqueous xanthan solution. We have not included two experiments which had extremely thick shelves, due to bending moments near the weir playing a role over a large section of the tank. Most likely, the correct thing to do is to subtract $\sim$10 cm from the front positions, to account for this region (as our theory only becomes valid beyond it). This is also suggested by the fact that, unlike all other experiments, the gradient on a log-log graph (of position against time) was actually increasing (rather than decreasing from 1 at early times towards $\sim$0.5), strongly suggesting a zero error. However, because we could not precisely determine what correction to use, we do not include such experiments. Also not included is the first experiment we conducted that had a sloped bed. This was partially exposed to air, which led to an unusual start to the experiment and loss of contact of the shelf with the sidewalls over a 5cm region near the front. Subsequent experiments had a much shallower ($\sim 10^{\circ}$ instead of $26^{\circ}$) sloped bed installed, as well as this being entirely submerged. Contact with the sidewalls was much improved as a result. Unsurprisingly, the additional length of tank used up by the sheet meant that convergence to self-similar propagation was harder to obtain (although it greatly improved the quality of the experiment). However, we still managed it on three occasions. As the sloped bed did not appear to have a significant effect on the shelf, we treat _all_ WL experiments in the same way. Errors were raised slightly by the lower viscosity of the fluid, which made it more prone to oscillations due to hydrostatic rebound. However, experiments with a sloped bed or with a flux below 7g/s were only slightly affected. This meant that only experiments WL2 and WL3 are noticeably affected. In what follows, we do not use either, because we could not average over enough oscillations. Our best estimate for the asymptotic behaviour of the shelf is that the front propagates as a power law in $t$ with exponent $0.565\pm 0.008$ This will require a value for $n$ of ${3.3}_{-0.4}^{+0.6}$ Getting independent values for $n$ proved difficult. In the end, we found values at lower concentrations of xanthan than we used, and extrapolated them to 0.5%. The value of $n$ at 0.4% was 3.33, and at 1% it is close to 5. Also, at 0.2% it is 2.83. Thus, we expect that at 0.5% $n$ should be about 3.8, making it consistent with our observations. The reference we used was: http://projekt.sik.se/nrs/conference/Old%20conferences/conf2003/Course2003/course_Taylor%20.pdf The value for ${{\eta}_{o}}$ we found in a similar way. At 0.2%, it is 0.57 Pa${s}^{\frac{1}{n}}$. All values of ${{\eta}_{o}}$ are given in these units. At 0.4%, it is 2.24 and at 1% it is about 10. Thus, we expect a value of about 3.5 at 0.5%. This allows us to check whether the intercepts are also consistent with our theory. The values of ${{\eta}_{o}}$ and $n$ could be verified by e.g. pumping the fluid into a dry Hele-Shaw cell, forming a lateral-shear dominated viscous gravity current. This is already a well-understood situation, so advantage could be taken of this fact. Other more advanced techniques are also possible. Experiment \| Ocean \| Flux \| Rescaled \| Error ---\|---\|---\|---\|--- (WL..) \| density \| (g/s) \| intercept \| 1 \| 1100 \| 6.62 \| $>-4.45$ \| 2 \| 1100 \| 12.05 \| $-4.18$ \| 0.1 3 \| 1100 \| 16.73 \| $-4.07$ \| 0.2 5 \| 1100 \| 3.96 \| $-4.32$ \| 0.06 B6 \| 1053 \| 11.79 \| $-4.31$ \| 0.06 7 \| 1053 \| 12.29 \| $-4.29$ \| 0.06 B10 \| 1029 \| 3.9 \| $-4.33$ \| 0.06 B11 \| 1100 \| 3.79 \| $>-4.8$ \| B12 \| 1100 \| 15.5 \| $>-5.4$ \| B13 \| 1029 \| 8.12 \| $-4.29$ \| 0.06 B14 \| 1053 \| 15.9 \| $>-4.6$ \| Table 5: Intercepts on a log-log graph obtained with $0.5\%$ xanthan experiments. These are rescaled so as to be what one would get for an experiment at 1$g/s$ and with $g^{\prime}=1m/{s}^{2}$, using SI units and base $e$. The density of xanthan is 995 $kg/{m}^{3}$. The theoretical value is about $-$4.42, with an error close to 0.05 (due to the uncertainties in extrapolating data). The mean value for the intercept that we obtain is: $-4.31\pm 0.03$ This is slightly greater than the theoretical value of -4.45. We have already mentioned one possible cause of the discrepancy - errors in the start time. However, although it is a systematic effect, its magnitude will likely be below the error budget quoted above. An error in the concentration of xanthan could also be to blame (a 15% reduction in $\eta_{o}$ leads to a 7% increase in ${x}_{n}$), as could the low shear rates in the experiments. Lateral thickness variations could also account for a further 2% discrepancy. Temperature could plausibly be a factor as well - the lighting we used was very inefficient and could definitely have heated a dark fluid. Combining all of these effects, the (relatively small) discrepancy between theory and observations could conceivably be explained. To test these ideas, the viscosity of the fluid we used, prepared in the same way as for the above experiments; should be measured directly. One possibility is to use a very narrow tank (a Hele-Shaw cell) and have no ocean, using the viscous gravity current theory to determine the viscosity parameters. Also possible is to use the technique outlined to get a better estimate for the start time. Repeating the experiments in a much narrower tank would make the start time clearer, because lateral variations in thickness would be sufficiently small that the area enclosed by the shelf in a photograph would be a good indicator of its volume. Figure 7: Blue dashes are from experiments without a sloped bed, while green dots are from ones which had a sloped bed installed. All data is rescaled according to how we expect changes in $Q$ and $g^{\prime}$ to affect the shelf. The thick red line is for theory, allowing for some errors. The black curves are from unreliable experiments $-$ we do not use these data when averaging. Note the extended period in all experiments where the gradient is 1, signifying a constant front speed (before sidewalls eventually make it decelerate). This strongly suggests that, without sidewalls, the front would advance at a constant rate. We observed that all experiments without a sloped bed converged over a length scale that is about $10L$. This suggests that the length of the flat portion of the shelf is only a tenth of the whole shelf, at the time when further convergence towards our similarity solution is no longer discernible in our data. It is possible to estimate $L$ from a photograph - the gradient transitions from 0 to that for the similarity solution over a length scale which is roughly the same as our theory predicts (see Figure 8 and note that the shelf is 90cm long). As discussed in Section 4.2, convergence should (and does) take longer in experiments with a sloped bed installed (when considering the length of the shelf only). ### 5.4 Thickness Profile Figure 8: Side view of a sidewall contact experiment (WL1). The dashed red line is drawn to fit the initial gradient of the lower surface of the xanthan. Notice how the shelf is slightly thinner than the red line predicts (i.e. $\left\|H^{\prime}\right\|$ rises slightly, as expected). Near the front, the shelf becomes flat. This is because, when this region was near the weir, the theory for ice tongues applied (as there was very little sidewall contact). The region should therefore be nearly flat (or parallel to the black line). Notice that this region is a few cm long (for scale, the shelf is 90cm long) - our equation for $L$ gives about 3cm. At later times, this region then got pushed along by the self-similar (triangular) region of the profile. Things being ‘pushed along’ in this manner is a characteristic feature of our model, because $\frac{\partial u}{\partial x}$ is very small. Figure 9: The region near the front (left) in the previous figure is enlarged here. Note that the black line is horizontal, while the red line has the same slope as in the previous figure. As expected, the real profile is not perfectly triangular, with the region shown here in fact being flat. However, this region eventually becomes an insignificant part of the entire shelf, as it doesn’t grow. ### 5.5 Particle Imaging Velocimetry Figure 10: Left: The shelf as it appears in a normal camera, viewed from underneath. Note that the central region is a little thicker, and also has more seeds than other regions. Also note that there was no sheet. Right: Arrows drawn in by DigiFlow as a result of comparing particle positions between frames. The background indicates the thickness of the shelf (it is in false colour). We attempted to determine directly whether the velocity profile across the channel in sidewall contact experiments was in agreement with our theoretical model. In order to do this, we put in a large number of poppy seeds into the xanthan (about 0.5% by mass). These were almost neutrally buoyant, so vertical motion was negligible during the course of our experiments. We used a high- sensitivity black and white camera with a resolution of $1024\times 1024$ pixels to capture photographs at 15 frames per second. Later, we used DigiFlow software to analyse these. PIV is a relatively new and difficult technique, so we got a relatively large amount of scatter. We therefore averaged over a 20 second period near the end of an experiment which lasted about 200 seconds. The thickness gradient in the shelf goes down as ${{t}^{-\frac{1}{2n+1}}}$, so we judged that it would hardly change over a 20 second period (and so velocities should hardly change). We also averaged over a 10cm region of the shelf just beyond the point of maximum thickness. Over this region, the change in $H^{\prime}$ was minimal (the shelf was about 8 times longer than this, and the shelf is in any case almost perfectly triangular) so $u$ should hardly vary within it. The theoretical curve drawn in Figure 11 is based on a maximum speed consistent with the PIV data. Because the fluid is clearly satisfying the no- slip boundary condition, the walls of the tank are clearly visible, 763 pixels apart (this corresponds to 15cm). For the maximum speed, we assume an error of at most 0.5 pixels/second, with a mean value of 12.5. Thus, the maximum velocity is $0.246\pm 0.010\text{ cm/s}$ The speed of the front at a time concurrent with these observations is found to be $0.27\pm 0.01\text{ cm/s}$ Errors are because the front decelerates during the 20s we averaged over. The theoretical change in $H^{\prime}$ between the front and rear of the profile is 2.8%, corresponding to an increase in $u$ of 11.1% between the rear and the front. This corresponds to a difference in $u$ of 0.027 cm/s, entirely consistent with our observations. Figure 11: The results of one of our attempts to determine the velocity profile of the shelf. The smooth blue line is for theory. Observations must lie between the jagged green curves (at $1\sigma$), with raw data along the central red curve. We conclude that both lateral and longitudinal variations in $u$ are accurately predicted by our theory. We also determined vertical velocity profiles in the sheet, to check whether a viscous gravity current model was accurate. We did this in an experiment with sidewall contact, to increase the thickness of the sheet and get more accurate measurements. Otherwise, only a very small number of particles fit in vertically, despite us using particles sufficiently small that a camera right next to the sheet could only just resolve them. We expected the viscous gravity current theory to be accurate only for locations sufficiently far downstream of the weir. Therefore, we were expecting to see a discrepancy between theory and observations sufficiently close to the weir. Figure 12: Velocity profiles obtained in the sheet are shown in red, near the weir (top) and further away (bottom). In blue, we try to fit the expected velocity profile for a viscous gravity current, by matching velocities at the top of the sheet. Note that the data are from an experiment with sidewall contact (this should not be important, as the sheet should be vertical shear dominated). We conclude that the viscous gravity current theory is an accurate description of the situation sufficiently far from the weir, but breaks down too close to the weir. This gives us confidence that we have a reasonable understanding of the sheet, suggesting the grounding line may also have been understood. ### 5.6 No Sidewalls We also conducted a series of experiments in which the shelf was not in contact with the sidewalls of the tank, at least for a while. The apparatus still looked the same, except for the weir. This now had a groove cut in the central 5cm, reduced to 3cm for the last 2 runs. The groove was just above sea level. For some of the experiments, we installed the sloped bed in the same way as before (i.e. totally submerged and at $\sim 10^{\circ}$). The front appeared to go at a constant velocity, except at very early times (when all the flux was not yet entering the shelf). This is consistent with our theory, as the experiments ran for much less time than that required for convergence to a self-similar mode of propagation. Consequently, the front propagated at constant speed. Combined with what appeared to be a constant width to the shelf, we suppose that the grounding line had reached a dynamic equilibrium. Due to severe technical difficulties, 6 experiments are not shown because the shelf rapidly hit a wall of our tank. Readers attempting to repeat our experiments should note that the inlet and seawater extraction pipes should be (extremely close to) vertical and in the middle of the tank and the whole tank (and any sloped bed inside it) should be level to the horizontal to within about 3 arc-minutes. The sloped bed, which needs to rest against the weir, also needs to have been manufactured to the correct working angle (within a few degrees). In what follows, we assume that the constant velocity of the shelf can be used to determine the thickness at the grounding line, given also the width of the shelf. One minor complication in measuring the width is that we used the bottom view of the shelf (obviously), whereas the length of the shelf came from the side view. This is because a thin shelf is hard to see from underneath, but it still provides 15cm of optical depth when viewed side on. However, the optical paths are different in the two cases, due to an extra reflection. This means the same physical distance appears as a greater number of pixels in the camera focal plane for the side view. Errors result from lateral variations in thickness, which make $d$ hard to determine. These variations were enhanced by the tendency of the fluid to spread sideways without any lateral confinement. We used the colour of the shelf in the bottom view to determine which regions were thin. These regions have been excluded from our measurements of $d$. Also, the grounding line is not at constant thickness laterally (i.e. it isn’t a ‘line’). This led to a systematic difference between theory and measurements. The reason is that the thick central regions of the sheet at the grounding line, which our theory addresses (because these regions make up most of the sheet), were thicker than the shelf far downstream. Apparently, after the grounding line, the shelf in these regions thins with distance until it becomes roughly the same thickness as the _thinnest_ parts of the sheet at the grounding line. Thus, the velocity measurements were essentially indicating how thick the thinnest regions of the grounding line were. This causes the theoretical grounding line thickness (for the thick central regions) to exceed the measured thickness (for the thin regions near the edge). This is an interesting phenomenon, and again a case of lateral structure in the flow outside direct consideration in our model having an influence on the shelf. As before, the effect is more pronounced with a more viscous fluid (more concentrated xanthan). Figure 13: This is a contrast-enhanced photograph of an experiment. The thicker regions appear darker in the bottom view. Note that downstream of the thickest region of the grounding line (in the middle), the shelf actually thins until it is approximately the same thickness as the thinnest regions of the grounding line. The thinning is evident from the side as well (circled region). Our experiments indicated that there was a large amount of lateral spreading in the sheet, mostly very close to the weir (even though the sloped bed was entirely underwater, which we believe reduces the spreading). However, it appeared that there was no noticeable lateral spreading in the shelf. In fact, the spreading appeared to have occurred well upstream of the grounding line in all of our experiments. Combined with the constant speed of the front of the shelf, this strongly suggests a constant thickness. We attempted to determine directly the thickness of the shelf at the grounding line. The resolution on this was relatively poor, because the thickness is only a few mm in most cases. Thus, it appeared in our photographs as about 20 pixels. Reflections from the ocean and a small amount of parallax made it extremely difficult to perform this sort of measurement (as a look at the photographs will show). The much lower accuracy prevented us from getting a reliable indication of whether our theory is correct using such measurements. However, they did indicate that measurements of ${H}_{G}$ made in this way are consistent with the results of measuring the downstream shelf velocity and width, which we use for testing our theory instead. Experiments were also conducted without a sloped bed. These indicated negligible lateral spreading, suggesting that ice tongues fed by sheets on steep slopes are unlikely to be much wider than the sheet. Figure 14: Results obtained from experiments without lateral friction. We have only a very basic understanding of experiments without a sloped bed (the first two listed above). Lateral thickness variations make our model work poorly for BH3. Note the minor impact of the change in $g^{\prime}$ between the last two experiments. We also attempted to understand experiments without a sloped bed. We believe, based on high-resolution photographs, that the angle of the upper surface of the xanthan behind the weir is always close to $30^{\circ}$. Combined with the no penetration condition, we can obtain a velocity profile analogous to those obtained previously for the sheet. The thickness required to drive flux $Q$ through the weir is then assumed to equal the downstream shelf thickness. However, it is almost completely certain that other phenomena are critical to understanding the weir. In particular, surface tension has implicitly been included in our model (in terms of fixing the angle of the free surface) but not explicitly in the force balance. This means that we can not expect the predictions based on this theory to be very accurate. On the other hand, it does give the right order of magnitude. Among other things, future work will need to predict an angle of the upper surface close to observed values. For experiments with a sloped bed, we find good agreement with 0.5% xanthan. At 1%, although there is only one experiment, it is again highly probable that lateral variations in thickness are more pronounced. As these are outside our model, they will lead to a reduction in the accuracy of our predictions. The prediction that $g^{\prime}$ does not significantly affect the grounding line thickness appears to be borne out by a comparison of experiments BL4 and BL5, the last of which nearly doubled $g^{\prime}$ relative to the others. However, both theory and experiments show a slight reduction in grounding line thickness as a result of making the xanthan nearly twice as buoyant. Another thing we note is that the shelf buckled in one of our experiments. This led to the front alternately hitting one wall and then the other. The reason for this is unclear, but it appears to be due to internal elasticity of the xanthan. We expect that this is unlikely to occur in ice. The buckling had a small effect on the speed of the front, but much smaller than the error in $d$, so we do not discuss it further. We now show that the effect of non-hydrostatic forces from the weir was dissipated against basal friction before the location of the grounding line. Xanthan is assumed to overflow the weir by an amount H. We assume that forces here are only hydrostatic, but that this gets converted to a non-hydrostatic force on the sheet due to the highly artificial geometry in the situation. Genuine non-hydrostatic forces at the weir may be calculated on the basis of $h^{\prime}$ always being approximately $30^{\circ}$. Such forces appear to be negligible in all of our experiments, compared with hydrostatic forces. The vertically integrated hydrostatic pressure is $\frac{1}{2}\rho g{{H}^{2}}$. The basal friction per unit length is ${{\eta}_{o}}{{\left(\frac{1}{2}\frac{\partial u}{\partial z}\right)}^{\frac{1}{n}-1}}\frac{\partial u}{\partial z}$. We assume that $\displaystyle\frac{\partial u}{\partial z}$ $\displaystyle\approx$ $\displaystyle\text{ }\frac{2\overline{u}}{H}\text{ }\text{ (vertical average indicated)}$ (61) $\displaystyle\approx$ $\displaystyle\frac{2Q}{{{H}^{2}}d}$ (62) This is an underestimate because the upper surface of the sheet is free so the base must have more than average shear. However, this will not affect the final result much if $n$ is large. We obtain that the non-hydrostatic force exerted by the supply mechanism will be dissipated over a length scale L, where $\displaystyle L=\frac{\rho g{{H}^{2+\frac{2}{n}}}}{{{4\eta}_{o}}}{{\left(\frac{d}{Q}\right)}^{\tfrac{1}{n}}}$ (63) For $H$ around 5mm at the weir (as suggested by photographs) in a 0.5% experiment, we get that $L<2$cm. This corresponds to a grounding line that needs to be at least 4.5mm thick (including an allowance for the sea level being 1mm higher than the top of the slope). We obtain similar conclusions for experiments at 1%, allowing the thickness at the weir to rise to as much as 9mm to account for the greater viscosity. However, the greater viscosity of the fluid also makes it easier for basal friction to dissipate the force exerted at the weir. Thus, noting that $L$ has been overestimated, we conclude that our grounding lines can not have been significantly affected by the force exerted at the weir. Therefore, a simple viscous gravity current model for the sheet should suffice for determining $H_{G}$. We also believe it unlikely that realistic natural situations will allow for such unusual configurations as we had in our experiments, especially anything resembling our weir. Therefore, ice sheets can likely be understood solely in terms of hydrostatic pressure gradients balancing basal friction. However, this should be confirmed based on real viscosity parameters and topography. A final note concerns the unusual corrugated pattern of the edge of the flow in our experiments. This is known to occur in nature. Our understanding is as follows: variations in flux, due to the oscillatory action of our peristaltic pump, lead to variations in the thickness of the shelf at the grounding line, presumably because a higher flux causes the sheet to spread laterally by an increased amount. Because the rest of the shelf essentially moves as a rigid body, the pattern remains permanently imprinted upon the shelf. This may also explain why the front of the shelf tapers - because it crossed the weir when the flux overflowing it was still rising towards $Q$. In natural situations, the effect can be due to seasonal or other changes in the flux entering the ice shelf. We note that the effect did not arise in those experiments in which we did not install a sloped bed. Thus, the effect is likely reduced if the terrain is steeper close to the grounding line. This affects the area in contact with the ocean, which may have important consequences. Figure 15: The Erebus ice tongue, showing a similar edge to our laboratory model for such systems. We believe that we have mimicked a seasonal variation in entry flux with the oscillatory action of our peristaltic pump, leading to the similar appearance. Note that our model shelf is much wider than the groove in the weir. This suggests that the shelf determines its own width, this being affected by $Q$ and likely also by $g^{\prime}$. Figure 16: In this run, there was no sloped bed. The corrugations are now absent, although the same pump was used for all of our experiments. The shelf is only as wide as the groove in the weir. ## 6 Towards The Natural System ### 6.1 Including Flux Non-Conservation Real ice shelves do not grow forever. This is because ice is lost from them. There are two major ways in which this occurs: iceberg formation and basal melting. We will neglect sublimation at the top compared to these processes. Melting on the underside of an ice shelf is an important process. We assume that it proceeds at a rate proportional to the surface area in contact with the ocean. Stability of this surface is assumed. As the ice shelf is nearly flat, we assume the area remains the same if projected vertically ($\cos(H^{\prime})\approx 1$). Calving of icebergs at the front of an ice shelf is still poorly understood. We treat it in a similar way to melting on its underside. Thus, the rate at which volume is lost is also proportional to the surface area of the front of the ice shelf. The exact mechanism by which this occurs will turn out to be important, in particular whether it is likely to rise by a similar amount to the rate of melting on the underside. For the moment, we assume that ocean water in contact with the front of an ice shelf slowly melts it. This may happen most efficiently near the surface. As the ice is melted away, a large overhang will be left. This will eventually collapse. If the melting primarily occurs in a narrow layer near sea level, then there will also be an ‘underhang’. Due to buoyancy forces, this will eventually break off as well. The most widely believed alternative to this mechanism is the action of waves at the front of an ice shelf. However, it seems likely that this mechanism will also lead to loss of volume being directly proportional to surface area. Our basic model therefore has two parameters. Using $F$ to denote the volume rate of loss of ice from the shelf (and $A$ for area), we define $\displaystyle c$ $\displaystyle\equiv$ $\displaystyle\frac{\partial F}{\partial A}\text{ (basal melting)}$ (64) $\displaystyle f$ $\displaystyle\equiv$ $\displaystyle\frac{\partial F}{\partial A}\text{ (calving of icebergs)}$ (65) ### 6.2 Ice Tongues Ice tongues are nearly flat due to the absence of substantial drag on any part of the shelf. Although there is theoretically a small amount of thinning between the grounding line and the front of the shelf (see Equation 20), we will assume here that ice tongues are perfectly flat owing to the very large value of $L$. The presence of basal melting will not substantially affect this, although it will require $u$ to decrease with $x$. This will lead to some longitudinal stress, but we assume that ice is sufficiently viscous that this does not much affect the force balance. It is also evident that a region of the shelf which is thinner than regions upstream of it will eventually thicken due to the increased flux entering this region. Thus, the ice shelf should be stable. The mass balance for an ice shelf will approximately be given by $\displaystyle Q=d(cx_{n}+H_{G}f)$ (66) We will assume that $d$ will not change much. Most likely, it is set by topography close to the grounding line. Thus, an increase in $c$ or in $f$ will lead to a reduction in $x_{n}$. However, assuming that conditions in the interior of the continent have not changed much, $Q$ will still be the same and so $H_{G}$ will remain unaltered $-$ see Equation 56. This means that a long ice tongue is not in imminent danger of collapse: it first needs to shorten. If such a collapse were to occur anyway, the force balance at the grounding line would be unaffected and so there would be no reason for the flux entering the ocean to change. Most likely, the ice tongue would re-establish itself. Under some circumstances, $f$ may become sufficiently high that the ice tongue ultimately has its length reduced to 0. In this case, a small further rise in $f$ will cause the grounding line to retreat. The force balance will now be affected. The resulting reduction in the integrated hydrostatic pressure of seawater will require a reduction in the pushing force (Equation 54). This means the flux entering the ocean must be reduced. However, with a continual supply of ice, it is not possible for the flux entering the ocean to actually be reduced. The ice sheet will attempt to steepen to keep the flux entering the ocean equal to that carried away by icebergs. This will require the upper surface to steepen, achieved by loss of ice near the grounding line. Such a loss is in any case required to maintain the flotation condition, which we believe will still hold at the grounding line. According to Equation 56, however, it is not possible for there to be an equilibrium solution at such a reduced grounding line thickness. The basic reason is that, for forces to balance, the upper surface of the sheet must be roughly parallel to the sloped bed. The extra surface gradient now present will not be sustainable in our model. We suppose that ice is also lost further upstream than the grounding line, thereby maintaining the gradient of the upper surface. This reduces the flux entering the ocean (see Equation 53). It also reduces the amount of ice lost to iceberg formation. However, it is clear that the latter effect is much smaller than the former. Consequently, the grounding line will be forced to retreat even further. Without doing detailed calculations (which may well give a different outcome), we speculate that the equilibrium solution is for the grounding line to be above sea level. Then, it is possible for the ice sheet to have zero thickness at the grounding line (buoyancy forces prevent this occurring for a grounding line below sea level). This will mean the pushing force at the front is zero, but some flux enters the ocean if the front is very steep (as will likely occur). An estimate of the timescale for reaching this equilibrium may be obtained by setting $Q=Hf$. Eventually, of course, the flux entering the ocean and that supplied to the sheet will be equal. Figure 17: The ice sheet is initially in equilibrium, with the flux entirely carried off by icebergs (so no shelf). The sheet runs nearly parallel to the sloped bed. A small further increase in $f$ causes recession of the grounding line. To maintain the flux, the front of the sheet steepens. The forces are no longer balanced at the grounding line, allowing an instability to develop whereby a steeper gradient leads to a higher flux and reduction in grounding line thickness (making the surface even steeper). This removes ice further upstream and so the sheet reverts to the previous surface slope. However, icebergs are carrying far too much ice away so the grounding line recedes further. We suppose that it ends up very close to sea level. ### 6.3 Laterally Confined Ice Shelves A laterally confined ice shelf requires a gradient in its thickness in order to flow. For the moment, we neglect the formation of icebergs and assume that sidewalls dominate the force balance. Setting $x=0$ at the front of the ice shelf and using $q=2dcx$ (assuming zero thickness at the front), we see that $H{H^{\prime}}^{n}\propto x$. Separating the variables, this implies that the shelf will be perfectly triangular. A quick look at any of our experiments suggests this to be perfectly reasonable, although none of them involved loss of fluid in this way. The entry flux fully determines the length of the shelf. $\displaystyle x_{n}$ $\displaystyle=$ $\displaystyle\frac{Q}{2cd}$ (67) $\displaystyle H^{\prime}$ $\displaystyle=$ $\displaystyle\frac{2cdH_{0}}{Q}$ (68) Combining this result with Equation 34, we see that $\displaystyle{{H}_{0}}=Q{{c}^{-\frac{n}{n+1}}}{{\left(\frac{n+2}{4}\right)}^{\frac{1}{n+1}}}{{\left(\frac{{{\eta}_{o}}}{\rho g^{\prime}}\right)}^{\frac{1}{n+1}}}{{d}^{-2}}$ (70) As with ice tongues, an increase in $c$ will force a reduction in $x_{n}$. This time, however, $H_{0}$ will also decrease. This will lead to grounding line retreat. To maintain the same entry flux, $H^{\prime}$ will be forced to increase. The solution we have found is self-consistent as no icebergs will form at the front. We now check whether this solution is stable. Suppose that a small region of shelf near the front broke off. We keep the origin of our co- ordinate system where the front previously was. The condition for stability is that more flux crosses the front than can be carried away by icebergs, thereby leading to a longer shelf and restoration of the lost region. This means that $\displaystyle 2dcx>2dfH^{\prime}x$ (71) If melting of ice is primarily responsible for the formation of icebergs at the front, then it is reasonable to have $c\propto f$. If other mechanisms are responsible, then $f$ will perhaps rise slowly or not at all despite large rises in $c$. The distinction will turn out to be critical. If $f\propto c$, then increases in these parameters will force an increase in $H^{\prime}$ and make Equation 71 harder to satisfy. The shelf will eventually become unstable. Unlike in the case of ice tongues, a large laterally confined ice shelf can suddenly become unstable and rapidly disintegrate. This will cause a drastic alteration in the force balance at the grounding line. The buttressing effect of the sidewalls upon the system is now gone. Consequently, the system will behave more like an ice tongue. This would most likely mean the new equilibrium thickness will be given by Equation 56. We suppose that, before the sheet has quite reached equilibrium, it will attempt to balance the forces across the grounding line. This will be achieved by having the ice run parallel to the bed very close to this point. This allows for a rough estimate of the flux entering the ocean, using Equation 53 and setting $h^{\prime}=\alpha$. One can of course calculate more precisely what flux is required for a particular value of $H$ to be the equilibrium grounding line thickness. As this flux will undoubtedly greatly exceed the flux supplied into the ice sheet, the grounding line will retreat. Our equations can thus be used to get a rough estimate of how long it will take to reach the new equilibrium configuration. The grounding line will most likely feed an ice tongue. We believe it unlikely that the rate of iceberg formation will be sufficiently high to prevent this occurring. Also, it appears unlikely that sidewall contact will be properly re-established. However, if $f$ was high enough, an ice tongue would not form at all. The equilibrium configuration would then not be as just discussed. Most likely, the grounding line would retreat all the way to sea level, with the retreat rate governed by the efficiency of iceberg formation ($Q=Hf$). ## 7 Conclusions Recent breakthroughs in understanding the force balance in a simplified laboratory model of a marine ice sheet have now been significantly extended. The theory we developed is valid for the case of a shear-thinning power law fluid, with arbitrary $n$ (not just for a Newtonian fluid, with $n=1$). Laboratory experiments confirm that our theory is valid to within the (very tight) experimental tolerances we achieved. The experiments themselves revealed additional aspects of the lab model that are still not fully understood, such as buckling and lateral thickness variations. However, we believe it unlikely that buckling could happen in natural ice shelves, although lateral thickness variations may play an important role. Our theory for ice tongues is based on a straightforward generalisation of Robison (2010). The equilibrium grounding line thickness is found in a similar way, but the additional complexity means there is no true analytic solution. We do, however, find an approximate analytic solution. The basis for our approximation scheme is that the upper surface of the sheet should be parallel to the lower surface (the bed). This is not always true, so we derive conditions for it to be a reasonable approximation (for ice in water, the bed should have a slope of about $9^{\circ}$). For shelves that are laterally confined, our theory is based on the assumption that the fluid undergoes generalised Poiseuille flow. This is true for sufficiently long shelves (assuming they do not lose contact with the sidewalls). We obtain constraints on what length is required, with observations indicating good agreement with our predictions and especially the correct dependence on parameters like the entry flux. Experiments confirm that the presence of a grounding line does not affect the asymptotic behaviour of laterally confined shelves. Therefore, our prediction for the shelf thickness at its start is equivalent to a prediction of the grounding line position. In this case, an important feature of our solution is that the grounding line advances for ever, although it decelerates. We combined our understanding of the forces with a basic model for loss of ice from a shelf due to melting on its underside and iceberg formation. This prevents the shelf growing for ever - an equilibrium configuration is attained. However, there was an instability peculiar to laterally confined ice shelves, which can suddenly collapse if oceanic conditions change in a particular way. Ice tongues seem more stable, but if conditions alter substantially and prevent it existing at all, then the sheet becomes unstable. This may lead to retreat of the grounding line to sea level, although this process may take a long time. The buttressing exerted by the shelf upon the sheet, which supposedly causes significant acceleration of the sheet if it is removed; actually comes from sidewall contact. If there is no sidewall contact, then the buttressing is equivalent to what would be provided by hydrostatic pressure of water alone, which means this will still be present with no shelf. Only the additional amount due to the shelf being in contact with sidewalls can be removed by melting the shelf, so in ice tongues we do not expect a sudden acceleration if the shelf were to break up. In this case, a reduction in viscosity (due to global warming) may still cause a significant acceleration of the flow (because it is very sensitive to $\eta_{o}$). However, this is not due to collapse of the ice shelf. Our model requires assumptions about poorly understood processes like iceberg formation. Detailed understanding of these processes will be essential at some point, if we are to fully understand events like the collapse of the Larsen B ice shelf (nearly ten years ago). Hopefully, these events can be understood in time to prepare for their consequences, something this work may help with.	arxiv-papers	2013-10-30T01:50:57	2024-09-04T02:49:53.073536	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "authors": "Indranil Banik and Justas Dauparas", "submitter": "Indranil Banik", "url": "https://arxiv.org/abs/1310.7998" }
1310.8055		arxiv-papers	2013-10-30T08:04:36	2024-09-04T02:49:53.095014	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "M\\\"ubariz Garayev", "submitter": "M\\\"ubariz Karaev", "url": "https://arxiv.org/abs/1310.8055" }
1310.8068	# HALO FORMATION IN NEUTRON RICH $\rm{Ca}$ NUCLEI M. Kaushik D. Singh H. L. Yadav [email protected] ###### Abstract We have investigated the halo formation in the neutron rich $\rm{Ca}$ isotopes within the framework of recently proposed relativistic mean-field plus BCS (RMF+BCS) approach wherein the single particle continuum corresponding to the RMF is replaced by a set of discrete positive energy states for the calculation of pairing energy. For the neutron rich $\rm{Ca}$ isotopes in the vicinity of neutron drip-line, it is found that further addition of neutrons causes a rapid increase in the neutron rms radius with a very small increase in the binding energy, indicating thereby the occurrence of halos. This is essentially caused by the gradual filling in of the loosely bound $3s_{1/2}$ state. Interesting phenomenon of accommodating several additional neutrons with almost negligible increase in binding energy is shown to be due to the pairing correlations. ###### pacs: 21.10.-k,21.10Ft, 21.10.Dr, 21.10.Gv, 21.60.-n, 21.60.Jz 116 Department of Physics, Rajasthan University, Jaipur 302004, India Received 22 April 2004 accepted 6 December 2004 ## 1 Introduction The availability of radioactive beam facilities has generated a spurt of activity devoted to the investigation of exotic drip line nuclei. The neutron rich nuclei away from the line of $\beta$-stability with unusually large isospin value are known to exhibit several interesting features. For nuclei close to the neutron drip-line, the neutron density distribution shows a much extended tail with a diffused neutron skin while the Fermi level lies close to the single particle continuum [1]. In some cases it may even lead to the phenomenon of neutron halo, as observed in the case of light nuclei [1, 2, 3, 4] , made of several neutrons outside a core with separation energy of the order of $\approx$ 100 keV or less. Interestingly, a theoretical discussion on the possibility of occurrence of such structures has been considered by Migdal[5] already in early 70’s. Obviously for such nuclei, due to the weak binding and large spatial dimension of the outermost nucleons, the role of continuum states and their coupling to the bound states become exceedingly important, especially for the pairing energy contribution to the total binding energy of the system. Theoretical investigations of such neutron rich nuclei have been carried out extensively within the framework of mean field theories [6, 7, 8, 9, 10, 11] and also employing their relativistic counterparts[3, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. Recently the effect of continuum on the pairing energy contribution has been studied by Grasso et al.[11] and Sandulescu et al.[10] within the HF+BCS+Resonant continuum approach. Similarly the effect of inclusion of positive energy resonant states on the pairing correlations has been investigated by Yadav et al.[30]. A detailed comparative study of the Hartree- Fock-Bogoliubov (HFB) approach with those of the HF+BCS+Resonant continuum calculations carried out by Grasso et al.[11] and Sandulescu et al.[10] has provided useful insight to the validity of different approaches for the treatment of drip-line nuclei. The interesting result of these investigation is that only a few low energy resonant states, especially those near the Fermi surface influence in an appreciable way the pairing properties of nuclei far from the $\beta$-stability. This finding is of immense significance because one can eventually make use of this for systematic studies of a large number of nuclei by employing a simpler HF+BCS approximation. Amongst the mean-field theoretic treatments, however, currently the relativistic mean field (RMF) theory is being extensively used for the study of unstable nuclei [3, 18, 19, 20, 21, 22, 23, 24, 25]. The advantage of the RMF approach is that it provides the spin-orbit interaction in the entire mass region in a natural way [12, 13, 14]. This indeed has proved to be very crucial for the study of unstable nuclei near the drip line, since the single particle properties near the threshold are prone to large changes as compared to the case of deeply bound levels in the nuclear potential. In addition to this, the pairing properties are equally important for nuclei near the drip line. In order to take into account the pairing correlations together with a realistic mean field, the framework of standard RHB approach is commonly used[24, 26]. In this connection, the finding above for the non-relativistic frameworks has turned out to be very important for the systematic work of unstable nuclei in the relativistic approach. This has been demonstrated recently by Yadav et al.[30, 31] for the chains of ${}^{48-98}\rm{Ni}$ and ${}^{96-176}\rm{Sn}$ isotopes covering the drip lines. Indeed the RMF+BCS scheme[30, 31] wherein the single particle continuum corresponding to the RMF is replaced by a set of discrete positive energy states yields results which are found to be in close agreement with the experimental data and with those of recent continuum relativistic Hartree-Bogoliubov (RCHB) and other similar mean-field calculations[24, 32]. With the success of the RMF+BCS approach for the prototype calculations of $\rm{Ni}$ and $\rm{Sn}$ isotopes[30, 31], detailed calculations for the chain of $\rm{Ca}$ isotopes and also those of $\rm{O,\,Ni,\,Zr,\,Sn}$ and $\rm{Pb}$ isotopes using the TMA[21] and the NL-SH[23] force parameterizations have been carried out. The results of these calculations[33] for the two neutron separation energy, neutron, proton, and matter rms radii, and single particle pairing gaps etc., and their comparison with the available experimental data and with the results of other mean-field approaches demonstrate the general validity of the RMF+BCS approach. In this paper, in continuation to our earlier publication[31], we present briefly the results for the chain of $\rm{Ca}$ isotopes but with a special emphasis on our findings for the possible halo formation in the neutron rich $\rm{Ca}$ isotopes within the RMF+BCS approach. It is shown that the resonant $1g_{9/2}$ and the $3s_{1/2}$ states which lie close to zero energy in continuum and gradually come down to become bound with increasing neutron number, play the crucial role. Evidently the concentration of the major part of the wave function of the resonant $1g_{9/2}$ state within the potential well and its proximity with the Fermi surface while being close to zero energy together provide a favorable condition for the existence of extremely neutron rich $\rm{Ca}$ isotopes, whereas the $3s_{1/2}$ state with a well spread wave function, due to the absence of a centrifugal barrier, helps to cause the occurrence of halos. The role of pairing correlations as described here is found to be consistent with the conclusions of non-relativistic HFB studies of neutron rich weakly bound nuclei discussed recently by Bennaceur et al. [4]. ## 2 Theoretical Formulation and Model Our RMF calculations have been carried out using the model Lagrangian density with nonlinear terms both for the ${\sigma}$ and ${\omega}$ mesons as described in detail in Refs. [21]. $\displaystyle{\cal L}$ $\displaystyle=$ $\displaystyle{\bar{\psi}}[\imath\gamma^{\mu}\partial_{\mu}-M]\psi$ (1) $\displaystyle+\frac{1}{2}\,\partial_{\mu}\sigma\partial^{\mu}\sigma-\frac{1}{2}m_{\sigma}^{2}\sigma^{2}-\frac{1}{3}g_{2}\sigma^{3}-\frac{1}{4}g_{3}\sigma^{4}-g_{\sigma}{\bar{\psi}}\sigma\psi$ $\displaystyle-\frac{1}{4}H_{\mu\nu}H^{\mu\nu}+\frac{1}{2}m_{\omega}^{2}\omega_{\mu}\omega^{\mu}+\frac{1}{4}c_{3}(\omega_{\mu}\omega^{\mu})^{2}-g_{\omega}{\bar{\psi}}\gamma^{\mu}\psi\omega_{\mu}$ $\displaystyle-\frac{1}{4}G_{\mu\nu}^{a}G^{a\mu\nu}+\frac{1}{2}m_{\rho}^{2}\rho_{\mu}^{a}\rho^{a\mu}-g_{\rho}{\bar{\psi}}\gamma_{\mu}\tau^{a}\psi\rho^{\mu a}$ $\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-e{\bar{\psi}}\gamma_{\mu}\frac{(1-\tau_{3})}{2}A^{\mu}\psi\,\,,$ where the field tensors $H$, $G$ and $F$ for the vector fields are defined by $\displaystyle H_{\mu\nu}$ $\displaystyle=$ $\displaystyle\partial_{\mu}\omega_{\nu}-\partial_{\nu}\omega_{\mu}$ $\displaystyle G_{\mu\nu}^{a}$ $\displaystyle=$ $\displaystyle\partial_{\mu}\rho_{\nu}^{a}-\partial_{\nu}\rho_{\mu}^{a}-2g_{\rho}\,\epsilon^{abc}\rho_{\mu}^{b}\rho_{\nu}^{c}$ $\displaystyle F_{\mu\nu}$ $\displaystyle=$ $\displaystyle\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\,\,,\ $ and other symbols have their usual meaning. The set of parameters appearing in the effective Lagrangian (1) have been obtained in an extensive study which provides a good description for the ground state of nuclei and that of the nuclear matter properties[21]. This set, termed as TMA, has an $A$-dependence and covers the light as well as heavy nuclei from ${}^{16}\rm{O}$ to ${}^{208}\rm{Pb}$. Table 1 lists the TMA set of parameters along with the results for the calculated bulk properties of nuclear matter. As mentioned earlier we have also carried out the RMF+BCS calculations using the NL-SH force parameters [23] in order to compare our results with those obtained in the RCHB calculations [32] using this force parameterizations. The NL-SH parameters are also listed in Table 1 together with the corresponding nuclear matter properties. Based on the single-particle spectrum calculated by the RMF described above, we perform a state dependent BCS calculations[34, 35]. As we already mentioned, the continuum is replaced by a set of positive energy states generated by enclosing the nucleus in a spherical box. Thus the gap equations have the standard form for all the single particle states, i.e. $\displaystyle\Delta_{j_{1}}$ $\displaystyle=$ $\displaystyle\,-\frac{1}{2}\frac{1}{\sqrt{2j_{1}+1}}\sum_{j_{2}}\frac{\left<{({j_{1}}^{2})\,0^{+}\,\|V\|\,({j_{2}}^{2})\,0^{+}}\right>}{\sqrt{\big{(}\varepsilon_{j_{2}}\,-\,\lambda\big{)}^{2}\,+\,{\Delta_{j_{2}}^{2}}}}\,\,\sqrt{2j_{2}+1}\,\,\,\Delta_{j_{2}}\,\,,$ (2) where $\varepsilon_{j_{2}}$ are the single particle energies, and $\lambda$ is the Fermi energy, whereas the particle number condition is given by $\sum_{j}\,(2j+1)v^{2}_{j}\,=\,{\rm N}$. In the calculations we use for the pairing interaction a delta force, i.e., $V=-V_{0}\delta(r)$ with the same strength $V_{0}$ for both protons and neutrons. The value of the interaction strength $V_{0}=350\,$ MeV fm3 was determined in ref. [30] by obtaining a best fit to the binding energy of $\rm{Ni}$ isotopes. We use the same value of $V_{0}$ for our present studies of isotopes Table 1: Parameters of the Lagrangian TMA[21] and NL-SH[23] together with the nuclear matter properties obtained with these effective forces. \| Force Parameters \| \| Nuclear Matter Properties \| ---\|---\|---\|---\|--- \| TMA \| NL-SH \| \| TMA \| NL-SH M (MeV) \| 938.9 \| 939.0 \| Saturation density \| \| mσ(MeV) \| 519.151 \| 526.059 \| $\rho_{0}$ (fm)-3 \| 0.147 \| 0.146 mω(MeV) \| 781.950 \| 783.0 \| Bulk binding energy/nucleon \| \| mρ(MeV) \| 768.100 \| 763.0 \| (E/A)∞ (MeV) \| 16.0 \| 16.346 gσ \| 10.055 + 3.050/A0.4 \| 10.444 \| Incompressibility \| \| gω \| 12.842 + 3.191/A0.4 \| 12.945 \| K (MeV) \| 318.0 \| 355.36 gρ \| 3.800 + 4.644/A0.4 \| 4.383 \| Bulk symmetry energy/nucleon \| \| g2 (fm)-1 \| -0.328 - 27.879/A0.4 \| -6.9099 \| asym (MeV) \| 30.68 \| 36.10 g3 \| 38.862 - 184.191/A0.4 \| -15.8337 \| Effective mass ratio \| \| c3 \| 151.590 - 378.004/A0.4 \| \| m∗/m \| 0.635 \| 0.60 of other nuclei as well. Apart from its simplicity, the applicability and justification of using such a $\delta$-function form of interaction has been recently discussed in Refs.[6] and [8], whereby it has been shown in the context of HFB calculations that the use of a delta force in a finite space simulates the effect of finite range interaction in a phenomenological manner ( see also [36] and [37] for more details ). The pairing matrix element for the $\delta$-function force is given by $\displaystyle\left<{({j_{1}}^{2})\,0^{+}\,\|V\|\,({j_{2}}^{2})\,0^{+}}\right>$ $\displaystyle=$ $\displaystyle\,-\,\frac{V_{0}}{8\pi}\sqrt{(2j_{1}+1)(2j_{2}+1)}\,\,I_{R}\,\,,$ (3) where $I_{R}$ is the radial integral having the form $\displaystyle I_{R}$ $\displaystyle=$ $\displaystyle\,\int\,dr\frac{1}{r^{2}}\,\left(G^{\star}_{j_{1}}\,G_{j_{2}}\,+\,F^{\star}_{j_{1}}\,F_{j_{2}}\right)^{2}$ (4) Here $G_{\alpha}$ and $F_{\alpha}$ denote the radial wave functions for the upper and lower components, respectively, of the nucleon wave function expressed as $\psi_{\alpha}={1\over r}\,\,\left({i\,\,\,G_{\alpha}\,\,\,{\mathcal{Y}}_{j_{\alpha}l_{\alpha}m_{\alpha}}\atop{F_{\alpha}\,{\sigma}\cdot\hat{r}\,\,{\mathcal{Y}}_{j_{\alpha}l_{\alpha}m_{\alpha}}}}\right)\,\,,$ (5) and satisfy the normalization condition $\displaystyle\int dr\,{\\{\|G_{\alpha}\|^{2}\,+\,\|F_{\alpha}\|^{2}}\\}\,=\,1$ (6) In Eq. (5) the symbol ${\mathcal{Y}}_{jlm}$ has been used for the standard spinor spherical harmonics with the phase $i^{l}$. The coupled field equations obtained from the Lagrangian density in (1) are finally reduced to a set of simple radial equations[14] which are solved self consistently along with the equations for the state dependent pairing gap $\Delta_{j}$ and the total particle number $\rm N$ for a given nucleus. Fig. 1. Upper panel: The RMF potential energy (sum of the scalar and vector potentials), for the nucleus ${}^{64}\rm{Ca}$ shown by the solid line as a function of radius $r$. The long dashed line represents the sum of RMF potential energy and the centrifugal barrier energy for the neutron resonant state $1g_{9/2}$. It also shows the energy spectrum of some important neutron single particle states along with the resonant $1g_{9/2}$ state at 0.16 MeV. Lower panel: Radial wave functions of a few representative neutron single particle states with energy close to the Fermi surface for the nucleus ${}^{64}\rm{Ca}$. The solid line shows the resonant $1g_{9/2}$ state. ## 3 Results and Discussion Our earlier calculations for chains of $\rm{Ni}$ and $\rm{Sn}$ isotopes[30, 31] and present investigations of $\rm{Ca}$ isotopes as well as of other nuclei indicate that the neutron rich $\rm{Ca}$ isotopes constitute the most interesting example of loosely bound system. For an understanding of such an exotic system the total pairing energy contribution to the binding energy plays a crucial role. This in turn implies the importance of the structure of single particle states near the Fermi level, as the scattering of particles from bound to continuum states and vice versa due to pairing interaction involves mainly these states. The last few occupied states near the Fermi level also provide an understanding of the radii of the loosely bound exotic nuclei. The neutron rich nuclei in which the last filled single particle state near the Fermi level is of low angular momentum ($s_{1/2}$ or $p_{1/2}$ state), especially the $l=0$ state, can have large radii due to large spatial extension of the $s_{1/2}$ state which has no centrifugal barrier. In order to demonstrate our results we have chosen ${}^{64}\rm{Ca}$ as a representative example of the neutron rich $\rm{Ca}$ isotopes. Moreover, since the results obtained with TMA and NL-SH forces are found to be almost similar, to save space we describe in details only the results for the TMA force, whereas the results obtained with the NL-SH force have been discussed at places for the purpose of comparison. The upper panel of Fig.1 shows the calculated RMF potential, a sum of scalar and vector potentials, along with the spectrum for the bound neutron single particle states for the neutron rich ${}^{64}\rm{Ca}$ obtained with the TMA force. The figure also shows the positive energy state corresponding to the first low-lying resonance $1g_{9/2}$, and other positive energy states, for example, $3s_{1/2}$, $2d_{5/2}$, $2d_{3/2}$ and $1g_{7/2}$ close to the Fermi surface which play significant role for the binding of neutron rich isotopes through their contributions to the total pairing energy. In contrast to other states in the box which correspond to the non-resonant continuum, the position of the resonant $1g_{9/2}$ state is not much affected by changing the box radius around $R=30$ fm. We have also depicted in this part of Fig.1 the total mean field potential for the neutron $1g_{9/2}$ state, obtained by adding the centrifugal potential energy. It is evident from the figure that the effective total potential for the $1g_{9/2}$ state has an appreciable barrier to form a quasi-bound or resonant state. Such a meta-stable state remains mainly confined to the region of the potential well and the wave function exhibits characteristics similar to that of a bound state. This is clearly seen in the lower panel of Fig.1 which depicts the radial wave functions of some of the neutron single particle states lying close to the Fermi surface, the neutron Fermi energy being $\lambda_{n}\,=\,-0.066$ MeV. These include the bound $2p_{1/2}$, and the continuum $3s_{1/2}$ and $2d_{5/2}$ states in addition to the state corresponding to the resonant $1g_{9/2}$. The wave function for the $1g_{9/2}$ state in Fig. 1 (lower panel) is clearly seen to be confined within a radial range of about 8 fm and has a decaying component outside this region, characterizing a resonant state. In contrast, the main part of the wave function for the non-resonant states, e.g. $2d_{5/2}$, is seen to be mostly spread over outside the potential region. This type of state thus has a poorer overlap with the bound states near the Fermi surface leading to small value for the pairing gap $\Delta_{2d_{5/2}}$. Further, the positive energy states lying much higher from the Fermi level, for example, $1h_{11/2}$, $1i_{13/2}$ etc. have a negligible contribution to the total pairing energy of the system.These features can be seen from Fig.2 (upper panel) which depicts pairing gap energy $\Delta_{j}$ for the neutron states in ${}^{64}\rm{Ca}$. The gap energy for the $1g_{9/2}$ state is seen to have a value close to $1$ MeV which is quantitatively similar to that of bound states $1f_{7/2}$ and $2p_{3/2}$ etc. The non-resonant states like $3s_{1/2}$ and $2d_{5/2}$ in continuum have much smaller gap energy. However, while approaching the neutron drip line nucleus ${}^{72}\rm{Ca}$, the single particle states $3s_{1/2}$, $1g_{9/2}$, $2d_{5/2}$ and $2d_{3/2}$ which lie near the Fermi level gradually come down close to zero energy, and subsequently the $1g_{9/2}$ and $3s_{1/2}$ states become bound. This helps in accommodating more and more neutrons with very little binding. In fact, the occupancy of the $3s_{1/2}$ state in these neutron rich isotopes causes the halo formation as will be seen later. In the lower panel of Fig.2 we have shown the contribution of pairing energy which plays an important role for the stability of nuclei and consequently in deciding the position of the neutron and proton drip lines. It is seen that the RMF+BCS calculations carried out with two different sets of force parameters, the TMA and NL-SH, yield almost similar results also for the pairing energies. The differences in the two results can be attributed to the difference in the detailed structure of single particle energies obtained with the TMA and NL-SH forces. One observes from Fig.2 (lower panel) that the pairing energy vanishes for the neutron numbers $N=14,20,28$ and $40$ indicating the shell closures. In particular the usual shell closure at $N=50$ is found to be absent for the neutron rich $\rm{Ca}$ isotopes and at $N=40$ a new shell closure appears. This reorganization of single particle energies with large values of N/Z ratio (for the neutron rich $\rm{Ca}$ isotopes N/Z $\geq 2$) has its origin in the deviation of the strength of spin-orbit splitting from the conventional shell model results for nuclei with not so large $N/Z$ ratio. Fig.2. Upper panel: Pairing gap energy $\Delta_{j}$ of neutron single particle states with energy close to the Fermi surface for the nucleus ${}^{64}\rm{Ca}$. The resonant $1g_{9/2}$ state at energy 0.165 MeV has the gap energy of about $1$ MeV which is close to that of bound states like $1f_{5/2}$, $1f_{7/2}$, $1d_{3/2}$ etc. Lower panel: Pairing energy for the $\rm{Ca}$ isotopes obtained with the TMA (open circles) and the NL-SH (open triangles) force parameters. The results for the ground state properties including the binding energy, two neutron separation energy, and the rms radii for the neutron, proton, matter and charge distributions for the $\rm{Ca}$ isotopes calculated with the TMA force have been listed in Table 2\. The table also lists the available experimental data[38] for the binding energies, two neutron separation energy as well as the binding energy per nucleon (B/A) for the purpose of comparison. The available experimental data for the radii are very sparse and have not been listed. It is interesting to note that the maximum value of B/A occurs for the ${}^{46}\rm{Ca}$ isotope and not the doubly magic isotopes ${}^{40}\rm{Ca}$ or ${}^{48}\rm{Ca}$. The shell structure as revealed by the pairing energies are also exhibited in the variation of two neutron separation energies $S_{2n}$ as shown in the lower panel of Fig.3. An abrupt increase in the $S_{2n}$ values for the isotopes next to magic numbers is clearly seen. This part of the figure also depicts the results of two neutron separation energy obtained in the RCHB approach and their comparison with available experimental data. The upper panel of Fig.3 depicts the difference between the experimental and calculated values. It is seen that the TMA and NL-SH forces yield similar results and the isotopes beyond A$=$66 have two neutron separation energy close to zero. The two neutron drip line is found to occur at A$=$70 (N$=$50) and A$=$72 (N$=$52) for the TMA and NL-SH forces, respectively. The isotopes with mass number $70<A<76$ for the TMA, and those with $72<A<76$ for the NL-SH case are found to be just unbound with negative separation energy very close to zero. Accordingly in Fig.3 we have shown the results up to N$=$56 to emphasize this point. Also, for our purpose we shall neglect this small difference in the position of drip line mentioned above. The similarity of different calculated results amongst themselves and their satisfactory comparison with data are well demonstrated in the upper panel. Table 2: Results for the ground state properties of Ca-isotopes calculated with the TMA force parameter set. Listed are the total binding energy, BE, the two neutron separation energy, S2n, binding energy per nucleon, B/A, and neutron, proton, matter and charge root mean square radii denoted by rn, rp, rm, rc, respectively. The available experimental data[38] on the binding energy, (BE)exp, and that of (S2n)exp and (B/A)exp are also listed for comparison. Nucleus \| (BE)exp \| BE \| (S2n)exp \| S2n \| (B/A)exp \| B/A \| rn \| rp \| rm \| rc ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- \| MeV \| MeV \| MeV \| MeV \| MeV \| MeV \| fm \| fm \| fm \| fm ${}^{34}Ca$ \| 245.625 \| 246.684 \| \| \| 7.224 \| 7.255 \| 3.039 \| 3.415 \| 3.266 \| 3.511 ${}^{36}Ca$ \| 281.360 \| 280.968 \| 35.735 \| 34.284 \| 7.816 \| 7.805 \| 3.168 \| 3.394 \| 3.296 \| 3.489 ${}^{38}Ca$ \| 313.122 \| 313.114 \| 31.762 \| 32.146 \| 8.240 \| 8.240 \| 3.262 \| 3.385 \| 3.327 \| 3.479 ${}^{40}Ca$ \| 342.052 \| 343.208 \| 28.930 \| 30.094 \| 8.551 \| 8.580 \| 3.337 \| 3.383 \| 3.360 \| 3.475 ${}^{42}Ca$ \| 361.895 \| 363.843 \| 19.843 \| 20.635 \| 8.617 \| 8.663 \| 3.426 \| 3.381 \| 3.404 \| 3.471 ${}^{44}Ca$ \| 380.960 \| 382.823 \| 19.065 \| 18.980 \| 8.658 \| 8.700 \| 3.501 \| 3.382 \| 3.447 \| 3.471 ${}^{46}Ca$ \| 398.769 \| 400.385 \| 17.809 \| 17.562 \| 8.669 \| 8.704 \| 3.565 \| 3.386 \| 3.488 \| 3.473 ${}^{48}Ca$ \| 415.991 \| 416.629 \| 17.222 \| 16.244 \| 8.666 \| 8.680 \| 3.621 \| 3.391 \| 3.527 \| 3.476 ${}^{50}Ca$ \| 427.491 \| 425.235 \| 11.500 \| 8.606 \| 8.550 \| 8.505 \| 3.759 \| 3.412 \| 3.624 \| 3.495 ${}^{52}Ca$ \| 436.600 \| 433.287 \| 9.109 \| 8.052 \| 8.396 \| 8.332 \| 3.872 \| 3.435 \| 3.710 \| 3.516 ${}^{54}Ca$ \| 443.800 \| 440.711 \| 7.200 \| 7.424 \| 8.219 \| 8.161 \| 3.966 \| 3.463 \| 3.788 \| 3.542 ${}^{56}Ca$ \| 449.600 \| 448.148 \| 5.800 \| 7.437 \| 8.029 \| 8.003 \| 4.082 \| 3.493 \| 3.882 \| 3.569 ${}^{58}Ca$ \| \| 455.380 \| \| 7.232 \| \| 7.851 \| 4.115 \| 3.522 \| 3.921 \| 3.596 ${}^{60}Ca$ \| \| 462.135 \| \| 6.755 \| \| 7.702 \| 4.173 \| 3.551 \| 3.977 \| 3.623 ${}^{62}Ca$ \| \| 462.704 \| \| 0.569 \| \| 7.463 \| 4.248 \| 3.573 \| 4.042 \| 3.643 ${}^{64}Ca$ \| \| 462.964 \| \| 0.260 \| \| 7.234 \| 4.361 \| 3.593 \| 4.137 \| 3.662 ${}^{66}Ca$ \| \| 463.032 \| \| 0.068 \| \| 7.016 \| 4.634 \| 3.609 \| 4.349 \| 3.675 ${}^{68}Ca$ \| \| 463.058 \| \| 0.026 \| \| 6.810 \| 4.851 \| 3.623 \| 4.525 \| 3.687 ${}^{70}Ca$ \| \| 463.075 \| \| 0.017 \| \| 6.615 \| 4.926 \| 3.640 \| 4.596 \| 3.703 ${}^{72}Ca$ \| \| 462.972 \| \| -0.103 \| \| 6.430 \| 5.007 \| 3.655 \| 4.671 \| 3.716 The rms radii for the proton and neutron , $r_{p,n}\,=\,(\langle r^{2}_{p(n)}\rangle\,)^{1/2}$ have been calculated from the respective density distributions. The experimental data for the rms charge radii are used to deduce the nuclear rms proton radii using the relation $r_{c}^{2}\,=\,r_{p}^{2}\,+\,0.64\,fm^{2}$ for the purpose of comparison. In the middle panel of Fig.4 we have shown the RMF+BCS results for the neutron and proton rms radii for the NL-SH force along with the RCHB results[32] also obtained using the NL-SH force for the purpose of comparison. These results are quite similar as can be seen from the differences plotted in the upper panel. The experimental data for the proton and neutron rms radii are available only for a few stable $\rm Ca$ isotopes. The lower panel of Fig.4 depicts a comparison of the RMF+BCS results using the TMA force with the available experimental data. It is seen from Fig.4 that the measured proton radii $r_{p}$ for the isotopes ${}^{40-48}\rm Ca$ are in excellent agreement with our RMF+BCS results. Similarly the neutron radii $r_{n}$ for the ${}^{40,42,44,48}\rm Ca$ isotopes are found to compare quite well as has been depicted in the lower panel of Fig.4. As described earlier, in the case of neutron rich $\rm Ca$ isotopes the neutron $1g_{9/2}$ state happens to be a resonant state having good overlap with the bound states near the Fermi level. This causes the pairing interaction to scatter particles from the neighboring bound states to the resonant state Fig.3. In the lower panel two neutron separation energies for the $\rm{Ca}$ isotopes calculated with the TMA (open circles) and the NL-SH (open squares) force parameters are compared with the RCHB calculations of Ref.[32] carried out with the NL-SH force (open triangles) and with the available experimental data[38]. The upper panel shows the difference in the RMF+BCS and the RCHB results as well as the difference between calculated results with respect to the available experimental data[38]. and vice versa. Thus, it is found that the resonant $1g_{9/2}$ state starts being partially occupied even before the lower bound single particle states are fully filled in. Further, for neutron rich $\rm Ca$ isotopes it is found that the neutron $3s_{1/2}$ state which lies close to the $1g_{9/2}$ state also starts getting partially occupied before the $1g_{9/2}$ state is completely filled. The neutron $3s_{1/2}$ state due to lack of centrifugal barrier contributes more to the neutron rms radius as compared to the $1g_{9/2}$ state, and thus one observes in Fig.4 (lower panel) a rapid increase in the neutron rms radius beyond the neutron number $N=42$ indicating the formation of halos. A comparison of the rms neutron radii with the $r_{n}=r_{0}N^{1/3}$ line shown in the figure suggests that these radii for the drip-line isotopes do not follow the $r_{0}N^{1/3}$ systematics. Similar calculations for the neutron radii of $\rm Ni$ and $\rm Sn$ isotopes[30, 31], however, do not exhibit halo formation as can be seen from Fig.5. As regards the possibility of halo formation in other nuclei, the following remark is pertinent. From our comprehensive calculations of chains of proton magic nuclei[33], it is found that the isotopes of $\rm O$ as well as $\rm Pb$ nuclei also do not exhibit the tendency of halo formations. On the other hand, the proton sub-magic neutron rich $\rm Zr$ isotopes do have a single particle structure that provides a favorable condition for the halos, albeit in a less pronounced manner. In this case the single particle $3p_{1/2}$ state lying close to the continuum threshold plays the crucial role. An important aspect of the heavy neutron rich nuclei is the formation of the neutron skin.[1] The neutron density distributions in the neutron rich ${}^{62-72}\rm{Ca}$ nuclei are found to be widely spread out in the space indicating the formation of neutron halos. This has been demonstrated in Fig.6 which shows the variation in the proton (upper panel) and neutron (lower panel) radial density distributions with increasing neutron number for the TMA force calculations. It may be emphasized that the density distributions obtained from the NL-SH force parameterizations are almost similar to those obtained using the TMA force and, therefore, here we have chosen to show only the results for the TMA force. Fig.4. Lower panel: The rms radii of neutron distribution $r_{n}$ (open circles), and that of proton distribution $r_{p}$ (open squares) obtained with the TMA force are compared with the available experimental data [40], shown by solid circles and solid squares, respectively. Middle panel: A comparison of RMF+BCS results (open squares) for rms radii $r_{n}$ and $r_{p}$ with that of RCHB (open triangles) from Ref.[32] obtained with the NL-SH force . Upper panel: Difference between the results obtained from RMF+BCS and the RCHB approaches for the rms radii using the NL-SH force shown in the middle panel. As depicted in the upper panel of the Fig.6, the proton distributions are observed to be confined to smaller distances. Moreover, these start to fall off rapidly already at smaller distances ( beyond $r>3$ fm.) as compared to those for the neutron density distributions shown in the lower panel. In the interior as well as at outer distances, as shown in the inset of the upper panel, the proton density values are larger for the proton rich $\rm{Ca}$ isotopes and decrease with increasing neutron number N. However, in the surface region, ($r\approx 4$ fm), the proton density values reverse their trend and increase with increasing neutron number. Due to this feature of the proton density distributions the proton radii are found to increase, albeit in a very small measure, with increasing neutron number. Similarly the lower panel of Fig.6, depicting the neutron density distribution, shows that for the magic numbers N = 14, 20, 28 and 40 the neutron densities fall off rapidly and have smaller tails as compared to the isotopes with other neutron numbers. The density distribution for the N = 50 case is seen to be very different from the above cases indicating thereby that for the neutron Fig.5. The RMF results[30, 31, 33] for the neutron rms radii $r_{n}$ for the isotopes of $\rm{Ca}$, $\rm{Ni}$ and $\rm{Sn}$ nuclei obtained with the TMA force. These are compared with a rough estimate of neutron distribution radius given by $r_{n}$ = $r_{0}N^{1/3}$ wherein the radius constant $r_{0}$ is chosen to provide the best fit to the theoretical results. Halo formation in the case of neutron rich $\rm{Ca}$ isotopes is clearly seen. rich $\rm{Ca}$ isotopes the N = 50 does not correspond to a magic number. The neutron densities of the isotopes having $N>40$ are found to exhibit especially widespread distributions out side the range of the interaction potential. This has been explicitly demonstrated in the inset of lower panel of Fig.6. Moreover, these results are also found to be very similar to those obtained using the RCHB approach[32]. In particular, for the isotopes with neutron shell closure corresponding to $N=$ 14, 20, 28 and 40 this similarity extends up to large radial distances. For the other isotopes, there are small deviations between the RMF+BCS and the RCHB approaches beyond the radial distance $r=8$ fm. However, beyond this distance the densities are already quite reduced ranging between $10^{-4}$ fm-3 to $10^{-8}$ fm-3. In Fig.6 (lower panel) it is interesting to note that the neutron density distributions, out side the nuclear surface and at large distances, for the neutron rich $\rm Ca$ isotopes with neutron number $N\geq 42$ are larger by several orders of magnitude as compared to the lighter isotopes. This behavior of the density distribution for the neutron rich $\rm Ca$ isotopes is quite different from the corresponding results, especially for the neutron rich isotopes of $\rm{Ni}$, $\rm{Sn}$ and $\rm{Pb}$ nuclei[30, 31, 33]. In the latter cases, as the neutron number is added the tail of the neutron density distributions for the neutron rich isotopes tend to saturate. The large relative enhancement in the tail region of neutron density distributions of the neutron rich $\rm{Ca}$ isotopes beyond $N=40$ (or $A=60$) gives rise to the halo formation in these isotopes. Essentially, it is caused due to weakly bound neutrons occupying the single particle states near the Fermi level which is itself almost close to zero energy for the neutron rich isotopes as has been shown in Fig.7. The large energy gaps between single particle levels $1d_{5/2}$ and $1d_{3/2}$ (not shown in Fig. 7), and the levels $2p_{1/2}$ and $3s_{1/2}$ etc. are responsible for the properties akin to shell or sub-shell closures in the ${}^{34-72}\rm{Ca}$ isotopes for the neutron number N = 14 and 40 apart from the traditional magic nos. N = 20 and 28. However the N = 50 shell closure is found to disappear due to absence of gaps between the $1g_{9/2}$ state and the states in the s-d shell. A better understanding of the halo formation in the neutron rich $(N>40)$ $\rm{Ca}$ isotopes is rendered if one looks into the detailed features of single particle spectrum and its variation shown in Fig.7 as one moves from the lighter isotope to heavier one . For example, the neutron Fermi energy which lies at $\epsilon_{f}=-19.90$ MeV in the neutron deficient ${}^{34}\rm{Ca}$ nucleus moves to $\epsilon_{f}=-0.21$ MeV in the neutron rich ${}^{62}\rm{Ca}$, and to $\epsilon_{f}=0.08$ MeV (almost at the beginning of the single particle continuum) in ${}^{70}\rm{Ca}$. The $1g_{9/2}$ state which lies at higher energy in continuum for the lighter isotopes, comes down gradually to become slightly bound for the neutron rich isotopes. Similarly, the $3s_{1/2}$ state which lies in continuum for the lighter isotopes (for example at $\epsilon=0.70$ MeV in ${}^{34}\rm{Ca}$) also comes down, though not so drastically, to become slightly bound (($\epsilon=-0.05$ MeV in ${}^{68}\rm{Ca}$) for the neutron rich isotopes. Fig.6. The upper and lower panels, respectively, show the proton and neutron density distributions for the $\rm{Ca}$ isotopes obtained with the TMA force. The numbers on the density distribution lines indicate the mass number of the $\rm{Ca}$ isotope.The insets show the results on a logarithmic scale up to rather large distances. In the case of ${}^{60}\rm{Ca}$ with shell closure for both protons and neutrons, the neutron single particle states are filled in up to the $2p_{1/2}$ state, while the next high lying states $3s_{1/2}$ and $1g_{9/2}$, separated by about $5$ MeV from the $2p_{1/2}$ level, are completely empty. Now on further addition of 2 neutrons, it is observed that the $1g_{9/2}$ is filled in first even though $3s_{1/2}$ state is slightly lower (by about 0.31 MeV) than the $1g_{9/2}$ state as has been shown in the lower panel of Fig.7. Still another addition of 2 neutrons is found to fill in the $1g_{9/2}$ state once again, though now the $1g_{9/2}$ state is higher to the $3s_{1/2}$ state merely by $0.08$ MeV. This preference for the $1g_{9/2}$ state stems from the fact that in contrast to the $3s_{1/2}$ state, the positive energy $1g_{9/2}$ state being a resonant state has its wave function entirely confined inside the potential well akin to a bound state as shown earlier in Fig.2 for the nucleus ${}^{64}\rm{Ca}$. For the neutron number N = 48, it is found that both the $1g_{9/2}$ and $3s_{1/2}$ states become bound and start to compete together to get occupied on further addition of neutrons as can be seen in Fig.7 (lower panel). Further it is observed from the figure that both of these states are completely filled in for the neutron number N = 52 and, thus, the neutron drip line is reached with a loosely bound ${}^{72}\rm{Ca}$ nucleus. The next single particle state, $2d_{5/2}$, is higher in energy by about 0.5 MeV in the continuum and further addition of neutrons does not produce a bound system. Fig.7.Upper panel: Variation of the neutron single particle energies obtained with the TMA force for the $\rm{Ca}$ isotopes with increasing neutron number. The Fermi level has been shown by filled circles connected by solid line to guide the eyes. Lower panel: Variation of the position and occupancy (no. of neutrons occupying the levels) of the neutron $1g_{9/2}$ and $3s_{1/2}$ single particle states in the neutron rich $\rm{Ca}$ isotopes. As mentioned above, the $1g_{9/2}$ state is mainly confined to the potential region, and hence its contribution to the neutron radii is similar to a bound state. In contrast, the $3s_{1/2}$ state which has no centrifugal barrier and, therefore, is spread over large spatial extension contributes substantially to the neutron density distribution at large distances. Due to this reason, for $N>42$ as the $3s_{1/2}$ state starts being occupied the neutron density distributions develop large tails, and the neutron radii for the neutron rich isotopes ($N>42$) grow abruptly as has been shown in figs. 4 and 5. Thus the filling in of the $3s_{1/2}$ single particle state with increasing neutron number in the ${}^{64-72}\rm{Ca}$ isotopes causes the formation of neutron halos in these nuclei. It is pertinent to point out that our results described above are consistent with the recent non-relativistic HFB calculations of Bennaceur et al. [4] demonstrating the effects of pairing correlations in the description of weakly bound neutron rich nuclei. The interesting result of these authors [4] is that the pairing correlations in the even-N nuclei, contrary to the case of odd-N nuclei (N being the neutron number) for the zero orbital angular momentum (l = 0) s-state, provide additional binding in a way to halt the unlimited increase in the rms radius as the single particle binding of the s-state tends to be zero. Further, it has also been demonstrated by these authors [4] that the low lying $l=0$ continuum states contribute significantly in generating large rms neutron radii of the neutron rich weakly bound nuclei, and that the exact position of l=0 orbital is not crucial for this enhancement. From our Fig. 7 it is seen that the $3s_{1/2}$ orbital whose positions varies only slightly for the neutron number $N=12$ to $N=52$ continues to be close to zero energy, starts being occupied only for $N>42$. This occupancy generates large increase in the neutron rms radii beyond $N=42$ as is readily observed in figs. 4 and 5 for the $Ca$ isotopes. Thus our results are in accord with those of ref. [4] whereby the occupied $l=0$ orbital provides a significant increase in the neutron rms radii and, moreover, this increase does not tend to become infinitely large even when the single particle energy of the s-state changes to lie very close to zero. In a similar calculation for deformed nuclei it may be difficult to discuss the results in terms of single particle wave functions as shown in fig. 1 for the spherical case. However, we believe that the main conclusions drawn here to characterize a favorable situation for the halo formation will not change significantly in a similar description for the even-even deformed nuclei . ## 4 Conclusion In conclusion, we have applied the BCS approach using a descretized continuum within the framework of relativistic mean-field theory to study the ground state properties of $\rm{Ca}$ isotopes up to the drip-lines, the main emphasis being on the possible formation of halos in the neutron rich $\rm{Ca}$ isotopes. Calculations have been performed using the popular TMA and the NL-SH sets of parameters for the effective mean-field Lagrangian. For the pairing energy a $\delta$-function interaction has been employed for the state dependent BCS calculations. It is found that from amongst the positive energy states, apart from the single particle states adjacent to the Fermi level, the dominant contribution to the pairing correlations is provided by a few states which correspond to low-lying resonances. An important result to be emphasized is the following. In the vicinity of neutron drip-line for the $\rm{Ca}$ isotopes, it is found that further addition of neutrons causes a rapid increase in the neutron rms radius with a very small increase in the binding energy, indicating thereby the occurrence of halos in the neutron rich $\rm{Ca}$ isotopes. The filling in of the resonant $1g_{9/2}$ state, that sets in even before it becomes bound, with a very little increase in the binding energy causes the existence of extremely neutron rich $\rm{Ca}$ isotopes, whereas the occupancy of loosely bound $3s_{1/2}$ state gives rise to the halo formation. Also, as in earlier prototype calculations[30, 31] for the $\rm{Ni}$ and $\rm{Sn}$ isotopes, our present RMF+BCS results for the two neutron separation energy, rms neutron and proton radii and pairing energies for the $\rm{Ca}$ isotopes compare well with the known experimental data[38]. Furthermore, detailed comparisons show that the RMF+BCS approach provide results almost similar to those obtained in the more complete relativistic continuum Hartree Bogoliubov (RCHB) treatment[32]. This is in accord with the conclusion of Grasso et al.[11] whereby the BCS approach is shown to be a good approximation to the Bogoliubov treatment in the context of non-relativistic mean-field studies. Moreover, the effects of pairing correlations observed in our treatment are found to be in agreement with those demonstrated recently by Bennaceur et al. 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1310.8197	EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN) CERN-PH-EP-2013-198 LHCb-PAPER-2013-058 December 4, 2013 Study of forward Z+jet production in pp collisions at $\sqrt{s}=7\,\text{TeV}$ The LHCb collaboration†††Authors are listed on the following pages. A measurement of the $\mathrm{Z}(\rightarrow\upmu^{+}\upmu^{-})+\text{jet}$ production cross-section in $\mathrm{p}\mathrm{p}$ collisions at a centre-of- mass energy $\sqrt{s}=7\text{ TeV}$ is presented. The analysis is based on an integrated luminosity of 1.0$\text{ fb}^{-1}$ recorded by the LHCb experiment. Results are shown with two jet transverse momentum thresholds, 10 and 20 GeV, for both the overall cross-section within the fiducial volume, and for six differential cross-section measurements. The fiducial volume requires that both the jet and the muons from the $\mathrm{Z}$ boson decay are produced in the forward direction ($2.0<\eta<4.5$). The results show good agreement with theoretical predictions at the second-order expansion in the coupling of the strong interaction. Submitted to JHEP © CERN on behalf of the LHCb collaboration, license CC-BY-3.0. LHCb collaboration R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24, S. Amato2, S. Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, M. Andreotti16,e, J.E. Andrews57, R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli37, A. Artamonov34, M. Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J. Back47, A. Badalov35, C. Baesso59, V. Balagura30, W. Baldini16, R.J. Barlow53, C. Barschel37, S. Barsuk7, W. Barter46, V. Batozskaya27, Th. Bauer40, A. Bay38, J. Beddow50, F. Bedeschi22, I. Bediaga1, S. Belogurov30, K. Belous34, I. Belyaev30, E. Ben-Haim8, G. Bencivenni18, S. Benson49, J. Benton45, A. 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Head37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, M. Heß60, A. Hicheur1, E. Hicks51, D. Hill54, M. Hoballah5, C. Hombach53, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54, D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten12, R. Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M. John54, D. Johnson54, C.R. Jones46, C. Joram37, B. Jost37, M. Kaballo9, S. Kandybei42, W. Kanso6, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, T. Ketel41, B. Khanji20, S. Klaver53, O. Kochebina7, I. Komarov38, R.F. Koopman41, P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,37,j, V. Kudryavtsev33, K. Kurek27, T. Kvaratskheliya30,37, V.N. La Thi38, D. Lacarrere37, G. Lafferty53, A. Lai15, D. Lambert49, R.W. Lambert41, E. Lanciotti37, G. Lanfranchi18, C. Langenbruch37, T. Latham47, C. Lazzeroni44, R. Le Gac6, J. van Leerdam40, J.-P. Lees4, R. Lefèvre5, A. Leflat31, J. Lefrançois7, S. Leo22, O. Leroy6, T. Lesiak25, B. Leverington11, Y. Li3, L. Li Gioi5, M. Liles51, R. Lindner37, C. Linn11, B. Liu3, G. Liu37, S. Lohn37, I. Longstaff50, J.H. Lopes2, N. Lopez-March38, H. Lu3, D. Lucchesi21,q, J. Luisier38, H. Luo49, E. Luppi16,e, O. Lupton54, F. Machefert7, I.V. Machikhiliyan30, F. Maciuc28, O. Maev29,37, S. Malde54, G. Manca15,d, G. Mancinelli6, J. Maratas5, U. Marconi14, P. Marino22,s, R. Märki38, J. Marks11, G. Martellotti24, A. Martens8, A. Martín Sánchez7, M. Martinelli40, D. Martinez Santos41,37, D. Martins Tostes2, A. Martynov31, A. Massafferri1, R. Matev37, Z. Mathe37, C. Matteuzzi20, E. Maurice6, A. Mazurov16,37,e, M. McCann52, J. McCarthy44, A. McNab53, R. McNulty12, B. McSkelly51, B. Meadows56,54, F. Meier9, M. Meissner11, M. Merk40, D.A. Milanes8, M.-N. Minard4, J. Molina Rodriguez59, S. Monteil5, D. Moran53, P. Morawski25, A. Mordà6, M.J. Morello22,s, R. Mountain58, I. Mous40, F. Muheim49, K. Müller39, R. Muresan28, B. Muryn26, B. Muster38, P. Naik45, T. Nakada38, R. Nandakumar48, I. Nasteva1, M. Needham49, S. Neubert37, N. Neufeld37, A.D. Nguyen38, T.D. Nguyen38, C. Nguyen-Mau38,o, M. Nicol7, V. Niess5, R. Niet9, N. Nikitin31, T. Nikodem11, A. Nomerotski54, A. Novoselov34, A. Oblakowska- Mucha26, V. Obraztsov34, S. Oggero40, S. Ogilvy50, O. Okhrimenko43, R. Oldeman15,d, G. Onderwater61, M. Orlandea28, J.M. Otalora Goicochea2, P. Owen52, A. Oyanguren35, B.K. Pal58, A. Palano13,b, M. Palutan18, J. Panman37, A. Papanestis48,37, M. Pappagallo50, C. Parkes53, C.J. Parkinson52, G. Passaleva17, G.D. Patel51, M. Patel52, C. Patrignani19,i, C. Pavel- Nicorescu28, A. Pazos Alvarez36, A. Pearce53, A. Pellegrino40, G. Penso24,l, M. Pepe Altarelli37, S. Perazzini14,c, E. Perez Trigo36, A. Pérez-Calero Yzquierdo35, P. Perret5, M. Perrin-Terrin6, L. Pescatore44, E. Pesen62, G. Pessina20, K. Petridis52, A. Petrolini19,i, E. Picatoste Olloqui35, B. Pietrzyk4, T. Pilař47, D. Pinci24, S. Playfer49, M. Plo Casasus36, F. Polci8, G. Polok25, A. Poluektov47,33, E. Polycarpo2, A. Popov34, D. Popov10, B. Popovici28, C. Potterat35, A. Powell54, J. Prisciandaro38, A. Pritchard51, C. Prouve7, V. Pugatch43, A. Puig Navarro38, G. Punzi22,r, W. Qian4, B. Rachwal25, J.H. Rademacker45, B. Rakotomiaramanana38, M.S. Rangel2, I. Raniuk42, N. Rauschmayr37, G. Raven41, S. Redford54, S. Reichert53, M.M. Reid47, A.C. dos Reis1, S. Ricciardi48, A. Richards52, K. Rinnert51, V. Rives Molina35, D.A. Roa Romero5, P. Robbe7, D.A. Roberts57, A.B. Rodrigues1, E. Rodrigues53, P. Rodriguez Perez36, S. Roiser37, V. Romanovsky34, A. Romero Vidal36, M. Rotondo21, J. Rouvinet38, T. Ruf37, F. Ruffini22, H. Ruiz35, P. Ruiz Valls35, G. Sabatino24,k, J.J. Saborido Silva36, N. Sagidova29, P. Sail50, B. Saitta15,d, V. Salustino Guimaraes2, B. Sanmartin Sedes36, R. Santacesaria24, C. Santamarina Rios36, E. Santovetti23,k, M. Sapunov6, A. Sarti18, C. Satriano24,m, A. Satta23, M. 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Teodorescu28, F. Teubert37, C. Thomas54, E. Thomas37, J. van Tilburg11, V. Tisserand4, M. Tobin38, S. Tolk41, L. Tomassetti16,e, D. Tonelli37, S. Topp-Joergensen54, N. Torr54, E. Tournefier4,52, S. Tourneur38, M.T. Tran38, M. Tresch39, A. Tsaregorodtsev6, P. Tsopelas40, N. Tuning40,37, M. Ubeda Garcia37, A. Ukleja27, A. Ustyuzhanin52,p, U. Uwer11, V. Vagnoni14, G. Valenti14, A. Vallier7, R. Vazquez Gomez18, P. Vazquez Regueiro36, C. Vázquez Sierra36, S. Vecchi16, J.J. Velthuis45, M. Veltri17,g, G. Veneziano38, M. Vesterinen37, B. Viaud7, D. Vieira2, X. Vilasis-Cardona35,n, A. Vollhardt39, D. Volyanskyy10, D. Voong45, A. Vorobyev29, V. Vorobyev33, C. Voß60, H. Voss10, R. Waldi60, C. Wallace47, R. Wallace12, S. Wandernoth11, J. Wang58, D.R. Ward46, N.K. Watson44, A.D. Webber53, D. Websdale52, M. Whitehead47, J. Wicht37, J. Wiechczynski25, D. Wiedner11, L. Wiggers40, G. Wilkinson54, M.P. Williams47,48, M. Williams55, F.F. Wilson48, J. Wimberley57, J. Wishahi9, W. Wislicki27, M. Witek25, G. Wormser7, S.A. Wotton46, S. Wright46, S. Wu3, K. Wyllie37, Y. Xie49,37, Z. Xing58, Z. Yang3, X. Yuan3, O. Yushchenko34, M. Zangoli14, M. Zavertyaev10,a, F. Zhang3, L. Zhang58, W.C. Zhang12, Y. Zhang3, A. Zhelezov11, A. Zhokhov30, L. Zhong3, A. Zvyagin37. 1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil 2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil 3Center for High Energy Physics, Tsinghua University, Beijing, China 4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France 5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont- Ferrand, France 6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France 7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France 8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot, CNRS/IN2P3, Paris, France 9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany 10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany 11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany 12School of Physics, University College Dublin, Dublin, Ireland 13Sezione INFN di Bari, Bari, Italy 14Sezione INFN di Bologna, Bologna, Italy 15Sezione INFN di Cagliari, Cagliari, Italy 16Sezione INFN di Ferrara, Ferrara, Italy 17Sezione INFN di Firenze, Firenze, Italy 18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy 19Sezione INFN di Genova, Genova, Italy 20Sezione INFN di Milano Bicocca, Milano, Italy 21Sezione INFN di Padova, Padova, Italy 22Sezione INFN di Pisa, Pisa, Italy 23Sezione INFN di Roma Tor Vergata, Roma, Italy 24Sezione INFN di Roma La Sapienza, Roma, Italy 25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Kraków, Poland 26AGH - University of Science and Technology, Faculty of Physics and Applied Computer Science, Kraków, Poland 27National Center for Nuclear Research (NCBJ), Warsaw, Poland 28Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania 29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia 30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia 31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia 32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN), Moscow, Russia 33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State University, Novosibirsk, Russia 34Institute for High Energy Physics (IHEP), Protvino, Russia 35Universitat de Barcelona, Barcelona, Spain 36Universidad de Santiago de Compostela, Santiago de Compostela, Spain 37European Organization for Nuclear Research (CERN), Geneva, Switzerland 38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland 39Physik-Institut, Universität Zürich, Zürich, Switzerland 40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands 41Nikhef National Institute for Subatomic Physics and VU University Amsterdam, Amsterdam, The Netherlands 42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine 43Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine 44University of Birmingham, Birmingham, United Kingdom 45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom 46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom 47Department of Physics, University of Warwick, Coventry, United Kingdom 48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom 49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom 50School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom 51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom 52Imperial College London, London, United Kingdom 53School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom 54Department of Physics, University of Oxford, Oxford, United Kingdom 55Massachusetts Institute of Technology, Cambridge, MA, United States 56University of Cincinnati, Cincinnati, OH, United States 57University of Maryland, College Park, MD, United States 58Syracuse University, Syracuse, NY, United States 59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to 2 60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11 61KVI - University of Groningen, Groningen, The Netherlands, associated to 40 62Celal Bayar University, Manisa, Turkey, associated to 37 aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, Russia bUniversità di Bari, Bari, Italy cUniversità di Bologna, Bologna, Italy dUniversità di Cagliari, Cagliari, Italy eUniversità di Ferrara, Ferrara, Italy fUniversità di Firenze, Firenze, Italy gUniversità di Urbino, Urbino, Italy hUniversità di Modena e Reggio Emilia, Modena, Italy iUniversità di Genova, Genova, Italy jUniversità di Milano Bicocca, Milano, Italy kUniversità di Roma Tor Vergata, Roma, Italy lUniversità di Roma La Sapienza, Roma, Italy mUniversità della Basilicata, Potenza, Italy nLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain oHanoi University of Science, Hanoi, Viet Nam pInstitute of Physics and Technology, Moscow, Russia qUniversità di Padova, Padova, Italy rUniversità di Pisa, Pisa, Italy sScuola Normale Superiore, Pisa, Italy ## 1 Introduction Measurements of electroweak boson production in the forward region are sensitive to parton distribution functions (PDFs) at low Bjorken-$x$ which are not particularly well constrained by previous results [1]. The LHCb experiment has recently presented measurements of inclusive $\mathrm{W}$ and $\mathrm{Z}$ boson111Throughout this article $\mathrm{Z}$ includes both the $\mathrm{Z}$ and the virtual photon ($\gamma^{}$) contribution. production in the muon decay channels [2] and inclusive $\mathrm{Z}$ boson production in the electron [3] and the tau lepton [4] decay channels. This article presents a measurement of the inclusive Z+jet production cross-section in proton-proton collisions at LHCb. These interactions typically involve the collision of a sea quark or gluon with a valence quark, and measurements of $\mathrm{Z}$ boson production in association with jets are sensitive to the gluon content of the proton [5]. LHCb is sensitive to a region of phase space in which both the $\mathrm{Z}$ boson and the jet are produced in the forward region. Measurements at LHCb are therefore complementary to those at ATLAS [6] and CMS [7, 8]. Hence, measurements of the $\mathrm{Z}$+jet production cross-section at LHCb enable comparisons of different PDF predictions and their relative performances in this previously unprobed region of phase space. The $\mathrm{Z}$+jet production cross-section, in addition to being sensitive to the PDFs at low Bjorken-$x$, is influenced by higher order contributions in perturbative quantum chromodynamics (pQCD). Studies of the Drell-Yan process in the forward region are sensitive to multiple radiation of partons [9]. Measurements in the forward region have not been used to tune generators and, consequently, studies of Z+jet production in the forward region can be used to test the accuracy of different models. Theoretical predictions for the Z+jet process are available at $\mathcal{O}(\alpha_{s}^{2})$ [10, 11, 12, 13, 14, 15, 16, 17], where $\alpha_{s}$ is the strong-interaction coupling strength. Similar analyses at ATLAS [6] and CMS [7, 8] have shown reasonable agreement between data and such predictions. This measurement of the cross-section of $\mathrm{Z}\rightarrow\upmu^{+}\upmu^{-}$ events with jets in the final state uses data corresponding to an integrated luminosity of $1.0\,\rm{fb}^{-1}$ taken by the LHCb experiment in $\mathrm{p}\mathrm{p}$ collisions at a centre- of-mass energy of 7 TeV. The analysis is performed in a fiducial region that closely corresponds to the kinematic coverage of the LHCb detector. For the dimuon decay of the $\mathrm{Z}$ boson, this requirement is the same as that in Ref. [2]. Both final state muons are required to have a transverse momentum222Throughout this article natural units, where $c=1$, are used., $p_{\text{T}}^{\mu}$, greater than 20 GeV, and to have pseudorapidity333The pseudorapidity is defined to be $\eta\equiv-\ln(\tan(\theta/2))$, where the polar angle $\theta$ is measured with respect to the beam axis. The rapidity of a particle is defined to be $y\equiv 0.5\ln[(E+p_{z})/(E-p_{z})]$, where the particle has energy $E$ and momentum $p_{z}$ in the direction of the beam axis. in the range $2.0<\eta^{\mu}<4.5$. The invariant mass of the dimuon system is required to be in the range $60<M_{\mu\mu}<120$ GeV. Jets are reconstructed using the anti-$k_{\text{T}}$ algorithm [18] with distance parameter $R=0.5$, and are required to be in the fiducial region $2.0<\eta^{\text{jet}}<4.5$, and to be separated from decay muons of the $\mathrm{Z}$ boson by $\Delta r(\mu,\text{jet})>0.4$. This separation is defined such that $\Delta r^{2}\equiv\Delta\phi^{2}+\Delta\eta^{2}$, where $\Delta\phi$ is the difference in azimuthal angle and $\Delta\eta$ the difference in pseudorapidity between the muon and the jet directions. Results are presented for two thresholds of the jet transverse momentum: $p_{\text{T}}^{\text{jet}}>20$ GeV and $p_{\text{T}}^{\text{jet}}>10$ GeV. Both the total $\mathrm{Z}$+jet cross-section and the cross-section ratio of $\mathrm{Z}$ +jet production to inclusive $\mathrm{Z}$ production are reported. In addition, six differential cross-sections for $\mathrm{Z}$+jet production are presented as a function of the $\mathrm{Z}$ boson rapidity and transverse momentum, the pseudorapidity and transverse momentum of the leading444The leading jet is defined to be the highest transverse momentum jet in the fiducial region. jet, and the difference in azimuthal angle and in rapidity between the $\mathrm{Z}$ boson and this jet. These differential measurements are presented normalised to the total $\mathrm{Z}$+jet cross- section. The data are compared to predictions at $\mathcal{O}(\alpha_{s})$ and $\mathcal{O}(\alpha_{s}^{2})$ using different PDF parametrisations. The remainder of this article is organised as follows: Sect. 2 describes the LHCb detector and the simulation samples used; Sect. 3 provides an overview of jet reconstruction at LHCb; Sect. 4 describes the selection and reconstruction of candidates and the determination of the background level; Sect. 5 describes the cross-section measurement; the associated systematic uncertainties are discussed in Sect. 1; the results are presented in Sect. 7; Sect. 8 concludes the article. ## 2 LHCb detector and simulation The LHCb detector [19] is a single-arm forward spectrometer covering the pseudorapidity range $2<\eta<5$, designed for the study of particles containing $\mathrm{b}$ or $\mathrm{c}$ quarks. The detector includes a high- precision tracking system consisting of a silicon-strip vertex detector (the VELO) surrounding the $\mathrm{p}\mathrm{p}$ interaction region, a large-area silicon-strip detector (the TT) located upstream of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift tubes placed downstream. The combined tracking system provides a momentum measurement with relative uncertainty that varies from 0.4 % at 5$\mathrm{\,Ge\kern-1.00006ptV}$ to 0.6 % at 100$\mathrm{\,Ge\kern-1.00006ptV}$, and impact parameter resolution of 20$\,\upmu\rm m$ for tracks with large transverse momentum. Charged hadrons are identified using two ring-imaging Cherenkov detectors [20]. Photon, electron and hadron candidates are identified by a calorimeter system consisting of scintillating-pad (SPD) and preshower detectors, an electromagnetic calorimeter and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers [21]. The trigger [22] consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction. To avoid the possibility that a few events with high occupancy dominate the CPU time of the software trigger, a set of global event cuts (GEC) is applied on the hit multiplicities of most subdetectors used in the pattern recognition algorithms. The dominant GEC in the trigger selection used in this analysis is the requirement that the hit multiplicity in the SPD, $n_{\text{SPD}}$, is less than 600. In the simulation, $\mathrm{p}\mathrm{p}$ collisions are generated using Pythia 6.4 [23] with a specific LHCb configuration [24], with the CTEQ6ll [25] parametrisation for the PDFs. Decays of hadronic particles are described by EvtGen [26], in which final state radiation is generated using Photos [27]. The interaction of the generated particles with the detector and its response are implemented using the Geant4 toolkit [28, Agostinelli:2002hh] as described in Ref. [30]. The main simulation sample used in this analysis is an $\mathcal{O}(\alpha_{s})$ prediction of the $\mathrm{Z}$+jet process, with the $\mathrm{Z}$ boson decaying to two muons. In addition, inclusive $\mathrm{Z}\rightarrow\upmu^{+}\upmu^{-}$ events are generated at leading order in pQCD, where all jets are produced by the parton shower, in order to study various stages of the analysis with an independent simulation sample. This simulation sample is hereafter referred to as the inclusive $\mathrm{Z}$ sample. ## 3 Jet reconstruction Inputs for jet reconstruction are selected using a particle flow algorithm. In order to benefit from the good momentum resolution of the LHCb tracking system, reconstructed tracks serve as charged particle inputs to the jet reconstruction. Tracks corresponding to the decay muons of the Z boson are excluded. The neutral particle inputs are derived from the energy deposits in the electromagnetic and hadronic calorimeters. If the deposits are matched to tracks, the expected calorimeter energies associated with the tracks are subtracted. The expected calorimeter energy is determined based on the likelihood that the track is associated with a charged hadron, a muon, or an electron, using information from the particle identification systems. If a significant energy deposit remains after the subtraction, the energy is associated with a neutral particle detected in the calorimeter. The use of the different particle identification hypotheses has negligible impact on the results presented in this article, since the jets studied here are mostly inititated by light quarks and gluons. Finally, in order to reduce the contribution from multiple proton-proton interactions, charged particles from tracks reconstructed within the VELO are not considered if they are associated to a different primary vertex to that of the $\mathrm{Z}$ boson. The charged particles and energy clusters are reconstructed into jets using the anti-$k_{\text{T}}$ algorithm [18], with distance parameter $R=0.5$, as implemented in Fastjet [31]. The same jet reconstruction algorithm is run on simulated Z+jet events. The anti-$k_{\text{T}}$ algorithm is also applied to these simulated events at the hadron-level using information that is available before the detector simulation is performed. The inputs for these ‘true’ jets are all stable final state particles, including neutrinos, from the same proton-proton interaction that produced the $\mathrm{Z}$ boson, that are not products of the $\mathrm{Z}$ boson decay. The transverse momentum of a reconstructed jet is scaled so that it gives an unbiased estimate of the true jet transverse momentum. The scaling factor, typically between $0.9$ and $1.1$, is determined from simulation and depends on the jet pseudorapidity and transverse momentum, the fraction of the jet transverse momentum measured with the tracking systems, and the number of proton-proton interactions in the event. The energy resolution of reconstructed jets varies with the jet energy. The half width at half maximum for the distribution of $p_{\text{T}}^{\text{reco}}/p_{\text{T}}^{\text{true}}$ is typically 10-15 % for jets with transverse momenta between 10 and 100 GeV. In simulation, 90 % of jets with at least 10 GeV transverse momentum are reconstructed with $\Delta r<0.13$ in $\eta-\phi$ space with respect to the true jet. At the $p_{\text{T}}$ threshold of 20 GeV the corresponding radius is 0.08. In order to reduce the number of spurious fake jets, and to select jets from the same interaction as that of the $\mathrm{Z}$ boson with a good estimate of the jet energy, additional jet identification requirements are imposed. Jets are required to contain at least two particles matched to the same primary vertex, to contain at least one track with $p_{\text{T}}>1.8$ GeV, and to contain no single particle with more than $75\,\%$ of the jet’s transverse momentum. ## 4 Selection and event reconstruction The $\mathrm{Z}\rightarrow\upmu^{+}\upmu^{-}$ selection follows that described in Ref. [2]. The events are initially selected by a trigger that requires the presence of at least one muon candidate with $p_{\text{T}}^{\upmu}>10~{}\text{GeV}$. Selected events are required to contain two reconstructed muons with $p_{\text{T}}^{\mu}>20~{}\text{GeV}$ and $2.0<\eta^{\mu}<4.5$, and one of these muons is required to have passed the trigger. The invariant mass of the dimuon pair must be in the range $60<M_{\mu\mu}<120\text{ GeV}$. The relative uncertainty on the measured momentum of each muon is required to be less than $10\,\%$ and the $\chi^{2}$ probability for the associated track larger than $0.1\,\%$. In total, $53\,182$ $\mathrm{Z}\rightarrow\upmu^{+}\upmu^{-}$ candidates are selected. A reconstructed jet with pseudorapidity in the range $2.0<\eta^{\text{jet}}<4.5$ is also required in the selection. The separation between each of the decay muons of the reconstructed $\mathrm{Z}$ boson and the jet is required to be $\Delta r>0.4$. Jets are reconstructed with transverse momentum above 7.5 GeV. Of the selected $\mathrm{Z}\rightarrow\upmu^{+}\upmu^{-}$ candidates, $4\,118$ contain a reconstructed jet with transverse momentum above 20 GeV, and $10\,576$ contain a jet with transverse momentum above 10 GeV. ### 4.1 Background The background contribution from random combinations of muons can come from semileptonic heavy flavour decays, $\mathrm{W}$ boson decays, or mesons that have decayed whilst passing through the detector and have been reconstructed as muons, or hadrons that have passed through the calorimeters without interacting. This background is determined from the number of events containing two muons of the same charge that would otherwise pass the selection requirements. No significant difference is found using events where both muons have a positive charge or events where both muons have a negative charge. This background source contributes $5\pm 2$ events for the 20 GeV jet transverse momentum threshold and $16\pm 4$ for the 10 GeV threshold, where the uncertainties are statistical. The production of diboson pairs and heavy flavour decays of $\mathrm{Z}$ bosons, where the heavy flavour decay products decay to muons, are found to contribute negligible background levels to this analysis. Decays from the $\mathrm{Z}\rightarrow\uptau^{+}\uptau^{-}$ process where both tau leptons decay to muons and neutrinos are another potential background source. This background is determined from simulation, and contributes $7\pm 3$ events for the 20 GeV transverse momentum threshold, and $12\pm 3$ events for the 10 GeV threshold, where the uncertainties are statistical. The background contribution from top quark pair production is also considered, where the top quark decay products include high transverse momentum muons. This background is determined from next-to-leading order (NLO) simulation to be $5\pm 2$ events, where the uncertainties are statistical. This background is largely independent of the 10 and 20 GeV jet transverse momentum thresholds as the top quark decays are associated with very high transverse momentum jets. The background associated with events where a jet above a threshold is reconstructed, despite there being no true jet above that threshold, is treated as a migration. This background is corrected for by unfolding the transverse momentum distribution (see Sect. 5). The total background contribution for the 20 GeV jet transverse momentum threshold is $17\pm 4$ events, and the contribution for the 10 GeV threshold is $33\pm 6$ events. This corresponds to a sample purity $\rho\equiv S/(S+B)$, where $S$ is the number of signal events and $B$ is the number of background events, of $(99.6\pm 0.1)\,\%$ for the 20 GeV threshold and $(99.7\pm 0.1)\,\%$ for the 10 GeV threshold. These purities are consistent with that found in the inclusive $\mathrm{Z}$ boson analysis [2]. The purity shows no significant dependence on other kinematic variables of interest. Since the purity is high and has little variation with the transverse momentum threshold it is treated as constant for this analysis. ### 4.2 Z detection and reconstruction efficiencies Following Ref. [3], the total Z boson detection efficiency is factorised into four separate components as $\varepsilon_{\mathrm{Z}}=\varepsilon_{\text{GEC}}\,\varepsilon_{\text{trigger}}\,\varepsilon_{\text{track}}\,\varepsilon_{\text{ID}}$, where the $\varepsilon_{\text{X}}$ factors correspond to the efficiency associated with the GEC, the trigger requirements, the muon track reconstruction and the muon identification, respectively. The GEC, applied in the trigger to stop very large events dominating processing time, cause signal events to be rejected. The associated inefficiency is obtained using the same method described in Ref. [3], where an alternative dimuon trigger requirement555This trigger route is not used elsewhere in this analysis as it has a lower efficiency than the single muon trigger. is used to determine the number of events that are rejected with $600<n_{\text{SPD}}<900$. The small number of events with $n_{\text{SPD}}>900$ is found by extrapolation using a fit with a gamma function. This approach is applied to determine the efficiency as a function of the number of reconstructed primary vertices and the number of jets reconstructed in the event. The average efficiency is $91\,\%$. The trigger efficiency for the single muon trigger is found using the same tag-and-probe method used in Ref. [2]. Events in which at least one muon from the $\mathrm{Z}$ boson decay passed the trigger are selected. The fraction of events where the other muon from the $\mathrm{Z}$ boson decay fired the trigger determines the muon trigger efficiency. This efficiency is found to be independent of the number of jets reconstructed in the event and is determined as a function of the muon pseudorapidity. The efficiencies for the two muons are then combined to determine the efficiency with which at least one of the two muons in the decay passes the trigger, $\varepsilon_{\text{trigger}}(\eta_{1},\eta_{2})=\varepsilon(\eta_{1})+\varepsilon(\eta_{2})-\varepsilon(\eta_{1})\varepsilon(\eta_{2})$. This combination assumes that the probability that one muon fires the trigger is independent of whether the other muon fired the trigger. This is confirmed with simulated data. The average of this combined efficiency is approximately $96\,\%$. The muon track reconstruction efficiency is determined using the tag-and-probe method, described in Refs. [2] and [32]. Well reconstructed tracks in the muon stations are linked to hits in the TT detector in events containing one other high-purity muon candidate. The invariant mass of this dimuon pair is required to lie within 10 GeV of the $\mathrm{Z}$ boson mass. The efficiency is determined as the fraction of events where the muon-station track is geometrically matched to a track in the tracking system that passes the track quality requirements. This efficiency depends on the muon pseudorapidity and the number of jets measured in the event, with an average efficiency of approximately $90\,\%$ for each muon. The muon identification efficiency is determined using the method described in Ref. [2]. Events containing two tracks with an invariant mass within 5 GeV of the $\mathrm{Z}$ boson mass are selected. One of the tracks is required to be identified as a muon. The fraction of events in which the other track is also identified as a muon defines the muon identification efficiency. This efficiency shows no dependence on the number of jets in the event and is found as a function of the muon pseudorapidity. The average muon identification efficiency is $99\,\%$. ### 4.3 Jet detection and reconstruction efficiencies The jet detection efficiency is determined from simulation and is defined as the efficiency for a jet to be reconstructed with transverse momentum greater than 7.5 GeV, satisfying the jet selection criteria, given that a true jet is reconstructed in the same event. This efficiency is determined as a function of the true jet transverse momentum and shows little variation in the central region of the LHCb detector. Reweighting the simulation to have the same jet pseudorapidity distribution as data has a negligible effect on the efficiency. This efficiency is about $75\,\%$ for jets with transverse momentum of about 10 GeV, but rises to about $96\,\%$ for high transverse momentum jets, as shown in Fig. 1. The drop in efficiency at low transverse momentum is mainly due to the jet identification requirements having a larger effect in this region. Figure 1: Jet identification efficiency as a function of the true jet $p_{\text{T}}$. The uncertainties shown are statistical. The zero on the vertical axis is suppressed. ## 5 Cross-section measurement Events are selected with reconstructed jet transverse momentum above 7.5 GeV. Migrations in the jet transverse momentum distribution are corrected for by unfolding the distribution using the method of D’Agostini [33], as implemented in RooUnfold [34]. Two iterations are chosen as this gives the best agreement between the unfolded distribution and the true distribution when the inclusive $\mathrm{Z}$ simulation sample is unfolded, using the same number of events in the inclusive $\mathrm{Z}$ simulation sample as are present in data. As a cross-check, the result is compared with the SVD unfolding method [35]. In these studies underflow bins are included in the unfolded distributions to account for the small number of events that lie below threshold after the unfolding procedure. Each event is assigned a weight for the Z boson reconstruction, detection and selection efficiency, $\varepsilon_{\mathrm{Z}}$. This enables the determination of the fraction of events within each bin of the unfolded jet transverse momentum, $N(p_{\text{T}}^{\text{unf}})$, corrected for the Z detection efficiency $N(p_{\text{T}}^{\text{unf}})=\sum_{\text{events}}\frac{M(p_{\text{T}}^{\text{unf}},p_{\text{T}}^{\text{reco}})}{\varepsilon_{\mathrm{Z}}},$ (1) where $M(p_{\text{T}}^{\text{unf}},p_{\text{T}}^{\text{reco}})$ is the element of the matrix, obtained from the unfolding, that gives the probability that an event containing a jet with reconstructed transverse momentum in the bin $p_{\text{T}}^{\text{reco}}$ contains a true jet with transverse momentum in the bin $p_{\text{T}}^{\text{unf}}$. For the differential distributions the matrix is determined for events restricted to the relevant bin in that differential distribution. This unfolding includes the correction for the background where a jet is reconstructed with $p_{\text{T}}$ above the threshold despite there being no true jet above that threshold in the event. In order to measure the cross-section, a correction is applied to account for the jet reconstruction efficiency, $\varepsilon_{\text{jet}}$. The correction is performed for each bin in each differential distribution separately. In differential measurements an additional factor $A_{\text{mig}}$ is applied to account for migration between different bins (for example, in the jet pseudorapidity distribution). These corrections are typically small ($2-3\,\%$) and are taken from simulation. The cross-section is determined by dividing the resulting event yield, corrected for migrations and the reconstruction acceptance, by the integrated luminosity, $\int\mathcal{L}\;\text{d}t$, as follows $\sigma=\frac{\rho}{\int\mathcal{L}\;\text{d}t}\sum_{p_{\text{T}}^{\text{unf}}>p_{\text{T}}^{\text{thr}}}\frac{A_{\text{mig}}}{\varepsilon_{\text{jet}}}N(p_{\text{T}}^{\text{unf}}),$ (2) where $p_{\text{T}}^{\text{thr}}$ is the relevant threshold, 20 or 10 GeV, and the sum is over the bins of the unfolded transverse momentum above this threshold. The purity of the sample, $\rho$, accounts for the presence of background as discussed in Sect. 4.1. The luminosity is determined as described in Ref. [36]. Measurements of the total $\mathrm{Z}$+jet cross-section are quoted at the Born level in QED; the correction factors for final state radiation (FSR) of the muons are calculated with Herwig++ [37]. Differential distributions are compared to theoretical predictions that include the effects of FSR, so they are not corrected for FSR from the muons. The differential distributions are also normalised to the total Z+jet cross-section above the relevant transverse momentum threshold, without corrections for FSR, so that their integral is unity. ## 6 Systematic uncertainties The different contributions to the systematic uncertainty are discussed below and are summarised in Table 1. Table 1: The relative uncertainty arising from each source of possible systematic uncertainties considered for the Z+jet cross-section for $p_{\text{T}}^{\text{jet}}>20\text{ GeV}$. The relative uncertainties are similar for the 10 GeV threshold. The contributions from the different sources are combined in quadrature. Source \| Relative uncertainty (%) ---\|--- Unfolding \| 1.5 Z detection and reconstruction \| 3.5 Jet-energy scale, resolution and reconstruction \| 7.8 Final state radiation \| 0.2 Total excluding luminosity \| 8.6 Luminosity \| 3.5 Two contributions associated with the unfolding are considered. The difference in the unfolded result between the SVD [35] and the D’Agostini [33] methods is assigned as an uncertainty. In addition, the unfolding process is carried out on the inclusive $\mathrm{Z}$ sample described in Sect. 2 (which is an independent simulation sample to that used to perform the unfolding), and the difference between the unfolded distribution and the true distribution is assigned as a systematic uncertainty. The number of events considered in the independent sample is the same as the number in data. The differences between the results found using the D’Agostini method with one iteration and those found using two iterations are less than the uncertainties assigned from the unfolding method. The systematic uncertainties for the muon identification and trigger efficiencies are obtained as in Ref. [2], where the statistical uncertainties on the tag-and-probe method are used as systematic uncertainties on the efficiency. The systematic uncertainty associated with the GEC efficiency is considered as in Ref. [3]. A variation in the fit model is applied and the change in efficiency is considered as a systematic uncertainty. In addition, the statistical uncertainty in the efficiency is assigned as a systematic uncertainty. The systematic uncertainty associated with the track reconstruction efficiency has two contributions. The uncertainty associated with the statistical precision of the efficiency determination is treated as in Ref. [2]. By comparing the tag-and-probe method applied to simulation with the true efficiency, the method is found to be accurate to $0.3\,\%$ for each muon. This sets the systematic uncertainty associated with the tag-and-probe method used to find the muon track reconstruction efficiency. Figure 2: Comparison between data (black points) and simulation (red line) in the $p_{\text{T}}^{\text{jet}}/p_{\text{T}}^{\mathrm{Z}}$ distribution for selected $\mathrm{Z}$+1-jet events where the $\mathrm{Z}$ boson and the jet are emitted azimuthally opposed. The uncertainties shown are statistical. The systematic uncertainty associated with the jet identification requirements is determined by tightening these requirements and comparing the fraction of events rejected in data and simulation. These are found to agree at the level of about $3\,\%$. This is therefore used as a systematic uncertainty. The efficiency is cross-checked on the independent inclusive $\mathrm{Z}$ sample, and the difference is taken as an additional systematic uncertainty. The efficiency associated with the jet reconstruction, neglecting the jet identification requirements, is found to be about $98.5\,\%$ at low transverse momentum, so an additional $1.5\,\%$ uncertainty is assigned to this reconstruction efficiency component of $\varepsilon_{\text{jet}}$ at low momentum. The jet-energy scale and resolution show no dependence on the separation of the Z and the jet in the azimuthal angle. The jet-energy scale and resolution uncertainties associated with how well the detector response to jets is modelled in simulated data are therefore determined by selecting $\mathrm{Z}$+1-jet events that are azimuthally opposed. In these events the $\mathrm{Z}$ boson and jet transverse momenta are expected to balance. Hence, the $\mathrm{Z}$ boson transverse momentum can be used as a proxy for the true jet transverse momentum. The $p_{\text{T}}^{\text{jet}}/p_{\text{T}}^{\mathrm{Z}}$ distribution in the selected events is shown in Fig. 2, and is also considered as a function of the jet pseudorapidity and transverse momentum. The mean is found to agree between data and simulation at the level of about $3\,\%$, consistent within the statistical precision. The width is consistent between data and simulation, and the resolution in simulation can be smeared at the level of about $10\,\%$ whilst maintaining this agreement. Based on these comparisons, systematic uncertainties to account for the reliability of the modelling are assigned to the jet-energy scale and resolution. In addition, a systematic uncertainty is assigned based on the difference in the jet-energy scale for gluon- and quark-initiated jets, and for the method used to correct the jet- energy scale. This contributes an additional $2\,\%$ systematic uncertainty on the jet-energy scale. These uncertainties are then propagated into the cross- sections and distributions measured. The contribution from the uncertainty on the jet-energy scale is the dominant uncertainty in most bins analysed. The systematic uncertainty on the FSR correction applied to the total cross- section is determined by comparing the correction taken from Herwig++ [37] and from Pythia [23] interfaced with Photos [27], as found in Ref. [2]. The difference in correction is at the level of $0.2\,\%$. The luminosity uncertainty is estimated to be $3.5\,\%$, as detailed in Ref. [36]. ## 7 Results The $\mathrm{Z}$+jet cross-section and the cross-section ratio $\sigma(\text{Z+jet})/\sigma(\text{Z})$ are measured at the Born level. For the $p_{\text{T}}^{\text{jet}}>20\text{ GeV}$ threshold the results are $\displaystyle\sigma(\text{Z+jet})$$\displaystyle\text{ }=\text{ }$ \| $\displaystyle 6.3\pm 0.1\,(\text{{stat.}})\pm 0.5\,(\text{{syst.}})\pm 0.2\,(\text{{lumi.}})\text{ pb,}$ ---\|--- $\displaystyle\frac{\sigma(\text{Z+jet})}{\sigma(\text{Z})}$$\displaystyle\text{ }=\text{ }$ \| $\displaystyle 0.083\pm 0.001\,(\text{{stat.}})\pm 0.007\,(\text{{syst.}}),$ and for the $\displaystyle p_{\text{T}}^{\text{jet}}>10\text{ GeV}$ threshold, $\displaystyle\sigma(\text{Z+jet})$$\displaystyle\text{ }=\text{ }$ \| $\displaystyle 16.0\pm 0.2\,(\text{{stat.}})\pm 1.2\,(\text{{syst.}})\pm 0.6\,(\text{{lumi.}})\text{ pb,}$ ---\|--- $\displaystyle\frac{\sigma(\text{Z+jet})}{\sigma(\text{Z})}$$\displaystyle\text{ }=\text{ }$ \| $\displaystyle 0.209\pm 0.002\,(\text{{stat.}})\pm 0.015\,(\text{{syst.}}),$ where the first uncertainty is statistical, the second is systematic and the third is the uncertainty due to the luminosity determination. The measured cross-sections are compared to theoretical predictions at $\mathcal{O}(\alpha_{s}^{2})$ calculated using Powheg[15, 38, 39, 40]. The parton shower development and hadronisation are simulated using Pythia 6.4 [23], with the Perugia 0 tune [41]. Jets are created out of all stable particles in the final state that are not produced by the decay of the $\mathrm{Z}$ boson. These predictions are computed with the renormalisation scale and factorisation scales set to the nominal value of the vector boson transverse momentum. The theoretical predictions are computed for three different NLO PDF parametrisations: MSTW08 [42], CTEQ10 [43] and NNPDF 2.3 [44]. For the differential distributions, the CTEQ10 and NNPDF 2.3 results are calculated at $\mathcal{O}(\alpha_{s}^{2})$. Results using the MSTW08 parametrisation are calculated at $\mathcal{O}(\alpha_{s})$ and $\mathcal{O}(\alpha_{s}^{2})$. For the ratio $\sigma_{\mathrm{Z}+\text{jet}}/\sigma_{\mathrm{Z}}$, the Z+jet cross-section is computed at $\mathcal{O}(\alpha_{s}^{2})$ and the Z cross- section at $\mathcal{O}(\alpha_{s})$ for the MSTW08, CTEQ10 and NNPDF 2.3 PDF parametrisations. To see the effect of higher orders in pQCD on the Z+jet cross-section, theoretical predictions are also computed by taking the ratio between the Z and Z+jet cross-sections at $\mathcal{O}(\alpha_{s})$, with the PDFs determined from the MSTW08 NLO parametrisation. In addition, the $\mathrm{Z}$+jet cross-section is computed using Fewz [13] at $\mathcal{O}(\alpha_{s}^{2})$, with the MSTW08 NLO PDF parametrisation. The cross-section for inclusive $\mathrm{Z}$ boson production is calculated using FEWZ at $\mathcal{O}(\alpha_{s})$, with the same PDF parametrisation. This theoretical prediction neglects effects from hadronisation and the underlying event, and so comparisons with the results and the other predictions are indicative of the size of these effects. For these calculations the renormalisation scale and factorisation scales are set to the nominal value of the vector boson mass. Uncertainties on all predictions are calculated by repeating the calculations with the renormalisation and factorisation scales simultaneously varied by a factor of two about their nominal values. The spread in predictions from the different PDF parametrisations is indicative of the PDF uncertainty. The cross-section ratios are compared in Fig. 3 to the Standard Model theoretical predictions discussed above. The results for the differential cross-sections, uncorrected for final state radiation from the muons, are presented in Figs. 4–9. For all cases reasonable agreement is seen between the Standard Model calculations and the data. The $\mathcal{O}(\alpha_{s}^{2})$ predictions tend to give better agreement with data than the $\mathcal{O}(\alpha_{s})$ prediction. This is most noticeably seen in the $\mathrm{Z}$ boson transverse momentum distribution, shown in Fig. 7. For high values of the boson transverse momentum, the $\mathcal{O}(\alpha_{s}^{2})$ predictions have a slope compatible with that in data, whereas the $\mathcal{O}(\alpha_{s})$ prediction is steeper than data. The $\mathcal{O}(\alpha_{s}^{2})$ predictions also match the data better for the $\Delta\phi$ distribution, as shown in Fig. 8. The $\mathcal{O}(\alpha_{s})$ prediction overestimates the number of events where the $\mathrm{Z}$ boson and jet are azimuthally opposed. Higher orders in pQCD are needed to simulate the production of $\mathrm{Z}$ bosons and jets that are not produced back-to-back as the parton shower tends to produce partons collinear with the parton produced in the hard interaction. Whilst the different PDF parametrisations studied agree with the data, there are hints of tension between the PDF sets in the $\Delta y$ distribution, shown in Fig. 9. Figure 3: Ratio of the Z+jet cross-section to the inclusive cross-section, for (top) $p_{\text{T}}^{\text{jet}}\nobreak>\nobreak 20\text{ GeV}$ and (bottom) $p_{\text{T}}^{\text{jet}}>10\text{ GeV}$. The bands show the LHCb measurement (with the inner band showing the statistical uncertainty and the outer band showing the total uncertainty). The points correspond to different theoretical predictions with the error bars indicating their uncertainties as described in the main text. These results are corrected for FSR from the final state muons from the $\mathrm{Z}$ boson decay. Figure 4: Cross-section for Z+jet production, differential in the leading jet $p_{\text{T}}$, for $p_{\text{T}}^{\text{jet}}>10\text{ GeV}$. The bands show the LHCb measurement (with the inner band showing the statistical uncertainty and the outer band showing the total uncertainty). The points correspond to different theoretical predictions with the error bars indicating their uncertainties as described in the main text. Predictions are displaced horizontally for presentation. These results are not corrected for FSR from the final state muons from the $\mathrm{Z}$ boson decay. Figure 5: Cross-section for Z+jet production, differential in the leading jet pseudorapidity, for (left) $p_{\text{T}}^{\text{jet}}\nobreak>\nobreak 20\text{ GeV}$ and (right) $p_{\text{T}}^{\text{jet}}>10\text{ GeV}$. The bands show the LHCb measurement (with the inner band showing the statistical uncertainty and the outer band showing the total uncertainty). Superimposed are predictions as described in Fig. 4. Figure 6: Cross-section for Z+jet production, differential in the $\mathrm{Z}$ boson rapidity, $y^{\mathrm{Z}}$, for (left) $p_{\text{T}}^{\text{jet}}\nobreak>\nobreak 20\text{ GeV}$ and (right) $p_{\text{T}}^{\text{jet}}>10\text{ GeV}$. The bands show the LHCb measurement (with the inner band showing the statistical uncertainty and the outer band showing the total uncertainty). Superimposed are predictions as described in Fig. 4. Figure 7: Cross-section for Z+jet production, differential in the $\mathrm{Z}$ boson transverse momentum, for (left) $p_{\text{T}}^{\text{jet}}\nobreak>\nobreak 20\text{ GeV}$ and (right) $p_{\text{T}}^{\text{jet}}>10\text{ GeV}$. The bands show the LHCb measurement (with the inner band showing the statistical uncertainty and the outer band showing the total uncertainty). Superimposed are predictions as described in Fig. 4. Figure 8: Cross-section for Z+jet production, differential in the difference in $\phi$ between the $\mathrm{Z}$ boson and the leading jet, for (left) $p_{\text{T}}^{\text{jet}}\nobreak>\nobreak 20\text{ GeV}$ and (right) $p_{\text{T}}^{\text{jet}}>10\text{ GeV}$. The bands show the LHCb measurement (with the inner band showing the statistical uncertainty and the outer band showing the total uncertainty). Superimposed are predictions as described in Fig. 4. Figure 9: Cross-section for Z+jet production, differential in the difference in rapidity between the $\mathrm{Z}$ boson and the leading jet, for (left) $p_{\text{T}}^{\text{jet}}\nobreak>\nobreak 20\text{ GeV}$ and (right) $p_{\text{T}}^{\text{jet}}>10\text{ GeV}$. The bands show the LHCb measurement (with the inner band showing the statistical uncertainty and the outer band showing the total uncertainty). Superimposed are predictions as described in Fig. 4. ## 8 Summary A measurement of the $\mathrm{p}\mathrm{p}\rightarrow\mathrm{Z}(\rightarrow\upmu^{+}\upmu^{-})+\text{jet}$ production cross-section at $\sqrt{s}=7$ TeV is presented, using a data sample corresponding to an integrated luminosity of $1.0\,$fb-1 recorded by the LHCb experiment. The measurement is performed within the kinematic acceptance, $p_{\text{T}}^{\mu}>20$ GeV, $2.0<\eta^{\mu}<4.5$, $60<M_{\mu\mu}<120$ GeV, $2.0<\eta^{\text{jet}}<4.5$ and $\Delta r(\mu,\text{jet})\nolinebreak>\nolinebreak 0.4$. The cross-sections are determined for jets with transverse momenta exceeding two thresholds, 20 and 10 GeV. The differential cross-sections are also measured as a function of various variables describing the Z boson kinematic properties, the jet kinematic properties, and the correlations between them. The measured cross- sections show reasonable agreement with expectations from $\mathcal{O}(\alpha_{s}^{2})$ calculations, for all the PDF parametrisations studied. Predictions at $\mathcal{O}(\alpha_{s}^{2})$ show better agreement with the $p_{\text{T}}$ and $\Delta\phi$ distributions, which are sensitive to higher order effects, than predictions at $\mathcal{O}(\alpha_{s})$. ## Acknowledgements We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We also acknowledge the support received from the ERC under FP7. The Tier1 computing centres are supported by IN2P3 (France), KIT and BMBF (Germany), INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom). We are thankful for the computing resources put at our disposal by Yandex LLC (Russia), as well as to the communities behind the multiple open source software packages that we depend on. ## References * [1] R. Thorne, A. Martin, W. Stirling, and G. 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Amhis,\n L. Anderlini, J. Anderson, R. Andreassen, M. Andreotti, J.E. Andrews, R.B.\n Appleby, O. Aquines Gutierrez, F. Archilli, A. Artamonov, M. Artuso, E.\n Aslanides, G. Auriemma, M. Baalouch, S. Bachmann, J.J. Back, A. Badalov, C.\n Baesso, V. Balagura, W. Baldini, R.J. Barlow, C. Barschel, S. Barsuk, W.\n Barter, V. Batozskaya, Th. Bauer, A. Bay, J. Beddow, F. Bedeschi, I. Bediaga,\n S. Belogurov, K. Belous, I. Belyaev, E. Ben-Haim, G. Bencivenni, S. Benson,\n J. Benton, A. Berezhnoy, R. Bernet, M.-O. Bettler, M. van Beuzekom, A. Bien,\n S. Bifani, T. Bird, A. Bizzeti, P.M. Bj{\\o}rnstad, T. Blake, F. Blanc, J.\n Blouw, S. Blusk, V. Bocci, A. Bondar, N. Bondar, W. Bonivento, S. Borghi, A.\n Borgia, T.J.V. Bowcock, E. Bowen, C. Bozzi, T. Brambach, J. van den Brand, J.\n Bressieux, D. Brett, M. Britsch, T. Britton, N.H. Brook, H. Brown, A.\n Bursche, G. Busetto, J. Buytaert, S. Cadeddu, R. Calabrese, O. Callot, M.\n Calvi, M. Calvo Gomez, A. Camboni, P. Campana, D. Campora Perez, A. Carbone,\n G. Carboni, R. Cardinale, A. Cardini, H. Carranza-Mejia, L. Carson, K.\n Carvalho Akiba, G. Casse, L. Castillo Garcia, M. Cattaneo, Ch. Cauet, R.\n Cenci, M. Charles, Ph. Charpentier, S.-F. Cheung, N. Chiapolini, M.\n Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic,\n H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins,\n A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti,\n B. Couturier, G.A. Cowan, D.C. Craik, M. Cruz Torres, S. Cunliffe, R. Currie,\n C. D'Ambrosio, J. Dalseno, P. David, P.N.Y. David, A. Davis, I. De Bonis, K.\n De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva,\n P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D.\n Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra, M. Dogaru, S.\n Donleavy, F. Dordei, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F. Dupertuis,\n P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U. Egede, V.\n Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U. Eitschberger, R.\n Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. Falabella, C. F\\\"arber, C.\n Farinelli, S. Farry, D. Ferguson, V. Fernandez Albor, F. Ferreira Rodrigues,\n M. Ferro-Luzzi, S. Filippov, M. Fiore, M. Fiorini, C. Fitzpatrick, M.\n Fontana, F. Fontanelli, R. Forty, O. Francisco, M. Frank, C. Frei, M.\n Frosini, E. Furfaro, A. Gallas Torreira, D. Galli, M. Gandelman, P. Gandini,\n Y. Gao, J. Garofoli, P. Garosi, J. Garra Tico, L. Garrido, C. Gaspar, R.\n Gauld, E. Gersabeck, M. Gersabeck, T. Gershon, Ph. Ghez, V. Gibson, L.\n Giubega, V.V. Gligorov, C. G\\\"obel, D. Golubkov, A. Golutvin, A. Gomes, P.\n Gorbounov, H. Gordon, M. Grabalosa G\\'andara, R. Graciani Diaz, L.A. Granado\n Cardoso, E. Graug\\'es, G. Graziani, A. Grecu, E. Greening, S. Gregson, P.\n Griffith, L. Grillo, O. Gr\\\"unberg, B. Gui, E. Gushchin, Yu. Guz, T. Gys, C.\n Hadjivasiliou, G. Haefeli, C. Haen, T.W. Hafkenscheid, S.C. Haines, S. Hall,\n B. Hamilton, T. Hampson, S. Hansmann-Menzemer, N. Harnew, S.T. Harnew, J.\n Harrison, T. Hartmann, J. He, T. Head, V. Heijne, K. Hennessy, P. Henrard,\n J.A. Hernando Morata, E. van Herwijnen, M. He\\ss, A. Hicheur, E. Hicks, D.\n Hill, M. Hoballah, C. Hombach, W. Hulsbergen, P. Hunt, T. Huse, N. Hussain,\n D. Hutchcroft, D. Hynds, V. Iakovenko, M. Idzik, P. Ilten, R. Jacobsson, A.\n Jaeger, E. Jans, P. Jaton, A. Jawahery, F. Jing, M. John, D. Johnson, C.R.\n Jones, C. Joram, B. Jost, M. Kaballo, S. Kandybei, W. Kanso, M. Karacson,\n T.M. Karbach, I.R. Kenyon, T. Ketel, B. Khanji, S. Klaver, O. Kochebina, I.\n Komarov, R.F. Koopman, P. Koppenburg, M. Korolev, A. Kozlinskiy, L. Kravchuk,\n K. Kreplin, M. Kreps, G. Krocker, P. Krokovny, F. Kruse, M. Kucharczyk, V.\n Kudryavtsev, K. Kurek, T. Kvaratskheliya, V.N. La Thi, D. Lacarrere, G.\n Lafferty, A. Lai, D. Lambert, R.W. Lambert, E. Lanciotti, G. Lanfranchi, C.\n Langenbruch, T. Latham, C. 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1310.8343	# Matter-wave interference with particles selected from a molecular library with masses exceeding 10 000 amu Sandra Eibenberger University of Vienna, Faculty of Physics, VCQ, QuNaBioS, Boltzmanngasse 5, 1090 Vienna (Austria) Stefan Gerlich University of Vienna, Faculty of Physics, VCQ, QuNaBioS, Boltzmanngasse 5, 1090 Vienna (Austria) Markus Arndt [email protected] University of Vienna, Faculty of Physics, VCQ, QuNaBioS, Boltzmanngasse 5, 1090 Vienna (Austria) Marcel Mayor [email protected] Department of Chemistry, University of Basel, St. Johannsring 19, 4056 Basel (Switzerland), Karlsruhe Institute of Technology (KIT), Institute of Nanotechnology, P.O. Box 3640, 76021 Karlsruhe (Germany) Jens Tüxen Department of Chemistry, University of Basel, St. Johannsring 19, 4056 Basel (Switzerland) ###### Abstract The quantum superposition principle, a key distinction between quantum physics and classical mechanics, is often perceived as a philosophical challenge to our concepts of reality, locality or space-time since it contrasts our intuitive expectations with experimental observations on isolated quantum systems. While we are used to associating the notion of localization with massive bodies, quantum physics teaches us that every individual object is associated with a wave function that may eventually delocalize by far more than the body’s own extension. Numerous experiments have verified this concept at the microscopic scale but intuition wavers when it comes to delocalization experiments with complex objects. While quantum science is the uncontested ideal of a physics theory, one may ask if the superposition principle can persist on all complexity scales. This motivates matter-wave diffraction and interference studies with large compounds in a three-grating interferometer configuration which also necessitates the preparation of high-mass nanoparticle beams at low velocities. Here we demonstrate how synthetic chemistry allows us to prepare libraries of fluorous porphyrins which can be tailored to exhibit high mass, good thermal stability and relatively low polarizability, which allows us to form slow thermal beams of these high-mass compounds, which can be detected in electron ionization mass spectrometry. We present successful superposition experiments with selected species from these molecular libraries in a quantum interferometer, which utilizes the diffraction of matter waves at an optical phase grating. We observe high- contrast quantum fringe patterns with molecules exceeding a mass of 10 000 amu and 810 atoms in a single particle. ††preprint: PCCP, DOI: 10.1039/C3CP51500A ## I Introduction Quantum physics has long been regarded as the science of small things , but experimental progress throughout the last two decades has led to the insight that it can also be observable for mesoscopic or even macroscopic objects. This applies for instance to the superposition of macroscopic numbers of electrons in superconducting quantum devices Mooij et al. (1999), the realization of large quantum degenerate atomic clouds in Bose-Einstein condensates Anderson et al. (1995), the cooling of micromechanical oscillators to their mechanical ground state. Quantum superposition studies with complex molecules Arndt et al. (1999) became possible with the advent of new matter wave interferometers Brezger et al. (2003); Gerlich et al. (2007); Haslinger et al. (2013) and techniques for slow macromolecular beams Deachapunya et al. (2008). These interferometers were for instance practically used to enable quantum enhanced measurements of internal molecular properties. The quantum fringe shift of a molecular interference pattern in the presence of external electric fields resulted in information for example on electric polarizabilities Berninger et al. (2007), dipole moments Eibenberger et al. (2011), or configuration changes Gring et al. (2010). Molecule interferometry can complement mass spectrometry Gerlich et al. (2008) and help to distinguish constitutional isomers Tüxen et al. (2010). In addition to their applications in chemistry, quantum interference experiments with massive molecules currently set the strongest bound on certain models that challenge the linearity of quantum mechanics Bassi et al. (2013). Further exploration of the frontiers of de Broglie coherence now profits from new capabilities in tailoring molecular properties to the needs of quantum optics. Our quantum experiment dictates the design of the molecules and the challenges increase with the number of atoms involved. In order to realize a molecular beam of sufficient intensity, the model compounds must be volatile, thermally stable and accessible in quantities of several hundred milligrams. Moreover, in order to minimize absorption at the wavelength of the optical diffraction (see below) they need to feature low absorption but sufficient polarizability at 532 nm. In reply to these needs, a dendritic library concept is particularly appealing since it can be scaled up to complex particles, once a suitable candidate has been identified. To meet these requirements we have functionalized organic chromophores with extended perfluoroalkyl chains. Such compounds show low inter-molecular binding and relatively high vapor pressures Stock et al. (2004); Krusic et al. (2005). They possess strong intra-molecular bonds and therefore sufficient thermal stability. In addition we start with a porphyrin core which is compatible with the required optical and electronic properties Meot-Ner et al. (1973). In the past, monodisperse fluorous porphyrins were generated by substituting the four para-fluorine substituents of tetrakis pentafluorophenylporphyrin (TPPF20) by dendritically branched fluorous moieties Tüxen et al. (2011). With this approach, molecules composed of 430 atoms were successfully synthesized and applied in quantum interference experiments Gerlich et al. (2011). With increasing complexity it becomes more challenging to purify monodisperse particles in sufficient amounts. Here we profit from the fact that our interferometer arrangement allows us to work with compound mixtures since each molecule interferes only with itself. By substituting some of the twenty fluorine atoms of TPPF20 with a branched, terminally perfluorinated alkylthiol (1), we obtain a mixture of compounds with molecular masses that differ exactly by an integer multiple of a particular value as molecular library. The molecular beam density is sufficiently low for the molecules not to interact with each other and the individual library compounds can be mass-specifically detected in a quadrupole mass spectrometer (QMS Extrel, 16 000 amu). Our synthetic approach is based on the fact that pentafluorophenyl moieties can be used to attach up to five polyfluoroalkyl substituents in nucleophilic aromatic substitution reactions. Substitutions at TPPF20 with its 20 potentially reactive fluorine substituents lead to a molecular library of derivatives with a varying number of fluorous side chains. ## II Results and Discussion We used sodium hydride as a base, microwave radiation as a heating source and diethylene glycol dimethyl ether (diglyme) as fluorophilic solvent. TPPF20, and a large excess of the thiol 1 (60 equivalents) and sodium hydride in diglyme were heated in a sealed microwave vial to 220 °C for 5 minutes.111 Synthetic protocol and analytical data of the porphyrin libraries L: General Remarks: All commercially available starting materials were of reagent grade and used as received. Microwave reactions were carried out in an Initiator 8 (400 W) from Biotage. Glass coated magnetic stirring bars were used during the reactions. The solvents for the extractions were of technical grade and distilled prior to use. Matrix Assisted Laser Desorption Ionization Time of Flight (MALDI-ToF) mass spectra were performed on an Applied Bio Systems Voyager-De™ Pro mass spectrometer or a Bruker microflex mass spectrometer. Significant signals are given in mass units per charge (m/z) and the relative intensities are given in brackets. Porphyrin library L: The thiol 1 was synthesized in seven reaction steps in an overall yield of 70% as reported elsewhere Tüxen et al. (2011). 5,10,15,20-Tetrakis(pentafluoro-phenyl)-porphyrin (TPPF20, 4.0 mg, 4.10 $\rm{mu}$mol, 1.0 eq.), thiol 1 (193 mg, 246 $\rm{mu}$mol, 60 eq.) and sodium hydride (60% dispersion in mineral oil, 14.8 mg, 369 $\rm{mu}$mol, 90 eq.) were added to diglyme (4 mL) in a microwave vial. The sealed tube was heated under microwave irradiation to 220 °C for 5 minutes. After cooling to room temperature the reaction mixture was quenched with water and subsequently extracted with diethylether. The organic layer was washed with brine and water, dried over sodium sulfate and evaporated to dryness. The resulting product mixture (183 mg) was analyzed by MALDI-ToF mass spectrometry. MS (MALDI-ToF, m/z): 12 403 (29%), 11 645 (52%), 10 884 (100%), 10 121 (78%), 9 339 (15%), 8 597 (7%). After aqueous workup the resulting mixture was analyzed by MALDI-ToF mass spectrometry (Figure 1) and subsequently used in our quantum interference experiments without further purification. We find up to 15 substituted fluorous thiol chains reaching to a molecular weight well beyond 10 000 amu. Figure 1: Synthetic scheme and MALDI-ToF mass spectrum of the fluorous porphyrin library textitL. High-mass matter-wave experiments were performed with component L12 of the library L. This structure is composed of 810 atoms and has a nominal molecular weight of 10 123 amu. In order to study the delocalized quantum wave nature of compounds in the fluorous library we utilize a Kapitza-Dirac-Talbot-Lau interferometer (KDTLI), which has already proven to be a viable tool with good mass scalability in earlier studies Gerlich et al. (2007, 2011). The interferometer is sketched in Figure 2: a molecular beam is created by thermal evaporation of the entire library in a Knudsen cell. The mixture traverses three gratings G1, G2, and G3, which all have the same period of d $\cong$ 266 nm. The molecules first pass the transmission grating G1, a SiNx mask with a slit opening of $s\approx$ 110 nm, where each molecule is spatially confined to impose the required spatial coherence by virtue of Heisenberg’s uncertainty principle Nairz et al. (2002). This is sufficient for the emerging quantum wavelets to cover several nodes of the optical phase grating G2, 10.5 cm further downstream. The standing light wave G2 is produced by retro-reflection of a green laser ($\lambda_{\rm{L}}$= 532 nm) at a plane mirror. Figure 2: KDTL interferometer setup: The molecules are evaporated in a furnace. Three height delimiters, D1 \- D1, define the particle velocity by selecting a flight parabola in the gravitational field. The interferometer consists of three gratings with identical periods of $d$ = 266 nm. G1 and G3 are SiNx gratings, whereas G2 is a standing light wave which is produced by retro-reflection of a green laser at a plane mirror. A phase modulation $\Phi\propto(\alpha\cdot P)/(v\cdot\omega_{y})$ is imprinted onto the molecular matter wave via the optical dipole force which is exerted by the light grating onto the molecular optical polarizability $\alpha_{opt}$. Here $P$ is the laser power, $v$ the molecular velocity, $\omega_{x}\cong\rm{18\,\mu m}$ and $\omega_{y}\cong\rm{945\,\mu m}$ the Gaussian laser beam waists. The transmitted molecules are detected using electron ionization quadrupole mass spectrometry after their passage through G3, which can be shifted along the z-axis to sample the interference pattern. When the molecular matter wave traverses the standing light wave, the dipole interaction between the electric field of power $P$ and the molecular optical polarizability $\alpha_{\rm{opt}}$ entails a periodic phase modulation $\Phi=\Phi_{0}\cdot\sin^{2}{(2\pi z/\lambda_{L})}$ with the maximum phase shift $\Phi_{0}=8\sqrt{2\pi}\alpha_{opt}P/(\hbar c\omega_{y}v_{x})$. Here $v_{x}$ is the forward directed molecular velocity, $\omega_{y}\cong$ 945 $\rm{\mu}$m is the Gaussian laser beam waist along the grating slits and $z$ is the coordinate along the laser beam. Interference of the molecular wavelets behind G2 leads to a molecular density pattern of the same period $d$ in front of the third grating. As long as we can neglect photon absorption by the molecules in the standing light wave G2 acts effectively as a pure phase grating. The expected fringe visibility $V$ is then given by Hornberger et al. (2009) $V=2(\sin(\pi f)/\pi f)^{2}J_{2}(-sgn(\Phi_{0}\sin(\pi L/L_{T}))\Phi_{0}\sin(\pi L/L_{T}))$. Here $f$ designates the grating open fraction, i.e. the ratio between the open slit width $s$ and the grating period $d$. $J_{2}$ is the Bessel function of second order, $L$ the separation of the gratings, $L_{T}=d^{2}/\lambda_{dB}$ the Talbot length and $\lambda_{dB}=h/(mv)$ the de Broglie wavelength with $h$ as Planck s quantum of action, $m$ the molecular mass and $v$ the modulus of its velocity. G3 is again a SiNx structure and lends spatial resolution to the detector. The interferogram is sampled by tracing the transmitted particle beam intensity as a function of the lateral ($z$) position of G3 and the mass selection is performed in the mass spectrometer behind this stage. Talbot-Lau interferometers Clauser and Li (1997) offer the important advantage over simple grating diffraction that the required grating period $d$ only weakly depends on the molecular de Broglie wavelength: $d\propto\sqrt{\lambda_{dB}}$. The setup accepts a wide range of velocities and low initial spatial coherence. This facilitates the use of dilute thermal molecular beams. Diffraction at the standing light wave G2 avoids the dephasing caused by the van der Waals interaction between the molecules and a dielectric wall. This is indeed present in G1 and G3 but can be neglected there since the molecular momentum distribution at G1 is wider than that caused by the grating and any phase shift in G3 is irrelevant if we are only interested in counting the particles that reach the mass spectrometer. Indistinguishability in all degrees of freedom is the basis for quantum interference Dirac (1958) and naturally given if a single molecule interferes only with itself. We only have to make sure that every molecule contributes to the final pattern in a similar way, which is true for all members of the library at about the same mass, independent of their internal state. Differences between various molecules, such as their isotopic distribution or the addition of a single atom are still acceptable. The KDTLI can tolerate a wavelength distribution of $\Delta\lambda_{dB}/\lambda_{dB}\leq\rm{20\,\%}$ and still produce a quantum fringe visibility in excess of the classical threshold. We here present quantum interference collected at the mass of one specific library compound, particularly for L12$=\rm{C_{284}H_{190}F_{320}N_{4}S_{12}}$ which has 12 fluorous side chains, a mass of 10 123 amu and 810 atoms bound in a single hot nanoparticle. All molecules of the library were evaporated at a temperature of about 600 K. We selected the velocity class around $v$ = 85 m/s ($\Delta v_{\rm{FWHM}}$ = 30 m/s) corresponding to a most probable de Broglie wavelength of approximately 500 fm. This is about four orders of magnitude smaller than the diameter of each individual molecule. We detected the signal by electron ionization quadrupole mass spectrometry. During the interference measurements the mass filter was set to the target mass of L12 and only this compound contributed to the collected interference pattern. The molecular beam was dilute enough to prevent classical interactions between any two molecules within the interferometer. Given that 80 mg of library L molecules were evaporated in 45 minutes, we estimate a flux at the source exit of $2\times 10^{15}$ particles per second. Including the acceptance angle of the instrument, the velocity selection as well as the grating transmission we estimate a molecular density inside the interferometer of 30 mm-3. This corresponds to a mean particle distance of about 300 $\rm{\mu}$m which is sufficient to exclude interactions with other neutral molecules in the beam. The average flight time of a molecule through the tightly focused standing light wave amounts to about 400 ns, i.e. much longer than the time scale of molecular vibrations (10-14 \- 10-12 s) and rotations ( 10-10 s). Therefore, the mean scalar polarizability governs the interaction with the standing light wave although the optical polarizability is generally described by a tensor. Thermal averaging also occurs for the orientation of any possibly existing molecular electric dipole moment Eibenberger et al. (2011). The internal molecular states are decoupled from de Broglie interference as long as we exclude effects of collisional or thermal decoherence Hackermüller et al. (2003, 2004) or external force fields Berninger et al. (2007). The thermal mixture of internal states is another reason why two-particle interference, i.e. mutual coherence of two macromolecules, is excluded in our experiments. The chances of finding two of them in the same indistinguishable set of all internal states electronic, vibrational and rotational levels, configuration, orientation and spin is vanishingly small. Figure 3: (a) Quantum interference pattern of L12 recorded at a laser power of $P\cong\rm{1\,W}$. The circles represent the experimental signal $s$ as a function of the position $z$ of the third grating. The solid line is a sinusoidal fit to the data, with a quantum fringe visibility of $V$ = 33(2) %. The shaded area represents the background signal of the detector. A classical picture predicts a visibility of only 8% for the same experimental parameters. b) Measured fringe visibility $V$ as a function of the diffracting laser power $P$. The expected contrast according to the quantum and the classical model are plotted as the blue and red lines, respectively Hornberger et al. (2009). The dashed blue lines correspond to the expected quantum contrast when the mean velocity is increased (reduced) by 5 ms-1. In Figure 3(a) we show a high contrast quantum interference pattern of L12. In contrast to far-field diffraction where the fringe separation is governed by the de Broglie wave-length of the transmitted molecules,Nairz et al. (2003); Juffmann et al. (2012) near-field interferometry of the Talbot-Lau type generates fringes of a fixed period, which are determined by the experimental geometry. Specifically, the expected interference figure in our KDTLI configuration is a sine curve whose contrast varies with the phase-shifting laser power as well as with the molecular beam velocity and polarizability. We distinguish the genuine quantum interferogram Brezger et al. (2003); Hornberger et al. (2009) from a classical shadow image by comparing the expected and experimental interference fringe visibility (contrast) with a classical model. The far off-resonance optical polarizability is assumed to be well approximated by the static value $\alpha_{opt}\cong\alpha_{stat}\cong\rm{410}\rm{\AA^{3}}\times\rm{4}\pi\epsilon_{0}$ as estimated using Gaussian G09 Frisch et al. (2009) with the 6-31 G basis set. The absorption cross section of L12 at 532 nm was estimated using the value of pure tetraphenylporphyrine dissolved in toluene Dixon et al. (2005) assuming that the perfluoroalkyl chains contribute at least an order of magnitude less to that value Gotsche et al. (2007). We thus find $\sigma_{532}\cong 1.7\times\rm{10^{-21}\,m^{2}}$. In Figure 3(b) we show the expected classical and quantum contrast as a function of the diffracting laser power. Our experimental contrast is derived from the recorded signal curves, such as shown in Figure 3(a) by $V=(S_{max}-S_{min})/(S_{max}+S_{min})$, where $S_{max}$ and $S_{min}$ are the maxima and minima of the sine curve fitted to the data. The contrast it is plotted as the black dots in Figure 3(b). These points are in good agreement with the quantum prediction (blue line). The classical effect, describing the shadow image by two material gratings and one array of dipole force lenses Hornberger et al. (2009) is shown as the red line. The experiment clearly excludes this classical picture. Since the amount of precious molecular material and thermal degradation processes in the source limited the measurement time we performed our experiments only in the parameter regime that maximizes the fringe visibility, to show the clear difference between the quantum and classical contrast. The experimental fringe visibility reproduces well the maximally expected quantum contrast. ## III Conclusion We have shown that a library approach towards stable and volatile high-mass molecules can substantially extend the complexity range of molecular de Broglie coherence experiments to masses in excess of 10 000 amu. Our data confirm the fully coherent quantum delocalization of single compounds composed of about 5 000 protons, 5 000 neutrons and 5 000 electrons. The internal complexity, number of vibrational modes and also the internal energy of each of these particles is higher than in any other matter-wave experiment so far. ###### Acknowledgements. This work was supported by the FWF programs Wittgenstein (Z149-N16) and CoQuS (W1210-2), the European Commission in the project NANOQUESTFIT (304 886), the Swiss National Science Foundation, the NCCR Nanoscale Science , and the Swiss Nanoscience Institute (SNI). 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1310.8461	# A conjecture on the primitive degree of Tensors 111P. Yuan’s research is supported by the NSF of China (Grant No. 11271142) and the Guangdong Provincial Natural Science Foundation(Grant No. S2012010009942), L. You’s research is supported by the Zhujiang Technology New Star Foundation of Guangzhou (Grant No. 2011J2200090) and Program on International Cooperation and Innovation, Department of Education, Guangdong Province (Grant No.2012gjhz0007). Pingzhi Yuan222Corresponding author: [email protected]., Zilong He, Lihua You 333Email address: [email protected]. (School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, P.R. China ) ###### Abstract In this paper, we prove: Let $\mathbb{A}$ be a nonnegative primitive tensor with order $m$ and dimension $n$. Then its primitive degree $\gamma(\mathbb{A})\leq(n-1)^{2}+1$, and the upper bound is sharp. This confirms a conjecture of Shao [7]. AMS classification: 05C50; 15A69 Keywords: tensor; product; primitive tensor; primitive degree. ## 1 Introduction In [1] and [2], Chang et al investigated the properties of the spectra of nonnegative tensors. They defined the irreducibility of tensors, and the primitivity of nonnegative tensors, and extended many important properties of primitive matrices to primitive tensors. Recently, as an application of the general tensor product defined by Shao [7], Shao presented a simple characterization of the primitive tensors in terms of the zero pattern of the powers of $\mathbb{A}$. He also proposed the following conjecture on the primitive degree. ###### Conjecture 1.1. When $m$ is fixed, then there exists some polynomial $f(n)$ on $n$ such that $\gamma(\mathbb{A})\leq f(n)$ for all nonnegative primitive tensors of order $m$ and dimension $n$. In this paper, we confirm the conjecture by proving the following theorem. ###### Theorem 1.2. Let $\mathbb{A}$ be a nonnegative primitive tensor with order $m$ and dimension $n$. Then its primitive degree $\gamma(\mathbb{A})\leq(n-1)^{2}+1$, and the upper bound is sharp. ## 2 Preliminaries An order $m$ dimension $n$ tensor $\mathbb{A}=(a_{i_{1}i_{2}\cdots i_{m}})_{1\leq i_{j}\leq n\hskip 5.69046pt(j=1,\cdots,m)}$ over the complex field $\mathbb{C}$ is a multidimensional array with all entries $a_{i_{1}i_{2}\cdots i_{m}}\in\mathbb{C}\,(i_{1},\cdots,i_{m}\in[n]=\\{1,\cdots,n\\})$. The majorization matrix $M(\mathbb{A})$ of the tensor $\mathbb{A}$ is defined as $(M(\mathbb{A}))_{ij}=a_{ij\cdots j},(i,j\in[n])$ by Pearson [4]. Let $\mathbb{A}$ (and $\mathbb{B}$) be an order $m\geq 2$ (and $k\geq 1$), dimension $n$ tensor, respectively. Recently, Shao [7] defined a general product $\mathbb{A}\mathbb{B}$ to be the following tensor $\mathbb{D}$ of order $(m-1)(k-1)+1$ and dimension $n$: $d_{i\alpha_{1}\cdots\alpha_{m-1}}=\sum\limits_{i_{2},\cdots,i_{m}=1}^{n}a_{ii_{2}\cdots i_{m}}b_{i_{2}\alpha_{1}}\cdots b_{i_{m}\alpha_{m-1}}\quad(i\in[n],\,\alpha_{1},\cdots,\alpha_{m-1}\in[n]^{k-1}).$ The tensor product possesses a very useful property: the associative law ([7], Theorem 1.1). With the general product, Shao [7] proved some results on the primitivity and primitive degree f nonnegative tensor. The following result will be used in Definition 2.3. ###### Proposition 2.1. ([7], Proposition 4.1) Let $\mathbb{A}$ be an order $m$ and dimension $n$ nonnegative tensor. Then the following three conditions are equivalent: (1). For any $i,j\in[n],a_{ij\cdots j}>0$ holds. (2). For any $j\in[n],\mathbb{A}e_{j}>0$ holds (where $e_{j}$ is the $j^{th}$ column of the identity matrix $I_{n}$). (3). For any nonnegative nonzero vector $x\in\mathbb{R}^{n},\mathbb{A}x>0$ holds. ###### Definition 2.2. ([4], Definition 3.1) A nonnegative tensor $\mathbb{A}$ is called essentially positive, if it satisfies (3) of Proposition 2.1. By Proposition 2.1, the following Definition 2.3 is eauivalent to Definition 2.2. ###### Definition 2.3. ([7], Definition 4.1) A nonnegative tensor $\mathbb{A}$ is called essentially positive, if it satisfies one of the three conditions in Proposition 2.1. In [2] and [4], Chang et al and Pearson define the primitive tensors as follows. ###### Definition 2.4. ([2, 5]) Let $\mathbb{A}$ be a nonnegative tensor with order $m$ and dimension $n$, $x=(x_{1},x_{2},\cdots,x_{n})^{T}\in\mathbb{R}^{n}$ a vector and $x^{[r]}=(x_{1}^{r},x_{2}^{r},\cdots,x_{n}^{r})^{T}$. Define the map $T_{\mathbb{A}}$ from $\mathbb{R}^{n}$ to $\mathbb{R}^{n}$ as: $T_{\mathbb{A}}(x)=(\mathbb{A}x)^{[\frac{1}{m-1}]}$. If there exists some positive integer $r$ such that $T_{\mathbb{A}}^{r}(x)>0$ for all nonnegative nonzero vectors $x\in\mathbb{R}^{n}$, then $\mathbb{A}$ is called primitive and the smallest such integer $r$ is called the primitive degree of $\mathbb{A}$, denoted by $\gamma(\mathbb{A})$. In [7], Shao show the following results and define the primitive degree by using the properties of tensor product and the zero patterns. ###### Proposition 2.5. ([7], Theorem 4.1) A nonnegative tensor $\mathbb{A}$ is primitive if and only if there exists some positive integer $r$ such that $\mathbb{A}^{r}$ is essentially positive. Furthermore, the smallest such $r$ is the primitive degree of $\mathbb{A}$. ###### Remark 2.6. Let $\mathbb{A}$ be a nonnegative tensor with order $m$ and dimension $n$. Then $\mathbb{A}$ is primitive if and only if there exists some positive integer $r$ such that $M(\mathbb{A}^{r})>0.$ Now we prove the following necessary conditions for a tensor to be primitive firstly. ###### Proposition 2.7. Let $\mathbb{A}$ be a nonnegative primitive tensor with order $m$ and dimension $n$, $M(\mathbb{A})$ the majorization matrix of $\mathbb{A}$. Then we have: (i). For each $j\in[n]$, there exists an integer $i\in[n]\backslash\\{j\\}$ such that $(M(\mathbb{A}))_{ij}>0$. (ii). There exist some $j\in[n]$ and integers $u,v$ with $1\leq u<v\leq n$ such that $(M(\mathbb{A}))_{uj}>0$ and $(M(\mathbb{A}))_{vj}>0$. ###### Proof. We prove the results via contradiction. (i) To obtain a contradiction, we suppose that there exists some integer $j\in[n]$ such that $(M(\mathbb{A}))_{ij}=0$ for every $i\in[n]\backslash\\{j\\}$. By definitions of the tensor product and the majorization matrix, for any $u\in[n]\backslash\\{j\\}$, we have $(M(\mathbb{A}))_{uj}=0$ and $(M(\mathbb{A}^{2}))_{uj}=\sum\limits_{j_{2},\cdots,j_{m}=1}^{n}a_{uj_{2}\cdots j_{m}}a_{j_{2}j\cdots j}\cdots a_{j_{m}j\cdots j}$ $=\sum\limits_{j_{2},\cdots,j_{m}=1}^{n}a_{uj_{2}\cdots j_{m}}(M(\mathbb{A}))_{j_{2}j}\cdots(M(\mathbb{A}))_{j_{m}j}$ $=(M(\mathbb{A}))_{uj}(M(\mathbb{A}))_{jj}^{m-1}=0$. Since $(M(\mathbb{A}^{r+1}))_{uj}=\sum\limits_{j_{2},\cdots,j_{m}=1}^{n}a_{uj_{2}\cdots j_{m}}(M(\mathbb{A}^{r}))_{j_{2}j}\cdots(M(\mathbb{A}^{r}))_{j_{m}j},$ hence, by induction on $k$, we conclude that $(M(\mathbb{A}^{k}))_{uj}=0$ holds for any positive integer $k$ and any $u\in[n]\backslash\\{j\\}$. This contradicts that $(M(\mathbb{A}^{\gamma(\mathbb{A})}))_{uj}>0$, where $\gamma(\mathbb{A})$ is the primitive degree of $\mathbb{A}$. (i) is proved. (ii) Suppose, to derive a contradiction, that there is at most one nonzero element in each of the following $n$ sets $\\{(M(\mathbb{A}))_{uj},u\in[n]\\},\quad j\in[n].$ Now we will show that for any positive integer $t$, there is at most one nonzero element in each of the following $n$ sets $\\{(M(\mathbb{A}^{t}))_{uj},u\in[n]\\},\quad j\in[n].$ We prove the above assertion by induction on $t$. Clearly, $t=1$ is obvious. Assume that the assertion holds for $t=k\geq 1$, that is, there is at most one nonzero element in each of the following $n$ sets $\\{(M(\mathbb{A}^{k}))_{uj},u\in[n]\\},\quad j\in[n],$ say, for any $j\in[n]$ and any $u\neq u_{j}$, $(M(\mathbb{A}^{k}))_{uj}=0$. Note that for any $v\in[n]$, $(M(\mathbb{A}^{k+1}))_{vj}=\sum\limits_{j_{2},\cdots,j_{m}=1}^{n}a_{vj_{2}\cdots j_{m}}(M(\mathbb{A}^{k}))_{j_{2}j}\cdots(M(\mathbb{A}^{k}))_{j_{m}j}=a_{vu_{j}\cdots u_{j}}((M(\mathbb{A}^{k}))_{u_{j}j})^{m-1},$ by the assumption, there is at most one $v\in[n]$ such that $a_{vu_{j}\cdots u_{j}}=(M(A))_{vu_{j}}>0$, therefore we have prove the assertion. It follows that $\mathbb{A}$ is not a primitive tensor, this contradiction proves (ii).∎ Follows from the proof of Proposition 2.7, we know that $(M(\mathbb{A}^{k+1}))_{uj}=\sum\limits_{j_{2},\cdots,j_{m}=1}^{n}a_{uj_{2}\cdots j_{m}}(M(\mathbb{A}^{k}))_{j_{2}j}\cdots(M(\mathbb{A}^{k}))_{j_{m}j},$ (2.1) is useful and used repeatly. By Equation (2.1), it is easy to prove the following assertions. ###### Corollary 2.8. Let $\mathbb{A}$ be a nonnegative primitive tensor with order $m$ and dimension $n$. Then we have (i). For each $u\in[n]$, there is at least one index $j_{2}\cdots j_{m}\in[n]^{m-1}$ such that $a_{uj_{2}\cdots j_{m}}>0$. (ii). Let $k\in[n]$ be fixed. Suppose that $T$ is a positive integer such that $(M(\mathbb{A}^{T}))_{uk}>0$ for all $u\in[n]$, then for any $t\geq T$, we have $(M(\mathbb{A}^{t}))_{uk}>0$ for all $u\in[n]$. ###### Proposition 2.9. Let $\mathbb{A}$ be a nonnegative tensor with order $m$ and dimension $n$, and let $a$ be a positive integer. Then $\mathbb{A}$ is primitive if and only if $\mathbb{A}^{a}$ is primitive. Let $\mathbb{A}$ be a nonnegative primitive tensor with order $m$ and dimension $n$, for any $j=j_{1}\in[n]$, by Proposition 2.7, there exists a sequence $j_{1},j_{2},\cdots,j_{s+1}$ such that $j_{k}\in[n],j_{k}\neq j_{k+1},1\leq k\leq s$ and $(M(\mathbb{A}))_{j_{k+1}j_{k}}>0$. ###### Definition 2.10. Let $\mathbb{A}$ be a nonnegative tensor with order $m$ and dimension $n$, if $j_{k}\in[n]$ for $1\leq k\leq t$ and $(M(\mathbb{A}))_{j_{k+1}j_{k}}>0$ for $1\leq k\leq t-1$, we say that $j_{1}\to j_{2}\to\cdots\to j_{t}$ is a $walk$ of $length$ $t-1$ of $M(\mathbb{A})$; if $j_{i}\not=j_{k}$ for any $i,k\in[t]$ with $i\not=k$, then we say the walk $j_{1}\to j_{2}\to\cdots\to j_{t}$ is a path of $M(\mathbb{A})$; if $j_{i}\not=j_{k}$ for any $i,k\in[t-1]$ with $i\not=k$ but $j_{1}=j_{t}$, then we say the walk $j_{1}\to j_{2}\to\cdots\to j_{t}$ is a cycle of $M(\mathbb{A})$. ###### Lemma 2.11. Let $\mathbb{A}$ be a nonnegative tensor with order $m$ and dimension $n$. Suppose that $t>1$ and $j_{1}\to j_{2}\to\cdots\to j_{t}$ is a $walk$ of $M(\mathbb{A})$ , then $(M(\mathbb{A}^{t-1}))_{j_{t}j_{1}}>0$. ###### Proof. We prove the assertion by induction on $t$. Clearly, $t=2$ is obvious. Assume that the result holds for $t=k\geq 2$, that is $(M(\mathbb{A}^{k-1}))_{j_{k}j_{1}}>0$. Since $(M(\mathbb{A}^{k}))_{j_{k+1}j_{1}}=\sum\limits_{i_{2},\cdots,i_{m}=1}^{n}a_{j_{k+1}i_{2}\cdots i_{m}}(M(\mathbb{A}^{k-1}))_{i_{2}j_{1}}\cdots(M(\mathbb{A}^{k-1}))_{i_{m}j_{1}}$ $>a_{j_{k+1}j_{k}\cdots j_{k}}((M(\mathbb{A}^{k-1}))_{j_{k}j_{1}})^{m-1}$ $=(M(A))_{j_{k+1}j_{k}}((M(\mathbb{A}^{k-1}))_{j_{k}j_{1}})^{m-1}>0,$ by assumption, thus the assertion holds for any integer $t>1$. We are done.∎ ###### Lemma 2.12. Let $\mathbb{A}$ be a nonnegative primitive tensor with order $m$ and dimension $n$. Then there exist an integer $j\in[n]$ and an integer $l\in[n-1]$ such that $(M(\mathbb{A}^{l}))_{jj}>0$. ###### Proof. By Proposition 2.7, we may assume that $j_{1}\to j_{2}\to\cdots\to j_{n+1}$ is a walk of length $n$ of $M(\mathbb{A})$. Since $j_{k}\in[n],1\leq k\leq n+1$, there exist at least two integers $u$ and $v$ such that $1\leq u<v\leq n+1$ and $j_{u}=j_{v}$. It follows from Lemma 2.11 that $(M(\mathbb{A}^{v-u}))_{j_{u}j_{u}}>0$. Case 1: There exist $u,v$ such that $(u,v)\neq(1,n+1)$. Then $v-u\leq n-1$ and we are done by taking $j=j_{u}$ and $l=v-u$. Case 2: $(u,v)=(1,n+1)$. Then $j_{1},j_{2},\ldots,j_{n}$ is a permutation of $1,2,\ldots,n$ and $j_{n+1}=j_{1}$. By (ii) of Proposition 2.7, there exist an integer $i\in[n]$ and two integers $p\neq q$ such that $1\leq p,q\leq n$, $(M(\mathbb{A}))_{pi}>0$ and $(M(\mathbb{A}))_{qi}>0$. Take $i=j_{t}$ for some $t\in[n]$, and assume that $p=j_{s}\neq j_{t+1}$. Thus $s\in[n]$, and $t\leq n-1$ when $s=1$. Subcase 2.1: $p\in\\{j_{1},\cdots,j_{t-1}\\}$. Then $1\leq s\leq t-1$ and $p=j_{s}\to\cdots j_{t-1}\to j_{t}\to p=j_{s}$ is a cycle of $M(\mathbb{A})$ with length $t+1-s\leq n-1$. We take $j=p$ and $l=t+1-s$. Subcase 2.2: $p=j_{t}$. Then $s=t$ and $j_{t}\to p(=j_{s})$ is a cycle of $M(\mathbb{A})$ with length $1$. We take $j=j_{t}$ and $l=1$. Subcase 2.3: $p\in\\{j_{t+2},\cdots,j_{n}\\}$. Then $t+2\leq s\leq n$ and $j_{1}\to j_{2}\to\cdots\to j_{t}\to j_{s}(=p)\to\cdots\to j_{n}\to j_{1}$ is a cycle of $M(\mathbb{A})$ and $t+n-s+1=n-(s-t-1)\leq n-1$. In this case we take $j=j_{1}$ and $l=n+t+1-s$. This proves the lemma.∎ ###### Remark 2.13. By the proof of Lemma 2.12, if $\mathbb{A}$ is a nonnegative primitive tensor with order $m$ and dimension $n$, then there exist an integer $t$ with $1\leq t\leq n-1$ and some integers $j_{1},j_{2},\cdots,j_{t}\in[n]$ such that $j_{1}\to j_{2}\to\cdots\to j_{t}\to j_{1}$ is a cycle of length $t$ of $M(\mathbb{A})$, and for any $k\in[t]$, $(M(\mathbb{A}^{t}))_{j_{k}j_{k}}>0$. Note that by (2.1), (ii) of Corollary 2.8 also holds when $\mathbb{A}$ is a nonnegative tensor with order $m$ and dimension $n$. Therefore it makes sense to consider the primitive degree of some column of a tensor. ###### Definition 2.14. Let $\mathbb{A}$ be a nonnegative tensor with order $m$ and dimension $n$. For a fixed integer $j\in[n]$, if there exists a positive integer $T$ such that $(M(\mathbb{A}^{T}))_{uj}>0,\hskip 5.69046pt{\mbox{f}or\,\,all}\,u,1\leq u\leq n,$ then $\mathbb{A}$ is called $j$-primitive and the smallest such integer $T$ is called the $j$-primitive degree of $\mathbb{A}$, denoted by $\gamma_{j}(\mathbb{A})$. By Corollary 2.8 and the above definition, we have the following result. ###### Proposition 2.15. Let $\mathbb{A}$ be a nonnegative primitive tensor with order $m$ and dimension $n$. Then $\gamma(\mathbb{A})=\max_{1\leq j\leq n}\\{\gamma_{j}(\mathbb{A})\\}.$ ## 3 Proof of the main results In this section, we will prove Theorem 1.2. We first prove the following special case of the theorem. ###### Theorem 3.1. Let $\mathbb{A}$ be a nonnegative primitive tensor with order $m$ and dimension $n$. Suppose that there is an integer $j\in[n]$ with $(M(\mathbb{A}))_{jj}>0$, then $\gamma_{j}(\mathbb{A})\leq n-1$. ###### Proof. Put $S_{k}=\\{u\in[n],(M(\mathbb{A}^{k}))_{uj}\\}>0,k=1,2,\ldots.$ Then $j\in S_{1}$ and there exists an integer $v\in[n]\setminus\\{j\\}$ such that $v\in S_{1}$ by (i) of Proposition 2.7. Thus $\|S_{1}\|\geq 2$. Since $(M(\mathbb{A}^{k+1}))_{uj}=\sum\limits_{i_{2},\cdots,i_{s}=1}^{n}(\mathbb{A}^{k})_{ui_{2}\cdots i_{s}}(M(\mathbb{A}))_{i_{2}j}\cdots(M(\mathbb{A}))_{i_{s}j}\geq(M(\mathbb{A}^{k}))_{uj}(M(\mathbb{A}))_{jj}^{s-1},$ we see that if $u\in S_{k}$, then $u\in S_{k+1}$. Therefore by induction on $k$, we can obtain that $S_{1}\subseteq S_{2}\subseteq\cdots\subseteq S_{l}\subseteq S_{l+1}\subseteq\cdots$. Note that $S_{k}\subseteq[n]$ for any $k=1,2,\cdots$, and $S_{1}\subseteq S_{2}\subseteq\cdots\subseteq S_{l}\subseteq S_{l+1}\subseteq\cdots$, hence the sequence $S_{1},S_{2},\ldots,S_{l},\ldots$ eventually terminates. Let $l$ be the smallest positive integer such that $S_{l}=S_{l+1}$, by (2.1) for $k=l,l+1$, we have $(M(\mathbb{A}^{l+1}))_{uj}=\sum\limits_{j_{2},\cdots,j_{m}=1}^{n}a_{uj_{2}\cdots j_{m}}(M(\mathbb{A}^{l}))_{j_{2}j}\cdots(M(\mathbb{A}^{l}))_{j_{m}j}$ and $(M(\mathbb{A}^{l+2}))_{uj}=\sum\limits_{j_{2},\cdots,j_{m}=1}^{n}a_{uj_{2}\cdots j_{m}}(M(\mathbb{A}^{l+1}))_{j_{2}j}\cdots(M(\mathbb{A}^{l+1}))_{j_{m}j}.$ It follows that $S_{l+1}=S_{l+2}=\cdots=S_{k}$ for all $k\geq l$. Since $\mathbb{A}$ is a nonnegative primitive tensor, hence $S_{l}=S_{\gamma(\mathbb{A})}=[n]$. Now by the above argumnets and the definition of $l$, we have $S_{1}\subsetneqq S_{2}\subsetneqq\cdots\subsetneqq S_{l-1}\subsetneqq S_{l},$ and thus $2\leq\|S_{1}\|<\|S_{2}\|<\cdots<\|S_{l}\|=n$. It follows that $l\leq n-1$ and $\gamma_{j}(\mathbb{A})\leq n-1$. ∎ We also need the following lemmas. ###### Lemma 3.2. Let $\mathbb{A}$ be a nonnegative primitive tensor with order $m$ and dimension $n$. Let $H=\\{i_{1},i_{2},\ldots,i_{s}\\}$ be the set of all elements $i\in[n]$ such that $i=k_{1}\to k_{2}\to\cdots\to k_{t}\to k_{1}=i$ is a cycle with length $t$ of $M(\mathbb{A})$ for some $t$ where $1\leq t\leq n-1$. Suppose there exists a positive integer $j$ with $j\in[n]\setminus H$, then $1\leq s\leq n-1$ and there exist positive integers $i$ and $l$ such that $i\in H$, $1\leq l\leq n-s$ and $(M(\mathbb{A}^{l}))_{ij}>0$. ###### Proof. It is obvious that $s\leq n-1$, and $s\geq 1$ by Lemma 2.12. By Proposition 2.7, we have the following walk of length $n-s$ of $M(\mathbb{A})$ starting with $j$: $j=j_{1}\to j_{2}\to\cdots\to j_{n-s+1}.$ If there exists an integer $w,1\leq w\leq n-s+1$ such that $j_{w}\in H$, then $(M(\mathbb{A}^{w-1}))_{j_{w}j}>0$ by Lemma 2.11, and we are done. Otherwise, we have $\\{j_{1},j_{2},\cdots,j_{n-s+1}\\}\cap H=\emptyset.$ Since $n-s+1+\|H\|=n+1$, so there exist two positive integers $u$ and $v$ such that $1\leq u<v\leq n-s+1$, $j_{u}=j_{v}$, and $j_{u}\to j_{u+1}\to\cdots\to j_{v-1}\to j_{v}=j_{u}$ is a cycle with length $v-u\leq n-s\leq n-1$ of $M(\mathbb{A})$. It follows that $j_{u}\in H$, a contradiction. This proves the lemma.∎ Let $Z(\mathbb{A})$ be the tensor obtained by replacing all the nonzero entries of $\mathbb{A}$ by one. Then $Z(\mathbb{A})$ is called the zero- nonzero pattern (or simply the zero pattern) of $\mathbb{A}$. Let $a$ be a complex number, we define $Z(a)=1$ if $a\not=0$ and $Z(a)=0$ if $a=0$. ###### Lemma 3.3. Let $\mathbb{A}$ be a nonnegative tensor with order $m$ and dimension $n$ such that $a_{ii_{2}\cdots i_{m}}=0$ if $i_{2}\cdots i_{m}\not=i_{2}\cdots i_{2}$ for any $i\in[n]$. Then for any positive $r$, $Z(M(\mathbb{A}^{r}))=Z((M(\mathbb{A}))^{r}).$ ###### Proof. We show the result by induction on $r$. Clearly, $r=1$ is obvious. Assume that the assertion holds for $r=k\geq 1$, then for any $i,j\in[n]$, $Z[(M(\mathbb{A}^{k+1}))_{ij}]=Z[\sum\limits_{i_{2},\cdots,i_{m}=1}^{n}a_{ii_{2}\cdots i_{m}}(M(\mathbb{A}^{k}))_{i_{2}j}\cdots(M(\mathbb{A}^{k}))_{i_{m}j}]$ $=Z[\sum\limits_{i_{2}=1}^{n}(M(\mathbb{A}))_{ii_{2}}((M(\mathbb{A}^{k}))_{i_{2}j})^{m-1}]$ $=Z[\sum\limits_{i_{2}=1}^{n}(M(\mathbb{A}))_{ii_{2}}(M(\mathbb{A}^{k}))_{i_{2}j}]$ $=Z[\sum\limits_{i_{2}=1}^{n}Z[(M(\mathbb{A}))_{ii_{2}}]Z[(M(\mathbb{A}^{k}))_{i_{2}j}]]$ $=Z[\sum\limits_{i_{2}=1}^{n}Z[(M(\mathbb{A}))_{ii_{2}}]Z[((M(\mathbb{A}))^{k})_{i_{2}j}]]$ $=Z[\sum\limits_{i_{2}=1}^{n}Z[(M(\mathbb{A}))_{ii_{2}}((M(\mathbb{A}))^{k})_{i_{2}j}]]$ $=Z[((M(\mathbb{A}))^{k+1})_{ij}]$ Thus $Z(M(\mathbb{A}^{k+1}))=Z((M(\mathbb{A}))^{k+1}).$ ∎ ###### Corollary 3.4. Let $\mathbb{A}$ be a nonnegative primitive tensor with order $m$ and dimension $n$ such that $a_{ii_{2}\cdots i_{m}}=0$ if $i_{2}\cdots i_{m}\not=i_{2}\cdots i_{2}$ for any $i\in[n]$. If $M(\mathbb{A})$ is primitive, then $\gamma(\mathbb{A})=\gamma(M(\mathbb{A})).$ Proof of Theorem 1.2: Let $H=\\{i_{1},i_{2},\ldots,i_{s}\\}$ be the set of all elements $i\in[n]$ such that $i=k_{1}\to k_{2}\to\cdots\to k_{t}\to k_{1}=i$ is a cycle with length $t$ of $M(\mathbb{A})$ and $1\leq t\leq n-1$. Case 1: $s=n$. Then for any $j\in[n]$, there exists an integer, say, $t_{j}$, such that there exists a cycle $j=k_{1}\to k_{2}\to\cdots k_{t_{j}}\to k_{1}=j$ with length $t_{j}$ where $t_{j}\in[n-1]$, so $(M(\mathbb{A}^{t_{j}}))_{jj}>0$. Note that $\mathbb{A}^{t_{j}}$ is primitive by Proposition 2.9 and $\mathbb{A}$ is primitive, we have $\gamma_{j}(\mathbb{A}^{t_{j}})\leq n-1$ by Theorem 3.1. Hence $\gamma_{j}(\mathbb{A})\leq t_{j}\gamma_{j}(\mathbb{A}^{t_{j}})\leq(n-1)^{2}.$ Thus $\gamma(\mathbb{A})\leq\max_{j\in[n]}\\{\gamma_{j}(\mathbb{A})\\}\leq(n-1)^{2}$ by Proposition 2.15. Case 2: $1\leq s\leq n-1$. Then there exists at least an integer $j\in[n]\backslash H$. Subcase 2.1: $j\in H$. Similar to Case 1, for any $j\in H$, there exists an integer, say, $t_{j}$, such that there exists a cycle $j=k_{1}\to k_{2}\to\cdots k_{t_{j}}\to k_{1}=j$ with length $t_{j}$ where $t_{j}\leq s$, so $(M(\mathbb{A}^{t_{j}}))_{jj}>0$. Note that $\mathbb{A}^{t_{j}}$ is primitive by Proposition 2.9 and $\mathbb{A}$ is primitive, we have $\gamma_{j}(\mathbb{A}^{t_{j}})\leq n-1$ by Theorem 3.1. Hence $\gamma_{j}(\mathbb{A})\leq t_{j}\gamma_{j}(\mathbb{A}^{t_{j}})\leq t_{j}(n-1)\leq s(n-1).$ Let $T=\max_{j\in H}\\{\gamma_{j}(\mathbb{A})\\}$, then $T\leq s(n-1)$ and $(M(\mathbb{A}^{T}))_{uj}>0$ for any $j\in H$ and all $u\in[n]$. Subcase 2.2: $j\in[n]\backslash H$. By Lemma 3.2, there exist positive integers $i$ and $l$ such that $i\in H$, $1\leq l\leq n-s$ and $(M(\mathbb{A}^{l}))_{ij}>0$. For any $u\in[n]$, since $(M(\mathbb{A}^{T+l}))_{uj}=\sum\limits_{j_{2},\cdots,j_{s}=1}^{n}(\mathbb{A}^{T})_{uj_{2}\cdots j_{s}}(M(\mathbb{A}^{l}))_{j_{2}j}\cdots(M(\mathbb{A}^{l}))_{j_{s}j}\geq(M(\mathbb{A}^{T}))_{ui}((M(\mathbb{A}^{l}))_{ij})^{s-1}>0,$ then $\gamma_{j}(\mathbb{A})\leq T+l\leq s(n-1)+n-s\leq(n-1)^{2}+1$. Thus combing Subcase 2.1 and Subcase 2.2, we have $\gamma(\mathbb{A})\leq\max_{j\in[n]}\\{\gamma_{j}(\mathbb{A})\\}\leq(n-1)^{2}+1$ by Proposition 2.15. Combing the above arguments, $\gamma(\mathbb{A})\leq\max\\{(n-1)^{2},(n-1)^{2}+1\\}=(n-1)^{2}+1$. Let $M_{1}=\left(\begin{array}[]{ccccc}0&0&\cdots&1&1\\\ 1&0&\cdots&0&0\\\ 0&1&\cdots&0&0\\\ 0&0&\ddots&0&0\\\ 0&0&\cdots&1&0\\\ \end{array}\right)$. It is well known that $M_{1}$ is primitive, and $\gamma(M_{1})=(n-1)^{2}+1$. Let $\mathbb{A}$ be a nonnegative primitive tensor with order $m$ and dimension $n$ such that $a_{ii_{2}\cdots i_{m}}=0$ if $i_{2}\cdots i_{m}\not=i_{2}\cdots i_{2}$ for any $i\in[n]$, and $M(\mathbb{A})=M_{1}$. Then by Corollary 3.4, we have $\gamma(\mathbb{A})=(n-1)^{2}+1.$ ∎ ###### Remark 3.5. It is well known $\gamma(\mathbb{A})\leq(n-1)^{2}+1$ for $m=2$ ($\mathbb{A}$ is a matrix). It implies that the upper bound on the primitive degree of primitive tensors and the upper bound on the primitive index of primitive matrices are coincident. ## References * [1] K. C. Chang, K. Pearson, and T. Zhang, Perron-Frobenius theorem for nonnegative tensors, Commun. Math. Sci. 6(2008), 507-520. * [2] K. C. Chang, K. Pearson, and T. Zhang, Primitivity, the convergence of the NQZ method, and the largest eigenvalue tensors, SIAM J. Matrix Anal. Appl. 32(2011), 806-819. * [3] L.H. Lim, Singular values and eigenvalues of tensors, a rational approach, in proceedings 1st IEEE international workshop on computational advances of adaptive processing (2005), 129-132. * [4] K. Pearson, Essentially positive tensors, Int. J. Algebra, 4(2010),421-427. * [5] K. Pearson, Primitive tensors and convergence of an iterative process for the eigenvalues of a primitive tensor, arXiv: 1004-2423v1, 2010. * [6] L. Qi, Eigenvalues of a real supersymmetric tensor, Symbolic Comput. 40(2005), 1302-1324. * [7] J.Y Shao, A general product of tensors with applications, Linear Algebra and its Appl. 439(2013), 2350-2366.	arxiv-papers	2013-10-31T11:10:10	2024-09-04T02:49:53.137894	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Pingzhi Yuany, Zilong He, Lihua You", "submitter": "Lihua You", "url": "https://arxiv.org/abs/1310.8461" }
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Mohanty Tata Institute of Fundamental Research, Mumbai 400005 T. Mori Graduate School of Science, Nagoya University, Nagoya 464-8602 R. Mussa INFN - Sezione di Torino, 10125 Torino Y. Nagasaka Hiroshima Institute of Technology, Hiroshima 731-5193 E. Nakano Osaka City University, Osaka 558-8585 M. Nakao High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 M. Nayak Indian Institute of Technology Madras, Chennai 600036 E. Nedelkovska Max-Planck- Institut für Physik, 80805 München C. Ng Department of Physics, University of Tokyo, Tokyo 113-0033 N. K. Nisar Tata Institute of Fundamental Research, Mumbai 400005 S. Nishida High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 O. Nitoh Tokyo University of Agriculture and Technology, Tokyo 184-8588 S. Ogawa Toho University, Funabashi 274-8510 S. Okuno Kanagawa University, Yokohama 221-8686 S. L. Olsen Seoul National University, Seoul 151-742 W. Ostrowicz H. Niewodniczanski Institute of Nuclear Physics, Krakow 31-342 G. Pakhlova Institute for Theoretical and Experimental Physics, Moscow 117218 C. W. Park Sungkyunkwan University, Suwon 440-746 H. Park Kyungpook National University, Daegu 702-701 H. K. Park Kyungpook National University, Daegu 702-701 T. K. Pedlar Luther College, Decorah, Iowa 52101 R. Pestotnik J. Stefan Institute, 1000 Ljubljana M. Petrič J. Stefan Institute, 1000 Ljubljana L. E. Piilonen CNP, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 M. Ritter Max-Planck-Institut für Physik, 80805 München M. Röhrken Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie, 76131 Karlsruhe A. Rostomyan Deutsches Elektronen–Synchrotron, 22607 Hamburg S. Ryu Seoul National University, Seoul 151-742 H. Sahoo University of Hawaii, Honolulu, Hawaii 96822 T. Saito Tohoku University, Sendai 980-8578 Y. Sakai High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 S. Sandilya Tata Institute of Fundamental Research, Mumbai 400005 D. Santel University of Cincinnati, Cincinnati, Ohio 45221 L. Santelj J. Stefan Institute, 1000 Ljubljana T. Sanuki Tohoku University, Sendai 980-8578 V. Savinov University of Pittsburgh, Pittsburgh, Pennsylvania 15260 O. Schneider École Polytechnique Fédérale de Lausanne (EPFL), Lausanne 1015 G. Schnell University of the Basque Country UPV/EHU, 48080 Bilbao Ikerbasque, 48011 Bilbao C. Schwanda Institute of High Energy Physics, Vienna 1050 D. Semmler Justus-Liebig-Universität Gießen, 35392 Gießen K. Senyo Yamagata University, Yamagata 990-8560 O. Seon Graduate School of Science, Nagoya University, Nagoya 464-8602 V. Shebalin Budker Institute of Nuclear Physics SB RAS and Novosibirsk State University, Novosibirsk 630090 C. P. Shen Beihang University, Beijing 100191 T.-A. Shibata Tokyo Institute of Technology, Tokyo 152-8550 J.-G. Shiu Department of Physics, National Taiwan University, Taipei 10617 B. Shwartz Budker Institute of Nuclear Physics SB RAS and Novosibirsk State University, Novosibirsk 630090 A. Sibidanov School of Physics, University of Sydney, NSW 2006 F. Simon Max- Planck-Institut für Physik, 80805 München Excellence Cluster Universe, Technische Universität München, 85748 Garching Y.-S. Sohn Yonsei University, Seoul 120-749 S. Stanič University of Nova Gorica, 5000 Nova Gorica M. Starič J. Stefan Institute, 1000 Ljubljana M. Steder Deutsches Elektronen–Synchrotron, 22607 Hamburg T. Sumiyoshi Tokyo Metropolitan University, Tokyo 192-0397 U. Tamponi INFN - Sezione di Torino, 10125 Torino University of Torino, 10124 Torino G. Tatishvili Pacific Northwest National Laboratory, Richland, Washington 99352 Y. Teramoto Osaka City University, Osaka 558-8585 K. Trabelsi High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 T. Tsuboyama High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 M. Uchida Tokyo Institute of Technology, Tokyo 152-8550 S. Uehara High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 T. Uglov Institute for Theoretical and Experimental Physics, Moscow 117218 Moscow Institute of Physics and Technology, Moscow Region 141700 Y. Unno Hanyang University, Seoul 133-791 S. Uno High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801 Y. Usov Budker Institute of Nuclear Physics SB RAS and Novosibirsk State University, Novosibirsk 630090 S. E. Vahsen University of Hawaii, Honolulu, Hawaii 96822 C. Van Hulse University of the Basque Country UPV/EHU, 48080 Bilbao P. Vanhoefer Max-Planck-Institut für Physik, 80805 München G. Varner University of Hawaii, Honolulu, Hawaii 96822 K. E. Varvell School of Physics, University of Sydney, NSW 2006 A. Vinokurova Budker Institute of Nuclear Physics SB RAS and Novosibirsk State University, Novosibirsk 630090 V. Vorobyev Budker Institute of Nuclear Physics SB RAS and Novosibirsk State University, Novosibirsk 630090 M. N. Wagner Justus-Liebig-Universität Gießen, 35392 Gießen C. H. Wang National United University, Miao Li 36003 P. Wang Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049 M. Watanabe Niigata University, Niigata 950-2181 Y. Watanabe Kanagawa University, Yokohama 221-8686 K. M. Williams CNP, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 E. Won Korea University, Seoul 136-713 J. Yamaoka University of Hawaii, Honolulu, Hawaii 96822 Y. Yamashita Nippon Dental University, Niigata 951-8580 S. Yashchenko Deutsches Elektronen–Synchrotron, 22607 Hamburg Y. Yook Yonsei University, Seoul 120-749 C. Z. Yuan Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049 Z. P. Zhang University of Science and Technology of China, Hefei 230026 V. Zhilich Budker Institute of Nuclear Physics SB RAS and Novosibirsk State University, Novosibirsk 630090 A. Zupanc Institut für Experimentelle Kernphysik, Karlsruher Institut für Technologie, 76131 Karlsruhe ###### Abstract The lifetime of the $\tau$-lepton is measured using the process $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$, where both $\tau$-leptons decay to $3\pi\nu_{\tau}$. The result for the mean lifetime, based on $711\,\mathrm{fb}^{-1}$ of data collected with the Belle detector at the $\Upsilon(4S)$ resonance and $60\,\mathrm{MeV}$ below, is $\tau=(290.17\pm 0.53(\mathrm{stat.})\pm 0.33(\mathrm{syst.}))\cdot 10^{-15}\,\mathrm{s}$. The first measurement of the lifetime difference between $\tau^{+}$ and $\tau^{-}$ is performed. The upper limit on the relative lifetime difference between positive and negative $\tau$-leptons is $\|\Delta\tau\|/\tau<7.0\times 10^{-3}$ at 90% CL. ###### pacs: 13.66.Jn, 14.60.Fq ††preprint: KEK Preprint 2013-44 Belle Preprint 2013-26 Submitted to PRL ## I Introduction High precision measurements of the mass, lifetime and leptonic branching fractions of the $\tau$-lepton can be used to test lepton universality LU , which is assumed in the Standard Model. Among the recent experimental results that may manifest the violation of the lepton universality in the case of the $\tau$-lepton, the combined measurement of the ratio of the branching fraction of $W$-boson decay to $\tau\nu_{\tau}$ to the mean branching fraction of $W$-boson decay to $\mu\nu_{\mu}$ and $e\nu_{e}$ by the four LEP experiments stands out: $2{\cal B}(W\rightarrow\tau\nu_{\tau})/({\cal B}(W\rightarrow\mu\nu_{\mu})+{\cal B}(W\rightarrow e\nu_{e}))=1.066\pm 0.025$ LEPEW , which differs from unity by 2.6 standard deviations. The present PDG value of the $\tau$-lepton lifetime $(290.6\pm 1.0)\cdot 10^{-15}\,\mathrm{s}$ PDG is dominated by the results obtained in the LEP experiments LEP . A high-statistics data sample collected at Belle allows us to select $\tau^{+}\tau^{-}$ events where both $\tau$-leptons decay to three charged pions and a neutrino. As explained later, for these events the directions of the $\tau$-leptons can be determined with an accuracy better than that given by the thrust axis of the event. At an asymmetric-energy collider, the laboratory frame angle between the produced $\tau$-leptons is not equal to 180 degrees, so their production point can be determined from the intersection of two trajectories defined by the $\tau$-lepton decay vertices and their momentum directions. The direction of each $\tau$-lepton in the laboratory system can be determined with twofold ambiguity. These special features of the asymmetric-energy B-factory experiments allow a high precision measurement of the $\tau$-lepton lifetime with systematic uncertainties that differ from those of the LEP experiments. Furthermore, Belle’s asymmetric-energy collisions provide a unique possibility to measure separately the $\tau^{+}$ and $\tau^{-}$ lifetimes, which allows us to test $CPT$ symmetry in $\tau$-lepton decays. ## II Description of the measurement method and selection criteria In the following, we use symbols with and without an asterisk for quantities in the $e^{+}e^{-}$ center-of-mass (CM) and laboratory frame, respectively. In the CM frame, $\tau^{+}$ and $\tau^{-}$ leptons emerge back to back with the energy $E^{}_{\tau}$ equal to the beam energy $E^{}_{\mathrm{beam}}$ if we neglect the initial- (ISR) and final-state radiation (FSR). We determine the direction of the $\tau$-lepton momentum in the CM frame as follows. If the neutrino mass is assumed to be zero for the hadronic decay $\tau\rightarrow X\nu_{\tau}$ ($X$ representing the hadronic system with mass $m_{X}$ and energy $E^{}_{X}$), the angle $\theta^{}$ between the momentum $\vec{P}^{}_{X}$ of the hadronic system and that of the $\tau$-lepton is given by: $\cos\theta^{}=\frac{2E^{}_{\tau}E^{}_{X}-m_{\tau}^{2}-m_{X}^{2}}{2P^{}_{X}\sqrt{(E^{}_{\tau})^{2}-m_{\tau}^{2}}}.$ (1) The requirement that the $\tau$-leptons be back to back in the CM can be written as a system of three equations: two linear and one quadratic. For the components $x^{}$, $y^{}$, $z^{}$ of the unit vector $\hat{n}^{}_{+}$ representing the direction of the positive $\tau$-lepton, we write: $\left\\{\begin{array}[]{l}x^{}\cdot P^{}_{1x}+y^{}\cdot P^{}_{1y}+z^{}\cdot P^{}_{1z}=\|P^{}_{1}\|\cos\theta^{}_{1}\\\ x^{}\cdot P^{}_{2x}+y^{}\cdot P^{}_{2y}+z^{}\cdot P^{}_{2z}=-\|P^{}_{2}\|\cos\theta^{}_{2}\\\ (x^{})^{2}+(y^{})^{2}+(z^{})^{2}=1\end{array}\right.$ (2) where $\vec{P}^{}_{1}$ and $\vec{P}^{}_{2}$ are the momenta of the hadronic systems in the CM and $\cos\theta^{}_{i}$ ($i=1,2$) are given by Eq. (1). Index 1 (2) is used for the positive (negative) $\tau$-lepton. There are two solutions for Eq. (2), so the direction $\hat{n}^{}_{+}$ is determined with twofold ambiguity. In the present analysis, we take the mean vector of the two solutions of Eq. (2) as the direction of the $\tau$-lepton in CM. The analysis of MC simulated events shows that there is no bias due to this choice. Figure 1: The schematic view of the $\tau^{+}\tau^{-}$ event in the laboratory frame. Each direction $\hat{n}_{\pm}^{}$ is converted to a four-momentum using the $e^{\pm}$ beam energy and the $\tau$ mass. Both four-momenta are then boosted into the laboratory frame, each passing through the corresponding $\tau$ decay vertex $\vec{V}_{i}$ that is determined by the three pion-daughter tracks (see Fig. 1). We approximate the trajectory of $\tau$-leptons in the magnetic field of the Belle detector with a straight line. Due to the finite detector resolution, these straight lines do not intersect at the $\tau^{+}\tau^{-}$ production point. The three-dimensional separation between these lines is characterized by the distance $dl$ between the two points ($\vec{V}_{01}$ and $\vec{V}_{02}$) of closest approach. The typical size of $dl$ is $\sim 0.01\,\mathrm{cm}$. For the production point of each $\tau$-lepton, we take the points $\vec{V}_{01}$ and $\vec{V}_{02}$. The flight distance $l_{1}$ ($l_{2}$) of the $\tau^{+}$ ($\tau^{-}$) in the laboratory frame is defined as the distance between the points $\vec{V}_{1}$ and $\vec{V}_{01}$ ($\vec{V}_{2}$ and $\vec{V}_{02}$). The proper time $t$ (the product of the speed of light and the decay time of $\tau$-lepton) for the positive $\tau$-lepton is equal to the distance $l_{1}$ divided by its relativistic kinematic factor $\beta\gamma$ in the laboratory frame: $t_{1}=l_{1}/(\beta\gamma)_{1}$. The corresponding parameter for the negative $\tau$-lepton is $t_{2}=l_{2}/(\beta\gamma)_{2}$. The analysis presented here is based on the data collected with the Belle detector Belle at the KEKB asymmetric-energy $e^{+}e^{-}$ ($3.5$ on $8\,\mathrm{GeV}$) collider KEKB operating at the $\Upsilon(4S)$ resonance and $60\,\mathrm{MeV}$ below. The total integrated luminosity of the data sample used in the analysis is $711\,\mathrm{fb}^{-1}$. Two inner detector configurations were used. A $2.0\,\mathrm{cm}$ beampipe and a 3-layer silicon vertex detector (SVD1) were used for the first sample of $157\,\mathrm{fb}^{-1}$, while a 1.5 cm beampipe, a 4-layer silicon detector (SVD2) and a small-cell inner drift chamber were used to record the remaining $554\,\mathrm{fb}^{-1}$ SVD2 . The integrated luminosity of the data sample at the energy below the $\Upsilon$(4S) resonance is about 10% of the total data sample. All analyzed distributions for the on- and off-resonance data coincide within the statistical uncertainties with each other; this justifies our combination of the on- and off-resonance $t$ distributions in the present analysis. The following requirements are applied for the selection of the $\tau^{+}\tau^{-}$ events where both $\tau$-leptons decay into three charged pions and a neutrino: 1\. there are exactly six charged pions with zero net charge and there are no other charged tracks; 2\. the $K^{0}_{S}$ mesons, $\Lambda$-hyperons and $\pi^{0}$ are found by $V^{0}$ V0 and $\pi^{0}$ reconstruction algorithms and the event is discarded if any of these are seen; 3\. the number of photons should be smaller than six and their total energy should be less than $0.7\,\mathrm{GeV}$; 4\. the thrust value of the event in the CM frame is greater than 0.9; 5\. the square of the transverse momentum of the $6\pi$ system is required to be greater than $0.25\,(\mathrm{GeV}/c)^{2}$ to suppress the $e^{+}e^{-}\rightarrow e^{+}e^{-}6\pi$ two-photon events; 6\. the mass $M(6\pi)$ of the $6\pi$ system should fulfill the requirement $4\,\mathrm{GeV}/c^{2}<M(6\pi)<10.25\,\mathrm{GeV}/c^{2}$ to suppress other background events; 7\. there should be three pions (triplet) with net charge equal to $\pm 1$ in each hemisphere (separated by the plane perpendicular to the thrust axis in the CM); 8\. the pseudomass (see the definition in Ref. taumass ) of each triplet of pions must be less than $1.8\,\mathrm{GeV}/c^{2}$; 9\. each pion-triplet vertex-fit quality must satisfy $\chi^{2}<20$; 10\. the discriminant $D$ of Eq. (2) should satisfy $D>-0.05$ (with slightly negative values arising from experimental uncertainties; if this happens, we use $D=0$ when solving the equation); 11\. the distance of closest approach must satisfy $dl<0.03\,\mathrm{cm}$ to reject events with large uncertainties in the reconstructed momenta and vertex positions. All of these selection criteria are based on a study of the signal and background Monte Carlo (MC) simulated events. For the signal MC sample, we use $\tau^{+}\tau^{-}$ events produced by the KKMC generator KKMC with the mean lifetime $\langle{\tau}\rangle=87.11\,\mu\mathrm{m}$ that are then fed to the full detector simulation based on GEANT 3 GEANT . These events are passed through the same reconstruction procedures as for the data. For the background estimation, we use the MC samples of events generated by the EVTGEN program EVTGEN , which correspond to the one-photon annihilation diagram $e^{+}e^{-}\rightarrow q\bar{q}$, where $q\bar{q}$ are $u\bar{u}$, $d\bar{d}$, $s\bar{s}$ ($uds$ events), $c\bar{c}$ (charm events), and $e^{+}e^{-}\rightarrow\Upsilon(4S)\rightarrow B^{+}B^{-},B^{0}\bar{B^{0}}$ (beauty events). All these events are passed through the full detector simulation and reconstruction procedures. The statistics in these MC samples are equivalent to the integrated luminosity of the data, i.e., the number of events of a given category is equal to the product of the integrated luminosity of the data and the expected cross section from theory. For the estimation of the background from the process $\gamma\gamma\rightarrow hadrons$ ($\gamma\gamma$ events), we use events generated by PYTHIA PYTHIA that are subjected to the afore-mentioned simulation and reconstruction procedures. In addition to the above MC samples, we also use two $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}$ MC samples, generated by KKMC, where both $\tau$-leptons are forced to decay into three charged pions and a neutrino. The mean lifetimes for these two samples are $84$ and $90\,\mu\mathrm{m}$, which are about $10\sigma$ below and above the PDG value. These two samples are also passed through the same detector simulation and reconstruction procedures. ## III Analysis of the experimental results In the measured proper time distribution, the exponential behavior is smeared by the experimental resolution. This resolution has been studied with MC simulation. The following samples are used, each one with a slightly different time resolution: with the SVD1 and SVD2 geometries and three different values of the mean $\tau$-lepton lifetime. For all MC samples, the resolution function is found to be described well by the expression: $\begin{array}[]{l}R(\Delta t)=(1-A\Delta t)e^{-(\Delta t-t_{0})^{2}/2\sigma^{2}}\mathrm{,where}\\\ \Delta t=t_{\mathrm{reconstructed}}-t_{\mathrm{true}},\quad\Delta t_{0}=\Delta t-t_{0},\\\ \sigma=a+b\|\Delta t_{0}\|^{1/2}+c\|\Delta t_{0}\|+d\|\Delta t_{0}\|^{3/2}\end{array}$ (3) The parameters $t_{0}$, $a$, $b$, $c$ and $d$ are allowed to vary freely in the fit, while the asymmetry $A=2.5\,\mathrm{cm}^{-1}$ is fixed because of its strong correlation with the lifetime parameter $\tau$. An example of the fitting of the resolution distribution for the MC sample with mean $\tau$-lepton lifetime equal to $87.11\,\mu\mathrm{m}$ and for the sum of the SVD1- and SVD2-geometry data sets by the function Eq. (3) is shown in Fig. 2. The goodness of fit is $\chi^{2}/ndf=770.8/794$. In an alternate fit where the parameter $A$ is allowed to vary freely, its best-fit value is $(2.5\pm 0.2)\,\mathrm{cm}^{-1}$. All of the other resolution distributions are described with the same level of quality. Figure 2: Distribution of the difference between the reconstructed and true $t$ values for $\tau$-leptons (obtained from an MC sample) for the combined SVD1- and SVD2-geometry data sets. The line is the result of the fit to Eq. (3). The distribution of residuals [(data–fit)/error] for the fit is shown in the bottom panel. After applying all the selection criteria, the contamination of the background in the data is about two percent. The dominant background arises from $uds$ events. For these events, all six pions emerge (typically) from one primary vertex and these $uds$ events are similar to the $\tau^{+}\tau^{-}$ events with zero lifetime. Using the MC, we check that the decay time distributions of $uds$-events that pass the selection criteria are well described by the resolution function of Eq. (3). The same behavior is found for $\gamma\gamma$ events, whose fraction in all the selected events is about $1.4\cdot 10^{-4}$. Other sources of background contribute to the selected data sample at the per mille level. The measured proper time distribution is parameterized by: $F(t)=N\int{e^{-t^{\prime}/\tau}R(t-t^{\prime})dt^{\prime}}+A_{uds}R(t)+B_{cb}(t),$ (4) where the resolution function $R(t)$ is given by Eq. (3), $A_{uds}$ is the normalization of the combined $uds$ and $\gamma\gamma$ background and $B_{cb}(t)$ is the background distribution due to charm and beauty events. The shapes and yields of the backgrounds ($B_{cb}(t)$, $A_{uds}$) are fixed from the MC simulation; the free parameters of the fit are the normalization $N$, the $\tau$-lepton lifetime $\tau$ and the five parameters of the resolution function $t_{0}$, $a$, $b$, $c$ and $d$. Figure 3: The measured proper time $t$ distribution for the data (filled circles with errors). The black line is the result of the fit by Eq. (4). The red histogram is the MC prediction for the sum of the $uds$ and $\gamma\gamma$ background contributions. The magenta line is the contribution for $uds+\gamma\gamma$ obtained in the fit. The blue histogram is the MC prediction for the sum of the charm and beauty background contributions. The blue line is the smoothed distribution of the charm and beauty contributions that is used in the fit. The distribution of residuals [(data–fit)/error] for the fit is shown in the bottom panel. The result of the fit of the experimental data to Eq. (4) is shown in Fig. 3, together with the contributions from the sum of $uds$ and $\gamma\gamma$ events and the sum of charm and beauty events. The curves on these contributions are the result of the fit with Eq. (4), $A_{uds}R(t)$ function (with fixed value of $A_{uds}$) for $uds$ plus $\gamma\gamma$ events and the fixed sum of two Gaussians for charm plus beauty events. The relation of the parameter $\tau$ in Eq. (4) to the generated value of the $\tau$-lepton mean lifetime is analyzed using three MC $\tau^{+}\tau^{-}$ samples with the mean lifetime values of $84$, $87.11$ and $90\,\mu\mathrm{m}$. The dependence of parameter $\tau$ on the input mean lifetime value $\langle{\tau}\rangle$ is found to be linear; $(\tau-87)=(0.97\pm 0.03)(\langle{\tau}\rangle-87)+(0.001\pm 0.07)$ [in units of $\,\mu\mathrm{m}$ ] with $\chi^{2}$ of 0.2. Bias arising from the selection criteria is checked by fitting the proper time distribution for the signal MC sample before and after applying cuts, and no bias is found for all the selection criteria listed above. To check that the fitting procedure gives the correct estimation of the input lifetime value for different resolution functions, we perform the fits of the decay time distributions for MC samples with a lifetime of $87.11\,\mu\mathrm{m}$ for the sum of the SVD1 and SVD2 samples, SVD1 and SVD2 samples separately, and for samples with lifetimes equal to $84$ and $90\,\mu\mathrm{m}$. In all cases, the value of the parameter $\tau$ is equal to the slope of the exponential distribution of the selected events at the generation level within the statistical error of the parameter $\tau$. The value of the parameter $\tau$ obtained from the fit to the real data is $86.99\pm 0.16\,\mu\mathrm{m}$. The conversion of this parameter to the value of the $\tau$-lepton mean lifetime using the straight-line parameters of the fit described above gives the same value: $86.99\pm 0.16\,\mu\mathrm{m}$. The error here is statistical. ## IV Analysis of systematic uncertainties The following sources of systematic uncertainties are considered and summarized in Table 1. Table 1: Summary of systematic uncertainties Source \| $\Delta\langle{\tau}\rangle$ ($\mu m$) ---\|--- SVD alignment \| 0.090 Asymmetry fixing \| 0.030 Beam energy and ISR/FSR description \| 0.024 Fit range \| 0.020 Background contribution \| 0.010 $\tau$-lepton mass \| 0.009 Total \| 0.101 A study of the influence of the SVD misalignment on the systematic shift in the $\tau$-lepton lifetime measurement is performed in the following way. We use $4.8$ M generated $\tau^{+}\tau^{-}$ events that decay with the $3\pi\nu_{\tau}-3\pi\nu_{\tau}$ topology and standard Belle SVD alignment. After all selection cuts, about $1.2$ M events remain (compared with $1.1$ M events in the data). We shift the sensitive elements of SVD along the $X/Y/Z$ axes by sampling from a Gaussian function with $\sigma=10\,\mu\mathrm{m}$ and rotation around these axes by sampling from a Gaussian function with $\sigma=0.1\,\mathrm{mrad}$. The values of $10\,\mu\mathrm{m}$ and $0.1\,\mathrm{mrad}$ are obtained from the dedicated studies of SVD alignment SVD2 . We prepare the following decay time MC distributions: with default alignment ($4.8$ M generated events), one sample with misalignment according to the aforementioned shifts and rotations ($4.8$ M generated events), several samples with misalignments according to these shifts and rotations with fewer generated events; all these samples have the same events at the generator level. The maximal difference of the parameter $\tau$ obtained in these fits is $0.07\,\mu\mathrm{m}$. This is due to the possible effect of misalignment and limited MC statistics. We also perform global SVD shifts and rotations with respect to the CDC by $20\,\mu\mathrm{m}$ and $1\,\mathrm{mrad}$, respectively. The values of $20\,\mu\mathrm{m}$ and $1\,\mathrm{mrad}$ are conservative estimates from the SVD alignment study. The variation of the $\tau$ parameter is within $0.06\,\mu\mathrm{m}$ for these shifts. We take the value $\sqrt{0.07^{2}+0.06^{2}}=0.09\,\mu\mathrm{m}$ for the systematics due to the SVD misalignment. For an additional check of the alignment of the tracking detectors, we divide our data sample into two non-intersecting samples by the azimuthal ($\phi$) angle of the momentum direction of the positive $\tau$-lepton. In the first sample (vertical), the direction of the positive $\tau$-lepton should have $\phi$ between $45$ and $135$ degrees or between $225$ and $315$ degrees. The second sample (horizontal) contains all the remaining events. The obtained $\tau$ parameters are the same within statistical errors, so we do not assign additional systematics due to the azimuthal dependence of the tracking system alignment. The systematic uncertainty due to fixing the parameter $A=2.5\,\mathrm{cm}^{-1}$ is estimated by removing the asymmetry term $(1-A\Delta t)$ in the resolution function in Eq. (3). The difference in the obtained lifetime, which is equal to $0.03\,\mu\mathrm{m}$, is taken as a systematic uncertainty. For the estimation of the accuracy of the initial and final state radiation description by the KKMC generator, we analyze the distributions of $M(\mu^{+}\mu^{-})c^{2}-2E^{}_{\mathrm{beam}}$ for $e^{+}e^{-}\rightarrow\mu^{+}\mu^{-}$ events for the data and KKMC events passed through the full Belle simulation and reconstruction procedure. Due to the ISR and FSR, these distributions are asymmetric and their maxima are shifted from zero to the left. If the KKMC description of ISR and FSR energy spectrum is harder or softer than for the data, we would observe the MC peak position shifted from the one in the data. The result of our comparison of the data and MC gives the difference of peak positions between the data and MC of $(3\pm 2)\,\mathrm{MeV}$. We take the relative error $3\,\mathrm{MeV}/10.58\,\mathrm{GeV}=2.8\cdot 10^{-4}$ as a combined uncertainty from the ISR and FSR description, beam energy calibration and the calibration of the tracking system. The variation of the fit range within about $30\%$ of that shown in Fig. 3 contributes an uncertainty on $\tau$ of $\pm 0.02\,\mu\mathrm{m}$. The demonstration of the stability of the obtained result to the choice of the selection cuts is shown in Fig. 4. Figure 4a shows the dependence of the fitted parameter $\tau$ on the value of the cut on $dl$ for data and MC. Figure 4b shows the measured value of the $\tau$-lepton lifetime as a function of the value of the $dl$-cut after the linear MC-determined calibration of the parameter $\tau$. One can see that this dependence in data is very well reproduced by MC. Figure 4: Stability when varying the value of the $dl$-cut. a) The dependence of the fitted parameter $\tau$ on the value of the $dl$-cut for data (filled black circles) and MC (open red squares); the errors are of the same size as the symbols. b) The measured value of the $\tau$-lepton lifetime as a function of the value of the $dl$-cut; the error bar is the statistical error of the data. During the fit of the real data, the level of the background contribution (parameter $A_{uds}$) is fixed to the nominal value predicted by the MC in a “nominal” fit. The contribution to the systematic error of the $\langle{\tau}\rangle$ value due to the uncertainty of the background level is tested by changing the background level in the range of the uncertainty of the $q\bar{q}$ continuum and other backgrounds, from $-50\%$ to $+150\%$. This range is estimated conservatively from the control sample with looser selection criteria. The maximal variation of the $\tau$ parameter is $0.01\,\mu\mathrm{m}$. The relative uncertainty due to the accuracy of the $\tau$-lepton mass PDG is $(0.16\,\mathrm{MeV}/c^{2})/$ $(1776.82\,\mathrm{MeV}/c^{2})=9.0\cdot 10^{-5}$. To check the stability of the result for the different periods of Belle operation and vertex detector geometries, we repeat the analysis for three subsamples of the data. The obtained results are consistent within statistical errors. ## V Lifetime difference between positive and negative $\tau$-leptons The present PDG listings provide only the average lifetime of the positive and negative $\tau$-leptons. Our measurement determines the lifetimes for positive and negative $\tau$-leptons separately. The difference of $\langle{\tau}\rangle$ for positive and negative $\tau$-leptons obtained in the corresponding fits is $(0.07\pm 0.33)\,\mu\mathrm{m}$. Most of the sources of systematic uncertainties affect the result for positive and negative $\tau$-leptons in the same way, so their contributions to the lifetime difference cancel. The upper limit on the relative lifetime difference is calculated according to Ref. FC as $\|\langle{\tau_{\tau^{+}}}\rangle-\langle{\tau_{\tau^{-}}}\rangle\|/\langle{\tau_{\tau}}\rangle<7.0\times 10^{-3}\mathrm{\,\,at\,\,90\%\,\,CL.}$ (5) The systematic uncertainty of the lifetime difference is at least one order of magnitude smaller than the statistical one, and is neglected. ## VI Conclusions In summary, the $\tau$-lepton lifetime has been measured using the technique of the direct decay time measurement in fully kinetically reconstructed $e^{+}e^{-}\rightarrow\tau^{+}\tau^{-}\rightarrow 3\pi\nu_{\tau}\ 3\pi\nu_{\tau}$ events. The obtained result for the product of the mean lifetime and speed of light is $\langle{\tau_{\tau}}\rangle=[86.99\pm 0.16(\mathrm{stat.})\pm 0.10(\mathrm{syst.})]\,\mu\mathrm{m},$ (6) or in units of seconds $(290.17\pm 0.53(\mathrm{stat.})\pm 0.33(\mathrm{syst.}))\cdot 10^{-15}\,\mathrm{s}.$ The first measurement of the lifetime difference between $\tau^{+}$ and $\tau^{-}$ is performed. The obtained upper limit on the relative lifetime difference between positive and negative $\tau$-leptons is $\|\langle{\tau_{\tau^{+}}}\rangle-\langle{\tau_{\tau^{-}}}\rangle\|/\langle{\tau_{\tau}}\rangle<7.0\times 10^{-3}$ at 90% CL. ## VII Acknowledgments We thank the KEKB group for excellent operation of the accelerator; the KEK cryogenics group for efficient solenoid operations; and the KEK computer group, the NII, and PNNL/EMSL for valuable computing and SINET4 network support. We acknowledge support from MEXT, JSPS and Nagoya’s TLPRC (Japan); ARC and DIISR (Australia); FWF (Austria); NSFC (China); MSMT (Czechia); CZF, DFG, and VS (Germany); DST (India); INFN (Italy); MEST, NRF, GSDC of KISTI, and WCU (Korea); MNiSW and NCN (Poland); MES and RFAAE (Russia); ARRS (Slovenia); IKERBASQUE and UPV/EHU (Spain); SNSF (Switzerland); NSC and MOE (Taiwan); and DOE and NSF (USA). ## References (1) Y.S. Tsai, Phys. Rev. D 4, 2821 (1971);H.B. Thacker and J.J. Sakurai, Phys. Lett. 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Sumisawa et al. (Belle Collaboration), Phys. Rev. Lett. 95, 061801 (2005). * (9) K. Belous et al. (Belle Collaboration), Phys. Rev. Lett. 99, 011801 (2007). * (10) S.Jadach, B.F.L.Ward and Z.Wa̧s, Comp. Phys. Commun. 130, 260 (2000). * (11) R. Brun et al. GEANT 3.21. Report No. CERN DD/EE/84-1 (1984). * (12) D.J. Lange, Nucl. Instr. and Meth. A 462, 152 (2001). * (13) T. Sjöstrand et al., Comp. Phys. Commun. 135, 238 (2001). * (14) G.J. Feldman and R.D. Cousins, Phys. Rev. D 57, 3873 (1998); J. Conrad et al., Phys. Rev. D 67, 012002 (2003).	arxiv-papers	2013-10-31T13:55:26	2024-09-04T02:49:53.145923	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Belle Collaboration: K. Belous, M. Shapkin, A. Sokolov, I. Adachi, H.\n Aihara, D. M. Asner, V. Aulchenko, A. M. Bakich, A. Bala, B. Bhuyan, A.\n Bobrov, A. Bondar, G. Bonvicini, A. Bozek, M. Bra\\v{c}ko, T. E. Browder, D.\n \\v{C}ervenkov, V. Chekelian, A. Chen, B. G. Cheon, K. Chilikin, R. Chistov,\n K. Cho, V. Chobanova, Y. Choi, D. Cinabro, J. Dalseno, Z. Dole\\v{z}al, D.\n Dutta, S. Eidelman, D. Epifanov, H. Farhat, J. E. Fast, T. Ferber, V. Gaur,\n S. Ganguly, A. Garmash, R. Gillard, Y. M. Goh, B. Golob, J. Haba, T. Hara, K.\n Hayasaka, H. Hayashii, Y. Hoshi, W.-S. Hou, T. Iijima, K. Inami, A. Ishikawa,\n R. Itoh, T. Iwashita, I. Jaegle, T. Julius, E. Kato, H. Kichimi, C. Kiesling,\n D. Y. Kim, H. J. Kim, J. B. Kim, M. J. Kim, Y. J. Kim, K. Kinoshita, B. R.\n Ko, P. Kody\\v{s}, S. Korpar, P. Kri\\v{z}an, P. Krokovny, T. Kuhr, A. Kuzmin,\n Y.-J. Kwon, J. S. Lange, S.-H. Lee, J. Libby, D. Liventsev, P. Lukin, D.\n Matvienko, H. Miyata, R. Mizuk, G. B. Mohanty, T. Mori, R. Mussa, Y.\n Nagasaka, E. Nakano, M. Nakao, M. Nayak, E. Nedelkovska, C. Ng, N. K. Nisar,\n S. Nishida, O. Nitoh, S. Ogawa, S. Okuno, S. L. Olsen, W. Ostrowicz, G.\n Pakhlova, C. W. Park, H. Park, H. K. Park, T. K. Pedlar, R. Pestotnik, M.\n Petri\\v{c}, L. E. Piilonen, M. Ritter, M. R\\\"ohrken, A. Rostomyan, S. Ryu, H.\n Sahoo, T. Saito, Y. Sakai, S. Sandilya, D. Santel, L. Santelj, T. Sanuki, V.\n Savinov, O. Schneider, G. Schnell, C. Schwanda, D. Semmler, K. Senyo, O.\n Seon, V. Shebalin, C. P. Shen, T.-A. Shibata, J.-G. Shiu, B. Shwartz, A.\n Sibidanov, F. Simon, Y.-S. Sohn, S. Stani\\v{c}, M. Stari\\v{c}, M. Steder, T.\n Sumiyoshi, U. Tamponi, G. Tatishvili, Y. Teramoto, K. Trabelsi, T. Tsuboyama,\n M. Uchida, S. Uehara, T. Uglov, Y. Unno, S. Uno, Y. Usov, S. E. Vahsen, C.\n Van Hulse, P. Vanhoefer, G. Varner, K. E. Varvell, A. Vinokurova, V.\n Vorobyev, M. N. Wagner, C. H. Wang, P. Wang, M. Watanabe, Y. Watanabe, K. M.\n Williams, E. Won, J. Yamaoka, Y. Yamashita, S. Yashchenko, Y. Yook, C. Z.\n Yuan, Z. P. Zhang, V. Zhilich, A. Zupanc", "submitter": "Mikhail Shapkin", "url": "https://arxiv.org/abs/1310.8503" }
1310.8506	# Latitudinal Connectivity of Ground Level Enhancement Events N. Gopalswamy1, P. Mäkelä1,2 1NASA Goddard Space Flight Center, Greenbelt, MD 20771, U.S.A. 2The Catholic University of America, Washington, DC 20064, U.S.A. ###### Abstract We examined the source regions and coronal environment of the historical ground level enhancement (GLE) events in search of evidence for non-radial motion of the associated coronal mass ejection (CME). For the 13 GLE events that had source latitudes $>$30∘ we found evidence for possible non-radial CME motion due to deflection by large-scale magnetic structures in nearby coronal holes, streamers, or pseudo streamers. Polar coronal holes are the main source of deflection in the rise and declining phases of solar cycles. In the maximum phase, deflection by large-scale streamers or pseudo streamers overlying high- latitude filaments seems to be important. The B0 angle reduced the ecliptic distance of some GLE source regions and increased in others with the net result that the average latitude of GLE events did not change significantly. The non-radial CME motion is the dominant factor that reduces the ecliptic distance of GLE source regions, thereby improving the latitudinal connectivity to Earth. We further infer that the GLE particles must be accelerated at the nose part of the CME-driven shocks, where the shock is likely to be quasi- parallel. ## 1 Introduction Ground level enhancement (GLE) in solar energetic particle (SEP) events represents the highest energy ($\sim$GeV) particles accelerated during large solar eruptions. GLE particles are thought to be accelerated by the flare process or the CME-driven shock. Extensive CME observations became available only during Solar Cycle (SC) 23 (Cliver 2006; Gopalswamy et al. 2012a). Among the 16 GLEs observed during SC 23, fast and wide CMEs were observed in all but one GLE. The 1998 August 24 GLE was the exception, occurring when the SOHO spacecraft was temporarily disabled (no CME observations). However, a fast interplanetary CME was observed, implying a fast CME near the Sun. Thus a one- to-one correspondence between GLEs and fast CMEs was established (Gopalswamy et al. 2012a). Based on metric type II radio burst onset, flare rise time, and CME speed, these authors found that the CME shocks form very close to the Sun ($<$0.5 solar radii above the solar surface) and that the shocks have sufficient time to accelerate particles to GeV energies. These observations strongly support the shock acceleration mechanism. Even though a large number of type II radio bursts were observed during SC 24 (indicating the occurrence of CME-driven shocks), there was only one GLE (on May 17, 2012) so far. Gopalswamy et al. (2013) suggested that poor latitudinal connectivity may be one of the main reasons for the very low occurrence rate of GLEs. In this paper, we present additional evidence in support of this connectivity constraint by considering all GLE events since their discovery in 1942 (Forbush, 1946). Heliographic coordinates of the associated flares are known for almost all CMEs (Cliver et al. 1982; Cliver, 2006; Gopalswamy et al. 2012a; Nitta et al. 2012; Miroshnichenko et al. 2013). CMEs were discovered in 1971, but routine information became available only after the launch of SOHO in December 1995. Therefore, we cannot determine CME non-radial motion for most GLEs. However, we identify coronal holes and streamers that can deflect the GLE-causing CMEs. Figure 1.: (a) A coronal hole map derived from the Kitt Peak Vacuum Telescope (KPVT) data. The blue and red patches correspond to coronal holes with positive and negative polarities, respectively. The flare location (N11W76) of the 2012 May 17 GLE at 02:40 UT is shown by the white circle. The large coronal hole to the north of the GLE source region is consistent with the southward (non-radial) motion of the GLE-associated CME. (b) The white-light CME in a SOHO/LASCO image taken by its C2 telescope at 01:48 UT. Superposed on the coronagraph image is a 193 Å EUV image taken by the Solar Dynamics Observatory s Atmospheric Imaging Assembly (AIA) showing the flare location close to the northeast limb. ## 2 The 2012 May 17 GLE Event The first and only GLE event of SC 24 as of this writing originated from NOAA active region 11476 in the northwest quadrant of the Sun (N11W76). Coronagraph images from SOHO and STEREO show that the CME moved non-radially, heading in the southwest direction (Gopalswamy et al. 2013). Forward-fitting of the coronagraph images using Thernisien (2011) flux rope model indicated that the effective location of the CME source region was S07W76, indicating that the CME was deflected to the south by about 18∘. When we examined the coronal environment of the GLE source a few hours before the eruption, we found a large coronal hole close to the active region in the northeast direction (see Fig. 1a). The southern end of the coronal hole was less than 10∘ away from the active region. The approximate centroid of the coronal hole was at N30W60. The CME motion is thus consistent with a deflection by the coronal hole (Fig. 1b). Such deflections are thought to cause “driverless” shocks observed at Earth in extreme cases (Gopalswamy et al. 2009; 2010) and decide whether a CME flux rope appears as a magnetic cloud or non-cloud ejecta in in-situ observations (Xie et al. 2013; Mäkelä et al. 2013). The configuration in Fig. 1 is the motivation for us to look at historical GLE events for possible non-radial motion of the associated CMEs. The SC 24 GLE was associated with only an M5.1 flare, which is well below the typical soft X-ray flare size for SC 23 GLEs (X3.8). Historically, there were only two other GLEs with flare size smaller than the SC 24 GLE: GLE 33 on 1979 August 21 with a C6 flare and GLE 35 on 1981 May 10 with a M1 flare (Cliver, 2006; Gopalswamy et al. 2012a). The CME associated with the SC 24 GLE was very fast ( 2000 km/s), capable of driving a strong shock, and hence produce GLE particles (Gopalswamy et al. 2013). By examining the source regions of several eruptions from SC 24 that had the same the source longitude range as this GLE event, Gopalswamy et al. (2013) concluded that the non-GLE source regions had latitudinal distance from the ecliptic: 32∘ compared to 9∘ for SC 23 GLEs from similar source longitude (13∘ when all SC 23 GLEs were considered). Unfavorable solar B0-angles and non- radial CME motions were found to be the reasons for the increased distance from the ecliptic. The B0 angle is the heliographic latitude of the central point of the solar disk and can vary from -7∘.23 to + 7∘.23. B0 represents the fact that the ecliptic plane (where Earth is located) and the Sun’s equatorial plane are not aligned. Thus for a northern (southern) CME source with negative (positive) B0 angle the magnetic connectivity to Earth worsens. B0 was -2∘.4 for the SC 24 GLE, making the latitudinal distance to the ecliptic as 4∘.6, which is less than the average distance for SC 23 GLE source regions (13∘). Thus the latitudinal connectivity of the shock nose is important for GLEs. ## 3 Source Regions of Historical GLE Events The historical GLE events are shown as a “butterfly” diagram in Fig. 2. The source locations were taken from the published literature (see e.g., Cliver et al. 1982; Cliver 2006; Gopalswamy et al. 2012a; Miroshnichenko et al. 2013). The first two GLEs were observed in SC 17 on 1942 February 28 and March 7, respectively. Forbush (1946) reported on these two GLEs and on the third GLE (1946 July 25), which occurred in SC 18. For the five cycles starting from SC 19, there were roughly a dozen GLEs per cycle, but we have only one GLE event in SC 24. Figure 2b shows that the source latitudes are below 40∘, as expected because the energy needed for the GLE events are available only in sunspot regions. The latitude distribution is bimodal because of the northern and southern active region belts. The average latitudes are -15∘.3 (south) and 19∘.6 (north). When we corrected for the B0 angle, the average ecliptic distances did not change significantly: -17∘.9 and 19∘.2 in the southern and northern hemispheres, respectively. These average latitudes are slightly larger than the 13∘ obtained for SC 23 events (after accounting for non-radial CME motion). Unfortunately, we have no CME observations for most of the pre- SOHO GLEs, but we shall show that the effective ecliptic distances are expected to be lower for pre-SOHO GLEs based on their coronal environment. The source longitudes of historical GLE events (Fig. 2d) range from E88 to W150, but there were only two events beyond E15 (E88 and E39). On the other hand, there were a dozen events behind the west limb. The longitude distribution suggests that the magnetic connectivity of Earth to the source region is very important. Figure 2.: (a) GLE latitude as a function of time for GLEs since their discovery in 1942. The approximate times of solar maxima of cycles 18–24 are denoted by vertical dashed lines (from http://sidc.oma.be/DATA/monthssn.dat). (b) The distribution of flare latitudes of the GLE events as in (a). (c) The latitude distribution corrected for the B0 angle. (d) The distribution of flare longitudes. The bin size is 6 degrees for all the distributions. In the latitude distribution, there are only 70 events because the latitude of the backside event 39 is unknown. The two GLEs with the easternmost source longitudes are on 1960 September 3 and 1978 April 29. In the longitude distribution, the $>$90∘ bin (the dark bin) includes all behind the west-limb events. ### 3.1 Higher-Latitude GLEs The GLE events with latitudes $>$30∘ are of interest because they clearly contradict our conclusion that the ecliptic distance of the source regions needs to be small for a GLE event. These higher-latitude GLEs are listed in Table 1 along with the flare location, B0 angle, latitudinal distance to the ecliptic ($\lambda$e), Carrington Rotation (CR) number with its starting day, and whether a deflecting structure (coronal hole, streamer, or pseudo streamer) existed poleward of the source region. We have included GLEs with source latitude $>$30∘ either before accounting for B0 angle or after. Before B0 correction, there were nine events with source latitude $>$30∘ and all were from the north. After B0 correction, GLEs 32 and 58 had $\lambda$e $<$30∘. GLE 15 had similar trend, but $\lambda$e $\sim$31∘. After B0 correction, 4 events from the south (GLEs 42–45) had $\lambda$e $>$30∘ (i.e, the connectivity worsened). The B0 angle was less than 2∘ and hence insignificant for six events (GLEs 47–52). GLEs with either flare latitude or $\lambda$e exceeding 30∘ fall into three groups depending on the SC phase: GLEs 15, 32, and 58 were rise-phase events, GLEs 51 and 52 were declining-phase events, and the remaining 8 were maximum- phase events. Accordingly, one expects different coronal environment for the two groups. For example, strong polar coronal holes (PCH) prevail in the rise phase and the sunspots appear at relatively higher latitudes. Table 1.: Historical GLE events with flare latitudes $>$30∘ GLE # \| Date \| Flare Loc. \| B0 \| $\lambda$e \| CR # \| Start Date \| Deflector ---\|---\|---\|---\|---\|---\|---\|--- 15 \| 1966/07/07 \| N35W48 \| +3.55 \| N31 \| 1509 \| 22.0747 \| PCH? 32 \| 1978/09/23 \| N35W50 \| +7.02 \| N28 \| 1673 \| 20.0572 \| PCH 42 \| 1989/09/29 \| S24W105 \| +6.80 \| S31 \| 1820 \| 11.5184 \| S+CH 43 \| 1989/10/19 \| S25E09 \| +5.54 \| S31 \| 1821 \| 8.7953 \| S 44 \| 1989/10/22 \| S27W32 \| +5.29 \| S32 \| 1821 \| 8.7953 \| S 45 \| 1989/10/24 \| S29W57 \| +5.11 \| S34 \| 1821 \| 8.7953 \| S 47 \| 1990/05/21 \| N34W37 \| -1.94 \| N36 \| 1829 \| 15.2476 \| PS 48 \| 1990/05/24 \| N36W76 \| -1.59 \| N38 \| 1829 \| 15.2476 \| PS 49 \| 1990/05/26 \| N35W103 \| -1.35 \| N36 \| 1829 \| 15.2476 \| PS 50 \| 1990/05/28 \| N35W120 \| -1.11 \| N36 \| 1829 \| 15.2476 \| PS 51 \| 1991/06/11 \| N32W15 \| +0.57 \| N31 \| 1843 \| 1.0619 \| PCH 52 \| 1991/06/15 \| N36W70 \| +1.07 \| N35 \| 1843 \| 1.0619 \| PCH 58 \| 1998/08/24 \| N35E09 \| +7.01 \| N28 \| 1939 \| 1.3214 \| PCH We examined coronal hole maps available from Kitt Peak National Observatory (KPNO): ftp://nsokp.nso.edu/kpvt/synoptic/choles/ for Carrington Rotations 1633 (1975 September 25) to 1987 (2002 March 2). We also used Yohkoh soft X-ray data for confirmation in SC 23. Finally, we also examined the H-alpha synoptic charts (ftp://ftp.ngdc.noaa.gov/STP/SOLAR_DATA/SGD_PDFversion/)from the Solar Geophysical Data to understand the environment of GLE source regions. Figure 3 shows the source regions of GLEs 51, 52, and 58 on He 10830 Å coronal hole maps. Clearly there were polar coronal holes (PCH) in each of the three cases. For GLE 58, there is additional confirmation from Yohkoh Soft X-ray Telescope (SXT) images, which show a prominent PCH in the north (see also Fig. 3 in Gopalswamy et al. 2012a). KPNO maps show north PCH for GLE 32 also. Sheeley (1980) reported on this PCH (his Figure 1), which shows an extension of the PCH in the direction of the GLE source. Note that GLEs 51 and 52 occurred right after the polarity reversal in the northern hemisphere during cycle 22 and the PCH started increasing area. On the other hand GLE 58 occurred during the rise phase of SC 23, when the PCH area is close to its peak. Figure 3.: Coronal hole synoptic maps with the GLE source regions superposed for Carrington rotations 1843 (GLEs 51, 52) and 1939 (GLE 58). The maps were derived from He 1083 nm spectroheliograms at KPNO. There was no coronal hole observations for GLE 15, but one can infer a PCH because the event occurred during the rise phase of SC 20. The cycle started in October 1964 and the GLE occurred within 2 years into the cycle. The situation is somewhat similar to GLE 58, which also occurred within 2 years into SC 23. The PCH attains its maximum area around the solar minimum, remaining high for two years on either side of the minimum. The GLE sources are generally at higher-latitude during the rise phase because sunspots originate at higher latitudes in the rise phase. Thus the combination of strong polar coronal holes and the higher-latitude source regions is conducive for CME deflections. This has been suggested as the reason for the offset between position angles of prominence eruptions and the corresponding CMEs (Gopalswamy et al. 2003; 2012b) and the larger fraction of flux-rope type CMEs (magnetic clouds) occurring during the rise phase of solar cycles (Gopalswamy et al. 2008). Thus we conclude that GLE 15 is consistent with a possible PCH deflection of the associated CME. The eight maximum-phase events occurred during SC 22. Since the PCHs disappear in this phase, we do not expect deflection of CMEs by coronal holes. Figure 4 shows that there was no coronal hole but an extended “switchback” filament to the north of the source region of GLEs. The leading and trailing branches of the switch back filaments can also be seen in the He 10830 Å synoptic map in Fig. 4. H-alpha synoptic chart available at the Solar Geophysical Data confirms the switch back filament (see the H-alpha solar synoptic chart for Carrington Rotation 1829 in ftp://ftp.ngdc.noaa.gov/STP/SOLAR_DATA/SGD_PDFversion/). The filament was very long, extending from Carrington longitude 170∘ to 360∘. The trailing and leading branches of the switchback were about 15∘ and 35∘ from the active region. One expects a pseudo-streamer type magnetic configuration immediately to the north of the eruption region. We suggest that the magnetic field of the streamer behaves similar to that of coronal holes in deflecting CMEs. For GLEs 42–45, the situation was very similar in the southern hemisphere. GLE 42 occurred from the same source region of GLEs 43–45, but one Carrington rotation earlier. In addition the long filament, GLE 42 had an isolated coronal hole to the southwest, which also might have contributed to the deflection (see H-alpha solar synoptic chart for Carrington Rotation 1820 and 1821 in ftp://ftp.ngdc.noaa.gov/STP/SOLAR_DATA/SGD_PDFversion/). The switch back was not as sharp as in Fig. 4, but the filament was even longer: 135∘ to 360∘ in Carrington longitude. The streamer nearest to the active region and overlying the filament is expected to be a normal streamer. Thus we conclude that the coronal environment is conducive for an equatorward deflection in all the GLEs listed in Table 1. The last column of Table 1 lists the magnetic structures that might have caused the deflection: a polar coronal hole (PCH), streamer (S), or pseudo streamer (PS). In one case, it is possible that a streamer and a coronal hole might have jointly caused the deflection. A preliminary examination of the remaining GLE events with CH observations indicate that small coronal holes were present at large distances, suggesting little influence on the CMEs. In some cases, the coronal holes were favorably located to improve the connectivity. There were a few cases in which the deflection would worsen the connectivity, but these need a detailed investigation to see if the ecliptic distance would increase substantially. A detailed report will be published elsewhere. Figure 4.: He 10830 Å synoptic map with the source region of GLEs 47 and 48 marked. In the map compact dark patches are active regions and thin elongated features are filaments. A long switch back filament can be seen to the north of the GLE source region. The arrow with the label switchback shows roughly the location where the two arms converge. The poleward arm is pointed by the upper arrow. ## 4 Discussion The main results of this paper are: (i) the non-radial motion of the SC 24 GLE seems to be due to the deflection by a coronal hole located to the northeast of the eruption region, (ii) the CMEs in historical GLE events with flare latitude $>$30∘ occurring in the rise and declining SC phases seem to be deflected toward the ecliptic by polar coronal holes, (iii) higher-latitude GLEs occurring in the maximum phase seem to deflected by a large-scale streamer or pseudo streamer structure overlying high-latitude filaments. The B0 angle reduced the ecliptic distance of the GLE source region in some cases and increased in others with the net result that the average latitude of GLE events did not change significantly. The non-radial motion seems to the dominant factor in reducing the ecliptic distance of GLE source regions and hence increasing the latitudinal connectivity to Earth. The coronal deflection of CMEs happens because of the enhanced magnetic content of the coronal holes (Gopalswamy et al. 2009; Gopalswamy et al. 2010; Shen et al. 2011). For polar coronal holes, this represents the solar dipolar field, which is the strongest during solar minima (see e.g., Svalgaard et al. 1978; Gopalswamy et al. 2003). One of the main properties of coronal holes is that the magnetic field at the photospheric level is enhanced and unipolar relative to the neighboring quiet regions. The field expands into the corona and represents magnetic pressure gradient between the coronal hole and the eruption region (e.g., Panasenco et al. 2012; Kay et al. 2013) and hence pushes the CME away from the coronal hole. We suggest that the same physical picture applies when a large-scale streamer or pseudo streamer is present on one side of the eruption region causing the magnetic pressure gradient. The equatorward deflection of CMEs associated with GLE events suggests that Earth may be connected to the nose part of the CME-driven shocks. Given the fact that GLE particles are released when the CME leading edge is at a heliocentric distance of $\sim$3 Rs (Gopalswamy et al. 2012a), we suggest that the GLE particles may be accelerated at the quasi-parallel section of CME-driven shocks. This is because the nose part is well above the source surface where the field lines in the ambient medium are expected to be radial. It must be pointed out that coronal hole deflection may not be the only reason for non-radial CME motion. In some active regions, it is possible that non-radial ejections happen due to the magnetic complexities in the source regions. It is also possible that CMEs involve a high-inclination flux rope so that the nose region is much extended in the north-south direction. We confirm that the ecliptic distance of the source region of a solar eruption is one of the factors that determine whether an SEP event becomes a GLE event. This may be one of the reasons why GLE events are so rare. Other reasons include the reduced number of energetic eruptions during SC 24 and the reduction in seed particles (and preceding CMEs) available to be accelerated by the CME-driven shocks. The reduced efficiency of particle acceleration by the shocks due to the change in physical conditions in the heliosphere (e.g., increase in the Alfvén speed of the ambient medium). Thus, GLE events require special conditions in terms of CME kinematics, coronal environment, and magnetic connectivity to Earth. ## 5 Summary We have confirmed that non-radial CME motion is likely to have happened in historical GLE events that occurred at latitudes $>$30∘ due to deflection by large-scale magnetic structures in coronal holes or in streamers. We also infer that the highest energy particles are produced at the nose part of the CME-driven shocks, where the shock strength is highest. Furthermore, the shock is likely to be quasi-parallel in the nose region because the GLE associated CMEs typically cross the potential field source surface at the time of GLE particle release. The special conditions needed to detect a GLE event at Earth coupled with the reduced frequency of energetic eruptions may be responsible for the paucity of GLE events in SC 24. #### Acknowledgments This work utilizes SOLIS data obtained by the NSO Integrated Synoptic Program (NISP), managed by the National Solar Observatory, which is operated by the Association of Universities for Research in Astronomy (AURA), Inc. under a cooperative agreement with the National Science Foundation. Work supported by NASA’s Living with a Star program. ## References * Cliver (2006) Cliver, E. W. 2006, Astrophys. J. 639, 1206 * Cliver et al. (1982) Cliver, E. W., Kahler, S. 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1311.0070	gbsn # Tunable slowing, storing and releasing of a weak microwave field Keyu Xia (夏可宇) [email protected] Centre for Engineered Quantum Systems, Department of Physics and Astronomy, Macquarie University, NSW 2109, Australia ###### Abstract We study the slowing, storing and releasing of microwave pulses in a superconducting circuits composed of two coplanar waveguide resonators and a superconducting transmon-type qubit. The quantum interference analogy to electromagnetically induced transparency is created in two coupled resonators. By tuning the resonance frequency of the transmon, we dynamically tune the effective coupling between the resonators. Via the modulation of the coupling, we show the tunable true time delay of microwave pulses at the single-photon level. We also store the microwave field in a high-Q resonator and release the signal from it to the output port. Our scheme promises applications in both quantum information processing and classical wireless communications. ###### pacs: 42.50.Gy, 41.20.Jb, 85.25.Cp, 42.60.Da ## I Introduction Since the electromagnetically induced transparency (EIT) was discovered as a quantum interference in atoms AEIT1 , it has been widely used to enhance nonlinear susceptibility AEIT2 ; Kerr , store AStore1 and slow light AEIT2 in an atomic medium. The light trapped in coupled resonators display the pathway interference OEIT1 ; OEIT2 ; OEIT3 ; OEIT4 ; OCavity1 similar to the quantum interference in atoms AEIT2 . This path interference induces the optical analogue of EIT in coupled optical resonators OEIT1 ; OEIT2 ; OEIT3 and optomechanics OEMIT1 ; OEMIT2 has been demonstrated for slow light. Although the optical analogue of EIT and its application for slowing light has been well studied, the tunable delay of a purely microwave pulse is still a challenging MWPhoton . On the other hand, a tunable control over true time delay of microwave signals is essential for the applications in radar and satellite communications MWPhoton . A design to indirectly delay the microwave signals modulates the optical signals by the microwave signal and then slow down the group velocity of the optical signal. In this way, the microwave signal added to an optical carrier is delayed MWPhoton . It is attractive for the wireless communications to slow down a purely microwave signal on a microscopic chip. Using the metamaterial analog of EIT, a microwave pulse is delayed by $40\%$ of the pulse width by two coupled metamaterial slabs MWDelay3 . In this configuration, the transparency window and the time delay are fixed once the setup is fabricated. Very recently, the tunable delay of microwave pulses has been demonstrated by using the microwave analog of EIT in electromechanics MWDelay1 ; MWDelay2 . The storage of microwave electromganetic waves has also been realized in the matamaterial analogue of EIT StoreMWinEIT . However, the performance of the EIT-based slowing of single-photon pulses in the quantum regime is unclear. The microwave field can also be stored in and retrieved from an electron spin ensemble MWStorage1 ; QMemory or an electromechanical resonator MWTransfer . It is essential to delay a single microwave photon for the superconducting- circuit-based quantum information processing. Yin et al. has reported the storage and retrieve of quantum microwave photons by dynamically tuning the coupling of a resonator to an open transmission line MWStorage2 . Moreover, the slowing microwave fields using the Aulter-Townes effects in “artificial” atoms is in progress AT1 ; AT2 ; AT3 . In this paper, We extend the basic idea directly controlling the photon-photon interaction between cavities in our previous work Idea to the superconducting circuits. We propose a scheme to directly tune the coupling between two coplanar waveguide (CPW) resonators and subsequently to create the tunable microwave analogue of EIT. This tunable EIT allows one to control the group delay in microwave pulse propagation. By switching on-off the coupling between two resonators, we can also store a microwave field in a high-Q resonator and then release the field to the output port on-demand. Using the single-photon scattering model, we show the time delay of the microwave propagation at the single-photon level. The propagation of single microwave photons in real space is presented as a numerical proof of our scheme working in the quantum regime. The paper is organized as follows. In Sec. II we introduce the setups of the systems and the relevant model for a microwave analogue of EIT. In Sec. III we then introduce the single-photon scattering model for numerical simulations of storing and slowing of microwave photons. The results are presented in Sec. IV and a conclusion of this work is given in Sec. V. ## II System and Model In our previous work Idea , we proposed a method to directly control the photon-photon interaction between two optical cavities using a $\Lambda-$type system. Here we apply the idea to control the coupling between two CPW resonators via tuning the transition frequency of a superconducting transmon qubit. Our system is shown in Fig. 1. Its structure is similar to the design by Schoelkopf et al. Design ; ExpSetup , but the transmon qubit with the excited state $\|e\rangle$ and the ground state $\|g\rangle$ simultaneously couples to two separate CPW resonators via their mutual capacitors $C_{1}$ and $C_{2}$, respectively. Besides, the first resonator capacitively couples to the transmission line yielding an external coupling $\kappa_{ex}$. While the second resonator only couples to the transmon. The incoming and outgoing microwave fields travel in the transmission line. The gate voltage $V_{g}$ is used to bias the qubit and dynamically tune the transition frequency $\omega_{q}$ of qubit. The transition frequency of the transmon can also be tuned by a biased flux TransmonFluxBias . The first setup shown in Fig. 1(a) is used for slowing, storing and releasing of microwave signals. A microwave pulse is incident into the resonator 1 and then reflected to the ouput port. The reflected pulse is isolated by a circulator from the input port. To do storage, first, the effective coupling $h_{e}$ is on and the resonator mode $\hat{a}_{2}$ is excited. Then $h_{e}$ is turned off and the photon is stored in the high-Q resonator 2. To retrieve the photon, we turn on $h_{e}$ again. The photon stored in the resonator 2 excites the resonator mode $\hat{a}_{1}$ and subsequently goes toward the output port. The second design shown in Fig. 1(b) is only used to slow down the microwave signals traveling in an open transmission line because the photons retrieved from the resonator 2 can enter both the input and output ports. So it can not be used to store and then release a photon to the targeted output port. Figure 1: (Color online) (a) direct(end)-coupling setup for storing and releasing of and (b) side-coupling setup for slowing of a weak microwave field. Two CPW resonator modes $\hat{a}_{1}$ and $\hat{a}_{2}$ couples to a superconducting transmon via their mutual capacitors $C_{1}$ and $C_{2}$. A gate voltage $V_{g}$ is used to bias the transmon via a gate capacitor $C_{g}$. The SQUIDs biased by magnetic fluxes are used to compensate the AC Stark shift induced by the transmon. In both setup, we apply superconducting quantum interference devices (SQUIDs) to tune the resonance frequency $\omega_{r_{j}}$ of the $j$the CPW resonator Tunefr1 and their frequency difference $\delta=\omega_{r_{2}}-\omega_{r_{1}}$, with $j\in\\{1,2\\}$. The resonators can also be tuned by a magnetic field with the help of a built-in SQUID Tunefr2 ; Tunefr3 . Detailedly, the SQUIDs dispersively couple with a strength $g_{s_{j}}$ to the nearby CPW resonator via their mutual capacitors. We assume that the SQUIDs can be modeled as a two-level system with the excited and ground states $\|e_{s_{j}}\rangle$ and $\|g_{s_{j}}\rangle$. The detuning between the $j$th SQUID and the $j$th resonator modes are $\Delta_{s_{j}}=\omega_{s_{j}}-\omega_{r_{j}}$ where $\omega_{s_{j}}$ denotes the transition frequency of the $j$th SQUID. In the non-resonant (dispersive) regime, the SQUIDs induce a shift equal to $-\|g_{s_{j}}\|^{2}/\Delta_{s_{j}}$ in the resonance frequency of resonators Tunefr1 . This small frequency shift is used to cancel the AC Stark shift due to the coupling to the transmon. We apply a magnetic flux $\Phi_{s_{j}}$ to tune the transition frequency $\omega_{s_{j}}$ and subsequently the detuning $\Delta_{s_{j}}$. In doing so, we can control the effective resonance frequencies of the resonators. In the following investigation, we simply assume the resonance frequencies of the resonators are tunable but neglect the process to change them. The quantum interference in our system in Fig. 1 can be understood in a $\Lambda-$type three ”Level” diagram shown in Fig. 2(a). In the dressed mode picture, by analogy to the dressed state view of EIT AEIT2 , it becomes clear that the absorption of the external driving by the two dressed modes composing of $\hat{a}_{1}$ and $\hat{a}_{2}$ is canceled, and the coupling $h_{e}$ can be used to switch the system from absorptive to transmittive/reflective in a narrow band around cavity resonance. In this case, the effective coupling between two cavity modes $\hat{a}_{1}$ and $\hat{a}_{2}$ creates an EIT-like transparency window for the external driving $\alpha_{in}$. The transmission profile depends on the resonance frequencies of modes $\hat{a}_{1}$ and $\hat{a}_{2}$, the frequency of driving $\alpha_{in}$, the coupling $h_{e}$ and the decay rates of two resonators. Figure 2: (Color online) (a) Level-diagram picture, showing three ”Levels” that represent the optical modes $\hat{a}_{1}$, $\hat{a}_{2}$ and the ”probe” of optical waveguide mode $\alpha_{in}$. (b) Level diagram, showing how to tune the effective coupling $h_{e}$ via a two-level superconducting transmon qubit simultaneously coupling to two microwave resonators. The small balls above the level $\|g\rangle$ indicates the population. Now we turn to explain the idea how to tune the coupling $h_{e}$ between two CPW resonators. As shown in Fig. 2(b), two cavity modes $\hat{a}_{1}$ and $\hat{a}_{2}$ dispersively couples to the transmon denoting the two-level’s transition frequency and associated quantum eigenstates as $\omega_{q}$ and $\|j\rangle$, $j\in\\{e,g\\}$. For simplicity, we assume that two modes couple identically to the transmon at a rate $g$, and $\|\delta\|\ll\|\Delta_{a}\|$. Both quantum fields $\hat{a}_{1}$ and $\hat{a}_{2}$ induce Stark shifts on levels $\|g\rangle$ and $\|e\rangle$. The value of this shift $\Delta_{Stark}=-\|g\|^{2}\langle(\hat{a}_{1}^{\dagger}+\hat{a}_{2}^{\dagger})(\hat{a}_{1}+\hat{a}_{2})\rangle/\Delta_{a}$. In the situation of $\|\Delta_{a}\|^{2}\gg\|g\|^{2}\langle\hat{a}_{1}^{\dagger}\hat{a}_{1}\rangle,\|g\|^{2}\langle\hat{a}_{2}^{\dagger}\hat{a}_{2}\rangle$, the excitation of the state $\|e\rangle$ is negligible, i.e. $\langle\hat{\sigma}_{ee}\rangle\sim 0$, where we set $\hat{\sigma}_{ij}=\|i\rangle\langle j\|$. This Stark shift effectively yields a coherent interaction $h_{e}(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{2}^{\dagger}\hat{a}_{1})$ between the two resonator modes with a strength $h_{e}=-\|g\|^{2}\langle\hat{\sigma}_{gg}\rangle/\Delta_{a}\sim-\|g\|^{2}/\Delta_{a}\,.$ (1) From Eq. (1), we see the manner to tune the effective coupling between two resonators is to tune the transition frequency $\omega_{q}$ of the transmon and subsequently the detuning $\Delta_{a}$. Applying the Eq. (1) to the model, the effective Hamiltonian after eliminating the transmon takes the form ($\hbar=1$) (Appendix A) $\begin{split}\hat{H}=&\Delta^{\prime}_{in}\hat{a}_{1}^{\dagger}\hat{a}_{1}+(\Delta^{\prime}_{in}+\delta)\hat{a}_{2}^{\dagger}\hat{a}_{2}+h_{e}(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{2}^{\dagger}\hat{a}_{1})\\\ &+i\sqrt{2\kappa_{ex}}(\alpha_{in}\hat{a}_{1}^{\dagger}-\alpha_{in}^{}\hat{a}_{1})\,,\end{split}$ (2) with $\Delta^{\prime}_{in}=\Delta_{in}-\|g\|^{2}/\Delta_{a}$. Then time- evolution of the system can be obtained by solving the Langevin equations $\begin{split}\frac{\partial\hat{a}_{1}}{\partial t}&=-(i\Delta^{\prime}_{in}+\kappa^{\prime}_{1})\hat{a}_{1}-ih_{e}\hat{a}_{2}+\sqrt{2\kappa_{ex}}\alpha_{in}\,,\\\ \frac{\partial\hat{a}_{2}}{\partial t}&=-(i\Delta^{\prime}_{in}+i\delta+\kappa_{2})\hat{a}_{2}-ih_{e}\hat{a}_{1}\,,\end{split}$ (3) where the decay rate $\kappa^{\prime}_{1}$ of resonator 1 includes two contributions: $\kappa^{\prime}_{1}=\kappa_{1}+\kappa_{ex}$. $\kappa_{1}$ is the intrinsic decay rate and $\kappa_{ex}$ denotes the loss rate due to the external coupling to the resonator. While $\kappa_{2}$ only arises from the intrinsic loss of the resonator 2. In the case of a weak coherent driving, the amplitude of intracavity fields can be approximated using the quantum average value of operators $\langle\hat{a}_{1}\rangle=\alpha$ and $\langle\hat{a}_{2}\rangle=\beta$, respectively. Using the input-output relation InputOutput1 ; InputOutput2 , the output field transmitted through or reflected by the transmon is given in terms of the input and intracavity fields as $\alpha_{out}=-\alpha_{in}+\sqrt{2\kappa_{ex}}\alpha\,.$ (4) Immediately, we have the output in the steady state $\alpha_{ss}=-\alpha_{in}+\frac{2\kappa_{ex}[\kappa_{2}+i(\delta+\Delta^{\prime}_{in})]\alpha_{in}}{h_{e}^{2}+(\kappa^{\prime}_{1}+i\Delta^{\prime}_{in})[\kappa_{2}+i(\delta+\Delta^{\prime}_{in})]}\,.$ (5) The transmission amplitude is defined here as $t=\alpha_{out}/\alpha_{in}$. The power transmission —the ratio of the power transmitted through the transmon divided by the input power —is calculated as $\mathcal{T}=\|t\|^{2}$. The coupling $h_{e}$ between two CPW resonators does not only induce a strong modulation of the transmission of the input field, at the same time it causes a rapid phase dispersion $\phi(\Delta^{\prime}_{in})=arg(t)$ leading to a ”group delay” $\tau_{g}$ given by $\tau_{g}=\frac{d\phi}{d\Delta^{\prime}_{in}}\,,$ (6) around the transparency window. Therefore, the tunability of the coupling $h_{e}$ allows one to control the group delay $\tau_{g}$. For identical cavities and on-resonance driving, i.e. $\Delta_{in}=0,\delta=0$ the group delay is given by $\tau_{g}=2\kappa_{ex}\frac{h_{e}^{2}-\kappa_{2}^{2}}{(h_{e}^{2}+\kappa^{\prime}_{1}\kappa_{2})^{2}}\,.$ (7) It is limited by the decay rate $\kappa_{2}$ of resonator 2. The maximum available group delay for $h_{e}^{2}\gg\kappa_{2}^{2},\kappa^{\prime}_{1}\kappa_{2}$ is $\tau_{g}=2\kappa_{ex}/h_{e}^{2}$. Clearly, the steady-state solution of the Langevin equation Eq. (3) can provide the EIT-like transmission or reflection spectrum. The solution is also useful to estimate the group delay $\tau_{g}$. ## III Single-photon scattering model Besides the transmission or reflection in the steady state, the propagation in a real space is also interesting. In this section, we provide the single- photon scattering model developed by Fan et al. OL30p2001 ; PRA79p023837 ; PRA79p023838 for the study of the slowing of a microwave single-photon pulse in the following section. We have discussed the manner to tune the photon-photon interaction between resonators Idea . Here we directly consider an tunable effective coupling $h_{e}$ but neglect how to modulate $h_{e}$ by tuning the transition frequency of the transmon. In this case, our reduced system only consists of the open transmission line and two CPW resonators with their tunable coupling $h_{e}$. For slowing light using the setup in Fig. 1(b), the transmission line supports two counter-propagating modes: the incoming waveguide mode, $\hat{c}^{\dagger}_{T}(x)$, from the input port and the other waveguide mode, $\hat{c}^{\dagger}_{R}(x)$, reflected by the CPW resonator 1. Both two traveling modes interact with the CPW resonator 1 coupling to the second resonator mediated by the transmon. $\hat{c}^{\dagger}_{T}(x)$ and $\hat{c}^{\dagger}_{R}(x)$ create the photon at $x$ traveling from and to the input port, respectively. Since we will be interested in a narrow bandwidth single-photon pulses with a central frequency $\omega_{in}$ corresponding to a wave vector $k_{0}$, we can linearize the dispersion of the transmission line in the vicinity of $\omega_{in}$. After the linearization, the effective Hamiltonian of the system we study then takes the form OL30p2001 ; PRA79p023837 ; PRA79p023838 after an energy shift of $\omega_{in}$ $\begin{split}\hat{H}_{eff}=&-iv_{g}\int dx\hat{c}_{T}(x)^{\dagger}\frac{\partial}{\partial x}\hat{c}_{T}(x)\\\ &+iv_{g}\int dx\hat{c}_{R}(x)^{\dagger}\frac{\partial}{\partial x}\hat{c}_{R}(x)\\\ &+(\Delta^{\prime}_{in}-i\kappa_{1})\hat{a}_{1}^{\dagger}\hat{a}_{1}+(\Delta^{\prime}_{in}+\delta-i\kappa_{2})\hat{a}_{2}^{\dagger}\hat{a}_{2}\\\ &+\int dxV\delta(x)(\hat{c}_{T}^{\dagger}(x)\hat{a}_{1}+\hat{a}_{1}^{\dagger}\hat{c}_{T}(x))\\\ &+\int dxV\delta(x)(\hat{c}_{R}^{\dagger}(x)\hat{a}_{1}+\hat{a}_{1}^{\dagger}\hat{c}_{R}(x))\\\ &+h_{e}(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{2}^{\dagger}\hat{c}_{1})\,,\end{split}$ (8) where $v_{g}$ is the group velocity of the traveling photon around $k_{0}$ in the transmission line. $V$ is the coupling strength between the transmission line and the resonator 1. Note that $V^{2}/v_{g}=\kappa_{ex}$ is the decay rate of resonator due to the external coupling to the transmission line and has the unit of frequency OL30p2001 ; PRA79p023837 ; PRA79p023838 . In general, an arbitrary single-photon state $\|\Phi(t)\rangle$ can be expressed as OL30p2001 ; PRA79p023837 ; PRA79p023838 $\begin{split}\|\Phi(t)\rangle=&\int dx[\tilde{\phi}_{T}(x,t)\hat{c}^{\dagger}_{T}(x)+\tilde{\phi}_{R}(x,t)\hat{c}^{\dagger}_{R}(x)]\|\varnothing\rangle\\\ &+\tilde{e}_{1}\hat{a}^{\dagger}_{1}\|\varnothing\rangle+\tilde{e}_{2}\hat{a}^{\dagger}_{2}\|\varnothing\rangle\,,\end{split}$ (9) where $\|\varnothing\rangle$ is the vacuum, which has zero photon and has the atom in the ground state. $\tilde{\phi}_{T/R}(x,t)$ is the single-photon wave function in the $T/R$ mode at the position $x$ and the time $t$. $\tilde{e}_{1/2}$ is the excitation amplitude of the CPW resonator modes. We consider the propagation of the single-photon wave packets which can be derived from the Schrödinger equation $i\hbar\frac{\partial\|\Phi(t)\rangle}{\partial t}=H_{eff}\|\Phi(t)\rangle\,.$ (10) Substitution of Eqs. (2) and (9) into Eq. (10) gives the following set of equations of motion in the real space $\displaystyle\frac{\partial\tilde{\phi}_{T}(x,t)}{\partial t}=$ $\displaystyle-v_{g}\frac{\partial\tilde{\phi}_{T}(x,t)}{\partial x}-iV\delta(x)\tilde{e}_{1}(t)\,,$ (11a) $\displaystyle\frac{\partial\tilde{\phi}_{R}(x,t)}{\partial t}=$ $\displaystyle v_{g}\frac{\partial\tilde{\phi}_{R}(x,t)}{\partial x}-iV\delta(x)\tilde{e}_{1}(t)\,,$ (11b) $\displaystyle\frac{\partial\tilde{e}_{1}(t)}{\partial t}=$ $\displaystyle-i(\Delta^{\prime}_{in}-i\kappa_{1})\tilde{e}_{1}(t)-ih_{e}(t)\tilde{e}_{2}(t)$ (11c) $\displaystyle- iV\delta(x)\tilde{\phi}_{T}(x,t)-iV\delta(x)\tilde{\phi}_{R}(x,t)\,,$ $\displaystyle\frac{\partial\tilde{e}_{2}(t)}{\partial t}=$ $\displaystyle-i(\Delta^{\prime}_{in}+\delta-i\kappa_{2})\tilde{e}_{2}(t)-ih_{e}(t)\tilde{e}_{1}(t)\,.$ (11d) For any given initial state $\|\Phi(t)\rangle$, the dynamics of the system can be obtained directly by integrating this set of equations. The dynamics of our system is different from those of the previous works PRA79p023838 ; PRA82p063839 . In our system, the first resonator mode $\hat{a}_{1}$ couples both traveling modes $\hat{c}_{T}$ and $\hat{c}_{R}$, while the second resonator mode $\hat{a}_{2}$ only interacts with the mode $\hat{a}_{1}$. Furthermore, the intermode interaction $h_{e}$ can be tuned by the transmon. For storage using the setup in Fig. 1(a), the microwave photons travels in the same transmission line before or after interaction with the resonator. The incoming pulse can be reflected by the transmon back to the transmission line. The microwave photon stored in the resonator can be also released to this branch of the transmission line. Following Shen and Fan PRA79p023837 , we define a field $\hat{c}_{T}^{\dagger}(x)$ for a traveling mode in the transmission line such that $\hat{c}_{T}^{\dagger}(x<x_{0})$ describes an incoming photon that is moving toward the resonator at $x$, and $\hat{c}_{T}^{\dagger}(x>x_{0})$ describes an outgoing photon leaving the resonator at $2x_{0}-x$. $x_{0}$ is the position of the resonator. In order to take into account the phase shift $\phi$ occurring during the reflection at the end of the waveguide, we write the Hamiltonian as $\begin{split}\hat{H}_{eff}=&-iv_{g}\int dx\hat{c}_{T}(x)^{\dagger}\frac{\partial}{\partial x}\hat{c}_{T}(x)\,,\\\ &-\int dxv_{g}\phi\frac{\partial f(x)}{\partial x}\hat{c}_{T}(x)^{\dagger}\hat{c}_{T}(x)\,,\\\ &+(\Delta^{\prime}_{in}-i\kappa_{1})\hat{a}_{1}^{\dagger}\hat{a}_{1}+(\Delta^{\prime}_{in}+\delta-i\kappa_{2})\hat{a}_{2}^{\dagger}\hat{a}_{2}\,,\\\ &+\int dxV\delta(x)(\hat{c}_{T}^{\dagger}(x)\hat{a}_{1}+\hat{a}_{1}^{\dagger}\hat{c}_{T}(x))\,,\\\ &+h_{e}(\hat{a}_{1}^{\dagger}\hat{a}_{2}+\hat{a}_{2}^{\dagger}\hat{c}_{1})\,,\end{split}$ (12) where $f(x)$ is a switch-on function with the general property that $lim_{x\rightarrow-\infty}f(x)=0$ and $lim_{x\rightarrow+\infty}f(x)=1$ in a short spatial extent. For computation purpose, we take $f(x)=\frac{1}{1+e^{-(x-x_{0})/f_{a}}}$ with $f_{a}=0.5$ spatial step, otherwise the specific form of $f(x)$ is unimportant. Consideration of a general single-excitation state $\|\Phi(t)\rangle=\int dx\tilde{\phi}(x,t)\hat{c}_{T}^{\dagger}(x)\|\varnothing\rangle+\tilde{e}_{1}\hat{a}^{\dagger}_{1}\|\varnothing\rangle+\tilde{e}_{2}\hat{a}^{\dagger}_{2}\|\varnothing\rangle\,$ (13) yields the equation of motion $\displaystyle\frac{\partial\tilde{\phi}(x,t)}{\partial t}=$ $\displaystyle- v_{g}\frac{\partial\tilde{\phi}(x,t)}{\partial x}-iV\delta(x)\tilde{e}_{1}(t)$ (14a) $\displaystyle+iv_{g}\phi\frac{\partial f}{\partial x}\tilde{\phi}(x,t)\,,$ $\displaystyle\frac{\partial\tilde{e}_{1}(t)}{\partial t}=$ $\displaystyle-i(\Delta^{\prime}_{in}-i\kappa_{1})\tilde{e}_{1}(t)-ih_{e}(t)\tilde{e}_{2}(t)$ (14b) $\displaystyle-iV\delta(x)\tilde{\phi}(x,t)\,,$ $\displaystyle\frac{\partial\tilde{e}_{2}(t)}{\partial t}=$ $\displaystyle-i(\Delta^{\prime}_{in}+\delta-i\kappa_{2})\tilde{e}_{2}(t)-ih_{e}(t)\tilde{e}_{1}(t)\,.$ (14c) The direct (end)-coupling model in Fig. 1(a) is essentially different from the side-coupling model in Fig. 1(b) PRA79p023837 . Therefore, the Hamiltonian Eq. (12) (Eq.(8)) and the differential equations Eq. (14) (Eq.(11)) only describes the interaction and propagation of microwave fields in the setup of Fig. 1(a) (1(b)). ## IV Results We use the Langevin equation to study the transmission and reflection spectrum of the system. Then we present the numerical validation of the capability of our system for slowing, storing and releasing of single-photon wave packets in the quantum regime. Throughout the following investigation, we apply a critical coupling $\kappa_{ex}=\kappa_{1}$ and assume that the decay rate of the resonator 2 is negligible, i.e. $\kappa_{2}=0$. This is practice if the resonator 1 has a low-Q with $Q_{1}\sim 10^{3}$ ExpSetup and the resonator 2 is optimized to be high-Q, $Q_{2}\sim 10^{6}$ Tunefr1 . For simplicity, we assume two identical resonators such that $\delta=0$ and $g_{1}=g_{2}=g$. A similar setup usable for our goals has been fabricated in experiment ExpSetup . In our numerical simulation of the single-photon scattering, we assume that $\kappa_{1}=2\pi\times 5$ M Hz . Thus, a coupling of $g=2\pi\times 100$ M Hz ExpSetup can yield $h_{e}=2\kappa_{1}$ for $\Delta_{1}=10g$ which is the maximum in our simulation. ### IV.1 Steady-state Solution When an array of resonators couple to each other, the analogue of EIT occurs due to the pathway interference OEIT1 ; OEIT2 ; OEIT3 ; OEIT4 ; OCavity1 . For our system here, the microwave field in the CPW resonators 1 capacitively couples to the traveling field in the transmission line and interacts with the mode in the CPW resonator 2 mediated by the transmon. As a result, the reflection spectrum in Fig. 1(a) and the transmission spectrum in Fig. 1(b) display EIT profiles (see Fig. 3(a)). This microwave analogue of EIT can be used to slow microwave pulses. Both of the transmission/reflection spectrum and the group delay (see Fig. 3) can be calculated by solving the Langevin equation (3). If the coupling between resonators is static, the group delay is fixed. Our system provides a manner to dynamically tune the effective coupling between the resonators. We tune the transition frequency of the transmon and subsequently the detuning $\Delta_{a}$ between the transmon and two resonators. Thus, the effective coupling $h_{e}=-\|g\|^{2}/\Delta_{a}$ can be modulated in time. In Fig. 3 the transmission/reflection spectrum and the group delay are compared for different couplings $h_{e}$. For $h_{e}=0$, no EIT-like profile appear. The microwave field is absorbed in the first setup (Fig. 1(a)) or reflected in the second setup (Fig. 1(b)) by the resonator 1. When $h_{e}>\kappa_{2}$, the EIT-like profile appear. For example, for $h_{e}=0.25\kappa^{\prime}_{1}$, a narrow transparent window opens and the group delay can be $\kappa^{\prime}_{1}\tau_{g}=16$, as shown in Fig. 3(b). The group delay decreases quickly as the coupling $h_{e}$ increases. For $h_{e}>\kappa^{\prime}_{1}$, the EIT-like window becomes broader but the delay of microwave pulses is very small. So, we use a small coupling $0<h_{e}<\kappa^{\prime}_{1}$ for the slowing microwave photons. Figure 3: (Color online) (a) Transmission $\mathcal{T}$ of the incoming waveguide mode and (b) the group delay of the transmitted microwave pules for different effective couplings $h_{e}/\kappa^{\prime}_{1}=0,0.25,1$, respectively. The dashed blue lines for $h_{e}=0$, red solid lines for $h_{e}=0.25\kappa^{\prime}_{1}$ and green dot-dashed lines for $h_{e}=\kappa^{\prime}_{1}$. ### IV.2 Single-photon Scattering Although the EIT-based slowing and storing of coherent microwave pulses has been demonstrated MWDelay2 ; MWDelay3 , it is unclear how well the EIT works for the slowing and storing of a microwave signal in the quantum regime. Here we study the storing and then releasing, and slowing of a single microwave photon by solving the motion of the single-photon scattering model. The initial input is a Gaussian pulse $\tilde{\phi}(x,0)=\sqrt[4]{\tau^{2}/\pi}e^{-(x-x_{0})^{2}/2\tau^{2}}$ where $\tau$ is the spatial duration of pulse. The input is normalized to yield a single excitation, $\int_{-\infty}^{+\infty}\tilde{\phi}^{}(x,0)\tilde{\phi}(x,0)dx=1$. Figure 4: (Color online) Storing and releasing of microwave single-photon pulses by solving Eq. (14). (a) the normalized time-evolution of excitation $\|\tilde{e}_{1,2}\|^{2}$ of the CPW resonators and the time-dependent effective coupling $h_{e}(t)$ (green dashed line) with the maximum $max(h_{e}(t))=2\kappa_{1}$. The blue solid line shows the excitation of the resonator mode $\hat{a}_{1}$, red solid line for the mode $\hat{a}_{2}$. Excitations are normalized by their maximal numbers. (b) the propagation of single-photon pulses showing the storage and retrieve of microwave pulses. The wave functions are normalized. The blue line is for the input pulse, green line for the transmitted factor of pulse and magenta line shows the retrieved pulse. $h_{e}=2\kappa_{1}$. The diamond indicates the position of the resonator at $x_{0}=1200$. In Fig. 4, we first tune on the effective coupling, $h_{e}=2\kappa_{1}$. The input pulse excites the resonator modes. Due to the transparent EIT window, a large fraction of the pulse is reflected by the resonator. Then the coupling is switched off within a short time. The excitation in the resonator 2 is stored for a time $\tau_{s}$ corresponding to a space delay $v_{g}\tau_{s}$. This fraction of the excitation is released to the transmission line again through the resonator 1 when the coupling turns on. It can be clearly seen that a retrieved pulse appears (magenta line) after the reflected one (green line). Because there is only a fraction of excitation can be stored and retrieved, the EIT-based scheme is useful for the storage of a coherent state. Now we go on to the slowing single-photon pulses. Both setups in Fig. 1 have the same capability to slow down the propagation of microwave pulses, shown in Fig. 5. The group delay is crucially dependent on the effective coupling $h_{e}$. For $h_{e}\geqslant\kappa_{1}$, the group delay is very small (green line). While $h_{e}\ll\kappa_{1}$ promises a considerable group delay. For example, $\tau_{g}$ can be larger than $50\%\tau$ if $h_{e}=0.25\kappa_{1}$. Although the slowing wave function has a smaller amplitude in comparison with the input pulse in space, the wave function broadens slightly in space. This can be seen from Fig. 5(b) because the transmitted pulse has a narrower spectrum. As a result, more than $86\%$ of excitation are reserved in the delayed pulse. Thus a cascade of two device can provide a group delay longer than the duration of pulse. Figure 5: (Color online) Slowing of microwave pulses at the single-photon level for various couplings $h_{e}$ by solving Eq. (11). (a) Propagation of the wave functions. All wave functions are normalized. The blue line is the input pulse. Red line for the free propagation in the absence of the resonators. Green dot-dashed line for $h_{e}=0.5\kappa_{1}$ and magenta dashed line for $h_{e}=0.25\kappa_{1}$. The diamond indicates the position of the resonator at $x_{0}=2100$. (b) Normalized spectrum of the input wave function (blue line) and the slowing one for $h_{e}=0.25\kappa_{1}$ (magenta dashed line). ## V Conclusion We proposed a scheme to directly control the coupling between two CPW resonators using a superconducting transmon-type qubit. This tunable interaction between microwave photons allows us to turn on/off the microwave analogue of the EIT. Using the single-photon scattering model, our simulations showed that we can tune the group delay of single-photon microwave pulses. We also can store and release a microwave pulse. Our scheme provides a manner for the slowing and storing of the microwave photons in both of the microwave- based quantum information processing and the classical wireless communications. ## Acknowledgements This work is supported by the ARC via the Centre of Excellence in Engineered Quantum Systems (EQuS), project number CE110001013. ## Appendix A Here we provide the detail regarding the intermediate steps to obtain Eq. (2) in the main text. The Hamiltonian of the full system including the transmon takes the form $\begin{split}\hat{H}=&\frac{\omega_{q}}{2}\sigma_{z}+\sum_{j=1,2}\omega_{r_{j}}\hat{a}_{j}^{\dagger}\hat{a}_{j}\\\ &+i\sqrt{2\kappa_{ex}}\left(\alpha_{in}e^{-i\omega_{in}t}\hat{a}_{1}^{\dagger}-\alpha_{in}^{}e^{i\omega_{in}t}\hat{a}_{1}\right)\\\ &+\sum_{j=1,2}\left(g_{j}^{}\hat{a}_{j}^{\dagger}\sigma_{-}+g_{j}\sigma_{+}\hat{a}_{j}\right)\end{split}$ (A1) In the frame defined by the unitary transformation $\hat{U}=exp\left\\{-i\left(\omega_{in}/2\sigma_{z}+\sum_{j=1,2}\omega_{in}\hat{a}_{j}^{\dagger}\hat{a}_{j}\right)t\right\\}\,,$ (A2) the Hamiltonian becomes $\begin{split}\hat{H}=&\frac{\Delta_{a}+\Delta_{in}}{2}\sigma_{z}+\Delta_{in}\hat{a}_{1}^{\dagger}\hat{a}_{1}+(\Delta_{in}+\delta)\hat{a}_{1}^{\dagger}\hat{a}_{1}\\\ &+i\sqrt{2\kappa_{ex}}\left(\alpha_{in}\hat{a}_{1}^{\dagger}-\alpha_{in}^{}\hat{a}_{1}\right)\\\ &+\sum_{j=1,2}\left(g_{j}^{}\hat{a}_{j}^{\dagger}\sigma_{-}+g_{j}\sigma_{+}\hat{a}_{j}\right)\,,\end{split}$ (A3) where the detunings are defined as $\Delta_{a}=\omega_{q}-\omega_{r_{1}}$, $\Delta_{in}=\omega_{r_{1}}-\omega_{in}$ and $\delta=\omega_{r_{2}}-\omega_{r_{1}}$. To use the projection-operator formalism Raman1 to derive the effective Hamiltonian, we divide the Hamiltonian Eq. (A3) into $\hat{H}=\hat{H_{e}}+\hat{H_{g}}+\hat{V}_{-}+\hat{V}_{+}+\hat{H}_{r}\,,$ with $\displaystyle\hat{H}_{e}=$ $\displaystyle\frac{\Delta_{a}+\Delta_{in}}{2}\|e\rangle\langle e\|\,,$ $\displaystyle\hat{H}_{g}=$ $\displaystyle-\frac{\Delta_{a}+\Delta_{in}}{2}\|g\rangle\langle g\|\,,$ $\displaystyle\hat{H}_{r}=$ $\displaystyle\Delta_{in}\hat{a}_{1}^{\dagger}\hat{a}_{1}+(\Delta_{in}+\delta)\hat{a}_{1}^{\dagger}\hat{a}_{1}$ $\displaystyle+i\sqrt{2\kappa_{ex}}\left(\alpha_{in}\hat{a}_{1}^{\dagger}-\alpha_{in}^{}\hat{a}_{1}\right)\,,$ $\displaystyle\hat{V}_{+}=$ $\displaystyle\sum_{j=1,2}\hat{V}_{+,j}\,,$ $\displaystyle\hat{V}_{-}=$ $\displaystyle\hat{V}_{+}^{\dagger}\,,$ where $\hat{H}_{r}$ is independent of the state of the transmon. The perturbative excitation $\hat{V}_{+,j}=g_{j}\sigma_{+}\hat{a}_{j}$ connects the transmon and the $j$th resonator. Assuming that $\|\Delta_{a}\|$ is much larger than $\|\delta\|$ and the decay rate of transmon, we have the inverse of Hamiltonian of the quantum jump formalism under two driving fields $\hat{a}_{1}$ and $\hat{a}_{2}$ $\displaystyle\left(\hat{H}_{NH}^{(1)}\right)^{-1}=$ $\displaystyle(\Delta_{a}+\Delta_{in}-\Delta_{in})^{-1}\|e\rangle\langle e\|=\|e\rangle\langle e\|/\Delta_{a}\,,$ $\displaystyle\left(\hat{H}_{NH}^{(2)}\right)^{-1}=$ $\displaystyle(\Delta_{a}+\Delta_{in}-\Delta_{in}-\delta)^{-1}\|e\rangle\langle e\|=\|e\rangle\langle e\|/(\Delta_{a}-\delta)\,.$ Thus, the effective Hamiltonian is given by $\begin{split}\hat{H}_{eff}&=-\frac{1}{2}\left[\hat{V}_{-}\sum_{j=1,2}\left(\hat{H}_{NH}^{(j)}\right)^{-1}\hat{V}_{+,j}\right]+\hat{H}_{g}+\hat{H}_{r}\\\ &=-\frac{1}{2}\left[\sum_{j=1,2}g_{j}^{}\hat{a}_{j}^{\dagger}\left(\frac{g_{1}\hat{a}_{1}}{\Delta_{a}}+\frac{g_{2}\hat{a}_{2}}{\Delta_{a}-\delta}\right)\|g\rangle\langle g\|+H.c.\right]+\hat{H}_{g}+\hat{H}_{r}\\\ &=\hat{H}_{g}+\hat{H}_{r}-\frac{\|g_{1}\|^{2}}{\Delta_{a}}\hat{a}_{1}^{\dagger}\hat{a}_{1}\|g\rangle\langle g\|-\frac{\|g_{2}\|^{2}}{\Delta_{a}-\delta}\hat{a}_{2}^{\dagger}\hat{a}_{2}\|g\rangle\langle g\|\quad-\frac{1}{2}\left(\frac{1}{\Delta_{a}}+\frac{1}{\Delta_{a}-\delta}\right)(g_{1}g_{2}^{}\hat{a}_{1}\hat{a}_{2}^{\dagger}+H.c.)\|g\rangle\langle g\|\,.\end{split}$ (A4) It is reasonable to assume $\hat{\sigma}_{ee}\sim 0$ and $\hat{\sigma}_{gg}\sim 1$ when $\hat{\tilde{\sigma}}_{+}$ varies slowly and the population in $\|e\rangle$ is small in the situation of $\|\Delta_{a}\|\gg\|g_{1}\|,\|g_{2}\|$. 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1311.0193	# Kinetic theory of acoustic-like modes in nonextensive pair plasmas E. Saberian11affiliation: Visiting Researcher in Department of Physics, Faculty of Basic Sciences, Azarbaijan Shahid Madani University, P.O.Box: 53714-161, Tabriz, Iran [email protected] Department of Physics, Faculty of Basic Sciences, University of Neyshabur, P.O.Box: 91136-599, Neyshabur, Iran A. Esfandyari-Kalejahi Department of Physics, Faculty of Basic Sciences, Azarbaijan Shahid Madani University, P.O.Box: 53714-161, Tabriz, Iran [email protected] ###### Abstract The low-frequency acoustic-like modes in a pair plasma (electron-positron or pair-ion) is studied by employing a kinetic theory model based on the Vlasov and Poisson’s equation with emphasizing the Tsallis’s nonextensive statistics. The possibility of the acoustic-like modes and their properties in both fully symmetric and temperature-asymmetric cases are examined by studying the dispersion relation, Landau damping and instability of modes. The resultant dispersion relation in this study is compatible with the acoustic branch of the experimental data [W. Oohara, D. Date, and R. Hatakeyama, Phys. Rev. Lett. 95, 175003 (2005)], in which the electrostatic waves have been examined in a pure pair-ion plasma. Particularly, our study reveals that the occurrence of growing or damped acoustic-like modes depends strongly on the nonextensivity of the system as a measure for describing the long-range Coulombic interactions and correlations in the plasma. The mechanism that leads to the unstable modes lies in the heart of the nonextensive formalism yet, the mechanism of damping is the same developed by Landau. Furthermore, the solutions of acoustic-like waves in an equilibrium Maxwellian pair plasma are recovered in the extensive limit ($q\rightarrow 1$), where the acoustic modes have only the Landau damping and no growth. Pair plasmas: Kinetic theory of plasma waves: Waves, oscillations, and instabilities in plasmas ## 1 Introduction Studying the pair plasmas has been an important challenge for many plasma physicists in two past decades. As we know, the difference between the electron and ion masses in an ordinary electron-ion plasma (in general, multi- component plasma with both light and heavy particles) gives rise to different time-space scales which are used to simplify the analysis of low- and high- frequency modes. Such time-space parity disappears when studying a pure pair plasma which consisting of only positive- and negative-charged particles with an equal mass, because the mobility of the particles in the electromagnetic fields is the same. Pair plasmas consisting of electrons and positrons have attracted an especial of interest because of their significant applications in astrophysics. In fact, electron-positron plasmas play an important role in the physics of a number of astrophysical situations such as active galactic nuclei (Begelman et al., 1984; Miller & Witta, 1987), pulsar and neutron star magnetosphere (Goldreich & Julian, 1969; Max & Perkins, 1972; Michel, 1982), solar atmosphere (Tandberg-Hansen & Emslie, 1988), accretion disk (Orsoz et al., 1997), black holes (Daniel & Tajima, 1998), the early universe (Misner et al., 1973; Gibbons et al., 1983) and many others. For example, the detection of circularly polarized radio emission from the jets of the archtypal quasar 3C297, indicates that electron-positron pairs are an important component of the jet plasma (Wardle et al., 1998). Similar detections in other radio sources suggest that, in general, extragalactic radio jets are composed mainly of an electron-positron plasma (Wardle et al., 1998). Furthermore, it has been suggested that the creation of electron-positron plasma in pulsars is essentially by energetic collisions between particles which are accelerated as a result of electric and magnetic fields in such systems (Sturrock, 1971; Michel, 1982, 1991). On the other hand, the successful achievements for creation of the electron-positron plasmas in laboratories have been frequently reported in the scientific literatures (Gibson et al., 1960; Gahn et al., 2000; Pedersen et al., 2003; Helander & Ward, 2003; Amoretti et al., 2003; Pedersen et al., 2004; Chen et al., 2009). In this regard, many authors have concentrated on the relativistic electron-positron plasmas (Berezhiani et al., 1993; Verheest, 1996; Gedalin et al., 1998; Keston et al., 2003; Muoz, 2004; Laing & Diver, 2006) because of its occurrence in astrophysics and encountering with positron as an antimatter in high-energy physics. However, there are many experiments that confirm the possibility of nonrelativistic electron-positron plasmas in laboratory (Trivelpiece, 1972; Boehmer et al., 1995). It has been observed that the annihilation time of electron-positron pairs in typical experiments is often long compared with typical confinement times (Surko & Murphy, 1990), showing that the lifetime of electron-positron pairs in the plasma is much longer than the characteristic time scales of typical oscillations. The long lifetime of electron-positron pairs against pair annihilation indicates that many collective modes can occur and propagate in an electron-positron plasma. Although pair plasmas consisting of electrons and positrons have been experimentally produced, however, because of fast annihilation and the formation of positronium atoms and also low densities in typical electron- positron experiments, the identification of collective modes in such experiments is practically very difficult. To resolve this problem, one may experimentally deal with a pure pair-ion plasma instead of a pure electron- positron plasma for identification of the collective modes. An appropriate experimental method has been developed by Oohara and Hatakeyama (Oohara & Hatakeyama, 2003) for the generation of pure pair-ion plasmas consisting only positive and negative ions with equal masses by using fullerenes $\mathrm{C}_{60}^{-}$ and $\mathrm{C}_{60}^{+}$. The fullerenes are molecules containing 60 carbon atoms in a very regular geometric arrangement, and so a fullerene pair plasma is physically akin to the an electron-positron plasma, without having to worry about fast annihilation. By drastically improving the pair-ion plasma source in order to excite effectively the collective modes, Oohara _et al._ (Oohara et al., 2005) have experimentally examined the electrostatic modes propagating along the magnetic-field lines in a fullerene pair plasma. In exploring the electrostatic modes in a pair plasma, most of authors have merely studied the high frequency Langmuir-type oscillation in a pure electron-positron plasma (Tsytovich & Wharton, 1978; Iwamoto, 1993; Zank & Greaves, 1995; Verheest, 2005) or in a pure pair-ion plasma (Vranjes & Poedts, 2005) via the theoretical studies. Particularly, Iwamato (Iwamoto, 1993) and Veranjes and Poedts (Vranjes & Poedts, 2005) have studied the longitudinal modes in a pair plasma only in the case in which the phase velocity of the wave is much larger than the thermal velocity of the particles which leads to the Langmuir-type waves. However, in the experiment of Oohara _et al._ (Oohara et al., 2005), three kinds of electrostatic modes have been observed from the obtained dispersion curves: a relatively low-frequency band with nearly constant group velocity (the acoustic waves), an intermediate-frequency backward-like mode (to our knowledge, with the lack of a satisfactory theoretical explanation), and the Langmuir-type waves in a relatively high- frequency band. This experiment indicates that in a pure pair plasma, besides of the Langmuir-type waves, the acoustic-like modes are possible in practice. There, Oohara _et al._ (Oohara et al., 2005) have briefly discussed some aspects of their experimental results by using a theoretical two-fluid model. Here, our goal is to investigate the possibility and the properties of the intriguing acoustic-like modes in a pair plasma (in both symmetric and asymmetric cases) by using a kinetic theory model and to argue some properties of these modes in a subtler manner. It is to be noted that the only asymmetry in a pure pair plasma may arise from a difference in temperatures of species. Physically, the temperature-asymmetry in a pair plasma may arise from the typical experimental procedure in which a pair plasma is produced in the laboratory. For example, an effective technique for creating an electron- positron plasma in the laboratory is as follows: at first, we may obtain a positron plasma through scattering from a buffer gas into a panning trap (Surko et al., 1989; Greaves et al., 1994); using this technique, the positrons can be stored at densities of order $10^{7}cm^{-3}$ and lifetime of order $10^{3}sec$ in the recent experiments. Then, an electron-positron plasma with sufficient stability can be produced by injecting a low-energy electron beam into the positrons (Greaves & Surko, 1995; Liang et al., 1998; Greaves & Surko, 2001; Surko & Greaves, 2004). For our purpose, we assume that the phase speed of the acoustic-like modes lies in the vicinity of the thermal velocities of the species (in fact, between the thermal velocities of the two species). The situation is somewhat similar to the case in which the possibility and properties of the electron-acoustic waves in a two-temperature (cold and hot) electron plasma is examined (Defler & Simonen, 1969; Watanabe & Taniuti, 1977; Dubouloz et al., 1991; Kakad et al., 2007; Amour et al., 2012). It is often observed that the physical distribution of particles in space plasmas as well as in laboratory plasmas are not exactly Maxwellian and particles show deviations from the thermal distribution (Huang & Driscoll, 1994; Liu et al., 1994). Presence of nonthermal particles in space plasmas has been widely confirmed by many spacecraft measurements (Montgomery et al., 1968; Feldman et al., 1975; Maksimovic et al., 1997; Zouganelis, 2008). In many cases, the velocity distributions show non-Maxwellian tails decreasing as a power-law distribution in particle speed. Several models for phase space plasma distributions with superthermal wings or other deviations from purely Maxwellian behavior have become rather popular in recent years, like the so- called kappa ($\kappa$) distribution which was introduced initially by Vasyliunas in 1968 (Vasyliunas, 1968) for describing plasmas out of the thermal equilibrium such as the magnetosphere environments and the Solar winds (Maksimovic et al., 1997), or the nonthermal model advanced by Cairns _et al._ in 1995 (Cairns et al., 1995) which was introduced at first for an explanation of the solitary electrostatic structures involving density depletions that have been observed in the upper ionosphere in the auroral zone by the Freja satellite (Dovner et al., 1994), and also the nonextensive model which go under the name of Tsallis. In the following we want to briefly review the formalism of the Tsallis model and to argue why it is preferred, rather than that of the Cairns and kappa model. From a statistical point of view, there are numerous studies indicating the breakdown of the Boltzmann-Gibbs (BG) statistics for description of many systems with long-range interactions, long-time memories and fractal space- time structures (see, e.g., Landsberg (1984); Tsallis et al. (1995); Tsallis (1995, 1999)). Generally, the standard BG extensive thermo-statistics constitutes a powerful tool when microscopic interactions and memories are short ranged and the environment is an Euclidean space-time, a continuous and differentiable manifold. Basically, systems subject to the long-range interactions and correlations and long-time memories are related to the nonextensive statistics where the standard BG statistics and its Maxwellian distribution do not apply. The plasma environments in the astrophysical and laboratorial systems are obviously subject to spatial and temporal long-range interactions evolving in a non-Euclidean space-time that make their behavior nonextensive. A suitable generalization of the Boltzmann-Gibbs-Shannon (BGS) entropy for statistical equilibrium was first proposed by Reyni (Reyni, 1955) and subsequently by Tsallis (Tsallis, 1988, 1994), preserving the usual properties of positivity, equiprobability and irreversibility, but suitably extending the standard extensivity or additivity of the entropy to nonextensivity. The nonextensive generalization of the BGS entropy which proposed by Tsallis in 1988 (Tsallis, 1988, 1994) is given by the following expression: $S_{q}=k_{B}\frac{1-\sum_{i}p_{i}^{q}}{q-1},$ (1) where $k_{B}$ is the standard Boltzmann constant, $\\{p_{i}\\}$ denotes the probabilities of the microstate configurations and $q$ is a real parameter quantifying the degree of nonextensivity. The most distinctive feature of $S_{q}$ is its pseudoadditivity. Given a composite system $A+B$, constituted by two subsystems $A$ and $B$, which are independent in the sense of factorizability of the joint microstate probabilities, the Tsallis entropy of the composite system $A+B$ satisfies $S_{q}(A+B)=S_{q}(A)+S_{q}(B)+(1-q)S_{q}(A)S_{q}(B)$. In the limit of $q\rightarrow 1$, $S_{q}$ reduces to the celebrated logarithmic Boltzmann- Gibbs entropy $S=-k_{B}\sum_{i}p_{i}\ln p_{i}$, and the usual additivity of entropy is recovered. Hence, $\|1-q\|$ is a measure of the lack of extensivity of the system. There are numerous evidences exhibiting that the nonextensive statistics, arising from $S_{q}$, is a better framework for describing many physical systems such as the galaxy clusters (Lavagno et al., 1998), the plasmas (Boghosian, 1996; Tsallis & de Souza, 1997), the turbulent systems (Arimitsu & Arimitsu, 2000; Beck, 2001; Beck et al., 2001), and so on, in which the system shows a nonextensive behavior as a result of long-range interactions and correlations. The experimental results in such systems display a non-Maxwellian velocity distribution for the particles (Huang & Driscoll, 1994; Liu et al., 1994). The functional form of the velocity distribution in the Tsallis formalism may be derived through a nonextensive generalization of the Maxwell ansatz (Silva et al., 1998), or through the maximizing Tsallis’ entropy under the constraints imposed by normalization and the energy mean value (Curado, 1999; Abe, 1999). Furthermore, from a nonextensive generalization of the “molecular chaos hypothesis”, it is shown that the equilibrium $q$-nonextensive distribution is a natural consequence of the _H_ theorem (Lima et al., 2001). It is to be noted that the empirically derived kappa distribution function in space plasmas is equivalent to the $q$-distribution function in Tsallis nonextensive formalism, in the sense that the spectrum of the velocity distribution function in both models show the similar behavior and, in fact, both the kappa distribution and the Tsallis $q$-nonextensive distribution describe deviations from the thermal distribution. Particularly, Leubner in 2002 (Leubner, 2002) showed that the distributions very close to the kappa distributions are a consequence of the generalized entropy favored by the nonextensive statistics, and proposed a link between the Tsallis nonextensive formalism and the kappa distribution functions. In fact, relating the parameter $q$ to $\kappa$ by formal transformation $\kappa=1/(1-q)$ (Leubner, 2002) provides the missing link between the $q$-nonextensive distribution and the $\kappa$-distribution function favored in space plasma physics, leading to a required theoretical justification for the use of $\kappa$-distributions from fundamental physics. Furthermore, Livadiotis and McComas in 2009 (Livadiotis & McComas, 2009) examined how kappa distributions arise naturally from the Tsallis statistical mechanics. On the other hand, the nonthermal distribution function introduced by Cairns et al. (Cairns et al., 1995) is a proposal function to model an electron distribution with a population of energetic particles. It is especially appropriate for describing the nonlinear propagation of large amplitude electrostatic excitations such as solitary waves and double layers which are very common in the magnetosphere. However, the lack of a statistical foundation behind this proposal function is clearly seen, leading to less attention to it rather than the kappa function and the Tsallis distribution. Anyway, the $q$-nonextensive formalism, with a powerful thermo-statistics foundation and numerous experimental evidences, may cover many features of the other nonthermal models and provide a good justification for its preference over the other models. It has considerably extended both statistical mechanics formalism and its range of applicability. The interested reader may refer to the Refs. (Plastino, 2004; Abe & Okamato, 2001; Gell-Mann & Tsallis, 2004; Tsallis, 2009) where the significance, historical background, physical motivations, foundations and applications of the nonextensive thermo- statistics have been discussed in detail. The problem of waves, Landau damping and instabilities in typical plasmas have been investigated by some authors in the framework of the Tsallis nonextensive statistics (Lima et al., 2000; Silva et al., 2005; Valentini, 2005; Liyan & Jiulin, 2008; Saberian & Esfandyari-Kalejahi, 2013). Particularly, it is to be noted that the physical state described by the $q$-nonextensive distribution in the Tsallis’s statistics is not exactly the thermodynamic equilibrium (Liyan & Jiulin, 2008). In fact, the deviation of $q$ from unity quantifies the degree of inhomogeneity of the temperature $T$ via the formula $k_{B}\nabla T+(1-q)Q_{\alpha}\nabla\phi=0$ (Du, 2004), where $Q_{\alpha}$ denotes the electric charge of specie $\alpha$, and $\phi$ is the electrostatic potential. In other words, the nonextensive statistics describes a system that have been evolved from a nonequilibrium stationary state with inhomogeneous temperature which contains a number of nonthermal particles. In the present work, we attempt to investigate the possibility of the acoustic-like modes in a field-free and collisionless pair plasma (electron- positron or pair-ion) and to discuss the damping and instability of modes in the context of the Tsallis’ nonextensive statistics. In Sec. 2, a kinetic theory model based on the linearized Vlasov and Poisson’s equations is applied for deriving the dielectric function ($D(k,\omega)$) for longitudinal waves in an unmagnetized pair plasma. We then find the solutions of $D(k,\omega)=0$ for the acoustic-like waves with the constraint of weak damping or growth by considering a $q$-nonextensive distribution for stationary state of the plasma, as demonstrated in Sec. 3. The dispersion relation, Landau damping and instability of the acoustic-like modes are discussed in Sec. 4. Finally, a summary of our results is given in Sec. 5. ## 2 The model equations In this section, we present a brief review of kinetic equations for describing the electrostatic collective modes specialized to a pair plasma (electron- positron or pair-ion) with the constraint of weak damping or growth. We consider a spatially uniform field-free pair plasma at the equilibrium state. If at a given time $t=0$ a small amount of charge is displaced in the plasma, the initial perturbation may be described by $f_{\alpha}(t=0)=f_{0,\alpha}(\vec{v})+f_{1,\alpha}(\vec{x},\vec{v},t=0),\ \ \ \ f_{1,\alpha}\ll f_{0,\alpha},$ where $f_{0,\alpha}$ corresponds to the unperturbed and time-independent stationary distribution and $f_{1,\alpha}$ is the corresponding perturbation about the equilibrium state. Here, $\alpha$ stands for electrons and positrons ($\alpha=e^{\pm}$) or fullerene pairs ($\alpha=\mathrm{C}_{60}^{\pm}$). We assume that the perturbation is electrostatic and the displacement of charge gives rise to a perturbed electric but no magnetic field. With this assumption, the time development of $f_{1,\alpha}(\vec{x},\vec{v},t)$ is given by the solution of the linearized Vlasov and Poisson’s equations as follows (Landau, 1946; Krall & Trivelpiece, 1973): $\frac{\partial f_{1,-}}{\partial t}+\vec{v}\cdot\frac{\partial f_{1,-}}{\partial\vec{x}}+\frac{e}{m}\nabla\phi_{1}\cdot\frac{\partial f_{0,-}}{\partial\vec{v}}\;=\;0,$ (2) $\frac{\partial f_{1,+}}{\partial t}+\vec{v}\cdot\frac{\partial f_{1,+}}{\partial\vec{x}}-\frac{e}{m}\nabla\phi_{1}\cdot\frac{\partial f_{0,+}}{\partial\vec{v}}\;=\;0,$ (3) $\nabla^{2}\phi_{1}=4\pi n_{0}e\int(f_{1,-}-f_{1,+})\,\mathrm{d}\vec{v},$ (4) where $e$, $m$ and $n$ denote, respectively, the absolute charge, mass and number density of the pairs and $\phi_{1}$ is the electrostatic potential produced by the perturbation. Here, we have labeled the distribution function of negative and positive pairs with the subscripts $\pm$. This set of linearized equations for perturbed quantities may be solved simultaneously to investigate the plasma properties for the time intervals shorter than the binary collision times. Specially, we can study the properties of the plasma waves whose oscillations period are much less than a binary collision time. The standard technique for simultaneously solving the differential equations (2)-(4) is the method of integral transforms, as developed for the first time by Landau in the case of an ordinary electron-ion plasma (Landau, 1946; Krall & Trivelpiece, 1973). Another simplified method of solving the Vlasov- Poisson’s equations for the longitudinal waves, with the frequency $\omega$ and the wave vector $\vec{k}$, is to assume that the solution has the form $\begin{array}[]{l}f_{1,\alpha}(\vec{x},\vec{v},t)=f_{1,\alpha}(\vec{v})e^{i(\vec{k}\cdot\vec{x}-\omega t)},\ \ \ \alpha=e^{\pm}\ \ \mathrm{or}\ \ \mathrm{C}_{60}^{\pm},\\\ \phi_{1}(\vec{x},t)=\phi_{1}e^{i(\vec{k}\cdot\vec{x}-\omega t)}.\end{array}$ (5) Without loss of the generality, we consider the $x$-axis to be along the direction of the wave vector $\vec{k}$, and let $v_{x}=u$. Then, by applying the Eq. (5) and solving the Eqs. (2)-(4) we find the dispersion relation for longitudinal waves in a pair plasma as follows $D(k,\omega)=1-\frac{4\pi n_{0}e^{2}}{mk^{2}}\int\frac{\frac{\partial}{\partial u}(f_{0,-}(u)+f_{0,+}(u))}{u-\frac{\omega}{k}}\,\mathrm{d}u=0,$ (6) where $D(k,\omega)$ is the dielectric function of a field-free pair plasma for the longitudinal oscillations. We then can investigate the response of the pair plasma to an arbitrary perturbation via the response dielectric function $D(k,\omega)$. In general, the frequency $\omega$ which satisfies the dispersion relation $D(k,\omega)=0$ is complex, i.e., $\omega=\omega_{r}+i\omega_{i}$. However, in many cases $Re[\omega(k)]\gg Im[\omega(k)]$, and the plasma responds to the perturbation a long time after the initial disturbance with oscillations at a range of the well-defined frequencies. These are the normal modes of the plasma, in the sense that they are the nontransient response of the plasma to an initial perturbation. We can determine the normal modes of the plasma via the dispersion relation $D[k,\omega(k)]=0$, which gives the frequency of the plasma waves as a function of the wave number $k$ or vice versa. It should be further mentioned that when we solve the Vlasov and Poisson’s equations as an initial valve problem, here via $f_{0,-}+f_{0,+}$, it is possible to obtain the solutions with negative or positive values of $\omega_{i}$, corresponding to the damped or growing waves, respectively. This can be explicitly seen from the electrostatic potential associated with the wave number $k$ of the excitation as follows: $\phi_{1}(x,t)=\phi_{1}e^{i(kx-\omega_{r}t)}e^{\omega_{i}t},$ (7) where a solution with negative $\omega_{i}$ displays a damped wave, while the solution with positive one corresponds to an unstable mode. When the damping or growth is weak we can expand the velocity integral in Eq. (6) around $\omega=\omega_{i}$ to find the zeros of $D(k,\omega)$. The dielectric function $D(k,\omega)$ is in general a complex function and thus the dispersion relation can be written as follows: $D(k,\omega_{r}+i\omega_{i})=D_{r}(k,\omega_{r}+i\omega_{i})+iD_{i}(k,\omega_{r}+i\omega_{i})=0,$ (8) where $D_{r}$ and $D_{i}$ are the real and imaginary parts of the dielectric function. Since we want to consider the weakly damped or growing waves, i.e., $\omega_{i}\ll\omega_{r}$, the Eq. (8) can be Taylor expanded in the small quantity $\omega_{i}$ as follows: $D_{r}(k,\omega_{r})+i\omega_{i}\frac{\partial D_{r}(k,\omega_{r})}{\partial\omega_{r}}+i[D_{i}(k,\omega_{r})+i\omega_{i}\frac{\partial D_{i}(k,\omega_{r})}{\partial\omega_{r}}],$ (9) where $D_{r}$ and $D_{i}$ read $D_{r}(k,\omega_{r})=1-\frac{4\pi n_{0}e^{2}}{mk^{2}}P.V.\int\frac{\frac{\partial}{\partial u}(f_{0,-}(u)+f_{0,+}(u))}{u-\frac{\omega_{r}}{k}}\,\mathrm{d}u,$ (10) $D_{i}(k,\omega_{r})=-\pi(\frac{4\pi n_{0}e^{2}}{mk^{2}})[\frac{\partial}{\partial u}(f_{0,-}(u)+f_{0,+}(u))]_{u=\frac{\omega_{r}}{k}}.$ (11) Here, we have made the analytic continuation of the velocity integral of the Eq. (6) over $u$, along the real axis, which passes under the pole at $u=\frac{\omega}{k}$ with the constraint of weakly damped waves, where $P.V.\int$ denotes the Cauchy principal value. With the assumption $\omega_{i}\ll\omega_{r}$, by balancing the real and imaginary parts of the Eq.(9) and neglecting the terms of order $(\frac{\omega_{i}}{\omega_{r}})^{2}$, we find that $\omega_{r}$ and $\omega_{i}$ can be computed, respectively, from the relations $\displaystyle D_{r}(k,\omega_{r})=0,$ (12a) $\displaystyle\omega_{i}=-\frac{D_{i}(k,\omega_{r})}{{\partial D_{r}(k,\omega_{r})}/{\partial\omega_{r}}}.$ (12b) ## 3 Acoustic modes with nonextensive stationary state Now, we want to obtain the formalism and some features of the acoustic-like modes in a pair (electron-positron or pair-ion) plasma in the context of the Tsallis nonextensive statistics. For this purpose we assume that the stationary state of the plasma obeys the $q$-nonextensive distribution function, instead of a Maxwellian one, which merely describes a fully equilibrium plasma. The $q$-nonextensive distribution function of stationary state for species $\alpha$ in one-dimension is given by (Silva et al., 1998; Curado, 1999; Abe, 1999; Lima et al., 2001) $f_{0\alpha}(u)=A_{\alpha,q}[1-(q-1)\frac{m_{\alpha}u^{2}}{2k_{B}T_{\alpha}}]^{\frac{1}{q-1}},$ (13) where $m_{\alpha}$ and $T_{\alpha}$ are, respectively, the mass and temperature of species $\alpha$ ($\alpha=e^{\pm}\ \mathrm{or}\ \mathrm{C}_{60}^{\pm}$) and $k_{B}$ is the standard Boltzmann constant. The normalization constant $A_{\alpha,q}$ can be written as $A_{\alpha,q}=L_{q}\sqrt{\frac{m_{\alpha}}{2\pi k_{B}T_{\alpha}}},$ (14) where the dimensionless $q$-dependent coefficient $L_{q}$ reeds $\displaystyle L_{q}=\frac{\Gamma(\frac{1}{1-q})}{\Gamma(\frac{1}{1-q}-\frac{1}{2})}\sqrt{1-q},\ \ \ \ \mathrm{for}\ \ -1<q\leq 1$ (15a) $\displaystyle L_{q}=(\frac{1+q}{2})\frac{\Gamma(\frac{1}{2}+\frac{1}{q-1})}{\Gamma(\frac{1}{q-1})}\sqrt{q-1}.\ \ \ \ \mathrm{for}\ \ q\geq 1$ (15b) One may examine that for $q>1$ , the $q$-distribution function (13) exhibits a thermal cutoff, which limits the velocity of particles to the values $u<u_{max}$, where $u_{max}=\sqrt{\frac{2k_{B}T_{\alpha}}{m_{\alpha}(q-1)}}$. For these values of the parameter $q$ we have $S_{q>1}(A+B)<S(A)+S(B)$ referred to the _subextensivity_. This thermal cutoff is absent when $q<1$ , that is, the velocity of particles is unbounded for these values of the parameter $q$. In this case, we have $S_{q<1}(A+B)>S(A)+S(B)$ referred to the _superextensivity_. Moreover, the $q$-nonextensive distribution (13) is unnormalizable for the values of the $q<-1$. Furthermore, the parameter $q$ may be further restricted by the other physical requirements, such as finite total number of particles and consideration of the energy equipartition for contribution of the total mean energy of the system. Interestingly, in the extensive limit $q\rightarrow 1$ where $S(A+B)=S(A)+S(B)$, and by using the formula $lim_{\mid z\mid\rightarrow\infty}z^{-a}[\frac{\Gamma(a+z)}{\Gamma(z)}]=1$ (Abramowitz & Stegun, 1972), the distribution function (13) reduces to the standard Maxwell- Boltzmann distribution $f_{0\alpha}(u)=\sqrt{\frac{m_{\alpha}}{2\pi k_{B}T_{\alpha}}}e^{-\frac{m_{\alpha}u^{2}}{2k_{B}T_{\alpha}}}$. In Fig. 1, we have depicted schematically the nonthermal behavior of the distribution function (13) for some values of the spectral index $q$ in which the velocity $u$ and the distribution function $f(u)$ have normalized by the standard thermal speed $v_{th}=\sqrt{\frac{2k_{B}T}{m}}$ and $\sqrt{\frac{m}{2\pi k_{B}T}}$, respectively. We can see that in the case of a superextensive distribution with $q<1$ [Fig. 1(a)], comparing with the Maxwellian limit (solid curve), there are more particles with the velocities faster than the thermal speed $v_{th}$. These are the so-called superthermal particles and we can see that the $q$-distribution with $q<1$ behave like the $\kappa$ distribution, the same as that introduced for the first time by Vasyliunas in 1968 to describe the space plasmas far from the thermal equilibrium (Vasyliunas, 1968). In fact, in a superthermal plasma modeled by a $\kappa$-like distribution (here, the cases in which $q<1$), the particles have distributed in a wider spectrum of the velocities, in comparison with a Maxwellian plasma. In other words, the low values of the spectral index $q$ correspond to a large fraction of superthermal particle populations in the plasma. On the other hand, in the case of a subextensive distribution with $q>1$ [Fig. 1(b)], comparing with the Maxwellian limit (solid curve), there is a large fraction of particles with the velocities slower than the thermal speed $v_{th}$. Moreover, for these values of the parameter $q$, we can explicitly see the mentioned thermal cutoff which limits the velocity of particles. In fact, the $q$-nonextensive distributions with $q>1$ are suitable for describing the systems containing a large number of low speed particles. The phase velocity of the acoustic modes in a pair plasma lies between the thermal velocities of the pairs. Here, we assume that $T_{+}<T_{-}$ and therefore the phase velocity of the acoustic waves lies in the frequency band $v_{th,+}<v_{\phi}<v_{th,-}$, where $v_{\phi}=\frac{\omega_{r}}{k}$ and $v_{th,\pm}=(\frac{k_{B}T_{\pm}}{m})^{\frac{1}{2}}$, respectively, denote the phase velocity of the wave and thermal speed of the pairs. It is to be noted that because of the symmetry involved in a pair plasma, the other case in which $T_{-}<T_{+}$ is physically identical to our assumption here. Moreover, it is reminded that because of the same dynamics of the species in a pure pair plasma, we do not make a considerable difference in temperatures of the pairs, but we assume that it is finite and small. As we mentioned earlier, we may postulate physically that this finite temperature-asymmetry in a pair plasma may arise from the typical experimental procedure in which the pair plasma is produced in the laboratory (Greaves & Surko, 1995; Liang et al., 1998; Greaves & Surko, 2001; Surko & Greaves, 2004). With $v_{th,+}<v_{\phi}<v_{th,-}$, the Cauchy principal value of Eq. (10) for the terms that are involving $f_{0,-}$ and $f_{0,+}$ may be evaluated by an expanding in $u$ as follows: $\displaystyle\int^{+u_{max}}_{-u_{max}}\frac{\frac{\partial}{\partial u}f_{0,-}(u)}{u-\frac{\omega_{r}}{k}}\,\mathrm{d}u=\int^{+u_{max}}_{-u_{max}}\frac{\partial f_{0,-}(u)}{\partial u}(\frac{1}{u}+\frac{1}{u^{2}}\frac{\omega_{r}}{k}+\frac{1}{u^{3}}\frac{\omega_{r}^{2}}{k^{2}}+...)\,\mathrm{d}u,$ (16a) $\displaystyle\int^{+u_{max}}_{-u_{max}}\frac{\frac{\partial}{\partial u}f_{0,+}(u)}{u-\frac{\omega_{r}}{k}}\,\mathrm{d}u=-\frac{k}{\omega_{r}}\int^{+u_{max}}_{-u_{max}}\frac{\partial f_{0,+}(u)}{\partial u}(1+\frac{k}{\omega_{r}}u+\frac{k^{2}}{\omega_{r}^{2}}u^{2}+\frac{k^{3}}{\omega_{r}^{3}}u^{3}+...)\,\mathrm{d}u.$ (16b) Here, in order to include both cases $q<1$ (superextensivity) and $q>1$ (subextensivity), we have denoted the integration limits in Eq. (16) by $\pm u_{max}$. In fact, as discussed earlier, the integration limits are unbounded, i.e., $\pm u_{max}=\pm\infty$, when $q<1$, and they are given by the $q$-dependent thermal cutoff $\pm u_{max}=\pm\sqrt{\frac{2k_{B}T_{\alpha}}{m_{\alpha}(q-1)}}$ when $q>1$. With the $q$-nonextensive distribution given in Eq. (13), noting that $f_{0\alpha}(u)$ is an even function with argument $u$ and $\frac{\partial f_{0\alpha}}{\partial u}$ is an odd function, we may calculate the real part of the dielectric function in Eq. (10) as follows: $D_{r}(k,\omega_{r})=1+\frac{4\pi n_{0}e^{2}}{mk^{2}}\frac{1}{v_{th,-}^{2}}(\frac{1+q}{2})-\frac{4\pi n_{0}e^{2}}{m\omega_{r}^{2}}[1+3(\frac{2}{3q-1})\frac{k^{2}}{\omega_{r}^{2}}v_{th,+}^{2}].$ (17) The integrals in Eq. (16) are computed by parts and there, we have calculated the average values of $u^{2}$ as follows: $<u^{2}>=\int^{+u_{max}}_{-u_{max}}u^{2}f_{\alpha 0}(u)\,\mathrm{d}u=\frac{2}{3q-1}\frac{k_{B}T_{\alpha}}{m_{\alpha}},$ (18) which requires that the parameter $q$ must restrict to the values of $q>\frac{1}{3}$. Note that for the values of $q$ equal or lower than the critical value $q_{c}=\frac{1}{3}$, the mean value of $u^{2}$ diverges. Therefore, we see that the parameter $q$ for the case $q<1$ is further restricted to the values $\frac{1}{3}<q<1$, in order that the physical requirement of energy equipartition is preserved. We emphasize that our results here are valid both for the case $\frac{1}{3}<q<1$ where the value of $u_{max}$ is unbounded and also in the case $q>1$ in which $u_{max}$ is given by the thermal cutoff $u_{max}=\sqrt{\frac{2k_{B}T_{\alpha}}{m_{\alpha}(q-1)}}$. Note that in both cases the integrals in Eq. (16) are evaluated by limits that are symmetric across the origin. The interested reader may easily check the validity of Eqs. (17) and (18) for all allowed values of $q$. Furthermore, in the extensive limit $q\rightarrow 1$, Eq. (18) reduces to the familiar energy equipartition theorem for each degree of freedom in the BG statistics as $<\frac{1}{2}m_{\alpha}u^{2}>=\frac{1}{2}k_{B}T_{\alpha}$. It is to be noted that the $q$ distribution given in Eq. (13) describes the stationary state of the species $\alpha$ in the framework of the Tsallis nonextensive formalism. The value of the spectral index $q$ is a measure that determines the slope of the energy spectrum of the nonthermal particles and measures the deviation from the standard thermal distribution (which is recovered at the limit $q\rightarrow 1$). The value of the spectral index $q$ is determined as a result of long-range interactions and correlations of the whole system. Therefore, a distinction between the pairs in $q$ can be or not, depend on the physics of the system under consideration. Here, following El- Tantawy _et al._ (El-Tantawy et al., 2012), we make no distinction between the pairs in $q$. The solution of the equation $D_{r}(k,\omega_{r})=0$ may yield the dispersion relation for the acoustic modes in a nonextensive pair plasma as follows: $\omega_{r}^{2}=k^{2}c_{s}^{2}[\frac{1}{(k\lambda_{D})^{2}(1+\frac{1}{\sigma})+(\frac{1+q}{2})}+3(\frac{2}{3q-1})\sigma],$ (19) where we have defined the sound-speed of the acoustic-like modes as $c_{s}={(\frac{k_{B}T_{-}}{m})}^{\frac{1}{2}}$. Here, $\sigma=\frac{T_{+}}{T_{-}}$ is the fractional temperature of positive to negative species and $\lambda_{D}$ is the Debye screening length and is given in a charge-neutral pair plasma by $\lambda_{D}^{-2}=\frac{4\pi n_{0}e^{2}}{k_{B}}(\frac{1}{T_{-}}+\frac{1}{T_{+}}).$ (20) By definition of the the natural oscillation frequency in a charge-neutral pair plasma as $\omega_{p}=(\frac{8\pi n_{0}e^{2}}{m})^{\frac{1}{2}}$ (Saberian & Esfandyari-Kalejahi, 2013), it is convenient to rewrite the linear dispersion relation for the later references as follows: $(\frac{\omega_{r}}{\omega_{p}})^{2}=(k\lambda_{D})^{2}[\frac{\frac{1}{2}(1+\frac{1}{\sigma})}{(k\lambda_{D})^{2}(1+\frac{1}{\sigma})+(\frac{1+q}{2})}+3(\frac{1}{3q-1})(1+\sigma)].$ (21) On the other hand, by using the Eq. (11) and applying the $q$-nonextensive distribution function (13), it is straightforward to obtain the imaginary part of the dielectric function as follows: $D_{i}(k,\omega_{r})=L_{q}\frac{\sqrt{\pi}}{k^{3}\lambda_{D}^{3}(1+\frac{1}{\sigma})^{\frac{3}{2}}}\frac{\omega_{r}}{\omega_{p}}\\{[1-(q-1)\frac{\omega_{r}^{2}}{k^{2}\lambda_{D}^{2}\omega_{p}^{2}(1+\frac{1}{\sigma})}]^{\frac{2-q}{q-1}}+\frac{1}{\sigma^{\frac{3}{2}}}[1-(q-1)\frac{\omega_{r}^{2}}{k^{2}\lambda_{D}^{2}\omega_{p}^{2}(1+\sigma)}]^{\frac{2-q}{q-1}}\\}.$ (22) By $D_{r}(k,\omega_{r})$ and $D_{i}(k,\omega_{r})$ given in Eqs. (17) and (22), we may obtain the explicit solution of the imaginary part of the frequency by using the relation (12b), noting that both $k\lambda_{D}$ and $\frac{\omega_{i}}{\omega_{r}}$ are assumed small. The result is as follows: $\displaystyle\omega_{i}=-\sqrt{\frac{\pi}{8}}\omega_{r}L_{q}(\frac{1}{(k\lambda_{D})^{2}(1+\frac{1}{\sigma})+(\frac{1+q}{2})}+3(\frac{2}{3q-1})\sigma)^{\frac{3}{2}}\times$ $\displaystyle\\{[1-(q-1)(\frac{\frac{1}{2}}{(k\lambda_{D})^{2}(1+\frac{1}{\sigma})+(\frac{1+q}{2})}+\frac{3}{2}(\frac{2}{3q-1})\sigma)]^{\frac{2-q}{q-1}}+$ $\displaystyle\frac{1}{\sigma^{\frac{3}{2}}}[1-(q-1)(\frac{\frac{1}{2\sigma}}{(k\lambda_{D})^{2}(1+\frac{1}{\sigma})+(\frac{1+q}{2})}+\frac{3}{2}(\frac{2}{3q-1}))]^{\frac{2-q}{q-1}}\\},$ (23) where $L_{q}$ is that given in Eq. (19). Note that in deriving the solutions (19) and (23) for the acoustic-like modes in a pair plasma, we have considered the condition $k\lambda_{D}\ll 1$ which indicates the regions with weak damping or growth (long wavelength limit). Moreover, the values of the parameter $\sigma$ (the fractional temperature of the species) must be considered at the vicinity of unit, in order that a suitable compatibility with the physical circumstances is preserved. In the extensive limit $q\rightarrow 1$, our results reduce to the solutions for the acoustic-like modes in a Maxwellian pair plasma as follows: $\omega_{r}^{2}=k^{2}c_{s}^{2}[\frac{1}{k^{2}\lambda_{D}^{2}(1+\frac{1}{\sigma})+1}+3\sigma]$ (24) $\frac{\omega_{i}}{\omega_{r}}=-\sqrt{\frac{\pi}{8}}(\frac{1}{k^{2}\lambda_{D}^{2}(1+\frac{1}{\sigma})+1}+3\sigma)^{\frac{3}{2}}\\{e^{-(\frac{\frac{1}{2}}{k^{2}\lambda_{D}^{2}(1+\frac{1}{\sigma})+1}+\frac{3}{2}\sigma)}+\frac{1}{\sigma^{\frac{3}{2}}}e^{-(\frac{\frac{1}{2\sigma}}{k^{2}\lambda_{D}^{2}(1+\frac{1}{\sigma})+1}+\frac{3}{2})}\\}$ (25) Note that in the extensive limit, the acoustic waves have only the (Landau) damping and no growth, because of the negative value of the imaginary part of the frequency, provided by Eq. (25). Furthermore, in the symmetric case $\sigma\rightarrow 1$, the dispersion relation of the acoustic waves in pair or pair-ion plasmas given in Eq.(24), reduce to Eq.(12) of Ref. (Kaladze et al., 2012). One basic feature of our work is the inclusion of the nonextensivity of the system, which is essentially as a result of the long- range Coulombic interactions of the charge particles in the plasma. The nonextensivity of the system is determined by the spectral index $q$ and may lead to positive or negative $\omega_{i}$ in Eq. (23). Therefore, depending on the nonextensivity of the plasma, both the damped and growing acoustic modes may be happened in a pair plasma. ## 4 Discussion ### 4.1 Dispersion relation The solutions (21) and (23) describe the acoustic-like modes in a nonextensive electron-positron plasma or pair-ion plasma at the limit of long wavelengths confirmed by $k\lambda_{D}\ll 1$. In Fig. (2a) we have plotted the dispersion relation of acoustic modes for some values of the nonextensivity index $q$. In the represented graph, the solid curve corresponds to the extensive limit $q=1$ and the other ones show the deviations from the Maxwellian limit. It is seen that for a given wavelength, the phase velocity of the acoustic modes increases with decreasing the value of $q$. The physical description can be discussed in the context of the nonextensive statistics as follows. As mentioned earlier, the $q$-distribution function with $q<1$, comparing with the Maxwellian one ($q=1$), indicates the systems with more superthermal particles, i.e., particles with the speed faster than the thermal speed $v_{th}=\sqrt{\frac{2k_{B}T}{m}}$ (superextensivity). On the other hand, the $q$-distribution with $q>1$ is suitable to describe systems containing a large number of low-speed particles (subextensivity). However, because of the long- range nature of Coulombic interactions in plasma environments and the presence of many superthermal particles in such systems, confirmed by many astrophysical measurements (Montgomery et al., 1968; Feldman et al., 1975; Maksimovic et al., 1997; Zouganelis, 2008), a $q$-distribution with $q<1$ is strongly suggested for the real plasma systems or superthermal plasmas. It is obvious that in a plasma with more superthermal particles ($q<1$), the phase velocity of the acoustic-like modes should be larger than the case with lack of superthemal particles ($q>1$), in agreement with our results here. In addition, we have illustrated the temperature-asymmetry effect, via $\sigma$, on the dispersion relation of acoustic modes in a pair plasma as shown in Fig. 2(b). There, the solid curve indicates the case in which the whole plasma is in a common thermal state with $T_{-}=T_{+}$, signifies a temperature-symmetric pair plasma, and the other curves show deviations from this symmetric case. We see that the temperature-asymmetry reduces the phase velocity of the acoustic modes in a pair plasma. However, our kinetic model confirms that the acoustic-like modes are possible in both symmetric and asymmetric pair plasmas, depart from a small shift in phase velocity. It is reminded that in this work we have specialized our study to the low- frequency band in which $v_{th,+}<v_{\phi}<v_{th,-}$ . Then, the Cauchy principal value of Eq. (10) is evaluated by an expanding in velocity in the form of Eq. (16). So, our calculations in this frequency band may lead to the acoustic modes and not to the Langmuir waves. On the other hand, considering a high frequency band in which the phase velocity of the wave is much larger than the thermal velocity of the particles ($v_{\phi}>>v_{th}$) may lead to the Langmuir-type waves, as studied in Ref. (Saberian & Esfandyari-Kalejahi, 2013). There, the dispersion relation for the Langmuir waves is given by $\omega_{r}^{2}=\omega_{p}^{2}[1+3(k\lambda_{D})^{2}\frac{2}{3q-1}].$ (26) However, for comparison of the acoustic modes and the Langmuir waves in a pair plasma with $T_{+}=T_{-}$, we have depicted both of the acoustic and Langmuir branches in Fig. 3. From this graph, we see explicitly that the acoustic waves belong to a low frequency band which tends to zero at the limit $k\rightarrow 0$, while the Langmuir waves occur at high frequencies above $\omega_{p}$. On the other hand, the experimental data presented by Oohara _et. al_ (Oohara et al., 2005) confirm the possibility of the acoustic-like modes in a pair plasma which is compatible with our results here. ### 4.2 Landau damping and unstable modes In Fig. 4 we have plotted the ratio $\omega_{i}/\omega_{r}$ with respect to the nonextensivity index $q$ for all allowed values of $q<1$ (referred to superextensivity) at the limit of long wavelengths (supported by, e.g., $k\lambda_{D}=0.1$). It is seen that both of the damped ($\omega_{i}<0$) and growing ($\omega_{i}>0$) acoustic-like modes are predicted in a nonextensive pair plasma with $q<1$. Our numerical analysis shows that in the $q$-region $0.34\lesssim q\lesssim 0.6$ the acoustic modes are unstable, due to the fact that $\omega$’s have positive imaginary parts and then the associated modes will grow in time (Eq. (7) is reminded). The mechanism which leads to this instability may explain as follows. As we expressed earlier, the $q$-nonextensive distribution with $q<1$ describes a system with a large number of superthermal particles. So, our solution for the Vlasov and Poisson’s equations with small values of $q<1$ indicates an evolution which has started from a stationary state with a large portion of superthermal particles. The acoustic-like waves may gain energy from these superthermal particles and results in growing waves in time. In other words, this instability arises from a stationary state which describes a superthermal plasma and, in fact, we have obtained a solution for acoustic-like modes in which the stationary sate of the plasma has started from a non-equilibrium distribution. However, our results have the flexibility to reduce to the equilibrium solutions in the limiting case of $q\rightarrow 1$ indicates a Maxwellian distribution. Furthermore, the acoustic-like modes have Landau damping in the $q$-region $0.6\lesssim q\lesssim 0.71$ because $\omega$’s have negative imaginary parts in these degrees of the nonextensivity (see Fig. 4). The Landau damping is a resonance phenomena between the plasma particles and the wave, for the particles moving with nearly the phase velocity of the wave (Landau, 1946; Krall & Trivelpiece, 1973). Noting that the $q$-distribution is a decreasing function with $u$, there are more particles moving slightly slower than the wave than the particles moving slightly faster than the wave; if the slower particles are accelerated by the wave, this must reduce the energy of the wave, and the wave damps. It is to be noted that our analysis shows that after $q=0.71$, the curve in Fig. 4 rises to positive values for a small interval of $q$ and then it returns to the negative values. The fluctuation of $\omega_{i}$ between the positive and negative values continues increasingly until to the limiting case at $q\rightarrow 1$. In fact, the curve in Fig. 4 don’t show a smooth behavior for the values $0.71<q<1$ and the analysis break down, until to the extensive limit at $q\rightarrow 1$, where our solutions reduce smoothly to that of a Maxwellian pair plasma given in Eqs. (24) and (25). This unsmooth behaviour is because of the existence of the terms $\Gamma(\frac{1}{1-q})$ and $\Gamma(\frac{1}{1-q}-\frac{1}{2})$ in our formalism supported by $L_{q}$. Indeed, this behavior is a mathematical consequence and there is not a physical justification for it. So, we have analyzed the problem in a well- defined interval of $q$, i.e, $1/3<q<0.71$, as shown in Fig. 4. We can also investigate the resonance between the plasma particles and the acoustic modes for the values of $q>1$ (referred to subextensivity). In Fig. 5, the ratio $\omega_{i}/\omega_{r}$ with respect to nonextensivity index $q$ is plotted for the values of $q>1$ at a typical long wavelength ($k\lambda_{D}=0.1$). From this graph, it is seen that the acoustic-like modes have only (Landau) damping and no growth for these degrees of the nonextensivity. Furthermore, the damping rate is relatively weak in these $q$-regions, in comparison with the case of a superthermal plasma ($q<1$). The reason is that the number of particles participating in the resonance with the wave is small for a stationary state with $q>1$. Strictly speaking, the slope of the velocity $q$-distribution function $f_{0\alpha}(u)$ given in Eq. (13) increases with $q$ and there is even a thermal cutoff in the case of $q>1$ [see Fig. 1(b)]. This corresponds to a weaker resonance with the wave, in comparison with the case $q<1$. Our analysis reveals that the acoustic-like modes are unstable in the $q$-region $0.34\lesssim q\lesssim 0.6$ (high superthermal $q$-region) yet, they are heavily damped in the $q$-region $0.6\lesssim q\lesssim 0.71$ (less superthermal $q$-region) and finally, they are relatively weakly damped for the values of $q>1$ (subextensive region). In Fig. 6, the damping and growing rates with respect to the wave number are plotted for some values of the nonextensive index $q$ for three cases of the heavily damped modes [Fig6(a)], weakly damped modes [Fig6(b)] and growing unstable modes [Fig6(c)]. We see that for the waves with longer wavelengths the rate of damping (or growth) becomes weaker. Moreover, our numerical analysis shows that in a pair plasma the acoustic-like modes have the maximum damping at the vicinity of $q=0.69$ [see Fig6(a)], and they have the maximum growth when the nonextensivity is at the vicinity of $q=0.55$ [see Fig6(c)]. In addition, we have included the Maxwellian limit ($q=1$) to the Fig. 6(b) which emphasizes that the acoustic- like modes in an equilibrium pair plasma are merely landau damped waves. For completing our discussion, we have examined the temperature-asymmetry effect, controlled by $\sigma$, on the Landau damping of the acoustic-like modes in a pair plasma, as plotted in Fig. 7. It is observed that the temperature-asymmetry in a pure pair plasma decreases the Landau damping. In other words, for a fixed value of $q$ and at a given wavelength, the Landau damping of the acoustic waves is maximum when a full symmetry in temperature of species is established, i.e, when $T_{-}=T_{+}$. ## 5 Conclusions In this paper, we have studied the acoustic-like modes in a collisionless and magnetic-field-free pair plasma on the basis of the nonextensive statistics. We have thereby used a kinetic theory model by employing the Vlasov and Poisson’s equations to obtain the response dielectric function of the pair plasma for the electrostatic waves. By using the dielectric function, we have investigated the acoustic-like modes whose phase speed lies between the thermal velocities of the species. The resultant dispersion relation in our study is compatible with the acoustic branch of the experimental data presented by Oohara _et. al_ (Oohara et al., 2005), in which the electrostatic waves have been examined in a pure pair-ion plasma. It has been shown that by decreasing the nonextensivity index $q$ the phase velocity of the acoustic modes increases, indicating to a plasma with a great deal of superthermal particles. Our kinetic model confirms the possibility of the acoustic modes in the case of a temperature-asymmetric and also symmetric pair plasma. However, it is found that the temperature-asymmetry in a pair plasma reduces the phase velocity of the acoustic modes. Furthermore, depending on the degree of nonextensivity of the plasma, both the damped and unstable acoustic modes are predicted in a collisionless pair plasma, arise from a resonance phenomena between the wave and nonthermal particles of the plasma. In the case of a superthemal plasma confirmed by $q<1$ (superextensivity), the heavily damped and growing unstable modes are predicted, while in the case $q>1$ (subextensivity) the acoustic-like modes have only damping and no growth. The mechanism that leads to the damping is the same as presented by Landau (Landau, 1946), arises from a decreasing velocity distribution function, but the mechanism of instability lies in the heart of the nonextensive formalism. We have postulated that the concerned instability can be associated with the presence of superthermal particles (in the case $q<1$), in the sense that in the process of the resonance they can give energy to the wave and then results in growing waves in time. This instability disappears in the case $q>1$, describing a plasma with plenty of the low-speed particles. Additionally, the damping rate is relatively weak in the case $q>1$, in comparison with the case of a superthermal plasma ($q<1$) with heavily damped modes. The reason is that the number of particles participating in the resonance with the wave is small for a stationary state with $q>1$. Moreover, our analysis indicates that the temperature-asymmetry in a pure pair plasma decreases the Landau damping of the acoustic-like modes. We emphasize that in the present work, we have considered an inhomogeneous plasma in a nonequilibrium thermal state by considering the $q$-nonextensive distribution for stationary state of the plasma. 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Figure 2: The linear dispersion relation of acoustic-like modes in a pair plasma. (a) The nonextensivity effect on dispersion relation with $\sigma=0.9$, where the solid curve corresponds to the extensive limit ($q=1$) and the other ones show the deviations from a Maxwellian pair plasma. (b) The effect of temperature-asymmetry on dispersion relation with $q=0.7$, where the solid curve corresponds to a temperature-symmetric pair plasma. Figure 3: The comparison of the acoustic-like modes and the Lanqmuir waves in a pair plasma with $T_{+}=T_{-}$. The acoustic waves belong to a low frequency band which tends to zero at the limit $k\rightarrow 0$, while the Langmuir waves occur in high frequencies above $\omega_{p}$. Figure 4: The imaginary part of the frequency with respect to the nonextensivity index for $q<1$, which shows the $q$-regions for the growing and heavily damped acoustic-like modes. Figure 5: The imaginary part of the frequency with respect to the nonextensivity parameter for $q>1$. For this values of the nonextensivity index $q$, the acoustic-like modes have only damping and no growth. Figure 6: The damping (growing) rate with respect to the wave number for (a) the heavily damped modes in the $q$-region $0.6\lesssim q\lesssim 0.71$, (b) the relatively weakly damped modes in the $q$-region $q>1$, and (c) the growing acoustic modes in the $q$-region $0.34\lesssim q\lesssim 0.6$, when $\sigma=0.9$. We have included the Maxwellian limit ($q=1$) to our results which emphasizes that the acoustic-like modes in an equilibrium pair plasma are merely the landau damped waves. Figure 7: The effect of temperature-asymmetry on Landau damping of the acoustic-like modes which indicates that the temperature- asymmetry in a pure pair plasma decreases the Landau damping rate.	arxiv-papers	2013-11-01T14:26:13	2024-09-04T02:49:53.187850	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "E. Saberian and A. Esfandyari-Kalejahi", "submitter": "Ehsan Saberian", "url": "https://arxiv.org/abs/1311.0193" }
1311.0276	# $B$-meson decay constants with domain-wall light quarks and nonperturbatively tuned relativistic $b$-quarks Center for Computational Science, Boston University, 3 Cummington Mall, Boston, MA 02215, USA E-mail ###### Abstract: We report on our progress to obtain the decay constants $f_{B}$ and $f_{B_{s}}$ from lattice-QCD simulations on the RBC-UKQCD Collaborations 2+1 flavor domain-wall Iwasaki lattices. Using domain-wall light quarks and relativistic $b$-quarks we analyze data with several partially quenched light- quark masses at two lattice spacings of $a\approx 0.11$ fm and $a\approx 0.08$ fm. ## 1 Motivation $B$-physics plays a central role in the global efforts to constrain the CKM unitarity triangle. The ratio of neutral $B$-meson mixing, e.g., is used in the unitarity triangle fits [1, 2, 3]. Neutral $B$-mesons mix with their anti- particle under the exchange of two $W$-bosons as depicted by the box-diagrams in Fig. 1. There $q$ denotes a light $d$\- or $s$-quark building either a $B$\- or a $B_{s}$-meson, respectively. In the experiments, e.g., BaBar, Belle, CDF or LHCb, $B_{q}$-mixing is measured in terms of the oscillation frequencies (mass differences) $\Delta M_{q}$ and in the Standard Model (SM) this process is parameterized by [4] Figure 1: Box-diagrams with top-quarks in the loop are the dominant contributions to neutral $B$-meson mixing. $q$ denotes either a $d$\- or $s$-quark. $\displaystyle\Delta M_{q}=\frac{G_{F}^{2}m^{2}_{W}}{6\pi^{2}}\eta_{B}S_{0}M_{B_{q}}{f_{B_{q}}^{2}B_{B_{q}}}\lvert V_{tq}^{}V_{tb}\rvert^{2},$ (1) where the QCD coefficient $\eta_{b}$ [4] and the Inami-Lim function $S_{0}$ [5] are computed perturbatively and a nonperturbative computation is needed for the leptonic $B_{q}$-meson decay constant $f_{B_{q}}$ and the bag parameter $B_{B_{q}}$ in order to extract the CKM matrix elements $V^{}_{tq}V_{tb}$. Experimentally $\Delta M_{q}$ is measured to subpercent accuracy [6], whereas the nonpeturbative (lattice) inputs contribute the dominant uncertainty (order few percent). Taking the ratio of neutral $B$-meson mixing $\displaystyle\frac{\Delta M_{s}}{\Delta M_{d}}=\frac{M_{B_{s}}}{M_{B_{d}}}\,{\xi^{2}}\,\frac{\lvert V_{ts}\rvert^{2}}{\lvert V_{td}\rvert^{2}},$ (2) the nonperturbative contribution is contained in the $SU(3)$ breaking ratio $\displaystyle\xi$ $\displaystyle=\frac{f_{B_{s}}\sqrt{B_{B_{s}}}}{f_{B_{d}}\sqrt{B_{B_{d}}}},$ (3) for which statistical and systematic uncertainties largely cancel [7]. Unfortunately $\xi$ still contributes the largest uncertainty. We therefore designed this project to compute neutral $B$-meson mixing matrix elements as well as the leptonic decay constants $f_{B}$ and $f_{B_{s}}$. The decay constants are important to further constrain new physics by allowing an alternative determination of $V_{ub}$ using the measurement of $B\to\tau\nu$ [8, 9, 10] or by allowing, e.g., to obtain predictions on rare decays like $B_{s}\to\mu_{+}\mu_{-}$ [11] which promise to be in particular sensitive to new physics. Computing $B$-physics quantities on the lattice faces the additional challenge to accommodate an additional scale given by the large $b$-quark mass. In our project we compute $B$-physics quantities using the RBC-UKQCD 2+1 flavor domain-wall Iwasaki gauge field configurations. We simulate the $b$-quarks with the relativistic heavy quark (RHQ) action and tune the action’s parameters nonperturbatively, while domain-wall fermions simulate the light $u,\,d,\,s$-quarks. Thus our project is an independent cross-check to published results by other groups based on 2-flavor [12], 2+1-flavor [13, 14, 15, 16, 17] or 2+1+1-flavor [18] gauge-field configurations. In these proceedings we focus on the computation of the $B$-meson decay constants $f_{B}$ and $f_{B_{s}}$. ## 2 Computational setup This computation uses the dynamical 2+1 flavor domain-wall Iwasaki gaugefield configurations generated by the RBC-UKQCD collaboration [19, 20] listed in Tab. 1. We use two coarser, $24^{3}$ ensembles with $a\approx 0.11$fm ($a^{-1}=1.729$ GeV) and three finer, $32^{3}$ ensembles with $a\approx 0.086$ fm ($a^{-1}=2.281$ GeV). On the coarser ensembles we place one source per configuration, whereas on the finer ensembles we place two time sources per configuration separated by half the temporal extent of the lattice. For each source we generate six domain-wall [21, 22] propagators with quark masses $am_{\text{val}}^{24}$ = 0.005, 0.010, 0.020, 0.030, 0.0343 and $0.040$ on the coarser $24^{3}$ ensembles and $am_{\text{val}}^{32}$ = 0.004, 0.006, 0.008, 0.025, 0.0272 and 0.030 on the finer $32^{3}$ ensembles. The masses of the three heaviest domain-wall propagators bracket the physical strange quark mass. Table 1: Lattice simulation parameters used in our $B$-physics program. The columns list the lattice volume, approximate lattice spacing, light ($m_{l}$) and strange ($m_{h}$) sea-quark masses, unitary pion mass, and number of configurations and time sources analyzed. $\left(L/a\right)^{3}\times\left(T/a\right)$ \| $\approx a$(fm) \| $am_{l}$ \| $am_{h}$ \| $M_{\pi}$(MeV) \| # configs. \| # time sources ---\|---\|---\|---\|---\|---\|--- $24^{3}\times 64$ \| 0.11 \| 0.005 \| 0.040 \| 329 \| 1636 \| 1 $24^{3}\times 64$ \| 0.11 \| 0.010 \| 0.040 \| 422 \| 1419 \| 1 $32^{3}\times 64$ \| 0.086 \| 0.004 \| 0.030 \| 289 \| 628 \| 2 $32^{3}\times 64$ \| 0.086 \| 0.006 \| 0.030 \| 345 \| 889 \| 2 $32^{3}\times 64$ \| 0.086 \| 0.008 \| 0.030 \| 394 \| 544 \| 2 We simulate the $b$-quarks using the the anisotropic Sheikholeslami-Wohlert (clover) action with the relativistic heavy-quark (RHQ) interpretation [23, 24]. The three parameters, $m_{0}a$, $c_{P}$, $\zeta$, are tuned nonperturbatively using the experimental inputs for the spin-averaged mass $\overline{M}$ and the hyperfine-splitting $\Delta_{M}$ in the $B_{s}$-meson system and demanding that the rest mass equals the kinetic mass, i.e., $M_{1}/M_{2}=1$ [25]. The parameters are tuned by probing seven points of the $(m_{0}a,\,c_{P},\,\zeta)$ parameter space and then we interpolate to the tuned value by matching to the experimental values. We use the same seven sets of RHQ parameters in our computation of the decay constants $f_{B}$ and $f_{B_{s}}$ because this allows us to cleanly propagate the statistical uncertainty of our tuning procedure to the final results. The decay constants are measured on the lattice by computing the decay amplitude $\Phi_{B}$ which is proportional to the vacuum-to-meson matrix element of the heavy-light axial vector current ${\cal A}_{\mu}=\bar{b}\gamma_{5}\gamma_{\mu}q$ and depicted in Fig. 3 Figure 2: Schematic computation of the decay amplitude $\Phi_{B_{q}}$ with $q$ denoting a $d$\- or $s$-quark. Figure 3: Schematic computation of the flavor-conserving renormalization factor $Z_{V}^{bb}$ using a $s$-quark as spectator. $\displaystyle\langle 0\|{\cal A}_{\mu}\|B_{q}(p)\rangle/\sqrt{M_{B_{q}}}=ip^{\mu}\Phi_{B_{q}}^{(0)}/M_{B_{q}}.$ (4) The mass of the $B_{q}$-meson is $M_{B_{q}}$ and $p^{\mu}$ denotes its four momentum. We reduce lattice discretization errors by $O(a)$-improving the axial vector current, $\Phi_{B_{q}}^{\text{imp}}=\Phi_{B_{q}}^{(0)}+c_{1}\Phi_{B_{q}}^{(1)}$, and compute the coefficient $c_{1}$ at 1-loop with mean-field improved lattice perturbation theory [26]. Finally we obtain the decay constant $f_{B_{q}}$ from $\Phi_{B_{q}}^{\text{imp}}$ by multiplying the renormalization factor $Z_{\Phi}$, the lattice spacing and the mass of the $B_{q}$-meson $\displaystyle f_{B_{q}}=Z_{\Phi}\Phi_{B_{q}}^{\text{imp}}a^{-3/2}/\sqrt{M_{B_{q}}}.$ (5) For the computation of the renormalization factor $Z_{\Phi}$ we follow the mostly nonperturbative method described in [27] and compute $Z_{\Phi}$ as product of the two nonperturbatively computed, flavor-conserving factors $Z_{V}^{ll}$ and $Z_{V}^{bb}$ and a perturbatively computed factor $\varrho_{bl}$ which is expected to be close to one and to have a more convergent series expansion in $\alpha_{s}$ $\displaystyle Z_{\Phi}=\varrho_{bl}\sqrt{Z_{V}^{bb}Z_{V}^{ll}}.$ (6) The perturbative factor $\rho_{bl}$ is computed at 1-loop with mean-field improved lattice perturbation theory [28] and the RBC-UKQCD collaboration already measured $Z_{V}^{ll}$ [20]. The factor $Z_{V}^{bb}$ is determined as part of this project [29]. ## 3 Preliminary results We determine $Z_{v}^{bb}$ by measuring the 3-point function describing a $B$-meson going to a $B$-meson with the insertion of a vector current between both $b$-quarks (see Fig. 3) $\displaystyle Z_{V}^{bb}\times\langle B\|V^{bb,0}\|B\rangle=2m_{B}.$ (7) Since $Z_{V}^{bb}$ does not explicitly depend on the spectator quark, it is advantageous to use a $s$-quark as spectator because it has smaller statistical uncertainties compared to a lighter quark. For this computation we simulate the $b$-quarks using a single set of tuned RHQ parameters [25]. We extract $Z_{V}^{bb}$ from a fit to the plateau of the above defined 3-pt function normalized by the corresponding $B_{s}$-meson 2-pt function for each of our five ensembles. Fig. 4 shows example data for $Z_{V}^{bb}$ on the finer, $32^{3}$ ensemble with light sea-quark mass $a_{32}m_{\text{sea}}^{l}=0.006$. The data form a long plateau and the fit interval is chosen such that excited state contamination present in the 2pt- data has decayed and is not affecting our signal. Plots for the other ensembles look similar. We list the values for $Z_{V}^{bb}$ for all our ensembles in Tab. LABEL:Tab:Zvbb. As expected we do not observe a dependence on the sea-quark mass. Furthermore we use the results to test the reliability of lattice perturbation theory used for different parts of this project, e.g., the factor $\varrho_{bl}$. We show the results for $Z_{V}^{bb}$ obtained at 1-loop mean-field improved lattice perturbation theory [26] and compare them to the averages of our nonperturbative determinations. We observe a better- than-expected agreement. The decay constants and the ratio are obtained by first fitting plateaus of the $O(a)$-improved and renormalized decay amplitudes, $\Phi_{B_{q}}^{\text{ren}}=Z_{\Phi}\Phi_{B_{q}}^{\text{imp}}$, for all six valence quark masses on our five ensembles. An example for $q=0.004$ on the $32^{3}$ ensemble with $am_{\text{sea}}^{l}=0.006$ is given in Fig. 6. We determine $f_{B_{s}}$ by performing a linear interpolation of the three strange-like data points to the physical value of the strange quark mass. Then we extrapolate the interpolated results on the five ensembles to the continuum with a function that is linear in $a^{2}$ (motivated by the leading scaling behavior of the light-quark and gluon actions) and independent of sea-quark mass and obtain $f_{B_{s}}=236(5)$ MeV (statistical error only). Figure 4: Example plot for the determination of the flavor-conserving renormalization factor $Z_{V}^{bb}$ on the finer, $32^{3}$ ensemble with $m_{\text{sea}}^{l}=0.006$. The physical value of the decay constant $f_{B}$ and the ratio $f_{B_{s}}/f_{B}$ are obtained from a combined chiral-continuum extrapolation using next-to-leading order SU(2) heavy meson chiral perturbation theory (HM$\chi$PT) [30, 31, 32, 33] $\displaystyle\Phi_{B}$ $\displaystyle=\Phi_{0}\left[1-\chi_{\text{SU(2)}}^{f_{B}}+c_{\text{sea}}m_{\text{sea}}^{l}2B/(4\pi f)^{2}+c_{\text{val}}m_{\text{val}}2B/(4\pi f)^{2}+c_{a}a^{2}/(a_{32}^{2}4\pi f)^{2}\right],$ (8) and $\displaystyle\Phi_{B_{s}}/\Phi_{B}$ $\displaystyle=R_{\Phi}\left[1-\chi_{\text{SU(2)}}^{\text{ratio}}+c_{\text{sea}}m_{\text{sea}}^{l}2B/(4\pi f)^{2}+c_{\text{val}}m_{\text{val}}2B/(4\pi f)^{2}+c_{a}a^{2}/(a_{32}^{2}4\pi f)^{2}\right].$ (9) The chiral logarithms, $\chi_{\text{SU(2)}}^{f_{B}}$ and $\chi_{\text{SU(2)}}^{\text{ratio}}$, are nonanalytic functions of the pseudo- Goldstone meson masses, and are given in the appendix of reference [33]. Our preliminary results are shown in Fig. 7. The fits are performed including partially-quenched data on all five sea-quark ensembles, but with valence- quark masses restricted to be below $M_{\pi}^{\text{val}}<350$ MeV. These extrapolations give us a preliminary value of $f_{B}=196(6)$ MeV and a SU(3) breaking ratio of $f_{B_{s}}/f_{B_{q}}$ = 1.21(2). Again only statistical uncertainties are quoted. We are finalizing our budget of systematic errors which will also include, e.g., heavy quark discretization errors. All our preliminary results are in agreement with the literature in particular if taking into account that systematic errors will be added. ## 4 Outlook We hope to complete and publish our analysis for $f_{B}$, $f_{B_{s}}$ and their ratio $f_{B_{s}}/f_{B}$ soon. We anticipate that our largest source of error will be from the chiral-continuum extrapolation. In the future, we will take advantage of the new Möbius domain-wall ensembles generated by the RBC- UKQCD collaboration which feature simulations at the physical pion mass. ## Acknowledgments We thank our colleagues of the RBC and UKQCD collaborations for useful help and discussions. Numerical computations for this work utilized USQCD resources at Fermilab, in part funded by the Office of Science of the U.S. Department of Energy, as well as computers at Brookhaven National Laboratory and Columbia University. O.W. acknowledges support at Boston University by the U.S. DOE grant DE-SC0008814. Figure 5: Example plot for the determination of the decay amplitude $\Phi_{B_{q}}^{ren}$ from a fit to the plateau for light valence quark $q=0.004$ on the $32^{3}$ ensemble with $am_{\text{sea}}^{l}=0.006$. Figure 6: Continuum extrapolation of $\Phi_{B_{s}}$. The different colored points at each lattice spacing correspond to different sea-quark ensembles, and are horizontally offset for clarity. Figure 7: Chiral-continuum extrapolation of $\Phi_{B_{q}}$ (left) and $\Phi_{B_{s}}/\Phi_{B_{q}}$ (right). For better visibility some data points are plotted with a small horizontal offset. Only the filled data points are included in the fit. The vertical gray bands indicate the physical values of the $u/d$\- and $s$-quark masses [19, 20]. ## References * [1] J. Charles _et al._ (CKMfitter Group), Eur.Phys.J. C41, 1 (2005), arXiv:hep-ph/0406184, http://ckmfitter.in2p3.fr/ * [2] M. Bona _et al._ (UTfit), JHEP 0507, 028 (2005), arXiv:hep-ph/0501199, http://utfit.roma1.infn.it/ * [3] J. Laiho, E. Lunghi, and R. S. Van de Water, Phys.Rev. D81, 034503 (2010), arXiv:0910.2928 [hep-ph], www.latticeaverages.org * [4] A. J. Buras, M. Jamin, and P. H. Weisz, Nucl. Phys. B347, 491 (1990) * [5] T. Inami and C. S. Lim, Prog. Theor. Phys. 65, 297 (1981) * [6] J. Beringer _et al._ (Particle Data Group), Phys.Rev. D86, 010001 (2012) * [7] C. W. Bernard, T. Blum, and A. Soni, Phys.Rev. 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Van de Water, and O. Witzel, PoS Lattice2012, 109 (2012), arXiv:1211.0956 [hep-lat] * [30] J. L. Goity, Phys. Rev. D46, 3929 (1992), arXiv:hep-ph/9206230 * [31] D. Arndt and C. D. Lin, Phys.Rev. D70, 014503 (2004), arXiv:hep-lat/0403012 * [32] C. Aubin and C. Bernard, Phys. Rev. D73, 014515 (2006), arXiv:hep-lat/0510088 * [33] C. Albertus _et al._ (RBC-UKQCD), Phys.Rev. D82, 014505 (2010), arXiv:1001.2023 [hep-lat]	arxiv-papers	2013-11-01T19:42:21	2024-09-04T02:49:53.199509	{ "license": "Public Domain", "authors": "Oliver Witzel", "submitter": "Oliver Witzel", "url": "https://arxiv.org/abs/1311.0276" }
1311.0378	# Comparative Performance Analysis of Intel Xeon Phi, GPU, and CPU George Teodoro1, Tahsin Kurc2,3, Jun Kong4, Lee Cooper4, and Joel Saltz2 1Department of Computer Science, University of Brasília, Brasília, DF, Brazil 2Department of Biomedical Informatics, Stony Brook University, Stony Brook, NY, USA 3Scientific Data Group, Oak Ridge National Laboratory, Oak Ridge, TN, USA 4Department of Biomedical Informatics, Emory University, Atlanta, GA, USA ###### Abstract We investigate and characterize the performance of an important class of operations on GPUs and Many Integrated Core (MIC) architectures. Our work is motivated by applications that analyze low-dimensional spatial datasets captured by high resolution sensors, such as image datasets obtained from whole slide tissue specimens using microscopy image scanners. We identify the data access and computation patterns of operations in object segmentation and feature computation categories. We systematically implement and evaluate the performance of these core operations on modern CPUs, GPUs, and MIC systems for a microscopy image analysis application. Our results show that (1) the data access pattern and parallelization strategy employed by the operations strongly affect their performance. While the performance on a MIC of operations that perform regular data access is comparable or sometimes better than that on a GPU; (2) GPUs are significantly more efficient than MICs for operations and algorithms that irregularly access data. This is a result of the low performance of the latter when it comes to random data access; (3) adequate coordinated execution on MICs and CPUs using a performance aware task scheduling strategy improves about 1.29$\times$ over a first-come-first-served strategy. The example application attained an efficiency of 84% in an execution with of 192 nodes (3072 CPU cores and 192 MICs). ## I Introduction Scientific computing using co-processors (accelerators) has gained popularity in recent years. The utility of graphics processing units (GPUs), for example, has been demonstrated and evaluated in several application domains [1]. As a result, hybrid systems that combine multi-core CPUs with one or more co- processors of the same or different types are being more widely employed to speed up expensive computations. The architectures and programming models of co-processors may differ from CPUs and vary among different co-processor types. This heterogeneity leads to challenging problems in implementing application operations and obtaining the best performance. The performance of an application operation will depend on the operation’s data access and processing patterns, and may vary widely from one co-processor to another. Understanding the performance characteristics of classes of operations can help in designing more efficient applications, choosing the appropriate co- processor for an application, and developing more effective task scheduling and mapping strategies. In this paper, we investigate and characterize the performance of an important class of operations on GPUs and Intel Xeon Phi Many Integrated Core (MIC) architectures. Our primary motivating application is digital Pathology involving the analysis of images obtained from whole slide tissue specimens using microscopy image scanners. Digital Pathology is a relatively new application domain and imaging modality compared to magnetic resonance imaging and computed tomography. Nevertheless, it is an important application domain because investigation of disease morphology at the cellular and sub-cellular level can reveal important clues about disease mechanisms that are not possible to capture by other imaging modalities. Analysis of a whole slide tissue image is both data and computation intensive because of the complexity of analysis operations and data sizes – a three-channel color image captured by a state-of-the-art scanner can reach 100K$\times$100K pixels in resolution. Compounding this problem is the fact that modern scanners are capable of capturing images rapidly, facilitating research studies to gather thousands of images. Moreover, an image dataset may be analyzed multiple times to look for different features or quantify sensitivity of analysis to input parameters. Although the microscopy image analysis is our main motivating application, we expect that our findings in this work will be applicable in other applications. Microscopy image analysis belongs to a class of applications that analyze low-dimensional spatial datasets captured by high resolution sensors. This class of applications include those that process data from satellites and ground-based sensors in weather and climate modeling; analyze satellite data in large scale biomass monitoring and change analyses; analyze seismic surveys in subsurface and reservoir characterization; and process wide field survey telescope datasets in astronomy [2, 3, 4, 5]. Datasets in these applications are generally represented in low-dimensional spaces (typically a 2D or 3D coordinate system); typical data processing steps include identification or segmentation of objects of interest and characterization of the objects (and data subsets) via a set of features. Table I lists the categories of common operations in these application domains and presents examples in microscopy image analysis. Operations in these categories produce different levels of data products that can be consumed by client applications. For example, a client application may request only a subset of satellite imagery data covering the east coast of the US. Operations from different categories can be chained to form analysis workflows to create other types of data products. The data access and processing patterns in these operation categories range from local and regular to irregular and global access to data. Local data access patterns correspond to accesses to a single data element or data elements within a small neighborhood in a spatial and temporal region (e.g., data cleaning and low-level transformations). Regular access patterns involve sweeps over data elements, while irregular accesses may involve accesses to data elements in a random manner (e.g., certain types of object classification algorithms, morphological reconstruction operations in object segmentation). Some data access patterns may involve generalized reductions and comparisons (e.g., aggregation) and indexed access (e.g., queries for data subsetting and change quantification). TABLE I: Operation Categories Operation Category \| Microscopy Image Analysis ---\|--- Data Cleaning and Low Level Transformations \| Color normalization. Thresholding of pixel and regional gray scale values. Object Segmentation \| Segmentation of nuclei and cells. Feature Computation \| Compute texture and shape features for each cell. Aggregation \| Aggregation of object features for per image features. Classification \| Clustering of nuclei and/or images into groups. Spatio-temporal Mapping and Registration \| Deformable registration of images to anatomical atlas. Data Subsetting, Filtering, and Subsampling \| Selection of regions within an image. Thresholding of pixel values. Change Detection and Comparison \| Spatial queries to compare segmented nuclei and features within and across images. Our work examines the performance impact of different data access and processing patterns on application operations on CPUs, GPUs, and MICs. The main contributions of the paper can be summarized as follows: (1) We define the data access and computation patterns of operations in the object segmentation and feature computation categories for a microscopy image analysis application. (2) We systematically evaluate the performance of the operations on modern CPUs, GPUs, and MIC systems. (3) The results show that the data access pattern and parallelization strategy employed by the operations strongly affect their performance. While the performance on a MIC of operations that perform regular data access is comparable or sometimes better than that on a GPU. GPUs are significantly more efficient than MICs for operations and algorithms that irregularly access data. This is a result of the low performance of the latter when it comes to random data access. Coordinated execution on MICs and CPUs using a performance aware task scheduling strategy improves about 1.29$\times$ over a first-come-first-served strategy. The example application attained an efficiency of 84% in an execution with of 192 nodes (3072 CPU cores and 192 MICs). ## II Example Application and Core Operations In this section we provide a brief overview of microscopy image analysis as our example application. Presently our work is focused on the development of operations in the object segmentation and feature computation categories, since these are the most expensive categories (or stages) in this application. We describe the operations in these stages, and present the operations’ data access and processing patterns. ### II-A Microscopy Image Analysis Along with advances in slide scanners, digitization of whole slide tissues, extracted from humans or animals, has become more feasible and facilitated the utility of whole slide tissue specimens in research as well as clinical settings. Morphological changes in tissues at the cellular and sub-cellular scales provide valuable information about tumors, complement genomic and clinical information, and can lead to a better understanding of tumor biology and clinical outcome [6]. Use of whole slide tissue images (WSIs) in large scale studies, involving thousands of high resolution images, is a challenging problem because of computational requirements and the large sizes of WSIs. The segmentation and feature computation stages may operate on images of 100K$\times$100K pixels in resolution and may identify about millions of micro-anatomic objects in an image. Cells and nuclei are detected and outlined during the segmentation stage. This stage applies a cascade of operations that include pixel value thresholds and morphological reconstruction to identify candidate objects, fill holes to remove holes inside objects, area thresholding to filter out objects that are not of interest. Distance transform and watershed operations are applied to separate objects that overlap. The feature computation stage calculates a set of spatial and texture properties per object that include pixel and gradient statistics, and edge and morphometry features. The next section describes the set of core operations in these two stages and their data and processing patterns. ### II-B Description of Core Operations The set of core operations in the segmentation and feature computation stages is presented in Table II. These operations are categorized according to the computation stage in which they are used (segmentation or feature computation), data access pattern, computation intensity, and the type of parallelism employed for speeding up computation. TABLE II: Core operations in segmentation and feature computation phases from microscopy image analysis. IWPP stands for Irregular Wavefront Propagation Pattern. Operations \| Description \| Data Access Pattern \| Computation \| Parallelism ---\|---\|---\|---\|--- Segmentation Phase Covert RGB to grayscale \| Covert a color RGB image into \| Regular, multi-channel \| Moderate \| Data grayscale intensity image \| local \| \| Morphological Open \| Opening removes small objects and \| Regular, neighborhood \| Low \| Data \| fills small holes in foreground \| (13x13 disk) \| \| Morphological \| Flood-fill a marker image that is limited by \| Irregular, neighborhood \| Low \| IWPP Reconstruction [7] \| a mask image. \| (4-/8-connected) \| \| Area Threshold \| Remove objects that are not within an area range \| Mixed, neighborhood \| Low \| Reduction FillHolles \| Fill holes in an image objects using a flood-fill \| Irregular, neighborhood \| Low \| IWPP \| in the background pixels starting at selected points \| (4-/8-connected) \| \| Distance Transform \| Computes the distance to the closest background \| Irregular, neighborhood \| Moderate \| IWPP pixel for each foreground pixel \| (8-connected) \| \| Connected Components \| Label with the same value pixels in components \| Irregular, global \| Low \| Union-find Labeling \| (objects) from an input binary image \| \| \| Feature Computation Phase Color Deconvolution [8] \| Used for separation of multi-stained \| Regular, multi-channel \| Moderate \| Data biological images into different channels \| local \| \| Pixel Statistics \| Compute vector of statistics (mean, median, \| Regular, access a set \| High \| Object max, etc) for each object in the input image \| of bounding-boxed areas \| \| Gradient Statistics \| Calculates magnitude of image gradient in x,y \| Regular, neighborhood \| High \| Object and derive same per object features \| and bounding-boxed areas \| \| Sobel Edge \| Compute vector of statistics (mean, median, \| Regular, access a set \| High \| Object max, etc) for each object in the input image \| of bounding-boxed areas \| \| The operations in the segmentation stage carry out computations on elements from the input data domain (pixels in the case of an image), while those in the feature computation stage additionally perform computations associated with objects. In regards to data access patterns, the core operations may be first classified as: 1) _regular operations_ that access data in contiguous regions such as data scans; or 2) _irregular operations_ in which data elements to be processed are irregularly distributed or accessed in the data domain. For some operations, data elements to be processed are only known during execution as runtime dependencies are resolved as a result of the computation. Examples of such operations include those that perform flood-fill and irregular wave front propagations. Data accessed in the computation of a given data element may be: 1) local for cases in which the computation of a data element depends only on its value; 2) multi-channel local, which is a variant of the former in operations that independently access data elements with the same index across multiple layers of the domain (e.g., multiple channels in an image); 3) within a neighborhood, which refers to cases when an operation performs computations on data elements in a spatial and/or temporal neighborhood. The neighborhood is often defined using structure elements such as 4-/8-connected components or discs; and 4) areas in a bounding-box, which are used in the feature computation phase in operations on objects that are defined within minimum bounding boxes. Parallel execution patterns exhibited by the core operations are diverse: 1) Data parallelism; 2) Object parallelism; 3) MapReduce [9] or generalized reduction; 4) Irregular wavefront propagation pattern (IWPP) [10, 11]; 5) Union-find [12]. The _data parallel_ operations are those that concurrently and independently process elements of the data domain. The _Object parallelism_ exists in operations that process multiple objects concurrently. Moreover, a _MapReduce-style pattern_ is also used in the “Area Threshold” operation. This operation maps elements from the input data according to their values (labels) before a reduction is performed to count the number of elements with the same label value (area of components). The area is then used to filter out components that are not within the desired size range. The _IWPP_ pattern is characterized by independent wavefronts that start in one or more elements of the domain. The structure of the waves is dynamic, irregular, data dependent, and only known at runtime as expansions are computed. The elements forming the front of the waves (active elements) work as sources of propagations to their neighbors, and only active elements are the ones that contribute to the output. Therefore, the efficient execution of this pattern relies on using a container structure, e.g., a queue or a set, to maintain the active elements and avoid computing areas of the data domain that are not effectively contributing to the output. This pattern is presented in Algorithm 1. A set of (active) elements from a multi-dimensional grid space ($D$) is selected to compose the wavefront ($S$). During the propagations, an element ($e_{i}$) from S is extracted and the algorithm tries to propagate $e_{i}$ value to its neighbors ($N_{G}$) in a structure $G$. If a propagation condition between the active element and each of the neighbors ($e_{j}\in Q$) is evaluated true, the propagation occurs and that neighbor receiving the propagation is added to the set of active elements ($S$). This process occurs until the set $S$ is not empty. The parallelization of this pattern heavily relies on an efficient parallel container to store the wavefront elements. In the parallel version of IWPP multiple elements from the active set may be computed in parallel as long as race conditions that may arise due parallel propagations that update the same element $e_{j}$ in the grid are avoided. Applications that use this pattern, in addition to our core operations, include: Watershed, Euclidean skeletons, skeletons by influence zones, Delaunay triangulations, Gabriel graphs and relative neighborhood graphs. Algorithm 1 Irregular Wavefront Propagation Pattern 1: $D\leftarrow$ data elements in a multi-dimensional space 2: {Initialization Phase} 3: $S\leftarrow$ subset active elements from $D$ 4: {Wavefront Propagation Phase} 5: while $S\neq\emptyset$ do 6: Extract $e_{i}$ from $S$ 7: $Q\leftarrow$ $N_{G}(e_{i})$ 8: while $Q\neq\emptyset$ do 9: Extract $e_{j}$ from $Q$ 10: if $PropagationCondition$($D(e_{i})$,$D(e_{j})$) $=$ true then 11: $D(e_{j})\leftarrow$ $Update$($D(e_{i})$) 12: Insert $e_{j}$ into $S$ The _union-find pattern_ [12] is used for manipulating disjoint-set data structures and is made up of three operations: 1) Find: determines the set in which a component is stored; 2) Union: merges two subsets into a single set; and 3) MakeSet: creates an elementary set containing a single element. This is the processing structure of the connected components labeling (CCL) operation in our implementation. The CCL first creates a forest in which each element (pixel) from the input data is an independent tree. It iteratively merges trees from adjacent elements in the data domain such that one tree becomes a branch in another tree. The condition for merging trees is that the neighbor elements must be foreground pixels in the original image. When merging two trees (Union), the label values of the root of the two trees are compared, and the root with the smaller value is selected as the root of the merged tree. After this process is carried out for all pixels, each connected component is assigned to a single tree, and the labeled output can be computed by flattening the trees and reading labels. ## III Implementation of Core Operations ### III-A Architectures and Programming Models We have implemented the operations listed in Table II for the new Intel Xeon Phi (MIC), CPUs, and GPUs. We briefly describe the MIC architecture only, because of space constraints. The MIC used in the experimental evaluation (SP10P) is built using 61 light-weight x86 cores clocked at 1090 MHz, with cache coherency among all cores that are connected using a ring network. The cores process instructions in-order, support a four-way simultaneous multi- threading (SMT), and execute 512-bit wide SIMD vector instructions. The MIC is equipped with 8GB of GDDR5 DRAM with theoretical peak bandwidth of 352GB/s. The programming tools and languages employed for code development for a MIC are the same as those used for CPUs. This is a significant advantage as compared to GPUs that alleviates code migration overhead for the co-processor. The MIC supports several parallel programming languages and models, such as OpenMP, POSIX Threads, and Intel Cilk Plus. In this work, we have implemented our operations using OpenMP on the MIC and CPU; we used CUDA111http://nvidia.com/cuda/. for the GPU implementations. The MIC supports two execution modes: native and offload. In the native mode the application runs entirely within the co-processor. This is possible because MICs run a specialized Linux kernel that provides the necessary services and interfaces to applications. The offload mode allows for the CPU to execute regions of the application code with the co-processor. These regions are defined using pragma tags and include directives for transferring data. The offload mode also supports conditional offload directives, which the application developer may use to decide at runtime whether a region should be offloaded to the coprocessor or should be executed on the CPU. This feature is used in our dynamic task assignment strategy for application execution using the CPU and the MIC cooperatively. ### III-B Parallel Implementation of Operations We present the details of the MIC and GPU implementations only, because the CPU runs the same code as the MIC. The _Data parallel_ operations were trivial to implement, since computing threads may be assigned for independent computation of elements from the input data domain. However, we had to analyze the results of the auto vectorization performed by the compiler for the MIC, because it was not be able to vectorize some of the loops when complex pointer manipulations were used. In these cases, however, we annotated the code with (#pragma simd) to guide the vectorization when appropriate. The parallelization of operations that use the _IWPP pattern_ heavily relies on the use of parallel containers to store the wavefront elements. The parallel computation of elements in the wavefront requires those elements be atomically updated, since multiple elements may concurrently update a third element $e_{j}$. In order to implement this operation in CUDA, we developed a complex hierarchical parallel queue to store wavefront elements [10]. This parallel queue exploits the multiple GPU memory levels and is implemented in a thread block basis, such that each block of threads has an independent instance of the queue to avoid synchronization among blocks. The implementation of the IWPP on the MIC was much simpler. The standard C++ queue container used in the sequential version of IWPP is also available with the MIC coprocessor. Thus, we instantiated one copy of this container per computing thread, which independently carries out propagations of a subset of wavefront elements. In both cases, atomic operations were used to update memory during a propagation to avoid race conditions and, as a consequence, the MIC vectorization was not possible since vector atomic instructions are not supported. The _MapReduce-style pattern_ , or reduction, is employed in object area calculations. The MIC and GPU implementations use a vector with an entry per object to accumulate its area, and threads concurrently scan pixels in the input data domain to atomically increment the corresponding entry in the reduction vector. Because the number of objects may be very high, it is not feasible to create a copy of this vector per thread and eliminate the use of atomic instructions. In the _Union-find pattern_ a forest is created in the input image, such that each pixel stores its neighbor parent pixel or itself when it is a tree root. For the parallelization of this pattern, we divided the input data into tiles that may be independently processed in parallel. A second phase was then executed to merge trees that cross tile boundaries. The MIC implementation assigns a single tile per thread and avoids the use of atomic instructions in the first phase. The GPU implementation, on the other hand, computes each tile using a thread block. Since threads computing a tile are in the same block, they can take advantage of fast shared-memory atomic instructions. The second phase of Union-find was implemented similarly for the MIC and the GPU. It uses atomic updates to guarantee consistency during tree merges across tile boundaries. Finally, the operations with _Object Parallelism_ can be independently carried out for each segmented object. Therefore, a single thread in the MIC or a block of threads in the GPU is assigned for the computation of features related to each object. All of the operations with this type of parallelism were fully vectorized. ## IV Cooperative Execution on Clusters of Accelerators The strategy for execution of a pipeline of segmentation and feature computation stages on a cluster system is based on a Manager-Worker model that combines a bag-of-tasks style execution with coarse-grain dataflow and makes use of function variants. A function variant represents multiple implementations of a function with the same signature. In our case, the function variant of an operation is the CPU, GPU, and MIC implementations. One Manager process is instantiated on the head node. Each computation node is designated as a Worker. The Manager creates tasks of the form (input image tile, processing stage), where processing stage is either the segmentation stage or the feature computation. Each of these tasks is referred to as a stage task. The Manager also builds the dependencies between stage task instances to enforce correct execution. The stage tasks are scheduled to the Workers using a demand-driven approach. A Worker may ask for multiple tasks from the Manager in order to keep all the computing devices on a node busy. A local Worker Resource Manager (WRM) on each computation node controls the CPU cores and co-processors (GPUs or MICs) used by a Worker. When the Worker receives a stage task, the WRM instantiates the operations in the stage task. It dynamically creates operation tasks, represented by tuples (input data, operation), and schedules them for execution as it resolves the dependencies between the operations – note that the segmentation and feature computation stages consist of pipelines of operations; hence there are dependencies between the operations. The set of stage tasks assigned to a Worker may create many operation tasks. The operations may have different performance characteristics on different computing devices. In order to account for this variability, a task scheduling strategy, called Performance Aware Task Scheduling (PATS), was employed in our implementation [13, 14]. PATS assigns tasks to CPU cores or co-processors based on an estimate of each task’s co- processor speedup and on the computational loads of the co-processors and CPUs. When an accelerator requests a task, PATS assigns the tasks with higher speedup to this processor. If the device available for computation is a CPU, the task to attain lower speedup on the accelerator is chosen. We refer reader to [13, 14] for a more detailed description of the PATS implementation. PATS also implements optimizations such as data reuse and overlap of data copy and computations to further reduce data processing overheads. ## V Experimental Evaluation We carried out the experimental evaluation using a distributed memory Linux cluster, called Stampede222https://www.xsede.org/tacc-stampede. Each compute node has dual socket Intel Xeon E5-2680 processors, an Intel Xeon Phi SE10P co-processor, and 32GB of main memory. We also used a node equipped with a single NVIDIA K20 GPUs. The nodes are connected using Mellanox FDR InfiniBand switches. The codes were compiled using Intel Compiler 13.1 with “-O3” flag. Since we execute the codes on the MIC using the offload mode, a computing core is reserved to run the offload daemon, and a maximum of 240 computing cores are launched. The images for the experiments were collected from brain tumor studies [6]. Each image is divided into 4K$\times$4K tiles which are processed concurrently on the cluster system. ### V-A Scalability of Operations on MIC This section evaluates the performance and scalability of the operations on the MIC. This analysis also considers the effects of thread affinity that determines the mapping of computing threads to computing cores. We examined three affinity strategies: compact, balanced, and scatter. Compact assigns threads to the next free thread context $n+1$, i.e., all four contexts in a physical core are used before threads are placed in the contexts of another core. Balanced allocates threads to new computing cores before contexts in the same core are used. Threads are balanced among computing cores and subsequent thread IDs are assigned to neighbor contexts or cores. Scatter allocates threads in a balanced way, like the balanced strategy, but it sets thread IDs such that neighbor threads are placed in different computing cores. We selected two operations with different data access and computation intensities for the experiments: (1) Morphological Open has a regular data access pattern and low computation intensity; (2) Distance Transform performs irregular data access and has a moderate computation intensity. OpenMP static loop scheduling was used for execution, because the dynamic version resulted in lower performance. Figure 1: Evaluation of scalability with respect to thread affinity type for selected operations on the MIC. The scalability results with respect to thread affinity are presented in Figure 1. As is shown, there is a great variability in speedups with different operations and different thread affinity strategies. Morphological Open and Distance Transform achieved the best performances when 120 and 240 threads were used, respectively (Figures 1 and 1). The graphs show that peaks in performance are reached when the number of threads is a multiple of the number of computing cores, i.e., 60, 120, 180, and 240 threads. In these cases, the number of threads allocated per computing core is the same; hence, computational work is better balanced among the physical cores. The performance of the Morphological Open operation scales until 120 threads are employed. Its performance significantly degrades when more threads are used. This behavior is a consequence of the MICs performance with memory intensive applications. As reported in [15, 16], the maximum memory bandwidth on the MIC is reached, measured using the STREAM benchmark [15] (regular data access), when one or two threads are instantiated per computing core (a total of 60 or 120 threads). When the number of threads increases to 180 and 240, there is a reduction in memory throughput due to congestion on the memory subsystem. Since Morphological Open is memory bound, this property of the MIC is a limiting factor on the performance of the operation when 120 and more threads are executed. The scalability of Distance Transform is presented in Figure 1. This operation fully benefited from the MIC’s 4-way hyperthreading and attained the best performance with 240 threads. Memory bandwidth also plays an important role in the scalability of this operation. However, this operations performs irregular access to data. Since random memory bandwidth is not typically included in devices specifications, we created a micro-benchmark to the MIC’s performance with random data access in order to better understand the operation’s performance. This benchmark consists of a program that randomly reads and writes elements from/to a matrix in parallel. The positions to be accessed in this matrix are stored into a secondary vector of indices, which is equally initialized in the CPU for all the devices. We observed that bandwidth attained by the MIC with random data access increases until the number of threads is 240. We conclude that this is the reason for the observed performance behavior of the Distance Transform operation. The scatter and balanced thread affinity strategies achieve similar performance, but the compact strategy fails to attain good performance with the Morphological Open operation. This is because the compact strategy uses all the 60 physical cores only when close to 240 threads are instantiated. On the other hand, Morphological Open achieves best performance with 120 threads. With this many threads the compact affinity uses only 30 physical cores. ### V-B Performance Impact of Vectorization on MIC This section analyzes the impact of using the Intel Xeon Phi SIMD capabilities. For this evaluation, we used the Gradient Stats operation, which makes full use of SIMD instructions to manipulate single-precision floating- point data. The scatter affinity strategy is used in the experiments because it was more efficient. Gradient Stats achieved speedups of 16.1$\times$ and 39.9$\times$, respectively, with the non-vectorized and vectorized versions, as compared to the single-threaded vectorized execution. The performance gains with vectorization are higher with lower number of threads – 4.1$\times$ for the single-threaded configuration. The performance gap between the two versions is reduced as the number of threads increases because of the better scalability of the non-vectorized version. At best, vectorization results in an improvement of 2.47$\times$ on top of the non-vectorized version. ### V-C Comparative Performance of MIC, GPU, and CPU This section evaluates the performance of the operations on the MIC, GPU, and CPU. The speedup values were calculated using the single core CPU execution as the baseline. While the CPU and MIC executables were generated from the same C++ source code annotated with OpenMP, the GPU programs were implemented using CUDA. The same parallelization strategy was employed in all of the implementations of an operation. Figure 2: Speedups achieved by operations on the CPU, MIC, and GPU, using the single core CPU version as a reference. The number above each dash refers to the number of threads that lead to the best performance on the MIC. The overall performance of the operations on different processors is presented in Figure 2. There is high variability in the speedup attained by each of the operations even when a single device is considered. In addition, the relative performance on the CPU, MIC, and GPU varies among the operations, which suggests that different computing devices are more appropriate for specific operations. In order to understand the reasons for the performance variations, we divided the operations into three disjoint groups that internally have similarities regarding the memory access pattern and execution strategy. The groups are: (1) Operations with regular data access: RGB2Gray, Morphological Open, Color Deconvolution, Pixel Stats, Gradient Stats, and Sobel Edge; (2) Operations with irregular data access: Morphological Reconstruction, FillHoles, and Distance Transform; (3) Operations that heavily rely on the use of atomic functions, which include the Area Threshold and Connected Component Labeling (CCL). To understand the performance of these operations, we measured their computation intensity and correlated it with each device’s capabilities using the notions of the Roofline model [17]. #### V-C1 Regular Operations The peak memory bandwidth and computation power of the processors are important to analyze the operations performance on each of them. The memory bandwidth with regular data access was measured using the STREAM benchmark [15] in which the K20 GPU, the CPU, and the MIC reached peak throughputs of 148GB/s, 78GB/s (combined for the two CPUs), and 160GB/s, respectively, with a single thread per core. Increasing the number of threads per core with the MIC results in a reduction of the bandwidth. Moreover, while the K20 GPU and the MIC are expected to deliver peak double precision performance of about 1 TFLOPS, the 2 CPUs together achieve 345 MFLOPS. The Morphological Open, RGB2Gray and Color Deconvolution operations are memory bound operations with low arithmetic-instruction-to-byte ratio. As presented in Figure 2, their performance on the GPU is about 1.25$\times$ higher than that on the MIC. Furthermore, the CPU scalability with this operations is low, because the memory bus is rapidly saturated. The improvements of the GPU on top of the CPU are consistent with their differences in memory bandwidth. The Color Deconvolution operation attains better raw speedups than other operations due to its higher computation intensity. While Morphological Open attains maximum performance with 120 threads because of its ability of reusing cached data (neighborhood in computation of different elements may overlap), the other two operations use 180 threads in order to hide the memory access latency. The remaining of the regular operations (Pixel Stats, Gradient Stats, and Sobel Edge) are compute bound due to their higher computation intensity. These operations achieve better scalability with all the devices. The performances of the GPU and MIC are similar with improvements of about 1.9$\times$ on top of the multicore CPU execution because of their higher computing capabilities. This set of compute intensive operations obtained the best performance with the MIC using 120 threads. Using more threads does not improve performance because the MIC threads can launch a vector instruction each two cycles, and compute intensive operations should maximize the hardware utilization with a 2-way hyperthreading. #### V-C2 Irregular Operations The operations with irregular data access patterns are Morphological Reconstruction, FillHoles, and Distance Transform. This set of operations strongly relies on the device performance to execute irregular (random) accesses to data. We used the same micro-benchmark described in Section V-A to measure each of the systems’ throughput in this context. TABLE III: Device Bandwidth with Random Data Accesses (MB/s). \| CPU \| MIC \| GPU ---\|---\|---\|--- Reading \| 305 \| 399 \| 895 Writing \| 74 \| 16 \| 126 The results are presented in Table III. The experiments were carried out by executing 10 million random reading or writing operations in a 4K$\times$4K matrix of integer data elements. As shown, the bandwidth attained by the processors is much lower than those with regular data access. The GPU significantly outperforms the other devices. The random writing bandwidth of the MIC processors is notably poor. This is in fact expected because this processor needs to maintain cache consistency among its many cores, which will result in a high data traffic and competition in its ring bus connecting caches. Due the low bandwidth attained by all the processors, all of our irregular operations are necessarily memory bound. As presented in Figure 2, the Distance Transform operation on the GPU is about 2$\times$ faster than on the MIC, whereas the MIC performance is not better than that of the CPU. This operation performs only irregular data access in all phases of its execution, and the differences in the random data access performances of the devices are crucial to its performance. The other two operations (Morphological Reconstruction and Fill Holes) in this category have a mixed data access patterns. These operations are based on the irregular wave front propagation pattern, and their most efficient execution is carried out with an initial regular propagation phase using raster/anti- raster scans, before the algorithm migrates to the second phase that irregularly access data and uses a queue to execute the propagations. Since the first phase of these operations is regular, it may be efficiently executed on the MIC. In the MIC execution, the algorithm will iterate several times over the data using the regular (initial) phase, before it moves to the irregular queue based phase. The MIC execution will only migrate to the irregular pattern after most of the propagations are resolved, which reduces the amount of time spent in the irregular phase. Hence, the performance gains on the GPU as compared to those on the MIC are smaller for both operations: about 1.33$\times$. We want to highlight that the same tuning regarding the appropriate moment to migrate from the regular to the irregular phase is also performed with the GPU version. #### V-C3 Operations that Rely on Atomic Instructions The Area Threshold and CCL operations heavily rely on the use of atomic add instructions to execute a reduction. Because the use of atomic instructions is critical for several applications and computation patterns, we analyze the performance of the evaluated devices with regard to execution of atomic instructions and its implications to the Area Threshold and CCL operations. To carry out this evaluation, we created a micro-benchmark in which computing threads concurrently execute atomic add instructions in two scenarios: (1) using a single variable that is updated by all threads and (2) an array indexed with the thread identifier. The first configuration intends to measure the worst case performance in which all threads try to update the same memory address, whereas the second case assesses performance with threads updating disjoint memory addresses. TABLE IV: Device Throughput with Atomic Adds (Millions/sec). \| CPU \| MIC \| GPU ---\|---\|---\|--- Single Variable \| 134 \| 55 \| 693 Array \| 2,200 \| 906 \| 38,630 The results are presented in Table IV. The GPU once again attained the best performance, and it is at least 5$\times$ faster than the other processors in both scenarios. The reduction in the GPU throughput from the configuration with an array to the single variable, however, is the highest among the processors evaluated. This drastic reduction in performance occurs because a GPU thread warp executes in a SIMD way and, hence, the atomic instructions launched by all threads in a warp will be serialized. In addition, the GPU launches a larger number of threads. This results in higher levels of concurrency and contention for atomic instructions. The CPU is about 2.4$\times$ faster than the MIC in both scenarios. The MIC is equipped with simpler computing core and typically relies on the use of vectorized operations to achieve high performance. However, it lacks support for vector atomic instructions, which poses a serious limitation. The introduction of atomic vector instructions, such as those proposed by Kumar et. al. [18] for other multiprocessors, could significantly improve the MIC performance. Because the Area Threshold and CCL operations greatly depend on atomic instructions, they attained better performance on the GPU. In both cases, the execution on the CPU is more efficient than on the MIC (Figure 2). ### V-D Multi-node Execution using CPUs and MICs This section evaluates application performance in using CPUs and MICs cooperatively on a distributed memory cluster. The example application is built from the core operations and is implemented as a hierarchical pipeline in which the first level is composed of segmentation and feature computation stages, and each of these stages is created as a pipeline of operations as described in Section IV. We evaluated four versions of the application: (1) CPU-only refers to the multi-core CPU version that uses all CPU cores available; (2) MIC-only uses a MIC per node to perform computations; (3) CPU- MIC FCFS uses all CPU cores and the MIC in coordination and distributes tasks to processors in each node using FCFS (First-Come, First-Served) fashion; (4) CPU-MIC PATS also uses all CPU cores and the MIC in coordination, but the tasks are scheduled to devices based on the expected speedup of each task on each device. The speedup estimates are those presented in Figure 2. Figure 3: Multi-node weak scaling evaluation: dataset size and the number of nodes increase proportionally. The weak scaling evaluation in which the dataset size and the number of nodes increase proportionally is presented in Figure 3. The experiments with 172 nodes used an input dataset with 68,284 4K$\times$4K image tiles (3.27TB of uncompressed data). All versions of the application scaled well as the number of nodes is increased from 4 to 192. The MIC-only execution was slightly faster than the multi-core CPU-only version. The cooperative CPU-MIC executions attained improvements of up to 2.06$\times$ on top of the MIC-only version. The execution using PATS is 1.29$\times$ faster than using FCFS. This is a result of PATS being able to map tasks to the more appropriate devices for execution. The efficiency of the fastest CPU-MIC PATS version is about 84%, when 192 computing nodes are used. The main factor limiting performance is the increasing cost of reading the input image tiles concurrently from disk as the number of nodes (and processes) grows. ## VI Related Work Efficient utilization of computing systems with co-processors requires the implementation of efficient and scalable application computing $kernels$, coordination of assignment of work to co-processors and CPUs, minimization of communication, and overlapping of communication and computation. Mars [19] and Merge [20] are designed to enable efficient execution of MapReduce computations on shared memory machines equipped with CPUs and GPUs. Qilin [21] implements an automated methodology to map computation tasks to CPUs and GPUs. PTask [22] provides OS abstractions for task based applications on GPU equipped systems. Other frameworks to support execution on distributed memory machines with CPUs and GPUs were also proposed [23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. DAGuE [23] and StarPU [29] support execution of regular linear algebra applications. They express the application tasks dependencies using a Directed Acyclic Graph (DAG) and provide different scheduling policies, including those that prioritize execution of tasks in the critical path. Ravi [24] and Hartley [25] proposed runtime techniques to auto-tune work partitioning among CPUs and GPUs. OmpSs [27] supports execution of dataflow applications created via compilation of annotated code. More recently, research groups have focused on applications that may benefit from Intel’s Xeon Phi co-processor [33, 34, 35, 36, 16]. Joó et al. [33] implemented a Lattice Quantum Chromodynamics (LQCD) method using Intel Phi processors. Linear algebra algorithms were also ported to MIC [34, 36]. Hamidouche et al. [37] proposed an automated approach to perform computation offloads on remote nodes. The use of OpenMP based parallelization on a MIC processor was evaluated in [35]. That work analyzed the overheads of creating and synchronizing threads, processor bandwidth, and improvements with the use of vector instructions. Saule et al. [16] implemented optimized sparse matrix multiplication kernels for MICs, and provided a comparison of MICs and GPUs for this operation. In our work, we perform a comparative performance evaluation of MICs, multi- core CPUs, and GPUs using an important class of operations. These operations employ diverse computation and data access patterns and several parallelization strategies. The comparative performance analysis correlates the performance of operations with co-processors characteristics using co- processor specifications or performance measured using micro-kernels. This evaluation provides a methodology and clues for application developers to understand the efficacy of co-processors for a class of applications and operations. We also investigate coordinated use of MICs and CPUs on a distributed memory machine and its impact on application performance. Our approach takes into account performance variability of operations to make smart task assignments. ## VII Conclusions Creating efficient applications that fully benefit from systems with co- processors is a challenging problem. New co-processors are being released with more processing and memory capacities, but application developers often have little information about which co-processors are more suitable for their applications. In this paper we provide a comparison of CPUs, GPUs, and MICs using operations, which exhibit different data access patterns (regular and irregular), computation intensity, and types of parallelism, from a class of applications. An array of parallelization strategies commonly used in several applications are studied. The experimental results show that different types of co-processors are more appropriate for specific data access patterns and types of parallelism, as expected. The MIC’s performance compares well with that of the GPU when regular operations and computation patterns are used. The GPU is more efficient for those operations that perform irregular data access and heavily use atomic operations. A strong performance variability exists among different operations, as a result of their computation patterns. This variability needs to be taken into account to efficiently execute pipelines of operations using co-processors and CPUs in coordination. Our results show that the example application can achieve 84% efficiency on a distributed memory cluster of 3072 CPU cores and 192 MICs using a performance aware task scheduling strategy. Acknowledgments. 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Prince, “Dense Linear Algebra Factorization in OpenMP and Cilk Plus on Intel’s MIC Architecture: Development Experiences and Performance Analysis,” in _TACC-Intel Highly Parallel Computing Symp._ , 2012. * [37] K. Hamidouche, S. Potluri, H. Subramoni, K. Kandalla, and D. K. Panda, “MIC-RO: enabling efficient remote offload on heterogeneous many integrated core (MIC) clusters with InfiniBand,” in _27th Int. ACM International Conference on Supercomputing_ , ser. ICS ’13, 2013, pp. 399–408.	arxiv-papers	2013-11-02T14:00:40	2024-09-04T02:49:53.208120	{ "license": "Public Domain", "authors": "George Teodoro and Tahsin Kurc and Jun Kong and Lee Cooper and Joel\n Saltz", "submitter": "George Teodoro", "url": "https://arxiv.org/abs/1311.0378" }
1311.0446	# Thermodynamic Treatment of High Energy Heavy Ion Collision Wedad AL-Harbi1 and Tarek Hussein 2 1 Physics Department- Sciences Faculty For girls, King Abdulaziz University 2 Physics Department, Faculty of Science, Cairo University, 12613 Giza, Egypt ###### Abstract The hadron production in heavy ion collision is treated in the framework of thermodynamic vision. Thermodynamic system formed during central collision of Pb-Pb at high energies is considered, through which binary collision is assumed among the valance quarks. The partition function of the system is calculated; accordingly the free available energy, the entropy and the chemical potential are calculated. The concept of string fragmentation and defragmentation are used to form the newly produced particles. The average multiplicity of the newly produced particles are calculated and compared with the recent experimental results. Keywords: Nuclear thermodynamic model, binary collision, quark-quark potential ## 1 Introduction In a wide energy range, there are some common aspects of heavy ion reaction dynamics. The energies are large enough and the masses of ions are also large, to consider the heavy ions as classical particles. Their De Broglie wavelength is much less than the typical nuclear sizes. Quantum effects influence the underlying microscopic dynamics only, which can be included in the equation of state, in the transport coefficients, or in the kinetic theory describing the reactions. During the collision and assuming straight line trajectories there can be target and projectile spectators in a collision [1-3] The rest of the nucleons may hit each other on the way forming a participant zone with both target and projectile nucleons in it. The most interesting phenomena and the new physics are in the participant zone. The spectator regions also provide us with interesting phenomena. Realistically, the nucleons do not propagate along exactly straight trajectories. Deviations from straight propagations are observed even at the highest energies of 200 GeV per nucleon. According to the fluid dynamical model, considerable collective sideward motion is generated [4]. We shall deal with the collision problem using statistical physics concepts according to the following vision: A heavy ion reaction is a dynamical system of a few hundred nucleons. This is a large number but still far from the continuum, so that deviations from infinite matter limit are important. On the other hand, the number of particles participating in a reaction is large enough that the signs of collective matter like behavior can be clearly observed. This is an interesting territory in statistical physics of small but collective systems. The methods developed in this field are unique and may also be applicable in other “small” statistical systems. However when Quark Gluon Plasma is formed the number of quanta increases to a large extent [5-6]. The plasma can already be considered as a continuum, and finite particle effects should be small. On the other hand the heavy ion reaction is a rapid dynamical process. The question of phase transitions in a dynamical system is still an open field of research. Heavy ion physics may contribute to this field at two points: i) the dynamics of the phase transitions in ”small” systems, and ii) the dynamics of the phase transitions in ultra-relativistic systems where the energy of the system is much higher than the rest mass of the particles. In this context, we shall deal with the problem from the statistical thermodynamic point of view. In other words a hypothetical model will be tailored according to the following concepts: a. During the heavy ion collisions, the participant nucleons form a thermodynamic system by a cylindrical cut of the projectile through the target nucleus. b. A fireball oriented trend is used considering valence quarks of the nucleons as the constituent particles of the overlap region. c. Binary collisions are assumed among the existing valence and the created sea quarks. Accordingly a significant increase in energy density of the system leads to an environment eligible to create more particles in a frame of grand canonical ensemble. d. A simple quark wave function is assumed to be used in calculating the partition function of the system. e. A power series expansion of the partition function is assumed and we will consider only the first few terms that fit the boundary conditions of the system. According to the above mentioned assumptions we calculate the newly created particle multiplicity produced in $Pb+Pb$ collisions to be compared with the recently published experimental data. In doing this we should take into consideration the following points: i) The leading baryon is hardly stopped, ii) In the rapidity region between the projectile and the target, secondary charged particles (mesons $\pi^{+}$; $\pi^{-}$; $\pi^{0}$; $\mathit{K}^{+};\mathit{K}^{-}$, etc.) are created through string mechanism [7-8]. The paper is organized in four sections after this soft introduction. In section 2 we represent the geometry and the formulation of the model. Results and discussion are given in section 3. Eventually conclusive remarks are given in section 4. ## 2 Thermodynamic Treatment of the nuclear System In the hard sphere model, one effective radius $r=0.4fm$ [9-10] is chosen for all quark collisions. Elastic and inelastic collisions are considered which are supposed to be dominant at temperatures $T\approx\|120-200MeV.$ The calculations are done in the framework of the Boltzmann equation with the Boltzmann statistical distribution functions and the real gas equation of state [1] , [11]. The Boltzmann equation is used to find the distribution function in phase space of the thermodynamic system. It is a function in space coordinate, momentum and time. In a system of large number of particles and a special form of interaction potential, it will be difficult to get perfect full solution of the equation. However many trials were done [1] to get approximate solution to the Boltzmann equation in a form of conversing power series. The first term represents the equilibrium state (zero order term). The next terms represent the higher order corrections to describe the shift due to the non-equilibrium state. These terms are time dependent and include a time parameter that measure how far from the equilibrium states the particles are produced. On putting the time parameter tends to zero, all the higher terms of the series vanishes and the system approach equilibrium and described only by the zero order term. Another trial to find phase space distribution function in a pre- equilibrium state is the following: Due to the none-equilibrium state the temperature is not homogeneous allover the system, but instead there will be a temperature gradient with minimum entropy. The approach is to divide the system into small stripes (subsystems) each has its local equilibrium with a specific temperature and local equilibrium distribution. The overall phase space distribution will be the sum over the distributions of the subsystems. This is what we did in our calculation. The Boltzmann statistical approximation allows one to conduct precise numerical calculations of transport coefficients in the hadron gas and to obtain some relatively simple relativistic analytical closed-form expressions. For particles having spin, the differential cross sections were averaged over the initial spin states and summed over the final ones. The local equilibrium distribution functions are: $f_{k}^{0}=e^{(\mu_{k}-p_{k}^{\mu}U_{\mu})/T}$ (2-1) Where, $\mu k$ is the chemical potential of the kth particle species, $T$ is the temperature and $U\mu$ is the relativistic flow 4-velocity such that $U\mu U^{\mu}=1$ with frequently used consequence. The distribution functions $f_{k}$ are found by solving the system of the Boltzmann equations approximately with the form [1]: $f_{k}=f_{k}^{0}+f_{k}^{1}=f_{k}^{0}+f_{k}^{0}\varphi(x,p_{k})$ (2-2) Because analytical expressions for the collision brackets are bulky the Mathematica was used for symbolical and some numerical manipulations [12]. The numerical calculations are done also for temperatures above $T=120MeV$. Let us consider the case of the binary collisions between quarks of the thermodynamic system that has been formed during the interaction of heavy ions. The total Hamiltonian is: $H_{tr}=\frac{1}{2M}\mathop{\displaystyle\sum}\limits_{i-1}^{N}P_{i}^{2}+\sum_{i\neq j}W_{ij}(r_{ij})$ (2-3) $p_{i}$ is the momentum of the ith quark and $W_{ij}$ is the binary interaction potential energy among quarks$i\&j.$ The static quark potential at fixed spatial separation has been obtained from an extrapolation of ratios of Wilson loops to infinite time separation. As we have to work on still rather coarse lattices and need to know the static quark potential at rather short distances (in lattice units) we have to deal with violations of rotational symmetry in the potential. In our analysis of the potential we take care of this by adopting a strategy used successfully in the analysis of static quark potentials [13] and heavy quark free energies [14]. This procedure removes most of the short distance lattice artifacts. It allows us to perform fits to the heavy quark potential with the 3-parameter approach, The quark-quark potential is given as $W_{qq}(r)=-\frac{\alpha}{r}+\sigma r+c$ (2-4) The quark potential is graphically represented in Fig. (1). It is formed by 3 terms. The first is a Coulomb like; the second is string repulsive potential that works in confinement the quarks inside the nucleon bag. In this model, the number of created particles depends on the available energy. Accordingly, we focus our calculation on getting information about the relation between the energy and the number of interacting particles and consequently the number of created particles The grand canonical ensemble considers large supersystem kept at constant $T$ and $P$ and consists of many subsystems that can exchange not only the energy but also the number of particles. A number of particles and their quantum numbers corresponding to their energy states specify a microstate in the grand canonical ensemble. The particle abundance during the heavy ion collision is much complicated and depends mainly on an environment in the presence of catalyst necessary for the particle creation. The available energy is necessary to create excess of quark-antiquark pairs. Not all the quarks have the chance to form particles. Only those quarks experiencing special conditions in presence of the colored field will form a particle that satisfies the selection rules. Moreover, the strength of the color field depends on the separation distance $r_{ij}$ . The number of parton pairs is ${\frac{1}{2}}N(N-1)$ and may be approximated as ${\frac{1}{2}}N$ for largevalues of $N$. $N$ depends mainly on the available energy required for creation of quqrk- quqrk pairs. We follow the thermodynamic regime that starts by calculating the partition function and The translational partition function is then, $Z_{tr}=\mathop{\displaystyle\int}...\mathop{\displaystyle\int}\mathop{\displaystyle\prod}\limits_{j}^{3N}\Psi_{j}^{\ast}e^{-H_{tr}/KT}\Psi_{j}dq_{j}/h$ (2-5) For the sake of indistinguishability we divide Eq. (2-5) by $N!$ and write $Z_{tr}=Z_{p}Z_{q}$ for the momentum $p$ and spatial space $q$. As usual, integration over $p$ we get $Z_{p}=\frac{1}{N!}\left(\frac{2\pi MkT}{h^{2}}\right)^{3N/2}\simeq\left(\frac{e}{N}\right)^{N}\left(\frac{2\pi MkT}{h^{2}}\right)^{3N/2}$ (2-6) While the integration over $q$, is not simply $V^{N}$ because of the presence of the potential energy $W_{ij}$. Which means that $W_{ij}$breaks the factorizability of $Z$. $Z_{q}=\mathop{\displaystyle\int}...\mathop{\displaystyle\int}\mathop{\displaystyle\prod}\limits_{j}^{3N}\Psi_{j}^{\ast}e^{-\sum W_{tr}/KT}\Psi_{j}dq_{j}/h$ (2-7) Notice that for the confinement property of the potential then the potential energy has trivial effect at small values of $r_{ij}$ so it is possible to write Eq.(2-7) as $Z_{q}=\mathop{\displaystyle\int}...\mathop{\displaystyle\int}\mathop{\displaystyle\prod}\limits_{j}^{3N}\Psi_{j}^{\ast}[1+Exp(-\sum W_{ij}/kT)-1]\Psi_{j}dq_{j}$ (2-8) $=V^{N}+\mathop{\displaystyle\int}...\mathop{\displaystyle\int}\mathop{\displaystyle\prod}\limits_{j}^{3N}\Psi_{j}^{\ast}[1+Exp(-\sum W_{ij}/kT)-1]\Psi_{j}dq_{j}$ For the case where $W_{ij}/kT\ll 1$ we can expand the exponential in Eq.(2-8) as $Exp[W_{ij}/kT]-1\approx-W_{ij}/kT$ this is applied when $r_{ij}\prec 2$ $r_{0}$ where $r_{0}$ is the quark radius ( $r_{0}=0.4fm$). On the other hand, if $W_{ij}/kT$ is not too small we can consider higher order terms in the expansion of the exponential in Eq. (2-8) .While, for $r_{ij}\succeq 2$ $r_{0}$ the potential is increasingly with $r$ i.e. positively large so that $Exp[W_{ij}/kT]\approx 0$ at extreme large values of $r$, then, $Z_{q}=V^{N}-\frac{1}{2}N^{2}V^{N-1}[4\pi\mathop{\displaystyle\int}\limits_{2r_{0}}^{R}\Psi_{1}^{\ast}[1+Exp(-W_{ij}/kT)-1]\Psi_{1}r_{1j}^{2}dr_{1j}$ (2-9) Now it is possible to calculate $F,S,U\ldots$ for this system in terms of $Z$. $F=-NkT(\ln Z-\ln N+1)$. The entropy $S=-(\partial F/\partial T)_{N,V}$, then $S=NkT\left(\frac{\partial\ln Z}{\partial T}\right)_{V}+Nk(\ln Z-\ln N+1)$ (2-10) The internal energy $U=F+TS$; then $U=NkT^{2}\left(\frac{\partial\ln Z}{\partial T}\right)_{V}$ (2-11) The total internal energy $U$ is the important physical quantity in the model and is directly related to the energy density and the particle multiplicity production. The wave function $\Psi_{j}$ is assumed to be very similar to that used by the parton model. Starting with the simplest parametric form of the quark wave function, $\widetilde{\Psi}_{a}(p)=Ce^{-\alpha p}\text{ \ \ \ \ \ \ \ \ }a\succ 0\text{ \ \ \ ,\ \ \ \ \ }p\succeq 0$ (2-12) p is the null momentum in the parton model [15],[16]. The Fourier transform of the quark wave function is then: $\Psi(r)=\frac{C^{{}^{\prime}}}{(r+ia)^{2}}$ (2-13) Where $C$ and $C\prime$ are normalization constants, while “$a$” is a fitting parameter. Again, inserting Eq. (2-13) in Eq. (2-5), this gives the total energy of the quark assembly of the nucleons. The range of the null momentum p extends from zero up to Pmax. It is more convenient to express the wave function and all other physical quantities in terms of the Bjorken scaling variable with $x=P/P_{\max}$ lies in the range $\ 0\prec x\prec 1.$Particle creation through formation and fragmentation of quarks is used through the string model [17] consequently the multiplicity of particle creation is calculated and compared with the recent results of Pb-Pb collisions at incident momentum per nucleon of $40,80$ and $158GeV$. (Experiment: CERN- NA-049 (TPC)) [18-21] ## 3 Results and Discussion We consider the case of central collisions in heavy ions of Pb-Pb. Let us assume that overlap region at impact parameter b has a cylindrical form. The number of nucleons and consequently the number of quarks Nn-part and Nq-part for Pb+Pb collisions are calculated as [22]: $N_{n-part}\|AB=\int d^{2}S\text{ }T_{A}(\overrightarrow{S})\left[1-\left(1-\frac{\sigma_{NN}^{inel}T_{B}(\overrightarrow{S}-\overrightarrow{b})}{B}\right)^{B}\right]$ (3-1) $\qquad\qquad\qquad\qquad\qquad+\int d^{2}ST_{B}(\overrightarrow{S})\left[1-\left(1-\frac{\sigma_{NN}^{inel}T_{A}(\overrightarrow{S}-\overrightarrow{b})}{A}\right)^{A}\right]$ Where $\U{3c3}$ ${}_{NN}^{inel}$ is the inelastic nucleon-nucleon cross section and $T(b)=\mathop{\displaystyle\int}\limits_{-\infty}^{\infty}dz$ $\ n_{A}(\sqrt{B^{2}+Z^{2}})$ is the thickness function. In calculating $N_{q-part}$; the density was increased three times and $\sigma_{NN}^{inel}$ is re placed by $\sigma_{qq}^{inel}$ We concentrate our calculation on the central collision region where $b\preceq R_{P}+R_{T}\char 126\relax 6fm$. On the average, the number of nucleons (quarks) in the considered region is $250$ $(750)$. Fig. (2) illustrates the participant number of nucleons (quarks) according to Glauber calculation [22]. The partition function for the $N$ particles is calculated according to Eq. (2-9) for separation distance $r_{ij}$ of the binary interacting quarks. The family of curves of the partition function represented in Fig (3) corresponds to temperature $20,40,60$ and $200MeV$. The temperature is very sensitive to the form of the nuclear density. The temperature increases as the number of participant nucleons from the projectile $N_{P}$ and the target nucleus $N_{T}$ come close $(N_{P}\longrightarrow N_{T})$. The temperature of the system is determined according to the impact parameter and consequently depends on the number of participating nucleons (quarks) [1] $\zeta_{cm}=3T+m\frac{K_{1}(m/T)}{K_{2}(m;T)}$ (3-2) $\zeta_{cm}=[m^{2}+2\eta(1-\eta)mt_{i}]^{1/2}$ (3-2) In Eq. (3-2), $\zeta_{cm}$ represents the relativistic form of the center of mass energy of a system of temperature $T$, (using system of units where the Boltzmann constant $K=1$). The first term on the LHS,$(3T$ $or$ $3kT)$ is the thermal energy. The second term is the relativistic correction [1]. In Eq. (3-3) ) $\zeta_{cm}=[m^{2}+2\eta(1-\eta)mt_{i}]^{1/2}$ describes also the center of mass energy for the particles in the overlap region of projectile and target having nuclear densities $\U{3c1}_{p}$ and $\U{3c1}_{T}$ respectively with relative projectile density $\eta(r,b)$ at position coordinate $r$ and impact parameter $b.$ [1] defined as $\eta(r,b)=\ \frac{\rho_{p}(r,b)}{\rho_{p}(r,b)+\rho_{I}(r,b)}$ (3-3) Solving the two equations (3-2) and (3-3), it is possible to find the temperature T at any $r$ and $b$. $t_{i}$ is the incident kinetic energy per nucleon. $K_{1}$ and $K_{2}$ are the Macdonald functions of first and second order. At low temperature $T\char 126\relax 20MeV$ the partition function $Z(r,T)$ approaches flat behavior just above $r_{ij}\simeq 2$ $fm$. At higher temperature $(T=40,60$ $MeV)$ the $r_{ij}$ dependence of $Z(r,T)$ becomes steeper. Approximate linear behavior is found at high temperature $T=200$ $MeV$. On the other hand, the smooth variation of $Z(r,T)$ over all the range of temperature $T,20\longrightarrow 200$ $MeV$ is given in Fig. (4) with family of curves corresponding to $r_{ij}\sim 2,3,4,5$ and $6$ $fm.$ Steep drop of $Z(r,T)$ is observed in the cold region $(T<20MeV)$ for all curves of $r_{ij}\sim 2,3,4,5$ and $6$ $fm$. This is followed by smooth increase toward the hot region up to $T\sim 200$ $MeV$. The changes of the partition function express the behavior of the quark chemical potential $\mu$ inside the hadronic system through the well known relation: $\mu=-kT\ln Z$ Fig (5) shows the change of quark chemical potential in a temperature range up to $200$ $MeV$. The total energy $U(r)$ dissipated in the nuclear interaction region due to the binary quark collisions is represented in Fig (6) in the temperature range $T<200$ MeV. The curves are plotted for quark-quark separation distances $r_{ij}\sim 2,3,4,5$ and $6$ $fm$. $U(r)$ has positive values in the small range $r_{ij}\sim 2,3,4$ $fm$, where the quarks are approximately free and can carry enough energy to create particles. However $U(r)$ has negative values in the large range $r_{ij}\sim 5$ and $6$ $fm$ where the quarks are mostly confined by the string potential. A Monte Carlo program is used to simulate the particle production in the frame of String Model [17]. The particle production is considered as a tunneling process in a colored field. Because of the 3-gluon coupling, the color flux lines will not spread out over the space as the electromagnetic field lines do but rather be constrained to a thin tube like region. Within this tube, new $q\overline{q}$ pairs can be created from the available field energy. The original system breaks into smaller pieces, until only ordinary hadrons remain. In the field behind the original outgoing quark $q_{0}$ a new quark pair $q_{1}$ $\overline{q}_{1}$ is produced so that the original one $q_{0}$ may join with a new one $\overline{q}_{1}$ to form a hadron $q_{0}$ $\overline{q}_{1}$ leaving $\overline{q}_{1}$unpaired. The production of another pair $q_{2}$ $\overline{q}_{2}$ will give a hadron $q_{1}$ $\overline{q}_{2}$ etc. From this assumption one may find the resulting particle spectra in a jet. The possible meson formation by the quark pairs are presented in Table (1). The simulation process allows the production of all types of mesons with all possible branching ratios taking into account the selection rules and the conservation laws. In Table (2) the prediction of the simulation shows overall fair agreement. In most cases at energies 20 and 30 A GeV the thermodynamic model prediction exceeds the measured value by 11- 15%. The prediction of the model for pions and keons at energy 158 A GeV gives values little bit under estimation with respect to the experimental values. However the calculated value for the $\phi$ particle gives unexpected result (double the experimental value) this may be due to the fact that the $\phi$ comes from the channel of $s\overline{s}\ \ $quarks. In our model the production of u, d and s pairs were considered equally probable; this seems in contrast to the real case. Unfortunately, the Monte Carlo code used in our calculation was designed by our research group since 1995 [17]. In this code the charged particles (mesons $\pi^{+}$; $\pi^{-}$; $\pi^{0}$; $K^{+};K^{-}$, etc.) are created through string mechanism and the recombination of the specific quarks, irrelevant of the mechanism of production whether due to the decay of resonance particle ($K\ast$,…) or not. We are working right now to develop this code, taking into consideration most of the recent information. On the other hand, the recent STAR measurements on the production of various strange hadrons (K0s, phi, Lambda, Xi and Omega) in $\sqrt{S_{NN}}$ = 7.7 - 39 GeV Au+Au collisions show that strange hadron productions are sensitive probes to the dynamics of the hot and dense matter created in heavy-ion collisions. The extracted chemical and kinetic freeze-out parameters with the thermal and blast wave models as a function of energy and centrality were studied and discussed by Xianglei ZHU [23-24] We also believe that hadron production in general is a good probe to study hadron formation mechanism in heavy ion collisions. At high transverse momentum, pT, the hard processes, which can be calculated with perturbative QCD, are expected to be the dominate mechanism for hadron productions. It was observed at RHIC that, at high pT, the RCP (the ratio of scaled particle yields in central collisions relative to peripheral collisions) of various particles [25] indicates dramatic energy loss of the scattered partons in the dense matter (jet quenching). RCP of hadrons have been measured also at SPS [26, 27] as well, though the limited statistics restricts the measurement at relatively lower pT (0.3 GeV/c). Measuring the nuclear modification factor in heavy ion collisions at this energy range, one can potentially pin down the beam energy at which interactions with the medium begin to affect hard partons [28]. ## 4 Conclusive remarks * • New particles need special environment to be produced during the heavy ion collisions. * • Multiple collisions among the quarks of the nuclear system should produce large enough energy compared with the particle chemical potential. The strong colored field plays the role of a catalyst parameter necessary for particle production. * • The free available energy U(r) has positive values in the small interaction distance where the quarks are approximately free and can carry enough energy to create particles. However U(r) has negative values in the large interaction distance where the quarks are mostly confined by the string potential. * • The string fragmentation and defragmentation is applied for the production of the different types of newly produced particles. * • Theoretical attempts to understand the energy dependence of the suppression were undertaken. Calculations were based on the Glauber-Gribov model, in which the energy-momentum conservation was implemented into the multiple soft parton re-scattering approach. * • The temperature is very sensitive to the form of the nuclear density. The temperature increases as the number of participant nucleons from the projectile $N_{P}$ and the target nucleus $N_{T}$ come close. * • The temperature of the system is determined according to the impact parameter and consequently depends on the number of participating nucleons (quarks). * • The thermodynamic model prediction exceeds the measured value by percentage 11- 15% * • The quark-hadron phase transition will be studied in a forthcoming article through temperature-quark chemical potential phase diagram. Acknowledgment This paper was funded by Deanship of Scientific (DSR), King abdulaziz University, Jeddah, under grant NO.(136-130-D1432). The authors, therefore, acknowledge with thanks DSR technical and financial support. The authors also would like to express deep thank to Prof. M.T. Ghoneim from Cairo University for his help in improving the language and overall style of the manuscript. Figure Captions Fig.(1) 3-parameter quark-quark binary potential Fig.(2) Number of participant nucleons (quarks) in the overlap region as a function of the impact parameter b for the Pb-Pb collision Fig.(3) The partition function Z(r,T) as a function of the separation distance between the interacting quarks. Different curves belong to temperature values of (20 (red), 40 (green), 60 (blue) and 200 (black) MeV). Fig.(4) The behavior of the partition function Z(r,T) in a temperature range up to 200 MeV Fig.(5) The change of quark chemical potential in a temperature range up to 200 MeV Fig.(6) The total potential energy formed inside the nuclear system in the temperature range T $<$ 200 MeV. The curves are plotted for quark separation distances rij $\sim$ 2 (red),3(green),4(blue),5 (yellow) and 6 (black) $fm$. Table Captions Table (1) All possible $q\overline{q}$ pairs with their probable meson type formation Table (2) Table (2) Yields of particle production at Pb-Pb collisions at 20 and 30 and 158 AGeV. The measured data are taken from ref [20, 21] for pions and keons only. The possible measured values are compared with the prediction of the present model. References [1] Mohamed Tarek Hussein, Nabila Mohamed Hassan, Naglaa El-Harby, Turk.J.Phys. 24(2000) 501; \- J. Gosset, H. H. Gutbrod, W. G. Meyer, A. M. Pokanger, A. Sandoval, R. Stock and G. D. Westfall, Phys. Rev., C16, (1977) 629 \- J. Gosset, J. I. Kapusta and G. D. Westafall,Phys. Rev., C18 (1978) 844 \- W. D. Myers, Nucl. Phys., A 296, (1978) 177 [2] M. T. Hussein, N. M. Hassan, N. Elharby, APH N.S. Heavy Ion Physics 13 (2001) 277 [3] Huichao Song, Steffen A Bass, Ulrich W Heinz, Tetsufumi Hirano, Chun Shen, Phys.Rev.C83 (2011) 054910 [4] ALICE Collaboration, Phys.Lett.B696, (2011), 328 [5] A.V. Nefediev, Yu. A.Simonov, Phys.Atom.Nucl.71 (2008) 171-179 [6] Jean-Paul Blaizot, J.Phys.G34 (2007) S243-252 [7] Qing-Guo Huang, Phys.Rev. D74 (2006) 063513 [8] M.N. Chernodub, F.V. Gubarev, Phys.Rev.D76 (2007) 016003 [9] S.N.Syritsyn, J.D.Bratt, M.F.Lin, H.B.Meyer, J.W.Negele, A.V. Pochinsky, M.Procura, M. Engelhardt, Ph. Hagler, T.R. Hemmert, W. Schroers, Phys.Rev.D81, (2010) 034507 [10] M. I. Gorenstein, M. Hauer, O. N. Moroz, Phys. Rev. C77, (2008) 024911 [11] Yuichi Mizutani, Tomohiro Inagaki, Prog.Theor.Phys.125 (2011) 933 [12] http://www.wolfram.com [13] R. Sommer, Nucl. Phys. B411 (1994) 839 [14] O. Kaczmarek, F. Karsch, F. Zantow and P. Petreczky, Phys. Rev. D 70, (2004) 074505 [15] M. T. Hussein, A. I. Saad; J. Mod. Phys., 2010, 1, 244-250 [16] Hussein, N. M. Hassan, and W. Elharbi, IJMPA Vol. 18, No. 4 (2003) 673-683 [17] M.T. Hussein, A. Rabea, A. El-Naghy and N.M. Hassan; Progress of Theoretical Physics, 93, 3 (1995), 585 [18] NA49 Collaboration (N. Davis et al.). Phys. Atom.Nucl. 75 (2012) 661. [19] NA49 Collaboration (G.L. Melkumov (Dubna, JINR) et al.). Nucl.Phys.Proc.Suppl. 219-220 (2011) 102 [20] NA49 Collaboration, Phys.Rev.C77, (2008) 024903 [21] Francesco Becattini et al; Phys. Rev. C 85, (2012) 044921 [22] M.K. Hegab, M.T. Hussein and N.M. Hassan, Z. Physics A 336, (1990) 345 [23] Xianglei ZHU, Acta Physica Polonica B Proceedings Supplement vol. 5 (2012) 213. [24] Xianglei Zhu, for the STAR Collaboration, Nucl.Phys.A830 (2009) 845c-848c [25] J. Rafelski and B. M¨uller, Phys. Rev. Lett. 48, (1982)1066 [26] B. I. Abelev et al., Phys. Rev. C 77, (2008) 044908 [27] J. Adams et al., Phys. Rev. Lett. 98, (2007) 062301 [28] X. Wang (STAR Collaboration), J. Phys. G 35, (2008) 104074	arxiv-papers	2013-11-03T09:47:55	2024-09-04T02:49:53.218975	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Wedad AL-Harbi and Tarek Hussein", "submitter": "Tarek Hussein", "url": "https://arxiv.org/abs/1311.0446" }
1311.0474	# Defect-induced conductivity anisotropy in MoS2 monolayers Mahdi Ghorbani-Asl,1 Andrey N. Enyashin,2,3 Agnieszka Kuc,1 Gotthard Seifert,2 and Thomas Heine1 [email protected] 1 School of Engineering and Science, Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany 2 Physical Chemistry, Technical University Dresden, Bergstr. 66b, 01062 Dresden, Germany 3 Institute of Solid State Chemistry UB RAS, Pervomayskaya Str. 91, 620990 Ekaterinburg, Russia ###### Abstract Various types of defects in MoS2 monolayers and their influence on the electronic structure and transport properties have been studied using the Density-Functional based Tight-Binding method in conjunction with the Green’s Function approach. Intrinsic defects in MoS2 monolayers significantly affect their electronic properties. Even at low concentration they considerably alter the quantum conductance. While the electron transport is practically isotropic in pristine MoS2, strong anisotropy is observed in the presence of defects. Localized mid-gap states are observed in semiconducting MoS2 that do not contribute to the conductivity but direction-dependent scatter the current, and that the conductivity is strongly reduced across line defects and selected grain boundary models. ## I Introduction The rise of grapheneNovoselov et al. (2005) launched the era of two- dimensional (2D) electronics, the manufacturing of electronic devices on substrates of one or few atomic layers thickness. Graphene shows exceptional mechanical and electronic properties as well as spectacular physical phenomena, as for example massless Dirac fermions.Geim and Novoselov (2007) However, as in 3D electronics, the successful manufacturing of a variety of devices requires the combination of conducting, insulating and semi-conducting materials with tunable properties. One class of 2D semiconductors and semimetals are transition-metal dichalcogenides (TMD). Its most prominent representative, molybdenum disulphide (MoS2), is a direct band gap semiconductor ($\Delta$ = 1.8 eV) in the monolayer (ML) form.Mak et al. (2010); Splendiani et al. (2010); Kuc et al. (2011) Pioneering measurements of MoS2-ML-based devices have shown that at room-temperature the mobility is about 200 cm2 V s-1, when exfoliated onto the HfO2 substrate, however, it decreases down to the 0.1–10 cm2 V s-1 range if deposited on SiO2.Radisavljevic et al. (2011) Various electronic devices have been fabricated on MoS2-ML, including thin film transistors,Radisavljevic et al. (2011); Pu et al. (2012); Kim et al. (2012) logical circuits,Wang et al. (2012) amplifiersRadisavljevic et al. (2012) and photodetectors.Lopez-Sanchez et al. (2013) It has been shown that the electronic properties of MoS2-ML can be easily tuned by doping,Ivanovskaya et al. (2006); Komsa et al. (2012); Dolui et al. (2013) bendingConley et al. (2013) or tube formation,Zibouche et al. (2012); Seifert et al. (2000) tensile strainGhorbani-Asl et al. (2013) or intrinsic defects.Ataca et al. (2011); Zhou et al. (2013); Enyashin et al. (2013); Van der Zande et al. (2013) The chemical and structural integrity of MoS2 depends on the manufacturing process. Monolayers can be produced, following the top-down approach, from natural MoS2 crystals by micromechanical exfoliation,Mak et al. (2010); Radisavljevic et al. (2011) intercalation based exfoliation,Ramakrishna Matte et al. (2010) or, on larger scale, by liquid-exfoliation techniques.Coleman et al. (2011) On the other hand, chemical vapour deposition (CVD) is a bottom-up procedure and it provides a controllable growth of the material with the desired number of layers on the substrate of interest, e.g. on SiO2Lee et al. (2012) or on graphene.Shi et al. (2012) MoS2-ML prepared in such different processes may contain numerous defects, including cationic or anionic vacancies, dislocations and grain boundaries. Those defects significantly influence transportLee et al. (2012) and optical propertiesTongay et al. (2013) of these materials. For example, it has been found that the maximum career mobility in CVD MoS2 can be up to 0.02 cm2 V-1 s-1,Lee et al. (2012) while mechanically exfoliated ML showed a mobility of 0.1–10 cm2 V-1 s-1.Novoselov et al. (2005); Radisavljevic et al. (2011) Tongay et al.Tongay et al. (2013) showed that point defects lead to a new photoemission peak and enhancement in photoluminescence intensity of MoS2-ML. These effects were attributed to their trapping potential for free charge carriers and to localized excitons. Defects may serve as means of engineering the MoS2 properties, similarly as chemical impurities in semiconductor doping. Zhou et al.Zhou et al. (2013) showed that S- and MoS3-vacancies can be generated in CVD MoS2-ML by extended electron irradiation. This suggests that controlled defect-engineering allows tailoring – even locally – the electronic properties of MoS2. Structural defects in the TMD layers can appear in various types, such as point vacancies, grain boundaries, or topological defects. The point vacancy is one of the native defects which have been investigated both in theoryKomsa et al. (2012); Zhou et al. (2013); Ma et al. (2011); Wei et al. (2012) and experiment.Zhou et al. (2013) The recent experiment by Zhou et al.Zhou et al. (2013) showed that divacancies are only randomly observed, while monovacancies occur more frequently in MoS2-ML. Intrinsic defects can be created without elimination of atoms from the lattice, e.g. by performing Stone-Wales rotations and reconstructing intralayer bonds.Zhou et al. (2013) First principles calculations by Zou et al.Zou et al. (2013) predicted that grain boundaries in MoS2-ML can be formed as odd- or even-fold rings, depending on the rotational angle and stoichiometry, what has been been confirmed in experiment.Van der Zande et al. (2013) Line defects, suggested by Enyashin et al.,Enyashin et al. (2013) introduce a mirror plane into the MoS2-ML, thus forming inversion domains. Yong et al.Yong et al. (2008) showed that a finite atomic line of sulphur vacancies created on a MoS2 surface can behave as a pseudo-ballistic wire for electron transport. So far, direct measurement of the defect influence on the electronic structure and transport properties have been impossible because of substrate-induced local potential variations and contact resistances. In order to fully understand and exploit defects in MoS2-ML, we study here the electronic properties and the quantum transport of several structural defects using the density-functional based methods. We will show that local defects introduce strongly localized mid-gap states in the electronic structure that act as scattering centers. These scattering states do not open new transport channels, but they introduce high anisotropy in the quantum conductance. ## II Methods All calculations have been carried out using the density-functional based tight-binding (DFTB) methodSeifert et al. (1996); Oliveira et al. (2009) as implemented in the deMonNano code.Koster et al. (2009) The structures of monolayers, that is, atomic positions and lattice vectors, have been fully optimized applying 3D periodic boundary conditions with a vacuum separation of 20 Å perpendicular to the MLs. The DFTB parameters for MoS2-ML have been validated and reported earlier.Seifert et al. (2000); Kaplan-Ashiri et al. (2006) The coherent electronic transport calculations were carried out using the DFTB method in conjunction with the Green’s function (GF) and the Landauer-Büttiker approach.Datta (2005); Di Carlo et al. (2002) Our in-house DFTB-GF software for quantum conductance has already been successfully applied to various nanostructures, including layered and tubular TMDs.Ghorbani-Asl et al. (2013); Miró et al. (2013); Ghorbani-Asl et al. (2013) The transport simulation setup consists of a finite defective MoS2-ML as scattering region, which is connected to two semi-infinite ideal MoS2-ML electrodes (Figure 1). The selected scattering region is at least 28 Å wide in order to prevent direct interaction between the electrodes. The whole system is two-dimensional, and we apply in-plane periodic boundary conditions perpendicular to the transport direction. Thus, unphysical edge effects and out-of-plane periodicity are avoided. Note that the electronic transport through the perfect monolayer represents the result for the bulk conductivity. The quantum conductance ($\mathcal{G}$) was calculated at zero-bias following the Landauer-Büttiker formula,Landauer (1970) where $\mathcal{G}$ is represented as:Fisher and Lee (1981) $\mathcal{G}(E)=\frac{2e^{2}}{h}trace\left[\hat{G}^{\dagger}\hat{\,\Gamma}_{R}\hat{\,G}\hat{\,\Gamma}_{L}\right],$ (1) where $\hat{G}$ denotes the total GF of the scattering region coupled to the electrodes and $\vphantom{}\hat{\Gamma}_{\alpha}=-2\mathrm{Im\mathit{(}{\hat{\Sigma}_{alpha})}}$ is the broadening function, self-energies ($\hat{\Sigma}_{L,R}$) are calculated following the iterative self-consistent approach.Sancho et al. (1985) Figure 1: (Color Online) Schematic representation of the electronic transport in the MoS2-ML with a point defect. Left and right electrodes (L and R) that consist of semi-infinite ideal MoS2-ML are highlighted. The scattering region (S) includes the defect. The transport direction is indicated by the arrow. The defective structures are shown in Figure 2. Besides the pristine monolayer (I) we have studied three types of point defects, namely vacancies (II-V and XIII), add-atoms (VI), and Stone-Wales rearrangements (VII-IX). Additionally, we have studied line defects formed from the vacancies (X, XI). In detail, we have considered non-stoichiometric single-atom vacancies of Mo and S atoms (II and III); multiple vacancies with dangling bonds (IV) or with reconstruction towards homonuclear bond formation (V); loops of line defects forming large triangular defects with homonuclear bond formation (XII and XIII). The addition of one MoS2 unit into the lattice (VI) causes rings of different oddity, such as ”4-8” rings, preserving the alternation of chemical bonds. A Stone-Wales rotation of a MoS2 unit by 180∘ (VII and VIII) results in hexagonal rings with homonuclear bonds, while rotation by only 90∘ (IX) forms ”5-7” rings, similar to the ones observed in graphene.Ma et al. (2009) Line defects can be formed by vacancies of S or Mo atoms along zigzag direction (X and XI), resulting in the formation Mo-Mo and S-S dimer bonds, respectively. Such systems have mirror symmetry along the defect lines, what imposes difficulties in the periodic model representation. Figure 2: (Color Online) Representation of local defects: Ideal (I) and defective MoS2-ML containing point (II - IX), line (X - XI) defects and grain boundaries (XII - XIII). The defective areas are highlighted. Mo and S are shown in red and yellow, respectively. These figures show partial sections of the super cells used in the simulations. The point defects were simulated using the supercell approach, where the MoS2 ML was expanded to 90 Mo and 180 S atoms. This supercell corresponds to the 5$\times$9 unit cells of the ideal lattice in rectangular representation. The line defects were optimized using Mo172S344 supercells and, in order to maintain the in-plane 2D periodicity, both types of defects were present simultaneously in the optimization setup. The transport calculations, however, were performed for each line defect separately, keeping the in-plane periodicity perpendicular to the transport direction. Along the transport axis the scattering region was connected to semi-infinite electrodes and the whole system was treated using periodic boundary conditions. Triangular island defects were represented using Mo303S586 and Mo293S606 supercells for S- and Mo-bridges, respectively. As MoS2-ML are produced under harsh conditions far from thermodynamic equilibrium, it can be assumed that a variety of defects are present in the samples. Thus, we are not considering the thermodynamic stability of the defects, and refer the reader to recent studies on this subject.Enyashin et al. (2013); Enyashin and Ivanovskii (2007) All defects considered in this work are modeled by fully relaxed structures. The geometry optimization did not reveal any considerable distortion of the layers and the defective structures preserved their integrity. Note that the DFTB method, in the present implementations, does not account for the spin-orbit coupling (SOC) and therefore, this effect has not been considered in the present studies. However, relativistic first-principles calculations including scalar relativistic effects and SO corrections showed that SOC in MoS2-ML accounts for a large valence band splitting of about 145–148 meV, while the conduction band is affected in a much lower degree, by $\sim$3 meV band splitting.Zhu et al. (2011); Kormányos et al. (2013) At the same time, the effective masses in the valence change by about 5%, while the effective masses in the conduction band are basically not affected by SOC. The fundamental band gap changes by $\sim$ 50 meV. Thus, the SOC effects will not significantly alter the results that are presented in the remainder of this work. ## III Results and Discussion We have investigated the electronic structure of defective MoS2-ML by calculating orbital-projected densities of state (PDOS). The results were compared with the perfect MoS2 system. Crystal orbitals are visualized corresponding to the states close to the Fermi level (EF) (Figure 3). The electronic structure of MoS2-ML suggests that the bottom of the conduction band is formed from empty Mo-4$d_{z^{2}}$ orbitals,Enyashin et al. (2013); Kuc et al. (2011) while the top of the valence band is composed of fully occupied $d_{xy}$ and $d_{x^{2}-y^{2}}$ orbitals, in agreement with the crystal-field splitting of trigonal prismatic systems. In pristine MoS2, the highest- occupied and lowest-unoccupied crystal orbitals (HOCO and LUCO) are delocalized and spread homogeneously throughout the system (Figure 2 I). The electronic band gap of MoS2-ML obtained at the DFTB level of about 1.5 eV is smaller than that obtained from DFTMak et al. (2010); Kuc et al. (2011); Splendiani et al. (2010) and experimentMak et al. (2010) due to the deviations in geometry. DFTB-estimated lattice vectors ($a$ = 3.32 Å) are by 5% larger than the experimental values ($a$ = 3.16 Å),Wilson and Yoffe (1969) and, as it has been discussed earlier, such a distortion in geometry causes the decrease in the band gap.Ghorbani-Asl et al. (2013) However, these discrepancies should not alter general trends and conclusions drawn here, as the relative change in the electronic structure is not influenced. Figure 3: (Color Online) (Left panel) Total (black), Mo-4$d$ (red) and S-3$p$ projected (green) densities of states (DOS) of selected defective MoS2-ML (labels as in Figure 2). (Right panel) Highest-occupied and lowest-unoccupied crystal orbitals (HOCO and LUCO), and delocalized conduction orbitals. As known from the literature,Komsa et al. (2012); Zhou et al. (2013) defects result in significant changes of the electronic structure close to EF and introduce mid-gap states. The mid-gap states are strongly localized in the vicinity of the defects and are mostly of 4$d$-Mo type, thus they act as scattering centers. Although defect states reduce the band gap significantly, the scattering character will prevent opening any new conduction channels close to EF. In case of a single Mo vacancy (Figure 3 II), the valence band maximum (HOCO) resembles the characteristics of perfect MoS2-ML, while the edge of conduction band is formed by two individual mid-gap states (states 808/809 and 810 in Figure 3 II), the first being the LUCO. The next delocalized states are located at around 1.2 eV above EF (state 814). A similar situation is observed for the single S vacancy (Figure 3 III), with the LUCO (degenerated states 808/809) composed of single strongly localized states. In this case, the next delocalized states are present only at about 1 eV above EF (810/811). For larger point defects, where both types of elements are removed from the lattice, the electronic structure changes even stronger, with HOCO states being no longer delocalized. In the case of Stone-Wales defects, the band gap reduces with the number of rotated bonds and introduces a larger number of mid-gap states. At the same time, the HOCO becomes more localized (Figure S1 and S2 in Supporting Information).SI Considering the triangular domain structures (XII and XIII), which contain Mo- Mo and S-S line defects, the PDOS shows interesting characteristics.Enyashin et al. (2013) The Mo-Mo bridges contribute primary with the 4$d$ to the HOCO and the LUCO. These are localized states at about 2.5 eV below EF, indicating the formation of strong Mo$-$Mo bonds. In contrast, the S-S line defects form S-3$p$ states, which do not contribute to the PDOS close to EF. These states can be found deep in the valence band region at about 3 eV below EF. The states in the vicinity of EF are, therefore, composed exclusively from the edge states of the MoS2 domains. The defect-induced variations in the electronic structure affect the electronic transport in the MoS2-ML. The transport through MoS2 should be direction dependent due to the structural anisotropy of the system. The pristine layer shows, however, very little anisotropy in the electron conductivity as reported earlier.Ghorbani-Asl et al. (2013) The two extremes are transport along armchair ($\mathcal{G}_{a}$) or zigzag ($\mathcal{G}_{z}$) directions. In order to ensure transferability of the results, we used a supercell with almost equal length and width along both transport directions (La = 28.75 Å and Lz = 29.88 Å). Figure 4 shows the electron conductivity of the MoS2-ML in the pristine form and in the presence of various point defects along the armchair and zigzag directions. The occurrence of defects reduces the conductivity (transmittance) in comparison with the pristine layer at 1.2 eV below and above EF. This is expected as the vacancy causes backscattering effects,Rutter et al. (2007); Deretzis et al. (2010) and, not surprisingly, the conductivity depends strongly on the type and concentration of the point defects. Noteworthy, in contrast to the pristine ML, the electron conductivity of the defective systems becomes strongly direction dependent and the conductivity is suppressed much stronger along the armchair direction. The only exception is the single S-vacancy, where the transport is rather direction independent. Figure 4: (Color Online) Electron conductivity of MoS2-ML with point defects. $\mathcal{G}_{a}$ (a) and $\mathcal{G}_{z}$ (b) denote electron conductivity along the armchair and zigzag direction, respectively. Labels are as in Figure 2. The directional dependence of the conductance might arise from different transmission pathwaysWang et al. (2009); Liu et al. (2013) and electron hopping within defective parts of MoS2-ML.Remskar et al. (2011) To date, only grain boundaries have been studied in experiment, and our results are consistent with the results reported by the Heinz group.Van der Zande et al. (2013) In case of transport in armchair direction, the electron conductance of MoS2 with one Mo-vacancy, corresponding to 1.11% structural defects, is suppressed by 75% compared with the pristine layer. The single S vacancy (0.55% structural defects) shows higher electron conductivity due to the electron injection directly to the conduction band. For the Stone-Wales defects (VII and IX) the conductance is reduced by less than 50% with respect to the pristine structure. Figure 5 shows the transport properties of the MoS2-ML with triangular grain boundaries along the armchair and zigzag directions. It is very interesting to notice that $\mathcal{G}$ in this case does not depend on the type of the defect and similar values are obtained for Mo–Mo and S–S bridges. The conductance is, however, strongly direction-dependent and again we observe that along the armchair lines it is more suppressed than along the zigzag ones. Figure 5: (Color Online) Electron conductivity of MoS2-ML with grain boundaries formed by inversion domains. $\mathcal{G}_{a}$ (a) and $\mathcal{G}_{z}$ (b) denote electron conductivity along the armchair and zigzag direction, respectively. Labels as in Figure 2. Our results indicate that local defects introduce spurious minor conductance peaks close to EF (see Figure S3 in Supporting InformationSI ). Because these electronic states are strongly localized, they do not contribute to the overall quantum transport, as they cannot generate additional conducting channels for a specific energy window within the perfect semi-infinite electrodes. We have also studied the conductance across line defects (X, XI) with respect to the length of the scattering region ls (see Figure 6). In this case, the structure is periodic along the line defects but in the perpendicular direction there is a mirror symmetry, which should be considered in the transport simulations and the choice of the electrodes. Here, our electrodes are still perfect MoS2-MLs, but represent mirror images with respect to each other. Therefore, we have decided to vary the length of ls and investigate its influence on the transport properties. Figure 6: (Color Online) Schematic representation of the electronic transport in the MoS2-ML with line defects X (a) and XI (b), and the corresponding electron conductance as function of the length of the scattering region, ls (c). ls in (a) and (b) correspond to the length of 6.3 nm. Labels as in Figure 2. Our results show that the conductance at about +/-1.5 eV from the Fermi level reduces with increasing the channel length, however the band gap does not change and no open channels are present close to the EF. ## IV Conclusion In summary, we investigated the coherent electron transport through MoS2-ML with various defects on the basis of the Green’s functions technique and the DFTB method. The presence of local defects leads to the occurrence of mid-gap states in semiconducting MoS2-ML. These states are localized and act as scattering centers. Our transport calculations show that single-atomic vacancies can significantly reduce the average conductance. The decrease of conductance depends on the type and concentration of the defects, and, surprisingly, on the transport direction. We find significant anisotropy of electron transfer in MoS2-ML with grain boundaries. Since structural and electronic properties of layered semiconducting TMD are comparable, we expect similar effects to occur in other defective TMD-ML. Our results indicate that structural defects and grain boundaries are principal contributors to the electronic transport properties of MoS2 monolayers, thus rationalizing the large variation of electronic conductivity in different samples. ## V Acknowledgements This work was supported by Deutsche Forschungsgemeinschaft (HE 3543/18-1), the Office of Naval Research Global (Award No N62909-13-1-N222) the European Commission (FP7-PEOPLE-2009-IAPP QUASINANO, GA 251149, and FP7-PEOPLE-2012-ITN MoWSeS, GA 317451) and ERC project (INTIF 226639). ## References * Novoselov et al. (2005) Novoselov, K. S.; Jiang, D.; Schedin, F.; Booth, T. J.; Khotkevich, V. V.; Morozov, S. V.; Geim, A. K. _Proc. Nat. Acad. Sci. USA_ 2005, _102_ , 10451–10453. * Geim and Novoselov (2007) Geim, A. K.; Novoselov, K. S. _Nat. Mater._ 2007, _6_ , 183–191. * Mak et al. (2010) Mak, K. F.; Lee, C.; Hone, J.; Shan, J.; Heinz, T. F. _Phys. Rev. Lett._ 2010, _105_ , 136805–1–136805–4. * Splendiani et al. 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1311.0790	# A Discontinuous Galerkin Time Domain Framework for Periodic Structures Subject To Oblique Excitation Nicholas C. Miller, Andrew D. Baczewski, John D. Albrecht, and Balasubramaniam Shanker ###### Abstract A nodal Discontinuous Galerkin (DG) method is derived for the analysis of time-domain (TD) scattering from doubly periodic PEC/dielectric structures under oblique interrogation. Field transformations are employed to elaborate a formalism that is free from any issues with causality that are common when applying spatial periodic boundary conditions simultaneously with incident fields at arbitrary angles of incidence. An upwind numerical flux is derived for the transformed variables, which retains the same form as it does in the original Maxwell problem for domains without explicitly imposed periodicity. This, in conjunction with the amenability of the DG framework to non-conformal meshes, provides a natural means of accurately solving the first order TD Maxwell equations for a number of periodic systems of engineering interest. Results are presented that substantiate the accuracy and utility of our method. ###### Index Terms: Periodic structures, Discontinuous Galerkin (DG) methods, time domain analysis. ## I Introduction Figure 1: Illustration of the $z$-plane of a doubly periodic structure with periods $\|\vec{\bf a}_{1}\|$ and $\|\vec{\bf a}_{2}\|$. The ellipses indicate that the structure is periodic in the $x$\- and $y$-directions. Periodic structures play a significant role in electromagnetics and optics in generating unique spectral responses that can be readily engineered. Applications of periodicity include frequency selective surfaces (FSS) [1], electromagnetic band gap (EBG) structures [2], biomimetic structures and metamaterials [3], [4], etc. Computational analysis of fields in increasingly intricate periodic unit cells plays a significant role in their design and optimization. In the frequency domain, Integral Equation (IE) [5], [6], Finite Element (FE) [7], [8], and Discontinuous Galerkin (DG) [9] methods have been successfully applied to a variety of periodic electromagnetic systems. Time- domain (TD) methods for studying periodic systems include FE [10],[11], IE [12], and Finite Difference Time Domain (FDTD) [13], while DG methods remain relatively unexplored. TD analysis of periodic structures provides a number of advantages, such as characterization of the broadband response of a structure in a single simulation, and treatment of nonlinearities. Both integral and differential formulations of the Maxwell problem have attendant disadvantages as well. For integral formulations, discretization yields a dense linear system. While fast and efficient [12], [14] methods have been applied to these problems, stable formulations of TDIEs remain a research problem, with much recent progress [15]. Recent work has also been presented on obtaining transient response using entire domain Laguerre polynomials that results a system wherein the time variable is completely avoided [16]. Alternatively, while differential formulations of the problem yield sparse linear systems and stability is better understood, the proper imposition of boundary conditions (BCs) becomes challenging. In particular, the asymptotic boundary condition on the fields receding to infinity must be enforced approximately with an absorbing boundary condition (ABC) or a perfectly matched layer (PML) [17]. Further, while periodic BCs at the perimeter of the unit cell are trivial to enforce for systems excited at normal incidence, there are well-known issues associated with causality at oblique incidence [10]. A set of field transformations that mitigate causality issues was introduced for FDTD in 1993 [18], and later adapted to an FETD framework in a sequence of papers in the mid-2000s [10], [11]. Here, the frequency domain Floquet- periodic boundary condition is exploited, wherein fields at the unit cell boundaries are related to one another by a phase shift that depends on the exciting wave vector and lattice vectors. The frequency domain Maxwell Equations are then posed in terms of a set of transformed variables, into which this phase shift is built, and an inverse transform is applied to return the equations to the time domain. Additional terms then appear in the TD Maxwell Equations for the transformed variables. In this work, we will apply these field transformations to a time domain Discontinuous Galerkin (DG) framework for the conservation form of the Maxwell equations for the first time. Time domain analysis of periodic structures with DG methods has received relatively little attention, with a few exceptions [9], [19]. The unique contributions of this paper are extensions of a time domain DG framework that permit the analysis of doubly periodic structures at oblique incidence. First, the field transformations that are used to remove causality issues are reviewed. We then demonstrate that the form of the upwind flux utilized in discretizing the transformed Maxwell Equations is invariant to whether or not one is utilizing the original or transformed fields. Issues addressing the use of non-conformal meshes across periodic boundaries are discussed, and relevant implementation details are provided. Finally, results are presented that validate the accuracy and utility of our method for a number of doubly periodic test cases. ## II Mathematical Formulation Consider a domain, $\Omega\subset\mathbb{R}^{3}$ depicted in Fig. 2, where a doubly periodic distribution of isotropic, lossless, dielectric and/or PEC scatterers reside. The periodicity of the system is described by a 2-lattice, $\mathcal{L}_{2}$, defined as: $\mathcal{L}_{2}=\\{\vec{\bf u}_{n}=n_{1}\vec{\bf a}_{1}+n_{2}\vec{\bf a}_{2}\|n_{1},n_{2}\in\mathbb{Z}\\}$ (1) Here, the subscript $n$ is defined as a multi-index, and $\vec{\bf a}_{i}$ are basis vectors for the lattice. These vectors will be orthogonal in this work, but extensions to non-orthogonal basis vectors are simply realized. Incident on the system is a planewave excitation $\vec{\bf E}_{i}(\vec{\bf r},t)$, with a wavevector $\hat{\bf k}_{i}=\sin\theta\cos\phi\hat{x}+\sin\theta\sin\phi\hat{y}+\cos\theta\hat{z}$. The incident wavevector, $\hat{\bf k}_{i}$, can be further decomposed into $\hat{\bf k}_{i}^{\parallel}$ and $\hat{\bf k}_{i}^{\perp}$, which are within and orthogonal to the span of $\mathcal{L}_{2}$, respectively. Figure 2: Illustration of a single unit cell of a doubly periodic structure with periods $\|\vec{\bf a}_{1}\|$ and $\|\vec{\bf a}_{2}\|$. The fields obey the following boundary conditions under spatial translation by a lattice vector in $\mathcal{L}_{2}$: $\displaystyle\vec{\bf E}(\vec{\bf r},t)=\vec{\bf E}(\vec{\bf r}+\vec{\bf u}_{n},t)\ast\delta\left(t+\frac{\hat{\bf k}_{i}^{\parallel}\cdot\vec{\bf r}}{c_{0}}\right)$ (2a) $\displaystyle\vec{\bf H}(\vec{\bf r},t)=\vec{\bf H}(\vec{\bf r}+\vec{\bf u}_{n},t)\ast\delta\left(t+\frac{\hat{\bf k}_{i}^{\parallel}\cdot\vec{\bf r}}{c_{0}}\right)$ (2b) Direct implementation of these periodic boundary conditions requires knowledge of future values of fields at one periodic boundary in order to update fields at the other periodic boundary. In the context of a time integration scheme in which fields are updated in time based upon a sequence of their previous values, this is not possible without extrapolation. Alternatively, transformed fields can be identified for which the periodic boundary conditions remain causal. As done in [10],[18], we introduce delayed auxiliary variables, $\vec{\bf P}(\vec{\bf r},\omega)$ and $\vec{\bf S}(\vec{\bf r},\omega)$ $\displaystyle\vec{\bf E}(\vec{\bf r},\omega)=\vec{\bf P}(\vec{\bf r},\omega)e^{-j\vec{\bf k}_{i}^{\parallel}\cdot\vec{\bf r}}$ (3a) $\displaystyle\vec{\bf H}(\vec{\bf r},\omega)=\vec{\bf S}(\vec{\bf r},\omega)e^{-j\vec{\bf k}_{i}^{\parallel}\cdot\vec{\bf r}}$ (3b) It can be shown trivially that these transformed fields obey $\displaystyle\vec{\bf P}(\vec{\bf r},t)=\vec{\bf P}(\vec{\bf r}+\vec{\bf u}_{n},t)$ (4a) $\displaystyle\vec{\bf S}(\vec{\bf r},t)=\vec{\bf S}(\vec{\bf r}+\vec{\bf u}_{n},t)$ (4b) As is evident from Eqns. (4a) and (4b), using these auxiliary field components is tantamount to zero phase propagation at the boundaries, i.e., there is no delay in boundaries of the unit cell. This is the time domain analog to cell- periodic Bloch functions typical of frequency analysis. Applying the field transformations to the first order time domain Maxwell Equations yields $\displaystyle\varepsilon\frac{\partial\vec{\bf P}(\vec{\bf r},t)}{\partial t}+\frac{\hat{\bf k}_{i}^{\parallel}}{c_{0}}\times\frac{\partial\vec{\bf S}(\vec{\bf r},t)}{\partial t}$ $\displaystyle=\nabla\times\vec{\bf S}(\vec{\bf r},t)$ (5a) $\displaystyle-\frac{\hat{\bf k}_{i}^{\parallel}}{c_{0}}\times\frac{\partial\vec{\bf P}(\vec{\bf r},t)}{\partial t}+\mu\frac{\partial\vec{\bf S}(\vec{\bf r},t)}{\partial t}$ $\displaystyle=-\nabla\times\vec{\bf P}(\vec{\bf r},t)$ (5b) It is these equations that we will now discretize within the DG framework. ## III The Discontinuous Galerkin Method ### III-A Discretization To allow a seamless extension from previous DG formulations [20], [21], [22], we write Eqns. (5a) and (5b) in conservation form: $Q\frac{\partial\vec{\bf q}(\vec{\bf r},t)}{\partial t}+\nabla\cdot\vec{\bf F}\left(\vec{\bf q}(\vec{\bf r},t)\right)=0$ (6) Here, the periodic/materials matrix $Q$, field six-vector $\vec{\bf q}(\vec{\bf r},t)$, and flux matrix $\vec{\bf F}\left(\vec{\bf q}(\vec{\bf r},t)\right)$ are defined as: $Q=\left(\begin{array}[]{cc}\varepsilon\mathcal{I}_{1}&c_{0}^{-1}\hat{\bf k}_{i}^{\parallel}\times\mathcal{I}_{1}\\\ -c_{0}^{-1}\hat{\bf k}_{i}^{\parallel}\times\mathcal{I}_{1}&\mu\mathcal{I}_{1}\end{array}\right),$ $\vec{\bf q}(\vec{\bf r},t)=\left(\begin{array}[]{c}\vec{\bf P}(\vec{\bf r},t)\\\ \vec{\bf S}(\vec{\bf r},t)\end{array}\right),\vec{\bf F}\left(\vec{\bf q}(\vec{\bf r},t)\right)=\left(\begin{array}[]{c}-\hat{e}_{i}\times\vec{\bf S}(\vec{\bf r},t)\\\ \hat{e}_{i}\times\vec{\bf P}(\vec{\bf r},t)\end{array}\right)$ here, $\hat{e}_{i}$ represents the ith Cartesian unit vector, $\varepsilon$ is the isotropic permittivity, $\mu$ is the isotropic permeability, and $\mathcal{I}_{1}$ is the 3x3 identity matrix. Solving this system of equations requires discretizing the domain using $k$ non-overlapping tetrahedra, where domains are denoted $\Omega^{k}$ with boundaries $\partial\Omega^{k}$ that are equipped with an outward pointing normal $\hat{n}$. The vector unknowns are expanded into a set of globally discontinuous nodal polynomials $\vec{\bf q}\left(\vec{\bf r},t\right)\approx\sum\limits_{i=1}^{N_{p}}\vec{\bf q}^{k}\left(\vec{\bf r}_{i},t\right)\ell_{i}^{k}\left(\vec{\bf r}\right)$. We use the nodal basis functions defined in [20]. Following standard DG practice [20], a strong form of the problem is obtained as: $\displaystyle\iiint\limits_{\Omega^{k}}\left(Q\frac{\partial\vec{\bf q}(\vec{\bf r},t)}{\partial t}+\nabla\cdot\vec{\bf F}\left(\vec{\bf q}(\vec{\bf r},t)\right)\right)\ell_{j}^{k}(\vec{\bf r})d\vec{\bf r}$ $\displaystyle=\iint\limits_{\partial\Omega^{k}}\vec{n}\cdot\left(\vec{\bf F}\left(\vec{\bf q}(\vec{\bf r},t)\right)-\vec{\bf F}^{}\left(\vec{\bf q}(\vec{\bf r},t)\right)\right)\ell_{j}^{k}(\vec{\bf r})d\vec{\bf r}$ (7) where $\vec{\bf F}^{}$ is called the numerical flux. We can rewrite the semi- discrete problem in Eqn. (7) as: $\frac{\partial\vec{\bf q}(\vec{\bf r},t)}{\partial t}=Q^{-1}\left(\mathcal{M}^{-1}\mathcal{S}\vec{\bf q}+\mathcal{M}^{-1}\mathcal{F}\left[\hat{n}\cdot\left(\vec{\bf F}-\vec{\bf F}^{}\right)\right]\right)$ (8) with the function of nodal values $\hat{n}\cdot\left(\vec{\bf F}-\vec{\bf F}^{}\right)$, defined on the element boundaries, replacing the flux matrix $\vec{\bf F}\left(\vec{\bf q}(\vec{\bf r},t)\right)$, the periodic/materials matrix $\mathcal{Q}$ re-defined as $Q=\left(\begin{array}[]{cccccc}\varepsilon\mathcal{I}_{2}&0&0&0&0&-\kappa_{y}\mathcal{I}_{2}\\\ 0&\varepsilon\mathcal{I}_{2}&0&0&0&\kappa_{x}\mathcal{I}_{2}\\\ 0&0&\varepsilon\mathcal{I}_{2}&\kappa_{y}\mathcal{I}_{2}&-\kappa_{x}\mathcal{I}_{2}&0\\\ 0&0&\kappa_{y}\mathcal{I}_{2}&\mu\mathcal{I}_{2}&0&0\\\ 0&0&-\kappa_{x}\mathcal{I}_{2}&0&\mu\mathcal{I}_{2}&0\\\ -\kappa_{y}\mathcal{I}_{2}&\kappa_{x}\mathcal{I}_{2}&0&0&0&\mu\mathcal{I}_{2}\end{array}\right)$ where $\hat{\bf k}_{i}^{\parallel}=\kappa_{x}\hat{x}+\kappa_{y}\hat{y}$ and $\mathcal{I}_{2}$ is the $N_{p}$x$N_{p}$ identity matrix. The mass matrix $\mathcal{M}$, stiffness matrix $\mathcal{S}$, and face matrix $\mathcal{F}$ are defined as $\displaystyle\mathcal{M}_{ij}=\iiint\limits_{\Omega^{k}}\ell_{i}^{k}(\vec{\bf r})\ell_{j}^{k}(\vec{\bf r})d\vec{\bf r}$ $\displaystyle\mathcal{S}_{ij}=\iiint\limits_{\Omega^{k}}\ell_{i}^{k}(\vec{\bf r})\nabla\ell_{j}^{k}(\vec{\bf r})d\vec{\bf r}$ $\displaystyle\mathcal{F}_{ij}=\iint\limits_{\partial\Omega^{k}}\ell_{i}^{k}(\vec{\bf r})\ell_{j}^{k}(\vec{\bf r})d\vec{\bf r}$ ### III-B Periodic Numerical Flux Choice of the nodal values $\hat{n}\cdot\left(\vec{\bf F}-\vec{\bf F}^{}\right)$ is at the heart of all DG formulations. Hesthaven and Warburton have proven that an upwind flux is both stable and convergent for Maxwell’s Equations [20]. For the non-periodic Maxwell’s Equations, the upwind flux takes the form $\hat{n}\cdot\left(\vec{\bf F}-\vec{\bf F}^{}\right)=\left(\begin{array}[]{c}-\bar{Z}^{-1}\hat{n}\times\left(Z^{+}\left[\left[\vec{\bf H}\right]\right]-\hat{n}\times\left[\left[\vec{\bf E}\right]\right]\right)\vspace{0.25cm}\\\ \bar{Y}^{-1}\hat{n}\times\left(Y^{+}\left[\left[\vec{\bf E}\right]\right]+\hat{n}\times\left[\left[\vec{\bf H}\right]\right]\right)\end{array}\right)$ (10) Here, the jump $\left[\left[\vec{\bf E}\right]\right]=\vec{\bf E}^{+}-\vec{\bf E}^{-}$ is defined in terms of nodal field values at the element boundaries, and the impedance $\bar{Z}=Z^{+}+Z^{-}$ is twice the average impedance shared at these boundaries. To derive the periodic numerical flux for $\vec{\bf P}(\vec{\bf r},t)$ and $\vec{\bf S}(\vec{\bf r},t)$, we note that $\vec{\bf E}=\vec{\bf P}\ast\delta\left(t-\frac{\hat{\bf k}_{i}^{\parallel}\cdot\vec{r}}{c_{0}}\right)$ and $\vec{\bf H}=\vec{\bf S}\ast\delta\left(t-\frac{\hat{\bf k}_{i}^{\parallel}\cdot\vec{r}}{c_{0}}\right)$. Using these in the conservation form of Maxwell’s equations $\displaystyle\left(\begin{array}[]{cc}\varepsilon\mathcal{I}_{1}&0\\\ 0&\mu\mathcal{I}_{1}\end{array}\right)\frac{\partial}{\partial t}\left(\begin{array}[]{c}\vec{\bf P}\ast\delta\left(t-\frac{\hat{\bf k}_{i}^{\parallel}\cdot\vec{r}}{c_{0}}\right)\\\ \vec{\bf S}\ast\delta\left(t-\frac{\hat{\bf k}_{i}^{\parallel}\cdot\vec{r}}{c_{0}}\right)\end{array}\right)$ $\displaystyle+\nabla\cdot\left(\begin{array}[]{c}-\hat{e}_{i}\times\vec{\bf S}\ast\delta\left(t-\frac{\hat{\bf k}_{i}^{\parallel}\cdot\vec{\bf r}}{c_{0}}\right)\\\ \hat{e}_{i}\times\vec{\bf P}\ast\delta\left(t-\frac{\hat{\bf k}_{i}^{\parallel}\cdot\vec{\bf r}}{c_{0}}\right)\end{array}\right)=0$ it is evident that this system has two distinct characteristic values, $\pm\left(\varepsilon\mu\right)^{-1/2}$. This implies that only three Rankine- Hugoniot jump conditions are needed to relate the fields across discontinuities [20], [23]. Using the convention in [24], integrating over a single element, and reducing integration limits to the faces of the elements, we arrive at the jump conditions for the equivalent transformed equations $\left[Z^{-}\left(\vec{\bf S}^{}-\vec{\bf S}^{-}\right)+\hat{n}\times\left(\vec{\bf P}^{}-\vec{\bf P}^{-}\right)\right]\ast\delta\left(t-\frac{\hat{\bf k}_{i}^{\parallel}\cdot\vec{\bf r}}{c_{0}}\right)=0$ $\left[Z^{+}\left(\vec{\bf S}^{}-\vec{\bf S}^{+}\right)+\hat{n}\times\left(\vec{\bf P}^{}-\vec{\bf P}^{+}\right)\right]\ast\delta\left(t-\frac{\hat{\bf k}_{i}^{\parallel}\cdot\vec{\bf r}}{c_{0}}\right)=0$ $\left[\hat{n}\times\left(\vec{\bf P}^{*}-\vec{\bf P}^{}\right)\right]\ast\delta\left(t-\frac{\hat{\bf k}_{i}^{\parallel}\cdot\vec{\bf r}}{c_{0}}\right)=0$ $\left[\hat{n}\times\left(\vec{\bf S}^{*}-\vec{\bf S}^{}\right)\right]\ast\delta\left(t-\frac{\hat{\bf k}_{i}^{\parallel}\cdot\vec{\bf r}}{c_{0}}\right)=0$ Since these equations hold for all time, the periodic numerical flux may now be written as [24] $\hat{n}\cdot\left(\vec{\bf F}-\vec{\bf F}^{}\right)=\left(\begin{array}[]{c}-\bar{Z}^{-1}\hat{n}\times\left(Z^{+}\left[\left[\vec{\bf S}\right]\right]-\hat{n}\times\left[\left[\vec{\bf P}\right]\right]\right)\vspace{0.25cm}\\\ \bar{Y}^{-1}\hat{n}\times\left(Y^{+}\left[\left[\vec{\bf P}\right]\right]+\hat{n}\times\left[\left[\vec{\bf S}\right]\right]\right)\end{array}\right)$ (13) In Eqn. 13, $\left[\left[\vec{\bf P}\right]\right]=\vec{\bf P}^{+}-\vec{\bf P}^{-}$ is the jump in the nodal field values at an element’s boundaries. ### III-C Boundary Conditions TABLE I: Boundary Condition Jumps B.C. \| $\left[\left[\vec{\bf P}\right]\right]$ \| $\left[\left[\vec{\bf S}\right]\right]$ ---\|---\|--- PEC: \| $-2\vec{\bf P}^{-}$ \| 0 ABC (TE): \| $-2\vec{\bf P}^{-}\left\|\cos\theta\right\|$ \| $-2\vec{\bf S}^{-}$ ABC (TM): \| $-2\vec{\bf P}^{-}$ \| $-2\vec{\bf S}^{-}\left\|\cos\theta\right\|$ TF/SF: \| $\vec{\bf P}^{+}-\vec{\bf P}^{-}\pm\vec{\bf P}^{inc}$ \| $\vec{\bf S}^{+}-\vec{\bf S}^{-}\pm\vec{\bf S}^{inc}$ Applying boundary conditions to the periodic system of equations requires constraining the jumps $\left[\left[\vec{\bf P}\right]\right]$ and $\left[\left[\vec{\bf S}\right]\right]$ across a face. We present a list of common DG jumps first presented in [21]. Here, TF/SF denotes total fields and scattered fields, respectively. The addition of the angle of incidence in the jumps for the planewave ABC allows the periodic numerical flux to satisfy the well-known Silver-Müller condition for the transformed fields $\displaystyle Z\hat{n}\times\vec{\bf S}=\left\|\cos\theta\right\|\hat{n}\times\hat{n}\times\vec{\bf P}$ $\displaystyle Y\hat{n}\times\vec{\bf P}=-\left\|\cos\theta\right\|\hat{n}\times\hat{n}\times\vec{\bf S}$ for TE and TM polarization, respectively. Here, $Z=1/Y$ is the impedance of the medium. We must also consider boundary conditions on the interfaces between unit cells. To implement Eqns. (4a) and (4b), a map must be created between the periodic planes of the unit cell. A natural first choice for creating these maps is to create a meshed unit cell in which the periodic planes are conformal, and set the jumps to be $\left[\left[\vec{\bf P}\right]\right]=\vec{\bf P}(\vec{\bf r}+\vec{\bf u}_{n},t)-\vec{\bf P}(\vec{\bf r},t)$ and $\left[\left[\vec{\bf S}\right]\right]=\vec{\bf S}(\vec{\bf r}+\vec{\bf u}_{n},t)-\vec{\bf S}(\vec{\bf r},t)$. Alternatively, it is significantly easier to generate a meshed unit cell without meticulous constraints on the periodic planes. The nodes of the periodic plane will not align, and information regarding the non-conformal triangles is generated. This interface is first decomposed into a list of the four different types of fragments: three-, four-, five-, and six-vertex fragments. A polygon clipping algorithm [25] is employed to generate this data. These fragments are defined to facilitate the definition of quadrature rules for numerically integrating surface terms. Figure 3: Reflection coefficient (in dB) of a planewave normally incident on periodically arranged PEC Minkowski Fractals. The unit cell dimensions for the fractal are $\|\vec{\bf a}_{1}\|=\|\vec{\bf a}_{2}\|=30$cm. Dimensions of the fractal are shown above. The ABC surfaces were placed $10$cm away from the PEC fractal in $\pm z$-direction. The electric field is $x$-polarized. Figure 4: Power reflected from a planewave obliquely incident on a nonmagnetic and lossless dielectric slab, $\theta=50^{\circ}$. Top: Power reflection over broadband frequency range for TE polarization (top left) and TM polarization (top right). Bottom: minimum edge length ($h$) and polynomial order ($P$) error convergence for TE polarization. ## IV Results To demonstrate the validity of our computational framework, we discuss several scattering results. In all cases, a low-storage fourth order Runge-Kutta integration [26] is used with a time step size determined by $c\Delta t=hP^{-2}$, where $h$ is the minimum edge length and $P$ is the polynomial order. Reflection or transmission data presented for each structure is obtained from Eqn. (14). $P_{r/t}(f)=\frac{\left\|\vec{\bf E}_{r/t}(f)\right\|^{2}}{\left\|\vec{\bf E}_{i}(f)\right\|^{2}}$ (14) Here, $\vec{\bf E}_{i}(f)$ is the Fourier transform of the planewave excitation. The reflected and transmitted field, denoted by $\vec{\bf E}_{r/t}(f)$, is calculated as the magnitude of the Fourier transform of the fundamental coefficient $\vec{\bf A}_{00}(t)$ given as $\vec{\bf A}_{00}(t)=\frac{1}{\|\vec{\bf a}_{1}\|\|\vec{\bf a}_{2}\|}\int\limits_{y=0}^{\|\vec{\bf a}_{2}\|}\int\limits_{x=0}^{\|\vec{\bf a}_{1}\|}\vec{\bf P}(x,y,z=z_{RT};t)dxdy$ (15) This coefficient is integrated over the $z=z_{RT}$ plane [10] located either below or above the scattering structure for reflection or transmission, respectively. Figure 5: (left) Illustration of the PEC rods oriented in the $y$-direction. The unit cell dimensions are $\|\vec{\bf a}_{1}\|=8$mm and $\|\vec{\bf a}_{2}\|=2$mm. The radius of both rods is $0.8$mm. (right) Power reflected from a normally (top) and obliquely (bottom, $\theta=30^{\circ}$) incident planewave. The electric field is $y$-polarized for both cases. The first result is scattering of a plane wave normally incident on a Minkowski fractal FSS. This result validates our implementation at normal incidence, and serves as a check of the non-conformal treatment of periodic boundary conditions independent of the oblique incidence framework. Fig. 3 displays an illustration of the fractal and its dimensions, and the unit cell dimensions were $\|\vec{\bf a}_{1}\|=\|\vec{\bf a}_{2}\|=30$cm. An air box was placed above and below the PEC fractal with heights of $10$cm. The DG-TD numerical results are displayed in Fig. 3. Reference data for the Minkowski fractal was drawn from [12]. The next structure is a simple dielectric slab of thickness $d=1.0$m and relative permittivity $\varepsilon_{r}=4.0$. This slab is lossless and nonmagnetic. The unit cell dimensions were chosen arbitrarily to be $\|\vec{\bf a}_{1}\|=\|\vec{\bf a}_{2}\|=0.35$m. The height of the air box above and below the slab was chosen to be $1.0$m. Fig. 4 displays the power reflected from the slab with the angle of incidence $\theta=50^{\circ}$. For this structure, we show excellent agreement between the theoretical and numerical power reflection coefficient across the frequency range. To demonstrate Figure 6: (left) Illustration of the nonmagnetic and lossless dielectric slab with periodically arranged PEC strips located at the center of the slab. The slab has a thickness of $2$mm, and the PEC strips are $2.5$mm by $5$mm, as shown in the illustration. (right) Power reflected from a normally (top) and obliquely (bottom, $\theta=30^{\circ}$) incident planewave on a nonmagnetic and lossless dielectric slab with periodically arranged PEC strips residing at the center of the slab’s thickness. The electric field is $y$-polarized for both cases. Figure 7: (left) Illustration of a single unit cell of periodically arranged dielectric slabs (outlined in black) in the $x$-direction with $\varepsilon_{r1}=2.56$ and $\varepsilon_{r2}=1.44$. The slab heights and widths were chosen based on the ratio $h/d=1.713$ and $d/2.0$, respectively. (right) Reflected power of an obliquely incident planewave ($\theta=45^{\circ}$). The electric field is $y$-polarized. the higher order accuracy of the computational framework, Fig. 4 displays the average absolute error between the numerically and theoretically calculated reflection over the frequency band. The next structure consists of two infinite PEC rods oriented in the $y$-direction. The unit cell dimensions, displayed in Fig. 5, are $8$mm by $2$mm in the $x$\- and $y$-direction, respectively. Length of the structure in the $y$-direction was chosen to reduce the number of unknowns, as it is infinite in the $y$-direction. The air boxes above and below the rods are $11$mm from the centers of the rods, and the centers of the rods were placed $8$mm apart. The radius of both rods is $0.8$mm. Fig. 5 displays the numerical results of the periodic DG-TD method compared against the numerical results of the periodic FEM-TD method. Our framework demonstrates excellent results compared to the FEM-TD framework. The effect of the planewave ABC past the next higher order Floquet mode is also captured. Our next structure is an array of PEC strips embedded in a dielectric slab. The dielectric slab is lossless and nonmagnetic, and the dimensions are shown in Fig. 6. An air box was placed above and below the dielectric slab with a height of $30$mm in the $\pm z$-direction. Reference data [10] agrees very well with the numerical results of the DG-TD code shown in Fig. 6. Again we see the effect of the planewave ABC much like the FEM-TD framework [10]. Our last validation structure consists of dielectric slabs with alternating dielectric constants. The dielectric slabs are lossless and nonmagnetic, and the unit cell is displayed in Fig. 7. Slab heights $h$ and width of the slabs $d$ are set based on the ratio $h/d=1.713$, and each slab’s width was set to $0.5d$. An air box was placed above and below the set of slabs with an arbitrarily chosen height of $0.5d$ above and $d$ below. The relative permittivity of each slab was $\varepsilon_{r1}=2.56$ and $\varepsilon_{r2}=1.44$. Results for this structure are shown in Fig. 7, with reference data drawn from [27]. Our results show good agreement with the reference data. We have shown several cases which validate this DGTD framework. The final topic of this work is addressing the stability of the explicit time integrator with respect to the planewave’s angle of incidence. The speed of Floquet modes is proportional to $cos^{-1}\theta$ [10], and therefore the CFL bound $c\Delta t\leq hP^{-2}$ is not sufficient for higher angles of incidence. The simplest solution of this problem is to scale the CFL condition as $c\Delta t=hP^{-2}V_{CFL}^{-1}$. Fig. 8 displays the smallest stable time step scale with respect to angle of incidence for a planewave passing through freespace. The unit cell dimensions for the freespace mesh were $\|\vec{\bf a}_{1}\|=\|\vec{\bf a}_{2}\|=\lambda_{min}/2$, the smallest edge length was $h=\lambda_{min}/10$, and the polynomial order was $P=2$. These parameters were held constant for each angle of incidence. The unit cell mesh was conformal with respect to the periodic boundaries. Figure 8: Angular dependence of time step scale $V_{CFL}$. Angles less than $\theta=20^{\circ}$ required unity scaling for stability. This simple result provides empirical evidence that the explicit time integration scheme is conditionally stable, even at near grazing angles of incidence. Satisfying the CFL condition at near grazing angles, however, requires scales of two orders of magnitude and thus increases the number of time steps accordingly. ## V Conclusion and Future Work In this paper, we have presented a higher-order three-dimensional Time Domain Discontinuous Galerkin Method for analyzing the interaction of obliquely incident planewaves with doubly periodic structures. We employed a field transformation to provide a formulation free from the well-known causality issues with periodic boundary conditions in time. The field transformations were applied to the first order Maxwell’s Equations, and a numerical flux was derived using an equivalent set of transformed equations. The computational framework was validated using existing results in the literature. While the particular examples elaborated in this paper employed a planewave ABC, we are currently developing an exact time domain Floquet radiation boundary condition. Future applications include the optimization of photonic band gap structures and complex frequency selective surfaces. ## VI Acknowledgment This work was supported by the National Science Foundation through grant CCF:1018576. The authors would like to thank General Electric (GE) for support, and acknowledge computing support from the HPC Center at Michigan State University, East Lansing. ## References [1] B. Munk. Frequency Selective Surfaces: Theory and Design. John Wiley & Sons, 2005. * [2] F. Yang and Y. Rahmat-samii. Electromagnetic Band Gap Structures in Antenna Engineering. 2007\. * [3] B. Munk. Metamaterials: Critique and Alternatives. John Wiley & Sons, 2009. * [4] F. Capolino. Theory and Phenomena of Metamaterials, volume 8. CRC Press, 2010. * [5] A.D. Baczewski, D.L. Dault, and B. Shanker. Accelerated Cartesian Expansions for the Rapid Solution of Periodic Multiscale Problems. IEEE Trans. Antennas Propagat., 60(9):4281–4290, 2012. * [6] A.D. Baczewski, N.C. Miller, and B. Shanker. Rapid analysis of scattering from periodic dielectric structures using accelerated Cartesian expansions. JOSA. A, 29(4):531–40, April 2012. * [7] E.W. Lucas and T.P. Fontana. A 3-D hybrid finite element/boundary element method for the unified radiation and scattering analysis of general infinite periodic arrays. IEEE Trans. Antennas Propagat., 43(2):145–153, 1995. * [8] P. Sotirelis and J.D. Albrecht. Numerical simulation of photonic crystal defect modes using unstructured grids and Wannier functions. Phys. Rev. B, 76(7):075123, August 2007. * [9] S. Chun. High-order Accurate Methods for solving Maxwell’s equations and their applications. (May), 2008. * [10] L.E.R. Petersson and J.M. Jin. Analysis of Periodic Structures via a Time-Domain Finite-Element Formulation With a Floquet ABC. IEEE Trans. Antennas Propagat., 54(3):933–944, March 2006. * [11] L.E.R. Petersson and J.M. Jin. A Three-Dimensional Time-Domain Finite-Element Formulation for Periodic Structures. IEEE Trans. Antennas Propagat., 54(1):12–19, January 2006. * [12] N.W. Chen, M. Lu, F. Capolino, B. Shanker, and E. Michielssen. Floquet wave-based analysis of transient scattering from doubly periodic, discretely planar, perfectly conducting structures. Radio Sci., 40(4), August 2005. * [13] P. Harms and R. Mittra. Implementation of the periodic boundary condition in the finite-difference time-domain algorithm for FSS structures. IEEE Trans. Antennas Propagat., 42(9):1317–1324, 1994. * [14] D.L. Dault, N. V. Nair, and B. Shanker. An O($N_{S}N_{t}log^{2}N_{t}$) method for evaluating convolutions with the time domain periodic Green’s function. In 2012 International Conference on Electromagnetics in Advanced Applications, pages 141–143. IEEE, September 2012. * [15] A.J. Pray, N.V. Nair, and B. Shanker. Stability Properties of the Time Domain Electric Field Integral Equation Using a Separable Approximation for the Convolution With the Retarded Potential. IEEE Trans. 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Phys., 181(1):186–221, September 2002. * [21] J. Niegemann, M. König, K. Stannigel, and K. Busch. Higher-order time-domain methods for the analysis of nano-photonic systems. Photonics and Nanostructures - Fundamentals and Applications, 7(1):2–11, February 2009. * [22] K. Busch, M. König, and J. Niegemann. Discontinuous Galerkin methods in nanophotonics. Laser & Photonics Reviews, 5(6):773–809, November 2011. * [23] R.J. LeVeque. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, 2002. * [24] A.H. Mohammadian, V. Shankar, and W.F. Hall. Computation of electromagnetic scattering and radiation using a time-domain finite-volume discretization procedure. Computer Physics Communications, 68(1-3):175–196, November 1991\. * [25] B.R. Vatti. A generic solution to polygon clipping. Communications of the ACM, 35(7):56–63, July 1992. * [26] M.H. Carpenter and A. Kennedy. Fourth-Order Kutta Schemes. 1994\. * [27] H.L. Bertoni and L.S. Cheo. Frequency Selective Reflection and Transmission by a Periodic Dielectric Layer. 31(1):78–83, 1989.	arxiv-papers	2013-11-04T17:48:27	2024-09-04T02:49:53.239758	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Nicholas C. Miller, Andrew D. Baczewski, John D. Albrecht, and\n Balasubramaniam Shanker", "submitter": "Nicholas Miller", "url": "https://arxiv.org/abs/1311.0790" }
1311.0838	# Electron Energy Loss Function of Silicene and Germanene Multilayers on Silver L. Rast [email protected] Applied Chemicals and Materials Division, National Institute of Standards and Technology, Boulder, CO 80305 V. K. Tewary Applied Chemicals and Materials Division, National Institute of Standards and Technology, Boulder, CO 80305 ###### Abstract We calculate electron energy loss spectra (EELS) for composite plasmonic structures based on silicene and germanene. A continued-fraction expression for the effective dielectric function is used to perform multiscale calculations of EELS for both silicene and germanene-based structures on silver substrates. A distinctive change in plasmonic response occurs for structures with a germanene or silicene surface coating of more than three layers. These differences may be exploited using spectroscopic characterization in order to determine if a few-layer coating has been successfully fabricated. ## I Introduction Silicene and germanene, two dimensional allotropes of silicon and germanium, have recently attracted attention as two-dimensional materials beyond graphene. (Bechstedt _et al._ , 2012; Chinnathambi _et al._ , 2012; Wei _et al._ , 2013; Jose, Nijamudheen, and Datta, 2013; Scalise _et al._ , 2013; Ni _et al._ , 2012) These materials possess predicted electron transport properties similar to graphene, (Tsai _et al._ , 2013) as well as the advantage of compatibility with existing silicon-based technology. Additionally, the inversion symmetry breaking imparted by the buckled lattice structure of both materials may be taken advantage of through the application of an external electrical field perpendicular to the plane for highly controllable band gap tunability. (Liu, Feng, and Yao, 2011; Liu, Jiang, and Yao, 2011; Tsai _et al._ , 2013; Ni _et al._ , 2012) The reactivity of silicene and germanene mean that they are more challenging to fabricate than graphene. Both materials bond easily with other materials and may oxidize rapidly in air, and are therefore fabricated using techniques such as epitaxial growth under ultra-high vacuum. Vogt _et al._ (2012) Electron energy loss spectroscopy (EELS) and optical absorption spectroscopy are convenient and broadly used materials characterization techniques. We demonstrate that distinctive differences in plasmonic response open up the possibility for the use of EELS or absorption spectroscopy to distinguish few- layer coating silicene or germanene on silver from bare samples as well as bulk coatings. Determination of the collective opto-electronic properties of composite multilayer structures, particularly those based on two dimensional materials, necessitate realistic treatment of both the individual material properties and the interactions among the constituent materials. (Rast, Sullivan, and Tewary, 2013) We detail and employ such a method for the calculation of electron energy loss spectra (EELS) of multilayer structures consisting of silicene and germanene layers on silver and silver/silicon substrates. This effective dielectric function is based on a specular reflection model, first derived by Lambin et al. Lambin, Vigneron, and Lucas (1985), and takes into account the boundary conditions across each layer in the stratified structure. The use of this efficient continued-fraction expression along with pre-prepared libraries of dielectric functions for the individual materials allows for extremely efficient calculation of a wide variety of configurations for multilayer composites. ## II EELS Calculation Details ### II.1 General Procedure Figure 1: Top and side views of monolayer crystal structures for (a) silicene and (b) germanene. Buckling amplitude d, obtained from the literature, is $.44\,$Åfor silicene and $.6\,$Åfor germanene. Figure 2: Multilayer structure: EELS are calculated for a variety of structures consisting of (a) silicene and germanene top layer(s) and semi-infinite Ag substrate and (b) silicene and germanene top layer(s), Ag middle layer, and SiO2/Si substrate. Individual complex dielectric functions are first obtained for each layer. The crystal structures used for the individual layer dielectric functions are depicted in 1. Then, EELS are calculated for a variety of multilayer sandwich structures as depicted in Fig. 2. We calculate the values of the silicene and germanene dielectric functions through ab-initio density functional theory (DFT) methods including excitonic effects. The buckling amplitude d, obtained from DFT structural calculations in (Scalise _et al._ , 2013), is $.44\,$Åfor silicene and $.6\,$Åfor germanene. These values are in good agreement with the values obtained by others. (Wei _et al._ , 2013) Empirical values from the literature are used for the silver and silicon substrate layers. These values are then stored for use as input to a continued-fraction algorithm, which yields the effective dielectric function. This algorithm is outlined in Section II.2. ### II.2 The Effective Dielectric Function As discussed in our previous work on graphene, (Rast, Sullivan, and Tewary, 2013) the effective dielectric function, $\xi(\omega,k,z)$, of the stratified structure in Fig. 2 is that of Lambin et al. Lambin, Vigneron, and Lucas (1985) The expression for $\xi$ was derived from EELS theory in a reflection geometry. This expression has been shown to be applicable to both phonons Lambin, Vigneron, and Lucas (1985) and polaritons Dereux _et al._ (1988) in stratified structures with histogram-like dielectric functions (continuous within each layer) and interacting interfaces. The expression and the formalism from which it is derived were first appplied to describing composite dielectric functions semiconducting materials, but are also applicable to surface plasmon resonance and phonon behavior of alternating of metal- insulator layers. Dereux _et al._ (1988); Economou (1969) This model has been shown to be in good agreement with the well-known Bloch hydrodynamic model in the small wave vector regime considered in this work. Ritchie and Marusak (1966) The $z$ coordinate is in the direction perpendicular to the free surface of the sample, extending from the $z=0$ surface to $-\infty$. $\mathbf{k}$ denotes the surface excitation (plasmon or phonon) wave vector and $\omega$ is the frequency of excitation. $\xi(\mathbf{k},\omega,z)=\frac{i\mathbf{D}(\mathbf{k},\omega,z)\cdot\mathbf{n}}{\mathbf{E}(\mathbf{k},\omega,z)\cdot\mathbf{k}/k},$ (1) where $\mathbf{D}(\mathbf{k},\omega,z)=\epsilon(\omega,z)\mathbf{E}(\mathbf{k},\omega,z)$, and $\epsilon(\omega,z$) is the long wavelength dielectric function (tensor) of the material at $z$. $\xi$ remains continuous even in the case of sharp interfaces parallel to the $x$-$y$ directions below the surface (as is the case in our multilayer system). This is due to the interface boundary conditions: continuity of $D_{\perp}$ and $E_{\parallel}$. The effective dielectric function $\xi_{0}(k,\omega)$ (Eq. 3) is a solution to the Riccati equation (Eq. 2), in the long-wavelength approximation $k\approx 0$, at the $z=0$ surface. Lambin, Vigneron, and Lucas (1985) We fix $k$ as $k=0.005\,$Å-1 for both the ab-initio calculations and the composite calculation. Eq. 2 was derived for heterogeneous materials made of a succession of layers (with homogeneous dielectric functions within each layer), the layers having parallel interfaces. $\epsilon(z)$ are complex functions, with positive imaginary parts at $z=0$. Lambin, Vigneron, and Lucas (1985) $\displaystyle\frac{1}{k}\frac{\mathrm{d}\xi(z)}{\mathrm{d}z}+\frac{\xi^{2}(z)}{\epsilon(z)}=\epsilon(z)$ (2) $\xi_{0}=a_{1}-\frac{b_{1}^{2}}{a_{1}+a_{2}-\frac{b_{2}^{2}}{a_{2}+a_{3}-\frac{b_{3}^{2}}{a_{3}+a_{4}-\cdots}}}$ (3) where $\displaystyle a_{i}=\epsilon_{i}\coth(kd_{i})$ (4) and $\displaystyle b_{i}=\epsilon_{i}/\sinh(kd_{i}).$ (5) Once individual dielectric functions are obtained, this procedure allows for the performance of mesoscale EELS calculations of a wide variety of layered structures. Layer thickness and material are easily substituted, with each EELS calculation running in a less than a second on a single processor (nearly independent of the spectral range). EELS are calculated directly from the effective dielectric function as $\mathrm{EELS}=\mathrm{Im}\left[\frac{-1}{\xi(\omega,k)+1}\right].$ (6) Inspection of Eq. 3 reveals that for $\mathrm{Im}[\epsilon_{i}]>0$, $\mathrm{Im}[\xi_{0}]>0$. EELS spectra given by Eq. 6 are then generally positive-valued. ### II.3 Silver Dielectric Function The silver dielectric functions are empirical values by Johnson and Christy Johnson and Christy (1972) obtained by reflection and transmission spectroscopy on vacuum-evaporated films at room temperature. Film-thickness in the Johnson and Christy study ranged from $185\,$Å– $500\,$Å. It was found that optical constants in the film-thickness range $250\,$Å – $500\,$Å did not vary appreciably. As in our previous work, (Rast, Sullivan, and Tewary, 2013) $340\,$Å film thickness is representative of bulk mode dominant (yet still nanoscale) metallic thin films. ### II.4 SiO2 and Si Dielectric Constants Relative static permittivities of 3.9 and 11.68 were chosen for the SiO2 and Si dielectric constants, respectively. These are reasonable and widely-used values obtained from the literature. Murarka (2003); Yi (2012) ### II.5 Silicene and Germanene Individual Layer Dielectric Functions Complex dielectric functions for silcene and germanene are displayed in Fig. 3 (a) and Fig. 3 (b), respectively. These ab-initio calculations use the time- dependent DFT with a GLLBSC exchange correlation functional, (Kuisma _et al._ , 2010) and are implemented in the Python code GPAW, a real-space electronic structure code using the projector augmented wave method. 111Certain commercial equipment, instruments, or materials are identified in this paper in order to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose., Mortensen, Hansen, and Jacobsen (2005); Enkovaara _et al._ (2010); Walter _et al._ (2008); Yan _et al._ (2011) Both silicene and germanene dielectric functions are calculated in the optical limit with a momentum transfer value of $0.005\,$Å-1, along the $\bar{\Gamma}$-$\bar{M}$ direction of the surface Brillouin zone. The $k$-point sampling with $20\times 20\times 1$ Monkhorst–Pack grid was chosen for the band-structure and EELS calculations for both silicene and germanene. We have chosen to employ both the GLLBSC functional and the Bethe Salpeter Equation (BSE) in order to calculate the individual layer dielectric functions due to the extreme accuracy of this method in predicting experimental values of dielectric functions and bandgaps for similar materials, such as a variety bulk semiconductors including silicon as well two dimensional materials graphene and hexagonal boron nitride. Yan, Jacobsen, and Thygesen (2012)The GLLBSC potential explicitly includes the derivative discontinuity of the xc-potential at integer particle numbers, critical for obtaining physically meaningful band structure via a DFT calculation. This functional has also been shown to have computational cost similar to the Local Density Approximation (LDA) with accuracy similar to methods such as the LDA-GW method. Yan, Jacobsen, and Thygesen (2012, 2011) The use of the BSE is important due to the inclusion of excitonic effects, an prominent spectral feature for both materials. (Wei _et al._ , 2013) A two dimensional Coulomb cutoff (Rozzi _et al._ , 2006) is employed in order to calculate the diectric function of the silicene and germanene monolayers. Our model utilizes dielectric functions due to surface parallel excitations only, as the effective dielectric function is derived in a specular reflection geometry. The dielectric functions we have obtained for silicene and germanene (see Fig. 3 (a)) agree well with previous calculations in the literature Chinnathambi _et al._ (2012); Bechstedt _et al._ (2012), and has particularly good agreement with the spectral profiles and peak positions in Wei _et al._ (2013), where the authors used the BSE to include excitonic effects. Figure 3: Complex relative dielectric function $\epsilon(\omega)$ for silicene (a) and germanene (b). Real and imaginary parts ($\epsilon^{\prime}(\omega)$ and $\epsilon^{\prime\prime}(\omega)$) are represented by solid and dotted lines, respectively. ## III RESULTS ### III.1 Silicene and Germanene on Silver/SiO2/Si Substrates: Varying the Noble Metal Layer Thickness Figure 4: EELS for single-layer silicene on Ag/SiO2/Si substrate : The effect of differing thickness for the Ag layer is demonstrated. The Ag layer thicknesses are $500\,$Å (solid line), $340\,$Å (long dashes), $200\,$Å (short dashes), $100\,$Å (dash-dot), and $40\,$Å (dotted line). Figure 5: EELS for single-layer germanene on Ag/SiO2/Si substrate : The effect of differing thickness for the Ag layer is demonstrated. The Ag layer thicknesses are $500\,$Å (solid line), $340\,$Å (long dashes), $200\,$Å (short dashes), $100\,$Å (dash-dot), and $40\,$Å (dotted line). Figures 4 and 5 demonstrate the effect of decreasing silver metallic layer thickness. As the Ag thickness is reduced, the so called _begrenzung effect_ is apparent. Enhanced surface-to-volume ratio in the metal causes stronger coupling to the surface resonance and decreased coupling to the bulk modes. Ritchie (1957); Osma and Garcia de Abajo (1997) In the case of a thin metallic slab, empirical models have been thoroughly explored. Upon the introduction of a boundary to an infinite metallic slab, a negative (begrenzung) peak is introduced at the same energy as the bulk peak, and a trailing surface peak appears. Ritchie (1957) The surface peak becomes more intense with decreasing thickness, as does the negative begrenzung peak, decreasing the net bulk-plasmon amplitude. Surface modes become dominant for silver thickness between $20\,$ and $10\,$nm. This is consistent with observations in the well-validated and widely-used empirical data by Johnson and Christy Johnson and Christy (1972) as well as observations in our previous work on graphene/noble metal multilayer systems. Rast, Sullivan, and Tewary (2013) Comparison with experimental results for very thin silver layers provides further verification of the model for a wide variety of silver metal layer thicknesses. Both bulk and surface peak locations and relative intensities for $4\,$nm are in excellent agreement with experimental results for EELS of $3.4\,$nm silver layers. (Nagao _et al._ , 2007) ### III.2 Silicene and Germanene Multilayers on Silver Figure 6: EELS for multilayers of silicene on a semi-infinite Ag substrate : The effect of differing numbers of silicene layers is demonstrated. The silicene layer numbers are 20 (solid line), 10 (long dashes), 3 (short dashes), 1 (dash-dot), and 0 (dotted line). Figures 6 and 7demonstrate the effect of varying numbers of silicene and germanene layers on a silver substrate, respectively. For up to three layers of silicene on silver, the bulk plasmon peak is diminished without significant broadening. This indicates an overall reduction in bulk losses. The effect is most notable when the silver slab is coated with a single layer of silicene, an effect which would be useful for determining successful fabrication of monolayer silicene on silver through spectroscopic characterization. At 10 layers and above, the system approaches the expected behavior for a bulk Si/Ag system, with a broad interfacial peak appearing at about 2.5 eV. (Rast, Sullivan, and Tewary, 2013) This peak broadens further and is enhanced in intensity with increasing number of silicene layers. Referring to Figure 3 (a), it is also apparent that at roughly 2.5 eV, the silicene dielectric function real part changes sign, and becomes increasingly positive up to nearly 4 eV. The silver diectric function real part is very negative in this regime, so the interfacial plasmon is expected at this energy. This is in contrast to the germanene dielectric function, which is only momentarily slightly positive (Figure 3 (b)) in this regime. As a result, figure 7 demonstrates that there is no well-defined interfacial plasmon for the germanene/silver system. Damping of the silver bulk plasmon for a few layers of two dimensional material, however, occurs in a very similar manner for the germanene/silver and silicene/silver systems. As in the case of the silicene/silver system, for 1-3 layers of germanene on silver, the bulk plasmon is diminished to a great extent without significant broadening of the peak. Figure 7: EELS for multilayers of silicene on a semi-infinite Ag substrate : The effect of differing numbers of silicene layers is demonstrated. The silicene layer numbers are 20 (solid line), 10 (long dashes), 3 (short dashes), 1 (dash-dot), and 0 (dotted line). ## IV DISCUSSION In this study we investigated the effect of varying numbers of silicene and germanene layers on an Ag substrate. For mono-, bi-, and tri-layer coatings of both silicene and germanene, bulk plasmon modes are significantly diminished without significant broadening, which would correspond to increased plasmonic losses. The significance of this is two-fold: (1) This marked reduction in bulk peak intensity should be of use for characterization of few-layer silicene and germanene systems on silver, as a few layers of either material leads to a diminishing of the bulk plasmon peak without a significant broadening or shift in peak position. In the case of silicene, an additional interfacial peak occurs and is enhanced for more than 10 layers, an indication that the silicene coating is approaching bulk thickness. In the case of germanene, the silver bulk plasmon is quenched at 20 layers. The obvious differences in behavior for uncoated, few-layer-coated systems, and and bulk coatings are useful as simple guidelines in the fabrication of these new materials.(2) The boundary physics for silicene and germanene, within the context of our mesoscopic model, is similar to our findings for graphene (Rast, Sullivan, and Tewary, 2013) — The addition of a graphene boundary layer on the metallic surface reduces coupling of excitations to bulk plasmons through the begrenzung effect. The origin of the begrenzung effect is a reduction of the degrees of freedom for excitations, and thus further surface confinement comes at the expense of bulk oscillations, leading to reduced losses. The mesoscopic model used in these calculations has some limitations that merit discussion. Results of this study are valid in the long-wavelength limit for which the continued fraction expression by Lambin et al. was derived. Additionally, coupling between layers is classical (via boundary conditions), and as a result inter-layer hopping is neglected. However, at least in the case of bilayer silicene, it has been shown that inter-layer hopping can be neglected. (Rui, Shaofeng, and Xiaozhi, 2013) It has been argued that the buckled silicene geometry, arising from mixing sp2 and sp3 hybridization, blocks interlayer hopping in bilayer silicene, thus preserving Dirac-type dispersion. (Rui, Shaofeng, and Xiaozhi, 2013) If this explanation is correct, the same argument may also apply to germanene bi-layers. In future work we plan to incorporate the effect of lattice strain on the optical properties of the composite for two reasons: (1) strain engineering is expected to provide a further means of plasmon tuning, Sciammarella _et al._ (2010) and (2) due to inherent lattice mismatch even in systems with epitaxial growth, strain effects are generally of interest for accurate prediction of plasmonic features in two dimensional and quasi-two dimensional material-based heterostructures. ###### Acknowledgements. The authors would like to thank Katie Rice, Ann Chiaramonti Debay, and Alex Smolyanitsky for helpful discussions. This research was performed while the first author held a National Research Council Research Associateship Award at the National Institute of Standards and Technology. 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B 83, 245122 (2011). * Yi (2012) Yi, G.-C., ed., _Semiconductor Nanostructures for Optoelectronic Devices: Processing, Characterization, and Applications_ (Springer, 2012).	arxiv-papers	2013-11-04T20:35:44	2024-09-04T02:49:53.248550	{ "license": "Public Domain", "authors": "L. Rast and V.K. Tewary", "submitter": "Lauren Rast", "url": "https://arxiv.org/abs/1311.0838" }
1311.1155	# Electronic properties of Mn decorated silicene on hexagonal boron nitride T. P. Kaloni1, S. Gangopadhyay2, N. Singh1, B. Jones2, and U. Schwingenschlögl1, [email protected],+966(0)544700080 1 PSE Division, KAUST, Thuwal 23955-6900, Kingdom of Saudi Arabia 2 IBM Almaden Research Center, San Jose, California 95120-6099, USA ###### Abstract We study silicene on hexagonal boron nitride, using first principles calculations. Since hexagonal boron nitride is semiconducting, the interaction with silicene is weaker than for metallic substrates. It therefore is possible to open a 50 meV band gap in the silicene. We further address the effect of Mn decoration by determining the onsite Hubbard interaction parameter, which turns out to differ significantly for decoration at the top and hollow sites. The induced magnetism in the system is analyzed in detail. Silicon based nanostructures, such as two-dimensional silicene (an analogue to graphene) and silicene nanoribbons are currently attracting the interest of many researchers due to materials properties that are similar to but richer than those of graphene verri ; olle . Moreover, they are advantageous to carbon based nanostructures, as they can be expected to be compatible with the existing semiconductor industry. It is observed that the electronic band structure of silicene shows a linear dispersion around the Dirac point, like graphene, and hence is a candidate for applications in nanotechnology. Due to an enhanced spin orbit coupling a band gap 1.55 meV is opened yao . A mixture of $sp^{2}$ and $sp^{3}$-type bonding results in a buckled structure, which leads to an electrically tunable band gap falko ; Ni . First-principles geometry optimization and phonon calculations as well as temperature dependent molecular dynamics simulations predict a stable low-buckled structure ciraci . Moreover, stability of silicene under biaxial tensile strain has been predicted up to 17% strain kaloni-jap . Silicene and its derivatives experimentally have been grown on Ag and ZrB2 substrates padova ; vogt ; ozaki , though there is still discussion about the quality of the results Lin . On a ZrB2 thin film an asymmetric buckling due to the interaction with the substrate is found, which opens a band gap. In general, transition metal decorated graphene has been studied extensively both in experiment and theory. It has been predicted that 5$d$ transition metal atoms show unique properties with topological transport effects. The large spin orbit coupling of 5$d$ transition metal atoms together with substantial magnetic moments leads to a quantum anomalous Hall effect blugel . A model study also has predicted the quantum anomalous Hall effect for transition metal decorated silicene nanoribbons ezawa . Energy arguments indicate that transition metal atoms bond to silicene much stronger than to graphene. As a result, a layer by layer growth of transition metals could be possible on silicene Ni1 The deposition of isolated transition metal atoms on layers of hexagonal boron nitride on a Rh(111) substrate has been studied in Ref. fabian . The authors have demonstrated a reversible switching between two states with controlled pinning and unpinning of the hexagonal boron nitride from the metal substrate. In the first state the interaction of the hexagonal boron nitride is reduced, which leads to a highly symmetric ring in scanning tunneling microscopy images, while the second state is imaged as a conventional adatom and corresponds to normal interaction. Motivated by this work, we present in the following a first-principles study of the transition metal decoration of silicene on hexagonal boron nitride. We will first address the interaction with the substrate and then will deal with the electronic and magnetic properties of the Mn decorated system, All calculations have been carried out using density functional theory in the generalized gradient approximation. We employ the Quantum-ESPRESSO package paolo , taking into account the van-der-Waals interaction grime . The calculations are performed with a plane wave cutoff energy of 816 eV, where a Monkhorst-Pack $8\times 8\times 1$ k-mesh is used to optimize the crystal structure and to obtain the self-consistent electronic structure. The atomic positions are relaxed until an energy convergence of 10-7 eV and a force convergence of 0.001 eV/Å are reached. To study the interaction of the silicene with the substrate, we employ a supercell consisting of a $2\times 2$ supercell of silicene on top of a $3\times 3$ supercell of hexagonal boron nitride. We have tested the convergence of the results with respect to the thickness of the substrate by taking into account 2, 3, 4, and 6 atomic layers of $h$-BN, finding only minor differences (in particular concerning the splitting and position of the Dirac cone) because of the inert nature of the substrate. A thin substrate consequently turns out to be fully sufficient in the calculations. Moreover, the $2\times 2$ supercell of silicene fits well on the $3\times 3$ supercell of the substrate with a lattice mismatch of only 2.8%. When we consider Mn decorated silicene we use a larger supercell that contains 16 Si in a layer over 18 B and 18 N. While the onsite Hubbard parameter for 3$d$ transition metal atoms is known to be several eV, we explicitly calculate the value in the present study for the different adsorption sites in order to obtain accurate results for the electronic and magnetic properties of the Mn decorated system. In general, the lattice mismatch of 2.8% between silicene and hexagonal boron nitride can be expected to be small enough to avoid experimental problems with a controlled growth. Moreover, accurate measurements of materials properties can be difficult to achieve on metallic substrates, whereas the interaction is reduced on semiconducting substrates. Our calculated binding energy for the interface between silicene and hexagonal boron nitride is only 100 meV per Si atom, as compared to typically 500 meV per Si atom for an interface to a metallic substrate. Experimental realizations of graphene based electronic devices using hexagonal boron nitride as substrate on a Si wafer support are subject to various limitations, such as a poor on/off ratio kim . However, on this substrate graphene exhibits the highest mobility dean and a sizable band gap Gweon ; Ruge ; jmc . Since silicene resembles the structure of graphene, synthesis on hexagonal boron nitride thus has great potential. The structural arrangement of the system under study is depicted in Fig. 1(b), together with the charge redistribution introduced by the interaction with the substrate. We obtain Si$-$Si bond lengths of 2.24 Å to 2.26 Å and a buckling of 0.48Å to 0.54 Å, which is slightly higher than in free-standing silicene yao ; ciraci . For the angle between the Si$-$Si bonds and the normal of the silicene sheet we observe values of 113∘ to 115∘, again close to the findings for free-standing silicene (116∘). The optimized distance between the silicene and hexagonal boron nitride sheets forming the interface turns out to be 3.57 Å, which is similar to the distance at the contact between graphene and hexagonal boron nitride. In addition, the interlayer distance within the hexagonal boron nitride amounts to be 3.40 Å, whereas in a bilayer configuration values of 3.30 Å to 3.33 Å have been reported Marini ; shi . The interaction between silicene and hexagonal boron nitride recently has been addressed by Liu and coworkers Zhao , who have reported a perturbation of the Dirac cone with an energy gap of 4 meV. This study has taken into account only a single layer of hexagonal boron nitride as substrate, so that a more realistic description may yield a different result. Indeed, we observe a perturbed Dirac cone with an energy gap of 50 meV in the band structure shown in Fig. 1(a). The $\pi$ and $\pi^{}$ bands forming the Dirac cone are due to the $p_{z}$ orbitals of the Si atoms, while the bands related to the B and N atoms are located about 0.5 eV above and 1 eV below the Fermi energy. We find a small but finite charge redistribution across the interface to the substrate; see the charge density difference isosurfaces plotted in Fig. 1(b). As a result the Dirac cone is perturbed and the 50 meV energy gap is realized, which can be interesting for nanoelectronic device applications, in particular because an external electric field can be used to tune the gap. The isosurface plot also demonstrates that the Si atom closest to a B atom is subject to the strongest charge transfer, while for all other Si atoms charge transfer effects are subordinate due to longer interatomic distances. Figure 1: (a) Electronic band structure and (b) charge transfer for silicene on a bilayer of BN (side view). The isosurfaces correspond to isovalues of $\pm 5\times 10^{-4}$ electrons/Å3. The black, blue, and red spheres denote Si, N, and B atoms, respectively. Red and blue isosurfaces refer to positive and negative charge transfer. Note the significant charge transfer of the Si closest to the BN. The possible decoration sites for a Mn atom on silicene can be classified as top, bridge, and hollow. Decoration at the bridge site is not considered in the following because the Mn atom immediately transfers to the top site. A side view of the relaxed structure for Mn decoration at the top site is given in Fig. 2(a), together with a spin density map. We obtain the onsite interaction parameter using a constraint-GGA method epl , and calculate the values of 3.8 eV for the Mn atom at the top site and 4.5 eV for the Mn atom at the hollow site. For the top site configuration, structural optimization reveals that the Mn atom moves close to an original Si position and thereby strongly displaces this Si atom, resulting in a short Mn$-$Si bond length of 2.43 Å. Moreover, the Mn atom is bound to three Si atoms with equal bond lengths of 2.45 Å. Si$-$Si bond lengths of 2.24 Å to 2.28 Å are observed, which corresponds to a slight modification as compared to the pristine configuration. The buckling of the silicene, on the other hand, is strongly altered, now amounting to 0.45 Å to 0.67 Å. Accordingly, angles of 113∘ to 117∘ are found between the Si$-$Si bonds and the normal of the silicene sheet. The height of the Mn atom above the silicene sheet is 1.30 Å. Finally, we note that the separation between the atomic layers of the hexagonal boron nitride is virtually not modified by the Mn decoration. A side view of the relaxed structure for Mn decoration on the hollow site is shown in Fig. 2(b). In this case, the Mn atom does not displace a specific Si atom but stays close to the center of the Si hexagon. It is bound equally to the neighboring Si atoms with bond lengths of 2.40 Å to the upper three and 2.77 Å to the lower three Si atoms. A buckling of 0.46 Å to 0.62 Å, Si$-$Si bond lengths of 2.23 Å to 2.28 Å, and angles to the normal of 112∘-117∘ are obtained. The Mn atom is located 1.01 Å above the silicene sheet and the separation between the atomic layers in the hexagonal boron nitride is slightly increased to 3.44 Å. In contrast, the distance between silicene and substrate here amounts to 3.55 Å and thus is significantly larger than in the case of decoration at the top site, because in the latter case one Si atom is displaced from the silicene sheet, which modifies the distance to the substrate. The calculated total energies indicate that decoration at the hollow site is by 33 meV favorable as compared to decoration at the top site. We find total magnetic moments of 4.56 $\mu_{B}$ and 3.50 $\mu_{B}$ per supercell for Mn decoration at the top and hollow sites, a reduction of spin from the free Mn value of 5.0 unpaired electrons. The magnetization reduction is notable on the hollow site, which can be seen from Fig. 2 and 3 to involve greater immersion in and hybridization with the Si than the top site. By far the largest contribution to the magnetic moment comes from the Mn atom and only small moments are induced on the Si atoms. This can be clearly seen in the spin density maps presented in Figs. 3(a) and (b). For Mn decoration at the top site we obtain a Mn moment of 4.40 $\mu_{B}$ and a total of 0.16 $\mu_{B}$ from all the Si atoms, whereas for decoration at the hollow site the Mn moment amounts to 4.16 $\mu_{B}$ and the Si atoms contribute a total $-0.66$ $\mu_{B}$. These results indicate that the Mn and Si moments are ordered ferromagnetically and antiferromagnetically for decoration at the top and hollow sites, respectively. Figure 2: The spin density map for silicene decorated by Mn at the (a) top and (b) hollow site of the $h$-BN substrate. The hollow site is energetically favorable. In Fig. 3 we address the density of states (DOS) for decoration at the (a) top and (b) hollow sites. The left panel of the figure shows the total DOS and the right panel the partial DOSs of the Mn $3d$ and $4s$ orbitals. In contrast to pristine silicene (band gap of 1.55 meV yao ), the DOSs show a region without states around 0.5 eV below the Fermi energy. This observation corresponds to an $n$-doping due to the mentioned charge transfer from Mn to silicene. Closer inspection of the partial DOSs for decoration at the top site shows that the spin majority $s$ and $d_{3r^{2}-z^{2}}$ as well as the spin minority $d_{3r^{2}-z^{2}}$, $d_{x^{2}-y^{2}}$, and $d_{xy}$ states contribute in the vicinity of the Fermi energy, while there are essentially no contributions from the $d_{zx}$ and $d_{zy}$ states. A sharp Mn peak is obsvered about 0.8 eV, which is due to the spin minority $d_{3r^{2}-z^{2}}$ states. For decoration at the hollow site almost exclusively the spin majority $s$ and spin minority $d_{x^{2}-y^{2}}$ and $d_{xy}$ states contribute around the Fermi energy. Two less pronounced DOS peaks appear 0.75 eV below the Fermi energy, contributed by the spin minority $d_{xy}$ and $d_{x^{2}-y^{2}}$ states. Figure 3: Total (left) and Mn partial (right) densities of states of silicene decorated by Mn at the (a) top and (b) hollow site of the $h$-BN substrate. In conclusion, we have employed density functional theory to discuss the structure and chemical bonding of silicene on hexagonal boron nitride. The interaction results in a band gap of 50 meV. Furthermore, we have calculated the onsite Hubbard interaction parameter for Mn decoration at the top and hollow sites of the silicene, finding values of 3.8 eV and 4.5 eV, respectively. The electronic and magnetic properties of Mn decorated silicene have been studied in detail. In particular, magnetic moments of 3.50 $\mu_{B}$ and 4.56 $\mu_{B}$, respectively, have been obtained for Mn decoration at the top and hollow sites. Interestingly, the orientation between the Mn and induced Si moments is ferromagnetic in the former and antiferromagnetic in the latter case. ## References (1) G. G. Guzmán-Verri and L. C. L. Y. Voon, Phys. Rev. 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1311.1222	# A Census of X-ray gas in NGC 1068: Results from 450ks of $Chandra$ HETG Observations T. Kallman NASA/GSFC Daniel A. Evans Harvard-Smithsonian CfA H. Marshall, C. Canizares, M. Nowak, N. Schulz MIT A. Longinotti European Space Astronomy Center of ESA, Madrid, Spain ###### Abstract We present models for the X-ray spectrum of the Seyfert 2 galaxy NGC 1068. These are fitted to data obtained using the High Energy Transmission Grating (HETG) on the $Chandra$ X-ray observatory. The data show line and radiative recombination continuum (RRC) emission from a broad range of ions and elements. The models explore the importance of excitation processes for these lines including photoionization followed by recombination, radiative excitation by absorption of continuum radiation and inner shell fluorescence. The models show that the relative importance of these processes depends on the conditions in the emitting gas, and that no single emitting component can fit the entire spectrum. In particular, the relative importance of radiative excitation and photoionization/recombination differs according to the element and ion stage emitting the line. This in turn implies a diversity of values for the ionization parameter of the various components of gas responsible for the emission, ranging from log($\xi$)=1 – 3. Using this, we obtain an estimate for the total amount of gas responsible for the observed emission. The mass flux through the region included in the HETG extraction region is approximately 0.3 M⊙ yr-1 assuming ordered flow at the speed characterizing the line widths. This can be compared with what is known about this object from other techniques. ## 1 Introduction X-ray spectra demonstrate that many compact sources are viewed through partially ionized gas. This gas manifests itself as a rich array of lines and bound-free features in the 0.1 - 10 keV energy range. The existence of this ‘warm absorber’ gas and the fact that it often shows Doppler shifts indicating outflows, has potential implications for the mass and energy budgets of these sources. These features have been used to infer mass outflow rates for many accreting sources, notably those in bright, nearby active galactic nuclei (AGN) (Turner & Miller, 2009; Crenshaw et al., 2003; Miller et al., 2009; Krongold et al., 2003; Kaastra et al., 2011; Kaspi et al., 2002). Intrinsic to the problem is the fact that the gas is not spherically distributed around the compact object. If it were, resonance lines would be absent or would have a P-Cygni character. P-Cygni profiles are observed in X-ray resonance lines from the X-ray binary Cir X-1 (Schulz et al., 2008); most AGN do not show this behavior clearly. Understanding the quantity of warm absorber gas, its origin and fate, are key challenges. Interpretation of absorption spectra is often based on the assumption that the residual flux in the line trough is solely due to finite optical depth or spectral resolution. That is, the effect of filling in by emission is neglected. Understanding the possible influence of emission is desirable, both from this point of view and since emission provides complementary information about a more extended region. Comparison between warm absorbers and the spectra of objects in which the gas is viewed in reflection rather than transmission provides an added test for our understanding of the geometry of the absorbing/emitting gas. Emission spectra also are possibly less affected by systematic errors associated with the observation, such as internal background or calibration errors which could affect the residual flux in the core of deep absorption features. Emission spectra are observed from compact objects when the direct line of sight to the central compact object is blocked. This can occur in X-ray binaries or in Seyfert galaxies where the obscuration comes from an opaque torus (Antonucci & Miller, 1985). A notable example is in Seyfert 2 galaxies, and the brightest such object is NGC 1068 (Bland-Hawthorn et al., 1997). This object has been observed by every X-ray observatory with sufficient sensitivity, most recently by the grating instruments on $Chandra$ and $XMM- Newton$. These reveal a rich emission line spectrum, including lines from highly ionized medium-Z elements, fluorescence from near neutral material, and radiative recombination continua (Liedahl et al., 1990) (RRCs) which are indicative of recombination following photoionization. Apparent emission can be caused by various physical mechanisms. These include cascades following recombination, electron impact excitation, inner shell fluorescence, and radiative excitation by absorption of the continuum from the central object in resonance lines. Radiative excitation produces apparent emission as a direct complement to the line features seen in warm absorbers; rates can exceed that due to other processes. It is also often referred to as resonance scattering, since it is associated with scattering of continuum photons in resonance lines. However, past treatments in the context of Seyfert galaxy X-ray spectra have been limited to consideration of a single scattering of an incident continuum photon by a resonance line, after which the photon is assumed to be lost. Since there is a considerable literature on the topic of transfer of photons in resonance lines, also sometimes referred to as resonance scattering, here and in what follows we will not use this term. Since radiative excitation preferentially affects lines with large oscillator strengths arising from the ground term of the parent ion, it affects line ratios such as the $n=$1 – 2 He-like lines, causing them to resemble the ratios from coronal plasmas. This was pointed out by Kinkhabwala et al. (2002), who presented high resolution spectra of NGC 1068 obtained using the reflection grating spectrometer (RGS) on the $XMM-Newton$ satellite. The RGS is most sensitive at wavelengths greater than approximately 10 $\AA$, and Kinkhabwala et al. (2002) showed that the He-like lines from N and O in NGC 1068 are affected by radiative excitation. They also showed that, since radiative excitation depends on pumping by continuum radiation from the central compact source, this process is affected by the attenuation of the continuum. This in turn depends on the column density of the line emitting gas; radiative excitation is suppressed when the continuum must traverse a high column density. Kinkhabwala et al. (2002) adopted a simple picture in which the gas is assumed to be of uniform ionization and opacity and were able to then constrain the column density in the NGC 1068 line emitting region from the observed line ratios. The High Energy Transmission Grating (HETG) on the $Chandra$ satellite is sensitive to wavelengths between 1.6 – 30 $\AA$, allowing study of the He-like ratios from elements Ne, Mg, Si and heavier, in addition to those from O. Ogle et al. (2003) used an HETG observation of NGC 1068 to show that the He-like ratios from Mg and Si show a stronger signature of radiative excitation than do O and Ne. They interpret this as being due to the fact that all the lines come from the same emitting region with a high column, and that attenuation of the continuum is stronger for Mg and Si owing to ionization effects. That is, that O and Ne are more highly ionized than the heavier elements, and point out that this is consistent with the results of photoionization models. The challenge of modeling X-ray emission spectra divides into parts: (i) modeling the excitation process which leads to emission; (ii) modeling the ionization of the gas; and (iii) summing over the spatial extent of the emitting gas leading to the observed spectrum. Kinkhabwala et al. (2002) include a detailed treatment of the population kinetics for emitting ions which takes into account the effect of the continuum radiation from the compact object on the populations, along with the attenuation of this radiation in a spatially extended region. Ogle et al. (2003) add to this a treatment of the ionization balance. Matt et al. (2004) have used the intensities of lines formed by fluorescence to infer the covering faction of low ionization, high column density gas and shown that this is consistent with the properties of the obscuring torus. A shortcoming of efforts so far is that none self-consistently treats the spatial dependence of the absorption effects affecting radiative excitation together with the corresponding effects on the ionization balance. That is, models such as Ogle et al. (2003) adopt temperature and ionization balance associated with optically thin photoionized gas. They then assume that this ionization is constant throughout the cloud and use this to calculate continuum attenuation and its effects on the population kinetics. They did not consider the spatial dependence of the ionization of the gas associated with attenuation of the incident continuum, along with the associated suppression of radiative excitation. Since the work of Kinkhabwala et al. (2002) and Ogle et al. (2003) there have been numerous studies of absorption dominated warm absorber spectra. In addition, a large campaign with the $Chandra$ HETG was carried out on NGC 1068 in 2008, resulting in a dataset with unprecedented statistical accuracy (Evans et al., 2010). For these reasons, and the reasons given above, we have carried out new modeling of the photoionized emission spectrum of NGC 1068. The questions we consider include: To what extent can photoionized emission models fit the observations? Which structures, known from other studies, can account for the observed X-ray line emission? Are there patterns in the elemental abundances in the photoionized emission spectrum which provide hints about the origin or fate of the X-ray gas? What mass of gas is associated with the X-ray emission, and what flow rate does this imply? In section 2 we present the observed spectrum and in section 3 we describe model fitting. A discussion is in section 4. ## 2 Data The dataset that we use in this paper consists of an approximately 400 ksec observing campaign on NGC 1068 during 2008 using the $Chandra$ HETG. A description of these observations was reported briefly in Evans et al. (2010), but no detailed description of the data has been published until now. In addition, we also incorporate the earlier 46 ksec observation with the same instrument which was carried out in 2000 and was published by Ogle et al. (2003). A log of the observations is given in table 1. The satellite roll angles during the observations were in the range 308 – 323 degrees. This corresponds to the dispersion direction being approximately perpendicular to the extended emission seen in both optical and X-rays (Young et al., 2001). The standard HEG and MEG extraction region size is 4.8 arcsec on the sky. This is comparable to the extent of the brightest part of the X-ray image, which is approximately 6 arcsec. It is possible to extract the spectra from regions which are narrower or wider in the direction perpendicular to the dispersion, and we will discuss this below. The spectrum was extracted using the standard Ciao tools, including detection of zero order, assigning grating orders, applying standard grade filters, gti filters, and making response files. The dispersion axes for all the observations were nearly perpendicular to the axis of the narrow line region (NLR; position angle $\simeq$ 40 degrees). As pointed out by Ogle et al. (2003) the width of the nuclear emission region in the dispersion direction for the zeroth order image is 0.81 – 0.66 arcsec, which corresponds to a smearing of FWHM = 0.015-0.018 Å over the 6-22 Å range in addition to the instrumental profile ( FWHM = 0.01, 0.02 Å for HEG, MEG). Events were filtered by grade according to standard filters, streak events were removed. First- order HEG and MEG spectra of the entire dataset were extracted using CIAO 4.4. Positive and negative grating orders were added. In this paper we analyze data extracted using the standard size extraction region from the HETG, which is 1.3 $\times 10^{-3}$ degrees or 4.8 arcsec. Significant line emission does originate from the ‘NE cloud’, located 3.2 arcsec from the nuclear region, and was discussed by Ogle et al. (2003). This emission is a factor 2-3 times weaker than the emission centered on the nucleus. In section 2.2 we briefly discuss analysis of spatially resolved data, i.e. data from smaller or larger extraction regions. A more complete examination of the spatial dependence of the spectrum will be carried out in a subsequent paper. We fit the spectrum with a model consisting of an absorbed power law continuum plus Gaussian lines. Table 2 lists 86 lines, of which 67 are distinct features detected using the criteria described below. Gaussian line fitting was carried out using an automated procedure which fits the spectrum in 2$\AA$ intervals. These are chosen for convenience: narrow enough that the continuum can be approximated by power law and containing a small number of features. Adjacent intervals are chosen to overlap in order to avoid artifacts associated with boundaries. Within each interval we fit a local power law continuum plus Gaussians. The Gaussians are added to the model at wavelengths corresponding to known lines. For each chosen centroid wavelength the width and normalization are varied using a $\chi^{2}$ minimization procedure until a best fit is obtained. The centroid wavelength is also allowed to vary by a moderate amount, twice the value of a fiducial thermal Doppler width, during this procedure. The trial line is considered to be detected if the $\chi^{2}$ improves by 10, corresponding to approximately 99$\%$ confidence for 3 interesting parameters (Avni, 1976). The fiducial thermal Doppler width is a free parameter in this procedure (but note it does not influence the final fitted width, only the searching procedure for the line center). For the results shown here we adopt a value of 200 km s-1 for this parameter. Most lines are free of blending so the results of the line search are independent of the value of this parameter. Also included in table 2 are tentative identifications for the lines, along with laboratory wavelengths. These come from the xstar database (Kallman and Bautista, 2001; Bautista & Kallman, 2001). We treat as potential identifications the (likely) strongest feature which lies within a wavelength range corresponding to a Doppler shift of $\leq$+5000 km s-1 from a known resonance line or RRC. Table 2 includes the values of the implied Doppler shift for these identifications. Note that our detection criterion does not guarantee accurate measurement of the other line parameters, i.e. centroid wavelength and width, so that weaker lines do not all have bounds on these quantities. Where they are given in table 2, errors on these quantities and the line flux represent 90$\%$ confidence ($\Delta\chi^{2}\simeq$3 for one parameter; Avni (1976)) limits. The NGC 1068 spectrum together with the model fits discussed in the next section are shown in Figures 1 – 4. These are plotted in 2 $\AA$ intervals and separated according to grating arm (HEG or MEG). We do not display wavelength regions where the grating arm has no sensitivity, or where there are no features. No rebinning or grouping is performed in these plots or in our fitting procedure. We do quote values of $\chi^{2}$ in section 3, and these were calculated using binned data. Also, when plotting the longer wavelength regions, we bin the spectrum is binned for plotting purposes only in order to avoid the dominance of the noisy bins with low counts at long wavelengths (i.e. $\geq$18 $\AA$). In addition, the iron K region, 1.5 – 2.5 $\AA$ is not plotted here, and is discussed in section 2.4. Notable line features in the spectrum include the 1s-np lines of H- and He- like ions of O, Ne, Mg, Si and S. Corresponding features are also present from Ar and Ca, though not clearly detected. The 21 – 23 $\AA$ wavelength region contains the He-like lines of oxygen. These are discussed in more detail in section 2.3 below. The 17 – 19 $\AA$ wavelength region contains the L$\alpha$ line from H-like line of O VIII, the most prominent line in the spectrum. In addition, lines from Fe XVII near 16.8 and 17.1 $\AA$, and n=1 – 3, 1–4 and 1 –5 lines from O VII near 17.4, 17.8 and 18.6 $\AA$, respectively. These lines are discussed in section 2.5. The 15 – 17 $\AA$ wavelength region contains the L$\beta$ lines from O VIII plus lines from ion stages of iron: Fe XVII – XIX. The n=1 – 2 lines from He-like Ne are at 13.5 – 13.8 $\AA$. The 11 – 13 $\AA$ wavelength region contains the L$\alpha$ lines from Ne IX at 12.16 $\AA$, the higher order n=1 – 3 and n=1 –4 lines from Ne IX at 11.6 and 11.0 $\AA$, respectively, and additional lines from Fe XVII – XIX. Also apparent between 11.15 and 11.35 $\AA$ are inner shell fluorescence lines from L shell ions of Mg: Mg V – Mg X. The 9 – 11 $\AA$ wavelength region contains the L$\beta$ line from Ne X plus the H- and He-like lines from Mg near 8.4 and 9.15 - 9.35 $\AA$, respectively. The RRC of Ne IX is near 10.35 $\AA$. The 5 – 7 $\AA$ wavelength region contains the n=1-2 lines from both H-like and He-like Si, near 6.2 $\AA$ and 6.7 $\AA$, respectively. The Mg XII L$\beta$ line is apparent at 7.15 $\AA$. Inner shell fluorescence lines from L shell ions of Si: Si VII – Si XII are between 6.85 – 7.05. Lines due to H- and He-like S are at 4.74 and 5.05 $\AA$, respectively. The fluxes in table 2 can be compared with those given by previous studies of the NGC 1068 X-ray spectrum, by Kinkhabwala et al. (2002) and Ogle et al. (2003). There is incomplete overlap between the lists of detected lines by us and these authors. They detect 40 and 60 lines, respectively, which can be compared with 86 in our table 2. Of these, some of the lines detected using the $XMM-Newton$ RGS by Kinkhabwala et al. (2002) are outside of the spectral range of the $Chandra$ HETG. There are additional discrepancies in the significance and identification of a small number of weak lines when compared with Ogle et al. (2003), though none of these is highly statistically significant in either our spectra or theirs. Since the Ogle et al. (2003) study was based on a much shorter observation than ours, it is not surprising that we detect a larger number of lines. We find general consistency between the fluxes in table 2 and those of Ogle et al. (2003). There are significant discrepancies in the comparison between our line fluxes and those of Kinkhabwala et al. (2002), generally in the sense that the Kinkhabwala et al. (2002) are greater than ours by factors of several, possibly due to the different characteristics of the $XMM-Newton$ telescope and the spatial extent of the NGC 1068 X-ray emission which allows flux from a more extended region to enter the spectrum. ### 2.1 Line wavelengths Figure 5 and Table 2 show the Doppler shifts of the line centroids for the strongest lines in Table 2. The shift is given in velocity units and is measured relative to the rest frame of the AGN. Laboratory wavelengths are taken from the xstar database (Bautista & Kallman, 2001); most are from NIST 111http://physics.nist.gov/PhysRefData/ASD/. Error bars are from the errors provided by the automated Gaussian fitting procedure described in the previous section. Wavelengths are only plotted if the errors on the wavelength are less than 2000 km s-1. Most wavelengths are consistent with a Doppler shift in the range 400 – 600 km s-1. Exceptions correspond to RRCs, eg. Ne IX 10.36 $\AA$, for which the tabulated wavelengths are less precise, and weak lines such as O VIII 1s – 5p which is blended with Fe XIX near 14.8 $\AA$. This figure shows that there is evidence for wavelength shifts which are larger than the errors on the line centroid determination. Examples of this include the O VIII L$\alpha$ line compared with the 1s2 – 1s2s${}^{3}S$ ($f$) line of O VII. Figure 5, lower panel, and Table 2 show the Gaussian widths of the lines for the strongest lines. The width is given in velocity units, i.e. $\sigma_{\varepsilon}c/\varepsilon$ where $\sigma_{\varepsilon}$ is the Gaussian full width at half maximum in energy units and $\varepsilon$ is the line energy. Error bars are from the confidence errors provided by our automated fitting procedure as described above. Wavelengths are only plotted in figure 5 if the errors on the wavelength are less than 2000 km s-1. This shows that there is no single value for the width which is consistent with all the lines. Examples of broader lines include the O VIII L$\alpha$ line, and contrast with the forbidden line $f$ of O VII, for which the width is smaller. The full width at half maximum which fits to most lines is approximately 1400 km s-1. This corresponds to a Gaussian sigma of $\simeq$ 600 km s-1 which we adopt in our numerical models discussed below. ### 2.2 Spatial Dependence The discussion so far has utilized data extracted using the standard size extraction region from the HETG, which has width 1.3 $\times 10^{-3}$ degrees or 4.8 arcsec. We have also explored possible spatial dependence of the spectrum in the direction perpendicular to the dispersion direction by extracting the data using regions which are half and double the angular size, i.e. 2.4 and 9.6 arcsec wide. We then carry the fitting steps described above. Figure 6 illustrates the positions of these regions superimposed on the zero order image. In this way we can study the influence of adding successive regions further from the dispersion axis to the spectrum. Figure 7 shows the ratios of the line fluxes in the standard extraction region (green hexagons) and the extraction double the standard size (red diamonds) compared to the line flux in the extraction region half the standard width. This shows that the line emission outside half width region is significant; the values of the ratios of essentially all the lines are greater than unity, and many are $\sim$2\. Plus, the emission from outside the standard 4.8 arcsec extraction region is not negligible; the values of the ratio for this region (red symbols) are on average greater than the values of the ratio for the standard region. The principal difference between standard region and the double size region is the importance of the radiative excitation emission, which is larger in the standard region. This is consistent with results from Ogle et al. (2003). For example, for the He-like lines of Ne IX, the G ratio is larger in the standard region spectrum than in the spectrum from the double size region. For the O VII lines, the G ratios are similar for the two regions, but the R ratio is lower in the standard region, implying higher density there. These results demonstrate that most of the line emission is contained within the standard extraction region, 4.8 arcsec wide, and most line ratios are unaffected when emission from outside the standard extraction region is included. It is worth noting that the physical length scale corresponding to this angular size is approximately 1 arcsec = 72 pc (Bland-Hawthorn et al., 1997), so the standard extraction region corresponds to a total size of 347 pc, or a maximum distance from the central source of approximately 174 pc. This is much larger than the region where the broad line region and the obscuring torus are likely to lie, which is $\simeq$ 2 – 3 pc (Jaffe et al., 2004). For the remainder of this paper we will discuss primarily analysis of data from the standard extraction region. ### 2.3 He-like lines The He-like lines in Table 2 provide sensitive diagnostics of emission conditions: excitation mechanism, density, and ionization balance. These challenges of fitting these lines are apparent from the region containing the H and He-like lines from the elements O, Ne, Mg, Si, and S. This shows the characteristic three lines from the n=2 – n=1 decay of these ions: the resonance ($r$), intercombination ($i$) and forbidden ($f$). These lines provide useful diagnostics of excitation and density. These are described in terms of the ratios $R=f/i$ and $G=(f+i)/r$ (Gabriel & Jordan, 1969a, b). $R$ is indicative of density, since the energy splitting between the upper levels of the two lines (23S and 23P) is small compared with the typical gas temperature and collisions can transfer ions from the 23S to the 23P when the density is above a critical value. $R$ can also be affected by radiative excitation from the 1s2s3S to the 1s2s3P levels, but this requires photon field intensities which are greater than what is anticipated for NGC 1068. In our modeling we include this process, using an extension of the global non- thermal power law continuum, and find that it is negligible. We do not consider the possibility of enhanced continuum at the energy of the 23S to the 23P provided by, eg., hot stars. $G$ is an indicator of temperature or other mechanism responsible for populating the various $n=2$ levels. At high temperature, or when the upper level of the $r$ line (21P) is excited in some other way, then $G$ can have a value $\leq$1; in the absence of such conditions, the 23S and 23P levels are populated preferentially by recombination and $G$ has a characteristic large value, $G\geq 4$. Table 2 shows that the various elements have common values of $R$, in the range 1 – 3, indicating moderate density. More interesting is the fact that there is a diversity of values for $G$: O and Ne have values 2.5 – 3 while the other elements all have smaller values, $G\leq$ 1.5. The $G$ ratio is most affected by the process responsible for excitation of the $r$ line. This can be either electron impact collisions or radiative excitation. Collisions are less likely in the case of NGC 1068 due to the presence of radiative recombination continua (RRCs) from H-like and He-like species in the observed spectra (Kinkhabwala et al., 2002). This indicates the presence of fully stripped and H-like ions, which produce the RRCs, but at a temperature which is low enough that the RRCs appear narrow. Typical RRC widths in NGC 1068 correspond to temperatures which are $\leq 10^{5}$K. A coronal plasma, in which electron impact collisions are dominant, would require a temperature $\geq 10^{6}$ K in order to produce the same ions. Radiative excitation can produce $G$ ratios as small as 0.1. Thus, the $G$ ratios for NGC 1068 indicate the importance of radiative excitation for Mg and Si, while O and Ne are dominated by recombination. Radiative excitation depends on the presence of strong unattenuated continuum from the central source in order to excite the $r$ line. A consequence of the large cross section for radiative excitation in the line is the fact that the line will saturate faster than continuum. At large column densities from the source, the radiation field is depleted in photons capable of exciting the $r$ line, while still having photons capable of ionizing or exciting other, weaker lines. Thus, the $G$ ratio is an indicator of column density. He-like lines formed after the continuum has traversed a large column density will have large $G$ ratios, corresponding to primarily recombination. He-like lines formed after the continuum has traversed a small column density will have small $G$ ratios, corresponding to radiative excitation (Kinkhabwala et al., 2002). Figure 8 and 9 display the G ratio for the elements O, Ne, Mg, and Si from the NGC 1068 HETG observations. These are shown with error bars as the red crosses in the centers of the various frames. This shows that the ratios differ between the elements: O and Ne have G$\simeq$2 – 3, while Si and Mg have G$\simeq$1\. This differing behavior suggests that the He-like O and Ne lines are emitted by recombination, while the He-like lines of Mg and Si are emitted by radiative excitation. Since radiative excitation is suppressed by large resonance line optical depths, this suggests small optical depths in Si and Mg and larger optical depths in the lines from other elements. The differences in optical depths are significant; as shown by Kinkhabwala et al. (2002) equivalent hydrogen columns of 1023 cm-2 or more are required to suppress radiative excitation. Also plotted in figures 8 and 9 are the values of these ratios produced by photoionization models consisting of a single slab of gas of given ionization parameter at the illuminated face and given column density. The solid curves correspond to constant total slab column density, where black= 1023.5, blue=1022.5, green=1021.5 and red=1020.5. The dashed curves correspond to constant ionization parameter in the range 1$\leq$log($\xi$)$\leq$3\. Here the ionization parameter is $\xi=4\pi F/n$ where $F$ is the ionizing energy flux in the 1 – 1000 Ry range and $n$ is the gas number density. The role of radiative excitation is apparent from the fact that the low column density models (red curves) produce smaller $G$ values than the high column density models (black curves). The results in figures 8 and 9 show that no single value of the photoionization model parameters can simultaneously account for all the ratios. O and Ne require log($\xi$)$\simeq$1 and column$\geq 10^{23}$cm-2, while Si requires log($\xi$)$\simeq$2 and column$\leq 10^{22}$cm-2 and Mg is intermediate in both $\xi$ and column. Also plotted in figures 8 and 9 are values for certain other ratios, in the same units as for the He-like lines, and also including the model values. These include the higher series allowed lines from some He-like ions, eg. 1-4/1-2 vs 1-3/1-2 for O VII, where 1-2 includes the $r$, $i$, and $f$ lines while 1-4 and 1-3 include only the resonance component. These ratios depend primarily on ionization parameter and only very weakly on column density. This is because recombination cascades make larger values of both ratios than radiative excitation, so recombination tends to dominate production of these lines. ### 2.4 Iron K Line Region The iron K lines from NGC 1068 have been the subject of previous studies by Matt et al. (2004); Iwasawa et al. (1997), who showed that the line consists of three components, corresponding to near neutral, and H- and He-like ion species. The HETG provides higher spectral resolution than these previous studies, but fewer total counts. Figure 10 shows the spectrum in the region between 1.5 – 2.5 $\AA$ and reveals the three components of the iron K lines. The most prominent are a feature at 6.37 keV (1.95 $\AA$), consistent with neutral or near-neutral iron, a feature at 6.67 (1.85 $\AA$) indicative of He- like iron, and a feature near 7 keV (1.75 $\AA$) which may be associated with a combination of the K absorption edge from near-neutral gas and emission from H-like iron. Matt et al. (2004) also separately identify emission from Be-like iron; we are not able to make such an identification. We do find emission at other nearby wavelengths, but this is suggestive of a broad component or blended emission from various other species. We model the Fe K line region using ‘analytic’ models which utilize the xstar database and subroutines. These include optically thin models which include radiative excitation (which we denote ‘scatemis’) and models which do not include this process (‘photemis’), plus power law continuum. Both these models are available for use as ‘analytic’ models in xspec from the xstar site 222http://heasarc.gsfc.nasa.gov/docs/software/xstar/xstar.html. Radiative excitation cannot excite the neutral line, and we model it as emission from gas at log($\xi$)=-3. The emission component with radiative excitation has log($\xi$)=3, corresponding to highly ionized gas in which the H- and He-like stages are the most abundant ionization stages of Fe. In the HETG data the He- like triplet is not fully resolved. Nonetheless, the centroid of the He-like component is closer to the wavelength of the Fe XXV $r$ line than it is to the wavelength of the $f$ or $i$ lines, so we find a better fit with a resonance excitation model. The fit is shown in figure 10. The best-fit parameters are summarized in table 3; we find $\chi^{2}$=103 for 206 degrees of freedom for the 1 – 2.5 $\AA$ (5 - 12.4 keV) spectral region. The flux in the neutral-like line is 3.8 ${}^{+}_{-}0.44\times 10^{-5}$ cm-2 s-1. We also point out what appears to be unresolved emission in the wavelength region near $\simeq$ 1.9 $\AA$, between the He-like line and the neutral-like line. We have not attempted to model this. It may be due to a broadened component of the modeled lines, or an unresolved blend from other high ionization stages of iron. These could include L shell ions between Li-like and Ne-like. The two ionization parameters which we use to fit the iron K lines are both higher and lower than the values used to fit the remainder of the emission spectrum, described in the next section. The high ionization parameter, log($\xi$)=3, may be an extension of the ionization parameter range needed to fit the lines in the energy band below $\sim$ 5keV. The low ionization parameter, log($\xi$)=-3, likely corresponds to the torus. Neither of these components produces significant emission below $\sim$ 5keV, so we will not discuss them further in the model fits described in the following section. The iron K line complex contains more flux than any other line in the spectrum. This is consistent with the large normalization for the low ionization component that is required to fit the spectrum. This was discussed by Krolik & Kallman (1987); Nandra (2006). We have not attempted to fit for a Compton shoulder on the on the red wing of the neutral-like iron line, although such a component may be present in figure 10 (Matt et al., 2004). We also test for the presence of Ni K$\alpha$ in our spectrum and find an upper limit of 1.5 $\times 10^{-5}$ cm-2 s-1. This is less than the flux for Ni K$\alpha$ claimed by Matt et al. (2004), which was 5.6${}^{+1.8}_{-1.0}\times 10^{-5}$ cm-2 s-1. One possible explanation could be that their data was extracted from a much larger spatial region, $\simeq$40 arcsec, and is also affected by a nearby point source. ### 2.5 Iron L lines The $Chandra$ HETG observation allows study of the lines from the L shell ions of iron, Fe XVII – XXIV, in more detail than has previously been possible. Table 2 and figures 1 – 4 show these in the wavelength range 10 – 17 $\AA$. Notable features include the well known ‘3C’ and ‘3D’ (Parkinson, 1973) lines of Fe XVII at 15.01 and 15.26 $\AA$ (826 and 812 eV), respectively. The intensity ratio of these lines is a topic of interest in the study of coronal plasmas; we find a ratio of 2.8 ${}^{+0.8}_{-0.5}$. This is consistent with several other lab measurements and astrophysical observations. It also reflects the current apparent discrepancy with calculations, which generally produce values of this ratio which are 3.5 or greater Bernitt et al. (2012). The xstar models, described below, reflect this situation and produce a value for this ratio which is $\simeq$3. Other strong lines in the spectrum include the Fe XVII – XVIII complex between 11 – 11.5 $\AA$, Fe XXII 11.77 $\AA$. Fe XVII 12.12 $\AA$ is blended with Ne X L$\alpha$. The Fe XVII 2p – 4d line at 12.26 $\AA$ is apparent, but the xstar models with radiative excitation (described in more detail in the following section) fail to produce as much flux as is observed. Fe XX 12.58 $\AA$ is well fit, as is Fe XX 12.81, 12.83 $\AA$. The 13.6 $\AA$ blend contains both Ne IX and Fe XIX 2p – 3d lines, produced primarily by radiative excitation. Also contributing at 13.82 $\AA$ is the Fe XVII 2s – 3p line; this line can only be produced efficiently by radiative excitation and it is clearly apparent on the red wing of the Ne IX $f$ line. The 2p-3d lines of Fe XVIII near 14.2 and out to 14.6 $\AA$ are fitted by the models. Longward of 15 $\AA$ the most notable feature is the Fe XVII line at 17.1 $\AA$, which is an indicator of recombination. Also plotted in figure 9 is the ratio 2p-4d/2p-3s vs 2p-3d/2p-3s for the Ne- like Fe XVII. The latter ratio is sensitive to the effects of radiative excitation since 2p-3d has a higher oscillator strength than 2p-3s, while 2p-3s is emitted efficiently via recombination cascade (Liedahl et al., 1990). Here it is clear from figures 1 – 4 that the 2p-3d line is strong, therefore requiring that radiative excitation be efficient and the cloud column density must be low for the gas producing the Fe XVII lines. ### 2.6 Fluorescence Lines Fluorescence lines are emitted during the cascade following K shell ionization. This is a likely signature of photoionization because, for ions with more than 3 electrons, the rate for K shell ionization by electron impact by a Maxwellian velocity distribution never exceeds the valence shell cross section. Fluorescence lines probe the existence of ions with valence shell energies far below the observable X-ray range, including neutral and near- neutral ions. In the NGC 1068 spectrum, fluorescence lines include the K$\alpha$ line of iron shown in figure 10, plus a series of Si lines near 6.8 $\AA$. These correspond to L shell ions Si, likely Si VIII – X. The conditions under which these ions are likely to be abundant, i.e. the ionization parameter in a photoionization equilibrium model, are not very different from those corresponding to N VII and O VII, the lowest ionization species in the spectrum. Thus they do not provide additional significant insight into the existence or quantity of material at low ionization in NGC 1068. ## 3 Model A goal of interpreting the X-ray spectrum of NGC 1068 is to understand the distribution of gas: its location relative to the central continuum source, its velocity, density, element abundances, temperature, ionization state. Fitting of models which predict these quantities to the data are the most effective way to derive this information, although there is a range of levels of detail and physical realism for doing this. We have examined this through the use of photoionization table models in xspec, which are described in the Appendix. Based on what was presented in section 2 we can summarize some of the criteria for a model for the NGC 1068 HETG spectrum: (i) The model must account for the H- and He-like line strengths from abundant elements from N to Ca (we exclude the iron region from this discussion); (ii) The model must account for the G ratios from the elements O, Ne, Mg, Si, which have G$\simeq$3 for O and Ne but G$\simeq$1 for Mg and Si; (iii) The model must account for 2p-3d/2p-3s ratio in Fe XVII which is indicative of recombination rather than radiative excitation; (iv) The model must account for the strengths of the RRCs, which have strengths comparable to the corresponding resonance lines for O VII. From criterion (i) it is clear that the emitting gas must have a range of ionization parameter; no single ionization parameter can provide these ions. From criterion (ii) it is likely that a range of gas column densities are needed. The hypothesis that the lower ion column densities associated with lower abundance elements (Mg and Si) can account for the diversity in $G$ ratios can be tested in this way. Criterion (iv) implies that some of the gas must have high column in order to have a detectable contribution from recombination. Our fit uses three components with varying ionization parameters; these are spread evenly between log($\xi$)=1 and log($\xi$)=2.6. The HETG spectrum is relatively insensitive to gas outside this range of ionization parameters, except for the iron K lines. Given the strength of both the neutral-like and the H- and He-like iron K lines, it is plausible that there is an approximately continuous distribution of ionization parameters which extends beyond the range considered here. The three ionization parameter components in our fit correspond crudely to the regions dominating the emission for the elements of varying nuclear charge: the low ionization parameter component dominates the emission for N and O, the intermediate component dominates the Fe L lines and Ne and Mg, and the high ionization parameter component dominates for Mg and Si. For each ionization parameter, we include models with two column densities: 3 $\times 10^{22}$ cm-2 and 3 $\times 10^{23}$ cm-2. In this way the models span the likely ionization parameter and column density for which the strongest lines we observe are emitted. We use a constant turbulent line width (sigma) of 600 km s-1 which adequately accounts for the widths the majority of features. Results of our fit are summarized in table 4. The red curves in figures 1 – 4 show that this model fits the strengths of almost all the strong features in the spectrum. This fit adopts a net redshift for the line emitting gas of 0.0023, corresponding to an outflow velocity of 450 km s-1. This is marginally less than typical speeds from UV and optical lines of $\leq$600 km s-1 (both blue- and red-shifted) (Crenshaw & Kraemer, 2000), but is consistent with the velocities plotted in figure 5. Free parameters of the fits include the elemental abundances and the normalizations for the six model components. Experimentation shows that the fit is not substantially improved when all the abundances are allowed to vary independently, so we force the abundances of N, Ne, Mg, Si, S, Ar and Ca to be the same for all the models. We allow the abundances of O and Fe to vary independently for all the models. Our best fit is shown in figures 1 – 4 has $\chi^{2}/\nu=2.34$ for 2561 degrees of freedom. More results are shown in figures 11 and 12, which show the elemental abundances from the various model components, and the masses of the various components (defined below) versus ionization parameter. In order to account for the diversity in the importance of radiative excitation between various ions and elements, in the low ionization parameter log($\xi$)=1 component, which is responsible for most of the O VII emission, the oxygen abundance in the low column density component (component 4 in table 4) is smaller than the oxygen abundance in the high column component (component 1 in table 4) by a factor 2.5. If not, the recombination emission into O VII would be stronger than observed. Iron abundances are similar between the low and high column density components, and are highest ($\sim$10) at low ionization parameter (green in figure 12) and lowest ($\sim$1) at high ionization parameter log($\xi$)=2.6. The normalizations and masses of the low column density components (components 4 – 6) is generally lower than for the high column components (components 1 – 3). The normalization for component 5 is consistent with zero, though we include it for completeness. Component 6 is needed to provide photoexcitation-dominated lines of Fe XIX – XXI. None of the abundances is significantly subsolar, except for O at log($\xi$)=2.6 and column=3 $\times 10^{22}$ cm-2 (component 6). This is necessary to avoid over-producing O VIII RRCs. It is worth noting that X-ray observations are not generally capable of determining abundances relative to H or He, but rather only abundances relative to other abundant metal elements such as O or Fe. Thus, an apparent underabundance of O, for example, is equivalent to an overabundance of other elements such as Si, S, and Fe, relative to O. The highest ionization parameter component is responsible for the Si and Mg emission. Although its mass is dominated by the 3$\times 10^{23}$ cm -2 component (component 3 in table 4), the abundances of these elements are lower than for lower Z elements, so the optical depths in the He-like resonance lines are small enough that radiative excitation dominates the He-like lines. It is notable that the differences in the importance of radiative excitation between the various He-like ions, and also Fe XVII, can essentially all be accounted for by the effects of varying amounts of resonance line optical depth which is produced solely as a result of ionization and elemental abundance effects. This is consistent with the hypothesis suggested by Ogle et al. (2003). An exception is for the lines from Ne IX, for which a large ion column density is needed in order to suppress the radiative excitation. Models with column less than 3 $\times 10^{22}$ cm-2 produce G$\leq$1.6, while the observations give G$\simeq$ 4${}^{+6}_{-2}$. Our model produces the Ne IX $f$ line with approximately half the the observed strength, while producing $r$ and $i$ lines which are close to the observed strengths. The Ne IX RRC is also too weak in our models. We have found, through experimentation, that an additional component with pure Ne and a column of 3 $\times 10^{24}$ cm-3, and this is able to provide the observed Ne recombination features, but we have not included this component in the fits in figures 1 – 4 or table 4. The table models are calculated assuming that the emitting gas fills a spherical shell defined by an inner radius $R_{i}=\left(\frac{L}{n\xi}\right)^{1/2}$ and outer radius $R_{o}$ defined by $\int_{R_{i}}^{R_{o}}n(R)dR=N$. Table 4 provides the normalizations $\kappa_{i}$ of the table models as derived from the xspec fits. This normalization can be interpreted in terms of the properties of the source as follows: $\kappa=f\frac{L_{38}}{D_{kpc}^{2}}$ where $L_{38}$ is the source ionizing luminosity integrated over the 1 - 1000 Ry energy range in units of 1038 erg s-1, and $D_{kpc}$ is the distance to the source in kpc. The normalization also includes a filling factor $f$ which accounts for the possibility that the gas does not fill the solid angle, and $f\leq 1$. We assume the density $n$ scales as $n=n_{i}(R/R_{i})^{-2}$. With this assumption the ionization parameter is independent of position for a given component and the total amount of gas in each emission component $j$, is: $M_{j}=4\pi f_{j}m_{H}n_{i}R_{i}^{2}(R_{o}-R_{i})$ (1) Similarly, the emission measure for each component can be written: ${\rm EM}_{j}=4\pi f_{i}n_{i}^{2}R_{i}^{4}\left(\frac{1}{R_{i}}-\frac{1}{R_{o}}\right)$ (2) Since $n_{i}R_{i}^{2}=L/\xi$ both the mass and the emission measure depend on the physical size of the emission region, but not directly on density. They also depend on the ionizing luminosity. Table 4 includes values for $f_{i}$, $M_{i}$ and EMi derived using these expressions, assuming $L=10^{44}$ erg s-1. The $Chandra$ HETG spectrum of NGC 1068 constrains the density to be $\leq 10^{10}$ cm-3, from the strength of the O VII forbidden line. In the results given in table 4 we have chosen the inner and outer radii of the emission region to reflect simple conclusions from our data: the inner radius is $R_{i}=$1 pc, based on physical arguments about the location of the obscuring torus (Krolik & Begelman, 1988), and the outer radius is chosen to be $R_{o}$=200 pc in order to reflect the observed extent of the X-ray emission, and this is roughly consistent with the extraction region width. The density is chosen such that the inner radius of the line emitting gas is at 1 pc for all three emission components. Total masses and emission measures are also given. The total emission measure is log(EM)=66.3 This is greater than would be required to simply emit the observed lines if the gas had an emissivity optimized for the maximum line emission, owing to the fact that the table models include the gas needed to create the column density responsible for shielding the photoionization dominated gas from the effects of photoexcitation. Furthermore, most of the mass and emission measure comes from hydrogen, while the $Chandra$ HETG spectrum constrains only metals. Values for mass and emission measure are calculated using the elemental abundances given in table 4. It is also apparent that the assumptions described so far are not fully self- consistent. That is, if density $\propto R^{-2}$ and if $R_{o}>>R_{i}$ then the column density is $\simeq n_{i}R_{i}$, and an inner radius of 1pc cannot produce an arbitrary column density and ionization parameter. A self consistent model with density $\propto R^{-2}$ requires different inner radii for the different ionization parameter and column density components, and results in a total mass which is greater than that given in table 4 by approximately a factor of 12. The emission measure is insensitive to these assumptions. Our subsequent discussion is based on the simple scenario given in table 4: $R_{i}$=1 pc, $R_{o}$=200 pc. Figures 1 – 4 show general consistency between most of the lines and a single choice for outflow velocity. This corresponds to a net redshift of 0.0023${}^{+}_{-}$0.0002, or an outflow velocity of 450${}^{+}_{-}$50 km s-1 relative to the systemic redshift of 0.00383 from Bland-Hawthorn et al. (1997). There are apparent discrepancies between some model line centroids and observed features when this redshift is adopted. lines which show apparent shifts which differ significantly from this, however. An example is Mg XII L$\beta$, which has an apparent central wavelength of 7.150 $\AA$, as shown in table 2. The laboratory wavelength of this line is 7.110 $\AA$ Drake (1971), which would correspond to an outflow velocity of 1688 km s-1. This is greater than other Doppler shifts in the spectrum, and the difference cannot be easily explained from uncertainties in the rest wavelength. Other possible line identifications include K$\alpha$ lines from Si ions such as Si III or Si IV. These would be associated with gas at lower ionization parameter than the majority of the other lines in the spectrum and we have not attempted to include them in our model. One result of global model fits is revealing the presence of features which are not obviously apparent from examination or Gaussian fits to the spectrum. These are features, primarily from Fe L shell ions, which are in the model, but which are blended or are not sufficiently statistically significant in the spectrum to require their inclusion in Gaussian fits. Therefore, these lines are not identified in table 2. Some are indicative of recombination vs. radiative excitation, in a sense similar to the Fe XVII lines discussed above. Examples can be seen from examination of figures 1 – 4: RRCs from Fe XVII and XVIII at 9.15 and 8.55 $\AA$, respectively, and Fe Lines at 10.55 (Fe XVII 2p-5d 10.523 $\AA$ in the lab), 10.62, 10.85 (Fe XIX 2p-4d 10.6323,10.8267 $\AA$), 11.05 (Fe XVII 2s-4p 11.043 $\AA$), 11.18, 11.28 (Fe XVII 2p-5d, 11.133, 11.253 $\AA$) 11.45 (Fe XVIII 2p-5s, 11.42$\AA$) 12.15 (Fe XVII 2p-4d 12.12 $\AA$), 12.95 (Fe XIX 2s-3p, 12.92 $\AA$) 13.49, 13.53 (Fe XIX 2p-3d, 13.46,13.5249 $\AA$) 14.3 (Fe XVIII 2p-3d, 14.208 $\AA$) 15.88 (Fe XVIII 2p-3s, 15.83 $\AA$). ## 4 Discussion It is possible to use the observed spatial extent of the X-ray emission to make further inferences about the X-ray emitting gas. For NGC 1068, the distance is such that 1 arcsec = 72 pc (Bland-Hawthorn et al., 1997). The brightness profile in the zero order image is such that a significant amount of flux is coming from outside an extraction region which is half the standard size, i.e. 2.4 arcsec. Thus, an approximate, but convenient, size scale for the extent of the observed X-rays is $\simeq$ 2.4 arcsec $\simeq$ 200 pc. If so, we can compare with the sizes derived from table 4. These estimates are based on an assumed ionizing luminosity for the nucleus of 1044 erg s-1; this is uncertain by a factor of a few, but is bounded by $L_{bol}=4\times 10^{44}$ erg s-1 (Bland-Hawthorn et al., 1997). The mass of material we infer based on the $Chandra$ HETG X-ray spectrum is $\simeq 3.7\times 10^{5}M_{\odot}$, though this depends on the assumptions about the inner and outer radii of the emission region and on the assumption that density scales with $R^{-2}$. This can be compared with typical masses for the entire narrow line region, which are likely to be $\sim 10^{6}M_{\odot}$, although gas with ionization parameters in the range log($\xi$) 1 – 3 will not radiate efficiently in typical optical/UV narrow lines. If this gas flows uniformly over a distance $\simeq$ 200 pc at the speed we derive, then the flow timescale is 4.4 $\times 10^{5}$ yr, and the mean mass loss rate is 0.3 $M_{\odot}$ yr-1. This is considerably greater than the mass flux needed to power the nucleus, which is $\simeq$1.7 $\times 10^{-2}M_{\odot}$ yr-1 $(\eta/0.1)^{-1}$ L/($10^{44}$ erg s-1) where $\eta$ is the efficiency of conversion of accreted mass into continuum luminosity. The filling factors in table 4 can be interpreted formally as the fraction of the total available volume in the spherical shell bounded by $R_{i}$ and $R_{o}$ which is filled with line emitting gas. The angular distribution of this gas is not constrained, so this can be interpreted as a conical region with fractional solid angle given by $f$ or as a spherically symmetric distribution in which the fractional covering of each component is given by $f$ when averaged over solid angle. In either case it is clear that the total covering fraction of the reprocessing gas is at most a few percent. This suggests that, if NGC 1068 were viewed from an arbitrarily chosen angle, and if the line of sight to the central continuum source were not blocked by the Compton thick torus, then the probability of observing a warm absorber similar to those seen in many Seyfert 1 galaxies would be small. If the fractional solid angle of the obscuring torus as seen from the nucleus is $\sim$50$\%$ of the total, then the probability of seeing a warm absorber would be $\sim$10$\%$. This estimate is inversely proportional to the value we have assumed for the ionizing luminosity of the nucleus in NGC 1068, which is $10^{44}$ erg s-1. A smaller value for this quantity, which is possible depending on the true spectral shape, would increase the inferred probability of seeing a warm absorber toward a value $\sim$50$\%$. Our model fits to the NGC 1068 HETG spectrum allow a simple phenomenological test of the scenario which has been widely used to interpret the spectra of Seyfert 1 objects. That is, Seyfert 1 X-ray spectra, which show prominent blueshifted line absorption, have been analyzed assuming that the absorber lies solely along the line of sight and neglecting any effect of emission filling in the lines (McKernan et al., 2007). This can only be exactly correct if the absorber subtends negligible solid angle as seen from the central source. This would in turn conflict with the apparent presence of absorption in $\sim$half of known Seyfert 1 objects (Reynolds, 1997). On the other hand, examination of the fits shown in figures 1 – 4 and the results in table 2 show a typical emission line flux for a strong line we measure is $\simeq$2 $\times 10^{-4}$ s-1 cm-2 for O VIII L$\alpha$. This can be compared with the amount of energy absorbed in the same line from a Seyfert 1 galaxy; for NGC 3783 this quantity is $\simeq$5 $\times 10^{-4}$ (Kaspi et al., 2002). Correcting for the difference in distance, this would imply that, if the same emission line gas seen in NGC 1068 is also present in NGC 3783, then the ratio of line flux absorbed from the observed spectrum to emission is $\sim$8\. That is, the apparent flux absorbed in the O VIII L$\alpha$ line, and likely other lines as well, is actually greater than observed by $\sim$ 10 – 15 $\%$. The O VIII L$\alpha$ line in NGC 3783 shows signs of this with an apparent weak P-Cygni emission on the red edge of the trough, though this is not apparent in most other lines in that spectrum. Although O VIII L$\alpha$ is one of the strongest lines in the NGC 1068 spectrum, it is possible that other lines in Seyfert 1 absorption spectra are more affected by filling in from emission. Accurate fitting of these spectra, used to derive mass outflow rates and other quantities, would need to account for this process. It is interesting to compare our results with those from UV and optical imaging and spectroscopy with the Faint Object Spectrograph (FOS) and the Spave Telescope Imaging Spectrograph (STIS) on the Hubble Spate Telescope (HST). These properties have been discussed by Kraemer et al. (1998); Crenshaw & Kraemer (2000); Kraemer & Crenshaw (2000); Crenshaw & Kraemer (2000b); Kraemer & Crenshaw (2000b) The strongest UV emission lines observed by the HST instruments, eg. C IV $\lambda$1550, Si IV $\lambda$1398, and the He II and hydrogen lines, comes from gas with a lower ionization parameter than the inferred from X-ray emitting gas. STIS provides spatially resolved kinematic information, which shows that the UV gas speed increases from the nucleus out to 130 pc, then the speed decreases at larger distances. Also seen are coronal lines from S XII, Fe XIV Ne V, Ne IV. These ions can exist at the same ionization parameter as the lowest ionization X-ray gas. The intensity as a function of distance from the nucleus shows apparent dilution as r${}^{-}2$, and the emission region appears conical. The continuum spectrum most closely resembles a nuclear power law, implying that it is scattered light from the obscured nucleus. The contribution from stars to the continuum is smaller than the non-thermal power law. The gas observed by the HETG extends to $\sim$200 pc from the nucleus, while the obscuration has a much smaller size $\sim$1 pc. The gas which extends beyond the torus may originate near the torus, resembling a warm absorber flow in a type 1 object, and flowing ballistically to larger distances. If so, the outflow speed would be expected to be constant or decrease with distance owing to gravitational forces, and the density would be subject to purely geometric dilution. Alternatively, the X-ray and UV gas could entrain gas from narrow line clouds, or gas evaporated from the narrow line clouds, or from some other source. Radiation pressure could play a role in energizing the flow. The X-ray gas could play a role in the apparent deceleration of the UV gas seen at $\sim$100 pc. There is little strong evidence for these latter scenarios from the HETG spectra, since we see no evidence for large changes in the ionization balance or speed of the gas in the spectra extracted from various width regions. Our results appear most nearly consistent with simple geometric dilution of the outflow, leading to constant ionization parameter with distance, and nearly constant outflow speed. ## 5 Summary The results presented in this paper can be summarized as follows: (i) No single ionization parameter and excitation mechanism can fit to all the lines in the NGC 1068 X-ray spectrum obtained with the $Chandra$ HETG. The table model fits require three components at ionization parameters ranging from log($\xi$)=1 to log($\xi$)=3. We are able to adequately fit all the strong identified lines in the spectrum with the exception of the lines from He-like Ne, which show a stronger recombination component than are produced by our models. (ii) The abundances required by all the models are approximately solar (Grevesse et al., 1996) or slightly greater. A notable exception is a large suppression of the abundance of oxygen at the highest ionization parameter. Otherwise the O VIII RRCs would be stronger than observed. (iii) The masses and emission measures of gas are greater than expected for optically thin emission, owing primarily to the need for shielding to provide the recombination dominated gas seen in the lines of Mg and Si. (iv) The combined constraints on the ionization parameter and column density constrain the location of the emitting gas relative to the continuum source, while the line strengths constrain the amount of emitting gas. Taken together, these require that the emitting gas have a volume filling factor less than unity; the values differ for the various components and range up to $\sim$0.01. The mass flux through the region included in the HETG extraction region is approximately 0.3 M⊙ yr-1 assuming ordered flow at the speed characterizing the line widths. (v) Limited experimentation with extracting spectra from various positions on the sky does not reveal a clear pattern which separates the various emitting components spatially. It appears that the emitting components coexist in the same physical region. ## Appendix A Appendix We make use of calculations using the xstar (Kallman and Bautista, 2001) modeling package 333http://heasarc.gsfc.nasa.gov/docs/software/xstar/xstar.html. xstar is freely available and distributed as part of the heasoft package. Models can be imported into xspec and other fitting packages as tables, or via the ’analytic’ model warmabs/photemis/scatemis. Our models are based on the assumption that the most plausible energy source for the X-ray lines and RRCs observed from NGC 1068 is reprocessing of the continuum from the innermost regions of the AGN. The continuum is presumably associated with the accretion disk and related structures close to the black hole, i.e. within a few $R_{G}\simeq 3\times 10^{11}M_{6}$ cm, where $M_{6}$ is the mass of the black hole in units of $10^{6}M_{\odot}$. The line and recombination emission we observe is likely formed at distances $>10^{6}R_{G}$, based on the geometry of the obscuring torus as indicated by high spatial resolution IR imaging (Jaffe et al., 2004). If it is assumed that the line emission is from gas where the heating, excitation and ionization are dominated by the continuum from the black hole, and that there is a local time steady balance between these processes and their inverses (radiative cooling, radiative decay and recombination), then the temperature, atomic level populations and ion fractions can be calculated. xstar carries out such a calculation and also calculates the associated X-ray emission and absorption. These emissivities and opacities are then applied to a simple one-dimensional solution of the equation of radiative transfer to derive the spectrum at a distant observer. Under the simple assumption that the gas is optically thin the important physical quantities depend most sensitively on the ratio of the ionizing X-ray flux to the gas density. We adopt the definition of this ionization parameter which is $\xi=4\pi F_{X}/n$ where $F_{X}$ is the ionizing (energy) flux between 1 and 1000Ry and $n$ is the gas number density. An important part of our results concerns the fact that the gas is not optically thin; this is key for explaining the varying amounts of radiative excitation seen in the line ratios. This in turn means that the column density, or the ion column densities, are also free parameters. We implement the models by direct fitting, in which xstar models are used to generate synthetic spectra which are compared to the observations within the fitting program xspec. xstar explicitly calculates the populations of all atomic levels associated with emission or absorption of radiation; it does not rely on the traditional ’nebular approximation’ which assumes that every excitation decays only to ground. Therefore it is straightforward to include radiative excitation, and this has been done. Interactive, iterative calculation of full xstar models within xspec and simultaneous fitting to data is not practical owing to computational limitations. We point out, parenthetically, that the simplest way to use xstar results within xspec is to use the associated xspec ‘analytic’ models. When called from xspec, these read stored tables of ionic level populations as a function of ionization parameter, and then calculate the opacity and emissivity ‘on the fly’. The physical quantity used the xspec model fit is an emitted flux or a transmission coefficient as a function of energy for a photoionized slab of given column density, abundances and ionization parameter. These are calculated from the opacity and emissivity by assuming these quantities are uniform throughout. Advantages of this procedure compared with the use of table models (described below) include: Ability to account for arbitrary element abundances, arbitrary spectral resolution, and arbitrary turbulent broadening. Limitations include the fact that it uses a saved file of level populations calculated for a grid of optically thin models for a fixed choice of ionizing spectrum rather than calculating the ionization balance self- consistently. It implicitly assumes that the absorber has uniform ionization even if the user specifies a large column, and that all emission freely escape. In this sense it is not self-consistent. In fact, analytic models fail when applied to the analysis of the NGC 1068 spectrum because, for column densities large enough to suppress radiative excitation, as demanded by eg. the Fe XVII 17$\AA$ line, the associated RRCs for Fe XVII are assumed to escape freely and are greatly over predicted. For this reason we will not discuss analytic models further in this paper. In order to self-consistently include the effects of absorption of the incident continuum in models for photoionized emission from NGC 1068 it is necessary to use optically thick models. These are calculated using xstar, by performing multiple model runs and using varying ionization parameter and cloud column density. The resulting emission spectra and transmission functions are stored as fits tables, binned in energy, using the table format prescribed by xspec. These can be read by xspec, selected according to parameter values, and interpolation between modeled parameter values is performed. xspec can fit the table models to the observed spectrum and thereby yield values for the cloud column density, ionization parameter, and the normalization, which is described in more detail below. Owing to the fact that the spectra are stored binned in energy the line width cannot be conveniently treated as a free parameter, and therefore it is important that the energy grid spacing used in constructing the table be no greater than than the intrinsic instrumental resolution. It is also possible to treat the element abundances as free parameters, by assuming that the escaping line and RRC fluxes fluxes depend linearly on the abundances. This ignores the coupling of the cloud ionization structure to abundance, via the temperature and the opacity of the model. In our fits we use this approximation when deriving element abundances. Within a given optically thick model, xstar calculates the effect of attenuation of the incident continuum using a simple single stream treatment of the radiative transfer. This takes into account the varying ionization and excitation through the model. The inclusion of radiative excitation necessitates a finer spatial grid than for continuum absorption alone, since the transfer must resolve the attenuation length of the photons in the resonance lines. Also, line opacity varies over a large range in a very narrow energy range and it is not feasible to accurately sample the relevant energies for all of the many lines in a photoionized plasma. We adopt a very simple treatment in which the line opacity is binned into continuum bins and used to calculate the attenuated flux. This flux in continuum bins is then used in the calculation of photoexcitation. The continuum bin size is typically $\Delta\varepsilon/\varepsilon=1.4\times 10^{-4}$ which corresponds to $\simeq 41$km s-1 Doppler width. This is comparable to the thermal Doppler width for, eg., oxygen at a temperature $\sim 10^{6}$K. Thus, we are not fully resolving most lines relevant to this study. The tables used to fit to the $Chandra$ HETG spectrum of NGC 1068 include emission from both the illuminated and unilluminated cloud faces, corresponding to the assumption that the the line emitting material is arranged with approximate spherical symmetry around the central continuum source. They are calculated assuming an ionizing spectrum which is a single power law with (photon number) spectral index $\Gamma$=2 and constant density 104 cm-3. These results are not expected to depend sensitively on the latter two assumptions; in particular $\Gamma$ values in the range 1.7 – 2.3 produce very similar ionization balance distributions. xstar makes use of atomic data compiled from various sources and described by (Bautista & Kallman, 2001). This has been extensively updated since that time, as described by Witthoeft et al. (2007, 2009, 2011); Palmeri et al. (2002, 2003a, 2003b, 2008a, 2008b, 2011, 2012); Mendoza et al. (2004); García et al. (2009); Bautista et al. (2003, 2004) The results presented here which flow from the model calculations are dependent on the assumptions, computational implementation and atomic data used by the models. The atomic data affects the line identifications and the outflow speeds, via the line rest wavelengths. It also affects the ionization balance, via the photoionization and recombination rates, and the line strengths. The quantitative uncertainty associated with the atomic data and the resulting uncertainty in the synthetic spectrum are difficult to estimate; this is a topic of interest for many problems beyond this one. We can point out that the emissivities of most of the lines from the H- and He-like ions in our study depend on two processes: recombination and photoexcitation. These depend in turn on photoionization cross sections and line oscillator strengths, respectively. These are quantities associated with radiative processes in relatively simple ions and therefore are generally more reliable than collisional rate coefficients which are important for ions with partially filled L shells in our models and in coronal plasmas. Foster et al. (2010) have shown that uncertainties of, say, 20$\%$ on the collision strengths for O VII can result in a range of a factor of $\simeq$3 in the inferred temperature in a coronal plasma based on the $G$ ratio. 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L. 2001, ApJ, 556, 6 Table 1: Observation log obsid \| time \| exposure ---\|---\|--- 332 \| 2000-12-04 18:11:52 \| 46290 9148 \| 2008-12-05 08:23:41 \| 80880 9149 \| 2008-11-19 04:49:51 \| 90190 9150 \| 2008-11-27 04:55:36 \| 41760 10815 \| 2008-11-20 16:23:22 \| 19380 10816 \| 2008-11-18 01:18:39 \| 16430 10817 \| 2008-11-22 17:36:37 \| 33180 10823 \| 2008-11-25 18:21:16 \| 35110 10829 \| 2008-11-30 20:16:45 \| 39070 10830 \| 2008-12-03 15:08:36 \| 43600 Table 2: Lines Found index \| wavelength ($\AA$) \| width (km/s) \| flux (cm-2s-1) \| lab ($\AA$) \| ion \| lower level \| upper level \| voff (km/s) ---\|---\|---\|---\|---\|---\|---\|---\|--- 1 \| 1.780 \| $\leq$ 2400 \| 1.3${}^{+0.1}_{-0.3}\times 10^{-5}$ \| 1.780 \| fe xxvi \| 1s${}^{1}(^{2}$S) \| 1s02p${}^{1}(^{2}$P) \| 1100${}^{+840}_{-40}$ 2 \| 1.855 \| $\leq$ 2000 \| 2.9${}^{+0.2}_{-0.2}\times 10^{-5}$ \| 1.869 \| fe xxv \| 1s${}^{2}(^{1}$S) \| 1s12p${}^{1}(^{1}$P) \| 3400${}^{+40}_{-400}$ 3 \| 1.945 \| 1700${}^{+17}_{-17}$ \| 5.6${}^{+0.2}_{-0.3}\times 10^{-5}$ \| 1.941 \| fe i \| K$\alpha$ \| \| 500${}^{+40}_{-800}$ 4 \| 2.550 \| $\leq$ 2500 \| 2.7${}^{+0.1}_{-1.6}\times 10^{-6}$ \| 2.549 \| ca xx \| 1s${}^{1}(^{2}$S) \| 1s03p${}^{1}(^{2}$P) \| 1100${}^{+2400}_{-40}$ 5 \| 3.030 \| 1900${}^{+550}_{-440}$ \| 3.9${}^{+0.1}_{-1.6}\times 10^{-6}$ \| 3.020 \| ca xx \| 1s${}^{1}(^{2}$S) \| 1s02p${}^{1}(^{2}$P) \| 150${}^{+40}_{-1000}$ 6 \| 3.175 \| $\leq$ 3600 \| 4.7${}^{+0.4}_{-0.4}\times 10^{-6}$ \| 3.150 \| ar xviii \| 1s${}^{1}(^{2}$S) \| 1s03p${}^{1}(^{2}$P) \| - 7 \| 3.195 \| $\leq$ 2600 \| 2.3${}^{+0.4}_{-0.4}\times 10^{-6}$ \| 3.180 \| ca xix \| 1s${}^{2}(^{1}$S) \| 1s12p${}^{1}(^{1}$P) \| - 8 \| 3.195 \| $\leq$ 2600 \| 2.3${}^{+0.4}_{-0.4}\times 10^{-6}$ \| 3.190 \| ca xix \| 1s${}^{2}(^{1}$S) \| 1s12p${}^{1}(^{3}$P) \| 670${}^{+3300}_{-40}$ 9 \| 3.215 \| $\leq$ 2000 \| 2.7${}^{+0.4}_{-0.4}\times 10^{-6}$ \| 3.210 \| ca xix \| 1s${}^{2}(^{1}$S) \| 1s12s${}^{1}(^{3}$S) \| 670${}^{+5200}_{-40}$ 10 \| 3.380 \| $\leq$ 2200 \| 5.5${}^{+0.4}_{-0.5}\times 10^{-6}$ \| 3.365 \| ar xvii \| 1s${}^{2}(^{1}$S) \| 1s13p${}^{1}(^{1}$P) \| - 11 \| 3.735 \| $\leq$ 2200 \| 4.0${}^{+0.1}_{-1.4}\times 10^{-6}$ \| 3.739 \| ar xviii \| 1s${}^{1}(^{2}$S) \| 1s02p${}^{1}(^{2}$P) \| 1500${}^{+840}_{-400}$ 12 \| 3.945 \| $\leq$ 2000 \| 5.1${}^{+0.2}_{-1.5}\times 10^{-6}$ \| 3.950 \| ar xvii \| 1s${}^{2}(^{1}$S) \| 1s12p${}^{1}(^{1}$P) \| 1500${}^{+1500}_{-40}$ 13 \| 4.000 \| $\leq$ 2600 \| 5.2${}^{+0.5}_{-0.4}\times 10^{-6}$ \| 3.990 \| ar xvii \| 1s${}^{2}(^{1}$S) \| 1s12p${}^{1}(^{3}$P) \| 380${}^{+1300}_{-40}$ 14 \| 4.000 \| $\leq$ 2600 \| 5.2${}^{+0.5}_{-0.4}\times 10^{-6}$ \| 3.990 \| s xvi \| 1s${}^{1}(^{2}$S) \| 1s03p${}^{1}(^{2}$P) \| 380${}^{+1300}_{-40}$ 15 \| 4.000 \| $\leq$ 2600 \| 5.2${}^{+0.5}_{-0.4}\times 10^{-6}$ \| 4.010 \| ar xvii \| 1s2.1S \| 1s1.2s1.3S \| 1900${}^{+1300}_{-40}$ 16 \| 4.755 \| $\leq$ 2100 \| 7.6${}^{+0.4}_{-0.9}\times 10^{-6}$ \| 4.730 \| s xvi \| 1s1.2S \| 1s0.2p1.2P \| - 17 \| 5.040 \| $\leq$ 2100 \| 1.4${}^{+0.1}_{-0.1}\times 10^{-5}$ \| 5.040 \| s xv \| 1s${}^{2}(^{1}$S) \| 1s12p${}^{1}(^{1}$P) \| 1100${}^{+1500}_{-40}$ 18 \| 5.100 \| $\leq$ 2000 \| 8.6${}^{+0.8}_{-0.7}\times 10^{-6}$ \| 5.070 \| s xv \| 1s${}^{2}(^{1}$S) \| 1s12p${}^{1}(^{3}$P) \| - 19 \| 5.100 \| $\leq$ 2000 \| 8.6${}^{+0.8}_{-0.7}\times 10^{-6}$ \| 5.084 \| si xiii \| rrc \| \| 180${}^{+2400}_{-160}$ 20 \| 5.225 \| $\leq$ 2000 \| 8.6${}^{+0.7}_{-0.6}\times 10^{-6}$ \| 5.212 \| si xiv \| 1s${}^{1}(^{2}$S) \| 1s03p${}^{1}(^{2}$P) \| 390${}^{+40}_{-320}$ 21 \| 5.690 \| 1100${}^{+120}_{-130}$ \| 7.0${}^{+0.3}_{-1.1}\times 10^{-6}$ \| 5.680 \| si xiii \| 1s${}^{2}(^{1}$S) \| 1s13p${}^{1}(^{1}$P) \| 600${}^{+160}_{-40}$ 22 \| 6.195 \| 1300${}^{+150}_{-27}$ \| 2.2${}^{+0.0}_{-0.2}\times 10^{-5}$ \| 6.180 \| si xiv \| 1s${}^{1}(^{2}$S) \| 1s02p${}^{1}(^{2}$P) \| 410${}^{+280}_{-120}$ 23 \| 6.660 \| 1500${}^{+120}_{-47}$ \| 3.1${}^{+0.0}_{-0.2}\times 10^{-5}$ \| 6.640 \| si xiii \| 1s${}^{2}(^{1}$S) \| 1s12p${}^{1}(^{1}$P) \| 230${}^{+480}_{-40}$ 24 \| 6.750 \| 1700${}^{+1300}_{-2300}$ \| 2.4${}^{+0.1}_{-0.1}\times 10^{-5}$ \| 6.690 \| si xiii \| 1s${}^{2}(^{1}$S) \| 1s12p${}^{1}(^{3}$P) \| - 25 \| 6.750 \| 1700${}^{+1300}_{-2300}$ \| 2.4${}^{+0.1}_{-0.1}\times 10^{-5}$ \| 6.744 \| si xiii \| 1s${}^{2}(^{1}$S) \| 1s12s${}^{1}(^{3}$S) \| 870${}^{+120}_{-120}$ 26 \| 6.870 \| $\leq$ 2000 \| 7.4${}^{+0.4}_{-0.3}\times 10^{-6}$ \| 6.860 \| si x \| K$\alpha$ \| \| 700${}^{+40}_{-120}$ 27 \| 6.995 \| $\leq$ 3600 \| 7.3${}^{+0.1}_{-2.2}\times 10^{-6}$ \| 6.930 \| si ix \| K$\alpha$ \| \| - 28 \| 7.005 \| 450 \| 2.9${}^{+2.5}_{-1.6}\times 10^{-7}$ \| 7.001 \| si viii \| K$\alpha$ \| \| 970${}^{+5600}_{-40}$ 29 \| 7.005 \| 450 \| 2.9${}^{+2.5}_{-1.6}\times 10^{-7}$ \| 7.040 \| mg xi \| rrc \| \| 2600${}^{+5600}_{-40}$ 30 \| 7.150 \| $\leq$ 2700 \| 2.2${}^{+0.1}_{-0.1}\times 10^{-5}$ \| 7.110 \| mg xii \| 1s${}^{1}(^{2}$S) \| 1s03p${}^{1}(^{2}$P) \| - 31 \| 7.320 \| $\leq$ 2000 \| 6.1${}^{+0.4}_{-0.3}\times 10^{-6}$ \| 7.310 \| mg xi \| 1s${}^{2}(^{1}$S) \| 1s15p${}^{1}(^{1}$P) \| 740${}^{+640}_{-40}$ 32 \| 7.485 \| 1400${}^{+14}_{-14}$ \| 8.1${}^{+0.2}_{-1.3}\times 10^{-6}$ \| 7.470 \| mg xi \| 1s${}^{2}(^{1}$S) \| 1s14p${}^{1}(^{1}$P) \| 530${}^{+40}_{-240}$ 33 \| 7.770 \| $\leq$ 2000 \| 9.6${}^{+0.4}_{-0.5}\times 10^{-6}$ \| 7.758 \| al xii \| 1s${}^{2}(^{1}$S) \| 1s13p${}^{1}(^{1}$P) \| 640${}^{+4400}_{-40}$ 34 \| 7.860 \| 1200${}^{+200}_{-12}$ \| 1.2${}^{+0.1}_{-0.0}\times 10^{-5}$ \| 7.850 \| mg xi \| 1s${}^{2}(^{1}$S) \| 1s13p${}^{1}(^{1}$P) \| 750${}^{+1200}_{-40}$ 35 \| 8.435 \| 1400${}^{+96}_{-43}$ \| 3.3${}^{+0.0}_{-0.3}\times 10^{-5}$ \| 8.420 \| mg xii \| 1s${}^{1}(^{2}$S) \| 1s02p${}^{1}(^{2}$P) \| 600${}^{+360}_{-40}$ 36 \| 9.190 \| 1500${}^{+230}_{-30}$ \| 4.2${}^{+0.0}_{-0.4}\times 10^{-5}$ \| 9.170 \| mg xi \| 1s${}^{2}(^{1}$S) \| 1s12p${}^{1}(^{1}$P) \| 480${}^{+200}_{-120}$ 37 \| 9.340 \| $\leq$ 2500 \| 3.5${}^{+0.1}_{-0.1}\times 10^{-5}$ \| 9.230 \| mg xi \| 1s${}^{2}(^{1}$S) \| 1s12p${}^{1}(^{3}$P) \| - 38 \| 9.340 \| $\leq$ 2500 \| 3.5${}^{+0.1}_{-0.1}\times 10^{-5}$ \| 9.320 \| mg xi \| 1s${}^{2}(^{1}$S) \| 1s12s${}^{1}(^{3}$S) \| 490${}^{+40}_{-760}$ 39 \| 9.465 \| $\leq$ 2000 \| 1.9${}^{+0.0}_{-0.3}\times 10^{-5}$ \| 9.475 \| fe xxi \| 2p${}^{2}(^{3}$P0) \| 2p14d${}^{1}(^{3}$D1) \| 1500${}^{+1900}_{-80}$ 40 \| 9.720 \| $\leq$ 2000 \| 1.9${}^{+0.1}_{-0.1}\times 10^{-5}$ \| 9.679 \| fe xix \| 2p${}^{4}(^{1}$D2) \| 2p35d${}^{1}(^{3}$D${}_{3}\\#3$) \| - 41 \| 9.720 \| $\leq$ 2000 \| 1.9${}^{+0.1}_{-0.1}\times 10^{-5}$ \| 9.800 \| fe xvii \| rrc \| \| 3600${}^{+160}_{-79}$ 42 \| 10.045 \| $\leq$ 2400 \| 2.5${}^{+0.0}_{-0.7}\times 10^{-5}$ \| 10.014 \| fe xix \| 2p${}^{4}(^{1}$D2) \| 2p35d${}^{1}(^{3}$D3) \| 230${}^{+40}_{-400}$ 43 \| 10.260 \| $\leq$ 2000 \| 3.9${}^{+0.1}_{-0.2}\times 10^{-5}$ \| 10.240 \| ne x \| 1s${}^{1}(^{2}$S) \| 1s03p${}^{1}(^{2}$P) \| 550${}^{+1600}_{-40}$ 44 \| 10.365 \| 1600${}^{+95}_{-120}$ \| 3.4${}^{+0.0}_{-0.7}\times 10^{-5}$ \| 10.388 \| ne ix \| rrc \| \| 1800${}^{+320}_{-40}$ 45 \| 10.670 \| $\leq$ 2200 \| 3.6${}^{+0.2}_{-0.1}\times 10^{-5}$ \| 10.641 \| fe xix \| 2p${}^{4}(^{3}$P2) \| 2p34d${}^{1}(^{3}$P2) \| 320${}^{+40}_{-1300}$ 46 \| 10.670 \| $\leq$ 2200 \| 3.6${}^{+0.2}_{-0.1}\times 10^{-5}$ \| 10.660 \| fe xvii \| \| \| 860${}^{+40}_{-1300}$ 47 \| 10.830 \| $\leq$ 2000 \| 2.3${}^{+0.1}_{-0.1}\times 10^{-5}$ \| 10.770 \| fe xvii \| 2p${}^{6}(^{1}$S0) \| 2p56d${}^{1}(^{1}$P1) \| - 48 \| 10.830 \| $\leq$ 2000 \| 2.3${}^{+0.1}_{-0.1}\times 10^{-5}$ \| 10.820 \| fe xix \| \| \| 860${}^{+240}_{-40}$ 49 \| 11.025 \| 1300${}^{+300}_{-13}$ \| 3.5${}^{+0.1}_{-0.4}\times 10^{-5}$ \| 11.000 \| ne ix \| 1s${}^{2}(^{1}$S) \| 1s14p${}^{1}(^{1}$P) \| 460${}^{+160}_{-80}$ 50 \| 11.555 \| 1400${}^{+40}_{-70}$ \| 4.0${}^{+0.0}_{-0.6}\times 10^{-5}$ \| 11.500 \| fe xviii \| \| \| - 51 \| 11.555 \| 1400${}^{+40}_{-70}$ \| 4.0${}^{+0.0}_{-0.6}\times 10^{-5}$ \| 11.547 \| ne ix \| 1s${}^{2}(^{1}$S) \| 1s13p${}^{1}(^{1}$P) \| 930${}^{+280}_{-40}$ 52 \| 11.800 \| $\leq$ 2000 \| 4.2${}^{+0.1}_{-0.2}\times 10^{-5}$ \| 11.762 \| fe xxi \| 2p${}^{2}(^{3}$P1) \| 2s12p23d${}^{1}(^{3}$P2) \| 170${}^{+520}_{-80}$ 53 \| 12.165 \| 1400${}^{+200}_{-14}$ \| 9.5${}^{+0.1}_{-0.8}\times 10^{-5}$ \| 12.100 \| ne x \| 1s${}^{1}(^{2}$S) \| 1s02p${}^{1}(^{2}$P) \| - 54 \| 12.300 \| $\leq$ 2400 \| 5.7${}^{+0.1}_{-1.6}\times 10^{-5}$ \| 12.264 \| fe xvii \| 2p${}^{6}(^{1}$S0) \| 2p54d${}^{1}(^{3}$D1) \| 260${}^{+40}_{-280}$ 55 \| 12.840 \| $\leq$ 2000 \| 5.5${}^{+0.2}_{-0.3}\times 10^{-5}$ \| 12.800 \| fe xx \| \| \| 200${}^{+1300}_{-40}$ 56 \| 12.840 \| $\leq$ 2000 \| 5.5${}^{+0.2}_{-0.3}\times 10^{-5}$ \| 12.812 \| fe xviii \| 2p${}^{5}(^{2}$P3/2) \| 2s12p53p${}^{1}(^{2}$D5) \| 480${}^{+1300}_{-40}$ 57 \| 13.509 \| $\leq$ 3600 \| 3.2${}^{+0.3}_{-0.5}\times 10^{-5}$ \| 13.447 \| ne ix \| 1s${}^{2}(^{1}$S) \| 1s12p${}^{1}(^{1}$P) \| - 58 \| 13.509 \| $\leq$ 3600 \| 3.2${}^{+0.3}_{-0.5}\times 10^{-5}$ \| 13.500 \| ne ix \| 1s${}^{2}(^{1}$S) \| 1s12p${}^{1}(^{3}$P) \| 940${}^{+6700}_{-40}$ 59 \| 13.725 \| $\leq$ 3000 \| 1.4${}^{+0.0}_{-0.1}\times 10^{-4}$ \| 13.700 \| ne ix \| 1s${}^{2}(^{1}$S) \| 1s12s${}^{1}(^{3}$S) \| 590${}^{+360}_{-40}$ 60 \| 14.215 \| $\leq$ 2000 \| 1.1${}^{+0.0}_{-0.2}\times 10^{-4}$ \| 14.206 \| fe xviii \| 2p${}^{5}(^{2}$P3/2) \| 2p53d${}^{1}(^{2}$D${}_{5/2}\\#$2) \| 950${}^{+960}_{-40}$ 61 \| 14.215 \| $\leq$ 2000 \| 1.1${}^{+0.0}_{-0.2}\times 10^{-4}$ \| 14.250 \| o viii \| rrc \| \| 1900${}^{+960}_{-40}$ 62 \| 14.415 \| $\leq$ 2100 \| 7.2${}^{+0.1}_{-2.2}\times 10^{-5}$ \| 14.394 \| fe xviii \| 2p${}^{5}(^{2}$P3/2) \| 2p43d${}^{1}(^{2}$D5/2) \| 710${}^{+40}_{-960}$ 63 \| 14.580 \| $\leq$ 2000 \| 5.5${}^{+0.4}_{-0.3}\times 10^{-5}$ \| 14.500 \| fe xx \| 2p${}^{3}(^{2}$P1/2) \| 2p23s${}^{1}(^{4}$P1/2) \| - 64 \| 14.835 \| 1600${}^{+410}_{-100}$ \| 3.5${}^{+0.1}_{-0.8}\times 10^{-5}$ \| 14.816 \| fe xix \| 2p${}^{4}(^{3}$P1) \| 2p33s${}^{1}(^{1}$D2) \| 740${}^{+920}_{-40}$ 65 \| 14.835 \| 1600${}^{+410}_{-100}$ \| 3.5${}^{+0.1}_{-0.8}\times 10^{-5}$ \| 14.832 \| o viii \| 1s1.2S \| 1s0.5p1.2P \| 1100${}^{+920}_{-40}$ 66 \| 15.040 \| $\leq$ 2700 \| 1.2${}^{+0.1}_{-0.0}\times 10^{-4}$ \| 15.015 \| fe xvii \| 2p${}^{6}(^{1}$S0) \| 2p53d1 \| 640${}^{+1600}_{-40}$ 67 \| 15.295 \| $\leq$ 3600 \| 8.5${}^{+0.4}_{-0.3}\times 10^{-5}$ \| 15.188 \| o viii \| 1s${}^{1}(^{2}$S) \| 1s04p${}^{1}(^{2}$P) \| - 68 \| 15.295 \| $\leq$ 3600 \| 8.5${}^{+0.4}_{-0.3}\times 10^{-5}$ \| 15.262 \| fe xvii \| 2p${}^{6}(^{1}$S0) \| 2p53d${}^{1}(^{3}$P1) \| 490${}^{+40}_{-990}$ 69 \| 15.295 \| $\leq$ 3600 \| 8.5${}^{+0.4}_{-0.3}\times 10^{-5}$ \| 15.418 \| fe xvii \| 2p${}^{6}(^{1}$S0) \| 2p53d${}^{1}(^{3}$P2) \| 3500${}^{+39}_{-980}$ 70 \| 15.670 \| $\leq$ 2800 \| 2.5${}^{+0.2}_{-0.2}\times 10^{-5}$ \| 15.627 \| fe xviii \| 2p${}^{5}(^{2}$P3/2) \| 2p43s${}^{1}(^{2}$P5/2) \| 310${}^{+200}_{-200}$ 71 \| 16.040 \| 1500${}^{+250}_{-62}$ \| 8.4${}^{+0.2}_{-1.0}\times 10^{-5}$ \| 16.007 \| fe xviii \| 2p${}^{5}(^{2}$P3/2) \| 2p43s${}^{1}(^{2}$P3/2) \| 520${}^{+400}_{-120}$ 72 \| 16.785 \| 1700${}^{+500}_{-17}$ \| 1.8${}^{+0.0}_{-0.3}\times 10^{-4}$ \| 16.777 \| fe xvii \| 2p${}^{6}(^{1}$S0) \| 2p${}^{5}(^{3}$S1) \| 990${}^{+40}_{-160}$ 73 \| 16.785 \| 1700${}^{+500}_{-17}$ \| 1.8${}^{+0.0}_{-0.3}\times 10^{-4}$ \| 16.777 \| o vii \| rrc \| \| 990${}^{+40}_{-160}$ 74 \| 17.135 \| $\leq$ 2000 \| 1.6${}^{+0.1}_{-0.1}\times 10^{-4}$ \| 17.050 \| fe xvii \| 2p${}^{6}(^{1}$S0) \| 2p${}^{5}(^{3}$S1) \| - 75 \| 17.420 \| 1100${}^{+300}_{-58}$ \| 3.1${}^{+0.2}_{-0.4}\times 10^{-5}$ \| 17.396 \| o vii \| 1s${}^{2}(^{1}$S) \| 1s15p${}^{1}(^{1}$P) \| 720${}^{+40}_{-200}$ 76 \| 17.785 \| 1400${}^{+200}_{-160}$ \| 5.2${}^{+0.3}_{-0.5}\times 10^{-5}$ \| 17.768 \| o vii \| 1s${}^{2}(^{1}$S) \| 1s14p${}^{1}(^{1}$P) \| 850${}^{+440}_{-40}$ 77 \| 18.635 \| 1700${}^{+470}_{-34}$ \| 1.2${}^{+0.0}_{-0.2}\times 10^{-4}$ \| 18.627 \| o vii \| 1s${}^{2}(^{1}$S) \| 1s13p${}^{1}(^{1}$P) \| 1000${}^{+40}_{-280}$ 78 \| 19.025 \| 1200${}^{+56}_{-59}$ \| 3.1${}^{+0.1}_{-0.2}\times 10^{-4}$ \| 18.968 \| o viii \| 1s${}^{1}(^{2}$S) \| 1s02p${}^{1}(^{2}$P) \| 240${}^{+80}_{-160}$ 79 \| 20.970 \| 1500${}^{+360}_{-190}$ \| 9.0${}^{+0.8}_{-1.4}\times 10^{-5}$ \| 20.910 \| n vii \| 1s${}^{1}(^{2}$S) \| 1s03p${}^{1}(^{2}$P) \| 280${}^{+40}_{-600}$ 80 \| 21.651 \| 1300${}^{+230}_{-38}$ \| 2.4${}^{+0.1}_{-0.2}\times 10^{-4}$ \| 21.602 \| o vii \| 1s${}^{2}(^{1}$S) \| 1s12p${}^{1}(^{1}$P) \| 460${}^{+440}_{-40}$ 81 \| 21.855 \| $\leq$ 2900 \| 1.7${}^{+0.1}_{-0.6}\times 10^{-4}$ \| 21.804 \| o vii \| 1s${}^{2}(^{1}$S) \| 1s12p${}^{1}(^{3}$P) \| 440${}^{+40}_{-1000}$ 82 \| 22.145 \| 560${}^{+27}_{-11}$ \| 5.8${}^{+0.5}_{-0.2}\times 10^{-4}$ \| 22.110 \| o vii \| 1s${}^{2}(^{1}$S) \| 1s12s${}^{1}(^{3}$S) \| 660${}^{+80}_{-40}$ 83 \| 23.625 \| $\leq$ 3600 \| 6.5${}^{+0.4}_{-2.1}\times 10^{-5}$ \| 23.440 \| o i \| 2p${}^{4}(^{1}$D2) \| 1s12s22p${}^{5}(^{1}$P1) \| - 84 \| 23.835 \| $\leq$ 2600 \| 8.6${}^{+0.6}_{-2.2}\times 10^{-5}$ \| 23.771 \| n vi \| 1s${}^{2}(^{1}$S) \| 1s14p${}^{1}(^{1}$P) \| 330${}^{+160}_{-720}$ 85 \| 24.840 \| 1300${}^{+210}_{-38}$ \| 3.1${}^{+0.1}_{-0.3}\times 10^{-4}$ \| 24.781 \| n vii \| 1s${}^{1}(^{2}$S) \| 1s02p${}^{1}(^{2}$P) \| 420${}^{+200}_{-40}$ 86 \| 24.840 \| 1300${}^{+210}_{-38}$ \| 3.1${}^{+0.1}_{-0.3}\times 10^{-4}$ \| 25.164 \| ar xv \| 2s${}^{2}(^{1}$S0) \| 2s13p${}^{1}(^{3}$P1) \| 5000${}^{+200}_{-39}$ Table 3: Model for Fe K region Parameters component \| log($\xi$) \| norm \| width (km/s) \| velocity (km/s) ---\|---\|---\|---\|--- photemis \| -3 \| 2.47${}^{+0.82}_{-0.61}\times 10^{8}$ \| 1.28${}^{+0.25}_{-0.23}\times 10^{3}$ \| 2.41${}^{+0.51}_{-0.67}\times 10^{2}$ km s-1 scatemis \| 3 \| 33.1${}^{+8.7}_{-9.7}$ \| 1.28${}^{+0.25}_{-0.23}\times 10^{3}$ \| 2.41${}^{+0.51}_{-0.67}\times 10^{2}$ km s-1 Table 4: Fitting Results: Table Models Component \| 1 \| 2 \| 3 \| 4 \| 5 \| 6 \| Total ---\|---\|---\|---\|---\|---\|---\|--- log(N) \| 23.5 \| 23.5 \| 23.5 \| 22.5 \| 22.5 \| 22.5 \| log($\xi$) \| 1 \| 1.8 \| 2.6 \| 1 \| 1.8 \| 2.6 \| N aaElemental abundances relative to solar (Grevesse et al., 1996) \| 0.99 $\pm{0.19}$ \| - \| - \| - \| - \| - \| O aaElemental abundances relative to solar (Grevesse et al., 1996) \| 2.48 $\pm{0.3}$ \| 0.86 $\pm{0.6}$ \| $\leq$0.09 \| 1.1 $\pm{1.5}$ \| $\leq$10 \| $\leq$0.1 \| Ne aaElemental abundances relative to solar (Grevesse et al., 1996) \| 1.91 $\pm{0.15}$ \| - \| - \| - \| - \| - \| Mg aaElemental abundances relative to solar (Grevesse et al., 1996) \| 1.06 $\pm{0.2}$ \| - \| - \| - \| - \| - \| Si aaElemental abundances relative to solar (Grevesse et al., 1996) \| 1.49 $\pm{0.1}$ \| - \| - \| - \| - \| - \| S aaElemental abundances relative to solar (Grevesse et al., 1996) \| 1.41 $\pm{0.3}$ \| - \| - \| - \| - \| - \| Ar aaElemental abundances relative to solar (Grevesse et al., 1996) \| 1.88 $\pm{0.5}$ \| - \| - \| - \| - \| - \| Ca aaElemental abundances relative to solar (Grevesse et al., 1996) \| 2.42 $\pm{0.9}$ \| - \| - \| - \| - \| - \| Fe aaElemental abundances relative to solar (Grevesse et al., 1996) \| $\leq$100 \| 2.7 $\pm{0.5}$ \| 1 $\pm{0.15}$ \| 8 $\pm{1}$ \| $\leq$100 \| 1.88 $\pm{1.68}$ \| $\kappa$ ($\times 10^{-6}$) \| 16.4 $\pm{3}$ \| 22\. $\pm{3}$ \| 83\. $\pm{6}$ \| 6.7 $\pm{1}$ \| $\leq$0.1 \| 3.9 $\pm{0.4}$ \| f ($\times 10^{-4}$) \| 34.0 $\pm{6.2}$ \| 45.6 $\pm{6.4}$ \| 172\. $\pm{12.4}$ \| 13.9 $\pm{2.1}$ \| $\leq$0.1 \| 8.0 $\pm{0.8}$ \| mass($\times 10^{4}M_{\odot}$)bbNote that these quantities depend on the density according to the equations in the text. \| 21.3 $\pm{3.9}$ \| 4.5 $\pm{0.6}$ \| 2.7 $\pm{0.2}$ \| 8.7 $\pm{1.3}$ \| $\leq$0.02 \| 0.13 $\pm{0.01}$ \| 37.3 $\pm{6.0}$ log(EM (cm-3)) \| 66.2 $\pm{0.07}$ \| 64.7 $\pm{0.06}$ \| 63.7 $\pm{0.03}$ \| 65.8 $\pm{0.06}$ \| 62.0 $\pm{0.3}$ \| 62.3 $\pm{0.04}$ \| 66.3 $\pm{0.88}$ Figure 1: Spectrum showing fits to table model described in the text. Vertical axis is counts s-1Hz-1 scaled according to the maximum flux in the panel. Figure 2: Spectrum showing fits to table model described in the text. Vertical axis is counts s-1Hz-1 scaled according to the maximum flux in the panel. Figure 3: Spectrum showing fits to table model described in the text. Vertical axis is counts s-1Hz-1 scaled according to the maximum flux in the panel. Figure 4: Spectrum showing fits to table model described in the text. Vertical axis is counts s-1Hz-1 scaled according to the maximum flux in the panel. Figure 5: Plot of the widths and velocity offsets of the lines shown in table 2 in km s-1. Only lines for which errors on the line width can be derived are plotted. Figure 6: Zero order image with positions of extraction regions shown. Lines corresponds to standard extraction region (4.8 arcsec width) plus half and double size regions as discussed in text. Only heg arm is shown for clarity. Figure 7: Values for the log of the ratios of line fluxes for regions with the standard size (4.8 arcsec) (green hexagons) and double this value (red diamonds) compared with line fluxes for a region half the standard size. Figure 8: Plot of the locus of points in the plane of the He/H line ratio vs. the G-ratio. See the text for definitions. The solid curves correspond to constant column density in where black= 1023.5, blue=1022.5, green=1021.5 and red=1020.5. The dashed curves correspond to constant ionization parameter in the range 1$\leq$log($\xi$)$\leq$3\. Red bars denote the range of measured values. Figure 9: Plot of the locus of points in the plane of the He/H line ratio vs. the G-ratio. See the text for definitions. The solid curves correspond to constant column density in where black= 1023.5, blue=1022.5, green=1021.5 and red=1020.5. The dashed curves correspond to constant ionization parameter in the range 1$\leq$log($\xi$)$\leq$3\. Red bars denote the range of measured values. Figure 10: Fe K line region. Lines are apparent from near-neutral Fe near 1.95 $\AA$, He-line near 1.85 $\AA$ and H-like near 1.75 $\AA$. The model consists of components with and without radiative excitation from the analytic model described in section 3, plus power law continuum. Figure 11: Distribution of mass among the emission components used to model the spectrum. Black points correspond to the component with column 3 $\times 10^{23}$ cm-2 and red corresponds to the component with column 3 $\times 10^{22}$ cm-2 Figure 12: Element abundances from best-fit models. Colors red, green and blue correspond to log($\xi$)=1, 1.8, 2.6, respectively; solid corresponds column 3 $\times 10^{23}$ cm-2 and dashed corresponds column 3 $\times 10^{22}$ cm-2. Black points are average over the best-fit model.	arxiv-papers	2013-11-05T21:11:41	2024-09-04T02:49:53.270919	{ "license": "Public Domain", "authors": "T. Kallman, D. A. Evans, H. Marshall, C. Canizares, A. Longinotti, M.\n Nowak, N. Schulz", "submitter": "T. Kallman", "url": "https://arxiv.org/abs/1311.1222" }
1311.1273	# Long-range string orders and topological quantum phase transitions in the one-dimensional quantum compass model Hai Tao Wang Centre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, The People’s Republic of China Sam Young Cho [email protected] Centre for Modern Physics and Department of Physics, Chongqing University, Chongqing 400044, The People’s Republic of China ###### Abstract In order to investigate the quantum phase transition in the one-dimensional quantum compass model, we numerically calculate non-local string correlations, entanglement entropy, and fidelity per lattice site by using the infinite matrix product state representation with the infinite time evolving block decimation method. In the whole range of the interaction parameters, we find that the four distinct string orders characterize the four different Haldane phases and the topological quantum phase transition occurs between the Haldane phases. The critical exponents of the string order parameters $\beta=1/8$ and the cental charges $c=1/2$ at the critical points show that the topological phase transitions between the phases belong to an Ising type of universality classes. In addition to the string order parameters, the singularities of the second derivative of the ground state energies per site, the continuous and singular behaviors of the von Neumann entropy, and the pinch points of the fidelity per lattice site manifest that the phase transitions between the phases are of the second-order, in contrast to the first-order transition suggested in pervious studies. ###### pacs: 75.10.Pq, 03.65.Vf, 03.67. Mn, 64.70.Tg ## I Introduction Transition metal oxides (TMOs) with orbital degeneracies have been intensively studied for quantum phase transitions (QPTs) because they have shown extremely rich phase diagrams due to competitions between orbital orderings and complex interplays between quantum fluctuations and spin interactions Brzezicki1 ; You1 ; Brzezicki2 ; Sun1 ; Eriksson ; Mahdavifar1 ; Liu1 ; Sun2 ; Wang ; Jafari1 ; Mahdavifar2 ; You2 ; Liu2 ; Wenzel ; Orus ; Jackeli . In order to mimic such competitions between orbital ordering in different directions and directional natures of the orbital states with twofold degeneracy in the language of the pseudospin-$1/2$ operators, Kugel and Khomskii Kugel first introduced the quantum compass model (QCM) in 1973. In this model, the pseudospin-1/2 operators characterize the orbital degrees of freedom, and the anisotropic couplings between these pseudospins simulate the competition between orbital orderings in different directions. Furthermore, such an idea has been implemented to describe some Mott insulators with orbital degeneracy Feiner ; Dorier , polar molecules in optical lattices Micheli and ion trap systems Milman , protected qubits for quantum computation in Josephson junction arrays Doucot , and so on. Based on the one-dimensional QCM, physical properties and QPTs in TMOs have been explored in the absence Brzezicki1 ; You1 ; Brzezicki2 ; Sun1 ; Eriksson ; Mahdavifar1 ; Liu1 or in the presence Sun2 ; Wang ; Jafari1 ; Mahdavifar2 ; You2 ; Liu2 of a transverse magnetic field. Especially for its criticality, in 2007, Brzezicki et al. Brzezicki1 used the Jordan-Wigner transformation mapping it to an Ising model, obtained an exact solution of the QCM, and suggested that the system has the first-order transition occurring between two disordered phases. In 2008, You and Tian You1 supported Brzezicki et al.’s result, i.e., the first-order transition by adopting the reflection positivity technique in the standard pseudospin representation. In 2009, in order to show that the phase transition is intrinsic to the system not an artifact originating from a singular parameterization of the exchange interactions, Brzezicki and Oleś Brzezicki2 revisited the model and reclaimed the first- order transition. In the same year, furthermore, Sun et al. Sun1 reached to a conclusion supporting the first-order phase transition between two different disordered phases by using the fidelity susceptibility and the concurrence. However, Sun and Chen Sun2 considered a transverse magnetic field on the QCM and found at the zero field that the phase transition is of the second-order from the finite-size scaling of the spin-spin correction as well as the fidelity susceptibility, the block entanglement entropy, and the concurrence. Also, following in Ref. Brzezicki1, but introducing one more tunable parameter, Eriksson and Johannesson Eriksson noticed the second-order phase transition rather than the first-order phase transition by using the concurrence and the block entanglement at the multicritical point in the one- dimensional extended quantum compass model (EQCM). In 2012, Liu et al. Liu1 numerically studied the EQCM by utilizing the matrix product state (MPS) with the infinite time-evolving block decimation (iTEBD) algorithm and, at the multicritical point, observed both features of the first-order and the second- order phase transition. Thus, the phase transition in the one-dimensional QCM is still not characterized clearly. Figure 1: (Color online) Groundstate phase diagram for the one-dimensional QCM in $J_{y}$-$J_{z}$ plane. The four topologically ordered phases are characterized by the four distinct string orders (defined in the text). The critical lines are (i) $J_{y}=J_{z}>0$ ($\theta=\pi/4$), (ii) $J_{y}=-J_{z}<0$ ($\theta=3\pi/4$), (iii) $J_{y}=J_{z}<0$ ($\theta=5\pi/4$), and (iv) $J_{y}=-J_{z}<0$ ($\theta=7\pi/4$). At the critical points, the central charges are $c=1/2$ and the critical exponent of each string order is $\beta=1/8$. The phase transitions between Haldane phases are a topological phase transition and belong to an Ising universality class. Here, the $\theta$ is the interaction parameter from the setting $J_{y}=J\cos\theta$ and $J_{z}=J\sin\theta$ for the numerical calculation. It seems to be believed that the phase transition occurs between two disordered phases in the one-dimensional QCM. Normally, disordered phases are not characterized by any local order parameter. This implies that the phase transition in the one-dimensional QCM would not be understood properly within the Landau paradigm of spontaneous symmetry breaking Sachdev . Consequently, such a controversy on the phase transition in the one-dimensional QCM would suggest us to consider non-local long-range orders for its proper characterization. In order to characterize the phase transition properly, in this paper, we investigate non-local string orders in the one-dimensional QCM. Actually, a string order as a non-local long-range order was introduced by Nijs and Rommelse Nijs and Tasaki Tasaki , and characterizes the Haldane phase in the spin-$1$ Heisenberg chain Yamamoto . To calculate non-local string orders directly Su , in contrast to an extrapolated value for finite- size lattices, we employ the infinite matrix product state (iMPS) Vidal1 ; Vidal2 representation with the iTEBD algorithm developed by Vidal Vidal2 . For a systematic study, the second derivative of ground state energy is calculated to reveal the phase transitions in the whole interaction parameter range. Its singularities indicate that there are the four phases separated by the second-order phase transitions. We find the four string order parameters that characterize each phases (see in Fig. 1), which means that all the four phases are a topologically ordered phase. Furthermore, the critical exponent from the string orders $\beta=1/8$ and the central charges $c\simeq 1/2$ at the critical points clarify that the topological quantum phase transitions (TQPTs) belong to the Ising-type phase transition. In addition, the continuous behaviors of the odd- and even-von Neumann entropies and the pinch points of the fidelity per lattice site (FLS) verify the second-order phase transitions between two topologically ordered phases. This paper is organized as follows. In Sec.II, we introduce the one- dimensional QCM and discuss the second derivative of the ground state energy per site. In Sec. III, we display string correlations and define properly the four string order parameters characterizing the four topologically ordered phases. The critical exponents are presented. The phase transitions are discussed by employing the von Neumann entropy in Sec. IV. The TQPTs are classified based on the central charge via the finite-entanglement scaling. In Sec. V, we discuss the pinch points of the FLS. Finally, our conclusion is given in Sec. VI. ## II Quantum Compass Model and groundstate energy We consider the one-dimensional spin-$1/2$ QCM Brzezicki1 written as $H=\sum_{i=-\infty}^{\infty}\left(J_{y}S^{y}_{2i-1}\cdot S^{y}_{2i}+J_{z}S^{z}_{2i}\cdot S^{z}_{2i+1}\right),$ (1) where $S^{y}_{i}$ and $S^{z}_{i}$ are the spin-$1/2$ operators on the $i$th site. $J_{y}$ and $J_{z}$ are nearest-neighbor exchange couplings on the odd and the even bonds, respectively. In order to cover the whole range of the parameter $J_{y}$ and $J_{z}$, we set $J_{y}=J\cos\theta$ and $J_{z}=J\sin\theta$. Figure 2: (Color online) (a) Groundstate energies per site on odd-/even-bonds $e_{0,odd/even}$, (b) average energy $e_{0}=(e_{0,odd}+e_{0,even})/2$, and (c) second derivative of the average energy $e_{0}$ as a function of the interaction parameter $\theta$. Here, the truncation dimension $\chi=40$ is chosen for the iMPS calculation. In (c), note that the singular behaviors of the second derivative occur at the points $\theta=\pi/4$, $\theta=3\pi/4$, $\theta=5\pi/4$, and $\theta=7\pi/4$. From the iMPS groundstate wavefunction, we obtain the groundstate energy of the QCM. In Figs. 2(a) and 2(b), we plot the groundstate energies $e_{0,odd}$ on the odd bond and $e_{0,even}$ on the even bond, and the groundstate energy per site $e_{0}$ as an average value of the energies $e_{0,odd}$ and $e_{0,even}$, i.e., $e_{0}=(e_{0,odd}+e_{0,even})/2$. Here, the truncation dimension is chosen as $\chi=40$. The energies are shown to be a periodic behavior as a function of the interaction parameter $\theta$. One way to know whether there is a phase transition is to check the non-analyticity of the groundstate energy on the system parameters. Thus, in order to see any possible phase transition, we calculate the derivative of the energies over the interaction parameter $\theta$. In the first derivative of the energies over the interaction parameter, no singular behavior is noticed in the whole parameter range. Then, in Fig. 2(c), we plot the second derivative of the energy $e_{0}$. Note that it exhibits the singular points at $\theta=\pi/4$, $\theta=3\pi/4$, $\theta=5\pi/4$, and $\theta=7\pi/4$. This result means that, at the singular points, the quantum phase transitions occur and they are of the second-order. As we introduced the controversy of the phase transition in the QCM, the critical point $\theta=\pi/4$ in our calculation corresponds to the the critical point $J_{y}=J_{z}$ investigated in previous studies. Consequently, our second derivative of the groundstate energy shows that the phase transition in the QCM should be of the second-order. Moreover, the critical lines separate the parameter space into the four regions [denoted by I, II, III, and IV in Fig. 1], which may indicate four possible phases. Then, in order to characterize the four possible phases, we discuss string correlations in the next section. ## III string order parameters and topological quantum phase transitions The QCM has the different strengthes of the spin exchange interaction depending on the odd and the even bonds. One can then define string correlations based on the bond alternation Hida ; Cho . Let us first consider the string correlations defined as $\displaystyle O^{\alpha}_{s,odd}\left(2i-1,2j\right)\\!\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\\!\left\langle S^{\alpha}_{2i-1}\exp\left[i\pi\sum_{k=2i}^{2j-1}S^{\alpha}_{k}\right]S^{\alpha}_{2j}\right\rangle$ (2a) $\displaystyle O^{\alpha}_{s,even}\left(2i,2j+1\right)\\!\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\\!\left\langle S^{\alpha}_{2i}\exp\left[i\pi\sum_{k=2i+1}^{2j}S^{\alpha}_{k}\right]S^{\alpha}_{2j+1}\right\rangle,$ (2b) where $\alpha=x$, $y$, and $z$. We observe numerically that the $x$ components of the string correlations $O^{x}_{str,odd/even}$ decrease to zero within the lattice distance $\|i-j\|=6$ in the whole parameter range. Figure 3: (Color online) (a) Behaviors of the odd string order parameters (indicated by the asterisk) in $J_{y}$-$J_{z}$ plane. (b) String correlations $O^{y/z}_{str,odd}$ for $\theta=0.2\pi$ and $\theta=0.8\pi$. In the insets, note that $O^{y}_{str,odd}$’s are saturated to a finite value, while $O^{z}_{str,odd}$’s decay to zero for very large distance. Behaviors of odd string correlations.$-$ In Fig. 3(a), we summarize the short- and long-distance behaviors of the odd string correlations $O^{y}_{str,odd}$ in $J_{y}$-$J_{z}$ plane. (i) For $\|J_{y}\|<\|J_{z}\|$ (the regions II and IV in Fig. 1), the odd string correlations $O^{y/z}_{str,odd}$ decrease to zero within the lattice distance $\|i-j\|=80$. (ii) For $\|J_{y}\|>\|J_{z}\|$ (the regions I and III in Fig. 1), the absolute value of $O^{y}_{str,odd}$ are saturated to a finite value while $O^{z}_{str,odd}$ decays to zero very slowly, which means $O^{y}_{str,odd}$ as a non-local long-range order parameter [indicated by the asterisk in Fig. 3(a)] can characterize a topologically ordered phase. Further, if $J_{y}>0$[region I] ($J_{y}<0$ [region III]), $O^{y}_{str,odd}$ shows a monotonic (oscillatory) saturation and $O^{z}_{str,odd}$ displays an oscillatory (monotonic) decaying to zero. As an example, in Fig. 3(b), we plot the odd string correlations $O^{y/z}_{str,odd}$ as a function of the lattice distance $\|i-j\|$ for $\theta=0.2\pi$ (the range I) and $\theta=0.8\pi$ (the region III). The string correlations are shown a very distinct behavior. For $\theta=0.2\pi$, the $O^{y}_{str,odd}$ has a minus sign, while the $O^{z}_{str,odd}$ has an alternating sign depending on the lattice distance. In contrast to the case of $\theta=0.2\pi$, for $\theta=0.8\pi$, the $O^{z}_{str,odd}$ has a minus sign, while the $O^{y}_{str,odd}$ has an alternating sign depending on the lattice distance. From the short-distance behaviors, as shown in Fig. 3(b), it is hard to see whether the string correlations decay to survive in the long distance limit (i.e., $\|i-j\|\rightarrow\infty$). In order to study the correlations in the limit of the infinite distance, one can then set a truncation error $\varepsilon$ rather than the lattice distance, i.e., $O(\|i-j\|)-O(\|i-j-1\|)<\varepsilon$. In this study, for instance, $\varepsilon=10^{-8}$ is set. The insets of Fig. 3(b) show the string correlations for relatively very large lattice distance. We see clearly that the $O^{y}_{str,odd}$’s have a finite value while the the $O^{z}_{str,odd}$’s decay to zero (around the lattice distance $\|i-j\|\sim 5\times 10^{4}$). As a consequence, the parameter regions I and III can be characterized by the odd string long-range order parameters. As discussed above, the odd string correlations have the two characteristic behaviors, i.e., one is a monotonic saturation for $\theta=0.2\pi$, the other is an oscillatory saturation for $\theta=0.8\pi$. Such a distinguishable behavior of the string correlations allows us to say that the region I ($J_{y}>0$) and the region III ($J_{y}<0$) are a different phase each other and we call the monotonic odd string order and the oscillatory odd string order, respectively. Figure 4: (Color online) (a) Behaviors of the even string order parameters (indicated by the asterisk) in $J_{y}$-$J_{z}$ plane. (b) String correlations $O^{y/z}_{str,even}$ for $\theta=0.3\pi$ and $\theta=1.3\pi$. In the insets, note that $O^{z}_{str,even}$’s are saturated to a finite value, while $O^{y}_{str,even}$’s decay to zero for very large distance. Behaviors of even string correlations.$-$ Similarly to the odd string correlations, the even string correlations show the two characteristic behaviors. In Fig. 4(a), we summarize the short- and long-distance behaviors of the even string correlations $O^{y}_{str,even}$ in $J_{y}$-$J_{z}$ plane. (i) For $\|J_{z}\|<\|J_{y}\|$ (the regions I and III in Fig. 1), the even string correlations $O^{y/z}_{str,even}$ decrease to zero within the lattice distance $\|i-j\|=80$. (ii) For $\|J_{z}\|>\|J_{y}\|$ (the regions II and IV in Fig. 1), the absolute value of $O^{z}_{str,even}$ are saturated to a finite value while $O^{y}_{str,even}$ decays to zero very slowly, which means $O^{z}_{str,even}$ as a non-local long-range order parameter [indicated by an asterisk in Fig. 4(a)] characterizes a topologically ordered phase. Further, if $J_{z}>0$[region II] ($J_{z}<0$ [region IV]), $O^{z}_{str,even}$ shows a monotonic (oscillatory) saturation and $O^{y}_{str,even}$ displays an oscillatory (monotonic) decaying to zero. As an example, in Fig. 4(b), we plot the even string correlations $O^{y/z}_{str,even}$ as a function of the lattice distance $\|i-j\|$ for $\theta=0.3\pi$ (the range II) and $\theta=1.3\pi$ (the region IV). For $\theta=0.3\pi$, the $O^{y}_{str,even}$ has an alternating sign depending on the lattice distance, while the $O^{z}_{str,even}$ has a minus sign. In contrast to the case of $\theta=0.3\pi$, for $\theta=1.3\pi$, the $O^{z}_{str,even}$ has an alternating sign depending on the lattice distance, while the $O^{y}_{str,even}$ has a minus sign. Similarly to the odd string correlations, the short distance behaviors of the even string correlations show to the difficulty to see which the string correlations survive in the long distance limit (i.e., $\|i-j\|\rightarrow\infty$). By using the truncation error $\varepsilon=10^{-8}$, we plot the string correlations for relatively very large lattice distance in the insets of Fig. 4(b). We see clearly that the $O^{z}_{str,even}$’s have a finite value while the the $O^{y}_{str,odd}$’s decay to zero (around the lattice distance $\|i-j\|\sim 2\times 10^{4}$). As a result, the parameter regions II and IV can be characterized by the even string long-range order parameters. The even string correlations also have the two characteristic behaviors, i.e., one is a monotonic saturation for $\theta=0.3\pi$, the other is an oscillatory saturation for $\theta=1.3\pi$. The region II ($J_{z}>0$) and the region IV ($J_{z}<0$) are a different phase each other and we call the monotonic even string order and the oscillatory even string order, respectively. Figure 5: (Color online) String order parameters (a) $O^{+/-,y}_{str,odd}$ and (b) $O^{+/-,z}_{str,even}$ as a function of $\theta$. The order parameters are defined in the text. Phase diagram from string order parameters.$-$ As we discussed, the even and odd string correlations have shown two characteristic behaviors, i.e., one is monotonic, the other is oscillatory. Then, one may define a proper long-range order based on the behaviors of the odd and the even string correlations. We define the long-range string order parameters as follows: $\displaystyle O^{+,y}_{str,odd}\\!\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\\!-\lim_{\|i-j\|\rightarrow\infty}O^{y}_{s,odd}\left(2i-1,2j\right),$ (3a) $\displaystyle O^{-,y}_{str,odd}\\!\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\\!-\lim_{\|i-j\|\rightarrow\infty}(-1)^{(j-i+1)}O^{y}_{s,odd}\left(2i-1,2j\right),$ (3b) $\displaystyle O^{+,z}_{str,even}\\!\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\\!-\lim_{\|i-j\|\rightarrow\infty}O^{z}_{s,even}\left(2i,2j+1\right),$ (3c) $\displaystyle O^{-,z}_{str,even}\\!\\!\\!$ $\displaystyle=$ $\displaystyle\\!\\!\\!-\lim_{\|i-j\|\rightarrow\infty}(-1)^{(j-i+1)}O^{z}_{s,even}\left(2i,2j+1\right),$ (3d) where, actually, the superscript $+$ ($-$) of the string order parameters denotes the monotonic behavior (the oscillatory behavior). The defined string orders are calculated from the iMPS groundstate wave function. In Fig. 5, we display the string order parameters as a function of the interaction parameter $\theta$. In Fig. 5(a), it is clearly shown that the odd string order parameters are finite for the region I ($-\pi/4<\theta<\pi/4$) and the region III ($3\pi/4<\theta<5\pi/4$). Further, the monotonic odd string order parameter $O^{+,y}_{str,odd}$ characterizes the region I and the oscillatory odd string order parameter $O^{-,y}_{str,odd}$ does the region III. Similarly to the odd string order parameters, the $z$ components of the even string order parameters $O^{+,z}_{str,even}$ and $O^{-,z}_{str,even}$ are finite for the region II ($\pi/4<\theta<3\pi/4$) and the region IV ($5\pi/4<\theta<7\pi/4$). The monotonic even string order parameter $O^{+,z}_{str,even}$ characterizes the region II and the oscillatory even string order parameter $O^{-,y}_{str,even}$ does the region IV. Consequently, the four regions in $J_{y}$-$J_{z}$ plane [Fig. 1] are characterized by the four string order parameters $O^{+/-,y}_{str,odd}$ and $O^{+/-,z}_{str,even}$, respectively, which implies that a different hidden $Z_{2}\times Z_{2}$ breaking symmetry occurs in each phase. Therefore, the one-dimensional QCM has the four distinct topologically ordered phases rather than disordered phases suggested in previous studies. The system undergoes a topological quantum phase transition between two topological ordered phases as the interaction parameter crosses the critical lines $\|J_{y}\|=\|J_{z}\|$. In addition, the continuous behaviors of the string order parameters across the critical lines show that the topological quantum phase transitions are of the continuous (second-order) phase transition rather than the discontinues (first-order) phase transition. In a previous study Liu1 on an EQCM, the existence of a string order has been noticed numerically for a relevant interaction parameter range. However, any characterization of phase has not been made in association with the one- dimensional QCM. However, the one-dimensional spin-$1/2$ Kitaev model Kitaev , which is equivalent to the one-dimensional QCM, has shown to have two string order parameters Feng based on the dual spin correlation function Pfeuty by using a dual transformation Fradkin ; Kohmoto mapping the model into a one- dimensional Ising model with a transverse field. The actual parameter range in the one-dimensional Kitaev model studied in Ref. Feng, corresponds to $J_{y}>0$ and $J_{z}>0$ in our one-dimensional QCM. The system was discussed to undergo a topological quantum phase transition at the critical point $J_{y}=J_{z}>0$ ($\theta=\pi/4$). In this sense, in the case of $J_{y},J_{z}>0$ in the one-dimensional QCM, we have numerically demonstrated and verified the existence of the string order parameters and the topological quantum phase transition as discussed in Ref. Feng, . Figure 6: (Color online) String order parameters (a) $O^{+,z}_{str,even}$ and (b) $O^{-,z}_{str,even}$ as a function of $\|\theta-\theta_{c}\|^{1/4}$ for $\theta_{c}=\pi/4$, $3\pi/4$, $5\pi/4$, and $7\pi/4$. Critical exponents.$-$ In the critical regimes, as the order parameters, the string orders should show a scaling behavior to characterize the phase transitions. We plot the string order parameters $O^{+,z}_{str,even}$ [Fig. 6(a)] and $O^{-,z}_{str,even}$ [Fig. 6(b)] as a function of $\|\theta-\theta_{c}\|^{1/4}$ with the critical points $\theta_{c}=\pi/4$, $3\pi/4$, $5\pi/4$, and $7\pi/4$. It is shown that all the string order parameters nearly collapse onto one scaling fitting function in the critical regimes, i.e., they scales as $O^{\pm,z/y}_{str,even/odd}\propto\|\theta-\theta_{c}\|^{1/4}$. As a result, the same critical exponents are given as $\beta=1/8$ via $O^{+/-,z}_{str,even}\propto\|\theta-\theta_{c}\|^{2\beta}$ Shelton , which reveals that the TQPTs belong the Ising-type phase transition. ## IV Entanglement entropy and central charge Quantum entanglement in many-body systems can be quantified by the von Neumann entropy that is a good measure of bipartite entanglement between two subsystems of a pure stateOsterloh ; Amico . Generally, for one-dimensional quantum spin lattices, at critical points,the von Neumann entropy exhibits its logarithmic scaling conforming conformal invariance. Its scaling is governed by a universal factor, i.e., a central charge $c$ of the associated conformal field theory. The central charge allows us to classify a universality class Cardy of quantum phase transition. In our iMPS representation, a diverging entanglement at quantum critical points gives simple scaling relations for (i) the von Neumann entropy $S$ and (ii) a correlation length $\xi$ with respect to the truncation dimension $\chi$ Tagliacozzo as follows: $\displaystyle\xi(\chi)$ $\displaystyle\propto$ $\displaystyle\xi_{0}\chi^{\kappa}$ (4a) $\displaystyle S(\chi)$ $\displaystyle\propto$ $\displaystyle\frac{c\kappa}{6}\log_{2}{\chi},$ (4b) where $\kappa$ is a so-called finite-entanglement scaling exponent and $\xi_{0}$ is a constant. Thus, one can calculate a central charge by using Eqs. (4a) and (4b). In order to obtain the von Neumann entropy, we partition the spin chain into the two parts denoted by the left semi-infinite chain $L$ and the right semi- infinite chain $R$. In terms of the reduced density matrix $\varrho_{L}$ or $\varrho_{R}$ of the subsystems $L$ and $R$, the von Neumann entropy can be defined as $S=-\mathrm{Tr}\varrho_{L}\log_{2}\varrho_{L}=-\mathrm{Tr}\varrho_{R}\log_{2}\varrho_{R}$. In the iMPS representation, the iMPS groundstate wavefunction can be written by the Schmidt decomposition $\|\Psi\rangle=\sum_{\alpha=1}^{\chi}\lambda_{\alpha}\|\phi^{L}_{\alpha}\rangle\|\phi^{R}_{\alpha}\rangle$, where $\|\phi^{L}_{\alpha}\rangle$ and $\|\phi^{R}_{\alpha}\rangle$ are the Schmidt bases for the semi-infinite chains $L(-\infty,\cdots,i)$ and $R(i+1,\cdots,\infty)$, respectively. $\lambda_{\alpha}^{2}$ are actually eigenvalues of the reduced density matrices for the two semi-infinite chains $L$ and $R$. In our four-site translational invariant iMPS representation, we have the four Schmidt coefficient matrices $\lambda_{A}$, $\lambda_{B}$, $\lambda_{C}$ and $\lambda_{D}$, which means that there are the four possible ways for the partitions. Due to the two-site translational invariance of the QCM, in fact, we have $\lambda_{A}=\lambda_{C}$ and $\lambda_{B}=\lambda_{D}$, i.e., one partition is on the odd sites, the other is on the even sites. From the $\lambda_{even}$ and $\lambda_{odd}$, one can obtain the two von Neumann entropies depending on the odd- or even-site partitions as $S_{even/odd}=-\sum_{\alpha=1}^{\chi}\lambda_{even/odd,\alpha}^{2}\log_{2}\lambda_{even/odd,\alpha}^{2},$ (5) where $\lambda_{even/odd,\alpha}$’s are diagonal elements of the matrix $\lambda_{even/odd}$. Figure 7: (Color online) Von Neumann entropies $S_{odd}$ and $S_{even}$ as a function of the interaction parameter $\theta$. Note that the entropy singular points at $\theta=\pi/4$, $3\pi/4$, $5\pi/4$, and $7\pi/4$ correspond to the critical points from the string order parameters. (b) Correlation length $\xi(\chi)$ as a function of the truncation dimension $\chi$ at the critical points $C_{1}(J_{x},J_{y})=(1,1)$, $C_{2}=(-1,1)$, $C_{3}=(-1,-1)$, and $C_{4}=(1,-1)$. (c) Von Neumann entropy $S(\chi)$ as a function of $\chi$ at the critical points. In Fig. 7(a), we plot the von Neumann entropies $S_{odd}(\theta)$ and $S_{even}(\theta)$ as a function of the control parameter $\theta$. One can easily notice that there are the four singular points $\theta=\pi/4$, $3\pi/4$, $5\pi/4$, and $7\pi/4$ in both the odd-bond and the even-bond entropies. The four singular points of the von Neumann entropies indicate a quantum phase transition at those points. It should be noted that the detected transition points from the von Neumann entropies correspond to the critical points from the second derivative of the groundstate energy and the string order parameters. The continuous behaviors of von Neumann entropies around critical points also indicate the occurrence of the continuous (second-order) quantum phase transition as the system crosses the transition points. Hence, it is shown that the von Neumann entropy can detect the topological quantum phase transitions. In Figs. 7(b) and 7(c), we plot the correlation length $\xi(\chi)$ as a function of the truncation dimension $\chi$ and the von Neumann entropy $S(\chi)$ as a function of $\chi$ at the critical points $C_{1}(J_{1},J_{2})=(1,1)$, $C_{2}=(-1,1)$, $C_{3}=(-1,-1)$, and $C_{4}=(1,-1)$, respectively. The truncation dimensions are taken as $\chi=12,16,20,24,28,32,40$, and $44$. The correlation length $\xi(\chi)$ and the von Neumann entropy $S(\chi)$ diverge as the truncation dimension $\chi$ increases. Using the numerical fitting function $\xi(\chi)=\xi_{0}\chi^{\kappa}$ in Eq. (4a), the fitting constants are obtained as (i) $\xi_{0}=0.04$ and $\kappa=2.071$ at $C_{1}$, (ii) $\xi_{0}=0.041$ and $\kappa=2.068$ at $C_{2}$, (iii) $\xi_{0}=0.039$ and $\kappa=2.087$ at $C_{3}$, and (iv) $\xi_{0}=0.041$ and $\kappa=2.065$ at $C_{4}$. In order to obtain the central charge, we use the numerical fitting function of the von Neumann entropy $S(\chi)=(c\kappa/6)\log_{2}\chi+S_{0}$. As shown in Figs. 7(c), the linear scaling behaviors of the entropies give (i) $c=0.5079$ with $S_{0}=0.331$ at $C_{1}$, (ii) $c=0.4992$ with $S_{0}=0.3464$ at $C_{2}$, (iii) $c=0.5048$ with $S_{0}=0.314$ at $C_{3}$, and (iv) $c=0.5983$ with $b=0.34$ at $C_{4}$. Our central charges are very close to the value $c=0.5$, respectively. Consequently, the topological quantum phase transitions at all the critical points belong to the same universality class, i.e., the Ising universality class. This result is consistent with the universality class from the critical exponent $\beta=1/8$ of the string order parameters. ## V Fidelity per lattice site Figure 8: (Color online) Fidelity per site $d(\theta,\theta^{\prime})$ surface as a function of the two parameters $\theta$ and $\theta^{\prime}$. The pinch points $\theta=\pi/4$, $3\pi/4$, $5\pi/4$, and $7\pi/4$ on the FLS surface indicate the occurrence of the continuous phase transitions. Similarly to the von Neumann entropy, the fidelity per lattice site (FLS) Zhou is known to enable us to detect a phase transition point as an universal indicator without knowing any order parameters. From our iMPS groundstate wave function $\|\Psi(\theta)\rangle$ with the interaction parameter $\theta$, we define the fidelity as $F(\theta,\theta^{\prime})=\|\langle\Psi(\theta)\|\Psi(\theta^{\prime})\rangle\|$. Following Ref. Zhou, , the ground-state FLS $d(\theta,\theta^{\prime})$ can then be defined as $\ln d(\theta,\theta^{\prime})=\lim_{L\rightarrow\infty}\frac{\ln F(\theta,\theta^{\prime})}{L},$ (6) where $L$ is the system size. In Fig. 8, the groundstate FLS $d(\theta,\theta^{\prime})$ is plotted in $\theta$-$\theta^{\prime}$ parameter space. The FLS surface reveals that there are the four pinch points $\theta=\pi/4$, $3\pi/4$, $5\pi/4$, and $7\pi/4$. Each pinch point corresponds to each phase transition point from the second- order derivative of the ground-state energy, the string order parameters, and the von Neumann entropy. In addition, the continuous behavior of the groundstate FLS verifies that the second-order quantum phase transitions occur at the pinch points. ## VI Conclusion We have investigated the quantum phase transition in the one-dimensional QCM by using the iMPS representation with the iTEBD algorithm. To characterize quantum phases in the one-dimensional QCM, we introduced the odd and the even string correlations based on the alternating strength of the exchange interaction. We have observed that there are the two distinct behaviors of the odd and the string correlations, i.e., one is of the monotonic, (ii) the other is of the oscillatory. Based on the topological characterization, we find that there are the four topologically ordered phases in the whole interaction parameter range [Fig. 1]. In the critical regimes, the critical exponents of the string order parameters are obtained as $\beta=1/8$, which implies that the topological quantum phase transitions belong to the Ising type of universality class. Consistently, we obtain the central charges $c=1/2$ from the entanglement entropy. 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Barjaktarevi$\check{\rm c}$, J. Phys. A: Math. Theor. 41, 412001 (2008); H.-Q. Zhou, R. Orús, and G. Vidal, Phys. Rev. Lett. 100, 080601 (2008).	arxiv-papers	2013-11-06T02:39:34	2024-09-04T02:49:53.286836	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Hai Tao Wang and Sam Young Cho", "submitter": "Haitao Wang", "url": "https://arxiv.org/abs/1311.1273" }
1311.1275	# A new optical field state as an output of diffusion channel when the input being number state ††thanks: This work was supported by the National Natural Science Foundation of China (Grant Nos. 11175113 and 11264018), and the Young Talents Foundation of Jiangxi Normal University. Hong-Yi Fan1† , Sen-Yue Lou1, Xiao-Yin Pan1 and Li-Yun Hu2 1Department of Physics, Ningbo University, Ningbo 315211, P. R. China 2Department of physics, Jiangxi Normal University, Nanchang, 330022 Correspondence authorCorrespondence author ###### Abstract We theoretically propose a new optical field state $\rho_{new}=\lambda\left(1-\lambda\right)^{l}\colon L_{k}\left(\frac{-\lambda^{2}a^{{\dagger}}a}{1-\lambda}\right)e^{-\lambda a^{{\dagger}}a}\colon$ (here $::$ denotes normal ordeing symbol) which is named Laguerre-polynomial- weighted chaotic field. We show that such state can be implemented, i.e., when a number state enters into a diffusion channel, the output state is just this kind of states. We solve the master equation describing the diffusion process by using the summation method within ordered product of operators and the entangled state representaion. The solution manifestly shows how a pure state evolves into a mixed state. The physical difference between the diffusion and the amplitude damping is pointed out. ## 1 Introduction In quantum optics theory there are some typical states, e.g., number state, coherent state, and squeezed state, these are pure states; there are also some mixed states, the typical one is the chaotic state described by $\rho_{c}=(1-e^{-\lambda})\exp\left(-\lambda a^{{\dagger}}a\right),$ (1) where $a\ $and $a^{{\dagger}}$ are photon annihilation and creation operators, obeying $\left[a,a^{\dagger}\right]=1$, $tr\rho_{c}=1.$ The normally orderd form of $\rho_{c}$ is $\rho_{c}=(1-e^{-\lambda})\colon\exp\left[\left(e^{-\lambda}-1\right)a^{{\dagger}}a\right]\colon,$ where the symbol $\colon$ $\colon$ denotes normal ordeing symbol. In this work we shall report that there exists another important mixed state which appears in normally ordered form $\rho_{new}=\lambda\left(1-\lambda\right)^{l}\colon L_{l}\left(\frac{-\lambda^{2}a^{{\dagger}}a}{1-\lambda}\right)e^{-\lambda a^{{\dagger}}a}\colon.$ (2) Here $L_{l}$ is the $l$-th Laguerre polynomial, $tr\rho_{new}=1$ (see Appendix 1). We show that this mixed state will appear experimently as it represents the output state of a diffusion process with the input state being a pure number state. When a pure state evolves into a mixed state the quantum decoherence happens. Decoherence is an important topic in quantum information processing. In nature, systems we concerned usually are surrounded by thermo reservoir, so some dissipative process or diffusion process naturally happen. An interesting question thus arises: when an input state for a diffusion channel is a number state $\left\|l\right\rangle\left\langle l\right\|,(\left\|l\right\rangle=\frac{a^{{\dagger}l}}{\sqrt{l!}}\left\|0\right\rangle),$ then how does it evolve with time? What kind of optical field will the output state be? The master equation describing the diffusion process is [1, 2] $\frac{d}{dt}\rho=-\kappa\left(a^{\dagger}a\rho-a\rho a^{\dagger}-a^{\dagger}\rho a+\rho a^{\dagger}a\right).$ (3) We shall first obtain $\rho\left(t\right)$ by deriving its infinite operator- sum form $\rho\left(t\right)=\sum\limits_{i,j}M_{i,j}\rho_{0}M_{i,j}^{\dagger},$ (4) where $M_{i,j}$ in general is named Kraus operator [3], whose concrete from will be derived for this diffusion problem, and then we examine how $\rho_{0}=$ $\left\|l\right\rangle\left\langle l\right\|$ evolves through the relation (4). We will employ the thermo entangled state representation and the technique of integration within an ordered product (IWOP) of operators [4-5] to realize our goal. ## 2 Solution of Eq. (3) obtained by entangled state representation and IWOP technique We begin with introducing the thermo entangled state [6] $\left\|\eta\right\rangle=\exp\left[-\frac{1}{2}\left\|\eta\right\|^{2}+\eta a^{\dagger}-\eta^{\ast}\tilde{a}^{\dagger}+a^{\dagger}\tilde{a}^{\dagger}\right]\left\|0\tilde{0}\right\rangle,$ (5) where $\tilde{a}^{\dagger}$ is a fictitious mode accompanying the real mode $a^{\dagger},$ $\left[\tilde{a},\tilde{a}^{\dagger}\right]=1$. $\left\|\eta\right\rangle$ obeys the eigenvector equations $\left(a-\tilde{a}^{\dagger}\right)\|\eta\rangle=\eta\|\eta\rangle,\text{ }\left(a^{\dagger}-\tilde{a}\right)\|\eta\rangle=\eta^{\ast}\|\eta\rangle,$ (6) $\left\langle\eta\right\|\left(a^{\dagger}-\tilde{a}\right)=\eta^{\ast}\left\langle\eta\right\|,\text{ }\left\langle\eta\right\|\left(a-\tilde{a}^{\dagger}\right)=\eta\left\langle\eta\right\|.$ (7) Using the normal ordering form of vacuum projector $\left\|0\tilde{0}\right\rangle\left\langle 0\tilde{0}\right\|=\colon e^{-a^{\dagger}a-\tilde{a}^{\dagger}\tilde{a}}\colon$, and the IWOP technique we can show the orthonormal and completeness relation $\left\langle\eta^{\prime}\right\|\left.\eta\right\rangle=\pi\delta\left(\eta^{\prime}-\eta\right)\delta\left(\eta^{\prime\ast}-\eta^{\ast}\right),$ (8) $\displaystyle 1$ $\displaystyle=$ $\displaystyle\int\frac{d^{2}\eta}{\pi}\left\|\eta\right\rangle\left\langle\eta\right\|$ $\displaystyle=$ $\displaystyle\int\frac{d^{2}\eta}{\pi}:\exp\left[-\|\eta\|^{2}+\eta a^{\dagger}-\eta^{\ast}\tilde{a}^{\dagger}+\eta^{\ast}a-\eta\tilde{a}+a^{\dagger}\tilde{a}^{\dagger}+a\tilde{a}-a^{\dagger}a-\tilde{a}^{\dagger}\tilde{a}\right]:=1.$ Let $\left\|\eta=0\right\rangle=e^{a^{\dagger}\tilde{a}^{\dagger}}\left\|0\tilde{0}\right\rangle\equiv\left\|I\right\rangle,$ (10) we have $a\left\|I\right\rangle=\tilde{a}^{\dagger}\left\|I\right\rangle,\text{ }a^{\dagger}\left\|I\right\rangle=\tilde{a}\left\|I\right\rangle,\text{ }(a^{\dagger}a)^{n}\left\|I\right\rangle=(\tilde{a}^{\dagger}\tilde{a})^{n}\left\|I\right\rangle.$ (11) Operating the two-sides of $(3)$ on $\left\|I\right\rangle,$ noting that the real field $\rho$ is independent of the fictitious mode, $\left[\rho,\tilde{a}\right]=0,$ $\left[\rho,\tilde{a}^{\dagger}\right]=0,$ and using (12) we have $\frac{d}{dt}\rho\left\|I\right\rangle=-\kappa\left(a^{\dagger}a\rho-a\tilde{a}\rho-a^{\dagger}\tilde{a}^{\dagger}\rho+\tilde{a}\tilde{a}^{\dagger}\rho\right)\left\|I\right\rangle.$ (12) Letting $\rho\left\|I\right\rangle\equiv\left\|\rho\right\rangle,$ we see $\frac{d}{dt}\left\|\rho\right\rangle=-\kappa\left(a^{\dagger}-\tilde{a}\right)\left(a-\tilde{a}^{\dagger}\right)\left\|\rho\right\rangle,$ (13) its formal solution is $\left\|\rho\right\rangle=\exp\left[-\kappa t\left(a^{\dagger}-\tilde{a}\right)\left(a-\tilde{a}^{\dagger}\right)\right]\left\|\rho_{0}\right\rangle,$ (14) where $\left\|\rho_{0}\right\rangle=\rho_{0}\left\|I\right\rangle.$ Projecting this equation onto the entanged state representation $\left\langle\eta\right\|$ and using the eigenvalue equation $(7)$ we have $\left\langle\eta\right.\left\|\rho\right\rangle=\left\langle\eta\right\|\exp\left[-\kappa t\left(a^{\dagger}-\tilde{a}\right)\left(a-\tilde{a}^{\dagger}\right)\right]\left\|\rho_{0}\right\rangle=e^{-\kappa t\|\eta\|^{2}}\left\langle\eta\right.\left\|\rho_{0}\right\rangle.$ (15) Multiplying the two-sides of (15) by $\int\frac{d^{2}\eta}{\pi}\left\|\eta\right\rangle$ and using the completeness relation (2) as well as the IWOP technique we obatin $\displaystyle\left\|\rho\right\rangle$ $\displaystyle=$ $\displaystyle\int\frac{d^{2}\eta}{\pi}e^{-\kappa t\|\eta\|^{2}}\left\|\eta\right\rangle\left\langle\eta\right.\left\|\rho_{0}\right\rangle$ $\displaystyle=$ $\displaystyle\int\frac{d^{2}\eta}{\pi}:e^{-\left(\kappa t+1\right)\|\eta\|^{2}+\eta a^{\dagger}-\eta^{\ast}\tilde{a}^{\dagger}+\eta^{\ast}a-\eta\tilde{a}+a^{\dagger}\tilde{a}^{\dagger}+a\tilde{a}-a^{\dagger}a-\tilde{a}^{\dagger}\tilde{a}}:\left\|\rho_{0}\right\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{1+\kappa t}\colon\exp\left[\frac{\kappa t}{1+\kappa t}\left(a^{\dagger}\tilde{a}^{\dagger}+a\tilde{a}-a^{\dagger}a-\tilde{a}^{\dagger}\tilde{a}\right)\right]\colon\left\|\rho_{0}\right\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{1+\kappa t}e^{\frac{\kappa t}{1+\kappa t}a^{\dagger}\tilde{a}^{\dagger}}\left(\frac{1}{1+\kappa t}\right)^{a^{\dagger}a+\tilde{a}^{\dagger}\tilde{a}}e^{\frac{\kappa t}{1+\kappa t}a\tilde{a}}\rho_{0}\left\|I\right\rangle,$ where we have noticed $\colon\exp\left[\frac{-\kappa t}{1+\kappa t}\left(a^{\dagger}a+\tilde{a}^{\dagger}\tilde{a}\right)\right]\colon=\left(\frac{1}{1+\kappa t}\right)^{a^{\dagger}a+\tilde{a}^{\dagger}\tilde{a}}$ (17) Using $\left[\tilde{a},\rho_{0}\right]=0,$ $\tilde{a}\left\|I\right\rangle=a^{\dagger}\left\|I\right\rangle$ we have $\displaystyle e^{\frac{\kappa t}{1+\kappa t}a\tilde{a}}\rho_{0}\left\|I\right\rangle$ $\displaystyle=$ $\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}\left(\frac{\kappa t}{1+\kappa t}a\right)^{n}\rho_{0}\tilde{a}^{n}\left\|I\right\rangle$ (18) $\displaystyle=$ $\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}\left(\frac{\kappa t}{1+\kappa t}\right)^{n}a^{n}\rho_{0}a^{\dagger n}\left\|I\right\rangle,$ After substituting $(18)$ into $(16)$ and then using the property that $\tilde{a}^{\dagger}\tilde{a}$ is commutable with all real field operators and $f(a^{\dagger}a)\left\|I\right\rangle=f(\tilde{a}^{\dagger}\tilde{a})\left\|I\right\rangle,$ we can put Eq.(2) into the following form $\displaystyle\left\|\rho\right\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{1+\kappa t}e^{\frac{\kappa t}{1+\kappa t}a^{\dagger}\tilde{a}^{\dagger}}\left(\frac{1}{1+\kappa t}\right)^{a^{\dagger}a+\tilde{a}^{\dagger}\tilde{a}}\sum_{n=0}^{\infty}\frac{1}{n!}\left(\frac{\kappa t}{1+\kappa t}\right)^{n}a^{n}\rho_{0}a^{\dagger n}\left\|I\right\rangle$ (19) $\displaystyle=$ $\displaystyle\frac{1}{1+\kappa t}e^{\frac{\kappa t}{1+\kappa t}a^{\dagger}\tilde{a}^{\dagger}}\left(\frac{1}{1+\kappa t}\right)^{a^{\dagger}a}\sum_{n=0}^{\infty}\frac{1}{n!}\left(\frac{\kappa t}{1+\kappa t}\right)^{n}a^{n}\rho_{0}a^{\dagger n}\left(\frac{1}{1+\kappa t}\right)^{\tilde{a}^{\dagger}\tilde{a}}\left\|I\right\rangle$ $\displaystyle=$ $\displaystyle\frac{1}{1+\kappa t}\sum_{m=0}^{\infty}\frac{1}{m!}\left(\frac{\kappa t}{1+\kappa t}\right)^{m}a^{\dagger m}\left(\frac{1}{1+\kappa t}\right)^{a^{\dagger}a}$ $\displaystyle\times\sum_{n=0}^{\infty}\frac{1}{n!}\left(\frac{\kappa t}{1+\kappa t}\right)^{n}a^{n}\rho_{0}a^{\dagger n}\left(\frac{1}{1+\kappa t}\right)^{a^{\dagger}a}\tilde{a}^{\dagger m}\left\|I\right\rangle.$ Finally, using $\tilde{a}^{\dagger m}\left\|I\right\rangle=a^{m}\left\|I\right\rangle$ we obtain $\displaystyle\rho\left(t\right)\left\|I\right\rangle$ $\displaystyle=$ $\displaystyle\sum_{m,n=0}^{\infty}\frac{\left(\kappa t\right)^{m+n}}{m!n!\left(\kappa t+1\right)^{m+n+1}}a^{\dagger m}\left(\frac{1}{1+\kappa t}\right)^{a^{\dagger}a}$ (20) $\displaystyle\times a^{n}\rho_{0}a^{\dagger n}\left(\frac{1}{1+\kappa t}\right)^{a^{\dagger}a}a^{m}\left\|I\right\rangle.$ It then follows the infinite sum form $\displaystyle\rho\left(t\right)$ $\displaystyle=$ $\displaystyle\sum\limits_{m,n=0}^{\infty}\frac{\left(\kappa t\right)^{m+n}}{m!n!\left(\kappa t+1\right)^{m+n+1}}a^{\dagger m}\left(\frac{1}{1+\kappa t}\right)^{a^{\dagger}a}a^{n}\rho_{0}a^{\dagger n}\left(\frac{1}{1+\kappa t}\right)^{a^{\dagger}a}a^{m}$ $\displaystyle\equiv$ $\displaystyle\sum\limits_{m,n=0}^{\infty}M_{m,n}\rho_{0}M_{m,n}^{\dagger}$ where $M_{m,n}=\sqrt{\frac{1}{m!n!}\frac{\left(\kappa t\right)^{m+n}}{\left(\kappa t+1\right)^{m+n+1}}}a^{\dagger m}\left(\frac{1}{1+\kappa t}\right)^{a^{\dagger}a}a^{n},$ (22) satisfying $\sum_{m,n=0}^{\infty}M_{m,n}^{\dagger}M_{m,n}=1$, which is trace conservative (see Appendix 2). Thus we have employed the entangled state representation to analytically derive the infinitive sum form of $\rho\left(t\right).$ ## 3 Diffusion of a number state We now consider the case that a number state undergoes the diffusion channel, i.e., let $\rho_{0}$ in Eq. (2) be $\left\|l\right\rangle\left\langle l\right\|$ and we begin with considering the part of summation over $n$ in Eq. $(21)$ $\displaystyle\mathfrak{I}$ $\displaystyle\equiv$ $\displaystyle\sum_{n=0}^{l}\frac{\left(\kappa t\right)^{n}}{n!\left(\kappa t+1\right)^{n}}\left(\frac{1}{1+\kappa t}\right)^{a^{\dagger}a}a^{n}\left\|l\right\rangle\left\langle l\right\|a^{\dagger n}\left(\frac{1}{1+\kappa t}\right)^{a^{\dagger}a}$ (23) $\displaystyle=$ $\displaystyle\sum_{n=0}^{l}\frac{1}{n!}\frac{\left[\kappa t\left(\kappa t+1\right)\right]^{n}}{\left(\kappa t+1\right)^{2l}}\frac{l!}{\left[\left(l-n\right)!\right]^{2}}\left(a^{\dagger}\right)^{l-n}\left\|0\right\rangle\left\langle 0\right\|a^{l-n}.$ Using the definition of the two-variable Hermite polynomials $H_{m,n}\left(x,y\right)=\sum_{l=0}^{\min(m,n)}\frac{m!n!(-1)^{l}}{l!(m-l)!(n-l)!}x^{m-l}y^{n-l},$ (24) and $\left\|0\right\rangle\left\langle 0\right\|=\colon e^{-a^{\dagger}a}\colon$, we see $\mathfrak{I}=\frac{1}{l!}\left(\frac{-\kappa t}{\kappa t+1}\right)^{l}\colon H_{l,l}\left(\frac{ia^{\dagger}}{\sqrt{\kappa t\left(\kappa t+1\right)}},\frac{ia}{\sqrt{\kappa t\left(\kappa t+1\right)}}\right)e^{-a^{\dagger}a}\colon,$ (25) then inserting (25) into (2) and using the summation method within ordered product of operators yields $\displaystyle\rho\left(t\right)$ $\displaystyle=$ $\displaystyle\frac{1}{l!}\left(\frac{-\kappa t}{\kappa t+1}\right)^{l}\sum_{m=0}^{\infty}\frac{1}{m!}\frac{\left(\kappa t\right)^{m}}{\left(\kappa t+1\right)^{m+1}}$ (26) $\displaystyle\times\colon a^{\dagger m}a^{m}H_{l,l}\left[\frac{ia^{\dagger}}{\sqrt{\kappa t\left(\kappa t+1\right)}},\frac{ia}{\sqrt{\kappa t\left(\kappa t+1\right)}}\right]e^{-a^{\dagger}a}\colon$ $\displaystyle=$ $\displaystyle\frac{\left(-\kappa t\right)^{l}}{l!\left(\kappa t+1\right)^{l+1}}\colon e^{\frac{-1}{\kappa t+1}a^{\dagger}a}H_{l,l}\left[\frac{ia^{\dagger}}{\sqrt{\kappa t\left(\kappa t+1\right)}},\frac{ia}{\sqrt{\kappa t\left(\kappa t+1\right)}}\right]\colon.$ Using the definition of Laguerre-polynomial $L_{l}\left(x\right)=\sum\binom{l}{l-k}\frac{\left(-x\right)^{k}}{k!}$ (27) and $L_{l}\left(xy\right)=\frac{\left(-1\right)^{l}}{l!}H_{l,l}(x,y),$ (28) then (26) becomes $\rho\left(t\right)=\frac{\left(\kappa t\right)^{l}}{\left(\kappa t+1\right)^{l+1}}\colon L_{l}\left(\frac{-a^{\dagger}a}{\kappa t\left(\kappa t+1\right)}\right)e^{\frac{-1}{\kappa t+1}a^{\dagger}a}\colon.$ (29) Noting $e^{\frac{-1}{\kappa t+1}a^{\dagger}a}$ represents a chaotic photon field, so $\rho\left(t\right)$ is a Laguerre-polynomial-weighted chaotic field. Thus we see $\left\|l\right\rangle\left\langle l\right\|$ evolves into the mixed state (29), so this diffusion process manifestly embodies quantum decoherence. As Eq. (29) is just in the type of Eq. (2), so we can confirm the state described by Eq. (2) indeed exists as an quantum optical field. Before we check $Tr\rho\left(t\right)=1$ for Eq. (29), let us present an integration formula $\int\frac{d^{2}\alpha}{\pi}e^{\lambda\|\alpha\|^{2}}\|\alpha\|^{2k}=\left(\frac{\partial}{\partial\lambda}\right)^{k}\int\frac{d^{2}\alpha}{\pi}e^{\lambda\|\alpha\|^{2}}=k!\left(\frac{-1}{\lambda}\right)^{k+1},$ (30) then we introduce the completeness relation of coherent state $\int\frac{d^{2}\alpha}{\pi}\left\|\alpha\right\rangle\left\langle\alpha\right\|=1$ (31) here $\left\|\alpha\right\rangle=\exp\left[\alpha a^{\dagger}-\|\alpha\|^{2}/2\right]\left\|0\right\rangle$. Due to $a\left\|\alpha\right\rangle=\alpha\left\|\alpha\right\rangle,\left\langle\alpha\right\|\colon f\left(a^{\dagger},a\right)\colon\left\|\alpha\right\rangle=f\left(\alpha^{\ast},\alpha\right),$ (32) we see $\displaystyle\left\langle\alpha\right\|\colon L_{l}\left(\frac{-a^{\dagger}a}{\kappa t\left(\kappa t+1\right)}\right)e^{\frac{-1}{\kappa t+1}a^{\dagger}a}\colon\left\|\alpha\right\rangle$ (33) $\displaystyle=$ $\displaystyle e^{\frac{-1}{\kappa t+1}\|\alpha\|^{2}}\sum_{k=0}^{l}\frac{l!(-1)^{k}}{k!\left[(l-k)!\right]^{2}}\left(\frac{-\|\alpha\|^{2}}{\kappa t\left(\kappa t+1\right)}\right)^{l-k}.$ Substituting (33) into $tr\rho\left(t\right)=\int\frac{d^{2}\alpha}{\pi}\left\langle\alpha\right\|\rho\left(t\right)\left\|\alpha\right\rangle$ we should calculate $\displaystyle tr\rho\left(t\right)$ $\displaystyle=$ $\displaystyle\int\frac{d^{2}\alpha}{\pi}\left\langle\alpha\right\|\rho\left(t\right)\left\|\alpha\right\rangle$ (34) $\displaystyle=$ $\displaystyle\int\frac{d^{2}\alpha}{\pi}\left\langle\alpha\right\|\frac{\left(\kappa t\right)^{l}}{\left(\kappa t+1\right)^{l+1}}\colon e^{\frac{-1}{\kappa t+1}a^{\dagger}a}L_{l}\left(\frac{-a^{\dagger}a}{\kappa t\left(\kappa t+1\right)}\right)\colon\left\|\alpha\right\rangle$ $\displaystyle=$ $\displaystyle\frac{\left(\kappa t\right)^{l}}{\left(\kappa t+1\right)^{l+1}}\sum_{k=0}^{l}\frac{l!}{k!\left[(l-k)!\right]^{2}}\int\frac{d^{2}\alpha}{\pi}e^{\frac{-1}{\kappa t+1}\left\|\alpha\right\|^{2}}\left(\frac{\|\alpha\|^{2}}{\kappa t\left(\kappa t+1\right)}\right)^{l-k}.$ By setting $\frac{\alpha}{\sqrt{\kappa t\left(\kappa t+1\right)}}=\alpha^{\prime}$ we reform the integration as $\displaystyle\int\frac{d^{2}\alpha}{\pi}e^{\frac{-1}{\kappa t+1}\left\|\alpha\right\|^{2}}\left(\frac{\|\alpha\|^{2}}{\kappa t\left(\kappa t+1\right)}\right)^{l-k}$ (35) $\displaystyle=$ $\displaystyle\kappa t\left(\kappa t+1\right)\int\frac{d^{2}\alpha^{\prime}}{\pi}e^{-\kappa t\left\|\alpha^{\prime}\right\|^{2}}\left(\|\alpha^{\prime}\|^{2}\right)^{l-k}$ $\displaystyle=$ $\displaystyle\kappa t\left(\kappa t+1\right)\left(l-k\right)!\left(\frac{1}{\kappa t}\right)^{l-k+1}.$ Substituting (35) into (34) we see $tr\rho\left(t\right)=\frac{\left(\kappa t\right)^{l+1}}{\left(\kappa t+1\right)^{l}}\sum_{k=0}^{l}\frac{l!}{k!(l-k)!}\left(\frac{1}{\kappa t}\right)^{l-k+1}=1$ (36) so it is trace conservative. ## 4 The photon number in the mixed state Then we evaluate the photon number for Eq. $(29)$ $\displaystyle Tr\left[\rho\left(t\right)a^{\dagger}a\right]$ $\displaystyle=$ $\displaystyle Tr\left[\rho\left(t\right)aa^{\dagger}\right]-1$ (37) $\displaystyle=$ $\displaystyle\frac{\left(\kappa t\right)^{l}}{\left(\kappa t+1\right)^{l+1}}Tr\left[\colon a^{\dagger}ae^{\frac{-1}{\kappa t+1}a^{\dagger}a}L_{l}\left(\frac{-a^{\dagger}a}{\kappa t\left(\kappa t+1\right)}\right)\colon\right]-1.$ By using the coherent state representation we have $\displaystyle Tr\left[\colon a^{\dagger}ae^{\frac{-1}{\kappa t+1}a^{\dagger}a}L_{l}\left(\frac{-a^{\dagger}a}{\kappa t\left(\kappa t+1\right)}\right)\colon\right]$ (38) $\displaystyle=$ $\displaystyle\int\frac{d^{2}\alpha}{\pi}e^{\frac{-1}{\kappa t+1}\left\|\alpha\right\|^{2}}\left\|\alpha\right\|^{2}L_{l}\left(\frac{-\left\|\alpha\right\|^{2}}{\kappa t\left(\kappa t+1\right)}\right)$ $\displaystyle=$ $\displaystyle\left[\kappa t\left(\kappa t+1\right)\right]^{2}\int\frac{d^{2}\alpha^{\prime}}{\pi}e^{-\kappa t\left\|\alpha^{\prime}\right\|^{2}}L_{l}\left(-\|\alpha^{\prime}\|^{2}\right)\left\|\alpha^{\prime}\right\|^{2}$ $\displaystyle=$ $\displaystyle\left[\kappa t\left(\kappa t+1\right)\right]^{2}\sum_{k=0}^{l}\frac{l!}{k!k!(l-k)!}\int\frac{d^{2}\alpha^{\prime}}{\pi}e^{-\kappa t\left\|\alpha^{\prime}\right\|^{2}}\left(\|\alpha^{\prime}\|^{2}\right)^{k+1}$ $\displaystyle=$ $\displaystyle\left(\kappa t+1\right)^{2}\sum_{k=0}^{l}\frac{l!}{k!(l-k)!}\left(k+1\right)\left(\frac{1}{\kappa t}\right)^{k},$ where $\displaystyle\sum_{k=0}^{l}\frac{l!}{k!(l-k)!}k\left(\frac{1}{\kappa t}\right)^{k}$ (39) $\displaystyle=$ $\displaystyle\frac{l}{\kappa t}\sum_{k=1}^{l}\frac{\left(l-1\right)!}{\left(k-1\right)!(l-k)!}\left(\frac{1}{\kappa t}\right)^{k-1}=\frac{l}{\kappa t}\left(\frac{\kappa t+1}{\kappa t}\right)^{l-1},$ and $\sum_{k=0}^{l}\frac{l!}{k!(l-k)!}\left(\frac{1}{\kappa t}\right)^{k}=\left(\frac{\kappa t+1}{\kappa t}\right)^{l}.$ (40) Substituting (39)-(40) into (38) we obtain $\displaystyle Tr\left[\colon a^{\dagger}ae^{\frac{-1}{\kappa t+1}a^{\dagger}a}L_{l}\left(\frac{-a^{\dagger}a}{\kappa t\left(\kappa t+1\right)}\right)\colon\right]$ (41) $\displaystyle=$ $\displaystyle\left(\kappa t+1\right)^{2}\left(\frac{\kappa t+1}{\kappa t}\right)^{l-1}\frac{l+\kappa t+1}{\kappa t}.$ Then substituting (41) into (37) we see $Tr\left[\rho\left(t\right)a^{\dagger}a\right]=l+\kappa t.$ (42) which tells that the photon number $l\rightarrow l+\kappa t.$ At the end of this work we point out that a diffusion process is quite different from the process in the amplitude dissipative channel (ADC) described by the following master equation [7] $\frac{d}{dt}\rho^{\prime}=\gamma\left(2a\rho^{\prime}a^{{\dagger}}-a^{{\dagger}}a\rho^{\prime}-\rho^{\prime}a^{{\dagger}}a\right)$ (43) where $\gamma$ is the rate of dissipation. The solution to Eq. (43) is [8] $\rho^{\prime}=\sum_{m=0}^{\infty}\frac{\left(1-e^{-2\gamma t}\right)^{n}}{n!}e^{-\gamma ta^{{\dagger}}a}a^{n}\rho_{0}^{\prime}a^{{\dagger}n}e^{-\gamma ta^{{\dagger}}a}.$ (44) In ADC an initial pure number state $\left\|l\right\rangle\left\langle l\right\|$ will evolve into a binomial state as shown in [9] $\sum_{m=0}^{l}\binom{l}{l-m}e^{-2\gamma mt}\left(1-e^{-2\gamma t}\right)^{l-m}\left\|m\right\rangle\left\langle m\right\|\equiv\rho_{b}^{\prime}$ (45) with photon number decaying $tr\left(a^{\dagger}a\rho_{b}^{\prime}\right)=le^{-2\gamma t}$. Comparing the diffusion master equation $(3)$ with the dissipation Eq. $(43)$ we realize that the term $a^{\dagger}\rho a$ may be responsible for diffusion. In summary, we theoretically propose a new optical field state $\rho_{new}=\lambda\left(1-\lambda\right)^{l}\colon L_{k}\left(\frac{-\lambda^{2}a^{{\dagger}}a}{1-\lambda}\right)e^{-\lambda a^{{\dagger}}a}\colon$ (46) which is named Laguerre-polynomial-weighted chaotic field. We show that such state can be implemented, i.e., when a number state enters into a diffusion channel, the output state is just this kind of states. We solve the master equation describing the diffusion process by using the summation method within ordered product of operators and the entangled state representaion. The solution manifestly shows how a pure state evolves into a mixed state. The physical difference between the diffusion and the amplitude damping is pointed out. Acknowledgements: This work was supported by the National Natural Science Foundation of China (Grant Nos. 11175113 and 11264018), and the Natural Science Foundation of Jiangxi Province of China (No 20132BAB212006). Appendix 1 For $\rho_{new}$ in Eq. (2) we prove $tr\rho_{new}=1.$ In fact, using the coherent state representation we have $\displaystyle tr\rho_{new}$ $\displaystyle=\int\frac{d^{2}\alpha}{\pi}\left\langle\alpha\right\|\lambda\left(1-\lambda\right)^{l}\colon L_{l}\left(\frac{-\lambda^{2}a^{{\dagger}}a}{1-\lambda}\right)e^{-\lambda a^{{\dagger}}a}\colon\left\|\alpha\right\rangle$ $\displaystyle=\lambda\left(1-\lambda\right)^{l}\int\frac{d^{2}\alpha}{\pi}e^{-\lambda\|\alpha\|^{2}}L_{l}\left(\frac{-\lambda^{2}\|\alpha\|^{2}}{1-\lambda}\right)$ $\displaystyle=\lambda\left(1-\lambda\right)^{l}\int_{0}^{\infty}d\left(\frac{\lambda-1}{\lambda^{2}}x\right)e^{-\frac{\lambda-1}{\lambda}x}L_{l}\left(x\right)=1,$ (A1) where we have used $\int_{0}^{\infty}e^{-bx}L_{l}\left(x\right)=\left(b-1\right)^{l}b^{-l-1}.$ (A2) Appendix 2 For $M_{m,n}$ in Eq. $\left(\ref{22}\right)$ we now prove $\sum_{m,n=0}^{\infty}M_{m,n}^{\dagger}M_{m,n}=1.$ Because $\vdots e^{xaa^{\dagger}}\vdots=\frac{1}{1-x}e^{a^{\dagger}a\ln\frac{1}{1-x}},$ (A3) where $\vdots$ $\vdots$ denotes anti-normal ordering, we have $\displaystyle\sum_{m,n=0}^{\infty}\frac{1}{m!}\frac{\left(\kappa t\right)^{m}}{\left(\kappa t+1\right)^{m}}a^{m}a^{\dagger m}$ $\displaystyle=\vdots\exp\left[\frac{\kappa t}{\kappa t+1}aa^{\dagger}\right]\vdots$ $\displaystyle=\left(\kappa t+1\right)e^{a^{\dagger}a\ln\left(\kappa t+1\right)}.$ (A4) Substituing it into the sum representation of $\rho\left(t\right)$ yields $\displaystyle\sum_{m,n=0}^{\infty}M_{m,n}^{\dagger}M_{m,n}$ $\displaystyle=\sum_{m,n=0}^{\infty}\frac{1}{m!n!}\frac{\left(\kappa t\right)^{m+n}}{\left(\kappa t+1\right)^{m+n+1}}a^{\dagger n}\left(\frac{1}{1+\kappa t}\right)^{a^{\dagger}a}a^{m}a^{\dagger m}\left(\frac{1}{1+\kappa t}\right)^{a^{\dagger}a}a^{n}$ $\displaystyle=\sum_{n=0}^{\infty}\frac{1}{n!}\frac{\left(\kappa t\right)^{n}}{\left(\kappa t+1\right)^{n}}a^{\dagger n}\left(\frac{1}{1+\kappa t}\right)^{2a^{\dagger}a}e^{a^{\dagger}a\ln\left(\kappa t+1\right)}a^{n}$ $\displaystyle=\sum_{n=0}^{\infty}\frac{1}{n!}\frac{\left(\kappa t\right)^{n}}{\left(\kappa t+1\right)^{n}}a^{\dagger n}e^{a^{\dagger}a\ln\left(\kappa t+1\right)}e^{2a^{\dagger}a\ln\frac{1}{1+\kappa t}}a^{n}$ $\displaystyle=\sum_{n=0}^{\infty}\frac{1}{n!}\frac{\left(\kappa t\right)^{n}}{\left(\kappa t+1\right)^{n}}a^{\dagger n}e^{a^{\dagger}a\left[\ln\left(\kappa t+1\right)+2\ln\frac{1}{1+\kappa t}\right]}a^{n}$ $\displaystyle=\sum_{n=0}^{\infty}\frac{1}{n!}\frac{\left(\kappa t\right)^{n}}{\left(\kappa t+1\right)^{n}}a^{\dagger n}e^{a^{\dagger}a\ln\frac{1}{1+\kappa t}}a^{n}$ $\displaystyle=\sum_{n=0}^{\infty}\frac{1}{n!}\frac{\left(\kappa t\right)^{n}}{\left(\kappa t+1\right)^{n}}a^{\dagger n}\colon e^{a^{\dagger}a\left(\frac{1}{1+\kappa t}-1\right)}\colon a^{n}$ $\displaystyle=\colon e^{a^{\dagger}a\frac{\kappa t}{1+\kappa t}}e^{a^{\dagger}a\ln\frac{-\kappa t}{1+\kappa t}}\colon=1.$ (A5) ## References * [1] Carmichael H J 1999 Statistical Methods in Ouantum Optics I, Master Equation and Fokker-Planck equations, Springer-Verlag, Berlin * [2] Orszag M 2000 Quantum Optics, Springer-Verlag, Berlin. * [3] Preskill J, 1998 Lecture Notes for Physics 229: Quantum Information and Computation, California Institution of Technology. * [4] Fan H Y, Lu H L, Fan Y 2006 Ann. Phys. 321, 480. * [5] Fan H Y 2003 J. Opt. B: Quantum Semiclass. Opt. 5 R147 * [6] Fan H Y, Klauder J R. 1994 Phys. Rev. A, 49 704 * [7] Gardiner C and Zoller P 2000 Quantum Noise (Springer Berlin). * [8] Fan H Y and Hu L Y 2009 Chin. Phys. B 18 1061 * [9] FAN Hong-Yi, REN Gang, Chin. Phys. Lett.. 2010, 27 (5): 050302	arxiv-papers	2013-11-06T02:54:46	2024-09-04T02:49:53.293982	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Hong-Yi Fan, Sen-Yue Lou, Xiao-Yin Pan and Li-Yun Hu", "submitter": "Cheng Da", "url": "https://arxiv.org/abs/1311.1275" }
1311.1351	# Combined action of the bound-electron nonlinearity and the tunnel-ionization current in low-order harmonic generation in noble gases Usman Sapaev, Anton Husakou and Joachim Herrmann ∗ Max-Born-Institute for Nonlinear optics and Fast Spectroscopy, Max-Born-Str. 2a, Berlin D-12489, Germany ∗[email protected] We study numerically low- order harmonic generation in noble gases pumped by intense femtosecond laser pulses in the tunneling ionization regime. We analyze the influence of the phase-mismatching on this process, caused by the generated plasma, and study in dependence on the pump intensity the origin of harmonic generation arising either from the bound-electron nonlinearity or the tunnel-ionization current. It is shown that in argon the optimum pump intensity of about 100 TW/cm2 leads to the maximum efficiency, where the main contribution to low-order harmonics originates from the bound-electron third and fifth order susceptibilities, while for intensities higher than 300 TW/cm2 the tunnel-ionization current plays the dominant role. Besides, we predict that VUV pulses at 133 nm can be generated with relatively high efficiency of about $1.5\times 10^{-3}$ by 400 nm pump pulses. ## References * [1] I. V. Hertel and W. Radloff, “Ultrafast dynamics in isolated molecules and molecular clusters,” Rep. Prog. Phys. 69, 1897–2003 (2006). * [2] S. Backus, J. Peatross, Z. Zeek, A. Rundquist, G. Taft, M. M. Murnane, and H. C. Kapteyn, “16-fs, 1- J ultraviolet pulses generated by third-harmonic conversion in air,” 21, 665–667 (1997). * [3] S. A. Trushin, K. Kosma, W. Fuß, and W. E. 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Herrmann, “Supercontinuum generation of higher-order solitons by fission in photonic crystal fibers,” 87, 203901 (2001). * [27] M. V. Ammosov, N. Delone, and V. P. Kraĭnov, “Tunnel ionization of complex atoms and of atomic ions in an alternating electromagnetic field,” Sov. Phys. JETP 91, 2008–2013 (1986). * [28] Z. Chang, Fundamentals of Attosecond Optics (Tayor and Francis Group, 2011). * [29] Z. Song, Y. Qin, G. Zhang, S. Cao, D. Pang, L. Chai, Q. Wanga, Z. Wangb, and Z. Zhang, “Femtosecond pulse propagation in temperature controlled gas-filled hollow fiber,” 281, 4109–4113 (2008). * [30] C. Bree, “Nonlinear optics in the filamentation regime,” Ph.D. Dissertation (2012), http://edoc.hu-berlin.de/dissertationen/bree-carsten-2011-09-21/PDF/bree.pdf. * [31] V. Loriot, E. Hertz, O. Faucher, and B. Lavorel, “Measurement of high order Kerr refractive index of major air components,” 17, 13429–13434 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-17-16-13429. * [32] J. Ni, J. Yao, B. Zeng, W. Chu, G. Li, H. Zhang, C. Jing, S. L. Chin, Y. Cheng and Z. Xu, “Comparative investigation of third- and fifth-harmonic generation in atomic and molecular gases driven by midinfrared ultrafast laser pulses,” 84, 063846 (2011). * [33] W. F. Chan, G. Cooper, X. Guo, G. R. Burton, and C. E. Brion, “Absolute optical oscillator strengths for the electronic excitation of atoms at high resolution. III. The photoabsorption of argon, krypton, and xenon,” 46, 149–171 (1992). ## 1 Introduction Low-order harmonic generation (LOHG) in gases pumped by ultrashort near-IR laser pulses is an important technique to generate ultraviolet (UV) and vacuum ultraviolet (VUV) femtosecond pulses for a wide variety of applications, in particular, for time-resolved spectroscopy of many molecules, clusters or biological specimens and for material characterization [1]. The use of noble gases as nonlinear medium instead of solid-state crystals is a preferable way to avoid strong dispersion, bandwidth limitations, low damage thresholds and strong absorption below 200 nm. In particular, by using different gases UV and VUV pulses with a duration down to 11 fs have been generated by third [2, 3] and fifth harmonic [4] conversion. Similar to other nonlinear frequency conversion processes, the efficiency of LOHG in gases is usually relatively low in practice. This is mainly caused by two factors: first, low conversion results from relatively small values of the third (and higher) order susceptibilities compared to crystalline media. A second problem is the realization of phase matching, which can be partially solved for various frequency transformation processes, e.g., by using the anomalous dispersion of hollow-core fibers [5, 6, 7], modulated hollow-core waveguides [8], a modulated third order nonlinearity by ultrasound [9] or by noncollinear four- wave mixing [10, 11]. In the intensity range below the ionization threshold the efficiency of frequency transformation increases with increasing pump intensity. However, as soon as the intensity rises above the ionization threshold, different additional processes play a role leading to a more complex dynamics. On the one hand the effective third-order nonlinearity decreases, since $\chi^{+(3)}$ of an ionized gas is lower than $\chi^{(3)}$ of a corresponding neutral gas [12]. On the other hand harmonics of the fundamental frequency emerge due to the ionization of the atoms and the interaction of the freed electrons with the intense pump field. The majority of studies of harmonic generation have been performed for relatively high orders of harmonics much in excess of the ionization potential which is well described by the three-step process of ionization, acceleration in the continuum and recombination with the parent ion (see e.g. [13]). Much less studied is an additional, physically different, mechanism of optical harmonic generation. In the regime of tunneling ionization the density of ionized electrons shows extremely fast, nearly stepwise increases at every half-cycle of the laser field. This stepwise modulation of the tunnel ionization current induces optical harmonic generation [14], which arises in the first stage of ionization and not in the final recombination stage in the three-step model. As shown in [15], the emission of the lowest harmonics up to about 9 are accounted for with the tunnel ionization current while higher orders are attributed with the recombination process. Further theoretical studies of this process has been published in [16, 17]. Note that the generation of THz pulses by two-color femtosecond pulses is also intrinsically connected to the optically induced step-wise increase of the plasma density due to tunneling ionization [18, 19, 20]. To date only few direct experimental observations of harmonic generation or frequency mixing due to the modulation of the tunnel ionization current has been reported [21, 22]. On the other hand the generation of third harmonics with efficiencies up to the range of $10^{-3}$ in a noble gas with pump intensities significantly larger than necessary for ionization has been reported in [2, 23]. Detections of this type of nonlinear response in the regime with intensites above the ionization threshold requires better understanding of the complex dynamics and insight into the competition of harmonic generation originating from atomic or ionic susceptibilities of bound states and the tunneling ionization current. The present paper is devoted to a theoretical study of this issue. Since here only low orders up to 7 are considered we neglect the recombination process and consider only the first stage of ionization. At least, up to our knowledge the combined occurrence of the two mechanisms by the bound-electron third (and higher) order nonlinearity and the tunnel-ionization current were not studied before. ## 2 Fundamentals Third harmonic generation (THG) in gases by focused beams for pump intensities below the ionization threshold has been studied theoretically already four decades ago [24, 25]. In the regime of tunneling ionization besides the nonlinearity due to bound electron states additional processes come into play which has to be accounted for. In particular, the sub-cycle temporal dynamic of the laser field plays an essential role in the ionization process. Therefore, in the theoretical description the slowly-varying envelope approximation requires the solution of a complicated, strongly coupled system of partial differential equations and would result in increased numerical errors due to relatively short (down to $8$ fs) durations of harmonic pulses. Since backward propagating field components are small we can use the unidirectional pulse propagation equation for the description of pulse propagation [26]. As will be seen later the effective propagation length is much smaller than the Rayleigh length, therefore we can neglect the diffraction term. Correspondingly the following basic equation for the electric field of linear polarized pulses will be used: $\displaystyle\partial_{z}\hat{E}(\omega)$ $\displaystyle=$ $\displaystyle ik(\omega)\hat{E}(\omega)+i\frac{\mu_{o}\omega^{2}}{2k(\omega)}\hat{P}_{NL}(\omega)$ (1) Here $\hat{E}(\omega)$ is the Fourier transform of the electric field $E(t)$; $k(\omega)=cn(\omega)/\omega$ is the frequency-dependent wavenumber, $\omega$ is the angular frequency, $c$ is the speed of light and $n(\omega)$ is the frequency-dependent refractive index of the chosen gas; $\mu_{o}$ is the vacuum permeability. The first term on the right-hand side of Eq. (1) describes linear dispersion of the gas. The nonlinear polarization is $\hat{P}_{NL}(\omega)=\hat{P}_{Bound}(\omega)+i\hat{J}_{e}(\omega)/\omega+i\hat{J}_{Loss}(\omega)/\omega$ with $\hat{P}_{Bound}(\omega)$ being the nonlinear polarization caused by the bound electron states, $\hat{J}_{e}(\omega)$ being the electron current and $\hat{J}_{Loss}(\omega)$ being the loss term due to photon absorption during ionization. The plasma dynamics is described by the free electron density $\rho(t)$, which can be calculated by: $\displaystyle\partial_{t}\rho(t)=W_{ST}(t)(\rho_{at}-\rho(t))$ (2) where $\rho_{at}$ is the neutral atomic density; $W_{ST}(t)$ is the quasistatic tunneling ionization rate for hydrogenlike atoms [27] $W_{ST}(t)=4\omega_{a}(r_{h})^{2.5}(\left\|E_{a}\right\|/E(t))$exp$(-2r^{1.5}\|E_{a}\|/3E(t))$, where $E_{a}=m_{e}^{2}q^{5}/(4\pi\epsilon_{o})^{3}\hbar^{4}$, $\omega_{a}=m_{e}q^{4}/(4\pi\epsilon_{o})^{2}\hbar^{3}$ and $r_{h}=U_{Ar}/U_{h}$, $U_{h}$ and $U_{Ar}$ are the ionization potentials of hydrogen and argon, correspondingly; $P_{Bound}(t)=\epsilon_{o}\chi^{(3)}(1-\rho(t)/\rho_{at})E(t)^{3}+\chi^{+(3)}(\rho(t)/\rho_{at})E(t)^{3}+\chi^{(5)}(1-\rho(t)/\rho_{at})E(t)^{5}$, $\epsilon_{o}$ is the vacuum permittivity; $m_{e}$ and $q$ being the electron mass and charge, respectively; $\chi^{(3)}$ and $\chi^{(5)}$ are third and fifth order susceptibilities of neutral gas, correspondingly, while $\chi^{+(3)}$ is that of ionized gas. In the following we consider nearly collimated beams with diameter corresponding to a Rayleigh length larger than the propagation lengths. In addition, for these parameters the pump power is below the self-focusing power. Therefore, we can neglect diffraction in the numerical model. The transverse macroscopic plasma current $J_{e}(t)$ is determined by [19]: $\displaystyle\partial_{t}J_{e}(t)+\nu_{e}J_{e}(t)=\frac{q^{2}}{m_{e}}E(t)\rho(t)$ (3) where $\nu_{e}$ is the electron collision rate (for argon $\nu_{e}\approx 5.7$ ps-1). Finally, the ionization energy loss is determined by $J_{Loss}(t)=W_{ST}(t)(\rho_{at}-\rho(t))U_{Ar}/E(t)$. A critical condition for an efficient frequency transfer to harmonics is the realization of phasematching which for intensities larger than the ionization threshold is sensitively influenced by the plasma contribution to the refraction index. The change of the linear refractive index of argon at the maximum of the pulse intensity $I^{\prime}$ (assuming Gaussian pulse shape), owing to the formation of laser plasma with a free electron density $\rho^{\prime}$ and the Kerr nonlinearity, is given by (see e.g., [28, 29]): $\displaystyle n(\omega,I^{\prime},\rho^{\prime})=n_{e}(\omega,\rho^{\prime})+\Delta n_{Kerr}(I^{\prime},\rho^{\prime})+\Delta n_{Plasma}(\omega,\rho^{\prime})$ (4) where $n_{e}(\omega,\rho^{\prime})=(n^{o}(\omega)-1)(1-\rho^{\prime}/\rho_{at})+1$, $\Delta n_{Kerr}(I^{\prime},\rho^{\prime})=I^{\prime}[n_{2}(1-\rho^{\prime}/\rho_{at})+n_{2}^{+}\rho^{\prime}+I^{\prime}n_{4}(1-\rho^{\prime}/\rho_{at})]$ and $\Delta n_{Plasma}(\omega,\rho^{\prime})=-q^{2}\rho^{\prime}/(2\epsilon_{o}m_{e}\omega^{2})$. The nonlinear susceptibility $\chi^{(3)}$ for argon is well known from many independent measurements, while only few experimental results exist for the higher-order susceptibilities. In [31, 32] coincident experimental date on $\chi^{(5)}$ for argon which also agrees (up to sign) with a theoretical estimation [30] can be found. On the other hand reported data for $\chi^{(7)}$ differ by orders of magnitudes. Correspondingly, neglecting the weak frequency dependence we assume in the following parameters $\chi^{(3)}=3.8\times 10^{-26}$ m2/V2 and $\chi^{(5)}=-2.02\times 10^{-47}$ m4/V4. Figure 1: Linear and nonlinear optical parameters of argon in the high- intensity regime (normal pressure P $=1$ atm) for a $20$ fs pulse at $800$ nm: (a) free electrons density at the trailing edge of the pulse normalized by the total number of atoms ($2.7\times 10^{25}$m-3); (b) contributions of the nonlinear refractive index, caused by Kerr nonlinearity, when only $n_{2}=1.1\times 10^{-23}$m${}^{2}/$W (solid), $n_{2}$ and $n_{4}=-0.36\times 10^{-41}$m${}^{4}/$W2 [31] (dashed), $n_{2}$, $n_{4}$ and ionized argon $n_{2}^{+}=0.6\times 10^{-23}$m${}^{2}/$W [12] (dotted) are taken into account; (c) change of the total refractive index of pump (black), third (red), fifth (green) and seventh (blue), harmonics; (d) change of coherent length of third (red), fifth (green) and seventh (blue) harmonics. Figure 1 shows some linear and nonlinear optical parameters of argon, calculated by using Eq. (4) for a $20$ fs (FWHM) pump pulse at $800$ nm in dependence on the pump intensity. Figure 1(a) shows the normalized plasma density after the pulse as a function of the pump intensity at the peak of the pulse. As can be seen full ionization at the trailing edge of the pulse occurs at around $450$ TW/cm2. Figure 1(b) shows changes of the Kerr-type nonlinear refractive index contribution $\Delta n_{Kerr}(I,\rho)$ taking into account (i) only n2 of neutral argon (solid curve), (ii) n2 and n4 of neutral argon (dashed line) and (iii) n2, n4 and n${}_{2}^{+}$ of neutral and ionized argon (dotted line). Figure 1(c) shows the change of the refractive indexes of the fundamental frequency (black), third (red), fifth (green) and seventh (blue) harmonics. As can be seen the refractive indexes of the fundamental frequency and third harmonic are decreased down to a value smaller than unity. For higher intensities, the difference between the refractive indexes of the fundamental and the harmonics becomes larger, i.e., the generation of plasma electrons decreases the phase-matching length. This can be seen in Fig. 1(d) demonstrating the change of the coherent lengths $(l_{coh}=\pi/\|\Delta k_{2r+1}\|)$ for different harmonics in dependence on the pump intensity. Here the coherent length strongly decreases above $100$ TW/cm2 for all harmonics. Below we show that this behavior appears also in our full numerical calculations of Eqs. (1)-(3). Neglecting bound electron contributions and the dependence of the pump intensity and plasma density on the propagation coordinate an analytical solution for the electric field of the harmonic has been derived in [14]. If we include the bound-electron contributions, the electric field for the harmonics with order of $2r+1$ can be expressed as: $\displaystyle E_{2r+1}(z)=\sqrt{A_{P}^{2}+\delta_{1r}(A_{3}+A_{51})^{2}+\delta_{2r}A_{52}^{2}}\sin(\Delta k_{2r+1}z/2)/(\Delta k_{2r+1}/2)$ (5) here $A_{p}=-\Phi k_{o}/(8\pi r(2r+1))(\omega_{pg}/\omega_{o})^{2}\left[{\rm{exp}(-3r^{2}/\xi)+r/(r+1){\rm{exp}(-3(r+1)^{2}/\xi)}}\right]E_{o}$, $A_{3}=3\mu_{o}c\epsilon_{o}\chi^{(3)}E_{o}^{3}\omega_{o}/8$, $A_{51}=15\mu_{o}c\epsilon_{o}\chi^{(5)}E_{o}^{5}\omega_{o}/2$, $A_{52}=5\mu_{o}c\epsilon_{o}\chi^{(5)}E_{o}^{5}\omega_{o}/32$; $\delta_{ij}$ is Kronecker’s symbol; $\omega_{pg}=4\pi\rho_{at}q^{2}/m_{e}^{2}$ is the plasma frequency associated with the initial gas density; $\Phi=8\sqrt{3\pi}(\omega_{a}/\omega_{o})\xi^{1/2}\rm{exp}(-2\xi/3)$, $\xi=E_{a}/E_{o}$, $E_{o}$, $\omega_{o}$ and $k_{o}$ are peak electric field, central angular frequency and wavenumber of pump, respectively; $\triangle k_{2r+1}$ is wave mismatch for $(2r+1)^{th}$ harmonic. The term $A_{51}$ describes the contribution of fifth-order susceptibility $\chi^{(5)}$ to the generation of third harmonic. We note that the relative phase of the fields generated by bound-electrons and the plasma current is $\pi/2$. In the following we compare this solution with the numerical solutions of the full model as presented in Eqs. (1)-(3). Figure 2: Numerical and analytical calculations for a 20-fs transform limited pump pulse with a 100 TW/cm2 peak intensity at 800 nm: (a) spectrum of the output pulse, calculated numerically with (red) and without (green) taking $\chi^{(3)}$ and $\chi^{(5)}$ into account; (b) efficiency conversion of third harmonic, calculated numerically (red) and analytically (black) for $\chi^{(3)}\neq 0$, $\chi^{(5)}\neq 0$, and $\chi^{(3)}=\chi^{(5)}=0$ (c); (d) efficiency conversion of fifth (green) and seventh (blue) harmonics; (e) normalized free electron density in time domain (red) and plasma current (blue). ## 3 Numerical results for 800-nm pump In this chapter we present numerical solutions of Eqs. (1) to (3) using the split-step method with fast Fourier transformation and the fifth-order Runge- Kutta method for 800-nm pump pulses with a 100 TW/cm2 peak intensity and 20-fs (FWHM) duration. The spectra in Fig. 2(a) calculated with (red) and without (green) contribution of $\chi^{(3)}$ and $\chi^{(5)}$ predict that LOHG is dominated by the bound electron contributions with the third and fifth order susceptibilities, since with $\chi^{(3)}=0$ and $\chi^{(5)}=0$ two order of magnitude lower efficiencies are predicted. In Fig. 2(b) and 2(c) analytical (black) and numerical (red) results for the efficiency of third harmonic conversion are compared for cases, when $\chi^{(3)}\neq 0$ and $\chi^{(5)}\neq 0$ (b) is included and for $\chi^{(3)}=\chi^{(5)}=0$ (c). Note that we calculated the efficiency by integration of the harmonic spectra. The analytical results are calculated by Eq. (5), using the wave vector mismatch with a constant pump intensity and a plasma density taken from the input parameters. These results confirm the conclusions drawn from Fig. 2(a) that at the optimum intensity of about 100 W/cm2 the bound-electron contribution is much larger than that of the tunnel ionization current. It should be noted that the maximum efficiency of the third harmonic of about 1.4 $\%$ appears at $0.4$ cm, which is approximately equal to the coherent length, as seen from Fig. 1(d). Figure 2(d) shows the efficiency of conversions to the fifth (green) and seventh (blue) harmonics with maximum values of about $10^{-4}$ and $10^{-7}$. In Fig. 2(e) the normalized plasma current and the plasma density are presented. Note the steplike nature of the density profile of free electrons (red curve), which explains the source of the harmonic generation due to the tunnel ionization current. Figure 3: Numerical calculations for a 20-fs transform limited pump pulse with 400 TW/cm2 peak intensity at 800 nm: (a) spectrum of the output pulse, calculated with (red) and without (green) taking $\chi^{(3)}$ and $\chi^{(5)}$ into account; (b) conversion efficiencies of third (red) and fifth harmonics (black); (c) time profile of the pump at the input (blue) and output (red); (d) normalized density of free electron distribution (red) and plasma current (blue). To study the regime where the tunnel-ionization current is dominant, in Fig. 3 results for LOHG are presented for a 20 fs pulse at 800 nm with an peak intensity of 400 TW/cm2. Due to the reduced coherence length the maximum conversion efficiencies are smaller than for the case of lower pump intensity in Fig. 2. Since the efficiencies are roughly the same independent on the inclusion of $\chi^{(3)}$ and $\chi^{(5)}$ in the model, we can conclude that the tunnel-ionization current is the main LOHG mechanism in this case. Due to the high intensity significant spectral broadening caused by self-phase modulation can be seen. The dependence of the efficiencies on the propagation distance indicates pump depletion rather than loss of coherence, since it exhibits no maximum. Here pump depletion, owing to ionization loss, appears mainly in the pulse center, as shown in Fig. 3(c). In order to analyze the roles of the $\chi^{(3)}$, $\chi^{(5)}$ and $\chi^{+(3)}$ nonlinearity and the tunnel-ionization current in dependence on the applied intensity range, we calculated the contribution to LOHG efficiencies of the two considered nonlinear optical processes in a large range of pump intensities. Figure 4 shows the conversion efficiencies in dependence on the pump intensity up to the 7th harmonic from a 800-nm pump with a 20-fs duration, calculated at the coherent length of third harmonic. The contribution of $\chi^{(3)}$ and $\chi^{(5)}$ dominates up to 300 TW/cm2, while after approximately 300 TW/cm2 the plasma current (red curves) becomes the main source for LOHG. The green curve in Fig. 4(a) shows results, which includes $\chi^{+(3)}$ of the ionized gas. For high intensities up to 500 TW/cm2 the efficiency of THG decreases down to the range of $10^{-5}$. Figure 4: Efficiency of LOHG in dependence on the pump intensity calculated for (a) third, (b) fifth and (c) seventh harmonic. Efficiencies were calculated with (blue) and without (red) taking $\chi^{(3)}$ and $\chi^{(5)}$ into account. Green curve in (a) shows results when $\chi^{(3)}$, $\chi^{(5)}$ and $\chi^{+(3)}$ were taken into account. In (d) the change of the coherent length of third harmonic and the length of its temporal walk-off from the fundamental frequency are shown. It seems to be surprising that even for relatively high intensities from $150$ to $300$ TW/cm2 the contribution of the $\chi^{(3)}$ and $\chi^{(5)}$ process remains in the same order as that of the tunnel-ionization current. This can be explained by the fact, that full ionization only occurs at the trailing edge of the pulses, while at the leading edge the atoms are not ionized and bound-electron contributions still play a significant role. The length of temporal walk-off between the fundamental and the third harmonic is shown in Fig. 4(d). It is much larger than the coherent lengths, and therefore does not play a significant role during propagation. A general observation arising from the results presented above is that the nonlinear susceptibilities $\chi^{(3)}$ and $\chi^{(5)}$ plays an important role in the formation of LOHG, especially for the third harmonic. Its contribution is dominant up to a pump intensity of 300 TW/cm2 for argon at normal pressure. As similar behavior can be expected for other noble gases, although the corresponding intensities will vary dependent on the properties of the noble gas. The tunnel-ionization current is a main source for LOHG for intensities larger than approximately 300 TW/cm2, especially for the fifth and seventh harmonics. As noted above, the high-intensity regime of pump can not support highly efficient LOHG because of the the contribution of the ionized electrons to the refraction index and the associated increased phase mismatch. Below 100 TW/cm2, the coherent length is roughly constant, but for larger intensities it shows a sharp decrease as visible in Fig. 1(d) and Fig. 4(d). This establishes a range around 100 TW/cm2 as optimum pump intensity for argon for the generation of the third and fifth harmonic. Figure 5: Results of numerical and analytical calculations for 400 nm pump pulses with 100 TW/cm2 [(a),(b)] and 300 TW/cm2 [(c), (d)]. In (a), (c) the spectrum of the output pulses, calculated with (red) and without (green) taking $\chi^{(3)}$ and $\chi^{(5)}$ are presented. In (b), (d) the efficiency of third harmonic, calculated numerically (red) and analytically (black) are shown. ## 4 Numerical results for 400-nm pump Nowadays, the generation of pump pulses at 400 nm with high energy by second harmonic generation in nonlinear crystals from near-IR ones is a standard method. Using THG with these pump pulses allows frequency conversion with relative high efficiency into the VUV spectral range at 133 nm. Figure 5 shows the results for such pump pulses with two different peak intensities for 100 TW/cm2 (a, b) and for 300 TW/cm2 (c, d) and the same pulse duration of 20 fs. The coherent length for the third harmonic is around a $0.05$ cm $(0.024)$ cm for 100 (300) TW/cm2, calculated by Eq. (4) and visible in Fig. 5(b) and 5(d). The tendency visible from Figs. 2 and 3 that for a lower intensity THG is caused by the third and fifth order susceptibilities while for higher intensities the tunnel ionization current plays the dominant role, is also observed for 400 nm as can be seen by comparison of Fig. 5(a) and 5(c). The maximum THG efficiency of about $1.5\times 10^{-3}$ for a pump intensity of 100 TW/cm(2) [Fig. 4(b)] is in the same range as in the case of a 800 nm pump pulses compare [Fig. 2(b)], but now a spectral transformation to the VUV range at 133 nm is realized. Higher harmonic orders above third, experience high linear loss of in the vacuum ultraviolet region for argon [33] due to the strong absorption band below 106 nm. ## 5 Conclusions In conclusion, we numerically studied the generation of low-order harmonics in argon in the high-intensity regime, when tunneling ionization takes place. The used numerical method is based on the unidirectional pulse propagation equation combined with the nonlinear response by the bound-electrons and a model for tunneling ionization and the associated plasma current. We analyzed LOHG in the regime of pump intensities up to 500 TW/cm2 arising either from the third- and fifth-order bound-electron nonlinearity or from the tunnel- ionization current. It was numerically observed that up to 300 TW/cm2 the formation of LOHG is caused mainly by the bound-electron nonlinearity, while for higher intensities the tunnel-ionization current plays the dominant role. It was also shown that a high intensity of the pump does not necessary lead to efficient LOHG, rather, due to the reduced coherence length by the plasma contribution to the refraction index an optimum around 100 TW/cm2 with efficiencies in the range of $3\times 10^{-3}$ and $10^{-4}$ for third and fifth harmonic generation, respectively, is predicted. Further on, we studied THG by intense pump pulses at 400 nm and predicted frequency transformation to the spectral range of 133 nm with maximum efficiency of about $1.5\times 10^{-3}$. ## Acknowledgments We acknowledge financial support by the German Research Foundation (DFG), project No. He $2083/17-1$.	arxiv-papers	2013-11-06T11:08:37	2024-09-04T02:49:53.304835	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "U Sapaev, A Husakou and J Herrmann", "submitter": "Usman Sapaev PhD", "url": "https://arxiv.org/abs/1311.1351" }
1311.1382	# New Periodic Solutions for Some Planar $N+3$-Body Problems with Newtonian Potentials ††thanks: Supported partially by NSF of China. Pengfei Yuan and Shiqing Zhang [email protected], [email protected] Department of Mathematics, Sichuan University, Chengdu 610064, China ###### Abstract For some planar Newtonian $N+3$-body problems, we use variational minimization methods to prove the existence of new periodic solutions satisfying that $N$ bodies chase each other on a curve, and the other $3$ bodies chase each other on another curve. From the definition of the group action in equations $(3.1)-(3.3)$, we can find that they are new solutions which are also different from all the examples of Ferrario and Terracini (2004)$[22]$. Key Words: $N+3$-body problems, periodic solutions, winding numbers, variational minimizers. 2000 Mathematicals Subject Classification: 34C15, 34C25, 58F ## 1 Introduction and Main Results In recent years, many authors used methods of minimizing the Lagrangian action on a symmetric space to study the periodic solutions for Newtonian $N$-body problem $([2],[4]-[6],[8]-[29],[31]-[40])$. Especially, A.Chenciner-R.Montgomery $[16]$ proved the existence of the remarkable figure eight type periodic solution for Newtonian three-body problem with equal masses, C.Simó $[32]$ discovered many new periodic solutions for Newtonian $N$-body problem using numerical methods. C.Machal $[27]$ studied the fixed- ends (Bolza) problem for Newtonian $N$-body problem and proved that the minimizer for the Lagrangian action has no interior collision; A.Chenciner $[12]$, D.Ferario and S.Terracini $[22]$ simplified and developed C.Marchal’s important works; S.Q.Zhang $[36]$, S.Q.Zhang, Q.Zhou $([37]-[40])$ decomposed the Lagrangian action for $N$-body problem into some sum for two-body problem and compared the lower bound for the lagrangian action on test orbits with the upper bound on collision set to avoid collisions under some cases. Motivated by the works of A.Chenciner and R.Montgomery, C.Simó, C.Marchal, S.Q.Zhang and Q.Zhou, K.C. Chen $([8]-[11])$ studied some planar $N$-body problems and got some new planar non-collision periodic and quasi-periodic solutions. The equations for the motion of the Newtonian $N$-body problem are: $\displaystyle m_{i}\ddot{q}_{i}=\frac{\partial U(q)}{\partial q_{i}},\quad i=1,\ldots,N,$ (1.1) where $q_{i}\in\mathbb{R}^{k}$ denotes the position of $m_{i}$, and the potential function is : $U=\sum_{1\leq i<j\leq N}\dfrac{m_{i}m_{j}}{\|q_{i}-q_{j}\|}.$ It is well known that critical points of the action functional $f$: $f(q)=\int_{0}^{T}(\frac{1}{2}\sum_{i=1}^{N}m_{i}\|\dot{q}_{i}\|^{2}+U(q))dt,\quad q\in E,$ $None$ are $T$ periodic solutions of the $N$-body problem $(1.1)$, where $E=\\{q=(q_{1},q_{2},\ldots,q_{N})\,\|\,q_{i}(t)\in W^{1,2}(\mathbb{R}/T\mathbb{Z},\mathbb{R}^{k}),\,\sum_{i=1}^{N}m_{i}q_{i}(t)=0,\,q_{i}(t)\neq q_{j}(t),\forall i\neq j,\forall t\in\mathbb{R}\\},$ $None$ $W^{1,2}(\mathbb{R}/T\mathbb{Z},\mathbb{R}^{k})=\\{x(t)\,\|\,x(t)\in L^{2}(\mathbb{R},\mathbb{R}^{k}),\dot{x}(t)\in L^{2}(\mathbb{R},\mathbb{R}^{k}),x(t+T)=x(t)\\}.$ $None$ Definition 1.1 Let $\Gamma:x(t),\,t\in[a,\,b]$ be a given oriented continuous closed curve, and $p$ a point of the plane, not on the curve. Then the mapping $\varphi:\Gamma\rightarrow S^{1}$ given by $\varphi(x(t))=\dfrac{x(t)-p}{\|x(t)-p\|},\quad t\in[a,b],$ $None$ is defined to be the position mapping of the curve $\Gamma$ relative to $p$. When the point on $\Gamma$ goes around the given oriented curve once, its image point $\varphi(x)$ will go around $S^{1}$ in the same direction with $\Gamma$ a number of times. When moving counter-clockwise or clockwise, we set the sign $+$ or $-$, and we denote it by $deg(\Gamma,\,\,p)$. If $p$ is the origin, we denote it by $deg(\Gamma)$. C.H.Deng and S.Q.Zhang $[20]$, X.Su and S.Q.Zhang $[33]$ studies periodic solutions for a class of planar $N+2$-body problems, they defined the following orbit spaces: $\displaystyle\Lambda_{0}=\\{$ $\displaystyle q\in E_{0}\,\|\,q_{i}(t+\dfrac{T}{r})=O(\dfrac{2\pi}{r})q_{i}(t),\quad i=1,\ldots,N+2;$ $\displaystyle q_{i+1}(t)=q_{i}(t+\dfrac{T}{N}),\,\,i=1,\ldots,N-1,\,\,q_{1}(t)=q_{N}(t+\dfrac{T}{N});$ $\displaystyle q_{i}(t+\dfrac{T}{N})=q_{i}(t),\,\,i=N+1,\,N+2,\forall\,t>0\\}$ (1.6) and $\displaystyle\Lambda=\\{$ $\displaystyle q\in\Lambda_{0}\,\|\,q_{i}(t)\neq q_{j}(t),\,\forall i\neq j,\forall t\in\mathbb{R};$ $\displaystyle deg(q_{i}(t)-q_{j}(t))=1,\,1\leq i\neq j\leq N,deg(q_{N+1}(t)-q_{N+2}(t))=k_{1}\\},$ (1.7) where $E_{0}=\\{q=(q_{1},q_{2},\ldots,q_{N+2})\,\|\,q_{i}(t)\in W^{1,2}(\mathbb{R}/T\mathbb{Z},\mathbb{R}^{2}),\,\sum_{i=1}^{N+2}m_{i}q_{i}(t)=0\\},$ $None$ $O(\theta)=\left(\begin{array}[]{cc}\cos{\theta}&-\sin{\theta}\\\ \sin{\theta}&\cos{\theta}\end{array}\right).$ Motivated by their work, we consider $N+3$-body problems($N>3$, $N$ and $3$ are coprime), the equations of the motion are: $m_{i}\ddot{q}_{i}(t)=\dfrac{\partial U(q)}{\partial q_{i}},\,\,\,i=1,\,\ldots,\,N+3.$ $None$ We define the following orbit spaces : $\displaystyle\Lambda_{1}=\\{q\in E_{1}\,\|\,$ $\displaystyle q_{i}(t+\frac{T}{r})=O(\frac{2\pi d}{r})q_{i}(t),\quad i=1,\,\ldots,\,N+3;$ $\displaystyle q_{i+1}(t)=q_{i}(t+\frac{T}{N}),\,\,i=1,\,\ldots,\,N,\,\,q_{1}(t)=q_{N}(t+\frac{T}{N});$ $\displaystyle q_{N+j}(t)=q_{N+j-1}(t+\frac{T}{3}),\,j=2,\,3,\,\,q_{N+1}(t)=q_{N+3}(t+\dfrac{T}{3});$ $\displaystyle q_{i}(t+\frac{T}{3})=q_{i}(t),\,i=1,\ldots,N;$ $\displaystyle q_{j}(t+\frac{T}{N})=q_{j}(t),\,j=N+1,N+2,N+3\\},$ (1.10) and $\displaystyle\Lambda_{2}=\\{q\in\Lambda_{1}\|$ $\displaystyle q_{i}(t)\neq q_{j}(t),\forall i\neq j,\forall t\in R;$ $\displaystyle deg(q_{i}(t)-q_{j}(t))=k_{1},1\leq i<j\leq N;$ $\displaystyle deg(q_{i^{\prime}}(t)-q_{j^{\prime}}(t))=k_{2},N+1\leq i^{\prime}<j^{\prime}\leq N+3\\},$ where $E_{1}=\\{q=(q_{1},q_{2},\ldots,q_{N+3})\|q_{i}(t)\in W^{1,2}(\mathbb{R}/T\mathbb{Z},\mathbb{R}^{2}),\,\sum_{i=1}^{N+3}m_{i}q_{i}(t)=0\\}.$ $None$ Notice that $r,k_{1},k_{2},d$ satisfy the following compatible conditions: $k_{1}=d(mod\,r)\,,k_{2}=d(mod\,r),k_{1}=3s_{1},k_{2}=Ns_{2},s_{1},s_{2}\in\mathbb{Z}.$ $None$ Since $N$ and $3$ are coprime, we have $(N,3)=1$. In this paper, we also require $r$ and $3$ coprime, so $(r,3)=1$. We get the following theorem: Theorem 1.1 $(1)$ Consider the seven-body problems $(1.9)$ of equal masses, for $r=7,\,k_{1}=3,\,k_{2}=-4,\,d=3$, then the global minimizer of $f$ on $\bar{\Lambda}_{2}$ is a non-collision periodic solution of $(1.9)$. $(2)$ Consider the eight-body problems $(1.9)$ of equal masses, for $r=8,\,k_{1}=3,\,k_{2}=-5,\,d=3$, then the global minimizer of $f$ on $\bar{\Lambda}_{2}$ is a non-collision periodic solution of $(1.9)$. $(3)$ Consider the ten-body problems $(1.9)$ of equal masses, for $r=10,\,k_{1}=3,\,k_{2}=-7,\,d=3$, then the global minimizer of $f$ on $\bar{\Lambda}_{2}$ is a non-collision periodic solution of $(1.9)$. ## 2 Some Lemmas ###### Lemma 2.1. (Eberlein-Shmulyan$[7]$) A Banach space $X$ is reflexive if and only if any bounded sequence in $X$ has a weakly convergent subsequence. ###### Lemma 2.2. $([7])$ Let $X$ be a real reflexive Banach space, $M\subset X$ is a weakly closed subset, $f:M\rightarrow R$ is weakly semi-continuous.If $f$ is coercive, that is, $f(x)\rightarrow+\infty$ as $\parallel x\parallel\rightarrow+\infty$, then $f(x)$ attains its infimum on $M$. ###### Lemma 2.3. $([30])$ Let $G$ be a group acting orthogonally on a Hilbert space $H$. Define the fixed point space $F_{G}=\\{x\in H\|g\cdot x=x,\forall g\in G\\}$, if $f\in C^{1}(H,R)$ and satisfies $f(g\cdot x)=f(x)$ for any $g\in G$ and $x\in H$, then the critical point of $f$ restricted on $F_{G}$ is also a critical point of $f$ on $H$. ###### Lemma 2.4. $([41])$ Let $q\in W^{1,2}(\mathbb{R}/T\mathbb{Z},\mathbb{R}^{n})$ and $\int_{0}^{T}q(t)\,\mathrm{d}t=0$, then we have $(i)$. Poincare-Wirtinger’s inequality: $\int_{0}^{T}\|\dot{q}(t)\|^{2}\,\mathrm{d}t\geq{\Big{(}\frac{2\pi}{T}\Big{)}}^{2}\int_{0}^{T}\|{q}(t)\|^{2}\,\mathrm{d}t.$ $None$ $(ii)$. Sobolev’s inequality: $\underset{0\leq t\leq T}{\max}\|q(t)\|=\parallel q\parallel_{\infty}\leq\sqrt{\frac{T}{12}}\big{(}\int_{0}^{T}\|\dot{q}(t)\|^{2}\mathrm{d}t\big{)}^{1/2}.$ $None$ ###### Lemma 2.5. (Gordon[24])(1) Let $x(t)\in W^{1,2}([t_{1},t_{2}],R^{k})$ and $x(t_{1})=x(t_{2})=0$, Then for any $a>0$, we have $\int_{t_{1}}^{t_{2}}(\frac{1}{2}\|\dot{x}\|^{2}+\frac{a}{\|x\|})dt\geq\frac{3}{2}(2\pi)^{\frac{2}{3}}a^{\frac{3}{2}}(t_{2}-t_{1})^{\frac{1}{3}}.$ $None$ (2)(Long and Zhang$[26]$) Let $x(t)\in W^{1,2}(R/TZ,R^{k})$, $\int_{0}^{T}xdt=0$, then for any $a>0$, we have $\int_{0}^{T}(\frac{1}{2}\|\dot{x}(t)\|^{2}+\frac{a}{\|x\|})dt\geq\frac{3}{2}(2\pi)^{\frac{2}{3}}a^{\frac{3}{2}}T^{\frac{1}{3}}.$ $None$ ## 3 Proof of Theorem 1.1 we consider the system $(1.9)$ of equal masses. Without loss of generality, we suppose that the masses $m_{1}=m_{2}=\cdots=m_{N+3}=1$, and the period $T=1$. Define $G=\mathbb{Z}_{r}\times\mathbb{Z}_{3}\times\mathbb{Z}_{N}$ and the group action $g=\langle g_{1}\rangle\times\langle g_{2}\rangle\times\langle g_{3}\rangle$ on the space $E_{1}$: $\displaystyle g_{1}(q_{1}(t),\ldots,q_{N+3}(t))=(O(-\frac{2\pi d}{r})q_{1}(t+\frac{1}{r}),\ldots,O(-\frac{2\pi d}{r})q_{N+3}(t+\frac{1}{r}))$ (3.1) $\displaystyle g_{2}(q_{1}(t),\ldots,q_{N+3}(t))$ $\displaystyle=(q_{1}(t+\frac{1}{3}),\ldots,q_{N}(t+\frac{1}{3}),q_{N+3}(t+\frac{1}{3}),q_{N+1}(t+\frac{1}{3}),q_{N+2}(t+\frac{1}{3}))$ (3.2) $\displaystyle g_{3}(q_{1}(t),\ldots,q_{N+3}(t))$ $\displaystyle=(q_{N}(t+\frac{1}{N}),q_{1}(t+\frac{1}{N})\ldots,q_{N-1}(t+\frac{1}{N}),q_{N+1}(t+\frac{1}{N}),q_{N+2}(t+\frac{1}{N}),q_{N+3}(t+\frac{1}{N}))$ (3.3) This implies that $\Lambda_{1}$ is the fixed point space of $g$ on $E_{1}$. Furthermore, for any $g_{i}$ and $q\in E_{1}$, we have $f(g_{i}\cdot q)=f(q)$ for $i=1,2,3$. Then the Palais symmetry principle implies that the critical point of $f$ restricted on $\Lambda_{1}$ is also a critical point of $f$ on $E_{1}$. ###### Lemma 3.1. The critical point of minimizing the Lagrangian functional $f$ restricted on $\Lambda_{2}$ (with winding number restriction) is also a critical point of $f$ on $\Lambda_{1}$, then it is also the solution of $(1.9)$. The proof is similar to that of Lemma $3.1$ in $[21]$, we omit it. By $q_{i}(t)=O(-\dfrac{2\pi d}{r})q_{i}(t+\dfrac{1}{r})(i=1,\cdots,N+3)\,$, we have $\int_{0}^{1}q_{i}(t)dt=0.$ Then the Lemma $2.4$ $\int_{0}^{1}\|\dot{q}_{i}(t)\|^{2}dt\geq(2\pi)^{2}\int_{0}^{1}\|q_{i}(t)\|^{2}dt.$ Hence $f(q)$ is coercive on $\bar{\Lambda}_{2}$. It is easy to see that $\bar{\Lambda}_{2}$ is a weakly closed subset.Fatou’s lemma implies that $f(q)$ is a weakly lower semi-continuous. Then by Lemma $2.2$, $f(q)$ attains $\inf{\\{f(q)\|q\in\bar{\Lambda}_{2}\\}}$. Similar to Lemma $3.2$ in $[21]$, we can obtain the following lemma. ###### Lemma 3.2. The limit curve $q(t)=(q_{1}(t),q_{2}(t),\ldots,q_{N+3}(t))\in\partial{\Lambda_{2}}$ of a sequence $q^{l}(t)=(q^{l}_{1}(t),q^{l}_{2}(t),\ldots,q^{l}_{N+3}(t))\in\Lambda_{2}$ may either have collisions between some two point masses or has the same winding number $(i.e.deg(q_{i}(t)-q_{j}(t))=k_{1},1\leq i\neq j\leq N;deg(q_{i^{\prime}}(t)-q_{j^{\prime}}(t))=k_{2},N+1\leq i^{\prime}\neq j^{\prime}\leq N+3).$ In the following, we prove that the minimizer of $f$ is a non-collision solutions of the system $(1.9).$ Since $\sum_{i=1}^{N+3}q_{i}=0$, by the Lagrangian identity, we have $f(q)=\frac{1}{N+3}\sum_{1\leq i<j\leq N+3}\int_{0}^{1}(\frac{1}{2}\,\|\dot{q}_{i}-\dot{q}_{j}\|^{2}+\frac{N+3}{\|q_{i}-q_{j}\|})dt$ $None$ Notice that each term on the right hand side of $(3.4)$ is a Lagrangian action for a suitable two body problem, which is a key step for the lower bound estimate on the collision set. We estimate the infimum of the action functional on the collision set. Since the symmetry for a two-body problem implies that the Lagrangian action on a collision solution is greater than that on the non-collision solution, and the more collisions there are, the greater the Lagrangian is. We only assume that the two bodies collide at some moment $t_{0}$, without loss of generality, let $t_{0}=0$, we will sufficiently use the symmetries of collision orbits. since $q\in\bar{\Lambda}_{2}$, we have $\displaystyle q_{i}(t+\dfrac{1}{r})=O(\dfrac{2\pi d}{r})q_{i}(t),\,i=1,\ldots,N+3;$ (3.5) $\displaystyle q_{i+1}(t)=q_{i}(t+\dfrac{1}{N}),i=1,\ldots,N-1,\,\,q_{1}(t)=q_{N}(t+\dfrac{1}{N});$ (3.6) $\displaystyle q_{N+2}(t)=q_{N+1}(t+\dfrac{1}{3}),\,q_{N+3}(t)=q_{N+2}(t+\dfrac{1}{3}),\,q_{N+1}(t)=q_{N+3}(t+\dfrac{1}{3});$ (3.7) $\displaystyle q_{i}(t+\dfrac{1}{3})=q_{i}(t),\,i=1,\ldots,N;$ (3.8) $\displaystyle q_{j}(t+\dfrac{1}{N})=q_{j}(t),\,j=N+1,N+2,N+3.$ (3.9) Case $1$: $q_{1},q_{2}$ collide at $t=0$. By $(3.5)$, we can deduce $q_{1},q_{2}$ collide at $t=\dfrac{i}{r},\,i=0,\ldots,r-1.$ Furthermore, by $(3.8)$, we can deduce $q_{1},q_{2}$ collide at $t=\dfrac{i}{r},\,\,\dfrac{i}{r}+\dfrac{1}{3},\,\,\dfrac{i}{r}+\dfrac{2}{3}\,(mod\,1).$ $None$ From $(3.6)$ and $(3.10)$, we have $\displaystyle q_{2},q_{3}\,\text{collide at}\,\dfrac{i}{r}+\dfrac{N-1}{N},\,\,\dfrac{i}{r}+\dfrac{1}{3}+\dfrac{N-1}{N},\,\,\dfrac{i}{r}+\dfrac{2}{3}+\dfrac{N-1}{N}(mod\,1),\,\,i=0,\ldots,r-1,$ $\displaystyle q_{3},q_{4}\,\text{collide at}\,\dfrac{i}{r}+\dfrac{N-2}{N},\,\,\dfrac{i}{r}+\dfrac{1}{3}+\dfrac{N-2}{N},\,\,\dfrac{i}{r}+\dfrac{2}{3}+\dfrac{N-2}{N}(mod\,1),\,\,i=0,\ldots,r-1,$ $\displaystyle\vdots$ $\displaystyle q_{N-1},q_{N}\,\text{collide at}\,\dfrac{i}{r}+\dfrac{2}{N},\,\,\dfrac{i}{r}+\dfrac{1}{3}+\dfrac{2}{N},\,\,\dfrac{i}{r}+\dfrac{2}{3}+\dfrac{2}{N}(mod\,1),\,i=0,\ldots,r-1,$ $\displaystyle q_{N},q_{1}\,\text{collide at}\,\dfrac{i}{r}+\dfrac{1}{N},\,\,\dfrac{i}{r}+\dfrac{1}{3}+\dfrac{1}{N},\,\,\dfrac{i}{r}+\dfrac{2}{3}+\dfrac{1}{N}(mod\,1),\,i=0,\ldots,r-1.$ ###### Lemma 3.3. $\forall\,\,0\leq i,\,j\leq r-1,\,0\leq k\leq 2,\,\,(i-j)^{2}+k^{2}\neq 0$, we have $\dfrac{i}{r}\neq\dfrac{j}{r}+\dfrac{k}{3}(mod\,1)$ $None$ ###### Proof. If there exist $0\leq i_{0},j_{0}\leq r-1,0\leq k_{0}\leq 2,(i_{0}-j_{0})^{2}+k_{0}^{2}\neq 0$ such that $\dfrac{i_{0}}{r}=\dfrac{j_{0}}{r}+\dfrac{k_{0}}{3}(mod\,1).$ Then we have $1\|(\dfrac{j_{0}}{r}+\dfrac{k_{0}}{3}-\dfrac{i_{0}}{r}).$ Since $\dfrac{j_{0}}{r}+\dfrac{k_{0}}{3}-\dfrac{i_{0}}{r}\geq-\dfrac{r-1}{r}=-1+\dfrac{1}{r}>-1,$ and $\dfrac{j_{0}}{r}+\dfrac{k_{0}}{3}-\dfrac{i_{0}}{r}\leq\dfrac{r-1}{r}+\dfrac{2}{3}<2,$ we can deduce $\dfrac{j_{0}}{r}+\dfrac{k_{0}}{3}-\dfrac{i_{0}}{r}=0$ or $\dfrac{j_{0}}{r}+\dfrac{k_{0}}{3}-\dfrac{i_{0}}{r}=1.$ If $\dfrac{j_{0}}{r}+\dfrac{k_{0}}{3}-\dfrac{i_{0}}{r}=0$, then $3(i_{0}-j_{0})=k_{0}r$. When $k_{0}=0$, we get $i_{0}=j_{0}$, which is a contradiction with our assumptions on the $i_{0},\,j_{0},\,k_{0}$; when $k_{0}\neq 0$, notice $0<k_{0}\leq 2$, we can deduce $3\|r$, which is a contradiction since $(r,3)=1.$ If $\dfrac{j_{0}}{r}+\dfrac{k_{0}}{3}-\dfrac{i_{0}}{r}=1$, then $3(j_{0}-i_{0})=(3-k_{0})r$. When $k_{0}=0$, we get $r=j_{0}-i_{0}$, which is a contradiction since $-r+1\leq j_{0}-i_{0}\leq r-1$; when $k_{0}\neq 0$, notice $1\leq 3-k_{0}<3$, we can deduce $3\|r$, which is also a contradiction since $(r,3)=1.$ ∎ By $(3.10)$ and Lemma $3.3$, we know that $q_{1},q_{2}$ collide at $t_{i}=\dfrac{i}{3r},\,\,i=0,\ldots,\,3r-1.$ $None$ Then by Lemma $2.5$, $(3.12)$, we have $\displaystyle\int_{0}^{1}(\dfrac{1}{2}\|\dot{q}_{1}(t)-\dot{q}_{2}(t)\|^{2}+\dfrac{N+3}{\|q_{1}(t)-q_{2}(t)\|})dt$ $\displaystyle=\sum_{i=0}^{3r-1}\int_{t_{i}}^{t_{i+1}}(\dfrac{1}{2}\|\dot{q}_{1}(t)-\dot{q}_{2}(t)\|^{2}+\dfrac{N+3}{\|q_{1}(t)-q_{2}(t)\|})dt$ $\displaystyle\geq\dfrac{3}{2}\times(2\pi)^{\frac{2}{3}}(N+3)^{\frac{2}{3}}3r(\dfrac{1}{3r})^{\frac{1}{3}}.$ (3.13) From $(3.6)$ and $(3.12)$, we have $\displaystyle q_{2},q_{3}\,\text{collide at}\,\dfrac{i}{3r}+\dfrac{N-1}{N}(mod\,1),\,\,i=0,\ldots,3r-1,$ $\displaystyle q_{3},q_{4}\,\text{collide at}\,\dfrac{i}{3r}+\dfrac{N-2}{N}(mod\,1),\,\,i=0,\ldots,3r-1,$ $\displaystyle\vdots$ $\displaystyle q_{N-1},q_{N}\,\text{collide at}\,\dfrac{i}{3r}+\dfrac{2}{N}(mod\,1),\,\,i=0,\ldots,3r-1,$ (3.14) $\displaystyle q_{N},q_{1}\,\text{collide at}\,\dfrac{i}{3r}+\dfrac{1}{N}(mod\,1),\,\,i=0,\ldots,3r-1.$ (3.15) ###### Lemma 3.4. $\forall\,0\leq i,i^{\prime}\leq 3r-1,1\leq j,j^{\prime}\leq N-1,(i-i^{\prime})^{2}+(j-j^{\prime})^{2}\neq 0$, we have $\dfrac{i}{3r}+\dfrac{j}{N}\neq\dfrac{i^{\prime}}{3r}+\dfrac{j^{\prime}}{N}(mod\,1).$ $None$ The proof is similar to Lemma $3.3$. Remark 3.1 From Lemma $3.4$, $\forall\,0\leq i,i^{\prime}\leq r-1,\,\,1\leq j,j^{\prime}\leq N-1,\,\,0\leq k,k^{\prime}\leq 2,(i-i^{\prime})^{2}+(j-j^{\prime})^{2}+(k-k^{\prime})^{2}\neq 0$, we have $\displaystyle\dfrac{i}{r}+\dfrac{j}{N}+\dfrac{k}{3}\neq\dfrac{i^{\prime}}{r}+\dfrac{j^{\prime}}{N}+\dfrac{k^{\prime}}{3}(mod\,1).$ By Lemma $2.5$, Lemma $3.4$ and $(3.15)$, we have $\displaystyle\int_{0}^{1}(\dfrac{1}{2}\|\dot{q}_{j+1}(t)-\dot{q}_{j+2}(t)\|^{2}+\dfrac{N+3}{\|q_{j+1}(t)-q_{j+2}(t)\|})dt$ $\displaystyle\geq\dfrac{3}{2}\times(2\pi)^{\frac{2}{3}}(N+3)^{\frac{2}{3}}3r(\dfrac{1}{3r})^{\frac{1}{3}},\,\,\,(j=1,\ldots,N-2),$ (3.17) $\displaystyle\int_{0}^{1}(\dfrac{1}{2}\|\dot{q}_{N}(t)-\dot{q}_{1}(t)\|^{2}+\dfrac{N+3}{\|q_{N}(t)-q_{1}(t)\|})dt$ $\displaystyle\geq\dfrac{3}{2}\times(2\pi)^{\frac{2}{3}}(N+3)^{\frac{2}{3}}3r(\dfrac{1}{3r})^{\frac{1}{3}}.$ (3.18) Let $\displaystyle M_{1}=\sum_{j=0}^{N-2}$ $\displaystyle\int_{0}^{1}(\dfrac{1}{2}\|\dot{q}_{j+1}(t)-\dot{q}_{j+2}(t)\|^{2}+\dfrac{N+3}{\|q_{j+1}(t)-q_{j+2}(t)\|})dt+$ $\displaystyle\int_{0}^{1}(\dfrac{1}{2}\|\dot{q}_{N}(t)-\dot{q}_{1}(t)\|^{2}+\dfrac{N+3}{\|q_{N}(t)-q_{1}(t)\|})dt.$ Then by $(3.13),(3.17),(3.18)$, Lemma $2.5$, and notice that $\forall\,1\leq i\leq N,\,N+1\leq j\leq N+3,\,\,\int_{0}^{\frac{1}{3}}q_{i}(t)dt=0,\,\,\int_{0}^{\frac{1}{N}}q_{j}(t)dt=0$, so we have $\displaystyle f(q)$ $\displaystyle=\frac{1}{N+3}\sum_{1\leq i<j\leq N+3}\int_{0}^{1}(\frac{1}{2}\,\|\dot{q}_{i}(t)-\dot{q}_{j}(t)\|^{2}+\frac{N+3}{\|q_{i}(t)-q_{j}(t)\|})dt$ $\displaystyle=\dfrac{1}{N+3}\\{\,\,M_{1}+[\sum_{1\leq i<j\leq N}\int_{0}^{1}(\frac{1}{2}\,\|\dot{q}_{i}(t)-\dot{q}_{j}(t)\|^{2}+\frac{N+3}{\|q_{i}(t)-q_{j}(t)\|})dt- M_{1}\,]+$ $\displaystyle\qquad\qquad\sum_{1\leq i\leq N,1\leq j\leq 3}\int_{0}^{1}(\frac{1}{2}\,\|\dot{q}_{i}(t)-\dot{q}_{N+j}(t)\|^{2}+\frac{N+3}{\|q_{i}(t)-q_{N+j}(t)\|})dt+$ $\displaystyle\qquad\qquad\sum_{N+1\leq i<j\leq N+3}\int_{0}^{1}(\frac{1}{2}\,\|\dot{q}_{i}(t)-\dot{q}_{j}(t)\|^{2}+\frac{N+3}{\|q_{i}(t)-q_{j}(t)\|})dt\,\,\\}$ $\displaystyle\geq\dfrac{3}{2}\times(\frac{4\pi^{2}}{N+3})^{\frac{1}{3}}[\,N\times 3r(\frac{1}{3r})^{\frac{1}{3}}+3\times(\frac{1}{3})^{\frac{1}{3}}(C_{N}^{2}-N)+3N+3N(\frac{1}{N})^{\frac{1}{3}}\,]$ $\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}A.$ (3.19) In the following cases, we firstly study the cases under $N$ is even. Case $2$: $q_{1},q_{k+2}(k=1,\ldots,\dfrac{N}{2}-2)$ collide at $t=0$. By $(3.5)$, we can deduce $q_{1},q_{k+2}(k=1,\ldots,\dfrac{N}{2}-2)$ collide at $t=\dfrac{i}{r},\,i=0,\ldots,r-1.$ Then by $(3.8)$ , $q_{1},q_{k+2}$ collide at $t=\dfrac{i}{r},\,\,\dfrac{i}{r}+\dfrac{1}{3},\,\,\dfrac{i}{r}+\dfrac{2}{3}\,(mod\,1),\,\,i=0,\cdots,r-1.$ $None$ From Lemma $3.3$, we get $q_{1},q_{k+2}$ collide at $t=\dfrac{i}{3r},i=0,\ldots,3r-1.$ $None$ Then by $(3.8)$, we have $\displaystyle q_{2},q_{k+3}\,\text{collide at}\,t=\dfrac{i}{3r}+\dfrac{N-1}{N}(mod\,1),\,i=0,\ldots,3r-1,$ $\displaystyle q_{3},q_{k+4}\,\text{collide at}\,t=\dfrac{i}{3r}+\dfrac{N-2}{N}(mod\,1),\,i=0,\ldots,3r-1,$ $\displaystyle\vdots$ $\displaystyle q_{N-k-1},q_{N}\,\text{collide at}\,t=\dfrac{i}{3r}+\dfrac{k+2}{N}(mod\,1),\,i=0,\ldots,3r-1,$ $\displaystyle q_{N-k},q_{1},\text{collide at}\,t=\dfrac{i}{3r}+\dfrac{k+1}{N}(mod\,1),\,i=0,\ldots,3r-1,$ $\displaystyle q_{N-k+1},q_{2}\text{collide at}\,t=\dfrac{i}{3r}+\dfrac{k}{N}(mod\,1),\,i=0,\ldots,3r-1,$ $\displaystyle\vdots$ $\displaystyle q_{N},q_{k+1}\text{collide at}\,t=\dfrac{i}{3r}+\dfrac{1}{N}(mod\,1),\,i=0,\ldots,3r-1.$ (3.22) Then by Lemma $2.5$, Lemma $3.3$, Lemma $3.4$, $(3.21)-(3.22)$, we have $\displaystyle f(q)$ $\displaystyle\geq\dfrac{3}{2}\times(\frac{4\pi^{2}}{N+3})^{\frac{1}{3}}[\,N\times 3r(\frac{1}{3r})^{\frac{1}{3}}+3\times(\frac{1}{3})^{\frac{1}{3}}(C_{N}^{2}-N)+3N+3N(\frac{1}{N})^{\frac{1}{3}}\,]$ $\displaystyle=A.$ (3.23) Case 3: $q_{1},q_{\frac{N}{2}+1}$ collide at $t=0$. By $(3.5),(3.6),\,\,(3.8)$, $q_{1},q_{\frac{N}{2}+1}$ collide at $\displaystyle t=$ $\displaystyle\dfrac{i}{r},\,\dfrac{i}{r}+\dfrac{1}{3},\,\,\dfrac{i}{r}+\dfrac{2}{3},\,$ $\displaystyle\dfrac{i}{r}+\dfrac{\frac{N}{2}}{N},\,\frac{i}{r}+\dfrac{1}{3}+\dfrac{\frac{N}{2}}{N},\,\,\dfrac{i}{r}+\dfrac{2}{3}+\dfrac{\frac{N}{2}}{N}(mod\,1),i=0,\ldots,r-1.$ (3.24) Simplify $(3.24)$ , we get $q_{1},q_{\frac{N}{2}+1}$ collide at $\displaystyle t=\dfrac{i}{r}+\dfrac{j}{6},\,i=0,\ldots,r-1,\,j=0,\ldots,5$ (3.25) ###### Lemma 3.5. $\forall\,0\leq i,i^{\prime}\leq r-1,0\leq j,j^{\prime}\leq 5,(i-i^{\prime})^{2}+(j-j^{\prime})^{2}\neq 0$, we have $\dfrac{i}{r}+\dfrac{j}{6}\neq\dfrac{i^{\prime}}{r}+\dfrac{j^{\prime}}{6}(mod\,1)$ $None$ ###### Proof. If there exist $0\leq i_{0},i_{1}\leq r-1,\,0\leq j_{0},j_{1}\leq 5,(i_{0}-i_{1})^{2}+(j_{0}-j_{1})^{2}\neq 0$ such that $\displaystyle\dfrac{i_{0}}{r}+\dfrac{j_{0}}{6}=\dfrac{i_{1}}{r}+\dfrac{j_{1}}{6}(mod1)$ (3.27) Since $\displaystyle\dfrac{i_{1}}{r}+\dfrac{j_{1}}{6}-\dfrac{i_{0}}{r}-\dfrac{j_{0}}{6}\geq-\dfrac{r-1}{r}-\dfrac{5}{6}>-2,$ $\displaystyle\dfrac{i_{1}}{r}+\dfrac{j_{1}}{6}-\dfrac{i_{0}}{r}-\dfrac{j_{0}}{6}\leq\dfrac{r-1}{r}+\dfrac{5}{6}<2,$ then we deduce $\dfrac{i_{1}}{r}+\dfrac{j_{1}}{6}-\dfrac{i_{0}}{r}-\dfrac{j_{0}}{6}=-1$ , or $\dfrac{i_{1}}{r}+\dfrac{j_{1}}{6}-\dfrac{i_{0}}{r}-\dfrac{j_{0}}{6}=0$, or $\dfrac{i_{1}}{r}+\dfrac{j_{1}}{6}-\dfrac{i_{0}}{r}-\dfrac{j_{0}}{6}=1$. If $\dfrac{i_{1}}{r}+\dfrac{j_{1}}{6}-\dfrac{i_{0}}{r}-\dfrac{j_{0}}{6}=-1$, we have $r(6+j_{1}-j_{0})=6(i_{0}-i_{1})$. When $i_{0}=i_{1}$, which is a contradiction since $r(6+j_{1}-j_{0})\neq 0$ ; when $i_{0}\neq i_{1}$ and $j_{0}=j_{1}$ , we can deduce $r=i_{0}-i_{1}$, which is a contradiction since $-r+1\leq i_{0}-i_{1}\leq r-1$; when $i_{0}\neq i_{1}$ and $j_{0}\neq j_{1}$, we can deduce $6\|r$, which is a contradiction since $(r,3)=1$. We can use similar arguments to prove $\dfrac{i_{1}}{r}+\dfrac{j_{1}}{6}-\dfrac{i_{0}}{r}-\dfrac{j_{0}}{6}\neq 0$ and $\dfrac{i_{1}}{r}+\dfrac{j_{1}}{6}-\dfrac{i_{0}}{r}-\dfrac{j_{0}}{6}\neq 1$. ∎ From $(3.25)$ and $(3.26)$, we can deduce $q_{1},q_{\frac{N}{2}+1}$ collide at $t_{i}=\dfrac{i}{6r},\,\,r=0,\ldots,6r-1.$ $None$ Then by Lemma $2.5$ and $(3.28)$, we have $\displaystyle\int_{0}^{1}(\dfrac{1}{2}\|\dot{q}_{1}(t)-\dot{q}_{\frac{N}{2}+1}(t)\|^{2}+\dfrac{N+3}{\|q_{1}(t)-q_{\frac{N}{2}+1}(t)\|})dt$ $\displaystyle=\sum_{i=0}^{6r-1}\int_{t_{i}}^{t_{i+1}}(\dfrac{1}{2}\|\dot{q}_{1}(t)-\dot{q}_{\frac{N}{2}+1}(t)\|^{2}+\dfrac{N+3}{\|q_{1}(t)-q_{\frac{N}{2}+1}(t)\|})dt$ $\displaystyle\geq\dfrac{3}{2}\times(2\pi)^{\frac{2}{3}}(N+3)^{\frac{2}{3}}6r(\dfrac{1}{6r})^{\frac{1}{3}}.$ (3.29) By $(3.6)$, $(3.28)$, we have $\displaystyle q_{2},q_{\frac{N}{2}+2},\,\text{collide at}\,t=\dfrac{i}{6r}+\dfrac{\frac{N}{2}-1}{N},\,i=0,\ldots,6r-1,$ $\displaystyle q_{3},q_{\frac{N}{2}+3},\,\text{collide at}\,t=\dfrac{i}{6r}+\dfrac{\frac{N}{2}-2}{N},\,i=0,\ldots,6r-1,$ $\displaystyle\vdots$ $\displaystyle q_{\frac{N}{2}},q_{N}\,\text{collide at}\,t=\dfrac{i}{6r}+\dfrac{1}{N},\,i=0,\ldots,6r-1.$ (3.30) ###### Lemma 3.6. $\forall\,0\leq i,i^{\prime}\leq 6r-1,1\leq j,j^{\prime}\leq\dfrac{N}{2}-1,\,(i-i^{\prime})^{2}+(j-j^{\prime})^{2}\neq 0$, we have $\dfrac{i}{6r}+\dfrac{j}{N}\neq\dfrac{i^{\prime}}{6r}+\dfrac{j^{\prime}}{N}.$ $None$ The proof is similar to Lemma $3.5$. By Lemma $2.5$, Lemma $3.6$, $(3.30)-(3.31)$, we have $\displaystyle\int_{0}^{1}(\dfrac{1}{2}\|\dot{q}_{j+1}(t)-\dot{q}_{\frac{N}{2}+j+1}(t)\|^{2}+\dfrac{N+3}{\|q_{j+1}(t)-q_{\frac{N}{2}+j+1}(t)\|})dt$ $\displaystyle\geq\dfrac{3}{2}\times(2\pi)^{\frac{2}{3}}(N+3)^{\frac{2}{3}}6r(\dfrac{1}{6r})^{\frac{1}{3}}\quad(j=1,\ldots,\frac{N}{2}-1).$ (3.32) Let $\displaystyle M_{2}=\sum_{j=0}^{\frac{N}{2}-1}\int_{0}^{1}(\frac{1}{2}\,\|\dot{q}_{j+1}(t)-\dot{q}_{\frac{N}{2}+j+1}(t)\|^{2}+\frac{N+3}{\|q_{j+1}(t)-q_{\frac{N}{2}+j+1}(t)\|})dt$ Then from Lemma $2.5$, Lemma $3.6$, $(3.29)$ and $(3.32)$, we obtain $\displaystyle f(q)$ $\displaystyle=\frac{1}{N+3}\sum_{1\leq i<j\leq N+3}\int_{0}^{1}(\frac{1}{2}\,\|\dot{q}_{i}(t)-\dot{q}_{j}(t)\|^{2}+\frac{N+3}{\|q_{i}(t)-q_{j}(t)\|})dt$ $\displaystyle=\dfrac{1}{N+3}\\{\,\,M_{2}+[\sum_{1\leq i<j\leq N}\int_{0}^{1}(\frac{1}{2}\,\|\dot{q}_{i}(t)-\dot{q}_{j}(t)\|^{2}+\frac{N+3}{\|q_{i}(t)-q_{j}(t)\|})dt- M_{2}]+$ $\displaystyle\quad\qquad\sum_{1\leq i\leq N,1\leq j\leq 3}\int_{0}^{1}(\frac{1}{2}\,\|\dot{q}_{i}(t)-\dot{q}_{N+j}(t)\|^{2}+\frac{N+3}{\|q_{i}(t)-q_{N+j}(t)\|})dt+$ $\displaystyle\quad\qquad\sum_{N+1\leq i<j\leq N+3}\int_{0}^{1}(\frac{1}{2}\,\|\dot{q}_{i}(t)-\dot{q}_{j}(t)\|^{2}+\frac{N+3}{\|q_{i}(t)-q_{j}(t)\|})dt\,\,\\}$ $\displaystyle\geq\dfrac{3}{2}\times(\frac{4\pi^{2}}{N+3})^{\frac{1}{3}}[\,\dfrac{N}{2}\times 6r(\frac{1}{6r})^{\frac{1}{3}}+3\times(\frac{1}{3})^{\frac{1}{3}}(C_{N}^{2}-\dfrac{N}{2})+3N+3N(\frac{1}{N})^{\frac{1}{3}}\,]$ $\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}B.$ (3.33) Finally, we study the cases under $N$ is odd. Case $2^{\prime}$: $q_{1},q_{k+2}(k=1,\ldots,{\frac{N+1}{2}}-2)$ collide at $t=0$. By $(3.5),(3.8)$, $q_{1},q_{k+2}(k=1,\ldots,\frac{N+1}{2}-2)$ collide at $t=\dfrac{i}{r},\,\,\dfrac{i}{r}+\dfrac{1}{3},\,\,\dfrac{i}{r}+\dfrac{2}{3}(mod\,1),\,i=0,\ldots,r-1,$ $None$ from Lemma $3.3$, we get $q_{1},q_{k+2}(k=1,\ldots,\frac{N+1}{2}-2)$ collide at $t=\dfrac{i}{3r},\,\,i=0,\ldots,3r-1,$ $None$ then by $(3.6)$, we have $\displaystyle q_{2},q_{k+3}\,\text{collide at}\,t=\dfrac{i}{3r}+\dfrac{N-1}{N}(mod\,1),\,i=0,\ldots,3r-1,$ $\displaystyle q_{3},q_{k+4}\,\text{collide at}\,t=\dfrac{i}{3r}+\dfrac{N-2}{N}(mod\,1),\,i=0,\ldots,3r-1,$ $\displaystyle\vdots$ $\displaystyle q_{N-k-1},q_{N}\,\text{collide at}\,t=\dfrac{i}{3r}+\dfrac{k+2}{N}(mod\,1),\,i=0,\ldots,3r-1,$ $\displaystyle q_{N-k},q_{1},\text{collide at}\,t=\dfrac{i}{3r}+\dfrac{k+1}{N}(mod\,1),\,i=0,\ldots,3r-1,$ $\displaystyle q_{N-k+1},q_{2}\text{collide at}\,t=\dfrac{i}{3r}+\dfrac{k}{N}(mod\,1),\,i=0,\ldots,3r-1,$ $\displaystyle\vdots$ $\displaystyle q_{N},q_{k+1}\text{collide at}\,t=\dfrac{i}{3r}+\dfrac{1}{N}(mod\,1),\,i=0,\ldots,3r-1.$ (3.36) Then by Lemma $2.5$, Lemma $3.4$, $(3.35),(3.36)$, we have $\displaystyle f(q)\geq$ $\displaystyle\dfrac{3}{2}\times(\dfrac{4\pi^{2}}{N+3})^{\frac{1}{3}}[\,N\times 3r(\dfrac{1}{3r})^{\frac{1}{3}}+3\times(\dfrac{1}{3})^{\frac{1}{3}}(C_{N}^{2}-N)+3N+3N(\dfrac{1}{N})^{\frac{1}{3}}\,]$ $\displaystyle=A.$ (3.37) Case $4$: $q_{N+1},q_{1}$ collide at $t=0$. By $(3.5)$, we have $q_{N+1},q_{1}$ collide at $t=\dfrac{i}{r},\quad i=0,\ldots,r-1.$ $None$ Then by Lemma $2.5$, $(3.37)$, we have $\displaystyle\int_{0}^{1}$ $\displaystyle(\dfrac{1}{2}\|\dot{q}_{1}(t)-\dot{q}_{N+1}(t)\|^{2}+\dfrac{N+3}{\|q_{1}(t)-q_{N+1}(t)\|})dt$ $\displaystyle=\sum_{i=0}^{r-1}\int_{t_{i}}^{t_{i+1}}(\dfrac{1}{2}\|\dot{q}_{1}(t)-\dot{q}_{N+1}(t)\|^{2}+\dfrac{N+3}{\|q_{1}(t)-q_{N+1}(t)\|})dt$ $\displaystyle\geq\dfrac{3}{2}\times(4\pi^{2})(N+3)^{\frac{2}{3}}r(\dfrac{1}{r})^{\frac{1}{3}}.$ (3.39) From $(3.38),(3.5)-(3.9)$, we can obtain $q_{N+2},q_{1},$ collide at $t=\dfrac{i}{r}+\dfrac{2}{3}(mod\,1)$, $q_{N+3},q_{1}$ collide at $t=\dfrac{i}{r}+\dfrac{1}{3}(mod\,1),\,\,i=0,\ldots,r-1,$ $q_{N+1},q_{2}$ collide at $\dfrac{i}{r}+\dfrac{N-1}{N}(mod\,1)$, $q_{N+2},q_{2}$ collide at $\dfrac{i}{r}+\dfrac{N-1}{N}+\dfrac{2}{3}(mod\,1)$, $q_{N+3},q_{2}$ collide at $\dfrac{i}{r}+\dfrac{N-1}{N}+\dfrac{1}{3}(mod\,1),\,i=0,\ldots,r-1,$ ⋮ $q_{N+1},q_{N-1}$ collide at $\dfrac{i}{r}+\dfrac{2}{N}(mod\,1)$, $q_{N+2},q_{N-1}$ collide at $\dfrac{i}{r}+\dfrac{2}{N}+\dfrac{2}{3}(mod\,1)$, $q_{N+3},q_{N-1}$ collide at $\dfrac{i}{r}+\dfrac{2}{N}+\dfrac{1}{3}(mod\,1),\,i=0,\ldots,r-1,$ $q_{N+1},q_{N}$ collide at $\dfrac{i}{r}+\dfrac{1}{N}(mod\,1)$, $q_{N+2},q_{N}$ collide at $\dfrac{i}{r}+\dfrac{1}{N}+\dfrac{2}{3}(mod\,1)$, $q_{N+3},q_{N}$ collide at $\dfrac{i}{r}+\dfrac{1}{N}+\dfrac{1}{3}(mod\,1),\,i=0,\ldots,r-1.$ Then by Lemma $2.5$, Lemma $3.3$, Remark $3.1$, we have $\forall\,0\leq i\leq r-1,1\leq j\leq 3,$ $\displaystyle\int_{0}^{1}$ $\displaystyle(\dfrac{1}{2}\|\dot{q}_{i}(t)-\dot{q}_{N+j}(t)\|^{2}+\dfrac{N+3}{\|q_{i}(t)-q_{N+j}(t)\|})dt$ $\displaystyle\geq\dfrac{3}{2}\times(4\pi^{2})(N+3)^{\frac{2}{3}}r(\dfrac{1}{r})^{\frac{1}{3}}.$ (3.40) So we get $\displaystyle f(q)$ $\displaystyle=\frac{1}{N+3}\sum_{1\leq i<j\leq N+3}\int_{0}^{1}(\frac{1}{2}\,\|\dot{q}_{i}(t)-\dot{q}_{j}(t)\|^{2}+\frac{N+3}{\|q_{i}(t)-q_{j}(t)\|})dt$ $\displaystyle=\dfrac{1}{N+3}(\,\,\sum_{\stackrel{{\scriptstyle 1\leq i\leq N}}{{1\leq j\leq 3}}}\int_{0}^{1}(\dfrac{1}{2}\|\dot{q}_{i}(t)-\dot{q}_{N+j}(t)\|^{2}+\dfrac{N+3}{\|q_{i}(t)-q_{N+j}(t)\|})dt+$ $\displaystyle\qquad\qquad\sum_{1\leq i<j\leq N}\int_{0}^{1}(\frac{1}{2}\,\|\dot{q}_{i}(t)-\dot{q}_{j}(t)\|^{2}+\frac{N+3}{\|q_{i}(t)-q_{j}(t)\|})dt+$ $\displaystyle\qquad\qquad\sum_{N+1\leq i<j\leq N+3}\int_{0}^{1}(\frac{1}{2}\,\|\dot{q}_{i}(t)-\dot{q}_{j}(t)\|^{2}+\frac{N+3}{\|q_{i}(t)-q_{j}(t)\|})dt\,\,)$ $\displaystyle\geq\dfrac{3}{2}\times(\frac{4\pi^{2}}{N+3})^{\frac{1}{3}}[\,3N\times r(\dfrac{1}{r})^{\frac{1}{3}}+3\times(\dfrac{1}{3})^{\frac{1}{3}}C_{N}^{2}+3N(\dfrac{1}{N})^{\frac{1}{3}}\,]$ $\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}C.$ (3.41) Case $5$: $q_{N+1},q_{N+2}$ collide at $t=0$. Then by $(3.5),(3.9)$, we deduce $q_{N+1},q_{N+2}$ collide at $\displaystyle t=\dfrac{i}{r}+\dfrac{j}{N}(mod\,1),\,i=0,\ldots r-1,\,j=0,\ldots,N-1.$ (3.42) From Remark $3.1$, and $(3.42)$, we can deduce $q_{N+1},q_{N+2}$ collide at $t_{i}=\dfrac{i}{Nr},\quad i=0,\ldots,Nr-1.$ $None$ Then we have $\displaystyle\int_{0}^{1}(\dfrac{1}{2}\|\dot{q}_{N+1}(t)-\dot{q}_{N+2}(t)\|^{2}+\dfrac{N+3}{\|q_{N+1}(t)-q_{N+2}(t)\|})dt$ $\displaystyle=\sum_{i=0}^{Nr-1}\int_{t_{i}}^{t_{i+1}}(\dfrac{1}{2}\|\dot{q}_{N+1}(t)-\dot{q}_{N+2}(t)\|^{2}+\dfrac{N+3}{\|q_{N+1}(t)-q_{N+2}(t)\|})dt$ $\displaystyle\geq\dfrac{3}{2}\times(4\pi^{2})(N+3)^{\frac{2}{3}}Nr(\dfrac{1}{Nr})^{\frac{1}{3}}.$ (3.44) By$(3.7)$, we deduce $q_{N+2},q_{N+3}$, collide at $\displaystyle t=\dfrac{i}{Nr}+\dfrac{2}{3},\quad i=0,\ldots,Nr-1,$ (3.45) $q_{N+3},q_{N+1}$ collide at $\displaystyle t=\dfrac{i}{Nr}+\dfrac{1}{3},\quad i=0,\ldots,Nr-1.$ (3.46) Then by Lemma $2.5$, Remark $3.1$, $(3.45)$, and $(3.46)$, we have $\displaystyle\int_{0}^{1}(\dfrac{1}{2}\|\dot{q}_{N+2}(t)-\dot{q}_{N+3}(t)\|^{2}+\dfrac{N+3}{\|q_{N+2}(t)-q_{N+3}(t)\|})dt$ $\displaystyle\geq\dfrac{3}{2}\times(4\pi^{2})(N+3)^{\frac{2}{3}}Nr(\dfrac{1}{Nr})^{\frac{1}{3}}$ (3.47) $\displaystyle\int_{0}^{1}(\dfrac{1}{2}\|\dot{q}_{N+3}(t)-\dot{q}_{N+1}(t)\|^{2}+\dfrac{N+3}{\|q_{N+3}(t)-q_{N+1}(t)\|})dt$ $\displaystyle\geq\dfrac{3}{2}\times(4\pi^{2})(N+3)^{\frac{2}{3}}Nr(\dfrac{1}{Nr})^{\frac{1}{3}}.$ (3.48) So, we obtain $\displaystyle f(q)$ $\displaystyle=\frac{1}{N+3}\sum_{1\leq i<j\leq N+3}\int_{0}^{1}(\frac{1}{2}\,\|\dot{q}_{i}(t)-\dot{q}_{j}(t)\|^{2}+\frac{N+3}{\|q_{i}(t)-q_{j}(t)\|})dt$ $\displaystyle=\dfrac{1}{N+3}(\sum_{N+1\leq i<j\leq N+3}\int_{0}^{1}(\frac{1}{2}\,\|\dot{q}_{i}(t)-\dot{q}_{j}(t)\|^{2}+\frac{N+3}{\|q_{i}(t)-q_{j}(t)\|})dt+$ $\displaystyle\qquad\qquad\qquad\sum_{\stackrel{{\scriptstyle 1\leq i\leq N}}{{1\leq j\leq 3}}}\int_{0}^{1}(\dfrac{1}{2}\|\dot{q}_{i}(t)-\dot{q}_{N+j}(t)\|^{2}\dfrac{N+3}{\|q_{i}(t)-q_{N+j}(t)\|})dt+$ $\displaystyle\qquad\qquad\qquad\sum_{1\leq i<j\leq N}\int_{0}^{1}(\frac{1}{2}\,\|\dot{q}_{i}(t)-\dot{q}_{j}(t)\|^{2}+\frac{N+3}{\|q_{i}(t)-q_{j}(t)\|})dt)$ $\displaystyle\geq\dfrac{3}{2}\times(\frac{4\pi^{2}}{N+3})^{\frac{1}{3}}[\,3\times Nr(\dfrac{1}{Nr})^{\frac{1}{3}}+3\times(\dfrac{1}{3})^{\frac{1}{3}}C_{N}^{2}+3N\,]$ $\displaystyle\stackrel{{\scriptstyle\triangle}}{{=}}D.$ (3.49) When $N$ is odd, let $\tilde{A}=\inf{\\{A,\,C,\,D\\}}$, then on the collision set, the action functional $f\geq\tilde{A}$. When $N$ is even, let $\tilde{B}=\inf{\\{A,\,B,\,C,\,D\\}}$, then on the collision set, the action functional $f\geq\tilde{B}$. (1)Take $N=4,d=3,r=7,k_{1}=3,k_{2}=-4$. We choose the following function as the test function: Let $a>0,\,\,b>0$, and $\displaystyle q_{i}=a(\cos{(6\pi t+\dfrac{2\pi(i-1)}{4})},\sin{(6\pi t+\dfrac{2\pi(i-1)}{4})}\,),\,\,i=1,\ldots,4,$ $\displaystyle q_{j}=b(\cos{(-8\pi t+\dfrac{2\pi(j-5)}{3})},\sin{(-8\pi t+\dfrac{2\pi(j-5)}{3})}),\,\,j=5,6,7.$ We choose $a=0.2300,\,\,b=0.0880$,then $\displaystyle A\approx 144.6215,\,B\approx 138.9586,\,$ $\displaystyle C\approx 170.7479,\,\,D\approx 139.2196,\,\,\tilde{B}=138.9586,$ $\displaystyle f(q)\approx 135.5123<\tilde{B}.$ This proves that the minimizer of $f(q)$ on the closure $\bar{\Lambda}_{2}$ is a non-collision solution of the seven-body problem. (2)Take $N=5,d=3,r=8,k_{1}=3,k_{2}=-5$. We choose the following function as the test function: Let $a>0,\,\,b>0$, and $\displaystyle q_{i}=a(\cos{(6\pi t+\dfrac{2\pi(i-1)}{5})},\,\sin{(6\pi t+\dfrac{2\pi(i-1)}{5})}\,),\,\,i=1,\ldots,5,$ $\displaystyle q_{j}=b(\cos{(-10\pi t+\dfrac{2\pi(j-6)}{3})},\,\sin{(-10\pi t+\dfrac{2\pi(j-6)}{3})}),\,\,j=6,7,8.$ We choose $a=0.2450,\,\,b=0.0760$, then $\displaystyle A\approx 193.5057,\,\,$ $\displaystyle C\approx 181.0305,\,\,D\approx 228.7437,\,\,\tilde{A}=181.0305,$ $\displaystyle f(q)\approx 175.2312<\tilde{A}.$ This proves that the minimizer of $f(q)$ on the closure $\bar{\Lambda}_{2}$ is a non-collision solution of the eight-body problem. (3)Take $N=7,d=3,r=10,k_{1}=3,k_{2}=-7$. We choose the following function as the test function: Let $a>0,\,b>0$, and $\displaystyle q_{i}=a(\cos{(6\pi t+\dfrac{2\pi(i-1)}{7})},\sin{(6\pi t+\dfrac{2\pi(i-1)}{7})}\,),\,\,i=1,\ldots,7,$ $\displaystyle q_{j}=b(\cos{(-14\pi t+\dfrac{2\pi(j-8)}{3})},\sin{(-14\pi t+\dfrac{2\pi(j-8)}{3})}),\,\,j=8,9,10.$ We choose $a=0.2500,\,\,b=0.0640$, then $\displaystyle A\approx 305.0645,\,\,$ $\displaystyle C\approx 274.1354,\,\,D\approx 360.6557,\,\,\tilde{A}=274.1354,$ $\displaystyle f(q)\approx 266.6297<\tilde{A}.$ This proves that the minimizer of $f(q)$ on the closure $\bar{\Lambda}_{2}$ is a non-collision solution of the ten-body problem. ## References * [1] G.Arioli1, V. Barutello, S.Terracini,A new branch of mountain pass solutions for the choreographical 3-Body problem, Commun. Math. Phys. 268(2006), 439-463. * [2] G. Arioli, F.Gazzola and S.Terracini, Minimization properties of Hill’s orbits and application to some N-body problems, Ann.Inst.Henri Poincaré Anal. 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1311.1613	# Crystallized and amorphous vortices in rotating atomic-molecular Bose- Einstein condensates Chao-Fei Liu1,2 Heng Fan1 Shih-Chuan Gou3 and Wu-Ming Liu1⋆ ###### Abstract Vortex is a topological defect with a quantized winding number of the phase in superfluids and superconductors. Here, we investigate the crystallized (triangular, square, honeycomb) and amorphous vortices in rotating atomic- molecular Bose-Einstein condensates (BECs) by using the damped projected Gross-Pitaevskii equation. The amorphous vortices are the result of the considerable deviation induced by the interaction of atomic-molecular vortices. By changing the atom-molecule interaction from attractive to repulsive, the configuration of vortices can change from an overlapped atomic- molecular vortices to carbon-dioxide-type ones, then to atomic vortices with interstitial molecular vortices, and finally into independent separated ones. The Raman detuning can tune the ratio of the atomic vortex to the molecular vortex. We provide a phase diagram of vortices in rotating atomic-molecular BECs as a function of Raman detuning and the strength of atom-molecule interaction. Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China School of Science, Jiangxi University of Science and Technology, Ganzhou 341000, China Department of Physics, National Changhua University of Education, Changhua 50058, Taiwan ⋆e-mail: [email protected] The realization of Bose-Einstein condensate (BEC) in dilute atomic gas is one of the greatest achievements for observing the intriguing quantum phenomena on the macroscopic scale. For example, this system is very suitable for observing the quantized vortex [1], and the crystallized quantized vortex lattice [2, 3]. Furthermore, it is found that vortex lattices in rotating single atomic BEC with dipole interaction can display the triangular, square, “stripe”, and “bubble” phases [4]. In two-component atomic BEC, the vortex states of square, triangular, double-core and serpentine lattices are showed according to the intercomponent coupling constant and the geometry of trap [5]. Considered two components with unequal atomic masses and attractive intercomponent interaction, the exotic lattices such as two superposed triangular, square lattices and two crossing square lattices tilted by $\pi/4$ are indicated [6]. Generally speaking, the crystallization of vortices into regular structures is common in the single BEC and the miscible multicomponent BECs under a normal harmonic trap. Vortices in atomic BECs have attracted much attentions [7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. However, it is not very clear the crystallization of vortices in atomic-molecular BECs [18, 17, 19, 27, 20, 28, 21, 22, 24, 25, 26, 23, 31, 32, 29, 30, 33]. The molecular BEC can be created by the magnetoassociation (Feshbach resonance) of cold atoms to molecules [20], and by the Raman photoassociation of atoms in a condensate [27, 28]. The atomic-molecular BEC provides a new platform for exploring novel vortex phenomena. It is shown recently that the coherent coupling can render a pairing of atomic and molecular vortices into a composite structure that resembles a carbon dioxide molecule [17]. Considering both attractive and repulsive atom-molecule interaction, Woo _et al._ have explored the structural phase transition of atomic-molecular vortex lattices by increasing the rotating frequency. They observed the Archimedean lattice of vortex with the repulsive atom-molecule interaction. In fact, atom-molecule interaction can be either attractive or repulsive with large amplitude by using the Feshbach resonance [20, 33]. In addition, we know that the population of atom and molecule in atomic-molecular BECs can be tuned by the Raman photoassociation [25, 31, 32, 29, 30, 33]. Then, we may wonder whether the combination control of Raman detuning and atom-molecule interaction may induce nontrivial vortex states and novel vortex phenomena. This seems not be well explored, especially in the grand canonical ensemble [36, 37, 38]. Furthermore, similarly to the normal system of two-component BECs [5], a phase diagram of vortices in rotating atomic-molecular BECs is required to provide a full realization of the nontrivial vortex phenomenon. In this report, we study the crystallized and amorphous vortices in rotating atomic-molecular BECs [19, 18, 27, 20, 28, 21, 22, 24, 25, 26, 23, 31, 32, 29, 30, 33]. Amorphous vortices are the result of the considerable deviation induced by the interaction of atomic-molecular vortices. The phase diagram indicates that atom-molecule interaction can control the atomic-molecular vortices to suffer a dramatic dissociation transition from an overlapped atomic-molecular vortices with interlaced molecular vortices to the carbon- dioxide-type atomic-molecular vortices, then to the atomic vortices with interstitial molecular vortices, and finally to the completely separated atomic-molecular vortices. This result is in accordance with the predicted dissociation of the composite vortex lattice in the flux-flow of two-band superconductors [39]. The Raman detuning adjusts the population of atomic- molecular BECs and the corresponding vortices. This leads to the imbalance transition among vortex states. This study shows a full picture about the vortex state in rotating atomic-molecular BECs. ## Results ### 0.1 The coupled Gross-Pitaevskii equations for characterizing atomic- molecular Bose-Einstein condensates. We ignore the molecular spontaneous emission and the light shift effect [28, 31, 32, 33]. According to the mean-field theory, the coupled equations of atomic-molecular BEC [33, 17, 40] can be written as $\displaystyle i\hbar\frac{\partial\Psi_{a}}{\partial t}=[-\frac{\hbar^{2}\nabla^{2}}{2M_{a}}+\frac{M_{a}\omega^{2}(x^{2}+y^{2})}{2}]\Psi_{a}-\Omega\widehat{L}_{z}\Psi_{a}$ $\displaystyle+(g_{a}\|\Psi_{a}\|^{2}+g_{am}\|\Psi_{m}\|^{2})\Psi_{a}+\sqrt{2}\chi\Psi_{a}^{}\Psi_{m},$ $\displaystyle i\hbar\frac{\partial\Psi_{m}}{\partial t}=[-\frac{\hbar^{2}\nabla^{2}}{2M_{m}}+\frac{M_{m}\omega^{2}(x^{2}+y^{2})}{2}]\Psi_{m}-\Omega\widehat{L}_{z}\Psi_{m}$ $\displaystyle+(g_{am}\|\Psi_{a}\|^{2}+g_{m}\|\Psi_{m}\|^{2})\Psi_{m}+\frac{\chi}{\sqrt{2}}\Psi_{a}^{2}+\varepsilon\Psi_{m},$ (1) where $\Psi_{j}(j=a,m)$ denotes the macroscopic wave function of atomic condensate and molecular condensate respectively, the coupling constants are, $g_{a}=\frac{4\pi\hbar^{2}a_{a}}{M_{a}}$, $g_{m}=\frac{4\pi\hbar^{2}a_{m}}{M_{m}}$, and $g_{am}=\frac{2\pi\hbar^{2}a_{am}}{M_{m}M_{a}/(M_{m}+M_{a})}$, also $M_{a}$ ($M_{m}$) is the mass of atom (molecule), $\omega$ is the trapped frequency, $\Omega$ is the rotation frequency, $\widehat{L}_{z}$ [$\widehat{L}_{z}=-i\hbar(x\partial_{y}-y\partial_{x})$] is the $z$ component of the orbital angular momentum. The parameter $\chi$ describes the conversions of atoms into molecules due to stimulated Raman transitions. $\varepsilon$ is a parameter to characterize Raman detuning for a two photon resonance [27, 28, 31, 32, 33]. In real experiment, it is observed that the coherent free-bound stimulated Raman transition can cause atomic BEC of 87Rb to generate a molecular BEC of 87Rb [27]. In numerical simulations, we use the parameters of atomic-molecular BECs of 87Rb system with $M_{m}=2M_{a}=2m$ ($m=144.42\times 10^{-27}Kg$), $g_{m}=2g_{a}$ ($a_{a}=101.8a_{B}$, where $a_{B}$ is the Bohr radius), $\chi=2\times 10^{-3}$, and the trapped frequency $\omega=100\times 2\pi$. Note that if the change in energy in converting two atoms into one molecule ($\Delta U=2U_{Ta}-U_{Tm}$) [27], not including internal energy, approaches zero, we can obtain the value $2g_{a}=g_{m}$. The unit of length, time, and energy correspond to $\sqrt{\hbar/(m\omega)}$ ($\approx 1.07\mu m$), $\omega^{-1}$ ($\approx 1.6\times 10^{-3}s$), and $\hbar\omega$, respectively. ### 0.2 Crystallized and amorphous vortices in rotating atomic-molecular Bose-Einstein condensates. With rotation frequency $\Omega=0.8\omega$, we study the influence of atom- molecule interaction on the formation of vortices. Figure 1 displays the densities and phases obtained under the equilibrium state with different atom- molecule interactions. The first and the second columns are the densities of the atomic BEC and the molecular BEC, respectively. The third column is the total density. The fourth and the fifth column are the corresponding phases of atomic and molecular BECs, respectively. The vortices can be identified in the phase image of BECs. The composite of atomic and molecular vortices locates at the trap center to lower the system’s energy. For the case of attractive atom-molecule interaction ($g_{am}=-0.87g_{a}$), vortices form a square lattice [see Fig. 1(a)]. Interestingly, we can observe that the size of the molecular vortices can be divided into two types: one is big, and the other is small. Each atomic vortex has approximately the same size. Figure 2(a) further plots the vortices, where the atomic vortices overlap with a molecular one. The overlapping of atomic and molecular vortices causes the size of molecular vortices to become big. Thus, we obtain different size of molecular vortices in the same experiment. It is easy to understand the size enlargement of molecular vortices which are overlapping with atomic ones. The size of vortex reflects the healing length [$\xi=\hbar(2mg\overline{n})^{-1/2}$, where $\overline{n}$ is the uniform density in a nonrotating cloud [41]] of the BEC because within this distance, the order parameter ‘heals’ from zero up to its bulk value. The attractive interspecies interaction implies that the densities of the two BECs would have a similar trend to decrease and increase. It also causes some molecular vortices to overlap with atomic ones. In addition, the density of atomic BEC forms local nonzero minima at the region of the left molecular vortex [see Fig. 2(a)]. Here, the size of atomic vortices is obvious bigger than that of molecular vortices. Therefore, the local density of molecular vortices follows that of atomic vortices and the size becomes big when molecular vortices overlap with atomic vortices. When $g_{am}=0$, atomic vortex lattices are triangular and the molecular vortices are amorphous state [see Fig. 1(b) and Fig. 2(d)]. Meanwhile, the total density (the third column) indicates that the molecular vortices and the atomic vortices form some structure like the carbon dioxide, which is also observed by a different method [17]. Figure 2(b) shows an enlarged configuration of the carbon dioxide vortices. Here, the size of atomic vortices is much larger than that of the molecular one. With repulsive interaction ($g_{am}=0.87g_{a}$), vortex lattices are approximately hexagonal with a little deviation [see Fig. 1(c)]. Increasing atom-molecule interaction up to $g_{am}=2g_{a}$ [see Fig. 1(d)], atomic-molecular BECs separate into two parts, molecular BEC locating at the center and atomic BEC rounding it. The results are understandable, since the mass of a molecule is twice as that of atom, molecular BEC tends to locate at the center. This is much different from that of the normal two-component BECs, where the same mass and intraspecies interactions are considered [5]. Figures 2(c)-2(f) further illuminate the position of vortices. Note that we do not point out the vortices where the densities of BECs are very low. We approximately view the vortex lattice as triangular, square, etc, although some vortices may deviate from the regular lattice slightly. The distance of adjacent lattice sites of atomic vortices is $\sqrt{2}$ times of that of molecular vortices [see Fig. 2(c)]. We have plotted a green circle to differentiate these vortices as two parts. The atomic vortices construct an approximately quadrangle lattice, especially near the center region, the atomic vortices overlap with a molecular one locating among four adjacent molecular vortices. Thus, vortex position indicates that vortices density of atomic BEC is half of that of molecular vortices. The atomic vortices expand over to the outskirts of the lattice where no overlapped molecular vortices appear [see Fig. 2(c)]. Figure 2(d) indicates that the carbon dioxide structure is not fixed in the same orientation. Similarly to Fig. 2(c), the carbon dioxide structure only exists at the center. However, the deviation of molecular vortices from the red lines d, e, and f is so large that we have to view the molecular vortices as an amorphous state. Vortex position in Fig. 2(e) shows that atomic vortices form the triangle lattice. All molecular vortices are distributed among atomic vortices, forming the hexagonal lattices without overlapping. Certainly, atomic vortices and molecular vortices are separated in Fig. 2(f) according to the immiscibility of atomic-molecular BECs with strong $g_{am}$. We can conclude that the strength of atom-molecule interaction can adjust the composite degrees of vortices, and cause the overlapping composite, carbon- dioxide-type composite, interstitial composite and separation. Furthermore, we find that the lattice configuration of vortices is very complex when atomic vortices and interstitial molecular vortices coexist. In Fig. 1(c), atomic vortices form the triangular lattice and interstitial molecular vortices display the honeycomb lattice. We further plot the densities of atomic BEC and molecular BEC at various cases in Fig. 3. When the number of atoms is much more than that of molecules, vortices in atomic BEC tend to form the triangular lattice, and vice versa. The lattice configurations are triangular in Figs. 3(a2), (b2), (e1) and (f1). Atomic vortices display square lattice in Figs. 3(a1)-3(c1). In all other subplots, the lattices are irregular and can be viewed as the amorphous state. For example, the number of adjacent molecular vortices which form bubbles [4] around some atomic vortices is not six but five in Figs. 3(d2)-(f2). In fact, the regular structures imply that both long-range order and short-range order should be remained. Thus, the observed random configuration is really amorphous. ### 0.3 The phase diagram of rotating atomic-molecular Bose-Einstein condensates. To explore the phase diagram of atomic-molecular vortices, we firstly show the modulation effect of Raman detuning for the number of vortices in rotating atomic-molecular BECs. Figures 4(a)-(d) show the relationship between vortices number and Raman detuning. Generally speaking, the number of molecular vortices decreases monotonously as Raman detuning increases. As we can see for $g_{am}=-0.87g_{a}$, 0, $0.87g_{a}$ and $2g_{a}$, the slope of the number of molecular vortices is $-2.7$, $-3.4$, $-4.2$ and $-7.8$, respectively. Increasing of the strength of atom-molecule interaction, the faster the number of molecular vortices decreases as Raman detuning increasing. The number of atomic vortices approaches to 40 as the Raman detuning increasing. Thus, the ratio of atomic vortices and molecular vortices is not fixed as the Raman detuning changes in atomic-molecular BECs. Furthermore, we calculate the number of composite vortices, i.e., the atom-vortex number $C^{\prime}_{a}$ and the molecule-vortex number $C^{\prime}_{m}$ in the green circles in Figs. 2(c)-(e), and define the parameter $P_{m}=100C^{\prime}_{a}/C^{\prime}_{m}$. $P_{a}$ in Figs. 4(a), (b), and (c) are almost around the value of 50, i.e, $C^{\prime}_{a}:C^{\prime}_{m}\approx 1:2$. Thus, the composite vortices keep the ratio 1:2 approximately. The deviation of $P_{a}$ from dash black line with the value of 50 mainly comes from vortices at the boundary. Certainly, vortex number in pure atomic BEC is independent of both atom-molecule interaction and Raman detuning. Now, we further indicate the modulation effect of Raman detuning for particle numbers of atomic-molecular BEC of 87Rb [see Figs. 4(e)-(h)]. The particle numbers in equilibrium state depend on the system itself. For attractive atom- molecule interaction $g_{am}=-0.87g_{a}$, both of atom number and molecule number decrease when the Raman detuning increases. Meanwhile, atom number always is greater than molecule number [see Fig. 4(e)]. For the limit case of $g_{am}=0$, atom number keeps unchanged and only molecule number decreases as the Raman detuning increases [see Fig. 4(f)]. When the interaction is repulsive ($g_{am}>0$), molecule number keeps decreasing but atom number increases [see Figs. 4(g) and (h)]. When the repulsive interaction is up to $g_{am}=2g_{a}$, the single molecular BEC or the single atomic BEC can be obtained by adjusting Raman detuning from $-4\hbar\omega$ to $14\hbar\omega$. The particle numbers can characterize the possible regions for the existence of atomic-molecular vortices. Figure 5 plots the phase diagram of atomic-molecular BECs. The stable atomic- molecular BECs system exists only when atom-molecule interaction is larger than $-\sqrt{g_{a}g_{m}}$. When the Raman detuning is large enough, single atomic BEC occurs. Oppositely, if the Raman detuning is low enough, production changes into pure molecular BEC. Between these two regions, it is atomic- molecular BECs, where AMBEC(I) denotes the miscible mixture and AMBEC(II) stands for the phase separated mixture. Therefore, to explore atomic-molecular vortices, we mainly focus on AMBEC(I) region. According to above analysis about atomic-molecular vortices and the corresponding atomic-molecular BECs, we calculate lots of other results and finally give a vortex phase diagram to summarize the vortex structures in Fig. 5. In (1) region [$-\sqrt{g_{a}g_{m}}<g_{am}<-0.1g_{a}$], atomic-molecular vortices form the square lattice where the overlapped atomic-molecular vortices and the molecular vortices interlacedly exist. The carbon-dioxide- type atomic-molecular vortices occur in (2) region [$-0.1g_{a}\leq g_{am}\leq 0.4g_{a}$]. Atomic vortices with interstitial molecular vortices emerge in region (3). In the AMBEC(II) region, atomic vortices and molecular vortices are separated. Certainly, in the region of atomic BEC (molecular BEC), atomic vortices (molecular vortices) favor to form the triangular lattice. The green region indicates that the created atomic and molecular vortices fully match with each other by roughly the ratio $1:2$. Above the green region, more atomic vortices occur. Below the green region, more molecular vortices appear. Table I shows a summary of the details of various vortices in rotating atomic- molecular BECs. Interestingly, the vortex phase diagram indicates some exotic transitions. (i) Imbalance transition: The increase of Raman detuning causes more atomic BEC. Thus, the pure molecular vortices change into carbon-dioxide-type atomic- molecular vortices (atom vortices with interstitial molecular vortices, and separated atomic-molecular vortices), and finally into single atomic vortices in region of $0<g_{am}\leq 0.5g_{a}$ ($0.5g_{a}<g_{am}\leq\sqrt{g_{a}g_{m}}$, and $g_{am}>\sqrt{g_{a}g_{m}}$, respectively). In the region of $-\sqrt{g_{a}g_{m}}<g_{am}<-0.1g_{a}$ ($-0.1g_{a}\leq g_{am}\leq 0$), the interlaced-overlapped atomic-molecular vortices (the carbon-dioxide-type atomic-molecular vortices) become into atomic vortices under a very high detuning parameter. (ii) Dissociation transition: By changing the atom- molecule interaction from attractive to repulsive, the composite atomic- molecular vortices change from overlapped to carbon-dioxide-type and finally into the independent separated ones. ## Discussion In this report, we focus on the strength of atom-molecule interaction and the Raman detuning term. The form of Hamiltonian in this paper is like that in the Ref. [17]. In fact, a real experiment would include lots of other factors such as the light shift effect [28, 31, 32, 33], decay due to spontaneous emission [31]. In Ref. [33], Gupta and Dastidar have considered a more complicated model when they study the dynamics of atomic and molecular BECs of 87Rb in a spherically symmetric trap coupled by stimulated Raman photoassociation process. In fact, the light shift effect almost has the same function as the Raman detuning term. Thus, it can be contributed to the Raman detuning term. This is the reason why we do not consider the light shift term in Hamiltonian like that in Ref. [33], but follows the form in Ref. [17]. In real experiment, it is believed that the single molecular BEC would occur when the Raman detuning goes to zero [31, 28]. However, the measure of the remaining fraction of atom does not reach the minimum when Raman detuning is zero [28]. With the adiabatic consideration, the dynamical study also agrees with this point [33]. In fact, they show the evolutionary process of creating a molecular BEC from a single atomic BEC. Thus, particle number of molecular BEC varies with time but not fixed. The resonance coupling would cause the atomic BEC to convert into a molecular one as much as possible, but the molecular BEC also will convert into the atomic one. Therefore, the results in Ref. [28, 33] only shows a temporary conversion of atoms into molecules. In fact, when we use single atomic BEC as the initial condition and set $\frac{\gamma_{j}}{k_{B}T}=0$, the temporary conversion of atomic BEC into molecular BEC can be observed with current damped projected Gross-Pitaevskii equations. It is obvious that the Raman detuning term in the Hamiltonian behaves just like the chemical potential to control the system’s energy. The external potential for atomic BEC is fixed to be $V_{a}(r)$ and molecular BEC experiences the trap potential $V_{m}(r)+\varepsilon$. Here, our method initially derives from the finite-temperature consideration: the system is divided into the coherent region with the energies of the state below $E_{R}$ and the noncoherent region with the energies of the state above $E_{R}$ [42, 43]. So, our method will behavior just likes to catch the particles with a shallow trap and exchange particles with an external thermal reservoir. But ultimately we remove the external thermal reservoir to get system to the ground state. Raman detuning changes the depth of shallow trap to $\mu_{m}-\varepsilon$. The molecular BEC will be converted by atoms until the system reaches the equilibrium state. Therefore, a maximum of creating molecular BEC does not occur at the equilibrium state when Raman detuning varies. Instead, molecule number decreases monotonously when Raman detuning increases. Why do atomic-molecular vortices display so rich lattice configurations? In fact, atomic vortices and molecular vortices tend to be attractive in region (1) and (2). Otherwise, the overlapped atomic-molecular vortices and the carbon-dioxide-type ones can not occur. The attractive force makes atomic vortices and molecular vortices behave similarly. Thus, both atomic and molecular vortex lattices in region (1) are square. In region (2), atomic vortices display the triangular lattice. Molecular vortices seem to follow the triangular lattice but the interaction among vortices causes the considerable deviation. Obviously, the $CO_{2}$-type structures do not follow the fixed direction, i.e., long-range order vanishes but there is still short-range order. Thus, we have to view molecular vortices as the amorphous state. In region (3), atomic vortices and molecular vortices can not form the carbon dioxide structure. Because the size of molecular vortices is smaller than that of atomic vortices, it tends to locate at the interval of the lattice of atomic vortices. When the number of one component is much more than that of the other, the vortices of this component dominate over the vortices of the other component. The former is easy to form the regular vortex lattice. The latter has to follow the interaction of the former and forms the vortex lattice. The amorphous state originates from the competition between atomic vortices and molecular vortices, especially when the number of atom and molecule has the considerable proportion [see Figs. 3(d1) and 3(d2)]. In that case, short-range order is only partly kept and ultimately long-range order is destroyed. Certainly, this also causes the distribution of vortices in one component is relatively regular and that in the other component is amorphous. The structural phase transitions of vortex lattices are explored through tuning the atom-molecule coupling coefficient and the rotational frequency of the system [17]. Certainly, the Archimedean lattice of vortices in Ref. [17] is one of the interstitial-composite-structures. Here, we show the crystallized and amorphous vortices by the combined control of Raman detuning and atom-molecule interaction. In fact, when we increase the value of $\chi$, the $CO_{2}$-type structure of vortices are easy to be created. Even the interstitial-composite structure we now obtain in Fig. 3 would transfer into the $CO_{2}$-type structure if $\chi$ is big enough. We have also considered the effect of rotation frequency. With the attractive interaction of atom- molecule ($g_{am}=-0.87g_{a}$), Figure 6 shows various rotation frequencies to produce the vortices. Figure 6(a) indicates that no vortex would occur with $\Omega=0$. For $\Omega=0.2\omega$, only one molecular vortex is induced. In atomic BEC, the phase indicates no vortex is created although there is a local minimum of density near the center. For $\Omega=0.4\omega$, the phase indicates that there is an atomic vortex. In fact, we find the atomic vortex is overlapped with a molecular vortex. Undoubtedly, more and more vortices emerge when rotation frequency increases. When the rotation frequency is up to $\Omega=0.8\omega$, we can obtain a regular square vortex lattice. Meanwhile, each atomic vortex is overlapped with a corresponding molecular vortex. Obviously, vortices and vortex lattice may not be induced with a slow rotation. This is the reason why we favor to investigate the vortices with a fast rotation in Figs. 1-4. We now show that ultracold Bose gases of 87Rb atoms are a candidate for observing the predicted atomic-molecular vortices. By initially using a large atomic BEC of 87Rb (the atom number is up to $3.6\times 10^{5}$ in Wynar’s experiment [27]), the Raman photoassociation of atoms [31, 27, 28, 34, 35] can produce the corresponding molecular BEC with partial of the atoms. By loading a pancakelike optical trap $V_{j=a,m}(x,y,z)=\frac{M_{j}[\omega^{2}(x^{2}+y^{2})+\omega_{z}z^{2}]}{2}$, with trapping frequencies $\omega_{z}\gg\omega$ [1, 2, 3], the 2D atomic- molecular BECs may be prepared. It is convenient to use the laser to rotate the atomic-molecular BECs and induce the atomic-molecular vortices. Meanwhile, the whole system should be further quenched to a lower temperature to approach the ground state by the evaporative cooling techniques. The resulting atomic- molecular vortices may be visualized by using the scanning probe imaging techniques. All the techniques are therefore within the reach of current experiments. In summary, we have observed various new atomic-molecular vortices and the lattices controlled by atom-molecule interaction and Raman detuning. Including the regular vortex lattices, we have displayed amorphous vortex state where vortices do not arrange regularly but like amorphous materials. We have obtained the vortex phase diagram as function of Raman detuning and atom- molecule interaction in the equilibrium state. Vortex configuration in atomic- molecular BECs includes the overlapped atomic-molecular vortices, the carbon- dioxide-type vortices, the atomic vortices with interstitial molecular vortices, and the completely separated atomic-molecular vortices. The lattice configuration of vortex mainly depends on atom-molecule interaction. For example, the overlapped atomic-molecular vortices display the square lattice. When the carbon-dioxide-type vortices occur, atomic vortices show the triangular lattice and molecular vortices show the amorphous state. Atomic vortices and interstitial molecular vortices can show several types of lattice, such as triangular, honeycomb, square and amorphous. And both atomic and molecular vortices show the triangular lattice in the incomposite region and in single BEC. Our results indicate that atom-molecule interaction can control the composite of atomic and molecular vortices, and can also cause novel dissociation transition of vortex state. Furthermore, the Raman detuning can control the numbers of particles in atomic-molecular BECs and approximately lead to the linear decrease of molecular vortices. This may induce the imbalance transition from atomic-molecular vortices to pure atomic (molecular) vortices. This study shows rich vortex states and exotic transitions in rotating atomic-molecular BECs. ## Methods We use the damped projected Gross-Pitaevskii equation (PGPE) [42] to obtain the ground state of atomic-molecular BEC. By neglecting the noise term according to the corresponding stochastic PGPE [43], the damped PGPE is described as $d\Psi_{j}=\mathcal{P}\\{-\frac{i}{\hbar}\widehat{H}_{j}\Psi_{j}dt+\frac{\gamma_{j}}{k_{B}T}(\mu_{j}-\widehat{H}_{j})\Psi_{j}dt\\},$ (2) where, $\widehat{H}_{j}\Psi_{j}=i\hbar\frac{\partial\Psi_{j}}{\partial t}$, $T$ is the final temperature, $k_{B}$ is the Boltzmann constant, $\mu_{j}$ is the chemical potential, and $\gamma_{j}$ is the growth rate for the $j$th component. The projection operator $\mathcal{P}$ is used to restrict the dynamics of atomic-molecular BEC in the coherent region. Meanwhile, we set the parameter $\frac{\gamma_{j}}{k_{B}T}=0.03$. The initial state of each $\Psi_{j}$ is generated by sampling the grand canonical ensemble for a free ideal Bose gas with the chemical potential $\mu_{m,0}=2\mu_{a,0}=8\hbar\omega$. 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Atom-molecule dephasing in an SU(1,1) interferometer based on the stimulated dissociation of a molecular Bose-Einstein condensate. Phys. Rev. A 80, 051604(R) (2009). * [41] Fischer U. R. and Baym G., Vortex States of Rapidly Rotating Dilute Bose-Einstein Condensates. Phys. Rev. Lett. 90 140402 (2003). * [42] Rooney, S. J., Bradley, A. S. & Blakie, P. B. Decay of a quantum vortex: Test of nonequilibrium theories for warm Bose-Einstein condensates. Phys. Rev. A 81, 023630 (2010). * [43] Bradley, A. S., Gardiner, C. W. & Davis, M. J. Bose-Einstein condensation from a rotating thermal cloud: Vortex nucleation and lattice formation. Phys. Rev. A 77, 033616 (2008). C. F. L. was supported by the NSFC under Grant No. 11247206, No. 11304130, No. 11365010 and the Science and Technology Project of Jiangxi Province, China (Grant No. GJJ13382). S.-C. G. was supported by the National Science Council, Taiwan, under Grant No. 100-2112-M-018-001-MY3, and partly by the National Center of Theoretical Science. W. M. L. is supported by the NKBRSFC under Grants No. 2011CB921502, No. 2012CB821305, the NSFC under Grants No. 61227902, No. 61378017, and No.11311120053. W.M.L. conceived the idea and supervised the overall research. C.F.L. and. S.C.G. designed and performed the numerical experiments. C.F.L. and H.F. wrote the paper with helps from all other co-authors. The authors declare that they have no competing financial interests. Correspondence and requests for materials should be addressed to Liu, Wu-Ming. Figure 1 The densities and phases of atomic-molecular BECs of 87Rb when the system reaches the equilibrium state. The rotation frequency is $\Omega=0.8\omega$. The strength of atom-atom interaction is $g_{a}$ with the scattering length $a_{a}=101.8a_{B}$. The strength of molecule-molecule interaction $g_{m}$ is twice as much as that of atom-atom interaction. The strength of atom-molecule interaction and Raman detuning is set as (a) $g_{am}=-0.87g_{a}$, $\varepsilon=14\hbar\omega$, (b) $g_{am}=0$, $\varepsilon=14\hbar\omega$, (c) $g_{am}=0.87g_{a}$, $\varepsilon=14\hbar\omega$ and (d) $g_{am}=2g_{a}$, $\varepsilon=7\hbar\omega$. Note that the fourth and fifth columns are the phases of atomic and molecular BECs, respectively. The unit of length is $1.07\mu m$. Figure 2 Vortex configurations and vortex position. (a) The scheme of composite vortices in Fig. 1(a). The size of the right molecular vortex which overlaps with an atomic vortex is bigger than the left one. (b) The scheme of carbon-dioxide-type vortex structure in Fig. 1(b). The size of the atomic vortex is bigger than that of the molecular vortex. The red, black and blue indicate the densities of atomic BEC, molecular BEC and the sum, respectively. (c), (d), (e) and (f) show the position of vortices in Figs. 1(a)-1(d), respectively. The circle ($\circ$) and asterisk ($\ast$) are the position of vortices formed by atomic BEC and molecular BEC, respectively. In (c), the red lines indicate that vortices can array in the square lattice. In (d), the blue lines show atomic vortices form the triangular lattice. While, the deviation of molecular vortices from the red lines indicates they form the amorphous state. In (e), atomic vortices form the triangular lattice and molecular vortices form the honeycomb lattices. Similarly, molecular vortices display the triangular lattice in (f). The unit of length is $1.07\mu m$. Figure 3 The effect of Raman detuning on the lattice of atomic vortices with interstitial molecular vortices at the equilibrium state. Here the strength of interactions are $g_{m}=2g_{a}$ and $g_{am}=0.87g_{a}$. The rotation frequency is $\Omega=0.8\omega$. The upper plots [(a1)-(f1)] show the densities of atomic BEC and the lower plots [(a2)-(f2)] indicate the corresponding densities of molecular BEC. The value of Raman detuning varies. (a) $\varepsilon=-2\hbar\omega$, (b) $\varepsilon=0\hbar\omega$, (c) $\varepsilon=2\hbar\omega$, (d) $\varepsilon=4\hbar\omega$, (e) $\varepsilon=7\hbar\omega$, and (f) $\varepsilon=10\hbar\omega$. (a1), (b1) and (c1) are the square lattice. (a2), (b2), (e1) and (f1) are the triangular lattice. Other plots show the amorphous state. The unit of length is $1.07\mu m$. Figure 4 The number of vortices and particles. (a)-(d) show the number of atomic vortices $C_{a}$ and molecular vortices $C_{m}$ in atomic-molecular BECs of 87Rb with the detuning parameter $\varepsilon$ when the system reaches the equilibrium state. (a) $g_{am}=-0.87g_{a}$, (b) $g_{am}=0$, (c) $g_{am}=0.87g_{a}$ and (d) $g_{am}=2g_{a}$. (e)-(h) indicate the corresponding particle number of atomic-molecular BECs of 87Rb, respectively. The rotation frequency is $\Omega=0.8\omega$, the strength of molecule-molecule interactions are $g_{m}=2g_{a}$ with the atom-atom scattering length $a_{a}=101.8a_{B}$, and the parameter $\chi$ is fixed to be $2\times 10^{-3}$. The unit of detuning parameter is $\hbar\omega$. Figure 5 Phase diagram of rotating atomic-molecular BECs of 87Rb when the system reaches the equilibrium state. AMBEC(I) denotes the miscible mixture of atomic-molecular BECs, and AMBEC(II) is immiscible atomic-molecular BEC. Furthermore, based on the phase diagram of atomic-molecular BECs, we further plot the phase diagram of atomic-molecular vortices when the atomic-molecular BECs of 87Rb reaches the equilibrium state. Then, the region of AMBEC(I) is divided into three parts: (1), (2), and (3). The overlapped atomic-molecular vortices, carbon-dioxide-type atomic-molecular vortices and atomic vortices with the interstitial molecular vortices occur in region (1), region (2) and region (3), respectively. In the green region, atomic and molecular vortices match fully with the rough ratio $1:2$. The parameters are $\Omega=0.8\omega$, $g_{m}=2g_{a}$ ($a_{a}=101.8a_{B}$), and $\chi=2\times 10^{-3}$. The units of detuning parameter and $g_{am}$ are $\hbar\omega$ and $g_{a}$, respectively. Figure 6 The densities and phases of the atomic-molecular BECs of 87Rb under various rotating frequencies when the system reaches the equilibrium state. The rotating frequencies are indicated at the title of the subplots. (a1)-(e1) are the densities of atomic BEC, (a2)-(e2) are the corresponding phases of atomic BEC, (a3)-(e3) are the densities of molecular BEC, and (a4)-(e4) are the corresponding phases of molecular BEC, respectively. The critical rotating frequencies for inducing molecular vortex and atomic vortex are about $0.1\omega$ and $0.3\omega$, respectively. The strength of atom-molecule interaction is $g_{am}=-0.87g_{a}$ with the atom-atom scattering length $a_{a}=101.8a_{B}$, molecule-molecule interactions is $g_{m}=2g_{a}$, the parameter $\chi$ is fixed to be $2\times 10^{-3}$ and Raman detuning is $\varepsilon=0\hbar\omega$. The unit of length is $1.07\mu m$. Table 1: A summary of the properties of vortices in the rotating atomic- molecular BECs of 87Rb when the system reaches the equilibrium state. The atomic-molecular vortices are composite in the matching region [inside the green circle in Figs. 2(c)-(e)]. Region (in Fig. 5) Vortex state (in the matching region) Lattice of atomic vortex Lattice of molecular vortex Vortex lattice out of the matching region $(1)$ Overlapped atomic-molecular vortices with interstitial molecular vortices square square triangular $(2)$ carbon-dioxide-type atomic-molecular vortices triangular amorphous triangular $(3)$ atomic vortices with interstitial molecular vortices square/ amorphous/ triangular triangular/ amorphous /honeycomb triangular AMBEC(II) separated atomic vortices and molecular vortices triangular triangular No atomic BEC pure atomic vortices triangular No No molecular BEC pure molecular vortices No triangular No	arxiv-papers	2013-11-07T09:09:36	2024-09-04T02:49:53.331655	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Chao-Fei Liu, Heng Fan, Shih-Chuan Gou, and Wu-Ming Liu", "submitter": "Chaofei Liu", "url": "https://arxiv.org/abs/1311.1613" }
1311.1739	# A general model on the simulation of the measurement-device independent quantum key distribution Qin Wang1 Xiang-Bin Wang2,3 [email protected] 1Institute of Signal Processing and Transmission, Nanjing University of Posts and Telecommunications, Nanjing 210003, China 2Department of Physics and State Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China 3Jinan Institute of Quantum Technology, Shandong Academy of Information Technology, Jinan, China ###### Abstract PACS number(s): 42.50.Ct, 78.67.Hc, 78.47.D- We present a general model on the simulation of the measurement-device independent quantum key distribution (MDI-QKD). It can be used to predict experimental observations of a MDI-QKD with linear channel loss, simulating corresponding values for the gains, the error rates in different basis, and also the final key rates. Our model can be applicable to the MDI-QKDs with whatever convex source states or using whatever coding schemes. Therefore, it is useful in characterizing and evaluating the performance of any MDI-QKD protocols, making it a valuable tool in studying the quantum key distributions. ## I Introduction There has been a long history between the attacks and the anti-attacks in the development of quantum key distributions (QKD) since the idea of BB84 (Bennett-Brassard 1984 BB84 ; GRTZ02 ) protocol was put forward, due to the conflictions between the ”in-principle” unconditional security and realistic implementations. Till today, there have been many different proposals for the secure QKD with realistic setups, such as the decoy-state method ILM ; H03 ; wang05 ; LMC05 ; AYKI ; qin1 ; qin2 ; haya ; peng ; wangyang ; rep ; njp which can rescue the QKD with imperfect single-photon sources PNS1 ; PNS , while the device-independent quantum key distribution ind1 ; gisin1 and the recently proposed measurement-device independent quantum key distribution (MDI-QKD) ind2 ; ind3 can further relieve the QKD even when the detectors are controlled by the eavesdropper lyderson . Most interestingly, the MDI-QKD is not only immune to any detector attacks, but also able to generate a significant key rate with existing technologies. Moreover, its security can still be maintained with imperfect single-photon sources ind2 ; wangPRA2013 ; tittel1 ; liuyang ; qin3 ; lopa ; curtty , and the effects of coding errors have also been studied wangPRA2013 ; kiyoshi . In developing practical QKDs, one important question is how to evaluate the performance of a proposal before really implementing it, since it is not realistic to experimentally test everything. Therefore, it is crucially important to make a thorough theoretical study and numerical simulation to predict the experimental results. In principle, it allows to use different kinds of sources in a decoy state MDI-QKD wangPRA2013 ; qin3 . Before experimentally testing all of them, one can choose to give a theoretical comparison with a reasonable model. In traditional decoy state methods H03 ; wang05 ; LMC05 , the models for calculation are relatively simple. However, for MDI-QKDs, it is not a simple job except for the special case of using weak coherent states. So far, there have been proposals with different sources, e.g., the heralded single-photon source (HSPS) _etc_ qin1 ; qin2 ; qin3 . And it has been shown that such a source can promise a longer secure distance than the weak coherent state. Nevertheless, it is unknown whether there are other sources which can present even better performance. Therefore, a general model on simulating the performance of arbitrary source states will be highly desirable. Here in this manuscript we solve the problem. For simplicity, we assume a linear lossy channel in our model. Note that the security does not depend on the condition of linear loss at all. We only use this model to predict: what values the gains and error rates would possibly be observed if one did the experiment in the normal case when there is no eavesdropper. Given these values, one can then calculate the low bound of the yield and the upper bound of the phase flip-error rates for single-photon pairs. The major goal here is to simulate the values of gains and error rates of different states in normal situations. Of course, they can be replaced with the observed values in real implementations. The paper is arranged as follows: In Sec. II we present the general model for the gains and error rates in a MDI-QKD, describing the detailed calculation processes. In Sec. III we proceed corresponding numerical simulations, comparing the different behaviors of MDI-QKDs when using different source states. Finally, discussions and summaries are given out in Sec. IV. ## II The general model on MDI-QKD Figure 1: (Color online) A schematic of the experimental setup for the collective measurements at the UTP. BS: beam-splitter; PBS: polarization beam- splitter; D1 - D4: single-photon detector; 1, 2: input port for photons. ### II.1 Setups and definitions Consider the schematic setup in Fig. 1 ind2 , there are three parties, the users-Alice and Bob, and the un-trusted third party (UTP)-Charlie. Alice and Bob send their polarized photon pulses to the UTP who will take collective measurement on the pulse-pairs. The collective measurement results at the UTP determine the successful events. They are two-fold click of detectors (1,4), (2,3), (1,2) or (3,4). The gain of any (two-pulse) source is determined by the number of successful events from the source. There are 4 detectors at the UTP, we assume each of them has the same dark count rate $d$, and the same detection efficiency $\xi$. In such a case, we can simplify our model by attributing the limited detection efficiency to the channel loss. Say, if the actual channel transmittance from Alice to Charlie is $\eta_{1}$, we shall assume perfect detection efficiency for Charlie’s detectors with channel transmittance of $\eta_{1}\xi$. Each detector will detect one of the 4 different modes, say $a_{H}^{\dagger},a_{V}^{\dagger},b_{H}^{\dagger},b_{V}^{\dagger}$ in creation operator. For simplicity, we denote them by $c_{i}^{\dagger}$, i.e., $c_{1}^{\dagger}=a_{H}^{\dagger},c_{2}^{\dagger}=a_{V}^{\dagger},c_{3}^{\dagger}=b_{H}^{\dagger},c_{4}^{\dagger}=b_{V}^{\dagger}$. In such a way, detector $D_{i}$ corresponds to mode $i$ exactly. To calculate the gains that would-be observed for different source states in the linear lossy channel, we need to model the probabilities of different successful events conditional on different states. Let’s first postulate some definitions before further study. Definition 1: event $(i,j)$. We define event $(i,j)$ as the event that both detector $i$ and detector $j$ click while other detectors do not click. Obviously, each $i,j$ must be from numbers $\\{1,2,3,4\\}$ and $i\not=j$ . For simplicity, we request $i<j$ throughout this paper. Definition 2: Output states and conditional probabilities of each events: notations $\rho_{out}$: the output state of the beam-splitter. $\|l_{i},l_{j}\rangle=\|l_{i}l_{j}\rangle$: the beam-splitter’s specific output state of $l_{i}$ photon in mode $i$, $l_{j}$ photon in mode $j$, and no photon in any other mode. Explicitly, $\|l_{i}l_{j}\rangle=\frac{1}{\sqrt{l_{i}!l_{j}!}}{c_{i}^{\dagger}}^{l_{i}}{c_{j}^{\dagger}}^{l_{j}}\|0\rangle$. $P(ij\|l_{i},l_{j})$ and $P(ij\|\rho_{out})$: the probability that event $(i,j)$ happens conditional on that the beam-splitter’s output state is $\|l_{i}l_{j}\rangle$ and $\rho_{out}$, respectively. Hereafter, we omit the comma between $l_{i}$ and $l_{j}$, i.e., we use $\|l_{i}l_{j}\rangle$ for $\|l_{i},l_{j}\rangle$, and $P(ij\|l_{i}l_{j})$ for $P(ij\|l_{i},l_{j})$. Definition 3: Events’ probability conditional on the beam-splitter’s input state: $p_{ij}^{\alpha\beta}(k_{1},k_{2})=p_{ij}^{\alpha\beta}(k_{1}k_{2})$. We denote $p_{ij}^{\alpha\beta}(k_{1},k_{2})$ as the probability of event $(i,j)$ conditional on that there are $k_{1}$ photons of polarization $\alpha$ for mode $a$ and $k_{2}$ photons of polarization $\beta$ for mode $b$ as the input state of the beam-splitter. Hereafter, we omit the comma between $k_{1}$ and $k_{2}$. $\alpha$ or $\beta$ indicate the photon polarization. Explicitly, $\alpha$ or $\beta$ can be $H,V,+,-$ for polarizations of horizontal, vertical, $\pi/4$ and $3\pi/4$, respectively. To indicate the corresponding polarization state, we simply put each of these symbols inside a ket. Definition 4: Events’ probability conditional on the two-pulse state of Alice and Bob’s source: $q_{ij}^{\alpha\beta}(\rho_{A}\otimes\rho_{B})$. It is the probability that event $(i,j)$ happens conditional on that Alice sends out photon-number state $\rho_{A}$ with polarization $\alpha$ and Bob sends out photon number state $\rho_{B}$ with polarization $\beta$. Sometimes we simply use $q_{ij}^{\alpha,\beta}$ for simplicity. ### II.2 Elementary formulas and outline for the model Given the definitions above, we now formulate various conditional probabilities. We start with the probability of event $(i,j)$ conditional on the output state $\|l_{i}l_{j}\rangle$. $\displaystyle P(ij\|l_{i}l_{j})=\left\\{\begin{array}[]{l}(1-d)^{2},\;{\rm if}\;l_{i}>0,l_{j}>0\\\ d(1-d)^{2},\;{\rm if}\;l_{i}\cdot\l_{j}=0\;{\rm and}\;l_{i}+l_{j}>0\\\ d^{2}(1-d)^{2},\;{\rm if}\;l_{i}=l_{j}=0\end{array}\right.$ (4) Here the detection efficiency does not appear because we put shall this into the channel loss and hence we assume perfect detection efficiency. The factor $(1-d)^{2}$ comes from the fact that we request detectors other than $i,j$ not to click. Also, the probability for event $(i,j)$ is 0 if any mode other than $i,j$ is not vacuum. Given these, we can now calculate probability distribution of the various two fold events given arbitrary input states of the beam-splitter. Therefore, for any output state of the beam-splitter $\rho_{out}$, the probability that event $(i,j)$ happens is $P(ij\|\rho_{out})=\sum_{l_{i},l_{j}}P(ij\|l_{i}l_{j})\langle l_{i}l_{i}\|\rho_{out}\|l_{i}l_{j}\rangle$ (5) Based on this important formula, we can calculate the probability of event $(i,j)$ for any input state by this formula. For the purpose, we only need to formulate $\rho_{out}$. Therefore, given the source state of the two pulses $\rho_{A}\otimes\rho_{B}$, we can use the following procedure to calculate the probability of event $(i,j)$, $p_{ij}(\rho_{A}\otimes\rho_{B})$: i) Using the linear channel loss model to calculate the two-pulse state when arriving at the beam-splitter. Explicitly, if the channel transmittance is $\eta$, any state $\|n\rangle\langle n\|$ is changed into $\|n\rangle\langle n\|\longrightarrow\sum C_{n}^{k}\eta^{k}(1-\eta)^{n-k}\|k\rangle\langle k\|.$ (6) ii) Using the transformation: $a_{H,V}^{\dagger}\longrightarrow\frac{1}{\sqrt{2}}(a_{H,V}^{\dagger}+b_{H,V}^{\dagger}$; $b_{H,V}^{\dagger}=\frac{1}{\sqrt{2}}(a_{H,V}^{\dagger}-b_{H,V}^{\dagger})$ to calculate the output state of the beam-splitter, $\rho_{out}$. iii) Using Eq.(5) to calculate the probability of event $(i,j)$. According to the protocol, we shall only be interested in the probabilities of successful events, $(1,2)$, $(3,4)$, $(1,4)$ and $(2,3)$. Below we will describe the detailed calculation processes in Z basis and X basis individually. In Z basis, all successful events correspond to correct bit values when Alice and Bob send out orthogonal polarizations, and they correspond to wrong bit values when Alice and Bob send out the same polarizations. The observed gain in $Z$ basis for photon-number state $\rho_{A}\otimes\rho_{B}$ is, $S^{Z}_{\rho_{A}\otimes\rho_{B}}=\frac{1}{4}\sum_{(i,j)\in Suc}\left[q_{ij}^{HV}(\rho_{A}\otimes\rho_{B})+q_{ij}^{VH}(\rho_{A}\otimes\rho_{B})+q_{ij}^{HH}(\rho_{A}\otimes\rho_{B})+q_{ij}^{VV}(\rho_{A}\otimes\rho_{B})\right]$ (7) and the set $Suc=\\{(1,2),(3,4),(1,4),(2,3)\\}$. Here, as defined in Definition 4, $q_{ij}^{\alpha\beta}(\rho_{A}\otimes\rho_{B})$ represents the probability of event $(i,j)$ whenever Alice sends her photon number state $\rho_{A}$ with polarization $\alpha$ and Bob sends his photon number state $\rho_{B}$ with polarization $\beta$. For simplicity, we shall omit $\rho_{A}\otimes\rho_{B}$ in brackets or in subscripts if there is no confusion. Meantime, the successful events caused by the same polarizations will be counted as wrong bits. These will contribute to the bit-flip rate by: $\tilde{E}^{Z}=\frac{\sum_{(i,j)\in Suc}\left[q_{ij}^{HH}+q_{ij}^{VV}\right]}{4S_{Z}}$ (8) In X basis, we should be careful that the situation is different from in Z basis, since now the successful events correspond to correct bits include two parts: 1) Alice and Bob send out the same polarizations ($++$ or $--$), and Charlie detects $\Phi^{+}$ ((1,2) or (3,4) events happen); 2) Alice and Bob send out orthogonal polarizations ($+-$ or $-+$), and Charlie detects $\Psi^{-}$ ((1,4) or (2,3) events happen). And the left successful events belong to wrong bits. Therefore, we have $S^{X}=\frac{1}{4}\sum_{(i,j)\in Suc}\left[q_{ij}^{+-}+q_{ij}^{-+}+q_{ij}^{++}+q_{ij}^{--}\right]$ (9) and $\tilde{E}^{X}=\frac{\sum_{(i,j)\in{(14),(23)}}\left[q_{ij}^{++}+q_{ij}^{--}\right]+\sum_{(i,j)\in{(12),(34)}}\left[q_{ij}^{+-}+q_{ij}^{-+}\right]}{4S_{X}}$ (10) Moreover, there are alignment errors which will cause a fraction ($E_{d}$) of states to be flipped. We then modify the error rate in different bases by $E^{Z}=E_{d}\cdot(1-2\tilde{E}^{Z})+\tilde{E}^{Z}$ (11) and $E^{X}=E_{d}\cdot(1-2\tilde{E}^{X})+\tilde{E}^{X}$ (12) Note that in the above two formulas above, we have considered this fact: before taking the alignment error into consideration, the successful events can be classified into two classes: one class has no error and the other class has an error rate of $50\%$, they are totally random bits. The second class takes a fraction of $2E^{Z}$ (or $2E^{X}$) among all successful events. Alignment error does not change the error rate of the second class of events, since they are random bits only. Given these, we can simulate the final key rate. In a model of numerical simulation, our goal is to deduce the probably would-be value for $S^{Z},S^{X}$ and $E^{Z},E^{X}$ in experiments. Given these, one can then calculate the yield of the single-photon pairs, $s_{11}$, the bit-flip rates in $Z$ basis and $X$ basis, and hence the final key rate. Now everything is reduced to calculate all $p_{ij}^{\alpha\beta}$ above. ### II.3 Conditional probabilities for beam-splitter’s incident state of $k_{1}$ photons in mode $a$ and $k_{2}$ photons in mode $b$ We consider the case that there are $k_{1}$ incident photons in mode $a$ and $k_{2}$ incident photons in mode $b$ of the beam-splitter. Each incident pulse of the beam-splitter has its own polarization and is indicated by a subscript. In general, we consider the state $\|k_{1}\rangle_{\alpha}\|k_{2}\rangle_{\beta}$ (13) We shall consider the conditional probabilities for various successful events, i.e. $p_{ij}^{\alpha\beta}(k_{1}k_{2})$. Since we only consider the incident state of $k_{1}$ photons in mode $a$ and $k_{2}$ photons in mode $b$, we shall simply use $p_{ij}^{\alpha\beta}$ for $p_{ij}^{\alpha\beta}(k_{1}k_{2})$ in what follows. i) in Z basis First, we consider the following two-mode state $\|k_{1}\rangle_{H}\|k_{2}\rangle_{V}=\frac{1}{\sqrt{k_{1}!k_{2}!}}{a_{H}^{\dagger}}^{k_{1}}{b_{V}^{\dagger}}^{k_{2}}\|0\rangle$ (14) as the input state of the beam-splitter. After BS, the output state $\|\psi\rangle$ is $\|\psi\rangle=\left(\frac{1}{\sqrt{2}}\right)^{k_{1}+k_{2}}\frac{1}{\sqrt{k_{1}!k_{2}!}}(a^{\dagger}_{H}+b^{\dagger}_{H})^{k_{1}}(a^{\dagger}_{V}-b^{\dagger}_{V})^{k_{2}}\|0\rangle$ (15) Therefore $\langle l_{1}l_{2}\|\rho_{out}\|l_{1}l_{2}\rangle=(1/2)^{k_{1}+k_{2}}\delta_{k_{1}l_{1}}\delta_{k_{2}l_{2}}$ (16) According to Eq.(5), the conditional probability for event (1,2) is $p_{12}^{HV}=P(12\|\rho_{out})=\sum_{l_{1},l_{2}}P(12\|l_{1}l_{2})(1/2)^{k_{1}+k_{2}}\delta_{k_{1}l_{1}}\delta_{k_{2}l_{2}}=(1/2)^{k_{1}+k_{2}}P(12\|k_{1}k_{2})$ (17) Similarly, we have $\displaystyle\begin{array}[]{l}p_{34}^{HV}=(1/2)^{k_{1}+k_{2}}P(34\|k_{1}k_{2})\\\ p_{14}^{HV}=(1/2)^{k_{1}+k_{2}}P(14\|k_{1}k_{2})\\\ p_{23}^{HV}=(1/2)^{k_{1}+k_{2}}P(23\|k_{2}k_{1})\end{array}$ (21) Note that here $P(ij\|k_{m}k_{n})$ is just $P(ij\|l_{i}=k_{m},l_{j}=k_{n})$ when $l_{1}=k_{1}$ as defined by our Definition 2 in previous section. For example, $P(23\|k_{2}k_{1})$ is actually $P(23\|l_{2}=k_{2},l_{3}=k_{1})$. Similarly, if the beam-splitter’s input state is $\|k_{1}\rangle_{V}\|k_{2}\rangle_{H}$, i.e. $k_{1}$ vertical photons in mode $a$ and $k_{2}$ horizontal photons in mode $b$, we have $\displaystyle\begin{array}[]{l}p_{12}^{VH}=(1/2)^{k_{1}+k_{2}}P(12\|k_{2}k_{1})\\\ p_{34}^{VH}=(1/2)^{k_{1}+k_{2}}P(34\|k_{2}k_{1})\\\ p_{14}^{VH}=(1/2)^{k_{1}+k_{2}}P(14\|k_{2}k_{1})\\\ p_{23}^{VH}=(1/2)^{k_{1}+k_{2}}P(23\|k_{1}k_{2})\end{array}$ (26) Next we consider the following two-mode state $\|k_{1}\rangle_{H}\|k_{2}\rangle_{H}=\frac{1}{\sqrt{k_{1}!k_{2}!}}{a_{H}^{\dagger}}^{k_{1}}{b_{H}^{\dagger}}^{k_{2}}\|0\rangle$ (27) as the input state of the beam-splitter. After the beam-splitter, it changes into: $\|\psi\rangle=\left(\frac{1}{\sqrt{2}}\right)^{k_{1}+k_{2}}\frac{1}{\sqrt{k_{1}!k_{2}!}}(a^{\dagger}_{H}+b^{\dagger}_{H})^{k_{1}}(a^{\dagger}_{H}-b^{\dagger}_{H})^{k_{2}}\|0\rangle$ (28) We have the following uniform formula for probabilities of any successful events: $\displaystyle p_{ij}^{HH}=\left\\{\begin{array}[]{l}\frac{(k_{1}+k_{2})!}{k_{1}!k_{2}!}(1/2)^{k_{1}+k_{2}}P(ij\|k_{1}+k_{2},0);\;{\rm for}\;i=1,\;j=2;{\rm or}\;i=3,\;j=4\\\ \frac{(k_{1}+k_{2})!}{k_{1}!k_{2}!}(1/2)^{k_{1}+k_{2}}P(ij\|0,k_{1}+k_{2});\;{\rm for}\;i=1,\;j=4;{\rm or}\;i=2,\;j=3\\\ \end{array}\right.$ (31) Similarly, when the beam-splitter’s input pulses are both vertical, we can find the value for $p_{ij}^{VV}$. ii) in $X$ basis We first consider the beam-splitter’s input state of $\|k_{1}\rangle_{+}\|k_{2}\rangle_{-}$, i.e., there are $k_{1}$ photon with $\pi/4$ polarization in mode $a$ and $k_{2}$ photons with $3\pi/4$ polarization in mode $b$. Note that $\|\pm\rangle=\frac{1}{\sqrt{2}}(\|H\rangle\pm\|V\rangle)$. The output state of the beam-splitter is $\|\psi\rangle=\frac{1}{2^{k_{1}+k_{2}}\sqrt{k_{1}!k_{2}!}}(a_{H}^{\dagger}+a_{V}^{\dagger}+b_{H}^{\dagger}+b_{V}^{\dagger})^{k_{1}}(a_{H}^{\dagger}-a_{V}^{\dagger}-b_{H}^{\dagger}+b_{V}^{\dagger})^{k_{2}}\|0\rangle$ (32) We have $\langle l_{i}l_{j}\|\psi\rangle=\frac{1}{2^{k_{1}+k_{2}}\sqrt{k_{1}!k_{2}!}}\sum_{s=\Delta_{1}}^{\Delta_{2}}\sqrt{l_{i}!l_{j}!}C_{k_{1}}^{s}C_{k_{2}}^{l_{1}-s}(-1)^{k_{2}-l_{i}+s}\delta_{l_{i}+l_{2},k_{1}+k_{2}}$ (33) where $\Delta_{1}=min\\{l_{i},k_{1}\\},\;\;\Delta_{2}=l_{i}-min\\{l_{i},k_{2}\\}$ (34) and $min\\{l_{i},k_{1}(k_{2})\\}$ is the smaller one of $l_{i}$ and $k_{1}$($k_{2}$). Thus we can calculate the conditional probabilities by $p_{ij}^{+-}=\sum_{l_{i}=0}^{k_{1}+k_{2}}\|\langle l_{i}l_{j}\|\psi\rangle\|^{2}$ Hence $p_{ij}^{+-}=\frac{1}{4^{k_{1}+k_{2}}k_{1}!k_{2}!}\sum_{l_{i}=0}^{k_{1}+k_{2}}\left\|\sum_{s=\Delta_{2}}^{\Delta_{1}}\sqrt{l_{1}!(k_{1}+k_{2}-l_{i})!}C_{k_{1}}^{s}C_{k_{2}}^{l_{i}-s}(-1)^{l_{i}-s}\right\|^{2}P(ij\|l_{i},k_{1}+k_{2}-l_{i})$ (35) for $i=1,j=2$ and $i=3,j=4$; and $p_{ij}^{+-}=\frac{1}{4^{k_{1}+k_{2}}k_{1}!k_{2}!}\sum_{l_{i}=0}^{k_{1}+k_{2}}\left\|\sum_{s=\Delta_{2}}^{\Delta_{1}}\sqrt{l_{i}!(k_{1}+k_{2}-l_{1})!}C_{k_{1}}^{s}C_{k_{2}}^{l_{i}-s}\right\|^{2}P(ij\|l_{i},k_{1}+k_{2}-l_{i})$ (36) for $i=1,j=4$ and $i=2,j=3$. Besides, it is easy to show $p_{ij}^{-+}=p_{ij}^{+-}$ (37) If the polarization of incident pulses of the beam-splitter are both $\pi/4$, then the output state is $\|\psi\rangle=\frac{1}{2^{k_{1}+k_{2}}\sqrt{k_{1}!k_{2}!}}(a_{H}^{\dagger}+a_{V}^{\dagger}+b_{H}^{\dagger}+b_{V}^{\dagger})^{k_{1}}(a_{H}^{\dagger}+a_{V}^{\dagger}-b_{H}^{\dagger}-b_{V}^{\dagger})^{k_{2}}\|0\rangle.$ (38) We find $p_{ij}^{++}=\frac{1}{4^{k_{1}+k_{2}}k_{1}!k_{2}!}\sum_{l_{i}=0}^{k_{1}+k_{2}}\left\|\sum_{s=\Delta_{2}}^{\Delta_{1}}\sqrt{l_{1}!(k_{1}+k_{2}-l_{i})!}C_{k_{1}}^{s}C_{k_{2}}^{l_{i}-s}\right\|^{2}P(ij\|l_{i},k_{1}+k_{2}-l_{i})$ (39) for $i=1,j=2$ and $i=3,j=4$; and $p_{ij}^{++}=\frac{1}{4^{k_{1}+k_{2}}k_{1}!k_{2}!}\sum_{l_{i}=0}^{k_{1}+k_{2}}\left\|\sum_{s=\Delta_{2}}^{\Delta_{1}}\sqrt{l_{i}!(k_{1}+k_{2}-l_{1})!}C_{k_{1}}^{s}C_{k_{2}}^{l_{i}-s}(-1)^{l_{i}-s}\right\|^{2}P(ij\|l_{i},k_{1}+k_{2}-l_{i})$ (40) for $i=1,j=4$ and $i=2,j=3$. Also, we have $p_{ij}^{--}=p_{ij}^{++}$ (41) ### II.4 Probabilities of events conditional on source states In the above subsection, we have formulated the probabilities of various events conditional on a pure input state $\|k_{1}\rangle\|k_{2}\rangle$. In fact, the results can be easily extended to the more general case when the beam-splitter’s input state is a mixed state. Say, $\left(\sum_{k_{1}}f_{k_{1}}\|k_{1}\rangle\langle k_{1}\right)\otimes\left(\sum_{k_{2}}f_{k_{2}}\|k_{2}\rangle\langle k_{2}\|\right)$ (42) Suppose the polarizations of mode $a,b$ are $\alpha,\beta$, respectively. We then have $p_{ij}^{\alpha\beta}=\sum_{k_{1},k_{2}}f_{k_{1}}f_{k_{2}}p_{ij}^{\alpha\beta}(k_{1}k_{2})$ (43) where $p_{ij}^{\alpha\beta}(k_{1}k_{2})$ is the same as defined in the previous subsection, for all possible polarizations $(\alpha,\beta)=(H,V),(V,H),(H,H),(V,V),(+,-),(-,+),(+,+),(-,-)$. To formulate the probabilities conditional on any source states, we only need to relate the source state with the beam-splitter’s input state. Suppose the source state in photon-number space is $\rho_{A}\otimes\rho_{B}$ and $\displaystyle\begin{array}[]{l}\rho_{A}=\sum_{n}a_{n}\|n\rangle\langle n\|\\\ \rho_{B}=\sum_{n}b_{n}\|n\rangle\langle n\|\end{array}$ (46) After some loss channel, the state changes into the beam-splitter’s input state as Eq.(42). Suppose the transmittance for the channel between Alice (Bob) and UTP is $\eta_{A}$ ($\eta_{B}$). Using the linear loss model of Eq. (6) we have $\displaystyle\begin{array}[]{l}f_{k_{1}}=\sum_{n\geq k_{1}}a_{n}\eta_{A}^{k_{1}}(1-\eta_{A})^{n-k_{1}}C_{n}^{k_{1}}\\\ f_{k_{2}}=\sum_{n\geq k_{2}}b_{n}\eta_{B}^{k_{2}}(1-\eta_{B})^{n-k_{2}}C_{n}^{k_{2}}\end{array}$ (49) We now arrive at our major conclusion: Major conclusion: Formulas of $p_{ij}^{\alpha\beta}(k_{1}k_{2})$ in the earlier subsection together with Eqs. (43,49) complete the model of probabilities of different events conditional on any source states, i.e., the gains. Using Eqs. (11,12), one can also model the observed error rates of any source states. ### II.5 3-intensity decoy-state MDI-QKD Using the Major conclusion above, we can model the gains and the error rates with a 3-intensity decoy-state MDI-QKD method wangPRA2013 ; qin3 . We assume that Alice (Bob) has three intensities in their source states, denoted as $0,\mu_{A},\mu_{A}^{\prime}$ ($0,\mu_{B},\mu_{B}^{\prime}$). Denote $\rho_{x}$ ($\rho_{y}$) as the density operator for source $x$ ($y$) at Alice’s (Bob’s) side, and $x$ ($y$) can take any value from $0,\mu_{A},\mu_{A}^{\prime}$ ($0,\mu_{B},\mu_{B}^{\prime}$). $\rho_{0}=\|0\rangle\langle 0\|;\\\ \rho_{\mu_{A}}=\sum_{k}a_{k}\|k\rangle\langle k\|;\,\,\rho_{\mu_{A}^{\prime}}=\sum_{k}a_{k}^{\prime}\|k\rangle\langle k\|;\\\ \rho_{\mu_{B}}=\sum_{k}b_{k}\|k\rangle\langle k\|;\,\,\rho_{\mu_{B}^{\prime}}=\sum_{k}b_{k}^{\prime}\|k\rangle\langle k\|$ (50) Then we have the expression for the low bound of the yield of single-photon pulse pairs $Y_{11}^{X}\geq{Y_{11}^{X,L}}\equiv\frac{a_{1}^{\prime}b_{2}^{\prime}(S_{\mu,\mu}^{X}-\tilde{S}_{0}^{X})-a_{1}b_{2}(S_{\mu^{\prime},\mu^{\prime}}^{X}-\tilde{S}_{0}^{\prime X})}{a_{1}^{\prime}a_{1}(b_{2}^{\prime}b_{1}-b_{2}b_{1}^{\prime})}$ (51) and their upper bound of the phase flip-error rate $e_{11}^{X}\leqslant e_{11}^{X,U}\equiv\frac{{E_{\mu,\mu}^{X}S_{\mu,\mu}^{X}-E_{\mu,0}^{X}S_{\mu,0}^{X}-E_{0,\mu}^{X}S_{0,\mu}^{X}+E_{0,0}^{X}S_{0,0}^{X}}}{{Y_{11}^{X}}}$ (52) With the results above, now we can calculate the key rate with the formula ind2 ; wangPRA2013 ; qin3 $R\geq a_{1}^{\prime}b_{1}^{\prime}Y_{11}^{Z}[1-H(e_{11}^{X})]-S_{\mu^{\prime}\mu^{\prime}}^{Z}f(E_{\mu^{\prime}\mu^{\prime}}^{Z})H(E_{\mu^{\prime}\mu^{\prime}}^{Z})$ (53) ## III Numerical simulations Using all the above correspondence, we can numerically simulate the gains and error rates of any source states. Taking as an example, we consider the source of a HSPS from parametric down-conversion processes qin3 . It originally has a Poissonian photon number distribution when pumped by a continuous wave (CW) laser explain1 , written as: $\left\|\psi\right\rangle{\text{ = }}\frac{{x^{n}}}{{n!}}e^{-x}$ (54) where $x$ is the the average intensity of the emission light. However, after chosen a proper gating time and triggered with a practical single photon detector, a sub-Poissonian distributed source state can be obtained, which can be expressed as: $\displaystyle\begin{array}[]{l}\rho=[{\text{P}}^{{\text{Cor}}}d_{i}+(1-{\text{P}}^{{\text{Cor}}})e^{-x}]\left\|0\right\rangle\left\langle 0\right\|+\sum\limits_{n=1}^{\infty}{{\text{[P}}^{{\text{Cor}}}e^{-x}\frac{{x^{n-1}}}{{(n-1)!}}+(1-{\text{P}}^{{\text{Cor}}})e^{-x}\frac{{x^{n}}}{{n!}}]}\left\|n\right\rangle\left\langle n\right\|\end{array}$ (56) where ${\text{P}}^{{\text{Cor}}}$ is the correlation rate of photon pairs, i.e., the probability that we can predict the existence of a heralded photon when a heralding one was detected; $d_{i}$ is the dark count rate of the triggering detector. Figure 2: (Color online) (a) The lower bound of $Y_{11}$ and (b) the upper bound of $e_{11}^{X}$ for different photon sources. The solid lines (W0) represent the results of using infinite-decoy state method, and the dashed or dotted lines (W1, P1 or S1) represent using three-decoy state method. Besides, W, P or S each corresponds to the scheme of using weak coherent sources ind2 , possonian heralded single photon sources qin2 or sub-possonian heralded single photon sources qin3 , individually. X or Z represent in X or Z basis respectively. Here at each point, we set $\mu=0.05$, and optimize the value for $\mu^{\prime}$. Figure 3: (Color online) (a) The gain and (b) the quantum error-bit rate in Z basis for different photon sources. The solid lines (W0) represent the results of using infinite-decoy state method, and the dashed or dotted lines (W1, P1 or S1) represent using three-decoy state method. Besides, W, P or S each corresponds to the scheme of using weak coherent sources, possonian heralded single photon sources qin2 or sub-possonian heralded single photon sources qin3 , individually. Here at each point, we set $\mu=0.05$, and optimize the value for $\mu^{\prime}$. Figure 4: (Color online) (a) The final key rate for different photon sources. The solid lines (W0) represent the results of using infinite-decoy state method, and the dashed or dotted lines (W1, P1 or S1) represent using three-decoy state method. Besides, W, P or S each corresponds to the scheme of using weak coherent sources, possonian heralded single photon sources qin2 or sub- possonian heralded single photon sources qin3 , individually. Here at each point, we set $\mu=0.05$, and optimize the value for $\mu^{\prime}$. In the following numerical simulations, for simplicity, we assume the UTP lies in the middle of Alice and Bob, and all triggering detectors (at Alice or Bob’s side) have the same detection efficiency ($75\%$) and the same dark count rate ($10^{-6}$). We also assume all triggered detectors (at the UTP’s side) have the same detection efficiency (they are attributed into the channel loss), and the same dark count rate ($3\times 10^{-6}$). Besides, we set the system misalignment probability to be $1.5\%$. Fig. 2(a) and (b) each show the low bound of $Y_{11}$ (in X or Z basis) and the upper bound of $e_{11}^{X}$ changing with channel loss for different source states, i.e., the weak coherent sources (W), the possonian heralded single photon sources (P) and the sub-possonian heralded single photon sources (S). The solid line represents the result of using infinite number of decoy state method (W0), and the dashed or dotted lines (P1 or S1) are the results of using three-decoy state method. Similar to Fig. 2(a) and (b), Fig. 3(a) and (b) each show corresponding values of the gains (${\text{S}}_{\mu^{\prime}\mu^{\prime}}^{Z}$) and the quantum bit-error rates (QBER) (${\text{E}}_{\mu^{\prime}\mu^{\prime}}^{Z}$) of signal pulses in Z basis for different source states. And Fig. 5 presents the final key rate changing with channel loss. See from Fig. 4, we find that the sub-possonian heralded single photon sources can generate the highest key rate at lower or moderate channel loss ($\leqslant 64$ dB). Because within this range, its signal state has a lower QBER than in the weak coherent sources, and a higher gain than in the possonian heralded single photon sources as simulated in Fig. 3 (a) and (b). However, at larger channel loss ($\geqslant 64$ dB), the possonian heralded single photon source shows better performance than the other two, this is mainly due to its much lower vacuum component which may play an essential role in the key distillation process when suffering from lager channel loss. ## IV Conclusions In summary, we have presented a general model for simulating the gains, the error rates and the key rates for MDI-QKDs, which can be applicable to the schemes of using arbitrary convex source states and any coding methods. 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Lo, C. H. F. Fung, and B. Qi, Phase encoding schemes for measurement device independent quantum key distribution and basis-dependent flaw. Phys. Rev. A 85, 042307 (2012). * (29) During a spontaneous parametric down conversion (SPDC) process, if a pulsed pump laser is used, as long as the coherence time of the emission, $\Delta t_{c}$, is much longer than the duration of the pump pulse, $\Delta t$, i.e., $\Delta t_{c}\gg\Delta t$, (in practice easily obtained by using ultrafast (fs) pulse pump lasers), a single emission process will take place, giving an thermal photon number distribution. In contrast, when a continuous wave (CW) laser is used, as long as $\Delta t_{c}$ is much shorter than the gating period of the detector, a large number of independent SPDC processes will be present, each thermally distributed, but collectively resulting in a Poisson distribution. However, the ”original” distribution can be altered by conditional gating. By choosing proper gating time and using an appropriate correlation rate, a sub-Poissonian distributed HSPS can be obtained as the result of postselections.	arxiv-papers	2013-11-07T16:54:55	2024-09-04T02:49:53.345579	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Qin Wang and Xiang-Bin Wang", "submitter": "Xiang-Bin Wang", "url": "https://arxiv.org/abs/1311.1739" }
1311.1812	Conditions for Bifurcations in a Non-Autonomous Scalar Differential Equation Sang-Mun Kim, Hyong-Chol O, Chol Kim and Gyong-Chol Kim Faculty of Mathematics, Kim Il Sung University, Pyongyang, D.P.R Korea e-mail address: [email protected] ###### Abstract In this paper is provided a sufficient condition to occur saddle-node and transcritical bifurcations for a non-autonomous scalar differential equation. Keywords: non-autonomous scalar differential equation, saddle-node bifurcation, forwards attracting, pullback attracting MSC(2010): 37B55, 37C75, 37G10, 34C23 ## 1 Introduction The concept of non-autonomous dynamical systems can be said to have been made from the study on skew product flows and random dynamical systems in 1990s in the viewpoint of topological dynamics. A lot of developments have been made together with considering problems of various concepts of attractiveness, existence and uniqueness of attracting sets and etc. [2]-[19]. In [13] they obtained sufficient conditions to occur transcritical, pitchfork and saddle-node bifurcations in a special type of non-autonomous differential equation generalized from a canonical form of autonomous differential equation where transcritical, pitchfork and saddle- node bifurcations occur. And then using them, they studied the conditions for similar bifurcations in the general scalar non-autonomous equation $\dot{x}=f(x,t,\lambda),$ where $\lambda$ is a parameter. By imposing conditions on the Taylor coefficients in the expansion of $f$ near $x=\lambda=0$, they proved various general theorems guaranteeing transcritical, pitchfork, and saddle-node bifurcations. In [16] they obtained a sufficient condition to occur transcritical bifurcation in non-autonomous differential equation $\dot{x}=a(t,\alpha)x+b(t,\alpha)x^{2}+r(t,x,\alpha)$ and a sufficient condition to occur pitchfork bifurcation in non-autonomous differential equation $\dot{x}=a(t,\alpha)x+b(t,\alpha)x^{3}+r(t,x,\alpha).$ Some particular examples have been analyzed in various settings. In [6, 8] using the framework of skew product flows has been considered a generalized notion of a Hopf bifurcation and in [19] studied almost periodic scalar non - autonomous differential equations. In [10] has been analyzed transcritical and pitchfork bifurcations in an almost periodic equation, [7] has considered a non-autonomous ‘two-step bifurcation’and [11] gave a nice discussion of the general problem in the context of skew product flows. On the other hand, conditions for bifurcations to occur in one-dimensional autonomous dynamical systems have been studied using higher order derivatives. In [20] sufficient conditions for transcritical, pitchfork, saddle-node and period doubling bifurcations to occur in one-dimensional maps with one parameter have been studied using higher order derivatives. In [1] sufficient conditions for cusp and period doubling bifurcations to occur in one- dimensional maps with two parameters have been studied using higher order derivatives. In this paper we consider some non-autonomous differential equations generalized from autonomous dynamical systems in [1, 20]. First we try to obtain a sufficient condition to occur saddle-node and transcritical bifurcations in the equations $\dot{x}=\mu^{2m-1}f(t)-g(t)x^{2n},~{}m,n\in\mathbf{N}$ (1) where the sufficient condition of [13] does not satisfied. And then we try to obtain sufficient conditions to occur saddle-node and transcritical bifurcations in more general equations $\dot{x}=G(x,t,\lambda),~{}(\lambda\text{ is a parameter})$ which include (1). ## 2 Preliminaries We consider the following initial value problem of non-autonomous differential equation $\dot{x}=f(t,x),x(s)=x_{0}$ (2) defined on a domain $D\subset\mathbf{R}^{m}$ of $x$. Let denote the solution to (2) by $x(t,s;x_{0})=S(t,s)x_{0}.$ Then $\\{S(t,s)\\}_{t\geq s}$ becomes a two-parameter family of transformations of $D$ satisfying the following properties [6, 16]: 1) $\forall t\in\textbf{R},S(t,t)$ is the identity of $D$. 2) $S(t,\tau)S(\tau,s)=S(t,s)~{}(\forall t,\tau,s\in\textbf{R})$. 3) $S(t,s)x_{0}$ is continuous on $t,s,x_{0}$. Through the whole paper, we assume that $\\{S(t,s):D\to D\\}_{t\geq s}$ preserves order [1]. For some basic concepts including complete orbits, time- varying family of sets, invariant sets, Hausdorf semi distance between sets, (local or global) forwards attracting sets, (local or global) pullback attracting sets, pullback Lyapunov stability, pullback Lyapunov instability, asymptotic instability and unstable sets of invariant sets, (local) pullback repelling sets, pullback attracting sets and the definitions on several types of bifurcations in non-autonomous differential equations, we refer to [2]. We note the following well-known facts as a remark: 1) If $\Sigma(\cdot)$ is forwards attracting in $D$, then it is locally forwards attracting in $D$. 2) If $D$ is a bounded set, any pullback attracting sets in $D$ locally pullback attracting in $D$. If $D$ is a unbounded set, global pullback attracting sets in $D$ might not be locally pullback attracting in $D$. 3) If an invariant set $\Sigma(\cdot)$ is pullback attracting set in $D$ and there is a $T$ such that $\bigcup_{t\leq T}\Sigma(t)$ is bounded, then $\Sigma(\cdot)$ is locally pullback attracting [2]. 4) If $x^{}(\cdot)$ is a complete orbit and locally pullback attracting, then it is pullback Lyapunov stable. 5) If $\Sigma(\cdot)$ is asymptotically instable, then it is pullback Lyapunov instable but it cannot be locally pullback attracting [13]. The following fact provides some information about attracting sets: Let $\\{K(t)\\}_{t\in\textbf{R}}$ be a family of non-empty compact sets and for all $t_{0}$ and compact set $B\subset D$, $\exists T=T(t_{0},B):\forall s\leq T,S(t_{0},s)B\subset K(t_{0}).$ Then there exists a pullback attracting set $A(t)$ which is a connect set for every $t\in\textbf{R}$ [15]. ## 3 Main Results ### 3.1 Saddle node Bifurcation First we consider a concrete example. ###### Theorem 1. Let consider the following non-autonomous differential equation $\dot{x}=\mu^{2m-1}f(t)-g(t)x^{2n},~{}m,n\in\mathbf{N}$ (3) Let assume that $f(t)$ and $g(t)$ satisfy $\displaystyle\int_{-\infty}^{t}f(s)ds=\int_{t}^{+\infty}f(s)ds=+\infty,$ (4) $\displaystyle\lim_{t\to\pm\infty}\textnormal{inf}~{}g(t)>0,~{}0<l\leq\lim_{t\to\pm\infty}\frac{f(t)}{g(t)}\leq M.$ (5) Then we have the following facts: 1) When $\mu\leq 0$, non-zero bounded complete orbits do not exist. When $\mu<0$, for any fixed $x_{s}$, we have $\exists\sigma:s\leq\sigma,\exists t^{}(s)<+\infty:\lim_{t\to t^{}(s)}x(t,s;x_{s})=-\infty$ and for any fixed $t$, we have $\exists s^{}(t)>-\infty:\lim_{s\to s^{}(t)}x(t,s;x_{s})=-\infty.$ 2) When $\mu=0$, the zero solution is locally pullback and forwards attracting in $[0,\infty)$. For the solution with initial value in $(-\infty,0)$, we have the same conclusions with the case of $\mu<0$. 3) When $\mu>0$, there exist two orbits $x^{}(t)$ and $y^{}(t)$ such that $x^{}(t)$ is pullback and forwards attracting and $y^{}(t)$ is pullback repelling and asymptotically instable. That is, we have $\displaystyle\lim_{s\to-\infty}S(t,s)x_{0}=x^{}(t),~{}x_{0}>\sqrt[-2n]{l\mu^{2m-1}},$ $\displaystyle\lim_{t\to+\infty}\textnormal{dist}\left[S(t,s)x_{0},x^{}(t)\right]=0,~{}x_{0}>\sqrt[-2n]{l\mu^{2m-1}},$ $\displaystyle\lim_{s\to+\infty}S(t,s)x_{0}=y^{}(t),~{}x_{0}<\sqrt[2n]{l\mu^{2m-1}},$ $\displaystyle\lim_{t\to-\infty}\textnormal{dist}\left[S(t,s)x_{0},y^{}(t)\right]=0,~{}x_{0}<\sqrt[2n]{l\mu^{2m-1}}.$ ###### Proof. In the case of $\mu<0$, by (5) we have $\exists T:\forall t\leq T\Rightarrow f(t)>0,~{}g(t)>0.$ In the case of $x_{s}<0$, by the above expression we have $\forall t\leq T,~{}\dot{x}\leq-g(t)x^{2n}$ and $\displaystyle\int_{s}^{t}\frac{\dot{x}}{x^{2n}}dr\leq-\int_{s}^{t}g(r)dr\Rightarrow\int_{x(s)}^{x(t)}\frac{1}{x^{2n}}dx\leq-\int_{s}^{t}g(r)dr$ $\displaystyle\left.\Rightarrow-\frac{1}{(2n-1)}x^{-(2n-1)}\right\|_{x=x(s)}^{x(t)}\leq-\int_{s}^{t}g(r)dr$ $\displaystyle\Rightarrow-\frac{1}{(2n-1)}x(t)^{-(2n-1)}\leq-\frac{1}{(2n-1)}x(s)^{-(2n-1)}-\int_{s}^{t}g(r)dr$ $\displaystyle\Rightarrow x(t)^{(2n-1)}\leq\left(x(s)^{-(2n-1)}+(2n-1)\int_{s}^{t}g(r)dr\right)^{-1}$ $\displaystyle\Rightarrow x(t)\leq\left(x(s)^{-(2n-1)}+(2n-1)\int_{s}^{t}g(r)dr\right)^{-\frac{1}{2n-1}}.$ For fixed $t,x_{s}^{-(2n-1)}<0$ and $g$ satisfies $\exists T^{}:s\leq T^{}\Rightarrow\int_{s}^{t}g(r)dr>0.$ Thus we have $\exists s^{}(t)>-\infty:\lim_{s\to s^{}(t)}x(t,s;x_{s})=-\infty.$ For fixed $x_{s}$, we have $\exists\sigma(t):s\leq\sigma(t),\exists t^{}(s)<+\infty:\lim_{t\to t^{}(s)}x(t,s;x_{s})=-\infty.$ Let consider the case of $x_{s}<0$. Then for $x_{s}=-1$, we have $\exists\sigma_{1}:s\leq\sigma_{1}\Rightarrow\exists t^{}(s)<+\infty,\lim_{t\to t^{}(s)}x(t,s;-1)=-\infty.$ (6) If $t\leq T$, then $\dot{x}\leq\mu^{(2n-1)}f(t)<0$ and thus we have $x(t,s;x_{s})\leq x_{s}+\mu^{(2n-1)}\int_{s}^{t}f(r)dr,~{}t\leq T.$ (7) Using $\mu<0$, (4) and (7), we have $\exists\sigma_{2}:s\leq\sigma_{2}\Rightarrow t\leq\sigma_{1},x(t,s;x_{s})\leq-1.$ From (6) and property of order preservation we know $t\leq\sigma_{1}$ and thus $\forall\tau>t,~{}x(\tau,t;x(t,s;x_{s}))\leq x(\tau,t;-1).$ When $\tau\to t^{}(s)$, we have $x(\tau,t;-1)\to-\infty$ and thus $x(\tau,s;x_{s})\to-\infty$. On the other hand, for fixed $t$, when $s\to s_{1}(t)>-\infty$, we have $x(t,s;-1)\to-\infty$. Using (7), we have $\exists s_{2}:s\leq s_{2},x(t,s;x_{s})\leq-1$ and from property of order perservation, we have $x(t,s;x_{s})\leq x(t,s;-1)$. When $s\to s_{1}(s_{2})$, we have $x(t,s;x_{s})\to-\infty$. Next consider the case of $\mu=0$. Then the solution of (3) is as follows: $x(t,s;x_{s})=\frac{1}{\left[x_{s}^{-(2n-1)}+(2n-1)\int_{s}^{t}g(r)dr\right]^{\frac{1}{2n-1}}}.$ If $x_{s}\geq 0$, then $x_{s}^{-(2n-1)}\geq 0$ and from (4) we have $s\to-\infty(t\to+\infty)$. Then $(2n-1)\int_{s}^{t}g(r)dr\to+\infty,~{}\left[x_{s}^{-(2n-1)}+(2n-1)\int_{s}^{t}g(r)dr\right]^{\frac{1}{2n-1}}\to+\infty.$ Thus we have $x(t,s;x_{s})\to 0(t\to+\infty,s\to-\infty).$ In order to show that the zero solution is locally pullback attracting, we must prove $\left[x_{s}^{-(2n-1)}+(2n-1)\int_{s}^{t}g(r)dr\right]^{\frac{1}{2n-1}}>0,~{}\forall\tau\in[s,t].$ If $x_{s}<\frac{1}{\sup_{\tau\in[T^{-},t]}\left\|\left[(2n-1)\int_{T^{-}}^{\tau}g(r)dr\right]^{\frac{1}{2n-1}}\right\|}=\delta(t),$ then the above expression holds. Thus the zero solution is locally pullback attracting. It is similar to prove that the zero solution is locally forwards attracting. If $x_{s}<0$, then we have the same result with the case when $\mu<0$ and $x_{s}<0$. Let consider the case of $\mu>0$. From the condition (5) we have $\displaystyle\exists T^{-}<0,\exists T^{+}>0;\forall t\leq T^{-},\forall t\geq T^{+}$ $\displaystyle\Rightarrow\dot{x}\leq\mu^{(2m-1)}Mg(t)-g(t)x^{2n}=g(t)\left[M\mu^{(2m-1)}-x^{2n}\right],$ $\displaystyle\dot{x}\geq\mu^{(2m-1)}lg(t)-g(t)x^{2n}=g(t)\left[l\mu^{(2m-1)}-x^{2n}\right].$ Thus we have $\displaystyle\dot{x}\leq g(t)\left[\sum_{k=1}^{n}\left(\sqrt[2n]{M\mu^{2m-1}}\right)^{2(n-k)}x^{2(k-1)}\right]\left[\left(\sqrt[2n]{M\mu^{2m-1}}\right)^{2}-x^{2}\right],$ $\displaystyle\dot{x}\geq g(t)\left[\sum_{k=1}^{n}\left(\sqrt[2n]{l\mu^{2m-1}}\right)^{2(n-k)}x^{2(k-1)}\right]\left[\left(\sqrt[2n]{l\mu^{2m-1}}\right)^{2}-x^{2}\right].$ Let $\displaystyle g_{1}(t):=g(t)\left[\sum_{k=1}^{n}\left(\sqrt[2n]{M\mu^{2m-1}}\right)^{2(n-k)}x^{2(k-1)}\right],$ $\displaystyle g_{2}(t):=g(t)\left[\sum_{k=1}^{n}\left(\sqrt[2n]{l\mu^{2m-1}}\right)^{2(n-k)}x^{2(k-1)}\right].$ Then $g_{1}(t)$ and $g_{2}(t)$ satisfy the condition (5) on $g(t)$. Therefore we have $\displaystyle\dot{x}\leq g_{1}(t)\left[\sqrt[2n]{M\mu^{2m-1}}+x\right]\left[\sqrt[2n]{M\mu^{2m-1}}-x\right],$ $\displaystyle\dot{x}\geq g_{2}(t)\left[\sqrt[2n]{l\mu^{2m-1}}+x\right]\left[\sqrt[2n]{l\mu^{2m-1}}-x\right].$ If $x_{0}>\sqrt[-2n]{l\mu^{2m-1}}$, then $\sqrt[2n]{l\mu^{2m-1}}\leq\lim_{\begin{subarray}{c}s\to-\infty\\\ t\to+\infty\end{subarray}}x(t,s;x_{0})\leq\sqrt[2n]{M\mu^{2m-1}}.$ Now let $x_{1}(t)$ and $x_{2}(t)$ be two different solutions of (3) and $z(t)=x_{1}(t)-x_{2}(t)$. Then $\dot{x}_{1}(t)=\mu^{2m-1}f(t)-g(t)x_{1}^{2n}(t),~{}\dot{x}_{2}(t)=\mu^{2m-1}f(t)-g(t)x_{2}^{2n}(t).$ Thus we have $\dot{z}(t)=-g(t)\left[x_{1}^{2n}-x_{2}^{2n}\right]=-g(t)\left[\sum_{k=1}^{n}x_{1}^{2(n-k)}(t)x_{2}^{2(k-1)}(t)\right][x_{1}+x_{2}]z(t).$ (8) Since $\forall t\leq T^{-}$ or $\forall t\geq T^{+},~{}g(t)\left(\sum_{k=1}^{n}x_{1}^{2(n-k)}(t)x_{2}^{2(k-1)}(t)\right)>0$ and $x_{1}(t),x_{2}(t)\geq\sqrt[2n]{l\mu^{2m-1}}$, thus we have $\forall t\leq T^{-}$ or $\forall t\geq T^{+},~{}x_{1}(t)=x_{2}(t)$. Therefore there exists a positive solution $x^{}(t)$ such that it (pullback, forwards) attracts all orbits with initial data greater than $\sqrt[-2n]{l\mu^{2m-1}}$. Now if $x_{0}<\sqrt[-2n]{M\mu^{2m-1}}$, then the solutions go to $-\infty$ (pullback, forwards). If $x_{0}<\sqrt[2n]{l\mu^{2m-1}}$, then $\sqrt[-2n]{M\mu^{2m-1}}\leq\lim_{\begin{subarray}{c}t\to-\infty\\\ s\to+\infty\end{subarray}}x(t,s;x_{0})\leq\sqrt[-2n]{l\mu^{2m-1}}$ and for the two different solutions $x_{1}(t)$ and $x_{2}(t)$ of (3), we have (8). Repeating the above arguments, we have the following conclusion: If $\forall t\leq T^{-},~{}\forall t\geq T^{+},~{}x_{1}(t),x_{2}(t)\leq\sqrt[-2n]{l\mu^{2m-1}}$, then $x_{1}(t)=x_{2}(t)$. Thus there exists a negative solution $y^{}(t)$ such that it (pullback, forwards) attracts all orbits with initial data less than $\sqrt[2n]{l\mu^{2m-1}}$ in the meaning of time inverse. That is, $y^{}(t)$ is pullback repelling. $\lim_{\begin{subarray}{c}s\to+\infty\\\ t\to-\infty\end{subarray}}x(t,s;x_{s})=y^{}(t),~{}x_{s}<\sqrt[2n]{l\mu^{2m-1}}.$ ∎ Now we consider general equations $\dot{x}=G(t,x,\mu).$ (9) Assume that $G$ is sufficiently smooth. The following is Taylor expansion of $G$ at $(t,0,0)$. $\displaystyle G(t,x,\mu)=G(t,0,0)+G_{x}(t,0,0)x+G_{\mu}(t,0,0)\mu+\frac{1}{2}G_{xx}(t,0,0)x^{2}$ $\displaystyle\qquad+G_{x\mu}(t,0,0)x\mu+\frac{1}{2}G_{\mu\mu}(t,0,0)\mu^{2}+\frac{1}{6}G_{xxx}(t,0,0)x^{3}+\frac{1}{2}G_{xx\mu}(t,0,0)x^{2}\mu$ $\displaystyle\qquad+\frac{1}{2}G_{x\mu\mu}(t,0,0)x\mu^{2}+\frac{1}{6}G_{\mu\mu\mu}(t,0,0)\mu^{3}+\cdots+\frac{1}{(2n)!}\left[\frac{\partial^{2n}}{\partial x^{2n}}G(t,0,0)x^{2n}\right.$ $\displaystyle\qquad+C_{2n}^{1}\frac{\partial^{2n}}{\partial x^{2n-1}\partial\mu}G(t,0,0)x^{2n-1}\mu+\cdots+C_{2n}^{2n-1}\frac{\partial^{2n}}{\partial x\partial\mu^{2n-1}}G(t,0,0)x\mu^{2n-1}$ $\displaystyle\qquad\left.+\frac{\partial^{2n}}{\partial\mu^{2n}}G(t,0,0)\mu^{2n}\right]+\frac{1}{(2n+1)!}\left[\frac{\partial^{2n+1}}{\partial x^{2n+1}}G(t,0,0)x^{2n+1}\right.$ $\displaystyle\qquad+C_{2n+1}^{1}\frac{\partial^{2n+1}}{\partial x^{2n}\partial\mu}G(t,0,0)x^{2n}\mu+\cdots+C_{2n+1}^{2n}\frac{\partial^{2n+1}}{\partial x\partial\mu^{2n}}G(t,0,0)x\mu^{2n}+\cdots$ $\displaystyle\qquad\left.+C_{2n+1}^{2n}\frac{\partial^{2n+1}}{\partial x\partial\mu^{2n}}G(t,0,0)x\mu^{2n}+\frac{\partial^{2n+1}}{\partial\mu^{2n+1}}G(t,0,0)\mu^{2n+1}\right].$ Here $n\in\mathbf{N}$. Now assume that $G$ satisfies the following conditions: $\displaystyle\textnormal{(i)}~{}G(t,0,0)=0,~{}\forall t\in\mathbf{R},$ $\displaystyle\textnormal{(ii)}~{}\frac{\partial}{\partial x}G(t,0,0)=\frac{\partial^{2}}{\partial x^{2}}G(t,0,0)=\cdots=\frac{\partial^{2n-1}}{\partial x^{2n-1}}G(t,0,0)=0.\qquad\qquad\qquad\qquad$ Then $G$ is provided as follows: $\displaystyle G(t,x,\mu)=\mu\left[G_{\mu}(t,0,0)+G_{x\mu}(t,0,0)x+\frac{1}{2}G_{\mu\mu}(t,0,0)\mu+\frac{1}{3}G_{xx\mu}(t,0,0)x^{2}\right.$ $\displaystyle\quad+\frac{1}{6}G_{\mu\mu}(t,0,0)\mu^{2}+\frac{1}{3}G_{x\mu\mu}(t,0,0)x\mu+\cdots+\frac{1}{(2n)!}C_{2n}^{1}\frac{\partial^{2n}}{\partial x^{2n-1}\partial\mu}$ $\displaystyle\quad\times~{}G(t,0,0)x^{2n-1}+\cdots+\frac{1}{(2n)!}C_{2n}^{2n-1}\frac{\partial^{2n}}{\partial x\partial\mu^{2n-1}}G(t,0,0)x\mu^{2n-2}$ $\displaystyle\quad\left.+\frac{1}{(2n)!}\frac{\partial^{2n}}{\partial\mu^{2n}}G(t,0,0)\mu^{2n-1}+\frac{1}{(2n+1)!}C_{2n+1}^{1}\frac{\partial^{2n+1}}{\partial x^{2n}\partial\mu}G(t,0,0)x^{2n}+\cdots\right]$ $\displaystyle\quad+\left[\frac{1}{(2n)!}\frac{\partial^{2n}}{\partial x^{2n}}G(t,0,0)+\frac{1}{(2n+1)!}\frac{\partial^{2n+1}}{\partial x^{2n+1}}G(t,0,0)x+\cdots\right]x^{2n}+\cdots.$ Let denote $f(t):=G_{\mu}(t,0,0),~{}g(t):=-\frac{1}{(2n)!}\frac{\partial^{2n}}{\partial x^{2n}}G(t,0,0)$. Then (9) can be rewritten as follows: $\dot{x}=\mu[f(t)+\phi(t,x,\mu)]-x^{2n}[g(t)+\psi(t,x)].$ (10) Here $\phi(t,0,0)=0,~{}\psi(t,0)=0$. ###### Theorem 2. Assume that $\lim_{t\to\pm\infty}\textnormal{inf}~{}g(t)>0,$ (11) $0<m=\lim_{t\to\pm\infty}\textnormal{inf}~{}\frac{f(t)}{g(t)}\leq\lim_{t\to\pm\infty}\textnormal{sup}~{}\frac{f(t)}{g(t)}=M<+\infty,$ (12) and there exists a positive valued function $h(t)$ such that $\|\phi(t,x,\mu)\leq h(t)[\|x\|+\|\mu\|],~{}~{}\|\phi_{x}(t,x,\mu)\|\leq h(t),$ (13) $\|\psi_{x}(t,x)\|\leq h(t),$ (14) $\lim_{t\to\pm\infty}\textnormal{sup}~{}\frac{h(t)}{g(t)}\leq k.$ (15) Then there occurs local saddle-node bifurcation when $\mu$ passes through 0. Furthermore, when $\mu>0$, locally attracting orbit $x_{\mu}(t)$ is forwards attracting in $(0,\varepsilon)$ and unstable orbits are pullback repelling in $(-\varepsilon,\delta)$. The main idea of the proof is similar to Theorem 1 and omitted. Now assume that $G$ satisfies the following conditions: $\displaystyle\textnormal{(iii)}~{}G(t,x,\mu)=G(t,x,0)+c(x)G(0,0,\mu),$ $\displaystyle\textnormal{(iv)}~{}G(t,0,0)=0,$ $\displaystyle\textnormal{(v)}~{}\frac{\partial}{\partial x}G(t,0,0)=\frac{\partial^{2}}{\partial x^{2}}G(t,0,0)=\cdots=\frac{\partial^{2n-1}}{\partial x^{2n-1}}G(t,0,0)=0,$ $\displaystyle\textnormal{(vi)}~{}\frac{\partial}{\partial\mu}G(t,0,0)=\frac{\partial^{2}}{\partial\mu^{2}}G(t,0,0)=\cdots=\frac{\partial^{2m-2}}{\partial\mu^{2m-2}}G(t,0,0)=0.\qquad\qquad\qquad\qquad$ Here $n,m\in\mathbf{N}$. Then $G$ is provided as follows: $\displaystyle G(t,x,\mu)=\mu^{2m-1}\left[x\mu^{-(2m-2)}\frac{\partial^{2}}{\partial x\partial\mu}G(t,0,0)+\frac{1}{2}x^{2}\mu^{-(2m-2)}\frac{\partial^{3}}{\partial x^{2}\partial\mu}G(t,0,0)\right.$ $\displaystyle\qquad+\frac{1}{2}x\mu^{-(2m-3)}\frac{\partial^{3}}{\partial x\partial\mu^{2}}G(t,0,0)+\cdots+\frac{1}{(2m-1)!}\frac{\partial^{2m-1}}{\partial\mu^{2m-1}}G(t,0,0)$ $\displaystyle\qquad+\frac{1}{(2m)!}C_{2m}^{1}x^{2m-1}\mu^{-(2m-2)}\frac{\partial^{2m}}{\partial x^{2m-1}\partial\mu}G(t,0,0)$ $\displaystyle\qquad\left.+\frac{1}{(2m)!}C_{2m}^{2}x^{2m-2}\mu^{-(2m-3)}\frac{\partial^{2m}}{\partial x^{2m-2}\partial\mu^{2}}G(t,0,0)+\cdots\right]$ $\displaystyle\qquad+x^{2n}\left[\frac{1}{(2n)!}\frac{\partial^{2n}}{\partial x^{2n}}G(t,0,0)+\frac{1}{(2n+1)!}x\cdot\frac{\partial^{2n+1}}{\partial x^{2n+1}}G(t,0,0)+\cdots\right].$ Let denote $f(t):=\frac{1}{(2m-1)!}\frac{\partial^{2m-1}}{\partial\mu^{2m-1}}G(t,0,0),~{}g(t):=-\frac{1}{(2n)!}\frac{\partial^{2n}}{\partial x^{2n}}G(t,0,0)$. Then we can rewritten (15) as follows: $\dot{x}=\mu^{2m-1}[f(t)+\phi(t,x,\mu)]-x^{2n}[g(t)+\psi(t,x)].$ Here $\phi(t,0,0)=0,~{}\psi(t,0)=0.$ ###### Theorem 3. Assume that $\lim_{t\to\pm\infty}\textnormal{inf}~{}g(t)>0,~{}0<m=\lim_{t\to\pm\infty}\textnormal{inf}\frac{f(t)}{g(t)}\leq\lim_{t\to\pm\infty}\textnormal{sup}~{}\frac{f(t)}{g(t)}=M<+\infty,$ $\|\phi(t,x,\mu)\leq h(t)[\|x\|+\|\mu\|^{-(2m-2)}],~{}~{}\|\phi_{x}(t,x,\mu)\|\leq h(t),$ $\|\psi_{x}(t,x)\|\leq h(t),~{}~{}\lim_{t\to\pm\infty}\textnormal{sup}~{}\frac{h(t)}{g(t)}\leq k.$ Then there occurs local saddle-node bifurcation when $\mu$ passes through 0. Furthermore, when $\mu>0$, locally attracting orbit $x_{\mu}(t)$ is forwards attracting in $(0,\varepsilon)$ and unstable orbits are pullback repelling in $(-\varepsilon,\delta)$. The proof is omitted. Example 1. In the equation $\dot{x}=\mu^{3}t^{2}-2t^{2}x^{4}$, saddle node bifurcation occurs when $\mu=0$. ### 3.2 Transcritical Bifurcation First we consider a concrete example. ###### Theorem 4. Let consider the non-autonomous differential equation (3). 1) Let assume that $f(t)$ and $g(t)$ satisfy $\displaystyle\forall t\in\mathbf{R},~{}\int_{-\infty}^{t}f(s)ds=+\infty,$ (16) $\displaystyle\exists T^{-}\in\mathbf{R}:\forall t(\leq T^{-}),~{}g(t)\geq r^{-}>0,$ (17) $\displaystyle\exists\mu_{0}(>0):$ $\displaystyle\quad\forall\mu(-\mu_{0}<\mu\leq 0),\forall t\in\mathbf{R},$ $\displaystyle\qquad\lim_{s\to-\infty}\textnormal{inf}\frac{e^{\mu^{(2m-1)}F(s)}}{\left[(2n-1)\int_{s}^{t}g(r)e^{(2n-1)\mu^{(2m-1)}F(r)}dr\right]^{\frac{1}{2n-1}}}\geq m_{\mu}>0,$ (18) $\displaystyle\quad\forall\mu(0<\mu<\mu_{0}),\forall t\in\mathbf{R},$ $\displaystyle\qquad 0<m_{\mu}\leq x_{\mu}(t)=\frac{e^{\mu^{(2m-1)}F(t)}}{\left[(2n-1)\int_{-\infty}^{t}g(r)e^{(2n-1)\mu^{(2m-1)}F(r)}dr\right]^{\frac{1}{2n-1}}}\leq M_{\mu}.\qquad$ (19) Here $F$ is an antiderivative of $f$. Then we have the following facts: When $-\mu_{0}<\mu\leq 0$, the zero solution to (3) is locally pullback attracting in $\mathbf{R}$. When $\mu=0$, the zero solution to (3) is asymptotically instable but locally pullback attracting in $\mathbf{R}^{+}$. When $0<\mu<\mu_{0}$, the zero solution to (3) is asymptotically instable and the orbit $x_{\mu}(t)$ is locally pullback attracting in $\mathbf{R}^{+}$. Furthermore $\forall t\in\mathbf{R},~{}x_{\mu}(t)\to 0~{}(\mu\to 0).$ 2) Let assume that $f(t)$ and $g(t)$ satisfy $\exists T^{+}:\forall t\geq T^{+},g(t)\geq r^{+}>0,~{}~{}\int_{t}^{+\infty}f(s)ds=+\infty.$ (20) Then there exists a $\mu_{0}(>0)$ such that the zero solution to (16) is forwards attracting for $-\mu_{0}<\mu\leq 0$ and the orbit $x_{\mu}(t)$ is forwards attracting for $0<\mu<\mu_{0}$. Furthermore the additional condition $\displaystyle\forall\mu<0,\forall t\in\mathbf{R},0<m_{\mu}\leq x_{\mu}(t)$ $\displaystyle\qquad\qquad\qquad=\frac{e^{\mu^{(2m-1)}F(t)}}{\left[(2n-1)\int_{t}^{\infty}g(r)e^{(2n-1)\mu^{(2m-1)}F(r)}dr\right]^{\frac{1}{2n-1}}}\leq M_{\mu}$ (21) is satisfied, then the orbit $x_{\mu}(t)$ is asymptotically instable and pullback repelling when $-\mu_{0}<\mu\leq 0$. And we have $\forall t\in\mathbf{R},~{}x_{\mu}(t)\to 0~{}(\mu\to 0).$ The proof is omitted. Now we consider general equations $\dot{x}=G(t,x,\mu)$. Assume that $G$ is sufficiently smooth. Then we obtain the following Taylor expansion of $G$ at $(t,0,0)$ as the above. $\displaystyle G(t,x,\mu)=G(t,0,0)+G_{x}(t,0,0)x+G_{\mu}(t,0,0)\mu+\frac{1}{2}G_{xx}(t,0,0)x^{2}$ $\displaystyle\qquad+G_{x\mu}(t,0,0)x\mu+\frac{1}{2}G_{\mu\mu}(t,0,0)\mu^{2}+\frac{1}{6}G_{xxx}(t,0,0)x^{3}+\frac{1}{2}G_{xx\mu}(t,0,0)x^{2}\mu$ $\displaystyle\qquad+\frac{1}{2}G_{x\mu\mu}(t,0,0)x\mu^{2}+\frac{1}{6}G_{\mu\mu\mu}(t,0,0)\mu^{3}+\cdots+\frac{1}{(2m-1)!}$ $\displaystyle\qquad\times\left[\frac{\partial^{2m-1}}{\partial x^{2m-1}}G(t,0,0)x^{2m-1}+C_{2m-1}^{1}\frac{\partial^{2m-1}}{\partial x^{2m-2}\partial\mu}G(t,0,0)x^{2m-2}\mu+\cdots\right.$ $\displaystyle\qquad\left.+C_{2m-1}^{2m-2}\frac{\partial^{2m-1}}{\partial x\partial\mu^{2m-2}}G(t,0,0)x\mu^{2m-2}+\frac{\partial^{2m-1}}{\partial\mu^{2m-1}}G(t,0,0)\mu^{2m-1}\right]$ $\displaystyle\qquad+\frac{1}{(2m)!}\left[\frac{\partial^{2m}}{\partial x^{2m}}G(t,0,0)x^{2m}+C_{2m}^{1}\frac{\partial^{2m}}{\partial x^{2m-1}\partial\mu}G(t,0,0)x^{2m-1}\mu+\cdots\right.$ $\displaystyle\qquad\left.+C_{2m}^{2m-1}\frac{\partial^{2m}}{\partial x\partial\mu^{2m-1}}G(t,0,0)x\mu^{2m-1}+\frac{\partial^{2m}}{\partial\mu^{2m}}G(t,0,0)\mu^{2m}\right]+\frac{1}{(2m+1)!}$ $\displaystyle\qquad\times\left[\cdots+C_{2m+1}^{2m}\frac{\partial^{2m+1}}{\partial x\partial\mu^{2m}}G(t,0,0)x\mu^{2m}+\frac{\partial^{2m+1}}{\partial\mu^{2m+1}}G(t,0,0)\mu^{2m+1}\right].$ Here $m\in\mathbf{N}$. Now assume that $G$ satisfies the following conditions: $\displaystyle\textnormal{(i)}~{}G(t,0,\mu)=0,~{}\forall t,\mu\in\mathbf{R},$ $\displaystyle\textnormal{(ii)}~{}G_{x}(t,0,0)=0,$ $\displaystyle\textnormal{(iii)}~{}\frac{\partial^{2}}{\partial x\partial\mu}G(t,0,0)=\frac{\partial^{3}}{\partial x\partial\mu^{2}}G(t,0,0)=\cdots=\frac{\partial^{2m-1}}{\partial x\partial\mu^{2m-2}}G(t,0,0)=0.\qquad\qquad\qquad\qquad$ From (i) and (ii) we have $\frac{\partial^{k}}{\partial\mu^{k}}G(t,0,0)=0,~{}\forall t\in\mathbf{R},\forall k\in\mathbf{Z}_{+}$ and thus $G$ is provided as follows: $\displaystyle G(t,x,\mu)=\mu^{2m-1}\left[\frac{1}{(2m)!}C_{2m}^{2m-1}\frac{\partial^{2m}}{\partial x\partial\mu^{2m-1}}G(t,0,0)+\frac{1}{(2m+1)!}\right.$ $\displaystyle\qquad\left.\times C_{2m+1}^{2m}\frac{\partial^{2m+1}}{\partial x\partial\mu^{2m}}G(t,0,0)\mu+\cdots\right]x+\left[\frac{1}{2}G_{xx}(t,0,0)+\frac{1}{6}G_{xxx}(t,0,0)x\right.$ $\displaystyle\qquad+\frac{1}{2}G_{xx\mu}(t,0,0)\mu+\cdots+\frac{1}{(2m-1)!}\frac{\partial^{2m-1}}{\partial x^{2m-1}}G(t,0,0)x^{2m-3}+\frac{1}{(2m-1)!}$ $\displaystyle\qquad\times C_{2m-1}^{1}\frac{\partial^{2m-1}}{\partial x^{2m-2}\partial\mu}G(t,0,0)x^{2m-4}\mu+\cdots+\frac{1}{(2m)!}\frac{\partial^{2m}}{\partial x^{2m}}G(t,0,0)x^{2m-2}$ $\displaystyle\qquad\left.+\frac{1}{(2m)!}C_{2m}^{1}\frac{\partial^{2m}}{\partial x^{2m-1}\partial\mu}G(t,0,0)x^{2m-3}\mu+\cdots\right]x^{2}.$ Let denote $f(t):=\frac{1}{(2m)!}C_{2m}^{2m-1}\frac{\partial^{2m}}{\partial x\partial\mu^{2m-1}}G(t,0,0),~{}g(t):=-\frac{1}{2}G_{xx}(t,0,0)$. Then (9) can be rewritten as follows: $\dot{x}=\mu^{2m-1}[f(t)+\mu\phi(t,\mu)]x-[g(t)+r(t,x,\mu)]x^{2}.$ (22) Here $\phi(t,0)=\frac{1}{(2m+1)!}C_{2m+1}^{2m}\frac{\partial^{2m+1}}{\partial x\partial\mu^{2m}}G(t,0,0)$. ###### Theorem 5. Assume that $r(t,0,0)=0,$ (23) $\lim_{t\to\pm\infty}\textnormal{inf}~{}g(t)>0,$ (24) $0<m=\lim_{t\to\pm\infty}\textnormal{inf}~{}\frac{f(t)}{g(t)}\leq\lim_{t\to\pm\infty}\textnormal{sup}~{}\frac{f(t)}{g(t)}=M<+\infty,$ (25) and there exists a positive valued function $h(t)$ such that $\|\phi(t,\mu)\|\leq h(t),~{}~{}\|r_{\mu}(t,x,\mu)\|\leq h(t),~{}~{}\|r_{x}(t,x,\mu)\|\leq h(t),$ (26) $\lim_{t\to\pm\infty}\textnormal{sup}~{}\frac{h(t)}{g(t)}\leq k.$ (27) Then there occurs local transcritical bifurcation when $\mu$ passes through 0. Furthermore, when $\mu<0$, a complete orbit $x_{\mu}(t)$ is pullback repelling in $(-\varepsilon,0)$; when $\mu=0$, the zero solution is locally forwards attracting in $\mathbf{R}^{+}$ and when $\mu>0$, pullback attracting orbit $x_{\mu}(t)$ is forwards attracting in $(0,\varepsilon)$. The proof is omitted. Now assume that G satisfies the following conditions: $\displaystyle\textnormal{(iv)}~{}G(t,x,\mu)=c(\mu)\cdot G(t,x,0)+x\cdot\frac{\partial}{\partial x}G(0,0,\mu),$ $\displaystyle\textnormal{(v)}~{}G(t,0,0)=0,$ $\displaystyle\textnormal{(vi)}~{}\frac{\partial}{\partial x}G(t,0,0)=\frac{\partial^{2}}{\partial x^{2}}G(t,0,0)=\cdots=\frac{\partial^{2n-1}}{\partial x^{2n-1}}G(t,0,0)=0,$ $\displaystyle\textnormal{(vii)}~{}\frac{\partial^{2}}{\partial x\partial\mu}G(t,0,0)=\frac{\partial^{3}}{\partial x\partial\mu^{2}}G(t,0,0)=\cdots=\frac{\partial^{2m-1}}{\partial x\partial\mu^{2m-2}}G(t,0,0)=0.\qquad\qquad\qquad\qquad$ Here $n,m\in\mathbf{N}$. Then $G$ is provided as follows: $\displaystyle G(t,x,\mu)=\mu^{2m-1}\left[\frac{1}{(2m)!}C_{2m}^{2m-1}\frac{\partial^{2m}}{\partial x\partial\mu^{2m-1}}G(t,0,0)+\frac{1}{(2m+1)!}\right.$ $\displaystyle\qquad\left.\times C_{2m+1}^{2m}\frac{\partial^{2m+1}}{\partial x\partial\mu^{2m}}G(t,0,0)\mu+\cdots\right]x+\left[\frac{1}{2}x^{-(2n-2)}\mu\frac{\partial^{3}}{\partial x^{2}\partial\mu}G(t,0,0)\right.$ $\displaystyle\qquad+\cdots+\frac{1}{(2m-2)!}C_{2m-2}^{1}x^{-(2n-2m+3)}\mu\frac{\partial^{2m-2}}{\partial x^{2m-3}\partial\mu}G(t,0,0)$ $\displaystyle\qquad+\cdots+\frac{1}{(2n)!}\frac{\partial^{2n}}{\partial x^{2n}}G(t,0,0)+\frac{1}{(2n)!}C_{2n}^{1}x^{-1}\mu\frac{\partial^{2n}}{\partial x^{2n-1}\partial\mu}G(t,0,0)$ $\displaystyle\qquad\left.+\frac{1}{(2n)!}C_{2n}^{2}x^{-2}\mu^{2}\frac{\partial^{2n}}{\partial x^{2n-2}\partial\mu^{2}}G(t,0,0)+\cdots\right]x^{2n},\qquad m,n\in\mathbf{N}.$ Let denote $\displaystyle f(t):=\frac{1}{(2m)!}C_{2m}^{2m-1}\frac{\partial^{2m}}{\partial x\partial\mu^{2m-1}}G(t,0,0),$ $\displaystyle g(t):=-\frac{1}{(2n)!}\frac{\partial^{2n}}{\partial x\partial\mu^{2n-1}}G(t,0,0).$ Then we can rewritten (9) as follows: $\dot{x}=\mu^{2m-1}[f(t)+\mu\phi(t,\mu)]x-[g(t)+r(t,x,\mu)]x^{2n}.$ Here $\phi(t,0)=\frac{1}{(2m+1)!}C_{2m+1}^{2m}\frac{\partial^{2m+1}}{\partial x\partial\mu^{2m}}G(t,0,0)$. ###### Theorem 6. Assume that $r(t,0,0)=0,~{}~{}\lim_{t\to\pm\infty}\textnormal{inf}~{}g(t)>0,$ $0<m=\lim_{t\to\pm\infty}\textnormal{inf}~{}\frac{f(t)}{g(t)}\leq\lim_{t\to\pm\infty}\textnormal{sup}~{}\frac{f(t)}{g(t)}=M<+\infty,$ $\|\phi(t,\mu)\|\leq h(t),~{}~{}\|r(t,x,\mu)\|\leq h(t)\left[\|x\|^{-(2n-2)}+\|\mu\|\right],$ $\lim_{t\to\pm\infty}\textnormal{sup}~{}\frac{h(t)}{g(t)}\leq k.$ Then we have the same conclusions with the Theorem 2. The proof is omitted. Example 2. In the equation $\dot{x}=\mu^{3}t^{2}x-2t^{2}x^{6}$, transcritical bifurcation occurs when $\mu=0$. ## References * [1] F. Balbrea, J.C. Valverde, Cusp and generalized flip bifurcations under higher degree conditions, Nonlinear Anal., 52 (2003) 405-419. * [2] D.N. Cheban, P.E. Kloeden, B. Schmalfuß, The relationship between pullback, forwards and global attractors of non-autonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 2 (2002) 9-28. * [3] V.V. Chepyzhov, M.I. 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Differential Equations, 72 (2002) 1-20. * [16] M. Rasmussen, Attractivity and bifurcation for nonautonomous dynamical systems. Ph.D. Thesis, University of Augsburg, 2007. * [17] B. Schmalfuß, Attractors for the non-autonomous dynamical systems, In: B. Fiedler, K. Groger, J. Sprekels (Eds.), Proceedings of Equadiff 99 Berlin, World Scientific, Singapore, 2000, pp. 684-689. * [18] G.R. Sell, Nonautonomous differential equations and topological dynamics, Trans. Amer. Math. Soc., 127 (1967) 241-283. * [19] W. Shen, Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc., 647 (1998). * [20] J.C. Valverde, Topological conjugations of normal forms in bifurcations, Acta Math. Hungar., 88(1-2) (2000) 113-121.	arxiv-papers	2013-11-07T07:01:17	2024-09-04T02:49:53.356890	{ "license": "Creative Commons Zero - Public Domain - https://creativecommons.org/publicdomain/zero/1.0/", "authors": "Sang-Mun Kim, Hyong-Chol O", "submitter": "Hyong-Chol O", "url": "https://arxiv.org/abs/1311.1812" }
1311.1836	# A New Stochastic Model of the Causal Interpretation of Quantum Theory on the Development of the Fundamental Concept of Mass Muhamad Darwis Umar [email protected] Department of Physics, Universitas Gadjah Mada, Sekip Utara BLS 21, 55281, Yogyakarta, Indonesia ###### Abstract In this paper we pose two fundamental ideas on the motion of an elementary particle supporting the internal ”spin motion” or _Zitterbewegung_ and a particle as concentrated energy. First, the particle moves randomly in a limited area (in a quantum-sized volume) like random vibrating system where the particle will diffuse in a quantum-sized volume when it absorbs or emits the quantized amount of energy. The quantum-sized volume can move too and plays role similar to the carrier amplitude, while the vibrational motion of the fundamental particle with frequency and amplitude represents a modulating signal. The current of diffusion process taking place in the quantum-sized volume will represent the emergence of a spin phenomenon shown by the existence of Clifford algebra. Second, the particle is pure energy concentrated on the surface of 3-dimensional sphere-form (2-manifold without boundary). Afterward we show that by defining the particle mass as an invariant quantity based on the two fundamental ideas, we can derive both the diffusion constant of the particle in the quantum-sized volume as $\beta=\hslash/2m$ and the Schrödinger equation. Furthermore, by posing that the vibrational motion of the particle limited on the quantum-sized volume plays fundamental role as time interval unit (proper time$/$particle clock), we show that the relativistic effects of the particle must represent atomic process. ###### pacs: 01.70.+w, 02.50.Ey, 03.30.+p, 03.65.Sq ††preprint: APS-PRA ## I INTRODUCTION Although there have been many attempts to interpret the causality principle of quantum theory IF-1946 ; IF-1952 ; WW-1954 ; DB-JPV-1954 ; EN-1966 ; AK-1985 ; RF-1933 ; KLC-ZZ-1985 ; DK-1964 ; KN-1986 ; MBJ-1988 ; THB-1970 , Nelson’s work EN-1966 stands alone JRB-2002 . Nelson approaches the Schrödinger equation in essence with stochastic mechanics as an open system, but he also used a non-friction model, thus there is no energy or mass transfers on average and the system can be kept as a closed system. Nonetheless, Grabert et. al HG-PH-TT-1979 showed that quantum mechanics is not equivalent to a markovian diffusion processes, followed later by Gillaspie DTG-1994 who also demonstrated that the measurable behavior of most quantum system cannot be modeled as a Markov process. Skorobogatov and Svertilov GAS-SIS-1988 also pointed out that the measurable behavior of elementary quantum system can be modeled by non-markovian stochastic processes. Stochastic interpretations, although they are based on the same fundamental idea on the existence of fluctuating field as background, have two major different viewpoints is considering the collapse of the wave function in the context of how we reconcile the probabilistic distribution of outcomes with deterministic form of Schrödinger equation when the measurement is made. First, one views that when the measurement is made, the system will become an open system. This proposal serves the main idea of decoherence and de Broglie- Bohm interpretations on ”measurement problem” is designated to keep the deterministic property of Schrödinger equation. Second, one poses that the state collapse into macroscopically unique state GJ-PP-JR-2004 occurs dynamically because the wave function (internal mass density) is coupled to a Brownian motion noise term. This approach assumes that quantum mechanics is not exact from the beginning (before measurement is made). From a fundamental point of view, decoherence and de Broglie-Bohm interpretations have a problem in describing ”reality” because it is just only one apparatus that exists in reality, while the dynamical collapse models do not explain the role of measurement well. Stochastic models with causality interpretations have been facing several challenges (questions) such as: 1. What is the origin of the non-classical force (related to osmotic velocity) acting on the particle or what is the origin of the quantum mechanical potential MPD-1979 ?, quantity that related to internal motion ER-GS-1998 ; 2. To derive stochastic models, one must introduce the diffusion constant by heuristic arguments, even though it has been shown that the Schrödinger equation and diffusion equations are equivalent in a mathematical structure MN-1993 . The constant of diffusion expressed by Planck constant and mass has delivered a fundamental question: what is the natural significance of the Planck constant and its connection with mass concept in determining diffusion constant? 3. A realistic explanation of quantum mechanics has to meet at least one of the three requirements that is the charge of a particle is concentrated in a small volume of space KJ-2009 . 4. Nelson’s model for stationery state contains the nodal surface caused by osmotic velocity that is inversely proportional to the probability density. 5. Stochastic models have not expressed yet an obvious picture of natural spin (a physical mechanism picture) that has been identified to play a fundamental role to determine the quantum behavior of micro-systems ER-GS-1998 . On the another word, stochastic models is not dealing with the existence of _Zitterbewegung_ that support proposal on splitting the motion of variables into two kinds of motion that are the motion of the center of mass to a chosen reference frame and internal motion with reference to the center of mass ER-GS-1998 . Besides the measurement problem in the foundation of quantum theory, mass, behind its role as the most fundamental notion underlying physics to perceive and to conceptualize relations among all physical phenomena, has not yet been fully understood and explained. Fundamentally, the definition of mass is not only present in classical-relativity with terms inertial mass and relativistic mass as well as gravitation mass, but also found in quantum theory in its connection with the density of probability or the wave function. The unclear ontology role of mass to unify our understanding on what truly goes on in physical realm indicates that the concepts of mass are simply not complete. It means that definition of mass still open to be redefined and extended for unified purposes of consistently creating a natural picture. The definition, of course, should be derived from a new theory with which we can explain the origin, the existence and the phenomenological properties of mass, as well as the fundamental source of unpredictability and the measurement problem in quantum theory at once. In this work, we pose a unified description on both the fundamental motion of particles and particle masses in the new stochastic process framework in order to produce a natural explanation of spin. From the proposal, we describe what happens to the particle when a measurement is made, what the relation between the deterministic evolution of the wave function and the probabilistic outcome from measurement is, and as well as what the connectivity between the particle and what its environment in the concentrated-energy context is. We also introduce a new meaning of special relativity theory in dynamics viewpoint and its connection with internal atomic mechanism. ## II THE FUNDAMENTAL MOTION OF A PARTICLE The debate on the interpretation of quantum mechanics has taken place since the 1920s and continues to this day in both the scientific and philosophical communities. One of issues that has become central to the debate is the possible presence of the role that causality plays in microscopic world governed by the laws of quantum mechanics PJR-2009 . Orthodox quantum theory known as the Copenhagen interpretation, which is the prevailing theory of quantum mechanics, confronts with the causality principle based on the notion of cause and effect. They, anti-realism, have a notion that in quantum realm, particles do not acquire some of their characteristics until they are observed by someone PJR-2009 . Contrary to orthodox quantum theory, attempts to understand quantum theory in the classic understanding with the causality principle still continue in varying models. The original stochastic model (a non-local theory) was first introduced by Bohm and Vigier DB-JPV-1954 describing the random motion of the particle in Madelung fluid. Afterward, Nelson EN-1966 attempted to discuss Schrödinger equation in the universal Brownian motion framework (a local theory). Stochastic interpretation by Nelson EN-1966 claimed that the quantum mechanical motion of particles governed by the Schrödinger equation can be equally understood as particles in classical Brownian motion in a vacuum which acts upon the particle as does heat in the theory of irreversible processes HG-PH-TT-1979 . Nelson’s approach is different from those of Fényes IF-1946 ; IF-1952 and Fürth RF-1933 in that it uses an imaginary diffusion coefficient. Contrary to the notion of Bohm-Vigier and Nelson that viewing the wave function evolves in deterministic way, Ghirardi, et. al. GCG-RA-TW-1986 ; GCG-PP-AR-1990 posed that the wave function interacts with background field fluctuations, therefore the probability outcome should include background field fluctuations. Actually, the proposal of Ghirardi, et. al. GCG-RA-TW-1986 ; GCG-PP-AR-1990 (known as GRW and CLS models) are not interpretation models of quantum theory, but rather, models to understand measurement problem. If we trace stochastic models, we will find out that almost all them use closely similar causal interpretations, which are 1. 1. The change in trajectory of an object (a particle) is continuous and that in instant of time it has some definite positions DB-1989 . 2. 2. Vacuum fluctuation plays a central role in determining diffusion and random processes. Here we introduce a new fundamental mechanism of the motion of a particle that is entirely different from the previous models. Although we build our model in terms of random motion (the change of motion trajectories), fundamentally, we have radically departed from traditional views. We pose that random motion in a quantum system is an intrinsic property of particles, and this motion is naturally caused by concentrated energy localized as a particle. It means that localized energy will play a role as quantum potential. Random motion takes place in a quantum-sized volume (it is similar to random vibrating motion) where this volume can move too, and then mass and charge distributed in the quantum-sized volume equal to the distribution of probability to find the particle in the quantum-sized volume. There are four mechanism introduced in our model, which are as follows. 1. 1. The particles will move randomly in a quantum-sized volume where the random motion serves an intrinsic property of particle and related to spin phenomena. In a stationer state, random motion will relatively take place forward and backward in time with markovian process. Nevertheless, because random motion takes place in limited area (the quantum-sized volume) that similar to random vibrating system, random motion does not present perspective on the diffusion process. 2. 2. Relative to random intrinsic motion, the quantum-sized volume can undergo translational movement, but it cannot take place backward in time at stationery state. The dynamics of the quantum-sized volume represent magnetic properties of angular or translational momentum, and it can represent de Broglie-Bohm theory on guiding particle. 3. 3. Every change of the speed of random intrinsic motion accompanied or not by the change of the velocity of translational motion of a quantum-sized volume will represent the transition between two states, and every change of the speed of random intrinsic motion will generate a diffusion process or Brownian motion perspective. The dissipative character of quantum systems in our model is similar to Langevin’s approach to Brownian motion but with a different mechanism. The difference lies in the cause of emission or absorption process. In our model, a particle will emit or absorb energy when the intrinsic speed of the particle and or the translational velocity of quantum-sized volume changes, and it is not related to a friction process. Internal diffusion process coincide with emission and absorption processes which also occurs in backward and forward in time with a non-markovian process (as long as the system does not undergo phase transition). 4. 4. For more any complex physical system (material), the quantum-sized volume can probably move in two ways that are transitional or random motions, and it will be determined by the potential forms of system (material). If we consider the kinematic aspects of the transition process (how the particle moves from one point to other point, i.e. how its position changes in time); for example, a case where transition processes include diffusion process and the movement of the quantum-sized volume, then the movement path of the particle at any time will be determined by the velocity of the quantum- sized volume, the distribution of the velocity field in the quantum-sized volume describing how the particle relatively moves from the center of quantum-sized volume, and the displacement due to the diffusion process. We can represent the forward time operator for describing how the average position of the particle changes in time when the transition process followed by emitting energy occurs (the speed of the intrinsic motion in the quantum- sized volume decreases) as EN-1966 : $\displaystyle D=\frac{\partial}{\partial t}+\mathbf{b}\cdot\nabla+\beta\nabla^{2}.$ (1) While the backward time operator for the case of absorbing energy followed by the increase of the speed of the quantum-sized volume is $\displaystyle D^{~{}}_{}=\frac{\partial}{\partial t}+\mathbf{b}^{~{}}_{}\cdot\nabla-\beta\nabla^{2},$ (2) where $\beta$ is the diffusion constant. If we consider a transition process followed by emitting energy. then $\mathbf{b}$ is the initial velocity field of a stationer state and $\mathbf{b}^{~{}}_{}$ is a relatively final velocity where $\left\|\mathbf{b}\|<\|\mathbf{b}^{~{}}_{}\right\|$. Next we consider Langevin’s equation describing energy emission when a particle undergoes a transition process generated by applying an external force (an external potential). We assume if the physical system does not undergo emission or absorption processes, then every particle of any physical system is in stationary states. If an external force (an external field) is applied to any physical system being in any stationary state where energy of the external field is equal to the gap energy between stationary states so that the interaction will produce emission or absorption processes (between) as long as the interaction don’t cause any phase transition. In such process, the set of possible stationary states of the physical system do not change by external field. But, if any applied external field do not generate transition process (diffusion processes: emission or absorption processes), then the physical system will respond to the presence of the external field by creating a new stationary state characterized by changing its particle positions and kinetics so that all new possible states produced by the external potential are different from previous possible stationary states (without any external field). Our starting point to consider any physical system at the stationery state is based on the historical facts that quantum theory was develop to understand the behaviors and the properties of atom and molecule as well as other stationer physical systems. We apply Langevin’s equation $\displaystyle m\,\mathbf{a}^{~{}}_{\mathrm{net}}$ $\displaystyle=\mathbf{F}^{~{}}_{\mathrm{qv}}\bm{(}(\mathbf{x}(t)\bm{)}-\xi\mathbf{b}\bm{(}\mathbf{F}^{~{}}_{\mathrm{external}}(t)\bm{)}$ $\displaystyle\qquad+\mathbf{F}^{~{}}_{\mathrm{external}}(t)+\mathbf{F}^{~{}}_{\mathrm{random\,force}}(t),$ (3) where $\mathbf{x}(t)=\mathbf{R}(t)+\mathbf{r}(t)$, $\xi$ is the friction coefficient, and $\mathbf{a}^{~{}}_{\mathrm{net}}(t)=[\ddot{\mathbf{R}}(t)+\ddot{\mathbf{r}}(t)]$ is the total acceleration where $\ddot{\mathbf{R}}(t)$ is the acceleration of the quantum-sized volume and $\ddot{\mathbf{r}}(t)$ is the acceleration of particle in the quantum-sized volume relative to the center of volume, $\mathbf{F}_{\mathrm{qv}}\bm{(}\mathbf{x}(t)\bm{)}$ is the force governing the stationer state of the quantum-sized volume [we can also consider an ideal condition in which $\mathbf{F}^{~{}}_{\mathrm{qv}}\bm{(}\mathbf{x}(t)\bm{)}=0$], $\mathbf{F}^{~{}}_{\mathrm{external}}(t)$ is the applied force for generating transition process, and $\mathbf{F}^{~{}}_{\mathrm{random\,force}}(t)$ is random force that in the conventional view is asserted to represent the effect of background noise, but in our model, by contrast, we pose this force only represent an intrinsic internal random motion. We can rewrite Eq. (II) in terms of velocity as $\displaystyle\mathrm{d}\mathbf{v}^{~{}}_{\mathrm{net}}$ $\displaystyle=-\frac{1}{m}\nabla V\bm{(}\mathbf{x}(t)\bm{)}\,\mathrm{d}t-\frac{\xi}{m}\mathbf{b}\bm{(}\mathbf{F}^{~{}}_{\mathrm{external}}(t)\bm{)}\,\mathrm{d}t$ $\displaystyle\qquad+\frac{1}{m}\mathbf{F}^{~{}}_{\mathrm{external}}(t)\,\mathrm{d}t+\frac{1}{m}\mathbf{F}^{~{}}_{\mathrm{random\,force}}(t)\,\mathrm{d}t,$ (4) where $\mathbf{v}^{~{}}_{\mathrm{net}}(t)=\dot{\mathbf{R}}(t)+\dot{\mathbf{r}}(t)$. Applying average forward time derivative to Eq. (II), we find that $\displaystyle D\mathbf{v}^{~{}}_{\mathrm{net}}$ $\displaystyle=-\frac{1}{m}\nabla V\bm{(}\mathbf{x}(t)\bm{)}-\frac{\xi}{m}\mathbf{b}\bm{(}\mathbf{F}^{~{}}_{\mathrm{external}}(t)\bm{)}$ $\displaystyle\qquad+\frac{1}{m}\mathbf{F}^{~{}}_{\mathrm{external}}(t).$ (5) Whereas for average back forward time derivative to Eq. (II), we find $\displaystyle D^{~{}}_{}\mathbf{v}_{\mathrm{net}}$ $\displaystyle=-\frac{1}{m}\nabla V\bm{(}\mathbf{x}(t)\bm{)}+\frac{\xi}{m}\mathbf{b}\bm{(}\mathbf{F}^{~{}}_{\mathrm{external}}(t)\bm{)}$ $\displaystyle\qquad+\frac{1}{m}\mathbf{F}^{~{}}_{\mathrm{external}}(t).$ (6) $\mathbf{b}\bm{(}\mathbf{F}^{~{}}_{\mathrm{external}}(t)\bm{)}$ in Eqs. (II) and (II) shows that diffusion processes representing emission and absorption processes are caused by $\mathbf{F}^{~{}}_{\mathrm{external}}(t)$, Thus, the absence of $\mathbf{F}^{~{}}_{\mathrm{external}}(t)$ represents the physical system is in stationary state. Seeing Eq. (II) and (II), it seems that the two equations describe ambiguous mechanisms. It is because the two equations use the same forces to describe two difference mechanisms. We can understand it with following description. Eq. (II) depicts external force applied on a particle occupying a stationary state with potential $V\bm{(}\mathbf{x}(t)\bm{)}$ and then makes the particle undergo a friction force (emission process). Basically, this process will create the change of internal potential where the change of internal potential will be the same as the external force. Thus we can still consider Eq. (II) as a backward process with the same internal potential and the same external force, but it takes place spontaneously (without external treatment). From this point of view, we can view the spontaneously emitting or absorbing processes as backward processes as long as the system does not undergo a phase transition. Since backward processes play a role as a natural tendency to bring final states (as the results of forward process) back to initial states (before there are external treatments) so that we pose that there are neither spontaneously emitting or absorbing processes without interaction between any physical system and external treatment or perturbation. Thus the information of any dynamic system must only be acquired and accessed by applying external treatment to any physical systems occupying stationary states. Using the mean forward derivative (1) and the mean backward derivative (2) introduced by EN-1966 , Eqs. (II) and (II) give $\displaystyle\frac{1}{2}\left(DD^{~{}}_{}+D^{~{}}_{}D\right)\mathbf{x}(t)$ $\displaystyle=\mathbf{a}^{~{}}_{\mathrm{net}}(t)$ $\displaystyle=-\frac{1}{m}\nabla V\bm{(}\mathbf{x}(t)\bm{)}+\frac{1}{m}\mathbf{F}^{~{}}_{\mathrm{external}}(t),$ (7) where $\mathbf{x}(t)=\mathbf{R}(t)+\mathbf{r}(t)$. For stationer state $\left[\mathbf{F}_{\mathrm{external}}(t)=0\right.$ and $\left.\pm v\mathbf{b}(t)=0\right]$, Eq. (II) becomes $\displaystyle\frac{1}{2}\left(DD^{~{}}_{}+D^{~{}}_{}D\right)\mathbf{x}(t)=\mathbf{a}_{\mathrm{net}}(t)=-\frac{1}{m}\nabla V\bm{(}\mathbf{x}(t)\bm{)}.$ (8) ## III THE NEW FUNDAMENTAL CONCEPT OF MASS If we assume that an obvious connection between the microscopic (quantum realm) and macroscopic worlds (classical picture) exists, then every proposed theory must not only reveal the unfinished understanding of how quantum realm is more obscure than our daily imagination and physical sense that perceive physical phenomena based on both conventional models about particle, interactions and concepts of mass, charge, field and energy in which all those physical concepts and models constitute physical pictures via mechanisms obeying causality principle. But, the proposal must also solve the polemic about whether mass and charge are no more than the abstract quantitative expression of facts that do not need to ’explain’ phenomena in terms of purposes or hidden causes like Weyl’s and Mach opinion on mass definition MJ-2000 . In our opinion, the existence of a hidden connection between the quantum and the classical realms may be caused by the fundamentally unenviable approach of describing the fundamental attributes of physical objects that have been causing and afterward shaping our incomplete perspective in understanding all physical phenomena and relations among them such as mass, time, force, field, interaction, etc. On of the fundamental concepts that has not been established yet is the concept of mass. How modern physics has both experimentally and theoretically contributed to a more profound understanding of the nature of mass has been comprehensively summarized and reported by Max Jammer in Ref. MJ-2000 . Mass was formally introduced by Newton in his second law of dynamic to depict the dynamics of physical object via the relation of force and acceleration. This description has provided the inertial feature of physical objects. Newton also presented the gravity mass to describe the ability of matter to generate the phenomenon of gravity force. Afterward, the meaning of mass was amended by Einstein when he introduced special relativity theory extends our understanding of mass through the differentiation of rest mass and relativistic mass. Nevertheless the debate on the rest mass versus relativistic mass still exists until now MJ-2000 . An important aspect of Einstein’s work has been the relation of energy and mass which leads to the perspective of a particle’s mass at the rest as concentrated energy. Einstein’s contribution to the perspective of mass also included a unified description of inertial mass and gravity mass when he worked on general relativity. In this context, mass is extended as a key factor governing the structure of space-time. Although Einstein had created a fundamental picture of the particle as concentrated energy using the term of the speed of light (the relation between mass and energy), the results cannot link the description of mass-energy with the origin of the electromagnetic field or the charge of the particle, another important physical concept which provides a perspective of the electromagnetic wave and its energy. The kinematic approach introduced by Einstein has not yet provided a complete description on what the meaning of electric and magnetic fields in relation to the mass feature of the charge particle is. Another missing explanation has been what the picture of the physical mechanisms on how electrical fields transform to magnetic fields or conversely when a charged particle moves (following the perspective of kinetic energy between two frames of reference)?; what is the effect of radiation (the production of photon) in regards to the mass of charged particle when the particle is accelerated?; how clock and length are connected with the atom and many body problems HRB-OP-2001 ; until how energy is stored as bounding state and how it is released as electromagnetic field or photon. We view that non-unified description of the electromagnetic and mechanical aspects of a particle is the source of an unfinished-description on electromagnetic interaction and also the source of a debate on what is the physically invariant quantity or not? Here, we pose a new description of mass that is a totally different from the conventional approaches. First, we give two backgrounds to our proposal. 1. 1. From our model on the fundamental motion of a particle, we can identify that random motion in the quantum-sized volume is the intrinsic property of the particle, hence the terminology of energy (quantum potential) generating intrinsic motion should be connected to concentrated energy ”rest mass” as in the special relativity theory perspective. 2. 2. Because the concentrated energy of the particle must decrease and increase by emission and absorption processes by external treatment, the amount of quantified-energy emitted or absorbed must relatively be representative of the structure and motion of the particle due to the existence of system (other particles/environment). This also means that energy related to transition processes will represent how other particles as an environment relatively evolves and changes with time towards the particle as a frame of reference. Conversely, transition processes will describe how the particle relatively evolves and changes with time toward their environment (how their potential and kinetic terms relatively change toward the other particle). To construct a new model based on the two possibilities above, we pose that: 1. 1. The elementary particle is 3-dimensional sphere-form (2-manifold without boundary) where natural energy will occupy at the surface (a concentrated- energy system). 2. 2. When there is no perspective about system (environment) ’particle in absolute vacuum’, energy occupying the surface of the particle makes the particle move randomly and rotate about its axis (spin motion) where the speeds of both translational random motion and internal angular motion is at the speed of light. The two kinds of fundamental particle motions take place in a quantum- sized volume as our previous model on fundamental of the particle. 3. 3. Random motion taking place in the quantum-sized volume allows the motion of the particle to be considered as a randomly vibrating-movement. 4. 4. We defined mass as the amount of energy per unit of surface area of the particle per its vibration and rotating frequencies, and the value of the mass of the particle is always constant for every state: $\displaystyle m=\frac{E}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{~{}}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}=\frac{E}{\mathbf{c}^{2}},$ where $\mathbf{c}$ is light velocity field, $m$ is mass tensor (in any time interval), $\nu_{\mathrm{vib}}$ and $\nu_{\mathrm{rot}}$ respectively are vibration and rotational frequencies, $4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)$ is the surface of the fundamental particle, and $\bm{\mathfrak{R}}$ is a vector field where its scalar represents the radius of sphere and its direction of random vibrating motion. If we pose that $E$ and $m$ are invariant to all frame of references so that $\mathbf{c}^{2}$ will be invariant quantity caused by the invariance of mass quantity. 5. 5. The concentrated energy of the particle at every state, for every physical system, must always be at a certain value that is less than that of its value when there is no interaction perspective (a particle is in absolute vacuum), whereas the loss of the concentrated energy of the particle will be transformed into the presence of any physical system as expressed by potential terms and kinetics terms and the changes of the speed of internal random motion to lower than the speed of light where the speed of the random motion in whatever kinds of physical systems must always be lower than the speed of light. Thus, interactions and transition states will only change the potential terms and or kinetics terms and or the speed of random motion. For example, the equation of the invariant mass of an atom consisting of $n$ electrons undergoing a transition process can be expressed by Eq. (9) and 10: Consider an electron of the atom is at a stationery state; so invariant mass principle requires $\displaystyle m=\frac{E-V^{~{}}_{1}(\mathbf{x})+V^{~{}}_{2}\left(\mathbf{x}^{~{}}_{1},\mathbf{x}^{~{}}_{2},\cdots,\mathbf{x}^{~{}}_{n}\right)+E^{~{}}_{k}\left(\mathbf{x},\mathbf{x}^{~{}}_{1},\mathbf{x}^{~{}}_{2},\cdots,\mathbf{x}^{~{}}_{n}\right)}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}=\frac{E-E^{~{}}_{0}}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}},$ (9) where $\mathbf{x}=\mathbf{R}+\mathbf{r}$ and $\mathbf{x}^{~{}}_{n}=\mathbf{R}^{~{}}_{n}+\mathbf{r}^{~{}}_{n}$. $V^{~{}}_{1}(\mathbf{x})$ is the Coulomb potential of the electron relatively to the nucleus, $V^{~{}}_{2}\left(\mathbf{x}^{~{}}_{1},\mathbf{x}^{~{}}_{2},\cdots,\mathbf{x}^{~{}}_{n}\right)$ is the Coulomb potential of an electron relative to the $n$ other electrons, and $E^{~{}}_{k}\left(\mathbf{x},\mathbf{x}^{~{}}_{1},\mathbf{x}^{~{}}_{2},\cdots,\mathbf{x}^{~{}}_{n}\right)$ is kinetic energy relative to both the nucleus and the $n$ other electrons. Afterward if the particle undergoes emitting or absorbing processes with transition energy $\pm\Delta E_{\mathrm{transition}}$, the velocity field will change to keep the invariance of mass and Eq. (9) becomes $\displaystyle m=\frac{E-E^{~{}}_{0}\pm\Delta E_{\mathrm{transition}}}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}},$ and the system will occupy a new state $\displaystyle m=\frac{E-V^{~{}}_{1}(\mathbf{x}^{\prime})+V^{~{}}_{2}\left(\mathbf{x}^{\prime}_{1},\mathbf{x}^{\prime}_{2},\cdots,\mathbf{x}^{\prime}_{n}\right)+E^{~{}}_{k}\left(\mathbf{x}^{\prime},\mathbf{x}^{\prime}_{1},\mathbf{x}^{\prime}_{2},\cdots,\mathbf{x}^{\prime}_{n}\right)}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}=\frac{E-E_{0}^{\prime}}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}},$ (10) $E^{~{}}_{0}$ and $E^{\prime}_{0}$ are the energies that have been transformed by the particle to create any physical system. When any transition process takes place with either emitting or absorbing processes, so the atom will rearrange itself through the change of potential terms and/or kinetic terms and the speed of random motion in the quantum-sized volume. Eqs. (9) and (10) represent the invariant mass principle when a transition process takes place, and propose that every $\Delta E^{~{}}_{\mathrm{transition}}$ can be generated by any combination of both a set $\left(V^{~{}}_{1},V^{~{}}_{2},\,\mathrm{and}\,E^{~{}}_{k}\right)$ and every combination of the subset of $\left(V^{~{}}_{1},V^{~{}}_{2},\,\mathrm{and}\,E^{~{}}_{k}\right)$, whereas every combination accompanying emitting or absorbing process will describe every possibility of interaction potential terms or coupling terms in the atom system. For the realistic model of an atom in which its concentrated-energy also represents the existence of environment expressed by fluctuating-field so that invariant mass principle requires that: $\displaystyle m$ $\displaystyle=\frac{E-E^{~{}}_{0}+V^{~{}}_{\mathrm{noise}}}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}$ $\displaystyle=\frac{E-E^{~{}}_{0}+V^{~{}}_{\mathrm{noise}}\pm\left(\Delta E\pm\delta E\right)_{\mathrm{transition}}}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}$ $\displaystyle=\frac{E-V^{~{}}_{1}(\mathbf{x}^{\prime})+V_{2}\left(\mathbf{x}^{\prime}_{1},\mathbf{x}^{\prime}_{2},\cdots,\mathbf{x}^{\prime}_{n}\right)+E^{~{}}_{k}\left(\mathbf{x}^{\prime},\mathbf{x}^{\prime}_{1},\mathbf{x}^{\prime}_{2},\cdots,\mathbf{x}^{\prime}_{n}\right)+V^{\prime}_{\mathrm{noise}}}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}$ $\displaystyle=\frac{E-E^{\prime}_{0}+V^{\prime}_{\mathrm{noise}}}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}.$ (11) Thus $\left(\Delta E\pm\delta E\right)_{\mathrm{transition}}$ can be transformed into any combination of either a set $\left(V^{~{}}_{1},V^{~{}}_{2},E^{~{}}_{k}\right.$, and $\left.V_{\mathrm{noise}}\right)$ and every combination of the subset of $\left(V^{~{}}_{1},V^{~{}}_{2},E^{~{}}_{k},\,\mathrm{and}\,V^{~{}}_{\mathrm{noise}}\right)$ where every combination accompanying the emitting or absorbing processes will describe every possibility of the interaction potential terms or coupling terms, and connection between $V^{~{}}_{\mathrm{noise}}$ and $V^{\prime}_{\mathrm{noise}}$ has been considered as either markovian or non- markovian processes. 6. 6. $1/\nu^{\prime}_{\mathrm{vib}}=T^{\prime}$ and $1/\nu^{\prime\prime}_{\mathrm{vib}}=T^{\prime\prime}$ are the period fields of the randomly vibrating-movement of the particle in the quantum-sized volume and it will play a role as the time interval unit for previous and new states. On of the most important things from invariant mass principle is that it is possible for every physical system to evolve in time without undergoing an emission or absorption process as long as the changes do not alter the total $E_{0}$ of each particle of the system, or the energy particle is constant, or even can take place by emission or absorption processes but the total amount of $E_{0}$ does change on average. From both proposal of the fundamental motion of the particle and the new concept of mass, it seems that there are two possible mechanisms that act as the sources of unpredictability of measurement outcome: 1. 1. Random motion taking place in the quantum-sized volume. (Internal random motion makes physical observables asserted and coupled with position will have statistical features). 2. 2. Macroscopic fluctuations, such as all process that happen in the universe including instrument states. Although the average macroscopic fluctuation do not change the stationary state of particles (any physical system), they will contribute to the outcome probabilities. ## IV THE DIFFUSION CONSTANT IN THE TRANSITION PROCESS To derive the diffusion constant of the particle, we consider a transition process where the particle emits energy to coincide with doing a diffusion process taking place in the quantum-sized volume. First, we consider that the particle is in a state without any kind of interaction or any kind of system or environment perspective, so that the relation between the concentrated- energy and the value of mass will be $\displaystyle m=\frac{E}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{~{}}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}=\frac{E}{\mathbf{c}^{2}}.$ When the particle makes the transition process to occupy a new state to form a new simple physical system consisting of two particles with no kinetic term of the quantum-sized volume, so we can write the process in a simple equation as $\displaystyle m=\frac{E-\Delta E^{~{}}_{\mathrm{transition}}}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}=\frac{E-V(\mathbf{R}+\mathbf{r})+E^{~{}}_{k}(\mathbf{r})}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}.$ (12) $-V(\mathbf{R}+\mathbf{r})$ shows the two particles have different types of charge and $E_{k}(\mathbf{r})$ is the kinetic energy of the particle in the quantum-sized volume. Because the particle undergoes a diffusion process in the quantum-sized volume, work that is equivalent to the displacement by diffusion process is expressed by $\displaystyle m\left(\frac{\mathrm{d}^{2}\mathbf{r}}{\mathrm{d}t^{2}}\right)\cdot\mathbf{r}$ $\displaystyle=-m\left[4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}\right]$ $\displaystyle=\Delta E^{~{}}_{\mathrm{transition}}-E.$ (13) Negative work means that the emission process is generated by the environment, such as the friction process in the perspective of Langevin’s equation. The process in Eq. (IV) can be considered from Langevin’s equation with force $\displaystyle\left(\frac{\mathrm{d}^{2}\mathbf{r}}{\mathrm{d}t^{2}}\right)$ $\displaystyle=-\frac{\xi}{m}\left(\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}\right)+\frac{1}{m}\mathbf{f}_{\mathrm{random}},$ (14a) $\displaystyle m\left(\frac{\mathrm{d}^{2}\mathbf{r}}{\mathrm{d}t^{2}}\right)$ $\displaystyle=-\xi\left(\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}\right)+\mathbf{f}_{\mathrm{random}}.$ (14b) Using the definition of work $\displaystyle m\left(\frac{\mathrm{d}^{2}\mathbf{r}}{\mathrm{d}t^{2}}\right)\cdot\mathbf{r}$ $\displaystyle=-\xi\left(\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}\right)\cdot\mathbf{r}+\mathbf{f}_{\mathrm{random}}\cdot\mathbf{r},$ (14c) with $\displaystyle\frac{1}{2}m\left(\frac{\mathrm{d}^{2}\mathbf{r}^{2}}{\mathrm{d}t^{2}}\right)=m\left(\frac{\mathrm{d}^{2}\mathbf{r}}{\mathrm{d}t^{2}}\right)\cdot\mathbf{r}+m\left(\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}\right)^{2},$ (14d) we obtain $\displaystyle m\left(\frac{\mathrm{d}^{2}\mathbf{r}}{\mathrm{d}t^{2}}\right)\cdot\mathbf{r}=-\frac{\xi}{2}\left(\frac{\mathrm{d}\mathbf{r}^{2}}{\mathrm{d}t}\right)=\frac{m}{2}\left(\frac{\mathrm{d}^{2}\mathbf{r}^{2}}{\mathrm{d}t^{2}}\right)-m\left(\frac{\mathrm{d\mathbf{r}}}{\mathrm{d}t}\right)^{2}.$ (14e) In Eq. (14a), $\mathbf{f}$ is the random forces that are produced by the collision between the particle and a medium, but because random motion, in our model, is an intrinsic properties of the particle, $\mathbf{f}=0$. This result is quantitatively no different from classical viewpoint because the length on average is null $\left(\left\langle\mathbf{r}\right\rangle=0\,\mathrm{or}\,\left\langle\mathbf{f}\cdot\mathbf{r}\right\rangle=0\right)$, while $\xi$ is a friction factor depending on geometry of molecules. We express $E$ in Eq. (IV) within differential form $\displaystyle E$ $\displaystyle=\left\langle m\,4\pi\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\nu^{~{}}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}\right\rangle$ $\displaystyle=\left\langle m\mathbf{c}^{2}\right\rangle$ $\displaystyle=m\mathbf{c}^{2}$ $\displaystyle=m\left(\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}\right)^{2}.$ (15) Because the geometry factor $\xi$ is associated with $\nu^{\prime}_{\mathrm{vib}}$ determining the amount of either emission or absorption energies, furthermore, we represent the vector field $\bm{\mathfrak{R}}$ with a displacement vector field $\mathbf{r}$ so that we can express $\displaystyle-m$ $\displaystyle\left(4\pi\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\,\nu^{\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}\right)$ $\displaystyle\qquad=-m\left(4\pi\bm{\mathfrak{R}}^{~{}}_{\mathrm{particle}}\nu^{\prime}_{\mathrm{vib}}\frac{\bm{\mathfrak{R}}^{~{}}_{\mathrm{particle}}}{T_{\mathrm{rot}}}\right)$ $\displaystyle\qquad\approx-m\left(4\pi\mathbf{r}\,\nu^{\prime}_{\mathrm{vib}}\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}\right)$ $\displaystyle\qquad\approx-2\pi\,m\,\nu^{\prime}_{\mathrm{vib}}\frac{\mathrm{d}\mathbf{r}^{2}}{\mathrm{d}t}.$ (16) where $T^{~{}}_{\mathrm{rot}}$ is the period of internal angular motion. Therefore we may write Eq. (IV) as $\displaystyle\Delta E^{~{}}_{\mathrm{transition}}-m\left(\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}\right)^{2}=-2\pi m\nu^{\prime}_{\mathrm{vib}}\frac{\mathrm{d}\mathbf{r}^{2}}{\mathrm{d}t}.$ (17) Eqs. (14e) and (17) represent the same process, therefore $\Delta E_{\mathrm{transition}}$ will be $\displaystyle\Delta E_{\mathrm{transition}}=\frac{m}{2}\left(\frac{\mathrm{d}^{2}\mathbf{r}^{2}}{\mathrm{d}t^{2}}\right),$ (18) and $\displaystyle\xi=4m\pi\nu^{\prime}_{\mathrm{vib}}.$ (19) Using energy relation, we have $\displaystyle E=\left\langle m\mathbf{c}^{2}\right\rangle=m\mathbf{c}^{2}=\left\langle m\left(\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t}\right)^{2}\right\rangle=h\nu.$ (20) From the invariant mass principle, it is clear that the emitting and absorbing processes will coincide with the decrease or increase of the vibration- frequency field, therefore the emission and absorption processes will correspond with positive or negative values of the change of $\nu^{\prime}_{\mathrm{vib}}$ or $\xi$ (excitation and de-excitation processes can be represented by the decrease or the increase of the speed of internal random motion). Use a new notation $\alpha=\left\langle\mathrm{d}\mathbf{r}^{2}\right\rangle/\mathrm{d}t$, and applying the condition that the random vibrating-frequency of the charged particle and the frequency of its field must be the same, so Eq. (14e) and (17) can be written as $\displaystyle\left(\frac{\mathrm{d}\alpha}{\mathrm{d}t}\right)+\frac{\xi}{m}\alpha=\frac{2h\nu}{m}.$ (21) Eq. (21) has a general solution $\displaystyle\alpha=\frac{2h\nu}{\xi}+C\exp\left(-\frac{\xi}{m}t\right).$ (22) Because the time interval of the transition process will much higher than the period of the vibration process taken place in the quantum-sized volume or $t\gg\xi/m$ or $\exp\left(-\xi t/m\right)\rightarrow 0$, so we obtain $\displaystyle\left\langle\mathbf{r}^{2}\right\rangle=\frac{2h\nu}{\xi}t.$ (23) Whereas according to Stokes-Einstein-Sutherland equation, diffusion constant is $\displaystyle\beta=\lim\limits_{t\rightarrow\infty}\frac{1}{2\Delta t}\left\langle\mathbf{r}^{2}\right\rangle.$ (24) Using the assumption that the time interval of transition process will much higher than the period of the vibration process or taking the perspective that a process taking place is sufficiently long time so that Eq. (24) is $\displaystyle 2t\beta$ $\displaystyle=\left\langle\mathbf{r}^{2}\right\rangle$ $\displaystyle\beta$ $\displaystyle=\frac{\left\langle\mathbf{r}^{2}\right\rangle}{2t}.$ (25) Using Eq. (23) and because $\xi=4\pi m\nu$, we find $\displaystyle\beta$ $\displaystyle=\frac{h}{4\pi m}$ $\displaystyle=\frac{\hslash}{2m}.$ (26) ## V DERIVING SCHRÖDINGER EQUATION We consider the dynamics aspect of our model and make interpretations due to the meaning behind Schrödinger equation. Applying Eq. (1) and (2) into Eq. (II) we have $\displaystyle\frac{1}{m}\mathbf{F}_{\mathrm{external}}-\frac{1}{m}\nabla V$ $\displaystyle=\frac{\partial^{2}\mathbf{R}}{\partial t^{2}}+\frac{1}{2}\frac{\partial}{\partial t}\left(\mathbf{b}+\mathbf{b}^{~{}}_{}\right)$ $\displaystyle\qquad+\frac{1}{2}\left[\left(\mathbf{b}+\mathbf{b}^{~{}}_{}\right)\cdot\nabla\right]\frac{\partial\mathbf{R}}{\partial t}$ $\displaystyle\qquad+\frac{1}{2}\left(\mathbf{b}\cdot\nabla\right)\mathbf{b}^{~{}}_{}+\frac{1}{2}\left(\mathbf{b}^{~{}}_{}\cdot\nabla\right)\mathbf{b}$ $\displaystyle\qquad-\frac{\hslash}{4m}\nabla^{2}\left(\mathbf{b}-\mathbf{b}^{~{}}_{}\right).$ (27) We rewrite Eq. (V) as $\displaystyle\frac{1}{m}\mathbf{F}^{~{}}_{\mathrm{external}}-\frac{1}{m}\nabla V$ $\displaystyle=\frac{\partial\mathbf{v}}{\partial t}+\frac{1}{2}\frac{\partial}{\partial t}\left(\mathbf{b}+\mathbf{b}^{~{}}_{}\right)$ $\displaystyle\qquad+\frac{1}{2}\left[\left(\mathbf{b}+\mathbf{b}^{~{}}_{}\right)\cdot\nabla\right]\mathbf{v}$ $\displaystyle\qquad+\frac{1}{2}\left(\mathbf{b}\cdot\nabla\right)\mathbf{b}^{~{}}_{}$ $\displaystyle\qquad+\frac{1}{2}\left(\mathbf{b}^{~{}}_{}\cdot\nabla\right)\mathbf{b}$ $\displaystyle\qquad-\frac{\hslash}{4m}\nabla^{2}\left(\mathbf{b}-\mathbf{b}^{~{}}_{}\right),$ (28) where $\partial\mathbf{R}/\partial t=\mathbf{v}$. This result is different from the Nelson model which does not include $\partial\mathbf{x}(t)/\partial t=\mathbf{v}$, in the proposal model, $\partial\mathbf{x}(t)/\partial t=\partial\mathbf{R}(t)/\partial t=\mathbf{v}$ represents the motion of a quantum-sized volume, however $\partial\mathbf{x}(t)/\partial t=\mathbf{v}$ may also describe the motion of vacuum (medium). According to our concept of mass, the particle must undergo random motion in the quantum-sized volume, thus the probability density to find a particle at any point in the quantum-sized volume will directly depend on the speed of the random motion and distribution of the random-velocity field for stationer state as well as depend on the diffusion process taking place in the quantum- sized volume for transition process. Because the proposed definition of mass covers the whole space of the quantum-sized volume, the distribution of mass in the quantum-sized volume is interchangeable with the probability distribution to find a particle in the quantum-sized volume, and how the probability density changes from one state to another can be interchangeably viewed as the change of either the probability density or the mass density or the charge density. Although the density of the mass and the charge of particle is always $m$ and $q$, which are equivalent to the condition that the total probability to find a particle in of the entire space of the quantum- sized volume is one. Defining $\rho\left(\mathbf{r}^{\prime},t\right)$ as the probability density to find the particle in the quantum-sized volume, so that $\rho\left(\mathbf{r}^{\prime},t\right)$ obeys $\displaystyle\int\rho\left(\mathbf{r}^{\prime},t\right)\,\mathrm{d}^{3}r^{\prime}=1.$ (29) This describes the probability of finding the particle in the entire space of the quantum-sized volume, which must be one. If we define $\rho^{~{}}_{e}\left(\mathbf{r}^{\prime},t\right)$ and $\rho_{m}\left(\mathbf{r}^{\prime},t\right)$ as charge and mass densities in the quantum-sized volume respectively, so that they obey $\displaystyle\int\rho^{~{}}_{e}\left(\mathbf{r}^{\prime},t\right)\,\mathrm{d}^{3}r^{\prime}=q,$ (30) $\displaystyle\int\rho^{~{}}_{m}\left(\mathbf{r}^{\prime},t\right)\,\mathrm{d}^{3}r^{\prime}=m.$ (31) From Eqs. (29) and (30) as well as (31), we can define the connection among the probability, charge and mass densities by $\displaystyle\rho^{~{}}_{e}\left(\mathbf{r}^{\prime},t\right)=q\rho\left(\mathbf{r}^{\prime},t\right),$ (32) $\displaystyle\rho^{~{}}_{m}\left(\mathbf{r}^{\prime},t\right)=m\rho\left(\mathbf{r}^{\prime},t\right).$ (33) From hydrodynamics viewpoint, we can imagine the mass density or charge density or probability density as fluid density. As shown in session (3), the fluid analogy is presented by the Langevin equation containing the coefficient friction describing interaction between the Brownian particle and the fluid (medium) particle. Based on this viewpoint, we can associate the properties of the field of internal motion velocity and emission or absorption processes with mechanical properties and thermodynamical-statistical properties of fluid. For instance, we can associate internal mass density (spreading over the quantum-sized volume) with fluid density. Internal random motion taking place in the quantum-sized volume, where the total mass and charge for the entire quantum-sized volume is always $m$ and $q$, makes us can consider a point in the quantum-sized volume as a particle of both mass $\rho^{~{}}_{m}\left(\mathbf{r}^{\prime},t\right)$ and charge $\rho^{~{}}_{e}\left(\mathbf{r}^{\prime},t\right)$. Because a particle of both mass $\rho^{~{}}_{e}\left(\mathbf{r}^{\prime},t\right)$ and charge $\rho^{~{}}_{e}\left(\mathbf{r}^{\prime},t\right)$ undergoes random motion where the density of the field of the internal random velocity is always relatively much higher than the time interval of measurement (observer), we will always see the current of the point particle with mass $\rho^{~{}}_{m}\left(\mathbf{r}^{\prime},t\right)$ and charge $\rho^{~{}}_{e}\left(\mathbf{r}^{\prime},t\right)$ for every time interval as the current density that will obey the continuity equation for emission and absorption processes for mass and charge and probability densities $\displaystyle\frac{\partial\rho^{~{}}_{m,e,p}\left(\mathbf{r}^{\prime},t\right)}{\partial t}$ $\displaystyle=-\nabla\cdot\left[\left(\mathbf{b}+\mathbf{v}\right)\rho_{m,e,p}\left(\mathbf{r}^{\prime},t\right)\right]$ $\displaystyle\qquad+\frac{\hslash}{2m}\nabla^{2}\rho^{~{}}_{m,e,p}\left(\mathbf{r}^{\prime},t\right),$ (34a) $\displaystyle\frac{\partial\rho^{~{}}_{m,e,p}\left(\mathbf{r}^{\prime},t\right)}{\partial t}$ $\displaystyle=-\nabla\cdot\left[\left(\mathbf{b}^{~{}}_{}+\mathbf{v}\right)\rho_{m,e,p}\left(\mathbf{r}^{\prime},t\right)\right]$ $\displaystyle\qquad-\frac{\hslash}{2m}\nabla^{2}\rho^{~{}}_{m,e,p}\left(\mathbf{r}^{\prime},t\right),$ (34b) where index $m,e,p$ respectively refer to mass, charge, and probability. Eq. (34) and (34) are forward and backward modified-Fokker equations. Eqs. (34) and (34) yield $\displaystyle\frac{\partial\rho}{\partial t}=-\nabla\cdot\left[\left(\bm{\upsilon}+\mathbf{v}\right)\rho\right],$ (35) and $\displaystyle\mathbf{u}=\beta\frac{\nabla\rho}{\rho},$ (36) where $\bm{\upsilon}$ is $\displaystyle\bm{\upsilon}=\frac{1}{2}\left(\mathbf{b}+\mathbf{b}^{~{}}_{}\right),$ (37) and we call $\bm{\upsilon}$ as the transition velocity. While $\mathbf{u}$ is defined by $\displaystyle\mathbf{u}=\frac{1}{2}\left(\mathbf{b}-\mathbf{b}^{~{}}_{}\right),$ (38) and we cal $\mathbf{u}$ as the stationer velocity. Computing $\partial\mathbf{u}/\partial t$ and applying (35), we obtain $\displaystyle\frac{\partial\mathbf{u}}{\partial t}=-\frac{\hslash}{2m}\nabla\left[\nabla\cdot\left(\bm{\upsilon}+\mathbf{v}\right)\right]-\nabla\left[\mathbf{u}\cdot\left(\bm{\upsilon}+\mathbf{v}\right)\right].$ (39) Applying (37) and (38) to (V), we find $\displaystyle\frac{\partial\left(\bm{\upsilon}+\mathbf{v}\right)}{\partial t}$ $\displaystyle=-\frac{1}{m}\left(\nabla V-\mathbf{F}^{~{}}_{\mathrm{external}}\right)-\left(\bm{\upsilon}\cdot\nabla\right)\left(\bm{\upsilon}+\mathbf{v}\right)$ $\displaystyle\qquad+\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}+\frac{\hslash}{2m}\nabla^{2}\mathbf{u}.$ (40) ### V.1 The real time-independent Schrödinger equation We consider the stationer state (i.e. no transition process). From the description of the stationer state in Eq. (8), the stationer state requires $\displaystyle\mathbf{F}_{\mathrm{external}}=0,$ whereas from definition of the stationer state at Eqs. (1) and (2), a stationery state requires $\displaystyle\mathbf{b}=-\mathbf{b}_{},$ (41) or $\bm{\upsilon}=0$ so that $\partial\bm{\upsilon}/\partial t=0$, furthermore $\partial\mathbf{u}/\partial t=0$ so that $\rho$ and $\mathbf{u}$ are independent of $t$. We have two cases for stationer states that are $\mathbf{v}=0$ and $\mathbf{v}\neq 0$. For $\mathbf{v}=0$ case, by applying the stationer conditions to Eq. (39) and (V), we obtain $\displaystyle\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}+\frac{\hslash}{2m}\nabla^{2}\mathbf{u}=\frac{1}{m}\nabla V$ (42) $\displaystyle\frac{1}{2}\nabla\mathbf{u}^{2}+\frac{\hslash}{2m}\nabla\left(\nabla\cdot\mathbf{u}\right)=\frac{1}{m}\nabla V.$ (43) According to the invariant mass principle, the total concentrated-energy of the particle before the physical system exists is $E=m\mathbf{c}^{2}$, and the system is the representation of how much the concentrated-energy has transformed to present potential terms, kinetic terms and the decrease of random motion to less than the speed of light. Furthermore, the transition energy represents how the system changes from any combination of potential and/or kinetic and random velocity terms to another combination. Therefore, the total amount of kinetic and potential energies at every stationer state must be equal to the total energy that is transformed by the particle, called $E^{~{}}_{0}$. If we integrate Eq. (43), the constant of integration should be the negative of $E^{~{}}_{0}$. We have $\displaystyle\frac{1}{m}E^{~{}}_{0}+\frac{\hslash}{2m}\left(\nabla\cdot\mathbf{u}\right)=-\frac{1}{2}\mathbf{u}^{2}+\frac{1}{m}V.$ (44) We notice $\mathbf{u}=\mathbf{b}=-\mathbf{b}_{}$ corresponds with the emission and absorption processes, and that the $\pm\left(\hslash/2m\right)\left(\nabla\cdot\mathbf{u}\right)$ terms represent either emitted-energy or absorbed-energy when the particle undergoes transition process. Therefore, $\left(\hslash/2\right)\left(\nabla\cdot\mathbf{u}\right)$ must be represent the difference between energy level of stationery state. Since $E$ is constant and $\left(\hslash/2\right)\left(\nabla\cdot\mathbf{u}\right)$ has definite value, every state must be quantified by both space variables due to the potential and intrinsic variable that represents random motion in the quantum- sized volume. Following Nelson’s work, Eq. (44) will be time independent Schrödinger equation $\displaystyle\left[-\left(\frac{\hslash^{2}}{2m}\right)\nabla^{2}+V\right]\Psi=E^{~{}}_{0}\Psi$ (45) with $\displaystyle\Psi=e^{R},$ (46) where $R$ in Eq. (46) satisfy $R=\left(\ln\rho\right)/2$. For $\mathbf{v}\neq 0$, form Eqs. (39) and (V), we obtain $\displaystyle\beta\nabla^{2}\mathbf{v}=-\nabla\left(\mathbf{u}\cdot\mathbf{v}\right),$ (47) $\displaystyle\mathbf{a}=-\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}-\beta\nabla^{2}\mathbf{u}.$ (48) Eqs. (47) and (48) determine that the energy of the emission or absorption processes is contributed to by the change of both the intrinsic and the quantum-sized volume velocities and can be presented by single equation $\displaystyle\left[-\left(\frac{\hslash^{2}}{2m}\right)\nabla^{2}+V\right]\Psi=E^{~{}}_{v}\Psi,$ (49) where $\Psi=e^{R+iS}$, $\nabla S=m\mathbf{v}/\hslash$, $E^{~{}}_{v}=E^{~{}}_{0}+E^{~{}}_{k}$, and $E^{~{}}_{k}=mv^{2}/2$. Eqs. (47) and (48) are the imaginary and real parts of Eq. (49) depicting that the emergence of the kinetic term will be compensated by the decrease of the amount of concentrated-particle energy. ### V.2 The real time-dependent Schrödinger equation There are two kinds of general time-dependent cases that are physical systems without or with motion of the quantum-sized volume. For systems with $\mathbf{v}=0$, Eqs. (39) and (V) become $\displaystyle\frac{\partial\mathbf{u}}{\partial t}=-\beta\nabla\cdot\bm{\upsilon}-\nabla\left(\mathbf{u}\cdot\bm{\upsilon}\right),$ (50) $\displaystyle\frac{\partial\bm{\upsilon}}{\partial t}=\mathbf{a}-\left(\bm{\upsilon}\cdot\nabla\right)\bm{\upsilon}+\left(\mathbf{u}\cdot\nabla\right)\mathbf{u}+\frac{\hslash}{2m}\nabla^{2}\mathbf{u}.$ (51) It has been shown by EN-1966 that Eq. (50) and (51) are equivalent to the Schrödinger equation: $\displaystyle i\hslash\frac{\partial\Psi}{\partial t}=-\frac{\hslash^{2}}{2m}\nabla^{2}\Psi+V_{\mathrm{net}}\Psi,$ (52) where $V^{~{}}_{\mathrm{net}}=V+V^{~{}}_{\mathrm{external}}$, and $\Psi=e^{R+iS}$ where $\nabla S=m\bm{\upsilon}/\hslash$. We can see that $\bm{\upsilon}$ in Eqs. (50) and (51) also presented in the modified-Fokker-Planck Eqs. (34) and (34) thus indicate that it is a field. In our proposal, the probability density to find a particle in the quantum-sized volume is equivalent to the mass density of the particle in the quantum-sized volume where it will be determined by the amount of concentrated-energy. While the kinetic term is one of the features of the physical system which represents the change in the amount of concentrated energy, hence the velocity of the quantum-sized volume will be a physical quantity which governs the probability of finding a particle in the quantum-sized volume. This means that when the quantum-sized volume moves from one state with certain velocity to another state with different velocity accompanied by the change of velocity of internal random motion, so the probability density will change simultaneously as well. The physical mechanism can be also understood from another perspective in that the quantum-sized volume will move from one point to another point with a certain probability. From a stochastic mechanics viewpoint, this can be considered as the Brownian motion of the quantum-sized volume from one point to another point that generates the field property of $\bm{\upsilon}$. For a system with $\mathbf{v}\neq 0$, Eqs. (39) and (V) will be equivalent to a new form of the Schrödinger equation $\displaystyle i\hslash\frac{\partial\Psi}{\partial t}$ $\displaystyle=-\frac{\hslash^{2}}{2m}\nabla^{2}\Psi$ $\displaystyle\qquad+\left[V^{~{}}_{\mathrm{net}}+E^{~{}}_{k}+\int m\left(\bm{\upsilon}\nabla\cdot\mathbf{v}\right)\,\mathrm{d}^{3}r^{\prime}\right]\Psi,$ (53) where $\Psi=e^{R+iS}$, with $\nabla S=m\left(\bm{\upsilon}+\mathbf{v}\right)/\hslash$, whereas $E^{~{}}_{k}=mv^{2}/2$ and $\mathrm{d}^{3}r^{\prime}$ is the volume element of the quantum-sized volume. For a special case when $\mathbf{u}$ is a solenoidal vector field ($\nabla\cdot\mathbf{v}=0$) then the Eq. (V.2) becomes $\displaystyle i\hslash\frac{\partial\Psi}{\partial t}=-\frac{\hslash^{2}}{2m}\nabla^{2}\Psi+\left(V_{\mathrm{net}}+E_{k}\right)\Psi.$ (54) In this case, the particle is not in any physical system (the particle is not governed by any potential). The Schrödinger equations for the cases $\mathbf{v}=0$ and $\mathbf{v}\neq 0$ are respectively $\displaystyle i\hslash\frac{\partial\Psi}{\partial t}=-\frac{\hslash^{2}}{2m}\nabla^{2}\Psi+V_{\mathrm{external}}\Psi,$ (55) $\displaystyle i\hslash\frac{\partial\Psi}{\partial t}$ $\displaystyle=-\frac{\hslash^{2}}{2m}\nabla^{2}\Psi$ $\displaystyle\qquad+\left[V^{~{}}_{\mathrm{net}}+E^{~{}}_{k}+\int m\left(\bm{\upsilon}\nabla\cdot\mathbf{v}\right)\,\mathrm{d}^{3}r^{\prime}\right]\Psi,$ (56) or for a special case when $\mathbf{u}$ is a solenoidal vector field ($\nabla\cdot\mathbf{v}=0$), Eq. (V.2) becomes $\displaystyle i\hslash\frac{\partial\Psi}{\partial t}=-\frac{\hslash^{2}}{2m}\nabla^{2}\Psi+\left(V_{\mathrm{external}}+E_{k}\right)\Psi.$ (57) ### V.3 The origin of spin on the new model of stochastic interpretation Recalling the modified-Fokker-Planck equations $\displaystyle\frac{\partial\rho}{\partial t}$ $\displaystyle=-\nabla\cdot\left[\left(\mathbf{b}+\mathbf{v}\right)\rho\right]+\frac{\hslash}{2m}\nabla^{2}\rho,$ $\displaystyle\frac{\partial\rho}{\partial t}$ $\displaystyle=-\nabla\cdot\left[\left(\mathbf{b}_{}+\mathbf{v}\right)\rho\right]-\frac{\hslash}{2m}\nabla^{2}\rho.$ Spin, in our proposal, is connected with the internal random motion that takes place in the quantum-sized volume. The particle moves by random walking with certain velocities will change its velocity field when every transition taking place coincides with forward or backward diffusions for either emitting or absorbing processes. The change of the velocity field of the random motion in the quantum-sized volume (limited motion) will generate internal spinning motion. Each transition process (emission and absorption processes), in our proposal, is represented in Eqs. (34) and (34). Eq. (34) describes if the particle occupies a certain general stationer state with both certain random velocity field $\mathbf{b}$ and the certain quantum-sized volume velocity $\mathbf{v}$, so when the transition process coincides with the diffusion process (emission process), the particle will evolve to a new stationer state with the velocity of random motion $\mathbf{b}_{}$. While Eq. (34) describes backward process corresponding to the absorption processes represented by negative coefficient diffusion. Because the change of the internal velocity field takes place in the quantum-size volume, it is very clear that this mechanism will generate spinning-motion perspective. The velocity field representing the change of internal random motion is shown by Eq. (36) $\displaystyle\mathbf{u}=\frac{\hslash}{2m}\frac{\nabla\rho}{\rho}.$ Now, we define the normal-direction field to $\mathbf{u}$ (the unit vectors that is perpendicular to the plane of $\mathbf{b}$ and $\mathbf{b}_{}$) as the direction field of internal spinning-motion and we can call it $\hat{\mathbf{s}}$. Using a description of $\hat{\mathbf{s}}$, we can approximately define velocity fields $\mathbf{b}$ and $\mathbf{b}_{}$ with $\mathbf{u}$ and $\hat{\mathbf{s}}$, that are $\displaystyle\mathbf{b}\approx\frac{\hslash}{2m}\frac{\nabla\rho}{\rho}\times\hat{\mathbf{s}},$ (58) $\displaystyle\mathbf{b}_{}\approx\frac{\hslash}{2m}\frac{\nabla\rho}{\rho}\times\left(-\hat{\mathbf{s}}\right).$ (59) For the stationer state, we rewrite the current density in the Eqs. (34) and (34) with $\displaystyle\mathbf{J}=\left[\left(\frac{\nabla\rho}{m\rho}\times\hat{\mathbf{s}}\right)+\mathbf{v}\right]\rho,$ (60) $\displaystyle\mathbf{J}=\left[-\left(\frac{\nabla\rho}{m\rho}\times\hat{\mathbf{s}}\right)+\mathbf{v}\right]\rho,$ (61) where $\mathbf{s}=\left(\hslash/2\right)\hat{\mathbf{s}}$ and $\mathbf{J}$ is the current density. Since $\hat{\mathbf{s}}$ is perpendicular to $\mathbf{u}=\left(\hslash/2m\right)\left(\nabla\rho\right)/\rho$ so that $\displaystyle\left[\frac{\nabla\rho}{m\rho}\times\mathbf{s}\right]^{2}=\left(\frac{\nabla\rho}{m\rho}\right)^{2}\mathbf{s}^{2}.$ (62) It is known from Clifford algebra to Dirac theory ER-GS-1998 . ### V.4 The relation among time interval, magnetic and electric fields as well as mass In this section, we will introduce a new approach to understand the origin of charge (fields) via the existence of magnetic and electric fields based on the proposed picture of mass. Mass, as having been described in the previous section, when the interaction does not exist, will connect to the amount of concentrated-energy contained by particle following equation $\displaystyle m=\frac{E}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{~{}}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}=\frac{E}{\mathbf{c}^{2}}.$ When an interaction exist to create any physical system – for instance, to create the simple physical system that is only governed by the electric potential and kinetic energy – the particle will emit energy through an emission process and thus cause the particle to evolve to occupy a state. In this process, the particle’s mass will be always constant to obey the invariant mass principle $\displaystyle m=\frac{E-\Delta E^{~{}}_{\mathrm{transition}}}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}=\frac{E-V(\mathbf{R}+\mathbf{r})+E^{~{}}_{k}(\mathbf{r})}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}.$ Through this simple process, we can view the electric potential or electric field as a representation of the particle’s tendency to have a minimum concentrated-energy after the emission process where to maintain a minimum energy, the two particles come closer to each other (for system that consist of two particles with opposite charge). Contrary to a system with two opposite charges, for simple system consisting of two particles with the same charge, the two particles will stay away from each other in order to maintain a minimum concentrated-energy. Thus, two particles with opposite charge, if interaction makes the perspective of kinetic energy between both references change while the perception on distance between them does not change, the perception on the change of kinetic energy perception (for instance, according to one frame of reference, the kinetic energy of another particle increase) will be responded as the loss of the particle’s tendency to come closer to each other (the loss of electric potential), or it represents the emergence of magnetic potential (field) between them. We see the change of the perspective of kinetic energy between two references represents the transformation of fields. In the same way, for a case, a particle experience the change of perception on both distance and velocity in the same time (for example when the particle makes a transition process from one state to another state) then this perception will be responded as the emergence of electromagnetic field between two terms of reference. Thus, electric and magnetic fields are relatively phenomena between two terms of reference. To show this view, we consider a Hydrogen atom in a simple closed-system (we ignore the fluctuating- environment field). According to the invariant mass principle, an electron at any state will describe how much energy of the particle has been transformed to potentials, kinetic energy, and the resulting decrease of internal random velocity to a speed less than speed of light $\displaystyle m$ $\displaystyle=\frac{E-E^{~{}}_{0}}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}$ $\displaystyle=\frac{E-V(\mathbf{R}+\mathbf{r})+E^{~{}}_{k}(\mathbf{R}+\mathbf{r})}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}.$ (63) When a transition process takes place by emission process, the particle will evolve to a new state $\displaystyle m$ $\displaystyle=\frac{E-E^{~{}}_{0}}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}$ $\displaystyle=\frac{E-V(\mathbf{R}+\mathbf{r})+E^{~{}}_{k}(\mathbf{R}+\mathbf{r})-\Delta E^{~{}}_{\mathrm{emission}}}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}$ $\displaystyle=\frac{E-V(\mathbf{R}^{\prime}+\mathbf{r}^{\prime})+E(\mathbf{R}^{\prime}+\mathbf{r}^{\prime})}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}.$ (64) Eq. (V.4) shows that, at every state, the mass of the particle will be always constant. However, mass is always constant at every state, but when a transition process takes place, there will be a difference in the perspective of mass between the new state and previous state. Eq. (V.4) shows simply that if a transition process occurs, the change of the potential term is higher than the emission energy. Hence the kinetic energy must increase to a level higher than the previous state. We can express the relation of both $\mathbf{v}^{\prime}+\mathbf{b}_{}$ for $E^{\prime}_{\mathrm{k}}$ (the total velocity at the new state) and $\mathbf{v}+\mathbf{b}$ for $E_{k}$ (total velocity at the previous state) as $\displaystyle\mathbf{v}^{\prime}+\mathbf{b}_{}=\left(\mathbf{v}+\mathbf{b}\right)+\Delta\mathbf{v}.$ (65) We consider a case where the vector $\Delta\mathbf{v}$ perpendicular to the direction of the stationer velocity of quantum-sized volume, we have $\left(\mathbf{v}+\mathbf{b}\right)\perp\Delta\mathbf{v}$, so we can rewrite Eq. (V.4) for a new state as $\displaystyle m=\frac{E-V(\mathbf{R}^{\prime}+\mathbf{r}^{\prime})+E^{~{}}_{k}(\mathbf{R}+\mathbf{r})+\Delta E^{~{}}_{k}}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}.$ (66) Relative to the previous state, the mass of particle at previous state will be $\displaystyle m_{0}=\frac{E-V(\mathbf{R}^{\prime}+\mathbf{r}^{\prime})+E^{~{}}_{k}(\mathbf{R}+\mathbf{r})}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}.$ (67) If we suppose that the time unit in every state must be represented by $\displaystyle T=\frac{1}{\nu^{~{}}_{\mathrm{vib}}},$ (68a) then the internal time unit will be different for every state. In every transition process, a particle emitting or absorbing energy will change the internal time unit (proper time) where the change of the internal time unit must coincide with the emergence of the kinetic term of the particle (the difference of velocity between two states). Then we assume the internal time unit in one state will relativistically relate to another state, for example the internal time units at two states in Eq. (48) and Eq. (49) relate each other following $\displaystyle T=\frac{1}{\nu^{~{}}_{\mathrm{vib}}}=\frac{1}{\nu^{\prime\prime}_{\mathrm{vib}}}=\frac{1}{\sqrt{1-(\Delta\mathbf{v})^{2}/c^{2}}}=\frac{T^{\prime\prime}}{\sqrt{1-(\Delta\mathbf{v})^{2}/c^{2}}}.$ (68b) Furthermore we assume the perspective on mass between an occupied state and an unoccupied state or vice versa will also relativistically relate following $\displaystyle\frac{E-V(\mathbf{R}^{\prime}+\mathbf{r^{\prime}})+E_{k}(\mathbf{R}+\mathbf{r})}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}\frac{1}{\sqrt{1-(\Delta\mathbf{v})^{2}/c^{2}}}$ $\displaystyle\qquad=m_{0}\frac{1}{\sqrt{1-(\Delta\mathbf{v})^{2}/c^{2}}}$ $\displaystyle\qquad=\frac{E-V(\mathbf{R}^{\prime}+\mathbf{r}^{\prime})+E_{k}(\mathbf{R}+\mathbf{r})+\Delta E_{k}}{4\pi\left(\bm{\mathfrak{R}}^{2}_{\mathrm{particle}}\right)\nu^{\prime\prime}_{\mathrm{vib}}\nu^{~{}}_{\mathrm{rot}}}$ $\displaystyle\qquad=m.$ (69) Thus, for the case of the emission process, energy relating to the relativistically difference of mass between the two states must represent both magnetic energy generated by the emergence of the kinetic term (the difference of velocity) between two states and emitted energy (radiated energy) when the particle does a transition process, or two following equations must be the same $\displaystyle\left(m-m^{~{}}_{0}\right)c^{2}$ $\displaystyle=m^{~{}}_{0}\left[\frac{1}{\sqrt{1-(\Delta\mathbf{v})^{2}/c^{2}}}-1\right]c^{2}$ $\displaystyle=\frac{1}{2}m^{~{}}_{0}(\Delta\mathbf{v})^{2}+\frac{3}{8}m^{~{}}_{0}\frac{(\Delta\mathbf{v})^{4}}{c^{2}}+\frac{5}{16}m^{~{}}_{0}\frac{(\Delta\mathbf{v})^{6}}{c^{4}}$ $\displaystyle\qquad+\frac{35}{128}m^{~{}}_{0}\frac{(\Delta\mathbf{v})^{8}}{c^{6}}+\cdots,$ (70a) $\displaystyle\left(m-m^{~{}}_{0}\right)c^{2}$ $\displaystyle=m^{~{}}_{0}\left[\frac{1}{\sqrt{1-(\Delta\mathbf{v})^{2}/c^{2}}}\right]c^{2}$ $\displaystyle=E^{~{}}_{\mathrm{magnetic}}+\Delta E^{~{}}_{\mathrm{radiation}}.$ (70b) For showing the equivalence of Eqs. (70) and (70), we use the wave function in Schrödinger picture to describe the distribution of mass/charge/probability inside the quantum-sized volume, while the wave function in Heisenberg picture is used to describe the distribution of mass/charge/probability outside the quantum sized volume and relates to classical fields. We then connect the meaning of electrical-magnetic fields with the states of concentrated energy (in the term of particle’s inclination to have minimum energy) in which the emergence of kinetic energy represents the increase of internal concentrated energy. Because the increase of internal concentrated energy serves the same effect as the decrease of the particle’s tendency to come near to one another (the decrease of electric potential energy for hydrogen consisted of electron and nucleus is equivalent to magnetic energy in the entire space of outside of the quantum-sized volume. Eq. (70) is the perspective of energy between two stationer states when the transition process takes place. Stationery current density at new state relatively to previous state is $\displaystyle\mathbf{J}(\mathbf{r}^{\prime})$ $\displaystyle=\left[\left(\mathbf{v}^{\prime}+\mathbf{b}^{~{}}_{}\right)-\left(\mathbf{v}-\mathbf{b}\right)\right]\rho_{e}$ (71) $\displaystyle\mathbf{J}(\mathbf{r}^{\prime})$ $\displaystyle=\left[2\mathbf{u}+\left(\mathbf{v}^{\prime}-\mathbf{v}\right)\right]\rho_{e},$ (72) where $\mathbf{u}=\left(\mathbf{b}^{~{}}_{}-\mathbf{b}\right)/2$. $\rho_{e}$ in Eqs. (71) and (72) satisfies Eq. (30) $\displaystyle\rho_{e}\left(\mathbf{r}^{\prime}\right)=q\rho\left(\mathbf{r}^{\prime}\right)=q\Psi^{}(\mathbf{r}^{\prime})\Psi(\mathbf{r}^{\prime})$ (73) with $q$ being the electron charge. Our model describes the random motion of the particle and allows us to see charge density and current density in Eqs. (72) and (73) as sources of classical electromagnetic field. $\displaystyle J^{\mu}=\left(c\rho,\mathbf{J}\right),\qquad r^{\mu}=\left(ct,\mathbf{r}\right).$ (74) Electromagnetic fields generated by these sources are $\displaystyle\partial_{\mu}F^{\mu\gamma}=\frac{4\pi}{c}J^{\gamma};\qquad F^{\mu\gamma}=\partial^{\mu}A^{\gamma}-\partial^{\gamma}A^{\mu},$ (75) $\displaystyle A^{\mu}=\left(U,\mathbf{A}\right).$ (76) Working in Lorentz gauge MPD-2004 $\left(\partial^{\mu}A^{~{}}_{\mu}=0\right)$, the classical result is $\displaystyle A^{\mu}\left(\mathbf{r},t\right)=\frac{1}{c}\int\frac{1}{R}\,J^{~{}}_{\mu}\left(\mathbf{r},t-\dfrac{R}{c}\right)\,\mathrm{d}^{3}r^{\prime}\quad;\quad R=\left\|\mathbf{r}-\mathbf{r}^{\prime}\right\|,$ (77) where $R$ is the position vector of consideration of the magnetic field from new state. The magnetic field that corresponds to the vector potential $\mathbf{A}$ is $\displaystyle\mathbf{B}\left(\mathbf{r},t\right)=\nabla\times\mathbf{A}=\nabla\times\frac{1}{c}\int\frac{1}{R}\,\mathbf{J}\left(\mathbf{r}^{\prime},t-\dfrac{R}{c}\right)\,\mathrm{d}^{3}r^{\prime},$ (78) where unit vector of position is $\displaystyle\hat{\mathbf{n}}=\frac{\mathbf{r}-\mathbf{r}^{\prime}}{\left\|\mathbf{r}-\mathbf{r}\right\|},$ (79) where $\mathbf{r}^{\prime}$ is the position vector from the reference frame of the previous state to the reference frame of new state, and $\mathbf{r}$ is position vector from the reference frame of previous state to the point of consideration. Using this vector we can rewrite Eq. (78) as $\displaystyle\mathbf{B}\left(\mathbf{r},t\right)=\frac{1}{c}\int\hat{\mathbf{n}}\times\frac{\partial}{\partial R}\left[\left(\frac{1}{R}\right)\,\mathbf{J}\left(\mathbf{r},t-\dfrac{R}{c}\right)\right]\,\mathrm{d}^{3}r^{\prime}.$ (80) For stationary current density, Eq. (80) becomes $\displaystyle\mathbf{B}\left(\mathbf{r},t\right)=\frac{1}{c}\int\hat{\mathbf{n}}\times\left[-\frac{1}{R^{2}}\,J\left(\mathbf{r}^{\prime},t-\frac{R}{c}\right)\right]\,\mathrm{d}^{3}r^{\prime}.$ (81) Applying Eq. (72) to Eq. (82), we have $\displaystyle\mathbf{B}\left(\mathbf{r}^{\prime},t\right)=\frac{1}{c}\int\left(-\hat{\mathbf{n}}\right)\times\frac{1}{R^{2}}\left[2\mathbf{u}+\left(\mathbf{v}^{\prime}-\mathbf{v}\right)\right]\rho_{e}\left(\mathbf{r}^{\prime}\right)\,\mathrm{d}^{3}r^{\prime}.$ (82) Using Eq. (73) to Eq. (82), we have $\displaystyle\mathbf{B}(\mathbf{r},t)$ $\displaystyle=\frac{1}{c}\int\left(-\hat{\mathbf{n}}\right)$ $\displaystyle\qquad\qquad\times\frac{q\Psi^{}\left(\mathbf{r}^{\prime}\right)\left[2\mathbf{u}+\left(\mathbf{v}^{\prime}-\mathbf{v}\right)\right]\Psi\left(\mathbf{r}^{\prime}\right)}{R^{2}}\,\mathrm{d}^{3}r^{\prime}.$ (83) The magnetic field generated by the new state from the previous state (V.4) becomes $\displaystyle\mathbf{B}\left(\mathbf{r},t\right)$ $\displaystyle=\frac{1}{c}\int\left(-\hat{\mathbf{n}}\right)\times\frac{q}{R^{2}}\Psi^{}\left(\mathbf{r}^{\prime}\right)(\Delta\mathbf{v})\Psi\left(\mathbf{r}^{\prime}\right)\,\mathrm{d}^{3}r^{\prime}$ (84) $\displaystyle\mathbf{B}\left(\mathbf{r},t\right)$ $\displaystyle=-\frac{1}{c}\hat{\mathbf{n}}\times\frac{q\left\langle\Delta\mathbf{v}\right\rangle}{R^{2}},$ (85) where $\mathbf{u}=\mathbf{b}-\mathbf{b}^{~{}}_{}$ and $\Delta\mathbf{v}$ are as in Eq. (65). Eq. (85) is the Biot-Savart law showing that magnetic field is the non isotropic field shown by its dependence on the sinus angle. It is the same as magnetic field depending on the angle between the position vector of location of a short considered segment of wire and the current density vector, the perception of kinetic energy between the previous state and the new state (relative velocity) depends on the angle between both velocities (trajectories) in the two states. When we considered the relation between velocities in the previous and new states as Eq. (65) $\displaystyle\mathbf{v}^{\prime}+\mathbf{b}^{~{}}_{}=\left(\mathbf{v}+\mathbf{b}\right)+\Delta\mathbf{v},$ we have assumed that $\displaystyle\left(\mathbf{v}+\mathbf{b}\right)\perp\Delta\mathbf{v}.$ It means that from every point in the previous state, the particle in the new state always move with direction perpendicular to the position vector of location of a short segment of wire. On another word, the condition of $(\mathbf{v}+\mathbf{b})\perp\Delta\mathbf{v}$ makes us consider the magnetic field as isotropic field. Assume the density of magnetic fields is $\displaystyle U^{~{}}_{\mathrm{mag}}=\frac{E^{~{}}_{\mathrm{mag}}}{V}=\frac{\mathbf{B}^{2}}{2\mu^{~{}}_{0}}.$ (86) We can write Eq. (86) as $\displaystyle\mathrm{d}E^{~{}}_{\mathrm{mag}}=\frac{\mathbf{B}^{2}}{2\mu^{~{}}_{0}}\,\mathrm{d}V.$ (87) Applying the magnetic field as an isotropic field and inserting Eq. (85) into Eq. (87), we have $\displaystyle\mathrm{d}E^{~{}}_{\mathrm{mag}}=\frac{q^{2}\left\langle\Delta\mathbf{v}\right\rangle^{2}}{2c^{2}\mu^{~{}}_{0}R^{4}}\,\mathrm{d}V.$ (88) Using spherical coordinates and assuming the quantum-sized volume has a spherical shape with radius $r^{~{}}_{\mathrm{min}}$ so magnetic energy in the whole space is $\displaystyle E^{~{}}_{\mathrm{mag}}=\left[\frac{q^{2}\left\langle\Delta\mathbf{v}\right\rangle^{2}}{2c^{2}\mu^{~{}}_{0}}\right]2\pi\int\limits_{0}^{\pi}\sin\theta\,\mathrm{d}\theta\int\limits_{r^{~{}}_{\mathrm{min}}}^{\infty}\frac{R^{2}}{R^{4}}\,\mathrm{d}R.$ (89) We find $\displaystyle E^{~{}}_{\mathrm{mag}}=\left[\frac{2\pi q^{2}\left\langle\Delta\mathbf{v}\right\rangle^{2}}{c^{2}\mu^{~{}}_{0}}\right]\frac{1}{r^{~{}}_{\mathrm{min}}}.$ (90) Using relation $1/c=\mu^{~{}}_{0}/4\pi$, Eq. (90) becomes $\displaystyle E^{~{}}_{\mathrm{mag}}=\left(\frac{\mu^{~{}}_{0}q^{2}}{8\pi}\right)\frac{1}{r^{~{}}_{\mathrm{min}}}\left\langle\Delta\mathbf{v}\right\rangle^{2}.$ (91) Eq. (91) must be equivalent to kinetic energy $\displaystyle E^{~{}}_{\mathrm{mag}}=\frac{1}{2}m^{~{}}_{0}\left\langle\Delta\mathbf{v}\right\rangle^{2}.$ So it should prevail that $\displaystyle m^{~{}}_{\mathrm{magnetic}}=\left(\frac{\mu^{~{}}_{0}q^{2}}{4\pi}\right)\frac{1}{r_{\mathrm{min}}}=m_{0}.$ (92) Substitute the values of physical constant to Eq. (92), where permeability in vacuum $(\mu^{~{}}_{0}\,=\,4\,\pi\,\times 10^{-7}\,$ $\mathrm{Wb/A\cdot m})$, electron charge $(q=1.602189\times 10^{-19}\,\mathrm{C})$, and the chosen $r_{\mathrm{min}}$ is the classical radius of electron $\left(r^{~{}}_{\mathrm{min}}=2.8179403\times 10^{-15}\,\mathrm{m}\right)$, and we obtain: $\displaystyle m_{\mathrm{mag}}=\left(\frac{\mu_{0}q^{2}}{4\pi}\right)\frac{1}{r_{\mathrm{min}}}=9.10952\times 10^{-31}\,\mathrm{kg}.$ (93) Eq. (93) shows that the magnetic mass is the same value as the rest mass, but this quantity is distributed to whole space of outside of the quantum-sized volume. The important result of this derivation is that classical radius of the electron must represent the radius of the quantum-sized volume. The particle that randomly move in the quantum-sized volume may be what Feynman refers to as a fuzzy ball RPF-RPL-MS-1963 . Furthermore, we will show that the terms after the term $m_{0}\left\langle\Delta\mathbf{v}\right\rangle^{2}/2$ $\displaystyle\frac{3}{8}m^{~{}}_{0}\frac{(\Delta\mathbf{v})^{4}}{c^{2}}+\frac{5}{16}m^{~{}}_{0}\frac{(\Delta\mathbf{v})^{6}}{c^{2}}+\frac{35}{128}m^{~{}}_{0}\frac{(\Delta\mathbf{v})^{8}}{c^{6}}+\cdots,$ (94) must represent the transition energy from previous state to new state. If we assume that pure-energy or the total emitted-energy in any transition process should be the same (invariant) from every relative-rest frame and the total emitted energy must be equivalent to the generated-electromagnetic-field energy spreading all over space from every relative-rest frame. Fundamentally, existing electromagnetic fields that coincide with the transition process should represent the change of the tendency of particles to move away or towards each other and this can be represented by the change of both position and velocity of electron when the electron makes the transition. It will be equivalent to the change of electric and magnetic fields in time and it will be observed as electromagnetic field. We consider the magnetic field from restively rest frame to both the previous and the new states. The magnetic field generated by the change of current when the transition process occurred, from Eq. (78), is $\displaystyle\mathbf{B}\left(\mathbf{r},t\right)=\frac{1}{c}\int\hat{\mathbf{n}}\times\left(-\frac{1}{Rc}\right)\mathbf{J}\left(\mathbf{r}^{\prime},t-\frac{R}{c}\right)\,\mathrm{d}^{3}r^{\prime}.$ (95) Following procedure and results by Davidson MPD-2004 having model from Schrödinger picture to Heisenberg picture to accommodate the evolution of momentum operator in time $\displaystyle\int\Psi^{}$ $\displaystyle\left(\mathbf{r}^{\prime},t\right)\,\mathbf{a}\,\Psi\left(\mathbf{r}^{\prime},t\right)\,\mathrm{d}^{3}t\,\,\,$ $\displaystyle\Rightarrow\int\Psi^{}\left(\mathbf{r}^{~{}}_{\mathrm{H}},t\right)\,\mathbf{a}\,\Psi\left(\mathbf{r}^{~{}}_{\mathrm{H}},t\right)\,\mathrm{d}^{3}r^{~{}}_{\mathrm{H}},$ (96) where $\mathbf{r}^{~{}}_{\mathrm{H}}$ symbolizing representation position outside of the quantum-sized volume and considering the first order of Larmor’s radiation, Davidson MPD-2004 finds that $\displaystyle\mathbf{B}\left(\mathbf{r},t\right)$ $\displaystyle=-\frac{q}{c^{2}R^{~{}}_{0}}\hat{\mathbf{n}}\times\int\Psi^{}\left(\mathbf{r}_{\mathrm{H}},t\right)\mathbf{a}\Psi\left(\mathbf{r}_{\mathrm{H}},t\right)\,\mathrm{d}^{3}r$ $\displaystyle=-\frac{q}{c^{2}R^{~{}}_{0}}\hat{\mathbf{n}}\times\left\langle\Psi\|\mathbf{a}\|\Psi\right\rangle.$ (97) In evaluating the radiation emitted, the limit where $R^{~{}}_{0}=\|\mathbf{r}\|\rightarrow\infty$ is taken MPD-2004 . The transition of the electron from one state with kinetic terms to another state with certain other kinetic terms, accompanied by the change of the velocity of both quantum-sized volume and internal random velocities, from observer frame, can be considered as the movement of an electrical circuit or medium. So, by applying Faraday’s law for the movement of a circuit, to the observer the electric field will be measured as RKW-1979 $\displaystyle\mathbf{E}=\mathbf{v}^{~{}}_{e}\times\mathbf{B},$ (98) where $\mathbf{v}^{~{}}_{e}$ is the velocity of a medium or circuit. The time interval of an electron for making transition process is $t^{~{}}_{\mathrm{transition}}=t^{~{}}_{i}+t^{~{}}_{j}$ where $t^{~{}}_{i}$ is the time interval needed by the electron to move from one point in the previous state to another point in the new state, and $t^{~{}}_{j}$ is the time interval needed by electron to complete a period of motion in the new state. $t^{~{}}_{j}$ consists of a motion period of the quantum-sized volume and or a period of internal random motion in new state. Thus, if $t^{~{}}_{i}$ and $t^{~{}}_{j}$ are the same order, electron making the transition process with interval time $t^{~{}}_{\mathrm{transition}}$ can be considered as the motion of circuit (wire) or medium with the time interval $t^{~{}}_{\mathrm{transition}}/2$ or with the velocity $\displaystyle\mathbf{v}^{~{}}_{e}\approx 2\Delta\mathbf{v}.$ (99) In regions which are far from any charge or current, the relation between electric and magnetic fields in vacuum is MPD-2004 $\displaystyle\mathbf{E}=-\hat{\mathbf{n}}\times\mathbf{B}.$ (100) Substitute Eqs. (99) and (100) into Eq. (V.4), and we obtain Poynting vector $\displaystyle\mathbf{S}=\frac{c}{4\pi}\mathbf{E}\times\mathbf{B}=\frac{q^{2}\|\Delta\mathbf{v}\|}{2\pi c^{3}R^{2}_{0}}\left\langle\Psi\|\mathbf{a}\|\Psi\right\rangle^{2}\left(\sin^{3}\theta\right)\hat{\mathbf{n}},$ (101) where $\theta$ is the angle between $\left\langle\Psi\|\mathbf{a}\|\Psi\right\rangle$ and $\hat{\mathbf{n}}$ which is the same angle as between $\Delta\mathbf{v}$ and $\mathbf{B}$. The total radiated-power is $\displaystyle P_{\mathrm{rad}}=\int\limits_{0}^{\pi}\frac{q^{2}\|\Delta\mathbf{v}\|}{2\pi c^{3}R^{2}_{0}}\left\langle\Psi\|\mathbf{a}\|\Psi\right\rangle^{2}\left(\sin^{3}\theta\right)\left(2\pi R^{2}_{0}\sin\theta\right)\,\mathrm{d}\theta.$ (102) Computing Eq. (102), we find $\displaystyle P^{~{}}_{\mathrm{rad}}=\frac{3}{8}\frac{q^{2}\|\Delta\mathbf{v}\|}{c^{3}}\left\langle\Psi\|\mathbf{a}\|\Psi\right\rangle^{2}.$ (103) To find the total radiated-energy, we integrate Eq. (103) covering total transition time $\displaystyle E^{~{}}_{\mathrm{rad}}=\frac{3}{8}\frac{q^{2}\|\Delta\mathbf{v}\|}{c^{3}}\int_{0}^{t^{~{}}_{\mathrm{transition}}}\left\langle\Psi\|\mathbf{a}\|\Psi\right\rangle^{2}\,\mathrm{d}t.$ (104) Since $t^{~{}}_{\mathrm{transition}}\approx r^{~{}}_{\mathrm{min}}/\|\Delta\mathbf{v}\|$, we can approximately approach the result of Eq. (104) with $\displaystyle E^{~{}}_{\mathrm{rad}}$ $\displaystyle\approx\frac{3}{8}\frac{q^{2}\|\Delta\mathbf{v}\|}{c^{3}}\left\langle\Psi\|\mathbf{a}\|\Psi\right\rangle^{2}\frac{r^{~{}}_{\mathrm{min}}}{\|\Delta\mathbf{v}\|}$ $\displaystyle\approx\frac{3}{8}\frac{q^{2}\|\Delta\mathbf{v}\|}{c^{3}}\left[\|\Delta\mathbf{v}\|\frac{\|\Delta\mathbf{v}\|}{r_{\mathrm{min}}}\right]^{2}\frac{r_{\mathrm{min}}}{\|\Delta\mathbf{v}\|}$ $\displaystyle\approx\left(\frac{3}{8}\frac{q^{2}\|\Delta\mathbf{v}\|^{4}}{c^{3}}\right)\frac{1}{r^{~{}}_{\mathrm{min}}}.$ (105) Applying relation $1/c=\mu^{~{}}_{0}/4\pi$, we rewrite Eq. (V.4) as $\displaystyle E^{~{}}_{\mathrm{rad}}\approx\frac{3}{8}\left(\frac{\mu^{~{}}_{0}q^{2}}{4\pi}\frac{1}{r^{~{}}_{\mathrm{min}}}\right)\frac{(\Delta\mathbf{v})^{4}}{c^{2}}=\frac{3}{8}m_{0}\frac{(\Delta\mathbf{v})^{4}}{c^{2}}.$ (106) Eq. (106) shows the first order of Larmor’s radiation represents the second terms of $\displaystyle\left(m-m^{~{}}_{0}\right)c^{2}$ $\displaystyle=m^{~{}}_{0}\left[\frac{1}{\sqrt{1-(\Delta\mathbf{v})^{2}/c^{2}}}-1\right]c^{2}$ $\displaystyle=\frac{1}{2}m^{~{}}_{0}(\Delta\mathbf{v})^{2}+\frac{3}{8}m^{~{}}_{0}\frac{(\Delta\mathbf{v})^{4}}{c^{2}}+\frac{5}{16}m^{~{}}_{0}\frac{(\Delta\mathbf{v})^{6}}{c^{4}}$ $\displaystyle\qquad+\frac{35}{128}m^{~{}}_{0}\frac{(\Delta\mathbf{v})^{2}}{c^{6}}+\cdots$ This result proves that Eq. (70), besides representing the magnetic energy caused by the emergence of kinetic energy (the loss of the particle’s tendency to approach each other is equivalent to the increase of the particle’s tendency to keep away each other) must also represent the radiation energy when the transition process takes place between two states. The general form of magnetic field of the accelerated particle MPD-2004 is $\displaystyle\mathbf{B}\left(\mathbf{r}^{\prime},t\right)$ $\displaystyle=-\hat{\mathbf{n}}\times\frac{1}{c^{2}R^{~{}}_{0}}\sum\limits_{p=1}^{\infty}\int\frac{1}{(p-1)!}$ $\displaystyle\qquad\qquad\left[\dfrac{\partial^{p}}{\partial t^{p}}\,\mathbf{J}\left(\mathbf{r}^{\prime},t-\dfrac{R^{~{}}_{0}}{c}\right)\right]\left[\dfrac{\hat{\mathbf{n}}\cdot\mathbf{r}^{\prime}}{c}\right]^{p-1}\,\mathrm{d}^{3}r^{\prime}.$ This equation can be rewritten with using electrical current observable as $\displaystyle\mathbf{B}\left(\mathbf{r},t\right)=-\hat{\mathbf{n}}\times\frac{1}{c^{2}R^{~{}}_{0}}\sum\limits_{p=1}^{\infty}\frac{1}{(p-1)!}\frac{\partial^{p}}{\partial t^{p}_{0}}\,\mathbf{I}^{~{}}_{p}\left(t^{~{}}_{0}\right)\left(\frac{1}{c}\right)^{p-1},$ (107) where $\displaystyle\mathbf{I}_{p}\left(t_{0}\right)=\int\mathbf{J}\left(\mathbf{r}^{\prime},t_{0}\right)\left(\frac{\hat{\mathbf{n}}\cdot\mathbf{r}^{\prime}}{c}\right)^{p-1}\,\mathrm{d}^{3}r^{\prime};\qquad t_{0}=t-\frac{R^{~{}}_{0}}{c}.$ (108) Doing transformation into Heisenberg picture, Davidson MPD-2004 found $\displaystyle\mathbf{I}_{p}\left(t^{~{}}_{0}\right)$ $\displaystyle=\left[\frac{q}{2M}\int\Psi^{}\left(\mathbf{r}^{~{}}_{\mathrm{H}},0\right)\left\\{\mathbf{P}^{~{}}_{u}\left(t^{~{}}_{0}\right)\left[\hat{\mathbf{n}}\cdot\mathbf{R}^{~{}}_{v}\left(t^{~{}}_{0}\right)\right]^{p-1}\right.\right.$ $\displaystyle\qquad+\left.\left.\left[\hat{\mathbf{n}}\cdot\mathbf{R}^{~{}}_{v}\left(t^{~{}}_{0}\right)\right]^{p-1}\mathbf{P}^{~{}}_{u}\left(t^{~{}}_{0}\right)\right\\}\Psi\left(\mathbf{r}^{~{}}_{\mathrm{H}},0\right)\,\mathrm{d}^{3}r^{~{}}_{\mathrm{H}}\right.\bm{\biggl{]}}$ $\displaystyle\qquad+\left[\frac{q}{M}\int\Psi^{}\left(\mathbf{r}^{~{}}_{\mathrm{H}},0\right)\left\\{P_{qv}\left[\hat{\mathbf{n}}\cdot\mathbf{R}^{~{}}_{v}\left(t^{~{}}_{0}\right)\right]^{p-1}\right\\}\right.$ $\displaystyle\qquad\qquad\Psi\left(\mathbf{r}^{~{}}_{\mathrm{H}},0\right)\,\mathrm{d}^{3}r^{~{}}_{\mathrm{H}}\bm{\bigg{]}}.$ (109) Linking Eq. (V.4) to the calculation of the second term of Eq. (85), we can conclude that the other terms $\displaystyle\frac{5}{16}m^{~{}}_{0}\frac{(\Delta\mathbf{v})^{6}}{c^{4}}+\frac{35}{128}m^{~{}}_{0}\frac{(\Delta\mathbf{v})^{2}}{c^{6}}+\cdots$ must be served by $\displaystyle\mathbf{B}\left(\mathbf{r}^{\prime},t\right)$ $\displaystyle=-\hat{\mathbf{n}}\times\frac{1}{c^{2}R^{~{}}_{0}}\sum\limits_{p=1}^{\infty}\int\frac{1}{(p-1)!}$ $\displaystyle\qquad\qquad\left[\dfrac{\partial^{p}}{\partial t^{p}}\,\mathbf{J}\left(\mathbf{r}^{\prime},t-\dfrac{R^{~{}}_{0}}{c}\right)\right]\left[\dfrac{\hat{\mathbf{n}}\cdot\mathbf{r}^{\prime}}{c}\right]^{p-1}\,\mathrm{d}^{3}r^{\prime},$ (110) with $p>1$. From applying the invariance of radiated energy, our proposal indicates that the quantization of a classical wave must represent the scheme of second quantization. ### V.5 Radiated-energy in spin transition Eq. (70) is the general term of radiation for the transition process including the change of both internal random motion and the quantum-sized volume velocities. If we only see the transition when the velocity of the quantum- sized volume does not change or the transition process only represents spin- state transition (the change of internal random motion), we get that magnetic and radiated-energy of transition process still keep the form $\displaystyle\left(m-m^{~{}}_{0}\right)$ $\displaystyle=m^{~{}}_{0}\left(1-\frac{1}{\sqrt{1-(\mathbf{b}_{}-\mathbf{b})^{2}/c^{2}}}\right)c^{2}$ $\displaystyle=\frac{1}{2}m^{~{}}_{0}\left(\mathbf{b}^{~{}}_{}-\mathbf{b}\right)^{2}+\frac{3}{8}m^{~{}}_{0}\frac{\left(\mathbf{b}^{~{}}_{}-\mathbf{b}\right)}{c^{2}}$ $\displaystyle\qquad+\frac{5}{16}m^{~{}}_{0}\frac{\left(\mathbf{b}^{~{}}_{}-\mathbf{b}\right)}{c^{4}}+\cdots,$ (111) where it still prevails that $\displaystyle E^{~{}}_{\mathrm{mag}}=\frac{1}{2}m^{~{}}_{0}\left\langle\mathbf{b}_{}-\mathbf{b}\right\rangle^{2}=\left(\frac{\mu^{~{}}_{0}q^{2}}{8\pi}\right)\frac{1}{r^{~{}}_{\mathrm{min}}}\left\langle\mathbf{b}^{~{}}_{}-\mathbf{b}\right\rangle.$ (112) with $r^{~{}}_{\mathrm{min}}$ is the classical radius of electron, and $\displaystyle E^{~{}}_{\mathrm{rad}}\approx\frac{3}{8}\left(\frac{\mu^{~{}}_{0}q^{2}}{4\pi}\frac{1}{r_{\mathrm{min}}}\right)^{~{}}_{\mathrm{rad}}\frac{\left\langle\mathbf{b}^{~{}}_{}-\mathbf{b}\right\rangle^{4}}{c^{2}}=\frac{3}{8}m^{~{}}_{0}\frac{\left\langle\mathbf{b}^{~{}}_{}-\mathbf{b}\right\rangle^{4}}{c^{2}}.$ (113) This prevails for firs order of radiated-energy. ### V.6 Spin-spin and spin-orbit interactions Here, we discuss the interaction potentials that accompany the process of transition between two states. We consider the case in which an emission process is accompanied by the emergence of internal spinning motion when the transition process that is taking place coincides with internal diffusion process. Spin or internal spinning motion is shown by the emergence of velocity $\displaystyle\mathbf{u}=\frac{\hslash}{2m}\frac{\nabla\rho}{\rho}.$ As in Eq. (38), this velocity describes the change of the velocity of internal motion in the quantum-sized volume when a transition process takes place. We see interaction potentials accompanying the transition process for the kinds of interaction in which the velocity of the quantum-sized volume does not change. There are two kinds of interactions for this transition. One is the energy-emitting process that makes the random internal velocity decrease (positive diffusion) and the other is the energy-absorbing process that makes the random internal velocity increase. From Eq. (71), relative to the nucleus, the current density of an electron is $\displaystyle\mathbf{J}^{~{}}_{0}\left(\mathbf{r}^{\prime},t\right)=\left(\mathbf{b}+\mathbf{v}\right)\rho_{e}\left(\mathbf{r}^{\prime},t\right).$ This current will generate the magnetic field $\displaystyle\mathbf{B}\left(\mathbf{r},t\right)$ $\displaystyle=\nabla\times\mathbf{A}$ $\displaystyle=\nabla\times\frac{1}{c}\int\frac{1}{R}\,\mathbf{J}^{~{}}_{0}\left(\mathbf{r}^{\prime},t-\dfrac{R}{c}\right)\,\mathrm{d}^{3}r^{\prime}.$ (114) Insert the current density into Eq. (V.6), and we have $\displaystyle\mathbf{B}\left(\mathbf{r},t\right)$ $\displaystyle=\nabla\times\mathbf{A}$ $\displaystyle=\frac{1}{c}\int\frac{1}{R}\,\left\\{\nabla\rho_{e}\left(\mathbf{r}^{\prime},t-\dfrac{R}{c}\right)\times\left(\mathbf{b}+\mathbf{v}\right)\right.$ $\displaystyle\qquad+\rho_{e}\left.\left(\mathbf{r}^{\prime},t-\dfrac{R}{c}\right)\left[\nabla\times\left(\mathbf{b}+\mathbf{v}\right)\right]\right\\}\,\mathrm{d}^{3}r^{\prime}.$ (115) We may rewrite Eq. (V.6) as $\displaystyle\mathbf{B}\left(\mathbf{r},t\right)$ $\displaystyle=\nabla\times\mathbf{A}$ $\displaystyle=\frac{1}{c}\int\frac{1}{R}\,\left[\nabla\rho_{e}\left(\mathbf{r}^{\prime},t-\dfrac{R}{c}\right)\times\left(\mathbf{b}+\mathbf{v}\right)\right.$ $\displaystyle\qquad+\rho_{e}\left.\left(\mathbf{r}^{\prime},t-\dfrac{R}{c}\right)\hat{\mathbf{n}}\times\frac{\partial}{\partial R}\left(\mathbf{b}+\mathbf{v}\right)\right]\,\mathrm{d}^{3}r^{\prime}.$ (116) If the electron emits energy (i.e. it undergoes a transition process) and its velocities $(\mathbf{b},\mathbf{v})$ do not change in relation to position $R$, so the magnetic field generated by the change of the distribution of charge density $\rho_{e}$ is $\displaystyle\mathbf{B}\left(\mathbf{r},t\right)$ $\displaystyle=\nabla\times\mathbf{A}$ $\displaystyle=\frac{1}{c}\int\frac{1}{R}\,\nabla\rho_{e}\left(\mathbf{r}^{\prime},t-\dfrac{R}{c}\right)\times\left(\mathbf{b}+\mathbf{v}\right)\,\mathrm{d}^{3}r^{\prime}.$ (117) Eq. (V.6) may be written as $\displaystyle\mathbf{B}\left(\mathbf{r},t\right)$ $\displaystyle=\nabla\times\mathbf{A}$ $\displaystyle=\frac{1}{c^{2}}\int\frac{1}{R}\,\dot{\rho}_{e}\left(\mathbf{r},t-\dfrac{R}{C}\right)\times\left(\mathbf{b}+\mathbf{v}\right)\,\mathrm{d}^{3}r^{\prime}.$ (118) If we only focus on the diffusion process that relates to the change of internal random motion (the density of diffusion current), with using (35), Eq. (V.6) becomes $\displaystyle\mathbf{B}\left(\mathbf{r},t\right)$ $\displaystyle=\nabla\times\mathbf{A}$ $\displaystyle=\frac{1}{c^{2}}\int\frac{1}{R}\,\left\\{\frac{\hslash}{2m}\nabla^{2}\left[\rho_{e}\left(\mathbf{r}^{\prime},t-\dfrac{R}{c}\right)\right]\right\\}$ $\displaystyle\qquad\qquad\times\left(\mathbf{b}+\mathbf{v}\right)\,\mathrm{d}^{3}r^{\prime}.$ (119) Recall Eq. (32) and (33) $\displaystyle\rho_{m}=m\Psi^{}\left(\mathbf{r}^{\prime},t\right)\Psi\left(\mathbf{r}^{\prime},t\right);\qquad\rho_{e}=q\Psi^{}\left(\mathbf{r}^{\prime},t\right)\Psi\left(\mathbf{r}^{\prime},t\right).$ (120) Applying (120) into Eq. (V.6), we obtain $\displaystyle\mathbf{B}\left(\mathbf{r},t\right)$ $\displaystyle=\nabla\times\mathbf{A}$ $\displaystyle=\frac{q}{mc^{2}}\int\frac{1}{R}\,\left\\{\frac{\hslash}{2m}\nabla^{2}\left[\rho_{m}\left(\mathbf{r}^{\prime},t-\dfrac{R}{c}\right)\right]\right\\}$ $\displaystyle\qquad\qquad\times\left(\mathbf{b}+\mathbf{v}\right)\,\mathrm{d}^{3}r^{\prime}.$ (121) Inserting $\mathbf{u}=\left(\hslash/2m\right)\left(\nabla\rho\right)/\rho$ into Eq. (V.6), we obtain $\displaystyle\mathbf{B}\left(\mathbf{r},t\right)$ $\displaystyle=\nabla\times\mathbf{A}$ $\displaystyle=\frac{q}{mc^{2}}\int\frac{1}{R}\,\left\\{\nabla\cdot\left[\mathbf{u}\rho_{m}\left(\mathbf{r}^{\prime},t-\dfrac{R}{c}\right)\right]\right\\}$ $\displaystyle\qquad\qquad\times\left(\mathbf{b}+\mathbf{v}\right)\,\mathrm{d}^{3}r^{\prime}.$ (122) If the distribution of the $\mathbf{u}$ field does not relatively change to the internal position, Eq. (V.6) may be rewritten by $\displaystyle\mathbf{B}\left(\mathbf{r},t\right)$ $\displaystyle=\nabla\times\mathbf{A}$ $\displaystyle=\frac{q}{mc^{2}}\int\frac{1}{R}\left\\{\mathbf{u}\cdot\left[\nabla\rho_{m}\left(\mathbf{r}^{\prime},t-\dfrac{R}{c}\right)\right]\right\\}$ $\displaystyle\qquad\qquad\times\left(\mathbf{b}+\mathbf{v}\right)\,\mathrm{d}^{3}r^{\prime}.$ (123) If we move from the Schrödinger picture to Heisenberg picture and ignore the distribution of mass in space, Eq. (V.6) may be written by $\displaystyle\mathbf{B}\left(\mathbf{r},t\right)$ $\displaystyle=\nabla\times\mathbf{A}$ $\displaystyle=\frac{q}{mc^{2}}\frac{h}{2}\frac{\nabla\rho}{\rho}\int\frac{1}{R}\,\left[\nabla\Psi^{}\left(\mathbf{r}^{~{}}_{\mathrm{H}},t\right)\Psi\left(\mathbf{r}^{~{}}_{\mathrm{H}},t\right)\right]$ $\displaystyle\qquad\qquad\qquad\times\left(\mathbf{b}+\mathbf{v}\right)\,\mathrm{d}^{3}r^{~{}}_{\mathrm{H}}.$ (124) Furthermore, we propose that $\displaystyle\int\nabla\Psi^{}\left(\mathbf{r}^{~{}}_{\mathrm{H}},t\right)\Psi\left(\mathbf{r}^{~{}}_{\mathrm{H}},t\right)\,\mathrm{d}^{3}r^{~{}}_{\mathrm{H}}=\frac{1}{4\pi\epsilon_{0}R^{2}}.$ (125) So we obtain $\displaystyle\mathbf{B}\left(\mathbf{r},t\right)$ $\displaystyle=\nabla\times\mathbf{A}$ $\displaystyle=\frac{q}{mc^{2}}\frac{1}{4\pi\epsilon^{~{}}_{0}R^{3}}\left(\dfrac{h}{2}\dfrac{\nabla\rho}{\rho}\right)\times\left(\mathbf{b}+\mathbf{v}\right).$ (126) From electron frame, the kinetic energy of both the nucleus and the observer will relatively increase because the emitting of energy makes its internal kinetic energy (internal motion velocity in the quantum-sized volume) decrease. The increase of the kinetic energy of nucleus or observer from electron frame will responded by electron with the loss of its tendency to come closer to nucleus where the loss of this tendency will represent the electron in magnetic field. By contrast, the nucleus and observer will see the decrease of the kinetic energy of electron as the increase of the particle’s tendency to get closer to the nucleus. Another possibility is that the nucleus and observer will see Eq. (V.6) as electric field $\displaystyle\mathbf{E}\left(\mathbf{r},t\right)=\frac{q}{mc^{2}}\frac{1}{4\pi\epsilon^{~{}}_{0}R^{3}}\left(\dfrac{h}{2}\dfrac{\nabla\rho}{\rho}\right)\times\left(\mathbf{b}+\mathbf{v}\right).$ (127) At the nucleus frame, interaction potential between the electron and nucleus will be the same as work $\displaystyle V^{\mathrm{interaction}}$ $\displaystyle=-\int_{\infty}^{R}\mathbf{F}^{~{}}_{c}\cdot\mathrm{d}\mathbf{r}$ $\displaystyle=\frac{q^{2}}{4\pi\epsilon^{~{}}_{0}mc^{2}}\left[\left(\frac{h}{2}\frac{\nabla\rho}{\rho}\right)\times\left(\mathbf{b}+\mathbf{v}\right)\right]\cdot\int_{\infty}^{R}\frac{\mathrm{d}\mathbf{r}}{R^{3}}$ $\displaystyle=-\frac{q}{mc^{2}}\left[\left(\frac{h}{2}\frac{\nabla\rho}{\rho}\right)\times\left(\mathbf{b}+\mathbf{v}\right)\right]\cdot\frac{1}{4\pi\epsilon^{~{}}_{0}}\frac{q}{R^{2}}\,\hat{\mathbf{r}}.$ (128) Eq. (V.6) can be rewritten by $\displaystyle V^{\mathrm{so}}=-\frac{q}{mc^{2}}\left[\left(\frac{h}{2}\frac{\nabla\rho}{\rho}\right)\times\left(\mathbf{b}+\mathbf{v}\right)\right]\cdot\mathbf{E}$ (129) We choose $\left(h/2\right)\left(\nabla\rho\right)/\rho=\mathbf{P}^{~{}}_{S}$ to be the internal momentum that emerges when any transition process takes place and connects to the transition velocity $\mathbf{u}$. Eq. (129) may be rewritten by $\displaystyle V^{\mathrm{interaction}}$ $\displaystyle=-\frac{q}{mc^{2}}\left[\mathbf{P}^{~{}}_{S}\times\left(\mathbf{b}+\mathbf{v}\right)\right]\cdot\mathbf{E}$ $\displaystyle=-\frac{q}{mc^{2}}\mathbf{P}^{~{}}_{S}\cdot\left[\left(\mathbf{b}+\mathbf{v}\right)\times\mathbf{E}\right].$ (130) Eq. (V.6) expresses Biot-Savart law $\displaystyle\mathbf{B}$ $\displaystyle=-\frac{\left[\left(\mathbf{b}+\mathbf{v}\right)\times\mathbf{E}\right]}{c}$ $\displaystyle=-\left[\frac{\left(\mathbf{v}\times\mathbf{E}\right)}{c}+\frac{\left(\mathbf{b}\times\mathbf{E}\right)}{c}\right]$ $\displaystyle=\mathbf{B}^{~{}}_{0}+\mathbf{B}^{~{}}_{s},$ (131) where $\mathbf{B}^{~{}}_{0}$ is the magnetic field generated by orbital motion, and $\mathbf{B}^{~{}}_{s}$ is the magnetic field generated by internal motion. We re-express Eq. (V.6) as $\displaystyle V^{\mathrm{interaction}}=V^{\mathrm{so}}+V^{\mathrm{ss}}=\frac{q}{mc}\mathbf{P}^{~{}}_{S}\cdot\mathbf{B}^{~{}}_{0}+\frac{q}{mc}\mathbf{P}^{~{}}_{S}\cdot\mathbf{B}^{~{}}_{s},$ (132) where $V^{\mathrm{so}}$ and $V^{\mathrm{ss}}$ respectively are potentials for spin-orbit and spin-spin interactions. ### V.7 The electron in external magnetic field In our model has proposed that before the external treatment, electrons have been at stationer state with internal potential $V$. When an external field is presen t, it will make an electron undergo the transition process demonstrated by the equation $\displaystyle\frac{1}{2}\left(DD^{~{}}_{}+D^{~{}}_{}D\right)\mathbf{x}(t)$ $\displaystyle=\mathbf{a}(t)$ $\displaystyle=-\frac{1}{m}\nabla\left(V+V^{~{}}_{\mathrm{external}}\right)$ $\displaystyle=\frac{1}{m}\left(\mathbf{F}^{~{}}_{\mathrm{system}}+\mathbf{F}^{~{}}_{\mathrm{external}}\right)$ If the electron has been in a physical system with the potential ($e\mathbf{E}$) such as in a Coulomb potential with a nucleus, when one applies an external magnetic field the electron will obey motion equations such as Eqs. (39) and (V) that are $\displaystyle\frac{\partial\mathbf{u}}{\partial t}=-\beta\nabla\left[\nabla\cdot\left(\bm{\upsilon}+\mathbf{v}\right)\right]-\nabla\left[\mathbf{u}\cdot\left(\bm{\upsilon}+\mathbf{v}\right)\right]$ (133) $\displaystyle\frac{\partial(\bm{\upsilon}+\mathbf{v})}{\partial t}$ $\displaystyle=\frac{e}{m}\left[\mathbf{E}+\left(\frac{1}{c}\right)\left(\bm{\upsilon}+\mathbf{v}\right)\times\mathbf{H}\right]+(\mathbf{u}\cdot\nabla)\mathbf{u}$ $\displaystyle\qquad\qquad+\frac{\hslash}{2m}\nabla\mathbf{u}-\left(\bm{\upsilon}\cdot\nabla\right)\left(\bm{\upsilon}+\mathbf{v}\right),$ (134) where $\displaystyle\mathbf{F}^{~{}}_{\mathrm{external}}=\mathbf{F}^{~{}}_{\mathrm{magnetic}}=\left(\frac{1}{c}\right)\left(\bm{\upsilon+\mathbf{v}}\right)\times\mathbf{H}.$ (135) Similar to the time-dependent Schrödinger equation, there are two general cases for this treatment. For the case in which $\mathbf{v}=0$, Eqs. (133) and (V.7) are equivalent to the Schrödinger equation $\displaystyle i\hslash\frac{\partial\Psi}{\partial t}=\frac{1}{2m}\left(-i\hslash\nabla+\frac{e}{c}\mathbf{A}\right)^{2}\Psi+e\phi\Psi,$ (136) with $\Psi=e^{R+iS}$ and $S$ fulfill $\nabla S=\left(m/\hslash\right)\left[\bm{\upsilon}+e\mathbf{A}/mc\right]$. Whereas for the cases where $\mathbf{v}=0$, Eqs. (133) and (V.7) are equivalent to the modified-Schrödinger equation $\displaystyle i\hslash\frac{\partial\Psi}{\partial t}$ $\displaystyle=\frac{1}{2m}\left(-i\hslash\nabla+\frac{e}{c}\mathbf{A}\right)^{2}\Psi$ $\displaystyle\qquad+\left[e\phi+E^{~{}}_{k}\int m\left(\bm{\upsilon}\nabla\cdot\mathbf{v}\right)\,\mathrm{d}^{3}r^{\prime}\right]\Psi,$ (137) where $E^{~{}}_{k}=mv^{2}/2$, $\Psi=e^{R+iS}$ with $\nabla S=\left(m/\hslash\right)$ $\left[\left(\bm{\upsilon}+\mathbf{v}\right)+\left(e/mc\right)\mathbf{A}\right]$ and $E^{~{}}_{k}=mv^{2}/2$. For a special case when $\mathbf{u}$ is a solenoidal vector field ($\nabla\cdot\mathbf{v}=0$) then the Eq. (V.7) becomes $\displaystyle i\hslash\frac{\partial\Psi}{\partial t}=\frac{1}{2m}\left(-i\hslash\nabla+\frac{e}{c}\mathbf{A}\right)^{2}\Psi\left(e\phi+E^{~{}}_{k}\right)\Psi$ (138) ## VI DISCUSSION The main point of the development of our model is that the displacement of the particle in a non-stationery case not only is contributed by both the diffusion process and the velocity field of internal random motion as classical Brownian motion $\displaystyle\mathrm{d}\mathbf{x}(t)=\mathbf{b}\bm{(}\mathbf{x}(t),t\bm{)}+\mathrm{d}\mathbf{W}(t),$ where $\mathbf{W}(t)$ is the Wiener process, but also is contributed to by the movement of the quantum-sized volume. As proposed by Recami and Salesi ER- GS-1998 , the movement of the quantum-sized volume is a ”classical” part of motion while the internal random motion is the ”quantum” part of motion. The internal random motion only changes when the transition processes (internal diffusion processes) occur where these processes are always accompanied by emitting or absorbing energies and will generate spinning motion in the quantum-sized volume. The internal random velocity field has maximum value at the speed of light when there is an interaction perspective (with the physical system or environment), which ensures that there are no nodal surface for every stationary state. In Nelson’s work EN-1966 , current velocity will be generated only by the diffusion process, so it makes the increase of current velocity strongly determine the smoothness of $\bm{\upsilon}$, $\mathbf{u}$, and $\rho$ (i.e. how the velocities spread in space). Additionally it indicates that the model faces difficulty in covering the transition process at a high velocity. The motion of the quantum-sized volume coupled with the change of internal mass/charge/probability density is actually posed in order to ensure the velocity fields ($\bm{\upsilon}$ and $\mathbf{u}$) and the density probability ($\rho$) become the smooth functions for both the stationer and the transition process. This approach is totally different from conventional Brownian motion models, which view the displacement of the particle for time-dependent cases and this is only generated by the diffusion process and velocity of Brownian motion. In our model, we also pose that random motion is limited in the quantum-sized volume. This is in order to support the realistic explanation of quantum mechanics that has to meet at least one of three requirements of the charge of a particle concentrated in a small volume of space as stated by Jung KJ-2009 . The probability density of finding the particle in the quantum-sized volume will change due to the transition process where it will be determined by the changes of the quantum-sized volume, random motion and diffusion velocities. However the total probability density of all space in the quantum-sized volume (the probability of finding the particle in the quantum-sized volume) as always one. Furthermore, because we can interchange probability density with charge and mass densities, the fact that the total number of probability density of finding the particle in the quantum-sized volume is always the same makes the total of mass and charge in the quantum-sized volume is always constant ($m$ and $q$), whatever the mass and charge density distributed in the quantum-sized volume. This property strongly supports our concepts on mass in that it must be invariant in every state. Mass, quantity, even though it is always constant at every state in both stationary and transition processes, can be different if we consider two states when the particle evolves from one stationary state to another. A new state occupied by a transited-electron with higher kinetic energy will get the rest-mass perspective relative to its left-state. The energy being equal to the difference of mass between previous state (the state that is left by the particle) and new state (the state is occupied by the particle) is $\displaystyle\left(m-m^{~{}}_{0}\right)c^{2}$ $\displaystyle=m^{~{}}_{0}\left(\frac{1}{\sqrt{1-(\Delta\mathbf{v})^{2}/c^{2}}}\right)c^{2}$ $\displaystyle=\frac{1}{2}m^{~{}}_{0}(\Delta\mathbf{v})^{2}+\frac{3}{8}m^{~{}}_{0}\frac{\left(\Delta\mathbf{v}\right)^{4}}{c^{2}}+\frac{5}{16}m^{~{}}_{0}\frac{\left(\Delta\mathbf{v}\right)^{6}}{c^{4}}$ $\displaystyle\qquad+\frac{35}{128}m^{~{}}_{0}\frac{\left(\Delta\mathbf{v}\right)^{8}}{c^{6}}+\cdots$ This energy, as shown above, will represent the magnetic energy that is equivalent with kinetic energy and the transition energy (emission and absorption energies) between a left state and an occupied state. $\displaystyle\left(m-m^{~{}}_{0}\right)c^{2}$ $\displaystyle=m^{~{}}_{0}\left(\frac{1}{\sqrt{1-(\Delta\mathbf{v})^{2}/c^{2}}}-1\right)c^{2}$ $\displaystyle=E^{~{}}_{\mathrm{magnetic}}+E^{~{}}_{\mathrm{transition}}$ This result shows that relativistic effects such as related to time interval should not be enough to be considered in the framework of kinematic aspects as established-understanding claimed by Einstein. Our interpretation shows that internal random motion exhibiting random vibrating system and determining internal time interval unit (proper time) for states must play a fundamental role in relativistic effects and it can depict the change of both potential and kinetic terms as well as the transition energy of the particle. Furthermore, from the seeking relation between mass and electric-magnetic fields, we have posed its tendency to attract particles of the same charges or repel different charges. Based on this viewpoint, we can understand the magnetic energy (magnetic field) as the lost of the particle’s tendency for having minimum concentrated-energy $E$ caused by the increase of the particle’s velocity (the kinetic energy of particles). We have also shown that Eq. (70) still prevail to picture the state transition of spin representing the increase or decrease of internal velocity ($\mathbf{b}$ or $\mathbf{b}_{}$) when diffusion process take places. The relativistic effect on to the mass when the transition process takes place must be generated by unit times relativistically changing by the emergence of the kinetic aspect. We also showed that spin-spin and spin-orbit interactions (for the case of hydrogen atom) must be caused by the difference perspective on kinetic energy between the electron and its nucleus. The increase of kinetic energy (the increase of magnetic field) that is felt by one particle can be felt as the decrease of kinetic energy (the increase of electric field) by another particle. In addition to the proposal of a new basic-fundamental connection between mass and electromagnetic fields, we also found that if we view absorbed-radiated energy in terms of light, our model indicates that electromagnetic fields should be not directly represented by the absorbed-radiated energy. Instead, electromagnetic fields should just represent the physical effects accompanying the process of absorbed-radiated energy generated by the transition process (or, how environment responds to the change of the transition process) where electromagnetic field can be used to determine the equivalence value of the amount of absorbed-radiated energy. We propose that if the electromagnetic wave must by the representation of wave properties of light, so must the pure radiated-absorbed energy represent the particle properties of the light. Furthermore, because the particle is modeled as energy localized on the surface of dimensional sphere-form (2-manifold without boundary) we can imagine that our real space may be composed of the space-particles sea where pure energy can be localized or propagated with particle properties. The mass feature of particles in our model is the physical condition generated by concentrated-energy located in the surface of 3-dimensional sphere-form. Using this viewpoint, we can understand the environment or the existence of system as the condition accompanying the loss of concentrated-energy. Thus, the total lost concentrated-energy from the particle must be compensated by the emergence of potentials and the kinetics terms as well as the change of velocity of internal random motion. The change of the concentrated-energy of the particle will describe the dynamics of the system relative to the particle or the dynamics of the particle relative to the system. Therefore the energy in Schrödinger Eq. (44) and (45) must represented the total energy radiated by the particle since there is no interaction perspective. We can also see that for the case of simple like an atom, the quantization of emitted absorbed energy will relate to states quantified by the potential and kinetic terms. The principle difference between our model and previous stochastic models lies in the causality of the emergence of random motion. In our model, we propose motion must be generated by concentrated-energy localized on the surface of a particle. Therefore, we do not only need quantum potential (external potential) to generate internal motion but also a bath (ether or vacuum fluctuation or a noise field background as the main source of fluctuation) to generated random motion and diffusion. In our proposal concepts on mass, we view the background noise field [Eq. (5)] that depicts environmental fluctuation must exist to interact with a stationer system. The interactions do not change on average, the energy of the stationary system and only affect the probabilistic character of outcome. In the context of the debate on how wave function evolves by the Schrödinger equation in a predictable deterministic way, despite when a physical quantity is measured, the outcome is not predictable in advance SLA-AB-2009 , our model demonstrate that the quantum-sized volume ”bringing” the probability density will evolve and coincide with transition processes in a predictable way. But, the information of physical quantity coupled with position (direction) is randomly stored in the quantum-sized volume, and because every measurement is a process to cause transition between states where this process is accompanied by the change of probability density to find the particle in the quantum-sized volume, thus, the measurement will determine the behavior of statistical outcome for every physical observable coupled to (defined) position (direction). However the statistical outcome can also be generated by a background noise field or environment fluctuation and it does not affect on average to a stationer system. In regards to superposition of states, our model supports the interaction of Schrödinger equation introduced by Bohm that at a definite time and position the particle only occupies one state but, the equation should cover all of the possible states of the particle. ###### Acknowledgements. The work was supported by grand Penelitian Dosen Muda in 2011 (contract no. LPPM-UGM /1506/BID.I/201), and I am indebted to Prof. Andrew Strominger, Wa Ode Kamaria and Sylvia Hase for their great supporting. ## References * [1] I. Fényes, Acta Bolyaina. 1, 5-7 (1946). * [2] I. Fényes, Z. Phys. 132, 81-106 (1952). * [3] W. Weizel, Z. Phys. 134, 264-285 (1954). * [4] D. Bohm and J.P. Vigier, Phys. Rev. 96, 1 208-216 (1954). * [5] E. Nelson, Phys. Rev. 150, 4 1079-1085 (1966). * [6] A. Kyprianidis, Problems in Quantum Physics (World Scientific, Gdansk, 1985). * [7] R. Fürth, Z. Phys. 81, 143-162 (1933). * [8] K. L. Chung and Z. Zhao From Brownian motion to Schrödinger equation (Springer, Berlin, 1985). * [9] D. Kershaw, Phys. Rev. 136, 1850-1856 (1964). * [10] K. Namsrai Non local quantum field theory and stochastic quantum mechanics, fundamental theories of physics (D. Reidel Publishing Company, Dordrecht, 1986). * [11] M. Baublitz, Jr. Prog. Theor. Phys. 80, 2 232-244 (1988). * [12] T. H. Boyer Ann. Phys. 56, 474-503 (1970). * [13] J. R. Bogan, arXiv: 02121.10. * [14] H. Grabert, et. al, Phys. Rev. A. 19, 6 2440-2445 (1970). * [15] D.T. Gillespie, Phys. Rev. A. 49, 3 1607-1612 (1994). * [16] G. A. Skorobogatov and S. I. Svertilov, Phys. Rev. A. 58, 5 3426-3432 (1988). * [17] G. Jones, P. Pearle, and J. Ring, Found. Phys. 34, 10 1467-1474 (2004). * [18] M. P. Davidson, J. Math. Phys. 20, 9 1865-1869 (1979). * [19] E. Recami and G. Salesi, Phys. Rev. A. 57, 1 97-105 (1998). * [20] M. Nagasawa, (Monograph in mathematics (Book 86), Bessel, Birkhäuser, 1993). * [21] K. Jung, Ann. Fond. Louis de Broglie. 34, 2 (2009). * [22] P. J. Riggs, Quantum Causality: Conceptual issues in the Causal Theory of Quantum Mechanics (Springer, New York, 2009). * [23] H. Graber, H. Hänggi, and T. Talkner, Phys. Rev. A.19, 2440-2445 (1979). * [24] G. C. Ghirardi, P. Pearle, T. Weber, Phys. Rev. D. 34, 470 (1986). * [25] G. C. Ghirardi, P. Pearle, and A. Rimini, Phys. Rev. A. 42, 78 (1990). * [26] D. Bohm, Quantum theory (Dover Publication, New York, 1989). * [27] M. Jammer, Concepts on mass in contemporary physics and philosophy (Princeton University Press, New Jersey, 2000) pp. 10-12, pp. 51-61. * [28] H. R. Brown and O. Pooley, Physics Meets Philosophy at the Planck Scale: Contemporary Theories in Quantum Gravity (Cambridge University Press, Cambridge, 2001). * [29] M. P. Davidson, Ann. Fond. Louis de Broglie. 29, 4 661-680 (2004). * [30] R. P. Feynman, R. P. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison-Wesley Publishing Company, USA, 1963). * [31] R. K. Wangness, Electromagnetic Field (John Wiley & Sons, USA, 1979) pp. 303-306. * [32] S. L. Adler and A. Bassi, A Quantum Theory: Exact or Approximate? http://www.sns.ias.edu/~adler/science.pdf ΨΨΨ (2009).	arxiv-papers	2013-11-06T08:10:18	2024-09-04T02:49:53.366560	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Muhamad Darwis Umar", "submitter": "Muhamad Darwis Umar", "url": "https://arxiv.org/abs/1311.1836" }
1311.1839	# An Efficiently Solvable Quadratic Program for Stabilizing Dynamic Locomotion Scott Kuindersma, Frank Permenter, and Russ Tedrake This work was supported by AFRL contract FA8750-12-1-0321 and NSF contract ERC-1028725, IIS-1161909, and IIS-0746194.The authors are with the Computer Science and Artificial Intelligence Laboratory at the Massachusetts Institute of Technology, Cambridge, MA, USA. {scottk,fpermenter,russt}@csail.mit.edu ###### Abstract We describe a whole-body dynamic walking controller implemented as a convex quadratic program. The controller solves an optimal control problem using an approximate value function derived from a simple walking model while respecting the dynamic, input, and contact constraints of the full robot dynamics. By exploiting sparsity and temporal structure in the optimization with a custom active-set algorithm, we surpass the performance of the best available off-the-shelf solvers and achieve 1kHz control rates for a 34-DOF humanoid. We describe applications to balancing and walking tasks using the simulated Atlas robot in the DARPA Virtual Robotics Challenge. ## I Introduction Achieving dynamically-stable locomotion in complex legged systems is a problem at the heart of modern robotics research. For humanoid systems in particular, nonlinear, underactuated, and high-dimensional dynamics conspire to make the control problem challenging. Optimization-based techniques must simultaneously reason about the dynamics, actuation limits, and contact constraints of the walking system. Model predictive control (MPC) is a popular approach to performing this type of constrained optimization iteratively over fixed horizons, but its computational complexity has hindered applications to high- dimensional systems. Furthermore, the hybrid dynamics of walking robots makes multi-step optimization difficult [1]. Successful examples of using MPC for humanoid control have therefore relied upon the use of low-dimensional linear models [2, 3] or relaxation of constraints to permit smooth optimization through discontinuous dynamics [4]. Several researchers have recently explored using quadratic programs (QPs) to control bipedal systems by exploiting the fact that the _instantaneous_ dynamics and contact constraints can be expressed linearly (effectively solving a horizon-1 MPC problem) [5, 6, 7, 8, 9, 10, 11, 12]. A key observation about these approaches in the context of balancing and locomotion tasks is that, during typical operation, the set of active inequality constraints changes very infrequently between consecutive control steps. We give a problem formulation and solution technique that explicitly take advantage of this observation. We describe a QP that exploits optimal control solutions for a simple unconstrained model of the walking system. Using time-varying LQR design, we compute the optimal cost-to-go for the simple model and use it as part of the objective function in a constrained optimization to compute inputs for the full robot. We describe the approach concretely in terms of a simulated bipedal system and zero-moment point (ZMP) dynamics. In addition to providing a principled and reliable way to stabilize walking trajectories, we show the resulting QP cost function contains low-dimensional structure that can be exploited to reduce solution time. To achieve real-time control rates, we designed a custom active-set solver that exploits consistency between subsequent solutions and outperforms the best available off-the-shelf solvers such as CVXGEN and Gurobi by a factor of 5 or more. Our analysis of solver performance during typical walking experiments suggests that the active set remains constant between consecutive control steps approximately 97% of the time, requiring only a _single linear system solve per step_. In our tests, we were able to achieve average control rates of 1kHz for a 34-DOF humanoid. We briefly summarize extensive simulation testing done with the Atlas robot as part of the DARPA Virtual Robotics Challenge. ## II LQR Design for ZMP Dynamics The planar center of mass (COM) and ZMP dynamics of a fully actuated rigid body system can be written in state space form as $\displaystyle\dot{\mathbf{x}}$ $\displaystyle=$ $\displaystyle\mathbf{A}\mathbf{x}+\mathbf{B}\mathbf{u}$ (5) $\displaystyle=$ $\displaystyle\left[\begin{array}[]{cc}0&\mathbf{I}\\\ 0&0\end{array}\right]\mathbf{x}+\left[\begin{array}[]{c}0\\\ \mathbf{I}\end{array}\right]\mathbf{u}$ $\displaystyle\mathbf{y}$ $\displaystyle=$ $\displaystyle\mathbf{C}\mathbf{x}-b(\mathbf{x},\dot{\mathbf{x}})\mathbf{u}$ (7) $\displaystyle=$ $\displaystyle\left[\begin{array}[]{cc}\mathbf{I}&0\end{array}\right]\mathbf{x}+\frac{z_{\rm com}}{\ddot{z}_{\rm com}+g}\mathbf{I}\mathbf{u},$ where $\mathbf{x}=[x_{\rm com},y_{\rm com},\dot{x}_{\rm com},\dot{y}_{\rm com}]^{T}$, $\mathbf{u}=[\ddot{x}_{\rm com},\ddot{y}_{\rm com}]^{T}$, $\mathbf{y}=[x_{\rm zmp},y_{\rm zmp}]^{T}$, $g$ is a constant gravitational acceleration, and $z_{\rm com}$ is the COM height. The ZMP is a well-studied quantity in the bipedal walking literature that defines the point on the ground plane at which the moment produced by inertial and gravitational forces is parallel to the surface normal (i.e., the robot is not tipping) [13]. Since dynamic balance is achieved when the contact forces directly oppose the gravitational and inertial forces, maintaining the ZMP within the contact support polygon can be an effective strategy for maintaining dynamic stability in legged locomotion. Given desired ZMP trajectory, $\mathbf{y}^{d}(t)$, we would like to compute an optimal tracking controller that takes into account the time- and state- varying constraints on $\mathbf{u}$ imposed by the dynamics, input limits, and contacts of the full walking system. Due to the prohibitive computational requirements of solving nonlinearly constrained optimal control problems of this scale, we instead solve an unconstrained time-varying LQR problem to compute the optimal cost-to-go, $J^{}$, which provides a control-Lyapunov function (CLF) for the ZMP dynamics. On each iteration, we select the control inputs to descend this ZMP CLF while reasoning about the instantaneous constraints of the full system. We begin by specifying a cost functional of the form $\displaystyle J=\bar{\mathbf{y}}(t_{f})^{T}\mathbf{Q}_{f}\bar{\mathbf{y}}(t_{f})+\int_{0}^{t_{f}}\bar{\mathbf{y}}(t)^{T}\mathbf{Q}\bar{\mathbf{y}}(t)dt,$ (8) where the coordinates $\bar{\mathbf{y}}(t)=\mathbf{y}(t)-\mathbf{y}^{d}(t)$, $\mathbf{Q}\succ 0$, and $\mathbf{Q}_{f}\succ 0$. In practice the COM height, $z_{\rm com}$, is often assumed to be constant, making the ZMP dynamics (7) linear [14]. More generally, if the COM height trajectory is constrained to be a known function of time, $(z_{\rm com}(t),\dot{z}_{\rm com}(t),\ddot{z}_{\rm com}(t))$, the ZMP dynamics are time-varying linear, $\displaystyle\mathbf{y}(t)=\mathbf{C}(t)\mathbf{x}(t)+\mathbf{D}(t)\mathbf{u}(t),$ (9) and therefore amenable to TVLQR design without explicit linearization. Solving the Riccati equation yields the optimal cost-to-go for the time- varying linear system, $\displaystyle J^{}(\bar{\mathbf{x}},t)=\bar{\mathbf{x}}^{T}\mathbf{S}(t)\bar{\mathbf{x}}+\mathbf{s}_{1}(t)^{T}\bar{\mathbf{x}}+s_{0}(t),$ and the linear optimal controller, $\displaystyle\bar{\mathbf{u}}^{}$ $\displaystyle=$ $\displaystyle-\mathbf{K}(t)\bar{\mathbf{x}}$ (10) $\displaystyle=$ $\displaystyle\arg\min_{\bar{\mathbf{u}}}\bar{\mathbf{y}}(t)^{T}\mathbf{Q}\bar{\mathbf{y}}(t)+\frac{\partial J^{}}{\partial\bar{\mathbf{x}}}\bigg{\|}_{\bar{\mathbf{x}}}\dot{\bar{\mathbf{x}}},$ where $\bar{\mathbf{x}}(t)=\mathbf{x}(t)-\mathbf{x}^{d}(t)$ and $\bar{\mathbf{u}}(t)=\mathbf{u}(t)-\mathbf{u}^{d}(t)$. In general, achieving $\bar{\mathbf{u}}^{}$ is not possible due to constraints imposed by the robot dynamics. For example, actuator saturations and contact friction properties can limit the possible magnitudes and directions of COM accelerations. Therefore, to compute control inputs we perform a constrained minimization using $\displaystyle V(\bar{\mathbf{x}},\bar{\mathbf{u}},t)=\bar{\mathbf{y}}(t)^{T}\mathbf{Q}\bar{\mathbf{y}}(t)+\frac{\partial J^{}}{\partial\bar{\mathbf{x}}}\bigg{\|}_{\bar{\mathbf{x}}}\dot{\bar{\mathbf{x}}}$ (11) as a surrogate value function. ## III QP Formulation Given the stabilizing solution for the ZMP dynamics, we design a QP to solve for control inputs for the full robot dynamics that minimizes (11) and a quadratic motion cost for walking subject to the instantaneous constraints. Consider the familiar rigid body dynamics, $\displaystyle\mathbf{H}(\mathbf{q})\ddot{\mathbf{q}}+\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})=\mathbf{B}(\mathbf{q},\dot{\mathbf{q}})\bm{\tau}+\bm{\Phi}(\mathbf{q})^{T}\bm{\lambda},$ (12) where $\mathbf{H}(\mathbf{q})$ is the system inertia matrix, $\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})$ captures the gravitational and Coriolis terms, $\mathbf{B}(\mathbf{q},\dot{\mathbf{q}})$ is the control input map, and $\bm{\Phi}(\mathbf{q})^{T}$ transforms external forces, $\bm{\lambda}$, into generalized forces. In our case, $\bm{\lambda}=[\begin{array}[]{ccc}\bm{\lambda}_{1}^{T}&\dots&\bm{\lambda}_{N_{c}}^{T}\end{array}]^{T}$ is a vector of ground-contact forces acting at $N_{c}$ contact points. The set of active contacts are determined by kinematic or force measurement classification at each control step. For floating-base systems such as humanoids, the dynamics can be partitioned into actuated and unactuated degrees of freedom [9], $\displaystyle\mathbf{H}_{f}\ddot{\mathbf{q}}+\mathbf{C}_{f}$ $\displaystyle=$ $\displaystyle\bm{\Phi}_{f}^{T}\bm{\lambda}$ (13) $\displaystyle\mathbf{H}_{a}\ddot{\mathbf{q}}+\mathbf{C}_{a}$ $\displaystyle=$ $\displaystyle\mathbf{B}_{a}\bm{\tau}+\bm{\Phi}_{a}^{T}\bm{\lambda},$ where we have dropped the explicit dependence on $\mathbf{q},\dot{\mathbf{q}}$ from our notation for conciseness. This separation permits the removal of $\bm{\tau}$ as a decision variable by including (13) as a constraint expressing $\bm{\tau}$ in terms of $\ddot{\mathbf{q}}$ and $\bm{\lambda}$: $\displaystyle{\bm{\tau}}=\mathbf{B}_{a}^{-1}\left[\mathbf{H}_{a}\ddot{\mathbf{q}}+\mathbf{C}_{a}-\bm{\Phi}_{a}^{T}\bm{\lambda}\right].$ We use a standard, conservative polyhedral approximation of the friction cone, $\hat{K}_{j}$, for each contact point, $\mathbf{c}_{j}$, $\displaystyle\hat{K}_{j}=\left\\{\sum_{i=1}^{N_{d}}\beta_{ij}\mathbf{v}_{ij}:\beta_{ij}\geq 0\right\\}.$ (14) The generating vectors, $\mathbf{v}_{ij}$, are computed as $\mathbf{v}_{ij}=\mathbf{n}_{j}+\mu_{j}\mathbf{d}_{ij}$, where $\mathbf{n}_{j}$ and $\mathbf{d}_{ij}$ are the contact-surface normal and $i^{\rm th}$ tangent vector for the $j^{\rm th}$ contact point, respectively, $\mu_{j}$ is the Coulomb friction coefficient, and $N_{d}$ is the number of tangent vectors used in the approximation [15]. Given the robot state, $\mathbf{q},\dot{\mathbf{q}}$, at time $t$, we solve the following quadratic program: ###### Quadratic Program 1 $\displaystyle\min_{\ddot{\mathbf{q}},\bm{\beta},\bm{\lambda},\bm{\eta}}V(\bar{\mathbf{x}},\bar{\mathbf{u}},t)+w_{\ddot{\mathbf{q}}}\|\|\ddot{\mathbf{q}}_{\rm des}-\ddot{\mathbf{q}}\|\|^{2}+\varepsilon\sum_{ij}\beta^{2}_{ij}+\|\|\bm{\eta}\|\|^{2}$ (15) subject to $\displaystyle\mathbf{H}_{f}\ddot{\mathbf{q}}+\mathbf{C}_{f}$ $\displaystyle=$ $\displaystyle\bm{\Phi}_{f}^{T}\bm{\lambda}$ (16) $\displaystyle\mathbf{J}\ddot{\mathbf{q}}+\dot{\mathbf{J}}\dot{\mathbf{q}}$ $\displaystyle=$ $\displaystyle-\alpha\mathbf{J}\dot{\mathbf{q}}+\bm{\eta}$ (17) $\displaystyle\mathbf{B}_{a}^{-1}(\mathbf{H}_{a}\ddot{\mathbf{q}}+\mathbf{C}_{a}-\bm{\Phi}_{a}^{T}\bm{\lambda})$ $\displaystyle\in$ $\displaystyle[\bm{\tau}_{\rm min},\bm{\tau}_{\rm max}]$ (18) $\displaystyle\forall_{j=\\{1\dots N_{c}\\}}~{}~{}\bm{\lambda}_{j}$ $\displaystyle=$ $\displaystyle\sum_{i=1}^{N_{d}}\beta_{ij}\mathbf{v}_{ij}$ (19) $\displaystyle\forall_{i,j}\beta_{ij}$ $\displaystyle\geq$ $\displaystyle 0$ (20) $\displaystyle\bm{\eta}$ $\displaystyle\in$ $\displaystyle[\bm{\eta}_{\rm min},\bm{\eta}_{\rm max}].$ (21) The constraints (16) and (18) ensure that the dynamics and input limits are respected, (17) is a no-slip constraint on the foot contacts requiring that their acceleration be negatively proportional to the velocity, and the constraints (19,20) together ensure that contact forces remain within $\hat{K}$. The parameter vector $\bm{\eta}$ allows bounded violations of the no-slip constraint to reduce the likelihood of infeasibility, $\varepsilon$ is a regularization constant typically set to a small value, e.g., $\varepsilon=10^{-8}$, and $\mathbf{J}=\partial\mathbf{c}/\partial\mathbf{q}$ is the Jacobian matrix for the vector of all contact points, $\mathbf{c}=[\begin{array}[]{ccc}\mathbf{c}_{1}^{T}&\dots&\mathbf{c}_{Nc}^{T}\end{array}]^{T}$. The weight parameter, $w_{\ddot{\mathbf{q}}}$, is used to balance the relative contribution of the desired motion cost with the ZMP tracking cost. To respect joint limits, the bounds $\ddot{q}_{i}\geq 0$ and $\ddot{q}_{i}\leq 0$ are added for all $i$ such that $q_{i}=q_{i}^{\rm MIN}$ and $q_{i}=q_{i}^{\rm MAX}$, respectively. ## IV Optimization We solve QP 1 at each control step using a simple active-set method. The method assumes the set of active inequality constraints remains constant for consecutive solutions. It then produces a candidate solution by solving a partial set of optimality conditions derived from the assumed active set. If the candidate solution satisfies the full set of optimality conditions, the assumption is correct and the algorithm terminates. Otherwise, the method updates the active set and repeats until a solution is found or a maximum number of iterations is reached. On rare occasions when no solution is found, the algorithm fails over to a more reliable (but on average slower) interior point solver. In our experiments, this lead to infrequent single-step input delays on the order of 3ms, which had no significant effect on walking performance. This contingency is required since finite termination cannot be guaranteed for the proposed method. In practice, however, instances of QP 1 are almost always solved in one iteration. The computational cost of each iteration is also very small. A candidate solution is produced by solving a structured system of linear equations and constraints are evaluated only once. ### IV-A Active-set method The QP solved at each control step can be written in the standard form, $\displaystyle\begin{array}[]{cl}\underset{\mathbf{z}}{\operatorname{min}}&\frac{1}{2}{\mathbf{z}}^{T}\mathbf{W}{\mathbf{z}}+\mathbf{g}^{T}{\mathbf{z}}\\\ \mbox{subject to}&\mathbf{A}\mathbf{z}=\mathbf{b}\\\ &\mathbf{P}\mathbf{z}\leq\mathbf{f},\end{array}$ (25) where the inequalities are defined by $\mathbf{P}=(\mathbf{p}_{1},\mathbf{p}_{2},\ldots,\mathbf{p}_{n})^{T}$ and $\mathbf{f}=(f_{1},f_{2},\ldots,f_{n})^{T}$. To solve this problem, it is assumed that $\mathbf{p}^{T}_{i}\mathbf{z}=f_{i}$ at optimality for each $i$ in a subset $\mathcal{A}\subseteq\\{1\ldots n\\}$ called the _active set_. For $t>0$, this subset equals the indices of the active inequalities from time $t-1$. With this assumption, the KKT conditions for the QP can be written in terms of $\mathbf{z}$, $\bm{\gamma}$, and $\bm{\alpha}$: $\displaystyle\begin{array}[]{rclc}\mathbf{W}\mathbf{z}+\mathbf{A}^{T}\bm{\alpha}+\sum_{i\in\mathcal{A}}\gamma_{i}\mathbf{p}_{i}&=&-\mathbf{g}\\\ \mathbf{A}\mathbf{z}&=&\mathbf{b}\\\ \mathbf{p}_{i}^{T}\mathbf{z}&=&f_{i}&\forall i\in\mathcal{A}\\\ \gamma_{i}&=&0&\forall i\neq\mathcal{A}\end{array}$ (30) $\displaystyle\begin{array}[]{rclc}\mathbf{P}\mathbf{z}&\leq&\mathbf{f}\\\ \gamma_{i}&\geq&0{}{}&\forall i\in\mathcal{A}.\\\ \end{array}$ (33) Our method solves the linear equations (30) and checks if the solution $(\mathbf{z},\bm{\gamma},\bm{\alpha})$ satisfies the inequalities (33). If the inequalities are satisfied, $\mathbf{z}$ solves the QP and the algorithm terminates. Otherwise, the algorithm adds index $i$ to $\mathcal{A}$ if $\mathbf{p}_{i}^{T}\mathbf{z}>f_{i}$ or removes index $i$ if $\gamma_{i}<0$ and resolves (30). The algorithm repeats this process until the inequalities (33) are satisfied or a until a specified maximum number of iterations is reached. The method is outline in Algorithm 1. Data: A QP of form (25) where the cost matrix $\mathbf{W}$ has the structure (42). A set of constraints $\mathcal{A}$ assumed to be active at optimality. Result: An optimal solution $\mathbf{z}$ with active set $\mathcal{A}$ or a flag indicating failure. 1 $iter\leftarrow 0$ 2 repeat 3 Compute candidate solution $\mathbf{z},\bm{\gamma},\bm{\alpha}$ from (36,39) 4 if _$\mathbf{p}^{T}_{i}\mathbf{z} >f_{i}$_ then 5 add $i$ to $\mathcal{A}$ 6 end if 7 if _$\gamma_{i} <0$_ then 8 remove $i$ from $\mathcal{A}$ 9 end if 10 $iter\leftarrow iter+1$ 11 if _$iter >iter_{\rm MAX}$_ then 12 return Failure 13 end if 14 15until _ $\mathbf{z}$ and $\bm{\gamma}$ satisfy (33) _ 16return $\mathcal{A}$ and $\mathbf{z}$ Algorithm 1 Active-set method for solving (25). The set $\mathcal{A}$ passed to the algorithm at time $t$ equals the set of constraints active at optimality for time $t-1$. ### IV-B Efficiently computing a candidate solution The structure of QP 1 admits an efficient solution of the linear system (30). In particular, one can cheaply compute $\mathbf{W}^{-1}$ and construct a smaller system for $\bm{\alpha}$ and $\bm{\gamma}$. Using a solution to this smaller system, one can then easily recover $\mathbf{z}$. To see this, first let $\mathbf{P}_{act}$ and $\mathbf{f}_{act}$ denote the rows of $\mathbf{P}$ and $\mathbf{f}$ indexed by $\mathcal{A}$ and let $\mathbf{R}=\begin{array}[]{cc}[\mathbf{A}^{T}&\mathbf{P}_{act}^{T}]^{T}\end{array}$ and $\mathbf{e}=[\begin{array}[]{cc}\mathbf{b}^{T}&\mathbf{f}^{T}_{act}\end{array}]^{T}$. A solution to (30) can be found by first solving the following system of equations for $\bm{\alpha}$ and $\bm{\gamma}$: $\displaystyle-\mathbf{R}\mathbf{W}^{-1}\mathbf{R}^{T}\left[\begin{array}[]{c}\bm{\alpha}\\\ \bm{\gamma}\end{array}\right]$ $\displaystyle=\mathbf{e}+\mathbf{R}\mathbf{W}^{-1}\mathbf{g}$ (36) Using a solution to this system, $\mathbf{z}$ can be recovered via $\displaystyle\mathbf{z}$ $\displaystyle=-\mathbf{W}^{-1}\left(\mathbf{g}+\mathbf{R}^{T}\left[\begin{array}[]{c}\bm{\alpha}\\\ \bm{\gamma}\end{array}\right]\right).$ (39) Efficient computation of $\mathbf{W}^{-1}$ arises from its block diagonal structure, $\displaystyle\mathbf{W}=\left[\begin{array}[]{cc}\mathbf{W}_{11}&0\\\ 0&\mathbf{W}_{22}\end{array}\right],$ (42) where $\mathbf{W}_{22}$ is diagonal and $\mathbf{W}_{11}=w_{\ddot{\mathbf{q}}}\mathbf{I}+\mathbf{U}^{T}\mathbf{Q}\mathbf{U}$. For the ZMP dynamics, $\mathbf{U}=\mathbf{D}(t)\mathbf{J}\in\mathbb{R}^{2\times n}$, where $\mathbf{J}$ is the COM$(x,y)$ Jacobian and $\mathbf{D}(t)$ is the input mapping defined in (9). Applying the matrix inversion lemma yields an expression for $\mathbf{W}_{11}^{-1}$ that involves computing the inverse of $2\times 2$ matrices: $\displaystyle\mathbf{W}_{11}^{-1}=\frac{1}{w_{\ddot{\mathbf{q}}}}\mathbf{I}-\frac{1}{w_{\ddot{\mathbf{q}}}^{2}}\mathbf{U}^{T}(\mathbf{Q}^{-1}+\frac{1}{w_{\ddot{\mathbf{q}}}}\mathbf{U}\mathbf{U}^{T})^{-1}\mathbf{U}.$ It should also be noted that $\mathbf{W}^{-1}$ is independent of $\mathcal{A}$ and thus only needs to be computed once per control step even if multiple solver iterations are required. The same holds for various sub-matrices in the expressions (36) and (39). ## V Application We implemented our controller using the 34-DOF Atlas humanoid model developed for the DARPA Virtual Robotics Challenge. Our evaluation of the controller included a variety of balancing and locomotion tasks using two independent simulation environments: Drake [16] and Gazebo [17]. As part of MIT’s entry into the DARPA Virtual Robotics Challenge (VRC), the controller was used to walk reliably over uneven terrain, through simulated knee-deep mud, and while carrying an unmodeled multi-link hose, all using imperfect state and terrain estimation (Figure 1).111Example simulation code is available at http://people.csail.mit.edu/scottk. Figure 1: Walking in simulation over obstacles, through simulated mud, and over rolling hills using state and terrain estimation. To design the balancing controller, we solved an infinite horizon LQR problem to regulate the ZMP at $(0,0)$. The cost functional took the form $\displaystyle J$ $\displaystyle=$ $\displaystyle\int_{0}^{\infty}\mathbf{y}^{T}\mathbf{Q}\mathbf{y}dt,$ $\displaystyle=$ $\displaystyle\int_{0}^{\infty}\left[\mathbf{x}^{T}\mathbf{C}^{T}\mathbf{C}\mathbf{x}+\mathbf{u}^{T}\mathbf{D}^{T}\mathbf{D}\mathbf{u}+2\mathbf{x}^{T}\mathbf{C}^{T}\mathbf{D}\mathbf{u}\right]dt,$ where $\mathbf{Q}=\mathbf{I}$. We assumed the COM height was constant while standing, thus making the ZMP dynamics linear. This had the advantage that it only required us to solve the LQR problem once. To see this, note that $J^{}(\bar{\mathbf{x}})=\bar{\mathbf{x}}^{T}\mathbf{S}\bar{\mathbf{x}}$, where $\mathbf{S}$ is the solution of the algebraic Riccati equation. Thus the QP cost had the form, $\displaystyle\bar{\mathbf{y}}^{T}\bar{\mathbf{y}}+2\bar{\mathbf{x}}^{T}\mathbf{S}(\mathbf{A}\bar{\mathbf{x}}+\mathbf{B}\mathbf{u})+w_{\ddot{\mathbf{q}}}\|\|\ddot{\mathbf{q}}_{\rm des}-\ddot{\mathbf{q}}\|\|^{2}+\varepsilon\sum_{ij}\beta^{2}_{ij},$ where new desired ZMP locations $\mathbf{k}=[\begin{array}[]{cc}k_{x}&k_{y}\end{array}]^{T}$ could be achieved by a change in coordinates, $\bar{\mathbf{y}}=\mathbf{y}-\mathbf{k}$, $\bar{\mathbf{x}}=\mathbf{x}-\mathbf{k}$, and $\mathbf{k}$ is, e.g., the point at the center of the foot support polygon. In practice, we found the constant COM height assumption has minimal practical effect on balancing performance, even when recovery motions included significant hip bends and arm motion. We computed $\ddot{\mathbf{q}}_{\rm des}$ via a simple PD control rule, $\ddot{\mathbf{q}}_{\rm des}=K_{p}({\mathbf{q}}_{\rm des}-\mathbf{q})-K_{d}(\dot{\mathbf{q}})$, using either a fixed nominal posture, ${\mathbf{q}}_{\rm des}$, for standing or a time-varying configuration trajectory for manipulation. We used the same scalar gains, $K_{p}$ and $K_{d}$, for all joints. Our planning implementation took desired foot trajectories as input and computed a ZMP plan, $\mathbf{y}^{d}(t)$, by linear interpolation between step locations. The footstep planner combined terrain map information with heuristics to select reasonable step locations and timing. We solved the TVLQR problem (8) for the linear ZMP dynamics using the Riccati solution for balancing as the final cost, $\mathbf{Q}_{f}=\mathbf{S}$. The corresponding COM$(x,y)$ trajectory, $\mathbf{x}^{d}(t)$, can be computed by simulating the COM dynamics (5) in a closed loop from time $t=0$ to $t=t_{f}$ with the optimal controller, $\bar{\mathbf{u}}^{}=-\mathbf{K}(t)\bar{\mathbf{x}}$. In practice, we were able to compute both $J^{}(\bar{\mathbf{x}},t)$ and $\mathbf{x}^{}(t)$ for a 10m walking plan in approximately $1/4$s using an unoptimized MATLAB implementation of the explicit ZMP Riccati solution described by Tedrake et al. [18]. The desired configuration, ${\mathbf{q}}_{\rm des}(t)$, was computed via inverse kinematics with constraints on the foot pose and COM position. Computation of ${\mathbf{q}}_{\rm des}(t)$ was done either offline for open- loop trajectory following or reactively inside the control loop by linearizing the forward kinematics at the current configuration and solving a second small QP to minimize the weighted $\ell^{2}$ distance to a nominal configuration while respecting foot pose, COM, and joint limit constraints. Qualitatively different motions could be achieved by varying the relative weights assigned to joints in the cost. For example, a smaller cost on back joints would tend to produce more torso sway to track the desired COM trajectory. We used a simplified 4-point contact representation for each foot. Active contacts were determined by a combination of the desired footstep plan and the estimated distance between the foot and terrain. If and only if the foot is perceived to be in contact and the plan agreed did we include the corresponding foot contact in the optimization. The requirement that both conditions be true was essential for breaking contact with the ground while walking. As with balancing, footstep and ZMP plans could be translated in three dimensions without additional computation by a simple change in coordinates in the QP cost. ### V-A Solver Performance We compared the solve time of our active-set algorithm against two general- purpose QP solvers, Gurobi [19] and CVXGEN [20]. For the Gurobi solver, we used the barrier (B) algorithm and dual simplex (DS) algorithm with both active constraint and solution warm-starting. Our CVXGEN problem formulation omitted the input saturation inequalities (18) to fit within the problem size requirements. These experiments were done on an i7 2.1GHz quad-core laptop. A comparison of average solve times while executing a fixed flat ground pattern is given in Table I. TABLE I: Comparison of average QP solve times while walking. \| Algorithm 1 \| Gurobi (DS) \| CVXGEN \| Gurobi (B) ---\|---\|---\|---\|--- Solve time \| 0.2 ms \| 1.0 ms \| 2.2 ms \| 3.1 ms The custom active-set method outperforms the next best solver by a factor of 5. The significant performance advantage of Algorithm 1 can be understood by considering the histogram in Figure 2. For an overwhelming percentage of control steps, the active set does not change and the solver succeeds in a single iteration. Thus, most of the time control inputs are computed by solving a single linear system of equations. For the active-set algorithm, the total controller computation time is largely spent setting up the QP, which involves computing the manipulator dynamics, contact surface normals, and kinematic quantities such as the COM and contact Jacobians. In our implementation, the average QP setup time is approximately $0.8$ms for the 34-DOF Atlas model, giving us a total control step time of $1$ms. Figure 2: Histogram of iterations needed to solve Quadratic Program 1 during a walking task. The method requires only one iteration approximately 97% of the time. The performance of the solver does have a subtle dependency on the problem formulation. We found that using the parameterization of the approximate friction cone (14) lead to fewer active set changes than the commonly used Stewart and Trinkle [21] parameterization, $\displaystyle\hat{K}_{\rm ST}=\left\\{z\mathbf{n}+\sum_{i=1}^{N_{d}}\beta_{i}\mathbf{d}_{i}:z\geq 0,\beta_{i}\geq 0,\sum_{i=1}^{N_{d}}\beta_{i}\leq\mu z\right\\},$ (43) where we have dropped the explicit contact point index, $j$. The parameterization (43) lead to approximately 50% more control steps requiring 2 iterations or more. Intuitively, this is a result of the fact that the active inequalities constraints, $\\{i:\beta_{i}=0\\}$, under parameterization (43) can change when forces inside the approximate friction cone change direction. By contrast, when using (14), the constraints on $\beta_{i}$ only become active on the surface of the polyhedron. This idea is illustrated in Figure 3. Figure 3: An illustration showing how different approximate friction cone parameterizations can affect active set stability. ## VI Related Work The controller design we proposed shares some properties with other horizon-1 MPC implementations. For example, the same flavor of dynamic, friction, and foot motion constraints have appeared in other QP formulations [5, 9, 11, 12]. Herzog et al. [9] proposed the idea of separating the manipulator equation into floating-base and actuated DOFs to remove $\bm{\tau}$ as a decision variable, which enabled them to achieve control rates of 1kHz for a 14-DOF biped. Polyhedral approximations are frequently used to linearize friction constraints, but to our knowledge no prior connection has been made between different parameterizations and solver performance. Ames et al. [22, 23] used CLFs for walking control design by solving QPs that minimize the input norm, $\|\|\mathbf{u}\|\|$, while satisfying constraints on the negativity of $\dot{V}_{\rm clf}$. By contrast, we placed no constraint on $\dot{V}_{\rm clf}$ and instead minimized an objective of the form $\ell(\mathbf{x},\mathbf{u})+\dot{V}_{\rm clf}$, where $\ell(\mathbf{x},\mathbf{u})$ is an instantaneous cost on $\mathbf{x}$ and $\mathbf{u}$. This approach gave us the significant practical robustness while making the QP less prone to infeasibilities. Other uses of active-set methods for MPC have exploited the temporal relationship between the QPs arising in MPC. Bartlett et al. compared active- set and interior-point strategies for MPC [24]. The described an active-set approach based on Schur complements for efficiently resolving KKT conditions after changes are made to the active set. This framework is analogous to the solution method we discuss in Section IV-B. In the discrete time setting, Wang and Boyd [25] describe an approach to quickly evaluating control-Lyapunov policies using explicit enumeration of active sets in cases where the number of states is roughly equal to the square of the number of inputs. Ferreau et al. [26] consider the MPC problems where the cost function and dynamic constraints are the same at each time step; i.e., the QPs solved at iteration differ only by a single constraint that enforces initial conditions. By smoothly varying the initial conditions from the previous to the current state, they were able to track a piecewise linear path traced by the optimal solution, where knot points in the path correspond to changes in the active set. Since the controller we considered had changing cost and constraint structure, this method would have been difficult to apply. ## VII Conclusion We described a stabilizing QP controller formulation for dynamic walking and solution technique that exploits consistency between active inequality constraints in subsequent control steps. In our experiments with a simulated Atlas robot, we were able to efficiently compute control inputs while walking by solving a single system of linear equations a high-percentage of the time, hence outperforming several popular general-purpose solvers used frequently in the literature. Although we have focused on humanoids and ZMP dynamics in this paper, the QP formulation we described is equally applicable to more general floating-base systems and other types of simple system models. Similarly, the active-set method used in this work could easily be applied to the various MPC formulations that exist in the literature. 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Boyd, “Fast evaluation of quadratic control-Lyapunov policy,” _IEEE Transactions on Control Systems Technology_ , vol. 19, no. 4, pp. 939–946, 2011. * [26] H. Ferreau, H. Bock, and M. Diehl, “An online active set strategy to overcome the limitations of explicit MPC,” _International Journal of Robust and Nonlinear Control_ , vol. 18, no. 8, pp. 816–830, 2008.	arxiv-papers	2013-11-07T22:13:21	2024-09-04T02:49:53.382133	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Scott Kuindersma and Frank Permenter and Russ Tedrake", "submitter": "Scott Kuindersma", "url": "https://arxiv.org/abs/1311.1839" }
1311.1905	A discrete integrability test based on multiscale analysis] A discrete integrability test based on multiscale analysis R. HERNÁNDEZ HEREDERO, D. LEVI and C. SCIMITERNA] R. HERNÁNDEZ HEREDERO${}^\dag$, D. LEVI${}^\diamond$ and C. SCIMITERNA${}^{\diamond}$ Departamento de Matemática Aplicada Universidad Politécnica de Madrid, Escuela Universitaria de Ingeniería Técnica de Telecomunicación, Campus Sur, Ctra. de Valencia Km. 7, 28031 Madrid, Spain ${}^\diamond$Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre and Sezione INFN, Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy In this article we present the results obtained applying the multiple scale expansion up to the order $\ep^6$ to a dispersive multilinear class of equations on a square lattice depending on 13 parameters. We show that the integrability conditions given by the multiple scale expansion give rise to 4 nonlinear equations, 3 of which are new, depending at most on 2 parameters and containing integrable sub cases. Moreover at least one sub case provides an example of a new integrable system. § INTRODUCTION Discrete equations play an important role in Mathematical Physics for its double role. From one side discrete space time seems to be basic in the description of fundamental phenomena of nature as provided by quantum gravity. From the other, from discrete equations one can easily by continuous limit obtain differential difference and differential equations and thus discrete equations may provide good numerical schemes for integrating differential equations. A classification of integrable partial difference equation has been given by Adler, Bobenko and Suris [2] in the particular case of equations defined on four lattice points using the consistency around the cube condition with some symmetry constrains to be able to get definite results. Due to the constraints introduced, this classification is partial and already new equations with respect to those contained in the ABS classification have been obtained [16, 14, 10, 7, 6, 1]. In this paper we provide necessary conditions for the integrability of a class of real, autonomous difference equations in the variable $u: \mathbb Z^2 \rightarrow \mathbb R$ defined on a $\mathbb{Z}^2$ square-lattice (u_n,m,u_n+ 1,m,u_n,m + 1, u_n + 1,m + 1; β_1, β_2,...)=0, where the $\beta_i$'s are real, autonomous parameters. Integrability conditions will be determined through a multiscale perturbative expansion. This approach has the distinctive advantage of providing criteria in a manner completely independent from other current approaches. Multiscale developments can be used to reinforce, enhance or augment our previous knowledge of discrete integrable systems given by other techniques. To be able to propagate in all the $\mathbb Z^2$ plain, we will suppose, as in [2], that (<ref>) is linear-affine in every variable, implying that the equation is invariant under the Möbius transformation $T$ \begin{equation}\label{eqMob} u_{n,m}\overset{T}{\mapsto} u_{n,m}'=\frac{Au_{n,m}+B}{Cu_{n,m}+D}. \end{equation} thus providing a geometrical significance to the classification. In this case, (<ref>) reduces to a polynomial equation in its variables with at most fourth order nonlinearity: _IV = f_0+a_00 u_00+ a_01 u_01 + a_10 u_10 + a_11 u_11+ (α_1-α_2) u_00 u_10 + (β_1-β_2) u_00 u_01 + d_1 u_00 u_11+ d_2 u_01 u_10 + (β_1+β_2) u_10 u_11 + (α_1+α_2) u_01 u_11 (τ_1-τ_3) u_00 u_01 u_10+ (τ_1+τ_3) u_00 u_10 u_11 + (τ_2+τ_4) u_00 u_01 u_11 (τ_2-τ_4) u_10 u_01 u_11 + f_1 u_00 u_01 u_10 u_11=0, where all coefficients are taken to be real and independent on $n$ and $m$. We consider here the multiple scale expansion around the dispersive solution u_n,m = K^n Ω^m, of the linearized equation of (<ref>). Rewriting the constants $K$ and $\Omega$ as $K=e^{\ri k}$ and $\Omega=e^{-\ri\omega}$, and introducing the solution (<ref>) into the linear part of Eq. (<ref>) we get a dispersion relation $\omega=\omega\left(k\right)$ ω=arctan[ a_00 a_01 + a_10 a_11 +(a_00 a_11 + a_01 a_10 ) cos(k)/(a_00 a_11 - a_10 a_01) sin(k) ], if $f_0=0$. The solution (<ref>) of (<ref>) with $f_0=0$ is dispersive if $\omega(k)$ is a real nonlinear function of the wave number $k$. This leads to the constraint a_00^2 -a_01^2+a_10^2-a_11^2 +2(a_00 a_10-a_01 a_11) cos(k) = 0 The constraint (<ref>) implies that one of the two following conditions must be satisfied: * $a_{00}=a_{11}\equiv a_1$, $a_{01}=a_{10}\equiv a_2$, * $a_{00}=-a_{11}\equiv a_1$, $a_{01}=-a_{10}\equiv a_2$. Then the dispersion relation (<ref>) reduces to: ω_±(k) =arctan[±2a_1 a_2 ±(a_1^2+a_2^2)cos(k)/(a_1^2-a_2^2)sin(k)] We denote the family of equations (<ref>) satisfying the condition (1) with dispersion relation $\omega_{+}(k)$ as $\CQ^+$ and the one with dispersion relation $\omega_{-}(k)$ as $\CQ^-$. In all the cases $a_1$ and $a_2$ cannot be zero and their ratio cannot be equal to $\pm 1$ to get a nontrivial dispersion relation. In the following we will consider the integrability conditions for the class of equations $\CQ^+$. The study of the class $\CQ^-$ is left to a future work. The result of this work are a series of integrability theorems and a table of equations, invariant under a restricted Möbius transformations that pass the very stringent integrability conditions obtained by considering the multiple scale expansion up to $\ep^6$ order. In Section <ref> we present the main result on the discrete multiscale integrability test and all the conditions up to order $\ep^6$ for a dispersive discrete equation $\CQ$ defined on a square lattice which at the lowest order gives a Nonlinear Schrödinger Equation (NLSE) and in Section <ref> we apply it to the classification of the dispersive multilinear equation $\CQ^+$. Section <ref> is devoted to some conclusive remarks. § THE DISCRETE MULTISCALE INTEGRABILITY TEST Let us consider a dispersive discrete equation of the form $\CQ$, which at the lowest perturbation order gives a NLSE. An example of such a case is given by $\CQ=\CQ^+$ however the results presented below will not be limited to such a case. In such a case the discrete multiscale integrability test may be summarized as follows: i. One considers a small amplitude solution of Eq. (<ref>) given by $u_{n,m}=\ep w_{n,m}$, $0 < \|\ep\| \ll 1$. In such a way (<ref>) will split into linear and nonlinear terms: = ∑_i=1^N ^i _i=0, where $N \in \mathbb{N}$ is the nonlinearity order. $N$ will be infinite only if the nonlinearity of Eq. (<ref>) comes from a non-polynomial function. In the case $\CQ=\CQ^+$ $N \le 4$. In the formal expansion (<ref>) each term $\CQ_i$ contains only homogeneous polynomials of degree $i$ in the field variables $w_{n,m}$ defined on the square. If the discrete equation is dispersive then the linear part $\CQ_1$ admits a solution w_{n,m}=\exp [\ri( \kappa n -\omega m)]=K^n \Omega^m, $\omega=\omega(\kappa)$, the dispersion relation, is a real function of $\kappa$. ii. The multiscale expansion of the basic field variable $w_{n,m}$ around the harmonic $K^n \Omega^m$reads w_n,m= ∑_ℓ=0^∞ ^ℓ∑_α=-ℓ-1^ℓ+1 K^αn Ω^αm u_ℓ+1^(α), where $u^{(\alpha)}_\ell =u^{(\alpha)}_\ell (n_1, \{m_j\})$ is a bounded slowly varying function of its arguments and $u^{(-\alpha)}_{\ell}=\bar u^{(\alpha)}_{\ell}$, $\bar u_{\ell}$ being the complex conjugate of $u_{\ell}$ as we are looking at real solutions. Here $n_1= \ep n$, $m_j = \ep^j m$ $j= 1, 2, \dots$ are the slow-varying lattice iii. The nearest-neighbors fields are expanded according to the following w_n + 1,m = ∑_ℓ=0^∞ ^ℓ∑_α=-ℓ-1^ℓ+1 K^α(n+ 1) Ω^αm ∑_j= max (0, \|α\|-1)^ℓ _ ℓ- j u_j+1^(α) , w_n ,m + 1 = ∑_ℓ=0^∞ ^ℓ∑_α=-ℓ-1^ℓ+1 K^αn Ω^α(m - 1) ∑_j= max (0, \|α\|-1)^ℓ _ ℓ- j u_j+1^(α) , w_n + 1 ,m + 1 = ∑_ℓ=0^∞ ^ℓ∑_α=-ℓ-1^ℓ+1 K^α(n + 1) Ω^ α(m - 1) ∑_j= max (0, \|α\|-1)^ℓ _ ℓ- j u_j+1^(α) , where the operators $\CA_i,\CB_i,\CC_i, are equal to one when $i=0$, while for the lowest values of $i$ they are presented in the following Table: & i=1 & i=2 & i=3& i=4\\ \hline\hline & & & & \\ \CA_i & \delta_{n_1} & \frac12\delta_{n_1}^2 & \frac16\delta_{n_1}^3 & \frac{1}{24}\delta_{n_1}^4\\ & & & & \\ \hline & & & & \\ \CB_i & \delta_{m_1} &\frac12\delta_{m_1}^2 + \delta_{m_2} & \frac16\delta_{m_1}^3+\delta_{m_1}\delta_{m_2}+\delta_{m_3} & \frac{1}{24}\delta_{m_1}^4+\frac12\delta_{m_1}^2\delta_{m_2}+\frac12\delta_{m_2}^2+\delta_{m_1}\delta_{m_3}+\delta_{m_4}\\ & & & & \\ \hline & & & & \\ \CC_i & \nabla & \frac12\nabla^2+\delta_{m_2} & \frac16 \nabla^3+\nabla\delta_{m_2}+\delta_{m_3} & \frac{1}{24}\nabla^4+\frac12\nabla^2\delta_{m_2}+\frac12\delta_{m_2}^2+\nabla\delta_{m_3}+\delta_{m_4}\\ & & & & \\ \hline \end{array} where $\delta_{k}$ are the formal derivatives with respect to the index $k$, $\delta_k\doteq\partial_k$ and $\nabla\doteq\delta_{m_1}+\delta_{n_1}$. The operator $\delta_k$ can always be expressed in terms of powers of the difference operators by the well known identity \delta_{k}= \sum_{i=1}^\infty \frac{(-1)^{i-1}}{i}\Delta_{k}^i, where $\Delta_{k}$ is the discrete first right difference operator with respect to the variable $k$, i.e. $\Delta_k u_k \doteq u_{k+1} - u_k$. A function $f_k$ will be a slow-varying function of order $L$ if $\Delta_k^{L+1} f_k \approx 0$. In such a case the $\delta_k$-operators, which in principle are formal series containing infinite powers of $\Delta_k$, when acting on slow-varying functions of finite order $L$ reduce to polynomials in $\Delta_k$ at most of order $L$. We shall assume here that we are dealing with functions of an infinite slow-varying order, i.e. $L=\infty$, so that the $\delta_k$-operators may be taken as differential operators acting on the indexes of the harmonics $u_j^{(\alpha)}$. iv. When we substitute the expansions (<ref>-<ref>) into (<ref>), we get an equation of the following ∑_j ^j ∑_α _j^(α)K^αn Ω^αm =0, i.e. we must have \CW_j^{(\alpha)}=0$ for all $\alpha$ and $j$. Let us notice that the equations $\CW_j^{(\alpha)}=0$ are equations for the slowly varying functions $u_{\ell+1}^{(\alpha)}$ with $\ell \leq j$. The multiscale expansion of the $\CQ$ equation for functions of infinite order will thus give rise to a set of continuous partial differential equations. By assumption, at lowest order (slow-time $m_{2}$) we get a NLSE. To define the values of the constants appearing in $\CQ$ for which the equation is integrable, we will consider the orders beyond that at which one obtains for the first harmonic $u^{\left(1\right)}_{1}$ the (integrable) NLSE. The first attempts to go beyond the NLSE order in the case of partial difference equations have been presented by Santini, Degasperis and Manakov in [4] and by Kodama and Mikhailov using normal forms[9]. In [4], the authors, starting from integrable models, through a combination of asymptotic functional analysis and spectral methods, succeeded in removing all the secular terms from the reduced equations, order by order. Their results are summarized in the following statements: * The number of slow-time variables required for the amplitudes $u^{\left(\alpha\right)}_{j}$s coincides with the number of nonvanishing coefficients $\omega_{j}\left(k\right)=\frac{1}{j!}\frac{d^j \omega(k)}{dk^j}$; * The amplitude $u^{\left(1\right)}_{1}$ evolves at the slow-times $t_{\sigma}$, $\sigma\geq 3$ according to the $\sigma$-th equation of the NLSE hierarchy; * The amplitudes of the higher perturbations of the first harmonic $u^{\left(1\right)}_{j}$, $j\geq 2$ evolve at the slow-times $t_{\sigma}$, $\sigma\geq 2$ according to certain linear, nonhomogeneous equations when taking into account some asymptotic boundary conditions. From the previous statements one can conclude that the cancellation at each stage of the perturbation process of all the secular terms is a sufficient condition to uniquely fix the evolution equations followed by every $u^{\left(1\right)}_{j}$, $j\geq 1$ for each slow-time $t_\sigma$. Conversely from [5] we can affirm that this expansion results secularity-free. In this way this procedure provides necessary and sufficient conditions to get secularity-free reduced equations. Following [5] we can state the following proposition: If a nonlinear dispersive partial difference equation is integrable, then under a multiscale expansion the functions $u^{\left(1\right)}_{l}$, $l\geq1$ satisfy the equations \begin{eqnarray} \partial_{t_{\sigma}}u^{\left(1\right)}_{1}=K_{\sigma}\left[u^{\left(1\right)}_{1}\right],\label{Valentia1}\ \ \ \ \ \ \ \ \ \ \ \ \ \\ M_{\sigma}u^{\left(1\right)}_{j}=f_{\sigma}(j),\ \ \ M_{\sigma}\doteq\partial_{t_{\sigma}}-K_{\sigma}^{\prime}\left[u^{\left(1\right)}_{1}\right],\label{Valentia2} \end{eqnarray} $\forall\ j,\ \sigma\geq 2$, where $K_{\sigma}\left[u^{\left(1\right)}_{1}\right]$ is the $\sigma$-th flow in the nonlinear Schrödinger hierarchy. All the other $u_{j}^{(\kappa)}$, $\kappa\geq 2$ are expressed in terms of differential monomials of $u_{\rho}^{(1)}$, $\rho\leq j$. In (<ref>) $f_{\sigma}(j)$ is a nonhomogeneous nonlinear forcing term depending on all the $u^{(1)}_{\kappa}$, $1\leq\kappa\leq j-1$, their complex conjugates and their $\xi$-derivatives, where $\xi$ is a variable depending on the group velocity and expressed through a linear combination of the slow-space and the first slow-time $t_{1}$, while $K_{\sigma}^{\prime}\left[u\right]v$ is the Frechet derivative of the nonlinear term $K_{\sigma}[u]$ along the direction $v$ defined by K_{\sigma}^{\prime}[u]v\doteq\frac{d} {ds}K_{\sigma}[u+sv]\mid_{s=0},\nonumber i.e. the linearization of the expression $K_{\sigma}[u]$ along the direction $v$ near the function $u$. In order to characterize the flows $K_{\sigma}\left[u^{\left(1\right)}_{1}\right]$ and the nonlinear forcing terms $f_{\sigma}(j)$, following [3], we introduce the finite dimensional vector spaces $\mathcal{P}_{\ell}$, $\ell\geq 2$, as being the set of all homogeneous, fully-nonlinear, differential polynomials in the functions $u_{j}^{(1)}$, $j\geq 1$, their complex conjugates and their $\xi$-derivatives of homogeneity degree $\ell$ in $\ep$ and $1$ in $e^{\ri\theta}$, where \mbox{order}_{\ep}\left(\partial_{\xi}^{\kappa}u^{(1)}_{j}\right)=\mbox{order}_{\ep}\left(\partial_{\xi}^{\kappa}\bar u^{(1)}_{j}\right)=\kappa+j,\quad \kappa\geq 0. We introduce the subspaces $\mathcal{P}_{ \ell}(\jmath)$ of $\mathcal{P}_{\ell}$, $\jmath\geq 1$, $\ell\geq 2$, whose elements are homogeneous, fully-nonlinear, differential polynomials in the functions $u_{k}^{(1)}$, their complex conjugates and their $\xi$-derivatives, with $1\leq k\leq \jmath$. From these definitions it follows that $\mathcal{P}_{\ell}=\mathcal{P}_{\ell}\left(\ell-2\right)$, that is $\jmath\leq \ell-2$. In fact the terms $u^{(1)}_{\ell}$ and $\bar u^{(1)}_{\ell}$, as well as $\partial_{\xi}u^{(1)}_{\ell-1}$ and $\partial_{\xi}\bar u^{(1)}_{\ell-1}$, are not included in $\mathcal{P}_{\ell}$ as any monomial should enter nonlinearly and terms like $u^{(1)}_{\ell-1}$ and $\bar u^{(1)}_{\ell-1}$ cannot be combined with any other of the monomials $u^{(1)}_{1}$ or $\bar u^{(1)}_{1}$ to give the right homogeneity degree in $e^{\ri\theta}$. For the same reasons, terms of the types $\partial_{\xi}^{\kappa}u^{(1)}_{\ell-\kappa}$, $\partial_{\xi}^{\kappa}\bar u^{(1)! }_{\ell-\kappa}$, $0\leq\kappa\leq \ell-1$ and $\partial_{\xi}^{\kappa}u^{(1)}_{\ell-\kappa-1}$, $\partial_{\xi}^{\kappa}\bar u^{(1)}_{\ell-\kappa-1}$, $0\leq\kappa\leq \ell-2$ cannot appear. So the space $\mathcal{P}_{\ell}(\jmath)$ is defined as that functional space generated by the base of monomials of the following types \begin{eqnarray} \prod_{\alpha,\beta,\gamma,\delta}\left(\partial_{\xi}^{\alpha}u^{(1)}_{\beta}\right)^{\rho\left(\alpha,\beta\right)}\left(\partial_{\xi}^{\gamma}\bar u^{(1)}_{\delta}\right)^{\sigma\left(\gamma,\delta\right)},\ \ \ \rho\left(\alpha,\beta\right)\geq 0,\ \ \forall\alpha,\beta,\ \ \ \sigma\left(\gamma,\delta\right)\geq 0,\ \ \forall\gamma,\delta,\nonumber \end{eqnarray} where the product is carried out for all $\alpha$, $\beta$, $\gamma$ and $\delta$ such that $1\leq\beta,\delta\leq \jmath\leq \ell-2$, $0\leq\alpha\leq \ell-\beta-2$ and $0\leq\gamma\leq \ell-\delta-2$, so that \begin{eqnarray} \sum_{\alpha,\beta,\gamma,\delta}\left(\alpha+\beta\right)\rho\left(\alpha,\beta\right)+\left(\gamma+\delta\right)\sigma\left(\gamma,\delta\right)=\ell,\nonumber\\ \sum_{\alpha,\beta,\gamma,\delta}\rho\left(\alpha,\beta\right)-\sigma\left(\gamma,\delta\right)=1\nonumber.\ \ \ \ \ \ \ \ \ \ \ \end{eqnarray} For $n\geq 3$ the subspaces $\mathcal{P}_{\ell}(\jmath)$, can be generated recursively starting from the lowest one, corresponding to $\ell=2$ by the following relation \begin{eqnarray} \mathcal{P}_{\ell}(\jmath)=\partial_{\xi}\mathcal{P}_{\ell-1}(\jmath)\cup\left\{\prod_{\beta,\delta}\left(u^{(1)}_{\beta}\right)^{\rho\left(\beta\right)}\left(\bar u^{(1)}_{\delta}\right)^{\sigma\left(\delta\right)}\right\},\nonumber \end{eqnarray} where $\rho\left(\beta\right)\geq 0$ $\forall\beta$, $\sigma\left(\delta\right)\geq 0$ $\forall\delta$ and the product is extended for $1\leq\beta,\delta\leq \jmath\leq \ell-2$, so that \begin{eqnarray} \sum_{\beta,\delta}\beta\rho\left(\beta\right)+\delta\sigma\left(\delta\right)=\ell,\ \ \ \sum_{\beta,\delta}\rho\left(\beta\right)-\sigma\left(\delta\right)=1.\nonumber \end{eqnarray} It is then clear that in general $K_{n}\left[u^{\left(1\right)}_{1}\right]\in\left\{\partial_{\xi}^{\ell}u^{\left(1\right)}_{1}\right\}\cup\mathcal{P}_{\ell+1}(1)$ and that $f_{\sigma}(j)\in\mathcal{P}_{\sigma+j}(j-1)$, $\forall\sigma$, $j\geq 2$. Eqs. (<ref>) are necessary conditions for integrability and represent a hierarchy of compatible evolutions for the function $u^{\left(1\right)}_{1}$ at different slow-times. The compatibility of (<ref>) implies some commutativity conditions among their r.h.s. $f_{\sigma}(j)$. If they are satisfied the operators $M_{\sigma}$ defined in Eq. (<ref>) commute among themselves. Once we fix the index $j\geq 2$ in the set of Eqs. (<ref>), this commutativity condition implies the following compatibility conditions \begin{eqnarray} M_{\sigma}f_{\sigma'}\left(j\right)=M_{\sigma'}f_{\sigma}\left(j\right),\ \ \ \forall\, \sigma,\sigma'\geq 2,\label{Lavinia} \end{eqnarray} where, as $f_{\sigma}\left(j\right)$ and $f_{\sigma'}\left(j\right)$ are functions of the different perturbations of the fundamental harmonic up to degree $j-1$, the time derivatives $\partial_{t_{\sigma}}$, $\partial_{t_{\sigma'}}$ of those harmonics appearing respectively in $M_{\sigma}$ and $M_{\sigma'}$ have to be eliminated using the evolution equations (<ref>) up to the index $j-1$. The commutativity conditions (<ref>) turn out to be an integrability test. We finally define the degree of integrability of a given equation: If the relations (<ref>) are satisfied up to the index $j$, $j\geq 2$, we say that our equation is asymptotically integrable of degree $j$ or $A_{j}$-integrable. Conjecturing that an $A_{\infty}$ degree of asymptotic integrability actually implies integrability, we have that under this assumption the relations (<ref>, <ref>) are a sufficient condition for the integrability or that integrability is a necessary condition to have a multiscale expansion where all the Eqs. (<ref>) are satisfied. So the multiscale integrability test tell us that $\CQ$ will be integrable if its multiscale expansion will follow all the infinite relations (<ref>, <ref>). The higher the degree of asymptotic integrability, the closer the equation will be to an integrable one. However, as we can test the conditions (<ref>, <ref>) only up to a finite order (actually $4$), from them we can only derive necessary conditions for integrability, so we will not be able to state with $100\%$ certainty that the discrete equation is integrable. The results obtained at a finite but sufficiently high o! rder will have a good probability to correspond to an integrable equation, but we need to use other techniques to prove it with $100\%$ certainty. Let us present for completeness in the following the lowest order conditions for asymptotic integrability of order $k$ or $A_{k}$-integrability conditions. To simplify the notation, we will use for $u^{\left(1\right)}_{j}$ the concise form $u(j)$, $j\geq 1$. Moreover, for convenience of the reader, we list the fluxes $K_{\sigma}\left[u\right]$ of the NLSE hierarchy for $u$ up to $\sigma=5$: \begin{align} K_{1}[u]&\doteq Au_{\xi},\\ K_{2}[u]&\doteq-\ri\rho_{1}\left[u_{\xi\xi}+\frac{\rho_{2}} {\rho_{1}}\|u\|^2u \right], \label{Rutuli1}\\ K_{3}[u]&\doteq B\left[u_{\xi\xi\xi}+\frac{3\rho_{2}} {\rho_{1}}\|u\|^2u_{\xi}\right],\label{Rutuli2}\\ K_{4}[u]&\doteq-\ri C\left\{u_{\xi\xi\xi\xi}+\frac{\rho_{2}} {\rho_{1}}\left[\frac{3\rho_{2}} {2\rho_{1}}\|u\|^4u+4\|u\|^2u_{\xi\xi}+3u_{\xi}^2\bar u+2\|u_{\xi}\|^2u+u^2\bar u_{\xi\xi}\right]\right\},\label{Rutuli3}\\ K_{5}[u]&\doteq D\bigg\{u_{\xi\xi\xi\xi\xi}\nonumber\\ &\qquad{}+\frac{5\rho_{2}}{\rho_{1}}\left[\frac{3\rho_{2}}{2\rho_{1}}\|u\|^4u_{\xi}+\|u_{\xi}\|^2u_{\xi}+u\bar u_{\xi}u_{\xi\xi}+2\bar uu_{\xi}u_{\xi\xi}+uu_{\xi}\bar u_{\xi\xi}+\|u\|^2u_{\xi\xi\xi}\right]\bigg\},\label{Rutuli4} \end{align} and the corresponding $K_{\sigma}^{'}[u]v$ up to $\sigma=4$: \begin{align} &K_{1}^{\prime}[u]v= Av_{\xi},\label{Arenta}\\ &K_{2}^{\prime}[u]v=-\ri\rho_{1}\left\{v_{\xi\xi}+\frac{\rho_{2}} {\rho_{1}}\left[u^2\bar v+2\|u\|^2v\right]\right\},\label{ArtemideEfesina}\ \ \ \ \ \ \\ &K_{3}^{\prime}[u]v=B\left\{v_{\xi\xi\xi}+\frac{3\rho_{2}} {\rho_{1}}\left[\|u\|^2v_{\xi}+\bar uu_{\xi}v+uu_{\xi}\bar v\right]\right\},\label{Abruzzo3}\\ &K_{4}^{\prime}\left[u\right]v=-iC\left\{v_{\xi\xi\xi\xi}+\frac{\rho_{2}}{\rho_{1}}\left[u^{2}\bar v_{\xi\xi}+4\|u\|^{2}v_{\xi\xi}+2uu_{\xi}\bar v_{\xi}+2u\bar u_{\xi}v_{\xi}+6\bar u u_{\xi}v_{\xi}+4uu_{\xi\xi}\bar v+\right.\right. \nonumber\\ &\qquad\quad\quad\qquad\left.\left.+3u_{\xi}^{2}\bar v+\frac{3\rho_{2}}{\rho_{1}}\|u\|^{2}u^{2}\bar v+4\bar u u_{\xi\xi}v+2u\bar u_{\xi\xi}v+\frac{9\rho_{2}}{2\rho_{1}}\|u\|^{4}v+2\|u_{\xi}\|^{2}v\right]\right\}, \end{align} where $A\not=0$, $\rho_{1}\not=0$, $\rho_{2}$, $B\not=0$, $C\not=0$ and $D\not=0$, if $\rho_{2}\not=0$, are arbitrary real constants. §.§ The $A_{1}$-integrability condition. The $A_{1}$-integrability condition is given by the reality of the coefficient $\rho_{2}$ of the nonlinear term in the NLSE. It is obtained commuting the NLSE flux $K_{2}[u]$ with the flux $B\left[u_{\xi\xi\xi}+\tau \|u\|^2u_{\xi}+\mu u^2\bar u_{\xi}\right]$ with $\tau$ and $\mu$ constants. This commutativity condition gives, if $\rho_{2}\not =0$, \begin{eqnarray} \operatorname{Im}\left[\rho_{2}\right]=\operatorname{Im}\left[B\right]=\operatorname{Im}\left[\rho_{1}\right]=0,\ \ \ \ \ \tau=3\rho_{2}/\rho_{1},\ \ \ \ \ \mu=0.\label{Montesiepi} \end{eqnarray} We remark that, when $\rho_{2}\not=0$, by the same method it is possible to determine all the coefficients of all the higher NLSE-symmetries (<ref>) together with the reality conditions of the coefficients $A$, $C$ and $D$. §.§ The $A_{2}$-integrability conditions. The $A_{2}$- integrability conditions are obtained choosing $j=2$ in the compatibility conditions (<ref>) with $\sigma=2$ and $\sigma'=3$ or alternatively $\sigma'=4$, respectively \begin{gather} \end{gather} In this case $f_{2}(2)$, $f_{3}(2)$ and $f_{4}(2)$ will be respectively identified by 2, ($a, b$), 5, ($\alpha, \beta, \gamma, \delta, \epsilon$), and 8, ($\theta_1, \cdots, \theta_8$), complex constants \begin{align} f_{2}(2)&\doteq au_{\xi}(1)\|u(1)\|^2+b\bar u_{\xi}(1)u(1)^2,\label{Abruzzo1}\\ f_{3}(2)&\doteq\alpha \|u(1)\|^4u(1)+\beta \|u_{\xi}(1)\|^2u(1)+\gamma u_{\xi}(1)^2\bar u(1)+\label{Abruzzo2}\\ &\qquad{}+\delta\bar u_{\xi\xi}(1)u(1)^2+\epsilon \|u(1)\|^2u_{\xi\xi}(1),\nonumber\\ f_{4}\left(2\right)&\doteq\theta_{1}\|u\left(1\right)\|^{4}u_{\xi}\left(1\right)+\theta_{2}\|u\left(1\right)\|^{2}u\left(1\right)^{2}\bar u_{\xi}\left(1\right)+\theta_{3}\|u_{\xi}\left(1\right)\|^{2}u_{\xi}\left(1\right)+\\ &\qquad{}+\theta_{4}u\left(1\right)\bar u_{\xi}\left(1\right)u_{\xi\xi}\left(1\right)+\theta_{5}\bar u\left(1\right)u_{\xi}\left(1\right)u_{\xi\xi}\left(1\right)+\theta_{6}u\left(1\right)u_{\xi}\left(1\right)\bar u_{\xi\xi}\left(1\right)+\nonumber\\ &\qquad{}+\theta_{7}\|u\left(1\right)\|^{2}u_{\xi\xi\xi}\left(1\right)+\theta_{8}u\left(1\right)^{2}\bar u_{\xi\xi\xi}\left(1\right).\nonumber \end{align} As $\rho_{2}\not=0$, eliminating from Eq. (<ref>) the derivatives of $u(1)$ with respect to the slow-times $t_{2}$ and $t_{3}$ using the evolutions (<ref>) with $\sigma=2$ and $\sigma'=3$ and equating term by term, we obtain the following 2 $A_{2}$-integrability conditions \begin{eqnarray} a=\bar a,\ \ \ b=\bar b.\label{CieloUrbico} \end{eqnarray} So we have two conditions obtained when requiring the reality of the coefficients $a$ and $b$. The expressions of $\alpha$, $\beta$, $\alpha$, $\delta$ in terms of $a$ and $b$ are: \begin{eqnarray} \alpha=\frac{3\ri Ba\rho_{2}} {4\rho_{1}^2},\ \ \ \beta=\frac{3\ri Bb} {\rho_{1}},\ \ \ \gamma=\frac{3\ri Ba} {2\rho_{1}},\ \ \ \delta=0,\ \ \ \epsilon=\gamma.\label{Molise} \end{eqnarray} The same integrability conditions (<ref>) can be derived using Eq. (<ref>). As in our analysis we will need them, here follow the explicit expressions of the coefficients of the forcing term $f_{4}\left(2\right)$ \begin{equation} \begin{gathered} \theta_{1}=\frac{6Ca\rho_{2}}{\rho_{1}^2},\ \ \ \theta_{2}=\frac{3Cb\rho_{2}}{\rho_{1}^2},\ \ \ \theta_{3}=\frac{\left(a+3b\right)C}{\rho_{1}},\ \ \ \theta_{4}=\frac{\left(a+4b\right)C}{\rho_{1}}, \\ \theta_{5}=\frac{5Ca}{\rho_{1}},\ \ \ \theta_{6}=\frac{\left(a+2b\right)C}{\rho_{1}},\ \ \ \theta_{7}=\frac{2Ca}{\rho_{1}},\ \ \ \theta_{8}=\frac{Cb}{\rho_{1}}. \end{gathered}\label{Moly} \end{equation} §.§ The $A_{3}$-integrability conditions. The $A_{3}$-integrability conditions are derived in a similar way setting $j=3$ in the compatibility conditions (<ref>) with $\sigma=2$ and $\sigma'=3$, so that $M_{2}f_{3}\left(3\right)=M_{3}f_{2}\left(3\right)$. In this case $f_{2}(3)$ and $f_{3}(3)$ will be respectively identified by 12 and 26 complex constants \begin{align} f_{2}(3)&\doteq\tau_{1}\|u(1)\|^4u(1)+\tau_{2}\|u_{\xi}(1)\|^2u(1)+\tau_{3}\|u(1)\|^2u_{\xi\xi}(1)+\tau_{4}\bar u_{\xi\xi}(1)u(1)^2\nonumber\\ &\qquad{}+\tau_{7}\bar u_{\xi}(2)u(1)^2+\tau_{8}u(2)^2\bar u(1)+\tau_{9}\|u(2)\|^2u(1)+\tau_{10}u(2)u_{\xi}(1)\bar u(1)\label{Lazio4}\\ &\qquad{}+\tau_{11}u(2)\bar u_{\xi}(1)u(1)+\tau_{12}\bar u(2)u_{\xi}(1)u(1)+\tau_{5}u_{\xi}(1)^2\bar u(1)+\tau_{6}u_{\xi}(2)\|u(1)\|^2,\nonumber\\ f_{3}(3)&\doteq\gamma_{1}\|u(1)\|^4u_{\xi}(1)+\gamma_{2}\|u(1)\|^2u(1)^2\bar u_{\xi}(1)+\gamma_{3}\|u(1)\|^2u_{\xi\xi\xi}(1)\nonumber\\ &\qquad{}+\gamma_{5}\|u_{\xi}(1)\|^2u_{\xi}(1)+\gamma_{6}\bar u_{\xi\xi}(1)u_{\xi}(1)u(1)+\gamma_{7}u_{\xi\xi}(1)\bar u_{\xi}(1)u(1)\nonumber\\ &\qquad{}+\gamma_{9}\|u(1)\|^4u(2)+\gamma_{10}\|u(1)\|^2u(1)^2\bar u(2)+\gamma_{11}\bar u_{\xi}(1)u(2)^2+\gamma_{12}u_{\xi}(1)\|u(2)\|^2\nonumber\\ &\qquad{}+\gamma_{13}\|u_{\xi}(1)\|^2u(2)+\gamma_{14}\|u(2)\|^2u(2)+\gamma_{15}u_{\xi}(1)^{2}\bar u(2)+\gamma_{16}\|u(1)\|^2u_{\xi\xi}(2)\nonumber\\ &\qquad{}+\gamma_{17}u(1)^2\bar u_{\xi\xi}(2)+\gamma_{18}u(2)\bar u_{\xi\xi}(1)u(1)+\gamma_{19}u(2)u_{\xi\xi}(1)\bar u(1)\nonumber\\ &\qquad{}+\gamma_{21}u(2)u_{\xi}(2)\bar u(1)+\gamma_{22}\bar u(2)u_{\xi}(2)u(1)+\gamma_{23}u_{\xi}(2)u_{\xi}(1)\bar u(1)\nonumber\\ &\qquad{}+\gamma_{25}\bar u_{\xi}(2)u_{\xi}(1)u(1)+\gamma_{26}\bar u_{\xi}(2)u(2)u(1)+\gamma_{4}u(1)^2\bar u_{\xi\xi\xi}(1) \nonumber \\ &\qquad{}+\gamma_{8}u_{\xi\xi}(1)u_{\xi}(1)\bar u(1)+\gamma_{20}\bar u(2)u_{\xi\xi}(1)u(1)+\gamma_{24}u_{\xi}(2)\bar u_{\xi}(1)u(1).\label{Lazio5} \end{align} Let us eliminate from Eq. (<ref>) with $j=3$ the derivatives of $u(1)$ with respect to the slow-times $t_{2}$ and $t_{3}$ using the evolutions (<ref>) respectively with $\sigma=2$ and $\sigma'=3$ and the derivatives of $u(2)$ using the evolutions (<ref>) with $\sigma=2$ and $\sigma'=3$. Let us equate the remaining terms term by term, if $\rho_{2}\not=0$, and, indicating with $R_{i}$ and $I_{i}$ the real and imaginary parts of $\tau_{i}$, $i=1,\ldots,12$, we obtain the following 15 $A_{3}$-integrability conditions \begin{equation} \begin{gathered} R_{1}=-\frac{aI_{6}} {4\rho_{1}},\qquad R_{3}=\frac{(b-a)I_{6}} {2\rho_{2}}-\frac{aI_{12}} {2\rho_{2}},\qquad R_{4}=\frac{R_{2}} {2}+\frac{(a-b)I_{6}} {4\rho_{2}}+\frac{aI_{12}} {4\rho_{2}},\\ R_{5}=\frac{R_{2}} {2}+\frac{(a-b)I_{6}} {4\rho_{2}}+\frac{(2b-a)I_{12}} {4\rho_{2}},\qquad R_{6}=-\frac{aI_{8}} {\rho_{2}},\qquad R_{7}=R_{12}+\frac{(a-b)I_{8}} {\rho_{2}},\\ R_{8}=R_{9}=0,\qquad R_{10}=R_{12},\qquad R_{11}=R_{12}+\frac{(a-2b)I_{8}} {\rho_{2}},\\ I_{4}=\frac{(b+a)R_{12}} {4\rho_{2}}+\frac{\rho_{1}I_{1}} {\rho_{2}}+\frac{I_{2}-I_{3}-2I_{5}} {4}+\frac{\left[2b(a-b)+a^2\right]I_{8}} {4\rho_{2}^2},\qquad I_{7}=0,\\ I_{9}=2I_{8},\qquad I_{10}=I_{12},\qquad I_{11}=I_{6}+I_{12}. \end{gathered}\label{Siculi} \end{equation} For completeness we give the expressions of $\gamma_{j}$, $j=1,\ldots,26$ as functions of $\tau_{i}$, $i=1,\ldots,12$: \begin{equation} \begin{gathered} \gamma_{1}=\frac{3B} {8\rho_{1}^2}\left[-2bR_{12}-8\rho_{1}I_{1}+2(I_{2}-2I_{3}-2I_{5})\rho_{2}+\ri (b-5a)I_{6}+\frac{2a^2I_{8}} {\rho_{2}}-3\ri aI_{12}\right], \\ \gamma_{2}=-\frac{3Ba} {4\rho_{1}^2}\left[\ri I_{6}+\frac{(a-2b)I_{8}} {\rho_{2}}+\tau_{12}\right],\quad \gamma_{3}=\frac{3\ri B\tau_{3}} {2\rho_{1}},\quad \gamma_{4}=0,\quad \gamma_{5}=\frac{3\ri B\tau_{2}} {2\rho_{1}}, \\ \gamma_{6}=\frac{3\ri B\tau_{4}} {\rho_{1}},\quad \gamma_{7}=\gamma_{5},\quad \gamma_{8}=\gamma_{3}+\frac{3\ri B\tau_{5}} {\rho_{1}},\quad \gamma_{9}=-\frac{3B(\rho_{2}I_{6}+3a\ri I_{8})} {4\rho_{1}^2}, \\ \gamma_{10}=\frac{3\ri B\rho_{2}R_{6}} {2\rho_{1}^2},\quad \gamma_{11}=0,\quad \gamma_{12}=\frac{3\ri B\tau_{9}} {2\rho_{1}},\quad \gamma_{13}=\frac{3\ri B\tau_{11}} {2\rho_{1}},\quad \gamma_{14}=0, \\ \gamma_{15}=\frac{3\ri B\tau_{12}} {2\rho_{1}},\quad\gamma_{16}=\frac{3\ri B\tau_{6}} {2\rho_{1}},\quad \gamma_{17}=\gamma_{18}=0,\quad \gamma_{19}=\frac{3\ri B\tau_{10}} {2\rho_{1}},\quad \gamma_{20}=\gamma_{15}, \\ \gamma_{21}=\frac{3\ri B\tau_{8}} {\rho_{1}},\quad\gamma_{22}=\gamma_{12},\quad \gamma_{23}=\gamma_{16}+\gamma_{19},\quad \gamma_{24}=\gamma_{13},\quad \gamma_{25}=\frac{3\ri B\tau_{7}} {\rho_{1}},\quad \gamma_{26}=0. \end{gathered}\label{gam} \end{equation} §.§ The $A_{4}$-integrability conditions. The $A_{4}$-integrability conditions are derived similarly from (<ref>) with $j=4$, that is $M_{2}f_{3}\left(4\right)=M_{3}f_{2}\left(4\right)$. Now $f_{2}(4)$ and $f_{3}(4)$ are respectively defined by 34 and 77 complex constants \begin{equation}\label{f24} \begin{aligned} f_{2}\left(4\right)&\doteq\eta_{1}\|u(1)\|^4u_{\xi}(1)+\eta_{2}\|u(1)\|^2u(1)^2\bar u_{\xi}(1)+\eta_{3}\|u(1)\|^2u_{\xi\xi\xi}(1)\\ &\qquad{}+\eta_{5}\|u_{\xi}(1)\|^2u_{\xi}(1)+\eta_{6}\bar u_{\xi\xi}(1)u_{\xi}(1)u(1)+\eta_{7}u_{\xi\xi}(1)\bar u_{\xi}(1)u(1)\\ &\qquad{}+\eta_{9}\|u(1)\|^4u(2)+\eta_{10}\|u(1)\|^2u(1)^2\bar u(2)+\eta_{11}\bar u_{\xi}(1)u(2)^2+\eta_{12}u_{\xi}(1)\|u(2)\|^2\\ &\qquad{}+\eta_{13}\|u_{\xi}(1)\|^2u(2)+\eta_{14}\|u(2)\|^2u(2)+\eta_{15}u_{\xi}(1)^{2}\bar u(2)+\eta_{16}\|u(1)\|^2u_{\xi\xi}(2)\\ &\qquad{}+\eta_{17}u(1)^2\bar u_{\xi\xi}(2)+\eta_{18}u(2)\bar u_{\xi\xi}(1)u(1)+\eta_{19}u(2)u_{\xi\xi}(1)\bar u(1)\\ &\qquad{}+\eta_{21}u(2)u_{\xi}(2)\bar u(1)+\eta_{22}\bar u(2)u_{\xi}(2)u(1)+\eta_{23}u_{\xi}(2)u_{\xi}(1)\bar u(1)\\ &\qquad{}+\eta_{25}\bar u_{\xi}(2)u_{\xi}(1)u(1)+\eta_{26}\bar u_{\xi}(2)u(2)u(1)+\eta_{4}u(1)^2\bar u_{\xi\xi\xi}(1) \\ &\qquad{}+\eta_{8}u_{\xi\xi}(1)u_{\xi}(1)\bar u(1)+\eta_{20}\bar u(2)u_{\xi\xi}(1)u(1)+\eta_{24}u_{\xi}(2)\bar u_{\xi}(1)u(1)+\\ &\qquad{}+\eta_{27}u(1)\bar u_{\xi}(1)u(3)+\eta_{28}\bar u(1)u_{\xi}(1)u(3)+\eta_{29}u(1)u_{\xi}(1)\bar u(3)+\\ &\qquad{}+\eta_{30}u(1)\bar u(2)u(3)+\eta_{31}\bar u(1)u(2)u(3)+\eta_{32}u(1)u(2)\bar u(3)+\\ &\qquad{}+\eta_{33}\|u(1)\|^2u_{\xi}(3)+\eta_{34}u(1)^2\bar u_{\xi}(3), \end{aligned} \end{equation} \begin{equation} \begin{aligned}\label{f34} f_{3}(4)&\doteq\kappa_{1}u(1)\|u(1)\|^6+\kappa_{2}\|u(1)\|^2\bar u(1)u_{\xi}(1)^2+\kappa_{3}\|u(1)\|^2u(1)\|u_{\xi}(1)\|^2+\kappa_{4}u(1)^3\bar u_{\xi}(1)^2\\ &\quad{}+\kappa_{5}\|u(1)\|^4u_{\xi\xi}(1)+\kappa_{6}\|u(1)\|^2u(1)^2\bar u_{\xi\xi}(1)+\kappa_{7}\|u_{\xi}(1)\|^2u_{\xi\xi}(1)+\kappa_{8}u_{\xi}(1)^2\bar u_{\xi\xi}(1)\\ &\quad{}+\kappa_{9}u(1)\|u_{\xi\xi}(1)\|^2+\kappa_{10}\bar u(1)u_{\xi\xi}(1)^2+\kappa_{11}\bar u(1)u_{\xi}(1)u_{\xi\xi\xi}(1)+\kappa_{12}u(1)\bar u_{\xi}(1)u_{\xi\xi\xi}(1)\\ &\quad{}+\kappa_{13}u(1)u_{\xi}(1)\bar u_{\xi\xi\xi}(1)+\kappa_{14}\|u(1)\|^2u_{\xi\xi\xi\xi}(1)+\kappa_{15}u(1)^2\bar u_{\xi\xi\xi\xi}(1)\\ &\quad{}+\kappa_{16}\|u(1)\|^2\bar u(1)u(2)^2+\kappa_{17}\|u(1)\|^2u(1)\|u(2)\|^2+\kappa_{18}u(1)^3\bar u(2)^2\\ &\quad{}+\kappa_{19}\|u(1)\|^2\bar u(1)u_{\xi}(1)u(2)+\kappa_{20}\|u(1)\|^2u(1)\bar u_{\xi}(1)u(2)+\kappa_{21}\|u(1)\|^2u(1)u_{\xi}(1)\bar u(2)\\ &\quad{}+\kappa_{22}u(1)^3\bar u_{\xi}(1)\bar u(2)+\kappa_{23}\bar u_{\xi}(1)u_{\xi\xi}(1)u(2)+\kappa_{24}u_{\xi}(1)\bar u_{\xi\xi}(1)u(2)\\ &\quad{}+\kappa_{25}u_{\xi}(1)u_{\xi\xi}(1)\bar u(2)+\kappa_{26}u(1)\bar u_{\xi\xi\xi}(1)u(2)+\kappa_{27}\bar u(1)u_{\xi\xi\xi}(1)u(2)\\ &\quad{}+\kappa_{28}u(1)u_{\xi\xi\xi}(1)\bar u(2)+\kappa_{29}\bar u_{\xi\xi}(1)u(2)^2+\kappa_{30}u_{\xi\xi}(1)\|u(2)\|^2+\kappa_{31}\|u(1)\|^4u_{\xi}(2)\\ &\quad{}+\kappa_{32}\|u(1)\|^2u(1)^2\bar u_{\xi}(2)+\kappa_{33}\|u_{\xi}(1)\|^2u_{\xi}(2)+\kappa_{34}u_{\xi}(1)^2\bar u_{\xi}(2)\\ &\quad{}+\kappa_{35}\bar u(1)u_{\xi\xi}(1)u_{\xi}(2)+\kappa_{36}u(1)\bar u_{\xi\xi}(1)u_{\xi}(2)+\kappa_{37}u(1)u_{\xi\xi}(1)\bar u_{\xi}(2)\\ &\quad{}+\kappa_{38}u(1)\bar u_{\xi}(1)u_{\xi\xi}(2)+\kappa_{39}\bar u(1)u_{\xi}(1)u_{\xi\xi}(2)+\kappa_{40}u(1)u_{\xi}(1)\bar u_{\xi\xi}(2)\\ &\quad{}+\kappa_{41}\|u(1)\|^2u_{\xi\xi\xi}(2)+\kappa_{42}u(1)^2\bar u_{\xi\xi\xi}(2)+\kappa_{43}\bar u_{\xi}(1)u(2)u_{\xi}(2)\\ &\quad{}+\kappa_{44}u_{\xi}(1)\bar u(2)u_{\xi}(2)+\kappa_{45}u_{\xi}(1)u(2)\bar u_{\xi}(2)+\kappa_{46}u(1)\|u_{\xi}(2)\|^2+\kappa_{47}\bar u(1)u_{\xi}(2)^2\\ &\quad{}+\kappa_{48}\bar u(1)u(2)u_{\xi\xi}(2)+\kappa_{49}u(1)\bar u(2)u_{\xi\xi}(2)+\kappa_{50}u(1)u(2)\bar u_{\xi\xi}(2)\\ &\quad{}+\kappa_{51}\|u(2)\|^2u_{\xi}(2)+\kappa_{52}u(2)^2\bar u_{\xi}(2)+\kappa_{53}\|u(1)\|^4u(3)+\kappa_{54}\|u(1)\|^2u(1)^2\bar u(3)\\ &\quad{}+\kappa_{55}\bar u(1)u(3)^2+\kappa_{56}u(1)\|u(3)\|^2+\kappa_{57}\|u(2)\|^2u(3)+\kappa_{58}u(2)^2\bar u(3)\\ &\quad{}+\kappa_{59}\|u_{\xi}(1)\|^2u(3)+\kappa_{60}u_{\xi}(1)^2\bar u(3)+\kappa_{61}u(1)\bar u_{\xi\xi}(1)u(3)+\kappa_{62}\bar u(1)u_{\xi\xi}(1)u(3)\\ &\quad{}+\kappa_{63}u(1)u_{\xi\xi}(1)\bar u(3)+\kappa_{64}u(1)\bar u_{\xi}(1)u_{\xi}(3)+\kappa_{65}\bar u(1)u_{\xi}(1)u_{\xi}(3)\\ &\quad{}+\kappa_{66}u(1)u_{\xi}(1)\bar u_{\xi}(3)+\kappa_{67}\|u(1)\|^2u_{\xi\xi}(3)+\kappa_{68}u(1)^2\bar u_{\xi\xi}(3)+\kappa_{69}u_{\xi}(1)\bar u(2)u(3)\\ &\quad{}+\kappa_{70}\bar u_{\xi}(1)u(2)u(3)+\kappa_{71}u_{\xi}(1)u(2)\bar u(3)+\kappa_{72}\bar u(1)u_{\xi}(2)u(3)+\kappa_{73}u(1)\bar u_{\xi}(2)u(3)\\ &\quad{}+\kappa_{74}u(1)u_{\xi}(2)\bar u(3)+\kappa_{75}u(1)\bar u(2)u_{\xi}(3)+\kappa_{76}\bar u(1)u(2)u_{\xi}(3)+\kappa_{77}u(1)u(2)\bar u_{\xi}(3). \end{aligned} \end{equation} If we indicate with $S_{j}$ and $T_{j}$ respectively the real and imaginary parts of $\eta_{j}$, $j=1$, …, $34$, when $\rho_{2}\not=0$, the $A_{4}$-integrability conditions are represented by $48$ real relations whose expressions are presented in the Appendix. To study the distance between an integrable partial differential equation and its discretizations other integrability conditions have been constructed in [13], corresponding to $M_{4}f_{2}\left(3\right)=M_{2}f_{4}\left(3\right)$ ($A_{3}$-integrability conditions) and to $M_{4}f_{2}\left(5\right)=M_{2}f_{4}\left(5\right)$ ($A_{5}$-integrability conditions) in the subspaces characterized by $u\left(2n\right)=0$, $n\geq1$ with purely imaginary coefficients. In this case the $A_{3}$-integrability conditions are given by one real relation which can be deduced from (<ref>) and corresponds to $I_{4}=\rho_{1}I_{1}/\rho_{2}+\left(I_{2}-I_{3}-2I_{5}\right)/4$. The next integrability condition, the $A_{5}$-integrability condition, is given by 14 real conditions, not included in the integrability conditions given here. The results presented in this Section will be used in the following Sections to classify integrable nonlinear equation on the square lattice. § DISPERSIVE AFFINE-LINEAR EQUATIONS ON THE SQUARE LATTICE The aim of this Section is to derive the necessary conditions for the integrability of the simplest class of $\mathbb{Z}^2$-lattice equations, that of dispersive and multilinear equations (<ref>) defined on the square lattice, satisfying the condition (1) with dispersion relation $\omega_{+}(k)$, i.e. \begin{equation} \begin{aligned} &\CQ^+\doteq a_1 (u_{n,m} + u_{n+1,m+1}) + a_2 (u_{n+1,m} + u_{n,m+1}) \\ & \qquad{}+ (\alpha_1-\alpha_2) \, u_{n,m}u_{n+1,m} + (\alpha_1+\alpha_2)\, u_{n,m+1}u_{n+1,m+1} \\ &\qquad + \, (\beta_1-\beta_2)\, u_{n,m}u_{n,m+1} + (\beta_1+\beta_2)\, u_{n+1,m}u_{n+1,m+1} \\ &\qquad{}+ \, \gamma_1 u_{n,m}u_{n+1,m+1} + \gamma_2 u_{n+1,m}u_{n,m+1} \\ &\qquad{}+ \, (\xi_1-\xi_3)\, u_{n,m}u_{n+1,m}u_{n,m+1} + (\xi_1+\xi_3)\, u_{n,m}u_{n+1,m}u_{n+1,m+1} \\ &\qquad{}+ \, (\xi_2-\xi_4)\, u_{n+1,m}u_{n,m+1}u_{n+1,m+1} + (\xi_2+\xi_4)\, u_{n,m}u_{n,m+1}u_{n+1,m+1} \\ &\qquad{}+ \, \zeta u_{n,m} u_{n+ 1,m} u_{n,m + 1} u_{n + 1,m + 1}=0, \label{q4} \end{aligned} \end{equation} where $a_1,a_2 \in \mathbb R\setminus \{0\}$, $\|a_1\| \neq \|a_2\|$, are the coefficients appearing in the linear part while $\alpha_1,\alpha_2,\beta_1,\beta_2,$ $\gamma_1,\gamma_2,$ $\xi_1, \xi_2, \xi_3, \xi_4, \zeta$ are real parameters which enter in the nonlinear part of the system. Here we will look, by using the multiscale procedure described in Section <ref> into the values of these coefficients such that the class $\CQ^+$ is $A_1$, $A_2$, $A_3$ and $A_4$ integrable. ( 0, 0)(1,0)100 ( 100, 0)(-1,0)100 ( 0,100)(1,0)100 ( 100,100)(-1,0)100 ( 0, 0)(0,1)100 ( 0, 100)(0,-1)100 (100, 0)(0,1)100 (100, 100)(0,-1)100 (0, 0)(1,1)100 (100, 0)(-1,1)100 (97, -3)$\bullet$ (-3, -3)$\bullet$ (-3, 97)$\bullet$ (97, 97)$\bullet$ Graphical representation of the quadratic nonlinearities of $\CQ^+$ To perform a classification of the equations $\CQ^+$, we need to find the set of transformations that leave them invariant, i.e. the equivalence group. As mentioned before, a generic multilinear equation of the form (<ref>) is invariant under a Möbius transformation (<ref>). The constant term $f_0$ and the differences $a_{00}-a_{11}$, $a_{01}-a_{10}$ transform according to f_0T↦ f_0'=D^4f_0+ 2 B^3D (ξ_1+ξ_2)+ B^2 D^2 a_00-a_11T↦a_00'-a_11'=Δ[D^2 (a_00-a_11)+B^2 (ξ_1-ξ_2-ξ_3+ξ_4)-2 B D a_01-a_10T↦ a_01'-a_10'=Δ[D^2 (a_01-a_10)-B^2 (ξ_1-ξ_2+ ξ_3-ξ_4)+2 B D with $\Delta=A D-B C$. These formulas allow to determine when a given linear-affine equation (<ref>) can be transformed into one belonging to class $\CQ^+$. For this to happen all three terms must be null, so setting the l.h.s. of (<ref>) to zero we get three polynomial equations for $B/D$ or $D/B$. If simultaneously solved (over the reals), we have an equation of the class $\CQ^+$. One could try to write the conditions over the coefficients of a general linear-affine equation (<ref>) by using resultant calculations on the three polynomial conditions, but they turn out to be too complicated. A solution of (<ref>) is given by restricted simultaneous Möbius transformations $R$ of the form u_n,m↦u_n,m'=u_n,m/(Cu_n,m+D), ∀ n, m which will be our equivalence transformation. Under (<ref>) the coefficients of Eq. (<ref>) undergo the following transformations: a_1R↦ a_1'= D^3a_1, a_2R↦ a_2'= D^3a_2, α_1R↦α_1'= D^2 [α_1+C(a_1+a_2) ], α_2R↦α_2'= D^2α_2, β_1R↦ β_1'= D^2 [β_1+C(a_1+a_2) ], β_2R↦β_2'= D^2β_2, γ_1R↦ γ_1'= D^2 (γ_1+2C a_1 ), γ_2R↦ γ_2'= D^2 (γ_2+2C a_2 ), ξ_1R↦ ξ_1'= D ξ_1+12 C D [3C +γ_1+γ_2+2 (α_1-α_2+β_1)], ξ_2R↦ ξ_2'= D ξ_2+12 C D [3C +γ_1+γ_2+2 (α_1+α_2+β_1)], ξ_3R↦ ξ_3'= D ξ_3+12 C D [C(a_1 -a_2)+γ_1-γ_2+2 β_2], ξ_4R↦ ξ_4'= D ξ_4+12 C D [C(a_1 -a_2)+γ_1-γ_2-2 β_2], ζR↦ ζ'= ζ+C^2 [2C +γ_1+γ_2+2 (α_1+β_1)]+2C We will indicate by $\CN$ the number of free parameters (although not all of them essential under $R$) appearing in each sub case of q4. Its maximum number is $\CN=13$, the number of free coefficients in q4. §.§ Classification at order $\ep^3$. By performing the multiscale expansion of Eq. (<ref>), the following statement holds for the $A_{1}$-asymptotic integrability The lowest order necessary conditions for the integrability of equations $\CQ^+$ read: * Case 1 ($\CN=9$): α_2 = β_2 = 0, ξ_1 = ξ_2, ξ_3 = ξ_4. * Case 2 ($\CN=7$) : α_2 = β_2, α_1 = β_1, a_1 =2 a_2 , γ_1 = 2 γ_2, a_1 (ξ_1 - ξ_2) = - a_1 (ξ_3 - ξ_4) = -2α_2 γ_2. * Case 3 ($\CN=7$): α_2 = - β_2, α_1 = β_1, a_2 =2 a_1 , γ_2 = 2 γ_1, a_1 (ξ_1 - ξ_2) = a_1 (ξ_3 - ξ_4)= - α_2 γ_1. * Case 4 ($\CN=8$): a_2α_1 = a_2β_1=1/2 (a_1+ a_1 (ξ_1 - ξ_2) = -α_2 γ_1, a_1 (ξ_3 - ξ_4) = β_2 γ_1 . * Case 5 ($\CN=8$): (a_2-a_1)β_2= (a_2+a_1) α_2, 2 a_1 a_2 (a_1 - a_2) α_1 = (a_1 + a_2) (γ_2 a_1^2-γ_1 2 a_1 a_2 β_1 = γ_1 a_2^2 + γ_2 a_1^2, (a_2- a_1)(ξ_1 - ξ_2)= (γ_1 - γ_2)α_2, (a_2 - a_1)^2(ξ_3 - ξ_4)= [ γ_2 (a_2 - 3 a_1 )- γ_1 (a_1 - 3 a_2 )] α_2 . * Case 6 ($\CN=8$): (a_2+a_1)β_2= (a_2- a_1) α_2, 2 a_1 a_2 α_1 = γ_1 a_2^2 + γ_2 a_1^2, 2 a_1 a_2 (a_1 - a_2) β_1 =(a_1 + a_2) (γ_2 a_1^2-γ_1 (a_2^2 - a_1^2)(ξ_1 - ξ_2)= [ γ_1 (a_1 - 3 a_2 )- γ_2 (a_2 - 3 a_1 )] α_2, (a_1+ a_2)(ξ_3 - ξ_4)= (γ_2 - γ_1)α_2. The obtained six subclasses of equation q4 are invariant under the restricted Möbius transformation (<ref>). Proof: Following the procedure described in Section <ref> we expand the fields appearing in equation $\CQ^+$ according to formulas The lowest order necessary conditions for the integrability of $\CQ^+$ are obtained by considering the equation $\CW_3$ (see (<ref>)), namely the order $\ep^3$ of the multiscale expansion. At this order we get the $m_2$-evolution equation for the harmonic $u_0^{(1)}$, that is a NLSE of the form δ_m_2 u_1^(1) + ρ_1δ_ξ^2 u_1^(1)+ ρ_2 u_1^(1) \|u_1^(1)\|^2=0, ξ≐n_1 - dω/d κ m_1, where the coefficients $\rho_1$ and $\rho_2$ will depend on the parameters of the equation $\CQ^+$ and on the wave parameters $\kappa$ and $\omega=\omega_+$, with $\omega_+$ expressed in terms of $\kappa$ through the dispersion relation (<ref>). According to our multiscale test the lowest order necessary condition for $\CQ^+$ to be an integrable lattice equation is that Eq. (<ref>) be integrable itself, namely $\rho_1$ and $\rho_2$ have to be real coefficients. Let us outline the construction of Eq. (<ref>). At $\CO(\ep)$ we get: * for $\alpha=1$ a linear equation which is identically satisfied by the dispersion relation (<ref>). * for $\alpha=0$ a linear equation whose solution is $u_1^{(0)}=0$. At $\CO(\ep^2)$, taking into account the dispersion relation (<ref>), we get: * for $\alpha=2$ an algebraic relation between $u_2^{(2)}$ and $u_1^{(1)}$. * for $\alpha=1$ a linear wave equation for $u_1^{(1)}$, whose solution is given by $u_1^{(1)}(n_1,m_1,m_2)=u_1^{(1)}(\xi,m_2)$, where $\xi\doteq n_1 - (d\omega/d \kappa) m_1$. * for $\alpha=0$ an algebraic relation between $u_2^{(0)}$ and $u_1^{(1)}$. Notice that from the $\CO(\ep^2)$ we find that the dependence of all the harmonics on the slow-variables $n_1$ and $m_1$ is given by $\xi$. At $\CO(\ep^3)$, for $\alpha=1$, by using the results obtained at the previous orders, one gets the NLSE (<ref>) with \rho_1 = \frac{a_1 a_2 (a_1^2-a_2^2) \sin \kappa }{ (a_1^2+a_2^2+2 a_1 a_2 \cos \kappa)^2}, \qquad \rho_2 = \CR_1 + \ri \CR_2, \label{r2} _1=sinκ[ _1^(0) + _1^(1) cosκ+ _1^(2) cos^2 κ+ _1^(3) cos^3 κ+_1^(4) cos^4 κ]/(a_1+a_2)(a_1^2+a_2^2+2 a_1 a_2 cosκ)^2 [(a_1-a_2)^2 + 2 a_1 a_2 cosκ(1+ cosκ) ], _2=_2^(0) + _2^(1) cosκ+ _2^(2) cos^2 κ+ _2^(3) cos^3 κ+_2^(4) cos^4 κ+_2^(5) cos^5 κ/(a_1+a_2)(a_1^2+a_2^2+2 a_1 a_2 cosκ)^2 [(a_1-a_2)^2 + 2 a_1 a_2 cosκ(1+ cosκ) ]. Here the coefficients $\CR_1^{(i)}$, $0 \leq i \leq 4$, and $\CR_2^{(i)}$, $0 \leq i \leq 5$, are polynomials depending on the coefficients $a_1,a_2 ,\alpha_1,\alpha_2,\beta_1,\beta_2,$ $\gamma_1,\gamma_2,$ $\xi_1,...,\xi_4$ and their expressions are cumbersome, so that we omit Note that $\rho_1$ is a real coefficient depending only on the parameters of the linear part of $\CQ^+$, while $\rho_2$ is a complex one. Hence the integrability of the NLSE (<ref>) is equivalent to the request $\CR_2 =0 \; \forall \, \kappa$, that is _2^(i)=0, 0≤i ≤5. Eqs. (<ref>) are a nonlinear algebraic system of six equations in twelve unknowns. By solving it one gets the six solutions contained in Proposition 1. These solutions are computed taking into account that $a_1,a_2 \in \mathbb R\setminus \{0\}$ with $\|a_1\| \neq \|a_2\|$. One can solve two of the six equations (<ref>) for $\xi_1$ and $\xi_3$, thus expressing them in terms of the remaining ten coefficients. The resulting system of four equations turns out to be $\xi_2$ and $\xi_4$-independent and linear in the four variables $\alpha_1$, $\beta_1$, $\gamma_1$ and $\gamma_2$. Therefore we may write the remaining four equations as a matrix equation with coefficients depending nonlinearly on $\alpha_2$, $\beta_2$, $a_1$ and $a_2$. The rank of the matrix is three. The six solutions are obtained by requiring that the matrix be of rank 3, 2, 1 and 0, and correspond to the six classes q4 that pass integrability conditions up to order $ \mathcal{O}(\varepsilon^3)$. A direct calculation proves the invariance of the six classes with respect to the restricted Möbius transformation $R$. If the coefficients $a_1,a_2 ,\alpha_1,\alpha_2,\beta_1,\beta_2,$ $\gamma_1,\gamma_2,$ $\xi_1,...,\xi_4$ of equation $\CQ^+$ do not satisfy one of the conditions given in (<ref>–<ref>) then $\CQ^+$ is not integrable. Quadratic difference equations are a subclass of $\CQ^+$ which have attracted a deal of attention. These equations are not Möbius invariant, but we can spot those that belong to the class $\CQ^+$ and pass our integrability conditions, just by inspection of (<ref>–<ref>). We have: For quadratic equations, when $\xi_1=\xi_2=\xi_3=\xi_4=\zeta=0$ in equations $\CQ^+$, the lowest order necessary conditions for the integrability of the resulting equation read: * Case Q1: $\alpha_2=\beta_2=0$; * Case Q2: $\alpha_2=\beta_2$, $\alpha_1=\beta_1$, $a_1=2a_2$, $\gamma_1=\gamma_2=0$, $\alpha_{j}\not=0$, $j=1$, $2$; * Case Q3: $\alpha_2=-\beta_2$, $\alpha_1=\beta_1$, $a_2=2a_1$, $\gamma_1=\gamma_2=0$, $\alpha_{j}\not=0$, $j=1$, $2$; * Case Q4: $\alpha_1=\beta_1=\gamma_1=\gamma_2=0$, $\left(\alpha_{2},\beta_{2}\right)\not=\left(0,0\right)$. §.§ Classification at order $\ep^4$. For what concerns the $A_{2}$-asymptotically integrable cases satisfying the integrability conditions (<ref>), the following statement holds At order $\ep^4$, the necessary conditions for the integrability of equations $\CQ^+$ read: * Case 1 ($\CN=9$): α_2 = β_2 = 0, ξ_1 = ξ_2, ξ_3 = ξ_4. * Case 4 ($\CN=8$): a_1 (ξ_1 - ξ_2) = -α_2 γ_1, a_1 (ξ_3 - ξ_4) = β_2 γ_1 , The corresponding two subclasses of equations are non overlapping and invariant under the restricted Möbius transformation (<ref>). As one can see, of the six $A_{1}$-asymptotically integrable cases listed in Proposition <ref>, Case 1 and Case 4 automatically satisfy the $A_{2}$-integrability conditions (<ref>), while the remaining four cases 2, 3, 5 and 6 reduce to some sub cases. For quadratic equations, when $\xi_1=\xi_2=\xi_3=\xi_4=\zeta=0$ in equations $\CQ^+$, the order $\left(\ep^4\right)$ necessary conditions for the integrability of the resulting equation read: * Case Q1: $\alpha_2=\beta_2=0$; * Case Q4: $\alpha_1=\beta_1=\gamma_1=\gamma_2=0$, $\left(\alpha_{2},\beta_{2}\right)\not=\left(0,0\right)$. As one can see, only two out the previous four quadratic cases in Remark 1 survive, the Cases Q1 and Q4: the first one is a sub case of Case 1, while the second is a sub case of Case 4. §.§ Classification at order $\ep^5$. It is possible to find all the cases satisfying the $A_{3}$-integrability conditions (<ref>). They are given by the following proposition The necessary and sufficient conditions for $\ep^{5}$ asymptotic integrability are: Case (a): ($\CN=4$) α_2=β_2=0, γ_2=α_1+β_1-γ_1, a_2=2a_1, (2α_1-3γ_1, 2β_1-3γ_1)≠(0, 0), ξ_1=ξ_2=α_1β_1/2a_1, ξ_3=ξ_4=-(α_1-γ_1)(β_1-γ_1)/2a_1, ζ=γ_1[3γ_1^2-3γ_1(α_1+β_1)+4α_1β_1]/4a_1^2; Case (b): ($\CN=4$) α_2=β_2=0, γ_1=α_1+β_1-γ_2, a_1=2a_2, (2α_1-3γ_2, 2β_1-3γ_2)≠(0, 0) ξ_1=ξ_2=α_1β_1/2a_2, ξ_3=ξ_4=(α_1-γ_2)(β_1-γ_2)/2a_2, ζ=γ_2[3γ_2^2-3γ_2(α_1+β_1)+4α_1β_1]/4a_2^2; Case (c): ($\CN=5$) α_1=β_1=(a_1+a_2)γ_1/2a_1, α_2=β_2=0, γ_2=a_2γ_1/a_1, ξ_1=ξ_2, ξ_3=ξ_4=(a_2-a_1)γ_1^2/4a_1^2-(a_2-a_1)/(a_2+a_1)ξ_2, ρ≐[8a_1^2ξ_2/(a_1+a_2)-3γ_1^2]1/(a_1+a_2)^2≠0; Case (d): ($\CN=5$) α_1=β_1=(a_1+a_2)γ_1/2a_1, α_2=β_2=0, γ_2=a_2γ_1/a_1, ξ_1=ξ_2, ξ_3=ξ_4=(a_1-a_2)γ_1^2/2a_1^2-(a_1-a_2)/(a_1+a_2)ξ_2, ρ≐[8a_1^2ξ_2/(a_1+a_2)-3γ_1^2]1/(a_1+a_2)^2≠0; Case (e): ($\CN=4$) α_1=β_1=γ_1+γ_2/2, α_2=β_2=0, γ_2≠a_2γ_1/a_1, a_2/a_1≠1/2, 2, ξ_1=ξ_2=3(γ_1+γ_2)^2/8(a_1+a_2), ξ_3=ξ_4=9(a_1-a_2)(a_1γ_2-a_2γ_1)^2/8a_1a_2(a_1+a_2)^2-a_1γ_2^2-a_2γ_1^2/8a_1a_2, Notes: In all of the cases $a_{2}/a_{1}\not=(0$, $\pm 1)$; the values $a_{2}/a_{1}=(2, \,\frac12)$ are excluded in Case (e) because they would provide a sub case of Case (a) or of Case (b). All the Cases (a)-(e) are sub cases of Case 1. So nothing survives out of Case 4 at order $\ep^5$. For quadratic equations, when $\xi_1=\xi_2=\xi_3=\xi_4=\zeta=0$ in equations $\CQ^+$, the $\ep^5$ order necessary conditions for the integrability of the resulting equation read: * Case Q$_{\alpha}$: $\alpha_{1}=\alpha_{2}=\beta_{2}=\gamma_{1}=0$, $\beta_{1}=\gamma_{2}\not=0$, $a_{2}=2a_{1}$; * Case Q$_{\beta}$: $\alpha_{1}=\alpha_{2}=\beta_{2}=\gamma_{2}=0$, $\beta_{1}=\gamma_{1}\not=0$, $a_{2}=2a_{1}$; * Case Q$_{\gamma}$: $\beta_{1}=\alpha_{2}=\beta_{2}=\gamma_{1}=0$, $\alpha_{1}=\gamma_{2}\not=0$, $a_{2}=2a_{1}$; * Case Q$_{\delta}$: $\beta_{1}=\alpha_{2}=\beta_{2}=\gamma_{2}=0$, $\alpha_{1}=\gamma_{1}\not=0$, $a_{2}=2a_{1}$; * Case Q$_{\eta}$: $\alpha_{1}=\alpha_{2}=\beta_{2}=\gamma_{1}=0$, $\beta_{1}=\gamma_{2}\not=0$, $a_{1}=2a_{2}$; * Case Q$_{\theta}$: $\alpha_{1}=\alpha_{2}=\beta_{2}=\gamma_{2}=0$, $\beta_{1}=\gamma_{1}\not=0$, $a_{1}=2a_{2}$; * Case Q$_{\kappa}$: $\beta_{1}=\alpha_{2}=\beta_{2}=\gamma_{1}=0$, $\alpha_{1}=\gamma_{2}\not=0$, $a_{1}=2a_{2}$; * Case Q$_{\lambda}$: $\beta_{1}=\alpha_{2}=\beta_{2}=\gamma_{2}=0$, $\alpha_{1}=\gamma_{1}\not=0$, $a_{1}=2a_{2}$. The Cases Q$_{\alpha}$-Q$_{\delta}$ are sub cases both of the Case Q1 and Case (a); the Cases Q$_{\eta}$-Q$_{\lambda}$ are sub cases both of the Case Q1 and Case (b). §.§.§ Canonical forms for $\varepsilon^5$ asymptotically integrable cases. We will now use the Möbius transformation to reduce the equation to normal form, i.e. to eliminate the maximum number of free parameters appearing in the nonlinear difference equation and reduce the coefficients of the linear part in $v_{n,m}$ and $v_{n+1,m+1}$ to 1. In the Case (a) of Proposition 4, performing the Möbius transformation β=0, γ=-γ_1δ/2, α=a_1δ, δ≠0, we obtain the canonical form: Case (a$^{\prime}$): ($\CN=2$) where $(\tau_{1}, \tau_{2})\doteq\left(\alpha_{1}-\frac{3\gamma_{1}}{2}, \beta_{1}-\frac{3\gamma_{1}}{2}\right)\not=(0, 0)$. Performing a further rescaling on (<ref>), we can fix, in all generality, the coefficients to either $\tau_{1}=0$ and $\tau_{2}=1$ or $\tau_{1}=1$ with $\tau_{2}$ arbitrary and we obtain the following two canonical forms respectively representing the two non overlapping subclasses of Case (a) defined respectively by the additional conditions $\alpha_{1}=\frac{3\gamma_{1}}{2}$ and $\alpha_{1}\not=\frac{3\gamma_{1}}{2}$. As under a restricted Möbious transformation $\tau_{2}$ is invariant, we see that two canonical forms (<ref>), specified by two invariants $\tau_{2a}$ and $\tau_{2b}$, form two disconnected components of the same conjugacy subclass unless $\tau_{2a}=\tau_{2b}$. In the Case (b) of Proposition 4, performing the Möbius transformation β=0, γ=-γ_2δ/2, α=a_2δ, δ≠0, we obtain the canonical form: Case (b$^{\prime}$): ($\CN=2$) where $(\tau_{1}, \tau_{2})\doteq\left(\alpha_{1}-\frac{3\gamma_{2}}{2}, \beta_{1}-\frac{3\gamma_{2}}{2}\right)\not=(0, 0)$. Performing a further rescaling on (<ref>) we can fix, in all generality, the parameters either to $\tau_{1}=0$ and $\tau_{2}=1$ or to $\tau_{1}=1$ with $\tau_{2}$ arbitrary and we obtain respectively the two canonical forms representing the two non overlapping subclasses of Case (b) defined respectively by the additional conditions $\alpha_{1}=\frac{3\gamma_{2}}{2}$ and $\alpha_{1}\not=\frac{3\gamma_{2}}{2}$. As $\tau_{2}$ is invariant under a restricted Möbious transformation, we see that two canonical forms (<ref>), specified by two invariants $\tau_{2a}$ and $\tau_{2b}$, form two disconnected components of the same conjugacy subclass unless $\tau_{2a}=\tau_{2b}$. In the Cases (c) and (d) of Proposition 4, performing the Möbius transformation α=2a_1δ/(a_1+a_2)√(\|ρ\|), β=0, γ=-γ_1δ/(a_1+a_2)√(\|ρ\|), δ≠0, we obtain the canonical forms: Case (c$^{\prime}$): ($\CN=2$) Case (d$^{\prime}$): ($\CN=2$) where $\epsilon\doteq a_{2}/a_{1}\not=0,\pm 1$ and $\zeta^{\prime}\doteq 8s\left\|\frac{\pi^2}{\rho^3}\right\|^{1/2}/\left(1+\epsilon\right)^2$, $\pi\doteq\left[\zeta-2\frac{\gamma_{1}} {a_{1}}\xi_{2}+\frac{\left(a_{1}+a_{2}\right)\gamma_{1}^3} {2a_{1}^3}\right]/\left(a_{1}+a_{2}\right)$ and $s\doteq\pm 1$. As under a restricted Möbius transformation $\rho\rightarrow\rho\left(\alpha/\delta\right)^2$ and $\pi\rightarrow\pi\left(\alpha/\delta\right)^3$, we see that the absolute value of $\zeta^{\prime}$ and $\operatorname{sgn}\left(\rho\right)$ are invariant under such a transformation. With another rescaling we can always fix $\zeta^{\prime}\geq 0$ and the two canonical forms, specified by the two set of invariants $\left(\epsilon_{a}, \operatorname{sgn}\left(\rho_{a}\right), \zeta^{\prime}_{a}\right)$ and $\left(\epsilon_{b}, \operatorname{sgn}\left(\rho_{b}\right), \zeta^{\prime}_{b}\right)$, form two disconnected components of the conjugacy class unless the two sets are the same. In the Case (e) of Proposition 4, performing the Möbius transformation β=0, γ=-(γ_1+γ_2)α/2(a_1+a_2), δ=(a_2γ_1-a_1γ_2)α/a_1(a_1+a_2), α≠0, we obtain the canonical form: Case (e$^{\prime}$): ($\CN=1$) where $\epsilon\doteq a_{2}/a_{1}\not=0,\pm 1,2,1/2$. As $\epsilon$ is invariant under a restricted Möbius transformation, we see that two canonical forms, specified by the two invariants $\epsilon_{a}$ and $\epsilon_{b}$, form two disconnected components of the conjugacy class unless $\epsilon_{a}=\epsilon_{b}$. §.§.§ Comparison with the ABS list. As our allowed transformations are sub cases of the full Möbius transformations allowed in the ABS approach [2], any conjugacy class of ours is either completely contained into one of the ABS classification or is totally disjointed from them. Considering that no one out of the canonical forms (a$^{\prime}$)-(e$^{\prime}$) possesses the invariance (up to an overall sign) under the transformation $v_{n,m}\leftrightarrow v_{n+1,m}$, $v_{n,m+1}\leftrightarrow v_{n+1,m+1}$, we can conclude that no intersection can exist between our classes and those generated by the $ABS$ list. Even more, no equation in our list is of Klein-type or, that is the same [11], a sub case of the $Q_{V}$ equation. We can enlarge our class of transformations by including also an exchange $n\leftrightarrow m$ between the two independent variables. The subclass (<ref>) can be discarded because under this exchange we would get it from subclass (<ref>) with $\tau_{2}=0$; similarly the subclass (<ref>) can be discarded because under this exchange we would get it from subclass (<ref>) with $\tau_{2}=0$; finally the subclasses (<ref>-<ref>) are invariant under this transformation. Let us include also the inversion $n\rightarrow -n$. Setting $\tilde v_{n,m}\doteq v_{-n,m}$, we have that, if $v_{n,m}$ satisfies (<ref>), then $\tilde v_{n,m}$ satisfies (<ref>); if $v_{n,m}$ satisfies (<ref>) with parameters $\epsilon$ and $\zeta^{\prime}$, then $\tilde v_{n,m}\doteq \operatorname{sgn}\left(\epsilon\right)v_{-n,m}$ satisfies (<ref>) with parameters $1/\epsilon$ and $\zeta^{\prime}/\left\|\epsilon\right\|$ and similarly for Eq. (<ref>); if $v_{n,m}$ satisfies (<ref>) with parameter $\epsilon$, then $\tilde v_{n,m}\doteq -v_{-n,m}/\epsilon$ satisfies (<ref>) with parameter $1/\epsilon$ (this implies that, if $v_{n,m}$ satisfies one of the four canonical forms (<ref>), (<ref>-<ref>), then also $\tilde v_{n,m}\doteq v_{-n,-m}$ does). As a consequence, under this enlarged class of transformations, (<ref>) can be discarded and in the case of (<ref>-<ref>) we can limit the p! arameter $\epsilon$ to the range $-1<\epsilon<1$, $\epsilon\not=0$ as the equation with parameters $1/\epsilon$ and $\zeta^{\prime}$ can be obtained from the corresponding with parameters $\epsilon$ and $\zeta^{\prime}\left\|\epsilon\right\|$. Remarque: The Cases (c') and (d'), when $\pi=0$, i.e. $\zeta^{\prime}=0$, reduce to the integrable cases analyzed in Levi-Yamilov [11] and Ramani-Grammaticos [14]. §.§ Classification at order $\ep^6$. Now we perform a multiscale reduction at order $\ep^6$ on the four canonical forms (<ref>), (<ref>-<ref>) and we find that all the so far obtained equations satisfy the $A_{4}$-integrability conditions (<ref>). Hence we can state the following proposition Up to a restricted Möbius transformations $\tilde v_{n,m}\doteq v_{n,m}/\left(\alpha v_{n,m}+\beta\right)$, exchanges $n\leftrightarrow m$ and inversions $n\rightarrow -n$, all the $A_{4}$-asymptotically integrable cases in the class ${\mathcal Q}^{+}$ are given by +τv_n,mv_n+1,mv_n,m+1v_n+1,m+1=0, -1<ϵ<1, ϵ≠0, δ≐±1, τ≥0; +τv_n,mv_n+1,mv_n,m+1v_n+1,m+1=0, -1<ϵ<1, ϵ≠0, δ≐±1, τ≥0; +(1-1/ϵ^2)v_n,mv_n+1,mv_n,m+1v_n+1,m+1=0, -1<ϵ<1, ϵ≠0, 1/2. If, when $\tau=0$ in (<ref>), we apply the (not allowed) transformation $v_{n,m}\doteq\sqrt{3}w_{n,m}-1$, we obtain which in the direction $n$ satisfies two necessary integrability conditions given in [11] but does't admit any three-points generalized symmetries either autonomous or not, while in the direction $m$ the integrability conditions given in [11] are not satisfied; If, when $\tau=1$ in (<ref>), we apply the (not allowed) transformation $v_{n,m}\doteq 2^{1/3}w_{n,m}-1$, we obtain an integrable equation considered in [12], which possesses a $3\times 3$ Lax pair and which is a degeneration of the discrete integrable Tzitzeica equation proposed by Adler in [1]. Finally, if we choose $\tau\not=0$, $1$ in (<ref>), we can apply the (not allowed) transformation $v_{n,m}\doteq \frac{1-\tau}{\tau}w_{n,m}-1$ and we obtain where $\chi\doteq\frac{\left(\tau-3\right)\tau^2}{\left(1-\tau\right)^3}$. This system is the sum of (<ref>), (<ref>) and an arbitrary constant and doesn't satisfy the integrability conditions given in [11] for three-points generalized symmetries either autonomous or not, either in the direction $n$ or $m$. If in (<ref>), (<ref>) we apply respectively the (not allowed) transformations $w_{n,m}\doteq\delta \operatorname{sgn}\left(\epsilon\right)/v_{n,m}$ and $\tilde w_{n,m}\doteq\delta/v_{n,m}$, we obtain Eqs.(<ref>, <ref>) are just an almost trivial looking modification of the two integrable systems discussed in [14], which are recovered when $\tau=0$. In [14] it was shown that, when $\tau=0$, (<ref>, <ref>) are mapped through a Möbious transformation respectively to the Hirota discrete sine-Gordon equation and to its potential form. After we replace $\epsilon\rightarrow 1/\epsilon$ in (<ref>) and $\delta\rightarrow s\delta$, with $s\doteq \operatorname{sgn}\left(\epsilon\right)$, in (<ref>) the precise form of the potentiation induced between them is Eqs.(<ref>, <ref>), if and only if $\tau=0$, satisfy the integrability conditions given in [11] for three-points generalized symmetries either autonomous or not which, in the $n$ direction, are given respectively by w_n,m,t = (δw_n,m^2-ϵ)(δϵw_n,m^2-1)(w_n+1,m-w_n-1,m)/(1+δw_n,mw_n+1,m)(1+δw_n,mw_n-1,m), w̃_n,m,t̃ = Y(δw̃_n,m^2-1)(w̃_n+1,m-w̃_n-1,m)/δw̃_n+1,mw̃_n-1,m-1+[(-1)^nκ+(-1)^mθ](δw̃_n,m^2-1), where $t$ and $\tilde t$ are two group parameters, and, in the $m$ direction, by expressions obtained changing $w_{n+1,m}\rightarrow w_{n,m+1}$ and $w_{n-1,m}\rightarrow w_{n,m-1}$. When $\tau=0$ (<ref>) admits a two parameters non autonomous point symmetry. Eqs. (<ref>, <ref>) are invariant under the transformation $w_{n,m}\doteq -v_{n,m}$; Eq. (<ref>) is covariant under the inversion $w_{n,m}\doteq 1/v_{n,m}$ as $\epsilon$ is changed into $1/\epsilon$, while (<ref>) is invariant. Eqs. (<ref>, <ref>), under the non autonomous transformation $w_{n,m}\doteq \left(-1\right)^{n+m}v_{n,m}$, are covariant as in the first case $\epsilon$ is changed into $-\epsilon$ and $\delta$ into $-\delta$ while in the second one $\epsilon$ is changed into $-\epsilon$, implying that we can limit ourselves to the range $0<\epsilon<1$. Eq. (<ref>) is invariant under the non autonomous transformation $\tilde w_{n,m}\doteq [v_{n,m}]^{\left(-1\right)^{n+m}}$ when $\delta=1$ ! and covariant when $\delta=-1$ as $\epsilon$ is changed into $\delta\epsilon$. Finally (<ref>, <ref>) are covariant under the transformation $w_{n,m}\doteq\ri v_{n,m}$ as $\delta$ is changed into $-\delta$, this implying that we can always take $\delta=1$ even if in general, allowing such a transformation, the solution will be no more real. If we apply the (not allowed) transformation $v_{n,m}\doteq\frac{\|\rho\|^{1/2}w_{0,0}+1}{\|\rho\|^{1/2}w_{0,0}-1/\epsilon}$, with $\rho\doteq\frac{-1+2\epsilon}{\epsilon\left(\epsilon-2\right)}\not=0$, to (<ref>) we obtain where $\delta\doteq \operatorname{sgn}\left(\rho\right)=\operatorname{sgn}\left(1/\epsilon-2\right)$, which, for $\delta=-1$, is a real discrete Tzitzeica equation with coefficient $c=1/\left(\epsilon\|\rho\|^{3/2}\right)$ and, for $\delta=1$, through the (not allowed) transformation $w_{n,m}\rightarrow\ri w_{n,m}$ becomes a complex Tzitzeica equation with coefficient $c=\ri/\left(\epsilon\|\rho\|^{3/2}\right)$. We remember that the discrete Tzitzeica equation is integrable and it admits a $3\times 3$ Lax pair [1]. We note that (<ref>), beside not being sub cases of the $Q_{V}$ equation, except for (<ref>, <ref>) with $\tau=0$, where a five-points generalized symmetry depending on the points $\left(n+1,m\right)$, $\left(n,m+1\right)$, $\left(n,m\right)$, $\left(n-1,m\right)$ and $\left(n,m-1\right)$ exists, is also not included into the Garifullin-Yamilov class [7]. § CONCLUDING REMARKS In this paper we have considered the application of a multiple scale expansion to the class of dispersive multilinear partial difference equation on the square lattice, $\CQ$. A great effort has been directed to extend the expansion up to order $\ep^6$ so as to be able to check the order $\ep^5$ results. The integrability conditions we obtain, when we require that the multiple scale expansion of the discrete class of equations is equivalent to the equations of the NLSE hierarchy, reduce the $13-$parameters initial class to four equations depending on few parameters. The $A_3$ integrable equations are invariant under the $A_4$ integrability conditions, indicating that we might have already filtered out all the nonintegrable cases and that the obtained equations might be integrable. Two open problems seem of great importance now: * The consideration of the second class of dispersive multilinear partial difference equations on the square lattice, $\CQ=\CQ^-$ is will provide by sure new classes of integrable equations, as in this case the lowest order integrability conditions appear already at order $\ep^2$ and will not be an equation of the NLSE type but more likely a coupled wave equations. * We may assume that (<ref>) are all integrable but in this article we are not able to prove it. We have to apply different techniques. In particular what remains to be analyzed are the four sub cases (<ref>), (<ref>) and (<ref>) when $\tau>0$. We are now working for proving that these systems have generalized symmetries, i.e. there exist some flows in the group parameter $\lambda$ commuting with our equations $$v_{n,m,\lambda}=g(v_{n+k,m}, v_{n+k-1,m}, \cdots, v_{n-k,m}, v_{n,m+\ell}, v_{n,m+\ell-1}, \cdots, v_{n,m-\ell}).$$ It is easy to prove that there are no symmetries with $k=\ell=1$. Thus it seems important to consider the case when $k=\ell\ge 2$. Work is in progress in both open problems. In particular in [15] one has proved the integrability of (<ref>) by constructing two generalized symmetries defined on five points, one with $k=2$ and $\ell=0$ and one with $k=0$ and $\ell=2$. § ACKNOWLEDGMENTS LD and SC have been partly supported by the Italian Ministry of Education and Research, PRIN “Nonlinear waves: integrable fine dimensional reductions and discretizations" from 2007 to 2009 and PRIN “Continuous and discrete nonlinear integrable evolutions: from water waves to symplectic maps" from 2010. RHH thanks the INFN, Sezione Roma Tre and the UPM for their support during his visits to Rome. We thank Matteo Petrera for many enlightening discussion in the first stage of this paper. § APPENDIX Here we present explicitly the $48$ conditions for $\varepsilon^6$ $S$-asymptotic integrability ($A_{4}$-integrability) involving the real ($S_{j}$) and imaginary parts ($T_{j}$) of the coefficients $\eta_{j}$, $j=1$,…, $34$ of the differential polynomial (<ref>). The expressions of the coefficients $\kappa_{m}$ , $m=1$,…, $77$ of the differential polynomial (<ref>) as functions of the $\eta_{j}$, $j=1$,…, $34$ are complicated, so we will omit them. These $48$ conditions are, as far as we know, presented here for the first time. \begin{align}\label{Phoenix} T_{9}&=T_{10}-\frac{I_{1}T_{32}}{2\rho_{2}}-\frac{I_{6}T_{33}}{4\rho_{1}},\qquad S_{11}=\frac{S_{22}}{2}-\left[\frac{R_{12}}{2}+\left(a-b\right)\frac{I_{8}}{\rho_{2}}\right]\frac{T_{32}}{2\rho_{2}},\qquad T_{11}=T_{22}+\frac{I_{8}T_{27}-\left(I_{6}+I_{12}\right)T_{32}}{2\rho_{2}},\nonumber\\ S_{12}&=-\frac{I_{8}S_{27}}{\rho_{2}}+\left(\frac{a}{2}-b\right)\frac{I_{8}T_{32}}{\rho_{2}^2},\qquad T_{12}=-\frac{I_{8}\left(T_{27}-T_{33}\right)}{\rho_{2}},\nonumber\\ S_{14}&=0,\qquad T_{14}=-\frac{I_{8}T_{32}}{2\rho_{2}},\nonumber\\ S_{16}&=-\frac{I_{6}S_{27}+aT_{22}}{2\rho_{2}}+\left(a-\frac{3b}{2}\right)\frac{I_{6}T_{32}}{2\rho_{2}^2}+\frac{R_{12}T_{33}}{2\rho_{2}},\qquad T_{16}=\frac{aS_{22}-I_{6}T_{27}}{2\rho_{2}}+\left(\frac{aR_{12}}{2\rho_{2}}-I_{3}\right)\frac{T_{32}}{2\rho_{2}}+\left(I_{6}+I_{12}\right)\frac{T_{33}}{2\rho_{2}},\nonumber\\ S_{17}&=S_{15}+\frac{I_{12}S_{27}}{2\rho_{2}}-\left(a-b\right)\frac{I_{8}T_{27}}{2\rho_{2}^2}+\left[\left(2a+3b\right)I_{6}+bI_{12}\right]\frac{T_{32}}{4\rho_{2}^2}+\left(a-\frac{5b}{4}\right)\frac{I_{8}T_{33}}{\rho_{2}^2},\qquad T_{17}=-\frac{aS_{22}}{2\rho_{2}}+\nonumber\\ T_{19}=T_{20},\qquad S_{21}=S_{22},\qquad T_{21}=T_{22},\nonumber\\ S_{26}&=-\frac{I_{8}S_{27}}{\rho_{2}}-\left(R_{12}+\frac{aI_{8}}{\rho_{2}}\right)\frac{T_{32}}{2\rho_{2}},\qquad T_{26}=T_{22}-\frac{I_{6}T_{32}-I_{8}T_{33}}{\rho_{2}},\qquad S_{30}=0,\qquad T_{30}=T_{32},\nonumber\\ S_{31}&=0,\qquad T_{31}=T_{32},\qquad S_{32}=0,\qquad S_{33}=-\frac{aT_{32}}{2\rho_{2}},\qquad S_{34}=S_{27}+\frac{bT_{32}}{2\rho_{2}},\qquad T_{34}=0,\nonumber\\ &\quad{}+2\left[2\rho_{2}\left(a-b\right)R_{12}+2\rho_{2}^2\left(I_{2}-I_{3}-2I_{5}\right)+a\left(b-2a\right)I_{8}\right]T_{33}=0,\qquad I_{6}T_{32}-I_{8}T_{33}=0.\nonumber \end{align} [1]V.E. Adler, On a discrete analog of the Tzitzeica equation, (arXiv:1103.5139); [2] Adler V.E., Bobenko A.I. and Suris Yu.B., Classification of integrable equations on quad-graphs. The consistency approach, Comm. Math. Phys. 233/3, 513–543 (2003); [3] A. Degasperis, Multiscale expansion and integrability of dispersive wave equations, lectures given at the Euro Summer School "What is integrability?", Isaac Newton Institute, Cambridge, U. K., 13-24 August 2001, in Integrability edited by A. V. Mikhailov, Lecture Notes in Physics 767, 215-244, Springer, Berlin-Heidelberg (2009); [4] A. Degasperis, S. V. Manakov and P.M. Santini, Multiple-scale perturbation beyond the nonlinear Schrödinger equation. I, Phys. D 100, 187–211 (1997); [5] A. Degasperis and M. Procesi, , "Asymptotic integrability", in A. Degasperis and G. Gaeta, Symmetry and Perturbation Theory (Rome, 1998), River Edge, NJ: World Scientific (1999), 23?37; [6]R.N. Garifullin, E.V. Gudkova, I.T. Habibullin, Method for searching higher symmetries for quad graph equations, J. Phys. A: Math. Theor. 44 (2011) 325202 (ArXiv:1104.0493); [7]R.N. Garifullin, R.I. Yamilov, Generalized symmetry classification of discrete equations of a class depending on twelve parameters, (arXiv:1203.4369); [8] R. Hernandez Heredero, D. Levi, C. Scimiterna, A discrete linearizability test based on multiscale analysis, Jour. Phys. A: Math. and Theor. 43 (2010), 502002; [9]Y. Hiraoka, Y. Kodama, Normal forms and solitons, in Integrability, A.V. Mikhailov editor, Lecture Notes in Physics Volume 767, Springer, Berlin 2009, 175-214; [10]D. Levi and R.I. Yamilov, On a nonlinear integrable difference equation on the square, Ufa Math. J. 1:2 (2009) 101-105. (arXiv:0902.2126v2); [11]D. Levi and R.I. Yamilov, Generalized symmetry integrability test for discrete equations on the square lattice, J. Phys. A: Math. Theor. 44 (2011) 145207; [12]A.V. Mikhailov and P. Xenitidis, Second order integrability conditions for difference equations. An integrable equation, arXiv:1305.4347; [13] Santini P.M., The multiscale expansions of difference equations in the small lattice spacing regime, and a vicinity and integrability test: I, Jour. Phys. A: Math. and Theor. 43 (2010), 045209; [14] C. Scimiterna, B. Grammaticos, A. Ramani, On two integrable lattice equations and their interpretation, Jour. Phys. A: Math. and Theor., 44, no. 3 (2011), 032002. [15] C. Scimiterna, M. Hay and D. Levi, On the integrability of the lattice equation $w_{n,m}w_{n+1,m}+w_{n+1,m}w_{n,m+1}+w_{n,m+1}w_{n+1,m+1}=1$, work in progress [16] C. Viallet, Integrable lattice maps: $Q_V$, a rational version of $Q_4$, Glasgow Math. J., 51A (2009) 157-163, (arXiv:0802.0294v1); \begin{multline} \text{I2.\ }2 a_2( u_{n,m+1}+ u_{n+1,m})+4 a_2d\, u u_{n+1,m+1}+2a_2d\, u_{n,m+1} \\ {}+3 a_2d\,( u_{n+1,m+1}+u_{n+1,m} u_{n+1,m+1} +u u_{n,m+1}+u \\ _2+\tau _4\right)u u_{n+1,m+1} ( _2-\tau _4\right) u_{n,m+1} u_{n+1,m} (u+u_{n+1,m+1}) \end{multline} \begin{multline} \text{I3.\ } \frac{1}{2} a_2 u u_{n+1,m+1}+2a_2d\, u_{n,m+1} \\ {}+\frac{3}{2} a_2d( u_{n+1,m} u_{n+1,m+1}+ u u_{n,m+1}+u \\ _2+\tau _4\right) u _2+\tau _4\right) u _2-\tau _4\right) u_{n,m+1} u_{n+1,m} u_{n+1,m+1} \\ {}+\left(\tau _2-\tau _4\right) u u_{n,m+1} \end{multline} * Case 1-4. $\left\{\alpha_2=\beta_2=0,\ \alpha _1= \beta _1 = \frac{1}{2} \left(\gamma_1+d_2\right),\ a_2d_1 = a_1d_2,\ \tau _1= \tau_2,\ \tau _3=\tau_4\right\}$ * Case 1-2. $\left\{\alpha_2=\beta_2=0,\ \alpha _1= \beta _1,\ d_1= 2 d_2,\ a_1= 2 a_2,\ \tau_1=\tau_2,\ \tau _3= \tau_4\right\}$ * Case 1-3. $\left\{\alpha_2=\beta_2=0,\ \alpha _1= \beta _1,\ 2d_1= d_2,\ 2a_1= a_2,\ \tau _1=\tau_2,\ \tau _3= \tau_4\right\}$ * Case 2-4. $\left\{\alpha _2= \beta _2,\ \alpha _1= \beta _1= \frac32 d_2,\ d_1= 2 d_2,\ a_1= 2 a_2,\ \tau _1= \tau _2-\frac{d_1 \alpha _2}{a_1},\tau _3= \frac{d_1 \alpha _2}{a_1}+\tau _4\right\}$ * Case 3-4. $\left\{\alpha _2= -\beta _2,\ \alpha _1= \beta _1= \frac34 d_2,\ 2d_1= d_2,\ 2a_1= a_2,\ \tau _1= \tau _2-\frac{d_1 \alpha _2}{a_1},\ \tau _3= \tau _4-\frac{d_1 \alpha * Case 1-5. $\left\{\alpha_2=\beta_2=0,\ \alpha _1= \frac{\left(a_1+a_2\right) \left(a_1^2 d_2-a_2^2 d_1\right)}{2 a_1 \left(a_1-a_2\right) a_2},\ \beta _1= \frac{a_1^2 d_2+a_2^2 d_1}{2 a_1 a_2},\ \tau _1= \tau _2,\ \tau _3= \tau _4\right\}$ * Case 1-6. $\left\{\alpha_2=\beta_2=0,\ \alpha _1= \frac{a_1^2 d_2+a_2^2 d_1}{2 a_1 a_2},\ \beta _1= \frac{\left(a_1+a_2\right) \left(a_1^2 d_2-a_2^2 d_1\right)}{2 a_1 \left(a_1-a_2\right) a_2},\ \tau _1= \tau _2,\ \tau _3= \tau _4\right\}$ * Case 2-5. $\left\{\alpha_2=\beta_2=0,\ \alpha _1=\beta_1= \frac32 d_2,\ d_1= 2 d_2,\ a_1=2 a_2,\ \tau _1=\tau_2,\ \tau _3= \tau * Case 2-6. $\left\{\alpha_2=\beta_2=0,\ \alpha _1=\beta_1= \frac32 d_2,\ d_1= 2 d_2,\ a_1=2 a_2,\ \tau _1=\tau_2,\ \tau _3= \tau * Case 3-5. $\left\{\alpha_2=\beta_2=0,\ \alpha _1=\beta _1= \frac34 d_2,\ 2d_1=d_2,\ 2a_1=a_2,\ \tau _1= \tau _2,\ \tau _3= \tau _4\right\}$ * Case 3-6. $\left\{\alpha_2=\beta_2=0,\ \alpha _1=\beta _1= \frac34 d_2,\ 2d_1=d_2,\ 2a_1=a_2,\ \tau _1= \tau _2,\ \tau _3= \tau _4\right\}$ * Case 4-5. $\left\{\alpha _2= \frac{a_2-a_1}{a_1+a_2}\beta _2,\ \alpha _1= \beta _1= \frac{a_1+a_2}{2 a_2}d_2,\ a_2d_1=a_1 d_2,\ \tau _1= \tau _2-\frac{d_2 \alpha _2}{a_2},\ \tau _3= \frac{d_2 \beta _2}{a_2}+\tau _4\right\}$ * Case 4-6. $\left\{\alpha _2= \frac{a_1+a_2 }{a_2-a_1}\beta _2,\ \alpha _1=\beta _1= \frac{a_1+a_2}{2 a_2}d_2,\ a_2d_1= a_1d_2,\ \tau _1= \tau _2-\frac{d_2 \alpha _2}{a_2},\ \tau _3= \frac{d_2 \beta _2}{a_2}+\tau _4\right\}$ * Case 1-2-5. $\left\{\alpha_2=\beta_2=0,\ \alpha _1= \beta _1= \frac{3 d_2}{4},\ 2d_1=d_2,\ 2a_1=a_2,\ \tau _1= \tau _2,\ \tau _3= \tau _4\right\}$ * Case 1-3-5. $\left\{\alpha_2=\beta_2=0,\ \alpha _1= \beta _1= \frac{3 d_2}{2},\ d_1=2d_2,\ a_1=2a_2,\ \tau _1= \tau _2,\ \tau _3= \tau _4\right\}$ * Case 4: \left\{ \begin{array}{l} {\displaystyle{\frac{a_2}{a_1} }}= -3.309 , \\ \alpha_1 = 0.081 \delta_2 - 0.887 \delta_1, \\ \beta_1 = -1.655 \delta_1 - 0.151 \delta_2, \\ {\displaystyle{ \frac{\beta_2}{\alpha_2}}} = 0.536, \\ {\displaystyle{ \frac{\tau_1 - \tau_2}{\alpha_2}}}= {\displaystyle{ \frac{ 0.232( \delta_2 - \delta_1) }{a_1}}}, \\ {\displaystyle{ \frac{\tau_3 - \tau_4}{\alpha_2}}}=- {\displaystyle{ \frac{0.588 \delta_1 + 0.339 \delta_2}{a_1}}}. \end{array} \right. * Case 5: \left\{ \begin{array}{l} {\displaystyle{\frac{a_2}{a_1} }}= 0.302 , \\ \alpha_1 = 0.081 \delta_1 - 0.887 \delta_2, \\ \beta_1 = -1.655 \delta_2 - 0.151 \delta_1, \\ {\displaystyle{ \frac{\beta_2}{\alpha_2}}} = -0.536, \\ {\displaystyle{ \frac{\tau_1 - \tau_2}{\alpha_2}}}= {\displaystyle{ \frac{ 0.768 ( \delta_2 - \delta_1) }{a_1}}}, \\ {\displaystyle{ \frac{\tau_3 - \tau_4}{\alpha_2}}}=- {\displaystyle{ \frac{1.124 \delta_1 + 1.984 \delta_2}{a_1}}}. \end{array} \right. * Case 6: \left\{ \begin{array}{l} {\displaystyle{\frac{a_2}{a_1} }}= -2.708 , \\ \alpha_1 = -0.185 \delta_2 - 1.354 \delta_1, \\ \beta_1 = 0.085 \delta_2 - 0.624 \delta_1, \\ {\displaystyle{ \frac{\beta_2}{\alpha_2}}} = 2.171, \\ {\displaystyle{ \frac{\tau_1 - \tau_2}{\alpha_2}}}= {\displaystyle{ \frac{ 1.440 \delta_1 +0.901 \delta_2 }{a_1}}}, \\ {\displaystyle{ \frac{\tau_3 - \tau_4}{\alpha_2}}}= {\displaystyle{ \frac{0.585 ( \delta_1 - \delta_2)}{a_1}}}. \end{array} \right. * Case 7: \left\{ \begin{array}{l} {\displaystyle{\frac{a_2}{a_1} }}= -0.369 , \\ \alpha_1 = -0.185 \delta_1 - 1.354 \delta_2, \\ \beta_1 = 0.085 \delta_1 - 0.624 \delta_2, \\ {\displaystyle{ \frac{\beta_2}{\alpha_2}}} = -2.171, \\ {\displaystyle{ \frac{\tau_1 - \tau_2}{\alpha_2}}}= -{\displaystyle{ \frac{ 2.440 \delta_1 + 3.901\delta_2 }{a_1}}}, \\ {\displaystyle{ \frac{\tau_3 - \tau_4}{\alpha_2}}}= {\displaystyle{ \frac{1.585 ( \delta_2 - \delta_1)}{a_1}}}. \end{array} \right.	arxiv-papers	2013-11-08T09:05:08	2024-09-04T02:49:53.392302	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "authors": "R. Hernandez Heredero, D. Levi and C. Scimiterna", "submitter": "Rafael Hern\\'andez Heredero", "url": "https://arxiv.org/abs/1311.1905" }
1311.1922	# Discrete Filters for Large Eddy Simulation of Forced Compressible MHD Turbulence Alexander A. Chernyshov Space Research Institute of Russian Academy of Sciences Profsoyuznaya 84/32, 117997 Moscow, Russia Email: [email protected] Kirill. V. Karelsky Space Research Institute of Russian Academy of Sciences Profsoyuznaya 84/32, 117997 Moscow, Russia Email: [email protected] Arakel. S. Petrosyan Address all correspondence to this author. Space Research Institute of Russian Academy of Sciences Profsoyuznaya 84/32, 117997 Moscow, Russia Moscow Institute of Physics and Technology State University Institutskiy Pereulok 9, 141700 Moscow Region, Dolgoprudny, Russia Email: [email protected] ###### Abstract In present study, we discuss results of applicability of discrete filters for large eddy simulation (LES) method of forced compressible magnetohydrodynamic (MHD) turbulent flows with the scale-similarity model. Influences and effects of discrete filter shapes on the scale-similarity model are examined in physical space using a finite-difference numerical schemes. We restrict ourselves to the Gaussian filter and the top-hat filter. Representations of this subgrid-scale model which correspond to various 3- and 5-point approximations of both Gaussian and top-hat filters for different values of parameter $\epsilon$ (the ratio of the mesh size to the cut-off lengthscale of the filter) are investigated. Discrete filters produce more discrepancies for magnetic field. It is shown that the Gaussian filter is more sensitive to the parameter $\epsilon$ than the top-hat filter in compressible forced MHD turbulence. The 3-point filters at $\epsilon=2$ and $\epsilon=3$ give the least accurate results and the 5-point Gaussian filter shows the best results at $\epsilon=2$. ## 1 Introduction Compressible turbulent flows in a magnetic fields are common both in engineering and applied areas and in physics of astrophysical and space processes. Among the engineering applications, possibility of boundary layer control and drug reduction, MHD flow in channel for steel-casting process, and in pipe for cooling of nuclear fusion reactors can be mentioned. Most of the applications demand understanding of turbulent flow at high Reynolds numbers with density fluctuations due to compressibility (like for example, in aerospace engineering design). The presence of velocity and magnetic field fluctuations in a wide range of space and time scales has been directly detected in the various turbulent flows in space processes. For example, there are strong indications of their presence in the solar corona, interplanetary medium, solar wind and others. Note that the MHD problems differ from those of the neutral fluid hydrodynamics. The MHD equations contain two fields which introduces considerably more freedom into the dynamics. Fundamental limitations of direct numerical simulation (DNS) method for a turbulence modeling and difficulties due to presence of compressibility and magnetic field demand development of new theoretical and computational methods and make important advancing of large eddy simulation (LES) method for such complex MHD flows. According to LES approach, the large-scale part of the flow is computed directly and only small-scale structures of turbulence are modeled. This scale separation is achieved by applying a filter. The small-scale motion is eliminated from the initial system of equations of motion by filtering procedures and its effect is taken into account by special closures referred to as the subgrid-scale (SGS) models [1, 2, 3, 4, 5, 6, 7]. Theoretical studies on SGS modeling are carried out by performing filtering operations that are defined as convolution products between the velocity field and the filter kernel. Such definition is suitable when dealing with numerical methods such as spectral or pseudo-spectral types, and is expensive when dealing with local methods (finite differences, finite volumes, finite elements). In practice, for local methods, discrete test filters with compact stencils based on weighted averages are used. The properties of the discrete filters differ substantially from those of the continuous filters, which are the basis of theoretical analysis. Hence, the need for the analysis of discrete filters, and for the choice of discrete filters with required properties in order to ensure a greater consistency between the continuous SGS model and its discretized version, which will be used for the computations. In the present paper, we deal with the question of the effects and influences of different filter shapes on scale-similarity model in LES method for compressible forced MHD turbulent flows using a finite-difference schemes. Recently, we have showed that the scale-similarity model for forced MHD turbulence can be used as a stand alone SGS model as opposed to decaying case [8]. The scale-similarity parametrization has evident advantages the main ones being to reproduce rightly the correlation between the tensors between actual and model turbulent stress tensor for isotropic flow as well as for anisotropic fluid flow, and the absence of special model constants in contrast to other SGS closures. However, the scale-similarity model does not dissipate energy enough and usually leads to inaccurate results in decaying turbulence or blows up the simulation. But the situation changes significantly when a forced turbulence is considered. In this case, subgrid modeling in LES approach provides correct stationary regime of the turbulence rather than to guarantee proper energy dissipation. It was shown that the scale-similarity model provides good accuracy and the results of this SGS model agree well with the DNS results. If differences between the results obtained by the scale- similarity model and the Smagorinsky closure for velocity field are insignificant, then the differences are considerable for magnetic field. For the magnetic field, discrepancies with the DNS results are substantially lower for scale-similarity model while the Smagorinsky parametrization for MHD case is more dissipative and the results of Smagorinsky model are worse in agreement with DNS[8]. The scale-similarity model is generally found to reproduce DNS results better. The present paper briefly summarizes results concerning discrete filters for LES of forced compressible MHD turbulence by the example of scale-similarity model. The structure of the paper is the following. The next section 2 describes the general features of LES technique in physical space. Influence and sensitivity of discrete filter shapes on scale-similarity model, test configurations and numerical analysis of the obtained results are specified in section 3. Finally, conclusion remarks are given in the last section 4. ## 2 Filtering procedure in large eddy simulation In this section, we formulate the general features of the theory of LES method for modeling of compressible forced MHD turbulent flows. To obtain the MHD equations governing the motion of the filtered (that is resolved) eddies, the large scales from the small are separated. LES approach is based on the definition of a filtering operation: a resolved (or large- scale) variable, denoted by an overbar in the present paper, is defined as $\bar{\zeta}(x_{i})=\int_{\Theta}\zeta(\acute{x_{i}})\xi(x_{i},\acute{x_{i}};\bar{\triangle})d\acute{x_{i}},$ (1) where $\xi$ is the filter function satisfying the normalization property, $\zeta$ is a flow parameter, $\Theta$ is the domain, $\bar{\triangle}$ is the filter-width associated with the wavelength of the smallest scale retained by the filtering procedure and $x_{j}=(x,y,z)$ are axes of Cartesian coordinate system. It is convenient to use the Favre filtration (it is also called mass-weighted filtration) to avoid additional SGS terms in compressible flow. Therefore, Favre filtering will be used further. Mass-weighted filtering is used for all parameters of charged fluid flow besides the pressure and magnetic fields. Favre filtering is determined as follows: $\tilde{\zeta}=\frac{\overline{{\rho}\zeta}}{\bar{\rho}}$ (2) where the tilde denotes the mass-weighted filtration. Thus, applying the Favre-filtering operation, we can rewrite the MHD equations for compressible fluid flow as [9, 8]: $\frac{\partial{\bar{\rho}}}{\partial t}+\frac{\partial{\bar{\rho}}\tilde{u_{j}}}{\partial x_{j}}=0;$ (3) $\frac{\partial{\bar{\rho}}\tilde{u_{i}}}{\partial t}+\frac{\partial}{\partial x_{j}}\left(\bar{\rho}\tilde{u_{i}}\tilde{u_{j}}+\frac{\bar{\rho}^{\gamma}}{\gamma M_{s}^{2}}\delta_{ij}-\frac{1}{Re}~{}\tilde{\sigma_{ij}}+\frac{\bar{B^{2}}}{2M_{a}^{2}}\delta_{ij}-\frac{1}{M^{2}_{a}}~{}\bar{B_{j}}\bar{B_{i}}\right)=-\frac{\partial\tau_{ji}^{u}}{\partial x_{j}}+\tilde{F_{i}^{u}};$ (4) $\frac{\partial\bar{B_{i}}}{\partial t}+\frac{\partial}{\partial x_{j}}\left(\tilde{u_{j}}\bar{B_{i}}-\tilde{u_{i}}\bar{B_{j}}\right)-\frac{1}{Re_{m}}\frac{\partial^{2}\bar{B}_{i}}{\partial x_{j}^{2}}=-\frac{\partial\tau_{ji}^{b}}{\partial x_{j}}+\tilde{F_{i}^{b}};$ (5) $\frac{\partial\bar{B_{j}}}{\partial x_{j}}=0,$ (6) Here $\rho$ is the density; $u_{j}$ is the velocity in the direction $x_{j}$; $B_{j}$ is the magnetic field in the direction $x_{j}$; $\sigma_{ij}=2\mu S_{ij}-\frac{2}{3}\mu S_{kk}\delta_{ij}$ is the viscous stress tensor; $S_{ij}=1/2\left(\partial u_{i}/\partial x_{j}+\partial u_{j}/\partial x_{i}\right)$ is the strain rate tensor; $\mu$ is the coefficient of molecular viscosity; $\eta$ is the coefficient of magnetic diffusivity; $\delta_{ij}$ is the Kronecker delta. The filtered magnetohydrodynamic equations (3) - (6) are written in the dimensionless form using the standard procedure [2] where $Re=\rho_{0}u_{0}L_{0}/\mu_{0}$ is the Reynolds number, $Re_{m}=u_{0}L_{0}/\eta_{0}$ is the magnetic Reynolds number. $M_{s}=u_{0}/c_{s}$ is the Mach number, where $c_{s}$ is the velocity of sound defined by the relation $c_{s}=\sqrt{\gamma p_{0}/\rho_{0}}$, and $M_{a}=u_{0}/u_{a}$ is the magnetic Mach number, where $u_{a}=B_{0}/(\sqrt{4\pi\rho_{0}})$ is the Alfvén velocity. To close the MHD equations (3) - (5) it is assumed that the relation between density and pressure is polytropic and has the following form: $p=\rho^{\gamma}$, where $\gamma$ is a polytropic index. The effect of the subgrid scales appears on the right-hand side of the governing MHD equations (4) - (5) through the SGS stresses: $\tau_{ij}^{u}=\bar{\rho}\left(\widetilde{u_{i}u_{j}}-\tilde{u_{i}}\tilde{u_{j}}\right)-\frac{1}{M_{a}^{2}}\left(\overline{B_{i}B_{j}}-\bar{B_{i}}\bar{B_{j}}\right);$ (7) $\tau_{ij}^{b}=\left(\overline{u_{i}B_{j}}-\tilde{u_{i}}\bar{B_{j}}\right)-\left(\overline{B_{i}u_{j}}-\bar{B_{i}}\tilde{u_{j}}\right).$ (8) Thus, the filtered system of magnetohydrodynamic equations contains the unknown turbulent tensors: $\tau_{ij}^{u}$ and $\tau_{ij}^{b}$. To determine their components special turbulent closures (models, parameterizations) based on large-scale values describing turbulent magnetohydrodynamic flow must be used. The main idea of any SGS closures used in LES is to reproduce the effects of the subgrid scale dynamics on the large-scale energy distribution, at that as a matter of fact Richardson turbulent cascade is simulated. In order to close the system of MHD equations, one should find such parameterizations for $\tau_{ij}^{u}$ and $\tau_{ij}^{b}$ that would relate these tensors to the known large-scale values of the flow parameters. There are external driving forces $F^{u}_{i}$ and $F^{b}_{i}$ on the right- hand sides of equations (4) - (5) respectively. Driving forces $F^{u}_{i}$ and $F^{b}_{i}$ which sustain turbulence are necessary to study statistically stationary flow and provide a stationary picture of the energy cascade and more statistical sampling. If energy is not injected into a turbulent flow, after some time this flow becomes laminar because of viscosity and diffusion. To sustain a three-dimensional turbulence, a driving force is employed to inject energy in the turbulent system to replace the energy which is dissipated on small spatial scales. Recently, ”linear forcing” was suggested and used for modeling of compressible MHD turbulence [9] with driving force in physical space. The idea essentially consists of adding a force proportional to the fluctuating velocity[10, 9, 11, 12]. Linear forcing resembles a turbulence when forced with a mean velocity gradient, that is, a shear. This force appears as a term in the equation for fluctuating velocity that corresponds to a production term in the equation of turbulent kinetic energy. In compressible MHD turbulence, system of MHD equations includes also the magnetic induction equation and in this case the driving force is proportional to the magnetic field in the magnetic induction equation [9]. Thus, linear external force can be interpreted as the production of magnetic energy due to the interaction between the magnetic field and the mean fluid shear. The determination of the driving forces $F^{u}_{i}$ and $F_{i}^{b}$ in the momentum conservation equation and in the magnetic induction equation, respectively, are: $F^{u}_{i}=\Theta\rho u_{i}$ (9) $F^{b}_{i}=\Psi B_{i}$ (10) where $\Theta$ in (9) is the coefficient which is determined from a balance of kinetic energy for a statistically stationary state. The forcing function $F^{u}_{i}=\Theta\rho u_{i}$ in the physical space is equivalent to force all the Fourier modes in the spectral space. This is in fact the only difference from the standard spectral forcing when energy is added in to system only in the range of small wave numbers (wavenumber shell), that is, in integrated (large) scale of turbulence. The coefficient $\Psi$ in the expression (10) is determined from the balance of magnetic energy for the statistically stationary state as well. More detailed derivation and information about linear forcing method in physical space for compressible MHD turbulent flows can be found in our article [9]. The scale-similarity model as a subgrid-scale closure for compressible MHD case is of the form[5]: $\tau_{ij}^{u}=\bar{\rho}\left(\widetilde{\tilde{u_{i}}\tilde{u_{j}}}-\tilde{\tilde{u_{i}}}\tilde{\tilde{u_{j}}}\right)-\frac{1}{M_{a}^{2}}\left(\overline{\bar{B_{i}}\bar{B_{j}}}-\bar{\bar{B_{i}}}\bar{\bar{B_{j}}}\right)$ (11) $\tau_{ij}^{b}=\left(\overline{\tilde{u_{i}}\bar{B_{j}}}-\tilde{\tilde{u_{i}}}\bar{\bar{B_{j}}}\right)-\left(\overline{\bar{B_{i}}\tilde{u_{j}}}-\bar{\bar{B_{i}}}\tilde{\tilde{u_{j}}}\right)$ (12) The scale-similarity model for MHD turbulence (11) - (12) can be calculated in a LES technique by means of the filtered variables in contrast to eddy- viscosity parameterizations. Model constants in (11) and (12) are not introduced as this would destroy the Galilean invariance of the expression[13]. ## 3 Numerical analysis of sensitivity of scale-similarity model on the filter shape Figure 1: Time dynamics of $b_{rms}$ for various filter shapes. The diamond line is the DNS results, the solid line is 5-point approximation of the Gaussian filter ($\epsilon=2$), the dashed line is 5-point approximation of the top-hat filter ($\epsilon=2$), the dash-dot line is 3-point approximation of the Gaussian (or top-hat) filter ($\epsilon=2$), the circle line is 5-point approximation of the Gaussian filter ($\epsilon=3$), the triangle line is 5-point approximation of the top-hat filter ($\epsilon=3$), and the plus line is 3-point approximation of the Gaussian (or top-hat) filter ($\epsilon=3$). In this section, influences and sensitivity of discrete filter shapes on scale-similarity model, test configurations and numerical analysis of obtained results are studied. The results obtained for LES are compared with the DNS results for three-dimensional forced compressible MHD turbulent flows. There is strong influence of the properties of the LES filter on the interactions between resolved and subgrid-scales. We examine the question of the effect of different filter shapes on scale-similarity model for forced compressible MHD turbulent flow using a finite-difference schemes. Several papers were devoted to this problem for a neutral fluids dynamics. Both theoretical and numerical studies have been carried out [14, 15, 16]. To our knowledge, the influence of discrete filters on the scale-similarity model for the case of compressible forced MHD turbulence is not studied. It should be remarked that the definition of filtering procedure (1) is too general. The real flows in the nature and in the experiments can be investigated with the help of some simpler appropriate filter. The operator defined by relation (1) is a priori non-local in physical space, and is then worst suited for computations performed with local numerical methods (e.g. finite differences, finite elements, finite volumes). Therefore, it is necessary to define some local discrete approximations for this operator. Since the finite-difference schemes for simulation of MHD turbulent flows are used in this paper, we consider the Gaussian filter and the top-hat (or box) filter. They are commonly applied when using non-spectral modeling techniques in physical space. The top-hat filter is defined as: $\xi(x,\acute{x})=\begin{cases}\frac{1}{\bar{\triangle}},\text{ if $\mid x-\acute{x}\mid\leq\frac{\bar{\triangle}}{2}$ }\\\ 0,\text{otherwise}\end{cases}$ (13) The Gaussian filter is: $\xi(x,\acute{x})=(\frac{6}{\pi\bar{\triangle}^{2}})^{1/2}exp(-\frac{6\mid x-\acute{x}\mid^{2}}{\bar{\triangle}^{2}})^{1/2})$ (14) Filter approach for hydrodynamics of neutral gas was analyzed by Sagaut and Grohens [15]. They were looking for an optimal shape of the filters which is consistent with the numerical scheme in use. They found by means of the Taylor series decomposition that the top-hat and Gaussian filters coincide exactly for second order accuracy numerical schemes (using 3 points): $\bar{\zeta_{i}}=\frac{1}{24}\epsilon^{2}\zeta_{i-1}+\frac{1}{12}(12-\epsilon^{2})\zeta_{i}+\frac{1}{24}\epsilon^{2}\zeta_{i+1}$ (15) Fourth order accuracy numerical schemes (using 5 point) consistent with different forms of these filters. Operator equivalent to the fourth-order Gaussian filter and top-hat filter respectively are: $\bar{\zeta_{i}}=\frac{\epsilon^{4}-4\epsilon^{2}}{1152}(\zeta_{i-2}+\zeta_{i+2})+\frac{16\epsilon^{2}-\epsilon^{4}}{288}(\zeta_{i-1}+\zeta_{i+1})+\frac{\epsilon^{4}-20\epsilon^{2}+192}{192}\zeta_{i},$ (16) $\bar{\zeta_{i}}=\frac{3\epsilon^{4}-20\epsilon^{2}}{5760}(\zeta_{i-2}+\zeta_{i+2})+\frac{80\epsilon^{2}-3\epsilon^{4}}{1440}(\zeta_{i-1}+\zeta_{i+1})+\frac{3\epsilon^{4}-100\epsilon^{2}+960}{690}\zeta_{i},$ (17) Here, $\zeta_{i}$ is the flow parameter in the point $i$ and the parameter $\epsilon$ represents the ratio of the mesh size to the cut-off lengthscale of the filter [15]. It is usually assumed that the parameter $\epsilon$ is equal to $2$ in the works where the fluid flows are modeled by means of LES approach. However, in order to study how this parameter affects the results of the calculation, we consider the cases when the parameter $\epsilon$ takes a different value, namely, $\epsilon=3$. Initially it should be noted that since the problem considered in this work is three-dimensional three dimensional filter (multidimensional one in the general case) must be constructed. Multidimensional filter can be constructed in two different ways [15]. The first one is a linear combination of one- dimensional filters, i.e. for every direction the flow parameter is filtered independently from the others $\xi^{n}=\frac{1}{n}\sum_{i=1}^{n}\xi^{i},$ (18) where $\xi^{i}$ is a one-dimensional filter in direction $i$, $n$ is the number of space dimensions. Linear combination represents simultaneous application of all one-dimensional filters in every spatial direction. The second approach is a product of one-dimensional filters. In that case the following can be written: $\xi^{n}=\prod_{i=1}^{n}\xi^{i}.$ (19) Such technique of determination of multidimensional filter $\xi^{n}$ represents non-simultaneous application of one-dimensional filters like in the first case but sequential one. The accuracy of constructed the multidimensional filters was tested by Sagaut and Grohens [15]. They showed that sequential product of filters gives more accurate results in comparison with linear combination of one-dimensional filters. Therefore, in this work the sequential product of filters (19) is used for three-dimensional filtration. Figure 2: Time dynamics of $u_{rms}$ for various filter shapes. Symbols as in Fig.1 Three-dimensional numerical simulations of forced compressible MHD turbulence in physical space are performed and the numerical code of the fourth order accuracy for MHD equations in the conservative form based on non-spectral finite-difference schemes is used in our work. The third order low-storage Runge-Kutta method is applied for time integration. The skew-symmetric form of nonlinear terms for modeling of turbulent flow is applied to reduce discretization errors. The skew-symmetric form is a form obtained by averaging divergent and convective forms of the nonlinear terms: $\Psi_{i}^{s}=\frac{1}{2}\left(\frac{(\partial\rho u_{i}u_{j})}{\partial x_{j}}+\rho u_{j}\frac{\partial u_{i}}{\partial x_{j}}+u_{i}\frac{(\partial\rho u_{j})}{\partial x_{j}}\right)$ (20) In spite of analytical equivalence of all three forms, their numerical realizations give different results and it was shown that skew-symmetric form improves computational accuracy for turbulent modeling. Periodic boundary conditions for all the three dimensions are applied. The similarity numbers in all simulations are: $Re\approx 300$, $Re_{M}\approx 50$, $M_{s}\approx 0.35$, $M_{a}\approx 1.4$, $\gamma=1.5$. The simulation domain is a cube $\pi\times\pi\times\pi$. The mesh with $64^{3}$ grid cells is used for LES and $256^{3}$ for DNS. The explicit LES method is used in this work. The initial isotropic turbulent spectrum close to $k^{-2}$ with random amplitudes and phases in all three directions was chosen for kinetic and magnetic energies in Fourier space. The choice of such spectrum as initial conditions is due to velocity perturbations with an initial power spectrum in Fourier space similar to that of developed turbulence [17]. This $k^{-2}$ spectrum corresponds to spectrum of Burgers turbulence. Initial conditions for the velocity and the magnetic field have been obtained in the physical space using inverse Fourier transform. The results obtained with LES technique are compared with DNS computations and performance of large eddy simulation is examined by difference between LES- and filtered DNS-results. The initial conditions for LES are obtained by filtering the initial conditions of DNS. Since our interest is on study scale-similarity SGS models which rely on the application of a filter to its discrete formulation, we consider various versions of scale-similarity closure that correspond to various 3- and 5-point approximations of both Gaussian and top-hat filters for $\epsilon=2$ and $\epsilon=3$. Time dynamics of root-mean-square magnetic field $b_{rms}$ and root-mean- square velocity $u_{rms}$ are shown in Fig.2 and in Fig.1 respectively. Here and below, in Fig.2 and Fig.1, the diamond line is the DNS results, the solid line is 5-point approximation of the Gaussian filter ($\epsilon=2$), the dashed line is 5-point approximation of the top-hat filter ($\epsilon=2$), the dash-dot line is 3-point approximation of the Gaussian (or top-hat) filter ($\epsilon=2$), the circle line is 5-point approximation of the Gaussian filter ($\epsilon=3$), the triangle line is 5-point approximation of the top- hat filter ($\epsilon=3$), and the plus line is 3-point approximation of the Gaussian (or top-hat) filter ($\epsilon=3$). In these plots, we can see that the use of the 5-point filters lead to a increase of the accuracy. The largest discrepancy with the DNS results is observed for scale-similarity results with the 3-point Gaussian (or top-hat) filters at different values of $\epsilon$. At the same time, the 5-point filters are in good agreement with the ”exact” results of DNS. One can notice that 5-point filters lead to similar results for the two values of parameter $\epsilon$ whereas a 3-point filter produces more discrepancies for magnetic field. The spectral distribution of the kinetic and the magnetic energies that shows redistribution of energy depending on wave number, i.e., at different scales. The investigation of inertial range properties is one of the main tasks in studies of scale-similarity spectra of MHD turbulence. Inertial range properties are defined as time averages over periods of stationary turbulence conditions. It is worth noting that the famous spectra of Iroshnikov-Kraichnan and Kolmogorov-Obukhov for MHD turbulence were obtained for the total energy. Total energy is the sum of kinetic and magnetic energy $E_{T}=E_{M}+E_{K}$. The spectra of total energy $E_{T}^{K}$ corresponding to these various cases are shown in Fig.3. As expected from the theory of LES method, the main differences in the results are concentrated on the small (unresolved) scales. In order to observe these differences better, for clarity sake, Figure 4 shows enlargement zone for large values of wave number $k$. It should be noted that the Gaussian filter is more sensitive to the parameter $\epsilon$ than the top-hat one for scale-similarity model in compressible MHD turbulence. From our calculations it can be seen that the 3-point filters give the worst results and the 5-point Gaussian filter demonstrates the best results (that is, best approximation to DNS) at $\epsilon=2$. However, the difference between these filters is still within $10\%$. Figure 3: Total energy spectrum $E_{T}^{K}$ for various filter shapes. Symbols as in Fig.1 Figure 4: Total energy spectrum $E_{T}^{K}$ in enlargement zone of large values of wave number $k$ for various filter shapes. Symbols as in Fig.1 ## 4 Concluding remarks It appears that parameter $\epsilon$ which represents the ratio of the mesh size to the cut-off lengthscale of the filter is an important parameter regarding the discrete filters of LES approach. The present study summarized results concerning discrete filters for LES method of forced compressible MHD turbulent flows with the scale-similarity model. Scale-similarity parametrization has evident advantages in forcing compressible turbulence. Influences and effects of discrete filter shapes on the scale-similarity model were examined in physical space using a finite-difference numerical schemes. In this paper, the obtained results of numerical computations for LES were compared with the DNS results of three-dimensional compressible forced MHD turbulent flows. The comparison between LES and DNS results was carried out regarding the time evolution of $b_{rms}$ and $u_{rms}$, and the total energy spectra of MHD turbulence. It was shown that the Gaussian filter is more sensitive to the parameter $\epsilon$ (the ratio of the mesh size to the cut- off lengthscale of the filter) than the top-hat filter for the scale- similarity model in compressible MHD turbulent fluid flow. Noteworthy result is that discrete filters produce more discrepancies for magnetic field. Therefore, it is important to choose correctly a filter using LES approach for modeling of forced compressible MHD turbulence. The 3-point filters at $\epsilon=2$ and $\epsilon=3$ give the least accurate results and the 5-point Gaussian filter demonstrates the best results at $\epsilon=2$. The difference between these filters is within $10\%$. As expected, the main differences in the results are concentrated on the small scales. The work was supported by the program P-22 of Russian Academy of Science Presidium ”Basic problems in solar system studies”. ## References * [1] Garnier, E., Adams, N., and Sagaut, P., 2009. Large Eddy Simulation for Compressible Flows. Springer Science+Business Media B.V., Netherlands. * [2] Biskamp, D., 2003. Magnetohydrodynamic turbulence. Cambridge University Press, United Kingdom. * [3] Chernyshov, A. A., Karelsky, K. V., and Petrosyan, A. S., 2006. “Large-eddy simulation of magnetohydrodynamic turbulence in compressible fluid”. Phys. Plasmas, 13(3), p. 032304. * [4] Chernyshov, A. A., Karelsky, K. V., and Petrosyan, A. S., 2008. “Modeling of compressible magnetohydrodynamic turbulence in electrically and heat conducting fluid using large eddy simulation”. Physics of Fluids, 20(8), p. 085106. * [5] Chernyshov, A. A., Karelsky, K. V., and Petrosyan, A. S., 2007. “Development of large eddy simulation for modeling of decaying compressible mhd turbulence”. Physics of Fluids, 19(5), p. 055106. * [6] Müller, W.-C., and Carati, D., 2002. “Dynamic gradient-diffudion subgrid models for incompressible magnetohydrodynamics turbulence”. Phys. Plasmas, 9(3), pp. 824–834. * [7] Chernyshov, A. A., Karelsky, K. V., and Petrosyan, A. S., 2009. “Validation of large eddy simulation method for study of flatness and skewness of decaying compressible magnetohydrodynamic turbulence”. Theor. Comput. Fluid Dyn., 23(6), pp. 451–470. * [8] Chernyshov, A. A., Karelsky, K. V., and Petrosyan, A. S., 2012. “Efficiency of scale-similarity model for study of forced compressible magnetohydrodynamic turbulence”. Flow, Turbulence and Combustion, 89(4), pp. 563–587. * [9] Chernyshov, A. A., Karelsky, K. V., and Petrosyan, A. S., 2010. “Forced turbulence in large eddy simulation of compressible magnetohydrodynamic turbulence”. Phys. Plasmas, 17(10), p. 102307. * [10] Lundgren, T. S., 2003. “Linearly forced isotropic turbulence”. Center for Turbulence Research Annual Research Briefs, pp. 461–473. * [11] Rosales, C., and Meneveau, C., 2005. “Linear forcing in numerical simulations of isotropic turbulence: Physical space implementations and convergence properties”. Physics of Fluids, 17(9), pp. 095106–+. * [12] Stefano, G. D., and Vasilyev, O. V., 2010. “Stochastic coherent adaptive large eddy simulation of forced isotropic turbulence”. J. Fluid Mechanics, 646, pp. 453–470. * [13] Speziale, G., 1985. “Galilean invariance of subgrid-scale stress models in les of turbulence”. J. Fluids Mech., 156, p. 55 62. * [14] Piomelli, U., Ferziger, J. H., and Moin, P., 1988. “Model consistency in large eddy simulation of turbulent channel flows”. Physics of Fluids, 31, pp. 1884–1891. * [15] Sagaut, P., and Grohens, R., 1999. “Discrete filters for large eddy simulation”. Int. J. Numer. Mech. Fluids, 31, pp. 1195–1220. * [16] Leslie, D. C., and Quarini, G. L., 1979. “The application of turbulence theory to the formulation of subgrid modelling procedures”. Journal of Fluid Mechanics, 91, pp. 65–91. * [17] Low, M.-M. M., Klessen, R. S., Burkert, A., and Smith, M. D., 1998. “Kinetic energy decay rates of supersonic and super-alfvenic turbulence in star-forming clouds”. Phys. Rev. Lett., 80, pp. 2754–2764.	arxiv-papers	2013-11-08T10:23:35	2024-09-04T02:49:53.406863	{ "license": "Public Domain", "authors": "Alexander A. Chernyshov, Kirill. V. Karelsky, Arakel. S. Petrosyan", "submitter": "Arakel Petrosyan", "url": "https://arxiv.org/abs/1311.1922" }
1311.2125	# Exact solution of the area reactivity model of an isolated pair Thorsten Prüstel Laboratory of Systems Biology National Institute of Allergy and Infectious Diseases National Institutes of Health Martin Meier-Schellersheim Laboratory of Systems Biology National Institute of Allergy and Infectious Diseases National Institutes of Health ###### Abstract We investigate the reversible diffusion-influenced reaction of an isolated pair in two space dimensions in the context of the area reactivity model. We compute the exact Green’s function in the Laplace domain for the initially unbound molecule. Furthermore, we calculate the exact expression for the Green’s function in the time domain by inverting the Laplace transform via the Bromwich contour integral. The obtained results should be useful for comparing the behavior of the area reactivity model with more conventional models based on contact reactivity. 11footnotetext: Email: [email protected], [email protected] ## 1 Introduction The Smoluchowski model is widely used in the theory of diffusion-influenced reactions [9, 7]. According to this picture, a pair of molecules separated by a distance $r$ may react when they encounter each other at a critical distance $r=a$ via their diffusive motion. Hence, reactive molecules can be modeled by solutions of the diffusion equation that satisfy certain types of boundary conditions (BC) at the encounter distance $r=a$. In the case of an isolated pair, exact expressions for Green’s functions (GF) in the time domain, describing irreversible and reversible reactions in one, two and three space dimensions, have been obtained [3, 2, 6, 5]. However, there are alternative approaches to describe the reversible diffusion-influenced reaction of an isolated pair. Ref. [4] discussed the so- called volume reactivity model that eliminates the distinct role of the encounter radius $r=a$ and instead postulates that the reaction can happen throughout the spherical volume $r\leq a$. In the present manuscript, we discuss the corresponding model in two dimensions (2D) and hence refer to it as the ”area reactivity” model. Diffusion in 2D is special from both a conceptual and technical point of view. Conceptually, it is the critical dimension regarding recurrence and transience of random walks [8]. Technically, the mathematical treatment appears to be more involved than in 1D and 3D [6]. A system of two molecules $A$ and $B$ with diffusion constants $D_{A}$ and $D_{B}$, respectively, can also be described as the diffusion of a point-like molecule with diffusion constant $D=D_{A}+D_{B}$ around a static disk. More precisely, the area-reactivity model assumes that the molecule undergoes free diffusion apart from inside the static ”reaction disk” of radius $r=a$, where it may react reversibly. Without loss of generality, we assume that the disk’s center is located at the origin. A central notion is the probability density function (PDF) $p(r,t\|r_{0})$ that gives the probability to find the molecule unbound at a distance equal to $r$ at time $t$, given that the distance was initially $r_{0}$ at time $t=0$. Note that in contrast to the contact reactivity model, $p(r,t\|r_{0})$ is also defined for $r<a$. Moreover, because the molecule may bind anywhere within the disk $r<a$, it makes sense to define another PDF $q(r,t\|r_{0})$, which yields the probability to find the molecule bound at a distance equal to $r<a$ at time $t$, given that the distance was initially $r_{0}$ at time $t=0$. The rates for association and dissociation are $\kappa_{r}\Theta(a-r)p(r,t\|r_{0})$ and $\kappa_{d}q(r,t\|r_{0})$, respectively, where $\Theta(x)$ refers to the Heaviside step-function that vanishes for $x<0$ and assumes unity otherwise. Furthermore, it is assumed that the dissociated molecule is released at the same point where it assumed its bound state. The equations of motion for the PDF $p(r,t\|r_{0})$ and $q(r,t\|r_{0})$ are coupled and read [4] $\displaystyle\frac{\partial p(r,t\|r_{0})}{\partial t}$ $\displaystyle=$ $\displaystyle\mathcal{L}_{r}p(r,t\|r_{0})-\kappa_{r}\Theta(a-r)p(r,t\|r_{0})+\kappa_{d}q(r,t\|r_{0}),\quad\quad$ (1.1) $\displaystyle\frac{\partial q(r,t\|r_{0})}{\partial t}$ $\displaystyle=$ $\displaystyle\kappa_{r}\Theta(a-r)p(r,t\|r_{0})-\kappa_{d}q(r,t\|r_{0}),$ (1.2) where $\mathcal{L}_{r}=D\biggl{(}\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial r}\biggr{)}.$ (1.3) The equations of motion have to be supplemented by BC at the origin and at infinity, respectively, $\displaystyle\lim_{r\rightarrow\infty}p(r,t\|r_{0})$ $\displaystyle=$ $\displaystyle 0,$ (1.4) $\displaystyle\lim_{r\rightarrow 0}r\frac{\partial p(r,t\|r_{0})}{\partial r}$ $\displaystyle=$ $\displaystyle 0.$ (1.5) In the present manuscript, we focus on the case of the initially unbound molecule. Therefore, the initial conditions are $\displaystyle 2\pi r_{0}p(r,0\|r_{0})$ $\displaystyle=$ $\displaystyle\delta(r-r_{0}),$ (1.6) $\displaystyle q(r,0\|r_{0})$ $\displaystyle=$ $\displaystyle 0.$ (1.7) ## 2 Exact Green’s function in the Laplace domain By applying the Laplace transform, Eqs. (1.1)-(1.2) become $\displaystyle s\tilde{p}(r,s\|r_{0})-p(r,0\|r_{0})$ $\displaystyle=$ $\displaystyle\mathcal{L}_{r}\tilde{p}(r,s\|r_{0})-\kappa_{r}\Theta(a-r)\tilde{p}(r,s\|r_{0})$ (2.1) $\displaystyle+\kappa_{d}\tilde{q}(r,s\|r_{0}),$ $\displaystyle s\tilde{q}(r,s\|r_{0})-q(r,0\|r_{0})$ $\displaystyle=$ $\displaystyle\kappa_{r}\Theta(a-r)\tilde{p}(r,s\|r_{0})-\kappa_{d}\tilde{q}(r,s\|r_{0}),$ (2.2) where $s$ denotes the Laplace space variable. We use Eq. (1.7) to obtain from Eq. (2.2) $\tilde{q}(r,s\|r_{0})=\frac{\kappa_{r}}{s+\kappa_{d}}\Theta(a-r)\tilde{p}(r,s\|r_{0}).$ (2.3) Now we can eliminate $\tilde{q}(r,s\|r_{0})$ from Eq. (2.1) $\bigg{[}\mathcal{L}_{r}-s-\frac{s\kappa_{r}}{s+\kappa_{d}}\Theta(a-r)\bigg{]}\tilde{p}(r,s\|r_{0})=-\frac{\delta(r-r_{0})}{2\pi r},$ (2.4) where we used Eq. (1.6). In the following, we will calculate the GF separately on the two different domains defined by $r>a$ and $r<a$. The two obtained solutions will still contain unknown constants. The GF can then be completely determined by matching both expressions upon continuity requirements at $r=a$. Henceforth, we will denote the GF within $r<a$ and outside $r>a$ the reactive disk by $p^{<}(r,t\|r_{0})$ and $p^{>}(r,t\|r_{0})$, respectively. Also, throughout this manuscript we assume that the molecule was initially located outside the reaction area $r_{0}>a$. Then, we make the following ansatz for the Laplace transform of the GF $p^{>}(r,t\|r_{0})$ outside the disk $r>a$, $\tilde{p}^{>}(r,s\|r_{0})=\tilde{p}_{0}(r,s\|r_{0})+\tilde{f}(r,s\|r_{0}),$ (2.5) where $\tilde{p}_{0}(r,s\|r_{0})=\frac{1}{2\pi D}\biggl{\\{}\begin{array}[]{lr}I_{0}(vr_{0})K_{0}(vr),&\text{$r>r_{0}$}\\\ I_{0}(vr)K_{0}(qr_{0}),&\text{$r<r_{0}$}\end{array}$ (2.6) is the Laplace transform of the free-space GF, cf. [3, Ch. 14.8, Eq. (2)]. $I_{0}(x),K_{0}(x)$ denote the modified Bessel functions of first and second kind, respectively, and zero order [1, Sect. 9.6]. The variable $v$ is defined by $v:=\sqrt{s/D}.$ (2.7) Note that the free GF takes into account the $\delta$ function term in Eq. (2.4) and therefore, the function $\tilde{f}(r,s\|r_{0})$ in Eq. (2.5) satisfies the Laplace transformed 2D diffusion equation [3, Ch. 14.8, Eq. (3)] $\frac{d^{2}\tilde{f}}{dr^{2}}+\frac{1}{r}\frac{d\tilde{f}}{dr}-v^{2}\tilde{f}=0.$ (2.8) The general solution to Eq. (2.8) is given by $\tilde{f}(r,v)=B(s,r_{0})I_{0}(vr)+C(s,r_{0})K_{0}(vr),$ (2.9) where $B(s,r_{0}),C(s,r_{0})$ are ”constants” that may depend on $s$ and $r_{0}$. Because we require the BC Eq. (1.4) and $\lim_{x\rightarrow\infty}I_{0}(x)\rightarrow\infty$, the coefficient $B(s,r_{0})$ has to vanish and the solution becomes, $\tilde{f}(r,v\|r_{0})=C(v,r_{0})K_{0}(vr).$ (2.10) Next, turning to the case $r<a$, the GF satisfies $\frac{d^{2}\tilde{p}^{<}}{dr^{2}}+\frac{1}{r}\frac{d\tilde{p}^{<}}{dr}-w^{2}\tilde{p}^{<}=0,$ (2.11) where $w$ is defined by $w:=v\sqrt{\frac{s+\kappa_{r}+\kappa_{d}}{s+\kappa_{d}}}.$ (2.12) Therefore, the general solution, which takes into account the BC Eq. (1.5) is $p^{<}(r,w\|r_{0})=A(s,r_{0})I_{0}(wr),$ (2.13) because $\lim_{x\rightarrow 0}xK_{1}(x)\neq 0$. The two ”constants” $A(s,r_{0})$ and $C(s,r_{0})$ can be determined by the requirement that the GF and its derivative have to be continuous at $r=a$ $\displaystyle\tilde{p}^{<}(r=a,s\|r_{0})$ $\displaystyle=$ $\displaystyle\tilde{p}^{>}(r=a,s\|r_{0})$ (2.14) $\displaystyle\frac{\partial\tilde{p}^{<}(r,s\|r_{0})}{\partial r}\bigg{\|}_{r=a}$ $\displaystyle=$ $\displaystyle\frac{\partial\tilde{p}^{>}(r,s\|r_{0})}{\partial r}\bigg{\|}_{r=a}$ (2.15) Using Eqs. (2.5), (2.6), (2.10), (2.13) as well as $\displaystyle I^{\prime}_{0}(x)$ $\displaystyle=$ $\displaystyle I_{1}(x),$ (2.16) $\displaystyle K^{\prime}_{0}(x)$ $\displaystyle=$ $\displaystyle- K_{1}(x),$ (2.17) $\displaystyle x^{-1}$ $\displaystyle=$ $\displaystyle I_{0}(x)K_{1}(x)+I_{1}(x)K_{0}(x),$ (2.18) [1, Eqs. (9.6.27), (9.6.15)], we obtain $\displaystyle A(s,r_{0})$ $\displaystyle=$ $\displaystyle\frac{K_{0}(vr_{0})}{2\pi aD\mathcal{N}},$ (2.19) $\displaystyle C(s,r_{0})$ $\displaystyle=$ $\displaystyle\frac{K_{0}(vr_{0})}{2\pi aDK_{0}(va)}\bigg{[}\frac{I_{0}(wa)}{\mathcal{N}}-aI_{0}(va)\bigg{]},$ (2.20) where we introduced $\displaystyle\mathcal{N}=vI_{0}(wa)K_{1}(va)+wI_{1}(wa)K_{0}(va).$ (2.21) ## 3 Exact Green’s function in the time domain To find the corresponding expressions for $p^{<}(r,t\|r_{0}),p^{>}(r,t\|r_{0})$ in the time domain, we apply the inversion theorem for the Laplace transformation $p^{<}(r,t\|r_{0})=\frac{1}{2\pi i}\int^{c+i\infty}_{c-i\infty}e^{st}\,\tilde{p}^{<}(r,s\|r_{0})ds.$ (3.1) We note that $\tilde{p}^{<}(r,s\|r_{0})$ has three branch points at $s=0,-\kappa_{d}$ and $s=-\kappa_{r}-\kappa_{d}\equiv-\varphi$. Therefore, to calculate the Bromwich integral, we use the contour of Fig. 1 with a branch cut along the negative real axis, cf. [3, Ch. 12.3, FIG. 40]. We arrive at $\displaystyle\int^{c+i\infty}_{c-i\infty}e^{st}\,\tilde{p}^{<}(r,s\|r_{0})ds=$ $\displaystyle-$ $\displaystyle\int_{\mathcal{C}_{2}}e^{ps}\,\tilde{p}^{<}(r,s\|r_{0})ds$ (3.2) $\displaystyle-$ $\displaystyle\int_{\mathcal{C}_{4}}e^{st}\,\tilde{p}^{<}(r,s\|r_{0})ds.$ To calculate the integral $\int_{\mathcal{C}_{2}}$, we choose $s=Dx^{2}e^{i\pi}.$ Then, $\displaystyle v$ $\displaystyle=$ $\displaystyle ix,\quad\text{for}\,\,s\in]-\infty,0[$ (3.3) $\displaystyle w$ $\displaystyle=$ $\displaystyle ix\sqrt{\frac{Dx^{2}-\varphi}{Dx^{2}-\kappa_{d}}}\equiv ix\xi_{1}\quad\text{for}\,\,s\in]-\infty,-\varphi[,$ (3.4) $\displaystyle w$ $\displaystyle=$ $\displaystyle x\sqrt{\frac{\varphi- Dx^{2}}{Dx^{2}-\kappa_{d}}}\equiv x\xi_{2}\quad\text{for}\,\,s\in]-\varphi,-\kappa_{d}[,$ (3.5) $\displaystyle w$ $\displaystyle=$ $\displaystyle ix\sqrt{\frac{\varphi- Dx^{2}}{\kappa_{d}-Dx^{2}}}=ix\xi_{1}\quad\text{for}\,\,s\in]-\kappa_{d},0[,$ (3.6) We now make use of [3, Append. 3, Eqs. (25), (26))] $\displaystyle I_{n}(xe^{\pm\pi i/2})$ $\displaystyle=$ $\displaystyle e^{\pm n\pi i/2}J_{n}(x),$ (3.8) $\displaystyle K_{n}(xe^{\pm\pi i/2})$ $\displaystyle=$ $\displaystyle\pm\frac{1}{2}\pi ie^{\mp n\pi i/2}[-J_{n}(x)\pm iY_{n}(x)].$ (3.9) $J_{n}(x),Y_{n}(x)$ refer to the Bessel functions of first and second kind, respectively [1, Sect. 9.1]. It follows that $\displaystyle\int_{\mathcal{C}_{2}}e^{st}\,\tilde{p}^{<}(r,s\|r_{0})ds=\frac{1}{\pi a}\bigg{[}\int^{\sqrt{\frac{\varphi}{D}}}_{\sqrt{\frac{\kappa_{d}}{D}}}e^{-Dx^{2}t}g^{(2)}(r,r_{0},x)dx$ $\displaystyle-\int^{\sqrt{\frac{\kappa_{d}}{D}}}_{0}e^{-Dx^{2}t}g^{(1)}(r,r_{0},x)dx-\int^{\infty}_{\sqrt{\frac{\varphi}{D}}}e^{-Dx^{2}t}g^{(1)}(r,r_{0},x)dx\bigg{]},\quad\quad\quad$ (3.10) where we introduced $\displaystyle g^{(2)}(r,r_{0},x)$ $\displaystyle\equiv$ $\displaystyle g^{(2)}_{R}(r,r_{0},x)+ig^{(2)}_{I}(r,r_{0},x),$ (3.11) $\displaystyle=$ $\displaystyle I_{0}(x\xi_{2}r)\frac{\eta(r_{0})+i\lambda(r_{0})}{\alpha^{2}+\beta^{2}},\quad\quad$ $\displaystyle g^{(1)}(r,r_{0},x)$ $\displaystyle\equiv$ $\displaystyle g^{(1)}_{R}(r,r_{0},x)+ig^{(1)}_{I}(r,r_{0},x),$ (3.12) $\displaystyle=$ $\displaystyle J_{0}(x\xi_{1}r)\frac{\omega(r_{0})+i\varkappa(r_{0})}{\gamma^{2}+\delta^{2}},\quad\quad$ and $\displaystyle\eta(r_{0})$ $\displaystyle=$ $\displaystyle\alpha Y_{0}(xr_{0})+\beta J_{0}(xr_{0}),$ (3.13) $\displaystyle\lambda(r_{0})$ $\displaystyle=$ $\displaystyle\alpha J_{0}(xr_{0})-\beta Y_{0}(xr_{0}),$ (3.14) $\displaystyle\alpha$ $\displaystyle=$ $\displaystyle\xi_{2}I_{1}(\xi_{2}xa)Y_{0}(xa)+I_{0}(\xi_{2}xa)Y_{1}(xa),$ (3.15) $\displaystyle\beta$ $\displaystyle=$ $\displaystyle\xi_{2}I_{1}(\xi_{2}xa)J_{0}(xa)+I_{0}(\xi_{2}xa)J_{1}(xa),$ (3.16) $\displaystyle\omega(r_{0})$ $\displaystyle=$ $\displaystyle\gamma Y_{0}(xr_{0})+\delta J_{0}(xr_{0}),$ (3.17) $\displaystyle\varkappa(r_{0})$ $\displaystyle=$ $\displaystyle\gamma J_{0}(xr_{0})-\delta Y_{0}(xr_{0}),$ (3.18) $\displaystyle\gamma$ $\displaystyle=$ $\displaystyle\xi_{1}J_{1}(\xi_{1}xa)Y_{0}(xa)-J_{0}(\xi_{1}xa)Y_{1}(xa),$ (3.19) $\displaystyle\delta$ $\displaystyle=$ $\displaystyle\xi_{1}J_{1}(\xi_{1}xa)J_{0}(xa)-J_{0}(\xi_{1}xa)J_{1}(xa).$ (3.20) Now, to calculate the integral along the contour $\mathcal{C}_{4}$, we choose $s=Dx^{2}e^{-i\pi}$ and after an analogous calculation one finds that $\int_{\mathcal{C}_{2}}e^{st}\,\tilde{p}^{<}(r,s\|r_{0})ds=-\bigg{(}\int_{\mathcal{C}_{4}}e^{st}\,\tilde{p}^{<}(r,s\|r_{0})ds\bigg{)}^{\ast},$ (3.21) where $\ast$ denotes complex conjugation. Thus, one obtains for the GF $p^{<}(r,t\|r_{0})$ on the domain $r<a$ $\displaystyle p^{<}(r,t\|r_{0})$ $\displaystyle=$ $\displaystyle-\frac{1}{\pi}\mathfrak{Im}\bigg{(}\int_{\mathcal{C}_{2}}e^{st}\,\tilde{p}^{<}(r,s\|r_{0})ds\bigg{)}$ (3.22) $\displaystyle=$ $\displaystyle-\frac{1}{\pi^{2}a}\bigg{[}\int^{\sqrt{\frac{\varphi}{D}}}_{\sqrt{\frac{\kappa_{d}}{D}}}e^{-Dx^{2}t}g^{(2)}_{I}(r,r_{0},x)dx$ $\displaystyle-$ $\displaystyle\int^{\sqrt{\frac{\kappa_{d}}{D}}}_{0}e^{-Dx^{2}t}g^{(1)}_{I}(r,r_{0},x)dx-\int^{\infty}_{\sqrt{\frac{\varphi}{D}}}e^{-Dx^{2}t}g^{(1)}_{I}(r,r_{0},x)dx\bigg{]},\quad\quad\quad$ Analogously, we can proceed to compute the GF for the region $r>a$. Therefore, we only give the result $\displaystyle p^{>}(r,t\|r_{0})=\frac{1}{4\pi Dt}e^{-(r^{2}+r^{2}_{0})/4Dt}I_{0}\bigg{(}\frac{rr_{0}}{2Dt}\bigg{)}$ $\displaystyle+\frac{1}{\pi^{2}a}\bigg{[}\int^{\sqrt{\frac{\kappa_{d}}{D}}}_{0}e^{-Dx^{2}t}h^{(1)}(r,r_{0},x)dx+\int^{\infty}_{\sqrt{\frac{\varphi}{D}}}e^{-Dx^{2}t}h^{(1)}(r,r_{0},x)dx\quad\quad\quad$ $\displaystyle-\int^{\sqrt{\frac{\varphi}{D}}}_{\sqrt{\frac{\kappa_{d}}{D}}}e^{-Dx^{2}t}h^{(2)}(r,r_{0},x)dx\bigg{]}-\frac{1}{2\pi}\int^{\infty}_{0}e^{-Dx^{2}t}h^{(3)}(r,r_{0},x)xdx,$ (3.23) where we defined $\displaystyle h^{(1)}(r,r_{0},x)$ $\displaystyle=$ $\displaystyle J_{0}(x\xi_{1}a)\frac{\rho(r)\omega(r_{0})+\psi(r)\varkappa(r_{0})}{[\gamma^{2}+\delta^{2}][J_{0}^{2}(xa)+Y_{0}^{2}(xa)]},$ (3.24) $\displaystyle h^{(2)}(r,r_{0},x)$ $\displaystyle=$ $\displaystyle I_{0}(x\xi_{2}a)\frac{\rho(r)\eta(r_{0})+\psi(r)\lambda(r_{0})}{[\alpha^{2}+\beta^{2}][J_{0}^{2}(xa)+Y_{0}^{2}(xa)]},$ (3.25) $\displaystyle h^{(3)}(r,r_{0},x)$ $\displaystyle=$ $\displaystyle J_{0}(xa)\frac{\Pi(r,r_{0})Y_{0}(xa)+\Omega(r,r_{0})J_{0}(xa)}{J_{0}^{2}(xa)+Y_{0}^{2}(xa)},$ (3.26) and $\displaystyle\rho(r)$ $\displaystyle=$ $\displaystyle J_{0}(xr)Y_{0}(xa)-Y_{0}(xr)J_{0}(xa),$ (3.27) $\displaystyle\psi(r)$ $\displaystyle=$ $\displaystyle J_{0}(xr)J_{0}(xa)+Y_{0}(xr)Y_{0}(xa),$ (3.28) $\displaystyle\Omega(r,r_{0})$ $\displaystyle=$ $\displaystyle J_{0}(xr)J_{0}(xr_{0})-Y_{0}(xr)Y_{0}(xr_{0}),$ (3.29) $\displaystyle\Pi(r,r_{0})$ $\displaystyle=$ $\displaystyle Y_{0}(xr)J_{0}(xr_{0})+J_{0}(xr)Y_{0}(xr_{0}).$ (3.30) Note that the first term appearing on the rhs of Eq. (3) is the inverse Laplace transform of Eq. (2.6), cf. [3, Ch. 14.8, Eq. (2)]. Finally, we can compute an exact expression for $q(r,t\|r_{0})$ by virtue of Eq. (2.3) and the convolution theorem of the Laplace transform. We obtain for $r<a$ $\displaystyle q(r,t\|r_{0})$ $\displaystyle=$ $\displaystyle-\frac{1}{\pi^{2}a}\bigg{[}\int^{\sqrt{\frac{\varphi}{D}}}_{\sqrt{\frac{\kappa_{d}}{D}}}\bigg{(}\frac{e^{-Dx^{2}t}-e^{-\kappa_{d}x^{2}t}}{\kappa_{d}-Dx^{2}}\bigg{)}g^{(2)}_{I}(r,r_{0},x)dx$ (3.31) $\displaystyle-$ $\displaystyle\int^{\sqrt{\frac{\kappa_{d}}{D}}}_{0}\bigg{(}\frac{e^{-Dx^{2}t}-e^{-\kappa_{d}t}}{\kappa_{d}-Dx^{2}}\bigg{)}g^{(1)}_{I}(r,r_{0},x)dx$ $\displaystyle-$ $\displaystyle\int^{\infty}_{\sqrt{\frac{\varphi}{D}}}\bigg{(}\frac{e^{-Dx^{2}t}-e^{-\kappa_{d}t}}{\kappa_{d}-Dx^{2}}\bigg{)}g^{(1)}_{I}(r,r_{0},x)dx\bigg{]}.\quad\quad\quad$ Clearly, $q(r,t\|r_{0})$ vanishes for $r>a$. The case of an initially unbound molecule with $r_{0}<a$ and the case of the initially bound molecule will be considered in a forthcoming manuscript. Figure 1: Integration contour used for calculating the GF in the time domain, Eq. (3.2). ## Acknowledgments This research was supported by the Intramural Research Program of the NIH, National Institute of Allergy and Infectious Diseases. ## References * [1] M. Abramowitz and I.A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York, 1965. * [2] N. Agmon. J. Chem. Phys., 81:2811, 1984. * [3] H.S. Carslaw and J.C. Jaeger. Conduction of Heat in Solids. Clarendon Press, New York, 1986. * [4] S.S. Khokhlova and N. Agmon. J. Chem. Phys., 137:184103, 2012. * [5] H. Kim and K.J. Shin. Phys. Rev. Lett., 82:1578, 1999. * [6] T. Prüstel and M. Meier-Schellersheim. J. Chem. Phys., 137:054104, 2012. * [7] S. A. Rice. Diffusion Limited Reactions. Elsevier, New York, 1985. * [8] D. Toussaint and F. Wilczek. J. Chem. Phys., 78:2642, 1983. * [9] M. von Smoluchowski. Z. Phys. Chem., 92:129, 1917.	arxiv-papers	2013-11-09T01:46:20	2024-09-04T02:49:53.423715	{ "license": "Public Domain", "authors": "Thorsten Pr\\\"ustel and Martin Meier-Schellersheim", "submitter": "Thorsten Pr\\\"ustel", "url": "https://arxiv.org/abs/1311.2125" }
1311.2172	# Entanglement of three cavity fields via resonant interactions with dressed three-level atoms Jinhua ZouAuthor for correspondence. [email protected] College of Physical Science and Technology, Yangtze University, Jingzhou, 434023, China ###### Abstract In this paper we show that three cavity fields can be entangled when they are tuned on resonance with an ensemble of dressed three-level atoms. The master equation for the three cavity modes is derived by using atomic dressed states and the inseparability of the three output cavity modes is described by using a sufficient criterion proposed by van Loock and Furusawa. The physical cause is the atomic coherence effects, by which the quantum correlations are created in the field dynamics. Keywords: continuous-variable entanglement, output tripartite entanglement, atomic coherence ###### pacs: PACS numbers: 42.50.Dv, 03.67.Mn ## I Introduction Atomic coherence lies in the center of many novel effects in quantum optics and laser physics. Electromagnetically induced transparency [1,2], coherent population trapping [2], Hanel-effect laser [3] and quantum beat laser [4] are such examples. Besides these, the correlation between the photons can also be induced by atomic coherence [5-12]. One such example is the generation of squeezed light in a three-level cascade laser using atomic coherence [5-8]. The atomic coherence can be created by preparing the atoms initially in a coherent superposition state of the two states which are dipole-forbidden [5-8] or driving the two states by a strong coherent field [9-11] or Raman coupling the two states through the third auxiliary atomic states [12]. For two-photon correlated-spontaneous-emission laser with injected atomic coherence, it exhibits complete spontaneous-emission noise quenching and phase squeezing simultaneously [5]. It has also been pointed out that atomic coherence in a two-photon correlated emission laser system can be used to generate a macroscopic two-mode entangled state and this system can be treated as an entanglement amplifier [12]. Recently, the topic of continuous-variable entanglement has attracted much attention as it is the base of all branches of quantum information and communication protocols [13]. Among various entanglement generation schemes, entanglement induced by atomic coherence has been extensively researched [14-16]. For a nondegenerate three-level cascade laser with a subthreshold nondegenerate parametric oscillator coupled to a vacuum reservoir, the entanglement and squeezing for the two cavity modes in this combined system is induced by the injected atomic coherence [14]. In a two-mode single-atom laser with the atomic coherence exhibited by two classical laser fields, entanglement between two field modes is demonstrated [15]. Later, it was shown that in a three-level $\Lambda$ or V atomic system with two classical driving fields and two cavity modes coupling corresponding transitions, by exploring the two-channel interaction mechanism and using the squeeze-transformed modes, continuous-variable entanglement between the two modes is obtained and the best achievable entangled state approaches the original EPR state [16]. The above work has mainly been confined to two-partite systems. With the progress in continuous-variable entanglement, the generation of more than two partite entanglement has been paid much attention as it may be the key ingredient for advanced multiparty quantum communication such as quantum teleportation network [17], telecloning [18] and controlled dense coding [19]. Among various generation schemes for tripartite systems, few work has been done to generate tripartite entanglement using atomic coherence. Most recently, a scheme to generate three-mode-entangled light fields via the interaction between the four-level atoms and the cavity has been proposed [20]. Three cavity modes are generated through three successive transitions in the four-level cascade atoms. In addition to the cavity modes, two strong classical fields drive a pair of two-photon transitions in the four-level atoms. They show that the entanglement could only be obtained in a short time as all the mean photons are amplified as time elapses. Thus at steady time, the entanglement does not exist. In this paper, we present a scheme to generate tripartite entanglement for three cavity modes via the interaction for the three-level lambda atoms with the three cavity modes and two classical fields. As the classical fields are strong, the effective interaction is resonant interaction in the dressed-state picture. We deduce the master equation of the three cavity modes by means of the atomic dressed states and linear theory. The sufficient inseparability criterion for continuous-variable entanglement is used to demonstrate the entanglement properties of the three cavity modes and the results show that our system can be used as a source to generate tripartite entangled light even at steady state. It is should be noted that, up to now many schemes have been proposed to generate tripartite entanglement using linear optics or nonlinearities [17, 21-26]. It was theoretically predicted that using single-mode squeezed state and linear optics, a truly N-partite entangled state can be generated [17]. Later, a continuous-variable tripartite entangled state was experimental realized by combing three independent squeezed vacuum states [21]. At first, the production of continuous-variable tripartite entanglement was presented by mixing squeezed beams on unbalanced beamsplitters [21,22]. Recently, generation of tripartite entanglement are focused on using cascade nonlinear interaction in an optical cavity [23-25] or in a quasiperiodic superlattice [26]. Among the latter are systems using parametric down-conversion with sum frequency generation [23,25,26] or using single nonlinearity [24]. During these nonlinear processes, the cavity modes couple with each other directly. As these nonlinear processes are related to the higher-order polarization, the efficiency of these processes are relatively small compared with the processes related to linear polarization. In this way, these nonlinear processes are not the best choice for the generation of high efficiency tripartite entangled states. Compared with the schemes based on the nonlinear processes [23-26], our scheme is more effective as the generation process is resonant interaction in the dressed states and it is only related to linear polarization. What’s more, the linear process provides much more parameters to choose than that of the nonlinear processes, as the atomic parameters can be varied. Compared with the scheme in Ref. [20], our scheme can provide steady state tripartite entanglement while the entanglement produced in scheme [20] is just kept in a quite limited time. And in Ref. [20], they use four-level cascade atomic system and the effective processes in the dressed states of the driving fields are all two-photon transitions. High excited states are involved in their scheme. In our scheme we use three-level $\Lambda$ atomic system, and the effective processes in the dressed states of the driving fields are all single-photon transitions. When we take into the account of the atomic spontaneous emission, their schemes seems to have more obstacles than ours. The paper is organized as follows. In Sec. II, we discuss the essential ingredients of the model and deduce the density-matrix equation for the cavity fields in a dressed-state picture. In Sec. III, we present the output correlation spectra by solving the equations of the cavity fields and analyze the output tripartite continuous-variable entanglement characteristics by using a sufficient criterion proposed by van Loock and Furusawa. In Sec. IV, we give a brief conclusion. ## II Model and equation We consider $N$ three-level lambda-type atoms in a three-mode cavity as shown in Fig. 1(a). Two laser fields of frequencies $\omega_{l1,l2}$ drive the transitions $\|1,2\rangle\leftrightarrow$ $\|3\rangle$, respectively. Two cavity modes $a_{1,2}$ of frequencies $\omega_{c1,c2}$ couple the atomic transition $\|1\rangle\leftrightarrow$ $\|3\rangle$, while the cavity mode $a_{3}$ with frequency $\omega_{c3}$ couples the transition $\|2\rangle\leftrightarrow$ $\|3\rangle$. $\gamma_{l}$ ($l=1,2$) are the atomic decay rates from level $\|3\rangle$ to levels $\|1,2\rangle$ and $\kappa_{l}$ ($l=1,2,3$) are the cavity loss rates. For simplicity, we assume that $\gamma_{1}=\gamma_{2}=\gamma$ and $\kappa_{1}=\kappa_{2}=\kappa_{3}=\kappa$. The three cavity modes are assumed to be in their vacuum state initially. In the frame of the frequencies of the laser fields and under the dipole and the rotating-wave approximations, the total Hamiltonian is Figure 1: (a) Atomic energy level scheme and the coupling of the cavity fields and the classic fields. (b) Equivalent resonant transitions in the picture dressed by the classical fields. $\displaystyle H$ $\displaystyle=$ $\displaystyle H_{1}+H_{2}+H_{3},$ $\displaystyle H_{1}$ $\displaystyle=$ $\displaystyle\sum_{j=1}^{3}\hbar\delta_{j}a_{j}^{\dagger}a_{j},$ $\displaystyle H_{2}$ $\displaystyle=$ $\displaystyle-\hbar\Delta\left(\sigma_{11}-\sigma_{22}\right)+\hbar\Omega\left(\sigma_{31}+\sigma_{32}+H.c.\right),$ (1) $\displaystyle H_{3}$ $\displaystyle=$ $\displaystyle i\hbar g\left(a_{1}\sigma_{31}+a_{2}\sigma_{31}+a_{3}\sigma_{32}\right)+H.c.,$ H.c. symbols the Hermitian conjugate. $H_{1}$ denotes the free energy for three cavity fields, $H_{2}$ describes the interaction of the laser fields with the atoms, and $H_{3}$ indicates the interaction of the cavity fields with the atoms. $\sigma_{jk}=\|j\rangle\langle k\|$ ($j,k=1,2,3$) are atomic dipole operators for $j\neq k$ and atomic projection operators for $j=k$. The cavity detunings are defined as $\delta_{j}=\omega_{cj}-\omega_{l1}$ ($j=1,2$), and $\delta_{3}=\omega_{c3}-\omega_{l2}$. The detunings of the laser fields are defined as $\Delta_{j}=\omega_{3j}-\omega_{lj}$ ($j=1,2$), where $\omega_{31}$ and $\omega_{32}$ are the resonance frequencies of transitions $\|1,2\rangle\leftrightarrow$ $\|3\rangle$. We have assumed equal coupling coefficients $g$ for three cavity modes, equal Rabi frequency $\Omega$ for the two laser fields, and opposite detunings of the two laser fields $\Delta_{1}=-\Delta_{2}=\Delta$. We assume that the laser fields are much stronger than the cavity fields, i.e., $\Omega\gg g\langle a_{l}\rangle$, ($l=1,2,3)$. The laser fields can be viewed as dressing fields for the atoms. Therefore, by diagonalizing the Hamiltonian $H_{2}$, we find the so-called semiclassical dressed states as $\displaystyle\|0\rangle$ $\displaystyle=$ $\displaystyle-\frac{c}{\sqrt{2}}\|1\rangle+\frac{c}{\sqrt{2}}\|2\rangle+s\|3\rangle,$ $\displaystyle\|+\rangle$ $\displaystyle=$ $\displaystyle\frac{1+s}{2}\|1\rangle+\frac{1-s}{2}\|2\rangle+\frac{c}{\sqrt{2}}\|3\rangle,$ (2) $\displaystyle\|-\rangle$ $\displaystyle=$ $\displaystyle\frac{1-s}{2}\|1\rangle+\frac{1+s}{2}\|2\rangle-\frac{c}{\sqrt{2}}\|3\rangle,$ where $c=\frac{\sqrt{2}\Omega}{d}$, $s=-\frac{\Delta}{d}$, and $d=\sqrt{\Delta^{2}+2\Omega^{2}}$. Now, we use the Hamiltonian $H_{0}=\hbar d\left(\sigma_{++}-\sigma_{--}\right)+H_{1}$ to perform the unitary dressing transformation. By choosing the cavity detunings as $\delta_{1}=d=-\delta_{2}=-\delta_{3}$, and neglecting the fast-oscillating terms such as $e^{\pm i2dt}$, we obtain the resonant interaction Hamiltonian as $V=ig\hbar\left(c_{3}a_{1}^{\dagger}+c_{2}a_{2}+c_{1}a_{3}\right)\sigma_{0+}+ig\hbar\left(c_{1}a_{1}-c_{3}a_{2}^{\dagger}+c_{3}a_{3}^{\dagger}\right)\sigma_{0-}+H.c.,$ (3) where $c_{1}=\frac{1}{2}s(1-s)$, $c_{2}=\frac{1}{2}s(1+s)$, and $c_{3}=\frac{1}{2}c^{2}$. The resonant transitions in the dressed states are shown in Fig. 1(b). The master equation for the cavity modes is obtained by using the usual approach [2], starting from $\frac{d}{dt}\rho=-\frac{i}{\hbar}\left[V,\rho\right]{\cal+L}_{a}\rho+{\cal L}_{c}\rho$, where ${\cal L}_{c}\rho=\frac{{}_{\kappa}}{2}\sum_{l=1}^{3}\left(2a_{l}\rho a_{l}^{\dagger}-a_{l}^{\dagger}a_{l}\rho-\rho a_{l}^{\dagger}a_{l}\right)$ and ${\cal L}_{a}\rho$ describes the atomic decay in the dressed states picture and its expression is very complicated. The detailed form of atomic decay term ${\cal L}_{a}\rho$ is given in Appendix A. The master equation for the cavity modes is obtained by tracing out the atomic states, which gives $\frac{d}{dt}\rho_{c}=g\left(c_{3}a_{1}^{\dagger}+c_{2}a_{2}+c_{1}a_{3}\right)\rho_{+0}+g\left(c_{1}a_{1}-c_{3}a_{2}^{\dagger}+c_{3}a_{3}^{\dagger}\right)\rho_{-0}+H.c$, where $\rho_{jk}=tr_{atom}(\sigma_{kj}\rho)$ ($j,k=0,+,-$). As the atomic variables vary much faster than the cavity fields, it is possible to express $\rho_{jk}=tr_{atom}(\sigma_{kj}\rho)$ ($jk=+0,-0,0+,0-$) in terms of $\rho_{c}$, $a_{l}$ and $a_{l}^{\dagger}$ ($l=$1-3) from the quasi-steady- state solution of the coupled equations for $\rho_{jk}=tr_{atom}(\sigma_{kj}\rho)$ ($jk=+0,-0,0+,0-$). By using $\rho_{jj}\simeq\rho_{jj}^{s}\rho_{c}$ ($j=0,+,-$) and $\rho_{+-}^{s}\simeq 0$, where “s” implies the steady-state solutions of the density matrix equations in the dressed state picture without the quantum fields $a_{l}$ and $a_{l}^{\dagger}$ ($l=$1-3). The steady state populations is obtained as $\rho_{00}^{s}=\frac{c^{4}}{1+3s^{4}}$ and $\rho_{++}^{s}=\rho_{--}^{s}=\frac{s^{2}\left(1+s^{2}\right)}{1+3s^{4}}$. The master equation for the cavity modes is obtained as $\displaystyle\frac{d}{dt}\rho_{c}$ $\displaystyle=$ $\displaystyle\sum_{l=1}^{3}\left\\{A_{ll}\left[a_{l}^{\dagger},\rho_{c}a_{l}\right]-\left(B_{ll}+\frac{\kappa_{l}}{2}\right)\left[a_{l}^{\dagger},a_{l}\rho_{c}\right]\right\\}$ $\displaystyle+\sum_{l=2}^{3}\left\\{A_{1l}\left[a_{1}^{\dagger},\rho_{c}a_{l}^{\dagger}\right]-B_{1l}\left[a_{1}^{\dagger},a_{l}^{\dagger}\rho_{c}\right]+A_{l1}\left[a_{l}^{\dagger},\rho_{c}a_{l}^{\dagger}\right]-B_{l1}\left[a_{l}^{\dagger},a_{1}^{\dagger}\rho_{c}\right]\right\\}$ $\displaystyle+\sum_{l,k=2;l\neq k}^{3}\left\\{A_{lk}\left[a_{l}^{\dagger},\rho_{c}a_{k}\right]-B_{lk}\left[a_{l}^{\dagger},a_{k}\rho_{c}\right]\right\\}+H.c..$ The explicit expressions for $A_{lk}$ and $B_{lk}$ ($l,k=1,2,3$) are given in Appendix B. Here the terms $A_{ll}$ ($l$=1-3) and $B_{ll}$ ($l$=1-3) represent the gain term and the absorption of mode $a_{l}$, respectively. And the terms $A_{lk}$ and $B_{lk}$ ($l\neq k$) represent the coupling between the two modes $a_{l}$ and $a_{k}$, and we will show that these quantities are responsible for entanglement among three cavity fields. It is easy to see that without these coupling terms between different cavity fields, the quantum correlation can not be introduced among the three cavity modes. Thus entanglement among the three cavity fields is attributed to the atomic coherence created through the interaction between the fields and the atoms. ## III Correlation spectra The master equation (4) enables us to derive equations of motion for the cavity modes: $\displaystyle\tau\frac{d}{dt}a_{1}^{\dagger}$ $\displaystyle=$ $\displaystyle\left(A_{11}-B_{11}-\frac{\kappa_{1}}{2}\right)a_{1}^{\dagger}+\left(A_{12}-B_{12}\right)a_{2}+\left(A_{13}-B_{13}\right)a_{3}+\sqrt{\kappa_{1}}a_{1}^{\dagger in},$ $\displaystyle\tau\frac{d}{dt}a_{2}$ $\displaystyle=$ $\displaystyle\left(A_{21}-B_{21}\right)a_{1}^{\dagger}+\left(A_{22}-B_{22}-\frac{\kappa_{2}}{2}\right)a_{2}+\left(A_{23}-B_{23}\right)a_{3}+\sqrt{\kappa_{2}}a_{2}^{in},$ (5) $\displaystyle\tau\frac{d}{dt}a_{3}$ $\displaystyle=$ $\displaystyle\left(A_{31}-B_{31}\right)a_{1}^{\dagger}+\left(A_{32}-B_{32}\right)a_{2}+\left(A_{33}-B_{33}-\frac{\kappa_{3}}{2}\right)a_{3}+\sqrt{\kappa_{3}}a_{3}^{in},$ where $\tau$ is the round-trip time of light in the cavity and assumed to be the same for three cavity modes. $a_{j}^{in}$ and $a_{j}^{\dagger in}$ ($j=$1-3) are annihilation and creation operators of the input fields to the cavity. This is a set of linear equations. In order to solve this equation, we use the Fourier transformation and the boundary conditions at the mirror between the output quantities and the input quantities $a_{j}^{in}+a_{j}^{out}=\sqrt{\kappa_{j}}a_{j}$ ($j=$1-3) to obtain the equation in the frequency domain as $a^{out}\left(\omega\right)=-\left(I+BD_{0}^{-1}B\right)a^{in}\left(\omega\right),$ (6) where $a^{out}\left(\omega\right)=\left(a_{1}^{\dagger out}\left(-\omega\right),a_{2}^{out}\left(\omega\right),a_{3}^{out}\left(\omega\right)\right)^{T},$ --- $a^{in}\left(\omega\right)=\left(a_{1}^{\dagger in}\left(-\omega\right),a_{2}^{in}\left(\omega\right),a_{3}^{in}\left(\omega\right)\right)^{T},$ (7) $D_{0}=\left(\begin{array}[]{ccc}A_{11}-B_{11}-\frac{\kappa_{1}}{2}-i\omega\tau&A_{12}-B_{12}&A_{13}-B_{13}\\\ A_{21}-B_{21}&A_{22}-B_{22}-\frac{\kappa_{2}}{2}-i\omega\tau&A_{23}-B_{23}\\\ A_{31}-B_{31}&A_{32}-B_{32}&A_{33}-B_{33}-\frac{\kappa_{3}}{2}-i\omega\tau\end{array}\right),$ (8) $B=\left(\begin{array}[]{lll}\sqrt{\kappa_{1}}&0&0\\\ 0&\sqrt{\kappa_{2}}&0\\\ 0&0&\sqrt{\kappa_{3}}\end{array}\right),\text{\qquad}I=\left(\begin{array}[]{lll}1&0&0\\\ 0&1&0\\\ 0&0&1\end{array}\right).$ (9) where T symbols the matrix transpose. Figure 2: The quantum correlations spectra $S_{123}^{out}\left(\omega^{\prime}\right)$, $S_{231}^{out}\left(\omega^{\prime}\right)$ and $S_{312}^{out}\left(\omega^{\prime}\right)$ versus the normalized analyzing frequency $\omega^{\prime}$ are plotted for (a) $\Delta=5$ and (b) $\Delta=10$ by solid, dashed and dotted line, respectively. The other parameters are $\Omega=35$, $g^{2}N=10$, $\gamma=1$ and $\kappa=0.1$. Figure 3: The quantum correlations spectra $S_{123}^{out}\left(\omega^{\prime}\right)$, $S_{231}^{out}\left(\omega^{\prime}\right)$ and $S_{312}^{out}\left(\omega^{\prime}\right)$ versus the detuning $\Delta$ for (a) $\omega^{\prime}=0$ and (b) $\omega^{\prime}=1.0$ by solid, dashed and dotted line, respectively. The other parameters are the same as those in Fig. 2. In order to study the entanglement properties of output cavity modes, we need to use quadrature amplitude and phase operators defined by $\displaystyle X_{j}^{out}$ $\displaystyle=$ $\displaystyle a_{j}^{out}\left(\omega\right)+a_{j}^{\dagger out}\left(-\omega\right),$ $\displaystyle Y_{j}^{out}$ $\displaystyle=$ $\displaystyle-i\left[a_{j}^{out}\left(\omega\right)-a_{j}^{\dagger out}\left(-\omega\right)\right],$ (10) Using Eq. (6) and $X_{j}^{in}=a_{j}^{in}\left(\omega\right)+a_{j}^{\dagger in}\left(-\omega\right)$, $Y_{j}^{in}=-i\left[a_{j}^{in}\left(\omega\right)+a_{j}^{\dagger in}\left(-\omega\right)\right]$, we can obtain the relationships between the input fields and the output fields as (13) (16) (19) where we have defined the normalized analyzing frequency $\omega^{\prime}=\omega\tau/\kappa$. The explicit expressions for $D_{jk}$ ($j,k$=1-3) are presented in Appendix C. The presence of entanglement between the three cavity modes can be investigated using the sufficient criterion for continuous-variable tripartite system proposed by van Loock and Furusawa [27]. The sufficient inseparability criterion for continuous variable tripartite entanglement is that if any one of the following inequalities is satisfied, genuine tripartite entanglement is demonstrated. The inequalities are $\displaystyle S_{123}$ $\displaystyle=$ $\displaystyle V\left[X_{1}+\left(X_{2}+X_{3}\right)/\sqrt{2}\right]+V\left[Y_{1}-\left(Y_{2}+Y_{3}\right)/\sqrt{2}\right]<4,$ $\displaystyle S_{231}$ $\displaystyle=$ $\displaystyle V\left[X_{2}+\left(X_{3}+X_{1}\right)/\sqrt{2}\right]+V\left[Y_{2}-\left(Y_{3}+Y_{1}\right)/\sqrt{2}\right]<4,$ (20) $\displaystyle S_{312}$ $\displaystyle=$ $\displaystyle V\left[X_{3}+\left(X_{1}+X_{2}\right)/\sqrt{2}\right]+V\left[Y_{3}-\left(Y_{1}+Y_{2}\right)/\sqrt{2}\right]<4,$ where $V\left(A\right)=<A^{2}>-<A>^{2}$. From the above definition, the correlation spectra of the quadratures of three output cavity fields are obtained as $\displaystyle S_{123}^{out}\left(\omega^{\prime}\right)$ $\displaystyle=$ $\displaystyle\|\sqrt{2}D_{11}-D_{21}-D_{31}\|^{2}+\|\sqrt{2}D_{12}-D_{22}-D_{32}\|^{2}+\|\sqrt{2}D_{13}-D_{23}-D_{33}\|^{2},$ $\displaystyle S_{231}^{out}\left(\omega^{\prime}\right)$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left(\|\sqrt{2}D_{21}-D_{11}-D_{31}\|^{2}+\|\sqrt{2}D_{22}-D_{12}-D_{32}\|^{2}+\|\sqrt{2}D_{23}-D_{13}-D_{33}\|^{2}\right)$ (21) $\displaystyle+\frac{1}{2}\left(\|\sqrt{2}D_{21}-D_{11}+D_{31}\|^{2}+\|\sqrt{2}D_{22}-D_{12}+D_{32}\|^{2}+\|\sqrt{2}D_{23}-D_{13}+D_{33}\|^{2}\right),$ $\displaystyle S_{312}^{out}\left(\omega^{\prime}\right)$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left(\|\sqrt{2}D_{31}-D_{11}-D_{21}\|^{2}+\|\sqrt{2}D_{32}-D_{12}-D_{22}\|^{2}+\|\sqrt{2}D_{33}-D_{13}-D_{23}\|^{2}\right)$ $\displaystyle+\frac{1}{2}\left(\|\sqrt{2}D_{31}+D_{11}-D_{21}\|^{2}+\|\sqrt{2}D_{32}-D_{12}+D_{22}\|^{2}+\|\sqrt{2}D_{33}-D_{13}+D_{23}\|^{2}\right).$ The quantum correlations spectra $S_{123}^{out}\left(\omega^{\prime}\right)$, $S_{231}^{out}\left(\omega^{\prime}\right)$ and $S_{312}^{out}\left(\omega^{\prime}\right)$ for three output cavity fields described in Eq. (13) versus the normalized analyzing frequency $\omega^{\prime}$ are plotted in Fig. $2$ for (a) $\Delta=5$ and (b) $\Delta=10$ by solid, dashed and dotted line, respectively. The other parameters are $\Omega=35$, $g^{2}N=10$, $\gamma=1$ and $\kappa=0.1$. The satisfaction of one of the three inequalities $S_{123}^{out}\left(\omega^{\prime}\right)<4$, $S_{231}^{out}\left(\omega^{\prime}\right)<4$ and $S_{312}^{out}\left(\omega^{\prime}\right)<4$ is sufficient to demonstrate genuine tripartite entanglement. In order to analyze the entanglement properties of the three cavity modes, we present all three correlations $S_{ijk}^{out}\left(\omega^{\prime}\right)$ and find that the indices of the three cavity modes are crucial. When the cavity modes are symmetric, the indices of the cavity modes are not important as the three correlations give the same result. But when the cavity modes are asymmetric, the indices are crucial in that the three correlations will give different results. As shown in Fig. 2(a), all three correlations are below 4 in a wide frequency range thus all three inequalities are satisfied. So the three output cavity modes are entangled. Among the three correlations, $S_{123}^{out}\left(\omega^{\prime}\right)$ gives the minimum values with the same parameters. When the inequalities are satisfied, the smaller the values of correlations are the larger the correlation degree. When we increase the detuning $\Delta$ to $10$ and keep other parameters unchanged as shown in Fig. 2(b), correlations $S_{123}^{out}\left(\omega^{\prime}\right)$ and $S_{231}^{out}\left(\omega^{\prime}\right)$ are always below $4$ in a wide frequency range while the correlation $S_{312}^{out}\left(\omega^{\prime}\right)$ is larger than $4$ in a frequency zone around the central analyzing frequency $\omega^{\prime}=0$. Thus tripartite entanglement is also demonstrated between the three output cavity modes. Compared with Fig. 2(a), the minimum value of $S_{123}^{out}\left(\omega^{\prime}\right)$ is smaller, which means that the correlation degree is also increased with the detuning. For both cases, we also see that the large correlation can be obtained at low analyzing frequency $\omega^{\prime}$. In Fig. $3$, we plot $S_{123}^{out}\left(\omega^{\prime}\right)$, $S_{231}^{out}\left(\omega^{\prime}\right)$ and $S_{312}^{out}\left(\omega^{\prime}\right)$ as a function of detuning $\Delta$ for (a) $\omega^{\prime}=0$ and (b) $\omega^{\prime}=1.0$ by solid, dashed and dotted line, respectively. The remain parameters are the same as those in Fig. 2. We also see that the correlation $S_{123}^{out}\left(\omega^{\prime}\right)$ gives the minimum values with the same parameters. It is seen from Fig. 3(a) and 3(b) that, correlations $S_{123}^{out}\left(\omega^{\prime}\right)$ and $S_{231}^{out}\left(\omega^{\prime}\right)$ always satisfy the inequalities while $S_{312}^{out}\left(\omega^{\prime}\right)$ only satisfy the inequality in a small frequency range. Thus tripartite entanglement between the three output cavity modes is demonstrated again. It is worthwhile to point out that when the analyzing frequency $\omega^{\prime}=0$, the system reaches its steady state. Thus at steady state, we can also obtain entangled tripartite light. This is in contrast with the results in Ref. [20], where the entanglement between the three cavity modes is time dependent. In that case all the mean photon numbers are amplified as time increases. Thus, the entanglement for three cavity modes can not be kept for a long time. And among the three correlations, $S_{123}^{out}\left(\omega^{\prime}\right)$ decreases with the increasing detuning, while correlations $S_{231}^{out}\left(\omega^{\prime}\right)$ and $S_{312}^{out}\left(\omega^{\prime}\right)$ first decrease than increase with increasing detuning. So correlation $S_{123}^{out}\left(\omega^{\prime}\right)$ is the best choice when we investigate the entanglement properties of the three cavity modes. Compared with Fig. 3(a) and 3(b), we find that the minimal values of correlations in Fig. 3(a) are smaller than those in Fig. 3(b). This indicates that the correlation degree is large when the analyzing frequency $\omega^{\prime}$ is small. ## IV Conclusion In conclusion, we have examined the entanglement properties of three cavity modes interacting with three-level $\Lambda$ atomic system coupled by two extra classical fields. As the classical fields are stronger than the cavity fields, we adopt the dressed-atom approach to calculate the equation for the cavity fields. After tracing out the atomic variables, we obtain the master equation of the cavity modes and analyze the entanglement properties of the output fields. The tripartite entanglement of the three output fields is demonstrated theoretically by a sufficient inseparability criterion and the entanglement characteristics are presented. This scheme of three-mode continuous variable entanglement generation using atomic coherence is useful in quantum information processing. Acknowledgments This work is supported by the Scientific Research Plan of the Provincial Education Department in Hubei (Grant No. Q20101304) and NSFC under Grant No. 11147153. ## References * (1) S.E. Harris, Phys. Today 50 (1997) 36. * (2) M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, Cambridge, England, 1997. * (3) J. Bergou, M. Orszag, M.O. Scully, Phys. 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Appendix A In this Appendix, we present the atomic decay term in terms of the dressed atomic states as $\displaystyle{\cal L}_{a}\rho$ $\displaystyle=$ $\displaystyle\sum_{j,k=0,+,-;j\neq k}\left({\cal L}_{jk}^{kj}\rho+{\cal L}_{ph}^{kj}\rho\right)+\sum_{j,k=+,-;j\neq k}{\cal L}_{in}^{kj}\rho,$ $\displaystyle{\cal L}_{jk}^{kj}\rho$ $\displaystyle=$ $\displaystyle\frac{{}_{\gamma_{jk}}}{2}\left(2\sigma_{p}^{kj}\rho\sigma_{p}^{kj}-\sigma_{p}^{kj}\sigma_{p}^{kj}\rho-\rho\sigma_{p}^{kj}\sigma_{p}^{kj}\right),$ $\displaystyle{\cal L}_{ph}^{kj}\rho$ $\displaystyle=$ $\displaystyle\epsilon_{kj}\frac{{}_{\gamma_{ph}^{kj}}}{4}\left(2\sigma_{jk}\rho\sigma_{kj}-\sigma_{kj}\sigma_{jk}\rho-\rho\sigma_{kj}\sigma_{jk}\right),$ () $\displaystyle\sigma_{p}^{kj}$ $\displaystyle=$ $\displaystyle\sigma_{kk}-\sigma_{jj},$ $\displaystyle{\cal L}_{in}^{kj}\rho$ $\displaystyle=$ $\displaystyle\gamma_{c}\left(\sigma_{j0}\rho\sigma_{k0}+\sigma_{0j}\rho\sigma_{0k}\right),$ with $\epsilon_{kj}=1$, for $k,j=0+,0-,+-$, otherwise $\epsilon_{kj}=0$. The parameters in the above equations are $\displaystyle\gamma_{+-}$ $\displaystyle=$ $\displaystyle\gamma_{-+}=\frac{\gamma}{4}c^{2}\left(1+s^{2}\right),$ $\displaystyle\gamma_{+0}$ $\displaystyle=$ $\displaystyle\gamma_{-0}=\frac{\gamma}{2}s^{2}\left(1+s^{2}\right),$ $\displaystyle\gamma_{0+}$ $\displaystyle=$ $\displaystyle\gamma_{0-}=\frac{\gamma}{2}c^{4},\gamma_{c}=\frac{\gamma}{2}c^{2}s^{2},$ () $\displaystyle\gamma_{ph}^{0+}$ $\displaystyle=$ $\displaystyle{\cal\gamma}_{ph}^{0-}=\gamma c^{2}s^{2},\gamma_{ph}^{+-}=\frac{\gamma}{2}c^{4}.$ Appendix B In this appendix, we present the explicit expressions for the coefficients $A_{jk}$ and $B_{jk}$ ($j,k=$1-3) in the equation of motion for the density operator $\rho_{c\text{ }}$of the cavity modes (Eq. ($4$) ): $\displaystyle A_{11}$ $\displaystyle=$ $\displaystyle g^{2}N\left(c_{1}e_{1}\rho_{00}^{s}+c_{3}e_{2}\rho_{++}^{s}\right)\text{, }B_{11}=g^{2}N\left(c_{3}e_{2}\rho_{00}^{s}+c_{1}e_{1}\rho_{--}^{s}\right),$ $\displaystyle A_{22}$ $\displaystyle=$ $\displaystyle g^{2}N\left(c_{2}e_{3}\rho_{00}^{s}+c_{3}e_{4}\rho_{--}^{s}\right)\text{, }B_{22}=g^{2}N\left(c_{2}e_{3}\rho_{++}^{s}+c_{3}e_{4}\rho_{00}^{s}\right),$ $\displaystyle A_{33}$ $\displaystyle=$ $\displaystyle g^{2}N\left(c_{1}e_{1}\rho_{00}^{s}+c_{3}e_{2}\rho_{--}^{s}\right)\text{, }B_{33}=g^{2}N\left(c_{1}e_{1}\tilde{\rho}_{++}+c_{3}e_{2}\rho_{00}^{s}\right),$ $\displaystyle A_{12}$ $\displaystyle=$ $\displaystyle g^{2}N\left(c_{2}e_{2}\rho_{++}^{s}-c_{3}e_{1}\rho_{00}^{s}\right)\text{, }B_{12}=g^{2}N\left(c_{2}e_{2}\rho_{00}^{s}-c_{3}e_{1}\rho_{--}^{s}\right),$ $\displaystyle A_{13}$ $\displaystyle=$ $\displaystyle g^{2}N\left(c_{1}e_{2}\rho_{++}^{s}+c_{3}e_{1}\rho_{00}^{s}\right)\text{, }B_{13}=g^{2}N\left(c_{1}e_{2}\rho_{00}^{s}+c_{3}e_{1}\rho_{--}^{s}\right),$ () $\displaystyle A_{21}$ $\displaystyle=$ $\displaystyle g^{2}N\left(c_{3}e_{3}\rho_{00}^{s}-c_{1}e_{4}\rho_{--}^{s}\right)\text{, }B_{21}=g^{2}N\left(c_{3}e_{3}\rho_{++}^{s}-c_{1}e_{42}\rho_{00}^{s}\right),$ $\displaystyle A_{23}$ $\displaystyle=$ $\displaystyle g^{2}N\left(c_{1}e_{3}\rho_{00}^{s}-c_{3}e_{4}\rho_{--}^{s}\right)\text{, }B_{23}=g^{2}N\left(c_{1}e_{3}\rho_{++}^{s}-c_{3}e_{4}\rho_{00}^{s}\right),$ $\displaystyle A_{31}$ $\displaystyle=$ $\displaystyle g^{2}N\left(c_{3}e_{1}\rho_{00}^{s}+c_{1}e_{2}\rho_{--}^{s}\right)\text{, }B_{31}=g^{2}N\left(c_{3}e_{1}\rho_{++}^{s}+c_{1}e_{2}\rho_{00}^{s}\right),$ $\displaystyle A_{32}$ $\displaystyle=$ $\displaystyle g^{2}N\left(c_{2}e_{1}\rho_{00}^{s}-c_{3}e_{2}\rho_{--}^{s}\right)\text{, }B_{32}=g^{2}N\left(c_{2}e_{1}\rho_{++}^{s}-c_{3}e_{2}\rho_{00}^{s}\right).$ where $e_{1}=\Gamma c_{1}-\gamma_{c}c_{3}$, $e_{2}=\Gamma c_{3}-\gamma_{c}c_{1}$, $e_{3}=$ $\Gamma c_{2}+\gamma_{c}c_{3}$ and $e_{4}=\Gamma c_{3}+\gamma_{c}c_{2}$ with $\Gamma=\gamma_{ph}^{0+}+\frac{1}{2}\left(\gamma_{+-}+\gamma_{+0}+\gamma_{-0}+\gamma_{0+}\right)+\frac{1}{4}\left(\gamma_{ph}^{0-}+\gamma_{ph}^{+-}\right)$. Appendix C In this appendix, we will give the explicit expressions for the coefficients $D_{jk}$ ($j,k=$1-3) in the relations between the input fields and the output fields in Eq. (11): $\displaystyle D_{11}$ $\displaystyle=$ $\displaystyle-1+\chi_{0}\left[\chi_{22}\chi_{33}-\left(A_{13}-B_{13}\right)\left(A_{32}-B_{32}\right)\right],$ $\displaystyle D_{22}$ $\displaystyle=$ $\displaystyle-1+\chi_{0}\left[\chi_{11}\chi_{33}-\left(A_{13}-B_{13}\right)\left(A_{31}-B_{31}\right)\right],$ $\displaystyle D_{33}$ $\displaystyle=$ $\displaystyle-1+\chi_{0}\left[\chi_{11}\chi_{22}-\left(A_{12}-B_{12}\right)\left(A_{21}-B_{21}\right)\right],$ $\displaystyle D_{12}$ $\displaystyle=$ $\displaystyle\chi_{0}\left[\left(A_{13}-B_{13}\right)\left(A_{32}-B_{32}\right)-\left(A_{12}-B_{12}\right)\chi_{33}\right],$ $\displaystyle D_{13}$ $\displaystyle=$ $\displaystyle\chi_{0}\left[\left(A_{12}-B_{12}\right)\left(A_{23}-B_{23}\right)-\left(A_{13}-B_{13}\right)\chi_{22}\right],$ () $\displaystyle D_{21}$ $\displaystyle=$ $\displaystyle\chi_{0}\left[\left(A_{23}-B_{23}\right)\left(A_{31}-B_{31}\right)-\left(A_{21}-B_{21}\right)\chi_{33}\right],$ $\displaystyle D_{23}$ $\displaystyle=$ $\displaystyle\chi_{0}\left[\left(A_{13}-B_{13}\right)\left(A_{21}-B_{21}\right)-\left(A_{23}-B_{23}\right)\chi_{11}\right],$ $\displaystyle D_{31}$ $\displaystyle=$ $\displaystyle\chi_{0}\left[\left(A_{32}-B_{32}\right)\left(A_{21}-B_{21}\right)-\left(A_{31}-B_{31}\right)\chi_{22}\right],$ $\displaystyle D_{32}$ $\displaystyle=$ $\displaystyle\chi_{0}\left[\left(A_{12}-B_{12}\right)\left(A_{31}-B_{31}\right)-\left(A_{32}-B_{32}\right)\chi_{11}\right],$ where $\displaystyle\|D_{0}\|$ $\displaystyle=$ $\displaystyle\chi_{11}\left[\chi_{22}\chi_{33}-\left(A_{23}-B_{23}\right)\left(A_{32}-B_{32}\right)\right]$ $\displaystyle+\left(A_{12}-B_{12}\right)\left[\left(A_{23}-B_{23}\right)\left(A_{31}-B_{31}\right)-\left(A_{21}-B_{21}\right)\chi_{33}\right]$ $\displaystyle+\left(A_{13}-B_{13}\right)\left[\left(A_{21}-B_{21}\right)\left(A_{32}-B_{32}\right)-\left(A_{31}-B_{31}\right)\chi_{22}\right].$ with the parameters $\chi_{jj}=\kappa\left(\frac{A_{jj}}{\kappa}-\frac{B_{jj}}{\kappa}-\frac{1}{2}-i\omega^{\prime}\right)$ ($j=$1-3) and $\chi_{0}=-\frac{\kappa}{\|D_{0}\|}$. And we have used the equation $\kappa_{1}=\kappa_{2}=\kappa_{3}=\kappa$.	arxiv-papers	2013-11-09T14:31:10	2024-09-04T02:49:53.431131	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Jinhua Zou", "submitter": "Jin-hua Zou", "url": "https://arxiv.org/abs/1311.2172" }
1311.2210	# On Interval Non-Edge-Colorable Eulerian Multigraphs Petros A. Petrosyan Department of Informatics and Applied Mathematics, Yerevan State University, 0025, Armenia Institute for Informatics and Automation Problems, National Academy of Sciences, 0014, Armenia E-mail: [email protected] ###### Abstract An edge-coloring of a multigraph $G$ with colors $1,\ldots,t$ is called an interval $t$-coloring if all colors are used, and the colors of edges incident to any vertex of $G$ are distinct and form an interval of integers. In this note, we show that all Eulerian multigraphs with an odd number of edges have no interval coloring. We also give some methods for constructing of interval non-edge-colorable Eulerian multigraphs. ## 1 Introduction In this note we consider graphs which are finite, undirected, and have no loops or multiple edges and multigraphs which may contain multiple edges but no loops. Let $V(G)$ and $E(G)$ denote the sets of vertices and edges of a multigraph $G$, respectively. The degree of a vertex $v\in V(G)$ is denoted by $d_{G}(v)$, the maximum degree of $G$ by $\Delta(G)$, and the chromatic index of $G$ by $\chi^{\prime}\left(G\right)$. A multigraph $G$ is Eulerian if it has a closed trail containing every edge of $G$. The terms and concepts that we do not define can be found in [7]. A proper edge-coloring of a multigraph $G$ is a coloring of the edges of $G$ such that no two adjacent edges receive the same color. If $\alpha$ is a proper edge-coloring of $G$ and $v\in V(G)$, then $S\left(v,\alpha\right)$ denotes the set of colors of edges incident to $v$. A proper edge-coloring of a multigraph $G$ with colors $1,\ldots,t$ is called an interval $t$-coloring if all colors are used, and for any vertex $v$ of $G$, the set $S\left(v,\alpha\right)$ is an interval of integers. A multigraph $G$ is interval colorable if it has an interval $t$-coloring for some positive integer $t$. The set of all interval colorable multigraphs is denoted by $\mathfrak{N}$. The concept of interval edge-coloring of multigraphs was introduced by Asratian and Kamalian [1]. In [1, 2], they proved the following result. Theorem 1. If $G$ is a multigraph and $G\in\mathfrak{N}$, then $\chi^{\prime}\left(G\right)=\Delta(G)$. Moreover, if $G$ is a regular multigraph, then $G\in\mathfrak{N}$ if and only if $\chi^{\prime}\left(G\right)=\Delta(G)$. Some results on interval edge-colorings of multigraphs were obtained in [5]. In [6], the authors described some methods for constructing of interval non- edge-colorable bipartite graphs and multigraphs. In this note we show that all Eulerian multigraphs with an odd number of edges have no interval coloring. We also give some methods for constructing of interval non-edge-colorable Eulerian multigraphs. ## 2 Results Let $G$ be a multigraph. For any $e\in E(G)$, by $G_{e}$ we denote the multigraph obtained from $G$ by subdividing the edge $e$. For a multigraph $G$, we define a multigraph $G^{\star}$ as follows: $V(G^{\star})=V(G)\cup\\{u\\}$, $u\notin V(G)$, $V(G^{\star})=E(G)\cup\\{uv:v\in V(G)~{}and~{}d_{G}(v)~{}is~{}odd\\}$. For a graph $G$, by $L(G)$ we denote the line graph of the graph $G$. We also need a classical result on Eulerian multigraphs. Euler’s Theorem. ([3]) A connected multigraph $G$ is Eulerian if and only if every vertex of $G$ has an even degree. Now we can prove our result. Theorem 2. If $G$ is an Eulerian multigraph and $\|E(G)\|$ is odd, then $G\notin\mathfrak{N}$. Proof Suppose, to the contrary, that $G$ has an interval $t$-coloring $\alpha$ for some $t$. Since $G$ is an Eulerian multigraph, $G$ is connected and $d_{G}(v)$ is even for any $v\in V(G)$, by Euler’s Theorem. Since $\alpha$ is an interval coloring and all degrees of vertices of $G$ are even, we have that for any $v\in V(G)$, the set $S\left(v,\alpha\right)$ contains exactly $\frac{d_{G}(v)}{2}$ even colors and $\frac{d_{G}(v)}{2}$ odd colors. Now let $m_{odd}$ be the number of odd colors in the coloring $\alpha$. By Handshaking lemma, we obtain $m_{odd}=\frac{1}{2}\sum\limits_{v\in V(G)}\frac{d_{G}(v)}{2}=\frac{\|E(G)\|}{2}$. Thus $\|E(G)\|$ is even, which is a contradiction. $\square$ Corollary 1. If $G$ is an Eulerian multigraph and $G\in\mathfrak{N}$, then $\|E(G)\|$ is even. Let us note that there are Eulerian graphs with an even number of edges that have no interval coloring. For example, the complete graph $K_{5}$ has no interval coloring. On the other hand, there are many Eulerian graphs with an even number of edges that have an interval coloring. In [4], Jaeger proved the following result. Theorem 3. If $G$ is a connected $r$-regular graph ($r\geq 2$), $\chi^{\prime}\left(G\right)=r$ and $\|E(G)\|$ is even, then $\chi^{\prime}\left(L(G)\right)=2r-2$. Since $G$ is a connected $r$-regular graph ($r\geq 2$) and $\|E(G)\|$ is even, we have that $L(G)$ is a connected $(2r-2)$-regular graph with an even number of edges. Moreover, by Theorems 1 and 3 and Euler’s Theorem, we obtain the following Corollary 2. If $G$ is a connected $r$-regular ($r\geq 2$) graph with an even number of edges and $G\in\mathfrak{N}$, then $L(G)$ is an Eulerian graph with an even number of edges and $L(G)\in\mathfrak{N}$. Let us note that Theorem 2 also gives some methods for constructing of interval non-edge-colorable Eulerian multigraphs from interval colorable multigraphs. Corollary 3. If $G$ is an Eulerian multigraph and $G\in\mathfrak{N}$, then for each $e\in E(G)$, $G_{e}\notin\mathfrak{N}$. Corollary 4. If $G$ is a connected multigraph with an odd number of edges and $G\in\mathfrak{N}$, then $G^{\star}\notin\mathfrak{N}$. ## References * [1] A.S. Asratian, R.R. Kamalian, Interval colorings of edges of a multigraph, Appl. Math. 5 (1987) 25-34 (in Russian). * [2] A.S. Asratian, R.R. Kamalian, Investigation on interval edge-colorings of graphs, J. Combin. Theory Ser. B 62 (1994) 34-43. * [3] L. Euler, Solutio problematis ad geometriam situs pertinentis, Commentarii Academiae Sci. I. Petropolitanae 8 (1736) 128-140. * [4] F. Jaeger, Sur l’indice chromatique du graphe representatif des aretes d’un graphe regulier, Discrete Math. 9 (1974) 161-172. * [5] R.R. Kamalian, Interval edge-colorings of graphs, Doctoral Thesis, Novosibirsk, 1990. * [6] P.A. Petrosyan, H.H. Khachatrian, Interval non-edge-colorable bipartite graphs and multigraphs, Journal of Graph Theory, 2013, http://onlinelibrary.wiley.com/doi/10.1002/jgt.21759/pdf * [7] D.B. West, Introduction to Graph Theory, Prentice-Hall, New Jersey, 1996.	arxiv-papers	2013-11-09T20:41:58	2024-09-04T02:49:53.439204	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Petros A. Petrosyan", "submitter": "Petros Petrosyan", "url": "https://arxiv.org/abs/1311.2210" }
1311.2213	# A first principles investigated optical spectra of oxizided graphene N. Singh1,2, T. P. Kaloni1, and Udo Schwingenschlögl1 [email protected] [email protected] 1 Physical Science & Engineering Division, KAUST, Thuwal 23955-6900, Kingdom of Saudi Arabia 2 Solar and Photovoltaic Energy Research Center (SPERC), KAUST, Thuwal 23955-6900, Kingdom of Saudi Arabia ###### Abstract The electronic and optical properties of mono, di, tri, and tetravacancies in graphene are studied in comparison to each other, using density functional theory. In addition, oxidized monovacancies are considered for different oxygen concentrations. Pristine graphene is found to be more absorptive than any defect configuration at low energy. We demonstrate characteristic differences in the optical spectra of the various defects for energies up to 3 eV. This makes it possible to quantify by optical spectroscopy the ratios of the defect species present in a sample. two-dimensional materials, graphene oxide, desity functional theory, and absorption spectrum While graphene is a zero band gap material, but a finite band gap is needed for various applications aiming at graphene based electronic devices schwierz ; kaloni3 . Functionalization is one of the possible methods to open a band gap for example by simple oxidation wu ; priya ; kaloni . Graphene oxide (GO) with epoxy, carbonyl, or hydroxyl groups could allow to tune the a band gap and therefor tailor the electronic, mechanical, and optical properties dai ; andre . The atomic structure of GO has been studied experimentally weiwei ; ajayan and theoretically sumit . Recently, GO nanostructures have created a lot of attention due to the fact that it paved the way for solution based synthesis of graphene sheets, low cost, easy processibility, and compatibility with various substrates Joung . The band gap of GO can be tunable by just varying the oxidation level. Fully oxidized GO can act as an electrical insulator and partially oxidized GO can act as a semiconductor Loh . Moreover, experiments demonstrat that GO nanostructures have promising applications in photocatalysis Krishnamoorthy . Reduction of GO may pave the way to mass production of graphene shenoy . While GO is usually insulating, a controlled deoxidation can lead to an electrically and optically active material that is transparent and conducting. Furthermore, in contrast to pristine graphene, GO is fluorescent over a broad range of wavelengths, owing to its heterogeneous electronic structure Loh . It can contain different chemical compositions of carbon, oxygen, and hydrogen weiwei ; lu . While commonly hydroxyl and epoxy groups are found, there can be small contributions of carbonyl and carboxyl groups. Experimentally, a coverage of between 25% and 75% has been observed, reflecting that typically a quarter of the C$-$C bonds are double bonds whereas the rest are single bonds Katsnelson . Adsorption behavior of oxygen atoms on the graphene sheets has been studied by using first-principles calculations ito and found that the lattice constant increases with the increase of the ratio of $O/C$ because of the formation of the epoxy group. At 50% $O/C$ ratio, a finite band gap of 3.39 eV is reported ito . The attachment of a carbonyl group leads to an almost planar $sp^{2}$ electronic configuration because the formation of C=O bonds induces little strain in the graphene sheet. On the contrary, the attachment of an epoxy group leads to a non-planar distorted $sp^{3}$ electronic configuration for those C atoms which are connected to O, which creates a significant strain on neighboring C$-$C bonds shenoy . The combination of $sp^{2}$ and $sp^{3}$ configurations as well as defects breaks the hexagonal symmetry of pristine graphene and a band gap is opened. The coexistence of $sp^{2}$ and $sp^{3}$ configurations is confirmed experimentally yun and theoretically shenoy . The defects associated with dangling bonds enhance the reactivity substantially. Figure 1: Calculated band structures of (a) a mono-vacancy (b) a di-vacancy, (c) tri-vacancy, and (d) tetra-vacancy in graphene along the path K’1 (0.6667 0.0000 0.0000), K2 (0.3333 -0.5774 0.0000), $\Gamma$ (0.0000 0.0000 0.0000), K’1 (0.6667 0.0000 0.0000), K1 (0.3333 0.5773 0.0000), M1 (0.500 0.2887 0.0000), and $\Gamma$ (0.0000 0.0000 0.0000) in unit of $\frac{2\pi}{a}$. Recently, a theoretical investigation of the electronic and optical properties of GO (without vacancies) for different functional groups and various compositions has been reported in Ref priya . The authors found that carbonyl groups are favourable for photoluminescense and that the optical gap of reduced GO is samaller than the optical gap of pristine and fully oxidize graphene. Theoretical and experimental studies Katsnelson1 ; Wang indicate that hydroxyl, carboxyl, and other functional groups can easily be attached to vacancies in graphene than to pristine graphene. Therefore, a study of the electronic and optical properties of defective GO becomes critical. Optical properties of oxidized mono-, di-, tri-, and tetra-vacancies in graphene have not been reported so far. In this work, we use first principles calculations of to provide insight into this topic. Our calculations are based on density functional theory and carried out using the generalized gradient approximation of Quantum Espresso pacakage, paolo ; pbe . All calculations are performed with a plane wave cutoff energy of 544 eV. We use a Monkhorst-Pack Monk of $8\times 8\times 1$ k-mesh for the Brillouin zone integration in order to relaxing the structures and achieving highly accurate electronic structure. We observe that a $5\times 5$ supercell of graphene is sufficiently large for monovacancies and oxidized monovacancies kaloni . Our supercell has a lattice constant of $a=12.2$ Å and extends $c=20$ Å in the perpendicular direction. It has been reported that nearby vacancies behave independently when they are separated by $\sim$7 Å andrey . Hence, we use a $6\times 6$ supercell for di-, tri-, and tetra-vacancies to avoid artificial interaction of the periodic images. This supercell has a lattice constant of $a=14.69$ Å and again $c=20$ Å. The cell parameters and atomic positions are fully relaxed until a force convergence of 0.05 eV/Å and an energy convergence of 10-7 eV are reached. The relaxed structures are used to calculate the optical properties by WIEN2k Wien2k code. For the optical calculations, a dense mesh of uniformely distributed k-points is required. Hence, the Brillouin zone integration is performed using tetrahedron method with 180 k-points in the irreducible part of the Brillouin zone. Well converged solutions are obtained for $R_{mt}\times K_{max}$ =7, where $K_{max}$ is the plane-wave cut-off and $R_{mt}$ is the smallest of all atomic sphere radii. The dielectric function ($\varepsilon(\omega)=\varepsilon_{1}(\omega)+i\varepsilon_{2}(\omega)$) is known to describe the optical response of the medium. The interband contribution to the imaginary part of the dielectric function $\varepsilon(\omega)$ is calculated by summing all transitions from occupied to unoccupied states (with fixed k) over the Brillouin zone, weighted with the appropriate matrix elements giving the probability for the transitions. The imaginary part of dielectric function $\varepsilon_{2}(\omega)$ is given as in wooten by $\epsilon_{2}(\omega)=\frac{4\pi^{2}e^{2}}{m^{2}\omega^{2}}\sum_{i,j}\int<i\|M\|j>^{2}\times f_{i}(1-f_{i})\delta(E_{f}-E_{i}-\omega)d^{3}k$ (1) Where M is dipole matrix, _i_ and j are the initial and final states, respectively, $\emph{f}_{i}$ is the Fermi distribution function for the $i_{t}h$ states, and $E_{i}$ is the energy of electron in the $i_{t}h$ state. The real part of the dielectric function can be extrated from the imaginary part using the Kramers-Kronig relation wooten ; Yu in the from $\epsilon_{1}(\omega)=1+\frac{2}{\pi}P\int_{0}^{\infty}\frac{\omega^{{}^{\prime}}\epsilon_{2}(\omega^{{}^{\prime}})}{\omega^{{}^{\prime}2}-\omega^{2}}d\omega^{{}^{\prime}}$ (2) Where P implies the principle value of the integral. The reflectivity spectra are derived from Fresnel’s formula for normal incidence assuming an orientation of the crystal surface parallel to the optical axis using the relation fox $R(\omega)=\|\frac{\sqrt{\varepsilon(\omega)}-1}{\sqrt{\varepsilon(\omega)}+1}\|^{2}$ (3) The knowledge of both real and imaginary parts of the dielectric tensor allows the calculations of the important optical functions. We calculate the absorption, the real part of optical conductivity, and the electron energy- loss spectrum using the following expressions fox ; delin $\alpha(\omega)=\sqrt{2}\omega(\sqrt{\varepsilon_{1}(\omega)^{2}+\varepsilon_{2}(\omega)^{2}}-\varepsilon_{1}(\omega))^{1/2}$ (4) $Re\sigma(\omega)=\frac{\omega\varepsilon_{2}}{4\pi}$ (5) $L(\omega)=\frac{\varepsilon_{2}(\omega)}{\varepsilon(\omega)^{2}+\varepsilon_{2}(\omega)^{2}}$ (6) This approach has been successfully applied to narrow band gap materials including rare earth Zintl compounds Zintl ; Zintl1 , and oxides singh . For a reliable integration, a set of 180 k-points in the irreducible wedge of the Brillouin zone is used. A Lorentzian broadening is used to simulate the effects of finite life-time and finite resolution of the optical measurement. Figure 2: Calculated optical absorption $\alpha(\omega)$ in $10^{4}$ cm-1, optical conductivity $\sigma(\omega)$, reflectivity $R(\omega)$ in %, and energy loss function $L(\omega)$ of pristine graphene, as compared to graphene with mono, di-, tri, and tetra-vacancies and oxidized monovacancy. The calculated values of formation energy of mono-, di-, tri-, and tetra- vacancies in graphene are 7.50, 6.94 eV, 11.45 eV, and 12.58 eV, respectively. This means the formation of a divacancy in graphene is more favourable than the formation of a single vacancy. Furthermore, a divacancy is known to be more stable than two isolated monovacancies (whose migration energy barrier is rather low), because the dangling C-C bonds of atoms next to the vacancy can be passivated by eachother Lee . The vacancies in graphene induce ferromagnetism with total magnetic moments of 1.35 $\mu_{B}$, 1.00 $\mu_{B}$, 2.00 $\mu_{B}$ for mono-, tri-, tetra-vacancies, respectively. Di-vacancies shows no spin-polarization. The results of our band structure (BS) calculations for mono-, di-, tri-, and tetra-vacancies are shown in Fig. 1 together with the corresponding DOS. For the sake of comparison we have included the BS (dotted lines) of pristine graphene. For an oxidized monovacancy, the magnetic and electronic properties have been reported in previous literature kaloni ; dai ; Yazyev . In $6\times 6$ supercell, the Dirac cone is shifted to the $\Gamma$-point due to Brillouin zone folding in $6\times 6$ supercell. In case of the tetra-vacancy, the BS shows that a single minority spin band crosses the Fermi energy ($E_{F}$) at the $\Gamma$-point and leaves the system metallic, whereas for the di-, tri- vacancies both majority and minority bands cross $E_{F}$. Moreover, an upward shift of the Dirac point is indicative of a hole-doped system. In case of the di-vacancy, the DOS is identical for the majority and minority spins, reflecting spin-degeneracy. It means pristine graphene becomes ferromagnetic by a single vacancy defect and can be non-magnetic metal by divacancy. The optical spectroscopy is a valuable tool in material science. Here, In the optical calculations, selfenergy and excitonic effects are not taken into account. It has been shown for graphene that in the energy range upto 3 eV, where the approximation of Dirac particles is valid, the influence of the many-particle effects is negligible. The calculated optical spectra of pristine graphene and its functionalized derivatives are addressed in Fig 1. The optical spectra show that for the two adsorbed O atoms, a band gap of 0.5 eV is opened due to the symmtery breaking and increased $sp^{3}$ characters. A semiconducting behaviour for this configuration is conformed by our previous calculations of BS and DOS kaloni . As comparised to pristine graphene, all defects are found to create metallic states as shown in Fig 1. In general, pristine graphene is more absorptive as compared to the other systems but for di-vacancy, absoption between 7.5 eV to 10 eV is higher than pristine graphene. The $\alpha(\omega)$ peak at 4.5 eV for pristine graphene splits into two peaks at 2.7 eV and 5 eV by the splitting of the Dirac cone for two attached O atoms. The splitting increasing with the O coverage. For di-vacancy and three adsorbed O atoms an additional sharp peak at 1.25 eV (visible region) is found which is absent for other systems. The pristine graphene have high reflectivity in the low energy as compared to other cases. The reflectivity is higher for the tetra-vacancy (most pronounced metallicity) as compare to mono-, di-, and tri-vacancies, but lower than for pristine graphene. The reflectivity in low energy range is the lowest for the case of monovacancy with two attached O atoms, due to its semiconducting nature and increases again for three and four attached O atoms. The most prominent peaks in $\sigma(\omega)$ become broaden and the magnitude also decreases as one moves from pristine graphene to tetra-vacancies. The $\sigma(\omega)$ is low for tri-, tetra-vacancies and monovacancy with four attached O atoms as compared to remaining systems. A large peak is observed at around 4.6 eV in all optical spectra for all the systems which is attributed to the $\pi-\pi^{}$ transitions of the aromatic C$-$C atoms. The maximum peak in energy loss spectra is at 4.9 eV, which is assigned to the energy of the volume plasmon $\hbar$ $\omega_{p}$. This maximum peak positions remain same for all systems. The peaks in optical spectra originates from the transition from valence band to conduction band. The dominating peak in the energy loss spectrum broadens from graphene to the oxidize vacancies with monovacancy. In conclusion, we have studied the optical properties of graphene derivatives (clean and oxidized vacancies) by means of density functional theory. We find that the formation of divacancies in graphene is energetically favourable. Divacancies are also exceptional in the sense that they do not lead to a local magnetic moment. Mono, di, tri, and tetravacancies are found to be metallic, while an oxidized monovacancy with two adsorbed oxygen atoms leads to a band gap of 0.5 eV (due to a splitting of the Dirac cone). Our optical spectra show that pristine graphene has the highest absorption in the energy range below 2.5 eV. In two cases (tetravacancy and monovacancy with three adsorbed oxygen atoms) a prominent absorption peak appears in the visible range. Our calculations suggest that the types of (oxidized) defects present in a graphene sample can be quantified by optical spectroscopy. With this knowledge, the electronic and optical properties of graphene derivatives can be tuned by controlled oxidation and reduction. ## References (1) F. Schwierz, Nat. Nanotechnol. 5, 487 (2010). * (2) T. P. Kaloni, Y. C. Cheng and U. 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Kaloni, and Udo Schwingenschl\\\"ogl", "submitter": "Thaneshwor Prashad Kaloni", "url": "https://arxiv.org/abs/1311.2213" }
1311.2555	# Hamiltonian gadgets with reduced resource requirements Yudong Cao [email protected] Department of Computer Science, Purdue University, 601 Purdue Mall, West Lafayette, IN 47907, USA Qatar Energy and Environment Research Institute (QEERI), Ar-Rayyān, P.O Box 5825, Doha, Qatar Ryan Babbush [email protected] Department of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138, USA Jacob Biamonte [email protected] ISI Foundation, Via Alassio 11/c, 10126, Torino, Italy Sabre Kais [email protected] Department of Computer Science, Purdue University, 601 Purdue Mall, West Lafayette, IN 47907, USA Qatar Energy and Environment Research Institute (QEERI), Ar-Rayyān, P.O Box 5825, Doha, Qatar Department of Chemistry, Physics and Birck Nanotechnology Center, Purdue University, 601 Purdue Mall, West Lafayette, IN 47907, USA Santa Fe Institute, 1399 Hyde Park Rd, Santa Fe, NM 87501, USA ###### Abstract Application of the adiabatic model of quantum computation requires efficient encoding of the solution to computational problems into the lowest eigenstate of a Hamiltonian that supports universal adiabatic quantum computation. Experimental systems are typically limited to restricted forms of 2-body interactions. Therefore, universal adiabatic quantum computation requires a method for approximating quantum many-body Hamiltonians up to arbitrary spectral error using at most 2-body interactions. Hamiltonian gadgets, introduced around a decade ago, offer the only current means to address this requirement. Although the applications of Hamiltonian gadgets have steadily grown since their introduction, little progress has been made in overcoming the limitations of the gadgets themselves. In this experimentally motivated theoretical study, we introduce several gadgets which require significantly more realistic control parameters than similar gadgets in the literature. We employ analytical techniques which result in a reduction of the resource scaling as a function of spectral error for the commonly used subdivision, 3- to 2-body and $k$-body gadgets. Accordingly, our improvements reduce the resource requirements of all proofs and experimental proposals making use of these common gadgets. Next, we numerically optimize these new gadgets to illustrate the tightness of our analytical bounds. Finally, we introduce a new gadget that simulates a $YY$ interaction term using Hamiltonians containing only $\\{X,Z,XX,ZZ\\}$ terms. Apart from possible implications in a theoretical context, this work could also be useful for a first experimental implementation of these key building blocks by requiring less control precision without introducing extra ancillary qubits. Although adiabatic quantum computation is known to be a universal model of quantum computation 2004quant.ph..5098A ; 2007PhRvL..99g0502M ; OT06 ; BL07 ; CL08 and hence, in principle equivalent to the circuit model, the mappings between an adiabatic process and an arbitrary quantum circuit require significant overhead. Currently the approaches to universal adiabatic quantum computation require implementing multiple higher order and non-commuting interactions by means of perturbative gadgets BL07 . Such gadgets arose in early work on quantum complexity theory and the resources required for their implementation are the subject of this study. Early work by Kitaev _et al_. KSV02 established that an otherwise arbitrary Hamiltonian restricted to have at most 5-body interactions has a ground state energy problem which is complete for the quantum analog of the complexity class NP (QMA-complete). Reducing the locality of the Hamiltonians from 5-body down to 2-body remained an open problem for a number of years. In their 2004 proof that 2-local Hamiltonian is QMA-Complete, Kempe, Kitaev and Regev formalized the idea of a perturbative gadget, which finally accomplished this task KKR06 . Oliveira and Terhal further reduced the problem, showing completeness when otherwise arbitrary 2-body Hamiltonians were restricted to act on a square lattice OT06 . The form of the simplest QMA-complete Hamiltonian is reduced to physically relevant models in BL07 (see also CM13 ), e.g. $H=\sum_{i}h_{i}Z_{i}+\sum_{i<j}J_{ij}Z_{i}Z_{j}+\sum_{i<j}K_{ij}X_{i}X_{j}.$ (1) Although this model contains only physically accessible terms, programming problems into a universal adiabatic quantum computer BL07 or an adiabatic quantum simulator sim11 ; 2014arXiv1401.3186V involves several types of $k$-body interactions (for bounded $k$). To reduce from $k$-body interactions to 2-body is accomplished through the application of gadgets. Hamiltonian gadgets were introduced as theorem-proving tools in the context of quantum complexity theory yet their experimental realization currently offers the only path towards universal adiabatic quantum computation. In terms of experimental constraints, an important parameter in the construction of these gadgets is a large spectral gap introduced into the ancilla space as part of a penalty Hamiltonian. This large spectral gap often requires control precision well beyond current experimental capabilities and must be improved for practical physical realizations. A perturbative gadget consists of an ancilla system acted on by Hamiltonian $H$, characterized by the spectral gap $\Delta$ between its ground state subspace and excited state subspace, and a perturbation $V$ which acts on both the ancilla and the system. $V$ perturbs the ground state subspace of $H$ such that the perturbed low-lying spectrum of the gadget Hamiltonian $\widetilde{H}=H+V$ captures the spectrum of the target Hamiltonian, $H_{\text{targ}}$, up to error $\epsilon$. The purpose of a gadget is dependent on the form of the target Hamiltonian $H_{\text{targ}}$. For example, if the target Hamiltonian is $k$-local with $k\geq 3$ while the gadget Hamiltonian is 2-local, the gadget serves as a tool for reducing locality. Also if the target Hamiltonian involves interactions that are hard to implement experimentally and the gadget Hamiltonian contains only interactions that are physically accessible, the gadget becomes a generator of physically inaccesible terms from accessible ones. For example the gadget which we introduce in Sec. VI might fall into this category. Apart from the physical relevance to quantum computation, gadgets have been central to many results in quantum complexity theory BDLT08 ; BL07 ; BDOT06 ; CM13 . Hamiltonian gadgets were also used to characterize the complexity of density functional theory Schuch09 and are required components in current proposals related to error correction on an adiabatic quantum computer Ganti2013 and the adiabatic and ground state quantum simulator sim11 ; 2014arXiv1401.3186V . Since these works employ known gadgets which we provide improved constructions of here, our results hence imply a reduction of the resources required in these past works. The first use of perturbative gadgets KKR06 relied on a 2-body gadget Hamiltonian to simulate a 3-body Hamiltonian of the form $H_{\text{targ}}=H_{\text{else}}+\alpha\cdot A\otimes B\otimes C$ with three auxiliary spins in the ancilla space. Here $H_{\text{else}}$ is an arbitrary Hamiltonian that does not operate on the auxiliary spins. Further, $A$, $B$ and $C$ are unit-norm operators and $\alpha$ is the desired coupling. For such a system, it is shown that it suffices to construct $V$ with $\\|V\\|<\Delta/2$ to guarantee that the perturbative self-energy expansion approximates $H_{\text{targ}}$ up to error $\epsilon$ OT06 ; KKR06 ; BDLT08 . Because the gadget Hamiltonian is constructed such that in the perturbative expansion (with respect to the low energy subspace), only virtual excitations that flip all 3 ancilla bits would have non-trivial contributions in the $1^{\text{st}}$ through $3^{\text{rd}}$ order terms. In JF08 Jordan and Farhi generalized the construction in KKR06 to a general $k$-body to 2-body reduction using a perturbative expansion due to Bloch bloch58 . They showed that one can approximate the low-energy subspace of a Hamiltonian containing $r$ distinct $k$-local terms using a 2-local Hamiltonian. Two important gadgets were introduced by Oliveira and Terhal OT06 in their proof that 2-local Hamiltonian on square lattice is QMA-Complete. In particular, they introduced an alternative 3- to 2-body gadget which uses only one additional spin for each 3-body term as well as a “subdivision gadget” that reduces a $k$-body term to a $(\lceil k/2\rceil+1)$-body term using only one additional spin OT06 . These gadgets, which we improve in this work, find their use as the de facto standard whenever the use of gadgets is necessitated. For instance, the gadgets from OT06 were used by Bravyi, DiVincenzo, Loss and Terhal BDLT08 to show that one can combine the use of subdivision and 3- to 2-body gadgets to recursively reduce a $k$-body Hamiltonian to $2$-body, which is useful for simulating quantum many-body Hamiltonians. We note that these gadgets solve a different problem than the type of many-body operator simulations considered previously cory99 ; cory00 for gate model quantum computation, where the techniques developed therein are not directly applicable to our situation. While recent progress in the experimental implementation of adiabatic quantum processors 2006cond.mat..8253H ; Boixo2012 ; BCM+13 ; Lidar2014 suggests the ability to perform sophisticated adiabatic quantum computing experiments, the perturbative gadgets require very large values of $\Delta$. This places high demands on experimental control precision by requiring that devices enforce very large couplings between ancilla qubits while still being able to resolve couplings from the original problem — even though those fields may be orders of magnitude smaller than $\Delta$. Accordingly, if perturbative gadgets are to be used, it is necessary to find gadgets which can efficiently approximate their target Hamiltonians with significantly lower values of $\Delta$. Results summary and manuscript structure. Previous works in the literature KKR06 ; OT06 ; BDOT06 ; BL07 ; BDLT08 choose $\Delta$ to be a polynomial function of $\epsilon^{-1}$ which is sufficient for yielding a spectral error $O(\epsilon)$ between the gadget and the target Hamiltonian. Experimental realizations however, will require a recipe for assigning the minimum $\Delta$ that guarantees error within specified $\epsilon$, which we consider here. This recipe will need to depend on three parameters: (i) the desired coupling, $\alpha$; (ii) the magnitude of the non-problematic part of the Hamiltonian, $\\|H_{\text{else}}\\|$; and (iii) the specified error tolerance, $\epsilon$. For simulating a target Hamiltonian up to error $\epsilon$, previous constructions OT06 ; BDOT06 ; BDLT08 use $\Delta=\Theta(\epsilon^{-2})$ for the subdivision gadget and $\Delta=\Theta(\epsilon^{-3})$ for the 3- to 2-body gadget. We will provide analytical results and numerics which indicate that $\Delta=\Theta(\epsilon^{-1})$ is sufficient for the subdivision gadget (Sec. II and III) and $\Delta=\Theta(\epsilon^{-2})$ for the 3- to 2-body gadget (Sec. IV and Appendix A), showing that the physical resources required to realize the gadgets are less than previously assumed elsewhere in the literature. In our derivation of the $\Delta$ scalings, we use an analytical approach that involves bounding the infinite series in the perturbative expansion. For the 3- to 2-body reduction, in Appendix A we show that complications arise when there are multiple 3-body terms in the target Hamiltonian that are to be reduced concurrently and bounding the infinite series in the multiple-bit perturbative expansion requires separate treatments of odd and even order terms. Furthermore, in the case where $\Delta=\Theta(\epsilon^{-2})$ is used, additional terms which are dependent on the commutation relationship among the 3-body target terms are added to the gadget in order to compensate for the perturbative error due to cross-gadget contributions (Appendix B). The next result of this paper, described in Sec. V, is a 3- to 2-body gadget construction that uses a 2-body Ising Hamiltonian with a local transverse field. This opens the door to use existing flux-qubit hardware 2006cond.mat..8253H to simulate $H_{\text{targ}}=H_{\text{else}}+\alpha Z_{i}Z_{j}Z_{k}$ where $H_{\text{else}}$ is not necessarily diagonal. One drawback of this construction is that it requires $\Delta=\Theta(\epsilon^{-5})$, rendering it challenging to realize in practice. For cases where the target Hamiltonian is diagonal, there are non- perturbative gadgets B08 ; WFB12 ; BOA13 that can reduce a $k$-body Hamiltonian to 2-body. In this work, however, we focus on perturbative gadgets. The final result of this paper in Sec. VI is to propose a gadget which is capable of reducing arbitrary real-valued Hamiltonians to a Hamiltonian with only XX and ZZ couplings. In order to accomplish this, we go to fourth-order in perturbation theory to find an XXZZ Hamiltonian which serves as an effective Hamiltonian dominated by YY coupling terms. Because YY terms are especially difficult to realize in some experimental architectures, this result is useful for those wishing to encode arbitrary QMA-Hard problems on existing hardware. This gadget in fact now opens the door to solve electronic structure problems on an adiabatic quantum computer. To achieve both fast readability and completeness in presentation, each section from Sec. II to Sec. VI consists of a Summary subsection and an Analysis subsection. The former is mainly intended to provide a high-level synopsis of the main results in the corresponding section. Readers could only refer to the Summary sections on their own for an introduction to the results of the paper. The Analysis subsections contain detailed derivations of the results in the Summary. ## I Perturbation theory In our notation the spin-1/2 Pauli operators will be represented as $\\{X,Y,Z\\}$ with subscript indicating which spin-1/2 particle (qubit) it acts on. For example $X_{2}$ is a Pauli operator $X=\|0\rangle\langle{1}\|+\|1\rangle\langle{0}\|$ acting on the qubit labelled as $2$. In the literature there are different formulations of the perturbation theory that are adopted when constructing and analyzing the gadgets. This adds to the challenge faced in comparing the physical resources required among the various proposed constructions. For example, Jordan and Farhi JF08 use a formulation due to Bloch, while Bravyi et al. use a formulation based on the Schrieffer- Wolff transformation BDLT08 . Here we employ the formulation used in KKR06 ; OT06 . For a review on various formulations of perturbation theory, refer to BDL11 . A gadget Hamiltonian $\tilde{H}=H+V$ consists of a penalty Hamiltonian $H$, which applies an energy gap onto an ancilla space, and a perturbation $V$. To explain in further detail how the low-lying sector of the gadget Hamiltonian $\tilde{H}$ approximates the entire spectrum of a certain target Hamiltonian $H_{\text{targ}}$ with error $\epsilon$, we set up the following notations: let $\lambda_{j}$ and $\|\psi_{j}\rangle$ be the $j^{\text{th}}$ eigenvalue and eigenvector of $H$ and similarly define $\tilde{\lambda}_{j}$ and $\|\tilde{\psi}_{j}\rangle$ as those of $\tilde{H}$, assuming all the eigenvalues are labelled in a weakly increasing order ($\lambda_{1}\leq\lambda_{2}\leq\cdots$, same for $\tilde{\lambda}_{j}$). Using a cutoff value $\lambda_{}$, let $\mathcal{L}_{-}=\text{span}\\{\|\psi_{j}\rangle\|\forall j:\lambda_{j}\leq\lambda_{}\\}$ be the low energy subspace and $\mathcal{L}_{+}=\text{span}\\{\|\psi_{j}\rangle\|\forall j:\lambda_{j}>\lambda_{}\\}$ be the high energy subspace. Let ${\Pi_{-}}$ and ${\Pi_{+}}$ be the orthogonal projectors onto the subspaces $\mathcal{L}_{-}$ and $\mathcal{L}_{+}$ respectively. For an operator $O$ we define the partitions of $O$ into the subspaces as $O_{-}={\Pi_{-}}O{\Pi_{-}}$, $O_{+}={\Pi_{+}}O{\Pi_{+}}$, $O_{-+}={\Pi_{-}}O{\Pi_{+}}$ and $O_{+-}={\Pi_{+}}O{\Pi_{-}}$. With the definitions above, one can turn to perturbation theory to approximate $\tilde{H}_{-}$ using $H$ and $V$. We now consider the operator-valued resolvent $\tilde{G}(z)=(z\openone-\tilde{H})^{-1}$. Similarly one would define $G(z)=(z\openone-H)^{-1}$. Note that $\tilde{G}^{-1}(z)-G^{-1}(z)=-V$ so that this allows an expansion in powers of $V$ as $\tilde{G}=(G^{-1}-V)^{-1}=G(\openone-VG)^{-1}=G+GVG+GVGVG+GVGVGVG+\cdots.$ (2) It is then standard to define the self-energy $\Sigma_{-}(z)=z\openone-({\tilde{G}}_{-}(z))^{-1}$. The self-energy is important because the spectrum of $\Sigma_{-}(z)$ gives an approximation to the spectrum of $\tilde{H}_{-}$ since by definition $\tilde{H}_{-}=z\openone-{\Pi_{-}}(\tilde{G}^{-1}(z)){\Pi_{-}}$ while $\Sigma_{-}(z)=z\openone-({\Pi_{-}}\tilde{G}(z){\Pi_{-}})^{-1}$. As is explained by Oliveira and Terhal OT06 , loosely speaking, if $\Sigma_{-}(z)$ is roughly constant in some range of $z$ (defined below in Theorem I.1) then $\Sigma_{-}(z)$ is playing the role of $\tilde{H}_{-}$. This was formalized in KKR06 and improved in OT06 where the following theorem is proven (as in OT06 we state the case where $H$ has zero as its lowest eigenvalue and a spectral gap of $\Delta$. We use operator norm $\\|\cdot\\|$ which is defined as $\\|M\\|\equiv\max_{\|\psi\rangle\in\mathcal{M}}\|\langle\psi\|M\|\psi\rangle\|$ for an operator $M$ acting on a Hilbert space $\mathcal{M}$): ###### Theorem I.1 (Gadget Theorem KKR06 ; OT06 ). Let $\\|V\\|\leq\Delta/2$ where $\Delta$ is the spectral gap of $H$ and let the low and high spectrum of $H$ be separated by a cutoff $\lambda_{}=\Delta/2$. Now let there be an effective Hamiltonian $H_{\text{eff}}$ with a spectrum contained in $[a,b]$. If for some real constant $\epsilon>0$ and $\forall z\in[a-\epsilon,b+\epsilon]$ with $a<b<\Delta/2-\epsilon$, the self-energy $\Sigma_{-}(z)$ has the property that $\\|\Sigma_{-}(z)-H_{\text{eff}}\\|\leq\epsilon$, then each eigenvalue $\tilde{\lambda}_{j}$ of $\tilde{H}_{-}$ differs to the $j^{\text{th}}$ eigenvalue of $H_{\text{eff}}$, $\lambda_{j}$, by at most $\epsilon$. In other words $\|\tilde{\lambda}_{j}-\lambda_{j}\|\leq\epsilon$, $\forall j$. To apply Theorem I.1, a series expansion for $\Sigma_{-}(z)$ is truncated at low order for which $H_{\text{eff}}$ is approximated. The 2-body terms in $H$ and $V$ by construction can give rise to higher order terms in $H_{\text{eff}}$. For this reason it is possible to engineer $H_{\text{eff}}$ from $\Sigma_{-}(z)$ to approximate $H_{\text{targ}}$ up to error $\epsilon$ in the range of $z$ considered in Theorem I.1 by introducing auxiliary spins and a suitable selection of 2-body $H$ and $V$. Using the series expansion of $\tilde{G}$ in Eq. 2, the self-energy $\Sigma_{-}(z)=z\openone-\tilde{G}_{-}^{-1}(z)$ can be expanded as (for further details see KKR06 ) $\Sigma_{-}(z)=H_{-}+V_{-}+V_{-+}G_{+}(z)V_{+-}+V_{-+}G_{+}(z)V_{+}G_{+}(z)V_{+-}+\cdots.$ (3) The terms of $2^{\text{nd}}$ order and higher in this expansion give rise to the effective many-body interactions. (a) (b) Figure 1: Numerical illustration of gadget theorem using a subdivision gadget. Here we use a subdivision gadget to approximate $H_{\text{targ}}=H_{\text{else}}+\alpha Z_{1}Z_{2}$ with $\\|H_{\text{else}}\\|=0$ and $\alpha\in[-1,1]$. $\epsilon=0.05$. “analytical” stands for the case where the value of $\Delta$ is calculated using Eq. 14 when $\|\alpha\|=1$. “numerical” represents the case where $\Delta$ takes the value that yield the spectral error to be $\epsilon$. In (a) we let $\alpha=1$. $z\in[-\max z,\max z]$ with $\max z=\\|H_{\text{else}}\\|+\max\alpha+\epsilon$. The operator $\Sigma_{-}(z)$ is computed up to the $3^{\text{rd}}$ order. Subplot (b) shows for every value of $\alpha$ in its range, the maximum difference between the eigenvalues $\tilde{\lambda}_{j}$ in the low-lying spectrum of $\tilde{H}$ and the corresponding eigenvalues $\lambda_{j}$ in the spectrum of $H_{\text{targ}}\otimes\|0\rangle\langle{0}\|_{w}$. ## II Improved Oliveira and Terhal subdivision gadget Summary. The subdivision gadget is introduced by Oliveira and Terhal OT06 in their proof that 2-local Hamiltonian on square lattice is QMA-Complete. Here we show an improved lower bound for the spectral gap $\Delta$ needed on the ancilla of the gadget. A subdivision gadget simulates a many-body target Hamiltonian $H_{\text{targ}}=H_{\text{else}}+\alpha\cdot A\otimes B$ ($H_{\text{else}}$ is a Hamiltonian of arbitrary norm, $\\|A\\|=1$ and $\\|B\\|=1$) by introducing an ancilla spin $w$ and applying onto it a penalty Hamiltonian $H=\Delta\|1\rangle\langle{1}\|_{w}$ so that its ground state subspace $\mathcal{L}_{-}=\text{span}\\{\|0\rangle_{w}\\}$ and its excited subspace $\mathcal{L}_{+}=\text{span}\\{\|1\rangle_{w}\\}$ are separated by energy gap $\Delta$. In addition to the penalty Hamiltonian $H$, we add a perturbation $V$ of the form $V=H_{\text{else}}+\|\alpha\|\|0\rangle\langle{0}\|_{w}+\sqrt{\frac{\|\alpha\|\Delta}{2}}(\text{sgn}(\alpha)A-B)\otimes X_{w}.$ (4) Hence if the target term $A\otimes B$ is $k$-local, the gadget Hamiltonian $\tilde{H}=H+V$ is at most $(\lceil{k/2}\rceil+1)$-local, accomplishing the locality reduction. Assume $H_{\text{targ}}$ acts on $n$ qubits. Prior work OT06 shows that $\Delta=\Theta(\epsilon^{-2})$ is a sufficient condition for the lowest $2^{n}$ levels of the gadget Hamiltonian $\widetilde{H}$ to be $\epsilon$-close to the corresponding spectrum of $H_{\text{targ}}$. However, by bounding the infinite series of error terms in the perturbative expansion, we are able to obtain a tighter lower bound for $\Delta$ for error $\epsilon$. Hence we arrive at our first result (details will be presented later in this section), that it suffices to let $\Delta\geq\left(\frac{2\|\alpha\|}{\epsilon}+1\right)(2\\|H_{\text{else}}\\|+\|\alpha\|+\epsilon).$ (5) In Fig. 2 we show numerics indicating the minimum $\Delta$ required as a function of $\alpha$ and $\epsilon$. In Fig. 2a the numerical results and the analytical lower bound in Eq. 5 show that for our subdivision gadgets, $\Delta$ can scale as favorably as $\Theta(\epsilon^{-1})$. For the subdivision gadget presented in OT06 , $\Delta$ scales as $\Theta(\epsilon^{-2})$. Though much less than the original assignment in OT06 , the lower bound of $\Delta$ in Eq. 5, still satisfies the condition of Theorem I.1. In Fig. 2 we numerically find the minimum value of such $\Delta$ that yields a spectral error of exactly $\epsilon$. (a)(b) Figure 2: Comparison between our subdivision gadget with that of Oliveira and Terhal OT06 . The data labelled as “numerical” represent the $\Delta$ values obtained from the numerical search such that the spectral error between $H_{\text{targ}}$ and $\widetilde{H}_{-}$ is $\epsilon$. The data obtained from the calculation using Eq. 5 are labelled as “analytical”. “[OT06]” refers to values of $\Delta$ calculated according to the assignment by Oliveira and Terhal OT06 . In this example we consider $H_{\text{targ}}=H_{\text{else}}+\alpha Z_{1}Z_{2}$. (a) Gap scaling with respect to $\epsilon^{-1}$. Here $\\|H_{\text{else}}\\|=0$ and $\alpha=1$. (b) The gap $\Delta$ as a function of the desired coupling $\alpha$. Here $\\|H_{\text{else}}\\|=0$, $\epsilon=0.05$. Analysis. The currently known subdivision gadgets in the literature assume that the gap in the penalty Hamiltonian $\Delta$ scales as $\Theta(\epsilon^{-2})$ (see for example OT06 ; BDLT08 ). Here we employ a method which uses infinite series to find the upper bound to the norm of the high order terms in the perturbative expansion. We find that in fact $\Delta=\Theta(\epsilon^{-1})$ is sufficient for the error to be within $\epsilon$. A variation of this idea will also be used to reduce the gap $\Delta$ needed in the $3$\- to 2-body gadget (see Sec. IV). The key aspect of developing the gadget is that given $H=\Delta\|1\rangle\langle{1}\|_{w}$, we need to determine a perturbation $V$ to perturb the low energy subspace $\mathcal{L}_{-}=\text{span}\\{\|\psi\rangle\otimes\|0\rangle_{w},\makebox[1.42271pt]{}\text{ $\|\psi\rangle$ is any state of the system excluding the ancilla spin $w$}\\}$ such that the low energy subspace of the gadget Hamiltonian $\tilde{H}=H+V$ approximates the spectrum of the entire operator $H_{\text{targ}}\otimes\|0\rangle\langle{0}\|_{w}$ up to error $\epsilon$. Here we will define $V$ and work backwards to show that it satisfies Theorem I.1. We let $V=H_{\text{else}}+\frac{1}{\Delta}({\kappa}^{2}A^{2}+{\lambda}^{2}B^{2})\otimes\|0\rangle\langle{0}\|_{w}+({\kappa}A+{\lambda}B)\otimes X_{w}$ (6) where ${\kappa}$, ${\lambda}$ are constants which will be determined such that the dominant contribution to the perturbative expansion which approximates $\tilde{H}_{-}$ gives rise to the target Hamiltonian $H_{\text{targ}}=H_{\text{else}}+\alpha\cdot A\otimes B$. In Eq. 6 and the remainder of the section, by slight abuse of notation, we use $\kappa A+\lambda B$ to represent $\kappa(A\otimes\openone_{\mathcal{B}})+\lambda(\openone_{\mathcal{A}}\otimes B)$ for economy. Here $\openone_{\mathcal{A}}$ and $\openone_{\mathcal{B}}$ are identity operators acting on the subspaces $\mathcal{A}$ and $\mathcal{B}$ respectively. The partitions of $V$ in the subspaces, as defined in Sec. I are $\begin{array}[]{c}\displaystyle V_{+}=H_{\text{else}}\otimes\|1\rangle\langle{1}\|_{w},\quad V_{-}=\left(H_{\text{else}}+\frac{1}{\Delta}({\kappa}^{2}A^{2}+{\lambda}^{2}B^{2})\openone\right)\otimes\|0\rangle\langle{0}\|_{w},\\\\[7.22743pt] V_{-+}=({\kappa}A+{\lambda}B)\otimes\|0\rangle\langle{1}\|_{w},\quad V_{+-}=({\kappa}A+{\lambda}B)\otimes\|1\rangle\langle{0}\|_{w}.\end{array}$ (7) We would like to approximate the target Hamiltonian $H_{\text{targ}}$ and so expand the self-energy in Eq. 3 up to $2^{\text{nd}}$ order. Note that $H_{-}=0$ and $G_{+}(z)=(z-\Delta)^{-1}\|1\rangle\langle{1}\|_{w}$. Therefore the self energy $\Sigma_{-}(z)$ can be expanded as $\begin{array}[]{ccl}\Sigma_{-}(z)&=&\displaystyle V_{-}+\frac{1}{z-\Delta}V_{-+}V_{+-}+\sum_{k=1}^{\infty}\frac{V_{-+}V_{+}^{k}V_{+-}}{(z-\Delta)^{k+1}}\\\\[7.22743pt] &=&\displaystyle\underbrace{\left(H_{\text{else}}-\frac{2{\kappa}{\lambda}}{\Delta}A\otimes B\right)\otimes\|0\rangle\langle{0}\|_{w}}_{H_{\text{eff}}}+\underbrace{\frac{z}{\Delta(z-\Delta)}({\kappa}A+{\lambda}B)^{2}\otimes\|0\rangle\langle{0}\|_{w}+\sum_{k=1}^{\infty}\frac{V_{-+}V_{+}^{k}V_{+-}}{(z-\Delta)^{k+1}}}_{\text{error term}}.\end{array}$ (8) By selecting ${\kappa}=\text{sgn}(\alpha)(\|\alpha\|\Delta/2)^{1/2}$ and ${\lambda}=-(\|\alpha\|\Delta/2)^{1/2}$, the leading order term in $\Sigma_{-}(z)$ becomes $H_{\text{eff}}=H_{\text{targ}}\otimes\|0\rangle\langle{0}\|_{w}$. We must now show that the condition of Theorem I.1 is satisfied i.e. for a small real number $\epsilon>0$, $\\|\Sigma_{-}(z)-H_{\text{eff}}\\|\leq\epsilon,\forall z\in[\min z,\max z]$ where $\max z=\\|H_{\text{else}}\\|+\|\alpha\|+\epsilon=-\min z$. Essentially this amounts to choosing a value of $\Delta$ to cause the error term in Eq. 8 to be $\leq\epsilon$. In order to derive a tighter lower bound for $\Delta$, we bound the norm of the error term in Eq. 8 by letting $z\mapsto\max z$ and from the triangle inequality for operator norms: $\begin{array}[]{rcl}\displaystyle\left\\|\frac{z}{\Delta(z-\Delta)}(\kappa A+\lambda B)^{2}\otimes\|0\rangle\langle{0}\|_{w}\right\\|&\leq&\displaystyle\frac{\max{z}}{\Delta(\Delta-\max{z})}\cdot 4\kappa^{2}=\frac{2\|\alpha\|\max{z}}{\Delta-\max{z}}\\\\[7.22743pt] \displaystyle\left\\|\sum_{k=1}^{\infty}\frac{V_{-+}V_{+}^{k}V_{+-}}{(z-\Delta)^{k+1}}\right\\|&\leq&\displaystyle\sum_{k=1}^{\infty}\frac{\\|V_{-+}\\|\cdot\\|V_{+}\\|^{k}\cdot\\|V_{+-}\\|}{(\Delta-\max{z})^{k+1}}\\\\[7.22743pt] &\leq&\displaystyle\sum_{k=1}^{\infty}\frac{2\|\kappa\|\cdot\\|H_{\text{else}}\\|^{k}\cdot 2\|\kappa\|}{(\Delta-\max{z})^{k+1}}=\sum_{k=1}^{\infty}\frac{2\|\alpha\|\Delta\\|H_{\text{else}}\\|^{k}}{(\Delta-\max{z})^{k+1}}.\end{array}$ (9) Using $H_{\text{eff}}=H_{\text{targ}}\otimes\|0\rangle\langle{0}\|_{w}$, from (8) we see that $\displaystyle\begin{array}[]{ccl}\\|\Sigma_{-}(z)-H_{\text{targ}}\otimes\|0\rangle\langle{0}\|_{w}\\|&\leq&\displaystyle\frac{2\|\alpha\|\max z}{\Delta-\max z}+\sum_{k=1}^{\infty}\frac{2\|\alpha\|\Delta\\|H_{\text{else}}\\|^{k}}{(\Delta-\max z)^{k+1}}\end{array}$ (11) $\displaystyle\begin{array}[]{ccl}&=&\displaystyle\frac{2\|\alpha\|\max z}{\Delta-\max z}+\frac{2\|\alpha\|\Delta}{\Delta-\max z}\cdot\frac{\\|H_{\text{else}}\\|}{\Delta-\max z-\\|H_{\text{else}}\\|}.\end{array}$ (13) Here going from Eq. 11 to Eq. 13 we have assumed the convergence of the infinite series in Eq. 11, which adds the reasonable constraint that $\Delta>\|\alpha\|+\epsilon+2\\|H_{\text{else}}\\|$. To ensure that $\\|\Sigma_{-}(z)-H_{\text{targ}}\otimes\|0\rangle\langle{0}\|_{w}\\|\leq\epsilon$ it is sufficient to let expression Eq. 13 be $\leq\epsilon$, which implies that $\Delta\geq\left(\frac{2\|\alpha\|}{\epsilon}+1\right)(\|\alpha\|+\epsilon+2\\|H_{\text{else}}\\|)$ (14) which is $\Theta(\epsilon^{-1})$, a tighter bound than $\Theta(\epsilon^{-2})$ in the literature BDLT08 ; KKR06 ; OT06 . This bound is illustrated with a numerical example (Fig. 1). From the data labelled as “analytical” in Fig. 1a we see that the error norm $\\|\Sigma_{-}(z)-H_{\text{eff}}\\|$ is within $\epsilon$ for all $z$ considered in the range, which satisfies the condition of the theorem for the chosen example. In Fig. 1b, the data labelled “analytical” show that the spectral difference between $\tilde{H}_{-}$ and $H_{\text{eff}}=H_{\text{targ}}\otimes\|0\rangle\langle{0}\|_{w}$ is indeed within $\epsilon$ as the theorem promises. Furthermore, note that the condition of Theorem I.1 is only sufficient, which justifies why in Fig. 1b for $\alpha$ values at $\max\alpha$ and $\min\alpha$ the spectral error is strictly below $\epsilon$. This indicates that an even smaller $\Delta$, although below the bound we found in Eq. 14 to satisfy the theorem, could still yield the spectral error within $\epsilon$ for all $\alpha$ values in the range. The smallest value $\Delta$ can take would be one such that the spectral error is exactly $\epsilon$ when $\alpha$ is at its extrema. We numerically find this $\Delta$ (up to numerical error which is less than $10^{-5}\epsilon$) and as demonstrated in Fig. 1b, the data labelled “numerical” shows that the spectral error is indeed $\epsilon$ at $\max(\alpha)$ and $\min(\alpha)$, yet in Fig. 1a the data labelled “numerical” shows that for some $z$ in the range the condition of the Theorem I.1, $\\|\Sigma_{-}(z)-H_{\text{targ}}\otimes\|0\rangle\langle{0}\|_{w}\\|\leq\epsilon$, no longer holds. In Fig. 1 we assume that $\epsilon$ is kept constant. In Fig. 2a we compute both analytical and numerical $\Delta$ values for different values of $\epsilon$. _Comparison with Oliveira and TerhalOT06 ._ We also compare our $\Delta$ assignment with the subdivision gadget by Oliveira and Terhal OT06 , where given a target Hamiltonian $H_{\text{targ}}=H_{\text{else}}+Q\otimes R$ it is assumed that $Q$ and $R$ are operators with finite norm operating on two separate spaces $\mathcal{A}$ and $\mathcal{B}$. The construction of the subdivision gadget in OT06 is the same as the construction presented earlier: introduce an ancillary qubit $w$ with energy gap $\Delta$, then the unperturbed Hamiltonian is $H=\Delta\|1\rangle\langle{1}\|_{w}$. In OT06 they add a perturbation $V$ that takes the form of (OT06, , Eq. 15) $V=H^{\prime}_{\text{else}}+\sqrt{\frac{\Delta}{2}}(-Q+R)\otimes{X_{w}}$ (15) where $H^{\prime}_{\text{else}}=H_{\text{else}}+Q^{2}/2+R^{2}/2$. Comparing the form of Eq. 15 and Eq. 6 we can see that if we redefine $Q=\sqrt{\|\alpha\|}A$ and $R=\sqrt{\|\alpha\|}B$, the gadget formulation is identical to our subdivision gadget approximating $H_{\text{targ}}=H_{\text{else}}+\alpha{A\otimes B}$ with $\alpha>0$. In the original work $\Delta$ is chosen as (OT06, , Eq. 20) $\Delta=\frac{(\\|H^{\prime}_{\text{else}}\\|+C_{2}r)^{6}}{\epsilon^{2}}$ where $C_{2}\geq\sqrt{2}$ and $r=\max\\{\\|Q\\|,\\|R\\|\\}$. In the context of our subdivision gadget, this choice of $\Delta$ translates to a lower bound $\Delta\geq\frac{(\\|H_{\text{else}}+\|\alpha\|\openone\\|+\sqrt{2\|\alpha\|})^{6}}{\epsilon^{2}}.$ (16) In Fig. 2a we compare the lower bound in Eq. 16 with our lower bound in Eq. 14 and the numerically optimized $\Delta$ described earlier. ## III Parallel subdivision and $k$\- to $3$-body reduction Summary. Applying subdivision gadgets iteratively one can reduce a $k$-body Hamiltonian $H_{\text{targ}}=H_{\text{else}}+\alpha\bigotimes_{i=1}^{k}\sigma_{i}$ to 3-body. Here each $\sigma_{i}$ is a single spin Pauli operator. Initially, the term $\bigotimes_{i=1}^{k}\sigma_{i}$ can be broken down into $A\otimes B$ where $A=\bigotimes_{i=1}^{r}\sigma_{i}$ and $B=\bigotimes_{i=r+1}^{k}\sigma_{i}$. Let $r=k/2$ for even $k$ and $r=(k+1)/2$ for odd $k$. The gadget Hamiltonian will be $(\lceil{k/2}\rceil+1)$-body, which can be further reduced to a $(\lceil{\lceil{k/2}\rceil+1}\rceil/2+1)$-body Hamiltonian in the same fashion. Iteratively applying this procedure, we can reduce a $k$-body Hamiltonian to $3$-body, with the $i^{\text{th}}$ iteration introducing the same number of ancilla qubits as that of the many-body term to be subdivided. Applying the previous analysis on the improved subdivision gadget construction, we find that $\Delta_{i}=\Theta(\epsilon^{-1}\Delta_{i-1}^{3/2})$ is sufficient such that during each iteration the spectral difference between $\widetilde{H}_{i}$ and $\widetilde{H}_{i-1}$ is within $\epsilon$. From the recurrence relation $\Delta_{i}=\Theta(\epsilon^{-1}\Delta_{i-1}^{3/2})$, we hence were able to show a quadratic improvement over previous $k$-body constructions BDLT08 . Analysis. The concept of parallel application of gadgets has been introduced in OT06 ; KKR06 . The idea of using subdivision gadgets for iteratively reducing a $k$-body Hamiltonian to 3-body has been mentioned in OT06 ; BDLT08 . Here we elaborate the idea by a detailed analytical and numerical study. We provide explicit expressions of all parallel subdivision gadget parameters which guarantees that during each reduction the error between the target Hamiltonian and the low-lying sector of the gadget Hamiltonian is within $\epsilon$. For the purpose of presentation, let us define the notions of “parallel” and “series” gadgets in the following remarks. ###### Remark III.1 (Parallel gadgets). Parallel application of gadgets refers to using gadgets on multiple terms $H_{\text{targ,i}}$ in the target Hamiltonian $H_{\text{targ}}=H_{\text{else}}+\sum_{i=1}^{m}H_{\text{targ,i}}$ concurrently. Here one will introduce $m$ ancilla spins $w_{1},\cdots,w_{m}$ and the parallel gadget Hamiltonian takes the form of $\tilde{H}=\sum_{i=1}^{m}H_{i}+V$ where $H_{i}=\Delta\|1\rangle\langle{1}\|_{w_{i}}$ and $V=H_{\text{else}}+\sum_{i=1}^{m}V_{i}$. $V_{i}$ is the perturbation term of the gadget applied to $H_{\text{targ,i}}$. ###### Remark III.2 (Serial gadgets). Serial application of gadgets refers to using gadgets sequentially. Suppose the target Hamiltonian $H_{\text{targ}}$ is approximated by a gadget Hamiltonian $\tilde{H}^{(1)}$ such that $\tilde{H}^{(1)}_{-}$ approximates the spectrum of $H_{\text{targ}}$ up to error $\epsilon$. If one further applies onto $\tilde{H}^{(1)}$ another gadget and obtains a new Hamiltonian $\tilde{H}^{(2)}$ whose low-lying spectrum captures the spectrum of $\tilde{H}^{(1)}$, we say that the two gadgets are applied in series to reduce $H_{\text{targ}}$ to $\tilde{H}^{(2)}$. Based on Remark III.1, a parallel subdivision gadget deals with the case where $H_{\text{targ,i}}=\alpha_{i}A_{i}\otimes B_{i}$. $\alpha_{i}$ is a constant and $A_{i}$, $B_{i}$ are unit norm Hermitian operators that act on separate spaces $\mathcal{A}_{i}$ and $\mathcal{B}_{i}$. Note that with $H_{i}=\Delta\|1\rangle\langle{1}\|_{w_{i}}$ for every $i\in\\{1,2,\cdots,m\\}$ we have the total penalty Hamiltonian $H=\sum_{i=1}^{m}H_{i}=\sum_{x\in\\{0,1\\}^{m}}h(x)\Delta\|x\rangle\langle{x}\|$ where $h(x)$ is the Hamming weight of the $m$-bit string $x$. This penalty Hamiltonian ensures that the ground state subspace is $\mathcal{L}_{-}=\text{span}\\{\|0\rangle^{\otimes{m}}\\}$ while all the states in the subspace $\mathcal{L}_{+}=\text{span}\\{\|x\rangle\|x\in\\{0,1\\}^{m},x\neq 00\cdots 0\\}$ receives an energy penalty of at least $\Delta$. The operator-valued resolvent $G$ for the penalty Hamiltonian is (by definition in Sec. I) $G(z)=\sum_{x\in\\{0,1\\}^{m}}\frac{1}{z-h(x)\Delta}\|x\rangle\langle{x}\|.$ (17) The perturbation Hamiltonian $V$ is defined as $V=H_{\text{else}}+\frac{1}{\Delta}\sum_{i=1}^{m}({\kappa_{i}^{2}}A_{i}^{2}+{\lambda_{i}^{2}}B_{i}^{2})+\sum_{i=1}^{m}({\kappa_{i}}A_{i}+{\lambda_{i}}B_{i})\otimes X_{u_{i}}$ (18) where the coefficients ${\kappa_{i}}$ and ${\lambda_{i}}$ are defined as ${\kappa_{i}}=\text{sgn}(\alpha_{i})\sqrt{{\|\alpha_{i}\|\Delta}/{2}},{\lambda_{i}}=-\sqrt{{\|\alpha_{i}\|\Delta}/{2}}$. Define $P_{-}=\|0\rangle^{\otimes m}\langle{0}\|^{\otimes m}$ and $P_{+}=\openone-P_{-}$. Then if $H_{\text{targ}}$ acts on the Hilbert space $\mathcal{M}$, $\Pi_{-}=\openone_{\mathcal{M}}\otimes P_{-}$ and $\Pi_{+}=\openone_{\mathcal{M}}\otimes P_{+}$. Comparing Eq. 18 with Eq. 6 we see that the projector to the low-lying subspace $\|0\rangle\langle{0}\|_{w}$ in Eq. 6 is replaced by an identity $\openone$ in Eq. 18. This is because in the case of $m$ parallel gadgets $P_{-}$ cannot be realized with only 2-body terms when $m\geq 3$. The partition of $V$ in the subspaces are $\begin{array}[]{ll}\displaystyle V_{-}=\left(H_{\text{else}}+\frac{1}{\Delta}\sum_{i=1}^{m}({\kappa_{i}^{2}}A_{i}^{2}+{\lambda_{i}^{2}}B_{i}^{2})\right)\otimes{P_{-}},&\displaystyle V_{+}=\left(H_{\text{else}}+\frac{1}{\Delta}\sum_{i=1}^{m}({\kappa_{i}^{2}}A_{i}^{2}+{\lambda_{i}^{2}}B_{i}^{2})\right)\otimes{P_{+}}\\\\[7.22743pt] \displaystyle V_{-+}=\sum_{i=1}^{m}({\kappa_{i}}A_{i}+{\lambda_{i}}B_{i})\otimes{P_{-}}X_{u_{i}}{P_{+}},&\displaystyle V_{+-}=\sum_{i=1}^{m}({\kappa_{i}}A_{i}+{\lambda_{i}}B_{i})\otimes{P_{+}}X_{u_{i}}{P_{-}}.\\\\[7.22743pt] \end{array}$ (19) The self-energy expansion in Eq. 3 then becomes $\begin{array}[]{ccl}\displaystyle\Sigma_{-}(z)&=&\displaystyle\left(H_{\text{else}}+\frac{1}{\Delta}\sum_{i=1}^{m}({\kappa_{i}^{2}}A_{i}^{2}+{\lambda_{i}^{2}}B_{i}^{2})\right)\otimes{P_{-}}+\frac{1}{z-\Delta}\sum_{i=1}^{m}({\kappa_{i}}A_{i}+{\lambda_{i}}B_{i})^{2}\otimes{P_{-}}\\\\[7.22743pt] &+&\displaystyle\sum_{k=1}^{\infty}V_{-+}(G_{+}V_{+})^{k}G_{+}V_{+-}.\end{array}$ (20) Rearranging the terms we have $\begin{array}[]{ccl}\displaystyle\Sigma_{-}(z)&=&\displaystyle\underbrace{\left(H_{\text{else}}+\sum_{i=1}^{m}\left(-\frac{2{\kappa_{i}}{\lambda_{i}}}{\Delta}A_{i}\otimes B_{i}\right)\right)\otimes{P_{-}}}_{H_{\text{eff}}}+\underbrace{\left(\frac{1}{\Delta}+\frac{1}{z-\Delta}\right)\sum_{i=1}^{m}({\kappa_{i}^{2}}A_{i}^{2}+{\lambda_{i}^{2}}B_{i}^{2})\otimes{P_{-}}}_{E_{1}}\\\\[7.22743pt] &+&\displaystyle\underbrace{\left(\frac{1}{\Delta}+\frac{1}{z-\Delta}\right)\sum_{i=1}^{m}2{\kappa_{i}}{\lambda_{i}}A_{i}\otimes B_{i}\otimes{P_{-}}}_{E_{2}}+\underbrace{\sum_{k=1}^{\infty}V_{-+}(G_{+}V_{+})^{k}G_{+}V_{+-}}_{E_{3}}\\\\[7.22743pt] \end{array}$ (21) where the term $H_{\text{eff}}=H_{\text{targ}}\otimes{P_{-}}$ is the effective Hamiltonian that we would like to obtain from the perturbative expansion and $E_{1}$, $E_{2}$, and $E_{3}$ are error terms. Theorem I.1 states that for $z\in[-\max(z),\max(z)]$, if $\\|\Sigma_{-}(z)-H_{\text{targ}}\otimes{P_{-}}\\|\leq\epsilon$ then $\tilde{H}_{-}$ approximates the spectrum of $H_{\text{targ}}\otimes P_{-}$ by error at most $\epsilon$. Similar to the triangle inequality derivation shown in (9), to derive a lower bound for $\Delta$, let $z\mapsto\max(z)=\\|H_{\text{else}}\\|+\sum_{i=1}^{m}\|\alpha_{i}\|+\epsilon$ and the upper bounds of the error terms $E_{1}$ and $E_{2}$ can be found as $\begin{array}[]{ccl}\\|E_{1}\\|&\leq&\displaystyle\frac{\max(z)}{\Delta-\max(z)}\sum_{i=1}^{m}\|\alpha_{i}\|\leq\frac{\max(z)}{\Delta-\max(z)}\left(\sum_{i=1}^{m}\|\alpha_{i}\|^{1/2}\right)^{2}\\\\[7.22743pt] \\|E_{2}\\|&\leq&\displaystyle\frac{\max(z)}{\Delta-\max(z)}\left(\sum_{i=1}^{m}\|\alpha_{i}\|^{1/2}\right)^{2}.\end{array}$ (22) From the definition in Eq. 17 we see that $\\|G_{+}(z)\\|\leq\frac{1}{\Delta-\max(z)}$. Hence the norm of $E_{3}$ can be bounded by $\begin{array}[]{ccl}\\|E_{3}\\|&\leq&\displaystyle\sum_{k=1}^{\infty}\frac{\\|\sum_{i=1}^{m}({\kappa_{i}}A_{i}+{\lambda_{i}}B_{i})\\|^{2}\\|H_{\text{else}}+\frac{1}{\Delta}\sum_{i=1}^{m}({\kappa_{i}^{2}}A_{i}^{2}+{\lambda_{i}^{2}}B_{i}^{2}){\openone}\\|^{k}}{(\Delta-\max(z))^{k+1}}\\\\[7.22743pt] &\leq&\displaystyle\sum_{k=1}^{\infty}\frac{2\Delta(\sum_{i=1}^{m}\|\alpha_{i}\|^{1/2})^{2}(\\|H_{\text{else}}\\|+\sum_{i=1}^{m}\|\alpha_{i}\|)^{k}}{(\Delta-\max(z))^{k+1}}\\\\[7.22743pt] &=&\displaystyle\frac{2\Delta(\sum_{i=1}^{m}\|\alpha_{i}\|^{1/2})^{2}}{\Delta-\max(z)}\frac{\\|H_{\text{else}}\\|+\sum_{i=1}^{m}\|\alpha_{i}\|}{\Delta-\max(z)-(\\|H_{\text{else}}\\|+\sum_{i=1}^{m}\|\alpha_{i}\|)}.\end{array}$ (23) Similar to the discussion in Sec. II, to ensure that $\\|\Sigma_{-}(z)-H_{\text{targ}}\otimes{P_{-}}\\|\leq\epsilon$, which is the condition of Theorem I.1, it is sufficient to let $\\|E_{1}\\|+\\|E_{2}\\|+\\|E_{3}\\|\leq\epsilon$: $\begin{array}[]{ccl}\\|E_{1}\\|+\\|E_{2}\\|+\\|E_{3}\\|&\leq&\displaystyle\frac{2\max(z)}{\Delta-\max(z)}\left(\sum_{i=1}^{m}\|\alpha_{i}\|^{1/2}\right)^{2}\\\\[7.22743pt] &+&\displaystyle\frac{2\Delta(\sum_{i=1}^{m}\|\alpha_{i}\|^{1/2})^{2}}{\Delta-\max(z)}\cdot\frac{\\|H_{\text{else}}\\|+\sum_{i=1}^{m}\|\alpha_{i}\|}{\Delta-\max(z)-(\\|H_{\text{else}}\\|+\sum_{i=1}^{m}\|\alpha_{i}\|)}\\\\[7.22743pt] &=&\displaystyle\frac{2(\sum_{i=1}^{m}\|\alpha_{i}\|^{1/2})^{2}(\max(z)+\\|H_{\text{else}}\\|+\sum_{i=1}^{m}\|\alpha_{i}\|)}{\Delta-\max(z)-(\\|H_{\text{else}}\\|+\sum_{i=1}^{m}\|\alpha_{i}\|)}\leq\epsilon\end{array}$ (24) where we find the lower bound of $\Delta$ for parallel subdivision gadget $\Delta\geq\left[\frac{2(\sum_{i=1}^{m}\|\alpha_{i}\|^{1/2})^{2}}{\epsilon}+1\right](2\\|H_{\text{else}}\\|+2\sum_{i=1}^{m}\|\alpha_{i}\|+\epsilon).$ (25) Note that if one substitutes $m=1$ into Eq. 25 the resulting expression is a lower bound that is less tight than that in Eq. 14. This is because of the difference in the perturbation $V$ between Eq. 18 and Eq. 6 which is explained in the text preceding Eq. 19. Also we observe that the scaling of this lower bound for $\Delta$ is $O(\text{poly}(m)/\epsilon)$ for $m$ parallel applications of subdivision gadgets, assuming $\|\alpha_{i}\|=O(\text{poly}(m))$ for every $i\in\\{1,2,\cdots,m\\}$. This confirms the statement in OT06 ; KKR06 ; BDLT08 that subdivision gadgets can be applied to multiple terms in parallel and the scaling of the gap $\Delta$ in the case of $m$ parallel subdivision gadgets will only differ to that of a single subdivision gadget by a polynomial in $m$. _Iterative scheme for $k$\- to 3-body reduction._ The following iterative scheme summarizes how to use parallel subdivision gadgets for reducing a $k$-body Ising Hamiltonian to 3-body (Here we use superscript (i) to represent the $i^{\text{th}}$ iteration and subscript i for labelling objects within the same iteration): $\begin{array}[]{c l}\tilde{H}^{(0)}=&H_{\text{targ}};\text{$H_{\text{targ}}$ acts on the Hilbert space $\mathcal{M}^{(0)}$.}\\\ \text{\bf while}&\text{$\tilde{H}^{(i)}$ is more than 3-body}\\\ &\text{Step 1: Find all the terms that are no more than 3-body (including $H_{\text{else}}$ from $\tilde{H}^{(0)}$) in $\tilde{H}^{(i-1)}$}\\\ &\text{$\qquad\quad$ and let their sum be $H_{\text{else}}^{(i)}$.}\\\ &\text{Step 2: Partition the rest of the terms in $\tilde{H}^{(i-1)}$ into $\alpha_{1}^{(i)}A_{1}^{(i)}\otimes B_{1}^{(i)}$, }\\\ &\text{$\qquad\quad$ $\alpha_{2}^{(i)}A_{2}^{(i)}\otimes B_{2}^{(i)}$, $\cdots$, $\alpha_{m}^{(i)}A_{m}^{(i)}\otimes B_{m}^{(i)}$. Here $\alpha_{j}^{(i)}$ are coefficients.}\\\ &\text{Step 3: Introduce $m$ ancilla qubits $w_{1}^{(i)}$, $w_{2}^{(i)}$, $\cdots w_{m}^{(i)}$ and construct $\tilde{H}^{(i)}$ using the}\\\ &\text{$\qquad\quad$ parallel subdivision gadget. Let $P^{(i)}_{-}=\|0\cdots 0\rangle\langle 0\cdots 0\|_{w_{1}^{(i)}\cdots w_{m}^{(i)}}$. Define $\Pi_{-}^{(i)}=\openone_{\mathcal{M}^{(i)}}\otimes P_{-}^{(i)}$.}\\\ &\text{$\qquad$ 3.1: Apply the penalty Hamiltonian $H^{(i)}=\sum_{x\in\\{0,1\\}}^{m}h(x)\Delta^{(i)}\|x\rangle\langle x\|$.}\\\ &\text{$\qquad\qquad$ Here $\Delta^{(i)}$ is calculated by the lower bound in Eq.\ \ref{eq:D_par_sub}.}\\\ &\text{$\qquad$ 3.2: Apply the perturbation $V^{(i)}=H_{\text{else}}^{(i)}+\sum_{j=1}^{m}\sqrt{\frac{\|\alpha_{j}^{(i)}\|\Delta^{(i)}}{2}}(\text{sgn}(\alpha_{j}^{(i)})A_{j}^{(i)}-B_{j}^{(i)})\otimes X_{w_{j}^{(i)}}$}\\\ &\text{$\qquad\qquad+\sum_{j=1}^{m}\|\alpha_{j}^{(i)}\|{\openone}$.}\\\ &\text{$\qquad$ 3.3: $\tilde{H}^{(i)}=H^{(i)}+V^{(i)}$ acts on the space $\mathcal{M}^{(i)}$ and the maximum spectral difference}\\\ &\text{$\qquad\qquad$ between $\tilde{H}^{(i)}_{-}=\Pi^{(i)}_{-}\tilde{H}^{(i)}\Pi^{(i)}_{-}$ and $\tilde{H}^{(i-1)}\otimes P^{(i)}_{-}$ is at most $\epsilon$.}\\\ &\text{$i\rightarrow{i+1}$}\\\ \text{\bf end}&\\\ \end{array}$ (26) ${S_{1}}{S_{2}}{S_{3}}{S_{4}}\|{S_{5}}{S_{6}}{S_{7}}$iteration (tree depth) $i$0.05(-0.5,3.55)(7.5,3.55) $i=1$${S_{1}}{S_{2}}{S_{3}}\|{S_{4}}X_{u_{1}}$$X_{u_{1}}{S_{5}}\|{S_{6}}{S_{7}}$0.05(-0.5,2.3)(7.5,2.3) $i=2$${S_{1}}{S_{2}}\|{S_{3}}X_{u_{2}}$$X_{u_{2}}{S_{4}}X_{u_{1}}$$X_{u_{1}}{S_{5}}{X_{u_{3}}}$$X_{u_{3}}{S_{6}}{S_{7}}$0.05(-0.5,1.2)(7.5,1.2) $i=3$${S_{1}}{S_{2}}X_{u_{4}}$$X_{u_{4}}{S_{3}}X_{u_{2}}$ (a) (b) (c) Figure 3: (a) Reduction tree diagram for reducing a 7-body term to 3-body using parallel subdivision gadgets. Each $S_{i}$ is a single-qubit Pauli operator acting on qubit $i$. The vertical lines $\|$ show where the subdivisions are made at each iteration to each term. (b) An example where we consider the target Hamiltonian $H_{\text{targ}}=\alpha S_{1}S_{2}S_{3}S_{4}S_{5}S_{6}S_{7}$ with $\alpha=5\times 10^{-3}$, $S_{i}=X_{i}$, $\forall i\in\\{1,2,\cdots,7\\}$, and reduce it to 3-body according to (a) up to error $\epsilon=5\times 10^{-4}$. This plot shows the energy gap applied onto the ancilla qubits introduced at each iteration. (c) The spectral error between the gadget Hamiltonian at each iteration $\tilde{H}^{(i)}$ and the target Hamiltonian $H_{\text{targ}}$. For both (b)(c) the data labelled as “numerical” correspond to the case where during each iteration $\Delta^{(i)}$ is optimized such that the maximum spectral difference between $\Pi_{-}^{(i)}\tilde{H}^{(i)}\Pi_{-}^{(i)}$ and $\tilde{H}^{(i-1)}\otimes P^{(i)}_{-}$ is $\epsilon$. For definitions of $\Delta^{(i)}$, $\tilde{H}^{(i)}$, $\Pi^{(i)}_{-}$ and $P^{(i)}_{-}$, see Eq. 26. Those labelled as ‘analytical’ correspond to cases where each iteration uses the gap bound derived in Eq. 25. We could show that after $s$ iterations, the maximum spectral error between $\Pi^{(s)}_{-}\tilde{H}^{(s)}\Pi^{(s)}_{-}$ and $\tilde{H}^{(0)}\bigotimes_{i=1}^{s}P^{(s)}_{-}$ is guaranteed to be within $s\epsilon$. Suppose we would like to make target Hamiltonian $\tilde{H}_{0}$, we construct a gadget $\tilde{H}=H^{(1)}+V^{(1)}$ according to algorithm (26), such that $\|\lambda(\tilde{H}^{(1)})-\lambda(\tilde{H}^{(0)})\|\leq\epsilon$ for low-lying eigenvalues $\lambda(\cdot)$. Note that in a precise sense we should write $\|\lambda(\Pi_{-}^{(1)}\tilde{H}^{(1)}\Pi_{-}^{(1)})-\lambda(\tilde{H}^{(0)}\otimes P_{-}^{(0)})\|$. Since the projectors $\Pi_{-}^{(i)}$ and $P_{-}^{(i)}$ do not affect the low-lying spectrum of $\tilde{H}^{(i)}$ and $\tilde{H}^{(i-1)}$, for simplicity and clarity we write only $\tilde{H}^{(i-1)}$ and $\tilde{H}^{(i)}$. After $\tilde{H}^{(1)}$ is introduced, according to algorithm (26) the second gadget $\tilde{H}^{(2)}$ is then constructed by considering the _entire_ $\tilde{H}^{(1)}$ as the new target Hamiltonian and introducing ancilla particles with unperturbed Hamiltonian $H^{(2)}$ and perturbation $V^{(2)}$ such that the low-energy spectrum of $\tilde{H}^{(2)}$ approximates the spectrum of $\tilde{H}^{(1)}$ up to error $\epsilon$. In other words $\|\lambda(\tilde{H}^{(1)})-\lambda(\tilde{H}^{(2)})\|\leq\epsilon$. With the serial application of gadgets we have produced a sequence of Hamiltonians $\tilde{H}^{(0)}\rightarrow\tilde{H}^{(1)}\rightarrow\tilde{H}^{(2)}\rightarrow\cdots\rightarrow\tilde{H}^{(k)}$ where $\tilde{H}^{(0)}$ is the target Hamiltonian and each subsequent gadget Hamiltonian $\tilde{H}^{(i)}$ captures the _entire_ previous gadget $\tilde{H}^{(i-1)}$ in its low-energy sector with $\|\lambda(\tilde{H}^{(i)})-\lambda(\tilde{H}^{(i-1)})\|\leq\epsilon$. Hence to bound the spectral error between the last gadget $\tilde{H}^{(k)}$ and the target Hamiltonian $\tilde{H}^{(0)}$ we could use triangle inequality: $\|\lambda(\tilde{H}^{(s)})-\lambda(\tilde{H}^{(0)})\|\leq\|\lambda(\tilde{H}^{(s)})-\lambda(\tilde{H}^{(s-1)})\|+\cdots+\|\lambda(\tilde{H}^{(1)})-\lambda(\tilde{H}^{(0)})\|\leq s\epsilon$. _Total number of iterations for a $k$\- to 3-body reduction._ In general, given a $k$-body Hamiltonian, we apply the following parallel reduction scheme at each iteration until every term is 3-body: if $k$ is even, this reduces it to two $(k/2+1)$-body terms; if $k$ is odd, this reduces it to a $(\frac{k+1}{2}+1)$\- and a $(\frac{k-1}{2}+1)$-body term. Define a function $f$ such that a $k$-body term needs $f(k)$ iterations to be reduced to 3-body. Then we have the recurrence $f(k)=\left\\{\begin{array}[]{cr}\displaystyle f\left(\frac{k}{2}+1\right)+1&\text{$k$ even}\\\\[7.22743pt] \displaystyle f\left(\frac{k+1}{2}+1\right)+1&\text{$k$ odd}\end{array}\right.$ (27) with $f(3)=0$ and $f(4)=1$. One can check that $f(k)=\lceil\log_{2}(k-2)\rceil$, $k\geq 4$ satisfies this recurrence. Therefore, using subdivision gadgets, one can reduce a $k$-body interaction to $3$-body in $s=\lceil\log_{2}(k-2)\rceil$ iterations and the spectral error between $\tilde{H}^{(s)}$ and $\tilde{H}^{(0)}$ is within $\lceil\log_{2}(k-2)\rceil\epsilon$. _Gap scaling._ From the iterative scheme shown previously one can conclude that $\Delta^{(i+1)}=\Theta(\epsilon^{-1}(\Delta^{(i)})^{3/2})$ for the $(i+1)^{\text{th}}$ iteration, which implies that for a total of $s$ iterations, $\Delta^{(s)}=\Theta\left(\epsilon^{-2[(3/2)^{s-1}-1]}(\Delta^{(1)})^{(3/2)^{s-1}}\right).$ (28) Since $s=\lceil\log_{2}(k-2)\rceil$ and $\Delta^{(1)}=\Theta(\epsilon^{-1})$ we have $\Delta^{(s)}=\Theta\left(\epsilon^{-3(\frac{1}{2}\lceil k-2\rceil)^{\log_{2}(3/2)}-2}\right)=\Theta\left(\epsilon^{-\text{poly}(k)}\right)$ (29) accumulating exponentially as a function of $k$. The exponential nature of the scaling with respect to $k$ agrees with results by Bravyi et al. BDLT08 . However, in our construction, due to the improvement of gap scaling in a single subdivision gadget from $\Delta=\Theta(\epsilon^{-2})$ to $\Theta(\epsilon^{-1})$, the scaling exponents in $\Delta^{(i+1)}=\Theta(\epsilon^{-1}(\Delta^{(i)})^{3/2})$ are also improved quadratically over those in BDLT08 , which is $\Delta^{(i+1)}=\Theta(\epsilon^{-2}(\Delta^{(i)})^{3})$. _Qubit cost._ Based on the reduction scheme described in Eq. 26 (illustrated in Fig. 3a for 7-body), the number of ancilla qubits needed for reducing a $k$-body term to 3-body is $k-3$. Suppose we are given a $k$-body target term $S_{1}S_{2}\cdots S_{k}$ (where all of the operators $S_{i}$ act on separate spaces) and we would like to reduce it to 3-body using the iterative scheme Eq. 26. At each iteration, if we describe every individual subdivision gadget by a vertical line $\|$ at the location where the partition is made, for example $S_{1}S_{2}S_{3}S_{4}\|S_{5}S_{6}S_{7}$ in the case of the first iteration in Fig. 3a, then after $\lceil\log_{2}(k-2)\rceil$ iterations all the partitions made to the $k$-body term can be described as $S_{1}S_{2}\|S_{3}\|S_{4}\|\cdots\|S_{k-2}\|S_{k-1}S_{k}$. Note that there are $k-3$ vertical lines in total, each corresponding to an ancilla qubit needed for a subdivision gadget. Therefore in total $k-3$ ancilla qubits are needed for reducing a $k$-body term to 3-body. _Example: Reducing 7-body to 3-body._ We have used numerics to test the reduction algorithm Eq. 26 on a target Hamiltonian $H_{\text{targ}}=\alpha S_{1}S_{2}S_{3}S_{4}S_{5}S_{6}S_{7}$. Here we let $S_{i}=X_{i}$, $\forall i\in\\{1,2,\cdots,7\\}$, $\epsilon=5\times 10^{-4}$ and $\alpha=5\times 10^{-3}$. During each iteration the values of $\Delta^{(i)}$ are assigned according to the lower bound in Eq. 25. From Fig. 3c we can see that the lower bounds are sufficient for keeping the total spectral error between $\tilde{H}_{-}^{(3)}$ and $\tilde{H}^{(0)}\bigotimes_{i=1}^{3}P^{(i)}_{-}$ within $3\epsilon$. Furthermore, numerical search is also used at each iteration to find the minimum value of $\Delta^{(i)}$ so that the spectral error between $\Pi_{-}^{(i)}\tilde{H}^{(i)}\Pi_{-}^{(i)}$ and $\tilde{H}^{(i-1)}\bigotimes_{j=1}^{i}P^{(j)}_{-}$ is $\epsilon$. The numerically found gaps $\Delta^{(i)}$ are much smaller than their analytical counterparts at each iteration (Fig. 3b), at the price that the error is larger (Fig. 3c). In both the numerical and the analytical cases, the error appears to accumulate linearly as the iteration proceeds. ## IV Improved Oliveira and Terhal 3- to 2-body gadget Summary. Subdivision gadgets cannot be used for reducing from 3- to 2-body; accordingly, the final reduction requires a different type of gadget KKR06 ; OT06 ; BDLT08 . Consider 3-body target Hamiltonian of the form $H_{\text{targ}}=H_{\text{else}}+\alpha A\otimes B\otimes C$. Here $A$, $B$ and $C$ are unit-norm Hermitian operators acting on separate spaces $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$. Here we focus on the gadget construction introduced in Oliveira and Terhal OT06 and also used in Bravyi, DiVincenzo, Loss and Terhal BDLT08 . To accomplish the 3- to 2-body reduction, we introduce an ancilla spin $w$ and apply a penalty Hamiltonian $H=\Delta\|1\rangle\langle{1}\|_{w}$. We then add a perturbation $V$ of form, $V=H_{\text{else}}+\mu C\otimes\|1\rangle\langle{1}\|_{w}+(\kappa A+\lambda B)\otimes X_{w}+V_{1}+V_{2}$ (30) where $V_{1}$ and $V_{2}$ are 2-local compensation terms (details presented later in this section): $\begin{array}[]{ccl}V_{1}&=&\displaystyle\frac{1}{\Delta}(\kappa^{2}+\lambda^{2})\|0\rangle\langle{0}\|_{w}+\frac{2\kappa\lambda}{\Delta}{A}\otimes{B}-\frac{1}{\Delta^{2}}(\kappa^{2}+\lambda^{2})\mu{C}\otimes\|0\rangle\langle{0}\|_{w}\\\\[7.22743pt] V_{2}&=&\displaystyle-\frac{2\kappa\lambda}{\Delta^{3}}\text{sgn}(\alpha)\bigg{[}(\kappa^{2}+\lambda^{2})\|0\rangle\langle{0}\|_{w}+2\kappa\lambda{A}\otimes{B}\bigg{]}.\end{array}$ (31) Here we let $\kappa=\text{sgn}(\alpha)\left({\alpha}/{2}\right)^{1/3}\Delta^{3/4}$, $\lambda=\left({\alpha}/{2}\right)^{1/3}\Delta^{3/4}$ and $\mu=\left({\alpha}/{2}\right)^{1/3}\Delta^{1/2}$. For sufficiently large $\Delta$, the low-lying spectrum of the gadget Hamiltonian $\widetilde{H}$ captures the entire spectrum of $H_{\text{targ}}$ up to arbitrary error $\epsilon$. In the construction of BDLT08 it is shown that $\Delta=\Theta(\epsilon^{-3})$ is sufficient. In KKR06 , $\Delta=\Theta(\epsilon^{-3})$ is also assumed, though the construction of $V$ is slightly different from Eq. 30. By adding terms in $V$ to compensate for the perturbative error due to the modification, we find that $\Delta=\Theta(\epsilon^{-2})$ is sufficient for accomplishing the 3- to 2-body reduction: $\Delta\geq\frac{1}{4}({-b+\sqrt{b^{2}-4c}})^{2}$ (32) where $b$ and $c$ are defined as $\begin{array}[]{ccl}b&=&\displaystyle-\left[\xi+\frac{2^{4/3}\alpha^{2/3}}{\epsilon}(\max{z}+\eta+\xi^{2})\right]\\\\[7.22743pt] c&=&\displaystyle-\left(1+\frac{2^{4/3}\alpha^{2/3}}{\epsilon}\xi\right)(\max{z}+\eta)\end{array}$ (33) with $\max z=\\|H_{\text{else}}\\|+\|\alpha\|+\epsilon$, $\eta=\\|H_{\text{else}}\\|+2^{2/3}\alpha^{4/3}$ and $\xi=2^{-1/3}\alpha^{1/3}+2^{1/3}\alpha^{2/3}$. From Eq. 32 we can see the lower bound to $\Delta$ is $\Theta({\epsilon^{-2}})$. Our improvement results in a power of $\epsilon^{-1}$ reduction in the gap. For the dependence of $\Delta$ on $\\|H_{\text{else}}\\|$, $\alpha$ and $\epsilon^{-1}$ for both the original OT06 and the optimized case, see Fig. 4. Results show that the bound in Eq. 32 is tight with respect to the minimum $\Delta$ numerically found that yields the spectral error between $\tilde{H}_{-}$ and $H_{\text{targ}}\otimes\|0\rangle\langle{0}\|_{w}$ to be $\epsilon$. Analysis. We will proceed by first presenting the improved construction of the 3- to 2-body gadget and then show that $\Delta=\Theta(\epsilon^{-2})$ is sufficient for the spectral error to be $\leq\epsilon$. Then we present the construction in the literature OT06 ; BDLT08 and argue that $\Delta=\Theta(\epsilon^{-3})$ is required for yielding a spectral error between $\tilde{H}$ and $H_{\text{eff}}$ within $\epsilon$ using this construction. In the improved construction we define the perturbation $V$ as in Eq. 30. Here the coefficients are chosen to be $\kappa=\Theta(\Delta^{3/4})$, $\lambda=\Theta(\Delta^{3/4})$ and $\mu=\Theta(\Delta^{1/2})$. In order to show that the assigned powers of $\Delta$ in the coefficients are optimal, we introduce a parameter $r$ such that ${\kappa}=\text{sgn}(\alpha)\left(\frac{\alpha}{2}\right)^{1/3}\Delta^{r},\qquad{\lambda}=\left(\frac{\alpha}{2}\right)^{1/3}\Delta^{r},\qquad{\mu}=\left(\frac{\alpha}{2}\right)^{1/3}\Delta^{2-2r}.$ (34) It is required that $\\|V\\|\leq\Delta/2$ (Theorem I.1) for the convergence of the perturbative series. Therefore let $r<1$ and $2-2r<1$, which gives $1/2<r<1$. With the definitions $\mathcal{L}_{-}$ and $\mathcal{L}_{+}$ being the ground and excited state subspaces respectively, $V_{-}$, $V_{+}$, $V_{-+}$, $V_{+-}$ can be calculated as the following: $\begin{array}[]{ccl}V_{-}&=&\displaystyle\left[H_{\text{else}}+\frac{1}{\Delta}(\kappa A+\lambda B)^{2}-\frac{1}{\Delta}(\kappa^{2}+\lambda^{2})\mu C-\frac{2\kappa\lambda}{\Delta^{3}}\text{sgn}(\alpha)(\kappa A+\lambda B)^{2}\right]\otimes\|0\rangle\langle{0}\|_{w}\\\\[7.22743pt] V_{+}&=&\displaystyle\left[H_{\text{else}}+\mu C+\frac{2\kappa\lambda}{\Delta}A\otimes B-\frac{4\kappa^{2}\lambda^{2}}{\Delta^{3}}\text{sgn}(\alpha)A\otimes B\right]\otimes\|1\rangle\langle{1}\|_{w}\\\\[7.22743pt] V_{-+}&=&\displaystyle(\kappa A+\lambda B)\otimes\|0\rangle\langle{1}\|_{w}\\\\[7.22743pt] V_{+-}&=&\displaystyle(\kappa A+\lambda B)\otimes\|1\rangle\langle{0}\|_{w}.\end{array}$ (35) The self-energy expansion, referring to Eq. 3, becomes $\begin{array}[]{ccl}\Sigma_{-}(z)&=&\displaystyle V_{-}+\frac{1}{z-\Delta}V_{-+}V_{+-}+\frac{1}{(z-\Delta)^{2}}V_{-+}V_{+}V_{+-}+\sum_{k=2}^{\infty}\frac{V_{-+}V_{+}^{k}V_{+-}}{(z-\Delta)^{k+1}}\\\\[7.22743pt] &=&\displaystyle\underbrace{H_{\text{else}}}_{(a)}+\underbrace{\frac{1}{\Delta}(\kappa A+\lambda B)^{2}}_{(b)}\underbrace{-\frac{1}{\Delta}(\kappa^{2}+\lambda^{2})\mu C}_{(c)}\underbrace{-\frac{2\kappa\lambda}{\Delta^{3}}\text{sgn}(\alpha)(\kappa A+\lambda B)^{2}}_{(d)}+\underbrace{\frac{1}{z-\Delta}(\kappa A+\lambda B)^{2}}_{(e)}\\\\[7.22743pt] &+&\displaystyle\frac{1}{(z-\Delta)^{2}}(\kappa A+\lambda B)\left[\underbrace{H_{\text{else}}}_{(f)}+\underbrace{\mu C}_{(g)}+\underbrace{\frac{2\kappa\lambda}{\Delta}A\otimes B}_{(h)}\underbrace{-\frac{4\kappa^{2}\lambda^{2}}{\Delta^{3}}\text{sgn}(\alpha)A\otimes B}_{(i)}\right](\kappa A+\lambda B)\\\\[7.22743pt] &+&\displaystyle\underbrace{\sum_{k=2}^{\infty}\frac{V_{-+}V_{+}^{k}V_{+-}}{(z-\Delta)^{k+1}}}_{(j)}.\end{array}$ (36) Now we rearrange the terms in the self energy expansion so that the target Hamiltonian arising from the leading order terms can be separated from the rest, whcih are error terms. Observe that term $(g)$ combined with the factors outside the bracket could give rise to a 3-body $A\otimes B\otimes C$ term: $\begin{array}[]{ccl}\displaystyle\frac{1}{(z-\Delta)^{2}}(\kappa A+\lambda B)^{2}\mu C&=&\displaystyle\underbrace{\frac{2\kappa\lambda\mu}{\Delta^{2}}A\otimes B\otimes C}_{(g_{1})}+\underbrace{\left(\frac{1}{(z-\Delta)^{2}}-\frac{1}{\Delta^{2}}\right)2{\kappa}{\lambda}{\mu}{A}\otimes{B}\otimes{C}}_{(g_{2})}\\\\[7.22743pt] &+&\displaystyle\underbrace{\frac{1}{(z-\Delta)^{2}}(\kappa^{2}+\lambda^{2})\mu C}_{(g_{3})}.\end{array}$ (37) Here $(g_{1})$ combined with term $(a)$ in (36) gives $H_{\text{targ}}$. $(g_{2})$ and $(g_{3})$ are error terms. Now we further rearrange the error terms as the following. We combine term $(b)$ and $(e)$ to form $E_{1}$, term $(c)$ and $(g_{3})$ to form $E_{2}$, term $(f)$ and the factors outside the bracket to be $E_{3}$. Rename $(g_{2})$ to be $E_{4}$. Using the identity $({\kappa}{A}+{\lambda}{B})({A}\otimes{B})({\kappa}{A}+{\lambda}{B})=\text{sgn}(\alpha)({\kappa}{A}+{\lambda}{B})^{2}$ we combine term $(d)$ and $(h)$ along with the factors outside the bracket to be $E_{5}$. Rename $(i)$ to be $E_{6}$ and $(j)$ to be $E_{7}$. The rearranged self-energy expanision reads $\begin{array}[]{ccl}\Sigma_{-}(z)&=&\displaystyle\bigg{[}\underbrace{H_{\text{else}}+\frac{2{\kappa}{\lambda}{\mu}}{\Delta^{2}}{A}\otimes{B}\otimes{C}}_{H_{\text{targ}}}+\underbrace{\left(\frac{1}{\Delta}+\frac{1}{z-\Delta}\right)({\kappa}{A}+{\lambda}{B})^{2}}_{E_{1}}\\\\[7.22743pt] &+&\displaystyle\underbrace{\left(\frac{1}{(z-\Delta)^{2}}-\frac{1}{\Delta^{2}}\right)({\kappa}^{2}+{\lambda}^{2}){\mu}{C}}_{E_{2}}+\underbrace{\frac{1}{(z-\Delta)^{2}}({\kappa}{A}+{\lambda}{B})H_{\text{else}}({\kappa}{A}+{\lambda}{B})}_{E_{3}}\\\\[7.22743pt] &+&\displaystyle\underbrace{\left(\frac{1}{(z-\Delta)^{2}}-\frac{1}{\Delta^{2}}\right)2{\kappa}{\lambda}{\mu}{A}\otimes{B}\otimes{C}}_{E_{4}}+\underbrace{\left(\frac{1}{(z-\Delta)^{2}}-\frac{1}{\Delta^{2}}\right)\frac{2{\kappa}{\lambda}}{\Delta}\text{sgn}(\alpha)({\kappa}{A}+{\lambda}{B})^{2}}_{E_{5}}\\\\[7.22743pt] &-&\underbrace{\frac{1}{(z-\Delta)^{2}}\cdot\frac{4{\kappa}^{2}{\lambda}^{2}}{\Delta^{3}}({\kappa}{A}+{\lambda}{B})^{2}}_{E_{6}}\bigg{]}\otimes\|0\rangle\langle{0}\|_{w}+\underbrace{\sum_{k=2}^{\infty}\frac{V_{-+}V_{+}^{k}V_{+-}}{(z-\Delta)^{k+1}}}_{E_{7}}.\end{array}$ (38) We bound the norm of each error term in the self energy expansion Eq. 38 by substituting the definitions of $\kappa$, $\lambda$ and $\mu$ in Eq. 34 and letting $z$ be the maximum value permitted by Theorem I.1 which is $\max z=\|\alpha\|+\epsilon+\\|H_{\text{else}}\\|$: $\\|E_{1}\\|\leq\displaystyle\frac{\max{z}{\cdot}2^{4/3}\alpha^{2/3}\Delta^{2r-1}}{\Delta-\max{z}}=\Theta(\Delta^{2r-2}),\qquad\\|E_{2}\\|\leq\displaystyle\frac{(2\Delta-\max{z})\max{z}}{(\Delta-\max{z})^{2}}\cdot\alpha=\Theta(\Delta^{-1})$ (39) $\\|E_{3}\\|\leq\displaystyle\frac{2^{4/3}\alpha^{2/3}\Delta^{2r}\\|H_{\text{else}}\\|}{(\Delta-\max{z})^{2}}=\Theta(\Delta^{2r-2}),\qquad\\|E_{4}\\|\leq\displaystyle\frac{(2\Delta-\max{z})\max{z}}{(\Delta-\max{z})^{2}}\cdot\alpha=\Theta(\Delta^{-1})$ (40) $\\|E_{5}\\|\leq\displaystyle\frac{(2\Delta-\max{z})\max{z}}{(\Delta-\max{z})^{2}}\cdot{2^{5/3}}\alpha^{4/3}\Delta^{4r-3}=\Theta(\Delta^{4r-4}),\qquad\\|E_{6}\\|\leq\displaystyle\frac{4\alpha^{2}\Delta^{6r-3}}{(\Delta-\max{z})^{2}}=\Theta(\Delta^{6r-5})$ (41) $\begin{array}[]{ccl}\\|E_{7}\\|&\leq&\displaystyle\sum_{k=2}^{\infty}\left\\|\frac{({\kappa}A+{\lambda}B)\left(H_{\text{else}}+{\mu}C+\frac{2{\kappa}{\lambda}}{\Delta}\left(1+\frac{2{\kappa}{\lambda}}{\Delta^{2}}\right)A\otimes B\right)^{k}({\kappa}A+{\lambda}B)}{(\Delta-\max{z})^{k+1}}\right\\|\\\\[7.22743pt] &\leq&\displaystyle\frac{2^{4/3}\alpha^{2/3}\Delta^{2r}}{(\Delta-\max{z})}\sum_{k=2}^{\infty}\frac{\left(\\|H_{\text{else}}\\|+2^{-1/3}\alpha^{1/3}\Delta^{2-2r}+2^{1/3}\alpha^{2/3}\Delta^{2r-1}+2^{2/3}\alpha^{4/3}\Delta^{4r-3}\right)^{k}}{(\Delta-\max{z})^{k}}\\\\[7.22743pt] &=&\displaystyle\Theta(\Delta^{\max\\{1-2r,6r-5,10r-9\\}}).\end{array}$ (42) Now the self energy expansion can be written as $\Sigma_{-}(z)=H_{\text{targ}}\otimes\|0\rangle\langle{0}\|_{w}+\Theta(\Delta^{f(r)})$ where the function $f(r)<0$ determines the dominant power in $\Delta$ from $\\|E_{1}\\|$ through $\\|E_{6}\\|$: $f(r)=\max\\{1-2r,6r-5\\},\quad\frac{1}{2}<r<1.$ (43) In order to keep the error $O(\epsilon)$, it is required that $\Delta=\Theta(\epsilon^{1/f(r)})$. To optimize the gap scaling as a function of $\epsilon$, $f(r)$ must take the minimum value. As is shown in Fig. 5b, when $r=3/4$, the minimum value $f(r)=-1/2$ is obtained, which corresponds to $\Delta=\Theta(\epsilon^{-2})$. We have hence shown that the powers of $\Delta$ in the assignments of $\kappa$, $\lambda$ and $\mu$ in Eq. 34 are optimal for the improved gadget construction. The optimal scaling of $\Theta(\epsilon^{-2})$ is also numerically confirmed in Fig. 4a. As one can see, the optimized slope $\text{d}\log\Delta/\text{d}\log\epsilon^{-1}$ is approximately 2 for small $\epsilon$. (a)(b) Figure 4: Comparison between our 3- to 2-body gadget with that of Oliveira and Terhal OT06 . As $\Delta$ is not explicitly assigned as a function of $\alpha$, $\\|H_{\text{else}}\\|$ and $\epsilon$ in OT06 , we numerically find the optimal $\Delta$ values for their constructions (marked as “[OT06]”). (a) shows the scaling of the gap $\Delta$ as a function of error tolerance $\epsilon$. (b) shows the gap $\Delta$ as a function of the desired coupling $\alpha$. For the meanings of the labels in the legend, see Fig. 2. The fixed parameters in each subplots are: (a) $\\|H_{\text{else}}\\|=0$, $\alpha=1$. (b) $\epsilon=0.01$, $\\|H_{\text{else}}\\|=0$. Note that our constructions have improved the $\Delta$ scaling for the ranges of $\alpha$ and $\epsilon$ considered. $r$0$f(r)$0.1(2,4)(6,0) $1-2r$1$\frac{1}{2}$0.1(4,0)(6,4) $4r-3$0.1(4.666,0)(6,4) $6r-5$$1$$\frac{2}{3}$0.03(4.6666666,1.3333333)(4.666,2) 0.03(4.6666666,1.3333333)(2,1.333) $-\frac{1}{3}$ \| $r$0$f(r)$0.1(2,4)(6,0) $1-2r$1$\frac{1}{2}$0.1(4.666,0)(6,4) $6r-5$$1$$\frac{3}{4}$0.03(5,1)(5,2) 0.03(5,1)(2,1) $-\frac{1}{2}$ ---\|--- (a) \| (b) Figure 5: The function $f(r)$ shows the dominant power of $\Delta$ in the error terms in the perturbative expansion. (a) When the error term $E_{4}$ in Eq. 51, which contributes to the $4r-3$ component of $f(r)$ in Eq. 53, is not compensated in the original construction by Oliveira and Terhal, the dominant power of $\Delta$ in the error term $f(r)$ takes minimum value of $-1/3$, indicating that $\Delta=\Theta(\epsilon^{-3})$ is required. (b) In the improved construction, $\min_{r\in(1/2,1)}f(r)=-1/2$ indicating that $\Delta=\Theta(\epsilon^{-2})$. One natural question to ask next is whether it is possible to further improve the gap scaling as a function of $\epsilon$. This turns out to be difficult. Observe that the $6r-5$ component of $f(r)$ in Eq. 43 comes from $E_{6}$ and $E_{7}$ in Eq. 38. In $E_{7}$, the $\Theta(\Delta^{6r-5})$ contribution is attributed to the term $\frac{1}{\Delta}({\kappa}{A}+{\lambda}{B})^{2}$ in $V_{1}$ of Eq. 31, which is intended for compensating the $2^{\text{nd}}$ order perturbative term and therefore cannot be removed from the construction. We now let $r=3/4$ be a fixed constant and derive the lower bound for $\Delta$ such that for given $\alpha$, $H_{\text{else}}$ and $\epsilon$, the spectral error between the effective Hamiltonian $H_{\text{eff}}=H_{\text{targ}}\otimes\|0\rangle\langle{0}\|_{w}$ and $\tilde{H}_{-}$ is within $\epsilon$. This amounts to satisfying the condition of Theorem I.1: $\\|\Sigma_{-}(z)-H_{\text{eff}}\\|\leq\epsilon.$ (44) Define the total error $E=\Sigma_{-}(z)-H_{\text{eff}}=E_{1}+\cdots+E_{7}$. For convenience we also define $\eta=\\|H_{\text{else}}\\|+2^{2/3}\alpha^{4/3}$ and $\xi=2^{-1/3}\alpha^{1/3}+2^{1/3}\alpha^{2/3}$. Then $\begin{array}[]{ccl}\\|E_{7}\\|&\leq&\displaystyle\frac{2^{4/3}\alpha^{2/3}\Delta^{3/2}}{\Delta-\max{z}}\sum_{k=2}^{\infty}\frac{(\eta+\xi\Delta^{1/2})^{k}}{(\Delta-\max{z})^{k}}=\frac{2^{4/3}\alpha^{2/3}\Delta^{3/2}}{\Delta-\max{z}-(\eta+\xi\Delta^{1/2})}\left(\frac{\eta+\xi\Delta^{1/2}}{\Delta-\max{z}}\right)^{2}.\end{array}$ (45) The upper bound for $\\|E\\|$ is then found by summing over Eq. 39, 40, 41 and 45: $\begin{array}[]{ccl}\\|E\\|&\leq&\displaystyle\frac{\max{z}{\cdot}2^{4/3}\alpha^{2/3}\Delta^{1/2}}{\Delta-\max{z}}+\frac{(2\Delta-\max{z})\max{z}}{(\Delta-\max{z})^{2}}\cdot{2}^{4/3}\alpha^{3/2}\xi+\frac{2^{4/3}\alpha^{2/3}\Delta^{3/2}\eta}{(\Delta-\max{z})^{2}}\\\\[14.45377pt] &+&\displaystyle\frac{2^{4/3}\alpha^{2/3}\Delta^{3/2}}{\Delta-\max{z}-(\eta+\xi\Delta^{1/2})}\left(\frac{\eta+\xi\Delta^{1/2}}{\Delta-\max{z}}\right)^{2}.\\\\[14.45377pt] \end{array}$ (46) By rearranging the terms in Eq. 46 we arrive at a simplified expression for the upper bound presented below. Requiring the upper bound of $\\|E\\|$ to be within $\epsilon$ gives $\begin{array}[]{ccl}\\|E\\|&\leq&\displaystyle 2^{4/3}\alpha^{2/3}\frac{(\max{z}+\eta+\xi^{2})\Delta^{1/2}+\xi(\max{z}+\eta)}{\Delta-\xi\Delta^{1/2}-(\max{z}+\eta)}\leq\epsilon.\end{array}$ (47) Eq. 47 is a quadratic constraint with respect to $\Delta^{1/2}$. Solving the inequality gives the lower bound of $\Delta$ given in Eq. 32. Note here that $\Delta=\Theta(\epsilon^{-2})$, which improves over the previously assumed $\Delta=\Theta(\epsilon^{-3})$ in the literature OT06 ; KKR06 ; BDLT08 . This bound is shown in Fig. 4b as the “analytical lower bound”. Comparison between the analytical lower bound and the numerically optimized gap in Fig. 4b indicates that the lower bound is relatively tight when $\\|H_{\text{else}}\\|=0$. _Comparison with Oliveira and TerhalOT06 ._ Given operators $Q$, $R$ and $T$ acting on separate spaces $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ respectively, the 3- to 2-body construction in OT06 ; KKR06 approximates the target Hamiltonian $H_{\text{targ}}=H_{\text{else}}+Q\otimes R\otimes T$. In order to compare with their construction, however, we let $\alpha=\\|Q\\|\cdot\\|R\\|\cdot\\|T\\|$ and define $Q=\alpha^{1/3}A$, $R=\alpha^{1/3}B$ and $T=\alpha^{1/3}C$. Hence the target Hamiltonian $H_{\text{targ}}=H_{\text{else}}+\alpha A\otimes B\otimes C$ with $A$, $B$ and $C$ being unit-norm Hermitian operators. Introduce an ancilla qubit $w$ and apply the penalty Hamiltonian $H=\Delta\|1\rangle\langle{1}\|_{w}$. In the construction by Oliveira and Terhal OT06 , the perturbation $V$ is defined as $V=H_{\text{else}}\otimes\openone_{w}+{\mu}C\otimes\|1\rangle\langle{1}\|_{w}+({\kappa}A+{\lambda}B)\otimes X_{w}+V^{\prime}_{1}$ (48) where the compensation term $V^{\prime}_{1}$ is $\displaystyle V^{\prime}_{1}=\frac{1}{\Delta}({\kappa}A+{\lambda}B)^{2}-\frac{1}{\Delta^{2}}(\kappa^{2}A^{2}+\lambda^{2}B^{2}){\mu}C.$ (49) Comparing Eq. 49 with the expression for $V_{1}$ in Eq. 31, one observes that $V_{1}$ slightly improves over $V^{\prime}_{1}$ by projecting 1-local terms to $\mathcal{L}_{-}$ so that $V$ will have less contribution to $V_{+}$, which reduces the high order error terms in the perturbative expansion. However, this modification comes at a cost of requiring more 2-local terms in the perturbation $V$. From the gadget construction shown in (OT06, , Eq. 26), the equivalent choices of the coefficients ${\kappa}$, ${\lambda}$ and ${\mu}$ are ${\kappa}=-\left(\frac{\alpha}{2}\right)^{1/3}\frac{1}{\sqrt{2}}\Delta^{r},\quad{\lambda}=\left(\frac{\alpha}{2}\right)^{1/3}\frac{1}{\sqrt{2}}\Delta^{r},\quad{\mu}=-\left(\frac{\alpha}{2}\right)^{1/3}\Delta^{2-2r}$ (50) where $r=2/3$ in the constructions used in OT06 ; BDLT08 . In fact this value of $r$ is optimal for the construction in the sense that it leads to the optimal gap scaling $\Delta=\Theta(\epsilon^{-3})$. Expanding the self-energy to $3^{\text{rd}}$ order, following a similar procedure as in (36), we have $\begin{array}[]{ccl}\Sigma_{-}(z)&=&\displaystyle\bigg{[}\underbrace{H_{\text{else}}+\frac{2{\kappa}{\lambda}{\mu}}{\Delta^{2}}{A}\otimes{B}\otimes{C}}_{H_{\text{targ}}}+\underbrace{\left(\frac{1}{\Delta}+\frac{1}{z-\Delta}\right)({\kappa}{A}+{\lambda}{B})^{2}}_{E_{1}}\\\\[7.22743pt] &+&\displaystyle\underbrace{\left(\frac{1}{(z-\Delta)^{2}}-\frac{1}{\Delta^{2}}\right)({\kappa}^{2}A^{2}+{\lambda}^{2}B^{2}){\mu}{C}}_{E_{2}}+\underbrace{\frac{1}{(z-\Delta)^{2}}({\kappa}{A}+{\lambda}{B})H_{\text{else}}({\kappa}{A}+{\lambda}{B})}_{E_{3}}\\\\[7.22743pt] &+&\displaystyle\underbrace{\frac{1}{(z-\Delta)^{2}}\cdot\frac{1}{\Delta}(\kappa A+\lambda B)^{4}}_{E_{4}}-\underbrace{\frac{1}{(z-\Delta)^{2}}\cdot\frac{1}{\Delta^{2}}(\kappa^{2}A^{2}+\lambda^{2}B^{2})\mu(\kappa A+\lambda B)^{2}\otimes C}_{E_{5}}\bigg{]}\otimes\|0\rangle\langle{0}\|_{w}\\\\[7.22743pt] &+&\displaystyle\underbrace{\sum_{k=2}^{\infty}\frac{V_{-+}V_{+}^{k}V_{+-}}{(z-\Delta)^{k+1}}}_{E_{6}}.\end{array}$ (51) Similar to the derivation of Eq. 39, 40, and 41 by letting $z\mapsto\max z$, where $\max z=\|\alpha\|+\epsilon+\\|H_{\text{else}}\\|$ is the largest value of $z$ permitted by the Theorem I.1, and using the triangle inequality to bound the norm, we can bound the norm of the error terms $E_{1}$ through $E_{6}$. For example, $\\|E_{1}\\|\leq\left(\frac{1}{\Delta-\max z}-\frac{1}{\Delta}\right)2^{2}\cdot\left(\frac{\alpha}{2}\right)^{2/3}\Delta^{2r}=\Theta(\Delta^{2r-2}).$ Applying the same calculation to $E_{2},E_{3},\cdots$ we find that $\\|E_{2}\\|=\Theta(\Delta^{-1})$, $\\|E_{3}\\|=\Theta(\Delta^{2r-2})$, $\\|E_{4}\\|=\Theta(\Delta^{4r-3})$, $\\|E_{5}\\|=\Theta(\Delta^{4r-4})$. The norm of the high order terms $E_{6}$ can be bounded as $\begin{array}[]{ccl}\\|E_{6}\\|&\leq&\displaystyle\sum_{k=2}^{\infty}\frac{\\|V_{-+}\\|\cdot\\|V_{+}\\|^{k}\cdot\\|V_{+-}\\|}{(\Delta-\max(z))^{k+1}}\leq\frac{4\left(\frac{\alpha}{2}\right)^{1/3}\Delta^{2r}}{\Delta-\max(z)}\sum_{k=2}^{\infty}\left(\frac{\rho}{\Delta-\max(z)}\right)^{k}\\\\[7.22743pt] &=&\displaystyle\frac{2^{4/3}\alpha^{2/3}\Delta^{2r}}{\Delta-\max(z)-\rho}\left(\frac{\rho}{\Delta-\max(z)}\right)^{2}=\Theta(\Delta^{2r-1+2\max\\{1-2r,2r-2\\}})=\Theta(\Delta^{\max\\{1-2r,6r-5\\}})\end{array}$ (52) where $\rho=\\|H_{\text{else}}\\|+2^{-1/3}\alpha^{1/3}\Delta^{2-2r}+2^{1/3}\alpha^{2/3}\Delta^{2r-1}$. If we again write the self energy expansion Eq. 51 as $\Sigma_{-}(z)=H_{\text{targ}}\otimes\|0\rangle\langle{0}\|_{w}+\Theta(\Delta^{f(r)}),$ the function $f(r)<0$, which determines the dominant power in $\Delta$ among $E_{1}$ through $E_{6}$, can be found as $f(r)=\max\\{1-2r,2r-2,4r-3,6r-5\\},\quad\frac{1}{2}<r<1.$ (53) Similar to the discussion after Eq. 43, the optimal scaling of $\Delta=\Theta(\epsilon^{1/f(r)})$ gives $r=\text{argmin}f(r)=2/3$, when $f(r)=-1/3$ and $\Delta=\Theta(\epsilon^{-3})$, as is shown in Fig. 5a. Note that the $4r-3$ component in $f(r)$, Eq. 53, comes from the error term $E_{4}$ in Eq. 51. The idea for improving the gadget construction comes from the observation in Fig. 5a that when we add a term in $V$ to compensate for $E_{4}$, the dominant power of $\Delta$ in the perturbation series, $f(r)$, could admit a lower minimum as shown in Fig. 5b. In the previous calculation we have shown that this is indeed the case and the minimum value of $f(r)$ becomes $-1/2$ in the improved case, indicating that $\Delta=\Theta(\epsilon^{-2})$ is sufficient for keeping the error terms $O(\epsilon)$. ## V Creating 3-body gadget from local X Summary. In general, terms in perturbative gadgets involve mixed couplings (e.g. $X_{i}Z_{j}$). Although such couplings can be realized by certain gadget constructions BL07 , physical couplings of this type are difficult to realize in an experimental setting. However, there has been significant progress towards experimentally implementing Ising models with transverse fields of the type 2006cond.mat..8253H : $H_{ZZ}=\sum_{i}\delta_{i}X_{i}+\sum_{i}h_{i}Z_{i}+\sum_{i,j}J_{ij}Z_{i}Z_{j}.$ (54) Accordingly, an interesting question is whether we can approximate 3-body terms such as $\alpha\cdot Z_{i}\otimes Z_{j}\otimes Z_{k}$ using a Hamiltonian of this form. This turns out to be possible by employing a perturbative calculation which considers terms up to $5^{\text{th}}$ order. Similar to the 3- to 2-body reduction discussed previously, we introduce an ancilla $w$ and apply the Hamiltonian $H=\Delta\|1\rangle\langle{1}\|_{w}$. We apply the perturbation $V=H_{\text{else}}+\mu(Z_{i}+Z_{j}+Z_{k})\otimes\|1\rangle\langle{1}\|_{w}+\mu\openone\otimes X_{w}+V_{\textrm{comp}}$ (55) where $\mu=\left(\alpha\Delta^{4}/6\right)^{1/5}$ and $V_{\textrm{comp}}$ is $\begin{array}[]{ccl}V_{\textrm{comp}}&=&\displaystyle\frac{\mu^{2}}{\Delta}\|0\rangle\langle{0}\|_{w}-\left(\frac{\mu^{3}}{\Delta^{2}}+7\frac{\mu^{5}}{\Delta^{4}}\right)\left(Z_{i}+Z_{j}+Z_{k}\right)\otimes\|0\rangle\langle{0}\|_{w}+\frac{\mu^{4}}{\Delta^{3}}\left(3\openone+2Z_{i}Z_{j}+2Z_{i}Z_{k}+2Z_{j}Z_{k}\right).\end{array}$ (56) To illustrate the basic idea of the $5^{\text{th}}$ order gadget, define subspaces $\mathcal{L}_{-}$ and $\mathcal{L}_{+}$ in the usual way and define $P_{-}$ and $P_{+}$ as projectors into these respective subspaces. Then the second term in Eq. 55 with $\otimes\|1\rangle\langle{1}\|_{w}$ contributes a linear combination $\mu Z_{i}+\mu Z_{j}+\mu Z_{k}$ to $V_{+}=P_{+}VP_{+}$. The third term in Eq. 55 induces a transition between $\mathcal{L}_{-}$ and $\mathcal{L}_{+}$ yet since it operates trivially on qubits 1-3, it only contributes a constant $\mu$ to the projections $V_{-+}=P_{-}VP_{+}$ and $V_{+-}=P_{+}VP_{-}$. In the perturbative expansion, the $5^{\text{th}}$ order contains a term $\frac{V_{-+}V_{+}V_{+}V_{+}V_{+-}}{(z-\Delta)^{4}}=\frac{\mu^{5}(Z_{i}+Z_{j}+Z_{k})^{3}}{(z-\Delta)^{4}}$ (57) due to the combined the contribution of the second and third term in Eq. 55. This yields a term proportional to $\alpha\cdot Z_{i}\otimes Z_{j}\otimes Z_{k}$ along with some 2-local error terms. These error terms, combined with the unwanted terms that arise at $1^{\text{st}}$ through $4^{\text{th}}$ order perturbation, are compensated by $V_{\text{comp}}$. Note that terms at 6${}^{\textrm{th}}$ order and higher are $\Theta(\Delta^{-1/5})$. This means in order to satisfy the gadget theorem of Kempe _et al._ ((KKR06, , Theorem 3), or Theorem I.1) $\Delta$ needs to be $\Theta(\epsilon^{-5})$. This is the first perturbative gadget that simulates a 3-body target Hamiltonian using the Hamiltonian Eq. 54. By rotating the ancilla space, subdivision gadgets can also be implemented using this Hamiltonian: in the $X$ basis, $Z$ terms will induce a transition between the two energy levels of $X$. Therefore $Z_{i}Z_{j}$ coupling could be used for a perturbation of the form in Eq. 4 in the rotated basis. In principle using the transverse Ising model in Eq. 54, one can reduce some diagonal $k$-body Hamiltonian to 3-body by iteratively applying the subdivision gadget and then to 2-body by using the 3-body reduction gadget. Analysis. Similar to the gadgets we have presented so far, we introduce an ancilla spin $w$. Applying an energy gap $\Delta$ on the ancilla spin gives the unperturbed Hamiltonian $H=\Delta\|1\rangle\langle{1}\|_{w}$. We then perturb the Hamiltonian $H$ using a perturbation $V$ described in (55). Using the same definitions of subspaces $\mathcal{L}_{+}$ and $\mathcal{L}_{-}$ as the previous 3-body gadget, the projections of $V$ into these subspaces can be written as $\begin{array}[]{ccl}V_{+}&=&\displaystyle\bigg{\\{}H_{\text{else}}+\mu(Z_{1}+Z_{2}+Z_{3})+\frac{{\mu}^{4}}{\Delta^{3}}\big{[}3{\openone}+2(Z_{1}Z_{2}+Z_{1}Z_{3}+Z_{2}Z_{3})\big{]}\bigg{\\}}\otimes\|1\rangle\langle{1}\|_{w}\\\\[7.22743pt] V_{-}&=&\displaystyle\bigg{\\{}H_{\text{else}}+\frac{{\mu}^{2}}{\Delta}{\openone}-\frac{{\mu}^{3}}{\Delta^{2}}(Z_{1}+Z_{2}+Z_{3}){\openone}+\frac{{\mu}^{4}}{\Delta^{3}}\big{[}3\openone+2(Z_{1}Z_{2}+Z_{1}Z_{3}+Z_{2}Z_{3})\big{]}\\\\[7.22743pt] &&\displaystyle-\frac{7{\mu}^{5}}{\Delta^{4}}\big{(}Z_{1}+Z_{2}+Z_{3}\big{)}\bigg{\\}}\otimes\|0\rangle\langle{0}\|_{w}\\\\[7.22743pt] V_{-+}&=&{\mu}{\openone}\otimes\|0\rangle\langle{1}\|_{w},\quad V_{+-}={\mu}{\openone}\otimes\|1\rangle\langle{0}\|_{w}.\\\\[7.22743pt] \end{array}$ (58) The low-lying spectrum of $\tilde{H}$ is approximated by the self energy expansion $\Sigma_{-}(z)$ below with $z\in[-\max{z},\max{z}]$ where $\max{z}=\\|H_{\text{else}}\\|+\|\alpha\|+\epsilon$. With the choice of $\mu$ above the expression of $V_{+}$ in Eq. 58 can be written as $V_{+}=\left(H_{\text{else}}+{\mu}(Z_{1}+Z_{2}+Z_{3})+O(\Delta^{1/5})\right)\otimes\|1\rangle\langle{1}\|_{w}.$ (59) Because we are looking for the $5^{\text{th}}$ order term in the perturbation expansion that gives a term proportional to $Z_{1}Z_{2}Z_{3}$, expand the self energy in Eq. 3 up to $5^{\text{th}}$ order: $\begin{array}[]{ccl}\Sigma_{-}(z)&=&\displaystyle V_{-}\otimes\|0\rangle\langle{0}\|_{w}+\frac{V_{-+}V_{+-}}{z-\Delta}\otimes\|0\rangle\langle{0}\|_{w}+\frac{V_{-+}V_{+}V_{+-}}{(z-\Delta)^{2}}\otimes\|0\rangle\langle{0}\|_{w}+\frac{V_{-+}V_{+}V_{+}V_{+-}}{(z-\Delta)^{3}}\otimes\|0\rangle\langle{0}\|_{w}\\\\[7.22743pt] &+&\displaystyle\frac{V_{-+}V_{+}V_{+}V_{+}V_{+-}}{(z-\Delta)^{4}}\otimes\|0\rangle\langle{0}\|_{w}+\sum_{k=4}^{\infty}\frac{V_{-+}V_{+}^{k}V_{+-}}{(z-\Delta)^{k+1}}\otimes\|0\rangle\langle{0}\|_{w}.\end{array}$ (60) Using this simplification as well as the expressions for $V_{-}$, $V_{-+}$ and $V_{+-}$ in Eq. 58, the self energy expansion Eq. 60 up to $5^{\text{th}}$ order becomes $\begin{array}[]{ccl}\Sigma_{-}(z)&=&\displaystyle\underbrace{\left(H_{\text{else}}+\frac{6\mu^{5}}{\Delta^{4}}Z_{1}Z_{2}Z_{3}\right)\otimes\|0\rangle\langle{0}\|_{w}}_{\text{$H_{\text{eff}}$}}+\underbrace{\left(\frac{1}{\Delta}+\frac{1}{z-\Delta}\right){\mu}^{2}{\openone}\otimes\|0\rangle\langle{0}\|_{w}}_{\text{$E_{1}$}}\\\\[7.22743pt] &+&\displaystyle\underbrace{\left(\frac{1}{(z-\Delta)^{2}}-\frac{1}{\Delta^{2}}\right)\mu^{3}(Z_{1}+Z_{2}+Z_{3})\otimes\|0\rangle\langle{0}\|_{w}}_{\text{$E_{2}$}}+\underbrace{\left(\frac{1}{\Delta^{3}}+\frac{1}{(z-\Delta)^{3}}\right)\cdot\mu^{4}\cdot(Z_{1}+Z_{2}+Z_{3})^{2}\otimes\|0\rangle\langle{0}\|_{w}}_{\text{$E_{3}$}}\\\\[7.22743pt] &+&\displaystyle\underbrace{\left(\frac{1}{(z-\Delta)^{4}}-\frac{1}{\Delta^{4}}\right)7{\mu}^{5}(Z_{1}+Z_{2}+Z_{3})\otimes\|0\rangle\langle{0}\|_{w}}_{\text{$E_{4}$}}+\underbrace{\frac{{\mu}^{2}}{(z-\Delta)^{2}}\cdot\frac{{\mu}^{4}}{\Delta^{3}}(Z_{1}+Z_{2}+Z_{3})^{2}\otimes\|0\rangle\langle{0}\|_{w}}_{\text{$E_{6}$}}\\\\[7.22743pt] &+&O(\Delta^{-2/5})+O(\\|H_{\text{else}}\\|\Delta^{-2/5})+O(\\|H_{\text{else}}\\|^{2}\Delta^{-7/5})+O(\\|H_{\text{else}}\\|^{3}\Delta^{-12/5})+\underbrace{\sum_{k=4}^{\infty}\frac{V_{-+}V_{+}^{k}V_{+-}}{(z-\Delta)^{k+1}}\otimes\|0\rangle\langle{0}\|_{w}}_{\text{$E_{7}$}}.\\\\[7.22743pt] \end{array}$ (61) Similar to what we have done in the previous sections, the norm of the error terms $E_{1}$ through $E_{7}$ can be bounded from above by letting $z\mapsto\max{z}$. Then we find that $\begin{array}[]{ccl}\\|\Sigma_{-}(z)-H_{\text{targ}}\otimes\|0\rangle\langle{0}\|_{w}\\|&\leq&\Theta(\Delta^{-1/5})\end{array}$ (62) if we only consider the dominant dependence on $\Delta$ and regard $\\|H_{\text{else}}\\|$ as a given constant. To guarantee that $\\|\Sigma_{-}(z)-H_{\text{targ}}\otimes\|0\rangle\langle{0}\|_{w}\\|\leq\epsilon$, we let the right hand side of Eq. 62 to be $\leq\epsilon$, which translates to $\Delta=\Theta(\epsilon^{-5})$. This $\Theta(\epsilon^{-5})$ scaling is numerically illustrated (Fig. 6a). Although in principle the $5^{\text{th}}$ order gadget can be implemented on a Hamiltonian of form Eq. 54, for a small range of $\alpha$, the minimum $\Delta$ needed is already large (Fig. 6b), rendering it challenging to demonstrate the gadget experimentally with current resources. However, this is the only currently known gadget realizable with a transverse Ising model that is able to address the case where $H_{\text{else}}$ is not necessarily diagonal. (a)(b) Figure 6: (a) The scaling of minimum $\Delta$ needed to ensure $\\|\Sigma_{-}(z)-H_{\text{eff}}\\|\leq\epsilon$ as a function of $\epsilon^{-1}$. Here we choose $\\|H_{\text{else}}\\|=0$, $\alpha=0.1$ and $\epsilon$ ranging from $10^{-0.7}$ to $10^{-2.3}$. The values of minimum $\Delta$ are numerically optimized footnote:num_op . The slope of the line at large $\epsilon^{-1}$ is $4.97\approx{5}$, which provides evidence that with the assignments of ${\mu}=(\alpha\Delta^{4}/6)^{1/5}$, the optimal scaling of $\Delta$ is $\Theta(\epsilon^{-5})$. (b) The numerically optimized footnote:num_op gap versus the desired coupling $\alpha$ in the target Hamiltonian. Here $\epsilon=0.01$ and $\\|H_{\text{else}}\\|=0$. ## VI YY gadget Summary. The gadgets which we have presented so far are intended to reduce the locality of the target Hamiltonian. Here we present another type of gadget, called “creation” gadgets BL07 , which simulate the type of effective couplings that are not present in the gadget Hamiltonian. Many creation gadgets proposed so far are modifications of existing reduction gadgets. For example, the ZZXX gadget in BL07 , which is intended to simulate $Z_{i}X_{j}$ terms using Hamiltonians of the form $\begin{array}[]{ccl}H_{ZZXX}&=&\displaystyle\sum_{i}\Delta_{i}X_{i}+\sum_{i}h_{i}Z_{i}+\sum_{i,j}J_{ij}Z_{i}Z_{j}+\sum_{i,j}K_{ij}X_{i}X_{j},\end{array}$ (63) is essentially a 3- to 2-body gadget with the target term $A\otimes B\otimes C$ being such that the operators $A$, $B$ and $C$ are $X$, $Z$ and identity respectively. Therefore the analyses on 3- to 2- body reduction gadgets that we have presented for finding the lower bound for the gap $\Delta$ are also applicable to this ZZXX creation gadget. Note that YY terms can be easily realized via bases rotation if single-qubit Y terms are present in the Hamiltonian in Eq. 63. Otherwise it is not _a priori_ clear how to realize YY terms using $H_{ZZXX}$ in Eq. 63. We will now present the first YY gadget which starts with a universal Hamiltonian of the form Eq. 63 and simulates the target Hamiltonian $H_{\text{targ}}=H_{\text{else}}+\alpha Y_{i}Y_{j}$. The basic idea is to use the identity $X_{i}Z_{i}=\iota Y_{i}$ where $\iota=\sqrt{-1}$ and induce a term of the form $X_{i}Z_{i}Z_{j}X_{j}=Y_{i}Y_{j}$ at the $4^{\text{th}}$ order. Introduce ancilla qubit $w$ and apply a penalty $H=\Delta\|1\rangle\langle{1}\|_{w}$. With a perturbation $V$ we could perform the same perturbative expansion as previously. Given that the $4^{\text{th}}$ order perturbation is $V_{-+}V_{+}V_{+}V_{+-}$ up to a scaling constant. we could let single $X_{i}$ and $X_{j}$ be coupled with $X_{w}$, which causes both $X_{i}$ and $X_{j}$ to appear in $V_{-+}$ and $V_{+-}$. Furthermore, we couple single $Z_{i}$ and $Z_{j}$ terms with $Z_{w}$. Then $\frac{1}{2}(\openone+Z_{w})$ projects single $Z_{i}$ and $Z_{j}$ onto the $+$ subspace and causes them to appear in $V_{+}$. For $H_{\text{targ}}=H_{\text{else}}+\alpha Y_{1}Y_{2}$, the full expressions for the gadget Hamiltonian is the following: the penalty Hamiltonian $H=\Delta\|1\rangle\langle{1}\|_{w}$ acts on the ancilla qubit. The perturbation $V=V_{0}+V_{1}+V_{2}$ where $V_{0}$, $V_{1}$, and $V_{2}$ are defined as $\begin{array}[]{ccl}V_{0}&=&\displaystyle H_{\text{else}}+\mu({Z_{1}+Z_{2}})\otimes{\|1\rangle\langle{1}\|_{w}}+\mu(X_{1}-\text{sgn}(\alpha)X_{2})\otimes X_{w}\\\\[3.61371pt] V_{1}&=&\displaystyle\frac{2\mu^{2}}{\Delta}(\openone\otimes\|0\rangle\langle{0}\|_{w}+X_{1}X_{2})\\\\[3.61371pt] V_{2}&=&\displaystyle-\frac{2\mu^{4}}{\Delta^{3}}Z_{1}Z_{2}.\end{array}$ (64) with $\mu=(\|\alpha\|\Delta^{3}/4)^{1/4}$. For a specified error tolerance $\epsilon$, we have constructed a YY gadget Hamiltonian of gap scaling $\Delta=O(\epsilon^{-4})$ and the low-lying spectrum of the gadget Hamiltonian captures the spectrum of $H_{\text{targ}}\otimes\|0\rangle\langle{0}\|_{w}$ up to error $\epsilon$. The YY gadget implies that a wider class of Hamiltonians such as $\begin{array}[]{ccl}H_{ZZYY}&=&\displaystyle\sum_{i}h_{i}X_{i}+\sum_{i}\Delta_{i}Z_{i}+\sum_{i,j}J_{ij}Z_{i}Z_{j}+\sum_{i,j}K_{ij}Y_{i}Y_{j}\end{array}$ (65) and $\begin{array}[]{ccl}H_{XXYY}&=&\displaystyle\sum_{i}h_{i}X_{i}+\sum_{i}\Delta_{i}Z_{i}+\sum_{i,j}J_{ij}X_{i}X_{j}+\sum_{i,j}K_{ij}Y_{i}Y_{j}\end{array}$ (66) can be simulated using the Hamiltonian of the form in Eq. 63. Therefore using the Hamiltonian in Eq. 63 one can in principle simulate any finite-norm real valued Hamiltonian on qubits. Although by the QMA-completeness of $H_{ZZXX}$ one could already simulate such Hamiltonian via suitable embedding, our YY gadget provides a more direct alternative for the simulation. Analysis. The results in BL07 shows that Hamiltonians of the form in Eq. 63 supports universal adiabatic quantum computation and finding the ground state of such a Hamiltonian is QMA-complete. This form of Hamiltonian is also interesting because of its relevance to experimental implementation 2006cond.mat..8253H . Here we show that with a Hamiltonian of the form in Eq. 63 we could simulate a target Hamiltonian $H_{\text{targ}}=H_{\text{else}}+\alpha Y_{1}Y_{2}$. Introduce an ancilla $w$ and define the penalty Hamiltonian as $H=\Delta\|1\rangle\langle{1}\|_{w}$. Let the perturbation $V=V_{0}+V_{1}+V_{2}$ be $\begin{array}[]{ccl}V_{0}&=&H_{\text{else}}+\kappa({Z_{1}+Z_{2}})\otimes{\|1\rangle\langle{1}\|_{w}}+\kappa(X_{1}-\text{sgn}(\alpha)X_{2})\otimes X_{w}\\\\[3.61371pt] V_{1}&=&2\kappa^{2}\Delta^{-1}[\|0\rangle\langle{0}\|_{w}-\text{sgn}(\alpha)X_{1}X_{2}]\\\\[3.61371pt] V_{2}&=&-4\kappa^{4}\Delta^{-3}Z_{1}Z_{2}.\end{array}$ (67) Then the gadget Hamiltonian $\tilde{H}=H+V$ is of the form in Eq. 63. Here we choose the parameter $\kappa=(\|\alpha\|\Delta^{3}/4)^{1/4}$. In order to show that the low lying spectrum of $\tilde{H}$ captures that of the target Hamiltonian, define $\mathcal{L}_{-}=\text{span}\\{\|\psi\rangle\text{ such that }\tilde{H}\|\psi\rangle=\lambda\|\psi\rangle,\lambda<\Delta/2\\}$ as the low energy subspace of $\tilde{H}$ and $\mathcal{L}_{+}=\openone-\mathcal{L}_{-}$. Define $\Pi_{-}$ and $\Pi_{+}$ as the projectors onto $\mathcal{L}_{-}$ and $\mathcal{L}_{+}$ respectively. With these notations in place, here we show that the spectrum of $\tilde{H}_{-}=\Pi_{-}\tilde{H}\Pi_{-}$ approximates the spectrum of $H_{\text{targ}}\otimes\|0\rangle\langle{0}\|_{w}$ with error $\epsilon$. To begin with, the projections of $V$ into the subspaces $\mathcal{L}_{-}$ and $\mathcal{L}_{+}$ can be written as $\begin{array}[]{ccl}V_{-}&=&\displaystyle\bigg{(}H_{\text{else}}+\underbrace{\frac{\kappa^{2}}{\Delta}(X_{1}-\text{sgn}(\alpha)X_{2})^{2}}_{(a)}\underbrace{-\frac{4\kappa^{4}}{\Delta^{3}}Z_{1}Z_{2}}_{(b)}\bigg{)}\otimes\|0\rangle\langle{0}\|_{w}\\\\[3.61371pt] V_{+}&=&\displaystyle\left(H_{\text{else}}+\kappa(Z_{1}+Z_{2})-\frac{2\kappa^{2}}{\Delta}\text{sgn}(\alpha)X_{1}X_{2}-\frac{4\kappa^{4}}{\Delta^{3}}Z_{1}Z_{2}\right)\otimes\|1\rangle\langle{1}\|_{w}\\\\[3.61371pt] V_{-+}&=&\kappa(X_{1}-\text{sgn}(\alpha)X_{2})\otimes\|0\rangle\langle{1}\|_{w}\\\\[3.61371pt] V_{+-}&=&\kappa(X_{1}-\text{sgn}(\alpha)X_{2})\otimes\|1\rangle\langle{0}\|_{w}\end{array}$ (68) Given the penalty Hamiltonian $H$, we have the operator valued resolvent $G(z)=(z\openone-H)^{-1}$ that satisfies $G_{+}(z)=\Pi_{+}G(z)\Pi_{+}=(z-\Delta)^{-1}\|1\rangle\langle{1}\|_{w}$. Then the low lying sector of the gadget Hamiltonian $\tilde{H}$ can be approximated by the perturbative expansion Eq. 3. For our purposes we will consider terms up to the $4^{\text{th}}$ order: $\Sigma_{-}(z)=V_{-}+\frac{1}{z-\Delta}V_{-+}V_{+-}+\frac{1}{(z-\Delta)^{2}}V_{-+}V_{+}V_{+-}+\frac{1}{(z-\Delta)^{3}}V_{-+}V_{+}V_{+}V_{+-}+\sum_{k=3}^{\infty}\frac{V_{-+}V_{+}^{k}V_{+-}}{(z-\Delta)^{k+1}}.$ (69) Now we explain the perturbative terms that arise at each order. The $1^{\text{st}}$ order is the same as $V_{-}$ in Eq. 68. The $2^{\text{nd}}$ order term gives $\frac{1}{z-\Delta}V_{-+}V_{+-}=\underbrace{\frac{1}{z-\Delta}\cdot\kappa^{2}(X_{1}-\text{agn}(\alpha)X_{2})^{2}}_{(c)}\otimes\|0\rangle\langle{0}\|_{w}.$ (70) At the $3^{\text{rd}}$ order, we have $\begin{array}[]{ccl}\displaystyle\frac{1}{(z-\Delta)^{2}}V_{-+}V_{+}V_{+-}&=&\displaystyle\bigg{(}\frac{1}{(z-\Delta)^{2}}\cdot\kappa^{2}(X_{1}-\text{agn}(\alpha)X_{2})H_{\text{else}}(X_{1}-\text{sgn}(\alpha)X_{2})\\\\[3.61371pt] &+&\displaystyle\underbrace{\frac{1}{(z-\Delta)^{2}}\frac{4\kappa^{4}}{\Delta}(X_{1}X_{2}-\text{sgn}(\alpha)\openone)}_{(d)}\bigg{)}\otimes\|0\rangle\langle{0}\|_{w}+O(\Delta^{-1/4}).\end{array}$ (71) The $4^{\text{th}}$ order contains the desired YY term: $\begin{array}[]{ccl}\displaystyle\frac{1}{(z-\Delta)^{3}}V_{-+}V_{+}V_{+}V_{+-}&=&\displaystyle\bigg{(}\underbrace{\frac{1}{(z-\Delta)^{3}}\cdot 2\kappa^{4}(X_{1}-\text{sgn}(\alpha)X_{2})^{2}}_{(e)}-\underbrace{\frac{1}{(z-\Delta)^{3}}4\kappa^{4}Z_{1}Z_{2}}_{(f)}\\\\[3.61371pt] &+&\displaystyle\frac{4\kappa^{4}\text{sgn}(\alpha)}{(z-\Delta)^{3}}Y_{1}Y_{2}\bigg{)}\otimes\|0\rangle\langle{0}\|_{w}+O(\\|H_{\text{else}}\\|\cdot\Delta^{-3/4})+O(\\|H_{\text{else}}\\|^{2}\cdot\Delta^{-1/2})\end{array}$ (72) Note that with the choice of $\kappa=(\|\alpha\|\Delta^{3}/4)^{1/4}$, all terms of $5^{\text{th}}$ order and higher are of norm $O(\Delta^{-1/4})$. In the $1^{\text{st}}$ order through $4^{\text{th}}$ order perturbations the unwanted terms are labelled as $(a)$ through $(f)$ in Eqs. 68, 70, 71, and 72. Note how they compensate in pairs: the sum of $(a)$ and $(c)$ is $O(\Delta^{-1/4})$. The same holds for $(d)$ and $(e)$, $(b)$ and $(f)$. Then the self energy is then $\Sigma_{-}(z)=(H_{\text{else}}+\alpha Y_{1}Y_{2})\otimes\|0\rangle\langle{0}\|_{w}+O(\Delta^{-1/4}).$ (73) Let $\Delta=\Theta(\epsilon^{-4})$, then by the Gadget Theorem (I.1), the low- lying sector of the gadget Hamiltonian $\tilde{H}_{-}$ captures the spectrum of $H_{\text{targ}}\otimes\|0\rangle\langle{0}\|_{w}$ up to error $\epsilon$. The fact that the gadget relies on $4^{\text{th}}$ order perturbation renders the gap scaling relatively larger than it is in the case of subdivision or 3- to 2-body reduction gadgets. However, this does not diminish its usefulness in various applications. ## Conclusion We have presented improved constructions for the most commonly used gadgets, which in turn implies a reduction in the resources for the many works which employ these current constructions. We presented the first comparison between the known gadget constructions and the first numerical optimizations of gadget parameters. Our analytical results are found to agree with the optimised solutions. The introduction of our gadget which simulates YY-interactions opens many prospects for universal adiabatic quantum computation, particularly the simulation of physics feasible on currently realizable Hamiltonians. ## Acknowledgements We thank Andrew Landahl for helpful comments. JDB and YC completed parts of this study while visiting the Institute for Quantum Computing at the University of Waterloo. RB was supported by the United States Department of Defense. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the official policies, either expressed or implied, of the U.S. Government. JDB completed parts of this study while visiting the Qatar Energy and Environment Research Institute and would like to acknowledge the Foundational Questions Institute (under grant FQXi-RFP3-1322) for financial support. ## Appendix A Parallel 3- to 2-body gadget Summary. In Sec. III we have shown that by using parallel subdivision gadgets iteratively, one can reduce a $k$-body target term to $3$-body. We now turn our attention to considering $H_{\text{targ}}=H_{\text{else}}+\sum_{i=1}^{m}\alpha_{i}A_{i}\otimes B_{i}\otimes C_{i}$, which is a sum of $m$ 3-body terms. A straightforward approach to the reduction is to deal with the 3-body terms in series _i.e._ one at a time: apply a 3-body gadget on one term, and include the entire gadget in the $H_{\text{else}}$ of the target Hamiltonian in reducing the next 3-body term. In this construction, $\Delta$ scales exponentially as a function of $m$. In order to avoid that overhead, we apply all gadgets in parallel, which means introducing $m$ ancilla spins, one for each 3-body term and applying the same $\Delta$ onto it. This poses additional challenges as the operator valued resolvent $G(z)$ now has multiple poles. Enumerating high order terms in the perturbation series requires consideration of the combinatorial properties of the bit flipping processes (Fig. 7). If we apply the current construction OT06 ; BDLT08 of 3-body gadgets in parallel, which requires $\Delta=\Theta(\epsilon^{-3})$, it can be shown BDLT08 that the cross-gadget contribution is $O(\epsilon)$. However, if we apply our improved construction of the 3- to 2-body gadget in parallel, the perturbation expansion will contain $\Theta(1)$ cross-gadget terms that are dependent on the commutation relations between $A_{i}$, $B_{i}$ and $A_{j}$, $B_{j}$. Compensation terms are designed to ensure that these error terms are suppressed in the perturbative expansion. With our improved parallel 3-body construction, $\Delta=\Theta({\epsilon^{-2}}\text{poly}(m))$ is sufficient. The combination of parallel subdivision with the parallel 3- to 2-body reduction allows us to reduce an arbitrary $k$-body target Hamiltonian $H_{\text{targ}}=H_{\text{else}}+\alpha\sigma_{1}\sigma_{2}\cdots\sigma_{k}$ to 2-body BDLT08 . In this paper we have improved both parallel 2-body and 3- to 2-body gadgets. When numerically optimized at each iteration, our construction requires a smaller gap than the original construction BDLT08 for the range of $k$ concerned. Analysis. In Sec. III we have shown that with subdivision gadgets one can reduce a $k$-body interaction term down to 3-body. To complete the discussion on reducing a $k$-body term to $2$-body, now we deal with reducing a 3-body target Hamiltonian of form $H_{\text{targ}}=H_{\text{else}}+\sum_{i=1}^{m}\alpha_{i}{A_{i}}\otimes{B_{i}}\otimes{C_{i}}$ where $H_{\text{else}}$ is a finite-norm Hamiltonian and all of $A_{i}$, $B_{i}$, $C_{i}$ are single-qubit Pauli operators acting on one of the $n$ qubits that $H_{\text{targ}}$ acts on. Here without loss of generality, we assume $A_{i}$, $B_{i}$ and $C_{i}$ are single-qubit Pauli operators as our construction depends on the commutation relationships among these operators. The Pauli operator assumption ensures that the commutative relationship can be determined efficiently a priori. We label the $n$ qubits by integers from 1 to $n$. We assume that in each 3-body term of the target Hamiltonian, ${A_{i}}$, ${B_{i}}$ and ${C_{i}}$ act on three different qubits whose labels are in increasing order i.e. if we label the qubits with integers from 1 to $n$, ${A_{i}}$ acts on qubit $a_{i}$, ${B_{i}}$ acts on $b_{i}$, ${C_{i}}$ on $c_{i}$, we assume that $1\leq a_{i}<b_{i}<c_{i}\leq n$ must hold for all values of $i$ from 1 to $m$. One important feature of this gadget is that the gap $\Delta$ scales as $\Theta(\epsilon^{-2})$ instead of the common $\Theta(\epsilon^{-3})$ scaling assumed by the other 3-body constructions in the literature KKR06 ; OT06 ; BDLT08 . To reduce the $H_{\text{targ}}$ to 2-body, introduce $m$ qubits labelled as $u_{1}$, $u_{2}$, $\cdots$, $u_{m}$ and apply an energy penalty $\Delta$ onto the excited subspace of each qubit, as in the case of parallel subdivision gadgets presented previously. Then we have $H=\sum_{i=1}^{m}\Delta\|1\rangle\langle{1}\|_{u_{i}}=\sum_{x\in\\{0,1\\}^{m}}h(x)\Delta\|x\rangle\langle x\|.$ (74) where $h(x)$ is the Hamming weight of the $m$-bit string $x$. In this new construction the perturbation $V$ is defined as $\begin{array}[]{ccl}V&=&\displaystyle H_{\text{else}}+\sum_{i=1}^{m}{\mu_{i}}{C_{i}}\otimes\|1\rangle\langle{1}\|_{u_{i}}+\sum_{i=1}^{m}({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})\otimes X_{u_{i}}+V_{1}+V_{2}+V_{3}\end{array}$ (75) where $V_{1}$ is defined as $V_{1}=\frac{1}{\Delta}\sum_{i=1}^{m}({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})^{2}-\frac{1}{\Delta^{2}}\sum_{i=1}^{m}({\kappa_{i}^{2}}+{\lambda_{i}^{2}}){\mu_{i}}{C_{i}}$ (76) and $V_{2}$ is defined as $V_{2}=-\frac{1}{\Delta^{3}}\sum_{i=1}^{m}({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})^{4}.$ (77) $V_{3}$ will be explained later. Following the discussion in Sec. IV, the coefficients ${\kappa_{i}}$, ${\lambda_{i}}$ and ${\mu_{i}}$ are defined as ${\kappa_{i}}=\text{sgn}(\alpha_{i})\left(\frac{\|\alpha_{i}\|}{2}\right)^{\frac{1}{3}}\Delta^{\frac{3}{4}},\quad{\lambda_{i}}=\left(\frac{\|\alpha_{i}\|}{2}\right)^{\frac{1}{3}}\Delta^{\frac{3}{4}},\quad{\mu_{i}}=\left(\frac{\|\alpha_{i}\|}{2}\right)^{\frac{1}{3}}\Delta^{\frac{1}{2}}.$ (78) However, as we will show in detail later in this section, a close examination of the perturbation expansion based on the $V$ in Eq. 75 shows that with assignments of ${\kappa_{i}}$, ${\lambda_{i}}$ and ${\mu_{i}}$ in Eq. 78 if $V$ has only $V_{1}$ and $V_{2}$ as compensation terms, the cross-gadget contribution in the expansion causes $\Theta(1)$ error terms to arise. In order to compensate for the $\Theta(1)$ error terms, we introduce the compensation $V_{3}=\sum_{i=1}^{m}\sum_{j=1,j\neq i}^{m}\bar{V}_{ij}$ into $V$ and $\bar{V}_{ij}$ is the compensation term for cross-gadget contribution footnote:cross . Before presenting the detailed form of $\bar{V}_{ij}$, let $s_{1}^{(i,j)}=s_{11}^{(i,j)}+s_{12}^{(i,j)}$ where $s_{11}^{(i,j)}=\left\\{\begin{array}[]{cl}1&\text{if }\left\\{\begin{tabular}[]{c}$[{A_{i}},{A_{j}}]\neq 0$\\\ $[{B_{i}},{B_{j}}]=0$\end{tabular}\right.\text{or }\left\\{\begin{tabular}[]{c}$[{B_{i}},{B_{j}}]\neq 0$\\\ $[{A_{i}},{A_{j}}]=0$\end{tabular}\right.\\\ 0&\text{otherwise}\end{array}\right.$ (79) $s_{12}^{(i,j)}=\left\\{\begin{array}[]{cl}1&\text{if $[{A_{i}},{B_{j}}]\neq 0$ or $[{B_{i}},{A_{j}}]\neq 0$}\\\\[7.22743pt] 0&\text{otherwise}\end{array}\right.$ (80) and further define $s_{2}^{(i,j)}$ as $s_{2}^{(i,j)}=\left\\{\begin{array}[]{cl}1&\text{if $[{A_{i}},{A_{j}}]\neq 0$ and $[{B_{i}},{B_{j}}]\neq 0$}\\\\[7.22743pt] 0&\text{otherwise.}\end{array}\right.$ (81) Then we define $\bar{V}_{ij}$ as $\begin{array}[]{ccl}\bar{V}_{ij}&=&\displaystyle- s_{1}^{(i,j)}\cdot\frac{1}{\Delta^{3}}({\kappa_{i}}{\kappa_{j}})^{2}{\openone}-s_{2}^{(i,j)}\bigg{(}\frac{2}{\Delta^{3}}({\kappa_{i}}{\kappa_{j}})^{2}{\openone}-\frac{2}{\Delta^{3}}{\kappa_{i}}{\kappa_{j}}{\lambda_{i}}{\lambda_{j}}{A_{i}}{A_{j}}{B_{i}}{B_{j}}\bigg{)}\end{array}$ (82) where $s_{1}^{(i,j)}$ and $s_{2}^{(i,j)}$ are coefficients that depend on the commuting relations between the operators in the $i$-th term and the $j$-th term. Note that in Eq. 82, although the term $A_{i}A_{j}B_{i}B_{j}$ is 4-local, it arises only in cases where $s_{2}^{(i,j)}=1$. In this case, an additional gadget with a new ancilla $u_{ij}$ can be introduced to generate the 4-local term. For succinctness we present the details of this construction in Appendix B. With the penalty Hamiltonian $H$ defined in Eq. 74, the operator-valued resolvent (or the Green’s function) can be written as $G(z)=\sum_{x\in\\{0,1\\}^{m}}\frac{1}{z-h(x)\Delta}\|x\rangle\langle{x}\|.$ (83) Define subspaces of the ancilla register $\mathcal{L}_{-}=\text{span}\\{\|00\cdots 0\rangle\\}$ and $\mathcal{L}_{+}=\text{span}\\{\|x\rangle\|x\neq 00\cdots 0\\}$. Define ${P_{-}}$ and ${P_{+}}$ as the projectors onto $\mathcal{L}_{-}$ and $\mathcal{L}_{+}$. Then the projections of $V$ onto the subspaces can be written as $\begin{array}[]{ccl}V_{+}&=&\displaystyle\bigg{(}H_{\text{else}}+\frac{1}{\Delta}\sum_{i=1}^{m}({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})^{2}-\frac{1}{\Delta^{2}}\sum_{i=1}^{m}({\kappa_{i}^{2}}+{\lambda_{i}^{2}}){\mu_{i}}{C_{i}}-\frac{1}{\Delta^{3}}\sum_{i=1}^{m}({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})^{4}+\sum_{i=1}^{m}\sum_{j=1,j\neq i}^{m}\bar{V}_{ij}\bigg{)}\otimes{P_{+}}\\\\[7.22743pt] &+&\displaystyle\sum_{i=1}^{m}{\mu_{i}}{C_{i}}\otimes{P_{+}}\|1\rangle\langle{1}\|_{u_{i}}{P_{+}}+\underbrace{\sum_{i=1}^{m}({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})\otimes{P_{+}}X_{u_{i}}{P_{+}}}_{V_{f}}=V_{s}+V_{f}\\\\[7.22743pt] V_{-+}&=&\displaystyle\sum_{i=1}^{m}({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})\otimes{P_{-}}X_{u_{i}}{P_{+}},\quad V_{+-}=\sum_{i=1}^{m}({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})\otimes{P_{+}}X_{u_{i}}{P_{-}}\\\\[7.22743pt] V_{-}&=&\displaystyle\bigg{(}H_{\text{else}}+\frac{1}{\Delta}\sum_{i=1}^{m}({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})^{2}-\frac{1}{\Delta^{2}}\sum_{i=1}^{m}({\kappa_{i}^{2}}+{\lambda_{i}^{2}}){\mu_{i}}{C_{i}}-\frac{1}{\Delta^{3}}\sum_{i=1}^{m}({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})^{4}+\sum_{i=1}^{m}\sum_{j=1,j\neq i}^{m}\bar{V}_{ij}\bigg{)}\otimes{P_{-}}.\end{array}$ (84) Here the $V_{+}$ projection is intentionally divided up into $V_{f}$ and $V_{s}$ components. $V_{f}$ is the component of $V_{+}$ that contributes to the perturbative expansion only when the perturbative term corresponds to flipping processes in the $\mathcal{L}_{+}$ subspace. $V_{s}$ is the component that contributes only when the perturbative term corresponds to transitions that involve the state of the $m$-qubit ancilla register staying the same. The projection of the Green’s function $G(z)$ onto $\mathcal{L}_{+}$ can be written as $G_{+}(z)=\sum_{x\neq 0\cdots 00}\frac{1}{z-h(x)\Delta}\|x\rangle\langle{x}\|.$ (85) We now explain the self energy expansion $\Sigma_{-}(z)=V_{-}+V_{-+}G_{+}V_{+-}+V_{-+}G_{+}V_{+}G_{+}V_{+-}+V_{-+}(G_{+}V_{+})^{2}G_{+}V_{+-}+V_{-+}(G_{+}V_{+})^{3}G_{+}V_{+-}+\cdots$ (86) in detail term by term. The $1^{\text{st}}$ order term is simply $V_{-}$ from Equation Eq. 84. The $2^{\text{nd}}$ order term corresponds to processes of starting from an all-zero state of the $m$ ancilla qubits, flipping one qubit and then flipping it back: $\begin{array}[]{ccl}V_{-+}G_{+}V_{+-}&=&\displaystyle\frac{1}{z-\Delta}\sum_{i=1}^{m}({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})^{2}\\\\[7.22743pt] \end{array}$ (87) The $3^{\text{rd}}$ order term corresponds to processes of starting from an all-zero state of the ancilla register, flipping one qubit, staying at the same state for $V_{+}$ and then flipping the same qubit back. Therefore only the $V_{f}$ component in $V_{+}$ in Equation Eq. 84 will contribute to the perturbative expansion: $\begin{array}[]{ccl}V_{-+}G_{+}V_{+}G_{+}V_{+-}&=&\displaystyle\frac{1}{(z-\Delta)^{2}}\sum_{i=1}^{m}({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})\bigg{[}H_{\text{else}}+{\mu_{i}}{C_{i}}+\frac{1}{\Delta}\sum_{j=1}^{m}({\kappa_{j}}{A_{j}}+{\lambda_{j}}{B_{j}})^{2}\\\\[7.22743pt] &+&\displaystyle\frac{1}{\Delta^{2}}\sum_{j=1}^{m}\bigg{[}(\kappa_{j}^{2}+\lambda_{j}^{2})\mu_{j}{C_{j}}-\frac{1}{\Delta^{3}}\sum_{j=1}^{m}({\kappa_{j}}{A_{j}}+{\lambda_{j}}{B_{j}})^{4}+\sum_{j=1}^{m}\sum_{l=1,l\neq j}^{m}\bar{V}_{jl}\bigg{]}\\\\[7.22743pt] &&\displaystyle({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}}).\\\\[7.22743pt] \end{array}$ (88) The $4^{\text{th}}$ order term is more involved. Here we consider two types of transition processes (for diagrammatic illustration refer to Fig. 7): 1. 1. Starting from the all-zero state, flipping one of the qubits, flipping another qubit, then using the remaining $V_{+}$ and $V_{+-}$ to flip both qubits back one after the other (there are 2 different possible sequences, see Fig. 7a). 2. 2. Starting from the all-zero state of the ancilla register, flipping one of the qubits, staying twice for the two $V_{+}$ components and finally flipping back the qubit during $V_{+-}$ (Fig. 7b). Therefore in the transition processes of type (1), $V_{+}$ will only contribute its $V_{f}$ component and the detailed form of its contribution depends on which qubit in the ancilla register is flipped. The two possibilities of flipping the two qubits back explains why the second term in Eq. 89 takes the form of a summation of two components. Because two qubits are flipped during the transition, $G_{+}$ will contribute a $\frac{1}{z-2\Delta}$ factor and two $\frac{1}{z-\Delta}$ factors to the perturbative term. In the transition processes of type (2), $V_{+}$ will only contribute its $V_{s}$ component to the $4^{\text{th}}$ order term since the states stay the same during both $V_{+}$ operators in the perturbative term. $G_{+}$ will only contribute a factor of $\frac{1}{z-\Delta}$ because the Hamming weight of the bit string represented by the state of the ancilla register is always 1. This explains the form of the first term in Eq. 89. $\begin{array}[]{ccl}V_{-+}(G_{+}V_{+})^{2}G_{+}V_{+-}&=&\displaystyle\frac{1}{(z-\Delta)^{3}}\sum_{i=1}^{m}({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})\bigg{[}H_{\text{else}}+{\mu_{i}}{C_{i}}+\frac{1}{\Delta}\sum_{j=1}^{m}({\kappa_{j}}{A_{j}}+{\lambda_{j}}{B_{j}})^{2}\\\\[7.22743pt] &-&\displaystyle\frac{1}{\Delta^{2}}\sum_{j=1}^{m}(\kappa_{j}^{2}+\lambda_{j}^{2})\mu_{j}{C_{j}}-\frac{1}{\Delta^{3}}\sum_{j=1}^{m}({\kappa_{j}}{A_{j}}+{\lambda_{j}}{B_{j}})^{4}+\sum_{j=1}^{m}\sum_{l=1,l\neq j}^{m}\bar{V}_{jl}\bigg{]}^{2}\\\\[7.22743pt] &&({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})\\\\[7.22743pt] &+&\displaystyle\frac{1}{(z-\Delta)^{2}(z-2\Delta)}\sum_{i=1}^{m}\sum_{j=1,j\neq i}^{m}\bigg{[}({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})({\kappa_{j}}{A_{j}}+{\lambda_{j}}{B_{j}})\\\\[7.22743pt] &&\makebox[132.30513pt]{}{}({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})({\kappa_{j}}{A_{j}}+{\lambda_{j}}{B_{j}})\\\\[7.22743pt] &+&\displaystyle({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})({\kappa_{j}}{A_{j}}+{\lambda_{j}}{B_{j}})({\kappa_{j}}{A_{j}}+{\lambda_{j}}{B_{j}})({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})\bigg{]}.\\\\[7.22743pt] \end{array}$ (89) Although the $4^{\text{th}}$ order does not contain terms that are useful for simulating the 3-body target Hamiltonian, our assignments of ${\kappa_{i}}$, ${\lambda_{i}}$ and ${\mu_{i}}$ values in Eq. 78 imply that some of the terms at this order can be $\Theta(1)$. Indeed, the entire second term in Eq. 89 is of order $\Theta(1)$ based on Eq. 78. Therefore it is necessary to study in detail what error terms arise at this order and how to compensate for them in the perturbation $V$. A detailed analysis on how to compensate the $\Theta(1)$ errors is presented in the Appendix B. The $5^{\text{th}}$ order and higher terms are errors that can be reduced by increasing $\Delta$: $\begin{array}[]{ccl}&&\displaystyle\sum_{k=3}^{\infty}V_{-+}(G_{+}V_{+})^{k}G_{+}V_{+-}.\end{array}$ (90) At first glance, with assignments of ${\kappa_{i}}$, ${\lambda_{i}}$ and ${\mu_{i}}$ in Eq. 78, it would appear that this error term is $\Theta(\Delta^{-1/4})$ since $\\|V_{-+}\\|=\Theta(\Delta^{3/4})$, $\\|V_{+-}\\|=\Theta(\Delta^{3/4})$, $\\|V_{+}\\|=\Theta(\Delta^{3/4})$ and $\\|G_{+}\\|=\Theta(\Delta^{-1})$, $\begin{array}[]{ccl}\displaystyle\sum_{k=3}^{\infty}V_{-+}(G_{+}V_{+})^{k}G_{+}V_{+-}&\leq&\displaystyle\sum_{k=3}^{\infty}\\|V_{-+}\\|\cdot\\|G_{+}V_{+}\\|^{k}\\|G_{+}\\|\cdot\\|V_{+-}\\|\\\\[7.22743pt] &=&\displaystyle\\|V_{-+}(G_{+}V_{+})^{3}G_{+}V_{+-}\\|\sum_{k=0}^{\infty}\\|G_{+}V_{+}\\|^{k}\\\\[7.22743pt] &=&\displaystyle O(\Delta^{-1/4})\end{array}$ (91) as $\sum_{k=0}^{\infty}\\|G_{+}V_{+}\\|^{k}=O(1)$. However, here we show that in fact this term in Eq. 90 is $\Theta(\Delta^{-1/2})$. Note that the entire term Eq. 90 consists of contributions from the transition processes where one starts with a transition from the all-zero state to a state $\|x\rangle$ with $x\in\\{0,1\\}^{m}$ and $h(x)=1$. If we focus on the perturbative term of order $k+2$: $V_{-+}(G_{+}V_{+})^{k}G_{+}V_{+-},$ after $k$ steps. During every step one can choose to either flip one of the ancilla qubits or stay in the same state of the ancilla register, the state of the ancilla register will go back to a state $\|y\rangle$ with $y\in\\{0,1\\}^{m}$ and $h(y)=1$. Finally the $\|1\rangle$ qubit in $\|y\rangle$ is flipped back to $\|0\rangle$ and we are back to the all-zero state which spans the ground state subspace $\mathcal{L}_{-}$. Define the total number of flipping steps to be $k_{f}$. Then for a given $k$, $k_{f}$ takes only values from $K(k)=\left\\{\begin{array}[]{cl}\\{k,k-2,\cdots,2\\}&\text{if $k$ is even}\\\\[7.22743pt] \\{k-1,k-3,\cdots,2\\}&\text{if $k$ is odd}.\end{array}\right.$ (92) ${\cal L_{-}}$${\cal L_{+}}$$\displaystyle G_{+}\left(z\right)=\frac{1}{z-\Delta}$$\displaystyle G_{+}\left(z\right)=\frac{1}{z-2\Delta}$$\|0\,.\,.\,.\\!\\!\\!\underbrace{0}_{i}\\!\\!\\!.\,.\,.\\!\\!\\!\underbrace{0}_{j}\\!\\!\\!.\,.\,.\,0\rangle$$\|0\ldots 1\ldots 0\ldots 0\rangle$$\|0\ldots 1\ldots 1\ldots 0\rangle$$\|0\ldots 0\ldots 1\ldots 0\rangle$$\small V_{-+}$$\small V_{+-}$$\small V_{+-}$$\small\,V_{-+}$$\small V_{f}\,$$\small\,\,V_{f}$$\small V_{f}\,$$\small\,\,V_{f}$ (a) ${\cal L_{-}}$${\cal L_{+}}$$\displaystyle G_{+}\left(z\right)=\frac{1}{z-\Delta}$$\|0\,.\,.\,.\\!\\!\\!\underbrace{0}_{i}\\!\\!\\!.\,.\,.\\!\\!\\!\underbrace{0}_{j}\\!\\!\\!.\,.\,.\,0\rangle$$\|0\ldots 1\ldots 0\ldots 0\rangle$$\|0\ldots 0\ldots 1\ldots 0\rangle$$\small V_{-+}$$\small V_{+-}$$\small V_{+-}$$\small\,V_{-+}$$\small V_{s}$$\small V_{s}$$\small V_{s}$$\small V_{s}$ (b) Figure 7: Diagrams illustrating the transitions that occur at 4th order. The two diagrams each represent a type of transition that occurs at 4th order. Each diagram is divided by a horizontal line where below the line is $\mathcal{L}_{-}$ space and above is $\mathcal{L}_{+}$ subspace. Each diagram deals with a fixed pair of ancilla qubits labelled $i$ and $j$. The diagram (a) has three horizontal layers connected with vertically going arrows. $V_{f}$ and $V_{s}$ are both components of $V_{+}$. In fact $V_{+}=V_{f}+V_{s}$ where $V_{f}$ is responsible for the flipping and $V_{s}$ contributes when the transition does not have flipping. At the left of each horizontal layer lies the expression for $G_{+}(z)$, which is different for states in $\mathcal{L}_{+}$ with different Hamming weights. The diagram (b) is constructed in a similar fashion except that we are dealing with the type of 4th order transition where the state stays the same for two transitions in $\mathcal{L}_{+}$, hence the $V_{s}$ symbols and the arrows going from one state to itself. The diagram (a) reflects the type of 4th order transition that induces cross-gadget contribution and given our gadget parameter setting, this contribution could be $O(1)$ when otherwise compensated. The diagram (b) shows two paths that don not interfere with each other and thus having no cross-gadget contributions. For the term of order $k+2$, all the transition processes that contribute non- trivially to the term can be categorized into two types: 1. 1. If $x=y$, the minimum number of flipping steps is 0. The contribution of all such processes to the $(k+2)$-th order perturbative term is bounded by footnote:comb $\begin{array}[]{cl}\leq&\displaystyle m^{k_{f}}\cdot\binom{k}{k_{f}}\cdot\\|V_{f}\\|^{k_{f}}\cdot\\|V_{s}\\|^{k-k_{f}}\cdot\frac{\\|V_{-+}\\|\cdot\\|V_{+-}\\|}{(\Delta-\max(z))^{k+1}}\end{array}$ (93) where the factor $m^{k_{f}}$ is the number of all possible ways of flipping $k_{f}$ times, each time one of the $m$ ancilla qubits. This serves as an upper bound for the number of transition processes that contribute non- trivially to the perturbative term. The factor $\binom{k}{k_{f}}$ describes the number of possible ways to choose which $(k-k_{f})$ steps among the total $k$ steps involve the state of the ancilla register staying the same. $\\|G_{+}\\|\leq\frac{1}{\Delta-\max(z)}$ is used in the upper bound. 2. 2. If $x\neq y$, the minimum number of flipping steps is 2. The contribution of all such processes to the $(k+2)$-th order perturbative term is bounded by $\begin{array}[]{cl}\leq&\displaystyle\binom{k}{k_{f}}\cdot\binom{k_{f}}{2}\cdot 2!\cdot\\|V_{f}\\|^{k_{f}}\\|V_{s}\\|^{k-k_{f}}\cdot m^{k_{f}-2}\cdot\frac{\\|V_{-+}\\|\cdot\\|V_{+-}\\|}{(\Delta-\max(z))^{k+1}}\end{array}$ (94) where the factor $\binom{k}{k_{f}}$ is the number of all possible ways to choose which $(k-k_{f})$ steps among the $k$ steps should the state remain the same. $\binom{k_{f}}{2}$ is the number of possible ways to choose from the $k_{f}$ flipping steps the 2 minimum flips. $2!$ is for taking into account the ordering of the 2 flipping steps. $\\|G_{+}\\|\leq\frac{1}{\Delta-\max(z)}$ is used in the upper bound. For a general $m$-qubit ancilla register, there are in total $m$ different cases of the first type of transition processes and $\binom{m}{2}$ different cases of the second type of transition processes. Therefore we have the upper bound to the norm of the $(k+2)$-th term (Fig. 8) $\begin{array}[]{ccl}\\|V_{-+}(G_{+}V_{+})^{k}G_{+}V_{+-}\\|&\leq&\displaystyle m\sum_{k_{f}\in K(k)}m^{k_{f}}\binom{k}{k_{f}}\cdot\\|V_{f}\\|^{k_{f}}\cdot\\|V_{s}\\|^{k-k_{f}}\frac{\\|V_{-+}\\|\cdot\\|V_{+-}\\|}{(\Delta-\max(z))^{k+1}}\\\\[7.22743pt] &&\displaystyle+\binom{m}{2}\sum_{k=3}^{\infty}\binom{k}{k_{f}}\cdot\binom{k_{f}}{2}\cdot 2!\cdot\\|V_{f}\\|^{k_{f}}\\|V_{s}\\|^{k-k_{f}}\cdot m^{k_{f}-2}\cdot\frac{\\|V_{-+}\\|\cdot\\|V_{+-}\\|}{(\Delta-\max(z))^{k+1}}\\\\[7.22743pt] &&\displaystyle=\sum_{k_{f}\in K(k)}\left(m+\frac{m-1}{m}\right)2^{k}\cdot\frac{\\|V_{-+}\\|\cdot(m\\|V_{f}\\|)^{k_{f}}\cdot\\|V_{s}\\|^{k-k_{f}}\cdot\\|V_{+-}\\|}{(\Delta-\max(z))^{k+1}}\\\\[7.22743pt] &&\displaystyle\leq\frac{\\|V_{-+}\\|\cdot\\|V_{+-}\\|}{\Delta-\max(z)}(m+1)\sum_{k=3}^{\infty}\left(\frac{\\|V_{s}\\|}{\Delta-\max(z)}\right)^{k}\sum_{k_{f}\in K(k)}\left(m\frac{\\|V_{f}\\|}{\\|V_{s}\\|}\right)^{k_{f}}.\end{array}$ (95) Figure 8: Numerical verification for the upper bound to the norm of the $(k+2)$-th order perturbative term in Eq. 95. Here we use the parallel 3-body gadget for reducing $H_{\text{targ}}=0.1X_{1}Z_{2}Z_{3}-0.2X_{1}X_{2}Z_{3}$ up to error $\epsilon=0.01$. The gap in the gadget construction is numerically optimized footnote:num_op . Here the calculation of the analytical upper bound uses the result in Eq. 95. The calculation is then compared with the norm of the corresponding perturbative term numerically calculated according to the self-energy expansion. Since $\\|\sum_{i=1}^{m}\sum_{j=1,j\neq i}^{m}\bar{V}_{ij}\\|$ is bounded by $\frac{1}{\Delta^{3}}\sum_{i=1}^{m}\sum_{j=1,j\neq i}^{m}8({\kappa_{i}}{\kappa_{j}})^{2}{\openone}$, from Eq. 84 we see that $\begin{array}[]{ccl}\\|V_{s}\\|&\leq&\displaystyle\\|H_{\text{else}}\\|+2^{-1/3}\Delta^{1/2}\sum_{i=1}^{m}\|\alpha_{i}\|^{1/3}+2^{4/3}\Delta^{1/2}\sum_{i=1}^{m}\|\alpha_{i}\|^{2/3}+\sum_{i=1}^{m}\|\alpha_{i}\|\\\\[7.22743pt] &&\displaystyle+2^{8/3}\sum_{i=1}^{m}\|\alpha_{i}\|^{4/3}+\sum_{i=1}^{m}\sum_{j=1,j\neq i}^{m}8\cdot 2^{-4/3}\|\alpha_{i}\|^{2/3}\|\alpha_{j}\|^{2/3}\equiv v_{s}\\\\[7.22743pt] \\|V_{f}\\|&\leq&\displaystyle 2^{2/3}\Delta^{3/4}\sum_{i=1}^{m}\|\alpha_{i}\|^{1/3}\equiv v_{f}.\end{array}$ (96) With bounds of $\\|V_{s}\\|$ and $\\|V_{f}\\|$ in Eq. 84, the summation in Equation Eq. 95 can be written as $\begin{array}[]{l}\displaystyle\\|\sum_{k=3}^{\infty}V_{-+}(G_{+}V_{+})^{k}G_{+}V_{+-}\\|\leq\frac{\\|V_{-+}\\|\cdot\\|V_{+-}\\|}{\Delta-\max(z)}(m+1)\\\\[7.22743pt] \displaystyle\bigg{[}\sum_{r=1}^{\infty}\left(\frac{2v_{s}}{\Delta-\max(z)}\right)^{2r+1}\sum_{s=1}^{r}\left(m\frac{v_{f}}{v_{s}}\right)^{2s}+\sum_{r=2}^{\infty}\left(\frac{2v_{s}}{\Delta-\max(z)}\right)^{2r}\sum_{s=1}^{r}\left(m\frac{v_{f}}{v_{s}}\right)^{2s}\bigg{]}.\end{array}$ (97) To guarantee convergence of the summation in Eq. 97 we require that $\Delta$ satisfies $\displaystyle\frac{2mv_{f}}{\Delta-\max(z)}<1$ (98) $\displaystyle m\left(\frac{v_{f}}{v_{s}}\right)>1,$ (99) both of which are in general satisfied. The summation in Eq. 97 can then be written as $\begin{array}[]{c}\displaystyle\\|\sum_{k=3}^{\infty}V_{-+}(G_{+}V_{+})^{k}G_{+}V_{+-}\\|\leq\frac{\\|V_{-+}\\|\cdot\\|V_{+-}\\|}{\Delta-\max(z)}\cdot\frac{\left(m\frac{v_{f}}{v_{s}}\right)^{2}}{\left(m\frac{v_{f}}{v_{s}}\right)^{2}-1}\\\\[7.22743pt] \displaystyle\frac{\left(\frac{2mv_{f}}{\Delta-\max(z)}\right)^{2}}{1-\left(\frac{2mv_{f}}{\Delta-\max(z)}\right)^{2}}(m+1)\left[\left(\frac{2mv_{f}}{\Delta-\max(z)}\right)^{2}+\frac{2v_{s}}{\Delta-\max(z)}\right]=\Theta(\Delta^{-1/2}),\end{array}$ (100) which shows that the high order terms are $\Theta(\Delta^{-1/2})$. This is tighter than the crude bound $\Theta(\Delta^{-1/4})$ shown in Eq. 91. The self-energy expansion Eq. 86 then satisfies $\\|\Sigma_{-}(z)-H_{\text{targ}}\otimes{P_{-}}\\|\leq\Theta(\Delta^{-1/2})$ (101) which indicates that $\Delta=\Theta(\epsilon^{-2})$ is sufficient for the parallel 3-body gadget to capture the entire spectrum of $H_{\text{targ}}\otimes{P_{-}}$ up to error $\epsilon$. We have used numerics to verify the $\Theta(\epsilon^{-2})$ scaling, as shown in Fig. 8. Furthermore, for a range of specified $\epsilon$, the minimum $\Delta$ needed for the spectral error between the gadget Hamiltonian and the target Hamiltonian is numerically found. In the optimized cases, the slope ${\rm d}\log\Delta/{\rm d}\log\epsilon^{-1}$ for the construction in BDLT08 is approximately 3, showing that $\Delta=\Theta(\epsilon^{-3})$ is the optimal scaling for the construction in BDLT08 . For our construction both the analytical bound and the optimized $\Delta$ scale as $\Theta(\epsilon^{-2})$ (see Fig. 9). Figure 9: Scaling of the spectral gap $\Delta$ as a function of error $\epsilon$ for the parallel 3-body example that is intended to reduce the target Hamiltonian $H_{\text{targ}}=Z_{1}Z_{2}Z_{3}-X_{1}X_{2}X_{3}$ to 2-body. Here $\epsilon=0.01$. We show both numerically optimized values (“numerical”) in our construction and the construction in BDLT08 , which is referred to as “[BDLT08]”. ## Appendix B Compensation for the 4-local error terms in parallel 3- to 2-body gadget Continuing the discussion in Appendix A, here we deal with $\Theta(1)$ error terms that arise in the $3^{\text{rd}}$ and $4^{\text{th}}$ order perturbative expansion when $V$ in Eq. 75 is without $V_{3}$ and in so doing explain the construction of $\bar{V}_{ij}$ in Eq. 82. From the previous description of the $3^{\text{rd}}$ and $4^{\text{th}}$ order terms, for each pair of terms $(i)$ and $(j)$ where $i$ and $j$ are integers between 1 and $m$, let $\begin{array}[]{ccl}M_{1}&=&({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})({\kappa_{j}}{A_{j}}+{\lambda_{j}}{B_{j}})\\\\[7.22743pt] M_{2}&=&({\kappa_{j}}{A_{j}}+{\lambda_{j}}{B_{j}})({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})\end{array}$ and then the $\Theta(1)$ error term arising from the $3^{\text{rd}}$ and $4^{\text{th}}$ order perturbative expansion can be written as $\frac{1}{(z-\Delta)^{2}}\bigg{[}\frac{1}{z-2\Delta}(M_{1}^{2}+M_{2}^{2})+\left(\frac{1}{\Delta}+\frac{1}{z-2\Delta}\right)(M_{1}M_{2}+M_{2}M_{1})\bigg{]}.$ (102) Based on the number of non-commuting pairs among ${A_{i}}$, ${A_{j}}$, ${B_{i}}$ and ${B_{j}}$, all possible cases can be enumerated as the following: $\begin{array}[]{ccl}\text{case 0:}&&[{A_{i}},{A_{j}}]=0,[{B_{i}},{B_{j}}]=0,[{A_{i}},{B_{j}}]=0,[{B_{i}},{A_{j}}]=0\\\\[7.22743pt] \text{case 1:}&1.1:&[{A_{i}},{A_{j}}]=0,[{B_{i}},{B_{j}}]=0,[{A_{j}},{B_{i}}]\neq 0\\\\[7.22743pt] &1.2:&[{A_{i}},{A_{j}}]=0,[{B_{i}},{B_{j}}]=0,[{A_{i}},{B_{j}}]\neq 0\\\\[7.22743pt] &1.3:&[{A_{i}},{A_{j}}]=0,[{B_{i}},{B_{j}}]\neq 0\\\\[7.22743pt] &1.4:&[{A_{i}},{A_{j}}]\neq 0,[{B_{i}},{B_{j}}]=0\\\\[7.22743pt] \text{case 2:}&&[{A_{i}},{A_{j}}]\neq 0,[{B_{i}},{B_{j}}]\neq 0.\end{array}$ (103) In case 0, clearly $M_{1}=M_{2}$. Then the $\Theta(1)$ error becomes $\frac{1}{(z-\Delta)^{2}}\left(\frac{1}{\Delta}+\frac{2}{z-2\Delta}\right)\cdot 2M_{1}^{2}=\Theta(\Delta^{-1})$ which does not need any compensation. In case 1, for example in the subcase 1.1, ${A_{j}}$ does not commute with ${B_{i}}$. Then $M_{1}$ and $M_{2}$ can be written as $\begin{array}[]{ccl}M_{1}&=&K+{\kappa_{j}}{\lambda_{i}}{B_{i}}{A_{j}}\\\\[7.22743pt] M_{2}&=&K+{\kappa_{j}}{\lambda_{i}}{A_{j}}{B_{i}}\end{array}$ where $K$ contains the rest of the terms in $M_{1}$ and $M_{2}$. Furthermore, $\begin{array}[]{c}M_{1}^{2}+M_{2}^{2}=2K^{2}-2({\kappa_{j}}{\lambda_{i}})^{2}{\openone}\\\\[7.22743pt] M_{1}M_{2}+M_{2}M_{1}=2K^{2}+2({\kappa_{j}}{\lambda_{i}})^{2}{\openone}.\end{array}$ Hence the $\Theta(1)$ term in this case becomes $\frac{1}{(z-\Delta)^{2}}\bigg{[}\left(\frac{1}{\Delta}+\frac{2}{z-2\Delta}\right)2K^{2}+\frac{1}{\Delta}\cdot 2({\kappa_{j}}{\lambda_{i}})^{2}{\openone}\bigg{]}$ (104) where the first term is $\Theta(\Delta^{-1})$ and the second term is $\Theta(1)$, which needs to be compensated. Similar calculations for cases 1.2, 1.3 and 1.4 will yield $\Theta(1)$ error with the same norm. In case 2, define $R={\kappa_{i}}{\lambda_{j}}{A_{i}}{B_{j}}+{\lambda_{i}}{\kappa_{j}}{B_{i}}{A_{j}}$ and $T={\kappa_{i}}{\kappa_{j}}{A_{i}}{A_{i}}+{\lambda_{i}}{\lambda_{j}}{B_{i}}{B_{i}}$. Then $\begin{array}[]{c}M_{1}^{2}+M_{2}^{2}=2(R^{2}+T^{2})\\\\[7.22743pt] M_{1}M_{2}+M_{2}M_{1}=2(R^{2}-T^{2}).\end{array}$ The $\Theta(1)$ error terms in the 3rd and 4th order perturbative expansion becomes $\frac{1}{(z-\Delta)^{2}}\bigg{[}\left(\frac{1}{\Delta}+\frac{2}{z-2\Delta}\right)\cdot 2R^{2}-\frac{1}{\Delta}\cdot 2T^{2}\bigg{]}$ (105) where the first term is $\Theta(\Delta^{-1})$ and hence needs no compensation. The second term is $\Theta(1)$. Define $s_{0}^{(i,j)}=\left\\{\begin{array}[]{ccl}1&&\text{if case 0}\\\\[7.22743pt] 0&&\text{Otherwise}\end{array}\right.$ (106) With the definitions of $s_{1}^{(i,j)}$ and $s_{2}^{(i,j)}$ in Eq. 79, Eq. 80 and Eq. 81, the contribution of the $i$-th and the $j$-th target terms to the $\Theta(1)$ error in the perturbative expansion $\Sigma_{-}(z)$ becomes $\begin{array}[]{cl}&\displaystyle s_{0}^{(i,j)}\cdot\frac{1}{(z-\Delta)^{2}}\left(\frac{1}{\Delta}+\frac{2}{z-2\Delta}\right)\cdot 2({\kappa_{i}}{A_{i}}+{\lambda_{i}}{B_{i}})^{2}({\kappa_{j}}{A_{j}}+{\lambda_{j}}{B_{j}})^{2}\\\\[7.22743pt] +&\displaystyle s_{1}^{(i,j)}\cdot\frac{1}{(z-\Delta)^{2}}\bigg{[}\left(\frac{1}{\Delta}+\frac{2}{z-2\Delta}\right)\cdot 2K_{ij}^{2}+\frac{1}{\Delta}\cdot 2({\kappa_{i}}{\kappa_{j}})^{2}{\openone}\bigg{]}\\\\[7.22743pt] +&\displaystyle s_{2}^{(i,j)}\cdot\frac{1}{(z-\Delta)^{2}}\bigg{[}\left(\frac{1}{\Delta}+\frac{2}{z-2\Delta}\right)\cdot 2R_{ij}^{2}+\frac{1}{\Delta}\cdot 2\\{[({\kappa_{i}}{\kappa_{j}})^{2}+({\lambda_{i}}{\lambda_{j}})^{2}]{\openone}\\\\[7.22743pt] &\displaystyle-2{\kappa_{i}}{\kappa_{j}}{\lambda_{i}}{\lambda_{j}}{A_{i}}{A_{j}}{B_{i}}{B_{j}}\\}\bigg{]}.\end{array}$ (107) The term proportional to $s_{0}^{(i,j)}$ in Eq. 107 does not need compensation since it is already $\Theta(\Delta^{-1})$. The term proportional to $s_{1}^{(i,j)}$ can be compensated by the corresponding term in $\bar{V}_{ij}$ in Eq. 82 that is proportional to $s_{1}^{(i,j)}$. Similarly, the $\Theta(1)$ error term proportional to $s_{2}^{(i,j)}$ can be compensated by the term in $\bar{V}_{ij}$ in Eq. 82 that is proportional to $s_{2}^{(i,j)}$. Now we deal with generating the 4-local term in $\bar{V}_{ij}$. Introduce an ancilla $u_{ij}$ and construct a gadget $\tilde{H}_{ij}=H_{ij}+V_{ij}$ such that $H_{ij}=\Delta\|1\rangle\langle{1}\|_{u_{ij}}$ and the perturbation $V_{ij}$ becomes $V_{ij}=(\kappa_{i}A_{i}+\lambda_{j}B_{j})\otimes X_{u_{ij}}+(\kappa_{j}A_{j}+\lambda_{i}B_{i})\otimes\|1\rangle\langle{1}\|_{u_{ij}}+V^{\prime}_{ij}$ (108) where $V^{\prime}_{ij}$ is defined as $V^{\prime}_{ij}=\frac{1}{\Delta}(\kappa_{i}A_{i}+\lambda_{j}B_{j})^{2}+\frac{1}{\Delta^{3}}\left[(\kappa_{j}^{2}+\lambda_{i}^{2})(\kappa_{i}A_{i}+\lambda_{j}B_{j})^{2}-2\kappa_{j}\lambda_{i}(\kappa_{j}^{2}+\lambda_{j}^{2})A_{j}B_{i}\right]$ (109) The self-energy expansion $\Sigma_{-}(z)$ is now $\Sigma_{-}(z)=\frac{1}{(z-\Delta)^{3}}4\kappa_{i}\kappa_{j}\lambda_{i}\lambda_{j}A_{i}A_{j}B_{i}B_{j}+O(\Delta^{-1/2})$ which is $O(\Delta^{-1/2})$ close to the 4-local compensation term in $\bar{V}_{ij}$. We apply the the gadget $\tilde{H}_{ij}$ for every pair of qubits with $s_{2}^{(i,j)}=1$. The cross-gadget contribution between the $\tilde{H}_{ij}$ gadgets as well as those cross-gadget contribution between $\tilde{H}_{ij}$ gadgets and gadgets based on ancilla qubits $u_{1}$ through $u_{m}$ both belong to the case 1 of the Eq. 103 and hence are easy to deal with using 2-body terms. ## References * [1] Dorit Aharonov, Julia Kempe, Seth Lloyd, Wim Van Dam, Zeph Landau, and Oded Regev. Adiabatic quantum computation is equivalent to standard quantum computation. SIAM Journal on Computing, 37:166–194, 2007. arXiv:quant-ph/0405098. * [2] A. Mizel, D. A. Lidar, and M. Mitchell. Simple proof of equivalence between adiabatic quantum computation and the circuit model. Physical Review Letters, 99(7):070502, August 2007. arXiv:quant-ph/0609067. * [3] R. Oliveira and B. Terhal. The complexity of quantum spin systems on a two-dimensional square lattice. Quant. Inf. and Comp., 8(10):0900–0924, 2008. arXiv:quant-ph/0504050. * [4] J. D. Biamonte and P. J. Love. 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Phys., 326(10), 2011. arXiv:1105.0675. * [27] The notion of ‘optimized case’ refers to the search for the gap $\Delta$ needed for yielding a spectral error of precisely $\epsilon$ between gadget and target Hamiltonian, which is described in Sec. II. * [28] As is shown by [11], for the gadget construction with the assignments of ${\kappa_{i}}$, ${\lambda_{i}}$ and ${\mu_{i}}$ all being $O(\Delta^{2/3})$, the cross-gadget contribution can be reduced by increasing $\Delta$, thus no cross-gadget compensation is needed. However, with our assignments of ${\kappa_{i}}$, ${\lambda_{i}}$ and ${\mu_{i}}$ in (78) there are cross-gadget error terms in the perturbative expansion that are of order $O(1)$, which cannot be reduced by increasing $\Delta$. This is why we need $\bar{V}_{ij}$. Since the $O(1)$ error terms are dependent on the commuting relations between $A_{i}$, $B_{i}$, $A_{j}$ and $B_{j}$ of each pair of $i$-th and $j$-th terms in the target Hamiltonian, $\bar{V}_{ij}$ depends on their commutation relations too. * [29] Here we use the notation ${\sf C}_{m}^{n}$ to represent the combinatorial number that is the number of ways to choose $n$ elements from a total of $m$ without distinguishing between different orderings.	arxiv-papers	2013-11-11T20:03:08	2024-09-04T02:49:53.466670	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Yudong Cao, Ryan Babbush, Jacob Biamonte and Sabre Kais", "submitter": "Yudong Cao", "url": "https://arxiv.org/abs/1311.2555" }
1311.2807	# Stability of germanene under tensile strain T. P. Kaloni and U. Schwingenschlögl [email protected],+966(0)544700080 Physical Science & Engineering Division, KAUST, Thuwal 23955-6900, Kingdom of Saudi Arabia ###### Abstract The stability of germanene under biaxial tensile strain and electronic properties are studied by means of density functional theory based calculations. Our results show that up to 16% biaxial tensile strain germanene lattice is stable and the Dirac cone shifts towards higher energy range with respect to the Fermi level as a result $p$-doped Dirac states are achieved. The realization of the $p$-doped Dirac states are due to the weakening of the Ge$-$Ge bonds and reduction of hybridization with the strain. We therefore calculate the phonon spectrum to demonstrate that the germanene is stable up to 16% under biaxial tensile strain. Our calculated Grüneisen parameter shows the similar trend to silicene and different trend to graphene under small biaxial tensile strain. Graphene is a two-dimensional (2D) honeycomb lattice of carbon atoms, currently a material of interest for many researcher due to the fact that its unique electronic properties, which is being proposed to be a great potential for future nanoelectronic applications geim . The mass production and band gap opening are real challenges as a result searching of a new materials which can be a counterpart of graphene is highly demanded. Recent years, the electronic properties of 2D hexagonal silicon and germanium also named as silicene and germanene, respectively, have been proposed as a potential alternative of graphene Topsakal . Experimentally, it has been demonstrated Ag and ZrB2 substrates padova ; vogt ; ozaki can be used to grow the silicene. However, the free standing silicene and germanene are not realized so far. C, Si, and Ge belongs to the same group in the periodic table whereas, Si and Ge have a larger ionic radius, which promotes $sp^{3}$ hybridization. The mixture of $sp^{2}$ and $sp^{3}$ hybridization in silicene and germanene results in a prominent buckling (0.46 Å and 0.68 Å for silicene and germanene) as compared to graphene, which opens an electrically tunable band gap falko ; Ni . As a consequences, its a huge advantage as compared to graphene. Germanene was proposed to be a poor metal Topsakal . In this study, the authors ignored the intrinsic spin orbit coupling. The magnitude of the spin orbit coupling is significantly larger in germanene and can not be neglected as it is a materials property. It was also noted that in-plane biaxial compressive strain turns germanene into a gapless semiconductor, by remain intact the linear energy dispersions at the K and K′ points Houssa . The magnitude of the intrinsic spin orbit coupling in Ge (6.3 meV) is stronger than that of Si (4 meV) and C (1.3 $\mu$eV) atoms Liu1 . It has been demonstrated that germanene can be a good candidate for the quantum spin hall effect with a sizable band gap at the Dirac points due to stronger spin orbit coupling and the higher buckling as compared to silicene Liu ; Liu1 . As a result, germanene can be a potential candidate for constructing promising spintronic devices. The absorption of F, Cl, Br and I has been studied ma and found that the intrinsic spin orbit coupling band gap in germanene is enhanced by absorption up to 162 meV, clearly higher than that for pristine germanene. Strain takes play a role when a crystal is compressed (stretched) from the equilibrium. The strain can affect the device performance, it can be applied intentionally to improve mobility. The biaxial tensile strain modify the crystal phonons, which usually resulting in mode softening. The rate of these changes is determined by the Grüneisen parameters, which also can determine the thermomechanical properties Mohiuddi . Graphene preserves the zero gap semiconducting nature even under huge strain of about 30% choi . However in silicene, the lattice stable up to 17% kaloni and shows self hole doped Dirac states Liu2 . Hence, it is an important issue whether the stability and electronic properties are modified under the biaxial strain. A comprehensive study of the effect of strain would be promising, which can provides detail information on the responses of germanene under biaxial strain and explores the possible typical properties. Hence, in this paper, based on first- principles calculations, we investigate the modification on the electronic structures and stability via phonon spectrum under the biaxial tensile strain for germanene. The phonon spectrum shows that germanene lattice can be stable up to 16% biaxial tensile strain. We also calculate the Grüneisen parameter and we find that the trend remain similar to silicene and behaves differently to graphene. The obtained results conclude that the biaxial tensile strain could bring an interesting $p$-doping phenomenon in germanene, which is consequences of the buckled structures and can not be possible in graphene. We carried out first-principles calculations using density functional theory as implemented in the QUANTUM-ESPRESSO package paolo . A full relativistic Rappe-Rabe-Kaxiras-Joannopoulos type rrkjus norm-conserving pseudopotential is employed together with the generalized gradient approximation in the Perdew, Burke, and Ernzerhof parametrization in order to include the spin orbit coupling. A Grimme scheme with scaling parameter 0.75 is considered in the calculations to include the van der Waals interaction grime ; kaloni-jmc . The calculations are performed with a plane wave cutoﬀ energy of 60 Ryd. A Monkhorst-Pack 16$\times$16$\times$1 k-mesh is employed for optimizing the crystal structure and calculating the electronic band structure. Moreover, we use a 24$\times$24$\times$1 k-mesh to calculate the phonon spectrum. The atomic positions are relaxed until an energy convergence of 10-9 eV and a force convergence of 10-4 eV/Å are achieved. To avoid artifacts of the periodic boundary conditions we use an interlayer spacing of 15 Å. The magnitude of the biaxial tensile strain is expressed as $\varepsilon=\frac{(a-a_{0})}{a_{0}}\times 100\%$, where $a$ and $a_{0}=4.06$ Å are the lattice parameters for strained and unstrained germanene, respectively. Strain is the efficient way to engineered the electronic properties of graphene Pereira . This allows to generate an all-graphene circuit, where all the elements of the circuit are made of graphene with different amounts and kinds of strain. It is also reported that the amounts and kinds of strain are equivalent to the magnetic field guinea , which indeed, produce pseudo- magnetic quantum Hall effect. Other way around, for graphene up to 10% strain is easily achievable Andresa . It enhances the reactivity of graphene about 5 times as a result H atoms are bound strongly to the strained graphene. Which is the important route for H storage in graphene. We expect similar effect could be applicable for germanene because of its 2D structure similar to graphene. Therefore, we study in the following the effects of strain on the electronic structure and phonon spectrum. We obtain a lattice parameter of $a_{0}=4.06$ Å and a buckling of 0.68 Å for unstrained germanene. These structural parameters are in good agreement with previous reports Ni ; ciraci . We address the dependence of the stress (in Gpa) on the applied strain (in %). The result is shown in Fig. 1. We find that the stress increases monotonically with the strain of 16% and remain constant up to 19% and decreases thereafter. Which in fact indicates that germanene is stable up to 16% biaxial tensile strain, which is very similar to silicene with a similar buckled geometry. Figure 1: Variation of the stress as a function of the applied biaxial tensile strain. In this section, we focus on the electronic band structure of germanene without/with variable biaxial tensile strain. We addressed the electronic band structure of the free standing germanene in Fig. 2(a). We have included spin orbit coupling in our calculations. It is noted that germanene behaves like a semiconductor with a band gap of 24 meV at the K point, see inset of Fig. 2(a), consistent with the previous findings Liu ; Liu1 . The $\pi$ and $\pi^{}$ bands of the Dirac cone are contributed by the $p_{z}$ orbitals of Ge, like as graphene and silicene. By the application of an external electric field, the magnitude of the band gap could be enhanced easily because this field breaks the sublattice symmetry as a result the band gap due to spin orbit coupling could be increases. Such a effect has been observed in case of silicene Ni . Due to its flexibility in the band gap opening, it can have a potential candidate in nano-electronic devices applications. Note that the the band gap of 24 meV in unstrained germanene becomes smaller (23 meV) for increasing strain. The reason is that the strain weakens the internal electric field because it reduces the magnitude of the buckling, which in fact reduces the strength of the intrinsic spin orbit coupling and thus the induced band gap is reduced. We find the Ge$-$Ge bond length is growing monotonically with the strain as a result the buckling decreases. For unstrained germanene the Ge$-$Ge bond length of 2.44 Å and buckling of 0.68 Å are obtained. For 10% strain these values change to 2.65 Å and 0.59 Å, and for 16% strain to 2.76 Å and 0.55 Å. The data for the variation of the Ge$-$Ge bond length and buckling under the biaxial strain are addressed in Table I. $\varepsilon$ (%) \| Ge$-$Ge \| $\Delta(\AA)$ \| $\theta^{\circ}$ \| $\Delta\omega_{G}$ (cm-1) \| ${\gamma_{G}}$ \| $s$ \| $p$ ---\|---\|---\|---\|---\|---\|---\|--- 5 \| 2.55 \| 0.63 \| 112 \| 363.2 \| 1.50 \| 1.55 \| 2.44 10 \| 2.65 \| 0.59 \| 114 \| 303.6 \| 1.45 \| 1.62 \| 2.36 16 \| 2.76 \| 0.55 \| 115 \| 243.8 \| 1.43 \| 1.69 \| 2.29 20 \| 2.85 \| 0.51 \| 116 \| 197.8 \| 1.34 \| 1.75 \| 2.22 Table 1: Strain, bond length, buckling, angle, and occupations. We define that the $p$-doped Dirac states by the shift of the Dirac cone with respect to the Fermi level under biaxial tensile strain. The calculated band structure addressed in Fig. 2(a-b) shows that the Dirac cone shifts towards the higher energy range with respect to the the Fermi level by inducing $p$-doped Dirac states. At a strain of 5%, we obtain a shifts of Dirac cone by 0.24 eV towards the higher energy range with respect to the Fermi level with a 23 meV band gap due to intrinsic spin orbit coupling, see inset of Fig. 2(b). The intrinsic spin orbit gap is decreases by 1 meV due to the decreasing the buckling and hence electric field become weaker, such effect already had been found for strained silicene kaloni . The conduction band at the $\Gamma$-point shifts towards the low energy range with respect to the Fermi level by 0.6 eV. This observation is well agree with the strained silicene Liu2 and germanene wang . Note that due to shifts the $\Gamma$-point towards the higher energy range with respect to the Fermi level by leaving $p$-doped Dirac sates, consistent with the recent observation for silicene kaloni and germanene wang , which is in contrast to graphene. This can be attributed due to the fact that graphene is planar structure as compared to silicene and germanene and thus changes the $s-p$ hybridization significantly in later case. The occupation of $s$ and $p$ orbitals changes for unstrained and strained germanene, which in fact reduce the hybridization of the $s$ and $p$ orbitals. For unstained germanene the occupation at $s$ and $p$ orbitals are 1.47 e and 2.51 e, while for a strain of 5% the occupation of $s/p$ orbitals increases/decreases (1.54/2.43 e), see Table I. The amount of the $p$-doped Dirac states are enhanced for increasing strain. The conduction band minimum at the $\Gamma$-point shifts further downwards and becomes more and more flat and occupied by increasing density of states at the Fermi level. At the strain of 10%, the Ge$-$Ge bond length is increases, buckling is decreases, and hence angle is increases. This is a consequence of the weakening of Ge$-$Ge bonds strength. Figure 2: Electronic band structure and partial densities of states for (a) unstrained germanene and germanene under biaxial tensile strain of (b) 5% and (c) 16%. The Dirac point lies at 0.3 eV towards higher energy range with respect to the Fermi level for strain of 16%, see Fig. 2(c). The $\pi$ and $\pi^{}$ bands of the Dirac cone are due to the $p_{z}$ orbitals of the Ge atoms with a minute contribution from the $p_{x}$ and $p_{y}$ orbitals, as it is expected. We obtain a gap of 22 meV, which in fact reduces as compared to unstrained germanene. The main reason for reducing the gap is the reduction of the buckling significantly (0.55 Å) with increasing bond length of 2.76 Å, and bond angle of $115^{\circ}$. The reduction of the buckling weakening the in built electric field as a result band gap is reduced, consistent with the strained silicene kaloni . For higher strain the conduction band minimum shifts further towards lower energy range and the Dirac cone accordingly to higher energy range with respect to Fermi level. This is due to the change in the occupation in the $s$ and $p$ orbitals, see Table I. Since, the number of the electrons in the system remain constant as a result the bands at the K/K′-points are depopulated and at the $\Gamma$-points are populated. Such a behavior is well agree with silicene but different from graphene, because the Ge$-$Ge bonds are much more flexible than the C$-$C bonds in graphene. Contrary to silicene and germanene, the electronic structure of graphene does not changes in the presence of strain, resulting a zero band gap semiconductor up to a very large strain (30%) choi . This indicates that there is not any possibilities to achieve $p$-doping in graphene by strain. However, it has been demonstrated that a $p$-doping can be achieved in graphene by the intercalation of F and Ge with the SiC substrate kaloni-epl ; cheng-apl . The lattice becomes instable beyond strain of 16% (for 20% strain the parameters are presented in Table I), we will prove this fact by performing phonon calculation in the next section. Figure 3: Phonon frequencies for (a) unstrained germanene and germanene under biaxial tensile strain of (b) 16% and (c) 20%. In this section, we discuss the phonon spectrum of germanene unstrained and under strain of 5%, 10%, 16%, and 20%, see Fig. 3. For unstrained germanene the obtained optical phonon frequencies are 3.7 times smaller than graphene (1580 cm-1 zabel ) and 1.28 times smaller than silicene (550 cm-1 kaloni ). This can be realized by the smaller force constant and weaker Ge$-$Ge bonds as compared to C$-$C and Si$-$Si bonds. Graphene shows a common features in the Raman spectra called G and D peaks, around 1580 cm-1 and 1360 cm-1 zabel . The G peak corresponds to the E2g phonon at the $\Gamma$-point of the Brillouin zone. The D peak correspond to the K-point Brillouin zone. On the based on our knowledge the the Raman spectra of germanene is unknown. So, we do believe that our study would be a reference for the experimental observation of the Raman spectra to get insight of Raman frequencies and identification of G and D peaks. We therefore focus on the G peak and D peak identification in our study. For unstrained germanene the calculated phonon frequencies of G and D peaks are 427 cm-1 and 366 cm-1, respectively, lower than that of silicene. For strained silicene a significant modification of the phonon frequencies is observed. At 5% strain the G and D peak frequencies found to be 363 cm-1 and 287 cm-1, respectively. Which reflects that the weakening of the Ge$-$Ge bond under biaxial tensile strain. It also can be understood by the fact that the optical bands shows a clear trend of softening, which is expected because Ge$-$Ge bond length increase uniformly. For the strain of 10% (16%) obtained phonon frequencies of G and D peak are 303 cm-1 (246 cm-1) and 212 cm-1 (150 cm-1), respectively. We conclude that the germanene lattice is stable up to strain of 16% because we still have positive frequencies along the $\Gamma$-M-K-$\Gamma$ path of the Brillouin zone. The germanene lattice becomes instable for the strain beyond 16%. For this purpose we calculate phonon spectrum for the strain of 20% and find a frequency of $-$62 cm-1, see Fig. 3(c). Which indicates that the lattice is instable. The effect of strain in 2D systems can be efficiently studied by Raman spectroscopy zabel . Since the strain modifies the crystal phonon frequencies. The rate of phonon mode softening or hardening described by the Grüneisen parameter, which in fact determines the thermomechanical properties fo the system. The Grüneisen parameter for G peak under biaxial strain is given by $\gamma_{G}=-\Delta\omega_{G}/2\omega_{G}^{0}\varepsilon,$ (1) where $\Delta\omega_{G}$ is the difference in the frequency for unstrained and strained germanene and $\omega_{G}^{0}$ is the frequency of the G peak in unstrained germanene. The Grüneisen parameter is difficult to study under uniaxial strain due to the fact that it require the Poisson ratio, which in fact depends on the choice of the substrate Mohiuddi . It is also reported that it is difficult to calculate the D and 2D Grüneisen parameters because under uniaxial strain the position of the Dirac cones changes. The biaxial tensile strain is suitable to calculation the Grüneisen parameter because it does not depend on the Poisson ratio as well as the position of the Dirac cone does not changes Mohiuddi . Experimentally, the Grüneisen parameter of graphene under biaxial strain has been demonstrated ding . Recently, in the Ref.kaloni , the Grüneisen parameter for silicene under biaxial tensile strain has been studied theoretically. Hence, we calculate the Grüneisen parameter for germanene and compare with the graphene and silicene. We find that the calculated Grüneisen parameter is decreasing with increasing the biaxial tensile strain. This variation is well agree with calculated Grüneisen parameter for silicene with a strain of 5% to 25% kaloni . For a biaxial tensile strain of 5%, 10%, and 20%, the obtained Grüneisen parameter are 1.50, 1.43, and 1.34, respectively, for silicene those values are 1.64, 1.62, and 1.34, respectively. The slight lowering of the Grüneisen parameter in germanene as compared to silicene can be attributed by the lowering of the respective phonon spectrum. However, for graphene this magnitude of the Grüneisen paramete can be obtained by a very low strain of 0.2% ding . The another reason for the lowering of the Grüneisen parameter with increasing strain is the buckling decreases monotonically with increasing the strain and Ge$-$Ge bond length. Such a behavior is essentially similar to silicene (buckled structure) and different from graphene (non-buckled structure). We call for an experimental observations for the confirmation our findings. In summary, we have performed first-principles calculations using density functional theory to study the effect of biaxial tensile train in germanene lattice, electronic properties, and phonon frequencies, and Grüneisen paramete. Our results show that up to 16% biaxial tensile strain germanene lattice is in stable and the Dirac cone shift towards the higher energy range with respect to Fermi level as a result $p$-doped Dirac states are achieved. The realization of the $p$-doped Dirac states is due to the weakening of the Ge$-$Ge bonds, well agree with strained silicene kaloni . We further calculate the phonon spectrum to demonstrate that germanene is stable up to 16% under biaxial tensile strain. The calculated Grüneisen parameter found to be similar to silicene and different from graphene as latter is non-buckled structure. 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1311.2897	# Exponential Stability of Homogeneous Positive Systems of Degree One With Time-Varying Delays Hamid Reza Feyzmahdavian, Themistoklis Charalambous, and Mikael Johansson H. R. Feyzmahdavian, T. Charalambous, and M. Johansson are with ACCESS Linnaeus Center, School of Electrical Engineering, KTH-Royal Institute of Technology, Stockholm, Sweden. Emails: {hamidrez, themisc, mikaelj}@kth.se. ###### Abstract While the asymptotic stability of positive linear systems in the presence of bounded time delays has been thoroughly investigated, the theory for nonlinear positive systems is considerably less well-developed. This paper presents a set of conditions for establishing delay-independent stability and bounding the decay rate of a significant class of nonlinear positive systems which includes positive linear systems as a special case. Specifically, when the time delays have a known upper bound, we derive necessary and sufficient conditions for exponential stability of (a) continuous-time positive systems whose vector fields are homogeneous and cooperative, and (b) discrete-time positive systems whose vector fields are homogeneous and order preserving. We then present explicit expressions that allow us to quantify the impact of delays on the decay rate and show that the best decay rate of positive linear systems that our bounds provide can be found via convex optimization. Finally, we extend the results to general linear systems with time-varying delays. ## I Introduction Positive systems are dynamical systems whose state variables are constrained to be nonnegative for all time whenever the initial conditions are nonnegative [1]. Due to their importance and wide applicability, the analysis and control of positive systems has attracted considerable attention from the control community (see, _e.g._ , [2, 3, 4, 5, 6, 7, 8] and references therein). Since time delays are omnipresent in engineering systems, the study of stability and control of dynamical systems with delayed states is essential and of practical importance. For general systems, the existence of time delays may impair performance, induce oscillations and even instability [9]. In contrast, positive linear systems have been shown to be insensitive to certain classes of time delays in the sense that a positive linear system is asymptotically stable for all bounded delays if and only if the corresponding delay-free system is asymptotically stable [10, 11, 12, 13]. In addition, if a positive linear system is asymptotically stable for an arbitrary constant delay and some positive initial conditions, the delay-free system is globally asymptotically stable [11]. Many important positive systems are nonlinear. It is thus natural to ask if the insensitivity properties of positive linear systems with respect to time delays will hold also for nonlinear positive systems. In [14], it was shown that for a particular class of nonlinear positive systems, homogeneous cooperative systems with constant delays, this is indeed the case. It is clear that constant delays is an idealized assumption as time delays are often time- varying in practice. However, to the best of our knowledge, there have been rather few studies on stability of nonlinear positive systems with time- varying delays. An important reason for this is that popular techniques for analyzing positive systems with constant delays, such as linear Lyapunov- Krasovskii functionals, cannot be applied or lead to excessive conservatism when the delays are time-varying. At this point, it is worth noting that the results for homogeneous cooperative systems and positive linear systems cited above concern asymptotic stability. However, there are processes and applications for which it is desirable that the system converges quickly enough to the equilibrium. While exponential stability of positive linear systems with constant delays was investigated in [15] using Lyapunov-Krasovskii techniques, extensions to time-varying delays are non-trivial. Moreover, although quantitative stability measures can be highly dependent on the magnitude of delays, no sharp characterization of how a maximum delay bound affects the guaranteed decay rate of a positive system exists to date. This paper addresses these issues. At the core of our paper is a set of powerful conditions for establishing exponential stability of a particular class of nonlinear continuous- and discrete-time positive systems with bounded time-varying delays. More specifically, we make the following contributions: * 1) We derive a _necessary and sufficient_ condition for exponential stability of continuous-time positive systems whose constituent vector fields are homogeneous of degree one and cooperative. * 2) For the case which the time delays have a known upper bound, we present an explicit expression that bounds the decay rate of the system. * 3) We demonstrate that the best decay rate of positive linear systems that our bound can provide can be found via convex optimization techniques. * 4) We extend our obtained results to general linear systems with time-varying delays. * 5) Finally, we provide the corresponding counterparts for discrete-time positive systems. The remainder of the paper is organized as follows. In Section II, we review some required background results and introduce the notation that will be used throughout the paper. The main results of this work for continuous- and discrete-time positive systems are stated in Sections III and IV, respectively. Illustrative examples are presented in Section V, justifying the validity of our results. Finally, concluding remarks are given in Section VI. ## II Notation and Preliminaries Vectors are written in bold lower case letters and matrices in capital letters. We let $\mathbb{R}$, $\mathbb{N}$, and $\mathbb{N}_{0}$ denote the set of real numbers, natural numbers, and the set of natural numbers including zero, respectively. The non-negative orthant of the n-dimensional real space $\mathbb{R}^{n}$ is represented by $\mathbb{R}^{n}_{+}$. The $i^{th}$ component of a vector $\bm{x}\in\mathbb{R}^{n}$ is denoted by $x_{i}$, and the notation $\bm{x}\geq\bm{y}$ means that $x_{i}\geq y_{i}$ for all components $i$. Given a vector $\bm{v}>\bm{0}$, the weighted $l_{\infty}$ norm is defined by $\displaystyle\\|\bm{x}\\|_{\infty}^{\bm{v}}$ $\displaystyle=\max_{1\leq i\leq n}{\frac{\|x_{i}\|}{v_{i}}}.$ For a matrix $A=[a_{ij}]\in\mathbb{R}^{n\times n}$, $a_{ij}$ denotes the entry in row $i$ and column $j$, and $\|A\|$ is the matrix whose elements are $\|a_{ij}\|$. A matrix $A\in\mathbb{R}^{n\times n}$ is said to be non-negative if $a_{ij}\geq 0$ for all $i$, $j$. It is called Metzler if $a_{ij}\geq 0$ for all $i\neq j$. For a real interval $[a,b]$, $\mathcal{C}\bigl{(}[a,b],\mathbb{R}^{n}\bigr{)}$ denotes the space of all real-valued continuous functions on $[a,b]$ taking values in $\mathbb{R}^{n}$. The upper-right Dini-derivative of a continuous function $h:\mathbb{R}\rightarrow\mathbb{R}$ is denoted by $D^{+}h(\cdot)$. Next, we review the key definitions and results necessary for developing the main results of this paper. We start with the definition of cooperative vector fields. ###### Definition 1 A continuous vector field $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ which is continuously differentiable on $\mathbb{R}^{n}\backslash\\{\bm{0}\\}$ is said to be cooperative if the Jacobian matrix $\frac{\partial f}{\partial x}(\bm{a})$ is Metzler for all $\bm{a}\in\mathbb{R}^{n}_{+}\backslash\\{\bm{0}\\}$. The next proposition provides an important property of cooperative vector fields. ###### Proposition 1 [16, Chapter 3, Remark 1.1] Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ be a cooperative vector field. For any two vectors $\bm{x}$ and $\bm{y}$ in $\mathbb{R}^{n}_{+}\backslash\\{\bm{0}\\}$ with $x_{i}=y_{i}$ and $\bm{x}\geq\bm{y}$, we have $f_{i}(\bm{x})\geq f_{i}(\bm{y})$. The following definition introduces homogeneous vector fields. ###### Definition 2 A vector field $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is called homogeneous of degree $\alpha$ if for all $\bm{x}\in\mathbb{R}^{n}$ and all real $\lambda>0$, $\bm{f}(\lambda\bm{x})=\lambda^{\alpha}\bm{f}(\bm{x})$. When $\alpha=1$, then $f$ is called the homogeneous of degree one. Finally, we recall the definition of an order-preserving vector field. ###### Definition 3 A vector field $g:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is said to be order-preserving on $\mathbb{R}^{n}_{+}$ if $\bm{g}(\bm{x})\geq\bm{g}(\bm{y})$ for any $\bm{x},\bm{y}\in\mathbb{R}^{n}_{+}$ such that $\bm{x}\geq\bm{y}$. ## III Continuous-Time Case Consider the continuous-time nonlinear dynamical system $\displaystyle{\mathcal{G}:}$ $\displaystyle\left\\{\begin{array}[l]{ll}\dot{\bm{x}}\bigl{(}t\bigr{)}=\bm{f}\bigl{(}\bm{x}(t)\bigr{)}+\bm{g}\bigl{(}\bm{x}(t-\tau(t))\bigr{)},&t\geq 0,\\\ \bm{x}\bigl{(}t\bigr{)}=\bm{\varphi}\bigl{(}t\bigr{)},&t\in[-\tau_{\max},0].\end{array}\right.$ (3) Here, $\tau_{\max}\geq 0$, $\bm{x}(t)\in\mathbb{R}^{n}$ is the system state, $f,g:~{}\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ are system vector fields with $\bm{f}(\bm{0})=\bm{g}(\bm{0})=\bm{0}$, and $\bm{\varphi}(\cdot)\in\mathcal{C}\bigl{(}[-\tau_{\max},0],\mathbb{R}^{n}\bigr{)}$ is the vector-valued initial function specifying the initial state of the system. The delay $\tau(\cdot)$ is assumed to be continuous with respect to time, not necessarily continuously differentiable, and satisfies $0\leq\tau(t)\leq\tau_{\max}$ for all $t\geq 0$. While no restriction on the derivative of $\tau(t)$ (such as $\dot{\tau}<1$) is imposed, causality of the state space for system (3) even under fast-varying delays is preserved, since $\tau(\cdot)$ is assumed to be bounded [17]. In the remainder of the section, vector fields $f$ and $g$ satisfy Assumption 1. ###### Assumption 1 The following properties hold. 1. a) $f$ and $g$ are continuous on $\mathbb{R}^{n}$, continuously differentiable on $\mathbb{R}^{n}\backslash\\{\bm{0}\\}$, and homogeneous; 2. b) $f$ is cooperative and $g$ is order-preserving on $\mathbb{R}^{n}_{+}$. Assumption 1a) implies that $f$ and $g$ are globally Lipschitz on $\mathbb{R}^{n}$ [14, Lemma 2.1]. Since $\bm{\varphi}(\cdot)$ and $\tau(\cdot)$ are continuous functions of time, it then follows that there exists a unique $\bm{x}(t)$ defined on $[-\tau_{\max},\infty)$ that coincides with $\bm{\varphi}(\cdot)$ on $[-\tau_{\max},0]$ and satisfies (3) for $t\geq 0$ [9, pp. 408–409]. The time-delay dynamical system $\mathcal{G}$ given by (3) is said to be positive if for every non-negative initial condition $\bm{\varphi}(\cdot)\in\mathcal{C}\bigl{(}[-\tau_{\max},0],\mathbb{R}^{n}_{+}\bigr{)}$, the corresponding state trajectory is non-negative, that is $\bm{x}(t)\in\mathbb{R}_{+}^{n}$ for all $t\geq 0$. It follows from [16, Chapter 5, Theorem 2.1] that Assumption 1b) ensures the positivity of system $\mathcal{G}$ given by (3). While $\bm{x}=\bm{0}$ is clearly an equilibrium point of the system (3), it is not necessarily stable. Moreover, the stability of general systems may depend on the magnitude and variation of the time delays. However, it was shown in [14, Theorem 4.1] that under Assumption 1, the positive system (3) with constant delays $(\tau(t)=\tau_{\max}\;\textup{for all}\;t\geq 0)$ is globally asymptotically stable for all $\tau_{\max}\geq 0$ if and only if the undelayed system $(\tau_{\max}=0)$ is globally asymptotically stable. Our main objectives are therefore $(i)$ to determine if a similar delay-independent stability result holds for the homogeneous cooperative system (3) with bounded time-varying delays; and $(ii)$ to determine how the decay rate of the positive system (3) depends on the magnitude of time delays. The following theorem establishes a necessary and sufficient condition for exponential stability of homogeneous cooperative systems with bounded time- varying delays and is our first key result. ###### Theorem 1 For system $\mathcal{G}$ given by (3), suppose Assumption 1 holds. The following statements are equivalent. * (a) There exists a vector $\bm{v}>\bm{0}$ such that $\displaystyle\bm{f}(\bm{v})+\bm{g}(\bm{v})<\bm{0}.$ (4) * (b) The positive system $\mathcal{G}$ is globally exponentially stable for all bounded time delays. In particular, every solution $\bm{x}(t)$ of $\mathcal{G}$ satisfies $\displaystyle\\|\bm{x}(t)\\|_{\infty}^{\bm{v}}\leq\\|\bm{\varphi}\\|e^{-\eta t},\quad t\geq 0,$ where $\\|\bm{\varphi}\\|=\sup_{-\tau_{\max}\leq s\leq 0}\\|\bm{\varphi}(s)\\|_{\infty}^{\bm{v}}$, $\eta\in\bigl{(}0,\min_{1\leq i\leq n}\eta_{i}\bigr{)}$, and $\eta_{i}$ is the unique positive solution of the equation $\displaystyle\left(\frac{f_{i}(\bm{v})}{v_{i}}\right)+\left(\frac{g_{i}(\bm{v})}{v_{i}}\right)e^{\eta_{i}\tau_{\max}}+\eta_{i}=0,\quad i=1,\ldots,n.$ (5) ###### Proof: See Appendix -A. ∎ ###### Remark 1 Equation (5) has three parameters: the positive vector $\bm{v}$, $\tau_{\max}$, and $\eta_{i}$. For any fixed $\bm{v}>\bm{0}$ and $\tau_{\max}\geq 0$, (5) is a nonlinear equation with respect to $\eta_{i}$. The left-hand side of (5) is strictly monotonically increasing in $\eta_{i}>0$ and, by (4), is smaller than the right-hand side for $\eta_{i}=0$. Therefore, (5) always admits a unique positive solution $\eta_{i}$. According to Theorem 1, the homogeneous cooperative system $\mathcal{G}$ given by (3) is globally exponentially stable for all bounded delays if and only if the the corresponding system without delay is stable. In other words, the exponential stability does not depend on the magnitude of the delays, but only on the vector fields. Moreover, any vector $\bm{v}>\bm{0}$ satisfying (4) can be used to find a guaranteed decay rate of the positive system $\mathcal{G}$ by computing the associated $\eta$. Note that $\eta_{i}$ in (5) is monotonically decreasing in $\tau_{\max}$ and approaches zero as $\tau_{\max}$ tends to infinity. Hence, the guaranteed decay rate deteriorates with increasing $\tau_{\max}$. ###### Remark 2 It has been shown in [14, Proposition 3.1] that (4) has a feasible solution $\bm{v}>\bm{0}$ if and only if there does not exist a non-zero vector $\bm{w}\geq\bm{0}$ satisfying $\bm{f}(\bm{w})+\bm{g}(\bm{w})\geq\bm{0}$. This result provides an alternative test for checking the global exponential stability of the homogeneous cooperative system $\mathcal{G}$ with time- varying delays. ###### Remark 3 The result in Theorem 1 can be easily extended to positive nonlinear systems with multiple delays of the form $\displaystyle\dot{\bm{x}}\bigl{(}t\bigr{)}$ $\displaystyle=\bm{f}\bigl{(}\bm{x}(t)\bigr{)}+\sum_{s=1}^{p}\bm{g}_{s}\bigl{(}\bm{x}(t-\tau_{s}(t))\bigr{)}.$ Here, $p\in\mathbb{N}$, $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is cooperative and homogeneous of degree one, $g_{s}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ for $s=1,\ldots,p$ are homogenous and order-preserving on $\mathbb{R}^{n}_{+}$, and $0\leq\tau_{s}(t)\leq\tau_{\max}$ for $t\geq 0$. In this case, the stability condition (4) becomes $\displaystyle\bm{f}(\bm{v})+\sum_{s=1}^{p}\bm{g}_{s}(\bm{v})<\bm{0}.$ We now discuss delay-independent exponential stability of a special case of (3), namely the continuous-time linear dynamical system $\mathcal{G}_{L}$ of the form $\displaystyle{\mathcal{G}}_{L}:$ $\displaystyle\left\\{\begin{array}[l]{ll}\dot{\bm{x}}\bigl{(}t\bigr{)}=A\bm{x}\bigl{(}t\bigr{)}+B\bm{x}\bigl{(}t-\tau(t)\bigr{)},&t\geq 0,\\\ \bm{x}\bigl{(}t\bigr{)}=\bm{\varphi}\bigl{(}t\bigr{)},&t\in[-\tau_{\max},0].\end{array}\right.$ (8) In terms of (3), $\bm{f}(\bm{x})=A\bm{x}$ and $\bm{g}(\bm{x})=B\bm{x}$. It is easy to verify that if $A$ is Metzler and $B$ is non-negative, Assumption 1 is satisfied. We then have the following special case of Theorem 1. ###### Theorem 2 Consider linear system $\mathcal{G}_{L}$ given by (8) where $A$ is Metzler and $B$ is non-negative. Then, there exists a vector $\bm{v}>\bm{0}$ such that $\displaystyle\bigl{(}A+B\bigr{)}\bm{v}<\bm{0},$ (9) if and only if the positive system $\mathcal{G}_{L}$ is globally exponentially stable for all bounded delays. The stability condition (9) is a linear programming problem in $\bm{v}$, and thus can be verified numerically in polynomial time. Clearly, the exponential bound on the decay rate of positive linear systems that our results can ensure depends on the choice of vector $\bm{v}$, and that an arbitrary feasible $\bm{v}$ not necessarily gives a tight bound on the actual decay rate. However, we will show that the best guaranteed decay rate can be found via convex optimization. To this end, we use the change of variables $z_{i}=\textup{ln}(v_{i})$, $i=1,\ldots,n$. Then, the search for $\bm{v}$ can be formulated as $\displaystyle\textbf{maximize}\hskip 14.22636pt\eta$ $\displaystyle\textbf{subject to}\hskip 8.5359pt\eta<\eta_{i},$ (10a) $\displaystyle\hskip 8.5359pta_{ii}+b_{ii}+\sum_{j\neq i}\bigl{(}a_{ij}+b_{ij}\bigr{)}e^{z_{j}-z_{i}}<0,$ (10b) $\displaystyle\hskip 8.5359pta_{ii}+\sum_{j\neq i}a_{ij}e^{z_{j}-z_{i}}+\sum_{j=1}^{n}b_{ij}e^{z_{j}-z_{i}+\eta_{i}\tau_{\max}}+\eta_{i}\leq 0,\quad i=1,\ldots,n,$ (10c) where the last two constraints are (9) and (5) in the new variables, respectively. The optimization variables are the decay rate $\eta$ and the vector $\bm{z}=[z_{1},\ldots,z_{n}]^{T}$. Since $a_{ij}\geq 0$ for all $i\neq j$ and $b_{ij}\geq 0$ for all $i,j$, the last two constraints in (10) are convex in $\eta$ and $z$. This implies that this is a convex optimization problem; hence, it can be efficiently solved. ###### Remark 4 A necessary and sufficient condition for asymptotic stability of the positive linear system (8) with time-varying delays has been established in [12]. Moreover, in [18], it has been shown that if (8) is asymptotically stable for all bounded delays, it is also exponentially stable for all bounded delays. However, the impact of delays on the decay rate of (8) was missing in [12, 18]. Thus, not only do we extend the result of [18] to general homogeneous cooperative systems (not necessarily linear), but we also provide an explicit exponential bound on the decay rate. We now extend Theorem 2 to general linear systems, not necessarily positive. ###### Theorem 3 Suppose that there exists a vector $\bm{v}>\bm{0}$ such that $\displaystyle\bigl{(}A^{M}+\|B\|\bigr{)}\bm{v}<\bm{0},$ (11) where $A^{M}=[a_{ij}^{M}]$ is a matrix with $a^{M}_{ii}=a_{ii}$ and $a^{M}_{ij}=\|a_{ij}\|$ for all $i\neq j$. Let $\eta_{i}$ be the unique positive solution of the equation $\displaystyle\biggl{(}a_{ii}+\sum_{j\neq i}\frac{1}{v_{i}}\|a_{ij}\|v_{j}\biggr{)}+\biggl{(}\sum_{j=1}^{n}\frac{1}{v_{i}}\bigl{\|}b_{ij}\|v_{j}\biggr{)}e^{\eta_{i}\tau_{\max}}+\eta_{i}=0.$ (12) Then, linear system $\mathcal{G}_{L}$ given by (8) is globally exponentially stable. Furthermore, $\displaystyle\\|\bm{x}(t)\\|_{\infty}^{\bm{v}}\leq\\|\bm{\varphi}\\|e^{-\eta t},\ \ t\geq 0,$ where $0<\eta<\min_{1\leq i\leq n}\eta_{i}$. ###### Proof: See Appendix -B. ∎ ###### Remark 5 The stability condition (11) does not include any information on the magnitude of delays, so it ensures delay-independent stability. Since $A^{M}$ is Metzler and $\|B\|$ is non-negative, $A^{M}+\|B\|$ is Metzler. It follows from [4, Proposition 2] that inequality (11) holds if and only if $A^{M}+\|B\|$ is Hurwitz, i.e., all its eigenvalues have negative real parts. ## IV Discrete-time Case Next, we consider the discrete-time analog of (3): $\displaystyle{\Sigma}:$ $\displaystyle\left\\{\begin{array}[l]{ll}\bm{x}\bigl{(}k+1\bigr{)}=\bm{f}\bigl{(}\bm{x}(k)\bigr{)}+\bm{g}\bigl{(}\bm{x}(k-d(k))\bigr{)},&k\in\mathbb{N}_{0}\\\ \bm{x}\bigl{(}k\bigr{)}=\bm{\phi}\bigl{(}k\bigr{)},&k\in\\{-d_{\max},\ldots,0\\}.\end{array}\right.$ (15) Here, $\bm{x}(k)\in\mathbb{R}^{n}$ is the state variable, $f,g:~{}\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$, $\bm{f}(\bm{0})=\bm{g}(\bm{0})=\bm{0}$, $d_{\max}\in\mathbb{N}_{0}$, $d(k)\in\mathbb{N}_{0}$ represents the time-varying delay satisfying $0\leq d(k)\leq d_{\max}$ for all $k\in\mathbb{N}_{0}$, and $\bm{\phi}(\cdot):\\{-d_{\max},\ldots,0\\}\rightarrow\mathbb{R}^{n}$ is the vector sequence specifying the initial state of the system. For the remainder of this section, Assumption 2 holds. ###### Assumption 2 $f$ and $g$ are continuous on $\mathbb{R}^{n}$, homogeneous of degree one, and order-preserving on $\mathbb{R}^{n}_{+}$. The time-delay dynamical System $\Sigma$ given by (15) is said to be positive if for every non-negative initial condition $\bm{\phi}(\cdot)\in\mathbb{R}^{n}_{+}$, the corresponding solution is non- negative, i.e., $\bm{x}(k)\geq\bm{0}$ for all $k\in\mathbb{N}$. Note that under Assumption 2, system $\Sigma$ is positive. Next theorem shows that homogeneous monotone systems are insensitive to bounded delays. ###### Theorem 4 For system $\Sigma$ given by (15), suppose Assumption 2 holds. Then, the following statements are equivalent. * (a) There exists a vector $\bm{v}>\bm{0}$ such that $\displaystyle\bm{f}(\bm{v})+\bm{g}(\bm{v})<\bm{v}.$ (16) * (b) The positive system $\Sigma$ is globally exponentially stable for all bounded time delays. In particular, every solution $\bm{x}(k)$ of $\Sigma$ satisfies $\displaystyle\\|\bm{x}(k)\\|_{\infty}^{\bm{v}}\leq\\|\bm{\phi}\\|\gamma^{k},\quad k\in\mathbb{N}_{0},$ (17) where $\\|\bm{\phi}\\|=\sup_{-d_{\max}\leq s\leq 0}\\|\bm{\phi}(s)\\|_{\infty}^{\bm{v}}$, $\gamma=\max_{1\leq i\leq n}\gamma_{i}$, and $\gamma_{i}\in(0,1)$ is the unique positive solution of the equation $\displaystyle\left(\frac{f_{i}(\bm{v})}{v_{i}}\right)+\left(\frac{g_{i}(\bm{v})}{v_{i}}\right)\gamma_{i}^{-d_{\max}}=\gamma_{i}.$ (18) ###### Proof: See Appendix -C. ∎ Theorem 4 provides a test for the global exponential stability of the homogeneous monotone system (15) with time-varying delays. In addition, for any vector $\bm{v}>\bm{0}$ that satisfies (16), this theorem provides an explicit bound on the impact that an increasing delay has on the decay rate. Note that $\gamma_{i}$ is monotonically increasing in $d_{\max}$, and approaches one as $d_{\max}$ tends to infinity. Hence, the guaranteed decay rate slows down as the delays increase in magnitude. Let $\bm{f}(\bm{x})=A\bm{x}$ and $\bm{g}(\bm{x})=B\bm{x}$ such that $A,B\in\mathbb{R}^{n\times n}$ are non-negative matrices. Then, homogeneous monotone system (15) reduces to the positive linear system $\Sigma_{L}$ of the form $\displaystyle{\Sigma}_{L}:$ $\displaystyle\left\\{\begin{array}[l]{ll}{\bm{x}}\bigl{(}k+1\bigr{)}=A\bm{x}\bigl{(}k\bigr{)}+B\bm{x}\bigl{(}k-d(k)\bigr{)},&k\in\mathbb{N}_{0}\\\ \bm{x}\bigl{(}k\bigr{)}=\bm{\phi}\bigl{(}k\bigr{)},&k\in\\{-d_{\max},\ldots,0\\}.\end{array}\right.$ (21) Theorem 4 helps us to derive a necessary and sufficient condition for exponential stability of discrete-time positive linear systems. Specifically, we note the following. ###### Theorem 5 Consider linear system $\Sigma_{L}$ given by (21) where $A$ and $B$ are non- negative. Then, there exists a vector $\bm{v}>\bm{0}$ such that $\displaystyle\bigl{(}A+B\bigr{)}\bm{v}<\bm{v},$ (22) if and only if the positive system $\Sigma_{L}$ is globally exponentially stable for all bounded delays. In order to find the best decay rate of the positive linear system (21) that our bound can provide, we use the logarithmic change of variables $z_{i}=\textup{ln}(v_{i})$ and $\bar{\gamma}_{i}=\textup{ln}(\gamma_{i})$. Note that these change of variables are valid since the variables $v_{i}$ and $\gamma_{i}$ are required to be positive for all $i$. Then, the search for vector $\bm{v}$ can be formulated as $\displaystyle\textbf{minimize}\hskip 14.22636pte^{\bar{\gamma}}$ $\displaystyle\textbf{subject to}\hskip 8.5359pte^{\bar{\gamma}_{i}-\bar{\gamma}}\leq 1,$ (23a) $\displaystyle\hskip 8.5359pt\sum_{j=1}^{n}\bigl{(}a_{ij}+b_{ij}\bigr{)}e^{z_{j}-z_{i}}<1,$ (23b) $\displaystyle\hskip 8.5359pt\sum_{j=1}^{n}a_{ij}e^{z_{j}-z_{i}-\bar{\gamma_{i}}}+\sum_{j=1}^{n}b_{ij}e^{z_{j}-z_{i}-\bar{\gamma_{i}}(d_{\max}+1)}\leq 1,\quad i=1,\ldots,n,$ (23c) where the last two constraints are (22) and (18) in the new variables, respectively. Here, the optimization variables are the vector $\bm{z}=[z_{1},\ldots,z_{n}]^{T}$ and $\bar{\gamma}$. Since the constraints in (23) define a convex set and the objective function is convex, (23) is a convex optimization problem. This implies that it can be solved globally and efficiently. We now give an extension of Theorem 5 to general linear systems with time- varying delays. ###### Theorem 6 Suppose that there exists a vector $\bm{v}>\bm{0}$ such that $\displaystyle\bigl{(}\|A\|+\|B\|\bigr{)}\bm{v}<\bm{v}.$ (24) Let $\gamma_{i}$ be the positive solution of the equation $\displaystyle\left(\sum_{j=1}^{n}\frac{1}{v_{i}}\bigl{\|}a_{ij}\|v_{j}\right)+\biggl{(}\sum_{j=1}^{n}\frac{1}{v_{i}}\bigl{\|}b_{ij}\|v_{j}\biggr{)}\gamma_{i}^{-d_{\max}}=\gamma_{i}.$ (25) Then, the discrete-time linear system (21) is globally exponentially stable. Moreover, $\displaystyle\\|\bm{x}(k)\\|_{\infty}^{\bm{v}}\leq\gamma^{k}\\|\bm{\phi}\\|,\quad k\in\mathbb{N}_{0},$ (26) where $\gamma=\max_{1\leq i\leq n}\gamma_{i}$. ###### Proof: See Appendix -D. ∎ ## V Illustrative Examples ### V-A Continuous-time Nonlinear Positive System Consider continuous-time nonlinear dynamical system $\mathcal{G}$ given by (3) with $\displaystyle\bm{f}(x_{1},x_{2})$ $\displaystyle=\begin{bmatrix}-3&6\\\ 2&-2\end{bmatrix}\begin{bmatrix}x_{1}\\\ x_{2}\end{bmatrix}-\sqrt{x_{1}^{2}+x_{2}^{2}}\begin{bmatrix}3\\\ 1\end{bmatrix},\quad\bm{g}(x_{1},x_{2})=\begin{bmatrix}\frac{x_{1}x_{2}}{\sqrt{x_{1}^{2}+x_{2}^{2}}}\\\ \frac{x_{1}x_{2}}{\sqrt{2x_{1}^{2}+3x_{2}^{2}}}\end{bmatrix}.$ (27) It is straightforward to verify that both $f$ and $g$ satisfy Assumption 1 [14, Example 4.1]. Moreover, $\bm{f}(1,1)+\bm{g}(1,1)<\bm{0}$. It follows from Theorem 1 that (27) is globally exponentially stable for all bounded time delays. For example, let $\tau(t)=5+\sin(t)$ and set $\tau_{\max}=6$. By using the vector $\bm{v}=~{}(1,1)$ together with $\tau_{\max}=6$, the solutions to the equation (5) can be obtained as $\eta_{1}=0.0825$ and $\eta_{2}=0.1705$. Thus, the decay rate of positive system (27) is upper bounded by $\eta\approx\min\\{0.0825,0.1705\\}=0.0825$. In particular, $\\|\bm{x}(t)\\|_{\infty}^{\bm{v}}\leq\\|\bm{\varphi}\\|e^{-0.0825t}$ for all $t\geq 0$. Figure 3 gives the simulation results of the actual decay rate of positive system (27), $x_{1}(t)$ and $x_{2}(t)$, and the theoretical upper bound $e^{-0.0825t}$ when the initial condition is $\bm{\varphi}(t)=(1,1)$ for $t\in[-6,0]$. Note that [14, Theorem 4.1] can not be applied in this example to ascertain the stability of homogeneous cooperative system (27), since the delay is assumed to be time-varying. Figure 1: Comparison of upper bound and actual decay rate of positive system (27) with bounded time-varying delays. ### V-B Continuous-time Linear Positive System Consider the continuous-time linear system (8) with $\displaystyle A$ $\displaystyle=\begin{bmatrix}-6&2\\\ 1&-3\end{bmatrix},\;B=\begin{bmatrix}3&0\\\ 0&0.5\end{bmatrix}.$ (28) The time-varying delay is given by $\displaystyle\tau(t)=5+\textup{sin}(t).$ Obviously, one may choose $\tau=6$ as an upper bound on the delay. Since $A$ is Metzler and $B$ is non-negative, the system (28) is _positive_. By Theorem 2, since $A+B$ is Hurwitz, (28) is exponentially stable for any bounded time-varying delays. Moreover, according to the linear inequality (9), the following inequality must be fulfilled $\displaystyle\begin{cases}\begin{bmatrix}-3&2\\\ 1&-2.5\end{bmatrix}\begin{bmatrix}v_{1}\\\ v_{2}\end{bmatrix}<\bm{0},\\\ \hskip 14.22636ptv_{1},v_{2}>0.\end{cases}$ (29) As discussed in Section III, any feasible solution $\bm{v}$ to these inequalities can be used to find a guaranteed rate of convergence of the system (28) by computing the associated $\eta$ in (5). One natural candidate for $\bm{v}$ can be found by considering the delay-free case. The solution of the positive system (28) with zero delay, $\dot{\bm{x}}(t)=(A+B)\bm{x}(t)$, satisfies $\displaystyle\\|\bm{x}(t)\\|_{\infty}^{\bm{v}}\leq\\|\bm{x}(0)\\|_{\infty}^{\bm{v}}\;e^{\mu_{\infty}^{\bm{v}}(A+B)t},\quad t\geq 0.$ For any vector $\bm{v}>\bm{0}$, since $A+B$ is Metlzer, $\pi(A+B)\leq\mu_{\infty}^{\bm{v}}(A+B)$. According to the Perron-Frobenius theorem for Metzler matrices [1, Theorem 17], if $A+B$ is Metzler and irreducible, then there exists an eigenvector $\bm{w}>\bm{0}$ such that $\displaystyle(A+B)\bm{w}$ $\displaystyle=\pi(A+B)\bm{w}.$ It is clear that the vector $\bm{w}$ satisfies $\pi(A+B)=\mu_{\infty}^{\bm{w}}(A+B)$. According to the above discussion, one natural candidate $\bm{v}$ can be the eigenvector of $A+B$ corresponding to $\pi(A+B)$ which gives the fast decay rate of solutions for the undelayed case. For the system (28), $\pi(A+B)=-1.3139$, and the corresponding eigenvector is $\bm{v}^{1}=\begin{bmatrix}0.7645&0.6446\end{bmatrix}^{T}.$ By using this solution together with $\tau=6$, the solutions to the nonlinear equation (5) can be obtained as $\eta_{1}=0.0583,\;\eta_{2}=0.1957.$ Thus, (28) is globally exponentially stable with decay rate $\eta=\min\\{0.0583,0.1957\\}=0.0583$. In particular, $\displaystyle\\|\bm{x}(t)\\|_{\infty}^{\bm{v}^{1}}\leq\sup_{-\tau\leq s\leq 0}\left\\{\\|\bm{\varphi}(s)\\|_{\infty}^{\bm{v}^{1}}\right\\}\;e^{-0.0583t},\ \ t\geq 0.$ The left-hand side of Figure 2 compares $\\|\bm{x}(t)\\|_{\infty}^{\bm{v}^{1}}$ obtained by simulating (28) from initial condition $\bm{\varphi}(t)=\bm{v}^{1}$ and the theoretical decay rate bound $e^{-0.0583t}$. Of course, $\bm{v}^{1}$ is only one of the possible solutions of (29). Next, by solving the convex optimization problem (10), we get $\bm{v}^{\star}=[0.9020,\;0.4317]^{T},\;\eta^{\star}=0.0838,$ which implies that the system (28) is globally exponentially stable with decay rate $0.0838$, and the solution $\bm{x}(t)$ satisfies $\displaystyle\\|\bm{x}(t)\\|_{\infty}^{\bm{v}^{\star}}\leq\sup_{-\tau\leq s\leq 0}\left\\{\\|\bm{\varphi}(s)\\|_{\infty}^{\bm{v}^{\star}}\right\\}\;e^{-0.0838t}.$ The right-hand side of Figure 2 gives the simulation results of $\\|\bm{x}(t)\\|_{\infty}^{\bm{v}^{\star}}$, and the theoretical upper bound $e^{-0.0838t}$ when the initial condition is $\bm{\varphi}(t)=\bm{v}^{\star}$. We can see that the linear inequalities (29) do not help us in guiding our search for a vector $\bm{v}$ which guarantees a fast decay rate. In contrast, solving the convex optimization problem (10) finds the best $\eta^{\star}$ that our bound can guarantee along with the associated $\bm{v}^{\star}$. The bound matches simulations very well and is a significant improvement over simply using the non-optimized $\bm{v}^{1}$. Figure 2: Comparison of upper bounds and actual decay rates of the solution $\bm{x}(t)$ without (left) and with (right) convex optimization for the system described by (28). ### V-C Discrete-time Linear Positive System Consider the discrete-time linear system (21) with $\displaystyle A$ $\displaystyle=\begin{bmatrix}0.4&0.1\\\ 0.2&0.6\end{bmatrix},\;B=\begin{bmatrix}0.3&0\\\ 0&0.1\end{bmatrix}.$ (30) The time-varying delay is given by $d(k)=4+\textup{sin}\left(\frac{k\pi}{2}\right),$ with an upper bound $d=5$. Since $A$ and $B$ are non-negative, the system (30) is _positive_. Since $\rho(A+B)<1$, Theorem 5 guarantees that the system (30) is exponentially stable and that the following set of inequalities have a solution $\displaystyle\begin{cases}\begin{bmatrix}-0.3&0.1\\\ 0.2&-0.3\end{bmatrix}\begin{bmatrix}v_{1}\\\ v_{2}\end{bmatrix}<\bm{0},\\\ \hskip 14.22636ptv_{1},v_{2}>0.\end{cases}$ (31) As in the continuous-time example, any feasible solution $\bm{v}$ of (31) yields a guaranteed decay rate of the system (30) by computing the associated $\gamma$ in (18). To find the optimal $\bm{v}$ for our bound, we solve the convex optimization problem (23), to find the vector $\bm{v}^{\star}$ and its guaranteed decay rate $\gamma^{\star}$: $\bm{v}^{\star}=[0.6884,\;0.7254]^{T},\;\gamma^{\star}=0.9320.$ Therefore, the solution $\bm{x}(k)$ satisfies $\displaystyle\\|\bm{x}(k)\\|_{\infty}^{\bm{v}^{\star}}\leq(0.9320)^{k}\\|\bm{\phi}\\|,\quad k\in\mathbb{N}.$ Figure 3 shows a comparison of $\\|\bm{x}(k)\\|_{\infty}^{\bm{v}^{\star}}$ and the theoretical bound $(0.9320)^{k}$, when the initial condition is $\bm{\phi}(k)=\bm{v}^{\star}$. Figure 3: Comparison of the upper bound and the actual decay rate of the solution $\bm{x}(k)$ for the discrete-time system described by (30). ## VI Conclusions In this paper, we have extended a fundamental property of positive linear systems to a class of nonlinear positive systems. Specifically, we have demonstrated that continuous-time homogeneous cooperative systems and discrete-time homogeneous monotone systems are insensitive to bounded time- varying delays. We have derived a set of necessary and sufficient conditions for establishing delay-independent exponential stability of such positive systems. When the time delays have a known upper bound, explicit expressions that bound the decay rate have been presented. We have further shown that the best bound on the decay rate of positive linear systems that our results can guarantee can be found via convex optimization. Finally, we have extended obtained results to general linear systems with time-varying delays. ### -A Proof of Theorem 1 $(a)\Rightarrow(b):$ Suppose that there exists a vector $\bm{v}>\bm{0}$ such that (4) holds. According to Remark 1, Equation (5) always admits a unique positive solution $\eta_{i}$. Pick a constant $\eta$ satisfying $0<\eta<\min_{1\leq i\leq n}\eta_{i}$. Since the left-hand side of (5) is strictly monotonically increasing in $\eta_{i}>0$, we have $\displaystyle\left(\frac{f_{i}(\bm{v})}{v_{i}}\right)+\left(\frac{g_{i}(\bm{v})}{v_{i}}\right)e^{\eta\tau_{\max}}+\eta<0,\quad\textup{for all}\;i.$ (32) Under Assumption 1, system (3) is positive. Hence, $x_{i}(t)\geq 0$ for all $i$ and all $t\geq 0$. Let $\displaystyle z_{i}(t)=\frac{x_{i}(t)}{v_{i}}-\\|\bm{\varphi}\\|e^{-\eta t}.$ (33) From the definition of $\\|\bm{\varphi}\\|$, $z_{i}(0)\leq 0$ for all $i$. To prove the exponential stability, we will show that $z_{i}(t)\leq 0$ for all $i$ and all $t\geq 0$. By contradiction, suppose this is not true. Then, there exist an index $m\in\\{1,\ldots,n\\}$ and $t_{1}\geq 0$ such that $z_{i}(t)\leq 0$ for $t\in[0,t_{1}]$, $z_{m}(t_{1})=0$, and $\displaystyle D^{+}z_{m}(t_{1})$ $\displaystyle\geq 0.$ (34) From (33), we have $x_{m}(t_{1})=\\|\bm{\varphi}\\|e^{-\eta t_{1}}v_{m}$, and $\bm{x}(t_{1})\leq\\|\bm{\varphi}\\|e^{-\eta t_{1}}\bm{v}$. Now, as $f$ is cooperative and homogeneous of degree one, it follows from Proposition 1 and the above observations that $\displaystyle f_{m}\bigl{(}\bm{x}(t_{1})\bigr{)}$ $\displaystyle\leq f_{m}\bigl{(}\\|\bm{\varphi}\\|e^{-\eta t_{1}}\bm{v}\bigr{)}=\\|\bm{\varphi}\\|e^{-\eta t_{1}}f_{m}\bigl{(}\bm{v}\bigr{)}.$ (35) Case 1) If $\tau(t_{1})\leq t_{1}$, then $t_{1}-\tau(t_{1})\in[0,t_{1}]$, and therefore $z_{i}\bigl{(}t_{1}-\tau(t_{1})\bigr{)}\leq 0$. As a result, $\displaystyle x_{i}\bigl{(}t_{1}-\tau(t_{1})\bigr{)}$ $\displaystyle\leq\\|\bm{\varphi}\\|e^{-\eta(t_{1}-\tau(t_{1}))}v_{i}$ $\displaystyle\leq\\|\bm{\varphi}\\|e^{-\eta(t_{1}-\tau_{\max})}v_{i},\quad i=1,\ldots,n,$ where we used the fact that $\tau(t_{1})\leq\tau_{\max}$ to get the second inequality. Further, as $g$ is order-preserving and homogeneous of degree one, this in turn implies $\displaystyle g_{m}\bigl{(}\bm{x}(t_{1}-\tau(t_{1}))\bigr{)}$ $\displaystyle\leq g_{m}\bigl{(}\\|\bm{\varphi}\\|e^{-\eta(t_{1}-\tau_{\max})}\bm{v}\bigr{)}=\\|\bm{\varphi}\\|e^{-\eta(t_{1}-\tau_{\max})}g_{m}\bigl{(}\bm{v}\bigr{)}.$ (36) The upper-right Dini-derivative of $z_{m}(t)$ along the trajectories of (3) at $t=t_{1}$ is given by $\displaystyle D^{+}z_{m}(t_{1})$ $\displaystyle=\frac{\dot{x}_{m}(t_{1})}{v_{m}}+\\|\bm{\varphi}\\|e^{-\eta t_{1}}\eta$ $\displaystyle=\frac{f_{m}\bigl{(}\bm{x}(t_{1})\bigr{)}+g_{m}\bigl{(}\bm{x}(t_{1}-\tau(t_{1}))\bigr{)}}{v_{m}}+\\|\bm{\varphi}\\|e^{-\eta t_{1}}\eta$ $\displaystyle\leq\\|\bm{\varphi}\\|e^{-\eta t_{1}}\left(\left(\frac{f_{m}(\bm{v})}{v_{m}}\right)+\left(\frac{g_{m}(\bm{v})}{v_{m}}\right)e^{\eta\tau_{\max}}+\eta\right),$ where we substituted (35) and (36) into the second equality. It now follows from (32) that $D^{+}z_{m}(t_{1})<0$. Case 2) If $\tau(t_{1})>t_{1}$, from the definition of $\\|\bm{\varphi}\\|$, we have $\\|\bm{x}(t_{1}-\tau(t_{1}))\\|_{\infty}^{\bm{v}}\leq\\|\bm{\varphi}\\|$. Thus, $\bm{x}(t_{1}-\tau(t_{1}))\leq\\|\bm{\varphi}\\|\bm{v}$, which implies that $g_{m}\bigl{(}\bm{x}(t_{1}-\tau(t_{1}))\bigr{)}\leq\\|\bm{\varphi}\\|g_{m}\bigl{(}\bm{v}\bigr{)}$. Then, $\displaystyle D^{+}z_{m}(t_{1})$ $\displaystyle\leq\\|\bm{\varphi}\\|e^{-\eta t_{1}}\left(\left(\frac{f_{m}(\bm{v})}{v_{m}}\right)+\left(\frac{g_{m}(\bm{v})}{v_{m}}\right)e^{\eta t_{1}}+\eta\right)$ $\displaystyle\leq\\|\bm{\varphi}\\|e^{-\eta t_{1}}\left(\left(\frac{f_{m}(\bm{v})}{v_{m}}\right)+\left(\frac{g_{m}(\bm{v})}{v_{m}}\right)e^{\eta\tau_{\max}}+\eta\right)$ $\displaystyle<0,$ where the second inequality follows from the fact that $t_{1}<\tau(t_{1})\leq\tau_{\max}$. In summary, we conclude that $D^{+}z_{m}(t_{1})<0$, which contradicts (34). Therefore, $z_{i}(t)\leq 0$ for all $t\geq 0$, and hence $\\|\bm{x}(t)\\|_{\infty}^{\bm{v}}\leq\\|\bm{\varphi}\\|e^{-\eta t}$ for $t\geq 0$. This completes the proof. $(b)\Rightarrow(a):$ Assume that system (3) is exponentially stable for all bounded time delays. Particularly, let $\tau(t)=0$. Then, $\dot{\bm{x}}\bigl{(}t\bigr{)}=\bm{f}\bigl{(}\bm{x}(t)\bigr{)}+\bm{g}\bigl{(}\bm{x}(t)\bigr{)}$ is exponentially stable, and hence is asymptotically stable. Since $f+g$ is cooperative and homogeneous of degree one, it follows from [14, Theorem 3.1] that there is some vector $\bm{v}>\bm{0}$ satisfying (4). ### -B Proof of Theorem 3 The proof is almost the same as that of Theorem 1. From (11), Equation (12) always has a unique positive solution $\eta_{i}$ for each $i$. Moreover, if $\eta\in\bigl{(}0,\min_{1\leq i\leq n}\eta_{i}\bigr{)}$, then $\displaystyle\biggl{(}a_{ii}+\sum_{j\neq i}\frac{1}{v_{i}}\|a_{ij}\|v_{j}\biggr{)}+\biggl{(}\sum_{j=1}^{n}\frac{1}{v_{i}}\bigl{\|}b_{ij}\|v_{j}\biggr{)}e^{\eta\tau_{\max}}+\eta<0,$ hold for all $i$. Let $z_{i}(t)=\|x_{i}(t)\|/{v_{i}}-\\|\bm{\varphi}\\|e^{-\eta t}$. We claim that $z_{i}(t)\leq 0$ for all $t\geq 0$. For each $i$, the upper-right Dini-derivative of $z_{i}(t)$ along the trajectories of (8) is given by $\displaystyle D^{+}z_{i}(t)$ $\displaystyle=\frac{\textup{sign}(x_{i})}{v_{i}}\biggl{\\{}\sum_{j=1}^{n}a_{ij}x_{j}\bigl{(}t\bigr{)}+\sum_{j=1}^{n}b_{ij}x_{j}\bigl{(}t-\tau(t)\bigr{)}\biggr{\\}}+\\|\bm{\varphi}\\|e^{-\eta t}\eta$ $\displaystyle=\frac{1}{v_{i}}\biggl{\\{}a_{ii}\|x_{i}(t)\|+\textup{sign}(x_{i})\sum_{j\neq i}a_{ij}x_{j}(t)+\textup{sign}(x_{i})\sum_{j=1}^{n}b_{ij}x_{j}\bigl{(}t-\tau(t)\bigr{)}\biggr{\\}}+\\|\bm{\varphi}\\|e^{-\eta t}\eta$ $\displaystyle\leq\frac{1}{v_{i}}\biggl{\\{}a_{ii}\bigl{\|}x_{i}(t)\bigr{\|}+\sum_{j\neq i}\bigl{\|}a_{ij}\bigr{\|}\bigl{\|}x_{j}(t)\bigr{\|}+\sum_{j=1}^{n}\bigl{\|}b_{ij}\bigr{\|}\bigl{\|}x_{j}(t-\tau(t))\bigr{\|}\biggr{\\}}+\\|\bm{\varphi}\\|e^{-\eta t}\eta.$ If there exists an index $m$ and $t_{1}\geq 0$ such that $z_{i}(t)\leq 0$ for $t\in[0,t_{1}]$ and $z_{m}(t_{1})=0$, then the same arguments as in the proof of Theorem 1 yields $D^{+}z_{m}(t_{1})<0$. The proof is complete. ### -C Proof of Theorem 4 $(a)\Rightarrow(b):$ First note that, for any fixed $d_{\max}\geq 0$ and any fixed $\bm{v}>\bm{0}$, Equation (18) always has a unique solution $\gamma_{i}\in(0,1)$ [19, pp. 444]. Let $\gamma=\max_{1\leq i\leq n}\gamma_{i}$. Since the left-hand side of (18) is strictly monotonically decreasing in $\gamma_{i}$, we have $\displaystyle\left(\frac{f_{i}(\bm{v})}{v_{i}}\right)+\left(\frac{g_{i}(\bm{v})}{v_{i}}\right)\gamma^{-d_{\max}}$ $\displaystyle\leq\gamma_{i}\leq\gamma,$ (37) for all $i$. We now use perfect induction to show that the desired relation (17) is true for all $k\in\mathbb{N}_{0}$. By the definition of $\\|\bm{\phi}\\|$, we have $\\|\bm{x}(0)\\|_{\infty}^{\bm{v}}\leq\\|\bm{\phi}\\|$, which implies that (17) holds for $k=0$. Assume that the induction hypothesis holds for all $k$ up to some $m$, i.e., $x(k)\leq\gamma^{k}\\|\bm{\phi}\\|\bm{v}$ for $k=1,\ldots,m$. Since $f$ and $g$ are homogeneous and order-preserving, it follows that $\displaystyle\begin{split}\bm{f}\bigl{(}\bm{x}(m)\bigr{)}&\leq\gamma^{m}\\|\bm{\phi}\\|\bm{f}\bigl{(}\bm{v}\bigr{)},\\\ \bm{g}\bigl{(}\bm{x}(m-d(m))\bigr{)}&\leq\gamma^{m-d_{\max}}\\|\bm{\phi}\\|\bm{g}\bigl{(}\bm{v}\bigr{)},\end{split}$ (38) where we used the fact that $\gamma<1$ and $d(m)\leq d_{\max}$ to get the second inequality. Using (37) and (38), we obtain $\displaystyle\frac{1}{v_{i}}x_{i}\bigl{(}m+1\bigr{)}$ $\displaystyle=\frac{1}{v_{i}}\bigl{(}f_{i}(\bm{x}(m))+g_{i}(\bm{x}(m-d(m)))\bigr{)}$ $\displaystyle\leq\gamma^{m}\\|\bm{\phi}\\|\left(\left(\frac{f_{i}(\bm{v})}{v_{i}}\right)+\left(\frac{g_{i}(\bm{v})}{v_{i}}\right)\gamma^{-d_{\max}}\right)$ $\displaystyle\leq\gamma^{m+1}\\|\bm{\phi}\\|,\quad i=1,\ldots,n.$ It follows from the definition of weighted $l_{\infty}$ norm that $\\|\bm{x}(m+1)\\|_{\infty}^{\bm{v}}\leq\gamma^{m+1}\\|\bm{\phi}\\|$. This completes the induction proof. $(b)\Rightarrow(a):$ Suppose (15) is globally exponentially stable for all bounded delays. Particularly, let $d(k)=0$. Then, system $\bm{x}\bigl{(}k+1\bigr{)}=\bm{f}\bigl{(}\bm{x}(k)\bigr{)}+\bm{g}\bigl{(}\bm{x}(k)\bigr{)}$ is globally asymptotically stable. Since $f+g$ is continuous, order- preserving, and $(\bm{f}+\bm{g})(\bm{0})=\bm{0}$ , the conclusion follows from [20, Propositions 5.2 and 5.6]. ### -D Proof of Theorem 6 We use perfect induction to prove that the desired relation (26) holds. For each $i$, we have $\displaystyle\frac{1}{v_{i}}\bigl{\|}x_{i}(k+1)\bigr{\|}$ $\displaystyle=\frac{1}{v_{i}}\biggl{\|}\sum_{j=1}^{n}a_{ij}x_{j}(k)+\sum_{j=1}^{n}b_{ij}x_{j}\bigl{(}k-d(k)\bigr{)}\biggr{\|}$ $\displaystyle\leq\frac{1}{v_{i}}\biggl{\\{}\sum_{j=1}^{n}\bigl{\|}a_{ij}\bigr{\|}\bigl{\|}x_{j}(k)\bigr{\|}+\sum_{j=1}^{n}\bigl{\|}b_{ij}\bigr{\|}\bigl{\|}x_{j}\bigl{(}k-d(k)\bigr{)}\bigr{\|}\biggr{\\}}.$ On the other hand, by (24), Equation (25) always admits a unique solution $\gamma_{i}\in(0,1)$ for each $i$. Let $\gamma=\max_{1\leq i\leq n}\gamma_{i}$. It follows that $\displaystyle\biggl{(}\sum_{j=1}^{n}\frac{1}{v_{i}}\bigl{\|}a_{ij}\|v_{j}\biggr{)}+\biggl{(}\sum_{j=1}^{n}\frac{1}{v_{i}}\bigl{\|}b_{ij}\|v_{j}\biggr{)}\gamma^{-d_{\max}}\leq\gamma,\quad i=1,\ldots,n.$ The rest of the proof is similar to the proof of Theorem 4 and is thus omitted. ## References * [1] L. Farina and S. Rinaldi, _Positive Linear Systems: Theory and Applications_. 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Verriest, “Inconsistencies in systems with time-varying delays and their resolution,” _IMA Journal of Mathematical Control and Information_ , vol. 28, no. 2, pp. 147–162, 2011. * [18] X. Liu and J. Lam, “Relationships between asymptotic stability and exponential stability of positive delay systems,” _International Journal of General Systems_ , vol. 42, no. 2, pp. 224–238, 2013. * [19] D. P. Bertsekas and J. N. Tsitsiklis, _Parallel and Distributed Computation_. Prentice-Hall, 1989. * [20] S. Dashkovskiy, B. Ruffer, and F. Wirth, “Discrete-time monotone systems: criteria for global asymptotic stability and applications,” _17th Int. Symp. Math. Theory of Networks and Systems (MTNS)_ , pp. 89–97, 2006.	arxiv-papers	2013-11-12T19:52:01	2024-09-04T02:49:53.504257	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Hamid Reza Feyzmahdavian, Themistoklis Charalambous, Mikael Johansson", "submitter": "Hamid Reza Feyzmahdavian", "url": "https://arxiv.org/abs/1311.2897" }
1311.3074	# The weak choice principle WISC may fail in the category of sets David Michael Roberts111Supported by the Australian Research Council (grant number DP120100106). This paper will appear in the journal _Studia Logica_. [email protected] ###### Abstract The set-theoretic axiom WISC states that for every set there is a _set_ of surjections to it cofinal in _all_ such surjections. By constructing an unbounded topos over the category of sets and using an extension of the internal logic of a topos due to Shulman, we show that WISC is independent of the rest of the axioms of the set theory given by a well-pointed topos. This also gives an example of a topos that is not a predicative topos as defined by van den Berg. ## 1 Introduction Well-known from algebra is the concept of a _projective object_ : in a finitely complete category this is an object $P$ such that any epimorphism with codomain $P$ splits. The axiom of choice (AC) can be stated as saying that every set is projective in the category of sets. Various constructive set theories seek to weaken this, and in particular the axiom known as PAx (Presentation Axiom) [2] or CoSHEP (Category of Sets Has Enough Projectives) asks merely that every set $X$ has an epimorphism $P\twoheadrightarrow X$ where $P$ is a projective set. Many results that seem to rely on the axiom of choice, such as the existence of enough projectives in module categories, may be proved instead with PAx. As a link with a more well-known axiom, PAx imples the axiom of dependent choice. There is, however, an even weaker option, here called WISC (to be explained momentarily). Consider the full subcategory $Surj/X\hookrightarrow\bm{\mathrm{set}}/X$ of surjections with codomain $X$, in some category $\bm{\mathrm{set}}$ of sets; clearly it is a large category. Then PAx implies the statement that $Surj/X$ has a _weakly initial object_ , namely an object with a map to any other object, not necessarily unique (the axiom of choice says $\mathrm{id}_{X}\colon X\to X$ is weakly initial in $Surj/X$). Another way to think of the presentation axiom is that for every set $X$ there is a ‘cover’ $P\twoheadrightarrow X$ such that any surjection $Y\twoheadrightarrow P$ splits. The axiom WISC (Weakly Initial Set of Covers), due to Toby Bartels and Mike Shulman, asks merely that the category $Surj/X$ has a weakly initial _set_ , for every $X$. This is a set $I_{X}$ of objects (that is, of surjections to $X$) such that for any other object (surjection), there is a map from _some_ object in $I_{X}$. To continue the geometric analogy, this is like asking that there is a set of covers of any $X$ such that each surjection $Y\twoheadrightarrow X$ splits locally over at least one cover in that set. An example implication of WISC is that the cohomology $H^{1}(X,G)$ defined by Blass in [3] is indeed a set. The assertion that $H^{1}(X,G)$ is a proper class seems to be strictly weaker than $\neg$WISC, but to the author’s knowledge no models have yet been produced where this is the case. The origin of the axiom WISC (see [9]) was somewhat geometric in flavour but the question naturally arises whether toposes, and in particular the category of sets, can fail to satisfy WISC. A priori, there is no particular reason why WISC should hold, so the burden is to supply an example where it fails. It goes without saying that neither AC nor PAx can hold in such an example. The first result in this direction was from van den Berg (see [12]222In that paper, WISC is used in a guise of an equivalent axiom called AMC, the Axiom of Multiple Choice. To avoid confusion with other axioms with that name, this paper sticks with the term ‘’WISC’.) who proved that WISC implies the existence of a proper class of regular cardinals, and so WISC must fail in Gitik’s model of ZF [5]. This model is constructed assuming the existence of a proper class of certain large cardinals, and it has no regular cardinals bigger than $\aleph_{0}$. Working in parallel to the early development of the current paper, Karagila [6] gave a model of ZF in which there is a proper class of incomparable sets (sets with no injective resp. surjective functions between them) surjecting onto the ordinal $\omega$. This gave a large- cardinal-free proof that WISC was independent of the ZF axioms, answering a question raised by van den Berg. The current paper started as an attempt to also give, via category-theoretic methods, a large-cardinals-free proof of the independence of WISC from ZF. Since the release of [6], this point is moot as far as independence from ZF goes. However, the proof in [6] relies on a symmetric submodel of a class- forcing model, which is rather heavy machinery. Thus this paper, while proving a slightly weaker result, does so with, in the opinion of the author, far less. The approach we take is to consider the negation of WISC in the _internal logic_ of a (boolean) topos. This allows us to interpret the theory of a well- pointed topos together with $\neg$WISC. However, since this internal version of WISC holds in any Grothendieck topos (assuming for example AC in the base topos of sets) [12], we necessarily consider a _non-bounded_ topos over the base topos of sets (recall that boundedness of a topos is equivalent to it being a Grothendieck topos). In fact the topos we consider is a variant on the ‘faux topos’ mentioned in [1, IV 2.8] (wherein ‘topos’ meant what we now call a Grothendieck topos). The reader familiar with such things may have already noticed that WISC or its negation is not the sort of sentence that can be written via the usual Kripke- Joyal semantics (see e.g. [8, §VI.6]) used for internal logic, as it contains unbounded quantifiers. As a result, we will be using an extension called the _stack semantics_ , given by Shulman [10], that permits their use. The majority of the proof is independent of the details of the stack semantics, which are only used to translate WISC from a statement in a well-pointed topos to a general topos (in fact a locally connected topos, as this is the only case we will consider). To summarise: starting from a well-pointed topos with natural number object we give a proper-class-sized group $\mathcal{Z}$ equipped with a certain topology, and consider the topos $\mathcal{Z}\bm{\mathrm{set}}$ of sets with a continuous action of this group. Of course, the preceeding sentence needs to be formalised appropriately, and we do this in terms of a base well-pointed topos and a large diagram of groups therein. We reduce the failure of WISC in the internal logic of $\mathcal{Z}\bm{\mathrm{set}}$ to simple group-theoretic statements. It should be pointed out that classical logic is used throughout, and all the toposes in this note are boolean. Finally, the topos constructed as in the previous paragraph is not a _predicative topos_ as defined in [11]. These are analogues of toposes that should capture predicative mathematics, as toposes capture the notion of intuitionistic mathematics. This apparent failure is understood and carefully discussed in _loc. cit._ ; the example given in this paper is hopefully of use as a foil in the development of predicative toposes. The author’s thanks go to Mike Shulman for helpful and patient discussions regarding the stack semantics. Thanks are also due to an anonymous referee who found an earlier version of this paper contained some critical errors. ## 2 WISC in the internal language We use the following formulation of WISC, equivalent to the usual statement in a well-pointed topos and due to François Dorais [4]. ###### WISC (in $\bm{\mathrm{set}}$). For every set $X$ there is a set $Y$ such that for every surjection $q\colon Z\to X$ there is a map $s\colon Y\to Z$ such that $q\circ s\colon Y\to X$ is a surjection. The aim of this paper is to show that an internal version of $\neg$WISC is valid in the (non-well-pointed) topos constructed in section 3 below. The internal logic of a topos, in the generality required here, is given by the _stack semantics_. We refer to [10, section 7] for more details on the stack semantics, recalling purely what is necessary for the translation of WISC into the internal logic of a topos $S$ (Shulman takes weaker assumptions on $S$, but this extra generality is not needed here). If $U$ is an object of $S$ we say that a formula of category theory $\phi$ with parameters in the category $S/U$ is a _formula over $U$_. We have333Technically, this is only after choosing a splitting of the fibred category $S^{\mathbf{2}}\to S$, but in practice one only deals with a finite number of instances so this can be glossed over. the base change functor $p^{}\colon S/U\to S/V$ for any map $p\colon V\to U$, and call the formula over $V$ given by replacing each parameter of $\phi$ by its image under $p^{}$ the _pullback_ of $\phi$ (denoted $p^{}\phi$). Note that the language of category theory is taken to be two-sorted, so there are quantifiers for both objects and arrows separately. Here and later $\twoheadrightarrow$ denotes a map that is an epimorphism. ###### Definition 1. (Shulman [10]) Given the topos $S$, and a sentence $\phi$ over $U$, we define the relation $U\Vdash\phi$ recursively as follows • $U\Vdash(f=g)\leftrightarrow f=g$ * • $U\Vdash\top$ always * • $U\Vdash\bot\leftrightarrow U\simeq 0$ * • $U\Vdash(\phi\wedge\psi)\leftrightarrow U\Vdash\phi$ and $U\Vdash\psi$ * • $U\Vdash(\phi\vee\psi)\leftrightarrow U=V\cup W$, where $i\colon V\hookrightarrow U$ and $j\colon W\hookrightarrow U$ are subobjects such that $V\Vdash i^{}\phi$ and $W\Vdash j^{}\psi$ * • $U\Vdash(\phi\Rightarrow\psi)\leftrightarrow$ for any $p\colon V\to U$ such that $V\Vdash p^{}\phi$, also $V\Vdash p^{}\psi$ * • $U\Vdash\neg\phi\leftrightarrow U\Vdash(\phi\Rightarrow\bot)$ * • $U\Vdash(\exists X)\phi(X)\leftrightarrow\exists p\colon V\twoheadrightarrow U$ and $A\in\operatorname{Obj}(S/V)$ such that $V\Vdash p^{}\phi(A)$ • $U\Vdash(\exists f\colon A\to B)\phi(f)\leftrightarrow\exists p\colon V\twoheadrightarrow U$ and $g\colon p^{}A\to p^{}B\in\operatorname{Mor}(S/V)$ such that $V\Vdash p^{}\phi(g)$ • $U\Vdash(\forall X)\phi(X)\leftrightarrow$ for any $p\colon V\to U$ and $A\in\operatorname{Obj}(S/V)$, $V\Vdash p^{}\phi(A)$ • $U\Vdash(\forall f\colon A\to B)\phi(f)\leftrightarrow$ for any $p\colon V\to U$ and $j\colon p^{}A\to p^{}B\in\operatorname{Mor}(S/V)$, $V\Vdash p^{}\phi(j)$ If $\phi$ is a formula over $1$ we say $\phi$ is _valid_ if $1\Vdash\phi$. Comparing with [8, §VI.6] one can recognise the Kripke-Joyal semantics as a fragment of the above, where attention is restricted to monomorphisms rather than arbitrary objects in slice categories, and all quantifiers are bounded. Since our intended model will be built using not just an arbitrary topos, but a locally connected and cocomplete one, the following lemma will simplify working in the internal logic. The proof follows that of lemma 7.3 in [10]. We recall that a locally connected topos $E$ is a topos over $\bm{\mathrm{set}}$ with an additional left adjoint $\pi_{0}$ to the inverse image part of the global section functor, and an object $A$ is called _connected_ if $\pi_{0}(A)=1$. ###### Lemma 2. Let $E$ be a locally connected cocomplete topos. Then then if for any _connected_ object $V$, arrow $p\colon V\to U$ and $A\in\operatorname{Obj}(S/V)$ we have $V\Vdash p^{}\phi(A)$, then $U\Vdash(\forall X)\phi(X)$. Here ‘locally connected cocomplete’ is relative to a base topos $\bm{\mathrm{set}}$ that is well-pointed (hence boolean) topos with natural number object (nno). We will refer to the objects of $\bm{\mathrm{set}}$ as ‘sets’, but without an implication that these arise from a particular collection of axioms. We will assume throughout that all toposes will come with an nno. For a locally connected and cocomplete topos the statement of WISC translates, using definition 1 and applying lemma 2, into the stack semantics as follows: $\displaystyle\forall\ X\to U,\ U\text{ connected,}$ (1) $\displaystyle\exists\ V\stackrel{{\scriptstyle p}}{{\twoheadrightarrow}}U,\ Y\to V,$ $\displaystyle\forall\ W\stackrel{{\scriptstyle q}}{{\to}}V,\ W\text{ connected,}\ Z\stackrel{{\scriptstyle g}}{{\twoheadrightarrow}}W\times_{U}X,$ $\displaystyle\exists\ T\stackrel{{\scriptstyle r}}{{\twoheadrightarrow}}W,\ T\times_{V}Y\xrightarrow{(\mathrm{pr}_{1},l)}T\times_{W}Z,$ $\displaystyle\text{the map}\ T\times_{V}Y\xrightarrow{(\mathrm{pr}_{1},l)}T\times_{W}Z\xrightarrow{r^{}(g)}T\times_{U}X\text{ is an epi}.$ Note also that “is an epi” is a proposition whose statement in the stack semantics is equivalent to the external statement (see discussion around example 7.10 of [10]). One does not need any knowledge of the stack semantics for the rest of this paper, and the uninitiated may choose to take (1) as the _definition_ of WISC in the internal language of a locally connected cocomplete topos, and ignore the stack semantics entirely. We will give a boolean $\bm{\mathrm{set}}$-topos $E$ that is locally connected and cocomplete and in which the following statement, the negation of (1), holds: $\displaystyle\exists\ X\to U,\ U\text{ connected,}$ (2) $\displaystyle\forall\ V\stackrel{{\scriptstyle p}}{{\twoheadrightarrow}}U,\ Y\to V,$ $\displaystyle\exists\ W\stackrel{{\scriptstyle q}}{{\to}}V,\ W\text{ connected,}\ Z\stackrel{{\scriptstyle g}}{{\twoheadrightarrow}}W\times_{U}X,$ $\displaystyle\forall\ T\stackrel{{\scriptstyle r}}{{\twoheadrightarrow}}W,\ T\times_{V}Y\xrightarrow{(\mathrm{pr}_{1},l)}T\times_{W}Z,$ $\displaystyle\text{the map}\ T\times_{V}Y\xrightarrow{(\mathrm{pr}_{1},l)}T\times_{W}Z\xrightarrow{r^{}(g)}T\times_{U}X\text{ is not epi}.$ We denote the natural number object of $E$ by $\mathbb{N}_{d}$, which is given by the image of the nno $\mathbb{N}$ of $\bm{\mathrm{set}}$ under the inverse image part of the geometric morphism $E\to\bm{\mathrm{set}}$. ###### Proposition 3. In a connected, locally connected cocomplete topos $E$ such that $\pi_{0}$ reflects epimorphisms, the statement $\displaystyle\forall\ Y\twoheadrightarrow V,\ V\text{ connected,}$ (3) $\displaystyle\exists\ \Omega\twoheadrightarrow\mathbb{N}_{d}\text{ with }\pi_{0}(\Omega)\simeq\pi_{0}(\mathbb{N}_{d}),$ $\displaystyle\forall\ T\twoheadrightarrow V,\ T\text{ connected, }T\times_{V}Y\xrightarrow{l}\Omega,$ $\displaystyle l\text{ is not epi}.$ implies (2), the negation of WISC in the internal language of $E$. ###### Proof. We give some facts about toposes that we will use in what follows. First, in a connected topos the terminal object is connected. Second, in a cocomplete topos one has infinitary extensivity, namely $A\times_{B}\coprod_{i\in I}C_{i}\simeq\coprod_{i\in I}A\times_{B}C_{i}$, and the initial object $0$ is _strict_ : any map to it is an isomorphism. Third, since $\pi_{0}$ is a left adjoint, it preserves epimorphisms. Combined with the hypothesis on $\pi_{0}$ this means a map $f$ in $E$ is an epimorphism if and only if $\pi_{0}(f)$ is an epimorphism. Similarly $\pi_{0}$ preserves initial objects and the hypotheses imply it also reflects initial objects. Now assume that (3) holds in $E$. In (2) take $X\to U$ to be $\mathbb{N}_{d}\to 1$ (using $1$ is connected). Given an epimorphism $V\twoheadrightarrow 1$, $V$ has a component as $\pi_{0}(V)\to 1$ is onto and $V=\coprod_{v\in\pi_{0}(V)}V_{v}$ (and $1$ is projective). Fix a component $V_{0}\hookrightarrow V$. Given any $Y\to V$, take $Y_{0}=V_{0}\times_{V}Y$ to get $Y_{0}\to V_{0}$. If $Y_{0}$ is initial, then (2) can be seen to hold by taking $W=V_{0}$ and $g=\mathrm{id}$ since $T\times_{V}Y=T\times_{V_{0}}Y_{0}=0$ and as $r$ is an epi and $W$ is connected, $T\times\mathbb{N}_{d}$ is not initial. Hence we can assume $Y_{0}$ is not initial, and hence has at least one component and so $Y_{0}\to V_{0}$ is an epi. Fix some $\Omega\twoheadrightarrow\mathbb{N}_{d}$ inducing an isomorphism $\pi_{0}(\Omega)\simeq\pi_{0}(\mathbb{N}_{d})$ such that the rest of (3) holds. In (2) take $q$ to be the inclusion $V_{0}\hookrightarrow V$ (hence $W=V_{0}$, which is connected), and $Z=V_{0}\times\Omega$ with the epimorphism $g$ the product of $\mathrm{id}_{V_{0}}$ and $\Omega\twoheadrightarrow\mathbb{N}_{d}$. Now take any $T$ and pair of maps $T\twoheadrightarrow V_{0}$ and $T\times_{V}Y=T\times_{V_{0}}Y_{0}\xrightarrow{(\mathrm{pr}_{1},l)}T\times_{V_{0}}Z=T\times\Omega$. We know that $T$ has a component by a similar argument to above, say $T_{0}\hookrightarrow T$. Then $T_{0}\to V_{0}$ is epi so (3) implies $T_{0}\times_{V_{0}}Y_{0}=T_{0}\times_{V}Y\to\Omega$ is not epi. This then implies $T_{0}\times_{V}Y\to\Omega\to\mathbb{N}_{d}$ is not epi, since if it were, $\pi_{0}(T_{0}\times_{V}Y)\to\pi_{0}(\Omega)\xrightarrow{\sim}\pi_{0}(\mathbb{N}_{d})$ would be epi, implying $\pi_{0}(T_{0}\times_{V}Y)\to\pi_{0}(\Omega)$ and hence $T_{0}\times_{V}Y\to\Omega$ was epi. Thus there is some component of $\mathbb{N}_{d}$ not in the image of this map, say indexed by $n\in\mathbb{N}$. Then $T_{0}\times_{V}Y\to T_{0}\times\mathbb{N}_{d}$ is not epi, as the component of $T_{0}\times\mathbb{N}_{d}$ indexed by $n$ (isomorphic to $T_{0}$, which has $T_{0}\to 1$ epi) is not in its image. It then follows that $T\times_{V}Y\to T\times\mathbb{N}_{d}$ is not epi, and so (2) holds. ∎ ## 3 The construction Given our base topos $\bm{\mathrm{set}}$, we can consider the category of objects in $\bm{\mathrm{set}}$ equipped with a linear order with no infinite descending chains, which we shall call ordinals, in analogy with material set theory. The usual Burali-Forti argument—which requires no Choice—tells us there is a large category $O$ with objects ordinals and arrows the order- preserving injections onto initial segments. This large category is a linear preorder and has no infinite strictly descending chains. That there are multiple representatives for a particular order type, that is, non-identical isomorphic ordinals, does not cause any problems. We also note that $O$ has small joins (defined up to isomorphism in $O$). Given a topological group $G$, the category of sets with a continuous $G$ action forms a cocomplete boolean topos $G\bm{\mathrm{set}}$. In practice, one specifies a filter $\mathcal{F}$ of subgroups of $G$ and then those $G$-sets all of whose stabiliser groups belong to $\mathcal{F}$ are precisely those with a continuous action for the topology generated by $\mathcal{F}$. For any group $G$, let $\mathcal{C}$ be a collection of finite-index subgroups closed under finite intersections. Then there is a filter $\mathcal{F}_{\mathcal{C}}$ with elements those subgroups $H\leq G$ containing a subgroup appearing in $\mathcal{C}$ (we say the filter is _generated_ by $\mathcal{C}$). The category of continuous $G$-sets is then a full subcategory of the category of $G$-sets with finite orbits. The internal hom $Y^{X}$ is given by taking the set $\bm{\mathrm{set}}(X,Y)$ then retaining only those functions whose stabiliser under the $G$-action $f\mapsto g\cdot\left(f(g^{-1}\cdot-)\right)$ belongs to $\mathcal{F}_{\mathcal{C}}$. The subobject classifier is the two-element set with trivial $G$-action. ###### Remark 4. Notice that every transitive $G$-set $X$ that is continuous with respect to the topology given by $\mathcal{F}_{\mathcal{C}}$ (all $G$-sets will be assumed continuous from now on) has an epimorphism from some $G/L$ where $L\in\mathcal{C}$. This is because any stabiliser $\operatorname{Stab}(x)\in\mathcal{F}_{\mathcal{C}}$, $x\in X$, is assumed to contain an element of $\mathcal{C}$. ###### Example 5. For $\alpha$ an ordinal, let $\mathbb{Z}^{\alpha}$ be the set of functions $\alpha\to\mathbb{Z}$, considered as a group by pointwise addition. Consider functions $d\colon\alpha\to\mathbb{N_{+}}=\\{1,2,3,\ldots\\}$ such that $d(i)\not=1$ for only finitely many $i\in\alpha$, which we shall call _local depth functions_. Such a function defines a subgroup $d\mathbb{Z}:=\prod_{i\in\alpha}d(i)\mathbb{Z}\leq\mathbb{Z}^{\alpha}$ of finite index. The intersection of two such subgroups, given by $d_{1}$ and $d_{2}$, is given by the function $i\mapsto\operatorname{lcm}\\{d_{1}(i)d_{2}(i)\\}$. The subgroups belonging to the filter generated by this collection will be called _bounded depth subgroups_. From now on $\mathbb{Z}^{\alpha}$ will be regarded as having the topology generated by this filter. If we are given a split open surjection $p\colon H\to G$ (with $p$ and its splitting continuous) there is a geometric morphism $(p^{}\dashv p_{})\colon H\bm{\mathrm{set}}\to G\bm{\mathrm{set}}$ with $p^{}$ fully faithful and possessing a left adjoint $p_{!}\dashv p^{}$. Here $p^{}$ sends a $G$-set to the same set with the $H$-action via $p$ and $p_{!}(X)=X/\ker(p)$ with the obvious $G$-action. The inverse image functor $p^{}$ is in this case also a _logical_ functor, meaning that it preserves the subobject classifier and internal hom, as well as finite limits. In the case that $G$ is the trivial group: $p^{}$ is denoted $(-)_{d}$ and sends a set to the same set with the trivial action; $p_{!}$ is denoted $\pi_{0}$ and $\pi_{0}(X)$ is the set of orbits of the $H$-action. ###### Example 6. For $\alpha\hookrightarrow\beta$ ordinals, there is a split open surjection $\mathbb{Z}^{\beta}\to\mathbb{Z}^{\alpha}$, projection being given by restriction of the domain, and the splitting given by extending a function by $0$. Note that a local depth function on $\alpha$ gives a local depth function on $\beta$ by extending it by $1$. Now consider a functor $\mathcal{G}\colon O^{op}\to\bm{\mathrm{Top}}\bm{\mathrm{Grp}}_{sos}$, where $\bm{\mathrm{Top}}\bm{\mathrm{Grp}}_{sos}$ is the category of topological groups and split open surjections. Define the category $\mathcal{G}\bm{\mathrm{set}}$ with objects pairs $(\alpha,X)$ where $\alpha$ is an ordinal and $X$ is an object of $\mathcal{G}(\alpha)\bm{\mathrm{set}}$, and arrows $\mathcal{G}\bm{\mathrm{set}}((\alpha,X),(\beta,Y))=\mathcal{G}(\gamma)(X_{\gamma},Y_{\gamma})$ where $\gamma=\max\\{\alpha,\beta\\}$ and $X_{\gamma},\ Y_{\gamma}$ are $X,Y$ considered as $\mathcal{G}(\gamma)$-sets via the inverse image functors as above. The hom-sets are defined without making any choices since $O$ is a linear preorder, and so $\gamma$ is either $\alpha$ or $\beta$ (and we can take $\gamma=\alpha$ if $\alpha\simeq\beta$). Composition is well defined due to the full faithfulness of the inverse image functors. The objects of $\mathcal{G}\bm{\mathrm{set}}$ will be referred to as $\mathcal{G}$-sets. Informally, this category is the colimit of the large diagram of inverse image functors. ###### Proposition 7. The category $\mathcal{G}\bm{\mathrm{set}}$ is a connected, locally connected, atomic and cocomplete boolean $\bm{\mathrm{set}}$-topos. Moreover, $\pi_{0}$ reflects epimorphisms. ###### Proof. Let us first show that we have a topos. Finite limits exist because they can be calculated in any $\mathcal{G}(\alpha)$ where $\alpha$ is greater than all ordinals appearing in the objects in the diagram, and when the universal property is checked in $\mathcal{G}(\beta)$ for $\beta>\alpha$, the limit is preserved by the inverse image functor. Likewise the internal hom $(\alpha,X)^{(\beta,Y)}$ is defined as $X_{\gamma}^{Y_{\gamma}}$ in $\mathcal{G}(\gamma)$ ($\gamma=\max\\{\alpha,\beta\\}$) and its universal property is satisfied due to inverse image functors preserving internal homs. The subobject classifier $\mathbf{2}$ in $\bm{\mathrm{set}}$ is preserved by all inverse image functors $\bm{\mathrm{set}}\to\mathcal{G}(\alpha)\bm{\mathrm{set}}$, so given any subobject in $\mathcal{G}\bm{\mathrm{set}}$ it has a classifying map to $\mathbf{2}$. Thus $\mathcal{G}\bm{\mathrm{set}}$ is a topos, and has a geometric morphism $((-)_{d}\dashv(-)^{\mathcal{G}})\colon\mathcal{G}\bm{\mathrm{set}}\to\bm{\mathrm{set}}$ as it is locally small ($(-)^{\mathcal{G}}:=\mathcal{G}\bm{\mathrm{set}}(1,-)$ is the global points functor). It is easy to check there is a functor $\pi_{0}$ sending a $\mathcal{G}(\alpha)$-set to its set of orbits and this is a left adjoint to $(-)_{d}$. Thus $\mathcal{G}\bm{\mathrm{set}}$ is locally connected. Since $(-)_{d}$ is fully faithful and logical $\mathcal{G}\bm{\mathrm{set}}$ is also connected and atomic respectively. Small colimits can be calculated in $\mathcal{G}(\alpha)$ where $\alpha$ is some small join of the ordinals appearing as the vertices of the diagram, and the universal property is verified since inverse image functors preserve all small colimits. Lastly, $\mathcal{G}\bm{\mathrm{set}}$ is boolean as $1\to\mathbf{2}\leftarrow 1$ is a coproduct cocone, using the definition of colimits and the fact it is such in $\bm{\mathrm{set}}$. To prove the last statement, suppose $X\to Y$ in $\mathcal{G}\bm{\mathrm{set}}$ (without loss of generality, take this in $\mathcal{G}(\alpha)\bm{\mathrm{set}}$ for some $\alpha$) is such that $\pi_{0}$ induces an epimorphism of connected components. Then for each orbit of $Y$ there is an orbit of $X$ mapping to it, and equivariant maps between orbits are onto, so $X\to Y$ is onto as a map of sets and hence an epi. ∎ The stack semantics in $\mathcal{G}\bm{\mathrm{set}}$ give a model of the structural set theory underlying $\bm{\mathrm{set}}$, minus any Choice that may hold in $\bm{\mathrm{set}}$ (see the discussion after lemma 7.13 in [10]). We will take a particular diagram of groups with the properties we need. ###### Corollary 8. The diagram $\mathcal{Z}\colon\alpha\mapsto\mathbb{Z}^{\alpha}$, where $\mathbb{Z}^{\alpha}$ is regarding as having the topology given by the filter of bounded depth subgroups, gives rise to a connected, locally connected boolean topos $\mathcal{Z}\bm{\mathrm{set}}$ such that $\pi_{0}$ reflects epimorphisms. If one is working in a setting that permits such reasoning, the proper class- sized group to which the introduction alludes is the colimit over the inclusions $\mathcal{Z}(\alpha)\hookrightarrow\mathcal{Z}(\beta)$ given by the splittings, for $\alpha\hookrightarrow\beta$. The rest of the paper will show that internal WISC fails in $\mathcal{Z}\bm{\mathrm{set}}$, and so WISC itself fails in the well-pointed topos given by the stack semantics of $\mathcal{Z}\bm{\mathrm{set}}$. ## 4 The failure of WISC We need some facts that hold in $\mathcal{Z}\bm{\mathrm{set}}$ regarding local depth functions. As a bit of notation, let us write $\mathcal{Z}/d\mathbb{Z}$ for the transitive $\mathcal{Z}$-set $\mathbb{Z}^{\alpha}/d\mathbb{Z}$ for $\alpha=\operatorname{dom}(d)$. ###### Lemma 9. Let $\mathcal{Z}/d_{1}\mathbb{Z}\to\mathcal{Z}/d_{2}\mathbb{Z}$ be an equivariant map of $\mathcal{Z}$-sets. Then for every $i\in\alpha$ we have $d_{2}(i)\mid d_{1}(i)$. ###### Proof. The existence of the map implies $d_{1}\mathbb{Z}$ is conjugate to a subgroup of $d_{2}\mathbb{Z}$, but all groups here are abelian so it _is_ a subgroup of $d_{2}\mathbb{Z}$. For the second statement, notice that the first statement implies $d_{1}(i)\mathbb{Z}\leq d_{2}(i)\mathbb{Z}\leq\mathbb{Z}$ for each $i\in\alpha$ and the result follows. ∎ We also need to consider what taking pullbacks looks like from the point of view of local depth functions. ###### Lemma 10. Any orbit in $\mathcal{Z}/(d_{1}\mathbb{Z}\cap d_{2}\mathbb{Z})\subset\mathcal{Z}/d_{1}\mathbb{Z}\times_{\mathcal{Z}/d_{3}\mathbb{Z}}\mathcal{Z}/d_{2}\mathbb{Z}$ is isomorphic to a transitive $\mathcal{Z}$-set with local depth function $d$ given by $d(i)=\operatorname{lcm}\\{d_{1}(i),d_{2}(i)\\},\quad\forall i\in\alpha$ where $\alpha=\max\\{\operatorname{dom}(d_{1}),\operatorname{dom}(d_{2})\\}$. ###### Proof. Notice that the fibred product as given is isomorphic to $\prod_{i\in\alpha}\mathbb{Z}/d_{1}(i)\mathbb{Z}\times_{\mathbb{Z}/d_{3}(i)\mathbb{Z}}\mathbb{Z}/d_{2}(i)\mathbb{Z}$ where the $\mathbb{Z}^{\alpha}$ action is such that the $i^{th}$ coordinate—a copy of $\mathbb{Z}$—acts diagonally on the $i^{th}$ factor of the preceeding expression. The stabiliser of any $(n_{i},n^{\prime}_{i})_{i\in\alpha}$ is then the product of the stabilisers of the $\mathbb{Z}$-action of the various $\mathbb{Z}/d_{1}(i)\mathbb{Z}\times_{\mathbb{Z}/d_{3}(i)\mathbb{Z}}\mathbb{Z}/d_{2}(i)\mathbb{Z}$. We thus only need to consider the simpler problem of determining the stabilisers for a $\mathbb{Z}$-set $\mathbb{Z}/k\mathbb{Z}\times_{\mathbb{Z}/m\mathbb{Z}}\mathbb{Z}/l\mathbb{Z}$. The stabiliser of $(0,0)$ is $\mathbb{Z}/(k\mathbb{Z}\cap l\mathbb{Z})$, from which the result follows by the description in example 5 of the intersection of subgroups given by local depth functions. We only then need to consider the stabilisers of $(0,n)$ for $n\in\mathbb{Z}/l\mathbb{Z}$ as all others are equal to one of these by abelianness – but $\operatorname{Stab}(0,n)$ is again $\mathbb{Z}/(k\mathbb{Z}\cap l\mathbb{Z})$ using abelianness. The statement regarding local depth functions then follows. ∎ We need a special collection of subgroups of $\mathbb{Z}^{\alpha}$ in the proof of theorem 11 below, namely those given by local depth functions $\delta[\alpha,n,i]\colon\alpha\to\mathbb{N_{+}}$ defined as $\delta[\alpha,n,i](k)=\begin{cases}n&\text{if $k=i$;}\\\ 1&\text{if $k\not=i$.}\end{cases}$ Note that the transitive $\mathcal{Z}$-set $\mathcal{Z}/\delta[\alpha,n,i]\mathbb{Z}$ has underlying set $\mathbb{Z}/n\mathbb{Z}$, and that $\Omega[\alpha,i]:=\coprod_{n\in\mathbb{N_{+}}}\mathcal{Z}/\delta[\alpha,n,i]\mathbb{Z}$ is an object of $\mathcal{Z}\bm{\mathrm{set}}$ for any $\alpha\in O$ and $i\in\alpha$. ###### Theorem 11. The statement of WISC in the stack semantics in $\mathcal{Z}\bm{\mathrm{set}}$ fails. ###### Proof. In the notation of proposition 3, taking transitive $\mathcal{Z}$-sets for connected objects, we need to show that for any $Y\twoheadrightarrow\mathcal{Z}/H$, there is an $\Omega$ such that for any $r\colon\mathcal{Z}/K\to\mathcal{Z}/H$, any $l\colon\mathcal{Z}/K\times_{\mathcal{Z}/H}Y\to\Omega$ is not an epimorphism. Let us write $Y=\coprod_{y\in\pi_{0}(Y)}Y_{y}$, and note that this coproduct, like all colimits in $\mathcal{Z}\bm{\mathrm{set}}$ takes place in some $\mathbb{Z}^{\alpha}\bm{\mathrm{set}}$. In particular, by remark 4 each $Y_{y}$ has an epimorphism from some $\mathcal{Z}/d_{y}\mathbb{Z}$ for a local depth function $d_{y}\colon\alpha\to\mathbb{N_{+}}$. As a result $H\leq\mathbb{Z}^{\alpha}$, so fix some $d_{H}\colon\alpha\to\mathbb{N_{+}}$ to get an epimorphism $\mathcal{Z}/d_{H}\mathbb{Z}\to\mathcal{Z}/H$. Define $\Omega=\Omega[\alpha+1,\top_{\alpha+1}]$, where $\top_{\alpha+1}$ is the top element of the ordinal $\alpha+1$. Given $\mathcal{Z}/K\to\mathcal{Z}/H$, fix a local depth function $d_{K}\colon\beta\to\mathbb{N_{+}}$ such that $d_{K}\mathbb{Z}\leq K$ (without loss of generality, we can assume $\alpha\leq\beta$). Since $\mathcal{Z}\bm{\mathrm{set}}$ is infinitary extensive, we have $\mathcal{Z}/K\times_{\mathcal{Z}/H}Y\simeq\coprod_{y\in\pi_{0}(Y)}\mathcal{Z}/K\times_{\mathcal{Z}/H}Y_{y}.$ Any map $l\colon\mathcal{Z}/K\times_{\mathcal{Z}/H}Y\to\Omega$ is then given by a collection of maps $l_{y}\colon\mathcal{Z}/K\times_{\mathcal{Z}/H}Y_{y}\to\Omega$. We need to show that this collection of maps is not jointly surjective, and will do this by showing the image of $l_{y}$, for arbitrary $y$, must be contained in a strict subobject of $\Omega$ that is independent of $y$. Given an epimorphism $\mathcal{Z}/d_{y}\mathbb{Z}\to Y_{y}$, consider, in $\mathcal{Z}/d_{K}\mathbb{Z}\times_{\mathcal{Z}/d_{H}\mathbb{Z}}\mathcal{Z}/d_{y}\mathbb{Z}$, an orbit $\mathcal{Z}/\delta_{y}\mathbb{Z}$ where $\delta_{y}(i)=\operatorname{lcm}\\{d_{K}(i),d_{y}(i)\\}$ for each $i\in\beta$, by lemma 10. In particular, we have that $\delta_{y}(\top_{\alpha+1})=d_{K}(\top_{\alpha+1})=:N_{0}$ is independent of $y$. Compose the inclusion $\mathcal{Z}/\delta_{y}\mathbb{Z}\hookrightarrow\mathcal{Z}/K\times_{\mathcal{Z}/H}Y_{y}$with $l_{y}$ to get a map $l^{\prime}_{y}\colon\mathcal{Z}/\delta_{y}\mathbb{Z}\to\Omega=\coprod_{n\in\mathbb{N_{+}}}\mathcal{Z}/\delta[\alpha,n,i]\mathbb{Z}.$ Applying lemma 9 to this map with $i=\top_{\alpha+1}$ we find that $n\mid N_{0}$ for any $n$ such that $\mathcal{Z}/\delta[\alpha,n,i]\mathbb{Z}\subset\operatorname{im}l^{\prime}_{y}$. Thus the image of any $l_{y}$ and hence of $l$ is contained in $\coprod_{n\mid N_{0}}\mathcal{Z}/\delta[\alpha,n,i]\mathbb{Z}\subsetneqq\Omega,$ hence $l$ is not an epimorphism. ∎ Recall that ETCS is a set theory defined by specifying the properties of the category of sets [7], namely that it is a well-pointed topos (with nno) satisfying the axiom of choice. We can likewise specify a choiceless version, which is the theory of a well-pointed topos (with nno). Given a model $\bm{\mathrm{set}}$ of ETCS, we have constructed a well-pointed topos in which WISC is false. Thus we have our main result. ###### Corollary 12. Assuming ETCS is consistent, so is the theory of a well-pointed topos with nno plus the negation of WISC. Finally, we recall the definition from [11] of a predicative topos: this is a $\Pi W$-pretopos satisfying WISC (or, as called there, AMC). ###### Corollary 13. The topos $\mathcal{Z}\bm{\mathrm{set}}$ is not a predicative topos. ## References [1] _SGA4 – Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos_ , Lecture Notes in Mathematics, Vol. 269, Springer-Verlag, Berlin, 1972. Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964 (SGA 4), Dirigé par M. Artin, A. Grothendieck, et J. L. Verdier. * [2] Aczel, Peter, ‘The type theoretic interpretation of constructive set theory’, in _Logic Colloquium ’77_ , vol. 96 of _Stud. Logic Foundations Math._ , North-Holland, 1978, pp. 55–66. * [3] Blass, Andreas, ‘Cohomology detects failures of the axiom of choice’, _Trans. Amer. Math. Soc._ , 279 (1983), 1, 257–269. * [4] Doraisx(http://mathoverflow.net/users/2000), François G., ‘On a weak choice principle’, MathOverflow, 2012. http://mathoverflow.net/a/99934/ (version: 2012-06-18). * [5] Gitik, M., ‘All uncountable cardinals can be singular’, _Israel J. Math._ , 35 (1980), 1-2, 61–88. * [6] Karagila, A., ‘Embedding orders into cardinals with $DC_{\kappa}$’, _Fundamenta Mathematicae_ , 226 (2014), 143–156. arXiv:1212.4396. * [7] Lawvere, F. William, ‘An elementary theory of the category of sets (long version) with commentary’, _Repr. Theory Appl. Categ._ , (2005), 11, 1–35. Reprinted and expanded from Proc. Nat. Acad. Sci. U.S.A. 52 (1964), With comments by the author and Colin McLarty. * [8] MacLane, S., and I. Moerdijk, _Sheaves in Geometry and Logic_ , Springer-Verlag, 1992. * [9] Roberts, D. M., ‘Internal categories, anafunctors and localisation’, _Theory Appl. Categ._ , 26 (2012), 29, 788–829. arXiv:1101.2363. * [10] Shulman, Michael, ‘Stack semantics and the comparison of material and structural set theories’, , 2010. arXiv:1004.3802. * [11] van den Berg, Benno, ‘Predicative toposes’, , 2012. arXiv:1207.0959. * [12] van den Berg, Benno, and Ieke Moerdijk, ‘The axiom of multiple choice and models for constructive set theory’, _Journal of Mathematical Logic_ , 14 (2014), 1. arXiv:1204.4045.	arxiv-papers	2013-11-13T10:36:25	2024-09-04T02:49:53.516384	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "David Michael Roberts", "submitter": "David Roberts", "url": "https://arxiv.org/abs/1311.3074" }
1311.3134	# Fredholm alternative, semilinear elliptic problems, and Wentzell boundary conditions Ciprian G. Gal , Gisele Ruiz Goldstein , Jerome A. Goldstein , Silvia Romanelli and Mahamadi Warma C. G. Gal, Department of Mathematics, University of Missouri, Columbia, MO 65211 (USA). [email protected] G. Ruiz Goldstein and J. A. Goldstein, Department of Mathematics University of Memphis, Memphis, TN 38152 (USA). [email protected] [email protected] S. Romanelli, Universit degli Studi di Bari Via E. Orabona 4 I-70125 Dipartimento di Matematica Bari, (Italy). [email protected] M. Warma, University of Puerto Rico, Department of Mathematics (Rio Piedras Campus), PO Box 23355 San Juan, PR 00931-3355 (USA). [email protected], [email protected] ###### Abstract. We give necessary and sufficient conditions for the solvability of some semilinear elliptic boundary value problems involving the Laplace operator with linear and nonlinear highest order boundary conditions involving the Laplace-Beltrami operator. ###### Key words and phrases: Laplace-Beltrami operator, global constraints, nonlinear elliptic boundary value problems at resonance, nonlinear boundary conditions, Fredholm alternative. ###### 2000 Mathematics Subject Classification: 35J20, 35J25, 35J60, 35J65, 49J27, 52A41. ## 1\. Introduction Let $\Omega\subset\mathbf{R}^{N},$ $N\geq 1,$ be a bounded domain with smooth boundary $\Gamma:=\partial\Omega$. Let $\alpha:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous monotone nondecreasing function with $\alpha\left(0\right)=0$ and consider the following boundary value problem: $\begin{cases}-\Delta u+\alpha\left(u\right)=f&\text{ in }\Omega,\\\ \frac{\partial u}{\partial n}=0&\text{ on }\Gamma,\end{cases}$ (1.1) where $f\in L^{2}\left(\Omega\right)$ is a given real function, $\frac{\partial u}{\partial n}$ denotes the outward normal derivative of $u$ on $\Gamma$ and $\Delta$ is the Laplace operator in $\Omega.$ Let us denote by $\left\|\Omega\right\|$ the Lebesgue measure of $\Omega.$ It is known that a necessary and sufficient condition for the existence of a solution of (1.1) is $\left\|\Omega\right\|^{-1}\int\limits_{\Omega}f\left(x\right)dx\in\mathcal{R}\left(\alpha\right).$ (1.2) Here all the functions are real valued. This result is due to J. Mawhin [15]. Earlier, Landesman and Lazer [12] obtained a similar result. This result lead to an enormous body of literature. Landesman and Lazer showed that (1.2) is a necessary condition, while a sufficient condition is $\left\|\Omega\right\|^{-1}\int\limits_{\Omega}f\left(x\right)dx\in int(\mathcal{R}\left(\alpha\right)),$ (1.3) where $int\left(I\right)$ denotes the interior of the set $I$. They also allowed for nonmonotone $\alpha,$ which was very important for later developments. Thus for them, $\alpha:\mathbb{R}\rightarrow\mathbb{R}$ is continuous, $\alpha\left(0\right)=0,$ and $\alpha\left(-\infty\right)=\lim_{x\rightarrow-\infty}\alpha\left(x\right)\leq\alpha\left(y\right)\leq\lim_{x\rightarrow+\infty}\alpha\left(x\right)=\alpha\left(+\infty\right)$ (1.4) for all $y\in\mathbb{R}$. They proved (1.2) is a necessary condition in this more general context of (1.4), while (1.3) is a sufficient condition. Prior to Mawhin’s work, Brezis and Haraux [2] put the [12] result in an abstract context and found a new, elegant proof for it. These works led to very much research, including major contributions by Brezis and Nirenberg [3] and many others. Brezis and Haraux worked in the context of subdifferentials of convex functionals on Hilbert spaces. We will explain the context and the abstract results, used in proving the assertion connecting (1.2) and (1.3), in Sections 2 and 4. But here we emphasize again that these results were inspired by the similar result of Landesman and Lazer [12] who, in giving necessary and sufficient conditions on $f$ for the solvability of certain elliptic problems of the form $Lu+Nu=f$ (with $L$ linear and $N$ nonlinear), established a sort of ”nonlinear Fredholm alternative” for the first time. When $\alpha\equiv 0$, the above result reduces to $-\Delta u=f\text{ \ in\ }\Omega,\text{ }\frac{\partial u}{\partial n}=0,\text{ on }\Gamma$ has a weak solution if and only if $\left\langle f,1\right\rangle_{L^{2}(\Omega)}=0,\text{ i.e}.\text{, }\int\limits_{\Omega}f\left(x\right)dx=0,$ which is exactly the Fredholm alternative since the null space of the Neumann Laplacian is the constants. Thus, Mawhin’s result (based on the work in [12]) is an exact nonlinear Fredholm alternative for the nonlinear problem (1.1). The goal of this paper is to establish similar results (comparable with (1.2), (1.3)) for the following boundary value problem with second order boundary conditions: $\begin{cases}-\Delta u+\alpha_{1}\left(u\right)=f\left(x\right)&\text{ in }\Omega,\\\ b\left(x\right)\frac{\partial u}{\partial n}+c\left(x\right)u-qb\left(x\right)\Delta_{\Gamma}u+\alpha_{2}\left(u\right)=g\left(x\right)&\text{ on }\Gamma,\end{cases}$ (1.5) where the functions appearing in (1.5) are real and satisfy $b\in C\left(\Gamma\right),$ $b>0,$ $c\in C\left(\Gamma\right),$ $c\geq 0$, $q$ is a nonnegative constant; $\alpha_{1},$ $\alpha_{2}:\mathbb{R}\rightarrow\mathbb{R}$ are continuous and monotone nondecreasing functions, such that $\alpha_{i}\left(0\right)=0$. Above, $\Delta_{\Gamma}$ is the Laplace-Beltrami operator on $\Gamma$, $f\in L^{2}\left(\Omega\right)$ and $g\in L^{2}\left(\Gamma\right)$ are given real functions. Thus, our emphasis is on the generality of the boundary conditions. We organize the paper as follows. In Sections 2 and 3, we discuss the auxiliary linear problems corresponding to (1.5), and in Section 4 we show the existence of weak solutions to (1.5) in case certain global constraints (similar to (1.2)) hold. In the same section, we will consider concrete examples as application of our results. Before we state our main result, we define the notion of weak solutions to (1.5). ###### Definition 1.1. A function $u\in H^{1}(\Omega)$ is said to be a weak solution to (1.5) if $\alpha_{1}(u)\in L^{1}(\Omega),$ $\alpha_{2}(tr(u))\in L^{1}(\Gamma)$, $tr\left(u\right):=u_{\mid\Gamma}\in H^{1}(\Gamma),$ if $q>0$, and $\displaystyle\int_{\Omega}fvdx+\int_{\Gamma}gv\frac{dS}{\beta}$ $\displaystyle=$ $\displaystyle\int_{\Omega}\nabla u\cdot\nabla vdx+\int_{\Omega}\alpha_{1}(u)vdx$ (1.6) $\displaystyle+\int_{\Gamma}\left(\alpha_{2}(u)v+cuv\right)\frac{dS}{\beta}+q\int_{\Gamma}\nabla_{\Gamma}u\cdot\nabla_{\Gamma}vdS,$ for all $v\in H^{1}(\Omega)\cap C\left(\overline{\Omega}\right),$ if $q=0$ and all $v\in H^{1}(\Omega)\cap C\left(\overline{\Omega}\right)$ with $tr(v)\in H^{1}(\Gamma),$ if $q>0$. Our main result is as follows. Let $\lambda_{1}=\int\limits_{\Omega}dx,\text{ }\lambda_{2}=\int\limits_{\Gamma}\frac{dS}{b},$ (1.7) and let $\widetilde{I}$ be the interval $\widetilde{I}=\lambda_{1}\mathcal{R}\left(\alpha_{1}\right)+\lambda_{2}\mathcal{R}\left(\alpha_{2}\right).$ Moreover, for each $i=1,2$, set $L_{i}(t):=\int_{0}^{t}\alpha_{i}(s)ds\text{ and }\Lambda_{i}(t):=\max\left\\{L_{i}\left(t\right),L_{i}\left(-t\right)\right\\},\text{ for all }t\in\mathbb{R}\text{.}$ (1.8) ###### Theorem 1.2. Let $c\equiv 0$ and let $\alpha_{i}:\mathbb{R}\rightarrow\mathbb{R}$ $(i=1,2)$ be continuous, monotone nondecreasing functions such that $\alpha_{i}\left(0\right)=0$. If (1.5) has a weak solution, then $\int\limits_{\Omega}f\left(x\right)dx+\int\limits_{\Gamma}g\left(x\right)\frac{dS}{b\left(x\right)}\in\widetilde{I}.$ (1.9) Conversely, if there exist positive constants $t_{i},$ $C_{i}>0$, such that the functions $\Lambda_{i}:\mathbb{R}\rightarrow[0,+\infty),$ $i=1,2,$ satisfy $\Lambda_{i}(2t)\leq C_{i}\Lambda_{i}(t),\;$for all $t\geq t_{i}$, and $\int\limits_{\Omega}f\left(x\right)dx+\int\limits_{\Gamma}g\left(x\right)\frac{dS}{b\left(x\right)}\in int(\widetilde{I}),$ (1.10) then (1.5) has a weak solution. ## 2\. The linear problem We need to introduce some notation and terminology. We first define the space $\mathbb{X}_{2}$ to be the real Hilbert space $L^{2}\left(\Omega,dx\right)\oplus L^{2}(\Gamma,dS/b),$ with norm $\left\\|u\right\\|_{\mathbb{X}_{2}}=\left(\int\limits_{\Omega}\left\|u\left(x\right)\right\|^{2}dx+\int\limits_{\Gamma}\left\|u\left(x\right)\right\|^{2}\frac{dS_{x}}{b\left(x\right)}\right)^{\frac{1}{2}}$ (2.1) for $u\in C\left(\overline{\Omega}\right)$, where $dS$ denotes the usual Lebesgue surface measure on $\Gamma$. Here, if $u\in C\left(\overline{\Omega}\right),$ we identify $u$ with the vector $U=\left(u\|_{\Omega},u\|_{\Gamma}\right)^{T}\in C\left(\Omega\right)\times C\left(\Gamma\right).$ We then note that $\mathbb{X}_{2}=L^{2}\left(\Omega,dx\right)\oplus L^{2}(\Gamma,dS/b)$ is the completion of $C\left(\overline{\Omega}\right)$ with respect to the norm $\left(2.1\right)$. In general, any vector $U\in\mathbb{X}_{2}$ will be of the form $\left(u_{1},u_{2}\right)^{T}$ with $u_{1}\in L^{2}\left(\Omega,dx\right)$ and $u_{2}\in L^{2}(\Gamma,dS/b),$ and there need be no connection between $u_{1}$ and $u_{2}.$ Here and below the superscript $T$ denotes transpose. Let $\left\langle\cdot,\cdot\right\rangle_{\mathbb{X}_{2}}$ denote the corresponding inner product on $\mathbb{X}_{2}$. For a complete discussion of this space, we refer the reader to [5]. We define the formal operator $A_{0}$ by $A_{0}U=\left(\left(-\Delta u\right)\|_{\Omega},\left(-\Delta u\right)\|_{\Gamma}\right)^{T},$ (2.2) for functions $U=\left(u\|_{\Omega},u\|_{\Gamma}\right)^{T}$ with $u\in C^{2}\left(\overline{\Omega}\right)$ that satisfy the Wentzell boundary condition $\,\Delta u+b\left(x\right)\frac{\partial u}{\partial n}+c\left(x\right)u-qb\left(x\right)\Delta_{\Gamma}u=0,$ (2.3) on $\Gamma.$ Here $\left(\Delta u\right)_{\|_{\Gamma}}$ stands for the trace of the function $\Delta u$ on the boundary $\Gamma$ and it should not be confused with the Laplace-Beltrami operator $\Delta_{\Gamma}u$. From now on, $tr\left(u\right)$ denotes the trace of $u$ on the boundary. We let $\displaystyle D\left(A_{0}\right)$ $\displaystyle=\left\\{U=\left(u_{1},u_{2}\right)^{T}\in\mathbb{X}_{2}:U\text{ corresponds to }u_{1}\in C^{2}\left(\overline{\Omega}\right),\right.$ (2.4) $\displaystyle\left.u_{2}=u_{1}\|_{\Gamma}=tr\left(u_{1}\right)\text{ and (\ref{2.3}) holds}\right\\}.$ For functions $u\in C^{2}\left(\overline{\Omega}\right)\subset\mathbb{X}_{2}$, $A_{0}U$ is defined by (2.2). For any functions $u,v$ belonging to $C^{2}\left(\overline{\Omega}\right),$ and each satisfying the boundary condition $\Delta\varpi+b\left(x\right)\frac{\partial\varpi}{\partial n}+c\left(x\right)\varpi-qb\left(x\right)\Delta_{\Gamma}\varpi=0$ on $\Gamma,$ we identify $u$ and $v$ with $U=\left(u\|_{\Omega},u\|_{\Gamma}\right)^{T}$ and $V=\left(v\|_{\Omega},v\|_{\Gamma}\right)^{T}$ and calculate $\left\langle A_{0}U,V\right\rangle_{\mathbb{X}_{2}}$ as follows: $\displaystyle\left\langle A_{0}U,V\right\rangle_{\mathbb{X}_{2}}$ $\displaystyle=\int\limits_{\Omega}\left(-\Delta u\right)vdx+\int\limits_{\Gamma}\left(-\Delta u\right)v\frac{dS}{b\left(x\right)}$ (2.5) $\displaystyle=\int\limits_{\Omega}\nabla u\cdot\nabla vdx+\int\limits_{\Gamma}\left(-\Delta u-b\left(x\right)\frac{\partial u}{\partial n}\right)v\frac{dS}{b\left(x\right)}$ $\displaystyle=\int\limits_{\Omega}\nabla u\cdot\nabla vdx+\int\limits_{\Gamma}\left(c\left(x\right)u-qb\left(x\right)\Delta_{\Gamma}u\right)v\frac{dS}{b\left(x\right)},$ since $-\Delta u-b\left(x\right)\frac{\partial u}{\partial n}=c\left(x\right)u-qb\left(x\right)\Delta_{\Gamma}u$ on $\Gamma.$ Furthermore, Stokes’ theorem applied in the last term of (2.5) yields $\left\langle A_{0}U,V\right\rangle_{\mathbb{X}_{2}}=\int\limits_{\Omega}\nabla u\cdot\nabla vdx+\int\limits_{\Gamma}c\left(x\right)uv\frac{dS}{b\left(x\right)}+q\int\limits_{\Gamma}\nabla_{\Gamma}u\cdot\nabla_{\Gamma}vdS,$ (2.6) where $\nabla_{\Gamma}$ stands for the tangential gradient on the surface $\Gamma.$ Finally, if we denote the right hand side of (2.6) by $\varrho\left(U,V\right)$, it is now clear that $\varrho\left(U,V\right)=\varrho\left(V,U\right)=\left\langle U,A_{0}V\right\rangle_{\mathbb{X}_{2}},$ therefore $A_{0}$ is symmetric on $\mathbb{X}_{2}$. Let us now consider a function $f\in C\left(\overline{\Omega}\right)\cup H^{1}\left(\Omega\right)$ such that $F=\left(f_{1},f_{2}\right)^{T}$ with $f_{1}:=f\|_{\Omega}$ and $f_{2}:=f\|_{\Gamma}.$ By the equality $A_{0}U=F,$ we mean the following boundary value problem: $-\Delta u=f_{1}\quad\text{ in }\quad\quad\Omega,$ (2.7) $-\Delta u=f_{2}\,\quad\text{on}\quad\quad\Gamma.$ (2.8) Using the Wentzell boundary condition (2.3) and replacing $f_{2}$ by $f_{\mid\Gamma},$ the boundary condition (2.8) becomes $b(x)\frac{\partial u}{\partial n}+c(x)u-qb(x)\Delta_{\Gamma}u=f_{2}\text{ on }\Gamma.$ (2.9) Any $u\in H^{s}\left(\Omega\right)$ has a trace $tr\left(u\right)=u\|_{\Gamma}$ in $H^{s-1/2}\left(\Gamma\right)$ for $s>1/2.$ More precisely, we recall that the linear map $tr:H^{s}\left(\Omega\right)\rightarrow H^{s-1/2}\left(\Gamma\right)$ is bounded and onto for $s>1/2$. We now define the ”Wentzell version of $A_{0}$”, $\widetilde{A}_{0},$ by $\widetilde{A}_{0}U=F=\left(f_{1},f_{2}\right)^{T}$ on $\displaystyle D(\widetilde{A}_{0})$ $\displaystyle=\left\\{U\in\mathbb{X}_{2}:U\text{ corresponds to }u\in H^{2}\left(\Omega\right),\right.$ (2.10) $\displaystyle\left.tr\left(u\right)\in H^{2}\left(\Gamma\right)\text{ if }q>0\text{, and (\ref{2.7}), (\ref{2.9}) holds}\right\\}.$ In this case, $f_{2}$ need not be the trace of $f_{1}$ on $\Gamma.$ Then, using the techniques as in [6], we can easily check that $\widetilde{A}_{0}$ is contained in the closure of $A_{0}$. Let $A=\overline{A}_{0}=\overline{\widetilde{A}}_{0}$. Then, $A$ is selfadjoint and nonnegative if $c\geq 0$ on $\Gamma$; $A$ is the operator associated with the nonnegative symmetric closed bilinear form $\varrho\left(U,V\right).$ We have $\left\langle AU,V\right\rangle_{\mathbb{X}_{2}}=\varrho\left(U,V\right),$ for all $U\in D\left(A\right)$ and all $V=\left(v\|_{\Omega},v\|_{\Gamma}\right)^{T}\in D(\varrho):=H^{1}\left(\Omega\right)\times H^{1}\left(\Gamma\right)$ (if $q>0$) and $V\in D(\varrho):=H^{1}\left(\Omega\right)\times H^{1/2}\left(\Gamma\right)$ if $q=0$. We emphasize that for $A=\overline{A}_{0},$ the equations (2.7) and (2.9) hold even if the vector $F=\left(f_{1},f_{2}\right)^{T}$ does not correspond to a function $f$ belonging to $C\left(\overline{\Omega}\right)\cup H^{1}\left(\Omega\right)$, that is, $f_{2}\neq f_{1}\|_{\Gamma}.$ For $U\in D\left(A\right),$ an operator matrix representation of $A$ is given by $A=\left(\begin{array}[]{cc}-\Delta&\qquad 0\\\ b\frac{\partial}{\partial n}&\qquad cI-qb\Delta_{\Gamma}\end{array}\right).$ (2.11) We will now give a concrete example when $q=0$ (that is, $\Delta_{\Gamma}$ does not appear in the boundary condition (2.9)). This is a simple example where $f_{2}\neq f_{1}\|_{\Gamma}$. ###### Example 2.1. Let $\Omega=\left(0,1\right)\subset\mathbb{R}$ and let $F=\left(0,k\right)$ where $\Gamma=\left\\{0,1\right\\}$ and $k\left(0\right)=a_{0},$ $k\left(1\right)=b_{0}$ with $\left(a_{0},b_{0}\right)\neq\left(0,0\right)$. Take $b\left(j\right)=c\left(j\right)=1$ for $j=0,1.$ Then $AU=F$ means $\left\\{\begin{array}[]{c}u^{{}^{\prime\prime}}=0\text{ in }\left[0,1\right],\\\ -u^{{}^{\prime}}\left(0\right)+u\left(0\right)=a_{0},\\\ u^{{}^{\prime}}\left(1\right)+u\left(1\right)=b_{0},\end{array}\right.$ (2.12) since $\partial/\partial n=\left(-1\right)^{j+1}d/dx$ at $x=j\in\left\\{0,1\right\\}$. Solving (2.12) gives $u\left(x\right)=\frac{1}{3}\left[\left(b_{0}-a_{0}\right)x+\left(2a_{0}+b_{0}\right)\right],\text{ }x\in\left[0,1\right].$ ## 3\. The domain of the Wentzell Laplacian We recall some facts from the theory of linear elliptic boundary value problems. The standard theory works for uniformly elliptic problems of even order $2m$; we shall restrict ourselves to the second order case, $m=1$. We shall treat the symmetric case, although this restriction is not needed for the results we present in this section. Our problem takes the form $\widehat{A}u=f$ in $\Omega,$ $\widehat{B}u=g$ on $\Gamma$, where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$ with boundary $\Gamma,$ $\left\\{\begin{array}[]{c}Au=-\nabla\cdot\mathcal{A}\left(x\right)\nabla u,\\\ Bu=b\partial_{n}^{\mathcal{A}}u+cu- qb\Delta_{\Gamma}u,\end{array}\right.$ (3.1) and $\widehat{A}=A+\lambda I,$ $\widehat{B}=B+\lambda I,$ for some $\lambda\in\mathbb{R}$. As the theory is based upon pseudo differential operator techniques, we make the standard assumption that $\Omega$, $\mathcal{A}$, $b$ and $c$ are all of class $C^{\infty}$ in addition to the assumptions that the $N\times N$ matrix function $\mathcal{A}$ is real, symmetric and uniformly positive definite, $b>0$, $c\geq 0$ and $q\in\left[0,+\infty\right)$. Let $s\in\mathbb{N}_{0}=\left\\{0,1,2,...\right\\}$ and $p\in\left(1,+\infty\right)$. We refer to Triebel [20] for the general case, where we use his notation; Lions-Magenes [13] treats the Hilbert space case ($p=2$). ###### Theorem 3.1. Let the above assumptions hold, with $q=0$. Then for all $\lambda>0$, with $\widehat{A}=A+\lambda I,$ $\widehat{B}=B+\lambda I$, the map $\Xi_{\lambda}:u\mapsto(\widehat{A}u,\widehat{B}u),$ viewed as a map from $W_{p}^{s+2}\left(\Omega\right)$ to $W_{p}^{2}\left(\Omega\right)\times B_{p,p}^{1+s-1/p}\left(\Gamma\right),$ is an isomorphism. This means that $\Xi_{\lambda}$ is a linear bijection, and there is a positive constant $C$, independent of $u$, such that $C^{-1}\left\\|u\right\\|_{W_{p}^{s+2}\left(\Omega\right)}\leq\left\\|\widehat{A}u\right\\|_{W_{p}^{2}\left(\Omega\right)}+\left\\|\widehat{B}u\right\\|_{B_{p,p}^{1+s-1/p}\left(\Gamma\right)}\leq C\left\\|u\right\\|_{W_{p}^{s+2}\left(\Omega\right)}$ (3.2) for all $u\in W_{p}^{s+2}\left(\Omega\right)$. Thus, the isomorphism is a linear homeomorphism, but it not need be isometric. Here $W_{p}^{r}\left(\Omega\right)$ is Triebel’s notation for the Sobolev space and $B_{p,p}^{r}\left(\Gamma\right)$ for the Besov space. For $s=0$ and $p=2,$ this reduces $\Xi_{\lambda}$ to being an isomorphism from $H^{2}\left(\Omega\right)$ to $L^{2}\left(\Omega\right)\oplus L^{2}\left(\Gamma,dS\right),$ which is equivalent to saying that $\Xi_{\lambda}$ is an isomorphism from $H^{2}\left(\Omega\right)$ to $\mathbb{X}_{2}$, since $L^{2}\left(\Gamma,dS\right)$ and $L^{2}\left(\Gamma,dS/b\right)$ are the same sets with equivalent inner products. It follows that, when $q=0$, the domain of the Wentzell Laplacian $A$ is exactly $H^{2}\left(\Omega\right)$. ###### Theorem 3.2. Let $H_{\ast}^{2}\left(\Omega\right)=\left\\{u\in H^{2}\left(\Omega\right):u_{\mid\Gamma}\in H^{2}\left(\Gamma\right)\right\\}.$ The domain of the Wentzell Laplacian $A$, the selfadjoint closure of $A_{0}$, defined by (2.7), (2.9), is exactly $D\left(A\right)=\left\\{\begin{array}[]{cc}H^{2}\left(\Omega\right)&\text{if }q=0,\\\ H_{\ast}^{2}\left(\Omega\right)&\text{if }q>0.\end{array}\right.$ (3.3) The same conclusion holds for the closure of the operator $A$ defined by (3.1). Before outlining the proof of this theorem we make some remarks. Theorem 3.2 gives the first ”simple” explicit characterization of $D\left(A\right)$, including the case of $q>0.$ Normally, knowing that $D\left(A_{0}\right)$ is a core for $A$ is enough for most purposes involving linear problems. But we need to know $D\left(A\right)$ exactly in order to apply the Brezis-Haraux result (see Proposition 4.14 below). Theorem 3.2 assumes that $\Gamma,$ $b$ and $c$ are $C^{\infty}$. Surely this much regularity is not needed. But the proof is based on pseudo differential operator techniques and this theory is always presented in the $C^{\infty}$ context, because to do otherwise would entail many complicated calculations requiring a lot of courage. So Theorem 3.2 should be valid if everything is $C^{2},$ but this is merely an educated guess (however, see Remark 3.1). We wish to recall the earlier work on this problem by Escher [4] (see also Fila and Quittner [7]). Escher proved Theorem 3.2 in the special case of $b\equiv 1$ and $q=0$. He worked in the $\mathbb{X}_{p}$ context for $1<p<+\infty,$ but, by focusing on the analytic semigroup aspect of the problem, he did not notice the selfadjointness of $A$. Moreover, his restriction to the case of $b\equiv 1$ avoids many interesting cases, since the coefficient $b$ has physical significance (cf. [8]). We now recall the strategy of the proof of Theorem 3.1. We outline the proof in several steps: Step 1. Treat the case of constant coefficients and take $\Omega$ to be a half-space. Step 2. Then localizing and using a partition of unity, this breaks the problem down into a large number of problems $\left\\{P_{j}\right\\},$ where the portion of $\Gamma$ is the subdomain corresponding to $P_{j}$ is almost flat and the coefficients are almost constants. Flatten out the boundary and solve each $P_{j},$ using Step 1, and the theory of pseudo differential operators (see, e.g., Taylor [19]). Finally, put everything together and complete the proof. The proof is quite long, technical and complicated, but it is now well understood and standard. For the moment we focus on Step 1 and, for simplicity, assume that $\mathcal{A}$ is the identity matrix, so that $A=-\Delta$. Then our problem (3.1) becomes the constant coefficient problem: $\left\\{\begin{array}[]{cc}\widehat{A}u=-\Delta u+\lambda u=f&\text{in }\mathbb{R}_{+}^{N},\\\ \widehat{B}u=b\partial_{n}u+cu+\lambda u-qb\Delta_{\Gamma}u=g&\text{on }\partial\mathbb{R}_{+}^{N}.\end{array}\right.$ (3.4) Here $\mathbb{R}_{+}^{N}=\\{x=\left(y,z\right):y\in\mathbb{R}^{N-1},$ $z\geq 0\\}$, $\partial\mathbb{R}_{+}^{N}=\\{x=\left(y,0\right):y\in\mathbb{R}^{N-1}\\}$ and the boundary condition of (3.4) is equivalent to $b\partial_{z}u+cu+\lambda u-qb\Delta_{y}u=g$ (3.5) on $\partial\mathbb{R}_{+}^{N}.$ For a function $h\left(y,z\right),$ let $\widehat{h}\left(\zeta,z\right)$ be the Fourier transform in the $\mathbb{R}^{N-1}$-variable with $z$ fixed: $\widehat{h}\left(\zeta,z\right)=\left(2\pi\right)^{\frac{1-N}{2}}\int\limits_{\mathbb{R}^{N-1}}e^{-i\zeta\cdot y}h\left(y,z\right)dy,\text{ }\left(\zeta,z\right)\in\mathbb{R}_{+}^{N}.$ Then, in Fourier space, the first equation of (3.4) and equation (3.5) become $\frac{\partial^{2}\widehat{u}}{\partial z^{2}}-\left(\left\|\zeta\right\|^{2}+\lambda\right)\widehat{u}=\widehat{f}\text{ in }\mathbb{R}_{+}^{N},$ (3.6) $b\frac{\partial\widehat{u}}{\partial z}+\left(c+\lambda+qb\left\|\zeta\right\|^{2}\right)\widehat{u}=\widehat{g}\text{ on }\partial\mathbb{R}_{+}^{N}.$ (3.7) We need $u$ to be an $L^{2}$ function. To solve (3.6), one finds the general solution of the homogeneous equation and adds to it a particular solution of (3.6), obtained by the variation of constants formula. The general solution of the homogenous version of (3.6) is $\widehat{u}\left(\zeta,z\right)=C_{1}e^{\gamma_{1}z}+C_{2}e^{\gamma_{2}z},$ (3.8) where $\gamma_{j}=\left(-1\right)^{j+1}\left(\left\|\zeta\right\|^{2}+\lambda\right)^{1/2},\text{ }j=1,2.$ Then for each $\zeta\in\mathbb{R}^{N-1}$, $\gamma_{2}<0<\gamma_{1}.$ Thus, the general $L^{2}$ solution of the homogeneous problem is given by (3.8), with $C_{2}$ an arbitrary constant and $C_{1}=0$. Next, (3.7) is of the form $\frac{\partial\widehat{u}}{\partial z}-p\left(\zeta\right)=m\left(\zeta\right),$ for $z=0,$ where $p\geq\varepsilon_{0}>0$ for all $\zeta$ (For more general problems, the corresponding inequality follows from uniform ellipticity). It follows that (3.4) (as well as (3.6), (3.7)) has a unique $L^{2}$ solution. Note that this works for $q>0$ as well as for $q=0$. For $q>0,$ we require that $\left\|\zeta\right\|^{2}\widehat{u}$ as well as $\widehat{u}$ is in $L^{2}$. If one studies the proof in [20] in detail, minor modifications of the tedious calculations lead to the proof of Theorem 3.2. More precisely, for $q>0$, we conclude that there is a positive constant $C=C\left(q,b,c,\lambda,\mathcal{A}\right),$ for every $\lambda>0,$ such that $C^{-1}\left\\|u\right\\|_{H_{\ast}^{2}\left(\Omega\right)}\leq\left\\|\left(\widehat{A}u,\widehat{B}u\right)^{T}\right\\|_{\mathbb{X}_{2}}\leq C\left\\|u\right\\|_{H_{\ast}^{2}\left(\Omega\right)}$ (3.9) for all $u\in H_{\ast}^{2}\left(\Omega\right).$ Moreover, the map $u\mapsto\left(\widehat{A}u,\widehat{B}u\right)^{T}$ is a surjective linear isomorphism of $H_{\ast}^{2}\left(\Omega\right)$ onto $\mathbb{X}_{2},$ for $q>0$. Above in (3.9), the norm in $H_{\ast}^{2}\left(\Omega\right)$ is defined as $\left\\|u\right\\|_{H_{\ast}^{2}\left(\Omega\right)}=\left(\left\\|u\right\\|_{H^{2}\left(\Omega\right)}^{2}+\left\\|tr\left(u\right)\right\\|_{H^{2}\left(\Gamma\right)}^{2}\right)^{1/2}.$ From this, the proof of Theorem 3.2 follows. $\square$ Remark 3.1. We note that the first inequality of (3.9) was already obtained in [16, Lemma A.1] for the weak solutions of (3.1), using standard Sobolev inequalities and assuming $b,$ $c\in C\left(\Gamma\right),$ $b,\lambda>0,$ $\mathcal{A}=I_{N\times N}$ and $\Gamma$ is of class $C^{2}$. Observe that (3.1) is also an elliptic boundary value problem in the sense specified in [11, 17], where similar estimates to (3.9) were also obtained. The second inequality of (3.9) is obvious and is based on the definition of $\widehat{A}$ and $\widehat{B}$. ## 4\. Convex analysis We begin with the following assumptions: (H1) The functions $\alpha_{i}:\mathbb{R}\rightarrow\mathbb{R}$, $i=1,2,$ are continuous, monotone nondecreasing with $\alpha_{i}(0)=0$. (H2) Let $\Lambda_{i}$ be as in (1.8) and suppose that they satisfy the _$\triangle_{2}$ -condition near infinity,_ in the sense that, there are positive constants $t_{i},$ $C_{i}>0$, $i=1,2,$ such that $\Lambda_{i}(2t)\leq C_{i}\Lambda_{i}(t),\;\mbox{ for all }\;t\geq t_{i}.$ (4.1) Let $\tilde{\alpha}_{i}:\;\mathbb{R}\rightarrow\mathbb{R}$ $(i=1,2)$ be the inverse of $\alpha_{i}$. Then $\tilde{\alpha}_{i}$ is a nondecresing function from $\mathbb{R}$ to $\mathbb{R}$, which is multivalued at its jumps and it is in $L_{loc}^{1}\left(\mathbb{R}\right)$. Its graph is a connected subset of $\mathbb{R}^{2}$. Let $\widetilde{L}_{i}:\mathbb{R}\rightarrow[0,+\infty),$ $i=1,2,$ be defined by $\widetilde{L}_{i}(t):=\int_{0}^{t}\widetilde{\alpha}_{i}(s)ds\text{ and }\widetilde{\Lambda}_{i}:=\max\left\\{\widetilde{L}_{i}(t),\widetilde{L}_{i}(-t)\right\\},\text{ for all }t\in\mathbb{R}.$ (4.2) All the functions given in (1.8) and (4.2) are convex and continuous on $\mathbb{R}$, nondecreasing on $\mathbb{R}_{+}$, and all vanish at the origin; $\Lambda_{i}$ and $\widetilde{\Lambda}_{i}$ are even functions and are complementary Young functions in the sense of [18, Chap. I, Section 1.3, Theorem 3], but they need not be $N$-functions. Note that $L_{i}^{{}^{\prime}}\left(t\right)=\alpha_{i}\left(t\right)$ on $\mathbb{R}$ and $\widetilde{L}_{i}^{{}^{\prime}}\left(t\right)=\widetilde{\alpha}_{i}\left(t\right)$ a.e.; $\left\|\Lambda_{i}^{{}^{\prime}}\left(t\right)\right\|\geq\left\|\alpha_{i}\left(t\right)\right\|$ and $\left\|\widetilde{\Lambda}_{i}^{{}^{\prime}}\left(t\right)\right\|\geq\left\|\widetilde{\alpha}_{i}\left(t\right)\right\|$ almost everywhere. It follows, from [18, Chap. I, Section 1.3, Theorem 3], that for all $s,$ $t\in\mathbb{R}$, $\left\|st\right\|\leq L_{i}(t)+\widetilde{L}_{i}(s)\leq\Lambda_{i}(t)+\widetilde{\Lambda}_{i}(s).$ (4.3) Suppose that $\Lambda_{i}\left(s\right)=L_{i}\left(\tau\right)$ and $\widetilde{\Lambda}_{i}\left(s\right)=\widetilde{L}_{i}\left(\sigma\right)$, where $\tau$ is $s$ or $-s$ and $\sigma$ is $t$ or $-t$. If $\tau=\widetilde{\alpha}_{i}(\sigma)$ or $\sigma=\alpha_{i}(\tau),$ then we also have equality, that is, $\widetilde{L}_{i}(\alpha_{i}(\tau))=\widetilde{\Lambda}_{i}(\alpha_{i}(\tau))=\tau\alpha_{i}(\tau)-\Lambda_{i}(\tau)=\tau\alpha_{i}(\tau)-L_{i}(\tau),\;\text{ }i=1,2.$ (4.4) Let now $\alpha_{i}:\;\mathbb{R}\rightarrow\mathbb{R}$, $i=1,2,$ satisfy (H1). Define the functional $J:\mathbb{X}_{2}\rightarrow[0,+\infty]$ by $J\left(U\right)=\frac{1}{2}\int\limits_{\Omega}\left\|\nabla u\right\|^{2}dx+\int\limits_{\Omega}L_{1}\left(u\right)dx+\int\limits_{\Gamma}j_{2}\left(x,u\right)\frac{dS}{b\left(x\right)},$ (4.5) for $U=\left(u,tr\left(u\right)\right)^{T},$ $u\in H^{1}\left(\Omega\right)$ such that all three integrals exist, and $tr\left(u\right)\in H^{1}\left(\Gamma\right)$ if $q>0$. We take $j_{2}\left(x,u\right)=c\left(x\right)\frac{u^{2}}{2}+qb\left(x\right)\frac{\left\|\nabla_{\Gamma}u\right\|^{2}}{2}+L_{2}\left(u\right).$ (4.6) The effective domain $\mathbb{D}_{q}:=D\left(J\right)$ of the functional $J$ is precisely $\mathbb{D}_{0}=\\{U=\left(u,tr\left(u\right))\right)^{T}:u\in H^{1}\left(\Omega\right),\;\int_{\Omega}\Lambda_{1}(u)dx+\int_{\Gamma}\Lambda_{2}(u)\frac{dS}{b\left(x\right)}<\infty\\}$ (4.7) if $q=0$, and $\mathbb{D}_{q}=\\{U=\left(u,tr\left(u\right))\right)^{T}\in\mathbb{D}_{0}:tr\left(u\right)\in H^{1}\left(\Gamma\right)\\}$ (4.8) if $q>0$, respectively. Define $J\left(U\right)=+\infty,$ for all $U\in\mathbb{X}_{2}\backslash\mathbb{D}_{q},$ $q\geq 0$. As before, for $u\in H^{1}\left(\Omega\right)$, we identify $u$ with $U=$ $\left(u,tr\left(u\right)\right)^{T}\in\mathbb{X}_{2}$. Then $J$ is proper, convex and lower semicontinuous on $\mathbb{X}_{2},$ as can be shown adapting the ideas of Brezis [1] (see also [6]). Suppose now that $\alpha_{i}$, $i=1,2$, satisfies assumptions (H1)-(H2). Then, by (4.1), the monotonicity (on $\mathbb{R}_{+}$) and the convexity of $\Lambda_{i},$ $i=1,2,$ we have that $\mathbb{D}_{q},$ $q\geq 0,$ is a vector space (see, e.g., [18, Chap. III, Section 3.1, Theorem 2]). In what follows, we shall compute the subdifferential of $J$. To this end, let $F:=(f,g)^{T}\in\mathbb{X}_{2}$ and $U=(u,tr(u))^{T}\in\mathbb{D}_{q}$. We claim that $F\in\partial J(U)$ if and only if $\displaystyle-\Delta u+\alpha_{1}(u)$ $\displaystyle=f\;\text{in }\mathcal{D}^{\prime}(\Omega),$ (4.9) $\displaystyle b(x)\frac{\partial u}{\partial n}+c(x)u-qb(x)\Delta_{\Gamma}u+\alpha_{2}(u)$ $\displaystyle=g\text{ on }\Gamma.$ First, assume that $F\in\partial J(U)$. Then, by definition, for every $V=(v,tr(v))^{T}\in\mathbb{D}_{q}$, we have $\displaystyle\int_{\Omega}f(v-u)dx+\int_{\Gamma}g(v-u)\frac{dS}{b}$ $\displaystyle\leq\frac{1}{2}\int_{\Omega}\left(\|\nabla v\|^{2}-\|\nabla u\|^{2}\right)dx$ (4.10) $\displaystyle+\int_{\Omega}\left(L_{1}(v)-L_{1}(u)\right)dx+\int_{\Gamma}\left(j_{2}(x,v)-j_{2}(x,u)\right)\frac{dS}{b},$ where, from (4.6), we find that $\displaystyle\int_{\Gamma}\left(j_{2}(x,v)-j_{2}(x,u)\right)\frac{dS}{b}$ $\displaystyle=\frac{1}{2}\int_{\Gamma}c\left(\|v\|^{2}-\|u\|^{2}\right)\frac{dS}{b}+q\frac{1}{2}\int_{\Gamma}\left(\|\nabla_{\Gamma}v\|^{2}-\|\nabla_{\Gamma}u\|^{2}\right)dS$ $\displaystyle+\int_{\Gamma}\left(L_{2}(v)-L_{2}(u)\right)\frac{dS}{b}.$ Let $W=(w,tr(w))^{T}\in\mathbb{D}_{q}$ be fixed and let $t\in[0,1]$. Choosing $V:=tW+(1-t)U\in\mathbb{D}_{q}$ in (4.10), dividing by $t$ and taking the limit as $t\rightarrow 0^{+},$ from (4.10), we obtain $\displaystyle\int_{\Omega}f(w-u)dx+\int_{\Gamma}g(w-u)\frac{dS}{b}$ (4.11) $\displaystyle\leq\int_{\Omega}\nabla u\cdot\nabla(w-u)dx+\int_{\Omega}\alpha_{1}(u)(w-u)dx+\int_{\Gamma}cu(w-u)\frac{dS}{b}$ $\displaystyle+q\int_{\Gamma}\nabla_{\Gamma}u\cdot\nabla_{\Gamma}(w-u)dS+\int_{\Gamma}\alpha_{2}(u)(w-u)\frac{dS}{b}.$ Here we used the definition of the functions $L_{i}$ ($i=1,2$) from (1.8) and the Lebesgue Dominated convergence theorem, which implies $\lim_{t\rightarrow 0^{+}}\int_{\Omega}\frac{L_{1}(u+t(w-u))-L_{1}(u)}{t}dx=\int_{\Omega}\alpha_{1}(u)(w-u)dx$ and $\lim_{t\rightarrow 0^{+}}\int_{\Gamma}\frac{L_{2}(u+t(w-u))-L_{2}(u)}{t}\frac{dS}{b}=\int_{\Gamma}\alpha_{2}(u)(w-u)\frac{dS}{b}.$ Letting $W=U\pm\Psi$ in (4.11), where $\Psi=(\psi,tr(\psi))^{T}$ is an arbitrary element of $\mathbb{D}_{q}$, we easily deduce $\displaystyle\int_{\Omega}f\psi dx+\int_{\Gamma}g\psi\frac{dS}{b}$ $\displaystyle=\int_{\Omega}\nabla u\cdot\nabla\psi dx+\int_{\Omega}\alpha_{1}(u)\psi dx$ (4.12) $\displaystyle+\int_{\Gamma}cu\psi\frac{dS}{b}+q\int_{\Gamma}\nabla_{\Gamma}u\cdot\nabla_{\Gamma}\psi\frac{dS}{b}+\int_{\Gamma}\alpha_{2}(u)\psi\frac{dS}{b}.$ Taking $\psi\in C_{0}^{\infty}(\Omega)$ in (4.12), one obtains the first equation of (4.9). A simple partial integration argument shows that one also has the second equation in (4.12). We shall now prove the converse. Let $U=(u,tr(u))^{T}\in\mathbb{D}_{q}$ be fixed and let $V=(v,tr(v))^{T}\in\mathbb{D}_{q}$ be arbitrary. On account of (4.3) and (4.4), we have $\displaystyle\alpha_{1}(u)(v-u)$ $\displaystyle=\alpha_{1}(u)v-\alpha_{1}(u)u$ (4.13) $\displaystyle\leq L_{1}(v)+\widetilde{L}_{1}(\alpha_{1}(u))-\alpha_{1}(u)u\text{,}$ $\displaystyle\leq L_{1}(v)-L_{1}(u)$ and $\displaystyle\alpha_{2}(u)(v-u)$ $\displaystyle=\alpha_{2}(u)v-\alpha_{2}(u)u$ (4.14) $\displaystyle\leq L_{2}(v)+\widetilde{L}_{2}(\alpha_{2}(u))-\alpha_{2}(u)u\text{,}$ $\displaystyle\leq L_{2}(v)-L_{2}(u).$ Therefore, by (4.5) and using (4.13)-(4.14), we have $\displaystyle J(V)-J(U)$ $\displaystyle=\frac{1}{2}\int_{\Omega}\left(\|\nabla v\|^{2}-\|\nabla u\|^{2}\right)dx+\int_{\Omega}\left(L_{1}(v)-L_{1}(u)\right)dx$ (4.15) $\displaystyle+\frac{1}{2}\int_{\Gamma}c\left(\|v\|^{2}-\|u\|^{2}\right)\frac{dS}{b}+\frac{q}{2}\int_{\Gamma}\left(\|\nabla_{\Gamma}v\|^{2}-\|\nabla_{\Gamma}u\|^{2}\right)dS$ $\displaystyle+\int_{\Gamma}\left(L_{2}(v)-L_{2}(u)\right)\frac{dS}{b}$ $\displaystyle\geq\int_{\Omega}\nabla u\cdot\nabla(v-u)dx+\int_{\Omega}\alpha_{1}(u)(v-u)dx$ $\displaystyle+\int_{\Gamma}cu(v-u)\frac{dS}{b}+q\int_{\Gamma}\nabla_{\Gamma}u\cdot\nabla_{\Gamma}(v-u)dS+\int_{\Gamma}\alpha_{2}(u)(v-u)\frac{dS}{b}.$ Thus, from Definition 1.1 and (4.15), for all $(V-U)\in\mathbb{D}_{q}$, it follows that $J(V)-J(U)\geq\int_{\Omega}f(v-u)dx+\int_{\Gamma}g(v-u)\frac{dS}{b}.$ This inequality is also true for $V=U+W\in\mathbb{D}_{q}$, for some arbitrary $W\in\mathbb{D}_{q}$. Indeed, let $W=(w,tr(w))^{T}\in\mathbb{D}_{q}$ be fixed, $w_{m}:=[w\wedge m]\vee(-m)$ and set $W_{m}:=(w_{m},tr(w_{m}))^{T}$. Let $W_{m,n}=(w_{m,n},tr(w_{m,n}))^{T}$ be a sequence in $\mathbb{D}_{q}$ such that $-m\leq w_{m,n}\leq m$, $w_{m,n}\rightarrow w_{m}$ in $H^{1}(\Omega)$ and $tr(w_{m,n})\rightarrow tr(w_{m})$ in $H^{1}(\Gamma),$ if $q>0,$ as $n\rightarrow\infty$. Then, $\displaystyle J(W_{m}+U)-J(U)$ $\displaystyle=\lim_{n\rightarrow\infty}J(W_{m,n}+U)-J(U)$ (4.16) $\displaystyle\geq\lim_{n\rightarrow\infty}\left(\int_{\Omega}fw_{m,n}dx+\int_{\Gamma}gw_{m,n}\frac{dS}{b}\right)$ $\displaystyle\geq\int_{\Omega}fw_{m}\;dx+\int_{\Gamma}gw_{m}\frac{dS}{b}.$ Passing to the limit as $m\rightarrow\infty$ in (4.16) in a standard way and using the fact $W\in\mathbb{D}_{q}$ is arbitrary, we immediately get $J(W+U)-J(U)\geq\int_{\Omega}fwdx+\int_{\Gamma}gw\frac{dS}{b}.$ (4.17) Since $\mathbb{D}_{q}$ is a vector space, we also obtain the corresponding inequality (4.17) when replacing $W+U$ by $V$. Hence, $F\in\partial J(U)$ and this completes the proof of the claim. We have shown that the (single-valued) subdifferential of the functional $J$ at $U$ is given by $D(\partial J)=\left\\{(u,tr\left(u\right))^{T}\in\mathbb{D}_{q}:-\Delta u+\alpha_{1}(u)\in L^{2}(\Omega),\text{ }b(x)\frac{\partial u}{\partial n}-qb(x)\Delta_{\Gamma}u+\alpha_{2}(u)\in L^{2}(\Gamma)\right\\}$ (4.18) and $\partial J(U)=\left(-\Delta u+\alpha_{1}\left(u\right),b\left(x\right)\frac{\partial u}{\partial n}+c\left(x\right)u-qb\left(x\right)\Delta_{\Gamma}u+\alpha_{2}\left(u\right)\right)^{T}.$ (4.19) Since the functional $J$ is proper, convex and lower-semicontinuous, it follows from Minty’s theorem [14] that the operator $B:=\partial J$ is maximal monotone (or $-B$ is m-dissipative), for our choice of the function $j_{2}\left(x,u\right)$ in (4.6). Thus, the first result of this section is the following. ###### Theorem 4.1. The operator $B$ is the subdifferential of a proper, convex, lower semicontinuous function on $\mathbb{X}_{2}$. Theorem 4.1 applies to both $A$, the negative Wentzell Laplacian (by taking both $\alpha_{1}$ and $\alpha_{2}$ to be zero) and to the operator governing (1.5) on $\mathbb{X}_{2}$. We remark that the above construction leads easily to a proof that the Wentzell Laplacian has a compact resolvent. Of course, this follows easily from the results quoted in Section $3$, but the compactness does not require $C^{\infty}$-regularity. Next, let $A_{2}U=\left(\alpha_{1}\left(u\right),\alpha_{2}\left(v\right)\right)^{T},$ for every $U\in D\left(A_{2}\right),$ where $D\left(A_{2}\right)=\left\\{\left(u,v)\right)^{T}\in\mathbb{X}_{2}:\left(\alpha_{1}\left(u\right),\alpha_{2}\left(v\right)\right)^{T}\in\mathbb{X}_{2}\right\\}.$ (4.20) Define the functional $J_{2}:\;\mathbb{X}_{2}\rightarrow[0,+\infty]$ by $J_{2}(U)=\begin{cases}\int_{\Omega}L_{1}(u)dx+\int_{\Gamma}L_{2}(v)\frac{dS}{b(x)},\;\;&\text{if }(u,v)^{T}\in D(J_{2})\\\ +\infty&\text{if }(u,v)^{T}\in\mathbb{X}_{2}\backslash D(J_{2}),\end{cases}$ with effective domain $D(J_{2}):=\\{(u,v)^{T}\in\mathbb{X}_{2}:\int_{\Omega}\Lambda_{1}(u)dx+\int_{\Gamma}\Lambda_{2}(v)\frac{dS}{b(x)}<\infty\\}.$ It is easy to see that, under the assumption (H1) on $\alpha_{i}$, the functional $J_{2}$ is proper, convex and lower-semicontinuous on $\mathbb{X}_{2}$. We have the following. ###### Lemma 4.2. Let $\alpha_{i}:\;\mathbb{R}\rightarrow\mathbb{R}$, $i=1,2,$ satisfy (H1)-(H2). Then the subdifferential $\partial J_{2}$ and the operator $A_{2}$ coincide, that is, $D(\partial J_{2})=D(A_{2})$ and, for all $U:=(u,v)^{T}\in D(A_{2}),$ we have $\partial J_{2}\left(U\right)=A_{2}U=\left(\alpha_{1}\left(u\right),\alpha_{2}\left(v\right)\right)^{T}.$ ###### Proof. Note that (H1) implies that $\partial J_{2}$ is a single valued operator. Let $U=(u,v)^{T}\in D(J_{2})$ and $(f,g)^{T}=\partial J_{2}(U)$. Then, by definition, $(f,g)^{T}\in\mathbb{X}_{2}$ and for every $V:=(u_{1},v_{1})^{T}\in D(J_{2}),$ we get $\int_{\Omega}f(u_{1}-u)dx+\int_{\Gamma}g(v_{1}-v)\frac{dS}{b(x)}\leq J_{2}(V)-J_{2}(U).$ (4.21) Next, let $W=(u,v)^{T}+t(u_{2},v_{2})^{T},$ with $(u_{2},v_{2})^{T}\in D(J_{2})$ and $0<t\leq 1$. Since (H2) implies that $D(J_{2})$ is a vector space, then $W\in D(J_{2})$. Now, replacing $V$ in (4.21) with $W$, dividing by $t$ and taking the limit as $t\rightarrow 0^{+}$ (where we make use of the Lebesgue Dominated Convergence theorem once again), we obtain $\int_{\Omega}fu_{2}dx+\int_{\Gamma}gv_{2}\frac{dS}{b(x)}\leq\int_{\Omega}\alpha_{1}(u)u_{2}dx+\int_{\partial\Omega}\alpha_{2}(v)v_{2}\,\frac{dS}{b(x)}.$ (4.22) Changing $(u_{2},v_{2})^{T}$ to $-(u_{2},v_{2})^{T}$ in (4.22) gives $\int_{\Omega}fu_{2}dx+\int_{\Gamma}gv_{2}\frac{dS}{b(x)}=\int_{\Omega}\alpha_{1}(u)u_{2}dx+\int_{\partial\Omega}\alpha_{2}(v)v_{2}\,\frac{dS}{b(x)}.$ In particular, taking $v_{2}=0$, for every $u_{2}\in C_{0}^{\infty}(\Omega)$, we have $\int_{\Omega}fu_{2}dx=\int_{\Omega}\alpha_{1}(u)u_{2}dx,$ and this shows that $\alpha_{1}(u)=f$. Similarly, one obtains $\alpha_{2}(v)=g$. We have shown that $U:=(u,v)^{T}\in D(A_{2})$ and $\partial J_{2}(U)=(\alpha_{1}(u),\alpha_{2}(v))^{T}$. Conversely, let $U=(u,v)^{T}\in D(A_{2})$ and set $(f,g)^{T}:=A_{2}U=(\alpha_{1}(u),\alpha_{2}(v))^{T}$. Observe preliminarily that, owing to (H2), there exist constants $t_{i}>0$ and $k_{i}\in(0,1]$ such that $k_{i}t\alpha_{i}(t)\leq\Lambda_{i}(t)\leq t\alpha_{i}(t)\text{, for all }\|t\|\geq t_{i},\text{ }i=1,2.$ (4.23) Since $(\alpha_{1}(u),\alpha_{2}(v))^{T}\in\mathbb{X}_{2},$ from (4.23), it follows that $\displaystyle\int_{\Omega}\Lambda_{1}(u)dx$ $\displaystyle=$ $\displaystyle\int_{\\{x\in\Omega:\;\|u(x)\|<t_{1}\\}}\Lambda_{1}(u)dx+\int_{\\{x\in\Omega:\;\|u(x)\|\geq t_{1}\\}}\Lambda_{1}(u)dx$ $\displaystyle\leq$ $\displaystyle\|\Omega\|(\Lambda_{1}(t_{1})+\Lambda_{1}(-t_{1}))+\int_{\Omega}u\alpha_{1}(u)dx<\infty,$ where a similar inequality holds for $\Lambda_{2}$. Hence $\int_{\Omega}\Lambda_{1}(u)dx+\int_{\partial\Omega}\Lambda_{2}(v)\frac{dS}{b(x)}<\infty$ and this shows that $(u,v)^{T}\in D(J_{2})$. Let $V=(u_{1},v_{1})^{T}\in D(J_{2})$. Note that by (4.13) and (4.14), we have once more that $\alpha_{1}(u)(u_{1}-u)\leq L_{1}(u_{1})-L_{1}(u)$ (4.24) and $\alpha_{2}(v)(v_{1}-v)\leq L_{2}(v_{1})-L_{2}(v).$ (4.25) Therefore, on account of (4.24)-(4.25), it follows that $\displaystyle\int_{\Omega}f(u_{1}-u)dx+\int_{\Gamma}g(v_{1}-v)\frac{dS}{b(x)}$ $\displaystyle=\int_{\Omega}\alpha_{1}(u)(u_{1}-u)dx+\int_{\partial\Omega}\alpha_{2}(v)(v_{1}-v)\frac{dS}{b(x)}$ $\displaystyle\leq J_{2}(V)-J_{2}(U).$ By definition, we have shown that $(\alpha_{1}(u),\alpha_{2}(v))^{T}=\partial J_{2}(U)$. Hence, $U\in D(\partial J_{2})$ and $A_{2}U=\partial J_{2}(U).$ This completes the proof. We will need the following results from semigroup theory and convex analysis. ###### Definition 4.3 ([2]). Let $\mathcal{H}$ be a real Hilbert space. Two subsets $K_{1}$ and $K_{2}$ are almost equal, written as $K_{1}\simeq K_{2},$ if $K_{1}$ and $K_{2}$ have the same closure and the same interior, that is, $\overline{K_{1}}=\overline{K_{2}}$ and $int\left(K_{1}\right)=int\left(K_{2}\right).$ The following result is contained in [2, pp.173–174]. ###### Theorem 4.4. Let $A$ and $B$ be subdifferentials of proper convex lower semicontinuous functionals $\varphi_{1}$ and $\varphi_{2}$, respectively, on a real Hilbert space $\mathcal{H}$ with $D(\varphi_{1})\cap D(\varphi_{2})\neq\emptyset$. Let $C$ be the subdifferential of the proper, convex lower semicontinuous functional $\varphi_{1}+\varphi_{2}$, that is, $C=\partial(\varphi_{1}+\varphi_{2})$. Then $\mathcal{R}(A)+\mathcal{R}(B)\subset\overline{\mathcal{R}(C)}\;\;\;\mbox{ and }\;\;\;\mbox{Int}\left(\mathcal{R}(A)+\mathcal{R}(B)\right)\subset\mbox{Int}\left(\mathcal{R}(C)\right).$ (4.26) In particular, if the operator $A+B$ is maximal monotone, then $\mathcal{R}\left(A+B\right)\simeq\mathcal{R}\left(A\right)+\mathcal{R}\left(B\right)$ (4.27) and this is the case, if $\partial(\varphi_{1}+\varphi_{2})=\partial\varphi_{1}+\partial\varphi_{2}$. Here by $\mathcal{R}\left(A\right)+\mathcal{R}\left(B\right)$ we mean $\displaystyle\cup\left\\{Af+Bg:f\in D\left(A\right),g\in D\left(B\right)\right\\}\newline $ $\displaystyle=$ $\displaystyle\cup\left\\{h+k:\left(f,h\right)\in A,\left(g,k\right)\in B\text{ for some }f,g\in\mathcal{H}\right\\}.$ We use the union symbol since $A$ and $B$ may be multi-valued. However, in our applications, $A$ and $B$ will be single valued. Let us recall that we want to solve the following problem: $\left\\{\begin{array}[]{c}-\Delta u+\alpha_{1}\left(u\right)=f_{1}\left(x\right)\text{ in }\Omega,\\\ b\left(x\right)\frac{\partial u}{\partial n}+c\left(x\right)u-qb\left(x\right)\Delta_{\Gamma}u+\alpha_{2}\left(u\right)=f_{2}\left(x\right)\text{ on }\Gamma.\end{array}\right.$ (4.28) In order to solve (4.28), recall that $A$ is the linear operator, defined in Section 2 (see (2.11)). More precisely, $A$ has the following operator representation: $A=\left(\begin{array}[]{cc}-\Delta&\qquad 0\\\ b\frac{\partial}{\partial n}&\qquad cI-qb\Delta_{\Gamma}\end{array}\right).$ (4.29) Denote the null space of $A$ by $\mathcal{N}\left(A\right).$ Then $U=(u,tr(u))^{T}\in\mathcal{N}\left(A\right)$ if and only if (by definition) $u$ is a weak solution of $\left\\{\begin{array}[]{c}-\Delta u=0\text{ in }\Omega,\\\ b\left(x\right)\frac{\partial u}{\partial n}+c\left(x\right)u-qb\left(x\right)\Delta_{\Gamma}u=0\text{ on }\Gamma,\end{array}\right.$ (4.30) that is, $u\in H^{1}(\Omega)$ with $tr(u)\in H^{1}(\Gamma)$ if $q>0$ and $\int_{\Omega}\nabla u\cdot\nabla vdx+\int_{\Gamma}cuv\frac{dS}{b}+q\int_{\Gamma}\nabla_{\Gamma}u\cdot\nabla_{\Gamma}vdS=0,$ (4.31) for all $v\in H^{1}(\Omega)$ with $tr(v)\in H^{1}(\Gamma)$ if $q>0$. In this case it is easy to see that $u$ is a weak solution of (4.30) if and only if $u\in H^{1}(\Omega)$ with $tr(u)\in H^{1}(\Gamma),$ if $q>0,$ and (4.31) holds for all $v\in H^{1}(\Omega)$ with $tr(v)\in H^{1}(\Gamma),$ if $q>0$. Hence, it is clear that the null space of $A$ is $\mathcal{N}\left(A\right)=\mathbb{R}\mathbf{1}=\left\\{C\mathbf{1:}\,C\in\mathbb{R}\right\\}$ if $c\equiv 0$ in (4.30), that is, $\mathcal{N}\left(A\right)$ consists of all the real constant functions on $\overline{\Omega}.$ We shall discuss this case first. From now on, let $A_{1}$ be the linear operator $A$ corresponding to the case of $c\equiv 0$. Moreover, let $A_{3}$ be the subdifferential $\partial J$ of the functional $J,$ defined in (4.5)-(4.6), that is, $A_{3}:=\partial J=\partial\left(J_{1}+J_{2}\right)$ (see (4.18)-(4.20)). It follows, from the assumptions on the functions $\alpha_{1},$ $\alpha_{2}$ and the results of Section 2, that $A_{i}=\partial J_{i}$, for each $i=1,2,3$, where each $J_{i}$ is a proper, convex and lower semicontinuous functional on $\mathbb{X}_{2}$. Let us recall the Fredholm alternative, which says that for any selfadjoint operator $B$ with compact resolvent and $0\notin\rho\left(B\right),$ we have that the range $\mathcal{R}\left(B\right)=\overline{\mathcal{R}\left(B\right)}=\mathcal{N}\left(B\right)^{\perp}.$ This is the case with our operator $A_{1},$ that is, we have $\mathcal{R}\left(A_{1}\right)=\mathcal{N}\left(A_{1}\right)^{\perp}=\mathbf{1}^{\perp}=\left\\{F\in\mathbb{X}_{2}:\int\limits_{\overline{\Omega}}Fd\mu=0\right\\},$ (4.32) where the measure $\mu$ is defined by $d\mu=dx\|_{\Omega}\oplus\frac{dS}{b}\|_{\Gamma}$ on $\overline{\Omega}$. Let us now define $\lambda_{1},$ $\lambda_{2}\in\mathbb{R}_{+}$ by $\lambda_{1}=\int\limits_{\Omega}dx,\text{ }\lambda_{2}=\int\limits_{\Gamma}\frac{dS}{b}$ (4.33) so that $\mu\left(\overline{\Omega}\right)=\lambda_{1}+\lambda_{2}.$ We also define the average of $F$ with respect to the measure $\mu$, as follows: $ave_{\mu}\left(F\right):=\frac{1}{\mu\left(\overline{\Omega}\right)}\int\limits_{\overline{\Omega}}Fd\mu=\frac{1}{\mu\left(\overline{\Omega}\right)}\left(\int\limits_{\Omega}f_{1}dx+\int\limits_{\Gamma}f_{2}\frac{dS}{b}\right),$ (4.34) for every $F=\left(f_{1},f_{2}\right)^{T}\in\mathbb{X}_{2}.$ We now restate Theorem 1.2. ###### Theorem 4.5. Let $\alpha_{i}:\;\mathbb{R}\rightarrow\mathbb{R}$, $i=1,2,$ satisfy (H1). Let $c\equiv 0$ in (4.28) and let $F=\left(f_{1},f_{2}\right)^{T}\in\mathbb{X}_{2}$. A necessary condition for the existence of a weak solution of (4.28) is $ave_{\mu}\left(F\right)\in\frac{\lambda_{1}\mathcal{R}\left(\alpha_{1}\right)+\lambda_{2}\mathcal{R}\left(\alpha_{2}\right)}{\lambda_{1}+\lambda_{2}},$ (4.35) while a sufficient condition is that $\alpha_{i}$ satisfies (H2) and $ave_{\mu}\left(F\right)\in int\left(\frac{\lambda_{1}\mathcal{R}\left(\alpha_{1}\right)+\lambda_{2}\mathcal{R}\left(\alpha_{2}\right)}{\lambda_{1}+\lambda_{2}}\right).$ (4.36) Assuming (H2), the condition (4.35) is both necessary and sufficient when $\lambda_{1}\mathcal{R}\left(\alpha_{1}\right)+\lambda_{2}\mathcal{R}\left(\alpha_{2}\right)$ is open, which holds if at least one of $\mathcal{R}\left(\alpha_{1}\right),$ $\mathcal{R}\left(\alpha_{2}\right)$ is open. ###### Proof. Let $\alpha_{i}:\;\mathbb{R}\rightarrow\mathbb{R}$, $i=1,2,$ satisfy (H1). Let $F=\left(f_{1},f_{2}\right)^{T}\in\mathbb{X}_{2}$ be given and let $u$ be a weak solution of (4.28) with $c\equiv 0$. Then (see Definition 1.1), $u\in H^{1}(\Omega)$, $\alpha_{1}(u)\in L^{1}(\Omega),$ $\alpha_{2}(tr(u))\in L^{1}(\Gamma)$, $tr(u)\in H^{1}(\Gamma)$ if $q>0$ and $\displaystyle\int_{\Omega}f_{1}vdx+\int_{\Gamma}f_{2}v\frac{dS}{b}$ $\displaystyle=\int_{\Omega}\nabla u\cdot\nabla vdx$ (4.37) $\displaystyle+\int_{\Omega}\alpha_{1}(u)vdx$ $\displaystyle+\int_{\Gamma}\alpha_{2}(u)v\frac{dS}{b}+q\int_{\Gamma}\nabla_{\Gamma}u\cdot\nabla_{\Gamma}vdS,$ for all $v\in H^{1}(\Omega)\cap C\left(\overline{\Omega}\right),$ if $q=0,$ and all $v\in H^{1}(\Omega)\cap C\left(\overline{\Omega}\right)$ with $tr(v)\in H^{1}(\Gamma),$ if $q>0$. Taking $v=1$ in (4.37), we obtain $\displaystyle\int_{\bar{\Omega}}Fd\mu=\int\limits_{\Omega}f_{1}dx+\int\limits_{\Gamma}f_{2}\frac{dS}{b}$ $\displaystyle=\int\limits_{\Omega}\alpha_{1}\left(u\right)dx+\int\limits_{\Gamma}\alpha_{2}\left(u\right)\frac{dS}{b}$ $\displaystyle\in\left(\lambda_{1}\mathcal{R}\left(\alpha_{1}\right)+\lambda_{2}\mathcal{R}\left(\alpha_{2}\right)\right),$ and so (4.35) holds. This proves the necessity. For the sufficiency, let (4.36) hold and assume that $\alpha_{i}$ satisfies (H2). To show that (4.28), with $c\equiv 0,$ has a weak solution $u$, it is enough to prove that $F:=(f_{1},f_{2})\in\mathcal{R}(A_{3})$. To this end, we will make use of (4.26) from Theorem 4.4 to show that $F\in int(\mathcal{R}(A_{1})+\mathcal{R}(A_{2}))\subset\mathcal{R}(A_{3})$. We know that $-A_{1},$ $-A_{2}$ and $-A_{3}$ are m-dissipative on $\mathbb{X}_{2}$ and $A_{i}=\partial J_{i},$ for every $i=1,2,3,$ where each $J_{i},$ $i=2,3,$ is a proper, convex and lower semicontinuous functional on $\mathbb{X}_{2}.$ Here, $J_{3}=J_{1}+J_{2}$ has the effective domain $D(J_{3})=D(J_{1})\cap D(J_{2})\neq\emptyset$. Let $c_{1},$ $c_{2}\in\mathbb{R}$, $C=\left(c_{1},c_{2}\right)^{T}\in\mathbb{X}_{2}$ and let $\mathcal{C}$ be the family of such vectors $C$ in $\mathbb{X}_{2}$. Let $Q:=\left\\{C\in\mathcal{C}:c_{i}\in\mathcal{R}\left(\alpha_{i}\right),\text{ }i=1,2\right\\}.$ Clearly $Q\subset\mathcal{R}\left(A_{2}\right),$ since $c_{i}=\alpha_{i}\left(d_{i}\right)$ for some constant function $d_{i}$ on $\Omega$ (if $i=1$) or on $\Gamma$ (if $i=2$). Now let (4.36) hold for $F\in\mathbb{X}_{2}$. We must show $F\in\mathcal{R}\left(A_{3}\right).$ By (4.36) we may choose $C=\left(c_{1},c_{2}\right)^{T}\in Q$ such that $ave_{\mu}\left(F\right)=\frac{\lambda_{1}c_{1}+\lambda_{2}c_{2}}{\lambda_{1}+\lambda_{2}}\in int\left(\frac{\lambda_{1}\mathcal{R}\left(\alpha_{1}\right)+\lambda_{2}\mathcal{R}\left(\alpha_{2}\right)}{\lambda_{1}+\lambda_{2}}\right),$ where $\lambda_{1},\lambda_{2}$ are given by (4.33). Then, for $F\in\mathbb{X}_{2},$ we have $F=\left[F-C\right]+C.$ First, $F-C\in\mathcal{R}\left(A_{1}\right)=\mathcal{N}\left(A_{1}\right)^{\perp}=\mathbf{1}^{\perp},$ since $\displaystyle\int\limits_{\overline{\Omega}}\left(F-C\right)d\mu$ $\displaystyle=\int\limits_{\overline{\Omega}}Fd\mu-\left(\lambda_{1}c_{1}+\lambda_{2}c_{2}\right)$ $\displaystyle=\int\limits_{\overline{\Omega}}\left[F-ave_{\mu}\left(F\right)\right]d\mu=0.$ Next, clearly $C\in\mathcal{R}\left(A_{2}\right).$ Thus, it is readily seen that $F\in\left(\mathcal{R}\left(A_{1}\right)+\mathcal{R}\left(A_{2}\right)\right)$. Let now $\varepsilon>0$ be given. We want $\varepsilon>0$ to be small enough, in particular, suppose $0<\varepsilon<\frac{1}{2}dist\left(\frac{\lambda_{1}c_{1}+\lambda_{2}c_{2}}{\lambda_{1}+\lambda_{2}},\mathbf{K}\right),$ where $\mathbf{K}$ consists of the endpoints of the interval $\widetilde{I}=\left(\lambda_{1}\mathcal{R}\left(\alpha_{1}\right)+\lambda_{2}\mathcal{R}\left(\alpha_{2}\right)\right)/\left(\lambda_{1}+\lambda_{2}\right).$ Let $\widetilde{F}=(\widetilde{f_{1}},\widetilde{f_{2}})^{T}\in\mathbb{X}_{2}$ satisfy $\left\\|F-\widetilde{F}\right\\|_{\mathbb{X}_{2}}<\varepsilon.$ We want to pick $\widetilde{C}=\left(\widetilde{c}_{1},\widetilde{c}_{2}\right)^{T}\in Q$ such that $\left\\|C-\widetilde{C}\right\\|_{\mathbb{X}_{2}}<\varepsilon\text{ and }ave_{\mu}\left(\widetilde{F}\right)=\frac{\lambda_{1}\widetilde{c}_{1}+\lambda_{2}\widetilde{c}_{2}}{\lambda_{1}+\lambda_{2}}.$ (4.38) To see how to do this, let $\mathcal{J}_{i}=\mathcal{R}\left(\alpha_{i}\right)$ for $i=1,2.$ Then $c_{i}\in\mathcal{J}_{i}$ and $ave_{\mu}\left(F\right)=\frac{\lambda_{1}c_{1}+\lambda_{2}c_{2}}{\lambda_{1}+\lambda_{2}}\in int\left(\frac{\lambda_{1}\mathcal{J}_{1}+\lambda_{2}J_{2}}{\lambda_{1}+\lambda_{2}}\right).$ (4.39) We may choose at least one of $\widetilde{c}_{1},$ $\widetilde{c}_{2},$ call it $\tilde{c}_{k},$ to be less than $c_{k},$ because $c_{k}$ cannot be the left hand end point of $\mathcal{J}_{k}$ for both $k=1,2,$ because of (4.36). In a similar way, we may choose one of $\widetilde{c}_{1},$ $\widetilde{c}_{2},$ call it $\widetilde{c}_{l},$ to be larger than $c_{l}.$ Next, $\left\|ave_{\mu}\left(F\right)-ave_{\mu}\left(\widetilde{F}\right)\right\|\leq\left\\|F-\widetilde{F}\right\\|_{\mathbb{X}_{2}}<\varepsilon,$ by the Schwarz inequality. By this observation and (4.38)-(4.39), we can find $\widetilde{C}=\left(\widetilde{c}_{1},\widetilde{c}_{2}\right)\in Q$ such that (4.38) holds. Thus, $\left(\mathcal{R}\left(A_{1}\right)+\mathcal{R}\left(A_{2}\right)\right)$ contains an $\varepsilon$-ball in $\mathbb{X}_{2},$ centered at $F,$ for sufficiently small $\varepsilon>0.$ Thus, $F\in int\left(\mathcal{R}\left(A_{1}\right)+\mathcal{R}\left(A_{2}\right)\right)\subset int\left(\mathcal{R}\left(A_{3}\right)\right)\subset\mathcal{R}\left(A_{3}\right),$ by (4.26). Consequently, problem (4.28) is (weakly) solvable in the sense of Definition 1.1, for any $f_{1}\in L^{2}\left(\Omega\right),$ $f_{2}\in L^{2}\left(\Gamma\right),$ if (4.36) holds. This completes the proof. We will now give some examples as applications of Theorem 4.5. ###### Example 4.6. Let $\alpha_{1}\left(s\right)$ or $\alpha_{2}\left(s\right)$ be equal to $\alpha\left(s\right)=r\left\|s\right\|^{p-1}s,$ where $r,$ $p>0$. Then, it is clear that $\alpha$ satisfies (H1) and that $L(s)=\Lambda(s)=\frac{r}{p+1}\|s\|^{p+1}$ also satisfies (H2). Note that $\mathcal{R}\left(\alpha\right)=\mathbb{R}$. Then, it follows that problem (4.28) with $c\equiv 0$ is solvable for any $f_{1}\in L^{2}\left(\Omega\right),$ $f_{2}\in L^{2}\left(\Gamma\right)$. ###### Example 4.7. Consider the case when $c=q=\alpha_{2}\equiv 0$ in (4.28), that is, consider the following boundary value problem: $\left\\{\begin{array}[]{c}-\Delta u+\alpha_{1}\left(u\right)=f_{1}\left(x\right)\text{ in }\Omega,\\\ b\left(x\right)\frac{\partial u}{\partial n}=f_{2}\left(x\right)\text{ on }\Gamma.\end{array}\right.$ Then, by Theorem 4.5, this problem has a weak solution if $\int\limits_{\Omega}f_{1}dx+\int\limits_{\Gamma}f_{2}\frac{dS}{b}\in\lambda_{1}int\left(\mathcal{R}\left(\alpha_{1}\right)\right),$ which yields the classical Landesman-Lazer result (see (1.3)) for $f_{2}\equiv 0$. ###### Example 4.8. Let us now consider the case when $\alpha_{1}\equiv 0$ and $\alpha_{2}\equiv\alpha,$ where $\alpha$ is a continuous, monotone nondecreasing function on $\mathbb{R}$ such that $\alpha\left(0\right)=0$. The problem $\left\\{\begin{array}[]{c}-\Delta u=f_{1}\left(x\right)\text{ in }\Omega,\\\ b\left(x\right)\frac{\partial u}{\partial n}-qb\left(x\right)\Delta_{\Gamma}u+\alpha\left(u\right)=f_{2}\left(x\right)\text{ on }\Gamma,\end{array}\right.$ (4.40) has a weak solution if $\int\limits_{\Omega}f_{1}dx+\int\limits_{\Gamma}f_{2}\frac{dS}{b}\in\lambda_{2}int\left(\mathcal{R}\left(\alpha\right)\right).$ (4.41) For example, if we choose $\alpha\left(s\right)=\arctan\left(s\right)$ in (4.40), (4.41) becomes the necessary and sufficient condition $\left\|\frac{1}{\lambda_{2}}\left(\int\limits_{\Omega}f_{1}dx+\int\limits_{\Gamma}f_{2}\frac{dS}{b}\right)\right\|<\frac{\pi}{2}.$ (4.42) Note that $\alpha(s)=\arctan(s)$ satisfies (H1) and that $L_{2}(s)=\Lambda_{2}(s)=s\arctan(s)-\ln\sqrt{1+s^{2}}$ satisfies (H2). Let us now turn to the case when $c>0$ on a set of positive $dS$-measure (that is, $c\left(x\right)$ is not identically zero on the boundary $\Gamma$) and consider $A_{1}^{1}$ to be the linear operator $A$ of (2.11) corresponding to this case. Since $A_{1}^{1}=\left(A_{1}^{1}\right)^{\ast}\geq 0$ and $A_{1}^{1}$ has compact resolvent, it has a ground state $Z=\left(z_{\mid\Omega},z_{\mid\Gamma}\right)^{T}.$ That is, $\lambda=\min\sigma\left(A_{1}^{1}\right)$ is a simple eigenvalue, $\lambda>0,$ and $\mathcal{N}\left(A_{1}^{1}-\lambda\right)=\left\\{CZ:C\in\mathbf{R}\right\\}$ for some positive function $Z$ on $\overline{\Omega}$. Before proceeding further, we find the ground state of $A_{1}^{1}$ in a simple one-dimensional example. Let $\Omega=\left(0,1\right),$ $\Gamma=\left\\{0,1\right\\}$, $b_{0}=b_{1}=1,$ $q=0$ and $c_{0},c_{1}$ will be specified in the sequel. Here $b_{j}=b\left(j\right)$ and $c_{j}=c\left(j\right).$ We will choose $c_{j}$, $j=0,1$ so that the smallest eigenvalue of $A_{1}^{1}$ is $\lambda=1.$ The required positive solution of $z^{{}^{\prime\prime}}+z=0$ has the form $z\left(x\right)=\cos\left(x-\delta\right)$ (times a constant, which we take to be $1$). We need to choose $\delta$ so that $z>0$ in $\left[0,1\right]$ and choose $c_{0},$ $c_{1}$ such that $z$ satisfies the correct boundary conditions. The boundary conditions are $-z\left(j\right)+\left(-1\right)^{j+1}z^{{}^{\prime}}\left(j\right)+c_{j}z\left(j\right)=0,$ (4.43) for $j=0,1,$ since $\partial/\partial n=\left(-1\right)^{j+1}\partial/\partial x$ and $z^{{}^{\prime\prime}}\left(j\right)=-z\left(j\right)$ at $x=j\in\left\\{0,1\right\\}$. Since $z\left(0\right)=\cos\left(\delta\right),$ $z^{{}^{\prime}}\left(0\right)=\sin\left(\delta\right),$ $z\left(1\right)=\cos\left(1-\delta\right)$ and $z^{{}^{\prime}}\left(1\right)=\sin\left(\delta-1\right).$ Then (4.43) implies $c_{0}=1+\tan\left(\delta\right),\text{ }c_{1}=1+\tan\left(1-\delta\right).$ (4.44) For $\delta\in\left(0.4,0.6\right),$ $c_{0}$ and $c_{1}$ are both positive. Next, for $x\in\left[0,1\right],$ we have $\left(x-\delta\right)\in\left(-1,1\right)\subset\left(-\frac{\pi}{2},\frac{\pi}{2}\right),$ whence $z$ is positive on $\left[0,1\right].$ Moreover, for $x\in\left[0,1\right],$ $x-\delta\in\left[-\delta,1-\delta\right],$ then choosing $\delta=1/2,$ we have $\left\|x-1/2\right\|\leq 1/2$, $\cos\left(x-1/2\right)\in\left[\cos\frac{1}{2},1\right]$ and $c_{0}=c_{1}=1+\tan\left(1/2\right).$ Finally, we can use the above results to prove our first result for a similar elliptic problem to (4.28) in this new case. As an application of (4.26) (see Theorem 4.4), we obtain the following. ###### Theorem 4.9. Let $c$ be a nonnegative function which is positive on $\Gamma_{1}\subset\Gamma,$ where $\int\limits_{\Gamma_{1}}dS>0$. Let $q=0$ and let $\alpha$ be a continuous, monotone nondecreasing function on $\mathbb{R}$ such that $\alpha\left(0\right)=0$, $\alpha\left(\pm\infty\right)=\underset{s\rightarrow\pm\infty}{\lim}\alpha\left(s\right)$. Let $F=\left(f_{1},f_{2}\right)^{T}\in\mathbb{X}_{2}.$ Also, suppose that $\lambda>0$ is the smallest eigenvalue of $A_{1}^{1}$ and let $Z$ be a positive member of the one-dimensional eigenspace of $A_{4}:=A_{1}^{1}-\lambda I.$ Here we view $Z\in\mathbb{X}_{2}$ as $Z=(z_{1},z_{2})^{T}:\overline{\Omega}\rightarrow\mathbb{R}$, and $Z$ corresponds to a $z_{1}\in C(\overline{\Omega})$, with $z_{2}=z_{1}\|_{\Gamma}$ and $z_{1}$ is a positive function on $\overline{\Omega}.$ A necessary condition for the existence of a weak solution for $\left\\{\begin{array}[]{c}-\Delta u-\lambda u+\alpha\left(u\right)=f_{1}\text{ in }\Omega,\\\ \Delta u+b\left(x\right)\frac{\partial u}{\partial n}+\left(c\left(x\right)+\lambda\right)u=f_{2}\text{ on }\Gamma\end{array}\right.$ (4.45) is $\alpha\left(-\infty\right)\left\langle Z,\boldsymbol{1}\right\rangle_{\mathbb{X}_{2}}\leq\left\langle F,Z\right\rangle_{\mathbb{X}_{2}}\leq\alpha\left(+\infty\right)\left\langle Z,\mathbf{1}\right\rangle_{\mathbb{X}_{2}},$ (4.46) while a sufficient condition is that $\alpha$ satisfies (H2) and $\frac{\alpha\left(-\infty\right)}{\min Z}<\left\langle F,Z\right\rangle_{\mathbb{X}_{2}}<\frac{\alpha\left(+\infty\right)}{\max Z}.$ (4.47) ###### Proof. For the necessity part, multiply the first equation of (4.45), the second equation of (4.45) by $z$ and integrate by parts; here $Z=\left(z\|_{\Omega},z\|_{\Gamma}\right)^{T}.$ Using the divergence theorem and the fact that $\mathcal{N}\left(A_{1}^{1}-\lambda\right)=span\left\\{Z\right\\},$ we obtain $\int\limits_{\Omega}\alpha\left(u\right)zdx+\int\limits_{\Gamma}\alpha\left(v\right)z_{\mid\Gamma}\frac{dS}{b}=\int\limits_{\Omega}f_{1}zdx+\int\limits_{\Gamma}f_{2}z_{\mid\Gamma}\frac{dS}{b},$ where $U=\left(u,v\right)^{T}$ with $v=tr\left(u\right)$ is the solution of (4.45) with $F=\left(f_{1},f_{2}\right)^{T}$. Since $Z>0$, this equation becomes $\frac{\left\langle F,Z\right\rangle_{\mathbb{X}_{2}}}{\left\langle Z,\boldsymbol{1}\right\rangle_{\mathbb{X}_{2}}}=\frac{\left\langle\alpha,Z\right\rangle_{\mathbb{X}_{2}}}{\left\langle Z,\boldsymbol{1}\right\rangle_{\mathbb{X}_{2}}}\in\left[\alpha\left(-\infty\right),\alpha\left(+\infty\right)\right],$ and the necessary condition (4.46) follows. If $\alpha\left(-\infty\right)<\alpha\left(r\right)$ for all $r\in\mathbb{R}$, then the endpoint $\alpha\left(-\infty\right)$ can be excluded. A similar remark applies to $\alpha\left(+\infty\right).$ The sufficiency proof is like that of Theorem 4.5, but $Z$ is not a constant. By the Fredholm alternative, we have $\mathcal{R}\left(A_{4}\right)=\mathcal{N}\left(A_{4}\right)^{\perp}=\left\\{F\in\mathbb{X}_{2}:\left\langle F,Z\right\rangle_{\mathbb{X}_{2}}=0\right\\}.$ (4.48) Let us also define the nonlinear operator $A_{5}U=\left(\alpha\left(u\right),0\right)^{T},$ for $\left(u,v\right)^{T}\in D\left(A_{5}\right)$ such that $D\left(A_{5}\right)=\left\\{\left(u,v\right)^{T}\in\mathbb{X}_{2}:u\text{ has a trace }tr\left(u\right)=v\text{ and }\left(\alpha\left(u\right),0\right)^{T}\in\mathbb{X}_{2}\right\\}.$ (4.49) Let us recall that, due to Theorem 4.1, we know that $-A_{4},$ $-A_{5},$ are m-dissipative on $\mathbb{X}_{2}$ and $A_{i}=\partial J_{i},$ for every $i=4,5$ and each $J_{i}$ is a proper, convex and lower semicontinuous functional on $\mathbb{X}_{2}$. Let $J_{6}:=J_{4}+J_{5}$ with domain $D(J_{6}):=D(J_{4})\cap D(J_{5})\neq\emptyset.$ Then $J_{6}$ is a proper, convex and lower semicontinuous functional on $\mathbb{X}_{2}$. Let $A_{6}:=\partial(J_{4}+J_{5})$. Then $-A_{6}$ is m-dissipative on $\mathbb{X}_{2}$. It follows, from (4.26), that $\mathcal{R}\left(A_{4}\right)+\mathcal{R}\left(A_{5}\right)\subset\overline{\mathcal{R}\left(A_{6}\right)}\;\text{and}\;int\left(\mathcal{R}\left(A_{4}\right)+\mathcal{R}\left(A_{5}\right)\right)\subset int\left(\mathcal{R}\left(A_{6}\right)\right).$ (4.50) Suppose now that $Z$ is a positive unit vector in $\mathcal{N}\left(A_{4}\right)$ (recall that $A_{4}=A_{1}^{1}-\lambda I$), that is, $\lambda=\min\sigma\left(A_{4}\right),$ $A_{1}^{1}Z=\lambda Z,$ $\left\\|Z\right\\|_{\mathbb{X}_{2}}=1$ and $Z>0.$ For $F\in\mathbb{X}_{2},$ we have $F=\left[F-\left\langle F,Z\right\rangle_{\mathbb{X}_{2}}Z\right]+\left\langle F,Z\right\rangle_{\mathbb{X}_{2}}Z\in\mathcal{R}\left(A_{4}\right)+\mathcal{R}\left(A_{5}\right),$ provided that $\alpha\left(-\infty\right)<\left\langle F,Z\right\rangle_{\mathbb{X}_{2}}Z<\alpha\left(+\infty\right)$ holds pointwise on $\overline{\Omega}$. But for, $\widetilde{F}=(\widetilde{f_{1}},\widetilde{f_{2}})^{T}\in\mathbb{X}_{2}$ and $\left\\|F-\widetilde{F}\right\\|_{\mathbb{X}_{2}}<\varepsilon,$ we have again $\displaystyle\left\\|\left\langle\widetilde{F},Z\right\rangle_{\mathbb{X}_{2}}Z-\left\langle F,Z\right\rangle_{\mathbb{X}_{2}}Z\right\\|_{\mathbb{X}_{2}}$ $\displaystyle=\left\\|\left\langle\widetilde{F}-F,Z\right\rangle_{\mathbb{X}_{2}}Z\right\\|_{\mathbb{X}_{2}}$ (4.51) $\displaystyle\leq\left\\|F-\widetilde{F}\right\\|_{\mathbb{X}_{2}}<\varepsilon,$ so then $\alpha\left(-\infty\right)<\left\langle\widetilde{F},Z\right\rangle_{\mathbb{X}_{2}}Z<\alpha\left(+\infty\right)$ on $\overline{\Omega},$ for $\varepsilon>0$ small enough. It follows that $F\in int\left(\mathcal{R}\left(A_{4}\right)+\mathcal{R}\left(A_{5}\right)\right)\subset int\left(\mathcal{R}\left(A_{6}\right)\right)\subset\mathcal{R}\left(A_{6}\right),$ by (4.50). This completes the proof of our theorem. ###### Remark 4.10. When $\lambda=0$ and $Z\equiv\mathbf{1,}$ we have, using a different normalization, $\left\\|Z\right\\|_{\mathbb{X}_{2}}^{2}=\mu\left(\overline{\Omega}\right)=\lambda_{1}+\lambda_{2},$ $\min Z=\max Z=1;$ in this case, it turns out that (4.47) reduces to (4.36). ###### Remark 4.11. Of course the result in Theorem 4.9 is interesting only when $\frac{\alpha\left(-\infty\right)}{\min Z}<\frac{\alpha\left(+\infty\right)}{\max Z}.$ But this always holds unless $\alpha\equiv 0.$ ###### Example 4.12. In the context of Theorem 4.9, let us now consider the one dimensional problem: $\left\\{\begin{array}[]{c}-u^{{}^{\prime\prime}}+u+\alpha\left(u\right)=f_{1}\text{ in }\Omega=\left(0,1\right),\\\ -u\left(j\right)+\left(-1\right)^{j+1}u^{{}^{\prime}}\left(j\right)+c_{j}u\left(j\right)=f_{2}^{j}\text{, }j=0,1,\end{array}\right.$ (4.52) where $c_{j}$ are given by (4.44) with $\delta=1/2.$ It follows from (4.47) that, for (4.52) to have at least one solution, it suffices to have $\frac{\alpha\left(-\infty\right)}{\cos\left(1/2\right)}<\int\limits_{0}^{1}f_{1}\left(x\right)\cos\left(x-1/2\right)dx+\left(f_{2}^{0}+f_{2}^{1}\right)\cos\left(1/2\right)<\alpha\left(+\infty\right).$ (4.53) Moreover, choosing $\alpha\left(u\right)=r\left\|u\right\|^{p-1}u,$ $r,p>0$ in the first equation of (4.52), then (4.53) yields at least one solution to (4.52) for any $f_{1}\in L^{2}\left(0,1\right)$ and $f_{2}^{j}\in\mathbb{R}$, $j=0,1.$ Finally, let us consider as an application of our main theorems, an example for which $q>0,$ that is, $\Delta_{\Gamma}$ is present in the boundary conditions for our nonlinear elliptic problems (4.45). For this purpose, let $\Omega$ be the two dimensional box $\left(0,1\right)^{2}\subset\mathbb{R}^{2}$, $b\left(x,y\right)\equiv 1,$ for all $\left(x,y\right)\in\Gamma=\Gamma_{1}\cup\Gamma_{2}\cup\Gamma_{3}\cup\Gamma_{4}$, $q>0$ and $c_{i}\left(x,y\right)$ will be determined in the sequel. The lines $\Gamma_{i}$ and $c_{i}$ will be defined below. We will choose $c_{i}\left(x,y\right),$ so that the smallest eigenvalue of $A_{1}^{1}$ is $\lambda=2$. The positive solution of $\Delta z+2z=0$ has the form $z\left(x,y\right)=\cos\left(x-1/2\right)\cos\left(y-1/2\right)$ (times a constant, which we take to be $1$). Note that $z\left(x,y\right)>0$ on $\overline{\Omega}=\left[0,1\right]^{2}$. Thus, we need to choose positive $c_{i}\left(x,y\right)$ for each $i=1,2,3,4$ such that $z\left(x,y\right)$ satisfies the correct boundary conditions. The boundary conditions are $\left\\{\begin{array}[]{c}-2z-z_{y}+c_{1}\left(x,y\right)z-qz_{yy}=0\text{ for }\left(x,y\right)\in\Gamma_{1}=\left\\{\left(x,0\right):x\in\left[0,1\right]\right\\},\\\ -2z+z_{x}+c_{2}\left(x,y\right)z-qz_{xx}=0\text{ for }\left(x,y\right)\in\Gamma_{2}=\left\\{\left(1,y\right):y\in\left[0,1\right]\right\\},\\\ -2z+z_{y}+c_{3}\left(x,y\right)z-qz_{yy}=0\text{ for }\left(x,y\right)\in\Gamma_{3}=\left\\{\left(x,1\right):x\in\left[0,1\right]\right\\},\\\ -2z-z_{x}+c_{4}\left(x,y\right)z-qz_{xx}=0\text{ for }\left(x,y\right)\in\Gamma_{4}=\left\\{\left(0,y\right):x\in\left[0,1\right]\right\\},\end{array}\right.$ (4.54) since $\partial/\partial n$ equals $\partial/\partial x$ and $\partial/\partial y$ along the lines $\Gamma_{2}$ and $\Gamma_{3},$ respectively and $\partial/\partial n$ equals $-\partial/\partial x$ and $-\partial/\partial y$ along the lines $\Gamma_{4}$ and $\Gamma_{1},$ respectively. Moreover, we note that $\Delta_{\Gamma}$ equals $\partial/\partial y^{2}$ along $\Gamma_{1}\cup\Gamma_{3}$ and $\partial/\partial x^{2}$ along $\Gamma_{2}\cup\Gamma_{4},$ respectively. Calculating in (4.54), we obtain, for any $q\in\left(0,q_{\pm}\right),$ $q_{\pm}=2\cos\left(1/2\right)\pm\tan\left(1/2\right),$ the functions $\left\\{\begin{array}[]{c}c_{1}\left(x,y\right)=q_{\pm}-q+d_{1}\left(y\right),\\\ c_{2}\left(x,y\right)=q_{\pm}-q+d_{2}\left(x\right),\\\ c_{3}\left(x,y\right)=q_{\pm}-q+d_{3}\left(y\right),\\\ c_{4}\left(x,y\right)=q_{\pm}-q+d_{4}\left(x\right),\end{array}\right.$ (4.55) where $d_{i}$ are nonnegative, continuous functions on $\left[0,1\right]$ such that $d_{1}\left(0\right)=d_{4}\left(0\right)=0$ and $d_{2}\left(1\right)=d_{3}\left(1\right)=0.$ Note that $c_{i}>0$ on $\Gamma_{i}$ for each $i.$ ###### Example 4.13. Let us now consider the boundary value problem in the open rectangle $\Omega=\left(0,1\right)^{2}$: $-\Delta u+2u+\alpha\left(u\right)=f_{1}\text{ in }\Omega,$ (4.56) endowed with the boundary conditions of (4.54), except that now the zero values on the right sides of these equalities are replaced by the functions $f_{2}^{1},$ $f_{2}^{2},$ $f_{2}^{3}$ and $f_{2}^{4},$ respectively. Let $c_{i}$ be the functions defined in (4.55). It follows from (4.47) that for (4.56) to have at least one solution, it suffices to have $\frac{\alpha\left(-\infty\right)}{\cos^{2}\left(1/2\right)}<\mathcal{J}<\alpha\left(+\infty\right),$ (4.57) where $\mathcal{J}=\int\limits_{0}^{1}\int\limits_{0}^{1}f_{1}\left(x,y\right)\cos\left(x-\frac{1}{2}\right)\cos\left(y-\frac{1}{2}\right)dxdy+\sum\limits_{i=1}^{4}\int\limits_{\Gamma_{i}}f_{2}^{i}zdS_{i}$ and each $\int\limits_{\Gamma_{i}}dS_{i}$ denotes the path integral corresponding to each line ${\Gamma_{i}}$. Moreover, choosing $\alpha\left(u\right)=r\left\|u\right\|^{p-1}u,$ $r,p>0$ in the (4.56), then (4.57) yields at least one solution to (4.56), for any $f_{1}\in L^{2}\left(\Omega\right)$ and $f_{2}^{i}\in L^{2}\left(\Gamma_{i}\right)$, $i=1,2,3,4.$ We conclude the paper by stating sufficient conditions for (4.27) (see Theorem 4.4) to hold. It is worth mentioning, however, that such conditions are not necessary to prove Theorem 4.5, but that the results below have an interest on their own. We consider the following growth conditions for a function $\alpha:\mathbb{R}\rightarrow\mathbb{R}$: (GC1) $N=1$. No growth condition on $\alpha.$ $\ N=2.$ The function $\alpha$ is bounded by a power: $\left\|\alpha\left(s\right)\right\|\leq C\left(1+\left\|s\right\|^{r}\right),\text{ for all }s\in\mathbb{R}\text{,}$ (4.58) where $C,$ $r$ are positive constants. $\ N=3$. (4.58) holds with $r=N/\left(N-2\right).$ (GC2) This is (GC1), modified by replacing $r=N/\left(N-2\right)$ by $r=\left(N-1\right)/\left(N-2\right)$ in the case $N\geq 3$ and $q>0,$ and replacing $r=N/\left(N-2\right)$ by $r=\left\\{\begin{array}[]{c}\text{any number, if }N=3\\\ \frac{N-1}{N-3}\text{, if }N\geq 4.\end{array}\right.$ We start with the following. ###### Proposition 4.14. Let $\alpha_{1},$ $\alpha_{2}:\mathbb{R}\rightarrow\mathbb{R}$ satisfy (H1). Assume that $(\alpha_{1}(u),\alpha_{2}(u))^{T}\in\mathbb{X}_{2},\;\text{for all}\;u\in H^{1}\left(\Omega\right),\;\text{if }q=0,$ (4.59) $(\alpha_{1}(u),\alpha_{2}(u_{\mid\Gamma}))^{T}\in\mathbb{X}_{2},\;\text{for all }(u,tr\left(u\right))^{T}\in H^{1}(\Omega)\times H^{1}(\Gamma),\;\text{if}\;q>0.$ (4.60) Let $A_{1}$, $A_{2}$ and $A_{3}$ be as in the proof of Theorem 4.5. Then $A_{1}+A_{2}=A_{3}\;\text{and}\;\mathcal{R}(A_{1})+\mathcal{R}(A_{2})\simeq\mathcal{R}(A_{3}).$ (4.61) ###### Proof. Let us first recall that, from Theorem 3.2, $D(A_{1})$ equals either $H^{2}(\Omega)$ or $H_{\ast}^{2}(\Omega),$ according to whether $q=0$ or $q>0$. Moreover, $A_{1}U=\left(-\Delta u,b(x)\partial_{n}u-qb(x)\Delta_{\Gamma}u\right)^{T}.$ The operators $A_{2},$ $A_{3}$ are given in (4.20) and (4.18)-(4.19), respectively. Since $A_{1}=\partial J_{1}$, $A_{2}=\partial J_{2}$ and $A_{3}=\partial J_{3}:=\partial(J_{1}+J_{2})$ with $D(J_{1})\cap D(J_{2})\neq\emptyset$, it follows that $A_{1}+A_{2}\subset A_{3}.$ Hence, $D\left(A_{1}\right)\cap D\left(A_{2}\right)\subset D\left(A_{3}\right)$. We claim that $A_{3}=A_{1}+A_{2}$. To show this we must prove $D\left(A_{3}\right)\subset D\left(A_{1}\right)\cap D\left(A_{2}\right).$ Assume (4.59) and let $U=(u,u_{\mid\Gamma})^{T}\in D(A_{3})$. Then $U\in\mathbb{D}_{0}$, and from (4.18), $-\Delta u+\alpha_{1}(u)\in L^{2}(\Omega),\text{ }\frac{\partial u}{\partial n}+\alpha_{2}(u)\in L^{2}(\Gamma)\text{, if }q=0.$ Therefore, $u\in H^{1}(\Omega)$, $\Delta u\in L^{2}(\Omega)$ and $\frac{\partial u}{\partial n}\in L^{2}(\Gamma)$. Since $\Omega$ is smooth, elliptic regularity implies that $u\in H^{2}(\Omega)$. Hence, $U\in D\left(A_{1}\right)\cap D\left(A_{2}\right),$ if $q=0$. If $q>0$, one also has that $\frac{\partial u}{\partial n}-qb(x)\Delta_{\Gamma}u+\alpha_{2}(u)\in L^{2}(\Gamma)$ and $tr\left(u\right)\in H^{1}(\Gamma)$. Since $u\in H^{2}(\Omega)$, and $\alpha_{2}(u)\in L^{2}(\Gamma),$ by (4.60), we also have that $\Delta_{\Gamma}u\in L^{2}(\Gamma)$. Elliptic regularity also implies that $tr\left(u\right)\in H^{2}(\Gamma)$. Hence, $U\in D\left(A_{1}\right)\cap D\left(A_{2}\right),$ if $q>0$. It is easy to verify that, for every $U\in D\left(A_{3}\right)=D\left(A_{1}\right)\cap D\left(A_{2}\right)$, $A_{3}U=A_{1}U+A_{2}U$. The statement (4.61) is a straightforward consequence of (4.27). The proof is finished. The following corollary is a consequence of Proposition 4.14. ###### Corollary 4.15. Let $\alpha_{1},$ $\alpha_{2}:\mathbb{R}\rightarrow\mathbb{R}$ be continuous, monotone nondecreasing functions satisfying the growth conditions (GC1)-(GC2). Then (4.59)-(4.60) are fulfilled and therefore, (4.61) holds. ###### Proof. To prove this result, we need the following properties of Sobolev spaces. Since the domain $\Omega$ has smooth boundary $\Gamma$, one has the following: 1. (1) If $N=1$, $H^{1}(\Omega)\hookrightarrow C(\bar{\Omega})$. 2. (2) If $N=2$, $H^{1}(\Omega)\hookrightarrow L^{p}(\Omega),$ for every $p\in[1,\infty)$ and $H^{1}(\Gamma)\hookrightarrow C(\Gamma)$. 3. (3) If $N\geq 3$, $H^{1}(\Omega)\hookrightarrow L^{\frac{2N}{N-2}}(\Omega)$. 4. (4) If $N=3$, $H^{1}(\Gamma)\hookrightarrow L^{q}(\Gamma),$ for every $q\in[1,\infty)$. 5. (5) If $N\geq 4$, $H^{1}(\Gamma)\hookrightarrow L^{\frac{2(N-1)}{N-3}}(\Gamma)$. Now, let $\widetilde{\Omega}$ denote either $\Omega$ or $\Gamma$ and suppose that $q\geq 0$. Then the regularity properties of $u\in$ $H^{1}\left(\Omega\right),$ if $q=0,$ $u_{\mid\Gamma}\in H^{1}\left(\Gamma\right),$ if $q>0$ given in the five points above, and $\left\|\alpha\left(s\right)\right\|\leq C\left(1+\left\|s\right\|^{r}\right)$ imply that $\alpha\left(u\right)\in L^{2}(\widetilde{\Omega}),$ provided that (GC1)-(GC2) are satisfied. In particular, it is easy to check that $\alpha_{i}\left(u\right)\in L^{2}(\widetilde{\Omega}),$ for $i=1,2$. This completes the proof. Acknowledgement. We are most grateful to Haim Brezis for his interest in this work and for his generous and helpful comments. We also thank Jean Mawhin for informing us about [15]. ## References * [1] H. Brezis, Propriétés régularisantes de certains semi-groupes non linéaires, Israel J. Math 9 (1971), 513–534. * [2] H. Brezis and A. 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Birkhäuser Verlag, Basel, 1983.	arxiv-papers	2013-11-13T14:05:36	2024-09-04T02:49:53.526872	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ciprian G. Gal, Gisele Ruiz Goldstein, Jerome A. Goldstein, Silvia\n Romanelli, Mahamadi Warma", "submitter": "Ciprian Gal", "url": "https://arxiv.org/abs/1311.3134" }
1311.3173	# Generalization of $([e],[e]\vee[c])$-Ideals Of BE-algebras Ahmad Fawad Ali Department of Basic Sciences, Riphah Internaional University Islamabad Pakistan. [email protected] , Saleem Abdullah Department of mathematics, Quaid-e-Aazam University Islamabad Pakistan. [email protected] , Muhammad Sarwar Kamran Department of Basic Sciences, Riphah Internaional University Islamabad Pakistan [email protected] and Muhammad Aslam Department of Mathematics, King Khlid University Saudi Arabia. ###### Abstract. In this paper, using $N$-structure, the notion of an $N$-ideal in a BE-algebra is introduced. Conditions for an $N$-structure to be an $N$-ideal are provided. To obtain a more general form of an $N$-ideal, a point $N$-structure which is ($k$ conditionally) employed in an $N$-structure is proposed. Using these notions, the concept of an $([e],[e]\vee[c_{k}])$-ideal is introduced and related properties are investigated. $([e],[e]\vee[c_{k}])$-ideal is a generalized form of $([e],[e]\vee[c])$-ideal. Characterizations of $([e],[e]\vee[c_{k}])$-ideals are discussed. ###### Key words and phrases: BE-algebra, (Transitive, self distributive) BE-algebra, Ideal, $N$-ideal, $([e],[e]\vee[c_{k}])$-ideal. ## 1\. Introduction A (crisp) set $A$ in a universe $X$ can be defined in the form of its characteristic function $\mu_{A}:X\rightarrow\\{0,1\\}$ yielding the value $1$ for elements belonging to the set $A$ and the value $0$ for elements excluded from the set $A$. So far most of the generalization of the crisp set have been conducted on the unit interval $[0,1]$ and they are consistent with the asymmetry observation. In other words, the generalization of the crisp set to fuzzy sets spread positive information that fit the crisp point $\\{1\\}$ into the interval $[0,1]$. Because no negative meaning of information is suggested, we now feel a need to deal with negative information. To do so, we also feel a need to supply a mathematical tool. To attain such an object, Jun et al.[7] introduced a new function which is called a negative-valued function, and constructed N-structures. They applied $N$-structures to BCK/BCI-algebras, and discussed $N$-ideals in BCK/BCI- algebras. In 1966, Imai and Iseki [3] and Iseki [4] introduced two classes of abstract algebras: BCK-algebras and BCI-algebras. It is known that the class of BCK-algebras is a proper subclass of the class of BCI-algebras. As a generalization of a BCK-algebra, Kim and Kim [5] introduced the notion of a BE-algebra, and investigated several properties. In Ahn and So [2] introduced the notion of ideals in BE-algebras. They considered several descriptions of ideals in BE-algebras. M.S. Kang, and Y.B. Jun [6], introduced the notion of an $N$-ideal of BE- algebra. In paper [6], a point $N$-structure which is (Conditionally) employed in an $N$-structure is proposed. The concept of $([e],[e]\vee[c])$-ideals and discussed the related properties. In this paper, the concept of an $([e],[e]\vee[c_{k}])$-ideal is introduced and related properties are investigated. $([e],[e]\vee[c_{k}])$-ideal is a generalized form of $([e],[e]\vee[c])$-ideal. In this paper, a point $N$-structure which is $(k$ Conditionally$)$ employed in an $N$-structure is introduced. Characterizations of $([e],[e]\vee[c_{k}])$-ideals are discussed. ## 2\. Preliminaries ###### Definition 1. $($[5]$)$ Let $K(\tau)$ be a class of type $\tau=(2,0)$. A system $(X;\ast,1)\in K(\tau)$ define a BE-Algebra if the following axioms hold: $(V_{1})$ $(\forall x\in X)$ $(\ x\ast x=1\ )$, $(V_{2})$ $(\forall x\in X)$ $(\ x\ast 1=1\ )$, $(V_{3})$ $(\forall x\in X)$ $(\ 1\ast x=x\ )$, $(V_{4})$ $(\forall x,y,z\in X)$ $(\ x\ast(y\ast z)=y\ast(x\ast z)\ )$. ###### Definition 2. $($[6]$)$ A relation ”$\leq$” on a BE-algebra $X$ is defined by $(\forall x,y\in X)$ $($ $x\leq y\Leftrightarrow x\ast y=1$ $)$. ###### Definition 3. $($[2]$)$ A BE-algebra $X$ is called Self-distributive if $x\ast(y\ast z)=(x\ast y)\ast(x\ast z)$ for all $x,y,z\in X$. ###### Definition 4. $($[5]$)$ A BE-algebra $(X;\ast,1)$ is said to be Transitive if it satisfies: $(\forall x,y,z\in X)$ $($ $y\ast z\leq(x\ast y)\ast(x\ast z)$ $)$. Result: ([2]) The converse of above proposition is not true in general. Note: ([6]) The collection of function from a set $X$ to $[-1,0]$ is denoted by $\tau(X,[-1,0])$. ###### Definition 5. $($[2]$)$ Let $I$ a non-empty subset of an BE-algebra $X$ then $I$ is called an Ideal of $X$ if; $(1)$ $(\forall x\in X$, $s\in I)$ $(\ x\ast s\in I\ )$, $(2)$ $(\forall x\in X$, $s,q\in I)$ $(\ (s\ast(q\ast x))\ast x\in I\ )$. ###### Lemma 1. $($[6]$)$ A non-empty subset $I$ of $X$ is an ideal of $X$ if and only if it satisfies: $(1)$ $1\in I$, $(2)$ $(\forall x,z\in X)$ $($ $\forall y\in I$ $)$ $(\ x\ast(y\ast z)\in I\Rightarrow x\ast z\in I$ $)\ )$. ## 3\. $N$-ideals of BE-algebra ###### Definition 6. $($[6]$)$ An element of $\tau(X,[-1,0])$ is called a Negative-valued function from $X$ to $[-1,0]$ $($briefly, $N$-function on $X)$. ###### Definition 7. $($[6]$)$ An ordered pair $(X$,$f)$ of $X$ and an $N$-function $f$ on $X$ is called an $N$-structure. ###### Definition 8. $($[6]$)$ For any $N$-structure $(X,f)$ the nonempty set $C(f;t):=\\{x\in X\text{ }\|\text{ }f(x)\leq t\\}$ is called a closed $(f,t)$-cut of $(X,f)$, where $t\in[-1,0]$. ###### Definition 9. $($[6]$)$ By an $N$-ideal of $X$ we mean an $N$-structure $(X,f)$ which satisfies the following condition: $(\forall t\in[-1,0])$ $(\ C(f;t)\in J(X)\cup\\{\emptyset\\}\ )$. Where $J(X)$ is a set of all ideal of $X$. ###### Example 1. Let $X=\\{1,\alpha,h,m,0\\}$ be a set with a multiplication table given by; $\ast$ \| $1$ \| $\alpha$ \| $h$ \| $m$ \| $0$ ---\|---\|---\|---\|---\|--- $1$ \| $1$ \| $\alpha$ \| $h$ \| $m$ \| $0$ $\alpha$ \| $1$ \| $1$ \| $\alpha$ \| $m$ \| $m$ $h$ \| $1$ \| $1$ \| $1$ \| $m$ \| $m$ $m$ \| $1$ \| $\alpha$ \| $h$ \| $1$ \| $\alpha$ $0$ \| $1$ \| $1$ \| $\alpha$ \| $1$ \| $1$ Then $(X;\ast,1)$ is a BE-algebra. Consider an $N$-structure $(X,f)$ in which $t$ is defined by; $f(y)=\left\\{\begin{array}[]{ll}-0.7&\text{if \ }y\in\\{1,\alpha,h\\}\\\ -0.2&\text{if \ }y\in\\{m,0\\}\end{array}\right.$ Then $C(f;t)=\left\\{\begin{array}[]{ll}\\{1,\alpha,h\\}&\text{if \ }t\in[-0.7,0]\\\ \emptyset&\text{if \ }t\in[-1,-0.7)\end{array}\right.$ Note that $\\{1,\alpha,h\\}$ is an ideals of BE-algebra $X$, and hence $(X,f)$ is an $N$-ideal of $X$. ###### Lemma 2. Each $N$-ideal $(X,f)$ of BE-algebra $X$ satisfies the condition: $(\forall x\in X)$ $(\ f(1)\leq f(x)\ )$. ###### Proof. Since in BE-algebra we have $x\ast x=1$, thus we have $f(1)=f(x\ast x)\leq f(x)$ for all $x\in X$. ###### Proposition 1. Each $N$-ideal $f$ of BE-algebra $X$ satisfies the condition: $(\forall x,y\in X)$ $(\ f((x\ast y)\ast y)\leq f(x)\ )$. ###### Proof. Straightforward. ###### Proposition 2. Each $N$-ideal $f$ of BE-algebra $X$ satisfies the condition; $(\forall x,y\in X)$ $(\ f(y)\leq\max\\{f(x),f(x\ast y)\\}\ )$. ###### Proof. It can be easily proved. ###### Corollary 1. If $x\leq y$, then each $N$-ideal $f$ of BE-algebra $X$ satisfies the condition; $f(y)\leq f(x)$. ###### Proof. Suppose $x\leq y$ for all $x,y\in X$. Then $x\ast y=1$, so $f(y)=f(1\ast y)=f((x\ast y)\ast y)$ By proposition 1, $f((x\ast y)\ast y)\leq f(x)$, hence $f(y)\leq f(x)$. ## 4\. $([e],[e]\vee[c_{k}])$-Ideals ###### Definition 10. $($[6]$)$ Let $f$ be an $N$-structure of of BE-algebra $X$ in wich $f$ is given by; $f(y)=\left\\{\begin{array}[]{ll}0&\text{if }y\neq x\\\ t&\text{if \ }y=x\end{array}\right.$ Where $\ t\in[-1,0)$, In this case, $f$ is represented by $\frac{x}{t}$. $(X,\frac{x}{t})$ is called Point $N$-structure. ###### Definition 11. $($[6]$)$ A Point $N$-structure $(X,\frac{x}{t})$ is called Employed in an $N$-structure $(X,f)$ of BE-algebra $X$ if $f(x)\leq t$ for all $x\in X$, and $t\in[-1,0)$. It is represented as $(X,\frac{x}{t})[e](X,f)$ or $\frac{x}{t}[e]f$. ###### Definition 12. A point $N$-structure $(X,\frac{x}{t})$ is called $(k$ Conditionally$)$ Employed in an $N$-structure $(X,f)$ if $f(x)+t+k+1<0$ for all $x\in X$, $t\in[-1,0)$ and $k\in(-1,0]$. It is denoted by $(X,\frac{x}{t})[c_{k}](X,f)$ or $\frac{x}{t}[c_{k}]f$. To say that $(X,\frac{x}{t})([e]\vee[c_{k}])(X,f)$ $($or briefly, $\frac{x}{t}([e]\vee[c_{k}])f)$ we mean $(X,\frac{x}{t})[e](X,f)$ or $(X,\frac{x}{t})[c_{k}](X,f)$ $($or briefly, $\frac{x}{t}[e]$ or $\frac{x}{t}[c_{k}]f)$. To say that $\frac{x}{t}\overline{\alpha}f$ we mean $\frac{x}{t}\alpha f$ does not hold for $\alpha\in\\{[e],[c_{k}],[e]\vee[c_{k}]\\}$. ###### Definition 13. An $N$-structure $(X,f)$ is called $([e],[e]\vee[c_{k}])$-ideal of $X$ if it satisfied; $(1)$ $\frac{y}{t}[e]f\Rightarrow\frac{x\ast y}{t}([e]\vee[c_{k}])f$, $(2)$ $\frac{x}{t}[e]f$, $\frac{y}{r}[e]f\Rightarrow\frac{(x\ast(y\ast z))\ast z}{\max\\{t,r\\}}([e]\vee[c_{k}])f$. for all $x,y,z\in X$, where $t,r\in[-1,0)$ and $k\in(-1,0]$. ###### Example 2. Let $X=\\{1,\gamma,0,m,\omega\\}$ be a set with a multiplication table given by; $\ast$ \| $1$ \| $\gamma$ \| $0$ \| $m$ \| $\omega$ ---\|---\|---\|---\|---\|--- $1$ \| $1$ \| $\gamma$ \| $0$ \| $m$ \| $\omega$ $\gamma$ \| $1$ \| $1$ \| $\gamma$ \| $m$ \| $m$ $0$ \| $1$ \| $1$ \| $1$ \| $m$ \| $m$ $m$ \| $1$ \| $\gamma$ \| $0$ \| $1$ \| $\gamma$ $\omega$ \| $1$ \| $1$ \| $\gamma$ \| $1$ \| $1$ Let $(X,f)$ be an $N$-structure. Then $f$ is defined in an $N$-structure $(X,f)$, as; $f=\left(\begin{array}[]{ccccc}1&\gamma&0&m&\omega\\\ -0.9&-0.8&-0.7&-0.9&-0.8\end{array}\right)$ and$\ t,r\in[-0.7,-0.3)$, also $k\in(-1,-0.4)$. for all $x,y,z\in X$, the followings $(1)$ $\frac{y}{t}[e]f\Rightarrow\frac{x\ast y}{t}([e]\vee[c_{k}])f$, $(2)$ $\frac{x}{t}[e]f$, $\frac{y}{r}[e]f\Rightarrow\frac{(x\ast(y\ast z))\ast z}{\max\\{t\text{, }r\\}}([e]\vee[c_{k}])f$. are hold. Hence, $f$ is an $([e],[e]\vee[c_{k}])$-ideal of $X$. ###### Theorem 1. For any $N$-structure $(X,f)$, the following are equivalent: $(1)$ $(X,f)$ is a $([e],[e]\vee[c_{k}])$-ideal of $X$. $(2)$ $(X,f)$ satisfies the following inequalities: $(2.1)$ $(\forall x,y\in X)$ $(\ f(x\ast y)\leq\max\\{f(y),\frac{-k-1}{2}\\}\ )$, $(2.2)$ $(\forall x,y,z\in X)$ $(\ f((x\ast(y\ast z))\ast z)\leq\max\\{f(x),f(y),\frac{-k-1}{2}\\}\ )$. where $k\in(-1,0]$. ###### Proof. Let $(X,f)$ be a $([e],[e]\vee[c_{k}])$-ideal of $X$. Suppose that $f(x\ast y)>\max\\{f(y),\frac{-k-1}{2}\\}$ for all $x,y\in X$. If we take $t_{y}:=\max\\{f(y),\frac{-k-1}{2}\\}$, $t_{y}\in[\frac{-k-1}{2},0]$, $\frac{y}{t_{y}}[e]f$ and $\frac{x\ast y}{t_{y}}[\overline{e}]f$. Also, $f(x\ast y)+t_{y}+k+1>2t_{y}+1\geq 0$, and so $\frac{x\ast y}{t_{y}}[\overline{c_{k}}]f$. This is a contradiction. Thus $f(x\ast y)\leq\max\\{f(y),\frac{-k-1}{2}\\}$ for all $x,y\in X$. Also suppose that $\ f((x\ast(y\ast z))\ast z)>\max\\{f(x),f(y),\frac{-k-1}{2}\\}$ for some $x,y,z\in X$. Take $t:=\max\\{f(x),f(y),\frac{-k-1}{2}\\}$. Then $t\geq\frac{-k-1}{2}$,$\frac{x}{t}[e]f$ and $\frac{y}{t}[e]f$, but $\frac{x\ast(y\ast z))\ast z}{t}[\overline{e}]f$. Also, $f((x\ast(y\ast z)\ast z)+t+k+1>2t+k+1\geq 0$, i.e., $\frac{x\ast(y\ast z))\ast z}{t}[\overline{c_{k}}]f$. This is a contradiction, and hence $f((x\ast(y\ast z))\ast z)\leq\max\\{f(x),f(y),\frac{-k-1}{2}\\}$ for all $x,y,z\in X$. Conversely, suppose that $(X,f)$ satisfies $(2.1)$ and $(2.2)$. Let $\frac{y}{t}[e]$ for all $y\in X$ and $t\in[-1,0)$. Then $f(y)\leq t$. Suppose that $\frac{x\ast y}{t}[\overline{e}]f$, i.e, $f(x\ast y)>t$. If $f(y)>\frac{-k-1}{2}$, then $f(x\ast y)\leq\max\\{f(y),\frac{-k-1}{2}\\}=f(y)\leq t$, which is a contradiction. Hence $f(y)\leq\frac{-k-1}{2}$, which implies that $f(x\ast y)+t+k+1<2f(x\ast y)+k+1\leq 2\max\\{f(y),\frac{-k-1}{2}\\}+k+1=0$, i.e., $\frac{x\ast y}{t}[c_{k}]f$. Thus $\frac{x\ast y}{t}([e]\vee[c_{k}])f$. Let $\frac{x}{t}[e]f$ and $\frac{y}{r}[e]f$ for all$\ x,y,z\in X$ and $t,r\in[-1,0)$. Then $f(x)\leq t$ and $f(y)\leq r$. Suppose that $\frac{(x\ast y\ast z))\ast z}{\max\\{t,r\\}}[\overline{e}]f$, i.e., $f((x\ast(y\ast z))\ast z)>\max\\{t,r\\}$. If $\max\\{f(x),f(y)\\}>\frac{-k-1}{2}$, then $f((x\ast(y\ast z))\ast z)\leq\max\\{f(x),f(y),\frac{-k-1}{2}\\}=\max\\{f(x),f(y)\\}\leq\max\\{t,r\\}\text{.}$ This is impossible, and so $\max\\{f(x),f(y)\\}\leq\frac{-k-1}{2}$. It follows that $f((x\ast(y\ast z))\ast z)+\max\\{t,r\\}+k+1<2f((x\ast(y\ast z))\ast z)+k+1\leq 2\max\\{f(x),f(y),\frac{-k-1}{2}\\}+k+1=0$ $\Rightarrow\frac{(x\ast y\ast z))\ast z}{\max\\{t,r\\}}[\overline{c_{k}}]f$. Hence $\frac{(x\ast y\ast z))\ast z}{\max\\{t,r\\}}([e]\vee[c_{k}])f$, and therefore $(X,f)$ is a $([e],[e]\vee[c_{k}])$-ideal of $X$. If $(k=0)$, then the followig holds. ###### Corollary 2. For any $N$-structure $(X,f)$, the following are equivalent: $(1)$ $(X,f)$ is a $([e],[e]\vee[c])$-ideal of $X$. $(2)$ $(X,f)$ satisfies the following inequalities: $(2.1)$ $(\forall x,y\in X)$ $(\ f(x\ast y)\leq\max\\{f(y),-0.5\\}\ )$, ###### Theorem 2. Every $([e],[e]\vee[c_{k}])$-ideal $(X,f)$ of an BE-algebra $X$ satisfies the following inequalities: $(1)$ $(\forall x\in X)$.$(\ f(1)\leq\max\\{f(x),\frac{-k-1}{2}\\}\ )$, $(2)$ $(\forall x,y\in X)$ $(\ f((x\ast y)\ast y)\leq\max\\{f(x),\frac{-k-1}{2}\\}\ )$. where $k\in(-1,0]$. ###### Proof. $(1)$: By using $(V_{1})$ and theorem 1$(2.1)$, we have $f(1)=f(x\ast x)\leq\max\\{f(x),\frac{-k-1}{2}\\}$ for all $x\in X$. $(2)$: By using $(V_{3})$, we have $f((x\ast y)\ast y)=f((x\ast(1\ast y))\ast y)$ for all $x,y\in X$ Then by using theorem 1$(2.2)$, we get $f((x\ast(1\ast y))\ast y)\leq\max\\{f(x),f(1),\frac{-k-1}{2}\\}=\max\\{f(x),\frac{-k-1}{2}\\}$, because by $(1)$ $f(1)\leq\max\\{f(x),\frac{-k-1}{2}\\}$ for all $x,y\in X$. Hence, $f((x\ast y)\ast y)\leq\max\\{f(x),\frac{-k-1}{2}\\}$ for all $x,y\in X$. If $(k=0)$, then the followig holds. ###### Corollary 3. Every $([e],[e]\vee[c])$-ideal $(X,f)$ of an BE-algebra $X$ satisfies the following inequalities: $(1)$ $(\forall x\in X)$.$(\ f(1)\leq\max\\{f(x),-0.5\\}\ )$, $(2)$ $(\forall x,y\in X)$ $(\ f((x\ast y)\ast y)\leq\max\\{f(x),-0.5\\}\ )$. ###### Corollary 4. Each $([e],[e]\vee[c_{k}])$-ideal $(X,f)$ satisfies the following condition; $(\forall x,y\in X)$ $(\ x\leq y\Rightarrow f(y)\leq\max\\{f(x),\frac{-k-1}{2}\\}\ )$. where $k\in(-1,0]$. ###### Proof. Let $x\leq y$ for all $x,y\in X$. Then $x\ast y=1$,and so $f(y)=f(1\ast y)=f((x\ast y)\ast y)\leq\max\\{f(x),\frac{-k-1}{2}\\}$ Hence, $f(y)\leq\max\\{f(x),\frac{-k-1}{2}\\}$. If $(k=0)$, then the followig holds. ###### Lemma 3. Each $([e],[e]\vee[c])$-ideal $(X,f)$ satisfies the following condition; $(\forall x,y\in X)$ $(\ x\leq y\Rightarrow f(y)\leq\max\\{f(x),-0.5\\}\ )$. ###### Proposition 3. Let $(X,f)$ be an $N$-structure such that $(1)$ $(\forall x\in X)$ $(\ f(1)\leq\max\\{f(x),\frac{-k-1}{2}\\}\ )$, $(2)$ $(\forall x,y,z\in X)$ $(\ f(x\ast z)\leq\max\\{f(x\ast(y\ast z)),f(y),\frac{-k-1}{2}\\}\ )$. Then the following implication is valid. $(\forall x,y\in X)$ $(\ x\leq y\Rightarrow f(y)\leq\max\\{f(x),\frac{-k-1}{2}\\}\ )$. where $k\in(-1,0]$. ###### Proof. Suppose $x\leq y$ for all $x,y\in X$. Then $x\ast y=1$, and by using $(1)$ we get $\displaystyle f(y)$ $\displaystyle=$ $\displaystyle f(1\ast y)\leq\max\\{f(1\ast(x\ast y)),f(x),\frac{-k-1}{2}\\}$ $\displaystyle=$ $\displaystyle\max\\{f(1\ast 1),f(x),\frac{-k-1}{2}\\}$ $\displaystyle=$ $\displaystyle\max\\{f(1),f(x),\frac{-k-1}{2}\\}$ $\displaystyle=$ $\displaystyle\max\\{f(x),\frac{-k-1}{2}\\}$ Hence, $f(y)\leq\max\\{f(x),\frac{-k-1}{2}\\}$. If $(k=0)$, then the followig holds. ###### Lemma 4. Let $(X,f)$ be an $N$-structure such that $(1)$ $(\forall x\in X)$ $(\ f(1)\leq\max\\{f(x),-0.5\\}\ )$, $(2)$ $(\forall x,y,z\in X)$ $(\ f(x\ast z)\leq\max\\{f(x\ast(y\ast z)),f(y),-0.5\\}\ )$. Then the following implication is valid. $(\forall x,y\in X)$ $(\ x\leq y\Rightarrow f(y)\leq\max\\{f(x),-0.5\\}\ )$. ###### Theorem 3. Let $(X,f)$ be an $N$-structure of transitive BE-algebra $X$. Then $(X,f)$ is an $([e],[e]\vee[c_{k}])$-ideal of $X$ if and only if it satisfies the following inequalities: $(1)$ $(\forall x\in X)$ $(\ f(1)\leq\max\\{f(x),\frac{-k-1}{2}\\}\ )$, $(2)$ $(\forall x,y,z\in X)$ $(\ f(x\ast z)\leq\max\\{f(x\ast(y\ast z)),f(y),\frac{-k-1}{2}\\}\ )$. where $k\in(-1,0]$. ###### Proof. Suppose that $(X,f)$ is an $([e],[e]\vee[c])$-ideal of $X$. From theorem 2$\left(1\right)$,it is easily seen that $f(1)\leq\max\\{f(x),\frac{-k-1}{2}\\}\text{.}$ Since $X$ is transitive, $((y\ast z)\ast z)\ast((x\ast(y\ast z))\ast(x\ast z))=1\text{ \ \ \ }\mathbf{(G)}$ for all $x,y,z\in X$. By using $(V_{3})$ and $\mathbf{(G)}$ $f(x\ast z)=f(1\ast(x\ast z))=f(((y\ast z)\ast z)\ast((x\ast(y\ast z))\ast(x\ast z))\ast(x\ast z))$ By using theorem 1$(2.2)$, 2$\left(2\right)$, we have $\displaystyle f(((y\ast z)\ast z)\ast((x\ast(y\ast z))\ast(x\ast z))\ast(x\ast z))$ $\displaystyle\leq$ $\displaystyle\max\\{f((y\ast z)\ast z),f(x\ast(y\ast z)),\frac{-k-1}{2}\\}$ $\displaystyle=$ $\displaystyle\max\\{f(x\ast(y\ast z)),f((y\ast z)\ast z),\frac{-k-1}{2}\\}$ $\displaystyle\leq$ $\displaystyle\max\\{f(x\ast(y\ast z)),f(y),\frac{-k-1}{2}\\}$ Hence $f(x\ast z)\leq\max\\{f(x\ast(y\ast z)),f(y),\frac{-k-1}{2}\\}$ for all $x,y,z\in X$. Conversly suppose that $(X,f)$ satisfies $(1)$ and $(2)$. By using $(2)$, $(V_{1})$, $(V_{2})$ and $(1)$ $\displaystyle f(x\ast y)$ $\displaystyle\leq$ $\displaystyle\max\\{f(x\ast(y\ast y)),f(y),\frac{-k-1}{2}\\}$ $\displaystyle=$ $\displaystyle\max\\{f(x\ast 1),f(y),\frac{-k-1}{2}\\}$ $\displaystyle=$ $\displaystyle\max\\{f(1),f(y),\frac{-k-1}{2}\\}$ $\displaystyle=$ $\displaystyle\max\\{f(y),\frac{-k-1}{2}\\}$ Also by using $(2)$ and $(1)$ we get $\displaystyle f((x\ast y)\ast y)$ $\displaystyle\leq$ $\displaystyle\max\\{f((x\ast y)\ast(x\ast y)),f(x),\frac{-k-1}{2}\\}$ $\displaystyle=$ $\displaystyle\max\\{f(1),f(x),\frac{-k-1}{2}\\}$ $\displaystyle=$ $\displaystyle\max\\{f(x),\frac{-k-1}{2}\\}$ for all $x,y\in X$. Now, since $(y\ast z)\ast z\leq(x\ast(y\ast z))\ast(x\ast z)$ for all $x,y,z\in X$, it follows that from proposition 3, we have $f((x\ast(y\ast z))\ast(x\ast z))\leq\max\\{f((y\ast z)\ast z),\frac{-k-1}{2}\\}$ So, from $(2)$, we have $\displaystyle f((x\ast(y\ast z))\ast z)$ $\displaystyle\leq$ $\displaystyle\max\\{f((x\ast(y\ast z))\ast(x\ast z)),f(x),\frac{-k-1}{2}\\}$ $\displaystyle\leq$ $\displaystyle\max\\{f((y\ast z)\ast z),f(x),\frac{-k-1}{2}\\}$ $\displaystyle\leq$ $\displaystyle\max\\{f(x),f(y),\frac{-k-1}{2}\\}$ for all $x,y,z\in X$. Using theorem 1, we conclude that $(X,f)$ is a $([e],[e]\vee[c])$-ideal of $X$. If $(k=0)$, then the followig holds. ###### Corollary 5. Let $(X,f)$ be an $N$-structure of transitive BE-algebra $X$. Then $(X,f)$ is an $([e],[e]\vee[c])$-ideal of $X$ if and only if it satisfies the following inequalities: $(1)$ $(\forall x\in X)$ $(\ f(1)\leq\max\\{f(x),-0.5\\}\ )$, $(2)$ $(\forall x,y,z\in X)$ $(\ f(x\ast z)\leq\max\\{f(x\ast(y\ast z)),f(y),-0.5\\}\ )$. ###### Theorem 4. Let $X$ be a transitive BE-algebra. If $(X,f)$ is a $([e],[e]\vee[c_{k}])$-ideal of $X$ such that $f(1)>\frac{-k-1}{2}$, then $(X,f)$ is an $N$-ideal of $X$. where $k\in(-1,0]$. ###### Proof. Suppose that $(X,f)$ is a $([e],[e]\vee[c_{k}])$-ideal of $X$ such that $\frac{-k-1}{2}<f(1)$. Then $\frac{-k-1}{2}<f(x)$ and so $\frac{-k-1}{2}<f(1)\leq f(x)$ for all $x\in X$ by theorem 3$(1)$ $f(1)\leq\max\\{f(x),\frac{-k-1}{2}\\}$ for all $x\in X$. It follows that from theorem 3$(2)$, $\displaystyle f(x\ast z)$ $\displaystyle\leq$ $\displaystyle\max\\{f(x\ast(y\ast z)),f(y),\frac{-k-1}{2}\\}$ $\displaystyle=$ $\displaystyle\max\\{f(x\ast(y\ast z)),f(y)\\}$ for all $x,y,z\in X$. Hence $(X,f)$ is an $N$-ideal of $X$. If $(k=0)$, then the followig holds. ###### Corollary 6. Let $X$ be a transitive BE-algebra. If $(X,f)$ is a $([e],[e]\vee[c])$-ideal of $X$ such that $f(1)>-0.5$, then $(X,f)$ is an $N$-ideal of $X$. ###### Theorem 5. If $(X,f)$ is a $([e],[e]\vee[c_{k}])$-ideal of a transitive BE-algebra $X$. Show that $(\forall t\in[-1,\frac{-k-1}{2}))\text{ }(Q(f;t)\in J(X)\cup\\{\emptyset\\})$ where $Q(f;t):=\\{x\in X$ $\|$ $\frac{x}{t}[c_{k}]f\\}$, $J(X)$ is a set of all ideal of $X$ and $k\in(-0.5,0]$. ###### Proof. ###### Corollary 7. Suppose that $Q(f;t)\neq\emptyset$ for all $t\in[-1,\frac{-k-1}{2})$. Then there exists $x\in Q(f;t)$, and so $\frac{x}{t}[c]f$, i.e., $f(x)+t+k+1<0$. Using theorem 3$(1)$, we have $\displaystyle f(1)$ $\displaystyle\leq$ $\displaystyle\max\\{f(x),\frac{-k-1}{2}\\}$ $\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}\frac{-k-1}{2}&\text{if \ }f(x)\leq\frac{-k-1}{2}\\\ f(x)&\text{if \ }f(x)>\frac{-k-1}{2}\end{array}\right.$ $\displaystyle<$ $\displaystyle-1-t-k$ which indicates that $1\in Q(f;t)$. Let $x\ast(y\ast z)\in Q(f;t)$ for all $x,y,z\in X$ here $y\in Q(f;t)$. Then $\frac{x\ast(y\ast z)}{t}[c_{k}]f$ and $\frac{y}{t}[c]f$, i.e., $f(x\ast(y\ast z))+t+k+1<0$ and $f(y)+t+k+1<0$. Using theorem 3$(2)$, we get $f(x\ast z)\leq\max\\{f(x\ast(y\ast z)),f(y),\frac{-k-1}{2}\\}$ Thus, if $\max\\{f(x\ast(y\ast z)),f(y)\\}>\frac{-k-1}{2}$, then $f(x\ast z)\leq\max\\{f(x\ast(y\ast z)),f(y)\\}<-1-t-k$ If $\max\\{f(x\ast(y\ast z)),f(y)\\}\leq\frac{-k-1}{2}$, then $f(x\ast z)\leq\frac{-k-1}{2}<-1-t-k$. This show that $\frac{x\ast z}{t}[c_{k}]f$ i.e., $x\ast z\in Q(f;t)$. By using lemma 1, we have $Q(f;t)$ is an ideal of $X$. If $(k=0)$, then the followig holds. ###### Corollary 8. If $(X,f)$ is a $([e],[e]\vee[c])$-ideal of a transitive BE-algebra $X$. Show that $(\forall t\in[-1,-0.5))\text{ }(Q(f;t)\in J(X)\cup\\{\emptyset\\})$ where $Q(f;t):=\\{x\in X$ $\|$ $\frac{x}{t}[c]f\\}$, and $J(X)$ is a set of all ideal of $X$ ###### Theorem 6. Let $X$ be a transitive BE-algebra. Then the followings are equivalent: $(1)$ An $N$-structure $(X,f)$ is a $([e],[e]\vee[c_{k}])$-ideal of $X$ $(2)$ $(\forall t\in[-1,0))$ $([f]_{t}\in J(X)\cup\\{\emptyset\\})$ where $[f]_{t}:=C(f;t)\cup\\{x\in X$ $\|$ $f(x)+t+k+1\leq 0\\}$, $J(X)$ is a set of all ideal of $X$, and $k\in(-1,0]$. ###### Proof. $(1)\Rightarrow(2)$: Suppose that $(1)$ satisfies. Let $[f]_{t}\neq\emptyset$, here $t\in[-1,0)$. Then there exists $x\in[f]_{t}$, and so $f(x)\leq t$ or $f(x)+t+k+1\leq 0$ for all $x\in X$ and $t\in[-1,0)$. If $f(x)\leq t$, then $\displaystyle f(1)$ $\displaystyle\leq$ $\displaystyle\max\\{f(x),\frac{-k-1}{2}\\}\leq\max\\{t,\frac{-k-1}{2}\\}$ $\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}t&\text{if \ }t>\frac{-k-1}{2}\\\ \frac{-k-1}{2}\leq-1-t-k&\text{if \ }t\leq\frac{-k-1}{2}\end{array}\right.$ By theorem 3$(1)$. Hence $1\in[f]_{t}$. If $f(x)+t+k+1\leq 0$, then $\displaystyle f(1)$ $\displaystyle\leq$ $\displaystyle\max\\{f(x),\frac{-k-1}{2}\\}\leq\max\\{-1-t-k,\frac{-k-1}{2}\\}$ $\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}-1-t-k&\text{if \ }t<\frac{-k-1}{2}\\\ \frac{-k-1}{2}\leq t&\text{if \ }t\geq\frac{-k-1}{2}\end{array}\right.$ And so $1\in[f]_{t}$. Let $x,y,z\in X$ be such that $y\in[f]_{t}$ and $x\ast(y\ast z)\in[f]_{t}$. Then $f(y)\leq t$ or $f(y)+t+k+1\leq 0$, and $f(x\ast(y\ast z))\leq t$ or $f(x\ast(y\ast z))+t+k+1\leq 0$. Thus we let the four cases: $(a_{1})$ $f(y)\leq t$ and $f(x\ast(y\ast z))\leq t$, $(a_{2})$ $f(y)\leq t$ and $f(x\ast(y\ast z))+t+k+1\leq 0$, $(a_{3})$ $f(y)+t+k+1\leq 0$ and $f(x\ast(y\ast z))\leq t$, $(a_{4})$ $f(y)+t+k+1\leq 0$ and $f(x\ast(y\ast z))+t+k+1\leq 0$. For case $(a_{1})$, theorem 3$(2)$, implies that $\displaystyle f(x\ast z)$ $\displaystyle\leq$ $\displaystyle\max\\{f(x\ast(y\ast z)),f(y),\frac{-k-1}{2}\\}\leq\max\\{t,\frac{-k-1}{2}\\}$ $\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}\frac{-k-1}{2}&\text{if \ }t<\frac{-k-1}{2}\\\ \text{ }t&\text{if \ }t\geq\frac{-k-1}{2}\end{array}\right.$ so that $x\ast z\in C(f;t)$ or $f(x\ast z)+t+k\leq\frac{-k-1}{2}+\frac{-k-1}{2}+k=-1$. Thus $x\ast z\in[f]_{t}$. For case $(a_{2})$, we have $\displaystyle f(x\ast z)$ $\displaystyle\leq$ $\displaystyle\max\\{f(x\ast(y\ast z)),f(y),\frac{-k-1}{2}\\}\leq\max\\{-1-t-k,t,\frac{-k-1}{2}\\}$ $\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}-1-t-k&\text{if \ }t<\frac{-k-1}{2}\\\ t&\text{if \ }t\geq\frac{-k-1}{2}\end{array}\right.$ Thus $x\ast z\in[f]_{t}$. For case $(a_{3})$, the prove is same to case $(a_{2})$. For case $(a_{4})$ we have, $\displaystyle f(x\ast z)$ $\displaystyle\leq$ $\displaystyle\max\\{f(x\ast(y\ast z)),f(y),\frac{-k-1}{2}\\}\leq\max\\{-1-t-k,\frac{-k-1}{2}\\}$ $\displaystyle=$ $\displaystyle\left\\{\begin{array}[]{ll}-1-t-k&\text{if \ }t<\frac{-k-1}{2}\\\ \frac{-k-1}{2}&\text{if \ }t\geq\frac{-k-1}{2}\end{array}\right.$ So that, $\ x\ast z\in[f]_{t}$. By using lemma 1, $[f]_{t}$ is an ideal of $X$. $(2)\Rightarrow(1)$: Suppose that $(2)$ hold. If $f(1)>\max\\{f(y),\frac{-k-1}{2}\\}$ for all $y\in X$, then $f(1)>t_{y}\geq\max\\{f(y),\frac{-k-1}{2}\\}$ for some $t_{y}\in[\frac{-k-1}{2},0)$. It follows that $x\in C(f;t_{y})\subseteq[f]_{t_{y}}$ but $1\notin C(f;t_{y})$. Also, $f(1)+t_{y}+k+1>2t_{y}+k+1\geq 0$. Hence $1\notin[f]_{t_{y}}$, which contradicts the supposition. So, $f(1)\leq\max\\{f(y),\frac{-k-1}{2}\\}$ for all $y\in X$. Suppose that for some $x,z\in X$, we have $f(x\ast z)>\max\\{f(x\ast(y\ast z)),f(y),\frac{-k-1}{2}\\}$ $\mathbf{(D)}$ Taking $t:=\max\\{f(x\ast(y\ast z)),f(y),\frac{-k-1}{2}\\}$ implies that $t\in[\frac{-k-1}{2},0)$, $x\in C(f;t)\subseteq[f]_{t}$, and $x\ast(x\ast z)\in C(f;t)\subseteq[f]_{t}$. Since $[f]_{t}$ is an ideal of $X$, we have $x\ast z\in[f]_{t}$, and so $f(x\ast z)\leq t$ or $f(x\ast z)+t+k+1\leq 0$. The inequality $\mathbf{(D)}$ induces $x\ast z\notin C(f;t)$, and $f(x\ast z)+t+k+1>2t+k+1\geq 0$. Thus $x\ast z\notin[f]_{t}$. It contradicts the supposition. Hence $f(x\ast z)\leq\max\\{f(x\ast(y\ast z)),f(y),\frac{-k-1}{2}\\}$ for all $x,y,z\in X$. Using theorem 3, we have, $(X,f)$ is a $([e],[e]\vee[c_{k}])$-ideal of $X$. If $(k=0)$, then the followig holds. ###### Corollary 9. Let $X$ be a transitive BE-algebra. Then the followings are equivalent: $(1)$ An $N$-structure $(X,f)$ is a $([e],[e]\vee[c])$-ideal of $X$ $(2)$ $(\forall t\in[-1,0))$ $([f]_{t}\in J(X)\cup\\{\emptyset\\})$ where $[f]_{t}:=C(f;t)\cup\\{x\in X$ $\|$ $f(x)+t+1\leq 0\\}$, and $J(X)$ is a set of all ideal of $X$ ## 5\. Conclusion: In this paper, we have investigated the $([e],[e]\vee[c_{k}])$-ideals of BE- algebra by using transitive and distributive BE-algebra, their related properties, and provide characterizations of $([e],[e]\vee[c_{k}])$-ideals in an $N$-structure $(X,f)$. Now by using these results we can deal with negative informations, also by using these results we will be able to solve the difficulties of theories such as probability theory, ideal theory, algebras theory. In this paper, we give the new mathematical tools for dealing with uncertainties. ## References * [1] K. S. So, and S. S. Ahn, On ideals and upper sets in BE-algebras, Sci. Math. Japo., Online 2008 351-357. * [2] K. S. So, and S. S. Ahn, On ideals and upper sets in BE-algerbas, Sci. Math. Japan 68 (2008), 279–285. * [3] K. Iseki, and Y. Imai, On axiom systems of propositional calculi XIV, Proc. Japan Academy 42 (1966), 19–22. * [4] K. Iseki, An algebra related with a propositional calculus, Proc. Japan Academy 42 (1966), 26–29. * [5] H.S. Kim and Y.H. Kim, On BE-algebras, Sci. Math. Japo., 66(1) (2007) 113-116. * [6] M.S. Kang, and Y.B. Jun, Ideal theory of BE-algebras based on N-structures Hacettepe Journal of Mathematics and Statistics volume 41(4) (2012), 435-447. * [7] K. J. Lee, S. Z. Song, and Y. B. Jun, $N$-ideals of BCK/BCI-algebras, J. Chungcheong Math. Soc. 22 (2009), 417–437. * [8] Molodtsov, D. Soft set theory - First results, Comput. Math. Appl. 37 (1999), 19–31.	arxiv-papers	2013-11-10T17:47:54	2024-09-04T02:49:53.541092	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ahmad Fawad Ali, Saleem Abdullah, Muhammad Sarwar Kamran and Muhammad\n Aslam", "submitter": "Saleem Abdullah", "url": "https://arxiv.org/abs/1311.3173" }
1311.3186	## Abstract Fast and accurate protein structure prediction is one of the major challenges in structural biology, biotechnology and molecular biomedicine. These fields require 3D protein structures for rational design of proteins with improved or novel properties. X-ray crystallography is the most common approach even with its low success rate, but lately NMR based approaches have gained popularity. The general approach involves a set of distance restraints used to guide a structure prediction, but simple NMR triple-resonance experiments often provide enough structural information to predict the structure of small proteins. Previous protein folding simulations that have utilised experimental data have weighted the experimental data and physical force field terms more or less arbitrarily, and the method is thus not generally applicable to new proteins. Furthermore a complete and near error-free assignment of chemical shifts obtained by the NMR experiments is needed, due to the static, or deterministic, assignment. In this thesis I present Chemshift, a module for handling chemical shift assignments, implemented in the protein structure determination program Phaistos. This module treats both the assignment of experimental data, as well as the weighing compared to physical terms, in a probabilistic framework where no data is discarded. Provided a partial assignment of NMR peaks, the module is able to improve the assignment with the intension to utilise this in the protein folding with little bias. ## Acknowledgements I’d like to thank my supervisor Jan H. Jensen for not bullying me as much as he does other students. Thanks to Casper S. Svendsen for inspiring me for future instructor work. Thanks to Anders S. Christensen for being perfect in every way. Thanks to Qian for not killing us, and lastly thanks to Jimmy for his good sense of humour. ###### Contents 1. 1 Introduction 2. 2 Background 3. 3 General Assignment Strategy 1. 3.1 Select Automated Assignment Methods 1. 3.1.1 Autoassign 2. 3.1.2 FLYA 4. 4 Theory 1. 4.1 Probabilistic Framework 5. 5 Computational Details 1. 5.1 Markov Chain Monte Carlo 2. 5.2 Chemshift implementation in Phaistos 1. 5.2.1 Monte Carlo Nuisance Parameter Moves 2. 5.2.2 Monte Carlo Assignment Moves 3. 5.2.3 Cashing 6. 6 Results 7. 7 Future Work 1. 7.1 Referencing Errors 1. 7.1.1 Model Validation 2. 7.2 Peak Intensities 8. 8 Summary and Outlook 9. 9 Appendix ## 1 Introduction To generalise there have been three branches in protein structure determination. X-ray crystallography is the most common approach, that gives very accurate structures and protein size is in general not an issue. It however has a very low success rate, since most proteins of interest does not easily crystallise. Another less popular experimental approach involves using NMR data to create a set of Nuclear Overhauser Effect (NOE) distance restraints. From these restraints the protein structure can be deduced, but protein size is a limiting factor and structures can in general not be inferred from large proteins. In the opposite end of the spectrum is the purely computational methods, that uses force fields to simulate protein mechanics. These methods uses a lot of approximations in order to provide results on a reasonable time-scale for large systems as proteins, which often hinder the correct conformers to be predicted. The quality of predictions from computational methods have recently been improved by including experimental chemical shifts alongside with force fields and chemical shift predictors in the structure prediction [1, 2, 3]. A necessary step between experiments and determining the protein structure is assignment of the measured chemical shifts, which for larger proteins can be very time consuming and is a major bottleneck. Several methods have been developed to automate this [4, 5, 6], but most still require a great deal of human intervention. Two methods that require minimal intervention is Autoassign [7] and FLYA [8]. The strengths of Autoassign is that it is a free service and that chemical shifts are analysed and assigned very quickly (typically less than a minute) with few wrong assignments. FLYA has been shown to perform better than Autoassign, but is slower and requires a license to use. A 2003 study estimated that 40% of all proteins in the Biological Magnetic Resonance Data Bank [9] (BMRB) contain at least one mis-assigned chemical shift [10]. The more severe errors might affect the predicted structures, since data is discarded if the structure calculations don’t converge. And even non-erroneous assignments might restrict the predicted conformers in cases where a protein has more than one native conformation. The purpose of this work is as follows: * • Remove the need for a manual assignment. * • Derive an energy function based on Bayesian inference principles for describing experimental data. * • Implement in the protein structure prediction program Phaistos a probabilistic method to include experimental data in structure prediction. * • Allow the assignment of chemical shifts to change during structure prediction, without discarding data. In this thesis the current state of the development of the Chemshift module in Phaistos is presented. Emphasis has been put on keeping the thesis short and readable, while presenting details of background, theory and computational implementation to an extent such that the thesis, along side with the code itself, can be used to maintain or recreate the module. To avoid any confusion, throughout the thesis a peak will refer to the chemical shifts from two or three linked nuclei. A spin system is the linked nuclei which give rise to a peak in the NMR spectrum. A spin system array is computationally the array that holds the assignment of peak. Each array belong to a specific type of experiment and spin system. When differences of chemical shifts is mentioned, only differences between chemical shifts from the same nuclei is assumed. ## 2 Background In atomic nuclei isotopes with non-zero magnetic moments, an energy difference due to Zeeman-splitting is observed between the different spin-states when a strong external magnetic field is applied. The local magnetic field these nuclei experience is slightly perturbed (shielded) by the local molecular environment, which causes the local environment to be reflected in the size of the energy-splitting. With Nuclear Magnetic Resonance (NMR) spectroscopy, the resonance frequency $\nu$ of the nucleus can be measured. But since this frequency is dependent of the field used, it is convenient to relate this to a reference frequency $\nu_{ref}$ as [11] $\delta=10^{6}\frac{\nu-\nu_{ref}}{\nu_{ref}},$ (1) where $\delta$, in units of ppm, is called the chemical shift. By utilising the coupling between neighbouring nuclei in a protein, one can correlate a nuclei chemical shift with another. One example is the two- dimensional HSQC-experiment which correlates a 15N nuclei with the neighbouring 1H nuclei and thus a peak for every H-N pair can be observed (See Figure 1 for an example). Figure 1: Contour plot of the ${}^{1}H$–${}^{15}N$ HSQC spectrum of recombinant human ubiquitin encapsulated in AOT reverse micelles dissolved in n-pentane [12] Several three-dimensional experiments can be performed as well. The most common ones couple H and N in a residue with one or more carbon nuclei from the same residue (refered to as intra or $i$), the preceding residue (inter or $i-1$) or both intra and inter. Seven of the NMR experiments often used in backbone chemical shift assignment are shown in Figure 2 for reference. (a) HSQC (b) HNcaCO (c) HNCA (d) HNcoCA (e) HNcoCACB (f) HNCACB (g) HNCO Figure 2: The subfigures show which spin systems produces a resonance peak in each experiment [13]. ## 3 General Assignment Strategy The NMR spectra contain no direct information about which residue each peak originates from. However using several experiments that probe different spin systems, it is possible to match identical chemical shifts in each experiment to the same nuclei. Furthermore inter and intra peaks can be matched together to form a ladder of chemical shift, as shown in Figure 3, only broken by Proline which doesn’t have a H-N pair and therefore are not represented in these spectra. Figure 3: Depiction of how matching of chemical shifts can be used to establish a ladder of peaks which corresponding residues must precede each other in the protein. CBCANNH and CBCA(CO)NNH are synonyms for HNCACB and HNcoCACB respectively [13] This is of course not as easy as it sounds since there might be overlapping peaks in the spectra, strong redundancy at a specific chemical shift value, missing peaks or peaks originating from noise or impurities etc.. When the prementioned ladders are formed, it is often possible to assign these uniquely to a part of the protein. This is possible since especially CA and CB chemical shifts contain information about which amino-acid they originate from. Protein databases such as the Biological Magnetic Resonance Data Bank [9] (BMRB) can be used to collect statistics about chemical shifts from each amino-acid which can be used to infer the likelihood of the assignment. (See Appendix for an example) When the spectra become more complex, for example with increased protein size, the assignment of the chemical shifts becomes increasingly more difficult, and in general complete assignments can’t be constructed and erroneous assignments might be made. A probabilistic framework can potentially remove the need for near 100% certainty in an assignment. The general idea of probabilistic methods is that sparse data is better than no data, and as explained in the introduction, the ability to change the assignment of chemical shifts during protein folding are important for two major reasons. Errors from using a deterministic assignment have less impact, and you get more information from an incomplete assignment than you otherwise would. ### 3.1 Select Automated Assignment Methods Two of the automated assignment methods that require the least human intervention is FLYA and Autoassign. This makes them suitable to use as alternatives to a manual assignment in the structure prediction, but they also provide a nice way to test how well an energy function describe these assignments of the data. #### 3.1.1 Autoassign The general assignment strategy of Autoassign [7] is to apply corrections to the chemical shift reference in each spectrum, to improve ”between-spectra” alignment. Then peaks from the 3D spectra, with H and N chemical shifts within a set tolerance, is mapped to peaks in the HSQC-spectrum, to create pseudo- residues with all intra- and intermolecular nuclei mapped to a base N-H pair. Peaks in HNCO with no corresponding peak in the HSQC spectrum, is used as a base and the previous step is repeated with these. It is argued that pseudo- residues which stems from side chain N-H pairs have low intensities in 3D experiments and thus pseudo-residues including less than three peaks from 3D spectra are recognised as side chains and are removed from backbone assignment. If more pseudo-residues are created than there are assignable residues in the protein, the pseudo-residues with weakest intensities are set aside. And the $C^{\alpha}$ and $C^{\beta}$ peaks in these pseudo-residues are used to create amino-acid probability scores. The most complete (containing most peaks) pseudo-residues intra and inter- peaks are paired and matched by a matching function. If the match is good and their combined amino-acid probability scores match a unique part of the sequence, the assignment is made. This is repeated with increasing tolerances until a full assignment is made or a upper bound on the constraints are reached. For the last step, the weaker pseudo-residues set aside earlier is analysed and assigned to one of the remaining missing residues if applicable or used to replace an already assigned one if it provide a better match. The Autoassign article reports 98% of backbone chemical shifts being assigned for 7 proteins below 150 residues in size with an error rate of 0.5%, using 9 different NMR spectra. #### 3.1.2 FLYA The assignment strategy in FLYA [8] is a mixture of deterministic and probabilistic approaches. A set of expected peak values is created based on sequence and chemical shift statistics. Each expected peak can be matched to only one experimental peak, but each experimental peak can be assigned multiple times. However if more peaks is found in a spectrum than 1.5 times the expected amount of peaks, the peaks with weakest intensities are removed. A scoring function to evaluate the quality of the assignment is used together with an evolutionary algorithm to find the best assignment. No mathematical basis for the scoring function is given, but the gist of their approach is that an ”external” part and an ”internal” part contributes to the score with certain hand-picked weights. The external part evaluates how well the expected chemical shift value agrees with the mean value of the chemical shifts assigned to the nuclei. The internal part evaluates the variance of the assigned peaks. This evaluation is based on a normal distribution where a discrepancy of less than 1.5 and 2.0 times some predefined standard deviation for the external and internal part respectively, will contribute positively to the score, while discrepancies higher than this will favor that the assignment isn’t made. The FLYA article reports 96-99% of backbone chemical shifts being assigned for three 100-150 residue proteins. A very large amount of NMR spectra was used, including NOE’s, but instead of manually picking the peaks from these spectra, the peaks were automatically picked by other programs. ## 4 Theory As mentioned previously, chemical shifts carry information about the protein structure, such as dihedral angles, side chain angles, ring current effects etc.. In the past chemical shifts have been used in a protein folding context, usually together with Nuclear Overhouser Effect (NOE) experiments to select conformers that provided the best match with the experimental data. In general the structures are selected by minimising a hybrid energy that connects a physical energy (e.g. from a forcefield) with experimental data $\mathrm{E}_{\mathrm{hybrid}}=\omega_{\mathrm{data}}\cdot\mathrm{E}_{\mathrm{data}}+\mathrm{E}_{\mathrm{phys}}.$ (2) However the methodology for evaluating agreement between structure and experimental data varies greatly, and is often somewhat arbitrary. Similarly the parameters and weights used for $\mathrm{E}_{\mathrm{data}}$ are often tweaked manually and optimal parameters seem to be based on trial and error. The inferential structure determination (ISD) approach [14, 15] uses a Bayesian formalism to handle these _nuisance parameters_ , such as the uncertainty and other model parameters, probabilistically as demonstrated by Olsson et al. [16] using a set of NOE restraints combined with a physical energy term. This section introduces the ISD formalism for the Markov Chain Monte Carlo method simulations used to simulate both chemical shift assignment and protein structure. ### 4.1 Probabilistic Framework The probability for event A given event B, $\mathrm{P}\left(A\mid B\right)$, is given by the chain rule $\mathrm{P}\left(A,B\right)=\mathrm{P}\left(A\mid B\right)\cdot\mathrm{P}\left(B\right),$ (3) where $P\left(A,B\right)$ is the probability for both $A$ and $B$, which often is written as $P\left(A\cap B\right)$. This, along with the equality $\mathrm{P}\left(A,B\right)=\mathrm{P}\left(B,A\right)$, leads directly to Bayes Theorem: $\mathrm{P}\left(A\mid B\right)=\frac{\mathrm{P}\left(B\mid A\right)\cdot\mathrm{P}\left(A\right)}{\mathrm{P}\left(B\right)}.$ (4) Using Bayes Theorem, we aim to find the most probable structure $X$, assignment $A$ and nuisance parameters $n$, given some experimental data $D$ and prior information $I$ (such as information used to generate the model describing the data, amino acid sequence etc.) $\mathrm{P}\left(X,A,n\mid D,I\right)=\frac{\mathrm{P}\left(D,I\mid X,A,n\right)\cdot\mathrm{P}\left(X,A,n\right)}{\mathrm{P}\left(D,I\right)}.$ (5) Since only $X$, $A$ and $n$ are changed in Monte Carlo moves, terms not involving these doesn’t need to be evaluated and can be disregarded, since the relative energy landscape is invariant of choice of normalisation constant. $\displaystyle\mathrm{P}\left(X,A,n\mid D,I\right)$ $\displaystyle\propto{\mathrm{P}\left(D,I\mid X,A,n\right)\cdot\mathrm{P}\left(X,A,n\right)}$ $\displaystyle={\mathrm{P}\left(D\mid I,X,A,n\right)\cdot\mathrm{P}\left(I\mid X,A,n\right)\cdot\mathrm{P}\left(X,A,n\right)}$ $\displaystyle={\frac{\mathrm{P}\left(D\mid I,X,A,n\right)\cdot\mathrm{P}\left(X,A,n\mid I\right)\cdot\mathrm{P}\left(I\right)\cdot\mathrm{P}\left(X,A,n\right)}{\mathrm{P}\left(X,A,n\right)}}$ (6) $\displaystyle\propto{\mathrm{P}\left(D\mid I,X,A,n\right)\cdot\mathrm{P}\left(X,A,n\mid I\right)}$ $\displaystyle=\mathrm{P}\left(D\mid I,X,A,n\right)\cdot\mathrm{P}\left(X\mid A,n,I\right)\cdot\mathrm{P}\left(A\mid n,I\right)\cdot\mathrm{P}\left(n\mid I\right)$ The prior distribution of $\mathrm{P}\left(n\mid I\right)$ is typically drawn from a log-normal distribution for purely positive parameters, and from a normal distribution if that’s not the case. The argument being that these are the least biasing distributions according to the principle of maximum entropy [17, 18]. $\mathrm{P}\left(X\mid A,n,I\right)$ is independent of $n$ and $A$. If a physical forcefield is used then the probability for a structure follows the usual Bolzmann distribution $\mathrm{P}\left(X\mid I\right)=\frac{1}{Z}\cdot\exp\left(-\frac{\mathrm{E_{phys}}}{\mathrm{k_{B}\cdot T}}\right),$ (7) Luckily we don’t have to evaluate the partition function $Z$ since it appears as just a normalisation constant. $\mathrm{P}\left(X\mid I\right)$ can also be introduced as a generative probabilistic model (GPM) such as Torus-dbn [19] and Basilisk [20] which replaces the physical term by a biased sampling of protein structure. These models are based on a large database of experimentally obtained structures backbone and side chain angles respectively. For describing $P\left(D\mid I,X,A,n\right)$ the normal distribution is used because it’s simple to work with mathematically and computationally. In addition, due to the Central Limit Theorem [21], the arithmetic mean of a large number of iterates of independent random variables will be approximately normal-distributed. A measured chemical shift $\delta_{i}$ will likely follow the distribution $g(\delta_{i};\mu,\hat{\sigma})=\frac{1}{\hat{\sigma}\sqrt{2\pi}}\,e^{-\frac{(\delta_{i}-\mu)^{2}}{2\hat{\sigma}^{2}}},$ (8) with $\mu$ being the population mean (or ”true” chemical shift) and $\hat{\sigma}$ being the standard deviation. The probability density of two independent measurements of a nuclei’s chemical shift, $\delta_{i}$ and $\delta_{j}$ is then: $\displaystyle f(\delta_{i},\delta_{j};\sigma_{i},\sigma_{j})$ $\displaystyle=\int_{-\infty}^{\infty}g(\delta_{i};\mu,\sigma_{i})g(\delta_{j};\mu,\sigma_{j})\pi\left(\mu\right)\mathrm{d}\mu$ (9) $\displaystyle\propto\left(\sigma_{i}^{2}+\sigma_{j}^{2}\right)^{-\frac{1}{2}}\exp\left(-\frac{\left(\delta_{i}-\delta_{j}\right)^{2}}{2\left(\sigma_{i}^{2}+\sigma_{j}^{2}\right)}\right)$ (10) Here $\mu$ has been integrated out using a uniform prior $\pi(\mu)$. Chemical shifts can be predicted using a forward model, such as SPARTA [22], PROSHIFT [23], SHIFTX [24], Camshift [25] etc., which relates a structure to a set of chemical shifts. If $\delta_{i}$ is a predicted chemical shift value, then the corresponding standard deviation will be much larger than the experimental error. Upon taking the negative logarithm $F_{pre}(\Delta_{ij};\sigma_{i})=\log\sigma_{i}+\frac{\Delta_{ij}^{2}}{2\sigma_{i}^{2}}$ (11) with $\Delta_{ij}=\delta_{i}-\delta_{j}$. If both $\delta_{i}$ and $\delta_{j}$ are obtained from experiment and the same variance is assumed, then we get $F_{exp}(\Delta_{ij};\sigma)=\log\sigma+\frac{\Delta_{ij}^{2}}{2\sigma^{2}}$ (12) with $\sigma=2\sigma_{i}=2\sigma_{j}$. When more than two measurements of the same nuclei’s chemical shift are used, things start to get more complex and some approximations are in order. For a predicted chemical shift $\delta_{i}$ and a set of experimentally obtained chemical shifts $\left\\{\delta_{j}\right\\}$, the following probability density is obtained $\displaystyle f(\delta_{i},$ $\displaystyle\left\\{\delta_{j}\right\\};\sigma_{i},\sigma_{j})=\int_{-\infty}^{\infty}g(\delta_{i};\mu,\sigma_{i})\prod_{j}^{N}g(\delta_{j};\mu,\sigma_{j})\pi\left(\mu\right)\mathrm{d}\mu$ $\displaystyle\mathrel{\vbox{ \offinterlineskip\halign{\hfil$#$\cr\propto\cr\kern 2.0pt\cr\sim\cr\kern-2.0pt\cr}}}\frac{1}{\sigma_{i}}\exp\left(-\frac{\sum_{j}^{N}\left(\delta_{i}-\delta_{j}\right)^{2}}{2N\sigma_{i}^{2}}\right)\frac{1}{\sigma_{j}^{N-1}}\exp\left(-\frac{\sum_{j}^{N}\sum_{k>j}^{N}\left(\delta_{j}-\delta_{k}\right)^{2}}{2N\sigma_{j}^{2}}\right)$ (15) $\displaystyle=\frac{1}{\sigma_{i}}\exp\left(-\frac{\chi_{pre}^{2}}{2N\sigma_{i}^{2}}\right)\frac{1}{\sigma_{j}^{N-1}}\exp\left(-\frac{\chi_{exp}^{2}}{2N\sigma_{j}^{2}}\right)$ (16) with $\chi_{pre}^{2}=\sum_{j}^{N}\left(\delta_{i}-\delta_{j}\right)^{2}$ and $\chi_{exp}^{2}=\sum_{j}^{N}\sum_{k>j}^{N}\left(\delta_{j}-\delta_{k}\right)^{2}$ where $k$ and $j$ refer to experimental chemical shifts. The middle expression in (4.1) is obtained by tedious algebra with the only approximation used being $\sigma_{i}\gg\sigma_{j}$. (4.1) can be approximated to the simpler form of (11) and (12) in order to simplify the calculations and reduce computational costs. Comparing these expressions, it is seen that if we make the approximation that every nuclei of the same type, have the same number of chemical shifts assigned to it, the negative logarithm of these expressions only differ by a normalisation factor. Using (11) to describe all interactions between the predicted chemical shift $\delta_{i}$ and the $N$ experimental ones $\left\\{\delta_{j}\right\\}$: $\displaystyle\sum_{j}^{N}F_{pre}(\Delta_{ij};\sigma_{i})$ $\displaystyle=\sum_{j}^{N}\left(\log\sigma_{i}+\frac{\Delta_{ij}^{2}}{2\sigma_{i}^{2}}\right)$ $\displaystyle=N\log\sigma_{i}+\frac{\chi_{pre}^{2}}{2\sigma_{i}^{2}}$ (17) Comparing this expression to (4.1) shows that the two equations differ by only a normalisation factor $\omega$: $\displaystyle\omega\left[N\log\sigma_{i}+\frac{\chi_{pre}^{2}}{2\sigma_{i}^{2}}\right]$ $\displaystyle=\log\sigma_{i}+\frac{\chi_{pre}^{2}}{2N\sigma_{i}^{2}}$ (18) $\displaystyle\omega$ $\displaystyle=\frac{1}{N}$ (19) Similarly, (12) can be used to describe all unique pairings of the experimental chemical shifts. For $N$ chemical shifts, there will be a total of $N\left(N-1\right)/2$ unique pairings (given by $\sum_{j}^{N}\sum_{k>j}^{N}$), resulting in: $\displaystyle\sum_{j}^{N}\sum_{k>j}^{N}F_{exp}(\Delta_{jk};\sigma_{j})$ $\displaystyle=\sum_{j}^{N}\sum_{k>j}^{N}\left(\log\sigma_{j}+\frac{\Delta_{jk}^{2}}{4\sigma_{j}^{2}}\right)$ $\displaystyle=\frac{N\left(N-1\right)}{2}\log\sigma_{j}+\frac{\chi_{exp}^{2}}{4\sigma_{j}^{2}}$ (20) where constant terms have been neglected. Note the factor of 4 in the denominator of the right-most term instead of a factor of 2, due to not replacing $\sigma_{j}$ with $\sigma$. Comparing with (4.1) to find the normalisation factor: $\displaystyle\omega\left[\frac{N\left(N-1\right)}{2}\log\sigma_{j}+\frac{\chi_{exp}^{2}}{4\sigma_{j}^{2}}\right]$ $\displaystyle=\left(N-1\right)\log\sigma_{i}+\frac{\chi_{exp}^{2}}{2N\sigma_{i}^{2}}$ (21) $\displaystyle\omega$ $\displaystyle=\frac{2}{N}$ (22) To summarise, considering only the disagreement between predicted and assigned chemical shifts, with a total of $N_{j}$ experimentally measured chemical shifts assigned to nuclei of the same type for $j\in\left\\{C^{\alpha},H,N,C,C^{\beta}\right\\}$, $\displaystyle\mathrm{P}_{pre}\left(D\mid X,A,\left\\{\sigma_{pre,j}\right\\},I\right)$ $\displaystyle\propto\prod_{j}\prod_{i}^{N_{j}}\left[\frac{1}{\sigma_{pre,j}}\exp\left(-\frac{\Delta_{ij}^{2}}{2\sigma_{pre,j}^{2}}\right)\right]^{\omega_{pre,j}}$ (23) $\displaystyle=\prod_{j}\left(\sigma_{pre,j}\right)^{-N_{j}\omega_{pre,j}}\exp\left(-\frac{\chi_{pre,j}^{2}\omega_{pre,j}}{2\sigma_{pre,j}^{2}}\right)$ (24) where $\Delta_{ijk}$ is the difference between chemical shift $i$ and the predicted chemical shift $k$ for nuclei type $j$, $\chi_{pre,j}^{2}=\sum_{i}^{N_{j}}\Delta_{ijk}^{2}$ and $\omega_{pre,j}$ is the weight for nuclei type $j$. Its exact weight can estimated from the number of contributions to $\chi_{pre,j}^{2}$ in the simulation. Likewise the disagreement between chemical shifts from different experiments assigned to the same atom is treated in the same manner, but with separate nuisance parameters $\left\\{\sigma_{exp,j}\right\\}$. $\mathrm{P}_{exp}\left(D\mid A,\left\\{\sigma_{exp,j}\right\\},I\right)\propto\prod_{j}\left(\sigma_{exp,j}\right)^{-m_{j}\omega_{exp,j}}\exp\left(-\frac{\chi_{exp,j}^{2}\omega_{exp,j}}{2\sigma_{exp,j}^{2}}\right)$ (25) with $\chi_{exp,j}^{2}$ containing a total of $m_{j}$ unique chemical shifts differences. $\mathrm{P}\left(A\mid n,I\right)$ basically describes the probability density for having $N_{j}$ chemical shifts assigned. Since a complete one to one assignment of all peaks usually is impossible, a model describing whether an assignment is better or worse than having no assignment at all is needed. Currently every ”missing” contribution to $\chi_{pre,j}^{2}$ is replaced by a chemical shift difference of $3\sigma_{pre,j}$. The effect of this is that assignment will be favoured if the chemical shift differences are lower than $3\sigma_{pre,j}$, and unassignment will be favoured if it is not. Likewise for $\chi_{exp,j}^{2}$, missing contributions is replaced by a difference of $4\sigma_{exp,j}$. These exact values were chosen since they seem to perform the best. Putting it all together when a physical force field is used, the probability distribution we aim to simulate will be: $\mathrm{P}\left(X,A,n\mid D,I\right)\propto\\\ \exp\left(-\frac{\mathrm{E_{phys}}}{\mathrm{k_{B}\cdot T}}\right)\prod_{j}\frac{\sigma_{pre,j}^{-N_{j}\omega_{pre,j}}}{\sigma_{exp,j}^{m_{j}\omega_{exp,j}}}\exp\left(-\frac{\chi_{pre,j}^{2}}{2\sigma_{pre,j}^{2}}-\frac{\chi_{exp,j}^{2}}{2\sigma_{exp,j}^{2}}\right)\cdot\mathrm{P}\left(n\mid I\right)$ (26) where $\mathrm{P}\left(n\mid I\right)$ will be removed as a bias in the acceptance rate (See Section 5.2.1). The associated hybrid energy is $\mathrm{E_{hybrid}}=E_{phys}\\\ +\mathrm{k_{B}T}\sum_{j}\left[\omega_{pre,j}\left(N_{j}\log\sigma_{pre,j}+\frac{\chi_{pre,j}^{2}}{2\sigma_{pre,j}^{2}}\right)+\omega_{exp,j}\left(m_{j}\log\sigma_{exp,j}+\frac{\chi_{exp,j}^{2}}{2\sigma_{exp,j}^{2}}\right)\right]$ (27) Since the structure $X$, assignment $A$ and parameters $n$ are all treated as variables, Monte Carlo moves are needed for each of these ’dimensions’ of the sampling space as described in the next section. ## 5 Computational Details Phaistos is a software framework for Markov chain Monte Carlo sampling for simulation, prediction, and inference of protein structure [26]. A large range of Monte Carlo moves is implemented for structure inference with selected physical force fields, and so is state of the art Monte Carlo methods and the forward model Camshift. In addition to this the probabilistic framework makes it easy to implement and treat empirical inferred models of experimental data together with physical forcefields in a rigid probabilistic fashion, which has been done previously for NOE’s [16]. ### 5.1 Markov Chain Monte Carlo Markov Chain Monte Carlo (MCMC) algorithms sample from probability distributions in the steady state, and are desirable to use when the distribution isn’t easily expressible analytically. The probability distribution of a set of variables $\left\\{x\right\\}$ can be approximated by this method, given that a function $f(\left\\{x\right\\})$ that’s proportional to the real distribution is known. The most common MCMC method is the Metropolis-Hastings algorithm [27]. Given the most recent sampled state $x_{t}$, a new state $x^{\prime}$ is proposed with a probability density that adhere to detailed balance $\mathrm{P}\left(x_{t}\right)\mathrm{P}\left(x_{t}\rightarrow x^{\prime}\right)=\mathrm{P}\left(x^{\prime}\right)\mathrm{P}\left(x^{\prime}\rightarrow x_{t}\right)$ (28) which in turn ensures that samples correspond to the steady state. If the probability for this state is greater than the previous state, the proposed new state is accepted and $x_{t+1}=x^{\prime}$. If the probability is lower, the Metropolis-Hastings acceptance criteria of the proposed state is given by $\mathrm{P_{acc}}=\min\left(1,\frac{f(x^{\prime})}{f(x_{t})}\right)$ (29) If the state is rejected the system will return to the previous state $x_{t+1}=x_{t}$. The Metropolis-Hastings algorithm is shown schematically in Figure 4 Figure 4: Flowchart showing the steps of the Metropolis-Hastings algorithm. Other more advanced MCMC methods is implemented in Phaistos, but all simulations run so far have been using the Metropolis-Hastings method. However since all implemented Monte Carlo moves in Chemshift uphold detailed balance, other methods can easily be used as well. ### 5.2 Chemshift implementation in Phaistos The Monte Carlo method requires both evaluation of energy and Monte Carlo moves that propose new values for the sampled parameters. The hybrid energy used is described in Section 4 and the Monte Carlo moves used for assignment is presented here. Each spectrum of the types, HSQC, HNCA, HNcoCA, HNcoCACB, HNCACB, HNCO and HNcaCO that are available, is parsed from their input files where each peak is split into the chemical shifts according to the originating nuclei, as shown below: $\left[C^{\alpha}_{i-1},H_{i-1},N_{i-1},C_{i-1},C^{\beta}_{i-1},C^{\alpha}_{i},H_{i},N_{i},C_{i},C^{\beta}_{i}\right]$ Unused sites in these constructed peak-lists are given a NAN value to be easily recognisable. If the peak is assigned to a specific spin system in the input file the same assignment is used in the module. All spin systems that have not been assigned a peak is assigned a list with only NAN values. This results in an array initially the same length of the protein. All the unassigned peaks is placed at the back of this array in an ”unassigned” region, where the energy isn’t evaluated. This procedure is repeated for all the spectra available. The spectra HNCA, HNcoCACB, HNCACB and HNcaCO contain peaks from more than one backbone spin system and an array is created for each spin system type. As an example HNCA is split into an inter-peak and intra- peak array. For HNCA and HNcaCO, unassigned peaks are placed randomly in the unassigned region of the inter and intra array, and for HNcoCACB the largest carbon chemical shifts is attributed to $C^{\alpha}$. For HNCACB, which contains four peaks per residue, peaks from $C^{\alpha}$ and $C^{\beta}$ are assumed to be of opposite phase, and the nuclei type can be uniquely identified. Whether a peak is placed in the nuclei specific inter or intra peak is random. #### 5.2.1 Monte Carlo Nuisance Parameter Moves $\sigma$ describes the always positive standard deviation, so the log-normal distribution is well suited to propose new values for this. However by imposing this distribution for the data, a small bias will be introduced in the acceptance criteria, since $\mathrm{P_{acc}}\propto\frac{\mathrm{P}\left(\sigma^{\prime}\mid I\right)}{\mathrm{P}\left(\sigma\mid I\right)}$ (30) From detailed balance (28) this bias is removed by multiplying with $\frac{\mathrm{P}\left(\sigma^{\prime}\rightarrow\sigma\right)}{\mathrm{P}\left(\sigma\rightarrow\sigma^{\prime}\right)}$ (31) whenever a move in the nuisance parameter space is made. The update_sigma move make changes to a single element in $\left\\{\sigma_{pre,j}\right\\}$ or $\left\\{\sigma_{exp,j}\right\\}$. Specifically this is done by drawing a factor $x$ from a log-normal distribution with parameters $\mu=0$ and $\sigma_{\sigma}=1$. $\mathrm{P}\left(x\right)\propto\frac{1}{x}\exp\left(\frac{\log^{2}x}{2}\right)$ (32) The proposed new value $\sigma^{\prime}$ for the standard deviation is $\sigma^{\prime}=\sigma\cdot x\quad\Leftrightarrow\quad\sigma=x^{-1}\sigma^{\prime}.$ (33) The corresponding bias that needs to be included in the acceptance criteria for the move is then $\displaystyle\frac{\mathrm{P}\left(\sigma^{\prime}\rightarrow\sigma\right)}{\mathrm{P}\left(\sigma\rightarrow\sigma^{\prime}\right)}$ $\displaystyle=\frac{\mathrm{P}\left(x^{-1}\right)}{\mathrm{P}\left(x\right)}$ (34) $\displaystyle=\frac{\left(x^{-1}\right)^{-1}\exp\left(-\frac{\left(\log{x^{-1}}\right)^{2}}{2}\right)}{\left(x\right)^{-1}\exp\left(-\frac{\left(\log{x}\right)^{2}}{2}\right)}$ (35) $\displaystyle=x^{2}$ (36) #### 5.2.2 Monte Carlo Assignment Moves To ensure that a specific assignment can be reached (at least in theory) in the simulation, it’s important to cover the entire assignment-space. This is done by the following five moves: move_single picks an array at random and interchanges two peaks in this array, providing the means to switch assignments, unassign previously assigned peaks and vice versa. move_HNCA works the same as above, but instead of interchanging two peaks in the same array, a peak from the inter HNCA array is interchanged with a peak in the intra HNCA array, followed by a reclassification of the chemical shift assigned from $C^{\alpha}_{i-1}$ to $C^{\alpha}_{i}$ and vice versa. move_HNcoCACB and move_HNcaCO are similar to the above, just with the arrays made from the HNcoCACB and HNcaCO spectra respectively. move_CA_HNCACB and move_CB_HNCACB moves between the spin systems $C^{\alpha}_{i-1}$ and $C^{\alpha}_{i}$ and likewise for $C^{\beta}$. Changing a $C^{\alpha}$ assignment to a $C^{\beta}$ assignment is not possible, since it is assumed that these are always distinguishable by their phase. During both a manual and simulated assignment, a ladder of spin systems connected through their intra and inter peaks can usually be constructed, where the created sequence of peaks matches very well. If these ladders are incorrectly assigned, it will be very difficult to reassign them with moves that only interchange two peaks at a time, due to a low acceptance rate. Because of this a set of moves that can reassign parts of or whole ladders is implemented. These moves are carried out in two functions, move_base and move_peak_blocks, with several Monte Carlo moves utilising these with different parameters. move_base is used by a range of Monte Carlo moves to reassign 1 to $N$ adjacent peaks from 1 to $M$ different spin system arrays simultaneously, but doesn’t change which array each peak is placed in. The number of arrays involved in the move depends entirely on arbitrary chosen weights. These weights will only affect how fast the simulation reaches convergence etc. and not the energy landscape as such. Because of this no rigorous optimisation of these parameters has been done. The probability for selecting a specific number of adjacent peaks is arbitrary as well, but smaller numbers are more probable than higher numbers, and the probability approximately follows an exponential decay with increasing ladder size. In the initialisation steps of the module, an array is generated with every possible placement for ladders of size 1 to $N$ which make $N$ equal to the size of the largest segment in the protein with no Prolines. The placement of Glycines in the protein is noted in this array as well to make sure no $C^{\beta}$ chemical shift are assigned there. The move itself, given a number of adjacent peaks to move in a number of spin system arrays, is often non problematic and two peak ”blocks” swap assignments. If a Glycine is present in one of these protein segments, any peak with a $C^{\beta}$ chemical shift that would wrongly be assigned to the Glycine is instead moved to the unassigned region. When a ladder is moved a smaller distance than the length of the ladder itself, the problem arises that the starting assignment of the ladder overlaps with the destination of the ladder. An example is shown below, with $i_{n}$ being peaks that are to be moved to sites $j_{n}$. $\left[\;i_{0}\;,\;i_{1}\;,\;i_{2}\;,\;i_{3}\;,\;i_{4}\;/\;j_{0}\;,\;i_{5}\;/\;j_{1}\;,\;j_{2}\;,\;j_{3}\;,\;j_{4}\;,\;j_{5}\;\right]$ For this situation special care is needed in order to conserve as much integrity of the moved ladders as possible.To achieve this one full ladder is selected at random from the two overlapping ones, and this ladder will be moved as it is, with the resulting assignment shown below $\left[\;j_{2}\;,\;j_{3}\;,\;j_{4}\;,\;j_{5}\;,\;i_{0}\;,\;i_{1}\;,\;i_{2}\;,\;i_{3}\;,\;i_{4}\;/\;j_{0}\;,\;i_{5}\;/\;j_{1}\;\right]$ move_peak_blocks is of similar construct, but interchanges two ladders from different spin system arrays, originating from the same experiment. Figure 5 shows a simplified flowchart of a Monte Carlo simulation with Chemshift. Figure 5: Flowchart showing the general strategy in a Monte Carlo simulation in Phaistos with the Chemshift module. Details in the text #### 5.2.3 Cashing The computational aspect of this project represents around 90% of the work done. Other than on implementation and development of the different aspects of the program, a considerate amount of time have been used on increasing the speed of the calculations. In the initialisation part of the program, starting guess values is set for the nuisance parameters, the Camshift predictions are created and the sum of all possible chemical shift differences ($\chi^{2}$) is calculated. This last step takes a very long time and would be a major bottleneck if it were to be run after each move. To reduce the time used, two functions, initialise_chi_sq_details and initialise_chi_sq_partial are employed. The first function scans through each spin system array and notes which chemical shift types the array contains, and stores all the possible permutations of chemical shift differences that can arise. That is it won’t try to check the $C^{\beta}$ differences between HNCA and HNCACB peaks, since the $C^{\beta}$ values will always be NAN in the HNCA as well as the $C^{\alpha}$ spin system arrays of HNCACB. The second function stores every contribution to $\chi^{2}$ separately instead of just storing the sum. In every iteration, information about what move is used, which spin system array change and which peaks are moved is stored, making it possible to both reverse the move made if it is rejected, instead of having to save and copy the complete assignment every iteration, but also to use the information from initialise_chi_sq_partial to only calculate the contributions that are changed. Knowing which spin systems the changed peaks were and became assigned to cuts down calculation cost dramatically. However further reducing the number of calculations done, to only include the spin system arrays that were moved in is a bit more complicated. When only changes are made in one spin system array, only the chemical shift differences between this array and all the others need to be updated (disregarding Camshift predicted chemical shifts for the moment, as calculation of these is trivial). If changes are made in all the spin system arrays, all terms have to be updated. However in between these extremes the computational part is a bit more complex, even though only the differences between just the changed spin system arrays, and the difference between the changed and the non-changed arrays need to be calculated. Because of this extra (but not easily recognised) computational cost, this procedure is only done on $H$ and $N$ chemical shifts, while all possible differences are calculated for the rest of the nuclei. The argument for doing it this way is that, given the spectra HSQC, HNCA, HNCACB, HNcoCACB, HNCO and HNcaCO, there will be 66 possible differences to be calculated for $H$ and $N$ each, 10 for $C^{\alpha}$ and 3 for $C$ and $C^{\beta}$ each. So carbon differences is only about 10% of all the contributions, and it didn’t seem like any noticeable benefit in computational cost would be gained. During these simulations, the assignment itself, as well as the nuisance parameters, $\chi^{2}$ and the list containing every contribution to $\chi^{2}$ need to be able to be returned to the previous state if the move is rejected. Just keeping and updating copies of these after every iteration would be a major bottleneck, so if a move is rejected, the moves are written such that the previous state can be regained by using the same move type, with the same parameters. The list with $\chi^{2}$ contributions, could be updated in a similar fashion, but a faster way is to keep a copy of the list, and instead of copying the full list every iteration, use the stored move information to only copy the terms that may have changed. Currently an average of 2.6 billion assignment moves per day can be done on the 101 residue protein S6 on a single 3.0 GHz Xeon core, with around 10% of the time spent being overhead from Phaistos itself. In comparison around 2.8 million Camshift predictions can be done per day, and further improvements to the speed of the program have been halted until it becomes a bottleneck in the protein folding process. ## 6 Results A range of simulations have been run on Ribosomal Protein S6, for the purpose of testing the accuracy and breaking points of the assignment model, given a crystal structure. S6 was chosen for the simple reason that it’s the only protein where a manual assignment, Autoassign assignment and FLYA assignment for individual peaks have been available to us. In these simulations no changes were being made to the structure. Figure 6: The 101 residue Ribosomal Protein S6 (PDB:1LOU) Using HSQC, HNCA, HNCO, HNcaCO, HNCACB and HNcoCACB spectra, the 101 residue protein could theoretically be assigned 1327 peaks, with 950 peaks being assigned in the manual assignment. The agreement between the manual assignment and assignments obtained via the simulations was investigated, for four different starting assignments. The manual assignment, the FLYA assignment, the Autoassign assignment and finally starting with a random assignment. Figure 7 shows the number of peaks correctly assigned as the simulation progresses. A peak is considered correctly assigned if all chemical shifts of the peak lies within 0.03 ppm for hydrogen and 0.4 ppm for the heavy nuclei compared to the manual assignment, which is the same criteria used in the FLYA paper. Figure 7: Simulation on S6 with assignment and nuisance parameter moves, with the initial assignment being done by Autoassign. Peaks were deemed correct if all chemical shifts of the peak were within the tolerance region of 0.03 ppm for Hydrogen and 0.4 ppm for the heavy nuclei, compared to the manual assignment The assignment by Autoassign agrees with the manual assignment for 575 peaks initially. As the simulation progresses, this number rises to around 770 while the number of peak assignments that disagrees with the manual assignment rose from 5 to around 80. The fact that a large number of chemical shifts is being incorrectly assigned isn’t as troublesome as it would be for a deterministic assignment, since each point in Figure 7 represents a snapshot of the assignment at a particular time. If the most probable assignment of a peak was taken from a histogram of all the assignment snapshots, the number of incorrect assignments would quite possibly be lower than what appears from the figure. However this trend would also be likely to be observed if the energy function used to describe the experimental data is of poor quality. Figure 8: Number of correcly assigned peaks with initial assignment done manually, by FLYA, by Autoassign and no initial assignment at all. Figure 8 shows the agreement of the simulation with the manual assignment, starting from different initial assignments. When starting from the manual assignment, the agreement went down as expected from 950 initially to around 924 peaks on average, with no incorrect assignments. FLYA experienced little change, going from 908 initially to 904 correct on average, with the number of incorrectly assigned peaks dropping from 18 initially to 14 on average. When a random initial assignment was given, the simulation was quickly stuck in a local minimum with very poor agreement on especially $H$ and $N$ nuclei chemical shifts, which could either be a sampling problem, or due to a poor model description. Investigating the energies of the different starting assignments, using only nuisance parameter moves (no changes being made to the assignment), the energies is expected to follow $E_{autoassign}>E_{FLYA}>=E_{manual}$, based on the correctness of the assignments. Surprisingly the energies were found as following $E_{FLYA}>E_{manual}>E_{Autoassign}$ as shown in Figure 9. Figure 9: Energies of three simulations on S6, with three different starting assignments, consisting of only nuisance parameter sampling. That Autoassign is lowest in energy strongly suggests that the model for describing unassigned chemical shifts needs to be improved. However the difference between the manual assignment and the FLYA assignment cannot be explained simply by this, since they should be very similar. Therefore it is very clear that improvements in general of the energy function is critical for improving upon the current assignment capabilities of the module. ## 7 Future Work The Chemshift module is as previously stated a work in progress, and in terms of module functionality, a number of improvements is planned. The most important being model improvements. In the following, planned improvements to the model, that have yet to be implemented, is presented. ### 7.1 Referencing Errors From the simulations on S6, it is clear that improvements to the energy function needs to be made. As shown in Figure 10 the current model describes actual data from the protein S6 somewhat poorly in some cases. (a) (b) (c) (d) Figure 10: Differences between chemical shifts assigned to the same nuclei from S6. Blue graph show the Kernel Density Estimate for the data, while green shows the best fit with a normal distribution The description of $H$ chemical shifts is especially poor and a likely cause of this is small perturbation differences to the reference nuclear shielding. In other words, the spectra used isn’t properly aligned. This alignment correction would correspond to a small correction to each chemical shift, depending on which spectra it originates from. The chemical shift difference for hydrogen from HSQC and HNCO would be $\left(\left(\delta_{HSQC}+\gamma_{HSQC}\right)-\left(\delta_{HNCO}+\gamma_{HNCO}\right)\right)$ instead of just $\left(\delta_{HSQC}-\delta_{HNCO}\right)$, with $\gamma_{i}$ representing the alignment offset of spectra $i$. These values of $\gamma_{i}$ could be treated as a nuisance parameter, with sampling done from a normal distribution. Correcting the S6 spectra, with values of $\gamma_{i}$ that maximises the model likelihood, the hydrogen differences obtained follow the simple Gaussian model much closer as seen in Figure 11. Figure 11: Differences between chemical shifts assigned to the same H nuclei, after alignment. #### 7.1.1 Model Validation When comparing different models, just a visual determination of the best model is prone to be erroneous. In addition adding parameters to be fitted will always improve a model, but might end up causing a low predictive validity due to over-fitting. To determine if the increase in goodness of fit outweighs the increase in complexity of the model (ignoring increased computational cost for the moment), Aikake’s Information Criterion (AIC) can be used [28]. AIC is a measure of the relative quality of a given model, and can be used for model selection, where the model with the minimum AIC value is prefered. The AICc is an improved version of the AIC that includes corrections for finite sample size, and should in general always be used instead of the AIC [29]. The AICc is given by: $AICc=2k-2\log\left(L\right)+\frac{2k\left(k+1\right)}{n-k-1},$ (37) with $k$ being the number of parameters in the model, $n$ being the sample size and $L$ being the maximum value of the likelihood function (the joint density function for all observations) for the estimated model. For the Gaussian model for $H$ differences with no alignment, the only parameter is the standard deviation. Maximising the likelihood of the S6 data yields an AICc value of -32060.97. Including alignment adds 5 new parameters when 6 spectra is used and yields an AICc value of -35858.28, which suggests that the improvement in goodness of fit is worth the information lost by increasing the number of parameters. ### 7.2 Peak Intensities In experiments containing both inter and intra peaks, the intra peak has a higher intensity on average than the inter peak, with an average ratio of around 1.5 having been reported [30]. But since there’s a large variance in this ratio, and ratio’s less than 1 often is observed, these intensities are often ignored by experimentalists. But for a probabilistic model, it should provide valuable information. Figure 12 shows these peak ratios for S6. Since the peak ratios approximately follow a log-normal distribution, it should be easy to implement this as an energy-term as well. Figure 12: Ratio of all intra- over inter peak intensities for carbon atoms in the S6 HNCACB, HNcaCO and HNCA spectra Of course the model selection will need to be validated on more than a single protein. Other model improvements that need to be investigated include describing data with different standard deviations depending on which spectrum it is from, using a function family other than the normal distribution, include possible correlation between different atom types and improving how unassigned chemical shifts is treated. ## 8 Summary and Outlook This thesis presents the current state of a new method for including experimental NMR data in protein structure determination, and the method has been implemented in the protein structure inference program Phaistos. The most noteworthy features is that 1) no peaks in the experimental spectra is discarded, providing more information about the structure than a regular deterministic assignment. 2) The assignment can change during protein folding, possibly giving a better description of the protein dynamics and reducing the effect of assignment errors. 3) The weight of experimental data relative to physical energy terms, is decided probabilistically instead of relying on arbitrary manual weights. By running simulations on the 101 residue Ribosomal Protein S6, some improvement to a partial assignment done by the program Autoassign has been made. By analysing the energies of assignments of differing qualities, it is clear that improvements need to be made to the proposed model. Improvements such as sampling the referencing errors between spectra and including additional energy terms related to peak intensities has been proposed based on statistical observations. Due to time restraints a proper validation of the method, by successfully folding a range of proteins, using unassigned chemical shift experiments, have yet to be done. However the entire framework for doing so has been created, and doing this is the intent of the project. Assuming that validation of the method is possible, the generated framework can easily be used to include assignment of protein side chain nuclei or to assign NOE’s at the same time as the chemical shifts. Furthermore histograms over the assignment of each peak could be generated to assist manual assignments. Over the next several months, work will continue on the Chemshift module, which will eventually be included in the official Phaistos release. ## 9 Appendix (a) (b) Figure 13: 1000 samples for each residue-type taken from normal approximations from BMRB to the distribution of chemical shifts. 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A refocused and optimized hnca: Increased sensitivity and resolution in large macromolecules. J. Biomol. NMR, 2(2):195–202, 1992.	arxiv-papers	2013-11-13T16:06:47	2024-09-04T02:49:53.554933	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Lars A. Bratholm", "submitter": "Lars Andersen Bratholm", "url": "https://arxiv.org/abs/1311.3186" }
1311.3253	# Modelling magnetism of C at O and B monovacancies in graphene T. P. Kaloni1, M. Upadhyay Kahaly1, R. Faccio2,3, and U. Schwingenschlögl1, Corresponding author. Tel.: +966 544700080. E-mail address: [email protected] (U. Schwingenschlogl) 1PSE Division, KAUST, Thuwal 23955-6900, Kingdom of Saudi Arabia 2Crystallography, Solid State and Materials Laboratory (Cryssmat-Lab), DETEMA, Facultad de Química, Universidad de la República, Gral. Flores 2124, P.O. Box 1157, Montevideo, Uruguay 3Centro NanoMat, Polo Tecnológico de Pando, Facultad de Química, Universidad de la República, Cno. Aparicio Saravia s/n, 91000, Pando, Canelones, Uruguay ###### Abstract The presence of defects can introduce important changes in the electronic structure of graphene, leading to phenomena such as C magnetism. In addition, vacancies are reactive and permit the incorporation of dopants. This paper discusses the electronic properties of defective graphene for O and B decoration. Phonon calculations allow us to address directly the stability of the systems under study. We show that it is possible to obtain magnetic solutions with and without dangling bonds, demonstrating that C magnetism can be achieved in the presence of B and O. 1\. Introduction C nanostructures have attracted the attention of the scientific community because of both unique fundamental properties and potential in technological applications. In particular, the high coherence length and large conductivity castro of C materials are promising in electronics and spintronics. On the other hand, also magnetism has been observed in this class of materials Palacio , generating huge interest of both experiment and theory han ; Setze ; rode ; lehtinen ; Flipse1 ; Ugeda ; mousumi . The structural, electronic, and magnetic properties of defective graphene have been studied in detail and it has been confirmed that vacancies are the source of magnetism new1 ; new2 ; new3 ; new4 . It has been demonstrated that point defects lead to notable paramagnetism but no magnetic ordering is achieved down to liquid helium temperature nair . In addition, the point defects carry magnetic moments irrespective of the vacancy concentration. In palacio1 the authors report that the extended $\pi$-band magnetism reduces to zero in the limit of monovacancies in graphene. For these reasons, it is important to evaluate the role of saturation and chemical modifications in the neighborhood of mono and multivacancies in order to understand how a persistent magnetic ordering can be achieved. Graphene samples in general are characterized by the presence of $sp^{2}$ hybridized C atoms, extended pores, and various attached functional groups, as revealed by scanning electron microscopy, atomic force microscopy, and Raman spectroscopy. It has been concluded that the appearance of magnetism requires the presence of defects, adatoms, or topological defects khana . Zagaynova and coworkers have prepared magnetic C by chemical oxidation mombr . B doping in general leads to different magnetic responses depending on the amount of doping. This behavior has been partially addressed by Faccio and coworkers pardo2 , who have demonstrated that magnetism requires the proximity of B and a monovacancy. However, when the B atom is too close to the vacancy all the dangling bonds are reconstructed and magnetism is suppressed. The authors of kaloni have addressed the situation of multiple O atoms attached to a monovacancy in graphene, demonstrating that vacancies are magnetic if and only if they are metallic and non-magnetic if and only if they are semiconducting. Metallicity and magnetism thus are simultaneously determined by the presence or absence of dangling C bonds after the oxidation. In our present work, we give a theoretical study of the effects of vacancies interacting with different amounts of B and O atoms, focusing on the structural, electronic, and magnetic properties. An experimental realization of the proposed structures in possible along the lines of Ref. carbon . 2\. Methods We employ density functional theory in the generalized gradient approximation (Perdew-Burke-Ernzerhof scheme) as implemented in the Quantum-ESPRESSO code paolo . All calculations are performed with a plane wave cutoff energy of 544 eV. A Monkhorst-Pack $8\times 8\times 1$ k-mesh is used to relax the structures and a $16\times 16\times 1$ k-mesh to calculate the density of states (DOS) with a high accuracy. We use a $5\times 5$ supercell of pristine graphene in our calculations cheng1 . This supercell size has shown in various studies to be sufficient for functionalization of graphene by atoms, small molecules, amine and nitrobezene groups, and others new12 ; new13 ; new14 ; new15 ; new16 ; new17 . Our supercell contains 50 C atoms and has a lattice constant of $a=12.2$ Å with a 20 Å thick vacuum layer on top. The atomic positions are relaxed upto an energy convergence of $10^{-7}$ eV and a force convergence of 0.005 eV/Å. For our vibrational study, the phonon frequencies and eigenvectors at the $\Gamma$-point are calculated with an energy error of less than $10^{-14}$ eV. Phonon frequencies at the $\Gamma$-point are determined by density functional perturbation theory for evaluating the structural stability Mod. . We note that the numerics break the symmetry of the dynamical matrix and introduce slight errors in the phonon frequencies of maximal 15 cm-1, which, however, are not critical. Figure 1: Crystal structures for 1 B and 2, 3, or 4 O atoms adsorbed at a monovacancy in graphene. The yellow, red, and gray spheres represent C, O, and B, respectively. Figure 2: Crystal structures for 2 O and 2, 3, 4, or 6 B atoms adsorbed at a monovacancy in graphene. The yellow, red, and gray spheres represent C, O, and B, respectively. In a first step, we create a monovacancy in our $5\times 5$ supercell and relax the system. Afterwards, we add O and B atoms in the vicinity of the monovacancy for different starting geometries and relax the system again. Several prototypical monovacancies are prepared in order to consider various possibilities with different O and B concentrations. We name these configurations giving first the number of O atoms and second the number of B atoms. For example, configuration 1/2 has 1 O and 2 B atoms in the vicinity of the monovacancy. All the configurations under study are shown in Figs. 1 and 2. Note that we have tested the required supercell size by addressing for the most stable configurations $10\times 10$ supercells and find no significant modification in the induced buckling of the graphene sheet after O and B adsorption. 3\. Results and discussion At a clean monovacancy the under-coordinated C atom is subject to Jahn-Teller distortion Jahn and therefore moves out of the graphene plane by 0.13 Å. We find a total magnetic moment of 1.35 $\mu_{B}$, where 1 $\mu_{B}$ is due to the localized dangling bond of one C atom and the remaining moment is carried by extended $\pi$ states kaloni . The vacancy formation energy, $E_{form}$, is determined by the expression $E_{form}=E_{vacancy}-\frac{N-1}{N}E_{graphene},$ (1) where $E_{vacancy}$ and $E_{graphene}$ are the total energies of defective and pristine graphene, respectively. Moreover, $N$ is the number of atoms in the pristine graphene. The obtained value of $E_{form}$ is 7.4 eV, which agrees well with previous reports pandey ; eggie . We first study a monovacancy decorated by 1 O atom to which 1 B atom is added. When forcing the O atom to occupy a position within the graphene plane connected to 3 C atoms the B atom is released and repelled, leading to configuration 1/1a. In configuration 1/1b the B atom connects to 1 O atom and 2 C atoms, saturating all dangling bonds. Moreover, configuration 1/1c deals with B adsorption far from the vacancy, for which we obtain a bridge position over a C$-$C bond with $d_{B-C}=1.87$ Å. The energetics of the doping process indicate that the chemical reaction between graphene-oxide and B can be evaluated as $E_{form}=E_{vacancy+B+O}-\frac{N-N_{C}}{N}E_{graphene}-\frac{N_{O}}{2}E_{O_{2}}-N_{B}E_{B,bulk},$ (2) where $E_{vacancy+B+O}$ is the total energy after B adsorption on graphene- oxide. In addition, $N_{C}$, $N_{O}$, and $N_{B}$ are the numbers of missing C atoms and additional O and B atoms in the decorated system. It turns out that configuration 1/1b is energetically favorable with $E_{form}=-6.9$ eV, see Table I, while for configurations 1/1a and 1/1c we obtain in each case a value of $-1.7$ eV. By the huge energy difference it is clear that only configuration 1/1b will be realized. Due to the B adsorption and the induced C and O reconstructions, this configuration becomes a metal, see Fig. 3(a). The Dirac cone, which is still due to the C $p_{z}$ orbitals with no contribution of O or B, splits and shifts above $E_{F}$ by about 0.6 eV. Interestingly, a magnetic moment of 0.2 $\mu_{B}$ is observed although no dangling bonds are present. System \| $E_{form}$ (eV) \| Magnetic moment ($\mu_{B}$) ---\|---\|--- 1/1a \| $-$1.7 \| 1.0 1/1b \| $-$6.9 \| 0.2 1/1c \| $-$1.7 \| 0.0 2/1a \| $-$7.0 \| 0.0 2/1b \| $-$7.4 \| 0.1 2/1c \| $-$5.3 \| 1.0 2/1d \| $-$4.8 \| 0.0 2/1e \| $-$6.9 \| 0.5∗ 3/1a \| $-$4.4 \| 0.2 3/1b \| $-$3.4 \| 1.0 3/1c \| $-$3.5 \| 1.0 4/1a \| 1.1 \| 0.0 4/1b \| $-$11.3 \| 0.0 4/1c \| 1.6 \| 0.0 2/2 \| $-$8.4 \| 1.0 2/3 \| $-$12.8 \| 1.0 2/4 \| $-$22.9 \| 0.0 2/6 \| $-$34.4 \| 0.0 Table 1: Formation energy and magnetic moment. ∗The magnetic moment is located on the released CO molecule. Figure 3: Electronic band structure and DOS for the structures (a) 1/1b, (b) 2/1b, (c) 3/1a, and (d) 4/1b. Turning to an oxidized monovacancy decorated by 2 O atoms, in configuration 2/1a the B atom establishes strong interaction with O and C, which originally was under-coordinated. We obtain $E_{form}=-7.0$ eV and no spin polarization due to saturation of all dangling C bonds. In configurations 2/1b and 2/1c the B atom is adsorbed at the boundary of the vacancy, leading to reconstructions with one and two C=O bonds, respectively. We obtain for $E_{form}$ values of $-7.4$ eV and $-5.3$ eV, where configuration 2/1c is less favorable due to the presence of a dangling C bond which induces a magnetic moment of 1 $\mu_{B}$. In configuration 2/1d the B atom is adsorbed far away from the monovacancy, which yields a surprising geometrical reconstruction: the two C=O bonds change into one ketone (C=O) and one ether group. However, with $E_{form}=-4.8$ eV this configuration is not favorable. Finally, in configuration 2/1e the B atom is located in the graphene plane and a CO molecule is released. We obtain $E_{form}=-6.9$ eV. In Fig. 3(b) we address the electronic band structure of the energetically favorable configuration 2/1b. We find that one spin majority band and one spin minority band cross $E_{F}$, both contributed by the C $p_{z}$ orbitals. The semiconducting state of a monovacancy decorated by 2 O atoms is transferred into a metallic state by B adsorption. For B adsorption at a monovacancy decorated by 3 O atoms, configuration 3/1a is characterized by three ketone groups, establishing a BO3 unit with an interatomic distance of $d_{B-O}=1.38$ Å. In configuration 3/1b the B atom is attached to one ketone group and one C terminated ether group, which changes the original $sp^{2}$ hybridization into a $sp^{3}$ hybridization, with interatomic distances of $d_{B-O}=1.35$ Å and $d_{B-C}=1.62$ Å. Since in configuration 3/1c the B atom substitutes a C atom near the vacancy, the structure does not change significantly. Configuration 3/1a is clearly favorable with $E_{form}=-4.4$ eV, as compared to values of $-3.4$ eV and $-3.5$ eV for configurations 3/1b and 3/1c. For configuration 3/1a we find a magnetic moment of 0.6 $\mu_{B}$. In addition, Fig. 3(c) shows that this configuration 3/1a is metallic with two distinct C $p_{z}$ derived bands crossing $E_{F}$. The observed localized states can be attributed to the under-coordination of the B atom, which clearly distinguishes this case from configurations 3/1b and 3/1c. In the case of 4 adsorbed O atoms, three configurations are found to be stable. Configuration 4/1a has two ketone groups and one peroxide ($-$O$-$O$-$) group and configuration 4/1b one ketone, one peroxide, and one ether group. The origin of the stability is the fact that the BO3 group itself is very stable new6 . In the former case the B atom is adsorbed far away from to the monovacancy, while in the latter it bonds with the 2 O atoms to form a BO3 group. Configuration 4/1c is similar to configuration 4/1a but with B substituting C at the boundary of the vacancy. Regarding the energetics, configuration 4/1b is strongly favorable with $E_{form}=-11.3$ eV, as compared to values of $1.1$ eV and $1.6$ eV, since a peroxide group is present. The electronic structure shown in Fig. 3(d) indicates that this configuration is semiconducting with a band gap of 0.5 eV at the K point, and not magnetic. We next evaluate the effect of the B concentration on the electronic structure of oxidized graphene by substituting 2 to 6 B atoms on C sites for a fixed O concentration (two adsorbed O atoms), see Fig. 2. In configuration 2/2 two B atoms replace two C atoms at the boundary of the vacancy. The system develops a magnetic moment of 1 $\mu_{B}$ due to the presence of a dangling C bond. In configuration 2/3 the substitution of an additional B atom does not change this situation. The fourth B atom in configuration 2/4 establishes a linear B$-$C=O bond, which is similar to the bond evolving in configuration 2/6. $E_{form}$ varies strongly between $-8.4$ eV and $-34.4$ eV due to the formation of B$-$B dimers. Our results show that there is a strong tendency to B adsorption at oxidized monovacancies. We note the appearance of fractional magnetic moments in several cases: 0.1 $\mu_{B}$ in configuration 2/1b and 0.2 $\mu_{B}$ in configurations 1/1b and 3/1a. For the remaining systems the moment is either 0 or 1 $\mu_{B}$, see Table I, and therefore results from localized dangling bonds. In all systems with fractional moments there are no contributions of the adsorped O and/or B atoms to the magnetization. Spin polarization is carried only by C atoms, where the atoms forming the boundary of the defect contribute less than those further away. While clean monovacancies in graphene do not give rise to extended $\pi$-band magnetism, the situation changes under O and B decoration. Figure 4: Phonon frequency at the $\Gamma$-point for (a) pristine graphene, the structures (b) 1/1b, (c) 2/1b, (d) 3/1a, (e) 4/1b, (f) 2/2, (g) 2/4, (h) 2/6, and (i) 2 O atoms adsorbed at a monovacancy. The calculation of $\Gamma$-point phonons allows us to address the structural stability after O and B doping. Phonon densities of states are shown as histograms for the most stable structures (1/1b, 2/1b, 3/1a, 4/1b, 2/2, 2/4, 2/6) in Fig. 4 together with results for pristine graphene and for 2 O atoms adsorbed at a monovacancy. Note that the reported frequencies refers to the $\Gamma$-point of the Brillouine zone of our $5\times 5$ supercell and due to backfolding therefore also include frequencies of other points of the standard graphene Brillouin zone. This phenomenon is well-known from carbon nanotubes new7 ; new8 ; new9 ; new10 ; new11 . We find all phonon frequencies to be positive and therefore conclude that all configurations addressed in Fig. 4 are stable. Pristine graphene shows a two peak phonon spectrum, where the gross shape is largely maintained under O and B adsorption. This finding is consistent with recent experimental results of Raman spectroscopy of B doped graphene new5 . However, there are characteristic differences evident in the high frequency range beyond 1500 cm-1. Except for configuration 1/1b, the modes in this energy range are suppressed and the high frequency peak shifts to the left. This fact cannot be a consequence of O adsorption, since an oxidized monovacancy without adsorbed B atoms rather comes along with an enhancement of the high energy modes, see Fig. 4(i). The softening of the G modes appears to be related to out-of-plane shifts of O atoms and substantial local distortions in the graphene plane induced by B adsorption. 4\. Conclusion We have performed first principles calculations to study the structural, electronic, and magnetic properties of O and B decorated graphene. We have identified the energetically favorable configurations for a variety of O and B concentrations and have demonstrated that there exist magnetic solutions with and without dangling bonds. Since vacancies in graphene are reactive and permit the incorporation of dopants, our calculations demonstrate that B doping of oxidized vacancies is a successful approach to induce extended $\pi$-band magnetism. By controlling the O and B concentrations, it is even possible to tune the magnetic state. A study of the $\Gamma$-point phonons has been performed to understand the structural stability of the decorated monovacancies. Acknowledgments We thank KAUST research computing for providing the computational resources used for this investigation. M. Upadhyay Kahaly thanks SABIC for financial support. R. 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Phonon dispersion of carbon nanotubes. Solid State Commun. 2002; 121: 471. * (37) Saito R, Takeya T, Kimura T, Dresselhaus G, Dresselhaus MS. Raman intensity of single-wall carbon nanotubes. Phys. Rev. B 1998; 57: 4145. * (38) Doorn SK, Zheng L, O’Connell MJ, Zhu Y, Huang S, Liu J. Raman spectroscopy and imaging of ultralong carbon nanotubes. Phys. Chem. 2005; 109: 3751. * (39) Kim YA, Fujisawa K, Muramatsu H, Hayashi T, Endo M, Fujimori T et al. Raman spectroscopy of boron-doped single-layer graphene. ACS Nano 2012; 6: 6293.	arxiv-papers	2013-11-13T19:08:54	2024-09-04T02:49:53.567057	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "T. P. Kaloni, M. Upadhyay Kahaly, R. Faccio, and U. Schwingenschl\\\"ogl", "submitter": "Thaneshwor Prashad Kaloni", "url": "https://arxiv.org/abs/1311.3253" }
1311.3262	# $\Lambda$CDM model with a scalar perturbation vs. preferred direction of the universe Xin Li [email protected] Hai-Nan Lin [email protected] Sai Wang [email protected] Zhe Chang [email protected] Institute of High Energy Physics, Theoretical Physics Center for Science Facilities, Chinese Academy of Sciences, 100049 Beijing, China ###### Abstract We present a scalar perturbation for the $\Lambda$CDM model, which breaks the isotropic symmetry of the universe. Based on the Union2 data, the least-$\chi^{2}$ fit of the scalar perturbed $\Lambda$CDM model shows that the universe has a preferred direction $(l,b)=(287^{\circ}\pm 25^{\circ},11^{\circ}\pm 22^{\circ})$. The magnitude of scalar perturbation is about $-2.3\times 10^{-5}$. The scalar perturbation for the $\Lambda$CDM model implies a peculiar velocity, which is perpendicular to the radial direction. We show that the maximum peculiar velocities at redshift $z=0.15$ and $z=0.015$ equal to $73\pm 28\rm km\cdot s^{-1}$ and $1099\pm 427\rm km\cdot s^{-1}$, respectively. They are compatible with the constraints on peculiar velocity given by Planck Collaboration. ###### pacs: 98.80.-k,98.80.Jk ## I Introduction The standard cosmological model, i.e., the $\Lambda$CDM model Sahni ; Padmanabhan has been well established. It is consistent with several precise astronomical observations that involve Wilkinson Microwave Anisotropy Probe (WMAP) Komatsu , Planck satellite Planck1 , Supernovae Cosmology Project Suzuki . One of the most important and basic assumptions of the $\Lambda$CDM model states that the universe is homogeneous and isotropic on large scales. However, such a principle faces challenges Perivolaropoulos . The Union2 SnIa data hint that the universe has a preferred direction $(l,b)=(309^{\circ},18^{\circ})$ in galactic coordinate system Antoniou . Toward this direction, the universe has the maximum expansion velocity. Astronomical observations Watkins found that the dipole moment of the peculiar velocity field on the direction $(l,b)=(287^{\circ}\pm 9^{\circ},8^{\circ}\pm 6^{\circ})$ in the scale of $50h^{-1}\rm Mpc$ has a magnitude $407\pm 81\rm km\cdot s^{-1}$. This peculiar velocity is much larger than the value $110\rm km\cdot s^{-1}$ given by WMAP5 WMAP5 . The recent released data of Planck Collaboration show deviations from isotropy with a level of significance ($\sim 3\sigma$) Planck2 . Planck Collaboration confirms asymmetry of the power spectrums between two preferred opposite hemispheres. These facts hint that the universe may have a preferred direction. Many models have been proposed to resolve the asymmetric anomaly of the astronomical observations. An incomplete and succinct list includes: an imperfect fluid dark energy Koivisto1 , local void scenario Alexander ; Garcia , noncommutative spacetime effect Akofor , anisotropic curvature in cosmology Koivisto2 , and Finsler gravity scenario Chang . In this paper, we present a scalar perturbation for the flat Friedmann- Robertson-Walker (FRW) metric Weinberg . Based on the Union2 data, the least-$\chi^{2}$ fit of the scalar perturbed $\Lambda$CDM model shows that the universe has a preferred direction. In the scalar perturbed $\Lambda$CDM model, the universe could be treated as a perfect fluid approximately. In comoving frame, however, the fluid has a small velocity $v$. It could be regarded as the peculiar velocity of the universe. The data of Planck Collaboration gives severe constraints on the peculiar velocity Planck3 . For the bulk flow of Local Group, it should be less than $254\rm km\cdot s^{-1}$. For bulk flow of galaxy clusters at $z=0.15$, it should be less than $800\rm km\cdot s^{-1}$. The paper is organized as follows. In Sec. II, we present a scalar perturbation for the FRW metric. Explicit relation between luminosity and redshift is obtained. In Sec. III, we show a least-$\chi^{2}$ fit of the scalar perturbed $\Lambda$CDM model to the Union2 SnIa data. The preferred direction is found $(l,b)=(287^{\circ}\pm 25^{\circ},11^{\circ}\pm 22^{\circ})$. The magnitude of the scalar perturbation is at the scale of $10^{-5}$. This perturbation implies a peculiar velocity with value $73\pm 28\rm km\cdot s^{-1}$ at $z=0.15$, and $1099\pm 427\rm km\cdot s^{-1}$ at $z=0.015$. The conclusions and remarks are given in Sec. IV. ## II Scalar perturbation for FRW metric The FRW metric describes the homogeneous and isotropic universe. In order to describe the deviation from isotropy, we try to add a scalar perturbation for the FRW metric. The scalar perturbed FRW metric is of the form $ds^{2}=(1-2\phi(\vec{x}))dt^{2}-a^{2}(t)(1+2\phi(\vec{x}))\delta_{ij}dx^{i}dx^{j}.$ (1) It should be noticed that the scalar perturbation field $\phi(\vec{x})$ is time-independent. And the scalar perturbation can be interpreted as a sort of space-dependent spatial curvature. By setting the scale factor $a(t)=1$, one can find that the spatial Ricci tensor of metric (1) is of the form $R_{ij}=-\delta_{ij}\phi_{,k,k}.$ (2) The nonvanishing components of Einstein tensor for the metric (1) are given as $\displaystyle G^{0}_{0}$ $\displaystyle=$ $\displaystyle 3(1+2\phi)H^{2}-2a^{-2}\phi_{,i,i}~{},$ (3) $\displaystyle G^{i}_{j}$ $\displaystyle=$ $\displaystyle\delta_{ij}(1+2\phi)\left(H^{2}+2\frac{\ddot{a}}{a}\right)~{},$ (4) $\displaystyle G^{0}_{j}$ $\displaystyle=$ $\displaystyle-2H\phi_{,j}~{},$ (5) where the commas denote the derivatives with respect to $x^{i}$, the dot denotes the derivatives with respect to cosmic time $t$ and $H\equiv\frac{\dot{a}}{a}$. The scalar perturbation breaks homogeneity and isotropy of the universe. Since $\phi$ is a perturbation, the cosmic inventory could be treated as a perfect fluid approximately. In comoving frame, however, the fluid has a perturbed velocity $v$. The energy-momentum tensor is given by $T^{\mu\nu}=(\rho+p)U^{\mu}U^{\nu}-pg^{\mu\nu},$ (6) where $\rho$ and $p$ are the energy density and pressure density of the fluid, respectively. Here, we set $U^{\mu}$ as $U^{0}=1,\frac{U^{i}}{U^{0}}\equiv v^{i}$, to first order in $v$. In this paper, we just investigate low redshift region of the universe, where the universe is dominated by matter and dark energy. Thus, the nonvanishing components of energy-momentum tensor are given as $\displaystyle T^{0}_{0}$ $\displaystyle=$ $\displaystyle\rho_{m}+\rho_{de}~{},$ (7) $\displaystyle T^{0}_{i}$ $\displaystyle=$ $\displaystyle\rho_{m}v_{i}~{},$ (8) $\displaystyle T^{i}_{j}$ $\displaystyle=$ $\displaystyle\delta^{i}_{j}\rho_{de}~{},$ (9) where $\rho_{m}$ and $\rho_{de}$ denote the energy density of matter and dark energy, respectively. Then, the Einstein field equation $G^{\mu}_{\nu}=8\pi GT^{\mu}_{\nu}$ gives three independent equations $\displaystyle(1+2\phi)H^{2}-\frac{2a^{-2}}{3}\phi_{,i,i}$ $\displaystyle=$ $\displaystyle\frac{8\pi G}{3}(\rho_{m}+\rho_{de})~{},$ (10) $\displaystyle(1+2\phi)(H^{2}+2\frac{\ddot{a}}{a})$ $\displaystyle=$ $\displaystyle 8\pi G\rho_{de}~{},$ (11) $\displaystyle H\phi_{,j}$ $\displaystyle=$ $\displaystyle-4\pi G\rho_{m}v_{j}~{}.$ (12) The energy-momentum conservation equation reads $\frac{\partial T^{\mu}_{\nu}}{\partial x^{\mu}}+\Gamma^{\mu}_{\alpha\mu}T^{\alpha}_{\nu}-\Gamma^{\alpha}_{\nu\mu}T^{\mu}_{\alpha}=0,$ (13) where $\Gamma^{\mu}_{\alpha\mu}$ is the Christoffel symbol. Then, following the theory of general relativity, we obtain the specific form of energy- momentum conservation equation for matter and dark energy in the perturbed FRW universe (1). It is as follows: $\displaystyle\frac{\partial\rho_{m}}{\partial t}+3H\rho_{m}+\frac{\partial\rho_{m}v^{i}}{\partial x^{i}}$ $\displaystyle=$ $\displaystyle 0,$ (14) $\displaystyle\frac{\partial\rho_{m}v_{i}}{\partial t}+3H\rho_{m}v_{i}-\phi_{,i}\rho_{m}$ $\displaystyle=$ $\displaystyle 0,$ (15) $\displaystyle\frac{\partial\rho_{de}}{\partial t}$ $\displaystyle=$ $\displaystyle 0,$ (16) $\displaystyle\frac{\partial\rho_{de}}{\partial x^{i}}$ $\displaystyle=$ $\displaystyle 0.$ (17) The equations (16) and (17) show that the energy density of dark energy remaining constant in our model. By making use of the field equation (12), we find from equation (14) that $\frac{\partial(\rho_{m}a^{3})}{\partial t}=-aH\frac{\phi_{,i,i}}{4\pi G}.$ (19) The solution of equation (19) reads $\rho_{m}a^{3}=-\frac{\phi_{,i,i}}{4\pi G}(a-1)+\rho_{m0},$ (20) where $\rho_{m0}$ denotes the energy density of matter at present. We have already used the initial condition that the present energy density of matter is constant to deduce continuity equation (20). The light propagation satisfies $ds=0$, which gives $\frac{dt}{a(t)}=(1+2\phi(\vec{x}))\delta_{ij}dx^{i}dx^{j}~{}.$ (21) The right-hand side of the equation (21) is time-independent. During a very short time, the location of a galaxy is unchanged. Then, we get $\int^{t_{0}}_{t_{1}}\frac{dt}{a(t)}=\int^{t_{0}+\delta t_{0}}_{t_{1}+\delta t_{1}}\frac{dt}{a(t)}~{}.$ (22) Thus, the redshift $z$ of galaxy satisfies $1+z=\frac{1}{a},$ (23) where we have set the scale factor $a(t)$ to be 1 at present. A particular form of $\phi(\vec{x})$ is necessary for deriving relation between luminosity distance and redshift. The form of $\phi(\vec{x})$ is determined by the perturbed energy-momentum tensor. However, the information of the perturbed energy-momentum tensor is unknown. On the contrary, we choose a specific form of $\phi$ to determine the perturbed energy-momentum tensor. It is given as $\phi=A\cos\theta~{},$ (24) where $A$ is a dimensionless parameter and $\theta$ is the angle between $\vec{r}$ and $z$-axis. By making use of (24), we reduce the equation (10) to $\frac{da}{dt}=H_{0}a(1-A\cos\theta)\sqrt{\Omega_{m0}a^{-3}+1-\Omega_{m0}-\frac{4A\cos\theta}{3r^{2}H_{0}^{2}}a^{-3}},$ (25) where $H_{0}$ is Hubble constant and $\Omega_{m0}\equiv 8\pi G\rho_{m0}/(3H_{0}^{2})$ is the energy density parameter for matter at present. Combining the equations (21), (23), (24), (25) and (20), and using the definition of luminosity distance Weinberg , we obtain the relation between luminosity distance and redshift $H_{0}d_{L}=(1+z)\int_{0}^{z}\frac{(1-A\cos\theta)dx}{\sqrt{\Omega_{m0}(1+x)^{3}+1-\Omega_{m0}-\frac{4A\cos\theta(1+x)^{5}}{3H_{0}^{2}d^{2}_{L0}}}}~{},$ (26) where $d_{L0}\equiv(1+z)\int_{0}^{z}\frac{dx}{H_{0}\sqrt{\Omega_{m0}(1+x)^{3}+1-\Omega_{m0}}}$. ## III Numerical results Our numerical studies are based on the Union2 SnIa data Amanullah . Our goal is to find whether the universe has a preferred direction or not. We perform a least-$\chi^{2}$ fit to the Union2 SnIa data $\chi^{2}\equiv\sum^{557}_{i=1}\frac{(\mu_{th}-\mu_{obs})^{2}}{\sigma_{\mu}^{2}},$ (27) where $\mu_{th}$ is theoretical distance modulus given by $\mu_{th}=5\log_{10}\frac{d_{L}}{\rm Mpc}+25~{}.$ (28) $\mu_{obs}$ and $\sigma_{\mu}$, given by the Union2 SnIa data, denote the observed values of the distance modulus and the measurement errors, respectively. The least-$\chi^{2}$ fit of the $\Lambda$CDM model gives $\Omega_{m0}=0.27\pm 0.02$ and $H_{0}=70.00\pm 0.35$ $\rm km\cdot s^{-1}\cdot Mpc^{-1}$. Before using our model to fit the Union2 SnIa data, we fix the values of $\Omega_{m}$ and $H_{0}$ as their mean values. Such an approach is valid for the scalar perturbed model, since it is just a perturbation for the $\Lambda$CDM model. Then, the least-$\chi^{2}$ fit of the formula (26) gives $A=(-2.34\pm 0.91)\times 10^{-5}$ and $(l,b)=(287^{\circ}\pm 25^{\circ},11^{\circ}\pm 22^{\circ})$. The preferred direction is plotted as point G of Fig.1. The preferred directions given by other models are plotted in Fig.1 for contrast. Kogut et al. Kogut1993 got $(l,b)=(276^{\circ}\pm 3^{\circ},30^{\circ}\pm 3^{\circ})$ is shown as point A, Antoniou et al. Antoniou got $(l,b)=({309^{\circ}}^{+23^{\circ}}_{-3^{\circ}},{18^{\circ}}^{+11^{\circ}}_{-10^{\circ}})$ is shown as point B, Cai and Tuo Cai and Tuo2012 got $(l,b)=({314^{\circ}}^{+20^{\circ}}_{-13^{\circ}},{28^{\circ}}^{+11^{\circ}}_{-33^{\circ}})$ is shown as point C, Kalus et al. Kalus:2013zu got $(l,b)=({325^{\circ}},{-19^{\circ}})$ is shown as point D, Cai et al. Cai:2013lja got $(l,b)=({306^{\circ}},{-13^{\circ}})$ is shown as point E, Chang et al. Chang got $(l,b)=(304^{\circ}\pm 43^{\circ},-27^{\circ}\pm 13^{\circ})$ is shown as point F. Within a level of significance ($1\sigma$), it is shown in Fig.2 that our results are consistent with the one of Kogut et al. Kogut1993 , Antoniou et al. Antoniou and Cai et al. Cai and Tuo2012 . Figure 1: The direction of preferred axis in galactic coordinate. The point G denotes our result, namely, $(l,b)=(287^{\circ}\pm 25^{\circ},11^{\circ}\pm 22^{\circ})$, which is obtained by fixing the parameters $\Omega_{m}=0.27$ and $H_{0}=70.00$ and doing the least-$\chi^{2}$ to the Union2 data for formula (26). The results for preferred direction in other models are presented for contrast. Point A denotes the result of Kogut et al. Kogut1993 , point B denotes the result of Antoniou et al. Antoniou , point C denotes the result of Cai and Tuo Cai and Tuo2012 , point D denotes the result of Kalus et al. Kalus:2013zu , point E denotes the result of Cai et al. Cai:2013lja , point F denotes the result of Chang et al. Chang . Figure 2: Contours figure for the preferred direction. The contours enclose 68% and 95% confidence regions of the scalar perturbed $\Lambda$CDM model. The scalar perturbation not only breaks the isotropy symmetry of the universe but also gives a peculiar velocity for the matter. By setting $\phi$ to be the form of (24), we find from (12) that $v\equiv\sqrt{\|v_{i}v^{i}\|}=a^{4}\frac{2H\|A\|\sin\theta}{3rH_{0}^{2}\Omega_{m0}}=\frac{2H\|A\|\sin\theta}{3H_{0}^{2}d_{L}\Omega_{m0}}(1+z)^{-3},$ (29) where we have used the relation $d_{L}=(1+z)r$ to obtain the second equation. Substituting the value of $H_{0}d_{L}$ and $H$ (given by the $\Lambda$CDM model) into formula (29), and setting $\sin\theta=1$ (the velocity $v$ is perpendicular to the preferred direction $(l,b)=(287^{\circ}\pm 25^{\circ},11^{\circ}\pm 22^{\circ})$), we could obtain the value of peculiar velocity $v$ for a given redshift. At $z=0.15$, we get $v\|_{z=0.15}\simeq 73\pm 28\rm km\cdot s^{-1}$. This peculiar velocity is compatible with the result of Planck Collaboration Planck3 . it gives a upper limit $800\rm km\cdot s^{-1}$ for peculiar velocity at $z=0.15$. It should be noticed that the peculiar velocity $v$ grows with time. We should check that if the peculiar velocity at the lowest redshift in the Union2 SnIa data is compatible with the result of Planck Collaboration Planck3 . The lowest redshift in the Union2 SnIa data is 0.015. At $z=0.015$, we get $v\|_{z=0.015}\simeq 1099\pm 427\rm km\cdot s^{-1}$. Planck Collaboration Planck3 gives an upper limit on the bulk flow for Local Group, which equals to $254\rm km\cdot s^{-1}$. Though, our result for peculiar velocity at $z=0.015$ larger than $254\rm km\cdot s^{-1}$, with a level of significance ($1\sigma$), it is still compatible with the upper limit $800\rm km\cdot s^{-1}$ given by Planck Collaboration. And our result for peculiar velocity at $z=0.015$ represents the upper limit of peculiar velocity in the scalar perturbed $\Lambda$CDM model. ## IV Conclusions and remarks We presented a scalar perturbation for the $\Lambda$CDM model that breaks the isotropic symmetry of the universe. Setting the scalar perturbation of the form $\phi=A\cos\theta$, we obtained a modified relation (26) between luminosity distance and redshift. The least-$\chi^{2}$ fit to the Union2 SnIa data showed that the universe has a preferred direction $(l,b)=(287^{\circ}\pm 25^{\circ},11^{\circ}\pm 22^{\circ})$, which is close to the results of Kogut et al. and Antoniou et al. and Cai et al. Kogut1993 ; Antoniou ; Cai and Tuo2012 . Also, the least-$\chi^{2}$ fit to the Union2 SnIa data showed that the magnitude of scalar perturbation $A$ equals to $(-2.34\pm 0.91)\times 10^{-5}$. The scalar perturbation has the same magnitude with the level of CMB anisotropy. The CMB anisotropy is a possible reason for the preferred direction of the universe. The peculiar velocity was obtained directly from the Einstein equation (12). The numerical calculations showed that the peculiar $v\|_{z=0.15}\simeq 73\pm 28\rm km\cdot s^{-1}$ and $v\|_{z=0.015}\simeq 1099\pm 427\rm km\cdot s^{-1}$. They are compatible with the results of Planck Collaboration Planck3 . It should be noticed that the peculiar velocity we obtained is perpendicular to the radial direction. Bianchi cosmology Rosquist has been studied for many years. It admits a set of anisotropic metrics such as Kasner metric Misner . The three dimensional space of Bianchi cosmology admits a set of Killing vectors $\xi_{i}^{(a)}$ which obey the following property $\left(\frac{\partial\xi_{i}^{(c)}}{\partial x^{k}}-\frac{\partial\xi_{k}^{(c)}}{\partial x^{i}}\right)\xi_{(a)}^{i}\xi_{(b)}^{k}=C^{c}_{ab},$ (30) where $C^{c}_{ab}$ is the structure constant of the symmetry group of the space. The scalar perturbation field $\phi(\vec{x})$ completely destroys the rotational symmetry of cosmic space. It means that no Killing vectors corresponding to the symmetry group of three dimensional cosmic space. Thus, there is no obvious relation between the Bianchi cosmology and our model. ###### Acknowledgements. We would like to thank Y. G. Jiang for useful discussions. Project 11375203 and 11305181 supported by NSFC. ## References * (1) V. Sahni, Class. Quant. Grav. 19, 3435 (2002). * (2) T. Padmanabhan, Phys. Rept. 380, 235 (2003). * (3) E. Komatsu, et al. (WMAP Collaboration), Astrophys. J. Suppl. 192, 18 (2011). * (4) Planck Collaboration, arXiv:1303.5062. * (5) N. Suzuki, et al., Astrophys. J. 746, 85 (2012). * (6) L. Perivolaropoulos, arXiv:1104.0539. * (7) I. Antoniou and L. Perivolaropoulos, JCAP 1012, 012 (2012). * (8) R. Watkins, H. A. Feldman and M. J. Hudson, Mon. Not. Roy. Astron. Soc. 392, 743 (2009). * (9) G. Hinshaw, et al., Astrophys. J. Suppl. 180, 225 (2009). * (10) Planck Collaboration, arXiv:1303.5083. * (11) T. Koivisto and D. F. Mota, Phys. Rev. D 73, 083502 (2006). * (12) S. Alexander, T. Biswas, A. Notari and D. Vaid, JCAP 0909, 025 (2009). * (13) J. Garcia-Bellido and T. Haugboelle, JCAP 0804, 003 (2008). * (14) E. Akofor, et al., JHEP 0805, 092 (2008). * (15) T. S. Koivisto, D. F. Mota, M. Quartin and T. G. Zlosnik, Phys. Rev. D 83, 023509 (2011). * (16) Z. Chang, M.-H. Li, X. Li and S. Wang, Eur. Phys. J. C 73, 2459 (2013). * (17) S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley & Sons, New York, 1972. * (18) Planck Collaboration, arXiv:1303.5090. * (19) S. Perlmutter, et al., Astrophys. J. 517, 565 (1999); A. G. Riess, et al., Astron. J. 116, 1009 (1998); Astron. J. 117, 707 (1999). * (20) R. Amanullah, et al., Astrophys. J. 716, 712 (2010). * (21) A. Kogut, et al., Astrophys. J. 419, 1 (1993). * (22) R.-G. Cai and Z.-L. Tuo, J. Cosmol. Astropart. Phys. 1202, 004 (2012). * (23) B. Kalus, et al., Astron. Astrophys. 553, A56 (2013). * (24) R. G. Cai, Y. Z. Ma, B. Tang and Z. L. Tuo, Phys. Rev. D 87, 123522 (2013). * (25) K. Rosquist and R. T. Jantzen, Phys. Rept. 166, 89 (1988). * (26) C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, W. H. Freeman and Company, San Francisco, 1973.	arxiv-papers	2013-11-12T04:04:49	2024-09-04T02:49:53.574344	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Xin Li, Hai-Nan Lin, Sai Wang and Zhe Chang", "submitter": "Xin Li", "url": "https://arxiv.org/abs/1311.3262" }
1311.3291	# Left-orderability and cyclic branched coverings Ying Hu Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803 [email protected] ###### Abstract. We provide an alternative proof of a sufficient condition for the fundamental group of the $n^{th}$ cyclic branched cover of $S^{3}$ along a prime knot $K$ to be left-orderable, which is originally due to Boyer-Gordon-Watson. As an application of this sufficient condition, we show that for any $(p,q)$ two- bridge knot, with $p\equiv 3\text{ mod }4$, there are only finitely many cyclic branched covers whose fundamental groups are not left-orderable. This answers a question posed by Da̧bkowski, Przytycki and Togha. ## 1\. Introduction ### 1.1. Background and results A group $G$ is called left-orderable if there exists a strict total ordering $<$ on the set of group elements, such that given any two elements $a$ and $b$ in $G$, if $a<b$ then $ca<cb$ for any $c\in G$. It is known that any connected, compact, orientable $3$-manifold with a positive first Betti number has a left-orderable fundamental group [BRW05, Theorem 1.1][HS85]. In contrast, for a rational homology sphere, the left- orderability of its fundamental group is a nontrivial property, which is closely related to the co-oriented taut foliations on the manifold [CD03]. Moreover, Boyer, Gordon and Watson conjectured that an irreducible rational homology $3$-sphere $M$ is an $L$-space [OS05a] if and only if its fundamental group $\pi_{1}(M)$ is not left-orderable [BGW13]. Let $X_{K}$ be the complementary space obtained by removing an open tubular neighborhood of the knot $K$ from the three sphere $S^{3}$ and $X_{K}^{(n)}$ be the $n$th cyclic branched cover of $S^{3}$ branched over the knot $K$. The first Betti number $b_{1}(X_{K}^{(n)})$ equals zero if no root of the Alexander polynomial $\Delta_{K}(t)$ is an $n^{th}$ root of unity. Hence, most of the cyclic branched covers along a knot are rational homology spheres. In particular, this is the case if $n$ is a prime power. For this class of rational homology spheres, the L-space conjecture [BGW13] has been verified in the following cases, where they are all L-spaces and have non-left-orderable fundamental groups: * • The twofold branched cover of any non-split alternating link [BGW13, Gre, Ito13, OS05b]. * • The $n^{th}$ cyclic branched cover of a $(p,q)$ two-bridge knot with $p/q=2m+\frac{1}{2k}$, $mk>0$ and $n$ arbitrary [DPT05, Pet]. * • The $3^{rd}$ and $4^{th}$ cyclic branched cover of a $(p,q)$ two-bridge knot with $p/q=n_{1}+\frac{1}{1+\frac{1}{n_{2}}}$ and $n_{1},n_{2}$ are positive odd integers (i.e. $p/q=2m+\frac{1}{2k}$, $mk<0$) [DPT05, GL, Pet, Ter]. The motivation of this paper is a question posed in [DPT05]: Given a two- bridge knot $K$, is $\pi_{1}(X_{K}^{(n)})$ always non-left-orderable whenever $b_{1}(X_{K}^{(n)})=0$ ? We answer this question negatively. In fact, we prove that for $(p,q)$ two-bridge knots with $p\equiv 3$ mod $4$, there are only finitely many cyclic branched covers that have non-left-orderable fundamental groups. At the end, we will present the knot $5_{2}$ as an example and show that the fundamental group $\pi_{1}(X_{5_{2}}^{(n)})$ is left-orderable if $n\geq 9$. Shortly after this article posted, Tran computed a lower bound (depending on the knot) on the order $n$ so that the $n^{th}$ cyclic branched cover has a non-left-orderable fundamental group for a large class of two- bridge knots [Tra]. A similar question for hyperbolic knots was also posed in [DPT05] and was first answered in [CLW13, Proposition 23]. They showed that the twofold branched cover of $S^{3}$ along the Conway knot, which is a non-alternating hyperbolic knot listed as 11n34 in the standard knot tables, has a left- orderable fundamental group and so do all even order cyclic branched covers. ### 1.2. Plan of the paper. Section 2 is devoted to proving Lemma 2.1, which is essential in our proof of Theorem 3.1. ###### Lemma (Lemma 2.1). Given a knot $K$ in $S^{3}$, denote by $Z$ a meridional element in the knot group $\pi_{1}(X_{K})$. Suppose that there exists a group homomorphism $\rho$ from $\pi_{1}(X_{K})$ to a group $G$ and $\rho(Z^{n})$ is in the center of $G$. Then $\rho$ induces a group homomorphism from $\pi_{1}(X_{K}^{(n)})$ to $G$. In particular, if $\rho$ is non-abelian, then the induced homomorphism is nontrivial. We finish the proof of Theorem 3.1 in Section 3. ###### Theorem (Theorem 3.1). Given any prime knot $K$ in $S^{3}$, denote by $Z$ a meridional element of $\pi_{1}(X_{K})$. If there exists a non-abelian representation $\pi_{1}(X_{K})$ to $SL(2,\mathbb{R})$ such that $Z^{n}$ is sent to $\pm I$ then the fundamental group $\pi_{1}(X_{K}^{(n)})$ is left-orderable. This result was first observed by Boyer-Gordon-Watson in [BGW13], where they showed the following: ###### Theorem (Theorem 6 in [BGW13]). Let $K$ be a prime knot in the $3$-sphere and suppose that the fundamental group of its twofold branched cyclic cover is not left-orderable. If $\rho:\pi_{1}(S^{3}\setminus K)\rightarrow Homeo_{+}(S^{1})$ is a homomorphism such that $\rho(\mu^{2})=1$ for some meridional class $\mu$ in $\pi_{1}(S^{3}\setminus K)$, then the image of $\rho$ is either trivial or isomorphic to $\mathbb{Z}_{2}$. Here we make two remarks in comparison to Theorem 3.1 with [BGW13, Theorem 6]. * • The proof of [BGW13, Theorem 6] naturally extends to the $n^{th}$ cyclic branched cover for arbitrary $n$. Since $PSL(2,\mathbb{R})$ is a subgroup of $Homeo_{+}(S^{1})$, the group of orientation preserving homeomorphisms of $S^{1}$, Theorem 3.1 is contained in [BGW13, Theorem 6] in this sense. * • On the other hand, if we replace the central extension $0\longrightarrow\mathbb{Z}\longrightarrow\widetilde{SL(2,\mathbb{R})}\longrightarrow SL(2,\mathbb{R})\longrightarrow 1$ that we used in the proof of Theorem 3.1 by the extension below [GHY01] $0\longrightarrow\mathbb{Z}\longrightarrow\widetilde{Homeo_{+}(S^{1})}\longrightarrow Homeo_{+}(S^{1})\longrightarrow 1$ the same statement with [BGW13, Theorem 6] can be achieved. Finally, in Section 4, we prove our main result in this paper. ###### Theorem (Theorem 4.3). Given a $(p,q)$ two-bridge knot $K$, with $p\equiv 3\text{ mod }4$, there are only finitely many cyclic branched covers, whose fundamental groups are not left-orderable. ## 2\. The fundamental groups of cyclic branched covers Given a Seifert surface $F$, one can present the knot group $\pi_{1}(X_{K})$ as an HNN extension of $\pi_{1}(S^{3}\setminus F)$ over the surface group $\pi_{1}(F)$, (the usual definition of the HNN extension requires $F$ to be incompressible, but we do not need it here). We then apply the Reidemeister- Schreier Method to the presentation of $\pi_{1}(X_{K})$ and obtain a presentation of $\pi_{1}(X_{K}^{(n)})$, from where Lemma 2.1 follows. More precisely, let $F$ be a Seifert surface of an oriented knot $K$. It has a regular neighborhood that is homeomorphic to $F\times[-1,1]$, where the positive direction is chosen so that the induced orientation on the boundary $\partial F$ is the same as the chosen orientation on the knot $K$. $K$$F_{-}$$F$$F_{+}$$P_{+}$$P_{-}$$C$$Z$ Figure 1. A cross-sectional view of a collar neighborhood of $F$ in the knot complement $X_{K}$, where $F_{\pm}$ represent $F\times\pm 1$, respectively. In addition, the point $P_{+}$ (resp. $P_{-}$) is the intersection point of the meridian $Z$ and $F_{+}$ (resp. $F_{-}$). Suppose that the free group $\pi_{1}(F_{-},P_{-})$ is generated by $\\{a^{-}_{i}\\}_{i=1,\dots,2g}$ and $\pi_{1}(F_{+},P_{+})$ is generated by $\\{a^{+}_{i}\\}_{i=1,\dots,2g}$, where $g$ is the genus of the Seifert surface $F$. We denote by $\alpha_{i}^{-}$ the image of $a_{i}^{-}$ under the inclusion map $\pi_{1}(F_{-},P_{-})\rightarrow\pi_{1}(S^{3}-F,P_{-})$ and denote by $\alpha_{i}^{+}$ the image of $a_{i}^{+}$ in $\pi_{1}(S^{3}-F,P_{-})$ under the composition map $\pi_{1}(F_{+},P_{+})\rightarrow\pi_{1}(S^{3}-F,P_{+})\rightarrow\pi_{1}(S^{3}-F,P_{-}),$ where the second map from $\pi_{1}(S^{3}-F,P_{+})$ to $\pi_{1}(S^{3}-F,P_{-})$ is the isomorphism induced by the arc $C$ connecting $P_{-}$ to $P_{+}$ as depicted in Figure $1$. By the Van Kampen Theorem, we have (2.1) $\displaystyle\pi_{1}(X_{K},P_{-})=$ $\displaystyle\quad\pi_{1}(S^{3}-F,P_{-})\ast<Z>/\ll Z\alpha_{i}^{+}Z^{-1}=\alpha_{i}^{-},i=1,\dots,2g\gg.$ If the complement of the Seifert surface $F$ in $S^{3}$ is also a handlebody, which is always the case when $F$ is constructed through Seifert’s algorithm, then the group $\pi_{1}(S^{3}-F,P_{-})$ is also free and we assume that $\pi_{1}(S^{3}-F,P_{-})=<x_{1},\dots,x_{2g}>.$ In this case, from (2.1), we obtain Lin’s presentation for the knot group $\pi_{1}(X_{K},P_{-})$ [Lin01, Lemma 2.1] as follows: (2.2) $\pi_{1}(X,P_{-})=<x_{1},x_{2},\dots,x_{2g-1},x_{2g},Z:Z\alpha_{i}^{+}Z^{-1}=\alpha_{i}^{-},i=1,\dots,2g>,$ where $\alpha_{i}^{\pm}$ are words in $x_{i}$ as described above. Let $\widetilde{X_{K}}^{(n)}$ be the $n^{th}$ cyclic cover of the knot complement $X_{K}$. Its fundamental group $\pi_{1}(\widetilde{X_{K}}^{(n)})\cong\text{Ker}(\pi_{1}(X_{K})\rightarrow\mathbb{Z}_{n})$ is an index $n$ subgroup of the knot group $\pi_{1}(X_{K})$. Choose $\\{Z^{i}\\}_{i=0,\dots,n-1}$ to be the representative from each coset. By applying the Reidemeister-Schreier Method [LS01] to the presentation (2.2), we obtain a presentation of the group $\pi_{1}(\widetilde{X_{K}}^{(n)})$ with * • generators: $Z^{n}$ and $Z^{k}x_{1}Z^{-k}$, … , $Z^{k}x_{2g}Z^{-k}$ for $k=0,\cdots,n-1$; * • relators: (2.3) $Z^{k+1}\alpha_{i}^{+}Z^{-(k+1)}=Z^{k}\alpha_{i}^{-}Z^{-k},\text{ for }k=0,\cdots,n-2\text{ and }i=1,\dots,2g,$ (2.4) $Z^{n}\cdot\alpha_{i}^{+}\cdot Z^{-n}=Z^{n-1}\alpha_{i}^{-}Z^{-(n-1)},\text{ for }i=1,\dots,2g.$ In the presentation above, $Z^{k}x_{i}Z^{-k}$ and $Z^{n}$ should be viewed as abstract symbols rather than products of $Z$ and $x_{i}$. Thus, words $Z^{k}\alpha_{i}^{+}Z^{-k}$ as in (2.3) are products of the generators $Z^{k}x_{i}Z^{-k}$ and the word $Z^{n}\cdot\alpha_{i}^{+}\cdot Z^{-n}$ in (2.4) is the product of $Z^{\pm n}$ and $x_{i}$. The notation is chosen to emphasize the fact that the isomorphism between the presented group and the subgroup $\text{Ker}(\pi_{1}(X_{K})\rightarrow\mathbb{Z}_{n})$ is given by sending the abstract symbol $Z^{k}x_{i}Z^{-k}$ in the presentation to the element $Z^{k}x_{i}Z^{-k}$ of the knot group $\pi_{1}(X_{K})$ for $k=0,\dots,n-1$ and $i=1,\dots,2g$. Intuitively, this presentation can be understood in the following way. The $n^{th}$ cyclic cover $\widetilde{X_{K}^{(n)}}$ can be constructed by gluing $n$ copies of $S^{3}-F\times(-1,1)$ together. We denote each copy by $Y_{k}$. Let $F_{k}$ be the Seifert surface associated with $Y_{k}$ and $F^{\pm}_{k}$ be $F_{k}\times\pm 1$ on $\partial Y_{k}$ for $k=0,\dots,n-1$. Then $Z^{k}x_{i}Z^{-k}$ are generator loops in $Y_{k}$ and each relation $Z^{k+1}\alpha_{i}^{+}Z^{-(k+1)}=Z^{k}\alpha_{i}^{-}Z^{-k}$ in (2.3) is due to the isomorphism between $\pi_{1}(F_{k}^{-})$ and $\pi_{1}(F_{k+1}^{+})$. In addition, the relation (2.4) is from the identification between $F_{0}^{+}$ and $F_{n-1}^{-}$. Now let’s look at the fundamental group of the $n^{th}$ cyclic branched cover $X_{K}^{(n)}$. From the construction of $X_{K}^{(n)}$, we have the following isomorphism $\pi_{1}(X_{K}^{(n)})\cong\text{Ker}(\pi_{1}(X_{K})\rightarrow\mathbb{Z}_{n})/\ll Z^{n}\gg.$ Therefore, the group $\pi_{1}(X_{K}^{(n)})$ inherits the presentation with * • generators: $Z^{k}x_{1}Z^{-k}$, … , $Z^{k}x_{2g}Z^{-k}$ for $k=0,\cdots,n-1$; * • relators: (2.5) $Z^{k+1}\alpha_{i}^{+}Z^{-(k+1)}=Z^{k}\alpha_{i}^{-}Z^{-k},\text{ for }k=0,\cdots,n-2\text{ and }i=1,\dots,2g,$ (2.6) $\alpha_{i}^{+}=Z^{n-1}\alpha_{i}^{-}Z^{-(n-1)},\text{ for }i=1,\dots,2g.$ ###### Lemma 2.1. Given a knot $K$ in $S^{3}$, denote by $Z$ a meridional element in the knot group $\pi_{1}(X_{K})$. Suppose that there exists a group homomorphism $\rho$ from $\pi_{1}(X_{K})$ to a group $G$ and $\rho(Z^{n})$ is in the center of $G$. Then $\rho$ induces a group homomorphism from $\pi_{1}(X_{K}^{(n)})$ to $G$. In particular, if $\rho$ is non-abelian, then the induced homomorphism is nontrivial. ###### Proof. Let $\rho\|_{ker}$ be the restriction of $\rho$ to the subgroup Ker$(\pi_{1}(X_{K})\rightarrow\mathbb{Z}_{n})$. We are going to show that the assignment $Z^{k}x_{i}Z^{-k}\mapsto\rho\|_{ker}(Z^{k}x_{i}Z^{-k})\text{ for }i=1,\dots,2g\text{ and }k=0,\dots,n-1$ also defines a homomorphism from $\pi_{1}(X_{K}^{(n)})$ to $G$. First of all, the relations in (2.3) which are the same as the relations in (2.5) automatically hold. It follows from (2.4) that $\rho\|_{ker}(Z^{n})\cdot\rho\|_{ker}(\alpha_{i}^{+})\cdot\rho\|_{ker}(Z^{-n})=\rho\|_{ker}(Z^{n-1}\alpha_{i}^{-}Z^{-(n-1)}).$ Since by assumption $\rho\|_{ker}(Z^{n})=\rho(Z^{n})$ is in the center of $G$, we have $\rho\|_{ker}(\alpha_{i}^{+})=\rho\|_{ker}(Z^{n})\cdot\rho\|_{ker}(\alpha_{i}^{+})\cdot\rho\|_{ker}(Z^{-n})=\rho\|_{ker}(Z^{n-1}\alpha_{i}^{-}Z^{-(n-1)}).$ That is, the relations in (2.6) hold as well. In addition, if $\rho$ is a non-abelian homomorphism, since the commutator subgroup $[\pi_{1}(X_{K}),\pi_{1}(X_{K})]$ is the normal subgroup generated by $\\{x_{1},\dots,x_{2g}\\}$, we have that $\rho(x_{i})$ is not equal to the identity in $G$ for some $i$. Therefore, the induced homomorphism from $\pi_{1}(X_{K}^{(n)})$ to $G$ is nontrivial. ∎ ## 3\. The left-orderability of the fundamental group $\pi_{1}(X_{K}^{(n)})$ We finish the proof of Theorem 3.1 in this section. ###### Theorem 3.1. Given any prime knot $K$ in $S^{3}$, denote by $Z$ a meridional element of $\pi_{1}(X_{K})$. If there exists a non-abelian representation $\pi_{1}(X_{K})$ to $SL(2,\mathbb{R})$ such that $Z^{n}$ is sent to $\pm I$ then the fundamental group $\pi_{1}(X_{K}^{(n)})$ is left-orderable. We will make use of the following criterion due to Boyer-Rolfsen-Wiest. ###### Theorem 3.2 ([BRW05]). Let $M$ be a compact, orientable, irreducible $3$-manifold. Then $\pi_{1}(M)$ is left-orderable, if there exists a nontrivial homomorphism from $\pi_{1}(M)$ to a left-orderable group. Note that the group $SL(2,\mathbb{R})$ itself is not left-orderable, but its universal covering group, denoted by $\widetilde{SL(2,\mathbb{R})}$, is left- orderable [Ber91]. Let $E$ be the covering map from $\widetilde{SL(2,\mathbb{R})}$ to $SL(2,\mathbb{R})$. Since $\widetilde{SL(2,\mathbb{R})}$ and $SL(2,\mathbb{R})$ are both connected, we have $\mathcal{Z}(\widetilde{SL(2,\mathbb{R})})=E^{-1}(\mathcal{Z}(SL(2,\mathbb{R}))),$ where $\mathcal{Z}(\widetilde{SL(2,\mathbb{R})})$ and $\mathcal{Z}(SL(2,\mathbb{R}))$ are the centers of the Lie groups $\widetilde{SL(2,\mathbb{R})}$ and $SL(2,\mathbb{R})$ respectively [HN12, p. 336]. Therefore, $\mathcal{Z}(\widetilde{SL(2,\mathbb{R})})=E^{-1}(\\{\pm I\\})$. ###### Lemma 3.3. Given any knot $K$ in $S^{3}$, let $Z$ be a meridional element in the knot group $\pi_{1}(X_{K})$. Suppose that there exists a non-abelian $SL(2,\mathbb{R})$ representation of $\pi_{1}(X_{K})$ such that $Z^{n}$ is sent to $\pm I$. Then this representation induces a nontrivial $\widetilde{SL(2,\mathbb{R})}$ representation of the fundamental group of the $n^{th}$ cyclic branched cover $\pi_{1}(X_{K}^{(n)})$ . ###### Proof. The kernel of the covering map $Ker(E)$ is isomorphic to $\pi_{1}(SL(2,\mathbb{R}))\cong\mathbb{Z}$ and we have the following central extension $0\longrightarrow\mathbb{Z}\longrightarrow\widetilde{SL(2,\mathbb{R})}\longrightarrow SL(2,\mathbb{R})\longrightarrow I.$ Suppose that $\rho$ is a representation of $\pi_{1}(X_{K})$ into $SL(2,\mathbb{R})$. Then the pullback $\widetilde{SL(2,\mathbb{R})}\times_{SL(2,\mathbb{R})}\pi_{1}(X_{K})=\\{(M,x)\in\widetilde{SL(2,\mathbb{R})}\times\pi_{1}(X_{K}):E(M)=\rho(x)\\},$ is a central extension of $\pi_{1}(X)$ by $\mathbb{Z}$. On the other hand, $H^{2}(\pi_{1}(X_{K}),\mathbb{Z})\cong H^{2}(X_{K},\mathbb{Z})=0,$ so every central extension of $\pi_{1}(X_{k})$ by $\mathbb{Z}$ splits. Hence, $\rho$ can be lifted to a representation into $\widetilde{SL(2,\mathbb{R})}$. That is, the composition of a splitting map with the projection from $\widetilde{SL(2,\mathbb{R})}\times_{SL(2,\mathbb{R})}\pi_{1}(X_{K})$ to $\widetilde{SL(2,\mathbb{R})}$ is a lifting of $\rho$ [Wei95] (also see [GHY01]). Now assume that the representation $\rho$ of the knot group $\pi_{1}(X_{K})$ satisfies the property $\rho(Z^{n})=\pm I$. We denote by $\tilde{\rho}$ a lifting of $\rho$. Since $\rho(Z^{n})=\pm I$, we have $\tilde{\rho}(Z^{n})$ is inside $E^{-1}(\pm I)$, which is equal to $\mathcal{Z}(\widetilde{SL(2,\mathbb{R})})$, the center of $\widetilde{SL(2,\mathbb{R})}$. $\tilde{\rho}$$\widetilde{SL(2,\mathbb{R})}$$E$$\pi_{1}(X_{K},P_{-})$$SL(2,\mathbb{R})$$\rho$ In addition, if $\rho$ is a non-abelian representation, then $\tilde{\rho}$ is non-abelian. By Lemma 2.1, the representation $\tilde{\rho}$ induces a nontrivial $\widetilde{SL(2,\mathbb{R})}$ representation of $\pi_{1}(X_{K}^{(n)})$. ∎ ###### Proof of Theorem 3.1 . Let $\rho$ be a non-abelian $SL(2,\mathbb{R})$ representation of the knot group $\pi_{1}(X_{K})$, with $\rho(Z^{n})=\pm I$. By Lemma 3.3, this representation induces a nontrivial $\widetilde{SL(2,\mathbb{R})}$ representation of the group $\pi_{1}(X_{K}^{(n)})$. The group $\widetilde{SL(2,\mathbb{R})}$ can be embedded inside the group of order-preserving homeomorphisms of $\mathbb{R}$, so it is left-orderable [Ber91]. Moreover, the $n^{th}$ cyclic branched cover $X_{K}^{(n)}$ is irreducible if $K$ is a prime knot [Plo84]. Thus, Theorem 3.1 follows from Theorem 3.2. ∎ ## 4\. An Application to $(p,q)$ two-bridge knots, with $p\equiv 3$ mod $4$ In this section we apply Theorem 3.1 to $(p,q)$ two-bridge knots with $p=3\text{ mod }4$. We show that given any two-bridge knot of this type, the fundamental group of the $n^{th}$ cyclic branched cover is left-orderable if $n$ is sufficiently large. Let $K$ be a $(p,q)$ two-bridge knot. From the Schubert normal form [Kaw96, p. 21], the knot group has a presentation of the following form: $\pi_{1}(X_{K})=<x,y:wx=yw>,$ $\text{ where }w=(x^{\epsilon_{1}}y^{\epsilon_{2}})\dots(x^{\epsilon_{p-2}}y^{\epsilon_{p-1}})\text{ and }\epsilon_{i}=\pm 1.$ Set $\rho:\pi_{1}(X_{K})\rightarrow SL(2,\mathbb{C})$ be a non-abelian representation of the knot group into $SL(2,\mathbb{C})$. Up to conjugation, we can assume that (4.1) $\rho(x)=\begin{pmatrix}m&1\\\ 0&m^{-1}\\\ \end{pmatrix},\quad\rho(y)=\begin{pmatrix}m&0\\\ s&m^{-1}\\\ \end{pmatrix}.$ Hence, $\rho(w)=\rho(x)^{\epsilon_{1}}\rho(y)^{\epsilon_{2}}\dots\rho(x)^{\epsilon_{p-2}}\rho(y)^{\epsilon_{p-1}}$ is a matrix with entries in $\mathbb{Z}[m^{\pm 1},s]$. Denote $\rho(w)=\begin{pmatrix}w_{11}&w_{12}\\\ w_{21}&w_{22}\end{pmatrix}$, $w_{ij}\in\mathbb{Z}[m^{\pm 1},s]$. From the group relation $wx=yw$, we have $\begin{pmatrix}w_{11}&w_{12}\\\ w_{21}&w_{22}\end{pmatrix}\begin{pmatrix}m&1\\\ 0&m^{-1}\\\ \end{pmatrix}=\begin{pmatrix}m&0\\\ s&m^{-1}\\\ \end{pmatrix}\begin{pmatrix}w_{11}&w_{12}\\\ w_{21}&w_{22}\end{pmatrix}.$ This is equivalent to (4.2) $\begin{pmatrix}0&w_{11}+(m^{-1}-m)w_{12}\\\ (m-m^{-1})w_{21}-sw_{11}&w_{21}-sw_{12}\end{pmatrix}=0$ and hence $s$ and $m$ must satisfy the equation $w_{11}+(m^{-1}-m)w_{12}=0.$ In [Ril84], it is shown that the above equation is also a sufficient condition. ###### Proposition 4.1 (Theorem 1 of [Ril84]). The assignment of $x$ and $y$ as in (4.1) defines a non-abelian $SL(2,\mathbb{C})$ representation of the knot group $\pi_{1}(X_{K})=<x,y:wx=yw>$ if and only if (4.3) $\varphi(m,s)\triangleq w_{11}+(m^{-1}-m)w_{12}=0.$ We need to make use of several properties of the polynomial $\varphi(m,s)$. All of these properties are either proven or claimed throughout Riley’s paper [Ril84]. For readers’ convenience, we organize them and provide a proof in the following lemma. ###### Lemma 4.2 (cf. [Ril84]). The polynomial $\varphi(m,s)$ in $\mathbb{Z}[m^{\pm 1},s]$ satisfies the following: 1. (1) As a polynomial in $s$ with coefficients in $\mathbb{Z}[m^{\pm 1}]$, $\varphi(m,s)$ has $s$-degree equal to $\frac{p-1}{2}$, with the leading coefficient $\pm 1$. 2. (2) $\varphi(1,0)\neq 0$. 3. (3) $\varphi(m,s)$ does not have repeated factors. 4. (4) $\varphi(m,s)=\varphi(m^{-1},s)$ and thus $\varphi(m,s)=f(m+m^{-1},s)$ where $f$ is a two-variable polynomial with coefficients in $\mathbb{Z}$. ###### Proof. 1. (1) Since we assign $\rho(x)=\begin{pmatrix}m&1\\\ 0&m^{-1}\end{pmatrix},\quad\rho(y)=\begin{pmatrix}m&0\\\ s&m^{-1}\end{pmatrix},$ through a direct computation we have $\rho(xy)=\begin{pmatrix}m^{2}+s&m^{-1}\\\ m^{-1}s&m^{-2}\end{pmatrix},\quad\rho(x^{-1}y)=\begin{pmatrix}1-s&-m^{-1}\\\ ms&1\end{pmatrix},$ $\rho(xy^{-1})=\begin{pmatrix}1-s&m\\\ -m^{-1}s&1\end{pmatrix},\quad\rho(x^{-1}y^{-1})=\begin{pmatrix}m^{-2}+s&-m\\\ -ms&m^{2}\end{pmatrix}.$ Say a matrix $A$ in $M_{2}(\mathbb{Z}[m^{\pm 1},s])$ has $s$-degree equal to $n$ if $A=\begin{pmatrix}\pm s^{n}+f_{11}(m,s)&f_{12}(m,s)\\\ f_{21}(m,s)&f_{22}(m,s)\end{pmatrix},\text{ where }$ the $s$-degrees of $f_{11}$, $f_{12}$ and $f_{22}$ are strictly less than $n$ and the $s$-degree of $f_{21}$ is less than or equal to $n$. Hence the matrices $\rho(xy)$, $\rho(x^{-1}y)$, $\rho(xy^{-1})$ and $\rho(x^{-1}y^{-1})$ all have $s$-degrees equal to $1$. Moreover, the product of an $s$-degree $n$ matrix and an $s$-degree $m$ matrix is an $s$-degree $m+n$ matrix. Since $w=(x^{\epsilon_{1}}y^{\epsilon_{2}})\dots(x^{\epsilon_{p-2}}y^{\epsilon_{p-1}}),\text{ with }\epsilon_{i}=\pm 1,$ we have that the matrix $\rho(w)=\begin{pmatrix}w_{11}&w_{12}\\\ w_{21}&w_{22}\end{pmatrix}$ is a product of $\frac{p-1}{2}$ $s$-degree $1$ matrices. Therefore, the matrix $\rho(w)$ has $s$-degree equal to $\frac{p-1}{2}$. That is, the entry $w_{11}$ has $\pm s^{n}$ as the leading term and the $s$-degree of $w_{12}$ is strictly less than $\frac{p-1}{2}$. As a result, $\varphi(m,s)=w_{11}+(m^{-1}-m)w_{12}$ has leading term equal to $\pm s^{n}$. 2. (2) Notice that as $m=1$ and $s=0$, we have $\rho(x)=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix},\quad\rho(y)=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}.$ This assignment can not define a representation of the knot group $\pi_{1}(X_{K})=<x,y:wx=yw>,$ because these two matrices $\rho(x)=\begin{pmatrix}1&1\\\ 0&1\end{pmatrix}$ and $\rho(y)=\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$ are not conjugate to each other. Therefore, $\varphi(1,0)\neq 0$ by the Proposition 4.3. 3. (3) Let $\Delta_{K}(t)$ be the Alexander polynomial of the knot $K$. It is shown in [Nag08, Proposition 1.1, Theorem 1.2] (also see [Lin01, BF08]) that any knot group has $\frac{\|\Delta_{K}(-1)\|-1}{2}$ irreducible $SL(2,\mathbb{C})$ metabelian representations up to conjugation and that these metabelian representations send meridional elements to matrices of eigenvalues $\pm i$. For a $(p,q)$ two-bridge knot, $p$ equals $\|\Delta_{K}(-1)\|$. This implies that the degree $\frac{p-1}{2}$ polynomial equation $\varphi(i,s)=0$ has $\frac{p-1}{2}$ distinguished roots. Therefore $\varphi(i,s)$ does not have repeated factors and so is $\varphi(m,s)$. Note that we can also use the fact that $\varphi(1,s)$ does not have any repeated factors to prove that $\varphi(m,s)$ has no repeated factors [Ril72, Theorem 3]. 4. (4) Assume that the assignment $\rho(x)=\begin{pmatrix}m&1\\\ 0&m^{-1}\end{pmatrix},\quad\rho(y)=\begin{pmatrix}m&0\\\ s&m^{-1}\end{pmatrix}$ defines a representation of the knot group $\pi_{1}(X_{K})=<x,y:wx=yw>.$ Then $\rho^{\prime}(x)=P\begin{pmatrix}m&1\\\ 0&m^{-1}\end{pmatrix}P^{-1}=\begin{pmatrix}m^{-1}&1\\\ 0&m\end{pmatrix}$ $\rho^{\prime}(y)=P\begin{pmatrix}m&0\\\ s&m^{-1}\end{pmatrix}P^{-1}=\begin{pmatrix}m^{-1}&0\\\ s&m\end{pmatrix}$ also defines a representation, where $P=\begin{pmatrix}1&(m^{-1}-m)/s\\\ m-m^{-1}&1\end{pmatrix}.$ The matrix $P$ is well-defined and invertible whenever $(m,s)$ is not in the finite set $S\triangleq\\{(m,s):s=0,\varphi(m,s)=0\\}\cup\quad\quad\quad\quad\quad$ $\quad\quad\quad\quad\quad\quad\\{(m,s):s=-(m-m^{-1})^{2},\varphi(m,s)=0\\}.$ The set $S$ is finite because neither $\varphi(m,0)$ nor $\varphi(m,-(m-m^{-1})^{2})$ is a zero polynomial. Otherwise, $(1,0)$ will be a solution for $\varphi(m,s)$, which contradicts part $(2)$. Denote by $V(g)$ the solution set of a polynomial $g$. As we described above, $V(\varphi(m,s))-S\subset V(\psi(m,s)),$ where $\psi(m,s)=\varphi(m^{-1},s)$. Points in $S$ are not isolated, since they are embedded inside the algebraic curve $V(\varphi(m,s))$. By continuity, we have $V(\varphi(m,s))\subset V(\psi(m,s)).$ By part $(3)$, neither of $\varphi(m,s)$ and $\psi(m,s)$ has repeated factors, so the ideal $<\psi(m,s)>$ is contained inside the ideal $<\varphi(m,s)>$ in $\mathbb{Z}[m^{\pm 1},s]$. On the other hand, both $\varphi(m,s)$ and $\psi(m,s)$ have the same leading term, which is either $s^{(p-1)/2}$ or $-s^{(p-1)/2}$, so $\varphi(m,s)=\psi(m,s)=\varphi(m^{-1},s)$. ∎ Now we are ready to prove the main result. ###### Theorem 4.3. Given a $(p,q)$ two-bridge knot $K$, with $p\equiv 3\text{ mod }4$, there are only finitely many cyclic branched covers, whose fundamental groups are not left-orderable. ###### Proof. We are going to show that for sufficiently large $n$, the group $\pi_{1}(X_{K})$ has a non-abelian $SL(2,\mathbb{R})$ representation with $x^{n}$ sent to $-I$. As before, we assign $\rho(x)=\begin{pmatrix}m&1\\\ 0&m^{-1}\end{pmatrix},\quad\rho(y)=\begin{pmatrix}m&0\\\ s&m^{-1}\end{pmatrix}.$ Let $m=e^{i\theta}$. Since $p=3\text{ mod }4$, by Lemma $4.2$, we have that $\varphi(e^{i\theta},s)$ is an odd degree real polynomial in $s$. So for any given $\theta$, the equation $\varphi(e^{i\theta},s)=0$ has at least one real solution for $s$. We assume that $s_{0}$ is a real solution of the equation $\varphi(1,s)=0$. It is known that the polynomial $\varphi(1,s)$ does not have repeated factors [Ril72, Theorem 3]. Hence, $\varphi_{s}(e^{i\theta},s)\|_{\theta=0,s=s_{0}}\neq 0$ and locally there exists a real function $s(\theta)$ such that $\varphi(e^{i\theta},s(\theta))=0$ and $s(0)=s_{0}$. Consider the following one-parameter family of non-abelian representations. $\rho\\{\theta\\}(x)=\begin{pmatrix}e^{i\theta}&1\\\ 0&e^{-i\theta}\end{pmatrix},\quad\rho\\{\theta\\}(y)=\begin{pmatrix}e^{i\theta}&0\\\ s(\theta)&e^{-i\theta}\end{pmatrix}.$ As $\theta\neq 0$, the representations $\rho\\{\theta\\}$ can be diagonalized to the following forms which we still denote by $\rho\\{\theta\\}$, (4.4) $\rho\\{\theta\\}(x)=\begin{pmatrix}e^{i\theta}&0\\\ 0&e^{-i\theta}\\\ \end{pmatrix},\quad\rho\\{\theta\\}(y)=\begin{pmatrix}e^{i\theta}-\frac{s(\theta)}{2\sin(\theta)}i&-1+\frac{s(\theta)}{4\sin^{2}(\theta)}\\\ s(\theta)&e^{-i\theta}+\frac{s(\theta)}{2\sin(\theta)}i\end{pmatrix}.$ According to [Kho03, p. 786], this representation can be conjugated to an $SL(2,\mathbb{R})$ representation if and only if (4.5) $\text{ either }s(\theta)<0\text{ or }s(\theta)>4\sin^{2}(\theta).$ We can verify this via a direction computation. In fact, when $s<0$ or $s>4\sin^{2}(\theta)$, the representation $\rho\\{\theta\\}$ is conjugate to an $SU(1,1)$ representation by the matrix $\begin{pmatrix}\sqrt{\frac{1}{\sqrt{t}}+t}&t\\\ \sqrt{t}&\sqrt{\sqrt{t}+t^{2}}\end{pmatrix},\text{ where }t=\frac{1}{4\sin^{2}(\theta)}-\frac{1}{s}\text{ is positive},$ and $SU(1,1)$ is conjugate to $SL(2,\mathbb{R})$ via the matrix $\begin{pmatrix}1&-i\\\ 1&i\end{pmatrix}$ in $GL(2,\mathbb{C})$. On the other hand, $\lim_{\theta\rightarrow 0}s(\theta)=s_{0},\text{ where $s_{0}$ is not equal to }0\text{ by Lemma \ref{lem:ril} part }(2).$ Hence, when $\theta$ is small enough, either $s(\theta)<0$ or $s(\theta)>4\sin^{2}(\theta)$. Now let $\theta=\pi/n$. For sufficiently large $n$, the non-abelian representation $\rho\\{\theta\\}$ as in (4.4) satisfies $\rho\\{\theta\\}(x)^{n}=-I$ and conjugates to an $SL(2,\mathbb{R})$ representation. Therefore, by Theorem 3.1, the conclusion follows. ∎ We are ending this paper by computing one specific example. ###### Example 4.4. We consider the two bridge knot $(7,4)$, which is listed as $5_{2}$ in Rolfsen’s table. The fundamental group $\pi_{1}(X_{5_{2}})$ has a presentation $\pi_{1}(X_{5_{2}})=<x,y:wx=yw>,$ where $w=xyx^{-1}y^{-1}xy$. From this presentation, we can compute the polynomial $\varphi(m,s)=s^{3}+(2(m^{2}+m^{-2})-3)s^{2}+((m^{4}+m^{-4})-3(m^{2}+m^{-2})+6)s+2(m^{2}+m^{-2})-3.$ as defined in (4.3). And $\varphi(e^{i\theta},s)=s^{3}+(4\text{ cos}(2\theta)-3)s^{2}+(2\text{ cos}(4\theta)-6\text{ cos}(2\theta)+6)s+4\text{ cos}(2\theta)-3,$ which is a real polynomial in $s$ with degree $3$. Hence, we can solve a closed formula for $s(\theta)$ such that $\varphi(e^{i\theta},s(\theta))=0$. Figure 2 is the graph of the solution $s(\theta)$ on the interval $\theta\in[0,1]$. Figure 2. In particular, when $n=9$, we have that $\frac{\pi}{9}\approx 0.349$ and $s(\frac{\pi}{9})\approx-0.03667$. The group $\pi_{1}(X_{5_{2}}^{(n)})$ is left-orderable when $n\geq 9$. For cyclic branched covers $X_{5_{2}}^{(n)}$ with $n<9$, the other known cases are $n=2,3$ [DPT05] and $n=4$ [GL], none of which has a left-orderable fundamental group. ### Acknowledgment The author would like to thank Oliver Dasbach for drawing her attention to the topic of the current paper and his consistent encouragement and support throughout her graduate study. Also, the author would like to give thanks to Michel Boileau, Tye Lidman and Neal Stoltzfus for helpful conversation and suggestions. Finally, she gives many thanks to the referee for pointing out the similarity between Theorem 3.1 and [BGW13, Theorem 6] and his or her many helpful comments. ## References * [Ber91] G. Bergman. Right orderable groups that are not locally indicable. Pacific J. Math., 147(2):243–248, 1991. * [BF08] H. Boden and S. Friedl. Metabelian ${S}{L}(n,\mathbb{C})$ representations of knot groups. Pacific J. Math., 238(1):7–25, 2008. * [BGW13] S. Boyer, C. Gordon, and L. Watson. On L-space and left-orderable fundamental groups. Mathematische Annalen, 356(4):1213–1245, 2013. * [BRW05] S. Boyer, D. Rolfsen, and B. Wiest. Orderable $3$-manifold groups. Annales de l’institut Fourier, 55(1):243–288, 2005. * [CD03] D. Calegari and N. Dunfield. Laminations and groups of homeomorphisms of the circle. Inventiones Mathematicae, 152:149–207, 2003. * [CLW13] A. Clay, T. Lidman, and L. Watson. Graph manifolds, left-orderability and amalgamation. Algebraic & Geometric Topology, 13:2347–2368, 2013. * [DPT05] M. Da̧bkowski, J. Przytycki, and A. Togha. Non-left-orderable $3$-manifold groups. Canadian Math. Bull., 48(1):32–40, 2005. * [GHY01] É. GHYS. Groups acting on the circle. Enseignement Mathematique, 47(3/4):329–408, 2001. * [GL] C. Gordon and T. Lidman. Taut foliations, left-orderability, and cyclic branched covers. Preprint. * [Gre] J. Greene. Alternating links and left-orderability. arXiv:1107.5232 [math.GT]. * [HN12] J. Hilgert and K. Neeb. Structure and geometry of Lie groups. Springer, 2012. * [HS85] J. Howie and H. Short. The band-sum problem. J. London Math. Soc., 31(2):571–576, 1985. * [Ito13] T. Ito. Non-left-orderable double branched coverings. Algebraic & Geometric Topology, 13:1937 –1965, 2013. * [Kaw96] A Kawauchi. Survey on Knot theory. Springer, 1996. * [Kho03] V. Khoi. A cut-and-paste method for computing the Seifert volumes. Mathematische Annalen, 326:759–801, 2003. * [Lin01] X. Lin. Representations of knot groups and twisted Alexander polynomials. Acta Mathematica Sinica, English Series, 17(3):361–380, July 2001\. * [LS01] R. Lyndon and P. Schupp. Combinatorial Group Theory. Springer-Verlag, New York, 2001. * [Nag08] F. Nagasato. Finiteness of a section of the ${S}{L}(2,\mathbb{C})$-character variety of the knot group. Kobe J. Math., 24(2):223–240, 2008. * [OS05a] P. Ozsváth and Z. Szabó. On knot Floer homology and lens space surgeries. Topology, 44:1281–1300, 2005. * [OS05b] P. Ozsváth and Z. Szabó. On the Heegaard Floer homology of branched double-covers. Adv. Math., 194:1–33, 2005. * [Pet] T. Peters. On L-spaces and non left-orderable $3$-manifold groups. Preprint. * [Plo84] S. Plotnick. Finite group actions and nonseparating 2-spheres. Proc. Amer. Math. Soc., 90(3):430–432, 1984. * [Ril72] R. Riley. Parabolic representations of knot groups. I. Proc. London Math. Soc., 24(3):217– 242, 1972. * [Ril84] R. Riley. Nonabelian representations of $2$-bridge knot groups. The Quarterly J. Math., 35(2):191–208, 1984. * [Ter] M. Teragaito. Four-fold cyclic branched covers of genus one two-bridge knots are L-spaces. Boletín de la Sociedad Matemática Mexicana. to appear. * [Tra] A. Tran. On left-orderablility and cyclic branched coverings. J. Math. Soc. Japan. to appear. * [Wei95] C. Weibel. An introduction to homological algebra. Cambridge university press, 1995.	arxiv-papers	2013-11-13T20:56:40	2024-09-04T02:49:53.581907	{ "license": "Public Domain", "authors": "Ying Hu", "submitter": "Ying Hu", "url": "https://arxiv.org/abs/1311.3291" }
1311.3368	# Anytime Belief Propagation Using Sparse Domains Sameer Singh University of Massachusetts Amherst MA 01003 [email protected] &Sebastian Riedel University College London London UK [email protected] &Andrew McCallum University of Massachusetts Amherst MA 01003 [email protected] For marginal inference on graphical models, belief propagation (BP) has been the algorithm of choice due to impressive empirical results on many models. These models often contain many variables and factors, however the domain of each variable (the set of values that the variable can take) and the neighborhood of the factors is usually small. When faced with models that contain variables with large domains and higher-order factors, BP is often intractable. The primary reason BP is unsuitable for large domains is the cost of message computations and representation, which is in the order of the cross-product of the neighbors’ domains. Existing extensions to BP that address this concern [1, 2, 4, 8, 9, 10, 13] use parameters that define the desired level of approximation, and return the approximate marginals at convergence. This results in poor _anytime_ behavior. Since these algorithms try to directly achieve the desired approximation, the marginals _during_ inference cannot be characterized, and are often inconsistent with each other. Further, the relationship of the parameter that controls the approximation to the quality of the intermediate marginals is often unclear. As a result, these approaches are not suitable for applications that require consistent, anytime marginals but are willing to trade-off error for speed, for example applications that involve real-time tracking or user interactions. There is a need for an anytime algorithm that can be interrupted to obtain consistent marginals corresponding to fixed points of a well-defined objective, and can improve the quality of the marginals over the execution period, eventually obtaining the BP marginals. In this work we propose a novel class of message passing algorithms that compute accurate, anytime-consistent marginals. Initialized with a sparse domain for each variable, the approach alternates between two phases: (1) augmenting values to sparse variable domains, and (2) converging to a fixed point of the approximate marginal inference objective as defined by these sparse domains. We tighten our approximate marginal inference objective by selecting the value to add to the sparse domains by estimating the impact of adding the value to the variational objective; this is an accurate prioritization scheme that depends on the instantiated domains and requires runtime computation. We also provide an alternate prioritization scheme based on the gradient of the primal objective that can be computed a priori, and provides constant time selection of the value to add. To converge to a fixed point of the approximate marginal objective, we perform message passing on the sparse domains. Since naive schedules that update messages in a round robin or random fashion are wasteful, we use residual-based dynamic prioritization [3]. Inference can be interrupted to obtain consistent marginals at a fixed point defined over the instantiated domains, and longer execution results in more accurate marginals, eventually optimizing the BP objective. ## 1 Marginal Inference for Undirected Graphical Models Let $\mathbf{x}$ be a random vector where each $x_{i}\in\mathbf{x}$ takes a value $v_{i}$ from domain $\mathcal{D}$. An assignment to a subset of variables $\mathbf{x}_{c}\subseteq\mathbf{x}$ is represented by $\mathbf{v}_{c}\in\mathcal{D}^{\|\mathbf{x}_{c}\|}$. A factor graph [6] is defined by a bipartite graph over the variables $\mathbf{x}$ and a set of factors $f\in\mathcal{F}$ (with neighborhood $\mathbf{x}_{f}\equiv\mathcal{N}(f)$). Each factor $f$ defines a scalar function $\boldsymbol{\phi}_{f}$ over the assignments $\mathbf{v}_{f}$ of its neighbors $\mathbf{x}_{f}$, defining the distribution: $\displaystyle p(\mathbf{v})\triangleq\frac{1}{Z}\exp\sum_{f\in\mathcal{F}}\boldsymbol{\phi}_{f}(\mathbf{v}_{f})\text{, where }Z\triangleq\sum_{\mathbf{v}^{\prime}\in\mathcal{D}^{n}}\exp\sum_{f\in\mathcal{F}}\boldsymbol{\phi}_{f}(\mathbf{v}^{\prime}_{f})$. Inference is used to compute the variable marginals $p(v_{i})=\sum_{\mathbf{v}/x_{i}}p(\mathbf{v})$ and the factor marginals $p(\mathbf{v}_{f})=\sum_{\mathbf{v}/\mathbf{x}_{f}}p(\mathbf{v})$. When performing approximate variational inference, we represent the approximate marginals $\boldsymbol{\mu}\equiv(\boldsymbol{\mu}_{\mathcal{X}},\boldsymbol{\mu}_{\mathcal{F}})$ that contain elements for every assignment to the variables $\boldsymbol{\mu}_{\mathcal{X}}\equiv\mu_{i}(v_{i}),\forall x_{i},v_{i}\in\mathcal{D}$ and factors $\boldsymbol{\mu}_{\mathcal{F}}\equiv\mu_{f}(\mathbf{v}_{f}),\forall f,\mathbf{v}_{f}\in\mathcal{D}^{\|\mathbf{x}_{f}\|}$. Minimizing the KL divergence between the desired and approximate marginals results in: $\displaystyle\max_{\boldsymbol{\mu}\in\mathcal{M}}\sum_{f\in\mathcal{F}}\sum_{\mathbf{v}_{f}}\mu_{f}(\mathbf{v}_{f})\boldsymbol{\phi}_{f}(\mathbf{v}_{f})+H\left(\boldsymbol{\mu}\right)$, where $\mathcal{M}$ is the set of _realizable_ mean vectors $\boldsymbol{\mu}$, and $H\left(\boldsymbol{\mu}\right)$ is the entropy of the distribution that yields $\boldsymbol{\mu}$. Both the polytope $\mathcal{M}$ and the entropy $H$ need to be approximated in order to efficiently solve the maximization. Belief propagation (BP) approximates $\mathcal{M}$ using the _local polytope_ : $\mathcal{L}\triangleq\biggl{\\{}\boldsymbol{\mu}\geq 0,~{}~{}~{}\forall f\in\mathcal{F}:\sum_{\mathbf{v}_{f}}\mu_{f}\left(\mathbf{v}_{f}\right)=1,\forall f,i\in\mathcal{N}\left(f\right),v_{s}:\sum_{\mathbf{v}^{\prime}_{f},v_{i}^{\prime}=v_{s}}\mu_{f}\left(\mathbf{v}^{\prime}_{f}\right)=\mu_{i}\left(v_{s}\right)\biggr{\\}}$ and entropy using Bethe approximation: $H_{B}\left(\boldsymbol{\mu}\right)\triangleq\sum_{f}H\left(\boldsymbol{\mu}_{f}\right)-\sum_{i}\left(d_{i}-1\right)H\left(\boldsymbol{\mu}_{i}\right)$, leading to: $\max_{\boldsymbol{\mu}\in\mathcal{L}}\sum_{f\in\mathcal{F}}\sum_{\mathbf{v}_{f}}\mu_{f}(\mathbf{v}_{f})\boldsymbol{\phi}_{f}(\mathbf{v}_{f})+H_{B}\left(\boldsymbol{\mu}\right)$ (1) The Lagrangian relaxation of this optimization is: $L_{\text{BP}}\left(\boldsymbol{\mu},\boldsymbol{\lambda}\right)\triangleq\sum_{f\in\mathcal{F}}\sum_{\mathbf{v}_{f}}\mu_{f}(\mathbf{v}_{f})\boldsymbol{\phi}_{f}(\mathbf{v}_{f})+H_{B}\left(\boldsymbol{\mu}\right)+\sum_{f}\lambda_{f}C_{f}\left(\boldsymbol{\mu}\right)+\sum_{f}\sum_{i\in N\left(f\right)}\sum_{v_{i}}\lambda_{fi}^{v_{i}}C_{f,i,v_{i}}\left(\boldsymbol{\mu}\right)$ (2) where $C_{f,i,v_{i}}=\mu_{i}(v_{i})-\sum_{\mathbf{v}_{f}/x_{i}}\mu_{f}(\mathbf{v}_{f})$ and $C_{f}=1-\sum_{\mathbf{v}_{f}}\mu_{f}(\mathbf{v}_{f})$ are the constraints that correspond to the local polytope $\mathcal{L}$. BP messages correspond to the dual variables, i.e. $\boldsymbol{m}_{fi}(v_{i})\propto\exp\lambda_{fi}^{v_{i}}$. If the messages converge, Yedidia et al. [12] show that the marginals correspond to a $\boldsymbol{\mu}^{}$ and $\boldsymbol{\lambda}^{}$ at a saddle point of $L_{\text{BP}}$, i.e. $\nabla_{\boldsymbol{\mu}}L_{\text{BP}}\left(\boldsymbol{\mu}^{},\boldsymbol{\lambda}^{}\right)=0$ and $\nabla_{\boldsymbol{\lambda}}L_{\text{BP}}\left(\boldsymbol{\mu}^{},\boldsymbol{\lambda}^{}\right)=0$. In other words: at convergence BP marginals are locally consistent and locally optimal. BP is not guaranteed to converge, or to find the global optimum if it does, however it often converges and produces accurate marginals in practice [7]. ## 2 Anytime Belief Propagation Graphical models are often defined over variables with large domains and factors that neighbor many variables. Message passing algorithms perform poorly for such models since the complexity of message computation for a factor is $\operatorname{O}\bigl{(}\|\mathcal{D}\|^{\|\mathcal{N}_{f}\|}\bigr{)}$ where $\mathcal{D}$ is the domain of the variables. Further, if inference is interrupted, the resulting marginals are not locally consistent, nor do they correspond to any fixed point of a well-defined objective. Here, we describe an algorithm that meets the following desiderata: (1) anytime property that results in consistent marginals, (2) more iterations improve the accuracy of marginals, and (3) convergence to BP marginals (as obtained at a fixed point of BP). Instead of directly performing inference on the complete model, our approach maintains _partial_ domains for each variable. Message passing on these sparse domains converges to a fixed point of a well-defined objective (Section 2.1). This is followed by incrementally _growing_ the domains (Section 2.2), and resuming message passing on the new set of domains till convergence. At any point, the marginals are close to a fixed point of the sparse BP objective, and we tighten this objective over time by growing the domains. If the algorithm is not interrupted, entire domains are instantiated, and the marginals converge to a fixed point of the complete BP objective. ### 2.1 Belief Propagation with Sparse Domains First we study the propagation of messages when the domains of each variables have been partially instantiated (and are assumed to be fixed here). Let $\mathcal{S}_{i}\subseteq D,\|\mathcal{S}_{i}\|\geq 1$ be the set of values associated with the instantiated domain for variable $x_{i}$. During message passing, we _fix_ the marginals corresponding to the non-instantiated domain to be zero, i.e. $\forall v_{i}\in\mathcal{D}-\mathcal{S}_{i},\mu_{i}(v_{i})=0$. By setting these values in the BP dual objective (2), we obtain the optimization defined only over the sparse domains: $\displaystyle L_{\text{SBP}}\left(\boldsymbol{\mu},\boldsymbol{\lambda},\mathcal{S}\right)\triangleq\sum_{f}\sum_{\mathbf{v}_{f}\in\mathcal{S}_{f}}\mu_{f}(\mathbf{v}_{f})\boldsymbol{\phi}_{f}(\mathbf{v}_{f})+H_{B}\left(\boldsymbol{\mu}\right)+\sum_{f}\lambda_{f}C_{f}\left(\boldsymbol{\mu}\right)+\sum_{f}\sum_{i\in N\left(f\right)}\sum_{v_{i}\in\mathcal{S}_{i}}\lambda_{fi}^{v_{i}}C_{f,i,v_{i}}\left(\boldsymbol{\mu}\right)$ (3) Note that $L_{\text{SBP}}(\boldsymbol{\mu},\boldsymbol{\lambda},\mathcal{D}^{n})=L_{\text{BP}}(\boldsymbol{\mu},\boldsymbol{\lambda})$. Message computations for this approximate objective, including the summations in the updates, are defined sparsely over the instantiated domains. In general, for a factor $f$, the computation of its outgoing messages requires $\operatorname{O}\bigl{(}\prod_{x_{i}\in\mathcal{N}_{f}}\|\mathcal{S}_{i}\|\bigr{)}$ operations, as opposed to $\operatorname{O}\bigl{(}\|D\|^{\|\mathcal{N}_{f}\|}\bigr{)}$ for whole domains. Variables for which $\|\mathcal{S}_{i}\|=1$ are treated as _observed_. ### 2.2 Growing the Domains As expected, BP on sparse domains is much faster than on whole domains, however it is optimizing a different, approximate objective. The approximation can be tightened by growing the instantiated domains, that is, as the sparsity constraints of $\mu_{i}(v_{i})=0$ are removed, we obtain more accurate marginals when message passing for newly instantiated domain converges. Identifying _which_ values to add is crucial for good anytime performance, and we propose two approaches here based on the gradient of the variational and the primal objectives. Dynamic Value Prioritization: When inference with sparse domains converges, we obtain marginals that are locally consistent, and define a saddle point of Eq (3). We would like to add the value $v_{i}$ to $\mathcal{S}_{i}$ for which removing the constraint $\mu_{i}(v_{i})=0$ will have the most impact on the approximate objective $L_{\text{SBP}}$. In other words, we select $v_{i}$ for which the gradient $\frac{\partial L_{\text{SBP}}}{\partial\mu_{i}(v_{i})}\|_{\mu_{i}(v_{i})=0}$ is largest. From (3) we derive $\frac{\partial L_{SBP}}{\partial\mu_{i}(v_{i})}=(d_{i}-1)(1+\log\mu_{i}(v_{i}))+\sum_{f\in\mathcal{N}(x_{i})}\lambda_{fi}^{v_{i}}$. Although $\log\mu_{i}(v_{i})\rightarrow-\infty$ when $\mu_{i}(v_{i})\rightarrow 0$, we ignore the term as it appears for all $i$ and $v_{i}$111Alternatively, approximation to $L_{\text{SBP}}$ that replaces the variable entropy $-\sum_{p}p\log p$ with its second order Taylor approximation $\sum_{p}p(1-p)$. The gradient at $\mu_{i}(v_{i})=0$ of the approximation is $d_{i}+\sum_{f\in\mathcal{N}(x_{i})}\lambda_{fi}^{v_{i}}$.. The rest of the gradient is the priority: $\pi_{i}(v_{i})=d_{i}+\displaystyle\sum_{f\in\mathcal{N}(x_{i})}\lambda_{fi}^{v_{i}}$. Since $\lambda_{fi}^{v_{i}}$ is undefined for $v_{i}\notin\mathcal{S}_{i}$, we estimate it by performing a single round of message update over the sparse domains. To compute priority of all values for a variable $x_{i}$, this computation requires an efficient $\operatorname{O}\bigl{(}\|\mathcal{D}\|\|\mathcal{S}\|^{\mathcal{N}_{f}-1}\bigr{)}$. Since we need to identify the value with the highest priority, we can improve this search by sorting factor scores $\boldsymbol{\phi}$, and further, we only update the priorities for the variables that have participated in message passing. Precomputed Priorities of Values: Although the dynamic strategy selects the value that improves the approximation the most, it also spends time on computations that may not result in a corresponding benefit. As an alternative, we propose a prioritization that precomputes the order of the values to add; even though this does not take the current beliefs into account, the resulting savings in speed may compensate. Intuitively, we want to add values to the domain that have the highest marginals in the final solution. Although the final marginals cannot be computed directly, we estimate them by enforcing a single constraint $\mu_{i}(v_{i})=\sum_{\mathbf{v}_{f}/x_{i}}\mu_{f}(\mathbf{v}_{f})$ and performing greedy coordinate ascent for each $f$ on the primal objective in (1). We set the gradient w.r.t. $\mu_{f}(\mathbf{v}_{f})$ to zero to obtain: $\displaystyle\pi_{i}(v_{i})=\hat{\mu}_{i}(v_{i})=\sum_{\mathbf{v}^{\prime}_{f},v_{i}^{\prime}=v_{s}}\hat{\mu}_{f}\left(\mathbf{v}^{\prime}_{f}\right)=\sum_{f\in\mathcal{N}(x_{i})}\log\sum_{\mathbf{v}_{f}\in\mathcal{D}_{f}\mathbf{x}_{i}}\exp\boldsymbol{\phi}_{f}(\mathbf{v}_{f})$. This priority can be precomputed and identifies the next value to add in constant time. ### 2.3 Dynamic Message Scheduling After the selected value has been added to its respective domain, we perform message passing as described in Section 2.1 to converge to a fixed point of the new objective. To focus message updates in the areas affected by the modified domains, we use dynamic prioritization amongst messages [3, 11] with the dynamic range of the change in the messages (_residual_) as the choice of the message norm [5]. Formally: $\displaystyle\pi(f)=\max_{x_{i}\in\mathcal{N}_{f}}\max_{v_{i},v_{j}\in S_{i}}\log\cfrac{e(v_{i})}{e(v_{j})},~{}~{}~{}e=\cfrac{\boldsymbol{m}_{fi}}{\boldsymbol{m}^{\prime}_{fi}}$. As shown by Elidan et al. [3], residuals of this form bound the reduction in distance between the factor’s messages and their fixed point, allowing their use in two ways: first, we pick the highest priority message since it indicates the part of the graph that is least locally consistent. Second, the maximum priority is an indication of convergence and consistency; a low max- residual implies a low bound on the distance to convergence. ## 3 Experiments Our primary baseline is _Belief Propagation_ (BP) using random scheduling. We also evaluate _Residual BP_ (RBP) that uses dynamic message scheduling. Our first baseline that uses sparsity, _Truncated Domain_ (TruncBP), is initialized with domains that contain a fixed fraction of values ($0.25$) selected according to precomputed priorities (Section 2.2) and are not modified during inference. We evaluate three variations of our framework. _Random Instantiation_ (Random) is the baseline that the value to be added at random, followed by priority based message passing. Our approach that estimates the gradient of the dual objective is _Dynamic_ , while the approach that precomputes priorities is _Fixed_. Grids: Our first testbed for evaluation consists of $5\times 5$ and $10\times 10$ grid models (with domain size of $L=10,20,50,100,250$), consisting of synthetically generated unary and pairwise factors. (a) Total Variation Distance (b) L2 Error (c) Average Residual in Messages Figure 1: Runtime Analysis: for 10$\times$10 grid with domain size of $100$, averaged over $10$ runs. The runtime error for our approaches compared against the marginals obtained by BP at convergence (Figure 1) is significantly better than BP; up to $\bf 12$ times faster to obtain $L_{2}$ error of $10^{-7}$. TruncBP is efficient, however converges to an inaccurate solution, suggesting that prefixed sparsity in domains is not desirable. Similarly, Random is initially fast, since adding _any_ value has a significant impact, however as the selections become crucial, the rate of convergence slows down considerably. Although both Fixed and Dynamic provide desirable trajectories, Fixed is much faster initially due to constant time growth of domains. However as messages and marginals become accurate, the dynamic prioritization that utilizes them eventually overtakes the Fixed approach. To examine the anytime local consistency, we examine the average residuals in Figure 1(c) since low residuals imply a consistent set of marginals for the objective defined over the instantiated domain. Our approaches demonstrate low residuals throughout, while the residuals for existing techniques remain significantly higher (note the log-scale), lowering only near convergence. When the total domain size is varied in Figure 2, we observe that although our proposed approaches are slower on problems with small domains, they obtain significantly higher speedups on larger domains ($\bf 25-40$ times on $250$ labels). Joint Information Extraction: Figure 2: Convergence time for different domains: to $L_{2}<10^{-4}$ over $10$ runs of $5\times 5$ grids. # Entities 4 6 8 # Vars 16 36 64 # Factors 28 66 120 BP 41,193 91,396 198,374 RBP 54,577 117,850 241,870 Fixed 24,099 26,981 49,227 Dynamic 24,931 36,432 41,371 Figure 3: Joint Information Extraction: Avg time taken (ms) to $L_{2}<0.001$ We also evaluate on the real-world task of joint entity type prediction and relation extraction for the entities that appear in a sentence. The domain sizes for entity types and relations is $42$ and $24$ respectively, resulting in $42,336$ neighbor assignments for joint factors (details omitted due to space). Figure 3 shows the convergence time averaged over $5$ runs. For smaller sentences, sparsity does not help much since BP converges in a few iterations. However, for longer sentences containing many more entities, we observe a significant speedup (up to $\bf 6$ times). ## 4 Conclusions In this paper, we describe a novel family of _anytime_ message passing algorithms designed for marginal inference on problems with large domains. The approaches maintain sparse domains, and efficiently compute updates that quickly reach the fixed point of an approximate objective by using dynamic message scheduling. Further, by growing domains based on the gradient of the objective, we improve the approximation iteratively, eventually obtaining the BP marginals. ## References * Coughlan and Shen [2007] James Coughlan and Huiying Shen. Dynamic quantization for belief propagation in sparse spaces. _Computer Vision and Image Understanding_ , 106:47–58, April 2007. ISSN 1077-3142. * Coughlan and Ferreira [2002] James M. Coughlan and Sabino J. Ferreira. Finding deformable shapes using loopy belief propagation. In _European Conference on Computer Vision (ECCV)_ , pages 453–468, 2002. * Elidan et al. [2006] G. Elidan, I. McGraw, and D. Koller. Residual belief propagation: Informed scheduling for asynchronous message passing. In _Uncertainty in Artificial Intelligence (UAI)_ , 2006. * Ihler and McAllester [2009] Alexander Ihler and David McAllester. Particle belief propagation. In _International Conference on Artificial Intelligence and Statistics (AISTATS)_ , pages 256–263, 2009. * Ihler et al. [2005] Alexander Ihler, John W. Fisher III, Alan S. Willsky, and Maxwell Chickering. 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Ihler, W. T. Freeman, and A. S. Willsky. Nonparametric belief propagation. In _Computer Vision and Pattern Recognition (CVPR)_ , 2003. * Sutton and McCallum [2007] Charles Sutton and Andrew McCallum. Improved dynamic schedules for belief propagation. In _Uncertainty in Artificial Intelligence (UAI)_ , 2007. * Yedidia et al. [2000] J.S. Yedidia, W.T. Freeman, and Y. Weiss. Generalized belief propagation. In _Neural Information Processing Systems (NIPS)_ , number 13, pages 689–695, December 2000. * Yu et al. [2007] Tianli Yu, Ruei-Sung Lin, B. Super, and Bei Tang. Efficient message representations for belief propagation. In _International Conference on Computer Vision (ICCV)_ , pages 1 –8, 2007. ## Appendix A Algorithm The proposed approach is outlined in Algorithm 1. We initialize the sparse domains using a single value for each variable with the highest priority. The domain priority queue ($Q_{d}$) contains the priorities for the rest of the values of the variables, which remain fixed or are updated depending on the prioritization scheme of choice (Section 2.2). Message passing uses dynamic message prioritization maintained in the message queue $Q_{m}$. Once message passing has converged to obtain locally-consistent marginals (according to some small $\epsilon$), we select another value to add to the domains using one of the value priority schemes, and continue till all the domains are fully-instantiated. If the algorithm is interrupted at any point, we return either the current marginals, or the last converged, locally-consistent marginals. We use a heap-based priority queue for both messages and domain values, in which update and deletion take $O(\log n)$, where $n$ is often smaller than the number of factors and total possible values. Algorithm 1 Anytime Belief Propagation 1:$\forall x_{i},\mathcal{S}_{i}\leftarrow\\{v_{i}\\}$ $\triangleright$ where $v_{i}=\displaystyle\operatorname*{arg\,max}_{v_{s}}\pi_{i}(v_{s})$ 2:$Q_{d}\oplus\left\langle(i,v_{i}),\pi_{i}(v_{i})\right\rangle$ $\triangleright$ $\forall x_{i},v_{i}\in\mathcal{D}_{i}-\mathcal{S}_{i}$ 3:$Q_{m}=\\{\\}$ 4:while $\|Q_{d}\|>0$ do $\triangleright$ Domains are still partial 5: GrowDomain($\mathcal{S}$, $Q_{d}$) $\triangleright$ Add a value to a domain, Algorithm 2 6: ConvergeUsingBP($\mathcal{S}$, $Q_{m}$)$\triangleright$ Converge to a fixed point, Algorithm 3 7:end while$\triangleright$ Converged on full domains Algorithm 2 Growing by a single value (Section 2.2) 1:procedure GrowDomain($\mathcal{S},Q_{d}$) 2: $(i,v_{p})\leftarrow Q_{d}.pop$ $\triangleright$ Select value to add 3: $\mathcal{S}_{i}\leftarrow\mathcal{S}_{i}\cup\\{v_{p}\\}$ $\triangleright$ Add value to domain 4: for $f\leftarrow\mathcal{N}(x_{i})$ do 5: $Q_{m}\oplus\langle f,\pi(f)\rangle$ $\triangleright$ Update msg priority 6: end for 7:end procedure Algorithm 3 BP on Sparse Domains (Sect. 2.1, 2.3) 1:procedure ConvergeUsingBP($\mathcal{S},Q_{m}$) 2: while $max(Q_{m})>\epsilon$ do 3: $f\leftarrow Q_{m}.pop$ $\triangleright$ Factor with max priority 4: Pass messages from $f$ 5: for $x_{j}\leftarrow\mathcal{N}_{f};f^{\prime}\leftarrow\mathcal{N}(x_{j})$ do 6: $Q_{m}\oplus\langle f^{\prime},\pi(f^{\prime})\rangle$ $\triangleright$ Update msg priorities 7: $Q_{d}\oplus\langle(k,v_{q}),\pi_{k}(v_{k})\rangle$$\triangleright$ $\forall x_{k}\in\mathcal{N}_{f^{\prime}},\forall v_{k}$ 8: end for 9: end while$\triangleright$ Converged on sparse domains 10:end procedure	arxiv-papers	2013-11-14T02:39:45	2024-09-04T02:49:53.591248	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Sameer Singh and Sebastian Riedel and Andrew McCallum", "submitter": "Sameer Singh", "url": "https://arxiv.org/abs/1311.3368" }
1311.3402	# Colossal Thermoelectric Power Factor in K7/8RhO2 Y. Saeed, N. Singh, and U. Schwingenschlgl PSE Division, KAUST, Thuwal 23955-6900, Kingdom of Saudi Arabia ###### Abstract We discuss the thermoelectric and optical properties of layered KxRhO2 (_x_ = 1/2 and 7/8) in terms of the electronic structure determined by first principles calculations as well as Boltzmann transport theory. Our optimized lattice constants differ significantly from the experiment, but result in optical and transport properties close to the experiment. The main contribution to the optical spectra are due to intra and inter-band transitions between the Rh 4 _d_ and O 2 _p_ states. We find a similar power factor for pristine KxRhO2 at low and high cation concentartions. Our transport results of hydrated KxRhO2 at room temperature show highest value of the power factor among the hole-type materials. Specially at 100 K, we obtain a value of 3$\times$10-3 K-1 for K7/8RhO2, which is larger than that of Na0.88CoO2 [M. Lee _et al_., Nat. Mater. 5, 537 (2006)]. In general, the electronic and optical properties of KxRhO2 are similar to NaxCoO2 with enhanced transport properties in the hydrated phase. PACS: 71.15.Mb, 71.20.Dg, 72.15.Jf, 78.20-e Keywords: Layered oxides, Density functional theory, Transport and Optical properties, Layered cobalt oxides are of technological interest because of their transport properties key-1 ; key-2 ; key-3 ; key-4 ; Nat.Mat.-2006 . By varying the Na concentration in NaxCoO2 the system changes its behaviour from metallic to insulating and becomes superconducting near _x_ = 0.3 key-3 ; key-4 . Recently, a strong enhancement of the Seebeck coefficient has been reported for Na0.88CoO2 at T = 80 K Nat.Mat.-2006 , which greatly improves the prospects for thermoelectric applications. The peak value of Seebeck coefficient and the power factor were found to be 200 $\mu$V/K and 1.8$\times$10-3 K-1, respectivily, which is among the highest values for hole- type materials below 100 K. This fact has promoted huge interest in the isostructural and isovalent families AxCoO2 (A = K, Rb, Cs). Angle-resolved photoemission spectroscopy points to similar electronic and optical properties of K1/2CoO2 and Na1/2CoO2, which is also confirmed by band structure calculations key-14 ; key-15 ; key-16 ; key-20 . The electronic structures of hydrated and unhydrated NaxCoO2 is studied by first principles calculations singh-h2o . Analogous compounds with Rh in place of Co are found to be good thermoelectric materials with reduced correlation effects key-20 ; key-23 ; key-25 ; key-26 ; key-27 . Shibasaki _et al_. Rh-2011 have shown that the substitution of Rh ions in La0.8Sr0.2Co1-xRhxO${}_{{}_{3-\delta}}$ diminishes the magnetic moment of Co, where the thermopower is enhanced by a factor of 10 at _x_ = 1/2 as compared to _x_ = 0 and 1\. The electronic structure, optical and thermoelectric properties of K0.49RhO2 is investigated by Okazaki _et al._ optic-2011 ; krho-40uv/k and found a qualitative similarities in the optical conductivity spectra as compared to NaxCoO2. The experimental Seebeck coefficient of 40 $\mu$V/K (300 K) is reported krho-40uv/k . The temperature dependence of transport properties is different from those of NaxCoO2, which also suggests that the correlation in the Rh oxides is weaker than in NaxCoO2. An enhancement of the transport properties for an increasing concentration of alkali cations has been reported for other layered oxides as well Li-PRB-2011 . Though, various 4 _d_ systems have been investigated, the electronic structure in general is poorly understood so far Axrho-1 ; Axrho-2 . In order to throw light on the inter connection between enhancement of thermoelectric properties and a high cation concentration, first principles calculations of electronic, optical, and thermoelectric properties for KxRhO2 (_x_ = 1/2 and 7/8) are performed. The experimental data of optical and transport properties is taken from Refs. krho-40uv/k ; optic-2011 . A comparison to NaxCoO2 is given in terms of the chemical nature of the Co 3 _d_ and Rh 4 _d_ orbitals. The optical transitions are explained by the electronic band structure (BS) and density of states (DOS). The influence of a high cation concentration on power factor is also discussed, which is key for thermoelectric devices. Figure 1: Volume optimization of K1/2RhO2 for different exchange correlation functionals. Our calculations are based on density functional theory (DFT), using the full- potential linearized augmented plane wave approach as implemented in the WIEN2k code wien2k . This method has been successfully applied to describe the electronic structure of oxides U1 ; U2 including the optical spectrum N.Singh . The transport is calculated by the semiclassical Boltzmann theory within the constant scattering approximation, as implemented in the BoltzTraP code BoltzTraP . Various exchange and correlation functionals (local density approximation (LDA), generalized gradient approximation (GGA), GGA-sol, and GGA-PBE0) are used for optimizing the _c/a_ ratio. Since the differences are small, we will discuss in following only GGA-sol results for the electronic, optical and transport properties. The unit cell is divided into non-overlapping atomic spheres centered at the atomic sites and an interstitial region. The convergence parameter RmtKmax, where Kmax is the plane-wave cut-off and Rmt is the smallest of all muffin-tin radii controls the size of the basis set. This convergance parameter is set to 7 together with Gmax=24. We use 66 k-points in the irreducible wedge of the Brillouin zone for calculating the electronic structure and a dense mesh of 480 k-points in the optical calculations. For the transport calculations, we use 4592 k-points. Self-consistency is assumed to be achieved for a total energy convergence of $10^{-5}$ Ryd. Figure 2: Energy band structures of (a) K1/2RhO2 and (b) K7/8RhO2. KxRhO2 crystallizes in the $\gamma$-NaxCoO2 structure (space group P6${}_{\text{3}}$/mmc) with the experimental lattice constants _a_ = 3.0647 and _c_ = 13.6 yubuta-exp a/c . The CdI${}_{\text{2}}$-type RhO${}_{\text{2}}$ layer and the K layer are stacked alternately along the _c_ -axis. The experimental lattice constants of K1/2RhO2 are used as a starting point of the optimization, obtaining the optimized volume and _c/a_ ratio presented in Fig. 1. Our LDA calculation yields a $\sim$ 14% reduction of the _c/a_ ratio from 4.44 (experimental) to 3.84 (calculated), which is close to other layered Co/Rh oxides (_c/a_ ratio $\sim$ 3.8). We obtain lattice constants of _a_ = 3.02 and _c_ = 11.63 . The bonding length between Rh and O is reduced from 2.1326 to 2.0395 . To confirm the large deviation of the _c/a_ ratio from the experimental structure parameters, we have optimized the __ structure __ by more involved exchange correlation functionals (GGA, GGA-sol and PBE0). However, we obtain almost the same results for all functionals. In addition, our optimized lattice parameters are confirmed by calculations of the optical and transport properties, see the discussion below. The calculated Seebeck coefficient is overestimated for the experimental lattice constants, while the optimized lattice constants yield a Seebeck coefficient close to the experiment. The calculated optical conductivity of K1/2RhO2 (with optimized lattice constants) at zero photon energy is found to be $\sim$ 2500 $\Omega^{-1}$cm-1,which is agrees excellently with the experiment. The observed difference of the _c_ length between KxRhO2 and NaxCoO2 could be attributed to the different ionic radii of K and Na. However, our optimized value of _c_ is similar to other isostructural layered oxides such as SrxRhO2 SrRhO2 ; sro2006 , NaxCoO2 key-15 , LixNbO2 LiNbO2 , and LaxCoO2 LaCoO2 . In addition, LiRhO2, NaRhO2, and KRhO2 layered oxides can form a hydrate (water intercalated) phase Axrho-1 with an increased _c_ length Park . Takada _et al._ key-4 showed that NaxCoO2 can be readily hydrated to form NaxCoO2 _y_ H${}_{\text{2}}$O, maintaining the CoO${}_{\text{2}}$ layers, but with a considerably expanded _c_ axis, to accomodate the intercalated water. In the following, the optimized lattice constants are used, unless stated otherwise. Figure 3: DOS obtained for K1/2RhO2 and K7/8RhO2 . In Figs. 2 (a) and (b) the calculated electronic BSs of K1/2RhO2 and K7/8RhO2is presented. The BS of K1/2RhO2 is similar to isostructural and isovalent Na1/2CoO2, except for a slightly larger pseudogap. Moreover, the strongly dispersive bands and high hole concentration in K7/8RhO2 give rise to a high thermoelectricity. The calculated DOS (Fig. 3) shows a crystal field splitting experienced by the Rh4+ ions (splitting into $e_{g}$ and $t_{2g}$ states) is similar to the Co3+ case, but with a larger bandwidth. The bandwidth of the $t_{2g}$ state is 1.52 eV for Rh4+ and 1.46 eV for Co3+ in the case of K1/2CoO2 (not shown here). A similar increment is also observed for the $e_{g}$ states, in agreement with the experiment optic-2011 . In addition, a weak hybridization is observed between the Rh 4 _d_ and O 2 _p_ states at/below the Fermi level. The O _p_ states lie deep in the valence band (below $-$2 eV) as compared to Na0.50CoO2. The DOS of K7/8RhO2 shows increased bandwidths of the $e_{g}$ and $t_{2g}$ states, which reflects a reduction of the electronic correlation effects in K7/8RhO2 as compared to NaxCoO2. Figure 4: Optical reflectivity and conductivity of K1/2RhO2 and K7/8RhO2 along with experimental data optic-2011 . Blue and black dots respresent the calculated and experimental values at zero photon energy. The optical properties of KxRhO2 (_x_ = 1/2 and 7/8) are studied and presented in Figs. 4(a) and 4(b). The experimental results are taken from Ref. [17]. The obtained reflectivities of K1/2RhO2 and K7/8RhO2 (Fig. 4(a)) are similar to each other with a maximum value of $\sim$ 90% near 0 eV. A Drude-like edge in the optical reflectivity of K1/2RhO2 is found at $\sim$ 1 eV (experiment: 1.2 eV), while for K7/8RhO2 this edge appears at $\sim$ 0.5 eV. At zero photon energy, the calculated optical conductivity of $\sigma\sim$ 2500 $\Omega^{-1}$cm-1 for K1/2RhO2 is obtained in excellent agreement with the experiment (blue and black dots in fig. 4(b)). Three well defined peaks are observed: (i) near 1 eV due to the intra-band transition of Rh ($t_{2g}$-$t_{2g}$), (ii) at $\sim$ 3 eV due to the inter-band transition of Rh 4 _d_ ($t_{2g}$-$e_{g}$), and (iii) around 5.5 eV due to the inter-band transition from the O 2 _p_ to the Rh 4 _d_ $e_{g}$ states. These peaks are also present in the experiment optic-2011 as well as for Na1/2CoO2 (0.5 eV, 1.6 eV, and 3 eV, respectively) key-14 , which again reflects the similarity between these isostructural and isovalent compounds. In the following, we will address both the experimental crystal structure (hydrated) and optimized structure of KxRhO2. We have calculated the Seebeck coefficient (S), thermal conductivity ($\kappa$), and power factor (Z). The results are plotted in Figs. 5(a), (b), and (c) as a function of the temperature from 0 to 700 K, and compared with the experimental Z of Na0.88CoO2 Nat.Mat.-2006 . Fig. 5(a) shows that the calculated S of pristine KxRhO2 is $\sim$ 50 $\mu$V/K at 300 K, in agreement with the experiment (40 $\mu$V/K) krho-40uv/k . The calculated S values of hydrated K1/2RhO2 and K7/8RhO2 are strongly enhanced, amounting to $\sim$ 100 $\mu$V/K and $\sim$ 140 $\mu$V/K, respectively. The calculated S of pristine KxRhO2 hardly depens on the K concentration upto 300 K, while for higher temperature for K7/8RhO2 increases stronger to reach a value of 80 $\mu$V/K at 700 K. In contrast to this behavior, the calculated S of hydrated KxRhO2 remains almost constant above 300 K. According to Fig. 5(b) the thermal conductivity is similar for the hydrated compounds and for pristine K7/8RhO2, while its much enhanced for pristine K1/2RhO2. Figure 5: Calculated thermoelectric properties of prestine (solid line) and hydrated (dashed line) KxRhO2. Experimental data from Ref. Nat.Mat.-2006 is included. In Fig. 5(c), the calculated power factor Z of pristine and hydrated KxRhO2 along with experimental data for Na0.88CoO2 Nat.Mat.-2006 is presented. Upto 25 K, power factor of hydrated K7/8RhO2 behaves similar to the experimental curve, but approaches a value of 3$\times$10-3 K-1 at 100 K, which is much higher than that of other hole-type materials (like Na0.88CoO2 Z = 1.8$\times$10-3 K-1 at 80 K) in this temperature range. Even at room temperature (300 K), the calculated Z value for hydrated KxRhO2 is much higher than Na0.88CoO2, while for pristine KxRhO2 is just slightly greater. The large power factor in hydrated KxRhO2 results from a decrease of the thermal conductivity, increase in the electrical conductivity as reported for the hydrated phase of NaRhO${}_{\text{2}}$ (Fig. 2 of Ref. Park ), and larger S as compared to that of pristine KxRhO2. Therefore, the transport properties of hydrated KxRhO2 are highly promising for technological applications. Figure 6: KRhO2-hydrated In conclusion, the electronic, optical, and transport properties of layered KxRhO2 (_x_ = 1/2 and 7/8) are calculated, and compared to the isostructural and isovalent compound NaxCoO2. Our optimized structure of K1/2RhO2 shows a huge deviation in the _c/a_ ratio from the experiment but gives a good agreement for the optical and transport properties. The large deviations, also in comparison the other related compounds, indicate that the experimental structure has been determined for the hydrated phase of KxRhO2. The Rh4+4 _d_ $e_{g}$ and $t_{2g}$ states of K1/2RhO2 are broader than the respective Co3+ states in NaxCoO2, which confirms previous reports. The calculated Seebeck coefficient of pristine KxRhO2 (_x_ = 1/2 and 7/8) is S $\sim$ 50 $\mu$V/K at room temperature, which is close to the experimental value of S = 40 $\mu$V/K. Our calculations also show large values of S and Z for hydrated KxRhO2 in whole temperature range from 0 to 700 K. At around 100 K, the calculated Z of hydrated K7/8RhO2 is 3$\times$10-3 K-1, which is the highest value in any hole-type material at this temperature. At room temperature, the calculated Z value of pristine KxRhO2 is $\sim$ 0.4$\times$10-3 K-1, which is also slightly greter than that of Na0.88CoO2. Therefore, our results suggest that hydration can be used to elongate the structure and inducing the doping by the formation of hydronium ions, results strong enhancement of thermoelectric properties of this class of layered oxides. 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1311.3407	# Half-metallicity and giant magneto-optical Kerr effect in N-doped NaTaO3 Y. Saeed1, N. Singh1,2, and U. Schwingenschlgl1 1Physical Science & Engineering division, KAUST, Thuwal 23955-6900, Kingdom of Saudi Arabia 2Solar and Photovoltaic Energy Research Center, KAUST, Thuwal 23955-6900, Kingdom of Saudi Arabia ###### Abstract We employ density functional theory using the modified Becke-Johnson (mBJ) approach to investigate the electronic and magneto-optical properties of N-doped NaTaO3. The mBJ results reveal a half metallic nature of NaTaO2N, in contrast to results obtained by the generalized gradient approximation. We find a giant polar Kerr rotation of 2.16∘ at 725 nm wave length (visible region), which is high as compared to other half metallic perovskites as well as to the prototypical half metal PtMnSb. Density Functional Theory, MBJ, Ferromagnetic Half Metal, Magneto-optical properties ## I Introduction Room temperature ferromagnetism has been reported for different doped oxides such as C/N-doped ZnO Pan ; Shen ; yang , TiO2 YangDai-apl ; YangDai-cpl ; Tao , SnO2 Rahman ; Xiao and confirmed recently Nagare-zno ; Bao-tio2 ; Hong-sio2 . Room temperature ferromagnetism with half metallicity is reported for N-doped SrTiO3 and BaTiO3 Liu-sto-n ; Tan-bto-n , in which the magnetic interactions between the nearest and next-nearest N dopants result in a strong ferromagnetic coupling Yang-2011-sto-n-ferro . Ferromagnetic half-metals have potential applications in spintronics devices Engen ; Groot and also show unusual magneto-optical effects due to a metallic state for one spin channel and an insulating state for the other. Yang _et al_. have reported that a high concentration of N is required for achieving a magnetic long rang order in perovskite oxide Yang-2011-sto-n-ferro . The perovskite oxide NaTaO3 (NTO) is a ferroelectric material with high permittivity and low dielectric loss, which suggests usage in microwave devices Rabe ; Geyer ; Axelsson . Several ab-initio calculations have been performed to described the electronic properties of bulk NTO Choi-2011 but a detailed study of the electronic structure and magneto-optical properties of N-doped NTO is missing in the literature. The magneto-optical Kerr effect of doped NTO is interesting for magneto-optical reading and recording devices Fiebig . The N-doped perovskite oxide NTO is a 5d system. Therefore, electron correlation effects are expected to be small as compared to 3d systems such as SrTiO3 and BaTiO3. In the following we establish a half metallic nature for NaTaO1-xNx ($x=0.04-0.33$) and discuss the electronic structure in comparison to the strongly correlated perovskites SrTiO3 and BaTiO3. We also address the magneto-optical Kerr effect in N-doped NTO. ## II computational method Our calculations are based on density functional theory, using the full- potential linearized augmented plane wave approach as implemented in the WIEN2k code wien2k . We use the modified Becke-Johnson (mBJ) exchange correlation potential mBJ . The popular generalized gradient approximation (GGA) GGA-PBE is employed to optimize the volume and the internal atomic coordinates. In general, the unit cell is divided into non-overlapping atomic spheres centered at the atomic sites and an interstitial region. The convergence parameter RmtKmax, where Kmax is the plane-wave cut-off and Rmt is the smallest muffin-tin radius, controls the size of the basis set. This convergence parameter is set to 7 together with G${}_{max}=24$. We use 66 k-points in the irreducible wedge of the Brillouin zone for calculating the electronic structure and a dense mesh of 480 k-points in the magneto-optical calculations. The cubic phase _Pm $\overline{3}$m_ ($a=b=c=3.93$ Å and $\alpha=\beta=\gamma=90^{\circ}$) of NTO cubic-nto is used in the present calculations for simplicity because the differences to the monoclinic phase $P2/m$ ($a=3.8995$ Å, $b=3.8965$ Å, and $c=3.8995$ Å, $\alpha=\gamma=90^{\circ}$, and $\beta=90.15^{\circ}$) are subordinate monoclinc-nto . As a consequence, the electronic band structures and density of states (DOS) are found to be very similar in both phases wang-JPCC2011 ; Lin-APL-phases . Figure 1: Calculated spin polarization and majority spin hole density _n_ as a function of N-concentration, as obtained by the GGA+SOC approximation. ## III Results and Discussion Our optimized lattice parameter (using the GGA) of cubic NTO is 3.98 Å, which is in good agreement with the experimental value of 3.93 Å cubic-nto . We replace one O with one N to form the oxynitrate (NaTaO2N). The optimized lattice parameters of NaTaO2N is slightly increased to 4.03 Å. In order to find the magnetic ground state, we construct a $1\times 1\times 2$ supercell using the optimized structure and replace two O atoms with N. We compare the ground state energies of ferromagnetic (FM) and anti-ferromagnetic (AFM) configurations. The magnetic energy $E_{mag}=E_{FM}-E_{AFM}=-51.3$ meV, and the N-N distance is $\sim$4 Å with a total magnetic moment of 2 $\mu_{B}$ per cell (or 1 $\mu_{B}$ per N atom) in FM case. The Curie temperature $T_{C}$ is calculated using the mean-field Heisenberg model, i.e., $T_{C}=(2/3)E_{mag}/k_{B}$ Kudrnovsk ; Maca . The calculated $T_{C}$ for NaTaO2N is 396 K, which is close to that of N-doped SrTiO3 and BaTiO3 at the same N-N distance Yang-2011-sto-n-ferro . In order to observe the long range FM order, we study a high N-doping of 33% by replacing one O by one N in a single unit cell. A magnetic moment is induced as the delocalized N _p_ states become polarized, where 0.15 $\mu_{B}$ come from the interstitial, 0.13 $\mu_{B}$ come from O and 0.61 $\mu_{B}$ come from N, summing upto 1 $\mu_{B}$ per N atom. To explain the induced spin-polarization in NaTaO2N, we analyzes DOS and electronic band structure obtained by GGA approximation (not shown here). The DOS shows a half-metallic character with a metallic state for the minority spin and an insulating state for the majority spin. To confirm the half- metallicity, we include spin orbit coupling (SOC) along with GGA in the calculations, finding that NaTaO2N becomes a metal since the majority spin states crosses the Fermi level. The spin polarization ($=\frac{N\uparrow-N\downarrow}{N\uparrow+N\downarrow}$, where N is the number of states at the Fermi level) of NaTaO2N is obtained $\sim$94%. In order to find the exact N-concentration at which the character of the system changes from a half-metal to metal, we construct a $3\times 3\times 3$ supercell and vary the N-concentration from 4% to 33% (including SOC in the calculations). In Fig. 1(a), we plot the spin-polarization as a function of the N-concentration. Below 16% N-doped, NTO shows a $\sim$99.8% spin-polarization which decreases sharply to $\sim 94\%$ at 33% N-doping. In Fig. 1(b), we plot the hole density (holes per volume) for the majority spin channel. Similar to the spin-polarization, the hole density increases rapidly upto 46.8$\times 10^{24}$ m-3 as the N-concentration increases to 33%, while the hole density is almost constant for low N-concentration. Figure 2: Band structure and DOS of NaTaO2N as obtained by the mBJ approximation. Recently, Guo _et al._Guo-mBJLDA 2011 have applied the mBJ approach successfully to improve the half-metallic ferromagnetism in zincblende MnAs, which turns into a half-metal without affecting the _d_ _t_ 2g bands. We apply the same method to NaTaO2N. The calculated band structure and DOS in Fig. 2 show a truly half-metallic nature for NaTaO2N. The majority spin bands are similar to pristine NTO with a gap of 3.96 eV, which is in excellent agreement with experiments Lin-APL-phases and the previous GW calculations wang- JPCC2011 . The minority spin channel is metallic due to a non-zero DOS at the Fermi level. The band splitting at the Fermi level along R-$\Gamma$ and M-$\Gamma$ is very small, while along $\Gamma$-X-M, it is large. The calculated plasma frequency $\omega_{p}$ from the minority spin channel due to metallic nature, is 2.7 eV, which is smaller in the ferromagnetic half-metal PtMnSb ($\omega_{p}=4.5$ eV) Picozzi , reflecting less dispersed bands. The calculated DOS shows that the valence bands (majority spin) are a combination of N 2 _p_ and O 2 _p_ states. The bottom of the conduction bands is composed of Ta 5 _d_ states (see Fig. 2). For the minority spin channel, the N 2 _p_ bands cross the Fermi level (with small O 2 _p_ contributions). The N 2$p^{\uparrow\downarrow}$ states split into $(p_{x}+p_{y})^{\uparrow\downarrow}$ and $p_{z}^{\uparrow\downarrow}$ bands. There is no shifting of peak position with respect to energy is observed at the Fermi level in N 2$(p_{x}+p_{y})^{\uparrow\downarrow}$ and N 2$p_{z}^{\uparrow\downarrow}$ states from the N-doped SrTiO3 and BaTiO3 Yang-2011-sto-n-ferro where N 2$p_{y}+p_{z}$ and N 2$p_{x}$ have different peak position at the Fermi level. There is a strong hybridization between the N 2$p^{\uparrow\downarrow}$ and O 2$p^{\uparrow\downarrow}$ states for the minority spin channel. The Ta 5 _$d^{\uparrow\downarrow}$_ bands do not change with the N-concentration. Figure 3: Calculated polar Kerr angle $\theta_{K}$ and Kerr ellipticity $\varepsilon_{K}$ of NaTaO2N. Intense search aim at materials with large magneto-optical peaks in the low wave-length region to be used for high-density storage Grundy-high disk . Both borates borates and Zintl compounds Zintl ; Zintl1 , can shows a remarkable Kerr signal in the low energy range. The Kerr rotation $\theta_{K}$ and Kerr ellipticity $\varepsilon_{K}$ of half metallic NaTaO2N are shown in Fig. 3. We find a value of $\theta_{K}=2.16^{\circ}$ at 1.71 eV ($\sim$725 nm), which is higher than in BiNiO3 ($\theta_{K}=1.28^{\circ}$) M. Q. Cai-BiNiO3 and the Heusler compound PtMnSb ($\theta_{K}=1.27^{\circ}$) van Engen-1983 ; Lobove-2012-ptmnsb . The high Kerr angle is an intraband effect, and not due to the SOC (which creates an imbalance in the optical transitions in PtMnSb and NiMnSb van Ek-1997 , for example. For the minority spin channel, the band structure of NaTaO2N shows a set of parallel bands across the Fermi level (R-$\Gamma$, $\Gamma$-X-M, and $\Gamma$-M) which consist of N 2$(p_{x}+p_{y})^{\downarrow}$ states. These parallel bands give rise to intraband transitions which contribute significantly to the Kerr spectrum in the low energy range. In NaTaO2N, the separation between these bands is much smaller than in PtMnSb van Ek-1997 . This past explain the higher magneto- optical Kerr effect in NaTaO2N. The calculated Kerr ellipticity $\varepsilon_{K}$ has a maximum of $\sim$1.7∘ at 1.6 eV. ## IV Conclusion In conclusion, we have presented first principles results of the band structure, DOS, and magneto-optical properties of N-doped NaTaO3, as obtained from density functional theory. Our results for NaTaO1-xNx ($x=0.04-0.33$) show that the GGA+SOC approach gives a 99% spin-polarization at low N-concentrations upto 16%. 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1311.3410	# Impact of lattice strain on the tunnel magneto-resistance in Fe/Insulator/Fe and Fe/Insulator/La0.67Sr0.33MnO3 magnetic tunnel junctions A. [email protected], +1-818-677-2782, Y. Saeed2, N. Singh2, N. Useinov3, U. Schwingenschlö[email protected], +966(0)544700080 1Department of Physics, California State University, Northridge, California 91330, USA 2PSE Division, King Abdullah University of Science and Technology, Thuwal 23955, Saudi Arabia 3Department of Solid State Physics, Kazan Federal University, Kazan, Russia ###### Abstract The objective of this work is to describe the tunnel electron current in single barrier magnetic tunnel junctions within a new approach that goes beyond the single-band transport model. We propose a ballistic multi-channel electron transport model that can explain the influence of in-plane lattice strain on the tunnel magnetoresistance as well as the asymmetric voltage behavior. We consider as an example single crystal magnetic Fe(110) electrodes for Fe/Insulator/Fe and Fe/Insulator/La0.67Sr0.33MnO3 tunnel junctions, where the electronic band structures of Fe and La0.67Sr0.33MnO3 are derived by ab- initio calculations. ###### pacs: 72.10.Fk, 73.40.Gk, 75.45.+j, 75.47.De ## I INTRODUCTION One of the fast growing directions in modern magnetic electronics (spintronics) is the field of magnetic tunnel junctions (MTJs) and their applications, for example, as basic elements in magnetic random access memories, read-heads of hard drives, and magnetic field sensors. Potential to realize memristors and vortex oscillators creates additional incentive for future investments in this area Sp1 ; Sp2 . MTJs such as FM/Insulator/FM and FM/Insulator/HM heterostructures, where FM is a ferromagnet (like Co, Fe, CoFeB), the insulator is ferroelectric (like BaTiO3, PbTiO3), and HM is a half-metal (like La0.67Sr0.33MnO3, Co2MnSn), are very promising, because they combine magnetic, ferroelectric, and spin filtering properties. Tunnel electroresistance and tunnel magnetoresistance (TMR) effects may coexist in these systems. The TMR arises from states of different resistance for parallel and antiparallel magnetic alignments, while the tunnel electroresistance relies on the polarization of the ferroelectric insulator. The insulating layer has to be thick enough to yield strong ferroelectricity, which usually rapidly disappears for decreasing thickness, and has to be thin enough for electron tunneling. Moreover, the ferroelectric polarization in thin ferroelectric films is conjugated with the magnitude of the lattice strain Sp3 ; Sp4 ; Sp5 . A high ferroelectric polarization is achieved by epitaxial film growth with an initially high difference between the in-plane lattice parameters of the substrate and the deposited layers. Obviously, the electronic band structures and transport properties of the strained FM and HM layers can be fundamentally different from those without strain. The objective of this work is to establish the interplay between the lattice strain and the magnitude of the TMR using a multi-band approach for the electron transport. We predict that for strained symmetric MTJs the TMR is reduced, because of changes in the electronic band structure under strain. In general, the tunnel electroresistance in ferroelectric TJs should logarithmically increase with strain (the ferroelectric polarization increases), as it was shown, for instance, in the works of Zhuravlev and coworkers Sp6 ; Sp7 . This means there is a balanced configuration of the insulator thickness (potential barrier thickness) and strain that provides the highest TMR and tunnel electroresistance. To calculate the tunnel current and TMR we have to go beyond the assumption of two conduction channels (single- band model) similar to Refs. Sp8 ; Sp9 ; Sp10 ; Sp11 ; Sp12 ; Sp13 ; Sp14 . Investigation of MTJs has a long history Sp15 ; Sp16 ; Sp17 ; Sp18 . In Ref. Sp15, Valet and Fert have introduced basic principles for the qualitative and quantitative interpretation of the spin polarized electron transport in magnetic multilayer structures, based on Boltzmann-like equations. An alternative theoretical approach of electronic transport through nanocontacts with and without domain walls between two FM electrodes has been developed in Ref. Sp19, . This theory utilizes quasiclassical as well as quantum mechanical ideas and is based on extended Boltzmann-like equations. Boundary conditions on the interfaces of the junction are taken into account as a key part of the solution. The theory can be adapted to the case of ballistic transport through single barrier Sp12 and double barrier Sp20 planar junctions. Using the universality of the above technique, we formulate a multi-channel (or multi-band) approach following the ideas of Ref. Sp21, . The tunneling conductance in MTJs can be written in terms of the averaged spin-dependent tunneling probabilities of the conduction channels for parallel (P) and antiparallel (AP) magnetizations. According to our ab-initio calculations for Fe, several minority and majority spin bands cross the Fermi level, representing different electron wave functions. We extract the dispersion relations along the tunneling direction (perpendicular to the Fe(110) interface) from the bulk band structure. For simplicity, the insulator is considered to be homogeneous. Our approach does not incorporate filtering effects inside the barrier, which are important in the case of MgO or for the splitting of the valence band in ${\rm{SrTiO}}_{3}$ and ${\rm{BaTiO}}_{3}$, for instance Sp16 ; Sp22 ; Sp23 . ## II THE MULTI-CHANNEL APPROACH Figure 1: Simplified schema of the multi-channel model of a single crystal MTJ for positive bias (electrons tunnel from left to right). The model assumes independent propagation channels, each being associated with a given spin and symmetry. The ideas of the multi-channel approach are demonstrated in Fig. 1. In this model each propagating channel is associated with a given spin and symmetry of the wave function. The emitter provides electrons with different Fermi vectors, which tunnel across the barrier into the states of the collector. We employ a formula for the current density originally derived for transport through a magnetic planar junction Sp12 . For the single-band model the current density is proportional to the integral of the product of the transmission coefficient, $D^{{\rm{P}}\left({{\rm{AP}}}\right)}$, and the cosine of the incidence angle of the electron trajectory, $\cos\left({\theta_{L}^{\uparrow,\downarrow}}\right)$. The angle ${\theta_{L}^{\uparrow,\downarrow}}$ is measured from the normal (transport direction) to the interface plane ($L$: left, $R$: right). The integral is taken over $d\Omega_{L}=\sin\left({\theta_{L}}\right)d\theta_{L}d\phi$: $J_{\uparrow,\downarrow}^{{\rm{P}}\left({{\rm{AP}}}\right)}=\frac{{e^{2}V\left({k_{L}^{\uparrow,\downarrow}}\right)^{2}}}{{4\pi^{2}\hbar}}\left\langle{\cos\left({\theta_{L}^{\uparrow,\downarrow}}\right)D_{\uparrow,\downarrow}^{{\rm{P}}\left({{\rm{AP}}}\right)}}\right\rangle_{\Omega_{L}}.$ (1) Here ${k_{L}^{\uparrow,\downarrow}}$ is the absolute value of the Fermi vector of the left-hand electrode and $\uparrow,\downarrow$ is the spin index. The transmission coefficient is a function of the applied bias voltage $V$, of $\theta_{L}^{\uparrow,\downarrow}=0...\arccos\left({\sqrt{\left\|{1-\left({k_{R}^{\uparrow,\downarrow}/k_{L}^{\uparrow,\downarrow}}\right)^{2}}\right\|}}\right)$, and of $k_{L\left(R\right)}^{\uparrow,\downarrow}$. With $x^{\uparrow,\downarrow}=\cos\left({\theta_{L}^{\uparrow,\downarrow}}\right)$ we can write $\left\langle{x^{\uparrow,\downarrow}D_{\uparrow,\downarrow}^{{\rm{P}}\left({{\rm{AP}}}\right)}}\right\rangle_{\Omega_{L}}=\int\limits_{X^{\uparrow,\downarrow}}^{1}{x^{\uparrow,\downarrow}D_{\uparrow,\downarrow}^{{\rm{P}}\left({{\rm{AP}}}\right)}dx^{\uparrow,\downarrow}},$ where the lower limit $X^{\uparrow,\downarrow}$ for the integration arises from the conservation of the projection of the Fermi vector in the xy-plane: $k_{\parallel}^{\uparrow,\downarrow}=k_{L}^{\uparrow,\downarrow}\sin\left({\theta_{L}^{\uparrow,\downarrow}}\right)=k_{R}^{\uparrow,\downarrow}\sin\left({\theta_{R}^{\uparrow,\downarrow}}\right)$. It equals zero when the electrons tunnel from the left minority into the right majority conduction band and $X^{\uparrow,\downarrow}=\sqrt{\left\|{1-\left({k_{R}^{\uparrow,\downarrow}/k_{L}^{\uparrow,\downarrow}}\right)^{2}}\right\|}$ when they tunnel from the left majority into the right minority conduction band. For the multi-band approach the majority and minority bands can be both spin up and down for any magnetic configuration. To achieve a multi-channel model (or model with multi-band tunnel relations) for single crystal junctions we redefine the current density in Eq. (1): $J_{\uparrow,\downarrow}^{\text{P}\left({\text{AP}}\right)}=\frac{{e^{2}V}}{{4\pi^{2}\hbar}}\sum\limits_{\eta=1}^{N}{\sum\limits_{\mu=1}^{M}{\left({k_{\eta}^{\uparrow,\downarrow}}\right)^{2}\left\langle{\cos\left({\theta_{\eta}}\right)D_{\eta,\mu}^{\text{P}\left({\text{AP}}\right)}\left({k_{\eta}^{\uparrow,\downarrow},k_{\mu}^{\uparrow,\downarrow\left({\downarrow,\uparrow}\right)}}\right)}\right\rangle_{\Omega_{L}}}}.$ (2) Here $\eta$ and $\mu$ are the indices of the left-hand and right-hand bands, respectively, and $N$ and $M$ are the numbers of bands. The combinations $\\{\eta,\mu\\}$, see Fig. 1, identify the conduction relations between the bands through the barrier. Equation (2) is valid for positive bias. The solution for negative bias is derived using symmetric relations of the system, i.e., the collector and emitter are exchanged ($k_{\eta}\to k_{\mu}$, $k_{\mu}\to k_{\eta}$). We assume that there is no spin flip leakage and that a conduction channel is available between any left-hand and right-hand bands with the same spin. Otherwise the electrons are reflected back, giving rise to a resistance. Note that the lowest conductance corresponds to the largest difference in the density of states at the Fermi level between the left and right electrodes. Regarding the transmission coefficient for the single barrier system, the basic mathematical expressions can be found in Ref. Sp12, , where an exact quantum mechanical solution has been derived employing Airy functions for the tunnel barrier. The band structures obtained from ab-initio calculations for bulk Fe (space goup Cmmm) and La0.67Sr0.33MnO3 are shown in Figs. 2 and 3, as derived using the WIEN2k package Sp24 . The exchange-correlation potential is parametrized in the generalized gradient approximation Sp25 . For the wave function expansion inside the atomic spheres a maximum value of the angular momentum of $\ell_{max}=12$ is employed and a plane-wave cutoff of $R_{mt}K_{max}=9$ with $G_{max}=24$ is used. Self-consistency is assumed when the total energy variation reaches less than 10-4 Ry. We use a mesh of $10\times 10\times 10$ $k$-points for calculating the electronic structure in order to describe the ground states of the compounds with high accuracy. Figure 2: (Color online) Electronic bands for bulk Fe along the $\Gamma$-Z direction. $E-E_{F}=0$ corresponds to zero bias. Data are derived for different lattice parameters, which correspond to different lattice strains. The four band structures refer to (a),(e) $a=3.875$ Å, $c=3.083$ Å; (b),(f) $a=3.937$ Å, $c=2.986$ Å; (c),(g) $a=3.999$ Å, $c=2.894$ Å; (d),(h) $a=4.030$ Å, $c=2.850$ Å. The $\Gamma$ point is located at $k_{z}^{\uparrow,\downarrow}=0$ and the Z point is shown by vertical dotted lines. Figure 2 shows the band structure of Fe for different in-plane lattice parameters $a=3.875$ Å, 3.937 Å, 3.999 Å, and 4.030 Å (bulk value), where red and green color represent the two spins. As an example, we consider the symmetric Fe/Insulator/Fe junction and demonstrate how to collect the conducting spin channels via the applied bias $V$. The bands of the left electrode (emitter) are the same as those of the right electrode (collector) and the Fermi energies $E_{F}^{L}=E_{F}^{R}={{E}_{F}}$ are equal at zero bias. Horizontal dashed lines represent $E_{F}$, which intersects with the bands at the Fermi vectors $k_{\eta\left(\mu\right)}^{\uparrow,\downarrow}$. In particular, in Figs. 2(a), 2(b) and Figs. 2(e), 2(f) the system has two $k^{\downarrow}$ and two $k^{\uparrow}$ vectors at zero bias, while in the case of Figs. 2(c), 2(d), 2(g), 2(h) $E_{F}$ is intersected by three spin up and three spin down bands. We thus have the Fermi vector set $\\{$$k_{\text{1}L\text{(}R\text{)}}^{\uparrow,\downarrow}$, $k_{\text{2}L\text{(}R\text{)}}^{\uparrow,\downarrow}$, $k_{3L\text{(}R\text{)}}^{\uparrow,\downarrow}$$\\}$. In the case of positive (negative) bias, by definition, $E_{F}$ of the left electrode shifts up (down) in energy, while for the right electrode it shifts down (up) by the same amount. The voltage drop is $\left\|E_{F}^{L}-E_{F}^{R}\right\|=\left\|e{V}\right\|$. As a result the Fermi vector set is changed. As an example, let us set $V=+0.8$ V with $E_{F}^{L}=0.4$ eV and $E_{F}^{R}=-0.4$ eV. According to Figs. 2(a) and (e), for the left electrode this results in the Fermi vector sets $\\{$0, 0, $k_{3L}^{\uparrow}\\}$ and $\\{k_{1L}^{\downarrow}$, $k_{2L}^{\downarrow}$, $k_{3L}^{\downarrow}\\}$ and for the right electrode in the sets $\\{k_{1R}^{\uparrow}$, $k_{2R}^{\uparrow}$, $k_{3R}^{\uparrow}$ and $k_{2R}^{\downarrow}$, $k_{3R}^{\downarrow}\\}$, which generates $1\times 3=3$ channels for spin up and $3\times 2=6$ channels for spin down, for the parallel magnetization. In contrast, $1\times 2=2$ channels for spin up and $3\times 3=9$ channels for spin down are generated in case of the antiparallel magnetization. Thus, the current can be represented by $3\times 3=9$ channels for each spin orientation (in the general case: $N\times M$). When the Fermi vectors vanish we have, of course, a non-conducting channel with vanishing current density. Figure 3: (Color online) Electronic bands for bulk La0.67Sr0.33MnO3 along the $\Gamma$-Z direction. $E-E_{F}=0$ corresponds to zero bias. Data are derived for different lattice parameters, which correspond to different lattice strains. The two band structures refer to (a),(c) $a=3.875$ Å, $c=23.250$ Å and (b),(d) $a=4.030$ Å, $c=21.496$ Å. The $\Gamma$ point is located at $k_{z}^{\uparrow,\downarrow}=0$ and the Z point is shown by vertical dotted lines. Figure 3 shows the band structure of La0.67Sr0.33MnO3 along the $\Gamma$-Z direction for two sets of lattice parameters: $a=3.875$ Å, $c=23.250$ Å and $a=4.030$ Å, $c=21.496$ Å. The Fermi vector, transmission coefficient, and current density for each band are derived as demonstrated before. However, some of the spin down bands are very flat with energy gaps between them, in contrast to the spin up bands. As a function of the bias the system therefore switches between a HM and FM. However, there are also energies at which neither spin up nor spin down states exist. ## III TUNNEL MAGNETORESISTANCE UNDER STRAIN Physical parameters that characterize the properties of MTJs are the total tunnel current density $J^{\text{P}\left(\text{AP}\right)}={{\left({{J}_{\uparrow}}+{{J}_{\downarrow}}\right)}^{\text{P}\left(\text{AP}\right)}}$, the $\text{TMR}={\left({{J}^{\text{P}}}-{{J}^{\text{AP}}}\right)}/{{{J}^{\text{AP}}}}\times 100\%$, the normalized ${\rm{TMR}}_{\rm{n}}=\left({J^{\rm{P}}-J^{{\rm{AP}}}}\right)/J^{{\rm{AP}}}\times{\rm{TMR}}^{-1}\left({V=0}\right)$, and the output voltage ${{V}_{\text{out}}}={{V}\left({{J}^{\text{P}}}-{{J}^{\text{AP}}}\right)}/{{{J}^{\text{AP}}}}$, which can be obtained from free-electron Sp11 ; Sp26 or tight-binding Sp13 ; Sp14 models. However, unfortunately these models do not reproduce the experimental effect of strain on the charge transport characteristics. A single-band approach is sufficient to model the TMR in amorphous sputtered MTJs Sp27 and can satisfactorily describe the ${\rm{TMR}}_{\rm{n}}$ and ${{V}_{\text{out}}}$ of epitaxial single and double barrier FeCoB/MgO junctions Sp28 . In our case we have to go beyond parabolic dispersions and the single-band model, however, keeping the simplicity of the approach. Figure 4: (Color online) TMR versus applied voltage for Fe/Insulator/Fe MTJs with the lattice parameters: $a=3.875$ Å, 3.937 Å, and 4.030 Å. The barrier parameters are $d=1.8$ nm and $U_{B}=2.8$ eV. For the Fermi vectors derived above as well as for typical parameters of an Al2O3 tunnel barrier, TMR results derived by Eq. (2) are shown in Figs. 4 to 6. The barrier thickness is set to $d=1.8$ nm, the barrier height above $E_{F}$ to $U_{B}=2.8$ eV, and the effective mass to $m_{B}=0.25$ Sp29 . In our calculations for metals the effective mass is equal to the free electron mass. Figure 4 presents the TMR as function of the bias for different lattice parameters, showing that the TMR, in general, behaves non-monotonically. For unstrained Fe ($a=4.030$ Å) a decreasing in-plane lattice parameter (increasing strain) leads to a lower TMR. Figure 5 gives the TMR as a function of the lattice parameter for 0.1 mV and 0.1 V bias. Interestingly, we observe deviations from a linear behavior: For almost zero bias the TMR increases up to 31.1% for $a=3.999$ Å, 28.6% for $a=4.030$ Å, and 27.3% for $a=3.968$ Å. This behavior is related to modifications in the reflection of the majority states at the Z point, where the Fermi vector achieves its maximal magnitude (Fig. 2, dashed rectangles). Note that these states give the main contribution to the tunnel current. The observed differences for different in-plane lattice parameters are explained by variations of the band structure. The dashed rectangles in Figs. 2(e-h) demonstrate the bands near the Z point. For $a=3.999$ Å, see Fig. 2(g), the majority band intersects the Fermi level at the Z point, favoring $J^{\rm{P}}$ over $J^{\rm{AP}}$, in contrast to the other lattice parameters. The maximal TMR value close to zero bias is in good agreement with the results of Yuasa and coworkers for Fe(110)/Al2O3/Fe50Co50, see Fig. 3(b) in Ref. Sp29, , and of Hauch and coworkers for Fe(110)/MgO(111)/Fe(110), 28% at $T=300$ K Sp30 . Figure 5: (Color online) TMR as function of the lattice parameter $a$ for Fe/Insulator/Fe MTJs. Black and red color refer to biases of 0.1 mV and 0.1 V, respectively. In the case of the Fe/Insulator/La0.67Sr0.33MnO3 MTJ our model gives a positive TMR for $V>0.11$ V as well as a negative TMR below, see Fig. 6. The TMR curves are qualitatively similar to those obtained experimentally for Co/SrTiO3/La0.7Sr0.3MnO3 Sp31 and agree with the room temperature TMR in ${\rm{Fe/MgO/Co}}_{2}{\rm{MnSn}}$ Sp32 (about $-5$% at a bias of 0.1 mV). However, according to these authors the TMR is suppressed in the voltage range $\|V\|\geq 0.5$ V, which is probably related to enhanced spin scattering for high bias. TMR curves are given in Fig. 6 for the in-plane lattice parameters $a=4.030$ Å and $a=3.875$ Å, where the latter corresponds to unstrained La0.67Sr0.33MnO3. For positive bias the magnitude of the TMR decreases with the Fe lattice strain, whereas for negative bias the situation is reversed. For the circled points in Fig. 6, where the TMR goes to zero, both spin channels are closed, compare the energy gaps in Fig. 3, because of $J^{\rm{P}}=J^{{\rm{AP}}}=0$. There are other points where the TMR is zero as $J^{\rm{P}}=J^{\rm{AP}}$. Variation of the effective mass in the tunnel barrier leads to a weak response of the TMR in symmetric (1.5% decrease) and a strong response in asymmetric (14% increase) junctions, for all lattice parameters close to zero bias, $m_{B}=1$. Figure 6: (Color online) TMR versus applied bias for the Fe/Insulator/La0.67Sr0.33MnO3 MTJ. The barrier parameters are $d=1.8$ nm, $U_{B}=2.8$ eV, and $m_{B}=0.25$. ## IV CONCLUSION We have extended an established quasi-classical ballistic transport model to multi-channel conductance, which has enabled us to investigate the role of the electronic band structure and the effect of strain on the transport properties of single crystal Fe/Insulator/Fe and Fe/Insulator/La0.67Sr0.33MnO3 MTJs. Our approach takes into account all bands of the FM and HM along the $\Gamma$-Z direction (direction of tunneling). We have demonstrated for typical parameters of an Al2O3 tunnel barrier a maximal TMR of 31.1% for the Fe/Insulator/Fe MTJ, which is in good agreement with the experiment. A negative TMR of 5% is found for the Fe/Insulator/La0.67Sr0.33MnO3 MTJ close to zero bias, where the dependence on the bias reproduces experimental findings. The developed technique thus has demonstrated great potential for further studies on transport properties (including the spin transfer torque) in simple and magnetic TJs. Strain effects on the TMR have been explored theoretically for the first time by a multi-band approach. For the Fe/Insulator/Fe MTJ it turnes out that for small bias the TMR decreases linearly with the in-plane strain at the interface, whereas in the case of the Fe/Insulator/La0.67Sr0.33MnO3 MTJ the strain effects strongly depend on the sign of the applied bias. For positive bias it is positive and maximal for unstrained Fe, while for negative bias it is negative and the amplitude increases with strain and bias. The observed relations between the strain and the TMR are explained by variations of the band structure. We have demonstrated that in-plane strain can increase and decrease the TMR and therefore makes it possible to obtain optimal regimes for MTJ applications. ## References * (1) A. Dussaux, B. Georges, J. 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1311.3415	# Influence of substitution on the optical properties of functionalized pentacene monomers and crystals: Experiment and theory Y. Saeed1, K. Zhao1, N. Singh1,2, R. Li1, J. E. Anthony3, A. Amassian1, and U. Schwingenschlögl1 1KAUST, Physical Science & Engineering division, Thuwal 23955-6900, Kingdom of Saudi Arabia 2Solar and Photovoltaic Energy Research Center, KAUST, Thuwal 23955-6900, Kingdom of Saudi Arabia 3Department of Chemistry, University of Kentucky, Lexington, Kentucky 40506-0055 ###### Abstract The influence of solubilizing substitutional groups on the electronic and optical properties of functionalized pentacene molecules and crystals have been investigated. Density functional theory is used to calculate the electronic and optical properties of pentacene, TIBS-CF3-pentacene, and TIPS- pentacene. The results are compared with experimental absorption spectra of solutions and the complex dielectric function of thin films in the 1 eV to 3 eV energy range. In all cases, the band gaps of the isolated molecules are found to be smaller than those of the crystals. The absorption spectra and dielectric function are interpreted in terms of the transitions between the highest occupied molecular orbitals and lowest unoccupied molecular orbitals. The bands associated to C and Si atoms connecting the functional side group to the pentacene in the (6,13) positions are found to be the main contributors to the optical transitions. The calculated dielectric functions of thin films agree with the experimental results. A redshift is observed in crystals as compared to molecules in experiment and theory both, where the amplitude depends on the packing structure. ## I Introduction The design of molecular semiconductors is increasingly important for the development of organic electronics and organic photovoltaics (OPV) Zaumseil ; WANG ; DODABALAPOR . Early on pentacene had proven to be one of the best performing molecular semiconductors, as vacuum-deposited organic thin film transistors have achieved a mobility as high as 6 cm2 V-1 s-1 Li ; Jurchescu ; Kim . However, it did not lend itself well to solution processing, which is believed to be key for low-cost manufacturing of organic semiconductors. In recent years, chemical modification of the acene has made it possible to overcome the low solubility and poor stability in solution, whilst maintaining or enhancing the inter-molecular orbital overlap Anthony ; Subramanian . Functional substitution of pentacene has been shown to induce favorable crystal packing motifs for both electronic and OPV applications Subramanian ; Shu . For example, pentacene without substitution shows a two-dimensional (2D) herringbone packing motif (see Fig. 1a), while the popular compound (6,13)-bis(tri-iso-propyl-silyl-ethynyl)-pentacene (TIPS-Pn) shows a brickwork 2D crystal packing (see Fig. 1c) Ostroverkhova . The latter is currently one of the organic semiconductors exhibiting the highest field-effect mobility Jackson , with recent carrier mobility reports exceeding 4 cm2 V-1 s-1 Bao- Nature . When changing substitution from tri-iso-propyl-silyl-ethynyl to tri- iso-butyl-silyl-ethynyl and introducing a tri-fluoro-methyl group on the acene backbone to modify the energy levels, to get 2-tri-fluoro-methyl-(6, 13)-bis-(tri-iso-butyl-silyl-ethynyl)-pentacene (TIBS-CF3-Pn), the crystal packing changes to one-dimensional (1D) sandwich herringbone (see Fig. 1b). This molecule is found to perform as one of the best non-fullerene acceptor molecules when mixed with P3HT donor polymer, yielding a power conversion efficiency of 1.28% Shu . The substitutional chemistry employed affects the electronic properties of the monomer as well as of the solid state material itself. Meng _et al._ Meng have demonstrated that adjusting the alkyl substitution to the four terminal positions (2, 3, 9, and 10) of the pentacene chromophore shifts the energies of both the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) without significantly changing the gap between these two. When substituting all hydrogen atoms of pentacene with fluorine atoms some interesting changes of the extinction coefficient are found as the optical band gap is redshifted Hinderhofer . Recently, Lim _et al._ lim have shown that the HOMO–LUMO energy levels can be tuned by varying the number of nitrile groups in cyano-pentacene substitution. From the above reports, a close relationship appears to exist between the substitution on the pentacene chromophore and its electronic and optical properties. To tailor and improve these properties, one should first understand the correlation between the chemical modification (like silyl-ethynyl substitution as in the cases of TIPS-Pn and TIBS-CF3-Pn) and the physical properties of the derivatives both in monomer and in crystalline states. The electronic and optical properties of pentacene in solution have been previously calculated using first principles methods Tiago-DFT . The calculated optical spectra of the vapor phase are found to be in agreement with the measurements performed on the thin film phase of pentacene. Doi _et al._ Doi-DFT have calculated the electronic band structures for both the single crystal and thin film polymorphs of pentacene and concluded that the effective mass of the electrons or holes is larger in the single crystal. A first principles simulation of the thin film phase of pentacene shows a crucial dependence of the bandwidths of the HOMO and LUMO and of the band gap on the molecular stacking angles Parisse . The electronic structures of iodine- and rubidium-doped pentacene molecular crystals have also been investigated by ab-initio calculations based on the ultrasoft pseudopotential method, predicting a metallic behavior Shichibu-DFT . Recently, the structural and electronic properties of pentacene multilayers on the Ag(111) surface have been studied, revealing that pentacene has no electronic contribution at the Fermi level Mete . Figure 1: Molecular structure (top) and crystal packing (bottom) of (a) pentacene, (b) TIBS-CF3-Pn, and (c) TIPS-Pn . Despite several experimental and theoretical investigations, the influence of the solubilizing chemical substitutions and the resulting changes of the crystal packing on the electronic and optical properties of pentacene have not been reported. In this work, we study and compare the theoretical (single molecule and single crystal) and experimental (dissolved and thin film polycrystal) optical properties of pentacene, TIPS-Pn, and TIBS-CF3-Pn. The results are analyzed in light of the calculated density of states (DOS) to see the influence of different alkyl-silyl groups on the electronic and optical properties of both monomer and crystal of these materials useful to in electronic and OPV applications. ## II Experiments and Characterization Pentacene, toluene (anhydrous 99.8%) and 1,3,5-trichlorobenzene (anhydrous 99%) were purchased from Sigma Aldrich and used without further purification. TIPS-Pn and TIBS-CF3-Pn were synthesized Chaung-exp-Anthony and purified by multiple recrystallization from acetone (TIPS-Pn) or ethanol (TIBS-CF3-Pn). Pentacene was dissolved in 1,3,5-trichlobenzene at 100∘C with a concentration of 0.5 wt.% and stirred overnight in the dark. TIPS-Pn and TIBS-CF3-Pn were dissolved in toluene at room temperature and stirred overnight in the dark. The solutions were filled in a 1 mm thick quartz cuvette and loaded in a Cary 5000 (Varian) instrument to aquire UV-vis absorption spectra. The measurements were performed over a spectral range from 300 nm to 2000 nm with a 2.0 nm slit width. Single crystal Si(100) wafers with a thermal oxide layer of 100 nm thickness were used as substrate for the thin film deposition. Prior to deposition, the substrates were cleaned in amonium hydroxide (30% NH4OH), hydrogen peroxide (30% H2O2) and Milli Q (1:1:5 ratio) for 15 min at 70∘C. Thin films of TIPS-Pn and TIBS-CF3-Pn were spin cast at 1000 rpm for 30 seconds in a N2-filled glove-box and left to dry in inert atmosphere at room temperature. The optical properties of the spin-coated films were measured using variable angle spectroscopic ellipsometry (VASE) based on the M-2000XI rotating compensator configuration (J. A. Woollam Co. Inc). VASE spectra ranging from 0.734 eV to 5.895 eV were recorded at a 18∘ angle of incidence with respect to the substrate normal from 45∘ to 80∘ with 2∘ increment. In the paper, we focus on the spectral range from 1 eV to 3 eV. Optical analysis of VASE data was performed using the EASETM and WVASE32 software packages (J. A. Woollam Co. Inc). Optical modeling was performed assuming a homogeneous thin film exhibiting uniaxial anisotropy. To describe the dielectric behavior, a general oscillator approach consisting of Gaussian peaks in the imaginary part of the dielectric function $\varepsilon_{2}(E)$ was applied (more detailed information about the fitting procedure and the Gaussian parameters can be found in the supporting information). All optical measurements were performed at room temperature in ambient air. ## III Simulations Our calculations are based on density functional theory, using the full- potential linearized augmented plane wave (FP-LAPW) approach as implemented in the WIEN2k code wien2k . This approach describes the ground state of the present compound with high accuracy udo1 . On the other hand, calculation of optical spectra, in principle, involves excited states. Thus, additional approximations have to be introduced, which, however, do not compromise the following line of reasoning nsingh1 . Exchange and correlation effects are treated within the local density approximation LDA . In the FP-LAPW method, the unit cell is divided into two parts: non-overlapping atomic spheres centered at the atomic sites and the interstitial region. The convergence parameter $R_{mt}\cdot K_{max}$, where $K_{max}$ is the plane wave cut-off and $R_{mt}$ is the smallest of the atomic sphere radii, controls the size of the basis set. It is set to $R_{mt}\cdot K_{max}=5$ with $G_{max}=24$. A mesh of 48 uniformly distributed k-points in the irreducible wedge of the Brillouin zone is used for calculating the electronic properties and a dense mesh of 112 k-points is used to calculate the optical properties. A total energy convergence of at least $10^{-5}$ Ry is achieved. The experimental lattice parameters of pentacene ($a=5.959$ Å, $b=7.596$ Å, $c=15.610$ Å, $\alpha=81.25^{\circ}$, $\beta=86.56^{\circ}$, and $\gamma=89.90^{\circ}$), TIPS-Pn ($a=7.565$ Å, $b=7.750$ Å, $c=16.835$ Å, $\alpha=89.15^{\circ}$, $\beta=78.42^{\circ}$, and $\gamma=83.63^{\circ}$), and TIBS-CF3-Pn ($a=17.203$ Å, $b=16.552$ Å, $c=18.168$ Å, $\alpha=90^{\circ}$, $\beta=113.34^{\circ}$, and $\gamma=90^{\circ}$) are used. Figure 2: Comparison of the pentacene DOS with TIBS-CF3-Pn and TIPS-Pn in both molecular and crystalline forms. ## IV Results and discussion In Fig. 1, we show the molecular and single crystal packing structures of pentacene, TIPS-Pn, and TIBS-CF3-Pn. Pentacene (C22H22) and TIPS-Pn (C44H54Si2) both exhibit triclinic ($P\bar{1}$) crystal symmetries, while TIBS-CF3-Pn (C57H65F3Si2) has monoclinic ($P21/c$) symmetry. In Fig. 2, we show the calculated projected DOS for pentacene, TIBS-CF3-Pn, and TIPS-Pn for the molecule and crystal in the energy range $\pm 3.5$ eV. The calculated band gaps of pentacene in the monomer and crystalline phases are found to be 0.84 eV and 0.74 eV, respectively, in agreement with previous calculations. The low band gap in single crystal pentacene may be due to the increase in the bandwidth of the HOMO and the LUMO as compared to the monomer (Fig. 2b). The band gaps in TIBS-CF3-Pn and TIPS-Pn are found to be 1 eV and 0.85 eV, respectively, in the monomer phases, and 0.80 eV and 0.45 eV, respectively, in the single crystal phases. The band gap of the TIPS-Pn single crystal is the lowest amongst these molecules owing to the largest LUMO bandwidth amongst the materials investigated. The calculated band gaps are lower than in experiments, due to the well known drawback of the local density approximation. The DOS shows more localized peaks for the pentacene molecule than its derivatives (Fig. 2). Due to the increase of the HOMO and LUMO bandwidths from monomers to crystals, the bands overlap (below $E_{F}$) in agreement with previous calculations for pentacene Doi-DFT . The HOMO and LUMO consist mainly of bands belonging to the C and Si atoms which connect the side group to the pentacene chromophore. The H bands appear 3.5 eV below $E_{F}$ for pentacene and its derivatives, while the F bands in TIBS-CF3-Pn lie between $-6.0$ eV and $-7.5$ eV (not shown). This means that the electronic states of the H and F atoms do not contribute to the optical transitions in the visible energy range and do not participate significantly in the conversion of sunlight into electricity via absorption. Figure 3: Spectra of $\alpha$, $\varepsilon_{2}$, and $\varepsilon_{1}$ for pentacene, TIBS-CF3-Pn, and TIPS-Pn in both molecule (monomer) and crystal form (experiment and simulation). To establish the effect of the various functional groups on the optical properties of pentacene and its derivatives, we have calculated their absorption spectra and dielectric function. The peaks in the optical spectra are determined by the electric-dipole transitions between the HOMO and LUMO. Since the local density approximation underestimates the band gap, the calculated spectra are shifted by difference between experimental and theoretical band gap to facilitate visual comparison with experimental spectra. In Fig. 3, we present the absorption coefficient $\alpha(E)$ as well as the photon energy-dependent complex dielectric function, $\widetilde{\varepsilon}=\varepsilon_{1}(E)-i\varepsilon_{2}(E)$. An average of the computed optical spectra along the three coordinate axes is taken and compared with the average of the experimental optical spectra along the axes parallel and perpendicular to the plane of substrate. The absorption coefficients, $\alpha(E)$, of pentacene and its derivatives are calculated for a single molecule and compared with experimental data of a dilute solution (see Figs. 3(a,b,c)). In the case of the pentacene solution, the experimental absorption spectrum shows several peaks at 2.13 eV, 2.30 eV, 2.47 eV, and 2.86 eV. TIBS-CF3-Pn and TIPS-Pn monomers exhibit peaks at 1.94 eV, 2.10 eV, 2.26 eV, 2.45 eV and 2.82 eV. The first absorption peak in pentacene (2.13 eV) is more intense than the other three peaks, while the first two peaks in TIBS- CF3-Pn and TIPS-Pn are more intense than the others. A red shift of 0.22 eV between the first absorption peak of pentacene and of both TIBS-CF3-Pn and TIPS-Pn, may be due to the effect of the substitutional groups in the latter two derivatives. The absorption spectrum of pentacene for the thin film exhibits four absorption peaks at 1.82 eV, 2.13 eV, 2.30 eV, and 2.86 eV which are redshifted as compare to the monomer. In the case of TIBS-CF3-Pn, the HOMO- LUMO absorption bands are shifted to lower energies with respect to the monomer spectrum, having peaks at 1.87 eV, 2.04 eV, 2.22 eV, and 2.81 eV. A similar phenomenon is observed for TIPS-Pn thin films which show peaks at 1.78 eV, 1.92 eV, 2.08 eV, and 2.76 eV. The shifts of the HOMO-LUMO absorption band are different in thin films of these materials with respect to the monomers, namely 0.20 eV for pentacene, 0.07 eV for TIBS-CF3-Pn, and 0.16 eV for TIPS- Pn. The different shifts may be attributed to the different crystalline packing structures. The CF3 group does not have any contribution because the C and F states (see the DOS) are well below the Fermi level. The calculated and experimental absorption coefficients also show a redshift between the monomer and the crystal. The calculations for monomers exhibit a single absorption peak, while the experiment shows more than one peak which is consistent with the DOS. The DOS of monomers shows the single sharp LUMO and HOMO bands (allow only one transition peak) while that of for thin film have wider LUMO and HOMO bands, which can have more transitions in absorption spectra. This may be due to the complete isolation of the molecule in the calculation, which may not be the case in solutions. The calculated $\varepsilon_{2}(E)$ spectra along with their experimental counterparts for thin film pentacene and its derivatives are presented in Figs. 3(d,e,f). The experimental $\varepsilon_{2}(E)$ spectra of TIBS-CF3-Pn and TIPS-Pn show two initial peaks at 1.85 eV and 1.91 eV, respectively, reflecting the optical band gap. The optical band gap is redshifted by 0.06 eV in TIPS-Pn as compared to TIBS-CF3-Pn. The third peak is situated at 2.26 eV and 2.19 eV for TIBS-CF3-Pn and TIPS-Pn, respectively. Another significant difference is the intensity of the third peak, which dominates in the $\varepsilon_{2}(E)$ spectrum of TIBS-CF3-Pn while in TIPS-Pn the first peak is most prominent. The $\varepsilon_{2}(E)$ spectrum changes dramatically by introducing a Si-branch in TIBS-CF3-Pn and TIPS-Pn, which is due to the modified crystal packing. The calculated $\varepsilon_{2}(E)$ spectra of TIBS- CF3-Pn and TIPS-Pn are in qualitative agreement with our experiments. The experimental $\varepsilon_{1}(E)$ spectra of TIBS-CF3-Pn and TIPS-Pn thin films show the first transition peaks at energies of 1.80 eV and 1.86 eV, respectively, while second and third peaks position remain at the same energies in both crystals. This reflects that the alky-silyl length results in changes of the energy state of the first transition peak in $\varepsilon_{1}(E)$. The subsequent peaks at 2.18 eV might be associated with a vibronic energy state between the Si HOMO and C LUMO. The calculated $\varepsilon_{1}(E)$ spectra of crystals and monomers of pentacene, TIBS- CF3-Pn, and TIPS-Pn show similar characteristics. The $\varepsilon_{1}(E)$ spectra of the TIBS-CF3-Pn and TIPS-Pn crystals demonstrate three peaks similar to the experimental results. Overall, the calculated $\varepsilon_{1}(E)$ spectra of single crystals of TIBS-CF3-Pn and TIPS-Pn are in agreement with our experimental thin film results. In conclusion, the effects of substitution on the electronic and optical properties have been discussed based on experiments and theoretical results. In the monomer state, the alkyl-silyl substitutions result in an energy shift of 0.22 eV (experimental) in TIBS-CF3-Pn and TIPS-Pn as compared to pentacene. In the crystal state, the alkyl-silyl substitution contributes to different packing structures, which leads to a redshift by 0.09 eV in TIPS-Pn as compared to TIBS-CF3-Pn. The HOMO-LUMO absorption band in thin films is shifted towards lower energies as compared to the monomer, by 0.07 eV and 0.16 eV for TIBS-CF3-Pn and TIPS-Pn, respectively. Our first principles calculation of optical spectra have been analyzed in terms of the calculated DOS. The optical transitions originate primarily from C and Si bands. A redshift is observed from monomer to crystal for all compounds, where the extent of redshift depends on the packing structure. 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B 23, 5048 (1981). * (26) R. D. McCullough, Adv. Mater. 10, 93 (1998) ## V Supporting Information The complex dielectric function $\widetilde{\varepsilon}=\varepsilon_{1}-i\varepsilon_{2}$ is related to the complex refraction index $\widetilde{n}=n-ik$ by the following equations: $\varepsilon_{1}=n^{2}-k^{2}$ and $\varepsilon_{2}=2nk$. Here, $n$ and $k$ are the refractive index and extinction coefficient, respectively. Kramers-Kronig transformation was used during the model fitting as a constraint: $\varepsilon_{1}(\omega)=1+\frac{2}{\pi}P\int_{0}^{\infty}\frac{\omega^{{}^{\prime}}\epsilon_{2}(\omega^{{}^{\prime}})}{\omega^{{}^{\prime}2}-\omega^{2}}d\omega^{{}^{\prime}}$ (1) $\varepsilon_{2}(\omega)=-\frac{2\omega}{\pi}P\int_{0}^{\infty}\frac{\epsilon_{1}(\omega^{{}^{\prime}})-1}{\omega^{{}^{\prime}2}-\omega^{2}}d\omega^{{}^{\prime}}$ (2) The mean square error was used to quantify the difference between experimental and model-generated data: $MSE=\sqrt{\frac{1}{3n-m}\sum_{i=1}^{n}\left[(N_{E_{i}}-N_{G_{i}})^{2}+(C_{E_{i}}-C_{G_{i}})^{2}+(S_{E_{i}}-S_{G_{i}})^{2}\right]}\times 1000$ (3) where $n$ is the number of wavelengths, $m$ is the number of fit parameters, and $N=\cos(2\Psi)$, $C=\sin(2\Psi)\cos(\Delta)$, $S=\sin(2\Psi)\sin(\Delta)$. Where, $\Psi$ and $\Delta$ are the amplitude ratio and phase shift, respectively. The $MSE$ generated is 12.12 and 15.5 for TIPS-Pn and TIBS-CF3-Pn with all angles variable from 45∘ to 80∘, with 2∘ increment, respectively. Gaussian oscillators produce a Gaussian line shape in $\varepsilon_{2}$: $\begin{split}&\varepsilon_{2}=\sum_{i}^{n}A_{n}\Bigg{(}\left[\Gamma(\dfrac{E-E_{n}}{\sigma_{n}})+\Gamma(\dfrac{E+E_{n}}{\sigma_{n}})\right]+\\\ &i\cdot\left(\exp\left[-(\dfrac{E-E_{n}}{\sigma_{n}})^{2}\right]+\exp\left[-(\dfrac{E+E_{n}}{\sigma_{n}})^{2}\right]\right)\Bigg{)}\end{split}$ (4) where $\sigma_{n}=B_{n}/(2\sqrt{\ln(2)})$ and $n$ is the oscillator number, $A_{n}=\varepsilon_{2}(E_{n})$ is the amplitude, $E_{n}$ (eV) is the center energy and $B_{n}$ (eV) is the full width at half maximum of the peak. The function $\Gamma$ is a convergence series that produces a Kramers-Kronig consistent line shape for $\varepsilon_{1}$. Table 1: Parameters of the modified Gaussian model obtained by fitting the imaginary part of dielectric function $\varepsilon_{2}(E)$ of TIBS-CF3-Pn. $\varepsilon_{2xx}(E)$=$\varepsilon_{2yy}(E)$ \| $\varepsilon_{2zz}(E)$ ---\|--- $\varepsilon_{\infty}$=1.901$\pm$0.006 \| $\varepsilon_{\infty}$=2.052$\pm$0.015 UV pole amplitude=11.361$\pm$0.315 \| UV pole amplitude=4.831$\pm$0.546 UV pole energy=6.883$\pm$0.024 \| UV pole energy=6.663$\pm$0.073 A1=0.887$\pm$0.009 \| B1=0.125$\pm$0.001 \| E1=1.868$\pm$0.001 \| A1=0.285$\pm$0.009 \| B1=0.159$\pm$0.007 \| E1=1.872$\pm$0.003 A2=0.395$\pm$0.002 \| B2=0.115$\pm$0.001 \| E2=2.040$\pm$0.000 \| A2=0.255$\pm$0.021 \| B2=0.127$\pm$0.013 \| E2=2.050$\pm$0.008 A3=0.156$\pm$0.002 \| B3=0.137$\pm$0.002 \| E3=2.204$\pm$0.001 \| A3=0.563$\pm$0.037 \| B3=0.155$\pm$0.011 \| E3=2.244$\pm$0.006 A4=0.123$\pm$0.002 \| B4=0.206$\pm$0.005 \| E4=2.368$\pm$0.004 \| A4=3.225$\pm$0.028 \| B4=0.670$\pm$0.003 \| E4=3.873$\pm$0.008 A5=0.133$\pm$0.003 \| B5=0.510$\pm$0.018 \| E5=2.838$\pm$0.006 \| A5=2.177$\pm$0.099 \| B5=0.125$\pm$0.034 \| E5=3.795$\pm$0.002 A6=0.110$\pm$0.015 \| B6=0.119$\pm$0.001 \| E6=3.304$\pm$0.006 \| A6=0.252$\pm$0.034 \| B6=0.337$\pm$0.017 \| E6=4.569$\pm$0.007 A7=0.453$\pm$0.002 \| B7=0.119$\pm$0.001 \| E7=3.744$\pm$0.000 \| A7=0.308$\pm$0.019 \| B7=0.657$\pm$0.081 \| E7=4.993$\pm$0.018 A8=0.629$\pm$0.001 \| B8=0.989$\pm$0.002 \| E8=4.187$\pm$0.002 \| A8=0.605$\pm$0.005 \| B8=0.975$\pm$0.069 \| E8=5.847$\pm$0.014 A9=0.235$\pm$0.002 \| B9=0.814$\pm$0.011 \| E9=5.076$\pm$0.006 \| \| \| A10=0.082$\pm$0.006 \| B10=0.145$\pm$0.013 \| E10=5.416$\pm$0.006 \| \| \| A11=0.617$\pm$0.004 \| B11=0.531$\pm$0.006 \| E11=5.649$\pm$0.003 \| \| \| Table 2: Parameters of the modified Gaussian model obtained by fitting the imaginary part of dielectric function $\varepsilon_{2}(E)$ of TIPS-Pn. $\varepsilon_{2xx}(E)$=$\varepsilon_{2yy}(E)$ \| $\varepsilon_{2zz}(E)$ ---\|--- $\varepsilon_{\infty}$=2.012$\pm$0.030 \| $\varepsilon_{\infty}$=2.251$\pm$0.023 UV pole amplitude=4.360$\pm$1.999 \| UV pole amplitude=3.889$\pm$0.443 UV pole energy=7.139$\pm$0.139 \| UV pole energy=6.190$\pm$0.033 A1=0.830$\pm$0.002 \| B1=0.080$\pm$0.002 \| E1=1.906$\pm$0.002 \| A1=1.087$\pm$0.027 \| B1=0.105$\pm$0.003 \| E1=1.891$\pm$0.001 A2=0.353$\pm$0.004 \| B2=0.080$\pm$0.001 \| E2=2.069$\pm$0.000 \| A2=0.666$\pm$0.028 \| B2=0.111$\pm$0.006 \| E2=2.053$\pm$0.006 A3=0.266$\pm$0.002 \| B3=0.497$\pm$0.005 \| E3=2.223$\pm$0.002 \| A3=1.566$\pm$0.055 \| B3=0.154$\pm$0.005 \| E3=3.503$\pm$0.002 A4=0.508$\pm$0.005 \| B4=0.198$\pm$0.003 \| E4=4.139$\pm$0.007 \| A4=2.709$\pm$0.203 \| B4=0.132$\pm$0.011 \| E4=3.886$\pm$0.005 A5=1.232$\pm$0.314 \| B4=0.204$\pm$0.003 \| E5=4.236$\pm$0.009 \| A5=1.532$\pm$0.141 \| B5=0.159$\pm$0.012 \| E5=4.256$\pm$0.013 A6=2.963$\pm$0.028 \| B6=0.467$\pm$0.001 \| E6=5.666$\pm$0.006 \| A6=1.344$\pm$0.047 \| B6=0.234$\pm$0.004 \| E6=4.132$\pm$0.016 A7=0.396$\pm$0.004 \| B7=0.963$\pm$0.022 \| E7=3.506$\pm$0.002 \| A7=0.467$\pm$0.005 \| B7=1.954$\pm$0.045 \| E7=5.419$\pm$0.006 A8=0.939$\pm$0.009 \| B8=0.586$\pm$0.007 \| E8=3.868$\pm$0.003 \| A8=0.400$\pm$0.045 \| B8=0.160$\pm$0.018 \| E8=5.737$\pm$0.008 \| \| \| A9=0.835$\pm$0.020 \| B9=0.453$\pm$0.141 \| E9=4.088$\pm$0.008	arxiv-papers	2013-11-14T08:29:06	2024-09-04T02:49:53.622393	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Y. Saeed, K. Zhao, N. Singh, R. Li, J. E. Anthony, A. Amassian, and U.\n Schwingenschl\\\"ogl", "submitter": "Yasir Saeed Mr.", "url": "https://arxiv.org/abs/1311.3415" }
1311.3496	Andrea Contu111The workshop was supported by the University of Manchester, IPPP, STFC, and IOP on behalf of The LHCb collaboration INFN Sezione di Cagliari, Italy and CERN, Switzerland > High-precision measurements performed by the LHCb collaboration have opened > a new era in charm physics. Several crucial measurements, particularly in > spectroscopy, rare decays and $CP$ violation, can benefit from the increased > statistical power of an upgraded LHCb detector. The upgrade of LHCb > detector, its software infrastructure, and the impact on charm physics are > discussed in detail. > PRESENTED AT > > > > > The 6th International Workshop on Charm Physics > (CHARM 2013) > Manchester, UK, 31 August – 4 September, 2013 ## 1 Introduction The LHC has performed excellently during its first years of operation allowing the four main experiments to collect large data samples at unprecedented centre-of-mass energies. The LHCb detector outperformed its design specification and played a crucial role in the advancement of charm physics. The LHCb measurements range from the charm cross-section at $\sqrt{s}=7\,\mathrm{TeV}$ [1], to direct and indirect $CP$ violation, neutral charm meson mixing, spectroscopy, and rare decays. These measurements exploit the large charm cross-section at the LHC and the outstanding performance of the trigger and reconstruction system of LHCb, which allowed unprecedented charm yields to be available for precision analyses. Charm physics plays a crucial role in the LHCb upgrade programme as well, in which the sensitivity for several key observables is expected to reach or exceed the theoretical precision. In this paper, the LHCb upgrade, both from the hardware and software point of view, is outlined. Prospects for charm physics in the LHCb upgrade era are discussed and extrapolations of the expected sensitivities for several observables are listed. The scientific value of these advances has been recognised by the CERN research board, which approved the upgrade of LHCb to be part of the long-term exploitation of the LHC. ## 2 The LHC upgrade schedule The first running phase of the LHC, with $pp$ centre of mass energy of 7 and 8 TeV, ended at the beginning of 2013. Currently, the LHC machine and the four experiments are in a 18-months shutdown (LS1) for maintenance and consolidation. Data taking will be resumed at the beginning of 2015 with a $pp$ center of mass energy of $13\textendash 14\,\mathrm{TeV}$. The spacing between consecutive proton bunches circulating in the accelerator is foreseen to go from $50\,\mathrm{ns}$ to the nominal $25\,\mathrm{ns}$, effectively doubling the $pp$ collision rate. From the beginning of 2018 a second long shutdown (LS2) is expected to last about a year, followed by three years of running up to 2022, after which a luminosity upgrade of the LHC is foreseen. It is noted that this schedule is likely to evolve with time. ## 3 The current LHCb detector and its upgrade The LHCb detector [2] is a single-arm forward spectrometer covering the pseudorapidity range $2<\eta<5$, designed for the study of particles containing $b$ or $c$ quarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the $pp$ interaction region, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift tubes placed downstream. The combined tracking system provides a momentum measurement with relative uncertainty that varies from 0.4% at 5 $\mathrm{GeV}/c$ to 0.6% at 100 $\mathrm{GeV}/c$, and impact parameter resolution of 20 $\mathrm{\mu m}$ for tracks with large transverse momentum. Different types of charged hadrons are distinguished by information from two ring-imaging Cherenkov detectors [3]. Photon, electron and hadron candidates are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic calorimeter and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers [4]. The upgraded LHCb detector is expected to be installed in 2018, during LS2, and is currently being designed to perform as well as or better than the current one at a higher instantaneous luminosity. The physics goal for the upgrade is to reach a sensitivity at the level of the theoretical prediction (or better) in several key observables. Therefore, in order to keep the same level of performance in harsher conditions, improvements in the trigger, reconstruction strategy, and detector technology are mandatory. The total integrated luminosity collected at the end of the LHCb upgrade data taking is expected to reach $70\,\mathrm{fb^{-1}}$. ### 3.1 Trigger strategy The current trigger scheme is based on a multi-stage approach with a first level, hardware-based, trigger and two software levels that have access to the full event information (see Figure 1(a)). The output rate of the first hardware-based trigger level, which uses information on transverse momentum, $p_{T}$, and transverse energy $E_{T}$, is limited by a maximum bandwidth of 1.1 MHz. At higher luminosity, this constraint would require using tighter $p_{T}$ and $E_{T}$ cuts in hadronic triggers in order for the computing infrastructure to cope with the increased event rate and size. This will also cause the trigger efficiency for hadronic channels to deteriorate, as shown in Figure 2. On the other hand, events that are selected by muonic triggers will be mostly unaffected since the muon system is already capable of sustaining a higher instantaneous luminosity to some extent. The effect is even more pronounced for charm hadrons, which are produced at a lower $p_{T}$ than $b$ hadrons. (a) Current trigger system (b) Trigger system in the upgrade Figure 1: Overview of current and planned LHCb trigger system. Figure 2: Trigger yield for several $B$ decays as a function of the instantaneous luminosity in the current trigger scheme. $B^{0}\to\pi^{+}\pi^{-}$ is represented by black squares, $B^{0}\to\phi\gamma$ by red triangles, $B^{0}_{s}\to J/\psi\phi$ by green upside-down triangles and $B_{s}^{0}\to D_{s}^{+}K^{-}$ by blue circles. The inefficiency in for the hadronic triggers will also affect the charm yield achievable. A new trigger strategy for the upgrade is being studied in which the first level hardware trigger is completely removed and the events are sent directly to a software trigger running on a larger and more powerful CPU farm, as shown in Figure 1(b). This new scheme is not affected by the “bandwidth bottleneck” after the first trigger level so that the event rate that can be processed and stored on disk depends only on the capabilities of the CPU farm. The final event output rate is expected to be a factor of four larger than the current one. ### 3.2 Tracking system and RICH upgrade One of the obvious effects of the increased instantaneous luminosity is a higher occupancy and radiation dose for all the subdetectors. Layout and technology improvements are needed to cope with the harsher conditions of the upgrade. In the following, the main changes introduced for the upgraded detector are described. Particular focus is given to the tracking system and the RICH detectors. At higher luminosity, the particle flux increases dramatically in the regions close to the beam axis, therefore a major upgrade is foreseen for the whole LHCb tracking system. The current VELO is based on semicircular silicon-strip sensors arranged in two rows that close around the interaction regions during data taking. While the moving layout will be kept, the baseline choice for the upgrade consists of silicon pixel sensors with an aggressive micro-channel cooling system. The new VELO sensor layout and the micro-channel cooling scheme are shown in Figure 3. The sensor choice is driven the necessity to reduce the occupancy allowing for a faster track reconstruction and low fake-track rate. (a) New VELO silicon pixel sensor layout (b) Micro-channel cooling technology Figure 3: The VELO sensors in the upgraded LHCb. The current Trigger Tracker (TT) will be replaced by the Upstream Tracker (UT). The UT is currently being designed to have a lower material budget (less than $5\%\,X_{0}$), and to have higher granularity and extended angular coverage compared with the TT. A comparison of the performance for tracks reconstructed using only information from the VELO and the UT (TT), the so- called upstream tracks, in the current detector and in the upgrade scenario, is shown in Figure 4. Figure 4: Transverse momentum resolution for tracks reconstructed using only information from the vertex detector and the upstream tracker. The performance of the current LHCb detector is shown in black, and the baseline upgrade configuration in green. It is noted that combination of information from the upgraded VELO and UT tracking leads to a considerable improvement in $p_{T}$ resolution compared with the current VELO+TT. The high track multiplicity in the central region also drives the upgrade of the current downstream tracking stations, located between the dipole magnet and the RICH2 detector. Several detector technologies are currently under study, with the baseline choice being the replacement of the entire inner tracking system (composed of a silicon strip tracker in the inner region and a straw tube outer tracker) with a design known as the Sci-Fi detector (see Figure 5). The Sci-Fi detector exploits scintillating fibres as the active material. The scintillation light from the fibres is read-out by silicon-based photo-multipliers. Figure 5: Options for the replacement of the current downstream tracking stations. From left to right: replacement of the silicon-strip detector and straw tubes in the central region (outer straw tubes are kept), scintillating fibres detector only in the central region (outer straw tubes are kept), entire downstream tracking station using scintillating fibres technology (baseline). The current RICH system is composed of two detectors, RICH1 and RICH2, located upstream and downstream of the dipole magnet, respectively. In order to cover a wide momentum range, three radiators are used: aerogel (solid) and $\mathrm{C_{4}F_{10}}$ (gaseous) in RICH1, and $\mathrm{CF}_{4}$ in RICH2. In the upgrade, due to increased occupancy the aerogel, which covers the low momentum range $1\textendash 10\,\mathrm{GeV}/c$, will be removed. Moreover, the current Hybrid-PhotoDetectors will be replaced by Multi-Anode photo- multipliers which will require new front-end electronics. The optics of both RICH1 and RICH2 will also be optimised. ## 4 Prospects for charm physics LHCb has a broad upgrade physics programme of which charm measurements are an important part. The large charm production cross-section at $\sqrt{s}=7\,\mathrm{TeV}$, recently measured at LHCb [1], is predicted to increase by a factor of 1.8 at $\sqrt{s}=14\,\mathrm{TeV}$. Exploratory studies indicate that improvements in the trigger strategy could provide an increase of a factor two for the trigger efficiency on charm hadronic decays. The improvement is even more pronounced in multibody decays. In the upgrade era, the charm signal yield is expected to increase by a factor of about 3.6 per $\mathrm{fb}^{-1}$. Since the integrated luminosity recorded per year is expected to also increase by a factor 3.5 per year, the total charm yield per year could increase by one order of magnitude. ### 4.1 Production and spectroscopy Charm production and spectroscopy are very active areas of research in LHCb. Recent studies of double-charm production observed as double-charmonium, charmonium and open charm, and double open charm [5] can in principle be extended to simultaneous charmonium and bottomonium production in the upgrade era. The search for new $D_{sJ}$ states [6] will also benefit enormously from an increased statistics. Improvements are also expected in studies of $\chi_{c(1,2,3)}$ production, $J/\psi$ polarisation, and charmed and doubly charmed baryons. ### 4.2 Rare decays Charm rare decays are very powerful means to search for new mediators and couplings. The current overview of for $D^{0}$ decays is shown in Figure 6. LHCb results on $D^{0}\to\mu^{+}\mu^{-}$ [7] (see Figure 6) and multibody decays, such as $D^{+}_{(s)}\to\pi^{+}\mu^{+}\mu^{-}$ and $D^{+}_{(s)}\to\pi^{-}\mu^{+}\mu^{+}$ [8] and $D^{0}\to\pi^{+}\pi^{-}\mu^{+}\mu^{-}$ [9], are already available and improved previous measurements by one or two orders of magnitude. Figure 6: Current limits on rare $D^{0}$ decays [10]. Multibody decays may proceed via an intermediate resonance, e.g. $D^{+}_{(s)}\to\pi^{+}\phi$ and then $\phi\to\mu^{+}\mu^{-}$. In this context the rare decay searches mentioned above are for the non-resonant modes. However, the resonant modes are themselves of interest for an angular analyses. There is particular interest in the study of forward-backward asymmetries, T-odd correlations and near-resonance effects. Decay modes with intermediate resonances in the dimuon mass can already be seen in the current LHCb data sample. The statistical precision required for angular analyses is expected to be available at the end of the LHCb upgrade. It is noted that hadronic modes are a dangerous background to rare decay searches, having a branching fractions $\mathcal{O}(10^{6})$ larger than typical predictions for electroweak $D$ meson decays in the SM. While this background is greatly reduced with information from the muon chambers, decays in flight of high momentum pions into muons can easily mimic a genuine muon directly from a $D$ decay in such a way that hadronic decays become an irreducible background. Since the discriminating power is currently reaching a limit, improvements in the muon identification in the upgrade are one key ingredient for the progress in this area. ### 4.3 Mixing Charm mixing is already established by a series of complementary measurements although considerable improvements are still needed in the precision with which the mixing parameters $x$ and $y$ are known. The LHCb collaboration, analysing data collected during the 2011 run only, made the first single measurement to exclude the no-mixing hypothesis to a level above five standard deviations [11]. The analysis, based on the study of the time-dependent ratio between wrong- (WS) and right-sign (RS) $D^{0}\to K^{\mp}\pi^{\pm}$, is a perfect demonstration of the LHCb’s statistical power. The updated analysis based on the complete Run 1 LHCb data sample ($3\,\mathrm{fb}^{-1}$), also contains the most precise determination of the mixing parameters $x^{\prime}$ and $y^{\prime}$ and a search for $CP$ violation [12]. Another observable which give access to the mixing parameters is $y_{CP}$, defined as the ratio between the effective lifetime for decays to $CP$-even eigenstate ($K^{+}K^{-}$ or $\pi^{+}\pi^{-}$) and Cabibbo-favoured decays to the $CP$-mixed final state $K^{-}\pi^{+}$. The current measurement from LHCb, based on a small data sample collected in 2010, proves the feasibility of the measurement at hadron machines [13]. An updated measurement, which uses the 2011 dataset, is in progress. The large yields available in the upgrade will allow a more refined treatment of backgrounds that will reduce the systematic uncertainty affecting the measurement. Other mixing measurement under study within the LHCb collaboration include: * • $x^{2}+y^{2}$ using the time integrated WS/RS ratio of $D^{0}\to K^{+}\mu^{-}\nu$ decays * • Direct access to $x$ and $y$ via a time-dependent Dalitz plot measurement of $D^{0}\to K_{S}hh$ decays * • Access to $x^{\prime\prime 2}$ and $y^{\prime\prime}$ via a time-dependent WS/RS Dalitz plot measurement of $D^{0}\to K^{+}\pi^{+}\pi^{0}$ The sensitivities expected for several mixing observables, extrapolated to an integrated luminosity of $50\,\mathrm{fb^{-1}}$ (note that the expected luminosity has increased since these estimates were made in Ref.[14]), are summarised in Table 1. Decay \| Observable \| Exp sensitivity $[\times 10^{-3}]$ (stat only) ---\|---\|--- $D^{0}\to KK$ \| $y_{CP}$ \| 0.04 $D^{0}\to\pi\pi$ \| $y_{CP}$ \| 0.08 $D^{0}\to K^{+}\pi^{-}$ \| $x^{\prime 2}$,$y^{\prime}$ \| 0.04,0.1 $D^{0}\to K_{S}\pi\pi$ \| $x$,$y$ \| 0.15,0.1 $D^{0}\to K^{+}\mu^{-}\nu$ \| $R_{M}=x^{2}+y^{2}$ \| 0.0001 Table 1: Projection of statistical sensitivities for mixing observables with $50\,\mathrm{fb^{-1}}$ [14]. ### 4.4 Indirect $CP$ violation As well as the $CPV$ search in the time-dependent wrong-sign $D^{0}\to K^{+}\pi^{-}$ decay mentioned previously, LHCb is carring out a search for indirect $CP$ violation in the charm sector through the measurement of $A_{\Gamma}$ [15]. The parameter $A_{\Gamma}$, defined as the asymmetry between the effective lifetimes of $D^{0}$ decays into a $CP$ eigenstate, is an almost clean measurement of indirect $CP$ violation and can expressed as $A_{\Gamma}=\frac{1}{2}(A_{m}+A_{d})y\cos\phi-x\sin\phi\approx-a^{ind}_{CP}-a^{dir}_{CP}y_{CP},$ (1) where $A_{m}=1-\|q/p\|$, $A_{d}=1-\|A_{f}/\overline{A}_{f}\|$ and $\phi$ is the relative $CP$ violating phase between $q/p$ and $\overline{A}_{f}/A_{f}$. In Eq. 1 it is manifest that this measurements benefits from a precise determination of the mixing parameters $x$ and $y$, which are expected to be constrained at a $10^{-4}$ level in the upgrade. Since the overall precision on $A_{\Gamma}$ at the end of the upgraded LHCb data-taking is expected to be better than $10^{-4}$, a precision independent measurement of the direct $CP$ violating component is necessary to probe the SM prediction for $A_{\Gamma}$ which is set to about $10^{-4}$. In addition to the mixing parameters, $D^{0}\to K_{S}h^{+}h^{-}$ decays which give also access to $CP$ violating quantities such as $\|q/p\|$ and $\phi$, making this a “golden-channel” for the LHCb upgrade. These parameters are accessible via a the time-dependent evolution in the $K_{S}\pi\pi$ Dalitz plane. Two strategies are possible: an unbinned, model-dependent measurement in which a full amplitude fit is performed, and a model-independent measurement that instead uses prior experimental measurements of the average strong phase difference in regions of the Dalitz-plot (e.g. from CLEOc and BESIII). Although such decays suffer from a relatively low reconstruction efficiency in LHCb, mainly due to the $K_{S}$ long lifetime, precise measurements of $x$, $y$, $q/p$ and $\phi$ can already be performed with the existing data samples and will be greatly improved in the upgrade. ### 4.5 Direct $CP$ violation Measurements of direct $CP$ violation are challenge for experiments at hadron colliders. In fact, several sources of asymmetry can bias the measurement such as the production asymmetry present in proton-proton collisions. Moreover, analyses can be affected by detection asymmetry biases. Therefore, independent measurements of production and detection asymmetries are a crucial ingredient for direct $CP$ violation searches in charm. These measurements are currently being performed within the LHCb collaboration [16, 17, 18] and will be pursued in the upgrade phase. It is interesting to note that if detection and production asymmetries are small, observables can be constructed in which they cancel at the first order. This fact is exploited in the measurement of $\Delta A_{CP}=A_{CP}(K^{+}K^{-})-A_{CP}(\pi^{+}\pi^{-})$ in prompt [19] and semileptonic [20] decays performed by LHCb. The improved detector and the larger statistics of the LHCb upgrade are therefore vital to reduce the statistical and systematic uncertainties and shed light on the still unclear picture of direct $CP$ violation in the charm sector. The sensitivities for several direct $CP$ violating observables are given in Table 2, assuming an integrated luminosity of $50\,\mathrm{fb^{-1}}$. Decay \| Observable \| Exp sensitivity $[\times 10^{-3}]$ (stat only) ---\|---\|--- $D^{0}\to KK,\pi\pi$ \| $\Delta A_{CP}$ \| 0.15 $D^{+}\to K_{S}K^{+}$ \| $A_{CP}$ \| 0.1 $D^{+}\to K^{-}K^{+}\pi^{+}$ \| $A_{CP}$ \| 0.05 $D^{+}\to\pi\pi\pi$ \| $x$,$y$ \| 0.08 $D^{+}\to hh\pi$ \| $CPV$ in phases \| $(0.01-0.10)^{\circ}$ $D^{+}\to hh\pi$ \| $CPV$ in fractions \| $0.1-1.0$ Table 2: Projection of statistical sensitivities for $CP$ observables with $50\,\mathrm{fb^{-1}}$ [14]. ## 5 Conclusions The LHCb detector is performing excellently and is already exceeding its design expectations confirming the feasibility of charm physics at hadron colliders. The collaboration is active in many complementary analysis in the charm sector, and in particular sub-percent measurements of several $CP$ quantities are expected to be already available before the upgrade and will reach or even exceed the current theoretical precision after the upgrade. In the upgrade era, these studies will be further improved thanks to the increased statistics and the improvements in the hardware and software infrastructure. In addition, the upgraded LHCb detector has tremendous potential for new measurements in charm rare decays, production and spectroscopy. In parallel, ongoing efforts are focused on reducing possible sources of systematic uncertainties that may limit the LHCb scope. Further, detailed and information on the LHCb upgrade is reported in [21, 22]. ## Acknowledgements The text below are the acknowledgements as approved by the collaboration board. Extending the acknowledgements to include individuals from outside the collaboration who have contributed to the analysis should be approved by the EB and, if possible, be included in the draft of first circulation. We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We also acknowledge the support received from the ERC under FP7. The Tier1 computing centres are supported by IN2P3 (France), KIT and BMBF (Germany), INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom). We are thankful for the computing resources put at our disposal by Yandex LLC (Russia), as well as to the communities behind the multiple open source software packages that we depend on. ## References * [1] R. Aaij et al. [LHCb Collaboration], Nucl. Phys. B 871 (2013) 1 * [2] LHCb collaboration, A. A. Alves Jr. et al., JINST 3 (2008) S08005 * [3] M. Adinolfi et al., Eur. Phys. J. C73 (2013) 2431, * [4] A. A. Alves Jr. et al., JINST 8 (2013) P02022 * [5] R. Aaij et al. [LHCb Collaboration], JHEP 1206 (2012) 141 * [6] R. Aaij et al. [LHCb Collaboration], JHEP 1309 (2013) 145 * [7] R. Aaij et al. [LHCb Collaboration], Phys. Lett. B 725 (2013) 15 * [8] R. Aaij et al. [LHCb Collaboration], Phys. Lett. B 724 (2013) 203 * [9] R. Aaij et al. [LHCb Collaboration], arXiv:1310.2535 [hep-ex]. * [10] Y. Amhis et al. [Heavy Flavor Averaging Group Collaboration], arXiv:1207.1158 [hep-ex]. * [11] R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 110 (2013) 10, 101802 * [12] R. Aaij et al. (LHCb Collaboration), arXiv:1309.6534 [hep-ex]. * [13] R. Aaij et al. [LHCb Collaboration], JHEP 1204 (2012) 129 * [14] R. Aaij et al. [LHCb Collaboration], Eur. Phys. J. C 73 (2013) 2373 * [15] R. Aaij et al. [LHCb Collaboration], arXiv:1310.7201 [hep-ex]. * [16] RAaij et al. [LHCb Collaboration], Phys. Lett. B 713 (2012) 186 * [17] RAaij et al. [LHCb Collaboration], Phys. Lett. B 718 (2013) 902 [arXiv:1210.4112 [Unknown]]. * [18] R. Aaij et al. [LHCb Collaboration], LHCb-CONF-2013-023 * [19] R. Aaij et al. [LHCb Collaboration], LHCb-CONF-2013-003 * [20] R. Aaij et al. [LHCb Collaboration], Phys. Lett. B 723 (2013) 33 * [21] CERN, Technical report, CERN-LHCC-2011-001. LHCC-I-018 * [22] I. Bediaga et al. [LHCb Collaboration], Technical report, CERN-LHCC-2012-007. LHCb-TDR-12	arxiv-papers	2013-11-14T13:25:02	2024-09-04T02:49:53.631455	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Andrea Contu", "submitter": "Andrea Contu", "url": "https://arxiv.org/abs/1311.3496" }
1311.3510	# Survival of New Physics: An Anomaly-free Neutral Gauge Boson at the LHC Ying Zhang1111Email address: [email protected], Qing Wang2,3222Email address: [email protected]. 1School of Science, Xi’an Jiaotong University, Xi’an, 710049, P.R.China 2Department of Physics, Tsinghua University, Beijing 100084, P.R.China 3Center for High Energy Physics, Tsinghua University, Beijing 100084, P.R.China ###### Abstract An anomaly-free $U(1)^{\prime}$ effective Lagrangian as the most common new physics beyond the standard model is proposed to survey the maximal parameter space constrained by electroweak precise measurements at the LEP and direct detection in dilepton decay channel at the LHC at $\sqrt{s}=7$ TeV. By the global fit of effective couplings of $Z$ boson to the SM fermions, $\Delta_{11},\Delta_{21},g_{2}\Delta_{31}$ related to mixings and $r$ related to $U(1)^{\prime}$ charge assignment are bounded. The allowed areas are plotted in the not only $r$-$g_{2}$ but also $m_{Z^{\prime}}$-$g_{2}$ planes, which show that a sub-TeV $Z^{\prime}$ is still permissible as long as the coupling $g_{2}\sim 0.01$. The results provides a prime requirement to an extra $U(1)^{\prime}$ gauge boson and hinds the direction of possible new physics beyond the standard model. The possible signal in dilepton decay channel at LHC at $\sqrt{s}=14$ TeV is also provided. ## I motivation When people were exciting for the found of Higgs-like particle with about 125GeV mass at the LHC, we have to worry about the survival of new physics (NP) right away. Almost all experiments are proving the Standard Model (SM), the space left to NP is less and less. There is an impending question we are dying to know that how much space can NP survive. It is a good checking point to choose a neutral gauge boson, which often appears in GUT and superstring model associated with $U(1)^{\prime}$ group, as a popular candidate of NP beyond the SM. There are many relative issues summarized by A. Leike and P. Langacker LeikeReview ; LangackerReview . However, duo to different motivations, $Z^{\prime}$ interactions to the SM fermions is set by models, which makes the results are highly model-dependent. The minimal mass of the vector boson is limited at about $1.8$ TeV. In the paper, we relax any possible motivations and roles coming from underlying theory or phenomenology, and construct a model-independent effective Lagrangian to describe $U(1)^{\prime}$ gauge boson $Z^{\prime}$ only following the requirement of gauge symmetry. In bosonic sector, all possible mass and kinetic mixings meeting gauge symmetry are investigated. And in fermionic sector, anomaly-free charge assignments is required to satisfy gauge symmetry. The interactions to fermions are dominated by a global coupling $g_{2}$ and a charge assignment parameter $r$. They are keys to realize model-independent description. We consider constrains from not only electroweak precise observables but also direct detection at LHC, and then decide the possible parameter space left to the simplest NP particle. The article is organized as following: firstly, anomaly-free $Z^{\prime}$ effective Lagrangian is constructed. And then, we diagonalize gauge bosons mixing matrixes to obtain mass eigenvalues of neutral bosons. The limit to parameters is studied based on electroweak precise measurements in LEP and direct search at the LHC at $\sqrt{s}=7$ TeV. The allowed area is shown in $m_{Z^{\prime}}$-$g_{2}$ plane. Finally, the possible signal in dilepton final states at $\sqrt{s}=14$ TeV at the LHC is predicted. ## II effective Lagrangian To construct-independent $Z^{\prime}$ effective theory, we will focus on parameterizations in $Z^{\prime}$ mixings and interactions. Denote $U(1)^{\prime}$ gauge eigenstate as $X_{\mu}$. On gauge eigenstates base $(W_{\mu}^{3},B_{\mu},X_{\mu})^{T}$, the covariant derivative has the form $D_{\mu}\hat{U}=\partial_{\mu}\hat{U}+igW_{\mu}\hat{U}-ig^{\prime}\hat{U}\frac{\tau_{3}}{2}B_{\mu}-i{g^{\prime\prime}}\hat{U}X_{\mu}.$ Here, $\hat{U}$ is non-linear realization of Goldstone bosons. $W_{\mu}$ and $B_{\mu}$ are respectively gauge field of $SU(2)_{L}$ and $U(1)_{Y}$ with gauge coupling $g$ and $g^{\prime}$. Using $SU(2)_{L}$ covariant building blocks $T\equiv\hat{U}\tau_{3}\hat{U}^{\dagger}$ and $V_{\mu}\equiv(D_{\mu}\hat{U})\hat{U}^{\dagger}$, mass terms arise from 4 operators in $p^{2}$ order: $tr[V_{\mu}V^{\mu}],tr[TV_{\mu}]^{2},tr[V_{\mu}]^{2}$ and $tr[TV_{\mu}]tr[V^{\mu}]$. The first operator corresponds to the electroweak standard model. The second one provides an extra mass correction to isospin third-component $W^{3}_{\mu}$. The third one generates non-standard mixing between $B_{\mu}$ and $W^{3}_{\mu}$. And the last one parameterizes $Z$-$Z^{\prime}$ mixing. However, the second and third one can be absorbed in the re-definition of gauge couplings $g$ and $g^{\prime}$, and are not independent OurCPC2012 . Similarly, kinetic mixing terms are also controlled by 4 operators: $tr[TW_{\mu\nu}]^{2}$, $tr[TW_{\mu\nu}]B^{\mu\nu}$, $tr[TW_{\mu\nu}]X^{\mu\nu}$ and $B_{\mu\nu}X^{\mu\nu}$ OurJHEP2008 ; OurJHEP2009 . The first operator corresponds a correction to $W_{\mu}^{3}$ kinetic term. The second one yields kinetic mixing between $W^{3}_{\mu}$ and $B_{\mu}$. The third and forth ones cause $U(1)^{\prime}$ boson $X_{\mu}$ kinetic mixings with $W^{3}_{\mu}$ and $B_{\mu}$, respectively. The first two operators are non-standard term beyond the SM and there is no any reason to neglect these invariant ones EWCL . Expressing these operators in obvious gauge fields, Lagrangian related to mixings is written as $\displaystyle\mathcal{L}_{mix}$ $\displaystyle=$ $\displaystyle\frac{m_{0}^{2}}{2}(c_{W}W_{\mu}^{3}-s_{W}B_{\mu})^{2}+\frac{m_{1}^{2}}{2}X_{\mu}X^{\mu}+2\beta m_{0}m_{1}X_{\mu}(c_{W}W_{\mu}^{3}-s_{W}B_{\mu})$ $\displaystyle-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}-\frac{1}{4}X_{\mu\nu}X^{\mu\nu}-\frac{1}{4}(1-\alpha_{b})(W_{\mu\nu}^{3})^{2}$ $\displaystyle+\frac{1}{2}\alpha_{a}B_{\mu\nu}W_{\mu\nu}^{3}+\alpha_{c}X^{\mu\nu}W_{\mu\nu}^{3}+\alpha_{d}X^{\mu\nu}B_{\mu\nu}$ with mass mixing $\beta$ OurCPC2012 . Here, $m_{0}$ and $m_{1}$ are $Z$ and $Z^{\prime}$ mass in gauge eigenstates and $s_{W},c_{W}$ are sine and cosine of Weinberg angle. To $Z^{\prime}$ interactions to EW gauge bosons $W^{\mu}$ and $Z$, some independent parameters control them, which may come from underlying theory and not vanish even though no $Z^{\prime}$ mixings. For example the decay channel $\Gamma(Z^{\prime}\rightarrow W^{+}W^{-})$ may arise from $Z$-$Z^{\prime}$ mixing or high order operator $X_{\mu\nu}tr[T[V^{\mu},V^{\nu}]]$. The former is suppressed by small $Z$-$Z^{\prime}$ mixing angle, and the later stands for possible NP which has no any promption from high energy experiments. For the similar reason, all these decays of $Z^{\prime}$ to EW bosons are expected to be small. The extra neutral current interaction is introduced like $\mathcal{L}_{int}=-g_{2}\sum_{f}\bar{f}\gamma_{\mu}(y^{\prime}_{Lf}P_{L}+{y^{\prime}_{Rf}}P_{R})fX^{\mu}$ with left-handed (right-handed) fermionic $U(1)^{\prime}$ charge $y^{\prime}_{Lf}$ ($y^{\prime}_{Rf}$). To keep gauge symmetry, $U(1)^{\prime}$ charge assignments must be anomaly-free. For the family universal case, there are 6 independent charges: $y^{\prime}_{l}$ and $y^{\prime}_{q}$ for left- handed leptons and quarks, $y^{\prime}_{u},y^{\prime}_{d},y^{\prime}_{\nu},y^{\prime}_{e}$ for right- handed up-quark, down-quark, neutrino and electron, respectively. From $[SU(3)_{C}]^{2}U(1)^{\prime}$, $[SU(2)]_{L}^{2}U(1)^{\prime}$ and $[U(1)_{Y}]^{2}U(1)^{\prime}$ cancellation requirements, we have $\displaystyle y^{\prime}_{l}=-3y^{\prime}_{q},~{}~{}~{}y^{\prime}_{d}=2y^{\prime}_{q}-y^{\prime}_{u},~{}~{}~{}y^{\prime}_{e}=-2y^{\prime}_{q}-y^{\prime}_{u}.$ $U(1)_{Y}[U(1)^{\prime}]^{2}$ anomaly is cancelled automatically. If and only if the number of right-handed neutrinos is 3, $[U(1)^{\prime}]^{3}$ anomaly and gravitational-gauge mixing anomaly can be satisfied simultaneously and the charge is $y^{\prime}_{\nu}=-4y^{\prime}_{q}+y^{\prime}_{u}$. Without loss of generality, the coupling $g_{2}$ is normalized so that $y^{\prime}_{u}=1$. Now, these couplings are dominated by two free parameters: coupling $g_{2}$ that controls global intensity and charge ratio $r(\equiv y^{\prime}_{q}/y^{\prime}_{u})$ that assigns flavor charges $\displaystyle y^{\prime}_{l}=-3r,~{},y^{\prime}_{q}=1,~{}y^{\prime}_{u}=1,~{}y^{\prime}_{d}=2r-1,~{}y^{\prime}_{e}=-2r-1,~{}y^{\prime}_{\nu}=-4r+1.$ Briefly, anomaly-free $Z^{\prime}$ effective theory, inspired by $U(1)^{\prime}$ gauge symmetry, can be parameterized by mass mixing $\beta$, kinetic mixing $\alpha_{c},\alpha_{d}$ in bosonic sector ($\alpha_{a}$ and $\alpha_{b}$ respectively parameterize non-standard $W^{3}_{\mu\nu}W^{3\mu\nu}$ and $W^{3}_{\mu\nu}B^{\mu\nu}$ electroweak bososic kinetic mixing terms, which are not relative to $Z^{\prime}$ boson), coupling $g_{2}$ and the charge ratio $r$ in fermionic sector. The full Lagrangian for $U(1)^{\prime}$ boson is $\mathcal{L}_{Z^{\prime}}=\mathcal{L}_{mix}+\mathcal{L}_{int}.$ ## III diagonalization matrix To obtain mass eigenstates, let’s make a rotation $U$ $\displaystyle\left(\begin{array}[]{c}W^{3}_{\mu}\\\ B_{\mu}\\\ X_{\mu}\end{array}\right)=U\left(\begin{array}[]{c}Z_{\mu}\\\ A_{\mu}\\\ Z^{\prime}_{\mu}\end{array}\right)$ (7) to diagonalize mass and kinetic mixings simultaneously. Considering rotation $U$ reducing to Weinberg’s rotation when no $Z^{\prime}$ mixings, $U$ may be expressed as the sum of Weinberg’s rotation and $Z^{\prime}$ mixing corrections $\displaystyle U=\left(\begin{array}[]{ccc}c_{W}+\Delta_{11}&s_{W}+\Delta_{12}&\Delta_{13}\\\ -s_{W}+\Delta_{21}&c_{W}+\Delta_{12}&\Delta_{23}\\\ \Delta_{31}&\Delta_{32}&1+\Delta_{33}\end{array}\right).$ Notice that 9 $\Delta_{ij}$ ($i=1..3,j=1..3$) are not independent, they are determined by only 7 phenomenological parameters: mass $m_{1}$ and $m_{2}$, mass mixing $\beta$, and kinetic mixings $\alpha_{a,b,c,d}$, i.e. $\Delta_{ij}=\Delta_{ij}(m_{1},m_{2},\beta,\alpha_{a},\alpha_{b},\alpha_{c},\alpha_{d})$. So, we must find two constraint conditions on $\Delta$s. One is $s_{W}\Delta_{22}=c_{W}\Delta_{12}$, which results from the requirement for a massless photon. Another constraint is $\Delta_{32}=0$ due to the requirement of keeping photon coupling vector-type StueckelbergNote . After the rotation (III), the mass eigenvalues of $Z$ and $Z^{\prime}$ are $\displaystyle m_{Z}^{2}$ $\displaystyle=$ $\displaystyle m_{0}^{2}(1+2c_{W}\Delta_{11}-2s_{W}\Delta_{21})+4\beta m_{0}m_{1}\Delta_{31}+\mathcal{O}(\Delta^{2})$ $\displaystyle m_{Z^{\prime}}^{2}$ $\displaystyle=$ $\displaystyle m_{1}^{2}(1+2\Delta_{33})+4\beta m_{0}m_{1}(c_{W}\Delta_{13}-s_{W}\Delta_{23})+\mathcal{O}(\Delta^{2}).$ ## IV constraints in LEP Due to the good fit of the SM to electroweak precise observables, $Z^{\prime}$ effective theory must be constrained by precise electroweak experiments. The heavy neutral boson contributes to low energy observables by two fashions: mixings and $Z^{\prime}$ exchange. They often coexist in phenomenology. However, corrections to fermionic couplings only come from mixings. A theoretical observables can be divided into the SM part and $Z^{\prime}$ contribution $\mathcal{O}_{th}=\mathcal{O}_{SM}+\Delta\mathcal{O}_{Z^{\prime}}.$ Due to the triumph of the SM, $\Delta\mathcal{O}_{Z^{\prime}}$ must be very small. Generally, it’s a safe precession to neglect high order effect of $Z^{\prime}$ LeikeReview . Using rotation (7), vector and axial-vector couplings in weak neutral current are given by $\displaystyle g^{f}_{V,A}=g^{f,0}_{V,A}+\Delta g^{f}_{V,A}$ $g^{f,0}_{V,A}$ is the SM couplings in tree level, $g^{f,0}_{V}=t_{3L}^{f}-2q^{f}s_{W}^{2}$ and $g^{f,0}_{A}=t_{3L}^{f}$. The effective couplings with radiative corrections to propagators and vertices in the SM can be found in LEP report LEP2006 . The $Z^{\prime}$ corrections are $\displaystyle\Delta g^{f}_{V,A}=c_{W}\Delta_{11}t_{3iL}+s_{W}\Delta_{21}(y_{iL}\pm y_{iR})+\frac{s_{W}c_{W}}{e}g_{2}\Delta_{31}(y^{\prime}_{iL}\pm y^{\prime}_{iR}).$ $Z^{\prime}$ corrections are constrained by $Z$-pole observables. The validity of constraint depends on two factors: experiments precision and calculation precision based on the SM. Although there are 14 observables in LEP/SLD at $Z$-pole, the SM calculation can be parameterized into only 4 radiate correction factors: $\Delta\rho,\Delta r_{W},\Delta\kappa$ and $\Delta\rho_{b}$ (or express into 4 new parameters $\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{b}$ introduced by Altarelli, et.al. LEP2006 ; EWepsilon ). Considering the number of independent measurements in experiment and the SM calculation, it’s a balanced treatment to choice pseudo observables, 8 effective coupling constants $g_{V,A}^{f}$ for $f=l,\nu,c,b$, to limit $Z^{\prime}$ parameters. We minimize $\displaystyle\chi^{2}=\sum_{f}\frac{(g_{V,A}^{f,exp}-g_{V,A}^{f,SM}-\Delta g_{V,A}^{f})^{2}}{(\delta g_{V,A}^{f,exp})^{2}}$ where supercript exp, SM respectively denote the corresponding experiment values and the SM fit values, and $\delta g_{V,A}^{f,exp}$ are their experimental errors. The four free parameters are $\Delta_{11},\Delta_{21},g_{2}\Delta_{31}$ and $r$. As we have mentioned, $\Delta_{ij}$ are the functions of mixing parameters. We must keep the fit parameters independent. It can be proved by calculating rotation matrixes invoked by mass mixing $\beta$, and kinetic mixings $\alpha_{c,d}$, respectively, even if EW boson kinetic mixing $\alpha_{a}$ and $\alpha_{b}$ vanishing. After detailed calculations, we arrive at the global fit results in Table. 1. $Z^{\prime}$ slight improves fit confidence level from 93% (about $1.8\sigma$) to 96% (about $2.1\sigma$). The parameter ranges are shown in Table. 2. Table 1: LEP experiment results on the effective coupling constants and the SM Z-pole fit. Data come from Table 7.9 and Table G.3 in literature LEP2006 . The last row represents the corresponding C.L.. coupling \| exp. \| SM fit \| $Z^{\prime}$ fit ---\|---\|---\|--- $g_{A}^{\nu}$ \| $+0.50075\pm 0.00077$ \| $+0.50199\pm^{0.00017}_{0.00020}$ \| $+0.50063$ $g_{A}^{l}$ \| $-0.50125\pm 0.00026$ \| $-0.50127\pm^{0.00020}_{0.00017}$ \| $-0.50116$ $g_{A}^{b}$ \| $-0.5144\pm 0.0051$ \| $-0.49856\pm^{0.00041}_{0.00020}$ \| $-0.49845$ $g_{A}^{c}$ \| $+0.5034\pm 0.0053$ \| $+0.50144\pm^{0.00017}_{0.00020}$ \| $+0.50013$ $g_{V}^{\nu}$ \| $+0.50075\pm 0.00077$ \| $+0.50199\pm^{0.00017}_{0.00020}$ \| $+0.50063$ $g_{V}^{l}$ \| $-0.03753\pm 0.00037$ \| $-0.03712\pm 0.00032$ \| $-0.03751$ $g_{V}^{b}$ \| $-0.3220\pm 0.0077$ \| $-0.34372\pm^{0.00049}_{0.00028}$ \| $-0.34267$ $g_{V}^{c}$ \| $+0.1873\pm 0.0070$ \| $+0.19204\pm 0.00023$ \| $+0.19185$ $\chi^{2}$/dof \| - \| $24.6/8$ \| $20.1/8$ P value \| - \| $93\%$ \| $96\%$ Table 2: global fit results. The corresponding errors come from diagonal elements of the inverse of Hessian matrix. The ranges in $2\sigma$ C.L. are listed in the last column. quantity \| fit result \| range in $2\sigma$ ---\|---\|--- $\Delta_{11}$ \| $-0.00067\pm 0.00040$ \| $(-0.00147,0.00013)$ $\Delta_{21}$ \| $0.0017\pm 0.0076$ \| $(-0.0135,0.0169)$ $g_{2}\Delta_{31}$ \| $-0.00044\pm 0.0018$ \| $(-0.00404,0.00316)$ $r$ \| $-0.015\pm 1.1$ \| $(-2.215,2.185)$ Notice that $Z^{\prime}$ mass does not been limited by effective couplings. Generally, $m_{Z^{\prime}}$ can be limited by $\rho$ or $Z$ mass correction by $Z-Z^{\prime}$ mixing. There are enough more results on the issue in literature zprimebound which shown that a small mass mixing corresponds a heavy $Z^{\prime}$ and vice versa. For the typical value $\beta\sim 10^{-3}$, $m_{Z^{\prime}}$ is several TeV. ## V search at the LHC The LHC has searched a vector resonance decaying into dilepton final states at $\sqrt{s}=7$ TeV ATLASZprime . No statistically significant excess above the SM expectation is observed, which strictly limits $Z^{\prime}$ couplings to fermions. Figure.1 shows the 95% C.L. allowed areas in $g_{2}$-$r$ plane with $m_{Z^{\prime}}=0.8,~{}1.0,~{}1.5,~{}2.0$ TeV respectively. The theoretical cross sections are calculated by Madgrapha5 ver1.5.12. Compared with observed limits on $\sigma(pp\rightarrow Z^{\prime}\rightarrow ll)$ at ATLAS, the values of $g_{2}$ and $r$ can be determined with fixed $m_{Z^{\prime}}$. It indicate that a light $Z^{\prime}$ with enough small coupling is not eliminated. Figure 1: $95\%$ C.L. possible allowed area in the $g_{2}$ vs $r$ plane at 7 TeV LHC. Exclusion lines correspond to excepted $2\sigma$ signal regions at ATLAS in dilepton decay channel in Fig.3 of ATLASZprime . Black vertical lines denote $r$ fit ranges in $1\sigma$ and $2\sigma$ C.L., respectively. More clearly, Fig. 2 plots $7$ TeV ATLAS allowed parameter space in the plane of $Z^{\prime}$ mass and coupling $g_{2}$, corresponding the different $r$ C.L.. A sub-TeV $Z^{\prime}$ is allowed when the coupling $g_{2}\sim 0.01$. Figure 2: allowed $Z^{\prime}$ mass by $7$ TeV ATLAS in $m_{Z^{\prime}}$-$g_{2}$ plane in different charge ratio $r$ fit range corresponding $1\sigma$, $2\sigma$ and $>2\sigma$ C.L.. A possible $\sigma(pp\rightarrow Z^{\prime}\rightarrow e^{+}e^{-},\mu^{+}\mu^{-})$ signal in dilepton final state at $\sqrt{s}=14$ TeV LHC is also calculated by Madgraph5, which is shown in Fig.3. For a significant coupling $g_{2}=0.05$, $Z^{\prime}$ may still exist at about $1.2$ TeV. Even for $g_{2}=0.1$, $Z^{\prime}$ with more that $1.6$ TeV mass is not eliminated. On the other hand, a light $Z^{\prime}$ at sub-TeV would be permissible as long as the coupling $g_{2}$ is enough small. Figure 3: theoretical signal for $\sigma(pp\rightarrow Z^{\prime}\rightarrow ll)$ at 14TeV LHC. The colorful massive shadow areas correspond $2\sigma$ fit range of charge ratio $r$. The red solid area on the top is eliminated by ATLAS direct detection in dilepton final states at $\sqrt{s}=7$ TeV. ## VI conclusion In conclusion, a model-independent effective theory for anomaly-free neutral boson has been presented. Based on electroweak precise measurements in LEP, four parameters related to $Z^{\prime}$ mixings and charge assignment have been constrained. Especially, the charge ratio $r$ has range $(-2.2,2.2)$ at 95% C.L.. To consistent with the LHC direct detection in dilepton decay channel at $7$ TeV, the limit areas to fixed $Z^{\prime}$ mass are shown in $r$-$g_{2}$ plane. More clearly, we map the possible parameter space to the plane of $m_{Z^{\prime}}$-$g_{2}$ at $2\sigma$ C.L. of $r$. in Fig. 2. The results show that a sub-TeV $Z^{\prime}$ with coupling $g_{2}\sim 0.01$ is still not eliminated in $2\sigma$ C.L. of $r$. It suggests a prime requirement to extra vector boson in NP. Further, a possible theoretical $\sigma(pp\rightarrow Z^{\prime}\rightarrow ll)$ signal at $\sqrt{s}=14$ TeV LHC is also calculated. ## Acknowledgments This work was supported by National Science Foundation of China (NSFC) under Grant No.11005084 and partly by the Fundamental Research Funds for the Central University. We thank Rong Li for useful discussions. ## References * (1) A. Leike, Phys. Rept. 317 (1999) 143-250. * (2) P. Langacker, Rev. Mod. Phys. 81 (2009) 1199-1228. * (3) Y. Zhang, Qing Wang, Chin. Phys. C36 (2012) 298-306. * (4) Y. Zhang, S.-Z. Wang, Q. Wang, JHEP 0803 (2008) 047. * (5) Y. Zhang, Q. Wang, JHEP 0907 (2009) 012. * (6) T. Appelquist, G.-H. Wu, Phys. Rev. D48 (1993) 3235-3241. * (7) When we expand covariant derivative to include Stueckelberg mechanism for $U(1)^{\prime}$ mass, $\Delta_{32}$ must be non-zero to diagonalize Stueckelberg mixing. In the case, vector-type photon coupling requires $y^{\prime f}_{L}=y^{\prime f}_{R}$ for any flavors. It yields the special charge assignment corresponding to charge ratio $y^{\prime}_{q}/y^{\prime}_{u}=r=1$ in anomaly-free case. So, Stueckelberg extended effective theory has the same phenomenology as $r=1$ case at the LHC. * (8) ALEPH, DELPHI, L3, OPAL, SLD, LEP electroweak working group, SLD electroweak group and SLD heavy flavour group collaborations, Phys. Rept. 427 (2006) 257-454. * (9) G. Altarelli, R. Barbieri. Phys. Lett. B253 (1991) 161-167. * (10) M.S. Carena, A. Daleo, B.A. Dobrescu, T.M.P. Tait, Phys. Rev. D70 (2004) 093009; Y. Umeda, Gi-Chol Cho, K. Hagiwara, Phys. Rev. D58 (1998) 115008; J. Erler, P. Langacker, S. Munir, E. Rojas, JHEP 0908 (2009) 017. * (11) ATLAS Collaboration, JHEP 1211 (2012) 138.	arxiv-papers	2013-11-14T14:21:27	2024-09-04T02:49:53.639646	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Ying Zhang and Qing Wang", "submitter": "Ying Zhang", "url": "https://arxiv.org/abs/1311.3510" }
1311.3514	# Expansions for a fundamental solution of Laplace’s equation on $\mathbb{R}^{3}$ in 5-cyclidic harmonics Howard S. Cohl1 and Hans Volkmer2 1Applied and Computational Mathematics Division, National Institute of Standards and Technology, Gaithersburg, MD 20899-8910, USA 2Department of Mathematical Sciences, University of Wisconsin–Milwaukee, P. O. Box 413, Milwaukee, WI 53201, USA ###### Abstract. We derive eigenfunction expansions for a fundamental solution of Laplace’s equation in three-dimensional Euclidean space in 5-cyclidic coordinates. There are three such expansions in terms of internal and external 5-cyclidic harmonics of first, second and third kind. The internal and external 5-cyclidic harmonics are expressed by solutions of a Fuchsian differential equation with five regular singular points. ## 1\. Introduction Expansions for a fundamental solution of Laplace’s equation on $\mathbb{R}^{3}$ in terms of solutions found by the method of separation of variables in a suitable curvilinear coordinate system are known for a long time. For example, when we choose spherical coordinates, we obtain the well- known expansion [21] (1.1) $\frac{1}{\\|\mathbf{r}-\mathbf{r}^{\prime}\\|}=\sum_{\ell=0}^{\infty}\frac{r^{\ell}}{(r^{\prime})^{\ell+1}}\sum_{m=-\ell}^{\ell}\frac{(\ell-m)!}{(\ell+m)!}{\mathsf{P}}_{\ell}^{m}(\cos\theta){\mathsf{P}}_{\ell}^{m}(\cos\theta^{\prime})e^{im(\phi-\phi^{\prime})},$ where $\\|\mathbf{r}\\|<\\|\mathbf{r}^{\prime}\\|$ ($\\|\mathbf{r}\\|$ denotes the Euclidian norm of $\mathbf{r}\in\mathbb{R}^{3}$), and $r,\theta,\phi$, $r^{\prime},\theta^{\prime},\phi^{\prime}$ are the spherical coordinates of $\mathbf{r}$ and $\mathbf{r}^{\prime}$, respectively. The expansion (1.1) contains the Ferrers function of the first kind (associated Legendre function of the first kind on-the-cut) ${\mathsf{P}}_{\ell}^{m}$ [22, (14.3.1)]. We may write expansion (1.1) in the more concise form (1.2) $\frac{1}{\\|\mathbf{r}-\mathbf{r}^{\prime}\\|}=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}G_{\ell}^{m}(\mathbf{r})\overline{H_{\ell}^{m}(\mathbf{r}^{\prime})},$ where $G_{\ell}^{m}:\mathbb{R}^{3}\to\mathbb{C}$ is the internal spherical harmonic (1.3) $G_{\ell}^{m}(\mathbf{r}):=\left(\frac{(\ell-m)!}{(\ell+m)!}\right)^{1/2}r^{\ell}{\mathsf{P}}_{\ell}^{m}(\cos\theta)e^{im\phi},$ and $H_{\ell}^{m}:\mathbb{R}^{3}\to\mathbb{C}$ is the external spherical harmonic (1.4) $H_{\ell}^{m}(\mathbf{r}^{\prime}):=\left(\frac{(\ell-m)!}{(\ell+m)!}\right)^{1/2}(r^{\prime})^{-\ell-1}{\mathsf{P}}_{\ell}^{m}(\cos\theta^{\prime})e^{im\phi^{\prime}}.$ In this paper we derive expansions analogous to (1.2) for the 5-cyclidic coordinate system [20, (6.24)] in place of spherical coordinates. The coordinate surfaces of 5-cyclidic coordinates are triply-orthogonal confocal cyclides. There are three kinds of internal and external 5-cyclidic harmonics, one for each family of coordinate surfaces, and three corresponding expansions. The authors already introduced internal 5-cyclidic harmonics in [13]. As far as we know, the definition of external 5-cyclidic harmonics and the expansions analogous to (1.2) are given in this paper for the first time. We also derive some needed additional properties of internal 5-cyclidic harmonics. In the definitions of internal and external spherical harmonics (1.3), (1.4) there appear only the associated Legendre functions apart from elementary functions. In the case of 5-cyclidic coordinates the definition of internal and external harmonics requires solutions of a Fuchsian differential equation with 5 regular singularities. The particular solutions of interest are eigenfunctions of two-parameter Sturm-Liouville eigenvalue problems; see [13]. In Maxime Bôcher’s 1891 dissertation, Ueber die Reihenentwickelungen der Potentialtheorie [3], it was shown that the 3-variable Laplace equation can be solved using separation of variables in seventeen conformally distinct quadric and cyclidic coordinate systems. These coordinates have coordinate surfaces which are zero sets for polynomials in $x,y,z$ with degree at most two and four respectively. The Helmholtz equation on $\mathbb{R}^{3}$ admits simply separable solutions in the same eleven quadric coordinate systems that the Laplace equation admits separable solutions [14]. The Laplace equation also admits ${R}$-separable solutions in an additional six conformally distinct coordinate systems [20, Table 17, page 210]. Unlike the Laplace equation, the Helmholtz equation does not admit solutions via ${R}$-separation of variables. The appearance of ${R}$-separation is intrinsic to the existence of conformal symmetries for a linear partial differential equation (see Boyer, Kalnins & Miller (1976) [4]), i.e. dilatations, special conformal transformations, inversions and reflections. The theory of separation of variables from a Lie group theoretic viewpoint has been treated in Miller (1977) [20]. In Miller’s book, separation of variables for the Laplace equation on $\mathbb{R}^{3}$ was treated and the general asymmetric ${R}$-separable 5-cyclidic coordinate system was introduced (see [20, Table 17, System 12]). In regard to this coordinate system, and the corresponding separable harmonic solutions, Miller indicates that “Very little is known about the solutions.” To the authors’ knowledge, eigenfunction expansions for the fundamental solution (the $1/r$ potential) have been obtained for the following coordinate systems. See [11, 16, 18, 21] for expansions in spherical, circular/parabolic/elliptic cylinder, oblate/prolate spheroidal, parabolic, bi-spherical and toroidal coordinates. The expansion in confocal ellipsoidal coordinates is treated in [2, 15]. This paper is a stepping-stone for derivations of eigenfunction expansions for a fundamental solution of Laplace’s equation in coordinate systems where these expansions are not known such as paraboloidal, flat-ring cyclide, flat-disk cyclidic, bi-cyclide, cap- cyclide and 3-cyclide [20, Table 17, System 13]) coordinates. The eigenfunction expansions are often connected with integral identities (such as the integral of Lipschitz [23, Section 13.2] and the Lipschitz-Hankel integral [23, Section 13.21] which appear in cylindrical coordinates), addition theorems (such as Neumann’s and Graf’s generalization of Neumann’s addition theorem [23, Section 11.1, Section 11.3] which appear in cylindrical coordinates and the addition theorem for spherical harmonics [24] which appears in spherical coordinates), generating functions for orthogonal polynomials (such as the generating function for Legendre polynomials [22, (18.12.4)] which appears in spherical coordinates), and special function expansion identities (such as Heine’s reciprocal square root identity [7, (3.11)] which appears in circular cylindrical coordinates). In this setting, one may perform eigenfunction expansions for a fundamental solution of Laplace’s equation in alternative separable coordinate systems to obtain new special function summation and integration identities which often have interesting geometrical interpretations (see for instance [5, 9, 10]). Eigenfunction expansions for fundamental solutions of elliptic partial differential equations have been extended to more general separable linear partial differential equations [6] and to partial differential equations on Riemannian manifolds of constant curvature [8]. The outline of this paper is as follows. The 5-cyclidic coordinate system $s_{1},s_{2},s_{3}$ is discussed in Section 2. In Section 3, we consider internal and external 5-cyclidic harmonics of the second kind which are related to the coordinate surfaces $s_{2}={\rm const}$. We start with functions of the second kind because they are slightly easier to treat than the harmonics of the first and third kind related to the coordinate surfaces $s_{1}={\rm const}$, $s_{3}={\rm const}$, respectively. In Section 4, as one of our main results, we obtain the expansion of the fundamental solution of Laplace’s equation in terms of internal and external 5-cyclidic harmonics of the second kind. The proof is based on $(a)$ an integral representation of the external harmonics in terms of internal harmonics given in Section 4, and $(b)$ the completeness property of internal harmonics obtained in [13]. In Sections 5,6 we treat 5-cyclidic harmonics of the first kind. In Sections 7,8 we treat 5-cyclidic harmonics of the third kind. ## 2\. 5-cyclidic coordinates We work on $\mathbb{R}^{3}$ with Cartesian coordinates $x,y,z$, and we use the notations $\mathbf{r}=(x,y,z)$ and $\\|\mathbf{r}\\|=(x^{2}+y^{2}+z^{2})^{1/2}$. Fix $a_{0}<a_{1}<a_{2}<a_{3}$. The 5-cyclidic coordinates of a point $\mathbf{r}\in\mathbb{R}^{3}$ are the solutions $s=s_{1},s_{2},s_{3}$ of the equation (2.1) $\frac{(\\|\mathbf{r}\\|^{2}-1)^{2}}{s-a_{0}}+\frac{4x^{2}}{s-a_{1}}+\frac{4y^{2}}{s-a_{2}}+\frac{4z^{2}}{s-a_{3}}=0$ (strictly speaking, this equation is multiplied by the common denominator of the left-hand side), where $a_{0}\leq s_{1}\leq a_{1}\leq s_{2}\leq a_{2}\leq s_{3}\leq a_{3};$ see [13, Section 4]. On the set (2.2) $R:=\\{\mathbf{r}:x,y,z>0,\,\\|\mathbf{r}\\|<1\\},$ the map $(x,y,z)\in R\mapsto(s_{1},s_{2},s_{3})\in(a_{0},a_{1})\times(a_{1},a_{2})\times(a_{2},a_{3})$ is bijective. The inverse map is given by (2.3) $x=\frac{x_{1}}{1+x_{0}},\quad y=\frac{x_{2}}{1+x_{0}},\quad z=\frac{x_{3}}{1+x_{0}},\\\ $ where (2.4) $x_{j}^{2}=\frac{\prod_{i=1}^{3}(s_{i}-a_{j})}{\prod_{j\neq i=0}^{3}(a_{i}-a_{j})},\quad x_{j}>0.$ We note that each $s_{i}$ is a continuous function on $\mathbb{R}^{3}$. Of particular interest are the sets $\displaystyle A_{1}$ $\displaystyle:=$ $\displaystyle\\{\mathbf{r}:s_{1}=s_{2}\\}=\\{(0,y,z):g_{1}(y,z)=0\\},$ $\displaystyle A_{2}$ $\displaystyle:=$ $\displaystyle\\{\mathbf{r}:s_{2}=s_{3}\\}=\\{(x,0,z):g_{2}(x,z)=0\\},$ where $\displaystyle g_{1}(y,z)$ $\displaystyle:=$ $\displaystyle\frac{(y^{2}+z^{2}-1)^{2}}{a_{1}-a_{0}}+\frac{4y^{2}}{a_{1}-a_{2}}+\frac{4z^{2}}{a_{1}-a_{3}},$ $\displaystyle g_{2}(x,z)$ $\displaystyle:=$ $\displaystyle\frac{(x^{2}+z^{2}-1)^{2}}{a_{2}-a_{0}}+\frac{4x^{2}}{a_{2}-a_{1}}+\frac{4z^{2}}{a_{2}-a_{3}}.$ Each set $A_{1},A_{2}$ consists of two closed curves; see Figures 1, 2. The function $s_{1}$ is (real-)analytic on $\mathbb{R}^{3}\setminus A_{1}$, $s_{2}$ is analytic on $\mathbb{R}^{3}\setminus(A_{1}\cup A_{2})$, and $s_{3}$ is analytic on $\mathbb{R}^{3}\setminus A_{2}$. We will also encounter the sets $\displaystyle K_{1}$ $\displaystyle:=$ $\displaystyle\\{\mathbf{r}:\\|\mathbf{r}\\|<1,s_{1}=a_{1}\\}=\\{(0,y,z):y^{2}+z^{2}<1,g_{1}(y,z)\geq 0\\},$ $\displaystyle L_{1}$ $\displaystyle:=$ $\displaystyle\\{\mathbf{r}:s_{2}=a_{1}\\}=\\{(0,y,z):g_{1}(y,z)\leq 0\\},$ $\displaystyle M_{1}$ $\displaystyle:=$ $\displaystyle\\{\mathbf{r}:\\|\mathbf{r}\\|>1,s_{1}=a_{1}\\}=\\{(0,y,z):y^{2}+z^{2}>1,g_{1}(y,z)\geq 0\\},$ $\displaystyle K_{2}$ $\displaystyle:=$ $\displaystyle\\{\mathbf{r}:z>0,s_{3}=a_{2}\\}=\\{(x,0,z):z>0,g_{2}(x,z)\leq 0\\},$ $\displaystyle L_{2}$ $\displaystyle:=$ $\displaystyle\\{\mathbf{r}:s_{2}=a_{2}\\}=\\{(x,0,z):g_{2}(x,z)\geq 0\\},$ $\displaystyle M_{2}$ $\displaystyle:=$ $\displaystyle\\{\mathbf{r}:z<0,s_{3}=a_{2}\\}=\\{(x,0,z):z<0,g_{2}(x,z)\leq 0\\}.$ The sets $A_{1},K_{1},L_{1},M_{1}$ are subsets of the plane $x=0$, and $A_{2},K_{2},L_{2},M_{2}$ are subsets of the plane $y=0$; see Figures 1, 2. Figure 1. Curves $A_{1}$ and regions $K_{1},L_{1},M_{1}$ for $a_{j}=j$. Figure 2. Curves $A_{2}$ and regions $K_{2},L_{2},M_{2}$ for $a_{j}=j$. We denote the inversion at the unit sphere on $\mathbb{R}^{3}$ by (2.5) $\sigma_{0}(\mathbf{r}):=\\|\mathbf{r}\\|^{-2}\mathbf{r},$ and the reflections at the coordinate planes by (2.6) $\sigma_{1}(x,y,z):=(-x,y,z),\,\sigma_{2}(x,y,z):=(x,-y,z),\,\sigma_{3}(x,y,z):=(x,y,-z).$ We note that the functions $s_{1},s_{2},s_{3}$ are invariant under $\sigma_{j}$, $j=0,1,2,3$. We define auxiliary functions $\chi_{j}:\mathbb{R}^{3}\to\mathbb{R}$, $j=0,1,2,3$, by $\displaystyle\chi_{0}(\mathbf{r})$ $\displaystyle:=$ $\displaystyle{\rm sgn}(1-\\|\mathbf{r}\\|)(s_{1}-a_{0})^{1/2},$ $\displaystyle\chi_{1}(\mathbf{r})$ $\displaystyle:=$ $\displaystyle{\rm sgn}(x)((s_{2}-a_{1})(a_{1}-s_{1}))^{1/2},$ $\displaystyle\chi_{2}(\mathbf{r})$ $\displaystyle:=$ $\displaystyle{\rm sgn}(y)((s_{3}-a_{2})(a_{2}-s_{2}))^{1/2},$ $\displaystyle\chi_{3}(\mathbf{r})$ $\displaystyle:=$ $\displaystyle{\rm sgn}(z)(a_{3}-s_{3})^{1/2}.$ ###### Lemma 2.1. The functions $\chi_{j}$, $j=0,1,2,3$, are continuous on $\mathbb{R}^{3}$. $\chi_{0},\chi_{2}$ are analytic on $\mathbb{R}^{3}\setminus A_{1}$, and $\chi_{1},\chi_{3}$ are analytic on $\mathbb{R}^{3}\setminus A_{2}$. Moreover, (2.7) $\chi_{j}\circ\sigma_{i}=\begin{cases}\chi_{j}&\text{if $i\neq j$},\\\ -\chi_{j}&\text{if $i=j$.}\end{cases}$ ###### Proof. Consider first $\chi_{3}$. The function $s_{3}$ is continuous, and $s_{3}=a_{3}$ if and only if $z=0$. Therefore, $\chi_{3}$ is continuous. In order to prove that $\chi_{3}$ is analytic on $\mathbb{R}^{3}\setminus A_{2}$, it is enough to show that $\chi_{3}$ is analytic at every point of the plane $z=0$. Let $\mathbf{r}_{0}=(x_{0},y_{0},0)$. There is $\epsilon\in(0,1)$ such that $s_{3}\neq a_{2}$ for $\mathbf{r}\in B_{\epsilon}(\mathbf{r}_{0})=\\{\mathbf{r}:\\|\mathbf{r}-\mathbf{r}_{0}\\|<\epsilon\\}$. Then (2.1) with $s=s_{3}$ implies $a_{3}-s_{3}=\frac{4z^{2}}{f(\mathbf{r})}\quad\text{for $\mathbf{r}\in B_{\epsilon}(\mathbf{r}_{0})$},$ where $f(\mathbf{r}):=\frac{(\\|\mathbf{r}\\|^{2}-1)^{2}}{s_{3}-a_{0}}+\frac{4x^{2}}{s_{3}-a_{1}}+\frac{4y^{2}}{s_{3}-a_{2}}$ is positive and analytic on $B_{\epsilon}(\mathbf{r}_{0})$. Therefore, we obtain $\chi_{3}(\mathbf{r})=\frac{2z}{(f(\mathbf{r}))^{1/2}}\quad\text{for $\mathbf{r}\in B_{\epsilon}(\mathbf{r}_{0})$},$ and this shows that $\chi_{3}$ is analytic at $\mathbf{r}_{0}$. $\chi_{0}$ is treated similarly. Consider next $\chi_{2}$. The functions $s_{2},s_{3}$ are continuous, and $(a_{2}-s_{2})(s_{3}-a_{2})=0$ if and only if $y=0$. Thus $\chi_{2}$ is continuous. In order to prove that $\chi_{2}$ is analytic on $\mathbb{R}^{3}\setminus A_{1}$, it is enough to show that $\chi_{2}$ is analytic at all points of the plane $y=0$ which do not lie in $A_{1}$. Suppose first $\mathbf{r}_{0}=(x_{0},0,z_{0})\in(K_{2}\cup M_{2})\setminus A_{2}$. There is $\epsilon>0$ such that $s_{3}\neq a_{3}$ and $s_{2}\neq a_{2}$ for $\mathbf{r}\in B_{\epsilon}(\mathbf{r}_{0})$. Then, by (2.1) with $s=s_{3}$, we obtain $s_{3}-a_{2}=\frac{4y^{2}}{g(\mathbf{r})},$ where $g(\mathbf{r}):=-\frac{(\\|\mathbf{r}\\|^{2}-1)^{2}}{s_{3}-a_{0}}-\frac{4x^{2}}{s_{3}-a_{1}}-\frac{4z^{2}}{s_{3}-a_{3}}$ is analytic on $B_{\epsilon}(\mathbf{r}_{0})$. Since $g(\mathbf{r}_{0})=-g_{2}(x_{0},z_{0})>0$, $g$ is also positive on $B_{\epsilon}(\mathbf{r}_{0})$ for sufficiently small $\epsilon>0$. Then $\chi_{2}(\mathbf{r})=(a_{2}-s_{2})^{1/2}\frac{2y}{(g(\mathbf{r}))^{1/2}}\quad\text{for $\mathbf{r}\in B_{\epsilon}(\mathbf{r}_{0})$.}$ This shows that $\chi_{2}$ is analytic at $\mathbf{r}_{0}$ provided that $\mathbf{r}_{0}\notin A_{1}$. In a similar way, by using (2.1) with $s=s_{2}$, we show that $\chi_{2}$ is analytic at all points $\mathbf{r}_{0}\in L_{2}\setminus A_{2}$. Finally, by subtracting equations (2.1) with $s=s_{2},s_{3}$ from each other, we show that $\chi_{2}$ is analytic at all points $\mathbf{r}_{0}\in A_{2}$. $\chi_{1}$ is treated similarly. The symmetries (2.7) follow from the definition of $\chi_{j}$. ∎ Solving the Laplace equation (2.8) $\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}+\frac{\partial^{2}u}{\partial z^{2}}=0$ by the method of separation of variables, we find solutions (2.9) $u(\mathbf{r}):=(\\|\mathbf{r}\\|^{2}+1)^{-1/2}w_{1}(s_{1})w_{2}(s_{2})w_{3}(s_{3}),\quad s_{i}\in(a_{i-1},a_{i}).$ Each function $w=w_{1},w_{2},w_{3}$ satisfies the Fuchsian equation (2.10) $\prod_{j=0}^{3}(s-a_{j})\left[w^{\prime\prime}+\frac{1}{2}\sum_{j=0}^{3}\frac{1}{s-a_{j}}w^{\prime}\right]+\left(\frac{3}{16}s^{2}+\lambda_{1}s+\lambda_{2}\right)w=0,$ where $\lambda_{1},\lambda_{2}$ are separation constants; see [13]. This equation has five regular singularities at $a_{0},a_{1},a_{2},a_{3},\infty$. The exponents at each finite singularity are $0$ or $\frac{1}{2}$. The function $u(\mathbf{r})$ defined in (2.9) is harmonic for all choices of solutions $w_{i}$ to (2.10). However, it is harmonic only in the open set obtained from $\mathbb{R}^{3}$ by removing the coordinate planes $x=0,y=0,z=0$ and the unit sphere $\\|\mathbf{r}\\|=1$. In order to obtain globally defined harmonic functions we have to select the Frobenius solutions $w$ at the finite singularities, that is, solutions that are either analytic at $a_{j}$ or of the form $(s-a_{j})^{1/2}g(s)$ with $g(s)$ analytic at $s=a_{j}$. It is impossible to choose the parameters $\lambda_{1},\lambda_{2}$ in such a way that each solution $w_{i}$, $i=1,2,3$, is a nontrivial Frobenius solution belonging to either one of the exponents $0$ or $\frac{1}{2}$ at both end points $a_{i-1}$, $a_{i}$. If this were possible (2.9) would define a function which is harmonic in the whole space $\mathbb{R}^{3}$ (as we see later) and converges to $0$ as $\\|\mathbf{r}\\|\to\infty$. But such a function would have to be identically zero. However, as shown in [13], we can determine special values of $\lambda_{1},\lambda_{2}$ (eigenvalues) such that two solutions (either (1) $w_{2}$, $w_{3}$, or (2) $w_{1}$, $w_{3}$, or (3) $w_{1}$, $w_{2}$) are nontrivial Frobenius solution at both end points simultaneously. These cases lead to 5-cyclidic harmonics of the first, second and third kind. If the remaining function $w_{i}$ in case $(i)$ is chosen appropriately, we obtain internal or external 5-cyclidic harmonics. ## 3\. 5-cyclidic harmonics of the second kind In [13, Section VII] we introduced special solutions $w_{i}(s_{i})=E^{(2)}_{i,\mathbf{n},\mathbf{p}}(s_{i})$ to equation (2.10) for eigenvalues $\lambda_{j}=\lambda^{(2)}_{j,\mathbf{n},\mathbf{p}}$, $j=1,2$, for every $\mathbf{n}\in\mathbb{N}_{0}^{2}$, $\mathbf{p}=(p_{0},p_{1},p_{2},p_{3})\in\\{0,1\\}^{4}$. If $\mathbf{n}=(n_{1},n_{3})$ then $n_{i}$ denotes the number of zeros of $E^{(2)}_{i,\mathbf{n},\mathbf{p}}$ in $(a_{i-1},a_{i})$ for $i=1,3$. The subscript $p_{j}$ describes the behavior of the solutions at the endpoint $a_{j}$: We have $E^{(2)}_{i,\mathbf{n},\mathbf{p}}(s_{i})=(s_{i}-a_{i-1})^{p_{i-1}/2}(a_{i}-s_{i})^{p_{i}/2}{\mathcal{E}}^{(2)}_{i,\mathbf{n},\mathbf{p}}(s_{i}),\quad s_{i}\in(a_{i-1},a_{i}),$ where ${\mathcal{E}}^{(2)}_{1,\mathbf{n},\mathbf{p}}$ is analytic on $[a_{0},a_{1}]$, ${\mathcal{E}}^{(2)}_{2,\mathbf{n},\mathbf{p}}$ is analytic on $[a_{1},a_{2})$ (but not at $a_{2}$), and ${\mathcal{E}}^{(2)}_{3,\mathbf{n},\mathbf{p}}$ is analytic on $[a_{2},a_{3}]$. According to (2.9) the function (3.1) $G^{(2)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}):=(\\|\mathbf{r}\\|^{2}+1)^{-1/2}E^{(2)}_{1,\mathbf{n},\mathbf{p}}(s_{1})E^{(2)}_{2,\mathbf{n},\mathbf{p}}(s_{2})E^{(2)}_{3,\mathbf{n},\mathbf{p}}(s_{3}),\quad\mathbf{r}\in R,$ is harmonic on $R$. In order to analytically extend $G^{(2)}_{\mathbf{n},\mathbf{p}}$ we use the functions $\chi_{j}$ introduced in Section 2. We set (3.2) $G^{(2)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}):=(\\|\mathbf{r}\\|^{2}+1)^{-1/2}\prod_{j=0}^{3}(\chi_{j}(\mathbf{r}))^{p_{j}}\prod_{i=1}^{3}{\mathcal{E}}^{(2)}_{i,\mathbf{n},\mathbf{p}}(s_{i})\quad\text{if $s_{2}\neq a_{2}$}$ which is consistent with (3.1). The condition $s_{2}\neq a_{2}$ is equivalent to $\mathbf{r}\in\mathbb{R}^{3}\setminus L_{2}$. We call $G^{(2)}_{\mathbf{n},\mathbf{p}}$ an internal 5-cyclidic harmonic of the second kind. ###### Theorem 3.1. Let $\mathbf{n}\in\mathbb{N}_{0}^{2}$ and $\mathbf{p}\in\\{0,1\\}^{4}$. Then $G^{(2)}_{\mathbf{n},\mathbf{p}}$ is harmonic on $\mathbb{R}^{3}\setminus L_{2}$. Moreover, (3.3) $G^{(2)}_{\mathbf{n},\mathbf{p}}(\sigma_{j}(\mathbf{r}))=(-1)^{p_{j}}G^{(2)}_{\mathbf{n},\mathbf{p}}(\mathbf{r})\quad\text{for $j=1,2,3$},$ and (3.4) $G^{(2)}_{\mathbf{n},\mathbf{p}}(\sigma_{0}(\mathbf{r}))=(-1)^{p_{0}}\\|\mathbf{r}\\|G^{(2)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}).$ ###### Proof. By (3.2) and Lemma 2.1, $G^{(2)}_{\mathbf{n},\mathbf{p}}$ is a composition of continuous functions, and thus it is continuous on $\mathbb{R}^{3}\setminus L_{2}$. As a composition of analytic functions, $G^{(2)}_{\mathbf{n},\mathbf{p}}$ is analytic and thus harmonic on $\mathbb{R}^{3}\setminus(A_{1}\cup L_{2})$. The set $A_{1}$ is a removable line singularity of $G^{(2)}_{\mathbf{n},\mathbf{p}}$. This can be seen in two different ways. 1) We may appeal to the general theory of harmonic functions. $A_{1}$ is a polar set, and we may apply [1, Cor. 5.2.3]. 2) We can show directly that $G^{(2)}_{\mathbf{n},\mathbf{p}}$ is analytic at each point of $A_{1}$ by the method used in the proof of [13, Lemma 6.1]. For example, take the simplest case $\mathbf{p}=(0,0,0,0)$. Then (3.1) holds for all $\mathbf{r}\in\mathbb{R}^{3}\setminus L_{2}$, and the product $E^{(2)}_{1,\mathbf{n},\mathbf{p}}(s_{1})E^{(2)}_{2,\mathbf{n},\mathbf{p}}(s_{2})$ is analytic at each point of $A_{1}$. This is because $E^{(2)}_{1,\mathbf{n},\mathbf{p}}(s)$ and $E^{(2)}_{2,\mathbf{n},\mathbf{p}}(s)$ are analytic extensions of each other, and $s_{1},$ $s_{2}$ enter symmetrically. Note that $s_{1}s_{2}$ and $s_{1}+s_{2}$ are analytic at each point of $A_{1}$ although $s_{1}$, $s_{2}$ are not analytic there. The symmetry properties of $G^{(2)}_{\mathbf{n},\mathbf{p}}$ also follow from (3.2) and Lemma 2.1. ∎ If $U(\mathbf{r})$ is a harmonic function then its Kelvin transformation $V(\mathbf{r})=\\|\mathbf{r}\\|^{-1}U(\sigma_{0}(\mathbf{r}))$ is also harmonic [17, page 232]. Equation (3.4) states that $G^{(2)}_{\mathbf{n},\mathbf{p}}$ is invariant or changes sign under the Kelvin transformation if $p_{0}=0$ or $p_{0}=1$, respectively. We see that $L_{2}$ is a “surface singularity” of $G^{(2)}_{\mathbf{n},\mathbf{p}}$ which is not removable (it is not a polar set). In fact, $G^{(2)}_{\mathbf{n},\mathbf{p}}$ cannot be harmonic on $\mathbb{R}^{3}$ because it would be identically zero otherwise. Let $F^{(2)}_{2,\mathbf{n},\mathbf{p}}$ be the Frobenius solution to the Fuchsian equation (2.10) (with $\lambda_{j}=\lambda^{(2)}_{j,\mathbf{n},\mathbf{p}}$) on $(a_{1},a_{2})$ belonging to the exponent $\frac{p_{2}}{2}$ at $s_{2}=a_{2}$, uniquely determined by the Wronskian condition (3.5) $\omega(s)\left(E^{(2)}_{2,\mathbf{n},\mathbf{p}}(s_{2})\frac{d}{ds_{2}}F^{(2)}_{2,\mathbf{n},\mathbf{p}}(s_{2})-F^{(2)}_{2,\mathbf{n},\mathbf{p}}(s_{2})\frac{d}{ds_{2}}E^{(2)}_{2,\mathbf{n},\mathbf{p}}(s_{2})\right)=1,$ where (3.6) $\omega(s):=\left\|(s-a_{0})(s-a_{1})(s-a_{2})(s-a_{3})\right\|^{1/2}.$ This definition is possible because we know that $E^{(2)}_{2,\mathbf{n},\mathbf{p}}(s_{2})$ is not a Frobenius solution belonging to the exponent $\frac{p_{2}}{2}$ at $s_{2}=a_{2}$. Now we define external 5-cyclidic harmonics of the second kind by (3.7) $H^{(2)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}):=(\\|\mathbf{r}\\|^{2}+1)^{-1/2}E^{(2)}_{1,\mathbf{n},\mathbf{p}}(s_{1})F^{(2)}_{2,\mathbf{n},\mathbf{p}}(s_{2})E^{(2)}_{3,\mathbf{n},\mathbf{p}}(s_{3}),\quad\mathbf{r}\in R.$ In order to analytically extend $H^{(2)}_{\mathbf{n},\mathbf{p}}$ we write $F^{(2)}_{2,\mathbf{n},\mathbf{p}}(s_{2})=(s_{2}-a_{1})^{p_{1}/2}(a_{2}-s_{2})^{p_{2}/2}{\mathcal{F}}^{(2)}_{2,\mathbf{n},\mathbf{p}}(s_{2}),\quad s_{2}\in(a_{1},a_{2}),$ where ${\mathcal{F}}^{(2)}_{2,\mathbf{n},\mathbf{p}}$ is analytic on $(a_{1},a_{2}]$ (but not at $a_{1}$). Then we define (3.8) $H^{(2)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}):=(\\|\mathbf{r}\\|^{2}+1)^{-1/2}\prod_{j=0}^{3}(\chi_{j}(\mathbf{r}))^{p_{j}}{\mathcal{E}}^{(2)}_{1,\mathbf{n},\mathbf{p}}(s_{1}){\mathcal{F}}^{(2)}_{2,\mathbf{n},\mathbf{p}}(s_{2}){\mathcal{E}}^{(2)}_{3,\mathbf{n},\mathbf{p}}(s_{3})\quad\text{if $s_{2}\neq a_{1}$}.$ The condition $s_{2}\neq a_{1}$ is equivalent to $\mathbf{r}\in\mathbb{R}^{3}\setminus L_{1}$. ###### Theorem 3.2. Let $\mathbf{n}\in\mathbb{N}_{0}^{2}$ and $\mathbf{p}\in\\{0,1\\}^{4}$. Then $H^{(2)}_{\mathbf{n},\mathbf{p}}$ is harmonic on $\mathbb{R}^{3}\setminus L_{1}$. The functions $H^{(2)}_{\mathbf{n},\mathbf{p}}$ share the symmetries (3.3), (3.4) with $G^{(2)}_{\mathbf{n},\mathbf{p}}$. Moreover, (3.9) $H^{(2)}_{\mathbf{n},\mathbf{p}}(\mathbf{r})=O(\\|\mathbf{r}\\|^{-1})\quad\text{as $\\|\mathbf{r}\\|\to\infty$},$ and (3.10) $\\|\nabla H^{(2)}_{\mathbf{n},\mathbf{p}}(\mathbf{r})\\|=O(\\|\mathbf{r}\\|^{-2})\quad\text{as $\\|\mathbf{r}\\|\to\infty$}.$ ###### Proof. The proof of analyticity and symmetry of $H^{(2)}_{\mathbf{n},\mathbf{p}}$ is similar to that given for $G^{(2)}_{\mathbf{n},\mathbf{p}}$ in Theorem 3.1, and is omitted. Estimates (3.9) and (3.10) follow easily from the observation that the Kelvin transformation of $H^{(2)}_{\mathbf{n},\mathbf{p}}$ is $\pm H^{(2)}_{\mathbf{n},\mathbf{p}}$ which is analytic at $\mathbf{0}\notin L_{1}$. ∎ ## 4\. Expansion of the reciprocal distance in 5-cyclidic harmonics of second kind For given $d_{2}\in(a_{1},a_{2})$ we consider the “5-cyclidic ring” (4.1) $D_{2}:=\\{\mathbf{r}\in\mathbb{R}^{3}:s_{2}<d_{2}\\},$ or, equivalently, (4.2) $D_{2}=\\{\mathbf{r}:\frac{(\\|\mathbf{r}\\|^{2}-1)^{2}}{d_{2}-a_{0}}+\frac{4x^{2}}{d_{2}-a_{1}}+\frac{4y^{2}}{d_{2}-a_{2}}+\frac{4z^{2}}{d_{2}-a_{3}}<0\\}.$ Note that each internal 5-cyclidic harmonic $G^{(2)}_{\mathbf{n},\mathbf{p}}$ is harmonic in $D_{2}$ (and on its boundary), and each external 5-cyclidic harmonic is harmonic on $\mathbb{R}^{3}\setminus D_{2}$ (and on its boundary). We represent external harmonics in terms of internal harmonics by a surface integral over the boundary $\partial D_{2}$ of the ring $D_{2}$ as follows. ###### Theorem 4.1. Let $d_{2}\in(a_{1},a_{2})$, $\mathbf{n}\in\mathbb{N}_{0}^{2}$, $\mathbf{p}\in\\{0,1\\}^{4}$. Then (4.3) $H^{(2)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}^{\prime})=\frac{1}{4\pi\omega(d_{2})\\{E^{(2)}_{2,\mathbf{n},\mathbf{p}}(d_{2})\\}^{2}}\int_{\partial D_{2}}\frac{G^{(2)}_{\mathbf{n},\mathbf{p}}(\mathbf{r})}{h_{2}(\mathbf{r})\\|\mathbf{r}-\mathbf{r}^{\prime}\\|}\,dS(\mathbf{r})$ for all $\mathbf{r}^{\prime}\in\mathbb{R}^{3}\setminus\bar{D}_{2}$. The scale factor $h_{2}$ is given by (4.4) $16\\{h_{2}(\mathbf{r})\\}^{2}=\frac{(\\|\mathbf{r}\\|^{2}-1)^{2}}{(d_{2}-a_{0})^{2}}+\frac{4x^{2}}{(d_{2}-a_{1})^{2}}+\frac{4y^{2}}{(d_{2}-a_{2})^{2}}+\frac{4z^{2}}{(d_{2}-a_{3})^{2}}.$ ###### Proof. Let $D$ be an open bounded subset of $\mathbb{R}^{3}$ with smooth boundary. For $u,v\in C^{2}(\bar{D})$, Green’s formula states that (4.5) $\int_{D}(u\Delta v-v\Delta u)\,d\mathbf{r}=\int_{\partial D}\left(u\frac{\partial v}{\partial\nu}-v\frac{\partial u}{\partial\nu}\right)\,dS,$ where $\frac{\partial u}{\partial\nu}$ is the outward normal derivative of $u$ on the boundary $\partial D$ of $D$. We apply (4.5) to the domain $D=D_{2}$, and functions $u=G=G^{(2)}_{\mathbf{n},\mathbf{p}}$, $v(\mathbf{r})=\frac{1}{4\pi\\|\mathbf{r}-\mathbf{r}^{\prime}\\|}$. Since $u,v$ are harmonic on an open set containing $\bar{D}_{2}$ we obtain (4.6) $0=\int_{\partial D_{2}}\left(G\frac{\partial v}{\partial\nu}-v\frac{\partial G}{\partial\nu}\right)dS.$ We now use (4.5) a second time. We choose $R>0$ so large that the ball $B_{R}(\mathbf{0})$ contains $\mathbf{r}^{\prime}$ and $\bar{D}_{2}$. Then we take $D=B_{R}(\mathbf{0})-\bar{D}_{2}-B_{\epsilon}(\mathbf{r}^{\prime})$ with small radius $\epsilon>0$. Take $u=H=H^{(2)}_{\mathbf{n},\mathbf{p}}$ and $v$ as before. Note that $u,v$ are harmonic on an open set containing $\bar{D}$. By a standard argument [19, Theorem 1, page 109], taking the limit $\epsilon\to 0$, we obtain (4.7) $H(\mathbf{r}^{\prime})=\int_{\partial B_{R}(\mathbf{0})}\left(H\frac{\partial v}{\partial\nu}-v\frac{\partial H}{\partial\nu}\right)dS-\int_{\partial D_{2}}\left(H\frac{\partial v}{\partial\nu}-v\frac{\partial H}{\partial\nu}\right)dS,$ where, in the second integral, $\frac{\partial}{\partial\nu}$ denotes the same derivative as in (4.6). The first integral in (4.7) tends to $0$ as $R\to\infty$ by (3.9), (3.10). Therefore, (4.8) $H(\mathbf{r}^{\prime})=-\int_{\partial D_{2}}\left(H\frac{\partial v}{\partial\nu}-v\frac{\partial H}{\partial\nu}\right)dS.$ We now multiply (4.6) by $F_{2}(d_{2})$, $F_{2}:=F^{(2)}_{2,\mathbf{n},\mathbf{p}}$, then multiply (4.8) by $E_{2}(d_{2})$, $E_{i}:=E^{(2)}_{i,\mathbf{n},\mathbf{p}}$, and add these equations. By (3.1) and (3.7) we have $F_{2}(d_{2})G(\mathbf{r})=E_{2}(d_{2})H(\mathbf{r}),\quad\mathbf{r}\in\partial D_{2},$ first for $\mathbf{r}\in\partial D_{2}\cap R$ but then for all $\mathbf{r}\in\partial D_{2}$ by shared symmetries (3.3), (3.4) of $G,H$. Therefore, we find (4.9) $E_{2}(d_{2})H(\mathbf{r}^{\prime})=\int_{\partial D_{2}}v\left(E_{2}(d_{2})\frac{\partial H}{\partial\nu}-F_{2}(d_{2})\frac{\partial G}{\partial\nu}\right)dS.$ The normal derivative and the derivative with respect to $s_{2}$ are related by $\frac{\partial}{\partial\nu}=\frac{1}{h_{2}}\frac{\partial}{\partial s_{2}},$ where $h_{2}$ is the scale factor of the 5-cyclidic coordinate $s_{2}$ given by (4.4); see [13, (22)]. Let $\mathbf{r}\in\partial D_{2}\cap R$ with 5-cyclidic coordinates $s_{1},s_{2}=d_{2},s_{3}$. Then $\displaystyle\left(E_{2}(d_{2})\frac{\partial H}{\partial\nu}-F_{2}(d_{2})\frac{\partial G}{\partial\nu}\right)(\mathbf{r})$ $\displaystyle\hskip 36.98866pt=E_{2}(d_{2})\frac{\partial(\\|\mathbf{r}\\|^{2}+1)^{-1/2}}{\partial\nu}E_{1}(s_{1})F_{2}(d_{2})E_{3}(s_{3})$ $\displaystyle\hskip 65.44142pt+E_{2}(d_{2})(\\|\mathbf{r}\\|^{2}+1)^{-1/2}h_{2}^{-1}E_{1}(s_{1})F_{2}^{\prime}(d_{2})E_{3}(s_{3})$ $\displaystyle\hskip 65.44142pt- F_{2}(d_{2})\frac{\partial(\\|\mathbf{r}\\|^{2}+1)^{-1/2}}{\partial\nu}E_{1}(s_{1})E_{2}(d_{2})E_{3}(s_{3})$ $\displaystyle\hskip 65.44142pt- F_{2}(d_{2})(\\|\mathbf{r}\\|^{2}+1)^{-1/2}h_{2}^{-1}E_{1}(s_{1})E_{2}^{\prime}(d_{2})E_{3}(s_{3})$ $\displaystyle\hskip 36.98866pt=h_{2}^{-1}(\\|\mathbf{r}\\|^{2}+1)^{-1/2}E_{1}(s_{1})\left\\{E_{2}(d_{2})F_{2}^{\prime}(d_{2})-E_{2}^{\prime}(d_{2})F_{2}(d_{2})\right\\}E_{3}(s_{3}).$ We now use (3.5) and obtain (4.10) $\left(E_{2}(d_{2})\frac{\partial H}{\partial\nu}-F_{2}(d_{2})\frac{\partial G}{\partial\nu}\right)(\mathbf{r})=\frac{G(\mathbf{r})}{h_{2}(\mathbf{r})\omega(d_{2})E_{2}(d_{2})},$ which holds for all $\mathbf{r}\in\partial D_{2}$ because $G$ and $H$ share the symmetries (3.3), (3.4). When we substitute (4.10) in (4.9) we arrive at (4.3) ∎ We obtain the expansion of the reciprocal distance in 5-cyclidic harmonics. ###### Theorem 4.2. Let $\mathbf{r},\mathbf{r}^{\prime}\in\mathbb{R}^{3}$ with 5-cyclidic coordinates $s_{2},s_{2}^{\prime}$, respectively. If $s_{2}<s_{2}^{\prime}$ then (4.11) $\frac{1}{\\|\mathbf{r}-\mathbf{r}^{\prime}\\|}=\pi\sum_{\mathbf{n}\in\mathbb{N}_{0}^{2}}\sum_{\mathbf{p}\in\\{0,1\\}^{4}}G^{(2)}_{\mathbf{n},\mathbf{p}}(\mathbf{r})H^{(2)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}^{\prime}).$ ###### Proof. We pick $d_{2}$ such that $s_{2}<d_{2}<s_{2}^{\prime}$, and consider the domain $D_{2}$ defined in (4.1). The function $f(\mathbf{q}):=\\|\mathbf{q}-\mathbf{r}^{\prime}\\|^{-1}$ is harmonic on an open set containing $\bar{D}_{2}$. Therefore, by [13, (95),(97)], we have (4.12) $\frac{1}{\\|\mathbf{r}-\mathbf{r}^{\prime}\\|}=\sum_{\mathbf{n}\in\mathbb{N}_{0}^{2}}\sum_{\mathbf{p}\in\\{0,1\\}^{4}}d^{(2)}_{\mathbf{n},\mathbf{p}}G^{(2)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}),$ where $d^{(2)}_{\mathbf{n},\mathbf{p}}:=\frac{1}{4\omega(d_{2})\\{E^{(2)}_{2,\mathbf{n},\mathbf{p}}(d_{2})\\}^{2}}\int_{\partial D_{2}}\frac{G^{(2)}_{\mathbf{n},\mathbf{p}}(\mathbf{q})}{h_{2}(\mathbf{q})\\|\mathbf{q}-\mathbf{r}^{\prime}\\|}\,dS(\mathbf{q}).$ Using Theorem 4.1, we obtain (4.11). ∎ ## 5\. 5-cyclidic harmonics of the first kind In [13, Section V] we introduced special solutions $w_{i}(s_{i})=E^{(1)}_{i,\mathbf{n},\mathbf{p}}(s_{i})$ to equation (2.10) for eigenvalues $\lambda_{j}=\lambda^{(1)}_{j,\mathbf{n},\mathbf{p}}$, $j=1,2$, for every $\mathbf{n}\in\mathbb{N}_{0}^{2}$, $\mathbf{p}=(p_{1},p_{2},p_{3})\in\\{0,1\\}^{3}$. These functions have the form $\displaystyle E^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1})$ $\displaystyle=$ $\displaystyle(a_{1}-s_{1})^{p_{1}/2}{\mathcal{E}}^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1}),\quad s_{1}\in(a_{0},a_{1}),$ $\displaystyle E^{(1)}_{i,\mathbf{n},\mathbf{p}}(s_{i})$ $\displaystyle=$ $\displaystyle(s_{i}-a_{i-1})^{p_{i-1}/2}(a_{i}-s_{i})^{p_{i}/2}{\mathcal{E}}^{(1)}_{i,\mathbf{n},\mathbf{p}}(s_{i}),\quad s_{i}\in(a_{i-1},a_{i}),i=2,3,$ where ${\mathcal{E}}^{(1)}_{1,\mathbf{n},\mathbf{p}}$ is analytic on $(a_{0},a_{1}]$ (but not at $a_{0}$) while ${\mathcal{E}}^{(1)}_{i,\mathbf{n},\mathbf{p}}$ is analytic on $[a_{i-1},a_{i}]$ for $i=2,3$. As in [13, Section VI] we define the internal 5-cyclidic harmonic of the first kind by (5.1) $G^{(1)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}):=(\\|\mathbf{r}\\|^{2}+1)^{-1/2}E^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1})E^{(1)}_{2,\mathbf{n},\mathbf{p}}(s_{2})E^{(1)}_{3,\mathbf{n},\mathbf{p}}(s_{3}),\quad\mathbf{r}\in R.$ According to (2.9), $G^{(1)}_{\mathbf{n},\mathbf{p}}$ is a harmonic function in the region $R$. In order to analytically extend $G^{(1)}_{\mathbf{n},\mathbf{p}}$ to a larger domain of definition, some preparations are necessary. Let $P^{(1)}_{1,\mathbf{n},\mathbf{p}}$ be the solution to (2.10) (with $\lambda_{j}=\lambda^{(1)}_{j,\mathbf{n},\mathbf{p}}$) on $(a_{0},a_{1})$ belonging to the exponent $0$ at $s=a_{0}$ and uniquely determined by the condition $P^{(1)}_{1,\mathbf{n},\mathbf{p}}(a_{0})=1$. We write $P^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1})=(a_{1}-s_{1})^{p_{1}/2}{\mathcal{P}}^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1}),\quad s_{1}\in(a_{0},a_{1}),$ where ${\mathcal{P}}^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1})$ is analytic on $[a_{0},a_{1})$. Then using the functions $\chi_{j}$ from Section 2 we define (5.2) $I^{(1)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}):=(\\|\mathbf{r}\\|^{2}+1)^{-1/2}\prod_{j=1}^{3}(\chi_{j}(\mathbf{r}))^{p_{j}}{\mathcal{P}}^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1})\prod_{i=2}^{3}{\mathcal{E}}^{(1)}_{i,\mathbf{n},\mathbf{p}}(s_{i})\quad\text{if $s_{1}\neq a_{1}$}.$ The condition $s_{1}\neq a_{1}$ is equivalent to $\mathbf{r}\in\mathbb{R}^{3}\setminus(K_{1}\cup M_{1})$; see Figure 1. Similarly, let $Q^{(1)}_{1,\mathbf{n},\mathbf{p}}$ be the solution to (2.10) (with $\lambda_{j}=\lambda^{(1)}_{j,\mathbf{n},\mathbf{p}}$) on $(a_{0},a_{1})$ belonging to the exponent $\frac{1}{2}$ at $s=a_{0}$ and uniquely determined by the condition $\lim_{s_{1}\to a_{0}^{+}}\omega(s_{1})\frac{d}{ds_{1}}Q^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1})=1$. We write $Q^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1})=(s_{1}-a_{0})^{1/2}(a_{1}-s_{1})^{p_{1}/2}{\mathcal{Q}}^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1}),\quad s_{1}\in(a_{0},a_{1}),$ where ${\mathcal{Q}}^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1})$ is analytic on $[a_{0},a_{1})$. Then we define (5.3) $J^{(1)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}):=(\\|\mathbf{r}\\|^{2}+1)^{-1/2}\chi_{0}(\mathbf{r})\prod_{j=1}^{3}(\chi_{j}(\mathbf{r}))^{p_{j}}{\mathcal{Q}}^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1})\prod_{i=2}^{3}{\mathcal{E}}^{(1)}_{i,\mathbf{n},\mathbf{p}}(s_{i})\quad\text{if $s_{1}\neq a_{1}$}.$ ###### Lemma 5.1. The functions $I^{(1)}_{\mathbf{n},\mathbf{p}}$ and $J^{(1)}_{\mathbf{n},\mathbf{p}}$ are harmonic on $\mathbb{R}^{3}\setminus(K_{1}\cup M_{1})$. They have the symmetries (5.4) $\displaystyle I^{(1)}_{\mathbf{n},\mathbf{p}}(\sigma_{0}(\mathbf{r}))$ $\displaystyle=$ $\displaystyle\\|\mathbf{r}\\|I^{(1)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}),$ (5.5) $\displaystyle I^{(1)}_{\mathbf{n},\mathbf{p}}(\sigma_{j}(\mathbf{r}))$ $\displaystyle=$ $\displaystyle(-1)^{p_{j}}I^{(1)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}),\quad j=1,2,3,$ (5.6) $\displaystyle J^{(1)}_{\mathbf{n},\mathbf{p}}(\sigma_{0}(\mathbf{r}))$ $\displaystyle=$ $\displaystyle-\\|\mathbf{r}\\|J^{(1)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}),$ (5.7) $\displaystyle J^{(1)}_{\mathbf{n},\mathbf{p}}(\sigma_{j}(\mathbf{r}))$ $\displaystyle=$ $\displaystyle(-1)^{p_{j}}J^{(1)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}),\quad j=1,2,3.$ ###### Proof. By definition (5.2), $I^{(1)}_{\mathbf{n},\mathbf{p}}$ is a composition of continuous functions provided $s_{1}\neq a_{1}$, that is, $I^{(1)}_{\mathbf{n},\mathbf{p}}$ is continuous on $\mathbb{R}^{3}\setminus(K_{1}\cup M_{1})$. $I^{(1)}_{\mathbf{n},\mathbf{p}}$ is also a composition of analytic functions provided $s_{1}\neq a_{1}$ and $s_{2}\neq s_{3}$, that is, $I^{(1)}_{\mathbf{n},\mathbf{p}}$ is analytic on $\mathbb{R}^{3}\setminus(K_{1}\cup M_{1}\cup A_{2})$. Thus it is also harmonic on $\mathbb{R}^{3}\setminus(K_{1}\cup M_{1}\cup A_{2})$. By the same argument as in the proof of Theorem 3.1, $A_{2}$ is a removable singularity of $I^{(1)}_{\mathbf{n},\mathbf{p}}$. Thus $I^{(1)}_{\mathbf{n},\mathbf{p}}$ is harmonic on $\mathbb{R}^{3}\setminus(K_{1}\cup M_{1})$. The proof that $J^{(1)}_{\mathbf{n},\mathbf{p}}$ is harmonic on $\mathbb{R}^{3}\setminus(K_{1}\cup M_{1})$ is analogous. The symmetry properties follow from (5.2), (5.3) and Lemma 2.1. ∎ Since $P^{(1)}_{1,\mathbf{n},\mathbf{p}},Q^{(1)}_{1,\mathbf{n},\mathbf{p}}$ form a fundamental system of solutions to (2.10) (with $\lambda_{j}=\lambda^{(1)}_{j,\mathbf{n},\mathbf{p}}$) on $(a_{0},a_{1})$, there are (nonzero) scalars $\alpha^{(1)}_{\mathbf{n},\mathbf{p}}$, $\beta^{(1)}_{\mathbf{n},\mathbf{p}}$ such that $E^{(1)}_{1,\mathbf{n},\mathbf{p}}=\alpha^{(1)}_{\mathbf{n},\mathbf{p}}P^{(1)}_{1,\mathbf{n},\mathbf{p}}+\beta^{(1)}_{\mathbf{n},\mathbf{p}}Q^{(1)}_{1,\mathbf{n},\mathbf{p}}.$ This leads us to the global definition of internal 5-cyclidic harmonics of the first kind (5.8) $G^{(1)}_{\mathbf{n},\mathbf{p}}:=\alpha^{(1)}_{\mathbf{n},\mathbf{p}}I^{(1)}_{\mathbf{n},\mathbf{p}}+\beta^{(1)}_{\mathbf{n},\mathbf{p}}J^{(1)}_{\mathbf{n},\mathbf{p}}$ which is consistent with (5.1). We also note that, if $\\|\mathbf{r}\\|<1$ and $\mathbf{r}\not\in K_{1}$, then (5.2), (5.3), (5.8) imply that (5.9) $G^{(1)}_{\mathbf{n},\mathbf{p}}(\mathbf{r})=(\\|\mathbf{r}\\|^{2}+1)^{-1/2}\prod_{j=1}^{3}(\chi_{j}(\mathbf{r}))^{p_{j}}\prod_{i=1}^{3}{\mathcal{E}}^{(1)}_{i,\mathbf{n},\mathbf{p}}(s_{i}).$ ###### Theorem 5.2. Let $\mathbf{n}\in\mathbb{N}_{0}^{2}$ and $\mathbf{p}=(p_{1},p_{2},p_{3})\in\\{0,1\\}^{3}$. Then $G^{(1)}_{\mathbf{n},\mathbf{p}}$ extends continuously to a harmonic function on $\mathbb{R}^{3}\setminus M_{1}$. Moreover, (5.10) $G^{(1)}_{\mathbf{n},\mathbf{p}}(\sigma_{j}(\mathbf{r}))=(-1)^{p_{j}}G^{(1)}_{\mathbf{n},\mathbf{p}}(\mathbf{r})\quad\text{for $j=1,2,3$}.$ ###### Proof. By Lemma 5.1, $G^{(1)}_{\mathbf{n},\mathbf{p}}$ is harmonic on $\mathbb{R}^{3}\setminus(K_{1}\cup M_{1})$. If $\\|\mathbf{r}\\|<1$ we have $s_{1}\neq a_{0}$. Therefore, the right-hand side of (5.9) is continuous on the ball $B_{1}(\mathbf{0})$ and harmonic on $B_{1}(\mathbf{0})\setminus(A_{1}\cup A_{2})$. Thus it is harmonic on $B_{1}(\mathbf{0})$ which proves the first part of the statement of the theorem. The symmetries follow from (5.5), (5.7). ∎ It will be useful to introduce another solution to (2.10) by (5.11) $F^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1}):=\gamma^{(1)}_{\mathbf{n},\mathbf{p}}\left(\alpha^{(1)}_{\mathbf{n},\mathbf{p}}P^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1})-\beta^{(1)}_{\mathbf{n},\mathbf{p}}Q^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1})\right),\quad s_{1}\in(a_{0},a_{1}).$ We determine $\gamma^{(1)}_{\mathbf{n},\mathbf{p}}$ from the Wronskian (5.12) $\omega(s_{1})\left(E^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1})\frac{d}{ds_{1}}F^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1})-F^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1})\frac{d}{ds_{1}}E^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1})\right)=1$ which is equivalent to $\gamma^{(1)}_{\mathbf{n},\mathbf{p}}=\frac{-1}{2\alpha^{(1)}_{\mathbf{n},\mathbf{p}}\beta^{(1)}_{\mathbf{n},\mathbf{p}}}.$ We define external 5-cyclidic harmonics of the first kind by (5.13) $H^{(1)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}):=\gamma^{(1)}_{\mathbf{n},\mathbf{p}}\\|\mathbf{r}\\|^{-1}G^{(1)}_{\mathbf{n},\mathbf{p}}(\sigma_{0}(\mathbf{r}))\quad\text{for $\mathbf{r}\in\mathbb{R}^{3}\setminus K_{1}$}.$ The reason to include the factor $\gamma^{(1)}_{\mathbf{n},\mathbf{p}}$ is that we aim for a simple form of the expansion formula (6.4). In particular, we have (5.14) $H^{(1)}_{\mathbf{n},\mathbf{p}}(\mathbf{r})=(\\|\mathbf{r}\\|^{2}+1)^{-1/2}F^{(1)}_{1,\mathbf{n},\mathbf{p}}(s_{1})E^{(1)}_{2,\mathbf{n},\mathbf{p}}(s_{2})E^{(1)}_{3,\mathbf{n},\mathbf{p}}(s_{3})\quad\text{for $\mathbf{r}\in R$}.$ We notice an important difference between 5-cyclidic harmonics of the first and second kind (considered in Section 3). The external 5-cyclidic harmonics of the first kind are simply the Kelvin transformations of the internal 5-cyclidic harmonics of the first kind up to a constant factor. There is no such simple relationship between internal and external 5-cyclidic harmonics of the second kind. ###### Theorem 5.3. Let $\mathbf{n}\in\mathbb{N}_{0}^{2}$ and $\mathbf{p}=(p_{1},p_{2},p_{3})\in\\{0,1\\}^{3}$. Then $H^{(1)}_{\mathbf{n},\mathbf{p}}$ is harmonic on $\mathbb{R}^{3}\setminus K_{1}$. The functions $H^{(1)}_{\mathbf{n},\mathbf{p}}$ share the symmetries (5.10) with $G^{(1)}_{\mathbf{n},\mathbf{p}}$. Moreover, (5.15) $H^{(1)}_{\mathbf{n},\mathbf{p}}(\mathbf{r})=O(\\|\mathbf{r}\\|^{-1})\quad\text{as $\\|\mathbf{r}\\|\to\infty$},$ and (5.16) $\\|\nabla H^{(1)}_{\mathbf{n},\mathbf{p}}(\mathbf{r})\\|=O(\\|\mathbf{r}\\|^{-2})\quad\text{as $\\|\mathbf{r}\\|\to\infty$}.$ ###### Proof. The proof of analyticity and symmetry follows directly from (5.13) and Theorem 5.2. Estimates (5.15) and (5.16) follow from the fact that the Kelvin transformation of $H^{(1)}_{\mathbf{n},\mathbf{p}}$ is analytic at the origin. ∎ ## 6\. Expansion of the reciprocal distance in 5-cyclidic harmonics of first kind For fixed $s\in(a_{0},a_{1})$ the coordinate surface (2.1) consists of two closed surfaces of genus $0$. One lies inside the unit ball $B_{1}(\mathbf{0})$ and the other one is obtained from it by inversion $\sigma_{0}$. We consider the region $D_{1}$ interior to the coordinate surface $s=d_{1}$ which lies in $B_{1}(\mathbf{0})$: (6.1) $D_{1}:=\\{\mathbf{r}\in\mathbb{R}^{3}:\\|\mathbf{r}\\|<1,s_{1}>d_{1}\\}.$ ###### Theorem 6.1. Let $d_{1}\in(a_{0},a_{1})$, $\mathbf{n}\in\mathbb{N}_{0}^{2}$, $\mathbf{p}\in\\{0,1\\}^{3}$. Then (6.2) $H^{(1)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}^{\prime})=\frac{1}{4\pi\omega(d_{1})\\{E^{(1)}_{1,\mathbf{n},\mathbf{p}}(d_{1})\\}^{2}}\int_{\partial D_{1}}\frac{G^{(1)}_{\mathbf{n},\mathbf{p}}(\mathbf{r})}{h_{1}(\mathbf{r})\\|\mathbf{r}-\mathbf{r}^{\prime}\\|}\,dS(\mathbf{r})$ for all $\mathbf{r}^{\prime}\in\mathbb{R}^{3}\setminus\bar{D}_{1}$. The scale factor $h_{1}$ is given by (6.3) $16\\{h_{1}(\mathbf{r})\\}^{2}=\frac{(\\|\mathbf{r}\\|^{2}-1)^{2}}{(d_{1}-a_{0})^{2}}+\frac{4x^{2}}{(d_{1}-a_{1})^{2}}+\frac{4y^{2}}{(d_{1}-a_{2})^{2}}+\frac{4z^{2}}{(d_{1}-a_{3})^{2}}.$ ###### Proof. The proof is similar to the proof of Theorem 4.1. We use (5.1), (5.14) and the Wronskian (5.12). ∎ We obtain the expansion of the reciprocal distance in 5-cyclidic harmonics of first kind. ###### Theorem 6.2. Let $\mathbf{r},\mathbf{r}^{\prime}\in\mathbb{R}^{3}$ with 5-cyclidic coordinates $s_{1},s_{1}^{\prime}$, respectively. If either (a) $\\|\mathbf{r}\\|,\\|\mathbf{r}^{\prime}\\|\leq 1$, $s_{1}>s_{1}^{\prime}$, or (b) $\\|\mathbf{r}\\|<1<\\|\mathbf{r}^{\prime}\\|$, or (c) $\\|\mathbf{r}\\|,\\|\mathbf{r}^{\prime}\\|\geq 1$, $s_{1}<s_{1}^{\prime}$, then (6.4) $\frac{1}{\\|\mathbf{r}-\mathbf{r}^{\prime}\\|}=2\pi\sum_{\mathbf{n}\in\mathbb{N}_{0}^{2}}\sum_{\mathbf{p}\in\\{0,1\\}^{3}}G^{(1)}_{\mathbf{n},\mathbf{p}}(\mathbf{r})H^{(1)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}^{\prime}).$ ###### Proof. Suppose (a) or (b) holds. Pick $d_{1}$ such that $s_{1}^{\prime}<d_{1}<s_{1}$ if (a) holds, or such that $a_{0}<d_{1}<s_{1}$ if (b) holds. Then consider the domain $D_{1}$ defined in (6.1). The function $f(\mathbf{q}):=\\|\mathbf{q}-\mathbf{r}^{\prime}\\|^{-1}$ is harmonic on an open set containing $\bar{D}_{1}$. Therefore, by [13, (71),(73)], we have (6.5) $\frac{1}{\\|\mathbf{r}-\mathbf{r}^{\prime}\\|}=\sum_{\mathbf{n}\in\mathbb{N}_{0}^{2}}\sum_{\mathbf{p}\in\\{0,1\\}^{3}}d^{(1)}_{\mathbf{n},\mathbf{p}}G^{(1)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}),$ where $d^{(1)}_{\mathbf{n},\mathbf{p}}=\frac{1}{2\omega(d_{1})\\{E^{(1)}_{1,\mathbf{n},\mathbf{p}}(d_{1})\\}^{2}}\int_{\partial D_{1}}\frac{G^{(1)}_{\mathbf{n},\mathbf{p}}(\mathbf{q})}{h_{1}(\mathbf{q})\\|\mathbf{q}-\mathbf{r}^{\prime}\\|}\,dS(\mathbf{q}).$ Using Theorem 6.1, we obtain (6.4). Now suppose (c) holds. Then the points $\sigma_{0}(\mathbf{r}^{\prime})$, $\sigma_{0}(\mathbf{r})$ in place of $\mathbf{r},\mathbf{r}^{\prime}$ satisfy (a), so, by what we already proved, $\frac{1}{\\|\sigma_{0}(\mathbf{r})-\sigma_{0}(\mathbf{r}^{\prime})\\|}=2\pi\sum_{\mathbf{n}\in\mathbb{N}_{0}^{2}}\sum_{\mathbf{p}\in\\{0,1\\}^{3}}G^{(1)}_{\mathbf{n},\mathbf{p}}(\sigma_{0}(\mathbf{r}^{\prime}))H^{(1)}_{\mathbf{n},\mathbf{p}}(\sigma_{0}(\mathbf{r})).$ This gives (6.4) by using (5.13) and observing that $\\|\mathbf{r}-\mathbf{r}^{\prime}\\|=\\|\mathbf{r}\\|\\|\mathbf{r}^{\prime}\\|\\|\sigma_{0}(\mathbf{r})-\sigma_{0}(\mathbf{r}^{\prime})\\|.$ ∎ ## 7\. 5-cyclidic harmonics of the third kind The 5-cyclidic harmonics of the third kind are treated analogously to the harmonics of the first kind. Therefore, we will omit all proofs in the following two sections. In [13, Section IX] we introduced special solutions $w_{i}(s_{i})=E^{(3)}_{i,\mathbf{n},\mathbf{p}}(s_{i})$ to equation (2.10) for eigenvalues $\lambda_{j}=\lambda^{(3)}_{j,\mathbf{n},\mathbf{p}}$, $j=1,2$, for every $\mathbf{n}\in\mathbb{N}_{0}^{2}$, $\mathbf{p}=(p_{0},p_{1},p_{2})\in\\{0,1\\}^{3}$. These functions have the form $\displaystyle E^{(3)}_{i,\mathbf{n},\mathbf{p}}(s_{i})$ $\displaystyle=$ $\displaystyle(s_{i}-a_{i-1})^{p_{i-1}/2}(a_{i}-s_{i})^{p_{i}/2}{\mathcal{E}}^{(3)}_{i,\mathbf{n},\mathbf{p}}(s_{i}),\quad s_{i}\in(a_{i-1},a_{i}),\ i=1,2,$ $\displaystyle E^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3})$ $\displaystyle=$ $\displaystyle(s_{3}-a_{2})^{p_{2}/2}{\mathcal{E}}^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3}),\quad s_{3}\in(a_{2},a_{3}),$ where ${\mathcal{E}}^{(3)}_{i,\mathbf{n},\mathbf{p}}$ is analytic on $[a_{i-1},a_{i}]$ for $i=1,2$ while ${\mathcal{E}}^{(3)}_{3,\mathbf{n},\mathbf{p}}$ is analytic on $[a_{2},a_{3})$. As in [13, Section X] we define the internal 5-cyclidic harmonic of the third kind by (7.1) $G^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}):=(\\|\mathbf{r}\\|^{2}+1)^{-1/2}E^{(3)}_{1,\mathbf{n},\mathbf{p}}(s_{1})E^{(3)}_{2,\mathbf{n},\mathbf{p}}(s_{2})E^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3}),\quad\mathbf{r}\in R.$ Let $P^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3})$ be the solution to (2.10) (with $\lambda_{j}=\lambda^{(3)}_{j,\mathbf{n},\mathbf{p}}$) on $(a_{2},a_{3})$ belonging to the exponent $0$ at $s=a_{3}$ and uniquely determined by the condition $P^{(3)}_{3,\mathbf{n},\mathbf{p}}(a_{3})=1$. We write $P^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3})=(s_{3}-a_{2})^{p_{2}/2}{\mathcal{P}}^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3}),\quad s_{3}\in(a_{2},a_{3}),$ where ${\mathcal{P}}^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3})$ is analytic on $(a_{2},a_{3}]$. Then we define (7.2) $I^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}):=(\\|\mathbf{r}\\|^{2}+1)^{-1/2}\prod_{j=0}^{2}(\chi_{j}(\mathbf{r}))^{p_{j}}\prod_{i=1}^{2}{\mathcal{E}}^{(3)}_{i,\mathbf{n},\mathbf{p}}(s_{i}){\mathcal{P}}^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3})\quad\text{if $s_{3}\neq a_{2}$}.$ The condition $s_{3}\neq a_{2}$ is equivalent to $\mathbf{r}\in\mathbb{R}^{3}\setminus(K_{2}\cup M_{2})$; see Figure 2. Similarly, let $Q^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3})$ be the solution to (2.10) (with $\lambda_{j}=\lambda^{(3)}_{j,\mathbf{n},\mathbf{p}}$) on $(a_{2},a_{3})$ belonging to the exponent $\frac{1}{2}$ at $s=a_{3}$ and uniquely determined by the condition $\lim_{s_{3}\to a_{3}^{-}}\omega(s_{3})\frac{d}{ds_{3}}Q^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3})=1$. We write $Q^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3})=(a_{3}-s_{3})^{1/2}(s_{3}-a_{2})^{p_{2}/2}{\mathcal{Q}}^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3}),\quad s_{3}\in(a_{2},a_{3}),$ where ${\mathcal{Q}}^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3})$ is analytic on $(a_{2},a_{3}]$. Then we define (7.3) $J^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}):=(\\|\mathbf{r}\\|^{2}+1)^{-1/2}\chi_{3}(\mathbf{r})\prod_{j=0}^{2}(\chi_{j}(\mathbf{r}))^{p_{j}}\prod_{i=1}^{2}{\mathcal{E}}^{(3)}_{i,\mathbf{n},\mathbf{p}}(s_{i}){\mathcal{Q}}^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3})\quad\text{if $s_{3}\neq a_{2}$}.$ ###### Lemma 7.1. The functions $I^{(3)}_{\mathbf{n},\mathbf{p}}$ and $J^{(3)}_{\mathbf{n},\mathbf{p}}$ are harmonic on $\mathbb{R}^{3}\setminus(K_{2}\cup M_{2})$. They have the symmetries (7.4) $\displaystyle I^{(3)}_{\mathbf{n},\mathbf{p}}(\sigma_{0}(\mathbf{r}))$ $\displaystyle=$ $\displaystyle(-1)^{p_{0}}\\|\mathbf{r}\\|I^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}),$ (7.5) $\displaystyle I^{(3)}_{\mathbf{n},\mathbf{p}}(\sigma_{j}(\mathbf{r}))$ $\displaystyle=$ $\displaystyle(-1)^{p_{j}}I^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}),\quad j=1,2,$ (7.6) $\displaystyle I^{(3)}_{\mathbf{n},\mathbf{p}}(\sigma_{3}(\mathbf{r}))$ $\displaystyle=$ $\displaystyle I^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}),$ (7.7) $\displaystyle J^{(3)}_{\mathbf{n},\mathbf{p}}(\sigma_{0}(\mathbf{r}))$ $\displaystyle=$ $\displaystyle(-1)^{p_{0}}\\|\mathbf{r}\\|J^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}),$ (7.8) $\displaystyle J^{(3)}_{\mathbf{n},\mathbf{p}}(\sigma_{j}(\mathbf{r}))$ $\displaystyle=$ $\displaystyle(-1)^{p_{j}}J^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}),\quad j=1,2,$ (7.9) $\displaystyle J^{(3)}_{\mathbf{n},\mathbf{p}}(\sigma_{3}(\mathbf{r}))$ $\displaystyle=-$ $\displaystyle J^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}).$ Since $P^{(3)}_{3,\mathbf{n},\mathbf{p}},Q^{(3)}_{3,\mathbf{n},\mathbf{p}}$ form a fundamental system of solutions to (2.10) (with $\lambda_{j}=\lambda^{(3)}_{j,\mathbf{n},\mathbf{p}}$) on $(a_{2},a_{3})$, there are (nonzero) scalars $\alpha^{(3)}_{\mathbf{n},\mathbf{p}}$, $\beta^{(3)}_{\mathbf{n},\mathbf{p}}$ such that $E^{(3)}_{3,\mathbf{n},\mathbf{p}}=\alpha^{(3)}_{\mathbf{n},\mathbf{p}}P^{(3)}_{3,\mathbf{n},\mathbf{p}}+\beta^{(3)}_{\mathbf{n},\mathbf{p}}Q^{(3)}_{3,\mathbf{n},\mathbf{p}}.$ This leads to the global definition of internal 5-cyclidic harmonics of the third kind (7.10) $G^{(3)}_{\mathbf{n},\mathbf{p}}:=\alpha^{(3)}_{\mathbf{n},\mathbf{p}}I^{(3)}_{\mathbf{n},\mathbf{p}}+\beta^{(3)}_{\mathbf{n},\mathbf{p}}J^{(3)}_{\mathbf{n},\mathbf{p}}.$ If $z>0$, we can write $G^{(3)}_{\mathbf{n},\mathbf{p}}$ as follows (7.11) $G^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r})=(\\|\mathbf{r}\\|^{2}+1)^{-1/2}\prod_{j=0}^{2}(\chi_{j}(\mathbf{r}))^{p_{j}}\prod_{i=1}^{3}{\mathcal{E}}^{(3)}_{i,\mathbf{n},\mathbf{p}}(s_{i}).$ ###### Theorem 7.2. Let $\mathbf{n}\in\mathbb{N}_{0}^{2}$ and $\mathbf{p}=(p_{0},p_{1},p_{2})\in\\{0,1\\}^{3}$. Then $G^{(3)}_{\mathbf{n},\mathbf{p}}$ extends continuously to a harmonic function on $\mathbb{R}^{3}\setminus M_{2}$. Moreover (7.12) $\displaystyle G^{(3)}_{\mathbf{n},\mathbf{p}}(\sigma_{0}(\mathbf{r}))$ $\displaystyle=$ $\displaystyle(-1)^{p_{0}}\\|\mathbf{r}\\|G^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}),$ (7.13) $\displaystyle G^{(3)}_{\mathbf{n},\mathbf{p}}(\sigma_{j}(\mathbf{r}))$ $\displaystyle=$ $\displaystyle(-1)^{p_{j}}G^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}),\quad j=1,2.$ We introduce another solution of (2.10) by (7.14) $F^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3})=\gamma^{(3)}_{\mathbf{n},\mathbf{p}}\left(\alpha^{(3)}_{\mathbf{n},\mathbf{p}}P^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3})-\beta^{(3)}_{\mathbf{n},\mathbf{p}}Q^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3})\right),\quad s_{3}\in(a_{2},a_{3}).$ We determine $\gamma^{(3)}_{\mathbf{n},\mathbf{p}}$ from the Wronskian (7.15) $\omega(s_{3})\left(E^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3})\frac{d}{ds_{3}}F^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3})-F^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3})\frac{d}{ds_{3}}E^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3})\right)=1,$ which is equivalent to $\gamma^{(3)}_{\mathbf{n},\mathbf{p}}=\frac{-1}{2\alpha^{(3)}_{\mathbf{n},\mathbf{p}}\beta^{(3)}_{\mathbf{n},\mathbf{p}}}.$ We define external 5-cyclidic harmonics of the third kind by (7.16) $H^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}):=\gamma^{(3)}_{\mathbf{n},\mathbf{p}}G^{(3)}_{\mathbf{n},\mathbf{p}}(\sigma_{3}(\mathbf{r}))\quad\text{for $\mathbf{r}\in\mathbb{R}^{3}\setminus K_{2}$}.$ In particular, we have (7.17) $H^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r})=(\\|\mathbf{r}\\|^{2}+1)^{-1/2}E^{(3)}_{1,\mathbf{n},\mathbf{p}}(s_{1})E^{(3)}_{2,\mathbf{n},\mathbf{p}}(s_{2})F^{(3)}_{3,\mathbf{n},\mathbf{p}}(s_{3})\quad\text{for $\mathbf{r}\in R$.}$ ###### Theorem 7.3. Let $\mathbf{n}\in\mathbb{N}_{0}^{2}$ and $\mathbf{p}=(p_{0},p_{1},p_{2})\in\\{0,1\\}^{3}$. Then $H^{(3)}_{\mathbf{n},\mathbf{p}}$ is harmonic on $\mathbb{R}^{3}\setminus K_{2}$. The functions $H^{(3)}_{\mathbf{n},\mathbf{p}}$ share the symmetries (7.12), (7.13) with $G^{(3)}_{\mathbf{n},\mathbf{p}}$. Moreover, (7.18) $H^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r})=O(\\|\mathbf{r}\\|^{-1})\quad\text{as $\\|\mathbf{r}\\|\to\infty$},$ and (7.19) $\\|\nabla H^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r})\\|=O(\\|\mathbf{r}\\|^{-2})\quad\text{as $\\|\mathbf{r}\\|\to\infty$}.$ ## 8\. Expansion of the reciprocal distance in 5-cyclidic harmonics of third kind For fixed $s\in(a_{2},a_{3})$ the coordinate surface (2.1) consists of two closed surfaces of genus $0$. One lies in the half-space $z>0$ and the other one is obtained from it by reflection at the plane $z=0$. We consider the region interior to the coordinate surface $s=d_{3}$ which lies in the half- space $\\{\mathbf{r}:z>0\\}$: (8.1) $D_{3}:=\\{\mathbf{r}\in\mathbb{R}^{3}:z>0,s_{3}<d_{3}\\}.$ ###### Theorem 8.1. Let $d_{3}\in(a_{2},a_{3})$, $\mathbf{n}\in\mathbb{N}_{0}^{2}$, $\mathbf{p}\in\\{0,1\\}^{3}$. Then (8.2) $H^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}^{\prime})=\frac{1}{4\pi\omega(d_{3})\\{E^{(3)}_{3,\mathbf{n},\mathbf{p}}(d_{3})\\}^{2}}\int_{\partial D_{3}}\frac{G^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r})}{h_{3}(\mathbf{r})\\|\mathbf{r}-\mathbf{r}^{\prime}\\|}\,dS(\mathbf{r})$ for all $\mathbf{r}^{\prime}\in\mathbb{R}^{3}\setminus\bar{D}_{3}$. The scale factor $h_{3}$ is given by (8.3) $16\\{h_{3}(\mathbf{r})\\}^{2}=\frac{(\\|\mathbf{r}\\|^{2}-1)^{2}}{(d_{3}-a_{0})^{2}}+\frac{4x^{2}}{(d_{3}-a_{1})^{2}}+\frac{4y^{2}}{(d_{3}-a_{2})^{2}}+\frac{4z^{2}}{(d_{3}-a_{3})^{2}}.$ We obtain the expansion of the reciprocal distance in 5-cyclidic harmonics of the third kind. ###### Theorem 8.2. Let $\mathbf{r}=(x,y,z),\mathbf{r}^{\prime}=(x^{\prime},y^{\prime},z^{\prime})\in\mathbb{R}^{3}$ with 5-cyclidic coordinates $s_{3},s_{3}^{\prime}$, respectively. If either (a) $z,z^{\prime}\geq 0$, $s_{3}<s_{3}^{\prime}$, or (b) $z^{\prime}<0<z$, or (c) $z,z^{\prime}\leq 0$, $s_{3}^{\prime}<s_{3}$, then (8.4) $\frac{1}{\\|\mathbf{r}-\mathbf{r}^{\prime}\\|}=2\pi\sum_{\mathbf{n}\in\mathbb{N}_{0}^{2}}\sum_{\mathbf{p}\in\\{0,1\\}^{3}}G^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r})H^{(3)}_{\mathbf{n},\mathbf{p}}(\mathbf{r}^{\prime}).$ ## References * [1] D. H. Armitage and S. J. Gardiner. Classical potential theory. Springer Monographs in Mathematics. Springer-Verlag London Ltd., London, 2001. * [2] J. Blimke, J. Myklebust, H. Volkmer, and S. Merrill. Four-shell ellipsoidal model employing multipole expansion in ellipsoidal coordinates. Medical & Biological Engineering & Computing, 46(9):859–869, Sep 2008. * [3] M. Bôcher. Ueber die Reihenentwickelungen der Potentialtheorie. B. G. 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Miller, Jr. Symmetry and separation of variables. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1977. With a foreword by Richard Askey, Encyclopedia of Mathematics and its Applications, Vol. 4. * [21] P. M. Morse and H. Feshbach. Methods of theoretical physics. 2 volumes. McGraw-Hill Book Co., Inc., New York, 1953. * [22] F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors. NIST handbook of mathematical functions. Cambridge University Press, Cambridge, 2010. * [23] G. N. Watson. A treatise on the theory of Bessel functions. Cambridge Mathematical Library. Cambridge University Press, Cambridge, second edition, 1944. * [24] Z. Y. Wen and J. Avery. Some properties of hyperspherical harmonics. Journal of Mathematical Physics, 26(3):396–403, 1985.	arxiv-papers	2013-11-14T14:31:49	2024-09-04T02:49:53.646686	{ "license": "Public Domain", "authors": "Howard S. Cohl and Hans Volkmer", "submitter": "Howard Cohl", "url": "https://arxiv.org/abs/1311.3514" }
1311.3586	# Spike-timing prediction with active dendrites Richard Naud1, Brice Bathellier2 and Wulfram Gerstner3 1 Department of Physics, University of Ottawa, 150 Louis Pasteur, ON, K1N 6N5, Canada. 2 Unit of Neuroscience Information and Complexity (UNIC) CNRS UPR-3239, 1 av. de la Terasse, Gif-sur-Yvette, 91198, France. 3 School of Computer and Communication Sciences and School of Life Sciences, Ecole Polytechnique Federale de Lausanne, Building AAB Lausane-EPFL, 1015, Switzerland. ###### Abstract A complete single-neuron model must correctly reproduce the firing of spikes and bursts. We present a study of a simplified model of deep pyramidal cells of the cortex with active dendrites. We hypothesized that we can model the soma and its apical tuft with only two compartments, without significant loss in the accuracy of spike-timing predictions. The model is based on experimentally measurable impulse-response functions, which transfer the effect of current injected in one compartment to current reaching the other. Each compartment was modeled with a pair of non-linear differential equations and a small number of parameters that approximate the Hodgkin-and-Huxley equations. The predictive power of this model was tested on electrophysiological experiments where noisy current was injected in both the soma and the apical dendrite simultaneously. We conclude that a simple two- compartment model can predict spike times of pyramidal cells stimulated in the soma and dendrites simultaneously. Our results support that regenerating activity in the dendritic tuft is required to properly account for the dynamics of layer 5 pyramidal cells under in-vivo-like conditions. ## I Introduction Partially neglected for a long time, dendrites have been recently shown to treat synaptic input in a surprising variety of modesStuart _et al._ (2007). One particularly striking example is found in pyramidal cells of deep cortical layers. In these cells, a coincidence between a back-propagating action potential and dendritic input can trigger voltage-sensitive ion channels situated on the apical dendrite more than 300 $\mu$m from the soma Larkum _et al._ (1999, 2001). The somatic membrane potential increases only after the activation of dendritic ion channels. This often resulting in a burst of action potentials. Bursts in these cells can therefore signal a coincidence of input from the soma (down) with inputs in the apical dendrites (top). Such top-down coincidence detection is one computation that is attributed to dendritic processes. Other allegedly dendritic computations include subtraction Gabbiani _et al._ (2002), direction selectivity Taylor _et al._ (2000), temporal sequence discrimination Branco _et al._ (2010), binocular disparity Archie and Mel (2000), gain modulation Larkum _et al._ (2004) and self-organization of neuron networks Legenstein and Maass (2011). Models of large pyramidal neurons that are active at the tuft of their apical dendrites were first described by Traub et al. (1991) Traub _et al._ (1991) for the hippocampus. This model of the large CA3 pyramidal neurons included voltage-dependent conductances on the dendrites. It is a model based on the Hodgkin-Huxley description of ion channels. Cable properties of dendrites are taken into account by segmenting the dendrite into smaller compartments. The resulting set of equations is solved numerically. A simplified version of this model was advanced by Pinsky and Rinzel (1994) Pinsky and Rinzel (1994). They have reduced the model to a dendritic compartment and a somatic compartment connected by an effective conductance. The model has a restricted set of five ion channels and accounts for bursting of CA3 pyramidal cells. Models specific to deep cortical cells have been described by extending the approach of Traub et al. (1991). Schaefer et al. (2003) Schaefer _et al._ (2003) used morphological reconstruction to define compartments. This model could reproduce the top-down coincidence detection. Using a simplified approach similar to Pinsky and Rinzel (1994) Pinsky and Rinzel (1994), Larkum et al. (2004) Larkum _et al._ (2004) have modelled dendrite-based gain modulation. The parameters in the model could be tuned to quantitatively reproduce the firing rate response of layer 5 pyramidal cells stimulated at the soma and the dendrites simultaneously. Larkum et al. (2004) concluded that a two-compartment model was sufficient to explain the time- averaged firing rate. A more stringent requirement for neuron model validation, however, is to predict spike times Keat _et al._ (2001); Pillow _et al._ (2005); Jolivet _et al._ (2006, 2008a, 2008b); Gerstner and Naud (2009). Given the low spike- time reliability of pyramidal neurons, spike time prediction is compared to the intrinsic reliability Jolivet _et al._ (2006). This approach can be seen as predicting the instantaneous firing rate Naud _et al._ (2011). Generalized integrate-and-fire models can predict instantaneous firing rate of layer 5 pyramidal neurons with substantial precisionJolivet _et al._ (2008a); Naud _et al._ (2009); Gerstner and Naud (2009) in the absence of dendritic stimulation. The question remains whether a neuron model can predict the spike times of layer 5 pyramidal neurons when both the dendrites and the soma are stimulated simultaneously. We present a study of a simplified model of layer 5 pyramidal cells of the cortex with active dendrites. Following Larkum et al. (2004) Larkum _et al._ (2004), we hypothesized that we can model the soma and its apical tuft with two compartments, without significant loss in the accuracy of spike-timing predictions. We introduce experimentally measurable impulse-response functions (Segev _et al._ , 1995), which transfer the effect of current injected in one compartment to current reaching the other. The impulse-response functions replace the instantaneous connection used in previous two-compartment models Pinsky and Rinzel (1994); Larkum _et al._ (2004) and acts as a third, passive, compartment. Each compartment was modeled with a pair of non-linear differential equations with a small number of parameters that approximate the Hodgkin-and-Huxley equations. The predictive power of this model was tested on electrophysiological experiments where noisy current was injected in both the soma and the apical dendrite simultaneously (Larkum _et al._ , 2004). ## II Methods Methods are separated in four parts. First we present the model, second the experimental protocol, then fitting methods and finally the analysis methods. Figure 1: Schematic representation of the two-compartment model. A Somatic and dendritic compartment communicate through passive and active propagation. Passive communication filters through a convolution (denoted by an asterisk) the current injected in the other compartment. Active communication in the soma introduces a perturbation proportional to the dendritic current $I_{Ca}$. Active communication to the dendrites introduces a stereotypical back- propagating action potential current (BAPC). The somatic compartment has spike-triggered adaptation and a moving threshold. The dendritic compartment has an activation current and recovery current. B Associated experimental protocol with current injection both in soma and apical dendrite of layer 5 pyramidal cells of the rat somato-sensory cortex. Variables are defined in the main text. ### II.1 Description of the Model Fig. 1 shows a schematic representation of the two-compartment model. In details, the model follows the system of differential equations: $\displaystyle C_{s}\frac{dV_{s}}{dt}$ $\displaystyle=$ $\displaystyle- g_{s}(V_{s}-E_{s})+\alpha m+I_{s}$ (1) $\displaystyle+\sum_{\\{\hat{t}_{i}\\}}I_{A}(t-\hat{t}_{i})+\epsilon_{ds}I_{d}$ $\displaystyle C_{d}\frac{dV_{d}}{dt}$ $\displaystyle=$ $\displaystyle- g_{d}(V_{d}-E_{d})+g_{1}m+g_{2}x+I_{d}$ (2) $\displaystyle+\sum_{\\{\hat{t}_{i}\\}}I_{BAP}(t-\hat{t}_{i})+\epsilon_{sd}I_{s}$ $\displaystyle\tau_{m}\frac{dm}{dt}$ $\displaystyle=$ $\displaystyle\frac{1}{1+\exp\left(-\frac{V_{d}-E_{m}}{D_{m}}\right)}-m$ (3) $\displaystyle\tau_{x}\frac{dx}{dt}$ $\displaystyle=$ $\displaystyle m-x$ (4) $\displaystyle\tau_{T}\frac{dV_{T}}{dt}$ $\displaystyle=$ $\displaystyle-(V_{T}-E_{T})+D_{T}\sum_{\\{\hat{t}_{i}\\}}\delta(t-\hat{t}_{i})$ (5) where $I_{s}$ is the current injected in the soma, $I_{d}$ the current injected in the dendrites, $V_{s}$ is the somatic voltage, $V_{d}$ is the dendritic voltage, $m$ is the level of activation of a putative calcium current ($I_{\rm Ca}=g_{1}m$), $x$ is the level of activation of a putative calcium-activated potassium current ($I_{\rm K(Ca)}=g_{2}x$), $V_{T}$ is the dynamic threshold for firing somatic spikes, $I_{A}$ is a spike-triggered current mediating adaptation, $I_{BAP}$ is the the current associated with the back-propagating action potential, $\epsilon_{sd}$ is the filter relating the current injected in the soma to the current arriving in the dendrite and $\epsilon_{ds}$ is the filter relating the current injected in the dendrite to the current arriving in the soma. The spikes are emitted if $V_{s}(t)>V_{T}(t)$ which results in $\hat{t}_{(last)}=t$ while $V_{s}\rightarrow E_{r}$ and $t\rightarrow t+\tau_{R}$. The parameters are listed in Table 1. Variable \| \| Value \| Units ---\|---\|---\|--- Somatic leak conductance \| $g_{s}$ \| 22 \| nS Somatic capacitance \| $C_{s}$ \| 379 \| pF Somatic reversal potential \| $E_{s}$ \| -73 \| mV Threshold baseline \| $E_{T}$ \| -53 \| mV Spike-triggered jump in threshold \| $D_{T}$ \| 2.0 \| mV Time-constant of dynamic threshold \| $\tau_{T}$ \| 27 \| ms Maximum ‘Ca’ current \| $g_{1}$ \| 567 \| pA Maximum effect of ‘Ca’ current in soma \| $\alpha$ \| 337 \| n.u. Dendritic leak conductance \| $g_{d}$ \| 22 \| nS Dendritic capacitance \| $C_{d}$ \| 86 \| pF Dendritic reversal potential \| $E_{d}$ \| -53 \| mV Time-constant for variable $m$ \| $\tau_{m}$ \| 6.7 \| ms Time-constant for variable $x$ \| $\tau_{x}$ \| 49.9 \| ms Sensitivity of ‘Ca’ Current \| $D_{m}$ \| 5.5 \| ms Maximum ‘K(Ca)’ Current \| $g_{2}$ \| -207 \| pA Half-activtion potential of ‘Ca’ current \| $E_{m}$ \| -0.6 \| mV Table 1: List of parameters and their fitted value for the two-compartment model. As a control, we also consider an entirely passive model of dendritic integration. In this model, the current injected in the dendrite is filtered passively to reach the soma. The generalized passive model has and instantaneous firing rate: $\lambda(t)=\lambda_{0}\exp\left(\kappa_{s}I_{s}+\kappa_{ds}I_{d}+\sum_{\\{\hat{t}_{i}\\}}\eta_{A}(t-\hat{t}_{i})\right)$ (6) where $\lambda_{0}$ is a constant related to the reversal potential, $\kappa_{s}$ somatic membrane filter, $\kappa_{ds}$ is the filter relating the current injected in the dendrite to the voltage change in the soma, and $\eta_{A}$ is the effective spike-triggered adaptation. ### II.2 Experimental Protocol Parasagittal brain slices of the somato-sensory cortex (300-350 m thick ) were prepared from 28-35 day-old Wistar rats. Slices were cut in ice-cold extracellular solution (ACSF), incubated at 34oC for 20 min and stored at room temperature. During experiments, slices were superfused with in ACSF at 34oC. The ACSF contained (in mM) 125 NaCl, 25 NaHCO3, 25 Glucose, 3 KCl, 1.25 NaH2PO4, 2 CaCl2 , 1 MgCl2 , pH 7.4, and was continuously bubbled with 5 % CO2 / 95 % O2. The intracellular solution contained (in mM) 115 K+-gluconate, 20 KCl, 2 Mg-ATP, 2 Na2-ATP, 10 Na2-phosphocreatine, 0.3 GTP, 10 HEPES, 0.1, 0.01 Alexa 594 and biocytin (0.2%), pH 7.2. Recording electrodes were pulled from thick-walled (0.25 mm) borosilicate glass capillaries and used without further modification (pipette tip resistance 5-10 M$\Omega$ for soma and 20-30M$\Omega$ for dendrites). Whole- cell voltage recordings were performed at the soma of a layer V pyramidal cell . After opening of the cellular membrane a fluorescent dye, Alexa 594 could diffuse in the entire neuron allowing to perform patch clamp recordings on the apical dendrite 600-700 $\mu$m from the soma. Both recordings were obtained using Axoclamp Dagan BVC-700A amplifiers (Dagan Corporation). Data was acquired with an ITC-16 board (Instrutech) at 10 kHz driven by routines written in the Igor software (Wavemetrics). The injection waveform consisted of 6 blocks of 12 seconds. Each block is made of three parts: 1) one second of low-variance colored noise injected only in the soma, 2) one second of low-variance colored noise injected only in the dendritic injection site, 3) ten seconds of high-variance colored noise whose injection site depends on the block: In the first block, the 10-second stimulus is injected only in the dendritic site, the second block delivers the 10-second stimulus in the soma only, and the four remaining blocks deliver simultaneous injections in the soma and the dendrites. The colored noise was simulated with MATLAB as an Ornstein-Uhlenbeck process with a correlation time of 3 ms. The six blocks make a 72 seconds stimulus that was injected repeatedly without redrawing the colored noise (frozen-noise). Twenty repetitions of the 72-second stimulus were carried out, separated by periods of 2-120 seconds. Out of the twenty repetitions, a set of seven successive repetitions were selected on the basis of high intrinsic reliability. ### II.3 Fitting Methods Figure 2: The two-compartment model fits qualitatively and quantitatively the electrophysiological recordings. A, B Overlay of the model (red) and experimental (black) somatic voltage trace. The dashed box indicates an area stretched out for higher precision. C, D The overlay of model (red) and experimental (blue) dendritic voltage is shown for the stretched sections in A and B. Left (A,C) and right (B,D) columns show two different injection regimes contrasting by the amount of dendritic activity which is high for A, C and medium for B, D. E Residuals from the linear regression are shown for the somatic (black) and dendritic (blue) compartment. F For each repetition the $\Gamma$ Coincidence factor is plotted against the intrinsic reliability of the cell. Grey points show the performance of the model on the test set and black points show the performance of the model on the training set. G Comparison of the inter-spike interval histogram for the model (red) and the experiment (black). H Comparison of the generalized passive (Pas), and the full two-compartment model (Full) with the intrinsic reliability (R) of the neuron in terms of the $\Gamma$ coincidence factor. The averaged $\Gamma$ factor is shown for the training set (black) and test set (Gray) Each kernel ($\kappa_{s}$, $\kappa_{ds}$, $\eta_{A}$, $\epsilon_{ds}$,$\epsilon_{sd}$, $I_{A}$, $I_{BAP}$) is expressed as a linear combination of nonlinear basis (i.e. $\kappa_{s}(t)=\sum_{i}a_{i}f_{i}(t)$). The rectangular function was chosen as the nonlinear basis. The parameters weighting the contributions of the different rectangular functions are then linear in the derivative of the membrane potential for the two-compartment model and generalized linear for the passive model. For the two-compartment model, we use a combination of regression methods and exhaustive search to maximize the mean square-error of the voltage derivative. The regression methods are similar to those previously used for estimating parameters with intracellular recordings. These methods are described in more details in Jolivet _et al._ (2006); Paninski _et al._ (2005); Mensi _et al._ (2012); Pozzorini _et al._ (2013). The fit of the somatic compartment essentially follows Jolivet et al. (2006) Jolivet _et al._ (2006) but using multi-linear regression to fit the linear parameters. The fit of the dendritic compartment needs to iterate through the restricted set of nonlinear parameters ($\tau_{m}$, $D_{m}$, $E_{m}$, $\tau_{x}$). All fits are performed only on the part of the data restricted for training the model. 1 Fit of the dendritic compartment, knowing the injected currents and the somatic spiking history: 1a Compute the first-order estimate of $dV_{d}/dt$; 1b Find the best estimates of the dendritic parameters linear in $dV_{d}/dt$ given a set of nonlinear parameters ($\tau_{m}$, $D_{m}$, $E_{m}$, $\tau_{x}$). The best estimates are chosen through multi-linear regression to minimize the mean square error of $dV_{d}/dt$. 1c Compute iteratively step 1b on a grid of the nonlinear parameters ($\tau_{m}$, $D_{m}$, $E_{m}$, $\tau_{x}$) and find the nonlinear parameters that yield the minimum mean square error of $dV_{d}/dt$. 2 Fit of the somatic compartment using the fitted dendritic compartment. 2a Compute the first-order estimate of $dV_{s}/dt$. 2b Find the best estimates of the somatic parameters linear in $dV_{s}/dt$ given a set of nonlinear parameters ($D_{T}$, $\tau_{T}$, $E_{T}$). The best estimates are chosen through linear regression to minimize the mean square error of $dV_{d}/dt$. 2c Compute iteratively step 2b on a grid of the nonlinear parameters and simulate the model with each set of nonlinear parameters in order to compute the coincidence rate $\Gamma$ (see Sect. II.4). 2d Take the parameters that yield the maximum $\Gamma$ coincidence factor. For the generalized linear model, we use maximum likelihood methods Paninski (2004); Pillow _et al._ (2005). Expressing the kernels as a linear combination of rectangular bases we recover the generalized linear model. Here the link-function is exponential so that the likelihood is convex. We therefore performed a gradient ascent of the likelihood to arrive at the optimal parameters. ### II.4 Analysis Methods When one focuses on spike timing, one may want to apply methods that compare spike trains in terms of a spike-train metric Victor and Purpura (1996) or the coincidence rate Kistler _et al._ (1997). Both measures can be used to compare a recorded spike train with a model spike train. A model which achieve an optimal match in terms of spike-train metrics will automatically account for global features of the spike train such as the interspike interval distribution. Here we used the averaged coincidence rate $\Gamma$ Kistler _et al._ (1997). It can be seen as a similarity measure between pairs of spike trains, averaged on all possible pairs. To compute the pairwise coincidence rate, one first finds the number of spikes from the model that fall within an interval of $\Delta=$4 ms after or before a spike from the real neuron. This is called the number of coincident events $N_{nm}$ between neuron repetition $n$ and model repetition $m$. The coincidence rate is the ratio of the number of coincident events over the averaged number of events 0.5($N_{n}$+$N_{m}$), where $N_{n}$ is the number of spikes in the neuron spike train and $N_{m}$ is the number of spikes in the model spike train. This ratio is then scaled by the number of chance coincidences $N_{\rm Poisson}=2\Delta N_{m}N_{n}/T$. This formula comes from the number of expected coincidences assuming a Poisson model at a fixed rate $N_{m}/T$ where $T$ is the time length of each individual spike trains. The scaled coincidence rate is $\Gamma_{nm}=\frac{N_{nm}-N_{\rm Poisson}}{0.5(1-N_{\rm Poisson}/N_{n})(N_{n}+N_{m})}.$ (7) The pairwise coincidence rate $\Gamma_{nm}$ is then averaged across all possible pairings of spike trains (trials) generated from the model with those from the neuron and gives the averaged coincidence rate $\Gamma$. Averaging across all possible pairings of spike trains from the neuron with a distinct repetition of the same stimulus given to the same neuron gives the intrinsic reliability $R$. ## III Results Dual patch-clamp recordings were performed in L5 Pyramidal cells of Wistar rats (see Experimental Methods). A simplified two-compartment model (see Model Description) was fitted on the first 36 seconds of stimulation for all repetitions. The rest of the data (36 sec) was reserved to evaluate the model’s predictive power. The predictive power of the two-compartment model with active dendrites was then compared to a model without activity in the dendrites (see Sect. II.1), the generalized linear passive model. Figure 3: Fitted kernels of the two-compartment model. A The kernel $I_{A}(t)$ for spike-triggered adaptation is negative and increases monotonically between 6 and 600 ms. B The back-propagating current $I_{\rm BAP}(t)$reaching the dendrites is a short (2ms) and strong (900 pA) pulse. C The convolution kernel $\epsilon_{ds}(t)$ linking the current injected in the dendrite to the current reaching the soma. D The convolution kernel $\epsilon_{sd}(t)$ linking the current injected in the soma to the current reaching the dendrite. Figure 4: The model reproduces the qualitative features of active dendrites reported in Larkum _et al._ (1999) and Larkum _et al._ (2004). A Dendritic non-linearity is triggered by somatic spiking above a critical frequency. Somatic spike- trains of 5 spikes are forced in the soma of the mathematical model at different firing frequencies. The normalized integral of the dendritic voltage is shown as a function of the somatic spiking frequency. B Dendritic injection modulates the slope of the somatic spiking-frequency vs. current curve. The slope of the frequency vs mean somatic current as measured between 5 and 50 Hz is plotted as a function of the mean dendritic current. Both somatic and dendritic currents injected are Ornstein-Uhlenbeck processes with a correlation time of 3 ms and a standard deviation of 300 pA. C Spike-triggered average of the current injected in the soma (black) and in the dendrites (blue). D Burst-triggered average of the current injected in the soma (black) and in the dendrites (blue). The fact that the blue curve is higher than the black curve, and that this relation is inverted in C, indicate that the two- compartment model performs a type of top-down coincidence detection with bursts. Figure 2 summarizes the predictive power of the two-compartment model. The somatic and dendritic voltage traces are well captured (Fig. 2 A-D). The main cause for erroneous prediction of the somatic voltage trace is extra or missed spikes (Fig. 2 A and B lower panels). The dendritic voltage trace of the model follows the recorded trace both in a low dendritic-input regime (Fig. 2 C) and in a high dendritic-input regime with dendritic ‘spikes’ (Fig. 2 D). The greater spread of voltage-prediction-error (Fig. 2) is mainly explained by the larger range of voltages in the dendrites (somatic voltage prediction is strictly subthreshold whereas dendritic voltage prediction ranges from -70 mV to +40 mV). The interspike interval distribution is well predicted by the model (Fig. 2 G). The generalized passive model does not predict as many spike times ( Fig. 2 H). The intrinsic variability in the test set was 68% and the two-compartment model predicted 50%. The prediction falls to 36 % in the absence of a dendritic non-linearity (Fig. 2 H). The fitted kernels show that spike triggered adaptation is a monotonically decaying current that starts very strongly and decays slowly for at least 500 ms (Fig. 3 A). The back-propagating action potential is mediated by a strong pulse of current lasting 2-3 ms (Fig. 3 B). The coupling $\epsilon_{ds}$ from dendrite to soma has a maximal response after 2-3 ms and then decays so as to be slightly negative after 35 ms (Fig. 3 C). The coupling $\epsilon_{sd}$ from soma to dendrite follows qualitatively $\epsilon_{ds}$ with smaller amplitudes and slightly larger delays for the maximum and minimum peaks (Fig. 3 D), consistent with the larger membrane time-constant in the soma than in the dendrites. The two-compartment model can reproduce qualitative features associated with the dendritic non-linearity in the apical tuft of L5 pyramidal neurons. We study two of these features: the critical frequency Larkum _et al._ (1999) and the gain modulation Larkum _et al._ (2004). The first relates to the critical somatic firing frequency above which a non-linear response is seen in the soma, reflecting calcium channel activation in the dendrites. To simulate the original experiment, we force 5 spikes in the soma at different frequencies and plot the integral of the dendritic voltage. The critical frequency for initating a non-linear increase in summed dendritic voltage is 138 Hz (Fig. 4 A). Perez-Garci et al. (2006) Pérez-Garci _et al._ (2006) reported a critical frequency of 105 Hz while Larkum et al. (1999) Larkum _et al._ (1999) reported 85 Hz. This appears to vary across different cells and pharmacological conditions. The model also appears to perform gain modulation as in Larkum _et al._ (2004) (Fig. 4 B). The relation between somatic firing rate and mean somatic current depends on the dendritic excitability. The onset (or shift) but also the gain (or slope) of the somatic frequency versus somatic current curve depend on the mean dendritic current. The gain modulation is attributed to a greater presence of bursts (Fig. 4 B) caused by dendritic calcium-current activation at higher dendritic input. The link between burst and dendritic activity is reflected in the burst- and spike-triggered average injected current (Fig. 4 C-D) similar to Ref. Larkum _et al._ (2004). The burst- triggered current is greater for the dendritic injection, whereas the spike- triggered current is larger for somatic injection. The greater correlation, relative to somatic current, of the dendritic current with the observation of bursts indicate that the two-compartment model performs a type of top-down coincidence detection with bursts. ## IV Conclusion Using a two-compartment model interconnected with temporal filters, we were able to predict a substantial fraction of spike times. The predicted spike trains achieved an averaged coincidence rate of 50%. The scaled coincidence rate obtained by dividing by the intrinsic reliability Jolivet _et al._ (2008a); Naud and Gerstner (2012) was 72%, which is comparable to the state- of-the performance for purely somatic current injection which reaches up to 76%Naud _et al._ (2009). Comparing with a passive model for dendritic current integration, we found that the predictive power decreased to a scaled coincidence rate of 53%. Therefore we conclude that regenerating activity in the dendritic tuft is required to properly account for the dynamics of layer 5 pyramidal cells under in-vivo-like conditions. ###### Acknowledgements. The authors would like to thank Matthew Larkum for helpful suggestions. ## References * Stuart _et al._ (2007) G. Stuart, N. Spruston, and M. Häusser, _Dendrites_ , 2nd ed. (Oxford University Press, Oxford, 2007). * Larkum _et al._ (1999) M. Larkum, J. Zhu, and B. Sakmann, Nature 398, 338 (1999). * Larkum _et al._ (2001) M. Larkum, J. Zhu, and B. Sakmann, J. 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Can We Predict Every Spike.	arxiv-papers	2013-11-14T17:28:21	2024-09-04T02:49:53.660424	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Richard Naud and Brice Bathellier and Wulfram Gerstner", "submitter": "Richard Naud", "url": "https://arxiv.org/abs/1311.3586" }
1311.3609	The search for strongly decaying exotic matter R.S. Longacrea aBrookhaven National Laboratory, Upton, NY 11973, USA ###### Abstract In this paper we explore the possibility of detecting strongly decaying exotic states. The dibaryon(2.15) $J^{P}$ = $2^{+}$ state which decays into d $\pi$ is the example we use in this report. ## 1 Introduction At the STAR experiment we can collect hundreds of million ultra-relativistic heavy ion collisions. Light nuclei and anti-nuclei emerge from these collisions during the last stage of the collision process. The quantum wave functions of the constituent nucleons or anti-nucleons, if close enough in momentum and coordinate space, will overlap to produce composite systems. The production rate for the systems with baryon or anti-baryon B is proportional to the baryon or anti-baryon density in momentum and coordinate space, raised to the power $\|$B$\|$, and therefore exhibits exponential behavior as a function of $\|$B$\|$. Figure 1 shows the exponential[1] invariant yields versus baryon number in $\sqrt{s_{NN}}$=200 GeV central Au+Au collisions. Empirically, the production rate reduces by a factor of 1.1 x $10^{3}$(1.6 x $10^{3}$) for each additional nucleon (anti-nucleon) added. The measurement of hundreds of million of events make it possible to probe up to a scale of five in baryon number. The baryon four data points come from a STAR measurement published in Ref.[2]. It should be noted that there are no baryon five nuclear fragments that live long enough or decay weakly such that they would have a displaced vertex[2]. The paper is organized in the following manner: Sec. 1 explores nuclear states that have been measured. Sec. 2 explores the possibility of detecting strongly decaying exotic states. The dibaryon(2.15) $J^{P}$ = $2^{+}$ state which decays into d $\pi$ is considered. Figure 1: Differential invariant yields as a function of baryon number B, evaluated at $p_{T}$/$\|$B$\|$ = 0.875 GeV/c, in central $\sqrt{s_{NN}}$=200 GeV Au+Au collisions. Yields for tritons ${}^{3}H$ (anti-tritons $\overline{{}^{3}H}$) lie close to the position for ${}^{3}He$ and $\overline{{}^{3}He}$. The lines represent fits with the exponential formula $\alpha$ $e^{-r\|B\|}$ for positive and negative particles separately, where $r$ is the production reduction factor. ## 2 Exotic States through strong decay. In the above section the states decayed by the weak interaction. The possibility of detecting strongly decaying exotic states is considered using the dibaryon(2.15) $J^{P}$ = $2^{+}$ state which decays into d $\pi$ as an example. In the QGP(Quark Gluon Plasma) six quarks or anti-quarks could come together to form a deuteron or anti-deuteron. However such states are loosely bound and easily destroyed in the hadronic phase. The cross section for $d$ $\pi$ scattering is 240 mb. This implies that the deuteron can only be formed in the final freeze-out of the hadronic system. At the time of freeze-out many hadrons scatter and coalesce into compound or excited states(see Figure 2). The dibaryon state interacts in three two-body scattering channels. Its mass is 2.15 GeV and has a strong interaction resonance decay width of 100 MeV. It interacts in the $N$ $N$ d-wave spin anti-aligned[3], $d$ $\pi$ p-wave spin aligned[4], and $\Delta$ $N$ s-wave spin aligned[5]. The dibaryon system mainly resonates in the s-wave $\Delta$ $N$ mode with a pion rotating in a p-wave about a spin aligned $N$ $N$ system which forms a isospin singlet. The pion moves back and forth forming $\Delta$ states with one nucleon and then the other(see Figure 3). All three isospin states of the pion can be achieved in this resonance. Thus we can have $\pi^{+}$ $d$, $\pi^{0}$ $d$, and $\pi^{-}$ $d$ states. If the pion is absorbed by any of the nucleons it under goes a spin flip producing a d-wave $N$ $N$ system. The resonance decays into $N$ $N$, $\pi$ $d$, or $\pi$ $N$ $N$. In a meson system an analogous resonance is formed where a pion is orbiting in a p-wave about a $K\overline{K}$ in a s-wave[6](see Figure 4). Both systems have a similar lifetime or width of $\sim$ .100 GeV. Figure 2: Above are a few of the compound or excited states that will form during the last stage of hadronic freeze-out. In order to predict the rate for dibaryon production we turn to a Monte Carlo heavy ion event generator[7]. This generator was a cradle to grave going from initial partons to final state hadrons. Figure 5 shows the time line in the center-of-mass frame for partons, then pre-hadrons and final hadrons. What is happening in the early times of the collision is of no importance for dibaryon formation, while the conditions of the hadrons at later times will determine the dibaryon production. For $\sqrt{s_{NN}}$=200 GeV central Au+Au collisions the spectrum is well measured. Therefore we can start the Monte Carlo at the intermediate times with a fireball of excited hadrons and let it evolve to the final state. We start with an expanding cylinder of radius 10.0 Fermi filled with excited hadrons with density and $p_{t}$ distribution that reproduces the $\sqrt{s_{NN}}$=200 GeV central Au+Au collisions. Figure 6 gives the measured Au+Au spectrum which we will tune for. Mesons in the fireball cascade include $\pi$, $K$, $\rho$, $\omega$, $a_{1}$, $\eta$, $\eta\prime$, $\phi$, $K^{}$, $K^{}(1420)$, $f_{0}(975)$, $a_{0}(980)$, $f_{2}(1270)$, $a_{2}(1320)$, $h_{1}(1170)$, $\rho(1700)$, $f_{2}(1800)$, $b_{1}(1235)$ and $f_{2}(1525)$. The cross section for $\pi\pi$ and $\pi K$ was determined from S-matrix phase shifts, while for $K\overline{K}$ we used the production of $\phi$, $f_{0}(975)$, $a_{0}(980)$, $f_{2}(1270)$, $a_{2}(1320)$,and $f_{2}(1525)$(see Figure 7). For $\rho\pi$ cross sections we used the production of $h_{1}(1170)$, $a_{1}(1260)$, and $a_{2}(1320)$, while for $K^{}\pi$ we used the production of $K^{}(1420)$. For $a_{1}\pi$ we used $\rho(1700)$ and $f_{2}(1800)$(see Figure 8). Finally for $\omega\pi$ we used $b_{1}(1235)$ and $\eta\pi$ we used $a_{0}(980)$ and $a_{2}(1320)$. For all other meson meson cross section we used the additive quark model. The particles produced from such scatterings were determined by a multi-pomeron chain model using a Field- Feymann algorithm(see Figure 9). Since we are detecting baryons and anti-baryons the $NN$, $N\pi$, $NK$, and $\Delta N$ cross section and scattering ratios are obtained from data and extracted S-matrix amplitudes(see Figure 10). All other cross sections for baryon meson and baryon baryon systems we use the additive quark model(see Figure 11). The particles produced from such scatterings are determined by a multi-pomeron chain model using a Field-Feymann algorithm. For baryon($B$) anti-baryon($\overline{B}$) scattering and cross section, data is used for $N\overline{N}$ annihilation and elastic scattering. For annihilation yields, we use a flavor consistent meson meson multi-pomeron chain model. For the rest of the yield a $B\overline{B}$ multi-pomeron chain model is used. The elastic scattering obtained by this method is close to the data for the $N\overline{N}$ system. Figure 3: The dibaryon system mainly resonates in the s-wave $\Delta$ $N$ mode with a pion rotating in a p-wave about a spin aligned $N$ $N$ system which forms a isospin singlet. The pion moves back and forth forming $\Delta$ states with one nucleon and then the other. Figure 4: The meson system mainly resonates in the s-wave $K^{}$ $\overline{K}$ and $K$ $\overline{K^{}}$ mode with a pion rotating in a p-wave about a $K$ $\overline{K}$ system which forms a isospin triplet. The pion moves back and forth forming $K^{}$ and $\overline{K^{}}$ states with one $K$ or $\overline{K}$. Figure 5: Time evolution of the total numbers of produced partons Np, pre- hadronic clusters Nc, and hadrons Nh during Au + Au collisions. The time refers to the center-of-mass frame of the colliding nuclei. Figure 6: The measured Au+Au spectrum which we will tune for. Figure 7: Cross sections for $\pi\pi$, $\pi K$, and $K\overline{K}$. Figure 8: Cross sections for $\rho\pi$, $K^{}\pi$, and $a_{1}\pi$. Figure 9: Cross sections for $\omega\pi$, $\eta\pi$, and others. Figure 10: Cross sections for $N\pi$, $NK$, $NN$ and others. Figure 11: The additive quark model calculates the cross section for the scattering of any two particles based on a product of geometric factors. The annihilation threshold effect is scaled to other $B\overline{B}$ scatterings using the $N\overline{N}$ ratios obtained in the above algorithm. We need to add the production of d’s into the Monte Carlo code. Let us assume that the formation of the $J^{p}=2^{+}$ dibaryon state is the driving source of d’s. We fit the d-wave $NN$ elastic scattering[3], p-wave $d\pi$ elastic scattering[4], and p-wave $d\pi$ to d-wave $NN$[4]. A three channel K-matrix was used to form a S-matrix, where the channels are d-wave $NN$, p-wave $d\pi$, and s-wave $\Delta N$ data. We are able to fit the above data if we use one K-matrix pole to generate the dibaryon 2.15 GeV state plus a far away pole and a flat none factorable background. Figure 12 shows the fit to elastic $NN$ scattering amplitude. Figure 13 shows the fit to $d\pi$ elastic scattering, while Figure 14 is the connection between $NN$ going to $d\pi$. The cross sections for $N$ $N$ $\rightarrow$ $\pi$ $d$, $\Delta$ $N$ $\rightarrow$ $\pi$ $d$, $\pi$ $d$ $\rightarrow$ $\Delta$ $N$ and $N$ $N$ where added to the hadron cascade part of the code. When we consider the known cross sections for $NN$ and $\Delta N$, the yield for charge pairs of $d\pi$ can be calculated and is plotted in Figure 15. In our hadron cascade these scatterings are the only source of d’s. The production of d’s and anti-d’s is close to the values measured in Figure 1. The value of d’s in the cascade would be much larger than the measured value except d’s are destroyed by interacting with pions. Figure 16 show the large $d\pi$ cross section of $\sim$ 250 mb. About 3/4 of these scattering remove the d’s from the cascade. We achieve the yield and spectrum for Au+Au $\sqrt{s_{NN}}$=200 GeV central collisions by adjusting the excited hadrons in our cylinder of radius 10.0 Fermi. We generate enough events at $\sqrt{s_{NN}}$=200 GeV central Au+Au collisions in order to obtain 1 million $d$ or $\overline{d}$ events in the STAR acceptance. Out of the 1 million events there were 230,000 pairs of either $d$ $\pi$ or $\overline{d}$ $\pi$ which decayed in the STAR acceptance. The effective mass distribution of these pairs are plotted as solid points in Figure 17. In order to obtain the mass spectrum from the data, We need to determined the uncorrelated background of either $d$ or $\overline{d}$ paired with a charge particle in a average event. For each of the 1 million events we can pair up either the $d$ or $\overline{d}$ with all charge particles(which then is assumed to be a pion) in that event and plot the total number of pairs as a function of effective mass. From this pair spectrum we then subtract the average uncorrelated spectrum times the number of events. We can determine this average uncorrelated spectrum by mixed event methods taking the same $d$ and $\overline{d}$ paired with the charged particles from other events. The subtracted effective mass spectrum is the open points of Figure 17. We see that we have recovered the mass spectrum. ## 3 Summary and Discussion In the first section of this manuscript we consider baryons and anti-baryons up to a baryon number five. These states decayed by the weak interaction. The exotic states that decay strongly is considered in the second section. In order to develop methods for such research we consider a dibaryon(2.15) $J^{P}$ = $2^{+}$ state which decays into d $\pi$. Figure 12: The real and imaginary parts of the elastic scattering T-matrix amplitude for $NN\rightarrow NN$ as a function of mass in GeV. Figure 13: The real and imaginary parts of the elastic scattering T-matrix amplitude for $d\pi\rightarrow d\pi$ as a function of mass in GeV. Figure 14: The modulus of the inelastic scattering T-matrix amplitude for $NN\rightarrow d\pi$ as a function of mass in GeV. Figure 15: The percentage of $d\pi$ charge pairs produced in $NN$ and $\Delta N$ scattering as a function of mass in GeV. Figure 16: The $d\pi$ total and elastic cross section in millibarns(mb) as a function of mass in GeV. Figure 17: The number of $d$ or $\overline{d}$ paired with charged pions coming from $10^{6}$ dibaryons decays within the STAR acceptance plotted as solid points. The open points are form by all $d$ and $\overline{d}$ paired with the charged particles in each event in the star acceptance minus the same $d$ and $\overline{d}$ paired with the charged particles from other events(mixed events). We create a Monte Carlo simulation that should give realistic events structure with realistic dibaryon production. With the ability to measure hundreds of million ultra-relativistic heavy ion collisions, we predicted that a clear dibaryon signal decaying into $d\pi$ should be measured. ## 4 Acknowledgments This research was supported by the U.S. Department of Energy under Contract No. DE-AC02-98CH10886. ## References [1] T.A. Armstrong et al., Phys. Rev. Lett. 83 (1999) 5431\. * [2] H. Agakishiev et al., Nature 473 (2011) 353. * [3] R.A. Arndt et al., Phys. Rev. C 76 (2007) 025209. * [4] C.H. Oh et al., Phys. Rev. C 56 (1997) 635. * [5] D. Schiff and J. Tran Thanh Van, Nucl. Phys. B5 (1968) 529. * [6] R. Longacre, Phys. Rev. D 42 (1990) 874. * [7] K. Geiger and R. Longacre, Heavy Ion Phys. 8 (1998) 41.	arxiv-papers	2013-11-14T19:08:53	2024-09-04T02:49:53.668916	{ "license": "Public Domain", "authors": "Ron S. Longacre", "submitter": "Ron S. Longacre", "url": "https://arxiv.org/abs/1311.3609" }
1311.3893	EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN) LHCb-DP-2013-003 7 January 2013 Performance of the LHCb Outer Tracker The LHCb Outer Tracker group R. Arink1, S. Bachmann2, Y. Bagaturia2, H. Band1, Th. Bauer1, A. Berkien1, Ch. Färber2, A. Bien2, J. Blouw2, L. Ceelie1, V. Coco1, M. Deckenhoff3, Z. Deng7, F. Dettori1, D. van Eijk1, R. Ekelhof3, E. Gersabeck2, L. Grillo2, W.D. Hulsbergen1, T.M. Karbach3,4, R. Koopman1, A. Kozlinskiy1, Ch. Langenbruch2, V. Lavrentyev1, Ch. Linn2, M. Merk1, J. Merkel3, M. Meissner2, J. Michalowski5, P. Morawski5, A. Nawrot6, M. Nedos3, A. Pellegrino1, G. Polok5, O. van Petten1, J. Rövekamp1, F. Schimmel1, H. Schuylenburg1, R. Schwemmer2,4, P. Seyfert2, N. Serra1, T. Sluijk1, B. Spaan3, J. Spelt1, B. Storaci1, M. Szczekowski6, S. Swientek3, S. Tolk1, N. Tuning1, U. Uwer2, D. Wiedner2, M. Witek5, M. Zeng7, A. Zwart1. 1Nikhef, Amsterdam, The Netherlands 2Physikalisches Institut, Heidelberg, Germany 3Technische Universität Dortmund, Germany 4CERN, Geneva, Switzerland 5H. Niewodniczanski Institute of Nuclear Physics, Cracow, Poland 6A. Soltan Institute for Nuclear Studies, Warsaw, Poland 7Tsinghua University, Beijing, China The LHCb Outer Tracker is a gaseous detector covering an area of $5\times 6$ m2 with 12 double layers of straw tubes. The detector with its services are described together with the commissioning and calibration procedures. Based on data of the first LHC running period from 2010 to 2012, the performance of the readout electronics and the single hit resolution and efficiency are presented. The efficiency to detect a hit in the central half of the straw is estimated to be 99.2%, and the position resolution is determined to be approximately 200 $\,\upmu\rm m$. The Outer Tracker received a dose in the hottest region corresponding to 0.12 C/cm, and no signs of gain deterioration or other ageing effects are observed. Published in JINST © CERN on behalf of the LHCb collaboration, license CC-BY-3.0. ###### Contents 1. 1 Introduction 2. 2 Services performance 1. 2.1 Gas system 2. 2.2 Gas monitoring 3. 2.3 Low voltage 4. 2.4 High voltage 3. 3 Commissioning and monitoring 1. 3.1 Quality assurance of detector modules and C-frame services 2. 3.2 Noise 3. 3.3 Threshold scans 4. 3.4 Delay scans 4. 4 Calibration 1. 4.1 Distance drift-time relation 2. 4.2 $t_{0}$ stability 3. 4.3 Geometrical survey 4. 4.4 Optical alignment with the Rasnik system 5. 4.5 Software alignment 5. 5 Performance 1. 5.1 Spillover and drift-time spectrum 2. 5.2 Occupancy 3. 5.3 Hit efficiency 4. 5.4 Hit resolution 5. 5.5 Monitoring of faulty channels 6. 5.6 Radiation tolerance 6. 6 Conclusions 7. Acknowledgements ## 1 Introduction The LHCb detector [1] is a single-arm forward spectrometer covering the pseudo-rapidity range $2<\eta<5$, designed for the study of particles containing $b$ or $c$ quarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the $pp$ interaction region, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about 4 Tm and three tracking stations located downstream. The area close to the beamline is covered by silicon-strip detectors, whereas the large area at more central rapidity is covered by the Outer Tracker (OT) straw-tube detector. Excellent momentum resolution is required for a precise determination of the invariant mass of the reconstructed $b$-hadrons. For example a mass resolution of 25 MeV/c2 for the decay $B_{s}^{0}\rightarrow\mu^{+}\mu^{-}$ translates into a required momentum resolution of $\delta p/p\approx 0.4\%$ [2]. Furthermore, the reconstruction of high-multiplicity $B$ decays demands a high tracking efficiency and at the same time a low fraction of wrongly reconstructed tracks. To achieve the physics goals of the LHCb experiment, the OT is required to determine the position of single hits with a resolution of 200 $\mu$m in the $x$-coordinate 111 The LHCb coordinate system is a right- handed coordinate system, with the $z$ axis pointing along the beam axis, $y$ is the vertical direction, and $x$ is the horizontal direction. The $xz$ plane is the bending plane of the dipole magnet., while limiting the radiation length to 3% $X_{0}$ per station (see Fig. 1b). A fast counting gas is needed to keep the occupancy below 10% at the nominal luminosity of $2\times 10^{32}$ cm-2s-1. The OT is a gaseous straw tube detector [3] and covers an area of approximately $5\times 6$ m2 with 12 double layers of straw tubes. The straw tubes are 2.4 m long with 4.9 mm inner diameter, and are filled with a gas mixture of Ar/CO2/O2 (70/28.5/1.5) which guarantees a fast drift-time below 50 ns. The anode wire is set to +1550 V and is made of gold plated tungsten of 25 $\mu$m diameter, whereas the cathode consists of a 40 $\mu$m thick inner foil of electrically conducting carbon doped Kapton-XC 222 Kapton® is a polyimide film developed by DuPont. and a 25 $\mu$m thick outer foil, consisting of Kapton-XC laminated together with a 12.5 $\mu$m thick layer of aluminium. The straws are glued to sandwich panels, using Araldite AY103-1 333Araldite® is a two component epoxy resin developed by Huntsman.. Two panels are sealed with 400 $\mu$m thick carbon fiber sidewalls, resulting in a gas-tight box enclosing a stand-alone detector module. A cross-section of the module layout is shown in Fig. 1(a). (a)(b) Figure 1: (a) Module cross section. (b) Arrangement of OT straw-tube modules in layers and stations. The modules are composed of two staggered layers (monolayers) of 64 drift tubes each. In the longest modules (type $F$) the monolayers are split in the middle into two independent readout sections composed of individual straw tubes. Both sections are read out from the outer ends. The splitting in two sections is done at a different position for the two monolayers to avoid insensitive regions in the middle of the module. $F$-modules have an active length of 4850 mm and contain twice 128 straws, in the upper and the lower half, respectively. Short modules (type $S$) have about half the length of $F$-type modules and are mounted above and below the beampipe. They contain 128 single drift tubes, and are read out only from the outer module end. The inner region not covered by the OT, $\|y\|<10(20)$ cm for $\|x\|<59.7(25.6)$ cm, is instrumented with silicon strip detectors [1]. One detector layer is built from 14 long and 8 short modules, see Fig. 1(b). The complete OT detector consists of 168 long and 96 short modules and comprises 53,760 single straw- tube channels. The detector modules are arranged in three stations. Each station consists of four module layers, arranged in an x-u-v-x geometry: the modules in the $x$-layers are oriented vertically, whereas those in the $u$ and $v$ layers are tilted by $+5^{o}$ and $-5^{o}$ with respect to the vertical, respectively. This leads to a total of 24 straw layers positioned along the $z$-axis. Each station is split into two halves, retractable on both sides of the beam line. Each half consists of two independently movable units, known as C-frames, see Fig. 1(b). The modules are positioned on the C-frames by means of precision dowel pins. The C-frames also provide routing for all detector services (gas, low and high voltage, water cooling, data fibres, slow and fast control). The OT C-frames are sustained by a stainless steel structure (OT bridge), equipped with rails allowing the independent movement of all twelve C-frames. At the top the C-frames hang on the rails, while at the bottom the C-frames are guided, but not supported by the rails, to constrain the movement in the $z$-coordinate. (a)(b) Figure 2: (a) Design and (b) photograph of the FE electronics mounted in a FE box. Only the boards that read out one monolayer of 64 straws are visible. In addition, the HV boards are not visible in the photograph as they are hidden by the ASDBLR boards. The front-end (FE) electronics measures the drift-times of the ionization clusters produced by charged particles traversing the straw-tubes with respect to the beam crossing (BX) signal [4]. The drift-times are digitized for each 25 ns (the LHC design value for the minimum bunch crossing interval) and stored in a digital pipeline to await the lowest-level trigger (L0) decision. On a positive L0 decision, the digitized data in a window of 75 ns is transmitted via optical links to TELL1 boards in the LHCb DAQ system [5]. As shown in Fig. 2, the FE electronics has a modular design, consisting of several interconnected boards housed inside a metallic box (FE box). The main components of the OT readout electronics are the high voltage (HV) board, the ASDBLR amplifier board, the OTIS digitization board, and the GOL auxiliary (GOL/AUX) board. Each ASDBLR board hosts two ASDBLR chips [6]. These are custom-made integrated circuits, providing the complete analog signal processing chain (amplification, shaping, baseline restoration, and discrimination) for the straw tube detectors. The hit outputs of two ASDBLR boards (32 channels) are connected to one OTIS board, which hosts one radiation-hard OTIS TDC chip for drift-time digitization [7, 8]. The time digitization is done through the 25 ns long Delay Locked Loop (DLL) using the 64 delay-stages of the DLL (64 time bins), giving a step size of about 0.4 ns. The drift-time data is stored in a pipeline memory with a depth of 164 events, corresponding to a latency of 4.1 $\mu$s. If a trigger occurs, the corresponding data words of up to 3 bunch crossings are transferred to a derandomizing buffer, able to store data from up to 16 consecutive triggers. Only the first hit in the 75 ns wide window of a given channel is stored. Later signals from multiple ionizations or reflections are thus not recorded. The OTIS boards in a FE box are connected to one GOL/AUX board. This board [9] provides the outside connections to the FE box: the power connection, the interface to the fast-control (beam crossing clock BX, triggers, resets) and the interface to the slow-control (I2C). These boxes are mounted at each end of the detector modules. A FE box is the smallest independent readout unit of the OT: the digitized data of the 128 channels of one module are sent via an optical link and received by the TELL1 board. High- and low-voltage, as well as fast- and slow-control signals are connected to each FE box individually. In total, 432 FE boxes are used to read out the OT detector. This paper describes the detector performance in the first LHC running period from 2010 to 2012, when the LHCb experiment collected data at stable conditions, corresponding to a typical instantaneous luminosity of about $3.5\,(4.0)\times 10^{32}$ cm-2s-1 in 2011 (2012), with a 50 ns bunch crossing scheme and a proton beam energy of 3.5 (4) TeV. The higher instantaneous luminosity, and only half of all bunches being filled, translates into a four times larger occupancy per event as compared to the conditions that correspond to the design parameters. In Sec. 2 the performance of the services is described in terms of the gas quality, and the low and high voltage stability. In Sec. 3 the performance of the electronics readout is discussed, in particular the noise, amplifier threshold uniformity and time-linearity. The drift-time calibration and position alignment is shown in Sec. 4. The final detector performance in terms of occupancy, single hit efficiency, resolution and radiation hardness is given in Sec. 5. ## 2 Services performance ### 2.1 Gas system The counting gas for the straw tube detectors of the OT was originally chosen as an admixture of Ar/CO2/CF4. Studies on radiation resistance first suggested to operate without CF4 [10], and subsequently with the addition of O2 [11], leading to the final mixture Ar/CO2/O2 (70/28.5/1.5). This choice is based on the requirement to achieve a reasonably fast charge collection to cope with the maximum bunch crossing rate of 40 MHz at the LHC, a good spatial resolution and to maximize the lifetime of the detectors. \| typical values \| specifications ---\|---\|--- Gas flow \| 800 - 850 l/h \| $<1000$ l/h Overpressure in detector \| $1.6$ mbar \| $<5$ mbar Impurity (H2O content) \| $<10$ ppm \| $<50$ ppm Table 1: Main parameters of the OT gas system. The gas is supplied by a gas system [12] operated in an open mode, without recycling of the gas. The gas system is a modular system, with the mixing module on the surface and two distribution modules, a pump module, an exhaust module and an analysis module in the underground area behind the shielding wall to allow access during beam operation. The gas is split between two distribution modules, each supplying a detector half using 36 individual gas lines. For each gas line the input flow can be adjusted and is measured continuously, as well as the output flow. Each distribution module regulates the pressure in the detector modules. The analysis module allows to sample each of the 36 lines individually at the detector inlet and outlet. An oxygen sensor and a humidity sensor are connected to the analysis rack. The measurement of one gas line takes a few minutes such that each line is measured approximately once every two hours. The main operational parameters of the gas system are shown in Table 1. The gas flow is kept low to prevent ageing effects observed in laboratory measurements (see Sec. 5.6). The detector modules have been tested to be sufficiently gas-tight, on average below $1.25\times 10^{-4}$ l/s (corresponding to 5% gas loss every 2 hours) [13], to prevent the accumulation of impurities from the environment. The level of impurities is monitored by measuring the water content in the counting gas, which is at a level below $10$ ppm. A system with pre-mixed bottles containing in total about 100 m3 of Ar/CO2/O2, is automatically activated in case of electrical power failures of the main gas system, ensuring a uninterupted flow through the detector at all times. The gas mixture, and the level of impurities (H2O) were stable during the whole operation from 2010 to 2012. ### 2.2 Gas monitoring The gas quality for the OT is crucial, as it directly affects the detector gain and stability, and potentially the hit efficiency and drift-time calibration. Moreover, a wrong gas mixture can lead to accelerated radiation damage or dangerously large currents. The gas gain is determined with the help of two custom built OT modules, of 1 m length, which are irradiated by a 55Fe source. LHCb OT Figure 3: Pressure calibration curve of the 55Fe spectrum, obtained from the dependence of the pulse height $P$ as a function of atmospheric pressure $p$. One of the monitoring modules that was used in 2011 was constructed using a particular glue, Trabond 2115, that does not provoke gain loss after long-term irradiation with the 55Fe source. The other module was built with the glue used in mass production, Araldite AY103-1. The modules were half-width modules containing 32 straw tubes. The readout electronics consists of a high voltage board carrying a number of single-channel charge pre-amplifiers. The 55Mn K-$\alpha$ line of the 55Fe source has an energy of 5.9 keV which is used as calibration reference. Two 55Fe sources with low intensity were used, resulting in a few events per second for both modules. The data acquisition system is based on a multi-functional readout box containing two fast ADC inputs to which the amplified signals are fed. In addition the atmospheric pressure is recorded. The 55Fe pulses are integrated over 15 minutes, and subsequently analyzed. A double Gaussian distribution is fitted to the 55Fe spectrum. The peak position is then corrected for the atmospheric pressure. The pressure is measured inside a buffer volume at the chambers input. The pressure correction is determined from a linear fit to the mean pulse-height as a function of the absolute atmospheric pressure, see Fig. 3. The resulting stability of the gas gain is within $\pm 2\%$ over 10 days. However, the gain loss due to ageing of the monitoring modules was about 5% over a period of about two months. The monitoring modules were therefore replaced at the end of 2011 by stainless steel modules, sealed with O-rings (instead of the standard construction with Rohacell panels with carbon-fiber facing, glued together to ensure the gas-tightness). The measurements of the gas gain were stable throughout the entire running period of 2012. ### 2.3 Low voltage The function of the low voltage (LV) distribution system is to provide the bias voltages to the front-end electronics. Each FE box (the GOL/AUX board) hosts three radiation-hard linear voltage regulators ($+2.5\,\mathrm{V}$, $+3\,\mathrm{V}$ and $-3\,\mathrm{V}$) biased by two main lines, $+6\,\mathrm{V}$ and $-6\,\mathrm{V}$. Two distribution boxes per C-frame split the $+6\,\mathrm{V}$ and $-6\,\mathrm{V}$ supply lines to the 18 FE boxes at the top and the 18 FE boxes at the bottom; all supply lines to the FE boxes are individually provided with slow fuses ($4\,\mathrm{A}$ for $+6\,\mathrm{V}$ and $2\,\mathrm{A}$ for $-6\,\mathrm{V}$) and LED’s showing their status. The low-voltage distribution systems worked reliably throughout the 2010 to 2012 data taking periods. In a few cases a single fuse of a FE box broke and was replaced in short accesses to the LHCb cavern. ### 2.4 High voltage The anode wires are supplied with +1550 V during operation, which corresponds to a gas gain of about $5\times 10^{4}$ [14]. Each FE box has four independent high voltage (HV) connections, one for each 32-channel HV board. Two mainframes 444CAEN SY1527LC ®., each equipped with four 28-channels supply boards 555A1833B PLC ®., are used as HV supply. Using an 8-to-1 distribution scheme a total of 1680 HV connections of the detector are mapped on 210 CAEN HV channels. The distribution is realized using a patch panel which offers the possibility to disconnect individual HV boards by means of an HV jumper. Both components, the HV supply as well as the patch panel, are located in the counting house. Access to the HV system during data taking is therefore possible. The typical current drawn by a single HV channel (supplying 256 detector channels) depends on the location in the detector and varies between 20 and 150 $\mu$A. The short-circuit trip value per HV channel was set to 200 $\mu$A with the exception of one channel were the current shows an unstable behaviour, and where the trip value was increased to 500 $\mu$A. The power supply can deliver a maximum current of 3 mA for a single HV channel. In the 2011 and 2012 running periods there were 8 single detector channels (wires) that showed a short-circuit, either due to mechanical damage, or due to a broken wire. During technical stops these single channels were disconnected, to allow the remaining 31 detector channels on the same HV board to be supplied with high voltage. ## 3 Commissioning and monitoring Quality assurance tests of the detector modules, the FE-boxes and C-frame services were performed prior to installation. Faulty components were repaired whenever possible. During commissioning and operating phases of the LHCb detector, the stability and the quality of the OT FE -electronics performances was monitored. Upon a special calibration trigger, sent by the readout supervisor, the FE -electronics generates a test-pulse injected via the ASDBLR test input [6, 15]. Test-pulse combinations can be generated, and is implemented such that only even or only odd numbered channels, or all channels simultaneously, are injected with charge. ### 3.1 Quality assurance of detector modules and C-frame services The quality of the detector modules was assured by measuring the wire tension, pitch, and leakage current (in air) prior to the module sealing. Subsequently, the gas tightness of the detector module was measured. Finally, the functionality of each wire was validated in the laboratory immediately after production, by measuring the response to radioactive sources (55Fe or 90Sr) [14]. Before and after shipment of the C-frames from Nikhef to CERN, the equipment of the services were checked, namely the gas tightness of the gas supply lines, the dark currents on the high-voltage cables, the voltage drop on the low-voltage supply cables and the power attenuation of the optical fibers [2, 16]. Following the installation of the modules on the C-frames in the LHCb cavern, the gas tightness of each module was confirmed. The response to 55Fe of approximately half of the straws was measured again, resulting in 12 noisy channels, 7 dead channels, and 4 straws with a smaller gas flow [16]. All FE-boxes were measured on a dedicated test-stand in the laboratory, and the faulty components were replaced. The tests performed on the test-stand are identical to the tests that are performed regularly during running periods. Three sequences of test runs are provided: * • runs with a random trigger at varying ASDBLR threshold settings to measure the noise rate; * • runs with test-pulse injected at the ASDBLR test input, at varying threshold settings to check the full readout chain for threshold uniformity and cross- talk; * • runs with test-pulse (at fixed threshold) at increasing test-pulse delay settings, to determine the time-linearity. ### 3.2 Noise The noise scan analysis aims at identifying channels that have an “abnormal” level of noise, which may be due to dark pulses from the detector, bad FE -electronics shielding, or bad grounding. In each channel the fraction of hits is determined for increasing values of the amplifier threshold, triggered randomly. The nominal value of the amplifier threshold is 800 mV, which corresponds to an input charge of about 4 fC. (a)(b) Figure 4: The 2d-hitmap histogram showing the noise occupancy, for each channel, and varying amplifier threshold (1 ADC count $\approx$ 10 mV) [17] for (a) a typical FE-box with good channels and (b) a FE-box with two groups of noisy channels. The typical noise occupancy for 128 channels in one FE box is shown in Fig. 4(a) for increasing amplifier threshold, where the occupancy is defined as the ratio of the number of registered hits in that channel over the total number of triggered events. A noise occupancy at the level of $10^{-4}$ is observed at nominal threshold, as expected from beam tests [18]. These results are representative for about 98% of all FE boxes in the detector. An example of a FE box with a few noisy channels is shown in Fig. 4(b), where two groups of about 5 noisy channels are identified. About 2% of the FE boxes exhibited this noise pattern during the 2011 running period. At the nominal threshold of 800 mV (4 fC) only 0.2% of the channels exhibited a noise occupancy larger than 0.1%. During the 2011/2012 winter shutdown, this noise pattern was understood and identified to be caused by imperfect grounding, and was subsequently solved. ### 3.3 Threshold scans The threshold scan records hits at fixed input charge (given by either a low or high test-pulse of 4 and 12 fC, respectively) and is aimed at monitoring the gain of the FE -electronics preamplifier, in order to locate dead channels, determine gain deteriorating effects and measure cross-talk. (a)(b) Figure 5: (a) Example of hit-efficiency as function of threshold for a fixed input charge (“high test-pulse”) [17]. (b) Stability of the half- efficiency point for channels in one FE-box (1 ADC count $\approx$ 10 mV). The ASDBLR chip selection prior to assembly of the FE -box components guarantees a good uniformity of the discriminators, such that a common threshold can be applied for the entire readout without loss of efficiency or increased noise levels [2]. An error function produced by the convolution of a step function (ideal condition in absence of noise) with Gaussian noise, is used to describe the hit-efficiency as a function of the threshold value. The stability of the half-efficiency point for all the channels was studied and the relative variation between channels is expected to be less than $\pm 60$ mV [2]. An example of the fit to the hit efficiency as a function of amplifier threshold is shown for one channel in Fig. 5(a), and the half-efficiency point for 128 channels in one FE box is shown in Fig. 5(b). Since the start of the data taking period in May 2010 the fraction of fully active channels has been 99.5% or more, see Sec. 5.5. ### 3.4 Delay scans (a)(b) Figure 6: (a) Example of a linear fit of the measured drift-time as a function of the test-pulse delay [17]. The slope corresponds to unity, if both axis are converted to ns (1 DAC count $\approx$ 0.1 ns, while 1 TDC count $\approx$ 0.39 ns). (b) The slope from the linear fit of the timing measurement for all 128 channels in one FE-box. The delay scan analysis aims at detecting defects in the timing of the OT channels, such as time offsets or non-linearities. An example of the time measurement as a function of the test-pulse delay is shown in Fig. 6(a), where each measurement corresponds to the average of 10,000 time measurements at a given test-pulse delay. The corresponding slope for all 128 channels of one FE box is shown in Fig. 6(b). No anomalous behaviour in the time measurement has been observed. Approximately 96% of all channels have a slope between 0.983 and 1.024 times the average value. Most of the remaining 4% of the channels suffer from insufficient test-pulse stability rather than from pre-amplifier or TDC shortcomings. ## 4 Calibration The position of the hits in the OT is determined by measuring the drift-time to the wire of the ionisation clusters created in the gas volume. The drift- time measurement can in principle be affected by variations in the time offset in the FE electronics, and is regularly monitored. The spatial position of the OT detector also affects the hit position, and the correct positioning of the detector modules is ensured by periodic alignment campaigns. ### 4.1 Distance drift-time relation The OT detector measures the arrival time of the ASDBLR signals with respect to the LHC clock, $T_{\mathrm{clock}}$, and is referred to as the TDC time, $t_{\mathrm{TDC}}$. This time is converted to position information to reconstruct the trajectory of the traversing charged particle, by means of the drift-time–distance relation, or TR-relation. The arrival time of the signal corresponds to the time of the $pp$ collision, $T_{\mathrm{collision}}$, increased by the time-of-flight of the particle, $t_{\mathrm{tof}}$, the drift-time $t_{\mathrm{drift}}$ of the electrons in the straw, the propagation time of the signal along the wire to the readout electronics, $t_{\mathrm{prop}}$, and the delay induced by the FE electronics, $t_{\mathrm{FE}}$. The various contributions to the TDC time are schematically shown in Fig. 7, and can be expressed as $t_{\mathrm{TDC}}=(T_{\mathrm{collision}}-T^{\mathrm{FE}}_{\mathrm{clock}})+t_{\mathrm{tof}}+t_{\mathrm{drift}}+t_{\mathrm{prop}}+t_{\mathrm{FE}}.$ (1) The phase of the clock at the TDC input, $T^{\mathrm{FE}}_{\mathrm{clock}}$, can be adjusted with a shift $t^{\mathrm{FE}}_{\mathrm{clock}}$. The expression for $t_{\mathrm{TDC}}$ can be rewritten as $t_{\mathrm{TDC}}=(T_{\mathrm{collision}}-T_{\mathrm{clock}})+t_{0}+t_{\mathrm{tof}}+t_{\mathrm{drift}}+t_{\mathrm{prop}},$ (2) where $t_{0}=t_{\mathrm{FE}}-t^{\mathrm{FE}}_{\mathrm{clock}}$. Variations in $t_{0}$ are discussed in the next section. The difference $t_{\mathrm{clock}}=T_{\mathrm{collision}}-T_{\mathrm{clock}}$ accounts for variations of the phase of the LHC clock received at the LHCb experiment control and is kept below 0.5 ns. (b)(a) Figure 7: (a) Sketch of the various contributions to the measured TDC time [19], as explained in the text. (b) Picture of a charged particle that traverses a straw. The TR-relation is the relation between the measured drift-time and the closest distance from the particle trajectory to the wire. The TR-relation is calibrated on data by fitting the distribution of drift-time as a function of the reconstructed distance of closest approach between the track and the wire, as shown in Fig. 8(a). At the first iteration the TR-relation obtained from beam tests was used. The line shows the currently used TR-relation [19], which has the following parameterization: (a)(b)(c) Figure 8: The (a) TR-relation distribution follows the shape of a second order polynomial distribution, which leads to a (b) falling drift-time spectrum (black), which, smeared with the time resolution (blue), leads to the shape of the (c) measured drift-time distribution. $t_{\mathrm{drift}}(r)=20.5\,\mathrm{ns}\cdot\frac{\|r\|}{R}+14.85\,\mathrm{ns}\cdot\frac{r^{2}}{R^{2}}\text{,}$ (3) where $r$ is the closest distance between the track and the wire and $R=2.45$ mm is the inner radius of the straw. This TR-relation is compatible with the one obtained from the beam test of 2005 [18], $t(r)=20.1\,\mathrm{ns}\cdot\frac{\|r\|}{R}+14.4\,\mathrm{ns}\cdot\frac{r^{2}}{R^{2}}$. The maximum drift-time extracted from the parameterization of the TR-relation is 35 ns. Due to the average drift-time resolution of 3 ns, and due to the variation in time-of-flight of the traversing particles, the drift-time distribution broadens, as illustrated in Fig. 8(b). The measured drift-time spectrum after $t_{0}$ calibration is shown in Fig. 8(c), and the start of the drift-time spectrum is thus set to 0 ns by construction. During operation, the start of the 75 ns wide readout gate was set to approximately $-9$ ns, to ensure that also the earliest hits are recorded. The varying number of entries in the subsequent time bins is a characteristic of the OTIS TDC chip known as the differential non-linearity (caused by variations of the digital delay bin sizes) and does not significantly affect the drift-time resolution [16]. ### 4.2 $t_{0}$ stability LHCb OT Figure 9: $t_{0}$ stability versus run number. Every point corresponds to one run that typically lasts one hour. The arrows indicate the adjustment of the $t_{clock}$ time. Figure 10: Distribution of differences between $t_{0}$ constants per FE box, for two different calibrations. The mean shift originates from a change of the overall $t_{clock}$ time, whereas the spread shows the stability of the delay $t_{\mathrm{FE}}$ induced by the FE electronics. The $t_{\mathrm{FE}}$ values need to be stable to a level better than the time resolution. There are two factors that contribute to the stability of $t_{0}$, usually referred to as the $t_{0}$ constants: one is the drift of the global LHCb clock and the second is the drift of FE electronic delays. The first can be extracted from the average over the whole OT of the drift-time residual distribution calculated for every run separately. The second can be estimated from the difference of $t_{0}$ values for two different calibrations, for each FE-box. Figure 9 shows the variation of the LHCb clock as a function of the run number, for the data taking period between May and July 2011\. The global LHCb clock is adjusted if it changes by more than 0.5 ns. As a result, the average value of the drift-time residual stays within the range of $\pm 0.5$ ns. Figure 10 shows the difference of the $t_{0}$ values per FE-box, for two different calibrations performed on runs 89350 and 91933, respectively. These runs correspond to the beginning of two data taking periods in May and July 2011. For most FE boxes the spread of the $t_{0}$ constants is smaller than 0.1 ns. The overall shift of 0.4 ns is due to the change of the global LHCb clock. The variation of $t_{0}$ is well below the time resolution of 3 ns and does therefore not contribute significantly to the detector resolution. ### 4.3 Geometrical survey The correct spatial positioning of the OT modules is ensured in three steps. First, the design and construction of the OT detector guarantees a mechanical stability of 100 (500) $\mu$m in the $x$($z$) direction. Secondly, an optical survey determined the position of all modules after installation. Finally, the use of reconstructed tracks allows to measure the position of the detector to the highest accuracy. By construction the anode wire is centered within 50 $\mu$m with respect to the straw tube. The detector modules are fixed with dowel-pins to the C-frames at the top and the bottom, with tolerances below 50 $\mu$m. The modules are not fixed at the center, making larger variations possible (see Sec. 12). Finally, the C-frames are mounted on rails, which fixes the $z$-coordinate at the top and at the bottom. First, the survey confirmed that the rails were straight within the few millimeters tolerance. Then, after installation, the position of the four corners of the C-frames were adjusted until all measured points on the dowel- pins at the top and bottom of the modules, and on the surface at the center of the modules, were within $\pm 1$ mm of their nominal position. The final survey coordinates provided the corrections to the nominal coordinates of the C-frames and modules [20]. The C-frames can be opened for maintenance, and the reproducibility of the C-frame positioning in the $x$-coordinate was checked to be better than the 200 $\mu$m precision of the optical survey. The shape of the modules in the $x$-coordinate is finally determined using reconstructed tracks, see Sec. 12. ### 4.4 Optical alignment with the Rasnik system The stability of the C-frame relative position during data taking is monitored by means of the Rasnik system [21, 22]. The Rasnik system consists of a CCD camera that detects a detailed pattern. The pattern, or “mask”, is mounted on the C-frame and a movement is detected by the CCD camera as a change of the pattern position. All four corners of the 12 C-frames are equipped with a Rasnik system. Together with two additional Rasnik lines to monitor movements of the suspension structure, this leads to a total of 50 Rasnik lines. Due to mechanical conflicts in the installation, only about 2/3 of the lines are used. The intrinsic resolution of the system perpendicular (parallel) to the beam axis is better than 10 (150) $\mu$m. The Rasnik measurements showed that the position of the C-frames is unchanged after opening and closing within $\pm 10$ $\mu$m, and unchanged within $\pm 20$ $\mu$m for data taking periods with opposite polarity of the LHCb dipole magnet. ### 4.5 Software alignment (a)LHCb OT(b)LHCb OT Figure 11: (a) Displacement of modules relative to the survey and (b) hit residuals in the first X-layer of station T2 before (dashed line) and after (continuous line) offline module alignment. To achieve optimal track parameter resolution the position and orientation of the OT modules must be known with an uncertainty that is negligible compared to the single hit resolution. The OT C-frames hang on rails and can be moved outside the LHCb acceptance to allow for maintenance work during technical stops of the LHC. Since no survey is performed after such operations, the reproducibility of the nominal position is important. Using track based alignment the reproducibility has been established to be better than 100 $\,\upmu\rm m$, consistent with the measurements done with the Rasnik system. The most precise alignment information is obtained with a software algorithm that uses charged particle trajectories [23]. For each module and C-frame the alignment is parametrized by three translations and three rotations. The algorithm selects high quality tracks and subsequently minimizes the total $\chi^{2}$ of those tracks with respect to the alignment parameters. Only a subset of parameters needs to be calibrated to obtain sufficient precision. For the alignment of modules inside each C-frame only the translation in $x$ and the rotation in the $xy$ plane are determined. For the C-frames themselves only the translations in $x$ and $z$ are calibrated. To constrain redundant degrees of freedom the survey measurements are used as constraints in the alignment procedure. Figure 11 illustrates the result of an alignment of module positions. For this alignment, tracks were fitted using only the OT hits. At least 18 hits per track were required. To remove poorly constrained degrees of freedom, modules in the first $x$ and stereo ($u$) layers of stations T1 and T3 were all fixed to their nominal position. Figure 11(a) shows the difference between the $x$-position of the module center relative to the survey. Statistical uncertainties in alignment parameters are negligible and the alignment is reproducible in data, taken under similar conditions, within about 20 $\,\upmu\rm m$. Figure 11(b) shows the hit residuals in one layer before and after alignment. A clear improvement is observed. The module displacements in Fig. 11 are larger than expected, based on the expected accuracy of the dowel pins that keep the modules in place. It is assumed that the disagreement can at least partially be explained by degrees of freedom that are not yet corrected for, such as module deformations and the positioning of straws within each module. Figure 12 shows an example of the average hit residual as a function of the coordinate along the wire for one module. A relative displacement of the two monolayers is observed, as well as jumps at the wire locators, which are placed at every 80 cm along the wire length. The effect on the final hit resolution is discussed in Sec. 5.4. LHCb OT Figure 12: Average hit residual as function of $y$ coordinate in one particular module (labelled T3L3Q1M7). The four curves show residuals for the four groups of 32 channels within one FE-module. The round markers correspond to one monolayer of 64 straws, whereas the square markers show the residuals of the second monolayer. The vertical dashed lines indicate the position of the wire locators, at every 80 cm along the wire [19]. ## 5 Performance The performance of the OT detector was stable in the entire first running period of the LHC between 2010 and 2012, as was shown in the previous sections. No significant failures in the LV, HV and gas systems occurred. The details of the data quality in terms of resolution and efficiency are described below. ### 5.1 Spillover and drift-time spectrum In order to register all charged particle hits produced in the $pp$ interaction, three consecutive intervals of $25\,\mathrm{ns}$ are readout upon a positive L0-trigger. Only the first hit in the readout window is recorded, as the first hit typically corresponds to the ionization cluster closest to the wire, and it is thus the best estimate for the radial distance to the wire. In the following, data are studied that are recorded in $75\,\mathrm{ns}$, $50\,\mathrm{ns}$ and $25\,\mathrm{ns}$ bunch-spacing data taking periods of the LHC. These varying conditions show the effect of so-called spillover hits on the drift-time spectrum and straw occupancies. Distributions obtained in the $75\,\mathrm{ns}$ bunch-crossing period are close to those observed with only one single bunch crossing in LHCb, and therefore they will be considered free of spillover. The drift-time spectrum and the occupancies presented here correspond to events with an average number of visible $pp$ interactions per bunch crossing of about 1.4, in accordance with the typical run conditions in 2011 and 2012. The events are triggered by any physics trigger, implying that most events contain $B$ or $D$-decays. The drift-time distributions for the $75\,\mathrm{ns}$, $50\,\mathrm{ns}$ and $25\,\mathrm{ns}$ bunch-spacing conditions are shown in Fig. 13. (a)LHCb OT(b)LHCb OT Figure 13: (a) Drift-time distribution in module 8, close to the beam, for $75\,\mathrm{ns},50\,\mathrm{ns},25\,\mathrm{ns}$ bunch- crossing spacing in red, black and blue, respectively. The vertical lines at 64 and 128 TDC counts correspond to 25 and 50 ns, respectively. The distributions correspond to all hits in 3000 events for each bunch-crossing spacing, recorded with an average number of overlapping events of $\mu=1.2,1.4$ and 1.2, for $75\,\mathrm{ns},50\,\mathrm{ns}$ and $25\,\mathrm{ns}$ conditions, respectively. (b) The drift-time distribution for empty events illustrates the contribution from spillover hits from “busy” previous bunch-crossings (red). The naive expectation of the spillover distribution is shown in black, and is obtained by shifting the nominal drift- time spectrum by $-50\,\mathrm{ns}$. The typical drift-time spectrum from the (spillover-free) distribution from the 75 ns running can be understood by inspecting Fig. 8 in Sec. 4.1. The projection of the TR-relation results in a linearly decreasing drift-time spectrum, assuming a flat distribution of the distance between the tracks and the wires. In addition, the number of earlier hits is slightly enhanced in the drift-time distribution, since late hits are hidden by earlier hits on the same straw, as only the first hit is recorded. The recording of the first hit only, induces a “digital dead-time” starting from the first hit until the end of the readout window at 192 TDC counts, or 75 ns. A second source of dead- time originates from the recovery time required by the amplifier. This “analog dead-time” lasts between 8 ns and 20 ns, depending on the signal pulse height, and is usually hidden by the digital dead-time. The black line in Fig. 13(a) correspond to the data recorded in the $50\,\mathrm{ns}$ bunch-spacing period. The contribution from hits from the next bunch-crossing, 50 ns later, is visible between 128 and 192 TDC counts. The relative contribution of these late hits from the next bunch crossing is determined by the average occupancy in the next bunch crossing, and thus depends on the run conditions. In principle it also depends on the occupancy of the triggered event, and thus on the trigger configuration, but in practice that is quite stable. The shape of the drift-time distribution of the late spillover hits corresponds to the nominal, spillover-free (i.e. 75 ns) drift- time spectrum, with a shift of $+50\,\mathrm{ns}$. The drift-time shape of the spillover hits from the previous $-50\,\mathrm{ns}$ bunch-crossing is more complex. It contains the late hits of the drift-time distribution from the previous bunch-crossing. Naively, the drift-time spectrum of these early hits can be modelled by a shift of the spillover-free distribution by $-50\,\mathrm{ns}$, as illustrated by the black line in Fig. 13(b). However, a traversing track can give rise to multiple hits, which are usually not detected due to the digital “dead-time”. These multiple-hits, or “double pulses” from the previous bunch-crossing now become visible, when they fall inside the readout window of the triggered bunch- crossing. In 30 to 40% of all hits, the first arriving ionization cluster produces a second hit that arrives about $30\,\mathrm{ns}$ later. Several effects, such as multiple ionizations, reflections [24] or photon feedback [25], can produce such a double pulse. The time-spectrum of late hits from the previous bunch- crossing, observed in the triggered bunch crossing, is clearly isolated by studying “empty” bunch-crossings with “busy” previous bunch-crossings. The empty and busy bunch-crossings are selected using the total activity in the calorimeter in the subsequent bunch-crossings. The resulting drift-time spectrum of late hits from busy previous bunch-crossings in empty triggered events is shown as the red line in Fig. 13(b). The large number of double- pulses around 40 TDC counts, or 15 ns explains the enhancement of hits between $0$ and $25\,\mathrm{ns}$ in the $50\,\mathrm{ns}$ bunch-crossing drift-time spectrum compared to the spillover-free drift-time spectrum from the $75\,\mathrm{ns}$ data, see Fig. 13(a). Finally, the drift-time spectrum corresponding to the 25 ns bunch spacing conditions (recorded in Dec 2012) is also overlayed in Fig. 13(a). An overall increase of the number of hits is seen for a comparable number of overlapping events, compared to the 50 and 75 ns running conditions. ### 5.2 Occupancy The occupancy per straw is shown in Fig. 14 for typical run conditions in 2011 and 2012, triggered by any physics trigger. The occupancy is shown for events with 25, 50 and 75 ns bunch-crossing conditions. In absence of spillover (i.e. the 75 ns case), the occupancy varies from about $15\%$ in the innermost modules to about $3\%$ in the outermost modules. For the data taken with $50\,\mathrm{ns}$ bunch-crossing spacing, about 30% of all hits originate from spillover, i.e. from the previous bunch crossing. Monte Carlo simulations demonstrate that most of the hits originate from secondary charged particles, produced in interactions with material. Figure 15 shows the fraction of hits that originate from a particle created at a given $z$ coordinate. The hits from tracks that originate from the genuine $pp$ interaction or a subsequent particle decay, are predominantly located close to the interaction region. They represent $27.7\%$ (resp. $27.1\%$ and $25.7\%$) of all hits seen in station T1 (resp. T2 and T3). The remaining hits originate from charged particles created in secondary interactions, mainly in the support of the beam pipe situated in the magnet or in the detectors located upstream of the detector layer (Vertex Locator, Ring Imaging Detector, Tracker Turicensis (TT), Inner Tracker (IT) and OT). LHCb OT Figure 14: Straw occupancy for $75\,\mathrm{ns},50\,\mathrm{ns},25\,\mathrm{ns}$ bunch-crossing spacing in red, black and blue, respectively, for typical run conditions with on average 1.2, 1.4 and 1.2 overlapping events per bunch crossing, respectively. One module contains in total 256 straws, whereas the width of one module is 340 mm. The steps in occupancy at the center of the detector correspond to the location of the shorter S-modules, positioned further from the beam in the $y$-coordinate. The data corresponding to $25\,\mathrm{ns}$ bunch-crossing spacing, was recorded with opposite LHCb-dipole polarity, as compared to the other two data sets shown here. (a)(b)LHCb OTLHCb OT Figure 15: Coordinate of the origin of charged particles that produce a hit in the OT detector. (a) The blue histogram peaks at $z=0$ and corresponds to hits from particles produced at the $pp$ interaction point and their daughters, while the hits from particles produced in secondary interactions (red) predominantly originate from $z>0$. (b) The longitudinal and transverse position of the origin of charged particles produced in secondary interactions, showing the structure corresponding to the material in the detector. ### 5.3 Hit efficiency A high single-hit efficiency is crucial, as it affects the tracking efficiency, and eventually the physics performance of the LHCb experiment. The efficiency is defined as the number of observed hits in a particular detector region over the number of expected hits in the same region. The number of expected hits is estimated by considering charged particle tracks in $pp$ collision data and extrapolating the charged particle trajectory to the monolayer under study. In order to determine the hit efficiency, good quality tracks have been selected, requiring a $\chi^{2}/ndf$ (where $ndf$ are the number of degrees of freedom) less than 2 and a minimum number of 21 hits in the OT detector. This corresponds to accepting about 87% of all good tracks. For each track, every OT monolayer has been considered, and a hit has been searched in the straw closest to the charged particle trajectory. Since a track is reconstructed by the same hits that are subsequently used for the efficiency estimation, the large number of required hits could bias the efficiency determination. This has been corrected for by not considering the monolayer under study, when counting the minimum number of hits per track. The hit efficiency is studied as a function of the distance between the predicted track position and the center of the considered straw. The resulting single-hit-efficiency profile is shown in Fig. 16(a), summed for all straws in the long modules closest to the beam-pipe (module 7). (a)(b)LHCb OTLHCb OT Figure 16: (a) Efficiency profile as a function of the distance between the predicted track position and the center of the straw, for straws in the long F-modules closest to the beampipe (module 7). The vertical lines represent the straw tube edge at $\|r\|=2.45$ mm. (b) Histogram of the average efficiencies per half module (128 channels), at the center of the straw, $\|r\|<1.25$ mm, for runs 96753, 96763 and 96768 on 22 July 2011. The shape of the efficiency profile can be understood by considering two effects. Near the straw tube edge, the path length of ionizing particles inside the gas volume is limited, resulting in a sizeable probability for not ionizing the gas. This can be described with a Poissonian distribution for the single-hit probability. Secondly, the finite track resolution smears the distribution at the edge of the straw tube, lowering the efficiency inside and increasing the efficiency outside of the straw. The finite probability to detect a hit outside straw tube originates from random hits unrelated to the track under study, and is proportional to the average occupancy in that part of the detector. The straw tube profile can thus be fitted with the following line shape, which describes the efficiency as a function of the distance $r$ from the center of the straw, $\displaystyle p(r)$ $\displaystyle=$ $\displaystyle 1-\Big{(}1-\varepsilon(r)\otimes\rm{Gauss}(r\|0,\sigma)\Big{)}\cdot(1-\omega),$ (4) $\displaystyle\mathrm{with}\,\,\,\,\,\,\varepsilon(r)$ $\displaystyle=$ $\displaystyle\varepsilon_{0}\left(1-e^{\frac{-2\sqrt{R^{2}-r^{2}}}{\lambda}}\right),$ where $R=2.45$ mm is the inner radius of the straw, $\omega$ is the average occupancy, $\lambda$ is the effective ionization length of a charged particle in the gas volume, and $\sigma$ is the track resolution. The deviation from the perfect efficiency is quantified in Eq. (4) by $\varepsilon_{0}$. However, in the following, an operationally straightforward definition of the single-hit efficiency is used. The single-hit efficiency per straw is defined as the average hit efficiency $\varepsilon_{hit}$ in the limited range close to the wire, $\|r\|<1.25$ mm. The inefficient regions between two straws lead to the maximum efficiency of 93%, integrated over the monolayer, and is calculated by taking the ratio of the straw diameter of 4.9 mm over the pitch, 5.25 mm. The inefficient regions are covered by the neighbouring monolayer in the same module, which is staggered by half a straw pitch. Table 2: Average single-hit efficiencies $\varepsilon_{hit}$ near the center of the straws, $\|r\|<1.25$ mm, for different module positions of the OT detector. Module position \| Efficiency (%) ---\|--- 1 \| $98.085\pm 0.011$ 2 \| $99.130\pm 0.005$ 3 \| $99.279\pm 0.003$ 4 \| $99.277\pm 0.003$ 5 \| $99.282\pm 0.002$ 6 \| $99.342\pm 0.002$ 7 \| $99.286\pm 0.002$ 8 \| $99.200\pm 0.002$ 9 \| $99.351\pm 0.003$ A fit to the straw efficiency profile, using Eq. (4), separately for the profiles of the nine module positions, yields the following average parameters, $\langle\lambda\rangle=0.79\pm 0.09$ mm, $\langle\varepsilon_{0}\rangle=0.993\pm 0.003$, $\langle\sigma\rangle=0.26\pm 0.06$ mm, $\langle\omega\rangle=0.07\pm 0.02$, where the quoted uncertainty is the standard deviation from the values obtained for the nine different module positions. Note that these parameters are averaged over the different module positions, corresponding to different conditions. For example, the measured occupancy $\omega$ varies from $4.7\%$ to $9.7\%$ depending on the distance of the module to the beam. Fits to the single-hit efficiency profiles show that the efficiency is close to maximal, i.e. the value for $\varepsilon_{0}$ is consistent with unity. For each half-module, corresponding to one FE box, the average single-hit efficiency $\varepsilon_{hit}$ has been calculated and the result is shown in Fig. 16(b). The efficiency distribution peaks around 99.5%, consistent with the measurements from beam tests [18]. The average of the distribution is about 99.2%. This value is consistent with the fit to the straw tube profile, shown in Fig. 16(a). The large hit efficiency is a prerequisite for large efficiency to reconstruct charged particle’s tracks in LHCb. The average tracking efficiency is approximately 95% in the region covered by the LHCb detector [26]. The modules that are located at the edge of the geometrical acceptance of LHCb, in particular the outermost modules in the first station, detect a relatively small number of tracks. All eight FE boxes in Fig. 16(b) with a value of the efficiency exactly equal to 1, and 11 out of the 14 FE boxes with $\varepsilon_{hit}<96\%$, are attached to modules located most distant from the beampipe (module 1), and suffer from few tracks in the efficiency determination. The remaining three FE boxes with $\varepsilon_{hit}<96\%$ suffer from hardware problems, representing $3/432=0.7\%$ of all FE boxes. In order to calculate the average efficiency for each module position, modules with few tracks, ie. with an efficiency lower than 96%, or with an efficiency equal to unity, have been discarded. The average efficiency thus obtained for each module position is listed in Table 2 where the reported uncertainties are statistical. As shown above, the decrease of the hit efficiency close to straw edge is partially due to the fact that the charged particle traverses a short distance through the straw volume. Hence, the probability to not form an ionization cluster increases towards the straw edge. Alternatively, the effective ionization length $\lambda$ can be probed by selecting only those tracks that pass close to the wire. In contrast to the first method exploiting Eq. 4, here the determination of the ionization length is not affected by absorption of drifting electrons. The larger the ionization length, the more hits will exhibit a large drift-time, as the ionization does not necessarily occur close to the wire. The effective ionization length $\lambda$ extracted from particles traversing the straw within $\|r\|<0.1$ mm amounts to about 0.7 mm [19, 27], consistent with $0.79\pm 0.09$ mm, as obtained above. ### 5.4 Hit resolution The single hit resolution is determined using good quality tracks, selected by requiring a momentum larger than 10 GeV, at least 16 OT hits and a track-fit $\chi^{2}/ndf<2$ (excluding the hit under study, and excluding any hit in the neighbouring monolayer in the same module). For a given track, the drift-time and the hit position in a straw are predicted, and compared with the measured drift-time and position, respectively. The resulting distribution of the drift-time residuals and hit position residuals are shown in Fig. 17. (a)(b) Figure 17: (a) Drift-time residual distribution and (b) hit distance residual distribution [19]. The core of the distributions (within $\pm 1\sigma$) are fitted with a Gaussian function and the result is indicated in the figures. (a)(b) Figure 18: Improvement in (a) drift-time residual distribution and (b) hit distance residual distribution, (red) before and (blue) after allowing for a different horizontal displacement per half monolayer, corresponding to 64 straws [19]. The drift-time residual distribution has a width of 3 ns which is dominated by the ionization and drift properties in the counting gas. The granularity of the step size of the TDC of 0.4 ns has a negligible impact on the drift-time resolution. The hits in the left tail of the drift-time residual distribution are early hits, that do not originate from the track under study, but instead are a combination of noise hits, hits from different tracks in the same bunch crossing, and hits from tracks from previous bunch crossings (spill-over hits). The hit distance residual distribution has a width of about 205 $\mu$m, which is close to the design value of 200 $\mu$m. An improvement of the hit position resolution is foreseen when the two monolayers within one detector module are allowed to be relatively displaced to each other in the global LHCb alignment procedure. By allowing a different average horizontal displacement per half monolayer, containing 64 straws, a single hit resolution of approximately 180 $\mu$m is in reach, see Fig. 18. Also allowing for a rotation of each half monolayer, improves the single hit resolution further to 160 $\mu$m. These values refer to a Gaussian width of the resolution, determined from a fit to the residual distribution, within two standard deviations of the mean. This is in good agreement with the hit resolution below 200 $\mu$m, as obtained in beam tests [18]. ### 5.5 Monitoring of faulty channels LHCb OT Figure 19: The evolution of number of dead and noisy channels as function of run number in the 2011 and 2012 running periods. The definition of dead and noisy channels is given in the text. The three periods with larger number of dead channels, correspond to periods with a problem affecting one entire front-end box. Noisy or dead channels due to malfunctioning front-end electronics are timely identified through the analysis of the calibration runs as described in Sec. 3. With the full offline data set available, the performance of individual channels is also monitored by comparing the occupancy to the expected value. First, the performance of entire groups of 32 channels is verified. Then, within a group of 32 channels, the occupancy is compared to the truncated mean, after correcting for the dependence of the occupancy on the distance to the beam. If the occupancy is above (below) 6 standard deviations from the truncated mean, the channel is declared “noisy” (“dead”). For a typical run recorded at the end of 2012 (run 133785), when all front-end modules were functioning properly, the OT contained 52 dead channels and 8 noisy channels, evenly distributed over the detector. The evolution of the number of bad channels throughout the 2011 and 2012 running periods is shown in Fig. 19. 666The three periods with larger number of dead channels correspond to a broken laser diode (VCSEL) between September and December 2011 at location T1L3Q0M2, a broken fuse in May 2012 at location T3L3Q0M8, and desynchronization problems between July and September 2012 at location T2L2Q0M9. Note that the front-end box at location M9 on the C-side reads out only 64 straws. ### 5.6 Radiation tolerance It was discovered that, in contrast to the excellent results of extensive ageing tests in the R&D phase, final production modules suffered from gain loss after moderate irradiation (i.e. moderate collected charge per unit time) in laboratory conditions. The origin of the gain loss was traced to the formation of an insulating layer on the anode wire [11], that contains carbon and is caused by outgassing inside the gas volume of the plastifier contained in the glue [28]. Remarkably, the gain loss was only observed upstream of the source position with respect to the gas flow. A negative correlation was observed between the ageing rate and the production of ozone [11], which suggests that the gain loss is prevented under and downstream of the source due to the formation of ozone in the avalanche region. As a consequence it was decided to add 1.5% O2 to the original gas mixture of Ar/CO2, to mitigate possible gain loss. In addition, a beneficial effect from large induced currents was observed, which removed the insulating layers from irradiation in the laboratory. These large currents can either be invoked by large values of the high voltage in the discharge regime (dark currents), or by irradiating the detector with a radioactive source [28]. (a)(b)LHCb OTAug 2010LHCb OTDec 2012 Figure 20: Hit efficiency as a function of amplifier threshold in (a) August 2010 and (b) December 2012 for the inner region, defined as $\pm 60\,\mathrm{cm}$ in $x$ and $\pm 60\,\mathrm{cm}$ in $y$ from the central beam pipe, summed over all OT layers. Note that the threshold value of 1350 mV, where the efficiency is 50%, is much higher than the operational threshold of 800 mV, and is equivalent to multiple times the corresponding average hit charge. No signs of gain loss have been observed in the 2010 to 2012 data taking period of LHCb, corresponding to a total delivered luminosity of 3.5 fb-1. Most of the luminosity was recorded in 2011 and 2012, corresponding to about $10^{7}$ s of running at an average instantaneous luminosity of $3.5\times 10^{32}$ cm-2s-1, and the region closest to the beam accumulated an integrated dose equivalent to a collected charge of 0.12 C/cm. Possible changes in the gain are studied by increasing the amplifier threshold value during LHC operation, and comparing the value where the hit efficiency drops, see Fig. 20. This value of the amplifier threshold can be converted to hit charge, which provides information on the change of the detector gain. This method to measure the gain variations is outlined in detail in Ref. [29]. ## 6 Conclusions The Outer Tracker has been operating in the 2010, 2011 and 2012 running periods of the LHC without significant hardware failures. The low voltage, high voltage and gas systems showed a reliable and stable performance. Typically 250 channels out of a total of 53,760 channels were malfunctioning, resulting in 99.5% working channels. The missing channels were mainly caused by problems in the readout electronics, whereas only a handful channels could not stand the high voltage on the detector. The occupancy of the Outer Tracker detector of typically 10% was larger than anticipated, due to twice larger instantaneous luminosity at LHCb with half the number of bunches in the LHC, compared to the design specifications. Despite these challenging conditions, the Outer Tracker showed an excellent performance with a single-hit efficiency of about 99.2% near the center of the straw, and a single hit resolution of about 200 $\mu$m. No signs of irradiation damage have been observed. ## Acknowledgements We wish to thank our colleagues of the CERN Gas Group for their continuous support of the Outer Tracker gas system. We also express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We also acknowledge the support received from the ERC under FP7. The Tier1 computing centres are supported by IN2P3 (France), KIT and BMBF (Germany), INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom). We are thankful for the computing resources put at our disposal by Yandex LLC (Russia), as well as to the communities behind the multiple open source software packages that we depend on. ## References * [1] LHCb collaboration, A. A. Alves Jr. et al., The LHCb detector at the LHC, JINST 3 (2008) S08005 * [2] E. Simioni, New physics from rare beauty, PhD thesis, Vrije Universiteit, Amsterdam, 2010, CERN-THESIS-2010-031 * [3] LHCb collaboration, P. Barbosa et al., Outer Tracker technical design report, , CERN-LHCC-2001-024 * [4] A. Berkien et al., The LHCb outer tracker front end electronics, CERN-LHCB-2005-025 * [5] G. Haefeli et al., The LHCb DAQ interface board TELL1, Nucl. Instrum. Meth. A560 (2006) 494 * [6] N. Dressnandt et al., Implementation of the ASDBLR and DTMROC ASICS for the ATLAS TRT in DMILL Technology, 6th Workshop on Electronics for LHC Experiments 2000 * [7] H. Deppe, U. Stange, U. Trunk, and U. Uwer, The OTIS reference manual, CERN-LHCB-2008-010 * [8] U. 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Amoraal, Alignment with Kalman filter fitted tracks and reconstruction of $B^{0}_{s}\to J/\psi\phi$ decays , PhD thesis, Vrije Universiteit, Amsterdam, 2011, CERN-THESIS-2011-011 * [21] H. Dekker et al., The RASNIK/CCD 3-dimensional alignment system, eConf C930928 (1993) 017, IWAA-1993-017 * [22] M. Adamus et al., Test results of the RASNIK optical alignment monitoring system for the LHCb outer tracker detector, LHCB-2001-004 * [23] J. Amoraal et al., Application of vertex and mass constraints in track-based alignment, Nucl. Instrum. Meth. A712 (2012) 48, arXiv:1207.4756 * [24] Y. Guz et al., Study of the global performance of an LHCb OT front-end, CERN-LHCB-2004-120 * [25] V. Suvorov, G. Van Apeldoorn, I. Gouz, and T. Sluijk, Avalanche and streamer production in $Ar/CO_{2}$ mixtures, CERN-LHCB-2005-038 * [26] LHCb collaboration, R. Aaij et al., Measurement of the track reconstruction efficiency at LHCb, LHCb-DP-2013-002. to be submitted to Nucl. Instrum. Meth. * [27] N. Tuning, Detailed performance of the Outer Tracker at LHCb, JINST 9 (2014) C01040 * [28] N. Tuning et al., Ageing in the LHCb outer tracker: Aromatic hydrocarbons and wire cleaning, Nucl. Instrum. Meth. A656 (2011) 45 * [29] D. van Eijk et al., Radiation hardness of the LHCb Outer Tracker, Nucl. Instrum. Meth. A685 (2012) 62	arxiv-papers	2013-11-15T15:58:52	2024-09-04T02:49:53.692931	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "LHCb Outer Tracker group: R. Arink, S. Bachmann, Y. Bagaturia, H.\n Band, Th. Bauer, A. Berkien, Ch. F\\\"arber, A. Bien, J. Blouw, L. Ceelie, V.\n Coco, M. Deckenhoff, Z. Deng, F. Dettori, D. van Eijk, R. Ekelhof, E.\n Gersabeck, L. Grillo, W.D. Hulsbergen, T.M. Karbach, R. Koopman, A.\n Kozlinskiy, Ch. Langenbruch, V. Lavrentyev, Ch. Linn, M. Merk, J. Merkel, M.\n Meissner, J. Michalowski, P. Morawski, A. Nawrot, M. Nedos, A. Pellegrino, G.\n Polok, O. van Petten, J. R\\\"ovekamp, F. Schimmel, H. Schuylenburg, R.\n Schwemmer, P. Seyfert, N. Serra, T. Sluijk, S. Tolk, B. Spaan, J. Spelt, B.\n Storaci, M. Szczekowski, S. Swientek, N. Tuning, U. Uwer, D. Wiedner, M.\n Witek, M. Zeng, A. Zwart", "submitter": "Niels Tuning", "url": "https://arxiv.org/abs/1311.3893" }
1311.3904	# On graded polynomial identities of $sl_{2}(F)$ over a finite field Luís Felipe Gonçalves Fonseca Departamento de Matemática, Universidade Federal de Viçosa - Campus Florestal, Rodovia LMG 818, km 06, Florestal, MG, Brazil [email protected] ###### Abstract. Let $F$ be a finite field of $charF>3$ and $sl_{2}(F)$ be the Lie algebra of traceless $2\times 2$ matrices over $F$. This paper aims for the following goals: * • Find a basis for the $\mathbb{Z}_{2}$-graded identities of $sl_{2}(F)$; * • Find a basis for the $\mathbb{Z}_{3}$-graded identities of $sl_{2}(F)$ when $F$ contains a primitive $3$rd root of one; * • Find a basis for the $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-graded identities of $sl_{2}(F)$. Keywords: Graded Identities; Graded Lie Algebras; Lie $A$-algebras; Levi decomposition; Monolithic and critical Lie algebras. To my family. ## 1\. Introduction The famous Ado-Iwasawa’ theorem (see for instance, chapter 6 in [10]) posits that any finite-dimensional Lie algebra over an arbitrary field has a faithful finite-dimensional representation. Briefly, any finite dimensional Lie algebra can be viewed as a subalgebra of a Lie algebra of square matrices under the commutator brackets. Thus, the study of Lie algebras of matrices is of considerable interest. A crucial task in PI-theory, an important branch of Ring theory, is to describe the identities of $sl_{2}(F)$, the Lie algebra of traceless $2\times 2$ matrices over a field $F$ of $charF\neq 2$. The first breakthrough in this area was made by Razmyslov [16] who described a basis for the identities of $sl_{2}(F)$ when $charF=0$. Vasilovsky [22] found a single-identity for the identities of $sl_{2}(F)$ when $F$ is an infinite field of $charF>2$ and Semenov [18] described a basis (with two identities) for the identities of $sl_{2}(F)$ when $F$ is a finite field of $charF>3$. The Lie algebra $sl_{2}(F)$ can be naturally graded by $\mathbb{Z}_{2}$ as follows: $sl_{2}(F)=\newline (sl_{2}(F))_{0}\oplus(sl_{2}(F))_{1}$ where $(sl_{2}(F))_{0},(sl_{2}(F))_{1}$ contain diagonal and off-diagonal matrices respectively. Moreover, the above Lie algebra can be naturally graded by $\mathbb{Z}_{3}$ and $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ as follows: $(sl_{2}(F))_{-1}=span_{F}\\{e_{21}\\},(sl_{2}(F))_{0}=span_{F}\\{e_{11}-e_{22}\\},(sl_{2}(F))_{1}=span_{F}\\{e_{21}\\}$. $(sl_{2}(F))_{(0,0)}=\\{0\\}$, $(sl_{2}(F))_{(1,0)}=span_{F}\\{e_{11}-e_{22}\\}$, $(sl_{2}(F))_{(0,1)}=span_{F}\\{e_{12}+e_{21}\\}$, and $(sl_{2}(F))_{(1,1)}=span_{F}\\{e_{12}-e_{21}\\}$. Here, $e_{ij}\subset gl_{2}(F)$ denotes the unitary matrix unit whose elements are $1$ in the positions $(ij)$ and $0$ otherwise. In his PhD thesis, Repin [17] proved, up to equivalence, that the above three gradings are unique abelian nontrivial gradings on $sl_{2}(F)$ (when $charF=0$ and $\overline{F}=F$). In one of the contributions in [2], Bahturin, Kochetov and Montgomery extended this Repin’s result, when $F=\overline{F}$ and $charF\neq 2$. Hereafter, we consider only $sl_{2}(F)$ graded by these three groups. A recent development in PI-theory is the description of the graded identities of $sl_{2}(F)$. Using invariant theory technics, P. Koshlukov [12] described the graded identities for $sl_{2}(F)$ when $F$ is an infinite field of $charF>2$. Two years later, Krasilnikov, Koshlukov and Silva [11] described the graded identities of $sl_{2}(F)$ when $F$ is an infinite field of $charF>2$, but from elementary methods. Giambruno and Souza [6] proved that the graded identities of $sl_{2}(F)$ have the Specht Property when $F$ is a field of characteristic zero. To date, no basis has been found for the graded identities of $sl_{2}(F)$ when $F$ is a finite field. This paper reports a basis for the graded identities $sl_{2}(F)$ when $F$ is a finite field of $charF>3$. The fundamentals in this paper have been adopted from can be found in [1], [5], [7], [10], [13], [14], [15], [18], [19], [20], and [21]. ## 2\. Preliminaries- part I Let $F$ be a fixed finite field of $charF>3$, let $\mathbb{N}_{0}=\\{1,2,\cdots,n,\cdots\\}$, let $G$ be a finite abelian group, and let $L$ be a Lie algebra over $F$. In this paper (unless otherwise mentioned), all vector spaces and Lie algebras are considered over $F$. The $+,\oplus,span_{F}\\{a_{1},\cdots,a_{n}\\},\langle a_{1},\cdots,a_{n}\rangle(a_{1},\cdots,a_{n}\in L)$ signs denote the direct sum of Lie algebras, direct sum of vector spaces, the vector space generated by $a_{1},\cdots,a_{n}$, and the ideal generated by $a_{1},\cdots,a_{n}$ respectively, while an associative product is represented by a dot: $``.^{\prime\prime}$. The commutator $([,])$ denotes the multiplication operation of a Lie algebra. We assume that all commutators are left-normed, i.e., $[x_{1},x_{2},\cdots,x_{n}]:=[[x_{1},x_{2},\cdots,x_{n-1}],x_{n}]\ \ n\geq 3$. We use the convention $[x_{1},x_{2}^{k}]=[x_{1},x_{2},\cdots,x_{2}]$, where $x_{2}$ appears $k$ times in the expanded commutator. The basic concepts of Lie algebra adopted in this paper can be found in chapters 1 and 2 of [9] and chapter 1 of [1]. We denote the center of $L$ by $Z(L)=\\{x\in L\|[x,y]=0\mbox{for all}y\in L\\}$. If $x\in L$, we denote by $adx$ the linear map with the function rule: $y\mapsto[x,y]$. $L$ is said to be metabelian if it is solvable with at most $2$. $L$ is said to be simple if $[L,L]\neq\\{0\\}$, and $L$ does not have any proper non-trivial ideals. $L$ is regarded as a Lie $A$-algebra if all of its nilpotent subalgebras are abelian. A Lie algebra $L$ is said to be $G$-graded (a graded Lie algebra or graded by $G$) when there exist subspaces $\\{L_{g}\\}_{g\in G}\subset L$ such that $L=\bigoplus_{g\in G}L_{g}$, and $[L_{g},L_{h}]\subset L_{g+h}$ for any $g,h\in G$. An element $a$ is called homogeneous when $a\in\bigcup_{g\in G}L_{g}$. We say that $a$ is a homogeneous element of $G$-degree $g$ when $a\in L_{g}$. A $G$-graded homomorphism of two $G$-graded Lie algebras $L_{1}$ and $L_{2}$ is a homomorphism $\phi:L_{1}\rightarrow L_{2}$ such that $\phi({L_{1}}_{g})\subset{L_{2}}_{g}$ for all $g\in G$. An ideal $I\subset L$ is graded when $I=\bigoplus_{g\in G}(I\cap L_{g})$. Likewise, if $I$ is a graded ideal of $L$, $C_{L}(I)=\\{a\in L\|[a,I]=\\{0\\}\\}$ is also a graded ideal of $L$. Furthermore, given $Z(L)=\\{a\in L\|[a,b]=0\ \ \mbox{for all}\ \ b\in L\\},L^{n}$ (the $n$-th term of a descending central series), and $L^{(n)}$ (the $n$-th derived series) are graded ideals of $L$. ###### Remark 2.1. Convention: $L^{(1)}=[L,L]$ and $L^{1}=L$. As is well-known if $L$ ($L$ over a finite field of $charF>3$) is a three- dimensional simple Lie algebra, then $L\cong sl_{2}(F)$. Let $L$ be a finite-dimensional Lie algebra. We denote by $Nil(L)$ the greatest nilpotent ideal of $L$ and let $Rad(L)$ be the greatest solvable ideal of $L$. Clearly, $Nil(L)$ is the unique maximal abelian ideal of $L$ when $L$ is a Lie $A$-algebra. Furthermore, every subalgebra and every factor algebra of $L$ is a Lie $A$-algebra when $L$ is also a Lie $A$-algebra (see Lemma 2.1 in [21] and Lemma 1 in [14]). Let $X=\\{X_{g}=\\{x_{1}^{g},\cdots,x_{n}^{g},\cdots\\}\|g\in G\\}$ be a class of pairwise-distinct enumerable sets, where $X_{g}$ denotes the variables of $G$-degree $g$. Let $L(X)\subset F\langle X\rangle^{(-)}$ (Lie algebra of $F\langle X\rangle$, the free associative unital algebra (over $F$) freely generated by $X$) be the subalgebra generated by $X$. $L(X)$ is known to be isomorphic to the free Lie algebra with a set of free generators $X$. The algebras $L(X)$ and $F\langle X\rangle$ have natural $G$-grading. A graded ideal $I\subset L(X)$ invariant under all graded endomorphisms is called a graded verbal ideal. Let $S\subset L(X)$ be a non-empty set. The graded verbal ideal generated by $S$, $\langle S\rangle_{T}$, is defined as the intersection of all verbal ideals containing $S$. A polynomial $f\in L(X)$ is called a consequence of $g\in L(X)$ when $f\in\langle g\rangle_{T}$ and is called a graded polynomial identity for a graded Lie algebra $L$ if $f$ vanishes on $L$ whenever the variables from $X_{g}$ are substituted by elements of $L_{g}$ for all $g\in G$. We denote by $V_{G}(L)$ the set of all graded identities of $L$. The variety determined by $S\subset L(X)$ is denoted by $Var(S)=\\{A\ \mbox{is a}\ \mbox{G-graded Lie algebra}\|V_{G}(A)\supset\langle S\rangle_{T}\\}$. The variety generated by a graded Lie algebra $L$ is denoted by $var_{G}(L)=\\{A\ \mbox{is a}\ \ \mbox{G-graded Lie algebra}\|V_{G}(L)\subset V_{G}(A)\\}$. We say that a class of graded Lie algebras $\\{L_{i}\\}_{i\in\Gamma}$, where $\Gamma$ is an index set, generates $Var(S)$ when $\langle S\rangle_{T}=\bigcap_{i\in\Gamma}var_{G}(A_{i})$. It is known that there exists a natural 1-1 correspondence between the graded verbal ideals of $L(X)$ and the varieties of graded Lie algebras. As is well-known a class of graded Lie algebras is a variety, if and only, if it is closed under the forming of subalgebras, homomorphic images and Cartesian products of its algebras (Birkhoff’s HSP theorem). Moreover, a variety of graded Lie algebras is generated by the finitely generated Lie algebras belong to a variety. All of the above described concepts have analogies in ordinary Lie algebras. We denote by $V(L)$ the set of all ordinary identities of a Lie algebra $L$, and by $var(L)$ the variety generated by $L$. The variety of metabelian Lie algebras over $F$ is denoted by $U^{2}$. A set $S\subset L(X)$ of ordinary polynomials (graded polynomials) is called a basis for the ordinary identities (graded identities) of a graded Lie algebra $A$ when $V(A)=\langle S\rangle_{T}$ (respectively $V_{G}(A)=\langle S\rangle_{T}$). ###### Remark 2.2. As is well-known, $V(gl_{2}(F))=V(sl_{2}(F))$ when $F$ is a field of $charF\neq 2$. In addition to $sl_{2}(F)$, $gl_{2}(F)$ can be naturally graded by $\mathbb{Z}_{2}$ ($gl_{2}(F)_{0}=span_{F}\\{e_{11},e_{22}\\}$ and $gl_{2}(F)_{1}=span_{F}\\{e_{12},e_{21}\\}$) and by $\mathbb{Z}$: ($gl_{2}(F)_{-1}=span_{F}\\{e_{21}\\},\newline gl_{2}(F)_{0}=span_{F}\\{e_{11},e_{22}\\}$ and $gl_{2}(F)_{1}=span_{F}\\{e_{12}\\}$). Similar to ordinary case, we have that: $V_{\mathbb{Z}_{2}}(gl_{2}(F))=V_{\mathbb{Z}_{2}}(sl_{2}(F))$ and $V_{\mathbb{Z}}(gl_{2}(F))=V_{\mathbb{Z}}(sl_{2}(F))$ when $F$ is field of $charF\neq 2$. ###### Definition 2.3. A finite-dimensional graded (ordinary) Lie algebra $L$ is critical if $var_{G}(L)$ ($var(L)$) is not generated by all proper subquotients of $L$. ###### Definition 2.4. A graded (ordinary) Lie algebra $L$ is monolithic if it contains a single graded (ordinary) minimal ideal. This single ideal is termed a monolith. ###### Proposition 2.5. Let $L$ be a graded (ordinary) critical Lie algebra. Then $L$ is monolithic. ###### Proof. It is sufficient to state verbatim in [13]. ∎ ###### Remark 2.6. Since [13] treats varieties of groups, some changes from that text are required here; namely, replacing a finite group by a finite-dimensional Lie algebra (graded Lie algebra), replacing a normal subgroup by an ideal, and the direct product of groups by the direct sum of Lie algebras. It is not difficult to see that if $L$ is a critical abelian (ordinary) graded Lie algebra, then $dimL=1$. Furthermore, $sl_{2}(F)$ is a critical Lie algebra. Following word for word the work of Silva in [20] (Proposition 1.36), we have the following lemma. ###### Lemma 2.7. : If $A$ and $B$ are two $G$-graded Lie algebras such that $V_{G}(A)\subset V_{G}(B)$, then $V(A)\subset V(B)$. Furthermore, if $V_{G}(A)=V_{G}(B)$, then $V(A)=V(B)$. : If $L=\oplus_{g\in G}L_{g}$ is not critical as a $G$-graded Lie algebra, then $L$ is not critical as an ordinary Lie algebra. Equivalently: if $L=\oplus_{g\in G}L_{g}$ is a critical ordinary Lie algebra, then $L$ is critical as a $G$-graded Lie algebra. ###### Definition 2.8. A locally finite Lie algebra is a Lie algebra for which every finitely generated subalgebra is finite. A variety of graded Lie algebras $var_{G}$ (or Lie algebras $var$) is said to be locally finite when every finitely generated has finite cardinality. ###### Corollary 2.9. Let $var_{G}$ be a locally finite variety of graded Lie algebras. Then $var_{G}$ is generated by its finite algebras. ###### Example 1. Let $L$ be a finite-dimensional graded Lie algebra. Then $var_{G}(L)$ is locally finite. ###### Proof. It is sufficient to state verbatim Theorem 2 in [1] (Chapter 4, page 99). ∎ The following proposition describes an application of critical algebras: ###### Proposition 2.10. Let $var_{G}$ be a locally finite variety of graded Lie algebras. Then $var_{G}$ is generated by its critical algebras. ###### Proof. It is sufficient to repeat verbatim the proof of Lemma 1 in [1]. ∎ An important consequence of Proposition 2.10 is the following corollary: ###### Corollary 2.11. Let $var_{1}\subset var_{2}$ be two varieties of graded Lie algebras. If $var_{2}$ is locally finite and its critical algebras belong to $var_{1}$, then $var_{1}=var_{2}$. The next result will prove useful for our purposes: ###### Theorem 2.12 (Chanyshev and Semenov, Proposition 2, [18]). Let $\mathcal{B}$ be a variety of (ordinary) Lie algebras over a finite field $F$. If there exists a polynomial $f(t)=a_{1}t+\cdots+a_{n}t^{n}\in F[t]$ such that $yf(adx):=a_{1}[y,x]+\cdots+a_{n}[y,x^{n}]\in V(\mathcal{B})$, then $\mathcal{B}$ is a locally finite variety. The next theorem describes the relationship between critical metabelian Lie $A$-algebras and monolithic Lie $A$-algebras. ###### Theorem 2.13 (Sheina, Theorem 1,[19]). A finite-dimensional monolithic Lie $A$-algebra $L$ over an arbitrary finite field is critical if, and only if, its derived algebra can not be represented as a sum of two ideals strictly contained within it. The next theorem is a structural result on solvable Lie $A$-algebras. ###### Theorem 2.14 (Towers, Theorem 3.5, [21]). Let $L$ be a (finite-dimensional) solvable Lie $A$-algebra (over an arbitrary field $F$) of derived length $n+1$ with nilradical $Nil(L)$. Also let $K$ be an ideal of $L$ and $A$ a minimal ideal of $L$. Then we have: * • $K=(K\cap A_{n})\oplus(K\cap A_{n-1})\oplus\cdots\oplus(K\cap A_{0})$; * • $Nil(L)=A_{n}+(A_{n-1}\cap Nil(L))+\cdots+(A_{0}\cap Nil(L))$; * • $Z(L^{(i)})=Nil(L)\cap A_{i}$ for each $0\leq i\leq n$; * • $A\subseteqq Nil(L)\cap A_{i}$ for some $0\leq i\leq n$. $A_{n}=L^{(n)}$, $A_{n-1},\cdots,A_{0}$ are abelian subalgebras of $L$ defined in the proof of Corollary 3.2 in [21]. ###### Corollary 2.15. Let $L=\bigoplus_{g\in G}L_{g}$ be a (finite-dimensional) solvable graded Lie $A$-algebra (over an arbitrary field $F$) of derived length $n+1$ with nilradical $Nil(L)$. Then $Nil(L)=\bigoplus_{g\in G}((Nil(L))\cap L_{g})$. ###### Proof. It is sufficient to notice that: $Nil(L)=A_{n}+(A_{n-1}\cap Nil(L))+\cdots+(A_{0}\cap Nil(L))=L^{(n)}+Z(L^{(n-1)})+\cdots+Z(L)$. ∎ ###### Corollary 2.16. Let $L=\bigoplus_{g\in G}L_{g}$ be a finite-dimensional monolithic (non- abelian) metabelian graded Lie $A$-algebra over an arbitrary field $F$. Then $Nil(L)=[L,L]$. ###### Proof. According to Theorem 2.14, we have that $Nil(L)=[L,L]+Z(L)$. By the hypothesis, $L$ is monolithic. Thus, $Z(L)=\\{0\\}$ and $Nil(L)=[L,L]$. ∎ A finite-dimensional Lie algebra $L$ is called semisimple if $RadL=\\{0\\}$. Recall that $L$ (finite-dimensional and non-solvable) has a Levi decomposition when there exist a semisimple subalgebra $S\neq\\{0\\}$ (termed a Levi subalgebra) such that $L$ is a semidirect product of $S$ and $Rad(L)$. We now present an important result. ###### Proposition 2.17 (Premet and Semenov, Proposition 2, [14]). Let $L$ be a finite-dimensional Lie $A$-algebra over a finite field $F$ of $charF>3$. Then: * • $[L,L]\cap Z(L)=\\{0\\}$. * • $L$ has a Levi decomposition. Moreover, each Levi subalgebra $S$ is represented as a direct sum of $F$-simple ideals in $S$, each one of which splits over some finite extension of the ground field into a direct sum of the ideals isomorphic to $sl_{2}(F)$. ###### Remark 2.18. In Proposition 2, $F(charF=p>3)$ is a field of cohomological dimension $\leq 1$ (see definition in section 6.1 of [8]). However, a finite field has cohomological dimension $1$ (see for instance: example 6.1.11, page 240, of [8]).The extension of $L$ over a field $\overline{F}$ ($\overline{F}$ is a field extension of $F$ such that $dim_{F}\overline{F}<\infty$) is $L\otimes_{F}\overline{F}$. Based on Proposition 2.17, if $L$ (finite-dimensional and non-solvable) is a $G$-graded Lie $A$-algebra over a finite field $F$ of $charF>3$, then $L$ has a Levi decomposition. Now, we extend that result to the setting of graded Lie algebras over a field $F$ that contains a primitive $\|G\|$th root of one and $charF\nmid\|G\|$. Let $\widehat{G}$ be the group of all irreducibles characters on $G$. Since $G$ is finite abelian, $G\cong\widehat{G}$. Any $\psi\in\widehat{G}$ acts on $L=\bigoplus_{g\in G}L_{g}$ by the automorphism $\psi.a_{g}=\psi(g)a_{g}$, where $a_{g}\in L_{g}$. A subspace $V\subset L$ is graded if and only if $V$ is $\widehat{G}$-stable. Analogously, if one defines the $\widehat{G}$-action on $L$ by automorphisms, then $L=\bigoplus_{g\in G}L_{g}$ is a $G$-grading on $L$, where $L_{g}=\\{v\in L\|\psi.v=\psi(g)v\ \ \mbox{for all}\ \psi\in\ \widehat{G}\\}$. Thus, there are a duality between $G$-gradings and $G$-actions on $L$ (see for instance [7], pages 63 and 64). Notice that $RadL$ is invariant under isomorphisms of $L$, so $RadL$ is a graded ideal. Following word by word the work of Zaicev et al in [15] (Proposition 3.1, items i and ii), we have the following proposition: ###### Proposition 2.19. Let $F$ be a finite field of $charF>3$ that contains a primitive $\|G\|$th root of one and $charF\nmid\|G\|$. Let $L$ be a (finite-dimensional) semisimple graded Lie $A$-algebra over $F$. Then $L$ is a direct sum of graded simple ideals. ###### Remark 2.20. For $G=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, every irreducible representation has dimension $1$. We can remove the assumption that $F$ contains a $4$-th root of one. The irreducible characters are: : $[\phi((0,0))=1,\phi((1,0))=1,\phi((0,1))=1,\phi((1,1))=1]$; : $[\phi((0,0))=1,\phi((1,0))=-1,\phi((0,1))=-1,\phi((1,1))=1]$; : $[\phi((0,0))=1,\phi((1,0))=1,\phi((0,1))=-1,\phi((1,1))=-1]$; : $[\phi((0,0))=1,\phi((1,0))=-1,\phi((0,1))=1,\phi((1,1))=-1]$. For $G=\mathbb{Z}_{2}$, the conclusions are the same as $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. The irreducible characters of $\mathbb{Z}_{2}$ are: : $[\phi(0)=1,\phi(1)=1]$; : $[\phi(0)=1,\phi(1)=-1]$. ###### Remark 2.21. For $G=\mathbb{Z}_{3}$, the inclusion of a $3$-rd primitive root of one ($\omega$) is needed to build the irreducible characters table. The irreducible characters of $\mathbb{Z}_{3}$ are: : $[\phi(0)=1,\phi(1)=1,\phi(-1)=1]$; : $[\phi(0)=1,\phi(1)=\omega,\phi(-1)=\omega^{2}]$; : $[\phi(0)=1,\phi(1)=\omega^{2},\phi(-1)=\omega]$. In [3], Khazal et al described all gradings on the set of all $2\times 2$ matrices over an arbitrary field. Following word by word the work Bahturin et al in [2] (Section 2 and Theorem 5.1), we have the following proposition: ###### Proposition 2.22. Let $F$ be a finite field of $charF=p>3$. Then, up to equivalence, the unique non-trivial $\mathbb{Z}_{2}$-grading and $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-grading on $sl_{2}(F)$ are those reported in the introduction. If $F$ contains a $3$-rd primitive root of $1$, then the unique non-trivial $\mathbb{Z}_{3}$-grading on $sl_{2}(F)$ is that reported in the introduction. ## 3\. Preliminaries- part II This section is based on chapter 7 of [1] (pages 225 and 226) and Chapter 5 of [13] (pages 162,163,164,165 and 166). Furthermore, we assume that all Lie algebras are finite-dimensional. ###### Definition 3.1. Let $L_{1}$ and $L_{2}$ be two graded Lie algebras, and $I_{1}\subset L_{1}$ and $I_{2}\subset L_{2}$ be graded ideals. We say that $I_{1}$ (in $L_{1}$) is similar to $I_{2}$ (in $L_{2}$) ( $I_{1}\trianglelefteq A_{1}\sim I_{2}\trianglelefteq A_{2}$) if there exist isomorphisms $\alpha_{1}:I_{1}\rightarrow I_{2}$ and $\alpha_{2}:\frac{L_{1}}{C_{L_{1}}(I_{1})}\rightarrow\frac{L_{2}}{C_{L_{2}}(I_{2})}$ such that for all $a\in I_{1}$ and $b+C_{L_{1}}(I_{1})\in\frac{L_{1}}{C_{L_{1}}(I_{1})}$: $\alpha_{1}([a,c])=[\alpha_{1}(a),d]\ \ c+C_{L_{1}}(I_{1})=b+C_{L_{1}}(I_{1}),d+C_{L_{2}}(I_{2})=\alpha_{2}(b+C_{L_{1}}(I_{1}))$. Note that the monolith of Lie algebras is well-defined. Moreover, by routine calculations, it is easily verified that $(I_{1}\unlhd L_{1})\sim(I_{2}\unlhd L_{2})$ is an equivalence relation. The next Lemma states a sequence of results that can be proved by repeating verbatim proofs cited in the square brackets for similar groups. ###### Lemma 3.2. * • If $I_{1},I_{2}\unlhd L_{1}$, and $I_{1}\cap I_{2}=\\{0\\}$, then $(I_{1}\trianglelefteq L_{1})\sim(\frac{I_{1}+I_{2}}{I_{2}}\trianglelefteq\frac{L_{1}}{I_{2}})$ (Lemma 53.13 in [13]); * • Let $S$ be a subalgebra of $L_{1}+L_{2}$ whose projection into $L_{2}$ is the whole $L_{2}$. If $I\trianglelefteq S$ and $I\subset S\cap L_{2}$, then $I\trianglelefteq L_{2}$, and $(I\trianglelefteq L_{2})\sim(I\trianglelefteq S)$ (Lemma 53.14 in [13]); * • If $I\subset L$ is a sum of minimal ideals and $I_{1}\trianglelefteq L(I_{1}\subset I)$, then there exists $I_{2}\trianglelefteq L(I_{2}\subset I)$ such that $I=I_{1}+I_{2}$ (Lemma 53.15 in [13]). ###### Proposition 3.3. If $I\subset L$ is a minimal ideal contained within a sum of similar minimal ideals $I_{1},I_{2},\cdots,I_{n}\subset L$, then $I$ is similar to all $I_{j}$, $j=1,2,\cdots,n$. ###### Proof. It is sufficient to repeat verbatim the proof of Lemma 53.13 in [13]. ∎ ###### Corollary 3.4. If $I\trianglelefteq L$ is contained within a sum of minimal ideals $I_{1},I_{2},\cdots,I_{n}\subset L$, then $I$ is a direct sum of minimal ideals, each one similar to $I_{1}$ (Lemma 53.17 in [13]). A set $D$ of graded Lie algebras is called factor closed if all factors in $D$ are itself in $D$. Consider $U=var_{G}(D)$, where $D$ is a finite factor closed set of graded finite-dimensional Lie algebras. If $L\in U$ is finite-dimensional (following the proof of Birkhoff’s HSP theorem; Theorem 1, on pages 98 and 99, in [1]), then it is contained in the factor of finite direct products of finite dimensional Lie algebras belonging to $D$. Mathematically, $L$ is represented as: (1) $L=\frac{B}{C},B\subset P=\prod_{i=1}^{n}A_{i},\ A_{1},\cdots,A_{n}\in D.$ $P$ is called a representation of $L$. From now on, we assume that $dimA_{1}\geq dimA_{2}\geq\cdots\geq dimA_{n}$ in the form of Equation (1). The projection of $P$ onto the component $A_{i}$ is denoted by $\pi_{i}$. ###### Definition 3.5. A representation of $L$ is called minimal when the $n$-tuple $\newline (dimA_{1},\cdots,dimA_{n})$ is (left) lexicographically at least possible. ###### Lemma 3.6. In the notation of Equation 1, we have: * • Each Lie algebra $A_{i}$ is critical, and if any Lie algebra $A_{i}$ is replaced by a proper factor, the resulting direct sum has no factor isomorphic to $L$ (Lemma 53.21 in [13]); * • $B=\pi_{1}(B)\times\cdots\times\pi_{n}(B)$ and $\pi_{i}(B)=A_{i}$ for each $i$ (Lemma 53.22 in [13]); * • A subalgebra $D\subset A_{i}$ is an ideal of $A_{i}$ if, and only if, $[B,D]\subset D$. Moreover, every non-trivial ideal of $A_{i}$ intersects $B$ non-trivially (Lemma 53.23 in [13]); * • $C\cap A_{i}=\\{0\\}$ for all $i=1,\cdots,n$ (Lemma 53.24 in [13]). ###### Lemma 3.7. If $L\in\mathcal{B}$ is a graded Lie algebra and $L=\frac{B}{C},B\subset P=\prod_{i=1}^{n}A_{i}$ is a minimal representation of $L$ in $\mathcal{B}$, then, for each $i$, $L$ contains a minimal ideal $W_{i}$ that is similar to the monolith $I_{i}$ in $A_{i}$. ###### Proof. It is sufficient to repeat verbatim the proof of Lemma 53.25 in [13]. ∎ According to Theorem 53.32 in [13] if two critical groups generate the same variety, then their monoliths are similar. For graded Lie algebras, we have the following proposition: ###### Proposition 3.8. If two critical graded Lie algebras $L_{1}$ and $L_{2}$ generate the same variety, then their monoliths are similar. ###### Proof. Let $L_{1}$ and $L_{2}$ be two critical Lie algebras such that $var_{G}(L_{1})=var_{G}(L_{2})$. Notice that $var_{G}(L_{1})$ is a locally finite variety. Thus, $var_{G}(L_{1})$ is generated by its critical algebras. Let us consider $L_{1}=\frac{B}{C}$ and $B\subset P=\prod_{i=1}^{n}B_{i}$, where $P$ is a minimal representation of $L_{1}$ in $var_{G}(L_{2})$. Due to the criticality of $L_{2}$, it is a component of $P$. So, the result follows by Lemma 3.7, because $L_{1}$ is a monolithic graded Lie algebra. ∎ ## 4\. $sl_{2}(F)$ graded by $\mathbb{Z}_{2}$ First, we investigate the $\mathbb{Z}_{2}$-graded identities of $sl_{2}(F)$ when $charF>3$. We denote by $h=e_{11}-e_{22}\in sl_{2}(F)$. ###### Lemma 4.1. The following polynomials are identities of $sl_{2}(F)$: $[y_{1},y_{2}],[z_{1},y_{1}^{q}]=[z_{1},y_{1}]$. ###### Proof. It is clear that $[y_{1},y_{2}]\in V_{\mathbb{Z}_{2}}(sl_{2}(F))$, because the diagonal is commutative. Choose $a_{i}=\lambda_{11,i}e_{11}-\lambda_{11,i}e_{22}$ and $b_{j}=\lambda_{12,j}e_{12}+\lambda_{21,j}e_{21}$, so: $[b_{j},a_{i}^{q}]=\lambda_{11,i}^{q}[b_{j},{h}^{q}]=\lambda_{11,i}^{q}((-2)^{q}\lambda_{12,j}e_{12}+2^{q}\lambda_{21,j}e_{21})=\lambda_{11,i}(-2\lambda_{12,j}e_{12}+2\lambda_{21,j}e_{21})=[b_{j},a_{i}]$. Thus, $[z_{1},y_{1}^{q}]=[z_{1},y_{1}]\in V_{\mathbb{Z}_{2}}(sl_{2}(F))$. The proof is complete. ∎ ###### Lemma 4.2. The polynomials $Sem_{1}(y_{1}+z_{1},y_{2}+z_{2})=(y_{1}+z_{1})f(ad(y_{2}+z_{2})),\ \ f(t)=t^{q^{2}+2}-t^{3}$ $Sem_{2}(y_{1}+z_{1},y_{2}+z_{2})=[y_{1}+z_{1},y_{2}+z_{2}]-[z_{1}+y_{1},z_{2}+y_{2},(z_{1}+y_{1})^{q^{2}-1}]\newline -[z_{1}+y_{1},(z_{2}+y_{2})^{q}]+[z_{1}+y_{1},z_{2}+y_{2},(z_{1}+y_{1})^{q^{2}-1},(z_{2}+y_{2})^{q-1}]$ $+[z_{1}+y_{1},z_{2}+y_{2},((z_{1}+y_{1})^{q^{2}}-(z_{1}+y_{1})),[z_{1}+y_{1},z_{2}+y_{2}]^{q-2},(z_{2}+y_{2})^{q^{2}}-(z_{2}+y_{2})]$ $-[z_{2}+y_{2},([(z_{1}+y_{1})^{q^{2}}-(z_{1}+y_{1}),z_{2}+y_{2}])^{q},((z_{2}+y_{2})^{q^{2}-2}-(z_{2}+y_{2})^{q-2})]$ are graded identities of $sl_{2}(F)$. ###### Proof. It is enough to repeat word for word the proof of Proposition 1 by [18]. ∎ From now on, we denote by $\mathcal{\beta}$ the variety determined by the identities: $Sem_{1}(y_{1}+z_{1},y_{2}+z_{2}),Sem_{2}(y_{1}+z_{1},y_{2}+z_{2}),[y_{1},y_{2}],\mbox{and}\ [z_{1},y_{1}^{q}]=[z_{1},y_{1}]$. ###### Corollary 4.3. The variety $\mathcal{\beta}$ is locally finite. ###### Proof. Recall that every Lie algebra can trivially graded by $\mathbb{Z}_{2}$. Furthermore, if $L=L_{0}\oplus L_{1}=L^{\prime}_{0}\oplus L^{\prime}_{1}$ are two different $\mathbb{Z}_{2}$-gradings on $L$, then $Sem_{1}(y_{1}+z_{1},y_{2}+z_{2})\in V_{\mathbb{Z}_{2}}(L_{0}\oplus L_{1})\Leftrightarrow Sem_{1}(y_{1}+z_{1},y_{2}+z_{2})\in V_{\mathbb{Z}_{2}}(L^{\prime}_{0}\oplus L^{\prime}_{1})$. By Lemma 4.2, we have that $Sem_{1}(y_{1}+z_{1},y_{2}+z_{2})\in V_{\mathbb{Z}_{2}}(\mathcal{\beta})$. Let $\mathcal{B}\supset\mathcal{\beta}$ be the variety of ordinary Lie algebras determined by the identity $Sem_{1}(y_{1}+z_{1},y_{2}+z_{2})$. Thus, since all Lie algebras belong to $\mathcal{B}$ this satisfy an identity of type $(y_{1}+z_{1})f(ad(y_{2}+z_{2}))$ for some polynomial $f(t)\in F[t]$ ($f(0)=0$). So, by Theorem 2.12, it follows that $\mathcal{B}$ is locally finite. Thus, $\mathcal{\beta}$ is locally finite too. ∎ ###### Corollary 4.4. Let $L\in\mathcal{\beta}$ be a finite-dimensional Lie algebra. Then every nilpotent subalgebra of $L$ is abelian. ###### Proof. Let $\mathcal{B}$ be the variety of ordinary Lie algebras determined by the identity $\newline Sem_{2}(y_{1}+z_{1},y_{2}+z_{2})\in V_{\mathbb{Z}_{2}}(\mathcal{\beta})$. Let $L\in\mathcal{B}$ be a finite-dimensional nilpotent Lie algebra. If $dimL\leq q+1$ or $L^{q+1}=\\{0\\}$, it is clear that $L$ is abelian. Suppose the statement is true for $q+2,\cdots,q+(dimL-1)$. By strong induction, we conclude that $\frac{L}{Z(L)}$ is abelian, and $L^{3}=\\{0\\}$. Consequently, $L$ is abelian. The proof is complete. ∎ It is well known that a verbal ideal (and, respectively a graded verbal ideal) over an infinite field is multi-homogeneous. In other words, if $V$ is a verbal ideal (and, respectively a graded verbal ideal) and $f\in V$ ($f\in V_{G}$), then each multi-homogeneous component of $f$ belongs to $V$ ($V_{G}$) as well (see for instance, Theorem 4, on pages 100 and 101). This fact can be weakened, as stated in the next lemma. ###### Lemma 4.5. Let $V_{G}$ be a graded verbal ideal over a field of size $q$. If $f(x_{1},\cdots,x_{n})\in V_{G}$ ($0\leq deg_{x_{1}}f,\cdots,deg_{x_{n}}f<q$), then each multi-homogeneous component of $f$ belong to $V_{G}$ as well. ###### Lemma 4.6. If $L=span_{F}\\{e_{11},e_{12}\\}\subset gl_{2}(F)$, then the $\mathbb{Z}_{2}$-graded identities of $L$ follow from: $[y_{1},y_{2}],[z_{1},z_{2}]$ and $[z_{1},y_{1}^{q}]=[z_{1},y_{1}]$. ###### Proof. It is clear that $L$ satisfies the identities $[y_{1},y_{2}],[z_{1},z_{2}]$ and $[z_{1},y_{1}^{q}]=[z_{1},y_{1}]$. We will prove that the reverse inclusion holds true. Let $\mathcal{\beta}$ be the variety determined by identities $[y_{1},y_{2}],[z_{1},z_{2}]$ and $[z_{1},y_{1}^{q}]=[z_{1},y_{1}]$. Let $f$ be a polynomial identity of $L$. We may write: $f=g+h$, where $h\in V_{\mathbb{Z}_{2}}(\mathcal{\beta})$, and $g(x_{1},\cdots,x_{n})\in V_{\mathbb{Z}_{2}}(L)\supset V_{\mathbb{Z}_{2}}(\mathcal{\beta})$, $0\leq deg_{x_{1}}g,\cdots,deg_{x_{n}}g<q$. In this way, we may suppose that $g$ is a multi-homogeneous polynomial. If $g(y_{1})=\alpha_{1}.y_{1}$ or $g(z_{1})=\alpha_{2}z_{1}$, we can easily see that $\alpha_{1}=\alpha_{2}=0$. In the other case, we may assume that: $g(z_{1},y_{1},\cdots,y_{l})=\alpha_{3}.[z_{1},y_{1}^{a_{1}},\cdots,y_{l}^{a_{l}}],1\leq a_{1},\cdots,a_{l}<q$. However, $g(e_{12},e_{11},\cdots,e_{11})$ is a non-zero multiple scalar of $e_{12}$, and consequently, $\alpha_{3}=0$. So $f=h$ and we are done. ∎ ###### Lemma 4.7. Let $L=L_{0}\oplus L_{1}\in U^{2}\cap\mathcal{\beta}$ be a critical Lie algebra. Then $\newline L\in var_{\mathbb{Z}_{2}}(span_{F}\\{e_{11},e_{12}\\})$. ###### Proof. According to Lemma 4.6, it is sufficient to prove that $L$ satisfies the identity $[z_{1},z_{2}]$. By assumption, $L$ is critical therefore $L$ is monolithic. If $L$ is abelian, then $dimL=1$. So $L\cong span_{F}\\{e_{11}\\}$ or $L\cong span_{F}\\{e_{12}\\}$. In the sequel, we suppose that $L$ is non-abelian. By Corollary 2.16, we have that $[L,L]=Nil(L)=[L_{1},L_{1}]\oplus[L_{0},L_{1}]$. Due to the identity $[z_{1},y_{1}]=[z_{1},y_{1}^{q}]$, $\\{0\\}=[L_{1},[L_{1},L_{1}]]=-[L_{1},L_{1},L_{1}]$. So, by the identity $Sem(z_{1},z_{2})$, we have that $[z_{1},z_{2}]\in V_{\mathbb{Z}_{2}}$, as required. The proof is complete. ∎ ###### Corollary 4.8. $U^{2}\cap var_{\mathbb{Z}_{2}}(sl_{2}(F)),U^{2}\cap\mathcal{\beta}\mbox{and}\ \ var(span_{F}\\{e_{11},e_{12}\\})$ coincide. ###### Proof. First, notice that $U^{2}\cap var_{\mathbb{Z}_{2}}(sl_{2}(F))\subset U^{2}\cap\mathcal{\beta}$ which is a locally finite variety. By Lemma 4.7, all critical algebras of $U^{2}\cap\mathcal{\beta}$ belong to $\newline var_{\mathbb{Z}_{2}}(span_{F}\\{e_{11},e_{12}\\})\subset U^{2}\cap var(sl_{2}(F))$. Therefore, $U^{2}\cap\mathcal{\beta}\subset var_{\mathbb{Z}_{2}}(span_{F}\\{e_{11},e_{12}\\})$ and it follows the result. ∎ ###### Lemma 4.9. Let $L$ be a critical solvable Lie algebra belonging to $\mathcal{\beta}$. Then $L$ is metabelian. ###### Proof. Let $L$ be a critical (non-abelian) solvable Lie algebra belongs to $\mathcal{\beta}$ with monolith $W$. By Proposition 2.17, we have $[L,L]\cap Z(L)=\\{0\\}$. Consequently, $Z(L)=\\{0\\}$ and $[L,Nil(L)]\neq\\{0\\}$. Notice that $Z(C_{L}(Nil(L)))=Nil(L)$. If $Nil(L)=L_{1}$, then $L$ is metabelian. Now, we assume that $(Nil(L))_{1}\varsubsetneq L_{1}$. We assert that $Nil(L)_{0}=\\{0\\}$. Suppose on the contrary that there exists $a\neq 0\in Nil(L)_{0}$. Hence, there exists $b\in L_{1}-Nil(L)_{1}$ such that $[b,a]\neq 0$, because $Z(L)=\\{0\\}$ and $[y_{1},y_{2}]\in V_{\mathbb{Z}_{2}}(L)$. However, $[b,a]=[b,a^{q}]=0$. This is a contradiction. Thus, $[L_{1},Nil(L)]=\\{0\\}$ and consequently $C_{L}(Nil(L))\supset L_{1}$. By Proposition 2.17: $Z(C_{L}(Nil(L)))\cap[C_{L}(Nil(L)),C_{L}(Nil(L))]=\\{0\\}$. On the other hand, $[C_{L}(Nil(L)),C_{L}(Nil(L))]=\\{0\\}$, because $L$ is monolithic. So $L^{(2)}=\\{0\\}$ and we are done. ∎ ###### Corollary 4.10. Let $L$ be a critical non-solvable Lie algebra belonging to $\mathcal{\beta}$. Then $L$ is simple. ###### Proof. According to Proposition 2.17, $L$ has a Levi decomposition, where each Levi subalgebra is a direct sum of simple ideals. By assumption, $L$ is non-solvable ($RadL\varsubsetneq L$ and there exists $n>0$ such that $L^{(n)}=L^{(n+1)}\neq\\{0\\}$) and it is critical (monolithic). Let $W$ be the monolith of $L$. Suppose on the contrary that $RadL\neq\\{0\\}$. Thus, $W\subset RadL\cap[L,L]$, $W$ is an abelian ideal and it is contained in $L^{(n)}$. According to Proposition 2.17, $[L,L]\cap Z(L)=\\{0\\}$, so $[L,W]=W$. Due to the identities $[y_{1},y_{2}],[z_{1},y_{1}]=[z_{1},y_{1}^{q}]$, we have that $W_{0}=\\{0\\}$ and $W_{1}\neq\\{0\\}$. Notice that $\\{[a_{i},a_{j}];a_{i},a_{j}\in L_{0}\cup L_{1}\\}$ spans $[L,L]$ and $[W,[L,L]]=\\{0\\}$. On the other hand, $Z(L^{(n)})\cap L^{(n)}=\\{0\\}$ (by Proposition 2.17). Consequently, $Z(L^{(n)})=\\{0\\}$. However, $\\{0\\}=[W,[L,L]]\supset[W,L^{(n)}]$. Thereby, $Z(L^{(n)})\supset W$. This is a contradiction. Therefore, $L$ is semisimple. In this situation, $L$ is a direct sum of graded simple ideals (by Proposition 2.19). Thus, $L$ is simple and we are done. ∎ ###### Lemma 4.11 (Jacobson’s book ([10]), Theorem 5, pages 40 and 41). Let $L$ be a nilpotent Lie algebra of linear transformations in a finite dimensional vector space $V$ over an arbitrary field $F$. Then we decompose $V=V_{1}\oplus\cdots\oplus V_{n}$ where $V_{i}$ is invariant under $L$ and the minimal polynomial of the restriction of every $A\in L$ to $V_{i}$ is a power of an irreducible polynomial. The next theorem was proved by Drensky in ([5] Lemma, page 991). We prove it again as follows: ###### Theorem 4.12. Let $V$ be a finite dimensional vector space over $F$ and let $A$ be an abelian Lie algebra of the linear transformations $\phi:V\rightarrow V$, where each one has the equality: $\phi^{q}=\phi$. Then, every $\phi\in A$ is diagonalizable. ###### Proof. Let $\phi\in A$. According to Lemma 4.11, $V$ can be decomposed as the direct sum $V=V_{1}\oplus\cdots\oplus V_{n}$, where $\phi(V_{i})\subset V_{i}$ for $i=1,\cdots,n$. Moreover, the minimal polynomial associated with $\phi$ on $V_{i}$ ($m_{\phi_{i}}$) is a power of an irreducible polynomial. Notice that $m_{\phi_{i}}\mid x^{q}-x.$ So, $m_{\phi_{i}}=(x-\alpha_{i})$ for some $\alpha_{i}\in F$. On the other hand, the minimal polynomial of $\phi$ on $V$ ($m_{\phi}$) is $lcm((m_{\phi_{1}}),\cdots,(m_{\phi_{n}}))$. Consequently, $m(\phi)$ splits in $F[x]$ and has distinct roots. Hence, $\phi\in A$ is diagonalizable, and we are done. ∎ ###### Definition 4.13. Let $L$ be a finite dimensional Lie algebra with a diagonalizable operator $T:L\rightarrow L$. We denote by $V(T)$ a basis of $L$ formed by the eigenvectors of $T$. Moreover, we denote by $V(T)_{\lambda}=\\{v\in V(T)\|T(v)=\lambda.v\\}$. If $w\in V(T)$, we denote by $EV(w)$ the eigenvalue associated with $w$. Let $L\in\mathcal{\beta}$ be a finite dimensional Lie algebra. Notice that $ad(L_{0}):L\rightarrow L$ is an abelian subalgebra of linear transformations of $L$. Moreover, $(ada_{0})^{p}=ada_{0}$ for all $a_{0}\in L_{0}$. By Theorem 4.12 and the identity $[y_{1},y_{2}]$, we have the following corollary: ###### Corollary 4.14. If $L\in\mathcal{\beta}$ (finite-dimensional) and $a_{0}\in L_{0}$, then there exists $\newline V(ad(a_{0})_{0})\subset L_{0}\cup L_{1}$. ###### Lemma 4.15. Let $L\in\mathcal{\beta}$ (finite-dimensional) be a simple Lie algebra. If there exists a diagonalizable operator $ada_{0}:L\rightarrow L$ ($a_{0}\in L_{0}$) and $V(ada_{0})\subset L_{0}\cup L_{1}$, then $V(ada_{0})_{0}\cap L_{1}=\\{\\}$. ###### Proof. According to this hypothesis, $L$ is simple and consequently $L=[L,L]=[L_{1},L_{1}]\oplus[L_{0},L_{1}]$. Notice that if $c_{1},c_{2}\in L_{1}\cap(V(ada_{0})-V(ada_{0})_{0})$ and $[c_{1},c_{2}]\neq 0$, then $EV(c_{1})=-EV(c_{2})$. Suppose by contradiction that $L_{1}\cap V(ada_{0})_{0}\supset\\{v\\}$. Let us define $W_{1}=\\{[b_{i},b_{j}]\neq 0\|b_{i},b_{j}\in L_{1}-V(ada_{0})_{0}\cap L_{1}\\}$ and $\newline W_{2}=\\{[b_{i},b_{j}]\neq 0\|b_{i},b_{j}\in L_{1}\cap V(ada_{0})_{0}\\}$. Seeing that $L$ is simple, we have that $\langle a_{0}\rangle=L$. In this way, $L$ is spanned by the following elements: $[a_{0},c_{i_{1}},\cdots,c_{i_{j}}]\neq 0$, where $j\geq 1$ and $c_{i_{1}},\cdots,c_{i_{j}}\in V(ada_{0})$. It is clear that $[a_{0},c_{i_{1}}]=0$ when $c_{i_{1}}\in(L_{1}\cap V(ada_{0})_{0})\cup W_{2}$. Applying an inductive argument, we can prove that: If $j>1,c_{i_{j}}\in(L_{1}\cap V(ada_{0})_{0})\cup W_{2}$, but $c_{i_{1}},\cdots,c_{i_{j-1}}\in W_{1}\cup(L_{1}-V(ada_{0})_{0}\cap L_{1})$, then $[a_{0},c_{i_{1}},\cdots,c_{i_{j}}]=0$. Bearing in mind that $W_{1}\cup W_{2}\cup(L_{1}\cap V)$ spans $L$, it follows that $v\in Z(L)$. This is a contradiction, because $L$ is simple. ∎ ###### Lemma 4.16. Let $L\in\mathcal{\beta}$ be a critical non-solvable algebra, then $L\cong sl_{2}(F)$. ###### Proof. First of all, notice that $dimL_{0}\geq 1$ and $dimL_{1}\geq 2$. Moreover, according to Corollary 4.10 $L$ is simple Let $a_{0}\in L_{0}$. By Corollary 4.14, there exists $V(ada_{0})\subset L_{0}\cup L_{1}$. Let $-\lambda_{1}\leq\cdots\leq-\lambda_{m}<0<\lambda_{m}\leq\cdots\leq\lambda_{1}$ associated with $V(ada_{0})$. Since $L$ is simple, $V(ada_{0})_{0}\cap L_{1}=\\{0\\}$ (Lemma 4.15) and $\bigcup_{i=1}^{m}[V(ada_{0})_{\lambda_{i}},\newline V(ada_{0})_{-\lambda_{i}}]\neq\\{0\\}$. So, $L_{0}=span_{F}\\{ad(a_{0})_{0}\\}=\sum_{i=1}^{m}[V(ada_{0})_{\lambda_{i}},V(ada_{0})_{-\lambda_{i}}]$. Without loss of generality, suppose that $[V(ada_{0})_{\lambda_{1}},V(ada_{0})_{-\lambda_{1}}]\neq\\{0\\}$. We assert that $[V(ada_{0})_{\lambda_{1}},V(ada_{0})_{-\lambda_{1}}]\oplus span_{F}\\{V(ada_{0})_{\lambda_{1}}\\}$ is a subalgebra of $L$. In fact, let $a\in V(ada_{0})_{\lambda_{1}}$ and $b\in[V(ada_{0})_{\lambda_{1}},V(ada_{0})_{-\lambda_{1}}]$. Consider: $[a,b]=\sum_{i=1}^{{m_{2}}}\alpha_{i}b_{i}$. So: $[a,b,a_{0}]=-\sum_{i=1}^{m_{2}}\alpha_{i}.EV(b_{i})b_{i}$. On the other hand, due to Jacobi’s identity: $[a,b,a_{0}]=-EV(b)[a,b]=-EV(b)(\sum_{j=1}^{m_{2}}\alpha_{j}b_{i})$. Hence: $(-EV(b_{j}).\alpha_{j}+EV(b).\alpha_{j})b_{j}=0$. Consequently, if $\alpha_{j}\neq 0$, then $EV(b)=EV(b_{j})$. Similarly: $[V(ada_{0})_{\lambda_{1}},V(ada_{0})_{-\lambda_{1}}]\oplus span_{F}\\{ad(ada_{0})_{-\lambda_{1}}\\}$ is a subalgebra. In this manner: $[V(ada_{0})_{\lambda_{1}},V(ada_{0})_{-\lambda_{1}}]\oplus span_{F}\\{V(ada_{0})_{\lambda_{1}}\\}\oplus span_{F}\\{V(ada_{0})_{-\lambda_{1}}\\}\trianglelefteq L$. Therefore, $L_{0}=span_{F}\\{V(ada_{0})_{0}\\}=[V(ada_{0})_{\lambda_{1}},V(ada_{0})_{-\lambda_{1}}]$ and $\newline L_{1}=V(ada_{0})_{\lambda_{1}}\oplus V(ada_{0})_{-\lambda_{1}}$. On the other hand, $span_{F}\\{V(ada_{0})_{\lambda_{1}}\\}$ is an irreducible $L_{0}$-module, because $L$ is a simple algebra. Moreover, it is not difficult to see that $L_{0}\oplus span_{F}\\{V(ada_{0})_{\lambda_{1}}\\}$ is a monolithic metabelian Lie algebra with monolith $span_{F}\\{V(ada_{0})_{\lambda_{1}}\\}$ when viewed as an ordinary Lie algebra. Notice that $[L_{0}\oplus span_{F}\\{V(ada_{0})_{\lambda_{1}}\\},L_{0}\oplus span_{F}\\{V(ada_{0})_{\lambda_{1}}\\}]=span_{F}\\{V(ada_{0})_{\lambda_{1}}\\}$, and $span_{F}\\{V(ada_{0})_{\lambda_{1}}\\}$ can not be represented by a sum of two ideals strictly contained within it. By Theorem 2.13, $L_{0}\oplus span_{F}\\{V(ada_{0})_{\lambda_{1}}\\}$ is critical when viewed as an ordinary Lie algebra. Thus, it is critical when viewed as a graded algebra as well (Lemma 2.7). Following word for word the proof of Lemma 4.6, we can prove that: $V_{\mathbb{Z}_{2}}(L_{0}\oplus span_{F}\\{V(ada_{0})_{\lambda_{1}}\\})=\langle[y_{1},y_{2}],[z_{1},y_{1}]=[z_{1},y_{1}^{q}],[z_{1},z_{2}]\rangle_{T}$. Consequently, it follows from Proposition 3.8 that $span_{F}\\{V(ada_{0})_{\lambda_{1}}\\}$ is a one-dimensional vector space. Analogously, we have that $dim(span_{F}\\{V(ada_{0})_{-\lambda_{1}}\\})=1$. Therefore, $[V(ada_{0})_{\lambda_{1}},V(ada_{0})_{-\lambda_{1}}]\oplus span_{F}\\{V(ada_{0})_{\lambda_{1}}\\}\oplus span_{F}\\{V(ada_{0})_{-\lambda_{1}}\\}\cong sl_{2}(F)$. ∎ Now, we prove the main theorem of section. ###### Theorem 4.17. Let $F$ be a field of $char(F)>3$ and size $\|F\|=q$. The $\mathbb{Z}_{2}$-graded identities of $sl_{2}(F)$ follow from: $[y_{1},y_{2}],Sem_{1}(y_{1}+z_{1},y_{2}+z_{2}),Sem_{2}(y_{1}+z_{1},y_{2}+z_{2}),[z_{1},y_{1}]=[z_{1},y_{1}^{q}]$. ###### Proof. It is clear that $var_{\mathbb{Z}_{2}}(sl_{2}(F))\subset\mathcal{\beta}$. To prove that the reverse inclusion holds, it is sufficient to prove that all critical algebras of $\mathcal{\beta}$ are also critical algebras of $var_{\mathbb{Z}_{2}}(sl_{2}(F))$. According to Corollary 4.8, $U^{2}\cap var(\mathcal{\beta})=U^{2}\cap(var(sl_{2}(F)))$. By Corollary 4.10 and Lemma 4.16, any critical non-metabelian of $\mathcal{\beta}$ is isomorphic to $sl_{2}(F)$. Therefore, $\mathcal{\beta}\subset var_{\mathbb{Z}_{2}}(sl_{2}(F))$, and we are done. ∎ ## 5\. $sl_{2}(F)$ graded by $\mathbb{Z}_{3}$ In this section, we describe the $\mathbb{Z}_{3}$-graded identities of $sl_{2}(F)$. We denote by $\newline X=\\{x_{1},\cdots,x_{n},\cdots\\},Y=\\{y_{1},\cdots,y_{n},\cdots\\},Z=\\{z_{1},\cdots,z_{n},\cdots\\}$ the variables of $\mathbb{Z}_{3}$-degree $-1,0,1$ respectively. ###### Lemma 5.1. The following polynomials are $\mathbb{Z}_{3}$-graded identities of $sl_{2}(F)$: $[y_{1},y_{2}],[x_{1},y_{1}^{q}]=[x_{1},y_{1}],[z_{1},y_{1}^{q}]=[z_{1},y_{1}]$ (Polynomials I). $Sem_{1}(x_{1}+y_{1}+z_{1},x_{2}+y_{2}+z_{2}),\mbox{and}\ Sem_{2}(x_{1}+y_{1}+z_{1},x_{2}+y_{2}+z_{2})$ (Polynomials II). ###### Proof. Repeating word for word the proof of Lemma 4.1, we can prove that the other Polynomials I are $\mathbb{Z}_{3}$-graded identities of $sl_{2}(F)$ as well. By Proposition 1 of [18], it follows that the Polynomials II are $\mathbb{Z}_{3}$-graded identities of $sl_{2}(F)$. ∎ Let $\mathcal{\beta}_{2}$ be the variety generated by $[x_{1},x_{2}],[z_{1},z_{2}]$, Polynomials I and II. Our aim is to prove that $var_{\mathbb{Z}_{3}}(sl_{2}(F))=\mathcal{\beta}_{2}$. Our strategy is going to be similar for case $sl_{2}(F)$ graded by $\mathbb{Z}_{2}$. ###### Corollary 5.2. The variety $\mathcal{\beta}_{2}$ is locally finite. ###### Proof. Let $\mathcal{B}\subset\mathcal{\beta}_{2}$ the variety of ordinary Lie algebras determined by $Sem_{1}(x_{1}+y_{1}+z_{1},x_{2}+y_{2}+z_{2})\in V_{\mathbb{Z}_{3}}(sl_{2}(F))$. By repeating word for word the proof of Corollary 4.3, we conclude that $\mathcal{B}$ is locally finite. Thus, $\mathcal{\beta}_{2}$ is also locally finite. ∎ ###### Corollary 5.3. Let $L\in\mathcal{\beta}_{2}$ be a finite-dimensional Lie algebra. Then every nilpotent subalgebra of $L$ is abelian. ###### Proof. It is sufficient to repeat word for word the proof of Corollary 4.4. ∎ We now determine the $\mathbb{Z}_{3}$-graded identities of the subalgebras $M_{1}=span_{F}\\{e_{11}-e_{22},e_{12}\\}\subset sl_{2}(F),M_{2}=span_{F}\\{e_{11}-e_{22},e_{21}\\}\subset sl_{2}(F)$, and $M=(M_{1},M_{2})$. ###### Lemma 5.4. The $\mathbb{Z}_{3}$-graded identities of $M_{1},M_{2}$ and $M$ respectively follow from: $z_{1}=0,[y_{1},y_{2}]=0,[x_{1},y_{1}^{q}]=[x_{1},y_{1}]$. $x_{1}=0,[y_{1},y_{2}]=0,[z_{1},y_{1}^{q}]=[z_{1},y_{1}]$. $[z_{1},z_{2}]=[x_{1},x_{2}]=[x_{1},z_{1}]=[y_{1},y_{2}]=0,[x_{1},y_{1}^{q}]=[x_{1},y_{1}],[z_{1},y_{1}^{q}]=[z_{1},y_{1}]$. ###### Proof. It is enough to repeat word for word the proof of Lemma 4.6. ∎ ###### Lemma 5.5. Let $L=\bigoplus_{i\in\mathbb{Z}_{3}}L_{i}$ be a critical metabelian Lie algebra belonging to $\mathcal{\beta}_{2}$. Then $L\in var_{\mathbb{Z}_{3}}(M)$. ###### Proof. By assumption, $L$ is critical, therefore $L$ is monolithic. If $L$ is abelian, then $dimL=1$, so $L\cong span_{F}\\{e_{11}-e_{22}\\}$, or $L\cong span_{F}\\{e_{12}\\}$, or $L\cong span_{F}\\{e_{21}\\}$. In the sequel, we assume that $L$ is not abelian. By Proposition 2.17, $Z(L)\cap[L,L]=\\{0\\}$; so $Z(L)=\\{0\\}$. By Corollary 2.16, we have $[L,L]=Nil(L)=[L_{-1},L_{0}]\oplus[L_{-1},L_{1}]\oplus[L_{0},L_{1}]$. Due to the identities $[y_{1},y_{2}]=0,[x_{1},y_{1}^{q}]=[x_{1},y_{1}],[z_{1},y_{1}^{q}]=[z_{1},y_{1}]$, it follows that $Nil(L)_{0}=[L_{1},L_{-1}]=\\{0\\}$. Consequently, $[x_{1},z_{1}]\in V_{\mathbb{Z}_{3}}(L)\supset V_{\mathbb{Z}_{3}}(M)$, and we are done. ∎ ###### Lemma 5.6. $U^{2}\cap var_{\mathbb{Z}_{3}}(sl_{2}(F)),U^{2}\cap\mathcal{\beta}_{2}$ and $V_{\mathbb{Z}_{3}}(M)$ coincide. ###### Proof. First, notice that $var_{\mathbb{Z}_{3}}(M)\subset U^{2}\cap var_{\mathbb{Z}_{3}}(sl_{2}(F))\subset U^{2}\cap\mathcal{\beta}_{2}$. By Lemma 5.5, if $L=\bigoplus_{i\in\mathbb{Z}_{3}}L_{i}$ is a critical metabelian Lie algebra belonging to $\mathcal{\beta}_{2}$, then $L\in var_{\mathbb{Z}_{3}}(M)$. So $var_{\mathbb{Z}_{3}}(M)=U^{2}\cap\mathcal{\beta}_{2}$, and we are done. ∎ ###### Lemma 5.7. Let $L=\bigoplus_{i\in\mathbb{Z}_{3}}L_{i}$ be a critical solvable Lie algebra belonging to $\mathcal{\beta}_{2}$. Then $L$ is metabelian. ###### Proof. The proof of this Lemma is similar to the demonstration of Lemma 4.9. In this case, we will have $Nil(L)_{0}=\\{0\\}$ and $C_{L}(Nil(L))\supset L_{-1}\cup L_{1}$. ∎ ###### Lemma 5.8. Let $L$ be a (finite-dimensional) graded Lie algebra belonging to $\mathcal{\beta}_{2}$. If $a\in L_{0}$, then $ad(a):L\rightarrow L$ is a diagonalizable operator. ###### Proof. It follows from Polynomials I and Theorem 4.12. ∎ ###### Lemma 5.9. Let $L\in\mathcal{\beta}_{2}$ (finite-dimensional) be a simple Lie algebra. If there exists a diagonalizable operator $ada_{0}:L\rightarrow L$ ($a_{0}\in L_{0}$) and $V(ada_{0})\subset L_{-1}\cup L_{0}\cup L_{1}$, then $V(ada_{0})_{0}\cap(L_{1}\cup L_{-1})=\\{\\}$. ###### Proof. It is sufficient to repeat word for word the proof of Lemma 4.15. ∎ From now on, in this section, we assume that $F$ contains a primitive $3$rd root of one. ###### Lemma 5.10. If $L\in\mathcal{\beta}_{2}$ is a critical non-metabelian, then $L\cong sl_{2}(F)$. ###### Proof. It is sufficient to repeat the proof of Corollary 4.10 and Lemma 4.16. In this case, we will have $V_{\mathbb{Z}_{3}}(M_{1})=V_{\mathbb{Z}_{3}}(L_{0}\oplus span_{F}\\{V(ada_{0})_{\lambda_{1}}\\})$ or $V_{\mathbb{Z}_{3}}(M_{2})=V_{\mathbb{Z}_{3}}(L_{0}\oplus span_{F}\\{V(ada_{0})_{\lambda_{1}}\\})$. ∎ ###### Theorem 5.11. Let $F$ be a field of $charF>3$ and $\|F\|=q$. Then the $\mathbb{Z}_{3}$-graded polynomial identities of $sl_{2}(F)$ follow from: $Sem_{1}(x_{1}+y_{1}+z_{1},x_{2}+y_{2}+z_{2}),Sem_{2}(x_{1}+y_{1}+z_{1},x_{2}+y_{2}+z_{2}),[y_{1},y_{2}],[z_{1},y_{1}^{q}]=[z_{1},y_{1}],[x_{1},y_{1}^{q}]=[x_{1},y_{1}],[x_{1},x_{2}],[z_{1},z_{2}]$. ###### Proof. It is clear that $var_{\mathbb{Z}_{3}}(sl_{2}(F))\subset\mathcal{\beta}_{2}$. According to Lemma 5.6, $U^{2}\cap\mathcal{\beta}_{2}=U^{2}\cap var_{\mathbb{Z}_{3}}(sl_{2}(F))$. On the other hand, by Lemma 5.10, we have that $L\cong sl_{2}(F)$ if $L$ is a critical non-metabelian Lie algebra. Consequently, $\mathcal{\beta}_{2}\subset var_{\mathbb{Z}_{3}}(sl_{2}(F))$ and we are done. ∎ ## 6\. $sl_{2}(F)$ graded by $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ In this section, we denote by $W=\\{w_{1},\cdots,w_{n},\cdots\\},X=\\{x_{1},\cdots,x_{n},\cdots\\},Y=\\{y_{1},\cdots,y_{n},\cdots\\}$, and $Z=\\{z_{1},\cdots,z_{n},\cdots\\}$ the variables of $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-degree $\newline (0,0),(0,1),(1,0),(1,1)$ respectively. ###### Lemma 6.1. The following polynomials are $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-graded identities of $sl_{2}(F)$: $w_{1},[y_{1},y_{2}],[x_{1},y_{1}^{q}]=[x_{1},y_{1}],[z_{1},y_{1}^{q}]=[z_{1},y_{1}]$ (Polynomials I). $Sem_{1}(w_{1}+x_{1}+y_{1}+z_{1},w_{2}+x_{2}+y_{2}+z_{2}),\ \mbox{and}\ Sem_{2}(w_{1}+x_{1}+y_{1}+z_{1},w_{2}+x_{2}+y_{2}+z_{2})$ (Polynomials II). ###### Proof. Repeating word for word the first part of the proof of Lemma 4.1, we can conclude that Polynomials I are $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-graded identities of $sl_{2}(F)$. Lastly, it follows from Proposition 1 by [18] that $Sem_{1},Sem_{2}\in V_{\mathbb{Z}_{2}\times\mathbb{Z}_{2}}(sl_{2}(F))$. ∎ Henceforth, we denote by $\mathcal{\beta}_{3}$ the variety determined by the Polynomials I and II. ###### Corollary 6.2. The variety $\mathcal{\beta}_{3}$ is locally finite ###### Proof. Let $\mathcal{B}\supset\mathcal{\beta}_{3}$ the variety of ordinary Lie algebras determined by $Sem_{1}(w_{1}+x_{1}+y_{1}+z_{1},w_{2}+x_{2}+y_{2}+z_{2})$. By repeating word for word the proof of Corollary 4.3, we have that $\mathcal{B}$ is locally finite. Thus, $\mathcal{\beta}_{3}$ is locally finite as well. ∎ ###### Corollary 6.3. Let $L\in\mathcal{\beta}_{3}$ be a finite-dimensional Lie algebra. Then every nilpotent subalgebra of $L$ is abelian. ###### Proof. It is sufficient to repeat word for word the proof of Corollary 4.4. ∎ The Lie algebra $N=(span_{F}(e_{11}-e_{22}),span_{F}(e_{12}-e_{21}),span_{F}(e_{12}+e_{21}))$ can be graded by $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ as follows: $N_{(1,0)}=(span_{F}\\{e_{11}-e_{22}\\},0,0),\newline N_{(1,1)}=(0,span_{F}\\{e_{12}-e_{21}\\},0)$, and $N_{(0,1)}=(0,0,span_{F}\\{e_{12}+e_{21}\\})$. ###### Lemma 6.4. The $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-graded identities of $N,N_{(1,0)},N_{(0,1)},N_{(1,1)}$ respectively follow from: $w_{1}=0,[a_{1},a_{2}]=0,\ \ a_{1},a_{2}\in X\cup Y\cup Z$. $w_{1}=x_{1}=z_{1}=0,$ and $[y_{1},y_{2}]=0$. $w_{1}=y_{1}=z_{1}=0,$ and $[x_{1},x_{2}]=0$. $w_{1}=y_{1}=x_{1}=0,$ and $[z_{1},z_{2}]=0$. ###### Lemma 6.5. Every critical solvable Lie algebra $L\in\mathcal{\beta}_{3}$ is abelian and consequently: $L\cong N_{(1,0)}$, or $L\cong N_{(0,1)}$ or $L\cong N_{(1,1)}$. ###### Proof. Suppose by contradiction that there exists a critical solvable Lie algebra $L=L_{(0,0)}\oplus L_{(0,1)}\oplus L_{(1,0)}\oplus L_{(1,1)}\in\mathcal{\beta}_{3}$ that is non-abelian. Let $W$ be the abelian monolith of $L$. Then, $Z(L)=\\{0\\}$ and $[L,W]=W$, because $Z(L)\cap[L,L]=\\{0\\}$ (Proposition 2.17). However, by applying Polynomials I, $Sem_{2}(y_{1},z_{1}),Sem_{2}(y_{1},x_{1})$, and $Sem_{3}(x_{1},z_{1})$, we will conclude that $W=\\{0\\}$. This is a contradiction. ∎ ###### Lemma 6.6. $U^{2}\cap var_{\mathbb{Z}_{2}\times\mathbb{Z}_{2}}(sl_{2}(F)),var_{\mathbb{Z}_{2}\times\mathbb{Z}_{2}}(N)$ and $U^{2}\cap\mathcal{\beta}_{2}$ coincide. ###### Proof. It is enough to repeat word for word the proof of Lemma 5.6. ∎ ###### Lemma 6.7. Let $L\in\mathcal{\beta}_{3}$ be a (finite-dimensional) graded Lie algebra. If $a\in L_{(1,0)}$, then $ad(a):L\rightarrow L$ is a diagonalizable operator. ###### Proof. It follows from Theorem 4.12 and the Polynomials I. ∎ Let $L\in\mathcal{\beta}_{3}$ a finite-dimensional Lie algebra and consider $a\in L_{(1,0)}$. Notice that $span_{F}\\{V(ada_{0})_{0}\\}$ is spanned by homogeneous elements of $L$, but if there exists $\lambda=EV(ada_{0})\neq 0$, we have that $span_{F}\\{V(ada_{0})_{\lambda}\\}$ is spanned by elements of type $b_{\lambda}^{(0,1)}+b_{\lambda}^{(1,1)}$, where $b_{\lambda}^{(0,1)}\in L_{(0,1)}$ and $b_{\lambda}^{(1,1)}\in L_{(1,1)}$. Furthermore, notice that $[L_{(0,1)}\oplus L_{(1,1)},L_{(0,1)}\oplus L_{(1,1)}]$. ###### Lemma 6.8. Let $L\in\mathcal{\beta}_{3}$ (finite-dimensional) be a simple Lie algebra. If there exists a diagonalizable operator $ada_{0}:L\rightarrow L$ ($a_{0}\in L_{(1,0)}$) and $V(ada_{0})_{0}\subset L_{(1,0)}\cup(L_{(0,1)}\oplus L_{(1,1)})$, then ${V(ada_{0})}_{0}\cap(L_{(0,1)}\oplus L_{(1,1)})=\\{\\}$. ###### Proof. It is enough to repeat word for word the proof of Lemma 4.15. ∎ ###### Lemma 6.9. If $L\in\mathcal{\beta}_{3}$ is a critical non-solvable Lie algebra, then $L\cong sl_{2}(F)$. ###### Proof. The proof of this Lemma is similar to the proof of Lemma 4.16. In this case, we have that $dimL_{(0,1)},dimL_{(1,0)},dimL_{(1,1)}\geq 1$. Furthermore: $L=span_{F}\\{V(ada_{0})_{-\lambda_{1}}\\}\oplus span_{F}\\{V(ada_{0})_{0}\\}\oplus span_{F}\\{V(ada_{0})_{\lambda_{1}}\\}$ ($a_{0}\in L_{(1,0)}$), and this decomposition is a $\mathbb{Z}$-grading on $L$ ($L_{-1}=span_{F}\\{V(ada_{0})_{-\lambda_{1}}\\},\newline L_{0}=span_{F}\\{V(ada_{0})_{0}\\}$ and $L_{1}=span_{F}\\{V(ada_{0})_{\lambda_{1}}\\}$). On the other hand, we have that $V_{\mathbb{Z}}(M_{1})=V_{\mathbb{Z}}(span_{F}\\{V(ada_{0})_{0}\\}\oplus span_{F}\\{V(ada_{0})_{\lambda_{1}}\\})$ or $V_{\mathbb{Z}}(M_{2})=(span_{F}\\{V(ada_{0})_{0}\\}\oplus span_{F}\\{V(ada_{0})_{\lambda_{1}}\\})$. Thus, $dim(span_{F}\\{V(ada_{0})_{0}\\})=dim(span_{F}\\{V(ada_{0})_{\lambda_{1}}\\})=dim(span_{F}\\{V(ada_{0})_{-\lambda_{1}}\\})=1$ and consequently $dimL=3$. Therefore $L\cong sl_{2}(F)$, and we are done. ∎ ###### Theorem 6.10. Let $F$ be a finite field of $charF>3$ and $\|F\|=q$. The $\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-graded polynomial identities of $sl_{2}(F)$ follow from: $w_{1},[y_{1},y_{2}],[x_{1},y_{1}^{q}]=[x_{1},y_{1}],[z_{1},y_{1}^{q}]=[z_{1},y_{1}]$ (Polynomials I). $Sem_{1}(w_{1}+x_{1}+y_{1}+z_{1},w_{2}+x_{2}+y_{2}+z_{2}),Sem_{2}(w_{1}+x_{1}+y_{1}+z_{1},w_{2}+x_{2}+y_{2}+z_{2})$ (Polynomials II). ###### Proof. It is clear that $var_{\mathbb{Z}_{2}\times\mathbb{Z}_{2}}(sl_{2}(F))\subset\mathcal{\beta}_{3}$. If $L\in\mathcal{\beta}_{3}$ is critical, it follows from Lemmas 6.5 and 6.9 that: $L\cong sl_{2}(F)$, or $L\cong span_{F}\\{e_{11}-e_{22}\\}$, or $L\cong span_{F}\\{e_{12}+e_{21}\\}$ or $L\cong span_{F}\\{e_{12}-e_{21}\\}$. Therefore, all critical algebras belonging to $\mathcal{\beta}_{3}$ are algebras belonging to $var_{\mathbb{Z}_{2}\times\mathbb{Z}_{2}}(sl_{2}(F))$ as well. So $var_{\mathbb{Z}_{2}\times\mathbb{Z}_{2}}(sl_{2}(F))\supset\mathcal{\beta}_{3}$, and we are done. ∎ ## References * [1] Yu. A. Bahturin. 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Finite basing of the identites of a matrix algebra of second order over a field of characteristic zero. Algebra and Logic 12, 43-63 (1973). * [17] D. V. Repin. Structure and identities of some graded Lie agebras. PhD thesis. Ulyanovsky State University (Russian) (2005). * [18] K.N. Semenov. Basis of identities of the algebra $sl_{2}(K)$ over a finite field. Matematicheskie Zametski 52, 114-119 (1992). * [19] G.V. Sheina. Metabelian Varieties of Lie A-algebras. Russian Math Surveys 33, 249-250 (1978). * [20] D.D.P.S. Silva. Identidades Graduadas em Álgebras não-Associativas. PhD thesis. Universidade de Campinas (Portuguese) (2010). * [21] D.A. Towers. Solvable Lie A-algebras. Journal of Algebra 340 (1), 1-12 (2011). * [22] S. Yu. Vasilovsky. Basis of identities of a three-dimensional simple Lie algebra over an infinite field. Algebra and Logic 28 (5), 355-368 (1990). * [23]	arxiv-papers	2013-11-15T16:24:01	2024-09-04T02:49:53.706124	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Lu\\'is Felipe Gon\\c{c}alves Fonseca", "submitter": "Lu\\'is Felipe Gon\\c{c}alves Fonseca Fonseca", "url": "https://arxiv.org/abs/1311.3904" }
1311.3930	11institutetext: University of Sussex, Department of Mathematics, University of Sussex, Brighton, GB-BN1 9QH England UK, [email protected]. 22institutetext: University of Reading, Department of Mathematics and Statistics, Whiteknights, PO Box 220, Reading, GB-RG6 6AX, England UK, [email protected]. # An adaptive finite element method for the infinity Laplacian Omar Lakkis 11 Tristan Pryer 22 ###### Abstract We construct a finite element method (FEM) for the infinity Laplacian. Solutions of this problem are well known to be singular in nature so we have taken the opportunity to conduct an a posteriori analysis of the method deriving residual based estimators to drive an adaptive algorithm. It is numerically shown that optimal convergence rates are regained using the adaptive procedure. ## 1 Introduction Nonlinear partial differential equations (PDEs) arise in many areas. Their numerical simulation is extremely important due to the additional difficulties arising in their classical solution [4]. One such example is that of the _infinity Laplace operator_ $\Delta_{\infty}$ defined by $\Delta_{\infty}u:=\frac{\sum_{i=1}^{d}\sum_{j=1}^{d}\partial_{i}u\partial_{j}u\partial_{{i}{j}}u}{\sum_{i=1}^{d}\\!\left({\partial_{i}u}\right)^{2}}=\frac{{\\!\left({\nabla u\otimes\nabla u}\right)}{:}{\mathrm{D}^{2}u}}{\left\|\nabla u\right\|^{2}},$ (1) for a twice-differentiable function $u:\Omega\to\mathbb{R}$, $\Omega\in\mathbb{R}^{d}$ open, bounded and connected, where $\nabla u:=\begin{bmatrix}\partial_{1}u\\\ \vdots\\\ \partial_{d}u\end{bmatrix},\quad\boldsymbol{x}\otimes\boldsymbol{y}:=\boldsymbol{x}{\boldsymbol{y}}^{{\boldsymbol{\intercal}}},\text{ and }{\boldsymbol{X}}{:}{\boldsymbol{Y}}:=\operatorname{trace}{{\boldsymbol{X}}^{{\boldsymbol{\intercal}}}\boldsymbol{Y}}$ (2) denote, respectively, the gradient, the (algebraic) tensor product of $\boldsymbol{x},\boldsymbol{y}\in\mathbb{R}^{d}$, and the Frobenius inner product of two matrices $\boldsymbol{X},\boldsymbol{Y}\in\mathbb{R}^{d\times d}$. This equation has been popular in classical studies [1, 3, e.g.] but is difficult to pose numerical schemes due to its nondivergence structure and general lack of classical solvability. The infinity Laplacian, which is in fact a misnomer (_homogeneous infinity Laplacian_ is more precise), occurs as the weighted formal limit of a variational problem. A more appropriate terminology would be that of _infinite harmonic_ function $u$ being one that solves $\Delta_{\infty}u=0$. This is justified, at least heuristically, as being the formal limit of the $p$-harmonic functions, $u_{p}$, $p\geq 1$, $p\to\infty$ where $\begin{split}0=\Delta_{p}u_{p}:=\operatorname{div}\\!\left({\left\|\nabla u_{p}\right\|^{p-2}\nabla u_{p}}\right)=\left\|\nabla u_{p}\right\|^{p-2}\Delta u_{p}+\\!\left({p-2}\right)\left\|\nabla u_{p}\right\|^{p-2}\Delta_{\infty}u_{p}.\end{split}$ (3) Multiplying by $\left\|\nabla u_{p}\right\|^{2-p}/(p-2)$ and taking the limit as $p\to\infty$ it follows that a would be limit $u=\lim_{p\to\infty}u_{p}$ is infinite harmonic. A rigorous treatment is provided in [5] and is based on the variational observation that the Dirichlet problem for the $p$–Laplacian is the Euler–Lagrange equation of the following _energy_ functional $\mathscr{L}_{p}[u]:=\frac{1}{p}\left\\|u\right\\|_{\operatorname{L}_{p}(\Omega)}^{p}=\int_{\Omega}\frac{1}{p}\left\|\nabla u\right\|^{p}\text{ for }p\in[1,\infty)$ (4) with appropriate Dirichlet boundary conditions. By analogy, setting $\mathscr{L}_{\infty}[u]:=\left\\|\nabla u\right\\|_{\operatorname{L}_{\infty}(\Omega)}=\operatorname{ess\,sup}_{\Omega}\left\|\nabla u\right\|,$ (5) we seek $u\in\operatorname{Lip}(\Omega)=\operatorname{W}^{1}_{\infty}(\Omega)$, the space of Lipschitz continuous functions over $\Omega$ (Rademacher), with $u=g$ on $\partial\Omega$ such that $\mathscr{L}_{\infty}[u]\leq\mathscr{L}_{\infty}[v]\quad\>\forall\>v\in\operatorname{Lip}(\Omega)\text{ and }v=g\text{ on }\partial\Omega.$ (6) Show that the solution exists and define it to be infinite harmonic. Such a solution is called _absolutely minimising Lipschitz extension of $g$_, we call it infinite harmonic. The infinity Laplacian is thus considered to be the paradigm of a variational problem in $\operatorname{W}^{1}_{\infty}(\Omega)$. If the solution is smooth, say in $\operatorname{C}^{2}$ and has no internal extrema, it can be shown to satisfy (3) classically. But an infinite harmonic function is generally not a classical solution (those in $\operatorname{C}^{2}(\Omega)$ satisfying (1) everywhere. Therefore solutions of (3) must be sought in a weaker sense. The notion of viscosity solution, introduced for second order PDEs in [6] turns out to be the correct setting to seek weaker solutions. Existence and uniqueness of a viscosity solution to the homogeneous infinity Laplacian (1) has been studied [11]. If the domain $\Omega$ is bounded, open and connected then (1) has a unique viscosity solution $u\in\operatorname{C}^{0}(\overline{\Omega})$. In the case $\Omega\subset\mathbb{R}^{2}$ this can be improved to $u\in\operatorname{C}^{1,\alpha}(\overline{\Omega})$ [9]. A study of existence and uniqueness of viscosity solutions to the inhomogeneous infinity Laplacian can be found in [13]. With $\Omega$ defined as before and in addition if $f\in\operatorname{C}^{0}(\Omega)$ and does not change sign, i.e., $\inf_{\Omega}f>0$ or $\sup_{\Omega}f<0$, one can find a unique viscosity solution. As to the topic of numerical methods to approximate the infinity Laplacian, to the authors knowledge only two methods exist. The first is based on finite differences [14]. The scheme involves constructing monotone sequences of schemes over concurrent lattices by minimising the discrete Lipschitz constant over each node of the lattice. The second is a finite element scheme named the vanishing moment method [10] in which the 2nd order nonlinear PDE is approximated via sequences of biharmonic quasilinear 4th order PDEs. In this paper we present a finite element method for the infinity Laplacian, without having to deal with the added complications of approximating a 4th order operator. It is based on the non-variational finite element method introduced in [12]. Roughly, this method involves representing the _finite element Hessian_ (see Definition 2.5) as an auxiliary variable in the formulation, to deal with the nonvariational structure. We also consider the problem as the steady state of an evolution equation making use of a _Laplacian relaxation_ technique (see Remark 2.1) [2, 8] to circumvent the degeneracy of the problem. The structure of the paper is as follows: In §2 we examine the linearisation of the PDE and present the necessary framework for the discretisation and state an a posteriori error indicator for the discrete problem. The estimator is of residual type and is used to drive an adaptive algorithm which is studied and used for numerical experimentation is §3. We choose our simulations in such a way that they can be compared with those given in [14, 10]. ## 2 Notation, linearisation and discretisation We consider the inhomogeneous Infinity Laplace problem with Dirichlet boundary conditions on a domain $\Omega\subset\mathbb{R}^{d}$. $\displaystyle\Delta_{\infty}u=f\quad\text{ in }\Omega\quad\text{ and }\quad u=g\quad\text{ on }\partial\Omega$ (7) with problem data $f,g\in\operatorname{C}^{0}(\Omega)$ chosen such that $f$ does not change sign throughout $\Omega$. In this case there exists a unique viscosity solution to (7) [13]. ### 2.0.1 Linearisation of the continuous problem (1) The application of a standard fixed point linearisation to (7) results in the following sequence of linear non-divergence PDEs: Given an initial guess $u^{0}$, for each $n\in\mathbb{N}$ find $u^{n+1}$ such that ${\frac{\\!\left({\nabla u^{n}\otimes\nabla u^{n}}\right)}{\left\|\nabla u^{n}\right\|^{2}}}{:}{\mathrm{D}^{2}u^{n+1}}=f.$ (8) Due to the degeneracy of the problem we introduce a slightly modified problem which utilises _Laplacian relaxation_ [2, 8], the problem is to find $u^{n+1}$ such that ${\\!\left({\frac{\nabla u^{n}\otimes\nabla u^{n}}{\left\|\nabla u^{n}\right\|^{2}}+\frac{\boldsymbol{I}}{\tau}}\right)}{:}{\mathrm{D}^{2}u^{n+1}}=f+\frac{\Delta u^{n}}{\tau}$ (9) where $\tau\in\mathbb{R}^{+}$. ###### Remark 2.1. The discretisation proposed in (9) is nothing but an implicit one stage discretisation of the following evolution equation $\partial_{t}{\\!\left({\Delta u}\right)}+\Delta_{\infty}u=f,$ (10) where $\Delta u$ is used as shorthand for $\Delta_{2}u$, the 2–Laplacian. With that in mind we must take care with our choice of $\tau$ which can be regarded as a timestep. We require a $\tau$ that is large enough to guarantee reaching the steady state and small enough such that we do not encounter stability problems. ### 2.0.2 Discretisation of the sequence of linear PDEs (9) Let $\mathscr{T}$ be a conforming, shape regular triangulation of $\Omega$, namely, $\mathscr{T}$ is a finite family of sets such that 1. 1. $K\in\mathscr{T}$ implies $K$ is an open simplex (segment for $d=1$, triangle for $d=2$, tetrahedron for $d=3$), 2. 2. for any $K,J\in\mathscr{T}$ we have that $\overline{K}\cap\overline{J}$ is a full sub-simplex (i.e., it is either $\emptyset$, a vertex, an edge, a face, or the whole of $\overline{K}$ and $\overline{J}$) of both $\overline{K}$ and $\overline{J}$ and 3. 3. $\bigcup_{K\in\mathscr{T}}\overline{K}=\overline{\Omega}$. We also define $\mathscr{E}$ to be the skeleton of the triangulation, that is the set of sub-simplexes of $\mathscr{T}$ contained in $\Omega$ but not $\partial\Omega$. For $d=2$, for example, $\mathscr{E}$ would consist of the set of edges of $\mathscr{T}$ not on the boundary. We also use the convention where $h(\boldsymbol{x}):=\max_{\overline{K}\ni\boldsymbol{x}}h_{K}$ to be the mesh-size function of $\mathscr{T}$. ###### Definition 2.2 (continuous and discontinuous FE spaces). Let $\mathbb{P}^{k}(\mathscr{T})$ denote the space of piecewise polynomials of degree $k$ over the triangulation $\mathscr{T}$ of $\Omega$. We introduce the _finite element spaces_ $\displaystyle\mathbb{V}_{D}\\!\left({k}\right)=\mathbb{P}^{k}(\mathscr{T})\qquad\mathbb{V}_{C}\\!\left({k}\right)=\mathbb{P}^{k}(\mathscr{T})\cap\operatorname{C}^{0}(\Omega)$ (11) to be the usual spaces of discontinuous and continuous piecewise polynomial functions over $\Omega$. ###### Remark 2.3 (generalised Hessian). Given a function $v\in\operatorname{H}^{1}(\Omega)$ and let $\boldsymbol{n}:\partial\Omega\to\mathbb{R}^{d}$ be the outward pointing normal of $\Omega$ then the _generalised Hessian_ of $v$, $\mathrm{D}^{2}v$ satisfies the following identity: $\left\langle\mathrm{D}^{2}v\,\|\,\phi\right\rangle=-\int_{\Omega}{\nabla v}\otimes{\nabla\phi}+\int_{\partial\Omega}{\nabla v}\otimes{\boldsymbol{n}\ \phi}\quad\>\forall\>\phi\in\operatorname{H}^{1}(\Omega),$ (12) where the final term is understood as a duality pairing between $\operatorname{H}^{-1/2}(\partial\Omega)\times\operatorname{H}^{1/2}(\partial\Omega)$. ###### Remark 2.4 (nonconforming generalised Hessian). The test functions applied to define the generalised Hessian in Remark 2.3 need not be $\operatorname{H}^{1}(\Omega)$. Suppose they are $\operatorname{H}^{1}(K)$ for each $K\in\mathscr{T}$ then it is clear that $\begin{split}\left\langle\mathrm{D}^{2}v\,\|\,\phi\right\rangle&=\sum_{K\in\mathscr{T}}\\!\left({-\int_{K}{\nabla v}\otimes{\nabla\phi}+\int_{\partial K}{\nabla v}\otimes{\boldsymbol{n}_{K}\phi}}\right)\\\ &=\sum_{K\in\mathscr{T}}-\int_{K}{\nabla v}\otimes{\nabla\phi}+\sum_{e\in\mathscr{E}}\int_{e}{\mathrel{\ooalign{\cr\kern 1.0pt$\\{$\cr\kern-0.5pt$\\{$}}\nabla v\mathrel{\ooalign{$\\}$\cr\kern-1.5pt$\\}$\cr\kern-1.0pt}}}\otimes{\left\llbracket\phi\right\rrbracket}+\sum_{e\in\partial\Omega}\int_{e}{\nabla v}\otimes{\boldsymbol{n}\ \phi},\end{split}$ (13) where $\left\llbracket\cdot\right\rrbracket$ and $\mathrel{\ooalign{\cr\kern 1.0pt$\\{$\cr\kern-0.5pt$\\{$}}\cdot\mathrel{\ooalign{$\\}$\cr\kern-1.5pt$\\}$\cr\kern-1.0pt}}$ denote the _jump_ and _average_ , respectively, over an element edge, that is, suppose $e$ is a $\\!\left({d-1}\right)$ subsimplex shared by two elements $K^{+}$ and $K^{-}$ with outward pointing normals $\boldsymbol{n}^{+}$ and $\boldsymbol{n}^{-}$ respectively, then $\left\llbracket\boldsymbol{\eta}\right\rrbracket={\eta\big{\|}_{K^{+}}}\boldsymbol{n}^{+}+{\eta\big{\|}_{K^{-}}}\boldsymbol{n}^{-}\text{ and }\mathrel{\ooalign{\cr\kern 1.0pt$\\{$\cr\kern-0.5pt$\\{$}}\boldsymbol{\xi}\mathrel{\ooalign{$\\}$\cr\kern-1.5pt$\\}$\cr\kern-1.0pt}}=\frac{1}{2}\\!\left({\boldsymbol{\xi}\big{\|}_{K^{+}}+\boldsymbol{\xi}\big{\|}_{K^{-}}}\right).$ (14) ###### Definition 2.5 (finite element Hessian). From Remark 2.3 and Remark 2.4 for $V\in\mathbb{V}_{C}\\!\left({k}\right)$ we define the _finite element Hessian_ , $\boldsymbol{H}[V]\in\\!\left[{\mathbb{V}_{D}\\!\left({k}\right)}\right]^{d\times d}$ such that we have $\int_{\Omega}{\boldsymbol{H}[V]}{\phi}=\left\langle\mathrm{D}^{2}V\,\|\,\phi\right\rangle\quad\>\forall\>\phi\in\mathbb{V}_{D}\\!\left({k}\right).$ (15) We discretise (9) utilising the non-variational Galerkin procedure proposed in [12]. We construct finite element spaces $\mathbb{V}:=\mathbb{V}_{C}\\!\left({k}\right)$ and $\mathbb{W}$ which can be taken as $\mathbb{V}_{C}\\!\left({k}\right)$, $\mathbb{V}_{D}\\!\left({k}\right)$ or $\mathbb{V}_{D}\\!\left({k-1}\right)$. Then given ${U^{0}}=\varLambda u^{0}$, for each $n\in\mathbb{N}_{0}$ we seek $\\!\left({U^{n+1},\boldsymbol{H}[U^{n+1}]}\right)\in\mathbb{V}\times\\!\left[{\mathbb{W}}\right]^{d\times d}$ such that $\begin{split}&\int_{\Omega}{{\\!\left({\frac{\nabla U^{n}\otimes\nabla U^{n}}{\left\|\nabla U^{n}\right\|^{2}}+\frac{\boldsymbol{I}}{\tau}}\right)}{:}{\boldsymbol{H}[U^{n+1}]}}{\Psi}=\int_{\Omega}\\!\left({f+\frac{\operatorname{trace}{\boldsymbol{H}[U^{n}]}}{\tau}}\right){\Psi}\\\ &\int_{\Omega}{\boldsymbol{H}[U^{n+1}]}{\Phi}=-\int_{\Omega}{\nabla U^{n+1}}\otimes{\nabla\Phi}+\sum_{e\in\mathscr{E}}\int_{e}{\mathrel{\ooalign{\cr\kern 1.0pt$\\{$\cr\kern-0.5pt$\\{$}}\nabla U^{n+1}\mathrel{\ooalign{$\\}$\cr\kern-1.5pt$\\}$\cr\kern-1.0pt}}}\otimes{\left\llbracket\Phi\right\rrbracket}\\\ &\qquad\qquad\qquad\qquad\qquad\qquad+\sum_{e\in\partial\Omega}\int_{e}{\nabla U^{n+1}}\otimes{\boldsymbol{n}\ \Phi}\quad\>\forall\>\\!\left({\Psi,\Phi}\right)\in\mathbb{V}\times\mathbb{W}.\end{split}$ (16) ###### Remark 2.6 (computational efficiency). Making use of a $\mathbb{V}_{D}\\!\left({k}\right)$ or $\mathbb{V}_{D}\\!\left({k-1}\right)$ space to represent the finite element Hessian allows us to construct a much faster algorithm in comparison to using a $\mathbb{V}_{C}{k}$ space for $\mathbb{W}$ due to the local representation of the $\operatorname{L}_{2}(\Omega)$ projection of discontinuous spaces [7]. ###### Theorem 2.7 (a posteriori residual upper error bound). Let $u$ be the solution to the infinity Laplacian (7) and $U^{n}$ be the $n$-th step in the linearisation defined by (LABEL:eq:2-0-NVFEM). Let $\boldsymbol{A}[v]:=\frac{\nabla v\otimes\nabla v}{\left\|\nabla v\right\|^{2}}+\frac{\boldsymbol{I}}{\tau},$ (17) then there exists a $C>0$ such that $\begin{split}\left\\|f+\frac{\Delta U^{n}}{\tau}-{\boldsymbol{A}[U^{n}]}{:}{\mathrm{D}^{2}U^{n+1}}\right\\|_{\operatorname{H}^{-1}(\Omega)}&\leq C\bigg{(}\sum_{K\in\mathscr{T}}h_{K}\left\\|\mathcal{R}[U^{n},U^{n+1},f]\right\\|_{\operatorname{L}_{2}(K)}\\\ &\qquad\qquad+\sum_{e\in\mathscr{E}}h_{K}^{1/2}\left\\|\mathcal{J}[U^{n},U^{n+1}]\right\\|_{\operatorname{L}_{2}(e)}\bigg{)}\end{split}$ (18) where the interior residual, $\mathcal{R}[U,\boldsymbol{A},f]$, over a simplex $K$ and jump residual, $\mathcal{J}[U,\boldsymbol{A}]$, over a common wall $e=\overline{K}^{+}\cap\overline{K}^{-}$ of two simplexes, $K^{+}$ and $K^{-}$ are defined as $\displaystyle\left\\|\mathcal{R}[U^{n},U^{n+1},f]\right\\|^{2}_{\operatorname{L}_{2}(K)}=\int_{K}\\!\left({{f-{\boldsymbol{A}[U^{n}]}{:}{\mathrm{D}^{2}U^{n+1}}}+\frac{\Delta U^{n}}{\tau}}\right)^{2},$ (19) $\displaystyle\left\\|\mathcal{J}[U^{n},U^{n+1}]\right\\|^{2}_{\operatorname{L}_{2}(e)}=\int_{e}\\!\left({\frac{\left\llbracket\nabla U^{n}\right\rrbracket}{\tau}-{\boldsymbol{A}[U^{n}]}{:}{\left\llbracket\nabla U^{n+1}\otimes\right\rrbracket}}\right)^{2},$ (20) with $\left\llbracket\boldsymbol{\xi}\otimes\right\rrbracket:=\boldsymbol{\xi}\|_{K^{+}}\otimes\boldsymbol{n}^{+}+\boldsymbol{\xi}\|_{K^{-}}\otimes\boldsymbol{n}^{-},$ (21) being defined as a _tensor jump_. ## 3 Numerical experiments All of the numerical experiments in this section are implemented using FEniCS and visualised with ParaView. Each of the tests are on the domain $\Omega=[-1,1]^{2}$, choosing the finite element spaces $\mathbb{V}=\mathbb{V}_{C}\\!\left({1}\right)$ and $\mathbb{W}=\mathbb{V}_{D}\\!\left({0}\right)$. This is computationally the quickest implementation of the non-variational finite element method and the lowest order stable pair of FE spaces for this class of problem. ### 3.0.1 Benchmarking and convergence – Classical solution To benchmark the numerical algorithm we choose the data $f$ and $g$ such that the solution is known and classical. In the first instance we choose $f\equiv 2$ and $g=\left\|\boldsymbol{x}\right\|^{2}$. It is easily verified that the exact solution is given by $u=\left\|\boldsymbol{x}\right\|^{2}$. Figure 1 details a numerical experiment on this problem. Figure 1: We benchmark the approximation of a classical solution to the inhomogeneous infinity Laplacian, plotting the log of the error together with its estimated order of convergence. We examine both $\operatorname{L}_{2}(\Omega)$ and $\operatorname{H}^{1}(\Omega)$ norms of the error together with the residual estimator given in Theorem 2.7. The linearisation tolerance is coupled to the mesh-size such that the linearisation is run until $\left\\|U^{n}-U^{n-1}\right\\|\leq 10h^{2}$. The convergence rates are optimal, that is, $\left\\|u-U^{N}\right\\|=\operatorname{O}(h^{2})$ and $\left\|u-U^{N}\right\|_{1}=\operatorname{O}(h)$. (a) Convergence rates (b) Finite element approximation ###### Remark 3.1 (on the value of $\tau$). The optimal values of the _timestep parameter_ or tuning parameter $\tau$ depend upon the regularity of the solution. For example, for a classical solution, one may choose $\tau$ large. In the numerical experiment above we took $\tau=1000$. Since the linearisation is nothing more than seeking the steady state of the evolution equation (9). The convergence (in $n$) is extremely quick taking no more than five iterations. For the examples below one must be careful choosing $\tau$, we will be looking at viscosity solutions that are not $\operatorname{C}^{2}(\Omega)$, in this case the lack of regularity of the solution will lead to an unstable linearisation for large $\tau$. In each of the cases below $\tau\in[1:10]$ was sufficiently small to achieve convergence of the linearisation in at most twenty iterations. ### 3.0.2 A known viscosity solution to the homogeneous problem To test the convergence of the method applied to a singular solution of the homogeneous problem we fix $\displaystyle f\equiv 0\text{ and }g=\left\|x\right\|^{4/3}-\left\|y\right\|^{4/3},$ (22) where $\boldsymbol{x}={\\!\left({x,y}\right)}^{{\boldsymbol{\intercal}}}$. A known viscosity solution of this equation is the Aronsson solution [1], $u(\boldsymbol{x})=\left\|x\right\|^{4/3}-\left\|y\right\|^{4/3}.$ (23) The function has singular derivatives about the coordinate axis, in fact $u\in\operatorname{C}^{1,1/3}(\Omega)$. Figure 2 details a numerical experiment on this problem. In Figure 3 we conduct an adaptive experiment based on the newest vertex bisection method. Figure 2: We benchmark problem (22), plotting the log of the error together with its estimated order of convergence. We examine both $\operatorname{L}_{2}(\Omega)$ and $\operatorname{H}^{1}(\Omega)$ norms of the error together with the residual estimator given in Theorem 2.7. We choose $\tau=1$ and the linearisation tolerance is coupled to the mesh-size such that the linearisation is run until $\left\\|U^{n}-U^{n-1}\right\\|\leq 10h^{2}$. The convergence rates are suboptimal due to the singularity, that is, $\left\\|u-U^{N}\right\\|\approx\operatorname{O}(h^{1.8})$ and $\left\|u-U^{N}\right\|_{1}\approx\operatorname{O}(h^{0.8})$. (a) Convergence rates (b) Finite element approximation Figure 3: This is an adaptive approximation of the viscosity solution $u=\left\|x\right\|^{4/3}-\left\|y\right\|^{4/3}$ from (22). The estimator tolerance was set at $0.1$ to coincide with the final estimate from the benchmark solution from Figure 2. The final number of degrees of freedom was $36,325$ compared to the uniform scheme which took $165,125$ degrees of freedom to reach the same tolerance. We chose $\tau=0.1$ as the timestep parameter. (a) The finite element approximation viewed from the top. (b) The underlying mesh ## References * [1] G. Aronsson, Construction of singular solutions to the $p$-harmonic equation and its limit equation for $p=\infty$, Manuscripta Math. 56:2 (1986), 135–158. * [2] G. Awanou, Pseudo time continuation and time marching methods for Monge–Ampère type equations, In revision - tech report available on http://www.math.niu.edu/$\sim$awanou/ (2012). * [3] E. N. Barron, L. C. Evans, and R. Jensen, The infinity Laplacian, Aronsson’s equation and their generalizations, Trans. Amer. Math. Soc. 360:1 (2008), 77–101. * [4] L. A. Caffarelli and X. Cabré, Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, American Mathematical Society, Providence, RI, 1995. * [5] M. G. Crandall, L. C. Evans, and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian, Calc. Var. Partial Differential Equations 13:2 (2001), 123–139. * [6] M. G. Crandall, H. Ishii, and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.) 27:1 (1992), 1–67. * [7] A. Dedner and T. Pryer, Discontinuous Galerkin methods for nonvariational problems, Submitted - tech report available on ArXiV http://arxiv.org/abs/1304.2265 (2013). * [8] S. Esedoglu and A. M. Oberman, Fast semi-implicit solvers for the infinity laplace and p-laplace equations, Arxiv (2011). * [9] L. C. Evans and O. Savin, $C^{1,\alpha}$ regularity for infinity harmonic functions in two dimensions, Calc. Var. Partial Differential Equations 32:3 (2008), 325–347. * [10] X. Feng and M. Neilan, Vanishing moment method and moment solutions for fully nonlinear second order partial differential equations, J. Sci. Comput. 38:1 (2009), 74–98. * [11] R. Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient, Arch. Rational Mech. Anal. 123:1 (1993), 51–74. * [12] O. Lakkis and T. Pryer, A finite element method for second order nonvariational elliptic problems, SIAM J. Sci. Comput. 33:2 (2011), 786–801. * [13] G. Lu and P. Wang, Inhomogeneous infinity Laplace equation, Adv. Math. 217:4 (2008), 1838–1868. * [14] A. M. Oberman, A convergent difference scheme for the infinity Laplacian: construction of absolutely minimizing Lipschitz extensions, Math. Comp. 74:251 (2005), 1217–1230 (electronic).	arxiv-papers	2013-11-15T17:27:29	2024-09-04T02:49:53.716499	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Omar Lakkis and Tristan Pryer", "submitter": "Omar Lakkis", "url": "https://arxiv.org/abs/1311.3930" }
1311.4098	# Dark Energy from holographic theories with hyperscaling violation Mariano Cadoni1,2 and Matteo Ciulu1 1 Dipartimento di Fisica, Università di Cagliari. 2 INFN, Sezione di Cagliari. Cittadella Universitaria, 09042 Monserrato, Italy. ###### Abstract We show that analytical continuation maps scalar solitonic solutions of Einstein-scalar gravity, interpolating between an hyperscaling violating and an Anti de Sitter (AdS) region, in flat FLRW cosmological solutions sourced by a scalar field. We generate in this way exact FLRW solutions that can be used to model cosmological evolution driven by dark energy (a quintessence field) and usual matter. In absence of matter, the flow from the hyperscaling violating regime to the conformal AdS fixed point in holographic models corresponds to cosmological evolution from power-law expansion at early cosmic times to a de Sitter (dS) stable fixed point at late times. In presence of matter, we have a scaling regime at early times, followed by an intermediate regime in which dark energy tracks matter. At late times the solution exits the scaling regime with a sharp transition to a dS spacetime. The phase transition between hyperscaling violation and conformal fixed point observed in holographic gravity has a cosmological counterpart in the transition between a scaling era and a dS era dominated by the energy of the vacuum. ###### Contents 1. I Introduction 2. II Dark energy, holographic theories and hyperscaling violation 3. III Exact cosmological solutions 4. IV Dark energy models 5. V Coupling to matter 1. V.1 Power-law evolution 2. V.2 Exponential evolution 3. V.3 Intermediate regime 6. VI Conclusions ## I Introduction Triggered by the anti-de Sitter/Conformal field theory (AdS/CFT) correspondence, recently we have seen several application of the holographic principle aimed to describe the strongly coupled regime of quantum field theory (QFT) Hartnoll et al. (2008a, b); Horowitz and Roberts (2008); Charmousis et al. (2009); Cadoni et al. (2010); Goldstein et al. (2010); Gouteraux and Kiritsis (2012). The most interesting example of these applications is represented by the holographic description of quantum phase transitions, such as those leading to critical superconductivity and hyperscaling violation Hartnoll et al. (2008a, b); Charmousis et al. (2009); Gubser and Rocha (2010); Cadoni et al. (2010); Goldstein et al. (2010); Dong et al. (2012); Cadoni and Pani (2011); Huijse et al. (2012); Cadoni and Mignemi (2012); Cadoni and Serra (2012); Narayan (2012); Cadoni et al. (2013). A general question that can be asked in this context is if these recent advances can be used to improve our understanding, not only of some holographic, strongly coupled dual QFT, but also of the gravitational interaction itself. After all the holographic principle in general and the AdS/CFT correspondence in particular, have been often used in this reversed direction. The most important example is without doubt the understanding of the statistical entropy of black holes by counting states in a dual CFT Strominger and Vafa (1996); Strominger (1998); Cadoni and Mignemi (1999). A challenge for any theory of gravity is surely cosmology and in particular the understanding of the present accelerated expansion of the universe and the related dark energy hypothesis Peebles and Ratra (2003); Padmanabhan (2003). It is not a priori self-evident that the recent developments on the holographic side may be useful for cosmology McFadden and Skenderis (2010). However, closer scrutiny reveals that key concepts used in the holographic description can be also used in cosmology. First of all the symmetries of the gravitational background. The AdS and de Sitter (dS) spacetime in $d$-dimensions share the same isometry group (the conformal group in $d-1$ dimensions). This fact has been the main motivation for the formulation of the dS/CFT correspondence Strominger (2001). Although this correspondence is problematic Goheer et al. (2003), it may be very useful to relate different gravitational backgrounds if one sees dS/CFT as analytical continuation $r\leftrightarrow it$ of AdS/CFT Cadoni and Carta (2004). Second, a domain wall/cosmology correspondence has been proposed Skenderis and Townsend (2006); Skenderis et al. (2007); Shaghoulian (2013). For every supersymmetric domain-wall, which is solution of some supergravity (SUGRA) model, there is a corresponding flat Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmology (which can be obtained by analytical continuation), of the same model but with opposite sign potential. This means that, although cosmologies in general cannot be supersymmetric they may allow for the existence of pseudo-Killing spinors. Third, the spacelike radial coordinate $r$ of a static asymptotically AdS geometry can be interpreted as an energy scale and the corresponding dynamics as a renormalization group (RG) flow. This flow drives the dual QFT from an ultraviolet (UV) conformal fixed point (corresponding to the AdS geometry) to some nontrivial near-horizon, infrared (IR) point where only some scaling symmetries are preserved (for instance one can have hyperscaling violation in the IR Cadoni et al. (2013)). By means of the analytic continuation the RG flow becomes the cosmological dynamics of a time-dependent gravitational background, driving the universe from a early time regime (corresponding to the IR) to a late time regime (corresponding to the UV) Kiritsis (2013). Last but not least, scalar fields play a crucial role both for holographic models and for cosmology. In the first case they are seen as scalar condensates triggering symmetry breaking and/or phase transitions in the dual QFT Hartnoll et al. (2008a, b); Cadoni et al. (2010). They are dual to relevant operators that drive the RG flow from the UV fixed point to the IR critical point. Moreover, they are the sources of scalar solitons, which are the gravitational background bridging the asymptotic AdS region and the near- horizon region. On the cosmological side it is well-known that scalar fields can be used to model dark energy (the so-called quintessence fields) Ford (1987); Wetterich (1988); Caldwell et al. (1998); Zlatev et al. (1999); Amendola and Tsujikawa (2010). In this paper we will consider a wide class of Einstein-scalar gravity model (parametrized by a potential $V$) that have scalar solitonic solution interpolating between an hyperscaling violating region and an AdS region. These models have been investigated for holographic applications Charmousis et al. (2009); Cadoni et al. (2010); Goldstein et al. (2010); Dong et al. (2012); Cadoni and Pani (2011); Cadoni and Mignemi (2012); Cadoni and Serra (2012); Narayan (2012); Cadoni et al. (2013). We show that an analytical continuation transforms the solitonic solution in a flat FLRW solution of a model with opposite sign of $V$. If the soliton has the AdS region in the UV (IR), the FLRW solution will have a dS epoch at late (early) times. Correspondingly, the FLRW solution will be characterized by power-law expansion at early (late) times ( Section II). Focusing on a particular Einstein-scalar model (parametrized by a parameter $\beta$) that has the AdS regime in the UV and for which exact solitonic solutions are known Cadoni et al. (2011), we generate (and characterize in detail) the corresponding flat FLRW exact solutions. For a broad range of $\beta$ the solutions describe a flat universe decelerating at early times but accelerating at late times (Section III). We proceed by showing that these solutions can be used as a model for dark energy, the scalar field playing the role of a quintessence field. The parameter of state describing dark energy decreases with cosmic time, from a positive value ($<1$) till $-1$ (Section IV). Finally, we discuss the cosmological dynamics in presence of matter in the form of a general perfect fluid. Although we are not able to solve exactly the coupled system, we give strong evidence that the universe naturally evolves from a scaling era at early times to a, cosmological constant dominated, de Sitter universe at late times. Moreover, the transition between the two regimes in not smooth and is the cosmological analogue of the hyperscaling violation/AdS spacetime phase transition of holographic models Cadoni et al. (2010, 2013); Gouteraux and Kiritsis (2012) (Section V). ## II Dark energy, holographic theories and hyperscaling violation We consider Einstein gravity coupled to a real scalar field $\phi$ in four dimensions: $I=\int d^{4}x\sqrt{-g}\left[{\cal{R}}-\frac{1}{2}\left(\partial\phi\right)^{2}-V(\phi)\right],$ (1) where ${\cal{R}}$ is the scalar curvature of the spacetime. The model is parametrized by the self-interaction potential $V(\phi)$ for the scalar field. For static, radially symmetric solutions with planar topology for the transverse space, one can use the following parametrization of the solution: $ds^{2}=-U(r)dt^{2}+\frac{dr^{2}}{U(r)}+R^{2}(r)(dx^{2}+dy^{2}),\quad\phi=\phi(r).$ (2) It is known that the theory (1) admits solutions (2) describing black branes with scalar hair, at least for specific choices of $V(\phi)$ Cadoni et al. (2011, 2012); Cadoni and Mignemi (2012); Cadoni et al. (2013). When the spacetime is asymptotically AdS $U=R^{2}=\left(\frac{r}{R_{0}}\right)^{2}$ (3) (where $R_{0}$ is the AdS length) or more generically scale-covariant $U=R^{2}=\left(\frac{r}{r_{-}}\right)^{\eta},$ (4) (where $r_{-}$ and $0\leq\eta\leq 2$ are parameters), usual no-hair theorems can be circumvented and regular, hairy black brane solutions of (1) are allowed Cadoni et al. (2011, 2012). Moreover, it has been shown that the zero-temperature extremal limit of these black brane solutions is necessarily characterized by $U=R^{2}$ in Eq. (2) Cadoni et al. (2011, 2013). The extremal limit describes a regular scalar soliton interpolating between an AdS spacetime and a scale-covariant metric. In particular, the behaviour of the potential at $r=\infty$ and in the near- horizon region determines the corresponding geometry. When the leading term of the potential is a constant $V(\phi)\sim-6/R_{0}^{2}$ the geometry is AdS. On the other hand if the potential behaves exponentially $V(\phi)\sim-e^{\lambda\phi}$ ($\lambda$ is some constant) we get a scale- covariant metric Cadoni et al. (2011). The AdS vacuum has isometries generated by the conformal group in three dimensions. In particular the AdS metric is invariant under scale transformations: $r\to\mu^{-1}r,\quad(t,x,y)\to\mu(t,x,y).$ (5) On the other hand the scale-covariant metric breaks some of the symmetries of the AdS metric. Under scale transformation the metric (4) is not invariant but only scale-covariant. For $\eta\neq 1$ we get $r\to\mu^{\frac{1}{1-\eta}}r,\quad(t,x,y)\to\mu(t,x,y),\quad ds^{2}\to\mu^{\frac{2-\eta}{1-\eta}}ds^{2}.$ (6) Depending on the form of the potential $V(\phi)$ we have two cases $1)$ AdS is the $r=\infty$ asymptotic geometry and the scale-covariant metric is obtained in the near-horizon region Cadoni et al. (2011, 2013). $2)$ The AdS spacetime appears in the near-horizon region whereas the scale- covariant metric is obtained as $r=\infty$ asymptotic geometry Cadoni et al. (2012); Cadoni and Mignemi (2012). This behaviour has a nice holographic interpretation and a wide range of application for describing dual strongly-coupled QFTs and quantum phase transitions Hartnoll et al. (2008a, b); Charmousis et al. (2009); Gubser and Rocha (2010); Cadoni et al. (2010); Cadoni and Pani (2011); Cadoni and Mignemi (2012); Cadoni and Serra (2012); Cadoni et al. (2013). In the dual QFT the two cases described on points $1)$ and $2)$ above correspond, respectively, to the following: $1)$ The dual QFT at zero temperature has an UV conformal fixed point. In the IR it flows to an hyperscaling violating phase where the conformal symmetry is broken, only the symmetry (6) is preserved and an IR mass-scale (the parameter $r_{-}$ in Eq.(4)) is generated Cadoni et al. (2010, 2011); Cadoni and Pani (2011); Cadoni et al. (2013). $2)$ The dual QFT at zero temperature has a conformal fixed point in the IR and flows in the UV to an hyperscaling violating phase Cadoni et al. (2012); Cadoni and Mignemi (2012); Cadoni and Serra (2012). When $U=R^{2}$ in Eq. (2) the field equations stemming from the action (1) become: $\frac{R^{\prime\prime}}{R}=-\frac{\phi^{\prime 2}}{4},\,\quad\quad\frac{d}{dr}(R^{4}\phi^{\prime})=R^{2}\frac{dV}{d\phi},\,\quad\quad(R^{4})^{\prime\prime}=-2R^{2}V(\phi),$ (7) where the prime denotes derivation with respect to $r$. Notice that only two of these equations are independent. In this paper we are interested in FLRW cosmological solutions with non trivial scalar field of the gravity theory (1). Such solutions have been widely used to describe the history of our universe. Depending on the model under consideration, the scalar field can be used to describe dark energy (quintessence models)Ford (1987); Wetterich (1988); Caldwell et al. (1998); Zlatev et al. (1999); Amendola and Tsujikawa (2010), the inflaton (inflationary models) and also dark matter Sahni and Wang (2000); Bertolami et al. (2012). Our main idea is to use the knowledge of effective holographic theories of gravity in the cosmological context. The key point is that once an exact static solution (2) with $U=R^{2}$ of the field equations (7) is known one can immediately generate a flat FLRW cosmological solution using the following transformation in (2) and (7), $r\to it,\quad t\to ir,\quad V(\phi)\to-V(\phi).$ (8) In fact this transformation maps the line element and the scalar field (2) into $ds^{2}=-R^{-2}(t)dt^{2}+R^{2}(t)(dr^{2}+dx^{2}+dy^{2}),\quad\phi=\phi(t).$ (9) describing a FLRW metric in which the curvature of the spatial sections is zero, i.e a flat universe with $R(t)$ playing the role of the scale factor. The same transformation (8) maps the field equations (7) into $\frac{\ddot{R}}{R}=-\frac{\dot{\phi}^{2}}{4},\,\quad\quad\frac{d}{dt}(R^{4}\dot{\phi})=-R^{2}\frac{dV_{c}}{d\phi},\,\quad\quad\ddot{(R^{4})}=2R^{2}V_{c}(\phi),$ (10) where the dot means derivation with respect to the time $t$ and $V_{c}=-V$. One can easily see that Eqs. (7) and (10) have exactly the same form, simply with the prime replaced by the dot. This means that once a zero-temperature static solution, describing a scalar soliton, of the theory (1) with potential $V$ is known, one can immediately write down a cosmological solution of the theory (1) with potential $V_{c}=-V$. The flip of the sign of the potential when passing from the static scalar soliton to the cosmological solution has important consequences. The AdS vacuum corresponding to constant negative potential $V=-6/R_{0}^{2}$ (a negative cosmological constant) will be mapped in the de Sitter spacetime, corresponding to $V_{c}=6/R_{0}^{2}$ ( (a positive cosmological constant), which describes an exponentially expanding universe. Correspondingly, the scale covariant static metric (4) will be mapped into a cosmological power-law solution $R\sim t^{\eta}$. It follows immediately that the scalar solitons corresponding to the cases $1)$ and $2)$ above will generate after the transformation (8) FLRW cosmological solutions with, respectively, the following properties: $1)$ The cosmological solution describes an universe evolving from a power-law scaling solution at early times to a de Sitter spacetime at late times. $2)$ The cosmological solution describes an universe evolving from a de Sitter spacetime at early times to a power-law solution at late times. It is interesting to notice that a universe evolving from a power-law solution at early times to an exponentially expanding phase at late times has an holographic counterpart in a QFT flowing from hyperscaling violation in the IR to an UV fixed point. Conversely, universe evolving from de Sitter at early times to the power-law behaviour al late times corresponds to a QFT flowing from an IR fixed point to hyperscaling violation in the UV. The FLRW solutions described in point $1)$ above are good candidates to model an universe, which is dominated at late times by dark energy. On the other hand, the cosmological solutions described in point $2)$ above are very promising to describe inflation. In this paper we will investigate in detail solutions of type $1)$. We will leave the investigation of solution of type $2)$ to a successive publication. Transformations like (8) mapping solitons into FLRW cosmologies have been already considered in the context of SUGRA theories. Skenderis and Townsend (2006); Skenderis et al. (2007); Shaghoulian (2013). They are known under the name of domain wall (DW)/cosmology correspondence. For every supersymmetric domain-wall, which is solution of some SUGRA model, we can obtain, by analytical continuation, a flat FLRW cosmology, of the same model but with opposite sign potential Skenderis and Townsend (2006). When the model (1) is the truncation to the metric and scalar sector of some supergravity theory (or more generally when the potential $V$ can be derived from a superpotential, i.e when we are dealing with a “fake” SUGRA model DeWolfe et al. (2000)) the transformation (8) describes exactly the DW/cosmology correspondence. However, in this paper we consider the transformation in the same spirit of effective holographic theories. We do not require the action (1) to come from a SUGRA model and we consider the transformation (8) in its most general form as a mapping between a generic scalar DW solution, i.e. a spacetime (2) with $U=R^{2}$ and cosmological solution (9) endowed with a non trivial time-dependent scalar field. The cosmological solution (9) is not written in terms of the usual cosmic time $\tau$. Using this time variable, solution (9) takes the form: $ds^{2}=-d\tau^{2}+R^{2}(\tau)(dr^{2}+dx^{2}+dy^{2}),\quad\phi=\phi(\tau),$ (11) and the coordinate time $t$ and cosmic time $\tau$ are related by $\tau=\int{\frac{dt}{R(t)}}.$ (12) Written in terms of $\tau$ the field equations (10) become the usual ones ${\dot{H}}=-\frac{\dot{\phi}^{2}}{4},\,\quad\ddot{\phi}+3H\dot{\phi}=-\frac{dV_{c}}{d\phi},\,\quad 3H^{2}=\frac{\dot{\phi}^{2}}{4}+\frac{V_{c}}{2},$ (13) where now the dot means derivation with respect to the cosmic time $\tau$ and $H$ is the Hubble parameter $H=\dot{R}/R$. ## III Exact cosmological solutions In the previous section we have described a general method that allows us to write down a flat FLRW solution with a nontrivial scalar field once a static scalar solitonic solution is known. In the recent literature dealing with holographic applications of gravity one can find several scalar solitons describing the flow from an scale-covariant metric in the IR to an AdS solution in the UV Cadoni et al. (2011, 2013). However, many of them are numeric solutions. An interesting class of exact analytic solutions with the above features have been derived in Ref. Cadoni et al. (2011) using a generating method. This generating method essentially consists in fixing the form of the scalar field. The metric part of the solution and the potential $V$ are found by solving a Riccati equation and a first order linear equation. This allows us to find a solution (2) of the theory (1) with potential Cadoni et al. (2011) $V_{c}(\phi)=\frac{2}{R_{0}^{2}}e^{2\gamma\beta\phi}\left[2-8\beta^{2}+(1+8\beta^{2})\cosh(\gamma\phi)-6\beta\sinh(\gamma\phi)\right]$ (14) where 111In this paper we are using a normalization of the kinetic term for the scalar, which differs from that used in Ref. Cadoni et al. (2011) by a factor of $4$. Correspondingly, $\gamma$ differs by a factor of $2$. $\frac{1}{2}\leq\|\beta\|,\quad\gamma^{-2}=1-4\beta^{2}.$ (15) The point $\phi=0$ is a maximum of the potential $V$, i.e we have $V^{\prime}(0)=0$ and $V^{\prime\prime}(0)=-2/R_{0}^{2}=m^{2}<0$, where $m$ is the mass of the scalar field. Notice that the squared-mass of the scalar is negative and depends only on the the value of the cosmological constant. The potential (14) contains as special cases, models resulting from truncation to the abelian sector of $N=8$, $D=4$ gauged supergravity Cadoni et al. (2011). In fact, for $\beta=0$ and $\beta=\pm 1/4$ Eq. (14) becomes $V_{c}(\phi,\beta=0)=\frac{2}{R_{0}^{2}}\left(2+\cosh\phi\right),\quad V_{c}(\phi,\beta=\pm 1/4)=\frac{6}{R_{0}^{2}}\cosh\left(\frac{\phi}{\sqrt{3}}\right).$ (16) The static, solitonic solutions (2) of the theory (1) with potential (14) are given by Cadoni et al. (2011) $\gamma\phi=\log X,\quad R=\frac{r}{R_{0}}X^{\beta+\frac{1}{2}},\quad X=1-\frac{r_{-}}{r},$ (17) where $r_{-}$ is an integration constant. In the $r=\infty$ asymptotic region, corresponding to $\phi=0$, the potential approaches to $-6/R_{0}^{2}$ and solutions becomes the AdS solution (3). In the near-horizon region, $r=r_{-}$, corresponding to $\phi=\pm\infty$ (depending on the sign of $\gamma$), the potential behaves exponentially and the metric becomes, after translation of the $r$ coordinate, the scale covariant solution (4) with $\eta=2\beta+1$. A FLRW solution can be now obtained applying the transformation (8) to Eqs. (17). We simply get $R(t)=\frac{t}{R_{0}}\left(1-\frac{t_{-}}{t}\right)^{\beta+\frac{1}{2}},\quad\gamma\phi=\log\left(1-\frac{t_{-}}{t}\right).$ (18) Solutions (18) is not defined for every real $t$. Moreover, the range of variation of $t$ is disconnected. For $t_{-}>0$ we have either $-\infty<t\leq 0$ (corresponding to $\gamma\phi>0$) or $t_{-}\leq t<\infty$ (corresponding to $\gamma\phi<0$). Conversely, for $t_{-}<0$ we have either $-\infty<t\leq t_{-}$ (corresponding to $\gamma\phi<0$) or $0\leq t<\infty$ (corresponding to $\gamma\phi>0$). Apart from the parameter $R_{0}$, which sets the value of the cosmological constant, the solution (18) depends on the parameters $\beta$ and $t_{-}$. The parameter $\gamma$ is not an independent parameter but, apart from the sign, it is determined by Eq. (15). The potential (14), hence the action (1), is invariant under the two groups of discrete transformations $(\phi\to-\phi,\,\gamma\to-\gamma)$ and $(\gamma\to-\gamma,\,\beta\to-\beta)$. This symmetries allow to restrict the range of variations of $\gamma,\beta$ to $\\{\gamma>0,\,\phi<0,\,-\frac{1}{2}\leq\beta\leq\frac{1}{2}\\}$. In terms of the time coordinate $t$ we are left with only two branches : $a)\,\\{t_{-}>0,\,t_{-}\leq t<\infty\\}$ and $b)\,\\{t_{-}<0,\,-\infty\leq t<t_{-}\\}$. However, one can easily realize that these two branches are related by the time reversal symmetry $t\to-t,\,t_{-}\to-t_{-}$ and are therefore physically equivalent. We are therefore allowed to restrict our consideration to the branch $a)$. The potential $V_{c}$ has a minimum at $\phi=0$. Near the minimum the potential behaves quadratically $V_{c}=\frac{6}{R_{0}^{2}}+\frac{1}{2}m^{2}\phi^{2}.$ (19) The squared mass of the scalar field is therefore positive and depends only on the cosmological constant $m^{2}=\frac{2}{R_{0}^{2}}=\frac{\Lambda}{3}.$ (20) As expected, for $t=\infty$ ($\phi=0$) $V_{c}$ approaches to a positive cosmological constant $V_{c}=\Lambda=6/R_{0}^{2}$ and the solution becomes the de Sitter spacetime. For $t\approx t_{-}$ ( $\phi\to\pm\infty$) the scale factor has a power-law form, $R\propto(t-t_{-})^{\beta+1/2}$ and the potential behaves exponentially. We get, respectively for $\phi=\pm\infty$, the asymptotic behaviour $\displaystyle V_{c}(\phi)$ $\displaystyle=$ $\displaystyle R_{0}^{-2}(1+8\beta^{2}-6\beta)e^{\gamma\phi(2\beta+1)},$ $\displaystyle V_{c}(\phi)$ $\displaystyle=$ $\displaystyle R_{0}^{-2}(1+8\beta^{2}+6\beta)e^{\gamma\phi(2\beta-1)}.$ (21) The range of variation of the parameter $\beta$ can be further constrained by some physical requirements that must be fulfilled if solution (18) has to describe the late-time acceleration of our universe. The usual way to achieve this is to considers quintessence models characterized by a slow roll of the scalar field. As we will see later in this paper the potential (14) does not satisfy the slow roll conditions, which are sufficient, but not necessary, for having late-time acceleration. We will use here a much weaker condition on the slope of the potential $V_{c}(\phi)$. The scalar field $\phi$ in Eq. (18) is a monotonic function of the time $t$ in the branch under consideration. Being the function $\phi(t)$ of Eq. (18) monotonic for $t_{-}>0$ and $t\in(t_{-},\infty)$ the simplest way to have a well-defined physical model (i.e a one-to-one correspondence $t\leftrightarrow V_{c}$) is to require also the potential to be a monotonic function inside the branch. This requirement restricts the range of variation of the parameter $\beta$ to $-\frac{1}{4}<\beta\leq\frac{1}{4}.$ (22) In fact, for $\frac{1}{4}<\|\beta\|\leq\frac{1}{2}$ the potential $V_{c}$ has other extrema. From the range of $\beta$, we have excluded the point $\beta=-1/4$ because in this case the potential (14) becomes exactly the same as for $\beta=1/4$. It is interesting to notice that the two simple models (16), arising from SUGRA truncations, appear as the two limiting cases of this range of variation. In conclusion, the FLRW solution (18) represents a well-behaved cosmological solution in the following range of the parameters and of the time coordinate $t$ $-\frac{1}{4}<\beta\leq\frac{1}{4},\quad 1\leq\gamma\leq\frac{2}{\sqrt{3}},\quad t_{-}>0,\quad t_{-}\leq t<\infty,\quad\phi<0.$ (23) Other branches are either physically equivalent to it (by using the discrete symmetries of the potential (14) or time-reversal transformations) or can be excluded by physical arguments. Let us now consider the Hubble parameter $H$ and the acceleration parameter $A$. We have for $H$ and $A$: $\displaystyle H$ $\displaystyle=$ $\displaystyle\frac{1}{R}\frac{dR}{d\tau}=\frac{dR}{dt}=\frac{X^{\alpha}}{R_{0}}\left[1+\alpha X^{-1}\left(\frac{t_{-}}{t}\right)\right],$ $\displaystyle A$ $\displaystyle=$ $\displaystyle\frac{1}{R}\frac{d^{2}R}{d\tau^{2}}=\left(\frac{dR}{dt}\right)^{2}+R\frac{d^{2}R}{dt^{2}}=\frac{X^{2\alpha-2}}{R_{0}^{2}t^{2}}\left[\left(t+(\alpha-1)t_{-}\right)^{2}+\alpha(\alpha-1)t_{-}^{2}\right].$ (24) where $\alpha=\beta+\frac{1}{2}$. An important physical requirements are the positivity of the Hubble parameter $H$. Moreover, the acceleration parameter $A$ must be positive, at least at late times, to describe late-time acceleration. One can easily check that in the range of variation of the parameter $\beta$ (23) we have always $H>0$. The behaviour of the acceleration parameter $A$ is more involved. $A$ becomes zero for $t_{12}=\left[1-\alpha\pm\sqrt{\alpha(1-\alpha)}\right]t_{-}$. For $t_{-}>0$ we have $t_{1}>t_{-},\,t_{2}<t_{-}$ for $-1/4<\beta\leq 0$, whereas $t_{1}<t_{-},\,t_{2}<0$ for $0\leq\beta\leq 1/4$. This means that in the branch under consideration for $\beta$ positive, the universe is always accelerating. For $\beta$ negative the universe will have a deceleration at early times (for $t_{-}<t<t_{1}$), whereas it will accelerate for $t>t_{1}$. Until now we have always used in our discussion the coordinate time $t$. The cosmic time $\tau$ is defined implicitly in terms of $t$ by Eq. (12). The correspondence $\tau\leftrightarrow t$ defined by Eq. (12) must be one-to-one, i.e $\tau(t)$ must be monotonic in the range (23). Let us show that this is indeed the case. Inserting the expression for $R$ given in Eq. (18) into (12) we get $\frac{\tau}{R_{0}}=\int\frac{dt}{t}\left(\frac{t}{t-t_{-}}\right)^{\beta+\frac{1}{2}}=-B_{z}(0,\frac{3}{2}-\beta),$ (25) where $B_{z}(0,\frac{3}{2}-\beta)$ is the incomplete beta function $B_{z}(p,q)$ and $z=t_{-}/t$. From the previous equation we get the leading behaviour of $\tau(t)$ near $t=t_{-}$ and $t=\infty$. We have, respectively, $\tau\propto(t-t_{-})^{\frac{1}{2}-\beta},\quad\tau\propto\log t.$ (26) From this equation we learn that $t=t_{-}$ and $t=\infty$ are mapped, respectively into $\tau=0$ and $\tau=\infty$. Moreover, from Eq. (25) one easily realises that $d\tau/dt$ is always strictly positive for $t_{-}\leq t<\infty$. When $\beta$ is a generic real number in $(-\frac{1}{4},\frac{1}{4})$ the function $\tau(t)$ cannot be expressed in terms of elementary functions. However, the integral (25) can be explicitly computed when $\beta$ is a rational number. The simplest example is given by $\beta=0$. In this case we get for the function $t=t(\tau)$, the scale factor $R$ and the scalar field $\phi$, $\frac{t}{t_{-}}=\cosh^{2}\frac{\tau}{2R_{0}},\quad R(\tau)=\frac{t_{-}}{2R_{0}}\sinh\frac{\tau}{R_{0}},\quad\phi=2\log\tanh\left(\frac{\tau}{2R_{0}}\right).$ (27) An other simple example is obtained for $\beta=1/4$. We get $\frac{\tau}{R_{0}}=-2\arctan Y-\log\frac{Y-1}{Y+1}+\pi,\,\quad\quad Y^{4}=\frac{t}{t-t_{-}}.$ (28) Let us conclude this section by giving a short description of the evolution of our universe described by Eq. (18). The universe starts from a curvature singularity at $\tau=0$, where the scale factor vanishes, $R=0$, and the scalar field, the Hubble parameter and the acceleration diverge. For $\tau>0$ the potential $V_{c}(\phi$) rolls down to its minimum at $\phi=0$ first following the exponential behavior given by Eq. (III). In this early stage the scale factor evolves following a power-law behaviour whereas the scalar field evolves logarithmically: $R\sim\tau^{\frac{1+2\beta}{1-2\beta}},\quad H\sim\frac{1}{\tau},\quad A\sim\frac{1}{\tau^{2}},\quad\phi\sim\log\tau.$ (29) The acceleration $A$ is positive for $\beta>0$ and negative for $\beta<0$. After a time-scale determined by $t_{-}$ the universe enters, for $\beta$ negative, in an accelerating phase, whereas for $\beta$ positive continues to accelerate. At late times, independently of the value of $\beta$, the potential approaches the quadratic minimum at $\phi=0$ and the universe has an exponential expansion described by de Sitter spacetime and a constant scalar. Therefore at late times the universe forgets about its initial conditions (the parameter $t_{-}$) and all the physical parameters are determined completely in terms of the cosmological constant. We have for the mass of the scalar field and for $H,A$: $m^{2}=2H^{2}=2A=\frac{2}{R_{0}^{2}}=\frac{\Lambda}{3}.$ (30) This behaviour is the cosmological counterpart of the flowing to an UV conformal fixed point of solitonic solutions in effective holographic theories with an hyperscaling violating phase. The dS solution corresponds to AdS vacuum (3) and is invariant under the scale symmetries (5) (obviously exchanging the $r,t$ coordinates). The power-law solution (29) corresponds to the scale covariant solution (4), it shares with it the scale symmetries (6). Thus, both class of solutions (the scalar soliton and the cosmological solutions) are characterized by the emergence of a mass-scale. In the case of the scalar soliton (17) this mass-scale is described by the the parameter $r_{-}$ and emerges in the IR of the dual QFT. In the case of the cosmological solution the mass-scale is described by the the parameter $t_{-}$, which characterizes the early-times cosmology. When the dual QFT flows in the UV fixed point, the conformal symmetry washes out all the information about the IR length $r_{-}$ which, characterizes the hyperscaling violating phase Dong et al. (2012); Cadoni and Mignemi (2012). Similarly, the cosmological evolution washes out all the information about the initial parameter $t_{-}$ and all the physical parameters are completely determined by the cosmological constant. In the next sections we will show how our cosmological solutions can be used to model dark energy. ## IV Dark energy models It is well known that dark energy can be considered a modified form of matter. The simplest way to model it, is by means of a scalar field (usually called quintessence) coupled to usual Einstein gravity, i.e with a model given by (1) with properly chosen potential. Modelling dark energy with a scalar field has many advantages. Unlike the cosmological constant scenario, the energy density of the scalar field at early times does not necessarily need to be small with respect to the other forms of matter. Cosmological evolution can be described as a dynamical system. It allows for the existence of attractor-like solutions (the so called “trackers”) in which the energy density of the scalar field is comparable with the the usual matter-fluid density for a wide range of initial conditions. This helps to solve the so-called coincidence problem of dark energy (see e.g. Amendola and Tsujikawa (2010)). The model described by Eq. (1) with the potential (14) is a good candidate for realizing a tracking behaviour. In fact, at early times the potential behaves exponentially (see Eq. (III)) giving the power-law cosmological solution (29). This kind of solution have been widely used to produce tracking behavior at early times. Moreover, at late times our model flows in a dS solution (i.e a solution modelling dark energy as a cosmological constant). This could help to explain the present accelerated expansion of the universe characterized by the tiny energy scale $\Lambda\approx 10^{-123}m_{pl}^{2}$. Obviously, to be realistic our models must pass all the tests coming from cosmological observations. The most stringent coming from the above value of the cosmological constant. In this section we will address the issues sketched above for our cosmological model (14). Being dark energy described as an exotic form of matter, useful information comes from its equation of state $p_{\phi}=w_{\phi}\rho_{\phi}$. For a quintessence model described by the action (1) one has $w_{\phi}=\frac{p_{\phi}}{\rho_{\phi}}=\frac{T(\phi)-V_{c}(\phi)}{T(\phi)+V_{c}(\phi)}=\frac{1-K(\phi)}{1+K(\phi)}.$ (31) where $T(\phi)=\dot{\phi}^{2}/2$ (the dot means derivation with respect the cosmic time $\tau$) is the kinetic energy of the scalar field and we have defined $K(\phi)=V_{c}/T$ as the ratio between potential and kinetic energy. The expression of $T$ and $K$ as a function of $\phi$ can be easily computed using Eq. (18) and (12). We have $\frac{t}{t_{-}}=(1-e^{\gamma\phi})^{-1}$ and $T(\phi)=\frac{2}{(R_{0}\gamma)^{2}}e^{2\gamma\beta\phi}\sinh^{2}(\gamma\phi/2)$. Whereas for $K$ we obtain $K(\phi)=\gamma^{2}\frac{2-8\beta^{2}+(1+8\beta^{2})\cosh\gamma\phi-6\beta\sinh\gamma\phi}{\sinh^{2}\frac{\gamma\phi}{2}}.$ (32) From these equations one can easily derive the time evolution of the parameter of state $w_{\phi}$. At $\tau=0$, corresponding to $\phi=-\infty$, both the kinetic and potential energy, as a function of $\phi$, diverge exponentially but their ratio is constant. $w_{\phi}$ takes the $\beta$-dependent value $w_{0}(\beta)=-\frac{1+10\beta}{3(1+2\beta)}.$ (33) In the range of variation of $\beta$ we have $-7/9\leq w_{0}<1$. In particular, for $0\leq\beta\leq 1/4$, $w_{0}(\beta)$ is always negative ($-7/9\leq w_{0}\leq-1/3$), whereas for $-1/4<\beta\leq 0$ , $w_{0}(\beta)$ goes from $-1/3$ to $1$. For $0<\tau<\infty$ (corresponding to $-\infty<\phi<0$) the ratio $K$ increases and, correspondingly, $w_{\phi}$ decreases, monotonically from $w_{\phi}=w_{0}(\beta)$ to $w_{\phi}=-1$. At $\tau=\infty$ ($\phi=0$) the potential energy goes to a minimum, the kinetic energy vanishes and the state parameter $w_{\phi}$ attains the value corresponding to a cosmological constant $w_{\phi}=-1$. As expected dark energy has an equation of state with $-1\leq w_{\phi}<1$ negative, but bigger than $-1$. The $w_{\phi}=-1$ value, corresponding to a cosmological constant, is attained when the potential rolls in its $\phi=0$ minimum at $\tau=\infty$. The behaviour of the parameter $w_{\phi}(t)$ is perfectly consistent with what we found for the acceleration parameter $A$. In fact, for $\beta$ positive $-1\leq w_{\phi}(t)<-1/3$ and the universe always accelerates. For $\beta$ negative, $-1\leq w_{\phi}(t)<1$ and we have a transition from early-times deceleration ($w_{\phi}(t)>-1/3$) to late-times acceleration ($w_{\phi}(t)<-1/3$). As we have mentioned in the previous section, in our model, late-time acceleration is not produced by the usual mechanism used in quintessence models, i.e by a slow-roll of the scalar field. Late-time acceleration requires $w_{\phi}<-1/3$ hence from Eq. (31), $K=V_{c}/T>2$. Sufficient conditions to satisfy the latter inequality is a slow evolution of the scalar field, which is guaranteed by the slow-roll conditions Bassett et al. (2006) $\epsilon=\left(\frac{1}{V_{c}}\frac{dV_{c}}{d\phi}\right)^{2}\ll 1,\quad\|\mu\|=2\left\|\frac{1}{V_{c}}\frac{d^{2}V_{c}}{d\phi^{2}}\right\|\ll 1.$ (34) In our model, the potential $V_{c}$ at late times behaves as Klein-Gordon potential (19), so that we have: $\epsilon=\mu=\frac{4}{\phi^{2}}$ (35) Obviously the slow-roll parameters 35 go to infinity at late time when $\phi$ approaches to $0$. However, the slow-roll conditions (34) are sufficient but not necessary for having late-time acceleration. In our model the condition $V_{c}>2T$ is satisfied by an alternative (freezing) mechanism: at late times the scalar field approaches its minimum at $\phi=0$ in which the potential energy $V_{c}$ is constant and non-vanishing whereas the kinetic energy $T$ is zero. ## V Coupling to matter Until now we have considered a quintessence model (1) with the potential (14) and shown that for a wide range of the parameter $\beta$ it can be consistently used to produce a late-time accelerating universe. The next step is to introduce matter fields in the action, in the form of a general perfect fluid (non-relativistic matter or radiation). Obviously, this is a crucial step because the key features of quintessence model (tracking behavior, stability etc.) are related to the presence of matter. In presence of matter the cosmological equations can be written as ${\dot{H}}=-\frac{1}{4}\left(\rho_{\phi}+\rho_{M}+p_{\phi}+p_{M}\right),\,\quad\ddot{\phi}+3H\dot{\phi}=-\frac{dV_{c}}{d\phi},\,\quad H^{2}=\frac{1}{6}\left(\rho_{\phi}+\rho_{M}\right),$ (36) where $\rho_{\phi}={\dot{\phi}}^{2}/2+V_{c},\,p_{\phi}={\dot{\phi}}^{2}/2-V_{c}$ are the density and pressure of the quintessence field, whereas $\rho_{M}$ and $p_{M}$ are those of matter, related by the equation of state $p_{M}=w_{M}\rho_{M}$. The cosmological dynamics following from Eqs. (36) can be recast in the form of a dynamical system. By defining $x={\dot{\phi}}/(\sqrt{12}H),\,y=\sqrt{V_{c}}/(\sqrt{6}H),\,N=\log R$, the cosmological equations (36) take the form (see e.g. Amendola and Tsujikawa (2010)): $\displaystyle\frac{dx}{dN}$ $\displaystyle=$ $\displaystyle-3x+\sqrt{\frac{3}{2}}\lambda y^{2}+\frac{3}{2}x\left[\left(1-w_{M}\right)x^{2}+\left(1+w_{M}\right)\left(1-y^{2}\right)\right]$ $\displaystyle\frac{dy}{dN}$ $\displaystyle=$ $\displaystyle-\sqrt{\frac{3}{2}}\lambda xy+\frac{3}{2}y\left[\left(1-w_{M}\right)x^{2}+\left(1+w_{M}\right)\left(1-y^{2}\right)\right]$ $\displaystyle\frac{d\lambda}{dN}$ $\displaystyle=$ $\displaystyle-\sqrt{6}\lambda^{2}\left(\Gamma-1\right)x,\quad\lambda=-\frac{\sqrt{2}}{V_{c}}\frac{dV_{c}}{d\phi},\quad\Gamma=V_{c}\frac{d^{2}V_{c}}{d\phi^{2}}\left(\frac{dV_{c}}{d\phi}\right)^{-2}.$ (37) This form of the dynamics is particularly useful for investigating the fixed points of the dynamics and their stability. In the case of a potential given by Eq. (14) neither $\lambda$ nor $\Gamma$ are constant and Eqs. (V) cannot be solved analytically. Even the characterization of the fixed points of the dynamical system is rather involved. To gain information about the cosmological dynamics we will use a simplified approach. We will first consider the dynamics in the two limiting regimes of small and large cosmic time, i.e $(1)\,\,\phi\to-\infty,\quad(2)\,\,\phi=0$ in which the potential behaves, respectively, exponentially (see Eq. (III) and quadratically (see Eq. (19)) and the scale factor evolves, respectively, as power-law and exponentially. After that we will describe qualitatively the cosmological evolution in the intermediate region $\phi\approx-1/\gamma$. ### V.1 Power-law evolution In the case of an exponential potential $\lambda=const$ in Eq. (V). Both the fixed points of the dynamical system (V) and their stability are well known Copeland et al. (1998); Neupane (2004); Amendola and Tsujikawa (2010)). Apart from fluid-dominated and quintessence-kinetic-energy-dominated fixed points, which are not interesting for our purposes, we have two fixed points in which the scale factor $R$ has a power-law behavior. The first fixed point is obtained for $x=\frac{\lambda}{\sqrt{6}},\quad y=\sqrt{1-\frac{\lambda^{2}}{6}},\quad\lambda=\sqrt{\frac{2(1-2\beta)}{1+2\beta}},$ (38) describes a quintessence-dominated solution with $\Omega_{\phi}=\frac{\rho_{\phi}}{6H^{2}}=1$ and a constant parameter of state $w_{\phi}=w_{0}{(\beta})$ with $w_{0}{(\beta})$ given by Eq. (33). This fixed point is stable for $\beta>\beta_{0}=-\frac{1+3w_{M}}{2(5+3w_{M})}.$ (39) Notice that if we take matter with $0\leq w_{M}<1$ we have $-1/4<\beta_{0}\leq-1/10$ so that the region of stability is inside the range of definition of the parameter $\beta$. One can easily realize that this solution is nothing but the previously found power-law solution (29) with the constant parameter of state $w_{0}(\beta)$ given by Eq. (33). Because $\Omega_{\phi}=1$ this solution cannot be obviously used to realize the radiation or matter-dominated epochs. Phenomenologically more interesting is the second fixed point of the dynamical system (V) with an exponential potential. This is the so-called scaling solution Copeland et al. (1998); Liddle and Scherrer (1999) and is given by $x=\sqrt{\frac{3}{2}}\frac{(1+w_{M})}{\lambda},\quad y=\sqrt{\frac{3(1-w_{M}^{2})}{2\lambda^{2}}},$ (40) where $\lambda$ is given as in Eq. (38). This scaling solution is characterized by a constant ratio $\Omega_{\phi}/\Omega_{M}$ and by the equality of the parameter of state for quintessence and matter $w_{\phi}=w_{M}$. Moreover we have $\Omega_{\phi}=x^{2}+y^{2}=\frac{3}{\lambda^{2}}(1+w_{M}).$ (41) The scale factor $R$ behaves also in this case as a power-law, with a $w_{M}$ dependent exponent, $R\propto\tau^{2/(3(1+w_{M}))}$. The scaling solution is a stable attractor for $\beta_{1}<\beta<\beta_{0}$, where $\beta_{0}$ is given as in Eq. (39) and $\beta_{1}=-\frac{12w_{M}^{2}+15w_{M}+5}{2(12w_{M}^{2}+33w_{M}+19)}.$ (42) Notice that for ordinary matter characterized $0\leq w_{M}<1$ we have $-1/4<\beta_{0}\leq-1/10$ and $-1/4<\beta_{1}\leq-5/38$. Hence, the range of stability of the scaling solution is well inside the range of definition of $\beta$. For $\beta>\beta_{0}$ the scaling solution is a saddle point, whereas for $\beta<\beta_{1}$ it is a stable spiral. The scaling solution has features that make it very appealing for describing the early-time universe. The ratio $\Omega_{\phi}/\Omega_{M}$ is constant and $\lambda$-dependent, in principle $\lambda$ can be chosen in such way that $\Omega_{\phi}$ and $\Omega_{M}$ have the same order of magnitude. Moreover the solution is an attractor making the dynamics largely independent of the initial conditions. These features allow to solve the coincidence problem. Cosmological evolution will be driven sooner or later to the scaling fixed point, allowing to have a value of density of the scalar field of the same order of magnitude of matter (or radiation) at the ending of inflation. Despite these nice features the scaling solution alone cannot be used to model the matter-dominated epoch of our universe for several reasons. Because $w_{\phi}=w_{M}$ it is not possible to realize cosmic acceleration using a scaling solution. The universe must therefore exit the scaling era, characterized by $\Omega_{\phi}=constant$, to connect to the accelerated epoch, but this is not possible if the parameters are within the range of stability of the solution. An other problem comes from nucleosynthesis constraints. They require $\Omega_{\phi}/\Omega_{M}<0.2$. However, in the range of the parameter $\beta$ where the scaling solution is a stable node the minimum value of the ratio is given by $\Omega_{\phi}/\Omega_{M}=(7+9w_{M})/(1-w_{M})$. In the most favourable case, $w_{M}=0$ (non-relativistic matter), we still have $\Omega_{\phi}=7\Omega_{M}$. The situation improves if we move in the region where the scaling solution is a stable spiral. Taking $w_{M}=0$ we find $1<\Omega_{\phi}/\Omega_{M}<7$, with $\Omega_{\phi}/\Omega_{M}\to 1$ for $\beta\to-1/4$. In the model under consideration some of these difficulties have the chance to be solved because the dynamics exits naturally the scaling era, at times when the exponential approximation $\lambda\approx const.$ is not anymore valid. ### V.2 Exponential evolution At late cosmic times the scalar field potential behaves as in Eq. (19) and the dynamics of the scalar field is governed by the equation: $\ddot{\phi}+3H\dot{\phi}+m^{2}\phi=0,$ (43) which is can be considered as describing a damped harmonic oscillator. In this analogy the scalar mass $m$ represents the pulsation of the oscillations and the Hubble parameter $H$ acts as a (Hubble) friction term. Two cases are possible Turner (1983); Dutta and Scherrer (2008): $(a)$ $r=3H/m>1$, the oscillations are suppressed by Hubble friction and $\phi$ goes to a constant value (overdamping); $(b)\,$ $r\ll 1$, the oscillating term dominates over Hubble friction and the scalar field oscillates around the minimum of the potential. Depending on the global dynamics of the system either case $(a)$ or case $(b)$ will be realized. Presently we do not have an exact control of this global dynamic. By studying the intermediate regime, however, we will give in Sect. V.3, strong evidence that cosmological evolution will be driven near to de Sitter point where $\phi\approx 0$ In the limit $\phi\to 0$ we have $V_{c}/\dot{\phi}^{2}\gg 1$ and the scalar field is frozen to a constant value and one can easily see that case $(a)$ is realized. This can be also checked directly. From Eq. (30) we can easily read out the ratio $r=\frac{3}{\sqrt{2}}>1$ for our mode model, so that we have overdamping. The absolute value of the scalar field decreases and approaches asymptotically the minimum of the potential where we can approximate $V_{c}(\phi)$ by a constant. Moreover, the value of the ratio $r$ does not depend on the parameter $\beta$. The value of the scalar field is completely determined by Eq. (43) and, in particular, is independent from the early time dynamics. This is again a manifestation of the conformal and scaling symmetries of the gravitational background: once the cosmological dynamics is driven near to the de Sitter vacuum any memory about the scaling regime is lost, the dynamics becomes universal and depends only on one mass-scale, that is set by the cosmological constant. This behaviour has to be compared with that pertinent to the previously discussed slow-roll conditions (34). They correspond to have $V_{c}\gg\dot{\phi^{2}}$ and $\|\ddot{\phi}\|\ll\|3H\dot{\phi}\|$ in (43). We can produce in this way late-time acceleration but the late-time dynamics is not universal but depends on the details of the model. Because of overdamping the cosmological evolution will be driven near to the minimum of the potential $V_{c}(\phi)$. In this region the potential at leading order can be approximated by a cosmological constant, $V_{c}(\phi)=6/R_{0}^{2}$. For a constant potential we have $\lambda=0$ in Eq. (V) and we can easily find the fixed points of the dynamical system. We have three fixed points $(1)$ $x=y=\Omega_{\phi}=w_{\phi}=0,\,\Omega_{M}=1,$ which represents a fluid- dominated solution. $(2)$ $y=0,\,x=\pm 1,\,\Omega_{\phi}=w_{\phi}=1,\,\Omega_{M}=0,$ which represents a solution dominated by the kinetic energy of the scalar field. $(3)$ $x=0,\,y=\pm 1,\,\Omega_{\phi}=1,\,w_{\phi}=-1,\,\Omega_{M}=0,$ which represents a a solution dominated by the energy of the vacuum (cosmological constant). Obviously, the only physical candidate for describing the late-time evolution of our universe is fixed point $(3)$. Neglecting the solution with negative $y$ (representing an exponentially shrinking universe), the solution with $y=1$ give the de Sitter spacetime, an exponentially expanding universe with $H=R_{0}^{-1}$, i.e. $R\propto{\rm e}^{\tau/R_{0}}$. By linearizing Eqs. (V) around the fixed point, one can easily find that the de Sitter solution is a stable node of the dynamical system. In fact the two eigenvalues of the matrix describing the linearized system are real and negative ($-3,-3(1+w_{M})$). Actually, for $\lambda=0$ one can go further and integrate exactly the dynamical system (V). After some calculation one finds $y=\frac{1}{\sqrt{1+cR^{-3(1+w_{M})}-a^{2}R^{-6}}},\quad x=\frac{aR^{-3}}{\sqrt{1+cR^{-3(1+w_{M})}-a^{2}R^{-6}}},$ (44) where $a,c$ are integration constants. Eq. (44) confirms that the dS spacetime is an attractor of the dynamical system. In fact, the two-parameter family of solutions (44) has a node at $x=0,y=1$ to which every member of the family approaches as $R\to\infty$. The three terms in the square root in the denominator represent, respectively, the contribution of the energy of the vacuum, the contribution of matter, and the contribution of the kinetic energy of the scalar field. One can easily see that at late times ($R\to\infty$) the vacuum energy always dominates over the other two contributions. Moreover, the scalar field kinetic energy contribution is always subdominant with respect to the matter contribution. In absence of matter ($c=0$) we have $HR_{0}=\sqrt{1-a^{2}R^{-6}},\quad\dot{\phi}^{2}\propto R^{-6}$, telling us that the kinetic energy of the scalar field falls off very rapidly as the scale factor $R$ increases. An explicit form of the time dependence of the scale factor can be derived from (44) only after fixing the parameter of state of matter. For dust ($w_{M}=0$) and radiation ($w_{M}=1/3$) we find, $R_{dust}(\tau)=c_{1}\left[\cosh\frac{3}{2R^{0}}(\tau-\tau_{0})\right]^{\frac{2}{3}},\quad R_{rad}(\tau)=c_{2}\left[\cosh\frac{2}{R^{0}}(\tau-\tau_{0})\right]^{\frac{1}{2}},$ (45) where $c_{1,2},\tau_{0}$ are constants. Summarizing, if cosmological evolution is such that the system is driven near to the minimum of the potential $V_{c}$, i.e the region where the potential can be approximated by a cosmological constant, then the universe will necessarily enter in the regime of exponential expansion described by the dS spacetime. Obviously, the crucial question is: will the system be driven to this near-minimum region? A definite answer to this question requires a full control of the global dynamics of the system (V). In the next subsection, by analyzing the intermediate region of the potential $V_{c}$, we will give strong indications that this is indeed the case. ### V.3 Intermediate regime A key role in discussing cosmological evolution in presence of dark energy is played by the so-called tracker solutions Steinhardt et al. (1999). These solutions are special attractor trajectories in the phase space of the dynamical system (V) characterized by having approximately constant $\lambda,\Omega_{\phi},w_{\phi}$. If the time-scale of the variation of $\lambda$ is much less then $H^{-1}$ we can consider these trajectories as build up from instantaneous fixed points changing in time Steinhardt et al. (1999); Amendola and Tsujikawa (2010). Thus, tracker solutions are very useful to solve the coincidence problem. During the matter dominate epoch dark energy tracks matter, the ratio $\Omega_{\phi}/\Omega_{M}$ remains almost constant and $w_{\phi}$ remains close to $w_{M}$ with $w_{\phi}<w_{M}$. Moreover, if the condition $\Gamma>1$ along the trajectory is satisfied, $\lambda$ decreases toward zero. Once the the value $\lambda^{2}=3(1+w_{M)}$ is reached the fixed point (38) with $\Omega_{\phi}=-1$ becomes stable and the universe exits the scaling phase to enter the accelerated phase. To check if our solutions behave as trackers let us first calculate the parameter $\Gamma$ of Eq. (V) for our potential (14). We get $\Gamma-1=\gamma^{2}\frac{1-16\beta^{2}+2(1+8\beta^{2})\cosh\gamma\phi-12\beta\sinh\gamma\phi}{\left(4\beta-4\beta\cosh\gamma\phi+\sinh\gamma\phi\right)^{2}}.$ (46) One can check analytically and numerically that for $-1/4<\beta\leq 1/4$ we have $\Gamma-1=0$ for $\phi=-\infty$. In the range $\phi\in(-\infty,0)$, $\Gamma-1$ monotonically increases and blows up to $\infty$ as $1/\phi^{2}$ for $\phi=0$. In Figs. 1 we show the plot of $\lambda$ and $\Gamma-1$ as a function of $\phi$ for selected values of the parameter $\beta$. The curves remain flat till the scalar field reaches values of order $-1/\gamma$. Moreover $\Gamma-1$ is exponentially suppressed as $\phi\to-\infty$ and stays flat, near to zero, till we reach values of $\|\phi\|$ of order $1/\gamma$. For instance for $-\infty<\phi<-10/\gamma$ we have $0<(\Gamma-1)<10^{-4}$. This shows that in the range $(-\infty,{\cal{O}}(1/\gamma))$, $\Gamma$ varies very slowly as a function of $\phi$. The same is true if we consider $\Gamma$ as a function of the number of e-foldings $N$. In fact we have $d\Gamma/dN=\sqrt{12}x(d\Gamma/d\phi)$ and because $x$ flows from the value $x=\sqrt{3/2}(1+w_{M})/\lambda$ at the scaling fixed point to $x=0$ at the dS fixed point we conclude that $\Gamma-1$ is also a slowly varying function of $N$. Notice that the previous features are not anymore true for $1/4<\|\beta\|<1/2$. This is because in these range of $\beta$ the denominator in Eq. (46) has a zero at finite negative values of $\phi$, namely for $\cosh\gamma\phi=-(1+16\beta^{2})/(1-16\beta^{2})$. Being $\Gamma$ nearly constant and $\Gamma>1$, we have a tracker behaviour of our solutions till the scalar field reaches values of order $1/\gamma$. In this region we have (see e.g. Amendola and Tsujikawa (2010)) $w_{\phi}=\frac{w_{M}-2(\Gamma-1)}{1+2(\Gamma-1)}.$ (47) Being $w_{\phi}<w_{M}$ dark energy evolves more slowly then matter. Also $\lambda$ and the ratio $\Omega_{\phi}/\Omega_{M}$ varies slowly. $\lambda$ decreases toward zero, whereas $\Omega_{\phi}/\Omega_{M}$ increases. The main difference between our model and the usual tracker solutions is the way in which the universe exits the scaling behaviour and produces the cosmic acceleration. In the usual scenario this happens when $\lambda$ reaches the lower bound for stability of the scaling solution, $\lambda^{2}=3(1+w_{M})$. One can easily check that for our models this happens instead when the system reaches the region where the approximation of slow varying $\lambda$ and $\Gamma$ does not hold anymore. The universe exits the scaling regime when it reaches the regions $\phi\sim-1/\gamma$ where $\Gamma$ and $\lambda$ vary very fast and we have a sharp transition to the dS phase. This transition is the cosmological counterpart of the hyperscaling violating/AdS phase transition in holographic theories of gravity Cadoni et al. (2010); Gouteraux and Kiritsis (2012). We are now in position of giving a detailed, albeit qualitative, description of the global behavior of our FLRW solutions. This behaviour depends on the range of variation of the parameter $\beta$. We have to distinguish three different cases: $(I):\beta<\beta_{1};\,(II):\beta_{1}<\beta<\beta_{0};\,(III):\beta>\beta_{0}$ with $\beta_{0,1}$ given by Eq. (39) and (42). In case $(I)$ the scaling solution, describing the universe at early times, is a stable spiral and $\Omega_{\phi}/\Omega_{M}\approx 1$. As the cosmic time increases, $\Omega_{\phi}/\Omega_{M}$ stays almost constant and $\lambda$ decreases toward the value $\lambda^{2}=24(1+w_{M})^{2}/(7+9w_{M})$ below which the scaling solution is a stable node. However, this value is not in the region of slow varying $\lambda$. Cosmological evolution undergoes a sharp transition to the dS accelerating phase. The behaviour of $\lambda$, $\Gamma-1$, $\Omega_{\phi}$ and $w_{\phi}$ as a function of $\phi$ for this class of solutions is explained in Figs. 1, 2, where we plot as representative element $\beta=-15/64$ and we take nonrelativistic matter, $w_{M}=0$. Notice that Figs 2 have been produced using the expressions (41), (47) respectively for $\Omega_{\phi}$ and $w_{\phi}$, which are valid in the region of slow variation of $\lambda$ and $\Gamma$. Therefore, the plots can be trusted only in the region $\phi<<-1/\gamma$. In case $(II)$ the scaling solution, describing the universe at early times, is a stable node and $\Omega_{\phi}/\Omega_{M}={\cal{O}}(1)$ but $\Omega_{\phi}>\Omega_{M}$. At early times $\lambda$ decreases very slowly. As explained above, there is no smooth transition to the accelerating scaling phase (38) with $\Omega_{\phi}=1$ but a sharp transition to the de Sitter phase. The behaviour of $\lambda$, $\Gamma-1$, $\Omega_{\phi}$ and $w_{\phi}$ as a function of $\phi$ for this class of solutions is explained in Figs. 1, 2, where we plot as representative element $\beta=-1/8$ and $w_{M}=0$. In case $(III)$ the scaling solution is a saddle point and at early times the accelerating, scalar-field dominated solution (38) is stable. We have $w_{\phi}<-1/3$ and $\Omega_{\phi}=1$. Here we have a transition from a power- law, accelerating universe at early times to the de Sitter solution at late times. Obviously, this case is not realistic because it cannot describe the matter dominated era. The plot of $\lambda$, $\Gamma-1$, $\Omega_{\phi}$ and $w_{\phi}$ as a function of $\phi$ for this class of solutions is depicted respectively in Figs. 1, 2 for $\beta=0$ and $w_{M}=0$. \| ---\|--- Figure 1: Plot of the function $\lambda(\phi)$ (left panel) and $\Gamma(\phi)-1$ (right panel) for selected values of the parameter $\beta$ representative of the three classes of solutions ($I,II$ and $III$) and for $w_{M}=0$. The thin, red lines are the plots relative to the model of class $I$ with $\beta=-15/64$. The thick blue lines give the plots of a model in class $II$ with $\beta=-1/8$. The green, dashed lines correspond to a model in class $III$ with $\beta=0$. \| ---\|--- Figure 2: Plot of the function $w_{\phi}(\phi)$ (left panel) and $\Omega_{\phi}(\phi)$ (right panel) for selected values of the parameter $\beta$ representative of the three classes of solutions ($I,II$ and $III$) and for $w_{M}=0$. The thin, red lines are the plots relative to the model of class $I$ with $\beta=-15/64$. The thick blue lines give the plot of a model in class $II$ $\beta=-1/8$. The green, dashed lines correspond to a model in class $III$ with $\beta=0$. Let us conclude this section with a brief, general discussion about the parameters entering in our model. Basically, apart from the Planck mass in the action (1) enter a dimensionless parameter $\beta$ and a length scale $R_{0}$ (Notice that in Eq. (1) we have set $\kappa^{2}=8\pi G=1/2$). In addition we have the integration constants of the differential equations (V), which have to be determined by solving the Cauchy problem. Some of these constants will be related to $t_{-}$ and $a,b$ characterizing respectively the power-law (29) and the exponential regime (44). However, the scale symmetries of the gravitational background together with the attractor behaviour of the scaling solution and of the de Sitter fixed point make the cosmological dynamics largely, if not completely, independent from the initial conditions. Cosmological evolution can be seen as a flow from a scaling fixed point to a conformal dS fixed point, in which the system looses any memory about initial conditions. The final state is therefore completely characterized by the length scale $R_{0}$, which determines everything (Hubble parameter, acceleration, cosmological constant and the mass for the scalar, see Eq. (30). The length scale $R_{0}$ can be fixed by the dark energy density necessary to explain the present acceleration of the universe, $\rho_{de}=10^{-123}m_{p}^{4}$. This gives a mass of the scalar $m\approx 10^{-33}eV$. On one side this uniqueness gives a lot of predictive power to the model, but on the other side the presence of an extremely light scalar excitation runs into the the well-known problems in the framework of particle physics, SUGRA theories and cosmological constant scenarios Peebles and Ratra (2003); Padmanabhan (2003). ## VI Conclusions In this paper we have shown that scalar solitonic solutions of holographic models with hyperscaling violation have an interesting cosmological counterpart, which can be obtained by analytical continuation and by flipping the sign of the potential for the scalar field. The resulting flat FLRW solutions can be used to model cosmological evolution driven by dark energy and usual matter. In absence of matter, the flow from the hyperscaling regime to the conformal AdS fixed point in holographic models correspond to cosmological evolution from power-law regime at early cosmic times to a dS fixed point at late times. In presence of matter, we have a scaling regime at early times, followed by an intermediate regime with tracking behaviour. At late times the solution exits the scaling regime with a sharp transition to a de Sitter spacetime. The phase transition between hyperscaling violation and conformal fixed point observed in holographic gravity has a cosmological analogue in the transition between a scaling, era and a dS era dominated by the energy of the vacuum. We have been able to solve exactly the dynamics only in absence of matter. When matter is present we do not have full control of the global solutions. Nevertheless, by writing the cosmological equations as a dynamical system and by investigating three approximated regimes we have given strong evidence that the above picture is realized. At the present stage our model for dark energy cannot be completely realistic. In the matter dominated epoch the ratio $\Omega_{\phi}/\Omega_{m}\approx 1$, so that we have a problem with nucleosynthesis. Moreover, the late-time cosmology shares the same problems of all cosmological constant scenarios. The vacuum energy is an unnaturally tiny free parameter of the model. The same is true for the mass of the scalar excitation associated to the quintessence field. There are several open questions, which are worth to be investigated in order to support the above picture. One should derive the exact full phase space description of the dynamical system in presence of matter to check the correctness of our results. In particular, having full control on the phase space would give a precise description of the sharp transition between the scaling and the dS regime. This would also help us to shed light on the analogy between the cosmological transition and the hyperscaling violation/ AdS holographic phase transition. 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1311.4113	# Quantum Anomalous Hall Effect in Magnetically Doped InAs/GaSb Quantum Wells Qingze Wang1, Xin Liu1, Hai-Jun Zhang2, Nitin Samarth1, Shou-Cheng Zhang2 and Chao-Xing Liu Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802-6300 2 Department of Physics, McCullough Building, Stanford University, Stanford, CA 94305-4045 ###### Abstract The quantum anomalous Hall effect has recently been observed experimentally in thin films of Cr doped (Bi,Sb)2Te3 at a low temperature ($\sim$ 30mK). In this work, we propose realizing the quantum anomalous Hall effect in more conventional diluted magnetic semiconductors with doped InAs/GaSb type II quantum wells. Based on a four band model, we find an enhancement of the Curie temperature of ferromagnetism due to band edge singularities in the inverted regime of InAs/GaSb quantum wells. Below the Curie temperature, the quantum anomalous Hall effect is confirmed by the direct calculation of Hall conductance. The parameter regime for the quantum anomalous Hall phase is identified based on the eight-band Kane model. The high sample quality and strong exchange coupling make magnetically doped InAs/GaSb quantum wells good candidates for realizing the quantum anomalous Hall insulator at a high temperature. ###### pacs: 73.20.-r,73.20.At,73.43.-f Introduction \- The quantum anomalous Hall (QAH) state in magnetic topological insulatorsHaldaneprl1988 ; Qiprb2006 ; Qiprb2008 ; Liucprl20082 ; Wuprl2008 ; Yusci2010 ; Dingprb2011 ; Zhangprl2012 ; Changsci2013 ; Wangzfprl2013 ; Wangjprl2013 ; Zhangscirep2013 possesses a quantized Hall conductance carried by chiral edge states, similar to the well-known quantum Hall stateKlitzingprl1980 . However, its physical origin is due to the exchange coupling between electron spin and magnetization, instead of the orbital effect of magnetic fields. Therefore, the QAH effect does not require any external magnetic field or the associated Landau levelsHaldaneprl1988 , and thus has great potential in the application of a new generation of electronic devices with low dissipation. Nevertheless, it is difficult to search for realistic systems of the QAH effect, mainly due to the stringent material requirements. To realize the QAH effect, the system should be an insulator with a topologically non-trivial band structure. Simultaneously, ferromagnetism is also required in the same system. In realistic materials, it is rare that ferromagnetism coexists with an insulating behavior. For example, the HgTe/CdTe quantum well (QW) is the first quantum spin Hall (QSH) insulator with a topologically non-trivial band structureBernevigsci2006 ; Konigsci2007 . With magnetization, it was also predicted to be a QAH insulatorLiucprl20082 . However, ferromagnetism cannot be developed spontaneously in this system. Thus, one has to apply an external magnetic field, obstructing the confirmation of the QAH effect. Alternatively, one can consider other two dimensional (2D) topological insulators with magnetic doping. It was realized that the non-trivial band structure in Bi2Te3 family of materials can significantly enhance spin susceptibility and may lead to ferromagnetism in the insulating stateYusci2010 . Consequently, these systems with magnetic doping become a suitable platform to observe the QAH effect. The recent experimental discoveryChangsci2013 confirmed this prediction by transport measurements on thin films of Cr doped (Bi,Sb)2Te3. In this experiment, the quantized Hall conductance was observed at a low temperature, around $\sim 30mK$, presumably due to the small band gap opened by exchange coupling and low carrier mobility $\sim 760cm^{2}/Vs$ . Therefore, searching for realistic systems with non-trivial band structures, strong exchange coupling and high sample quality is essential to realize the QAH effect at a higher temperature. The QAH effect has also been theoretically predicted in other types of systemsZhangprl2012 ; Wuprl2008 ; Dingprb2011 ; Zhangscirep2013 ; Liuxprl2013 ; Hsuprb2013 ; Zhangharxiv2013 . Here we propose a new system for the QAH effect, magnetically doped InAs/GaSb type II QWs. The InAs/GaSb QW is predicted to be a QSH insulator with a certain range of well thicknessLiucprl2008 . Transport experiments have indeed observed a stable longitudinal conductance plateau with a value of $2e^{2}/h$ in this systemKnezprb2010 ; Knezprl2011 ; Suzukiprb2013 ; Dularxiv2013 . Moreover, Mn-doped InAs and GaSb are known diluted magnetic semiconductorsvon1991 ; Ohnoprl1992 ; NishitaniPE2010 . High quality heterostructures integrating (In,Mn)As and GaSb have been fabricated by molecular beam epitaxymunekata1993 . Unlike the case of Mn-doped GaAs where debate continues about the origin of ferromagnetic ordering samarth2012 ; chapler2012 ; fujii2013 , the ferromagnetism in Mn-doped InAs is consistent with free hole mediated ferromagnetism within a mean field Zener model Dietlsci2000 ; macdonald2005 ; Jungwirthrmp2006 ; Dietlnm2010 . Magnetically- doped InAs/GaSb heterostructures are also attractive for optical control koshihara1997 and electric field control ohno2000 of carrier mediated ferromagnetism. Therefore, it is natural to ask whether the QAH effect can be realized in this system. In this work, we find the Curie temperature of ferromagnetism can be significantly enhanced due to the non-trivial band structure. The quantized Hall conductance appears in a wide regime of parameters below the Curie temperature. Therefore, magnetically doped InAs/GaSb QWs provide an promising platform to search for the QAH effect with a high critical temperature. Figure 1: (Color online) (a) Illustration of an AlSb/InAs/GaSb/AlSb Type-II semiconductor QW. The widths of AlSb, InAs and GaSb are denoted as d3, d2 and d1, respectively. In the inverted regime, hole subbands of GaSb are located above electron subbands of InAs, leading to an electric charge transfer between these two layers. (b), (c) and (d) Band structures for InAs/GaSb QW with $A=0eV\cdot\AA{}$, $A=0.3eV\cdot\AA{}$ and $A=2eV\cdot\AA{}$, respectively; A is the coupling strength between the first hole subband of GaSb and the first electron subbband of InAs. InAs/GaSb quantum wells and ferromagnetism \- A schematic plot is presented in Fig.1 (a) for InAs/GaSb QWs, in which InAs and GaSb together serve as well layers and AlSb layers are for barriersChangss1980 ; Yangprl1997 ; Chaoprb2000 . The unique feature of InAs/GaSb QWs is that the conduction band minimum of InAs has lower energy than the valence band maximum of GaSb, due to the large band-offset. Consequently, when the well thickness is large enough, the first electron subband of InAs layers, denoted as $\|E_{1}\rangle$, lies below the first hole subband of GaSb layers, denoted as $\|H_{1}\rangle$. This special band alignment is similar to that in HgTe QWs, known as an “inverted band structure”, which is essential for the QSH effect Liucprl2008 ; Knezprl2011 . Since the $\|H_{1}\rangle$ state has a higher energy than the $\|E_{1}\rangle$ state, there is an intrinsic charge transfer between InAs and GaSb layers. This can be seen from the energy dispersion in Fig.1 (b), where the Fermi energy (dashed line) crosses both the $\|E_{1}\rangle$ band in InAs layer (blue line) and the $\|H_{1}\rangle$ band in GaSb layer (red line) if we neglect the coupling between two layers. Clearly, InAs layer is electron doped while GaSb is hole doped. With magnetic doping, free carriers in InAs and GaSb layers are able to mediate exchange coupling between magnetic moments through Ruderman- Kittel-Kasuya-Yosida (RKKY) interactionDietlsci2000 ; Dietlprb2001 ; Jungwirthrmp2006 ; Satormp2010 , leading to ferromagnetism in these systems with a Curie temperature $T_{c}\sim 25K$NishitaniPE2010 . Therefore, we expect that magnetically doped InAs/GaSb QWs in the metallic phase of Fig.1 (b) should also be ferromagnetic. The problem is complicated by the coupling between two layers, which induces a hybridization gap, as shown in Fig.1 (c) or (d). Therefore, it is natural to ask what happens to ferromagnetism for the insulating regime with the Fermi energy in the hybridization gap. Below we will answer this question by studying a four band model. The low energy physics of magnetically doped InAs/GaSb QWs can be well described by a four band model, which was first developed by Bernevig, Hughes and Zhang (BHZ) for HgTe QWsBernevigsci2006 ; Liucprl2008 . In the basis of $\|E_{1}+\rangle$, $\|H_{1}+\rangle$, $\|E_{1}-\rangle$ and $\|H_{1}-\rangle$, the effective Hamiltonian can be expressed as $\displaystyle H=H_{0}+H_{BIA}+H_{SIA}+H_{ex}.$ (1) The BHZ Hamiltonian $H_{0}$ is given by $\displaystyle H_{0}=\epsilon_{k}\mathbf{1}_{4\times 4}+\mathcal{M}(\vec{k})\sigma_{0}\otimes\tau_{z}+Ak_{x}\sigma_{z}\otimes\tau_{x}-Ak_{y}\sigma_{0}\otimes\tau_{y}$ where $\epsilon_{k}=C-D(k^{2}_{x}+k^{2}_{y})$, $\mathcal{M}(\vec{k})=M_{0}-B(k^{2}_{x}+k^{2}_{y})$ , $\mathbf{1}_{4\times 4}$ is a 4 by 4 identity matrix, $\sigma$ and $\tau$ are Pauli matrices, representing spin and sub-bands, respectively. All the parameters used below are given in the appendixWangappendix2013 for InAs/GaSb QWs. The linear term ($A$ term) couples electron subbands of InAs and hole subbands of GaSb and opens a hybridization gap. $H_{BIA}$ and $H_{SIA}$ describe bulk inversion asymmetry and structural inversion asymmetryLiucprl2008 ; Wangappendix2013 . $H_{ex}$ describes the exchange coupling between magnetic moments and electron spin. In the basis of the four band model, we can write $H_{ex}$ as $\displaystyle H_{ex}=\sum_{\vec{R}_{n}}{\bf S}_{M}(\vec{R}_{n})\cdot\tilde{\bf s},$ (2) where ${\bf S}_{M}(\vec{R}_{n})$ denotes magnetic impurity spin at the position $\vec{R}_{n}$ and $\tilde{\bf s}$ is regarded as an effective spin operator of the four band model. We should emphasize that both the total angular momentum and exchange coupling constants are included in $\tilde{\bf s}$ for simplicity (See appendix for detailsWangappendix2013 ). We only consider magnetization along the z-direction and the operator $\tilde{s}_{z}$ can be decomposed into $\displaystyle\tilde{s}_{z}=s_{1}\sigma_{z}\otimes\tau_{z}+s_{2}\sigma_{z}\otimes\tau_{0},$ (3) where $(s_{1}+s_{2})\sigma_{z}$ ($(-s_{1}+s_{2})\sigma_{z}$) describes the effective spin operator for conduction (valence) band in the four band model. Next we determine the Curie temperature of ferromagnetism in this system through the standard mean field theory. With linear response theory, the spin susceptibility for the operator $\tilde{s}_{z}$ is given by $\displaystyle\tilde{\chi}_{s}={\lim_{q\rightarrow 0}}Re[\sum_{i,j,\sigma,\sigma^{\prime},\vec{k}}$ $\displaystyle\frac{\|\langle u_{i\sigma,\vec{k}}\|\tilde{s}\|u_{j\sigma^{\prime},\vec{k}+\vec{q}}\rangle\|^{2}(f_{i\sigma}(\vec{k})-f_{j\sigma^{\prime}}(\vec{k}+\vec{q}))}{E_{j\sigma^{\prime}}(\vec{k}+\vec{q})-E_{i\sigma}(\vec{k})+i\Gamma}]$ (4) where $i,j$ denote conduction and valence bands, $\sigma,\sigma^{\prime}$ are spin indices, $u_{i\sigma}$ is the eigen wave function with the energy Eiσ, $f_{i\sigma}(\vec{k})$ is the Fermi-Dirac distribution function and $\Gamma$ is band broadening, estimated as $\sim 10^{-4}$eVKnezprb2010 . The susceptibility of magnetic moment takes the form $\tilde{\chi}_{M}=\frac{S_{0}(S_{0}+1)}{3k_{B}T}$, which is obtained from the dilute limit of Curie-Weiss behavior. $S_{0}$ is the spin magnitude of magnetic impurities. The Curie temperature can be determined by the condition $N_{0}x_{eff}\tilde{\chi}_{s}(T_{c})\tilde{\chi}_{M}(T_{c})=1$ Dietlprb2001 ; Wangappendix2013 in the mean field approximation, where $N_{0}$ is the cation concentration and $x_{eff}$ is the effective composition of magnetic atoms. As described above, in the absence of the coupling between two layers, the system must be in a ferromagnetic phase. This corresponds to the case with $A=0$ in the four band model. Therefore, we treat $A$ as a parameter and plot the calculated Curie temperature $T_{c}$ as a function of $A$ and Fermi energy $E_{f}$, as shown in Fig. 2(a). When the Fermi energy lies in the hybridization gap (around 0meV), the Curie temperature first increases and then decreases with the increasing of $A$. Therefore, we find surprisingly that the opening of a small hybridization gap will enhance ferromagnetism. Figure 2: (Color online). (a) Curie temperatures as a function of the parameter A and Fermi energy. (b) Total spin susceptibility $\chi_{s}$ as a function of Fermi energy $E_{f}$ and temperature. (c) Different contribution (intra-band and inter-band contribution) to spin susceptibility $\chi_{s}$ at $T=0K$, respectively. (d) Density of states for InAs/GaSb quantum well with a hybridization gap. To understand the underlying physics, we consider different origins of spin susceptibility. According to Eq. (4), spin susceptibility can be separated into two parts: the intra-band contribution ($i=j$), and inter-band contribution ($i\neq j$). The intra-band contribution originates from the states near Fermi energy and mediates the RKKY type of coupling between magnetic moments. It is the main origin for ferromagnetism in metallic systems. Indeed, our calculation shows that the intra-band contribution has a maximum around band edge because of the singularity of density of states (Fig. 2(d))WangjarXiv2012 ; xu2013 , but is significantly reduced when the Fermi energy is tuned into the band gap (the blue line in Fig. 2(c)). On the other hand, the inter-band contribution mainly originates from the hybridization of wave functions between conduction and valence bands due to the inverted band structure, as discussed in Ref. Yusci2010, . Therefore, the inter-band contribution shows a peak in the insulating regime and diminishes as the Fermi energy goes away from the band gap. Taking into account both contributions, we find sharp peaks of the total spin susceptibility near band edges at low temperatures (below 4 K), as shown in Fig.2(b). With increasing temperatures, both peaks are smeared and the spin susceptibility reveals a broad peak structure around band gap, leading to the enhancement of ferromagnetic Curie temperature $T_{c}$ in the insulating regime. It is necessary to compare magnetic mechanism in magnetically doped InAs/GaSb QWs with that in Mn doped HgTe QWs and Cr doped (Bi,Sb)2Te3. The low energy effective Hamiltonian of HgTe QWs takes the same form as the Hamiltonian in Eq. Quantum Anomalous Hall Effect in Magnetically Doped InAs/GaSb Quantum Wells. However, there is one essential difference: the parameter $A$ is much larger in HgTe QWs because electron and hole subbands are in the same layer and coupled strongly. For a large $A$, spin susceptibility will be suppressed due to the large band gap and the disappearance of band edge singularity, as shown in Fig.1 (d). Consequently, ferromagnetism is not favorable in Mn doped HgTe QWsLiucprl20082 . The strong interband contribution in our case is similar to that in Cr doped (Bi,Sb)2Te3. The $s_{1}$ term of the effective spin operator (Eq. (3)) takes the same form as that in the effective model of Cr doped (Bi,Sb)2Te3 (See Ref. Yusci2010, ), which mainly contributes to the inter-band spin susceptibility. Eq. (3) includes an additional part $s_{2}\sigma_{z}\otimes\tau_{0}$, which dominates the intra-band contribution. Our calculation shows that both parts of the spin operator have a significant contribution to spin susceptibility in the case of a small hybridization gap and a band edge singularity. Therefore, in the regime favorable for QAH, a relatively high Curie temperature for ferromagnetism ($T_{C}\sim 30$ K ) is expected for magnetically doped InAs/GaSb QWs in comparison with the Cr-doped Bi chalcogenides. Quantized Hall transport and realistic systems \- Our calculations clearly show that ferromagnetism can be developed in magnetically doped InAs/GaSb QWs. Below $T_{c}$, magnetic moments align and induce a Zeeman type spin splitting for both conduction and valence bands due to exchange coupling. To realize the QAH states, spin splitting needs to exceed the band gap. This situation is similar to that of Mn-doped HgTe QWs. From Ref. Liucprl20082, , we find that two conditions for the QAH effect should be satisfied: (1) one spin block becomes a normal band ordering while the other spin block remains in an inverted band ordering; and (2) the system stays in an insulating state Wangappendix2013 . The first condition is satisfied in InAs/GaSb QWs by controlling magnetic dopingChangss1980 ; Yangprl1997 ; Chaoprb2000 , while the second condition can be achieved by tuning well thickness. Once these two conditions are satisfied, the QAH effect is expected. At a low temperature, the average spin $\langle S_{M}\rangle$ of magnetic atoms and the average effective spin polarization $\langle\tilde{s}_{z}\rangle$ can be numerically calculated self- consistentlyJungwirthprb1999 ; Wangappendix2013 . The magnetization of magnetic dopants as a function of the Fermi level and temperature is shown in Fig.3 (a). The critical temperature for ferromagnetic order determined in Fig. 3 (a) is consistent with the early calculation based on spin susceptibility. With the obtained magnetization, we compute the Hall conductivity at $T=1K$ with the standard Kubo formulaThoulessprl1982 ; Sinitsynprl2006 . As seen in Fig. 3 (b), the Hall conductance is quantized at a value of $e^{2}/h$ when the Fermi energy falls in band gap and decreases in the metallic regime. Figure 3: (Color online). (a) Magnetization of Mn as a function of temperature and the Fermi energy. (b) The Hall conductance as a function of Fermi energy $E_{F}$ at the temperature $T=1K$. Two key ingredients in the above analysis are the small hybridization gap and band edge singularity, which have been observed in transport experiments of InAs/GaSb QWs Dularxiv2013 ; Knezfp2012 ; Knezprl2011 . Therefore, although our calculation is based on a simple four band model, all the arguments should remain valid qualitatively in realistic materials. Quantitatively, to determine the regime of the QAH effect, we perform an electronic band structure calculation with an eight-band Kane modelli2009 ; zakharova2001 . The band gap as a function of $d_{InAs}$ and spin of magnetic atom $S_{M}$ is plotted in Fig.4, from which we can extract the phase diagram. With increasing magnetization, we find a gap closing line in the phase diagram, at which the energy dispersion reveals a single Dirac cone type of band crossing, as shown in the inset of Fig. 4. The Hall conductance will change by $\pm e^{2}/h$ across a Dirac cone type of transition. Therefore, the system is in the QAH phase for large magnetization. With an experimentally achievable magnetic doping concentrationWangappendix2013 , our calculation gives a band gap as high as 10 meV for the QAH phase. The exchange coupling induced band gap is large enough to host the QAH effect at a high temperature. Figure 4: (Color online). The band gap is shown as a function of the well thickness of InAs layer and the spin of magnetic impurities. The blue color in the figure shows a gap closing line separating the QAH phase from the normal insulator (NI) phase. The inset image shows a Dirac dispersion at the transition point. All the parameters for Kane model in the calculation are shown in appendix Wangappendix2013 . Discussion and Conclusion \- In conclusion, we have proposed a promising material system for the observation of QAH states at relatively high temperatures ($T\sim 30$ K). The materials involved – Mn-doped InAs/GaSb QWs – are already well-known to show carrier-mediated ferromagnetism Ohnoprl1992 ; AbePE2000 ; NishitaniPE2010 ; Dietlnm2010 . In the absence of Mn-doping and at zero bias, InAs/GaSb QWs have electron-type carriers with a concentration of $\sim 7\times 10^{11}$ cm-2Knezprb2010 . Since Mn-doping adds holes, we estimate a doping of 0.014% Mn atoms to compensate electron carriers and to shift the chemical potential to the hybridization gap. Additional Mn doping will introduce p-type carriers, which can be diminished by tuning the front and back gates. To further increase Mn doping, a compensation doping might be required. Another advantage of InAs/GaSb QWs is the high sample quality with potentially a large mobility of 6,000$cm^{2}/Vs$ for p-type carriers in non- magnetic heterostructures Dularxiv2013 , although it is expected to be somewhat smaller in Mn-doped samples matsuda2004 . Due to the strong exchange coupling, the band gap of the QAH state is able to reach 10 meV, far above the Curie temperature of ferromagnetism (around 30 K). Thus, a well-defined quantized Hall conductance plateau will be expected when the temperature is below Curie temperature. The corresponding experiment is feasible in the present experimental condition. Our calculation based on the standard Zener model has shown a high critical temperature for the QAH effect in magnetically doped InAs/GaSb QWs, which will provide a basis for new spintronics devices with low dissipation. We would like to thank Kai Chang, Rui-Rui Du, Fu-Chun Zhang, Jian-Hua Zhao and Yi Zhou for useful discussions. This work is supported by the Defense Advanced Research Projects Agency Microsystems Technology Office, MesoDynamic Architecture Program (MESO) through the contract numbers N66001-11-1-4105 and N66001-11-1-4110, and in part by FAME, one of six centers of STARnet, a Semiconductor Research Corporation program sponsored by MARCO and DARPA. ## References * (1) F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988) * (2) X.-L. Qi, Y.-S. Wu, and S.-C. Zhang, Phys. Rev. B 74, 085308 (2006) * (3) X.-L. Qi, T. L. Hughes, and S.-C. Zhang, Phys. Rev. B 78, 195424 (2008) * (4) C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and S.-C. Zhang, Phys. Rev. Lett. 101, 146802 (2008) * (5) C. Wu, Phys. Rev. Lett. 101, 186807 (2008) * (6) R. Yu, W. Zhang, H.-J. Zhang, S.-C. Zhang, X. Dai, and Z. Fang, Science 329, 61 (2010) * (7) J. Ding, Z. Qiao, W. Feng, Y. Yao, and Q. Niu, Phys. Rev. B 84, 195444 (2011) * (8) H. Zhang, C. Lazo, S. Blügel, S. Heinze, and Y. Mokrousov, Phys. Rev. Lett. 108, 056802 (2012) * (9) C.-Z. 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Mermin, _Solid State Physics_ (Cengage Learning, 1976) ## Appendix A 8-band Kane model The 8-band Kane model in the bulk basis $\displaystyle\|\Gamma^{6},1/2\rangle=\|S\rangle\|\uparrow\rangle$ $\displaystyle\|\Gamma^{6},-1/2\rangle=\|S\rangle\|\downarrow\rangle$ $\displaystyle\|\Gamma^{8},3/2\rangle=-\frac{1}{\sqrt{2}}\|X+iY\rangle\|\uparrow\rangle$ $\displaystyle\|\Gamma^{8},1/2\rangle=\frac{1}{\sqrt{6}}(2\|Z\rangle\|\uparrow\rangle-\|X+iY\rangle\|\downarrow\rangle$ $\displaystyle\|\Gamma^{8},-1/2\rangle=\frac{1}{\sqrt{6}}(2\|Z\rangle\|\downarrow\rangle+\|X-iY\rangle\|\uparrow\rangle$ $\displaystyle\|\Gamma^{8},-3/2\rangle=\frac{1}{\sqrt{2}}\|X-iY\rangle\|\downarrow\rangle$ $\displaystyle\|\Gamma^{7},1/2\rangle=-\frac{1}{\sqrt{3}}(\|Z\rangle\|\uparrow\rangle+\|X+iY\rangle\|\downarrow\rangle$ $\displaystyle\|\Gamma^{7},1/2\rangle=-\frac{1}{\sqrt{3}}(-\|Z\rangle\|\downarrow\rangle+\|X-iY\rangle\|\uparrow\rangle$ (5) can be written as $\displaystyle H_{Kane}=\left(\begin{array}[]{cccccccc}T&0&-\frac{1}{\sqrt{2}}Pk_{+}&\sqrt{\frac{2}{3}}Pk_{z}&\frac{1}{\sqrt{6}}Pk_{-}&0&-\frac{1}{\sqrt{3}}Pk_{z}&-\frac{1}{\sqrt{3}}Pk_{-}\\\ 0&T&0&-\frac{1}{\sqrt{6}}Pk_{+}&\sqrt{\frac{2}{3}}Pk_{z}&\frac{1}{\sqrt{2}}Pk_{-}&-\frac{1}{\sqrt{3}}Pk_{+}&\frac{1}{\sqrt{3}}Pk_{z}\\\ -\frac{1}{\sqrt{2}}Pk_{-}&0&U+V&-\bar{S}_{-}&R&0&\frac{1}{\sqrt{2}}\bar{S}_{-}&-\sqrt{2}R\\\ \sqrt{\frac{2}{3}}Pk_{z}&-\frac{1}{\sqrt{6}}Pk_{-}&-\bar{S}^{{\dagger}}_{-}&U-V&C&R&\sqrt{2}V&-\sqrt{\frac{3}{2}}\tilde{S}_{-}\\\ \frac{1}{\sqrt{6}}Pk_{+}&\sqrt{\frac{2}{3}}Pk_{z}&R^{{\dagger}}&C^{{\dagger}}&U-V&\bar{S}^{{\dagger}}_{+}&-\sqrt{\frac{3}{2}\tilde{S}_{+}}&-\sqrt{2}V\\\ 0&\frac{1}{\sqrt{2}}Pk_{+}&0&R^{{\dagger}}&\bar{S}_{+}&U+V&\sqrt{2}R^{\dagger}&\frac{1}{\sqrt{2}}\bar{S}_{+}\\\ -\frac{1}{\sqrt{3}}Pk_{z}&-\frac{1}{\sqrt{3}}Pk_{-}&\frac{1}{\sqrt{2}}\bar{S}^{\dagger}_{-}&\sqrt{2}V&-\sqrt{\frac{3}{2}}\tilde{S}^{\dagger}_{+}&\sqrt{2}R&U-\Delta&C\\\ -\frac{1}{\sqrt{3}}Pk_{+}&\frac{1}{\sqrt{3}}Pk_{z}&-\sqrt{2}^{\dagger}&-\sqrt{\frac{3}{2}}\tilde{S}^{\dagger}_{-}&-\sqrt{2}V&\frac{1}{\sqrt{2}}\bar{S}^{\dagger}_{+}&C^{\dagger}&U-\Delta\end{array}\right)$ (14) where $\displaystyle T=E_{c}(z)+\frac{\hbar^{2}}{2m_{0}}[(2F+1)k^{2}_{\|\|}+k_{z}(2F+1)k_{z}]$ $\displaystyle U=E_{v}(z)-\frac{\hbar^{2}}{2m_{0}}(\gamma_{1}k^{2}_{\|\|}+k_{z}\gamma_{1}k_{z})$ $\displaystyle V=-\frac{\hbar^{2}}{2m_{0}}(\gamma_{2}k^{2}_{\|\|}-2k_{z}\gamma_{2}k_{z})$ $\displaystyle R=-\frac{\hbar^{2}}{2m_{0}}(\sqrt{3}\mu k^{2}_{+}-\sqrt{3}\bar{\gamma}k^{2}_{-})$ $\displaystyle\bar{S}_{\pm}=-\frac{\hbar^{2}}{2m_{0}}\sqrt{3}k_{\pm}(\\{\gamma_{3},k_{z}\\}+[\kappa,k_{z}])$ $\displaystyle\tilde{S}_{\pm}=-\frac{\hbar^{2}}{2m_{0}}\sqrt{3}k_{\pm}(\\{\gamma_{3},k_{z}\\}-\frac{1}{3}[\kappa,k_{z}])$ $\displaystyle C=\frac{\hbar^{2}}{2m_{0}}k_{-}[\kappa,k_{z}]$ (15) Here $\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$, $\bar{\gamma}=(\gamma_{2}+\gamma_{3})/2$ and $\mu=(\gamma_{3}-\gamma_{2})/2$ are parameters depending on materials; $\\{$,$\\}$ and $[,]$ are commutative and anticommutative operators. The bulk inversion asymmetrical Hamiltonian is expressed as $\displaystyle H_{BIA}=\left(\begin{array}[]{cccccccc}0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\ 0&0&0&-\frac{1}{2}C_{k}k_{-}&C_{k}k_{z}&-\frac{\sqrt{3}}{2}C_{k}k_{-}&\frac{1}{2\sqrt{2}}C_{k}k_{+}&\frac{1}{\sqrt{2}}C_{k}k_{z}\\\ 0&0&-\frac{1}{2}C_{k}k_{-}&0&\frac{\sqrt{3}}{2}C_{k}k_{+}&-C_{k}k_{z}&0&-\frac{\sqrt{3}}{2\sqrt{2}}C_{k}k_{+}\\\ 0&0&C_{k}k_{z}&-\frac{\sqrt{3}}{2}C_{k}k_{+}&0&-\frac{1}{2}C_{k}k_{+}&\frac{\sqrt{3}}{2\sqrt{2}}C_{k}k_{-}&0\\\ 0&0&-\frac{\sqrt{3}}{2}C_{k}k_{+}&-C_{k}k_{z}&-\frac{1}{2}C_{k}k_{-}&0&\frac{1}{\sqrt{2}}C_{k}k_{z}&-\frac{1}{2\sqrt{2}}C_{k}k_{-}\\\ 0&0&\frac{1}{2\sqrt{2}}C_{k}k_{-}&0&\frac{\sqrt{3}}{2\sqrt{2}}C_{k}k_{+}&\frac{1}{\sqrt{2}}C_{k}k_{z}&0&0\\\ 0&0&\frac{1}{\sqrt{2}}C_{k}k_{z}&-\frac{\sqrt{3}}{2\sqrt{2}}C_{k}k_{-}&0&-\frac{1}{2\sqrt{2}}C_{k}k_{+}&0&0\end{array}\right)$ (24) where $C_{k}$ depends on materials. For magnetically doped semiconductors, the sp-d exchange coupling is described by the phenomenological Kondo-like Hamiltonian, $\displaystyle H_{ex}$ $\displaystyle=-\sum_{\vec{R}_{n}}J(r-\vec{R}_{n}){\bf S_{M}}(\vec{R}_{n})\cdot{\bf s}_{z}$ (25) where $J(r-\vec{R}_{n})$ is the phenomenological coupling parameters for the magnetic impurity dopant at random site $\vec{R}_{n}$, ${\bf S_{M}}(\vec{R_{n}})$ is the spin of impurity atoms and $\bf{s}_{z}$ is the electron spin located at $\vec{r}$. The exchange coupling elements in the mean-field approximation is expressed as $\displaystyle\langle\mu\sigma\|H_{ex}\|\mu^{\prime}\sigma^{\prime}\rangle$ $\displaystyle=-\sum_{\vec{R}_{n}}\langle\mu\|J(r-\vec{R}_{n})\|\mu^{\prime}\rangle{\bf S_{M}}(\vec{R}_{n})\cdot\langle\sigma\|{\bf s}_{z}\|\sigma^{\prime}\rangle=-\sum_{\vec{R}_{n}}\delta_{\mu\mu^{\prime}}\langle\mu\|J\|\mu\rangle S_{M}(\vec{R}_{n})\vec{e}\cdot\langle\sigma\|{\bf s}_{z}\|\sigma^{\prime}\rangle$ (26) where $\mu$, $\mu^{\prime}$ are the band orbital indices, $\sigma$ and $\sigma^{\prime}$ denote spins, and $\vec{e}$ is the magnetization direction of ${\bf S_{M}}$. The exchange parameter $\langle\mu\|J\|\mu\rangle$ depends on the symmetry on the band orbital $\|\mu\rangle$ that could be simplified as $\alpha=\langle S\|J\|S\rangle$ and $\beta=\langle X\|J\|X\rangle=\langle Y\|J\|Y\rangle=\langle Z\|J\|Z\rangle$. In this report, we take $N_{0}\alpha=0.5$ eV and $N_{0}\beta=-0.98$ eV for both InAs and GaSb layersBurchjmmm2008 with $N_{0}$ taking the value of cation concentration. In the basis as presented in Eq. (5), the exchange coupling Hamiltonian reads $\displaystyle H_{ex}=\sum_{\vec{R}_{n}}S_{M}(\vec{R}_{n})\left(\begin{array}[]{cccccccc}-\hat{e}_{z}\alpha&-\hat{e}_{-}\alpha&0&0&0&0&0&0\\\ -\hat{e}_{+}\alpha&\hat{e}_{z}\alpha&0&0&0&0&0&0\\\ 0&0&-\hat{e}_{z}\beta&-\frac{\sqrt{3}}{3}\hat{e}_{-}\beta&0&0&-\frac{\sqrt{6}}{3}\hat{e}_{-}\beta&0\\\ 0&0&-\frac{\sqrt{3}}{3}\hat{e}_{+}\beta&-\frac{1}{3}\hat{e}_{z}\beta&-\frac{2}{3}\hat{e}_{-}\beta&0&\frac{4}{3\sqrt{2}}\hat{e}_{z}\beta&-\frac{\sqrt{2}}{3}\hat{e}_{-}\beta\\\ 0&0&0&-\frac{2}{3}\hat{e}_{+}\beta&\frac{1}{3}\hat{e}_{z}\beta&-\frac{\sqrt{3}}{3}\hat{e}_{-}\beta&\frac{\sqrt{2}}{3}\hat{e}_{-}\beta&\frac{4}{3\sqrt{2}}\hat{e}_{z}\beta\\\ 0&0&0&0&-\frac{\sqrt{3}}{3}\hat{e}_{+}\beta&\hat{e}_{z}\beta&0&\frac{\sqrt{6}}{3}\hat{e}_{+}\beta\\\ 0&0&-\frac{\sqrt{6}}{3}\hat{e}_{+}\beta&\frac{4}{3\sqrt{2}}\hat{e}_{z}\beta&\frac{\sqrt{2}}{3}\hat{e}_{-}\beta&0&\frac{1}{3}\hat{e}_{z}\beta&\frac{1}{3}\hat{e}_{-}\beta\\\ 0&0&0&-\frac{\sqrt{2}}{3}\hat{e}_{+}\beta&\frac{4}{3\sqrt{2}}\hat{e}_{z}\beta&\frac{\sqrt{6}}{3}\hat{e}_{-}\beta&\frac{1}{3}\hat{e}_{+}\beta&-\frac{1}{3}\hat{e}_{z}\beta\end{array}\right)$ (35) where $\hat{e}$ denotes the magnetization direction and $\hat{e}_{\pm}=\hat{e}_{x}\pm i\hat{e}_{y}$. ## Appendix B 4-band effective Hamiltonian ### B.1 BIA and SIA Hamiltonian The 4-band effective Hamiltonian $H_{eff}$ could be obtained by projecting the 8-band Hamiltonian on the following four subbands: $\|E_{1}+\rangle$, $\|H_{1}+\rangle$, $\|E_{1}-\rangle$ and $\|H_{1}-\rangle$, denoted as $\|A\rangle$, $\|B\rangle$, $\|C\rangle$ and $\|D\rangle$, correspondingly. According to symmetry, one can write the above four bases as $\displaystyle\|E_{1}+\rangle=f_{A,1}\|\Gamma^{6},1/2\rangle+f_{A,4}\|\Gamma^{8},1/2\rangle$ $\displaystyle\|H_{1}+\rangle=f_{B,3}\|\Gamma^{8},3/2\rangle$ $\displaystyle\|E_{1}-\rangle=f_{C,2}\|\Gamma^{6},-1/2\rangle+f_{C,5}\|\Gamma^{8},-1/2\rangle$ $\displaystyle\|H_{1}+\rangle=f_{D,6}\|\Gamma^{8},-3/2\rangle$ (36) . Since there are two different atoms in one unit cell that breaks the bulk inversion symmetry, we consider the bulk inversion asymmetry (BIA) contributionWinkler2003 . The BIA Hamiltonian with projection on $\|E_{1}+\rangle$, $\|H_{1}+\rangle$, $\|E_{1}-\rangle$ and $\|H_{1}-\rangle$ bands reads $\displaystyle H_{BIA}=\left(\begin{array}[]{cccc}0&0&\Delta_{e}k_{+}&-\Delta_{0}\\\ 0&0&\Delta_{0}&\Delta_{h}k_{-}\\\ \Delta_{e}k_{-}&\Delta_{0}&0&0\\\ -\Delta_{0}&\Delta_{h}k_{+}&0&0\end{array}\right)$ (41) where the parameters $\Delta_{e}$, $\Delta_{\theta}$, and $\Delta_{h}$ can be determined by the QW geometry. Because of lack of inversion symmetry along the growth direction of InAs/GaSb QW, we also consider the structural inversion asymmetry (SIA) contribution, which takes form of $\displaystyle H_{SIA}=\left(\begin{array}[]{cccc}0&0&i\xi_{e}k_{-}&0\\\ 0&0&0&0\\\ -i\xi^{*}_{e}k_{+}&0&0&0\\\ 0&0&0&0\end{array}\right)$ (46) where $\xi_{e}$ is another parameter that depends on the QW geometry. BHZ model with BIA and SIA corrections confirms the existence of QSH states in InAs/GaSb quantum wells with a set of appropriate parameters. ### B.2 Exchange Hamiltonian With projection of 8-band exchange Hamiltonian on $\|E_{1}+\rangle$, $\|H_{1}+\rangle$, $\|E_{1}-\rangle$ and $\|H_{1}-\rangle$ bands, we can obtain the effective 4-band exchange Hamiltonian expressed as $\displaystyle H_{ex}=\sum_{\vec{R}_{n}}S_{M}(\vec{R}_{n})\left(\begin{array}[]{cccc}\hat{e}_{z}(\alpha F_{1}+\frac{\beta}{3}F_{4})&0&\hat{e}_{-}(\alpha F_{1}+\frac{2\beta}{3}F_{4})&0\\\ 0&\hat{e}_{z}\beta&0&0\\\ \hat{e}_{+}(\alpha F_{1}+\frac{2\beta}{3}F_{4})&0&-\hat{e}_{z}(\alpha F_{1}+\frac{\beta}{3}F_{4})&0\\\ 0&0&0&-\hat{e}_{z}\beta\end{array}\right)$ (51) where $F_{1}=\langle f_{A,1}\|f_{A,1}\rangle=\langle f_{C,2}\|f_{C,2}\rangle=\langle f_{A,1}\|f_{C,2}\rangle$ and $F_{4}=\langle f_{A,4}\|f_{A,4}\rangle=\langle f_{C,5}\|f_{C,5}\rangle=\langle f_{A,4}\|f_{C,5}\rangle$. Since we are interested in the QW growth direction ($\hat{e}_{z}$ direction), we take the effective exchange Hamiltonian as $\displaystyle H_{ex}=sum_{\vec{R}_{n}}S_{M}(\vec{R}_{n})\left(\begin{array}[]{cccc}\alpha F_{1}+\frac{\beta}{3}F_{4}&0&0&0\\\ 0&\beta&0&0\\\ 0&0&-(\alpha F_{1}+\frac{\beta}{3}F_{4})&0\\\ 0&0&0&-\beta\end{array}\right)$ (56) One could write Eq. 56 as $H_{ex}=\sum_{\vec{R}_{n}}S_{M}(\vec{R}_{n})\tilde{s}_{z}$ where $\tilde{s}_{z}$ is the effective spin operator and can be expressed as $\displaystyle\tilde{s}_{z}=\left(\begin{array}[]{cccc}\alpha F_{1}+\frac{\beta}{3}F_{4}&0&0&0\\\ 0&\beta&0&0\\\ 0&0&-(\alpha F_{1}+\frac{\beta}{3}F_{4})&0\\\ 0&0&0&-\beta\end{array}\right)$ (61) ## Appendix C Susceptibility and self-consistent calculation We will derive effective susceptibilities of carriers and magnetic dopants starting from $H_{ex}=\sum_{\vec{R}_{n}}S_{M}(\vec{R}_{n})\tilde{s}_{z}$. One can treat the average magnetization of magnetic impurities as an effective magnetic field $h_{e}=\sum_{\vec{R}_{n}}S_{M}(\vec{R}_{n})=N_{mag}\langle S_{M}\rangle$ that is felt by electron spin $\tilde{s}_{z}$. Here $\langle S_{M}\rangle$ gives the average magnetization of a magnetic impurity and $N_{mag}=N_{0}x_{eff}$, where $N_{0}$ is the cation concentration and $x_{eff}$ is the effective composition of magnetic dopants. Thus, the effective spin susceptibility for carriers is defined as $\langle\tilde{s}_{z}\rangle=\tilde{\chi}_{s}h_{e}$, given by $\displaystyle\tilde{\chi}_{s}=$ $\displaystyle{\lim_{q\rightarrow 0}}Re[\sum_{i,j,\sigma,\sigma^{\prime},\vec{k}}$ (62) $\displaystyle\frac{\|\langle u_{i\sigma,\vec{k}}\|\tilde{s}_{z}\|u_{j\sigma^{\prime},\vec{k}+\vec{q}}\rangle\|^{2}(f_{i\sigma}(\vec{k})-f_{j\sigma^{\prime}}(\vec{k}+\vec{q}))}{E_{j\sigma^{\prime}}(\vec{k}+\vec{q})-E_{i\sigma}(\vec{k})+i\Gamma}]$ from the second order perturbation theoryDietlprb2001 ; Yusci2010 , where $i,j$ denote conduction and valence bands, $\sigma,\sigma^{\prime}$ are spin indices, $u_{i\sigma}$ is the wave function corresponding to energy state Eiσ with spin index $\sigma$, $f_{i\sigma}(\vec{k})$ is the Fermi-Dirac distribution function and $\Gamma$ is band broadening. Similarly, the average value of $\langle\tilde{s}_{z}\rangle$ provides an effective magnetic field $H_{M}=\langle\tilde{s}_{z}\rangle$ for the spin $S_{M}$ of magnetic impurities. The corresponding susceptibility of $S_{M}$ is defined as $\langle S_{M}\rangle=\tilde{\chi}_{M}H_{M}$. In the diluted limit, $\langle S_{M}\rangle$ can be expressed in an empirical form: $\langle S_{M}\rangle=S_{0}B_{s}(\frac{S_{0}H_{M}}{k_{B}T})$, where $B_{s}$ denotes the Brilluoin function $B_{S}(x)=\frac{2S+1}{S}coth(\frac{2S+1}{2S}x)-\frac{1}{2S}coth(\frac{x}{2S})\approx\frac{S+1}{3S}x-\frac{(S+1)(2S^{2}+2S+1)}{900S^{3}}x^{3}$, $S_{0}$ is the spin magnitude of magnetic dopants, $k_{B}$ is the Boltzmann constant and T represents temperatureAshcroftssp . At temperatures close to critical temperature, $\langle S_{M}\rangle\approx\frac{S_{0}(S_{0}+1)H_{M}}{3k_{B}T}$. Therefore, the susceptibility of diluted distributed magnetic dopants reads $\displaystyle\tilde{\chi}_{M}=\frac{S_{0}(S_{0}+1)}{3k_{B}T}$ (63) Finally, One can obtain a self-consistent equation for effective susceptibilities of carriers and magnetic dopants as $N_{mag}\tilde{\chi}_{s}(T_{c})\tilde{\chi}_{M}(T_{c})=1$Dietlprb2001 . The $T_{c}$ can be solved self-consistently according to the above equation. ## Appendix D Conditions for quantum anomalous Hall state In order to obtain QAH state, one need to bring one spin block into a right band order while keeping the other spin block still in an inverted band order. By writing $H_{ex}=Diag(G_{E},G_{H},-G_{E},-G_{H})$, one arrives at $\|G_{E}-G_{H}\|>2\|M_{0}\|$. Another requirement for the realization of QAH state is that the system is still being in an insulating state. The second condition reads $2\|G_{E}\|>\|B+D\|k^{2}_{c}-\|Ak_{c}\|$, with the intersection momenta $k^{2}_{c}=\frac{\|\|2M_{0}\|+\|G_{E}-G_{H}\|\|}{2\|B\|}$. The above two conditions can be written in function of $s_{1}$ and $s_{2}$ used in the paper:(1)$N_{0}x_{eff}\|s_{1}\langle S_{M}\rangle\|>\|M_{0}\|$ and (2) $2N_{0}x_{eff}\langle S_{M}\rangle\|s_{1}+s_{2}\|>\frac{B+D}{2B}\|\|2M_{0}\|+N_{0}x_{eff}\|s_{1}\|\langle S_{M}\rangle\|-A\sqrt{\|\|2M_{0}\|+N_{0}x_{eff}\|s_{1}\|\langle S_{M}\rangle\|/2\|B\|}$. ## Appendix E Parameters The parameters in 8-band Kane model calculation are listed in Table 1, which can be found in Ref. Chaoprb2000, . The band offset between InAs and GaSb layers is about 150 meV. The valence band difference between InAs and AlSb layers is taken as 180 meV. Table 1: The parameters of Kane model for InAs, GaSb and AlSb. \| a[$\AA{}$] \| $E_{g}$[eV] \| $\Delta_{so}$[eV] \| P[$eV\cdot\AA$] \| $\gamma_{1}$ \| $\gamma_{2}$ \| $\gamma_{3}$ \| $\kappa$ \| $C_{k}$ \| F ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- InAs \| 6.058 \| 0.41 \| 0.38 \| 9.19 \| 1.62 \| -0.65 \| 0.27 \| -0.005 \| -0.01 \| -0.005 GaSb \| 6.082 \| 0.8128 \| 0.752 \| 9.23 \| 2.61 \| -0.56 \| 0.67 \| 0.33 \| -0.23 \| 0.333 AlSb \| 6.133 \| 2.32 \| 0.75 \| 8.43 \| 1.46 \| -0.33 \| 0.41 \| -0.92 \| -0.23 \| 0.465 The parameters presented in Table 2 are used in the effective four band calculation. The compositions of magnetic dopants for InAs and GaSb layers we used are 6 % and 1 %, respectively. The effective composition is taken as 3%. Table 2: The parameters of the four band model for InAs/GaSb quantum wells. A[$eV\cdot\AA{}$] \| B[$eV\cdot\AA{}^{2}$] \| C[eV] \| D[eV] \| $M_{0}$[eV] \| $\Delta_{z}$[eV] ---\|---\|---\|---\|---\|--- 0.3 \| -40 \| -$2.97\times 10^{-3}$ \| -30 \| -$5\times 10^{-3}$ \| $2\times 10^{-4}$ $\Delta_{e}$[$eV\cdot\AA{}$] \| $\Delta_{h}$[$eV\cdot\AA{}$] \| $\chi_{e}$[$eV\cdot\AA{}$] \| $F_{1}$ \| $F_{4}$ \| $\Gamma$[eV] $6.6\times 10^{-4}$ \| $6\times 10^{-4}$ \| -$8\times 10^{-4}$ \| 0.52 \| 0.48 \| $3\times 10^{-4}$	arxiv-papers	2013-11-17T03:24:58	2024-09-04T02:49:53.742693	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Qingze Wang, Xin Liu, Hai-Jun Zhang, Nitin Samarth, Shou-Cheng Zhang\n and Chao-Xing Liu", "submitter": "Qingze Wang", "url": "https://arxiv.org/abs/1311.4113" }
1311.4136	MEE Validity of covariance models for the analysis of geographical variation Gilles Guillot111Applied Mathematics and Computer Science Department, Technical University of Denmark, Richard Petersens Plads, Bygning 321, 2800 Lyngby, Denmark. email: [email protected]., René L. Schilling222Technische Universität Dresden, Institut für Mathematische Stochastik, 01062 Dresden, Germany. email: [email protected]., Emilio Porcu333Universidad Federico Santa Maria, Department of Mathematics, Valparaiso, Chile. email: [email protected]. and Moreno Bevilacqua444Universidad de Valparaiso, Department of Statistics, Valparaiso, Chile. email: [email protected]. Summary 1. 1. Due to the availability of large molecular data-sets, covariance models are increasingly used to describe the structure of genetic variation as an alternative to more heavily parametrised biological models. 2. 2. We focus here on a class of parametric covariance models that received sustained attention lately and show that the conditions under which they are valid mathematical models have been overlooked so far. 3. 3. We provide rigorous results for the construction of valid covariance models in this family. 4. 4. We also outline how to construct alternative covariance models for the analysis of geographical variation that are both mathematically well behaved and easily implementable. Keywords: isolation by distance, isolation by ecology, landscape genetics, geostatistics, positive-definite function. ## Background The spatial auto-covariance function quantifies the linear statistical dependence between observations of a variable measured repeatedly across space. It has long been considered a useful tool in studies that involve spatially structured variables in ecology and evolution. It is indeed used at an exploratory and descriptive stage to identify characteristic scales of variation of the data (Levin, 1992; Jackson & Caldwell, 1993; Perry et al., 2002), it plays a central role in methods for spatial prediction (Robertson, 1987; Liebhold et al., 1993; Hay et al., 2009) and it is also involved in regression-type analyses where an explicit spatial model is used as a way to avoid confounding effects due to spatial auto-correlation (Diniz-Filho et al., 2003; Diggle et al., 2007; Rahbek et al., 2007). In recent years, the advent of new genotyping techniques has triggered a flood of population genetics data in ecology. These data-sets are large and of ever increasing sizes, therefore they can not be handled with heavily parametrised models. This situation has rekindled interest in approaches based on the covariance structure of data. Indeed, although of rather descriptive nature compared to biologically explicit models, covariance-based approaches can capture characteristic scales in a parcimonious way and offer computationally efficient ways to recover information about evolutionary processes. In a recent paper, Bradburd et al. (2013) introduced a method to quantify the relative effects of geographic and ecological isolation on genetic differentiation, making it possible to investigate the role of these two factors on migration and gene flow. In the model considered, a sample of individuals from a locality is indexed by its geographic coordinates $x$ and a quantitative environmental variable $e$. The frequency of an allele $f(x,e)$ is assumed to be a suitable transform of a Gaussian random variable $y(x,e)$. One of the key assumptions of the method is that the covariance structure of $y(x,e)$ is of the form: $\textrm{Cov}\left[y(x,e),y(x^{\prime},e^{\prime})\right]=C(h,u)=\frac{1}{\alpha_{0}}\exp\left[-\left(\alpha_{G}h+\alpha_{E}u\right)^{\alpha_{2}}\right]$ (1) hereafter referred to as BRC model. In the formula (1), $h$ and $u$ denote the geographic and environmental distances between samples indexed by $(x,e)$ and $(x^{\prime},e^{\prime})$. The parameters $\alpha_{0},\alpha_{G},\alpha_{E}$ and $\alpha_{2}$ are positive numbers which have to be inferred from the data. The ratio $\alpha_{E}/\alpha_{G}$ can be interpreted as the geographic distance equivalent to a unit environmental distance. Plots of the spatial margins of this covariance function are shown in Figure 1. This model is an extension of a simpler model which is known as the stable (or powered exponential) covariance (Chilès & Delfiner, 1999; Diggle & Ribeiro, 2007) and defined as $K(h)=\frac{1}{\alpha_{0}}\exp[-(\alpha_{G}h)^{\alpha_{2}}].$ (2) The latter has been used by Wasser et al. (2004, 2007) and Rundel et al. (2013) to perform spatial continuous assignment from genetic data, by Novembre & Stephens (2008) to investigate the pattern in principal components of geographically structured population genetics data and by Guillot & Santos (2009) to assess the effect of spatial sampling on the performances of spatial clustering methods. \| ---\|--- Figure 1: Cross-sections of the BRC covariance function $C(h,u)=1/\alpha_{0}\exp\left[-\left(\alpha_{G}h+\alpha_{E}u\right)^{\alpha_{2}}\right]$ with $\alpha_{0}=1$, $\alpha_{G}=1/20$ and $\alpha_{E}=2$. Left panel $\alpha_{2}=0.3$, right panel: $\alpha_{2}=0.9$. The use of spatial covariance functions has a long tradition in statistics and the model and method proposed by Bradburd et al. (2013) can be advocated as well grounded alternative to the widely criticized partial Mantel test (Guillot & Rousset, 2013). The stable covariance and the BRC extension in particular can capture complex patterns of genetic variation, yet they depend on a small number of parameters; as such, they are potentially useful tools for modelling spatial variation in ecology and evolution. Despite its apparent simplicity, this family of covariance functions contains a subtle, but crucial, difficulty: not every function is a covariance function. In this note, we first clarify what is involved in the specification of a covariance model and show that some of the models used earlier are not valid. Then, standing on a firm mathematical footing, we provide results on the range of validity of the models defined above and outline alternative way of constructing valid covariance models. We conclude by discussing implications of our findings for earlier works. ## A covariance model must be a positive-definite function ### Theoretical aspects Considering values $y(x_{i},e_{i})$ at $n$ locations in the geographical $\times$ environmental domain, the variance of a weighted sum can be written $\textrm{Var}\bigg{[}\sum_{i=1}^{n}\lambda_{i}y(x_{i},e_{i})\bigg{]}=\sum_{i=1}^{n}\sum_{j=1}^{n}\lambda_{i}\lambda_{j}\textrm{Cov}[y(x_{i},e_{i}),y(x_{j},e_{j})]$ (3) and it is $\geqslant 0$ for any combination of weights $\lambda_{1},\ldots,\lambda_{n}$. Using a mathematical phrasing: the covariance function $\textrm{Cov}[y(x_{i},e_{i}),y(x_{j},e_{j})]$ is a positive-definite function. Consequently, if one intends to use a certain covariance function considered suitable (e.g. for modelling or computational reasons), one has to make sure that it is positive-definite, i.e. the expression in Equation (3) has to be non-negative. A scientist using a covariance model without this property is likely to face negative variances and undefined probability densities when embedding this covariance function into a Gaussian model. This would also thwart any simulation algorithm based on the Choleski decomposition. In other words, this model would make little sense. It is therefore important to know whether the functions $C$ and $K$ defined by Equations (1-2) are valid in this respect, or in mathematical parlance: When are $C$ and $K$ positive-definite functions? This question has been overlooked so far and holds a number of subtleties, among others the fact that (i) validity in a certain dimension does not imply validity in higher dimensions, and importantly here, (ii) the answer depends on the way distances are measured (for example Euclidean in the plan vs. geodesic distance on the earth’s surface). ### A worked example: spatial prediction of tree abundance data with an invalid covariance model We illustrate some of the consequences of using an invalid covariance model on abundance data for a tree genus in the moist forest of the Congo basin. These data have been published by Mortier et al. (2013) and made publicly available via the R package SCGLR. The variable considered here consists of abundance in thousand 8km by 8 km plots. The location of sampling sites and abundance data are shown in Figure 2. --- Figure 2: Study area and tree abundance data in the tropical forest of the Congo-Basin in thousand 8km$\times$8km plots. The empirical covariance function for this variable displays a regular decrease and the exponential covariance $C(h)=\alpha_{0}^{-1}\exp(-\alpha_{G}\|h\|)$ provides a reasonably good fit as shown in Figure 3. Since the decrease of the empirical covariance is approximately linear, one may want to use a function of the form $C(h)=\alpha_{0}^{-1}(1-\alpha_{G}\|h\|)_{+}$, where $(a)_{+}$ denotes positive part of $a$, that is $C(h)=\alpha_{0}^{-1}\left(1-\alpha_{G}\|h\|\right)$ whenever $\quad\|h\|<\frac{1}{\alpha_{G}}$ and $0$ elsewhere. This covariance is known as the triangle model in the Geostatistics literature. This function provides visually an even better fit (Fig. 3). Figure 3: Empirical and theoretical covariances for the tree abundance data. Distances are in kilometers. Unfortunately, this covariance is valid in one dimension but not in two dimensions (Chilès & Delfiner, 1999), which has consequences illustrated below. Using the exponential covariance as a covariance model for tree abundance (which is a valid model in any dimension) enables us to perform spatial prediction (Fig. 4 top left panel) and to derive an assessment of the error realized by the prediction known as kriging variance (Fig. 4 top right panel). Both maps are well behaved and seem to make sense ecologically and statistically. Using the triangle model to compute spatial prediction and kriging variance does not bring any difficulty computer-wise. The fact that the triangle function is not positive-definite shows up in the kriging variance: the latter displays spatial variation that does not mirror the location of the sampling sites, it is negative in several areas (Fig. 4 bottom right panel) and takes a minimum of $\sigma^{2}_{K}=-5720$. For short, using the triangle covariance in 2 dimensions leads to non-sensical results. Figure 4: Spatial prediction of tree abundance data in the tropical forest of the Congo-Basin. Top: computations with an exponential covariance function. Bottom: computations with a triangular function. Left: abundance map obtained by simple kriging. Right: kriging variance (white areas in bottom right panel correspond to negative kriging variances). Eastings and Northings in kilometers. ## Validity of the stable and the BRC models In addition to $\alpha_{2}$, the models we consider involve three or four parameters. Positive-definiteness is, however, not influenced by $\alpha_{0},\alpha_{G}$ and $\alpha_{E}$ as long as they are positive. Therefore, without loss of generality on the mathematical side, we assume from now on that $\alpha_{0}=\alpha_{G}=\alpha_{E}=1$. ### Euclidean distance If $h$ is $\\|x-x^{\prime}\\|=\sqrt{(x_{1}-x_{1}^{\prime})^{2}+\ldots+(x_{d}-x_{d}^{\prime})^{2}}$ (the Euclidean distances in ${{\mathds{R}}^{d}}$) the stable covariance $K(h)=\exp\left[-h^{\alpha_{2}}\right]$ is a valid covariance model in ${{\mathds{R}}^{d}}$ if and only if $\alpha_{2}\in[0,2]$. Arguments proving these results are given by Schoenberg (1938). For $u$ defined as $\|e-e^{\prime}\|$, the BRC model defined by $C(h,u)=\exp\left[-\left(h+u\right)^{\alpha_{2}}\right]$ is a valid covariance model on ${{\mathds{R}}^{d}}\times\mathds{R}$ if and only if $\alpha_{2}\in[0,1]$. We give a proof of this original result in the Appendix. ### Geodesic distance We denote by $\mathbb{S}^{d-1}$ the unit sphere in ${{\mathds{R}}^{d}}$ and define now $h$ as $\arccos\Big{(}\sum_{i=1}^{d}x_{i}x_{i}^{\prime}\Big{)}$ (geodesic or great circle distance on the sphere) while keeping $u=\|e-e^{\prime}\|$. The stable model is a valid covariance model in $\mathbb{S}^{d-1}$ if and only if $\alpha_{2}\in[0,1]$. Arguments proving this result are given by Gneiting (2013). For the general BRC model on $\mathbb{S}^{d-1}\times\mathds{R}$, we found counter-examples showing that for $\alpha_{2}=1.001$, the model is not valid. Using a continuity argument, this means that no model with $\alpha_{2}\geqslant 1.001$ will be valid. An instance is as follows: we consider three points on the sphere with (Lon,Lat) coordinates $x_{1}=(-60.0,60)$, $x_{2}=(-60.1,60)$, $x_{3}=(-60.2,60)$ and values $e_{1}=0.1$, $e_{2}=0.2$, $e_{3}=0.3$ of an environmental variable. We also set $\alpha_{0}=1$, $\alpha_{G}=\alpha_{e}=1/300$, and $\alpha_{2}=1.01$. Under the BRC model the covariance matrix associated to this configuration is a $9\times 9$ matrix whose minimum eigenvalue is approximately $-1.84\times 10^{-5}$, which shows that the matrix is not positive-definite. A general theoretical result similar to the case of Euclidean distances is still lacking, but we conjecture that the BRC model on $\mathbb{S}^{d-1}\times\mathds{R}$ is valid if and only if $\alpha_{2}\in[0,1]$. ### Other distances in the plan or the sphere It is a common practice in ecology to measure distances in terms of cumulative cost for an individual to move from a geographical location to another. This is referred to as cost or resistance distance. There is considerable flexibility in the way such a distance can be obtained and the validity of the BRC model should be checked on a case by case basis. From the previous paragraphs, it is clear that the choice of the distance is not innocuous and that a distance that makes sense ecologically may not lead to a model that is well behaved mathematically. We note also that if the cost distance is obtained via numerical values (without a mathematical expression), there is little hope for proving the validity of a covariance model as this would involve checking all possible sums of the form given in Equation (3). ## Alternate covariance models for applications in evolutionary biology ### Valid gluing of the Euclidean geographical distance and the environmental distances If the distance on ${{\mathds{R}}^{d}}\times\mathds{R}$ is defined as $d[(x,e),(x^{\prime},e^{\prime})]=\sqrt{\sum_{i=1}^{d}(x_{i}-x_{i}^{\prime})^{2}+(e-e^{\prime})^{2}}$ (4) then any valid covariance model on ${{\mathds{R}}^{d}}\times\mathds{R}$ can be used. In particular, $\exp(-d^{\;\alpha_{2}})$ is a valid model for $\alpha_{2}\in(0,2]$. See classical textbooks by Chilès & Delfiner (1999) and Diggle & Ribeiro (2007) for alternative choices. With a valid model in hands, quantifying the relative effect of distance and environment variables as suggested by Bradburd et al. (2013) can be done by re-scaling the distance as $\sqrt{\sum_{i=1}^{d}\alpha_{G}(x_{i}-x_{i}^{\prime})^{2}+\alpha_{E}(e-e^{\prime})^{2}}$. For data gathered at large scale, one has to use geographic distances on the sphere and there seems to be no straightforward way to combine the geodesic distance with the environmental distance along this line to obtain a valid model. #### Sums and products of valid models If $C_{G}(h)$ is a valid model on ${{\mathds{R}}^{d}}$ or $\mathbb{S}^{d-1}$ and $C_{E}(u)$ is a valid model on $\mathds{R}$, then $C_{1}(h,u)=C_{G}(h)+C_{E}(u)$ (5) and $C_{2}(h,u)=C_{G}(h)\times C_{E}(u)$ (6) are valid models for which we give examples in Table 1. ### Space-time covariance models Covariance models developed to handle spatio-temporal data can be used readily for the analysis of data of the form considered by Bradburd et al. (2013). The list of such models on ${{\mathds{R}}^{d}}\times\mathds{R}$ or $\mathbb{S}^{d-1}\times\mathds{R}$ is still limited but it comes with clear guidelines about the valid range of parameters. We refer interested readers to recent spatial statistics books Gelfand et al. (2010) and Porcu et al. (2010). Model name \| Covariance function \| Parameter range ---\|---\|--- Stable \| $C(h)=\exp\left(-h^{\alpha}\right)$ \| $\alpha\in(0,2]$ on $\mathds{R}^{d}$ \| \| $\alpha\in(0,1]$ on $\mathbb{S}^{d-1}$ BRC \| $C(h,u)=\exp\left(-(h+u)^{\alpha}\right)$ \| $\alpha\in(0,1]$ on $\mathds{R}^{d}\times\mathds{R}$ \| \| Unknown for $\mathbb{S}^{d-1}\times\mathds{R}$ Modified BRC \| $C((x,e),(x^{\prime},e^{\prime}))=$ \| \| $\exp\left(-\sqrt{\sum_{i=1}^{d}(x_{i}-x_{i}^{\prime})^{2}+(e-e^{\prime})^{2}}^{\;\alpha}\right)$ \| $\alpha\in(0,2]$ on $\mathds{R}^{d}\times\mathds{R}$ Sum of stable \| $C(h,u)=\exp\left(-h^{\alpha})+\exp(-u^{\beta}\right)$ \| $(\alpha,\beta)\in(0,2]\times(0,2]$ on $\mathds{R}^{d}\times\mathds{R}$ models \| \| $(\alpha,\beta)\in(0,1]\times(0,2]$ on $\mathbb{S}^{d-1}\times\mathds{R}$ Product of stable \| $C(h,u)=\exp\left(-h^{\alpha})\times\exp(-u^{\beta}\right)$ \| $(\alpha,\beta)\in(0,2]\times(0,2]$ on $\mathds{R}^{d}\times\mathds{R}$ models \| \| $(\alpha,\beta)\in(0,1]\times(0,2]$ on $\mathbb{S}^{d-1}\times\mathds{R}$ Table 1: Summary of covariance models with range of validity. In the table, $u$ is the environmental distance $\|e-e^{\prime}\|$ while $h$ refers to the Euclidean distance $\\|x-x^{\prime}\\|=\sqrt{\sum_{i=1}^{d}(x_{i}-x_{i}^{\prime})^{2}}$ on $\mathds{R}^{d}$, and to the geodesic distance $\arccos\Big{(}\sum_{i=1}^{d}x_{i}x_{i}^{\prime}\Big{)}$ on the unit sphere $\mathbb{S}^{d-1}$ of ${{\mathds{R}}^{d}}$. ## Conclusion There are limitations on the parameter range for the stable and the BRC models and they depend on the way distances are measured. We provide clear guidelines for the case of Euclidean distances while the case of geodesic distances still requires more work. For cost distances, a general theoretical statement is not possible and checking the validity for numerically-derived distances seems out of reach. We recommend users to be cautious when using cost distances in this context. These limitations have remained un-noticed so far and some of the earlier works making use of these models have been based on invalid parameter ranges. 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A map $\gamma$ from $\mathds{R}^{d}\times\mathds{R}$ into $\mathds{R}$ is called a _variogram_ if it represents the variance of the increments of an intrinsically stationary random field, i.e. $\gamma\left(x_{j}-x_{i},e_{j}-e_{i}\right)=\textrm{Var}\left(Z(x_{j},e_{j})-Z(x_{i},e_{i})\right).$ Variograms are real-valued _negative definite functions_ , i.e. for any finite family of points $\\{(x_{i},e_{i})\\}_{i=1}^{N}$ and constants $\\{a_{i}\\}_{i=1}^{N}$ with $\sum_{i=1}^{N}a_{i}=0$, we have $\sum_{i=1}^{N}\sum_{j=1}^{N}\gamma\left(x_{j}-x_{i},e_{j}-e_{i}\right)a_{i}a_{j}\leq 0.$ The connection between variograms and covariance functions is due to Schoenberg (1938): $C:{{\mathds{R}}^{d}}\times\mathds{R}\to\mathds{R}$ is a covariance function if and only if $C(x,e)=\exp(-r\gamma(x,e))$ where $\gamma(x,e)$ is a variogram. Thus, we can re-cast the question about the valid range of parameter in the following way: for which $\alpha_{2}>0$ is the function $(h,u)\mapsto\left(h+u\right)^{\alpha_{2}}$ a variogram? (7) As before, $h=\\|x\\|=\sqrt{x_{1}^{2}+\ldots+x_{d}^{2}}$ is the Euclidean distance (taken from the origin) in ${{\mathds{R}}^{d}}$ and $u=\|e\|$ is the ecological distance (in $\mathds{R}$, also relative to the origin). In order to simplify the notation, we write $\alpha$ instead of $\alpha_{2}$. It is known that every _continuous_ variogram on ${{\mathds{R}}^{n}}$ is given by a _Lévy–Khintchine formula_ : $\gamma(\eta)=\frac{1}{2}\eta\cdot Q\eta+\int_{y\neq 0}\Big{(}1-\cos\Big{(}{\textstyle\sum\limits_{i=1}^{n}\eta_{i}y_{i}}\Big{)}\Big{)}\,\nu(dy),\quad\eta\in{{\mathds{R}}^{n}},$ (8) where $Q$ is a symmetric positive semi-definite $n\times n$ matrix, and $\nu$ is a measure on ${{\mathds{R}}^{n}}\setminus\\{0\\}$ such that $\int_{y\neq 0}\\|y\\|^{2}/(1+\\|y\\|^{2})\,\nu(dy)<\infty$; $\gamma$ is uniquely determined by $(Q,\nu)$ and vice versa. Typical examples of continuous variograms on ${{\mathds{R}}^{n}}$ are $\\|\eta\\|^{2},\quad\eta\cdot Q\eta,\quad 1-\cos y\cdot\eta,\quad\log(1+\\|\eta\\|^{2}),\quad\\|\eta\\|^{\alpha}\;(0<\alpha<2).$ A good source for variograms (which are also known as negative definite real functions) are the monographs by Berg & Forst (1975) and R.L. Schilling & Vondraček (2012). We only need the following properties. (A) Subadditivity: If $\gamma(\eta)$ is a continuous variogram, then $\sqrt{\gamma(\eta+\eta)}\leqslant\sqrt{\gamma(\eta)}+\sqrt{\gamma(\eta)}$. In particular, $\gamma(\eta)$ grows at most like $\\|\eta\\|^{2}$ as $\\|\eta\\|\to\infty$. (B) Closure under pointwise limits: If $\gamma_{j}(\eta),j=1,2,\ldots$ are continuous variograms such that the limit $\gamma(\eta):=\lim_{j\to\infty}\gamma_{j}(\eta)$ exists and is continuous, then $\gamma(\eta)$ is a continuous variogram. (C) Let $\eta\mapsto\gamma(\eta)$ be a continuous variogram on $\mathds{R}^{d}$ and write $\eta=(\eta^{\prime},\eta^{\prime\prime})$ where $\eta^{\prime}\in{{\mathds{R}}^{n}}$, $\eta^{\prime\prime}\in\mathds{R}^{d-n}$. Then $\eta^{\prime}\mapsto\gamma(\eta^{\prime},0)$ is a continuous variogram on ${{\mathds{R}}^{n}}$. (D) Let $\gamma(\eta^{\prime})$, $\psi(\eta^{\prime\prime})$ be continuous variograms on ${{\mathds{R}}^{n}}$ and $\mathds{R}^{m}$, respectively. Then $(\eta^{\prime},\eta^{\prime\prime})\mapsto\gamma(\eta^{\prime})+\psi(\eta^{\prime\prime})$ is a continuous variogram on ${{\mathds{R}}^{d}}=\mathds{R}^{n+m}$. The variogram property is also preserved under a technique called _Bochner’s subordination_ , cf. R.L. Schilling & Vondraček (2012). At the level of the random variables this corresponds to a mixture of the processes with a further infinitely divisible random variable, at the level of variograms this is just a composition with the class of so-called _Bernstein functions_. These are also given by a Lévy–Khintchine formula $f(\lambda)=b\lambda+\int_{0+}^{\infty}(1-e^{-s\lambda})\,\mu(ds),\quad\lambda\geqslant 0,$ where $b\geqslant 0$ and $\mu$ is a measure on $(0,\infty)$ such that $\int_{0}^{\infty}s(1+s)^{-1}\,\mu(ds)<\infty$. Typical examples of Bernstein functions are $\lambda,\quad\lambda^{\alpha}\;(0<\alpha<1),\quad\log(1+\lambda).$ ###### Theorem 1. If $\gamma(\eta)$ is a continuous variogram and $f$ is a Bernstein function, then $f(\gamma(\eta))$ is again a continuous variogram. We now have all ingredients for the ###### Proof of the valid parameter range. Note that $\psi(\eta)=\\|\eta\\|=\sqrt{\eta_{1}^{2}+\ldots+\eta_{d}^{2}}$ and $\phi(\tau)=\|\tau\|$ are continuous variograms in $\mathds{R}^{d}$ and $\mathds{R}$, respectively. Moreover, take the Bernstein function $f(\lambda)=\lambda^{\alpha}$, $\lambda>0$; the corresponding mixing random variables are one-sided $\alpha$-stable random variables (if $0<\alpha<1$) or a deterministic drift (if $\alpha=1$). By property (D) and subordination, $(\eta,\tau)\mapsto\gamma_{\alpha}(\eta,\tau):=(\\|\eta\\|+\|\tau\|)^{\alpha},\quad 0<\alpha\leqslant 1,$ (9) is a continuous variogram. On the other hand, by the quadratic growth property, see (A), it is clear that $\gamma_{\alpha}(\eta,\tau)$ is not a variogram if $\alpha>2$. Let us now consider the case where $\alpha\in(1,2]$. Assume first that $\alpha=2$. Then $(\\|\eta\\|+\|\tau\|)^{2}=\\|\eta\\|^{2}+2\,\\|\eta\\|\cdot\|\tau\|+\tau^{2}.$ Since $\\|\eta\\|^{2}+\tau^{2}$ would appear in the Lévy–Khintchine formula (8) as part of the expression involving the matrix $Q$, it is enough to prove or disprove that the mixed term $c(\eta,\tau):=\\|\eta\\|\cdot\|\tau\|$ is a continuous variogram. But $\sqrt{\\|\eta\\|\cdot\|\tau\|}=\sqrt{c(\eta,\tau)}\geqslant\sqrt{c(\eta,0)}+\sqrt{c(0,\tau)}=0,$ which means that $\sqrt{c(\eta,\tau)}$ is _not_ sub-additive, violating the subadditivity property (A), i.e. $(\eta,\tau)\mapsto(\\|\eta\\|+\|\tau\|)^{2}\quad\text{is not a variogram}.$ Now we use the property (B): Clearly, $\lim_{j\to\infty}(\\|\eta\\|+\|\tau\|)^{2-1/j}=(\\|\eta\\|+\|\tau\|)^{2}$. Since variograms are preserved under pointwise limits, we conclude from this, and the subordination argument, that there is some $1\leqslant b<2$ such that $(\eta,\tau)\mapsto(\\|\eta\\|+\|\tau\|)^{\alpha}\quad\text{is\ \ }\begin{cases}\text{\ a continuous variogram if}&0<\alpha\leqslant b\\\ \text{\ not a continuous variogram if}&\alpha>b.\end{cases}$ We conclude the proof by showing that necessarily $b=1$. Use Property (C) above, and suppose that the function in Equation (9) is a variogram on $\mathds{R}^{d}$. Then the function $\tilde{\gamma}(\eta_{1},\tau):=\gamma_{\alpha}\left((\eta_{1},0,\ldots,0),\tau\right)$ is a variogram on $\mathds{R}\times\mathds{R}$. Arguments by Zastavnyi (2000) show that this is true if and only if $\alpha\leq 1$, which completes the proof. ∎	arxiv-papers	2013-11-17T09:35:43	2024-09-04T02:49:53.754588	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Gilles Guillot, Ren\\'e Schilling, Emilio Porcu, Moreno Bevilacqua", "submitter": "Gilles Guillot", "url": "https://arxiv.org/abs/1311.4136" }
1311.4145	# Derivation of reaction cross sections from experimental elastic backscattering probabilities V.V.Sargsyan1,2, G.G.Adamian1, N.V.Antonenko1, and P.R.S.Gomes3 1Joint Institute for Nuclear Research, 141980 Dubna, Russia 2International Center for Advanced Studies, Yerevan State University, 0025 Yerevan, Armenia 3Instituto de Fisica, Universidade Federal Fluminense, Av. Litorânea, s/n, Niterói, R.J. 24210-340, Brazil ###### Abstract The relationship between the backward elastic scattering probabilities and reaction cross sections is derived. This is a a very simple and useful method to extract reaction cross sections for heavy ion systems. We compare the results of our method with those using the traditional full elastic scattering angular distributions, for several systems, at energies near and above the Coulomb barrier. From the calculated reaction and capture cross sections using the present method, we derive the cross sections of other mechanisms for nearly spherical systems. ###### pacs: 25.70.Jj, 24.10.-i, 24.60.-k Key words: reaction cross section, elastic scattering probability at backward angle, quasielastic scattering excitation function at backward angle, capture cross section ## I Introduction For a long time, measurements of elastic scattering angular distributions covering full angular ranges and optical model analysis have been used for the determination of reaction cross sections. The traditional method consists in deriving the parameters of the complex optical potentials which fit the experimental elastic scattering angular distributions and then to derive the reaction cross sections predicted by these potentials. This can be done because there is well known and clear relationship between the reaction and the elastic scattering processes due to the conservation of the total reaction flux. Any loss from the elastic scattering channel directly contributes to the reaction channel and vice versa. The direct measurement of the reaction cross section is a very difficult task, since it would require the measurement of individual cross sections of all reaction channels, and most of them could be reached only by specific experiments. This would require different experimental setups not always available at the same laboratory and, consequently, such direct measurements would demand a large amount of beam time and would take probability some years to be reached. On the other hand, the measurement of elastic scattering angular distributions is much simpler than that. Even so, both the experimental part and the analysis of this latter method are not so simple. In the present work, as an extension of previous works of our group Sargsyan13a ; Sargsyan13b , we present a much simpler method to determine reaction cross sections than the one using full elastic scattering angular distribution data. It consists of measuring only elastic scattering at one backward angle and from that the extraction of the reaction cross sections can be easily done. The paper is organized in the following way. In Sec. II we derive the formula for the extraction of the reaction cross sections by employing the experimental elastic scattering excitation function at backward angle. In Sec. III we use this formula to extract the reaction cross sections for several systems and then we compare the results with those extracted from the experimental elastic scattering angular distributions for the same systems (4He + 92Mo, 4He + 110,116Cd, 4He + 112,120Sn, 6,7Li + 64Zn, and 16O + 208Pb). In this section we also show the comparison of calculated and experimental capture cross section for the 6,7Li + 64Zn systems, and we predict the approximate cross sections for transfer + inelastic processes for those systems. In Sec. IV the paper is summarized. ## II Relationship between reaction cross sections and elastic scattering excitation function at backward angle Quasi-elastic scattering is defined as the sum of elastic scattering, inelastic excitations and a few nucleon transfer reactions. So, one defines the quasi-elastic scattering probability as $P_{qe}(E_{\mathrm{c.m.}},J)=P_{el}(E_{\mathrm{c.m.}},J)+P_{in}(E_{\mathrm{c.m.}},J)+P_{tr}(E_{\mathrm{c.m.}},J),$ (1) where $P_{el}$, $P_{in}$, and $P_{tr}$ are the elastic scattering, inelastic and transfer probabilities, respectively. The total reaction probability may be written as $P_{R}(E_{\mathrm{c.m.}},J)=P_{in}(E_{\mathrm{c.m.}},J)+P_{tr}(E_{\mathrm{c.m.}},J)+P_{cap}(E_{\mathrm{c.m.}},J)+P_{BU}(E_{\mathrm{c.m.}},J)+P_{DIC}(E_{\mathrm{c.m.}},J),$ (2) where $P_{R}$ refers to the non-elastic reaction channel probability, $P_{cap}$ is the capture probability (sum of evaporation-residue formation, fusion-fission, and quasi-fission probabilities or sum of fusion and quasi- fission probabilities), $P_{DIC}$ is the deep inelastic collision probability, and $P_{BU}$ is the breakup probability, important particularly when weakly bound nuclei are involved in the reaction Canto06 . Note that the deep inelastic collision process is only important at large energies above the Coulomb barrier. The deep inelastic collision process one can neglect because we are concerned with low energy region. From the conservation of the total reaction flux one can write Sargsyan13a ; Canto06 the expression $P_{el}(E_{\mathrm{c.m.}},J)+P_{R}(E_{\mathrm{c.m.}},J)=1$ (3) or $P_{qe}(E_{\mathrm{c.m.}},J)+P_{cap}(E_{\mathrm{c.m.}},J)+P_{BU}(E_{\mathrm{c.m.}},J)=1.$ (4) Here and in the following of this paper, we neglect the deep inelastic collision, since we are concerned with low energies. Thus, one can extract the reaction probability $P_{R}(E_{\mathrm{c.m.}},J=0)$ at $J=0$ from the experimental elastic scattering probability $P_{el}(E_{\mathrm{c.m.}},J=0)$ at $J=0$: $P_{R}(E_{\mathrm{c.m.}},J=0)=1-P_{el}(E_{\mathrm{c.m.}},J=0)=1-d\sigma_{el}(E_{\mathrm{c.m.}})/d\sigma_{Ru}(E_{\mathrm{c.m.}}).$ (5) Here, the elastic scattering probability Canto06 ; Timmers ; Timmers1 ; Timmers2 ; Zhang $P_{el}(E_{\mathrm{c.m.}},J=0)=d\sigma_{el}/d\sigma_{Ru}$ (6) for angular momentum $J=0$ is given by the ratio of the elastic scattering differential cross section and Rutherford differential cross section at 180 degrees. Furthermore, one can approximate the $J$ dependence of the reaction probability $P_{R}(E_{\mathrm{c.m.}},J)$ at a given energy $E_{\mathrm{c.m.}}$ by shifting the energy Bala : $P_{R}(E_{\mathrm{c.m.}},J)\approx P_{R}(E_{\mathrm{c.m.}}-\frac{\hbar^{2}\Lambda}{2\mu R_{b}^{2}}-\frac{\hbar^{4}\Lambda^{2}}{2\mu^{3}\omega_{b}^{2}R_{b}^{6}},J=0),$ (7) where $\Lambda=J(J+1)$, $R_{b}=R_{b}(J=0)$ is the position of the Coulomb barrier at $J=0$, $\mu=m_{0}A_{1}A_{2}/(A_{1}+A_{2})$ is the reduced mass ($m_{0}$ is the nucleon mass), and $\omega_{b}$ is the curvature of the s-wave potential barrier. Employing Eqs. (5) and (7), converting the sum over the partial waves $J$ into an integral, and expressing $J$ by the variable $E=E_{\mathrm{c.m.}}-\frac{\hbar^{2}\Lambda}{2\mu R_{b}^{2}}$, we obtain the following simple expression: $\sigma_{R}(E_{\mathrm{c.m.}})=\frac{\pi R_{b}^{2}}{E_{\mathrm{c.m.}}}\int_{0}^{E_{\mathrm{c.m.}}}dE[1-d\sigma_{el}(E)/d\sigma_{Ru}(E)][1-\frac{4(E_{\mathrm{c.m.}}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}].$ (8) The formula (8) relates the reaction cross section with elastic scattering excitation function at backward angle. By using the experimental elastic scattering probabilities $P_{el}(E_{\mathrm{c.m.}},J=0)$ and Eq. (8) one can obtain the reaction cross sections. It is important to mention that since the generalized form of the optical theorem connects the reaction cross section and forward elastic scattering amplitudeCanto06 , from our method we show that the forward and backward elastic scattering amplitudes are related with each other. ## III Results of calculations ### III.1 Reaction cross sections In the following, we show the results of our method to extract the reaction cross section, using Eq. (8). To calculate the position $R_{b}$ and frequency $\omega_{b}$ of the Coulomb barrier, we use the nucleus-nucleus interaction potential $V(R,J)$ of Ref. Pot . For the nuclear part of the nucleus-nucleus potential, the double-folding formalism with the Skyrme-type density-dependent effective nucleon-nucleon interaction is employed Pot . To confirm the validity of our method of the extraction of $\sigma_{R}$, firstly we compare the obtained reaction cross sections with those extracted from the traditional experimental elastic scattering angular distributions plus optical potential method. The results from our method are shown as solid (red color on-line) and dashed (blue color on-line) lines in all figures from Fig. 1 to Fig. 6, whereas the results obtained from the traditional full elastic scattering angular distribution data are shown by solid squares. As the backscattering elastic data were not taken at 180 degree, but rather at backward angles in the range from 150 to 170 degrees, the corresponding center of mass energies were corrected by the centrifugal potential at the experimental angle, as suggested by Timmers et al. Timmers . In figures 7 and 8 we also show results of our calculations for the capture cross sections, and other curves are shown. Figure 1: (Color on line) The extracted reaction cross sections employing Eq. (8) (solid line) for the 4He + 92Mo reaction. The used experimental elastic scattering probabilities at backward angle are from Ref. hemo . The reaction cross sections extracted from the experimental elastic scattering angular distribution with optical potential are presented by squares hemo . Figure 2: (Color on line) The extracted reaction cross sections employing Eq. (8) (lines) for the 4He + 110Cd reaction. The used experimental elastic scattering probabilities at backward angle are from Refs. hecd1 ; hecd3 (solid line) and Ref. hecd2 (dashed lines). The reaction cross sections extracted from the experimental elastic scattering angular distribution with optical potential are presented by squares hemo . Figure 3: (Color on line) The same as in Fig. 2 but for the 4He + 116Cd reaction. Figure 4: (Color on line) The extracted reaction cross sections employing Eq. (8) (solid line) for the 4He + 112Sn reaction. The used experimental elastic scattering probabilities at backward angle are from Ref. hemo . The reaction cross sections extracted from the experimental elastic scattering angular distribution with optical potential are presented by squares hemo . Figure 5: (Color on line) The same as in Fig. 4 but for the 4He + 120Sn reaction. Figure 6: (Color on line) The extracted reaction cross sections employing Eq. (8) (solid line) for the 16O + 208Pb reaction. The used experimental elastic scattering probabilities at backward angle are from Ref. opb . The reaction cross sections extracted from the experimental elastic scattering angular distribution with optical potential are presented by squares opb . As it can be observed in Figs. 1–8, there is a good agreement between reaction cross sections extracted from experimental elastic scattering at backward angle and from the experimental elastic scattering angular distributions with optical potential for the reactions 4He + 92Mo, 4He + 110,116Cd, 4He + 112,120Sn, 16O + 208Pb, and 6,7Li + 64Zn at energies near and above the Coulomb barrier. One can see that the used formula (8) is suitable not only for almost spherical nuclei, but also for the reactions with slightly deformed target-nuclei. The deformation effect is effectively contained in the experimental $P_{el}$. For very deformed nuclei, it is not possible experimentally to separate elastic events from the low lying inelastic excitations. In our calculations, to obtain better agreement for the reactions 16O+208Pb and 6Li+64Zn, the extracted reaction cross sections were shifted in energy by 0.3 MeV to higher energies and 0.4 MeV to lower energies with respect to the measured experimental data, respectively. There is no clear physical justification for the energy shift. The most probable reason might be related with the uncertainty associated with the elastic scattering data. ### III.2 Capture and transfer plus breakup plus inelastic cross sections By using a similar formalism as the one presented in Section II and Eq. (4), the capture cross section can be written, if one assumes that $P_{BU}=0$, since it is much smaller than $P_{qe}$, as Sargsyan13b $\sigma_{cap}(E_{\mathrm{c.m.}})=\frac{\pi R_{b}^{2}}{E_{\mathrm{c.m.}}}\int_{E_{\mathrm{c.m.}}-\frac{\hbar^{2}\Lambda_{cr}}{2\mu R_{b}^{2}}}^{E_{\mathrm{c.m.}}}dE[1-d\sigma_{qe}(E)/d\sigma_{Ru}(E)][1-\frac{4(E_{\mathrm{c.m.}}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}],$ (9) where in $\Lambda_{cr}=J_{cr}(J_{cr}+1)$, $J_{cr}$ is the critical angular momentum at which potential pocket in the nucleus-nucleus interaction potential $V(R,J)$ vanishes and capture does not occur. So, the capture cross sections can be extracted from the experimental quasielastic scattering probabilities $P_{qe}(E_{\mathrm{c.m.}},J=0)=d\sigma_{qe}/d\sigma_{Ru}$, as it was already demonstrated in Ref. Sargsyan13b . Figure 7: (Color on line) The extracted reaction (solid line) and capture (dashed line) cross sections employing Eqs. (8) and (9) for the 6Li + 64Zn reaction. The used experimental elastic and quasielastic scattering probabilities at backward angle are from Ref. Torresi ; Pietro . The reaction cross sections extracted from the experimental elastic scattering angular distribution with optical potential and capture (fusion) cross sections are presented by circles Torresi ; Pietro , triangles Gomes034 ; GomesPLB04 and squares Gomes034 ; GomesPLB04 , stars Torresi ; Pietro , respectively. Figure 8: (Color on line) The same as in Fig. 7, but for the 7Li + 64Zn reaction. The reaction cross sections extracted from the experimental elastic scattering angular distribution with optical potential and capture (fusion) cross sections are presented by circles Gomes034 ; GomesPLB04 and squares Gomes034 ; GomesPLB04 , respectively. In figures 7 and 8 we also show the results of our calculations for capture cross sections of the 6,7Li+64Zn systems, for which the fusion process can be considered to exhaust the capture cross section. Figure 7 shows that the extracted and experimental capture cross sections are in good agreement for the 6Li+64Zn reaction at energies near and above the Coulomb barrier for the data taken in Refs. Torresi ; Pietro . Note that the extracted capture excitation function is shifted in energy by 0.7 MeV to higher energies with respect to the experimental data. This could be the result of different energy calibrations in the experiments on the capture measurement and quasielastic scattering. The data taken in Refs. Gomes034 ; GomesPLB04 are below our predictions. This fact was already observed and commented in Ref. Gomes09 , and the reason given for the low fusion cross sections was as owing to experimental problems with the high electronic threshold of the events, when the data were taken. Figure 8 shows that the capture cross section for the 7Li+64Zn system, obtained in the same works of Refs. Gomes034 ; GomesPLB04 is also below our predictions. The same reason for this behavior as for the 6Li+64Zn system was given in the same Ref. Gomes09 , since the 6Li and 7Li data were taken at the same experiment. The extraction of reaction (capture) cross sections from the experimental elastic (quasielastic) backscattering probabilities leads to uncertainties of the order of 10% at energies above the Coulomb barrier. At energies below the barrier the uncertainties are larger because a deviation of the elastic (quasielastic) backscattering cross section from the Rutherford cross section is comparable with the experimental uncertainties. Those overall uncertainties are comparable with the ones obtained from the traditional method using full elastic scattering angular distributions. For the 7Li+64Zn reaction, the $Q$-value of the one neutron stripping transfer is positive and this process should have a reasonable high probability to occur, whereas for the 6Li+64Zn reaction, $Q$-values of neutron transfers are negative. Therefore, one might expect that transfer cross sections for 7Li+64Zn are larger than for 6Li+64Zn. Concerning breakup, since 6Li has a smaller threshold energy for breakup than 7Li, one might expect that breakup cross sections for 6Li+64Zn are larger than for 7Li+64Zn. Actually, in Fig. 9 one can observe that our calculations show that $\sigma(^{7}\mathrm{Li}+^{64}\mathrm{Zn})>\sigma(^{6}\mathrm{Li}+^{64}\mathrm{Zn})$, where $\sigma=\sigma_{R}-\sigma_{cap}\approx\sigma_{tr}+\sigma_{in}$ since $\sigma_{tr}+\sigma_{in}\gg\sigma_{BU}$ for these light systems at energies close and below the Coulomb barrier ($\sigma_{tr}$, $\sigma_{in}$, and $\sigma_{BU}$ are the transfer, inelastic scattering, and breakup cross sections, respectively). So, our present method of extracting reaction and capture cross sections from backward elastic scattering data allows the approximate determination of the sum of transfer and inelastic scattering cross sections, or $\sigma_{tr}+\sigma_{in}+\sigma_{BU}$ in systems where $P_{BU}$ can not be neglected. For both systems investigated, the values of these cross sections are shown to increase with $E_{\mathrm{c.m.}}$, reach a maximum slightly above the Coulomb barrier energy and after decrease. The difference between the two curves in Fig. 9 may be considered approximately as the difference of $\sigma_{tr}$ between the two systems, since $\sigma_{in}$ should be similar for both systems with the same target, apart from the excitation of the bound excited state of 7Li. Because $\sigma_{tr}(^{7}\mathrm{Li}+^{64}\mathrm{Zn})\gg\sigma_{tr}(^{6}\mathrm{Li}+^{64}\mathrm{Zn})$ one can find $\sigma_{tr}(^{7}\mathrm{Li}+^{64}\mathrm{Zn})\approx\sigma(^{7}\mathrm{Li}+^{64}\mathrm{Zn})-\sigma(^{6}\mathrm{Li}+^{64}\mathrm{Zn})$. The maximum absolute value of the transfer cross section $\sigma_{tr}$ at energies near the Coulomb barrier is about 30 mb. Fig. 9 also shows that the difference between transfer cross sections for 7Li and 6Li are much more important than the possible larger $\sigma_{BU}$ for 6Li than for 7Li. Figure 9: The extracted $\sigma_{R}-\sigma_{cap}$ for the reactions 6Li + 64Zn (dashed line) and 7Li + 64Zn (solid line). ## IV Summary We propose a new and very simple way to determine reaction cross sections, through a relation (8) between the elastic scattering excitation function at backward angle and reaction cross section. We show, for several systems, that this method works well and that the elastic backscattering technique could be used as an important and simple tool in the study of the reaction cross sections. The extraction of reaction (capture) cross sections from the elastic (quasielastic) scattering at backward angle is possible with reasonable uncertainties as long as the deviation between the elastic (quasielastic) scattering cross section and the Rutherford cross section exceeds the experimental uncertainties significantly. The behavior of the transfer+inelastic excitation function extracted from the experimental probabilities of the elastic and quasielastic scatterings at backward angle was also shown. We are grateful to G. Kiss, R. Lichtenthäler, P. Mohr, and M. Zadro for providing us the experimental data. P.R.S.G. acknowledges the partial financial support from CNPq and FAPERJ. This work was supported by DFG, NSFC, RFBR, and JINR grants. The IN2P3(France)-JINR(Dubna) and Polish - JINR(Dubna) Cooperation Programmes are gratefully acknowledged. ## References * (1) V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, W. Scheid, and H.Q. Zhang, Eur. Phys. J. A 49, 19 (2013). * (2) V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, and P.R.S. Gomes, Phys. Rev. C 87, 044611 (2013). * (3) L.F. Canto, P.R.S. Gomes, R. Donangelo, and M.S. Hussein, Phys. Rep. 424, 1 (2006). * (4) H. Timmers, J.R. Leigh, M. Dasgupta, D.J. Hinde, R.C. Lemmon, J.C. Mein, C.R. Morton, J.O. Newton, and N. Rowley, Nucl. Phys. A584, 190 (1995). * (5) H. Timmers et al., J. Phys. G 23, 1175 (1997). * (6) H. Timmers, Ph.D. thesis, Australian National University (1996). * (7) H.Q. Zhang, F. Yang, C. Lin, Z. Liu, and Y. Hu, Phys. Rev. C 57, R1047 (1998). * (8) A.B. Balantekin, A.J. DeWeerd, and S. Kuyucak, Phys. Rev. C 54, 1853 (1996). * (9) G.G. Adamian, N.V. Antonenko, R.V. Jolos, S.P. Ivanova, and O.I. Melnikova, Int. J. Mod. Phys. E 5, 191 (1996); V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, W. Scheid, and H.Q. Zhang, Phys. Phys. C 84, 064614 (2011). * (10) P. Mohr et al., Phys. Rev. C 82, 047601 (2010). * (11) J.S. Lilley, M.A. Nagarajan, D.W. Banes, B.R. Fulton, and I.J. Thompson, Nucl. Phys. A 463, 710 (1987). * (12) M. Miller, A.M. Kleinfeld, A. Bockisch, and K. Bharuth-Ram, Z. Phys. A 300, 97 (1981). * (13) G.G. Kiss et al., Phys. Rev. C 83, 065807 (2011). * (14) I. Badawy, B. Berthier, P. Charles, M. Dost, B. Fernandez, J. Gastebois, and S.M. Lee, Phys. Rev. C 17, 978 (1978). * (15) D. Torresi et al., Eur. Phys. J. Conf 17, 16018 (2011). * (16) A. Di Pietro et al., Phys. Rev. C 87, 064614 (2013) * (17) P.R.S. Gomes et al., Phys. Rev. C 71, 034608 (2005). * (18) P.R.S. Gomes et al., Phys. Lett. B 601, 20 (2004) * (19) P.R.S. Gomes, J. Lubian, and L. F. Canto, Phys. Rev. C 79, 027606 (2009).	arxiv-papers	2013-11-17T11:00:14	2024-09-04T02:49:53.762059	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "V.V.Sargsyan, G.G.Adamian, N.V.Antonenko, and P.R.S.Gomes", "submitter": "Vazgen Sargsyan Dr.", "url": "https://arxiv.org/abs/1311.4145" }
1311.4176	11institutetext: Computer Science & Engineering University of South Florida, Tampa, FL, USA 22institutetext: Electrical and Computer Engineering University of Alabama, Tuscaloosa, Alabama, USA [email protected],[email protected] # The Network of Faults: A Complex Network Approach to Prioritize Test Cases for Regression Testing Imrul Kayes 11 Jacob Chakareski 22 ###### Abstract Regression testing is performed to provide confidence that changes in a part of software do not affect other parts of the software. An execution of all existing test cases is the best way to re-establish this confidence. However, regression testing is an expensive process—there might be insufficient resources (e.g., time, workforce) to allow for the re-execution of all test cases. Regression test prioritization techniques attempt to re-order a regression test suite based on some criteria so that highest priority test cases are executed earlier. In this study, we want to prioritize test cases for regression testing based on the dependency network of faults. In software testing, it is common that some faults are consequences of other faults (leading faults). Moreover, dependent faults can be removed if and only if the leading faults have been removed. Our goal is to prioritize test cases so that test cases that exposed leading faults (the most central faults in the fault dependency network) in the system testing phase, are executed first in regression testing. We present ComReg, a test case prioritization technique based on the dependency network of faults. We model a fault dependency network as a directed graph and identify leading faults to prioritize test cases for regression testing. We use a centrality aggregation technique which considers six network representative centrality metrics to identify leading faults in the fault dependency network. We also discuss the use of fault communities to select an arbitrary percentage of the test cases from a prioritized regression test suite. We conduct a case study that evaluates the effectiveness and applicability of the proposed method. We obtain a fault dependency network from the development of a vocabulary learning software. We found that the fault network is a small-world graph with distinguishable community structure. The leading faults are common in all centralities and a re-ordering of test cases is feasible for regression testing based on those leading faults. Our method outperforms traditional regression testing prioritization techniques in detecting fault dependencies. Our modeling of the network of faults provides insights into the requirement of recognizing fault dependencies while re-ordering regression test suites for both research and practice. The dependency model needs further evaluation and improvement considering associated resources (e.g., man-hours). ###### Keywords: Software testing, Regression testing, Test case prioritization ## 1 Introduction Regression testing is performed after a software is modified. The purpose of regression testing is to test the modified software with some test cases in order to re-establish our confidence that the software will perform according to the modified specification and the newly introduced changes do not hinder the behavior of the unchanged part of the software. In a development cycle, regression testing may begin after the detection and correction of faults in a tested software [1]. Regression test suite ensures that the evolution of an application does not result in a low quality software product. However, regression testing has become a complex procedure because of recent trends in software development paradigms. For example, short and iterative “Agile” software development imposes restrictions and constraints on how regression testing can be performed within limited resources [2]. Intuitively, the best way to re-gain confidence from regression testing is to execute all existing test cases from a test suite. Unfortunately, regression testing is often directly associated with high costs. Beizer [3] points out that regression testing accounts for as much as one-half the cost of software maintenance. One industrial collaborator of Elbaum et al. [4] reports that for one of their products of about 20,000 lines of code, the entire test suite requires seven weeks to run. Some of the most well-studied software failures, for example, the Ariane-5 rocket was blamed on the failure to test changes in a software system [5]. In general, test case prioritization techniques seek to schedule test cases in an order so that the tester obtains maximum benefit, even if the testing is prematurely halted at some arbitrary point [2]. Regression test prioritization aims to re-order a regression test suite so that those tests with highest priorities, according to some established criterion, are executed earlier in the process of regression testing than those with lower priorities [6]. Researchers have proposed various techniques for test case prioritization to re-order the test cases for regression testing. These techniques focus on various aspects of product development, such as coverage-based approaches [7, 8, 9, 6], requirement-based approaches [10, 11] and constraint-based approaches [12, 13, 14]. However, none of the solutions addressed dependencies among faults while prioritization. In software testing, it is known that some faults are the consequences of other faults (commonly termed as leading faults). Experience shows that in a software development process, mutually independent faults can be directly detected and removed, but dependent faults can be removed if and only if leading faults have been removed [15]. In worst cases, fault dependencies can create a cascade of faults that can severely effect a software system. For example, in 1990, a fault in the failure recovery code of the AT&T led to cascading faults, which costs 9 hours of downtime and at least $60$ million in lost revenue [16]. Another example of cascading faults is the escalation of a divide-by-zero exception into a Navy ship’s network that left the smart ship dead in the water [17]. Researchers hint that the Internet is also at risk of cascading failures [18]. We argue that test case that reveals leadings faults should be executed first in a regression testing process in order to get an early confirmation that the software is free from dependent faults. We attempted a first step to prioritize regression testing based on fault dependency in [19]. We proposed an algorithm to prioritize test cases based on fault dependency. However, in [19], we only considered $1$-hop neighborhood or dependencies of faults. This paper uses a fault dependency network to prioritize test cases for regression testing. We leverage faults’ positions in the network to determine leading faults (central faults in the network). The contributions of this work are: * • First, we describe ComReg, which leverages fault dependency network to prioritize test cases for regression testing. We present a directed graph model for the fault dependency network and identify leading faults (central faults) to prioritize test cases. Our identification of leading faults is based on a centrality aggregation technique. Centralities can represent the position of a fault in a fault network. We propose an aggregation of different representative centrality metrics (indegree, betweenness, closeness, eigenvector, pagerank, and hub centrality) into a final leading score to identify leading faults. * • Second, we discuss the use of fault communities to select $X\%$ of the test cases from a prioritized regression test suite. * • Finally, we present a case study from the development of a subject software “Tarantula”. We discuss the test cases written for the software, the faults it exposed after testing, and the fault network from the exposed faults. We show the identification of leading faults for prioritization and compare the effectiveness with traditional techniques. We also show fault communities for a selection of test cases from the prioritized regression test suite. The rest of the paper is organized as follows. Section 2 introduces fault dependency-aware test case prioritization technique, ComReg. Section 3 presents a case study. Section 4 reviews related work and Section 5 concludes. ## 2 Fault Dependency-Aware Test Case Prioritization ### 2.1 Problem Statement Based on Elbaum et al. [9], we define a prioritization of test cases for regression testing as follows. Given a test suite $T$, the set of permutations of $T$ as $PT$ and a function from $PT$ to the real numbers as $f$, a prioritization of test cases for regression testing solution provides an ordered test suite $T^{\prime}$ such that for all $T^{\prime\prime}$, $f(T^{\prime})\geq f(T^{\prime\prime})$, where $T^{\prime}\in PT$ and $T^{\prime\prime}\in PT$. $PT$ represents the set of all possible prioritization (orderings) of $T$ and $f$ is an utility function that, applied to any such ordering, yields an award value to that ordering. For example, let us we have $n$ test cases as $(T_{1},T_{2},T_{3},\dots,T_{n})\in T$. From those test cases, $n!$ orderings are possible. Test case prioritization techniques attempt to find an order from $n!$ number of orderings such that the order maximizes the utility function $f$. Let us we have $t$ test cases $(T_{1},T_{2},T_{3},\dots,T_{t})$ in the test suite $T$. After running those test cases for a system testing, we get $n$ faults such as $(F_{1},F_{2},F_{3},\dots,F_{n})$ as $F$. There exists a relation from $T$ to $F$, $R:T\rightarrow F$, such that for each test case $t\in T$ there exists none, single or multiple faults $f\in F$. Our goal is to prioritize the test suite and select $X\%$ of the test cases for regression testing. ### 2.2 Our Approach-ComReg We propose ComReg, a fault dependency-aware test case prioritization for regression testing. ComReg is based on the fact that mutually independent faults can be directly detected and removed, but dependent faults can be removed if and only if the leading faults have been removed [15]. A leading fault is the fault that causes dependent faults to occur. For example, consider a simple dictionary program, which has a load functionality to read all words and their meanings from text files, a next word functionality that allows users to browse words and a random number generator for generating a number for an arbitrary selection of a word list. The next word functionality is dependent on the load functionality in that if the system fails to read words and meanings, there is no way to browse the words. So, consider three following faults that occur. 1. 1. Fault F1: words and meanings upload failure. 2. 2. Fault F2: does not find the next word. 3. 3. Fault F3: random generator does not show a random number. Figure 1 shows the faults. We can draw an arrow from fault F2 to fault F1 to show the dependency of F2 on F1. In this case, F1 is a leading fault and F2 is a dependent fault. However, the fault F3 is an independent fault (no arrow to or from F3 in Figure 1). Leading faults might be limited in numbers. For example, Microsoft reports that 80 percent of the errors and crashes in Windows and Office are caused by 20 percent of the entire pool of faults [20]. We propose to prioritize a regression test suite based on leading faults and to run $X\%$ of the test cases which contain leading faults. So, in our fault dependency-aware test case prioritization, an order of the test cases attempts to maximize the utility function $f$ that determines the number of the leading faults. Fault dependencies could be data, control or module dependent. Figure 1: Fault dependency. However, the problem of detecting leading faults is not trivially solvable. The challenge is due to the fact that faults not only have local effects (e.g., Fault A is dependent on fault B, so A could not be removed before removing B), but also faults have global effects too (e.g., Fault A is dependent on fault B and Fault B is dependent on Fault C. Fault A could not be removed before removing Fault B and Fault C). We present various scenarios of fault dependencies considering two faults F1 and F2 in Figure 2. Some of the scenarios are listed below. * • Fault F1 is dependent on Fault F2, or vice versa * • a) Fault F1 is dependent on Fault F2; b) other faults are dependent on Fault F1; c) vice versa of (a) and (b) * • a) Fault F1 is dependent on Fault F2; b) other faults are dependent on Fault F2; c) vice versa of (a) and (b) * • a) Fault F1 is dependent on Fault F2; b) other faults are dependent on Fault F1; c) other faults are dependent on Fault F2 (d) vice versa of (a), (b) and (c) So, it appears that if we consider all faults and their dependencies, the situation becomes very complex. All faults and their dependencies can be captured by a complex network as shown in Figure 3. The network is comprised of $77$ faults (shown as nodes) and $254$ dependencies (shown as edges). There is an edge from Fault A to Fault B if Fault A is dependent on Fault B. The leading faults in the network are those who occupy central positions. Formally, we model a fault dependency network as a directed graph $F=(V,E)$, where a node $v\in V$ is a fault and an edge $e_{ij}\in E$ from $v_{i}\in V$ to $v_{j}\in V$ denotes that the fault $v_{i}$ is dependent on the fault $v_{j}$. The number of nodes and edges are $\|V\|=n$ and $\|E\|=m$ respectively. The directed graph can be represented by a $nn$ matrix $F_{nn}$, where an entry $F(i,j):$ $F(i,j)=\begin{cases}1&\text{if $e_{ij}\in E$}\\\ 0,&\text{otherwise}\end{cases}$ (1) Figure 2: Examples of various fault dependencies considering two faults: Fault F1 and Fault F2. (a) Fault F1 is dependent on Fault F2 (b) Fault F1 is dependent on Fault F2 and other faults are dependent on Fault F1 (c) Fault F1 is dependent on Fault F2 and other faults are dependent on Fault F2 (d) Fault F1 is dependent on Fault F2 and other faults are dependent on both Faults F1 and F2. Figure 3: A fault dependency network of $77$ nodes and $254$ dependencies. Node size is proportional to in-degree. The position of a node (fault) in the network can be represented in network analysis by different centrality metrics. For example, the larger the number connections a fault receive from its direct neighbors, the higher number of other faults depend on the fault. Without removing the faults all dependent faults could not be removed. Alternatively, the larger the number of paths between other pairs of faults a fault is part of, the more it can control the fault propagation between distant faults. Based on this intuition, we conjecture that a fault’s position is determined by and manifests via its centrality in the fault network. We propose to aggregate different representative centrality metrics into a final leading score to identify the leading faults based on [21]. In [21], the authors used centrality aggregation technique to identify influential bloggers in a blogging network. We define the leading score of a fault (node) as the average of the positions of that node in decreasing order of centrality scores over various centrality metrics. Specifically, each centrality metric assigns each node a score that can be used to order the nodes in decreasing order of importance (according to that centrality). This allows each fault to receive a rank according to each centrality metric: the first ranked fault will be the most central one, the last ranked will be the one with the lowest centrality score. Faults having the same centrality score are given the same rank. A fault’s final rank is the average rank over all centrality measures. We selected six representative centrality metrics as the focus of our study: indegree, betweenness, closeness, eigenvector, pagerank, and hub centrality. Degree centrality is defined as the number of links that a node has. In a directed graph like fault dependency graph, two types of degree centralities are possible: indegree and outdegree centrality. For a node, the number of direct incoming connections is characterized as indegree of the node. On the other hand, the number of direct outgoing connections is characterized as out degree of the node. Although simple, indegree centrality intuitively captures an important aspect of a fault’s potential leading position: faults who have many incoming connections from many other faults are those that make other faults to depend. In our fault dependency graph $F$, the indegree centrality of a fault $i$ can be represented by the following equation. $indegree(i)=\sum_{1\leq j\leq n}F_{ji}$ (2) Betweenness centrality, which measures the extent to which a node lies on the shortest paths between other nodes, was introduced as a measure for quantifying the control of a human on the communication between other humans in a social network [22]. Faults with high betweenness centrality may have considerable influence within a fault dependency network by virtue of their control over fault propagation among other faults. The nodes with the highest betweenness are also the ones whose removal from the network will most disrupt communications between other nodes because they lie on the largest number of paths taken by faults [23]. Formally, the betweenness centrality of a node is the sum of the fraction of all-pairs shortest paths that pass through : $C(v)=\sum_{s,t\in V}\frac{\sigma(s,t\|v)}{\sigma(s,t)}$ (3) where v is the set of nodes, $\sigma(s,t)$ is the number of shortest $(s,t)$ paths, and $(s,t\|v)$ is the number of those paths passing through some nodes $v$ other than $s,t.$ If $s=t$, $\sigma(s,t)=1,$ and if $v\in s,t,\sigma(s,t\|v)=0$. Our implementation of betweenness for this research is based on the Brandes algorithm [24]. Closeness centrality measures the mean distance from a node to other nodes, assuming that faults propagate along the shortest paths. Formally, the closeness centrality $(C(x))$ of a node $x$ is defined as follows: $C(x)=\frac{n-1}{\sum_{y\in U,y\neq x}d(x,y)}$ (4) where $d(x,y)$ is the distance between node $x$ and node $y$; $U$ is the set of all nodes; $d$ is the average distance between $x$ and the other nodes. In our fault dependency network, this centrality measure estimates the amount of faults a fault may have access to compared to other faults. Specifically, a fault with lower mean distance to others can reach others faster. The centrality of a node does not only depend on the number of its adjacent nodes, but also on their relative importance. Eigenvector centrality allocates relative scores to all nodes in the network such that high-score neighbors contribute more to the score of the node. Formally, Bonacich [25] defines the eigenvector centrality $C(v)$ of a node $v$ as the function of the sum of the eigenvector centralities of the adjacent nodes, i.e. $C(v)=1/\lambda\sum_{(v,t)\in E}c(t)$ (5) where $\lambda$ is a constant. This can be rewritten in vector notation, resulting in an eigenvector equation with well-known solutions. Originally designed as an algorithm to rank web pages [26], PageRank computes a ranking of the nodes in a graph based on the structure of the incoming links. The algorithm assigns a numerical weighting to each node of a network with the purpose of “measuring” its relative importance within the network. Hubs and authorities are other relevant centralities for the fault network context. In a graph, authorities are nodes that contain useful information on a topic of interest; hubs are nodes that know where the best authorities are to be found [23]. A high authority centrality node is pointed to by many hubs, i.e., by many other nodes with high hub centrality. A high hub centrality node points to many nodes with high authority centrality. These two centralities can play a significant role also in our work of finding leading faults. They can infer that the faults that have high hub and authority centrality are not only leading but also they are connected with leading faults. ### 2.3 Fault Communities to Select $X\%$ of Test Cases A common approach of regression testing is to select and run $X\%$ of the test cases from a prioritized test suite. However, an optimal selection is always challenging. On one hand, a few selections of test cases might remain a significant portion of the software virtually untested. On the other hand, too many selections of test cases will require to test the entire system again. However, in a fault network, fault communities could be leveraged to select an $X\%$ of the test cases. Complex networks show communities in them: a community is a subset of nodes within which node to node connections are dense, but between which connections are less dense [27]. Communities are natural outcomes of real-world networks. For example, e-mail network [28], social application network [29], mobile communication network [30], blogging network [31], and yeast protein-protein interaction network [32] revealed community structures. Figure 3 shows communities in a fault network; nodes in a community are colored the same. Newman proposed a community detection algorithm [33] based on modularity maximization. Modularity is a utility function that computes the quality of a particular division of a network into communities. It is defined as the fraction of the edges that fall within the given community minus the expected such fraction if the edges were distributed at random. $Q=(E_{1}-E_{2})$ (6) where $E_{1}$= fraction of edges within communities and $E_{2}$=expected fraction of such edges. The expected fraction of edges is typically evaluated within a random graph conditioned on the degree sequence of the original network. In that random graph, the probability of an edge between two nodes $i$ and $j$ is $(k_{i}k_{j})/2m$, where $k_{i}$ is the degree of node $i$ and $m$ is the total number of edges in the network. The modularity can then be written $Q=\frac{1}{2m}\sum_{ij}\left({A_{ij}-\frac{k_{i}k_{j}}{2m}}\right)\delta(c_{i},c_{j})$ (7) $\delta(c_{i},c_{j})=\begin{cases}1&\text{if if i and j belong to the same community}\\\ 0,&\text{otherwise}\end{cases}$ (8) where $A_{ij}$ is the matrix representation of the graph, $\delta$ is the Kronecker delta, $c_{i}$ is the label of the community to which node $i$ is assigned. The authors describe the modularity for an undirected graph. However, the modularity can be extended for a directed graph such as fault dependency network. In a random directed graph, the probability of an edge from node $j$ to node $i$ is $(k_{i}^{out}k_{i}^{in})/m$. Then for the fault dependency network the above equation could be written as $Q=\frac{1}{m}\sum_{ij}\left({F_{ij}-\frac{k_{i}^{out}k_{i}^{in}}{m}}\right)\delta(c_{i},c_{j})$ (9) Where $F$ is a fault dependency matrix and $F_{ij}$ is 1 if there is an edge from $j$ to $i$ and zero otherwise. We propose to apply the community detection algorithm to uncover communities of the faults. After detecting communities, the faults in the same communities with a leading fault could be identified and corresponding test cases could be executed as a regression test. Moreover, all modules in a software are not the same in terms of fault tolerance. For example, login credential authentication or a module that processes financial transaction are more crucial than a module that prints documents. Furthermore, Pareto principle also (known as 80-20 rule) applies to software systems. The Pareto principle [34] states that for many events, roughly 80% of the effects come from 20% of the causes. The Standish Group’s report shows that in a software system, 45% of features are never used, 19% of features are rarely used, 19% of features are used sometimes, 13% of features are used often and only 7% of features are always used [35]. So, in sum, only 20% of software features are often and always used. It becomes apparent that ensuring quality of those 20% of software features is vital. Fault communities could be leveraged to ensure the quality of prioritized features (e.g., 20% of software features). The leading faults and faults from their communities revealed by the test cases (which target prioritized features) could be used in selecting regression test cases. This way a regression testing can ensure a high customer satisfaction. ## 3 Case Study and Evaluation The goal of the case study is to prioritize a test suite of size $N$ and identify $X\%$ of the test cases from the suite for regression testing. To accomplish the goal, we developed a medium-scale English vocabulary learning software, “Tarantula”. In the process of development, we wrote test cases, executed the test cases, applied our method ComReg to prioritize the test cases and identified $X\%$ of the test cases for regression testing. In this section, we first give a detailed overview of the functionalities of “Tarantula”. Then we describe the test cases and relevant faults that we have got after applying the test cases in an initial product. We explain the features of the Tarantula, so that readers can understand the underlying goals of the test cases. Finally, we present the leading faults using our centrality aggregation method and compare ComReg with traditional approaches. We also show and discuss the relevance of using community detection techniques in the fault network. ### 3.1 An Overview of Tarantula The Tarantula is an English vocabulary learning software, which is built targeting high frequency GRE words. When a user installs and runs the software, she is presented with $50$ word lists. Figure 4 shows the first window of Tarantula. The word lists consist of $4054$ words and when a user hovers a mouse on a word list icon, it shows the first and the last word in the word list. Users can click on a word list icon to exercise various features of that word list. However, users can use the random button (upper right corner in Figure 4) to select a random word list (see Figure 5). When users select a word list, relevant features of the word list are shown in a separate option window as shown in Figure 6. There are several options available on the option window for users to learn and practice words of a word list. The options are: “Learn WordList”, “Multiple Choice”, “Reverse Challenge”, “Words Jam” and “Flip Words”. Users can click a radiobutton to select an option. The “Learn WordList” feature shows the words and their meanings from the word list serially (see in Figure 7). Users can use “Next” and “Prev” buttons to view next and previous word respectively. There is a counter which indicates the serial number or position of the word in the word list. Figure 4: First window of Tarantula. Figure 5: Random window in Tarantula. Figure 6: Option window of Tarantula. Figure 7: Learn wordlist feature of Tarantula. The “Multiple Choice” feature shows a random word from the word list with five possible meanings (see Figure 8). Meanings are from the same word list taken randomly, and of them one is appropriate for the word shown. When users click a meaning of the word, the “Result” label shows whether the selection is right or wrong. There is a timer label which increases in each second to show how much time a user is taking. The “Count” label shows the number of words a user has practiced. Users can click on a “Next” button to get a new word. The “Reverse Challenge” feature is the opposite of “Multiple Choice” feature. This feature shows a random meaning from the word list with five possible words (see Figure 9). These words are from the same word list, taken randomly, and of them one is appropriate for the meaning shown. When users click a word of the meaning, the “Result” label shows whether the selection is right or wrong. ‘Count”, ‘Next” and “Timer” functionalities are similar to the functionalities of “Multiple Choice” feature. Figure 8: Multiple Choice feature of Tarantula. Figure 9: Reverse Challenge Choice feature of Tarantula. The “Words Jam” feature shows ten words from the word list on one side and their meanings from the word list on the other side. Words’ and their meanings’ positions on each side is random (see Figure 10). Users have to click a word and then the meaning of the word (or vice versa). If the meaning of the word is right then both of them will be disappeared from the Jam. Users can load a new Jam by clicking “Load Next Jam” button. A counter shows the number of the Jam a user is practicing. The “Flip Words” feature shows a random word from the word list to guess (see Figure 11). Users can click “Flip” button to see the meaning of the word. Users can use “Next button” for a new word. A counter also counts the number of words the user has seen. Figure 10: Words jam feature in Tarantula. Figure 11: Flip Words feature in Tarantula. ### 3.2 Development of Tarantula The Tarantula is a desktop application, written in C# programming language and we used Microsoft Visual Studio $2010$ platform. The software consists of $19390$ lines of code. It can be installed from GitHub 111https://github.com/ImrulKayes/Tarantula/ blob/master/Tarantula1.0.msi. The code of Tarantula is also publicly available at GitHub 222https://github.com/ImrulKayes/Tarantula. We first developed a web crawler using Python programming language to crawl a subset of HTML pages from [36]. These pages have all the words and their meanings. We parsed the crawled HTML pages using a Parser (also written in Python). The Parser went through all HTML pages and extracted words and meanings using regular expressions. Then we created the repository of fifty word lists ($4054$ words) from the extracted words and meanings. Finally, we used the repository as a word database for Tarantula. ### 3.3 Test Cases and Faults Based on the required features, we wrote fifty test cases before the development. We ran the test cases after finishing an iteration of the development cycle. Sixteen of the test cases revealed twenty three faults. The test cases and the faults are below. * • Test Case #$1$ Action: click on a Word List icon to enable the system to load the words of the list with their meanings. Expected result: the Word List should be loaded with features. Fault #$1$: the Word List is unavailable due to missing of the file. * • Test Case #2 Action: select the _Learn Word List_ option from the Radiobuttons of a word list. Expected result: a random word from the word list and its meaning should be shown. Fault #$2$: the word in the selected word list is not generated. Fault #$3$: the meaning in the selected world list is not available. * • Test Case #$3$: Action: click on the _Next_ button on the _Learn Word List_ feature. Expected result: a new random word from the word list and its meaning should be shown. Fault #$4$: _Next_ button does not generate a random word. Fault #$5$: _Next_ button does not generate a meaning. * • Test Case #$4$: Action: Click on the _Previous_ button event in the _Learn Word List_ feature. Expected result: a new random word from the word list and its meaning should be shown. Fault #$6$: _Previous_ button does not generate a random word. Fault #$7$: _Previous_ button does not generate a meaning. * • Test Case #$5$: Action: check the _Count_ functionality in the _Learn Word List_ feature by clicking on the _Next_ and _Previous_ buttons. Expected result: _Count_ should be increased by one on clicking _Next_ button and count should be decreased by one on clicking _Previous_ button. Fault #$8$: _Count_ does not increase after clicking the _Next_ button. Fault #$9$: _Count_ does not decrease after clicking the _Previous_ button. * • Test Case #$6$: Action: select the _Multiple Choice_ option from the Radiobuttons of a word list. Expected result: a random word from the wordlist and its possible choices of meanings should be shown. The meanings are also from the same wordlist. Fault #$10$: the word is not generated. Fault #$11$: meanings are not available. * • Test Case #$7$: Action: verify the functionality of the _Multiple Choice_ option. Select the right meaning of the word. Select a wrong meaning of the word. Expected result: the system should show “Correct” if the choice is right, otherwise it will show a message saying that the choice is wrong. Fault #$12$: the “wrong” message is not shown. * • Test Case #$8$: Action: verify _Timer_ functionality of the _Multiple Choice_ option. Select the _Multiple Choice_ option from the Radiobuttons of a word list. Then click the _Next_ button. Expected result: The _Timer_ should start from a zero value. It will increase by one after each second. Clicking the _Next_ button should set it a zero value. Fault #$13$: the _Counter_ does not increase. * • Test Case #$9$: Action: select the _Words Jam_ option from the Radiobuttons of a word list. Expected result: ten words and their meaning should be shown for matching from the word list. Fault #$14$: words in _Words Jam_ are missing. Fault #$15$: meanings in _Words Jam_ are missing. * • Test Case #$10$: Action: click a word and then click its meaning in the _Words Jam_ feature. Click a meaning and then click it’s corresponding word in the _Words Jam_ feature. Expected result: the word and the meaning should be disappeared. Fault #$16$: the word does not disappear. * • Test Case #$11$: Action: in _Words Jam_ feature, click a word and click a wrong meaning of the word. Expected result: the word and the meaning should not be disappeared. Fault #$17$: the word disappears. * • Test Case #$12$: Action: click _Load Next Jam_ in _Words Jam_ feature. Expected result: ten words and their meanings should be shown to match and _Jam counts_ should be increased by one. Fault #$18$: _Jam Count_ does not increase. * • Test Case #$13$: Action: select _Flip Words_ option from the Radiobuttons of a word list. Expected result: a random word should be shown which will allow the users to guess the meaning of the word. Fault #$19$: the word is not generated. * • Test Case #$14$: Action: click the _Flip_ button in _Flip Words_ feature. Expected result: The meaning of the word should be shown and the text “Flip” of the button should be changed as “Next”. Fault #$20$: meaning is not available. Fault #$21$: text does not change. * • Test Case #$15$: Action: check the Radom word list generator, click the _Rand_ button. Expected result: a random word list number should be generated. Fault #$22$: the random generator does not generate a random number. * • Test Case #$16$: Action: click the _Go to Word List_ button of the random wordlist generator. Expected result: the word list should be loaded with features. Fault #$23$: the word list is not loaded. ### 3.4 Properties of the Fault Network As discussed in Section 2.2, a fault dependency network is a directed graph $F=(V,E)$, where a node $v\in V$ is a fault and an edge $e_{ij}\in E$ from $v_{i}\in V$ to $v_{j}\in V$ denotes that the fault $v_{i}$ is dependent on the fault $v_{j}$. The directed graph can be represented by a $nn$ matrix $F_{nn}$, where an entry $F(i,j):$ $F(i,j)=\begin{cases}1&\text{if $e_{ij}\in E$}\\\ 0,&\text{otherwise}\end{cases}$ (10) The fault dependency matrix can be constructed after the system testing is done. For example, in a Scrum process, a fault review is usually done before the regression testing by examining reported faults on the Dashboard. In our case study, running the test cases we have got a fault dependency matrix $F$ shown in Table 1. The fault dependency matrix has $23$ faults and we associated relevant dependencies from 3.3. Figure 12 shows the largest component of the fault network ($22$ faults), where node size is proportional to in-degree of the node. We used Gephi (https://gephi.org/) to visualize and obtain structural properties of the network. The structural properties of the fault network (largest component) are presented in Table 2. Faults$\downarrow$$\rightarrow$ \| 1 \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 8 \| 9 \| 10 \| 11 \| 12 \| 13 \| 14 \| 15 \| 16 \| 17 \| 18 \| 19 \| 20 \| 21 \| 22 \| 23 ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 2 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 3 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 4 \| 1 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 5 \| 1 \| 1 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 6 \| 1 \| 1 \| 1 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 7 \| 1 \| 1 \| 1 \| 1 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 8 \| 1 \| 1 \| 1 \| 1 \| 1 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 9 \| 1 \| 1 \| 1 \| 1 \| 1 \| 1 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 10 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 11 \| 1 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 12 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 13 \| 1 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 14 \| 1 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 15 \| 1 \| 1 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 16 \| 1 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 17 \| 1 \| 1 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 18 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 1 \| 0 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 19 \| 1 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 20 \| 1 \| 1 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 21 \| 1 \| 1 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 1 \| 0 \| 0 \| 0 22 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 23 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 Table 1: Fault dependency matrix. Figure 12: The fault dependency network. Node size is proportional to in-degree. Nodes in a community are colored the same. Nodes \| $22$ ---\|--- Edges \| $97$ Average In-degree \| $3.95$ Average Path length \| $1.074$ Average Clustering Coefficient \| $0.416$ Table 2: Structural properties of the fault network. A notable characteristic of the fault network is the high clustering coefficient. Given a network $G=(V,E)$, the clustering coefficient $C_{i}$ of a node $i\in V$ is the proportion of all the possible edges between neighbors of the node that actually exist in the network [23]. The clustering coefficient is defined as: $C=\frac{3\text{Number of triangles}}{\text{Number of connected triples of the nodes}}$ (11) In the fault network, for a node $v_{i}\in V$ there could be $k_{i}(k_{i}-1)$ links exist among the neighborhood of $v_{i}$, where $k_{i}$ is the number of neighbor of $v_{i}$. So, the local clustering coefficient of fault $v_{i}$ in the fault network is: $C=\frac{\|\\{e_{jk}:v_{j},v_{k}\in\text{Neighbor}(V_{i}),e_{jk}\in E\\}\|}{k_{i}(k_{i}-1)}$ (12) The clustering coefficient for the whole network is the average of the local clustering coefficients of all the nodes $n$ [37]: $C=\frac{1}{n}\sum_{i=1}^{n}C_{i}$ (13) A high clustering coefficient in the fault networks implies that a faults’s connections are interconnected and have a greater effect on one another. The small average path length ($1.074$), comparable with that of the corresponding random graph of the same size ($1.675$), together with the high average clustering coefficient (the fault network has average clustering coefficient $0.416$, where a same size random graph has $0.295$), places the fault network in the category of small-world graphs [37]. ### 3.5 Leading Faults and Prioritized Test Cases As described in Section 2, we use centrality metrics to rank leading faults in the network. To manage the fault network and compute centralities, we used Python 2.7 with the NetworkX333http://networkx.github.io/ library. Faults that appear among the top $10$ in multiple centrality metrics are represented in color in Table 3.5. The average rank of the top $10$ leading faults and their average ranks considering all centralities are shown in Table 13. Out of the $10$ leading faults listed on each centrality, $6$ faults (60%) are common in all the centralities. To observe more closely, we plot the ranks of top $10$ of the faults assigned by all centralities, showed in Figure 13. As expected, a fault’s assigned ranks from centralities form a cluster and together with all the clusters we can visualize a straight line. This shows that all the centralities tend to rank the same fault in the top. Note Pareto principle described in Section 2—for many events, roughly 80% of the effects come from 20% of the causes. Pareto principle also holds for the fault dependency network. In the fault dependency network, $78$ out of $97$ ($80.41\%$) edges are incident on top $5$ nodes out of $23$ nodes (21.73%). It shows that $80.41\%$ of the fault dependencies are due to 21.73% of faults—almost equal figures from Pareto principle. Table 3: Top $10$ faults according to each centrality measurement, sorted in increasing order by rank from left to right. IDC: in-degree centrality, BC: betweenness centrality, CC: closeness centrality, EC: eigenvector centrality, PG: page-rank centrality, and HC: hub centrality. Faults common to all centralities are colored the same. IDC \| $Fault\\#1$ \| $Fault\\#2$ \| $Fault\\#3$ \| $Fault\\#4$ \| $Fault\\#5$ \| $Fault\\#14$ \| $Fault\\#15$ \| $Fault\\#6$ \| $Fault\\#13$ \| $Fault\\#7$ ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- BC \| $Fault\\#1$ \| $Fault\\#2$ \| $Fault\\#3$ \| $Fault\\#4$ \| $Fault\\#15$ \| $Fault\\#14$ \| $Fault\\#17$ \| $Fault\\#21$ \| $Fault\\#21$ \| $Fault\\#13$ CC \| $Fault\\#1$ \| $Fault\\#2$ \| $Fault\\#3$ \| $Fault\\#4$ \| $Fault\\#15$ \| $Fault\\#5$ \| $Fault\\#6$ \| $Fault\\#7$ \| $Fault\\#8$ \| $Fault\\#9$ EC \| $Fault\\#1$ \| $Fault\\#2$ \| $Fault\\#3$ \| $Fault\\#4$ \| $Fault\\#5$ \| $Fault\\#6$ \| $Fault\\#7$ \| $Fault\\#8$ \| $Fault\\#9$ \| $Fault\\#15$ PC \| $Fault\\#1$ \| $Fault\\#2$ \| $Fault\\#3$ \| $Fault\\#4$ \| $Fault\\#5$ \| $Fault\\#13$ \| $Fault\\#14$ \| $Fault\\#15$ \| $Fault\\#6$ \| $Fault\\#11$ HC \| $Fault\\#1$ \| $Fault\\#2$ \| $Fault\\#3$ \| $Fault\\#4$ \| $Fault\\#5$ \| $Fault\\#6$ \| $Fault\\#7$ \| $Fault\\#8$ \| $Fault\\#9$ \| $Fault\\#15$ Figure 13: Assigned rank of top ten most central faults from all centralities. Table 4: Average rank of the top ten faults. Faults’ ID \| Average Rank ---\|--- $Fault\\#1$ \| $1.00$ $Fault\\#2$ \| $1.33$ $Fault\\#3$ \| $2.33$ $Fault\\#4$ \| $3.33$ $Fault\\#15$ \| $5.16$ $Fault\\#5$ \| $5.33$ $Fault\\#6$ \| $6.00$ $Fault\\#14$ \| $6.17$ $Fault\\#7$ \| $6.5$ $Fault\\#17$ \| $7.00$ Finally, our prioritized ordering of the test cases for regression testing based on leadings faults’ exposure in test cases is: T1, T2, T3, T4, T9, T11, T5, T8, T14, T6, T10, T12, T13, T7, T16, T15 (T denotes a test case). ### 3.6 Effectiveness of ComReg We used three techniques to prioritize our regression test suite and compared them to ComReg. We want to observe which method has faster fault dependency detection rate. The techniques are the following: 1. 1. Prioritization using relevant slices (ReSl): ReSl prioritizes test cases taking into account the coverage requirements present in the relevant slices of the outputs of test cases [38]. 2. 2. Prioritization based on Function Call Path (FuCa): FuCa leverages function call-level paths and prioritizes test cases based on those coverage paths [39]. 3. 3. Random Prioritization (Random): Random prioritizes test cases based on a randomization algorithm. We used a metric, APFDD (Average of the Percentage Fault Dependency Detected) [19], to measure effectiveness of ComReg to the techniques described above. The APFDD quantifies how rapidly a prioritized test suite can detect dependency among faults The values of the APFDD range from 0 to 100; higher value implies faster fault dependency detection. Figures 14, 15, 16 show the percentage of test cases executed and the percentage of fault dependency detected for the test cases prioritized by ComReg and other methods (random, ReSl and FuCa respectively). The areas under the curves represent the weighted average of the percentage of the fault dependency detected (APFDD). From the figures we see that the random prioritization method performed the worst, yielded only 45.32% APFDD. The ReSl and FuCa methods performed moderately, both were better than the random prioritization method with APFDD 54.10% and 66.73% respectively. Our method ComReg provided the best value of the APFDD (85.10%), hence outperformed the random, ReSl and FuCa methods in rapidly detecting fault dependencies. Figure 14: Average percentage of fault dependency detected (APFDD) for the prioritized test cases using ComReg and random techniques. Figure 15: Average percentage of fault dependency detected (APFDD) for the prioritized test cases using ComReg and ReSl techniques [38]. Figure 16: Average percentage of fault dependency detected (APFDD) for the prioritized test cases using ComReg and Function Call Path techniques [39]. ### 3.7 Community Detection Techniques We used a popular modularity maximization approach, Louvain method [40], to detect fault communities in the network. Louvain method is a greedy optimization method that attempts to optimize the modularity of a partition of the fault network. The optimization is performed in two steps: modularity maximization and community aggregation. In modularity maximization step, the method looks for “small” communities by optimizing modularity locally. In community aggregation step, it aggregates nodes who belong to the same community and builds a new network whose nodes are the communities. Two steps are repeated iteratively until a maximum of modularity is attained and a hierarchy of communities is produced. Applying the algorithm on the fault network, we have got three fault communities: community #1 (pink color nodes in Figure 12) has seven faults, community #2 (green color nodes in Figure 12) has nine faults and community #3 (violet color nodes in Figure 12) has six faults. Leading faults are distributed among communities. For example, community #1, community #2 and community #3 have one, two and two top leading faults respectively out of top five leading faults. As discussed in the Section 2, leading faults and faults from their communities revealed by the test cases (which target prioritized features) could be used in selecting regression test cases. For example, leading fault Fault # 1’s community has eight faults originated from seven test cases (46.66% of test cases). ## 4 Related Work Different solutions have been proposed to prioritize test cases for regression testing. In this section, we discuss test case prioritization techniques from the literature. _Coverage-based prioritization_ techniques aim to achieve higher fault detection rates by maximizing early coverage. The solutions are inspired by the intuition that early maximization of structural coverage will also maximize early fault detection. Rothermel et al. proposed a family of techniques [7, 8] for test-case prioritization based on several coverage criteria. They considered different types of coverages: branch-total, branch- additional, statement-total, statement-additional, Fault Exposing Potential (FEP)-total, and FEP-additional. A branch-total coverage solution prioritizes test cases according to the number of branches covered by individual test cases. On the other hand, branch-additional prioritizes test cases according to the additional number of branches covered by individual test cases. Statement-total and statement-additional coverage based solutions are similar to previous two approaches, but rather than considering branches, they consider statements. The FEP-total and FEP-additional are based on program mutation. Program mutation produce a mutant version of the program by introducing modifications to the program source. The prioritization techniques prioritize the test cases such that the test cases can reveal the difference between the original program and the mutant. The authors introduced a metic Average Percentage of Fault Detection (APFD) to quantify the success of a prioritization. Elbaum et al. [9, 6] further proposed prioritization techniques covering coverage criterion at the function level, while Do et al. [41] considered the coverage criteria at the block level. Korel et al. discussed several model-based test prioritization heuristics in [42, 43]. Their coverage criteria is system model; they identified elements of the model related to source-code modifications and applied heuristics to prioritize test cases so that early fault detection in the modified system is maximized. Jones and Harrold described a fine-grain coverage criterion in [44], which considers a modified condition/decision coverage. _Requirement-based approaches_ consider a software’s requirements as a basis for prioritization of test cases. Srikanth et al. [10] prioritized test cases based on four factors: requirements volatility, customer priority, implementation complexity, and fault proneness of the requirements. Krishnamoorthi at al. [11] adopted a similar approach. Their prioritization is based on six factors: customer priority, changes in requirement, implementation complexity, completeness, traceability and fault impact. However, a potential weakness of requirement-based approaches is that requirement properties are subjective and thus estimations might be biased. _Constraint-based approaches_ consider different constraints and practical complications in test case prioritization. Kim et at. [45] consider resource and time constraints. The resource and time constraint do not allow the execution of the entire test suite for a regression testing. They proposed a heuristic that uses historical information to do test case prioritization. Alspaugh et al. [14] consider a situation when regression testing is performed in a time constrained environment. They empirically compared seven Knapsack solvers (e.g., greedy, dynamic programming and the core algorithm) and identified a test suite reordering that rapidly covers the test requirements and always terminates within a specified testing time limit. Walcott et al. [13] proposed a genetic algorithm-based time-ware test case prioritization technique and empirically compared the approach with the initial ordering, the reverse ordering and two control techniques (random prioritization and fault- aware prioritization). They defined a metric to evaluate the effectiveness of prioritization in a time-constrained environment. Zhang et al. [12] also studied time-aware test case prioritization problem. Their proposed test case prioritization is based on integer linear programming. They empirically showed that their two proposed techniques outperform genetic algorithms-based time- aware test case prioritization and four other traditional techniques for test- case prioritization. Researchers used a number of other criteria to prioritize test cases. Sherriff et al. prioritized test cases based on historical change records in [46]. They proposed a methodology for determining the effect of a software feature change and then prioritized regression test cases by gathering software change records and analyzing them through singular value decomposition. Leon et al. [47] introduced distribution-based filtering and prioritized test cases based on the distribution of the profiles of test cases in the multi-dimensional profile space. Sampath et al. [48] prioritized test cases for web applications. They prioritized test suites by test lengths, frequency of appearance of request sequences and systematic coverage of parameter-values and their interactions. Rummel et al. [49] introduced a prioritization technique based on data-flow analysis. They focused on the definition and use of program variables for the data-flow analysis. Jeffrey et al. [38] prioritized test cases using relevant slices. Qu et al. [50] prioritized test cases in a black box environment. However, none of the above solutions considered dependencies among faults in prioritizing test cases for regression testing. In software testing, it is known that some faults are the consequences of other faults (leading faults). So, intuitively, test cases that revealed the leadings faults should be executed first in a regression testing in order to get an early confirmation that software is free from dependent faults. In [19] we took the first step to prioritize regression testing based on fault dependency. We proposed an algorithm to prioritize test cases based on fault dependency. We also proposed a metric Average Average Percentage Fault Dependency Detected (APFDD) to quantify how rapidly a prioritized test suite can detect dependencies among faults. However, that work only considered $1$-hop neighborhood or dependencies of faults. This paper leverages a fault network for prioritization. ## 5 Summary and Discussions In this paper, we have presented ComReg, which uses a fault dependency network to prioritize test cases for regression testing. We have modeled a fault dependency network as a directed graph and identified leading faults to prioritize test cases. We have leveraged a network centrality aggregation technique in the fault dependency network to identify leading faults. The centrality aggregation technique considers six representative centrality metrics such as indegree, betweenness, closeness, eigenvector, pagerank and hub centrality and offers a final leading score to identify the leading faults. Our discussions on fault communities shed light on selecting X% of the test cases from a prioritized regression test suite. Finally, we have presented a case study. In the case study, we have developed an English vocabulary learning software, “Tarantula” and identified leading faults from a fault network after running a set of test cases at the end of the first phase of the development. We have showed the fault communities in the fault network for test case selection from a prioritized regression test suite. The fault dependency network might not be a connected graph. For example, in our fault dependency network of Tarantula consists of two components. However, small-world networks tend to have giant components(e.g., [51, 52, 53]). A giant component is a connected subgraph that contains a majority of the entire graph’s nodes [54]. The giant component fills most of the network—usually more than half and not infrequently over 90%—while the rest of the network is divided into a large number of small components disconnected from the rest [23]. Our small-world fault dependency network also has one giant component ($22$ nodes). So, if a fault network has a large number of nodes and if it shows a large number of connected components, the giant component could be leveraged to detect the leading faults. Our work has multiple limitations. First, we built a subject software (“Tarantula”) to present a case study and show the effectiveness of our prioritization technique. The Tarantula is a medium-scale software, which lacks the rigorous development cycle of a typical commercial software. This leads to a higher number of faults in system testing, even using a small number of test cases. 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1311.4238	# Nonlinear Acoustics - Perturbation Theory and Webster’s Equation Rogério Jorge Departamento de Física, Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal [email protected] ###### Abstract Webster’s horn equation (1919) offers a one-dimensional approximation for low- frequency sound waves along a rigid tube with a variable cross-sectional area S(x). It can be thought as a wave equation with a source term that takes into account the nonlinear geometry of the tube. In this document we derive this equation using a simplified fluid model of an ideal gas. By a simple change of variables, we convert it to a Schrödinger equation and use the well-known variational and perturbative methods to seek perturbative solutions. As an example, we apply these methods to the ”Gabriel’s Horn” geometry, deriving the first order corrections to the linear frequency. An algorithm to the harmonic modes in any order for a general horn geometry is derived. ## I Introduction We study the propagation of a wave in a narrow but long, tubular domain of finite length whose cross-sections are circular and of varying area. In this case, the wave equation has a classical approximation depending on a single spatial variable in the long direction of the domain. This approximation is known as Webster’s equation (2). The geometry of the tube is represented by the area function $S(x)$ whose values are cross-sectional areas of the domain. We derive this result in section II. As the name suggests, this equation was derived by Webster in 1919 webster but, citing Edward Eisner (referring to P. A. Martin article in onwebster ) - ”we see that there is little justification for this name. Daniel Bernoulli, Euler, and Lagrange all derived the equation and did most interesting work on its solution, more than 150 years before Webster.” In section III, the link between the Schrödinger’s equation and eq. 2 is shown, offering an effective Hamiltonian and a potential energy that can be thought as a perturbation to the ”free” hamiltonian. The key feature of this document is the analysis in section V, where we obtain the first order harmonic corrections using perturbation theory on a well known geometry - Gabriel’s horn (discussed in section IV). In section VI, an algorithm to obtain this frequency corrections in any order and geometry is provided. With this analysis, we can infer how much the instrument (in the wind or brass family) will be out of tune only by its geometry. ## II Physical Model ### II.1 Extended Derivation From fluid mechanics, the material derivative $\frac{Dm}{Dt}$ is given by Reynolds’ transport theorem. It can be stated as $\frac{Dm}{Dt}=\partial_{t}\int_{V}\rho dV+\int_{S}(\vec{v}.\vec{n})\rho dS=0,$ (1) where $m$ is the mass of the gas inside the tube (which is constant), $\rho$ is its mass density, $V$ and $S$ are the volume and surfaces of integration along the tube and $\vec{v}$ is the velocity of the gas in across the surface of integration. Choosing these domains of integration and reference axes, we refer to fig. 1. Figure 1: Fluid model for a varying cross section. Image taken from Fundamentals of Physical Acoustics by David T. Blackstock For a consistent and rigorous derivation, we assume the following conditions: * • The mass density is constant throughout the cross-section area but is time dependent $\rho=\rho(t)$ (later we will simplify this assumption). * • The tube shape is fixed i.e. independent of time but not constant in $x$, which by eq. 1 implies $(\partial_{t}\rho)S=-\rho(\partial_{x}vS)$. * • The ideal gas law $P=k_{B}Tn$ holds, where $P$ is the pressure, $T$ the temperature (assumed constant), $k_{B}$ the Boltzmann constant and $n=\frac{\rho}{m}$, being $m$ the mass of each particle (assumed equal). * • By Newton’s second law - $\rho\frac{\partial v}{\partial t}=-\frac{\partial P}{\partial x}$. * • The fluid is irrotational, meaning $\nabla\times\vec{v}=0$. Hence, differential calculus tells us that we can always find a velocity potential $\phi$, such that (in one dimension) $v=-\frac{\partial\phi}{\partial x}$. This offers a closed system of equations. Solving all equations for $P$, we discard at the end the time derivative of $\rho$. This is justified assuming that for long tubes, the local pressure variation is much larger than the local density fluctuation, obtaining Webster’s equation (eq. 2). $\frac{1}{S}\frac{\partial}{\partial x}\left(S\frac{\partial P}{\partial x}\right)=\frac{1}{c^{2}}\frac{\partial^{2}P}{\partial t^{2}},$ (2) where $c^{2}=\frac{k_{B}T}{m}$. Physically, the measurable quantity in the laboratory is $P$, justifying the form of eq. 2. ### II.2 Alternative Derivation Using the concept of bulk modulus, we can easily derive (but lacking physical intuiton) equation (2). The differential volume for the gas section is $dV=dx\frac{\partial}{\partial x}\left(S\zeta\right)$, where $\zeta=\zeta(x,t)$ is the displacement of surfaces with equal pressure. By the ideal gas law, we can use the definition of bulk modulus $B$ (assumed constant) to obtain $P(x)=-\frac{B}{S}\partial_{x}\left(S\zeta\right).$ (3) From Newton’s second law we compute the gas volume acceleration due to pressure variation along $x$ $S\rho dx\frac{\partial^{2}\zeta}{\partial t^{2}}=-\frac{\partial P}{\partial z}Sdx.$ (4) Substituting into the previous equation we have Webster‘s equation with $c^{2}=\frac{B}{\rho}$. This is also the procedure used in hornfunction . ## III Relation with Schrödinger Equation Starting with Webster’s equation (2) previously derived, we apply the following change of variables (as in hornfunction ) * • $\psi=P\sqrt{S}$, * • $r=\sqrt{S}$, * • The time dependence on $P$ is given by $e^{iwt}$, * • $k=\frac{w}{c}$, which in turn implies $-\frac{d^{2}\psi}{dx^{2}}+\frac{r^{\prime\prime}}{r}\psi=k^{2}\psi.$ (5) This is equivalent to the Schrödinger equation for one dimensional scattering, where the particle’s energy is now $k^{2}$ and the potential energy function ise replaced by $\frac{r^{\prime\prime}}{r}$. In literature, this equation is often called the horn function. The ”potential energy” can be thought as a normalized curvature and $r$ is (apart from a numerical factor) the radius of the horn. We can even infer a Hamiltonian operator from (5) $\hat{H}=\hat{T}+\hat{V}=-\frac{d^{2}}{dx^{2}}+\frac{r^{\prime\prime}(x)}{r(x)},$ (6) since $\psi$ has no time dependence. In our case, the tube is long (compared with its radius), so we can treat the potential $\frac{r^{\prime\prime}(x)}{r(x)}$ as a perturbation of the ”free” hamiltonean. ## IV Gabriel’s Horn Gabriel’s horn, also called Torricelli’s trumpet, is the surface of revolution of $y=\frac{1}{x}$ about the x-axis for $x\geq 1$. It is therefore given by the following parametric equations (wolfram ) $x=u,\hskip 5.69046pty=\frac{a\hskip 2.84544pt\cos(\nu)}{u},\hskip 5.69046ptz=\frac{a\hskip 2.84544pt\sin(\nu)}{u},$ (7) where $a$ is the radius of the surface on $x=1$. It is easy to show that this surface has finite volume, but infinite surface area. Figure 2: Gabriel‘s Horn with $a=1$ (computed with the Wolfram Alpha platform). We can see that, for fig. 2, we have $r(x)=\frac{a}{x},\hskip 5.69046ptS(x)=\frac{\pi a^{2}}{x^{2}},$ (8) so Webster’s and Horn equations (2 and 5) become respectively (10) and (9). $\frac{d^{2}P(x)}{dx^{2}}-\frac{2}{x}\frac{dP(x)}{dx}+w^{2}P(x)=0$ (9) $\frac{d^{2}\psi}{dx^{2}}+\left(k^{2}-\frac{2}{x^{2}}\right)\psi=0.$ (10) The solution for these equations are, respectively $P(x)=\sqrt{\frac{2}{\pi w^{3}}}\left[(Awx+B)\cos(wx)+(Bwx-A)\sin(wx)\right]$ (11) $\psi(x)=\sqrt{\frac{2}{\pi k^{3}}}\left[\left(Ak+\frac{B}{x}\right)\cos(kx)+\left(Bk-\frac{A}{x}\right)\sin(kx)\right].$ (12) Clearly, these can’t be expressed as a Fourier sum. This could be achieved by expanding over a small parameter and we will do so in the next section. It is also hard to find the quantized frequencies without the use of a numerical method. As we will see, the perturbative method offers much more insight and simpler expressions to use on real situations. ## V Perturbative Methods As stated in section III, eq. 6 is the hamiltonian of the system, written as a sum of a ”free” hamiltonian plus a perturbation. The latter term, for Gabriel’s horn, is written as $\frac{2}{x^{2}}$. The solution of eq. 10 for the free wave (neglecting the potential) is $\psi_{0}=A\cos(kx)+B\sin(kx)$. From the boundary conditions $k$ will be quantized, so a sum over $k_{n}$ is performed, obtaining a Fourier decomposition as we would expect. $\psi_{0}=\sum_{n}A_{n}\cos(k_{n}x)+B_{n}\sin(k_{n}x).$ (13) The tube is open on both sides, which means, the pressure must be $0$ on $x=1$ and $x=L$ (defined as the constant ambient pressure). The origin is chosen so that the tube length is $L-1$ and the potential is regular. At zeroth order, the surface is constant. Reverting the change of variables and denoting $S_{0}$ as the surface area at the origin, we have $P_{0}(x)=\frac{1}{\sqrt{S_{0}}}\sum_{n}(A_{n}\cos(k_{n}x)+B_{n}\sin(k_{n}x))$. The boundary conditions impose the quantization $\tan(k)=\tan(kL)$ and the form (14) $\psi_{0}=\sum_{n}B_{n}\left(\sin(k_{n}x)-\tan(k)\cos(k_{n}x)\right),\hskip 5.69046ptk_{n}=\frac{n\pi}{L-1},$ (14) which in turn offers the expected result $f_{n}=\frac{c}{2(L-1)}n$, where $f$ is the frequency of the sound wave and $n$ is an integer $>0$ and $L-1$ is the real length of the tube. In the quantum mechanics formalism (as the one outlined below), the wave function must be normalized so that an explicit expression for $B_{n}$ can be found. This is the merging point of the classical and quantum treatment so it must be carefully done. This can be done by the following algorithm: * • Obtain the pressure profile and the length $L-1$ of the horn. With this, compute $\lambda^{2}=\int_{L}P^{2}(x)dx$. Normalize the pressure profile by $\lambda$. * • Performing the previous integral analitically, the factor $B_{n}$ will depend on $\lambda$. By the argument above, we can set $\lambda=1$ in our model. Normalizing the square of the wave function over the tube results in $B_{n}=\sqrt{\frac{2}{L-1}}\cos\left(\frac{n\pi}{L-1}\right).$ (15) ### V.1 Variational Method This model requires a test function and a minimizing parameter $\delta$. As expected, our test function will be (14) (the free wave). The parameter, as defined in eq. 16) is expected to minimize $<H>$ near $\delta=1$. The function $\frac{d}{d\delta}<H>$ is shown in fig. 3. The results are valid for all $n$, as fig. 3 implies (for bigger $n$ the derivative explodes). For $\delta>0$, there are no roots of $\frac{d}{d\delta}<H>$, so the method can’t be applied in this framework. Incidentally, the variational method only provides a correction to the ”ground state”, so we can’t calculate to an arbitrary order the corrections to the frequency. $k_{n}(\delta)=\frac{\pi}{L-1}n^{\delta}$ (16) Figure 3: $\frac{d}{d\delta}<H(\delta)>$ for a Gabriel horn with $L=20$ u.l., $\delta\in[0,2]$ and $n\in[0,3]$. ### V.2 Non-degenerate time independendent perturbation theory Perturbation theory tells us that the difference in energy $k^{2}$ from $k_{0}^{2}$ in first order is $\Delta k^{2}=\int_{1}^{L}\psi_{0}^{\dagger}\hat{H}\psi_{0}dx=\int_{1}^{L}\frac{2\psi_{0}^{2}(x)}{x^{2}}dx,$ (17) where $\psi_{0}$ is the unperturbed wave function. Performing the integration on (18), an analytical expression for $\Delta k^{2}=k^{2}-k_{n}^{2}$ is obtained. A plot of $f-f_{n}$ is shown on fig. 4. $\Delta k^{2}=\frac{4\cos^{2}\left(\frac{n\pi}{L-1}\right)}{L-1}\int_{1}^{L}\frac{\left[\sin(kx)-\tan(k)\cos(kx)\right]^{2}}{x^{2}}.$ (18) Figure 4: Difference of the total frequency to the unperturbed one up to $n=50$ for a $L=20$ Gabriel horn, with $c=344$ m/s in first order perturbation theory. ## VI Concluding Remarks We have derived the expression for the perturbation on the frequency spectrum of a horn with varying cross section using time-independent perturbation theory in first order. Physically, the wave is an infinite sum of $n$ modes. Analyzing our results, the perturbation convergence is secured, as the correction is smaller for smaller values of $n$. As a general procedure, one could find how much the geometry of the horn ”constraints” the non-linearity of the frequency harmonics. * • Express the radius of the horn in terms of the cylindrical coordinate $x$ \- $r=r(x)$ and the length of the horn as $L-1$. * • On that particular horn, measure the value of $\lambda^{2}=\int_{L}P^{2}(x)dx$ and normalize the pressure profile by $\lambda$. * • Using $B_{n}$ as in eq. 15 with $\lambda=1$, $\psi_{0}=B_{n}\left[\sin(k_{0}x)-\tan(k_{0})\cos(k_{0}x)\right]$ and $k_{0}=\frac{n\pi}{L-1}$, calculate the integral $\Delta{k^{2}}=\int_{1}^{L}\psi_{0}^{2}(x)\frac{r^{\prime\prime}(x)}{r(x)}dx$. * • The correction to the wave number in first order is $k=\sqrt{k_{0}^{2}+\Delta{k^{2}}}$. We hope that this approach serves both the acoustical science community and the curious physicist, providing an interesting application to the quantum mechanical methods within a classical framework. ###### Acknowledgements. I would like to thank prof. Henrique Oliveira for fruitful discussions and prof. Filipe Joaquim for essential corrections to the text and guidance. ## References * (1) Weisstein, Eric W. ”Gabriel’s Horn.” From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/GabrielsHorn.html * (2) Webster, A. G. (1919) “Acoustical impedance, and the theory of horns and of the phonograph”, Proc. Natl. Acad. Sci. U.S.A. 5, 275–282. * (3) Ting, L., and Miksis, M. J. (1983) “Wave propagation through a slender curved tube”, J. Acoust. Soc. Am. 74, 631–639. * (4) P. A. Martin, (2004) ”On Webster’s horn equation and some generalizations”, Acoustical Society of America. [DOI: 10.1121/1.1775272] * (5) Bjørn Kolbrek, (2008) ”Horn Theory: An Introduction, Part 1”, article prepared for www.audioxpress.com * (6) David Berners and Julius O. Smith III (1994) ”On the use of Schrödinger’s equation in the analytic determination of horn reflectance”, ICMC Preceedings in Sound Synthesis Techniques * (7) D. J. Griffiths (2005) ”Introduction to Quantum Mechanics”, 2nd Ed. Prentice Hall	arxiv-papers	2013-11-18T01:19:05	2024-09-04T02:49:53.782623	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Rog\\'erio Jorge", "submitter": "Rog\\'erio Jorge", "url": "https://arxiv.org/abs/1311.4238" }
1311.4280	# A little more Gauge Mediation and the light Higgs mass V. Suryanarayana Mummidi [email protected] Sudhir K Vempati [email protected] Centre for High Energy Physics, Indian Institute of Science, Bangalore 560012 ###### Abstract We consider minimal models of gauge mediated supersymmetry breaking with an extra $U(1)$ factor in addition to the Standard Model gauge group. A $U(1)$ charged, Standard Model singlet is assumed to be present which allows for an additional NMSSM like coupling, $\lambda H_{u}H_{d}S$. The U(1) is assumed to be flavour universal. Anomaly cancellation in the MSSM sector requires additional coloured degrees of freedom. The $S$ field can get a large vacuum expectation value along with consistent electroweak symmetry breaking. It is shown that the lightest CP even Higgs boson can attain mass of the order of 125 GeV. ###### pacs: 73.21.Hb, 73.21.La, 73.50.Bk ## I Introduction Gauge mediated supersymmetry breaking Dine:1993yw ; Dine:1994vc ; Dine:1995ag ; Giudice:1998bp (for earlier works, please see, Dine:1981za ; Dine:1981gu ; Dine:1982zb ; Dimopoulos:1982gm ; Dimopoulos:1981au ; Nappi:1982hm ; AlvarezGaume:1981wy ) attractive due to several interesting features (i) flavour blind supersymmetry breaking soft terms (ii) very few parameters determine the entire spectrum (iii) different collider phenomenology compared to gravity mediated models as typically gravitino is the lightest supersymmetric particle (LSP) etc. However, phenomenologically111For an early phenomenology of these models, please see, Agashe:1997kn ; Bagger:1996bt ; Baer:1996hx . the minimal versions of gauge mediation are severely constrained due to the discovery of a Higgs particle with a mass around 125 GeV. In MSSM, for the lightest CP even Higgs to be around 125 GeV would require, stop mixing parameter $X_{t}$ to be large, $X_{t}\sim\sqrt{6}M_{S}$, where $M_{S}=\sqrt{m_{\tilde{t}_{1}}m_{\tilde{t}_{2}}}$. While this holds true as long as stops are light $\sim 1~{}\text{TeV}$, for very heavy stops $\gtrsim 4\,\text{TeV}$, the mixing parameter can be smaller. This would however push stops out of the reach of the LHC. In spite of theoretically appealing features, unfortunately, in minimal gauge mediation, the only way to fit a light Higgs mass $\sim 125~{}\text{GeV}$ is by making stops very heavy. The typical trilinear couplings in these models are very small at the mediation scale $\sim 0$. Renormalisation group (RG) effects do generate them at the weak scale, however their magnitude is not large enough unless one makes gluinos ultra heavy $\sim$ several TeV Draper:2011aa . It should be noted that the constraints from 125 GeV Higgs boson are stronger even if one moves away from minimal mediation models to general gauge mediation models as long as $A_{t}$ remains zero at the messenger scale Grajek:2013ola . Several possible solutions have been explored in the literature Albaid:2012qk ; Frank:2013yta ; Evans:2013kxa ; Craig:2013wga ; Calibbi:2013mka ; Byakti:2013ti ; Evans:2011bea ; Evans:2012hg ; Craig:2012xp ; Yanagida:2012ef ; Jelinski:2011xe ; Abdullah:2012tq ; Perez:2012mj ; Endo:2012rd ; Martin:2012dg ; Fischler:2013tva ; Bhattacharyya:2013xma ; Kang:2012ra . One of the directions which is popular with many authors is to introduce direct Yukawa couplings between messenger fields and the MSSM fields in addition to gauge interactions Chacko:2001km ; Chacko:2002et . In some cases, these interactions could also violate flavour Shadmi:2011hs . In most of the models it is possible to generate large enough $A_{t}$ at the weak scale to fit the 125 GeV light CP even Higgs boson mass. In a recent survey Evans:2013kxa ; davidtalk it has been pointed out that a particular class of messenger-matter interactions, messenger- stop mixings, has the least fine tuning of all the possible models which fit the light Higgs mass. Another direction which has been considered is to add additional vector like quarks close to the weak scale which couple to the Higgs superfields. These lead to additional corrections to the light Higgs boson thus lifting its mass without the need of increasing the stop masses (see for example, Martin:2012dg ; Endo:2012rd ; Fischler:2013tva ). In the following we would like to take an alternate route. We would like to keep the minimal mediation structure in tact, thus would not like to introduce direct couplings between matter and messenger fields. Adding an additional singlet field, like in NMSSM could help to raise the light Higgs mass. There are however, problems with electroweak symmetry breaking while incorporating NMSSM in minimal gauge mediation. These are well documented in literature deGouvea:1997cx ; Dine:1996xk . There are ways out, either by adding additional matter fields or dynamics through which NMSSM can be made viable with minimal gauge mediation Langacker:1999hs ; Ellwanger:2008py ; Liu:2008pa ; Hamaguchi:2011nm ; Hamaguchi:2011kt ; Dimopoulos:1997je ; Morrissey:2008gm ; Yanagida:1997yf ; deBlas:2011cr ; deBlas:2011hs . Post 125 GeV Higgs boson, a model within this class has been explored in Yanagida:2012ef . In the present work, we will consider an additional U(1) gauge group under which the ‘singlet’ of the NMSSM is charged. This U(1) factor also participates in gauge mediation. Anomaly cancellation requires additional vector like matter to be present. Such vector like matter is typically introduced to generate correct electroweak symmetry breaking while incorporating NMSSM in minimal mediation models Dine:1996xk . In the present case, it is motivated from anomaly cancellation requirements. It should be noted that this kind of model has been considered earlier by the authors of Ref. Langacker:1999hs . Ours is a more explicit realisation of it in the sense that we have taken care of $U(1)$ charges and anomaly cancellation conditions. Furthermore, we have performed a more detailed analysis of the Higgs masses in the light of 125 GeV Higgs discovery. We found that it is possible to find an appropriate set of rational U(1) charges which satisfy the anomaly cancellation conditions as well as allow the correct set of terms in the superpotential. Electroweak symmetry breaking is possible as the U(1) charged singlet can achieve a reasonable vacuum expectation value (vev). Two factors contribute to the raise in the lightest CP even Higgs mass: the effective $\mu$ term is sufficiently large $\sim 0.5-1\,\text{TeV}$ and secondly the RG generated $A_{t}$ term is large compared to minimal gauge mediation. The later is because at the 1-loop level, the $SU(3)$ beta function, $b_{3}$ is zero in this model and the 2-loop $b_{3}$ is not sufficiently large. Together they result in sufficient $X_{t}$ to ensure large mixing in the stop mass matrix. It is possible to find reasonable parameter space which gives correct lightest CP even Higgs mass and satisfy direct constraints from LHC as well as constraints from $Z-Z^{\prime}$ mixing. The rest of the paper is organised as follows: In the next section particle spectrum and the model are presented. The details of supersymmetric spectrum and various constraints on the parameter space are discussed in section 3. Numerical results are presented in section 4. We close with an outlook in section 5. ## II Model and the Particle Spectrum The basic premise of the model is that the singlet of the NMSSM should no longer be a singlet, but instead, it is charged under an extension of the Standard Model gauge group such that it receives non-zero supersymmetry breaking contributions at the mediation scale. As it will be detailed in the next section this would help in attaining a large enough vacuum expectation value for the field ‘S’. In this present work, we try to do this by considering the simplest extension in terms of a $U(1)$. The relevant field $S$ is singlet under the Standard Model gauge group, but charged under the extra $U(1)$ ; as a consequence of which all the Standard Model fields are charged under the $U(1)$. The total gauge group is $G_{SM+A}=SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}\times U(1)_{A}$ (1) where the first three represent the usual Standard Model gauge group and the additional gauge group is represented by a subscript A. $U(1)_{A}$ is a chiral gauge group and hence introduces an extra set of anomalies which need to be canceled for a consistent quantum field theory. This imposes a set of conditions on the $U(1)_{A}$ charges; they are listed in Appendix A. We insist that the anomalies cancel independently for the NMSSM sector and the Messenger sector. It is easily verified that the MSSM particle spectrum along with the new field $S$ is not sufficient to cancel all the anomalies. In particular, from $(U(1)_{A}-[SU(3)_{c}]^{2})$ anomaly condition we get $A_{1}(exotics)=-3s$ (2) where $A_{1}(exotics)$ is the contribution of the new exotic fields which need to be added and $s$ is the $U(1)_{A}$ charge of the field $S$. The $U(1)_{A}$ charge $s$ cannot be zero as per our requirements. Furthermore, to generate the effective $\mu$ term ($\lambda SH_{u}H_{d}$) in the super-potential, the charge $s$ should be equal to $s=-(h_{1}+h_{2})\neq 0$ (3) where $h_{1}$ and $h_{2}$ are the $U(1)_{A}$ charges of $H_{1}$ and $H_{2}$ respectively. We thus need coloured exotics to satisfy $U(1)_{A}-[SU(3)_{c}]^{2}$ anomaly. The number of the exotics is fixed by other anomaly conditions as well as by the $U(1)_{A}$ gauge invariance of the super-potential. It turns out that one possible minimal set of exotic fields would be three families of $SU(2)_{L}$ singlet coloured exotics. We introduce a pair of colour fundamental and anti-fundamentals $D_{i}$ and $\bar{D}_{i}$, which are $SU(2)$ singlets, for each of the three families. In addition to the QCD interactions they are allowed to couple with the field $S$ in the super- potential. The total particle spectrum and their corresponding representations and the $U(1)_{A}$ charges , in the order of Eq. (1) are given in the table below. $\begin{array}[]{rlrlrl}Q_{i}:&(3,2,\frac{1}{6},q_{i})&~{}U^{c}_{i}:&({\bar{3}},1,-\frac{2}{3},u_{i}),&D^{c}_{i}:&({\bar{3}},1,\frac{1}{3},d_{i}),\\\ L_{i}:&(1,2,-\frac{1}{2},l_{i}),&~{}E^{c}_{i}:&(1,1,1,e_{i}),&&\\\ H_{1}:&(1,2,-\frac{1}{2},h_{1})&~{}H_{2}:&(1,2,\frac{1}{2},h_{2}),&S:&(1,1,0,s),\\\ D_{i}:&(3,1,y_{i},z_{i})&\bar{D}_{i}&(\bar{3},1,-y_{i},\bar{z}_{i}),&&\end{array}$ (4) where $i$ represents the generation index running from 1 to 3. In the rest of the paper, we will consider all the $U(1)_{A}$ charges to be universal over all the generations and thus suppress the generation index. The only exception to this rule is the $U(1)_{A}$ charges of exotics $z_{i}$. We will consider them to be different for each of the generation, subject to the constraint that in each generation, $z_{i}+\bar{z}_{i}=-s$. The super-potential is given by $W=Y_{E}L{E^{c}}H_{1}+Y_{D}Q{D^{c}}H_{1}+Y_{U}Q{U^{c}}H_{2}+\lambda SH_{1}H_{2}+\kappa_{i}SD_{i}{\bar{D}}_{i}$ (5) where $Y_{E},\,Y_{D},\,Y_{U},\lambda$, $\kappa_{i}$ are Yukawa couplings and we have suppressed generation and colour indices. Note that the field S does not have cubic self interactions. We will consider a minimal set of messengers communicating the effect of spontaneous supersymmetry breaking in the hidden sector. The spurion $X$ couples to the messengers with the super-potential $W=\eta X\Phi\bar{\Phi}$ (6) where $\Phi$ are messengers in fundamental representation of an $SU(N)\supset G_{SM+A}$ gauge group and $\eta$ is some Yukawa coupling. The resultant soft terms can easily be generalised with the extra $U(1)_{A}$ and can be verified with the wave-function methods of Refs. Giudice:1997ni ; ArkaniHamed:1998kj . The mass terms for the gauginos and soft mass squared terms for the scalars at the mediation scale, $X$ are given as follows222In writing the formulae Eq.(7) we have suppressed the 1-loop and 2-loop functions. They are however taken in to account in the numerical analysis: $\displaystyle M_{i}(X)$ $\displaystyle\approx$ $\displaystyle\frac{\Lambda}{16\pi^{2}}\sum_{i}\left(g_{i}^{2}(X)\right)$ $\displaystyle m^{2}_{\tilde{f}}(X)$ $\displaystyle\approx$ $\displaystyle\frac{2\Lambda^{2}}{(16\pi^{2})^{2}}\sum_{i}\left(g_{i}^{4}(X)~{}C_{i}(f)\right)$ (7) where through an abuse of notation, we have expanded the spurion as $<X>=X+\theta^{2}F_{X}$ and defined $\Lambda={F_{X}/X}$. $C_{i}(f)$ are quadratic Casimirs for the fields $f$ under the four gauge groups. The index $i$ here runs over all the four gauge groups of Eq.(1). We denote the gauge coupling corresponding to $U(1)_{A}$ as $g_{4}$ and we can see, the soft mass of $S$ has the following non-zero value at the $X$ scale : $m_{S}^{2}(X)\approx~{}2s^{2}~{}\tilde{\alpha}_{4}^{2}(X)~{}\Lambda^{2},$ (8) where we used the standard notation of $\tilde{\alpha}_{i}=\alpha_{i}/(4\pi)$ and $\alpha_{i}=g_{i}^{2}/(4\pi)$. Similarly, we christen $M_{4}$ to be the neutral gaugino corresponding to $U(1)_{A}$ group. It’s mass is given by $M_{4}\approx\tilde{\alpha}_{4}(X)~{}\Lambda$ (9) The presence of additional $U(1)_{A}$ also introduces additional splittings between the mass squared terms at the mediation scale $X$. For example, the slepton doublets and the Higgs which are degenerate at the high scale in Minimal case, get split as: $\displaystyle m_{L}^{2}(X)-m_{H_{1,2}}^{2}(X)$ $\displaystyle=$ $\displaystyle 2(l^{2}-h_{1,2}^{2})~{}\tilde{\alpha}_{4}^{2}(X)~{}\Lambda^{2}$ $\displaystyle m_{H_{1}}^{2}(X)-m_{H_{2}}^{2}(X)$ $\displaystyle=$ $\displaystyle 2(h_{1}^{2}-h_{2}^{2})~{}\tilde{\alpha}_{4}^{2}(X)~{}\Lambda^{2}$ (10) However, as we will see later the freedom of these splits is limited as the choice of $U(1)_{A}$ is quite restricted due to phenomenological constraints and anomaly cancellation conditions. Finally, just as in the minimal messenger model, the trilinear $A$ -terms and bilinear $B$ terms remain zero at the mediation scale $X$. ## III Weak Scale Spectrum The soft terms at the weak scale can be evaluated by using the relevant Renormalisation Group (RG) equations with the above boundary conditions, Eq.(7). One interesting aspect about the one loop beta functions for the gauge couplings is that the beta function of $SU(3)$, $b_{3}^{(1)}=0$. This is due to the presence of the additional colour triplets $D,\bar{D}$ in three generations333 We have not explored in the present work about the possibility of making this model finite in the UV (see for example Babu:2002ki ).. As the $\alpha_{s}$ does not run at the 1-loop level, most coloured particles receive larger corrections in RGE running, compared to the Minimal messenger model. This has consequences for the running of $y_{t}$ and subsequently to all the parameters which depend on $y_{t}$ or $A_{t}$. We have used 1-loop RGE for the soft terms and added 2-loop RGE’s for the gauge couplings and Yukawa couplings in this analysis. The relevant RGE for this model are given in Appendix C. Before proceeding further, a comment about kinetic mixing is in order. The U(1) gauge fields can mix through the kinetic terms of the type $\chi\int d\theta~{}\mathcal{W}^{A}\mathcal{W}_{Y}$. The current bounds on $\chi$ limit it to $10^{-3}$Hook:2010tw . We expect that the implications on the phenomenology to be discussed in our paper will be minimally affected due to the presence of the kinetic mixing. For this reason, we will neglect all its effects in the subsequent discussion. At the weak scale, $M_{SUSY}\sim 1\,\text{TeV}$, we impose electroweak symmetry breaking conditions along with the $U(1)_{A}$ breaking. The neutral Higgs scalar potential is given by $V_{0}=V_{F}+V_{D}+V_{\rm soft}$ (11) where $\displaystyle V_{F}$ $\displaystyle=$ $\displaystyle\|\lambda H_{2}\cdot H_{1}\|^{2}+\|\lambda S\|^{2}\left(\|H_{1}\|^{2}+\|H_{2}\|^{2}\right),$ (12) $\displaystyle V_{D}$ $\displaystyle=$ $\displaystyle\frac{(g_{1}^{2}+g_{2}^{2})}{8}\left(\|H_{1}\|^{2}-\|H_{2}\|^{2}\right)^{2}+\frac{g_{2}^{2}}{2}\left(\|H_{1}\|^{2}\|H_{2}\|^{2}-\|H_{2}\cdot H_{1}\|^{2}\right)$ (13) $\displaystyle+$ $\displaystyle{{g_{4}}^{2}\over 2}\left(h_{1}\|H_{1}\|^{2}+h_{2}\|H_{2}\|^{2}+s\|S\|^{2}\right)^{2}$ (14) $\displaystyle V_{\rm soft}$ $\displaystyle=$ $\displaystyle m_{1}^{2}\|H_{1}\|^{2}+m_{2}^{2}\|H_{2}\|^{2}+m_{s}^{2}\|S\|^{2}+\left(A_{\lambda}SH_{2}\cdot H_{1}+h.c.\right).$ (15) The neutral components of the Higgs fields $H_{1}$ and $H_{2}$ get vacuum expectation values (VEV) at the weak scale, $\frac{v_{1}}{\sqrt{2}}$ and $\frac{v_{2}}{\sqrt{2}}$. The field $S$ also gets a VEV, $\frac{v_{s}}{\sqrt{2}}$ at the weak scale, breaking the $U(1)_{A}$ symmetry spontaneously. At the minima of the potential, the vevs and the soft terms along with the other parameters of the model get related. These minimisation conditions are given as $\displaystyle m_{1}^{2}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\left[\frac{G^{2}}{4}+h_{1}^{2}{g_{4}}^{2}\right]v_{1}^{2}+\frac{1}{2}\left[\frac{G^{2}}{4}-\lambda^{2}-h_{1}h_{2}{g_{4}}^{2}\right]v_{2}^{2}-\frac{1}{2}\left[\lambda^{2}+h_{1}s{g_{4}}^{2}\right]{v_{s}}^{2}$ (16) $\displaystyle+\ \frac{A_{\lambda}}{\sqrt{2}}\frac{v_{2}v_{s}}{v_{1}},$ $\displaystyle m_{2}^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left[\frac{G^{2}}{4}-\lambda^{2}-h_{1}h_{2}{g_{4}}^{2}\right]v_{1}^{2}-\frac{1}{2}\left[\frac{G^{2}}{4}+h_{2}^{2}{g_{4}}^{2}\right]v_{2}^{2}-\frac{1}{2}\left[\lambda^{2}+h_{2}s{g_{4}}^{2}\right]{v_{s}}^{2}$ (17) $\displaystyle+\ \frac{A_{\lambda}}{\sqrt{2}}\frac{v_{1}v_{s}}{v_{2}},$ $\displaystyle m_{s}^{2}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\left[\lambda^{2}+h_{1}s{g_{4}}^{2}\right]v_{1}^{2}-\frac{1}{2}\left[\lambda^{2}+h_{2}s{g_{4}}^{2}\right]v_{2}^{2}-\frac{1}{2}s^{2}{g_{4}}^{2}v_{s}^{2}+\frac{A_{\lambda}}{\sqrt{2}}\frac{v_{1}v_{2}}{v_{s}},$ (18) where $G^{2}=g_{1}^{2}+g_{2}^{2}$. The minimisation conditions are modified compared to the standard NMSSM case due to the presence of terms proportional to $g_{4}$. Subsequently, we can see from Eq. (18), that in the limit $v_{s}\gg v_{1},v_{2}$($v_{s}$ is required to be large which is discussed later in this section), we have $v_{s}^{2}\approx-{2~{}m_{s}^{2}\over s^{2}g_{4}^{2}},$ which is the typical vev one expects in extra U(1) models Barger:2008wn ; Langacker:1999hs . At the high scale, $X$, $m_{S}^{2}$ which is positive and proportional to $\tilde{\alpha}_{4}^{2}\Lambda^{2}$ can be driven negative at the electroweak scale by the Yukawa couplings of the exotics $k_{1},k_{2},k_{3}$ . This should be contrasted with the vev in minimal gauge mediation, without the $U(1)$ factor. See for example,Refs.[ deGouvea:1997cx ; Ellwanger:2009dp ]. From the minimization conditions of NMSSM, we get $v_{s}^{2}\approx-\frac{1}{2\kappa^{2}}\left(\lambda^{2}(v_{1}^{2}+v_{2}^{2})+2m_{s}^{2}-2\lambda\kappa v_{1}v_{2}\right)$ (19) which is too small to get $\mu_{eff}$ ($\frac{\lambda v_{s}}{\sqrt{2}}$) of the order of electroweak symmetry breaking. To achieve a significant value either $\lambda$ has to be very large ($>1$) or $\kappa$ has to be too small. In both the cases, achieving electroweak symmetry breaking is highly constrained Delgado:2007rz . We now turn our attention to the Higgs sector. The CP-even tree-level Higgs mass squared matrix, $\Psi^{\dagger}\mathcal{M}_{+}^{2}\Psi$, where $\Psi^{T}=\\{H_{1}^{0},H_{2}^{0},S\\}$, and the elements of the matrix are given as: $\displaystyle\left({\mathcal{M}_{+}^{0}}\right)_{11}^{2}$ $\displaystyle=$ $\displaystyle\left[\frac{G^{2}}{4}+h_{1}^{2}{g_{4}}^{2}\right]v_{1}^{2}+\frac{A_{\lambda}}{\sqrt{2}}\frac{v_{2}v_{s}}{v_{1}}$ $\displaystyle\left({\mathcal{M}_{+}^{0}}\right)_{12}^{2}$ $\displaystyle=$ $\displaystyle-\left[\frac{G^{2}}{4}-\lambda^{2}-h_{1}h_{2}{g_{4}}^{2}\right]v_{1}v_{2}-\frac{A_{\lambda}}{\sqrt{2}}v_{s}$ $\displaystyle\left({\mathcal{M}_{+}^{0}}\right)_{13}^{2}$ $\displaystyle=$ $\displaystyle\left[\lambda^{2}+h_{1}s{g_{4}}^{2}\right]v_{1}v_{s}-\frac{A_{\lambda}}{\sqrt{2}}v_{2}$ $\displaystyle\left({\mathcal{M}_{+}^{0}}\right)_{22}^{2}$ $\displaystyle=$ $\displaystyle\left[\frac{G^{2}}{4}+h_{2}^{2}{g_{4}}^{2}\right]v_{2}^{2}+\frac{A_{\lambda}}{\sqrt{2}}\frac{v_{1}v_{s}}{v_{2}}$ $\displaystyle\left({\mathcal{M}_{+}^{0}}\right)_{23}^{2}$ $\displaystyle=$ $\displaystyle\left[\lambda^{2}+h_{2}s{g_{4}}^{2}\right]v_{2}v_{s}-\frac{A_{\lambda}}{\sqrt{2}}v_{1}$ $\displaystyle\left({\mathcal{M}_{+}^{0}}\right)_{33}^{2}$ $\displaystyle=$ $\displaystyle s^{2}{g_{4}}^{2}v_{s}^{2}+\frac{A_{\lambda}}{\sqrt{2}}\frac{v_{1}v_{2}}{v_{s}}$ (20) Figure 1: The determinant of the CP even Higgs mass matrix is shown as a function of $g_{4}$ and $\lambda$. In the shaded region, the determinant is negative, thus electroweak symmetry breaking is not possible. The $U(1)$ charges used are presented in Table 1 and tan$\beta$ is chosen to be 10. Given that the physical Higgs spectrum should be non-tachyonic, we can derive constraints on the parameter space of the model. Firstly the sign of the determinant of the matrix, in the limit $v_{s}>>v_{1,2}$ is crucially dependent on the sign of the $A_{\lambda}$. This is obvious, by considering the full determinant of the $3\times 3$ mass matrix, which is given by $\displaystyle Det[(\mathcal{M}_{+}^{0})^{2}]$ $\displaystyle\approx$ $\displaystyle\frac{A_{\lambda}v_{s}^{3}}{4\sqrt{2}v_{1}v_{2}}\left[G^{2}\,g_{4}^{2}\,s^{2}\,(v_{1}^{2}-v_{2}^{2})^{2}+4\,\left(g_{4}^{4}\,h_{1}^{2}\,s^{2}\,v_{1}^{4}-(\,l^{4}+2\,g_{4}^{2}\,l^{2}\,(\,h_{2}-s\,)\,s+g_{4}^{4}\,h_{2}\right.\right.$ $\displaystyle\left.\left.(-2\,h_{1}+h_{2}\,)\,s^{2})\,v_{1}^{2}v_{2}^{2}+g_{4}^{4}h_{2}^{2}s^{2}v_{2}^{4}\right)\right]$ For $A_{\lambda}>0$, the region in which the sign of the determinant of the Higgs mass matrix changes is plotted in $\lambda,g_{4}$ plane by taking $h_{1}=-\frac{1}{2},h_{2}=-\frac{5}{2},s=3$, and $\tan{\beta}=10$. Electroweak symmetry breaking is not possible for the shaded region ($Det<0$) in the parameter space. From the figure, it is seen that for $g_{4}\lesssim 0.1$, large values of $\lambda\gtrsim 0.6$ are disfavoured as they do not allow electroweak symmetry breaking. The question then arises, whether $A_{\lambda}>0$ ?. Typically the A terms are negative due to the RG running from the high scale. However, in this case, $A_{\lambda}$ turns out to be $\mathcal{O}(10)$ and positive at the weak scale. This positive $A_{\lambda}$ ensures us a safe electroweak vacuum. This is shown in the left panel of Figure 2 , where we have plotted $A_{\lambda}$ with respect to running scale. As we see from the figure 2, $A_{\lambda}$ initially turns negative and then increases turning positive at the weak scale. This happens because of the complicated coupling between $A_{t}$ and $A_{\lambda}$ RGE. The RGE of these parameters are presented in the Appendix C along with the other parameters. In the below, we reproduce them: $\displaystyle{dA_{t}\over dt}$ $\displaystyle\approx$ $\displaystyle\frac{y_{t}}{16\pi^{2}}\left[2y_{b}A_{b}+A_{\lambda}\lambda+{32\over 3}g_{3}^{2}M_{3}+6g_{2}^{2}M_{2}+{26\over 9}g_{1}^{2}M_{1}+4(q^{2}+u^{2}+h_{2}^{2})g_{4}^{2}M_{4}\right]$ $\displaystyle\frac{dA_{\lambda}}{dt}$ $\displaystyle\approx$ $\displaystyle\frac{\lambda}{16\pi^{2}}\left[6y_{t}A_{t}+6y_{b}A_{b}+2A_{\tau}y_{\tau}+6(A_{k_{1}}k_{1}+A_{k_{2}}k_{2}+A_{k_{3}}k_{3})\right.$ $\displaystyle\left.+6g_{2}^{2}M_{2}+2g_{1}^{2}M_{1}+4(s^{2}+h_{2}^{2}+h_{1}^{2})g_{4}^{2}M_{4}\right]$ Compared to the minimal gauge mediated models, the running effects on the parameter $A_{t}$ are very large as $\alpha_{3}$ barely runs in this models. As mentioned above, $b_{3}=0$ at 1-loop and is very small, at the 2-loop. For this reason, after the SUSY threshold $M_{S}\sim 1\text{TeV}$, $\alpha_{s}$ barely runs all the way to the mediation scale. Due to this $Y_{t}$ and $A_{t}$ receive comparatively large corrections due to the relatively large $\alpha_{s}$. Additional corrections from $g_{4},k_{i}$ and $A_{k_{i}}$ also contribute in the running of the $A_{\lambda}$. This feeds into $A_{\lambda}$, making it positive at the weak scale. In the right panel of the Fig [2], we show the running of the $A_{t}$ for the same parameters Figure 2: $A_{\lambda}$ and $A_{t}$ are plotted as a function of the energy scale, where free parameters are fixed as $\lambda=0.394$, $g_{4}=0.137$, $k_{1}=0.016$, $k_{2}=1.07$, $k_{3}=0.117$,$\tan{\beta}=3.7$ Let us focus our attention to the lightest Higgs mass eigenvalue. The matrix Eq.(20) gives an upper bound on the tree level lightest Higgs mass. In the present model, it has additional contribution from $\lambda$ and $g_{4}$ which is given as ${m_{h_{0}}}^{2}\leq M_{Z}^{2}\left[\cos{2\beta}^{2}+\frac{\lambda^{2}}{2g^{2}}\sin{2\beta}^{2}+\frac{g_{4}^{2}}{g^{2}}(h_{1}+h_{2}+(h_{1}-h_{2})\cos{2\beta})^{2}\right]$ (22) In the NMSSM, it is well known that the tree level contribution can be appreciably enhanced from the MSSM tree level values only for large values of $\lambda\gtrsim 0.7$. The above bound is thus saturated only for special values of the parameters. For most of the parameter space, however the actual eigenvalue is far below the above bound. As in MSSM, one loop corrections would play a major role. The total number of parameters are $\Lambda$, $M_{X}$, $g_{4}$, $\lambda$ and the $U(1)$ charges. Before proceeding to present the numerical results, we discuss the possible constraints on the various parameters. The first constraint we discuss is from the neutral gauge boson mixing. The neutral gauge bosons $Z$ and $Z^{\prime}$ mix with their mass matrix given by $\mathcal{L}~{}\supset~{}\chi^{T}\mathcal{M}_{Z^{\prime}Z}^{2}\chi$ where $\chi^{T}=\\{Z^{\prime},Z\\}$ with $\mathcal{M}_{Z^{\prime}Z}^{2}=\left(\begin{array}[]{cc}M^{2}_{Z^{\prime}Z^{\prime}}&M^{2}_{Z^{\prime}Z}\\\ M^{2}_{Z^{\prime}Z}&M_{ZZ}^{2}\end{array}\right)$ (23) where $\displaystyle M^{2}_{Z^{\prime}Z^{\prime}}$ $\displaystyle=$ $\displaystyle{g_{4}}^{2}(h_{1}^{2}v_{1}^{2}+h_{2}^{2}v_{2}^{2}+s^{2}v_{s}^{2}),$ $\displaystyle M^{2}_{ZZ^{\prime}}$ $\displaystyle=$ $\displaystyle{g_{4}}\sqrt{g_{1}^{2}+g_{2}^{2}}\left(v_{1}^{2}h_{1}-v_{2}^{2}h_{2}\right),$ $\displaystyle M^{2}_{ZZ}$ $\displaystyle=$ $\displaystyle{({g_{1}^{2}+g_{2}^{2}})\left(v_{1}^{2}+v_{2}^{2}\right)\over 4}.$ (24) The mixing of the matrix is given by $\Theta_{ZZ^{\prime}}={1\over 2}\tan^{-1}\left({2M^{2}_{ZZ^{\prime}}\over M_{Z^{\prime}}^{2}-M_{Z}^{2}}\right).$ (25) The current limits on $M_{Z^{\prime}}$ require it to be greater than 1 TeV Chatrchyan:2012ku . For $g_{4}\sim g_{1}$, these limits already push $v_{s}$ to be much larger than 1 TeV. $\Theta_{ZZ^{\prime}}$ is constrained by electroweak precision data, it should be less than $O(10^{-3})$ Hook:2010tw . As $v_{s}$ is already very heavy with $M_{Z}^{\prime}$ of a mass of TeV order, the constraint on mixing angle is avoided easily. A second constraint comes from the mass spectrum of the scalar super-partners. The D-terms due to the new $U(1)_{A}$ group play an important role in determining the sfermion mass spectrum due to the large vev of the $S$ field. The strongest effects are felt in the stau mass matrix which is given as: $\displaystyle\mathcal{M}_{\widetilde{\tau}}^{2}=\left(\begin{array}[]{cc}m_{\widetilde{L_{3}}}^{2}+m_{\tau}^{2}+D_{L}&{1\over{\sqrt{2}}}(A_{\tau}v_{1}-{\mu}y_{\tau}v_{2})\\\ {1\over{\sqrt{2}}}(A_{\tau}v_{1}-{\mu}y_{\tau}v_{2})&m_{\widetilde{e_{3}}}^{2}+m_{\tau}^{2}+D_{e}\end{array}\right),$ (28) where $\displaystyle D_{L}$ $\displaystyle=$ $\displaystyle{1\over 8}(v_{1}^{2}-v_{2}^{2})(-g_{2}^{2}+g_{1}^{2})+{1\over 2}g_{4}^{2}l~{}(h_{1}v_{1}^{2}+h_{2}v_{2}^{2}+sv_{s}^{2})$ (29) $\displaystyle D_{e}$ $\displaystyle=$ $\displaystyle-{1\over 4}(v_{1}^{2}-v_{2}^{2})~{}g_{1}^{2}+{1\over 2}g_{4}^{2}e(h_{1}v_{1}^{2}+h_{2}v_{2}^{2}+sv_{s}^{2}).$ (30) Notice that for the $D_{L}$ and $D_{e}$ to have positive values, the products of the $U(1)_{A}$ charges, $ls$ and $es$ should always be positive. This is because unlike $m_{\widetilde{Q}}^{2},m_{\widetilde{u}}^{2}$ and $m_{\widetilde{u}}^{2}$, the value of $m_{\widetilde{L}}^{2}$ at electroweak scale due to running is very low, as it should be, owing to the fact that $y_{\tau}\ll y_{t}$. So the sign of the diagonal terms in the stau mass squared matrix depends on the $D_{L}$ and $D_{e}$ which in turn depends on the dominant term $l\,s\,g_{4}^{2}\,v_{s}^{2}$. If we choose $U(1)_{A}$ charges $l$ and $s$ of different signs we expect tachyonic masses for stau’s. The chargino mass matrix remains unaltered compared to the MSSM whereas the neutralino mass matrix is now expanded to include the neutral gauging of $U(1)_{A}$ as well as the fermionic partner of the $S$ field. Note that the fermonic partner of the $S$ is not exactly the singlino as it carries a $U(1)_{A}$ charge unlike the NMSSM case. To summarise the constraints, we have : * • For consistent electroweak breaking : we need, $\lambda$ $=\sqrt{2}\,\,\frac{\mu_{eff}}{v_{s}}$ and $A_{\lambda}>0$. So $\lambda$ cannot be arbitrarily large for a given $g_{4}$ which is evident from the Figure 1 * • From $Z-Z^{\prime}$ mixing: we require that $v_{s}\sim$ $O(TeV)\gg v_{1},v_{2}$ * • Sfermion masses: From the $D$-terms of the sfermion mass matrices, we require that $U(1)_{A}$ charges $l$ and $s$ should have opposite signs * • Landau pole:the new gauge coupling $g_{4}<2\pi\sqrt{\frac{2}{b_{4}\log\frac{M_{X}}{M_{z}}}}.\\\ $ $g_{4}\approx 0.28$ for $b_{4}=145$ and $M_{X}=100$ TeV ## IV Numerical Results Table 1: $U(1)_{A}$ charges of the fields q \| u \| d \| l \| e \| $h_{1}$ \| $h_{2}$ \| s \| $z_{1}$ \| $z_{2}$ \| $z_{3}$ \| $\bar{z_{1}}$ \| $\bar{z_{2}}$ \| $\bar{z_{3}}$ ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- ${1\over 6}$ \| ${1\over 3}$ \| ${7\over 3}$ \| ${1\over 2}$ \| ${2}$ \| $-{5\over 2}$ \| $-{1\over 2}$ \| ${3}$ \| -3 \| ${-1}$ \| ${-1}$ \| ${0}$ \| -2 \| -2 To compute the sparticle spectrum at the weak scale, we use a modified version of the publicly available code SuSeFLAV Chowdhury:2011zr with 2-loop RGE for the gauge couplings and the Yukawa couplings. The RGE for the rest of the soft parameters are evaluated at the 1-loop level. For the Higgs spectrum, we compute the full 1-loop effective potential corrections presented in Appendix D. These corrections come from stop-top loop and the exotic quarks loop. Stop- top loop correction is the dominant contributor to the Higgs mass at the one loop. The correction due the exotic quarks is significant. It changes Higgs mass by few percent and we have checked that it is possible to get Higgs mass of 125 GeV by adding both the corrections, although we have not considered exotic quarks loop correction to the Higs mass in the numerical analysis. The free parameters are $\Lambda$, $\tan{\beta}$, $\lambda$, $g_{4}$, $k_{1}$, $k_{2}$ and $k_{3}$. These are randomly fixed at the low energy scale, for each set of these parameters, using RGEs we obtain corresponding values at the GMSB scale $X\simeq\Lambda$. Now along with the boundary conditions for the soft masses and A-terms, the same parameters are run down to the electroweak scale to check whether they satisfy minimization conditions given in section (2) and other constraints presented in section (3). This process is repeated several times to obtain a parameter space which satisfy electroweak symmetry breaking conditions. Subsequent to this, we impose phenomenological constraints from direct SUSY searches at LHC atlas:12 ; cms:12 as well as the flavour constraints from $b\to s+\gamma$ and $b\to s+\mu^{+}\mu^{-}$. In the numerical analysis, we fix the $U(1)_{A}$ charges to be as given in Table 1. It should be noted that these are not the only solutions available from anomaly cancellation conditions. A list of five solutions is presented in Appendix A. Of the remaining parameters, we have fixed $\tan\beta=10$ and varied the remaining parameters within a range presented in Table (2). Table 2: Ranges for the various Parameters Parameter \| Range ---\|--- $\Lambda$ \| $1\times 10^{5}\,-\,5\times 10^{7}[GeV]$ $g_{4}$ \| $0.01-2.5$ $\lambda$ \| $0.1-0.9$ $\kappa_{1}$ \| $0.1-0.9$ $\kappa_{2}$ \| $0.1-0.9$ $\kappa_{3}$ \| $0.1-0.9$ Instead of presenting the results in terms of regions of allowed parameter space, we present the correlations of the parameters with the lightest CP even Higgs boson mass. In Fig. (3), we present the correlation of the light Higgs mass with respect to the $A_{t}$ generated at the weak scale. The left panel presents the total Higgs mass whereas the right panel shows the 1-loop correction to the light Higgs mass. As expected we see that as $\|A_{t}\|$ increases, the 1-loop correction to the Higgs mass increases so does the total mass. It is also surprising to see larger values for $A_{t}\sim 900~{}\,GeV$ possible in this case and accordingly the higher values for Higgs mass $\sim 140~{}\,GeV$. Of course, the heavier Higgs masses correspond to heavier stops. Note that we have considered only dominant 1-loop corrections to the light Higgs mass. Two loop contributions Martin:1993zk can be important and they would give a more precise number for the light Higgs mass. However, it is clear that one can easily achieve a light Higgs mass of $\mathcal{O}(125)\,\text{GeV}$. Figure 3: Higgs mass, including one-loop correction, and only one loop correction are plotted against $A_{t}$. The U(1) charges are taken from Table 1. In Fig. (4), we present the correlation between $m_{h}$ and $\lambda$ in the left panel and $m_{h}$ and $g_{4}$ in the right panel. We find a surprising relation between $\lambda$ and $m_{h}$. The Higgs mass seems to be lower for higher values of $\lambda$. This is contrary to expectations based on NMSSM. This is because for higher values of $\lambda$ achieving electroweak symmetry breaking becomes harder. Similarly, larger values of $\lambda$ typically mean lighter values of $v_{s}$. Similarly, larger values of $g_{4}$ are not preferred by the data as they can lead to Landau poles. This can be seen from the right panel of Fig.(4). Thus, the regular NMSSM like enhancement of the tree level Higgs mass is not possible in this model. Figure 4: Higgs mass, including one-loop correction is plotted against $\lambda$ and $g_{4}$ From the allowed parameter space, we now present a representative point, Point(A) which give the lightest Higgs mass to be around 125 GeV. In this point, the next to lightest supersymmetry particle (NLSP) is the A-ino, the supersymmetric partner for the extra $U(1)_{A}$ gauge boson. Point (A): The various parameters for this point are : $v_{s}=2225.53\text{GeV}$, $\tan(\beta)=3.26$, $\lambda=0.3439$, $g_{4}=0.1198$, $M_{X}=194.22~{}\text{TeV}$, $\Lambda=97.112\text{TeV}$, $\kappa_{1}=0.1368$, $\kappa_{2}=0.7865$, $\kappa_{3}=0.7813$ Parameter \| mass(GeV) \| Parameter \| mass(GeV) \| Parameter \| mass(GeV) ---\|---\|---\|---\|---\|--- $\widetilde{t_{1}}$ \| 773.35 \| ${\chi}_{1}^{0}$ \| 37.00 \| $h_{1}^{0}$ \| 127.1 $\widetilde{t_{2}}$ \| 882.39 \| ${\chi}_{2}^{0}$ \| 122.26 \| $h_{2}^{0}$ \| 244 $\widetilde{b_{1}}$ \| 847.4 \| ${\chi}_{3}^{0}$ \| 544.8 \| $h_{3}^{0}$ \| 802.8 $\widetilde{b_{2}}$ \| 1002.5 \| ${\chi}_{4}^{0}$ \| 554.19 \| $A^{0}$ \| 370.99 $\widetilde{{\tau}_{1}}$ \| 294.25 \| ${\chi}_{5}^{0}$ \| 799.7 \| ${\chi}_{1}^{\pm}$ \| 123.16 $\widetilde{{\tau}_{2}}$ \| 460.58 \| ${\chi}_{6}^{0}$ \| 806.8 \| ${\chi}_{2}^{\pm}$ \| 549.94 $\tilde{g}$ \| 911.5 \| $A_{\lambda}$ \| 10.1 \| $A_{t}$ \| -279.3 ## V Outlook The discovery of a Higgs boson at 125 GeV has led to strong constraints on the gauge mediated supersymmetry breaking models. Most of the present models have concentrated on generating the required large trilinear $A_{t}$ coupling through messenger matter interactions. In the present work, we tried a different approach of combining the ideas of an extra $U(1)$ factor and NMSSM like models. Anomaly cancellation requirement automatically determines the extra particle spectrum of the model. The coloured particles barely run in this model from the weak scale to the mediation scale due to the small value of the strong beta function. This ‘stagnation’ of $\alpha_{s}$ between $M_{SUSY}$ and $M_{mess}$ and the presence of additional U(1) couplings helps for a larger value of the $A_{t}$ at the $M_{SUSY}$ even though one starts with zero at the mediation scale. Together with a reasonable value for the $\mu_{eft}=\lambda v_{s}$, this generates the required $X_{t}$ at the weak scale for the light stops. While we have focussed on getting the right Higgs mass, the rest of the spectrum of the model is also quite interesting. There are heavy exotic coloured particles, new neutralinos which are combinations of the Standard Model singlino and the fermion of the $U(1)_{A}$ gauge boson. The lightest neutralino is still the LSP and could be the dark matter candidate. A study of collider signatures and dark matter issues could be interesting and will be pursued in a future work. Finally, we have not concentrated on the issue of fine tuning in this model. Though we have not explicitly measured it, it is expected that it could be large as long as $M_{X}$ and $\Lambda$ are close as we have chosen. A reasonable separation between the scales can perhaps reduce the fine tuning (see for example, discussion in Komargodski:2008ax ). Acknowledgments We acknowledge discussions and important inputs from E. J. Chun. We also acknowledge discussions with P. Bandopadhyaya. We also thank L. Calibbi to bringing to our notice a reference. SKV is supported by DST Ramanujan Grant SR/S2/2008/RJN-25 of Govt of India. VSM is supported by CSIR fellowship 09/079(2377)/2010-EMR-1. ## Appendix A Anomaly Conditions In the following we present the anomaly cancellation conditions and U(1) charges which are solutions to them. More elaborate analysis of anomaly cancellations pertinent to U(1) extensions of MSSM has been presented in Lee:2007fw . To begin with, the $U(1)_{A}$ gauge invariance of the superpotential Eq.(5) leads to the below equations which should be satisfied by the $U(1)_{A}$ charges. $\displaystyle h_{1}+q+d$ $\displaystyle=$ $\displaystyle 0$ (31) $\displaystyle h_{2}+q+u$ $\displaystyle=$ $\displaystyle 0$ (32) $\displaystyle h_{1}+l+e$ $\displaystyle=$ $\displaystyle 0$ (33) $\displaystyle s+h_{1}+h_{2}$ $\displaystyle=$ $\displaystyle 0$ (34) In addition, the following five anomaly cancellation conditions should also be satisfied. $\displaystyle\mathcal{A}_{1}$ $\displaystyle:$ $\displaystyle U(1)_{A}-[SU(3)_{C}]^{2}$ $\displaystyle\mathcal{A}_{2}$ $\displaystyle:$ $\displaystyle U(1)_{A}-[SU(2)_{L}]^{2}$ $\displaystyle\mathcal{A}_{3}$ $\displaystyle:$ $\displaystyle U(1)_{A}-[U(1)_{Y}]^{2}$ $\displaystyle\mathcal{A}_{4}$ $\displaystyle:$ $\displaystyle U(1)_{Y}-[U(1)_{A}]^{2}$ $\displaystyle\mathcal{A}_{5}$ $\displaystyle:$ $\displaystyle U(1)_{A}^{3}$ In the following, we analyse each of these conditions and the corresponding solutions for $U(1)_{A}$ charges. #### A.0.1 Anomaly $\mathcal{A}_{1}(U(1)_{A}-[SU(3)_{C}]^{2})$ $3(2q+u+d)+\mathcal{A}_{1}(exotics)=0$ (35) Here first term is the contribution from three generations of the quarks in the MSSM without considering the exotic $D,\bar{D}$ quarks presented in section (1). We can show in the limit $\mathcal{A}_{1}(exotics)~{}=0$, the $S$ field $U(1)_{A}$ would go to zero. This can be easily seen by considering the combination of the equations: Eq.(35) - 3 Eq.(32) - 3 Eq.(31) + 3 Eq.(34), gives us $\mathcal{A}_{1}(exotics)=-3s$ (36) We assume that the exotics are triplets and anti triplets of $SU(3)_{c}$ with equal and opposite $U(1)_{Y}$ hypercharges $\pm y_{i}$. Eq. (35) now becomes $3(2q+u+d)+\Sigma_{i}(z_{i}+{\bar{z}}_{i})=0$ (37) where $z_{i}$ are the $U(1)_{A}$ charges of the exotics. The coupling between the exotic vector like quarks the singlet is allowed under $U(1)_{A}$ symmetry which gives $s+z_{i}+{\bar{z}}_{i}=0$ (38) Finally, to derive the number of families of exotic quarks one should add, consider the combination Eq. (37) - 3 Eq. (32) - 3 Eq. (31) + 3 Eq. (34)- $\Sigma_{i}$ Eq. (34). We have $(3-N_{k})s=0$, where $N_{k}$ is the number of exotic families which ends up being equal to three. #### A.0.2 Anomaly $\mathcal{A}_{2}(U(1)_{A}-[SU(2)_{L}]^{2})$ The constraint here is given as $9\,q+3\,l+h_{1}+h_{2}=0$ (39) From Eqs. (31), (32), (33), (34) and (39) we have 5 constraints. Without the $U(1)_{A}$ charges of the exotics, we have eight unknowns. Using the constraints, a general solution can be written in terms of $l,h_{1},s$ as $\displaystyle\left(\begin{array}[]{c}q\\\ u\\\ d\\\ e\\\ h_{2}\\\ \end{array}\right)={l\over 3}\left(\begin{array}[]{c}-1\\\ 1\\\ 1\\\ -3\\\ 0\\\ \end{array}\right)+h_{1}\left(\begin{array}[]{c}0\\\ 1\\\ -1\\\ -1\\\ -1\\\ \end{array}\right)+{s\over 9}\left(\begin{array}[]{c}1\\\ 8\\\ -1\\\ 0\\\ -9\\\ \end{array}\right)$ (60) #### A.0.3 Anomaly $\mathcal{A}_{3}(U(1)_{A}-[U(1)_{Y}]^{2})$ This anomaly condition puts constraints on the hypercharges of the exotic fields. The anomaly condition is give by $q+8\,u+2\,d+3\,l-6\,e+h_{1}+h_{2}-6\,s\Sigma_{i}y_{i}^{2}=0$ (61) By taking the combination of Eqs. (61) + (39) - 8 (32) + 2 (31) - 6 (33) + 6 (34), we get $\Sigma_{i}y_{i}^{2}=1$ (62) which has several solutions. In the present work, we choose $y_{i}=\\{-{1\over 3},{2\over 3},{2\over 3}\\}$ #### A.0.4 Anomalies $\mathcal{A}_{4}(U(1)_{Y}-[U(1)_{A}]^{2})$ and $A_{5}[U(1)_{A}]^{3}$ The final two anomalies do not have simple algebraic solutions. These are given as $\mathcal{A}_{4}:$ $3q^{2}-6u^{2}+3d^{2}-3l^{2}+3e^{2}-h_{1}^{2}+~{}h_{2}^{2}+3~{}\Sigma_{i}y_{i}(z_{i}^{2}-{\bar{z}_{i}}^{2})=0$ (63) $\mathcal{A}_{5}:$ $18q^{3}+9u^{3}+9d^{3}+6l^{3}+3e^{3}+2h_{1}^{3}+2h_{2}^{3}+s^{3}+3~{}\Sigma_{i}(z_{i}^{3}+{\bar{z}_{i}}^{3})=0$ (64) We looked for integer solutions for the $U(1)_{A}$ charges. We could not find any as long as the charges are restricted to lie below 10. We then resorted to rational charges. There are several solutions which have been found. In Table 3, we present five sample solutions which satisfy the anomaly conditions as well as the superpotential requirements. In addition to this set of charges, one can also find sets where all the $z_{i}$ and $\bar{z}_{i}$ are equal. It should also be noted that each of the set of the charges has a completely different phenomenology. This is because the charges decide the $U(1)_{A}$ one loop beta function, $b_{4}$, which could vary drastically. This in turn modifies the values of $\lambda$ and $\kappa_{i}$ allowed and their respective ranges. Table 3: q \| u \| d \| l \| e \| $h_{1}$ \| $h_{2}$ \| s \| $z_{1}$ \| $z_{2}$ \| $z_{3}$ \| $\bar{z_{1}}$ \| $\bar{z_{2}}$ \| $\bar{z_{3}}$ ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- ${1\over 6}$ \| ${1\over 3}$ \| ${7\over 3}$ \| ${1\over 2}$ \| ${2}$ \| $-{5\over 2}$ \| $-{1\over 2}$ \| ${3}$ \| -3 \| ${-1}$ \| ${-1}$ \| ${0}$ \| -2 \| -2 $-{1\over 18}$ \| $-{5\over 18}$ \| ${11\over 9}$ \| $-{3\over 2}$ \| $-{5\over 6}$ \| ${7\over 6}$ \| ${1\over 3}$ \| $-{3\over 2}$ \| ${1\over 3}$ \| ${1\over 3}$ \| ${1\over 3}$ \| $-{7\over 6}$ \| $-{7\over 6}$ \| $-{7\over 6}$ $-{1\over 27}$ \| ${10\over 27}$ \| $-{8\over 27}$ \| $-{1\over 3}$ \| ${0}$ \| ${1\over 3}$ \| $-{1\over 3}$ \| ${2\over 3}$ \| $-{14\over 27}$ \| $-{14\over 27}$ \| $-{14\over 27}$ \| ${4\over 27}$ \| ${4\over 27}$ \| ${4\over 27}$ ${1\over 27}$ \| ${5\over 27}$ \| $-{22\over 27}$ \| ${2\over 9}$ \| ${5\over 9}$ \| $-{7\over 9}$ \| $-{2\over 9}$ \| ${1}$ \| -${2\over 9}$ \| -${2\over 9}$ \| -${2\over 9}$ \| -${7\over 9}$ \| -${7\over 9}$ \| -${7\over 9}$ ## Appendix B One loop corrections to the CP even Higgs mass matrix In the following we present the one loop corrections to the CP even Higgs mass matrix. There are two main contributions, one from the stop-top sector and the second one from from the vector like exotic quarks.To derive the one loop corrections, we use the well known effective potential methods. The one loop effective potential is given by Coleman:1973jx $\displaystyle V^{1}$ $\displaystyle=$ $\displaystyle\frac{3}{32\pi^{2}}\left[\sum_{j=1}^{2}m_{\widetilde{f}_{j}}^{4}\left(\ln\frac{m_{\widetilde{f}_{j}}^{2}}{Q^{2}}-\frac{3}{2}\right)-2\bar{m}_{f}^{4}\left(\ln\frac{\bar{m}_{f}^{2}}{Q^{2}}-\frac{3}{2}\right)\right].$ (65) where $m_{\widetilde{f}_{1,2}}^{2}$ are the eigenvalues of the field dependent sfermion mass matrix. $\bar{m}_{f}$ is the corresponding fermion mass. The corrections to the CP even mass matrices can be written as $\left({\mathcal{M}}_{+}^{1}\right)_{ij}=\left.\frac{\partial^{2}V^{1}}{\partial\phi_{i}\partial\phi_{j}}\right\|_{0}-\left.\delta_{ij}\frac{1}{v_{i}}\frac{\partial V^{1}}{\partial\phi_{i}}\right\|_{0}$ (66) By denoting $\displaystyle\frac{\partial^{2}m_{\widetilde{f}_{l}}^{2}}{\partial\phi_{i}\partial\phi_{j}}$ $\displaystyle=$ $\displaystyle{A}_{ij}^{\prime}\pm{A}_{ij}$ $\displaystyle\frac{\partial m_{\widetilde{f}_{l}}^{2}}{\partial\phi_{i}}$ $\displaystyle=$ $\displaystyle{B}_{i}^{\prime}\pm{B}_{i}$ mass matrix can be written as $\displaystyle\left({\mathcal{M}}_{+}^{1}\right)_{ij}$ $\displaystyle=$ $\displaystyle 2\,k\left[{\mathcal{F}}_{\tilde{f}}\,(A_{ij}^{\prime}-{\delta_{ij}\over H_{j}}B_{j}^{\prime})+{\mathcal{G}}_{\tilde{f}}\,(A_{ij}-{\delta_{ij}\over\phi_{j}}B_{j})+{\mathcal{FF}}_{\tilde{f}}\,(B_{i}^{\prime}B_{j}^{\prime}+B_{i}B_{j})+{\mathcal{GG}}_{\tilde{f}}\,(B_{i}^{\prime}B_{j}+B_{i}B_{j}^{\prime})\right.$ (67) $\displaystyle\left.-8\,{\mathcal{H}}_{f}\,y_{f}^{4}\,\langle\phi\rangle^{2}\right]$ where $\displaystyle{\mathcal{F}}_{\tilde{f}}$ $\displaystyle=$ $\displaystyle-(m_{\tilde{f}_{2}}^{2}+m_{\tilde{f}_{1}}^{2})+(m_{\tilde{f}_{2}}^{2}\log{m_{\tilde{f}_{2}}^{2}\over Q^{2}}+m_{\tilde{f}_{1}}^{2}\log{m_{\tilde{f}_{1}}^{2}\over Q^{2}})$ $\displaystyle{\mathcal{G}}_{\tilde{f}}$ $\displaystyle=$ $\displaystyle(m_{\tilde{f}_{2}}^{2}-m_{\tilde{f}_{1}}^{2})+(m_{\tilde{f}_{2}}^{2}\log{m_{\tilde{f}_{2}}^{2}\over Q^{2}}-m_{\tilde{f}_{1}}^{2}\log{m_{\tilde{f}_{1}}^{2}\over Q^{2}})$ $\displaystyle{\mathcal{FF}}_{\tilde{f}}$ $\displaystyle=$ $\displaystyle\log{m_{\tilde{f}_{1}}^{2}m_{\tilde{f}_{2}}^{2}\over Q^{4}}$ $\displaystyle{\mathcal{GG}}_{\tilde{f}}$ $\displaystyle=$ $\displaystyle\log{m_{\tilde{f}_{2}}^{2}\over m_{\tilde{f}_{1}}^{2}}$ $\displaystyle{\mathcal{H}}_{\tilde{f}}$ $\displaystyle=$ $\displaystyle\log{m_{f}^{2}\over Q^{2}}$ and $k={3\over 32\pi^{2}}$ To include corrections to the Higgs mass matrix from the stop-top loop and all the three exotic quarks, we need to calculate 67 in each case separately and add them. We have presented below corrections from the stop-top loop and one exotic quark. ### B.1 Top-Stop correction Dominant one loop correction to the Higgs mass matrix comes from the top and stop loop. The stop mass squared matrix is given as $\displaystyle\mathcal{M}_{\widetilde{t}}^{2}=\left(\begin{array}[]{cc}M_{\widetilde{Q}}^{2}+y_{t}^{2}\|H_{2}\|^{2}&X_{t}\\\ (X_{t})^{\dagger}&M_{\widetilde{U}}^{2}+y_{t}^{2}\|H_{2}\|^{2}\end{array}\right),$ (70) where $X_{t}=({A_{t}H_{2}}-\mu{\rm{}_{eff}}H_{1}y_{t})$ and $m_{t}=y_{t}H_{2}$ $\displaystyle A_{11}$ $\displaystyle=$ $\displaystyle\mu_{eff}^{2}y_{t}^{2}\left[{2\over m_{\tilde{t}_{2}}^{2}-m_{\tilde{t}_{1}}^{2}}-{8X_{t}^{2}\over{m_{\tilde{t}_{2}}^{2}-m_{\tilde{t}_{1}}^{2}}^{3}}\right]$ $\displaystyle A_{12}$ $\displaystyle=$ $\displaystyle-\,\mu_{eff}\,y_{t}\,A_{t}\left[{2\over m_{\tilde{t}_{2}}^{2}-m_{\tilde{t}_{1}}^{2}}-{8X_{t}^{2}\over{m_{\tilde{t}_{2}}^{2}-m_{\tilde{t}_{1}}^{2}}^{3}}\right]$ $\displaystyle A_{13}$ $\displaystyle=$ $\displaystyle\left[\frac{-2A_{t}H_{2}\lambda y_{t}}{m_{\tilde{t}_{2}}^{2}-m_{\tilde{t}_{1}}^{2}}-\frac{8X_{t}^{2}\mu_{eff}\lambda H_{1}y_{t}^{2}}{{m_{\tilde{t}_{2}}^{2}-m_{\tilde{t}_{1}}^{2}}}^{3}\right]$ $\displaystyle A_{22}$ $\displaystyle=$ $\displaystyle A_{t}^{2}\left[{2\over m_{\tilde{t}_{2}}^{2}-m_{\tilde{t}_{1}}^{2}}-{8X_{t}^{2}\over{m_{\tilde{t}_{2}}^{2}-m_{\tilde{t}_{1}}^{2}}^{3}}\right]$ $\displaystyle A_{23}$ $\displaystyle=$ $\displaystyle-\mu_{eff}y_{t}A_{t}\left[{2\over m_{\tilde{t}_{2}}^{2}-m_{\tilde{t}_{1}}^{2}}-{8X_{t}^{2}\over{m_{\tilde{t}_{2}}^{2}-m_{\tilde{t}_{1}}^{2}}^{3}}\right]$ $\displaystyle A_{33}$ $\displaystyle=$ $\displaystyle{\lambda^{2}y_{t}^{2}H_{1}^{2}}\left[{2\over m_{\tilde{t}_{2}}^{2}-m_{\tilde{t}_{1}}^{2}}-{8X_{t}^{2}\over{m_{\tilde{t}_{2}}^{2}-m_{\tilde{t}_{1}}^{2}}^{3}}\right]$ $\displaystyle A_{i2}^{\prime}$ $\displaystyle=$ $\displaystyle\delta_{i2}2y_{t}^{2}$ $\displaystyle B_{1}$ $\displaystyle=$ $\displaystyle{-2\,X_{t}\,\mu_{eff}\,y_{t}\over m_{\tilde{t}_{2}}^{2}-m_{\tilde{t}_{1}}^{2}}$ $\displaystyle B_{2}$ $\displaystyle=$ $\displaystyle{2\,X_{t}\,A_{t}\over m_{\tilde{t}_{2}}^{2}-m_{\tilde{t}_{1}}^{2}}$ $\displaystyle B_{3}$ $\displaystyle=$ $\displaystyle{-2\,X_{t}\,\lambda H_{1}\,y_{t}\over m_{\tilde{t}_{2}}^{2}-m_{\tilde{t}_{1}}^{2}}$ $\displaystyle B_{i}^{\prime}$ $\displaystyle=$ $\displaystyle\delta_{i2}=2y_{t}^{2}H_{2}$ ### B.2 Correction due to Exotic quarks The one loop correction due to the exotic quarks changes Higgs mass by few percent. The exotic quark mass matrix given by $\displaystyle\mathcal{M}_{\widetilde{D}_{i}}^{2}=\left(\begin{array}[]{cc}M_{\tilde{D_{i}}}^{2}+k_{i}^{2}\|S\|^{2}&X_{d_{i}}\\\ (X_{d_{i}})^{\dagger}&M_{\widetilde{\bar{D}_{i}}}^{2}+k_{i}^{2}\|S\|^{2}\end{array}\right),$ (73) where $X_{d_{i}}=({A_{k_{i}}S}-\lambda\,k_{i}H_{1}H_{2})$ and $m_{D_{i}}=k_{i}S$ $\displaystyle A_{11}$ $\displaystyle=$ $\displaystyle(\lambda k_{i}H_{2})^{2}\left[{2\over m_{\tilde{D}_{2}}^{2}-m_{\tilde{D}_{1}}^{2}}-{8X_{d_{i}}^{2}\over{m_{\tilde{D}_{2}}^{2}-m_{\tilde{D}_{1}}^{2}}^{3}}\right]$ $\displaystyle A_{22}$ $\displaystyle=$ $\displaystyle(\lambda k_{i}H_{1})^{2}\left[{2\over m_{\tilde{D}_{2}}^{2}-m_{\tilde{D}_{1}}^{2}}-{8X_{d_{i}}^{2}\over{m_{\tilde{D}_{2}}^{2}-m_{\tilde{D}_{1}}^{2}}^{3}}\right]$ $\displaystyle A_{33}$ $\displaystyle=$ $\displaystyle A_{k_{i}}^{2}\left[{2\over m_{\tilde{D}_{2}}^{2}-m_{\tilde{D}_{1}}^{2}}-{8X_{d_{i}}^{2}\over{m_{\tilde{D}_{2}}^{2}-m_{\tilde{D}_{1}}^{2}}^{3}}\right]$ $\displaystyle A_{12}$ $\displaystyle=$ $\displaystyle\left[{\lambda^{2}k_{i}^{2}H_{1}H_{2}-2\lambda k_{i}A_{k_{i}}S\over m_{\tilde{D}_{2}}^{2}-m_{\tilde{D}_{1}}^{2}}-{2X_{d_{i}}^{2}\lambda^{2}k_{i}^{2}\,H_{1}H_{2}\over{m_{\tilde{D}_{2}}^{2}-m_{\tilde{D}_{1}}^{2}}^{3}}\right]$ $\displaystyle A_{13}$ $\displaystyle=$ $\displaystyle\lambda k_{i}H_{2}A_{k_{i}}\left[{2\over m_{\tilde{D}_{2}}^{2}-m_{\tilde{D}_{1}}^{2}}-{8X_{d_{i}}^{2}\over{m_{\tilde{D}_{2}}^{2}-m_{\tilde{D}_{1}}^{2}}^{3}}\right]$ $\displaystyle A_{23}$ $\displaystyle=$ $\displaystyle\lambda k_{i}H_{1}A_{k_{i}}\left[{2\over m_{\tilde{D}_{2}}^{2}-m_{\tilde{D}_{1}}^{2}}-{8X_{d_{i}}^{2}\over{m_{\tilde{D}_{2}}^{2}-m_{\tilde{D}_{1}}^{2}}^{3}}\right]$ $\displaystyle A_{i3}^{\prime}$ $\displaystyle=$ $\displaystyle\delta_{i3}\,2k_{i}^{2}$ $\displaystyle B_{1}$ $\displaystyle=$ $\displaystyle-2\,\lambda k_{i}H_{2}{X_{d_{i}}\over m_{\tilde{D}_{2}}^{2}-m_{\tilde{D}_{1}}^{2}}$ $\displaystyle B_{2}$ $\displaystyle=$ $\displaystyle-2\,\lambda k_{i}H_{1}{X_{d_{i}}\over m_{\tilde{D}_{2}}^{2}-m_{\tilde{D}_{1}}^{2}}$ $\displaystyle B_{3}$ $\displaystyle=$ $\displaystyle 2A_{k_{i}}{X_{d_{i}}\over m_{\tilde{D}_{2}}^{2}-m_{\tilde{D}_{1}}^{2}}$ $\displaystyle B_{i}^{\prime}$ $\displaystyle=$ $\displaystyle\delta_{i3}2k_{i}^{2}S$ ## Appendix C RG Equations In the last section of the appendix we present the renormalisation equations for the various superpotential and gauge parameters as well as soft terms. To derive the formulae we use the standard formulae available in the literatureFalck:1985aa ; Martin:1993zk . The notation we use is $t=Log({\mu\over M_{susy}})$. $\displaystyle\frac{dg_{i}}{dt}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi^{2}}{\beta_{i}}^{(1)}+\frac{1}{(16\pi^{2})^{2}}{\beta_{i}}^{(2)}$ (74) $\displaystyle\frac{dy_{i}}{dt}$ $\displaystyle=$ $\displaystyle\frac{y_{i}}{16\pi^{2}}{\gamma_{i}}^{(1)}+\frac{y_{i}}{(16\pi^{2})^{2}}{\gamma_{i}}^{(2)}$ (75) $\beta_{a}^{(1)}=b_{a}g_{a}^{3},$ (76) where $~{}b_{a}=~{}\\{17,1,0\\}$ and $b4=18q^{2}+6l^{2}+9(u^{2}+d^{2})+3e^{2}+s^{2}+2(h_{1}^{2}+h_{2}^{2})+3(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+(s+z_{1})^{2}+(s+z_{2})^{2}+(s+z_{3})^{2})$, $\displaystyle{\beta_{1}}^{(2)}$ $\displaystyle=$ $\displaystyle 4g_{1}^{3}\left(\frac{287}{36}g_{1}^{2}+\frac{9}{4}g_{2}^{2}+\frac{46}{3}g_{3}^{2}+(q^{2}/2+d^{2}+4u^{2}+3l^{2}/2+(h_{1}^{2}+h_{2}^{2})/2+3e^{2}\right.$ (78) $\displaystyle\left.+{1\over 3}(z_{1}^{2}+(s+z_{1})^{2}+4(z_{2}^{2}+z_{3}^{2}+(s+z_{2})^{2}+(s+z_{3})^{2}))-{1\over 4}(\frac{26}{3}y_{t}^{2}+\frac{14}{3}y_{b}^{2}+6y_{\tau}^{2}\right.$ $\displaystyle\left.+2\lambda^{2}+\frac{4}{3}k_{1}^{2}+\frac{16}{3}(k_{2}^{2}+k_{3}^{2}))\right)$ $\displaystyle{\beta_{2}}^{(2)}$ $\displaystyle=$ $\displaystyle 4g_{2}^{5}+g_{2}^{3}\big{(}3g_{1}^{2}+4g_{2}^{2}+24g_{3}^{2}+g_{4}^{2}(18q^{2}+6l^{2}+4(h_{1}^{2}+h_{2}^{2}))-6(y_{t}^{2}+y_{b}^{2})-2(y_{\tau}^{2}+\lambda^{2}))$ $\displaystyle{\beta_{3}}^{(2)}$ $\displaystyle=$ $\displaystyle-54g_{3}^{5}+4g_{3}^{3}\left(\frac{47}{12}g_{1}^{2}+\frac{9}{4}g_{2}^{2}+21g_{3}^{2}+g_{4}^{2}(3q^{2}+{3\over 2}(u^{2}+d^{2})+{1\over 2}(z_{1}^{2}+z_{2}^{2}\right.$ $\displaystyle\left.+z_{3}^{2}+(s+z_{1})^{2}+(s+z_{2})^{2}+(s+z_{3})^{2})-4(y_{t}^{2}+y_{b}^{2})-{4\over 3}\lambda^{2}-3(k_{1}^{2}+k_{2}^{2}+k_{3}^{2}))\right)$ $\displaystyle{\beta_{4}}^{(2)}$ $\displaystyle=$ $\displaystyle 4g_{4}^{3}\left(g_{1}^{2}({q^{2}\over 2}+4u^{2}+d^{2}+{3l^{2}\over 2}+3e^{2}+{1\over 2}(h_{1}^{2}+h_{2}^{2})+{3\over 9}(z_{1}^{2}+(s+z_{1})^{2}+4(z_{2}^{2}+z_{3}^{2}+(s+z_{2})^{2}\right.$ $\displaystyle\left.+(s+z_{3})^{2}))+g_{2}^{2}({27\over 2}q^{2}+{9\over 2}l^{2}+{3\over 2}(h_{1}^{2}+h_{2}^{2}))+g_{3}^{2}(24q^{2}+12(u^{2}+d^{2})+4(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+(s+z_{1})^{2}\right.$ $\displaystyle\left.+(s+z_{2})^{2}+(s+z_{3})^{2}))+g_{4}^{2}(18q^{4}+9(u^{4}+d^{4})+6l^{4}+3e^{4}+2(h_{1}^{4}+h_{2}^{4})+s^{4}+3(z_{1}^{4}+z_{2}^{4}+z_{3}^{4}\right.$ $\displaystyle\left.+(s+z_{1})^{4}+(s+z_{2})^{4}+(s+z_{3})^{4}))-{1\over 4}(12y_{t}^{2}(q^{2}+u^{2}+h_{2}^{2})+12y_{b}^{2}(q^{2}+d^{2}+h_{1}^{2})+4y_{\tau}^{2}(l^{2}+e^{2}+h_{1}^{2})\right.$ $\displaystyle\left.+4\lambda^{2}(s^{2}+h_{1}^{2}+h_{2}^{2})+6k_{1}^{2}(s^{2}+z_{1}^{2}+(s+z_{1})^{2})+6k_{2}^{2}(s^{2}+z_{2}^{2}+(s+z_{2})^{2})+6k_{3}^{2}(s^{2}+z_{3}^{2}+(s+z_{3})^{2})))\right)$ $\displaystyle{\gamma_{t}}^{(1)}$ $\displaystyle=$ $\displaystyle\left[\lambda^{2}+6y_{t}^{2}+y_{b}^{2}-{16\over 3}g_{3}^{2}-3g_{2}^{2}-{13\over 9}g_{1}^{2}-2g_{4}^{2}(q^{2}+u^{2}+h_{2}^{2})\right]$ $\displaystyle{\gamma_{b}}^{(1)}$ $\displaystyle=$ $\displaystyle\left[\lambda^{2}+6y_{b}^{2}+y_{t}^{2}+{y_{\tau}}^{2}-{16\over 3}g_{3}^{2}-3g_{2}^{2}-{7\over 9}g_{1}^{2}-2g_{4}^{2}(q^{2}+d^{2}+h_{1}^{2})\right]$ $\displaystyle{\gamma_{\tau}}^{(1)}$ $\displaystyle=$ $\displaystyle\left[\lambda^{2}+3y_{b}^{2}+4{y_{\tau}}^{2}-3g_{2}^{2}-3g_{1}^{2}-2g_{4}^{2}(l^{2}+e^{2}+h_{1}^{2})\right]$ $\displaystyle{\gamma_{\lambda}}^{(1)}$ $\displaystyle=$ $\displaystyle\left[4\lambda^{2}+3(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})+3(y_{t}^{2}+y_{b}^{2})+{y_{\tau}}^{2}-g_{1}^{2}-2g_{4}^{2}(s^{2}+h_{2}^{2}+h_{1}^{2})\right]$ $\displaystyle{\gamma_{k_{1}}}^{(1)}$ $\displaystyle=$ $\displaystyle\left[2\lambda^{2}+5k_{1}^{2}-{16\over 3}g_{3}^{2}-{4\over 9}g_{1}^{2}-2g4^{2}(s^{2}+z_{1}^{2}+(s+z_{1})^{2})\right]$ $\displaystyle{\gamma_{k_{2}}}^{(1)}$ $\displaystyle=$ $\displaystyle\left[2\lambda^{2}+5k_{2}^{2}-{16\over 3}g_{3}^{2}-{8\over 9}g_{1}^{2}-2g4^{2}(s^{2}+z_{2}^{2}+(s+z_{2})^{2})\right]$ $\displaystyle{\gamma_{k_{3}}}^{(1)}$ $\displaystyle=$ $\displaystyle\left[2\lambda^{2}+5k_{3}^{2}-{16\over 3}g_{3}^{2}-{8\over 9}g_{1}^{2}-2g4^{2}(s^{2}+z_{3}^{2}+(s+z_{3})^{2})\right]$ $\displaystyle{\gamma_{t}}^{(2)}$ $\displaystyle=$ $\displaystyle\left[-22y_{t}^{4}-5y_{b}^{4}-y_{t}^{2}(3\lambda^{2}+5y_{b}^{2})-y_{b}^{2}y_{\tau}^{2}-3\lambda^{4}-4y_{b}^{2}\lambda^{2}-\lambda^{2}y_{\tau}^{2}\right.$ $\displaystyle\left.-3\lambda^{2}(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})+y_{t}^{2}(2g_{1}^{2}+6g_{2}^{2}+16g_{3}^{2}+g_{4}^{2}(8q^{2}+4u^{2}))+y_{b}^{2}({2\over 3}g_{1}^{2}+2g_{4}^{2}(d^{2}\right.$ $\displaystyle\left.+h_{1}^{2}-q^{2}))+2\lambda^{2}g4^{2}(h_{1}^{2}+s^{2}-h_{2}^{2})+\frac{3679}{162}g_{1}^{4}+\frac{15}{2}g_{2}^{4}+\frac{416}{9}g_{3}^{4}+g_{4}^{4}(2s_{4}(q^{2}+u^{2}+h_{2}^{2})\right.$ $\displaystyle\left.+4(q^{4}+u^{4}+h_{2}^{4}))+\frac{5}{3}g_{1}^{2}g_{2}^{2}+\frac{136}{27}g_{1}^{2}g_{3}^{2}+8({h_{2}^{2}\over 4}+{q^{2}\over 36}+{4u^{2}\over 9})g_{1}^{2}g_{4}^{2}+8g_{2}^{2}g_{3}^{2}+6g_{2}^{2}g_{4}^{2}\right.$ $\displaystyle\left.(q^{2}+h_{2}^{2})+\frac{32}{3}(q^{2}+u^{2})g_{3}^{2}g_{4}^{2}\right]$ $\displaystyle{\gamma_{b}}^{(2)}$ $\displaystyle=$ $\displaystyle\left[-22y_{b}^{4}-5y_{t}^{4}-4y_{t}^{2}\lambda^{2}-y_{b}^{2}(3\lambda^{2}+5y_{t}^{2}+y_{\tau}^{2}-3\lambda^{4}-3y_{\tau}^{4}-3\lambda^{2}(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})\right.$ $\displaystyle\left.+y_{b}^{2}({4\over 3}g_{1}^{2}+{9\over 2}g_{2}^{2}+16g_{3}^{2}+g_{4}^{2}(6q^{2}+6d^{2}+2h_{1}^{2}))+2y_{t}^{2}({4\over 3}g_{1}^{2}+2g_{4}^{2}(u^{2}+h_{2}^{2}-q^{2}))+2\lambda^{2}g_{4}^{2}(s^{2}\right.$ $\displaystyle\left.+h_{2}^{2}-h_{1}^{2})+y_{\tau}^{2}(2g_{1}^{2}+2g_{4}^{2}(l^{2}+e^{2}-h_{1}^{2}))+\frac{1939}{162}g_{1}^{4}+\frac{15}{2}g_{2}^{4}+\frac{416}{9}g_{3}^{4}+g_{4}^{4}(2s_{4}(q^{2}+d^{2}+h_{1}^{2})\right.$ $\displaystyle\left.+4(q^{4}+d^{4}+h_{1}^{4}))+\frac{5}{3}g_{1}^{2}g_{2}^{2}+\frac{40}{27}g_{1}^{2}g_{3}^{2}+8({h_{1}^{2}\over 4}+{q^{2}\over 36}+{4d^{2}\over 9})g_{1}^{2}g_{4}^{2}+8g_{2}^{2}g_{3}^{2}+6g_{2}^{2}g_{4}^{2}(q^{2}+h_{1}^{2})\right.$ $\displaystyle\left.+\frac{32}{3}(q^{2}+d^{2})g_{3}^{2}g_{4}^{2}\right]$ $\displaystyle{\gamma_{\tau}}^{(2)}$ $\displaystyle=$ $\displaystyle\left[-9y_{b}^{4}-3\lambda^{4}-10y_{\tau}^{4}-3y_{t}^{2}y_{b}^{2}-3\lambda^{2}y_{t}^{2}-3_{\tau}^{2}(\lambda^{2}+3y_{b}^{2})-3\lambda^{2}(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})+y_{b}^{2}(-{2\over 3}g_{1}^{2}+16g_{3}^{2}\right.$ $\displaystyle\left.+6g_{4}^{2}(q^{2}+d^{2}-h_{1}^{2}))+2\lambda^{2}g_{4}^{2}(s^{2}+h_{2}^{2}-h_{1}^{2})+y_{\tau}^{2}(2g_{1}^{2}+6g_{2}^{2}+4g_{4}^{2}(l^{2}+h_{1}^{2}))+\frac{99}{2}g_{1}^{4}\right.$ $\displaystyle\left.+\frac{15}{2}g_{2}^{4}+g_{4}^{4}(2s_{4}(e^{2}+l^{2}+h_{1}^{2})+4(l^{4}+h_{1}^{4}+e^{4}))+3g_{1}^{2}g_{2}^{2}+g_{1}^{2}g_{4}^{2}(2h_{1}^{2}+2l^{2}+8e^{2})+6g_{2}^{2}g_{4}^{2}(h_{1}^{2}+l^{2})\right]$ $\displaystyle{\gamma_{\lambda}}^{(2)}$ $\displaystyle=$ $\displaystyle\left[-9y_{t}^{4}-9y_{b}^{4}-10\lambda^{4}-3y_{\tau}^{4}-6y_{t}^{2}y_{b}^{2}-\lambda^{2}(9y_{b}^{2}+9y_{t}^{2}+3y_{\tau}^{2}+6(k_{1}^{2}+k_{2}^{2}+k_{3}^{2}))-6(k_{1}^{4}+k_{2}^{4}+k_{3}^{4})\right.$ $\displaystyle\left.+y_{t}^{2}({3\over 2}g_{1}^{2}+16g_{3}^{2}+6g_{4}^{2}(u^{2}+q^{2}-h_{2}^{2}))+\lambda^{2}(2g_{1}^{2}+6g_{2}^{2}+4g_{4}^{2}(h_{1}^{2}+h_{2}^{2}))+k_{1}^{2}({4\over 3}g_{1}^{2}\right.$ $\displaystyle\left.+16g_{3}^{2}+6g_{4}^{2}(z_{1}^{2}+(s+z_{1})^{2}-s^{2}))+k_{2}^{2}({16\over 3}g_{1}^{2}+16g_{3}^{2}+6g_{4}^{2}(z_{2}^{2}+(s+z_{2})^{2}-s^{2}))+k_{3}^{2}({16\over 3}g_{1}^{2}\right.$ $\displaystyle\left.+16g_{3}^{2}+6g_{4}^{2}(z_{3}^{2}+(s+z_{3})^{2}-s^{2}))+y_{b}^{2}(-{2\over 3}g_{1}^{2}+3g_{2}^{2}+16g_{3}^{2}+6g_{4}^{2}(q^{2}+d^{2}-h_{1}^{2}))+2y_{\tau}^{2}(g_{1}^{2}+g_{4}^{2}\right.$ $\displaystyle\left.(l^{2}+e^{2}-h_{1}^{2}))+\frac{34}{3}g_{1}^{4}+\frac{15}{2}g_{2}^{4}+g_{4}^{4}(2s_{4}(h_{1}^{2}+s^{2}+h_{2}^{2})+4(h_{1}^{4}+s^{4}+h_{2}^{4}))+3g_{1}^{2}g_{2}^{2}\right.$ $\displaystyle\left.+2g_{1}^{2}g_{4}^{2}(h_{1}^{2}+h_{2}^{2})+6g_{2}^{2}g_{4}^{2}(h_{1}^{2}+h_{2}^{2})\right]$ $\displaystyle{\gamma_{k_{1}}}^{(2)}$ $\displaystyle=$ $\displaystyle\left[-6k_{1}^{2}\lambda^{2}-6k_{1}^{4}-4\lambda^{4}-\lambda^{2}(2y_{\tau}^{2}+6y_{b}^{2}+6y_{t}^{2})-6k_{1}^{2}(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})+k_{1}^{2}({4\over 3}g_{1}^{2}+16g_{3}^{2}+2g_{4}^{2}(z_{1}^{2}\right.$ $\displaystyle\left.+(s+z_{1})^{2}-s^{2})+\lambda^{2}(2g_{1}^{2}+6g_{2}^{2}+2g_{4}^{2}(h_{1}^{2}+h_{2}^{2}-s^{2}))+\frac{542}{81}g_{1}^{4}+\frac{416}{9}g_{3}^{4}+g_{4}^{4}(2s_{4}(z_{1}^{2}\right.$ $\displaystyle\left.+(s+z_{1})^{2})+4(z_{1}^{4}+(s+z_{1})^{4}))+\frac{64}{27}g_{1}^{2}g_{3}^{2}+\frac{8}{9}(z_{1}^{2}+(s+z_{1})^{2})g_{1}^{2}g_{4}^{2}+\frac{32}{3}(z_{1}^{2}+(s+z_{1})^{2})g_{4}^{2}g_{3}^{2}\right]$ $\displaystyle{\gamma_{k_{2}}}^{(2)}$ $\displaystyle=$ $\displaystyle\left[-6k_{2}^{2}\lambda^{2}-6k_{2}^{4}-4\lambda^{4}-\lambda^{2}(2y_{\tau}^{2}+6y_{b}^{2}+6y_{t}^{2})-6k_{2}^{2}(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})+k_{2}^{2}({16\over 3}g_{1}^{2}+16g_{3}^{2}+2g_{4}^{2}(z_{2}^{2}\right.$ $\displaystyle\left.+(s+z_{2})^{2}-s^{2})+\lambda^{2}(2g_{1}^{2}+6g_{2}^{2}+2g_{4}^{2}(h_{1}^{2}+h_{2}^{2}-s^{2}))+\frac{2168}{81}g_{1}^{4}+\frac{416}{9}g_{3}^{4}+g_{4}^{4}(2s_{4}(z_{2}^{2}\right.$ $\displaystyle\left.+(s+z_{2})^{2})+4(z_{2}^{4}+(s+z_{2})^{4}))+\frac{256}{27}g_{1}^{2}g_{3}^{2}+\frac{32}{9}(z_{2}^{2}+(s+z_{2})^{2})g_{1}^{2}g_{4}^{2}+\frac{32}{3}(z_{2}^{2}+(s+z_{2})^{2})g_{4}^{2}g_{3}^{2}\right]$ $\displaystyle{\gamma_{k_{3}}}^{(2)}$ $\displaystyle=$ $\displaystyle\left[-6k_{3}^{2}\lambda^{2}-6k_{3}^{4}-4\lambda^{4}-\lambda^{2}(2y_{\tau}^{2}+6y_{b}^{2}+6y_{t}^{2})-6k_{3}^{2}(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})+k_{3}^{2}({16\over 3}g_{1}^{2}+16g_{3}^{2}+2g_{4}^{2}(z_{3}^{2}\right.$ $\displaystyle\left.+(s+z_{3})^{2}-s^{2})+\lambda^{2}(2g_{1}^{2}+6g_{2}^{2}+2g_{4}^{2}(h_{1}^{2}+h_{2}^{2}-s^{2}))+\frac{2168}{81}g_{1}^{4}+\frac{416}{9}g_{3}^{4}+g_{4}^{4}(2s_{4}(z_{3}^{2}\right.$ $\displaystyle\left.+(s+z_{3})^{2})+4(z_{3}^{4}+(s+z_{3})^{4}))+\frac{256}{27}g_{1}^{2}g_{3}^{2}+\frac{32}{9}(z_{3}^{2}+(s+z_{3})^{2})g_{1}^{2}g_{4}^{2}+\frac{32}{3}(z_{3}^{2}+(s+z_{3})^{2})g_{4}^{2}g_{3}^{2}\right]$ where $s_{4}=18q^{2}+9(u^{2}+d^{2})+6l^{2}+2(h_{1}^{2}+h_{2}^{2})+3(z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+(s+z_{1})^{2}+(s+z_{2})^{2}+(s+z_{3})^{2})+e^{2}+s^{2}$, $\displaystyle\frac{dm_{q_{3}}^{2}}{dt}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi^{2}}\left[2y_{t}^{2}(m_{q_{3}}^{2}+m_{u_{3}}^{2}+m_{2}^{2})+2A_{t}^{2}+2y_{b}^{2}(m_{q_{3}}^{2}+m_{d_{3}}^{2}+m_{1}^{2})+2A_{b}^{2}\right.$ $\displaystyle\left.-{32\over 3}g_{3}^{2}M_{3}^{2}-6g_{2}^{2}M_{2}^{2}-{2\over 9}g_{1}^{2}M_{1}^{2}-8q^{2}g_{4}^{2}M_{4}^{2}+{1\over 3}g_{1}^{2}\xi+2qg_{4}^{2}\xi^{\prime}\right]$ $\displaystyle\frac{dm_{u_{3}}^{2}}{dt}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi^{2}}\left[4y_{t}^{2}(m_{q_{3}}^{2}+m_{u_{3}}^{2}+m_{2}^{2})+4A_{t}^{2}-{32\over 3}g_{3}^{2}M_{3}^{2}-8{4\over 9}g_{1}^{2}M_{1}^{2}-8u^{2}g_{4}^{2}M_{4}^{2}-{4\over 3}g_{1}^{2}\xi+2ug_{4}^{2}\xi^{\prime}\right]$ (97) $\displaystyle\frac{dm_{d_{3}}^{2}}{dt}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi^{2}}\left[4y_{b}^{2}(m_{q_{3}}^{2}+m_{d_{3}}^{2}+m_{1}^{2})+4A_{b}^{2}-{32\over 3}g_{3}^{2}M_{3}^{2}-8{1\over 9}g_{1}^{2}M_{1}^{2}-8d^{2}g_{4}^{2}M_{4}^{2}+{2\over 3}g_{1}^{2}\xi+2dg_{4}^{2}\xi^{\prime}\right]$ (98) $\displaystyle\frac{dm_{l_{3}}^{2}}{dt}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi^{2}}\left[2y_{\tau}^{2}(m_{l_{3}}^{2}+m_{\tau}^{2}+m_{1}^{2})+2A_{\tau}^{2}-6g_{2}^{2}M_{2}^{2}-8{1\over 4}g_{1}^{2}M_{1}^{2}-8l^{2}g_{4}^{2}M_{4}^{2}-g_{1}^{2}\xi+2lg_{4}^{2}\xi^{\prime}\right]$ (99) $\displaystyle\frac{dm_{\tau}^{2}}{dt}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi^{2}}\left[4y_{\tau}^{2}(m_{l_{3}}^{2}+m_{\tau}^{2}+m_{1}^{2})+4A_{\tau}^{2}-8g_{1}^{2}M_{1}^{2}-8e^{2}g_{4}^{2}M_{4}^{2}+2g_{1}^{2}\xi+2eg_{4}^{2}\xi^{\prime}\right]$ $\displaystyle\frac{dm_{1}^{2}}{dt}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi^{2}}\left[6y_{b}^{2}(m_{d_{3}}^{2}+m_{q_{3}}^{2}+m_{1}^{2})+6A_{b}^{2}+2y_{\tau}^{2}(m_{\tau}^{2}+m_{l_{3}}^{2}+m_{1}^{2})+2A\tau\right.$ $\displaystyle\left.+2\lambda^{2}(m_{1}^{2}+m_{2}^{2}+m_{s}^{2})+2A\lambda-6g_{2}^{2}M_{2}^{2}-2g_{1}^{2}M_{1}^{2}-8h_{1}^{2}g_{4}^{2}M_{4}^{2}-g_{1}^{2}\xi+2h_{1}g_{4}^{2}\xi^{\prime}\right]$ $\displaystyle\frac{dm_{2}^{2}}{dt}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi^{2}}\left[6y_{t}^{2}(m_{u_{3}}^{2}+m_{q_{3}}^{2}+m_{2}^{2})+6A_{t}^{2}+2\lambda^{2}(m_{1}^{2}+m_{2}^{2}+m_{s}^{2})\right.$ $\displaystyle\left.+2A\lambda-6g_{2}^{2}M_{2}^{2}-2g_{1}^{2}M_{1}^{2}-8h_{1}^{2}g_{4}^{2}M_{4}^{2}+g_{1}^{2}\xi+2h_{1}g_{4}^{2}\xi^{\prime}\right]$ $\displaystyle\frac{dm_{s}^{2}}{dt}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi^{2}}\left[6k_{1}^{2}(m_{s}^{2}+m_{D_{1}}^{2}+m_{{\bar{D}}_{1}}^{2})+6A_{k_{1}}^{2}+6k_{2}^{2}(m_{s}^{2}+m_{D_{2}}^{2}+m_{{\bar{D}}_{2}}^{2})\right.$ $\displaystyle\left.+6A_{k_{2}}^{2}+6k_{3}^{2}(m_{s}^{2}+m_{D_{3}}^{2}+m_{{\bar{D}}_{3}}^{2})+6A_{k_{3}}^{2}+4\lambda^{2}(m_{1}^{2}+m_{2}^{2}+m_{s}^{2})+4A\lambda-8s^{2}g_{4}^{2}M_{4}^{2}+2sg_{4}^{2}\xi^{\prime}\right]$ $\displaystyle\frac{dm_{D_{1}}^{2}}{dt}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi^{2}}\left[2k_{1}^{2}(m_{s}^{2}+m_{D_{1}}^{2}+m_{{\bar{D}}_{1}}^{2})+2A_{k_{1}}^{2}-{32\over 3}g_{3}^{2}M_{3}^{2}-{8\over 9}g_{1}^{2}M_{1}^{2}-8z_{1}^{2}g_{4}^{2}M_{4}^{2}-{2\over 3}g_{1}^{2}\xi+2z_{1}g_{4}^{2}\xi^{\prime}\right]$ $\displaystyle\frac{dm_{D_{2}}^{2}}{dt}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi^{2}}\left[2k_{2}^{2}(m_{s}^{2}+m_{D_{2}}^{2}+m_{{\bar{D}}_{2}}^{2})+2A_{k_{2}}^{2}-{32\over 3}g_{3}^{2}M_{3}^{2}-{32\over 9}g_{1}^{2}M_{1}^{2}-8z_{2}^{2}g_{4}^{2}M_{4}^{2}+{4\over 3}g_{1}^{2}\xi+2z_{2}g_{4}^{2}\xi^{\prime}\right]$ $\displaystyle\frac{dm_{D_{3}}^{2}}{dt}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi^{2}}\left[2k_{3}^{2}(m_{s}^{2}+m_{D_{3}}^{2}+m_{{\bar{D}}_{3}}^{2})+2A_{k_{3}}^{2}-{32\over 3}g_{3}^{2}M_{3}^{2}-{32\over 9}g_{1}^{2}M_{1}^{2}-8z_{3}^{2}g_{4}^{2}M_{4}^{2}+{4\over 3}g_{1}^{2}\xi+2z_{3}g_{4}^{2}\xi^{\prime}\right]$ (106) $\displaystyle\frac{dm_{{\bar{D}}_{1}}^{2}}{dt}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi^{2}}\left[2k_{1}^{2}(m_{s}^{2}+m_{D_{1}}^{2}+m_{{\bar{D}}_{1}}^{2})+2A_{k_{1}}^{2}-{32\over 3}g_{3}^{2}M_{3}^{2}-{8\over 9}g_{1}^{2}M_{1}^{2}-8(s+z_{1})^{2}g_{4}^{2}M_{4}^{2}+{2\over 3}g_{1}^{2}\xi\right.$ $\displaystyle\left.+2(s+z_{1}g_{4}^{2}\xi^{\prime}\right]$ $\displaystyle\frac{dm_{{\bar{D}}_{2}}^{2}}{dt}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi^{2}}\left[2k_{2}^{2}(m_{s}^{2}+m_{D_{2}}^{2}+m_{{\bar{D}}_{2}}^{2})+2A_{k_{2}}^{2}-{32\over 3}g_{3}^{2}M_{3}^{2}-{32\over 9}g_{1}^{2}M_{1}^{2}-8(s+z_{2})^{2}g_{4}^{2}M_{4}^{2}-{4\over 3}g_{1}^{2}\xi\right.$ $\displaystyle\left.+2(s+z_{2})g_{4}^{2}\xi^{\prime}\right]$ $\displaystyle\frac{dm_{{\bar{D}}_{3}}^{2}}{dt}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi^{2}}\left[2k_{3}^{2}(m_{s}^{2}+m_{D_{3}}^{2}+m_{{\bar{D}}_{3}}^{2})+2A_{k_{3}}^{2}-{32\over 3}g_{3}^{2}M_{3}^{2}-{32\over 9}g_{1}^{2}M_{1}^{2}-8(s+z_{3})^{2}g_{4}^{2}M_{4}^{2}-{4\over 3}g_{1}^{2}\xi\right.$ $\displaystyle\left.+2(s+z_{3})g_{4}^{2}\xi^{\prime}\right]$ $\displaystyle\frac{dA_{t}}{dt}$ $\displaystyle=$ $\displaystyle\frac{A_{t}}{16\pi^{2}}\left[18y_{t}^{2}+y_{b}^{2}+\lambda^{2}-{16\over 3}g_{3}^{2}-3g_{2}^{2}-{13\over 9}g_{1}^{2}-2(q^{2}+u^{2}+h2^{2})g_{4}^{2}\right]$ $\displaystyle+\frac{y_{t}}{16\pi^{2}}\left[2y_{b}A_{b}+A_{\lambda}\lambda+{32\over 3}g_{3}^{2}M_{3}+6g_{2}^{2}M_{2}+{26\over 9}g_{1}^{2}M_{1}+4(q^{2}+u^{2}+h_{2}^{2})g_{4}^{2}M_{4}\right]$ $\displaystyle\frac{dA_{b}}{dt}$ $\displaystyle=$ $\displaystyle\frac{A_{b}}{16\pi^{2}}\left[18y_{b}^{2}+y_{t}^{2}+y_{\tau}^{2}+\lambda^{2}-{16\over 3}g_{3}^{2}-3g_{2}^{2}-{7\over 9}g_{1}^{2}-2(q^{2}+d^{2}+h_{1}^{2})g_{4}^{2}\right]$ $\displaystyle+\frac{y_{b}}{16\pi^{2}}\left[2y_{t}A_{t}+2A_{\tau}y_{\tau}+2A_{\lambda}\lambda+{32\over 3}g_{3}^{2}M_{3}+6g_{2}^{2}M_{2}+{14\over 9}g_{1}^{2}M_{1}+4(q^{2}+d^{2}+h_{1}^{2})g_{4}^{2}M_{4}\right]$ $\displaystyle\frac{dA_{\tau}}{dt}$ $\displaystyle=$ $\displaystyle\frac{A_{\tau}}{16\pi^{2}}\left[12y_{\tau}^{2}+3y_{b}^{2}+\lambda^{2}-3g_{2}^{2}-3g_{1}^{2}-2(l^{2}+e^{2}+h1^{2})g_{4}^{2}\right]$ $\displaystyle+\frac{y_{t}au}{16\pi^{2}}\left[6y_{b}A_{b}+2A_{\lambda}\lambda+6g_{2}^{2}M_{2}+6g_{1}^{2}M_{1}+4(l^{2}+e^{2}+h1^{2})g_{4}^{2}M_{4}\right]$ $\displaystyle\frac{dA_{\lambda}}{dt}$ $\displaystyle=$ $\displaystyle\frac{A_{\lambda}}{16\pi^{2}}\left[3y_{b}^{2}+3y_{t}^{2}+y_{\tau}^{2}+12\lambda^{2}+3(k_{1}^{2}+k_{2}^{2}+k_{3}^{2})-3g_{2}^{2}-g_{1}^{2}-2(s^{2}+h_{2}^{2}+h_{1}^{2})g_{4}^{2}\right]$ $\displaystyle+\frac{\lambda}{16\pi^{2}}\left[6y_{t}A_{t}+6y_{b}A_{b}+2A_{\tau}y_{\tau}+6(A_{k_{1}}k_{1}+A_{k_{2}}k_{2}+A_{k_{3}}k_{3})\right.$ $\displaystyle\left.+6g_{2}^{2}M_{2}+2g_{1}^{2}M_{1}+4(s^{2}+h_{2}^{2}+h1^{2})g_{4}^{2}M_{4}\right]$ $\displaystyle\frac{dA_{k_{1}}}{dt}$ $\displaystyle=$ $\displaystyle\frac{A_{k_{1}}}{16\pi^{2}}\left[3k_{1}^{2}+\lambda^{2}-{16\over 3}g_{3}^{2}-{4\over 9}g_{1}^{2}-2(s^{2}+z_{1}^{2}+(s+z_{1})^{2})g_{4}^{2}\right]$ $\displaystyle+\frac{k_{1}}{16\pi^{2}}\left[4\lambda A_{\lambda}+{32\over 3}g_{3}^{2}M_{3}+{8\over 9}g_{1}^{2}M_{1}+4(s^{2}+z_{1}^{2}+(s+z_{1})^{2})g_{4}^{2}M_{4}\right]$ $\displaystyle\frac{dA_{k_{2}}}{dt}$ $\displaystyle=$ $\displaystyle\frac{A_{k_{2}}}{16\pi^{2}}\left[3k_{2}^{2}+\lambda^{2}-{16\over 3}g_{3}^{2}-{16\over 9}g_{1}^{2}-2(s^{2}+z_{2}^{2}+(s+z_{2})^{2})g_{4}^{2}\right]$ $\displaystyle+\frac{k_{2}}{16\pi^{2}}\left[4\lambda A_{\lambda}+{32\over 3}g_{3}^{2}M_{3}+{32\over 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1311.4317	# Mobile Multimedia Streaming Techniques: QoE and Energy Consumption Perspective Mohammad Ashraful Hoque Matti Siekkinen Jukka K. Nurminen Aalto University School of Science, [email protected] Mika Aalto Nokia Solutions and Networks, [email protected] Sasu Tarkoma University of Helsinki, [email protected] ###### Abstract Multimedia streaming to mobile devices is challenging for two reasons. First, the way content is delivered to a client must ensure that the user does not experience a long initial playback delay or a distorted playback in the middle of a streaming session. Second, multimedia streaming applications are among the most energy hungry applications in smartphones. The energy consumption mostly depends on the delivery techniques and on the power management techniques of wireless access technologies (Wi-Fi, 3G, and 4G). In order to provide insights on what kind of streaming techniques exist, how they work on different mobile platforms, their efforts in providing smooth quality of experience, and their impact on energy consumption of mobile phones, we did a large set of active measurements with several smartphones having both Wi-Fi and cellular network access. Our analysis reveals five different techniques to deliver the content to the video players. The selection of a technique depends on the mobile platform, device, player, quality, and service. The results from our traffic and power measurements allow us to conclude that none of the identified techniques is optimal because they take none of the following facts into account: access technology used, user behavior, and user preferences concerning data waste. We point out the technique with optimal playback buffer configuration, which provides the most attractive trade-offs in particular situations. ###### keywords: Performance analysis, measurement, power consumption, wireless multimedia , Quality of Experience (QoE). ††journal: Pervasive and Mobile Computing ## 1 Introduction Digital video content is increasingly consumed using mobile devices [1]. At the same time, the playback quality experienced by the user and the battery life of smartphones have become critical factors in user satisfaction. Consequently, it is essential that mobile video streaming not only provides a good viewing experience but also avoids excessive energy consumption. Multimedia streaming services consider a number of challenges while sending content to the streaming clients for providing smooth playback, such as initial playback delay, clients with different kinds of connectivity, and the bandwidth variation between a server and a client [2]. While consuming multimedia streaming content, energy consumption of smartphones is also considered as an important issue and consequently a significant number of research work focused on reducing energy consumption of mobile devices using streaming applications [3]. The aforementioned streaming services have adopted various techniques to deliver video content to mobile users, such as encoding rate streaming, rate throttling, buffer adaptive streaming, fast caching, and rate adaptive streaming over HTTP. Encoding rate streaming is used to deliver content at the encoding rate. Throttling and fast aching send video content at a higher rate than the encoding rate. Buffer adaptive mechanisms work based on the playback buffer status of a client player. In this case, the client receives content from the server only when playback buffer drains to a specific lower threshold. Fast caching allows the player to download the whole content at the very beginning. Rate adaptive mechanisms adapt video quality according to the end-to-end bandwidth between a server and the client. There has been work on analyzing the merits of these streaming techniques from the server performance point of view. For example, fast caching reduces start- up delay at the client and guards against bandwidth fluctuation, but it also consumes a lot of CPU and memory at the streaming server [2]. Although most of the techniques are understood by research community, a thorough study of these streaming techniques is still required from the perspective of the mobile device and the user. Even though some studies have looked at the traffic pattern of video streaming services with Android, iOS devices, and desktop users [4, 5, 6], at present it is not well understood how the different techniques are chosen, how they compare to each other, and what are the optimal techniques to use in different contexts. Most importantly, the effect of these streaming techniques on user satisfaction on playback quality, Wi-Fi and cellular network usage, and on the energy consumption of mobile devices is yet to be fully uncovered. Such knowledge is imperative before one can design a streaming service that satisfies users demands in terms of quality of experience and battery life of their smartphones. We actively captured traffic of more than five hundred video streaming sessions, from YouTube, Vimeo, Dailymotion and Netflix, via Wi-Fi, HSPA, and LTE. During those sessions we estimated the joining time. From the captured traffic we computed the playback buffer status. We also measured the energy consumption of smartphones during the streaming sessions. Our main observations are the following: * 1. In general, fast caching and throttling are applied by the server, whereas video players enforce encoding rate and buffer adaptive mechanism by exploiting TCP’s flow control mechanism, hence, overriding the server selected mechanisms. In encoding rate streaming, the player unintentionally triggers TCP flow control because the player has too small playback buffer compared to the amount of content the server offers. The buffer adaptive mechanisms deliberately pause and resume download, and these techniques are applied only by the video players in Android phones. (Section 4) * 2. Our analysis reveals that in smartphones different techniques are applied with little or no consensus: different techniques are used by different clients to access the same service in the same context. For example, Android devices use three different techniques for YouTube videos. The selection of those techniques depends on the quality of the video and the player. However, the strategy selection does not depend on the wireless interface being used for streaming and, thus, network operators do not play any role. (Section 4) * 3. The joining time (a.k.a. initial playback delay) varies according to the wireless interface being used for streaming, the quality of the content, and the video service. The players experience shorter delay when streaming via Wi- Fi than HSPA or LTE. From the quality perspective, low quality videos are played with a shorter initial delay. Among the targeted video services, the Netflix players experience the longest delay. However, most of the streaming strategies are optimized for providing uninterrupted playback by allowing the players to keep a large amount of data in the playback buffer. (Section 5) * 4. There is a large variation in playback energy consumption between different types of players and containers on the same device. The differences are due to inefficient player implementation. However, the video quality (resolution) does not seem to have a large impact on energy consumption. (Section 6.1) * 5. When the user views the entire video clip, fast caching and throttling are the most optimized techniques for providing uninterrupted playback at the client. At the same time, they are the most energy efficient. If the user is likely to interrupt the video viewing, buffer adaptive streaming is more attractive as the player generates ON-OFF traffic pattern and less energy is consumed for wireless communication during an OFF period. However, the ON period duration should be adjusted to match fast start period in order to avoid server rate throttling. Similarly, the duration of the OFF period should also be optimized so that the player does not suffer from playback buffer starvation. However, none of the identified techniques alone is optimal because they do not adapt to the wireless access technology, user behavior, and preferences. (Section 6.3) We structure our paper as follows. In the next section, we briefly describe the energy consumption characteristics of wireless communication in smartphones, explain the characteristics of mobile video streaming. In Section 3, we describe our measurement and data collection methodology. In Section 4, we investigate the different streaming techniques. Section 5 examines the effort of the streaming techniques in providing uninterrupted playback. Section 6 is devoted to presenting the results from the energy consumption measurements. In section 7, we discuss the tradeoff between energy savings and potential playback buffer underrun. Finally, we contrast our work with earlier research in Section 8 before concluding the paper. ## 2 Background Smartphones allow users to access Internet via Wi-Fi and mobile broadband access. Mobile broadband experience is enabled by the latest 3G and 4G technologies such as EV-DO, HSPA, and LTE. The most widely deployed mobile broadband technology is currently HSPA, while LTE is the fastest ever growing cellular and mobile broadband technology. In this section, we first review the power consumption characteristics of Wi-Fi and cellular interfaces that we use in this study. Then, we explain the characteristics of mobile streaming services and the metrics to assess the quality of experience of the users. ### 2.1 Power Saving Mechanisms for Wireless Network Interfaces #### 2.1.1 Wi-Fi Smartphones implement 802.11 Power Saving Mechanism (PSM) to manage the power consumption of Wi-Fi. There are four states; transmit, receive, idle and sleep. PSM allows the interface to be in sleep when there is not data activity. However, the client periodically powers up the interface to receive a traffic indication map (TIM) frame from the access point (AP). This interval is usually 100ms and also called listen interval. The TIM frame tells a mobile client whether the AP has some buffered data for the mobile device or not. If the AP has data for the client, the client sends PS-Poll frame in return to receive the buffered data. Otherwise, the client goes back to sleep. Modern devices usually implement a timer which keeps the interface in idle state for a few hundred milliseconds after the transmission or reception of packets, which improves especially the performance of short TCP connections. This is also known as PSM adaptive [7]. Figure 1: WCDMA/HSPA RRC states with typical values of the inactivity timers and power consumption. #### 2.1.2 WCDMA/HSPA 3GPP standards specify the efficient usage of the radio resources considering the mobility and power consumption of smartphones via a resource control protocol (RRC). Figure 1 shows that there are a number of states and inactivity timers in 3GPP RRC protocol. These timers ensure that if a certain resource is not utilized for a certain period of time in a particular state, the resource must be released. For example, high volume data transmission happens in CELL_DCH state and small packet transmission is possible in CELL_FACH state. A mobile device switches from CELL_DCH to CELL_FACH in absence of data activity for a period of T1 seconds. These timers have static values and they are operator controlled. If the mobile device and network both support Rel 8.0 Fast Dormancy (FD) [8], CELL_DCH$\rightarrow$ CELL_PCH transition happens. For non standard FD, the transition is CELL_DCH$\rightarrow$ IDLE (Figure 1) which releases the RRC connection altogether. RRC protocol has a large impact on the energy spending of smartphones. Figure 1 also shows that average current consumption in CELL_DCH is 200mA, in CELL_FACH is 150mA, and in CELL_PCH is 50mA approximately. The potential consequence especially with long inactivity timers is high power consumption at the mobile device. To learn more about different cellular network configurations and their effect on energy consumption, readers can follow [9]. Figure 2: LTE DRX Cycles and timers. #### 2.1.3 LTE The radio resource control protocol for LTE specifies only two states; RRC_IDLE and RRC_CONNECTED. Similar to the HSPA RRC protocol, an inactivity timer (RRCidle) controls the connected to idle state transition. LTE includes a discontinuous transmission and reception (DTX/DRX) mechanism that enables a mobile device to consume low power even being in the RRC_CONNECTED state. DRX in the connected state is also called connected mode DRX or cDRX, and the associated inactivity timer is DRXidle. Figure 2 shows that if there is no data activity for DRXidle time, then a DRX cycle, DRXc, is initiated. The length of a such cycle can vary from 20ms to few seconds. The device checks data activity during the on period, DRXon, of the cycle. If the data inactivity continues for a long time, RRCidle, the network commands the device to switch from RRC_CONNECTED to RRC_IDLE state. Then the device enters in the paging monitoring mode in the IDLE state. ### 2.2 Mobile Video Streaming Streaming Services \| YouTube, Vimeo, Dailymotion, and Netflix ---\|--- Players \| Native Application, Flash, and HTML5 Video Quality \| LD (240p), SD (270-480p), HD (720-1080p) Containers \| 3GPP, MP4, WebM, X-FLV, ismv Table 1: Streaming services, the players used by the clients for playback, the quality of the content and the containers to deliver the content. Today mobile streaming services deliver content using HTTP over TCP. Smartphone users can access these services using either a native app or a browser. The browser may load a Flash, HTML5 or Microsoft Silverlight player. The quality of the video played is often denoted with a p-notation, such as 240p, which refers to the resolution of the video. 240p usually refers to 360x240 resolution. Different services use also low, standard, and high definition (LD, SD, HD) notations but the resolutions that each one refers to varies between services. Therefore, we define 240p videos as LD, 270-480p videos as SD and 720-1080p or higher resolution videos as HD. MP4, WebM, and X-FLV are the default containers for the players. The native apps of YouTube, Dailymotion and Vimeo also play MP4 and 3GPP videos. Netflix players play ismv videos. WebM and X-FLV are the default containers for the HTML5, and Flash player respectively. Table 1 shows the examples of examples of different video services, the types of video players, video qualities, and containers. ### 2.3 Quality of Experience The quality of streaming perceived by a user is influenced by the network condition, content quality (e.g. HD or SD), user’s preference on the content, and the context in which the user is viewing a video. The network condition translates to network congestion caused by the bottleneck point in between a streaming client and the server. This network congestion is evidenced by the reduced available bandwidth and packet loss. The impact is realised by the user as long initial playback delay and pauses or playback starvation during playback. In wireless networks, the bottleneck situation can arise when multiple users share the common resources and the throughput per user is reduced so much that user experience is degraded. The bottleneck also can be caused by the radio conditions, i.e. in cell edge the available bit rate is much lower than peak HSPA/LTE bit rates even in an empty cell. The state transition of the WNIs can introduce additional delays. For dealing with various network conditions, video services apply a number of strategies; i) Encoding rate streaming, ii) Throttling, iii) Buffer Adaptive Streaming, iv) Rate Adaptive Streaming, and v) Fast Caching. A common feature of all streaming services is an initial buffering of multimedia content at the client. This initial buffering is also referred to as Fast Start. The name comes from the fact that a player downloads content using all the available bandwidth. Fast caching is similar to Fast Start, the only difference is that fast caching lasts longer until the whole content is downloaded. These techniques are used by the services for constant bit rate streaming, except rate adaptive streaming. The most prevalent forms of rate adaptive streaming are HTTP live streaming (HLS), Microsoft Smooth Streaming (MSS). Figure 3: Capturing traffic at the Gn interface between SGSN and GGSN in the test HSPA Network. Figure 4: Capturing traffic at the S1 interface between the eNB (Base Station) and EPC (Evolved Packet Core) in the LTE network. ## 3 Measurement and Data Collection ### 3.1 Properties of the multimedia content Compared with our earlier work [10], we excluded previous results for Meego, Symbian, and WP7.5 platforms. We included three latest smartphones; iPhone5, Galaxy S3 LTE (GS3 LTE) and Lumia825. All the video services, YouTube, Dailymotion, Vimeo and Netflix, have the native applications for the target mobile platforms. The desktop edition of YouTube was used only in the Android platforms as it provides the opportunity to use both Flash and HTML5 players. Our target video services, players, and smartphones are listed in Table 3. Whenever available for the particular smartphone and player, we streamed videos of multiple qualities that range from LD to HD. The average duration of the videos was 10 minutes. Config Name \| Parameters ---\|--- noDRX \| RRC_${idle}$=10 s DRX80ms \| RRCidle=10 s, DRXcycle=80 ms, DRXon=10 ms, DRX160ms \| RRCidle=10 s, DRXcycle=160 ms, DRXon=10 ms, DRX640ms \| RRCidle=10 s, DRXcycle=640 ms, DRXon=10 ms, Table 2: LTE network configurations. \| iPhone4S \| iPhone5 \| Galaxy S3/Galaxy S3 LTE(Android-4.0.4) \| Lumia825 ---\|---\|---\|---\|--- \| iOS 5.0 \| (iOS 7.0) \| \| (WP8) YouTubeStreaming \| (App) ThrottlingFactor=2.0 \| (App) ThrottlingFactor=1.25 \| (Flash) Encoding rate(HD),Throttling($<$HD)Factor=1.25 \| (App& HTML5) ON-OFF-M \| (App)Fast Caching Quality \| LD(240p), SD(360p), HD(720p) \| LD(240p), SD(360p), HD(720p) \| LD(240p),SD(360,480p),HD(720,1080p) \| LD(240p),SD(360,480p),HD(720p) \| SD(270p), HD(720p) Container \| MP4(360,720p) \| MP4(360,720p)3GPP(240p) \| XFLV \| MP4($>$240p)WebM($>$240p)3GPP(270p) \| MP4(720p)3GPP(270p) VimeoStreaming \| (App)HLSChunk Size=10s \| (App)ON-OFF-M \| (App) ON-OFF-S \| (App)Fast Caching Quality \| * \| SD(270,480p), HD(720p) \| SD(270p), HD(720p) \| HD(720p) Container \| MP4 \| MP4 \| MP4 \| MP4 DailymotionStreaming \| (App)ThrottlingFactor=1.25 \| (App)HSLChunk Size=10s \| (App) Fast Caching(288p), ON-OFF-S($>$288p) \| (App)ThrottlingFactor=1.25 Quality \| LD(240) \| * \| SD(288,480p),HD(720p) \| SD(288p) Container \| MP4 \| MP4 \| MP4 \| MP4 NetflixStreaming \| (App)HLSChunk Size=10s \| (App)HLSChunk Size=10s \| (App) ON-OFF-S \| (App)MSSChunk Size=4s Quality \| * \| * \| HD(720p) \| * Container \| isma, ismv \| isma, ismv \| MP4 \| isma, ismv Table 3: Streaming techniques for popular video streaming services to mobile phones of three major platforms. The selection of a streaming technique does not depend on the wireless interface being used for, rather depends on the player, video quality, device and the video service provider. ### 3.2 Network Setup We watched videos from the video services in the smartphones via Wi-Fi, HSPA, and LTE. In the case of Wi-Fi, a 802.11 b/g access point was used. The access point was connected to the Internet via 100 Mbps Ethernet. AirPcap111AirPcap:www.cacetech.com/documents/AirPcap%20Nx%20Datasheet.pdf was used to capture the Wi-Fi traffic. HSPA network measurements were conducted in the Nokia Solutions and Networks test networks. The network parameters, i.e. states and inactivity timers, were configured according to the vendor recommendation. The values of the inactivity timers were from few seconds to few minutes; T1=8s,T2=3s,T3=29min. The CELL_PCH state was enabled in the network. We captured traffic of the streaming clients at the Gn interface between SGSGN and GGSN (see Figure 3). The LTE measurements were conducted with connected mode DRX enabled in the network. Traffic capture is taken at the S1 interface between the eNB and EPC. We measured power consumption with three sets of DRX profiles. The DRX profiles are described in Table 2. ### 3.3 Power Measurement We used Monsoon222Monsoon Power Monitor : www.msoon.com and another custom power monitor for measuring the energy consumption of the smartphones during multimedia streaming. We removed the battery of most of the mobile phones and powered them using the measurement devices. Only the iPhones get power from the battery. All the devices were in automatic brightness settings during the power measurements. ## 4 Streaming Techniques From traffic traces we inferred manually the type of streaming technique used for each of the different combinations of device, service, stream quality, player type, and access network type. These findings are summarized in Table 3 and discussed below. Figure 5: Interaction between playback buffer and TCP receive buffer for encoding rate streaming. ### 4.1 Encoding Rate Streaming Encoding rate technique is exclusively applied by the streaming clients. The server sends content using fast caching and the player has a small playback buffer. Therefore, the playback buffer and TCP receive buffer become full at the very beginning. Since the player decodes content at the encoding rate, the same amount of buffer is freed from the playback buffer and also from the TCP receive buffer. The client again can receive the same amount of content from the server. The mechanism is illustrated in Figure 5. From Table 3, we can see that the Flash player in Android devices receives HD videos from YouTube at the encoding rate. ### 4.2 Throttling Throttling is a server-side streaming technique. In this case, the server sends content at a limited constant rate, which is higher than the encoding rate. Therefore, the content is downloaded at the client at a faster pace than the encoding rate. The multiple of the encoding rate is referred to as the throttle factor. The throttling factor can vary depending on the video service or even on the player type for the same service. For instance, the native YouTube application receives content at a throttled factor of 2.0 in iPhone4S, whereas the Dailymotion application receives at a factor of 1.25. The Flash player in Android devices and the native app in iPhone5 specify the throttling factor in the request URL (e.g., algorithm = throttle-factor and factor = 1.25) or a service specific default throttle factor is used. (a) CDF of the chunk sizes YouTube. (b) CDF of the chunk intervals. Figure 6: YouTube server sends content in small chunks and periodic manner when throttling the sending rate. #### 4.2.1 Single TCP connection In general, throttling is carried over a single TCP connection and the data is sent in small chunks. Figure 6(a) shows that in the YouTube player in iPhone receives a LD video in 64KB chunks. This observation is similar to those explored in [11] and [4] for YouTube. The chunk size increases to 192KB when receiving the same video of HD quality. We observed variable chunk sizes when streaming to Samsung Galaxy S3 (see Figure 6(a)). However, these chunks are sent by the streaming servers at some periodic intervals to the streaming clients. The interval increases as the encoding rate or quality of the video decreases. Figure 6(b) shows that the chunks are separated by few hundred milliseconds to 1.2s. This burstiness is independent of the wireless interface being used at the client to receive the content. Nevertheless, this kind of burstiness was absent in Dailymotion and Vimeo traffic. #### 4.2.2 Multiple TCP connections In iPhone4S, the YouTube application uses a significant number of TCP connections to receive HD quality videos. In an example video session, we found that the player downloads a HD video in 66 connections. The player maintains a 25MB size playback buffer. At the beginning, the player receives content at the throttled rate. Since the playback continues at the encoding rate, there is always some extra content in the buffer. Therefore, this playback buffer becomes full at some point and the player closes the existing TCP connection. Whenever some buffer is freed, the player initiates another HTTP partial content request over TCP. In this way, the player actually receives more data from the server than the actual size of the content. Finamore et al. [5] also reported similar observation. From traffic traces, we identified that a YouTube server always sends media content from the beginning of a key frame for any partial content request. The reason is that the player is unable to keep track of the ending position of the current key frame or the beginning of the next key frame. Therefore, it may terminate the connection when receiving a key frame. In addition, the player must support the forward and backward seeking during playback. Subsequently, each time the player requests content from the beginning of a key frame, which it has received partially for the previous request. As a result, the player wastes all the data of the partially received key frame. From traffic traces, we calculated that the player received 160MB data in total for for a 76MB video. (a) Buffer adaptive streaming over a single TCP connection activates TCP flow control during an OFF period. (b) The growth of the TCP persist timer at the streaming servers during an OFF period. Figure 7: ON-OFF-S mechanism and interaction with TCP flow control. ### 4.3 Buffer Adaptive Streaming Buffer adaptive techniques represent smart player implementation. The players maintain two thresholds of buffer level: a lower and an upper. During a streaming session, the player stops downloading content when the playback buffer is filled to the upper threshold value and it resumes downloading when the buffer drains to the lower threshold. The video players apply buffer adaptation in two different ways and generate ON-OFF traffic pattern. Some video players apply the buffer adaptation over a single TCP connection. We refer this kind as ON-OFF-S. The others use multiple TCP connections and we refer as ON-OFF-M. #### 4.3.1 Single Persistent TCP Connection (ON-OFF-S) The native applications of Dailymotion, Vimeo, and Netflix video services apply buffer adaptation over a single TCP connection in the Android devices (see Table 3). The players stop reading from the TCP socket and an OFF period begins. Figure 7(a) illustrates that TCP flow control packets are exchanged during an OFF period. Figure 8: The YouTube player in Galaxy S3 downloads a video by initiating multiple TCP connections. The duration of an OFF period can be very long. The older Android devices (e.g., Samsung Nexus S) use an upper threshold of 5MB [10]. Therefore, the duration of the OFF period is almost equivalent to the $\frac{5MB}{Encodingrate}$s. On the other hand, in latest devices the duration is $\frac{20MB}{Encodingrate}$s. However, from traffic traces we found that the TCP persist timer at the server grows only to maximum 5s. The reason is that the players intentionally reset the persist timer after every 16s by receiving 64KB data from the server. This behavior was absent in the case of Netflix. Figure 7(b) shows how the TCP persist timer values grow at the servers of different video streaming services. In the case of Netflix, the OFF period is always 30s and the persist timer increases to maximum 10s. Later in Section 6.3, we will see how the TCP flow control messages and TCP persist timer affects the power consumption of smartphones. #### 4.3.2 Non-persistent TCP connections (ON-OFF-M) Only the native app and HTML5 player for YouTube in Android devices use multiple TCP connections for buffer adaptation. The players maintain dynamic lower and upper thresholds of playback buffer. When the playback buffer is filled to the upper threshold, the player closes the TCP connection and an OFF period begins. The ON period begins after a fixed 60s OFF period (see Figure 8). The recent version of Vimeo player in iPhone5 also uses multiple TCP connections. Unlike the YouTube player, the Vimeo player downloads 30MB during the Fast Start and downloads rest of the content in 5MB chunks. Therefore, the duration of an OFF period is equal to $\frac{5MB}{Encodingrate}$s. (a) The Vimeo player in iPhone4S, using HLS, discards content of low quality from the playback buffer upon switching to a higher quality. (b) The Netflix player in iPhone5, using HLS, downloads audio and video chunks asynchronously. Figure 9: Joining observed for the video services and when streaming via wireless network interfaces. ### 4.4 Fast Caching Fast caching refers to downloading the whole content in one go at the very beginning of the streaming using the maximum available bandwidth. The players continue playback and at the same time maintains very large playback buffer. YouTube Flash player uses ratebypass=yes parameter in the HTTP request to deactivate any rate control at the server side. For example, the YouTube player of Lumia825 downloaded a 10-minute long $720p$ video within 120s via LTE or HSPA in our experiments. Lumia825 also receives video content from Vimeo at possible maximum rate. ### 4.5 Rate Adaptive Streaming The streaming techniques we discussed so far are for streaming constant quality content during a streaming session. The players or the servers cannot change the quality on the fly, unless the user interrupts the playback. On the other hand, Dynamic Adaptive Streaming over HTTP (DASH [12])-like rate adaptive mechanisms are able to change the quality on the fly for adapting with bandwidth fluctuations. The quality switching algorithms are implemented in the client players. A player estimates the bandwidth continuously and transitions to a lower or to a higher quality stream if the bandwidth permits. We identified two kinds of rate adaptive streaming; (i) HTTP Live Streaming (HLS) and (ii) Microsoft Smooth Streaming (MSS). #### 4.5.1 HTTP Live Streaming The Netflix and Vimeo players in iPhone4S, and the Dailymotion player in iPhone5 use HTTP Live Streaming and downloads content in 10s chunks. At the beginning, a player receives the media description files, which contain the chunk duration, encoding rates and the bandwidth requirements for the chunk download. The player begins by downloading seven 10s chunks of the SD quality. After that, the player downloads chunks after every ten seconds. In this way, the player always keeps 60s playback content in the buffer when streaming via Wi-Fi. In case of transitioning to a higher quality, the player discards the downloaded lower quality content in order to provide instant response to the quality change to the user. One streaming scenario via Wi-Fi is illustrated in Figure 9(a), where the player switches from a SD to HD quality at 232s and downloads from 23rd to 29th segments of HD quality. In the case of HSPA, the player wastes 20s content. This observation can change with bandwidth variation. Similarly, the Netflix player uses HLS in iPhones. However, the Netflix downloads the audio and video chunks separately, where the chunks are of 10s. From multiple traces, we verified that the audio and video chunk downloading are not synchronized. Figure 9(b) shows that after the Fast Start phase, the interval between an audio and a video chunk is approximately five seconds. There were also some cases where an audio chunk appears very close to the next video chunk. Another interesting observation is that the server specifies it’s TCP parameters in the HTTP response header, as for example X-TCP- Info:rtt=11625;snd_cwnd=217201;rcv_wnd=1049800. The reason is likely that the streaming server lets the client player to calculate the bandwidth and to decide the quality accordingly. #### 4.5.2 Microsoft Smooth Streaming The Netflix player in Lumia825 uses Microsoft’s smooth streaming. The player receives video content in 4s chunks over a single TCP connection. The same connection is used to receive audio content chunks also. However, the audio chunks are received after every sixteen seconds, i.e. after four consecutive video chunks. Unlike the Vimeo and Dailymotion rate adaptive players in iPhones, Netflix is aggressive in providing the highest quality of the stream in Lumia825. In traffic traces, we noticed that the player begins with the lowest quality, and then switches to the maximum quality within the first few seconds of streaming. During this period, the player downloads 60s playback content. However, unlike the desktop player [13], the mobile version requests different filenames for different qualities and specifies the byte range in the URL GET (abc).ismv/range/0-40140/. In response, the server sends the chunks of the corresponding quality. The server also sends a .bif file which contains information about the frames, which is used by the player upon forward or backward seeking by the user. We also found that the Netflix server sends TCP parameters to the player. ### 4.6 Summary Table 3 summarizes our findings on the usage of different techniques in different mobile platforms with four video services. Figure 10 illustrates how the client app behaviour leads to the choice of particular streaming technique for constant quality streaming. We sum up our main observations below: Figure 10: The choice of a streaming technique by the client player for constant bit rate streaming. (a) Streaming Service (b) Wi-Fi (c) HSPA (d) LTE Figure 11: Joining time observed for the video services and when streaming via wireless network interfaces. * 1. Streaming servers use either throttling or fast caching to deliver constant bit rate video to mobile devices. The choice between these two is influenced by client player’s request. For instance, YouTube Vimeo servers use both throttling and fast caching. The Dailymotion servers use throttling. Netflix servers use fast caching for constant bit rate streaming, and MSS or HSL for rate adaptive streaming. Some native mobile apps continuously pause and resume downloading leading to ON-OFF traffic patterns. Encoding rate streaming is the result of small playback buffer at the client buffer and fast caching streaming by the server. * 2. For constant bit rate streaming, the relevance of a technique depends on the mobile platforms to some extent. Buffer adaptive streaming is commonly used by all the video streaming services in the Android platforms. However, the only exception is Dailymotion. The reason is that the videos are small in size and the throttling rate is also small compared with YouTube and Vimeo. Therefore, the player does not get enough buffer filled to apply the adaptation. Fast caching is prevalent only in Windows-based devices. * 3. None of the video services apply rate adaptive streaming in Android mobile devices. The Netflix, Vimeo, and Dailymotion players use HLS in iPhones. In iOS devices, Netflix receives audio and video chunks in separately streams. MSS is used only in Windows. This also reflects the influence of platforms on the choice of steaming techniques. * 4. Although the streaming strategies can vary based on the quality of the video, platforms, and video services, we could not find any evidence that the strategies vary according to the wireless interface being used for streaming. (a) Encoding rate streaming and playback buffer status (in second). (b) Playback buffer status when using throttling, buffer adaptive and Fast Caching (c) Rate adaptive streaming and playback buffer status Figure 12: Playback buffer status of the streaming clients during multimedia streaming sessions using different techniques. * 5. The amount of data wasted by the YouTube player in iPhone4S is significant even when a user watches the complete video. Although this problem could be solved with a smarter player implementation, the YouTube player in the latest iOS version sends request with a throttle factor of 1.25. As a result, the playback buffer never becomes full, and consequently, there is no data waste. However, potential data waste is also possible when the whole video is downloaded and the user abandons watching earlier. ## 5 Streaming Techniques and Quality of Experience The key metrics that characterize the QoE perceived by a user while streaming video are the initial playback delay which is called joining time, and the occurrence and frequency of playback pause events experienced [14, 15]. As we discussed earlier that playback pause events are the results of bandwidth variation due to various network conditions. In this section, we take a look at the joining time and the performance of different strategies in providing smooth playback during short/long term bandwidth changes. Figure 11 shows the joining time experienced by the players according the video service and the WNI. Although, all the streaming services use fast start at the beginning of streaming, it is shown in Figure 11(a) that the YouTube players take less time than the other players. On the other hand, the other services have longer joining time. The reason is that YouTube caching servers are extensively spread around the globe. Therefore, the content is served from the CDN that is very close to the user. We validated this by measuring the roundtrip time from the captured traces. Our observation is similar to [14], in which the authors also proposed to serve content from the nearby CDN to improve the playback experience. However, in the case of Vimeo and Netflix two other facts also contribute in higher joining time. The Vimeo player always receives the HD quality video and the Netflix player always decides the maximum quality at the beginning of streaming, which take more time than the players of other services. We explained earlier that quality of the stream affects the initial start-up time. The boxplots in Figure 11(b), 11(c), 11(d) illustrate the similar findings. There are two observations. First, streaming via Wi-Fi experiences less joining time than streaming via HSPA and LTE. The joining time is the largest when HSPA is used. We investigated and found that the wireless latency plays the role when streaming via HSPA and LTE. This is because, at the beginning of a streaming session, the HSPA interface transitions from IDLE/CELL_PCH to CELL_DCH state and the LTE interface switches from IDLE to the CONNECTED state. The transition latencies for LTE and HSPA are 120ms and 2.0s respectively. In the case of Wi-Fi, the transition latency from sleep to active state transition is few milliseconds which is negligible. The other observation is that the rate adaptive players experience more delay in the joining. This observation is biased because of the Netflix’s rate switching strategy. Next, we looked at the prefetching behavior of the players by studying how much content they maintain in the playback buffer throughout a streaming session. This analysis requires the time series of content consumption and arrival. The arrival time series is computed by extracting timestamps and playload sizes of received packets from the traffic traces considering the joining time. Although there are findings that YouTube-like video services stream constant bit rate content [16], we found that the video services use variable bit rate encoding for streaming HD videos. Hence, we replayed each video using a VLC player and extracted the instantaneous encoding rate of the content from VLC’s web interface module using a shell script. Finally, we compute the amount of buffered content as a function of time by taking the difference of the cumulative sums of the arrival and consumption time series. Figure 12 shows the playback buffer status during the streaming sessions using different streaming techniques. Using encoding rate streaming, a player always keeps 30-40s equivalent content in the playback buffer. Hence, even if the player receives content at the negligible rate after the fast start phase, the player can provide playback for that 30-40s period. Throttling and fast caching continuously accumulate more content into the buffer and therefore are more robust also towards longer periods of low available bandwidth. From 12(b), we can see that when the playback is at 50s, the player already has content for next 50s using throttling. In case of fast caching, the player has 200 s worth of content in the buffer. When using the ON-OFF strategies, the buffer is periodically filled up and drained in between. ON-OFF-M begins refilling the buffer 40s earlier. A surprising result is that ON-OFF-S (in Android 2.3.6) nearly dries the buffer before new content is prefetched. Therefore, the possibility of playback starvation increases, when streaming via HSPA. The rate adaptive players maintain 60-100s playback buffer, and at the same time they can select to a lower quality (see Figure 12(c)). Nevertheless, the streaming strategies provide the best effort in guarding short term and long term bandwidth fluctuations. ## 6 Streaming Services and Power Consumption We also measured the total current consumed by the smartphones during the streaming sessions. We separated the total current drawn into the average video playback and wireless interface current consumption. The playback current includes decoding and display current. We can identify this current draw at the end of the power trace of each streaming session when the content has been fully delivered but playback still continues, since some of content is always buffered at the end regardless of streaming technique used. During this time, the WNIs are in the lowest power consuming states according to their own power savings protocols. We computed the average wireless communication current, which we refer to as streaming current, by subtracting the average playback current from the total current. The results presented in this section are the average of repeated measurements. (a) Avg. playback current draw when streaming $240-1080p$ YouTube videos to the app and browser in Galaxy S3. (b) Amount of CPU used by different video players in Galaxy S3 while playing different quality videos of different containers. (c) Avg. playback current consumption while playing different quality videos of different containers. Figure 13: Playback current consumption of Galaxy S3 and CPU usage with different qualities, players and containers. ### 6.1 Playback Power Consumption #### 6.1.1 Video Quality In Figure 13(a), we can see that playback current draw of Galaxy S3 increases as the quality of YouTube video increases as long as the same container is used. We also observed similar pattern for watching Dailymotion videos in iPhone4S and Galaxy S3. It is logical that high quality videos have more information to present than low quality videos and, therefore, more current is drawn. However, in some cases even doubling the resolution adds a relatively small increment to the average playback current. (a) Wi-Fi and HSPA (b) LTE Figure 14: Current consumption of wireless network interfaces in smartphones. #### 6.1.2 Video Player For playing YouTube LD, SD and HD videos, the browser loads a Flash player. Flash has support for different kind of codecs and containers, such as X-FLV, MP4 and H.264. The browser loads HTML5 player to play WebM videos. Figure 13(a) compares the energy consumption when using different players for streaming. It is noticeable that the native YouTube application consumes the least amount of energy. In contrast, browser-based players can draw even more than the double current compared with the app when playing the same video. We discovered that during playback the Flash player does not leverage any native system support to decode the video but consumes a significant amount of more CPU than the native application (see Figure 13(b)). Although the HTML5 player takes native system support, it consumes 60% of CPU even during the playback of a $480p$ video. It seems that HTML5 player is required to go through further optimization to be used in mobile platforms. #### 6.1.3 Video Container We already showed how the videos of different quality and different players affect the energy consumption of smartphones. In Figure 13(a), we can see that playback of a $240p$ 3GPP video requires less energy than that of an X-FLV video of the same quality. It is also illustrated that the same $240p$ X-FLV requires more current than a $720p$ MP4 video. Although from Figure 13(a) we can infer that 3GPP is the least and WebM is the most energy consuming containers, it is difficult to isolate the effect of the corresponding video containers since some videos can be played only using browsers. Besides, the energy consumption of the browser-based players are very high. Therefore, we downloaded some YouTube videos of X-FLV and WebM formats and then measured energy consumption during playback. The results are shown in Figure 13(c). This figure also illustrates that playback energy consumption does not change significantly when the quality of video changes with the same container category. ### 6.2 Device Variation Before discussing the impact of different streaming strategies on the streaming power consumption, we investigate the power consumption of individual WNI in smartphones. In Section 2.1, we described the standard power saving mechanisms applied by different WNIs. We also discussed that there are a number of states and a mobile device consumes different amount of energy in different states. Consequently, we explore what kind of power saving mechanism are applied by our target smartphones and the variation among them in consuming energy. In Figure 14(a), we can see that the Wi-Fi interfaces in iOS phones consume lowest energy. Android devices consume more current when the Wi-Fi interface is active, whereas the Wi-Fi interface in Lumia825 consumes the maximum energy. However, all of them use PSM adaptive. iOS devices use an aggressive idle period of 50ms. The other devices use 200ms. The power consumption during this idle state is half of the active state power consumption. Figure 14(a) shows the power consumption of HSPA interface during data transfer in CELL_DCH state. In this case of also iOS devices consume the lowest energy when the HSPA interface is active. Lumia825 is the second least. On the other hand Android devices consume the maximum energy. However, all the devices use Fast Dormancy with an inactivity timer of 5s, except iPhone5 which uses an inactivity timer of 8s. We measured power consumption of the LTE interface with four different network configurations; DRX is disabled, DRX is enabled with a short DRX cycle (80ms), with DRX cycles of 160ms and 640ms respectively. From the results presented in Figure 14(b), we find that the smartphones consume the maximum energy when DRX is not enabled in the network. If DRX is enabled in the network, the smartphones consume less power. This is because the devices periodically wake up to check data activity according to the DRX cycles in the connected state. This Figure also depicts that Lumia825 consumes the lowest current when LTE is active. (a) DRX cycle length 80 ms. (b) DRX cycle length 640 ms. Figure 15: Current consumption of GS3 LTE with different DRX cycles. Figure 16: Avg. streaming current consumption of smartphones when streaming a 600 s long constant bit rate video using the streaming strategies. Figure 14(b) also depicts that iPhone5 is the most and Lumia825 is the least energy consuming device when DRX is enabled. From power traces we identified that even though the DRXon was configured to 10ms, iPhone5 spends 60ms. On the other hand, Lumia825 and GS3 LTE spend 30 and 45ms respectively in the on period of the DRX cycle. From Figure 14(b), we can also see that the devices consume more current when the cycle lengths are shorter. For instance, when DRX cycle is of 80ms, GS3 LTE and Lumia825 consume around 120mA current. If the cycle length is increased to 640ms, the power consumption is decreased by a factor of three approximately. The first reason is that when short DRX cycles are in action, a mobile device will spend more time in the on period of the cycles as there will be more cycles when the RRC inactivity timer is active. Second, the LTE chipset is not optimized yet to operate on such small cycles. They cannot efficiently shutdown the power consumption during the DRX sleep phase. Figure 15(a) shows that current consumption of GS3 LTE is stable at $\approx$220mA from 132 to 142s even though the DRX is active. Current consumption during short DRX cycles does not scale down like when DRX cycle is of 640ms (from 245 to 255s in Figure 15(b)). This pattern is also consistent with iPhone5 and Lumia825 (Figure 14(b)). However, power consumption of these interfaces can vary according to the downloading rate. The deviation can be $\pm 50$mA. ### 6.3 Impact of Streaming Techniques In the previous section, we showed the basic power consumption characteristics of different WNI. In this section, we discuss the effect of streaming techniques on the energy consumption in smartphones. Since all the techniques are not available in a single platform, it is difficult to compare the energy efficiency of the techniques. Therefore, we compare only the current consumed by the wireless interfaces of the smartphones and exclude the playback current in order to provide a comparison ground. In the case of LTE, the DRX was enabled in the network and we used a single DRX profile with DRX cycle of 80ms, as this profile is used by the network operators in Finland. We compare them in Figure 16. #### 6.3.1 Encoding Rate Streaming In this case, the content is delivered continuously throughout the entire streaming session and the wireless interface is active all the time. For example, downloading a 6 minute video would require approximately six minutes. As a consequence, the average streaming current drawn by Galaxy S3 LTE is very high for the YouTube videos. Figure 16 also shows that Galaxy S3 LTE (GS3 LTE) consumes around 77mA for Wi-Fi, 200mA and 310mA for HSPA and LTE respectively (HD video using browser). The high current consumption of HSPA/LTE is expected, since these interfaces are constantly in the highest power consuming state. However, power consumption over Wi-Fi is low with respect to the usage of the interface. This is because, the Android devices use DVFS when streaming via Wi-Fi. #### 6.3.2 Throttling In Section 4.2, we discussed that in case of throttling, the throttle factor defines the amount of time is used to deliver the content to the client players. The higher is the throttle factor, the lower is the time required at the client to download the content. Therefore, this factor also determines the amount of time the wireless radio will be powered on and hence it also determines power consumption at smartphones. Energy consumption for two throttled sessions is presented in Figure 16. In the first case, the server uses the throttle factor 1.25 for iPhone5. The second session is for iPhone4S, where the factor is 2. iPhone5 consumes more current than iPhone4S for streaming via Wi-Fi and 3G. The obvious reason is that iPhone4S downloads at a faster rate. And both smartphones consume less current than the GS3 LTE which downloads video at the encoding rate. Therefore, throttling delivers energy savings over encoding rate streaming as interface usage time is reduced. #### 6.3.3 Buffer Adaptive Streaming Figure 16 shows that GS3 LTE consumes more current in streaming a Vimeo video than the Netflix video via any WNI. This is because of the player behavior in resetting TCP persist timer. We described in Section 4.3.1 that the Vimeo player resets TCP persist timer after every 16 seconds. Therefore, the maximum interval between TCP control packets from Vimeo can be 5s. On the other hand, the Netflix player rests after every 30s and the maximum interval between TCP control packets from Netflix is 10s. Therefore, the interfaces can spend more time in low power consuming states when streaming from Netflix than streaming from Vimeo. However, the average streaming current consumption is less than the encoding rate streaming. Figure 16 also includes a case where GS3 LTE receives content from YouTube in multiple TCP connections. Since the duration of such an OFF period is 60s, the wireless interfaces can be in sleep or the lowest power consuming states for very long time. As a result, GS3 LTE consumes roughly 50% less energy when using ON-OFF-M than the encoding rate. However, it can be seen that ON-OFF-M does not outperform throttling (iPhone4S) in current consumption as the player receives content at the same throttled rate in each TCP connection. Figure 17: Avg. streaming current consumption of smartphones for rate adaptive streaming techniques, HTTP Live Streaming, Microsoft Smooth Streaming and Netflix’s own adaptive mechanism in iPhone5. #### 6.3.4 Fast Caching Fast caching is used to download content at the client with as high throughput as possible. As a result the wireless interface is maximally utilized for as little time as possible. Figure 16 shows that GS3 LTE consumes the least current, if the YouTube player downloads the whole video using Fast Caching. #### 6.3.5 Rate Adaptive Streaming Similar to the ON-OFF-M mechanism, the quality or rate adaptive players also receive content in chunks over a single or multiple TCP connections. The duration of a chunk varies from a minimum four seconds to maximum ten seconds depending on the service. Figure 17 shows the current consumption of the WNIs when streaming Netflix and Dailymotion videos in iPhone5 and Lumia825. In both devices, power consumption of the Wi-Fi interface is about 30mA. The players in iPhone receive content in 10s chunks. Therefore, the HSPA interface avails the lower states rarely as the FD timer is 8s and consequently current consumption is high. The LTE interface also consumes significant current even though the DRX was enabled. This is because, the LTE interface in the iPhone5 takes long time in the ON period of the DRX cycle. iPhone5 consumes more current when streaming Netflix than the Dailymotion via cellular networks. The reason is that the Netflix player downloads audio and video chunks separately and their downloading was not synchronized. Compared with iPhone5, Lumia825 consumes less current when the Netflix player streams via LTE as the interface spends lesser time in the ON state of the DRX cycles when DRX is active. ### 6.4 summary From Section 6.1, we learned that native apps are the most energy efficient. Since, HTML5 is an important technology at this moment, optimizing the HTML5-based player implementations would be an important future work. We also noticed that video container/codec also has significant impact on the energy consumption (3GPP seems more efficient than MP4), while video quality has a small impact. Therefore, the focus should be choosing an optimal codec or container. Concerning the current consumption of wireless network interfaces, Wi-Fi is the the most energy efficient interface. When using LTE, the smartphones are not optimized yet for 80-160ms DRX cycles. Therefore, the network operators should use longer DRX cycles in the network to improve the battery life time of smartphones. The main lesson concerning the different streaming techniques is that encoding rate streaming causes clearly the largest amount of energy consumption. Fast caching is the most energy efficient technique. An effective ON-OFF-M technique should deliver content without any rate control. Although the rate adaptive techniques are similar to ON-OFF-M, higher chunk size and synchronization between audio/video chunks would reduce energy consumption significantly. ## 7 QoE and Energy Consumption Tradeoffs In Section 5, we found that most of the video services use optimized methods so that streaming quality does not deteriorate user experience by enabling the players in providing uninterrupted playback as long as possible. From this perspective, fast caching and throttling are the most efficient techniques. However, if the user does not watch the whole video, the downloaded data is wasted. Furthermore, using the cellular access to download unnecessarily content is problematic for users having small quota in their data plan and for the network resources. For example, Finamore et al. analyzed YouTube traffic to desktop computers and iOS devices accessed via Wi-Fi and discovered that 60% of videos were watched for less than 20% of their duration [5]. Therefore, ON-OFF mechanisms are attractive considering the unnecessary content download. Figure 18: Average draw of current as a function of viewing time for HSPA access. From the energy consumption point of view, the downloading energy is also wasted to retrieve the unwanted content. In Figure 18, we plot the average current draw for fast caching and ON-OFF-M techniques as a function of percentage of watched video computed out of the complete power traces. We see that abandoning the video watching early on would cause a hefty penalty in terms of wasted energy in both cases but the penalty gets smaller faster with the ON-OFF-M streaming making it a more attractive technique, since it is common not to watch the video completely. Figure 19: Relative power draw as a function of dynamic buffer size for HSPA access. S is the stream encoding rate and C is the available bandwidth to download content. Since ON-OFF-M is the balanced technique in providing both less data waste and less energy consumption, a tradeoff between the buffer thresholds and energy consumption must be understood. Assuming that the upper threshold is fixed, i.e. the player allocates a fixed amount of memory for the playback buffer in the beginning of a streaming session, the lower threshold determines how large chunks of content will be downloaded at a time, i.e. what is the duration and frequency of the ON periods. The lower the threshold, the less frequent are the buffer refill events (ON periods), and the less power is consumed on the average. On the other hand, the lower the lower threshold is set, the higher is also the chance that there is a playback pause event when the buffer refilling begins in case a transient period of low bandwidth happens to coincide. For this reason, there is a tradeoff between risking a buffer underrun event and the power consumption which is controlled by the lower buffer threshold. We plot in Figure 19 the average power draw as a function of the dynamic buffer size. The dynamic buffer size is directly determined by the lower threshold if we keep the upper threshold fixed. We notice that if there is plenty of spare bandwidth available compared to the stream encoding rate, then the buffer size should be set at least to a value around 40-50s, but setting the buffer to a larger value than that no longer reduces the power consumption significantly. The current YouTube players in Android that use the ON-OFF-M strategy set the upper threshold to a value equalling $100s\times r_{s}$ and the lower one to $40s\times r_{s}$ where $r_{s}$ is the average encoding rate. These thresholds translate to a $60s$ dynamic buffer size which, in light of Figure 19, strikes a good balance. Those players using ON-OFF-S technique in newer versions of Android use a 20MB buffer size. Assuming a lower threshold at zero, the dynamic buffer size would translate to 400s and 80s for videos having encoding rate of 400 kbps and 2 Mbps, respectively. With the higher quality video, the lower threshold could be set to $30-40s\times r_{s}$ in order to safeguard from buffer underrun events, and that configuration would still provide good energy efficiency when using HSPA. ## 8 Related Work The diverse nature of existing popular mobile streaming services in delivering better user experience, and the resulting energy consumption characteristics have so far not been completely uncovered. Krishnan et al. [17] studied the effect of initial joining time and playback pause events on the engagement in watching videos for fixed host users. Their findings were such that users cannot tolerate more than 2 seconds of joining delay and if a pause event persists more than 1% of total duration of the video the engagement decreases. Balachandran et al. [15] proposed a machine learning approach which tries to improve the engagement further by selecting the appropriate CDN according to the bit rate of the content. Many papers have studied the energy efficiency of multimedia streaming over Wi-Fi and developed custom protocols or scheduling mechanisms to optimize the behavior. Examples of such work range from proxy based traffic shaping and scheduling to traffic prediction and adaptive buffer management [3]. However, streaming over HSPA and the specific nature of the streaming services and client apps provide new challenges that these solutions cannot overcome. Balasubramanian et al. [18] studied 3G power characteristics in general and quantified the so called tail energy concept. The most popular streaming services, especially YouTube, have been subject to numerous measurement studies in recent few years. Xiao et al. [19] measured the energy consumption of different Symbian based Nokia devices while using a YouTube application over both Wi-Fi and 3G access. A similar study was done by Trestian et al. [20] for Android platform. They investigated energy consumption while streaming over Wi-Fi at different network conditions and studied the effect of video quality on energy consumption. However, these studies did not consider the details of traffic patterns and their impact on the energy consumption. In a measurement study, Rao et al. [4] studied YouTube and Netflix traffic to different smartphones (iOS and Android) and web browsers accessed via Wi-Fi interface. They found three different traffic patterns of YouTube. In a similar passive measurement study, Finamore et al. [5] also analyzed YouTube traffic to PCs and iOS devices accessed via Wi-Fi and demonstrated that iPhone and iPad employ chunk based streaming. Qian et al. [21] explored RRC state machine settings in terms of inactivity timers using real network traces from different operators and proposed a traffic shaping solution for YouTube which closely resembles the ON-OFF streaming technique. Liu et al. [22] studied power consumption of different streaming services. However, the scope of their study is considerably different from ours. They limit their study to streaming over Wi-Fi and performed experiments with only iPod, while we explored all the major mobile platforms and contrasted Wi-Fi and HSPA energy consumption in [10]. In contrast to these studies, our contributions are the followings. (i) We investigated the traffic pattern of the streaming techniques and the characteristics which influence the choice of a streaming technique. (ii) We measured the initial joining time that varies according to the service, quality of the content and wireless access. (iii) We examined the playback buffer status of the players during playback to understand to which extent they can avoid a playback pause event in case of spurious network condition. (iv) We also studied the impact of the streaming techniques on the energy consumption on different smartphones using Wi-Fi, HSPA and LTE. (v) Finally, we proposed playback buffer configurations for ON-OFF mechanism, which can ensure significant energy savings, reduce data waste, and can tolerate bandwidth fluctuations to some moderate extent. ## 9 Conclusions We analyzed the performance of four video services in tolerating bandwidth fluctuation and the energy consumption of smartphones. Based on he measurements with the latest smartphones, we identified five different streaming techniques. 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Spatscheck, Characterizing radio resource allocation for 3g networks, in: Proceedings of IMC 2010, ACM, New York, NY, USA, 2010, pp. 137–150. * [22] Y. Liu, L. Guo, F. Li, S. Chen, An empirical evaluation of battery power consumption for streaming data transmission to mobile devices, in: Proceedings of the 19th ACM international conference on Multimedia, MM ’11, ACM, New York, NY, USA, 2011, pp. 473–482.	arxiv-papers	2013-11-18T10:20:25	2024-09-04T02:49:53.803991	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Mohammad Ashraful Hoque, Matti Siekkinen, Jukka K. Nurminen, Mika\n Aalto, Sasu Tarkoma", "submitter": "Mohammad Ashraful Hoque Mohammad Ashraful Hoque", "url": "https://arxiv.org/abs/1311.4317" }
1311.4351	# Production of exotic isotopes in complete fusion reactions with radioactive beams V.V. Sargsyan1,2, A.S. Zubov1, G.G. Adamian1, N.V. Antonenko1, and S. Heinz3 1Joint Institute for Nuclear Research, 141980 Dubna, Russia 2International Center for Advanced Studies, Yerevan State University, 0025 Yerevan, Armenia 3GSI Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany ###### Abstract The isotopic dependence of the complete fusion (capture) cross section is analyzed in the reactions 130,132,134,136,138,140,142,144,146,148,150Xe+48Ca with stable and radioactive beams. It is shown for the first time that the very neutron-rich nuclei 186-191W can be reached with relatively large cross sections by complete fusion reactions with radioactive ion beams at incident energies near the Coulomb barrier. A comparison between the complete fusion and fragmentation reactions for the production of neutron-rich W and neutron- deficient Rn isotopes is performed. ###### pacs: 25.70.Hi, 24.10.-i, 24.60.-k Key words: Complete fusion reactions; Neutron-rich and neutron-deficient nuclei; Radioactive beams; Sub-barrier capture ## I Introduction The new generation of radioactive ion beam facilities will provide high intensity ($>10^{9}$ ions/s) exotic beams (for example, 88-94Kr, 126-132Sn, 138-144Xe or 119-132Cs). One of the most interesting areas of research with radioactive beams will be the study of the complete fusion process Love where the fusion experiments with exotic beams can be performed to synthesize and study new isotopes of existing elements. The central issue is whether the capture and fusion cross sections will be enhanced due to the large deformation of the neutron-rich or neutron-deficient projectile-nucleus. However, one should bear in mind the smaller intensity of these beams in comparison with the intensity of stable beams. Our aim is to find the global trend in the production cross section of exotic nuclei as a function of the charge (mass) number of the projectile in complete fusion reactions. Based on this trend one can find a consensus between the cross sections resulting from a certain beam and the intensity of this beam. The goal of the present paper is to compare the fusion of stable 130,132,134,136Xe and radioactive 138,140,142,144,146,148,150Xe beams with the same target, 48Ca, in order to study the effects of the neutron excess and neutron transfer on the fusion process. The target 48Ca is ideal for this purpose since this nucleus has the largest possible neutron excess and the systems 138,140,142,144,146,148,150Xe+48Ca have positive neutron transfer $Q$-values while all the corresponding reactions 130,132,134,136Xe+48Ca display negative $Q$-values. In the present paper we demonstrate for the first time the possibilities for producing neutron-rich isotopes of 186-191W in the complete fusion reactions 146,148Xe+48Ca with rather large cross sections. The nucleus 190W was the heaviest isotope which has been synthesized in ($n$,$n2p$) and ($p$,$3p$) reactions NPN . In these experiments the chemical extraction of 190W was possible after long irradiation. Another method to produce the neutron-rich nuclei is fragmentation reactions Ben ; Pod . Cross- sections smaller than 0.4$\mu$b were measured for the isotopes 190-192W in cold fragmentation of 950 MeV/nucleon 197Au beams on Be targets Ben . However, the production cross section decreases strongly with increasing neutron number. The most neutron-rich W isotopes, up to 197W, were observed in projectile fragmentation of 238U at 1 GeV/nucleon on Be targets at the Fragment Separator (FRS) at GSI Kurc . Here, cross-sections smaller than 5 nb were measured for W isotopes with mass numbers A $\geq$ 190 where the cross- section decreases by approximately one order of magnitude for every two neutrons more in the residual nucleus. In the present paper we also compare the complete fusion reactions 146Xe+48Ca with fragmentation reactions leading both to the production of neutron-rich W isotopes. Additionally, we will compare the complete fusion reactions 123Cs+69Ga which lead to neutron- deficient Rn isotopes with the respective yields from the fragmentation reactions. ## II Model Because the capture cross section is equal to the fusion cross section for the reactions AXe+48Ca treated in the present paper, the quantum diffusion approach EPJSub ; EPJSub1 for the capture is applied to study the complete fusion. With this approach many heavy-ion capture reactions at energies above and well below the Coulomb barrier have been successfully described EPJSub ; EPJSub1 ; PRCPOP . Since the details of our theoretical treatment were already published in Refs. EPJSub ; EPJSub1 ; PRCPOP , the model will be only shortly described. In the quantum diffusion approach EPJSub ; EPJSub1 the collisions of nuclei are described with a single relevant collective variable: the relative distance between the colliding nuclei. This approach takes into consideration the fluctuation and dissipation effects in collisions of heavy ions which model the coupling with various channels (for example, coupling of the relative motion with low-lying collective modes such as dynamical quadrupole and octupole modes of the target and projectile Ayik333 ). We have to mention that many quantum-mechanical and non-Markovian effects accompanying the passage through the Coulomb barrier are taken into consideration in our formalism EPJSub ; EPJSub1 ; PRCPOP . The diffusion models, which include the quantum statistical effects, were also proposed in Refs. Hofman . The nuclear deformation effects are taken into account through the dependence of the nucleus-nucleus potential on the deformations and mutual orientations of the colliding nuclei. To calculate the nucleus-nucleus interaction potential $V(R)$, we use the procedure presented in Ref. EPJSub1 . For the nuclear part of the nucleus-nucleus potential, the double-folding formalism with a Skyrme- type density-dependent effective nucleon-nucleon interaction is used Adamian96 . The nucleon densities of the projectile and target nuclei are specified in the form of the Woods-Saxon parameterization, where the nuclear radius parameter is $r_{0}=1.15$ fm and the diffuseness parameter takes the values $a=0.55$ fm for all nuclei. The absolute values of the quadrupole deformation parameters $\beta_{2}$ of deformed nuclei were taken from Refs. Ram and MN for the known and unknown nuclei, respectively. For the magic 48Ca and semimagic 136Xe nuclei in the ground state, we set $\beta_{2}=0$ and $\beta_{2}=0.05$, respectively. The capture cross section is the sum of the partial capture cross sections EPJSub ; EPJSub1 $\displaystyle\sigma_{cap}(E_{\rm c.m.})$ $\displaystyle=$ $\displaystyle\sum_{J}\sigma_{\rm cap}(E_{\rm c.m.},J)=$ (1) $\displaystyle=$ $\displaystyle\pi\lambdabar^{2}\sum_{J}(2J+1)\int_{0}^{\pi/2}d\theta_{1}\sin\theta_{1}\int_{0}^{\pi/2}d\theta_{2}\sin\theta_{2}P_{\rm cap}(E_{\rm c.m.},J,\theta_{1},\theta_{2}),$ where $\lambdabar^{2}=\hbar^{2}/(2\mu E_{\rm c.m.})$ is the reduced de Broglie wavelength, $\mu=m_{0}A_{1}A_{2}/(A_{1}+A_{2})$ is the reduced mass ($m_{0}$ is the nucleon mass), and the summation is over the possible values of the angular momentum $J$ at a given bombarding energy $E_{\rm c.m.}$. Knowing the potential of the interacting nuclei for each orientation with the angles $\theta_{i}(i=1,2)$, one can obtain the partial capture probability $P_{\rm cap}$ which is defined by the probability to penetrate the potential barrier in the relative distance coordinate $R$ at a given $J$. The value of $P_{\rm cap}$ is obtained by integrating the propagator $G$ from the initial state $(R_{0},P_{0})$ at time $t=0$ to the final state $(R,P)$ at time $t$ ($P$ is the momentum): $\displaystyle P_{\rm cap}$ $\displaystyle=$ $\displaystyle\lim_{t\to\infty}\int_{-\infty}^{r_{\rm in}}dR\int_{-\infty}^{\infty}dP\ G(R,P,t\|R_{0},P_{0},0)$ (2) $\displaystyle=$ $\displaystyle\lim_{t\to\infty}\frac{1}{2}{\rm erfc}\left[\frac{-r_{\rm in}+\overline{R(t)}}{{\sqrt{\Sigma_{RR}(t)}}}\right].$ Here, $r_{\rm in}$ is an internal turning point. The second line in (2) is obtained by using the propagator $G=\pi^{-1}\|\det{\bf\Sigma}^{-1}\|^{1/2}\exp(-{\bf q}^{T}{\bf\Sigma}^{-1}{\bm{q}})$ (${\bf q}^{T}=[q_{R},q_{P}]$, $q_{R}(t)=R-\overline{R(t)}$, $q_{P}(t)=P-\overline{P(t)}$, $\overline{R(t=0)}=R_{0}$, $\overline{P(t=0)}=P_{0}$, $\Sigma_{kk^{\prime}}(t)=2\overline{q_{k}(t)q_{k^{\prime}}(t)}$, $\Sigma_{kk^{\prime}}(t=0)=0$, $k,k^{\prime}=R,P$) calculated for an inverted oscillator which approximates the nucleus-nucleus potential $V$ in the variable $R$ as follows. At given $E_{\rm c.m.}$ and $J$, the classical action is calculated for the realistic nucleus-nucleus potential. Then the realistic nucleus-nucleus potential is replaced by an inverted oscillator which has the same barrier height and classical action. So, the frequency $\omega(E_{\rm c.m.},J)$ of this oscillator is set to obtain an equality of the classical actions in the approximated and realistic potentials. The action is calculated in the WKB approximation which is the accurate at the sub-barrier energies. Usually in the literature the parabolic approximation with $E_{\rm c.m.}$-independent $\omega$ is employed which is not accurate at the deep sub- barrier energies. Our approximation is well justified for the reactions and energy range considered here EPJSub ; EPJSub1 . We assume that the sub-barrier capture mainly depends on the two-neutron transfer with positive $Q$-value. Our assumption is that, just before the projectile is captured by the target-nucleus (just before the crossing of the Coulomb barrier), the transfer occurs and leads to the population of the first excited collective state in the recipient nucleus SSzilner . So, the motion to the $N/Z$ equilibrium starts in the system before the capture because it is energetically favorable in the dinuclear system in the vicinity of the Coulomb barrier. For the reactions under consideration, the average change of mass asymmetry is connected to the two-neutron transfer. Since after the transfer the mass numbers, the isotopic composition and the deformation parameters of the interacting nuclei, and, correspondingly, the height $V_{b}=V(R_{b})$ and shape of the Coulomb barrier are changed, one can expect an enhancement or suppression of the capture. If after the neutron transfer the deformations of the interacting nuclei increase (decrease), the capture probability increases (decreases). When the isotopic dependence of the nucleus-nucleus potential is weak and after the transfer the deformations of the interacting nuclei do not change, there is no effect of the neutron transfer on the capture. In comparison with Ref. Dasso , we assume that the negative transfer $Q-$values do not play a visible role in the capture process. Our scenario was verified in the description of many reactions EPJSub1 . The primary neutron-rich products of the complete fusion reactions AXe+48Ca of interest are excited and transformed into the secondary products with a smaller number of neutrons. Since neutron emission is the dominant deexcitation channel in the neutron-rich isotopes of interest, the production cross sections of the secondary nuclei are the same as those of the corresponding primary nuclei. This seems to be evident without special statistical treatment. The calculations are performed by employing the predicted values of the mass excesses and the neutron separation energies $S_{n}(Z,N)$ for unknown nuclei from the finite range liquid drop model MN . ## III Results of the calculations ### III.1 Complete fusion reactions AXe+48Ca To analyze the isotopic trend of the fusion cross section, it is useful to use the so called universal fusion function (UFF) representation GomesUFF . The advantage of this representation appears clearly when one wants to compare fusion cross sections for systems with different Coulomb barrier heights and positions. In the reactions where the capture and fusion cross sections coincide, the elimination of the influence of the nucleus-nucleus potential on the fusion cross section with the UFF representation allows us to conclude about the role of deformation of the colliding nuclei and the nucleon transfer between interacting nuclei in the capture and fusion. Figure 1: (Color online) Calculated dependencies of $F(x)=\frac{2E_{\rm c.m.}}{\hbar\omega_{b}R_{b}^{2}}\sigma$ on $x=\frac{E_{\rm c.m.}-V_{b}}{\hbar\omega_{b}}$ for the indicated reactions. Figure 2: Calculated dependence of fusion cross section $\sigma$ on $A$ for the reactions AXe+48Ca at fixed bombarding energies $E_{\rm c.m.}=V_{b}-5$ MeV (triangles), $V_{b}$ (stars), $V_{b}+10$ MeV (circles). In Ref. GomesUFF the reduction procedure consists of the following transformations: $E_{\rm c.m.}\rightarrow x=\frac{E_{\rm c.m.}-V_{b}}{\hbar\omega_{b}},\qquad\sigma\rightarrow F(x)=\frac{2E_{\rm c.m.}}{\hbar\omega_{b}R_{b}^{2}}\sigma.$ The frequency $\omega_{b}=\sqrt{\|V^{{}^{\prime\prime}}(R_{b})\|/\mu}$ is related with the second derivative $V^{{}^{\prime\prime}}(R_{b})$ of the nucleus-nucleus potential $V(R)$ at the barrier radius $R_{b}$ and the reduced mass parameter $\mu$. With these replacements one can compare the cross sections for different reactions. Figure 3: The expected evaporation residue cross sections $\sigma_{ER}$ for the indicated neutron-rich isotopes 186-189W produced in the 146Xe+48Ca reaction. The vertical dashed lines show the range of energies for the production of given isotope. Figure 4: The expected evaporation residue cross sections $\sigma_{ER}$ for the indicated neutron-rich isotopes 188-191W produced in the 148Xe+48Ca reaction. The vertical dashed lines show the range of energies for the production of given isotope. In Fig. 1, one can see the comparison of the calculated functions $F(x)$ for the reactions 130,132,134,136,138,140,142,144Xe+48Ca with stable and radioactive beams. As expected, at sub-barrier energies the enhancement of the complete fusion (capture) cross section is larger in the case of reactions with strongly quadrupole deformed projectile-nuclei and after neutron transfer. The quadrupole deformation parameter $\beta_{2}$ of the projectile nucleus increases with changing mass number from $A$=136 to $A$=130 or to $A$=144. For the reaction 136Xe+48Ca with spherical target and projectile and without neutron transfer the cross section is the smallest one at $x<0$. The sub-barrier cross sections for the reactions 138,140,142,144,146,148,150Xe+48Ca with neutron transfer (positive $Q$-values) are larger than those for the reactions 130,132,134,136Xe+48Ca, where the neutron transfer is suppressed (negative $Q$-values). Since after two-neutron transfer the mass numbers and the deformation parameters of the interacting nuclei are changed and the height of the Coulomb barrier decreases, one can expect an enhancement of the capture. For example, after the neutron transfer in the reaction 144Xe($\beta_{2}=0.18$)+48Ca($\beta_{2}=0$)$\to^{142}$Xe($\beta_{2}=0.15$)+50Ca($\beta_{2}=0.25$), the deformation of the target-nucleus increases and the mass asymmetry of the system decreases, and, thus, the value of the Coulomb barrier decreases and the capture cross section becomes larger (Fig. 1). We observe the same behavior in the reactions with the projectiles 138,140,142,146,148,150Xe. The complete fusion (capture) cross sections for the reactions 130,132,134,136,138,140,142,144,146,148,150Xe+48Ca at different bombarding energies are presented in Fig. 2. The behaviour of the curves in Fig. 2 is determined by the quadrupole deformation and neutron transfer effects. The isotopic dependency is rather weak at energies above the corresponding Coulomb barriers. At sub-barrier energies the fusion cross section decreases by about one order of magnitude with increasing mass number $A$ of the projectile from $A=130$ up to $A=136$ ($N=82$). For $A>136$ a steep increase can be observed for beam energies of 5 MeV below the corresponding Coulomb barriers. At energies near the Coulomb barrier the cross section changes in a similar way but the curve shows a much flatter slope. Figure 5: The expected evaporation residue cross sections $\sigma_{xn}$ for the indicated neutron-deficient isotopes of Rn produced in the $xn$-channels ($x$=2-4) of the 125Cs+69Ga reaction. Figure 6: The same as in Fig. 5, but for the 123Cs+69Ga reaction. In Figs. 3 and 4 we present the possibilities for future experiments to produce the neutron-rich isotopes 186-191W in complete fusion reactions of 146,148Xe+48Ca with radioactive beams. The production cross sections of the neutron-rich 190,191W isotopes, for example, are between the 10 $\mu$b and 100 mb levels meaning that they can be observed with rather low beam intensities and with the present experimental techniques. The calculated cross sections are more than two orders of magnitude larger than in fragmentation reactions Benlliure . Note also, that when the neutron number approaches the drip-line the production cross section in complete fusion decreases not so fast as in fragmentation reactions. ### III.2 Comparison between complete fusion and fragmentation reactions The availability of heavy radioactive beams at Coulomb barrier energies at future facilities like FAIR, HIE-ISOLDE or SPIRAL-II will enable the experimental utilization of the above discussed effects for fusion reactions. Another competing method to produce heavy exotic isotopes is projectile fragmentation at relativistic energies which is for example used at the Fragment Separator (FRS) at GSI. In the following, we give some comparative considerations on both methods since, depending on the region of the nuclear chart, fragmentation can lead to high yields of exotic nuclei. As an example, we consider the isotope 189W. Cross-sections of up to about 2 mb are predicted for its production in the complete fusion reactions of 146Xe+48Ca at E${}_{cm}=110$ MeV. Cross-sections on the same order are also measured in the fragmentation reactions leading to yields of $10^{4}$ ions/s. At the future Super-FRS facility even yields of $2\times 10^{6}$ ions/s are predicted. In order to obtain at least the same yields of $10^{4}$ ions/s in fusion reactions, 146Xe beams with intensities of at least $10^{13}$ ions/s are required. The largest intensities for neutron-rich Xe beams are expected at SPIRAL-II where $10^{5}$ of 146Xe projectiles per second are predicted which is, however, still eight orders of magnitude less than needed for an efficient application of fusion reactions to reach 189W. As an other example, we discuss in the following the synthesis of neutron deficient Rn ($Z=86$) isotopes in the complete fusion reactions. Figures 5 and 6 show the calculated excitation functions for fusion reactions of 123,125Cs beams with 69Ga target. The survival probabilities of the excited compound nuclei in the neutron evaporation channels $xn$ ($x=2-4$) are calculated by employing the modified statistical code GROGIF GROGIF with the same parameters as in Ref. AZ . The capture cross sections and fusion probabilities are calculated with the quantum diffusion approach EPJSub ; EPJSub1 and the dinuclear system fusion model AZ , respectively. Radioactive Cs beams are already now available with high intensities for a broad variety of isotopes and are therefore favourable projectiles. At REX-ISOLDE for example, the isotopes 122-129Cs are provided with intensities around $10^{10}$ ions/s and for the future HIE-ISOLDE facility ten times higher intensities are expected at beam energies of $\geq$ 5.5 MeV/nucleon. A comparison of the predicted yields for neutron deficient Rn isotopes at the SuperFRS facility with the expected yields from fusion evaporation reactions with 123Cs beams at intensities of $10^{10}$ ions/s leads to the conclusion that the complete fusion is not superior to fragmentation for ARn isotopes with $188\leq A\leq 190$. For these mass numbers at least 2-7 times lower yields can be obtained in the fusion reactions with the presently available beam intensities. ## IV Summary Because of deformation and neutron transfer effects, a strong dependence of the sub-barrier complete fusion (capture) cross section on the isospin was found for the reactions 130,132,134,136,138,140,142,144,146,148,150Xe+48Ca. At fixed bombarding energy, the cross section increases with changing mass number of the projectile-nucleus from $A$=136 to $A$=130 or to $A$=150. The 136Xe+48Ca reaction with magic and semimagic nuclei has the smallest cross section. The complete fusion (capture) cross sections for the reactions 130,132,134,136Xe+48Ca without neutron transfer are smaller than those for the reactions 138,140,142,144,146,148,150Xe+48Ca with neutron transfer. We demonstrated the possibilities for producing neutron-rich isotopes of 186-191W with relatively large cross sections for future experiments in the complete fusion reactions 146,148Xe+48Ca with radioactive beams. However, we found that for the production of neutron-rich W the fragmentation reactions are more preferable than the complete fusion reactions. Even if we consider here the formation of neutron-rich W isotopes as an example, our findings have general validity and are not restricted to specific isotopes. Exotic nuclei with large deformations which could be used as projectiles can equally be found in wide regions on the neutron-rich as well as on the neutron-deficient side of the nuclear chart. 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1311.4353	# Neutron pair transfer in sub-barrier capture process V.V.Sargsyan1,2, G. Scamps3, G.G.Adamian1, N.V.Antonenko1, and D. Lacroix4 1Joint Institute for Nuclear Research, 141980 Dubna, Russia 2International Center for Advanced Studies, Yerevan State University, M. Manougian 1, 0025, Yerevan, Armenia 3GANIL, 14076 Caen Cedex, France 4Institut de Physique Nucléaire, IN2P3-CNRS, Université Paris-Sud, F-91406 Orsay Cedex, France ###### Abstract The sub-barrier capture reactions following the neutron pair transfer are proposed to be used for the indirect study of neutron-neutron correlation in the surface region of nucleus. The strong effect of the dineutron-like clusters transfer stemming from the surface of magic and non-magic nuclei 18O, 48Ca, 64Ni, 94,96Mo, 100,102,104Ru, 104,106,108Pd, and 112,114,116,118,120,124,132Sn is demonstrated. The dominance of two-neutron transfer channel at the vicinity of the Coulomb barrier is further supported by time-dependent mean-field approaches. ###### pacs: 25.70.Jj, 24.10.-i, 24.60.-k Key words: sub-barrier capture, neutron pair transfer, pairing correlation, quantum diffusion approach, time-dependent mean-field approach ## I Introduction Two-neutron transfer reactions such as ($p$,$t$) or ($t$,$p$) have been used for many years in order to study the nucleon pairing correlations in the stable nuclei BohrNathan ; vonOertzen . The corresponding pair transfer modes are usually described in terms of pairing vibrations or pairing rotations Broglia ; Kart , which are associated with the pair correlation. It has been established that the two-neutron transfer amplitude is influenced by collective modes caused by the Cooper-pair superfluidity vonOertzen . In the superfluid nuclei 18O, 206,210Pb, and 114Sn, the Cooper pair with short range space correlation has been theoretically predicted past . The size of the Cooper pair is estimated to be comparable to the average inter-nucleon distance past . Recently, there is a renewal of interest on experimental nucleon pair, alpha cluster, and more generally multinucleon transfer channels at bombarding energies above and below the Coulomb barriers Corradi ; Corradi2 ; Simenel . The effect of correlations between nucleons on the nuclear breakup or decay mechanism has been studied both experimentally and theoretically Marques ; Kolata ; Ershov ; Lacroix0 ; Grigor ; Spyrou . Studies of pairing effects in both finite nuclei and nuclear matter have intensified interests in the recent years Fortunato ; Volya ; Dean ; Khan ; Saper1 ; Matsuo ; Grasso ; Broglia2 ; Lacroix ; Lacroix2 ; Lacroix3 ; Sambataro . Attention has been paid to the properties of the pair correlation in the neutron-rich nuclei with the neutron skin and the neutron halo Zhukov ; Mar ; Barranco ; Mat . The ($p$,$t$) reactions on light-mass neutron-rich nuclei such as 6,8He and 11Li point out the importance of the pair correlations in these typical halo or skin nuclei. The experimental signatures of a spatial two-neutron correlation or the di- neutron correlation between two weakly bound neutrons forming the halo in 6,8He and 11Li have been reported in Refs. Ter-Akopian ; DeYoung ; Nakamura ; Moeller ; Chatterjee . There exists also several studies demonstrating enhancement of the pair correlation in the nuclear surface and exterior regions of the neutron-rich nuclei Khan ; Saper1 ; Matsuo ; Grasso ; Lacroix ; Saper2 ; Hag05 . A possible link between the pair transfer and the surface enhancement of the pairing in medium and heavy neutron-rich nuclei has been suggested in Ref. Dobaczewski and more recently discussed in Matsuo ; Broglia2 ; Grasso ; Lacroix ; Gra13 It has been argued in Ref. Khan that the pair transfer can be a possible probe of different models of the pairing interaction. In literature Schuck , the origin of the small size of Cooper pair on the nuclear surface is still under discussions. It can be a consequence of the enhanced pairing correlations or of the finiteness of the single-particle wave functions. A strong spatial correlation between the nucleons gives rise to specific features like dineutron or alpha clustering formation and to the possibility of a contribution to the transfer from the simultaneous one-step pair transfer mechanism. By describing the capture (fusion) reactions at sub-barrier energies within the quantum diffusion approach, we want to demonstrate indirectly the strong dineutron spatial correlations in the surface region of stable nuclei. We will consider the capture reactions with the negative one- neutron transfer ($Q_{1n}<0$) and the positive two-neutron transfer ($Q_{2n}>0$) (before crossing the Coulomb barrier), where the one-step neutron pair transfer is expected to be dominant. The study of this process is one of the important points in the understanding of pairing correlations in nuclei. The distinction between two-step sequential and one-step cluster transfer is a great challenge, not only in nuclear physics but also in electron transfer between ions or atomic cluster collisions vonOertzen . Note that the capture (fusion) reaction following the neutron pair transfer is the indirect way of the study of pairing effects. ## II Model In the quantum diffusion approach EPJSub ; EPJSub1 ; EPJSub2 ; EPJSub3 the collisions of nuclei are treated in terms of a single collective variable: the relative distance between the colliding nuclei. The nuclear deformation effects are taken into consideration through the dependence of the nucleus- nucleus potential on the deformations and mutual orientations of the colliding nuclei. Our approach takes into account the fluctuation and dissipation effects in the collisions of heavy ions which model the coupling with various channels (for example, coupling of the relative motion with the non-collective single-particle excitations and low-lying collective modes such as dynamical quadrupole and octupole excitations of the target and projectile Ayik333 ). We have to mention that many quantum-mechanical and non-Markovian effects accompanying the passage through the potential barrier are considered in our formalism EPJSub ; our through the friction and diffusion. The two-neutron transfer with the positive $Q_{2n}$-value was taken into consideration in EPJSub ; EPJSub2 . Our assumption is that, just before the projectile is captured by the target-nucleus (i.e. just before the crossing of the Coulomb barrier), the two-neutron transfer occurs and can lead to the population of the first excited collective state in the recipient nucleus Corradi2 ; SSzilner (the donor nucleus remains in the ground state). So, the motion to the $N/Z$ equilibrium starts in the system before the capture because it is energetically favorable in the dinuclear system in the vicinity of the Coulomb barrier. For the reactions under consideration, the average change of mass asymmetry is connected to the two-neutron transfer ($2n$-transfer). Since after the transfer the mass numbers, the isotopic composition and the deformation parameters of the interacting nuclei, and, correspondingly, the height $V_{b}=V(R_{b})$ [$R=R_{b}$ is the position of the Coulomb barrier] and shape of the Coulomb barrier change, one can expect an enhancement or suppression of the capture. If after the neutron transfer the deformations of interacting nuclei increase (decrease), the capture probability increases (decreases). When the isotopic dependence of the nucleus-nucleus potential is weak and after the transfer the deformations of interacting nuclei do not change, there is no effect of the neutron transfer on the capture. In comparison with Ref. Dasso , we assume that the negative transfer $Q-$values do not play visible role in the capture process. Our scenario was verified in the description of many reactions EPJSub2 . The calculated results for all reactions are obtained with the same set of parameters as in Refs. EPJSub1 ; EPJSub2 and are rather insensitive to the reasonable variation of them. One should note that the diffusion models, which include quantum statistical effects, were also treated in Refs. Hofman ; Ayik ; Hupin . The capture cross section is the sum of the partial capture cross sections EPJSub ; EPJSub1 ; EPJSub2 $\displaystyle\sigma_{cap}(E_{\rm c.m.})$ $\displaystyle=$ $\displaystyle\sum_{J}\sigma_{\rm cap}(E_{\rm c.m.},J)=$ (1) $\displaystyle=$ $\displaystyle\pi\lambdabar^{2}\sum_{J}(2J+1)\int_{0}^{\pi/2}d\theta_{1}\sin(\theta_{1})\int_{0}^{\pi/2}d\theta_{2}\sin(\theta_{2})P_{\rm cap}(E_{\rm c.m.},J,\theta_{1},\theta_{2}),$ where $\lambdabar^{2}=\hbar^{2}/(2\mu E_{\rm c.m.})$ is the reduced de Broglie wavelength, $\mu=m_{0}A_{1}A_{2}/(A_{1}+A_{2})$ is the reduced mass ($m_{0}$ is the nucleon mass), and the summation is over the possible values of the angular momentum $J$ at a given bombarding energy $E_{\rm c.m.}$. Knowing the potential of the interacting nuclei for each orientation with the angles $\theta_{i}(i=1,2)$, one can obtain the partial capture probability $P_{\rm cap}$ which is defined by the probability to penetrate the potential barrier in the relative distance coordinate $R$ at a given $J$. The value of $P_{\rm cap}$ is obtained by integrating the propagator $G$ from the initial state $(R_{0},P_{0})$ at time $t=0$ to the final state $(R,P)$ at time $t$ ($P$ is the momentum): $\displaystyle P_{\rm cap}$ $\displaystyle=$ $\displaystyle\lim_{t\to\infty}\int_{-\infty}^{r_{\rm in}}dR\int_{-\infty}^{\infty}dP\ G(R,P,t\|R_{0},P_{0},0)$ (2) $\displaystyle=$ $\displaystyle\lim_{t\to\infty}\frac{1}{2}{\rm erfc}\left[\frac{-r_{\rm in}+\overline{R(t)}}{{\sqrt{\Sigma_{RR}(t)}}}\right].$ Here, $r_{\rm in}$ is an internal turning point. The second line in (2) is obtained by using the propagator $G=\pi^{-1}\|\det{\bf\Sigma}^{-1}\|^{1/2}\exp(-{\bf q}^{T}{\bf\Sigma}^{-1}{\bm{q}})$ (${\bf q}^{T}=[q_{R},q_{P}]$, $q_{R}(t)=R-\overline{R(t)}$, $q_{P}(t)=P-\overline{P(t)}$, $\overline{R(t=0)}=R_{0}$, $\overline{P(t=0)}=P_{0}$, $\Sigma_{kk^{\prime}}(t)=2\overline{q_{k}(t)q_{k^{\prime}}(t)}$, $\Sigma_{kk^{\prime}}(t=0)=0$, $k,k^{\prime}=R,P$) calculated for an inverted oscillator which approximates the nucleus-nucleus potential $V$ in the variable $R$. At given $E_{\rm c.m.}$ and $J$, the classical action is calculated for the realistic nucleus-nucleus potential. Then the realistic nucleus-nucleus potential is replaced by an inverted oscillator which has the same barrier height and classical action. So, the frequency $\omega(E_{\rm c.m.},J)$ of this oscillator is set to obtain an equality of the classical actions in the approximated and realistic potentials. The action is calculated in the WKB approximation which is the accurate at the sub-barrier energies. Usually in the literature the parabolic approximation with $E_{\rm c.m.}$-independent $\omega$ is employed which is not accurate at the deep sub- barrier energies. Our approximation is well justified for the reactions and energy range considered here EPJSub ; EPJSub1 ; EPJSub2 . Finally, one can find the expression for the capture probability: $\displaystyle P_{\rm cap}$ $\displaystyle=$ $\displaystyle\frac{1}{2}{\rm erfc}\left[\left(\frac{\pi s_{1}(\gamma- s_{1})}{2\hbar\mu(\omega_{0}^{2}-s_{1}^{2})}\right)^{1/2}\frac{\mu\omega_{0}^{2}R_{0}/s_{1}+P_{0}}{\left[\gamma\ln(\gamma/s_{1})\right]^{1/2}}\right],$ (3) where $\gamma$ is the internal-excitation width, $\omega_{0}^{2}=\omega^{2}\\{1-\hbar\tilde{\lambda}\gamma/[\mu(s_{1}+\gamma)(s_{2}+\gamma)]\\}$ is the renormalized frequency in the Markovian limit, the value of $\tilde{\lambda}$ is related to the strength of linear coupling in the coordinates between collective and internal subsystems. The non-Markovian effects appear in the calculations through $\gamma$. Here, $\hbar\gamma$=15 MeV. The $s_{i}$ are the real roots ($s_{1}\geq 0>s_{2}\geq s_{3}$) of the following equation EPJSub ; EPJSub1 ; EPJSub2 : $\displaystyle(s+\gamma)(s^{2}-\omega_{0}^{2})+\hbar\tilde{\lambda}\gamma s/\mu=0.$ (4) As shown in Refs. EPJSub ; EPJSub1 , the nuclear forces start to play a role at $R_{int}=R_{b}+1.1$ fm where the nucleon density of the colliding nuclei approximately reaches 10% of the saturation density. If the value of $r_{\rm ex}$ corresponding to the external turning point is larger than the interaction radius $R_{int}$, we take $R_{0}=r_{\rm ex}$ and $P_{0}=0$ in Eq. (3). For $r_{\rm ex}<R_{int}$, it is natural to start our treatment with $R_{0}=R_{int}$ and $P_{0}$ defined by the kinetic energy at $R=R_{0}$. In this case the friction hinders the classical motion to proceed towards smaller values of $R$. If $P_{0}=0$ at $R_{0}>R_{int}$, the friction almost does not play a role in the transition through the barrier. Thus, two regimes of interaction at sub-barrier energies differ by the action of the nuclear forces and the role of friction at $R=r_{\rm ex}$. To calculate the nucleus-nucleus interaction potential $V(R)$, we use the procedure described in Refs. EPJSub ; EPJSub1 ; EPJSub2 ; AAShobzor . For the nuclear part of the nucleus-nucleus potential, the double-folding formalism with the Skyrme-type density-dependent effective nucleon-nucleon interaction is used. The parameters of the potential were adjusted to describe the experimental data at energies above the Coulomb barrier corresponding to spherical nuclei. The absolute values of the quadrupole deformation parameters $\beta_{2}$ of even-even deformed nuclei and of the first excited collective states of nuclei were taken from Ref. Ram . For the nuclei deformed in the ground state, the $\beta_{2}$ in the first excited collective state is similar to the $\beta_{2}$ in the ground state. For the double magic nuclei, we take $\beta_{2}=0$ in the ground state. For the rest of nuclei, we used the ground- state quadrupole deformation parameters extracted in Ref. EPJSub2 from a comparison of the calculated capture cross sections with the existing experimental data. ## III Influence of neutron pair transfer on capture The choice of the projectile-target combination is crucial in the understanding of pair transfer phenomenon in the capture process. In the capture reactions with $Q_{1n}<0$ and $Q_{2n}>0$, the two-step sequential transfer is almost closed before capture. So, choosing properly the reaction combination, one can reduce the successive transfer in the process. For the systems studied, one can make unambiguous statements regarding the neutron transfer process with a positive $Q_{2n}$ value when the interacting nuclei are double magic or semimagic nuclei. In this case one can disregard the strong nuclear deformation effects before the neutron transfer. Figure 1: The calculated (lines) and experimental (symbols) trotta40ca48ca ; Stefanini40ca116124sn capture cross sections vs $E_{\rm c.m.}$ for the reactions 40Ca+48Ca (a) and 40Ca+116,124Sn (b). The calculated capture cross sections without taking into account the neutron pair transfer are shown by dotted lines. Figure 2: The same as in Fig. 1, but for the reactions 58Ni+64Ni (a) and 64Ni+132Sn (b). The experimental data are from Refs. Beckerman58Ni5864Ni74Ge ; LiangNi64Sn132 . In Figs. 1 and 2 the calculated capture cross sections for the reactions 40Ca + 48Ca ($Q_{1n}=-1.6$ MeV, $Q_{2n}=2.6$ MeV), 40Ca + 116Sn ($Q_{1n}=-1.2$ MeV, $Q_{2n}=2.8$ MeV), 40Ca + 124Sn ($Q_{1n}=-0.1$ MeV, $Q_{2n}=5.4$ MeV), 58Ni + 64Ni ($Q_{1n}=-0.66$ MeV, $Q_{2n}=3.9$ MeV), and 64Ni + 132Sn ($Q_{1n}=-1.21$ MeV, $Q_{2n}=2.5$ MeV) are in a good agreement with the available experimental data trotta40ca48ca ; Stefanini40ca116124sn ; Beckerman58Ni5864Ni74Ge ; LiangNi64Sn132 . In all reactions $1n$-neutron transfer is closed ($Q_{1n}<0$) and $Q_{2n}$-values for the $2n$-transfer processes are positive. Thus, the $2n$-neutron transfer is more important for a good description of the experimental data than the $1n$-neutron transfer. The influence of the $2n$-neutron transfer on the capture cross section occurs due to the change of the isotopic composition and the deformations of the reaction partners. The $2n$-transfer indirectly influence the quadrupole deformation of the nuclei. When after the neutron transfer (just before the crossing of the Coulomb barrier) in the reactions 40Ca($\beta_{2}=0$)+48Ca($\beta_{2}=0$)$\to^{42}$Ca($\beta_{2}=0.247$)+46Ca($\beta_{2}=0$), 40Ca($\beta_{2}=0$)+116Sn($\beta_{2}=0.112$)$\to^{42}$Ca($\beta_{2}=0.247$)+114Sn($\beta_{2}=0.121$), 40Ca($\beta_{2}=0$)+124Sn($\beta_{2}=0.095$)$\to^{42}$Ca($\beta_{2}=0.247$)+122Sn($\beta_{2}=0.104$), 58Ni($\beta_{2}=0.05$)+64Ni($\beta_{2}=0.087$)$\to^{60}$Ni($\beta_{2}=0.207$)+62Ni($\beta_{2}=0.087$), and 64Ni($\beta_{2}=0.087$)+132Sn($\beta_{2}=0$)$\to^{66}$Ni($\beta_{2}=0.158$)+130Sn($\beta_{2}=0$) the deformations of nuclei increase, the values of the corresponding Coulomb barriers decrease. As a result, the two-neutron transfer enhances the capture process in these reactions at the sub-barrier energies. The enhancement becomes stronger with decreasing bombarding energy (Figs. 1 and 2). Previously, the importance of the neutron pair transfer in the capture (fusion) process was stressed in Refs. Dasso ; Pengo ; Stef . Figure 3: The same as in Fig. 1, but for the reactions 32S+106Pd (a) and 32S+104Pd (b). The experimental data are from Ref. Pengo . Figure 4: The same as in Fig. 1, but for the reactions 32S+104Ru (a) and 32S+102Ru (b). The experimental data are from Ref. Pengo . Since $Q_{1n}<0$ in these reactions, the enhancement arises not from the coherent successive transfer of two single neutrons, but from the direct transfer of one spatially correlated pair (the simultaneous transfer of two neutrons). Our results show that the capture (fusion) cross section of the reactions under consideration can be described by assuming the preformed dineutron-like clusters in the ground state of the nuclei 48Ca, 64Ni, and 116,124,132Sn. Note that the strong spatial two-neutron correlation and the strong surface enhancement of the neutron pairing in the cases of a slab, a semi-infinite nuclear matter, and the finite superfluid nuclei are well known and it is well established that nuclear superfluidity of the Cooper pairs is mainly a surface effect past ; Dean ; Matsuo . Figure 5: The calculated capture cross section vs $E_{\rm c.m.}-V_{b}$ for the reactions 58Ni+62Ni (a) 40Ca + 64Ni (b). The results with and without taking into consideration the neutron pair transfer are shown by solid and dotted lines, respectively. Figure 6: (Color online) The calculated one- (symbols connecting by solid lines) and two-neutron (symbols connecting by dotted lines) transfer probabilities vs $B_{0}-E_{\rm c.m.}$ for the reactions 40Ca+116Sn (circles), 40Ca+124Sn (triangles), and 40Ca+130Sn (squares). Our calculations also show that the neutron pair transfer has to be taken into consideration in the description of the reactions 58Ni+112,114,116,118,120Sn, 32S+94,96Mo,100,102,104Ru,104,106,108Pd, and 18O+112,118,124Sn (for example, see Figs. 3 and 4) EPJSub2 . In Figs. 3 and 4 one can see that after neutron pair transfer in the reactions 32S($\beta_{2}=0.312$)+106Pd($\beta_{2}=0.229$)$\to^{34}$S($\beta_{2}=0.252$)+104Pd($\beta_{2}=0.209$), 32S($\beta_{2}=0.312$)+104Pd($\beta_{2}=0.209$)$\to^{34}$S($\beta_{2}=0.252$)+102Pd($\beta_{2}=0.196$) or 32S($\beta_{2}=0.312$)+104Ru($\beta_{2}=0.271$)$\to^{34}$S($\beta_{2}=0.252$)+102Ru($\beta_{2}=0.24$), 32S($\beta_{2}=0.312$)+102Ru($\beta_{2}=0.24$)$\to^{34}$S($\beta_{2}=0.252$)+100Ru($\beta_{2}=0.215$) the deformations of the nuclei decrease and the values of the corresponding Coulomb barriers increase and, respectively, the capture cross sections decrease at the sub-barrier energies. These results indicate again the strong spatial two-neutron correlations in the surface of the stable nuclei 18O, 94,96Mo, 100,102,104Ru, 104,106,108Pd, and 112,114,116,118,120Sn. Since the dominance of the dineutron-like clusters is found in the surface of double magic, semimagic, and nonmagic nuclei, one can conclude that this effect is general for all stable and radioactive nuclei. Figure 7: (Color online) Focus on the calculated one- (black filled triangles), two-(blue filled squares) and three-(red filled circles) neutron transfer probabilities as a function of $B_{0}-E_{\rm c.m.}$ for the reactions 40Ca+116Sn (a), 40Ca+124Sn (b), and 40Ca+130Sn (c). In each case, the gray area indicates the energy region where the two-particle channel dominates. One can make unambiguous statements regarding the neutron pair transfer process in the reactions 40Ca + 62Ni ($Q_{1n}=-2.23$ MeV, $Q_{2n}=1.43$ MeV), 40Ca + 64Ni ($Q_{1n}=-1.29$ MeV, $Q_{2n}=3.45$ MeV), 40Ca + 114Sn ($Q_{1n}=-1.94$ MeV, $Q_{2n}=1.8$ MeV), 40Ca + 118Sn ($Q_{1n}=-1.55$ MeV, $Q_{2n}=3.56$ MeV), 40Ca + 120Sn ($Q_{1n}=-0.75$ MeV, $Q_{2n}=4.25$ MeV), 40Ca + 122Sn ($Q_{1n}=-0.45$ MeV, $Q_{2n}=4.86$ MeV), 58Ni + 62Ni ($Q_{1n}=-1.6$ MeV, $Q_{2n}=1.94$ MeV), 60Ni + 64Ni ($Q_{1n}=-1.84$ MeV, $Q_{2n}=1.95$ MeV), 64Ni + 128Sn ($Q_{1n}=-1.8$ MeV, $Q_{2n}=1.6$ MeV), and 64Ni + 130Sn ($Q_{1n}=-1.52$ MeV, $Q_{2n}=2.1$ MeV). As seen in Fig. 5, there is a considerable difference between the sub-barrier capture cross sections with and without taking into consideration the neutron pair transfer in these reactions. After two-neutron transfer, the deformation of light nucleus strongly increases and the capture cross section enhances. The neutron pair transfer induces the effect of the quadrupole deformation in the light nucleus. The study of the capture reactions following the neutron transfer will provide a good test for the effects of the neutron pair transfer. ## IV Neutron pair transfer phenomenon in heavy-ion sub-barrier reactions The Time-Dependent Hartree-Fock (TDHF) plus BCS approach Simenelnew ; Scamps has been recently used Scamps to extract the one-, two-, three-neutrons transfer probabilities ($P_{1n}$, $P_{2n}$, $P_{3n}$) in heavy-ion scattering reactions. It was shown that, when the energy is well below the Coulomb barrier, the one-nucleon channel largely dominates. This is further illustrated here for the reactions 40Ca + 116,124,130Sn that have been discussed above and where the tin isotopes are superfluid. In Fig. 6, the one- and two-neutron transfer probabilities are displayed as functions of $B_{0}-E_{\rm c.m.}$ for the sub- and near-barrier binary collisions of 40Ca and tin isotopes. The Coulomb barrier (capture threshold energy) $B_{0}$ is deduced from the mean-field transport theory. This barrier are equal to $116.41\pm 0.07$ (116Sn), $114.69\pm 0.04$ (124Sn) and $113.92\pm 0.02$ (130Sn) MeV. It was found that the calculated $B_{0}$ are insensitive to the introduction of pairing and in a good agreement with the barriers extracted from the experimental data Scamps . Note that the presented calculation are shown for the mixed pairing interaction only. The use of other interaction (surface or volume) leads to similar conclusions. Figure 6 gives an interesting insight in the one- and two-neutron transfers. As seen, a strong enhancement of $P_{1n}$ and $P_{2n}$ occurs with increasing bombarding energy. Since the enhancement of $P_{2n}$ is stronger than that of $P_{1n}$, these probabilities become close to each other with decreasing $B_{0}-E_{\rm c.m.}$. This is indeed observed experimentally in Refs. Corradi ; Corradi2 ; Simenel where it was found that $P_{2n}$ grows faster than $P_{1n}$ with decreasing $B_{0}-E_{\rm c.m.}$ at energy relatively far below the Coulomb barrier. In Fig. 7, a closer look is made on the one-, two- and three-neutrons transfer channels at the vicinity of the Coulomb barrier for the different tin isotopes. In all cases, as the energy approaches the capture barrier energy, there exist an energy range where $P_{2n}>P_{1n}$ dominates (shaded area). We also note that the energy windows where the two-nucleon channel becomes dominant increases as the neutron nucleus become more exotic. This evidently supports our assumption about important role of the two-neutron transfer (compared to the one-neutron transfer) in the capture process, because in the TDHF calculation the scattering trajectory of two heavy ions at energy near the Coulomb barrier is close to the capture trajectory. Note that in the capture process the system trajectory crosses the barrier position $R=R_{b}$ at any energies. The results of our calculations predict that there is the crossing point of $P_{2n}$ and $P_{1n}$ at energy very close to the Coulomb barrier. Just before reaching $R_{b}$ the neutron-pair transfer becomes the dominant channel. Thus, our assumption about two-neutron transfer before the capture is correct. The transfer more than two neutrons mainly occurs at $R<R_{b}$, i.e., just after the capture. ## V Summary Within the quantum diffusion approach it turns out that the sub-barrier capture (fusion) reactions with $Q_{1n}<0$ and $Q_{2n}>0$ may help us understanding of the neutron pair transfer and of the pair correlation phenomenon on the surface of a nucleus. In these reactions the main contribution to transfer is due to the dineutron-like cluster component. In the capture process, the transfer of neutron pair before the crossing of the Coulomb barrier is a clear signature of the strong correlations between the transferred nucleons and the surface character of pairing interaction. Our results indicate the dominance of the dineutron structure (of the preformed dineutron-like clusters) in the surface of the stable and unstable nuclei 18O, 48Ca, 64Ni, 94,96Mo, 100,102,104Ru, 104,106,108Pd, and 112,114,116,118,120,124,132Sn. Measurements of sub-barrier capture cross sections in various reactions can be utilized to study the role of pairing correlations between the transferred nucleons. The information obtained from the sub-barrier capture (fusion) reactions is complementary to that obtained from the two-neutron transfer reactions such as ($p$,$t$) or ($t$,$p$) and the multinucleon transfer reactions. 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C 87, 014605 (2013).	arxiv-papers	2013-11-18T12:13:45	2024-09-04T02:49:53.824852	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "V.V.Sargsyan, G. Scamps, G.G.Adamian, N.V.Antonenko, and D. Lacroix", "submitter": "Vazgen Sargsyan Dr.", "url": "https://arxiv.org/abs/1311.4353" }
1311.4762	mtvDisplayMath magenta!43!cyan!10!white # Software Uncertainty in Integrated Environmental Modelling: the role of Semantics and Open Science Daniele de Rigo European Commission, Joint Research Centre, Institute for Environment and Sustainability Via E. Fermi 2749, I-21027 Ispra (VA), Italy Politecnico di Milano, Dipartimento di Elettronica, Informazione e Bioingegneria Via Ponzio 34/5, I-20133 Milano, Italy Copyright © 2013 Daniele de Rigo. This work is licensed under a Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0/). See: http://www.egu2013.eu/abstract_management/license_and_copyright.html This is the author’s version of the work. The definitive version has been published in the Vol. 15 of Geophysical Research Abstracts (ISSN 1607-7962) and presented at the European Geosciences Union (EGU) General Assembly 2013, Vienna, Austria, 07–12 April 2013 http://www.egu2013.eu/ Cite as: de Rigo, D., 2013. Software Uncertainty in Integrated Environmental Modelling: the role of Semantics and Open Science. Geophys Res Abstr 15, 13292+ Author’s version DOI: 10.6084/m9.figshare.155701 , arXiv: 1311.4762 Computational aspects increasingly shape environmental sciences [1]. Actually, transdisciplinary modelling of complex and uncertain environmental systems is challenging computational science (CS) and also the science-policy interface [2, 3, 4, 5, 6, 7]. Large spatial-scale problems falling within this category – i.e. wide-scale transdisciplinary modelling for environment (WSTMe) [8, 9, 10] – often deal with factors (a) for which deep-uncertainty [2, 11, 7, 12] may prevent usual statistical analysis of modelled quantities and need different ways for providing policy-making with science-based support. Here, practical recommendations are proposed for tempering a peculiar – not infrequently underestimated – source of uncertainty. Software errors in complex WSTMe may subtly affect the outcomes with possible consequences even on collective environmental decision-making. Semantic transparency in CS [2, 8, 10, 13, 14] and free software [15, 16] are discussed as possible mitigations (b). Software uncertainty, black-boxes and free software Integrated natural resources modelling and management (INRMM) [17] frequently exploits chains of nontrivial data-transformation models (D-TM), each of them affected by uncertainties and errors. Those D-TM chains may be packaged as monolithic specialized models, maybe only accessible as black-box executables (if accessible at all) [18]. For end- users, black-boxes merely transform inputs in the final outputs, relying on classical peer-reviewed publications for describing the internal mechanism. While software tautologically plays a vital role in CS, it is often neglected in favour of more theoretical aspects. (a) Complexity= { Transdisciplinary integration (e.g. systems of systems) Environmental system(s) heterogeneity (e.g. geospatial fragmentation) Data heterogeneity (formats, definitions, spatiotemporal density, …) Software complexity (algorithms, dependencies, languages, interfaces, …) Uncertainty= { Incomplete scientific knowledge (e.g. climate scenarios [19, 20, 21], tipping points [22, 23, 24], … ) Modelling assumptions and simplifications [25, 26, 27] Uncertainty of measured/derived data Software uncertainty Dynamicbehaviour= { Uncertainty propagation via: Propagation in the network of interconnected WSTMe components [2, 14, 28, 17, 29, 30, 31, 32, 33] Iterations within nonlinear optimization steps [5, 34, 35, 36, 37, 38, 39, 40] Data fusion, harmonization, integration [9, 41, 42, 43, 44] Steps for computing and aggregating criteria and indices [6, 11, 7, 45, 46, 47, 48] This paradox has been provocatively described as “the invisibility of software in published science. Almost all published papers required some coding, but almost none mention software, let alone include or link to source code” [49]. Recently, this primacy of theory over reality [50, 51, 52] has been challenged by new emerging hybrid approaches [53] and by the growing debate on open science and scientific knowledge freedom [2, 54, 55, 56, 57]. In particular, the role of free software has been underlined within the paradigm of reproducible research [18, 56, 57, 58]. In the spectrum of reproducibility, the free availability of the source code is emphasized [56] as the first step from non-reproducible research (only based on classic peer- reviewed publications) toward reproducibility. Y = f^( X ) = f( θ^ , X ) Theoretic D-TM whose algorithm is typically described in peer reviewed publications. The D-TM may e.g. implement a given WSTMe as instance of a suitable family of functions $f$ by means of selected parameters $\theta^{}$. $\theta^{}$ may be the result of an optimization (regression, control problem, …). Y = f^$\zeta$ = f( θ^$\zeta$ , X , $\zeta$ ) Real D-TM where the software uncertainty $\zeta$ may affect both the function family $f$ and the optimality of the selected parameters $\theta^{\zeta}$. ::\|f( θ , X , $\zeta$ ) \|::^^sem Semantically enhanced D-TM (e.g. SemAP). The D-TM is subject to the semantic checks $sem$ as pre-, post-conditions and invariants on inputs, outputs and the D-TM itself: Y = ::\|f(θ,X,ζ ) \|::^^sem⇔{Y=f(θ,X,ζ )□sem( Y, f, θ,X,ζ )(b)where {X is the input array of data X = { X_1, X_2, ⋯X_i ⋯X_n }X_i ∈C^N_i1 ×⋯×N_in_i is a multi-dimensional array (e.g. a two-dimensional raster layer)Y is analogously the output array of data the modal/deontic logic operator □p means: it ought to be that p . Applying this paradigm to WSTMe, an alternative strategy to black-boxes would suggest exposing not only final outputs but also key intermediate layers of data and information along with the corresponding free software D-TM modules. “Software errors in complex WSTMe may subtly affect the outcomes with possible consequences even on collective environmental decision-making” “The chain of free-software modules should be transparent” A concise, semantically-enhanced modularization [13, 14] may help not only to see the code (as a very basic prerequisite for semantic transparency) but also to understand – and correct – it [59]. Semantically-enhanced, concise modularization is e.g. supported by semantic array programming (SemAP) [13, 14] and its extension to geospatial problems [8, 10]. Some WSTMe may surely be classified in the subset of software systems which “are growing well past the ability of a small group of people to completely understand the content”, while “data from these systems are often used for critical decision making” [50]. In this context, the further uncertainty arising from the unpredicted “(not to say unpredictable)” [51] behaviour of software errors propagation in WSTMe should be explicitly considered as software uncertainty [60, 61] (see b). The data and information flow of a black-box D-TM is often a (hidden) composition of D-TM modules: This chain of free-software D-TM modules (each of them semantically-enhanced) should be transparent: Semantics and design diversity Silent faults [62] are a critical class of software errors altering computation output without evident symptoms – such as computation premature interruption (exceptions, error messages, …), obviously unrealistic results or computation patterns (e.g. noticeably shorter/longer or endless computations). As it has been underlined, “many scientific results are corrupted, perhaps fatally so, by undiscovered mistakes in the software used to calculate and present those results” [63]. “Semantic modularization might help to catch at least a subset of silent faults, when misusing intermediate data outside the expected semantic context” “Where the complexity and scale may lead to deep uncertainty, techniques such as ensemble modelling may be recommendable” Despite the ubiquity of software errors [60, 61, 62, 63, 64, 65, 66, 67, 68], the structural role of scientific software uncertainty seems dramatically underestimated [2, 51]. Semantic D-TM modularization might help to catch at least a subset of silent faults, when misusing intermediate data outside the expected semantic context of a given D-TM module (b). Where the complexity and scale of WSTMe may lead unavoidable software-uncertainty to induce or worsen deep-uncertainty [2], techniques such as ensemble modelling may be recommendable [11, 7, 12]. Adapting those techniques for glancing at the software-uncertainty of a given WSTMe would imply availability of multiple instances (implementations) of the same abstract WSTMe. Independently re-implementing the same WSTMe (design diversity [69]) might of course be extremely expensive. However, partly independent re- implementations of critical D-TM modules may be more affordable and examples of comparison between supposedly equivalent D-TM algorithms seem to corroborate the interest of this research option [57, 70, 49]. ## References * [1] Casagrandi, R., Guariso, G., 2009. Impact of ICT in environmental sciences: A citation analysis 1990-2007. _Environmental Modelling & Software 24_ (7), 865-871. 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1311.4764	# Large-scale analysis of frequency modulation in birdsong databases Dan Stowell and Mark D. Plumbley Centre for Digital Music, Queen Mary University of London [email protected] ###### Abstract Birdsong often contains large amounts of rapid frequency modulation (FM). It is believed that the use or otherwise of FM is adaptive to the acoustic environment, and also that there are specific social uses of FM such as trills in aggressive territorial encounters. Yet temporal fine detail of FM is often absent or obscured in standard audio signal analysis methods such as Fourier analysis or linear prediction. Hence it is important to consider high resolution signal processing techniques for analysis of FM in bird vocalisations. If such methods can be applied at big data scales, this offers a further advantage as large datasets become available. We introduce methods from the signal processing literature which can go beyond spectrogram representations to analyse the fine modulations present in a signal at very short timescales. Focusing primarily on the genus Phylloscopus, we investigate which of a set of four analysis methods most strongly captures the species signal encoded in birdsong. In order to find tools useful in practical analysis of large databases, we also study the computational time taken by the methods, and their robustness to additive noise and MP3 compression. We find three methods which can robustly represent species-correlated FM attributes, and that the simplest method tested also appears to perform the best. We find that features representing the extremes of FM encode species identity supplementary to that captured in frequency features, whereas bandwidth features do not encode additional information. Large-scale FM analysis can efficiently extract information useful for bioacoustic studies, in addition to measures more commonly used to characterise vocalisations. ## 1 Introduction Frequency modulation (FM) is an important component of much birdsong: various species of bird can discriminate the fine detail of frequency-chirped signals (Dooling _et al._ , 2002; Lohr _et al._ , 2006), and use fine FM information as part of their social interactions (Trillo & Vehrencamp, 2005; de Kort _et al._ , 2009). Use of FM is also strongly species-dependent, in part due to adaptation of birds to their acoustic environment (Brumm & Naguib, 2009; Ey & Fischer, 2009) . Songbirds have specific musculature around the syrinx which endows them with independent fine control over frequency (Goller & Riede, 2012). They can control the two sides of their syrinx largely independently: a sequence of two tones might be produced by each side separately, or by one side alone, a difference shown by the absence/presence of brief FM “slurs” between notes (Marler & Slabbekoorn, 2004, e.g. Figure 9.8). Therefore, if we can analyse bird vocalisation recordings to characterise the use of FM across species and situations, this information could cast light upon acoustic adaptations and communicative issues in bird vocalisations. As Slabbekoorn _et al._ (2002) concluded, “Measuring note slopes [FM], as well as other more traditional acoustic measures, may be important for comparative studies addressing these evolutionary processes in the future.” Frequency analysis of birdsong is typically carried out using the short-time Fourier transform (STFT) and displayed as a spectrogram. FM can be observed implicitly in spectrograms, especially at slower modulation rates. However, FM data are rarely explicitly quantified in bioacoustics analyses of birdsong (one exception is Gall _et al._ (2012)), although the amount of FM is partly implicit in measurements such as the rate of syllables and the bandwidth (e.g. in Podos (1997); Vehrencamp _et al._ (2013)). The relative absence of fine FM analysis may be due to the difficulty in extracting good estimates of FM rates from spectrograms, especially with large data volumes. Some previous work has indicated that the FM data extracted from a chirplet representation can improve the accuracy of a bird species classifier (Stowell & Plumbley, 2012). However, there exists a variety of signal processing techniques which can characterise frequency-modulated sounds, and no formal study has considered their relative merits for bird vocalisation analysis. In the present work we aim to facilitate the use of direct FM measurements in bird bioacoustics, by conducting a formal comparison of four methods for characterising FM. Each of these methods goes beyond the standard spectrogram analysis to capture detail of local modulations in a signal on a fine time- scale. To explore the merits of these methods we will use the machine learning technique of feature selection (Witten & Frank, 2005) for a species classification task. In the present work our focus is on methods that can be used with large bird vocalisation databases. Many hypotheses about vocalisations could be explored using FM information, most fruitfully if data can be analysed at relatively large scale. For this reason, we will describe an analysis workflow for audio which is simple enough to be fully automatic and to run across a large number of files. We will consider the runtime of the analysis techniques as well as the characteristics of the statistics they extract. The genus Phylloscopus (leaf warblers) has been studied previously for evidence of adaptive song variation. For example Irwin _et al._ (2008) studied divergence of vocalisation in a ring species (Phylloscopus trochiloides), suggesting that stochastic genetic drift may be a major factor in diversity of vocalisations. Mahler & Gil (2009) found correlations between aspects of frequency range and body size across the Phylloscopus genus. They also considered character displacement effects, which one might expect to cause the song of sympatric species to diverge, but found no significant such effect on the song features they measured. Linhart _et al._ (2012) studied Phylloscopus collybita, also finding a connection between song frequency and body size. Such research context motivated our choice to use Phylloscopus as our primary focus in this study, in order to develop signal analysis methods that might provide further data on song structure. However, we also conducted a larger- scale FM analysis using a database with samples representing species across the wider order of Passeriformes. We first discuss the FM analysis methods to be considered. ### 1.1 FM analysis methods For many purposes, the standard representation of audio signals is the spectrogram, calculated from the magnitudes of the windowed short-time Fourier transform (STFT). The STFT is applied to each windowed “frame” of the signal (of duration typically 10 or 20 ms), resulting in a representation of variations across time and frequency. The spectrogram is widely used in bioacoustics, and a wide variety of measures are derived from this, manually or automatically: it is common to measure the minimum and maximum frequencies in each recording or each syllable, as well as durations, amplitudes and so forth (Marler & Slabbekoorn, 2004). Notable for the present work is the FM rate measure of Gall _et al._ (2012), derived from manual identification of frequency inflection points (i.e. points at which the modulation changes from upward to downward, or downward to upward) on a spectrogram. Trillo & Vehrencamp (2005) characterise “trill vigour” in a related manner but applicable only to trilled syllables. For fully automatic analysis, in Section 2.2 we will describe a method related to that of Gall _et al._ (2012) but with no manual intervention. The spectrogram is a widespread tool, but it does come with some limitations. Analysing a 10 or 20 ms frame with the STFT implies the assumption that the signal is locally stationary (or pseudo-stationary), meaning it is produced by a process whose parameters (such as the fundamental frequency) do not change across the duration of the individual frame (Mallat, 1999, Section 10.6.3). However, many songbirds sing with very dramatic and fast FM (as well as AM), which may mean that the local stationarity assumption is violated and that there is fine-resolution FM which cannot be represented with a standard spectrogram. Signal analysis is under-determined in general: many different processes can in principle produce the same audio signal. Hence the representations derived by STFT and LPC analysis are but two families of possible “explanation” for the observed signal. A large body of research in signal processing has considered alternative representations, tailored to various classes of signal including signals with fast FM. One recent example which was specifically described in the context of birdsong is that of Stowell & Plumbley (2012), which uses a kind of chirplet analysis to add an extra chirp-rate dimension to a spectrogram. A “chirplet” is a short-time packet of signal having a central frequency, amplitude, and a parametric chirp-rate which modulates the frequency over time. More generally, the field of sparse representations allows one to define a “dictionary” of a large number of elements from which a signal may be composed, and then to analyse the signal into a small number of components selected from the dictionary (Plumbley _et al._ , 2010). For the present purposes, notable is the method of Gribonval (2001) which applies an accelerated version of a technique known as Matching Pursuit specifically adapted to analyse a signal as a sparse combination of chirplets. Alternative paradigms are also candidates for performing high-resolution FM analysis. One paradigm is that of spectral reassignment, based on the idea that after performing an STFT analysis it is possible to “reassign” the resulting list of frequencies and magnitudes to shift them to positions which are in some sense a better fit to the evidence (Fulop & Fitz, 2006). The distribution derivative method (DDM) of Muševič (2013, Chapter 10) is one such approach which is able to reassign a spectrum to find the best-matching parameters on the assumption that the signal is composed of amplitude- and frequency-modulated sinusoids. Another approach is that of Badeau _et al._ (2006) which uses a subspace model to achieve high-resolution characterisation of signals with smooth modulations. However, there may be limitations on the rate of FM that can be reflected faithfully: this method relies on a smoothness assumption in the frame-to-frame evolution of the sound which means that it is most suited to relatively moderate rates of FM, such as the vibrato in human singing. In the following we will apply a selection of analysis techniques to birdsong recordings, and study whether the FM information extracted is a reliable signal of species identity. This is not the only application for which FM information is relevant: our aim is that this exploration will encourage other researchers to add high-resolution FM analysis to their toolbox. ## 2 Materials and methods ### 2.1 Data We first collected a set of recordings of birds in the genus Phylloscopus from a dataset made available by the Animal Sound Archive in Berlin.111http://www.animalsoundarchive.org/ This consisted of 45 recordings over 5 species, in WAV format, with durations ranging from 34 seconds to 19 minutes. In the following we will refer to this dataset as PhyllASA. As a second dataset, we also considered a broader set of audio from the Animal Sound Archive, not confined to Phylloscopus but across the order Passeriformes (762 recordings over 84 species). We will refer to this as PassaASA. Thirdly we collected a larger Phylloscopus dataset from the online archive Xeno Canto.222http://www.xeno-canto.org/ This consisted of 1390 recordings across 56 species, ranging widely in duration from one second to seven minutes. Our criteria for selecting files from the larger Xeno Canto archive were: genus Phylloscopus; quality level A or B (the top two quality ratings); not flagged as having uncertain species identity. In the following we will refer to this dataset as PhyllXC. Note that the “crowdsourced” Xeno Canto dataset is qualitatively different from PhyllASA. Firstly it was compiled from various contributors online, and so is not as tightly controlled. The noise conditions and recording quality can vary widely. Secondly, all audio content is compressed in MP3 format (with original uncompressed audio typically unavailable). The MP3 format reduces filesize by discarding information which is considered unnecessary for audio quality as judged by human perception (International Standards Organisation, 1993). However, human and avian audition differ in important ways, including time and frequency resolution, and we cannot assume that MP3 compression is “transparent” regarding the species-specific information that might be important in bird communication. Hence in our study we used this large crowdsourced MP3 dataset only after testing experimentally the impact of compression and signal degradation on the features we measured (using the PhyllASA data). For each dataset considered here, we resampled audio files to 48 kHz mono WAV format before processing, and truncated long files to a maximum duration of 5 minutes. All of the datasets contain an uneven distribution, with some species represented in more recordings than others (Table 1). This is quite common but carries implications for the evaluation of automatic classification, as will be discussed below. Table 1: Counts of species occurrence in our three datasets. Note that PhyllASA is a subset of PassaASA, as reflected in the counts. Species \| PhyllASA \| PassaASA \| PhyllXC ---\|---\|---\|--- Acrocephalus arundinaceus \| \| 9 \| Acrocephalus palustris \| \| 12 \| Acrocephalus schoenobaenus \| \| 3 \| Acrocephalus scirpaceus \| \| 5 \| Aegithalos caudatus \| \| 1 \| Alauda arvensis \| \| 8 \| Anthus pratensis \| \| 1 \| Anthus trivialis \| \| 74 \| Apalis chariessa \| \| 3 \| Calcarius lapponicus \| \| 1 \| Carduelis carduelis \| \| 1 \| Carduelis chloris \| \| 3 \| Carduelis spinus \| \| 4 \| Certhia brachydactyla \| \| 3 \| Certhia familiaris \| \| 1 \| Corvus corax \| \| 1 \| Corvus corone \| \| 3 \| Cyanocitta cristata \| \| 2 \| Delichon urbica \| \| 4 \| Emberiza calandra \| \| 4 \| Emberiza citrinella \| \| 34 \| Emberiza hortulana \| \| 94 \| Emberiza pusilla \| \| 1 \| Emberiza rustica \| \| 3 \| Emberiza schoeniclus \| \| 11 \| Emberiza spodocephala \| \| 2 \| Erithacus rubecula \| \| 14 \| Ficedula albicollis \| \| 1 \| Ficedula hypoleuca \| \| 4 \| Ficedula parva \| \| 7 \| Fringilla coelebs \| \| 87 \| Fringilla montifringilla \| \| 9 \| Garrulax subunicolor \| \| 1 \| Garrulus glandarius \| \| 2 \| Hippolais icterina \| \| 19 \| Hirundo rustica \| \| 3 \| Lanius collurio \| \| 4 \| Locustella fluviatilis \| \| 5 \| Locustella lanceolata \| \| 1 \| Locustella luscinioides \| \| 3 \| Locustella naevia \| \| 6 \| Loxia curvirostra \| \| 1 \| Lullula arborea \| \| 6 \| Luscinia calliope \| \| \| Luscinia luscinia \| \| 10 \| Luscinia megarhynchos \| \| 26 \| Luscinia svecica \| \| 3 \| Species \| PhyllASA \| PassaASA \| PhyllXC ---\|---\|---\|--- Motacilla alba \| \| 1 \| Motacilla flava \| \| 3 \| Muscicapa striata \| \| 1 \| Nucifraga caryocatactes \| \| 20 \| Panurus biarmicus \| \| 1 \| Parus ater \| \| 5 \| Parus caeruleus \| \| 8 \| Parus major \| \| 9 \| Parus montanus \| \| 4 \| Parus palustris \| \| 3 \| Perisoreus infaustus \| \| 1 \| Phoenicurus ochruros \| \| 3 \| Phoenicurus phoenicurus \| \| 22 \| Phylloscopus affinis \| \| \| 7 Phylloscopus amoenus \| \| \| 2 Phylloscopus armandii \| \| \| 6 Phylloscopus bonelli \| 3 \| 3 \| 71 Phylloscopus borealis \| \| \| 25 Phylloscopus borealoides \| \| \| 1 Phylloscopus budongoensis \| \| \| 1 Phylloscopus calciatilis \| \| \| 9 Phylloscopus canariensis \| \| \| 11 Phylloscopus cantator \| \| \| 6 Phylloscopus cebuensis \| \| \| 4 Phylloscopus chloronotus \| \| \| 10 Phylloscopus claudiae \| \| \| 15 Phylloscopus collybita \| 12 \| 12 \| 323 Phylloscopus coronatus \| \| \| 6 Phylloscopus davisoni \| \| \| 2 Phylloscopus emeiensis \| \| \| 4 Phylloscopus examinandus \| \| \| 3 Phylloscopus forresti \| \| \| 14 Phylloscopus fuligiventer \| \| \| 4 Phylloscopus fuscatus \| 5 \| 5 \| 33 Phylloscopus griseolus \| \| \| 6 Phylloscopus hainanus \| \| \| 3 Phylloscopus herberti \| \| \| 4 Phylloscopus humei \| \| \| 51 Phylloscopus ibericus \| \| \| 42 Phylloscopus ijimae \| \| \| 2 Phylloscopus inornatus \| \| \| 53 Phylloscopus kansuensis \| \| \| 4 Phylloscopus laetus \| \| \| 1 Phylloscopus maculipennis \| \| \| 16 Phylloscopus magnirostris \| \| \| 13 Phylloscopus makirensis \| \| \| 7 Phylloscopus neglectus \| \| \| 3 Species \| PhyllASA \| PassaASA \| PhyllXC ---\|---\|---\|--- Phylloscopus nigrorum \| \| \| 7 Phylloscopus nitidus \| \| \| 9 Phylloscopus occisinensis \| \| \| 5 Phylloscopus ogilviegranti \| \| \| 15 Phylloscopus olivaceus \| \| \| 2 Phylloscopus orientalis \| \| \| 5 Phylloscopus plumbeitarsus \| \| \| 10 Phylloscopus poliocephalus \| \| \| 8 Phylloscopus presbytes \| \| \| 15 Phylloscopus proregulus \| \| \| 17 Phylloscopus pulcher \| \| \| 6 Phylloscopus reguloides \| \| \| 26 Phylloscopus ricketti \| \| \| 1 Phylloscopus ruficapilla \| \| \| 7 Phylloscopus sarasinorum \| \| \| 11 Phylloscopus schwarzi \| \| \| 16 Phylloscopus sibilatrix \| 11 \| 11 \| 105 Phylloscopus sindianus \| \| \| 7 Phylloscopus subviridis \| \| \| 1 Phylloscopus tenellipes \| \| \| 28 Phylloscopus trivirgatus \| \| \| 16 Phylloscopus trochiloides \| \| \| 61 Phylloscopus trochilus \| 14 \| 14 \| 208 Phylloscopus tytleri \| \| \| 1 Phylloscopus umbrovirens \| \| \| 5 Phylloscopus xanthoschistos \| \| \| 25 Phylloscopus yunnanensis \| \| \| 11 Prunella modularis \| \| 2 \| Pyrrhula pyrrhula \| \| 1 \| Regulus ignicapillus \| \| 3 \| Regulus regulus \| \| 2 \| Saxicola rubetra \| \| 2 \| Sitta europaea \| \| 6 \| Smithornis capensis \| \| 1 \| Sturnus vulgaris \| \| 1 \| Sylvia atricapilla \| \| 14 \| Sylvia borin \| \| 10 \| Sylvia communis \| \| 9 \| Sylvia curruca \| \| 2 \| Sylvia nisoria \| \| 2 \| Troglodytes troglodytes \| \| 11 \| Turdus iliacus \| \| 2 \| Turdus merula \| \| 36 \| Turdus philomelos \| \| 21 \| Turdus pilaris \| \| 4 \| Turdus viscivorus \| \| 7 \| ### 2.2 Method For all analysis methods we used a frame size of 512 samples (10.7 milliseconds, at 48 kHz), with Hann windowing for STFT, and the frequency range of interest was restricted to 2–10 kHz. For each recording in each dataset, we applied a fully automatic analysis using each of four signal processing techniques. Our requirement of full automation excludes a preprocessing step of manually segmenting of birdsong syllables from the background. We chose to use the simplest form of automatic segmentation, simply to select the 10% of highest-energy frames in each recording. More sophisticated procedures can be applied in future; however, as well as simplicity this method has an advantage of speed when analysing large databases. We analysed each recording using each of the following techniques (which we assign two-letter identifiers for reference): ss: a spectrographic method related to the method of Gall _et al._ (2012) but with no manual intervention, as follows. Given a sample of birdsong, for every temporal frame we identify the frequency having peak energy, within the frequency region of interest. We calculate the absolute value of the first difference, i.e. the magnitude of the frequency jump between successive frames. We then summarise this by the median or other statistics, to characterise the distribution over the depth of FM present in each recording. This method relies on the peak-energy within each frame rather than manual identification of inflection points in the pitch trace, which means that it is potentially susceptible to noise and other corruptions, but it remains a relatively robust technique which can be applied to a standard spectrogram representation. In the following we will refer to this method as the “simple spectrographic” method. rm: the heterodyne (ring modulation) chirplet analysis of Stowell & Plumbley (2012), taking information from the peak-energy detection in each frame.333Python source code for the method of Stowell & Plumbley (2012) is available at https://code.soundsoftware.ac.uk/projects/chirpletringmod. mp: the Matching Pursuit technique of Gribonval (2001), implemented using the open-source Matching Pursuit ToolKit (MPTK) v0.7.444Available at http://mptk.irisa.fr/. For this technique the 10% highest-energy threshold is not applicable, since the method is iterative and could return many more results than there are signal frames: we automatically set a threshold at a number of results which recovers roughly the same amount of signal as the 10% threshold. dd: the distribution derivative method (DDM) of Muševič (2013, Chapter 10), taking information from the peak-energy sinusoid detected in each frame.555Matlab/Octave source code for the method of Muševič (2013) is available at https://code.soundsoftware.ac.uk/projects/ddm. We also conducted a preliminary test with the subspace method of Badeau _et al._ (2006), but this proved to be inappropriate for the rapid FM modulations found in birdsong because of an assumption of smooth FM variation inherent in the method (Badeau, pers. comm.). Each of these methods resulted in a list of “frames” or “atoms” for a recording, each with an associated frequency and FM rate. In order to characterise each recording as a whole, we selected summary statistics over these frames in a recording to use as features. We summarised the frequency data by their median, and by their 5- and 95-percentiles. The 5- and 95-percentiles are robust measures of minimum and maximum frequency; we also calculated the “bandwidth” as the difference between the 5- and 95-percentile. We summarised the FM data by their median, and also by their 75- and 95-percentiles. These percentiles were chosen to explore whether information about the relative extremes of FM found in the recording provide useful extra information. So, for each recording and each analysis method we can extract a set of frequency and FM summary features. It remains to determine which of these features might be most useful in looking for signals of species identity in recorded bird vocalisations. We explored this through two interrelated approaches: feature selection, and automatic classification experiments. Through these two approaches, we were able to compare the different features against each other, and also compare the features as extracted by each of the four signal-processing techniques given above. One approach that has been used to explore the value of different features is principal components analysis (PCA) applied to the features, to determine axes that represent the strongest dimensions of variance in the features (see e.g. Mahler & Gil (2009); Handford & Lougheed (1991)). This method is widespread and well-understood. However, it is a purely linear analysis which may fail to reflect nonlinear information-carrying patterns in the data; and more importantly for our purposes, PCA does not take into account the known species labels, and so can only ever serve as indirect illumination on questions about which features might carry such information. In the field of data mining/machine learning, researchers instead use feature selection techniques to evaluate directly the predictive power that a feature (or a set of features) has with respect to some attribute (Witten & Frank, 2005). We used an information-theoretic feature selection technique from that field. In information gain feature selection, each of our features is evaluated by measuring the information gain with respect to the species label, which is the amount by which the feature reduces our uncertainty in the label: $\text{IG}(\text{Species},\text{Feature})=H(\text{Species})-H(\text{Species}\|\text{Feature})$ where $H(\cdot)$ is the Shannon entropy. The value $H(\text{Species})$ represents the number of binary bits of information that must typically be conveyed in order to identify the species of an individual (from a fixed set of species). The information gain $\text{IG}(\text{Species},\text{Feature})$ then tells us how many of those binary bits are already encoded in a particular feature, i.e. the extent to which that feature reduces the uncertainty of the species identity. If a feature is repeatedly ranked highly, this means that it contains a stronger signal of species identity than lower- ranked features and thus suggests it should be a useful measure. The approach just described is reminiscent of the information-theoretic method introduced by Beecher (1989), except that his concern was with signals of individual identity rather than species identity. Having performed feature selection, we were then able to choose promising subsets of features which might concisely represent species information. To evaluate these subsets concretely we conducted an experiment in automatic species classification. For this we used a leading classification algorithm, the Support Vector Machine (SVM), implemented in the libsvm library version 3.1, choosing the standard radial basis function SVM classifier. The evaluation statistic we used was the weighted “area under the receiver operating characteristics curve” (the weighted AUC), which summarises the rates of true-positive and false-positive detections made (Fawcett, 2006). This measure is more appropriate than raw accuracy, when analysing datasets with wide variation in numbers per class as in the present case (ibid.). The AUC yields the same information as the Wilcoxon signed-rank statistic (Hanley & McNeil, 1982). The feature selection and classification experiments were all performed using Weka 3.6.0 (Witten & Frank, 2005), and analysed using R version 2.13.1 (R Development Core Team, 2010). An important issue when considering automatic feature extraction is the robustness of the features to corruptions that may be found in audio databases, such as background noise or MP3 compression artifacts. This has particular pertinence for the crowdsourced PhyllXC dataset, as discussed above. For this reason, we also studied our first dataset after putting the audio files through two corruption processes: added white noise ($-45$ dB relative to full-scale, judged by ear to be noticeable but not overwhelming), and MP3 compression (64 kbps, using the lame software library version 3.99.5). To quantify whether an audio feature was badly impacted by such corruption, we measured the Pearson correlations of the features measured on the original dataset with their corrupted equivalent. This test does not depend on species identity as in our main experimental tests, but simply on the numerical stability of the summary statistics we consider. In this study we focussed on frequency and FM characteristics of sounds, both of which can be extracted completely automatically from short time frames. We did not include macro-level features such as syllable lengths or syllable rates, because reliable automatic extraction of these is complex. Rather, we compared the fine-detail FM analyses against frequency measures, the latter being common in the bioacoustics literature: our feature set included features corresponding to the lower, central and upper frequency, and frequency bandwidth. ## 3 Results Figure 1: Standard spectrogram for a short excerpt of Chiffchaff (Phylloscopus collybita). The FM can be seen by eye but is not explicit in the underlying data, being spread across many “pixels”. Figure 2: Time-frequency plots of the “chirp” data recovered by each method, for the same excerpt as in Figure 1. We first illustrate the data which is produced by the analysis methods tested, using a recording of Phylloscopus collybita (Chiffchaff) from PhyllASA as an example. Figure 1 shows a conventional spectrogram plot for our chosen excerpt. We can infer FM characteristics visually, but the underlying data (a grid of intensity “pixels”) does not directly present FM for analysis. Figure 2 represents the same excerpt analysed by each of the methods we consider. Each of the plots appears similar to a conventional spectrogram, showing the presence of energy at particular time and frequency locations. However, instead of a uniform grid the image is created from a set of line segments, each segment having a location in time and frequency but also a slope. It is clear from Figure 2 that each of the methods can build up a portrait of the birdsong syllables, although some are more readable than others. The plot from mp appears more fragmented than the others. This can be traced back to the details of the method used, but for now we merely note that the apparent neatness of each representation does not necessarily indicate which method most usefully captures species-specific FM characteristics. Table 2: Time taken to run each analysis method on our first dataset PhyllASA, expressed as a proportion of the total duration of the audio files (so that any number below 1 indicates faster than real-time processing). Times were measured on a laptop with Intel i5 2.5 GHz processor. Method \| Time taken (relative to audio duration) ---\|--- ss \| 0.02 rm \| 0.40 mp \| 0.58 dd \| 1.22 The relative speeds of the analysis methods described here are given in Table 2. The simple spectrogram method is by far the fastest, as is to be expected given its simplicity. All but one of the methods run much faster than realtime, though the difference in speed between the simple spectrogram and the more advanced methods is notable, and certainly pertinent when considering the analysis of large databases. Figure 3: Squared Pearson correlation between audio features and their values after applying audio degradation, across the PhyllASA dataset. Each point represents one feature; features are grouped by analysis method and degradation type. We inspected the variation according to feature, and found no general tendencies; therefore features are collapsed into a single column per analysis method in order to visualise the differences in range. Note that the vertical axis is warped to enhance visibility at the top end of the scale. Features extracted by methods ss rm and dd were highly robust to the noise and MP3 degradations applied, in all cases having a correlation with the original features better than 0.95 (Figure 3). Method rm showed particularly strong robustness. The mp method, on the other hand, yielded features of very low robustness: correlation with the original features was never above 0.95, in some cases going as low as to be around zero. This indicates that features from the mp method may be generally unreliable when applied to the PhyllXC dataset considered next. Our feature selection experiments revealed notable trends in the information gain (IG) values associated with certain features, with broad commonalities across the three datasets tested (see Appendix for details). In particular, the bandwidth features achieve very low IG values in all cases. Conversely, the median frequency feature performs strongly for all datasets and all methods. The FM features perform relatively strongly on PhyllASA, appearing generally stronger than frequency features, but this pattern does not persist into the other (larger) datasets. However, the 75-percentile of FM did generally rank highly in the feature selection results. Based on the results of feature selection, we chose to take the following four feature sets forward to the classification experiment: * • Three FM features (fm_med, fm_75pc, fm_95pc); * • Three frequency-based features (freq_05pc, freq_med, freq_95pc); * • The “Top-2” performing features (freq_med, fm_75pc); * • All six FM and frequency-based features together. We did not include the poorly-performing bandwidth features. This yielded an advantage that the FM and frequency-based features had the same cardinality, ensuring the fairness of our experimental comparison of the two feature types. Figure 4: Performance of species classification across 56 species, evaluated using datasets PassaASA (upper) and PhyllXC (lower). Results are shown for each analysis method, and for four different subsets of the available features (see text for details). The horizontal dashed line indicates the baseline chance performance at 50%. Table 3: Marginal mean of the weighted area under the curve (AUC) scores for the results shown in Figure 4. Dataset \| Method \| AUC (%) ---\|---\|--- PassaASA \| ss \| 67.6 \| dd \| 67.2 \| rm \| 64.3 \| mp \| 62.2 PhyllXC \| ss \| 66.5 \| dd \| 65.3 \| rm \| 63.2 \| mp \| 61.6 Dataset \| Feature set \| AUC (%) PassaASA \| FM+Freq \| 69.6 \| Top-2 \| 65.8 \| Freq \| 66.9 \| FM \| 58.9 PhyllXC \| FM+Freq \| 69.5 \| Top-2 \| 64.4 \| Freq \| 63.6 \| FM \| 59.1 Results for the classification experiment with different extraction methods and different feature subsets are shown in Figure 4 and Table 3. This is a difficult classification task (across 56 species), and the average AUC score in this case peaks at around 70%. A repeated-measures factorial ANOVA confirmed, for both datasets, a significant effect on accuracy for both feature set ($p<2\times 10^{-16}$) and method ($p\leq 1.2\times 10^{-6}$), with no significant interaction term found ($p>0.07$). We conducted post-hoc tests for differences in AUC between pairs of methods and pairs of feature-sets, using paired t-tests with Bonferroni correction for all pairwise comparisons (this is a repeated-measures alternative to the Tukey HSD test). Means were found to be different ($p<0.0035$) for all pairs of methods except ss vs. dd (ss $\approx$ dd $>$ rm $>$ mp ). For the choice of feature set, means were found to be different ($p<2.2\times 10^{-6}$) for all pairs of feature sets except Top-2 vs. Freq (FM+Freq $>$ Freq $\approx$ Top-2 $>$ FM). ## 4 Discussion The fine detail of frequency modulation (FM) is known to be used by various songbird species to carry information (Marler & Slabbekoorn (2004, Chapter 7); Brumm & Naguib (2009); Sprau _et al._ (2010); Vehrencamp _et al._ (2013)), but automatic tools for analysis of such FM are not yet commonly used. Our experiments have demonstrated that FM information can be extracted efficiently from large datasets, in a fashion which captures species-related information despite the simplicity of method (we used no source-separation, syllable segmentation or pitch tracking). This was explicitly designed for application on large collections: our experiments used up to 1390 individual recordings, larger numbers than in many bioacoustic studies. Our results show an effect of the choice of summary features, both for frequency and for FM data. The consistently strongest-performing summary feature was the median frequency, which is similar to measures of central tendency used elsewhere in the literature and can be held to represent a bird’s central “typical” frequency. On the contrary, we were surprised to find that bandwidth measurements as implemented in our study showed rather little predictive power for species identity, since bandwidth has often been discussed with respect to the variation in vocal capacities across avian species (Podos, 1997; Trillo & Vehrencamp, 2005; Mahler & Gil, 2009). In our case the upper frequency extent alone (represented by the 95-percentile) appears more reliable, which may reflect the importance of production limits in the highest frequencies in song. The FM features, taken alone, were not as predictive of species identity as were the frequency features. However, they provided a significant boost in predictive power when appended to the frequency features. This tells us not only that FM features encode aspects of species identity, but they encode complementary information which is not captured in the frequency measurements. In light of our results we note that Trillo & Vehrencamp (2005) explored a measure of “trill vigour”: “Because of the known production constraint trade- off between note rate and bandwidth of trilled songs (Podos 1997), we derived an index of trill vigour by multiplying the standardized scores of these two parameters” (Trillo & Vehrencamp, 2005, p. 925). This index was not further pursued since in their study it yielded similar results as the raw bandwidth data. However, if we assume for the moment that each note in the trills studied by Trillo & Vehrencamp is one full sweep of the bandwidth of the trill (this is the case for all except “hooked” trills), then multiplying the bandwidth (in Hz) by the note rate (in sec-1) yields exactly the mean value of the instantaneous absolute FM rate (in Hz/sec). This “trill vigour” calculation is thus very close in spirit to our measurement of the median FM rate. Their comparison of bandwidth features against trill vigour features served for them as a kind of feature selection, although in their case the focus was on trills in a single species. A further aspect of our study is the comparison of four different methods for extracting FM data. A clear result emerges from this, which is that the simplest method (ss) attains the strongest classification results (tied with method dd), and is sufficiently robust to the degradations we tested. This should be taken together with the observation that it runs at least 20 times faster than any of the other methods on the same audio data, to yield a strong recommendation for the ss method. This outcome came as a surprise to us, especially considering the simplifying assumptions implicit in the ss method. It considers the peak-amplitude frequencies found in adjacent STFT frames (i.e. in adjacent “slices” of a spectrogram), which may in many cases relate to the fundamental frequency of the bird vocalisation, but can often happen to relate to a harmonic, or a chance fluctuation in background noise. It contains no larger-scale corrections for continuity, as might be used in pitch-tracking-type methods (though note that as we found with the method of Badeau _et al._ (2006), those methods can incur difficulties tracking fast modulations). The statistical strength of simple methods has been studied elsewhere in the literature. For example Kershenbaum _et al._ (2013) found that bottlenose dolphin signature whistles could usefully be summarised by a strongly decimated representation of the pitch track: a so-called “Parsons code” based on whether the pitch is rising or falling at a particular timescale, and which completely omits the magnitude of such rises or falls. The method is not analogous to ours, but has in common that it uses suprisingly simple statistics to summarise temporal variation. Audio “fingerprinting” systems such as Shazam (Wang, 2003) also rely on highly-reduced summary data, customised to the audio domain of interest. Our ss method relies on finding a temporal difference between adjacent frames, as does that of Kershenbaum _et al._ (2013). This is partly reminiscent of the “delta” features often added to MFCCs to reflect how they may be changing. Such deltas are common in speech recognition and are also used in some automatic species classification (for example Trifa _et al._ (2008)). However note that MFCC “deltas” represent differences in magnitude, not in frequency. Separately from the classification experiment, we studied the effects of noise and MP3 degradation on our summary features. Such issues are pertinent for crowdsourced datasets such as PhyllXC. Measures such as minimum and maximum frequency carry some risk of dependence on recording conditions, particularly when derived from manual inspection of spectrograms (Zollinger _et al._ , 2012; Cardoso & Atwell, 2012). We have demonstrated that our automatic FM measures using methods rm, dd or ss are robust against two common types of degradation (noise and compression), with rm particularly robust. They are therefore suitable tools to explore the variation in songbirds’ use of FM in the laboratory and in the field. Future work: in this study we did not use any higher-level temporal modelling such as the temporal structure of trill syllables, nor did we use advanced methods for segmenting song/call syllables from background. We have demonstrated the utility of fully automatic extraction of fine temporal structure information, and in future work we aim to combine this with richer modelling of other aspects of vocalisation. We also look forward to combining fine FM analysis with physiological models of the songbird vocal production mechanism—as has already been done with linear prediction for the source- filter model (Markel, 1972)—but explicitly accounting for songbirds’ capacity for rapid nonstationary modulation and their use of two separate sound sources in the syrinx. ## 5 Conclusions In much research involving acoustic analysis of birdsong, frequency modulation (FM) has been measured manually, described qualitatively or left implicit in other measurements such as bandwidth. We have demonstrated that it is possible to extract data about FM on a fine temporal scale, from large audio databases, in fully automatic fashion, and that this data encodes aspects of ecologically pertinent information such as species identity. Further, we have demonstrated that a relatively simple technique based on spectrogram data is sufficient to extract information pertinent to species, which one might expect could only be extracted with more advanced signal-processing techniques. Our study provides evidence that researchers can and should measure such FM characteristics when analysing the acoustic characteristics of bird vocalisations. ## Acknowledgments DS & MP are supported by an EPSRC Leadership Fellowship EP/G007144/1. 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The overall pattern of IG values shows broad similarities between PhyllASA and PhyllXC, indicating commonalities in species-dependent features. The IG values evaluated on PhyllXC are consistently lower than those in PhyllASA, suggesting that the species information in the former may be affected by noise and MP3 effects. However, the tendency to lower IG values may also be an artefact of differences in species distribution within each dataset. Table 4: Ranked results of information-gain (IG) feature selection applied to each of our three datasets. Features are ranked in order of how strongly they predict species identity. Left to right: PhyllASA, PassaASA, PhyllXC. Rank \| IG \| Feature ---\|---\|--- 1 \| 1.5667 \| fm_med_mp 2 \| 1.3878 \| fm_75pc_rm 3 \| 1.3591 \| fm_75pc_mp 4 \| 1.2131 \| fm_95pc_rm 5 \| 1.1928 \| fm_75pc_ss 6 \| 1.1874 \| fm_75pc_dd 7 \| 1.1516 \| freq_med_rm 8 \| 1.1266 \| fm_95pc_ss 9 \| 1.0786 \| fm_med_rm 10 \| 1.0224 \| freq_med_ss 11 \| 0.9984 \| freq_med_dd 12 \| 0.9213 \| freq_med_mp 13 \| 0.8461 \| fm_med_dd 14 \| 0.8084 \| freq_95pc_ss 15 \| 0.7994 \| fm_med_ss 16 \| 0.7754 \| freq_05pc_rm 17 \| 0.7754 \| freq_05pc_dd 18 \| 0.6906 \| freq_05pc_ss 19 \| 0.6587 \| freq_95pc_dd 20 \| 0.6556 \| freq_05pc_mp 21 \| 0.6165 \| fm_95pc_dd 22 \| 0.5314 \| fm_95pc_mp 23 \| 0.4748 \| freq_95pc_rm 24 \| 0.4396 \| freq_bw_dd 25 \| 0.4273 \| freq_95pc_mp 26 \| 0.3998 \| freq_bw_rm 27 \| 0 \| freq_bw_mp 28 \| 0 \| freq_bw_ss Rank \| IG \| Feature ---\|---\|--- 1 \| 1.3133 \| freq_med_dd 2 \| 1.2701 \| freq_med_rm 3 \| 1.2387 \| freq_med_ss 4 \| 1.0457 \| freq_med_mp 5 \| 0.9629 \| freq_95pc_rm 6 \| 0.9432 \| freq_95pc_ss 7 \| 0.8563 \| fm_med_ss 8 \| 0.8533 \| fm_med_dd 9 \| 0.8353 \| freq_05pc_dd 10 \| 0.7708 \| freq_95pc_dd 11 \| 0.7343 \| freq_95pc_mp 12 \| 0.6424 \| fm_75pc_rm 13 \| 0.5923 \| fm_75pc_dd 14 \| 0.5648 \| fm_75pc_ss 15 \| 0.5194 \| fm_med_rm 16 \| 0.5098 \| fm_med_mp 17 \| 0.5079 \| fm_95pc_dd 18 \| 0.4964 \| fm_95pc_ss 19 \| 0.4767 \| freq_05pc_ss 20 \| 0.4747 \| fm_75pc_mp 21 \| 0.43 \| freq_05pc_rm 22 \| 0.4039 \| freq_bw_dd 23 \| 0 \| freq_bw_rm 24 \| 0 \| fm_95pc_rm 25 \| 0 \| freq_bw_mp 26 \| 0 \| fm_95pc_mp 27 \| 0 \| freq_bw_ss 28 \| 0 \| freq_05pc_mp Rank \| IG \| Feature ---\|---\|--- 1 \| 0.83 \| freq_med_ss 2 \| 0.752 \| freq_med_dd 3 \| 0.669 \| fm_75pc_rm 4 \| 0.653 \| freq_med_rm 5 \| 0.603 \| fm_75pc_ss 6 \| 0.558 \| fm_75pc_dd 7 \| 0.541 \| freq_med_mp 8 \| 0.525 \| fm_med_ss 9 \| 0.494 \| fm_med_rm 10 \| 0.474 \| freq_95pc_rm 11 \| 0.467 \| freq_95pc_dd 12 \| 0.459 \| fm_95pc_ss 13 \| 0.449 \| fm_95pc_dd 14 \| 0.428 \| freq_95pc_ss 15 \| 0.427 \| fm_med_mp 16 \| 0.412 \| freq_95pc_mp 17 \| 0.38 \| fm_75pc_mp 18 \| 0.336 \| fm_med_dd 19 \| 0.331 \| fm_95pc_rm 20 \| 0.29 \| freq_05pc_ss 21 \| 0.286 \| freq_05pc_rm 22 \| 0.286 \| freq_05pc_dd 23 \| 0.238 \| fm_95pc_mp 24 \| 0 \| freq_bw_dd 25 \| 0 \| freq_bw_mp 26 \| 0 \| freq_05pc_mp 27 \| 0 \| freq_bw_ss 28 \| 0 \| freq_bw_rm Figure 5: Overview of information gain (IG) values calculated during feature selection; as in Table 4 but ordered by feature type.See Table 4 for numerical values	arxiv-papers	2013-11-19T15:02:55	2024-09-04T02:49:53.860817	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Dan Stowell and Mark D. Plumbley", "submitter": "Dan Stowell", "url": "https://arxiv.org/abs/1311.4764" }
1311.4799	# Adaptive Hierarchical Data Aggregation using Compressive Sensing (A-HDACS) for Non-smooth Data Field Xi Xu Department of Electrical and Computer Engineering University of Illinois at Chicago Chicago,Illinois,60607 Email:[email protected] Rashid Ansari Department of Electrical and Computer Engineering University of Illinois at Chicago Chicago,Illinois,60607 Email: [email protected] Ashfaq Khokhar Department of Electrical and Computer Engineering University of Illinois at Chicago Chicago,Illinois,60607 Email:[email protected] ###### Abstract Compressive Sensing (CS) has been applied successfully in a wide variety of applications in recent years, including photography, shortwave infrared cameras, optical system research, facial recognition, MRI, etc. In wireless sensor networks (WSNs), significant research work has been pursued to investigate the use of CS to reduce the amount of data communicated, particularly in data aggregation applications and thereby improving energy efficiency. However, most of the previous work in WSN has used CS under the assumption that data field is smooth with negligible white Gaussian noise. In these schemes signal sparsity is estimated globally based on the entire data field, which is then used to determine the CS parameters. In more realistic scenarios, where data field may have regional fluctuations or it is piecewise smooth, existing CS based data aggregation schemes yield poor compression efficiency. In order to take full advantage of CS in WSNs, we propose an Adaptive Hierarchical Data Aggregation using Compressive Sensing (A-HDACS) scheme. The proposed schemes dynamically chooses sparsity values based on signal variations in local regions. We prove that A-HDACS enables more sensor nodes to employ CS compared to the schemes that do not adapt to the changing field. The simulation results also demonstrate the improvement in energy efficiency as well as accurate signal recovery. ###### Index Terms: Data Aggregation, Compressive Sensing, Wireless Sensor Networks, Hierarchy, Power Efficient Algorithm, Non-Smooth Data Field ## I Introduction Energy efficiency is a major target in the design of wireless sensor networks due to limited battery power of the sensor nodes. Also, at times it is difficult to replenish battery power depending on the application area. Since data communication is the most basic but high energy consuming task in sensor networks, a plethora of research work has been done to improve its energy consumption [1] [2] [3] [4]. Compressive Sensing (CS) [5] [6] has emerged as a promising technique to reduce the amount of data communicated in WSNs. It has been also applied in other application areas such as photography, shortwave infrared cameras, optical system research, facial recognition, MRI, etc. [7]. Luo et. al. [8] proposed the use of CS random measurements to replace raw data communication in data aggregation tasks in WSNs, thus reducing the amount of data transmitted. However, their technique introduced redundant data communication in nodes that were farther away from the root node of the data aggregation tree. Xiang et. al.[9] [10] optimized this scheme by reducing the data transmission redundancy. In our previous work, We further improved CS based data aggregation by proposing a Hierarchical Data Aggregation using Compressive Sensing (HDACS) [11] that introduced a hierarchy of clusters into CS data aggregation model and achieved significant energy efficiency. However, most of the previous work has used CS under the assumption that data field is smooth with negligible white Gaussian noise. In these schemes, signal sparsity is calculated globally based on the entire data field. In more realistic scenarios, where data field may have regional fluctuations or it is piecewise smooth, existing CS based data aggregation schemes will yield poor compression efficiency. The sparsity constant $K$ is usually a big number, with large probability, when the field consists of bursts or bumps. In such cases, the number of CS measurements $M=K\log N$ is bigger than $N$, where $N$ is local cluster size. In order to take full advantage of CS for its great compression capability, we propose an Adaptive Hierarchical Data Aggregation using Compressive Sensing (A-HDACS) scheme.The proposed schemes adaptively chooses sparsity values based on signal variations in local regions. Our solution is based on the observation that the number of CS random measurements from any region (spatial or temporal) should correspond to the local sparsity of the data field, instead of global sparsity. Intuitively, it should work well because the nodes are more correlated with each other in a local area than the entire global area. Also, it is easy to compute the local sparsity, particularly when a data aggregation scheme is based on a hierarchical clustering scheme. Also, in order to compute global sparsity, apriori knowledge of the data field is required. We show that the proposed A-HDACS scheme enables more sensor nodes to utilize compressive sensing compared to the HDACS scheme [11] that employs global sparsity based compressive sensing. Using the SIDnet-SWANS [12] sensor simulation platform for our experiments, we demonstrate the effectiveness of the proposed scheme for different types of data fields and network sizes. For the smooth data field with multiple Gaussian bumps, A-HDACS reduces energy consumption by $\approx 6\%$ to $10\%$, depending on the network size. Similarly, for the piecewise smooth data field, it reduces energy consumption by $\approx 23.36\%$ to $30.17\%$ depending on the network size. We observe higher gains in larger network sizes. The experimental results are consistent with our theoretical analysis. The rest of paper is organized as the follows: Section II gives an overview of the existing CS based data aggregation schemes. In Section III, the details of the proposed A-HDACS scheme are presented. The analysis of the data field sparsity and its effect on CS in both HDACS and A-HDACS is given in Section IV. Section V shows the simulation evaluation. ## II Related Work Any conventional data collection scheme that does not involve pre-processing of data usually employs $O(N^{2})$ data transmissions in an $N-$node routing path. Lou et al. [8] were the first to examine the use of Compressive Sensing (CS) [5] [6] in data gathering applications for large scale WSNs. Their scheme reduced the required number of transmissions to $O(NM)$, where $M<<N$. According to CS [5], $M=K\log{N}$ and $K$ is the signal sparsity, representing the number of nonzero entries of the signal. We refer to this scheme as the plain CS aggregation scheme (PCS). PCS requires all sensors to collectively provide to the sink the same amount of random measurements, i.e. $M$, regardless of their location in the network. Note that when PCS is applied in a large scale network, $M$ may still be a large number. Moreover, in the initial data aggregation phase in [8], nodes placed on or closer to the leaves of aggregation tree also transmit $M$ measurements, which is in excess of their single readings and therefore introduces redundancy in data aggregation. The hybrid CS (HCS) aggregation [9][10] eliminated the data aggregation redundancy in the initial phase by combining conventional data aggregation with PCS. It optimizes the data aggregation cost by setting a global threshold $M$ and applying CS at only those nodes where the number of accumulated data samples equals to, or exceeds $M$; otherwise all other nodes communicate just raw data. The major drawback of HCS is that only a small fraction of the sensors are able to utilize the advantage of CS scheme, and the required amount of data that need to be transmitted for even these nodes is still large. Thus, an energy-efficient technique: Hierarchical Data Aggregation using Compressive Sensing (HDACS) [11] was presented based on a multi- resolution hierarchical clustering architecture and HCS. The central idea was to configure sensor nodes so that instead of one sink node being targeted by all sensors, several nodes, arranged in a way to yield a hierarchy of clusters, are designated for the intermediate data collection. The amount of data transmitted by each sensor is determined based on the local cluster size at different levels of the hierarchy rather than the entire network, which, therefore, leads to reduction in the data transmitted, with an upper bound of $O(K\log{N})$. In other words, in HDACS the value of $N$ is different for different nodes. But HDACS has its own limitation. It can only solve the data aggregation problem when the data field is globally smooth with negligible variations, since its data field sparsity is assumed as a single constant $K$ derived from the whole data field. It is more desirable that we can consider more realistic scenarios when the data field is not relatively flat, i.e. sparsity of the data field is different for different regions of the network. In this work, our attention will mainly focus on how the fluctuations of the data field affects HDACS and we propose Adaptive HDACS (A-HDACS) to solve this problem. ## III Proposed Adaptive HDACS (A-HDACS) Scheme The basic idea behind A-HDACS is that CS random measurements for each sensor that need to be communicated are determined by the sparsity of data field within each clusters at different levels of the data aggregation tree. For consistency, we adopt the same notations as in [11], showed in Table I. TABLE I: Parameters Definition $N$ \| The network size ---\|--- $T$ \| The total level of the hierarchy $N_{i}^{(l)}$ \| The cluster size at level $i$ in cluster $l$ $M_{i}^{(l)}$ \| The amount of data transmitted after performing CS at level $i$ in cluster $l$ $C_{i}$ \| The collection of clusters at level $i$ $\|C_{i}\|$ \| The number of cluster at level $i$ in cluster $l$ \| where $\|C_{i}\|=n^{T-i}$ In order to capture varying sparsity of the data field based on local regions, we also define some new variables. * • $K_{T}$: the whole data field sparsity * • $K_{i\\_T}$: threshold defined as $K_{i\\_T}=\max_{\begin{subarray}{c}l\in C_{i}\end{subarray}}\\{\frac{N_{i}^{(l)}}{\log{N_{i}^{(l)}}}\\}$ at level $i$ * • $K_{i}^{(l)}$: sparsity of the data field in cluster $l$ at level $i$ Besides, we also define two types of nodes: CS-enabled nodes and CS-disable nodes. In CS-enabled nodes the data collected is large and sparse enough that CS pays off. On the other hand, in CS-disabled nodes the data collected is small and/or not sparse enough to yield the benefits of CS. The salient steps of A-HDACS implemented on the multi-resolution data collection hierarchy are as follows: 1. 1. At level one, leaf nodes send their single sensed data to their cluster heads without applying CS. The cluster head receives the data and performs the conventional transformation to obtain the signal representation and its sparsity factor $K_{1}^{(l)}$. Then it compares $K_{1}^{(l)}$ to $\frac{N_{1}^{(l)}}{\log{N_{1}^{(l)}}}$. If $K_{1}^{(l)}<\frac{N_{1}^{(l)}}{\log{N_{1}^{(l)}}}$, it becomes the CS-enabled sensor and takes the CS random measurements. The amount of data that need to be transmitted is $M_{1}^{(l)}=K_{1}^{(l)}\log{N_{1}^{(l)}}$; otherwise, it disables itself as CS-disabled node and transmits $N_{1}^{(l)}$ data directly to its parent clusters. 2. 2. At level $i$ ($i\geq 2$), cluster head receives packets from its children nodes. If it receives packets with CS random measurements, the CS recovery algorithm will be performed firstly to recover all the data. After cluster head gets all the data from the children nodes, it projects the whole data into transformation domain to obtain the signal representation and its sparsity factor $K_{i}^{(l)}$. If $K_{i}^{(l)}<\frac{N_{i}^{(l)}}{\log{N_{i}^{(l)}}}$, cluster head turns itself as CS-enabled node and performs the process of taking CS random measurements with length $M_{i}^{(l)}=K_{i}^{(l)}\log{N_{i}^{(l)}}$; otherwise, it becomes CS-disabled node and send the data directly. 3. 3. Repeat step 2 ) until the cluster head at the top level $T$ obtains and recovers the whole data. ## IV Analysis of Data Field Sparsity ###### Proposition 1 In HDACS, if $K_{T}>K_{i\\_T}$, all the nodes at the level equal to and below $i$ are all CS-disabled nodes. ###### Proof: Define: $f(x)=\frac{x}{\log{x}}$. since $f^{\prime}(x)=\frac{\log{x}-\frac{1}{\ln{2}}}{(\log{x})^{2}}>0\text{ when }x>3$. Therefore, $f(x)$ is a monotonous increasing function when $x>3$. 1. 1. At level $i$, if $K_{T}>K_{i\\_T}$ then $K_{T}>\frac{N_{i}^{(l)}}{\log{N_{i}^{(l)}}}$. In HDACS, CS requires the amount of data to be transmitted $M_{i}^{(l)}=K_{T}\log{N_{i}^{(l)}}$. Therefore, $M_{i}^{(l)}>N_{i}^{(l)}\text{ for }\forall j\in C_{i}$. Thus clusters at level $i$ are all CS-disabled nodes. 2. 2. At level $j$ and $j<i$, since $N_{j}^{(l)}<N_{i}^{(p)}\text{ for }\forall l\in C_{j}\text{ and }\forall p\in C_{i}$, $K_{i\\_T}>K_{j\\_T}$. So $K_{T}>K_{j\\_T}>\frac{N_{j}^{(l)}}{\log{N_{j}^{(l)}}}$ and $M_{j}^{(l)}=K_{T}\log{N_{j}^{(l)}}>N_{j}^{(l)}$. Thus the nodes at levels below $i$ are also all CS-disabled nodes. ∎ On the other hand, if $\exists l\in C_{i}$ s.t. $K_{T}>K_{i\\_T}>\frac{N_{i}^{(l)}}{\log{N_{i}^{(l)}}}>K_{i}^{(l)}$ at level $i$. In A-HDACS, since $M_{i}^{(l)}=K_{i}^{(l)}\log{N_{i}^{(l)}}<N_{i}^{(l)}$, CS can be utilized. Let’s define $C_{i}^{\prime}$ consisting of all the clusters as CS-disabled nodes at level $i$ in A-HDACS, $\rho_{i}$ the percentage of CS-disabled clusters at level $i$. In cluster $l$, $\sigma_{i}^{(l)}$ is defined as the percentage of the CS-disabled children clusters in a CS-disabled cluster at level $i$, where $\sigma_{i}^{(l)}\in\\{\frac{1}{n},\frac{2}{n},\cdots,\frac{n}{n}\\}$. We get $\rho_{i}=\frac{\|C_{i}^{\prime}\|}{\|C_{i}\|}$ at level $i$; and $\rho_{i-1}=\frac{\sum_{l=1}^{\|C_{i}^{\prime}\|}n\sigma_{i}^{(l)}}{\|C_{i-1}\|}$ at level $i-1$. ###### Proposition 2 If $K_{T}>K_{i\\_T}$, the CS-disabled nodes of A-HDACS at the level equal to and below $i$ are only small percentage of that of HDACS. ###### Proof: Let’s define $\sigma_{i}=\frac{1}{\|C_{i}^{\prime}\|}\sum_{l=1}^{\|C_{i}^{\prime}\|}\sigma_{i}^{(l)}$, which shows the average ratio of CS-disabled children clusters to their parent clusters. Therefore, we get $\rho_{i-1}=\frac{n\|C_{i}^{\prime}\|\sigma_{i}}{\|C_{i-1}\|}=\frac{\|C_{i}^{\prime}\|\sigma_{i}}{\|C_{i}\|}=\rho_{i}\sigma_{i}$. Follow the same derivation, $\rho_{i-2}=\rho_{i}\sigma_{i}\sigma_{i-1},\rho_{i-3}=\rho_{i}\sigma_{i}\sigma_{i-1}\sigma_{i-2},\cdots,\rho_{1}=\rho_{i}\sigma_{i}\sigma_{i-1}\cdots\sigma_{2}$. In summary, the ratio of CS-disabled clusters in HDACS at level $i$ and below level $i$ is: $\zeta=\frac{\sum_{j=1}^{i}\|C_{j}\|\rho_{j}}{\sum_{j=1}^{i}\|C_{j}\|}=\frac{\sum_{j=1}^{i}\|C_{j}\|\rho_{i}(\sigma_{i}\sigma_{i-1}\cdots\sigma_{j+1})}{\sum_{j=1}^{i}\|C_{j}\|}$ Since $\rho_{i}$ and $\sigma_{i}$ are equal to or less than 1, $\zeta$ is strictly less than 1. Thus, it proves that only $\zeta$ percent of the nodes at the level equal to and below $i$ are CS-disabled nodes for A-HDACS. ∎ At the level higher than $i$, i.e. $i<t<T$, the conditions are more diversified and we summarize them as follows: 1. 1. If $\frac{N_{t}^{(l)}}{\log{N_{t}^{(l)}}}>K_{t}^{(l)}>K_{T}$, HDACS and A-HDACS both enable CS. HDACS requires fewer measurements than A-HDACS. But the problem is whether or not HDACS can guarantee the recovery accuracy when a local area has significantly more data variations compared to the global area. 2. 2. If $K_{t}^{(l)}>\frac{N_{t}^{(l)}}{\log{N_{t}^{(l)}}}>K_{T}$, HDACS enables CS and A-HDACS requires direct data transmission. But it has the the same problem as condition 1). 3. 3. If $K_{T}>\frac{N_{t}^{(l)}}{\log{N_{t}^{(l)}}}>K_{t}^{(l)}$, A-HDACS enables CS but HDACS does not. 4. 4. If $\frac{N_{t}^{(l)}}{\log{N_{t}^{(l)}}}>K_{T}>K_{t}^{(l)}$, both HDACS and A-HDACS enable CS. But HDACS requires more measurements. 5. 5. The remaining conditions disable CS for both aggregation models. (a) A smooth data field with fluctuations (b) HDACS logical tree (c) A-HDACS logical tree Figure 1: An example of a smooth data field with fluctuations and its corresponding logical tree in HDACS and A-HDACS To better understand this analysis, Fig.1(a) gives a simple example of a smooth data field with a few variations measured by the sensor network in a data aggregation task. Fig.1(b) and Fig.1(c) are its corresponding logical hierarchical trees in HDACS and A-HDACS. The local variations in data field lead to the large value of global sparsity constant $K_{T}$ of the data field, and in HDACS it leads to plenty of nodes to be classified as CS-disabled nodes. However, in the same situation, since in A-HDACS sparsity constants $K_{i}$s are computed based on local variations in each cluster $i$, a large fraction of the CS-disabled nodes in HDACS become CS-enabled nodes in A-HDACS. ## V Performance Evaluation ### V-A Simulation Settings We evaluate the performance of the proposed A-HDACS scheme using SIDnet-SWANS [12], a JAVA based sensor network simulation platform. In our experiments we have used multiple network sizes, ranging from 300 to 800 sensor nodes, populated in a fixed field size of $40004000m^{2}$ area. The average nodes distribution density varies from $18.75/\text{km}^{2}$ to $50/\text{km}^{2}$. Fig. 2(a) shows a constant data field filled with randomly located Gaussian bumps. It has the maximum height 10 units and decays with 0.01 exponential rate. Fig. 2(b) depicts a smooth data field with a discontinuity along the line $x=y$, where the readings from smooth area are either 10 or 20 plus independent Gaussian noise with zero mean and 0.01 variance. (a) Smooth data field filled with Gaussian bumps (b) Piecewise data field (c) DCT domain of smooth data field filled with Gaussian bumps (d) DCT domain of piecewise data field Figure 2: Data Fields and their corresponding DCT Domain Besides, Discrete Cosine Transform (DCT) has been used to represent the data field in the transform domain. DCT is a suboptimal transform for sparse signal representation and approaches the ideal optimal transform when the correlation coefficient between adjacent data elements approaches unity [13]. Fig. 2(c) and Fig. 2(d) show the results when data fields are projected into DCT space. As we can see, most signal energy is captured in a very few coefficients, and the magnitudes decay rapidly. Also, note that the DCT signal corresponding to the piecewise data field, shown in Fig. 2(d), has less fluctuations than the signal corresponding to the smooth data field with Gaussian bumps, shown in Fig. 2(c). ### V-B The Nodes Distribution Fig. 3 shows the SIDnet simulation results of A-HDACS and HDACS for two types of data fields with network size 400, where black nodes denote CS-enabled nodes, gray nodes denote that are unable to use CS, and white nodes are the leaf nodes at level one of the aggregation tree. As we can see in Fig3(a), due to the scattered fluctuations present in the data field with Gaussian bumps it is very difficult to obtain sparse signal representation, therefore there are only a few CS-enabled nodes. But still for the clusters in local smooth data areas A-HDACS is able to utilize CS. Fig. 3(b) shows that piecewise data field has a large percent of CS-enabled nodes. CS-disabled nodes are mainly placed around the discontinuity of the line $x=y$. And the clusters away from this line can fully utilize CS. Fig. 3(c) and Fig. 3(d) depict the nodes distribution for both data fields using HDACS. The results are identical: almost no CS can be performed at the lower level except a few nodes at top levels. It demonstrates the significant improvement of CS-enabled nodes in A-HDACS and it is consistent with theoretical analysis in Section IV. (a) A-HDACS: smooth data field filled with Gaussian bumps (b) A-HDACS: piecewise data field (c) HDACS: smooth data field filled with Gaussian bumps (d) HDACS: piecewise data field Figure 3: The SIDnet simulation results of A-HDACS and HDACS with network size 400: black nodes denote CS-enabled nodes, gray nodes denote CS-disabled nodes, white nodes are the leaf nodes on level one, and red node denotes the sink. ### V-C Data Recovery Results Common signals are usually K-compressive – K entries with significant magnitudes and the other entries rapidly decaying to zero. Since K-sparse signal is one requirement of CS, it is necessary to perform signal truncation process. In the simulation, we tested different signal truncation thresholds so as to get as many CS-enabled nodes as possible without compromising too much signal recovery accuracy. Based on the characteristic of DCT signal, truncation threshold is set up as the percentage of the first dominant magnitude. In the evaluation, Mean Square Error (MSE) of recovered signal in the root node (sink) is defined as the average difference between recovered data and actual reading values for all the sensors. Fig. 4 depicts MSE versus DCT truncation threshold for two types of data field with network size 400. Since small truncation threshold filters out fewer significant entries than larger thresholds, it obtains better MSE. Fig. 4 shows that MSE of the smooth data field with Gaussian bumps is below 0.066 when DCT thresholds are smaller than 0.0225, and it increases dramatically when DCT thresholds are large. In the case of the field with Gaussian bumps, fluctuations in the signal cause increase in the number of DCT coefficients that has significant magnitudes, therefore truncation process is less effective. Relatively, piecewise field has more smooth clustering area with only a few significant entries. Its MSE is under a negligible value when DCT threshold is in the range of $[0.005,0.03]$. Figure 4: MSE versus DCT truncation threshold with network size 400 In the simulation results presented here onwards, DCT magnitudes bigger than $1\%$ of the first dominant coefficient are preserved. Figs. 5(a) and 5(b) show MSE at each level of the aggregation tree for the two data fields. In both cases, MSE results deteriorate with the increase of levels. This is because the signal truncation errors propagate in the data aggregation hierarchy. In the meanwhile, comparing Fig. 5(a) with Fig. 5(b), overall piecewise data field has smaller errors than the smooth data field with Gaussian bumps. It is due to relatively less fluctuations in the piecewise smooth data field. (a) Smooth data field filled with Gaussian bumps (b) Piecewise data field Figure 5: Data recovery mean square error (MSE) results at each level ### V-D Energy Consumption (a) Smooth data field filled with Gaussian bumps (b) Piecewise data field Figure 6: Total Transmission Energy Cost versus Different Network Sizes Since communication operations consumes majority of the battery power, we start counting energy consumption only when data aggregation begins. Fig. 6(a) and Fig. 6(b) show energy consumption versus networks size for two types of data field. A-HDACS consumes only $90.1\%\sim 94.20\%$ energy of HDACS in all the network sizes. Although plenty of fluctuations in the data field affects A-HDACS to apply CS in a certain degree, it still captures the sparsity feature within a few cluster area. But HDACS is insensitive to the local area, when the data field slightly change, it loses its data compression capability. This advantage is obvious, when it comes to the piecewise data field. Fig. 6(b) shows that A-HDACS can save around $23.36\%\sim 30.17\%$ battery power, compared to HDACS. The results demonstrate that significant energy efficiency can be obtained by the proposed technique. ## VI Conclusion and Future Work In this paper, Adaptive Hierarchical Data Aggregation using Compressive Sensing (A-HDACS) has been proposed to perform data aggregation in non-smooth multimodal data fields. Existing CS based data aggregation schemes for WSNs are inefficient for such data fields, in terms of energy consumed and amount of data transmitted. The A-HDACS solution is based on computing sparsity coefficients using signal sparsity of data gathered in local clusters. We analytically prove that A-HDACS enables more clusters to use CS compared to conventional HDACS. The simulation evaluated on SINnet-SWANS also demonstrates the feasibility and robustness of A-HDACS and its significant improvement of energy efficiency as well as accurate data recovery results. In the future work, more factors will be considered to strength A-HDACS. For example, in our implementations the cluster size is fixed at each level of the hierarchy. It may be possible to further improve communication cost if cluster size itself is also set up depending on the local density of the nodes. Besides, temporal correlations in the data may be exploited to further reduce the amount of data transmitted. Finally, other distributed computing tasks beyond data aggregation, such DFT, DWT, will also be implemented using A-HDACS framework, to take advantage of its power-efficient execution. ## References [1] R. Rajagopalan and P. K. Varshney, “Data aggregation techniques in sensor networks: A survey,” _IEEE Communications Surveys and Tutorials_ , vol. 8, no. 4, 2006. * [2] H. 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Clarke, “Relation between the karhunen loã¨ve and cosine transforms,” _Communications, Radar and Signal Processing, IEE Proceedings F_ , vol. 128, no. 6, pp. 359 – 360, Nov 1981.	arxiv-papers	2013-11-19T16:36:56	2024-09-04T02:49:53.870929	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Xi Xu, Rashid Ansari and Ashfaq Khokhar", "submitter": "Xi Xu", "url": "https://arxiv.org/abs/1311.4799" }
1311.4823	EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH (CERN) CERN-PH-EP-2013-207 LHCb-PAPER-2013-056 19 November 2013 Studies of beauty baryon decays to $D^{0}ph^{-}$ and $\mathchar 28931\relax^{+}_{c}h^{-}$ final states The LHCb collaboration†††Authors are listed on the following pages. Decays of beauty baryons to the $D^{0}ph^{-}$ and $\mathchar 28931\relax^{+}_{c}h^{-}$ final states (where $h$ indicates a pion or a kaon) are studied using a data sample of $pp$ collisions, corresponding to an integrated luminosity of 1.0$\mbox{\,fb}^{-1}$, collected by the LHCb detector. The Cabibbo-suppressed decays $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}$ and $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}$ are observed and their branching fractions are measured with respect to the decays $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ and $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-}$. In addition, the first observation is reported of the decay of the neutral beauty-strange baryon $\mathchar 28932\relax_{b}^{0}$ to the $D^{0}pK^{-}$ final state, and a measurement of the $\mathchar 28932\relax_{b}^{0}$ mass is performed. Evidence of the $\mathchar 28932\relax_{b}^{0}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}$ decay is also reported. Submitted to Phys. Rev. D © CERN on behalf of the LHCb collaboration, license CC-BY-3.0. LHCb collaboration R. Aaij40, B. Adeva36, M. Adinolfi45, C. Adrover6, A. Affolder51, Z. Ajaltouni5, J. Albrecht9, F. Alessio37, M. Alexander50, S. Ali40, G. Alkhazov29, P. Alvarez Cartelle36, A.A. Alves Jr24, S. Amato2, S. Amerio21, Y. Amhis7, L. Anderlini17,f, J. Anderson39, R. Andreassen56, M. Andreotti16,e, J.E. Andrews57, R.B. Appleby53, O. Aquines Gutierrez10, F. Archilli37, A. Artamonov34, M. Artuso58, E. Aslanides6, G. Auriemma24,m, M. Baalouch5, S. Bachmann11, J.J. Back47, A. Badalov35, V. Balagura30, W. Baldini16, R.J. Barlow53, C. 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Cenci57, M. Charles8, Ph. Charpentier37, S.-F. Cheung54, N. Chiapolini39, M. Chrzaszcz39,25, K. Ciba37, X. Cid Vidal37, G. Ciezarek52, P.E.L. Clarke49, M. Clemencic37, H.V. Cliff46, J. Closier37, C. Coca28, V. Coco40, J. Cogan6, E. Cogneras5, P. Collins37, A. Comerma-Montells35, A. Contu15,37, A. Cook45, M. Coombes45, S. Coquereau8, G. Corti37, B. Couturier37, G.A. Cowan49, D.C. Craik47, M. Cruz Torres59, S. Cunliffe52, R. Currie49, C. D’Ambrosio37, J. Dalseno45, P. David8, P.N.Y. David40, A. Davis56, I. De Bonis4, K. De Bruyn40, S. De Capua53, M. De Cian11, J.M. De Miranda1, L. De Paula2, W. De Silva56, P. De Simone18, D. Decamp4, M. Deckenhoff9, L. Del Buono8, N. Déléage4, D. Derkach54, O. Deschamps5, F. Dettori41, A. Di Canto11, H. Dijkstra37, M. Dogaru28, S. Donleavy51, F. Dordei11, P. Dorosz25,n, A. Dosil Suárez36, D. Dossett47, A. Dovbnya42, F. Dupertuis38, P. Durante37, R. Dzhelyadin34, A. Dziurda25, A. Dzyuba29, S. Easo48, U. Egede52, V. Egorychev30, S. Eidelman33, D. van Eijk40, S. Eisenhardt49, U. Eitschberger9, R. Ekelhof9, L. Eklund50,37, I. El Rifai5, Ch. Elsasser39, A. Falabella14,e, C. Färber11, C. Farinelli40, S. Farry51, D. Ferguson49, V. Fernandez Albor36, F. Ferreira Rodrigues1, M. Ferro-Luzzi37, S. Filippov32, M. Fiore16,e, M. Fiorini16,e, C. Fitzpatrick37, M. Fontana10, F. Fontanelli19,i, R. Forty37, O. Francisco2, M. Frank37, C. Frei37, M. Frosini17,37,f, E. Furfaro23,k, A. Gallas Torreira36, D. Galli14,c, M. Gandelman2, P. Gandini58, Y. Gao3, J. Garofoli58, P. Garosi53, J. Garra Tico46, L. Garrido35, C. Gaspar37, R. Gauld54, E. Gersabeck11, M. Gersabeck53, T. Gershon47, Ph. Ghez4, V. Gibson46, L. Giubega28, V.V. Gligorov37, C. Göbel59, D. Golubkov30, A. Golutvin52,30,37, A. Gomes2, H. Gordon37, M. Grabalosa Gándara5, R. Graciani Diaz35, L.A. Granado Cardoso37, E. Graugés35, G. Graziani17, A. Grecu28, E. Greening54, S. Gregson46, P. Griffith44, L. Grillo11, O. Grünberg60, B. Gui58, E. Gushchin32, Yu. Guz34,37, T. Gys37, C. Hadjivasiliou58, G. Haefeli38, C. Haen37, T.W. Hafkenscheid62, S.C. Haines46, S. Hall52, B. Hamilton57, T. Hampson45, S. Hansmann-Menzemer11, N. Harnew54, S.T. Harnew45, J. Harrison53, T. Hartmann60, J. He37, T. Head37, V. Heijne40, K. Hennessy51, P. Henrard5, J.A. Hernando Morata36, E. van Herwijnen37, M. Heß60, A. Hicheur1, E. Hicks51, D. Hill54, M. Hoballah5, C. Hombach53, W. Hulsbergen40, P. Hunt54, T. Huse51, N. Hussain54, D. Hutchcroft51, D. Hynds50, V. Iakovenko43, M. Idzik26, P. Ilten55, R. Jacobsson37, A. Jaeger11, E. Jans40, P. Jaton38, A. Jawahery57, F. Jing3, M. John54, D. Johnson54, C.R. Jones46, C. Joram37, B. Jost37, N. Jurik58, M. Kaballo9, S. Kandybei42, W. Kanso6, M. Karacson37, T.M. Karbach37, I.R. Kenyon44, T. Ketel41, B. Khanji20, S. Klaver53, O. Kochebina7, I. Komarov38, R.F. Koopman41, P. Koppenburg40, M. Korolev31, A. Kozlinskiy40, L. Kravchuk32, K. Kreplin11, M. Kreps47, G. Krocker11, P. Krokovny33, F. Kruse9, M. Kucharczyk20,25,37,j, V. 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Zvyagin37. 1Centro Brasileiro de Pesquisas Físicas (CBPF), Rio de Janeiro, Brazil 2Universidade Federal do Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil 3Center for High Energy Physics, Tsinghua University, Beijing, China 4LAPP, Université de Savoie, CNRS/IN2P3, Annecy-Le-Vieux, France 5Clermont Université, Université Blaise Pascal, CNRS/IN2P3, LPC, Clermont- Ferrand, France 6CPPM, Aix-Marseille Université, CNRS/IN2P3, Marseille, France 7LAL, Université Paris-Sud, CNRS/IN2P3, Orsay, France 8LPNHE, Université Pierre et Marie Curie, Université Paris Diderot, CNRS/IN2P3, Paris, France 9Fakultät Physik, Technische Universität Dortmund, Dortmund, Germany 10Max-Planck-Institut für Kernphysik (MPIK), Heidelberg, Germany 11Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany 12School of Physics, University College Dublin, Dublin, Ireland 13Sezione INFN di Bari, Bari, Italy 14Sezione INFN di Bologna, Bologna, Italy 15Sezione INFN di Cagliari, Cagliari, Italy 16Sezione INFN di Ferrara, Ferrara, Italy 17Sezione INFN di Firenze, Firenze, Italy 18Laboratori Nazionali dell’INFN di Frascati, Frascati, Italy 19Sezione INFN di Genova, Genova, Italy 20Sezione INFN di Milano Bicocca, Milano, Italy 21Sezione INFN di Padova, Padova, Italy 22Sezione INFN di Pisa, Pisa, Italy 23Sezione INFN di Roma Tor Vergata, Roma, Italy 24Sezione INFN di Roma La Sapienza, Roma, Italy 25Henryk Niewodniczanski Institute of Nuclear Physics Polish Academy of Sciences, Kraków, Poland 26AGH - University of Science and Technology, Faculty of Physics and Applied Computer Science, Kraków, Poland 27National Center for Nuclear Research (NCBJ), Warsaw, Poland 28Horia Hulubei National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania 29Petersburg Nuclear Physics Institute (PNPI), Gatchina, Russia 30Institute of Theoretical and Experimental Physics (ITEP), Moscow, Russia 31Institute of Nuclear Physics, Moscow State University (SINP MSU), Moscow, Russia 32Institute for Nuclear Research of the Russian Academy of Sciences (INR RAN), Moscow, Russia 33Budker Institute of Nuclear Physics (SB RAS) and Novosibirsk State University, Novosibirsk, Russia 34Institute for High Energy Physics (IHEP), Protvino, Russia 35Universitat de Barcelona, Barcelona, Spain 36Universidad de Santiago de Compostela, Santiago de Compostela, Spain 37European Organization for Nuclear Research (CERN), Geneva, Switzerland 38Ecole Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland 39Physik-Institut, Universität Zürich, Zürich, Switzerland 40Nikhef National Institute for Subatomic Physics, Amsterdam, The Netherlands 41Nikhef National Institute for Subatomic Physics and VU University Amsterdam, Amsterdam, The Netherlands 42NSC Kharkiv Institute of Physics and Technology (NSC KIPT), Kharkiv, Ukraine 43Institute for Nuclear Research of the National Academy of Sciences (KINR), Kyiv, Ukraine 44University of Birmingham, Birmingham, United Kingdom 45H.H. Wills Physics Laboratory, University of Bristol, Bristol, United Kingdom 46Cavendish Laboratory, University of Cambridge, Cambridge, United Kingdom 47Department of Physics, University of Warwick, Coventry, United Kingdom 48STFC Rutherford Appleton Laboratory, Didcot, United Kingdom 49School of Physics and Astronomy, University of Edinburgh, Edinburgh, United Kingdom 50School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom 51Oliver Lodge Laboratory, University of Liverpool, Liverpool, United Kingdom 52Imperial College London, London, United Kingdom 53School of Physics and Astronomy, University of Manchester, Manchester, United Kingdom 54Department of Physics, University of Oxford, Oxford, United Kingdom 55Massachusetts Institute of Technology, Cambridge, MA, United States 56University of Cincinnati, Cincinnati, OH, United States 57University of Maryland, College Park, MD, United States 58Syracuse University, Syracuse, NY, United States 59Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rio de Janeiro, Brazil, associated to 2 60Institut für Physik, Universität Rostock, Rostock, Germany, associated to 11 61National Research Centre Kurchatov Institute, Moscow, Russia, associated to 30 62KVI - University of Groningen, Groningen, The Netherlands, associated to 40 63Celal Bayar University, Manisa, Turkey, associated to 37 aP.N. Lebedev Physical Institute, Russian Academy of Science (LPI RAS), Moscow, Russia bUniversità di Bari, Bari, Italy cUniversità di Bologna, Bologna, Italy dUniversità di Cagliari, Cagliari, Italy eUniversità di Ferrara, Ferrara, Italy fUniversità di Firenze, Firenze, Italy gUniversità di Urbino, Urbino, Italy hUniversità di Modena e Reggio Emilia, Modena, Italy iUniversità di Genova, Genova, Italy jUniversità di Milano Bicocca, Milano, Italy kUniversità di Roma Tor Vergata, Roma, Italy lUniversità di Roma La Sapienza, Roma, Italy mUniversità della Basilicata, Potenza, Italy nAGH - University of Science and Technology, Faculty of Computer Science, Electronics and Telecommunications, Kraków, Poland oLIFAELS, La Salle, Universitat Ramon Llull, Barcelona, Spain pHanoi University of Science, Hanoi, Viet Nam qUniversità di Padova, Padova, Italy rUniversità di Pisa, Pisa, Italy sScuola Normale Superiore, Pisa, Italy ## 1 Introduction Although there has been great progress in studies of beauty mesons, both at the $B$ factories and hadron machines, the beauty baryon sector remains largely unexplored. The quark model predicts seven ground-state ($J^{P}=\frac{1}{2}^{+}$) baryons involving a $b$ quark and two light ($u$, $d$, or $s$) quarks [1]. These are the $\mathchar 28931\relax^{0}_{b}$ isospin singlet, the $\mathchar 28934\relax_{b}$ triplet, the $\mathchar 28932\relax_{b}$ strange doublet, and the doubly strange state $\mathchar 28938\relax^{-}_{b}$. Among these states, the $\mathchar 28934\relax_{b}^{0}$ baryon has not been observed yet, while for the others the quantum numbers have not been experimentally established, very few decay modes have been measured, and fundamental properties such as masses and lifetimes are in general poorly known. Moreover, the $\mathchar 28934\relax_{b}^{\pm}$ and $\mathchar 28932\relax_{b}^{0}$ baryons have been observed by a single experiment [2, 3]. It is therefore of great interest to study $b$ baryons, and to determine their properties. The decays of $b$ baryons can be used to study $C\\!P$ violation and rare processes. In particular, the decay $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}\mathchar 28931\relax$ has been proposed to measure the Cabibbo- Kobayashi-Maskawa (CKM) unitarity triangle angle $\gamma$ [4, 5, 6] following an approach analogous to that for $B^{0}\rightarrow DK^{0}$ decays [7]. A possible extension to the analysis of the $D^{0}\mathchar 28931\relax$ final state is to use the $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}$ decay, with the $pK^{-}$ pair originating from the $\mathchar 28931\relax^{0}_{b}$ decay vertex. Such an approach can avoid limitations due to the lower reconstruction efficiency of the $\mathchar 28931\relax$ decay. In addition, if the full phase space of the three-body decay is used, the sensitivity to $\gamma$ may be enhanced, in a similar manner to the Dalitz plot analysis of $B^{0}\rightarrow DK^{+}\pi^{-}$ decays, which offers certain advantages over the quasi-two-body $B^{0}\rightarrow DK^{0}$ analysis [8, 9]. This paper reports the results of a study of beauty baryon decays into $D^{0}p\pi^{-}$, $D^{0}pK^{-}$, $\mathchar 28931\relax^{+}_{c}\pi^{-}$, and $\mathchar 28931\relax^{+}_{c}K^{-}$ final states.111The inclusion of charge- conjugate processes is implied. A data sample corresponding to an integrated luminosity of 1.0$\mbox{\,fb}^{-1}$ is used, collected by the LHCb detector [10] in $pp$ collisions with centre-of-mass energy of 7$\mathrm{\,Te\kern-1.00006ptV}$. Six measurements are performed in this analysis, listed below. The decay mode $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ is the Cabibbo-favoured partner of $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}$ with the same topology and higher rate. We measure its rate using the mode $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-}$ for normalisation. To avoid dependence on the poorly measured branching fraction of the $\mathchar 28931\relax^{+}_{c}\rightarrow pK^{-}\pi^{+}$ decay, we quote the ratio $R_{\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}}\equiv\frac{{\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-})\times{\cal B}(D^{0}\rightarrow K^{-}\pi^{+})}{{\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-})\times{\cal B}(\mathchar 28931\relax^{+}_{c}\rightarrow pK^{-}\pi^{+})}\,.$ (1) The $D^{0}$ meson is reconstructed in the favoured final state $K^{-}\pi^{+}$ and the $\mathchar 28931\relax^{+}_{c}$ baryon in the $pK^{-}\pi^{+}$ mode. In this way, the $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-}$ and $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ decays have the same final state particles, and some of the systematic uncertainties, in particular those related to particle identification (PID), cancel in the ratio. The branching fraction of the Cabibbo-suppressed $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}$ decay mode is measured with respect to that of $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ $R_{\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}}\equiv\frac{{\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-})}{{\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-})}\,.$ (2) The Cabibbo-suppressed decay $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}$ is also studied. This decay has been considered in various analyses as a background component [11, 12], but a dedicated study has not been performed so far. We measure the ratio $R_{\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}}\equiv\frac{{\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-})}{{\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-})}\,.$ (3) The heavier beauty-strange $\mathchar 28932\relax_{b}^{0}$ baryon can also decay into the final states $D^{0}pK^{-}$ and $\mathchar 28931\relax^{+}_{c}K^{-}$ via $b\rightarrow c\overline{}ud$ colour-suppressed transitions. Previously, the $\mathchar 28932\relax_{b}^{0}$ baryon has only been observed in one decay mode, $\mathchar 28932\relax_{b}^{0}\rightarrow\mathchar 28932\relax_{c}^{+}\pi^{-}$ [3], thus it is interesting to study other final states, as well as to measure its mass more precisely. Here we report measurements of the ratios of rates for $\mathchar 28932\relax_{b}^{0}\rightarrow D^{0}pK^{-}$, $R_{\mathchar 28932\relax_{b}^{0}\rightarrow D^{0}pK^{-}}\equiv\frac{f_{\mathchar 28932\relax_{b}^{0}}\times{\cal B}(\mathchar 28932\relax_{b}^{0}\rightarrow D^{0}pK^{-})}{f_{\mathchar 28931\relax^{0}_{b}}\times{\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-})}\,,$ (4) and $\mathchar 28932\relax_{b}^{0}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}$ decays, $R_{\mathchar 28932\relax_{b}^{0}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}}\equiv\frac{{\cal B}(\mathchar 28932\relax_{b}^{0}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-})\times{\cal B}(\mathchar 28931\relax^{+}_{c}\rightarrow pK^{-}\pi^{+})}{{\cal B}(\mathchar 28932\relax_{b}^{0}\rightarrow D^{0}pK^{-})\times{\cal B}(D^{0}\rightarrow K^{-}\pi^{+})}\,,$ (5) where $f_{\mathchar 28932\relax_{b}^{0}}$ and $f_{\mathchar 28931\relax^{0}_{b}}$ are the fragmentation fractions of the $b$ quark to $\mathchar 28932\relax_{b}^{0}$ and $\mathchar 28931\relax^{0}_{b}$ baryons, respectively. The difference of $\mathchar 28932\relax_{b}^{0}$ and $\mathchar 28931\relax^{0}_{b}$ masses, $m_{\mathchar 28932\relax_{b}^{0}}-m_{\mathchar 28931\relax^{0}_{b}}$, is also measured. ## 2 Detector description The LHCb detector [10] is a single-arm forward spectrometer covering the pseudorapidity range $2<\eta<5$, designed for the study of particles containing $b$ or $c$ quarks. The detector includes a high-precision tracking system consisting of a silicon-strip vertex detector surrounding the $pp$ interaction region, a large-area silicon-strip detector located upstream of a dipole magnet with a bending power of about $4{\rm\,Tm}$, and three stations of silicon-strip detectors and straw drift tubes placed downstream. The combined tracking system provides a momentum measurement with relative uncertainty that varies from 0.4% at 5${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ to 0.6% at 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, and impact parameter (IP) resolution of 20$\,\upmu\rm m$ for tracks with high transverse momentum ($p_{\rm T}$). Charged hadrons are identified using two ring-imaging Cherenkov (RICH) detectors [13]. Photon, electron and hadron candidates are identified by a calorimeter system consisting of scintillating-pad and preshower detectors, an electromagnetic calorimeter and a hadronic calorimeter. Muons are identified by a system composed of alternating layers of iron and multiwire proportional chambers [14]. The trigger [15] consists of a hardware stage, based on information from the calorimeter and muon systems, followed by a software stage, which applies a full event reconstruction. Events used in this analysis are required to satisfy at least one hardware trigger requirement: a final state particle has to deposit energy in the calorimeter system above a certain threshold, or the event has to be triggered by any of the requirements not involving the signal decay products. The software trigger requires a two-, three-, or four-track secondary vertex with a high sum of $p_{\rm T}$ of the tracks and a significant displacement from the primary $pp$ interaction vertices (PVs). At least one track should have $\mbox{$p_{\rm T}$}>1.7{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and $\chi^{2}_{\rm IP}$ with respect to any PV greater than 16, where $\chi^{2}_{\rm IP}$ is defined as the difference in $\chi^{2}$ of a given PV reconstructed with and without the considered track. A multivariate algorithm [16] is used for the identification of secondary vertices consistent with the decay of a $b$ hadron. In the simulation, $pp$ collisions are generated using Pythia 6.4 [17] with a specific LHCb configuration [18]. Decays of hadronic particles are described by EvtGen [19]; the interaction of the generated particles with the detector and its response are implemented using the Geant4 toolkit [20, Agostinelli:2002hh] as described in Ref. [22]. ## 3 Selection criteria The analysis uses four combinations of final-state particles to form the $b$-baryon candidates: $\mathchar 28931\relax^{+}_{c}\pi^{-}$, $D^{0}p\pi^{-}$, $\mathchar 28931\relax^{+}_{c}K^{-}$, and $D^{0}pK^{-}$. The $D^{0}$ mesons are reconstructed in the $K^{-}\pi^{+}$ final state, and $\mathchar 28931\relax^{+}_{c}$ baryons are reconstructed from $pK^{-}\pi^{+}$ combinations. In addition, the combinations with the $D^{0}$ meson of opposite flavour (i.e. $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}p\pi^{-}$ and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}pK^{-}$ with $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}\pi^{-}$) are selected to better constrain the shape of the combinatorial background in $D^{0}ph^{-}$ final states. These decay modes correspond to either doubly Cabibbo-suppressed decays of the $D^{0}$, or to $b\rightarrow u$ transitions in the $\mathchar 28931\relax^{0}_{b}$ and $\mathchar 28932\relax_{b}^{0}$ decays, and are expected to contribute a negligible amount of signal in the current data sample. The selection of $b$-baryon candidates is performed in two stages: the preselection and the final selection. The preselection is performed to select events containing a beauty hadron candidate with an intermediate charm state. It requires that the tracks forming the candidate, as well as the beauty and charm vertices, have good quality and are well separated from any PV, and the invariant masses of the beauty and charm hadrons are in the region of the known values of the masses of the corresponding particles. The preselection has an efficiency 95–99% for the signal depending on the decay mode. Two different sets of requirements are used for the final selection. The ratio $R_{\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}}$ is measured by fitting the invariant mass distribution for candidates obtained with a loose selection to minimise the systematic uncertainty. The signal yields of these decays are large and the uncertainty in the ratio is dominated by systematic effects. The ratios $R_{\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}}$ and $R_{\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}}$ are less affected by systematic uncertainties since the topologies of the decays are the same. A tight multivariate selection is used in addition to the loose selection requirements when measuring these ratios, as well as the ratios of the $\mathchar 28932\relax_{b}^{0}$ decay rates. The loose selection requires that the invariant masses of the intermediate $\mathchar 28931\relax^{+}_{c}$ and $D^{0}$ candidates are within 25${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ of their known masses [1], and the decay time significance of the $D^{0}$ meson from the $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ decay is greater than one standard deviation. The decay time significance is defined as the measured decay time divided by its uncertainty for a given candidate. The final-state particles are required to satisfy PID criteria based on information from the RICH detectors [13]. Pion candidates are required to have a value $\mathrm{DLL}_{K\pi}<5$ for the difference of logarithms of likelihoods between the kaon and pion hypotheses; the efficiency of this requirement is about 95%. The requirement for kaon candidates of $\mathrm{DLL}_{K\pi}>0$ is about 97% efficient. The protons are required to satisfy $\mathrm{DLL}_{p\pi}>5$ and $\mathrm{DLL}_{pK}>0$. The corresponding efficiency is approximately 88%. The momentum of each final-state track is required to be less than 100${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$, corresponding to the range of good separation between particle types. For candidates passing the above selections, a kinematic fit is performed [23]. The fit employs constraints on the decay products of the $\mathchar 28931\relax^{0}_{b}$, $\mathchar 28931\relax^{+}_{c}$, and $D^{0}$ particles to originate from their respective vertices, the $\mathchar 28931\relax^{0}_{b}$ candidate to originate from the PV, and the $\mathchar 28931\relax^{+}_{c}$ and $D^{0}$ invariant masses to be equal to their known values [1]. A momentum scale correction is applied in the kinematic fit to improve the mass measurement as described in Ref. [24]. The momentum scale of the detector has been calibrated using inclusive ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}\rightarrow\mu^{+}\mu^{-}$ decays to account for the relative momentum scale between different data taking periods, while the absolute calibration is performed with $B^{+}\rightarrow{J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}K^{+}$ decays. The tight selection is based on a boosted decision tree (BDT) [25] trained with the gradient boost algorithm. The $D^{0}ph^{-}$ selection is optimised using simulated $D^{0}pK^{-}$ signal events, and combinations with opposite- flavour $D^{0}$ candidates ($\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}pK^{-}$) in data as a background estimate. The optimisation of the $\mathchar 28931\relax^{+}_{c}h^{-}$ selection is performed with a similar approach, with the $\mathchar 28931\relax^{+}_{c}K^{+}$ candidates as the background training sample. The optimisation criteria for the BDTs are the maximum expected statistical significances of the $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}$ and $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}$ signals, $S_{\rm stat}=N_{\rm sig}/\sqrt{N_{\rm sig}+N_{\rm bck}}$, where $N_{\rm sig}$ and $N_{\rm bck}$ are the expected numbers of signal and background events. The expected number of events for the optimisation is taken from the observed yields in the $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-}$ and $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ modes scaled by the Cabibbo suppression factor. The variables that enter the BDT selection are the following: the quality of the kinematic fit ($\chi^{2}_{\rm fit}/{\rm ndf}$, where ${\rm ndf}$ is the number of degrees of freedom in the fit); the minimum IP significance $\chi^{2}_{\rm IP}$ of the final-state and intermediate charm particles with respect to any PV; the lifetime significances of the $\mathchar 28931\relax^{0}_{b}$ and intermediate charm particles; and the PID variables ($\mathrm{DLL}_{p\pi}$ and $\mathrm{DLL}_{pK}$) for the proton candidate. The $D^{0}ph^{-}$ selection has a signal efficiency of 72% on candidates passing the loose selection while retaining 11% of the combinatorial background. The $\mathchar 28931\relax^{+}_{c}h^{-}$ selection is 99.5% efficient and retains 65% of the combinatorial background. In approximately 2% of events more than one candidate passes the selection. In these cases, only the candidate with the minimum $\chi_{\rm fit}^{2}/{\rm ndf}$ is retained for further analysis. Several vetoes are applied for both the loose and tight selections to reduce backgrounds. To veto candidates formed from $J/\psi\rightarrow\mu^{+}\mu^{-}$ combined with two tracks, at least one of the pion candidates in $\mathchar 28931\relax^{+}_{c}\pi^{-}$ and $D^{0}p\pi^{-}$ combinations is required not to have hits in the muon chambers. For $D^{0}ph^{-}$ combinations, a $\mathchar 28931\relax^{+}_{c}\rightarrow p\pi^{+}h^{-}$ veto is applied: the invariant mass of the $p\pi^{+}h^{-}$ combination is required to differ from the nominal $\mathchar 28931\relax^{+}_{c}$ mass by more than 20${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. This requirement rejects the background from $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}$ decays. Cross-feed between $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}ph^{-}$ and $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-}$ decays does not occur since the invariant mass of the $D^{0}p$ combination in $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}ph^{-}$ decays is greater than the $\mathchar 28931\relax^{+}_{c}$ invariant mass. ## 4 Determination of signal yields The signal yields are obtained from extended maximum likelihood fits to the unbinned invariant mass distributions. The fit model includes signal components ($\mathchar 28931\relax^{0}_{b}$ only for $\mathchar 28931\relax^{+}_{c}\pi^{-}$ and $D^{0}p\pi^{-}$ final states, and both $\mathchar 28931\relax^{0}_{b}$ and $\mathchar 28932\relax_{b}^{0}$ for $D^{0}pK^{-}$ and $\mathchar 28931\relax^{+}_{c}K^{-}$ final states), as well as various background contributions. The ratio $R_{\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}}$ is obtained from the combined fit of the $\mathchar 28931\relax^{+}_{c}\pi^{-}$ and $D^{0}p\pi^{-}$ invariant mass distributions of candidates that pass the loose selection, while the other quantities are determined from the simultaneous fit of the $\mathchar 28931\relax^{+}_{c}h^{-}$, $D^{0}ph^{-}$, and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}ph^{-}$ ($h=\pi$ or $K$) invariant mass distributions passing the tight BDT-based selection requirements. The shape of each signal contribution is taken from simulation and is parametrised using the sum of two Crystal Ball (CB) functions [26]. In the fit to data, the widths of each signal component are multiplied by a common scaling factor that is left free. This accounts for the difference between the invariant mass resolution observed in data and simulation. The masses of the $\mathchar 28931\relax^{0}_{b}$ and $\mathchar 28932\relax_{b}^{0}$ states are also free parameters. Their mean values as reconstructed in the $D^{0}ph^{-}$ and $\mathchar 28931\relax^{+}_{c}h^{-}$ spectra are allowed to differ by an amount $\Delta M$ (which is the same for $\mathchar 28931\relax^{0}_{b}$ and $\mathchar 28932\relax_{b}^{0}$ masses) to account for possible imperfect calibration of the momentum scale in the detector. The mass difference $\Delta M$ obtained from the fit is consistent with zero. The background components considered in the analysis are subdivided into three classes: random combinations of tracks, or genuine $D^{0}$ or $\mathchar 28931\relax^{+}_{c}$ decays combined with random tracks (combinatorial background); decays where one or more particles are incorrectly identified (misidentification background); and decays where one or more particles are not reconstructed (partially reconstructed background). The combinatorial background is parametrised with a quadratic function. The shapes are constrained to be the same for the $D^{0}ph^{-}$ signal and $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}ph^{-}$ background combinations. The $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}p\pi^{-}$ fit model includes only the combinatorial background component, while in the $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}pK^{-}$ model, the $\mathchar 28931\relax^{0}_{b}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}pK^{-}$ signal and partially reconstructed background are included with varying yields to avoid biasing the combinatorial background shape. The two contributions are found to be consistent with zero, as expected. Contributions of charmed $B$ decays with misidentified particles are studied using simulated samples. The $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}h^{-}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}h^{-}$ decay modes are considered as $\mathchar 28931\relax^{+}_{c}h^{-}$ backgrounds, while $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{0}\pi^{+}\pi^{-}$, $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{0}K^{+}K^{-}$ [27], and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{0}K^{+}\pi^{-}$ [28] are possible backgrounds in the $D^{0}ph^{-}$ spectra. These contributions to $D^{0}ph^{-}$ modes are found to be negligible and thus are not included in the fit model, while the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow D^{+}_{(s)}\pi^{-}$ component is significant and is included in the fit. The ratio between $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{s}\rightarrow D^{+}_{s}\pi^{-}$ and $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}\rightarrow D^{+}\pi^{-}$ contributions is fixed from the measured ratio of their event yields [29]. Contributions to $D^{0}pK^{-}$ and $\mathchar 28931\relax^{+}_{c}K^{-}$ spectra from the $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ and $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-}$ modes, respectively, with the pion misidentified as a kaon ($K/\pi$ misidentification backgrounds) are obtained by parametrising the simulated samples with a CB function. In the case of the $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ background, the squared invariant mass of the $D^{0}p$ combination, $M^{2}(D^{0}p)$, is required to be smaller than $10{\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$. This accounts for the dominance of events with low $D^{0}p$ invariant masses observed in data. In the case of the $\mathchar 28931\relax^{+}_{c}\pi^{-}$ spectrum, the $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}$ contribution with the kaon misidentified as a pion is also included. In all cases, the nominal selection requirements, including those for PID, are applied to the simulated samples. Partially reconstructed backgrounds, such as $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$, $D^{0}\rightarrow D^{0}\,\pi^{0}/\gamma$ decays, or $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28934\relax_{c}^{+}\pi^{-}$, $\mathchar 28934\relax_{c}^{+}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{0}$ decays, contribute at low invariant mass. Simulation is used to check that these backgrounds are well separated from the signal region. However, their mass distribution is expected to depend strongly on the unknown helicity structure of these decays. Therefore, an empirical probability density function (PDF), a bifurcated Gaussian distribution with free parameters, is used to parametrise them. The shapes of the backgrounds are constrained to be the same for the $D^{0}pK^{-}$ and $D^{0}p\pi^{-}$ decay modes, as well as for the $\mathchar 28931\relax^{+}_{c}K^{-}$ and $\mathchar 28931\relax^{+}_{c}\pi^{-}$ decay modes. Backgrounds from partially reconstructed $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ and $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28934\relax_{c}^{+}\pi^{-}$ decays with the pion misidentified as a kaon contribute to the $D^{0}pK^{-}$ and $\mathchar 28931\relax^{+}_{c}K^{-}$ mass spectra, respectively. These backgrounds are parametrised with CB functions fitted to samples simulated assuming that the amplitude is constant across the phase space. Their yields are constrained from the yields of partially reconstructed components in the $D^{0}p\pi^{-}$ and $\mathchar 28931\relax^{+}_{c}\pi^{-}$ spectra taking into account the $K/\pi$ misidentification probability. Charmless $\mathchar 28931\relax^{0}_{b}\rightarrow pK^{-}\pi^{+}h^{-}$ backgrounds, which have the same final state as the signal modes but no intermediate charm vertex, are studied with the $\mathchar 28931\relax^{0}_{b}$ invariant mass fit to data from the sidebands of the $D^{0}\rightarrow K^{-}\pi^{+}$ invariant mass distribution: $50<\|M(K^{-}\pi^{+})-m_{D^{0}}\|<100$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$. Similar sidebands are used in the $\mathchar 28931\relax^{+}_{c}\rightarrow pK^{-}\pi^{+}$ invariant mass. A significant contribution is observed in the $D^{0}p\pi^{-}$ mode. Hence, for the $D^{0}ph^{-}$ combinations, the $D^{0}$ vertex is required to be downstream of $\mathchar 28931\relax^{0}_{b}$ vertex and the $D^{0}$ decay time must differ from zero by more than one standard deviation. The remaining contribution is estimated from the $\mathchar 28931\relax^{0}_{b}$ invariant mass fit in the sidebands. The $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ yield obtained from the fit is corrected for a small residual charmless contribution, while in other modes the contribution of this background is consistent with zero. The $\mathchar 28931\relax^{+}_{c}\pi^{-}$ and $D^{0}p\pi^{-}$ invariant mass distributions obtained with the loose selection are shown in Fig. 1 with the fit result overlaid. The $\mathchar 28931\relax^{0}_{b}$ yields obtained from the fit to these spectra are presented in Table 1. Figures 2 and 3 show the invariant mass distributions for the $D^{0}ph^{-}$ and $\mathchar 28931\relax^{+}_{c}h^{-}$ modes after the tight BDT-based selection. The $\mathchar 28931\relax^{0}_{b}$ and $\mathchar 28932\relax_{b}^{0}$ yields, as well as their masses, obtained from the fit are given in Table 2. The raw masses obtained in the fit are used to calculate the difference of $\mathchar 28932\relax_{b}^{0}$ and $\mathchar 28931\relax^{0}_{b}$ masses, $m_{\mathchar 28932\relax_{b}^{0}}-m_{\mathchar 28931\relax^{0}_{b}}=174.8\pm 2.3{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, which is less affected by the systematic uncertainty due to knowledge of the absolute mass scale. (a)(b) Figure 1: Distributions of invariant mass for (a) $\mathchar 28931\relax^{+}_{c}\pi^{-}$ and (b) $D^{0}p\pi^{-}$ candidates passing the loose selection (points with error bars) and results of the fit (solid line). The signal and background contributions are shown. Table 1: Results of the fit to the invariant mass distribution of $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-}$ and $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ candidates passing the loose selection. The uncertainties are statistical only. Decay mode \| Yield ---\|--- $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ \| $3383\pm 94$ $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-}$ \| $50\,301\pm 253$ (a)(b) Figure 2: Distributions of invariant mass for (a) $D^{0}p\pi^{-}$ and (b) $D^{0}pK^{-}$ candidates passing the tight selection (points with error bars) and results of the fit (solid line). The signal and background contributions are shown. (a)(b) (c)(d) Figure 3: Distributions of invariant mass for (a) $\mathchar 28931\relax^{+}_{c}\pi^{-}$ and (b) $\mathchar 28931\relax^{+}_{c}K^{-}$ candidates passing the tight selection (points with error bars) and results of the fit (solid line). The signal and background contributions are shown. The same distributions are magnified in (c) and (d) to better distinguish background components and $\mathchar 28932\relax_{b}^{0}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}$ signal. Table 2: Results of the fit to the invariant mass distributions of $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}h^{-}$ and $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}ph^{-}$ candidates passing the tight selection. The uncertainties are statistical only. Decay mode \| Yield ---\|--- $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ \| $\,00$$2452\pm 58$$0$ $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-}$ \| $0$$50\,072\pm 253$ $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}$ \| $\,000$$163\pm 18$$0$ $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}$ \| $\,00$$3182\pm 66$$0$ $\mathchar 28932\relax_{b}^{0}\rightarrow D^{0}pK^{-}$ \| $\,0000$$74\pm 13$$0$ $\mathchar 28932\relax_{b}^{0}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}$ \| $\,0000$$62\pm 20$$0$ Particle \| Mass $[{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}]\vphantom{D^{0^{0}}}$ $\mathchar 28931\relax^{0}_{b}$ \| $5618.7\pm 0.1$ $\mathchar 28932\relax_{b}^{0}$ \| $5793.5\pm 2.3$ Figures 4 and 5 show the Dalitz plot of the three-body decay $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$, and the projections of the two invariant masses, where resonant contributions are expected. In the projections, the background is subtracted using the sPlot technique [30]. The distributions show an increased density of events in the low-$M(D^{0}p)$ region where a contribution from excited $\mathchar 28931\relax^{+}_{c}$ states is expected. The $\mathchar 28931\relax_{c}(2880)^{+}$ state is apparent in this projection. Structures in the $p\pi^{-}$ combinations are also visible. The Dalitz plot and projections of $D^{0}p$ and $pK^{-}$ invariant masses for the $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}$ mode are shown in Fig. 6. The distributions for the $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}$ mode exhibit similar behaviour with the dominance of a low-$M(D^{0}p)$ contribution and an enhancement in the low-$M(pK^{-})$ region. (a)(b)(c) Figure 4: Dalitz plot of $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ candidates in (a) the full phase space region, and magnified regions of (b) low $M^{2}(D^{0}p)$ and (c) low $M^{2}(p\pi^{-})$. (a)(b)(c)(d) Figure 5: Background-subtracted distributions of (a,b) $M(p\pi^{-})$ and (c,d) $M(D^{0}p)$ invariant masses in $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ decays, where (b) and (d) are versions of (a) and (c), respectively, showing the lower invariant mass parts of the distributions. The distributions are not corrected for efficiency. (a)(b)(c) Figure 6: (a) $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}$ Dalitz plot and background-subtracted distributions of (b) $M(pK^{-})$ and (c) $M(D^{0}p)$ invariant masses. The distributions are not corrected for efficiency. ## 5 Calculation of branching fractions The ratios of branching fractions are calculated from the ratios of yields of the corresponding decays after applying several correction factors $R=\frac{N^{i}}{N^{j}}\frac{\varepsilon^{j}_{\rm sel}}{\varepsilon^{i}_{\rm sel}}\frac{\varepsilon^{j}_{\rm PID}}{\varepsilon^{i}_{\rm PID}}\frac{\varepsilon^{j}_{\rm PS}}{\varepsilon^{i}_{\rm PS}},$ (6) where $N^{i}$ is the yield for the $i^{\mathrm{th}}$ decay mode, $\varepsilon^{i}_{\rm sel}$ is its selection efficiency excluding the PID efficiency, $\varepsilon^{i}_{\rm PID}$ is the efficiency of the PID requirements, and $\varepsilon^{i}_{\rm PS}$ is the phase-space acceptance correction defined below. The trigger, preselection and final selection efficiencies that enter $\varepsilon_{\rm sel}$ are obtained using simulated signal samples. The selection efficiency is calculated without the PID requirements applied, except for the proton PID in the tight selection, which enters the multivariate discriminant. Since the multiplicities of all the final states are the same, and the kinematic distributions of the decay products are similar, the uncertainties in the efficiencies largely cancel in the quoted ratios of branching fractions. The efficiencies of PID requirements for kaons and pions are obtained with a data-driven procedure using a large sample of $D^{+}\rightarrow D^{0}\pi^{+}$, $D^{0}\rightarrow K^{-}\pi^{+}$ decays. The calibration sample is weighted to reproduce the kinematic properties of the decays under study taken from simulation. For protons, however, the available calibration sample $\mathchar 28931\relax\rightarrow p\pi^{-}$ does not cover the full range in momentum- pseudorapidity space that the protons from the signal decays populate. Thus, in the case of the calculation of the ratio of $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-}$ and $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ branching fractions, the ratio of proton efficiencies is taken from simulation. For the calculation of the ratios ${\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-})/{\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-})$ and ${\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-})/{\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-})$, where the kinematic properties of the proton track for the decays in the numerator and denominator are similar, the efficiencies are taken to be equal. The simulated samples used to obtain the selection efficiency are generated with phase-space models for the three-body $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}ph^{-}$ and $\mathchar 28931\relax^{+}_{c}\rightarrow pK^{-}\pi^{+}$ decays. The three-body distributions in data are, however, significantly non-uniform. Therefore, the efficiency obtained from the simulation has to be corrected for the dependence on the three-body decay kinematic properties. In the case of $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ decays, the relative selection efficiency as a function of $D^{0}p$ and $p\pi^{-}$ squared invariant masses $\varepsilon[M^{2}(D^{0}p),M^{2}(p\pi^{-})]$ is determined from the phase- space simulated sample and parametrised with a polynomial function of fourth order. The function $\varepsilon[M^{2}(D^{0}p),M^{2}(p\pi^{-})]$ is normalised such that its integral is unity over the kinematically allowed phase space. The efficiency correction factor $\varepsilon_{\rm PS}$ is calculated as $\varepsilon_{\rm PS}=\frac{\sum_{i}w_{i}}{\sum_{i}w_{i}/\varepsilon[M_{i}^{2}(D^{0}p),M_{i}^{2}(p\pi^{-})]},$ (7) where $M^{2}_{i}(D^{0}p)$ and $M_{i}^{2}(p\pi^{-})$ are the squared invariant masses of the $D^{0}p$ and $p\pi^{-}$ combinations for the $i^{\mathrm{th}}$ event in data, and $w_{i}$ is its signal weight obtained from the $M(D^{0}ph^{-})$ invariant mass fit. The correction factor for the $\mathchar 28931\relax^{+}_{c}\rightarrow pK^{-}\pi^{+}$ decay is calculated similarly. Since the three-body decays $\mathchar 28931\relax^{+}_{c}\rightarrow pK^{-}\pi^{+}$ and $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}ph^{-}$ involve particles with non-zero spin in the initial and final states, the kinematic properties of these decays are described by angular variables in addition to the two Dalitz plot variables. The variation of the selection efficiency with the angles can thus affect the measurement. We use three independent variables to parametrise the angular phase space, similar to those used in Ref. [31] for the analysis of the $\mathchar 28931\relax^{+}_{c}\rightarrow pK^{-}\pi^{+}$ decay. The variables are defined in the rest frame of the decaying $\mathchar 28931\relax^{0}_{b}$ or $\mathchar 28931\relax^{+}_{c}$ baryons, with the $x$ axis given by their direction in the laboratory frame, the polarisation axis $z$ given by the cross product of the beam and $x$ axes, and the $y$ axis by the cross product of the $z$ and $x$ axes. The three variables are the cosine of the polar angle $\theta_{p}$ of the proton momentum in this reference frame, the azimuthal angle $\phi_{p}$ of the proton momentum in the reference frame, and the angle between the $D^{0}h^{-}$-plane (for $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}ph^{-}$) or $K^{-}\pi^{+}$-plane (for $\mathchar 28931\relax^{+}_{c}\rightarrow pK^{-}\pi^{+}$) and the plane formed by the proton and polarisation axis. The angular acceptance corrections are calculated from background-subtracted angular distributions obtained from the data. The distributions are similar to those obtained from the simulation of unpolarised $\mathchar 28931\relax^{0}_{b}$ decays, supporting the observation of small $\mathchar 28931\relax^{0}_{b}$ polarisation in $pp$ collisions [32]. The angular corrections are found to be negligible and are not used in the calculation of the ratios of branching fractions. Table 3: Efficiency correction factors used to calculate the ratios of branching fractions. Correction factor \| $R_{\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}}$ \| $R_{\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}}$ \| $R_{\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}}$ \| $R_{\mathchar 28932\relax_{b}^{0}\rightarrow D^{0}pK^{-}}$ \| $R_{\mathchar 28932\relax_{b}^{0}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}}$ ---\|---\|---\|---\|---\|--- $\varepsilon^{i}_{\rm sel}$/$\varepsilon^{j}_{\rm sel}$ \| 1.18 \| 1.01 \| 0.99 \| 0.97 \| 0.68 $\varepsilon^{i}_{\rm PID}$/$\varepsilon^{j}_{\rm PID}$ \| 0.98 \| 1.06 \| 1.17 \| – \| 1.07 $\varepsilon^{i}_{\rm PS}$/$\varepsilon^{j}_{\rm PS}$ \| 1.03 \| 1.02 \| – \| – \| 0.92 The values of the efficiency correction factors are given in Table 3. The values of the branching fraction ratios defined in Eqs. (2–5) obtained after corrections as described above, and their statistical uncertainties, are given in Table 4. Table 4: Measured ratios of branching fractions, with their statistical and systematic uncertainties in units of $10^{-2}$. \| $R_{\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}}$ \| $R_{\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}}$ \| $R_{\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}}$ \| $R_{\mathchar 28932\relax_{b}^{0}\rightarrow D^{0}pK^{-}}$ \| $R_{\mathchar 28932\relax_{b}^{0}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}}$ ---\|---\|---\|---\|---\|--- Central value \| $<\;$8.06 \| 7.27 \| $<\;$7.31 \| $<\;$44.3 \| $<\;$57 Statistical uncertainty \| $<\;$0.23 \| 0.82 \| $<\;$0.16 \| $<0$9.2 \| $<\;$22 Systematic uncertainties \| \| \| \| \| Signal model \| $<\;$0.03 \| 0.03 \| $<\;$0.05 \| $<0$0.2 \| $<0$3 Background model \| $<\;$0.07 \| ${}^{+0.34}_{-0.54}$ \| $<\;$0.09 \| $<0$5.0 \| $<\;$20 Trigger efficiency \| $<\;$0.01 \| 0.08 \| $<\;$0.07 \| $0$$<0.1$ \| $0$$<1$ Reconstruction efficiency \| $<0.01$ \| 0.04 \| $<\;$0.04 \| $0$$<0.1$ \| $0$$<1$ Selection efficiency \| $<\;$0.12 \| 0.01 \| $<0.01$ \| $0$$<0.1$ \| $0$$<1$ Simulation sample size \| $<\;$0.06 \| 0.07 \| $<\;$0.08 \| $<0$0.6 \| $0$$<1$ Phase space acceptance \| $<\;$0.07 \| 0.04 \| $<\;$– \| $0$$<0.1$ \| $0$$<1$ Angular acceptance \| $<\;$0.15 \| 0.29 \| $<\;$– \| $<0$3.5 \| $<0$4 PID efficiency \| $<\;$0.26 \| 0.11 \| $<\;$0.04 \| $<0$– \| $<0$1 Total systematic uncertainty \| $<\;$0.35 \| ${}^{+0.48}_{-0.64}$${}^{0^{0^{0}}}$ \| $<\;$0.16 \| $<0$6.0 \| $<\;$21 ## 6 Systematic uncertainties The systematic uncertainties in the measurements of the ratios of branching fractions are listed in Table 4. The uncertainties due to the description of signal and background contributions in the invariant mass fit model are estimated as follows: • The uncertainty due to the parametrisation of the signal distributions is obtained by using an alternative description based on a double-Gaussian shape, or a triple-Gaussian shape in the case of $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-}$. * • To determine the uncertainty due to the combinatorial background parametrisation, an alternative model with an exponential distribution is used instead of the quadratic polynomial function. * • The uncertainty in the parametrisation of the backgrounds from $B$ meson decays with misidentified particles in the final state is estimated by removing the $\kern 1.79993pt\overline{\kern-1.79993ptB}{}^{0}_{(s)}\rightarrow D^{+}_{(s)}\pi^{-}$ contribution. The uncertainty due to the parametrisaton of the $K/\pi$ misidentification background is estimated by using the shapes obtained without the PID requirements and without rejecting the events with the $D^{0}p$ invariant mass squared greater than 10${\mathrm{\,Ge\kern-1.00006ptV^{2}\\!/}c^{4}}$ in the fit to the simulated sample. * • The uncertainty due to the partially reconstructed background is estimated by fitting the invariant mass distributions in the reduced range of 5500–5900${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, and by excluding the contributions of partially reconstructed backgrounds with $K/\pi$ misidentification from the fit for $D^{0}pK^{-}$ and $\mathchar 28931\relax^{+}_{c}K^{-}$ combinations. * • The uncertainty due to the charmless background component $\mathchar 28931\relax^{0}_{b}\rightarrow pK^{-}\pi^{+}h^{-}$ is estimated from the fit of the $D^{0}ph^{-}$ ($\mathchar 28931\relax^{+}_{c}h^{-}$) invariant mass distributions in the sidebands of the $D^{0}$ ($\mathchar 28931\relax^{+}_{c}$) candidate invariant mass. A potential source of background that is not included in the fit comes from $\mathchar 28932\relax_{b}^{0}$ baryon decays into $D^{0}pK^{-}$ or similar final states, which differ from the reconstructed $D^{0}pK^{-}$ state by missing low-momentum particles. Such decays can contribute under the $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}$ signal peak. The possible contribution of these decays is estimated assuming that ${\cal B}(\mathchar 28932\relax_{b}^{0}\rightarrow D^{0}pK^{-})/{\cal B}(\mathchar 28932\relax_{b}^{0}\rightarrow D^{0}pK^{-})$ is equal to ${\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-})/{\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-})$ and that the selection efficiencies for $\mathchar 28932\relax_{b}^{0}$ and $\mathchar 28931\relax^{0}_{b}$ decays are the same. The one-sided systematic uncertainty due to this effect is added to the background model uncertainty for the $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}$ decay mode. The trigger efficiency uncertainty is dominated by the difference of the transverse energy threshold of the hardware-stage trigger observed between simulation and data. It is estimated by varying the transverse energy threshold in the simulation by 15%. In the case of measuring the ratios $R_{\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}}$ and $R_{\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}}$, one also has to take into account the difference of hadronic interaction cross section for kaons and pions before the calorimeter. This difference is studied using a sample of $B^{+}\rightarrow\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\pi^{+}$, $\kern 1.99997pt\overline{\kern-1.99997ptD}{}^{0}\rightarrow K^{+}\pi^{-}$ decays that pass the trigger decision independent of the final state particles of these decays. The difference was found to be 4.5% for $D^{0}ph^{-}$ and 2.5% for $\mathchar 28931\relax^{+}_{c}h^{-}$. Since only about 13% of events are triggered exclusively by the $h^{-}$ particle, the resulting uncertainty is low. The uncertainty due to track reconstruction efficiency cancels to a good approximation for the quoted ratios since the track multiplicities of the decays are the same. However, for the ratios $R_{\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}}$ and $R_{\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}}$, the difference in hadronic interaction rate for kaons and pions in the tracker can bias the measurement. A systematic uncertainty is assigned taking into account the rate of hadronic interactions in the simulation and the uncertainty on the knowledge of the amount of material in the LHCb tracker. The uncertainty in the selection efficiency obtained from simulation is evaluated by scaling the variables that enter the offline selection. The scaling factor is chosen from the comparison of the distributions of these variables in simulation and in a background-subtracted $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-}$ sample. In addition, the uncertainty due to the finite size of the simulation samples is assigned. The uncertainty of the phase-space efficiency correction includes four effects. The statistical uncertainty on the correction factor is determined by the data sample size and variations of the efficiency over the phase space. The uncertainty in the parametrisation of the efficiency shape is estimated by using an alternative parametrisation with a third-order rather than a fourth- order polynomial. The correlation of the efficiency shape and invariant mass of $\mathchar 28931\relax^{0}_{b}$ ($\mathchar 28932\relax_{b}^{0}$) candidates is estimated by calculating the efficiency shape in three bins of $\mathchar 28931\relax^{0}_{b}$ ($\mathchar 28932\relax_{b}^{0}$) mass separately and using one of the three shapes depending on the invariant mass of the candidate. The uncertainty due to the difference of the $\mathchar 28931\relax^{0}_{b}$ ($\mathchar 28932\relax_{b}^{0}$) kinematic properties between simulation and data is estimated by using the efficiency shape obtained after weighting the simulated sample using the momentum distribution of $\mathchar 28931\relax^{0}_{b}$ ($\mathchar 28932\relax_{b}^{0}$) from background-subtracted $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-}$ data. Corrections due to the angular acceptance in the calculation of ratios of branching fractions are consistent with zero. The central values quoted do not include these corrections, while the systematic uncertainty is evaluated by taking the maximum of the statistical uncertainty for the correction, determined by the size of the data sample, and the deviation of its central value from unity. The uncertainty in the PID response is calculated differently for the ratio of $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ and $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-}$ branching fractions using loose selection, and for the measurements using tight BDT- based selections. For the ratio of $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ and $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-}$ branching fractions, $R_{\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}}$, the uncertainty due to the pion and kaon PID requirements is estimated by scaling the PID variables within the limits given by the comparison of distributions from the reweighted calibration sample and the background-subtracted $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-}$ data. The dominant contribution to the PID uncertainty comes from the uncertainty in the proton PID efficiency ratio, which is caused by the difference in kinematic properties of the proton from $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}$ and $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-}$ decays. The proton efficiency ratio in this case is taken from simulation, and the systematic uncertainty is estimated by taking this ratio to be equal to one. In the case of measuring the ratios $R_{\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}}$ and $R_{\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}}$, the uncertainty due to the proton PID and the tracks coming from the $D^{0}$ or $\mathchar 28931\relax^{+}_{c}$ candidates is negligible due to similar kinematic distributions of the decays in the numerator and denominator. The dominant contribution comes from the PID efficiency ratio for the kaon or pion track from the $\mathchar 28931\relax^{0}_{b}$ vertex; this is estimated by scaling the PID distribution as described above. In addition, there are contributions due to the finite size of the PID calibration sample, and the uncertainty due to assumption that the PID efficiency for the individual tracks factorises in the total efficiency. The latter is estimated with simulated samples. Since the results for the $\mathchar 28931\relax^{0}_{b}$ decay modes are all ratios to other $\mathchar 28931\relax^{0}_{b}$ decays, there is no systematic bias introduced by the dependence of the efficiency on the $\mathchar 28931\relax^{0}_{b}$ lifetime, and the fact that the value used in the simulation ($1.38{\rm\,ps}$) differs from the latest measurement [33]. We also do not assign any systematic uncertainty due to the lack of knowledge of the $\mathchar 28932\relax_{b}^{0}$ lifetime, which is as-yet unmeasured (a value of $1.42{\rm\,ps}$ is used in the simulation). The dominant systematic uncertainties in the measurement of the $\mathchar 28932\relax_{b}^{0}$ and $\mathchar 28931\relax^{0}_{b}$ mass difference (see Table 5) come from the uncertainties of the signal and background models, and are estimated from the same variations of these models as in the calculation of branching fractions. The uncertainty due to the momentum scale calibration partially cancels in the quoted difference of $\mathchar 28932\relax_{b}^{0}$ and $\mathchar 28931\relax^{0}_{b}$ masses; the residual contribution is estimated by varying the momentum scale factor within its uncertainty of 0.3% [24]. Table 5: Systematic uncertainties in the measurement of the mass difference $m_{\mathchar 28932\relax_{b}^{0}}-m_{\mathchar 28931\relax^{0}_{b}}$. Source \| Uncertainty (${\mathrm{Me\kern-1.00006ptV\\!/}c^{2}}$) ---\|--- Signal model \| 0.19 Background model \| 0.50 Momentum scale calibration \| 0.03 Total \| 0.54 ## 7 Signal significance and fit validation The statistical significance of the $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}$, $\mathchar 28932\relax_{b}^{0}\rightarrow D^{0}pK^{-}$, and $\mathchar 28932\relax_{b}^{0}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}$ signals, expressed in terms of equivalent number of standard deviations ($\sigma$), is evaluated from the maximum likelihood fit as $S_{\rm stat}=\sqrt{-2\Delta\ln\mathcal{L}},$ (8) where $\Delta\ln\mathcal{L}$ is the difference in logarithms of the likelihoods for the fits with and without the corresponding signal contribution. The fit yields the statistical significance of the $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}$, $\mathchar 28932\relax_{b}^{0}\rightarrow D^{0}pK^{-}$, and $\mathchar 28932\relax_{b}^{0}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}$ signals of $10.8\,\sigma$, $6.7\,\sigma$, and $4.7\,\sigma$, respectively. The validity of this evaluation is checked with the following procedure. To evaluate the significance of each signal, a large number of invariant mass distributions is generated using the result of the fit on data as input, excluding the signal contribution under consideration. Each distribution is then fitted with models that include background only, as well as background and signal. The significance is obtained as the fraction of samples where the difference $\Delta\ln\mathcal{L}$ for the fits with and without the signal is larger than in data. The significance evaluated from the likelihood fit according to Eq. (8) is consistent with, or slightly smaller than that estimated from the simulated experiments. Thus, the significance calculated as in Eq. (8) is taken. The significance accounting for the systematic uncertainties is evaluated as $S_{\rm stat+syst}=S_{\rm stat}\left/\sqrt{1+\sigma^{2}_{\rm syst}/\sigma^{2}_{\rm stat}}\right.,$ (9) where $\sigma_{\rm stat}$ is the statistical uncertainty of the signal yield and $\sigma_{\rm syst}$ is the corresponding systematic uncertainty, which only includes the relevant uncertainties due to the signal and background models. As a result, the significance for the $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}$, $\mathchar 28932\relax_{b}^{0}\rightarrow D^{0}pK^{-}$, and $\mathchar 28932\relax_{b}^{0}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}$ signals is calculated to be $9.0\,\sigma$, $5.9\,\sigma$, and $3.3\,\sigma$, respectively. The fitting procedure is tested with simulated experiments where the invariant mass distributions are generated from the PDFs that are a result of the data fit, and then fitted with the same procedure as applied to data. No significant biases are introduced by the fit procedure in the fitted parameters. However, we find that the statistical uncertainty on the $\mathchar 28932\relax_{b}^{0}$ mass is underestimated by 3% in the fit and the uncertainty on the $\mathchar 28932\relax_{b}^{0}\rightarrow D^{0}pK^{-}$ yield is underestimated by 5%. We apply the corresponding scale factors to the $\mathchar 28932\relax_{b}^{0}\rightarrow D^{0}pK^{-}$ yield and $\mathchar 28932\relax_{b}^{0}$ mass uncertainties to obtain the final results. ## 8 Conclusion We report studies of beauty baryon decays to the $D^{0}ph^{-}$ and $\mathchar 28931\relax^{+}_{c}h^{-}$ final states, using a data sample corresponding to an integrated luminosity of 1.0$\mbox{\,fb}^{-1}$ collected with the LHCb detector. First observations of the $\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}$ and $\mathchar 28932\relax_{b}^{0}\rightarrow D^{0}pK^{-}$ decays are reported, with significances of 9.0 and 5.9 standard deviations, respectively. The decay $\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}$ is observed for the first time; the significance of this observation is greater than 10 standard deviations. The first evidence for the $\mathchar 28932\relax_{b}^{0}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}$ decay is also obtained with a significance of 3.3 standard deviations. The combinations of branching and fragmentation fractions for beauty baryons decaying into $D^{0}ph^{-}$ and $\mathchar 28931\relax^{+}_{c}h^{-}$ final states are measured to be $\begin{split}R_{\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-}}\equiv\frac{{\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-})\times{\cal B}(D^{0}\rightarrow K^{-}\pi^{+})}{{\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-})\times{\cal B}(\mathchar 28931\relax^{+}_{c}\rightarrow pK^{-}\pi^{+})}&=0.0806\pm 0.0023\pm 0.0035,\\\ R_{\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-}}\equiv\frac{{\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-})}{{\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}p\pi^{-})}&=0.073\pm 0.008\,^{+0.005}_{-0.006},\\\ R_{\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}}\equiv\frac{{\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-})}{{\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow\mathchar 28931\relax^{+}_{c}\pi^{-})}&=0.0731\pm 0.0016\pm 0.0016,\\\ R_{\mathchar 28932\relax_{b}^{0}\rightarrow D^{0}pK^{-}}\equiv\frac{f_{\mathchar 28932\relax_{b}^{0}}\times{\cal B}(\mathchar 28932\relax_{b}^{0}\rightarrow D^{0}pK^{-})}{f_{\mathchar 28931\relax^{0}_{b}}\times{\cal B}(\mathchar 28931\relax^{0}_{b}\rightarrow D^{0}pK^{-})}&=0.44\pm 0.09\pm 0.06,\\\ R_{\mathchar 28932\relax_{b}^{0}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-}}\equiv\frac{{\cal B}(\mathchar 28932\relax_{b}^{0}\rightarrow\mathchar 28931\relax^{+}_{c}K^{-})\times{\cal B}(\mathchar 28931\relax^{+}_{c}\rightarrow pK^{-}\pi^{+})}{{\cal B}(\mathchar 28932\relax_{b}^{0}\rightarrow D^{0}pK^{-})\times{\cal B}(D^{0}\rightarrow K^{-}\pi^{+})}&=0.57\pm 0.22\pm 0.21,\\\ \end{split}$ where the first uncertainty is statistical and the second systematic. The ratios of the Cabibbo-suppressed to Cabibbo-favoured branching fractions for both the $D^{0}ph^{-}$ and the $\mathchar 28931\relax^{+}_{c}h^{-}$ modes are consistent with the those observed for the $B\rightarrow Dh$ modes [1]. In addition, the difference of $\mathchar 28932\relax_{b}^{0}$ and $\mathchar 28931\relax^{0}_{b}$ baryon masses is measured to be $m_{\mathchar 28932\relax_{b}^{0}}-m_{\mathchar 28931\relax^{0}_{b}}=174.8\pm 2.4\pm 0.5{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}.$ Using the latest LHCb measurement of the $\mathchar 28931\relax^{0}_{b}$ mass $m_{\mathchar 28931\relax^{0}_{b}}=5619.53\pm 0.13\pm 0.45{\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$ [24], the $\mathchar 28932\relax_{b}^{0}$ mass is determined to be $m_{\mathchar 28932\relax_{b}^{0}}=5794.3\pm 2.4\pm 0.7$${\mathrm{\,Me\kern-1.00006ptV\\!/}c^{2}}$, in agreement with the measurement performed by CDF [3] and twice as precise. ## Acknowledgements We express our gratitude to our colleagues in the CERN accelerator departments for the excellent performance of the LHC. We thank the technical and administrative staff at the LHCb institutes. We acknowledge support from CERN and from the national agencies: CAPES, CNPq, FAPERJ and FINEP (Brazil); NSFC (China); CNRS/IN2P3 and Region Auvergne (France); BMBF, DFG, HGF and MPG (Germany); SFI (Ireland); INFN (Italy); FOM and NWO (The Netherlands); SCSR (Poland); MEN/IFA (Romania); MinES, Rosatom, RFBR and NRC “Kurchatov Institute” (Russia); MinECo, XuntaGal and GENCAT (Spain); SNSF and SER (Switzerland); NAS Ukraine (Ukraine); STFC (United Kingdom); NSF (USA). We also acknowledge the support received from the ERC under FP7. The Tier1 computing centres are supported by IN2P3 (France), KIT and BMBF (Germany), INFN (Italy), NWO and SURF (The Netherlands), PIC (Spain), GridPP (United Kingdom). We are thankful for the computing resources put at our disposal by Yandex LLC (Russia), as well as to the communities behind the multiple open source software packages that we depend on. ## References [1] Particle Data Group, J. Beringer et al., Review of particle physics, Phys. Rev. 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Cauet, R.\n Cenci, M. Charles, Ph. Charpentier, S.-F. Cheung, N. Chiapolini, M.\n Chrzaszcz, K. Ciba, X. Cid Vidal, G. Ciezarek, P.E.L. Clarke, M. Clemencic,\n H.V. Cliff, J. Closier, C. Coca, V. Coco, J. Cogan, E. Cogneras, P. Collins,\n A. Comerma-Montells, A. Contu, A. Cook, M. Coombes, S. Coquereau, G. Corti,\n B. Couturier, G.A. Cowan, D.C. Craik, M. Cruz Torres, S. Cunliffe, R. Currie,\n C. D'Ambrosio, J. Dalseno, P. David, P.N.Y. David, A. Davis, I. De Bonis, K.\n De Bruyn, S. De Capua, M. De Cian, J.M. De Miranda, L. De Paula, W. De Silva,\n P. De Simone, D. Decamp, M. Deckenhoff, L. Del Buono, N. D\\'el\\'eage, D.\n Derkach, O. Deschamps, F. Dettori, A. Di Canto, H. Dijkstra, M. Dogaru, S.\n Donleavy, F. Dordei, P. Dorosz, A. Dosil Su\\'arez, D. Dossett, A. Dovbnya, F.\n Dupertuis, P. Durante, R. Dzhelyadin, A. Dziurda, A. Dzyuba, S. Easo, U.\n Egede, V. Egorychev, S. Eidelman, D. van Eijk, S. Eisenhardt, U.\n Eitschberger, R. Ekelhof, L. Eklund, I. El Rifai, Ch. Elsasser, A. 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Nikitin, T. Nikodem, A.\n Nomerotski, A. Novoselov, A. Oblakowska-Mucha, V. Obraztsov, S. Oggero, S.\n Ogilvy, O. Okhrimenko, R. Oldeman, G. Onderwater, M. Orlandea, J.M. Otalora\n Goicochea, P. Owen, A. Oyanguren, B.K. Pal, A. Palano, M. Palutan, J. Panman,\n A. Papanestis, M. Pappagallo, L. Pappalardo, C. Parkes, C.J. Parkinson, G.\n Passaleva, G.D. Patel, M. Patel, C. Patrignani, C. Pavel-Nicorescu, A. Pazos\n Alvarez, A. Pearce, A. Pellegrino, G. Penso, M. Pepe Altarelli, S. Perazzini,\n E. Perez Trigo, A. P\\'erez-Calero Yzquierdo, P. Perret, M. Perrin-Terrin, L.\n Pescatore, E. Pesen, G. Pessina, K. Petridis, A. Petrolini, E. Picatoste\n Olloqui, B. Pietrzyk, T. Pila\\v{r}, D. Pinci, S. Playfer, M. Plo Casasus, F.\n Polci, G. Polok, A. Poluektov, E. Polycarpo, A. Popov, D. Popov, B. Popovici,\n C. Potterat, A. Powell, J. Prisciandaro, A. Pritchard, C. Prouve, V. Pugatch,\n A. Puig Navarro, G. Punzi, W. Qian, B. Rachwal, J.H. Rademacker, B.\n Rakotomiaramanana, M.S. Rangel, I. Raniuk, N. Rauschmayr, G. Raven, S.\n Redford, S. Reichert, M.M. Reid, A.C. dos Reis, S. Ricciardi, A. Richards, K.\n Rinnert, V. Rives Molina, D.A. Roa Romero, P. Robbe, D.A. Roberts, A.B.\n Rodrigues, E. Rodrigues, P. Rodriguez Perez, S. Roiser, V. Romanovsky, A.\n Romero Vidal, M. Rotondo, J. Rouvinet, T. Ruf, F. Ruffini, H. Ruiz, P. Ruiz\n Valls, G. Sabatino, J.J. Saborido Silva, N. Sagidova, P. Sail, B. Saitta, V.\n Salustino Guimaraes, B. Sanmartin Sedes, R. Santacesaria, C. Santamarina\n Rios, E. Santovetti, M. Sapunov, A. Sarti, C. Satriano, A. Satta, M. Savrie,\n D. Savrina, M. Schiller, H. Schindler, M. Schlupp, M. Schmelling, B. Schmidt,\n O. Schneider, A. Schopper, M.-H. Schune, R. Schwemmer, B. Sciascia, A.\n Sciubba, M. Seco, A. Semennikov, K. Senderowska, I. Sepp, N. Serra, J.\n Serrano, P. Seyfert, M. Shapkin, I. Shapoval, Y. Shcheglov, T. Shears, L.\n Shekhtman, O. Shevchenko, V. Shevchenko, A. Shires, R. Silva Coutinho, M.\n Sirendi, N. Skidmore, T. Skwarnicki, N.A. Smith, E. Smith, E. Smith, J.\n Smith, M. Smith, M.D. Sokoloff, F.J.P. Soler, F. Soomro, D. Souza, B. Souza\n De Paula, B. Spaan, A. Sparkes, P. Spradlin, F. Stagni, S. Stahl, O.\n Steinkamp, S. Stevenson, S. Stoica, S. Stone, B. Storaci, S. Stracka, M.\n Straticiuc, U. Straumann, V.K. Subbiah, L. Sun, W. Sutcliffe, S. Swientek, V.\n Syropoulos, M. Szczekowski, P. Szczypka, D. Szilard, T. Szumlak, S.\n T'Jampens, M. Teklishyn, G. Tellarini, E. Teodorescu, F. Teubert, C. Thomas,\n E. Thomas, J. van Tilburg, V. Tisserand, M. Tobin, S. Tolk, L. Tomassetti, D.\n Tonelli, S. Topp-Joergensen, N. Torr, E. Tournefier, S. Tourneur, M.T. Tran,\n M. Tresch, A. Tsaregorodtsev, P. Tsopelas, N. Tuning, M. Ubeda Garcia, A.\n Ukleja, A. Ustyuzhanin, U. Uwer, V. Vagnoni, G. Valenti, A. Vallier, R.\n Vazquez Gomez, P. Vazquez Regueiro, C. V\\'azquez Sierra, S. Vecchi, J.J.\n Velthuis, M. Veltri, G. Veneziano, M. Vesterinen, B. Viaud, D. Vieira, X.\n Vilasis-Cardona, A. Vollhardt, D. Volyanskyy, D. Voong, A. Vorobyev, V.\n Vorobyev, C. Vo\\ss, H. Voss, J.A. de Vries, R. Waldi, C. Wallace, R. Wallace,\n S. Wandernoth, J. Wang, D.R. Ward, N.K. Watson, A.D. Webber, D. Websdale, M.\n Whitehead, J. Wicht, J. Wiechczynski, D. Wiedner, L. Wiggers, G. Wilkinson,\n M.P. Williams, M. Williams, F.F. Wilson, J. Wimberley, J. Wishahi, W.\n Wislicki, M. Witek, G. Wormser, S.A. Wotton, S. Wright, S. Wu, K. Wyllie, Y.\n Xie, Z. Xing, Z. Yang, X. Yuan, O. Yushchenko, M. Zangoli, M. Zavertyaev, F.\n Zhang, L. Zhang, W.C. Zhang, Y. Zhang, A. Zhelezov, A. Zhokhov, L. Zhong, A.\n Zvyagin", "submitter": "Anton Poluektov", "url": "https://arxiv.org/abs/1311.4823" }
1311.4831	# Dynamical evolution and spin-orbit resonances of potentially habitable exoplanets. The case of GJ 667C Valeri V. Makarov & Ciprian Berghea United States Naval Observatory, 3450 Massachusetts Ave. NW, Washington DC, 20392-5420 [email protected] ###### Abstract We investigate the dynamical evolution of the potentially habitable super- earth GJ 667Cc in the multiple system of at least two exoplanets orbiting a nearby M dwarf, paying special attention to its spin-orbital state. The published radial velocities for this star are re-analyzed and evidence is found for additional periodic signals, which could be taken for two additional planets on eccentric orbits. Such systems are not dynamically viable and break up quickly in numerical integrations. The nature of the bogus signals in the available data remains unknown. Limiting the scope to the two originally detected planets, we assess the dynamical stability of the system and find no evidence for bounded chaos in the orbital motion, unlike the previously investigated planetary system of GJ 581. The orbital eccentricity of the planets b and c is found to change cyclicly in the range 0.06 - 0.28 and 0.05 - 0.25, respectively, with a period of approximately 0.46 yr, and a semimajor axis that little varies. Taking the eccentricity variation into account, numerical integrations are performed of the differential equations modeling the spin-orbit interaction of the planet GJ 667Cc with its host star, including fast oscillating components of both the triaxial and tidal torques and assuming a terrestrial composition of its mantle. Depending on the interior temperature of the planet, it is likely to be entrapped in the 3:2 (probability 0.51) or even higher spin-orbit resonance. It is less likely to reach the 1:1 resonance (probability 0.24). Similar capture probabilities are obtained for the inner planet GJ 667Cb. The estimated characteristic spin-down times are quite short for the two planets, i.e., within 1 Myr for planet c and even shorter for planet b. Both planet arrived at their current and, most likely, ultimate spin-orbit states a long time ago. The planets of GJ 667C are most similar to Mercury of all the Solar System bodies, as far as their tidal properties are concerned. However, unlike Mercury, the rate of tidal dissipation of energy is formidably high in the planets of GJ 667, estimated at $10^{23.7}$ and $10^{26.7}$ J yr-1 for c and b, respectively. This raises a question of how such relatively massive, close super-Earths could survive overheating and destruction. planet-star interactions — planets and satellites: dynamical evolution and stability — celestial mechanics — planets and satellites: detection — planets and satellites: individual (GJ 667) ## 1 Introduction According to a recent comprehensive study based on long-term spectroscopic observations with HARPS, super-Earth exoplanets in the habitable zones of nearby M dwarfs are probably very abundant, with an estimated rate of $\eta_{\rm Earth}=0.41^{+0.54}_{-0.13}$ per host star (Bonfils et al., 2013). In exoplanet research, super-Earths usually designate planets appreciably more massive than the Earth, but smaller than 10 Earth masses. From the same study, the frequency of super-Earths with orbital periods between 10 and 100 days is $0.52^{+0.50}_{-0.16}$. The current observational data thus do not rule out the possibility that there is a habitable super-Earth in almost every M-type stellar system. This puts nearby M-stars into the focus of exoplanet atmospheres and habitability research. In particular, detecting molecular tracers of biological life no longer seems a merely speculative proposition. The observations of planetary transits with Kepler, on the other hand, are better suited to the detection of planets orbiting larger, solar-type stars. Traub (2012) estimates that the frequency of terrestrial planets in the habitable zones of FGK stars is $\eta_{\rm Earth}=0.34\pm{0.14}$. This estimate was obtained by extrapolation of the observational statistics for planets with shorter periods ($<42$ days) collected from the first 136 days of Kepler observations. Terrestrial planets may not be as common as ice giants, but ubiquitous enough to investigate in earnest how extraterrestrial life can thrive in other planetary systems. Planet’s rotation is one of the issues that have a bearing on habitability and atmosphere properties. The major planets of the solar system, with the notable exception of Venus and Mercury, rotate at significantly faster rates than their orbital motion. They are far enough from the Sun for the tidal dissipation to be so slow that the characteristic times of spin-down are comparable to, or longer, than the life time. The Earth’s spin rate, for example, does slow down, but it will take billions of years for the solar day to become appreciably longer. There is a common understanding that the significance of tidal interactions is markedly larger in exoplanet systems. At the same time, most publications on this topic were based on the two simplified, “toy” models, one called the constant-Q model and the other the constant time-lag model, which are either inaccurate for terrestrial bodies or completely wrong (Efroimsky & Makarov, 2013). The stability of super-Earth’s atmospheres around M dwarfs was investigated by Heng & Kopparla (2012). Their starting assumption, not justified by any dynamical consideration, was that such planets are either synchronized (i.e., are captured into a 1:1 spin-orbit resonance) or pseudosynchronized. The latter stable equilibrium state, when a planet rotates somewhat faster than the mean orbital motion, is a direct prediction of the two above-mentioned simplified tidal models. Using a more realistic tidal model developed in (Efroimsky & Lainey, 2007; Efroimsky & Williams, 2009), Makarov & Efroimsky (2013) showed that a pseudosynchronous state is inherently unstable for celestial bodies of terrestrial composition (the situation is less clear for gaseous stars and liquid planets). Thus, a misassumption on the rotational state of a planet can nullify the impact of an otherwise timely and significant theoretical study on planetary atmospheres. The assumption that a close super-Earth should be synchronously rotating appears to be hardly assailable. Do we not have a close analogy in the Moon, always facing the Earth with the same side? Hut (1980) demonstrated that synchronously rotating components on a circular orbit is the only long-term stable equilibrium in a two-body system with tides. This statement implies that given sufficiently long time, all interacting systems should become synchronized and circularized. It appears entirely plausible that the close super-Earths have had sufficient time to reach this terminal point of interaction. This widely accepted point was critically reviewed in (Makarov et al., 2013) where the possibly habitable super-Earth GJ 581c was discussed. Compared to the rocky planets and moons of the Solar System, super-Earths orbiting in the habitable zones of M stars are more massive, usually closer to their host stars, have mostly higher orbital eccentricities, and are likely to be more axially symmetric. This combination of characteristics makes them a very interesting class of objects for tidal dynamics. We would argue that the closest analogs of super-Earths in our system are Mercury and Venus, rather than the Moon, Phobos or Titan. We found for GJ 581d that the probabilities of capture into the supersynchronous resonances is so high with the observed eccentricity, that the planet practically has no chance of reaching the 1:1 resonance, if its initial rotation was prograde. The most likely state within a range of estimated parameters is a 2:1 spin-orbit resonance, when the planet makes exactly 2 complete sidereal rotations during one orbital revolution. In this paper, we offer a similar dynamical study of another exoplanet system, GJ 667C, which probably includes at least two super-Earth planets. The composition of the system and the difficulties arising in the interpretation of the spectroscopic data are discussed in §2. ## 2 The orbits of GJ 667C planets For our analysis of the orbital configuration of the GJ 667C system, we use the precision radial velocity data published by Delfosse et al. (2013), collected with the HARPS instrument. The measurements span 2657 days and have a median error of 1.3 m s-1. The RV data obviously exhibit a systematic trend across the interval of observation, rising by approximately 20 m s-1. Delfosse et al. (2013) note that the observed trend is consistent with a gravitational acceleration exerted by the inner pair of stars (A and B) in this multiple star system. Our planet detection algorithm is described in more detail in (Makarov et al., 2013). Briefly, it is based on well-known and well-tested iterative periodogram analysis and brute-force grid search. The possible points of distinction with respect to other algorithms used in exoplanet studies are: 1) we do not subtract the mean RV from the data but instead fit a constant term (and a trend when needed) into the data in each iteration; 2) the original data are never altered in the process as each new signal is fitted along with all the previous signals, gradually building up the model; 3) the nonlinear parameters of the Keplerian fit (eccentricity and phase) are optimized by a brute force grid search minimizing the reduced $\chi^{2}$ statistics of the residuals. The confidence of each detection is computed through the well- tested F-statistic, with the number of degrees of freedom decreasing by 7 with each planet signal added to the model. Only detections with a confidence level above 0.99 are normally accepted. The algorithm was originally intended to be a major part of a web-accessible multi-purpose tool for simulation, visualization and analysis of exoplanet systems, constructed at NExScI (CalTech) in support of the Space Interferometry Mission. It was designed to be very fast and to work on massive sets of data, optionally fitting combined RV and astrometric data. Error estimation was not a built-in function of the algorithm, but rather was realized as a separate task based on the Minimum Variance Bound estimators. For this reason, formal errors are not automatically provided for orbital fits, unless a dedicated statistical comparison is in order. Figure 1: The generalized $\chi^{2}$ periodogram of the GJ 667C system prior to orbital fitting based on the HARPS data (Delfosse et al., 2013). The dips corresponding to the two confidently detected planets are indicated with arrows and planet names. Fig. 1 shows the initial $\chi^{2}$ periodogram of the HARPS data prior to any planet detection. The dips corresponding to the planets detected from these data are marked with arrows and planet designations. The strongest signal is associated with planet b, which is detected first. The second strongest signal corresponds to planet c. The periodogram is riddled with a multitude of less significant dips of approximately equal strength, some of which are formally accepted by the algorithm as bona fide planetary signals. Table 1: Orbital parameters of the two-planet GJ 667C system and additional signals detected by our RVfit algorithm and by Delfosse et al. (2013). Planet \| $P$ \| Mass \| $a$ \| $e$ \| $\omega$ \| $M_{0}$ \| Note ---\|---\|---\|---\|---\|---\|---\|--- \| d \| $M_{\earth}$ \| AU \| \| $\arcdeg$ \| $\arcdeg$ \| $b$ \| $7.20$ \| $5.80$ \| $0.050$ \| 0.15 \| 344 \| 104 \| 1 \| $7.199\pm 0.001$ \| 5.46 \| 0.0504 \| $0.09\pm 0.05$ \| $-4\pm 33$ \| \| 2 $c$ \| $28.1$ \| $4.28$ \| $0.125$ \| 0.29 \| 152 \| 212 \| 1 \| $28.13\pm 0.03$ \| 4.25 \| 0.1251 \| $0.34\pm 0.10$ \| $166\pm 20$ \| \| 2 $signal1$ \| $91.5$ \| $5.02$ \| $0.275$ \| 0.35 \| 193 \| 252 \| 1 $signal2$ \| $53.3$ \| $3.36$ \| $0.192$ \| 0.49 \| 122 \| 156 \| 1 $signal3$ \| $35.3$ \| $2.27$ \| $0.145$ \| 0.35 \| 84 \| 248 \| 1 Note. — 1: Our work; 2: Delfosse et al. (2013) The orbital and physical parameters estimated from the fits of the five detected planets are given in Table 1. For the two safely detected planets b and c, our periods are in close agreement with the estimates by Delfosse et al. (2013) and Anglada-Escudé et al. (2012). A similarly very good agreement among the three studies is found for other parameters of b and c, including the projected mass and semimajor axis, except for the eccentricity. As the eccentricity of planet b is almost the same in this study and in Anglada- Escudé et al. (2012) (0.15 and 0.17, respectively), it is appreciably smaller in Delfosse et al. (2013) in their 3-planet fit (Table 1, $0.09\pm 0.05$). On the contrary, planet’s c eccentricity in Anglada-Escudé et al. (2012) is restricted to smaller values ($<0.27$) than those found in the other two studies (0.29 and 0.34). We conclude that there is a comfortable degree of certainty about the properties of planets b and c, but the realistic uncertainty of eccentricities is probably up to 0.1. This consistency breaks up completely when it comes to the interpretation of the remaining periodic signals in the HARPS data. Different planet detection algorithms seem to pick up different sets of prominent features in the periodograms as the most significant signals. Anglada-Escudé et al. (2012) denote the residual feature as (d?) and find a period of 74.79 d for it, forcing both the eccentricity and periastron longitude at zero. Delfosse et al. (2013) offer a set of plausible two-planet solutions with a linear trend, with the period of planet c taking values of 28, 90, 106, 124, 186, and 372 d, all yielding similar final $\chi^{2}$ statistics on the residuals. Their baseline three-planet solution (with a trend) has a third planet d with a period of 106.4 d and eccentricity 0.68. As we found out in our N-body simulations, such a system is not dynamically viable. The authors also express strong doubts in the existence of a third planet, because they find a strong peak at a period of $\simeq 105$ d in the activity diagnostics correlated with the rotation. A rotational period of 105 d is long, but not uncommon for inactive stars. Finally, our present analysis results in a set of features (labeled signal 1–3) with periods of 91.5, 53.3, and 35.3 d. Only the former periodicity seems to be in common with one of the tentative two-planet solutions of (Delfosse et al., 2013). However, we note that the 53.3 and 35.5 d periodicities are the second and third harmonics of the 106 d oscillation, which we do not explicitly detect in our analysis. It would be interesting to investigate if a long-lived photospheric feature rotating with the star and crossing the visible hemisphere with a period of 106 d could generate a whole set of harmonic signals in the RV periodogram. It may be considered somewhat worrying that the period of planet c (28 d) is close to the fourth harmonic of this period. The star GJ 667C is the tertiary component of a hierarchical multiple system, which is known to also include the K-dwarfs GJ 667A and B. According to the Washington Double Star catalog (WDS), the separation between the A and C components was measured to 29.4 and 32.7 arcsec in 1875 and 2010, respectively. Söderhjelm (1999) obtained a much improved visual double star solution for components A and B, using the Hipparcos observational data, and determined a semimajor axis of 1.81 arcsec, period 42.15 yr, and eccentricity 0.58. The D component listed in the WDS, on the other hand, is certainly optical, with a proper motion discordant with the fast-moving GJ 667 system (William Hartkopf, priv. comm.). Thus, no companions more distant than C are known in this system. The considerable eccentricity of the AB pair could be pumped by the C component via the Kozai-cycle mechanism of orbital interaction. On the other hand, the observed trend in the RV of the C component can be caused by its orbital acceleration around the AB pair. ## 3 Long-term evolution of the orbits Figure 2: Simulated eccentricity variations of planets GJ 667Cc and GJ 667Cd. Using the values listed in Table 3 as initial parameters and assuming the orbits to be coplanar we performed multiple simulations of the dynamical evolution of a 2-body GJ 667C system. The parameters given in this Table are consistent with the data, considered to be the best Keplerian solution in (Anglada-Escudé et al., 2012), but neglecting the third, tentative planet d. The only change made is the eccentricity of planet c, which was set to 0.20. We chose to adopt the results from ibid, because our own Keplerian fit (Table 1) did not seem to provide more reliable or accurate information about the reliably detected planets b and c. In order to investigate the dynamical stability of such a three-body system, we employed the symplectic integrator HNBody, version 1.0.7, (Rauch & Hamilton, 2002) with the hybrid symplectic option. The time step was set at 0.036 hours and the orbits of the planets were integrated for 5 million years or longer. The mass of the star was assumed to be $0.33\,M_{\sun}$. We find that the system of two planets (b and c) is perfectly stable. As found already by (Anglada-Escudé et al., 2012), the system is stabilized by a close 4:1 mean-motion resonance (MMR) with the arguments of periastron librating around $180\arcdeg$. The semimajor axes vary little. The eccentricities show significant periodic variations with opposite phases, so that when planet b reaches the largest eccentricity, planets c assumes the smallest eccentricity, and vice versa, see Fig. 2. These variations for both planets for the first 20000 years of integration are shown in Fig. 2. The range of variations of eccentricity for planet b is between 0.06 and 0.27, and for planet c, between 0.05 and 0.25. The period of libration in eccentricity is approximately 0.47 yr. Table 2: Orbital parameters of the two-planet GJ 667C system used for orbital integration in this paper. Planet \| $P$ \| Mass \| $a$ \| $e$ \| $\omega$ \| $M_{0}$ ---\|---\|---\|---\|---\|---\|--- \| d \| $M_{\earth}$ \| AU \| \| $\arcdeg$ \| $\arcdeg$ $b$ \| $7.20$ \| $5.68$ \| $0.049$ \| 0.17 \| 344 \| 107 $c$ \| $28.16$ \| $4.54$ \| $0.123$ \| 0.20 \| 238 \| 144 We also numerically probed for evidence of chaos in the orbital motion of GJ 667C, employing the sibling simulation technique proposed by Hayes (2007, 2008) to investigate the behavior of the outer Solar system. The orbits of the two planets were integrated with slightly different initial conditions for up to 5 million years. Two sibling trajectories are generated by perturbing the initial semi-major axis of the planet GJ 667Cb by a factor of $10^{-14}$. The distance between the unperturbed planets and their siblings is then computed as a function of time. A chaotic motion manifests itself as an exponential divergence between the trajectories. We did not find any signs of chaos in the motion of this system, the difference between the sibling trajectories being polynomial over the entire span of integration (Fig. 3, left). This marks a significant difference with the previously studied case of GJ 581, where a rapid onset of chaos with Lyapunov times of tens to a hundred years was unambiguously detected. The important difference between the system of GJ 581 and the system of GJ 667C is that the former includes more than two planets, which may significantly interact with one another, whereas the latter, as integrated in this paper, includes only two planets in 4:1 mean motion resonance. This result should thus be taken with a caveat, because we do not know if no other planets orbit GJ 667C. Recalling that the host star is a tertiary in a hierarchical triple system, we investigated if the presence of other distant and massive bodies could invoke dynamical chaos in the resonant planetary system. The pair of stars A and B was replaced in these simulation by a single point mass of double the estimated mass of the C component. This primary mass was placed at 297 AU from C with some randomly selected initial mean anomaly, an inclination of $30\arcdeg$ and zero eccentricity. Still no chaos was detected with the sibling method (Fig. 3, right), at least within the first $8\times 10^{5}$ yr. To rule out possible technical errors, this result was verified with an independent integration technique, for which the well-tested Mercury code was selected (Chambers & Migliorini, 1997). The conclusion drawn from these numerical simulations is that GJ 667C with its two confidently detected planets represents a remarkably stable system in the 4:1 MMR with a rapid periodical exchange of angular momentum between the two planets. Figure 3: Absolute distance between two sibling integrations of the orbits of planets GJ 667Cc and GJ 667Cd. Left: three bodies simulated in the system, i.e., the host star GJ 667C and the planets. Right: four bodies simulated, a single primary body replacing the close pair of stars GJ 667 A and B, the host star GJ 667C, and the two planets. Table 3: Default parameters GJ 667C planets b and c. Parameter \| b \| c ---\|---\|--- $\xi$ \| $0.4$ \| 0.4 $R$ \| $1.79R_{\rm Earth}$ \| $1.65R_{\rm Earth}$ $M_{2}$ \| $5.7M_{\rm Earth}$ \| $4.5M_{\rm Earth}$ $a$ \| $0.049$ AU \| $0.12$ AU $e$ \| 0.27 \| 0.25 $(B-A)/C$ \| $5\cdot 10^{-5}$ \| $5\cdot 10^{-5}$ $P_{\rm orb}$ \| $7.2$ d \| 28.1 d $\tau_{M}$ \| 50 yr \| 50 yr $\mu$ \| $0.8\cdot 10^{11}$ kg m-1 s-2 \| $0.8\cdot 10^{11}$ kg m-1 s-2 $\alpha$ \| $0.2$ \| 0.2 ## 4 The likely spin-orbit state of GJ 667Cc The polar torque acting on a rotating planet is the sum of the gravitational torque, caused by the triaxial permanent shape and the corresponding quadrupole inertial momentum, and the tidal torque, caused by the dynamic deformation of its body. The former torque is often considered to be strictly periodical, resulting in the net zero acceleration of rotation. An important violation of this rule is discussed in (Makarov, 2012), namely, that when the planet is locked in a spin-orbit resonance, a compensating nonzero secular torque emerges. The suitable equation for the oscillating triaxial torque can be found in, e.g., (Danby, 1962; Goldreich & Peale, 1966, 1968). For the tidal component of torque, we are using an advanced model based on the development of Kaula and Darwin’s harmonic decomposition in (Efroimsky & Lainey, 2007; Efroimsky & Williams, 2009; Efroimsky, 2011b, a), combined with a rheological model approximately known for the Earth. The rheological model derives from two principles, which we believe are valid for any silicate or icy bodies: 1) in the low-frequency limit, the mantle’s behavior should be close to that of the Maxwell body because of the dominant mechanism of dissipation, namely, the lattice diffusion; 2) at frequencies higher than a certain threshold (which is $\simeq 1$ yr-1 for Earth), the mantle behaves as the Andrade body due to the pinning-unpinning of dislocations. The threshold may move depending on the temperature of the mantle, which is unknown for exoplanets. Other rheological parameters may vary too, depending on the average temperature and chemical constituency. However, the qualitative shape of the frequency-dependence of tidal response shown in Fig. 4 (left) should be universal for rocky or icy planets. Possible deviations from this model may be caused by extensive internal or surface oceans, a subject outside the scope of this paper. As has been emphasized in the previous publications about this theory, profound implications for planets and moons of terrestrial composition call for a significant review of currently accepted views and assumptions (Makarov & Efroimsky, 2013; Efroimsky & Makarov, 2013). In particular, capture into higher-order, supersynchronous resonances is much easier than in the previously widely exploited models of linear or constant (frequency- independent) torque. This is due to the kink-shape of the secular tidal torque component, which acts as an efficient trap abruptly arresting the gradual spin-down. Fig. 4(left) shows the dependence of the secular tidal acceleration for GJ 667Cc with normalized spin rate $\dot{\theta}/n$ in a narrow vicinity of the 5:2 resonance. The default parameters used in this calculation, and in most of our other simulations, are given in Table 3. Figure 4: Left: Secular tidal acceleration of GJ 667Cc versus normalized spin rate in the vicinity of the 5:2 spin-orbit resonance. Right: A simulated capture of GJ 667Cc into the 5:2 resonance. As explained in more detail in (Makarov, 2012; Noyelles et al., 2013), the structure of secular torque in the vicinity of a spin-orbit resonance can be considered to consist of two functionally different components: the resonant kink, which is perfectly symmetrical around the point of resonance, and a nearly constant bias. In fact, the bias is nothing other than the sum of the distant parts of all other resonant kinks, separated by $1\,n$ in $\dot{\theta}$. For low and moderate eccentricities, the amplitude of the kinks is a rapidly decreasing function of the resonance order $q$, the synchronous kink ($lmpq=2200$) being by far the dominating one. Therefore, the bias at the 1:1 resonance is positive, being the sum of the positive left wings of the higher-order kinks, whereas the bias at the higher-order order resonances is negative, being dominated by the negative right wing of the overarching 1:1 kink. As a result, the secular action of tides is to decelerate a planet rotating faster than $1\,n$, and to accelerate a slower rotating planet (including a retrograde rotation). On the example shown in Fig 4 (left) for the 5:2 resonance, we can see that the negative bias can well be larger in absolute value than the amplitude of the corresponding kink, in which case the tidal torque is negative (decelerating) everywhere around this resonance. However, this does not preclude the possibility of the planet to be captured into this resonance. Fig. 4 (right) depicts such a capture, simulated with these initial conditions: mean anomaly at time zero, ${\cal M}(0)=0$, initial rate of rotation, $\dot{\theta}(0)=2.52\,n$, sidereal angle of rotation, $\theta(0)=0$111All other notations used in this formula and throughout the paper are listed in Table 4.. Thus, the capture of GJ 667Cc into a 5:2 resonance is possible with the set of estimated default parameters (if not probable). Following the ideas in (Makarov et al., 2013), we herewith apply two different, independent methods to estimate the probabilities of capture into the 5:2 resonance. The first method is a brute-force integration of the ODE on a grid of initial conditions $\theta(0)$ for a fixed $\dot{\theta}(0)$ and a zero initial mean anomaly. The other method is to use a semi-analytical analogue of the capture probability formula derived by (Goldreich & Peale, 1968) for the simple constant- and linear-torque models. Using the first method, we performed 40 short-term integrations similar to the one depicted in Fig. 4 (right), with a fixed eccentricity $e=0.25$ and for a grid of initial rotation angles $\theta(0)=\pi\,j/40$, $j=0,1,\ldots,39$. The time of integration in each case was 1000 yr, and the variable step of integration not larger than $1.5\times 10^{-3}$ yr. Only 4 out of these 40 integrations resulted in capture, the others traversing this resonance. The estimated probability of capture for the given set of parameters is $0.10\pm 0.03$. The uncertainty of this and other numerically estimated probabilities are simply the formal error on a Poisson-distributed sample estimate, given here only as a guidance. The uncertainty associated with the input parameters may be more significant. Table 4: Explanation of notations Notation \| Description ---\|--- $\xi$ \| moment of inertia coefficient $R$ \| radius of planet $T$ \| torque $M_{2}$ \| mass of planet $M_{1}$ \| mass of star $a$ \| semimajor axis of planet $r$ \| instantaneous distance of planet from star $\nu$ \| true anomaly of planet $e$ \| orbital eccentricity $M$ \| mean anomaly of planet $B$ \| second moment of inertia $A$ \| third moment of inertia $n$ \| mean motion, i.e. $2\pi/P_{\rm orb}$ ${\cal G}$ \| gravitational constant, $=66468$ m3 kg-1 yr-2 $\tau_{M}$ \| Maxwell time $\mu$ \| unrelaxed rigidity modulus $\alpha$ \| tidal lag responsivity Figure 5: Left: Simulated passage of GJ 667Cc’s spin rate through the 5:2 spin-orbit resonance. Only two free libration cycles are shown, one immediately preceding the passage, and the other following it. Right: The same two libration cycles as in the left panel, but as a separatrix trajectory in the $\\{\gamma,\dot{\gamma}\\}$ parameter plane. The actual, accurately integrated trajectory is depicted with the bold line, and the assumed separatrix in the semi-analytical calculation of capture probability with a dashed line. The other method is the semi-analytical calculation based on the energy balance consideration proposed by Goldreich & Peale (1966, 1968). An estimate of capture probability is derived from the consideration of two librations around the point of resonance $\dot{\gamma}=-\chi_{220q^{\prime}}=2\dot{\theta}-(2+q^{\prime})n=0$, i.e., the last libration with positive $\dot{\gamma}$ and the first libration with negative $\dot{\gamma}$ for a slowing down planet. The angular parameter $\gamma=2\theta-(2+q^{\prime}){\cal M}$ is introduced for convenience. Goldreich & Peale (1966) assumed that the energy offset from zero at the beginning of the last libration above the resonance is uniformly distributed between $0$ and $\Delta E=\int\langle T\rangle\dot{\gamma}dt$. Then the probability of capture is $P_{\rm capt}=\frac{\delta E}{\Delta E},$ (1) with $\delta E$ being the total change of kinetic energy at the end of the libration below the resonance. Thus, $\langle T\rangle\dot{\gamma}$ should be integrated over one cycle of libration to obtain $\Delta E$, and over two librations symmetric around the resonance $\chi_{220q^{\prime}}=0$ to obtain $\delta E$. As a result, the odd part of the tidal torque at $q=q^{\prime}$ doubles in the integration for $\delta E$, whereas the bias vanishes; both these components are involved in the computation of $\Delta E$. Denoting the bias and the kink components, respectively, $V$ and $W(\dot{\gamma})$, the capture probability is $P_{\rm capt}=\frac{2}{1+2\pi V/\int_{-\pi}^{\pi}W(\dot{\gamma})d\gamma}$ (2) The integral in this equation can be computed if we further assume that the trajectory in the vicinity of resonance follows the singular separatrix solution of zero energy $\dot{\gamma}=2\,n\,\left[\frac{3(B-A)}{C}G_{20q^{\prime}}(e)\right]^{\frac{1}{2}}\cos\frac{\gamma}{2},$ (3) where $G_{lpq}(e)$ is the eccentricity function. The combination of Eqs. 2 and 3 makes for a fast way of estimating the capture probability without the need of performing multiple integrations of differential equations. It should also work for a rising spin rate, i.e., for an accelerating rotation. Computation using this method with the default parameters of GJ 667Cc ($\tau_{M}=50$ yr, $e=0.25$) for the 5:2 resonance yield a capture probability of 0.19. This estimate is significantly higher than the number (0.10) we obtained by brute-force computations. The large discrepancy indicates that some of the assumptions used for one, or both, of the methods is invalid or inaccurate. We investigated this problem in depth, and came to the following conclusion. The weakest assumption in the semi-analytical method is the shape of the separatrix, Eq. 3. It assumes that the libration curves begin and end at the resonant $\dot{\gamma}=0$. This may be a good approximation for a slow tidal dissipation case, but it breaks for fast spin-downs, such as the one we are dealing with here. This conclusion is illustrated by Fig. 5. The left panel of this Figure shows the variation of $\dot{\gamma}$ for $q^{\prime}=3$ (i.e., 5:2 resonance) while the planet is traversing the resonance, obtained by numerical integration. For a better detail, only the two critical libration oscillations are displayed. We observe that the starting $\dot{\gamma}$ of the pre-resonance libration is significantly above the resonance, whereas the ending $\dot{\gamma}$ of the post-resonance libration is significantly below it. These shifts are due to the substantial secular tidal torque acting on the planet, and the ensuing fast deceleration. The shape of the separatrix trajectory is more clearly shown in the right part of Fig. 5 for the same pair of libration cycles. The pre-resonance libration is positive in this parametric plot, the post-resonance libration is negative, and the planet moves clockwise along this trajectory. The actual, accurately computed trajectory is shown with the solid red line, whereas the approximate trajectory described by Eq. 3 is shown as the dashed line. The actual trajectory does not make a closed loop, as the start and the end points are separated by a gap. This discontinuity of the separatrix is always present, of course, due to the finite tidal dissipation of kinetic energy during the two critical libration cycles. But in many cases, e.g., most of the bodies in the Solar system, the tidal dissipation is so small that the departure from a closed, symmetric separatrix can be neglected. Furthermore, for a constant or slowly varying with frequency tidal torque, this departure results in a small error, which can be neglected. This is not the case for GJ 667Cc with our tidal model. As seen in Fig. 4 (left), the kink function is rapidly decreasing with tidal frequency on either side of the resonance. The ”missing” part of the integrand $W(\dot{\gamma})\dot{\gamma}$ due to the gap may therefore bring about a significant change of estimated probability. In order to test this idea, we extracted the appropriate segment of the $\dot{\gamma}$-curve from the integrated solution and used this numerically quantified function in Eq. 2 to compute the probability of capture. The resulting probability is 0.14, which is much closer to the value $0.10\pm 0.03$ estimated by brute-force integration. In the following analysis of probabilities of capture and of resonance end- states, we rely on the first, entirely numerical way of estimation. For a given resonance $(2+q^{\prime})$:2, $q^{\prime}=0,1,\ldots$, and with a fixed $e$, we ran 40 simulations of the spin rate, starting from a value of $\dot{\theta}$ above the resonant value, ${\cal M}(0)=0$, and 40 initial values of $\theta$ evenly distributed between 0 and $\pi$. If $N_{c}$ is the number of captures detected in a set of forty, the estimated probability of capture is $N_{c}/40$. We further had to take into account that the eccentricity varies in a wide range (Fig. 2). When the planet’s spin rate crosses a particular resonance, any phase value of the eccentricity oscillation can be assumed equally probable. The probability of eccentricity to have a certain value at this time can be approximated by dividing the full oscillation period into a number of intervals of equal length and computing the median eccentricity for each interval. Table 5 gives the quantized probability distribution of $e$ estimated in this fashion, using the results of numerical simulations described in §3. The top-range values are more likely than the bottom-range values because the eccentricity curve is flat at the top. For each of the characteristic values of $e$, we performed a set of 40 integrations of the spin-orbit differential equation on a regular grid of initial $\theta(0)$ and counted the number of captures. The estimated probability of capture for the given eccentricity was then one-fortieth of this number. Each estimated probability of capture was multiplied by the corresponding probability of $e$, and the sum of these 10 numbers was the overall probability of capture for a random realization of $e$. This procedure was repeated for the 5:2, 2:1, and 3:2 resonances, resulting in a total of 1200 simulations. Using this somewhat laborious method, we arrived at these probabilities of capture (on a single trial): 0.03 into 5:2, 0.23 into 2:1, and 0.68 into 3:2. These numbers were obtained for the default $\tau_{M}=50$ yr and $(B-A)/C=5\times 10^{-5}$. If we are more interested in the current-state probabilities, these numbers need to be recomputed, taking into account that a planet locked into a higher resonance, e.g., 5:2, can not ever reach a lower resonance, e.g., 2:1. The current-state probabilities are, obviously, 0.03 for 5:2, 0.22 for 2:1, and 0.51 for 3:2. The remaining trials, at a probability of 0.24, are certain to end up in the 1:1 resonance. Table 5: Quantized probability distributions for planet GJ 667Cc and b. planet c \| planet b ---\|--- $e$ \| $P_{e}$ \| $e$ \| $P_{e}$ $0.061$ \| 0.146 \| 0.074 \| 0.118 $0.080$ \| 0.072 \| 0.095 \| 0.060 $0.100$ \| 0.070 \| 0.116 \| 0.061 $0.119$ \| 0.070 \| 0.138 \| 0.061 $0.138$ \| 0.069 \| 0.159 \| 0.059 $0.158$ \| 0.071 \| 0.180 \| 0.069 $0.177$ \| 0.071 \| 0.202 \| 0.075 $0.197$ \| 0.088 \| 0.223 \| 0.093 $0.216$ \| 0.109 \| 0.244 \| 0.114 $0.236$ \| 0.232 \| 0.266 \| 0.289 The probabilities of capture are known to depend on the degree of elongation $(B-A)/C$ and the Maxwell time $\tau_{M}$, which are quite uncertain. From our previous study on GJ 581d, we knew that the dependence on $(B-A)/C$ is much weaker than on $\tau_{M}$. Bodies with smaller $(B-A)/C$, i.e., more spherical or axially symmetric, are more easily captured into super-synchronous resonances. Generally, larger planets have smaller elongation parameters than the smaller planets or moons. Our choice of this parameter is deemed conservative in terms of the capture probability estimation. On the other hand, a warmer, less viscous planet with a smaller value of $\tau_{M}$ is much more likely to be captured into super-synchronous equilibria. The average viscosity of the mantle is poorly known even for the Solar system bodies, including the Moon. The amount of partial melt, in particular, may be crucially significant for the spin-orbit evolution. The reverse is also true, in that the spin-orbit interactions define the amount of tidal heat production, resulting, under favorable conditions, in a partial melt-down of the mantle, or in a significant warming over the course of billions years. ## 5 Characteristic time of spin-down Assuming that in the distant past, the planet was rotating very fast in the prograde sense, how long does it take to spin down and fall into one of the spin-orbit resonances? This can be assessed through a parameter called the characteristic spin-down time, customarily defined as $\tau_{\rm spin-down}=\frac{\dot{\theta}}{\|\ddot{\theta}\|},$ (4) where $\ddot{\theta}$ is the angular acceleration caused by the secular component of the tidal torque. This time parameter should not be confused with the actual time for the planet to decelerate from a certain initial spin rate and fall into a resonance, which is normally shorter. Indeed, Eq. 4 allows us to quickly compute the instantaneous angular acceleration $\ddot{\theta}$ and the corresponding spin-down time for a given spin rate $\dot{\theta}$ using known analytical equations for the secular tidal torque. But the angular acceleration by itself is a nonlinear function of $\dot{\theta}$; therefore, the rate of spin-down grows faster as the planet decelerates. The instantaneous characteristic spin-down times as functions of the spin rate are shown in Fig. 6 for two values of $e$, which bracket the range of its variation, the upper curve corresponding to $e=0.06$ and the lower curve to $e=0.24$. In both cases, we assumed the present-day observed values of semimajor axis (Table 3). We find that for $\dot{\theta}<8\,n$, the spin-down times are well within 1 Myr, which is much shorter than the presumed life time of the planet. The sign of $\ddot{\theta}$ changes to positive for spin rates slower than the mean orbital motion, $\dot{\theta}<1\,n$, i.e., the planet spins up with such slow rotation rates, including retrograde rotation. The apparent discontinuities of the lower curve correspond to main supersynchronous resonances, which the planet either traverses quite quickly or becomes entrapped in. The short spin-down times suggest that the planet was captured into the current resonant state as long as a few Gyr ago. This may also indicate that the semimajor axes and the separation between the planets were different when this capture happened, because the energy for tidal dissipation is drawn from the orbital motion when the spin rate is locked. The orbital evolution of two-planet systems with significant tidal dissipation locked both in a MMR and a spin-orbit resonance is a complex problem, which lies beyond the scope of this paper. Figure 6: Characteristic times of tidal spin-down of the planet GJ 667Cc for two values of orbital eccentricity: 0.06 (upper curve) and 0.24 (lower curve). ## 6 Likely spin-orbit state of GJ 667Cb According to the data in Table 3, the planet b is much closer to the host star and is somewhat more massive than the planet c. As the polar component of the tidal torque scales as $T_{z}\propto R^{5}/r^{6}$ (Efroimsky & Makarov, 2013), where $R$ is the radius of the planet and r is the distance to the host star, the tidal forces on planet b should be at least 200 times stronger than on planet c. This, however, does not necessarily imply higher probabilities of capture into super-synchronous spin-orbit resonances. To understand why this is the case, the analogy of the capture process to a rotating driven pendulum with damping may be useful Goldreich & Peale (1968); Makarov (2012). The overall (negative) bias of the tidal torque acts as a weak driving force against the initial prograde rotation of the pendulum, whereas the frequency- dependent, and highly nonlinear in our case, component of the torque acts as friction. The pendulum gradually slows down in its rotation and, inevitably, it is no longer able to come over the top. On the first backward swing, the bias will assist it in passing the top in the opposite direction, while the friction will further diminish the amplitude of oscillation. The subtle balance between these components become crucial in whether the pendulum can traverse the top point and commence rotating in the retrograde sense, or it becomes locked in the gradually diminishing swings around the point of stable equilibrium. The probabilistic nature of these outcomes originates from the finite range of possible positions when it is stalled in the vicinity of the top point. A much stronger tidal force increase the bias and the frequency- dependent (friction) components in the same proportion, but the balance between them is mostly affected by the orbital eccentricity. Since the eccentricity of the two planets are not too different on average (Fig. 2), the capture probabilities may be close too. Here we compute the planet b capture probabilities for the resonances 3:2, 2:1 and 5:2 essentially repeating the steps described in §4. We assume the same Maxwell time, $\tau_{M}=50$ yr, as for planet c. On a grid of regularly spaced points in eccentricity between 0.074 and 0.266, 40 simulations with uniformly distributed initial libration angles are performed, starting with a spin rate above the resonant value. The number of captures is counted in each batch of 40 simulations, with the total number of batches being 300\. The estimated probabilities are weighted with the binned probabilities of eccentricity given in Table 5 and summed up for each resonance. The probabilities of individual capture events are: 0.82 for 3:2, 0.32 for 2:1, and 0.10 for 5:2 resonances. Capture into the 1:1 resonance is certain. If the planet evolved from high prograde spin rates, which seems to be the likeliest scenario judging from the Solar planets, it can reach a lower resonant state only if it traversed all the higher resonances. Taking into account the compounding conditional probability, the probabilities of the end-states are: 0.10 for 1:1, 0.51 for 3:2, 0.29 for 2:1, and 0.10 for 5:2 resonances. Again, as in the case of planet c, the most likely state for planet b is the 3:2 spin-orbit resonance. The distribution of probabilities for b is shifted toward higher orders of resonance because of the slightly larger average eccentricity. If the planet originally had a retrograde spin (such as Venus in the Solar System), the only long-term stable state is the complete synchronization of rotation. ## 7 The rate of tidal heating An exoplanet of terrestrial composition captured into a spin-orbit resonance continues to dissipate the orbital kinetic energy through the tidal friction inside its body. If the resonance is not synchronous, the spin rate differs from the orbital rate, resulting in a constant drift of the tidally-raised bulge across the surface of the planet. The internal shifts of the material cause the mantle to warm up. Naively, one would expect that faster motions of the tidal bulge should bring about higher rates of dissipation and, therefore, more vigorous production of tidal heat. This indeed follows from the commonly used Constant Time Lag (CTL) model of tides (e.g., Leconte et al., 2010), which is also commonly misapplied to rocky planets. There are two crucial defects of such models as applied to rocky exoplanets: 1) the actual rheology of earth-like solids unambiguously implies a declining with frequency kvalitet222Kvalitet stands for ”quality” in Danish, which we use here as a more general term for the customary tidal quality factor $Q$ in the literature. at high perturbation frequencies; 2) self-gravity strongly limits the amplitudes of tides on large exoplanets and stars. On the other hand, the tidal bulge is stationary with respect to the planet’s body if the planet is locked into a synchronous rotation and both the orbital eccentricity and obliquity of the equator are exactly zero; in such a hypothetical case, the tide is on, but there is no tidal dissipation of energy or heat production. We have determined that the most likely state of GJ 667Cc is a 3:2 resonance. In this state, the planet makes a full turn around its axis 3 times for every 2 orbital periods with respect to distant stars, but only one turn for every two orbital periods with respect to the host star. In other words, the day on this planet is likely to be $28.1\times 2=56.2$ d long. The main semi-diurnal tidal mode will have a period of 28.1 d. Apart from the average prograde motion of the tidal bulge, relatively small oscillations of the tidal perturbation should be expected from the longitudinal and latitudinal librations. Physically, the librations can be separated into two categories, the forced librations caused by a variable perturbing force, and the free librations caused by an initial excess of kinetic energy. The latter kind of librations is expected to damp relatively quickly for Mercury-like planets (Peale, 2005), with a characteristic damping time orders of magnitude shorter than the planet’s life time. The remaining forced librations in a two-body system are solely due to the periodical orbital acceleration of the perturber on an eccentric orbit, and their amplitude is quite small for massive super- earths. Additional harmonics of forced libration should be expected for GJ 667C planets due to their mutual interaction. It turns out that the periodical terms of the tidal torque are completely insignificant in comparison with the secular modes in this case. The formula to compute the rate of energy dissipation is taken from Makarov (2013, Eq. 2), which is an adaptation of the seminal work by Peale & Cassen (1978, in particular, Eq. 31) for the synchronously rotating Moon. The latter equation was essentially obtained following the geometrical consideration by Kaula (1961). This formalism can be used for any eccentricity and any spin rate, as long as the secular components of the tidal dissipation are concerned. It should be noted that the $lmpq$ terms in the series given by Peale & Cassen (1978) appear to come with different signs. In practice, their absolute values should be used instead, because each tidal mode can only increase the dissipation, independent of the sign of the corresponding tidal torque. Alternatively, one can consider the factor $Q_{lmpq}$ to be an odd function of tidal frequency. With these important corrections, the resulting dissipation rate is significantly higher than what would have been obtained with the approximate equation truncated to $O(e^{2})$. Fig. 7 shows the rate of energy dissipation computed as a function of spin rate in the vicinity of the 3:2 resonance, for the default parameters of GJ 667Cc and $e=0.15$. The gently sloping curve is only marked with a tiny dent at the resonant spin rate. Therefore, the exact shape and amplitude of longitudinal librations is not important for the estimated rate of dissipation. The dependence on eccentricity is not strong either, ranging from $10^{23.53}$ J yr-1 for $e=0.05$ to $10^{23.77}$ J yr-1 for $e=0.22$. For the median value of eccentricity $e=0.18$, the estimated energy dissipation rate is $10^{23.7}$ J yr-1. Figure 7: The rate of tidal energy dissipation inside GJ 667Cc in the vicinity of the 3:2 spin-orbit resonance, which we find the most likely state for this planet. We are adopting a value of 1200 J kg-1 K-1 for the planet’s heat capacity from (Běhounková et al., 2011). The average rise of temperature for the entire planet is $1.6\cdot 10^{-5}$ K yr-1. At this rate, the mantle should reach the melting point of silicates in less than $0.1$ Gyr. More accurate calculations than we are able to carry out in this paper would require a careful modeling of the mantle convection effects, radiogenic heating, and heat transfer. But given the warm-up time that is much shorter than the life time of M dwarfs, it will not be too bold to say that a considerable degree of melting and structural stratification should have occurred on planet GJ 667Cc. In that respect, the situation is reminiscent of Mercury in the Solar System, which has a massive molten core extending up to 0.8 of its radius. The spin-down time for Mercury is of the order of 107 yr Noyelles et al. (2013), suggesting that the planet has been in the current 3:2 spin-orbit resonance for billions of years. The molten core therefore formed after the capture into this resonance. The tidal dissipation rate at a super-synchronous rotation resonance becomes of an overarching importance for our understanding of the structure and destiny of inner planets subject to relatively strong tidal forces. If the tidal dissipation rate suggests that at least a partial melt should have occurred on the planet GJ 667Cc, similar calculations for the planet GJ 667Cb leave little doubt that the planet should be completely molten. For a median eccentricity $e=0.18$ and the same heat capacity, the estimated heat production is 1.1 K in just 100 yr. The planet should quickly become a ball of molten magma. The most likely scenario for such close-in planets trapped in resonances seems to be overheating and destruction. But something obviously has prevented this planet from complete evaporation. This problem requires a dedicated and accurate study; here, we only offer some possibilities that could change the conclusions. When the temperature of a rocky planet rises, its rheology changes too. In particular, the Maxwell time in the expression for kvalitet is quite sensitive to the average temperature. The probabilities of capture into spin-orbit resonances estimated in Sections 4 and 6 can only become higher for a warmer planet due to the shortening of its Maxwell time. If the planets were already heated up by the time their spin rates reached the lower commensurabilities with the orbital motions, they are even more likely to have been captured and to have remained in the higher spin-orbit resonances. For partially molten or partially liquid bodies, the Maxwell time may be comparable to the period of rotation, or shorter, which is probably the case for Titan (F. Nimmo, priv. comm.). This may drastically change the frequency-dependent terms of tidal dissipation. At the extreme, a ball of water or a planet with a massive ocean is likely to have a principally different tidal response than solid bodies (Tyler, 2013). Besides, a liquified planet can lose its “permanent” figure and become nearly perfectly spherical or oblate, radically changing the conditions of continuous entrapment in the resonance. A combination of these events, speculatively, can create a seesaw effect, when an eccentric solid planet is liquified by the tidal dissipation, which causes the rate of dissipation to drop by orders of magnitude, followed by a slow cooling, solidification of the surface layers, acquiring a permanent figure shaped by the tidal interaction, bringing up an episode of strong tidal heating and melt-down, and so on, ad infinitum. ## 8 Conclusions The rapidly growing class of detected super-Earths, which may reside in the habitable zone and, thus, harbor life in the biological forms familiar to us, sets new objectives and motivations for the interpretation of planetary spin- orbital dynamics and the theory of tides. The history of planet’s rotation and the tidal dissipation of kinetic energy in the long past is certain to play a crucial role in the formation (or the absence of such) of liquid oceans and gaseous atmospheres. This study is ridden with uncertainties of both observational and theoretical kind. Earthlings are blessed with a rapid and stable rotation of their home planet, but are the potentially habitable super- Earths similarly hospitable in this respect? Looking at the better known Solar planets and satellites, which objects are the closest analogy to the massive super-Earths orbiting near their M-type hosts? Judging from the observational data we have today, and making use of the much improved tidal model for rocky planets, the super-Earths seem to be more similar to the tiny Mercury than to the Earth, as far as their spin-orbit dynamics is concerned. Mercury, making exactly 3 sidereal rotations per one orbital period (Pettengill & Dyce, 1965), is the only planet in the Solar system captured into a supersychronous resonance. This resonance happens to be the most likely outcome of Mercury’s spin-orbit evolution even without the assistance of a liquid core friction, due to the relative proximity of the planet to the Sun and its considerable eccentricity, which could have reached even higher values in the past (Correia & Laskar, 2004). Mercury also has rather short characteristic times of spin-down of the order of $10^{7}$ yr (Noyelles et al., 2013), which indicate that the massive molten core was formed after the capture event. The same circumstances define a supersynchronous resonant rotation as the most probable state of both GJ 667C planets. The characteristic spin-down time for the planet GJ 667Cc is of the order of 1 Myr, which is even shorter than that for Mercury. The planet is certain to be in one of the low-order resonances, that is, its current rotation with respect to the host star is very slow, if any. Due to the massiveness of this exoplanet, the longitudinal forced librations are likely insignificant for the considerations of its habitability. However, being locked into a mean-motion resonance with planet b, the orbit undergoes rapid, high amplitude variations having, undoubtedly, a significant impact on the circulation of the hypothetical atmosphere and climate. The slow relative rotation and a long solar day generally imply rather harsh conditions on the surface; at the same time, a combination of the significant eccentricity, obliquity of the equator (which is ignored in this paper), and longitude of the vernal equinox may provide for well-shielded areas on the surface with favorably stable and moderate insolation (Dobrovolskis, 2011). Perhaps even more threatening in terms of habitability is the high rate of tidal heating that we estimate for the planet GJ 667Cc. The estimated rise of average temperature by 1.6 K per $10^{5}$ yr is likely to cause a partial or complete melting of the planet’s mantle. It remains to be investigated if there are any safety mechanisms in the physics of tidal dissipation that can automatically prevent a super-Earth on an eccentric orbit from overheating. The rate of tidal dissipation in resonantly spinning super-Earths happens to be weakly dependent on orbital eccentricity, which is not very accurately determined by the RV planet detection technique. For example, the rate of dissipation at $e=0.05$ is smaller by just $dex(0.1)$ compared to that at $e=0.15$. Likewise, the unknown parameters $\tau_{M}$ and $(B-A)/C$ have a very limited impact on the accuracy of this estimation. 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1311.4843	C[1]>m#1 # Supplementary material for: Joint analysis of functional genomic data and genome-wide association studies of 18 human traits Joseph K. Pickrell1,2 1 New York Genome Center, New York, NY 2 Department of Biological Sciences, Columbia University, New York, NY Correspondence to: [email protected] ###### Contents 1. 1 GWAS data 1. 1.1 GIANT data 2. 1.2 GEFOS data 3. 1.3 IIBDGC data 4. 1.4 MAGIC data 5. 1.5 Global lipid genetics consortium data 6. 1.6 Red blood cell trait data 7. 1.7 Platelet traits 2. 2 Functional genomic data 1. 2.1 DNase-I hypersensitivity data 2. 2.2 Chromatin state data 3. 2.3 Gene models 3. 3 Imputation of summary statistics 4. 4 Details of application of the hierarchical model 1. 4.1 Simulations 2. 4.2 Robustness to choice of prior and window size 3. 4.3 Quantifying the relative roles of coding versus non-coding changes in each phenotype 4. 4.4 Interaction effects in annotation models 5. 4.5 Calibrating a ``significance" threshold 6. 4.6 Identification of novel loci ## 1 GWAS data ### 1.1 GIANT data We downloaded summary statistics from large GWAS of height [Lango-Allen et al.,, 2010] and BMI [Speliotes et al.,, 2010] from http://www.broadinstitute.org/collaboration/giant/index.php/GIANT_consortium. The height summary statistics consisted of 2,469,635 SNPs either directly genotyped or imputed in an average of 129,945 individuals. We removed all SNPs with a sample size of less than 120,000 individuals. The BMI summary statistics consisted of 2,471,516 summary statistics either directly genotyped or imputed in an average of 120,569 individuals. We removed all SNPs with a sample size of less than 110,000 individuals. We then imputed summary statistics at SNPs identified in the 1000 Genomes Project as described in Section 3. ### 1.2 GEFOS data We downloaded summary statistics from large GWAS of bone mineral density [Estrada et al.,, 2012] from http://www.gefos.org/?q=content/data-release. There are two traits in these data: bone density measured in the femoral neck and bone density measured in the lumbar spine. The femoral neck bone density GWAS consisted of 2,478,337 SNPs, and the lumbar spine bone density consisted of 2,468,080 SNPs. Because the sample size at each SNP was not reported, we used the overall study sample sizes of 32,961 and 31,800 as approximations of the sample size at each SNP, and imputed summary statistics as described in Section 3. ### 1.3 IIBDGC data We downloaded summary statistics from a large GWAS of Crohn's disease [Jostins et al.,, 2012] from http://www.ibdgenetics.org/downloads.html. The downloaded data consisted of 953,242 SNPs. Because the sample size at each SNP was not reported, we used the overall study sample sizes of 6,299 cases and 15,148 controls as approximations of the sample size at each SNP, and imputed summary statistics as described in Section 3. Note that summary statistics from a GWAS of ulcerative colitis were also available from this site; however, these data contain a number of false positive associations that were filtered by Jostins et al., [2012] using criteria that were not available to us. We thus only used the Crohn's disease association study. ### 1.4 MAGIC data We downloaded summary statistics from a large GWAS of fasting glucose levels [Manning et al.,, 2012] from http://www.magicinvestigators.org/downloads/. The downloaded data consisted of 2,628,880 SNPs. Because the sample size at each SNP was not reported, we used the overall study sample size of 58,074 as an approximation of the sample size at each SNP, and imputed summary statistics as described in Section 3. ### 1.5 Global lipid genetics consortium data We downloaded summary statistics from a large GWAS of lipid traits [Teslovich et al.,, 2010] from http://www.sph.umich.edu/csg/abecasis/public/lipids2010/. These data consist of summary statistics for association studies of four traits: LDL cholesterol, HDL cholesterol, trigylcerides, and total cholesterol. The HDL data consisted of 2,692,429 SNPs genotyped or imputed in an average of 88,754 individuals, the LDL data consisted of 2,692,564 SNPs genotyped or imputed in an average of 84,685 individuals, the total cholesterol data consisted of 2,692,413 SNPs genotyped or imputed in an average of 89,005 individuals, and the triglycerides data consisted of 2,692,560 SNPs genotypes or imputed in an average of 85,691 individuals. For all traits, we removed SNPs with a sample size less than 80,000 individuals, and imputed summary statistics as described in Section 3. To calibrate significance thresholds, we additionally used summary statistics from Global Lipids Genetics Consortium et al., [2013]. These were downloaded from http://www.sph.umich.edu/csg/abecasis/public/lipids2013/. ### 1.6 Red blood cell trait data We obtained summary statistics from a large GWAS of red blood cell traits [van der Harst et al.,, 2012] from the European Genome-Phenome Archive (accession number EGAS00000000132). We downloaded summary statistics from association studies of six traits: hemoglobin levels, mean cell hemoglobin (MCH), mean corpuscular hemoglobin concentration (MCHC), mean cell volume (MCV), packed cell volume (PCV), and red blood cell count (RBC). The hemoglobin level data consisted of 2,593,078 SNPs genotyped or imputed in 50,709 individuals, the MCH data consisted of 2,586,785 SNPs genotyped or imputed in an average of 43,127 individuals, the MCHC data consisted of 2,588,875 SNPs genotyped or imputed in an average of 46,469 individuals, the MCV data consisted of 2,591,132 SNPs genotyped or imputed in an average of 47,965 individuals, the PCV data consisted of 2,591,079 SNPs genotyped or imputed in an average of 44,485 individuals, and the RBC data consisted of 2,589,454 SNPs genotyped or imputed in an average of 44,851 individuals. We removed all SNPs with a sample size of less than 50,000 individuals (for hemoglobin levels) or 40,000 individuals (for the other traits), and imputed summary statistics as described in Section 3. ### 1.7 Platelet traits Summary statistics from a large GWAS of platelet traits [Gieger et al.,, 2011] were generously provided to us by Nicole Soranzo. The data consist of summary statistics from association studies of two traits: platelet counts and mean platelet volume. The platelet count data consisted of 2,705,636 SNPs genotyped or imputed in an average of 44,217 individuals, and the platelet volume data consisted of 2,690,858 SNPs genotyped or imputed in an average of 16,745 individuals. We removed all SNPs with sample sizes less than 40,000 (for platelet counts) or 15,000 (for platelet volume), and imputed summary statistics as described in Section 3. ## 2 Functional genomic data ### 2.1 DNase-I hypersensitivity data We downloaded DNase-I hypersensitivity data from two sources. The first was a set of regions defined as DNase-I hypersensitive by Maurano et al., [2012] in 349 samples. We downloaded .bed files for 349 samples from http://www.uwencode.org/proj/Science_Maurano_Humbert_et_al/ on February 13, 2013. These samples include 116 samples from cell lines or sorted blood cells, and 333 samples from primary fetal tissues. These latter samples were sampled from several tissues at various time points; we treated each track as independent rather than pooling data from tissues, since different experiments may have slightly different properties. The tissues in this latter group are fetal heart, fetal brain, fetal lung, fetal kidney, fetal intestine (large and small), fetal muscle, fetal placenta, and fetal skin. The second was a set of regions defined as DNase-I hypersensitive by the Crawford lab in the context of the ENCODE project [Thurman et al.,, 2012]. We downloaded .bed files for 53 samples from http://ftp.ebi.ac.uk/pub/databases/ensembl/encode/integration_data_jan2011/byDataType/openchrom/jan2011/fdrPeaks/ on March 29, 2013. We restricted ourselves to the files labeled as being generated at Duke University. Each experiment defined a set of regions of open chromatin in a particular cell type or cell line. The ``Duke" DNase-I hypersensitive sites are all of exactly 150 bases in length, and each annotation covers approximately 1% of the genome (range: 0.4 - 1.9 % of the genome). The ``Maurano" DNase-I hypersensitive sites are on average 514 bases long, and each covers on average 2.7% of the genome (range: 0.9-5.1 % of the genome). ### 2.2 Chromatin state data We downloaded the ``genome segmentations" of the six ENCODE cell lines [Hoffman et al.,, 2013] from http://ftp.ebi.ac.uk/pub/databases/ensembl/encode/integration_data_jan2011/byDataType/segmentations/jan2011/hub/ on December 18, 2012. We used the ``combined" segmentation from two algorithms. This segmentation splits the genome into non-overlapping regions described as CTCF binding sites, enhancers, promoter-flanking regions, repressed chromatin, transcribed regions, transcription start sites, and weak enhancers. This segmentation was done independently in each of six cell lines, for a total of 42 annotations. Overall the ``repressed chromatin" mark covers the largest fraction of the genome, on average 66% (ranging from 60% for HUVEC cells to 70% for H1 ES cells). The ``transcribed" mark covers on average 13% of the genome, the ``CTCF" mark 1% of the genome, the ``enhancer" mark 0.9% of the genome, the ``TSS" mark 0.7% of the genome, the ``weak enhancer" mark 0.4% of the genome, and the ``promoter-flanking" mark 0.2% of the genome. The remainder of the genome is not mappable by short reads and it thus excluded from these annotations. ### 2.3 Gene models We downloaded the Ensembl gene annotations from the UCSC genome browser on May 21. Annotations of nonsynonymous and synonymous status for all SNPs in phase 1 of the 1000 Genomes Project were obtained from ftp://ftp- trace.ncbi.nih.gov/1000genomes/ftp/phase1/analysis_results/functional_annotation/annotated_vcfs/. Coding exons cover about 3% of the genome, while 3' UTRs and 5' UTRs cover 2% and 0.6% of the genome, respectively. ## 3 Imputation of summary statistics We used ImpG v1.0 [Pasaniuc et al.,, 2013] under the default settings to impute summary statistics from all GWAS. As a reference panel, we used all haplotypes from European individuals in phase 1 of the 1000 Genomes Project, and only used SNPs with a minor allele frequency greater than 2%. The reference haplotype files were derived from the 1000 Genomes integrated phase 1 v3.20101123 calls, downloaded from ftp://ftp- trace.ncbi.nih.gov/1000genomes/ftp/phase1/analysis_results/integrated_call_sets/. We used all 379 individuals labeled as ``European". After imputation, we removed all imputed SNPs with a predicted accuracy (in terms of correlation with the true summary statistics) less than 0.8. Overall, for each GWAS, we successfully imputed about 75-80% of SNPs with a minor allele frequency over 10% (Figure 1). To verify that imputation did not induce inflation of the test statistics, we computed the genomic control inflation factor $\lambda_{GC}$ [Bacanu et al.,, 2002] before and after imputation (Supplementary Table 1). In all studies, inflation decreased after imputation, sometimes leading to a marked deflation in the test statistics. This is consistent with previous observations using this software [Pasaniuc et al.,, 2013]. The reason for this deflation is the shrinkage prior used in the imputation, which leads to conservative estimates of significance (imposed to strictly avoid false positive associations). ## 4 Details of application of the hierarchical model ### 4.1 Simulations To test the performance of the model, we performed simulations using a GWAS of height [Lango-Allen et al.,, 2010]. Using the imputed summary statistics, we split the genome into blocks of 5,000 SNPs, then extracted the blocks with a genome-wide significant SNP reported in Lango-Allen et al., [2010]. In each block, we had a reported Z-score for each SNP. To simulate annotations, we called the SNP with the smallest P-value in the region the ``causal" SNP. We then simulated annotations by placing all non-``casual" SNPs in an annotation with rate $r_{1}$, and all ``casual" SNPs in the annotation with rate $r_{2}$. We also varied the numbers of blocks included in the model. In each simulation, we randomly assigned SNPs to annotations according to determined rates, then ran our model under the assumption that $\Pi_{k}=1$, that is, all blocks contain a causal SNP. We then calculated power as the fraction of simulations in which the confidence intervals of the annotation effect did not overlap zero. We chose parameter settings of $r_{1}$ and $r_{2}$ such that the enrichment factors were similar to those in observed data (log-enrichment of 0.98 and 1.80). We chose $r_{1}$ to be either 0.2 and 0.1. For each set of parameters, we simulated 100 annotations and ran the model separately on each. Shown in Figure 2 is the power of the model. As expected, power increased as $r_{1}$ or the effect size increased, and as the number of loci increased. ### 4.2 Robustness to choice of prior and window size There are two parameters in the model that are set by the user–the prior variance $W$ on the effect size and the window size defining ``independent" blocks of the genome. We empirically tested the robustness of the model to variation in these parameters using the Crohn's disease dataset. We ran the model on each annotation using $W=0.1$ and $W=0.5$, additionally including–as in our main analyses–region-level parameters for regions in the top third and bottom third of gene density and SNP-level parameters for SNPs located from 0-5kb from a transcription start site and SNPs 5-10kb from a transcription start site. Plotted in Figure 16A are these annotation parameter estimates for all annotations where the 95% confidence intervals did not overlap 0 in at least one run. The estimates from the two runs with different priors are highly correlated. We additionally tested window sizes of 5,000 SNPs and 10,000 SNPs (both with $W=0.1$). The annotation effect estimates from these two window sizes are plotted in Figure 16B, and again are highly correlated. ### 4.3 Quantifying the relative roles of coding versus non-coding changes in each phenotype To generate Figure 3 in the main text, we fit a model to each GWAS where we included region-level annotations for regions in the top third and bottom third of the distribution of gene density, and SNP-level annotations for non- synonymous SNPs and SNPs within 5kb of a transcription start site. Shown in Figure 3A in the main text are the estimates of the enrichment parameter for non-synonymous SNPs. At each SNP, the result of this model is the posterior probability that the SNP is casual (see Equation 19 in the main text). If we let this posterior probability at SNP $i$ be $PPA_{i}$, then the fraction of causal SNPs that are non-synonymous, $f_{NS}$ is: $f_{NS}=\frac{\sum_{i}PPA_{i}I^{NS}_{i}}{\sum_{i}PPA_{i}},$ (1) where $I^{NS}_{i}$ is an indicator variable that takes value one if SNP $i$ is non-synonymous and zero otherwise. To get error bars on this fraction, we performed a block jackknife. We split the genome into 20 blocks with equal numbers of SNPs. If $f^{j}_{NS}$ is the estimate of the fraction of casual SNPs that are non-synonymous excluding block $j$, then: $SE=\sqrt{\frac{19}{20}\sum\limits_{j=1}^{20}(f^{j}_{NS}-\bar{f}_{NS})^{2}},$ (2) where $\bar{f}_{NS}=\frac{1}{20}\sum\limits_{i=1}^{20}f^{i}_{NS}$. In Supplementary Figure 3, we show the corresponding results for synonymous SNPs. ### 4.4 Interaction effects in annotation models As noted in the main text, there were two cases in which the sign of the annotation effect flipped between the single annotation models and the combined models. These were Crohn's disease (Supplementary Table 6) and red blood cell count (Supplementary Table 18). In the main text we discuss the Crohn's disease example. For the red blood cell count example, note that SNPs influencing this trait are enriched in the annotation of DNAse-I hypersensitive sites in the fetal renal pelvis when this annotation is considered alone (log2 enrichment of 2.48, 95% CI [0.04, 4.17]). This annotation is correlated with the fetal stomach annotation, which has a log2 enrichment of 4.83 (95% CI [3.30, 6.45]) when treated alone. The SNPs in both of these annotations have a log2 enrichment of 2.41 (95% CI [-1.83, 4.23]), which leads to the interaction effect. Essentially the signal in the fetal stomach is driven by those SNPs that fall in DNase-I hypersensitive sites in the fetal stomach but _not_ the fetal renal pelvis. This suggests that there are a subset of DNase-I hypersensitive sites that are of particular interest for this phenotype. The interpretation of the Crohn's disease example is similar. ### 4.5 Calibrating a ``significance" threshold For each genomic region, our method estimates the posterior probability that the region contains a SNP associated with a trait. If the model were a perfect description of reality, this probability could be interpreted literally. Since the model is not perfect, however, we sought a more empirical calibration. We used the fact that we initially ran the method on the GWAS data reported by Teslovich et al., [2010] on four lipid traits. Since then, a GWAS with more individuals (though at a considerably smaller number of SNPs) has been reported for these four traits [Global Lipids Genetics Consortium et al.,, 2013]. This latter study contains many of the individuals from the former (which had approximately 90,000 individuals), as well as about 80,000 more individuals. However, the additional individuals were genotyped in the Metabochip [Voight et al.,, 2012], which has less than 200,000 markers, rather than the more dense standard GWAS arrays. This means that some regions of the genome do not benefit from the larger sample size. For each region of the genome for each of the four traits, we built a table containing the minimum P-value from Teslovich et al., [2010], the posterior probability of association in the region (computed using the data from Teslovich et al., [2010]), the minimum P-value from Global Lipids Genetics Consortium et al., [2013], and the sample size used to get this minimum P-value (from Global Lipids Genetics Consortium et al., [2013]). We discarded regions where sample size at the SNP with the minimum P-value in the replication data set was smaller than 120,000 (since in these regions there is essentially no new data). We then coded each region as a ``true positive" if the minimum P-value from Global Lipids Genetics Consortium et al., [2013] was less than $5\times 10^{-8}$ and a ``true negative" otherwise. In Figure 15, we plot the number of ``true positives" and ``false negatives" that exceed various P-value and PPA thresholds. Note that since the data in Global Lipids Genetics Consortium et al., [2013] is not independent of that in Teslovich et al., [2010], this comparison is not appropriate for evaluating the relative performance of P-values versus the PPA. Our goal was simply to find a PPA threshold with similar performance in terms of reducing the number of false positives as the standard P-value threshold of $5\times 10^{-8}$. By visual inspection we set a PPA threshold at 0.9 (Figure 15). At this threshold, we identify 45 ``true positives" and zero ``false positives" for HDL, 43 and 1 for LDL, 47 and zero for total cholesterol, and 27 and zero for triglycerides. These are similar to the numbers for a P-value threshold of $5\times 10^{-8}$ (Supplementary Table 21). Combining the loci identified by both methods leads to 48 loci for HDL (versus 43 using a P-value threshold), 44 for LDL (versus 40), 51 for TC (versus 51) and 30 for TG (versus 29). This is on average an increase of 6% in the number of loci identified. Note that this number is likely a lower bound, since the P-values in the replication study are naturally highly correlated to those in the initial study since they use many of the same individuals. A proper comparison would use a completely separate, large set of individuals to determine ``true positives" and ``true negatives", but such samples are not yet available. ### 4.6 Identification of novel loci For each fitted model (using the parameters from Supplementary Tables 3-20 estimated using the penalized likelihood), we calculated the posterior probability of association in each genomic region. We then identified all regions with a PPA greater than 0.9 but that had a minimum P-value less than $5\times 10^{-8}$. For each remaining region, we identified the ``lead" SNP as the SNP with the largest posterior probability of being the causal SNP in the region. If this SNP was within 500kb of a SNP with $P<5\times 10^{-8}$ (this can happen because we use non-overlapping windows and sometimes the best SNP is at the edge of the region), we removed it. We also manually removed two regions (surrounding rs8076131 in Crohn's disease and surrounding rs11535944 in HDL), where the ``new" association was in LD with a previously reported SNP over 500kb away. In Supplementary Table 22, we show the remaining SNPs; these regions are high-confidence associations that did not reach traditional genome-wide significance. Figure 1: . Proportion of SNPs in the 1000 Genomes Project either genotyped or successfully imputed. For each trait, we split all SNPs in phase 1 of the 1000 Genomes Project into bins based on their minor allele frequency in the European population. Bin sizes were of 5% frequency. Shown are the proportions of SNPs in each bin that were either genotyped or successfully imputed for each trait (the points are at the lower ends of the bins, such that the point at 45% frequency contains all SNPs from 45%-50% minor allele frequency). Labeled are the traits with the lowest and highest coverage. HB = hemogobin levels, FNBMD = femoral neck bone mineral density. Figure 2: . Power to detect a significant annotation. We simulated GWAS data under different levels of enrichment of causal SNPs in an annotation (see Supplementary Text), then evaluated the power of the method to detect the enrichment with different numbers of loci. In red and pink are log2-enrichments of 2.6, and in black and grey are log2-enrichments of 1.4. Figure 3: . Estimated role of synonymous polymorphisms in each trait. A. Estimated enrichment of synonymous SNPs. For each trait, we fit a model including an effect of synonymous SNPs and an effect of SNPs within 5kb of a TSS. Shown are the estimated enrichments parameters and 95% confidence intervals for the synonymous SNPs. B. Estimated proportion of GWAS hits driven by synonymous SNPs. For each trait, using the model fit in A., we estimated the proportion of GWAS signals driven by synonymous SNPs. Shown is this estimate and its standard error. Figure 4: . Annotation effects in the bone mineral density data. We estimated an enrichment parameter for each annotation individually in the GWAS for A. bone density in the femoral neck and B. bone density in the lumbar spine. Shown are the maximum likelihood estimates and 95% confidence intervals. Annotations are ranked according to how much each improves the fit of the model; shown are the 50 annotations that most improve the model (or if there were less than 50 significant annotations, all of the significant annotations). In red are the annotations included in the combined model, and in pink are annotations that are statistically equivalent to those in the combined model. Figure 5: . Annotation effects in the GIANT data. We estimated an enrichment parameter for each annotation individually in the GWAS for A. BMI and B. height. Shown are the maximum likelihood estimates and 95% confidence intervals. Annotations are ranked according to how much each improves the fit of the model; shown are the 50 annotations that most improve the model (or if there were less than 50 significant annotations, all of the significant annotations). In red are the annotations included in the combined model, and in pink are annotations that are statistically equivalent to those in the combined model. Figure 6: . Annotation effects in the Crohn's disease and fasting glucose data. We estimated an enrichment parameter for each annotation individually in the GWAS for A. Crohn's disease and B. fasting glucose. Shown are the maximum likelihood estimates and 95% confidence intervals. Annotations are ranked according to how much each improves the fit of the model; shown are the 50 annotations that most improve the model (or if there were less than 50 significant annotations, all of the significant annotations). In red are the annotations included in the combined model, and in pink are annotations that are statistically equivalent to those in the combined model. Figure 7: . Annotation effects in the red blood cell data. We estimated an enrichment parameter for each annotation individually in the GWAS for A. hemoglobin levels and B. mean cellular hemoglobin. Shown are the maximum likelihood estimates and 95% confidence intervals. Annotations are ranked according to how much each improves the fit of the model; shown are the 50 annotations that most improve the model (or if there were less than 50 significant annotations, all of the significant annotations). In red are the annotations included in the combined model, and in pink are annotations that are statistically equivalent to those in the combined model. Figure 8: . Annotation effects in the red blood cell data. We estimated an enrichment parameter for each annotation individually in the GWAS for A. mean corpuscular hemoglobin concentration and B. mean red cell volume. Shown are the maximum likelihood estimates and 95% confidence intervals. Annotations are ranked according to how much each improves the fit of the model; shown are the 50 annotations that most improve the model (or if there were less than 50 significant annotations, all of the significant annotations). In red are the annotations included in the combined model, and in pink are annotations that are statistically equivalent to those in the combined model. Figure 9: . Annotation effects in the red blood cell data. We estimated an enrichment parameter for each annotation individually in the GWAS for A. packed cell volume and B. mean red cell count. Shown are the maximum likelihood estimates and 95% confidence intervals. Annotations are ranked according to how much each improves the fit of the model; shown are the 50 annotations that most improve the model (or if there were less than 50 significant annotations, all of the significant annotations). In red are the annotations included in the combined model, and in pink are annotations that are statistically equivalent to those in the combined model. Figure 10: . Annotation effects in the lipids data. We estimated an enrichment parameter for each annotation individually in the GWAS for A. triglyceride levels and B. total cholesterol. Shown are the maximum likelihood estimates and 95% confidence intervals. Annotations are ranked according to how much each improves the fit of the model; shown are the 50 annotations that most improve the model (or if there were less than 50 significant annotations, all of the significant annotations). In red are the annotations included in the combined model, and in pink are annotations that are statistically equivalent to those in the combined model. Figure 11: . Annotation effects in the lipids data. We estimated an enrichment parameter for each annotation individually in the GWAS for A. HDL levels and B. LDL levels. Shown are the maximum likelihood estimates and 95% confidence intervals. Annotations are ranked according to how much each improves the fit of the model; shown are the 50 annotations that most improve the model (or if there were less than 50 significant annotations, all of the significant annotations). In red are the annotations included in the combined model, and in pink are annotations that are statistically equivalent to those in the combined model. Figure 12: . Annotation effects in the platelet data. We estimated an enrichment parameter for each annotation individually in the GWAS for A. mean platelet volume and B. platelet count. Shown are the maximum likelihood estimates and 95% confidence intervals. Annotations are ranked according to how much each improves the fit of the model; shown are the 50 annotations that most improve the model (or if there were less than 50 significant annotations, all of the significant annotations). In red are the annotations included in the combined model, and in pink are annotations that are statistically equivalent to those in the combined model. Figure 13: . Correlated patterns of enrichment across traits. We estimated an enrichment parameter for each of 450 annotations for each of the 18 traits. For each pair of traits, we then estimated the Spearman correlation coefficient between the enrichment parameters. Plotted are these correlation coefficients. Orders of rows and columns were chosen by hierarchical clustering in R [R Core Team,, 2013]. Figure 14: . Combined models for nine traits. For each trait, we built a combined model of annotations using the algorithm presented in the Methods from the main text. Shown are the maximum likelihood estimates and 95% confidence intervals for all annotations included in each model. Note that though these are the maximum likelihood estimates, model choice was done using a penalized likelihood. In parentheses next to each annotation (expect for those relating to distance to transcription start sites), we show the total number of annotations that are statistically equivalent to the included annotation in a conditional analysis. For the other nine traits, see Figure 4 in the main text. This annotation of DNase-I hypersensitive sites in fetal kidney (renal pelvis) has a positive effect when treated alone; see Supplementary Text for discussion. Figure 15: . Calibrating a PPA threshold similar to a P-value threshold. For each of the four phenotypes in the lipids data, we plot the number of ``true positives" and ``false positives" obtained by different statistical thresholds; see Supplementary Text for details. Points show the positions of the thresholds used in the paper. Figure 16: . Robustness of parameter estimates to preset parameters. A. Prior variance on effect size. We estimated an enrichment parameter for each annotation in Crohn's disease using prior variances of 0.1 or 0.5. Shown are the estimates for all annotations with 95% confidence intervals that did not overlap 0 in at least one of the two runs. In red is the $y=x$ line. B. Window size. We estimated an enrichment parameter for each annotation in Crohn's disease using window sizes of 5,000 and 10,000 SNPs. Shown are the estimates for all annotations with 95% confidence intervals that did not overlap 0 in at least one of the two runs. In red is the $y=x$ line. Phenotype \| $\lambda_{GC}$ (before imputation) \| $\lambda_{GC}$ (after imputation) ---\|---\|--- Height \| 1.04 \| 0.99 BMI \| 1.04 \| 0.97 BMD (femoral neck) \| 1.0 \| 0.92 BMD (lumbar spine) \| 1.0 \| 0.93 Crohn's \| 1.27 \| 0.71 FG \| 1.08 \| 0.97 HB \| 1.07 \| 0.99 MCH \| 1.13 \| 1.0 MCHC \| 1.07 \| 0.85 MCV \| 1.13 \| 1.0 PCV \| 1.09 \| 0.97 RBC \| 1.14 \| 1.01 TC \| 1.0 \| 0.93 TG \| 1.0 \| 0.92 HDL \| 1.0 \| 0.94 LDL \| 1.0 \| 0.93 PLT \| 1.08 \| 1.01 MPV \| 1.04 \| 0.96 Table 1: : Genomic control inflation factors before and after imputation. We show $\lambda_{GC}$ [Bacanu et al.,, 2002] before and after imputation for all 18 GWAS included in this study. Phenotype \| Proportion [95% CI] ---\|--- BMI \| 0.022 [0.013, 0.032] FNBMD \| 0.028 [0.019, 0.040] LSBMD \| 0.028 [0.019, 0.041] Crohn's \| 0.078 [0.059, 0.10] FG \| 0.020 [0.012, 0.03] HB \| 0.010 [0.006, 0.015] HDL \| 0.034 [0.026, 0.044] Height \| 0.131 [0.111, 0.153] LDL \| 0.034 [0.026, 0.045] MCH \| 0.035 [0.025, 0.047] MCHC \| 0.018 [0.011, 0.027] MCV \| 0.046 [0.034, 0.059] MPV \| 0.025 [0.017, 0.035] PCV \| 0.003 [0.002, 0.005] PLT \| 0.036 [0.028, 0.047] RBC \| 0.023 [0.016, 0.033] TC \| 0.052 [0.040, 0.067] TG \| 0.023 [0.015, 0.032] Table 2: : Estimates of the fraction of regions containing an associated SNP for each phenotype. We show the estimates of $\frac{1}{1+e^{-\kappa}}$, the proportion of regions from the middle third of the distribution of gene density that contain associated SNPs (see Equation 7 in the main text), along with the 95% confidence interval of this parameter. Annotation \| Description \| log2(Effect) [95% CI] \| Penalized effect \| Marginal effect [95% CI] ---\|---\|---\|---\|--- BE2_C-DS14625 \| DNAse-I in BE(2)-C neuroblastoma cell line \| 5.25 [3.09, 7.10] \| 5.15 \| 5.60 [3.51, 7.47] HUVEC PF \| Genome segmentation in HUVEC cells: promoter-flanking \| 7.47 [3.90,9.91] \| 7.18 \| 8.51 [5.41, 10.62] Table 3: : Combined model learned for BMI. Shown are the exact annotation names and parameters learned for BMI, along with the penalized effect sizes and the effect of each annotation in a single-annotation model. Annotation \| Description \| log2(Effect) [95% CI] \| Penalized effect \| Marginal effect [95% CI] ---\|---\|---\|---\|--- High gene density \| Regional annotation: top 1/3 of gene density \| 0.89 [0.01, 1.62] \| 0.88 \| NA Low gene density \| Regional annotation: bottom 1/3 of gene density \| -1.05 [-2.68, 0.12] \| -0.95 \| NA fHeart-DS12810 \| DNase-I in fetal heart \| 2.83 [1.11, 4.40] \| 2.45 \| 4.83 [3.08, 6.43] fHeart-DS16621 \| DNase-I in fetal heart \| 2.12 [0.50, 3.64] \| 2.21 \| 4.47 [2.76, 6.03] Table 4: : Combined model learned for bone mineral density (femur). Shown are the exact annotation names and parameters learned for FNBMD, along with the penalized effect sizes and the effect of each annotation in a single-annotation model. Annotation \| Description \| log2(Effect) [95% CI] \| Penalized effect \| Marginal effect [95% CI] ---\|---\|---\|---\|--- High gene density \| Regional annotation: top 1/3 of gene density \| 0.52 [-0.40, 1.27] \| 0.53 \| NA Low gene density \| Regional annotation: bottom 1/3 of gene density \| -1.49 [-3.65, -0.13] \| -1.33 \| NA HSMM_D-DS15542 \| DNase-I in skeletal muscle myoblasts \| 4.23 [2.24, 5.97] \| 3.75 \| 3.90 [1.98, 5.58] Table 5: : Combined model learned for bone mineral density (spine). Shown are the exact annotation names and parameters learned for LSBMD, along with the penalized effect sizes and the effect of each annotation in a single-annotation model. Annotation \| Description \| log2(Effect) Effect [95% CI] \| Penalized effect \| Marginal effect [95% CI] ---\|---\|---\|---\|--- High gene density \| Regional annotation: top 1/3 of gene density \| 1.18 [0.61, 1.72] \| 1.18 \| NA Low gene density \| Regional annotation: bottom 1/3 of gene density \| -2.18 [-4.10, -0.92] \| -2.03 \| NA fSkin_fibro_upper_back-DS19696 \| DNase-I in fetal skin fibroblasts from the upper back \| 5.21 [4.08, 6.20] \| 4.78 \| 3.84 [2.63, 4.89] gm12878.combined.R \| Genome segmentation of GM12878: repressed \| -1.83 [-3.06, -0.78] \| -1.79 \| -2.35 [-4.50, -1.05] fSkin_fibro_abdomen-DS19561 \| DNase-I in fetal skin fibroblasts from abdomen \| -2.34 [-3.85, -1.18] \| -1.86 \| 2.77 [1.27, 3.94] huvec.combined.T \| Genome segmentation of HUVEC: transcribed \| 1.20 [0.25, 2.15] \| 1.17 \| 1.63 [0.61, 2.65] Distance to TSS [0-5 kb] \| From 0-5 kb from a TSS \| 1.18 [0.17, 2.15] \| 1.17 \| NA Distance to TSS [5-10 kb] \| From 5-10 kb from a TSS \| 0.45 [-1.38, 1.75] \| 0.40 \| NA Table 6: : Combined model learned for Crohn's disease. Shown are the exact annotation names and parameters learned for Crohn's disease, along with the penalized effect sizes and the effect of each annotation in a single-annotation model. Annotation \| Description \| log2(Effect) [95% CI] \| Penalized effect \| Marginal effect [95% CI] ---\|---\|---\|---\|--- fStomach-DS17878 \| DNase-I in fetal stomach \| 3.66 [1.66, 5.31] \| 3.62 \| 3.82 [1.66, 5.50] Nonsynonymous \| nonsynonymous SNPs \| 4.28 [1.53, 6.10] \| 4.13 \| 4.95 [1.40, 6.95] Distance to TSS [0-5 kb] \| From 0-5 kb from a TSS \| 1.83 [0.22, 3.40] \| 1.75 \| NA Distance to TSS [5-10 kb] \| From 5-10 kb from a TSS \| 2.68 [0.76, 4.28] \| 2.54 \| NA Table 7: : Combined model learned for fasting glucose. Shown are the exact annotation names and parameters learned for FG, along with the penalized effect sizes and the effect of each annotation in a single-annotation model. Annotation \| Description \| log2(Effect) 95% CI] \| Penalized effect \| Marginal effect [95% CI] ---\|---\|---\|---\|--- High gene density \| Regional annotation: top 1/3 of gene density \| 2.78 [1.98, 3.46] \| 2.80 \| NA Low gene density \| Regional annotation: bottom 1/3 of gene density \| -40.6 [$-\inf$, -0.72] \| -5.44 \| NA HMVEC_dAd-DS12957 \| DNase-I in microvascular endothelium \| 4.91 [3.09, 6.52] \| 4.86 \| 4.43 [2.60, 6.02] k562.combined.T \| Genome segmentation of K562: transcribed \| 2.15 [0.49, 3.77] \| 2.12 \| 1.82 [0.01, 3.55] Table 8: : Combined model learned for hemoglobin levels. Shown are the exact annotation names and parameters learned for hemoglobin levels, along with the penalized effect sizes and the effect of each annotation in a single-annotation model. Annotation \| Description \| log2(Effect) [95% CI] \| Penalized effect \| Marginal effect [95% CI] ---\|---\|---\|---\|--- High gene density \| Regional annotation: top 1/3 of gene density \| 1.69 [1.13, 2.19] \| 1.56 \| NA Low gene density \| Regional annotation: bottom 1/3 of gene density \| -1.17 [-0.13, 0.69] \| -0.20 \| NA hepg2.combined.R \| Genome segmentation of HepG2: repressed \| -1.83 [-3.12, -0.68] \| -1.79 \| -3.35 [-4.63, -2.19] hepg2.combined.TSS \| Genome segmentation of HepG2: TSS \| 3.10 [1.79, 4.20] \| 2.84 \| 5.09 [3.91, 6.16] ens_coding_exons \| Ensembl: coding exons \| 3.16 [1.51, 4.40] \| 2.73 \| 4.31 [2.73, 5.55] k562.combined.R \| Genome segmentation of K562: repressed \| -1.43 [-2.65, -0.30] \| -1.43 \| -2.90 [-4.08, -1.79] Table 9: : Combined model learned for HDL levels. Shown are the exact annotation names and parameters learned for HDL, along with the penalized effect sizes and the effect of each annotation in a single-annotation model. Annotation \| Description \| log2(Effect) [95% CI] \| Penalized effect \| Marginal effect [95% CI] ---\|---\|---\|---\|--- High gene density \| Regional annotation: top 1/3 of gene density \| 1.50 [1.13, 1.86] \| 1.49 \| NA Low gene density \| Regional annotation: bottom 1/3 of gene density \| -0.95 [-1.62, -0.36] \| -0.94 \| NA helas3.combined.R \| Genome segmentation of HeLa: repressed \| -1.50 [-2.39, -0.71] \| -1.50 \| -2.74 [-3.78, -1.85] fMuscle_lower_limb-DS18174 \| DNase-I in fetal muscle from lower limb \| 2.27 [1.50, 3.02] \| 2.24 \| 3.61 [2.81, 4.40] Nonsynonymous \| Nonsynonymous SNPs \| 3.74 [2.55, 4.65] \| 3.58 \| 4.27 [2.77, 5.32] fLung-DS15573 \| DNase-I in fetal lung \| 2.09 [1.30, 2.80] \| 2.05 \| 3.77 [2.97, 4.50] huvec.combined.T \| Genome segmentation of HUVEC: transcribed \| 1.27 [0.52, 1.96] \| 1.24 \| 1.63 [0.89, 2.34] ens_utr3_exons \| Ensembl: 3' UTRs \| 1.57 [0.00, 2.64] \| 1.54 \| 2.93 [1.34, 3.98] Table 10: : Combined model learned for height. Shown are the exact annotation names and parameters learned for height, along with the penalized effect sizes and the effect of each annotation in a single-annotation model. Annotation \| Description \| log2(Effect) [95% CI] \| Penalized effect \| Marginal effect [95% CI] ---\|---\|---\|---\|--- High gene density \| Regional annotation: top 1/3 of gene density \| 1.77 [1.21, 2.27] \| 1.72 \| NA Low gene density \| Regional annotation: bottom 1/3 of gene density \| -0.72 [-1.98, 0.25] \| -0.71 \| NA hepg2.combined.R \| Genome segmentation of HepG2: repressed \| -2.78 [-4.36, -1.51] \| -2.70 \| -3.04 [-4.70, -1.76] Nonsynonymous \| Nonsynonymous SNPs \| 4.24 [2.74, 5.40] \| 3.97 \| 4.89 [3.48, 6.02] Distance to TSS [0-5 kb] \| From 0-5 kb from a TSS \| 3.13 [1.96, 4.56] \| 2.84 \| NA Distance to TSS [5-10 kb] \| From 5-10 kb from a TSS \| 1.63 [-0.65, 3.12] \| 1.17 \| NA Table 11: : Combined model learned for LDL levels. Shown are the exact annotation names and parameters learned for LDL, along with the penalized effect sizes and the effect of each annotation in a single-annotation model. Annotation \| Description \| log2(Effect) [95% CI] \| Penalized effect \| Marginal effect [95% CI] ---\|---\|---\|---\|--- High gene density \| Regional annotation: top 1/3 of gene density \| 1.56 [0.94, 2.11] \| 1.51 \| NA Low gene density \| Regional annotation: bottom 1/3 of gene density \| -1.17 [-2.80, -0.01] \| -1.10 \| NA k562.combined.E \| Genome segmentation of K562: enhancers \| 3.68 [2.47, 4.75] \| 3.53 \| 5.67 [4.49, 6.74] k562.combined.R \| Genome segmentation of K562: repressed \| -3.17 [-4.80, -1.86] \| -2.97 \| -3.94 [-5.57, -2.61] hTH17-DS11039 \| DNase-I in Th17 T cells \| 2.21 [0.35, 3.51] \| 2.06 \| 4.53 [2.93, 5.74] Table 12: : Combined model learned for mean cell hemoglobin. Shown are the exact annotation names and parameters learned for MCH, along with the penalized effect sizes and the effect of each annotation in a single-annotation model. Annotation \| Description \| log2(Effect) [95% CI] \| Penalized effect \| Marginal effect [95% CI] ---\|---\|---\|---\|--- High gene density \| Regional annotation: top 1/3 of gene density \| 1.11 [0.09, 1.93] \| 1.17 \| NA Low gene density \| Regional annotation: bottom 1/3 of gene density \| -1.57 [-4.41, 0.10] \| -1.36 \| NA k562.combined.R \| Genome segmentation of K562: repressed \| -3.81 [-7.43, -1.79] \| -3.42 \| -4.34 [-8.94, -2.27] K562-DS9767 \| DNase-I in K562 cells \| 2.67 [0.61, 4.44] \| 2.47 \| 4.46 [2.60, 6.22] Nonsynonymous \| Nonsynonymous SNPs \| 4.66 [1.90, 6.52] \| 4.03 \| 4.27 [0.97, 6.25] Table 13: : Combined model learned for mean corpuscular hemoglobin concentration. Shown are the exact annotation names and parameters learned for MCHC, along with the penalized effect sizes and the effect of each annotation in a single-annotation model. Annotation \| Description \| log2(Effect) [95% CI] \| Penalized effect \| Marginal effect [95% CI] ---\|---\|---\|---\|--- High gene density \| Regional annotation: top 1/3 of gene density \| 1.36 [0.76, 1.86] \| 1.31 \| NA Low gene density \| Regional annotation: bottom 1/3 of gene density \| -1.51 [-3.06, -0.39] \| -1.46 \| NA k562.combined.R \| Genome segmentation of K562: repressed \| -3.91 [-6.25, -2.38] \| -3.69 \| -5.24 [-7.76, -3.59] k562.combined.E \| Genome segmentation of K562: enhancer \| 3.10 [1.86, 4.15] \| 2.96 \| 5.67 [4.47, 6.77] hTH17-DS11039 \| DNase-I in Th17 T cells \| 2.31 [0.81, 3.48] \| 2.25 \| 5.40 [4.21, 6.46] Nonsynonymous \| Nonsynonymous SNPs \| 4.54 [2.34, 5.92] \| 4.13 \| 5.11 [3.26, 6.39] CMK-DS12393 \| DNase-I in CMK leukemia line \| 1.28 [0.04, 2.35] \| 1.34 \| 4.52 [3.30, 5.64] Distance to TSS [0-5 kb] \| From 0-5 kb from a TSS \| 0.38 [-1.59, 0.65] \| -0.33 \| NA Distance to TSS [5-10 kb] \| From 5-10 kb from a TSS \| 0.89 [-0.40, 1.83] \| 0.84 \| NA Table 14: : Combined model learned for mean red cell volume. Shown are the exact annotation names and parameters learned for MCV, along with the penalized effect sizes and the effect of each annotation in a single-annotation model. Annotation \| Description \| log2(Effect) [95% CI] \| Penalized effect \| Marginal effect [95% CI] ---\|---\|---\|---\|--- High gene density \| Regional annotation: top 1/3 of gene density \| 1.95 [1.30, 2.52] \| 1.88 \| NA Low gene density \| Regional annotation: bottom 1/3 of gene density \| -2.06 [-4.73, -0.40] \| -1.63 \| NA CD34-DS12274 \| DNase-I in CD34+ cells \| 3.02 [1.69, 4.26] \| 2.76 \| 4.37 [2.99, 5.64] gm12878.combined.T \| Genome segmentation of GM12878: transcribed \| 2.35 [1.07, 3.53] \| 1.83 \| 1.86 [0.59, 3.04] helas3.combined.E \| Genome segmentation of HeLa: enhancer \| 2.80 [0.75, 4.23] \| 2.27 \| 3.35 [0.16, 5.09] fSpleen-DS17448 \| DNase-I in fetal spleen \| 1.93 [0.59, 3.15] \| 1.88 \| 3.65 [2.22, 4.92] Table 15: : Combined model learned for mean platelet volume. Shown are the exact annotation names and parameters learned for MPV, along with the penalized effect sizes and the effect of each annotation in a single-annotation model. Annotation \| Description \| log2(Effect) [95% CI] \| Penalized effect \| Marginal effect [95% CI] ---\|---\|---\|---\|--- High gene density \| Regional annotation: top 1/3 of gene density \| 4.24 [3.36, 4.95] \| 3.72 \| NA Low gene density \| Regional annotation: bottom 1/3 of gene density \| -40.60 [-$\inf$, 0.94] \| -1.92 \| NA Nonsynonymous \| Nonsynonymous SNPs \| 4.11 [1.34, 6.07] \| 3.61 \| 5.34 [2.83, 7.23] fStomach-DS17172 \| DNase-I in fetal stomach \| 3.90 [1.40, 6.17] \| 3.48 \| 4.78 [2.54, 7.03] Table 16: : Combined model learned for packed red cell volume. Shown are the exact annotation names and parameters learned for PCV, along with the penalized effect sizes and the effect of each annotation in a single-annotation model. Annotation \| Description \| log2(Effect) [95% CI] \| Penalized effect \| Marginal effect [95% CI] ---\|---\|---\|---\|--- High gene density \| Regional annotation: top 1/3 of gene density \| 1.67 [2.81, 2.64] \| 2.14 \| NA Low gene density \| Regional annotation: bottom 1/3 of gene density \| -1.63 [-3.40, -0.38] \| -1.51 \| NA k562.combined.R \| Genome segmentation in K562: repressed \| -1.60 [-2.63, -0.66] \| -1.60 \| -2.60 [-3.65, -1.64] CD34-DS12274 \| DNase-I in CD34+ cells \| 1.82 [0.59, 2.86] \| 1.80 \| 3.39 [2.24, 4.43] Nonsynonymous \| Nonsynonymous SNPs \| 3.38 [1.31, 4.79] \| 3.00 \| 3.98 [2.02, 5.38] huvec.combined.E \| Genome segmentation in HUVEC: enhancers \| 1.67 [0.16, 2.84] \| 1.59 \| 3.27 [1.82, 4.41] helas3.combined.R \| Genome segmentation in HeLa: repressed \| -1.17 [-2.37, -0.13] \| -1.14 \| -2.18 [-3.40, -1.11] Table 17: : Combined model learned for platelet count. Shown are the exact annotation names and parameters learned for PLT, along with the penalized effect sizes and the effect of each annotation in a single-annotation model. Annotation \| Description \| log2(Effect) [95% CI] \| Penalized effect \| Marginal effect [95% CI] ---\|---\|---\|---\|--- High gene density \| Regional annotation: top 1/3 of gene density \| 1.99 [1.33, 2.58] \| 1.96 \| NA Low gene density \| Regional annotation: bottom 1/3 of gene density \| -2.74 [-6.97, -0.59] \| -2.18 \| NA fStomach-DS17878 \| DNAse-I in fetal stomach \| 5.31 [3.87, 6.91] \| 4.83 \| 4.83 [3.30, 6.45] k562.combined.E \| Genome segmentation of K562: enhancer \| 1.53 [-0.04, 2.83] \| 1.56 \| 4.28 [1.41, 5.90] fKidney_renal_pelvis_R-DS18663 \| DNase-I in fetal renal pelvis \| -3.49 [-7.68, -1.56] \| -2.80 \| 2.48 [0.04, 4.17] K562-DS9767 \| DNase-I in K562 leukemia line \| 2.28 [0.97, 3.58] \| 2.25 \| 4.50 [2.97, 5.97] Table 18: : Combined model learned for red blood cell count. Shown are the exact annotation names and parameters learned for RBC, along with the penalized effect sizes and the effect of each annotation in a single-annotation model. Annotation \| Description \| log2(Effect) [95% CI] \| Penalized effect \| Marginal effect [95% CI] ---\|---\|---\|---\|--- High gene density \| Regional annotation: top 1/3 of gene density \| 1.05 [0.48, 1.56] \| 1.04 \| NA Low gene density \| Regional annotation: bottom 1/3 of gene density \| -1.40 [-2.74, -0.39] \| -1.34 \| NA hepg2.combined.R \| Genome segmentation of HepG2: repressed \| -2.90 [-4.36, -1.72] \| -2.84 \| -3.19 [-4.76, -1.95] Nonsynonymous \| Nonsynonymous SNPs \| 4.36 [2.90, 5.48] \| 4.18 \| 4.89 [3.51, 5.99] Distance to TSS [0-5 kb] \| From 0-5 kb from a TSS \| 2.76 [1.62, 4.15] \| 2.58 \| NA Distance to TSS [5-10 kb] \| From 5-10 kb from a TSS \| 1.88 [-0.27, 3.29] \| 1.56 \| NA Table 19: : Combined model learned for total cholesterol. Shown are the exact annotation names and parameters learned for total cholesterol, along with the penalized effect sizes and the effect of each annotation in a single-annotation model. Annotation \| Description \| log2(Effect) [95% CI] \| Penalized effect \| Marginal effect [95% CI] ---\|---\|---\|---\|--- High gene density \| Regional annotation: top 1/3 of gene density \| 1.56 [0.85, 2.18] \| 1.49 \| NA Low gene density \| Regional annotation: bottom 1/3 of gene density \| -0.82 [-2.65, 0.40] \| -0.78 \| NA hepg2.combined.R \| Genome segementation of HepG2: repressed \| -4.24 [-6.68, -2.47] \| -3.75 \| -4.56 [-7.11, -2.76] ens_utr3_exons \| Ensembl: 3' UTRs \| 3.87 [2.11, 5.28] \| 3.46 \| 4.60 [2.86, 6.03] Table 20: : Combined model learned for triglyceride levels. Shown are the exact annotation names and parameters learned for triglycerides, along with the penalized effect sizes and the effect of each annotation in a single- annotation model. \| PPA \| P-value \| combined ---\|---\|---\|--- Phenotype \| True positives \| False positives \| True positives \| False positives \| True positives \| False positives HDL \| 45 \| 0 \| 43 \| 1 \| 48 \| 1 LDL \| 43 \| 1 \| 40 \| 0 \| 44 \| 1 TC \| 47 \| 0 \| 51 \| 0 \| 51 \| 0 TG \| 27 \| 0 \| 29 \| 0 \| 30 \| 0 Table 21: : Comparison of loci identified in the lipids data with different methods. We ranked genomic regions in GWAS of four lipid traits according to their minimum P-value or posterior probability of association from Teslovich et al., [2010]. We then evaluated false positives and false negatives by comparison to a larger GWAS [Global Lipids Genetics Consortium et al.,, 2013]. See Supplementary Text for details. trait \| region (hg19) \| Regional PPA \| lead SNP (P-value) \| Nearest gene \| Successful replication (SNP, $r^{2}$ with lead) ---\|---\|---\|---\|---\|--- BMI \| chr13:27,75,5426-29,745,954 \| 0.94 \| rs9512699 ($6\times 10^{-8}$) \| MTIF3 \| [Speliotes et al.,, 2010] (rs4771122, 0.73) BMD (femur) \| chr1:170,892,281-173,086,517 \| 0.93 \| rs6701929 ($2\times 10^{-7}$) \| DNM3 \| [Estrada et al.,, 2012] (rs479336, 0.93) HDL \| chr1:25,427,217-29,426,896 \| 0.96 \| rs6659176 ($1.5\times 10^{-6}$) \| NR0B2 \| [Global Lipids Genetics Consortium et al.,, 2013] (rs12748152, 0.85) HDL \| chr1:93,534,311-95,828,501 \| 0.93 \| rs2297707 ($1\times 10^{-6}$) \| TMED5 \| [Global Lipids Genetics Consortium et al.,, 2013] (rs12133576, 0.79) HDL \| chr1:108,743,042-111,481,349 \| 0.97 \| rs12740374 ($6\times 10^{-8}$) \| CELSR2 \| [Global Lipids Genetics Consortium et al.,, 2013] (rs12740374) HDL \| chr2:85,349,339-88,736,950 \| 0.98 \| rs1044973 ($1.5\times 10^{-7}$) \| TGOLN2 \| No (sample size not increased in Global Lipids Genetics Consortium et al., [2013]) HDL \| chr10:45,535,916-50,321,467 \| 0.93 \| rs10900223 ($1.4\times 10^{-7}$) \| MARCH8 \| [Global Lipids Genetics Consortium et al.,, 2013] (rs970548, 0.99) MCV \| chr3:139,060,509-141,377,851 \| 0.98 \| rs13059128 ($3.8\times 10^{-7}$) \| ZBTB38 \| [van der Harst et al.,, 2012] (rs6776003, 0.48) MCV \| chr9:134,164,493-136,620,584 \| 0.90 \| rs8176662 ($7.5\times 10^{-7}$) \| ABO \| NA MCV \| chr20:24,615,239-30,836,608 \| 0.98 \| rs6088962 ($7.5\times 10^{-7}$) \| BCL2L1 \| NA TG \| chr16:31,050,033-49,644,030 \| 0.95 \| rs1549293 ($2.7\times 10^{-7}$) \| KAT8 \| [Global Lipids Genetics Consortium et al.,, 2013] (rs749671, 0.80) LDL \| chr1:91,146,258-93,672,688 \| 0.97 \| rs7542747 ($2.3\times 10^{-7}$) \| RPAP2 \| [Global Lipids Genetics Consortium et al.,, 2013] (rs4970712, 0.75) LDL \| chr1:146,751,272-152,014,485 \| 0.98 \| rs267733 ($7\times 10^{-8}$) \| ANXA9 \| [Global Lipids Genetics Consortium et al.,, 2013] (rs267733) LDL \| chr2:116,901,934-119,001,466 \| 0.98 \| rs1052639 ($6.6\times 10^{-8}$) \| DDX18 \| [Global Lipids Genetics Consortium et al.,, 2013] (rs10490626, 0.53) LDL \| chr13:31,693,235-34,119,073 \| 0.93 \| rs4942505 ($9.8\times 10^{-8}$) \| BRCA2 \| [Global Lipids Genetics Consortium et al.,, 2013] (rs4942505) LDL \| chr17:7,456,344-9,908,665 \| 0.92 \| rs4791641 ($2.6\times 10^{-7}$) \| PFAS \| No ($P=1.3\times 10^{-7}$ in [Global Lipids Genetics Consortium et al.,, 2013]) MCHC \| chr7:76,062,644-78,334,941 \| 0.93 \| rs58176556 ($5.4\times 10^{-8}$) \| PHTF2 \| NA Height \| chr2:240,701,166-243,060,642 \| 0.98 \| rs13006939 ($3.9\times 10^{-7}$) \| SEPT2 \| [Lango-Allen et al.,, 2010] (rs12694997, 0.99) Height \| chr3:11,167,568-13,294,698 \| 0.98 \| rs2276749 ($3.0\times 10^{-6}$) \| VGLL2 \| NA Height \| chr3:13,294,698-15,353,840 \| 0.93 \| rs2597513 ($1.1\times 10^{-7}$) \| HDAC11 \| [Lango-Allen et al.,, 2010] (rs2597513) Height \| chr3:55,068,506-57,000,141 \| 0.94 \| rs7637449 ($1.3\times 10^{-6}$) \| CCDC66 \| [Lango-Allen et al.,, 2010] (rs9835332, 0.87) Height \| chr4:72,048-2,570,837 \| 0.98 \| rs3958122 ($6.0\times 10^{-8}$) \| SLBP \| [Lango-Allen et al.,, 2010] (rs2247341, 0.99) Height \| chr5:71,376,237-73,712,303 \| 0.98 \| rs34651 ($2.5\times 10^{-7}$) \| TNPO1 \| NA Height \| chr6:108,017,102-110,694,347 \| 0.95 \| rs1476387 ($2.2\times 10^{-6}$) \| SMPD2 \| [Lango-Allen et al.,, 2010] (rs1046943, 0.93) Height \| chr7:22,074,248-23,998,552 \| 0.99 \| rs12534093 ($5.6\times 10^{-8}$) \| IGF2BP3 \| [Lango-Allen et al.,, 2010] (rs12534093) Height \| chr7:46,327,426-48,083,339 \| 0.97 \| rs12538905 ($2.6\times 10^{-7}$) \| IGFBP3 \| NA Height \| chr9:87,279,007-89,667,667 \| 0.90 \| rs405761 ($1.3\times 10^{-7}$) \| ZCCHC6 \| [Lango-Allen et al.,, 2010] (rs8181166, 0.82) Height \| chr11:12,559,691-14,685,886 \| 1.0 \| rs7926971 ($7.3\times 10^{-8}$) \| TEAD1 \| [Lango-Allen et al.,, 2010] (rs7926971) Height \| chr11:14,685,886-17,491,336 \| 0.93 \| rs757081 ($2.2\times 10^{-6}$) \| NUCB2 \| [Lango-Allen et al.,, 2010] (rs1330, 0.60) Height \| chr15:62,349,517-64,370,301 \| 0.97 \| rs7178424 ($2.2\times 10^{-7}$) \| C2CD4A \| [Lango-Allen et al.,, 2010] (rs7178424) Height \| chr17:19,924,256-26,838,292 \| 0.96 \| rs9895199 ($3.6\times 10^{-7}$) \| KCNJ12 \| [Lango-Allen et al.,, 2010] (rs4640244, 0.79) Height \| chr17:45,331,502-47,944,460 \| 0.99 \| rs9904645 ($2.2\times 10^{-7}$) \| ATP5G1 \| NA Height \| chr22:32,075,899-33,846,972 \| 0.97 \| rs1012366 ($6.9\times 10^{-8}$) \| SYN3 \| [Lango-Allen et al.,, 2010] (rs4821083 [not in 1000 Genomes]) Crohn's \| chr2:42,522,756-44,575,426 \| 1.0 \| rs17031095 ($2.6\times 10^{-7}$) \| THADA \| [Jostins et al.,, 2012] (rs10495903, 0.95) Crohn's \| chr10:59,615,595-61,881,674 \| 1.0 \| rs1832556 ($2.0\times 10^{-7}$) \| IPMK \| [Jostins et al.,, 2012] (rs2790216, 0.94) Crohn's \| chr11:61,269,649-64,734,682 \| 0.98 \| rs174568 ($2.8\times 10^{-7}$) \| FADS2 \| [Jostins et al.,, 2012] (rs4246215, 0.86) Crohn's \| chr13:99,900,420-102,096,823 \| 0.94 \| rs3742130 ($2.3\times 10^{-5}$) \| GPR18 \| [Jostins et al.,, 2012] (rs9557195, 0.91) Crohn's \| chr15:67,140,517-70,199,927 \| 0.93 \| rs11639295 ($6.4\times 10^{-7}$) \| SMAD3 \| [Jostins et al.,, 2012] (rs17293632, 0.10) Crohn's \| chr17:17,986,955-26,038,545 \| 0.92 \| rs2945406 ($4.1\times 10^{-7}$) \| KSR1 \| [Jostins et al.,, 2012] (rs2945412, 0.13) PLT \| chr1:44,022,121-47,087,366 \| 0.99 \| rs4468203 ($3.2\times 10^{-7}$) \| GPBP1L1 \| NA PLT \| chr9:90,221,450-92,241,847 \| 0.90 \| rs9410382 ($1.9\times 10^{-6}$) \| S1PR3 \| NA PLT \| chr11:32,343,164-34,501,064 \| 0.93 \| rs7481878 ($7.2\times 10^{-8}$) \| QSER1 \| NA MCH \| chr4:86,147,717-88,340,969 \| 0.98 \| rs6819155 ($2.3\times 10^{-7}$) \| APP1 \| NA MCH \| chr14:102,971,016-107,289,436 \| 0.93 \| rs17616316 ($1.5\times 10^{-7}$ \| EIF5 \| [van der Harst et al.,, 2012] (rs17616316) HB \| chr15:75,349,145-78,654,148 \| 0.90 \| rs1874953 ($4.2\times 10^{-7}$) \| NRG4 \| [van der Harst et al.,, 2012] (rs11072566, 0.93) BMD (spine) \| chr17:43,556,652-46,084,026 \| 0.99 \| rs117504376 ($3.1\times 10^{-7}$) \| MAPT (chr17 inversion) \| [Estrada et al.,, 2012] (rs1864325, 0.99) RBC \| chr20:54,899,828-57,013,873 \| 0.96 \| rs737092 ($4.5\times 10^{-7}$) \| MIR5095 \| [van der Harst et al.,, 2012] (rs737092) MPV \| chr14:67,315,438-69,802,709 \| 0.91 \| rs117823369 ($3.9\times 10^{-6}$ \| DCAF5 \| NA FG \| chr9:111,051,626 - 112,662,634 \| 0.96 \| rs76817627 ($3.4\times 10^{-7}$) \| FAM206A \| NA Table 22: : Sub-threshold associations with high posterior probability. In each GWAS, we identified regions of the genome with a posterior probability of association greater than 0.9 but with no P-values less than $5\times 10^{-8}$. Shown are the positions of these regions for each trait. See Supplementary Text for details. LD between lead SNPs and replication SNPs was computed from the 1000 Genomes Project haplotypes in Europeans; the exact file versions are listed in Section 3. ## References Bacanu et al., [2002] Bacanu, S.-A., Devlin, B., and Roeder, K., 2002. Association studies for quantitative traits in structured populations. Genetic epidemiology, 22(1):78–93. * Estrada et al., [2012] Estrada, K., Styrkarsdottir, U., Evangelou, E., Hsu, Y.-H., Duncan, E. L., Ntzani, E. E., Oei, L., Albagha, O. M., Amin, N., Kemp, J. P., _et al._ , 2012\. Genome-wide meta-analysis identifies 56 bone mineral density loci and reveals 14 loci associated with risk of fracture. Nature genetics, 44(5):491–501. * Gieger et al., [2011] Gieger, C., Radhakrishnan, A., Cvejic, A., Tang, W., Porcu, E., Pistis, G., Serbanovic-Canic, J., Elling, U., Goodall, A. H., Labrune, Y., _et al._ , 2011\. New gene functions in megakaryopoiesis and platelet formation. Nature, 480(7376):201–208. * Global Lipids Genetics Consortium et al., [2013] Global Lipids Genetics Consortium, Willer, C. J., Schmidt, E. M., Sengupta, S., Peloso, G. M., Gustafsson, S., Kanoni, S., Ganna, A., Chen, J., Buchkovich, M. L., _et al._ , 2013. Discovery and refinement of loci associated with lipid levels. Nat Genet, 45(11):1274–83. * Hoffman et al., [2013] Hoffman, M. M., Ernst, J., Wilder, S. P., Kundaje, A., Harris, R. S., Libbrecht, M., Giardine, B., Ellenbogen, P. M., Bilmes, J. A., Birney, E., _et al._ , 2013. Integrative annotation of chromatin elements from ENCODE data. Nucleic acids research, 41(2):827–841. * Jostins et al., [2012] Jostins, L., Ripke, S., Weersma, R. K., Duerr, R. H., McGovern, D. P., Hui, K. Y., Lee, J. C., Schumm, L. P., Sharma, Y., Anderson, C. A., _et al._ , 2012\. Host-microbe interactions have shaped the genetic architecture of inflammatory bowel disease. Nature, 491(7422):119–124. * Lango-Allen et al., [2010] Lango-Allen, H., Estrada, K., Lettre, G., Berndt, S. I., Weedon, M. N., Rivadeneira, F., Willer, C. J., Jackson, A. U., Vedantam, S., Raychaudhuri, S., _et al._ , 2010. Hundreds of variants clustered in genomic loci and biological pathways affect human height. Nature, 467(7317):832–838. * Manning et al., [2012] Manning, A. K., Hivert, M.-F., Scott, R. A., Grimsby, J. L., Bouatia-Naji, N., Chen, H., Rybin, D., Liu, C.-T., Bielak, L. F., Prokopenko, I., _et al._ , 2012. A genome-wide approach accounting for body mass index identifies genetic variants influencing fasting glycemic traits and insulin resistance. Nature genetics, 44(6):659–669. * Maurano et al., [2012] Maurano, M. T., Humbert, R., Rynes, E., Thurman, R. E., Haugen, E., Wang, H., Reynolds, A. P., Sandstrom, R., Qu, H., Brody, J., _et al._ , 2012. Systematic localization of common disease-associated variation in regulatory DNA. Science, 337(6099):1190–5. * Pasaniuc et al., [2013] Pasaniuc, B., Zaitlen, N., Shi, H., Bhatia, G., Gusev, A., Pickrell, J., Hirschhorn, J., Strachan, D. P., Patterson, N., and Price, A. L., _et al._ , 2013. Fast and accurate imputation of summary statistics enhances evidence of functional enrichment. arXiv preprint arXiv:1309.3258, . * R Core Team, [2013] R Core Team, 2013. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. * Speliotes et al., [2010] Speliotes, E. K., Willer, C. J., Berndt, S. I., Monda, K. L., Thorleifsson, G., Jackson, A. U., Allen, H. L., Lindgren, C. M., Luan, J., Mägi, R., _et al._ , 2010. Association analyses of 249,796 individuals reveal 18 new loci associated with body mass index. Nature genetics, 42(11):937–948. * Teslovich et al., [2010] Teslovich, T. M., Musunuru, K., Smith, A. V., Edmondson, A. C., Stylianou, I. M., Koseki, M., Pirruccello, J. P., Ripatti, S., Chasman, D. I., Willer, C. J., _et al._ , 2010. Biological, clinical and population relevance of 95 loci for blood lipids. Nature, 466(7307):707–713. * Thurman et al., [2012] Thurman, R. E., Rynes, E., Humbert, R., Vierstra, J., Maurano, M. T., Haugen, E., Sheffield, N. C., Stergachis, A. B., Wang, H., Vernot, B., _et al._ , 2012\. The accessible chromatin landscape of the human genome. Nature, 489(7414):75–82. * van der Harst et al., [2012] van der Harst, P., Zhang, W., Leach, I. M., Rendon, A., Verweij, N., Sehmi, J., Paul, D. S., Elling, U., Allayee, H., Li, X., _et al._ , 2012. Seventy-five genetic loci influencing the human red blood cell. Nature, 492(7429):369–375. * Voight et al., [2012] Voight, B. F., Kang, H. M., Ding, J., Palmer, C. D., Sidore, C., Chines, P. S., Burtt, N. P., Fuchsberger, C., Li, Y., Erdmann, J., _et al._ , 2012. The metabochip, a custom genotyping array for genetic studies of metabolic, cardiovascular, and anthropometric traits. PLoS Genet, 8(8):e1002793.	arxiv-papers	2013-11-19T19:12:08	2024-09-04T02:49:53.905338	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Joseph K. Pickrell", "submitter": "Joseph Pickrell", "url": "https://arxiv.org/abs/1311.4843" }
1311.4889	Lattice Strong Dynamics (LSD) Collaboration # Two-Color Theory with Novel Infrared Behavior T. Appelquist Department of Physics, Sloane Laboratory, Yale University, New Haven, Connecticut 06520, USA R. C. Brower Department of Physics, Boston University, Boston, Massachusetts 02215, USA M. I. Buchoff Institute for Nuclear Theory, Box 351550, Seattle, WA 98195-1550, USA M. Cheng Center for Computational Science, Boston University, Boston, Massachusetts 02215, USA G. T. Fleming Department of Physics, Sloane Laboratory, Yale University, New Haven, Connecticut 06520, USA J. Kiskis Department of Physics, University of California, Davis, California 95616, USA M. F. Lin Computational Science Center, Brookhaven National Laboratory, Upton, NY 11973, USA E. T. Neil Department of Physics, University of Colorado, Boulder, CO 80309, USA RIKEN- BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA J. C. Osborn Argonne Leadership Computing Facility, Argonne, Illinois 60439, USA C. Rebbi Department of Physics, Boston University, Boston, Massachusetts 02215, USA D. Schaich Department of Physics, Syracuse University, Syracuse, New York 13244, USA C. Schroeder Lawrence Livermore National Laboratory, Livermore, California 94550, USA S. Syritsyn RIKEN-BNL Research Center, Brookhaven National Laboratory, Upton, NY 11973, USA G. Voronov Department of Physics, Sloane Laboratory, Yale University, New Haven, Connecticut 06520, USA P. Vranas Lawrence Livermore National Laboratory, Livermore, California 94550, USA O. Witzel Center for Computational Science, Boston University, Boston, Massachusetts 02215, USA ###### Abstract Using lattice simulations, we study the infrared behavior of a particularly interesting $\mathrm{SU}(2)$ gauge theory, with six massless Dirac fermions in the fundamental representation. We compute the running gauge coupling derived non-perturbatively from the Schrödinger functional of the theory, finding no evidence for an infrared fixed point up through gauge couplings $\bar{g}^{2}$ of order $20$. This implies that the theory either is governed in the infrared by a fixed point of considerable strength, unseen so far in non-supersymmetric gauge theories, or breaks its global chiral symmetries producing a large number of composite Nambu-Goldstone bosons relative to the number of underlying degrees of freedom. Thus either of these phases exhibits novel behavior. ###### pacs: 11.10.Hi, 11.15.Ha, 11.25.Hf, 12.60.Nz, 11.30.Qc ## Introduction A new sector, described by a strongly interacting gauge theory, could play a key role in physics beyond the Standard Model. With the recent discovery of a 125 GeV Higgs-like scalar Aad et al. (2012); Chatrchyan et al. (2012), SU(2) vector-like gauge theories provide attractive candidates. Due to the pseudo reality of the fundamental representation of SU(2), two-color theories with $N_{f}$ massless Dirac fermions in this representation have an enhanced chiral symmetry, a novel symmetry breaking pattern SU($2N_{f}$) $\rightarrow$ Sp($2N_{f}$), and, therefore, a relatively large number of Nambu-Goldstone bosons (NGB) Peskin (1980); Preskill (1981). This feature has motivated SU(2)-based models of a composite Higgs boson Galloway et al. (2010); Katz et al. (2005) and of dark matter Lewis et al. (2012); Hietanen et al. (2013); Buckley and Neil (2013). These models take $N_{f}=2$, but new intriguing possibilities emerge for larger $N_{f}$. With $N_{f}$ just below the value at which asymptotic freedom is lost, a conformal window opens up, with the theory initially governed by a weakly-coupled infrared fixed point (IRFP). As $N_{f}$ is decreased, the strength of the fixed point increases. Below some critical value $N_{f}^{c}$, chiral symmetry is broken and the theory confines. This critical value defines the lower edge of the conformal window Caswell (1974); Banks and Zaks (1982). Knowing the extent of the window and the behavior of theories in it and near it could be crucial for building a successful model of BSM physics. The extent of the conformal window is also interesting from a more theoretical point of view, and this is particularly true of the two-color theory. For example, a general notion about quantum field theories, as first applied to second-order phase transitions and critical phenomena, is that the renormalization group (RG) flow toward the infrared (IR) should result in a thinning of the degrees of freedom. This can provide an important constraint on IR behavior if it can be shown that the IR count cannot exceed the UV count. One implementation of this idea, much studied recently Cardy (1988); Komargodski and Schwimmer (2011), defines the degree-of-freedom count through the coefficient $a$ entering the trace of the energy momentum tensor on an appropriate space-time manifold. Although a UV-IR inequality can perhaps be proven, it does not seem to lead to useful constraints. Another approach Appelquist et al. (1999) defines the degree-of-freedom count via the thermodynamic free energy $F\left(T\right)$, using the temperature $T$ as the RG scale. The dimensionless quantity $f\left(T\right)\equiv 90F\left(T\right)/\pi^{2}T^{4}$ is $T$-independent for a free massless theory, leading to $f=2N_{V}+(7/2)N_{F}+N_{S}$, where $N_{V}$, $N_{F}$, and $N_{S}$ count the gauge, Dirac-fermion, and real-scalar fields. The conjectured inequality of Ref. Appelquist et al. (1999) is that for an asymptotically free theory, $f_{IR}\equiv f(0)\leq f_{UV}\equiv f(\infty)$. In the case of an IR phase with broken chiral symmetry and confinement, $f_{IR}$ counts the number of NGBs. For a vector-like SU($N$) gauge theory with $N\geq 3$ and $N_{f}$ Dirac fermions, this count is $N_{f}^{2}-1$. Also, in the UV, $N_{V}=N^{2}-1$ and $N_{F}=NN_{f}$. The above inequality then demands $N_{f}^{c}<\frac{1}{4}\left(7N+\sqrt{81N^{2}-16}\right)$. This is a testable constraint, and it has been satisfied by recent lattice simulations Neil (2011). For $N=2$ on the other hand, the enhanced chiral symmetry, the different pattern of symmetry breaking, and the resultant enhanced NGB count ($2N_{f}^{2}-N_{f}-1$) Peskin (1980) lead to a significantly reduced bound on $N_{f}$ for the broken phase: $N_{f}^{c}<(4+\sqrt{30})/2\approx 4.7$. Crude estimates of the edge of the conformal window, based on quasi- perturbative methods, also exist. Gap-equation methods Cohen and Georgi (1989) provide an estimate of the gauge coupling strength, and therefore maximum value of $N_{f}$, required to induce spontaneous chiral symmetry breaking. For any SU($N$) gauge theory, these notions lead to the estimate $N_{f}^{c}\approx 4N$. While this is nicely compatible with the inequality for $N\geq 3$, it clearly disagrees with it for $N=2$. This tension suggests that the $N_{f}=6$ theory could be particularly worthy of study. Early lattice calculations attempted to explore the two-color conformal window by studying the lattice theory at strong bare coupling Iwasaki et al. (2004); Nagai et al. (2010). Recent efforts have primarily searched for an IRFP with non-perturbative running coupling calculations. Evidence that $N_{f}=10$ ($N_{f}=4$) is inside (outside) the conformal window is presented in Ref. Karavirta et al. (2012). Additionally, Ohki et al. argue that $N_{f}=8$ is inside the conformal window Ohki et al. (2010). The case $N_{f}=6$, arguably the most interesting, while tackled by many groups Karavirta et al. (2012); Bursa et al. (2011); Voronov (2011, 2012); Hayakawa et al. (2013), has remained inconclusive. Here we study the $N_{f}=6$ theory, drawing on larger computational resources than in all previous work, to determine whether $N_{f}=6$ has an IRFP by calculating the Schrödinger Functional (SF) Luscher et al. (1992) running coupling. We use the stout-smeared Morningstar and Peardon (2004) Wilson fermion action, which suppresses coupling the fermions to unphysical fluctuations of the gauge field on the scale of the lattice spacing. This improved action reduces lattice artifacts and allows us to search for an IRFP up through a large and interesting range of running couplings. Smeared actions have also been used in SF running coupling studies of other theories DeGrand et al. (2010, 2013). ## Preliminaries A stout-smeared fermion action replaces “thin” gauge links by “fat” links which are averaged with nearby gauge links. To define a stout-smeared Morningstar and Peardon (2004) link is we start with $C_{\mu}\left(x\right)$, the weighted sum of staples about the link $(x,x+\hat{\mu})$: $\displaystyle C_{\mu}\left(x\right)$ $\displaystyle=$ $\displaystyle\sum_{\nu\neq\mu}\rho_{\mu\nu}\left(U_{\nu}\left(x\right)U_{\mu}\left(x+\hat{\nu}\right)U_{\nu}^{\dagger}\left(x+\hat{\mu}\right)\right.$ $\displaystyle\left.+U_{\nu}^{\dagger}\left(x-\hat{\nu}\right)U_{\mu}\left(x-\hat{\nu}\right)U_{\nu}\left(x-\hat{\nu}+\hat{\mu}\right)\right).$ We want our fat links to be elements of SU($N$). This is guaranteed by taking the smearing kernel to be of form $e^{iQ}$ with $Q$ an element of the Lie algebra $\mathfrak{su}\left(N\right)$. We take $\displaystyle Q_{\mu}\left(x\right)$ $\displaystyle=$ $\displaystyle\frac{i}{2}\left(\Omega_{\mu}^{\dagger}\left(x\right)-\Omega_{\mu}\left(x\right)\right)$ (2) $\displaystyle-\frac{i}{2N}\mathrm{Tr}\left(\Omega_{\mu}^{\dagger}\left(x\right)-\Omega_{\mu}\left(x\right)\right),$ with $\Omega_{\mu}\left(x\right)=C_{\mu}\left(x\right)U_{\mu}^{\dagger}\left(x\right)$ ($\mu$ is not summed over). Then a fat link is defined by $U_{\mu}^{\left(n+1\right)}\left(x\right)=\exp\left(iQ_{\mu}^{\left(n\right)}\left(x\right)\right)U_{\mu}^{\left(n\right)}\left(x\right).$ (3) This smearing procedure may be applied iteratively, say $n_{\rho}$ times, to produce stout links $\tilde{U}=U^{\left(n_{\rho}\right)}$. It has the advantage that it is analytic and can therefore be used in conjunction with molecular dynamics (MD) updating schemes such as Gottlieb et al. (1987). The formulas required to implement this smearing procedure in an MD algorithm are derived for the case of SU(3) links in Morningstar and Peardon (2004). We have derived the relevant formulas for the SU(2) case. Recently, another group implemented two-color stout-smearing as well Catterall and Veernala (2013). We use only one level of stout-smearing with an isotropic smearing parameter $\rho_{\mu\nu}=\rho=0.25$. As all calculations in this work are done with Dirichlet boundary conditions (BC) in the time directions, there is some ambiguity in how to implement the smearing of the gauge field near this boundary. We choose to not smear the boundary links with bulk links and vice versa. This choice results in a simpler running-coupling observable (which will be defined in the next section). The Wilson fermion action contains an additional irrelevant operator that lifts the mass of the fermion doublers to the cutoff scale so they decouple from the calculation. This additional term explicitly breaks chiral symmetry, and as a result the fermion mass is additively renormalized. The bare mass $m_{0}$ therefore must be carefully tuned in order to restore chiral symmetry. The critical value of the bare mass (as a function of the bare coupling) $m_{c}(g_{0}^{2})$ is defined as the bare mass value that results in a zero renormalized quark mass Luscher et al. (1997). In practice, $m_{c}$ is determined, at fixed bare gauge coupling $g_{0}^{2}$ and lattice volume $\left(L/a\right)^{3}\times 2L/a$, as the root of a fitted linear function to measurements of the renormalized quark mass versus the bare quark mass. This is done for a range of bare couplings and lattice volumes and the results are fit to a polynomial given by $m_{c}^{\mbox{fit}}\left(g_{0}^{2},\frac{a}{L}\right)=\sum_{i=1}^{n}g_{0}^{2i}\left[a_{i}+b_{i}\left(\frac{a}{L}\right)\right].$ (4) Then, $m_{c}^{\mbox{fit}}\left(g_{0}^{2},0\right)$ is used in the running coupling calculations. All data used to fit $m_{c}^{\mbox{fit}}\left(g_{0}^{2},a/L\right)$ and $m_{c}^{\mbox{fit}}\left(g_{0}^{2},0\right)$ are shown in Figure 1. Figure 1: Bare masses that result in zero PCAC mass at lattice volumes $8^{3}\times 16$, $10^{3}\times 20$, $12^{3}\times 24$, $14^{3}\times 28$, and $16^{3}\times 32$. All data points fit to $m_{c}^{\mbox{fit}}\left(g_{0}^{2},\frac{a}{L}\right)$ and the continuum extrapolation $m_{c}^{\mbox{fit}}\left(g_{0}^{2},0\right)$ (black dashed line) are shown. $m_{c}^{\mbox{fit}}\left(g_{0}^{2},0\right)$ determines masses used in running coupling simulations. Additionally the peak in the plaquette susceptibility (turquoise xs) is shown. We collect all running coupling data along the critical mass line on the weak coupling side of the phase transition line. In order to guarantee that we can take a continuum limit, we need to obtain data only from the weak-coupling side of any spurious lattice phase transition. With this in mind, we scan through the bare parameter space and locate peaks in the plaquette susceptibility on a $L/a=10$ lattice. This search indicates a line in the $m_{0}-g_{0}^{2}$ plane of first order phase transitions that ends at a critical point at around $g_{0}^{2}\approx 2.2$. For $g_{0}^{2}\lesssim 2.2$, we see crossover behavior. In Figure 1, we show the above transition line plotted along with $m_{c}^{\mbox{fit}}(g_{0}^{2},0)$. Figure 1 indicates that our action has a sensible continuum limit only for $g_{0}^{2}\lesssim 2.175$. Therefore, we examine the running coupling only on lattices with a bare coupling within this range. ## Running Coupling To define a non-perturbative renormalized coupling, we employ the Schrödinger functional (SF) Luscher et al. (1992). It is given by a path integral over gauge and fermion fields that reside within a four-dimensional Euclidean box of spatial extent $L$ with periodic BC’s in spatial directions and Dirichlet BC’s in the time direction. We choose gauge BC’s Luscher et al. (1993), $\left.U\left(x,\mathrm{k}\right)\right\|_{x^{0}=0}=\exp\left[-i\eta\frac{a}{L}\tau_{3}\right]\mbox{ and }\left.U\left(x,\mathrm{k}\right)\right\|_{x^{0}=L}=\exp\left[-i\left(\pi-\eta\right)\frac{a}{L}\tau_{3}\right],$ and fermion BC’s Sint (1994), $\left.P_{+}\psi\right\|_{x^{0}=0}=\left.\bar{\psi}P_{-}\right\|_{x^{0}=0}=\left.P_{-}\psi\right\|_{x^{0}=L}=\left.\bar{\psi}P_{+}\right\|_{x^{0}=L}=0.$ These BC’s classically induce a constant chromoelectric background field whose strength is characterized by the dimensionless parameter $\eta$. With these BC’s the SF is given by $\mathcal{Z}(\eta,L)=\int D\left[U,\psi,\bar{\psi}\right]e^{-S[U,\psi,\bar{\psi};\eta]}.$ The running coupling is then defined by, $\frac{k}{\bar{g}^{2}\left(g_{0}^{2},\frac{L}{a}\right)}=\left.\frac{\partial}{\partial\eta}\log\mathcal{Z}\right\|_{\eta=\pi/4}=\left\langle\frac{\partial S}{\partial\eta}\right\rangle,$ (5) with $k=-24\left(L/a\right)^{2}\sin\left[\left(a/L\right)^{2}\left(\pi/2\right)\right]$ so that the renormalized coupling agrees with the bare coupling at tree-level. The first two perturbative coefficients of the SF beta function are the universal coefficients given in Caswell (1974). This renormalization scheme has the virtue that it is fully non-perturbative and it is amenable to a lattice calculation. We calculate the SF renormalized coupling over a range of bare couplings and lattice volumes. Lattice perturbation theory gives $g_{0}^{2}/\bar{g}^{2}$ as an expansion in powers of $g_{0}^{2}$. This motivates an interpolating fit Appelquist et al. (2009), $\frac{1}{g_{0}^{2}}-\frac{1}{\bar{g}^{2}\left(g_{0}^{2},\frac{L}{a}\right)}=\sum_{i=0}^{N_{L/a}}a_{i,L/a}g_{0}^{2i}.$ (6) We choose the lowest possible $N_{L/a}$ to give a reasonable $\chi^{2}$ per dof (in practice, values in the range $\chi^{2}/\mathrm{dof}\in\left[0.7,1.5\right]$), finding $N_{L/a\leq 12}=6$ and $N_{L/a>12}=5$. This procedure produces smooth functions, one for each lattice volume $L/a$, of the renormalized coupling versus the bare coupling. Before using this interpolation for further analysis, it is worth noting that there is no hint of an IRFP in the lattice data and therefore in the interpolating curves. At any fixed $g_{0}^{2}$, the running coupling $\bar{g}^{2}\left(g_{0}^{2},\frac{L}{a}\right)$ is seen only to increase as a function of $L/a$ in the range of the data. The question is whether a careful continuum extrapolation will indicate otherwise. A step scaling Luscher et al. (1991) analysis allows us to address this issue and to study the renormalized coupling over a large range of scales in computationally feasible manner. The continuum step scaling function $\sigma\left(u,s\right)$ is defined by $\int_{u}^{\sigma\left(u,s\right)}\frac{d\bar{g}^{2}}{\beta\left(\bar{g}^{2}\right)}=2\log s.$ (7) It is the renormalized coupling at a length scale $sL$ given that the running coupling $\bar{g}^{2}=u$ at a length scale $L$. On the lattice we calculate the discrete step scaling function, $\Sigma\left(u,\frac{a}{L},s\right)\equiv\left.\bar{g}^{2}\left(g_{0_{}}^{2},\frac{sL}{a}\right)\right\|_{\bar{g}^{2}\left(g_{0_{}}^{2},\frac{L}{a}\right)=u}.$ (8) It is the value of the renormalized coupling on a lattice volume of $(sL/a)^{4}$ and bare coupling tuned such that we have a renormalized coupling of $u$ on a lattice of volume $\left(L/a\right)^{4}$. We arrive back at a continuum step scaling function by taking the continuum limit: $\sigma\left(u,s\right)=\underset{a/L\rightarrow 0}{\lim}\Sigma\left(u,\frac{a}{L},s\right).$ (9) From here we use $s=2$ and drop reference to this from our notation. To extract $\sigma$ as a function of $u$, we first use the interpolating fits, given by Eq. 6, to evaluate $\Sigma$ at each fixed value of $u$ and $L/a=5,\mbox{ }6,\mbox{ }7,\mbox{ }8,\mbox{ }9,\mbox{ }10,\mbox{ and }12$. We take the continuum limit, at each $u$ independently, by fitting $\Sigma\left(u,a/L\right)$ to a polynomial in $a/L$, and extrapolating to $a/L\rightarrow 0$. Our result, shown in Fig 2, displays several plots of the quantity $\left(\sigma\left(u\right)-u\right)/u$ versus $u$. This quantity is a finite-difference version of the continuum beta function. In one curve (red), we fit $\Sigma\left(u,a/L\leq 1/6\right)$ to a quadratic polynomial and then extrapolate the result to $a/L\rightarrow 0$. Additionally, we show, $\Sigma\left(u,a/L\leq 1/5\right)$ extrapolated from a cubic polynomial fit (green). We see that these two curves are consistent, but the errors of the cubic extrapolation become large at $u\approx 8$. The remaining (blue) curve is obtained with a constant extrapolation to the continuum using only the three points with $a/L\leq 1/9$. To asses the goodness-of-fit of any particular functional form for continuum extrapolation of $\Sigma$ we examine $\chi^{2}/\mathrm{dof}$ over the entire range of $u$. For the constant extrapolation (blue) in Fig. 2 for $L/a\geq 9$, $\chi^{2}/\mathrm{dof}$ varies from 0.5-2. A quadratic extrapolation (red) for $L/a\geq 6$ and a cubic extrapolation for $L/a\geq 5$ have comparable $\chi^{2}/\mathrm{dof}$ ranging from 0.5-4 throughout the range of $u$. The constant (quadratic and cubic) extrapolation relies on fits with two (three) degrees-of-freedom. These various extrapolations all perform well at reproducing the perturbative two-loop curve (magenta) at small values of $u$. If the resulting curves were to cross zero at some larger $u$, this would be indicative of an IRFP. We see no indication of this; in fact we see, regardless of which extrapolation we use, the running coupling grow up to and beyond estimates of the critical coupling required to induce spontaneous chiral symmetry breaking Cohen and Georgi (1989). We see no evidence even of an inflection point, which would hint at an IRFP at a stronger coupling strength. Figure 2: $\left(\sigma(u)-u\right)/u$ vs $u$ for three different extrapolations to the continuum. A contour at $\bar{g}^{2}=20$ is shown to provide a measure of the strength of renormalized coupling explored here. The 2-loop perturbative result is also shown here (dot-dashed magenta). We next compare these three continuum extrapolations more carefully and comment also on extrapolation via a linear polynomial in $a/L$. For each $u$, $\Sigma\left(u,a/L\right)$, evaluated at $L/a=5,\mbox{ }6,\mbox{ }7,\mbox{ }8,\mbox{ }9,\mbox{ }10,\mbox{ and }12$, is fit to a cubic polynomial, $p\left(a/L\right)=\sum_{i=0}^{3}\alpha_{i}\left(a/L\right)^{i}$. For several values of $a/L$, the relative sizes of the constant, O$(a/L)$, O$(a/L)^{2}$, and O$(a/L)^{3}$ terms in the polynomial are plotted vs $u$. We can then assess the validity of some truncation of the polynomial continuum extrapolation within some window in $a/L$. We show the results of such an analysis in Fig. 3 for $L/a=6,\mbox{ }9,\mbox{ and }12$. A number of interesting features are evident. At weak coupling the lattice artifacts are small, and a constant extrapolation adequately describes the continuum limit. But at intermediate and strong coupling ($u\gtrsim 6$), lattice artifacts become significant. Throughout the coupling range, the linear and quadratic lattice artifacts are comparable for $a/L\geq 1/9$ and hence we can not perform a reliable linear extrapolation to the continuum. The cubic contribution, however, is small for $a/L\leq 1/6$ and $u\lesssim 8$, indicating that a quadratic extrapolation to the continuum is reliable at least up to this input coupling strength. This indicates that the running coupling reaches a $\bar{g}^{2}$ of order $20$ without encountering an IRFP . Figure 3: Plots of relative magnitudes of low order contributions to the continuum extrapolation. We fit $s=2$ steps at $L/a=5,\mbox{ }6,\mbox{ }7,\mbox{ }8,\mbox{ }9,\mbox{ }10,\mbox{ and }12$ to a polynomial $\sum_{i=0}^{3}\alpha_{i}\left(\frac{a}{L}\right)^{i}$. Then $\left\|\alpha_{0}\right\|/T$ (blue), $\left\|\alpha_{1}\left(\frac{a}{L}\right)\right\|/T$ (red), $\left\|\alpha_{2}\left(\frac{a}{L}\right)^{2}\right\|/T$ (green), and $\left\|\alpha_{3}\left(\frac{a}{L}\right)^{3}\right\|/T$ (cyan) are plotted versus $u$, at various values of $a/L$, with $T=\sum_{i=0}^{3}\left\|\alpha_{i}\left(\frac{a}{L}\right)^{i}\right\|$. Insight may also be gleaned by plotting the extrapolation to the continuum at fixed coupling strength $u$. We show in Fig. 4 the example of $u=7.5$. We plot $\Sigma\left(u,a/L\right)$ vs $a/L$, along with a quadratic and cubic polynomial fit, as well as a constant extrapolation based on the three smallest $a/L$ values. These correspond to the fits used in Fig. 2. Fig. 4 demonstrates that a constant extrapolation to the continuum is reasonable. Taking the larger $a/L$ points into account shows the presence of significant non-linear lattice artifacts, in fact suggesting that the constant extrapolation significantly underestimates $\sigma\left(u\right)$ for $u\gtrsim 7$. It is also evident that the quadratic and cubic fits extrapolate to a value of $\sigma$ that is well above the smallest-$a/L$ points. It is likely that the true extrapolated value is somewhere between the constant and quadratic extrapolations. Figure 4: Plot of $\Sigma\left(u=7.5,a/L\right)$ vs $a/L$ with various extrapolations to the continuum. The continuum limit of the quantity is obtained by fitting these points to a polynomial in $a/L$. Recently Hayakawa et al. claim to see evidence of an IRFP in the two-color six-flavor theory Hayakawa et al. (2013). They employ the SF method as we do but with the unimproved Wilson fermion action and a linear extrapolation to the continuum. It is reasonable to expect that for large enough $L/a$ the linear term will be the dominant lattice artifact but it is difficult to quantify how large an $L/a$ is necessary outside of perturbation theory. Other extrapolation forms, including quadratic terms can be used to fit their data with a comparable or slightly better $\chi^{2}/\mathrm{dof}$. When this is done, we cannot conclude that an IRFP exists. Moreover, from our data set, sampling many more bare couplings and lattice volumes, we are able to study the relative contributions of different lattice artifacts. In Figure 3, we see that in the strong coupling regime, the quadratic term becomes significant in the $a/L$ range studied by Hayakawa et al. and by us. With the caveat that we use a different lattice action, the relative importance of the quadratic term suggests that concluding the existence of an IRFP from a linear extrapolation to the continuum is premature. To summarize, for an SU(2) gauge theory with six massless fermions in the fundamental representation, we find no evidence of an infrared fixed point in the running gauge coupling as defined in the Schrödinger Functional scheme. Our simulations reach well into a strong-coupling range, potentially capable of triggering chiral symmetry breaking and confinement. We conclude that this theory either flows to a very strong infrared fixed point, so-far unseen in non-supersymmetric theories, or it breaks chiral symmetry and confines, producing a large number (65) of Nambu-Goldstone bosons, well above the number of underlying fermionic and gauge degrees of freedom. Thus either of these (zero-temperature) phases exhibits novel behavior. In the latter case, the finite-temperature phase transition can be expected to have interesting features. We could in principle probe even larger couplings than presented here, but the computational challenges and lattice-artifact difficulties grow with coupling strength. Other approaches, such as the computation of correlation functions and the particle spectrum, will be important to firmly establish the infrared nature of this theory. ## Acknowledgments We thank Robert Shrock for helpful discussions. We would like to acknowledge our use of the Chroma Edwards and Joo (2005) software package for all calculations performed here. We thank the Lawrence Livermore National Laboratory (LLNL) Institutional Computing Grand Challenge program for computing time on the LLNL Sierra, Hera, Atlas, and Zeus computing clusters. We thank LLNL for funding from LDRD10-ERD-033 and LDRD13-ERD- 023\. Several of us (T. A., G. F., R. B., M. C., E. N., M. L., and D. S.) thank the Aspen Center for Physics (supported by NSF grant PHYS-1066293) for its hospitality while some of the research reported here was being done. This work has been supported by the U. S. Department of Energy under Grants DE-FG02-00ER41132 (M.I.B.), DE-FG02-91ER40676 (R.C.B., M.C., C.R.), DE-FG02-92ER-40704 (T.A.), DE-FC02-12ER41877 (D. S.), DE-FG02-85ER40231 (D. S.), and Contracts DE- AC52-07NA27344 (LLNL), DE-AC02-06CH11357 (Argonne Leadership Computing Facility), and by the National Science Foundation under Grant Nos. NSF PHY11-00905 (G.F., M.L., G.V.) and PHY11-25915 (Kavli Institute for Theoretical Physics). We thank USQCD for computer time on FNAL and JLab clusters. We thank XSEDE for computer time on Kraken under grant TG-MCA08X008. ## References * Aad et al. (2012) G. Aad et al. (ATLAS Collaboration), Phys.Lett. B716, 1 (2012), eprint 1207.7214. * Chatrchyan et al. (2012) S. Chatrchyan et al. (CMS Collaboration), Phys.Lett. 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Syritsyn\n (8), G. Voronov (1), P. Vranas (11), O. Witzel (4) ((1) Yale University, (2)\n Boston University, (3) INT Seattle WA, (4) Center for Computational Science,\n Boston University, (5) UC Davis, (6) Computational Science Center, BNL, (7)\n UC Boulder, (8) RIKEN-BNL Research Center, BNL, (9) Argonne Leadership\n Computing Facility, ANL, (10) Syracuse University, (11) LLNL)", "submitter": "George T. Fleming", "url": "https://arxiv.org/abs/1311.4889" }
1311.4949	# Mid-Infrared Imaging of the Bipolar Planetary Nebula M2-9 from _SOFIA_ M. W. Werner11affiliation: Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91107 USA. [email protected] , R. Sahai11affiliation: Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91107 USA. [email protected] , J. Davis 11affiliation: Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91107 USA. [email protected] , J. Livingston 11affiliation: Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91107 USA. [email protected] , F. Lykou 22affiliation: Institute for Astronomy, University of Vienna, Turkenschanzstrasse 17, A-1180, Vienna, Austria , J. de Buizer 33affiliation: USRA SOFIA Science Center, M/S 211-3, NASA Ames Research Center, Moffett Field, CA 94035 USA , M. R. Morris 44affiliation: Division of Astronomy, PO Box 951547, UCLA, Los Angeles, CA 90095 USA , L. Keller 55affiliation: Department of Physics, Ithaca College, Ithaca, NY 14850 USA , J. Adams 66affiliation: Department of Astronomy, Cornell University, Ithaca, NY 14853 USA , G. Gull 66affiliation: Department of Astronomy, Cornell University, Ithaca, NY 14853 USA , C. Henderson 66affiliation: Department of Astronomy, Cornell University, Ithaca, NY 14853 USA , T. Herter 66affiliation: Department of Astronomy, Cornell University, Ithaca, NY 14853 USA , J. Schoenwald 66affiliation: Department of Astronomy, Cornell University, Ithaca, NY 14853 USA ###### Abstract We have imaged the bipolar planetary nebula M2-9 using _SOFIA_ ’s FORCAST instrument in six wavelength bands between 6.6 and 37.1 $\mu m$. A bright central point source, unresolved with _SOFIA_ ’s $\sim$ 4′′-to-5′′ beam, is seen at each wavelength, and the extended bipolar lobes are clearly seen at 19.7 $\mu m$ and beyond. The photometry between 10 and 25 $\mu m$ is well fit by the emission predicted from a stratified disk seen at large inclination, as has been proposed for this source by Lykou et al and by Smith and Gehrz. The principal new results in this paper relate to the distribution and properties of the dust that emits the infrared radiation. In particular, a considerable fraction of this material is spread uniformly through the lobes, although the dust density does increase at the sharp outer edge seen in higher resolution optical images of M2-9. The dust grain population in the lobes shows that small ($<$ 0.1 $\mu m$) and large ($>$ 1 $\mu m$) particles appear to be present in roughly equal amounts by mass. We suggest that collisional processing within the bipolar outflow plays an important role in establishing the particle size distribution. planetary nebulae: individual (M2-9) ††slugcomment: To appear in the Astrophysical Journal ## 1 INTRODUCTION Although planetary nebulae (PNs) evolve from (initially) spherically–symmetric mass-loss envelopes around AGB stars, modern ground-based and Hubble Space Telescope (HST) imaging surveys have shown that the vast majority of PNs deviate strongly from spherical symmetry (e.g., Schwarz, Corradi, & Melnick 1992, Sahai, Morris, & Villar 2011b). The morphologically unbiased survey of young PNs with HST (Sahai & Trauger 1998, Sahai et al. 2011b) shows that PNs with bipolar and multipolar morphologies represent almost half of all PNs, as was previously pointed out by Zuckerman and Aller (1986). The significant changes in the circumstellar envelope morphology during the evolutionary transition from the AGB to the PN phase require a primary physical agent or agents which can break the spherical symmetry of the radiatively-driven, dusty mass-loss phase. Sahai and Trauger (1998) proposed that the primary agent for breaking spherical symmetry is a jet or collimated, fast wind (CFW) operating during the early post-AGB or late AGB evolutionary phase. The nature of the central engine that can produce such CFW’s is poorly understood, although a number of theoretical models, most of them requiring the central star to be a binary, have been considered (Morris, 1987; Garcia-Segura, 1997; Balick & Frank, 2002; Matt, Frank & Blackman, 2006). Detailed multi-wavelength studies of individual bipolar and multipolar PNs that can constrain the physical properties of the lobes produced by the CFW’s and the central regions are needed to test these models. M2-9 is a well-studied bipolar nebula at many wavelengths from the optical (e.g., Solf 2000) to the radio (Kwok et al. 1985), and is usually classified as a planetary nebula, although it has also been suggested that this object has a symbiotic star at its center (e.g., Schmeja & Kimeswerner 2001), or belongs to the compact planetary nebula (cPNB[e]) subclass of B[e] stars (Frew & Parker 2010). Each of the bipolar lobes appears in optical, emission-line images as a collimated, limb-brightened structure with an “inner” lobe of radial extent about $29{{}^{\prime\prime}}$ and width $12{{}^{\prime\prime}}$, and a much fainter “outer” lobe of radial extent about $62{{}^{\prime\prime}}$ and similar narrow width (e.g., Corradi, Balick, & Santander-Garcia 2011, hereafter Co11). Proper motion of bright ansae at the tips of the faint lobes implies a radial expansion speed of 147 $km\,s^{-1}$ (Co11). One of the most striking phenomena observed in M2-9 is a pattern of emission-line knots in each lobe that appears to rotate with a period of 90 yr, interpreted as resulting from either a pair of rotating light-beams (Livio & Soker 2001) or jets (e.g., Co11). Livio & Soker (2001) propose a model for producing the light-beams in which jets clear a path which allows ionizing radiation from a white-dwarf companion of the primary AGB (or post-AGB) star, to irradiate the knot regions. The dense, dusty waist separating the two lobes was first mapped with a $3{{}^{\prime\prime}}\times 5{{}^{\prime\prime}}$ beam in the CO(J=2–1) line with the Plateau de Bure millimeter-wave interferometer (PdBI) by Zweigle et al. (1997) revealing the presence of a large ($\sim\,6{{}^{\prime\prime}}$ diameter) ring structure with a mass of $\sim$ 0.01$M_{\odot}$. More recent PdBI mapping with a $0\farcs 8\times 0\farcs 4$ beam reveals a second inner ring that is almost three times smaller than the outer one (Castro-Carrizo et al. 2012). The rings are co-planar and seen almost edge-on, with their axes being inclined at $\sim\,19\arcdeg$ to the sky-plane, similar to the inclination of the axis of the bipolar lobes (Solf 2000). ## 2 PREVIOUS INFRARED STUDIES OF M2-9 The mid-infrared observations from _SOFIA_ reported here build on previous studies of M2-9 by Smith and Gehrz (2005; hereafter SG05) and Lykou et al. (2011; hereafter Lyk11). SG05 imaged M2-9 at 8.8, 17.9, and 24.5 $\mu m$, using the _IRTF_. Their results for the flux from the central point source, using a smaller effective aperture, are consistent with ours. They also detect the lobes at all three wavelengths, although the images they present of the extended emission are less extensive and of lower signal-to-noise than the present results. More recently, Lyk11 present an extensive study of M2-9, reporting spectroscopy from both $ISO$ and _Spitzer_ out to $\sim$ 35 $\mu m$ and summarizing previous measurements as well. They also present ground-based interferometric observations in the 10 $\mu m$ region which identify a disk of dimension $\sim$ 40 $mas$ within the central unresolved point source. We have adapted the model for the central source used by Lyk11 (see also Chesneau et al., 2007) to the analysis of our photometry. Lagadec et al. (2011) have also published recent photometry of M2-9 in the 8-13 $\mu m$ region, reporting fluxes $\sim$ 20-to-30% higher than found here or given by Lyk11. This discrepancy may be due in part to the various photometric bands used for the different measurements. Finally, Sanchez-Contreras et al. (1998; hereafter SC98) present an image of M2-9 at 1.3 $mm$ which suggests a significant amount of very cold dust associated with the lobes of the nebula. In addition, M2-9 was measured in the $IRAS$, $AKARI$, and $WISE$ surveys. ## 3 OBSERVATIONS We observed M2-9 with _SOFIA_ ’s FORCAST instrument (Herter et al., 2012), which provides imaging capability in multiple spectral bands between 5 and 40 $\mu m$, using two $256\times 256$ blocked impurity band detector arrays. For many observations, the arrays are used simultaneously with a dichroic beam splitter. Wavelengths from 5-25 $\mu m$ are directed to a Si:As array, while the 25-40 $\mu m$ wavelengths pass through to a Si:Sb array. After correction for focal plane distortion, FORCAST effectively samples at 0.768 $arcsec\,pixel^{-1}$, which yields a $3.2arcmin\times 3.2arcmin$ instantaneous field of view in each camera. We elected to observe M2-9 in six bands at 6.6, 11.1, 19.7, 24.2, 33.6, and 37.1 $\mu m$. Each of these filters has a bandwidth of $\sim$ 4-to-30% (see Herter et al. 2012 for further information about the FORCAST instrument). The observations were made on two separate _SOFIA_ flights on 11 May and 2 June 2011. The 6.6 and 11.1 $\mu m$ data were taken sequentially with a mirror in place of the dichroic, while the 24.2 $\mu m$ data were taken simultaneously with the 37.1 $\mu m$ data, and the 19.7 $\mu m$ data with the 33.6 $\mu m$ data. The chopping secondary on _SOFIA_ was configured to chop east-west with a 30′′ amplitude on the sky (perpendicular to the long axis of the nebula, which is very close to north-south) to cancel atmospheric emission. The telescope was nodded east-west every 30 sec with a $30{{}^{\prime\prime}}$ throw to facilitate subtraction of (predominantly) telescope radiative offsets. This chopping and nodding strategy made it possible to keep an image of the nebula within the field of view of the array continually during the observations. The total observation time in each filter for the observations presented here is about 10 minutes, with data being written to a FITS image approximately every second, and with approximately half the total exposure time in each filter coming from each of the two flights. The data were reduced and calibrated with pipeline software at the _SOFIA_ Science Center and a series of FITS files were posted in the _SOFIA_ Archive for download by the investigator team. ## 4 RESULTS ### 4.1 Images In Figure 1 we show the final _SOFIA_ images of M2-9 in all six bands. Also shown, for comparison, is a composite line emission image of M2-9 from HST, as well as a continuum image from HST in the 547 $nm$ filter; the HST images were obtained on 1997 Aug 7 with WFPC2 as part of GO program 6502 (PI: B. Balick). In each of the _SOFIA_ images, a compact central source is apparent, and the extended emission lobes are clearly seen at 19.7 $\mu m$ and longward. N is to the top and E to the left in these images, so the position angle on the sky of M2-9 is very close to N-S. Therefore, we refer to scans parallel and perpendicular to the outflow-lobes as N-S and E-W scans, respectively. In Figure 2, we show North-South scans at 19.7 and 37.1 $\mu m$ extending more than 20′′ from the central compact source. At the longer wavelength, the compact source is superposed on the emission from the lobes, which contribute a much larger fraction of the total flux than at the short wavelengths (see Table 1 and Figure 1). We emphasize that the extended wings due to the lobe emission seen at 37.1 $\mu m$ in Figure 2 are not seen in the point source PSF (cf. Figure 7). ### 4.2 Photometry Photometry of the central source has been carried out at each wavelength using the point source photometry routine in ATV, which also provides an estimate of the FWHM of the point source. Based on the scans shown in Figure 2, we set the aperture radius for this photometry to be $5.4{{}^{\prime\prime}}$ (7 pixels) and the reference sky annulus to be between $5.4{{}^{\prime\prime}}$ and 6.9′′ (9 pixels). This choice helped us to determine the compact source flux with minimum contamination from the surrounding plateau of emission. These results are tabulated in Table 1. Also given in Table 1 is the total flux at each wavelength, as determined by integrating the total flux within the $20{{}^{\prime\prime}}\times 40{{}^{\prime\prime}}$ area shown in Figure 1 and subtracting the average sky brightness determined from $20{{}^{\prime\prime}}\times 40{{}^{\prime\prime}}$ areas N, S, E, and W of M2-9 on the images. The third column in the table gives the difference between the compact source flux and the total flux, which is an estimate of the flux in the extended lobes plus any extended component in the EW plane of the compact source. Note that although the lobes are not readily visible in the images at 6.6 and 11.1 $\mu m$, they are detected at these wavelengths in the integrated emission from the source; the flux tabulated for the extended component at these wavelengths in Table 1 is consistent with that which can be estimated for the lobes at 8.8 $\mu m$ from the images presented by SG05. For completeness, we include fluxes at the longer wavelengths as measured by $IRAS$ and by SC98 as well as the flux measured by ISO shortward of 5 $\mu m$ as reported by Lyk11; the central compact source was not resolved by ISO’s 1′′ to 2′′ beam at these short wavelengths. The point source FWHM reported by ATV varied from 3.7′′ at 6.6 and 11.1 $\mu m$ to 4.9′′ at 37.1 $\mu m$. At all wavelengths, the observed FWHM agrees with the recommended value for this flight series provided by the _SOFIA_ Science Center (Table 1). Thus there is no evidence that _SOFIA_ has resolved the central source at any wavelength. In Figure 3 we plot the total flux from the $20{{}^{\prime\prime}}\times 40{{}^{\prime\prime}}$ area of Figure 1. Note from Figure 3 that most of the flux measured by $IRAS$ shortward of 60 $\mu m$ comes from this area. The S/N of our _SOFIA_ measurements is quite high; for both lobes and point source the principal uncertainty in almost all cases is the $\pm 20\%\,(3\sigma)$ calibration uncertainty (Herter, 2012). ## 5 ANALYSIS AND INTERPRETATION ### 5.1 Emission Mechanism and Total Luminosity The spectral and spatial characteristics of the emission from the lobes are suggestive of emission from dust. Fine structure emission lines appear in _Spitzer_ spectra of the lobes (Lyk11), but they are not strong enough to contribute substantial flux to that measured in _SOFIA_ ’s broad filters: The strongest lines which lie in any of our band passes are [SIII] at 18.7 $\mu m$, [OIV] at 26.4 $\mu m$, and [SiII] at 35 $\mu m$, but comparison of the line intensity with that of the adjacent continuum shows that the lines contribute only 1-to-2% of the total flux measured with _SOFIA_. The weak PAH emission seen at 11.3 $\mu m$ does not contribute significantly to the integrated flux in the 11.1 $\mu m$ band, and no PAH emission is seen in the ISO spectra centered on the central point source. There is ample evidence from previous studies of scattered light and polarization that the lobes contain dust. We thus interpret the radiation from both lobes and the point source as being due to emission from dust. Integrating over the SEDs tabulated in Table 1, and assuming isotropic emission and a distance to the source of 1200 $pc$ (see below), we find that the total 2.5-40 $\mu m$ infrared luminosity of the compact source is $\sim$ 840 $L_{\odot}$, that of the lobes $\sim$ 390 $L_{\odot}$. The total 2.5-120 $\mu m$ luminosity of M2-9, including the $IRAS$ measurements at longer wavelengths, is $\sim$ 1530 $L_{\odot}$. The observed luminosity at shorter wavelengths is no more than a few percent of that seen in the infrared. ### 5.2 Modeling the Central Point Source At each wavelength the central point source appears unresolved, with a measured FWHM close to the value recommended by the _SOFIA_ Science Center for the flight series during which M2-9 was observed. However, the SED of the central source is much broader than a blackbody, suggesting that a range of dust temperatures is being sampled, as would be the case for an optically thin or somewhat face-on disk-like geometry. This type of geometry has previously been proposed for this central source by SG05, based on similar arguments. Lyk11 report interferometric imaging of a compact $\sim$ $0.037{{}^{\prime\prime}}\times 0.046{{}^{\prime\prime}}$ dust disk at the center of M2-9, and they have produced a model of a circumbinary disk (see Chesneau et al. 2007 for details), suggesting that the interferometric measurements sample the warm inner regions of the disk. This model has now been updated to fit the _SOFIA_ data on the central point source. As is shown in Figure 4, the fit is excellent at wavelengths from 11.1 to 24.2 $\mu m$ over which most of the energy from this source is observed. The parameters for this model are detailed in Table 2. Neither our observations nor those of Lyk11 constrain the disk outer radius. The distance to M2-9 is quite uncertain, as is often the case for planetary nebulae. The present observations do not constrain the distance, so we have adopted D=1200 $pc$ for consistency with Lyk11. This in agreement with the recent careful estimate of $1.3\pm 0.12$ $kpc$, based on kinematic analysis of motions of features in the lobes (Co11). We emphasize, however, that the principal new results of this work, which relate to the spatial distribution and particle size distribution of the dust, are derived directly from the observations and are independent of the adopted distance. At a distance of 1200 $pc$, the angular extent of the disk modeled in Figure 4 is less than 2′′. Thus there is ample room for cooler material exterior to this disk, perhaps associated with the inner CO-emitting disk most recently discussed by Castro-Carrizo et al. (2012), which could produce the radiation seen at 33.6 and 37.1 $\mu m$ in excess of the model prediction without producing a spatially resolved source at these wavelengths. The flux measured with _SOFIA_ at 6.6 $\mu m$ and by $ISO$ from 2.5-to-6 $\mu m$ is also in excess of the predictions of the model, and may be due to additional scattered or thermal emission leaking outwards from warm dust close to the star that resides in a different geometrical component than the disk. A plausible origin of this component may be a dusty wind from the disk (see 5.3.2). There is no discrepancy between the observed luminosity, 1530 $L_{\odot}$, and the modeled stellar luminosity, 2500 $L_{\odot}$. As shown by the bipolar geometry, this object is markedly asymmetric. It is likely that more power emerges perpendicular to the disk, that is, in the plane of the sky along the general direction of the outflow, than is radiated into our direction. In addition, it appears that the lobes may not be optically thick to the heating radiation (see below). These facts could account for the (less than a factor of two) difference between the two luminosity estimates. ### 5.3 Extended Emission #### 5.3.1 Particle Size and Composition The SED and temperature of the emission from the lobes shows that the dust particles are considerably smaller than the heating wavelengths around 1 $\mu m$. This is apparent from Figure 5 and Table 1, which show that the emission from the lobes peaks at a wavelength of 35-to-40 $\mu m$, corresponding to a grain temperature of around 100 $K$. A black particle at a projected distance of 10′′ from the point source (assumed to have a luminosity of 2500 $L_{\odot}$ and to be 1200 $pc$ from Earth) would have a temperature of around 20 $K$. Therefore, the particles in the lobes that produce the emission seen by _SOFIA_ at 19.7 $\mu m$ and beyond have to be small. Based on this observation we have fit the SED of the extended source of M2-9, defined as the total observed flux minus that of the point source. We have used the DUSTY spherically-symmetric dust radiative transfer code (Ivezic et al. 1999) to fit the SED longward of $\sim$ 20 $\mu m$. We assumed a central illuminating blackbody with an effective temperature $T_{b}=5000\,K$ and luminosity $L\sim\,2500\,L_{\odot}$ surrounded by a shell with an $r^{-2}$ radial-density distribution of $\sim$ 0.1 $\mu m$ amorphous carbon particles (dust-type amC in the code). We justify the use of carbon dust, rather than the (oxygen-rich) silicate dust used in the Lyk11 model, later in this section. We have roughly accounted for the non-spherical geometry of the emitting region in our modeling as follows. We approximate the emitting region as covering a solid angle $2\pi$ (instead of the $4\pi$ covered by a spherical shell), and therefore scale the model output flux by a factor 0.5 when fitting to the observed fluxes. Our derived model parameters below are not too sensitive to the geometry because they are constrained by the mid- and far- infrared emission, which is optically-thin. The results of the fit are shown in Figure 5; data from _AKARI_ , _WISE_ , and _IRAS_ are used in addition to the _SOFIA_ data. We find that the grains in the shell are warm, with equilibrium temperatures varying from 95 to 40 K from the inner to outer radius of the shell (i.e., from 5$\farcs$4 to $20{{}^{\prime\prime}}$). The radial optical depth of the shell, in the visible, is $\tau_{V}=1$. The total dust mass of the shell is estimated using Eqn. 2 of Sarkar & Sahai (2008), to be 0.001$M_{\odot}$, assuming $\kappa_{60\,\mu m}$=150 cm2 g-1 (Jura 1986). The shell mass scales linearly with the outer radius. Our model flux falls increasingly below the observed values for wavelengths longer than $\sim$ 70 $\mu m$. However, simply increasing the outer radius is not adequate for decreasing this discrepancy as shown by models with outer radii $>20{{}^{\prime\prime}}$; a population of cooler grains is needed that does not reside in the lobes. We suggest that such grains may reside in the low-latitude regions of the dusty equatorial waist of the nebula, perhaps associated with the molecular rings, and/or beyond their radial extent. We have also not attempted to fit the lobe flux shortward of 20 $\mu m$, where the model fluxes are considerably less than observed. It is possible that this excess is due to single photon heating as described below. Consideration of the 24.2/37.1 $\mu m$ flux ratio dictated our choice of using carbon dust. We found that although we can construct models with silicate dust that reproduce the SED of the extended source in M2-9 just as well as those with carbon dust, the silicate-dust models are not able to produce the observed 24.2/37.1 $\mu m$ flux ratio at the inner radius of the lobes - the model ratio is about 0.22, significantly lower than observed [cf. Figure 6]. This is because the silicate grains at this radius are cooler than carbon grains. Although the model fit to the SED over the range of peak emission from _SOFIA_ looks excellent, further exploration shows that the model is not totally adequate. In Figure 6, we show the 24.2/37.1 $\mu m$ flux ratio as a function of position along the midline of the lobes, moving northwards from the central source. Close to the central source, the observed ratio agrees well with the predictions of the model, decreasing as expected with distance from the star. However, starting at about 8′′ from the source the observed ratio levels off and no further decrease is seen. A similar effect is seen in the 19.7/37.1 $\mu m$ flux ratio. We suggest that this is due to transient heating of the small grains by single photons becoming the dominant heating mechanism in the outer portions of the nebula. This naturally leads to a distance-independent reradiated spectrum. A complete treatment of this idea is beyond the scope of this paper. However, we note that Castelaz, Sellgren, & Werner (1987) show that transient heating is important out at least to 25 $\mu m$ within a few arc minutes of 23 Tau in the Pleiades, at an assumed distance of 125 $pc$. This supports our suggestion that we have observed this phenomenon within 10′′ of a star that has comparable luminosity but is ten times further away. While further modeling would be warranted, we note that the models shown in Figures 4 and 5 provide satisfactory fits to the data at the wavelengths where both the compact source and the lobes emit most of their energy as seen from Earth. However, one further point merits discussion, which is the long wavelength emission seen by SC98 at 1.3 $mm$. Our model flux falls far below this measurement at 1.3 $mm$. Although free-free emission is present both towards the center and in the lobes, it does not dominate the 1.3 $mm$ flux. In the model of free-free emission of this object by Kwok et al. (1985), the spectrum of the lobes turns over at $\sim$ 1 GHz, limiting its contribution to be less than a factor 10 of the measured $mm$-wave flux from the lobes. We conclude, in agreement with SC98, that there is a substantial component of rather cold, large ($>1\,\mu m$) grains in the lobes. SC98 estimate that (adjusted to the 1200 $pc$ distance we have adopted for M2-9) the lobes contain $\sim$ 0.0015 $M_{\odot}$ of cold dust particles with radii of 1.5-to-20 $\mu m$ if the emission is attributed to amorphous carbon. It is noteworthy that this is comparable to the $\sim$ 0.001 $M_{\odot}$ of small particles required to fit the shorter wavelength radiation, as discussed above. SG05, using the IRAS data at wavelengths $\gtrsim$ 25um, derive a mass for the dust producing the emission from the lobes of $\sim$ 0.005 $M_{\odot}$ for “carbon” grains. The five-fold discrepancy with our value of $\sim$ 0.001 $M_{\odot}$ is largely due to the fact that they attribute the emission to graphite grains, and adopt a 5$\times$ lower mass absorption coefficient than the 160 $cm^{2}$ $gm^{-1}$ adopted here for amorphous carbon. The recent PdBI observations by Castro-Carrizo et al. (2012) with a $0\farcs 8\times 0\farcs 4$ beam reveal an unresolved source of 1.3 $mm$ continuum emission with flux 240 mJy associated with the central disk. This is in agreement with the $\sim$ 210 mJy estimated for the central source at this wavelength by SC98. By extrapolating the ionized-wind model of Kwok et al. (1985), we estimate that the contribution of the emission from ionized gas to the core flux at 1.3 $mm$ is about 90 $mJy$ or less. Hence the thermal dust emission from the core is at least 150 $mJy$, and since the (extrapolated) disk model flux at 1.3 $mm$ falls far below this value, we infer that like the lobes, the central region must also contain a substantial population of large grains. This agrees with the increasing observational evidence for the presence of large grains in the central regions of post-AGB objects (Sahai et al. 2011a). #### 5.3.2 Particle Size Distribution It is striking that comparable masses of large (radii $>$ 1 $\mu m$) and small (radii $<$ 0.1 $\mu m$) grains are present in the M2-9 lobes. We speculate that large grains are present in the central disk source (as suggested by the 1.3 $mm$ flux of the central source), and grain-grain collisions between these produce small particles. Both small and large particles are driven out of the disk by radiation pressure by the starlight, forming a disk wind. This mechanism has been proposed by Jura et al. (2001) to explain the far-infrared excesses observed towards the red giant SS Lep. Sputtering of large grains by shocks due to the interaction of high-velocity outflows with slowly-expanding circumstellar material may further enhance the population of small grains in the lobes. The grain size distribution in M2-9 may be far from the equilibrium power law established in a collisional cascade. This is consistent with the short time scales which characterize this source, which has a dynamical age of $\sim$ 2500 years (Co11). #### 5.3.3 Spatial Distribution of the Emission In Figure 1 we show an image of M2-9 in the HST 547 $nm$ filter, which samples continuum emission due, presumably, to scattered light. In Figure 7 we compare scans through the Northern lobe at a position 10′′ N of the point source at 24.2 and 37.1 $\mu m$ with the corresponding scan through the HST 547 $nm$ image. The emission measured with _SOFIA_ shows much less structure and much greater symmetry than seen in the HST image, which shows significant limb brightening, but only on the Eastern edge of the lobe. Thus the thermal emission seen in the mid-infrared is more uniformly distributed than the scattered light seen in the visible. Although one might expect the mid-infrared emission to be produced by dust located exterior to the lobes, where it might be associated with material that confines the outflow, our results show quite clearly that a good fraction – or perhaps all – of the mid-infrared radiation seen by _SOFIA_ arises interior to the lobes as traced by optical images. We adopt a simple model in which the emission arises in an annular cylinder aligned with the observed lobes and convolve the resultant profile with the _SOFIA_ beam to compare with the data. We do a 1-dimensional convolution in the EW direction. The source is uniform enough in the NS direction to make this an appropriate approach. We wish to compare an EW scan across the lobe 10′′ N of the central point source with the predictions of this model. At this position, the HST 547 $nm$ continuum image shows that the FWZI of the observed lobe is 12′′. We take half of this, or 6′′, as the radius of the annular cylinder at this position. We compare the data at 37.1 $\mu m$, averaged over 3 pixels ($\sim$ 2′′) in the NS direction to improve the S/N, with the predictions of the model. For simplicity, we neglect any possible temperature dependence of the emitting material, so we are actually modeling the volume emissivity distribution and assuming that it is equivalent to the dust distribution. The use of the data at 37.1 $\mu m$, our longest and therefore least temperature-sensitive wavelength, should make this an acceptable approximation for an initial calculation, particularly if single-photon heating is important at this wavelength. The _SOFIA_ scans in Figure 7 show two separate peaks along the scan, with a small depression in the middle, particularly at 24.2 $\mu m$ where the resolution is slightly better than at 37.1 $\mu m$. Thus it is obvious that a uniformly filled lobe cannot fit the data; this is shown in Figure 8a, which compares the scan at 37.1 $\mu m$ with the prediction for a uniformly filled lobe. On the other hand, a model in which the dust is confined to the outer regions of the lobe, as might be the case if material is piled up at the interface between the lobe and its exterior environment, also does not fit the data, as is shown in Figure 8b for the case where the dust occupies only the outer 5% of the lobe. Figures 8a and 8b together suggest that a simple linear combination of a uniform dust distribution with one which is concentrated towards the edge of the lobe might provide a good fit to the data. This proves to be the case; in fact several such combinations provide an adequate fit because with a lobe width of $\sim$ 12′′ and a beam width close to 5′′ (cf. Figure 7), we do not have many statistically independent points in the comparison. As one interesting example, we show in Figure 8c a model which combines the distributions shown in Figures 8a and 8b in such a way that 30% of the material lies in a uniform distribution all the way to the edge of the lobe while an additional 70% is confined to the outer 5% of the lobe. Assuming that the dust and gas are well-mixed, this model could be consistent with the limb brightening seen in some of the optical emission lines, as the density in the outer, narrow annulus would be about 20 times that in the central regions. Note, however, that a model in which the outer emission is confined to a narrow annulus exterior to the visible wavelength lobe provides an equally good fit to the data; the implications of having the increased dust density exterior to the visible lobe are substantially different from those of having the increase interior to the lobe. Higher resolution observations, perhaps from JWST, will be required to distinguish between these possibilities. The basic conclusion of this discussion – that an appreciable fraction of the infrared emission comes from well inside the lobes, implying as well that the dust is similarly distributed – is well-established, however. #### 5.3.4 Comparing Visible and Infrared Images Large Scale Morphology The connection – or lack of connection – between the dust producing the scattered light at visible wavelengths and that producing the infrared radiation is puzzling. Although the limb brightening on the Eastern edge of the lobe at 547 $nm$ (Figure 7c) might be consistent with the model shown in Figure 8c, a similar brightening expected from the dust distribution is not seen on the Western edge; there is no evidence in the symmetrical infrared images for a preferential brightening of the Eastern limb of the lobe. The time interval between the HST 547 $nm$ image and our SOFIA measurements is about 14 years. It is possible that the bright region has moved away from the limb in a manner similar to the motion of the features in the emission line images presented by Co11. However, with an overall period $\sim$ 90 years, during this 14 year interval the bright region would have moved (in projection) less than half of the distance to the center line of the lobe. It should thus be visible near the Eastern edge of the infrared scan if it were as bright relative to the Western half of the lobe in the infrared as it is in the visible. The average 547 $nm$ surface brightness of the Western half of the lobe shown in Figure 7c is about 20 $\mu Jy\,arcsec^{-2}$, while that at 37.1 $\mu m$ is about 170 $mJy\,arcsec^{-2}$. The corresponding power [$\nu F_{\nu}$] of the infrared radiation is almost 100 times that of the visible, suggesting that the scattering grains in the West have very low net albedo. Note that we are explicitly assuming at this point that the starlight absorbed by these grains heats them to produce the radiation seen by _SOFIA_ , while that scattered is seen at 0.547 $\mu m$ by HST and that the stellar temperature is about 5000 $K$ as suggested by Table 2. This observation suggests a possible explanation for the apparent decoupling of the infrared and visible light distributions at the Eastern limb. Because of the broad distribution of dust particle sizes in M2-9, it is possible that the scattered light at the Eastern limb comes from an admixture of grains – perhaps larger and considerably colder than those seen to the West – that scatter very effectively, increasing the visible brightness with little impact at _SOFIA_ wavelengths. It is also conceivable that the marked asymmetry in the scattered light as compared to the high degree of symmetry shown in the infrared images could be due to a foreground absorbing cloud with $\tau_{V}$ $\sim$ 1 which happens to bisect the nebula. Apart from the improbability of such an alignment, however, there is no evidence for such variable extinction in the HST line emission image shown in Figure 1. We can further investigate the issue of the low observed optical surface- brightness, compared to that observed with SOFIA at long wavelengths (e.g., 37.1 $\mu m$) using our DUSTY modeling. Our model (described in 5.3.1) gives a 37.1 $\mu m$ surface brightness of $S(37.1\,\mu m)=175\,mJy\,arcsec^{-2}$ at a radial offset of $10{{}^{\prime\prime}}$, in good agreement with the observed value, but predicts the optical surface brightness is $S(0.55\,\mu m)=0.65\,mJy\,arcsec^{-2}$, much higher than observed. In order to check that this discrepancy is not simply a problem associated with the optical imaging (e.g., its calibration), we have examined near-infrared images of M2-9, and find that a discrepancy exists there as well. Hora & Latter (1994) present ground-based spectroscopy and imaging of M2-9 in multiple filters in the near-infrared. Their 2.26 $\mu m$ filter image indicates a total (line+continuum) surface brightness S(2 $\mu m$)$\sim\,1.5\times 10^{-4}\,Jy\,arcsec^{-2}$ at a distance of $10{{}^{\prime\prime}}$, and although their spectrum shows that there is weak line emission included within the filter bandpass, there is a weak continuum present as well. We also found an archival near-IR HST image, obtained with NICMOS (NIC2) using the F215N filter on 1998 May 19, as part of GO program 7365 (PI: W. B. Latter). This filter spans 2.14 to 2.16 $\mu m$, and only has a very weak H2 line within this range. Using the HST pipeline photometry from the image file header, we find S(2 $\mu m)\,\sim\,3\times 10^{-4}\,Jy\,arcsec^{-2}$, at a radial offset of $10{{}^{\prime\prime}}$ offset in the N-lobe. The model-predicted value of S(2 $\mu m$) is $0.24\times 10^{-4}\,Jy\,arcsec^{-2}$, i.e., significantly lower than either the ground- based or the HST value. Hence the observed optical and near-IR surface brightnesses are discrepant from the model ones, but in opposite directions, suggesting that the radiation heating the grains in the lobes is redder than the 5000 $K$ of our standard model. This reddening of the heating radiation can be achieved in two ways (i) assuming a lower value of $T_{eff}$ for the central star, and (ii) increasing the extinction in the inner region of the model dust shell. We have computed models to examine both these effects and find that by lowering $T_{eff}$ from $5000\,K$ to $3000\,K$ and raising $\tau_{V}$ from 1 to 3, we obtain $S(0.55\,\mu m)=35\,\mu Jy\,arcsec^{-2}$ and $S(2\,\mu m)=1.2\times 10^{-4}\,Jy\,arcsec^{-2}$, in better agreement with their observed values. Both of the above changes can be accommodated in our models of the central source and the lobes. The central disk model is not very sensitive to the adopted value of $T_{eff}$ of the central star. An increase in $\tau_{V}$, together with no significant change in the total far-infrared model fluxes and the input luminosity, can be achieved with a decrease in the solid angle of the dust shell subtended at the center, by a factor 3 from its value of 2$\pi$ in our standard model – such a decrease is not unreasonable (and may in fact be desirable), given that the lobes in M 2-9 are very highly collimated. In this scenario as well, the increased brightness at the Eastern limb suggests an additional population of larger and colder grains. Infrared Detection of the Optical Knots As illustrated most recently by Co11, the optical images of M2-9 show persistent structures in the form of knots and arcs. Most pronounced are the knots N3 and S3 which lie along the center line about 15′′ N and S of the central source (Figure 1). Although their morphologies have varied somewhat with time, these knots persist over the 1999-to-2010 time period sampled by Co11 and can also be seen as far back as the images presented by Allen and Swings (1972). We have searched for these knots by examining scans along the axis of the outflow; the scans at 19.7, 24.2, and 37.1 $\mu m$ are shown in Figure 9. Both N3 and S3 are seen very clearly at 19.7 $\mu m$, approximately equidistant from the central source. The separation of the two knots is about 28′′, very close to the 29′′ estimated by eye for these somewhat diffuse structures from the Co11 images. The brighter knot, N3, is seen less strongly at 24.2 $\mu m$ than at 19.7 $\mu m$, but neither is seen at 37.1 $\mu m$. Because the knots are seen very strongly in visible emission lines of OIII, NII, and HI, it is tempting to conclude that the enhancement at 19.7 $\mu m$ is due to localized emission from the SIII line at 18.7 $\mu m$, which falls well within the broad 19.7 $\mu m$ filter. This line is very strong in the _Spitzer_ spectrum of the Northern Lobe, which included the knot N3 in the slit; there might also be a contribution from an FeII line at 18 $\mu m$. However, the flux in the narrow 18.7 $\mu m$ line over the entire $4.7{{}^{\prime\prime}}\times 11.3{{}^{\prime\prime}}$ _Spitzer_ slit (Lyk11) is about an order of magnitude less than the excess flux seen over the broad 19.7 $\mu m$ filter in the feature $\sim$ 14′′ North of the central source. Thus another cause, possibly related to a localized population of very small grains at the position of the knot, or perhaps to mechanical heating of the dust grains if the knots are produced by shocks where highly collimated outflows impact slowly moving ambient material, must be sought for the 19.7 $\mu m$ excess. ### 5.4 M2-9 and PN Shaping Jet-sculpting models for producing bipolar PNe (e.g., Lee & Sahai 2003) require a fast, collimated outflow expanding inside a pre-existing AGB circumstellar envelope (CSE), resulting in collimated lobes with dense walls, as seen in M2-9 and other bipolar PNe. The true nature of M2-9 has been debated, i.e., whether or not it is a normal PN (i.e., an object in which the optically-visible nebula seen prominently in forbidden line emission, consists of matter ejected by a central star during its AGB phase, that is then photoionized by the same star after it has evolved off the AGB and become much hotter). The current consensus appears to be that M2-9 is not a normal PN, but that its central star is an AGB or young post-AGB star with a hot white-dwarf companion (e.g., Livio & Soker 2001), and thus represents the “long-period interacting binary” evolutionary channel for the formation of PN-like nebulae (Frew & Parker 2010). In either case, one would expect to see the presence of the mass ejected during the AGB phase, in the form of a circumstellar envelope around the bipolar lobes. So it appears somewhat surprising that we have seen no direct observational evidence of the AGB CSE in M2-9 so far. However, our inability to detect material outside the lobes in M2-9, e.g., via scattered light at optical wavelengths, or thermal emission from dust, or molecular-line emission, may simply be a sensitivity issue, if the AGB mass-loss rate in M2-9 was relatively low. Note that the lateral expansion of the lobes is set by the sound speed in ionized gas ($\sim$ 10 $km/s$ at 104 K), which is small compared to the much higher axial speed of the lobes (145 $km/s$, Co11), hence even if the confining pressure of the ambient medium is relatively low for a low AGB mass-loss rate, the lobes would maintain their collimated shapes. The extensive optical imaging survey of young PNe with HST by Sahai et al. (2011b) found faint halos in a significant fraction, but not all, of their sample. Deep optical imaging of M2-9 with HST would be very useful. Molecular material in the halo may be difficult to detect since photodissociation by the general interstellar UV field might be significant, especially if the AGB mass-loss rate was relatively low. ### 5.5 Directions for Further Work The main new results of this paper refer to the spatial and particle size distribution of the dust seen in the thermal infrared and at mm-wavelengths. Along the way, however, we have identified several additional areas where the work to date poses interesting, unanswered questions: Firstly, the identification of transient heating of small particles as an important contribution to the radiation from the lobes beyond $\sim\,10{{}^{\prime\prime}}$ from the central source calls for an analysis of the extended emission which would go beyond the simple DUSTY model reported here and include transient heating and explore a range of grain materials and sizes. Such modeling could also address the uncertainty in the stellar temperature discussed in 5.3.4; it would also be appropriate to use cylindrical coordinates in this improved analysis. Secondly, we note that the model for the central compact source given in Table 2 is based on silicate grains rather than the amorphous carbon which we chose to describe the grains in the lobes. This dichotomy is consistent with the fact that the spectra of M2-9 show silicate absorption in the central source and PAH emission in the lobes (Lyk11). It therefore appears that M2-9 belongs to the well-known subclass of post-AGB objects that have been labeled as having “mixed-chemistry” (e.g., Morris 1990; Waters et al. 1998a,b; Cohen et al. 1999, 2002). Since, during the AGB phase, a star may evolve from being oxygen rich to being carbon rich (due to the 3rd dredge up), a popular hypothesis for this phenomenon is that the disk formed (e.g., by gravitational capture of the stellar wind around a close companion) when the central star was still oxygen rich, whereas the extended emission is due to a more recent carbon-rich outflow. But a difficulty with applying this hypothesis to M2-9 is that the gaseous nebula in this object is known to be O-rich, with C/O $<$ 0.5 (Liu et al. 2001). Guzman-Ramirez et al. (2011) propose an alternative hypothesis, based on the strong correlation between the presence of a dense torus and mixed-chemistry in their sample of 40 objects. They argue that the popular hypothesis cannot explain the widespread presence of the mixed-chemistry phenomenon among PNe in the Galactic bulge (Perea-Calderon et al. 2009, Guzman-Ramirez et al. 2011), as these old, low-mass stars should not go through the 3rd dredge-up. They suggest that the mixed-chemistry phenomenon in Galactic bulge planetary nebulae may be due to hydrocarbon chemistry in an UV-irradiated, dense torus that produces long-carbon chain hydrocarbons that then produce PAHs. Given that PAH features have been observed in the lobes of M2-9, it is plausible that the UV-irradiation hypothesis is responsible for the presence of small carbon-rich grains in the lobes. We suggest that the new data presented here on the spatial and size distributions of the grains, together with our suggestions concerning large grains in both the lobes and the disk, make M2-9 a detailed astrophysical laboratory for further study of the processes which produce these mixed-chemistry objects. ## 6 CONCLUSIONS We have presented and analysed images of M2-9 with $\sim$ 4′′-to-5′′ resolution in six infrared bands at wavelengths between 6.6 and 37.1 $\mu m$. The principal new results from these _SOFIA_ observations of M2-9 center around the spatial and size distribution of the grains which produce the infrared radiation from the outflow lobes in this bipolar nebula. The spatial distribution of the emission implies that the lobes are fairly uniformly filled with dust, with a marked increase in the dust density in a relatively narrow cylindrical annulus – with width order 5% of the lobe radius – at the outer edge of the lobes. We caution that the spatial resolution of the _SOFIA_ observations, in comparison with the width of the lobes, does not permit the model parameters to be pinned down definitively; for example, we can not determine whether this outer annulus lies within or exterior to the optically visible lobes. However, the result that dust is well mixed over the interior of the lobe is well-established. The side to side asymmetry seen in the HST continuum image of M2-9 is not seen in the infrared images, although it would have been apparent at _SOFIA_ ’s resolution. The reason for this puzzling discrepancy is unclear, but it may be related to the range of grain sizes which characterize this source; the M2-9 lobes have comparable masses of particles with radii $>1$ $\mu m$ and with radii $<0.1$ $\mu m$. The other principal results of this work are: 1\. At wavelengths from 6.6 to 37.1 $\mu m$, the image of M2-9 is dominated by a bright central point source, which is not definitely resolved at any wavelength with _SOFIA_ ’s $\sim$ 4-to-5′′ beams. 2\. The extended bipolar lobes are clearly seen at wavelengths from 19.7 to 37.1 $\mu m$; the integrated emission from the lobes is detected down to 6.6 $\mu m$. 3\. The infrared emission of the point source at wavelengths between 11.1 and 24.2 $\mu m$ agrees extremely well with the predictions of a disk model based on Lyk11. The fit suggests a distance of 1200 $pc$ to M2-9 and a luminosity of 2500 $L_{\odot}$ for the star which powers it, although the distance and hence the luminosity are not well-constrained. Emission from the point source in excess of the model is seen shortward of 11.1 $\mu m$, likely contributed by warm dust close to the star that resides in a different geometrical component than the disk, such as the inner part of a disk wind. Assuming isotropic emission, we find that the total 2.5-40 $\mu m$ infrared luminosity of the compact source is $\sim$ 840 $L_{\odot}$, and that of the lobes $\sim$ 390 $L_{\odot}$. The total observed 2.5-120 $\mu m$ luminosity of M2-9, including the $IRAS$ measurements at longer wavelengths, and assuming isotropic emission, is $\sim$ 1530 $L_{\odot}$. Because the emission is clearly not isotropic, this is not inconsistent with the 2500 $L_{\odot}$ inferred from the disk model. The _SOFIA_ photometry agrees well with that obtained from other platforms, including $ISO$, $WISE$ and $IRAS$. This work shows that compact planetary nebulae are ideal targets for study from _SOFIA_ , not only photometrically but with other capabilities, most notably grism spectroscopy, now becoming available on this new airborne observatory. We thank the staff and crew of the _SOFIA_ observatory for their support in carrying out these observations at an epoch when _SOFIA_ was still in its development phase. We thank Bruce Balick for encouragement and useful discussions and comments, and the referee for a very useful report. Portions of the work were carried out at the Jet Propulsion Laboratory operated by the California Institute of Technology under a contract with NASA. J. Davis was the Charles and Valerie Elachi SURF Fellow, under Caltech’s Summer Undergraduate Research Fellow program, while working on this project. 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Table 1: M2-9 Photometry and Size Data Wavelength \| Flux \| FWHM \| Beam Size ---\|---\|---\|--- $\mu$m \| Jy \| arcsec \| arcsec \| Point Source \| Total Flux \| Extended Component \| \| 3 \| 4.6 \| \| \| \| 3.7 \| 7.9 \| \| \| \| 4.5 \| 15 \| \| \| \| 6.6 \| 24 \| 30.2 \| 6.2 \| 3.7 \| 3.68 11.1 \| 32 \| 46.9 \| $14.9\pm 5.4$ \| 3.7 \| 3.85 19.7 \| 58 \| 87.5 \| 29.5 \| 3.9 \| 3.76 24.2 \| 55 \| 93.4 \| 38.9 \| 4.1 \| 4.19 33.6 \| 63 \| 157.7 \| 94.7 \| 4.5 \| 4.42 37.1 \| 48 \| 138.4 \| 90.4 \| 4.9 \| 4.51 12 \| \| 50 \| \| \| 25 \| \| 110 \| \| \| 60 \| \| 123 \| \| \| 100 \| \| 76 \| \| \| 1300 \| 0.21 \| 0.36 \| 0.15 \| \| Note. — Infrared photometry of M2-9. Data from 6.6 to 37.1 $\mu$m, is from this paper. 3-4.5 $\mu$m data is from Lyk11. The central source is unresolved at these wavelengths by _ISO_ ’s 1′′ to 2′′ PSF. Also included is 12-100 $\mu$m data from the _IRAS_ Point Source Catalog (beam size $\gtrsim$ 1′) and the 1.3 $mm$ measurements of SC98. The right two columns compare the FWHM of the compact central source in our _SOFIA_ images with the beam size determined for our flight series by the _SOFIA_ Science Center. The quoted ($2\sigma$) uncertainties in this beam size are $>10$% at all wavelengths. Error determined from variations in brightness of reference positions. For all other _SOFIA_ measurements, the statistical errors are smaller than the $\sim$ 20% calibration uncertainty. Table 2: M2-9 – Best-Fit Parameters for Central Source Model Parameter \| Value \| Comment ---\|---\|--- Disk inner radius \| $15\pm 1$ $au$ \| Disk outer radius \| $800\pm 100$ $au$ \| Not well constrained Mass of dust \| $1\pm 0.1\times 10^{-5}$ $M_{\odot}$ \| Draine and Lee astronomical silicates, sizes 0.01-to-5 $\mu m$. \| \| (Mathis et al. 1977 size distribution) $\alpha$ \| $2.2\pm 0.05$ \| Density within disk falls radially as $r^{-\alpha}$ $\beta$ \| $1.23\pm 0.02$ \| Scale height varies radially as $r^{\beta}$. Disk is flared $h_{100}$ \| $37\pm 3$ $au$ \| Fiducial scale height at 100 $au$ $T_{eff}$ \| 5000 $K$ \| Temperature of central star doing most of the heating $L$ \| 2500 $L_{\odot}$ \| Luminosity of central star Note. — The disk model assumed here can be described by the following density law, described further in Lyk11, where $r$ is the radial and $z$ the vertical coordinate in a cylindrical coordinate system centered on the star, and $R_{}$ is the radius of the star: $\rho(r,z)=\rho_{0}(\frac{R_{}}{r})^{\alpha}exp(-\frac{1}{2}(\frac{z}{h(r)})^{2})$; $h(r)=h_{0}(\frac{r}{R_{*}})^{\beta}$ Figure 1: Images of M2-9 at six _SOFIA_ wavelengths, augmented with data from _HST_ in the two right hand panels. The images are oriented with N to the top and E to the left. The rectangle on the 33.6 $\mu m$ image is 20${{}^{\prime\prime}}\times 40{{}^{\prime\prime}}$ in size. The dotted lines 10′′ to the N of the central point source trace the path along which the simulations described in the text were calculated. The HST data, 0.547 $nm$ in the upper right, and a multiband optical emission line image in the lower right, were taken in 1997. Figure 2: N-S scans through the M2-9 point source at 19.7 [left] and 37.1 $\mu m$. The data are averaged over 5 pixels ($\sim$ 3.8′′) in the E-W direction at each wavelength. Figure 3: 2.5-120 $\mu m$ SED of M2-9 including data from _ISO_ , _IRAS_ , and _SOFIA_. Figure 4: _SOFIA_ and _ISO_ photometry of the central point source of M2-9 compared with the predictions of the model described in Table 2. Figure 5: Observed (symbols) and model (curve) SED of the M2-9 lobes. The photometric data shown are as follows: filled squares – _SOFIA_ , open squares – _IRAS_ , open triangles – _WISE_ , open circles – _AKARI_ , filled triangle – _IRAM_ 30m. In all cases, the photometry has been corrected for the point source contribution based on the model shown in Figure 4. Figure 6: 24.2-to-37.1 $\mu m$ ratio as a function of position N of the central point source. Solid line – Data, averaged over 11 pixels in the E-W direction. Dotted line – Prediction of the DUSTY model shown in Figure 5. Figure 7: Normalized E-W scans 10′′ N of the point source through the images of M2-9 at 24.2 (a) and 37.1 (b) $\mu m$. Overlaid on each _SOFIA_ scan is the beam profile at that wavelength. For comparison, a similar scan through the 547 $nm$ HST image is shown in panel (c). The 37.1 $\mu m$ data is averaged over three N-S pixels, and the HST data over 20 N-S pixels (the HST pixels are $\sim$ $0.1{{}^{\prime\prime}}\times 0.1{{}^{\prime\prime}}$ in size). The relative uncertainty in the _SOFIA_ data is just a few percent, as shown by the small point-to-point scatter. Figure 8: Three models for the dust distribution in the Northern lobe are compared with the data at 37.1 $\mu m$. Panel (a): a uniform dust distribution which fills the entire lobe. Panel (b): the dust is confined radially to the outer 5% of the optically visible lobe. Panel (c): 30% of the dust is in the uniform distribution and 70% is confined to the outer 5% of the lobe. Note that what is actually modeled is the volume emissivity, which should be very close to the dust distribution as described in the text. Figure 9: N-S scans along the axis of the M2-9 lobes [cf. Figure 2] are displayed on an expanded scale. In all cases the data are averaged over 5 pixels in the E-W direction. Panel a. (top) shows that the knots $\sim$ 15 arcsec N and S of the central point source are detected at 19.7 $\mu m$ but not at 37.1 $\mu m$; only the northern knot is detected at 24.2 $\mu m$. Panel b. (bottom) shows the 19.7 $\mu m$ scan in units of $Jy/pixel$ to illustrate the absolute brightness of the emission.	arxiv-papers	2013-11-20T04:00:49	2024-09-04T02:49:53.925682	{ "license": "Public Domain", "authors": "M. W. Werner, R. Sahai, J. Davis, J. Livingston, F. Lykou, J. de\n Buizer, M. R. Morris, L. Keller, J. Adams, G. Gull, C. Henderson, T. Herter,\n J. Schoenwald", "submitter": "John Livingston", "url": "https://arxiv.org/abs/1311.4949" }
1311.4997	# No universal group in a cardinal Saharon Shelah Einstein Institute of Mathematics Edmond J. Safra Campus, Givat Ram The Hebrew University of Jerusalem Jerusalem, 91904, Israel and Department of Mathematics Hill Center - Busch Campus Rutgers, The State University of New Jersey 110 Frelinghuysen Road Piscataway, NJ 08854-8019 USA [email protected] http://shelah.logic.at (Date: April 21, 2014) ###### Abstract. For many classes of models there are universal members in any cardinal $\lambda$ which “essentially satisfies GCH, i.e. $\lambda=2^{<\lambda}$”. But if the class is “complicated enough”, e.g. the class of linear orders, we know that if $\lambda$ is “regular and not so close to satisfying GCH” then there is no universal member. Here we find new sufficient conditions (which we call the olive property), not covered by earlier cases (i.e. fail the so-called ${\rm SOP}_{4}$). The advantage of those conditions is witnessed by proving that the class of groups satisfies one of those conditions. ###### Key words and phrases: model theory, universal models, the olive property, group theory, non- structure, classification theory ###### 2010 Mathematics Subject Classification: Primary: 03C55, 20A15; Secondary: 03C45, 03E04 The author thanks the Israel Science Foundation for partial support of this work. Grant No. 1053/11. The author thanks Alice Leonhardt for the beautiful typing. First typed March 27, 2013. Publication 1029. Anotated Content §0 Introduction, (labels y,z), pg. ‣ 0\. Introduction §1 The olive property, (label d), pg.1 1. [We give definitions of some versions of the olive property and give an example failing the ${\rm SOP}_{4}$. We phrase relevant set theoretic conditions like ${\rm Qr}_{1}$ (slightly weaker than those used earlier). Then we give complete proof using ${\rm Qr}_{1}(\chi_{2},\chi_{1},\lambda)$ to deduce ${\rm Univ}(\chi_{1},\lambda,\mathbb{k})\geq\chi_{2}$ so no universal in the class $\mathbb{k}$ in the cardinal $\lambda$, when $\mathbb{k}$ has the olive property.] §2 The class of groups have the olive property, (label s), pg.2 1. [We prove the stated result. We also deal with the non-existence of universal structures for pairs of classes, e.g. the pair (locally finite groups, groups).] §3 Concluding Remarks, (label m), pg. 3 1. [We consider some generalizations of the properties, but no clear gain.] ## 0\. Introduction ### 0(A). Background and open questions On history see Kojman-Shelah [KjSh:409] and later Dzamonja [Mir05]. Recall that if $\lambda=2^{<\lambda}>\aleph_{0}$ then many classes have a universal in $\lambda$, so assuming GCH, we know when there is a universal model in every $\lambda>\aleph_{0}$. For transparency we consider a first order countable $T$. Recall that on the one hand Kojman-Shelah [KjSh:409] show that if $T$ is the theory of dense linear orders or just $T$ has the strict order property, then $T$ fails (in a strong way) to have a universal in regular cardinals in which cardinal arithmetic is “not close to GCH”; (for regular $\lambda$ this means there is a regular $\mu$ such that $\mu^{+}<\lambda<2^{\mu}$, for singular $\lambda$ we need of course $\lambda<2^{<\lambda}$ and a very weak pcf condition). By [Sh:500], we can weaken “the strict order property” to the 4-strong order property ${\rm SOP}_{4}$. Natural questions are (we shall address some of them): ###### Question 0.1. 1) Is there a weaker condition (on $T$) than ${\rm SOP}_{4}$ which suffice? 2) Can we find a best one? 3) Can we find such a condition satisfied for some theory $T$ which is ${\rm NSOP}_{3}$? ###### Question 0.2. 1) Is there $T$ with the class ${\rm Univ}(T)\backslash(2^{\aleph_{0}})^{+}$ strictly smaller than the one for linear order, see 0.12(3); we better restrict ourselves to regular cardinals above $2^{\aleph_{0}}$? 2) Can we get the above to be $\\{\lambda:\lambda=2^{<\lambda}\\}$? 3) What about singular cardinals? ###### Question 0.3. 1) Is it consistent that the class of linear order has a universal member in $\lambda$ such that $2^{<\lambda}>\lambda>2^{\aleph_{0}}$ (for $\lambda=\aleph_{1}<2^{\aleph_{0}}$, yes, [Sh:100]). 2) Similarly for some theory with ${\rm SOP}_{4}$ or the olive property. Recall that by Shelah-Usvyatsov [ShUs:789] the class of groups has ${\rm NSOP}_{4}$ but has ${\rm SOP}_{3}$, so it was not clear where it stands. ###### Question 0.4. 1) Where does the class of group stand (concerning the existence of a universal member in a cardinal)? 2) Is it consistent that there is a universal locally finite group of cardinality $\aleph_{1}$? of cardinality $\beth_{\omega}$? of other cardinality $\lambda<\lambda^{\aleph_{0}}$? Recall (Grossberg-Shelah [GrSh:174]) if $\mu$ is strong limit of cofinality $\aleph_{0}$ above a a compact cardinal, then there is a universal locally finite group of cardinality $\mu$ but if $\mu=\mu^{\aleph_{0}}$ then there is no one. Concerning singulars ###### Question 0.5. Does $\theta={\rm cf}(\theta)$ and $\theta^{+2}<{\rm cf}(\lambda)<\lambda<2^{\theta}$ implies $\lambda<{\rm univ}(\lambda,T)$? ###### Question 0.6. 0) Characterize the failure of the criterion of [Sh:457], Džamonja-Shelah [DjSh:614](for consistency). 1) Does ${\rm SOP}_{3}$ (or something weaker) suffice for no universal in $\lambda$ when $\mu=\mu^{<\mu}\ll\lambda<2^{\mu}$? 2) Which theories $T$ fails to have a universal in $\lambda$ when $\lambda=\mu^{++}=2^{\mu}<2^{\mu^{+}}$. 3) Weaker properties of $T$ for no universal in $\lambda,\mu=\mu^{<\mu}\ll\lambda<2^{\mu}$. 4) Sort out the variants of the olive property. ###### Discussion 0.7. The case $\lambda=\mu^{+},\lambda<2^{\mu}$ and $2^{<\mu}\leq\lambda$ (e.g. for transparency $\mu=\mu^{<\mu}$) is not resolved as we do not necessarily have $\bar{C}=\langle C_{\delta}:\delta\in S^{\lambda}_{\mu}\rangle$ guessing clubs. Earlier if $\mu=2^{\kappa}$, so $\mu$ not strong limit, in the case of failure there was a sequence $\langle\Lambda_{\delta}:\delta\in S^{\lambda}_{\mu}\rangle,\Lambda_{\delta}\subseteq{}^{(C_{\delta})}\mu$ of cardinality $\lambda$ such that for every sequence $\langle\eta_{\delta}\subseteq{}^{(C_{\delta})}\mu:\delta\in S^{\lambda}_{\mu}\rangle$ for some club $E$ of $\lambda$ for every $\delta\in E\cap S^{\lambda}_{\mu}$ for some $\nu\in\Lambda_{\delta}$ the functions $\eta_{\delta},\nu$ agree on $E\cap{\rm nacc}(C_{\delta})$. Using more complicated $T$ we can replace ${}^{C_{\delta}}\mu$ by ${}^{(C_{\delta}\times D_{\delta})}\mu$ so the agreement above is on $(E\cap{\rm nacc}(C_{\delta}))\times(E\cap{\rm nacc}(C_{\delta}))$ but of unclear value. See lately [Sh:F1330], more on consistency (after the present work) see [Sh:F1414] on 0.2, and more on ZFC results in [Sh:F1425]. ### 0(B). What is accomplished What do we achieve? We introduce the “olive property” which suffice for the class to have a universal member in $\lambda$ only if $\lambda$ is “close to satisfying G.C.H.”, similarly to the linear order case. This condition is weaker than ${\rm SOP}_{4}$, hence gives a positive answer to 0.8(1). But the condition implies ${\rm SOP}_{3}$ so it does not answer 0.8(3), also it is totally unclear whether it is best in any sense and whether its negation has interesting consequences. However, it answers 0.4(1) to a large extent because the class of groups has the olive property and we can also deal with locally finite groups; see §2. Also we try to formalize conditions sufficient for non-existence, see 1.6 and see more in §3. As the reader may find the definition of the (variants of the) olive property opaque, we define a simple case used for the class of groups, and the reader then may look first at the class of groups in §2. ###### Definition 0.8. A (first order) universal theory $T$ has the olive property when there are $(\varphi_{0},\varphi_{1},\psi)$ and model ${\mathfrak{C}}$ of $T$ such that: 1. $(a)$ for some $m,\varphi_{0}=\varphi_{0}(\bar{x}_{[m]},\bar{y}_{[m]}),\varphi_{1}=\varphi_{1}(\bar{x}_{[m]},\bar{y}_{[m]}),\psi=\psi(\bar{x}_{[m]},\bar{y}_{[m]},\bar{z}_{[m]})$ are quantifier free formulas (and $\bar{x}_{[m]},\bar{y}_{[m]},\bar{z}_{[m]}$ are $m$-tuples of variables, see 0.10 below) 2. $(b)$ for every $k$ and $\bar{f}=\langle f_{\alpha}:\alpha<k\rangle,f_{\alpha}$ is a function from $\alpha$ to $\\{0,1\\}$ we can find $\bar{a}_{\alpha}\in{}^{m}{{\mathfrak{C}}}$ for $\alpha<k$ such that: 1. $(\alpha)$ $\varphi_{\iota}[\bar{a}_{\alpha},\bar{a}_{\beta}]$ for $\alpha<\beta<k$ when $\iota=f_{\beta}(\alpha)$ 2. $(\beta)$ $\psi[\bar{a}_{\alpha},\bar{a}_{\beta},\bar{a}_{\gamma}]$ when $\alpha<\beta<\lambda$ and $f_{\gamma}{\restriction}[\alpha,\beta]$ is constantly 0 3. $(c)$ there are no $\bar{a}_{\ell}\in{}^{m}{\mathfrak{C}}$ for $\ell=0,1,2,3$ such that the following are111in the class of groups, in clause $(\alpha),\varphi_{0}[\bar{a}_{0},\bar{a}_{1}),\varphi_{1}[a_{1},a_{2}],\varphi_{1}[a_{1},\bar{a}_{3}]$ suffice satisfied in ${\mathfrak{C}}$ 1. $(\alpha)$ $\varphi_{0}[\bar{a}_{0},\bar{a}_{\ell}]$ for $\ell=1,2,3,\varphi_{1}[\bar{a}_{1},\bar{a}_{\ell}]$ for $\ell=1,2$ and $\varphi_{0}[\bar{a}_{2},\bar{a}_{3}]$ 2. $(\beta)$ $\psi[\bar{a}_{0},\bar{a}_{2},\bar{a}_{3}]$ ###### Concluding Remarks 0.9. Concerning some things not addressed here. 1) Concerning the proof here of “there is no universal” we can carry it via defining invariants parallel to Kojman-Shelah [KjSh:409] such that (for transparency $\lambda$ is regular uncountable, see 0.11(5),(7)) 1. $()$ $(a)\quad$ if $M\in{\rm Mod}_{T,\lambda}$ then ${\rm INV}_{\lambda}(M)$ is a set of cardinality $\leq\lambda$ or just $\leq\chi<2^{\lambda}$ 2. $(b)\quad$ if $M_{1},M_{2}\in{\rm Mod}_{T,\lambda}$ and $M_{1}$ is embeddable into $M_{2}$ then ${\rm INV}_{\lambda}(M_{1})\subseteq{\rm INV}_{\lambda}(M_{2})$ 3. $(c)\quad$ there is a set of $2^{\lambda}$ objects $\mathbb{x}$ such that $(\exists M\in{\rm Mod}_{T,\lambda})(\mathbb{x}\in{\rm INV}_{\lambda}(M))$. 2) We can use more complicated versions of the olive property. In the proof we use one $\delta$ and then one $\alpha\in{\rm nacc}(C_{\delta})\cap E$ (or less), but we may use several $\alpha$’s getting more complicated versions. This will become more pressing if we have a complimentary property, guaranteeing “no universal” or some variant. ### 0(C). Preliminaries ###### Notation 0.10. 1) Let $\bar{x}_{[I]}=\langle x_{t}:t\in I\rangle$ and similarly $\bar{y}_{[I]},\bar{x}_{[I],\alpha}$, etc. and $\bar{x}_{[I],\ell}=\langle x_{t,\ell}:t\in I\rangle$. 2) For a first order complete $T,{\mathfrak{C}}_{T}$ is the “monster model of $T$”. ###### Definition 0.11. 1) For a set $A,\|A\|$ is its cardinality but for a structure $M$ its cardinality is $\\|M\\|$ while its universe is $\|M\|$; this apply e.g. to groups. 2) We use $G,H$ for groups, $M,N$ for general models. 3) Let ${\mathfrak{k}}$ denote a pair $(K_{{\mathfrak{k}}},\leq_{{\mathfrak{k}}})$, may say a class ${\mathfrak{k}}$, where: 1. $(a)$ $K_{{\mathfrak{k}}}$ is a class of $\tau_{{\mathfrak{k}}}$-structures where $\tau_{{\mathfrak{k}}}$ is a vocabulary 2. $(b)$ $\leq_{{\mathfrak{k}}}$ is a partial order on $K_{{\mathfrak{k}}}$ such that $M\leq_{{\mathfrak{k}}}N\Rightarrow M\subseteq N$ 3. $(c)$ both $K_{{\mathfrak{k}}}$ and $\leq_{{\mathfrak{k}}}$ are closed under isomorphisms. 4) We say $f:M\rightarrow N$ is a $\leq_{{\mathfrak{k}}}$-embedding when $f$ is an isomorphism from $M$ onto some $M_{1}\leq_{{\mathfrak{k}}}N$. 5) If $T$ is a first order theory then ${\rm Mod}_{T}$ is the pair $(\text{Mod}_{T},\leq_{T})$ where $\text{ mod}_{T}$ is the class of models of $T$ and $\leq_{T}$ is: $\prec$ if $T$ is complete, $\subseteq$ if $T$ is not complete. 6) We may write $T$ instead of ${\rm Mod}_{T}$, e.g. in Definition 0.12 below. 7) For a class $K$ of structures $K_{\lambda}=\\{M\in K:\\|M\\|=\lambda\\}$. ###### Definition 0.12. 1) For a class ${\mathfrak{k}}$ and a cardinal $\lambda$, a set $\\{M_{i}:i<i^{}\\}$ of models from $K_{{\mathfrak{k}}}$, is jointly $(\lambda,{\mathfrak{k}})$-universal when for every $N\in K_{{\mathfrak{k}}}$ of size $\lambda$, there is an $i<i^{}$ and an $\leq_{{\mathfrak{k}}}$-embedding of $N$ into $M_{i}$. 2) For ${\mathfrak{k}}$ and $\lambda$ as above, let (if $\mu=\lambda$ we may omit $\mu$) $\begin{array}[]{clcr}{\rm univ}(\lambda,\mu,{\mathfrak{k}}):=&\min\\{\|{\mathscr{M}}\|:{\mathscr{M}}\text{ is a family of members of }K_{{\mathfrak{k}}}\text{ each}\\\ &\text{ of cardinality }\leq\mu\text{ which is jointly}\\\ &{\mathfrak{k}}\text{-universal for }\lambda\\}\end{array}$ Let ${\rm Univ}({\mathfrak{k}})=\\{\lambda:{\rm univ}(\lambda,{\mathfrak{k}})=1\\}$. 3) For a pair $\bar{{\mathfrak{k}}}=({\mathfrak{k}}_{1},{\mathfrak{k}}_{2})$ of classes with ${\mathfrak{k}}_{\iota}=(K_{{\mathfrak{k}}_{\iota}},\leq_{{\mathfrak{k}}_{\iota}})$ as in 0.11(3) for $\iota=1,2$ such that $K_{{\mathfrak{k}}_{1}}\subseteq K_{{\mathfrak{k}}_{2}}$, let ${\rm univ}(\lambda,\mu,\bar{{\mathfrak{k}}})$ be the minimal $\|{\mathscr{M}}\|$ such that ${\mathscr{M}}$ is a family of members of $K_{{\mathfrak{k}}_{2}}$ each of cardinality $\mu$ such that every $M\in K_{{\mathfrak{k}}_{1}}$ of cardinality $\lambda$ can be $\leq_{{\mathfrak{k}}_{2}}$-embedded into some member of ${\mathscr{M}}$. Dealing with a.e.c.’s (see [Sh:h]) ###### Definition 0.13. 1) We say that a formula $\varphi=\varphi(\bar{x}_{[I]})$, in any logic, is ${\mathfrak{k}}$-upward preserved when $\tau_{\varphi}\subseteq\tau_{{\mathfrak{k}}}$ and if $M\leq_{{\mathfrak{k}}}N$ and $\bar{a}\in{}^{I}M$ then $M\models\varphi[\bar{a}]$ implies $N\models\varphi[\bar{a}]$. 2) For $\bar{{\mathfrak{k}}}$ as in 0.12(3) we say a pair $\bar{\varphi}(\bar{x}_{[I]})=(\varphi_{1}(\bar{x}_{[I]}),\varphi_{2}(\bar{x}_{[I]}))$ is $\bar{{\mathfrak{k}}}$-upward preserving when $\tau_{\varphi_{1}}\cup\tau_{\varphi_{2}}\subseteq\tau_{{\mathfrak{k}}_{\iota}}$ and if $M_{\iota}\in K_{{\mathfrak{k}}_{\iota}}$ for $\iota=1,2,\bar{a}\in{}^{I}(M_{1})$ and $M_{1}\leq_{{\mathfrak{k}}_{2}}M_{2}$ then $M_{1}\models\varphi_{1}[\bar{a}]$ implies $M_{2}\models\varphi_{1}[\bar{a}]$. 3) In part (2), if $\varphi_{0}=\varphi_{1}$ then we may write $\varphi$ instead of $\bar{\varphi}$. Saying a sequence $\bar{\psi}$ is ${\mathfrak{k}}$-upward preserving means every formula appearing in $\bar{\psi}$ is ${\mathfrak{k}}$-upward preserving. ###### Definition 0.14. 1) For an ideal $J$ on a set $A$ and a set $B$ let $\mathbb{U}_{J}(B)={\rm Min}\\{\|{{\mathscr{P}}}\|:{{\mathscr{P}}}$ is a family of subsets of $B$, each of cardinality $\leq\|A\|$ such that for every function $f$ from $A$ to $B$ for some $u\in{{\mathscr{P}}}$ we have $\\{a\in A:f(a)\in u\\}\in J^{+}\\}$. 2) For an ideal $J$ on a set $A$, cardinal $\theta$ and set $B$ let $\mathbb{U}^{\theta}_{J}(B)={\rm Min}\\{\|{\mathscr{P}}\|:{\mathscr{P}}\subseteq[B]^{\leq\|A\|}$ and if $f\in{}^{A}({}^{\theta}B)$ then for some $u\in{\mathscr{P}}$ we have $\\{a\in A:{\rm Rang}(f(a))\subseteq u\\}\in J^{+}\\}$. So $\mathbb{U}^{\theta}_{J}(B)\leq\mathbb{U}_{J}(\|N\|^{\theta})$. 3) Clearly only $\|B\|$ matters so we normally write $\mathbb{U}_{J}(\lambda)$, (see on it [Sh:589]). ## 1\. The Olive property ###### Definition 1.1. 1) (Convention) 1. $(a)$ Let $T$ be a first order theory and ${\mathfrak{C}}={\mathfrak{C}}_{T}$ a monster for $T$ 2. $(b)$ $(\alpha)\quad\Delta\subseteq{\mathbb{L}}(\tau_{T})$ a set of formulas 3. $(\beta)\quad$ omitting $\Delta$ means $\Delta={\mathbb{L}}(\tau_{T})$ if $T$ is complete, $\Delta$ = set of quantifiers free formula otherwise, and we may write ${\rm qf}$ instead of $\Delta$ 4. $(c)$ $(\alpha)\quad m$ and $n\geq k_{\iota}\geq 2$ for $\iota=0,1,n\geq k_{0}+k_{1}\geq 3,\eta\in{}^{n}2$ are such that $\eta(0)=0$ and $\eta^{-1}\\{0\\}$ is not an initial segment 5. $(\beta)\quad$ if $\eta(\ell)=\ell\mod 2$ for $\ell<k$ we may write $n$ instead of $\eta$ 6. $(d)$ $(\alpha)\quad$ if $\bar{k}=(k_{0},k_{1}),k_{1}\leq k_{0}+1\leq k_{1}+1$ we may write $k_{0}+k_{1}$ instead of $\bar{k}$ and let $k(\iota)=k_{\iota}$ for $\iota=0,1$. 7. $(\beta)\quad$ omitting $m$ means some $m$ 8. $(\gamma)\quad$ omitting $n,\eta,\bar{k}$ means $n=3,\eta=\langle 0,1,0\rangle,\bar{k}=(2,1)$) so for some $m$ 9. $(e)$ $(\alpha)\quad$ below we may write $\psi_{\iota}=\psi_{\iota,k_{\iota}}$ and $\varphi_{\iota}=\psi_{\iota,1}$ for $\iota=0,1$ 10. $(\beta)\quad$ if $\varphi_{0}=\varphi_{1}=\varphi$ we may write $\varphi$; 11. $(\gamma)\quad$ we may omit $\psi_{3,k}$ when it is a logically true formula. 2) We say $T$ has the $(\Delta,\eta,\bar{k},m)$-olive property when there is a pair $(\bar{\psi}_{0},\bar{\psi}_{1})$ of sequences of formulas from $\Delta$ witnessing it, see (3). 3) We say $(\bar{\psi}_{0},\bar{\psi}_{1})$ witness the $(\Delta,\eta,\bar{k},m)$-olive property (for $T$, with the convention above) when : 1. $(a)$ $\bar{\psi}_{\iota}=\langle\psi_{\iota,k}(\bar{x}_{[m],0},\dotsc,\bar{x}_{[m],k}):k=1,\dotsc,k_{\iota}\rangle$ for $\iota=0,1$ with $\psi_{\iota,k}\in\Delta$ 2. $(b)_{\lambda}$ for every $\bar{f}=\langle f_{\alpha}:\alpha<\lambda\rangle$ with $f_{\alpha}$ a function from $\alpha$ to $\\{0,1\\}$, we can find $\bar{a}_{\alpha}\in{}^{m}{{\mathfrak{C}}}$ for $\alpha<\lambda$ such222Actually clause $(\alpha)$ is a specific case of clause $(\beta)$ provided that in clause $(\beta)$ we allow $k=1$. Similarly for clauses $(c)(\alpha),(\beta)$. that: 1. $(\alpha)$ $\varphi_{\iota}[\bar{a}_{\alpha},\bar{a}_{\beta}]$ for $\alpha<\beta<\lambda$ when $\iota=f_{\beta}(\alpha)$, see 1.1$(1)(e)(\beta)$ 2. $(\beta)$ $\psi_{\iota,k}(\bar{a}_{\alpha_{0}},\dotsc,\bar{a}_{\alpha_{k-1}},\bar{a}_{\beta})$ when $k\in\\{2,\dotsc,k_{\iota}-1\\}$ and $\alpha_{0}<\ldots<\alpha_{k-1}<\beta<\lambda$ and $f_{\beta}{\restriction}[\alpha_{0},\alpha_{k-1}]$ is constantly $\iota$, so when $k=1$, it holds trivially 3. $(c)$ there are no $\bar{a}_{\ell}\in{}^{m}{{\mathfrak{C}}}$ for $\ell<n+1$ such that: 1. $(\alpha)$ $\varphi_{\iota}[\bar{a}_{i},\bar{a}_{j}]$ for $i<j<n+1$ and $\eta(i)=\iota$ 2. $(\beta)$ if $\iota\in\\{0,1\\},k\in\\{2,\dotsc,k_{\iota}\\}$ and $\ell_{0}<\ldots<\ell_{k-1}$ are from $\\{\ell<n:\eta(\ell)=\iota\\}$ and $\ell_{k-1}<\ell\leq n$ then $\psi_{\iota,k}[\bar{a}_{\ell_{0}},\dotsc,\bar{a}_{\ell_{k-1}},\bar{a}_{\ell}]$. ###### Remark 1.2. This fits the classification of properties of such $T$ in [Sh:702, 5.15-5.23]. ###### Definition 1.3. 1) Let $K$ be a universal class of $\tau$-models. We say $K$ has the $\lambda-(\eta,\bar{k},m)$-olive property when that some quantifier free $(\bar{\psi}_{0},\bar{\psi}_{1})$ witnessing it, that is, $(a)+(b)_{\lambda}+(c)$ holds (replacing ${\mathfrak{C}}_{T}$ by “in some $M\in K$”). 2) We say that an a.e.c. ${\mathfrak{k}}=(K_{{\mathfrak{k}}},\leq_{{\mathfrak{k}}})$ has the $\lambda-(\eta,\bar{k},m)$-property when : there are $\bar{\psi}_{0},\bar{\psi}_{1}$ which are ${\mathfrak{k}}$-upward preserved formulas in any logic (see 0.13) and $(a)+(b)_{\lambda}+(c)$ of 1.1 holds, replacing ${\mathfrak{C}}={\mathfrak{C}}_{T}$ by “some ${\mathfrak{C}}\in K_{{\mathfrak{k}}}$ of cardinality $\lambda$”. ###### Remark 1.4. 1) Note that for $T$ first order complete, ${\mathfrak{k}}={\rm Mod}_{T}=(\text{mod}_{T},\prec)$, Definition 1.3(2) gives Definition 1.1 and for $T$ first order universal not complete, ${\mathfrak{k}}={\rm Mod}_{T}=(\text{mod}_{T},\subseteq)$, Definition 1.3(2) gives Definition 1.1. Similarly for Definition 1.3(1). 2) Of course, for $T$ first order, the $\lambda$ does not matter. ###### Claim 1.5. Assume $n\geq k_{0}+k_{1}\geq 3,\eta\in{}^{n}2$ and $\|\eta^{-1}\\{\iota\\}\|\geq k_{\iota}\geq 1$ for $\iota=0,1$ then there is a complete first order countable $T$ having the $(\eta,\bar{k},1)$-olive property but $T$ is ${\rm NSOP}_{4}$ and is categorical in $\aleph_{0}$. ###### Proof.. Let $\tau=\\{P,Q_{0},Q_{1}\\}$ where $P$ is a binary predicate and $Q_{\iota}$ is a $(k_{\iota}+1)$-place predicates. Let $T^{0}_{\eta,\bar{k}}$ be the following universal theory in ${\mathbb{L}}(\tau)$: 1. $()_{1}$ a $\tau$-model $M$ is a model of $T^{0}_{\eta,\bar{k}}$ iff we cannot embed $N^{}_{\eta,\bar{k}}$ into $M$ where 2. $\oplus\quad N^{}_{\eta,\bar{k}}$ is the $\tau$-model with universe $\\{a_{0},\dotsc,a_{n}\\}$ as in $(c)(\alpha),(\beta)$ from Definition 1.1(3) for $\varphi(x_{0},x_{1})=P(x_{0},x_{1}),\psi_{\iota}(x_{0},\dotsc,x_{k(\iota)})=$ $Q_{\iota}(x_{0},\dotsc,x_{k(\iota)})$ recalling 1.1$(1)(e)(\gamma)$. Now 1. $()_{2}$ $T^{0}_{\eta,\bar{k}}$ has the JEP and amalgamation property by disjoint union. [Why? Assume $M_{0}\subseteq M_{1},M_{0}\subseteq M_{2}$ are models of $T_{0}$ (but abusing notation we allow $M_{0}$ to be empty) and $\|M_{1}\|\cap\|M_{2}\|=\|M_{0}\|$, we define $M=M_{1}\cup M_{2}$ that is 1. $()_{2.1}$ $(a)\quad\|M\|=\|M_{1}\|\cup\|M_{2}\|$ 2. $(b)\quad P^{M}=P^{M_{1}}\cup P^{M_{2}}$ 3. $(c)\quad Q^{M}_{\iota}=Q^{M_{1}}_{\iota}\cup Q^{M_{2}}_{\iota}$ for $\iota=1,2$. So $M$ is a $\tau$-model, it is a model of $T$ as in (b) any pair of elements belongs to a relation.] 1. $()_{3}$ $T_{\eta,\bar{k}}$, the model completion of $T^{0}_{\eta,\bar{k}}$, is well defined and has elimination of quantifiers. [Why? As $\tau$ is finite with no function symbols and $()_{2}$.] 1. $()_{4}$ $T_{\eta,\bar{k}}$ is ${\rm NSOP}_{4}$ (see [Sh:500, 2.5]). [Why? Because 1. $()_{4.1}$ if (A) then (B) where: 1. $(A)$ $(a)\quad A_{0},A_{1},A_{2},A_{3}$ are disjoint sets 2. $(b)\quad M_{\ell}$ is a model of $T^{0}_{\eta,\bar{k}}$ with universe $A_{\ell}$ for $\ell=0,1,2,3$ 3. $(c)\quad$ if $\\{\ell(1),\ell(2)\\}\in{\mathscr{W}}:=\\{\\{0,1\\},\\{1,2\\},\\{2,3\\},\\{3,4\\}\\}$ then $M_{\\{\ell(1),\ell(2)\\}}$ is a model of $T^{0}_{k,n}$ with universe $A_{\ell(1)}\cup A_{\ell(2)}$ extending $M_{\ell(1)}$ and $M_{\ell(2)}$ 4. $(B)$ $M=\cup\\{M_{\\{\ell(1),\ell(2)\\}}:\\{\ell(1),\ell(2)\\}\in{\mathscr{W}}\\}$ where the union is defined as in the proof of $()_{2}$, is a model of $T^{0}_{k,n}$ extending all of them.] [Why? Clearly $M$ is a $\tau$-model and if $f$ embeds $N^{}_{\eta,\bar{k}}$ into $M$, as in $()_{2}$ we have ${\rm Rang}(f)\subseteq M_{\ell(1),\ell(2)}$ for some $\\{\ell(1),\ell(2)\\}\in{\mathscr{W}}$, contradiction.] 1. $()_{5}$ $T_{k,n}$ (and ${\rm Mod}_{T^{0}_{\eta,\bar{k}}}$) has the $(\eta,\bar{k})$-olive property as witnessed by $\varphi(x_{0},x_{1})=P(x_{0},x_{1}),\psi_{\iota}(x_{0},\dotsc,x_{k(\iota)})=Q_{\iota}(x_{0},\dotsc,x_{k(\iota)})$. [Why? In Definition 1.1(3), clause (a) holds trivially and clause (c) is obvious from the choice of $T^{0}_{\eta,\bar{k}}$. For clause $(b)_{\lambda}$ we are given $\langle f_{\alpha}:\alpha<\lambda\rangle$ with $f_{\alpha}$ a function from $\alpha$ to $\\{0,1\\}$ and we have to find $M$ as there. We define a $\tau$-model $M$ with: 1. $\bullet$ universe $\\{a^{}_{\alpha}:\alpha<\lambda\\}$ such that $\alpha<\beta\Rightarrow a^{}_{\alpha}\neq a^{}_{\beta}$ 2. $\bullet$ $P^{M}=\\{(a^{}_{\alpha},a^{}_{\beta}):\alpha<\beta<\lambda\\}$ 3. $\bullet$ $Q^{M}_{\iota}=\\{(a^{}_{\alpha_{0}},\dotsc,a^{}_{\alpha_{k(\iota)-1}},a_{\beta}):\alpha_{0}<\ldots<\alpha_{k(\iota)-1}<\beta$ and $f_{\beta}{\restriction}[\alpha_{0},\alpha_{k(\iota)-1}]$ is constantly $\iota\\}$. It suffices to prove that $M$ is a model of $T^{0}_{\eta,\bar{k}}$. So toward a contradiction assume $h$ embeds $N^{}_{\eta,\bar{k}}$ into $M$, so let $h(a^{}_{\ell})=a_{g(\ell)}$ where $g:\\{0,\dotsc,n\\}\rightarrow\lambda$; necessarily $g$ is a one-to-one function. For $\ell<n$, recall $N^{}_{\eta,\bar{k}}\models``P(a^{}_{\ell},a^{}_{\ell+1})"$ but $h$ is an embedding so $M\models``P[a^{}_{g(\ell)},a^{}_{g(\ell+1)}]"$, but if $g(\ell)\geq g(\ell+1)$ this fails by the choice of $P^{M}$ hence $g(\ell)<g(\ell+1)$. Now let $i_{}=\min\\{i:\eta(i)=1\\}$. Let $i_{0}<\ldots<i_{k(0)-1}$ be from $\eta^{-1}\\{0\\}$ such that $i_{0}=0$ and $i_{k(\iota)-1}$ is maximal hence $i_{}\in[i_{0},i_{k(0)-1})$. Now $N^{}_{\eta,\bar{k}}\models Q_{0}[a^{}_{i_{0}},\dotsc,a^{}_{i_{k(0)-1}},a^{}_{n}]$ hence $M\models Q_{0}[a^{}_{g(i_{0})},\dotsc,a^{}_{g(i_{k(0)-1})},a^{}_{g(n)}]$ and this implies that $f_{g(n)}{\restriction}[g(i_{0}),g(i_{k(0)-1)})]$ is constantly $0$ hence $f_{g(n)}(g(i_{}))=0$. Similarly let $j_{0}<\ldots<j_{k(1)-1}$ be from $\eta^{-1}\\{1\\}$ such that $j_{0}=i_{}$; now $N^{}_{\eta,\bar{k}}\models Q_{1}[a^{}_{j_{0}},\dotsc,a^{}_{j_{k(1)-1}},a^{}_{n}]$ hence $M\models Q_{1}[a_{g(0)},\dotsc,a_{g(j_{k(1)-1})},a_{n}]$ hence $f_{g(n)}{\restriction}[g(j_{0}),g(j_{k(1)-1})]$ is constantly 1, hence $f_{g(n)}(g(i_{}))=1$, contradiction. ∎ As in earlier cases we apply a kind of guessing of clubs (almost suitable also for them i.e. for the proof with strict order and ${\rm SOP}_{4}$). An unexpected gain here is that here we use a weaker version: there is no requirement $\alpha<\lambda\Rightarrow\lambda>\|\\{C_{\delta}\cap\alpha:\delta\in S$ satisfies $\alpha\in{\rm nacc}(C_{\delta})\\}\|$ but not clear how this helps. Also here the use of the pair $(\bar{{\mathscr{A}}},\bar{\mathbb{g}})$ may be helpful. ###### Definition 1.6. 1) For $\lambda$ regular uncountable and $\chi_{2}>\chi_{1}\geq\lambda$ let ${\rm Qr}_{1}(\chi_{2},\chi_{1},\lambda)$ mean that there are $S,\bar{C},I,\bar{{\mathscr{A}}},\bar{\mathbb{g}}$ witnessing it, this means (note: if $\chi_{1}=\lambda$ then $I=\\{S\\}$): 1. $\boxplus$ $(a)\quad S\subseteq\lambda$ and $I$ an ideal on $S$ 2. $(b)\quad\bar{C}=\langle C_{\delta}:\delta\in S\rangle$ 3. $(c)\quad C_{\delta}\subseteq\delta$, note that possibly $\sup(C_{\delta})<\delta$ 4. $(d)(\alpha)\quad\bar{\mathbb{g}}=\langle\bar{g}_{j}:j<\chi_{2}\rangle$ 5. $\hskip 10.0pt(\beta)\quad\bar{g}_{j}=\langle g_{j,\delta}:\delta\in S\rangle$ 6. $\hskip 10.0pt(\gamma)\quad g_{j,\delta}:C_{\delta}\rightarrow\\{0,1\\}$ 7. $(f)(\alpha)\quad\bar{{\mathscr{A}}}=\langle\bar{{\mathscr{A}}}_{j}:j<\chi_{2}\rangle$ 8. $\hskip 10.0pt(\beta)\quad\bar{{\mathscr{A}}}_{j}=\langle{\mathscr{A}}_{j,\delta}:\delta\in S\rangle$ 9. $\hskip 10.0pt(\gamma)\quad{\mathscr{A}}_{j,\delta}\subseteq{\mathscr{P}}({\rm nacc}(C_{\delta}))$ 10. $(g)\quad\mathbb{U}_{I}(\chi_{1})<\chi_{2}$; see Definition 0.14, if $\chi_{1}=\lambda$ then we stipulate $\mathbb{U}_{I}(\chi_{1})=\chi_{1}$ hence this means $\chi_{1}<\chi_{2}$ 11. $(h)\quad$ if $j_{1}\neq j_{2},\delta\in S,A_{1}\in{\mathscr{A}}_{j_{1},\delta}$ and $A_{2}\in{\mathscr{A}}_{j_{2},\delta}$ then there is $\gamma\in A_{1}\cap A_{2}$ such that $g_{j_{1},\delta}(\gamma)\neq g_{j_{2},\delta}(\gamma)$ 12. $(i)\quad$ if $j<\chi_{2}$ and $E$ is a club of $\lambda$ then for some $Y\in I^{+}$ hence $Y\subseteq S$ for every $\delta\in Y$ we have ${\rm nacc}(C_{\delta})\cap E\in{\mathscr{A}}_{j,\delta}$. 2) For $\ell=1,2,3$ let ${\rm Qr}_{\ell}(\chi_{2},\chi_{1},\lambda)$ be defined by: 1. $\bullet$ if $\ell=1$ as above 2. $\bullet$ if $\ell=2$ as above but there is a sequence $\langle J_{\delta}:\delta\in S\rangle$ of ideals on ${\rm nacc}(C_{\delta})$ such that ${\mathscr{A}}_{j,\delta}=\\{{\rm nacc}(C_{\delta})\backslash X:X\in J_{\delta}\\}$ 3. $\bullet$ if $\ell=3$ we use clauses (a)-(g) from part (1) and 4. $(h)^{-}\quad$ if $E_{j}$ is a club of $\lambda$ for $j<\chi_{2}$ and $\langle\xi_{j}:j<\chi_{2}\rangle$ is a sequence of ordinals with $\sup\\{\xi_{j}:j<\chi_{2}\\}<\chi_{2}$ then we can find $j_{1}<j_{2}<\chi_{2},\delta\in S$ and $\gamma\in{\rm nacc}(C_{\delta})$ such that $\xi_{j_{1}}=\xi_{j_{2}},\gamma\in E_{j_{1}}\cap E_{j_{2}}$ and $g_{j_{1},\delta}(\gamma)\neq g_{j_{2},\delta}(\gamma)$. 3) ${\rm Qr}_{\ell,\iota}(\chi_{2},\chi_{1},\lambda)$ is defined as in ${\rm Qr}_{\ell}(\chi_{2},\chi_{1},\lambda)$ but $g_{j,\delta}:{\rm nacc}(C_{\delta})\rightarrow\iota_{}$, etc. ###### Remark 1.7. Can we weaken the conclusion of clause (i), etc. to: 1. $\bullet$ $\\{\alpha\in{\rm nacc}(C_{\delta}):\sup(\alpha\cap E)>\max(C_{\delta}\cap\alpha)\\}\in{\mathscr{A}}_{j,\delta}$. That is, this suffices in 1.9 but there is no clear gain so have not looked into it. ###### Fact 1.8. 1) ${\rm Qr}_{2}(\chi_{2},\chi_{1},\lambda)\Rightarrow{\rm Qr}_{1}(\chi_{2},\chi_{1},\lambda)\Rightarrow{\rm Qr}_{3}(\chi_{2},\chi_{1},\lambda)$. 2) We have ${\rm Qr}_{1}(\chi_{2},\chi_{1},\lambda)$ and even ${\rm Qr}_{2}(\chi_{2},\chi_{1},\lambda)$ when 1. $(a)$ $\kappa^{+}<\lambda\leq\chi_{1}<\chi_{2}<2^{\kappa}$ 2. $(b)$ $\kappa={\rm cf}(\kappa),\lambda={\rm cf}(\lambda)$ 3. $(c)$ $\mathbb{U}_{I}(\chi_{1})<\chi_{2},I$ an ideal on $S$ so $S\notin I$ 4. $(d)$ $S\subseteq S^{\lambda}_{\kappa}$ is stationary, $\bar{C}=\langle C_{\delta}:\delta\in S\rangle$ guess clubs, $C_{\delta}\subseteq\delta,{\rm otp}(C_{\delta})=\kappa$ 5. $(e)$ $I=\\{A\subseteq S$: for some club $E$ of $\lambda$ for no $\delta\in S$ do we have ${\rm nacc}(C_{\delta})\cap E\in J^{{\rm bd}}_{C_{\delta}}\\}$ ###### Proof.. 1) Easy. 2) Clause (d) follows by [Sh:420, §2]. The proof itself is straightforward. ∎ ###### Theorem 1.9. 1) If $T$ is complete, with the $(\eta,\bar{k})$-olive property and $\lambda>\kappa^{+}$ and $\lambda,\kappa$ are regular, $2^{\kappa}>\lambda\geq\kappa^{++}+\|T\|$ then $T$ has no universal in $\lambda$ (for $\prec$). 2) If $T$ is complete, with the $(\eta,\bar{k},m)$-olive property and $\lambda={\rm cf}(\lambda)\geq\|T\|$ and ${\rm Qr}_{1}(\chi_{2},\chi_{1},\lambda)$ then ${\rm univ}(\chi_{1},\lambda,T)\geq\chi_{2}$. 3) Similarly for a.e.c. see 1.3(2), so e.g. for universal $K$ with the ${\rm JEP}$ and the $\lambda-({\rm qf},\eta,\bar{k})$-olive property. 4) We can weaken ${\rm Qr}_{1}(\chi_{2},\chi_{1},\lambda)$ to ${\rm Qr}_{1,\theta}(\chi_{2},\chi_{1},\lambda)$ when $\theta=2^{\partial},\chi_{1}=\chi^{\partial}_{1},\partial<\lambda$. ###### Remark 1.10. 1) We can use ${\rm Qr}_{3}$ instead of ${\rm Qr}_{1}$ by the same proof but the gain is not clear. 2) If e.g. $\lambda=\mu^{+},\mu=\mu^{<\mu}=2^{\partial},\chi_{1}=\lambda=\chi_{2}$ (so have a universal in $\lambda$), failure of ${\rm Qr}_{1}(\lambda,\lambda,\lambda)$ implies: there is ${\mathscr{F}}\subseteq{}^{\mu}\mu$ such that $(\forall\eta\in{}^{\mu}\mu)(\exists\nu\in{\mathscr{F}})(\exists^{\mu}i<\mu)(\eta(i)=\nu(i))$. ###### Proof.. 1) It follows from (2) by 1.8(2). 2) Let $(\bar{\psi}_{0},\bar{\psi}_{1})$, i.e. $\bar{\psi}_{\iota}=\langle\psi_{\iota,k}(\bar{x}_{0},\dotsc,\bar{x}_{k}):k=1,\dotsc,k_{\iota}\rangle$ for $\iota=0,1$ witness the $(\eta,\bar{k},m)$-olive property. For simplicity we can, without loss of generality assume that $m=1$ and $T$ has elimination of quantifiers and only predicates and its vocabulary is finite. Let $S,\bar{C},\bar{{\mathscr{A}}},\bar{\mathbb{g}}$ witness ${\rm Qr}_{1}(\chi_{2},\chi_{1},\lambda)$. For each $j<\chi_{2}$ we define $\bar{f}_{j}$ by: 1. $()_{1}$ $(a)\quad\bar{f}_{j}=\langle f_{j,\alpha}:\alpha<\lambda\rangle$ 2. $(b)\quad f_{j,\alpha}:\alpha\rightarrow\\{0,1\\}$ is defined by: 1. $(\alpha)\quad$ if $\beta<\alpha\in S$ then $f_{j,\alpha}(\beta)=g_{j,\alpha}(\min(C_{\alpha}\backslash\beta))$ 2. $(\beta)\quad$ if $\beta<\alpha\in\lambda\backslash S$ then $f_{j,\alpha}(\beta)=0$. For each $j<\chi_{2}$ we can find $M_{j}\models T$ of cardinality $\lambda$ and pairwise distinct elements $\langle a_{j,\alpha}:\alpha<\lambda\rangle$ satisfying $(b)_{\lambda}$ of Definition 1.1 for $\bar{f}_{j}$. Let $M_{j,\alpha}=M_{j}{\restriction}\cup\\{a_{j,\beta}:\beta<\alpha\\}$. Let the function $h^{0}_{j}:\lambda\rightarrow M_{j}$ be defined by $h^{0}_{j}(\alpha)=a_{j,\alpha}$. Let ${\mathscr{P}}\subseteq[\chi_{1}]^{\lambda}$ witness $\mathbb{U}_{J}(\chi_{1})<\chi_{2}$ and for $u\in{\mathscr{P}}$ or just $u\in[\chi_{1}]^{\lambda}$ let $h^{1}_{u}$ be one to one from $u$ onto $\lambda$. Toward contradiction assume that there are $\xi_{}<\chi_{2}$ and a sequence $\langle{\mathfrak{A}}_{\xi}:\xi<\xi_{}\rangle$ of models of $T$ each of cardinality $\leq\chi_{1}$ witnessing ${\rm univ}(\chi_{1},\lambda,T)<\chi_{2}$, even equal to $\|\xi_{}\|$. Without loss of generality the universe of each ${\mathfrak{A}}_{\xi}$ is $\alpha_{\xi}\leq\chi_{1}$. So for every $j<\chi_{2}$ there are $\xi=\xi_{j}<\xi_{}$ and an (elementary) embedding $h^{2}_{j}$ of $M_{j}$ into ${\mathfrak{A}}_{\xi}$, hence there is $u_{j}\in{\mathscr{P}}$ such that $W_{j}:=\\{\alpha\in S:h^{2}_{j}(a_{j,\alpha})\in u_{j}\\}\in I^{+}$ and let $v_{j}\supseteq u_{j}\cup{\rm Rang}(h^{2}_{j})$ be such that $v_{j}\in[\chi_{1}]^{\lambda}$ and ${\mathfrak{A}}_{j}{\restriction}v_{j}\prec{\mathfrak{A}}_{j}$ and let $\langle\gamma_{j,\alpha}:\alpha<\lambda\rangle$ list the members of $v_{j}$. Let $h_{j}=h^{1}_{v_{j}}\circ h^{2}_{j}\circ(h^{0}_{j}{\restriction}W_{j})$ so a function from $W_{j}$ into $\lambda$. Let $N_{j}=({\mathfrak{A}}_{\xi_{j}}{\restriction}v_{j},P^{N_{j}}_{})$ be the expansion of ${\mathfrak{A}}_{\xi_{j}}{\restriction}v_{j}$ by the relation $P^{N_{j}}_{}={\rm Rang}(h_{j})$ and let $E_{j}=\\{\delta<\lambda:\delta$ is a limit ordinal, $(\forall\alpha<\lambda)(h^{1}_{v_{j}}(\alpha)\in\\{\gamma_{j,\beta}:\beta<\delta\\}\equiv\alpha<\delta)$ and $N_{j}{\restriction}\\{\gamma_{j,\alpha}:\alpha<\delta\\}\prec N_{j}\\}$, clearly a club of $\lambda$. Hence by clause (i) of Definition 1.6(1) there is $\delta_{j}\in E_{j}\cap S$ such that $A_{j}:={\rm nacc}(C_{\delta})\cap E_{j}$ belongs to ${\mathscr{A}}_{j,\delta}$. As $\xi_{}<\chi_{2},\|{\mathscr{P}}\|<\chi_{2}$ and $\|\\{h_{j}(a_{j,\delta}):j<\chi_{2},\delta\in S\\}\|<\sup\\{\\|{\mathfrak{A}}_{\xi}\\|:\xi<\xi_{}\\}\leq\chi_{1}<\chi_{2}$ by the pigeon hull principle there are: 1. $()_{2}$ $(a)\quad j_{1}=j(1)<j_{2}=j(2)$ 2. $(b)\quad\xi_{j(1)}=\xi_{j(2)}$ 3. $(c)\quad\delta_{j_{1}}=\delta_{j_{2}}$ call it $\delta$ (so $\delta\in S$) 4. $(d)\quad u_{j_{1}}=u_{j_{2}}$ call it $u$, so $u=u_{j_{\iota}}\subseteq\|N_{j_{\iota}}\|$ for $\iota=0,1$ 5. $(e)\quad h_{j_{1}}(a_{j_{1},\delta})=h_{j_{2}}(a_{j_{2},\delta})$ call it $b$, so $b\in{\rm Rang}(h^{2}_{j_{1}})\cap{\rm Rang}(h^{2}_{j_{2}})$. By clause (h) of Definition 1.6(1) there is $\gamma\in A_{j_{1}}\cap A_{j_{2}}$ such that $g_{j_{1},\delta}(\gamma)\neq g_{j_{2},\delta}(\gamma)$. Now we shall choose $\alpha_{\ell}$ by induction on $\ell<n$ such that: 1. $()_{3}$ $(a)\quad\alpha_{\ell}\in W_{j_{\eta(\ell)}}$ 2. $(b)\quad\alpha_{\ell}<\gamma$ but $\alpha_{\ell}>\sup(C_{\delta}\cap\gamma_{\ell})$ 3. $(c)\quad\langle\alpha_{0},\dotsc,\alpha_{\ell}\rangle$ is increasing 4. $(d)\quad$ in the model $N_{j_{\eta(\ell)+1}}$ the elements $h^{2}_{j_{\eta(\ell)+1}}(a_{j_{1},\delta})=b,h^{1}_{j_{\eta(\ell)+1}}(a_{j_{1},\alpha_{\ell}})$ realize the same quantifier type over $\\{h_{j_{\eta(\ell(1))+1}}(a_{j_{\eta},\alpha_{\ell(1)}})$: $\ell(1)<\ell\\}$ or at least for all relevant (finitely many) formulas. If we succeed, then in the model ${\mathfrak{A}}_{\xi_{}}$ which extends $N_{j_{1}}$ and $N_{j_{2}}$ the sequence $\langle h^{2}_{j_{\eta(\ell)}}(a_{j_{\eta(\ell)},\alpha_{\ell}}):\ell<n\rangle\char 94\relax\langle b\rangle$ realizes the “forbidden” type that is the one from clause (c) of Definition 1.1, contradiction. As $\delta\in W_{j}\cap E_{j_{\eta(\ell)}}$ by the choice of $E_{j_{\eta(\ell)}}$ we can carry the induction. 3) similarly. 4) As in [Sh:457] and the above, just use $\partial$-tuples of $\bar{a}$’s. ∎ A sufficient condition for cases of ${\rm Qr}_{i}$ is ###### Definition 1.11. Let ${\rm Qr}_{4}(\lambda)$ mean: $\lambda=\mu^{+}$ and $\langle C_{\delta},D_{\delta}:\delta\in S\rangle$ satisfies $C_{\delta}\subseteq\delta,D_{\delta}$ a filter on ${\rm nacc}(C_{\delta})$ such that ${\mathscr{P}}({\rm nacc}(C_{\delta}))/D_{\delta}$ satisfies the $2^{\mu}$-c.c. and for every club $E$ of $\lambda$ for some $\delta\in S,E\cap{\rm nacc}(C_{\delta})\in D^{+}_{\delta}$. ## 2\. The class of Groups have the olive property We shall try to prove that the class of groups has a universal member almost only when cardinal arithmetic is close to G.C.H. This is done by ###### Theorem 2.1. The class of groups has the olive property, see Definition 0.8 or 1.1(1)(d)$(\gamma)$, in fact, the $(\eta,\bar{k},m)$-olive property, where $\eta=\langle 0,1,0\rangle,\bar{k}=(2,1),m=6$. We break the proof into a series of definitions and claims; we may replace the use of HNN extension (in 2.14) and free amalgamation (in 2.13) by the proof of 2.15. ###### Definition 2.2. Let $\bar{\psi}=\bar{\psi}^{{\rm grp}}_{{\rm olive}}$ be $(\varphi_{0,1},\varphi_{0,2},\varphi_{1,1})$ defined as follows (letting $m=6$): 1. $(a)$ $\psi_{0,1}=\varphi_{0}=\varphi_{0}(\bar{x}_{[m]},\bar{y}_{[m]})=y^{-1}_{5}x_{0}y_{5}=x_{2}$ 2. $(b)$ $\psi_{1,1}=\varphi_{1}=\varphi_{1}(\bar{x}_{[m]},\bar{y}_{[m]})=x^{-1}_{5}y_{1}x_{5}=y_{3}\wedge x^{-1}_{5}y_{4}x_{5}=y_{4}$ 3. $(c)$ $\psi_{0,2}=\psi(\bar{x}_{[m]},\bar{y}_{[m]},\bar{z}_{[m]})=\sigma_{}(x_{0},y_{1},z_{4})=e\wedge\sigma_{}(x_{2},y_{3},z_{4})\neq e$, on $\sigma_{}$ see below. ###### Definition/Claim 2.3. There is a $\sigma_{}=\sigma_{}(x,y,z)$ such that: 1. $(a)$ $\sigma_{}$ is a group word 2. $(b)$ for some group $G$ and $a,b,c\in G$ we have “$\sigma_{}(a,b,c)\neq e_{G}$” 3. $(c)$ for any group $G$ and $a,b,c\in G$ we have $e\in\\{a,b,c\\}\Rightarrow\sigma^{G}(a,b,c)=e_{G}$ ###### Remark 2.4. Earlier we intend to use [LS77] hence add 1. $(d)$ for no two distinct interval $\sigma_{1},\sigma_{2}$ of some cyclic permutation $\sigma^{\prime}$ of $\sigma$ do we have $\sigma_{2}\in\\{\sigma_{1},\sigma^{-1}_{2}\\}$ and $\ell g(\sigma_{1})>\ell g(\sigma)/6$; it seems not necessary 2. $(e)$ $(\alpha)\quad\sigma_{}$ is cyclically reduced. But this is not necessary. ###### Proof.. Straightforward, e.g. $(x^{-1}y^{-1}x^{-1}y)^{-1}z^{-1}(x^{-1}y^{-1}xy)z$. ∎ ###### Claim 2.5. The $\bar{\psi}$ from 2.2 satisfies clause (c) of Definition 0.8 or 1.1(3), i.e. for no group $G$ and $\bar{a}_{\ell}\in{}^{m}G$ for $\ell<4$ do the formulas there hold. ###### Remark 2.6. We prove more: there are no group $G$ and $\bar{a}_{\ell}\in{}^{m}G$ for $\ell=0,1,2,3$ such that $\varphi_{0}[\bar{a}_{0},\bar{a}_{1}],\varphi_{1}[\bar{a}_{1},\bar{a}_{2}],\varphi_{1}[\bar{a}_{1},\bar{a}_{3}]$ and $\psi[\bar{a}_{0},\bar{a}_{2},\bar{a}_{3}]$. ###### Proof.. Assume toward contradiction that $G,\langle\bar{a}_{\ell}:\ell<4\rangle$ forms a counterexample; now conjugation by $a_{1,5}$ is an automorphism of $G$ which we call $g$. Now: 1. $\bullet$ $g(a_{0,0})=a_{0,2}$ by (a) of 2.2 as $G\models\varphi_{0}[\bar{a}_{0},\bar{a}_{1}]$ 2. $\bullet$ $g(a_{2,1})=a_{2,3}$ by first conjunct of (b) of 2.2 as $G\models\varphi_{1}[\bar{a}_{1},\bar{a}_{2}]$ 3. $\bullet$ $g(a_{3,4})=a_{3,4}$ by the second conjunct of (b) of 2.2 as $G\models\varphi_{1}[\bar{a}_{1},\bar{a}_{3}]$. Together 1. $\bullet$ $g(\sigma_{}(a_{0,0},a_{2,1},a_{3,4}))=\sigma_{}(a_{0,2},a_{2,3},a_{3,4})$ but this contradicts $G\models\psi[\bar{a}_{0},\bar{a}_{2},\bar{a}_{3}]$, see clause (c) of 2.2. ∎ ###### Definition 2.7. Let $\bar{f}\in\mathbb{F}_{\lambda}$, i.e. $\bar{f}=\langle f_{\alpha}:\alpha<\lambda\rangle,f_{\alpha}:\alpha\rightarrow\\{0,1\\}$. 1) Let $X_{\bar{f}}=X_{\bar{f},m}$ where we let $X_{\bar{f},k}=\\{x_{\alpha,\ell}:\alpha<\lambda,\ell<k\\}$ for $k\leq m$; recall that here $m=6$. 2) Let $\bar{x}_{\alpha,k}=\langle x_{\alpha,\ell}:\ell<k\rangle$ for $k\leq m$ and let $\bar{x}_{\alpha}=\bar{x}_{\alpha,m}$. 3) For $\ell=0,1$ we define the set $\Gamma^{\ell}_{\bar{f}}$ of equations (pedantically, for $\ell=0$ conjunctions of two equations): $\\{\varphi_{\ell}(\bar{x}_{\alpha},\bar{x}_{\beta}):\alpha<\beta<\lambda\text{ and }f_{\beta}(\alpha)=\ell\\}.$ 4) We define the set $\Gamma^{2}_{\bar{f}}$ of equations $\\{\sigma_{}(x_{\alpha,0},x_{\beta,1},x_{\gamma,4})=e:\alpha<\beta<\gamma<\lambda\text{ and }f_{\gamma}{\restriction}[\alpha,\beta]\text{ is constantly }0\\}.$ 5) Let $G^{5}_{\bar{f}}$ be the group generated by $X_{\bar{f},5}$ freely except the equations in $\Gamma^{2}_{\bar{f}}$, note that the $x_{\alpha,5}$’s are not mentioned in $\Gamma^{2}_{\bar{f}}$. 6) Let $G^{6}_{\bar{f}}$ be the group generated by $X_{\bar{f},6}$ freely except the equations in $\Gamma^{0}_{\bar{f}}\cup\Gamma^{1}_{\bar{f}}\cup\Gamma^{2}_{\bar{f}}$. ###### Discussion 2.8. For our purpose we have to show that for $\alpha<\beta<\gamma$ (and $\bar{f}\in\mathbb{F}_{\lambda}$) we have: $G_{\bar{f},6}\models``\psi[\bar{x}_{\alpha},\bar{x}_{\beta},\bar{x}_{\gamma}]"$ iff $f_{\gamma}{\restriction}[\alpha,\beta]=0_{[\alpha,\beta]}$. For proving the “if” implication, assume $f_{\gamma}{\restriction}[\alpha,\beta]=0_{[\alpha,\beta]}$. Now the satisfaction of “$\sigma_{}(x_{\alpha,0},x_{\beta,1},x_{\gamma,4})=e"$ is obvious by the role of $\Gamma^{2}_{\bar{f}}$, the analysis below is intended to prove the other half “$\sigma_{}(x_{\alpha,2},x_{\beta,3},x_{\gamma,4})\neq e$”. For proving the “only if” implication it suffices to prove that “$\sigma_{}(x_{\alpha,0},x_{\beta,1},x_{\gamma,4})\neq e"$ when $f_{\gamma}{\restriction}[\alpha,\beta]\neq 0_{[\alpha,\beta]}$. For both cases, we prove that this holds in $G^{5}_{\bar{f}}$ and then prove that $G^{5}_{\bar{f}}\subseteq G^{6}_{\bar{f}}$ in the natural way. ###### Claim 2.9. 1) If $\alpha<\beta<\gamma<\lambda$ and $f_{\gamma}{\restriction}[\alpha,\beta]\neq 0_{[\alpha,\beta]}$ then $G^{5}_{\bar{f}}\models``\sigma_{}[x_{\alpha,0},x_{\beta,1},x_{\gamma,4}]\neq e"$. 2) If $\alpha<\beta<\gamma<\lambda$ then $G^{5}_{\bar{f}}\models``\sigma_{}(x_{\alpha,2},x_{\beta,3},x_{\gamma,4})\neq e"$. ###### Proof.. 1) Use 2.10 below with $X=\\{X_{\xi,\ell}:\xi\in\\{\alpha,\beta,\gamma\\}$ and $\ell<5\\}$. 2) Use 2.10(2) below with $X=\\{x_{\xi,\ell}:\xi<\lambda,\ell<5$ and $\ell>0\\}$. ∎ ###### Observation 2.10. 1) If $x_{\alpha,\ell},x_{\beta,k}\in X^{5}_{\bar{f}}$ and $(\alpha,\ell)\neq(\beta,k)$ then $G^{5}_{\bar{f}}\models``x_{\alpha,\ell}\neq x_{\beta,k}"$. 2) If $X\subseteq X^{5}_{\bar{f}}$ and $(\sigma_{}(x_{\alpha,0},x_{\beta,1},x_{\gamma,4})=e)\in\Gamma^{2}_{\bar{f}}\Rightarrow\\{x_{\alpha,0},x_{\beta,1},x_{\gamma,4}\\}\nsubseteq X$ then $X$ generates freely a subgroup of $G^{5}_{\bar{f}}$. ###### Proof.. 1) Let $G^{\prime}=\oplus\\{{\mathbb{Z}}x:x\in X^{5}_{\bar{f}}\\}$, it is an abelian group; let $G^{\prime\prime}=\oplus\\{{\mathbb{Z}}x_{\alpha,i}:\alpha<\lambda,i\notin\\{\ell,k\\}\\}$ a subgroup. So $G^{\prime}/G^{\prime\prime}$ by clause (c) of Definition 2.3, satisfies all the equations in $\Gamma^{2}_{\bar{f}}$ and it satisfies the desired inequality. As $G^{5}_{\bar{f}}$ is generated by $X^{5}_{\bar{f}}$ freely except the equations in $\Gamma^{2}_{\bar{f}}$ the desired result follows. Alternatively use part (2). 2) Let $H=H_{X}$ be the group generated by $X$ freely. We define a function $F$ from $X^{5}_{\bar{f}}$ into $H$ by 1. $\bullet$ $F(x)$ is $x$ if $x\in X$ and is $e_{H}$ if $x\in X^{5}_{\bar{f}}\backslash X$. Now $F$ respects every equation form $\Gamma^{2}_{\bar{f}}$ by clause (c) of 2.5, hence $f$ induces a homomorphism from $G^{5}_{\bar{f}}$ into $H$, really onto. Hence the desired conclusion follows. ∎ ###### Definition 2.11. For $\beta<\lambda$ we define a partial function $F_{\beta}$ from $X^{5}_{\bar{f}}$ to $X^{5}_{\bar{f}}$: 1. $\bullet$ if $\alpha<\beta$ and $f_{\beta}(\alpha)=0$ then $F_{\beta}(x_{\alpha,0})=x_{\alpha,2}$ 2. $\bullet$ if $\gamma>\beta$ and $f_{\gamma}(\beta)=1$ then $F_{\beta}(x_{\gamma,1})=x_{\gamma,3},F_{\beta}(x_{\gamma,4})=x_{\gamma,4}$. ###### Claim 2.12. 1) $F_{\beta}$ is a well defined partial one-to-one function from $X^{5}_{\bar{f}}$ to $X^{5}_{\bar{f}}$. 2) The domain and the range of $F_{\beta}$ satisfies the criterion of 2.10(2). ###### Proof.. 1) It is a function as no $x_{\alpha,\ell}$ appears in two cases. Also if $F_{\beta}(x_{\alpha_{1},\ell})=x_{\alpha_{2},k}$ then $\alpha_{1}=\alpha_{2}\wedge(\ell,k)\in\\{(0,2),(1,3),(4,4)\\}$ so $F_{\beta}$ is one to one. 2) Assume $[\sigma_{}(x_{\alpha_{1},0},x_{\alpha_{2},1},x_{\alpha_{3},4})=e]\in\Gamma^{2}_{\bar{f}}$ so 1. $()_{1}$ $\alpha_{1}<\alpha_{2}<\alpha_{3}$ and 2. $()_{2}$ $f_{\alpha,3}{\restriction}[\alpha_{1},\alpha_{2}]=0_{[\alpha_{1},\alpha_{2}]}$. First, toward contradiction assume $\\{x_{\alpha_{1},0},x_{\alpha_{2},1},x_{\alpha_{3},4}\\}\subseteq{\rm Dom}(F_{\beta})$. Now if $\alpha_{1}\geq\beta$ then $x_{\alpha_{1},0}\notin{\rm Dom}(F_{\beta})$, just inspect Definition 2.11 so necessarily $\alpha_{1}<\beta$ and similarly $f_{\beta}(\alpha_{1})=0$ (but not used). If $\alpha_{2}\leq\beta$ then $x_{\alpha_{2},1}\notin{\rm Dom}(F_{\beta})$, so $\beta<\alpha_{2}$ and similarly $f_{\alpha_{2}}(\beta)=1$ (again not used) so together $\alpha_{1}<\beta<\alpha_{2}$. Also as $x_{\alpha_{3},4}\in{\rm Dom}(F_{\beta})$ it follows that ($\beta<\alpha_{3}$ which follows by earlier inequalities and) $f_{\alpha_{3}}(\beta)=1$, so together $\beta$ witness that $f_{\alpha_{3}}{\restriction}[\alpha_{1},\alpha_{2}]$ is not constantly zero contradiction to $[\sigma(x_{\alpha_{1},0},x_{\alpha_{2},0},x_{\alpha_{3},0})=e]\in\Gamma^{2}_{\bar{f}}$. Second, toward contradiction assume $\\{x_{\alpha_{2},0},x_{\alpha_{1},2},x_{\alpha_{3},4}\\}\subseteq X\subseteq{\rm Rang}(F_{\beta})$, but “$x_{\alpha_{2},0}\in{\rm Rang}(F_{\beta})"$ is impossible by Definition 2.11. ∎ ###### Claim 2.13. To prove $G^{5}_{\bar{f}}\subseteq G^{6}_{\bar{f}}$ any of the following conditions suffice: 1. $(a)$ there are a group $H$ extending $G^{5}_{\bar{f}}$ and $y_{\zeta}\in G$ for $\zeta<\lambda$ such that $\zeta<\lambda\wedge F_{\zeta}(x_{\varepsilon_{1},\ell_{1}})=x_{\varepsilon_{2},\ell_{2}}\Rightarrow H\models``y^{-1}_{\zeta}x_{\varepsilon_{1},\ell_{1}}y_{\zeta}=x_{\varepsilon_{2},\ell_{2}}"$ 2. $(b)$ for each $\zeta<\lambda$ there is a group $H$ extending $G^{5}_{\bar{f}}$ and $y\in G$ such that $F_{\zeta}(x_{\varepsilon_{1},\ell_{1}})=x_{\varepsilon_{2},\ell_{2}}\Rightarrow H\models``y^{-1}x_{\varepsilon_{1},\ell_{1}}y=x_{\varepsilon_{2},\ell_{2}}"$. ###### Proof.. Clause (a) suffice: We define a function $F$ from $X^{6}_{\bar{f}}$ into $H$ by: 1. $\bullet$ $F(x_{\varepsilon,\ell})$ is $x_{\varepsilon,\ell}\in G^{5}_{\bar{f}}\subseteq H$ if $\ell<5\wedge\varepsilon<\lambda$ is $y_{\zeta}$ if $\varepsilon=\zeta\wedge\ell=5$. Check that the mapping $F$ respects the equations in $\Gamma^{0}_{\bar{f}}\cup\Gamma^{1}_{\bar{f}}\cup\Gamma^{2}_{\bar{f}}$ hence it induces a homomorphism $F^{1}$ from $G^{6}_{\bar{f}}$ into $H$, for every group word $\sigma=\sigma(\ldots,x_{\varepsilon_{i},\ell_{i}},\ldots)_{i<n},x_{\varepsilon_{i},\ell_{i}}\in X^{5}_{\bar{f}}$, we have $G^{6}_{\bar{f}}\models``\sigma=e"\Rightarrow G^{5}_{\bar{f}}\models``\sigma=e"$, so we are done. Clause (b) suffice: Let $(H_{\zeta},y_{\zeta})$ for $\zeta<\lambda$ be as guaranteed by the assumption, i.e. clause (b). Without loss of generality $\zeta\neq\xi<\lambda=G_{\zeta}\cap G_{\xi}=G^{5}_{\bar{f}}$. Now clause (a) follows by using free amalgamation of $\langle H_{\zeta}:\zeta<\lambda\rangle$ over $G^{5}_{\bar{f}}$, we know it is as required in clause (a), see e.g. [LS77]. ∎ ###### Claim 2.14. 1) Clause (b) of 2.13 holds. 2) The conclusion of claim 2.13 holds. 3) The conclusions of 2.9 hold also for $G^{6}_{\bar{f}}$. ###### Proof.. 1) By the theorems on HNN extensions see [LS77] applied with the group being $G^{5}_{\bar{f}}$ and the partial automorphism $\pi_{\zeta}$ being the one $F_{\zeta}$ induced, i.e. 1. $\bullet$ ${\rm Dom}(\pi_{\zeta})$ is the subgroup of $G^{5}_{\bar{f}}$ generated by ${\rm Dom}(F_{\zeta})$ 2. $\bullet$ $\pi_{\zeta}(x_{\varepsilon,\ell})=F_{\zeta}(x_{\varepsilon,\ell})$ for $x_{\varepsilon,\ell}\in{\rm Dom}(F_{\zeta})$. By claim 2.12(2) and 2.10(2) we know that $\pi_{\zeta}$ is indeed an isomorphism. 2) Follows by 2.13 and 2.14. 3) By 2.9 and part (2). ∎ Proof of 2.1: Should be clear by now. $\qquad\qquad$ ###### Claim 2.15. The pair $(K_{{\rm lfgr}},K_{{\rm gr}})$ of classes, i.e. (locally finite groups, groups), has the olive property, as witnessed by $\bar{\varphi}$ from 2.2. ###### Proof.. We rely on observation 2.16 below and use its notation. Let $J=\\{(\alpha,\beta,\gamma):\alpha<\beta<\gamma<\lambda$ and $f_{\gamma}{\restriction}[\alpha,\beta]=0_{[\alpha,\beta]}\\}$. Let $G^{5}_{\bar{f}},G^{6}_{\bar{f}}$ be as in the proof of 2.1, that in Definition 2.7. Now for $\bar{\alpha}=(\alpha_{0},\alpha_{1},\alpha_{2})\in J$ let $\pi^{5}_{\bar{\alpha}}$ be the function from $X_{\bar{f},5}$ (see Definition 2.7) into $K$ defined by 1. $()_{1}$ $\pi^{5}_{\bar{\alpha}}(x_{\beta,k})$ is 1. $\bullet$ $e_{K}$ if $\beta\notin\\{\alpha_{0},\alpha_{1},\alpha_{2}\\}$ 2. $\bullet$ $z_{\ell,k}$ if $\beta=\alpha_{\ell}$. Now 1. $()_{2}$ $\pi^{5}_{\bar{\alpha}}$ respects the equations from $\Gamma^{2}_{\bar{f}}$. [Why? The equation $\sigma_{}(x_{\alpha_{0},0},x_{\alpha_{1},1},x_{\alpha_{2},4})=e$ holds as $K$ satisfies $\sigma_{}(z_{0,0},z_{1,1},z_{2,4})=e$. For the other equations see 2.3(c).] Let $\pi^{6}_{\bar{\alpha}}$ be the following function from $X^{6}_{\bar{f}}$ into $K$: 1. $()_{3}$ $\pi^{6}_{\bar{\alpha}}(x)$ is 1. $\bullet$ $\pi^{5}_{\bar{\alpha}}(x)$ when $x\in X^{5}_{\bar{f}}$ 2. $\bullet$ $z_{s}$ when $x=x_{\beta,5},\beta<\lambda$ and $s=s_{\bar{\alpha},\beta}:=(\\{\ell\leq 2:\alpha_{\ell}<\beta$ and $f_{\beta}(\alpha_{\ell})=0\\},\\{\ell\leq 2:\beta<\alpha_{\ell}$ and $f_{\alpha_{\ell}}(\beta)=1\\})$. [Why is $\pi^{6}_{\bar{\alpha}}$ as required? The least obvious point is: why $s\in S_{}$? Let $s=(u_{1},u_{1})$, now $\ell_{1}\in u_{1}\wedge\ell_{2}\in u_{2}\Rightarrow\alpha_{\ell_{1}}<\beta<\alpha_{\ell_{2}}\Rightarrow\ell_{1}<\ell_{2}$ and $(\\{0\\},\\{1,2\\})\neq s$ because $f_{\alpha_{2}}{\restriction}[\alpha_{0},\alpha_{1}]$ is constantly zero.] 1. $()_{4}$ $\pi^{6}_{\bar{\alpha}}$ respects the equations in $\Gamma^{0}_{\bar{f}}\cup\Gamma^{1}_{\bar{f}}$. [Why? Check the definitions.] By $()_{2},()_{4}$ there is a homomorphism $\pi_{\bar{\alpha}}$ from $G_{\bar{f}}$ into $K$. Let $G_{}$ be the product of $J$-copies of $K$, i.e. 1. $()_{5}$ $(a)\quad$ the set of elements of $G_{}$ is the set of functions $g$ from $J$ into $K$ 2. $(b)\quad G_{}\models``g_{1}g_{2}=g_{3}"$ iff $\bar{\alpha}\in J\Rightarrow K\models``g_{1}(\bar{\alpha})g_{2}(\bar{\alpha})=g_{3}(\bar{\alpha})"$ 3. $()_{6}$ $G_{}$ is a locally finite group 4. $()_{7}$ for $\alpha<\lambda,k<m$ let $\bar{g}_{\beta}=\langle g_{\beta,k}:k<m\rangle$ where $g^{}_{\beta,k}\in G_{}$ be defined by $(g_{\beta,k}(\bar{\alpha}))(x)=\pi^{6}_{\bar{\alpha}}(x_{\beta,k})$ 5. $()_{8}$ $G_{},\langle\bar{g}_{\beta}:\beta<\lambda\rangle$ witnesses the olive property. [Why? Check.] So we are done. ∎ ###### Observation 2.16. There are $K,z_{i,k}(i<3,k<m)$ and $\langle\pi_{s}:s\in S_{}\rangle$ such that: 1. $(a)$ $K$ is a finite group 2. $(b)$ $z_{i,k}\in K$ 3. $(c)$ $\sigma_{}(z_{0,0},z_{1,2},z_{2,4})=e$ but $\sigma_{}(z_{0,2},z_{1,3},z_{2,4})\neq e$ 4. $(d)$ $S_{}=\\{(u_{1},u_{2}):u_{1},u_{2}\subseteq\\{0,1,2\\}$ and $(\forall\ell_{1}\in u_{1})(\forall\ell_{2}\in u_{2})(\ell_{1}<\ell_{2})$ but $(u_{1},u_{2})\neq(\\{0\\},\\{1,2\\})$ 5. $(e)$ for $s=(u_{1},u_{2})\in S_{}$ we have: $\pi_{s}$ is a partial isomorphism of $K$ such that: 1. $(\alpha)$ if $\ell\in u_{1}$ then $\pi_{s}(x_{\ell,0})=x_{\ell,2}$ 2. $(\beta)$ if $\ell\in u_{2}$ then $\pi_{s}(x_{\ell,1})=x_{\ell,2},\pi_{s}(z_{\ell,4})=z_{\ell,4}$ 6. $(f)$ moreover there are $z_{s}\in K$ for $s\in S_{}$ such that $(\forall x\in{\rm Dom}(\pi_{s}))(\pi_{s}(x)=z^{-1}_{s}xz_{s})$. ###### Proof.. First, we ignore clause (f). We use finite nilpotent groups. Let $n_{2}=3m,n_{1}=\binom{n_{2}}{2},n_{0}=\binom{n_{1}}{2}$, let $f_{\ell}:[n_{\ell+1}]^{2}\rightarrow n_{\ell}$ be one-to-one for $\ell=0,1$. Let $K_{1}$ be the group generated by $\\{y_{j,\ell}:j\leq 2,\ell<n_{j}\\}$ freely except the equations 1. $()_{1}$ $(a)\quad y_{j,\ell}\cdot y_{j,\ell}=e$ 2. $(b)\quad[y_{j+1,\ell_{1}},y_{j+1,\ell_{2}}]=y_{j,f\\{\ell_{1},\ell_{2}\\}}$, i.e. $y^{-1}_{j+1,\ell_{1}}y^{-1}_{j+1,\ell_{2}}y_{j+1,\ell_{2}}y_{j+1,\ell_{2}}=y_{j,f\\{\ell_{1},\ell_{2}\\}}$ when $j<2,\ell_{1}<\ell_{2}<n_{j+1}$ 3. $(c)\quad[y_{j_{1},\ell_{1}},y_{j_{2},\ell_{2}}]=e$ when $(j_{1}=0=j_{2})\vee(j_{1}\neq j_{2}\leq 2)$ and $\ell_{1}<n_{j_{1}},\ell_{2}<n_{j_{2}}$. Clearly $K_{1}$ is finite. Let $z_{i,\ell}=y_{2,6i+\ell}$ for $i<3,\ell<m$, let $\ell_{}$ be such that $[[z_{0,0},z_{1,1}],z_{2,4}]=y_{0,\ell_{}}$. Let $K_{0}$ be the subgroup $\\{e,y_{0,\ell_{}}\\}$ of $K$, it is a normal subgroup as it is included in the center of $K_{1}$ and let $K_{2}=K_{1}/K_{0}$ and we define $z_{i,\ell}$ as $y_{i,\ell}/K_{0}$. Now 1. $()_{2}$ $K_{2},\langle z_{i,\ell}:i\leq 2,\ell<m\rangle$ are as required in (a)-(e) of the claim. [Why? We should just check that for $s\in S_{}$ there is $\pi_{s}$ as required, i.e. that some subgroups of $K_{2}$ generated by subsets of $\langle z_{i,\ell}:i\leq 2,\ell<m\rangle$ are isomorphic, but as none of them included $\\{z_{0,0},z_{1,1},z_{2,4}\\}$ and the way $K_{2}$ was defined this is straightforward.] Lastly, there is a finite group $K$ extending $K_{2}$ and $z_{s}\in K$ for $s\in S$ such that $x\in{\rm Dom}(\pi_{s})\Rightarrow z^{-1}_{s}xz_{s}=\pi_{s}(x)$. Why? Simply $K_{2}$ can be considered as a group of permutations of the set $K_{2}$ (e.g. multiplying from the right), and it is easy to find $z_{s}\in{\rm Sym}(K_{2})$ as required. ∎ ###### Conclusion 2.17. Assume ${\rm Qr}_{1}(\chi_{1},\chi_{2},\lambda)$. Then there is no sequence $\langle G_{\alpha}:\alpha<\alpha_{}\rangle$ of length $<\chi_{2}$ of groups of cardinality $\leq\chi_{1}$ such that any locally finite group $H$ of cardinality $\lambda$ can be embedded into at least one of them. So, e.g. ###### Conclusion 2.18. 1) If $\mu={\rm cf}(\mu),\mu^{+}<\lambda={\rm cf}(\lambda)<2^{\mu}$ then there is no group of cardinality $\lambda$ universal for the class of locally finite groups. 2) E.g. if $\aleph_{2}\leq\lambda={\rm cf}(\lambda)<2^{\aleph_{0}}$ this applies. ## 3\. Concluding remarks We may like to weaken the model theoretic condition but add to the property ${\rm Qr}$ of the relevant cardinals that “the $C_{\delta}$’s has few branches”. It is not clear whether there will be any gain. ###### Definition 3.1. $T$ has the $(\eta,\bar{k},m)--\Delta$-olive property when $\Delta\subseteq{\mathbb{L}}(\tau_{T})$ and for some $(\bar{\varphi}_{1},\bar{\varphi}_{2})$ we have (for every $\lambda$) 1. $(a)$ for $\iota=0,1$ we have $\bar{\varphi}_{\iota}=\langle\varphi_{\iota,\ell}(\bar{x}_{0},\dotsc,\bar{x}_{\ell-1}):\ell=2,\dotsc,k_{\iota}\rangle$ with $\varphi_{\iota,\ell}\in\Delta$ and $m=\ell g(\bar{x}_{0})=\ldots\ell g(\bar{x}_{k-1})$ 2. $(b)_{\lambda}$ for every $I\in K_{{\rm etr}}$ (see 3.4, old $\boxplus_{1}$ of the proof of LABEL:n2) we can find $\bar{\mathbb{a}}$ such that 1. $(\alpha)$ $\bar{\mathbb{a}}=\langle\bar{a}_{\eta}:\eta\in P_{I}\rangle$ 2. $(\beta)$ $\bar{a}_{\eta}={}^{m}{\mathfrak{C}}$ where ${\mathfrak{C}}={\mathfrak{C}}_{T}$ 3. $(\gamma)$ if $\iota<2,\ell\in\\{2,\dots,k\\}$ and $\bar{\eta}=\langle\eta_{0},\dotsc,\eta_{\ell-1}\rangle$ is an $(I,\iota)$-sequence (i.e. $\eta_{i}\in P_{I},F_{I,\iota}(\eta_{i})<^{{\rm tr}}_{I}\eta_{i+1}$ (when defined) then ${\mathfrak{C}}\models\varphi_{\iota,\ell}[\bar{a}_{\eta_{0}},\dotsc,\bar{a}_{\eta_{\ell-1}}]$ 3. $(c)$ there are no $\bar{a}_{i}\in{}^{m}{\mathfrak{C}}$ for $\ell<n+1$ such that: 4. $\bullet\quad$ if $\iota\in\\{0,1\\},\ell\in\\{2,\dotsc,k\\},i_{0}<\ldots<i_{k-1}<n$ and $\ell<k\Rightarrow\eta(\ell)=\iota$ then ${\mathfrak{C}}\models\varphi_{\iota,\ell}[\bar{a}_{i_{0}},\dotsc,\bar{a}_{i_{\ell-1}},\bar{a}_{n}]$. ###### Definition 3.2. We say an a.e.c. ${\mathfrak{k}}$ has the $(\eta,\bar{k},<\sigma)$-olive when : there are pairs of sequences of formulas $(\bar{\varphi}_{0},\bar{\varphi}_{1})$ which are ${\mathfrak{k}}$-upward invariant (see 0.13) with $\ell g(x_{\zeta})=\varepsilon<\sigma$ such that for every $I\in K_{{\rm etr}}$ (see LABEL:n2) of cardinality $\lambda$ there is $M\in K_{{\mathfrak{k}}}$ of cardinality $\lambda$ and $\bar{a}_{\eta}\in{}^{\varepsilon}M$ for $\eta\in P_{\eta}$ such that the parallel of 3.1 holds. ###### Discussion 3.3. The intention is to have a parallel of §1 with somewhat weaker version of the olive here, but the price is a somewhat stronger set theoretic condition. ###### Definition 3.4. 1) Let $K_{{\rm etr}}$ (expanded tree) be the class of structures $I=({\mathscr{T}},<_{{\rm lin}},<_{{\rm tr}},P,F_{0},F_{1})=({\mathscr{T}}_{I},<^{{\rm lin}}_{I},P_{I},F_{I,0},F_{I,1})$ satisfying: 1. $(a)$ ${\mathscr{T}}_{I}=({\mathscr{T}},<_{{\rm tr}})=({\mathscr{T}},\leq_{{\mathscr{T}}})=({\mathscr{T}}_{I},\leq^{{\rm tr}}_{I})$ is a partial order; moreover a well founded tree 2. $(b)$ $P\subseteq I$ 3. $(c)$ $F_{I,\ell}$ is a one-to-one function from $P_{I}$ into $I\backslash P_{I}$, for $\ell=0,1$ 4. $(d)$ ${\mathscr{T}}$ is the disjoint union of $P_{I},{\rm Rang}(F_{I,0}),{\rm Rang}(F_{I,1})$ 5. $(e)$ if $\ell<2$ and $t\in P_{I}$ then $F_{I,\ell}(t)$ is a successor of $t$, i.e. $t<_{I}F_{I,\ell}(t)$ and $\neg(\exists s\in I)(t<_{{\mathscr{T}}}s<_{{\mathscr{T}}}F_{{\mathscr{T}},\ell}(t))$ 6. $(f)$ if $t\in P_{I}$ and $t<_{{\mathscr{T}}}s$ then $\bigvee\limits_{\ell>n}F_{\ell}(t)\leq_{{\mathscr{T}}}s$ 7. $(g)$ $({\mathscr{T}},<_{{\rm lin}})=({\mathscr{T}},<^{{\rm lin}}_{I})$ is a linear order 8. $(h)$ if $\ell<2,s\in P_{I}$ and $F_{\ell}(s)\leq_{k}t_{\ell}$ for $\ell=0,1$ then $t_{0}<_{{\rm lin}}s<_{{\rm lin}}t_{s}$ 2) We define $K_{{\rm ftr}}$ as $\\{J_{I}:I\in K_{{\rm etr}}\\}$ where for $I\in K_{{\rm etr}}$ let $J_{I}$ be the structure $(P_{I},<^{{\rm tr}}_{I}{\restriction}P_{I},<^{{\rm lin}}_{I}{\restriction}P_{I},Q_{I,0},Q_{I,1}$ where $Q^{\iota}_{I}=\\{(\eta,\nu):\eta\in P_{I},\nu\in P_{I}$ and $F_{I,\iota}(\eta)\leq^{{\rm tr}}_{I}\nu\\}$. ###### Definition 3.5. 1) Let “$T$ have the $(\Delta,\eta,\bar{k},m)$-olive property where $\eta\in{}^{n}(\iota_{}),\bar{k}=\langle k_{\iota}=k(\iota):\iota<\iota_{}\rangle$, is defined (similarly to Definition 1.1 but 2 is replaced by $\iota_{}$ and $\bar{k}=(k_{0},k_{1})$ by $\bar{k}=\langle k_{\iota}:\iota<\iota_{}\rangle$) when there are formulas $\varphi(\bar{x}_{[m],0},\bar{x}_{[m],1}),\bar{\psi}_{\iota}=\langle\psi_{\iota,k}(\bar{x}_{[m],0},\dotsc,\bar{x}_{[m],k(\iota)}):k=1,\dotsc,k_{\iota}\rangle$ for $\iota<\iota_{}$ in $\Delta$ such that for every $\lambda$ 1. $(a)_{\lambda}$ for every $\bar{f}=\langle f_{\alpha}:\alpha<\lambda\rangle,f_{\alpha}:\alpha\rightarrow\iota_{}$ we can find $\bar{a}_{\alpha}\in{}^{m}{\mathfrak{C}}$ for $\alpha<\lambda$ such that 1. $(\alpha)$ $\psi_{\iota,1}[\bar{a}_{\alpha},\bar{a}_{\beta}]$ for $\alpha<\beta<\lambda$ such that $f_{\beta}(\alpha)=\iota$ 2. $(\beta)$ $\psi_{\iota,k}(\bar{a}_{\alpha_{0}},\dotsc,\bar{a}_{\alpha_{k(\iota)-1}},\bar{a}_{\beta})$ when $\iota<\iota_{},k=2,\dotsc,k_{\iota}$ and $\alpha_{0}<\ldots<\alpha_{k(\iota)-1}<\beta<\lambda$ and $f_{\beta}{\restriction}[\alpha_{0},\alpha_{k(\iota)-1}]$ is contantly $\iota$ 2. $(b)$ there are no $\bar{a}_{0},\dotsc,\bar{a}_{n}\in{}^{m}{\mathfrak{C}}$ such that: 1. $(\alpha)$ $\psi_{\iota,1}(\bar{a}_{\ell(1)},\bar{a}_{\ell(2)})$ when $\ell(1)<\ell(2)$ 2. $(\beta)$ $\psi_{\iota,k}(\bar{a}_{\ell(0)},\bar{a}_{\ell(1)},\dotsc,\bar{a}_{\ell(k-1)},\bar{a}_{n})$ when $\ell(1)<\ell(2)<\ldots<\ell(k(\iota)-1)<n$ and $\iota=\eta(\ell(0))=\eta(\ell(1))$. 2) Relatives are as in Definitions 1.1, 1.3. ###### Remark 3.6. To apply 3.5 we may replace $\mathbb{F}_{\lambda}$ by: for $\mathbb{n}\leq\omega$ 1. $()$ $F^{\mathbb{n}}_{\lambda,\iota}$ is the set of $\bar{f}=\langle f_{\alpha}:\alpha<\lambda\rangle$ such that $f_{\alpha}:[\alpha]^{<\mathbb{n}}\rightarrow\iota$. ###### Remark 3.7. Note that it does not matter if we use 1. $(a)$ $T$ universal with JEP and amalgamation, $\Delta\subseteq{\rm qf}$, no function symbols or 2. $(b)$ $T$ compelte, $\Delta={\mathbb{L}}(\tau_{T})$. Why? 1) Given a complete $T$, let $T^{\prime}$ be $T\cup\\{(\forall\bar{x}_{[m]})(\varphi(\bar{x}_{[m]})\equiv R_{\varphi(\bar{y}_{[m]})}(\bar{x}_{[m]})):\varphi(\bar{x})\in{\mathbb{L}}(\tau_{T})\\}$ where $\langle R_{\varphi(\bar{x}_{[m]})}:\varphi(\bar{x}_{[m]})\in{\mathbb{L}}(\tau_{T})\rangle$ are new with no repetitions. Let $T^{\prime\prime}$ be the universal theory in the vocabulary $\tau^{\prime\prime}=\\{R_{\varphi(\bar{x}_{[m]})}:\varphi(\bar{x}_{[m]})\in{\mathbb{L}}(\tau_{T})\\}$ such that ${\rm Mod}_{T^{\prime\prime}}=\\{N^{\prime\prime}$: there is $N^{\prime}\models T^{\prime}$ such that $N^{\prime\prime}\subseteq(N^{\prime}{\restriction}\tau^{\prime\prime})\\}$. So $T^{\prime}$ is complete with elimination of quantifiers and $T^{\prime\prime}$ universal with amalgamation and JEP with no function symbols and ${\rm univ}(\chi_{1},\lambda,T)={\rm univ}(\chi_{1},\lambda,T^{\prime})={\rm univ}(\chi_{1},\lambda,T^{\prime\prime})$, recalling the first is for $\prec$, elementary embeddings and the second and third for $\subseteq$, embeddings. 2) If $T$ is universal (not complete) with the JEP (otherwise univerality is a dull question) and amalgamation let $T^{\prime}={\rm Th}(M)$ for some $M\in{\rm Mod}_{T}$ which is existentially closed. Now 1. $(a)$ ${\rm univ}(\chi_{1},\lambda,T^{\prime})\leq{\rm univ}(\chi_{1},\lambda,T)$. [Why? Let $\chi_{2}={\rm univ}(\chi_{1},\lambda,T)$ and $\langle M_{\alpha}:\alpha<\chi_{2}\rangle$ exemplify it. For each $i$ there is $N_{\alpha}$ such that $M_{\alpha}\subseteq N_{\alpha}\in{\rm Mod}_{T_{i}}$ and without loss of generality $\\|N_{\alpha}\\|=\lambda$, etc.] 1. $(b)$ ${\rm univ}(\chi_{1},\lambda,T)\leq{\rm univ}(\chi_{1},\lambda,T)$. [Why? Also easy.] ## References * [LS77] Roger C. Lyndon and Paul E. Schupp, _Combinatorial group theory_ , Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 89, Springer-Verlag, Berlin–Heidelberg–New York, 1977. * [Mir05] Džamonja Mirna, _Club guessing and the universal models_ , Notre Dame J. Formal Logic 46 (2005), 283–300. * [Sh:h] Saharon Shelah, _Classification Theory for Abstract Elementary Classes_ , Studies in Logic: Mathematical logic and foundations, vol. 18, College Publications, 2009. * [Sh:100] by same author, _Independence results_ , The Journal of Symbolic Logic 45 (1980), 563–573. * [GrSh:174] Rami Grossberg and Saharon Shelah, _On universal locally finite groups_ , Israel Journal of Mathematics 44 (1983), 289–302. * [KjSh:409] Menachem Kojman and Saharon Shelah, _Non-existence of Universal Orders in Many Cardinals_ , Journal of Symbolic Logic 57 (1992), 875–891, math.LO/9209201. * [Sh:420] Saharon Shelah, _Advances in Cardinal Arithmetic_ , Finite and Infinite Combinatorics in Sets and Logic, Kluwer Academic Publishers, 1993, N.W. Sauer et al (eds.). 0708.1979, pp. 355–383. * [Sh:457] by same author, _The Universality Spectrum: Consistency for more classes_ , Combinatorics, Paul Erdős is Eighty, vol. 1, Bolyai Society Mathematical Studies, 1993, Proceedings of the Meeting in honour of P.Erdős, Keszthely, Hungary 7.1993; A corrected version available as ftp: //ftp.math.ufl.edu/pub/settheory/shelah/457.tex. math.LO/9412229, pp. 403–420. * [Sh:500] by same author, _Toward classifying unstable theories_ , Annals of Pure and Applied Logic 80 (1996), 229–255, math.LO/9508205. * [Sh:589] by same author, _Applications of PCF theory_ , Journal of Symbolic Logic 65 (2000), 1624–1674, math.LO/9804155. * [DjSh:614] Mirna Dzamonja and Saharon Shelah, _On the existence of universal models_ , Archive for Mathematical Logic 43 (2004), 901–936, math.LO/9805149. * [Sh:702] Saharon Shelah, _On what I do not understand (and have something to say), model theory_ , Mathematica Japonica 51 (2000), 329–377, math.LO/9910158. * [ShUs:789] Saharon Shelah and Alex Usvyatsov, _Banach spaces and groups - order properties and universal models_ , Israel Journal of Mathematics 152 (2006), 245–270, math.LO/0303325. * [Sh:F1330] Saharon Shelah, _Universal models consistently exist: revisited_. * [Sh:F1414] by same author, _Minimal universality spectrum_. * [Sh:F1425] by same author, _Theorem with minimal universality spectrum_.	arxiv-papers	2013-11-20T10:10:37	2024-09-04T02:49:53.937652	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/4.0/", "authors": "Saharon Shelah", "submitter": "Saharon Shelah", "url": "https://arxiv.org/abs/1311.4997" }
1311.5006	Alma Mater Studiorum $\cdot$ Università di Bologna FACOLTÀ DI SCIENZE MATEMATICHE, FISICHE E NATURALI Corso di Laurea Magistrale in Informatica INDAGINI IN DEEP INFERENCE Tesi di Laurea in Tipi e Linguaggi di Programmazione Relatore: Chiar.mo Prof. Simone Martini Presentata da: Andrea Simonetto II Sessione Anno Accademico 2009/2010 _Alla venerata memoria di mio nonno, Gino Simonetto._ _Wir müssen wissen. Wir werden wissen. (David Hilbert)_ ## Introduzione La matematica ama parlare di sé stessa. Dalla teoria dei numeri – ritenuta da Gauss la “regina della matematica” – all’analisi – l’hilbertiana “sinfonia coerente dell’universo” – la matematica pura è abituata a porsi ad oggetto delle proprie speculazioni. Anche la _logica matematica_ possiede spiccate tendenze narcisistiche111Secondo Locke, la logica è “l’anatomia del pensiero”. Occorre tuttavia precisare che usando il termine “logica”, Locke intendeva quella parte di filosofia che studia il ragionamento e l’argomentazione; la logica matematica cominciò a fiorire solo un secolo e mezzo dopo la sua morte.; nondimeno, vista la sua propensione ad affrontare questioni di fondamento, è certamente una delle branche che ama maggiormente parlare di matematica. Tra i vari contributi è doveroso citare, come esempi eccellenti, l’ _aritmetica di Peano_ e la _teoria assiomatica degli insiemi_ di Zermelo–Fraenkel. Uno dei concetti cardine di tutta la matematica è quello di _dimostrazione_. Per studiare questi oggetti, i logici hanno sviluppato un’infrastruttura nota come _teoria della dimostrazione_. In questa tesi faremo una panoramica sulla teoria della dimostrazione, concentrandoci su uno degli sviluppi più recenti, una tecnica nota come _deep inference_. La deep inference è una nuova _metodologia_ in proof theory utile a progettare _famiglie di sistemi formali_ (o _formalismi_) con buone proprietà, quali: * • l’ _efficienza_ nella rappresentazione delle dimostrazioni: alcuni sistemi rendono disponibili dimostrazioni più brevi di quanto possano fare altri (questi aspetti sono studiati in _complessità delle dimostrazioni_ o _proof complexity_); * • _analiticità_ : alcuni sistemi vengono naturalmente con algoritmi di _ricerca delle dimostrazioni_ (o di _proof search_) _efficienti_ , altri no, altri ancora solo con alcuni accorgimenti. L’analiticità è la proprietà chiave per ottenere algoritmi di proof search efficienti; * • l’abilità di esprimere dimostrazioni che sono matematicamente naturali, cioè senza artefatti sintattici “amministrativi” (si parla in questi casi di _burocrazia delle dimostrazioni_). Uno dei problemi di ricerca principali in proof theory è trovare una buona corrispondenza tra le dimostrazioni e il loro significato. In particolare, il problema dell’ _identità delle dimostrazioni_ è prominente, e consiste nel trovare nozioni di equivalenza tra dimostrazioni non banali, supportate da semantiche appropriate alle dimostrazioni e ai sistemi formali. I _formalismi_ controllano, in larga parte, la progettazione delle regole d’inferenza. Per esempio, la deduzione naturale prescrive che, per ogni connettivo, siano date due regole: una d’introduzione e una di eliminazione. In tutti i formalismi tradizionali (ma anche in quelli più moderni derivati dai primi), viene adottato un meccanismo noto come _shallow inference_ o _inferenza di superficie_ , nel quale le regole di inferenza operano sui connettivi più prossimi alla radice delle formule – quando vengono viste come alberi, cioè quando ci si concentra sulla loro _struttura sintattica_. L’inferenza di superficie è una metodologia molto naturale, poiché permette di procedere per induzione strutturale diretta sulle formule. Tuttavia non è ottimale riguardo alcune proprietà dei sistemi formali, quali quelle sopra menzionate. In particolare: * • sembra sia incapace di fornire formalismi analitici che siano efficienti riguardo la complessità delle dimostrazioni; * • le dimostrazioni tendono ad avere molta burocrazia, cioè rappresentazioni sintatticamente complesse degli argomenti matematici. Inoltre la shallow inference fatica a relazionarsi con le logiche modali. Teorie logiche modali possono essere definite nei sistemi di Frege-Hilbert, ma ottenere analiticità per essi è una sfida molto ardua, in alcuni casi ancora irrisolta. In più è altrettanto difficile (se non addirittura impossibile) esprimere sistemi formali per logiche non-commutative. Nel seguito mostreremo alcuni tra i maggiori risultati di teoria della dimostrazione ottenuti mediante un formalismo deep inference, noto come _calcolo delle strutture_. Il calcolo delle strutture è un contributo importante nello sviluppo della metodologia deep inference, per la sua semplicità e la sua somiglianza coi formalismi tradizionali. Grazie al calcolo delle strutture, sono stati ottenuti i seguenti risultati: * • la logica classica, intuizionista, lineare a alcune logiche modali possono essere espresse in sistemi che godono di analiticità; * • è possibile esprimere logiche lineari munite di _operazioni_ _non-commuta- tive_ in sistemi analitici, e si dimostra che queste logiche non possono essere formalizzate analiticamente nel calcolo dei sequenti; inoltre questi sistemi logici sono in forte corrispondenza con le algebre di processo; * • sono state sviluppate tecniche nuove e generali di normalizzazione, e sono state scoperte _nozioni del tutto nuove_ di normalizzazione, in aggiunta alla tradizionale _cut elimination_ ; * • la maggior parte dei sistemi sviluppati sono costituiti interamente da regole d’inferenza _locali_ ; una regola d’inferenza locale ha _complessità computazionale costante_. La località è una proprietà difficile da conseguire, e non è ottenibile nel calcolo dei sequenti per la logica classica; * • i sistemi ottenuti sono estremamente modulari; questo significa una _forte indipendenza tra le regole d’inferenza_ ; * • moti sistemi sono stati implementati, grazie a tecniche che producono regole d’inferenza atte a migliorare l’efficienza senza sacrificare le proprietà teoriche; * • tutti i sistemi ottenuti sono semplici, nel senso che le regole d’inferenza sono _contenute e intelligibili_. Il calcolo delle strutture è una generalizzazione di molti formalismi shallow inference, in particolare del calcolo dei sequenti. Questo significa che ogni dimostrazione data in questi formalismi shallow inference, può essere “mimata” nel calcolo delle strutture, preservandone la complessità e senza perdita di proprietà strutturali. ###### Contents 1. 1 Formalismi e metodologie 1. 1.1 Linguaggi formali 2. 1.2 Meta-livello 3. 1.3 Sistemi formali e formalismi 4. 1.4 Metodologie: shallow _versus_ deep inference 2. 2 Logica classica proposizionale 1. 2.1 Eliminazione del taglio 2. 2.2 Deep inference e simmetria 1. 2.2.1 Sistema SKS generalizzato 2. 2.2.2 Località: il Sistema SKS 3. 2.2.3 Rompere la simmetria: il Sistema KS 3. 3 Logica lineare 1. 3.1 Calcolo dei sequenti lineari 2. 3.2 Sistema LBV 1. 3.2.1 Eliminazione del taglio 2. 3.2.2 Un’interpretazione operazionale 4. Conclusioni ###### List of Figures 1. 1.1 Definizioni induttive di _saturazione_ e _ordine_ di un contesto 2. 1.2 Sistema formale in shallow inference ed esempio di formula 3. 1.3 Sistema formale in deep inference ed esempio di dimostrazione 4. 2.1 Equivalenza tra formule di SKSg 5. 2.2 Sistema deduttivo SKSg 6. 2.3 Regole del Sistema _locale_ SKS 7. 2.4 Regole del Sistema KS 8. 3.1 Definizione di negazione e implicazione lineari 9. 3.2 Sistema deduttivo per CLL 10. 3.3 Sistema MLL+mix 11. 3.4 Sistema LBV+cut, equivalenza tra formule e negazione 12. 3.5 Definizione dell’operatore di _merge_ ## Chapter 1 Formalismi e metodologie La teoria della dimostrazione nacque sul finire del XIX secolo, ad opera di David Hilbert e dei suoi collaboratori. Fu un periodo in cui la matematica visse una crisi senza precedenti, intaccata in prossimità dei suoi fondamenti logici dal manifestarsi di una serie di paradossi, proprio mentre la scuola intuizionista di Brouwer ne metteva in dubbio alcuni princìpi filosofici basilari, fatti che sommati minavano alla base la maggior parte della matematica esistente. Tuttavia, forti evidenze empiriche suffragavano la matematica conosciuta, e molti matematici rifiutarono di abbandonarla o rifondarla: tra loro, Hilbert avanzò un programma di salvataggio completo. Egli propose di formulare la matematica classica come teoria formale assiomatica, e in seguito di provarne la _consistenza_ (ossia la non contraddittorietà). Prima della proposta di Hilbert, la consistenza di teorie assiomatiche veniva provata esibendo un “modello”: data una teoria assiomatica, un _modello_ è un sistema di oggetti, presi da qualche altra teoria, tali da soddisfare gli assiomi, cioè, ad ogni oggetto o nozione primitiva della teoria assiomatica, viene fatto corrispondere un oggetto o una nozione dell’altra teoria, in modo tale che gli assiomi corrispondano a teoremi nell’altra teoria. Se l’altra teoria è consistente, anche quella assiomatica deve esserlo. Un esempio famoso è dato dalla dimostrazione di Beltrami (1868) della consistenza della geometria iperbolica: egli provò che le rette nel piano non-euclideo della geometria iperbolica, potevano essere rappresentate dalle geodesiche su una superficie di curvatura costante e negativa nello spazio euclideo. Da questo concluse che il piano iperbolico dev’essere consistente, a patto che la geometria euclidea lo sia. È chiaro che il metodo del modello è relativo. La teoria assiomatica è consistente solo se il suo modello lo è. Ma per provare l’assoluta consistenza della matematica classica, il metodo dei modelli non offriva speranze: nessuna teoria matematica era accettabile come modello, poiché da ognuna di esse sarebbe fatalmente riemerso il problema di partenza, cioè dimostrarne la consistenza. Hilbert propose di affrontare il problema in maniera diretta: per provare la consistenza di una teoria, si deve dimostrare al suo interno una proposizione sulla teoria stessa, cioè un teorema su tutte le possibili dimostrazioni della teoria. La branca di matematica che si occupa di questi aspetti di formalizzazione e riflessione, venne battezzata da Hilbert “metamatematica” o “teoria della dimostrazione”. Purtroppo il sogno di Hilbert s’infranse nel 1931 con il Secondo Teorema d’Incompletezza di Gödel [1931], che enuncia l’impossibilità per sistemi abbastanza espressivi da formalizzare l’aritmetica di dimostrare la propria consistenza: purtroppo la quasi totalità della matematica da salvare, passava per l’aritmetica. Tuttavia la teoria della dimostrazione sopravvisse a questo scossone, diventando importante in vari ambiti, tra i quali l’informatica. In informatica, i dimostratori automatici di teoremi richiedono uno studio della struttura combinatoriale delle dimostrazioni, mentre nella programmazione logica la deduzione è usata come fondamento della computazione. Inoltre esistono forti connessioni tra sistemi logici e linguaggi di programmazione funzionali, e tecniche di proof theory sono state utilizzate per porre dei vincoli di complessità computazionale ad alcuni di questi linguaggi (ad esempio in Girard [1995a]). Uno dei princìpi fondamentali in proof theory è che la formalizzazione di una teoria richiede una totale astrazione dal significato, cioè un _sistema formale_ dovrebbe essere una mera manipolazione simbolica spogliata di ogni interpretazione semantica. Dato un sistema formale, distinguiamo il livello rigoroso del sistema stesso (o _livello oggetto_), dal livello in cui esso viene studiato (il _meta-livello_) espresso nel linguaggio della matematica intuitiva e informale. Inoltre, per essere convincenti, gli strumenti usati nelle meta-teorie dovrebbero essere ristretti a tecniche – chiamate _finitarie_ dai formalisti, o, in un accezione più moderna, _combinatorie_ – che impiegano solo oggetti intuitivi e processi effettivi (in accordo con la scuola intuizionista). Nessuna classe infinita di oggetti deve poter essere trattata come un “tutto”; le prove di esistenza dovrebbero sempre esibire, almeno implicitamente, un testimone. La proof theory è dunque una collezione di meta-teorie finitarie, espresse nel linguaggio ordinario e con l’ausilio di simboli matematici – come variabili di meta-livello (o _meta-variabili_) introdotte ove necessario – tali da caratterizzare le proprietà dei vari sistemi formali. In questo capitolo introdurremo le nozioni di base della teoria della dimostrazione, partendo da insiemi finiti e generando quelli infiniti con procedure effettive, calcolabili. Quella di cui abbiamo discusso finora, va oggi sotto il nome di teoria della dimostrazione _strutturale_ , cioè un’analisi combinatoriale della struttura delle dimostrazioni formali; gli strumenti centrali sono il _Teorema di eliminazione del taglio_ e quello di _normalizzazione_. Il percorso che seguiremo in questo capitolo è liberamente ispirato a Kleene [1952], e si articola in quattro sezioni, le prime tre piuttosto standard, mentre la quarta aggiunta appositamente per trattare il tema della tesi, ossia la deep inference. Nell’ordine: 1. 1. si definirà uno strumento linguistico formale, in grado di produrre dei _linguaggi oggetto_ che costituiranno gli elementi base della logica da indagare; 2. 2. sarà introdotto un livello linguistico formale superiore (o _meta-livello_), tale da permetterci di ragionare sui vari linguaggi oggetto, e saranno date le definizioni e gli strumenti di indagine basilari; 3. 3. verranno formalizzati i concetti di _deduzione_ e di _dimostrazione_ , a partire da un generico _linguaggio oggetto_ , usando gli strumenti del _meta- livello_ , e saranno presi in esame i _formalismi_ (i.e. le famiglie di sistemi) esprimibili con tali strumenti; 4. 4. si definirranno e si metteranno a confronto le due _metodologie di inferenza_ : di superficie e di profondità (shallow _versus_ deep inference). Affronteremo il tutto sempre tenedo presente il vincolo di effettiva costruibilità delle procedure e la caratterizzazione combinatoria delle tecniche e degli strumenti via via introdotti, in pieno stile formalista. ### 1.1 Linguaggi formali In questa sezione svilupperemo le basi di linguaggi formali che utilizzeremo da qui in avanti. Alcuni concetti saranno forniti in maniera intuitiva, altri in modo più preciso: per approfondimenti su linguaggi formali e grammatiche, si rimanda ad Aho and Ullman [1972]; Aho et al. [2006]. ###### Definizione 1.1.1 (Alfabeti, stringhe, linguaggi, sottolinguaggi). Un _alfabeto_ $\Sigma$ è un insieme finito non vuoto di _simboli_. Una _stringa su un alfabeto_ 111Per convenzione useremo le lettere minuscole prese dall’inizio dell’alfabeto inglese per denotare i simboli, e le lettere minuscole alla fine dell’alfabeto, di solito $w,x,y,z$, per denotare le stringhe. $\Sigma$ è una sequenza finita di simboli scelti da $\Sigma$. La _stringa vuota_ $\epsilon$ è la stringa composta da zero simboli; essa è una stringa su qualunque alfabeto. Siano $x=a_{1}\cdots a_{n}$ e $y=b_{1}\cdots b_{m}$ due stringhe su un alfabeto $\Sigma$: la loro _concatenazione_ (si denota giustapponendo $x$ a $y$) è la stringa $xy=a_{1}\cdots a_{n}b_{1}\cdots b_{m}$. In particolare, per ogni stringa $w$, si ha $\epsilon w=w\epsilon=w$. Dato un alfabeto $\Sigma$, definiamo: $\begin{array}[]{lclll}\Sigma^{0}&=&\\{\epsilon\\}&&\mbox{(Stringa di \emph{lunghezza} $0$)}\\\ \Sigma^{n+1}&=&\\{aw\>\|\>a\in\Sigma,w\in\Sigma^{n}\\}&&\mbox{(Stringhe di \emph{lunghezza} $n+1$)}\\\ \Sigma^{}&=&\bigcup_{n\in\mathbb{N}}\Sigma^{n}&&\mbox{(Stringhe su $\Sigma$)}\end{array}$ Un _linguaggio_ $\mathscr{L}$ _su un alfabeto_ $\Sigma$ è un’insieme di stringhe su $\Sigma$ (cioè $\mathscr{L}\subseteq\Sigma^{}$). Infine, dato un linguaggio $\mathscr{L}$, si definisce _sottolinguaggio di_ $\mathscr{L}$ qualunque insieme di stringhe $\mathscr{L}^{\prime}\subseteq\mathscr{L}$. ###### Definizione 1.1.2 (Grammatiche, linguaggio generato). Una grammatica è una quadrupla $G=(\Sigma,\mathcal{C},S,\mathcal{P})$, dove: * • $\Sigma$ è un alfabeto di _simboli grammaticali_ ; * • $\mathcal{C}\subseteq\Sigma$ è un insieme di simboli, detti _categorie sintattiche_ (o _simboli non terminali_ , in contrapposizione con gli altri simboli grammaticali $\Sigma{\smallsetminus}\mathcal{C}$, chiamati _simboli terminali_); * • $S\in\mathcal{C}$ è una particolare categoria sintattica, chiamata _simbolo iniziale_ , che rappresenta il linguaggio da definire; * • $\mathcal{P}\subseteq\Sigma^{}{\times}\Sigma^{}$ è un insieme di coppie di stringhe $(\alpha,\beta)\in\Sigma^{}{\times}\Sigma^{}$, chiamate _produzioni grammaticali_. $\alpha$ è chiamata _testa della produzione_ , mentre $\beta$ è il _corpo della produzione_. Data una grammatica $(\Sigma,\mathcal{C},S,\mathcal{P})$, la _riscrittura ad un passo_ ($\leadsto$) è un’applicazione di una delle produzioni in $\mathcal{P}$. Formalmente: siano $\alpha,\beta,x,y\in\Sigma^{}$, allora: $x\alpha y\leadsto x\beta y\quad\mbox{ sse }\quad\mbox{esiste }(\alpha,\beta)\in\mathcal{P}$ La _riscrittura multipasso_ ($\leadsto^{}$) è la chiusura riflessiva e transitiva di quella ad un passo: $w_{1}\leadsto^{}w_{n}\quad\mbox{ sse }\quad w_{1}\leadsto\cdots\leadsto w_{n}\quad\mbox{per qualche $n\geq 0$}$ In particolare $w\leadsto^{}w$ per ogni $w\in\Sigma^{}$. Il _linguaggio generato da una grammatica_ $G=(\Sigma,\mathcal{C},S,\mathcal{P})$ (denotato con $\mathscr{L}_{G}$) è l’insieme delle stringhe di simboli terminali ottenibili tramite riscrittura multipasso a partire dal simbolo iniziale. In simboli: $\mathscr{L}_{G}=\\{w\in(\Sigma{\smallsetminus}\mathcal{C})^{}\>\|\>S\leadsto^{}w\\}$ ###### Definizione 1.1.3 (Grammatiche context-free, BNF). Sia $G=(\Sigma,\mathcal{C},S,\mathcal{P})$ una grammatica. Una _regola di produzione_ è _context-free_ se è della forma $(P,\beta)$ con $P\in\mathcal{C}$ e $\beta\in\Sigma^{}$. Una _grammatica_ si dice _context- free_ se ogni sua regola di produzione è context-free. Un modo compatto ed elegante per scrivere le regole di produzione grammaticale, è quello di usare la _forma di Backus-Naur_ o BNF (Backus et al. [1960]). Sia $G=(\Sigma,\mathcal{C},S,\mathcal{P})$ una grammatica e sia: $\mathcal{P}=\\{(\alpha_{1},\beta_{1,1}),\ldots,(\alpha_{1},\beta_{1,n_{1}}),\ldots,(\alpha_{k},\beta_{k,1}),\ldots,(\alpha_{k},\beta_{k,n_{k}}),\ldots\\}$ il suo insieme di produzioni. Allora $\mathcal{P}$ in BNF si rappresenta come segue: $\displaystyle\alpha_{1}$ $\displaystyle::=$ $\displaystyle\beta_{1,1}\>\|\>\beta_{1,2}\>\|\>\cdots\>\|\>\beta_{1,n_{1}}$ $\displaystyle\cdots$ $\displaystyle\cdots$ $\displaystyle\cdots$ $\displaystyle\alpha_{k}$ $\displaystyle::=$ $\displaystyle\beta_{k,1}\>\|\>\cdots\>\|\>\beta_{k,n_{k}}$ $\displaystyle\cdots$ $\displaystyle\cdots$ $\displaystyle\cdots$ L’ultimo (ma non ultimo) strumento sintattico che consideriamo, serve per fare emergere le _profonde simmetrie_ che soggiacciono ai sistemi logici formali, ed è uno degli strumenti più usati in proof theory: il sequente. I sequenti sono una notazione sintattica, finalizzata ad inserire le formule logiche in ambienti adatti al ragionamento logico-deduttivo. Più precisamente: ###### Definizione 1.1.4 (Sequente). Dato un linguaggio $\mathscr{L}$, un _sequente_ è un espressione del tipo: $\Gamma\vdash\Delta$ dove $\Gamma,\Delta$ sono _liste finite_ (eventualmente vuote) di stringhe di $\mathscr{L}$ – con le usuali operazioni definite sulle liste: $\Gamma,P$ è l’aggiunta di una stringa $P$ in coda ad una lista $\Gamma$, mentre $\Gamma,\Gamma^{\prime}$ è la concatenazione delle liste $\Gamma$ e $\Gamma^{\prime}$; non ci sono simboli per la lista vuota. Il simbolo $\vdash$ è noto come _turnstile_ o _tornello_ e fu originariamente introdotto in Frege [1879]. L’idea intuitiva è che il sequente afferma (ipotizza) la deducibilità di almeno una formula logica in $\Delta$ a partire dalle premesse in $\Gamma$. Se $\Gamma=P_{1},\ldots,P_{n}$ e $\Delta=Q_{1},\ldots,Q_{m}$, il sequente $\Gamma\vdash\Delta$ è da intendersi come: Se $P_{1}$ _e_ $\cdots$ _e_ $P_{n}$ allora $Q_{1}$ _oppure_ $\cdots$ _oppure_ $Q_{m}$ dove i significati di _“Se … allora …”_ , _“e”_ ed _“oppure”_ devono essere resi espliciti in maniera formale. Il sequente avente una lista vuota alla _destra_ del tornello ($\Gamma\vdash\leavevmode\nobreak\ $), afferma l’ _inconsistenza delle premesse_ , quello avente la lista vuota alla _sinistra_ del tornello ($\leavevmode\nobreak\ \vdash\Delta$), afferma che $\Delta$ è un _teorema_ , ossia che è vero a prescindere da ogni premessa. Il _sequente vuoto_ (cioè avente liste vuote alla _destra_ ed alla _sinistra_ del tornello) _afferma il falso_ (se in un sistema logico-formale si è in grado di _dimostrare il falso_ , allora esso è _inconsistente_). ### 1.2 Meta-livello In questa sezione introdurremo i principali strumenti meta-linguistici: in genere nei testi di logica questo aspetto è lasciato perlopiù ad un livello intuitivo. Ho cercato, con questa presentazione originale, di renderli più formali perché, sebbene spesso sottovalutati, ritengo che offrano alcuni interessanti spunti di riflessione. Osserviamo come, data una grammatica: $G=(\Sigma,\mathcal{C}=\\{S_{1},\ldots,S_{n}\\},S_{i},\mathcal{P})$ al variare di $i\in\\{1,\ldots,n\\}$ si producano linguaggi $\mathscr{L}_{i}$ diversi, seppur correlati tra loro, in funzione di quale categoria sintattica scegliamo come simbolo iniziale. ###### Definizione 1.2.1 (Meta-variabili, meta-linguaggi, (sotto)formule). Per ogni categoria sintattica $S_{i}$, definiamo un insieme finito $\mathscr{M}_{i}$ di _meta-variabili_ di categoria $S_{i}$, che sono dei “segnaposti” per una qualche stringa di $\mathscr{L}_{i}$. Grazie alle meta- variabili, possiamo imporre dei vincoli sulla forma delle stringhe di $\mathscr{L}_{G}$. Il _meta-livello linguistico_ $\mathscr{L}_{G}^{\prime}$ è quello in cui, giunti ad un certo passo di riscrittura, sostituiamo ad ogni occorrenza di simboli non terminali, una meta-variabile di categoria corrispondente. Il _meta-alfabeto_ è composto dai terminali e dalle meta-variabili: $\Sigma^{\prime}=\Sigma{\smallsetminus}\mathcal{C}\cup\bigcup_{1\leq i\leq n}\mathscr{M}_{i}$ mentre il _meta-linguaggio_ di $G$ è definito da: $\mathscr{L}_{G}^{\prime}=\bigcup\\{\varphi(w)\>\|\>S\leadsto^{}w\\}$ dove $\varphi(w)$ sta per “qualunque sostituzione di metavariabili al posto dei simboli non terminali in $w$”. In simboli, se $a$ denota un terminale: $\displaystyle\varphi$ $\displaystyle:$ $\displaystyle\Sigma^{}\rightarrow\mathscr{P}(\Sigma^{\prime})$ $\displaystyle\varphi(\epsilon)$ $\displaystyle=$ $\displaystyle\\{\epsilon\\}$ $\displaystyle\varphi(aw)$ $\displaystyle=$ $\displaystyle\\{ay\>\|\>y\in\varphi(w)\\}$ $\displaystyle\varphi(S_{i}w)$ $\displaystyle=$ $\displaystyle\\{\sigma y\>\|\>\sigma\in\mathscr{M}_{i},y\in\varphi(w)\\}$ Usiamo l’appellativo _formula_ per riferirci alle stringhe del meta-linguaggio $\mathscr{L}_{G}^{\prime}$. Data una formula $w$ appartenente a $\mathscr{L}_{G}^{\prime}$, per _sottoformula_ s’intende una qualunque porzione di $w$, che sia a sua volta compresa in $\mathscr{L}_{G}^{\prime}$. È immediato dimostrare che, data una grammatica $G$, il meta-linguaggio è più ricco del linguaggio, cioè che, per ogni $G$: $\mathscr{L}_{G}\subset\mathscr{L}_{G}^{\prime}$ Infatti, al meta-linguaggio appartengono banalmente tutte le stringhe di $\mathscr{L}_{G}$ (se spingiamo la riscrittura fino a produrre stringhe di terminali, la funzione $\varphi$ non fa niente), mentre in $\mathscr{L}_{G}$ non ci sono le meta-variabili e quindi è strettamente incluso. Le meta-variabili si _istanziano_ a stringhe del linguaggio oggetto tramite _unificazione_ , grazie alla quale è anche possibile eseguire delle _istanze parziali_ tra meta-variabili e altre formule del meta-linguaggio. Osserviamo che i linguaggi sono insiemi infiniti, così come il loro meta- livello genera insiemi infiniti. Tuttavia la base da cui sono generati questi insiemi (la grammatica) è finita e la procedura di generazione è concreta. Inoltre per grammatiche context-free verificare se una stringa appartiene o meno al linguaggio generato è un problema decidibile (in tempo polinomiale), e l’operazione di unificazione è anch’essa effettiva. Insomma, tutti gli strumenti dati fin qui sono _finitari_ , in pieno stile formalista. Il meta-linguaggio ci permette di ragionare induttivamente (ricorsivamente) sulla struttura delle stringhe di un linguaggio. Normalmente l’induzione è concentrata sulla parte più esterna delle formule, cioè su quella di superficie: ma la metodologia deep inference aggiunge qualcosa in più. ###### Definizione 1.2.2 (Contesti, saturazione, ordine). Data una grammatica $G=(\Sigma,\mathcal{C},S,\mathcal{P})$, il _linguaggio dei contesti su_ $G$, denotato con $\Xi_{G}$, è definito come il linguaggio generato dalla grammatica aumentata $(\Sigma\cup\\{\bullet\\},\mathcal{C},S,\mathcal{P}\cup\\{(S,\bullet)\\})$, dove $\bullet\not\in\Sigma$ è un nuovo simbolo terminale chiamato _contesto vuoto_. Un _contesto (generico)_ è una stringa di $\mathbb{C}\in\Xi_{G}$ (si indica spesso con $\mathbb{C}\\{\bullet\\}$ per enfatizzare il fatto che è un contesto). Intuitivamente un contesto è una stringa di $\mathscr{L}_{G}$ con alcuni “buchi” (denotati da $\bullet$) che possono a loro volta essere riempiti con stringhe di $\mathscr{L}_{G}$. L’operazione di _saturazione di un contesto_ $\mathbb{C}$ _con una stringa_ $w\in\mathscr{L}_{G}$ si indica con $\mathbb{C}\\{w\\}$, e consiste nella sostituzione testuale di $w$ al posto di tutte le occorrenze di $\bullet$ dentro $\mathbb{C}$; l’ _ordine di un contesto_ (in simboli $\\|\mathbb{C}\\|$) è il numero di occorrenze di $\bullet$ al suo interno. Ambedue si definiscono formalmente per induzione sulla struttura di $\mathbb{C}$ come mostrato in Figura 1.1. Saturazione contesti \| \| Ordine contesti ---\|---\|--- $\epsilon\\{w\\}$ \| $=$ \| $\epsilon$ \| \| $\\|\epsilon\\|$ \| $=$ \| $0$ $(\bullet y)\\{w\\}$ \| $=$ \| $w(y\\{w\\})$ \| \| $\\|\bullet y\\|$ \| $=$ \| $1+\\|y\\|$ $(ay)\\{w\\}$ \| $=$ \| $a(y\\{w\\})$ \| \| $\\|ay\\|$ \| $=$ \| $\\|y\\|$ Figure 1.1: Definizioni induttive di _saturazione_ e _ordine_ di un contesto Infine, sia $n\geq 0$ un numero naturale: il _linguaggio dei contesti di ordine_ $n$ _su_ $G$ (i.e. $\Xi_{G}^{n}$) è così definito: $\Xi_{G}^{n}=\\{\mathbb{C}\in\Xi_{G}\>\|\>\\|\mathbb{C}\\|=n\\}$ Data una grammatica $G=(\Sigma,\mathcal{C},S,\mathcal{P})$, osserviamo che si ha $\Xi_{G}^{0}=\mathscr{L}_{G}$, poiché per produrre le stringhe di $\Xi_{G}^{0}$ non si usa mai la regola di produzione aggiuntiva $(S,\bullet)$, ma solo quelle in $\mathcal{P}$, esattamente come accade per $\mathscr{L}_{G}$. Usando un argomento, analogo è possibile dimostrare che: $\mathscr{L}_{G}=\\{\mathbb{C}\\{w\\}\>\|\>\mathbb{C}\in\Xi_{G}^{1},w\in\mathscr{L}_{G}\\}$ Infatti i contesti di $\Xi_{G}^{1}$ sono generati usando una (e una sola) volta la regola di produzione aggiuntiva $(S,\bullet)$; in $G$, dove questa regola non è presente, tutto quello che è possibile fare è riscrivere $S$ con una delle altre produzioni di $\mathcal{P}$ per $S$, che equivale a sostituire _quella_ occorrenza di $S$ con una delle stringhe del linguaggio generato da $G$, cioè proprio $\mathscr{L}_{G}$. Per $n>1$ non è possibile ottenere risultati analoghi su $\Xi_{G}^{n}$, poiché se da un lato è possibile riscrivere occorrenze diverse di $S$ in modi diversi, dall’altro la saturazione di un contesto ammette un solo parametro in $\mathscr{L}_{G}$ (che viene replicato sempre uguale $n$ volte). Inoltre $n=0$ è un caso triviale, perché la saturazione dei contesti in $\Xi_{G}^{0}$ non produce effetti (non si fanno sostituzioni). L’unico caso degno di nota è $n=1$: esso rappresenta il punto di contatto tra il concetto di saturazione di un contesto – i.e. “sostituzione di _una_ variabile (fresca)” – e quello più generale di riscrittura: saturazione di un contesto e riscrittura multipasso coincidono per $n\leq 1$, cioè quando il processo di riscrittura è sostituito da quello di saturazione _al più in un singolo punto_. I contesti di ordine $1$ su una grammatica sono uno strumento molto potente che ci consente di focalizzare l’attenzione su una porzione specifica di una stringa del linguaggio che dipende dalla sua struttura sottostante, astraendoci dal resto. Per la loro rilevanza, d’ora in poi quando parleremo di _contesti_ intenderemo sempre quelli di ordine $1$. Anche i contesti sono strumenti di meta-livello, perché trascendono il linguaggio oggetto, per permetterci di ragionare su esso. Per di più, un contesto può essere saturato con una qualunque formula di meta-livello: considerare arbitrari contesti ci permette di ragionare su classi di formule molto estese, ossia su formule _immerse_ in contesti arbitrari, cioè ad _arbitrari livelli di profondità_ , concetto cardine di tutta la deep inference. Usando i contesti e il meta-linguaggio, possiamo ragionare per induzione strutturale sulla stringhe del linguaggio _a qualsiasi livello di profondità_. Inoltre, anche i contesti si possono _unificare_ con una procedura effettiva. Al meta-livello ragioniamo su sequenti definiti sul meta-linguaggio. Le definizioni e le tecniche viste in precedenza per singole formule, si estendono in maniera naturale alle liste di formule e ai sequenti: in particolare, è possibile l’ _unificazione di sequenti_ e considerare sequenti composti da formule _immerse in contesti arbitrari_ (con procedure effettive). ### 1.3 Sistemi formali e formalismi Le dimostrazioni sono l’oggetto di studio della proof theory; in questa sezione esplicitiamo la nozione di dimostrazione. Il termine “dimostrare” deriva dal latino _demonstrare_ , composto dalla radice _de-_ (di valore intensivo) e da _monstrare_ (“mostrare”, “far vedere”), da cui il significato di _rendere manifesto con fatti e con segni certi_. In matematica una dimostrazione è un _processo di deduzione_ che, partendo da _premesse_ assunte come valide (ipotesi) o da proposizioni dimostrate in virtù di tali premesse, determina la necessaria validità di una nuova _proposizione_ in funzione della (sola) _coerenza formale_ del ragionamento. Le proposizioni saranno dunque stringhe appartenenti ad un linguaggio formale; il processo di deduzione sarà scandito dalla corretta applicazione di alcune regole di base in qualche modo riconosciute come elementari e la coerenza formale dovrà essere opportunamente formalizzata e fungerà da argomento a sostegno della bontà delle regole scelte. ###### Definizione 1.3.1 (Regole, derivazioni, dimostrazioni). Un _sistema formale_ è una coppia $(\mathscr{L},\mathscr{S})$ composta da un _linguaggio_ $\mathscr{L}$ (generato da qualche grammatica $G$, tipicamente – ma non necessariamente – context-free) e da un insieme di regole d’inferenza (o _sistema di deduzione_) $\mathscr{S}$. Date le formule $P_{1},\ldots,P_{n},Q\in\mathscr{L}^{\prime}$, una _regola di inferenza_ $(\mathsf{\rho})$ è un’espressione della forma: $P_{1}$ $\cdots$ $P_{n}$ $(\mathsf{\rho})$ $Q$ dove $P_{1},\ldots,P_{n}$ sono chiamate _premesse della regola $(\mathsf{\rho})$_ mentre $Q$ ne è la _conclusione_. Una regola di inferenza senza premesse (i.e. avente $n=0$) è chiamata _assioma_ , mentre, per $n>0$, è detta _regola d’inferenza propria_. In genere le premesse e la conclusione di $(\mathsf{\rho})$ sono formule (o sequenti) aventi una certa forma di superficie – ed eventualmente, nell’approccio deep inference, immerse in arbitrari contesti. A questo modo di procedere, cioè di specificare la _forma_ delle (eventuali) ipotesi e della conclusione delle regole d’inferenza, ci si riferisce spesso in letteratura col termine _schema_ (p.e. dicendo “schema d’assioma”). Un _passo d’inferenza_ o _applicazione_ o _istanza_ di una regola d’inferenza $(\mathsf{\rho})$ è un’espressione della forma: $P_{1}^{\prime}$ $\cdots$ $P_{n}^{\prime}$ $(\mathsf{\rho})$ $Q^{\prime}$ dove $P_{1}^{\prime},\ldots,P_{n}^{\prime},Q^{\prime}\in\mathscr{L}^{\prime}$ sono formule ottenute rispettivamente per unificazione (anche parziale, al meta-livello) con $P_{1},\ldots,P_{n},Q\in\mathscr{L}^{\prime}$. Le stringhe $P_{1}^{\prime},\ldots,P_{n}^{\prime}$ sono chiamate _premesse dell’applicazione di $(\mathsf{\rho})$_ mentre $Q^{\prime}$ ne è la _conclusione_. Indicheremo anche il nome della regola accanto alla barra orizzontale di derivazione, quando questo sarà d’aiuto alla comprensione e non sarà fonte d’ambiguità. Una _derivazione_ $\Phi$ da una lista di premesse $P_{1},\ldots,P_{n}$ ad una conclusione $Q$ è un albero di istanze di regole in $\mathscr{S}$, avente $Q$ come radice e $P_{1},\ldots,P_{n}$ come foglie, e indicato con: $P_{1}$$\cdots$$P_{n}$ $\textstyle{\scriptstyle\Phi,\mathscr{S}}$ $Q$ Nel seguito ometteremo $\Phi$ e/o $\mathscr{S}$ quando questo non comporterà ambiguità. Infine, una _dimostrazione_ è una derivazione avente per come premesse $P_{1},\ldots,P_{n}$ solo istanze di assiomi. La indicheremo con: $\scriptstyle-$ $\scriptstyle\scriptstyle\Phi\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\mathscr{S}$ $Q$ omettendo $\Phi$ e/o $\mathscr{S}$ quando e se necessario. La _derivabilità_ in un sistema formale (“da un insieme di formule $\Gamma$ è possibile derivare la formula $P$”, o anche “$P$ è derivabile da $\Gamma$”) è un concetto sintattico, così come lo è la _dimostrabilità_ – in contrapposizione al concetto di _verità_ e a quello di _modello_ , che sono invece di natura semantica. Finora non abbiamo mai parlato di verità: in questa sede non ci occuperemo degli aspetti semantici legati ai sistemi formali, che non sono oggetto di studio di proof theory, rimandando per questi ad Abramsky et al. [1992]; Barwise [1977]; Chang et al. [1973]. Il concetto di derivabilità si estende anche alle regole dei sistemi formali: un regola è derivabile quando è ottenibile tramite altre regole. Ma una regola può anche essere _ammissibile_ (o _eliminabile_): questo accade quando la sua presenza all’interno del sistema non altera l’insieme di formule dimostrabili, ossia eliminando la regola dal sistema, si riescono a dimostrare _le stesse cose_. Questo vale anche per le regole derivabili, ma mentre in quel caso era sufficiente sostituire la regola con la sua derivazione, per regole ammissibili occorre ristrutturare l’albero di prova. ###### Definizione 1.3.2 (Regole derivabili e ammissibili). Una regola $(\mathsf{\rho})$ è _derivabile_ per un sistema $\mathscr{S}$ se, per ogni istanza di $(\mathsf{\rho})$: $P_{1}$ $\cdots$ $P_{n}$ $(\mathsf{\rho})$ $Q$ esiste una derivazione: $P_{1}$$\cdots$$P_{n}$ $\textstyle{\scriptstyle\mathscr{S}}$ $Q$ Una regola $(\mathsf{\rho})$ è _ammissibile_ (o _eliminabile_) per un sistema $\mathscr{S}$ se, per ogni dimostrazione $\scriptstyle-$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\mathscr{S}\cup\\{(\mathsf{\rho})\\}$ $Q$ esiste una dimostrazione $\scriptstyle-$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\mathscr{S}$ $Q$ . In teoria della dimostrazione ci concentriamo (al meta-livello) sulle proprietà dei sistemi formali, cioè le proprietà di cui godono le dimostrazioni espresse in qualche sistema formale. Ma _quale_ sistema formale? Seguendo Troelstra and Schwichtenberg [1996], possiamo raggrupparli in alcune grandi famiglie chiamate _formalismi_ : • _Sistemi assiomatici_ o _sistemi alla Frege-Hilbert_ (Hilbert and Ackermann [1928]; Frege [1879]): in questo approccio accettiamo un numero molto ristretto di regole d’inferenza proprie (p.e. nella logica proposizionale solo una, il _modus ponens_) mentre il resto del sistema deduttivo sarà composto da assiomi; le derivazioni in questo formalismo furono originariamente concepite per rispecchiare le dimostrazioni espresse in linguaggio naturale, obiettivo ambizioso e scarsamente raggiunto da questo approccio in cui le dimostrazioni tendono invece ad essere molto dettagliate e “pedanti”. È l’approccio più datato e grazie ad esso è stato possibile formalizzare con successo sistemi deduttivi rilevanti quali: la logica classica, quella intuizionista ed alcune logiche modali; * • _Deduzione naturale_ : introdotta nel celebre Gentzen [1935] (assieme, come vedremo, al _calcolo dei sequenti_) questa famiglia di sistemi è pensata per mimare il ragionamento logico-deduttivo umano (da qui l’aggettivo “naturale”). Non ci sono assiomi e le formule possono essere composte e decomposte (usando il gergo tecnico, rispettivamente _introdotte_ ed _eliminate_). Un’operazione comune nella pratica matematica è quella di _ragionare per assunzioni_ : la deduzione naturale mette a disposizione un artificio per compiere questa operazione, e i sistemi espressi in deduzione naturale godono di buone proprietà, relativamente semplici da dimostrare (vedi il classico Prawitz [1965]); * • _Calcolo dei sequenti_ : dovuto a Gentzen, questo è lo strumento preferito in teoria della dimostrazione per le ottime proprietà di cui gode. Fa tipicamente uso di pochi assiomi (p.e. nella logica proposizionale solo uno, l’assioma _identità_) e molte regole d’inferenza proprie, che permettono di comporre nuove formule a partire dalle premesse (usando la terminologia della deduzione naturale, sono presenti le regole di _introduzione_ ma non quelle di _eliminazione_). Volendo aderire ad una visione “proof theoretical”, ci concentreremo in seguito sul calcolo dei sequenti. È tuttavia doveroso osservare che questi formalismi consentono di definire sistemi formali aventi il medesimo potere espressivo: in altre parole, nessuno prevale a priori sugli altri, dipende dal _setting_ in cui ci poniamo. Inoltre questi sono solo i formalismi “classici”; esistono altre famiglie di sistemi formali, che possono essere usate per mettere in evidenza altri aspetti importanti del processo deduttivo e delle dimostrazioni. Tra questi è doveroso citare le _Proof Nets_ , introdotte in Girard [1987] allo scopo di far emergere alcune simmetrie dei sistemi formali che erano “oscurate” dal calcolo dei sequenti. Le tre famiglie sopra descritte adottano tutte shallow inference come specifica delle regole d’inferenza (per quanto questa sia una scelta del tutto arbitraria). Ma l’approccio deep inference apre la via ad (almeno) un quarto formalismo: * • _Calcolo delle strutture_ : introdotto in Guglielmi [2002], i sistemi deduttivi in calcolo delle strutture constano di un piccolo numero di assiomi e di regole d’inferenza proprie, e godono di una notevole quantità di proprietà, sostanzialmente estendendo quelle studiate per il calcolo dei sequenti. Le nuove prospettive aperte nell’ambito del calcolo delle strutture, ne fanno uno strumento di grande interesse e in continuo sviluppo da parte della comunità scientifica. ### 1.4 Metodologie: shallow _versus_ deep inference Le metodologie guidano la progettazione dei sistemi deduttivi e il processo di inferenza: le due metodologie conosciute allo stato dell’arte sono _shallow_ e _deep inference_ e le esamineremo a turno. ###### Definizione 1.4.1 (Shallow inference). Per _shallow inference_ o _inferenza di superficie_ , intendiamo la _metodologia_ d’inferenza che interpreta l’insieme delle regole d’inferenza come _schemi_ che disciplinano il comportamento della deduzione _in funzione del connettivo principale_ delle formule. Regole d’inferenza $P\vdash P\quad(\mathsf{ax})$ $\Gamma,P\vdash R$ $(\mathsf{\wedge_{l.1}})$ $\Gamma,P\wedge Q\vdash R$ \| $\Gamma,Q\vdash R$ $(\mathsf{\wedge_{l.2}})$ $\Gamma,P\wedge Q\vdash R$ ---\|--- $\Gamma,P\vdash Q$ $(\mathsf{\rightarrow_{r}})$ $\Gamma\vdash P\rightarrow Q$ \| $\Gamma\vdash P$ $\Gamma\vdash Q$ $(\mathsf{\wedge_{r}})$ $\Gamma\vdash P\wedge Q$ $\Gamma\vdash P$ $\Gamma,Q\vdash R$ $(\mathsf{\rightarrow_{l}})$ $\Gamma,P\rightarrow Q\vdash R$ \| Gramm. linguaggio $P::=a\>\|\>P\wedge P\>\|\>P\rightarrow P$ (con $a\in\mathcal{A}$ infinità numerabile di simboli proposizionali) Struttura formula $\rightarrow$$\wedge$$a$$b$$\wedge$$b$$a$ Figure 1.2: Sistema formale in shallow inference ed esempio di formula Essendo l’approccio più datato, è anche il più usato in letteratura, come in _deduzione naturale_ (i sistemi NK ed NJ usano shallow inference) e nel _calcolo dei sequenti_ (sistemi LK, LJ). Ad esempio: per derivare $\vdash(a\wedge b)\rightarrow(b\wedge a)$ con le regole d’inferenza in Figura 1.2, consideriamo la struttura della formula $(a\wedge b)\rightarrow(b\wedge a)$ ed osserviamo che il connettivo principale è $\rightarrow$. A questo punto l’unica regola applicabile (i.e. istanziabile, ricordiamo che le regole d’inferenza sono _schemi_) in shallow inference è $(\mathsf{\rightarrow_{r}})$. In questo modo otteniamo: $a\wedge b\vdash b\wedge a$ $\vdash(a\wedge b)\rightarrow(b\wedge a)$ Procedendo in maniera analoga osserviamo che ci sono tre regole applicabili per derivare $a\wedge b\vdash b\wedge a$, cioè: * • $(\mathsf{\wedge_{l.1}})$ produce la derivazione: $a\vdash b\wedge a$ $a\wedge b\vdash b\wedge a$ $\vdash(a\wedge b)\rightarrow(b\wedge a)$ da cui è applicabile solo $(\mathsf{\wedge_{r}})$ che produce una derivazione bloccata (cioè un albero le cui foglie non sono istanze di assiomi, né sono derivabili dalle regole del sistema); * • $(\mathsf{\wedge_{l.2}})$ analogo al precedente; * • $(\mathsf{\wedge_{r}})$ produce la derivazione: $(\mathsf{1})$ $a\wedge b\vdash b$ $(\mathsf{2})$ $a\wedge b\vdash a$ $a\wedge b\vdash b\wedge a$ $\vdash(a\wedge b)\rightarrow(b\wedge a)$ in cui è possibile applicare $(\mathsf{\wedge_{l.1}})$ o $(\mathsf{\wedge_{l.2}})$ sia alla formula $(\mathsf{1})$ che alla $(\mathsf{2})$. L’unica combinazione che porta ad una conclusione – cioè in cui ogni foglia è un’istanza di $(\mathsf{ax})$ – è un’applicazione di $(\mathsf{\wedge_{l.2}})$ a $(\mathsf{1})$ e di $(\mathsf{\wedge_{l.1}})$ a $(\mathsf{2})$, ottenendo così: $b\vdash b$ $a\wedge b\vdash b$ $a\vdash a$ $a\wedge b\vdash a$ $a\wedge b\vdash b\wedge a$ $\vdash(a\wedge b)\rightarrow(b\wedge a)$ La procedura descritta nell’esempio è nota come _proof search_ ed è automatizzabile (p.e. si può basare sulla risoluzione come avviene in PROLOG) per sistemi _in cui la regola di taglio è ammissibile_. ###### Definizione 1.4.2 (Deep inference). Per _deep inference_ o _inferenza di profondità_ intendiamo la metodologia in cui le regole d’inferenza si possono applicare ad arbitrari contesti, e quindi ad arbitrari livelli di profondità, in contrapposizione a quanto avviene nell’inferenza di superficie o _shallow inference_. Il ruolo dei contesti è quello di permettere l’accesso alla struttura delle formule senza dover usare alberi di derivazione (cioè senza decomposizione strutturale delle formule). Per questa ragione le regole d’inferenza in deep inference hanno al più una premessa: pertanto le derivazioni prendono la forma di liste. Per enfatizzare il fatto che le derivazioni sono _alberi degeneri_ (i.e. ogni nodo ha al più un figlio), usiamo la notazione: $P$ $\scriptstyle\scriptstyle\Phi\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\mathscr{S}$ $Q$ per indicare una derivazione $\Phi$ che usa le regole in $\mathscr{S}$, e avente premessa $P$ e conclusione $Q$. Dimostriamo l’analogo della formula di prima, usando il sistema deep inference in Figura 1.3. Invece dell’implicazione, qui abbiamo solo la negazione sugli atomi, quindi la formula di prima diventa: $(\overline{a}\vee\overline{b})\vee(b\wedge a)$. Regole logiche $\mathsf{t}\qquad(\mathsf{ax})$ $\mathbb{C}\\{\mathsf{t}\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{id})$ $\mathbb{C}\\{a\vee\overline{a}\\}$ \| \| $\mathbb{C}\\{P\wedge(Q\vee R)\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{s})$ $\mathbb{C}\\{(P\wedge Q)\vee R\\}$ ---\|---\|--- Grammatica linguaggio $P::=\mathsf{t}\>\|\>\mathsf{f}\>\|\>a\>\|\>\overline{a}\>\|\>P\vee P\>\|\>P\wedge P$ (con $a\in\mathcal{A}$ infinità numerabile di simboli proposizionali) Regole strutturali $\mathbb{C}\\{P\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{\wedge_{\mathsf{t}}})$ $\mathbb{C}\\{P\wedge\mathsf{t}\\}$ \| \| $\mathbb{C}\\{P\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{\vee_{\mathsf{f}}})$ $\mathbb{C}\\{P\vee\mathsf{f}\\}$ ---\|---\|--- $\mathbb{C}\\{P\wedge Q\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{\wedge_{com}})$ $\mathbb{C}\\{Q\wedge P\\}$ \| \| $\mathbb{C}\\{(P\wedge Q)\wedge R\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{\wedge_{as}})$ $\mathbb{C}\\{P\wedge(Q\wedge R)\\}$ $\mathbb{C}\\{P\vee Q\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{\vee_{com}})$ $\mathbb{C}\\{Q\vee P\\}$ \| \| $\mathbb{C}\\{(P\vee Q)\vee R\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{\vee_{as}})$ $\mathbb{C}\\{P\vee(Q\vee R)\\}$ Dimostrazione d’esempio $\mathsf{t}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{\wedge_{\mathsf{t}}})$ $\mathsf{t}\wedge\mathsf{t}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{id})$ $(a\vee\overline{a})\wedge\mathsf{t}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{id})$ $(a\vee\overline{a})\wedge(b\vee\overline{b})$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{s})$ $((a\vee\overline{a})\wedge b)\vee\overline{b}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{\wedge_{com}})$ $(b\wedge(a\vee\overline{a}))\vee\overline{b}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{s})$ $((b\wedge a)\vee\overline{a})\vee\overline{b}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{\vee_{as}})$ $(b\wedge a)\vee(\overline{a}\vee\overline{b})$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{\vee_{com}})$ $(\overline{a}\vee\overline{b})\vee(b\wedge a)$ Figure 1.3: Sistema formale in deep inference ed esempio di dimostrazione In questo caso abbiamo raggruppato le regole d’inferenza in “regole logiche” (simili a quelle viste prima) e “regole strutturali”. Quest’ultime servono a formalizzare il fatto che i connettivi di congiunzione e di disgiunzione godono della proprietà commutativa – rispettivamente regole $(\mathsf{\wedge_{com}})$ e $(\mathsf{\vee_{com}})$ – e di quella associativa – regole $(\mathsf{\wedge_{as}})$ e $(\mathsf{\vee_{as}})$ – e che inoltre l’atomo $\mathsf{t}$ (risp. $\mathsf{f}$) è elemento neutro per il connettivo di congiunzione (risp. disgiunzione). A parte per la proprietà commutativa, che è intrinsecamente simmetrica, per le altre bisognerebbe specificare anche le regole opposte; ad esempio, per $(\mathsf{\wedge_{as}})$ servirebbe una regola: $\mathbb{C}\\{P\wedge(Q\wedge R)\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{\wedge_{as}^{-1}})$ $\mathbb{C}\\{(P\wedge Q)\wedge R\\}$ Per questa ragione, in deep inference, si è soliti _sostituire le regole strutturali con una relazione d’equivalenza_ tra formule. Come possiamo vedere, la dimostrazione della medesima formula di prima è completamente mutata: innanzitutto osserviamo che le regole induttive, nel calcolo delle strutture, hanno sempre e solo una premessa ed una conclusione. Questo fa sì che le dimostrazioni si sviluppino solo in altezza, collassando il lavoro strutturale svolto dagli alberi, all’interno dei contesti. Inoltre il sistema formale è meno rigido di quello visto nell’esempio precedente, nel senso che la metodologia deep inference permette un’applicazione più capillare delle regole d’inferenza, e quindi in generale un maggior grado di libertà e di non-determinismo. ## Chapter 2 Logica classica proposizionale In questo capitolo presenteremo una serie di definizioni e risultati tradizionali in proof theory di logica classica proposizionale, e studieremo le proprietà formali del suo corrispettivo in deep inference, chiamato Sistema SKS. Il Sistema che andremo a studiare (chiamato LKp) è il frammento proposizionale del Sistema LK di Gentzen [1935], il cui nome è l’acronimo di _Logik Klassische_ ossia “logica classica” (mentre la “p” sta appunto per proposizionale). Il linguaggio $\mathscr{L}_{\mathsf{LKp}}$ è quello dei sequenti in Definizione 1.1.4, aventi per formule quelle generate dalla grammatica: $G_{\mathsf{LKp}}=(\\{\neg,\wedge,\vee,\rightarrow,[a{-}z],S\\},\\{S\\},S,\mathcal{P})$ dove $[a{-}z]$ è una notazione abbreviata per i caratteri dell’alfabeto inglese, e le produzioni in $\mathcal{P}$, sono: $S::=[a{-}z]{+}\>\|\>\neg S\>\|\>S\wedge S\>\|\>S\vee S\>\|\>S\rightarrow S$ dove $[a{-}z]{+}$ appartiene alla _notazione EBNF_ (_BNF estesa_) e significa semplicemente “qualunque stringa non vuota di caratteri alfabetici”. Solitamente s’introduce un’ulteriore generalizzazione, considerando la produzione $(S,a)$ dove $a$ è una meta-variabile appartenente ad un insieme $\mathcal{A}$ composto da un’infinità numerabile di stringhe (alfabetiche, alfa-numeriche, indicizzate con apici, pedici, …), chiamate genericamente “simboli proposizionali”. Osserviamo che le stringhe di $\mathcal{A}$ sarebbero facilmente ottenibili da una grammatica context-free, questa semplificazione serve solo ad alleggerire la notazione, mentre preserva intatto il carattere finitario del linguaggio oggetto. Il sistema deduttivo $\mathscr{S}_{\mathsf{LKp}}$ è dato usando forma di superficie dei sequenti. Le regole d’inferenza si possono dividere in quattro gruppi: assiomi, taglio, regole strutturali e regole logiche. Le regole strutturali sono di fondamentale importanza, perché permettono di manipolare l’ordine ed il numero delle formule del sequente. Sono tre: 1. 1. L’ _ordine_ delle premesse (e delle conclusioni) _non è rilevante_. Da qui otteniamo le regole di _permutazione_ : $\Gamma,P,Q,\Gamma^{\prime}\vdash\Delta$ $(\mathsf{perm_{l}})$ $\Gamma,Q,P,\Gamma^{\prime}\vdash\Delta$ $\Gamma\vdash\Delta,P,Q,\Delta^{\prime}$ $(\mathsf{perm_{r}})$ $\Gamma\vdash\Delta,Q,P,\Delta^{\prime}$ 2. 2. Assumere due volte la stessa premessa (o la stessa conclusione) è equivalente ad assumerla una volta sola. Questa osservazione ci conduce alle regole di _contrazione_ : $\Gamma,P,P\vdash\Delta$ $(\mathsf{cont_{l}})$ $\Gamma,P\vdash\Delta$ $\Gamma\vdash Q,Q,\Delta$ $(\mathsf{cont_{r}})$ $\Gamma\vdash Q,\Delta$ 3. 3. È sempre possibile sia aggiungere nuove ipotesi (rafforzare l’antecedente), sia aggiungere nuove conclusioni (indebolire il conseguente). In generale il sequente ne risulterà indebolito (in un caso serve un’ipotesi in più affinché funzioni, nell’altro a parità di ipotesi dimostra una cosa più vaga, con più possibili conseguenze). Questa è pertanto chiamata regola di _indebolimento_ : $\Gamma\vdash\Delta$ $(\mathsf{w_{l}})$ $\Gamma,P\vdash\Delta$ $\Gamma\vdash\Delta$ $(\mathsf{w_{r}})$ $\Gamma\vdash Q,\Delta$ Per individuare gli assiomi, ci poniamo la seguente domanda: quando si può sostenere che un sequente $\Gamma\vdash\Delta$ è _evidentemente_ vero? Chiaramente quando $\Gamma\cap\Delta\not=\varnothing$, cioè quando almeno una delle premesse in $\Gamma$ compare tra le conclusioni in $\Delta$. Questo sarà l’unico assioma del nostro Sistema, non ci sono altri criteri evidenti per passare dalle premesse alle conclusioni senza fare inferenza. In virtù delle regole strutturali, sappiamo che l’ordine non conta: pertanto dimostrare che esiste un $P\in\Gamma$ tale che $P\in\Delta$, si può scrivere: $P,\Gamma\vdash P,\Delta$. Inoltre possiamo sempre applicare la regola d’indebolimento a sinistra e a destra, per ottenere alfine: $P\vdash P\quad(\mathsf{ax})$ Quella di taglio è un’altra regola fondamentale, che ci si aspetta che sia soddisfatta da ogni sistema deduttivo. Il suo scopo è garantire la _componibilità_ delle dimostrazioni; questa proprietà è ampiamente sfruttata nella pratica matematica: per provare un teorema complesso, si può cominciare dimostrando dei lemmi più semplici, che possono essere poi composti per ottenere il risultato cercato. La formulazione della _regola di taglio_ è pertanto la seguente: $\Gamma\vdash P,\Delta$ $\Gamma^{\prime},P\vdash\Delta^{\prime}$ $(\mathsf{cut})$ $\Gamma,\Gamma^{\prime}\vdash\Delta,\Delta^{\prime}$ Infine le regole logiche sono quelle che specificano il comportamento dei connettivi logici. Come abbiamo già avuto modo di menzionare, nel calcolo dei sequenti è solo possibile _introdurre_ nuovi connettivi ma mai di _eliminarli_. Questo fatto è alla base di una proprietà molto importante, detta _della sottoformula_ (vedi Definizione 2.0.1). Le regole d’introduzione dei connettivi saranno classificate in _destre_ (indicate con una “$r$” a pedice) o _sinistre_ (indicate con “$l$”) a seconda che permettano d’introdurre il connettivo a destra oppure a sinistra del tornello. Vediamole rapidamente, ci sono quattro connettivi nel nostro linguaggio: 1. 1. Congiunzione: introdurre una congiunzione a sinistra significa rafforzare la premessa aggiungendo un’ipotesi. Come abbiamo detto in precedenza, il significato intuitivo del sequente va specificato formalmente, e queste regole chiarificano come le formule a sinistra del turnstile siano da considerarsi in congiunzione tra loro: $\Gamma,P\vdash\Delta$ $(\mathsf{\wedge_{l.1}})$ $\Gamma,P\wedge Q\vdash\Delta$ $\Gamma,Q\vdash\Delta$ $(\mathsf{\wedge_{l.2}})$ $\Gamma,P\wedge Q\vdash\Delta$ A destra invece, cioè per concludere che una congiunzione $P\wedge Q$ vale, sotto un certo insieme di ipotesi, dobbiamo aver dimostrato separatamente i due rami $P$ e $Q$ a partire dalle stesse assunzioni, cioè: $\Gamma\vdash P,\Delta$ $\Gamma\vdash Q,\Delta$ $(\mathsf{\wedge_{r}})$ $\Gamma\vdash P\wedge Q,\Delta$ 2. 2. Disgiunzione: il ragionamento e le regole seguono in maniera perfettamente simmetrica quanto visto per la congiunzione. Non c’è da sorprendersi, poiché la disgiunzione è il connettivo duale alla congiunzione, e poiché il sequente è fatto in modo da rispettare naturalmente tale simmetria. A destra del tornello, le formule sono da considerarsi in disgiunzione tra loro, e quindi abbiamo: $\Gamma\vdash P,\Delta$ $(\mathsf{\vee_{r.1}})$ $\Gamma\vdash P\vee Q,\Delta$ $\Gamma\vdash Q,\Delta$ $(\mathsf{\vee_{r.2}})$ $\Gamma\vdash P\vee Q,\Delta$ mentre a sinistra, se da un set comune di ipotesi $\Gamma$ unito ad un’ipotesi $P$ riusciamo a concludere che valgono certe conclusioni $\Delta$, e indipendentemente, dallo stesso set di premesse $\Gamma$ unito stavolta ad una formula $Q$, siamo in grado di concludere le medesime conclusioni $\Delta$, allora da $\Gamma$ e $P\vee Q$ possiamo concludere che vale $\Delta$, cioè: $\Gamma,P\vdash\Delta$ $\Gamma,Q\vdash\Delta$ $(\mathsf{\vee_{l}})$ $\Gamma,P\vee Q\vdash\Delta$ 3. 3. Implicazione: se la virgola a sinistra e a destra del tornello si comportano rispettivamente come una congiunzione e come una disgiunzione, il tornello stesso è l’implicazione. Questo fatto è reso evidente dalla regola d’introduzione destra della freccia. Infatti, la regola è: $\Gamma,P\vdash Q,\Delta$ $(\mathsf{\rightarrow_{r}})$ $\Gamma\vdash P\rightarrow Q,\Delta$ cioè afferma che se otteniamo una certa conclusione $Q$ supponendo $\Gamma$ e $P$, con solo $\Gamma$ è possibile concludere che “se vale $P$ allora $Q$”, cioè proprio $P\rightarrow Q$. Le altre conclusioni in $\Delta$ non giocano alcun ruolo intuitivo per questa regola, se non di preservare una certa omogeneità nella forma del sequente. Abbiamo visto come una formula $P$ sia in grado di passare da sinistra a destra del tornello, tramutandosi in un’implicazione. Anche il passaggio inverso è possibile: $\Gamma\vdash P,\Delta$ $\Gamma,Q\vdash\Delta$ $(\mathsf{\rightarrow_{l}})$ $\Gamma,P\rightarrow Q\vdash\Delta$ Qui si è dimostrato che da $\Gamma$ si deriva $P$ e, indipendentemente, che da $\Gamma$ unito all’ipotesi aggiuntiva $Q$ si conclude $\Delta$. Allora da $\Gamma$ e supponendo che $P$ implichi $Q$ è possibile concludere $\Delta$. 4. 4. Negazione: in virtù di quanto visto finora, il comportamento della negazione dovrebbe risultare piuttosto semplice. Infatti, se consideriamo una singola formula, passare da una parte all’altra del tornello significa introdurre una negazione (la negazione di una formula è equivalente ad un’implicazione in cui dalla validità della formula si conclude il falso). Formalmente: $\Gamma\vdash P,\Delta$ $(\mathsf{\neg_{l}})$ $\Gamma,\neg P\vdash\Delta$ $\Gamma,P\vdash\Delta$ $(\mathsf{\neg_{r}})$ $\Gamma\vdash\neg P,\Delta$ Facciamo un esempio di _regola derivabile_ nel Sistema LKp: scriviamo la regola _destra di congiunzione_ e la regola di _destra di congiunzione generalizzata_ : $\Gamma\vdash P,\Delta$ $\Gamma\vdash Q,\Delta$ $(\mathsf{\wedge_{r}})$ $\Gamma\vdash P\wedge Q,\Delta$ $\Gamma\vdash P,\Delta$ $\Gamma^{\prime}\vdash Q,\Delta^{\prime}$ $(\mathsf{\wedge_{r}^{gen}})$ $\Gamma,\Gamma^{\prime}\vdash P\wedge Q,\Delta,\Delta^{\prime}$ * • $(\mathsf{\wedge_{r}})$ è banalmente derivabile da $(\mathsf{\wedge_{r}^{gen}})$, infatti basta porre $\Gamma^{\prime}=\Gamma$ e $\Delta^{\prime}=\Delta$ per ottenere: $\Gamma\vdash P,\Delta$ $\Gamma\vdash Q,\Delta$ $(\mathsf{\wedge_{r}^{gen}})$ $\Gamma,\Gamma\vdash P\wedge Q,\Delta,\Delta$ $\Gamma\vdash P\wedge Q,\Delta$ dove la doppia barra orizzontale indica un certo numero applicazioni di regole strutturali, in questo caso _permutazione_ e _contrazione_. D’ora in avanti useremo sempre questa convenzione. * • $(\mathsf{\wedge_{r}^{gen}})$ è derivabile da $(\mathsf{\wedge_{r}})$: $\Gamma\vdash P,\Delta$ $\Gamma,\Gamma^{\prime}\vdash P,\Delta,\Delta^{\prime}$ $\Gamma^{\prime}\vdash Q,\Delta^{\prime}$ $\Gamma,\Gamma^{\prime}\vdash Q,\Delta,\Delta^{\prime}$ $(\mathsf{\wedge_{r}})$ $\Gamma,\Gamma^{\prime}\vdash P\wedge Q,\Delta,\Delta^{\prime}$ Ragionamenti analoghi valgono per la regole sinistre di disgiunzione e implicazione (generalizzate). Per quanto riguarda le regole _ammissibili_ , avremo modo nel seguito di dimostrare _l’ammissibilità della regola di taglio_ in LKp. ###### Definizione 2.0.1 (Proprietà della sottoformula). Si dice che _una regola d’inferenza $(\mathsf{\rho})$ gode della proprietà della sottoformula_ sse per ogni sua istanza: $P_{1}$ $\cdots$ $P_{n}$ $Q$ si ha che $P_{1},\ldots,P_{n}$ sono sottoformule di $Q$. Questa definizione si estende naturalmente alle regole del calcolo dei sequenti, imponendo che tutte le formule nelle premesse (sia a destra che a sinistra del turnstile) siano sottoformule di quelle presente nel sequente conclusione. Inoltre si dice che _un sistema formale gode della proprietà della sottoformula_ quando tutte le sue regole d’inferenza ne godono. La proprietà della sottoformula è molto rilevante, perché conferisce ai sistemi una natura “costruttiva”, il che ha molte importanti ripercussioni sulla meccanizzazione del processo inferenziale e sulla proof search. Un risultato classico è il seguente: ###### Teorema 2.0.2 (Consistenza). Sia dato un sistema formale, espresso mediante il calcolo dei sequenti, non triviale (che non contiene il sequente vuoto tra gli assiomi). Allora, se gode della proprietà della sottoformula, esso è consistente (cioè non permette di derivare il sequente vuoto). ###### Proof. La dimostrazione è immediata, poiché, se il sequente vuoto non è fra gli assiomi del sistema, dev’essere derivato con una regola d’inferenza propria $(\mathsf{\rho})$. Ma per la proprietà della sottoformula, la regola d’inferenza $(\mathsf{\rho})$ può avere per premesse solo sottoformule di quelle nel sequente vuoto, cioè non può avere premesse, ma $(\mathsf{\rho})$ è propria per ipotesi. Pertanto il sequente vuoto non è derivabile ed il sistema è consistente. ∎ Da una rapida ispezione alle regole del Sistema LKp, ci accorgiamo che la proprietà della sottoformula vale per tutte le regole d’inferenza tranne che per la regola di taglio. Infatti $(\mathsf{cut})$ introduce un’arbitraria formula $P$ tra le premesse. Onde preservare la proprietà della sottoformula, seguiamo i passi di Gentzen, dimostrando uno dei teoremi centrali della proof theory, noto come “Gentzen Hauptsatz”, che ci garantisce che la regola di taglio è ammissibile all’interno del Sistema. ### 2.1 Eliminazione del taglio Ci accingiamo a dimostrare una proprietà essenziale per la logica classica (e non solo), chiamata Hauptsatz, o teorema di eliminazione del taglio. L’Hauptsatz presumibilmente traccia il confine tra la logica e la nozione più ampia di sistema formale. Per sottolinearne l’importanza, Girard usa il motto: > _“A sequent calculus without cut elimination is like a car without engine”_ > – Girard [1995b] ###### Definizione 2.1.1 (Grado, altezza derivazioni). Il _grado di una formula_ $\delta(P)$ è definito per induzione strutturale come segue: * • $\delta(a)=1$ per $a$ simbolo proposizionale * • $\delta(P\wedge Q)=\delta(P\vee Q)=\delta(P\rightarrow Q)=1+\max\\{\delta(P),\delta(Q)\\}$ * • $\delta(\neg P)=1+\delta(P)$ Il _grado di un’applicazione della regola di taglio_ è definito come il grado della formula che elimina. Il _grado_ $\delta(\Phi)$ _di una derivazione_ è il massimo tra i gradi delle regole di taglio che vi compaiono. In particolare $\delta(\Phi)=0$ se $\Phi$ non fa uso della regola di taglio. Infine, l’ _altezza_ $h(\Phi)$ _di una derivazione_ è quella associata all’albero $\Phi$: se la regola conclusiva di $\Phi$ ha come premesse le derivazioni $\Phi_{1},\ldots,\Phi_{n}$, allora $h(\Phi)=1+\max\\{h(\Phi_{1}),\ldots,h(\Phi_{n})\\}$ (mentre se $n=0$, cioè se $\Phi$ è istanza di un assioma, allora $h(\Phi)=0$). ###### Lemma 2.1.2. Sia $\Phi$ una derivazione della forma seguente: $P_{1}$ $\cdots$ $P_{n}$ $(\mathsf{\rho_{l}})$ $\Gamma\vdash P,\Delta$ $P_{n+1}$ $\cdots$ $P_{n+m}$ $(\mathsf{\rho_{r}})$ $\Gamma^{\prime},P\vdash\Delta^{\prime}$ $(\mathsf{cut})$ $\Gamma,\Gamma^{\prime}\vdash\Delta,\Delta^{\prime}$ in cui $(\mathsf{\rho_{l}})$ (la premessa di sinistra del cut) è una regola logica “destra”, mentre $(\mathsf{\rho_{r}})$ (premessa destra del cut) è una regola logica “sinistra”, tali da introdurre entrambe la formula $P$. Allora esiste una derivazione: $P_{1}^{\prime}$$\cdots$$P_{k}^{\prime}$ $\textstyle{\scriptstyle\Psi}$ $\Gamma,\Gamma^{\prime}\vdash\Delta,\Delta^{\prime}$ con $\\{P_{1}^{\prime},\ldots,P_{k}^{\prime}\\}\subseteq\\{P_{1},\ldots,P_{n+m}\\}$ e tale che $\delta(\Psi)<\delta(\Phi)$. ###### Proof. Procediamo per casi sulla premessa sinistra dell’applicazione di $(\mathsf{cut})$. Il fatto di concentrarci sulla premessa sinistra è del tutto irrilevante, poiché grazie alla simmetria delle regole logiche del Sistema LKp, se nella premessa sinistra s’introduce un certo connettivo (con una regola logica “destra”), questo dovrà essere introdotto anche nella premessa di destra (con una regola logica simmetrica “sinistra”). 1. 1. $(\mathsf{\wedge_{r}})$ e $(\mathsf{\wedge_{l.1}})$: qui abbiamo $P=Q\wedge R$. $\Gamma\vdash Q,\Delta$ $\Gamma\vdash R,\Delta$ $(\mathsf{\wedge_{r}})$ $\Gamma\vdash Q\wedge R,\Delta$ $\Gamma^{\prime},Q\vdash\Delta^{\prime}$ $(\mathsf{\wedge_{l.1}})$ $\Gamma^{\prime},Q\wedge R\vdash\Delta^{\prime}$ $(\mathsf{cut})$ $\Gamma,\Gamma^{\prime}\vdash\Delta,\Delta^{\prime}$ La derivazione $\Phi$ sopra si trasforma in $\Psi$ come segue: $\Gamma\vdash Q,\Delta$ $\Gamma^{\prime},Q\vdash\Delta^{\prime}$ $(\mathsf{cut})$ $\Gamma,\Gamma^{\prime}\vdash\Delta,\Delta^{\prime}$ Osserviamo che $\delta(\Psi)=\delta(\Phi)-\delta(R)$, cioè il grado di $\Psi$ è diminuito di un fattore $\delta(R)>0$. 2. 2. $(\mathsf{\wedge_{r}})$ e $(\mathsf{\wedge_{l.2}})$: in maniera simmetrica, qui abbiamo: $\Gamma\vdash Q,\Delta$ $\Gamma\vdash R,\Delta$ $(\mathsf{\wedge_{r}})$ $\Gamma\vdash Q\wedge R,\Delta$ $\Gamma^{\prime},R\vdash\Delta^{\prime}$ $(\mathsf{\wedge_{l.2}})$ $\Gamma^{\prime},Q\wedge R\vdash\Delta^{\prime}$ $(\mathsf{cut})$ $\Gamma,\Gamma^{\prime}\vdash\Delta,\Delta^{\prime}$ che si trasforma nuovamente in $\Psi$ di grado inferiore: $\Gamma\vdash R,\Delta$ $\Gamma^{\prime},R\vdash\Delta^{\prime}$ $(\mathsf{cut})$ $\Gamma,\Gamma^{\prime}\vdash\Delta,\Delta^{\prime}$ 3. 3. $(\mathsf{\vee_{r.1}})$ e $(\mathsf{\vee_{l}})$: qui abbiamo $P=Q\vee R$. Questo è il duale del caso 1: $\Gamma\vdash Q,\Delta$ $(\mathsf{\vee_{r.1}})$ $\Gamma\vdash Q\vee R,\Delta$ $\Gamma^{\prime},Q\vdash\Delta^{\prime}$ $\Gamma^{\prime},R\vdash\Delta^{\prime}$ $(\mathsf{\vee_{l}})$ $\Gamma^{\prime},Q\vee R\vdash\Delta^{\prime}$ $(\mathsf{cut})$ $\Gamma,\Gamma^{\prime}\vdash\Delta,\Delta^{\prime}$ che si trasforma in $\Psi$ come segue: $\Gamma\vdash Q,\Delta$ $\Gamma^{\prime},Q\vdash\Delta^{\prime}$ $(\mathsf{cut})$ $\Gamma,\Gamma^{\prime}\vdash\Delta,\Delta^{\prime}$ col grado di $\Psi$ diminuito di un fattore $\delta(R)$. 4. 4. $(\mathsf{\vee_{r.2}})$ e $(\mathsf{\vee_{l}})$: $P=Q\vee R$, caso simmetrico al precendente e duale a 2, produciamo una derivazione $\Psi$ avente grado pari a $\delta(\Phi)-\delta(Q)$, a partire da $\Phi$: $\Gamma\vdash R,\Delta$ $(\mathsf{\vee_{r.2}})$ $\Gamma\vdash Q\vee R,\Delta$ $\Gamma^{\prime},Q\vdash\Delta^{\prime}$ $\Gamma^{\prime},R\vdash\Delta^{\prime}$ $(\mathsf{\vee_{l}})$ $\Gamma^{\prime},Q\vee R\vdash\Delta^{\prime}$ $(\mathsf{cut})$ $\Gamma,\Gamma^{\prime}\vdash\Delta,\Delta^{\prime}$ nel modo seguente: $\Gamma\vdash R,\Delta$ $\Gamma^{\prime},R\vdash\Delta^{\prime}$ $(\mathsf{cut})$ $\Gamma,\Gamma^{\prime}\vdash\Delta,\Delta^{\prime}$ 5. 5. $(\mathsf{\neg_{r}})$ e $(\mathsf{\neg_{l}})$: qui abbiamo $P=\neg Q$. La derivazione $\Phi$ pertanto è: $\Gamma,Q\vdash\Delta$ $(\mathsf{\neg_{r}})$ $\Gamma\vdash\neg Q,\Delta$ $\Gamma^{\prime}\vdash Q,\Delta^{\prime}$ $(\mathsf{\neg_{l}})$ $\Gamma^{\prime},\neg Q\vdash\Delta^{\prime}$ $(\mathsf{cut})$ $\Gamma,\Gamma^{\prime}\vdash\Delta,\Delta^{\prime}$ Costruiamo $\Psi$ scambiando le premesse di $\Phi$ e applicando direttamente il taglio, per ottenere una derivazione di grado $\delta(\Phi)-1$, come segue: $\Gamma^{\prime}\vdash Q,\Delta^{\prime}$ $\Gamma,Q\vdash\Delta$ $(\mathsf{cut})$ $\Gamma^{\prime},\Gamma\vdash\Delta^{\prime},\Delta$ $\Gamma,\Gamma^{\prime}\vdash\Delta,\Delta^{\prime}$ 6. 6. $(\mathsf{\rightarrow_{r}})$ e $(\mathsf{\rightarrow_{l}})$: $P=Q\rightarrow R$. Allora $\Phi$: $\Gamma,Q\vdash R,\Delta$ $(\mathsf{\rightarrow_{r}})$ $\Gamma\vdash Q\rightarrow R,\Delta$ $\Gamma^{\prime}\vdash Q,\Delta^{\prime}$ $\Gamma^{\prime},R\vdash\Delta^{\prime}$ $(\mathsf{\rightarrow_{l}})$ $\Gamma^{\prime},Q\rightarrow R\vdash\Delta^{\prime}$ $(\mathsf{cut})$ $\Gamma,\Gamma^{\prime}\vdash\Delta,\Delta^{\prime}$ si trasforma in $\Psi$ come segue: $\Gamma^{\prime}\vdash Q,\Delta^{\prime}$ $\Gamma,Q\vdash R,\Delta$ $(\mathsf{cut})$ $\Gamma^{\prime},\Gamma\vdash\Delta^{\prime},R,\Delta$ $\Gamma,\Gamma^{\prime}\vdash R,\Delta,\Delta^{\prime}$ $\Gamma^{\prime},R\vdash\Delta^{\prime}$ $\Gamma,\Gamma^{\prime},R\vdash\Delta,\Delta^{\prime}$ $(\mathsf{cut})$ $\Gamma,\Gamma^{\prime}\vdash\Delta,\Delta^{\prime}$ osserviamo che in quest’ultimo caso il problema è stato risolto usando _due_ tagli, entrambi di grado inferiore. ∎ ###### Definizione 2.1.3 (Rimozione). Sia $P$ una formula e $\Gamma$ una lista di formule: allora $\Gamma{\smallsetminus}P$ denota $\Gamma$ in cui _tutte le occorrenze_ della formula $P$ sono state _rimosse_. Il seguente lemma dice che una (eventuale) applicazione della regola di taglio finale può essere eliminata. La sua complessa formulazione tiene conto delle regole strutturali che possono interferire col taglio. ###### Lemma 2.1.4. Sia $P$ una formula di grado $d$, e siano $\Phi,\Phi^{\prime}$ rispettivamente le dimostrazioni di $\Gamma\vdash\Delta$ e di $\Gamma^{\prime}\vdash\Delta^{\prime}$ ambedue di grado minore di $d$. Allora è possibile costruire una dimostrazione $\Psi$ di $\Gamma,\Gamma^{\prime}{\smallsetminus}P\vdash\Delta{\smallsetminus}P,\Delta^{\prime}$ di grado minore di $d$. ###### Proof. $\Psi$ è costruito per induzione su $h(\Phi)+h(\Phi^{\prime})$, ma sfortunatamente non in maniera simmetrica rispetto $\Phi$ e $\Phi^{\prime}$: ad un certo punto la preferenza sarà data a $\Phi$ od a $\Phi^{\prime}$, e $\Psi$ sarà irreversibilmente affetta da questa scelta. Siano $\Phi$ e $\Phi^{\prime}$ rispettivamente: $\begin{array}[]{ccc}{{{{}{}{}{}{}{}{}{}{}{}}{{{}}}}{}{{{}{}{}{}{}{}{}{}{}{}}{{{}}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 4.70284pt\hbox{\hbox{\kern 7.55553pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{1}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 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0.0pt\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to86.76706pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 81.21156pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{\rho})$}}$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 30.25858pt\hbox{$\Gamma\vdash\Delta$}\kern 30.25858pt}}}\kern 15.72566pt}}}&&{{{{}{}{}{}{}{}{}{}{}{}}{{{}}}}{}{{{}{}{}{}{}{}{}{}{}{}}{{{}}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 4.70284pt\hbox{\hbox{\kern 7.55553pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox 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1.43518pt\hbox{\hbox{$\Gamma_{1}^{\prime}\vdash\Delta_{1}^{\prime}$}}}}\kern 0.0pt}\kern 10.00002pt}\hbox{$\cdots$}\kern 10.00002pt}\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 5.30765pt\hbox{\hbox{\kern 9.06758pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{m}^{\prime}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 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31.663pt\hbox{$\Gamma^{\prime}\vdash\Delta^{\prime}$}\kern 31.663pt}}}\kern 17.26566pt}}}\end{array}$ e siano $i\in\\{1,\ldots,n\\}$ e $j\in\\{1,\ldots,m\\}$. Ci sono vari casi da considerare: 1. 1. $\Phi$ è un assioma. Ci sono due sottocasi: 1. (a) $\Phi$ prova $P\vdash P$. Allora la dimostrazione $\Psi$ di $P,\Gamma^{\prime}{\smallsetminus}P\vdash\Delta^{\prime}$ è ottenuta da $\Phi^{\prime}$ mediante l’applicazione di regole strutturali. 2. (b) $\Phi$ prova $Q\vdash Q$, con $Q\not=P$. Anche in questo caso applichiamo regole strutturali a $\Phi^{\prime}$ per ottenere $Q,\Gamma^{\prime}{\smallsetminus}Q\vdash Q,\Delta^{\prime}$. 2. 2. $\Phi^{\prime}$ è un assioma. Questo caso è del tutto analogo al precedente; è interessante notare che se $\Phi$ e $\Phi^{\prime}$ sono entrambi assiomi, abbiamo arbitrariamente privilegiato $\Phi$ (e questo potrebbe avere delle ripercussioni sulla complessità di $\Psi$). 3. 3. $(\mathsf{\rho})$ è una regola strutturale. L’ipotesi induttiva per $\Phi_{1}$ e $\Phi^{\prime}$ ci danno una dimostrazione $\Psi_{1}$ per $\Gamma_{1},\Gamma^{\prime}{\smallsetminus}P\vdash\Delta_{1}{\smallsetminus}P,\Delta^{\prime}$. Allora $\Psi$ è ottenuto da $\Psi_{1}$ mediante regole strutturali. Questo è possibile perché, qualunque sia la regola strutturale $(\mathsf{\rho})$, questa gode della proprietà della sottoformula, e quindi $\Gamma_{1}$ è composto esclusivamente di sottoformule di $\Gamma$, così come $\Delta_{1}{\smallsetminus}P$ è composto solo di sottoformule di $\Delta{\smallsetminus}P$. Quindi per ottenere il sequente conclusivo di $\Psi$, non dovrà essere tolta alcuna formula presente nella conclusione di $\Psi_{1}$, ma al massimo lo si dovrà _indebolire_. 4. 4. $(\mathsf{\rho^{\prime}})$ è una regola strutturale: analogo al precedente. 5. 5. $(\mathsf{\rho})$ è una regola logica, tranne una regola logica destra che introduce $P$. L’ipotesi induttiva per $\Phi_{i}$ e $\Phi^{\prime}$ ci da $n$ dimostrazioni $\Psi_{i}$ di $\Gamma_{i},\Gamma^{\prime}{\smallsetminus}P\vdash\Delta_{i}{\smallsetminus}P,\Delta^{\prime}$. Poiché la regola $(\mathsf{\rho})$ non introduce nuove occorrenze di $P$ a destra del tornello, questa è applicabile alle $\Psi_{i}$ per ottenere $\Psi$: $\Gamma,\Gamma^{\prime}{\smallsetminus}P\vdash\Delta{\smallsetminus}P,\Delta^{\prime}$. 6. 6. $(\mathsf{\rho^{\prime}})$ è una regola logica: analogo al precedente. 7. 7. Sia $(\mathsf{\rho})$ che $(\mathsf{\rho^{\prime}})$ sono regole logiche: $(\mathsf{\rho})$ è una regola logica destra che introduce $P$, mentre $(\mathsf{\rho^{\prime}})$ è una regola logica sinistra che introduce $P$. Questo è l’ultimo caso rimanente, nonché l’unico interessante, ed è simmetrico. Per ipotesi induttiva, applicata a: 1. (a) $\Phi_{i}$ e $\Phi^{\prime}$, otteniamo le dimostrazioni $\Psi_{i}$ di $\Gamma_{i},\Gamma^{\prime}{\smallsetminus}P\vdash\Delta_{i}{\smallsetminus}P,\Delta^{\prime}$; ora, applicando $(\mathsf{\rho})$ alle $\Psi_{i}$, e usando delle regole strutturali, otteniamo la dimostrazione $\Upsilon$ di $\Gamma,\Gamma^{\prime}{\smallsetminus}P\vdash P,\Delta{\smallsetminus}P,\Delta^{\prime}$; 2. (b) $\Phi$ e $\Phi_{j}^{\prime}$, otteniamo le dimostrazioni $\Psi_{j}^{\prime}$ di $\Gamma,\Gamma_{j}^{\prime}{\smallsetminus}P\vdash\Delta{\smallsetminus}P,\Delta_{j}^{\prime}$; ora, applicando $(\mathsf{\rho^{\prime}})$ alle $\Psi_{j}^{\prime}$, e con l’ausilio di regole strutturali, otteniamo la dimostrazione $\Upsilon^{\prime}$ di $\Gamma,\Gamma^{\prime}{\smallsetminus}P,P\vdash\Delta{\smallsetminus}P,\Delta^{\prime}$. Ora abbiamo due dimostrazioni, $\Upsilon$ e $\Upsilon^{\prime}$, che si concludono come richiesto, se non per un’occorrenza di troppo della formula $P$. Applicando la regola di taglio ad $\Upsilon$ e $\Upsilon^{\prime}$, otteniamo una dimostrazione $\Upsilon^{\prime\prime}$ di: $\scriptstyle-$ $\scriptstyle\scriptstyle\Upsilon\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\Gamma,\Gamma^{\prime}{\smallsetminus}P\vdash P,\Delta{\smallsetminus}P,\Delta^{\prime}$ $\scriptstyle-$ $\scriptstyle\scriptstyle\Upsilon^{\prime}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\Gamma,\Gamma^{\prime}{\smallsetminus}P,P\vdash\Delta{\smallsetminus}P,\Delta^{\prime}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{cut})$ $\Gamma,\Gamma^{\prime}{\smallsetminus}P,\Gamma,\Gamma^{\prime}{\smallsetminus}P\vdash\Delta{\smallsetminus}P,\Delta^{\prime},\Delta{\smallsetminus}P,\Delta^{\prime}$ che con semplici manipolazioni strutturali è riducibile a: $\Gamma,\Gamma^{\prime}{\smallsetminus}P\vdash\Delta{\smallsetminus}P,\Delta^{\prime}$ Tuttavia il grado del taglio usato in $\Upsilon^{\prime\prime}$ è troppo elevato (è proprio di grado $d$). Ma questo è precisamente il caso in cui si applica il Lemma 2.1.2, grazie al quale il taglio in $\Upsilon^{\prime\prime}$ può essere rimpiazzato con una derivazione di grado minore di $d$, e avente la stessa conclusione, dalla quale, mediante regole strutturali, possiamo ottenere $\Psi$. ∎ Il prossimo lemma, che ci condurrà al risultato finale, afferma che è sempre possibile trasformare una dimostrazione in modo tale da diminuirne il grado. Formalmente: ###### Lemma 2.1.5. Sia $\Phi$ una dimostrazione di grado $d>0$ per un certo sequente. Allora è possibile costruire una dimostrazione $\Psi$ per il medesimo sequente, avente grado inferiore. ###### Proof. Per induzione sull’altezza $h(\Phi)$ della dimostrazione iniziale. Sia $(\mathsf{\rho})$ l’ultima regola applicata in $\Phi$ e siano $\Phi_{i}$ le premesse di $(\mathsf{\rho})$. Abbiamo due casi: 1. 1. $(\mathsf{\rho})$ non è un taglio di grado $d$. Per ipotesi induttiva, abbiamo $\Psi_{i}$ di grado minore di $d$, a cui possiamo applicare $(\mathsf{\rho})$ per ottenere $\Psi$; 2. 2. $(\mathsf{\rho})$ è un taglio di grado $d$: $\scriptstyle-$ $\scriptstyle\scriptstyle\Phi_{1}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\Gamma\vdash P,\Delta$ $\scriptstyle-$ $\scriptstyle\scriptstyle\Phi_{2}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\Gamma^{\prime},P\vdash\Delta^{\prime}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{cut})$ $\Gamma,\Gamma^{\prime}\vdash\Delta,\Delta^{\prime}$ Osserviamo che poiché il grado di questo $(\mathsf{cut})$ è $d$, abbiamo $\delta(P)=d$. Per ipotesi induttiva: $\begin{array}[]{ccc}{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 8.47395pt\hbox{\hbox{\kern 7.94443pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Psi_{1}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 0.0pt}}\kern 16.41838pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\Gamma\vdash P,\Delta$}}}}\kern 0.0pt}}}&&{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 10.5695pt\hbox{\hbox{\kern 7.94443pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Psi_{2}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 0.0pt}}\kern 18.51393pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\Gamma^{\prime},P\vdash\Delta^{\prime}$}}}}\kern 0.0pt}}}\end{array}$ hanno grado minore di $d$, e possiamo applicarvi il Lemma 2.1.4 per produrre $\Gamma,\Gamma^{\prime}{\smallsetminus}P\vdash\Delta{\smallsetminus}P,\Delta^{\prime}$ di grado inferiore a $d$; con alcune applicazioni di regole strutturali, otteniamo infine $\Psi$. ∎ ###### Teorema 2.1.6 (Gentzen Hauptsatz). La regola di taglio è ammissibile nel Sistema LKp. ###### Proof. È sufficiente iterare l’applicazione del lemma precedente per trasformare una dimostrazione di grado strettamente positivo, in una di grado nullo, e quindi esente da applicazioni della regola di taglio. ∎ Il processo di eliminazione dei tagli fa esplodere l’altezza delle dimostrazioni. Infatti il Lemma 2.1.4 fa crescere l’altezza della prova in modo lineare nel caso peggiore (di un fattore $\kappa=4$, senza considerare le applicazioni delle regole strutturali). Il Lemma 2.1.5 comporta una crescita esponenziale nel caso pessimo, cioè ridurre il grado di $1$ può accrescere l’albero di prova da $h$ a $\kappa^{h}$, poiché usando il Lemma 2.1.4 moltiplichiamo per $\kappa$ ad ogni unità di altezza. Quindi, mettendo tutto assieme, applicare l’Hauptsatz comporta una crescita iperesponenziale. Partendo da una dimostrazione di grado $d$ e altezza $h$ se ne ottiene una avente altezza $\mathcal{H}(d,h)$, dove: $\displaystyle\mathcal{H}(0,h)$ $\displaystyle=$ $\displaystyle h$ $\displaystyle\mathcal{H}(d+1,h)$ $\displaystyle=$ $\displaystyle\kappa^{\mathcal{H}(d,h)}$ L’Hauptsatz – in varie forme, come la normalizzazione nel $\lambda$-calcolo – è utilizzabile come fondamento teorico per la computazione. Per esempio, consideriamo un editor di testo: può essere visto come un insieme di lemmi generici (corrispondenti alle varie procedure di formattazione, impaginazione, …) che possono essere applicati a input concreti, come una pagina scritta da qualche utente. Il numero di input possibili è chiaramente infinito e infatti i lemmi sono fatti per trattare infiniti casi; ma quando eseguiamo il programma su un certo input – ad esempio per produrre in output una visualizzazione del testo – i riferimenti a queste infinità scompaiono. Concretamente, questa eliminazione dell’infinito è effettuata sostituendo sistematicamente le variabili (gli input dei lemmi) con il testo inserito dall’utente, in altre parole, eseguendo il programma. Questo è esattamente quello che fa l’algoritmo di eliminazione del taglio. Ecco perché la struttura della procedure di cut elimination è così importante (osservazione fatta nel Lemma 2.1.4). La strategia adottata nel ristrutturare la dimostrazione, effettuando le sostituzioni, produce delle scelte che sono, in generale, irreversibili. Questo può essere un problema, e si può risolvere ad esempio usando, al posto del calcolo dei sequenti, la deduzione naturale, che gode della proprietà di _confluenza_ (o _proprietà di Church-Rosser_), la quale garantisce che _le scelte fatte sono sempre reversibili_. Purtroppo la deduzione naturale soffre di altri problemi, e specialmente non gode della proprietà della sottoformula, e non si relaziona bene con la simmetria classica (ha molte premesse ma una sola conclusione). L’approccio deep inference può offrire diversi vantaggi nei confronti di ambedue questi formalismi. ### 2.2 Deep inference e simmetria L’eliminazione del taglio è un’idea centrale della proof theory. Se spostiamo tutto alla destra del turnstile e applichiamo qualche regola strutturale, la regola di taglio diventa: $\vdash P,\Delta$$\vdash\neg P,\Delta^{\prime}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{cut^{1}})$ $\vdash\Delta,\Delta^{\prime}$ Quando letta dal basso verso l’alto, la regola di taglio introduce una formula arbitraria $P$, insieme alla sua negazione $\neg P$. Osserviamo ora la regola d’identità, manipolata nuovamente per portare tutto a destra del tornello: $\vdash P,\neg P\qquad(\mathsf{id^{1}})$ Ci accorgiamo che, quando letta dall’alto al basso, anch’essa introduce una formula arbitraria assieme alla sua negazione. È chiaro che le due regole sono intimamente correlate. Tuttavia, la loro dualità è oscurata dal fatto che le simmetrie verticali sono nascoste nel calcolo dei sequenti: le derivazioni sono alberi, e gli alberi sono verticalmente asimmetrici. Per rivelare la dualità tra le due regole, occorre ripristinare questa simmetria verticale. La forma ad albero delle derivazioni nel calcolo dei sequenti è dovuta alla presenza di regole d’inferenza con due premesse. Per esempio la regola destra di congiunzione, nella versione ad un lato diventa: $\vdash P,\Delta$$\vdash Q,\Delta^{\prime}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{\wedge_{r}^{1}})$ $\vdash P\wedge Q,\Delta,\Delta^{\prime}$ in cui è presente un’asimmetria: due premesse ma solo una conclusione. O per meglio dire: un connettivo nella conclusione, ma nessuno tra le premesse. Questa asimmetria può essere riparata. Sappiamo che la virgola a destra del turnstile corrisponde alla disgiunzione, e che i diversi rami dell’albero di derivazione corrispondono a congiunzioni; pertanto la regola $(\mathsf{\wedge_{r}^{1}})$ può essere riscritta come: $\vdash(P\vee\Delta)\wedge(Q\vee\Delta^{\prime})$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{\wedge_{r}^{1.1}})$ $\vdash(P\wedge Q)\vee\Delta\vee\Delta^{\prime}$ In tal modo andiamo ad indentificare una parte del livello oggetto (i connettivi tra le formule) con il meta-livello (i rami dell’albero di derivazione). Così facendo rendiamo il sistema “incompleto”, poiché uno degli scopi degli alberi di derivazione è quello di permettere alle regole d’inferenza di essere applicate in profondità, seguendo la struttura sintattica delle formule. Consideriamo la derivazione: $\cdots$ $\vdash P,\Delta$$\vdash Q,\Delta^{\prime}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\wedge_{r}^{1})$ $\vdash P\wedge Q,\Delta,\Delta^{\prime}$ $\vdash\mathbb{C}\\{P\wedge Q\\}$ in cui la conclusione contiene la sottoformula $P\wedge Q$. Leggendola dal basso all’alto, il motivo per cui la regola $(\mathsf{\wedge_{r}^{1}})$ può essere applicata, è che le applicazioni nella derivazione sottostante decompongono il contesto $\mathbb{C}\\{\bullet\\}$ e ne distribuiscono il contenuto tra le foglie dell’albero di derivazione. Se vogliamo eliminare la forma ad albero delle derivazioni per ottenere un sistema completamente simmetrico, dobbiamo in qualche modo riconferire alle derivazioni l’abilità di accedere alle sottoformule: questo può essere fatto direttamente, usando la metodologia deep inference. In questo modo, l’assioma d’identità e la regola di taglio diventano: $\mathsf{t}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{id})$ $P\vee\neg P$ $P\wedge\neg P$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{cut})$ $\mathsf{f}$ da cui è evidente il carattere duale delle due: una può essere ottenuta dall’altra scambiando e negando la premessa e la conclusione. A questa nozione di dualità ci si riferisce con l’aggettivo _contrappositiva_. Avremo modo di osservare una profonda simmetria, tutte le regole d’inferenza si raggrupperanno in coppie duali, come identità e taglio. Questa dualità si estenderà naturalmente alle derivazioni: per ottenere la duale di una derivazione, basterà negare ogni formula e “girare la derivazione sottosopra”, cioè leggerla dal basso verso l’alto. #### 2.2.1 Sistema SKS generalizzato Presentiamo un sistema formale per la logica classica proposizionale, che da una parte segue la tradizione del calcolo dei sequenti, in particolare possiede una regola di taglio e la sua ammissibilità è dimostrata, mentre dall’altra, in contrasto col calcolo dei sequenti, ha regole che si applicano a profondità arbitraria nelle formule e le derivazioni sono alberi degeneri (i.e. liste, le regole hanno al più una premessa). In questo Sistema potremo osservare una simmetria verticale nelle regole che mancava nel calcolo dei sequenti. ###### Definizione 2.2.1 (Linguaggio SKSg, equivalenza). Sia $\mathcal{P}$ un insieme infinito enumerabile di _simboli proposizionali_. L’insieme degli _atomi_ $\mathcal{A}$ è così definito: $\mathcal{A}=\\{p,\overline{p}\>\|\>p\in\mathcal{P}\\}$ dove $\overline{\cdot}$ è una _funzione di negazione primitiva sui simboli proposizionali_. La negazione si estende facilmente a tutti gli atomi definendo $\overline{\overline{p}}=p$ per ogni simbolo proposizionale negato $\overline{p}$. Siano $\mathsf{t},\mathsf{f}\not\in\mathcal{A}$ simboli costanti o _unità_ (che denotano rispettivamente _il vero_ ed _il falso_) e sia $a\in\mathcal{A}$. Il _linguaggio di SKSg_ è definito dalle seguenti regole BNF di produzione: $\begin{array}[]{llll}T&::=&\mathsf{t}\>\|\>\mathsf{f}\>\|\>a&\quad\mbox{(termini)}\\\ P&::=&T\>\|\>\textnormal{{(}}P,P\textnormal{{)}}\>\|\>\textnormal{{[}}P,P\textnormal{{]}}&\quad\mbox{(formule)}\end{array}$ dove $\textnormal{{(}}P_{1},P_{2}\textnormal{{)}}$ e $\textnormal{{[}}P_{1},P_{2}\textnormal{{]}}$ denotano rispettivamente la _congiunzione_ e la _disgiunzione_ delle formule $P_{1}$ e $P_{2}$. Come prima, dato un contesto $\mathbb{C}\\{\bullet\\}$ e una formula $P$, indichiamo con $\mathbb{C}\\{P\\}$ la formula ottenuta saturando il contesto $\mathbb{C}$ con la formula $P$. Ad esempio, sia $\mathbb{C}\\{\bullet\\}=\textnormal{{[}}a,\textnormal{{(}}\bullet,c\textnormal{{)}}\textnormal{{]}}$: allora $\mathbb{C}\\{\overline{b}\\}=\textnormal{{[}}a,\textnormal{{(}}\overline{b},c\textnormal{{)}}\textnormal{{]}}$ mentre $\mathbb{C}\\{\textnormal{{(}}b_{1},b_{2}\textnormal{{)}}\\}=\textnormal{{[}}a,\textnormal{{(}}\textnormal{{(}}b_{1},b_{2}\textnormal{{)}},c\textnormal{{)}}\textnormal{{]}}$; in quest’ultimo caso possiamo adottare la convenzione di _omettere le parentesi graffe attorno ai termini composti_ e pertanto di scrivere semplicemente $\mathbb{C}\textnormal{{(}}b_{1},b_{2}\textnormal{{)}}$. Consideriamo due formule equivalenti quando appartengono alla relazione indotta dalle equazioni in Figura 2.1. Infine, una formula è in _forma normale negata_ quando la negazione occorre solo sui simboli proposizionali. Associatività $\textnormal{{[}}\textnormal{{[}}P,Q\textnormal{{]}},R\textnormal{{]}}=\textnormal{{[}}P,\textnormal{{[}}Q,R\textnormal{{]}}\textnormal{{]}}$ $\textnormal{{(}}\textnormal{{(}}P,Q\textnormal{{)}},R\textnormal{{)}}=\textnormal{{(}}P,\textnormal{{(}}Q,R\textnormal{{)}}\textnormal{{)}}$ Unità $\textnormal{{[}}\mathsf{t},\mathsf{t}\textnormal{{]}}=\mathsf{t}$ \| \| $\textnormal{{[}}\mathsf{f},P\textnormal{{]}}=P$ ---\|---\|--- $\textnormal{{(}}\mathsf{f},\mathsf{f}\textnormal{{)}}=\mathsf{f}$ \| \| $\textnormal{{(}}\mathsf{t},P\textnormal{{)}}=P$ Chiusura contestuale $P=Q$ $\mathbb{C}\\{P\\}=\mathbb{C}\\{Q\\}$ Commutatività $\textnormal{{[}}P,Q\textnormal{{]}}=\textnormal{{[}}Q,P\textnormal{{]}}$ $\textnormal{{(}}P,Q\textnormal{{)}}=\textnormal{{(}}Q,P\textnormal{{)}}$ Negazione $\overline{\mathsf{f}}$ \| $=$ \| $\mathsf{t}$ ---\|---\|--- $\overline{\mathsf{t}}$ \| $=$ \| $\mathsf{f}$ $\overline{\textnormal{{[}}P,Q\textnormal{{]}}}$ \| $=$ \| $\textnormal{{(}}\overline{P},\overline{Q}\textnormal{{)}}$ $\overline{\textnormal{{(}}P,Q\textnormal{{)}}}$ \| $=$ \| $\textnormal{{[}}\overline{P},\overline{Q}\textnormal{{]}}$ $\overline{\overline{P}}$ \| $=$ \| $P$ Equivalenza $P=P$ $P=Q$ $Q=P$ $P=Q$ $Q=R$ $P=R$ Figure 2.1: Equivalenza tra formule di SKSg In virtù della proprietà associativa, adottiamo la seguente convenzione: $\displaystyle\textnormal{{[}}P_{1},P_{2},\ldots,P_{n}\textnormal{{]}}$ $\displaystyle=$ $\displaystyle\textnormal{{[}}P_{1},\textnormal{{[}}P_{2},\textnormal{{[}}\ldots,P_{n}\textnormal{{]}}\ldots\textnormal{{]}}\textnormal{{]}}$ $\displaystyle\textnormal{{(}}P_{1},P_{2},\ldots,P_{n}\textnormal{{)}}$ $\displaystyle=$ $\displaystyle\textnormal{{(}}P_{1},\textnormal{{(}}P_{2},\textnormal{{(}}\ldots,P_{n}\textnormal{{)}}\ldots\textnormal{{)}}\textnormal{{)}}$ cioè scriviamo rispettivamente liste di disgiunzioni e congiunzioni senza curarci di come le sottoformule siano associate tra loro, assumendo quando non specificato che associno a destra. Osserviamo inoltre che l’equivalenza ci permette di spingere la negazione all’interno delle formule fino a livello degli atomi, scambiando ogni volta congiunzione e disgiunzione in stile De Morgan. ###### Definizione 2.2.2 (Dualità, simmetria). Il _duale di una regola_ d’inferenza si ottiene scambiando la premessa con la conclusione e negando ambedue. Ad esempio: $\mathsf{t}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{i}{\downarrow})$ $\textnormal{{[}}P,\overline{P}\textnormal{{]}}$ $\textnormal{{(}}P,\overline{P}\textnormal{{)}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{i}{\uparrow})$ $\mathsf{f}$ Un _sistema deduttivo_ è _simmetrico_ se per ogni regola d’inferenza esso contiene anche la duale. Il Sistema deduttivo SKSg è riportato in Figura 2.2. Il suo nome è un acronimo, in cui la prima “S” indica che è simmetrico, la “K” sta per “Klassisch” (come nel Sistema LK) e la “S” finale dice che il Sistema è espresso nel calcolo delle strutture (il termine “struttura” è usato per indicare una lista di formule in congiunzione o in disgiunzione). La “g” minuscola indica che il Sistema è _generalizzato_ , che significa che le regole non sono ristrette alla forma atomica. È possibile dimostrare (Brünnler [2004]) che questo sistema formale cattura tutte le dimostrazioni esprimibili in LKp, passando per un sistema intermedio chiamato calcolo dei sequenti “ad un lato” o calcolo dei sequenti di Gentzen- Schütte (Schütte [1950]; Troelstra and Schwichtenberg [1996]). $\mathsf{t}\quad(\mathsf{ax})$ \| $\mathbb{C}\\{\mathsf{t}\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{i}{\downarrow})$ $\mathbb{C}\textnormal{{[}}P,\overline{P}\textnormal{{]}}$ \| $\mathbb{C}\\{\mathsf{f}\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{w}{\downarrow})$ $\mathbb{C}\\{P\\}$ \| $\mathbb{C}\textnormal{{[}}P,P\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{c}{\downarrow})$ $\mathbb{C}\\{P\\}$ \| $\mathbb{C}\textnormal{{(}}P,\textnormal{{[}}Q,R\textnormal{{]}}\textnormal{{)}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{s})$ $\mathbb{C}\textnormal{{[}}\textnormal{{(}}P,Q\textnormal{{)}},R\textnormal{{]}}$ ---\|---\|---\|---\|--- (nessuna regola per $\mathsf{f}$) \| $\mathbb{C}\textnormal{{(}}P,\overline{P}\textnormal{{)}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{i}{\uparrow})$ $\mathbb{C}\\{\mathsf{f}\\}$ \| $\mathbb{C}\\{P\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{w}{\uparrow})$ $\mathbb{C}\\{\mathsf{t}\\}$ \| $\mathbb{C}\\{P\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{c}{\uparrow})$ $\mathbb{C}\textnormal{{(}}P,P\textnormal{{)}}$ \| Figure 2.2: Sistema deduttivo SKSg Le regole $(\mathsf{s})$, $(\mathsf{w}{\downarrow})$ e $(\mathsf{c}{\downarrow})$ sono chiamate rispettivamente _scambio_ , _indebolimento_ e _contrazione_. Le duali portano lo stesso nome, con l’aggiunta del prefisso “co-”, ad esempio $(\mathsf{w}{\uparrow})$ è chiamata _co-indebolimento_. La regola di scambio è duale a sé stessa, o _auto-duale_. La sua funzione è quella di modellare il comportamento duale di congiunzione e disgiunzione. La sua semantica intesa non è facile da cogliere: per una spiegazione dettagliata, si rimanda alla Sezione 3.2.2 del presente volume. Mentre è immediato osservare la corrispondenza tra $(\mathsf{w}{\downarrow})$ e $(\mathsf{c}{\downarrow})$ in SKS e le regole di indebolimento e contrazione nel calcolo dei sequenti, le loro duali non hanno corrispettivi in LKp. Il loro ruolo è quello di assicurare la simmetria del Sistema; se non siamo interessati alla simmetria, si può dimostrare che queste regole (e anche il taglio, cioè tutte le regole aventi la freccia rivolta verso l’alto) sono ammissibili. Infatti la nozione di dimostrazione è inerentemente asimmetrica: il duale di una dimostrazione _non è_ una dimostrazione, bensì è una derivazione che si conclude con l’unità $\mathsf{f}$, ossia una _refutazione_. Il primo meta-teorema che andremo a dimostrare, ci dà una caratterizzazione del Sistema SKSg, mettendo in relazione il concetto di derivazione con quello di dimostrazione. ###### Teorema 2.2.3 (Deduzione). Esiste una derivazione $P$ $\scriptstyle\scriptstyle\Psi\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ SKSg $Q$ se e solo se esiste una dimostrazione $\scriptstyle-$ $\scriptstyle\scriptstyle\Phi\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ SKSg $\textnormal{{[}}\overline{P},Q\textnormal{{]}}$ . ###### Proof. La dimostrazione $\Phi$ può essere ottenuta, data una derivazione $\Psi$, come segue: $\mathsf{t}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{i}{\downarrow})$ $\textnormal{{[}}\overline{P},P\textnormal{{]}}$ $\scriptstyle\scriptstyle\textnormal{{[}}\overline{P},\Psi\textnormal{{]}}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ SKSg $\textnormal{{[}}\overline{P},Q\textnormal{{]}}$ Osserviamo che grazie alla metodologia deep inference, è stato possibile racchiudere l’intera derivazione $\Psi$ all’interno del contesto $\mathbb{C}\\{\bullet\\}=\textnormal{{[}}\overline{P},\bullet\textnormal{{]}}$. La derivazione $\Psi$ si ottiene da $\Phi$ come segue: $P=\textnormal{{(}}P,\mathsf{t}\textnormal{{)}}$ $\scriptstyle\scriptstyle\textnormal{{(}}P,\Phi\textnormal{{)}}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ SKSg $\textnormal{{(}}P,\textnormal{{[}}\overline{P},Q\textnormal{{]}}\textnormal{{)}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{s})$ $\textnormal{{[}}\textnormal{{(}}P,\overline{P}\textnormal{{)}},Q\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{i}{\uparrow})$ $\textnormal{{[}}\mathsf{f},Q\textnormal{{]}}=Q$ Anche in questo caso la trasformazione ha avuto successo perché è stato possibile usare la dimostrazione $\Phi$ nel contesto $\mathbb{C^{\prime}}\\{\bullet\\}=\textnormal{{(}}P,\bullet\textnormal{{)}}$. ∎ #### 2.2.2 Località: il Sistema SKS Le regole d’inferenza che duplicano una quantità illimitata di informazione sono problematiche dal punto di vista della complessità e dell’implementazione, ad esempio, della proof search. Nel calcolo dei sequenti, la regola di contrazione: $\Gamma,P,P\vdash\Delta$ $(\mathsf{cont_{l}})$ $\Gamma,P\vdash\Delta$ quando letta dall’alto al basso duplica una formula $P$ di dimensione arbitraria. Qualunque sia il meccanismo effettivo che compie questa duplicazione, esso necessita di una visione _globale_ delle copie di $P$ presenti: se ad esempio pensiamo di implementare la contrazione su un sistema distribuito, in cui ogni processore ha una quantità limitata di memoria locale, la formula $P$ potrebbe essere replicata in processori diversi. In questo caso nessun processore avrebbe una visone globale delle copie di $P$, e bisognerebbe usare un meccanismo _ad hoc_ per gestire questa situazione. Chiamiamo _locali_ le regole d’inferenza che non necessitano di una visione globale su formule di dimensione arbitraria, e _non-locali_ le altre. Mentre è possibile utilizzare tecniche per risolvere questa situazione nelle implementazioni, una questione interessante è trovare un approccio teorico che sia in grado di eliminare le regole non-locali. Questo è possibile, riducendo le regole non-locali alla loro forma atomica. Ad esempio, l’identità: $\mathbb{C}\\{\mathsf{t}\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{i}{\downarrow})$ $\mathbb{C}\textnormal{{[}}P,\overline{P}\textnormal{{]}}$ è sostituita dalla regola $\mathbb{C}\\{\mathsf{t}\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{ai}{\downarrow})$ $\mathbb{C}\textnormal{{[}}a,\overline{a}\textnormal{{]}}$ dove $a$ è un simbolo proposizionale. Operazioni analoghe possono essere fatte anche nel calcolo dei sequenti; l’unica regola problematica è, appunto, la contrazione. Essa non può semplicemente essere ristretta alla forma atomica nel Sistema SKSg. Il problema si risolve inserendo nel Sistema una nuova regola, introdotta in Brünnler and Tiu [2001] e chiamata _mediale_ : $\mathbb{C}\textnormal{{[}}\textnormal{{(}}P,Q\textnormal{{)}},\textnormal{{(}}R,S\textnormal{{)}}\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{m})$ $\mathbb{C}\textnormal{{(}}\textnormal{{[}}P,R\textnormal{{]}},\textnormal{{[}}Q,S\textnormal{{]}}\textnormal{{)}}$ Questa regola non ha analoghi nel calcolo dei sequenti, ma è chiaramente corretta, poiché è derivabile da $\\{(\mathsf{c}{\downarrow}),(\mathsf{w}{\downarrow})\\}$: $\textnormal{{[}}\textnormal{{(}}P,Q\textnormal{{)}},\textnormal{{(}}R,S\textnormal{{)}}\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{w}{\downarrow})$ $\textnormal{{[}}\textnormal{{(}}P,Q\textnormal{{)}},\textnormal{{(}}R,\textnormal{{[}}Q,S\textnormal{{]}}\textnormal{{)}}\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{w}{\downarrow})$ $\textnormal{{[}}\textnormal{{(}}P,Q\textnormal{{)}},\textnormal{{(}}\textnormal{{[}}P,R\textnormal{{]}},\textnormal{{[}}Q,S\textnormal{{]}}\textnormal{{)}}\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{w}{\downarrow})$ $\textnormal{{[}}\textnormal{{(}}P,\textnormal{{[}}Q,S\textnormal{{]}}\textnormal{{)}},\textnormal{{(}}\textnormal{{[}}P,R\textnormal{{]}},\textnormal{{[}}Q,S\textnormal{{]}}\textnormal{{)}}\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{w}{\downarrow})$ $\mathbb{C}\textnormal{{[}}\textnormal{{(}}\textnormal{{[}}P,R\textnormal{{]}},\textnormal{{[}}Q,S\textnormal{{]}}\textnormal{{)}},\textnormal{{(}}\textnormal{{[}}P,R\textnormal{{]}},\textnormal{{[}}Q,S\textnormal{{]}}\textnormal{{)}}\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{c}{\downarrow})$ $\mathbb{C}\textnormal{{(}}\textnormal{{[}}P,R\textnormal{{]}},\textnormal{{[}}Q,S\textnormal{{]}}\textnormal{{)}}$ $\mathsf{t}\quad(\mathsf{ax})$ \| \| $\mathbb{C}\\{\mathsf{t}\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{ai}{\downarrow})$ $\mathbb{C}\textnormal{{[}}a,\overline{a}\textnormal{{]}}$ \| \| $\mathbb{C}\\{\mathsf{f}\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{aw}{\downarrow})$ $\mathbb{C}\\{a\\}$ \| \| $\mathbb{C}\textnormal{{[}}a,a\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{ac}{\downarrow})$ $\mathbb{C}\\{a\\}$ ---\|---\|---\|---\|---\|---\|--- (nessuna regola per $\mathsf{f}$) \| \| $\mathbb{C}\textnormal{{(}}a,\overline{a}\textnormal{{)}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{ai}{\uparrow})$ $\mathbb{C}\\{\mathsf{f}\\}$ \| \| $\mathbb{C}\\{a\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{aw}{\uparrow})$ $\mathbb{C}\\{\mathsf{t}\\}$ \| \| $\mathbb{C}\\{a\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{ac}{\uparrow})$ $\mathbb{C}\textnormal{{(}}a,a\textnormal{{)}}$ $\mathbb{C}\textnormal{{(}}P,\textnormal{{[}}Q,R\textnormal{{]}}\textnormal{{)}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{s})$ $\mathbb{C}\textnormal{{[}}\textnormal{{(}}P,Q\textnormal{{)}},R\textnormal{{]}}$ $\mathbb{C}\textnormal{{[}}\textnormal{{(}}P,Q\textnormal{{)}},\textnormal{{(}}R,S\textnormal{{)}}\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{m})$ $\mathbb{C}\textnormal{{(}}\textnormal{{[}}P,R\textnormal{{]}},\textnormal{{[}}Q,S\textnormal{{]}}\textnormal{{)}}$ Figure 2.3: Regole del Sistema _locale_ SKS Il prossimo teorema ci garantisce la derivabilità del Sistema locale KS in Figura 2.3: ###### Teorema 2.2.4. Le regole $(\mathsf{i}{\downarrow})$, $(\mathsf{w}{\downarrow})$ e $(\mathsf{c}{\downarrow})$ sono derivabili, rispettivamente da $\\{(\mathsf{ai}{\downarrow}),(\mathsf{s})\\}$, $\\{(\mathsf{aw}{\downarrow}),(\mathsf{s})\\}$, $\\{(\mathsf{ac}{\downarrow}),(\mathsf{m})\\}$. Dualmente, le regole $(\mathsf{i}{\uparrow})$, $(\mathsf{w}{\uparrow})$ e $(\mathsf{c}{\uparrow})$ sono risp. derivabili da $\\{(\mathsf{ai}{\uparrow}),(\mathsf{s})\\}$, $\\{(\mathsf{aw}{\uparrow}),(\mathsf{s})\\}$, $\\{(\mathsf{ac}{\uparrow}),(\mathsf{m})\\}$. ###### Proof. Data un’istanza di una delle seguenti regole: ${\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 3.07118pt\hbox{\hbox{$\mathbb{C}\\{\mathsf{t}\\}$}}\kern 3.07118pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to27.25352pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 21.69803pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{i}{\downarrow})$}}$}\hss}}\kern 1.43518pt\hbox{\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\textnormal{{[}}P,\overline{P}\textnormal{{]}}$}\kern 0.0pt$}}}}\kern 18.33331pt}}}\qquad,\qquad{\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 2.37672pt\hbox{\hbox{$\mathbb{C}\\{\mathsf{f}\\}$}}\kern 2.37672pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to25.0313pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 19.4758pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{w}{\downarrow})$}}$}\hss}}\kern 1.43518pt\hbox{\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\\{P\\}$}\kern 0.0pt$}}}}\kern 22.77776pt}}}\qquad,\qquad{\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\textnormal{{[}}P,P\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to31.72917pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 26.17368pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{c}{\downarrow})$}}$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 3.34894pt\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\\{P\\}$}\kern 0.0pt$}\kern 3.34894pt}}}\kern 19.99997pt}}}$ costruiamo una nuova derivazione per induzione strutturale su $P$: * • $P$ è un atomo. Allora l’istanza di una regola generale è anche un’istanza della corrispettiva in forma atomica. * • $P=\mathsf{t}$ o $P=\mathsf{f}$. Allora l’istanza di una regola generale è un’istanza della relazione d’equivalenza, con l’eccezione dell’indebolimento quando $P=\mathsf{f}$. Allora la regola d’indebolimento generale è sostituita da: $\mathbb{C}\\{\mathsf{f}\\}=\mathbb{C}\textnormal{{(}}\mathsf{f},\textnormal{{[}}\mathsf{t},\mathsf{t}\textnormal{{]}}\textnormal{{)}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{s})$ $\mathbb{C}\textnormal{{[}}\textnormal{{(}}\mathsf{f},\mathsf{t}\textnormal{{)}},\mathsf{t}\textnormal{{]}}=\mathbb{C}\\{\mathsf{t}\\}$ * • $P=\textnormal{{[}}Q,R\textnormal{{]}}$. Per ipotesi induttiva, usando rispettivamente le sole regole $\\{(\mathsf{ai}{\downarrow}),(\mathsf{s})\\}$, $\\{(\mathsf{aw}{\downarrow}),(\mathsf{s})\\}$ e $\\{(\mathsf{ac}{\downarrow}),(\mathsf{m})\\}$, abbiamo: ${{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 3.23509pt\hbox{\hbox{\kern 0.43991pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\\{\mathsf{t}\\}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 10.55557pt\hbox{$\kern 3.33333pt$}\kern 10.55557pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 10.55557pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{Q}^{(\mathsf{i}{\downarrow})}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 10.55557pt}\kern-1.43518pt\hbox{\kern 10.55557pt\hbox{\kern 3.33333pt}\kern 10.55557pt}}}\kern 0.0pt}}\kern 3.675pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\mathbb{C}\textnormal{{[}}Q,\overline{Q}\textnormal{{]}}$}}}}\kern 0.0pt}}}\quad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 3.21155pt\hbox{\hbox{\kern 0.34575pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\\{\mathsf{t}\\}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 10.55557pt\hbox{$\kern 3.33333pt$}\kern 10.55557pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 10.55557pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{R}^{(\mathsf{i}{\downarrow})}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 10.55557pt}\kern-1.43518pt\hbox{\kern 10.55557pt\hbox{\kern 3.33333pt}\kern 10.55557pt}}}\kern 0.0pt}}\kern 3.5573pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\mathbb{C}\textnormal{{[}}R,\overline{R}\textnormal{{]}}$}}}}\kern 0.0pt}}}\quad,\quad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 1.56842pt\hbox{\hbox{\kern 0.85657pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\\{\mathsf{f}\\}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 10.13892pt\hbox{$\kern 3.33333pt$}\kern 10.13892pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 10.13892pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{Q}^{(\mathsf{w}{\downarrow})}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 10.13892pt}\kern-1.43518pt\hbox{\kern 10.13892pt\hbox{\kern 3.33333pt}\kern 10.13892pt}}}\kern 0.0pt}}\kern 2.42499pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\mathbb{C}\\{Q\\}$}}}}\kern 0.0pt}}}\quad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 1.54488pt\hbox{\hbox{\kern 0.7624pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\\{\mathsf{f}\\}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 10.13892pt\hbox{$\kern 3.33333pt$}\kern 10.13892pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 10.13892pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{R}^{(\mathsf{w}{\downarrow})}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 10.13892pt}\kern-1.43518pt\hbox{\kern 10.13892pt\hbox{\kern 3.33333pt}\kern 10.13892pt}}}\kern 0.0pt}}\kern 2.30728pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\mathbb{C}\\{R\\}$}}}}\kern 0.0pt}}}\quad,\quad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\textnormal{{[}}Q,Q\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 16.51665pt\hbox{$\kern 3.33333pt$}\kern 16.51665pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 16.51665pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{Q}^{(\mathsf{c}{\downarrow})}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 16.51665pt}\kern-1.43518pt\hbox{\kern 16.51665pt\hbox{\kern 3.33333pt}\kern 16.51665pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 3.95274pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 3.95274pt}\kern 1.43518pt\hbox{\kern 3.95274pt\hbox{$\mathbb{C}\\{Q\\}$}\kern 3.95274pt}}}\kern 0.0pt}}}\quad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\textnormal{{[}}R,R\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 16.28125pt\hbox{$\kern 3.33333pt$}\kern 16.28125pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 16.28125pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{R}^{(\mathsf{c}{\downarrow})}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 16.28125pt}\kern-1.43518pt\hbox{\kern 16.28125pt\hbox{\kern 3.33333pt}\kern 16.28125pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 3.83505pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 3.83505pt}\kern 1.43518pt\hbox{\kern 3.83505pt\hbox{$\mathbb{C}\\{R\\}$}\kern 3.83505pt}}}\kern 0.0pt}}}$ da cui è possibile derivare: ${{{}{}{}{}{}{}{}{}{}{}}{{{{{}{}{}{}{}{}{}{}{}{}}{{{}}}}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 16.73058pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 4.04486pt\hbox{\hbox{\kern 0.34575pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\\{\mathsf{t}\\}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 10.55557pt\hbox{$\kern 3.33333pt$}\kern 10.55557pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 10.55557pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{R}^{(\mathsf{i}{\downarrow})}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 10.55557pt}\kern-1.43518pt\hbox{\kern 10.55557pt\hbox{\kern 3.33333pt}\kern 10.55557pt}}}\kern 0.0pt}}\kern 4.39061pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\mathbb{C}\textnormal{{[}}\overline{R},R\textnormal{{]}}$}}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 14.94618pt\hbox{$\kern 3.33333pt$}\kern 14.94618pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 14.94618pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{Q}^{(\mathsf{i}{\downarrow})}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 14.94618pt}\kern-1.43518pt\hbox{\kern 14.94618pt\hbox{\kern 3.33333pt}\kern 14.94618pt}}}\kern 0.0pt}}\kern 16.73058pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\mathbb{C}\textnormal{{(}}\textnormal{{[}}\overline{Q},Q\textnormal{{]}},\textnormal{{[}}\overline{R},R\textnormal{{]}}\textnormal{{)}}$}}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to63.35352pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 57.79802pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{s})$}}$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\textnormal{{[}}\textnormal{{(}}\overline{R},\textnormal{{[}}\overline{Q},Q\textnormal{{]}}\textnormal{{)}},R\textnormal{{]}}$}\kern 0.0pt$}}}}\kern 14.49995pt}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to63.35352pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 57.79802pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{s})$}}$}\hss}\kern 14.49995pt}\kern 1.43518pt\hbox{\kern 0.83331pt\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\textnormal{{[}}\textnormal{{(}}\overline{Q},\overline{R}\textnormal{{)}},\textnormal{{[}}Q,R\textnormal{{]}}\textnormal{{]}}$}\kern 0.0pt$}\kern 15.33327pt}}}\kern 0.0pt}}}\qquad,\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{{{}{}{}{}{}{}{}{}{}{}}{{{}}}}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\\{\mathsf{f}\\}=\mathbb{C}\textnormal{{[}}\mathsf{f},\mathsf{f}\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 24.86118pt\hbox{$\kern 3.33333pt$}\kern 24.86118pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 24.86118pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{Q}^{(\mathsf{w}{\downarrow})}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 24.86118pt}\kern-1.43518pt\hbox{\kern 24.86118pt\hbox{\kern 3.33333pt}\kern 24.86118pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 11.60283pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 11.60283pt}\kern 1.43518pt\hbox{\kern 11.60283pt\hbox{$\mathbb{C}\textnormal{{[}}Q,\mathsf{f}\textnormal{{]}}$}\kern 11.60283pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 11.60283pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 11.60283pt}\kern 1.43518pt\hbox{\kern 24.86118pt\hbox{$\kern 3.33333pt$}\kern 24.86118pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 24.86118pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{R}^{(\mathsf{w}{\downarrow})}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 24.86118pt}\kern-1.43518pt\hbox{\kern 24.86118pt\hbox{\kern 3.33333pt}\kern 24.86118pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 8.46222pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 8.46222pt}\kern 1.43518pt\hbox{\kern 8.46222pt\hbox{$\mathbb{C}\textnormal{{[}}Q,R\textnormal{{]}}$}\kern 8.46222pt}}}\kern 0.0pt}}}\qquad,\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{{{}{}{}{}{}{}{}{}{}{}}{{{}}}}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\textnormal{{[}}Q,Q,R,R\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 28.63121pt\hbox{$\kern 3.33333pt$}\kern 28.63121pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 28.63121pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{Q}^{(\mathsf{c}{\downarrow})}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 28.63121pt}\kern-1.43518pt\hbox{\kern 28.63121pt\hbox{\kern 3.33333pt}\kern 28.63121pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 6.17497pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 6.17497pt}\kern 1.43518pt\hbox{\kern 6.17497pt\hbox{$\mathbb{C}\textnormal{{[}}Q,R,R\textnormal{{]}}$}\kern 6.17497pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 6.17497pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 6.17497pt}\kern 1.43518pt\hbox{\kern 28.63121pt\hbox{$\kern 3.33333pt$}\kern 28.63121pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 28.63121pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{R}^{(\mathsf{c}{\downarrow})}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 28.63121pt}\kern-1.43518pt\hbox{\kern 28.63121pt\hbox{\kern 3.33333pt}\kern 28.63121pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 12.23225pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 12.23225pt}\kern 1.43518pt\hbox{\kern 12.23225pt\hbox{$\mathbb{C}\textnormal{{[}}Q,R\textnormal{{]}}$}\kern 12.23225pt}}}\kern 0.0pt}}}$ * • $P=\textnormal{{(}}Q,R\textnormal{{)}}$. L’ipotesi induttiva è identica a quella del caso precedente, da cui è possibile derivare: ${{{}{}{}{}{}{}{}{}{}{}}{{{{{}{}{}{}{}{}{}{}{}{}}{{{}}}}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 0.83331pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 15.89726pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 3.21155pt\hbox{\hbox{\kern 0.34575pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\\{\mathsf{t}\\}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 10.55557pt\hbox{$\kern 3.33333pt$}\kern 10.55557pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 10.55557pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{R}^{(\mathsf{i}{\downarrow})}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 10.55557pt}\kern-1.43518pt\hbox{\kern 10.55557pt\hbox{\kern 3.33333pt}\kern 10.55557pt}}}\kern 0.0pt}}\kern 3.5573pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\mathbb{C}\textnormal{{[}}R,\overline{R}\textnormal{{]}}$}}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 14.11287pt\hbox{$\kern 3.33333pt$}\kern 14.11287pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 14.11287pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{Q}^{(\mathsf{i}{\downarrow})}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 14.11287pt}\kern-1.43518pt\hbox{\kern 14.11287pt\hbox{\kern 3.33333pt}\kern 14.11287pt}}}\kern 0.0pt}}\kern 15.89726pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\mathbb{C}\textnormal{{(}}\textnormal{{[}}Q,\overline{Q}\textnormal{{]}},\textnormal{{[}}R,\overline{R}\textnormal{{]}}\textnormal{{)}}$}}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to60.02026pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 54.46477pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{s})$}}$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\textnormal{{[}}\textnormal{{(}}R,\textnormal{{[}}Q,\overline{Q}\textnormal{{]}}\textnormal{{)}},\overline{R}\textnormal{{]}}$}\kern 0.0pt$}}}}\kern 14.49995pt}}}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to61.68689pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 56.1314pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{s})$}}$}\hss}\kern 13.66664pt}\kern 1.43518pt\hbox{\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\textnormal{{[}}\textnormal{{(}}Q,R\textnormal{{)}},\textnormal{{[}}\overline{Q},\overline{R}\textnormal{{]}}\textnormal{{]}}$}\kern 0.0pt$}\kern 13.66664pt}}}\kern 0.83331pt}}}\qquad,\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{{{}{}{}{}{}{}{}{}{}{}}{{{}}}}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\\{\mathsf{f}\\}=\mathbb{C}\textnormal{{(}}\mathsf{f},\mathsf{f}\textnormal{{)}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 25.97229pt\hbox{$\kern 3.33333pt$}\kern 25.97229pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 25.97229pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{Q}^{(\mathsf{w}{\downarrow})}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 25.97229pt}\kern-1.43518pt\hbox{\kern 25.97229pt\hbox{\kern 3.33333pt}\kern 25.97229pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 11.60283pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 11.60283pt}\kern 1.43518pt\hbox{\kern 11.60283pt\hbox{$\mathbb{C}\textnormal{{(}}Q,\mathsf{f}\textnormal{{)}}$}\kern 11.60283pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 11.60283pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 11.60283pt}\kern 1.43518pt\hbox{\kern 25.97229pt\hbox{$\kern 3.33333pt$}\kern 25.97229pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 25.97229pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{R}^{(\mathsf{w}{\downarrow})}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 25.97229pt}\kern-1.43518pt\hbox{\kern 25.97229pt\hbox{\kern 3.33333pt}\kern 25.97229pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 8.46222pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 8.46222pt}\kern 1.43518pt\hbox{\kern 8.46222pt\hbox{$\mathbb{C}\textnormal{{(}}Q,R\textnormal{{)}}$}\kern 8.46222pt}}}\kern 0.0pt}}}\qquad,\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{{{}{}{}{}{}{}{}{}{}{}}{{{}}}}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\textnormal{{[}}\textnormal{{(}}Q,R\textnormal{{)}},\textnormal{{(}}Q,R\textnormal{{)}}\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to71.15141pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 65.59592pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{m})$}}$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 1.11111pt\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\textnormal{{(}}\textnormal{{[}}Q,Q\textnormal{{]}},\textnormal{{[}}R,R\textnormal{{]}}\textnormal{{)}}$}\kern 0.0pt$}\kern 1.11111pt}}}\kern 18.88887pt}}}\kern 1.43518pt\hbox{\kern 1.11111pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 19.99998pt}\kern 1.43518pt\hbox{\kern 35.5757pt\hbox{$\kern 3.33333pt$}\kern 54.46457pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 35.5757pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{Q}^{(\mathsf{c}{\downarrow})}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 54.46457pt}\kern-1.43518pt\hbox{\kern 35.5757pt\hbox{\kern 3.33333pt}\kern 54.46457pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 10.06387pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 28.95274pt}\kern 1.43518pt\hbox{\kern 10.06387pt\hbox{$\mathbb{C}\textnormal{{(}}Q,\textnormal{{[}}R,R\textnormal{{]}}\textnormal{{)}}$}\kern 28.95274pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 10.06387pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 28.95274pt}\kern 1.43518pt\hbox{\kern 35.5757pt\hbox{$\kern 3.33333pt$}\kern 54.46457pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 35.5757pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Phi_{R}^{(\mathsf{c}{\downarrow})}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 54.46457pt}\kern-1.43518pt\hbox{\kern 35.5757pt\hbox{\kern 3.33333pt}\kern 54.46457pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 18.06563pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 36.9545pt}\kern 1.43518pt\hbox{\kern 18.06563pt\hbox{$\mathbb{C}\textnormal{{(}}Q,R\textnormal{{)}}$}\kern 36.9545pt}}}\kern 0.0pt}}}$ I casi duali si dimostrano allo stesso modo, “girando sottosopra” le dimostrazioni e negando tutte le formule. ∎ Questo è un risultato molto significativo e difficilmente ottenibile usando il calcolo dei sequenti. Inoltre la dimostrazione è costruttiva e modulare, caratteristiche che ci permetteranno in seguito di utilizzare regole generalizzate con la consapevolezza di poterle sempre sostituire con una procedura effettiva con le loro versioni atomiche. #### 2.2.3 Rompere la simmetria: il Sistema KS Dimostriamo che nel Sistema SKS le regole con la freccia rivolta in alto $(\mathsf{\rho}{\uparrow})$ sono _ammissibili_ (e quindi in particolare anche la regola di taglio lo è). Il Sistema risultante dall’eliminazione delle regole $(\mathsf{\rho}{\uparrow})$ è chiamato Sistema KS, ed è riportato in Figura 2.4. In questa sezione seguiamo la dimostrazione di Brünnler [2004], a cui ho apportato alcune modifiche personali di carattere tecnico. ###### Lemma 2.2.5. Ogni regola di SKS è derivabile usando solo la sua duale, _identità_ , _taglio_ e _switch_. ###### Proof. Le regole $(\mathsf{s})$ e $(\mathsf{m})$ sono auto-duali, e pertanto banalmente derivabili. Un’istanza di una regola $\mathbb{C}\\{P\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{\rho}{\uparrow})$ $\mathbb{C}\\{Q\\}$ può essere sostituita da: $\mathbb{C}\\{P\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{i}{\downarrow})$ $\mathbb{C}{\textnormal{{(}}P,\textnormal{{[}}\overline{Q},Q\textnormal{{]}}\textnormal{{)}}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{s})$ $\mathbb{C}\textnormal{{[}}\textnormal{{(}}P,\overline{Q}\textnormal{{)}},Q\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{\rho}{\downarrow})$ $\mathbb{C}\textnormal{{[}}\textnormal{{(}}P,\overline{P}\textnormal{{)}},Q\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{i}{\uparrow})$ $\mathbb{C}\\{Q\\}$ e lo stesso vale per le regole $(\mathsf{\rho}{\downarrow})$. ∎ Prima di proseguire con la cut elimination, occorre stabilire una semplice proposizione, valida per la maggior parte dei sistemi espressi col calcolo delle strutture. ###### Proposizione 2.2.6. Per ogni struttura $P,Q$ e contesto $\mathbb{C}$, esiste una derivazione $\mathbb{C}\textnormal{{[}}P,Q\textnormal{{]}}$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\\{(\mathsf{s})\\}$ $\textnormal{{[}}\mathbb{C}\\{P\\},Q\textnormal{{]}}$ . ###### Proof. Per induzione sulla dimensione del contesto $\mathbb{C}$. 1. 1. Il caso base è $\mathbb{C}\\{\bullet\\}=\bullet$, da cui si deriva che esiste una derivazione (vuota) per $\textnormal{{[}}P,Q\textnormal{{]}}$. 2. 2. $\mathbb{C}\\{\bullet\\}=\textnormal{{[}}R,\mathbb{C^{\prime}}\\{\bullet\\}\textnormal{{]}}$. Allora, per ipotesi induttiva, esiste una derivazione: $\mathbb{C^{\prime}}\textnormal{{[}}P,Q\textnormal{{]}}$ $\scriptstyle\scriptstyle\Pi\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\\{(\mathsf{s})\\}$ $\textnormal{{[}}\mathbb{C^{\prime}}\\{P\\},Q\textnormal{{]}}$ che può essere usata per costruire: $\textnormal{{[}}R,\mathbb{C^{\prime}}\textnormal{{[}}P,Q\textnormal{{]}}\textnormal{{]}}$ $\scriptstyle\scriptstyle\textnormal{{[}}R,\Pi\textnormal{{]}}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\\{(\mathsf{s})\\}$ $\textnormal{{[}}R,\mathbb{C^{\prime}}\\{P\\},Q\textnormal{{]}}$ e $\textnormal{{[}}R,\mathbb{C^{\prime}}\\{P\\},Q\textnormal{{]}}$ è proprio uguale a $\textnormal{{[}}\mathbb{C}\\{P\\},Q\textnormal{{]}}$. 3. 3. $\mathbb{C}\\{\bullet\\}=\textnormal{{(}}R,\mathbb{C^{\prime}}\\{\bullet\\}\textnormal{{)}}$. Qui l’ipotesi induttiva ci dà: $\mathbb{C^{\prime}}\textnormal{{[}}P,Q\textnormal{{]}}$ $\scriptstyle\scriptstyle\Pi\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\\{(\mathsf{s})\\}$ $\textnormal{{[}}\mathbb{C^{\prime}}\\{P\\},Q\textnormal{{]}}$ che può essere usata per costruire: $\textnormal{{(}}R,\mathbb{C^{\prime}}\textnormal{{[}}P,Q\textnormal{{]}}\textnormal{{)}}$ $\scriptstyle\scriptstyle\textnormal{{(}}R,\Pi\textnormal{{)}}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\\{(\mathsf{s})\\}$ $\textnormal{{(}}R,\textnormal{{[}}\mathbb{C^{\prime}}\\{P\\},Q\textnormal{{]}}\textnormal{{)}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{s})$ $\textnormal{{[}}\textnormal{{(}}R,\mathbb{C^{\prime}}\\{P\\}\textnormal{{)}},Q\textnormal{{]}}=\textnormal{{[}}\mathbb{C}\\{P\\},Q\textnormal{{]}}$ ∎ ###### Definizione 2.2.7 (Taglio atomico di superficie). Un’istanza della regola di taglio atomica $(\mathsf{ai}{\uparrow})$ è chiamata _shallow_ (o _taglio atomico di superficie_) quando è della forma: $\textnormal{{[}}S,\textnormal{{(}}a,\overline{a}\textnormal{{)}}\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{ai}{\uparrow})$ $S$ ###### Lemma 2.2.8. La regola di taglio atomica $(\mathsf{ai}{\uparrow})$ è derivabile usando _taglio atomico di superficie_ e _switch_. ###### Proof. Ogni formula $\mathbb{C}\textnormal{{(}}a,\overline{a}\textnormal{{)}}$ è equivalente a $\mathbb{C}\textnormal{{[}}\mathsf{f},\textnormal{{(}}a,\overline{a}\textnormal{{)}}\textnormal{{]}}$. Per la Proposizione 2.2.6, esiste una derivazione: $\mathbb{C}\textnormal{{[}}\mathsf{f},\textnormal{{(}}a,\overline{a}\textnormal{{)}}\textnormal{{]}}$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\\{(\mathsf{s})\\}$ $\textnormal{{[}}\mathbb{C}\\{\mathsf{f}\\},\textnormal{{(}}a,\overline{a}\textnormal{{)}}\textnormal{{]}}$ Pertanto basta porre $S=\mathbb{C}\\{\mathsf{f}\\}$ per effettuare la trasformazione: ${\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\textnormal{{(}}a,\overline{a}\textnormal{{)}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to28.06372pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 22.50822pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{ai}{\uparrow})$}}$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 3.89294pt\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\\{\mathsf{f}\\}$}\kern 0.0pt$}\kern 3.89294pt}}}\kern 23.33333pt}}}\qquad\rightsquigarrow\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 3.63194pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\textnormal{{(}}a,\overline{a}\textnormal{{)}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 14.03186pt\hbox{$\kern 3.33333pt$}\kern 14.03186pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 14.03186pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle\;\\{(\mathsf{s})\\}$}\hss}\kern 14.03186pt}\kern-1.43518pt\hbox{\kern 14.03186pt\hbox{\kern 3.33333pt}\kern 14.03186pt}}}\kern 6.95145pt}}}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 3.3195pt}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}S,\textnormal{{(}}a,\overline{a}\textnormal{{)}}\textnormal{{]}}$}\kern 3.3195pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to35.3276pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 29.77211pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{ai}{\uparrow})$}}$}\hss}\kern 3.3195pt}\kern 1.43518pt\hbox{\kern 14.30965pt\hbox{$\kern 1.64584pt\hbox{$S$}\kern 1.64584pt$}\kern 17.62915pt}}}\kern 20.01382pt}}}$ ∎ ###### Lemma 2.2.9. Ogni dimostrazione $\scriptstyle-$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\mathsf{KS}$ $\mathbb{C}\\{a\\}$ può essere trasformata in $\scriptstyle-$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\mathsf{KS}$ $\mathbb{C}\\{\mathsf{t}\\}$ . ###### Proof. Risalendo la dimostrazione, sostituiamo nelle regole l’occorrenza di $a$ e le sue copie prodotte per contrazione, con l’unità $\mathsf{t}$. Le istanze delle regole $(\mathsf{s})$ e $(\mathsf{m})$ rimangono intatte, le istanze di $(\mathsf{ac}{\downarrow})$ si riducono ad applicazioni della relazione d’equivalenza $=$. Le altre applicazioni vengono sostituite dalle seguenti derivazioni: $\begin{array}[]{ccc}{\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 1.11516pt\hbox{\hbox{$\mathbb{C}\\{\mathsf{f}\\}$}}\kern 1.11516pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to22.50815pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 16.95265pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{aw}{\downarrow})$}}$}\hss}}\kern 1.43518pt\hbox{\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\\{a\\}$}\kern 0.0pt$}}}}\kern 27.49998pt}}}&\rightsquigarrow&{\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 0.41666pt\hbox{\hbox{$\mathbb{C}\\{\mathsf{f}\\}=\mathbb{C}\textnormal{{(}}\mathsf{f},\textnormal{{[}}\mathsf{t},\mathsf{t}\textnormal{{]}}\textnormal{{)}}$}}\kern 0.41666pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to65.83351pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 60.27802pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{s})$}}$}\hss}}\kern 1.43518pt\hbox{\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\textnormal{{[}}\textnormal{{(}}\mathsf{f},\mathsf{t}\textnormal{{)}},\mathsf{t}\textnormal{{]}}=\mathbb{C}\\{\mathsf{t}\\}$}\kern 0.0pt$}}}}\kern 14.49995pt}}}\\\ \nobreak\leavevmode\hfil&&\nobreak\leavevmode\hfil\\\ {\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 2.36516pt\hbox{\hbox{$\mathbb{C}\\{\mathsf{t}\\}$}}\kern 2.36516pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to25.84149pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 20.286pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{ai}{\downarrow})$}}$}\hss}}\kern 1.43518pt\hbox{\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\textnormal{{[}}a,\overline{a}\textnormal{{]}}$}\kern 0.0pt$}}}}\kern 23.33333pt}}}&\rightsquigarrow&{\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\\{\mathsf{t}\\}=\mathbb{C}\textnormal{{[}}\mathsf{t},\mathsf{f}\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to51.38902pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 45.83353pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{aw}{\downarrow})$}}$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 13.47226pt\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\textnormal{{[}}\mathsf{t},\overline{a}\textnormal{{]}}$}\kern 0.0pt$}\kern 13.47226pt}}}\kern 27.49998pt}}}\end{array}$ ∎ thbs $\mathsf{t}\quad(\mathsf{ax})\qquad\qquad{\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 2.36516pt\hbox{\hbox{$\mathbb{C}\\{\mathsf{t}\\}$}}\kern 2.36516pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to25.84149pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 20.286pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{ai}{\downarrow})$}}$}\hss}}\kern 1.43518pt\hbox{\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\textnormal{{[}}a,\overline{a}\textnormal{{]}}$}\kern 0.0pt$}}}}\kern 23.33333pt}}}\qquad\qquad{\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 1.11516pt\hbox{\hbox{$\mathbb{C}\\{\mathsf{f}\\}$}}\kern 1.11516pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to22.50815pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 16.95265pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{aw}{\downarrow})$}}$}\hss}}\kern 1.43518pt\hbox{\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\\{a\\}$}\kern 0.0pt$}}}}\kern 27.49998pt}}}\qquad\qquad{\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\textnormal{{[}}a,a\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to27.79399pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 22.2385pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{ac}{\downarrow})$}}$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.64291pt\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\\{a\\}$}\kern 0.0pt$}\kern 2.64291pt}}}\kern 24.99998pt}}}$ $\mathbb{C}\textnormal{{(}}P,\textnormal{{[}}Q,R\textnormal{{]}}\textnormal{{)}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{s})$ $\mathbb{C}\textnormal{{[}}\textnormal{{(}}P,Q\textnormal{{)}},R\textnormal{{]}}$ $\mathbb{C}\textnormal{{[}}\textnormal{{(}}P,Q\textnormal{{)}},\textnormal{{(}}R,S\textnormal{{)}}\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{m})$ $\mathbb{C}\textnormal{{(}}\textnormal{{[}}P,R\textnormal{{]}},\textnormal{{[}}Q,S\textnormal{{]}}\textnormal{{)}}$ Figure 2.4: Regole del Sistema KS ###### Teorema 2.2.10. Ogni dimostrazione $\scriptstyle-$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\mathsf{SKS}$ $P$ può essere trasformata in una dimostrazione $\scriptstyle-$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\mathsf{KS}$ $P$ . ###### Proof. Grazie al Lemma 2.2.5, sappiamo che l’unica regola da eliminare è il taglio $(\mathsf{ai}{\uparrow})$. Grazie al Lemma 2.2.8 possiamo sostituire tutti i tagli con tagli di superficie. Partendo dall’alto, selezioniamo la prima istanza della regola di taglio: $\scriptstyle-$ $\scriptstyle\scriptstyle\Pi\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\mathsf{KS}$ $\textnormal{{[}}R,\textnormal{{(}}a,\overline{a}\textnormal{{)}}\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{ai}{\uparrow})$ $R$ $\scriptstyle\scriptstyle\Phi\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\mathsf{KS}\cup\\{(\mathsf{ai}{\uparrow})\\}$ $P$ Applicando due volte il Lemma 2.2.9 a $\Pi$, otteniamo: ${{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 1.4503pt\hbox{\hbox{\kern 7.74997pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Pi_{1}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle\;\mathsf{KS}$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 9.83333pt}}}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.63306pt}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}R,a\textnormal{{]}}$}\kern 0.63306pt}}}\kern 0.0pt}}}\qquad,\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 0.47404pt\hbox{\hbox{\kern 7.74997pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Pi_{2}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle\;\mathsf{KS}$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 9.83333pt}}}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 1.60931pt}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}R,\overline{a}\textnormal{{]}}$}\kern 1.60931pt}}}\kern 0.0pt}}}$ Partendo dalla conclusione e risalendo la dimostrazione $\Pi_{1}$, sostituiamo l’occorrenza di $a$ e le sue copie prodotte per contrazione, con la formula $R$. Le istanze delle regole $(\mathsf{m})$ e $(\mathsf{s})$ rimangono intatte, mentre le istanze di $(\mathsf{ac}{\downarrow})$ e $(\mathsf{aw}{\downarrow})$ vengono sostituite dalle loro versioni generalizzate: $\begin{array}[]{ccc}{\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\textnormal{{[}}a,a\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to27.79399pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 22.2385pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{ac}{\downarrow})$}}$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.64291pt\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\\{a\\}$}\kern 0.0pt$}\kern 2.64291pt}}}\kern 24.99998pt}}}&\rightsquigarrow&{\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\textnormal{{[}}R,R\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to32.56252pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 27.00702pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{c}{\downarrow})$}}$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 3.83505pt\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\\{R\\}$}\kern 0.0pt$}\kern 3.83505pt}}}\kern 19.99997pt}}}\\\ \nobreak\leavevmode\hfil&&\nobreak\leavevmode\hfil\\\ {\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 1.11516pt\hbox{\hbox{$\mathbb{C}\\{\mathsf{f}\\}$}}\kern 1.11516pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to22.50815pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 16.95265pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{aw}{\downarrow})$}}$}\hss}}\kern 1.43518pt\hbox{\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\\{a\\}$}\kern 0.0pt$}}}}\kern 27.49998pt}}}&\rightsquigarrow&{\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 2.30728pt\hbox{\hbox{$\mathbb{C}\\{\mathsf{f}\\}$}}\kern 2.30728pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to24.89241pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 19.33691pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{w}{\downarrow})$}}$}\hss}}\kern 1.43518pt\hbox{\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\\{R\\}$}\kern 0.0pt$}}}}\kern 22.77776pt}}}\end{array}$ Le istanze di $(\mathsf{ai}{\downarrow})$ sono sostituite da $\mathbb{C}\\{\Pi_{2}\\}$: ${\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 2.36516pt\hbox{\hbox{$\mathbb{C}\\{\mathsf{t}\\}$}}\kern 2.36516pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to25.84149pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 20.286pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{ai}{\downarrow})$}}$}\hss}}\kern 1.43518pt\hbox{\hbox{$\kern 0.0pt\hbox{$\mathbb{C}\textnormal{{[}}a,\overline{a}\textnormal{{]}}$}\kern 0.0pt$}}}}\kern 23.33333pt}}}\quad\quad\rightsquigarrow\quad\quad\quad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 14.52773pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\mathbb{C}\\{\mathsf{t}\\}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 10.55557pt\hbox{$\kern 3.33333pt$}\kern 10.55557pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 10.55557pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\mathbb{C}\\{\Pi_{2}\\}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle\;\mathsf{KS}$}\hss}\kern 10.55557pt}\kern-1.43518pt\hbox{\kern 10.55557pt\hbox{\kern 3.33333pt}\kern 10.55557pt}}}\kern 1.5555pt}}\kern 2.0018pt}\kern 1.43518pt\hbox{\kern 10.97043pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\kern 10.97043pt\hbox{$\mathbb{C}\textnormal{{[}}R,\overline{a}\textnormal{{]}}$}}}}\kern 0.0pt}}}$ Il risultato di questa sostitituzione di $\Pi_{2}$ dentro $\Pi_{1}$ è una dimostrazione $\Pi_{3}$, grazie alla quale possiamo costruire: $\scriptstyle-$ $\scriptstyle\scriptstyle\Pi_{3}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\mathsf{KS}$ $\textnormal{{[}}R,R\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{c}{\downarrow})$ $R$ $\scriptstyle\scriptstyle\Phi\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\mathsf{KS}\cup\\{(\mathsf{ai}{\uparrow})\\}$ $P$ Ora basta procedere induttivamente verso il basso per rimuovere le rimanenti istanze di $(\mathsf{ai}{\uparrow})$. Alla fine di questo procedimento, le regole generalizzate possono essere rimosse usando la procedura descritta nella dimostrazione del Teorema 2.2.4. ∎ ## Chapter 3 Logica lineare La logica lineare è un’estensione della logica classica ideata da Jean-Yves Girard verso la fine degli anni ’80 (Girard [1987]; Girard et al. [1989]; Girard [1995b]). La caratteristica peculiare della logica lineare è che tratta l’implicazione come _fenomeno causale_ anziché (com’è pratica comune in matematica) come _concetto stabile_ : $\mbox{se }A\mbox{ e }A{\Rightarrow}B\mbox{ allora }B\mbox{, \emph{ma $A$ \\`{e} ancora valida.}}$ Un’implicazione causale non può essere reiterata, poiché le condizioni iniziali sono modificate dopo il suo utilizzo; questo processo di modifica delle premesse (condizioni) è noto in fisica come _reazione_ 111Quello di reazione è un concetto base anche della teoria dei modelli concorrenti, vedi ad esempio Milner et al. [1992]; Sangiorgi and Walker [2001].. Per esempio, se $A$ è “spendere una moneta nel distributore automatico di bevande (o DAB)” e $B$ è “prendere un caffè”, la moneta viene persa nel processo, che quindi non si può ripetere una seconda volta. Esistono tuttavia casi, sia in matematica che nella vita reale, in cui le reazioni non esistono o sono trascurabili: ad esempio un lemma che resta sempre vero, o un tecnico che possiede la chiave del DAB e può recuperare ogni volta la sua moneta. Questi sono i casi che Girard chiama _situazioni_ , cioè condizioni durature e immutevoli (o verità stabili), e sono comunque gestibili in logica lineare tramite speciali connettivi (gli _esponenziali_ , “!” e “?”). Gli esponenziali esprimono la reiterabilità di un’azione, ossia l’assenza di reazioni; tipicamente $!A$ significa “spendere quante monete si vogliono”. Usiamo il simbolo $\multimap$ per denotare l’implicazione causale (o _implicazione lineare_); vale la seguente equazione: $A\Rightarrow B\quad=\quad(!A)\multimap B$ cioè $B$ è causato da un certo numero d’iterazioni di $A$. Una _azione di tipo $A$_ consisterà nel tirare fuori una certa moneta dalla tasca di qualcuno (ci potrebbero essere diverse azioni di questo tipo, poiché potremmo disporre di diverse monete). Analogamente saranno disponibili un certo numero di caffè nel distributore automatico, perciò ci saranno diverse _azioni di tipo $B$_. La logica lineare apre nuovi interessanti scenari sulla visione dei connettivi classici: ad esempio esistono _due_ congiunzioni ($\otimes$ o “per”, inteso in senso di moltiplicazione, ed $\with$ o “con”) corrispondenti a due usi radicalmente differenti della parola “e”. Ambedue le congiunzioni esprimono la disponibilità di due azioni; ma nel caso di $\otimes$, saranno fatte tutt’e due, mentre nel caso di $\with$, solo una delle due sarà eseguita (ma noi potremo decidere quale). Ad esempio, siano $A$, $B$, $C$: $A$ \| : \| spendere una moneta nel DAB ---\|---\|--- $B$ \| : \| prendere un caffè $C$ \| : \| prendere un tè Data un’azione di tipo $A\multimap B$ e una di tipo $A\multimap C$, non sarà possibile formare un’azione di tipo $A\multimap B\otimes C$, poiché per una moneta non si potrà mai avere ciò che ne costa due (sarà invece possibile formare un’azione di tipo $A\otimes A\multimap B\otimes C$, cioè avere due bevande in cambio di due monete). Comunque potremmo sempre produrre un’azione di tipo $A\multimap B\with C$ come sovrapposizione delle due. Per eseguire quest’ultima azione dovremmo prima scegliere tra le possibili azioni che vogliamo produrre e in seguito effettuare quella scelta. Questo è analogo a quanto accade col costrutto $\mathsf{if}\leavevmode\nobreak\ \dots\leavevmode\nobreak\ \mathsf{then}\leavevmode\nobreak\ \dots\leavevmode\nobreak\ \mathsf{else}\leavevmode\nobreak\ \dots$ ben noto in informatica: infatti, sia la parte $\mathsf{then}$ … che quella $\mathsf{else}$ … sono disponibili, ma solo una di esse verrà eseguita. Per quanto “$\with$” abbia delle ovvie caratteristiche disgiuntive, sarebbe tecnicamente errato vederlo come disgiunzione: infatti in logica lineare sia $A\with B\multimap A$, sia $A\with B\multimap B$ sono dimostrabili. In logica lineare, in maniera del tutto speculare, abbiamo due disgiunzioni, che sono $\oplus$ o “più”, e $\parr$ o “par” (mnemonico per _parallelo_). $\oplus$ è il duale di “$\with$” ed esprime la presenza di due opzioni: in questo caso però, non sarà possibile scegliere quale delle due eseguire. La differenza tra $\with$ e $\oplus$ è la stessa che c’è in informatica tra nondeterminismo esterno ed interno. Infine $\parr$ è il duale di $\otimes$. Il più importante connettivo lineare è la _negazione lineare_ $\;\overline{\cdot}\;$ o “nil”. Poiché l’implicazione lineare si può sempre riscrivere come $\overline{A}\parr B$, “nil” è l’unica operazione negativa della logica lineare. La negazione lineare si comporta come la trasposizione in algebra lineare, esprime cioè _dualità_ , ovverosia un cambio di prospettiva: _azione di tipo_ $A$ = _reazione di tipo_ $\overline{A}$ La proprietà principale di “nil” è che, come accade in logica classica, $\overline{\overline{A}}$ può essere identificato con $A$ stesso. A differenza della logica classica però, la logica lineare gode di una _semplice interpretazione costruttiva_. Il carattere involutivo di “nil” assicura il comportamento _alla De Morgan_ per tutti i connettivi ed i quantificatori, ad esempio: $\exists x.A\quad=\quad\overline{(\forall x.\overline{A})}$ che può sembrare insolito ad un primo sguardo, specialmente se consideriamo che l’esistenziale in logica lineare è un operatore _effettivo_ : tipicamente si dimostra $\exists x.A$ dimostrando $A[t/x]$ per un certo termine $t$. Questo comportamento di “nil” deriva dal fatto che $\overline{A}$ nega (cioè _reagisce con_) una singola azione di tipo $A$, mentre la negazione classica nega solo alcune (non specificate) iterazioni di $A$, che tipicamente porta ad una disgiunzione di lunghezza non specificata. La negazione lineare è da un lato più primitiva, e dall’altro più forte (e anche più difficile da trattare) di quella classica. Grazie alla presenza degli esponenziali, la logica lineare è espressiva quanto quella classica o quella intuizionista. Di fatto è più espressiva. Qui bisogna essere cauti: è lo stesso problema della logica intuizionista, che è anch’essa “più espressiva” di quella classica. Tecnicamente il potere espressivo è equivalente: ma i connettivi della logica lineare possono esprimere in maniera primitiva cose che in logica classica possono essere espresse solo tramite complesse traduzioni _ad hoc_. L’introduzione di nuovi connettivi è quindi la chiave di volta verso formalizzazioni più semplici ed efficaci; la restrizione a vari frammenti apre le frontiere a linguaggi con specifico potere espressivo, ad esempio con una complessità computazionale nota (Girard [1998]; Lafont [2002]; Dal Lago and Baillot [2006]). Un notevole problema aperto è quello di trovare una versione convincente di logica lineare non-commutativa. Anche se molti convengono sul fatto che la non-commutatività ha ragione d’esser considerata a questo livello (esistono svariati esempi di problemi intrinsecamente non-commutativi, si pensi all’operatore di prefisso del $\pi$-calcolo), semantiche non trivali di non- commutatività non sono note. Unite all’introduzione di una semantica naturale, le metodologie per raggiungere un sistema non-commutativo potrebbero comportare un effettivo guadagno di potere espressivo, in relazione al caso commutativo. ### 3.1 Calcolo dei sequenti lineari Definiamo la sintassi della logica lineare classica (o CLL, acronimo di Classical Linear Logic): ###### Definizione 3.1.1 (Linguaggio CLL). Sia $\mathcal{P}$ un insieme infinito enumerabile di _simboli proposizionali_. L’insieme degli _atomi_ $\mathcal{A}$ è così definito: $\mathcal{A}=\\{p,\overline{p}\>\|\>p\in\mathcal{P}\\}$ dove $\overline{\cdot}$ è una _funzione di negazione primitiva sui simboli proposizionali_. La negazione si estende facilmente a tutti gli atomi definendo $\overline{\overline{p}}=p$ per ogni simbolo proposizionale negato $\overline{p}$. Siano $\mathsf{1},\bot,\mathsf{0},\top\not\in\mathcal{A}$ simboli costanti o _unità_ , e sia $a\in\mathcal{A}$. Il _linguaggio CLL delle formule lineari classiche_ è così definito: $\begin{array}[]{llll}T&::=&\mathsf{1}\>\|\>\bot\>\|\>\mathsf{0}\>\|\>\top\>\|\>a&\quad\mbox{(termini)}\\\ P&::=&T\>\|\>P\oplus P\>\|\>P\with P\>\|\>P\otimes P\>\|\>P\parr P\>\|\>\oc P\>\|\>\wn P&\quad\mbox{(formule)}\end{array}$ I connettivi $\otimes$, $\parr$, $\multimap$, insieme agli elementi neutri $\mathsf{1}$ (relativamente a $\otimes$) e $\bot$ (relativamente a $\parr$) sono chiamati _moltiplicativi_ ; i connettivi $\with$, $\oplus$, insieme agli elementi neutri $\top$ (relativamente a $\with$) e $\mathsf{0}$ (relativamente a $\oplus$) sono chiamati _additivi_ ; i connettivi $\oc$ e $\wn$ sono chiamati _esponenziali_. Questa notazione è stata scelta perché facile da memorizzare: infatti essa suggerisce che $\otimes$ sia moltiplicativo e congiuntivo, con elemento neutro $\mathsf{1}$, mentre $\oplus$ è additivo e disgiuntivo, con elemento neutro $\mathsf{0}$; inoltre, anche la distributività di $\otimes$ su $\oplus$ è suggerita dalla notazione. La negazione si estende alle formule, come mostrato in Figura 3.1; inoltre l’implicazione lineare è definita con l’ausilio di negazione e connettivo “par”. $\begin{array}[]{rclcrcl}\overline{\mathsf{1}}&=&\bot&&\overline{\bot}&=&\mathsf{1}\\\ \overline{\top}&=&\mathsf{0}&&\overline{\mathsf{0}}&=&\top\\\ \overline{P\otimes Q}&=&\overline{P}\parr\overline{Q}&&\overline{P\parr Q}&=&\overline{P}\otimes\overline{Q}\\\ \overline{P\with Q}&=&\overline{P}\oplus\overline{Q}&&\overline{P\oplus Q}&=&\overline{P}\with\overline{Q}\\\ \overline{\oc P}&=&\wn\overline{P}&&\overline{\wn P}&=&\oc\overline{P}\end{array}$ $P\multimap Q=\overline{P}\parr Q$ Figure 3.1: Definizione di negazione e implicazione lineari Procediamo mostrando in Figura 3.2 un sistema deduttivo per la logica lineare reminiscente il calcolo dei sequenti di Gentzen [1935]. Come visto in precedenza, un sequente è un’espressione $\Gamma\vdash\Delta$, in cui $\Gamma=P_{1},\dots,P_{n}$ e $\Delta=Q_{1},\dots,Q_{m}$ sono sequenze finite di formule. Il significato inteso di $\Gamma\vdash\Delta$ è: $P_{1}\mbox{ e }\dots\mbox{ e }P_{n}\quad\mbox{ implica }\quad Q_{1}\mbox{ oppure }\dots\mbox{ oppure }Q_{m}$ dove il senso di “e”, “implica” e “oppure” devono essere specificati formalmente. I sequenti lineari sono ad un lato, cioè della forma $\vdash\Gamma$; sequenti nella forma generale $\Gamma\vdash\Delta$ si possono “mimare” usando $\vdash\overline{\Gamma},\Delta$. Identità / taglio $\vdash P,\overline{P}\quad(\mathsf{id})$ $\vdash\Gamma,P$ $\vdash\overline{P},\Delta$ $(\mathsf{cut})$ $\vdash\Gamma,\Delta$ Regole strutturali $\vdash\Gamma,P,Q,\Delta$ $(\mathsf{perm})$ $\vdash\Gamma,Q,P,\Delta$ Regole logiche $\vdash\mathsf{1}\quad(\mathsf{one})$ $\vdash\Gamma$ $(\mathsf{false})$ $\vdash\Gamma,\bot$ $\vdash\Gamma,P$ $\vdash\Delta,Q$ $(\mathsf{\otimes})$ $\vdash\Gamma,\Delta,P\otimes Q$ $\vdash\Gamma,P,Q$ $(\mathsf{\parr})$ $\vdash\Gamma,P\parr Q$ $\vdash\Gamma,\top\quad(\mathsf{true})$ (nessuna regola per $\mathsf{0}$) $\vdash\Gamma,P$ $\vdash\Gamma,Q$ $(\mathsf{\with})$ $\vdash\Gamma,P\with Q$ $\vdash\Gamma,P$ $(\mathsf{\oplus_{l}})$ $\vdash\Gamma,P\oplus Q$ $\vdash\Gamma,Q$ $(\mathsf{\oplus_{r}})$ $\vdash\Gamma,P\oplus Q$ $\vdash\wn\Gamma,P$ $(\mathsf{\oc})$ $\vdash\wn\Gamma,\oc P$ $\vdash\Gamma$ $(\mathsf{weak})$ $\vdash\Gamma,\wn P$ $\vdash\Gamma,P$ $(\mathsf{drlc})$ $\vdash\Gamma,\wn P$ $\vdash\Gamma,\wn P,\wn P$ $(\mathsf{cntr})$ $\vdash\Gamma,\wn P$ Figure 3.2: Sistema deduttivo per CLL Nel calcolo dei sequenti lineari abbiamo rimosso le regole strutturali di indebolimento (è sempre possibile aggiungere una formula nella premessa o nella conclusione del sequente) e contrazione (la molteplicità di una formula non conta) in virtù delle critiche mosse dalla scuola lineare. La possibilità di utilizzare queste operazioni è tuttavia ripristinata grazie all’introduzione degli operatori $\oc$ e $\wn$. Identità e taglio restano invariate rispetto al Sistema LKp, così come la regola di permutazione. La situazione è diversa per quanto riguarda la congiunzione: come avveniva in precedenza, per dimostrare una congiunzione tra $P$ e $Q$ bisogna aver dimostrato separatamente sia $P$ che $Q$, ma in assenza della regola d’indebolimento, possiamo distinguere il caso in cui le dimostrazioni di $P$ e $Q$ siano fatte nello stesso ambiente ($\Gamma$ nella regola $(\mathsf{\with})$), o in ambienti diversi ($\Gamma$ e $\Delta$ in $(\mathsf{\otimes})$). Un ragionamento analogo vale per la disgiunzione: in logica lineare possiamo infatti distinguere il caso in cui, nel dimostrare la disgiunzione di $P$ e $Q$, disponiamo solo di $P$ (regola $(\mathsf{\oplus_{l}})$), solo di $Q$ (regola $(\mathsf{\oplus_{r}})$), e quello in cui abbiamo ambedue (regola $(\mathsf{\parr})$). È possibile introdurre nuove formule, indebolendo il sequente, a patto che queste siano “marcate” con l’operatore $\oc$; per questa classe di formule (chiamate formule “perché non” o “why not”) la molteplicità non è rilevante: inoltre ogni formula può essere trasformata in una _why not_ grazie alla regola di _derelizione_ $(\mathsf{drlc})$. La regola di _promozione_ $(\mathsf{\oc})$ permette “aumentare” la molteplicità di una formula di una quantità arbitraria. Il Sistema così ottenuto gode di buone proprietà, oltre ad avere una granularità più fine rispetto alla logica classica. Per CLL è possibile dimostrare la _cut elimination_ : ###### Teorema 3.1.2 (Hauptsatz lineare). La regola di taglio lineare è eliminabile da CLL. ###### Proof. La dimostrazione segue un argomento del tutto analogo a quello visto per la logica classica nel Teorema 2.1.6, con alcune semplificazioni dovute al fatto di non dover trattare le usuali regole strutturali. ∎ Nuovamente, la dimostrazione risultante dalla procedura di cut elimination non è univocamente determinata, a causa della _permutazione delle regole_. Ad esempio, nella derivazione: $\vdash\Gamma,P$ $(\mathsf{\rho})$ $\vdash\Gamma^{\prime},P$ $\vdash\overline{P},\Delta$ $(\mathsf{\sigma})$ $\vdash\overline{P},\Delta^{\prime}$ $(\mathsf{cut})$ $\vdash\Gamma^{\prime},\Delta^{\prime}$ non c’è nessun modo ovvio di eliminare l’applicazione di $(\mathsf{cut})$, poiché le regole $(\mathsf{\rho})$ e $(\mathsf{\sigma})$ non agiscono su $P$ e $\overline{P}$. Quindi l’idea è di “spingere il cut verso l’alto”: $\vdash\Gamma,P$ $\vdash\overline{P},\Delta$ $(\mathsf{cut})$ $\vdash\Gamma,\Delta$ $(\mathsf{\rho})$ $\vdash\Gamma^{\prime},\Delta$ $(\mathsf{\sigma})$ $\vdash\Gamma^{\prime},\Delta^{\prime}$ ma così facendo abbiamo arbitrariamente privilegiato la regola $(\mathsf{\rho})$ rispetto alla $(\mathsf{\sigma})$, mentre l’altra scelta: $\vdash\Gamma,P$ $\vdash\overline{P},\Delta$ $(\mathsf{cut})$ $\vdash\Gamma,\Delta$ $(\mathsf{\sigma})$ $\vdash\Gamma,\Delta^{\prime}$ $(\mathsf{\rho})$ $\vdash\Gamma^{\prime},\Delta^{\prime}$ sarebbe stata altrettanto legittima. La scelta compiuta in questo passo della cut elimination è in generale irreversibile: a meno che $(\mathsf{\rho})$ o $(\mathsf{\sigma})$ non siano successivamente eliminate, non sarà più possibili scambiarle. Per eliminare questa fonte di non-determinismo, fu introdotto in Girard [1987] un nuovo formalismo, basato sulla teoria dei grafi, e chiamato _Proof Nets_. Il Sistema CLL non è l’unico rappresentante della classe delle logiche lineari. Vista l’ampia gamma di regole che possiede, questo Sistema può essere suddiviso in moduli con interessanti proprietà computazionali: il punto è proprio che la logica lineare è in grado di trattare naturalmente con le risorse (rappresentate dalla _molteplicità_ delle formule), e per questo ci si riferisce ad essa con l’appellativo _resource-conscious_ ; in informatica avere coscienza delle risorse significa saper distinguere varie classi di complessità. Tra i vari sottosistemi, quelli che maggiormente divergono dalla logica classica (e intuizionista), sono chiamati LLL (Light Linear Logic) ed ELL (Elementary Linear Logic) – Girard [1995a]; Danos and Joinet [2001]. Essi seguono dalla scoperta che, in assenza degli esponenziali, la procedura di eliminazione dei tagli può essere eseguita in tempo lineare. Per i nostri scopi, ci occuperemo esclusivamente del frammento moltiplicativo: questo è il più semplice ed il più piccolo frammento di logica lineare (fu anche il primo che venne trasposto nelle Proof Nets, per via della sua semplicità). Nella fattispecie tratteremo d’ora in avanti il Sistema in Figura 3.3, chiamato MLL+mix, cioè _Multiplicative Linear Logic_ con l’aggiunta della regola mix che “fonde” i sequenti provenienti da due diversi sottoalberi di derivazione. La negazione è definita dalle leggi di De Morgan: $\displaystyle\overline{P\otimes Q}$ $\displaystyle=$ $\displaystyle\overline{P}\parr\overline{Q}$ $\displaystyle\overline{P\parr Q}$ $\displaystyle=$ $\displaystyle\overline{P}\otimes\overline{Q}$ Grammatica di MLL+mix $P::=a\>\|\>\overline{P}\>\|\>P\otimes P\>\|\>P\parr P$ (con $a\in\mathcal{A}$ infinità numerabile di simboli proposizionali) Sistema deduttivo $\vdash P,\overline{P}\;(\mathsf{id})$ $\vdash\Gamma,P$ $\vdash Q,\Delta$ $(\mathsf{\otimes})$ $\vdash\Gamma,\Delta,P\otimes Q$ $\vdash\Gamma,P,Q$ $(\mathsf{\parr})$ $\vdash\Gamma,P\parr Q$ $\vdash\Gamma$ $\vdash\Delta$ $(\mathsf{mix})$ $\vdash\Gamma,\Delta$ Figure 3.3: Sistema MLL+mix Avremo modo di osservare una natuale corrispondenza di questo Sistema ed il suo corrispettivo in deep inference: il Sistema LBV di Guglielmi [2002]. ### 3.2 Sistema LBV È il più semplice Sistema deep inference concepibile: un calcolo proposizionale composto da _due operatori duali_ (del tutto simili a congiunzione e disgiunzione classici), una _negazione auto-duale_ alla De Morgan e una _unità logica_. Come nel caso classico, le formule sono considerate uguali modulo una relazione di equivalenza. Le regole sono l’assioma $(\mathsf{ax})$ per l’unità e la regola di scambio $(\mathsf{s})$; inoltre la regola d’identità (chiamata anche _regola d’interazione_) e quella di taglio (o _regola di co-interazione_), nella versione generalizzata: $\mathbb{C}\\{\circ\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{i}{\downarrow})$ $\mathbb{C}\textnormal{{[}}P,\overline{P}\textnormal{{]}}$ $\mathbb{C}\textnormal{{(}}P,\overline{P}\textnormal{{)}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{i}{\uparrow})$ $\mathbb{C}\\{\circ\\}$ che tuttavia, come prima, possono essere ridotte alla loro forma atomica, dando origine al Sistema di Figura 3.4. ###### Teorema 3.2.1 (Località di LBV+cut). La regola $(\mathsf{i}{\downarrow})$ è derivabile da $\\{(\mathsf{ai}{\downarrow}),(\mathsf{s})\\}$. Dualmente, la regola $(\mathsf{i}{\uparrow})$ è derivabile da $\\{(\mathsf{ai}{\uparrow}),(\mathsf{s})\\}$. ###### Proof. Data l’istanza $\mathbb{C}\\{\circ\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{i}{\downarrow})$ $\mathbb{C}\textnormal{{[}}P,\overline{P}\textnormal{{]}}$ procediamo per induzione strutturale su $P$. Il caso duale $(\mathsf{i}{\uparrow})$ si dimostra allo stesso modo. Casi base 1. 1. $P=\circ$. Ovvio, poiché $\mathbb{C}\textnormal{{[}}P,\overline{P}\textnormal{{]}}=\mathbb{C}\\{\circ\\}$. 2. 2. $P$ è un atomo: Allora $(\mathsf{i}{\downarrow})$ è un’istanza di $(\mathsf{ai}{\downarrow})$. Casi induttivi 1. 3. $P=\textnormal{{[}}R,S\textnormal{{]}}$. Per ipotesi induttiva, abbiamo due derivazioni $\Phi_{R}$ e $\Phi_{S}$: $\mathbb{C}\\{\circ\\}$ $\scriptstyle\scriptstyle\Phi_{R}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\\{(\mathsf{ai}{\downarrow}),(\mathsf{s})\\}$ $\mathbb{C}\textnormal{{[}}R,\overline{R}\textnormal{{]}}$ $\mathbb{D}\\{\circ\\}$ $\scriptstyle\scriptstyle\Phi_{S}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\\{(\mathsf{ai}{\downarrow}),(\mathsf{s})\\}$ $\mathbb{D}\textnormal{{[}}S,\overline{S}\textnormal{{]}}$ da cui è possibile ottenere: $\mathbb{C}\\{\circ\\}$ $\scriptstyle\scriptstyle\Phi_{R}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\\{(\mathsf{ai}{\downarrow}),(\mathsf{s})\\}$ $\mathbb{C}\textnormal{{[}}R,\overline{R}\textnormal{{]}}=\mathbb{C}\textnormal{{(}}\textnormal{{[}}R,\overline{R}\textnormal{{]}},\circ\textnormal{{)}}$ $\scriptstyle\scriptstyle\Phi_{S}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\\{(\mathsf{ai}{\downarrow}),(\mathsf{s})\\}$ $\mathbb{C}\textnormal{{(}}\textnormal{{[}}R,\overline{R}\textnormal{{]}},\textnormal{{[}}S,\overline{S}\textnormal{{]}}\textnormal{{)}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{s})$ $\mathbb{C}\textnormal{{[}}R,\textnormal{{(}}\overline{R},\textnormal{{[}}S,\overline{S}\textnormal{{]}}\textnormal{{)}}\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{s})$ $\mathbb{C}\textnormal{{[}}R,S,\textnormal{{(}}\overline{R},\overline{S}\textnormal{{)}}\textnormal{{]}}$ 2. 4. Infine, il caso $P=\textnormal{{(}}R,S\textnormal{{)}}$ è analogo al precedente. ∎ Sintassi $P::=\circ\>\|\>a\>\|\>\overline{a}\>\|\>\textnormal{{[}}P,P\textnormal{{]}}\>\|\>\textnormal{{(}}P,P\textnormal{{)}}$ (con $a\in\mathcal{A}$ infinità numerabile di simboli proposizionali) Sistema deduttivo $\circ\quad(\mathsf{ax})$ $\mathbb{C}\\{\circ\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{ai}{\downarrow})$ $\mathbb{C}\textnormal{{[}}a,\overline{a}\textnormal{{]}}$ $\mathbb{C}\textnormal{{(}}a,\overline{a}\textnormal{{)}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{ai}{\uparrow})$ $\mathbb{C}\\{\circ\\}$ $\mathbb{C}\textnormal{{(}}P,\textnormal{{[}}Q,R\textnormal{{]}}\textnormal{{)}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{s})$ $\mathbb{C}\textnormal{{[}}\textnormal{{(}}P,Q\textnormal{{)}},R\textnormal{{]}}$ Associatività $\displaystyle\textnormal{{[}}\textnormal{{[}}P,Q\textnormal{{]}},R\textnormal{{]}}$ $\displaystyle=$ $\displaystyle\textnormal{{[}}P,\textnormal{{[}}Q,R\textnormal{{]}}\textnormal{{]}}$ $\displaystyle\textnormal{{(}}\textnormal{{(}}P,Q\textnormal{{)}},R\textnormal{{)}}$ $\displaystyle=$ $\displaystyle\textnormal{{(}}P,\textnormal{{(}}Q,R\textnormal{{)}}\textnormal{{)}}$ Commutatività $\displaystyle\textnormal{{[}}P,Q\textnormal{{]}}$ $\displaystyle=$ $\displaystyle\textnormal{{[}}Q,P\textnormal{{]}}$ $\displaystyle\textnormal{{(}}P,Q\textnormal{{)}}$ $\displaystyle=$ $\displaystyle\textnormal{{(}}Q,P\textnormal{{)}}$ Unità $\textnormal{{[}}P,\circ\textnormal{{]}}=\textnormal{{(}}P,\circ\textnormal{{)}}=P$ Negazione $\displaystyle\overline{\circ}$ $\displaystyle=$ $\displaystyle\circ$ $\displaystyle\overline{\textnormal{{[}}P,Q\textnormal{{]}}}$ $\displaystyle=$ $\displaystyle\textnormal{{(}}\overline{P},\overline{Q}\textnormal{{)}}$ $\displaystyle\overline{\textnormal{{(}}P,Q\textnormal{{)}}}$ $\displaystyle=$ $\displaystyle\textnormal{{[}}\overline{P},\overline{Q}\textnormal{{]}}$ $\displaystyle\overline{\overline{P}}$ $\displaystyle=$ $\displaystyle P$ Congruenza $P=P$ $P=R$ $R=Q$ $P=Q$ $P=Q$ $Q=P$ $P=Q$ $\mathbb{C}\\{P\\}=\mathbb{C}\\{Q\\}$ Figure 3.4: Sistema LBV+cut, equivalenza tra formule e negazione Esiste una corrispondenza 1:1 tra Sistema MLL+mix e Sistema LBV. ###### Definizione 3.2.2 (Trasformazioni LBV$\leftrightarrow$MLL). $\begin{array}[]{rclcrcl}\lx@intercol\hfil\underline{\cdot}\>:\>\mathscr{L}_{\mathsf{MLL}}\rightarrow\mathscr{L}_{\mathsf{LBV}}\hfil\lx@intercol&&\lx@intercol\hfil\underline{\underline{\cdot}}\>:\>\mathscr{L}_{\mathsf{LBV}}\rightarrow\mathscr{L}_{\mathsf{MLL}}\hfil\lx@intercol\\\ \underline{a}&=&a&\nobreak\leavevmode\hfil&\underline{\underline{a}}&=&a\\\ \underline{P\parr Q}&=&\textnormal{{[}}\underline{P},\underline{Q}\textnormal{{]}}&\nobreak\leavevmode\hfil&\underline{\underline{\textnormal{{[}}P,Q\textnormal{{]}}}}&=&\underline{\underline{P}}\parr\underline{\underline{Q}}\\\ \underline{P\otimes Q}&=&\textnormal{{(}}\underline{P},\underline{Q}\textnormal{{)}}&\nobreak\leavevmode\hfil&\underline{\underline{\textnormal{{(}}P,Q\textnormal{{)}}}}&=&\underline{\underline{P}}\otimes\underline{\underline{Q}}\end{array}$ Inoltre la definizione di $\underline{\cdot}$ si estende facilmente ai sequenti: $\underline{\vdash P_{1},\ldots,P_{n}}=\textnormal{{[}}\underline{P_{1}},\ldots,\underline{P_{n}}\textnormal{{]}}$ per $n>0$; per $n=0$ si pone $\underline{\vdash}=\circ$. ###### Teorema 3.2.3 (Equivalenza di LBV e MLL+mix). 1. i) Se il sequente $\vdash P$ è dimostrabile in MLL+mix, allora la struttura $\underline{P}$ è dimostrabile in LBV. 2. ii) Se la struttura $P$ (in forma normale, con $P\not=\circ$) è dimostrabile in LBV, allora il sequente $\vdash\underline{\underline{P}}$ è dimostrabile in MLL+mix. Questo teorema (dimostrato per la prima volta in Guglielmi [2002]) stabilisce una correlazione tra calcolo dei sequenti e calcolo delle strutture; come già accennato, è possibile conseguire un risultato analogo per il Sistema SKS, ma, ad esempio, la proprietà di località non vale per la logica classica proposizionale nel calcolo dei sequenti. #### 3.2.1 Eliminazione del taglio L’argomento calssico per dimostrare l’eliminazione del taglio nel calcolo dei sequenti, risiede nel fatto che, quando le formule principali del taglio sono introdotte in entrambi i rami, esse determinano che regole saranno applicate immediatamente sopra a quella di taglio. Questo è conseguenza del fatto che le formule hanno un connettivo principale, e le regole logiche si basano solo su quello, e su nessun’altra proprietà delle formule. Questo fatto non vale nel calcolo delle strutture. Per dimostrare la cut elimination nel Sistema LBV, occorre appoggiarsi ad un’altra proprietà, scoperta in Guglielmi [2002], e chiamata _scissione_ o _splitting_. Essa è una generalizzazione della tecnica vista nella dimostrazione di eliminazione del taglio per il sistema SKS. Si consideri la dimostrazione del sequente: $\vdash\mathbb{C}\\{P\otimes Q\\},\Gamma$ dove $\mathbb{C}\\{P\otimes Q\\}$ è una formula contenente la sottoformula $P\otimes Q$. Sappiamo per certo che nella dimostrazione ci deve essere un’istanza della regola $(\mathsf{\otimes})$ che scinde $P$ da $Q$ assieme ai rispettivi contesti. Siamo nella seguente situazione: $\textstyle{\scriptstyle\Pi_{1}}$ $\vdash P,\Gamma^{\prime}$ $\textstyle{\scriptstyle\Pi_{2}}$ $\vdash Q,\Gamma^{\prime\prime}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{\otimes})$ $\vdash P\otimes Q,\Gamma^{\prime},\Gamma^{\prime\prime}$ $\textstyle{\scriptstyle\Psi}$ $\vdash\mathbb{C}\\{P\otimes Q\\},\Gamma$ corrispondente a $\scriptstyle-$ $\scriptstyle\scriptstyle\Pi_{2}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{[}}Q,\Gamma^{\prime\prime}\textnormal{{]}}$ $\scriptstyle\scriptstyle\Pi_{1}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{(}}\textnormal{{[}}P,\Gamma^{\prime}\textnormal{{]}},\textnormal{{[}}Q,\Gamma^{\prime\prime}\textnormal{{]}}\textnormal{{)}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{s})$ $\textnormal{{[}}\textnormal{{(}}\textnormal{{[}}P,\Gamma^{\prime}\textnormal{{]}},Q\textnormal{{)}},\Gamma^{\prime\prime}\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{s})$ $\textnormal{{[}}\textnormal{{(}}P,Q\textnormal{{)}},\Gamma^{\prime},\Gamma^{\prime\prime}\textnormal{{]}}$ $\scriptstyle\scriptstyle\Psi\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{[}}\mathbb{C}\textnormal{{(}}P,Q\textnormal{{)}},\Gamma\textnormal{{]}}$ La derivazione $\Psi$ implementa lo splitting, che è ottenuto in due passi: 1. 1. riduzione del contesto: se $\mathbb{C}\\{P\\}$ è dimostrabile, allora $\mathbb{C}$ può essere ridotto, risalendo nella dimostrazione, ad un contesto $\textnormal{{[}}\bullet,S\textnormal{{]}}$, per un $S$ opportuno, tale che $\textnormal{{[}}P,S\textnormal{{]}}$ è dimostrabile. Nell’esempio sopra, $\textnormal{{[}}\mathbb{C}\\{\bullet\\},\Gamma\textnormal{{]}}$ è ridotto a $\textnormal{{[}}\bullet,S\textnormal{{]}}$ per un certo $S$; 2. 2. scissione di superficie: se $\textnormal{{[}}\textnormal{{(}}R,T\textnormal{{)}},P\textnormal{{]}}$ è dimostrabile, allora $P$ può essere ridotto, risalendo nella dimostrazione, ad una struttura $\textnormal{{[}}P_{1},P_{2}\textnormal{{]}}$ tali che $\textnormal{{[}}R,P_{1}\textnormal{{]}}$ e $\textnormal{{[}}T,P_{2}\textnormal{{]}}$ sono dimostrabili. Nell’esempio, $S$ è scisso in $\textnormal{{[}}\Gamma^{\prime},\Gamma^{\prime\prime}\textnormal{{]}}$. Grazie al Teorema di splitting, abbiamo la capacità di scindere un copar in due dimostrazioni, una per ogni rispettiva sottoformula: l’importanza di tale capacità ai fini della cut elimination, diventa chiara se consideriamo la regola di taglio nel Sistema LBV: $\mathbb{C}\textnormal{{(}}a,\overline{a}\textnormal{{)}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{ai}{\uparrow})$ $\mathbb{C}\\{\circ\\}$ Il contesto $\mathbb{C}$ viene scisso in due componenti $S_{1}$ e $S_{2}$ tali che esistono le dimostrazioni $\scriptstyle-$ $\scriptstyle\scriptstyle\Pi_{1}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{[}}a,S_{1}\textnormal{{]}}$ e $\scriptstyle-$ $\scriptstyle\scriptstyle\Pi_{2}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{[}}\overline{a},S_{2}\textnormal{{]}}$ . Ora possiamo sfruttare il fatto che gli atomi $a$ e $\overline{a}$ possono essere introdotti, rispettivamente nelle conclusioni di $\Pi_{1}$ e $\Pi_{2}$, solo mediante applicazioni della regola $(\mathsf{ai}{\downarrow})$ (fatto non vero, ad esempio, nel Sistema KS). A questo punto siamo in grado di isolare il segmento di dimostrazione che introduce gli atomi che verranno in seguito rimossi dal taglio, e possiamo pertanto trasformare questa sezione per bloccare il flusso degli atomi diretti al taglio sul nascere. ###### Teorema 3.2.4 (Shallow splitting). Se $\textnormal{{[}}\textnormal{{(}}R,T\textnormal{{)}},P\textnormal{{]}}$ è dimostrabile in LBV, allora esistono $P_{1}$ e $P_{1}$ tali che: $\textnormal{{[}}P_{1},P_{2}\textnormal{{]}}$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\mathsf{LBV}$ $P$ e $\textnormal{{[}}R,P_{1}\textnormal{{]}}$ e $\textnormal{{[}}T,P_{2}\textnormal{{]}}$ siano entrambi dimostrabili in LBV. ###### Proof. Consideriamo l’ordinamento lessicografico sui naturali: $(m^{\prime},n^{\prime})\prec(m,n)\quad\mbox{ sse }\quad m^{\prime}<m\mbox{ oppure }(m^{\prime}=m\mbox{ e }n^{\prime}<n)$ Vogliamo procedere per induzione completa su due quantità: l’altezza della dimostrazione di $\textnormal{{[}}\textnormal{{(}}R,T\textnormal{{)}},P\textnormal{{]}}$ e la _lunghezza delle formule_ , definita induttivamente da: $\begin{array}[]{ccccccc}\|\>{\circ}\>\|&=&0&&\|\>{\textnormal{{[}}P,Q\textnormal{{]}}}\>\|&=&\|\>{P}\>\|+\|\>{Q}\>\|\\\ \|\>{a}\>\|&=&1&&\|\>{\textnormal{{(}}P,Q\textnormal{{)}}}\>\|&=&\|\>{P}\>\|+\|\>{Q}\>\|\\\ \|\>{\overline{P}}\>\|&=&\|\>{P}\>\|\end{array}$ Dato il meta-enunciato: $\begin{array}[]{lll}C(m,n)&=&\forall R,T,P.\\\ &\nobreak\leavevmode&\quad\forall(m^{\prime},n^{\prime})\preceq(m,n).\\\ \nobreak\leavevmode&\nobreak\leavevmode&\quad\quad\left(m^{\prime}=\big{\|}\>{\textnormal{{[}}\textnormal{{(}}R,T\textnormal{{)}},P\textnormal{{]}}}\>\big{\|}\quad\bigwedge\quad\mbox{esiste }{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 20.18927pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 0.0pt}}\kern 20.18927pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}\textnormal{{(}}R,T\textnormal{{)}},P\textnormal{{]}}$}}}}\kern 0.0pt}}}\mbox{ con altezza $n^{\prime}$}\;\right)\\\ \nobreak\leavevmode&\nobreak\leavevmode&\quad\quad\mbox{\scalebox{1.5}{$\Rightarrow$}}\quad\exists P_{1},P_{2}.\left(\;{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\textnormal{{[}}P_{1},P_{2}\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 15.60902pt\hbox{$\kern 3.33333pt$}\kern 15.60902pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 15.60902pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 15.60902pt}\kern-1.43518pt\hbox{\kern 15.60902pt\hbox{\kern 3.33333pt}\kern 15.60902pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 11.70451pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 11.70451pt}\kern 1.43518pt\hbox{\kern 11.70451pt\hbox{$P$}\kern 11.70451pt}}}\kern 0.0pt}}}\quad\bigwedge\>\quad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 11.86185pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 0.0pt}}\kern 11.86185pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}R,P_{1}\textnormal{{]}}$}}}}\kern 0.0pt}}}\>\quad\bigwedge\>\quad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 11.36531pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 0.0pt}}\kern 11.36531pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}T,P_{2}\textnormal{{]}}$}}}}\kern 0.0pt}}}\;\right)\end{array}$ il teorema è equivalente a $\forall m,n.C(m,n)$. Per ipotesi induttiva possiamo suppore di avere una dimostrazione di $C(m^{\prime},n^{\prime})$ per ogni $(m^{\prime},n^{\prime})\prec(m,n)$. La lunghezza di $\textnormal{{[}}\textnormal{{(}}R,T\textnormal{{)}},P\textnormal{{]}}$ è $m$ e l’altezza della sua dimostrazione è $n$. Consideriamo l’istanza dell’ultima regola di questa dimostrazione: $\scriptstyle-$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $Q$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{\rho})$ $\textnormal{{[}}\textnormal{{(}}R,T\textnormal{{)}},P\textnormal{{]}}$ Procediamo per casi su $(\mathsf{\rho})$ (assumiamo sempre $P\not=\circ$ e $R\not=T\not=\circ$, perché in questi casi il teorema vale banalmente): 1. 1. $(\mathsf{\rho})=(\mathsf{ai}{\downarrow})$. Questa regola si può applicare in tre diversi modi: 1. 1.1. all’interno di $R$, cioè: $\scriptstyle-$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{[}}\textnormal{{(}}R^{\prime},T\textnormal{{)}},P\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{ai}{\downarrow})$ $\textnormal{{[}}\textnormal{{(}}R,T\textnormal{{)}},P\textnormal{{]}}$ Per ipotesi induttiva esistono $P_{1}$, $P_{2}$ tali che: ${{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\textnormal{{[}}P_{1},P_{2}\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 15.60902pt\hbox{$\kern 3.33333pt$}\kern 15.60902pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 15.60902pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 15.60902pt}\kern-1.43518pt\hbox{\kern 15.60902pt\hbox{\kern 3.33333pt}\kern 15.60902pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 11.70451pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 11.70451pt}\kern 1.43518pt\hbox{\kern 11.70451pt\hbox{$P$}\kern 11.70451pt}}}\kern 0.0pt}}}\qquad,\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 12.63184pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 0.0pt}}\kern 12.63184pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}R^{\prime},P_{1}\textnormal{{]}}$}}}}\kern 0.0pt}}}\qquad,\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 11.36531pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 0.0pt}}\kern 11.36531pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}T,P_{2}\textnormal{{]}}$}}}}\kern 0.0pt}}}$ È sufficiente applicare $(\mathsf{ai}{\downarrow})$ in coda alla dimostrazione di $\textnormal{{[}}R^{\prime},P_{1}\textnormal{{]}}$ per ottenere: $\scriptstyle-$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{[}}R^{\prime},P_{1}\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{ai}{\downarrow})$ $\textnormal{{[}}R,P_{1}\textnormal{{]}}$ 2. 1.2. all’interno di $T$. Analogo al caso precedente. 3. 1.3. all’interno di $P$, cioè: $\scriptstyle-$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{[}}\textnormal{{(}}R,T\textnormal{{)}},P^{\prime}\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{ai}{\downarrow})$ $\textnormal{{[}}\textnormal{{(}}R,T\textnormal{{)}},P\textnormal{{]}}$ Anche in questo caso procediamo per induzione diretta, a meno di un’applicazione di $(\mathsf{ai}{\downarrow})$ in: $\textnormal{{[}}P_{1},P_{2}\textnormal{{]}}$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ (per I.H.) $P^{\prime}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{ai}{\downarrow})$ $P$ 2. 2. $(\mathsf{\rho})=(\mathsf{s})$. Se la regola si applica all’interno di $R$, $T$ o $P$, procediamo in maniera del tutto analoga a quanto fatto nel caso $(\mathsf{\rho})=(\mathsf{ai}{\downarrow})$. Esistono altre due possibilità: 1. 2.1. $R=\textnormal{{(}}R^{\prime},R^{\prime\prime}\textnormal{{)}}$, $T=\textnormal{{(}}T^{\prime},T^{\prime\prime}\textnormal{{)}}$, $P=\textnormal{{[}}P^{\prime},P^{\prime\prime}\textnormal{{]}}$ e: $\scriptstyle-$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{[}}\textnormal{{(}}\textnormal{{[}}\textnormal{{(}}R^{\prime},T^{\prime}\textnormal{{)}},P^{\prime}\textnormal{{]}},R^{\prime\prime},T^{\prime\prime}\textnormal{{)}},P^{\prime\prime}\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{s})$ $\textnormal{{[}}\textnormal{{(}}R^{\prime},R^{\prime\prime},T^{\prime},T^{\prime\prime}\textnormal{{)}},P^{\prime},P^{\prime\prime}\textnormal{{]}}$ Possiamo applicare l’ipotesi induttiva per ottenere: ${{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\textnormal{{[}}P_{1}^{\prime},P_{2}^{\prime}\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 15.60902pt\hbox{$\kern 3.33333pt$}\kern 15.60902pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 15.60902pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 15.60902pt}\kern-1.43518pt\hbox{\kern 15.60902pt\hbox{\kern 3.33333pt}\kern 15.60902pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 10.16452pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 10.16452pt}\kern 1.43518pt\hbox{\kern 10.16452pt\hbox{$P^{\prime\prime}$}\kern 10.16452pt}}}\kern 0.0pt}}}\qquad,\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 22.27602pt\hbox{\hbox{\kern 7.74997pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Pi_{1}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 0.0pt}}\kern 30.02599pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}\textnormal{{(}}R^{\prime},T^{\prime}\textnormal{{)}},P^{\prime},P_{1}^{\prime}\textnormal{{]}}$}}}}\kern 0.0pt}}}\qquad,\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 16.9193pt\hbox{\hbox{\kern 7.74997pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Pi_{2}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 0.0pt}}\kern 24.66927pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}\textnormal{{(}}R^{\prime\prime},T^{\prime\prime}\textnormal{{)}},P_{2}^{\prime}\textnormal{{]}}$}}}}\kern 0.0pt}}}$ Ora possiamo applicare nuovamente l’ipotesi induttiva su $\Pi_{1}$ e su $\Pi_{2}$: infatti, anche se non conosciamo l’altezza di queste due dimostrazioni, sappiamo che la loro dimensione è inferiore a: $\big{\|}\>{\textnormal{{[}}\textnormal{{(}}R^{\prime},R^{\prime\prime},T^{\prime},T^{\prime\prime}\textnormal{{)}},P^{\prime},P^{\prime\prime}\textnormal{{]}}}\>\big{\|}$ perché per ipotesi, l’istanza di $(\mathsf{s})$ non è triviale. Pertanto abbiamo per ipotesi induttiva: $\begin{array}[]{ccccc}{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\textnormal{{[}}P_{1}^{\prime\prime},P_{2}^{\prime\prime}\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 15.60902pt\hbox{$\kern 3.33333pt$}\kern 15.60902pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 15.60902pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 15.60902pt}\kern-1.43518pt\hbox{\kern 15.60902pt\hbox{\kern 3.33333pt}\kern 15.60902pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 0.63pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.63pt}\kern 1.43518pt\hbox{\kern 0.63pt\hbox{$\textnormal{{[}}P^{\prime},P_{1}^{\prime}\textnormal{{]}}$}\kern 0.63pt}}}\kern 0.0pt}}}&\quad,&{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 12.63184pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 0.0pt}}\kern 12.63184pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}R^{\prime},P_{1}^{\prime\prime}\textnormal{{]}}$}}}}\kern 0.0pt}}}&\quad,&{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 12.41309pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 0.0pt}}\kern 12.41309pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}T^{\prime},P_{2}^{\prime\prime}\textnormal{{]}}$}}}}\kern 0.0pt}}}\\\ \\\ {{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\textnormal{{[}}P_{1}^{\prime\prime\prime},P_{2}^{\prime\prime\prime}\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 15.60902pt\hbox{$\kern 3.33333pt$}\kern 15.60902pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 15.60902pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 15.60902pt}\kern-1.43518pt\hbox{\kern 15.60902pt\hbox{\kern 3.33333pt}\kern 15.60902pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 10.3045pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 10.3045pt}\kern 1.43518pt\hbox{\kern 10.3045pt\hbox{$P_{2}^{\prime}$}\kern 10.3045pt}}}\kern 0.0pt}}}&,&{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 13.40184pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 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to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}R^{\prime\prime},P_{1}^{\prime\prime\prime}\textnormal{{]}}$}}}}\kern 0.0pt}}}&,&{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 13.18309pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 0.0pt}}\kern 13.18309pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}T^{\prime\prime},P_{2}^{\prime\prime\prime}\textnormal{{]}}$}}}}\kern 0.0pt}}}\end{array}$ Ora, ponendo $P_{1}=\textnormal{{[}}P_{1}^{\prime\prime},P_{1}^{\prime\prime\prime}\textnormal{{]}}$ e $P_{2}=\textnormal{{[}}P_{2}^{\prime\prime},P_{2}^{\prime\prime\prime}\textnormal{{]}}$ otteniamo: ${{{}{}{}{}{}{}{}{}{}{}}{{{{{}{}{}{}{}{}{}{}{}{}}{{{{{}{}{}{}{}{}{}{}{}{}}{{{}}}}}}}}}\vbox{\hbox{\kern 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0.0pt}}}\kern 1.43518pt\hbox{\kern 8.15672pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 8.15672pt}\kern 1.43518pt\hbox{\kern 30.66248pt\hbox{$\kern 3.33333pt$}\kern 30.66248pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 30.66248pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to8.0pt{ \vbox to8.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 2.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 30.66248pt}\kern-1.43518pt\hbox{\kern 30.66248pt\hbox{\kern 3.33333pt}\kern 30.66248pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 15.54346pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 15.54346pt}\kern 1.43518pt\hbox{\kern 15.54346pt\hbox{$\textnormal{{[}}P^{\prime},P^{\prime\prime}\textnormal{{]}}$}\kern 15.54346pt}}}\kern 0.0pt}}}\quad,\quad{{{}{}{}{}{}{}{}{}{}{}}{{{{{}{}{}{}{}{}{}{}{}{}}{{{}}}}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 21.02069pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 13.40184pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 0.0pt}}\kern 13.40184pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}R^{\prime\prime},P_{1}^{\prime\prime\prime}\textnormal{{]}}$}}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 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1.43518pt\hbox{\hbox{$\textnormal{{[}}\textnormal{{(}}\textnormal{{[}}R^{\prime},P_{1}^{\prime\prime}\textnormal{{]}},R^{\prime\prime}\textnormal{{)}},P_{1}^{\prime\prime\prime}\textnormal{{]}}$}}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to73.40054pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 67.84505pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{s})$}}$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.77779pt\hbox{$\kern 0.0pt\hbox{$\textnormal{{[}}\textnormal{{(}}R^{\prime},R^{\prime\prime}\textnormal{{)}},P_{1}^{\prime\prime},P_{1}^{\prime\prime\prime}\textnormal{{]}}$}\kern 0.0pt$}\kern 2.77779pt}}}\kern 14.49995pt}}}\quad,\quad{{{}{}{}{}{}{}{}{}{}{}}{{{{{}{}{}{}{}{}{}{}{}{}}{{{}}}}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 20.80194pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 13.18309pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 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to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 15.46083pt}\kern-1.43518pt\hbox{\kern 15.46083pt\hbox{\kern 3.33333pt}\kern 15.46083pt}}}\kern 0.0pt}}\kern 20.80194pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}\textnormal{{(}}\textnormal{{[}}T^{\prime},P_{2}^{\prime\prime}\textnormal{{]}},T^{\prime\prime}\textnormal{{)}},P_{2}^{\prime\prime\prime}\textnormal{{]}}$}}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\smash{\lower 0.0pt\hbox{$$}}$}}\vbox{\vbox to0.4pt{ \vfill\hbox to72.52554pt{\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\hss\xleaders\vbox to0.0pt{\vss\hbox{\kern-1.3pt\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}\kern-1.3pt} \vss}\hskip 66.97005pt\hss\vbox to0.0pt{\vss\hbox{$\scriptstyle-$}\vss}}\vfill}}\hbox to0.0pt{\hbox{$\smash{\lower 0.0pt\hbox{$\;(\mathsf{s})$}}$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.77779pt\hbox{$\kern 0.0pt\hbox{$\textnormal{{[}}\textnormal{{(}}T^{\prime},T^{\prime\prime}\textnormal{{)}},P_{2}^{\prime\prime},P_{2}^{\prime\prime\prime}\textnormal{{]}}$}\kern 0.0pt$}\kern 2.77779pt}}}\kern 14.49995pt}}}$ 2. 2.2. $P=\textnormal{{[}}\textnormal{{(}}P^{\prime},P^{\prime\prime}\textnormal{{)}},U^{\prime},U^{\prime\prime}\textnormal{{]}}$ e: $\scriptstyle-$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{[}}\textnormal{{(}}\textnormal{{[}}\textnormal{{(}}R,T\textnormal{{)}},P^{\prime},U^{\prime}\textnormal{{]}},P^{\prime\prime}\textnormal{{)}},U^{\prime\prime}\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{s})$ $\textnormal{{[}}\textnormal{{(}}R,T\textnormal{{)}},\textnormal{{(}}P^{\prime},P^{\prime\prime}\textnormal{{)}},U^{\prime},U^{\prime\prime}\textnormal{{]}}$ Per ipotesi induttiva abbiamo: ${{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\textnormal{{[}}U_{1},U_{2}\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 15.71802pt\hbox{$\kern 3.33333pt$}\kern 15.71802pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 15.71802pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 15.71802pt}\kern-1.43518pt\hbox{\kern 15.71802pt\hbox{\kern 3.33333pt}\kern 15.71802pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 10.21901pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 10.21901pt}\kern 1.43518pt\hbox{\kern 10.21901pt\hbox{$U^{\prime\prime}$}\kern 10.21901pt}}}\kern 0.0pt}}}\qquad,\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 35.49171pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle\;\Pi$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 5.74997pt}}\kern 29.74174pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}\textnormal{{(}}R,T\textnormal{{)}},P^{\prime},U^{\prime},U_{1}\textnormal{{]}}$}}}}\kern 0.0pt}}}\qquad,\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 13.52577pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 0.0pt}}\kern 13.52577pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}P^{\prime\prime},U_{2}\textnormal{{]}}$}}}}\kern 0.0pt}}}$ È di nuovo è possibile applicare l’ipotesi induttiva su $\Pi$ poiché: $\big{\|}\>{\textnormal{{[}}\textnormal{{(}}R,T\textnormal{{)}},P^{\prime},U^{\prime},U_{1}\textnormal{{]}}}\>\big{\|}<\textnormal{{[}}\textnormal{{(}}\textnormal{{[}}\textnormal{{(}}R,T\textnormal{{)}},P^{\prime},U^{\prime}\textnormal{{]}},P^{\prime\prime}\textnormal{{)}},U^{\prime\prime}\textnormal{{]}}$ per ottenere: ${{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 6.37572pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\textnormal{{[}}P_{1},P_{2}\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 15.60902pt\hbox{$\kern 3.33333pt$}\kern 15.60902pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 15.60902pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 15.60902pt}\kern-1.43518pt\hbox{\kern 15.60902pt\hbox{\kern 3.33333pt}\kern 15.60902pt}}}\kern 0.0pt}}\kern 6.37572pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}P^{\prime},U^{\prime},U_{1}\textnormal{{]}}$}}}}\kern 0.0pt}}}\qquad,\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 11.86185pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 0.0pt}}\kern 11.86185pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}R,P_{1}\textnormal{{]}}$}}}}\kern 0.0pt}}}\qquad,\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 11.36531pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 0.0pt}}\kern 11.36531pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}T,P_{2}\textnormal{{]}}$}}}}\kern 0.0pt}}}$ Ora possiamo costruire: $\textnormal{{[}}P_{1},P_{2}\textnormal{{]}}$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{[}}P^{\prime},U^{\prime},U_{1}\textnormal{{]}}$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{[}}\textnormal{{(}}P^{\prime},\textnormal{{[}}P^{\prime\prime},U_{2}\textnormal{{]}}\textnormal{{)}},U^{\prime},U_{1}\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{s})$ $\textnormal{{[}}\textnormal{{(}}P^{\prime},P^{\prime\prime}\textnormal{{)}},U^{\prime},U_{1},U_{2}\textnormal{{]}}$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{[}}\textnormal{{(}}P^{\prime},P^{\prime\prime}\textnormal{{)}},U^{\prime},U^{\prime\prime}]\textnormal{{]}}$ ∎ ###### Teorema 3.2.5 (Riduzione del contesto). Per ogni struttura $P$ ed ogni contesto $\mathbb{C}$ tale che $\mathbb{C}\\{P\\}$ è dimostrabile in LBV, esiste una struttura $C$ tale che, per ogni struttura $X$ esistono le derivazioni: $\textnormal{{[}}C,X\textnormal{{]}}$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\mathsf{LBV}$ $\mathbb{C}\\{X\\}$ e $\scriptstyle-$ $\scriptstyle\scriptstyle\mathsf{LBV}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{[}}C,P\textnormal{{]}}$ ###### Proof. Per induzione sulla dimensione di $\mathbb{C}\\{\bullet\\}$. Il caso base è triviale, $C=\circ$. I casi induttivi sono: 1. 1. $\mathbb{C}\\{\bullet\\}=\textnormal{{(}}\mathbb{C^{\prime}}\\{\bullet\\},Q\textnormal{{)}}$, per qualche $Q\not=\circ$. Se $\mathbb{C}\\{P\\}$ è dimostrabile, allora devono esistere le dimostrazioni di $\mathbb{C^{\prime}}\\{P\\}$ e di $Q$. Applicando l’ipotesi induttiva su $\mathbb{C^{\prime}}\\{P\\}$, otteniamo $C$ tale che, per ogni $X$: $\textnormal{{[}}C,X\textnormal{{]}}$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\mathbb{C^{\prime}}\\{X\\}$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{(}}\mathbb{C^{\prime}}\\{X\\},Q\textnormal{{)}}$ e tale che $\textnormal{{[}}C,P\textnormal{{]}}$ è dimostrabile in LBV. Lo stesso argomento si applica quando $\mathbb{C}\\{\bullet\\}=\textnormal{{(}}Q,\mathbb{C^{\prime}}\\{\bullet\\}\textnormal{{)}}$ con $Q\not=\circ$. 2. 2. $\mathbb{C}\\{\bullet\\}=\textnormal{{[}}\mathbb{C^{\prime}}\\{\bullet\\},Q\textnormal{{]}}$, per qualche $Q\not=\circ$. Assumiamo che $\mathbb{C^{\prime}}\\{\bullet\\}$ non sia un par: questa ipotesi non è limitativa, perché è sempre possibile far “rientrare” il parallelo in $Q$, lasciando $\mathbb{C^{\prime}}$ come copar. Se alla fine di questo processo otteniamo $\mathbb{C^{\prime}}\\{\bullet\\}=\bullet$, il teorema è banalmente provato. Quindi $\mathbb{C^{\prime}}\\{\bullet\\}=\textnormal{{(}}\mathbb{C^{\prime\prime}}\\{\bullet\\},Q^{\prime}\textnormal{{)}}$ con $Q^{\prime}\not=\circ$. Per il Teorema 3.2.4, esistono: ${{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\textnormal{{[}}Q_{1},Q_{2}\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 15.70554pt\hbox{$\kern 3.33333pt$}\kern 15.70554pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 15.70554pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle\;\mathsf{LBV}$}\hss}\kern 15.70554pt}\kern-1.43518pt\hbox{\kern 15.70554pt\hbox{\kern 3.33333pt}\kern 15.70554pt}}}\kern 1.65552pt}}}\kern 1.43518pt\hbox{\kern 11.75276pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 13.40828pt}\kern 1.43518pt\hbox{\kern 11.75276pt\hbox{$Q$}\kern 13.40828pt}}}\kern 0.0pt}}}\qquad,\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 16.38069pt\hbox{\hbox{\kern 5.74997pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Pi\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle\;\mathsf{LBV}$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 15.08331pt}}\kern 7.04735pt}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}\mathbb{C^{\prime\prime}}\\{P\\},Q_{1}\textnormal{{]}}$}}}}\kern 0.0pt}}}\qquad,\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 12.79779pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle\;\mathsf{LBV}$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 15.08331pt}}}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 2.28552pt}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}Q^{\prime},Q_{2}\textnormal{{]}}$}\kern 2.28552pt}}}\kern 0.0pt}}}$ Ora, applicando l’ipotesi induttiva su $\Pi$, otteniamo: $\textnormal{{[}}C,X\textnormal{{]}}$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{[}}\mathbb{C^{\prime\prime}}\\{X\\},Q_{1}\textnormal{{]}}$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{[}}\textnormal{{(}}\textnormal{{[}}Q^{\prime},Q_{2}\textnormal{{]}},\mathbb{C^{\prime\prime}}\\{X\\}\textnormal{{)}},Q_{1}\textnormal{{]}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{s})$ $\textnormal{{[}}\textnormal{{(}}\mathbb{C^{\prime\prime}}\\{X\\},Q^{\prime}\textnormal{{)}},Q_{1},Q_{2}\textnormal{{]}}$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{[}}\textnormal{{(}}\mathbb{C^{\prime\prime}}\\{X\\},Q^{\prime}\textnormal{{)}},Q\textnormal{{]}}=\textnormal{{(}}\mathbb{C^{\prime}}\\{X\\},Q\textnormal{{)}}=\mathbb{C}\\{X\\}$ e $\scriptstyle-$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{[}}C,P\textnormal{{]}}$ Analogamente si dimostra $\mathbb{C^{\prime}}\\{\bullet\\}=\textnormal{{(}}Q^{\prime},\mathbb{C^{\prime\prime}}\\{\bullet\\}\textnormal{{)}}$ e si usa lo stesso argomento per $\mathbb{C}\\{\bullet\\}=\textnormal{{[}}Q,\mathbb{C^{\prime}}\\{\bullet\\}\textnormal{{]}}$. ∎ ###### Corollario 3.2.6 (Splitting). Per ogni struttura $P$ e $Q$ e contesto $\mathbb{C}$, se $\mathbb{C}\textnormal{{(}}P,Q\textnormal{{)}}$ è dimostrabile in LBV, allora esistono due strutture $S_{1}$ e $S_{2}$ tali che, per ogni struttura $X$, esistono le derivazioni: ${{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\textnormal{{[}}X,S_{1},S_{2}\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 20.98744pt\hbox{$\kern 3.33333pt$}\kern 20.98744pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 20.98744pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle\;\mathsf{LBV}$}\hss}\kern 20.98744pt}\kern-1.43518pt\hbox{\kern 20.98744pt\hbox{\kern 3.33333pt}\kern 20.98744pt}}}\kern 0.0pt}}}\kern 1.43518pt\hbox{\kern 7.8416pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 7.8416pt}\kern 1.43518pt\hbox{\kern 7.8416pt\hbox{$\mathbb{C}\\{X\\}$}\kern 7.8416pt}}}\kern 0.0pt}}}\qquad,\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 10.82536pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle\;\mathsf{LBV}$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 15.08331pt}}}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 4.25795pt}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}P,S_{1}\textnormal{{]}}$}\kern 4.25795pt}}}\kern 0.0pt}}}\qquad,\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 11.42918pt\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle\;\mathsf{LBV}$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 15.08331pt}}}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 3.65413pt}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}Q,S_{2}\textnormal{{]}}$}\kern 3.65413pt}}}\kern 0.0pt}}}$ ###### Proof. Prima si applica il Teorema 3.2.4, poi il Teorema 3.2.5. ∎ Infine, prima di passare alla cut elimination, occorre enunciare un ultimo semplice risultato. ###### Proposizione 3.2.7. Per ogni struttura $P,Q$ e contesto $\mathbb{C}$, esiste una derivazione: $\mathbb{C}\textnormal{{[}}P,Q\textnormal{{]}}$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\\{(\mathsf{s})\\}$ $\textnormal{{[}}\mathbb{C}\\{P\\},Q\textnormal{{]}}$ ###### Proof. Per induzione sulla dimensione del contesto $\mathbb{C}$. Questa dimostrazione è uguale a quella della Proposizione 2.2.6, che stabilisce la stessa proposizione per il Sistema KS. ∎ ###### Teorema 3.2.8 (Cut elimination). La regola $(\mathsf{ai}{\uparrow})$ è ammissibile in LBV. ###### Proof. Consideriamo la dimostrazione: $\scriptstyle-$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\;\mathsf{LBV}$ $\mathbb{C}\textnormal{{(}}a,\overline{a}\textnormal{{)}}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{ai}{\uparrow})$ $\mathbb{C}\\{\circ\\}$ Per il Corollario 3.2.6, esistono $S_{1}$ e $S_{2}$ tali che esistono le derivazioni: ${{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{$\textnormal{{[}}S_{1},S_{2}\textnormal{{]}}$}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 14.5083pt\hbox{$\kern 3.33333pt$}\kern 14.5083pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 14.5083pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle\;\mathsf{LBV}$}\hss}\kern 14.5083pt}\kern-1.43518pt\hbox{\kern 14.5083pt\hbox{\kern 3.33333pt}\kern 14.5083pt}}}\kern 2.85275pt}}}\kern 1.43518pt\hbox{\kern 3.39717pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 6.24992pt}\kern 1.43518pt\hbox{\kern 3.39717pt\hbox{$\mathbb{C}\\{\circ\\}$}\kern 6.24992pt}}}\kern 0.0pt}}}\qquad,\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 2.36938pt\hbox{\hbox{\kern 7.74997pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Pi_{1}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle\;\mathsf{LBV}$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 15.08331pt}}}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 4.96396pt}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}a,S_{1}\textnormal{{]}}$}\kern 4.96396pt}}}\kern 0.0pt}}}\qquad,\qquad{{{}{}{}{}{}{}{}{}{}{}}{{{}}}\vbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\kern 2.22644pt\hbox{\hbox{\kern 7.74997pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\kern 0.0pt\hbox{\vbox{\offinterlineskip\hbox{\hbox{\hbox{\vbox to0.0pt{\vss\kern 0.0pt\kern-0.2pt\hbox{$\scriptstyle-$}\vss}}}}\kern 1.43518pt\hbox{\kern 0.0pt\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 0.0pt}\kern 1.43518pt\hbox{\kern 2.27774pt\hbox{$\kern 3.33333pt$}\kern 2.27774pt}}}\kern 0.0pt}}}\kern-1.43518pt\kern 0.0pt\kern 0.0pt\hbox{\kern 2.27774pt\hbox to0.0pt{\hss\hbox{$\scriptstyle\scriptstyle\Pi_{2}\;$}}\vbox{\hbox{$\vbox{\vbox{\offinterlineskip\hbox{$\vbox to16.0pt{ \vbox to16.0pt{\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\vss\xleaders\vbox{\kern-1.0pt\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern-1.0pt} \vskip 10.0pt} \vss\hbox{$\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$}\kern 0.0pt}$}}}$}}\hbox to0.0pt{\hbox{$\scriptstyle\;\mathsf{LBV}$}\hss}\kern 2.27774pt}\kern-1.43518pt\hbox{\kern 2.27774pt\hbox{\kern 3.33333pt}\kern 2.27774pt}}}\kern 15.08331pt}}}\kern 1.43518pt\hbox{\hbox to0.0pt{\hss\hbox{$\scriptstyle$}}\vbox{\hbox{\hfil}}\hbox to0.0pt{\hbox{$\scriptstyle$}\hss}\kern 5.1069pt}\kern 1.43518pt\hbox{\hbox{$\textnormal{{[}}\overline{a},S_{2}\textnormal{{]}}$}\kern 5.1069pt}}}\kern 0.0pt}}}$ Vogliamo individuare, nella dimostrazione $\Pi_{1}$, il punto in cui l’atomo $a$ viene introdotto. Certamente deve esistere un contesto $\mathbb{C^{\prime}}$ tale che $S_{1}=\mathbb{C^{\prime}}\\{\overline{a}\\}$. Inoltre, deve esistere un contesto $\mathbb{C^{\prime\prime}}$ tale che: $\scriptstyle-$ $\scriptstyle\scriptstyle\Pi_{1}^{\prime\prime}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\mathbb{C^{\prime\prime}}\\{\circ\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{ai}{\downarrow})$ $\mathbb{C^{\prime\prime}}\textnormal{{[}}a,\overline{a}\textnormal{{]}}$ $\scriptstyle\scriptstyle\Pi_{1}^{\prime}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{[}}a,\mathbb{C^{\prime}}\\{\overline{a}\\}\textnormal{{]}}$ sia la dimostrazione $\Pi_{1}$ in cui abbiamo individuato l’applicazione della regola $(\mathsf{ai}{\downarrow})$. Ora, sostituendo in $\Pi_{1}^{\prime}$ le occorrenze di $a$ e $\overline{a}$ con $\circ$, otteniamo una dimostrazione $\Psi_{1}^{\prime}$, grazie alla quale è possibile dimostrare: $\scriptstyle-$ $\scriptstyle\scriptstyle\Pi_{1}^{\prime\prime}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\mathbb{C^{\prime\prime}}\\{\circ\\}$ $\scriptstyle\scriptstyle\Psi_{1}^{\prime}\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\mathbb{C^{\prime}}\\{\circ\\}$ Analogamente possiamo trasformare la dimostrazione di $\Pi_{2}$ in una dimostrazione di $\mathbb{D^{\prime}}\\{\circ\\}$ dove $S_{2}=\mathbb{D^{\prime}}\\{a\\}$. Ora possiamo concludere, esibendo la seguente dimostrazione: $\scriptstyle-$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\mathbb{C^{\prime}}\\{\circ\\}$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\mathbb{C^{\prime}}\\{\mathbb{D^{\prime}}\\{\circ\\}\\}$ $\scriptstyle-$ $\scriptstyle-$ $\scriptstyle-$ $\;(\mathsf{ai}{\downarrow})$ $\mathbb{C^{\prime}}\\{\mathbb{D^{\prime}}\textnormal{{[}}a,\overline{a}\textnormal{{]}}\\}$ $\scriptstyle\scriptstyle\Phi\;$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\textnormal{{[}}\mathbb{C^{\prime}}\\{\overline{a}\\},\mathbb{D^{\prime}}\\{a\\}\textnormal{{]}}$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\scriptstyle\mskip-4.0mu\mathchar 781\relax\mskip-4.0mu$ $\mathbb{C}\\{\circ\\}$ in cui $\Phi$ è ottenuta applicando due volte la Proposizione 3.2.7. Possiamo ripetere induttivamente l’argomento per ogni dimostrazione di $\mathsf{LBV}\cup\\{(\mathsf{ai}{\uparrow})\\}$, partendo dall’alto, ed eliminare una per una tutte le istanze di $(\mathsf{ai}{\uparrow})$. ∎ #### 3.2.2 Un’interpretazione operazionale Quando lette dal basso all’alto, cioè nel verso della _proof search_ , le regole d’inferenza possono essere direzionate, trasponendole in _regole di riscrittura_. Questo è reso possibile dal fatto che, nel calcolo delle strutture, ogni regola ha sempre _al più una premessa_. Riscriviamo le regole del Sistema LBV sotto questa nuova prospettiva; la regola d’interazione atomica diventa: $\textnormal{{[}}a,\overline{a}\textnormal{{]}}\rightarrow\circ\qquad(\mathsf{ai})$ che dice che due atomi di polarità opposta messi in parallelo possono interagire. Abbiamo omesso la chiusura contestuale; nei sistemi di riscrittura si è soliti fattorizzare la chiusura contestuale con una regola: $P\rightarrow Q$ $(\mathsf{di})$ $\mathbb{C}\\{P\\}\rightarrow\mathbb{C}\\{Q\\}$ che per noi corrisponde moralmente all’impiego della metodologia deep inference. L’assioma del Sistema LBV non viene trasposto: infatti in questa interpretazione, significa solo che l’unità non può essere riscritta, e che essa rappresenta l’unico _valore_ del Sistema. Procedendo nel processo di riscrittura, potremo in generale imbatterci in situazioni in cui nessuna regola è applicabile; l’unico caso “accettato” è quello di una computazione che termina sul simbolo $\circ$. Negli altri casi, il processo di riscrittura non avrà individuato una dimostrazione, bensì una derivazione _bloccata_. L’equivalenza tra formule, definita come in Figura 3.4, è nota nel mondo dei sistemi di riscrittura, come _riscrittura modulo_ una certa relazione d’equivalenza, come riportato in Baader and Nipkow [1998]. Un modo di esprimerla è usare la regola: $P=P^{\prime}$ $P^{\prime}\rightarrow Q^{\prime}$ $Q^{\prime}=Q$ $(\mathsf{eq})$ $P\rightarrow Q$ La riscrittura modulo associatività, commutatività e identità è delicata, perché non è sempre terminante. Per garantire la terminazione occorre imporre dei vincoli sull’applicabilità della regola $(\mathsf{eq})$, come mostrato in Baird et al. [1989]. Inoltre Kahramanoğulları [2006] ha sviluppato una tecnica per ridurre il non-determinismo dettato dalla fine grana delle regole di LBV. Se la regola di switch è di difficile comprensione quando la si considera come regola d’inferenza, la sua trasposizione in regola di riscrittura offre una prospettiva molto più intuitiva. Nell’approccio operazionale, consideriamo le due formule all’interno di un “par” come due processi paralleli, che girano simultaneamente e che si “conoscono” a vicenda (cioè che hanno modo di individuarsi, ad esempio possiedono le reciproche coordinate all’interno di una rete), e pertanto sono in grado di interagire. I processi in “copar” girano anch’essi in parallelo, ma non hanno la possibilità di comunicare, perché non possiedono l’informazione sulle reciproche coordinate. Oltre all’interazione, l’unica altra operazione possibile per i processi, è “dimenticarsi”: un processo all’interno di un “par” può decidere di eliminare l’informazione sulle coordinate di un’altro processo, o in altre parole, _disconoscerlo_. L’effetto di questo comportamento, è inibire ogni possibilità di interazione tra i processi coinvolti. Questo meccanismo dà luogo a una serie di casi. Siano $P$ e $Q$ due processi in un’ambiente “par”: $\textnormal{{[}}P,Q\textnormal{{]}}{\;}\rightarrow{\;}?$ Cosa possono fare $P$ e $Q$ a parte interagire? 1. 1. possono lavorare per conto loro, cioè, ai fini del comportamento concorrente, non fare niente. Questo caso va contemplato per completezza dell’interpretazione: $\textnormal{{[}}P,Q\textnormal{{]}}\rightarrow\textnormal{{[}}P,Q\textnormal{{]}}$; 2. 2. $P$ può disconoscere $Q$ o viceversa: in entrambi in casi $\textnormal{{[}}P,Q\textnormal{{]}}\rightarrow\textnormal{{(}}P,Q\textnormal{{)}}$; 3. 3. se $P$ è una composizione parallela $\textnormal{{[}}P_{1},P_{2}\textnormal{{]}}$: 1. 3.1. $P_{1}$ può disconoscere $Q$: $\textnormal{{[}}P_{1},P_{2},Q\textnormal{{]}}\rightarrow\textnormal{{[}}\textnormal{{(}}P_{1},Q\textnormal{{)}},P_{2}\textnormal{{]}}$; 2. 3.2. $P_{2}$ può disconoscere $Q$: $\textnormal{{[}}P_{1},P_{2},Q\textnormal{{]}}\rightarrow\textnormal{{[}}\textnormal{{(}}P_{2},Q\textnormal{{)}},P_{1}\textnormal{{]}}$; 3. 3.3. se $P_{1}$ è una composizione parallela $\textnormal{{[}}P_{11},P_{12}\textnormal{{]}}$ … 4. 3.4. se $P_{1}$ è un “copar” $\textnormal{{(}}P_{11},P_{12}\textnormal{{)}}$ … 5. 3.5. se $P_{2}$ è una composizione parallela $\textnormal{{[}}P_{21},P_{22}\textnormal{{]}}$ … 6. 3.6. se $P_{2}$ è un “copar” $\textnormal{{(}}P_{21},P_{22}\textnormal{{)}}$ … 4. 4. se $P$ è un “copar” $\textnormal{{(}}P_{1},P_{2}\textnormal{{)}}$: 1. 4.1. $P_{1}$ può disconoscere $Q$: $\textnormal{{[}}\textnormal{{(}}P_{1},P_{2}\textnormal{{)}},Q\textnormal{{]}}\rightarrow\textnormal{{(}}\textnormal{{[}}P_{2},Q\textnormal{{]}},P_{1}\textnormal{{)}}$; 2. 4.2. $P_{2}$ può disconoscere $Q$: $\textnormal{{[}}\textnormal{{(}}P_{1},P_{2}\textnormal{{)}},Q\textnormal{{]}}\rightarrow\textnormal{{(}}\textnormal{{[}}P_{1},Q\textnormal{{]}},P_{2}\textnormal{{)}}$; 3. 4.3. se $P_{1}$ è una composizione parallela $\textnormal{{[}}P_{11},P_{12}\textnormal{{]}}$ … 4. 4.4. … 5. 5. se $P$ è una composizione parallela $\textnormal{{[}}P_{1},P_{2}\textnormal{{]}}$, si procede in maniera simmetrica rispetto al caso 3.; 6. 6. se $P$ è un “copar” $\textnormal{{(}}P_{1},P_{2}\textnormal{{)}}$, simmetricamente rispetto a 4.. $P\>{\diamond}\>Q=\\{\textnormal{{[}}P,Q\textnormal{{]}},\textnormal{{(}}P,Q\textnormal{{)}}\\}\cup P\>\underline{\diamond}\>Q\cup Q\>\underline{\diamond}\>P$ dove: \| $\circ\>\underline{\diamond}\>Q=a\>\underline{\diamond}\>Q$ \| $=$ \| $\varnothing$ ---\|---\|---\|--- \| $\textnormal{{[}}S,T\textnormal{{]}}\>\underline{\diamond}\>Q$ \| $=$ \| $\\{\textnormal{{[}}S,X\textnormal{{]}}\>\|\>X\in T\>{\diamond}\>Q\\}\cup\\{\textnormal{{[}}X,T\textnormal{{]}}\>\|\>X\in S\>{\diamond}\>Q\\}$ \| $\textnormal{{(}}S,T\textnormal{{)}}\>\underline{\diamond}\>Q$ \| $=$ \| $\\{\textnormal{{(}}S,X\textnormal{{)}}\>\|\>X\in T\>{\diamond}\>Q\\}\cup\\{\textnormal{{(}}X,T\textnormal{{)}}\>\|\>X\in S\>{\diamond}\>Q\\}$ Figure 3.5: Definizione dell’operatore di _merge_ Questa struttura, chiaramente ricorsiva, porta alla definizione di Guglielmi [2002] di _merge set_ , riportata in Figura 3.5. Due processi $P$ e $Q$ immersi in un contesto parallelo, possono muovere ad un processo $R$ appartenente al merge set, come prescritto dalla _regola di merge_ : $\textnormal{{[}}P,Q\textnormal{{]}}\rightarrow R\;\mbox{ se }R\in P\>{\diamond}\>Q\qquad(\mathsf{g})$ Questa regola è abbastanza pesante, poiché necessita il ricalcolo del merge set ogni volta che dev’essere applicata. Ecco perché facciamo ricorso alla regola di switch: $\textnormal{{[}}\textnormal{{(}}P,Q\textnormal{{)}},R\textnormal{{]}}\rightarrow\textnormal{{(}}P,\textnormal{{[}}Q,R\textnormal{{]}}\textnormal{{)}}\qquad(\mathsf{s})$ ###### Teorema 3.2.9. La regola di merge è eliminabile in presenza di $(\mathsf{s})$. ###### Proof. Siano $P$, $Q$ ed $R$ processi, tali che $R\in P\>{\diamond}\>Q$. Allora: $\textnormal{{[}}P,Q\textnormal{{]}}\rightarrow R\qquad(\mathsf{g})$ Vogliamo dimostrare che esiste un cammino: $\textnormal{{[}}P,Q\textnormal{{]}}\rightarrow^{}R$ composto di sole applicazioni di $(\mathsf{s})$, di $(\mathsf{eq})$ e di $(\mathsf{di})$. Procediamo per induzione strutturale su $R$, usando la definizione di merge set di Figura 3.5: 1. 1. $R=\circ$. Allora dev’essere $P=Q=\circ$, e quindi $\textnormal{{[}}P,Q\textnormal{{]}}=R$; 2. 2. $R=a$. Allora dev’essere $P=a$ e $Q=\circ$ o $P=\circ$ e $Q=a$. In entrambi i casi $\textnormal{{[}}P,Q\textnormal{{]}}=a=R$; 3. 3. $R=\textnormal{{[}}P,Q\textnormal{{]}}$ vale banalmente, in quanto $\textnormal{{[}}P,Q\textnormal{{]}}\rightarrow^{}\textnormal{{[}}P,Q\textnormal{{]}}$ per ogni $P$, $Q$; 4. 4. $R=\textnormal{{(}}P,Q\textnormal{{)}}$. Allora: $\textnormal{{[}}P,Q\textnormal{{]}}=\textnormal{{[}}\textnormal{{(}}P,\circ\textnormal{{)}},Q\textnormal{{]}}$ $\textnormal{{[}}\textnormal{{(}}P,\circ\textnormal{{)}},Q\textnormal{{]}}\rightarrow\textnormal{{(}}P,\textnormal{{[}}\circ,Q\textnormal{{]}}\textnormal{{)}}$ $\textnormal{{(}}P,\textnormal{{[}}\circ,Q\textnormal{{]}}\textnormal{{)}}=\textnormal{{(}}P,Q\textnormal{{)}}$ $\textnormal{{[}}P,Q\textnormal{{]}}\rightarrow\textnormal{{(}}P,Q\textnormal{{)}}$ 5. 5. $R=\textnormal{{[}}R^{\prime},R^{\prime\prime}\textnormal{{]}}$. Ci sono due sottocasi da considerare: 1. 5.a. $P=\textnormal{{[}}R^{\prime},P^{\prime}\textnormal{{]}}$ e $R^{\prime\prime}\in P^{\prime}\>{\diamond}\>Q$. Per ipotesi induttiva, sappiamo che $\textnormal{{[}}P^{\prime},Q\textnormal{{]}}\rightarrow^{}R^{\prime\prime}$ senza usare la regola $(\mathsf{g})$. Allora possiamo concludere, esibendo: $\textnormal{{[}}P^{\prime},Q\textnormal{{]}}\rightarrow^{}R^{\prime\prime}$ $(\mathsf{di})$ $\textnormal{{[}}R^{\prime},P^{\prime},Q\textnormal{{]}}\rightarrow^{}\textnormal{{[}}R^{\prime},R^{\prime\prime}\textnormal{{]}}$ e osservando che $\textnormal{{[}}P,Q\textnormal{{]}}=\textnormal{{[}}R^{\prime},P^{\prime},Q\textnormal{{]}}$; 2. 5.b. $P=\textnormal{{[}}P^{\prime\prime},R^{\prime\prime}\textnormal{{]}}$ e $R^{\prime}\in P^{\prime\prime}\>{\diamond}\>Q$. Allora, come prima: $\textnormal{{[}}P^{\prime\prime},Q\textnormal{{]}}\stackrel{{\scriptstyle\mathsf{I.H.}}}{{\rightarrow^{}}}R^{\prime}$ $(\mathsf{di})$ $\textnormal{{[}}P^{\prime\prime},Q,R^{\prime\prime}\textnormal{{]}}\rightarrow^{}\textnormal{{[}}R^{\prime},R^{\prime\prime}\textnormal{{]}}$ e $\textnormal{{[}}P,Q\textnormal{{]}}=\textnormal{{[}}P^{\prime\prime},Q,R^{\prime\prime}\textnormal{{]}}$. 6. 6. $R=\textnormal{{(}}R^{\prime},R^{\prime\prime}\textnormal{{)}}$. Di nuovo, seguendo la definizione di merge set, ci sono due sottocasi possibili: 1. 6.a. $P=\textnormal{{(}}R^{\prime},P^{\prime}\textnormal{{)}}$ e $R^{\prime\prime}\in P^{\prime}\>{\diamond}\>Q$. Poiché $\textnormal{{[}}P,Q\textnormal{{]}}=\textnormal{{[}}\textnormal{{(}}R^{\prime},P^{\prime}\textnormal{{)}},Q\textnormal{{]}}$, possiamo applicare la regola di switch per ottenere: $\textnormal{{[}}\textnormal{{(}}R^{\prime},P^{\prime}\textnormal{{)}},Q\textnormal{{]}}\rightarrow\textnormal{{(}}R^{\prime},\textnormal{{[}}P^{\prime},Q\textnormal{{]}}\textnormal{{)}}$ da cui, per ipotesi induttiva immersa nel contesto $\textnormal{{(}}R^{\prime},\bullet\textnormal{{)}}$, possiamo concludere esibendo: $\textnormal{{[}}P^{\prime},Q\textnormal{{]}}\stackrel{{\scriptstyle\mathsf{I.H.}}}{{\rightarrow^{}}}R^{\prime\prime}$ $(\mathsf{di})$ $\textnormal{{(}}R^{\prime},\textnormal{{[}}P^{\prime},Q\textnormal{{]}}\textnormal{{)}}\rightarrow^{}\textnormal{{(}}R^{\prime},R^{\prime\prime}\textnormal{{)}}$ 2. 6.b. $P=\textnormal{{(}}P^{\prime\prime},R^{\prime\prime}\textnormal{{)}}$ e $R^{\prime}\in P^{\prime\prime}\>{\diamond}\>Q$. Allora, grazie alle regole $(\mathsf{s})$ ed $(\mathsf{eq})$: $\textnormal{{[}}P,Q\textnormal{{]}}=\textnormal{{[}}\textnormal{{(}}R^{\prime\prime},P^{\prime\prime}\textnormal{{)}},Q\textnormal{{]}}\rightarrow\textnormal{{(}}R^{\prime\prime},\textnormal{{[}}P^{\prime\prime},Q\textnormal{{]}}\textnormal{{)}}=\textnormal{{(}}\textnormal{{[}}P^{\prime\prime},Q\textnormal{{]}},R^{\prime\prime}\textnormal{{)}}$ da cui concludiamo, grazie all’ipotesi induttiva e a: $\textnormal{{[}}P^{\prime\prime},Q\textnormal{{]}}\stackrel{{\scriptstyle\mathsf{I.H.}}}{{\rightarrow^{}}}R^{\prime}$ $(\mathsf{di})$ $\textnormal{{(}}\textnormal{{[}}P^{\prime\prime},Q\textnormal{{]}},R^{\prime\prime}\textnormal{{)}}\rightarrow^{}\textnormal{{(}}R^{\prime},R^{\prime\prime}\textnormal{{)}}$ ∎ Grazie a questo risultato, possiamo appoggiarci sulla più pratica (nonché locale) regola di switch, eliminando quella di merge. Il sistema così ottenuto, costituisce una personale interpretazione operazionale del Sistema LBV, reminiscente le algebre di processo, che può costituire un (ulteriore) ponte tra il mondo della proof theory e quello dei modelli concorrenti, oltre ad un punto di partenza per indagini riguardanti le proprietà di complessità della proof search. ## Conclusioni La deep inference offre una prospettiva nuova e moderna in teoria della dimostrazione. Grazie a questa metodologia, il lavoro strutturale svolto dagli alberi in shallow inference, viene collassato nell’uso dei contesti, che sono un concetto fondamentale in questo approccio. A questo proposito, è interessante osservare come questo metta in relazione diretta il modo di operare tipico in proof theory (con alberi di derivazione) con il mondo dei sistemi di riscrittura. Infatti le derivazione nel calcolo delle strutture possono essere linearizzate, leggendole dal basso in alto (verso della proof search), e le regole d’inferenza si possono vedere come regole di riscrittura; quali corrispondenze si possono trovare in questo senso? A quali sistemi di riscrittura corrispondono i sistemi in calcolo delle strutture, e di quali proprietà godono? Inoltre, osserviamo come, nelle procedure di cut elimination per il calcolo delle strutture, l’attenzione sia posta sugli atomi da eliminare, a conseguenza del fatto che questi sistemi godono di località. Una sotto- procedura invariante nella cut elimination è la discesa nella dimostrazione alla ricerca del taglio, per poi risalire seguendo il flusso degli atomi coinvolti. Da questa osservazione nasce un nuovo filone di ricerca in deep inference che tratta i cosiddetti “flussi atomici” o “atomic flows”; per una introduzione all’argomento, vedere Gundersen [2009]. Infine, esistono molti problemi rilevanti riguardanti la complessità delle dimostrazioni, alcuni dei quali ancora aperti, altri già risolti, ad esempio in Jeřábek [2009]; Bruscoli and Guglielmi [2009]; Bruscoli et al. [2009]. Il calcolo delle strutture è un formalismo molto espressivo, ma difficile da trattare a causa del forte non-determinismo che comporta la fine grana (i.e. la vasta applicabilità) delle sue regole; Kahramanoğulları [2006] ha ideato una tecnica capace di ridurre questo non-determinismo. Tutte le fonti e le informazioni riguardanti le ricerche in deep inference, sono reperibili online sulla pagina di Alessio Guglielmi, uno dei maggiori promotori di questo approccio, all’indirizzo: http://alessio.guglielmi.name/res/cos/ ## References Abramsky et al. [1992] Abramsky, S., Gabbay, D. M., and Maibaum, T., editors 1992. Handbook of Logic in Computer Science. Oxford University Press, Oxford. * Aho et al. [2006] Aho, A. V., Lam, M. S., Sethi, R., and Ullman, J. D. 2006. 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Proof complexity of the cut-free calculus of structures. Journal of Logic and Computation, 19(2):323–339. http://www.math.cas.cz/~jerabek/papers/cos.pdf. * Kahramanoğulları [2006] Kahramanoğulları, O. 2006. Reducing nondeterminism in the calculus of structures. In Hermann, M. and Voronkov, A., editors, LPAR 2006, volume 4246 of Lecture Notes in Computer Science, pages 272–286. Springer-Verlag. http://dx.doi.org/10.1007/11916277_19. * Kleene [1952] Kleene, S. C. 1952. Introduction to metamathematics. North-Holland, Amsterdam. * Lafont [2002] Lafont, Y. 2002. Soft linear logic and polynomial time. Theoretical computer science, 318. * Milner et al. [1992] Milner, R., Parrow, J., and Walker, D. 1992. A calculus of mobile processes. Information and Computation, 100(1):1–77. * Prawitz [1965] Prawitz, D. 1965. Natural Deduction: a proof-theoretical study. Dover Publications. * Sangiorgi and Walker [2001] Sangiorgi, D. and Walker, D. 2001. $\pi$-calculus: A Theory of Mobile Processes. Cambridge University Press, New York, NY, USA. * Schütte [1950] Schütte, K. 1950. Schlussweisen-Kalküle der Prädikatenlogik. Mathematische Annalen, 122:47–65. * Troelstra and Schwichtenberg [1996] Troelstra, A. S. and Schwichtenberg, H. 1996. Basic Proof Theory. Cambridge University Press, New York, NY, USA. ## Ringraziamenti Ringrazio anzitutto i mei genitori Carmen e Roberto, senza i quali tutto questo non sarebbe stato possibile. Con loro, ringrazio tutta la mia famiglia per l’amore che mi hanno dato dacché sono al mondo. Ringrazio i miei amici per le ore passate a discutere insieme, per aver ascoltato pazientemente i miei vaneggianti sproloqui, ma soprattutto per avermi dato la certezza di aver sempre qualcuno su cui contare. Infine ringrazio i miei professori, per avermi ascoltato e per la pazienza che hanno avuto nel sopportare questo tremendo rompiscatole. Senza i vostri insegnamenti, ma non solo, senza il vostro esempio, non sarei quello che sono. Grazie di cuore a tutti quanti, grazie a chi ha sempre creduto in me, grazie a chi non ci ha creduto mai, grazie agli amici ma anche ai nemici, grazie al contributo di tutti perché mi è stato indispensabile per raggiungere, oggi, questo risultato.	arxiv-papers	2013-11-20T10:46:35	2024-09-04T02:49:53.955864	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Andrea Simonetto", "submitter": "Andrea Simonetto", "url": "https://arxiv.org/abs/1311.5006" }
1311.5012	# Derivation of capture and reaction cross sections from experimental quasi- elastic and elastic backscattering probabilities V.V.Sargsyan1,2, G.G.Adamian1, N.V.Antonenko1, and P.R.S.Gomes3 1Joint Institute for Nuclear Research, 141980 Dubna, Russia 2International Center for Advanced Studies, Yerevan State University, 0025 Yerevan, Armenia 3Instituto de Fisica, Universidade Federal Fluminense, Av. Litorânea, s/n, Niterói, R.J. 24210-340, Brazil ###### Abstract We suggest simple and useful methods to extract reaction and capture (fusion) cross sections from the experimental elastic and quasi-elastic backscattering data. ###### pacs: 25.70.Jj, 24.10.-i, 24.60.-k Key words: sub-barrier capture, neutron transfer, quantum diffusion approach ## I Introduction The direct measurement of the reaction or capture (fusion) cross section is a difficult task since it would require the measurement of individual cross sections of many reaction channels, and most of them could be reached only by specific experiments. This would require different experimental set-ups not always available at the same laboratory and, consequently, such direct measurements would demand a large amount of beam time and would take probably some years to be reached. Because of that, the measurements of elastic scattering angular distributions that cover full angular ranges and optical model analysis have been used for the determination of reaction cross sections. This traditional method consists in deriving the parameters of the complex optical potentials which fit the experimental elastic scattering angular distributions and then of deriving the reaction cross sections predicted by these potentials. Even so, both the experimental part and the analysis of this latter method are not so simple. In the present work we present a much simpler methods to determine reaction and capture (fusion) cross sections. They consist of measuring only elastic or quasi-elastic scattering at one backward angle, and from that, the extraction of the reaction or capture cross sections can easily be performed. ## II Relationship between capture and quasi-elastic backscattering, and relationship between reaction and elastic backscattering From the conservation of the total reaction flux one can write Sargsyan13b ; Sargsyan13c ; Sargsyan13a ; Canto06 the expressions $P_{el}(E_{\mathrm{c.m.}},J)+P_{R}(E_{\mathrm{c.m.}},J)=1$ (1) and $P_{qe}(E_{\mathrm{c.m.}},J)+P_{cap}(E_{\mathrm{c.m.}},J)+P_{BU}(E_{\mathrm{c.m.}},J)+P_{DIC}(E_{\mathrm{c.m.}},J)=1.$ (2) Quasi-elastic scattering probability $P_{qe}(E_{\mathrm{c.m.}},J)=P_{el}(E_{\mathrm{c.m.}},J)+P_{in}(E_{\mathrm{c.m.}},J)+P_{tr}(E_{\mathrm{c.m.}},J)$ (3) is defined as the sum of elastic scattering $P_{el}$, inelastic excitations $P_{in}$ and a few nucleon transfer $P_{tr}$ probabilities. The reaction probability may be written as $P_{R}(E_{\mathrm{c.m.}},J)=P_{in}(E_{\mathrm{c.m.}},J)+P_{tr}(E_{\mathrm{c.m.}},J)+P_{cap}(E_{\mathrm{c.m.}},J)+P_{BU}(E_{\mathrm{c.m.}},J)+P_{DIC}(E_{\mathrm{c.m.}},J),$ (4) where $P_{cap}$ is the capture probability (sum of evaporation-residue formation, fusion-fission, and quasi-fission probabilities or sum of fusion and quasi-fission probabilities), $P_{DIC}$ is the deep inelastic collision probability, and $P_{BU}$ is the breakup probability, important particularly when weakly bound nuclei are involved in the reaction Canto06 . Note that the deep inelastic collision process is only important at large energies above the Coulomb barrier. Here we neglect the deep inelastic collision process, since we are concerned with low energies. Thus, one can extract the reaction $P^{ex}_{R}(E_{\mathrm{c.m.}},J=0)=1-P_{el}(E_{\mathrm{c.m.}},J=0)$ (5) and capture $P^{ex}_{cap}(E_{\rm c.m.},J=0)=1-[P_{qe}(E_{\rm c.m.},J=0)+P_{BU}(E_{\rm c.m.},J=0)]$ (6) probabilities at $J=0$ from the experimental elastic backscattering probability $P_{el}(E_{\mathrm{c.m.}},J=0)$ and quasi-elastic backscattering probability $P_{qe}(E_{\mathrm{c.m.}},J=0)$ plus breakup probability $P_{BU}(E_{\rm c.m.},J=0)$ at backward angle, respectively. Here, the elastic or quasi-elastic scattering or breakup probability Canto06 ; Timmers ; Timmers1 ; Timmers2 ; Zhang $P_{el,qe,BU}(E_{\mathrm{c.m.}},J=0)=d\sigma_{el,qe,BU}/d\sigma_{Ru}$ (7) for angular momentum $J=0$ is given by the ratio of the elastic or quasi- elastic scattering or breakup differential cross section and Rutherford differential cross section at 180 degrees. Furthermore, one can approximate the $J$ dependence of the reaction $P_{R}(E_{\mathrm{c.m.}},J)$ and capture $P_{cap}(E_{\mathrm{c.m.}},J)$ probabilities at a given bombarding energy $E_{\mathrm{c.m.}}$ by shifting the energy Sargsyan13b ; Sargsyan13c : $P_{R}(E_{\mathrm{c.m.}},J)\approx P^{ex}_{R}(E_{\mathrm{c.m.}}-\frac{\hbar^{2}\Lambda}{2\mu R_{b}^{2}}-\frac{\hbar^{4}\Lambda^{2}}{2\mu^{3}\omega_{b}^{2}R_{b}^{6}},J=0)$ (8) and $\displaystyle P_{cap}(E_{\rm c.m.},J)\approx P^{ex}_{cap}(E_{\rm c.m.}-\frac{\hbar^{2}\Lambda}{2\mu R_{b}^{2}}-\frac{\hbar^{4}\Lambda^{2}}{2\mu^{3}\omega_{b}^{2}R_{b}^{6}},J=0),$ (9) where $\Lambda=J(J+1)$, $R_{b}=R_{b}(J=0)$ is the position of the Coulomb barrier at $J=0$, $\mu=m_{0}A_{1}A_{2}/(A_{1}+A_{2})$ is the reduced mass ($m_{0}$ is the nucleon mass), and $\omega_{b}$ is the curvature of the s-wave potential barrier. Here we used the expansion of the height $V_{b}(J)$ of the Coulomb barrier up to second order in $\Lambda$ Sargsyan13b ; Sargsyan13c : $V_{b}(J)=V_{b}(J=0)+\frac{\hbar^{2}\Lambda}{2\mu R_{b}^{2}}+\frac{\hbar^{4}\Lambda^{2}}{2\mu^{3}\omega_{b}^{2}R_{b}^{6}}.$ Employing formulas for the reaction $\displaystyle\sigma_{R}(E_{\rm c.m.})=\pi\lambdabar^{2}\sum_{J=0}^{\infty}(2J+1)P^{ex}_{R}(E_{\rm c.m.}-\frac{\hbar^{2}\Lambda}{2\mu R_{b}^{2}}-\frac{\hbar^{4}\Lambda^{2}}{2\mu^{3}\omega_{b}^{2}R_{b}^{6}},J=0)$ (10) and capture $\displaystyle\sigma_{cap}(E_{\rm c.m.})=\pi\lambdabar^{2}\sum_{J=0}^{J_{cr}}(2J+1)P^{ex}_{cap}(E_{\rm c.m.}-\frac{\hbar^{2}\Lambda}{2\mu R_{b}^{2}}-\frac{\hbar^{4}\Lambda^{2}}{2\mu^{3}\omega_{b}^{2}R_{b}^{6}},J=0)$ (11) cross sections, converting the sum over the partial waves $J$ into an integral, and expressing $J$ by the variable $E=E_{\mathrm{c.m.}}-\frac{\hbar^{2}\Lambda}{2\mu R_{b}^{2}}$, we obtain the following simple expressions Sargsyan13b ; Sargsyan13c : $\sigma_{R}(E_{\mathrm{c.m.}})=\frac{\pi R_{b}^{2}}{E_{\mathrm{c.m.}}}\int_{0}^{E_{\mathrm{c.m.}}}dEP^{ex}_{R}(E,J=0)[1-\frac{4(E_{\mathrm{c.m.}}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}]$ (12) and $\displaystyle\sigma_{cap}(E_{\rm c.m.})=\frac{\pi R_{b}^{2}}{E_{\rm c.m.}}\int_{E_{\rm c.m.}-\frac{\hbar^{2}\Lambda_{cr}}{2\mu R_{b}^{2}}}^{E_{\rm c.m.}}dEP^{ex}_{cap}(E,J=0)[1-\frac{4(E_{\rm c.m.}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}],$ (13) where $\lambdabar^{2}=\hbar^{2}/(2\mu E_{\rm c.m.})$ is the reduced de Broglie wavelength, $\Lambda_{cr}=J_{cr}(J_{cr}+1)$, and $J=J_{cr}$ is the critical angular momentum. For values $J$ greater than $J_{cr}$, the potential pocket in the nucleus-nucleus interaction potential vanishes and the capture is not occur. To calculate the critical angular momentum $J_{cr}$ and the position $R_{b}$ of the Coulomb barrier, we use the nucleus-nucleus interaction potential $V(R,J)$ of Ref. Pot . For the nuclear part of the nucleus-nucleus potential, the double-folding formalism with the Skyrme-type density-dependent effective nucleon-nucleon interaction is employed Pot . For the systems with $Z_{1}\times Z_{2}<2000$ ($Z_{1,2}$ are the atomic numbers of interacting nuclei), the critical angular momentum $J_{cr}$ is large enough and Eq. (13) can be approximated with good accuracy as: $\displaystyle\sigma_{cap}(E_{\rm c.m.})\approx\frac{\pi R_{b}^{2}}{E_{\rm c.m.}}\int_{0}^{E_{\rm c.m.}}dEP^{ex}_{cap}(E,J=0)[1-\frac{4(E_{\rm c.m.}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}].$ (14) The formula (12) [(13)] relates the reaction [capture] cross section with elastic [quasi-elastic] scattering excitation function at a backward angle. By using the experimental $P_{el}(E_{\mathrm{c.m.}},J=0)$ [$P_{qe}(E_{\mathrm{c.m.}},J=0)$+$P_{BU}(E_{\mathrm{c.m.}},J=0)$] and Eq. (12) [(13)], one can obtain the reaction [capture] cross sections. It is important to mention that since the generalized form of the optical theorem connects the reaction cross section and forward elastic scattering amplitudeCanto06 , we show that the forward and backward elastic scattering amplitudes are related to each other. Using the extracted $\sigma_{cap}$ and the experimental $P_{qe}$, one can find the average angular momentum $\displaystyle<J>=\frac{\pi R_{b}^{2}}{E_{\rm c.m.}\sigma_{cap}(E_{\rm c.m.})}\int_{E_{\rm c.m.}-\frac{\hbar^{2}\Lambda_{cr}}{2\mu R_{b}^{2}}}^{E_{\rm c.m.}}$ $\displaystyle dEP^{ex}_{cap}(E,J=0)[1-\frac{5(E_{\rm c.m.}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}]$ (15) $\displaystyle\times[(\frac{2\mu R_{b}^{2}}{\hbar^{2}}(E_{\rm c.m.}-E)+\frac{1}{4})^{1/2}-\frac{1}{2}]$ and the second moment of the angular momentum $\displaystyle<J(J+1)>=\frac{2\pi\mu R_{b}^{4}}{\hbar^{2}E_{\rm c.m.}\sigma_{cap}(E_{\rm c.m.})}\int_{E_{\rm c.m.}-\frac{\hbar^{2}\Lambda_{cr}}{2\mu R_{b}^{2}}}^{E_{\rm c.m.}}$ $\displaystyle dEP^{ex}_{cap}(E,J=0)[1-\frac{6(E_{\rm c.m.}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}]$ (16) $\displaystyle\times[E_{\rm c.m.}-E]$ of the captured system Sargsyan13b . ## III Results of calculations As the elastic, quasi-elastic, and breakup data were not taken at 180 degrees, but rather at backward angles in the range from 150 to 170 degrees, the corresponding center of mass energies were corrected by the centrifugal potential at the experimental angle Timmers . Figure 1: The extracted capture cross sections for the reactions 16O + 120Sn (a) and 18O + 124Sn (b) by employing Eq. (13) (solid line) and Eq. (14) (dotted line). These lines are almost coincide. The used experimental quasi- elastic backscattering data are from Ref. Sinha . The experimental capture (fusion) data (symbols) are from Refs. Sinha ; JACOBS . Figure 2: The same as in Fig. 1, but for the reactions 16O + 208Pb(a),144Sm(b). The used experimental quasi-elastic backscattering data are from Refs. Timmers2 ; Timmers . For the 16O + 208Pb reaction, the experimental capture (fusion) data are from Refs. Pbcap (open squares), Pbcap1 (open circles), Pbcap2 (closed stars), and Pbcap3 (closed triangles). For the 16O + 144Sm reaction, the experimental capture (fusion) data are from Refs. SmCap1 (closed squares) and SmCap2 (open squares). ### III.1 Capture cross sections For the verification of our method of the extraction of $\sigma_{cap}$, first we compare the extracted capture cross sections with experimental one for the reactions with toughly bound nuclei [$P_{BU}(E_{\mathrm{c.m.}},J=0)=0$]. In Figs. 1 and 2 one can see good agreement between the extracted and directly measured capture cross sections for the reactions 16O + 120Sn, 18O + 124Sn, 16O + 208Pb, and 16O + 144Sm at energies above the Coulomb barrier. The results on the sub-barrier energy region are discussed later on. To extract the capture cross section, we use both Eq. (13) (solid lines) and Eq. (14) (dotted lines). The used values of critical angular momentum are $J_{cr}$=54, 56, 57, and 62 for the reactions 16O + 120Sn, 18O + 124Sn, 16O + 144Sm, and 16O + 208Pb, respectively. The difference between the results of Eqs. (13) and (14) is less than 5$\%$ at the highest energies. At low energies, Eqs. (13) and (14) lead to the same values of $\sigma_{cap}$. The factor $1-\frac{4(E_{\rm c.m.}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}$ in Eqs. (13) and (14) very weakly influences the results of the calculations for the systems and energies considered. Hence, one can say that for the relatively light systems the proposed method of extracting the capture cross section is model independent (particular, independent on the potential used): $\sigma_{cap}(E_{\rm c.m.})\approx\frac{\pi R_{b}^{2}}{E_{\rm c.m.}}\int_{0}^{E_{\rm c.m.}}dEP^{ex}_{cap}(E,J=0).$ One can see that the used formulas are suitable not only for almost spherical nuclei (Figs. 1 and 2) but also for the reactions with strongly deformed target- or projectile-nucleus (Figs. 3 and 4). So, the deformation effect is effectively contained in the experimental $P_{qe}$. $J_{cr}=58$, 68, 74, and 76 for the reactions 16O+154Sm, 32S+90Zr, 32S+96Zr, and 20Ne+208Pb, respectively. The results obtained by employing the formula (14) are almost the same and not presented in Figs. 3 and 4. Figure 3: The same as in Fig. 1, but for the reactions 20Ne + 208Pb and 16O + 154Sm. The used experimental quasi-elastic backscattering data are from Refs. Piasecki ; Timmers . The experimental capture (fusion) data (symbols) are from Refs. SmCap2 ; Piasecki . For the 16O + 154Sm reaction, the dashed line is obtained from the shift of the solid line by 1.7 MeV to higher energies. Figure 4: The same as in Fig. 1, but for the reactions 32S + 90Zr (a) and 32S + 96Zr (b). For the 32S+90Zr reaction, we show the extracted capture cross sections, increasing the experimental $P_{qe}$ by 1% (dashed line), 2% (dotted line), and 3% (dash-dotted line). The used experimental quasi-elastic backscattering data are from Ref. Zhang3 . The experimental capture (fusion) data (symbols) are from Ref. ZhangS32Zn9096 . For the 32S + 96Zr reaction, the energy scale for the extracted capture cross sections is adjusted to that of the direct measurements. For the reactions 16O+154Sm and 32S+96Zr, the extracted capture cross sections are shifted in energy by 1.7 and 1.9 MeV with respect to the measured capture data, respectively. This could be the result of different energy calibrations in the experiments on the capture measurement and on the quasi-elastic scattering. Because of the lack of systematics in these energy shifts, their origin remains unclear and we adjust the Coulomb barriers in the extracted capture cross sections to the values following the experiments. Note that the extracted and experimental capture cross sections deviate from each other in the reactions 16O+208Pb, 16O+144Sm, and 32S+90Zr at energies below the Coulomb barrier. Probably this deviation (the mismatch between quasi-elastic backscattering and fusion (capture) experimental data) is a reason for the large discrepancies in the diffuseness parameter extracted from the analyses of the quasi-elastic scattering and fusion (capture) at deep sub- barrier energies. One of the possible reasons for the overestimation of the capture cross section from the quasi-elastic data at sub-barrier energies is the underestimation of the total reaction differential cross section taken as the Rutherford differential cross section. Indeed, for the 32S+90Zr reaction, the increase of $P_{qe}$ within 2–3% is needed in order to obtain the agreement between the extracted and measured capture cross sections at the sub-barrier energies [Fig. 4(a)]. As seen in Fig. 5, the extracted capture cross sections $\sigma_{cap}(E_{\rm c.m.})$ (solid line) for the 6Li+208Pb reaction with weakly bound nucleus [$P_{BU}(E_{\rm c.m.},J=0)\neq 0$] are rather close to those found in the direct measurements Li6Pbcap at energies above the Coulomb barrier. Figure 5: (Color online) The extracted capture cross sections $\sigma_{cap}(E_{\rm c.m.})$ (solid line) and $\sigma^{noBU}_{cap}(E_{\rm c.m.})$ (dotted line) for the 6Li+208Pb reaction. The used experimental quasi- elastic backscattering and quasi-elastic backscattering plus breakup at the backward angle data are from Ref. Li6Pb . The experimental capture cross sections (solid squares) are from Ref. Li6Pbcap . The energy scale for the extracted capture cross sections is adjusted to that of the direct measurements. It appears that at energies near and below the Coulomb barrier the extracted $\sigma_{cap}(E_{\rm c.m.})$ deviates from the direct measurements. It is similarly possible to calculate the capture excitation function $\displaystyle\sigma^{noBU}_{cap}(E_{\rm c.m.})=\frac{\pi R_{b}^{2}}{E_{\rm c.m.}}\int_{E_{\rm c.m.}-\frac{\hbar^{2}\Lambda_{cr}}{2\mu R_{b}^{2}}}^{E_{\rm c.m.}}dEP^{noBU}_{cap}(E,J=0)[1-\frac{4(E_{\rm c.m.}-E)}{\mu\omega_{b}^{2}R_{b}^{2}}]$ (17) in the absence of the breakup process (Fig. 5, dotted line) by using the following formula for the capture probability in this case Nash : $\displaystyle P^{noBU}_{cap}(E_{\rm c.m.},J=0)=1-\frac{P_{qe}(E_{\rm c.m.},J=0)}{1-P_{BU}(E_{\rm c.m.},J=0)}.$ (18) By employing the measured excitation functions $P_{qe}$ and $P_{BU}$ at the backward angle Li6Pb , Eqs. (13), (17), and the formula $\displaystyle<P_{BU}>(E_{\rm c.m.})=1-\frac{\sigma_{cap}(E_{\rm c.m.})}{\sigma^{noBU}_{cap}(E_{\rm c.m.})},$ (19) we extract the mean breakup probability $<P_{BU}>(E_{\rm c.m.})$ averaged over all partial waves $J$ (Fig. 6). Figure 6: The extracted mean breakup probability $<P_{BU}>(E_{\rm c.m.})$ [Eq. (19)] as a function of bombarding energy $E_{\rm c.m.}$ for the 6Li+208Pb reaction. The used experimental quasi-elastic backscattering and quasi-elastic backscattering plus breakup at the backward angle data are from Ref. Li6Pb . The value of $<P_{BU}>$ has a maximum at $E_{\rm c.m.}-V_{b}\approx 4$ MeV ($<P_{BU}>$=0.26) and slightly (sharply) decreases with increasing (decreasing) $E_{\rm c.m.}$. The experimental breakup excitation function at backward angle has the similar energy behavior Li6Pb . By comparing the calculated capture cross sections in the absence of breakup and experimental capture (complete fusion) data, the opposite energy trend is found in Ref. Nash , where $<P_{BU}>$ has a minimum at $E_{\rm c.m.}-V_{b}\approx 2$ MeV ($<P_{BU}>$=0.34) and globally increases in both sides from this minimum. It is also shown in Refs. Nash ; PRSGomes4 that there are no systematic trends of breakup in the complete fusion reactions with the light projectiles 9Be, 6,7,9Li, and 6,8He at near-barrier energies. Thus, by employing the experimental quasi-elastic backscattering, one can obtain the additional information about the breakup process. Figure 7: The extracted $<J>$ and $<J^{2}>$ for the reactions 16O + 208Pb (a) and 16O + 154Sm (b) by employing Eqs. (15) and (16). The used experimental quasi-elastic backscattering data are from Ref. Timmers2 . The experimental data of $<J^{2}>$ and $<J>$ are from Refs. Vand (open squares) and Gil ; Vand2 (open squares and circles), respectively. By using the Eqs. (15) and (16) and experimental $P_{qe}$, we extract $<J>$ and $<J^{2}>$ of the captured system for the reactions 16O + 154Sm and 16O + 208Pb, respectively (Fig. 7). The agreements with the results of direct measurements of the $\gamma-$multiplicities in the corresponding complete fusion reactions are quite good. For the 16O + 208Pb reaction at sub-barrier energies, the difference between the extracted and experimental angular momenta is related with the deviation of the extracted capture excitation function from the experimental one (see Fig. 2). ### III.2 Reaction cross sections As can be observed in Figs. 8–15, there is a good agreement between the reaction cross sections extracted from the experimental elastic scattering at backward angle and from the experimental elastic scattering angular distributions with optical potential for the reactions 4He + 92Mo, 4He + 110,116Cd, 4He + 112,120Sn, 16O + 208Pb, and 6,7Li + 64Zn at energies near and above the Coulomb barrier. One can see that the used formula (12) is suitable not only for almost spherical nuclei, but also for the reactions with slightly deformed target-nuclei. The deformation effect is effectively contained in the experimental $P_{el}$. For very deformed nuclei, it is not possible experimentally to separate elastic events from the low-lying inelastic excitations. In our calculations, to obtain better agreement for the reactions 16O+208Pb and 6Li+64Zn, the extracted reaction cross sections were shifted in energy by 0.3 MeV to higher energies and 0.4 MeV to lower energies with respect to the measured experimental data, respectively. There is no clear physical justification for the energy shift. The most probable reason might be related with the uncertainty associated with the elastic scattering data. Figure 8: (Color online) The extracted reaction cross sections (solid line) for the 4He + 92Mo reaction by employing Eq. (12). The used experimental elastic scattering probabilities at the backward angle are from Ref. hemo . The reaction cross sections extracted from the experimental elastic scattering angular distribution with optical potential are presented by squares hemo . Figure 9: (Color online) The extracted reaction cross sections (lines) for the 4He + 110Cd reaction by employing Eq. (12). The used experimental elastic scattering probabilities at the backward angle are from Refs. hecd1 ; hecd3 (solid line) and Ref. hecd2 (dashed line). The reaction cross sections extracted from the experimental elastic scattering angular distribution with optical potential are presented by squares hemo . By using Eq. (13), the capture cross sections of the reactions 6,7Li+64Zn can be extracted, if one assumes that $P_{BU}=0$, since it is much smaller than $P_{qe}$. In Figs. 14 and 15 we also show the results of our calculations for capture cross sections of the 6,7Li+64Zn systems, for which the fusion process can be considered to exhaust the capture cross section. Figure 14 shows that the extracted and experimental capture cross sections are in good agreement for the 6Li+64Zn reaction at energies near and above the Coulomb barrier for the data taken in Refs. Torresi ; Pietro . Note that the extracted capture excitation function is shifted in energy by 0.7 MeV to higher energies with respect to the experimental data. This could be the result of different energy calibrations in the experiments on the capture measurement and quasi-elastic scattering. Figure 10: (Color online) The same as in Fig. 9, but for the 4He + 116Cd reaction. Figure 11: (Color online) The extracted reaction cross sections (solid line) for the 4He + 112Sn reaction by employing Eq. (12). The used experimental elastic scattering probabilities at the backward angle are from Ref. hemo . The reaction cross sections extracted from the experimental elastic scattering angular distribution with optical potential are presented by squares hemo . Figure 12: (Color online) The same as in Fig. 11 but for the 4He + 120Sn reaction. Figure 13: (Color online) The extracted reaction cross sections (solid line) for the 16O + 208Pb reaction by employing Eq. (12). The used experimental elastic scattering probabilities at the backward angle are from Ref. opb . The reaction cross sections extracted from the experimental elastic scattering angular distribution with optical potential are presented by squares opb . Figure 14: (Color online) The extracted reaction (solid line) and capture (dashed line) cross sections for the 6Li + 64Zn reaction by employing Eqs. (12) and (13). The used experimental elastic and quasi-elastic backscattering probabilities are from Refs. Torresi ; Pietro . The reaction cross sections extracted from the experimental elastic scattering angular distribution with optical potential and capture (fusion) cross sections are presented by circles Torresi ; Pietro , triangles Gomes034 ; GomesPLB04 and stars Torresi ; Pietro , respectively. For the 7Li+64Zn reaction, the $Q$-value of the one neutron stripping transfer is positive and this process should have a reasonable high probability to occur, whereas for the 6Li+64Zn reaction, $Q$-values of neutron transfers are negative. Therefore, one might expect that transfer cross sections for 7Li+64Zn are larger than for 6Li+64Zn. With concern for breakup, since 6Li has a smaller threshold energy for breakup than 7Li, one might expect that breakup cross sections for 6Li+64Zn are larger than for 7Li+64Zn. Actually, in Fig. 16, one can observe that our calculations show that $\sigma(^{7}\mathrm{Li}+^{64}\mathrm{Zn})>\sigma(^{6}\mathrm{Li}+^{64}\mathrm{Zn}),$ where $\sigma=\sigma_{R}-\sigma_{cap}\approx\sigma_{tr}+\sigma_{in}$ since $\sigma_{tr}+\sigma_{in}\gg\sigma_{BU}$ for these light systems at energies close and below the Coulomb barrier ($\sigma_{tr}$, $\sigma_{in}$, and $\sigma_{BU}$ are the transfer, inelastic scattering, and breakup cross sections, respectively). So, our present method of extracting reaction and capture cross sections from backward elastic scattering data allows the approximate determination of the sum of transfer and inelastic scattering cross sections or $\sigma_{tr}+\sigma_{in}+\sigma_{BU}$ in systems where $P_{BU}$ cannot be neglected. For both systems investigated, the values of these cross sections are shown to increase with $E_{\mathrm{c.m.}}$, reach a maximum slightly above the Coulomb barrier energy, and after, decrease. The difference between the two curves in Fig. 16 may be considered approximately as the difference of $\sigma_{tr}$ between the two systems, since $\sigma_{in}$ should be similar for both systems with the same target, apart from the excitation of the bound excited state of 7Li. Because $\sigma_{tr}(^{7}\mathrm{Li}+^{64}\mathrm{Zn})\gg\sigma_{tr}(^{6}\mathrm{Li}+^{64}\mathrm{Zn})$, one can find $\sigma_{tr}(^{7}\mathrm{Li}+^{64}\mathrm{Zn})\approx\sigma(^{7}\mathrm{Li}+^{64}\mathrm{Zn})-\sigma(^{6}\mathrm{Li}+^{64}\mathrm{Zn}).$ The maximum absolute value of the transfer cross section $\sigma_{tr}$ at energies near the Coulomb barrier is about 30 mb. Figure 16 also shows that the difference between transfer cross sections for 7Li+64Zn and 6Li+64Zn are much more important than the possible larger $\sigma_{BU}$ for 6Li than for 7Li. Figure 15: (Color online) The same as in Fig. 14, but for the 7Li + 64Zn reaction. The reaction cross sections extracted from the experimental elastic scattering angular distribution with optical potential are presented by circles Gomes034 ; GomesPLB04 . Figure 16: The extracted $\sigma_{R}-\sigma_{cap}$ for the reactions 6Li + 64Zn (dashed line) and 7Li + 64Zn (solid line). ## IV Summary We propose a new and very simple ways to determine reaction and capture (fusion) cross sections, through the relation (12) between the elastic backscattering excitation function and reaction cross section and through the relation (13) between the quasi-elastic scattering excitation function at the backward angle and capture cross section. We show, for several systems, that these methods work well and that the elastic and quasi-elastic backscattering technique could be used as an important and simple tools in the study of the reaction and capture cross sections in the reactions with toughly and weakly bound nuclei. The extraction of reaction (capture) cross sections from the elastic (quasi-elastic) backscattering is possible with reasonable uncertainties as long as the deviation between the elastic (quasi-elastic) scattering cross section and the Rutherford cross section exceeds the experimental uncertainties significantly. By employing the quasi-elastic backscattering data, one can extract the moments of the angular momentum of the captured system. The behavior of the transfer plus inelastic excitation function extracted from the experimental probabilities of the elastic and quasi-elastic scatterings at the backward angle also was shown. We thank H.Q. Zhang for fruitful discussions and suggestions. We are grateful to G. Kiss, R. Lichtenthäler, C.J. Lin, P. Mohr, E. Piasecki, M. Zadro, and H.Q. Zhang for providing us the experimental data. P.R.S.G. acknowledges the partial financial support from CNPq and FAPERJ. This work was supported by DFG, NSFC, RFBR, and JINR grants. The IN2P3(France)-JINR(Dubna) and Polish - JINR(Dubna) Cooperation Programmes are gratefully acknowledged. ## References * (1) V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, and P.R.S. Gomes, Phys. Rev. C 87, 044611 (2013). * (2) V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, and P.R.S. Gomes, Phys. Rev. C 88, 044606 (2013). * (3) V.V. Sargsyan et al., Eur. Phys. J. A 49, 19 (2013). * (4) L.F. Canto, P.R.S. Gomes, R. Donangelo, and M.S. Hussein, Phys. Rep. 424, 1 (2006). * (5) H. Timmers et al., Nucl. Phys. A584, 190 (1995). * (6) H. Timmers et al., J. Phys. G 23, 1175 (1997). * (7) H. Timmers, Ph.D. thesis, Australian National University (1996). * (8) H.Q. Zhang, F. Yang, C. Lin, Z. Liu, and Y. Hu, Phys. Rev. C 57, R1047 (1998). * (9) G.G. Adamian et al., Int. J. Mod. Phys. E 5, 191 (1996); V.V. Sargsyan et al., Phys. Phys. C 84, 064614 (2011). * (10) S. Sinha et al., Phys. Rev. C 64, 024607 (2001). * (11) P. Jacobs, Z. Fraenkel, G. Mamane, and L. Tserruya, Phys. Lett. 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Gil et al., Phys. Rev. C 43, 701 (1991). * (28) R. Vandenbosch et al., Phys. Rev. C 28, 1161 (1983). * (29) P. Mohr et al., Phys. Rev. C 82, 047601 (2010). * (30) J.S. Lilley et al., Nucl. Phys. A 463, 710 (1987). * (31) M. Miller et al., Z. Phys. A 300, 97 (1981). * (32) G.G. Kiss et al., Phys. Rev. C 83, 065807 (2011). * (33) I. Badawy et al., Phys. Rev. C 17, 978 (1978). * (34) D. Torresi et al., Eur. Phys. J. Conf 17, 16018 (2011). * (35) A. Di Pietro et al., Phys. Rev. C 87, 064614 (2013) * (36) P.R.S. Gomes et al., Phys. Rev. C 71, 034608 (2005). * (37) P.R.S. Gomes et al., Phys. Lett. B 601, 20 (2004)	arxiv-papers	2013-11-20T11:12:06	2024-09-04T02:49:53.980032	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "V.V.Sargsyan, G.G.Adamian, N.V.Antonenko, and P.R.S.Gomes", "submitter": "Vazgen Sargsyan Dr.", "url": "https://arxiv.org/abs/1311.5012" }
1311.5020	# Fusion at near-barrier energies within quantum diffusion approach V.V.Sargsyan1, G.G.Adamian1, N.V.Antonenko1, W. Scheid2, and H.Q.Zhang3 1Joint Institute for Nuclear Research, 141980 Dubna, Russia 2Institut für Theoretische Physik der Justus–Liebig–Universität, D–35392 Giessen, Germany 3China Institute of Atomic Energy, Post Office Box 275, Beijing 102413, China ###### Abstract Within the quantum diffusion approach the role of neutron transfer in the fusion (capture) reactions with toughly and weakly bound nuclei is discussed. The breakup process is analyzed. New methods for the study of the breakup probability are suggested. ###### pacs: 25.70.Jj, 24.10.-i, 24.60.-k Key words: sub-barrier capture, neutron transfer, quantum diffusion approach ## I Introduction The nuclear deformation and neutron-transfer process have been identified as playing a major role in the magnitude of the sub-barrier fusion (capture) cross sections Gomes . There are a several experimental evidences which confirm the importance of nuclear deformation on the fusion. The influence of nuclear deformation is straightforward. If the target nucleus is prolate in the ground state, the Coulomb field on its tips is lower than on its sides, that then increases the capture or fusion probability at energies below the barrier corresponding to the spherical nuclei. The role of neutron transfer reactions is less clear. The importance of neutron transfer with positive $Q$-values on nuclear fusion (capture) originates from the fact that neutrons are insensitive to the Coulomb barrier and therefore they can start being transferred at larger separations before the projectile is captured by target- nucleus. Therefore, it is generally thought that the sub-barrier fusion cross section will increase because of the neutron transfer. The fusion (capture) dynamics induced by loosely bound radioactive ion beams is currently being extensively studied. However, the long-standing question whether fusion (capture) is enhanced or suppressed with these beams has not yet been answered unambiguously. The study of the fusion reactions involving nuclei at the drip-lines has led to contradictory results. ## II Quantum diffusion approach for capture In the quantum diffusion approach EPJSub ; EPJSub02 ; EPJSub1 ; EPJSub2 ; EPJSub3 the capture of the projectile by the target-nucleus is described with a single relevant collective variable: the relative distance between the colliding nuclei. This approach takes into consideration the fluctuation and dissipation effects in collisions of heavy ions which model the coupling of the relative motion with various channels (for example, the non-collective single-particle excitations, low-lying collective dynamical modes of the target and projectile). The nuclear static deformation effects are taken into account through the dependence of the nucleus-nucleus potential on the deformations and mutual orientations of the colliding nuclei. We have to mention that many quantum-mechanical and non-Markovian effects accompanying the passage through the potential barrier are taken into consideration in our formalism EPJSub ; EPJSub1 . The capture cross section is a sum of partial capture cross sections EPJSub ; EPJSub1 $\displaystyle\sigma_{cap}(E_{\rm c.m.})$ $\displaystyle=$ $\displaystyle\sum_{J}\sigma_{\rm cap}(E_{\rm c.m.},J)=$ (1) $\displaystyle=$ $\displaystyle\pi\lambdabar^{2}\sum_{J}(2J+1)\int_{0}^{\pi/2}d\theta_{1}\sin(\theta_{1})\int_{0}^{\pi/2}d\theta_{2}\sin(\theta_{2})P_{\rm cap}(E_{\rm c.m.},J,\theta_{1},\theta_{2}),$ where $\lambdabar^{2}=\hbar^{2}/(2\mu E_{\rm c.m.})$ is the reduced de Broglie wavelength, $\mu=m_{0}A_{1}A_{2}/(A_{1}+A_{2})$ is the reduced mass ($m_{0}$ is the nucleon mass), and the summation is over the possible values of angular momentum $J$ at a given bombarding energy $E_{\rm c.m.}$. Knowing the potential of the interacting nuclei for each orientation with the angles $\theta_{i}(i=1,2)$, one can obtain the partial capture probability $P_{\rm cap}$ which is defined by the passing probability of the potential barrier in the relative distance $R$ coordinate at a given $J$. The value of $P_{\rm cap}$ is obtained by integrating the propagator $G$ from the initial state $(R_{0},P_{0})$ at time $t=0$ to the final state $(R,P)$ at time $t$ ($P$ is a momentum): $\displaystyle P_{\rm cap}=\lim_{t\to\infty}\int_{-\infty}^{r_{\rm in}}dR\int_{-\infty}^{\infty}dP\ G(R,P,t\|R_{0},P_{0},0)=\lim_{t\to\infty}\frac{1}{2}{\rm erfc}\left[\frac{-r_{\rm in}+\overline{R(t)}}{{\sqrt{\Sigma_{RR}(t)}}}\right].$ (2) The second line in (2) is obtained by using the propagator $G=\pi^{-1}\|\det{\bf\Sigma}^{-1}\|^{1/2}\exp(-{\bf q}^{T}{\bf\Sigma}^{-1}{\bf q})$ (${\bf q}^{T}=[q_{R},q_{P}]$, $q_{R}(t)=R-\overline{R(t)}$, $q_{P}(t)=P-\overline{P(t)}$, $\overline{R(t=0)}=R_{0}$, $\overline{P(t=0)}=P_{0}$, $\Sigma_{kk^{\prime}}(t)=2\overline{q_{k}(t)q_{k^{\prime}}(t)}$, $\Sigma_{kk^{\prime}}(t=0)=0$, $k,k^{\prime}=R,P$) calculated for an inverted oscillator which approximates the nucleus-nucleus potential $V$ in the variable $R$. The frequency $\omega$ of this oscillator with an internal turning point $r_{\rm in}$ is defined from the condition of equality of the classical actions of approximated and realistic potential barriers of the same hight at given $J$. This approximation is well justified for the reactions and energy range, which are here considered. We assume that the sub-barrier capture mainly depends on the optimal one- neutron ($Q_{1n}>Q_{2n}$) or two-neutron ($Q_{2n}>Q_{1n}$) transfer with the positive $Q$-value. Our assumption is that, just before the projectile is captured by the target-nucleus (just before the crossing of the Coulomb barrier) which is a slow process, the transfer occurs and can lead to the population of the first excited collective state in the recipient nucleus SSzilner (the donor nucleus remains in the ground state). So, the motion to the $N/Z$ equilibrium starts in the system before the capture because it is energetically favorable in the dinuclear system in the vicinity of the Coulomb barrier. For the reactions under consideration, the average change of mass asymmetry is connected to the one- or two-neutron transfer ($1n$\- or $2n$-transfer). Since after the transfer the mass numbers, the isotopic composition and the deformation parameters of the interacting nuclei, and, correspondingly, the height $V_{b}=V(R_{b})$ and shape of the Coulomb barrier are changed, one can expect an enhancement or suppression of the capture. If after the neutron transfer the deformations of interacting nuclei increase (decrease), the capture probability increases (decreases). When the isotopic dependence of the nucleus-nucleus potential is weak and after the transfer the deformations of interacting nuclei do not change, there is no effect of the neutron transfer on the capture. In comparison with Ref. Dasso , we assume that the negative transfer $Q-$values do not play visible role in the capture process. Our scenario was verified in the description of many reactions EPJSub1 ; EPJSub2 ; EPJSub3 . ## III Results of calculations Because the capture cross section is equal to the complete fusion cross section for the reactions treated, the quantum diffusion approach for the capture is applied to study the complete fusion. All calculated results are obtained with the same set of parameters as in Ref. EPJSub . Realistic friction coefficient in the relative distance coordinate $\hbar\lambda$=2 MeV is used. Its value is close to that calculated within the mean-field approaches obzor . For the nuclear part of the nucleus-nucleus potential, the double-folding formalism with the Skyrme-type density-dependent effective nucleon-nucleon interaction is used EPJSub ; EPJSub1 . The parameters of the nucleus-nucleus interaction potential $V(R)$ are adjusted to describe the experimental data at energies above the Coulomb barrier corresponding to spherical nuclei. The absolute values of the experimental quadrupole deformation parameters $\beta_{2}$ of even-even deformed nuclei in the ground state and of the first excited collective states of nuclei are taken from Ref. Ram . For the nuclei deformed in the ground state, the $\beta_{2}$ in the first excited collective state is similar to the $\beta_{2}$ in the ground state. For the quadruple deformation parameter of an odd nucleus, we choose the maximal value from the deformation parameters of neighboring even-even nuclei (for example, $\beta_{2}$(231Th)=$\beta_{2}$(233Th)=$\beta_{2}$(232Th)=0.261). For the double magic and neighboring nuclei, we take $\beta_{2}=0$ in the ground state. Since there are uncertainties in the definition of the values of $\beta_{2}$ in light-mass nuclei, one can extract the ground-state quadrupole deformation parameters of these nuclei from a comparison of the calculated capture cross sections with the existing experimental data. By describing the reactions 12C+208Pb, 18O+208Pb, 32,36S+90Zr, 34S+168Er, 36S+90,96Zr, 58Ni + 58Ni, and 64Ni + 58Ni, where there are no neutron transfer channels with positive $Q$-values, we extract the ground-state quadrupole deformation parameters $\beta_{2}$=-0.3, 0.1, 0.312, 0.1, 0, 0.05, and 0.087, for the nuclei 12C, 18O, 32S, 34S, 36S, 58Ni, and 64Ni, respectively, which are used in our calculations. ### III.1 Role of neutron transfer in capture process at sub-barrier energies After the neutron transfer in the reaction 40Ca($\beta_{2}=0$) + 96Zr($\beta_{2}=0.08$)$\to^{42}$Ca($\beta_{2}=0.247$) + 94Zr($\beta_{2}=0.09$) (Fig. 1) or 40Ca($\beta_{2}=0$) + 124Sn($\beta_{2}=0.095$)$\to^{42}$Ca($\beta_{2}=0.247$) + 122Sn($\beta_{2}=0.1$) (Fig. 1) the deformation of the nuclei increases and the mass asymmetry of the system decreases, and, thus, the value of the Coulomb barrier decreases and the capture cross section becomes larger (Fig. 1). In Fig. 2, we observe the same behavior in the reactions 58Ni($\beta_{2}=0.05$) + 132Sn($\beta_{2}=0$)$\to^{60}$Ni($\beta_{2}=0.207$) + 130Sn($\beta_{2}=0$) ($Q_{2n}=7.8$ MeV), 58Ni($\beta_{2}=0.05$) + 130Te($\beta_{2}=0$)$\to^{60}$Ni($\beta_{2}=0.207$) + 128Te($\beta_{2}=0$) ($Q_{2n}=5.9$ MeV), 64Ni($\beta_{2}=0.087$) + 132Sn($\beta_{2}=0$)$\to^{66}$Ni($\beta_{2}=0.158$) + 130Sn($\beta_{2}=0$) ($Q_{2n}=2.5$ MeV), and 64Ni($\beta_{2}=0.087$) + 130Te($\beta_{2}=0$)$\to^{66}$Ni($\beta_{2}=0.158$) + 128Te($\beta_{2}=0$) ($Q_{2n}=0.55$ MeV). One can see a good agreement between the calculated results and the experimental data Liang ; TimmersCa40Zr96 ; Stefanini40ca116124sn . So, the observed capture enhancement at sub-barrier energies in the reactions mentioned above is related to the two-neutron transfer channel. One can see that at energies above and near the Coulomb barrier the cross sections with and without two-neutron transfer are almost similar. Since the two-neutron transfer causes a larger change of the deformations of the nuclei in the reactions 58Ni + 132Sn,130Te than in the reactions 64Ni + 132Sn,130Te, at sub-barrier energies the capture enhancement in the reactions with 58Ni is larger than in the reactions with 64Ni (Fig. 2). Figure 1: The calculated capture cross sections versus $E_{\rm c.m.}$ for the indicated reactions 40Ca + 96Zr (solid line), 40Ca + 90Zr (dashed line), and 48Ca + 124Sn (solid line). For the reactions 40Ca + 96Zr,124Sn, the calculated capture cross sections without the neutron transfer process are shown by dotted lines. The experimental data (symbols) are from Refs. TimmersCa40Zr96 ; Stefanini40ca116124sn . Figure 2: The same as in Fig. 1, for the reactions 58,64Ni + 132Sn (solid lines) and 58,64Ni + 130Te (dashed lines). The experimental data (symbols) are from Refs. Liang ; Liang0 . For the reactions 58,64Ni + 132Sn (dotted lines) and 58,64Ni + 130Te (dash-dotted lines), the calculated capture cross sections without the neutron transfer are shown. Figure 3: (Color online) The calculated reduced capture cross sections versus $(E_{\rm c.m.}-V_{b})/(\hbar\omega_{b})$ in the reactions 40Ca+124Sn (solid line), 48Ca+124Sn (dashed line), 48Ca+124Sn (dotted line), and 48Ca+132Sn (dash-dotted line). Figure 4: (Color online) The calculated capture cross sections versus $E_{\rm c.m.}$ for the reactions 40Ca+124Sn (solid line) and 48Ca+124Sn (dashed line). The experimental data for the reactions 40Ca+124Sn (solid squares) and 48Ca+124Sn (open squares) are from Ref. Kol . In the calculations the barriers were adjusted to the experimental values. Figure 5: (Color online) The calculated capture cross sections versus $E_{\rm c.m.}$ for the reactions 40Ca+132Sn (solid line) and 48Ca+132Sn (dashed line). The experimental data for the reactions 40Ca+132Sn (solid squares) and 48Ca+132Sn (open squares) are from Ref. Kol . In the calculations the barriers were adjusted to the experimental values. One can make unambiguous statements regarding the neutron transfer process with a positive $Q$-value when the colliding nuclei are double magic or semi- magic. In this case one can disregard the deformation and orientation effects before the neutron transfer. To eliminate the influence of the nucleus-nucleus potential on the capture (fusion) cross section and to make conclusions about the role of deformation of colliding nuclei and the nucleon transfer between interacting nuclei in the capture (fusion) cross section, a reduction procedure is useful Gomes2 . It consists of the following transformations: $E_{\rm c.m.}\rightarrow x=\dfrac{E_{\rm c.m.}-V_{b}}{\hbar\omega_{b}},\qquad\sigma_{cap}\rightarrow\sigma_{cap}^{red}=\dfrac{2E_{\rm c.m.}}{\hbar\omega_{b}R_{b}^{2}}\sigma_{cap},$ where $\sigma_{cap}=\sigma_{cap}(E_{\rm c.m.})$ is the capture cross section at bombarding energy $E_{\rm c.m.}$. The frequency $\omega_{b}=\sqrt{V^{{}^{\prime\prime}}(R_{b})/\mu}$ is related with the second derivative $V^{{}^{\prime\prime}}(R_{b})$ of the total nucleus-nucleus potential $V(R)$ (the Coulomb + nuclear parts) at the barrier position $R_{b}$. With these replacements we compared the reduced calculated capture (fusion) cross sections $\sigma_{cap}^{red}$ for the reactions 40,48Ca+124,132Sn (Fig. 3). The choice of the projectile-target combination is crucial, and for the systems studied one can make unambiguous statements regarding the neutron transfer process with a positive $Q$-value when the interacting nuclei are double magic or semi-magic spherical nuclei. In this case one can disregard the strong direct nuclear deformation effects. In Fig. 3, one can see that the reduced capture cross sections in the reactions 40Ca+124,132Sn with the positive $Q_{2n}$-values strongly deviate from those in the reactions 48Ca+124,132Sn, where the neutron transfers are suppressed because of the negative $Q$-values. After two-neutron transfer in the reactions 40Ca($\beta_{2}=0$)+124Sn($\beta_{2}=0.1$)$\to^{42}$Ca($\beta_{2}=0.25$)+122Sn($\beta_{2}=0.1$) ($Q_{2n}$=5.4 MeV) and 40Ca($\beta_{2}=0$)+132Sn($\beta_{2}=0$)$\to^{42}$Ca($\beta_{2}=0.25$)+130Sn($\beta_{2}=0$) ($Q_{2n}$=7.3 MeV) the deformation of the light nucleus increases and the mass asymmetry of the system decreases and, thus, the value of the Coulomb barrier decreases and the capture cross section becomes larger (Fig. 3). So, because of the transfer effect the systems 40Ca+124,132Sn show large sub-barrier enhancements with respect to the systems 48Ca+124,132Sn. We observe that the $\sigma_{cap}^{red}$ in the 40Ca+124Sn (48Ca+124Sn) reaction are larger than those in the 40Ca+132Sn (48Ca+132Sn) reaction. The reason of that is the nonzero quadrupole deformation of the heavy nucleus 124Sn. It should be stressed that there are almost no difference between $\sigma_{cap}^{red}$ in the reactions 40,48Ca+124,132Sn at energies above the Coulomb barrier. In Figs. 4 and 5 one can see a good agreement between the calculated results and the experimental data in the reactions 40,48Ca+124,132Sn. This means that the observed capture enhancements in the reactions 40Ca+124,132Sn at sub- barrier energies are related to the two-neutron transfer effect. Note that the slope of the excitation function strongly depends on the deformations of the interacting nuclei and, respectively, on the neutron transfer effect. To describe the reactions 40,48Ca+132Sn and 48Ca+124,132Sn (Figs. 4 and 5), we extracted the values of the corresponding Coulomb barrier $V_{b}$ for the spherical nuclei. There are differences between the calculated and extracted $V_{b}$. From the direct calculations of the nucleus-nucleus potentials (with the same set of parameters), we obtained $V_{b}$(40Ca+124Sn)-$V_{b}$(48Ca+124Sn)=2.3 MeV, $V_{b}$(40Ca+132Sn)-$V_{b}$(48Ca+132Sn)=2.2 MeV, $V_{b}$(40Ca+124Sn)-$V_{b}$(40Ca+132Sn)=1.3 MeV, and $V_{b}$(48Ca+124Sn)-$V_{b}$(48Ca+132Sn)=1.2 MeV. From the extractions, we got $V_{b}$(40Ca+124Sn)-$V_{b}$(48Ca+124Sn)=1.1 MeV, $V_{b}$(40Ca+132Sn)-$V_{b}$(48Ca+132Sn)=1.0 MeV, $V_{b}$(40Ca+124Sn)-$V_{b}$(40Ca+132Sn)=-0.3 MeV, and $V_{b}$(48Ca+124Sn)-$V_{b}$(48Ca+132Sn)=-0.4 MeV, which seem to be unrealistically small. However, these differences of $V_{b}$ do not influence the slopes of the excitation functions but only lead to the shifting of the energy scale. With realistic isospin trend of $V_{b}$ $\sigma_{cap}$(40Ca+124Sn)$<\sigma_{cap}$(48Ca+124Sn) and $\sigma_{cap}$(40Ca+132Sn)$<\sigma_{cap}$(48Ca+132Sn) at energies above the corresponding Coulomb barriers. Figure 6: (Color online) The same as in Fig. 1, for the indicated reactions 60Ni + 100Mo,150Nd (solid lines), and 64Ni + 100Mo,150Nd (dashed lines). For the reactions 60Ni + 100Mo and 60Ni + 150Nd, the calculated capture cross sections without the neutron transfer are shown by dotted lines. The experimental data for the reactions 60Ni + 100Mo (closed squares) and 64Ni + 100Mo (open squares) are from Ref. Scarlassara . One can find reactions with a positive $Q$-values of the two-neutron transfer where the transfer weakly influences or even suppresses the capture process. This happens if after the transfer the deformations of the nuclei do not change much or even decrease. For instance, in the reactions 60Ni($\beta_{2}\approx 0.1$) + 100Mo($\beta_{2}=0.231$)$\to^{62}$Ni($\beta_{2}=0.198$) + 98Mo($\beta_{2}=0.168$) ($Q_{2n}=4.2$ MeV), 64Ni($\beta_{2}\approx 0.087$) + 100Mo($\beta_{2}=0.231$)$\to^{66}$Ni($\beta_{2}=0.158$) + 98Mo($\beta_{2}=0.168$) ($Q_{2n}=0.94$ MeV), and 60Ni($\beta_{2}\approx 0.1$) + 150Nd($\beta_{2}=0.285$)$\to^{62}$Ni($\beta_{2}=0.198$) + 148Nd($\beta_{2}=0.204$) ($Q_{2n}=6$ MeV) we expect a weak dependence of the capture cross section on the neutron transfer (Fig. 6). There is the experimental evidence Scarlassara of such an effect for the 60Ni + 100Mo reaction. So, the two-neutron transfer channel with large positive $Q_{2n}$-value weakly influences the fusion (capture) cross section. The reduced capture cross sections in the reactions 60Ni + 100Mo,150Nd are close to each other in contrast to those in the reactions 58,64Ni + 132Sn,130Te. The 60Ni + 150Nd reaction has even a small suppression due to the neutron transfer. Figure 7: The calculated capture cross sections vs $E_{\rm c.m.}$ for the reactions 32S+108,110Pd (dashed lines) and 36S+106,108Pd (solid lines) (a,b). For the 32S+110Pd reaction (a), the calculated capture cross section without the neutron transfer process is shown by a dotted line. For the reactions 32S+110Pd, the experimental data from Pengo and Stefanini3236s110pd are marked by open squares and stars, respectively. Figure 8: The same as in Fig. 7, for the reactions 32S+104,106Pd (dashed lines) and 36S+104,110Pd (solid lines) (a,b). The dotted lines correspond to the reactions 32S+104,106Pd when the neutron transfer is disregarded. The experimental data (symbols) are from Ref. Pengo . Figure 9: The calculated capture cross sections vs $E_{\rm c.m.}$ for the reactions 32S+100,102,104Ru (dashed lines) (a,b,c) and 36S+100,102,104Ru (solid lines) (a,b). The dotted lines correspond to the reactions 32S+102,104Ru (a,c) when the neutron transfer is disregarded. The experimental data (symbols) are from Ref. Pengo . Figures 7-9 show the capture excitation function for the reactions 32,36S+Pd,Ru as a function of the bombarding energy. One can see a relatively good agreement between the calculated results and the experimental data Pengo . The $Q_{2n}$-values for the $2n$-transfer processes are positive (negative) for all reactions with 32S (36S). At energies above and near the Coulomb barrier the cross sections with and without two-neutron transfer are almost similar. After the $2n$-transfer (before the capture) in the reactions 32S($\beta_{2}=0.312$)+110Pd($\beta_{2}=0.257$)$\to^{34}$S($\beta_{2}=0.252$)+108Pd($\beta_{2}=0.243$), 32S($\beta_{2}=0.312$)+108Pd($\beta_{2}=0.243$)$\to^{34}$S($\beta_{2}=0.252$)+106Pd($\beta_{2}=0.229$), 32S($\beta_{2}=0.312$)+106Pd($\beta_{2}=0.229$)$\to^{34}$S($\beta_{2}=0.252$)+104Pd($\beta_{2}=0.209$), 32S($\beta_{2}=0.312$)+104Pd($\beta_{2}=0.209$)$\to^{34}$S($\beta_{2}=0.252$)+102Pd($\beta_{2}=0.196$), or 32S($\beta_{2}=0.312$)+104Ru($\beta_{2}=0.271$)$\to^{34}$S($\beta_{2}=0.252$)+102Ru($\beta_{2}=0.24$), 32S($\beta_{2}=0.312$)+102Ru($\beta_{2}=0.24$)$\to^{34}$S($\beta_{2}=0.252$)+100Ru($\beta_{2}=0.215$), 32S($\beta_{2}=0.312$)+100Ru($\beta_{2}=0.215$)$\to^{34}$S($\beta_{2}=0.252$)+98Ru($\beta_{2}=0.195$) the deformations of the nuclei decrease and the values of the corresponding Coulomb barriers increase. As a result, the transfer suppresses the capture process in these reactions at the sub-barrier energies. The suppression becomes stronger with decreasing energy (Figs. 7-9). As seen in Fig. 7, the capture cross sections calculated without two-neutron transfer are larger than those calculated with two-neutron transfer in the case of the 32S+110Pd reaction. The enhancement of the sub-barrier fusion for the reactions with 32S with respect to the reactions with 36S is related to a larger deformation of 34S in comparison with 36S. We observe the same behavior in the reactions 32,36S+94,96,98,100Mo. Figure 10: (Color online) The calculated (solid line) capture cross sections vs $E_{\rm c.m.}$ for the reactions 16O+76Ge and 18O+74Ge (the curves coincide). For the 18O+74Ge reaction, the calculated capture cross sections without neutron transfer are shown by dotted line. The experimental data for the reactions 16O+76Ge (open circles) and 18O+74Ge (open squares) are from Ref. Jia . The experimental data for the 16O+76Ge reaction (solid circles) are from Ref. 16OAGe . Figure 11: The calculated capture cross sections vs $E_{\rm c.m.}$ for the reactions 18S+112,118,124Sn (solid, dashed and dotted lines, respectively) (a) and 32S+112,116,120Sn (solid, dashed and dotted lines, respectively) (b). The experimental data (symbols) are from Ref. AOASn ; Tripathi . Figures 10 and 11 show the excitation functions for the reactions 18O+74Ge,112,118,124Sn and 32S+112,116Sn. For the 32S-induced reactions, $Q_{2n}>0$. For the projectile 18O there is a large range of positive $Q_{2n}$-values, for example, varying from 1.4 MeV for 18O+124Sn up to 5.5 MeV for 18O+112Sn. The agreement between the calculated results and the experimental data Jia ; AOASn is rather good. As seen in Fig. 11, the cross sections increase systematically with the target mass number and run nearly similarly down to the lowest energy treated. In the reactions 32S($\beta_{2}=0.312$)+112Sn($\beta_{2}=0.123$)$\to^{34}$S($\beta_{2}=0.252$)+110Sn($\beta_{2}=0.122$), 32S($\beta_{2}=0.312$)+116Sn($\beta_{2}=0.112$)$\to^{34}$S($\beta_{2}=0.252$)+114Sn($\beta_{2}=0.121$), 18O($\beta_{2}=0.1$) + 74Ge($\beta_{2}=0.283$)$\to^{16}$O($\beta_{2}=0$) + 76Ge($\beta_{2}=0.262$), 18O($\beta_{2}=0.1$)+112Sn($\beta_{2}=0.123$)$\to^{16}$O($\beta_{2}=0$)+114Sn($\beta_{2}=0.121$), 18O($\beta_{2}=0.1$)+118Sn($\beta_{2}=0.111$)$\to^{16}$O($\beta_{2}=0$)+120Sn($\beta_{2}=0.104$), and 18O($\beta_{2}=0.1$)+124Sn($\beta_{2}=0.095$)$\to^{16}$O($\beta_{2}=0$)+126Sn($\beta_{2}=0.09$) the $2n$-transfer suppresses the capture process (Figs. 10 and 11). The sub- barrier capture cross sections for the systems 18O+ASn studied here do not show any strong dependence on the mass number of the target isotope. Our results show that cross sections for reactions 16O+76Ge (16O+114,120,126Sn) [$Q_{2n}<0$] and 18O+74Ge (18O+112,118,124Sn) are very similar (Fig. 10). Just the same behavior was observed in the recent experiments 16,18O+76,74Ge Jia . ### III.2 Neutron transfer in reactions with weakly bound nuclei After the neutron transfer in the reactions 13C+232Th($\beta_{2}=0.261$)$\to^{14}$C($\beta_{2}=-0.36$)+231Th($\beta_{2}=0.261$) ($Q_{1n}=1.74$ MeV), 15C+232Th($\beta_{2}=0.261$)$\to^{14}$C($\beta_{2}=-0.36$)+233Th($\beta_{2}=0.261$) ($Q_{1n}=3.57$ MeV) the deformations of the target or projectile nuclei in these reactions and in the 14C+232Th($\beta_{2}=0.261$) ($Q_{1n,2n}<0$) reaction are the same. In Fig. 12 the calculated cross sections slightly increase with the mass number of C, and are nearly parallel down to the lowest energy treated. There is a relatively good agreement between the calculated results EPJSub3 and the experimental data Alcorta ; CTh for the reactions 12,13,14C+232Th, but the experimental enhancement of the cross section in the 15C+232Th reaction at sub-barrier energies cannot be explained with our and other Alcorta models. Because we take into account the neutron transfer (15C$\to^{14}$C), one can suppose that this discrepancy is attributed to the influence of the breakup channel Gomes which is not considered in our model. However, it is unclear why the breakup process influences only two experimental points at lowest energies. Different deviations of these points in energy from the calculated curve in Fig. 12 create doubt in an influence of the breakup on the kinetic energy. So, additional experimental and theoretical investigations are desirable. Figure 12: (Color online) The calculated (lines) and experimental (symbols) capture cross sections vs $E_{\rm c.m.}$ for the reactions 12C+232Th (dash- dotted line, solid triangles), 13C+232Th (dotted line, open triangles), 14C+232Th (solid line, open squares), and 15C+232Th (dashed line, solid squares). The experimental data are from Refs. Alcorta ; CTh . Figure 13: The calculated (lines) and experimental (symbols) capture cross sections vs $E_{\rm c.m.}$ for the reactions 12C+208Pb (dash-dotted line), 13C+208Pb (dotted line), 14C+208Pb (solid line), and 15C+208Pb (dashed line). The experimental data (solid squares) for the 12C+208Pb reaction are from Ref. 12C208Pb . The question is whether the fusion of nuclei involving weakly bound neutrons is enhanced or suppressed at low energies. This question can been addressed to the systems 12-15C+208Pb Alamanos3 . After the neutron transfer in the reactions 13C+208Pb($\beta_{2}=0$)$\to^{14}$C($\beta_{2}=-0.36$)+207Pb($\beta_{2}=0$) ($Q_{1n}=1.74$ MeV), 15C+208Pb($\beta_{2}=0$)$\to^{14}$C($\beta_{2}=-0.36$)+209Pb($\beta_{2}=0.055$) ($Q_{1n}=3.57$ MeV) the deformations of the light nuclei are the same as in the 14C+208Pb($\beta_{2}=0$) ($Q_{1n,2n}<0$) reaction. The heavy nuclei are almost spherical. This means that the slopes of the excitation functions are almost the same (Fig. 13). As in the case of the 15C+232Th reaction, we do not expect enhancement of the capture cross section in the 15C+208Pb reaction owing to the neutron transfer. The same effect was observed in Ref. Alamanos3 . The study of the reactions 15C+208Pb,232Th at sub-barrier energies provides a good test for the verification of the effect of weakly bound nuclei on fusion and capture because it reveals the role of other effects besides neutron transfer. Figure 14: The calculated (solid line) and experimental (symbols) capture cross sections vs $E_{\rm c.m.}$ for the reaction 9Li+70Zn. The experimental data are from Ref. Vino . By assuming that the $2n$-transfer process takes place and the break-up channels are closed, one can predict almost the same capture cross sections for the reaction with large positive $Q_{2n}$ value 6He+206Pb (9Li+68Zn) and for the complemented reaction 4He+208Pb (7Li+70Zn). Indeed, after the transfer in the reactions 6He+206Pb$\to^{4}$He($\beta_{2}=0$)+208Pb($\beta_{2}=0.055$) ($Q_{2n}=13.13$ MeV), 9Li+86Zn$\to^{7}$Li($\beta_{2}\approx 0.4$)+70Zn($\beta_{2}=0.248$) ($Q_{2n}=9.60$ MeV) they become equivalent to the reactions 4He+208Pb and 7Li+70Zn. Therefore, the slopes of the excitation functions in the reactions with 6He (9Li) and 4He (7Li) should be similar. This conclusion supports the experimental data of Ref. Wolski , where the authors concluded that the fusion enhancement in the 6He+206Pb reaction (with respect to the 4He+208Pb reaction) is rather small or absent. By assuming that the $2n$-transfer process occurs, we calculated the capture cross sections for the 9Li+70Zn reaction (Fig. 14). The agreement with the experimental data of Ref. Vino is quite satisfactory. At lowest energies, the calculated cross section is by factor of $\sim 5$ less than the experimental value. The experimental data are well reproduced by the model Bala where two- neutron transfer from the 70Zn leads to 11Li halo structure and molecular bond between the nuclei in contact enhances the fusion cross section. Note that two-neutron transfer 9Li+70Zn$\to^{7}$Li+72Zn with $Q_{2n}=8.6$ MeV is much energetically favorable than the two-neutron transfer 9Li+70Zn$\to^{11}$Li+68Zn with $Q_{2n}=-15.4$ MeV. These observations deserve further experimental and theoretical investigations including the breakup channel. ### III.3 Breakup probabilities The difference between the calculated capture cross section $\sigma_{cap}^{th}$ in the absence of breakup and the experimental complete fusion cross section $\sigma_{fus}^{exp}$ can be ascribed to the breakup effect with the probability EPJSub4 $\displaystyle P_{\rm BU}=1-\sigma_{fus}^{exp}/\sigma_{c}^{th}.$ (3) If at some energy $\sigma_{fus}^{exp}>\sigma_{cap}^{th}$, the values of $\sigma_{cap}^{th}$ was normalized so to have $P_{\rm BU}\geq 0$ at any energy. Note that $\sigma_{fus}^{exp}=\sigma_{fus}^{noBU}+\sigma_{fus}^{BU}$ contains the contribution from two processes: the direct fusion of the projectile with the target ($\sigma_{fus}^{noBU}$), and the breakup of the projectile followed by the fusion of the two projectile fragments with the target ($\sigma_{fus}^{BU}$). A more adequate estimate of the breakup probability would then be: $\displaystyle P_{\rm BU}=1-\sigma_{fus}^{noBU}/\sigma_{cap}^{th},$ (4) which leads to larger values of $P_{\rm BU}$ than the expression employed by us. However, the ratio between $\sigma_{fus}^{noBU}$ and $\sigma_{fus}^{BU}$ cannot be measured experimentally but can be estimated with the approach suggested in Ref. Maximka . The parameters of the potential are taken to fit the height of the Coulomb barrier obtained in our calculations. The parameters of the breakup function Maximka are set to describe the value of $\sigma_{fus}^{exp}$. As shown in Ref. Maximka and in our calculations, in the 8Be+208Pb reaction the fraction of $\sigma_{fus}^{BU}$ in $\sigma_{fus}^{exp}$ does not exceed few percents at $E_{\rm c.m.}-V_{b}<$4 MeV. This fraction rapidly increases and reaches about 12–20%, depending on the reaction, at $E_{\rm c.m.}-V_{b}\approx$10 MeV. Because we are mainly interested in the energies near and below the barrier, the estimated $\sigma_{fus}^{BU}$ does not exceed 20% of $\sigma_{fus}^{exp}$ at $E_{\rm c.m.}-V_{b}<$10 MeV. The results for $P_{\rm BU}$ are presented, taking $\sigma_{fus}^{noBU}$ into account in Eq. (4). Figure 15: (Color online) The dependence of the extracted breakup probability $P_{BU}$ vs $E_{c.m.}-V_{b}$ for the indicated reactions with 9Be-projectiles in %. Formula (4) was used. Figure 16: (Color online) The same as in Fig. 15, but for the indicated reactions with 6,7,9Li-projectiles. As seen in Figs. 15 and 16, at energies above the Coulomb barriers the values of $P_{\rm BU}$ vary from 0 to 84%. In the reactions 9Be+144Sm,208Pb,209Bi the value of $P_{\rm BU}$ increases with charge number of the target at $E_{\rm c.m.}-V_{b}>3$ MeV. This was also noted in Ref. PRSGomes5 . However, the reactions 9Be+89Y,124Sn are out of this systematics. In the reactions 6Li+144Sm,198Pt,209Bi the value of $P_{\rm BU}$ decreases with increasing charge number of the target at $E_{\rm c.m.}-V_{b}>3$ MeV. While in the reactions 9Be+89Y,144Sm,208Pb,209Bi the value of $P_{\rm BU}$ has a minimum at $E_{\rm c.m.}-V_{b}\approx 0$ and a maximum at $E_{\rm c.m.}-V_{b}\approx-(1-3)$ MeV, in the 9Be+124Sn reaction the value of $P_{\rm BU}$ steadily decreases with energy. In the reactions 6Li+144Sm,198Pt,209Bi, 7Li+208Pb,209Bi, and 9Li+208Pb there is maximum of $P_{\rm BU}$ at $E_{\rm c.m.}-V_{b}\approx-(0-1)$ MeV. However, in the reactions 6Li+208Pb and 7Li+165Ho $P_{\rm BU}$ has a minima $E_{\rm c.m.}-V_{b}\approx 2$ MeV and no maxima at $E_{\rm c.m.}-V_{b}\approx 0$. For 9Be, the breakup threshold is slightly larger than for 6Li. Therefore, we cannot explain a larger breakup probability at smaller $E_{\rm c.m.}-V_{b}$ in the case of 9Be. ## IV Quasi-elastic and elastic backscattering - tools for search of breakup process in reactions with weakly bound projectiles The lack of a clear systematic behavior of the complete fusion suppression as a function of the target charge requires new additional experimental and theoretical studies. The quasi-elastic backscattering has been used Timmers ; EPJSub4 as an alternative to investigate fusion (capture) barrier distributions, since this process is complementary to fusion. Since the quasi- elastic experiment is usually not as complex as the capture (fusion) and breakup measurements, they are well suited to survey the breakup probability. There is a direct relationship between the capture, the quasi-elastic scattering and the breakup processes, since any loss from the quasi-elastic and breakup channel contributes directly to capture (the conservation of the total reaction flux): $\displaystyle P_{qe}(E_{\rm c.m.},J)+P_{cap}(E_{\rm c.m.},J)+P_{BU}(E_{\rm c.m.},J)=1,$ (5) where $P_{qe}$ is the reflection quasi-elastic probability, $P_{BU}$ is the breakup probability, and $P_{cap}$ is the capture probability. The quasi- elastic scattering ($P_{qe}$) is the sum of all direct reactions, which include elastic ($P_{el}$), inelastic ($P_{in}$), and a few nucleon transfer ($P_{tr}$) processes. In Eq. (5) we neglect the deep inelastic collision process, since we are concerned with low energies. Equation (5) can be rewritten as $\displaystyle\frac{P_{qe}(E_{\rm c.m.},J)}{1-P_{BU}(E_{\rm c.m.},J)}+\frac{P_{cap}(E_{\rm c.m.},J)}{1-P_{BU}(E_{\rm c.m.},J)}=P_{qe}^{noBU}(E_{\rm c.m.},J)+P_{cap}^{noBU}(E_{\rm c.m.},J)=1,$ (6) where $P_{qe}^{noBU}(E_{\rm c.m.},J)=\frac{P_{qe}(E_{\rm c.m.},J)}{1-P_{BU}(E_{\rm c.m.},J)}$ and $P_{cap}^{noBU}(E_{\rm c.m.},J)=\frac{P_{cap}(E_{\rm c.m.},J)}{1-P_{BU}(E_{\rm c.m.},J)}$ are the quasi-elastic and capture probabilities, respectively, in the absence of the breakup process. From these expressions we obtain the useful formulas $\displaystyle\frac{P_{qe}(E_{\rm c.m.},J)}{P_{cap}(E_{\rm c.m.},J)}=\frac{P_{qe}^{noBU}(E_{\rm c.m.},J)}{P_{cap}^{noBU}(E_{\rm c.m.},J)}=\frac{P_{qe}^{noBU}(E_{\rm c.m.},J)}{1-P_{qe}^{noBU}(E_{\rm c.m.},J)}=a.$ (7) Using Eqs. (5) and (7), we obtain the relationship between breakup and quasi- elastic processes: $\displaystyle P_{BU}(E_{\rm c.m.},J)=1-P_{qe}(E_{\rm c.m.},J)[1+1/a]=1-\frac{P_{qe}(E_{\rm c.m.},J)}{P_{qe}^{noBU}(E_{\rm c.m.},J)}.$ (8) The reflection quasi-elastic probability $P_{qe}(E_{\rm c.m.},J=0)=d\sigma_{qe}/d\sigma_{Ru}$ for bombarding energy $E_{\rm c.m.}$ and angular momentum $J=0$ is given by the ratio of the quasi-elastic differential cross section $\sigma_{qe}$ and Rutherford differential cross section $\sigma_{Ru}$ at 180 degrees Timmers . Employing Eq. (8) and the experimental quasi-elastic backscattering data with toughly and weakly bound isotopes- projectiles and the same compound nucleus, one can extract the breakup probability of the exotic nucleus. For example, using Eq. (8) at backward angle, the experimental $P_{qe}^{noBU}$[4He+AX] of the 4He+AX reaction with toughly bound nuclei (without breakup), and $P_{qe}$[6He+A-2X] of the 6He+A-2X reaction with weakly bound projectile (with breakup), and taking into consideration $V_{b}$(4He+AX)$\approx V_{b}$(6He+A-2X) for the very asymmetric systems, one can extract the breakup probability of the 6He: $\displaystyle P_{BU}(E_{\rm c.m.},J=0)=1-\frac{P_{qe}(E_{\rm c.m.},J=0)[^{6}He+^{A-2}{\rm X}]}{P_{qe}^{noBU}(E_{\rm c.m.},J=0)[^{4}He+^{A}{\rm X}]}.$ (9) Comparing the experimental quasi-elastic backscattering cross sections in the presence and absence of breakup data in the reaction pairs 6He+68Zn and 4He+70Zn, 6He+122Sn and 4He+124Sn, 6He+236U and 4He+238U, 8He+204Pb and 4He+208Pb, 8Li+207Pb and 7Li+208Pb, 7Be+207Pb and 10Be+204Pb, 9Be+208Pb and 10Be+207Pb, 11Be+206Pb and 10Be+207Pb, 8B+208Pb and 10B+206Pb, 8B+207Pb and 11B+204Pb, 9B+208Pb and 11B+206Pb, 15C+204Pb and 12C+207Pb, 15C+206Pb and 13C+208Pb, 15C+207Pb and 14C+208Pb, 17F+206Pb and 19F+208Pb, leading to the same corresponding compound nuclei, one can analyze the role of the breakup channels in the reactions with the light weakly bound projectiles 6,8He, 8Li, 7,9,11Be, 8,9B, 15C, and 17F at near and above the barrier energies. On other side, the experimental uncertainties could be probably smaller when the same target-nucleus AX is used in the reactions with weakly and toughly bound isotopes. Then, one can extract the breakup probability of the 6He [$\Delta E=V_{b}(^{4}{\rm He}+^{A}{\rm X})-V_{b}(^{6}{\rm He}+^{A}{\rm X})]$: $\displaystyle P_{BU}(E_{\rm c.m.},J=0)=1-\frac{P_{qe}(E_{\rm c.m.},J=0)[^{6}{\rm He}+^{A}{\rm X}]}{P_{qe}^{noBU}(E_{\rm c.m.}+\Delta E,J=0)[^{4}{\rm He}+^{A}{\rm X}]}.$ (10) For the very asymmetric systems, one can neglect $\Delta E$. Using the conservation of the total reaction flux, analogously one can find the following expression $\displaystyle P_{BU}(E_{\rm c.m.},J)=1-\frac{P_{el}(E_{\rm c.m.},J)}{P_{el}^{noBU}(E_{\rm c.m.},J)},$ (11) which relates the breakup and elastic scattering processes. $P_{el}^{noBU}(E_{\rm c.m.},J)$ is the elastic scattering probability in the absence of the breakup process. So, one can extract the breakup probability of the 6He at the backward angle: $\displaystyle P_{BU}(E_{\rm c.m.},J=0)=1-\frac{P_{el}(E_{\rm c.m.},J=0)[^{6}{\rm He}+^{A-2}{\rm X}]}{P_{el}^{noBU}(E_{\rm c.m.},J=0)[^{4}{\rm He}+^{A}{\rm X}]}$ (12) or $\displaystyle P_{BU}(E_{\rm c.m.},J=0)=1-\frac{P_{el}(E_{\rm c.m.},J=0)[^{6}{\rm He}+^{A}{\rm X}]}{P_{el}^{noBU}(E_{\rm c.m.}+\Delta E,J=0)[^{4}{\rm He}+^{A}{\rm X}]}.$ (13) One concludes that the quasi-elastic or elastic backscattering technique could be a very important tool in breakup research. We propose to extract the breakup probability directly from the quasi-elastic or elastic backscattering probabilities of systems mentioned above. ## V Summary The quantum diffusion approach was applied to study the role of the neutron transfer with positive $Q$-value in the capture reactions at sub-, near- and above-barrier energies. We demonstrated a good agreement of the theoretical calculations with the experimental data. We found, that the change of the magnitude of the capture cross section after the neutron transfer occurs due to the change of the deformations of nuclei. The effect of the neutron transfer is an indirect effect of the quadrupole deformation. When after the neutron transfer the deformations of nuclei do not change or slightly decrease, the neutron transfer weakly influences or suppresses the capture cross section. Good examples for this effect are the capture reactions 60Ni + 100Mo,150Nd, 18O + 64Ni,112,114,116,118,120,122,124Sn,204,206Pb, and 32S+96Zr,94,96,98,100Mo,100,102,104Ru,104,106,108,110Pd,112,114,116,118,120,122,124Sn. at sub-barrier energies. Thus, the general point of view that the sub-barrier capture (fusion) cross section strongly increases because of the neutron transfer with a positive $Q$-values has to be revised. The neutron transfer effect can lead to a weak influence of halo-nuclei on the capture. Comparing the capture cross sections calculated without the breakup effect and experimental complete fusion cross sections, the breakup was analyzed in reactions with weakly bound projectiles. A trend of a systematic behavior for the complete fusion suppression as a function of the target charge and bombarding energy is not achieved. The quasi-elastic or elastic backscattering was suggested to be an useful tool to study the behavior of the breakup probability. We thank P.R.S. Gomes and A. Lépina-Szily for fruitful discussions and suggestions. This work was supported by DFG, NSFC, RFBR, and JINR grants. The IN2P3(France)-JINR(Dubna) and Polish - JINR(Dubna) Cooperation Programmes are gratefully acknowledged. ## References * (1) L.F. Canto, P.R.S. Gomes, R. Donangelo, and M.S. Hussein, Phys. Rep. 424, (2006) 1. * (2) V.V. Sargsyan, G.G. Adamian, N.V. Antonenko, and W. Scheid, Eur. Phys. J. A 45, 125 (2010). * (3) V.V. Sargsyan et al., Eur. Phys. J. 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C 65, 054611 (2002).	arxiv-papers	2013-11-20T11:33:55	2024-09-04T02:49:53.987296	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "V.V.Sargsyan, G.G.Adamian, N.V.Antonenko, W. Scheid, and H.Q.Zhang", "submitter": "Vazgen Sargsyan Dr.", "url": "https://arxiv.org/abs/1311.5020" }
1311.5024	11footnotetext: CNRS, CMAP, Ecole Polytechnique, 91120 Palaiseau, France.22footnotetext: Department of Mathematics, Technion, I.I.T, Haifa 32000, Israel.33footnotetext: Email: [email protected] 44footnotetext: Email: [email protected]: Supported by the Mathematical Sciences Institute – The Australian National University. # Minimax rate of convergence and the performance of ERM in phase recovery Guillaume Lecué1,3 Shahar Mendelson2,4,5 ###### Abstract We study the performance of Empirical Risk Minimization in noisy phase retrieval problems, indexed by subsets of $\mathbb{R}^{n}$ and relative to subgaussian sampling; that is, when the given data is $y_{i}=\bigl{<}a_{i},x_{0}\bigr{>}^{2}+w_{i}$ for a subgaussian random vector $a$, independent noise $w$ and a fixed but unknown $x_{0}$ that belongs to a given subset of $\mathbb{R}^{n}$. We show that ERM produces $\hat{x}$ whose Euclidean distance to either $x_{0}$ or $-x_{0}$ depends on the gaussian mean-width of the indexing set and on the signal-to-noise ratio of the problem. The bound coincides with the one for linear regression when $\\|x_{0}\\|_{2}$ is of the order of a constant. In addition, we obtain a minimax lower bound for the problem and identify sets for which ERM is a minimax procedure. As examples, we study the class of $d$-sparse vectors in $\mathbb{R}^{n}$ and the unit ball in $\ell_{1}^{n}$. ## 1 Introduction Phase retrieval has attracted much attention recently, as it has natural applications in areas that include X-ray crystallography, transmission electron microscopy and coherent diffractive imaging (see, for example, the discussion in [2] and references therein). In phase retrieval, one attempts to identify a vector $x_{0}$ that belongs to an arbitrary set $T\subset\mathbb{R}^{n}$ using noisy, quadratic measurements of $x_{0}$. The given data is a sample of cardinality $N$, $(a_{i},y_{i})_{i=1}^{N}$, for vectors $a_{i}\in\mathbb{R}^{n}$ and $y_{i}=\|\bigl{<}a_{i},x_{0}\bigr{>}\|^{2}+w_{i},$ (1.1) for a noise vector $(w_{i})_{i=1}^{N}$. Our aim is to investigate phase retrieval from a theoretical point of view, relative to a well behaved, random sampling method. To formulate the problem explicitly, let $\mu$ be an isotropic, $L$-subgaussian measure on $\mathbb{R}^{n}$ and set $a$ to be a random vector distributed according to $\mu$. Thus, for every $x\in\mathbb{R}^{n}$, $\mathbb{E}\bigl{<}x,a\bigr{>}^{2}=\\|x\\|_{2}^{2}$ (isotropicity) and for every $u\geq 1$, $Pr(\|\bigl{<}x,a\bigr{>}\|\geq Lu\\|\bigl{<}x,a\bigr{>}\\|_{L_{2}})\leq 2\exp(-u^{2}/2)$ ($L$-subgaussian). Given a set $T\subset\mathbb{R}^{n}$ and a fixed, but unknown $x_{0}\in T$, $y_{i}$ are the random noisy measurements of $x_{0}$: for a sample size $N$, $(a_{i})_{i=1}^{N}$ are independent copies of $a$ and $(w_{i})_{i=1}^{N}$ are independent copies of a mean-zero variable $w$ that are also independent of $(a_{i})_{i=1}^{N}$. Clearly, due to the nature of the given measurements, $x_{0}$ and $-x_{0}$ are indistinguishable, and the best that one can hope for is a procedure that produces $\hat{x}\in T$ that is close to one of the two points. The goal here is to find such a procedure and identify the way in which the distance between $\hat{x}$ and either $x_{0}$ or $-x_{0}$ depends on the structure of $T$, the measure $\mu$ and the noise. The procedure studied here is empirical risk minimization (ERM), which produces $\hat{x}$ that minimizes the empirical risk in $T$: $P_{N}\ell_{x}=\frac{1}{N}\sum_{i=1}^{N}\big{(}\bigl{<}a_{i},x\bigr{>}^{2}-y_{i}\big{)}^{2}.$ The loss is the standard squared loss functional, which, in this case,satisfies $\ell_{x}(a,y)=(f_{x}(a)-y)^{2}=(\bigl{<}x,a\bigr{>}^{2}-\bigl{<}x_{0},a\bigr{>}^{2}-w)^{2}=(\bigl{<}x-x_{0},a\bigr{>}\bigl{<}x+x_{0},a\bigr{>}-w)^{2}.$ Comparing the empirical and actual structures on $T$ is a vital component in the analysis of ERM. In phase recovery, the centered empirical process that is at the heart of this approach is defined for any $x\in T$ by, $P_{N}(\ell_{x}-\ell_{x_{0}})=\frac{1}{N}\sum_{i=1}^{N}\bigl{<}x-x_{0},a_{i}\bigr{>}^{2}\bigl{<}x+x_{0},a_{i}\bigr{>}^{2}-\frac{2}{N}\sum_{i=1}^{N}w_{i}\bigl{<}x-x_{0},a_{i}\bigr{>}\bigl{<}x+x_{0},a_{i}\bigr{>}.$ Both the first and second components are difficult to handle directly, even when the underlying measure is subgaussian, because of the powers involved (an effective power of $4$ in the first component and of $3$ in the second one). Therefore, rather than trying to employ the concentration of empirical means around the actual ones, which might not be sufficiently strong in this case, one uses a combination of a small-ball estimate for the ‘high order’ part of the process, and a more standard deviation argument for the low-order component (see Section 3 and the formulation of Theorem A and Theorem B). We assume that linear forms satisfy a certain small-ball estimate, and in particular, do not assign too much weight to small neighbourhoods of $0$. ###### Assumption 1.1 There is a constant $\kappa_{0}>0$ satisfying that for every $s,t\in\mathbb{R}^{n}$, $\mathbb{E}\|\bigl{<}a,s\bigr{>}\bigl{<}a,t\bigr{>}\|\geq\kappa_{0}\\|s\\|_{2}\\|t\\|_{2}.$ Assumption 1.1 is not very restrictive and holds for many natural choices of random vectors in $\mathbb{R}^{n}$, like the gaussian measure or any isotropic log-concave measure on $\mathbb{R}^{n}$ (see, for example, the discussion in [2]). It is not surprising that the error rate of ERM depends on the structure of $T$, and because of the subgaussian nature of the random measurement vector $a$, the natural parameter that captures the complexity of $T$ is the gaussian mean-width associated with normalizations of $T$. ###### Definition 1.1 Let $G=(g_{1},...,g_{n})$ be the standard gaussian vector in $\mathbb{R}^{n}$. For $T\subset\mathbb{R}^{n}$, set $\ell(T)=\mathbb{E}\sup_{t\in T}\Big{\|}\sum_{i=1}^{n}g_{i}t_{i}\Big{\|}.$ The normalized sets in question are $\displaystyle T_{-,R}$ $\displaystyle=\left\\{\frac{t-s}{\\|t-s\\|_{2}}\ :\ t,s\in T,\ \ R<\\|t-s\\|_{2}\\|t+s\\|_{2}\right\\},$ $\displaystyle T_{+,R}$ $\displaystyle=\left\\{\frac{t+s}{\\|t+s\\|_{2}}\ :\ t,s\in T,\ \ R<\\|t-s\\|_{2}\\|t+s\\|_{2}\right\\},$ which have been used in [2], or their ‘local’ versions, $\displaystyle T_{-,R}(x_{0})$ $\displaystyle=\left\\{\frac{t-x_{0}}{\\|t-x_{0}\\|_{2}}\ :\ t\in T,\ \ R<\\|t-x_{0}\\|_{2}\\|t+x_{0}\\|_{2}\right\\},$ $\displaystyle T_{+,R}(x_{0})$ $\displaystyle=\left\\{\frac{t+x_{0}}{\\|t+x_{0}\\|_{2}}\ :\ t\in T,\ \ R<\\|t-x_{0}\\|_{2}\\|t+x_{0}\\|_{2}\right\\}.$ The sets in question play a central role in the exclusion argument that is used in the analysis of ERM. Setting ${\cal L}_{x}=\ell_{x}-\ell_{x_{0}}$, the excess loss function associated with $\ell$ and $x\in T$, it is evident that $P_{N}{\cal L}_{\hat{x}}\leq 0$ (because ${\cal L}_{x_{0}}=0$ is a possible competitor). If one can find an event of large probability and $R>0$ for which $P_{N}{\cal L}_{x}>0$ if $\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2}\geq R$, then on that event, $\\|\hat{x}-x_{0}\\|_{2}\\|\hat{x}+x_{0}\\|_{2}\leq R$, which is the estimate one is looking for. The normalization allows one to study ‘relative fluctuations’ of $P_{N}{\cal L}_{x}$ – in particular, the way these fluctuations scale with $\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2}$. This is achieved by considering empirical means of products of functions $\bigl{<}u,\cdot\bigr{>}\bigl{<}v,\cdot\bigr{>}$, for $u\in T_{+,R}(x_{0})$ and $v\in T_{-,R}(x_{0})$. The obvious problem with the ‘local’ sets $T_{+,R}(x_{0})$ and $T_{-,R}(x_{0})$ is that $x_{0}$ is not known. As a first attempt of bypassing this problem, one may use the ‘global’ sets $T_{+,R}$ and $T_{-,R}$ instead, as had been done in [2]. Unfortunately, this global approach is not completely satisfactory. Roughly put, there are two types of subsets of $\mathbb{R}^{n}$ one is interested in, and that appear in applications. The first type consists of sets for which the ‘local complexity’ is essentially the same everywhere, and the sets $T_{+,R},T_{-,R}$ are not very different from the seemingly smaller $T_{+,R}(x_{0})$, $T_{-,R}(x_{0})$, regardless of $x_{0}$. A typical example of such a set is $d$-sparse vectors – a set consisting of all the vectors in $\mathbb{R}^{n}$ that are supported on at most $d$-coordinates. For every $x_{0}\in T$ and $R>0$, the sets $T_{+,R}(x_{0}),T_{-,R}(x_{0})$, and $T_{+,R},T_{-,R}$ are contained in the subset of the sphere consisting of $2d$-sparse vectors, which is a relatively small set. For this kind of set, the ‘global’ approach, using $T_{+,R}$ and $T_{-,R}$, suffices, and the choice of the target $x_{0}$ does not really influence the rate in which $\left\\|\hat{x}-x_{0}\right\\|_{2}\left\\|\hat{x}+x_{0}\right\\|_{2}$ decays to $0$ with $N$. In contrast, sets of the second type one would like to study, have vastly changing local complexity, with the typical example being a convex, centrally symmetric set (i.e. if $x\in T$ then $-x\in T$). Consider, for example, the case $T=B_{1}^{n}$, the unit ball in $\ell_{1}^{n}$. It is not surprising that for small $R$, the sets $T_{+,R}(0)$ and $T_{-,R}(0)$ are very different from $T_{-,R}(e_{1})$ and $T_{+,R}(e_{1})$: the ones associated with the centre $0$ are the entire sphere, while for $e_{1}=(1,0,....,0)$, $T_{+,R}(e_{1})$ and $T_{-,R}(e_{1})$ consist of vectors that are well approximated by sparse vectors (whose support depend on $R$), and thus are rather small subsets of the sphere . The situation that one encounters in $B_{1}^{n}$ is generic for convex centrally-symmetric sets. The sets become locally ‘richer’ the closer the centre is to $0$, and at $0$, for small enough $R$, $T_{+,R}(0)$ and $T_{-,R}(0)$ are the entire sphere. Since the sets $T_{+,R}$ and $T_{-,R}$ are blind to the location of the centre, and are, in fact, the union over all possible centres of the local sets, they are simply too big to be used in the analysis of ERM in convex sets. A correct estimate on the performance of ERM for such sets requires a more delicate local analysis and additional information on $\\|x_{0}\\|_{2}$. Moreover, the rate of convergence of ERM truly depends on $\\|x_{0}\\|_{2}$ in the phase recovery problem via the signal-to- noise ratio $\left\\|x_{0}\right\\|_{2}/\sigma$. We begin by formulating our results using the ‘global’ sets $T_{+,R}$ and $T_{-,R}$. Let $T_{+}=T_{+,0}$ and $T_{-}=T_{-,0}$, set $E_{R}=\max\\{\ell(T_{+,R}),\ell(T_{-,R})\\},\ \ \ E=\max\\{\ell(T_{+}),\ell(T_{-})\\}$ and observe that as nonempty subsets of the sphere $\ell(T_{-,r}),\ell(T_{+,r})\geq\mathbb{E}\|g\|=\sqrt{2/\pi}$. The first result presented here is that the error rate of ERM for the phase retrieval problem in $T$ depends on the fixed points $r_{2}(\gamma)=\inf\left\\{r>0:E_{r}\leq\gamma\sqrt{N}r\right\\}$ and $r_{0}(Q)=\inf\left\\{r>0:E_{r}\leq Q\sqrt{N}\right\\},$ for constants $\gamma$ and $Q$ that will be specified later. Recall that the $\psi_{2}$-norm of a random variable $w$ is defined by $\\|w\\|_{\psi_{2}}=\inf\\{c>0:\mathbb{E}\exp(w^{2}/c^{2})\leq 2\\}$ and set $\sigma=\\|w\\|_{\psi_{2}}$. Theorem A. For every $L>1$, $\kappa_{0}>0$ and $\beta>1$, there exist constants $c_{0},c_{1}$ and $c_{2}$ that depend only on $L$, $\kappa_{0}$ and $\beta$ for which the following holds. Let $a$ and $w$ be as above and assume that $w$ has a finite $\psi_{2}$-norm. If $\ell$ is the squared loss and $\hat{x}$ is produced by ERM, then with probability at least $1-2\exp(-c_{0}\min\\{\ell^{2}(T_{+,r_{2}^{}}),\ell^{2}(T_{-,r_{2}^{}})\\})-2N^{-\beta+1},$ $\\|\hat{x}-x_{0}\\|_{2}\\|\hat{x}+x_{0}\\|_{2}\leq r_{2}^{}:=\max\\{r_{0}(c_{1}),r_{2}(c_{2}/\sigma\sqrt{\log N})\\}.$ When $\left\\|w\right\\|_{\infty}<\infty$ the term $\sigma\sqrt{\log N}$ may be replaced by $\left\\|w\right\\|_{\infty}$. The upper estimate of $\max\\{r_{0},r_{2}\\}$ in Theorem A represents two ranges of noise. It follows from the definition of the fixed points that $r_{0}$ is dominant if $\sigma\leq r_{0}/\sqrt{\log N}$. As explained in [7] for linear regression, $r_{0}$ captures the difficulty of recovery in the noise free case, when the only reason for errors is that there are several far-away functions in the class that coincide with the target on the noiseless data. When the noise level $\sigma$ surpasses that threshold, errors occur because of the interaction class members have with the noise, and the dominating term becomes $r_{2}$. Of course, there are cases in which $r_{0}=0$ for $N$ sufficiently large. This is precisely when exact recovery is possible in the noise-free environment. And, in such cases, the error of ERM tends to zero with $\sigma$. The behavior of ERM in the noise-free case is one of the distinguishing features of sets with well behaved ‘global complexity’ – because $E$ is not too large. Since $E_{R}\leq E$ for every $R>0$, it follows that when $N\gtrsim E^{2}$, $r_{0}=0$ and that $r_{2}(\gamma)\leq E/(\gamma\sqrt{N})$. Therefore, on the event from Theorem A, $\\|\hat{x}-x_{0}\\|_{2}\\|\hat{x}+x_{0}\\|_{2}\lesssim\sigma\frac{E}{\sqrt{N}}\sqrt{\log N}.$ This estimate suffices for many applications. For example, when $T$ is the set of $d$-sparse vectors, one may show (see, e.g. [2]) that $E\lesssim\sqrt{d\log(en/d)}.$ Hence, by Theorem A, when $N\gtrsim d\log\big{(}en/d\big{)}$, with high probability, $\\|\hat{x}-x_{0}\\|_{2}\\|\hat{x}+x_{0}\\|_{2}\lesssim\sigma\sqrt{\frac{d\log(en/d)}{N}}\sqrt{\log N}.$ The proof of this observation regarding $d$-sparse vectors, and that this estimate is sharp in the minimax sense (up to the logarithmic term) may be found in Section 6. One should note that Theorem A improves the main result from [2] in three ways. First of all, the error rate (the estimate on $\\|\hat{x}-x_{0}\\|_{2}\\|\hat{x}+x_{0}\\|_{2}$) established in Theorem A is $\sim E/\sqrt{N}$ (up to logarithmic factors), whereas in [2], it scaled like $c/N^{1/4}$ for very large values of $N$. Second, the error rate scales linearly in the noise level $\sigma$ in Theorem A. On the other hand, the rate obtained in [2] does not decay with $\sigma$ for $\sigma\leq 1$. Finally, the probability estimate has been improved, though it is still likely to be suboptimal. Although the main motivation for [2] was dealing with phase retrieval for sparse classes, and for which Theorem A is well suited, we next turn to the question of more general classes, the most important example of which is a convex, centrally-symmetric class. For such a class, the global localization is simply too big to yield a good bound. ###### Definition 1.2 Let $r_{N}^{}(Q)=\inf\left\\{r>0:\ell(T\cap rB_{2}^{n})\leq Qr\sqrt{N}\right\\},$ $s_{N}^{}(\eta)=\inf\left\\{s>0:\ell(T\cap sB_{2}^{n})\leq\eta s^{2}\sqrt{N}\right\\},$ and $v_{N}^{}(\zeta)=\inf\left\\{v>0:\ell(T\cap vB_{2}^{n})\leq\zeta v^{3}\sqrt{N}\right\\},$ The parameters $r_{N}^{}$ and $s_{N}^{}$ have been used in [7] to obtain a sharp estimate on the performance of ERM for linear regression in an arbitrary convex set, and relative to $L$-subgaussian measurements. This result is formulated below in a restricted context, analogous to the phase retrieval setup: a linear model $z=\bigl{<}a,x_{0}\bigr{>}+w$, for an isotropic, $L$-subgaussian vector $a$, independent noise $w$ and $x_{0}\in T$. Let $\hat{x}$ be the output of ERM using the data $(a_{i},z_{i})_{i=1}^{N}$ and set $\\|w\\|_{\psi_{2}}=\sigma$. ###### Theorem 1.3 For every $L\geq 1$ there exist constants $c_{1},c_{2},c_{3}$ and $c_{4}$ that depend only on $L$ for which the following holds. Let $T\subset\mathbb{R}^{d}$ be a convex set, put $\eta=c_{1}/\sigma$ and set $Q=c_{2}$. 1\. If $\sigma\geq c_{3}r_{N}^{}(Q)$ then with probability at least $1-4\exp(-c_{4}N\eta^{2}(s_{N}^{}(\eta))^{2})$, $\\|x-x_{0}\\|_{2}\leq s_{N}^{}(\eta).$ 2\. If $\sigma\leq c_{3}r_{N}^{}(Q)$ then with probability at least $1-4\exp(-c_{4}NQ^{2})$, $\\|x-x_{0}\\|_{2}\leq r_{N}^{}(Q).$ Our main result is a phase retrieval version of Theorem 1.3. Theorem B. For every $L\geq 1$, $\kappa_{0}>0$ and $\beta$ there exist constants $c_{1},c_{2},c_{3}$, $c_{4},c_{5}$ and $Q$ that depend only on $L$ and $\kappa_{0}$ and $\beta$ for which the following holds. Let $T\subset\mathbb{R}^{d}$ be a convex, centrally-symmetric set, and let $a$ and $w$ be as in Theorem A. Assume that $(\sigma/\\|x_{0}\\|_{2})\geq c_{0}r_{N}^{}(Q)/\sqrt{\log N}$, set $\eta=c_{1}\\|x_{0}\\|_{2}/(\sigma\sqrt{\log N})$ and let $\zeta=c_{1}/(\sigma\sqrt{\log N})$. 1\. If $\\|x_{0}\\|_{2}\geq v_{N}^{}(c_{2})$, then with probability at least $1-2\exp(-c_{3}N\eta^{2}(s_{N}^{}(\eta))^{2})-2N^{-\beta+1}$, $\min\\{\\|x-x_{0}\\|_{2},\\|x+x_{0}\\|_{2}\\}\leq c_{4}s_{N}^{}(\eta).$ 2\. If $\\|x_{0}\\|_{2}\leq v_{N}^{}(c_{2})$ then with probability at least $1-2\exp(-c_{3}N\zeta^{2}(v_{N}^{}(\zeta))^{2})-2N^{-\beta+1}$, $\max\\{\\|x-x_{0}\\|_{2},\\|x+x_{0}\\|_{2}\\}\leq c_{4}v_{N}^{}(\zeta).$ If $(\sigma/\\|x_{0}\\|_{2})\geq c_{0}r_{N}^{}(Q)/\sqrt{\log N}$ the same assertion as in 1. and 2. holds, with an upper bound of $r_{N}^{}(Q)$ replacing $s_{N}^{}(\eta)$ and $v_{N}^{}(\zeta)$. Theorem B follows from Theorem A and a more transparent description of the localized sets $T_{-,R}(x_{0})$ and $T_{+,R}(x_{0})$ (see Lemma 4.1). To put Theorem B in some perspective, observe that $v_{N}^{}$ tends to zero. Indeed, since $\ell(T\cap rB_{2}^{n})\leq\ell(T)$, it follows that $v_{N}^{}(\zeta)\leq(\ell(T)/\sqrt{N}\zeta)^{1/3}$. Hence, for the choice of $\zeta\sim(\sigma\sqrt{\log N})^{-1}$ as in Theorem B, $v_{N}^{}\leq\left(\sigma\ell(T)\sqrt{\frac{\log N}{N}}\right)^{1/3},$ which tends to zero when $\sigma\to 0$ and when $N\to\infty$. Therefore, if $x_{0}\not=0$, the first part of Theorem B describes the ‘long term’ behaviour of ERM. Also, and using the same argument, $r_{N}^{}(Q)\leq\frac{\ell(T)}{Q\sqrt{N}}.$ Thus, for every $\sigma>0$ the problem becomes ‘high noise’ in the sense that the condition $(\sigma/\\|x_{0}\\|_{2})\geq c_{0}r_{N}(Q)/\sqrt{\log N}$ is satisfied when $N$ is large enough. In the typical situation, which is both ‘high noise’ and ‘large $\\|x_{0}\\|_{2}$’, the error rate depends on $\eta=c_{1}\\|x_{0}\\|_{2}/\sigma\sqrt{\log N}$. We believe that the $1/\sqrt{\log N}$ factor is an artifact of the proof, but the other term, $\\|x_{0}\\|_{2}/\sigma$ is the signal-to-noise ratio, and is rather natural. Although Theorem A and Theorem B clearly improve the results from [2], it is natural to ask whether these are optimal in a more general sense. The final result presented here is that Theorem B is close to being optimal in the minimax sense. The formulation and proof of the minimax lower bound is presented in Section 5. Finally, we end the article with two examples of classes that are of interest in phase retrieval: $d$-sparse vectors and the unit ball in $\ell_{1}^{n}$. The first is a class with a fixed ‘local complexity’, and the second has a growing ‘local complexity’. ## 2 Preliminaries Throughout this article, absolute constants are denoted by $C,c,c_{1},...$ etc. Their value may change from line to line. The fact that there are absolute constants $c,C$ for which $ca\leq b\leq Ca$ is denoted by $a\sim b$; $a\lesssim b$ means that $a\leq cb$, while $a\sim_{L}b$ means that the constants depend only on the parameter $L$. For $1\leq p\leq\infty$, let $\\|\cdot\\|_{p}$ be the $\ell_{p}^{n}$ norm endowed on $\mathbb{R}^{n}$, and for a function $f$ (or a random variable $X$) on a probability space, set $\\|f\\|_{L_{p}}$ to be its $L_{p}$ norm. Other norms that play a significant role here are the Orlicz norms. For basic facts on these norms we refer the reader to [9, 15]. Recall that for $\alpha\geq 1$, $\\|f\\|_{\psi_{\alpha}}=\inf\\{c>0:\mathbb{E}\exp(\|f\|^{\alpha}/c^{\alpha})\leq 2\\},$ and it is straightforward to extend the definition for $0<\alpha<1$. Orlicz norms measure the rate of decay of a function. One may verify that $\\|f\\|_{\psi_{\alpha}}\sim\sup_{p\geq 1}\\|f\\|_{L_{p}}/p^{1/\alpha}$. Moreover, for $t\geq 1$, $Pr(\|f\|\geq t)\leq 2\exp(-ct^{\alpha}/\\|f\\|_{\psi_{\alpha}}^{\alpha})$, and $\\|f\\|_{\psi_{\alpha}}$ is equivalent to the smallest constant $\kappa$ for which $Pr(\|f\|\geq t)\leq 2\exp(-t^{\alpha}/\kappa^{\alpha})$ for every $t\geq 1$. ###### Definition 2.1 A random variable is $L$-subgaussian if it has a bounded $\psi_{2}$ norm and $\\|X\\|_{\psi_{2}}\leq L\\|X\\|_{L_{2}}$. Observe that for $L$-subgaussian random variables, all the $L_{p}$ norms are equivalent and their tails exhibits a faster decay than the corresponding gaussian. Indeed, if $X$ is $L$-subgaussian, $\\|X\\|_{L_{p}}\lesssim\sqrt{p}\\|X\\|_{\psi_{2}}\lesssim L\sqrt{p}\\|X\\|_{L_{2}},$ and for every $t\geq 1$, $Pr(\|X\|>t)\leq 2\exp(-ct^{2}/\\|X\\|_{\psi_{2}}^{2})\leq 2\exp(-ct^{2}/(L^{2}\\|X\\|_{L_{2}}^{2}))$ for a suitable absolute constant $c$. It is standard to verify that for every $f,g$, $\\|fg\\|_{\psi_{1}}\lesssim\\|f\\|_{\psi_{2}}\\|g\\|_{\psi_{2}}$, and that if $X_{1},...,X_{N}$ are independent copies of $X$ and $1\leq\alpha\leq 2$, then $\\|\max_{1\leq i\leq N}X_{i}\\|_{\psi_{\alpha}}\lesssim\\|X\\|_{\psi_{\alpha}}\log^{1/\alpha}N.$ (2.1) An additional feature of $\psi_{\alpha}$ random variables is concentration, namely that if $(X_{i})_{i=1}^{N}$ are independent copies of a $\psi_{\alpha}$ random variable $X$, then $N^{-1}\sum_{i=1}^{N}X_{i}$ concentrates around $\mathbb{E}X$. One example of such a concentration result is the following Bernstein-type inequality (see, e.g., [15]). ###### Theorem 2.2 There exists an absolute constant $c_{0}$ for which the following holds. If $X_{1},...,X_{N}$ are independent copies of a $\psi_{1}$ random variable $X$, then for every $t>0$, $Pr\left(\left\|\frac{1}{N}\sum_{i=1}^{N}X_{i}-\mathbb{E}X\right\|>t\\|X\\|_{\psi_{1}}\right)\leq 2\exp(-c_{0}N\min\\{t^{2},t\\}).$ One important example of a probability space considered here is the discrete space $\Omega=\\{1,...,N\\}$, endowed with the uniform probability measure. Functions on $\Omega$ can be viewed as vectors in $\mathbb{R}^{N}$ and the corresponding $L_{p}$ and $\psi_{\alpha}$ norms are denoted by $\\|\cdot\\|_{L_{p}^{N}}$ and $\\|\cdot\\|_{\psi_{\alpha}^{N}}$. A significant part of the proofs of Theorem A has to do with the behaviour of a monotone non-increasing rearrangement of vectors. Given $v\in\mathbb{R}^{N}$, let $(v_{i}^{})_{i=1}^{N}$ be a non-increasing rearrangement of $(\|v_{i}\|)_{i=1}^{N}$. It turns out that the $\psi_{\alpha}^{N}$ norm captures information on the coordinates of $(v_{i}^{})_{i=1}^{N}$. ###### Lemma 2.3 For every $1\leq\alpha\leq 2$ there exist constants $c_{1}$ and $c_{2}$ that depend only on $\alpha$ for which the following holds. For every $v\in\mathbb{R}^{N}$, $c_{1}\sup_{i\leq N}\frac{v_{i}^{}}{\log^{1/\alpha}(eN/i)}\leq\\|v\\|_{\psi_{\alpha}^{N}}\leq c_{2}\sup_{i\leq N}\frac{v_{i}^{}}{\log^{1/\alpha}(eN/i)}.$ Proof. We will prove the claim only for $\alpha=2$ as the other cases follow an identical path. Let $v\in\mathbb{R}^{N}$ and denote by $Pr$ the uniform probability measure on $\Omega=\\{1,\ldots,N\\}$. By the tail characterization of the $\psi_{2}$ norm, $N^{-1}\|\\{j:\|v_{j}\|>t\\}\|=Pr(\|v\|>t)\leq 2\exp(-ct^{2}/\\|v\\|_{\psi_{2}^{N}}^{2}).$ Hence, for $t_{i}=c^{-1/2}\\|v\\|_{\psi_{2}^{N}}\sqrt{\log(eN/i)}$, $\|\\{j:\|v_{j}\|>t_{i}\\}\|\leq 2i/e\leq i$, and for every $1\leq i\leq N$, $v_{i}^{}\leq t_{i}$. Therefore, $\sup_{i\leq N}\frac{v_{i}^{}}{\sqrt{\log(eN/i)}}\leq c^{-1/2}\\|v\\|_{\psi_{2}^{N}},$ as claimed. In the reverse direction, consider ${\cal B}=\big{\\{}\beta>0:\forall\ 1\leq i\leq N,\ \\|v\\|_{\psi_{2}^{N}}\geq\beta v_{i}^{}/\sqrt{\log(eN/i)}\big{\\}}.$ It is enough to show that ${\cal B}$ is bounded by a constant that is independent of $v$. To that end, fix $\beta\in{\cal B}$ and without loss of generality, assume that $\beta>2$. Set $B=\sup_{i\leq N}\beta v_{i}^{}/\sqrt{\log(eN/i)}$ and since $\beta\in{\cal B}$, $\\|v\\|_{\psi_{2}^{N}}\geq B$. Also, since $1/\beta^{2}<1$, $\sum_{i=1}^{N}\left(\frac{1}{i}\right)^{1/\beta^{2}}\leq 1+\int_{1}^{N}\left(\frac{1}{x}\right)^{1/\beta^{2}}dx\leq\frac{N^{1-1/\beta^{2}}}{1-1/\beta^{2}}.$ Therefore, $\displaystyle\sum_{i=1}^{N}\exp(v_{i}^{2}/B^{2})=\sum_{i=1}^{N}\exp((v_{i}^{})^{2}/B^{2})\leq\sum_{i=1}^{N}\exp(\beta^{-2}\log(eN/i))$ $\displaystyle\leq$ $\displaystyle\sum_{i=1}^{N}\left(\frac{eN}{i}\right)^{1/\beta^{2}}\leq(eN)^{1/\beta^{2}}\cdot\frac{N^{1-1/\beta^{2}}}{1-1/\beta^{2}}\leq\frac{Ne^{1/\beta^{2}}}{1-1/\beta^{2}}<2N,$ provided that $\beta\geq c_{1}$. Thus, if $\beta\geq c_{1}$, $\\|v\\|_{\psi_{2}^{N}}<B$ which is a contradiction, showing that ${\cal B}$ is bounded by $c_{1}$. ### 2.1 Empirical and Subgaussian processes The sampling method used here is isotropic and $L$-subgaussian, meaning that the vectors $(a_{i})_{i=1}^{N}$ are independent and distributed according to a probability measure $\mu$ on $\mathbb{R}^{n}$ that is both isotropic and $L$-subgaussian [15]: ###### Definition 2.4 Let $\mu$ be a probability measure on $\mathbb{R}^{n}$ and let $a$ be distributed according to $\mu$. The measure $\mu$ is isotropic if for every $t\in\mathbb{R}^{n}$, $\mathbb{E}\bigl{<}a,t\bigr{>}^{2}=\\|t\\|_{2}^{2}$. It is $L$-subgaussian if for every $t\in\mathbb{R}^{n}$ and every $u\geq 1$, $Pr(\|\bigl{<}a,t\bigr{>}\|\geq Lu\\|\bigl{<}t,a\bigr{>}\\|_{2})\leq 2\exp(-u^{2}/2)$. Given $T\subset\mathbb{R}^{n}$, let $d_{T}=\sup_{t\in T}\\|t\\|_{2}$ and put $k_{}(T)=(\ell(T)/d_{T})^{2}$. The latter appears naturally in the context of Dvoretzky type theorems, and in particular, in Milman’s proof of Dvoretzky’s Theorem (see, e.g., [11]). ###### Theorem 2.5 [10] For every $L\geq 1$ there exist constants $c_{1}$ and $c_{2}$ that depend only on $L$ and for which the following holds. For every $u\geq c_{1}$, with probability at least $1-2\exp(-c_{2}u^{2}k_{}(T))$, for every $t\in T$ and every $I\subset\\{1,...,N\\}$, $\left(\sum_{i\in I}\bigl{<}t,a_{i}\bigr{>}^{2}\right)^{1/2}\leq Lu^{3}\left(\ell(T)+d_{T}\sqrt{\|I\|\log(eN/\|I\|)}\right).$ For every integer $N$, let $j_{T}$ be the largest integer $j$ in $\\{1,...,N\\}$ for which $\ell(T)\geq d_{T}\sqrt{j\log(eN/j)}.$ It follows from Theorem 2.5 that if $t\in T$ and $\|I\|\leq j_{T}$, $(\sum_{i\in I}\bigl{<}t,a_{i}\bigr{>}^{2})^{1/2}\lesssim_{L,u}\ell(T),$ and if $\|I\|\geq j_{T}$, $(\sum_{i\in I}\bigl{<}t,a_{i}\bigr{>}^{2})^{1/2}\lesssim_{L,u}d_{T}\sqrt{\|I\|\log(eN/\|I\|)}.$ Therefore, if $v=(\bigl{<}t,a_{i}\bigr{>})_{i=1}^{N}$ and $(v_{i}^{})_{i=1}^{N}$ is a monotone non-increasing rearrangement of $(\|v_{i}\|)_{i=1}^{N}$, then $v_{i}^{}\leq\left(\frac{1}{i}\sum_{j=1}^{i}(v_{j}^{})^{2}\right)^{1/2}\lesssim_{L,u}\left\\{\begin{array}[]{cc}\frac{\ell(T)}{\sqrt{i}}&\mbox{if}\ \ i\leq j_{T}\\\ &\\\ d_{T}\sqrt{\log(eN/i)}&\mbox{otherwise}.\end{array}\right.$ (2.2) This observation will be used extensively in what follows. The next fact deals with product processes. ###### Theorem 2.6 [10] There exist absolute constants $c_{0},c_{1}$ and $c_{2}$ for which the following holds. If $T_{1},T_{2}\subset\mathbb{R}^{n}$, $1\leq 2^{j}\leq N$ and $u\geq c_{0}$, then with probability at least $1-2\exp(-c_{1}u^{2}2^{j})$, $\displaystyle\sup_{t\in T_{1},\ s\in T_{2}}\left\|\sum_{i=1}^{N}\bigl{<}a_{i},t\bigr{>}\bigl{<}a_{i},s\bigr{>}-\mathbb{E}\bigl{<}a,t\bigr{>}\bigl{<}a,s\bigr{>}\right\|$ $\displaystyle\leq$ $\displaystyle c_{2}L^{2}u^{2}\left(\ell(T_{1})\ell(T_{2})+u\sqrt{N}\left(\ell(T_{1})d(T_{2})+\ell(T_{2})d(T_{1})+2^{j/2}d(T_{1})d(T_{2})\right)\right)$ and $\displaystyle\sup_{t\in T_{1},\ s\in T_{2}}\left\|\sum_{i=1}^{N}\|\bigl{<}a_{i},t\bigr{>}\bigl{<}a_{i},s\bigr{>}\|-\mathbb{E}\|\bigl{<}a,t\bigr{>}\bigl{<}a,s\bigr{>}\|\right\|$ $\displaystyle\leq$ $\displaystyle c_{2}L^{2}u^{2}\left(\ell(T_{1})\ell(T_{2})+u\sqrt{N}\left(\ell(T_{1})d(T_{2})+\ell(T_{2})d(T_{1})+2^{j/2}d(T_{1})d(T_{2})\right)\right).$ ###### Remark 2.7 Let $(\varepsilon_{i})_{i=1}^{N}$ be independent, symmetric, $\\{-1,1\\}$-valued random variables. It follows from the results in [10] that with the same probability estimate in Theorem 2.6 and relative to the product measure $(\varepsilon\otimes X)^{N}$, $\displaystyle\sup_{t\in T_{1},\ s\in T_{2}}\left\|\sum_{i=1}^{N}\varepsilon_{i}\bigl{<}a_{i},t\bigr{>}\bigl{<}a_{i},s\bigr{>}\right\|$ $\displaystyle\lesssim$ $\displaystyle L^{2}u^{2}\left(\ell(T_{1})\ell(T_{2})+u\sqrt{N}\left(\ell(T_{1})d(T_{2})+\ell(T_{2})d(T_{1})+2^{j/2}d(T_{1})d(T_{2})\right)\right).$ Assume that $(k^{}(T_{1}))^{1/2}=\ell(T_{1})/d(T_{1})\geq\ell(T_{2})/d(T_{2})$. Setting $2^{j/2}=\ell(T_{1})/d(T_{1})$, Theorem 2.6 and Remark 2.7 yield that with probability at least $1-2\exp(-c_{1}u^{2}k_{}(T_{1}))$, $\sup_{t\in T_{1},\ s\in T_{2}}\left\|\sum_{i=1}^{N}\bigl{<}a_{i},t\bigr{>}\bigl{<}a_{i},s\bigr{>}-\mathbb{E}\bigl{<}a,t\bigr{>}\bigl{<}a,s\bigr{>}\right\|\lesssim L^{2}u^{2}\ell(T_{1})\left(\ell(T_{2})+u\sqrt{N}d(T_{2})\right),$ $\sup_{t\in T_{1},\ s\in T_{2}}\left\|\sum_{i=1}^{N}\|\bigl{<}a_{i},t\bigr{>}\bigl{<}a_{i},s\bigr{>}\|-\mathbb{E}\|\bigl{<}a,t\bigr{>}\bigl{<}a,s\bigr{>}\|\right\|\lesssim L^{2}u^{2}\ell(T_{1})\left(\ell(T_{2})+u\sqrt{N}d(T_{2})\right)$ and $\sup_{t\in T_{1},\ s\in T_{2}}\left\|\sum_{i=1}^{N}\varepsilon_{i}\bigl{<}a_{i},t\bigr{>}\bigl{<}a_{i},s\bigr{>}\right\|\lesssim L^{2}u^{2}\ell(T_{1})\left(\ell(T_{2})+u\sqrt{N}d(T_{2})\right).$ (2.3) One case which is of particular interest is when $T_{1}=T_{2}=T$, and then, with probability at least $1-2\exp(-c_{1}u^{2}k_{}(T))$, $\sup_{t\in T}\left\|\sum_{i=1}^{N}\bigl{<}a_{i},t\bigr{>}^{2}-\mathbb{E}\bigl{<}a,t\bigr{>}^{2}\right\|\lesssim L^{2}u^{2}\left(\ell^{2}(T)+u\sqrt{N}d(T)\ell(T)\right).$ ### 2.2 Monotone rearrangement of coordinates The first goal of this section is to investigate the coordinate structure of $v\in\mathbb{R}^{m}$, given information on its norm in various $L_{p}^{m}$ and $\psi_{\alpha}^{m}$ spaces. The vectors we will be interested in are of the form $(\bigl{<}a_{i},t\bigr{>})_{i=1}^{N}$ for $t\in T$, and for which, thanks to the results presented in Section 2.1, one has the necessary information at hand. It is standard to verify that if $\\|v\\|_{\psi_{\alpha}^{m}}\leq A$, then $\\|v\\|_{p}\lesssim_{p,\alpha}A\cdot m^{1/p}$. Thus, $\\|v\\|_{L_{p}^{m}}\lesssim_{p}\\|v\\|_{\psi_{\alpha}^{m}}$. It turns out that if the two norms are equivalent, $v$ is regular in some sense. The next lemma, which is a version of the Paley-Zygmund Inequality, (see, e.g. [4]), describes the regularity properties needed here in the case $p=\alpha=1$. ###### Lemma 2.8 For every $\beta>1$ there exist constants $c_{1}$ and $c_{2}$ that depend only on $\beta$ and for which the following holds. If $\\|v\\|_{\psi_{1}^{m}}\leq\beta\\|v\\|_{L_{1}^{m}}$, there exists $I\subset\\{1,...,m\\}$ of cardinality at least $c_{1}m$, and for every $i\in I$, $\|v_{i}\|\geq c_{2}\\|v\\|_{L_{1}^{m}}$. Proof. Recall that $\\|v\\|_{\psi_{1}^{m}}\sim\sup_{1\leq i\leq m}v_{i}^{}/\log(em/i)$. Hence, for every $1\leq j\leq m$, $\sum_{\ell=1}^{j}v_{\ell}^{}\lesssim\\|v\\|_{\psi_{1}^{m}}\sum_{\ell=1}^{j}\log(em/\ell)\lesssim\beta\\|v\\|_{L_{1}^{m}}j\log(em/j).$ Therefore, $m\\|v\\|_{L_{1}^{m}}=\sum_{\ell=1}^{m}\|v_{\ell}\|=\sum_{\ell\leq j}v_{\ell}^{}+\sum_{\ell=j+1}^{m}v_{\ell}^{}\leq c_{0}\beta\\|v\\|_{L_{1}^{m}}j\log(em/j)+\sum_{\ell=j+1}^{m}v_{\ell}^{}.$ Setting $c_{1}(\beta)\sim 1/(\beta\log(e\beta))$ and $j=c_{1}(\beta)m$, $c_{0}\beta\\|v\\|_{L_{1}^{m}}j\log(em/j)\leq(m/2)\\|v\\|_{L_{1}^{m}}.$ Thus, $\sum_{\ell=j+1}^{m}v_{\ell}^{}\geq(m/2)\\|v\\|_{L_{1}^{m}}$, while $v_{j+1}^{}\leq\frac{1}{j+1}\sum_{\ell\leq j+1}v_{\ell}^{}\lesssim\beta\log(e\beta)\\|v\\|_{L_{1}^{m}}.$ Let $I$ be the set of the $m-j$ smallest coordinates of $v$. Fix $\eta>0$ to be named later, put $I_{\eta}\subset I$ to be the set of coordinates in $I$ for which $\|v_{i}\|\geq\eta\\|v\\|_{L_{1}^{m}}$ and denote by $I_{\eta}^{c}$ its complement in $I$. Therefore, $\displaystyle(m/2)\\|v\\|_{L_{1}^{m}}\leq$ $\displaystyle\sum_{\ell\geq j+1}v_{\ell}^{}=\sum_{\ell\in I_{\eta}}\|v_{\ell}\|+\sum_{\ell\in I_{\eta}^{c}}\|v_{\ell}\|\leq v_{j+1}^{}\|I_{\eta}\|+\eta\\|v\\|_{L_{1}^{m}}\|I_{\eta}^{c}\|$ $\displaystyle\lesssim$ $\displaystyle\\|v\\|_{L_{1}^{m}}\|I\|\left(\beta\log(e\beta)\frac{\|I_{\eta}\|}{\|I\|}+\eta\frac{\|I_{\eta}^{c}\|}{\|I\|}\right).$ Hence, $\frac{m}{2}\lesssim\|I\|\left(\beta\log(e\beta)\frac{\|I_{\eta}\|}{\|I\|}+\eta\left(1-\frac{\|I_{\eta}\|}{\|I\|}\right)\right)\lesssim m\left(\left(\beta\log(e\beta)-\eta\right)\frac{\|I_{\eta}\|}{\|I\|}+\eta\right).$ If $\eta=\min\\{1/4,(\beta/2)\log(e\beta)\\}$, then $\|I_{\eta}\|\geq(\eta/2)\|I\|\geq c_{2}(\beta)m$, as claimed. Next, let us turn to decomposition results for vectors of the form $(\bigl{<}a_{i},t\bigr{>})_{i=1}^{N}$. Recall that for a set $T\subset\mathbb{R}^{N}$, $j_{T}$ is the largest integer for which $\ell(T)\geq d_{T}\sqrt{j\log(eN/j)}$. ###### Lemma 2.9 For every $L>1$ there exist constants $c_{1}$ and $c_{2}$ that depend only on $L$ and for which the following holds. Let $T\subset\mathbb{R}^{n}$ and set $W=\\{t/\\|t\\|_{2}:t\in T\\}\subset S^{n-1}$. With probability at least $1-2\exp(-c_{1}\ell^{2}(W))$, for every $t\in T$, $(\bigl{<}a_{i},t\bigr{>})_{i=1}^{N}=v_{1}+v_{2}$ and $v_{1},v_{2}$ have the following properties: 1\. The supports of $v_{1}$ and $v_{2}$ are disjoint. 2\. $\\|v_{1}\\|_{2}\leq c_{2}\ell(W)\\|t\\|_{2}$ and $\|{\rm supp}(v_{1})\|\leq j_{W}$. 3\. $\\|v_{2}\\|_{\psi_{2}^{N}}\leq c_{2}\\|t\\|_{2}$. Proof. Fix $t\in T$ and let $J_{t}\subset\\{1,...,N\\}$ be the set of the largest $j_{W}$ coordinates of $(\|\bigl{<}a_{i},t\bigr{>}\|)_{i=1}^{N}$. Set ${\bar{v}}_{1}=(\bigl{<}a_{j},t/\\|t\\|_{2}\bigr{>})_{j\in J_{t}}\ \ {\rm and}\ \ {\bar{v}}_{2}=(\bigl{<}a_{j},t/\\|t\\|_{2}\bigr{>})_{j\in J_{t}^{c}}.$ By Theorem 2.5 and the characterization of the $\psi_{2}^{N}$ norm of a vector using the monotone rearrangement of its coordinates (Lemma 2.3), $\\|\bar{v}_{1}\\|_{2}\lesssim L\ell(W),\ \ {\rm and}\ \ \\|\bar{v}_{2}\\|_{\psi_{2}^{N}}\lesssim L.$ To conclude the proof, set $v_{1}=\\|t\\|_{2}\bar{v}_{1}$ and $v_{2}=\\|t\\|_{2}{\bar{v}}_{2}$. Recall that for every $R>0$, $T_{+,R}=\left\\{\frac{t+s}{\\|t+s\\|_{2}}:\ t,s\in T,\ \\|t+s\\|_{2}\\|t-s\\|_{2}\geq R\right\\},$ and a similar definition holds for $T_{-,R}$. Set $j_{+,R}=j_{T_{+,R}}$, $j_{-,R}=j_{T_{+,R}}$ and $E_{R}=\max\\{\ell(T_{+,R}),\ell(T_{-,R})\\}$. Combining the above estimates leads to the following corollary. ###### Corollary 2.10 For every $L>1$ there exist constants $c_{1},c_{2},c_{3}$ and $c_{4}$ that depend only on $L$ for which the following holds. Let $T\subset\mathbb{R}^{n}$ and $R>0$, and consider $T_{+,R}$ and $T_{-,R}$ as above. With probability at least $1-4\exp(-c_{1}L^{2}\min\\{\ell^{2}(T_{+,R}),\ell^{2}(T_{-,R})\\})$, for every $s,t\in T$ for which $\\|t-s\\|_{2}\\|t+s\\|_{2}\geq R$, 1\. $(\bigl{<}s-t,a_{i}\bigr{>})_{i=1}^{N}=v_{1}+v_{2}$, for vectors $v_{1}$ and $v_{2}$ of disjoint supports satisfying $\|{\rm supp}(v_{1})\|\leq j_{-,R},\ \ \\|v_{1}\\|_{2}\leq c_{2}\ell(T_{-,R})\\|s-t\\|_{2}\ \ {\rm and}\ \ \\|v_{2}\\|_{\psi_{2}^{N}}\leq c_{2}\\|s-t\\|_{2}.$ 2\. $(\bigl{<}s+t,a_{i}\bigr{>})_{i=1}^{N}=u_{1}+u_{2}$, for vectors $u_{1}$ and $u_{2}$ of disjoint supports satisfying $\|{\rm supp}(u_{1})\|\leq j_{+,R},\ \ \\|u_{1}\\|_{2}\leq c_{2}\ell(T_{+,R})\\|s+t\\|_{2}\ \ {\rm and}\ \ \\|u_{2}\\|_{\psi_{2}^{N}}\leq c_{2}\\|s+t\\|_{2}.$ 3\. If $h_{s,t}(a)=\bigl{<}\frac{s+t}{\\|s+t\\|_{2}},a\bigr{>}\bigl{<}\frac{s-t}{\\|s-t\\|_{2}},a\bigr{>}$, then $\left\|\frac{1}{N}\sum_{i=1}^{N}\|h_{s,t}(a_{i})\|-\mathbb{E}\|h_{s,t}\|\right\|\leq c_{3}\left(\frac{E_{R}}{\sqrt{N}}+\frac{E_{R}^{2}}{N}\right).$ In particular, recalling that for every $s,t\in T$, $\mathbb{E}\|\bigl{<}s+t,a\bigr{>}\bigl{<}s-t,a\bigr{>}\|\geq\kappa_{0}{\\|s+t\\|_{2}\\|s-t\\|_{2}},$ it follows that if $\sqrt{N}\geq c_{4}(L)E_{R}/\kappa_{0}$ then 4. $\frac{\kappa_{0}}{2}\\|s+t\\|_{2}\\|s-t\\|_{2}\leq\frac{1}{N}\sum_{i=1}^{N}\|\bigl{<}s+t,a_{i}\bigr{>}\bigl{<}s-t,a_{i}\bigr{>}\|\lesssim_{L}\\|s+t\\|_{2}\\|s-t\\|_{2}.$ (2.4) From here on, denote by $\Omega_{1,R}$ the event on which Corollary 2.10 holds for the sets $T_{+,R}$ and $T_{-,R}$ and samples of cardinality $N\gtrsim_{L}E_{R}^{2}/\kappa_{0}^{2}$. ###### Lemma 2.11 There exist constants $c_{0}$ depending only on $L$ and $c_{1},\kappa_{1}$ that depend only on $\kappa_{0}$ and $L$ for which the following holds. If $N\geq c_{0}E_{R}^{2}/\kappa_{0}^{2}$, then for $(a_{i})_{i=1}^{N}\in\Omega_{1,R}$, for every $s,t\in T$ for which $\\|s-t\\|_{2}\\|s+t\\|_{2}\geq R$, there is $I_{s,t}\subset\\{1,...,N\\}$ of cardinality at least $\kappa_{1}N$, and for every $i\in I_{s,t}$, $\|\bigl{<}s-t,a_{i}\bigr{>}\bigl{<}s+t,a_{i}\bigr{>}\|\geq c_{1}\\|s-t\\|_{2}\\|s+t\\|_{2}.$ Lemma 2.11 is an empirical “small-ball” estimate, as it shows that with high probability, and for every pair $s,t$ as above, many of the coordinates of $(\|\bigl{<}a_{i},s-t\bigr{>}\|\cdot\|\bigl{<}a_{i},s+t\bigr{>}\|)_{i=1}^{N}$ are large. Proof. Fix $s,t\in T$ as above and set $y=(\bigl{<}s-t,a_{i}\bigr{>})_{i=1}^{N},\ \ {\rm and}\ \ x=(\bigl{<}s+t,a_{i}\bigr{>})_{i=1}^{N}.$ Let $y=v_{1}+v_{2}$ and $x=u_{1}+u_{2}$ as in Corollary 2.10. Let $j_{0}=\max\\{j_{-,R},j_{+,R}\\}$ and put $J={\rm supp}(v_{1})\cup{\rm supp}(u_{1})$. Observe that $\|J\|\leq 2j_{0}$ and that $\displaystyle\sum_{j\in J}\|y(j)\|\cdot\|x(j)\|\leq\sum_{j\in{\rm supp}(v_{1})}\|v_{1}(j)x(j)\|+\sum_{j\in{\rm supp}(u_{1})}\|y(j)u_{1}(j)\|$ $\displaystyle\leq$ $\displaystyle\\|v_{1}\\|_{2}\left(\sum_{i=1}^{2j_{0}}(x^{2}(j))^{}\right)^{1/2}+\\|u_{1}\\|_{2}\left(\sum_{i=1}^{2j_{0}}(y^{2}(j))^{}\right)^{1/2}$ $\displaystyle\lesssim_{L}$ $\displaystyle\ell(T_{-,R})\\|s-t\\|_{2}\cdot\sqrt{j_{0}\log(eN/j_{0})}\\|s+t\\|_{2}$ $\displaystyle+$ $\displaystyle\ell(T_{+,R})\\|s+t\\|_{2}\cdot\sqrt{j_{0}\log(eN/j_{0})}\\|s-t\\|_{2}$ $\displaystyle\lesssim_{L}$ $\displaystyle E_{R}^{2}\\|s-t\\|_{2}\\|s+t\\|_{2}\leq\frac{\kappa_{0}N}{4}\\|s-t\\|_{2}\\|s+t\\|_{2},$ because, by the definition of $j_{0}$, $\sqrt{j_{0}\log(eN/j_{0})}\lesssim\max\\{\ell(T_{-,R}),\ell(T_{+,R})\\}$ and since $N\geq c_{0}E_{R}^{2}/\kappa_{0}^{2}$ for $c_{0}=c_{0}(L)$ large enough. Thus, by (2.4), $\sum_{j\in J^{c}}\|y(j)x(j)\|\geq N\kappa_{0}\\|s-t\\|_{2}\\|s+t\\|_{2}/4.$ Set $m=\|J^{c}\|$ and let $z=(y(j)x(j))_{j\in J^{c}}=(v_{2}(j)u_{2}(j))_{j\in J^{c}}$. Since $N\gtrsim_{L}E_{R}^{2}/\kappa_{0}^{2}$, it is evident that $j_{0}\leq N/2$; thus $N/2\leq m\leq N$ and $\\|z\\|_{L_{1}^{m}}=\frac{1}{m}\sum_{j\in J^{c}}\|y(j)x(j)\|\geq\frac{N}{4m}\kappa_{0}\\|s-t\\|_{2}\\|s+t\\|_{2}\gtrsim\kappa_{0}\\|s-t\\|_{2}\\|s+t\\|_{2}.$ On the other hand, $\\|z\\|_{\psi_{1}^{m}}\leq\\|(v_{2}u_{2}(j))_{j\in J_{c}}\\|_{\psi_{1}^{m}}\lesssim\\|v_{2}\\|_{\psi_{2}^{m}}\\|u_{2}\\|_{\psi_{2}^{m}}\lesssim_{L}\\|s-t\\|_{2}\\|s+t\\|_{2},$ and $z$ satisfies the assumption of Lemma 2.8 for $\beta=c_{1}(L,\kappa_{0})$. The claim follows immediately from that lemma. ## 3 Proof of Theorem A It is well understood that when analyzing properties of ERM relative to a loss $\ell$, studying the excess loss functional is rather natural. The excess loss shares the same empirical minimizer as the loss, but it has additional qualities: for every $x\in T$, $\mathbb{E}{\cal L}_{x}\geq 0$ and ${\cal L}_{x_{0}}=0$. Since $0$ is a potential minimizer of $\\{P_{N}{\cal L}_{x}:x\in T\\}$, the minimizer $\hat{x}$ satisfies that $P_{N}{\cal L}_{\hat{x}}\leq 0$, giving one a way of excluding parts of $T$ as potential empirical minimizers. One simply has to show that with high probability, those parts belong to the set $\\{x:P_{N}{\cal L}_{x}>0\\}$, for example, by showing that $P_{N}{\cal L}_{x}$ is equivalent to $\mathbb{E}{\cal L}_{x}$, as the latter is positive for points that are not true minimizers. The squared excess loss has a simple decomposition to two processes: a quadratic process and a multiplier one. Indeed, given a class of functions $F$ and $f\in F$, $\big{(}f(a)-y\big{)}^{2}-\big{(}f^{}(a)-y\big{)}^{2}=\big{(}f(a)-f^{}(a)\big{)}^{2}-2\big{(}f(a)-f^{}(a)\big{)}\big{(}f^{}(a)-y\big{)}.$ where, as always, $f^{}$ is a minimizer of the functional $\mathbb{E}\big{(}f(a)-y\big{)}^{2}$ in $F$. In the phase retrieval problem, $y=\bigl{<}x_{0},a\bigr{>}^{2}+w$ for a noise $w$ that is independent of $a$, and each $f_{x}\in F$ is given by $f_{x}=\bigl{<}x,\cdot\bigr{>}^{2}$. Thus, $\displaystyle{\cal L}_{x}(a,y)$ $\displaystyle=\ell_{x}(a,y)-\ell_{x_{0}}(a,y)=\big{(}f_{x}(a)-y\big{)}^{2}-\big{(}f_{x_{0}}(a)-y\big{)}^{2}$ $\displaystyle=\left(\bigl{<}x-x_{0},a\bigr{>}\bigl{<}x+x_{0},a\bigr{>}\right)^{2}-2w\bigl{<}x-x_{0},a\bigr{>}\bigl{<}x+x_{0},a\bigr{>}.$ Since $w$ is a mean-zero random variable that is independent of $a$, and by Assumption 1.1, $\mathbb{E}{\cal L}_{x}(a,y)=\mathbb{E}\|\bigl{<}x-x_{0},a\bigr{>}\bigl{<}x+x_{0},a\bigr{>}\|\geq\kappa_{0}^{2}\\|x-x_{0}\\|_{2}^{2}\\|x+x_{0}\\|_{2}^{2}.$ Therefore, $\mathbb{E}\big{(}f_{x}(a)-y\big{)}^{2}$ has a unique minimizer in $F$: $f^{}=f_{x_{0}}=f_{-x_{0}}$. To show that $P_{N}{\cal L}_{x}>0$ on a large subset $T^{\prime}\subset T$, it suffices to obtain a high probability lower bound on $\inf_{x\in T^{\prime}}\frac{1}{N}\sum_{i=1}^{N}\left(\bigl{<}x-x_{0},a_{i}\bigr{>}\bigl{<}x+x_{0},a_{i}\bigr{>}\right)^{2}$ that dominates a high probability upper bound on $\sup_{x\in T^{\prime}}\left\|\frac{2}{N}\sum_{i=1}^{N}w_{i}\bigl{<}x-x_{0},a_{i}\bigr{>}\bigl{<}x+x_{0},a_{i}\bigr{>}\right\|.$ The set $T^{\prime}$ that will be used is $T_{R}=\\{x\in T:\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2}\geq R\\}$ for a well-chosen $R$. ###### Theorem 3.1 There exist a constant $c_{0}$ depending only on $L$, and constants $c_{1},\kappa_{1}$ depending only on $\kappa_{0}$ and $L$ for which the following holds. For every $R>0$ and $N\geq c_{0}E_{R}^{2}/\kappa_{0}^{2}$, with probability at least $1-4\exp(-c_{1}L^{2}\min\\{\ell^{2}(T_{+,R}),\ell^{2}(T_{-,R})\\}),$ for every $x\in T_{R}$, $\frac{1}{N}\sum_{i=1}^{N}\bigl{<}x_{0}-x,a_{i}\bigr{>}^{2}\bigl{<}x_{0}+x,a_{i}\bigr{>}^{2}\geq c_{1}\\|x_{0}-x\\|_{2}^{2}\\|x_{0}+x\\|_{2}^{2}.$ Theorem 3.1 is an immediate outcome of Lemma 2.11 ###### Theorem 3.2 There exist absolute constants $c_{1}$ and $c_{2}$ for which the following holds. For every $\beta>1$, with probability at least $1-2\exp(-c_{1}L^{2}\min\\{\ell^{2}(T_{+,R}),\ell^{2}(T_{-,R})\\})-2N^{-(\beta-1)},$ for every $x\in T_{R}$, $\left\|\frac{1}{N}\sum_{i=1}^{N}w_{i}\bigl{<}x-x_{0},a_{i}\bigr{>}\bigl{<}x+x_{0},a_{i}\bigr{>}\right\|\leq c_{2}\sqrt{\beta}\\|w\\|_{\psi_{2}}\sqrt{\log{N}}\cdot\frac{E_{R}}{\sqrt{N}}\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2}.$ Proof. By standard properties of empirical process, and since $w$ is mean-zero and independent of $a$, it suffices to estimate $\sup_{x\in T_{R}}\left\|\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{i}\|w_{i}\|\bigl{<}x-x_{0},a_{i}\bigr{>}\bigl{<}x+x_{0},a_{i}\bigr{>}\right\|,$ for independent signs $(\varepsilon_{i})_{i=1}^{N}$. By the contraction principle for Bernoulli processes (see, e.g., [9]), it follows that for every fixed $(w_{i})_{i=1}^{N}$ and $(a_{i})_{i=1}^{N}$, $\displaystyle Pr_{\varepsilon}\left(\sup_{x\in T_{R}}\left\|\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{i}\|w_{i}\|\bigl{<}\frac{x-x_{0}}{\\|x-x_{0}\\|_{2}},a_{i}\bigr{>}\bigl{<}\frac{x+x_{0}}{\\|x+x_{0}\\|_{2}},a_{i}\bigr{>}\right\|>u\right)$ $\displaystyle\leq$ $\displaystyle 2Pr_{\varepsilon}\left(\max_{i\leq N}\|w_{i}\|\cdot\sup_{x\in T_{R}}\left\|\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{i}\bigl{<}\frac{x-x_{0}}{\\|x-x_{0}\\|_{2}},a_{i}\bigr{>}\bigl{<}\frac{x+x_{0}}{\\|x+x_{0}\\|_{2}},a_{i}\bigr{>}\right\|>\frac{u}{2}\right).$ Applying Remark 2.7, if $N\gtrsim_{L}E_{R}$ then with $(\varepsilon\otimes a)^{N}$-probability of at least $1-2\exp(-c_{1}L^{2}\min\\{\ell^{2}(T_{+,R}),\ell^{2}(T_{-,R})\\})$, $\sup_{x\in T_{R}}\left\|\frac{1}{N}\sum_{i=1}^{N}\varepsilon_{i}\bigl{<}\frac{x-x_{0}}{\\|x-x_{0}\\|_{2}},a_{i}\bigr{>}\bigl{<}\frac{x+x_{0}}{\\|x+x_{0}\\|_{2}},a_{i}\bigr{>}\right\|\leq c_{2}L^{2}\frac{E_{R}}{\sqrt{N}}.$ Also, because $w$ is a $\psi_{2}$ random variable, $Pr(w_{1}^{}\geq t\\|w\\|_{\psi_{2}})\leq 2N\exp(-t^{2}/2),$ and thus, $w_{1}^{}\leq\sqrt{2\beta\log N}\\|w\\|_{\psi_{2}}$ with probability at least $1-2N^{-\beta+1}$. Combining the two estimates and a Fubini argument, it follows that with probability at least $1-2\exp(-c_{1}L^{2}\min\\{\ell^{2}(T_{+,R}),\ell^{2}(T_{-,R})\\})-2N^{-\beta+1}$, for every $x\in T_{R}$, $\left\|\frac{1}{N}\sum_{i=1}^{N}w_{i}\bigl{<}x-x_{0},a_{i}\bigr{>}\bigl{<}x+x_{0},a\bigr{>}\right\|\leq c_{3}L^{2}\sqrt{\beta}\\|w\\|_{\psi_{2}}\sqrt{\log{N}}\frac{E_{R}}{\sqrt{N}}\cdot\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2}.$ On the intersection of the two events appearing in Theorem 3.1 and Theorem 3.2, if $N\gtrsim_{\kappa_{0},L}E_{R}^{2}$ and setting $\rho=\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2}\geq R\geq r_{2}(c_{1}\kappa_{0}/(c_{2}L^{2}\sqrt{\beta})),$ then for every $x\in T_{R}$, $\displaystyle P_{N}{\cal L}_{x}\geq$ $\displaystyle\left(c_{1}\kappa_{0}^{2}\rho- c_{2}L^{2}\sqrt{\beta}\\|w\\|_{\psi_{2}}\sqrt{\log{N}}\frac{E_{R}}{\sqrt{N}}\right)\rho$ $\displaystyle\geq$ $\displaystyle\left(c_{1}\kappa_{0}^{2}R-c_{2}L^{2}\sqrt{\beta}\\|w\\|_{\psi_{2}}\sqrt{\log{N}}\frac{E_{R}}{\sqrt{N}}\right)R.$ Therefore, if $N\gtrsim_{L,\kappa_{0}}E_{R}^{2}$ and $E_{R}\leq c_{3}(L,\kappa_{0})\frac{R}{\\|w\\|_{\psi_{2}}}\sqrt{\frac{N}{\beta\log N}},$ (3.1) then $P_{N}{\cal L}_{x}>0$ and $\hat{x}\not\in T_{R}$. Theorem A follows from the definition of $r_{2}(\gamma)$ for a well chosen $\gamma$. ## 4 Proof of Theorem B Most of the work required for the proof of Theorem B has been carried out in Section 3. A literally identical argument, in which one replaces the sets $T_{+,R}$ and $T_{-,R}$ with $T_{+,R}(x_{0})$ and $T_{-,R}(x_{0})$ may be used, leading to an analogous version of Theorem A, with the obvious modifications: the complexity parameter is $\max\\{\ell(T_{+,R}(x_{0})),\ell(T_{-,R}(x_{0}))\\}$ for the right choice of $R$, and the probability estimate is $1-2\exp(-c_{0}\min\\{\ell^{2}(T_{+,R}(x_{0})),\ell^{2}(T_{-,R}(x_{0}))\\})-N^{-\beta+1}$. All that remains to complete the proof of Theorem B is to analyze the structure of the local sets and identify the fixed points $r_{0}$ and $r_{2}$. A first step in that direction is the following: ###### Lemma 4.1 There exist absolute constants $c_{1}$ and $c_{2}$ for which the following holds. For every $R>0$ and $\\|x_{0}\\|_{2}\geq\sqrt{R}/4$, 1\. If $\\|x_{0}\\|_{2}\min\\{\\|x-x_{0}\\|_{2},\\|x+x_{0}\\|_{2}\\}\geq R$ then $\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2}\geq c_{1}R$. 2\. If $\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2}\geq R$ then $\\|x_{0}\\|_{2}\min\\{\\|x-x_{0}\\|_{2},\\|x+x_{0}\\|_{2}\\}\geq c_{2}R$. Moreover, if $\\|x_{0}\\|_{2}\leq\sqrt{R}/4$ then $\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2}\geq R$ if and only if $\\|x\\|_{2}\gtrsim\sqrt{R}$. Proof. Assume without loss of generality that $\\|x-x_{0}\\|_{2}\leq\\|x+x_{0}\\|_{2}$. If $\\|x-x_{0}\\|_{2}\leq\\|x_{0}\\|_{2}$ then $\\|x_{0}\\|_{2}\leq 2\\|x_{0}\\|_{2}-\\|x-x_{0}\\|_{2}\leq\\|x+x_{0}\\|_{2}\leq\\|x-x_{0}\\|_{2}+2\\|x_{0}\\|_{2}\leq 3\\|x_{0}\\|_{2}.$ Hence, $\\|x_{0}\\|_{2}\sim\\|x+x_{0}\\|_{2}$, and $\\|x_{0}\\|_{2}\min\\{\\|x-x_{0}\\|_{2},\\|x+x_{0}\\|_{2}\\}\sim\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2}.$ Otherwise, $\\|x-x_{0}\\|_{2}>\\|x_{0}\\|_{2}$. If, in addition, $\\|x_{0}\\|_{2}\geq(\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2})^{1/2}/4,$ then $4\\|x_{0}\\|_{2}\geq(\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2})^{1/2}\geq\\|x_{0}\\|_{2}^{1/2}\\|x+x_{0}\\|_{2}^{1/2},$ and thus $\\|x+x_{0}\\|_{2}\leq 16\\|x_{0}\\|_{2}$. Since $\\|x_{0}\\|_{2}<\\|x-x_{0}\\|_{2}\leq\\|x+x_{0}\\|_{2}$, it follows that $\\|x+x_{0}\\|_{2}\sim\\|x-x_{0}\\|_{2}\sim\\|x_{0}\\|_{2}$, and again, $\\|x_{0}\\|_{2}\min\\{\\|x-x_{0}\\|_{2},\\|x+x_{0}\\|_{2}\\}\sim\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2}.$ Therefore, the final case, and the only one in which there is no point-wise equivalence between $\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2}$ and $\\|x_{0}\\|_{2}\min\\{\\|x-x_{0}\\|_{2},\\|x+x_{0}\\|_{2}\\}$, is when $\min\\{\\|x-x_{0}\\|_{2},\\|x+x_{0}\\|_{2}\\}\geq\\|x_{0}\\|_{2}$ and $\\|x_{0}\\|_{2}\leq(\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2})^{1/2}/4$. In that case, if $\\|x_{0}\\|_{2}\geq\sqrt{R}/4$ then $\\|x_{0}\\|_{2}\min\\{\\|x-x_{0}\\|_{2},\\|x+x_{0}\\|_{2}\\}\geq\\|x_{0}\\|_{2}^{2}\geq R/16,$ and $\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2}\geq 4\\|x_{0}\\|_{2}^{2}\geq R/4,$ from which the first part of the claim follows immediately. For the second one, observe that $\\|x\\|_{2}^{2}-2\\|x_{0}\\|_{2}\\|x\\|_{2}\leq\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2}\leq\\|x\\|_{2}^{2}+2\\|x\\|_{2}\\|x_{0}\\|_{2}+\\|x_{0}\\|_{2}^{2},$ and if $\\|x_{0}\\|_{2}\leq\sqrt{R}/4$, the equivalence is evident. In view of Lemma 4.1, the way the product $\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2}$ relates to $\min\\{\\|x-x_{0}\\|_{2},\\|x+x_{0}\\|_{2}\\}$ depends on $\\|x_{0}\\|_{2}$. If $\\|x_{0}\\|_{2}\geq\sqrt{R}/4$, then $\\{x\in T:\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2}\leq R\\}\subset\\{x\in T:\min\\{\\|x-x_{0}\\|_{2},\\|x+x_{0}\\|_{2}\\}\leq c_{1}R/\\|x_{0}\\|_{2}\\},$ and if $\\|x_{0}\\|_{2}\leq\sqrt{R}/4$, $\\{x\in T:\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2}\leq R\\}\subset\\{x\in T:\\|x\\|_{2}\leq c_{1}\sqrt{R}\\},$ for a suitable absolute constant $c_{1}$. When $T$ is convex and centrally-symmetric, the corresponding complexity parameter - the gaussian average of $T_{+,R}(x_{0})=T_{-,R}(x_{0})$ is $E_{R}(x_{0})\lesssim\begin{cases}\frac{\\|x_{0}\\|_{2}}{R}\cdot\ell(2T\cap(c_{1}R/\\|x_{0}\\|_{2})B_{2}^{n})&\mbox{if}\ \ \\|x_{0}\\|_{2}\geq\sqrt{R},\\\ \\\ \frac{1}{\sqrt{R}}\ell(2T\cap c_{1}\sqrt{R}B_{2}^{n})&\mbox{if}\ \ \\|x_{0}\\|<\sqrt{R}.\end{cases}$ The fixed point conditions appearing in Theorem A now become $r_{0}=\inf\\{R:E_{R}(x_{0})\leq c_{2}\sqrt{N}\\}$ (4.1) and $r_{2}(\gamma)=\inf\\{R:E_{R}(x_{0})\leq\gamma\sqrt{N}R\\},$ (4.2) where one selects the slightly suboptimal $\gamma=c_{2}/\sigma\sqrt{\log N}$. The assertion of Theorem A is that with high probability, ERM produces $\hat{x}$ for which $\\|\hat{x}-x_{0}\\|_{2}\\|\hat{x}+x_{0}\\|_{2}\leq\max\\{r_{2}(\gamma),r_{0}\\}.$ If $\\|x_{0}\\|_{2}\geq\sqrt{R}$, the fixed-point condition (4.1) is $\ell(2T\cap(c_{1}R/\\|x_{0}\\|_{2})B_{2}^{n})\leq c_{3}\left(\frac{R}{\\|x_{0}\\|_{2}}\right)\sqrt{N}$ (4.3) while (4.2) is, $\frac{\\|x_{0}\\|_{2}}{R}\ell(2T\cap(c_{1}R/\\|x_{0}\\|_{2})B_{2}^{n})\leq(c_{4}/\sigma\sqrt{\log N})\cdot\sqrt{N}R.$ (4.4) Recall that $r_{N}^{}(Q)=\inf\\{r>0:\ell(T\cap rB_{2}^{n})\leq Qr\sqrt{N}\\},$ and $s_{N}^{}(\eta)=\inf\\{s>0:\ell(T\cap sB_{2}^{n})\leq\eta s^{2}\sqrt{N}\\}.$ Therefore, it is straightforward to verify that $r_{0}=2\\|x_{0}\\|_{2}r_{N}^{}(c_{3})\ \ {\rm and}\ \ r_{2}\big{(}c_{2}/(\sigma\sqrt{\log N})\big{)}=2\\|x_{0}\\|_{2}s_{N}^{}(c_{4}\\|x_{0}\\|_{2}/\sigma\sqrt{\log{N}}).$ Setting $R=2\\|x_{0}\\|_{2}\max\\{r_{N}^{}(c_{3}),s_{N}^{}(c_{4}\\|x_{0}\\|_{2}/\sigma\sqrt{\log{N}})\\}$, it remains to ensure that $\\|x_{0}\\|_{2}^{2}\geq R$; that is, $2\max\\{s_{N}^{}(c_{4}\\|x_{0}\\|_{2}/\sigma\sqrt{\log N}),r_{N}^{}(c_{3})\\}\leq\\|x_{0}\\|_{2}.$ (4.5) Observe that if $r_{N}^{}(c_{3})\leq\frac{c_{3}\sigma}{c_{4}\\|x_{0}\\|_{2}}\sqrt{\log N},$ (4.6) then $r_{N}^{}(c_{3})\leq s_{N}^{}(c_{4}\\|x_{0}\\|_{2}/\sigma\sqrt{\log N})$. Indeed, applying the convexity of $T$, it is standard to verify that $r_{N}^{}(Q)$ is attained and $r_{N}^{}(Q)\leq\rho$ if and only if $\ell(T\cap\rho B_{2}^{n})\leq Q\rho\sqrt{N}$ – with a similar statement for $s_{N}^{}$ (see, e.g., the discussion in [7]). Therefore, $s_{N}^{}(\eta)\geq r_{N}^{}(Q)$ if and only if $\ell(T\cap r_{N}^{}(Q))\geq\eta(r_{N}^{}(Q))^{2}\sqrt{N}$. The latter is evident because $\ell(T\cap r_{N}^{}(Q))=Qr_{N}^{}(Q)\sqrt{N}$ and recalling that $Q=c_{3}$ and $\eta=c_{4}\\|x_{0}\\|_{2}/\sigma\sqrt{\log N}$. Under (4.6), an assumption which has been made in the formulation of Theorem B, (4.5) becomes $2s_{N}^{}(c_{4}\\|x_{0}\\|_{2}/\sigma\sqrt{\log N})\leq\\|x_{0}\\|_{2}$ and, by the definition of $s_{N}^{}$, this is the case if and only if $\ell(T\cap\\|x_{0}\\|_{2}B_{2}^{n})\leq\frac{c_{4}\\|x_{0}\\|_{2}}{\sigma\sqrt{\log N}}\cdot\\|x_{0}\\|_{2}^{2}\sqrt{N};$ that is simply when $\\|x_{0}\\|_{2}\geq v_{N}^{}(\zeta)$ for $\zeta=c_{4}/\sigma\sqrt{\log N}$. Hence, by Theorem A, combined with Lemma 4.1, it follows that with high probability, $\min\\{\\|\hat{x}-x_{0}\\|_{2},\\|\hat{x}+x_{0}\\|_{2}\\}\leq 2s_{N}^{}(c_{4}\\|x_{0}\\|_{2}/\sigma\sqrt{\log N}).$ The other cases, when either $\\|x_{0}\\|_{2}$ is ‘small’, or when $r_{0}$ dominates $r_{2}$ are treated is a similar fashion, and are omitted. ## 5 Minimax lower bounds In this section we study the optimality of ERM as a phase retrieval procedure, in the minimax sense. The estimate obtained here is based on the maximal cardinality of separated subsets of the class with respect to the $L_{2}(\mu)$ norm. ###### Definition 5.1 Let $B$ be the unit ball in a normed space. For any subset $A$ of the space, let $M(A,rB)$ be the maximal cardinality of a subset of $A$ that is $r$-separated with respect to the norm associated with $B$. Observe that if $M(A,rB)\geq L$ there are $x_{1},...,x_{L}\in A$ for which the sets $x_{i}+(r/3)B$ are disjoint. A similar statement is true in the reverse direction. Let $F$ be a class of functions on $(\Omega,\mu)$ and let $a$ be distributed according to $\mu$. For $f_{0}\in F$ and a centred gaussian variable $w$, which has variance $\sigma$ and is independent of $a$, consider the gaussian regression model $y=f_{0}(a)+w.$ (5.1) Any procedure that performs well in the minimax sense, must do so for any choice of $f_{0}\in F$ in (5.1). Following [7], there are two possible sources of ‘statistical complexity’ that influence the error rate of gaussian regression in $F$. 1\. Firstly, that there are functions in $F$ that, despite being far away from $f_{0}$, still satisfy $f_{0}(a_{i})=f(a_{i})$ for every $1\leq i\leq N$, and thus are indistinguishable from $f_{0}$ on the data. This statistical complexity is independent of the noise, and for every $f_{0}\in F$ and ${\mathbb{A}}=(a_{i})_{i=1}^{N}$, it is captured by the $L_{2}(\mu)$ diameter of the set $K(f_{0},{\mathbb{A}})=\\{f\in F:(f(a_{i}))_{i=1}^{N}=(f_{0}(a_{i}))_{i=1}^{N}\\},$ which is denoted by $d_{N}^{}({\mathbb{A}})$. 2\. Secondly, that the set $(F-f_{0})\cap rD=\\{f-f_{0}:f\in F,\\|f-f_{0}\\|_{L_{2}}\leq r\\}$ is ‘rich enough’ at a scale that is proportional to its $L_{2}(\mu)$ diameter $r$. The richness of the set is measured using the cardinality of a maximal $L_{2}(\mu)$-separated set. To that end, let $D$ be the unit ball in $L_{2}(\mu)$, set $C(r,\theta_{0})=\sup_{f_{0}\in F}r\log^{1/2}M(F\cap(f_{0}+\theta_{0}rD),rD)$ and put $q^{}_{N}(\eta)=\inf\big{\\{}r>0:C(r,\theta_{0})\leq\eta r^{2}\sqrt{N}\big{\\}}.$ (5.2) ###### Theorem 5.2 [7] For every $f_{0}\in F$ let $\mathbb{P}_{f_{0}}^{\otimes N}$ be the probability measure that generates samples $(a_{i},y_{i})_{i=1}^{N}$ according to (5.1). For every $\theta_{0}\geq 2$ there exists a constant $\theta_{1}>0$ for which $\inf_{\hat{f}}\sup_{f_{0}\in F}\mathbb{P}_{f_{0}}^{\otimes N}\Big{(}\left\\|f_{0}-\hat{f}\right\\|_{2}\geq\max\\{q_{N}^{}(\theta_{1}/\sigma),(d_{N}^{}({\mathbb{A}})/4)\\}\Big{)}\geq 1/5$ (5.3) where $\inf_{\hat{f}}$ is the infimum over all possible estimators constructed using the given data. Earlier versions of this minimax bound may be found in [14], [16] and [1]. To apply this general principle to the phase recovery problem, note that the regression function is $f_{0}(x)=\bigl{<}x_{0},x\bigr{>}^{2}:=f_{x_{0}}(x)$ for some unknown vector $x_{0}\in T\subset\mathbb{R}^{n}$, while the estimators are $\hat{f}=\bigl{<}\hat{x},\cdot\bigr{>}^{2}$. Also, observe that for every $x_{1},x_{2}\in T$, $\left\\|f_{x_{0}}-f_{x_{1}}\right\\|_{L^{2}(\mu)}^{2}=\mathbb{E}\big{(}\bigl{<}x_{0},a\bigr{>}^{2}-\bigl{<}x_{1},a\bigr{>}^{2}\big{)}^{2}=\mathbb{E}\bigl{<}x_{0}-x_{1},a\bigr{>}^{2}\bigl{<}x_{0}+x_{1},a\bigr{>}^{2}$ and therefore, one has to identify the $L_{2}$ structure of the set $F-f_{x_{0}}=\\{\bigl{<}x,\cdot\bigr{>}^{2}-\bigl{<}x_{0},\cdot\bigr{>}^{2}:x\in T\\}.$ To obtain the desired bound, it suffices to assume the following: ###### Assumption 5.1 There exist constants $C_{1}$ and $C_{2}$ for which, for every $s,t\in\mathbb{R}^{n}$, $C_{1}^{2}\\|s-t\\|_{2}^{2}\\|s+t\\|_{2}^{2}\leq\mathbb{E}\bigl{<}s-t,a\bigr{>}^{2}\bigl{<}s+t,a\bigr{>}^{2}\leq C_{2}^{2}\\|s-t\\|_{2}^{2}\\|s+t\\|_{2}^{2}.$ It is straightforward to verify that if $a$ is an $L$-subgaussian vector on $\mathbb{R}^{n}$ that satisfies Assumption 1.1, then it automatically satisfies Assumption 5.1. The norm $\left\\|x_{0}\right\\|_{2}$ plays a central role in the analysis of the rates of convergence of the ERM in phase recovery. Therefore, the minimax lower bounds presented here are not only for the entire model $T$ but for every shell $V_{0}=T\cap R_{0}S^{n-1}$. A minimax lower bound over $T$ follows by taking the supremum over all possible choices of $R_{0}$. To apply Theorem 5.2, observe that by Assumption 5.1, for every $u,v\in T$, $C_{1}\\|u-v\\|_{2}\\|u+v\\|_{2}\leq\\|f_{v}-f_{u}\\|_{L_{2}}\leq C_{2}\\|u-v\\|_{2}\\|u+v\\|_{2}.$ Fix $R_{0}>0$ and consider $V_{0}=T\cap R_{0}S^{n-1}$. Clearly, for every $r>0$ and every $x_{0}\in V_{0}$, $\Big{\\{}u\in V_{0}:\\|u-x_{0}\\|_{2}\\|u+x_{0}\\|_{2}\leq\frac{\theta_{0}r}{C_{2}}\Big{\\}}\subset\\{u\in V_{0}:f_{u}\in F^{\prime}\cap(f_{x_{0}}+\theta_{0}rD)\\}$ (5.4) where $F^{\prime}=\\{f_{u}:u\in V_{0}\\}$. Fix $\theta_{0}>2$ to be named later, and let $\theta_{1}$ be as in Theorem 5.2. If there are $x_{0}\in V_{0}$ and $\\{x_{1},...,x_{M}\\}\subset V_{0}$ that satisfy 1\. $\\|x_{i}-x_{0}\\|_{2}\\|x_{i}+x_{0}\\|_{2}\leq\theta_{0}r/C_{2}$, 2\. for every $1\leq i<j\leq M$, $\\|x_{i}-x_{j}\\|_{2}\\|x_{i}+x_{j}\\|_{2}\geq r/C_{1}$, and 3\. $\log M>N(\theta_{1}r/\sigma)^{2}$, then $\sup_{f_{0}\in F^{\prime}}r\log^{1/2}M(F^{\prime}\cap(f_{0}+\theta_{0}rD),rD)>\theta_{1}r\sqrt{N}/\sigma$, and the best possible rate in phase recovery in $V_{0}$ is larger than $r$. Fix $x_{0}\in V_{0}$ and $r>0$, and let $R=r/C_{2}$. We will present two different estimates, based on $R_{0}=\left\\|x_{0}\right\\|_{2}$, the ‘location’ of $x_{0}$. Centre of ‘small norm’. Recall that $\theta_{0}>2$ and assume first that $R_{0}=\\|x_{0}\\|_{2}\leq\sqrt{\theta_{0}R}/4$. Note that $V_{0}\cap(\sqrt{R}/8)B_{2}^{n}\subset\Big{\\{}u\in V_{0}:\\|u-x_{0}\\|_{2}\\|u+x_{0}\\|_{2}\leq\frac{\theta_{0}r}{C_{2}}\Big{\\}},$ and thus it suffices to constructed a separated set in $V_{0}\cap(\sqrt{R}/8)B_{2}^{n}$. Set $x_{1},...,x_{L}$ to be a maximal $c_{3}\sqrt{R}$-separated subset of $V_{0}\cap(\sqrt{R}/8)B_{2}^{n}$ for a constant $c_{3}$ that depends only on $C_{1}$ and $C_{2}$ and which will be specified later; thus, $L=M(V_{0}\cap(\sqrt{R}/8)B_{2}^{n},c_{3}\sqrt{R}B_{2}^{n})$. ###### Lemma 5.3 There is a subset $I\subset\\{1,...,L\\}$ of cardinality $M\geq L/2-1$ for which $(x_{i})_{i\in I}$ satisfies 1. and 2.. Proof. Since $x_{i}\in(\sqrt{R}/8)B_{2}^{n}$ and $\\|x_{0}\\|_{2}\leq\sqrt{\theta_{0}R}/4$, $\\|x_{i}-x_{0}\\|_{2}\\|x_{i}+x_{0}\\|_{2}\leq((\sqrt{\theta_{0}}+1)\sqrt{R}/4)^{2}\leq\theta_{0}R=\theta_{0}r/C_{2},$ and thus 1. is satisfied for every $1\leq i\leq L$. To show that 2. holds for a large subset, it suffices to find $I\subset\\{1,...,L\\}$ of cardinality at least $L/2-1$ such that for every $i,j\in I$, $\\|x_{i}-x_{j}\\|_{2}\geq c_{3}\sqrt{R}/2\ \ {\rm and}\ \ \\|x_{i}+x_{j}\\|_{2}\geq c_{3}\sqrt{R}/2$ for a well-chosen $c_{3}$. To construct the subset, observe that if there are distinct integers $1\leq i,j,k\leq L$ for which $\\|x_{i}+x_{j}\\|_{2}<c_{3}\sqrt{R}/2$ and $\\|x_{i}+x_{k}\\|_{2}<c_{3}\sqrt{R}/2$, then $\\|x_{j}-x_{k}\\|_{2}<c_{3}\sqrt{R}$, which is impossible. Therefore, for every $x_{i}$ there is at most a single index $j\in\\{1,...,L\\}\backslash\\{i\\}$ satisfying that $\\|x_{i}+x_{j}\\|_{2}<c_{3}\sqrt{R}/2$. With this observation the set $I$ is constructed inductively. Without loss of generality, assume that $I=\\{1,...,M\\}$ for $M\geq L/2-1$. If $i\not=j$ and $1\leq i,j\leq M$, $\\|x_{i}-x_{j}\\|_{2}\\|x_{i}+x_{j}\\|_{2}\geq c_{3}^{2}R/4\geq 2r/C_{1},$ for the right choice of $c_{3}$, and thus $(x_{i})_{i=1}^{M}$ satisfies 2.. Centre of ‘large norm’. Next, assume that $R_{0}=\\|x_{0}\\|_{2}\geq\sqrt{\theta_{0}R}/4$. By Lemma 4.1, there is an absolute constant $c_{4}<1/32$, for which, if $\\|x_{0}\\|_{2}\geq\sqrt{\rho}/4$ and $\\|x_{0}\\|_{2}\min\\{\\|x-x_{0}\\|_{2},\\|x+x_{0}\\|_{2}\\}\leq c_{4}\rho$, then $\\|x-x_{0}\\|_{2}\\|x+x_{0}\\|_{2}\leq\rho$. Therefore, applied to the choice $\rho=\theta_{0}R$, $\displaystyle\left(V_{0}\cap(x_{0}+(c_{4}\theta_{0}R/\\|x_{0}\\|_{2})B_{2}^{n})\right)\cup\left(V_{0}\cap(-x_{0}+(c_{4}\theta_{0}R/\\|x_{0}\\|_{2})B_{2}^{n})\right)$ $\displaystyle\subset$ $\displaystyle\left\\{u\in V_{0}:\\|x_{0}-u\\|_{2}\\|x_{0}+u\\|_{2}\leq\theta_{0}R\right\\},$ and it suffices to a find a separated set in the former. Note that if $x\in V_{0}\cap(x_{0}+(c_{4}\theta_{0}R/\\|x_{0}\\|_{2})B_{2}^{n})$ then $\\|x\\|_{2}\geq\\|x_{0}\\|_{2}-c_{4}\theta_{0}R/\\|x_{0}\\|_{2}\geq\\|x_{0}\\|_{2}/2\geq\sqrt{\theta_{0}R/4}/4$ because one can choose $c_{4}\leq 1/32$, and $\\|x\\|_{2}\leq 3\\|x_{0}\\|_{2}/2.$ Moreover, if $x_{1},x_{2}\in V_{0}\cap(x_{0}+(c_{4}\theta_{0}R/\\|x_{0}\\|_{2})B_{2}^{n})$, then $\\|x_{1}+x_{2}\\|_{2}\geq 2\\|x_{0}\\|_{2}-2c_{4}\theta_{0}R/\\|x_{0}\\|_{2}\geq\\|x_{0}\\|_{2}.$ Applying Lemma 4.1, there is an absolute constant $c_{5}$, for which, if $\\|x_{0}\\|_{2}\min\left\\{\\|x_{i}-x_{j}\\|_{2},\\|x_{i}+x_{j}\\|_{2}\right\\}\geq\theta_{0}R/4,$ (5.5) then $\\|x_{i}-x_{j}\\|_{2}\\|x_{i}+x_{j}\\|_{2}\geq c_{5}\theta_{0}R/4.$ Hence, if $x_{1},...,x_{M}\in V_{0}\cap(x_{0}+(c_{4}\theta_{0}R/\\|x_{0}\\|_{2})B_{2}^{n})$ is $\theta_{0}R/4\\|x_{0}\\|_{2}$-separated, then (5.5) holds, and $\\|x_{i}-x_{j}\\|_{2}\\|x_{i}+x_{j}\\|_{2}\geq c_{5}\theta_{0}R/4\geq r/C_{1},$ provided that $\theta_{0}$ is a sufficiently large constant that depends only on $C_{1}$ and $C_{2}$. ###### Corollary 5.4 There exist absolute constants $\theta_{1}$, $c_{1},c_{2}$ and $c_{3}$ that depend only on $C_{1}$ and $C_{2}$ and for which the following holds. Let $R_{0}>0$ and set $V_{0}=T\cap R_{0}S^{n-1}$. Define $q^{\prime}(r)=\left\\{\begin{array}[]{cc}\log M(V_{0}\cap c_{1}\sqrt{r}B_{2}^{n},c_{2}\sqrt{r}B_{2}^{n})&\mbox{if}\ \ R_{0}\leq c_{3}\sqrt{r},\\\ &\\\ \sup_{x_{0}\in V_{0}}\log M\left(V_{0}\cap\left(x_{0}+c_{1}\left(\frac{r}{R_{0}}\right)B_{2}^{n}\right),c_{2}\frac{r}{R_{0}}B_{2}^{n}\right)&\mbox{if}\ \ R_{0}>c_{3}\sqrt{r}\end{array}\right.$ If $q^{\prime}(r)\geq\theta_{1}(r/\sigma)^{2}N$, then the minimax rate in $V_{0}$ is larger than $r$. Now, a minimax lower bound for the risk $\min\\{\left\\|\tilde{x}-x_{0}\right\\|_{2},\left\\|\tilde{x}+x_{0}\right\\|_{2}\\}$ for all shells $V_{0}$ (and therefore, for $T$) may be derived using Lemma 4.1. To that end, let $c_{0}$ be a large enough absolute constant and $C(R_{0},r)=\sup_{x_{0}\in T:\left\\|x_{0}\right\\|_{2}=R_{0}}r\log^{1/2}M\big{(}(T\cap R_{0}S^{n-1})\cap(x_{0}+c_{0}rB_{2}^{n}),rB_{2}^{n}).$ (5.6) ###### Definition 5.5 Fix $R_{0}>0$. For every $\alpha,\beta>0$ set $q_{N}^{}(\alpha)=\inf\big{\\{}r>0:C(R_{0},r)\leq\alpha r^{2}\sqrt{N}\big{\\}}$ and put $t_{N}^{}(\beta)=\inf\big{\\{}r>0:C(R_{0},r)\leq\beta r^{3}\sqrt{N}\big{\\}}.$ Note that for any $c>0$, if $t_{N}^{}(c)\leq 1$ then $t_{N}^{}(c)\geq q_{N}^{}(c)$ and $q_{N}^{}\big{(}cR_{0}/\sigma\big{)}\geq t_{N}^{}(c/\sigma)$ if and only if $q_{N}^{}\big{(}cR_{0}/\sigma\big{)}\geq R_{0}$. Theorem C. There exists an absolute constant $c_{1}$ for which the following holds. Let $R_{0}>0$. 1. 1. If $R_{0}\geq t_{N}^{}\big{(}c_{1}/\sigma\big{)}$, then for any procedure $\tilde{x}$, there exists $x_{0}\in T$ with $\left\\|x_{0}\right\\|_{2}=R_{0}$ and for which, with probability at least $1/5$, $\left\\|\tilde{x}-x_{0}\right\\|_{2}\left\\|\tilde{x}+x_{0}\right\\|_{2}\geq\left\\|x_{0}\right\\|_{2}q_{N}^{}\Big{(}\frac{c_{1}\left\\|x_{0}\right\\|_{2}}{\sigma}\Big{)}$ and $\min\\{\left\\|\tilde{x}-x_{0}\right\\|_{2},\left\\|\tilde{x}+x_{0}\right\\|_{2}\\}\geq q_{N}^{}\Big{(}\frac{c_{1}\left\\|x_{0}\right\\|_{2}}{\sigma}\Big{)}.$ 2. 2. If $R_{0}\leq t_{N}^{}\big{(}c_{1}/\sigma\big{)}$ then for any procedure $\tilde{x}$ there exists $x_{0}\in T$ with $\left\\|x_{0}\right\\|_{2}=R_{0}$ for which, with probability at least $1/5$, $\left\\|\tilde{x}-x_{0}\right\\|_{2}\left\\|\tilde{x}+x_{0}\right\\|_{2}\geq\Big{(}t_{N}^{}\Big{(}\frac{c_{1}}{\sigma}\Big{)}\Big{)}^{2}$ and $\left\\|\tilde{x}\right\\|_{2},\left\\|\tilde{x}-x_{0}\right\\|_{2},\left\\|\tilde{x}+x_{0}\right\\|_{2}\geq t_{N}^{}\Big{(}\frac{c_{1}}{\sigma}\Big{)}.$ Theorem C is a general minimax bound, and although it seems strange at first glance, the parameters appearing in it are very close to those used in Theorem C. Following the same path as in [7], let us show that Theorem C and Theorem C are almost sharp, under some mild structural assumptions on $T$. First, recall Sudakov’s inequality (see, e.g., [9]): ###### Theorem 5.6 If $W\subset\mathbb{R}^{n}$ and $\varepsilon>0$ then $c\varepsilon\log^{1/2}M(W,\varepsilon B_{2}^{n})\leq\ell(W),$ where $c$ is an absolute constant. Fix $R_{0}>0$ set $V_{0}=T\cap R_{0}B_{2}^{n}$, and put $s_{N}^{}=s_{N}^{}\big{(}c_{1}R_{0}/(\sigma\sqrt{\log N})\big{)}\ \ {\rm and}\ \ v_{N}^{}=v_{N}^{}\big{(}c_{1}R_{0}/(\sigma\sqrt{\log N})\big{)}.$ Assume that there is some $x_{0}\in T$, with the following properties: 1\. $\left\\|x_{0}\right\\|_{2}=R$. 2\. The localized sets $V_{0}\cap(x_{0}+s_{N}^{}B_{2}^{n})$ and $V_{0}\cap(x_{0}+v_{N}^{}B_{2}^{n})$ satisfy that $\ell(V_{0}\cap(x_{0}+s_{N}^{}S^{n-1}))\sim\ell(T\cap s_{N}^{}B_{2}^{n})$ and that $\ell(V_{0}\cap(x_{0}+v_{N}^{}S^{n-1}))\sim\ell(T\cap v_{N}^{}B_{2}^{n}),$ which is a mild assumption on the complexity structure of $T$. 3\. Sudakov’s inequality is sharp at the scales $s_{N}^{}$ and $v_{N}^{}$, namely, $s_{N}^{}\log^{1/2}M(V_{0}\cap(x_{0}+c_{0}s_{N}^{}B_{2}^{n}),s_{N}^{}B_{2}^{n})\sim\ell(V_{0}\cap(x_{0}+s_{N}^{}B_{2}^{n}))$ (5.7) and a similar assertion holds for $v_{N}^{}$. In such a case, the rates of convergence obtained in Theorem B are minimax (up to the an extra $\sqrt{\log N}$ factor) thanks to Theorem C. When Sudakov’s inequality is sharp as in (5.7), we believe that ERM should be a minimax procedure in phase recovery, despite the logarithmic gap between Theorem B an Theorem C. A similar conclusion for linear regression was obtained in [7]. Sudakov’s inequality is sharp in many cases - most notably, when $T=B_{1}^{n}$, but not always. It is not sharp even for standard sets like the unit ball in $\ell_{p}^{n}$ for $1+(\log n)^{-1}<p<2$. ## 6 Examples Here, we will present two simple applications of the upper and lower bounds on the performance of ERM in phase recovery. Naturally, there are many other examples that follow in a similar way and that can be derived using very similar arguments. The choice of the examples has been made to illustrate the question of the optimality of Theorem A, B and C, as well as an indication of the similarities and differences between phase recovery and linear regression. Since the estimate used in these examples are rather well known, some of the details will not be presented in full. ### 6.1 Sparse vectors The first example we consider represents classes with a local complexity that remains unchanged, regardless of the choice of $x_{0}$. Let $T=W_{d}$ be the set of $d$-sparse vectors in $\mathbb{R}^{n}$ (for some $d\leq N/4$) – that is, vectors with at most $d$ non-zero coordinates. Clearly, for every $R>0$ $T_{+,R},T_{-,R}\subset W_{2d}\cap S^{n-1}$. Also, for any $x_{0}\in T$ and any $I\subset\\{1,...,n\\}$ of cardinality $d$ that is disjoint of ${\rm supp}(x_{0})$, $(1/\sqrt{2})S^{I}\subset\left\\{\frac{(x-x_{0})_{i\in I}}{\\|x-x_{0}\\|_{2}}:x\in W_{d}\right\\},\left\\{\frac{(x+x_{0})_{i\in I}}{\\|x+x_{0}\\|_{2}}:x\in W_{d}\right\\},$ where $S^{I}$ is the unit sphere supported on the coordinates $I$. With this observation, a straightforward argument shows that, $\ell(T_{+,R}),\ell(T_{-,R})\sim\sqrt{d\log(en/d)}.$ Applying Theorem A, if follows that for $N\gtrsim d\log\big{(}en/d\big{)}$, with probability at least $1-2\exp(-c(d\log(en/d)))-N^{-\beta+1}$, ERM produces $\hat{x}$ that satisfies $\\|\hat{x}-x_{0}\\|_{2}\\|\hat{x}+x_{0}\\|_{2}\lesssim_{\kappa_{0},L,\beta}\sigma\sqrt{\frac{d\log(en/d)}{N}}\sqrt{\log N}=().$ (6.1) Moreover, with the same probability estimate, if $\left\\|x_{0}\right\\|_{2}^{2}\gtrsim()$ then by Lemma 4.1, $\min\\{\\|\hat{x}-x_{0}\\|_{2},\\|\hat{x}+x_{0}\\|_{2}\\}\lesssim_{\kappa_{0},L,\beta}\frac{\sigma}{\left\\|x_{0}\right\\|_{2}}\sqrt{\frac{d\log(en/d)}{N}}\sqrt{\log N}$ (6.2) and if $\left\\|x_{0}\right\\|_{2}^{2}\lesssim()$ then $\left\\|\hat{x}\right\\|_{2}^{2},\\|\hat{x}-x_{0}\\|_{2}^{2},\\|\hat{x}+x_{0}\\|_{2}^{2}\lesssim_{\kappa_{0},L}\sigma\sqrt{\frac{d\log(en/d)}{N}}\sqrt{\log N}.$ (6.3) In particular, when $\left\\|x_{0}\right\\|_{2}$ is of the order of a constant, the rate of convergence in (6.1) and (6.2) is identical to the one obtained in [7] in the linear regression (up to a $\sqrt{\log N}$ term). In the latter, ERM achieves the minimax rate (with the same probability estimate) of $\\|\hat{x}-x_{0}\\|_{2}\lesssim_{L}\sigma\sqrt{\frac{d\log(en/d)}{N}}.$ Otherwise, when $\left\\|x_{0}\right\\|_{2}$ is large, the rate of convergence of the ERM in (6.2) is actually better than in linear regression, but, when $\left\\|x_{0}\right\\|_{2}$ is small, it is worse - deteriorating to the square root of the rate in linear regression (up to logarithmic terms). When the noise level $\sigma$ tends to zero, the rates of convergence in linear regression and phase recovery tend to zero as well. In particular, exact reconstruction happens – that is $\hat{x}=x_{0}$ in linear regression and $\hat{x}=x_{0}$ or $\hat{x}=-x_{0}$ in phase recovery – when $N\gtrsim d\log\big{(}en/d\big{)}$. For the lower bound, it is well known that $\log^{1/2}M(W_{d}\cap c_{0}rB_{2}^{n},rB_{2}^{n})\sim\sqrt{d\log(en/d)}$ for every $r>0$ (and $c_{0}\geq 2$). Combined with the results of the previous section, this suffices to show that the rate obtained in Theorem A is the minimax one (up to a $\sqrt{\log N}$ term in the “large noise” regime) and that the ERM is a minimax procedure for the phase retrieval problem when the signal $x_{0}$ is known to be $d$-sparse and $N\gtrsim d\log\big{(}en/d\big{)}$. ### 6.2 The unit ball of $\ell_{1}^{n}$ Consider the set $T=B_{1}^{n}$, the unit ball of $\ell_{1}^{n}$. Being convex and centrally symmetric, it is a natural example of a set with changing ‘local complexity’ – which becomes very large when $x_{0}$ is close to $0$. Moreover, it is an example in which one may obtain sharp estimates on $\ell(B_{1}^{n}\cap rB_{2}^{n})$ at every scale. Indeed, one may show (see, for example, [5]) that $\ell\big{(}B_{1}^{n}\cap rB_{2}^{n}\big{)}\sim\left\\{\begin{array}[]{cc}\sqrt{\log(enr^{2})}&\mbox{ if }r^{2}n\geq 1\\\ &\\\ r\sqrt{n}&\mbox{ otherwise.}\end{array}\right.$ It follows that for $B_{1}^{n}$, one has $r_{N}^{}(Q)\ \ \left\\{\begin{array}[]{cc}\sim\Big{(}\frac{1}{Q^{2}N}\log\Big{(}\frac{n}{Q^{2}N}\Big{)}\Big{)}^{1/2}&\mbox{ if }n\geq C_{0}Q^{2}N\\\ &\\\ \lesssim\frac{1}{N}&\mbox{ if }C_{1}Q^{2}N\leq n\leq C_{0}Q^{2}N\\\ &\\\ =0&\mbox{ if }n\leq C_{1}Q^{2}N.\end{array}\right.$ where $C_{0}$ and $C_{1}$ are absolute constants. The only range in which this estimate is not sharp is when $n\sim Q^{2}N$, because in that range, $r_{N}^{}(Q)$ decays to zero very quickly. A more accurate estimate on $\ell(B_{1}^{n}\cap rB_{2}^{n})$ can be performed when $n\sim Q^{2}N$ (see [8]), but since it is not our main interest, we will not pursue it further, and only consider the cases $n\leq C_{1}Q^{2}N$ and $n\geq C_{0}Q^{2}N$. A straightforward computation shows that the two other fixed points satisfy: $s_{N}^{}(\eta)\sim\left\\{\begin{array}[]{cc}\Big{(}\frac{1}{\eta^{2}N}\log\Big{(}\frac{n^{2}}{\eta^{2}N}\Big{)}\Big{)}^{1/4}&\mbox{ if }n\geq\eta\sqrt{N}\\\ \\\ \sqrt{\frac{n}{\eta^{2}N}}&\mbox{ if }n\leq\eta\sqrt{N}\end{array}\right.$ and $v_{N}^{}(\zeta)\sim\left\\{\begin{array}[]{cc}\Big{(}\frac{1}{\zeta^{2}N}\log\Big{(}\frac{n^{3}}{\zeta^{2}N}\Big{)}\Big{)}^{1/6}&\mbox{ if }n\geq\zeta^{2/3}N^{1/3}\\\ \\\ \Big{(}\frac{n}{\zeta^{2}N}\Big{)}^{1/4}&\mbox{ if }n\leq\zeta^{2/3}N^{1/3}.\end{array}\right.$ The estimates above will be used to derive rates of convergence for the ERM $\hat{x}$ (for the squared loss) of the form $\min\\{\left\\|\hat{x}-x_{0}\right\\|_{2},\left\\|\hat{x}+x_{0}\right\\|_{2}\\}\leq rate.$ Upper bounds on the rate of convergence ${\it rate}$ follow from Theorem B, and hold with high probability as stated in there. For the sake of brevity, we will not present the probability estimates, but those can be easily derived from Theorem B. Thanks to Theorem B, obtaining upper bounds on ${\it rate}$ involves the study of several different regimes, depending on $\left\\|x_{0}\right\\|_{2}$, the noise level $\sigma$ and the way the number of observations $N$ compares with the dimension $n$. The noise-free case: $\sigma=0$. In this case, rate is upper bounded by $r_{N}^{}(Q)$, for $Q$ that is an absolute constant. In particular, when $n\geq C_{0}Q^{2}N$, the rate is less than $\big{(}N^{-1}\log\big{(}n/N\big{)}\big{)}^{1/2}.$ At this point, it is natural to wonder whether there is a procedure that outperforms ERM in the noise-free case. The minimax lower bound $d_{N}^{}({\mathbb{A}})$ in Theorem 5.2 may be used to address this question, as no algorithm can do better than $d_{N}^{}({\mathbb{A}})/4$, with probability greater than $1/5$. In the phase recovery problem and using the notation of section 5, one has $\displaystyle d_{N}^{}({\mathbb{A}})$ $\displaystyle=\sup\left\\{\left\\|f_{x}-f_{x_{0}}\right\\|_{2}:x\in B_{1}^{n},\ f_{x}(a_{i})=f_{x_{0}}(a_{i})\ ,i=1,\ldots,N\right\\}$ $\displaystyle\sim\sup\left\\{\left\\|x-x_{0}\right\\|_{2}\left\\|x+x_{0}\right\\|_{2}:\ x\in B_{1}^{n},\ \|\bigl{<}a_{i},x\bigr{>}\|=\|\bigl{<}a_{i},x_{0}\bigr{>}\|,\ i=1,\ldots,N\right\\}$ $\displaystyle\gtrsim\inf_{L:\mathbb{R}^{n}\rightarrow\mathbb{R}^{N}}\sup\left\\{\left\\|x-x_{0}\right\\|_{2}\left\\|x+x_{0}\right\\|_{2}:\ x\in B_{1}^{n},\ L(x)=L(x_{0})\right\\}$ with an infimum taken over all linear operators $L:\mathbb{R}^{n}\to\mathbb{R}^{N}$. By Lemma 4.1, for $x_{0}=(1/2,0,\ldots,0)\in B_{1}^{n}$ (in fact, any vector $x_{0}$ in $B_{1}^{n}$ for which $\left\\|x_{0}\right\\|_{2}$ is a positive constant smaller than $1/2$ would do) $\displaystyle d_{N}^{}({\mathbb{A}})$ $\displaystyle\gtrsim\inf_{L:\mathbb{R}^{n}\rightarrow\mathbb{R}^{N}}\sup_{x\in B_{1}^{n}\cap({\rm ker}L-x_{0})}\min\\{\left\\|x-x_{0}\right\\|_{2},\left\\|x+x_{0}\right\\|_{2}\\}$ $\displaystyle\gtrsim\inf_{L:\mathbb{R}^{n}\rightarrow\mathbb{R}^{N}}\sup_{x,y\in B_{1}^{n}\cap{\rm ker}L}\left\\|x-y\right\\|_{2}=c_{N}(B_{1}^{n})$ which is the Gelfand $N$-width of $B_{1}^{n}$. By a result due to Garnaev and Gluskin (see [3]), $c_{N}(B_{1}^{n})\sim\left\\{\begin{array}[]{cc}\min\Big{\\{}1,\sqrt{\frac{1}{N}\log\Big{(}\frac{en}{N}\Big{)}}\Big{\\}}&\mbox{ if }N\leq n\\\ \\\ 0&\mbox{ otherwise}.\end{array}\right.$ which is of the same order as $r_{N}^{}$ (except when $n\sim N$, which is not treated here). Therefore, no algorithm can outperform ERM and ERM is a minimax procedure in this case. Note that when $n\leq c_{1}Q^{2}N$, exact reconstruction of $x_{0}$ or $-x_{0}$ is possible and it can happens only in that case (i.e. $\sigma=0$ and $n\leq c_{1}Q^{2}N$) because of the minimax lower bound provided by $d_{N}^{}({\mathbb{A}})$. The noisy case: $\sigma>0$. According to Theorem B, the rate of convergence rate depends on $r_{N}^{}=r_{N}^{}(Q)$ for some absolute constant $Q$, $s_{N}^{}=s_{N}^{}(\eta)$ for $\eta=c_{1}\left\\|x_{0}\right\\|_{2}/(\sigma\sqrt{\log N})$ and on $v_{N}^{}=v_{N}^{}(\zeta)$ for $\zeta=c_{1}/(\sigma\sqrt{\log N})$. The outcome of Theorem B is presented in Figure 1. rate $\lesssim$ \| $\sigma/\left\\|x_{0}\right\\|_{2}\leq c_{0}r_{N}^{}/\sqrt{\log N}$ \| $\sigma/\left\\|x_{0}\right\\|_{2}\geq c_{0}r_{N}^{}/\sqrt{\log N}$ ---\|---\|--- $\left\\|x_{0}\right\\|_{2}\leq v_{N}^{}$ \| $r_{N}^{}$ \| $s_{N}^{}$ $\left\\|x_{0}\right\\|_{2}\geq v_{N}^{}$ \| $r_{N}^{}$ \| $v_{N}^{}$ Figure 1: High probability bounds on the rate of convergence of the ERM $\hat{x}$ for the square loss in phase recovery: $\min\\{\left\\|\hat{x}-x_{0}\right\\|_{2},\left\\|\hat{x}+x_{0}\right\\|_{2}\\}\leq rate$. As the proof of all the estimates is similar, a detailed analysis is only presented when $\zeta^{2/3}N^{1/3}\leq\eta\sqrt{N}\leq C_{1}Q^{2}N$, which is equivalent to $\Big{(}\frac{\sigma^{2}\log N}{c_{1}N}\Big{)}^{1/6}\leq\left\\|x_{0}\right\\|_{2}\leq\frac{c_{1}Q^{2}\sigma\sqrt{N\log N}}{c_{1}}.$ The upper bounds on rate change according to the way the number of observations $N$ scales relative to $n$: 1. 1. $n\geq C_{0}Q^{2}N$. In this situation, $r_{N}^{}\sim\big{(}\log(n/N)/N\big{)}^{1/2}$. Therefore, if $\sigma/\left\\|x_{0}\right\\|_{2}\lesssim\sqrt{\log(n/N)/(N\log N)}$ then $rate\leq\big{(}\log(n/N)/N\big{)}^{1/2}$, and if $\sigma/\left\\|x_{0}\right\\|_{2}\gtrsim\sqrt{\log(n/N)/(N\log N)}$, $rate\leq\left\\{\begin{array}[]{cc}\Big{(}\frac{\sigma^{2}\log N}{\left\\|x_{0}\right\\|_{2}^{2}N}\log\Big{(}\frac{\sigma^{2}n^{2}}{\left\\|x_{0}\right\\|_{2}^{2}N}\Big{)}\Big{)}^{1/4}&\mbox{ if }\left\\|x_{0}\right\\|_{2}\geq\Big{(}\frac{\sigma^{2}\log N}{N}\log\Big{(}\frac{\sigma^{2}n^{3}}{N}\Big{)}\Big{)}^{1/6}\\\ \\\ \Big{(}\frac{\sigma^{2}\log N}{N}\log\Big{(}\frac{\sigma^{2}n^{3}}{N}\Big{)}\Big{)}^{1/6}&\mbox{ otherwise}.\end{array}\right.$ (6.4) 2. 2. $c_{1}\left\\|x_{0}\right\\|_{2}/(\sigma\sqrt{\log N})\sqrt{N}\leq n\leq C_{1}Q^{2}N$. In that case $r_{N}^{}=0$. In particular $\sigma/\left\\|x_{0}\right\\|_{2}>c_{0}r_{N}^{}/\sqrt{\log N}$ and therefore, the rate is upper bounded as in (6.4). 3. 3. $\big{(}c_{1}/(\sigma\sqrt{\log N})\big{)}^{2/3}N^{1/3}\leq n\leq c_{1}\left\\|x_{0}\right\\|_{2}/(\sigma\sqrt{\log N})\sqrt{N}$. Again, in this case, $r_{N}^{}=0$. Therefore, one is in the situation of the small signal- to-noise ratio and $rate\leq\left\\{\begin{array}[]{cc}\frac{\sigma}{\left\\|x_{0}\right\\|_{2}}\sqrt{\frac{n\log N}{N}}&\mbox{ if }\left\\|x_{0}\right\\|_{2}\geq\Big{(}\frac{\sigma^{2}\log N}{N}\log\Big{(}\frac{\sigma^{2}n^{3}}{N}\Big{)}\Big{)}^{1/6}\\\ \\\ \Big{(}\frac{\sigma^{2}\log N}{N}\log\Big{(}\frac{\sigma^{2}n^{3}}{N}\Big{)}\Big{)}^{1/6}&\mbox{ otherwise}.\end{array}\right.$ 4. 4. $n\leq\big{(}c_{1}/(\sigma\sqrt{\log N})\big{)}^{2/3}N^{1/3}$. Once again, $r_{N}^{}=0$, and $rate\leq\left\\{\begin{array}[]{cc}\frac{\sigma}{\left\\|x_{0}\right\\|_{2}}\sqrt{\frac{n\log N}{N}}&\mbox{ if }\left\\|x_{0}\right\\|_{2}\geq\Big{(}\sigma\sqrt{\frac{n\log N}{N}}\Big{)}^{1/2}\\\ \\\ \Big{(}\sigma\sqrt{\frac{n\log N}{N}}\Big{)}^{1/2}&\mbox{ otherwise}.\end{array}\right.$ One may ask whether these estimates are optimal in the minimax sense, or perhaps there is another procedure that can outperform ERM. It appears that (up to an extra $\sqrt{\log N}$ factor), ERM is indeed optimal. To see that, it is enough to apply Theorem C and verify that Sudakov’s inequality is sharp in the following sense: (see the discussion following Theorem C): that if $\\|x_{0}\\|_{1}\leq 1/2$, then for every $\varepsilon<1/4$ $\varepsilon\log^{1/2}M(B_{1}^{n}\cap(x_{0}+c_{0}\varepsilon B_{2}^{n}),\varepsilon B_{2}^{n})\sim\ell\big{(}B_{1}^{n}\cap\varepsilon B_{2}^{n}\big{)}.$ This fact is relatively straightforward to verify (see, e.g., Example 2 in [10]). Therefore, up to the extra $\sqrt{\log N}$ factor, which we believe is parasitic, ERM is a minimax phase-recovery procedure in $B_{1}^{n}$. ## References [1] L. Birgé, Approximation dans les espaces métriques et théorie de l’estimation, Z. Wahrsch. Verw. Gebiete (2) vol. 65, 1983 * [2] Y. C. Eldar, S. Mendelson, Phase Retrieval: Stability and Recovery Guarantees, preprint. * [3] A. Yu. Garnaev and E. D. Gluskin. The widths of a Euclidean ball. Dokl. Akad. Nauk SSSR, 277(5):1048–1052, 1984. * [4] E. Giné, V. de la Peña, Decoupling: From Dependence to Independence, Springer-Verlag, 1999. * [5] Y.Gordon, A.E. Litvak, S. Mendelson and A. Pajor, Gaussian averages of interpolated bodies and applications to approximate reconstruction, J. Approx. Theory, 149 (1), 2007 * [6] Vladimir Koltchinskii. Oracle inequalities in empirical risk minimization and sparse recovery problems, volume 2033 of Lecture Notes in Mathematics. Springer, Heidelberg, 2011. 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Statist., 27(5):1564–1599, 1999.	arxiv-papers	2013-11-20T11:53:42	2024-09-04T02:49:53.997348	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Guillaume Lecu\\'e and Shahar Mendelson", "submitter": "Guillaume Lecu\\'e", "url": "https://arxiv.org/abs/1311.5024" }
1311.5073	Degenerate twistor spaces for hyperkähler manifolds Misha Verbitsky111Partially supported by RFBR grants 12-01-00944- , 10-01-93113-NCNIL-a, and AG Laboratory NRI-HSE, RF government grant, ag. 11.G34.31.0023, and the Simons-IUM fellowship grant. Abstract Let $M$ be a hyperkähler manifold, and $\eta$ a closed, positive (1,1)-form with $\operatorname{rk}\eta<\dim M$. We associate to $\eta$ a family of complex structures on $M$, called a degenerate twistor family, and parametrized by a complex line. When $\eta$ is a pullback of a Kähler form under a Lagrangian fibration $L$, all the fibers of degenerate twistor family also admit a Lagrangian fibration, with the fibers isomorphic to that of $L$. Degenerate twistor families can be obtained by taking limits of twistor families, as one of the Kähler forms in the hyperkähler triple goes to $\eta$. ###### Contents 1. 1 Introduction 1. 1.1 Complex structures obtained from non-degenerate closed 2-forms 2. 1.2 Degenerate twistor families and Teichmüller spaces 3. 1.3 Semipositive (1,1)-forms, degenerate twistor families and SYZ conjecture 4. 1.4 Degenerate twistor spaces and Lagrangian fibrations 2. 2 Basic notions of hyperkähler geometry 1. 2.1 Hyperkähler manifolds 2. 2.2 The Bogomolov-Beauville-Fujiki form 3. 2.3 The hyperkähler SYZ conjecture 4. 2.4 Cohomology of hyperkähler manifolds 3. 3 Degenerate twistor space 1. 3.1 Integrability of almost complex structures and Cartan formula 2. 3.2 Semipositive (1,1)-forms on hyperkähler manifold 3. 3.3 Positive $(p,p)$-forms 4. 3.4 Positive $(p,p)$-forms and holomorphic symplectic forms 5. 3.5 Degenerate twistor space: a definition ## 1 Introduction ### 1.1 Complex structures obtained from non-degenerate closed 2-forms The degenerate twistor spaces (3.5) are obtained through the following construction. Definition 1.1: A complex-valued 2-form $\Omega$ on a real manifold $M$ is called non-degenerate if $\Omega(v,\cdot)\neq 0$ for any non-zero tangent vector $v\in T_{m}M$. Complex structures on $M$ can be obtained from complex sub-bundles $B=T^{1,0}M\subset TM\otimes_{\mathbb{R}}{\mathbb{C}}$ satisfying $B\oplus\overline{B}=TM\otimes_{\mathbb{R}}{\mathbb{C}},\ \ [B,B]\subset B$ (1.1) (3.1). To obtain such $B$, take a non-degenerate (1.1), closed 2-form $\Omega\in\Lambda^{2}(M,{\mathbb{C}})$, satisfying $\Omega^{n+1}=0$, where $4n=\dim_{\mathbb{R}}M$. Then $\ker\Omega:=\\{v\in T_{m}M\otimes_{\mathbb{R}}{\mathbb{C}}\ \ \|\ \ \Omega(v,\cdot)=0\\}$ satisfies the conditions of (1.1) (see 3.1). Degenerate twistor spaces are obtained by constructing a family $\Omega_{t}$ of such 2-forms, parametrized by $t\in{\mathbb{C}}$, on hyperkähler manifolds. The relation $\Omega_{t}^{n+1}=0$ follows from the properties of cohomology of hyperkähler manifolds, most notably the Fujiki formula, computation of cohomology performed in [V2], and positivity (see Subsection 3.5). ### 1.2 Degenerate twistor families and Teichmüller spaces In this subsection, we provide a motivation for the term “degenerate twistor family”. We introduce the twistor families of complex structures on hyperkähler manifolds and the corresponding rational curves in the moduli, called the twistor lines. A degenerate twistor family is a family ${\cal Z}$ of deformations of a holomorphically symplectic manifold $(M,\Omega)$ associated with a positive, closed, semidefinite form $\eta$ satisfying $\eta^{n-i}\wedge\Omega^{i+1}=0$, for all $i=0,1,...,n$, where $\dim_{\mathbb{C}}M=2n$ (3.2). In this subsection, we define a twistor family of a hyperkähler manifold, and explain how these families can be obtained as limits of twistor deformations. Throughout this paper, a hyperkähler manifold is a compact, holomorphically symplectic manifold $M$ of Kähler type. It is called simple (2.2) if $\pi_{1}(M)=0$ and $H^{2,0}(M)={\mathbb{C}}$. We shall (sometimes silently) assume that all hyperkähler manifolds we work with are simple. A hyperkähler metric is a metric $g$ compatible with three complex structures $I,J,K$ satisfying the quaternionic relations $IJ=-JI=K$, which is Kähler with respect to $I,J,K$. By the Calabi-Yau theorem, any compact, holomorphically symplectic manifold of Kähler type admits a hyperkähler metric, which is unique in each Kähler class (2.1). A hyperkähler structure is a hyperkähler metric $g$ together with the compatible quaternionic action, that is, a triple of complex structures satisfying the quaternionic relations and Kähler. For any $(a,b,c)\in S^{2}\subset{\mathbb{R}}^{3}$, the quaternion $L:=aI+bJ+cK$ defines another complex structure on $M$, also Kähler with respect to $g$. This can be seen because the Levi-Civita connection $\nabla$ of $(M,g)$ preserves $I,J,K$, hence $\nabla L=0$, and this implies integrability and Kählerness of $L$. Such a complex structure is called induced complex structure. The ${\mathbb{C}}P^{1}$-family of induced complex structures obtained this way is in fact holomorphic (Subsection 2.1). It is called the twistor deformation. The twistor families can be described in terms of periods of hyperkähler manifolds as follows. Definition 1.2: Let $M$ be a compact complex manifold, and $\operatorname{Diff}_{0}(M)$ a connected component of its diffeomorphism group (also known as the group of isotopies). Denote by $\operatorname{\sf Comp}$ the space of complex structures on $M$, equipped with topology induced from the $C^{\infty}$-topology on the space of all tensors, and let $\operatorname{Teich}:=\operatorname{\sf Comp}/\operatorname{Diff}_{0}(M)$. We call it the Teichmüller space. Definition 1.3: Let $\operatorname{\sf Per}:\;\operatorname{Teich}{\>\longrightarrow\>}{\mathbb{P}}H^{2}(M,{\mathbb{C}})$ map $J$ to a line $H^{2,0}(M,J)\in{\mathbb{P}}H^{2}(M,{\mathbb{C}})$. The map $\operatorname{\sf Per}$ is called the period map. For a simple hyperkähler manifold, an important bilinear symmetric form $q\in\operatorname{Sym}^{2}H^{2}(M,{\mathbb{Q}})^{}$ is defined, called Bogomolov-Beauville-Fujiki form (2.2). This form is a topological invariant of the manifold $M$, allowing one to describe deformations of a complex structure very explicitly. Recall that two points $x,y$ on a topological space are called non-separable, if all their neighbourhoods $U_{x}\ni x$, $U_{y}\ni y$ intersect. We denote the corresponding symmetric relation in $\operatorname{Teich}$ by $x\sim y$. D. Huybrechts has shown that $x\sim y$ for $x,y\in\operatorname{Teich}$ implies that the corresponding complex manifolds $(M,x)$ and $(M,y)$ are bimeromorphic ([H1]). In [V5] it was shown that $\sim$ defines an equivalence relation on $\operatorname{Teich}$; the corresponding quotient space $\operatorname{Teich}/\sim$ is called the birational Teichmüller space, and denoted $\operatorname{Teich}_{b}$. Define the period space $\operatorname{{\mathbb{P}}\sf er}$ as $\operatorname{{\mathbb{P}}\sf er}:=\\{l\in{\mathbb{P}}(H^{2}(M,{\mathbb{C}}))\ \ \|\ \ q(l,l)=0,q(l,\overline{l})>0\\}.$ The global Torelli theorem ([V5]) can be stated as follows. Theorem 1.4: Let $M$ be a simple hyperkähler manifold, $\operatorname{Teich}_{b}$ the birational Teichmüller space, and $\operatorname{\sf Per}:\;\operatorname{Teich}_{b}{\>\longrightarrow\>}{\mathbb{P}}(H^{2}(M,{\mathbb{C}}))$ the period map. Then $\operatorname{\sf Per}$ maps $\operatorname{Teich}_{b}$ to $\operatorname{{\mathbb{P}}\sf er}$, inducing a diffeomorphism of each connected component of $\operatorname{Teich}_{b}$ with $\operatorname{{\mathbb{P}}\sf er}$. Proof: See [V5]. Remark 1.5: The period space $\operatorname{{\mathbb{P}}\sf er}$ is equipped with a transitive action of $SO(H^{2}(M,{\mathbb{R}}))$. Using this action, one can identify $\operatorname{{\mathbb{P}}\sf er}$ with the Grassmann space of 2-dimensional, positive, oriented planes $\operatorname{Gr}_{{}_{+,+}}(H^{2}(M,{\mathbb{R}}))=SO(b_{2}-3,3)/SO(2)\times SO(b_{2}-3,1)$. Indeed, for each $l\in{\mathbb{P}}H^{2}(M,{\mathbb{C}})$, the space generated by $\langle\operatorname{Im}l,\operatorname{Re}l\rangle$ is 2-dimensional, because $q(l,l)=0,q(l,\overline{l})\neq 0$ implies that $l\cap H^{2}(M,{\mathbb{R}})=0$. This produces a point of $\operatorname{Gr}_{{}_{+,+}}(H^{2}(M,{\mathbb{R}}))$ from $l\in\operatorname{{\mathbb{P}}\sf er}$. To obtain the converse correspondence, notice that for any 2-dimensional positive plane $V\in H^{2}(M,{\mathbb{R}})$, the quadric $\\{l\in V\otimes_{\mathbb{R}}{\mathbb{C}}\ \ \|\ \ q(l,l)=0\\}$ consists of two lines $l\in\operatorname{{\mathbb{P}}\sf er}$. A choice of one of two lines is determined by the orientation in $V$. We shall describe the Teichmüller space and the moduli of hyperkähler structures in the same spirit, as follows. Recall that any hyperkähler structure $(M,I,J,K,g)$ defines a triple of Kähler forms $\omega_{I},\omega_{J},\omega_{K}\in\Lambda^{2}(M)$ (Subsection 2.1). A hyperkähler structure on a simple hyperkähler manifold is determined by a complex structure and a Kähler class (2.1). We call hyperkähler structures equivalent if they can be obtained by a homothety and a quaternionic reparametrization: $(M,I,J,K,g)\sim(M,hIh^{-1},hJh^{-1},hKh^{-1},\lambda g),$ for $h\in{\mathbb{H}}^{}$, $\lambda\in{\mathbb{R}}^{>0}$. Let $\operatorname{Teich}^{\cal H}$ be the set of equivalence classes of hyperkähler structrues up to the action of $\operatorname{Diff}_{0}(M)$, and $\operatorname{Teich}^{\cal H}_{b}$ its quotient by $\sim$ (the non- separability relation). Theorem 1.6: Consider the period map $\operatorname{\sf Per}_{\cal H}:\;\operatorname{Teich}^{\cal H}_{b}{\>\longrightarrow\>}\operatorname{Gr}_{+++}(H^{2}(M,{\mathbb{R}}))$ associating the plane $\langle\omega_{I},\omega_{J},\omega_{K}\rangle$ in the Grassmannian of 3-dimensional positive oriented planes to an equivalence class of hyperkähler structures. Then $\operatorname{\sf Per}_{\cal H}$ is injective, and defines an open embedding on each connected component of $\operatorname{Teich}^{\cal H}_{b}$. Proof: As follows from global Torelli theorem (1.2) and 1.2, a complex structure is determined (up to diffeomorphism and a birational equivalence) by a 2-plane $V\in\operatorname{Gr}_{{}_{+,+}}(H^{2}(M,{\mathbb{R}}))=SO(b_{2}-3,3)/SO(2)\times SO(b_{2}-3,1)$, where $V=\langle\operatorname{Re}\Omega,\operatorname{Im}\Omega\rangle$, and $\Omega$ a holomorphically symplectic form (defined uniquely up to a multiplier). Let $\omega\in H^{1,1}(M,I)=V^{\bot}$ be a Kähler form. The corresponding hyperkähler structure gives an orthogonal triple of Kähler forms $\omega_{J},\omega_{K}\in V,\omega_{I}:=\omega\in V^{\bot}$ satisfying $q(\omega_{I},\omega_{I})=q(\omega_{J},\omega_{J})=q(\omega_{K},\omega_{K})=C$. The group $SU(2)\times{\mathbb{R}}^{>0}$ acts on the set of such orthogonal bases transitively. Therefore, a hyperkähler structure is determined (up to equivalence of hyperkähler structures and non-separability) by a 3-plane $W=\langle\omega_{I},\omega_{J},\omega_{K}\rangle\subset H^{2}(M,{\mathbb{R}})$. We have shown that $\operatorname{\sf Per}_{\cal H}$ is injective. To finish the proof of 1.2, it remains to show that $\operatorname{\sf Per}_{\cal H}$ is an open embedding. However, for a sufficiently small $v\in\langle\omega_{J},\omega_{K}\rangle^{\bot}=H^{1,1}_{\mathbb{R}}(M,I)$, the form $v+\omega_{I}$ is also Kähler (the Kähler cone is open in $H^{1,1}_{\mathbb{R}}(M,I)$), hence $W^{\prime}=\langle\omega_{I}+v,\omega_{J},\omega_{K}\rangle$ also belongs to an image of $\operatorname{\sf Per}_{\cal H}$. This implies that the differential $D(\operatorname{\sf Per}_{\cal H})$ is surjective. Every hyperkähler structure induces a whole 2-dimensional sphere of complex structures on $M$, as follows. Consider a triple $a,b,c\in{\mathbb{R}}$, $a^{2}+b^{2}+c^{2}=1$, and let $L:=aI+bJ+cK$ be the corresponding quaternion. Quaternionic relations imply immediately that $L^{2}=-1$, hence $L$ is an almost complex structure. Since $I,J,K$ are Kähler, they are parallel with respect to the Levi-Civita connection. Therefore, $L$ is also parallel. Any parallel complex structure is integrable, and Kähler. We call such a complex structure $L=aI+bJ+cK$ a complex structure induced by the hyperkähler structure. The corresponding complex manifold is denoted by $(M,L)$. There is a holomorphic family of induced complex structures, parametrized by $S^{2}={\mathbb{C}}P^{1}$. The total space of this family is called the _twistor space_ of a hyperkähler manifold; it is constructed as follows. Let $M$ be a hyperkähler manifold. Consider the product $\operatorname{Tw}(M)=M\times S^{2}$. Embed the sphere $S^{2}\subset{\mathbb{H}}$ into the quaternion algebra ${\mathbb{H}}$ as the set of all quaternions $J$ with $J^{2}=-1$. For every point $x=m\times J\in X=M\times S^{2}$ the tangent space $T_{x}\operatorname{Tw}(M)$ is canonically decomposed $T_{x}X=T_{m}M\oplus T_{J}S^{2}$. Identify $S^{2}$ with ${\mathbb{C}}P^{1}$, and let $I_{J}:T_{J}S^{2}\to T_{J}S^{2}$ be the complex structure operator. Consider the complex structure $I_{m}:T_{m}M\to T_{m}M$ on $M$ induced by $J\in S^{2}\subset{\mathbb{H}}$. The operator $I_{\operatorname{Tw}}=I_{m}\oplus I_{J}:T_{x}\operatorname{Tw}(M)\to T_{x}\operatorname{Tw}(M)$ satisfies $I_{\operatorname{Tw}}\circ I_{\operatorname{Tw}}=-1$. It depends smoothly on the point $x$, hence it defines an almost complex structure on $\operatorname{Tw}(M)$. This almost complex structure is known to be integrable (see e.g. [Sal], [Kal]). Definition 1.7: The space $\operatorname{Tw}(M)$ constructed above is called the twistor space of a hyperkähler manifold. The twistor space defines a family of deformations of a complex structire on $M$, called the twistor family; the corresponding curve in the Teichmüller space is called the twistor line. Let $(M,I,J,K)$ be a hyperkähler structure, and $W=\langle\omega_{I},\omega_{J},\omega_{K}\rangle$ the corresponding 3-dimensional plane. The twistor family gives a rational line ${\mathbb{C}}P^{1}\subset\operatorname{Teich}$, which can be recovered from $W$ as follows. Recall that by global Torelli theorem, each component of $\operatorname{Teich}$ is identified (up to gluing together non-separable points) with the Grassmannian $\operatorname{Gr}_{{}_{+,+}}(H^{2}(M,{\mathbb{R}}))$. There is a ${\mathbb{C}}P^{1}$ of oriented 2-dimensional planes in $W$; this family is precisely the twistor family associated with the hyperkähler structure corresponding to $W$. In the present paper, we consider what happens if one takes a 3-dimensional plane $W\subset H^{2}(M,{\mathbb{R}})$ with a degenerate metric of signature $(+,+,0)$. Instead of a ${\mathbb{C}}P^{1}$ worth of complex structures, as happens when $W$ is positive, the set of positive 2-planes in $W\subset H^{2}(M,{\mathbb{R}})$ is parametrized by ${\mathbb{C}}={\mathbb{R}}^{2}$. It turns out that the corresponding family can be constructed explicitly from an appropriate semipositive form on a manifold, whenever such a form exists. Moreover, this family (called a degenerate twistor family; see 3.5) is holomorphic and has a canonical smooth trivialization, just as the usual twistor family. ### 1.3 Semipositive (1,1)-forms, degenerate twistor families and SYZ conjecture Let $(M,I,\Omega)$ be a simple holomorphically symplectic manifold of Kähler type (that is, a hyperkähler manifold), and $\eta\in\Lambda^{1,1}(M,I)$ a real, positive, closed $(1,1)$-form. By Fujiki formula, either $\eta$ is strictly positive somewhere, or at least half of the eigenvalues of $\eta$ vanish (3.2). In the latter case, the form $\Omega_{t}:=\Omega+t\eta$ is non- degenerate and satisfies the assumption $\Omega_{t}^{n+1}=0$ for all $t$, hence defines a complex structure (3.2). This is used to define the degenerate twistor space (3.5). Positive, closed forms $\eta\in\Lambda^{1,1}(M)$ with $\int_{M}\eta^{\dim_{\mathbb{C}}M}=0$ are called semipositive. Such forms necessarily lie in the boundary of a Kähler cone; this implies that their cohomology classes are nef (3.2). Notice that we exclude strictly positive forms from this definition. Remark 1.8: The conventions for positivity of differential forms and currents are intrinsically confusing. Following the French tradition, one says “positive form” meaning really “non-negative”, and “strictly positive” meaning “positive definite”. On top of it, for $(n-k,n-k)$ forms on $n$-manifold, with $2\leqslant k\leqslant n-2$, there are two notions of positive forms, called “strongly positive” and “weakly positive”; this creates monsters such that “stricly weakly positive” and “non-strictly stronly positive”. The various notions of positivity in this paper are taken (mostly) from [D], following the French conventions as explained. The study of nef classes which satisfy $\int_{M}\eta^{\dim_{\mathbb{C}}M}=0$ (such classes are called parabolic) is one of the central themes of hyperkähler geometry. One of the most important conjectures in this direction is the so-called hyperkähler SYZ conjecture, due to Tyurin-Bogomolov-Hassett- Tschinkel-Huybrechts-Sawon ([HT], [Saw], [Hu2]; for more history, please see [V3]). This conjecture postulates that any rational nef class $\eta$ on a hyperkähler manifold is semiample, that is, associated with a holomorphic map $\varphi:\;M{\>\longrightarrow\>}X$, $\eta=\varphi^{}\omega_{X}$, where $\omega_{X}$ is a Kähler class on $X$. For nef classes which satisfy $\int_{M}\eta^{\dim_{\mathbb{C}}M}>0$ (such nef classes are known as big), semiampleness follows from the Kawamata base point free theorem ([Kaw]), but for parabolic classes it is quite non-trivial. If a parabolic class $\eta$ is semiample, it can obviously be represented by a smooth, semipositive differential form. The converse implication is not proven. However, in [V3] it was shown that whenever a rational parabolic class can be represented by a semipositive form, it is ${\mathbb{Q}}$-effective (that is, represented by a rational effective divisor). Existence of a smooth semipositive form in a given nef class is a separate (and interesting) question of hyperkähler geometry. The following conjecture is supported by empirical evidence obtained by S. Cantat and Dinh-Sibony ([C1], [C2, Theorem 5.3], [DS, Corollary 3.5]). Conjecture 1.9: Let $\eta$ be a parabolic nef class on a hyperkähler manifold. Then $\eta$ can be represented by a semipositive closed form with mild (say, Hölder) singularities. Notice that $\eta$ can be represented by a closed, positive current by compactness of the space of positive currents with bounded mass; however, there is no clear way to understand the singularities of this current. If this conjecture is true, a cohomology class is ${\mathbb{Q}}$-effective whenever it is nef and rational ([V3], [V4]); this would prove a part of SYZ conjecture. One of the ways of representing a nef class by a semipositive form is based on reverse-engineering the construction of degenerate twistor spaces. Let $\eta$ be a parabolic nef class on a hyperkähler manifold $(M,I)$, $\Omega$ its holomorphic symplectic form, and $W:=\langle\eta,\operatorname{Re}\Omega,\operatorname{Im}\Omega\rangle$ the corresponding 3-dimensional subspace in $H^{2}(M,{\mathbb{R}})$. Clearly, the Bogomolov-Beauville-Fujiki form on $W$ is degenerate of signature $(+,+,0)$. The set $S$ of positive, oriented 2-dimensional planes $V\subset W$ is parametrized by ${\mathbb{C}}$. Identifying the Grassmannian $\operatorname{Gr}_{++}(H^{2}(M,{\mathbb{R}}))$ with a component of $\operatorname{Teich}_{b}$ as in 1.2, we obtain a deformation ${\cal Z}{\>\longrightarrow\>}S$; as explained in Subsection 1.2, this family can be obtained as a limit of twistor families. The twistor families are split as smooth manifolds: $\operatorname{Tw}(M)=M\times{\mathbb{C}}P^{1}$; this gives an Ehresmann connection $\nabla$ on the twistor family $\operatorname{Tw}(M){\>\longrightarrow\>}{\mathbb{C}}P^{1}$. This connection satisfies $\nabla\Omega_{t}=\lambda\omega_{I}$, that is, a derivative of a holomorphically symplectic form is proportional to a Kähler form. If this connection converges to a smooth connection $\nabla_{0}$ on the limit family ${\cal Z}{\>\longrightarrow\>}{\mathbb{C}}$, we would obtain $\nabla\Omega_{t}=\lambda\eta$, where $\eta$ is a limit of Kähler forms, hence semipositive. This was the original motivation for the study of degenerate twistor spaces. ### 1.4 Degenerate twistor spaces and Lagrangian fibrations The main source of examples of degenerate twistor families comes from Lagrangian fibrations. Let $(M,\Omega)$ be a simple holomorphically symplectic Kähler manifold, and $\varphi:\;M{\>\longrightarrow\>}X$ a surjective holomorphic map, with $0<\dim X<\dim M$. Matsushita (2.3) has shown that $\varphi$ is a Lagrangian fibration, that is, the fibers of $\varphi$ are Lagrangian subvarieties in $M$, and all smooth fibers of $\varphi$ are Lagrangian tori. It is not hard to see that $X$ is projective ([Mat2]). Let $\omega_{X}$ be the Kähler form on $X$. Then $\eta:=\varphi^{}\omega_{X}$ is a semipositive form, and 3.2 together with 3.1 imply existence of a degenerate twistor family ${\cal Z}{\>\longrightarrow\>}{\mathbb{C}}$, with the fibers holomorphically symplectic manifolds $(M,\Omega+t\eta)$, $t\in{\mathbb{C}}$. For each fiber $Y:=\varphi^{-1}(y)$, the restriction $\eta{\left\|{}_{{\phantom{\|}\\!\\!}_{Y}}\right.}$ vanishes, because $\eta=\varphi^{}\omega_{X}$. Therefore, the complex structure induced by $\Omega_{t}=\Omega+t\eta$ on $Y$ does not depend on $t$. This implies that the fibers of $\varphi$ remain holomorphic and independent from $t\in{\mathbb{C}}$. Theorem 1.10: Let $M$ be a simple hyperkähler manifold equipped with a Lagrangian fibration $\varphi:\;M{\>\longrightarrow\>}X$, and $(M_{t},\Omega_{t})$ the degenerate twistor deformation associated with the family of non-degenerate 2-forms $\Omega+t\eta$, $\eta=\varphi^{}\omega_{X}$ as in 3.2. Then the fibration $M_{t}\stackrel{{\scriptstyle\varphi_{t}}}{{{\>\longrightarrow\>}}}X$ is also holomorphic, and for any fixed $x\in X$, the fibers of $\varphi_{t}$ are naturally isomorphic: $\varphi_{t}^{-1}(x)\cong\varphi^{-1}(x)$ for all $t\in{\mathbb{C}}$. Proof: The complex structure on $M_{t}$ is determined from $T^{0,1}M_{t}=\ker\Omega_{t}$. Let $Z:=\varphi^{-1}(x)$. Since $\eta(v,\cdot)=0$ for each $v\in T_{z}Z$, one has $TZ\cap\ker\Omega_{t}=T^{0,1}Z$, hence the complex structure on $Z$ is independent from $t$. Since $Z$ is Lagrangian in $M_{t}$, its normal bundle is dual to $TZ$ and trivial when $Z$ is a torus (that is, for all smooth fibers of $\varphi$). Therefore, the complex structure on $NZ$ is independent from $t\in{\mathbb{C}}$. This implies that the projection $M_{t}\stackrel{{\scriptstyle\varphi}}{{{\>\longrightarrow\>}}}X$ is holomorphic in the smooth locus of $\varphi$ for all $t\in{\mathbb{C}}$. To extend it to the points where $\varphi$ is singular, we notice that a map is holomorphic whenever its differential is complex linear, and complex linearity of a given tensor needs to be checked only in an open dense subset. Remark 1.11: In [Mar], Eyal Markman considered the following procedure. One starts with a Lagrangian fibration $\pi$ on a hyperkähler manifold and takes a 1-cocycle on the base of $\pi$ taking values in fiberwise automorphisms of the fibration. Twisting the $\pi$ by such a cocycle, one obtains another Lagrangian fibration with the same base and the respective fibers isomorphic to that of $\pi$. Markman calls this procedure “the Tate-Shafarevich twist”. In this context, degenerate twistor deformations associated with semipositive forms $\eta$, $[\eta]\in H^{2}(M,{\mathbb{Z}})$, occur very naturally; Markman calls them “Tate-Shafarevich lines”. One can view $\eta=\varphi^{}\omega_{X}$ as lying in $\varphi^{}H^{1,1}(X)=\varphi^{}H^{1}(X,\Omega^{1}X)\subset H^{1}(M,\varphi^{}\Omega^{1}X)=H^{1}(M,T_{M/X}),$ where $T_{M/X}$ is the fiberwise tangent bundle, and $\varphi^{}\Omega^{1}X=T_{M/X}$ because $M{\>\longrightarrow\>}X$ is a Lagrangian fibration. Of course, this cocycle comes from $X$ so it is constant in the fibre direction; it describes the deformation infinitesimally. Integrating the vector field then gives a 1-cocycle on $X$ taking values in the bundle of fibrewise automorphisms. This is the 1-cocycle giving the ”Tate- Shafarevich twist”. Remark 1.12: The degenerate twistor family constructed in 3.5 consists of a family of complex structures, but it is not proven that all fibers, which are complex manifolds, are also Kähler (hence hyperähler). As is, the Kähler property is known only over a small open subset in the base (affine line), since the condition of being Kähler is open. We expect all members of the degenerate twistor family to be Kähler, but there is no obvious way to prove this. However, it is easy to show that the set of points on the base affine line corresponding to non-Kähler complex structures is closed and countable. ## 2 Basic notions of hyperkähler geometry ### 2.1 Hyperkähler manifolds Definition 2.1: Let $(M,g)$ be a Riemannian manifold, and $I,J,K$ endomorphisms of the tangent bundle $TM$ satisfying the quaternionic relations $I^{2}=J^{2}=K^{2}=IJK=-\operatorname{Id}_{TM}.$ The triple $(I,J,K)$ together with the metric $g$ is called a hyperkähler structure if $I,J$ and $K$ are integrable and Kähler with respect to $g$. Consider the Kähler forms $\omega_{I},\omega_{J},\omega_{K}$ on $M$: $\omega_{I}(\cdot,\cdot):=g(\cdot,I\cdot),\ \ \omega_{J}(\cdot,\cdot):=g(\cdot,J\cdot),\ \ \omega_{K}(\cdot,\cdot):=g(\cdot,K\cdot).$ (2.1) An elementary linear-algebraic calculation implies that the 2-form $\Omega:=\omega_{J}+\sqrt{-1}\>\omega_{K}$ (2.2) is of Hodge type $(2,0)$ on $(M,I)$. This form is clearly closed and non- degenerate, hence it is a holomorphic symplectic form. In algebraic geometry, the word “hyperkähler” is essentially synonymous with “holomorphically symplectic”, due to the following theorem, which is implied by Yau’s solution of Calabi conjecture ([Bea, Bes]). Theorem 2.2: Let $M$ be a compact, Kähler, holomorphically symplectic manifold, $\omega$ its Kähler form, $\dim_{\mathbb{C}}M=2n$. Denote by $\Omega$ the holomorphic symplectic form on $M$. Assume that $\int_{M}\omega^{2n}=\int_{M}(\operatorname{Re}\Omega)^{2n}$. Then there exists a unique hyperkähler metric $g$ within the same Kähler class as $\omega$, and a unique hyperkähler structure $(I,J,K,g)$, with $\omega_{J}=\operatorname{Re}\Omega$, $\omega_{K}=\operatorname{im}\Omega$. ### 2.2 The Bogomolov-Beauville-Fujiki form Definition 2.3: A hyperkähler manifold $M$ is called simple if $\pi_{1}(M)=0$, $H^{2,0}(M)={\mathbb{C}}$. In the literature, such manifolds are often called irreducible holomorphic symplectic, or irreducible symplectic varieties. This definition is motivated by the following theorem of Bogomolov ([Bo1]). Theorem 2.4: ([Bo1]) Any hyperkähler manifold admits a finite covering which is a product of a torus and several simple hyperkähler manifolds. Theorem 2.5: ([F]) Let $\eta\in H^{2}(M)$, and $\dim M=2n$, where $M$ is a simple hyperkähler manifold. Then $\int_{M}\eta^{2n}=\lambda q(\eta,\eta)^{n}$, for some integer quadratic form $q$ on $H^{2}(M)$, and $\lambda\in{\mathbb{Q}}$ a positive rational number. Definition 2.6: This form is called Bogomolov-Beauville-Fujiki form. It is defined by this relation uniquely, up to a sign. The sign is determined from the following formula (Bogomolov, Beauville; [Bea], [Hu2], 23.5) $\displaystyle\lambda q(\eta,\eta)$ $\displaystyle=(n/2)\int_{X}\eta\wedge\eta\wedge\Omega^{n-1}\wedge\overline{\Omega}^{n-1}-$ $\displaystyle-(1-n)\frac{\left(\int_{X}\eta\wedge\Omega^{n-1}\wedge\overline{\Omega}^{n}\right)\left(\int_{X}\eta\wedge\Omega^{n}\wedge\overline{\Omega}^{n-1}\right)}{\int_{M}\Omega^{n}\wedge\overline{\Omega}^{n}}$ where $\Omega$ is the holomorphic symplectic form, and $\lambda$ a positive constant. Remark 2.7: The form $q$ has signature $(3,b_{2}-3)$. It is negative definite on primitive forms, and positive definite on the space $\langle\operatorname{Re}\Omega,\operatorname{Im}\Omega,\omega\rangle$ where $\omega$ is a Kähler form, as seen from the following formula $\mu q(\eta_{1},\eta_{2})=\\\ \int_{X}\omega^{2n-2}\wedge\eta_{1}\wedge\eta_{2}-\frac{2n-2}{(2n-1)^{2}}\frac{\int_{X}\omega^{2n-1}\wedge\eta_{1}\cdot\int_{X}\omega^{2n-1}\wedge\eta_{2}}{\int_{M}\omega^{2n}},\ \ \mu>0$ (2.3) (see e. g. [V2], Theorem 6.1, or [Hu2], Corollary 23.9). Definition 2.8: Let $[\eta]\in H^{1,1}(M)$ be a real (1,1)-class in the closure of the Kähler cone of a hyperkähler manifold $M$. We say that $[\eta]$ is parabolic if $q([\eta],[\eta])=0$. ### 2.3 The hyperkähler SYZ conjecture Theorem 2.9: (D. Matsushita, see [Mat1]). Let $\pi:\;M{\>\longrightarrow\>}X$ be a surjective holomorphic map from a simple hyperkähler manifold $M$ to a complex variety $X$, with $0<\dim X<\dim M$. Then $\dim X=1/2\dim M$, and the fibers of $\pi$ are holomorphic Lagrangian (this means that the symplectic form vanishes on the fibers).111Here, as elsewhere, we silently assume that the hyperkähler manifold $M$ is simple. Definition 2.10: Such a map is called a holomorphic Lagrangian fibration. Remark 2.11: The base of $\pi$ is conjectured to be rational. J.-M. Hwang ([Hw]) proved that $X\cong{\mathbb{C}}P^{n}$, if $X$ is smooth and $M$ projective. D. Matsushita ([Mat2]) proved that it has the same rational cohomology as ${\mathbb{C}}P^{n}$ when $M$ is projective. Remark 2.12: The base of $\pi$ has a natural flat connection on the smooth locus of $\pi$. The combinatorics of this connection can be (conjecturally) used to determine the topology of $M$ ([KZ1], [KZ2], [G]). Remark 2.13: Matsushita’s theorem is implied by the following formula of Fujiki. Let $M$ be a hyperkähler manifold, $\dim_{\mathbb{C}}M=2n$, and $\eta_{1},...,\eta_{2n}\in H^{2}(M)$ cohomology classes. Then $C\int_{M}\eta_{1}\wedge\eta_{2}\wedge...=\frac{1}{(2n)!}\sum_{\sigma}q(\eta_{\sigma_{1}}\eta_{\sigma_{2}})q(\eta_{\sigma_{3}}\eta_{\sigma_{4}})...q(\eta_{\sigma_{2n-1}}\eta_{\sigma_{2n}})$ (2.4) with the sum taken over all permutations, and $C$ a positive constant, called Fujiki constant. An algebraic argument (see e.g. 2.4) allows to deduce from this formula that for any non-zero $\eta\in H^{2}(M)$, one would have $\eta^{n}\neq 0$, and $\eta^{n+1}=0$, if $q(\eta,\eta)=0$, and $\eta^{2n}\neq 0$ otherwise. Applying this to the pullback $\pi^{}\omega_{X}$ of the Kähler class from $X$, we immediately obtain that $\dim_{\mathbb{C}}X=n$ or $\dim_{\mathbb{C}}X=2n$. Indeed, $\omega_{X}^{\dim_{\mathbb{C}}X}\neq 0$ and $\omega_{X}^{\dim_{\mathbb{C}}X+1}=0$. This argument was used by Matsushita in his proof of 2.3. The relation (2.4) is another form of Fujiki’s theorem (2.2), obtained by differentiation of $\int_{M}\eta^{2n}=\lambda q(\eta,\eta)^{n}$, ### 2.4 Cohomology of hyperkähler manifolds Further on in this paper, some basic results about cohomology of hyperkähler manifolds will be used. The following theorem was proved in [V2], using representation theory. Theorem 2.14: ([V2]) Let $M$ be a simple hyperkähler manifold, and $H^{}_{r}(M)$ the part of cohomology generated by $H^{2}(M)$. Then $H^{}_{r}(M)$ is isomorphic to the symmetric algebra (up to the middle degree). Moreover, the Poincare pairing on $H^{}_{r}(M)$ is non-degenerate. This brings the following corollary. Corollary 2.15: Let $\eta_{1},...\eta_{n+1}\in H^{2}(M)$ be cohomology classes on a simple hyperkähler manifold, $\dim_{\mathbb{C}}M=2n$. Suppose that $q(\eta_{i},\eta_{j})=0$ for all $i,j$. Then $\eta_{1}\wedge\eta_{2}\wedge...\wedge\eta_{n+1}=0$. Proof: See e.g. [V4, Corollary 2.15]. This equation also follows from (2.4). ## 3 Degenerate twistor space ### 3.1 Integrability of almost complex structures and Cartan formula An almost complex structure on a manifold is a section $I\in\operatorname{End}(TM)$ of the bundle of endomorphisms, satisfying $I^{2}=-\operatorname{Id}$. It is called integrable if $[T^{1,0}M,T^{1,0}M]\subset T^{1,0}M$, where $T^{1,0}M\subset TM\otimes_{\mathbb{R}}{\mathbb{C}}$ is the eigenspace of $I$, defined by $v\in T^{1,0}M\Leftrightarrow I(v)=\sqrt{-1}\>v.$ Equivalently, $I$ is integrable if $[T^{0,1}M,T^{0,1}M]\subset T^{0,1}M$, where $T^{0,1}M\subset TM\otimes_{\mathbb{R}}{\mathbb{C}}$ is a complex conjugate to $T^{1,0}M\subset TM\otimes_{\mathbb{R}}{\mathbb{C}}$. One of the ways of making sure a given almost complex structure is integrable is by using the Cartan formula expressing the de Rham differential through commutators of vector fields. Proposition 3.1: Let $(M,I)$ be a manifold equipped with an almost complex structure, and $\Omega\in\Lambda^{2,0}(M)$ a non-degenerate $(2,0)$-form (3.1). Assume that $d\Omega=0$. Then $I$ is integrable. Proof: Let $X\in T^{1,0}M$ and $Y,Z\in T^{0,1}(M)$. Since $\Omega$ is a (2,0)-form, it vanishes on $(0,1)$-vectors. Then Cartan formula together with $d\Omega=0$ implies that $0=d\Omega(X,Y,Z)=\Omega(X,[Y,Z]).$ (3.1) From the non-degeneracy of $\Omega$ we obtain that unless $[Y,Z]\in T^{0,1}(M)$, for some $X\in T^{1,0}M$, one would have $\Omega(X,[Y,Z])\neq 0$. Therefore, (3.1) implies $[Y,Z]\in T^{0,1}(M)$, for all $Y,Z\in T^{0,1}(M)$, which means that $I$ is integrable. Remark 3.2: It is remarkable that the closedness of $\Omega$ is in fact unnecessary. The proof 3.1 remains true if one assumes that $d\Omega\in\Lambda^{3,0}(M)\oplus\Lambda^{2,1}(M)$. Notice that the sub-bundle $T^{1,0}M\subset TM\otimes_{\mathbb{R}}{\mathbb{C}}$ uniquely determines the almost complex structure. Indeed, $I(x+y)=\sqrt{-1}\>x-\sqrt{-1}\>y$, for all $x\in T^{1,0}M,y\in T^{0,1}M=\overline{T^{1,0}M}$, and we have a decomposition $T^{1,0}M\oplus T^{0,1}M=TM\otimes_{\mathbb{R}}{\mathbb{C}}$. This decomposition is the necessarily and sufficient ingredient for the reconstruction of an almost complex structure: Claim 3.3: Let $M$ be a smooth, $2n$-dimensional manifold. Then there is a bijective correspondence between the set of almost complex structures, and the set of sub-bundles $T^{0,1}M\subset TM\otimes_{\mathbb{R}}{\mathbb{C}}$ satisfying $\dim_{\mathbb{C}}T^{0,1}M=n$ and $T^{0,1}M\cap TM=0$ (the last condition means that there are no real vectors in $T^{1,0}M$). The last two statements allow us to define complex structures in terms of complex-valued 2-forms (see 3.1 below). For this theorem, any reasonable notion of non-degeneracy would suffice; for the sake of clarity, we state the one we would use. Definition 3.4: Let $\Omega\in\Lambda^{2}(M,{\mathbb{C}})$ be a smooth, complex-valued 2-form on a $2n$-dimensional manifold. $\Omega$ is called non- degenerate if for any real vector $v\in T_{m}M$, the contraction $\Omega\hskip 2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip 2.0ptv$ is non-zero. Theorem 3.5: Let $\Omega\in\Lambda^{2}(M,{\mathbb{C}})$ be a smooth, complex- valued, non-degenerate 2-form on a $4n$-dimensional real manifold. Assume that $\Omega^{n+1}=0$. Consider the bundle $T^{0,1}_{\Omega}(M):=\\{v\in TM\otimes{\mathbb{C}}\ \ \|\ \ \Omega\hskip 2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip 2.0ptv=0\\}.$ Then $T^{0,1}_{\Omega}(M)$ satisfies assumptions of 3.1, hence defines an almost complex structure $I_{\Omega}$ on $M$. If, in addition, $\Omega$ is closed, $I_{\Omega}$ is integrable. Proof: Integrability of $I_{\Omega}$ follows immediately from 3.1. Let $v\in TM$ be a non-zero real tangent vector. Then $\Omega\hskip 2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip 2.0ptv\neq 0$, hence $T^{0,1}_{\Omega}(M)\cap TM=0$. To prove 3.1, it remains to show that $\operatorname{rk}T^{0,1}_{\Omega}(M)\geqslant 2n$. Clearly, $\Omega$ is non- degenerate on $\frac{TM\otimes{\mathbb{C}}}{T^{0,1}_{\Omega}(M)}$, hence its rank is equal to $4n-\operatorname{rk}T^{0,1}_{\Omega}(M)$. From $\Omega^{n+1}=0$ it follows that rank of $\Omega$ cannot exceed $2n$, hence $\operatorname{rk}T^{0,1}_{\Omega}(M)\geqslant 2n$. ### 3.2 Semipositive (1,1)-forms on hyperkähler manifold Definition 3.6: Let $\eta\in\Lambda^{1,1}(M,{\mathbb{R}})$ be a real (1,1)-form on a complex manifold $(M,I)$. It is called semipositive if $\eta(x,Ix)\geqslant 0$ for any $x\in TM$, but it is nowhere positive definite. Remark 3.7: Fix a Hermitian structure $h$ on $(M,I)$. Clearly, any semipositive (1,1)-form is diagonal in some $h$-orthonormal basis in $TM$. The entries of its matrix in this basis are called eigenvalues; they are real, non-negative numbers. The maximal number of positive eigenvalues is called the rank of a semipositive (1,1)-form. Definition 3.8: A closed semipositive form $\eta$ on a compact Kähler manifold $(M,I,\omega)$ is a limit of Kähler forms $\eta+\varepsilon\omega$, hence its cohomology class is nef (belongs to the closure of the Kähler cone). Its cohomology class $[\eta]$ is parabolic, that is, satisfies $\int_{M}[\eta]^{\dim_{\mathbb{C}}M}=0$. However, not every parabolic nef class can be represented by a closed semipositive form ([DPS]). Proposition 3.9: On a simple hyperkähler manifold $M$, $\dim_{\mathbb{C}}M=2n$, any semipositive (1,1)-form has rank $0$ or $2n$. Proof: This assertion easily follows from 2.4. Indeed, if $q(\eta,\eta)\neq 0$, one has $\int_{M}\eta^{2n}=\lambda q(\eta,\eta)^{n}\neq 0$, hence its rank is $4n$. If $q(\eta,\eta)=0$, its cohomology class $[\eta]$ satisfies $[\eta]^{n}\neq 0$ and $[\eta]^{n+1}=0$ (2.4). Since all eigenvalues of $\eta$ are non-negative, its rank is twice the biggest number $k$ for which one has $\eta^{k}\neq 0$. However, since $\eta^{k}$ is a sum of monomials of an orthonormal basis with non-negative coefficients, $\int_{M}\eta^{k}\wedge\omega^{2n-k}=0$ $\Leftrightarrow$ $\eta^{k}=0$ for any Kähler form $\omega$ on $(M,I)$. Then $[\eta]^{n}\neq 0$ and $[\eta]^{n+1}=0$ imply that the rank of $\eta$ is $2n$. The main technical result of this paper is the following theorem. Theorem 3.10: Let $(M,\Omega)$ be an simple hyperkähler manifold, $\dim_{\mathbb{R}}M=4n$, and $\eta\in\Lambda^{1,1}(M,I)$ a closed, semipositive form of rank $2n$. Then the 2-form $\Omega+t\eta$ satisfies the assumptions of 3.1 for all $t\in{\mathbb{C}}$: namely, $\Omega+t\eta$ is non- degenerate, and $(\Omega+t\eta)^{n+1}=0$. Proof: Non-degeneracy of $\Omega_{t}:=\Omega+t\eta$ is clear. Indeed, let $v:=\|t\|t^{-1}$, and let $\omega_{v}:=\operatorname{Re}v\omega_{K}-\operatorname{im}v\omega_{J}$. Then $\omega_{v}$ is a Hermitian form associated with the induced complex structure $\operatorname{Im}vJ-\operatorname{Re}vK$, hence it is non-degenerate. However, the imaginary part of $v\Omega_{t}$ is equal to $\omega_{v}$ (see (2.1)). Then $\operatorname{Im}(\Omega_{t}\hskip 2.0pt\raisebox{1.0pt}{\text{$\lrcorner$}}\hskip 2.0ptv)\neq 0$ for each non- zero real vector $v\in TM$. To see that $(\Omega+t\eta)^{n+1}=0$, we observe that this relation is true in cohomology; this is implied from [V1] using the same argument as was used in the proof of 3.2. Each Hodge component of $(\Omega+t\eta)^{n+1}$ is proportional to $\Omega^{n-p}\wedge\eta^{p+1}$, and it is sufficient to prove that $\Omega^{n-p}\wedge\eta^{p+1}=0$ for all $p$. We deduce this from two observations, which are proved further on in this section. Lemma 3.11: Let $(M,\Omega)$, $\dim_{\mathbb{R}}M=4n$ be a holomorphically symplectic manifold, and $\eta\in\Lambda^{1,1}(M,I)$ a closed, semipositive form of rank $2n$. Assume that $\Omega^{n-p}\wedge\eta^{p+1}$ is exact. Then $\Omega^{n-p}\wedge\overline{\Omega}^{n-p}\wedge\eta^{p+1}=0,$ for all $p$. Proof: See Subsection 3.3. Lemma 3.12: Let $(M,\Omega)$, $\dim_{\mathbb{R}}M=4n$, be a holomorphically symplectic manifold and $\rho\in\Lambda^{p+1,p+1}(M,I)$ a strongly positive form (3.3). Suppose that $\Omega^{n-p}\wedge\overline{\Omega}^{n-p}\wedge\rho=0$. Then $\Omega^{n-p}\wedge\rho=0$. Proof: See Subsection 3.4. ### 3.3 Positive $(p,p)$-forms We recall the definition of a positive $(p,p)$-form (see e.g. [D]). Definition 3.13: Recall that a real $(p,p)$-form $\eta$ on a complex manifold is called weakly positive if for any complex subspace $V\subset TM$, $\dim_{\mathbb{C}}V=p$, the restriction $\rho{\left\|{}_{{\phantom{\|}\\!\\!}_{V}}\right.}$ is a non-negative volume form. Equivalently, this means that $(\sqrt{-1}\>)^{p}\rho(x_{1},\overline{x}_{1},x_{2},\overline{x}_{2},...,x_{p},\overline{x}_{p})\geqslant 0,$ for any vectors $x_{1},...x_{p}\in T_{x}^{1,0}M$. A real $(p,p)$-form on a complex manifold is called strongly positive if it can be locally expressed as a sum $\eta=(\sqrt{-1}\>)^{p}\sum_{i_{1},...i_{p}}\alpha_{i_{1},...i_{p}}\xi_{i_{1}}\wedge\overline{\xi}_{i_{1}}\wedge...\wedge\xi_{i_{p}}\wedge\overline{\xi}_{i_{p}},\ \ $ running over some set of $p$-tuples $\xi_{i_{1}},\xi_{i_{2}},...,\xi_{i_{p}}\in\Lambda^{1,0}(M)$, with $\alpha_{i_{1},...,i_{p}}$ real and non-negative functions on $M$. The following basic linear algebra observations are easy to check (see [D]). All strongly positive forms are also weakly positive. The strongly positive and the weakly positive forms form closed, convex cones in the space $\Lambda^{p,p}(M,{\mathbb{R}})$ of real $(p,p)$-forms. These two cones are dual with respect to the Poincare pairing $\Lambda^{p,p}(M,{\mathbb{R}})\times\Lambda^{n-p,n-p}(M,{\mathbb{R}}){\>\longrightarrow\>}\Lambda^{n,n}(M,{\mathbb{R}})$ For (1,1)-forms and $(n-1,n-1)$-forms, the strong positivity is equivalent to weak positivity. Finally, a product of a weakly positive form and a strongly positive one is always weakly positive (however, a product of two weakly positive forms may be not weakly positive). Clearly, an exact weakly positive form $\eta$ on a compact Kähler manifold $(M,\omega)$ always vanishes. Indeed, the integral $\int_{M}\eta\wedge\omega^{\dim M-p}$ is strictly positive for a non-zero weakly positive $\eta$, because the convex cones of weakly and strongly positive forms are dual, and $\omega^{\dim M-p}$ sits in the interior of the cone of strongly positive forms. However, by Stokes’ formula, this integral vanishes whenever $\eta$ is exact. Now we are in position to prove 3.2. The form $\Omega^{n-p}\wedge\overline{\Omega}^{n-p}\wedge\eta^{p+1}$ is by assumption of this lemma exact, but it is a product of a weakly positive form $\Omega^{n-p}\wedge\overline{\Omega}^{n-p}$ and a strongly positive form $\eta^{p+1}$, hence it is weakly positive. Being exact, this form must vanish. Remark 3.14: A form is strongly positive if it is generated by products of $dz_{i}\wedge d\overline{z}_{i}$ with positive coefficients; hence $\eta$ and all its powers are positive. The form $\Omega\wedge\overline{\Omega}$ and its powers are positive on all complex spaces of appropriate dimensions, which can be seen by using Darboux coordinates. This means that this form is weakly positive. ### 3.4 Positive $(p,p)$-forms and holomorphic symplectic forms Now we shall prove 3.2. This is a linear-algebraic statement, which can be proven pointwise. Fix a complex vector space $V$, equipped with a non- degenerate complex linear 2-form $\Omega$. Every strongly positive form $\rho$ on $V$ is a sum of monomials $(\sqrt{-1}\>)^{p}\xi_{i_{1}}\wedge\overline{\xi}_{i_{1}}\wedge...\wedge\xi_{i_{p}}\wedge\overline{\xi}_{i_{p}}$ with positive coefficients, and the equivalence $\Omega^{n-p}\wedge\rho\neq 0\Leftrightarrow\Omega^{n-p}\wedge\overline{\Omega}^{n-p}\wedge\rho\neq 0$ is implied by the following sublemma. Sublemma 3.15: Let $V$ be a complex vector space, equipped with a non- degenerate complex linear 2-form $\Omega\in\Lambda^{2,0}V$. Then for any monomial $\rho=(\sqrt{-1}\>)^{p}\xi_{i_{1}}\wedge\overline{\xi}_{i_{1}}\wedge...\wedge\xi_{i_{p}}\wedge\overline{\xi}_{i_{p}}$ for which $\Omega^{n-p}\wedge\rho$ is non-zero, the form $\Omega^{n-p}\wedge\overline{\Omega}^{n-p}\wedge\rho$ is non-zero and weakly positive. Proof: Let $\xi_{j_{1}},\xi_{j_{1}},...,\xi_{j_{n-p}}$ be the elements of the basis in $V$ complementary to $\xi_{i_{1}},\xi_{i_{1}},...,\xi_{i_{p}}$, and $W\subset V$ the space generated by $\xi_{j_{1}},\xi_{j_{1}},...,\xi_{j_{n-p}}$. Clearly, a form $\alpha$ is non- zero on $W$ if and only if $\alpha\wedge\rho$ is non-zero, and positive on $W$ if and only if $\alpha\wedge\rho$ is positive. Now, 3.4 is implied by the following trivial assertion: for any $(n-p)$-dimensional subspace $W\subset V$ such that $\Omega^{n-p}{\left\|{}_{{\phantom{\|}\\!\\!}_{W}}\right.}$ is non-zero, the restriction $\Omega^{n-p}\wedge\overline{\Omega}^{n-p}{\left\|{}_{{\phantom{\|}\\!\\!}_{W}}\right.}$ is non-zero and positive. This proves 3.4, and 3.2 follows as indicated. As a corollary of the vanishing of the forms $\Omega^{n-p}\wedge\eta^{p+1}$, we prove the following statement, used further on. Lemma 3.16: Let $(M,\Omega)$ be a simple holomorphically symplectic manifold, $\dim_{\mathbb{R}}M=4n$ and $\eta\in\Lambda^{1,1}(M,I)$ a closed, semipositive form of rank $2n$. Let $I_{t}$ be the complex structure on $M$ defined by $\Omega+t\eta$, as in 3.2. Then $\eta\in\Lambda^{1,1}(M,I_{t})$. Proof: By construction, $(M,I_{t})$ is a holomorphically symplectic manifold, with the holomorphic symplectic form $\Omega_{t}:=\Omega+t\eta$. For a holomorphic symplectic manifold $(M,\Omega_{t})$, $\dim_{\mathbb{R}}M=4n$, there exist an elementary criterion allowing one to check whether a given 2-form $\eta$ is of type (1,1): one has to have $\eta\wedge\Omega_{t}^{n}=0$ and $\eta\wedge\overline{\Omega}_{t}^{n}=0$. However, from 3.2 it follows immediately that $\eta\wedge\Omega_{t}^{n}=0$ and $\eta\wedge\overline{\Omega}_{t}^{n}=0$, hence $\eta$ is of type (1,1). ### 3.5 Degenerate twistor space: a definition Just as it is done with the usual twistor space, to define a degenerate twistor space we construct a certain almost complex structure, and then prove it is integrable. The proof of integrability is in fact identical to the argument which could be used to prove that the usual twistor space is integrable. Definition 3.17: Let $(M,\Omega)$ be an irreducible holomorphically symplectic manifold, $\dim_{\mathbb{R}}M=4n$ and $\eta\in\Lambda^{1,1}(M,I)$ a closed, semipositive form of rank $2n$. Consider the product $\operatorname{Tw}_{\eta}(M):={\mathbb{C}}\times M$, equipped with the almost complex structure ${\cal I}$ acting on $T_{t}{\mathbb{C}}\oplus T_{m}M$ as $I_{\mathbb{C}}\oplus I_{t}$, where $I_{\mathbb{C}}$ is the standard complex structure on ${\mathbb{C}}$ and $I_{t}$ is the complex structure recovered from the form $\Omega+t\eta$ using 3.2 and 3.1. The almost complex manifold $(\operatorname{Tw}_{\eta}(M),{\cal I})$ is called a degenerate twistor space of $M$. Theorem 3.18: The almost complex structure on a degenerate twistor space is always integrable. Proof: We introduce a dummy variable $w$, and consider a product $\operatorname{Tw}_{\eta}(M)\times{\mathbb{C}}$, equipped with the (2,0)-form $\widetilde{\Omega}:=\Omega+t\eta+dt\wedge dw$. Here, $\Omega$ is a holomorphic symplectic form on $M$ lifted to $M\times{\mathbb{C}}\times{\mathbb{C}}$, and $t$ and $w$ are complex coordinates on ${\mathbb{C}}\times{\mathbb{C}}$. Clearly, $\widetilde{\Omega}$ is a non-degenerate (2,0)-form. From 3.4 we obtain that $d\widetilde{\Omega}=\eta\wedge dt\in\Lambda^{2,1}(\operatorname{Tw}_{\eta}(M)\times{\mathbb{C}})$. Now, 3.1 implies that $\widetilde{\Omega}$ defines an integrable almost complex structure on $\operatorname{Tw}_{\eta}(M)\times{\mathbb{C}}$. However, on $\operatorname{Tw}_{\eta}(M)\times\\{w\\}$ this almost complex structure coincides with the one given by the degenerate twistor construction. Acknowledgements: I am grateful to Eyal Markman and Jun-Muk Hwang for their interest and encouragement. Thanks to Ljudmila Kamenova for her suggestions and to the organizers of the Quiver Varieties Program at the Simons Center for Geometry and Physics, Stony Brook University, where some of the research for this paper was performed. Also much gratitude to the anonymous referee for important suggestions. ## References [Bea] Beauville, A. Varietes Kähleriennes dont la première classe de Chern est nulle. J. Diff. 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Misha Verbitsky Laboratory of Algebraic Geometry, Faculty of Mathematics, NRU HSE, 7 Vavilova Str. Moscow, Russia [email protected], also: Kavli IPMU (WPI), the University of Tokyo.	arxiv-papers	2013-11-20T14:51:13	2024-09-04T02:49:54.012301	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Misha Verbitsky", "submitter": "Misha Verbitsky", "url": "https://arxiv.org/abs/1311.5073" }
1311.5179	# Sparse PCA via Covariance Thresholding Yash Deshpande and Andrea Montanari ###### Abstract In sparse principal component analysis we are given noisy observations of a low-rank matrix of dimension $n\times p$ and seek to reconstruct it under additional sparsity assumptions. In particular, we assume here that the principal components ${\mathbf{v}}_{1},\dots,{\mathbf{v}}_{r}$ have at most $k_{1},\cdots,k_{r}$ non-zero entries respectively, and study the high- dimensional regime in which $p$ is of the same order as $n$. In an influential paper, Johnstone and Lu [JL04] introduced a simple algorithm that estimates the support of the principal vectors ${\mathbf{v}}_{1},\dots,{\mathbf{v}}_{r}$ by the largest entries in the diagonal of the empirical covariance. This method can be shown to succeed with high probability if $k_{q}\leq C_{1}\sqrt{n/\log p}$, and to fail with high probability if $k_{q}\geq C_{2}\sqrt{n/\log p}$ for two constants $0<C_{1},C_{2}<\infty$. Despite a considerable amount of work over the last ten years, no practical algorithm exists with provably better support recovery guarantees. Here we analyze a covariance thresholding algorithm that was recently proposed by Krauthgamer, Nadler and Vilenchik [KNV13]. We confirm empirical evidence presented by these authors and rigorously prove that the algorithm succeeds with high probability for $k$ of order $\sqrt{n}$. Recent conditional lower bounds [BR13] suggest that it might be impossible to do significantly better. The key technical component of our analysis develops new bounds on the norm of kernel random matrices, in regimes that were not considered before. ## 1 Introduction In the spiked covariance model proposed by [JL04], we are given data ${\mathbf{x}}_{1},{\mathbf{x}}_{2},\dots,{\mathbf{x}}_{n}$ with ${\mathbf{x}}_{i}\in\mathbb{R}^{p}$ of the form111Throughout the paper, we follow the convention of denoting scalars by lowercase, vectors by lowercase boldface, and matrices by uppercase boldface letters.: $\displaystyle{\mathbf{x}}_{i}$ $\displaystyle=\sum_{q=1}^{r}\sqrt{\beta_{q}}\,u_{q,i}\,{\mathbf{v}}_{q}+{\mathbf{z}}_{i}\,,$ (1) Here ${\mathbf{v}}_{1},\dots,{\mathbf{v}}_{r}\in\mathbb{R}^{p}$ is a set of orthonormal vectors, that we want to estimate, while $u_{q,i}\sim{\sf N}(0,1)$ and ${\mathbf{z}}_{i}\sim{\sf N}(0,{\rm I}_{p})$ are independent and identically distributed. The quantity $\beta_{q}\in\mathbb{R}_{>0}$ quantifies the signal-to-noise ratio. We are interested in the high-dimensional limit $n,p\to\infty$ with $\lim_{n\to\infty}p/n=\alpha\in(0,\infty)$. In the rest of this introduction we will refer to the rank one case, in order to simplify the exposition, and drop the subscript $q=\\{1,2,\dots,r\\}$. Our results and proofs hold for general bounded rank. The standard method of principal component analysis involves computing the sample covariance matrix ${\mathbf{G}}=n^{-1}\sum_{i=1}^{n}{\mathbf{x}}_{i}{\mathbf{x}}_{i}^{{\sf T}}$ and estimates ${\mathbf{v}}={\mathbf{v}}_{1}$ by its principal eigenvector ${\mathbf{v}}_{\mbox{\tiny{\sc PC}}}({\mathbf{G}})$. It is a well-known fact that, in the high dimensional asymptotic $p/n\to\alpha>0$, this yields an inconsistent estimate [JL09]. Namely $\\|{\mathbf{v}}_{\mbox{\tiny{\sc PC}}}-{\mathbf{v}}\\|_{2}\not\to 0$ in the high-dimensional asymptotic limit, unless $\alpha\to 0$ (i.e. $p/n\to 0$). Even worse, Baik, Ben-Arous and Péché [BBAP05] and Paul [Pau07] demonstrate a phase transition phenomenon: if $\beta<\sqrt{\alpha}$ the estimate is asymptotically orthogonal to the signal $\langle{\mathbf{v}}_{\mbox{\tiny{\sc PC}}},{\mathbf{v}}\rangle\to 0$. On the other hand, for $\beta>\sqrt{\alpha}$, $\langle{\mathbf{v}}_{\mbox{\tiny{\sc PC}}},{\mathbf{v}}\rangle$ remains strictly positive as $n,p\to\infty$. This phase transition phenomenon has attracted considerable attention recently within random matrix theory [FP07, CDMF09, BGN11, KY13]. These inconsistency results motivated several efforts to exploit additional structural information on the signal ${\mathbf{v}}$. In two influential papers, Johnstone and Lu [JL04, JL09] considered the case of a signal ${\mathbf{v}}$ that is sparse in a suitable basis, e.g. in the wavelet domain. Without loss of generality, we will assume here that ${\mathbf{v}}$ is sparse in the canonical basis ${\mathbf{e}}_{1}$, …${\mathbf{e}}_{p}$. In a nutshell, [JL09] proposes the following: 1. 1. Order the diagonal entries of the Gram matrix ${\mathbf{G}}_{i(1),i(1)}\geq{\mathbf{G}}_{i(2),i(2)}\geq\dots\geq{\mathbf{G}}_{i(p),i(p)}$, and let $J\equiv\\{i(1),i(2),\dots,i(k)\\}$ be the set of indices corresponding to the $k$ largest entries. 2. 2. Set to zero all the entries ${\mathbf{G}}_{i,j}$ of ${\mathbf{G}}$ unless $i,j\in J$, and estimate ${\mathbf{v}}$ with the principal eigenvector of the resulting matrix. Johnstone and Lu formalized the sparsity assumption by requiring that ${\mathbf{v}}$ belongs to a weak $\ell_{q}$-ball with $q\in(0,1)$. Instead, here we consider a strict sparsity constraint where ${\mathbf{v}}$ has exactly $k$ non-zero entries, with magnitudes bounded below by $\theta/\sqrt{k}$ for some constant $\theta>0$. Amini and Wainwright [AW09] studied the more restricted case when every entry of ${\mathbf{v}}$ has equal magnitude of $1/\sqrt{k}$. Within this model, it was proved that diagonal thresholding successfully recovers the support of ${\mathbf{v}}$ provided ${\mathbf{v}}$ is sparse enough, namely $k\leq C\sqrt{n/\log p}$ with $C=C(\alpha,\beta)$ a constant [AW09]. (Throughout the paper we denote by $C$ constants that can change from point to point.) This result is a striking improvement over vanilla PCA. While the latter requires a number of samples scaling as the number of parameters222Throughout the introduction, we write $f(n)\gtrsim g(n)$ as a shorthand of _‘ $f(n)\geq C\,g(n)$ for a some constant $C=C(\beta,\alpha)$’._ Further $C$ denotes a constant that may change from point to point. $n\gtrsim p$, sparse PCA via diagonal thresholding achieves the same objective with a number of samples scaling as the number of _non-zero_ parameters, $n\gtrsim k^{2}\log p$. At the same time, this result is not as strong as might have been expected. By searching exhaustively over all possible supports of size $k$ (a method that has complexity of order $p^{k}$) the correct support can be identified with high probability as soon as $n\gtrsim k\log p$. On the other hand, no method can succeed for much smaller $n$, because of information theoretic obstructions [AW09]. Over the last ten years, a significant effort has been devoted to developing practical algorithms that outperform diagonal thresholding, see e.g. [MWA05, ZHT06, dEGJL07, dBG08, WTH09]. In particular, d’Aspremont et al. [dEGJL07] developed a promising M-estimator based on a semidefinite programming (SDP) relaxation. Amini and Wainwright [AW09] carried out an analysis of this method and proved that, if _(i)_ $k\leq C(\beta)\,n/\log p$, and _(ii)_ if the SDP solution has rank one, then the SDP relaxation provides a consistent estimator of the support of ${\mathbf{v}}$. At first sight, this appears as a satisfactory solution of the original problem. No procedure can estimate the support of ${\mathbf{v}}$ from less than $k\log p$ samples, and the SDP relaxation succeeds in doing it from –at most– a constant factor more samples. This picture was upset by a recent, remarkable result by Krauthgamer, Nadler and Vilenchik [KNV13] who showed that the rank-one condition assumed by Amini and Wainwright does not hold for $\sqrt{n}\lesssim k\lesssim(n/\log p)$. This result is consistent with recent work of Berthet and Rigollet [BR13] demonstrating that sparse PCA cannot be performed in polynomial time in the regime $k\gtrsim\sqrt{n}$, under a certain computational complexity conjecture for the so-called planted clique problem. In summary, the sparse PCA problem demonstrates a fascinating interplay between computational and statistical barriers. From a statistical perspective, and disregarding computational considerations, the support of ${\mathbf{v}}$ can be estimated consistently if and only if $k\lesssim n/\log p$. This can be done, for instance, by exhaustive search over all the $\binom{p}{k}$ possible supports of ${\mathbf{v}}$. (See [VL12, CMW+13] for a minimax analysis.) From a computational perspective, the problem appears to be much more difficult. There is rigorous evidence [BR13, MW13] that no polynomial algorithm can reconstruct the support unless $k\lesssim\sqrt{n}$. On the positive side, a very simple algorithm (Johnstone and Lu’s diagonal thresholding) succeeds for $k\lesssim\sqrt{n/\log p}$. Of course, several elements are still missing in this emerging picture. In the present paper we address one of them, providing an answer to the following question: > _Is there a polynomial time algorithm that is guaranteed to solve the sparse > PCA problem with high probability for $\sqrt{n/\log p}\lesssim > k\lesssim\sqrt{n}$?_ We answer this question positively by analyzing a covariance thresholding algorithm that proceeds, briefly, as follows. (A precise, general definition, with some technical changes is given in the next section.) 1. 1. Form the empirical covariance matrix ${\mathbf{G}}$ and set to zero all its entries that are in modulus smaller than $\tau/\sqrt{n}$, for $\tau$ a suitably chosen constant. 2. 2. Compute the principal eigenvector $\mathbf{\widehat{v}}_{1}$ of this thresholded matrix. 3. 3. Denote by ${\sf B}\subseteq\\{1,\dots,p\\}$ be the set of indices corresponding to the $k$ largest entries of $\mathbf{\widehat{v}}_{1}$. 4. 4. Estimate the support of ${\mathbf{v}}$ by ‘cleaning’ the set ${\sf B}$. (Briefly, ${\mathbf{v}}$ is estimated by thresholding ${\mathbf{G}}\mathbf{\widehat{v}}_{{\sf B}}$ with $\mathbf{\widehat{v}}_{{\sf B}}$ obtained by zeroing the entries outside ${\sf B}$.) Such a covariance thresholding approach was proposed in [KNV13], and is in turn related to earlier work by Bickel and Levina [BL08]. The formulation discussed in the next section presents some technical differences that have been introduced to simplify the analysis. Notice that, to simplify proofs, we assume $k$ to be known: This issue is discussed in the next two sections. The rest of the paper is organized as follows. In the next section we provide a detailed description of the algorithm and state our main results. Our theoretical results assume full knowledge of problem parameters for ease of proof. In light of this, in Section 3 we discuss a practical implementation of the same idea that does not require prior knowledge of problem parameters, and is entirely data-driven. We also illustrate the method through simulations. The complete proofs are available in the accompanying supplement, in Sections 4, 5 and 6 respectively. ## 2 Algorithm and main result Algorithm 1 Covariance Thresholding 1:Input: Data $({\mathbf{x}}_{i})_{1\leq i\leq 2n}$, parameters $k_{q}\in{\mathbb{N}}$, $\tau,\rho\in\mathbb{R}_{\geq 0}$; 2:Compute the empirical covariance matrices ${\mathbf{G}}\equiv\sum_{i=1}^{n}{\mathbf{x}}_{i}{\mathbf{x}}_{i}^{{\sf T}}/n$ , ${\mathbf{G}}^{\prime}\equiv\sum_{i=n+1}^{n}{\mathbf{x}}_{i}{\mathbf{x}}_{i}^{\sf T}/n$; 3:Compute ${\mathbf{\widehat{\Sigma}}}={\mathbf{G}}-{\rm I}_{p}$ (resp. ${\mathbf{\widehat{\Sigma}}}^{\prime}={\mathbf{G}}^{\prime}-{\rm I}_{p}$); 4:Compute the matrix $\eta({\mathbf{\widehat{\Sigma}}})$ by soft-thresholding the entries of ${\mathbf{\widehat{\Sigma}}}$: $\displaystyle\eta({\mathbf{\widehat{\Sigma}}})_{ij}$ $\displaystyle=\begin{cases}{\mathbf{\widehat{\Sigma}}}_{ij}-\frac{\tau}{\sqrt{n}}&\mbox{if ${\mathbf{\widehat{\Sigma}}}_{ij}\geq\tau/\sqrt{n}$,}\\\ 0&\mbox{if $-\tau/\sqrt{n}<{\mathbf{\widehat{\Sigma}}}_{ij}<\tau/\sqrt{n}$,}\\\ {\mathbf{\widehat{\Sigma}}}_{ij}+\frac{\tau}{\sqrt{n}}&\mbox{if ${\mathbf{\widehat{\Sigma}}}_{ij}\leq-\tau/\sqrt{n}$,}\end{cases}$ 5:Let $(\mathbf{\widehat{v}}_{q})_{q\leq r}$ be the first $r$ eigenvectors of $\eta({\mathbf{\widehat{\Sigma}}})$; 6:Define ${\mathbf{s}}_{q}\in\mathbb{R}^{p}$ by $s_{q,i}=\widehat{v}_{q,i}\mathbb{I}(\left\lvert{\widehat{v}_{q,i}\geq\theta/2\sqrt{k_{q}}}\right\rvert)$; 7:Output: ${\widehat{\sf Q}}=\\{i\in[p]:\;\exists\,q\text{ s.t. }\|({\mathbf{\widehat{\Sigma}}}^{\prime}{\mathbf{s}}_{q})_{i}\|\geq\rho\\}$. For notational convenience, we shall assume hereafter that $2n$ sample vectors are given (instead of $n$): $\\{{\mathbf{x}}_{i}\\}_{1\leq i\leq 2n}$. These are distributed according to the model (1). The number of spikes $r$ will be treated as a known parameter in the problem. We will make the following assumptions: 1. A1 The number of spikes $r$ and the signal strengths $\beta_{1},\dots,\beta_{r}$ are fixed as $n,p\to\infty$. Further $\beta_{1}>\beta_{2}>\dots\beta_{r}$ are all _distinct_. 2. A2 Let ${\sf Q}_{q}$ and $k_{q}$ denote the support of ${\mathbf{v}}_{q}$ and its size respectively. We let ${\sf Q}=\cup_{q}{\sf Q}_{q}$ and $k=\sum_{q}k_{q}$ throughout. Then the non-zero entries of the spikes satisfy $\|v_{q,i}\|\geq\theta/\sqrt{k_{q}}$ for all $i\in{\sf Q}_{q}$ for some $\theta>0$. Further, for any $q,q^{\prime}$ we assume $\left\lvert{v_{q,i}/v_{q^{\prime},i}}\right\rvert\leq\gamma$ for every $i\in{\sf Q}_{q}\cap{\sf Q}_{q^{\prime}}$, for some constant $\gamma$. As before, we are interested in the high-dimensional limit of $n,p\to\infty$ with $p/n\to\alpha$. A more detailed description of the covariance thresholding algorithm for the general model (1) is given in Table 1. We describe the basic intuition for the simpler rank-one case (omitting the subscript $q\in\\{1,2,\dots,r\\}$), while stating results in generality. We start by splitting the data into two halves: $({\mathbf{x}}_{i})_{1\leq i\leq n}$ and $({\mathbf{x}}_{i})_{n<i\leq 2n}$ and compute the respective sample covariance matrices ${\mathbf{G}}$ and ${\mathbf{G}}^{\prime}$ respectively. As we will see, the matrix ${\mathbf{G}}$ is used to obtain a good estimate for the spike ${\mathbf{v}}$. This estimate, along with the (independent) second part ${\mathbf{G}}^{\prime}$, is then used to construct a consistent estimator for the supports of ${\mathbf{v}}$. Let us focus on the first phase of the algorithm, which aims to obtain a good estimate of ${\mathbf{v}}$. We first compute ${\mathbf{\widehat{\Sigma}}}={\mathbf{G}}-{\rm I}$. For $\beta>\sqrt{\alpha}$, the principal eigenvector of ${\mathbf{G}}$, and hence of ${\mathbf{\widehat{\Sigma}}}$ is positively correlated with ${\mathbf{v}}$, i.e. $\lim_{n\to\infty}\langle\mathbf{\widehat{v}}_{1}({\mathbf{\widehat{\Sigma}}}),{\mathbf{v}}\rangle>0$. However, for $\beta<\sqrt{\alpha}$, the noise component in ${\mathbf{\widehat{\Sigma}}}$ dominates and the two vectors become asymptotically orthogonal, i.e. for instance $\lim_{n\to\infty}\langle\mathbf{\widehat{v}}_{1}({\mathbf{\widehat{\Sigma}}}),{\mathbf{v}}\rangle=0$. In order to reduce the noise level, we must exploit the sparsity of the spike ${\mathbf{v}}$. Denoting by ${\mathbf{X}}\in\mathbb{R}^{n\times p}$ the matrix with rows ${\mathbf{x}}_{1}$, …${\mathbf{x}}_{n}$, by ${\mathbf{Z}}\in\mathbb{R}^{n\times p}$ the matrix with rows ${\mathbf{z}}_{1}$, …${\mathbf{z}}_{n}$, and letting ${\mathbf{u}}=(u_{1},u_{2},\dots,u_{n})$, the model (1) can be rewritten as $\displaystyle{\mathbf{X}}$ $\displaystyle=\sqrt{\beta}\,{\mathbf{u}}\,{\mathbf{v}}^{{\sf T}}+{\mathbf{Z}}\,.$ (2) Hence, letting $\beta^{\prime}\equiv\beta\\|u\\|^{2}/n\approx\beta$, and ${\mathbf{w}}\equiv\sqrt{\beta}{\mathbf{Z}}^{{\sf T}}{\mathbf{u}}/n$ $\displaystyle{\mathbf{\widehat{\Sigma}}}$ $\displaystyle=\beta^{\prime}\,{\mathbf{v}}{\mathbf{v}}^{{\sf T}}+{\mathbf{v}}\,{\mathbf{w}}^{{\sf T}}+{\mathbf{w}}\,{\mathbf{v}}^{{\sf T}}+\frac{1}{n}{\mathbf{Z}}^{{\sf T}}{\mathbf{Z}}\;\;-{\rm I}_{p},.$ (3) For a moment, let us neglect the cross terms $({\mathbf{v}}{\mathbf{w}}^{{\sf T}}+{\mathbf{w}}{\mathbf{v}}^{{\sf T}})$. The ‘signal’ component $\beta^{\prime}\,{\mathbf{v}}{\mathbf{v}}^{{\sf T}}$ is sparse with $k^{2}$ entries of magnitude $\beta/k$, which (in the regime of interest $k=\sqrt{n}/C$) is equivalent to $C\beta/\sqrt{n}$. The ‘noise’ component ${\mathbf{Z}}^{{\sf T}}{\mathbf{Z}}/n-{\rm I}_{p}$ is dense with entries of order $1/\sqrt{n}$. Assuming $k/\sqrt{n}$ a small enough constant, it should be possible to remove most of the noise by thresholding the entries at level of order $1/\sqrt{n}$. For technical reasons, we use the soft thresholding function $\eta:\mathbb{R}\times\mathbb{R}_{\geq 0}\to\mathbb{R},\,\eta(z;\tau)={\operatorname{\rm{sgn}}}(z)(\left\lvert{z}\right\rvert-\tau)_{+}$. We will omit the second argument from $\eta(\cdot;\cdot)$ wherever it is clear from context. Classical denoising theory [DJ94, Joh02] provides upper bounds the estimation error of such a procedure. Note however that these results fall short of our goal. Classical theory measures estimation error by (element- wise) $\ell_{p}$ norm, while here we are interested in the resulting principal eigenvector. This would require bounding, for instance, the error in operator norm. Since the soft thresholding function $\eta(z;\tau/\sqrt{n})$ is affine when $z\gg\tau/\sqrt{n}$, we would expect that the following decomposition holds approximately (for instance, in operator norm): $\displaystyle\eta({\mathbf{\widehat{\Sigma}}})$ $\displaystyle\approx\eta\left(\beta^{\prime}{\mathbf{v}}{\mathbf{v}}^{\sf T}\right)+\eta\left(\frac{1}{n}{\mathbf{Z}}^{\sf T}{\mathbf{Z}}-{\rm I}_{p}\right).$ (4) The main technical challenge now is to control the operator norm of the perturbation $\eta({\mathbf{Z}}^{\sf T}{\mathbf{Z}}/n-{\rm I}_{p})$. It is easy to see that $\eta({\mathbf{Z}}^{\sf T}{\mathbf{Z}}/n-{\rm I}_{p})$ has entries of variance $\delta(\tau)/n$, for $\delta(\tau)\to 0$ as $\tau\to\infty$. If entries were independent with mild decay, this would imply –with high probability– $\displaystyle\left\lVert{\eta\left(\frac{1}{n}{\mathbf{Z}}^{\sf T}{\mathbf{Z}}-{\rm I}_{p}\right)}\right\rVert_{2}\lesssim C\delta(\tau),$ (5) for some constant $C$. Further, the first component in the decomposition (4) is still approximately rank one with norm of the order of $\beta^{\prime}\approx\beta$. Consequently, with standard linear algebra results on the perturbation of eigenspaces [DK70], we obtain an error bound $\left\lVert{\mathbf{\widehat{v}}-{\mathbf{v}}}\right\rVert\lesssim\delta(\tau)/C^{\prime}\beta$. Our first theorem formalizes this intuition and provides a bound on the estimation error in the principal components of $\eta({\mathbf{\widehat{\Sigma}}})$. ###### Theorem 1. Under the spiked covariance model Eq. (1) satisfying Assumption A1, let $\mathbf{\widehat{v}}_{q}$ denote the $q^{\text{th}}$ eigenvector of $\eta({\mathbf{\widehat{\Sigma}}})$ using threshold $\tau$. For every $\alpha,(\beta_{q})_{q=1}^{r}\in(0,\infty)$, integer $r$ and every ${\varepsilon}>0$ there exist constants, $\tau=\tau({\varepsilon},\alpha,(\beta_{q})_{q=1}^{r},r,\theta)$ and $0<c_{}=c_{}({\varepsilon},\alpha,(\beta_{q})_{q=1}^{r},r,\theta)<\infty$ such that, if $\sum_{q}k_{q}=\sum_{q}\|{\rm supp}({\mathbf{v}}_{q})\|\leq c_{}\sqrt{n})$, then $\displaystyle{\mathbb{P}}\Big{\\{}\min(\left\lVert{\mathbf{\widehat{v}}_{q}-{\mathbf{v}}_{q}}\right\rVert,\left\lVert{\mathbf{\widehat{v}}_{q}+{\mathbf{v}}_{q}}\right\rVert)\leq{\varepsilon}\;\;\forall q\in\\{1,\dots,r\\}\Big{\\}}\geq 1-\frac{\alpha}{n^{4}}\,.$ (6) It is clear from the discussion above that the proof of Theorem 1 requires a formalization of Eq. (5). Indeed, the spectral properties of random matrices of the type $f({\mathbf{Z}}^{\sf T}{\mathbf{Z}}/n-{\rm I}_{p})$ , called inner-product kernel random matrices, have attracted recent interest within probability theory [EK10a, EK10b, CS12]. In this literature, the asymptotic eigenvalue distribution of a matrix $f({\mathbf{Z}}^{\sf T}{\mathbf{Z}}/n-{\rm I}_{p})$ is the object of study. Here $f:\mathbb{R}\to\mathbb{R}$ is a kernel function and is applied entry-wise to the matrix ${\mathbf{Z}}^{\sf T}{\mathbf{Z}}/n-{\rm I}_{p}$, with ${\mathbf{Z}}$ a matrix as above. Unfortunately, these results do not suffice to prove Theorem 1 for the following reasons: • The results [EK10a, EK10b] are perturbative and provide conditions under which the spectrum of $f({\mathbf{Z}}^{\sf T}{\mathbf{Z}}/n-{\rm I}_{p})$ is close to a rescaling of the spectrum of $({\mathbf{Z}}^{\sf T}{\mathbf{Z}}/n-{\rm I}_{p})$ (with rescaling factors depending on the Taylor expansion of $f$ close to $0$). We are interested instead in a non-perturbative regime in which the spectrum of $f({\mathbf{Z}}^{\sf T}{\mathbf{Z}}/n-{\rm I}_{p})$ is very different from the one of $({\mathbf{Z}}^{\sf T}{\mathbf{Z}}/n-{\rm I}_{p})$ (and the Taylor expansion is trivial). * • The authors of [CS12] consider $n$-dependent kernels, but their results are asymptotic and concern the weak limit of the empirical spectral distribution of $f({\mathbf{Z}}^{\sf T}{\mathbf{Z}}/n-{\rm I}_{p})$. This does not yield an upper bound on the spectral norm333Note that [CS12] also provide a non- asymptotic bound for the spectral norm of $f({\mathbf{Z}}^{\sf T}{\mathbf{Z}}/n-{\rm I}_{p})$ via the moment method, but this bound diverges with $n$ and does not give a result of the type of Eq. (5). of $f({\mathbf{Z}}^{\sf T}{\mathbf{Z}}/n-{\rm I}_{p})$. Our approach to prove Theorem 1 follows instead the so-called ${\varepsilon}$-net method: we develop high probability bounds on the maximum Rayleigh quotient: $\displaystyle\max_{{\mathbf{y}}\in S^{p-1}}\langle{\mathbf{y}},\eta({\mathbf{Z}}^{\sf T}{\mathbf{Z}}/n-{\rm I}_{p}){\mathbf{y}}\rangle$ $\displaystyle=\max_{{\mathbf{y}}\in S^{p-1}}\sum_{i,j}\eta\left(\frac{\langle{\mathbf{\tilde{z}}}_{i},{\mathbf{\tilde{z}}}_{j}\rangle}{n};\frac{\tau}{\sqrt{n}}\right)y_{i}y_{j},$ where $S^{p-1}=\\{{\mathbf{y}}\in\mathbb{R}^{p}:\\|{\mathbf{y}}\\|=1\\}$ is the unit sphere. Since $\eta({\mathbf{Z}}^{\sf T}{\mathbf{Z}}/n-{\rm I}_{p})$ is not Lipschitz continuous in the underlying Gaussian variables ${\mathbf{Z}}$, concentration does not follow immediately from Gaussian isoperimetry. We have to develop more careful (non-uniform) bounds on the gradient of $\eta({\mathbf{Z}}^{\sf T}{\mathbf{Z}}/n-{\rm I}_{p})$ and show that they imply concentration as required. While Theorem 1 guarantees that $\mathbf{\widehat{v}}$ is a reasonable estimate of the spike ${\mathbf{v}}$ in $\ell_{2}$ distance (up to a sign flip), it does not provide a consistent estimator of its support. This brings us to the second phase of our algorithm. Although $\mathbf{\widehat{v}}$ is not even expected to be sparse, it is easy to see that the largest entries of $\mathbf{\widehat{v}}$ should have significant overlap with ${\rm supp}({\mathbf{v}})$. Steps 6, 7 and 8 of the algorithm exploit this property to construct a consistent estimator ${\widehat{\sf Q}}$ of the support of the spike ${\mathbf{v}}$. Our second theorem guarantees that this estimator is indeed consistent. ###### Theorem 2. Consider the spiked covariance model of Eq. (1) satisfying Assumptions A1, A2. For any $\alpha,(\beta_{q})_{q\leq r}\in(0,\infty)$, $\theta,\gamma>0$ and integer $r$, there exist constants $c_{},\tau,\rho$ dependent on $\alpha,(\beta_{q})_{q\leq r},\gamma,\theta,r$, such that, if $\sum_{q}k_{q}=\|{\rm supp}({\mathbf{v}}_{q})\|\leq c_{}\sqrt{n}$, the Covariance Thresholding algorithm of Table 1 recovers the union of supports of ${\mathbf{v}}_{q}$ with high probability. Explicitly, there exists a constant $C>0$ such that $\displaystyle{\mathbb{P}}\Big{\\{}{\widehat{\sf Q}}=\cup_{q}{\rm supp}({\mathbf{v}}_{q})\Big{\\}}\geq 1-\frac{C}{n^{4}}\,.$ (7) Before passing to the proofs of Theorem 1 and Theorem 2 (respectively in Sections 6 and 5 of the Supplementary Material), it is useful to pause for a few remarks. ###### Remark 2.1. We focus on a consistent estimation of the union of the supports $\cup_{q}{\rm supp}({\mathbf{v}}_{q})$ of the spikes. In the rank-one case, this obviously corresponds to the standard support recovery. In the general case, once the union is correctly estimated, estimating the individual supports poses no additional difficulty: indeed, since $\|\cup_{q}{\rm supp}({\mathbf{v}}_{q}))\|=O(\sqrt{n})$ an extra step with $n$ fresh samples ${\mathbf{x}}_{i}$ restricted to ${\widehat{\sf Q}}$ yields consistent estimates for ${\mathbf{v}}_{q}$, hence ${\rm supp}({{\mathbf{v}}_{q}})$. ###### Remark 2.2. Recovering the signed supports ${\sf Q}_{q,+}=\\{i\in[p]:v_{q,i}>0\\}$ and ${\sf Q}_{q,-}=\\{i\in[p]:v_{q,i}<0\\}$ is possible using the same technique as recovering the supports ${\rm supp}({\mathbf{v}}_{q})$ above, and poses no additional difficulty. ###### Remark 2.3. Assumption A2 requires $\|v_{q,i}\|\geq\theta/\sqrt{k_{q}}$ for all $i\in{\sf Q}_{q}$. This is a standard requirement in the support recovery literature [Wai09, MB06]. The second part of assumption A2 guarantees that when the supports of two spikes overlap, their entries are roughly of the same order. This is necessary for our proof technique to go through. Avoiding such an assumption altogether remains an open question. Our covariance thresholding algorithm assumes knowledge of the correct support sizes $k_{q}$. Notice that the same assumption is made in earlier theoretical, e.g. in the analysis of SDP-based reconstruction by Amini and Wainwright [AW09]. While this assumption is not realistic in applications, it helps to focus our exposition on the most challenging aspects of the problem. Further, a ballpark estimate of $k_{q}$ (indeed of $\sum_{q}k_{q}$) is actually sufficient. Indeed consider the algorithm obtained by replacing steps 7 and 8 as following. * 7: Define ${\mathbf{s}}^{\prime}_{q}\in\mathbb{R}^{p}$ by $\displaystyle s^{\prime}_{q,i}=\begin{cases}\widehat{v}_{q,i}&\mbox{ if }\|\widehat{v}_{q,i}\|>\theta/(2\sqrt{k_{0}})\\\ 0&\mbox{ otherwise.}\end{cases}$ (8) * 8: Return $\displaystyle{\widehat{\sf Q}}=\cup_{q}\\{i\in[p]:\;\|({\mathbf{\widehat{\Sigma}}}^{\prime}{\mathbf{s}}^{\prime}_{q})_{i}\|\geq\rho\\}\,.$ (9) The next theorem shows that this procedure is effective even if $k_{0}$ overestimates $\sum_{q}k_{q}$ by an order of magnitude. Its proof is deferred to Section 5. ###### Theorem 3. Consider the spiked covariance model of Eq. (1). For any $\alpha,\beta\in(0,\infty)$, let constants $c_{},\tau,\rho$ be given as in Theorem 2. Further assume $k=\sum_{q}\|{\rm supp}({\mathbf{v}}_{q})\|\leq c_{}\sqrt{n}$, and $\sum_{q}k\leq k_{0}\leq 20\,\sum_{q}k_{q}$. Then, the Covariance Thresholding algorithm of Table 1, with the definitions in Eqs. (8) and (9), recovers the union of supports of ${\mathbf{v}}_{q}$ successfully, i.e. $\displaystyle{\mathbb{P}}\Big{(}{\widehat{\sf Q}}=\cup_{q}{\rm supp}({\mathbf{v}}_{q})\Big{)}\geq 1-\frac{C}{n^{4}}\,.$ (10) ## 3 Practical aspects and empirical results Specializing to the rank one case, Theorems 1 and 2 show that Covariance Thresholding succeeds with high probability for a number of samples $n\gtrsim k^{2}$, while Diagonal Thresholding requires $n\gtrsim k^{2}\log p$. The reader might wonder whether eliminating the $\log p$ factor has any practical relevance or is a purely conceptual improvement. Figure 1 presents simulations on synthetic data under the strictly sparse model, and the Covariance Thresholding algorithm of Table 1, used in the proof of Theorem 2. The objective is to check whether the $\log p$ factor has an impact at moderate $p$. We compare this with Diagonal Thresholding. Figure 1: The support recovery phase transitions for Diagonal Thresholding (left) and Covariance Thresholding (center) and the data-driven version of Section 3 (right). For Covariance Thresholding, the fraction of support recovered correctly _increases_ monotonically with $p$, as long as $k\leq c\sqrt{n}$ with $c\approx 1.1$. Further, it appears to converge to one throughout this region. For Diagonal Thresholding, the fraction of support recovered correctly _decreases_ monotonically with $p$ for all $k$ of order $\sqrt{n}$. This confirms that Covariance Thresholding (with or without knowledge of the support size $k$) succeeds with high probability for $k\leq c\sqrt{n}$, while Diagonal Thresholding requires a significantly sparser principal component. We plot the empirical success probability as a function of $k/\sqrt{n}$ for several values of $p$, with $p=n$. The empirical success probability was computed by using $100$ independent instances of the problem. A few observations are of interest: $(i)$ Covariance Thresholding appears to have a significantly larger success probability in the ‘difficult’ regime where Diagonal Thresholding starts to fail; $(ii)$ The curves for Diagonal Thresholding appear to decrease monotonically with $p$ indicating that $k$ proportional to $\sqrt{n}$ is not the right scaling for this algorithm (as is known from theory); $(iii)$ In contrast, the curves for Covariance Thresholding become steeper for larger $p$, and, in particular, the success probability increases with $p$ for $k\leq 1.1\sqrt{n}$. This indicates a sharp threshold for $k={\rm const}\cdot\sqrt{n}$, as suggested by our theory. In terms of practical applicability, our algorithm in Table 1 has the shortcomings of requiring knowledge of problem parameters $\beta_{q},r,k_{q}$. Furthermore, the thresholds $\rho,\tau$ suggested by theory need not be optimal. We next describe a principled approach to estimating (where possible) the parameters of interest and running the algorithm in a purely data- dependent manner. Assume the following model, for $i\in[n]$ $\displaystyle{\mathbf{x}}_{i}$ $\displaystyle={\boldsymbol{\mu}}+\sum_{q}\sqrt{\beta_{q}}u_{q,i}{\mathbf{v}}_{q}+\sigma{\mathbf{z}}_{i},$ where ${\boldsymbol{\mu}}\in\mathbb{R}^{p}$ is a fixed mean vector, $u_{q,i}$ have mean $0$ and variance $1$, and ${\mathbf{z}}_{i}$ have mean $0$ and covariance ${\rm I}_{p}$. Note that our focus in this section is not on rigorous analysis, but instead to demonstrate a principled approach to applying covariance thresholding in practice. We proceed as follows: Estimating ${\boldsymbol{\mu}}$, $\sigma$: We let $\widehat{\boldsymbol{\mu}}=\sum_{i=1}^{n}{\mathbf{x}}_{i}/n$ be the empirical mean estimate for $\mu$. Further letting $\overline{\mathbf{X}}={\mathbf{X}}-\mathbf{1}\widehat{{\boldsymbol{\mu}}}^{\sf T}$ we see that $pn-(\sum_{q}k_{q})n\approx pn$ entries of $\overline{\mathbf{X}}$ are mean $0$ and variance $\sigma^{2}$. We let $\widehat{\sigma}={{\rm MAD}(\overline{\mathbf{X}})}/{\nu}$ where ${\rm MAD}(\,\cdot\,)$ denotes the median absolute deviation of the entries of the matrix in the argument, and $\nu$ is a constant scale factor. Guided by the Gaussian case, we take $\nu=\Phi^{-1}(3/4)\approx 0.6745$. Choosing $\tau$: Although in the statement of the theorem, our choice of $\tau$ depends on the SNR $\beta/\sigma^{2}$, we believe this is an artifact of our analysis. Indeed it is reasonable to threshold ‘at the noise level’, as follows. The noise component of entry $i,j$ of the sample covariance (ignoring lower order terms) is given by $\sigma^{2}\langle{\mathbf{z}}_{i},{\mathbf{z}}_{j}\rangle/n$. By the central limit theorem, $\langle{\mathbf{z}}_{i},{\mathbf{z}}_{j}\rangle/\sqrt{n}{\,\stackrel{{\scriptstyle\mathrm{d}}}{{\Rightarrow}}\,}{\sf N}(0,1)$. Consequently, $\sigma^{2}\langle{\mathbf{z}}_{i},{\mathbf{z}}_{j}\rangle/n\approx{\sf N}(0,\sigma^{4}/n)$, and we need to choose the (rescaled) threshold proportional to $\sqrt{\sigma^{4}}=\sigma^{2}$. Using previous estimates, we let $\tau=\nu^{\prime}\cdot\widehat{\sigma}^{2}$ for a constant $\nu^{\prime}$. In simulations, a choice $3\lesssim\nu^{\prime}\lesssim 4$ appears to work well. Estimating $r$: We define ${\mathbf{\widehat{\Sigma}}}=\overline{\mathbf{X}}^{\sf T}\overline{\mathbf{X}}/n-\sigma^{2}{\rm I}_{p}$ and soft threshold it to get $\eta({\mathbf{\widehat{\Sigma}}})$ using $\tau$ as above. Our proof of Theorem 1 relies on the fact that $\eta({\mathbf{\widehat{\Sigma}}})$ has $r$ eigenvalues that are separated from the bulk of the spectrum. Hence, we estimate $r$ using $\widehat{r}$: the number of eigenvalues separated from the bulk in $\eta({\mathbf{\widehat{\Sigma}}})$. The edge of the spectrum can be computed numerically using the Stieltjes transform method as in [CS12]. Estimating ${\mathbf{v}}_{q}$: Let $\mathbf{\widehat{v}}_{q}$ denote the $q^{\text{th}}$ eigenvector of $\eta({\mathbf{\widehat{\Sigma}}})$. Our theoretical analysis indicates that $\mathbf{\widehat{v}}_{q}$ is expected to be close to ${\mathbf{v}}_{q}$. In order to denoise $\mathbf{\widehat{v}}_{q}$, we assume $\mathbf{\widehat{v}}_{q}\approx(1-\delta){\mathbf{v}}_{q}+{\boldsymbol{{\varepsilon}}}_{q}$, where ${\boldsymbol{{\varepsilon}}}_{q}$ is additive random noise. We then threshold ${\mathbf{v}}_{q}$ ‘at the noise level’ to recover a better estimate of ${\mathbf{v}}_{q}$. To do this, we estimate the standard deviation of the “noise” ${\boldsymbol{{\varepsilon}}}$ by $\widehat{\sigma_{{\boldsymbol{{\varepsilon}}}}}={{\rm MAD}({\mathbf{v}}_{q})}/{\nu}$. Here we set –again guided by the Gaussian heuristic– $\nu\approx 0.6745$. Since ${\mathbf{v}}_{q}$ is sparse, this procedure returns a good estimate for the size of the noise deviation. We let $\eta_{H}(\mathbf{\widehat{v}}_{q})$ denote the vector obtained by hard thresholding $\mathbf{\widehat{v}}_{q}$: set $(\eta_{H}(\mathbf{\widehat{v}}_{q}))_{i}=\mathbf{\widehat{v}}_{q,i}\text{ if }\left\lvert{\widehat{v}_{q,i}}\right\rvert\geq\nu^{\prime}\widehat{\sigma}_{{\boldsymbol{{\varepsilon}}}_{q}}$ and $0\text{ otherwise.}$ We then let $\mathbf{\widehat{v}}^{}_{q}=\eta(\mathbf{\widehat{v}}_{q})/\left\lVert{\eta(\mathbf{\widehat{v}}_{q})}\right\rVert$ and return $\mathbf{\widehat{v}}^{}_{q}$ as our estimate for ${\mathbf{v}}_{q}$. Note that –while different in several respects– this empirical approach shares the same philosophy of the algorithm in Table 1. On the other hand, the data- driven algorithm presented in this section is less straightforward to analyze, a task that we defer to future work. Figure 1 also shows results of a support recovery experiment using the ‘data- driven’ version of this section. Covariance thresholding in this form also appears to work for supports of size $k\leq\text{const}\sqrt{n}$. Figure 2 shows the performance of vanilla PCA, Diagonal Thresholding and Covariance Thresholding on the “Three Peak” example of Johnstone and Lu [JL04]. This signal is sparse in the wavelet domain and the simulations employ the data- driven version of covariance thresholding. A similar experiment with the “box” example of Johnstone and Lu is provided in the supplement. These experiments demonstrate that, while for large values of $n$ both Diagonal Thresholding and Covariance Thresholding perform well, the latter appears superior for smaller values of $n$. Figure 2: The results of Simple PCA, Diagonal Thresholding and Covariance Thresholding (respectively) for the “Three Peak” example of Johnstone and Lu [JL09] (see Figure 1 of the paper). The signal is sparse in the ‘Symmlet 8’ basis. We use $\beta=1.4,p=4096$, and the rows correspond to sample sizes $n=1024,1625,2580,4096$ respectively. Parameters for Covariance Thresholding are chosen as in Section 3, with $\nu^{\prime}=4.5$. Parameters for Diagonal Thresholding are from [JL09]. On each curve, we superpose the clean signal (dotted). ## 4 Proof preliminaries In this section we review some notation and preliminary facts that we will use throughout the paper. ### 4.1 Notation We let $[m]=\\{1,2,\dots,m\\}$ denote the set of first $m$ integers. We will represent vectors using boldface lower case letters, e.g. ${\mathbf{u}},{\mathbf{v}},{\mathbf{x}}$. The entries of a vector ${\mathbf{u}}\in\mathbb{R}^{n}$ will be represented by $u_{i},i\in[n]$. Matrices are represented using boldface upper case letters e.g. ${\mathbf{A}},{\mathbf{X}}$. The entries of a matrix ${\mathbf{A}}\in\mathbb{R}^{m\times n}$ are represented by ${\mathbf{A}}_{ij}$ for $i\in[m],j\in[n]$. Given a matrix ${\mathbf{A}}\in\mathbb{R}^{m\times n}$, we generically let ${\mathbf{a}}_{1}$, ${\mathbf{a}}_{2},\dots,{\mathbf{a}}_{m}$ denote its rows, and ${\mathbf{\tilde{a}}}_{1}$, ${\mathbf{\tilde{a}}}_{2},\dots,{\mathbf{\tilde{a}}}_{n}$ its columns. For $E\subseteq[m]\times[n]$, we define the projector operator ${\cal P}_{E}:\mathbb{R}^{m\times n}\to\mathbb{R}^{m\times n}$ by letting ${\cal P}_{E}({\mathbf{A}})$ be the matrix with entries $\displaystyle{\cal P}_{E}({\mathbf{A}})_{ij}=\begin{cases}{\mathbf{A}}_{ij}&\mbox{if $(i,j)\in E$,}\\\ 0&\mbox{otherwise.}\end{cases}$ (11) If $E=E_{1}\times E_{2}$, we write ${\cal P}_{E_{1},E_{2}}$ for ${\cal P}_{E_{1}\times E_{2}}$. In the case $E=E_{1}\times E_{2}$ we also define a projection operator ${\widetilde{\cal P}}_{E_{1},E_{2}}:\mathbb{R}^{m\times n}\to\mathbb{R}^{\|E_{1}\|\times\|E_{2}\|}$ that returns the $E_{1}\times E_{2}$ submatrix. If $m=n$, and $E$ is the diagonal, we write ${\mathcal{P}_{\sf d}}$ for ${\cal P}_{E}$. If instead $E$ is the complement of the diagonal, we write ${\mathcal{P}_{\sf nd}}$. For a matrix ${\mathbf{A}}\in\mathbb{R}^{m\times n}$, and a set $E\subseteq[n]$, we define its column restriction ${\mathbf{A}}_{E}\in\mathbb{R}^{m\times n}$ to be the matrix obtained by setting to $0$ columns outside $E$: $\displaystyle({\mathbf{A}}_{E})_{ij}$ $\displaystyle=\begin{cases}{\mathbf{A}}_{ij}&\text{ if }j\in E,\\\ 0&\text{otherwise. }\end{cases}$ Similarly ${\mathbf{y}}_{E}$ is obtained from ${\mathbf{y}}$ by setting to zero all indices outside $E$. The operator norm of a matrix ${\mathbf{A}}$ is denoted by $\left\lVert{{\mathbf{A}}}\right\rVert$ (or $\left\lVert{{\mathbf{A}}}\right\rVert_{2}$) and its Frobenius norm by $\left\lVert{{\mathbf{A}}}\right\rVert_{F}$. We write $\left\lVert{{\mathbf{x}}}\right\rVert$ for the standard $\ell_{2}$ norm of a vector ${\mathbf{x}}$. We let ${\sf Q}_{q}$ denotes the support of the $q^{\text{th}}$ spike ${\mathbf{v}}_{q}$. Also, we denote the union of the supports of ${\mathbf{v}}_{q}$ by ${\sf Q}=\cup_{q}{\sf Q}_{q}$. The complement of a set $E\in[n]$ is denoted by $E^{c}$. We write $\eta(\cdot;\cdot)$ for the soft-thresholding function. By $\partial\eta(\cdot;\tau)$ we denote the derivative of $\eta(\cdot;\tau)$ with respect to the _first_ argument, which exists Lebesgue almost everywhere. In the statements of our results, consider the limit of large $p$ and large $n$ with $p/n\to\alpha$. This limit will be referred to either as “$n$ large enough” or “$p$ large enough” where the phrase “large enough” indicates dependence of $p$ (and thereby $n$) on specific problem parameters. ### 4.2 Preliminary facts Let $S^{n-1}$ denote the unit sphere in $n$ dimensions, i.e. $S^{n-1}=\\{{\mathbf{x}}:\left\lVert{{\mathbf{x}}}\right\rVert=1\\}$. We use the following definition (see [Ver12]) of the ${\varepsilon}$-net of a set $X\subseteq\mathbb{R}^{n}$: ###### Definition 4.1 (Nets, Covering numbers). A subset $T^{\varepsilon}(X)\subseteq X$ is called an ${\varepsilon}$-net of $X$ if every point in $X$ may be approximated by one in $T^{\varepsilon}(X)$ with error at most ${\varepsilon}$. More precisely: $\displaystyle\forall x\in X,\quad\inf_{y\in T^{\varepsilon}(X)}\left\lVert{x-y}\right\rVert$ $\displaystyle\leq{\varepsilon}.$ The minimum cardinality of an ${\varepsilon}$-net of $X$, if finite, is called its covering number. The following two facts are useful while using ${\varepsilon}$-nets to bound the spectral norm of a matrix. For proofs, we refer the reader to [Ver12]. ###### Lemma 4.2. Let $S^{n-1}$ be the unit sphere in $n$ dimensions. Then there exists an ${\varepsilon}$-net of $S^{n-1}$, $T^{\varepsilon}(S^{n-1})$ satisfying: $\displaystyle\|T^{\varepsilon}(S^{n-1})\|\leq\left(1+\frac{2}{{\varepsilon}}\right)^{n}.$ ###### Lemma 4.3. Let ${\mathbf{A}}\in\mathbb{R}^{n\times n}$ be a symmetric matrix. Then: $\displaystyle\left\lVert{{\mathbf{A}}}\right\rVert_{2}=\sup_{{\mathbf{x}}\in S^{n-1}}\|\langle{\mathbf{x}},{\mathbf{A}}{\mathbf{x}}\rangle\|\leq(1-2{\varepsilon})^{-1}\sup_{{\mathbf{x}}\in T^{\varepsilon}(S^{n-1})}\|\langle{\mathbf{x}},{\mathbf{A}}{\mathbf{x}}\rangle\|.$ In particular, if ${\mathbf{A}}$ is a random matrix, then for $\Delta>0$ we have: $\displaystyle{\mathbb{P}}\left\\{\left\lVert{{\mathbf{A}}}\right\rVert_{2}\geq\Delta\right\\}$ $\displaystyle\leq\left(1+\frac{2}{{\varepsilon}}\right)^{n}\sup_{{\mathbf{x}}\in T^{\varepsilon}(S^{n-1})}{\mathbb{P}}\left\\{\left\lvert{\langle{\mathbf{x}},{\mathbf{A}}{\mathbf{x}}\rangle}\right\rvert\geq\Delta(1-2{\varepsilon})\right\\}.$ Throughout the paper we will denote by $T^{\varepsilon}_{n}$ the _minimum cardinality_ ${\varepsilon}$-net on the unit sphere $S^{n-1}$, which naturally satisfies Lemma 4.2. Further, for a non-zero vector ${\mathbf{y}}\in\mathbb{R}$, we define the set $S^{n-1}_{\mathbf{y}}=\\{{\mathbf{x}}:\langle{\mathbf{x}},{\mathbf{y}}\rangle=0,\left\lVert{{\mathbf{x}}}\right\rVert=1\\}$ and let its minimum cardinality ${\varepsilon}$-net be denoted by $T^{\varepsilon}_{n}({\mathbf{y}})$. Since $S^{n-1}_{\mathbf{y}}$ is isometric to $S^{n-2}$, Lemma 4.2 holds for $T^{\varepsilon}_{n}({\mathbf{y}})$ as well. We now state some measure concentration results that we will use at various points in the proofs of Theorems 1 and 2. ###### Lemma 4.4. Consider ${\mathbf{z}}\sim{\sf N}(0,{\rm I}_{N})$ be a vector of $N$ i.i.d. standard normal random variables on a probability space $(\Omega,{\cal F},{\mathbb{P}})$. Suppose $F:\mathbb{R}^{N}\to\mathbb{R}$ is a $\mathbb{R}$-valued, continuous, a.e. differentiable function and $G\in{\mathcal{B}}_{\mathbb{R}^{N}}$ is a closed convex set satisfying: $\displaystyle\left\lVert{{\nabla}F({\mathbf{z}})}\right\rVert\mathbb{I}({\mathbf{z}}\in G)$ $\displaystyle\leq L\quad{\mathbb{P}}\emph{-a.e.}$ $\displaystyle{\mathbb{P}}\left\\{G\right\\}$ $\displaystyle\geq 1-q.$ Then, there exists a function $F_{L}:\mathbb{R}^{N}\to\mathbb{R}$ such that $F_{L}$ is $L$-Lipschitz throughout and $F_{L}$ coincides with $F$ on the set $G$. Further for each $\Delta>0$ we have that: $\displaystyle{\mathbb{P}}\left\\{\|F({\mathbf{z}})-{\mathbb{E}}F({\mathbf{z}})\|\geq\Delta\right\\}$ $\displaystyle\leq q+2\exp\left(-\frac{\widetilde{\Delta}^{2}}{2L^{2}}\right),$ where $\widetilde{\Delta}=\Delta-\|{\mathbb{E}}F({\mathbf{z}})-{\mathbb{E}}F_{L}({\mathbf{z}})\|$. ###### Proof. For any ${\mathbf{y}},{\mathbf{y}}^{\prime}\in G$ we have that: $\displaystyle F({\mathbf{y}}^{\prime})$ $\displaystyle=F({\mathbf{y}})+\int_{0}^{1}\langle{\nabla}F(t{\mathbf{y}}^{\prime}+(1-t){\mathbf{y}}),{\mathbf{y}}^{\prime}-{\mathbf{y}}\rangle\mathrm{d}t.$ From this we obtain that $\|F({\mathbf{y}}^{\prime})-F({\mathbf{y}})\|\leq L\left\lVert{{\mathbf{y}}^{\prime}-{\mathbf{y}}}\right\rVert$ using the bound on ${\nabla}{F}$ in $G$ and the convexity of $G$. By Kirszbraun’s theorem, there exists an $L$-Lipschitz extension $F_{L}$ of $F$ to $\mathbb{R}^{N}$. Indeed we may take $F_{L}({\mathbf{y}})=\inf_{{\mathbf{y}}^{\prime}\in G}F({\mathbf{y}})+L\left\lVert{{\mathbf{y}}-{\mathbf{y}}^{\prime}}\right\rVert$. Then: $\displaystyle{\mathbb{P}}\left\\{\|F({\mathbf{z}})-{\mathbb{E}}F({\mathbf{z}})\|\geq\Delta\right\\}$ $\displaystyle={\mathbb{P}}\left\\{\|F({\mathbf{z}})-{\mathbb{E}}F({\mathbf{z}})\|\geq\Delta;{\mathbf{z}}\in G\right\\}+{\mathbb{P}}\left\\{\|F({\mathbf{z}})-{\mathbb{E}}F({\mathbf{z}})\|\geq\Delta;{\mathbf{z}}\in G^{c}\right\\}$ $\displaystyle\leq{\mathbb{P}}\\{\|F_{L}({\mathbf{z}})-{\mathbb{E}}F_{L}({\mathbf{z}})\|\geq\widetilde{\Delta}\\}+{\mathbb{P}}\\{G^{c}\\}$ Applying Gaussian concentration of measure [Led01] to $F_{L}$ finishes the proof. ∎ For further reference, we define the following: ###### Definition 4.5. For a function $F:\mathbb{R}^{N}\to\mathbb{R}$, a constant $L>0$ and a measurable set $G$, we call $F_{L}(\cdot)$ the _$G,L$ -Lipschitz extension_ of $F(\cdot)$. It is given by: $\displaystyle F_{L}\left({\mathbf{y}}\right)$ $\displaystyle=\inf_{{\mathbf{y}}^{\prime}\in G}\left(F({\mathbf{y}}^{\prime})+L\left\lVert{{\mathbf{y}}-{\mathbf{y}}^{\prime}}\right\rVert\right).$ ###### Lemma 4.6. Let ${\mathbf{A}}\in\mathbb{R}^{M\times N}$ be a matrix with i.i.d. standard normal entries, i.e. ${\mathbf{A}}_{ij}\sim{\sf N}(0,1)$. Then, for every $t\geq 0$: $\displaystyle{\mathbb{P}}\left\\{\left\lVert{{\mathbf{A}}}\right\rVert_{2}\geq\sqrt{M}+\sqrt{N}+t\right\\}$ $\displaystyle\leq\exp\left(-\frac{t^{2}}{2}\right).$ The proof of this result can be found in [Ver12]. ## 5 Proof of Theorems 2 and 3 In this section we prove Theorem 2 and Theorem 3, assuming that Theorem 1 holds. The proof of the latter can be found in Section 6. ### 5.1 Proof of Theorem 2 For any fixed ${\varepsilon}>0$, and assume $\sum_{q}k_{q}\leq\sqrt{n\log\tau/\tau^{3}}$, where $\tau=\tau({\varepsilon},{\underline{\beta}},\alpha)$ as per Theorem 1. Then we have for every $q$, $\left\lVert{\mathbf{\widehat{v}}_{q}-{\mathbf{v}}_{q}}\right\rVert\leq{\varepsilon}$ with probability at least $1-C/n^{4}$ for some constant $C>0$. Throughout the proof, we will work on this favorable event of Theorem 1, namely use $\displaystyle{\mathbb{P}}\Big{(}{\widehat{\sf Q}}\neq\cup_{q}{\rm supp}({\mathbf{v}}_{q})\Big{)}\leq{\mathbb{P}}\Big{(}{\widehat{\sf Q}}\neq\cup_{q}{\rm supp}({\mathbf{v}}_{q});\;\;\left\lVert{\mathbf{\widehat{v}}_{q}-{\mathbf{v}}_{q}}\right\rVert^{2}\leq{\varepsilon}^{2}\Big{)}+\frac{C}{n^{4}}\,,$ (12) hence focusing on bounding the first term on the right hand side. It is convenient to isolate the following lemma. ###### Lemma 5.1. Assume $\\|\mathbf{\widehat{v}}_{q}-{\mathbf{v}}_{q}\\|^{2}\leq{\varepsilon}^{2}$ and that $\left\lvert{v_{q,i}}\right\rvert\geq\theta/\sqrt{k_{q}}$. Let ${\sf B}_{q}\equiv{\rm supp}({\mathbf{s}}_{q})$ with ${\mathbf{s}}_{q}$ defined as per Algorithm 1, step 7. Then $\|{\sf B}_{q}\triangle{\sf Q}_{q}\|\leq 4{\varepsilon}^{2}k_{q}/\theta^{2}$ and hence $\|{\sf B}_{q}\cap{\sf Q}_{q}\|\geq(1-4{\varepsilon}^{2}/\theta^{2})k_{q}$. (Here $\triangle$ denotes the symmetric set-difference.) Further $\min(\left\lVert{{\mathbf{s}}_{q}-{\mathbf{v}}_{q}}\right\rVert^{2},\left\lVert{{\mathbf{s}}_{q}+{\mathbf{v}}_{q}}\right\rVert^{2})\leq 5{\varepsilon}^{2}$. ###### Proof. Recall that $s_{q,i}=\widehat{v}_{q,i}\mathbb{I}(\left\lvert{\widehat{v}_{q,i}}\right\rvert\geq\theta/2\sqrt{k_{q}})$. Since $\left\lvert{v_{q,i}}\right\rvert\geq\theta/\sqrt{k_{q}}$: $\displaystyle{\sf B}_{q}\triangle{\sf Q}_{q}$ $\displaystyle\subseteq\left\\{i:\left\lvert{v_{q,i}-\widehat{v}_{q,i}}\right\rvert\geq\frac{\theta}{2\sqrt{k_{q}}}\right\\}.$ Thus $\left\lvert{{\sf B}_{q}\triangle{\sf Q}_{q}}\right\rvert\leq 4k_{q}\left\lVert{\mathbf{\widehat{v}}_{q}-{\mathbf{v}}_{q}}\right\rVert^{2}/\theta^{2}\leq 4{\varepsilon}^{2}k_{q}/\theta^{2}$. Now we bound the error $\left\lVert{{\mathbf{s}}_{q}-{\mathbf{v}}_{q}}\right\rVert$, assuming that $\left\lVert{\mathbf{\widehat{v}}_{q}-{\mathbf{v}}_{q}}\right\rVert\leq{\varepsilon}$. The other case is handled in an analogous fashion: $\displaystyle\left\lVert{{\mathbf{s}}_{q}-{\mathbf{v}}_{q}}\right\rVert^{2}$ $\displaystyle=\sum_{i\in{\sf Q}_{q}}(\widehat{v}_{q,i}\mathbb{I}(\|\widehat{v}_{q,i}\|\geq\theta/2\sqrt{k_{q}})-v_{q,i})^{2}+\sum_{i\in{\sf Q}_{q}^{c}}(\widehat{v}_{q,i})^{2}\mathbb{I}(\left\lvert{\widehat{v}_{q,i}}\right\rvert\geq\theta/2\sqrt{k_{q}})$ $\displaystyle=\sum_{i\in{\sf Q}_{q}}v_{q,i}^{2}\mathbb{I}(\left\lvert{\widehat{v}_{q,i}}\right\rvert\leq\theta/2\sqrt{k_{q}})+\sum_{i\in{\sf Q}_{q}}(\widehat{v}_{q,i}-v_{q,i})^{2}\mathbb{I}(\left\lvert{\widehat{v}_{q,i}}\right\rvert\geq\theta/2\sqrt{k_{q}})+\sum_{i\in{\sf Q}_{q}^{c}}(\widehat{v}_{q,i})^{2}\mathbb{I}(\left\lvert{\widehat{v}_{q,i}}\right\rvert\geq\theta/2\sqrt{k_{q}})$ $\displaystyle\leq\sum_{i\in{\sf Q}_{q}}v_{q,i}^{2}\mathbb{I}(\left\lvert{\widehat{v}_{q,i}-v_{q,i}}\right\rvert\geq\|{v_{q,i}\|-\theta/(2\sqrt{k_{q}})})+\left\lVert{\mathbf{\widehat{v}}_{q}-{\mathbf{v}}_{q}}\right\rVert^{2}$ $\displaystyle\leq\sum_{i\in{\sf Q}_{q}}\frac{v_{q,i}^{2}}{(\left\lvert{v_{q,i}}\right\rvert-\theta/2\sqrt{k_{q}})^{2}}(\widehat{v}_{q,i}-v_{q,i})^{2}+\left\lVert{\mathbf{\widehat{v}}_{q}-{\mathbf{v}}_{q}}\right\rVert^{2}$ $\displaystyle\leq 5\left\lVert{\mathbf{\widehat{v}}_{q}-{\mathbf{v}}_{q}}\right\rVert^{2}\leq 5{\varepsilon}^{2}.$ The first inequality above follows from triangle inequality as $\left\lvert{\widehat{v}_{q,i}}\right\rvert\geq\left\lvert{v_{q,i}}\right\rvert-\left\lvert{\widehat{v}_{q,i}-v_{q,i}}\right\rvert$. The second inequality employs $\mathbb{I}(z\geq z^{\prime})\leq(z/z^{\prime})^{2}$. The final inequality uses the fact that $\left\lvert{v_{q,i}}\right\rvert\geq\theta/2\sqrt{k_{q}}$ implies $\left\lvert{v_{q,i}}\right\rvert/(\left\lvert{v_{q,i}}\right\rvert-\theta/2\sqrt{k_{q}})\leq 2$. ∎ Now we are in position to prove the main theorem. Without loss of generality, we will assume that $\langle\mathbf{\widehat{v}}_{q},{\mathbf{v}}_{q}\rangle>0$ for every $q$. The other case is treated in the same way. Recall that ${\mathbf{\widehat{\Sigma}}}^{\prime}$ was formed from the samples $({\mathbf{x}}_{i})_{n<i\leq 2n}$, which are independent of $\mathbf{\widehat{v}}_{q}$ and hence ${\sf B}_{q}$. We let ${\mathbf{X}}^{\prime}\in\mathbb{R}^{n\times p}$ denote the matrix with rows $({\mathbf{x}}_{i})_{n<i\leq 2n}$ we have, in the same fashion as Eq. (2), ${\mathbf{X}}^{\prime}=\sum_{q}\sqrt{\beta_{q}}{\mathbf{u}}^{\prime}_{q}({\mathbf{v}}_{q})^{\sf T}+{\mathbf{Z}}^{\prime}$. We let ${\mathbf{\tilde{z}}}^{\prime}_{i},1\leq i\leq p$ denote the columns of ${\mathbf{Z}}^{\prime}$. For any $i$: $\displaystyle({\mathbf{\widehat{\Sigma}}}^{\prime}{\mathbf{s}}^{1})_{i}$ $\displaystyle=\frac{\beta_{1}\left\lVert{{\mathbf{u}}^{\prime}_{1}}\right\rVert^{2}\langle{\mathbf{v}}_{1},{\mathbf{s}}_{1}\rangle v_{1,i}}{n}+\sum_{q\neq 1}\frac{\beta_{q}\left\lVert{{\mathbf{u}}^{\prime}_{q}}\right\rVert_{2}^{2}\langle{\mathbf{v}}_{q},{\mathbf{s}}_{q}\rangle v_{q,i}}{n}+\sum_{q\geq 1}\frac{\sqrt{\beta_{q}}}{n}(\langle{\mathbf{Z}}^{\prime{\sf T}}{\mathbf{u}}^{\prime}_{q},{\mathbf{s}}_{1}\rangle v_{q,i}+\langle{\mathbf{v}}_{q},{\mathbf{s}}_{1}\rangle({\mathbf{Z}}^{\prime{\sf T}}{\mathbf{u}}_{q})_{i})$ $\displaystyle+\sum_{q^{\prime}>q}\frac{\sqrt{\beta_{q}\beta_{q^{\prime}}}}{n}\langle{\mathbf{u}}^{\prime}_{q},{\mathbf{u}}^{\prime}_{q^{\prime}}\rangle(v_{q,i}\langle{\mathbf{v}}_{q^{\prime}},{\mathbf{s}}_{1}\rangle+v_{q^{\prime},i}\langle{\mathbf{v}}_{q},{\mathbf{s}}_{1}\rangle)+\frac{1}{n}\sum_{j\in{\sf B}^{1},j\neq i}\langle{\mathbf{\tilde{z}}}^{\prime}_{j},{\mathbf{\tilde{z}}}^{\prime}_{i}\rangle s_{1,j}+\bigg{(}\frac{\lVert{\mathbf{\tilde{z}}}^{\prime}_{i}\rVert^{2}}{n}-1\bigg{)}s_{1,i}$ Let $T_{1},T_{2}\dots T_{5}$ denote the terms above. Firstly, by a standard calculation $n/2\leq\left\lVert{{\mathbf{u}}^{\prime}_{q}}\right\rVert_{2}^{2}\leq 2n$ and $\max_{q\neq q^{\prime}}\|\langle{\mathbf{u}}^{\prime}_{q},{\mathbf{u}}^{\prime}_{q}\rangle\|\leq\sqrt{Cn\log n}$ with probability at least $1-rn^{-10}$ for some constant $C$. Further, using Lemma 5.1 and Cauchy-Schwarz we have that $\langle{\mathbf{v}}_{1},{\mathbf{s}}_{1}\rangle\geq(1-5{\varepsilon}^{2})$ and $\|\langle{\mathbf{v}}_{q},{\mathbf{s}}_{1}\rangle\|\leq\left\lVert{{\mathbf{v}}_{1}-{\mathbf{s}}_{1}}\right\rVert\leq 3{\varepsilon}$. This implies that: $\displaystyle\left\lvert{T_{1}}\right\rvert$ $\displaystyle\geq\frac{\beta_{1}(1-5{\varepsilon}^{2})\left\lvert{v_{1,i}}\right\rvert}{2},$ $\displaystyle\left\lvert{T_{2}}\right\rvert$ $\displaystyle\leq 6{\varepsilon}\sum_{q>1}\beta_{q}\left\lvert{v_{q,i}}\right\rvert,$ $\displaystyle\left\lvert{T_{4}}\right\rvert$ $\displaystyle\leq C((\beta_{q})_{q\leq r})\sqrt{\frac{\log n}{n}}.$ Now consider the term $T_{5}=\sum_{j\in{\sf B}_{1}\backslash i}\langle{\mathbf{\tilde{z}}}^{\prime}_{i},{\mathbf{\tilde{z}}}^{\prime}_{j}\rangle s_{1,j}/n=\langle{\mathbf{\tilde{z}}}^{\prime}_{i},\sum_{j\in{\sf B}_{1}\backslash i}s_{1,j}{\mathbf{\tilde{z}}}^{\prime}_{j}\rangle/n$. Thus, $T_{5}{\,\stackrel{{\scriptstyle\mathrm{d}}}{{=}}\,}Y_{ij}\equiv\langle{\mathbf{\tilde{z}}}_{i}^{\prime},{\mathbf{\tilde{z}}}_{j}^{\prime}\left\lVert{{\mathbf{s}}_{1}}\right\rVert\rangle/n$ for $j\neq i$. Conditional on ${\mathbf{\tilde{z}}}^{\prime}_{j}$, $Y_{ij}\sim{\sf N}(0,\lVert{{\mathbf{\tilde{z}}}^{\prime}_{j}}\rVert^{2}\left\lVert{{\mathbf{s}}_{1}}\right\rVert^{2}/n^{2})$. Using the Chernoff bound , $\left\lVert{{\mathbf{\tilde{z}}}^{\prime}_{i}}\right\rVert^{2}\leq 2n$ with probability at leat $1-\exp(-n/8)$ and, conditional on this event, $\left\lvert{Y_{ij}}\right\rvert\leq\sqrt{C^{\prime}\log n/n}$ with probability at least $1-n^{-10}$ for some absolute constant $C^{\prime}$. It follows from the union bound that $\left\lvert{T_{5}}\right\rvert\leq\sqrt{C^{\prime}\log n/n}$ with probability at least $1-2n^{-10}$ for $n$ large enough. Using a similar calculation $\left\lvert{T_{3}}\right\rvert\leq\sqrt{C^{\prime}((\beta_{q})q)\log n/n}$ with probability exceeding $1-n^{-10}$. Finally using Proposition 6.4 below, we have that $\displaystyle\left\lvert{T_{5}}\right\rvert$ $\displaystyle\leq\left\lVert{{\mathbf{s}}_{1}}\right\rVert\max_{i}\bigg{(}\frac{\left\lVert{{\mathbf{\tilde{z}}}_{i}}\right\rVert^{2}}{n}-1\bigg{)}$ $\displaystyle\leq\sqrt{\frac{C^{\prime\prime}\log n}{n}},$ with probability at least $1-n^{-10}$. Here we used the fact that $\left\lVert{{\mathbf{s}}_{1}}\right\rVert\leq\left\lVert{\mathbf{\widehat{v}}_{1}}\right\rVert=1$. By Assumption A2, and the above estimates, we have with probability at least $1-n^{-9}$: $\displaystyle\text{For }i\in{\sf Q}_{1},\quad\left\lvert{({\mathbf{\widehat{\Sigma}}}{\mathbf{s}}_{1})_{i}}\right\rvert$ $\displaystyle\geq\frac{\beta_{1}}{2}(1-5{\varepsilon}^{2}-12{\varepsilon}\gamma\sum_{q}\beta_{q})\left\lvert{v_{1,i}}\right\rvert-\sqrt{C\log n/n}$ $\displaystyle\geq\frac{\beta_{1}(1-5{\varepsilon}^{2}-12{\varepsilon}\gamma\sum_{q}\beta_{q})\theta}{4\sqrt{k_{1}}}-\sqrt{\frac{C\log n}{n}},$ $\displaystyle\text{For }i\in[p]\backslash(\cup_{q}{\sf Q}_{q}),\quad\left\lvert{({\mathbf{\widehat{\Sigma}}}{\mathbf{s}}_{1})_{i}}\right\rvert$ $\displaystyle\leq\sqrt{\frac{C\log n}{n}}.$ Choosing ${\varepsilon}={\varepsilon}((\beta_{q})_{q\leq r},r,\theta,\gamma)$ small enough and using threshold $\rho=\min_{q}(\beta_{q}\theta/4\sqrt{k_{q}})$ we have that ${\sf Q}_{1}\subseteq{\widehat{\sf Q}}$ and ${\widehat{\sf Q}}\subseteq\cup_{q}{{\sf Q}_{q}}$. The analogous guarantees for all $1\leq q\leq r$ imply Theorem 2. ### 5.2 Proof of Theorem 3 Analogously to the previous proof, we fix ${\varepsilon}>0$, and observe that $\sum_{q}{k_{q}}\leq\sqrt{n\log\tau/\tau^{3}}$, where $\tau=\tau({\varepsilon},{\underline{\beta}},\alpha,\theta)$, and per Theorem 1. Then we have that $\left\lVert{\mathbf{\widehat{v}}_{q}-\mathbf{\widehat{v}}}\right\rVert^{2}\leq{\varepsilon}/20$ with probability at least $1-C/n^{4}$ for some constant $C>0$. We then use $\displaystyle{\mathbb{P}}\Big{(}{\widehat{\sf Q}}\neq\cup_{q}{\rm supp}({\mathbf{v}}_{q})\Big{)}\leq{\mathbb{P}}\Big{(}{\widehat{\sf Q}}\neq\cup_{q}{\rm supp}({\mathbf{v}}_{q});\;\;\left\lVert{\mathbf{\widehat{v}}_{q}-{\mathbf{v}}}\right\rVert^{2}\leq\frac{{\varepsilon}}{20r}\Big{)}+\frac{C}{n^{4}}\,,$ (13) and bound the first term. The key change with respect to the proof of theorems 2 is that we need to replace Lemma 5.1 with the following lemma, whose proof follows exactly the same argument as that of Lemma 5.1. ###### Lemma 5.2. Assume $\\|{\mathbf{v}}_{q}-\mathbf{\widehat{v}}_{q}\\|^{2}\leq{\varepsilon}/20$, and let ${\sf B}^{\prime}\equiv{\rm supp}({\mathbf{s}}^{\prime})$ with ${\mathbf{s}}^{\prime}$ defined as per Eq. (8). Further assume $k\leq k_{0}\leq 20\,k$. Then $\left\lVert{{\mathbf{s}}_{q}-{\mathbf{v}}_{q}}\right\rVert^{2}\leq 5{\varepsilon}^{2}$. The rest of the proof of Theorem 3 is identical to the one of Theorem 2 in the previous section. ## 6 Proof of Theorem 1 Since ${\mathbf{\widehat{\Sigma}}}={\mathbf{X}}^{{\sf T}}{\mathbf{X}}/n-{\rm I}_{p}$, we have: $\displaystyle{\mathbf{\widehat{\Sigma}}}$ $\displaystyle=\sum_{q=1}^{r}\left\\{\frac{\beta_{q}\left\lVert{{\mathbf{u}}_{q}}\right\rVert^{2}}{n}{\mathbf{v}}_{q}({\mathbf{v}}_{q})^{\sf T}+\frac{\sqrt{\beta_{q}}}{n}({\mathbf{v}}_{q}({\mathbf{u}}_{q})^{\sf T}{\mathbf{Z}}+{\mathbf{Z}}^{\sf T}{\mathbf{u}}_{q}{\mathbf{v}}^{\sf T})\right\\}$ $\displaystyle\quad+\sum_{q\neq q^{\prime}}\left\\{\frac{\sqrt{\beta_{q}\beta_{q^{\prime}}}\langle{\mathbf{u}}_{q},{\mathbf{u}}_{q^{\prime}}\rangle}{n}{\mathbf{v}}_{q}({\mathbf{v}}_{q^{\prime}})^{\sf T}\right\\}+\frac{{\mathbf{Z}}^{\sf T}{\mathbf{Z}}}{n}-{\rm I}_{p}.$ (14) We let ${\sf D}=\\{(i,i):i\in[p]\backslash\cup_{q}{\sf Q}_{q}\\}$ be the diagonal entries not included in any support and ${\sf Q}=\cup_{q}{\sf Q}_{q}$ denote the union of the supports. Further let ${\sf E}=\cup_{q}({\sf Q}_{q}\times{\sf Q}_{q})$, ${\sf F}=({\sf Q}^{c}\times{\sf Q}^{c})\backslash{\sf D}$, and ${\sf G}=[p]\times[p]\backslash({\sf D}\cup{\sf E}\cup{\sf F})$. Since these are disjoint we have: $\displaystyle\eta({\mathbf{\widehat{\Sigma}}})$ $\displaystyle=\underbrace{{\cal P}_{{\sf E}}\left\\{\eta({\mathbf{\widehat{\Sigma}}})\right\\}}_{{\mathbf{S}}}+\underbrace{{\cal P}_{{\sf F}}\left\\{\eta\left(\frac{1}{n}{\mathbf{Z}}^{\sf T}{\mathbf{Z}}\right)\right\\}}_{{\mathbf{N}}}+\underbrace{{\cal P}_{{\sf G}}\left\\{\eta({\mathbf{\widehat{\Sigma}}})\right\\}}_{{\mathbf{R}}_{1}}+\underbrace{{\cal P}_{{\sf D}}\left\\{\eta({\mathbf{\widehat{\Sigma}}})\right\\}}_{{\mathbf{R}}_{2}}.$ (15) The first term corresponds to the ‘signal’ component while the last three terms correspond to the ‘noise’ component. Theorem 1 is a direct consequence of the next four propositions. The first of these proves that the signal component is preserved, while the others demonstrate that the noise components are small. ###### Proposition 6.1. Let ${\mathbf{S}}$ denote the first term in Eq. (15): $\displaystyle{\mathbf{S}}$ $\displaystyle={\cal P}_{\sf E}\left\\{\eta({\mathbf{\widehat{\Sigma}}})\right\\}.$ (16) Then with probability at least $1-3\exp(-n^{2/3}/4)$: $\displaystyle\left\lVert{{\mathbf{S}}-\sum_{q=1}^{r}\beta_{q}{\mathbf{v}}_{q}({\mathbf{v}}_{q})^{\sf T}}\right\rVert_{2}$ $\displaystyle\leq\frac{\tau\sum_{q}{k_{q}}}{\sqrt{n}}+\kappa_{n}.$ Here $\kappa_{n}=16(\sqrt{r\alpha}+r\sqrt{\beta_{1}})n^{-1/6}$. ###### Proposition 6.2. Let ${\mathbf{N}}$ denote the second term of Eq. (15): $\displaystyle{\mathbf{N}}$ $\displaystyle={\cal P}_{{\sf F}}\left\\{\eta\left(\frac{1}{n}{\mathbf{Z}}^{\sf T}{\mathbf{Z}}\right)\right\\}.$ Then there exists $\tau_{1}=\tau_{1}(\alpha)$ such that for any $\tau\geq\tau_{1}$ and all $p$ large enough, we have $\displaystyle\left\lVert{{\mathbf{N}}}\right\rVert_{2}\leq C_{1}(\alpha)\sqrt{\frac{\log\tau}{\tau}}\,,$ (17) with probability at least $1-2\exp(-c_{1}(\tau)p)$. The constants can be taken as $\tau_{1}=100\max(1,\alpha^{2}\log\alpha)$, $c_{1}(\tau)=1/4\tau$ and $C_{1}(\alpha)=5000\max(1,\alpha^{3/2})$. ###### Proposition 6.3. Let ${\mathbf{R}}_{1}$ denote the matrix corresponding to the third term of Eq. (15): $\displaystyle{\mathbf{R}}_{1}$ $\displaystyle={\cal P}_{{\sf G}}\left\\{\eta({\mathbf{\widehat{\Sigma}}})\right\\}.$ Then there exists $\tau_{2}=\tau_{2}(\alpha,\beta_{1},r)$ such that for $\tau\geq\tau_{2}$ and every $p$ large enough we have: $\displaystyle\left\lVert{{\mathbf{R}}_{1}}\right\rVert_{2}$ $\displaystyle\leq C_{2}(\alpha,r,\beta_{1})\sqrt{\frac{\log\tau}{{\tau}}}.$ (18) with probability at least $1-\exp(-c_{2}(\tau)p)$. Here we may take $c_{2}(\tau)=c_{1}(\tau)=1/4\tau$. ###### Proposition 6.4. Let ${\mathbf{R}}_{2}$ denote the matrix corresponding to the third term of Eq. (15): $\displaystyle{\mathbf{R}}_{2}$ $\displaystyle={\cal P}_{{\sf D}}\left\\{\eta({\mathbf{\widehat{\Sigma}}})\right\\}.$ Then with probability at least $1-\alpha n^{-C/6+1}$ for every $n$ large enough: $\displaystyle\left\lVert{{\mathbf{R}}_{2}}\right\rVert_{2}$ $\displaystyle\leq\sqrt{\frac{C\log n}{n}}.$ (19) We defer the proofs of Propositions 6.1, 6.2, 6.3 and 6.4 to Sections 6.1, 6.2, 6.3 and 6.4 respectively. ###### Proof of Theorem 1. Using these results we now proceed to prove Theorem 1. We will assume that the events in these proposition hold, and control the probability of their complement via the union bound. Denote by $k$ the sum of the support sizes, i.e. $\sum_{q}k_{q}$. From Propositions 6.1, 6.2, 6.3, 6.4 and the triangle inequality we have: $\displaystyle\left\lVert{\eta({\mathbf{\widehat{\Sigma}}})-\sum_{q}\beta_{q}{\mathbf{v}}_{q}({\mathbf{v}}_{q})^{\sf T}}\right\rVert$ $\displaystyle\leq\frac{k\tau}{\sqrt{n}}+\max(C_{1},C_{2})\sqrt{\frac{\log\tau}{\tau}},$ for every $\tau\geq\max(\tau_{1},\tau_{2})$ with probability at least $1-\alpha n^{-4}$. Setting $k\leq\sqrt{n\log\tau/\tau^{3}}$, the right hand side above is bounded by $\delta(\tau)=2\max(C_{1},C_{2})\sqrt{\log\tau/\tau}$. Further define ${\underline{\beta}}\equiv\min_{q\neq q^{\prime}\leq r}(\beta_{q},\left\lvert{\beta_{q}-\beta_{q^{\prime}}}\right\rvert)$. Employing the Davis-Kahan $\sin\theta$ theorem [DK70] we have: $\displaystyle\min(\left\lVert{\mathbf{\widehat{v}}_{q}-{\mathbf{v}}_{q}}\right\rVert,\left\lVert{\mathbf{\widehat{v}}_{q}+{\mathbf{v}}_{q}}\right\rVert)$ $\displaystyle\leq\sqrt{2}\sin\theta(\mathbf{\widehat{v}}_{q},{\mathbf{v}}_{q})$ $\displaystyle\leq\frac{\sqrt{2}\delta(\tau)}{{\underline{\beta}}-\delta(\tau)}.$ Choosing $\tau\geq(8\max(C_{1},C_{2})/{\underline{\beta}}{\varepsilon})^{4}$ yields that $\delta(\tau)/({\underline{\beta}}-\delta(\tau))\leq{\varepsilon}$. Letting $\tau$ be the largest of $\tau_{1}$, $\tau_{2}$ and $(8\max(C_{1},C_{2})/{\underline{\beta}}{\varepsilon})^{4})$ gives the desired result. ∎ ### 6.1 Proof of Proposition 6.1 The proof proceeds in two steps. In the first lemma we bound $\left\lVert{{\mathbb{E}}\\{{\mathbf{S}}\\}-\sum_{q}\beta_{q}{\mathbf{v}}_{q}({\mathbf{v}}_{q})^{\sf T}}\right\rVert$ and in the second we control $\left\lVert{{\mathbf{S}}-{\mathbb{E}}\\{{\mathbf{S}}\\}}\right\rVert$. ###### Lemma 6.5. Consider ${\mathbf{S}}$ as defined in Proposition 6.1. Then $\displaystyle\left\lVert{{\mathbb{E}}\\{{\mathbf{S}}\\}-\sum_{q}\beta_{q}{\mathbf{v}}_{q}({\mathbf{v}}_{q})^{\sf T}}\right\rVert$ $\displaystyle\leq\frac{\tau\sum_{q}k_{q}}{\sqrt{n}}.$ ###### Proof. Notice that ${\mathbb{E}}\\{{\mathbf{S}}\\}$ is supported on a set of indices $\cup_{q}{\sf Q}_{q}\times\cup_{q}{\sf Q}_{q}$ which has size at most $(\sum_{q}k_{q})^{2}$. Hence $\displaystyle\left\lVert{{\mathbb{E}}\\{{\mathbf{S}}\\}-\sum_{q}{\beta_{q}}{\mathbf{v}}_{q}({\mathbf{v}}_{q})^{\sf T}}\right\rVert$ $\displaystyle\leq(\sum_{q}k_{q})\left\lVert{{\mathbb{E}}\\{{\mathbf{S}}\\}-\sum_{q}\beta_{q}{\mathbf{v}}_{q}({\mathbf{v}}_{q})^{\sf T}}\right\rVert_{\infty},$ where the last term denotes the entrywise $\ell_{\infty}$ norm of the matrix. Since ${\mathbf{S}}$ and $\sum_{q}\beta_{q}{\mathbf{v}}_{q}({\mathbf{v}}_{q})^{\sf T}$ have common support and since $\|\eta(z;\tau/\sqrt{n})-z\|\leq\tau/\sqrt{n}$ we obtain that: $\displaystyle\left\lVert{{\mathbb{E}}\\{{\mathbf{S}}\\}-\sum_{q}\beta_{q}{\mathbf{v}}_{q}({\mathbf{v}}_{q})^{\sf T}}\right\rVert_{\infty}$ $\displaystyle\leq\left\lVert{{\mathbb{E}}\\{{\cal P}_{{\sf E}}(\eta({\mathbf{\widehat{\Sigma}}}))\\}-\sum_{q}\beta_{q}{\mathbf{v}}_{q}{\mathbf{v}}_{q}^{\sf T}}\right\rVert_{\infty}$ $\displaystyle\leq\frac{\tau}{\sqrt{n}}.$ The thesis then follows directly. ∎ ###### Lemma 6.6. Let ${\mathbf{S}}$ be as defined in Proposition 6.1. Then: $\displaystyle\left\lVert{{\mathbf{S}}-{\mathbb{E}}\\{{\mathbf{S}}\\}}\right\rVert$ $\displaystyle\leq\kappa_{n},$ with probability at least $1-\exp(-n^{2/3}/4)$ where we define $\kappa_{n}\equiv 16(\sqrt{r\alpha}+r\sqrt{\beta_{1}})n^{-1/6}$. Proposition 6.1 follows directly from these two lemmas since we have by triangle inequality: $\displaystyle\left\lVert{{\mathbf{S}}-\sum_{q}\beta_{q}{\mathbf{v}}_{q}({\mathbf{v}}_{q})^{\sf T}}\right\rVert$ $\displaystyle\leq\left\lVert{{\mathbf{S}}-{\mathbb{E}}\\{{\mathbf{S}}\\}}\right\rVert+\left\lVert{{\mathbb{E}}\\{{\mathbf{S}}\\}-\sum_{q}\beta_{q}{\mathbf{v}}_{q}({\mathbf{v}}_{q})^{\sf T}}\right\rVert.$ This completes the proof of Proposition 6.1 conditional on Lemma 6.6. In the next subsection we prove Lemma 6.6. #### 6.1.1 Proof of Lemma 6.6 Let ${\mathbf{y}}\in\mathbb{R}^{p}$ denote a vector supported on $\cup_{q}{\sf Q}_{q}$. Recall that ${\sf Q}=\cup_{q}{\sf Q}_{q}$. Fix an $\ell\in{\sf Q}$. The gradient of the Rayleigh quotient ${\nabla}_{{\mathbf{\tilde{z}}}_{\ell}}\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle$ reads: $\displaystyle{\nabla}_{{\mathbf{\tilde{z}}}_{\ell}}\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle$ $\displaystyle=\frac{1}{n}\sum_{i:(i,\ell)\in\cup_{q}{\sf Q}_{q}\times{\sf Q}_{q}}2\partial\eta\left({\mathbf{\widehat{\Sigma}}}_{i\ell};\frac{\tau}{\sqrt{n}}\right)({\mathbf{\tilde{z}}}_{i}+\sum_{q}\sqrt{\beta_{q}}v^{q}_{i}{\mathbf{u}}_{q})y_{i}y_{\ell}.$ Define the vector ${\boldsymbol{\sigma}}^{\ell}({\mathbf{y}})\in\mathbb{R}^{p}$ as follows: $\displaystyle\sigma^{\ell}_{i}({\mathbf{y}})$ $\displaystyle=\begin{cases}\partial\eta\left({\mathbf{\widehat{\Sigma}}}_{i\ell};\frac{\tau}{\sqrt{n}}\right)y_{i},&\text{ if }(i,\ell)\in\cup_{q}({\sf Q}_{q}\times{\sf Q}_{q})\\\ 0\text{ otherwise.}\end{cases}$ where the left hand side denotes the $i^{\text{th}}$ entry of ${\boldsymbol{\sigma}}^{\ell}({\mathbf{y}})$. Recall that ${\mathbf{Z}}_{E}$ is the matrix obtained from ${\mathbf{Z}}$ by setting to zero all columns with indices outside $E\subseteq[p]$. Using this, we can now rewrite the gradient in the following form: $\displaystyle{\nabla}_{{\mathbf{\tilde{z}}}_{\ell}}\langle{\mathbf{y}},{\mathbf{S}}^{q}{\mathbf{y}}\rangle$ $\displaystyle=\frac{2y_{\ell}}{n}({\mathbf{Z}}_{{\sf Q}}+\sum_{q}\sqrt{\beta_{q}}{\mathbf{u}}_{q}({\mathbf{v}}_{q})^{\sf T}){\boldsymbol{\sigma}}^{\ell}({\mathbf{y}}).$ Since $\partial\eta(\cdot;\cdot)\in\\{0,1\\}$, we see that $\left\lVert{{\boldsymbol{\sigma}}^{\ell}({\mathbf{y}})}\right\rVert\leq\left\lVert{{\mathbf{y}}}\right\rVert=1$. Consequently, we have that: $\displaystyle\left\lVert{{\nabla}_{{\mathbf{\tilde{z}}}_{\ell}}\langle{\mathbf{y}},{\mathbf{S}}^{q}{\mathbf{y}}\rangle}\right\rVert$ $\displaystyle\leq\frac{\left\lvert{2y_{\ell}}\right\rvert}{n}\left\lVert{{\mathbf{Z}}_{{\sf Q}}+\sum_{q}\sqrt{\beta_{q}}{\mathbf{u}}_{q}({\mathbf{v}}_{q})^{\sf T}}\right\rVert\left\lVert{{\boldsymbol{\sigma}}^{\ell}({\mathbf{y}})}\right\rVert$ $\displaystyle\leq\frac{2\left\lvert{y_{\ell}}\right\rvert}{n}\left(\left\lVert{{\mathbf{Z}}_{{\sf Q}}}\right\rVert+\sum_{q}\sqrt{\beta_{q}}\left\lVert{{\mathbf{u}}_{q}({\mathbf{v}}_{q})^{\sf T}}\right\rVert\right)$ $\displaystyle=\frac{2\left\lvert{y_{\ell}}\right\rvert}{n}\left(\left\lVert{{\mathbf{Z}}_{{\sf Q}}}\right\rVert+\sum_{q}\sqrt{\beta_{q}}\left\lVert{{\mathbf{u}}_{q}}\right\rVert\right),$ Squaring and summing over $\ell$: $\displaystyle\left\lVert{{\nabla}_{{\mathbf{Z}}_{\sf Q}}\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle}\right\rVert^{2}$ $\displaystyle\leq\frac{4}{n^{2}}(\left\lVert{{\mathbf{Z}}_{{\sf Q}}}\right\rVert+\sum_{q}\beta_{q}\left\lVert{{\mathbf{u}}_{q}}\right\rVert)^{2}.$ The gradient above is with respect to all the variables ${\mathbf{\tilde{z}}}_{\ell},\ell\in{\sf Q}_{q}$ and the norm is the standard vector $\ell_{2}$ norm. Let $G:\\{{\mathbf{Z}},({\mathbf{u}}_{q})_{q\leq r}:\left\lVert{{\mathbf{Z}}_{{\sf Q}}}\right\rVert\leq(2+\sqrt{r\alpha})\sqrt{n},\left\lVert{{\mathbf{u}}_{q}}\right\rVert\leq 4\sqrt{n}\\}$. Clearly $G$ is a closed, convex set. Further, using Lemma 4.6 we can bound the probability of $G^{c}$: $\left\lVert{{\mathbf{Z}}_{{\sf Q}}}\right\rVert\leq(\sqrt{n}+\sqrt{\sum_{q}k_{q}}+\sqrt{n})\leq(2+\sqrt{r\alpha})\sqrt{n}$ with probability at least $1-\exp(-n/2)$. Also, with probability at least $1-r\exp(-n/2)$, for every $q$ $\left\lVert{{\mathbf{u}}_{q}}\right\rVert\leq 4\sqrt{n}$. Thus, on the set $G$ we have: $\displaystyle\left\lVert{{\nabla}\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle}\right\rVert^{2}\mathbb{I}\\{({\mathbf{Z}},{\mathbf{u}}_{1}\cdots{\mathbf{u}}_{r})\in G\\}$ $\displaystyle\leq\frac{64}{n}(2+\sqrt{r\alpha}+\sqrt{\beta})^{2}$ $\displaystyle{\mathbb{P}}\\{G^{c}\\}$ $\displaystyle\leq 2\exp\left(-\frac{n}{4}\right).$ Define $L$ and $\kappa_{n}$ as follows: $\displaystyle L$ $\displaystyle\equiv\frac{8(2+\sqrt{r\alpha}+r\sqrt{\beta_{1}})}{\sqrt{n}}$ $\displaystyle\kappa_{n}$ $\displaystyle\equiv 16(2+\sqrt{r\alpha}+r\sqrt{\beta_{1}})n^{-1/6}=2Ln^{1/3}.$ Also let $F_{L}({\mathbf{Z}}_{{\sf Q}})$ denote the $G,L$-Lipschitz extension of $\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle$. We prove the following remark in Appendix A: ###### Remark 6.7. For every $n$ large enough, $\|{\mathbb{E}}\left\\{\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle- F_{L}({\mathbf{Z}}_{{\sf Q}})\right\\}\|\leq n^{-1}$. Now employing Lemma 4.4: $\displaystyle{\mathbb{P}}\left\\{\|\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle-{\mathbb{E}}\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle\|\geq\kappa_{n}/2\right\\}$ $\displaystyle\leq 2\exp\left(-\frac{n^{2/3}}{2}\right)+2r\exp\left(-\frac{n}{4}\right)$ $\displaystyle\leq 3\exp\left(-\frac{n^{2/3}}{2}\right),$ for every $n$ large enough. Then using ${\mathbf{y}}$ as a vector in the $1/4$-net $T^{1/4}_{\|{\sf Q}\|}$ embedded in $\mathbb{R}^{p}$ via the union of supports ${\sf Q}$, we use Lemma 4.3 to obtain that: $\displaystyle\left\lVert{{\mathbf{S}}-{\mathbb{E}}\\{{\mathbf{S}}\\}}\right\rVert\leq\kappa_{n},$ with probability at least $1-3\cdot 9^{\left\lvert{{\sf Q}}\right\rvert}\exp(-n^{2/3}/2)\geq 1-\exp(-n^{2/3}/4)$ since $\left\lvert{{\sf Q}}\right\rvert\leq\sum_{q}{k_{q}}=O(\sqrt{n})\leq n^{2/3}/2$ for large enough $n$. ### 6.2 Proof of Proposition 6.2 It suffices to bound the norm of ${\mathbf{\widetilde{N}}}$ defined as $\displaystyle{\mathbf{\widetilde{N}}}$ $\displaystyle={\mathcal{P}_{\sf nd}}\left\\{\eta\left(\frac{1}{n}{\mathbf{Z}}^{\sf T}{\mathbf{Z}}\right)\right\\}.$ We use a variant of the ${\varepsilon}$-net argument. For a set of indices $E\subseteq[p]$, recall that ${\mathbf{y}}_{E}\in\mathbb{R}^{p}$ denotes the vector coinciding with ${\mathbf{y}}$ on $E$, and zero outside $E$. By decomposing the Rayleigh quotient: $\displaystyle{\mathbb{P}}\left\\{\left\lVert{{\mathbf{\widetilde{N}}}}\right\rVert_{2}\geq\Delta\right\\}$ $\displaystyle\leq{\mathbb{P}}\left\\{\sup_{{\mathbf{y}}\in T^{\varepsilon}_{p}}\langle{\mathbf{y}},{\mathbf{\widetilde{N}}}{\mathbf{y}}\rangle\geq\Delta(1-2{\varepsilon})\right\\}$ $\displaystyle\leq{\mathbb{P}}\left\\{\sup_{{\mathbf{y}}\in T^{\varepsilon}_{p}}\langle{\mathbf{y}}_{E},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E}\rangle+\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle+2\langle{\mathbf{y}}_{E},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle\geq\Delta(1-2{\varepsilon})\right\\}.$ We let $E=\\{i\in[p]:\|y_{i}\|>\sqrt{A/p}\\}$ for the constant $A=A(\tau)=\tau\log\tau$. Since $\left\lVert{{\mathbf{y}}}\right\rVert=1$, it follows that $\|E\|\leq p/A$. The following lemma allows to bound the term $\langle{\mathbf{y}}_{E},{\mathbf{\widetilde{N}}},{\mathbf{y}}_{E}\rangle$ uniformly over all subsets $E$ smaller than $p/A$. ###### Lemma 6.8. Fix $A\geq 180\max(\sqrt{\alpha},1)$. Then, for every $p$ large enough, the following holds with probability at least $1-\exp(-p\log A/4A)$: $\displaystyle\sup_{E\subseteq[p],\|E\|\leq p/A}\left\lVert{{\widetilde{\cal P}}_{E,E}({\mathbf{\widetilde{N}}})}\right\rVert_{2}$ $\displaystyle\leq 32\sqrt{\alpha\frac{\log A}{A}}.$ The proof of this lemma is provided in subsection 6.2.1. Denoting by ${\cal E}$ the favorable event of Lemma 6.8, we obtain: $\displaystyle{\mathbb{P}}\left\\{\left\lVert{{\mathbf{\widetilde{N}}}}\right\rVert_{2}\geq\Delta\right\\}$ $\displaystyle\leq{\mathbb{P}}\left\\{{\cal E}^{c}\right\\}+{\mathbb{P}}\left\\{\sup_{{\mathbf{y}}\in T^{\varepsilon}_{p}}\left(\langle{\mathbf{y}}_{E},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E}\rangle+\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle+2\langle{\mathbf{y}}_{E},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle\right)\geq\Delta(1-2{\varepsilon}),{\cal E}\right\\}$ $\displaystyle\leq{\mathbb{P}}\left\\{{\cal E}^{c}\right\\}+{\mathbb{P}}\left\\{\sup_{{\mathbf{y}}\in T^{\varepsilon}_{p}}\left(\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle+2\langle{\mathbf{y}}_{E},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle\right)\geq\widetilde{\Delta}\right\\},$ where $\widetilde{\Delta}=\Delta(1-2{\varepsilon})-16\sqrt{2\alpha\log A/A}$. Further, using the union bound and Lemma 4.2: $\displaystyle{\mathbb{P}}\left\\{\left\lVert{{\mathbf{\widetilde{N}}}}\right\rVert_{2}\geq\Delta\right\\}$ $\displaystyle\leq{\mathbb{P}}\left\\{{\cal E}^{c}\right\\}+\left\lvert{T^{\varepsilon}_{p}}\right\rvert\sup_{{\mathbf{y}}\in T^{\varepsilon}_{p}}{\mathbb{P}}\left\\{\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle\geq\frac{\widetilde{\Delta}}{3}\right\\}$ $\displaystyle\quad+\left\lvert{T^{\varepsilon}_{p}}\right\rvert\sup_{{\mathbf{y}}\in T^{\varepsilon}_{p}}{\mathbb{P}}\left\\{\langle{\mathbf{y}}_{E},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle\geq\frac{\widetilde{\Delta}}{3}\right\\}.$ (20) ${\mathbb{P}}\left({\cal E}^{c}\right)$ is bounded in Lemma 6.8. We now proceed to bound the latter two terms. For the second term, the gradient ${\nabla}_{{\mathbf{\tilde{z}}}_{\ell}}\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle$ reads, for any fixed $\ell\in E^{c}$: $\displaystyle{\nabla}_{{\mathbf{\tilde{z}}}_{\ell}}\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle$ $\displaystyle=\frac{2y_{\ell}}{n}\sum_{i\in E^{c}\backslash\ell}\partial\eta\left(\frac{\langle{\mathbf{\tilde{z}}}_{i},{\mathbf{\tilde{z}}}_{\ell}\rangle}{n};\frac{\tau}{\sqrt{n}}\right)y_{i}{\mathbf{\tilde{z}}}_{i}.$ Let ${\boldsymbol{\sigma}}^{\ell}({\mathbf{y}})\in\mathbb{R}^{p}$ be a vector defined by: $\displaystyle\sigma_{i}^{\ell}({\mathbf{y}})$ $\displaystyle=\begin{cases}\partial\eta\left(\frac{\langle{\mathbf{\tilde{z}}}_{i},{\mathbf{\tilde{z}}}_{\ell}\rangle}{n};\frac{\tau}{\sqrt{n}}\right)y_{i}&\text{ if }i\in E^{c}\backslash\ell,\\\ 0&\text{ otherwise.}\end{cases}$ With this definition we can represent the norm of the gradient as: $\displaystyle\left\lVert{{\nabla}_{{\mathbf{\tilde{z}}}_{\ell}}\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle}\right\rVert$ $\displaystyle=\frac{2\left\lvert{y_{\ell}}\right\rvert}{n}\left\lVert{{\mathbf{Z}}_{E^{c}}{\boldsymbol{\sigma}}^{\ell}({\mathbf{y}})}\right\rVert$ $\displaystyle\leq\frac{2\left\lvert{y_{\ell}}\right\rvert}{n}\left\lVert{{\mathbf{Z}}_{E^{c}}}\right\rVert\left\lVert{{\boldsymbol{\sigma}}^{\ell}({\mathbf{y}})}\right\rVert$ $\displaystyle\leq\frac{2\left\lvert{y_{\ell}}\right\rvert}{n}\left\lVert{{\mathbf{Z}}}\right\rVert\left\lVert{{\boldsymbol{\sigma}}^{\ell}({\mathbf{y}})}\right\rVert.$ For ${\boldsymbol{\sigma}}^{\ell}({\mathbf{y}})$: $\displaystyle\left\lVert{{\boldsymbol{\sigma}}^{\ell}({\mathbf{y}})}\right\rVert^{2}$ $\displaystyle=\sum_{i\in E^{c}\backslash\ell}\partial\eta\left(\frac{\langle{\mathbf{\tilde{z}}}_{i},{\mathbf{\tilde{z}}}_{\ell}\rangle}{n};\frac{\tau}{\sqrt{n}}\right)^{2}y_{i}^{2}$ $\displaystyle\leq\sum_{i\in E^{c}\backslash\ell}\frac{\langle{\mathbf{\tilde{z}}}_{i},{\mathbf{\tilde{z}}}_{\ell}\rangle^{2}}{n\tau^{2}}y_{i}^{2}$ $\displaystyle\leq\frac{A}{\tau^{2}np}\langle{\mathbf{\tilde{z}}}_{\ell},{\mathbf{Z}}_{E^{c}\backslash\ell}^{\sf T}{\mathbf{Z}}_{E^{c}\backslash\ell}{\mathbf{\tilde{z}}}_{\ell}\rangle$ $\displaystyle\leq\frac{A\left\lVert{{\mathbf{\tilde{z}}}_{\ell}}\right\rVert^{2}\left\lVert{{\mathbf{Z}}}\right\rVert^{2}}{np\tau^{2}}.$ Here the first line follows from $\partial\eta(x;y)=\mathbb{I}(\left\lvert{x}\right\rvert\geq y)\leq\left\lvert{x}\right\rvert/y$. The second line follows from the choice of $E$ whereby $\left\lvert{y_{i}}\right\rvert\leq\sqrt{A/p}$ and the last line from Cauchy-Schwarz. For any $\ell\in E$, ${\nabla}_{{\mathbf{\tilde{z}}}_{\ell}}\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle=0$. Now, fix $\Gamma=5$, $\gamma=\Gamma\max(\alpha^{-1},1)\geq\Gamma$ and let $G=\\{{\mathbf{Z}}:\left\lVert{{\mathbf{Z}}}\right\rVert\leq 2\sqrt{\gamma p},\forall\ell,\left\lVert{{\mathbf{\tilde{z}}}_{\ell}}\right\rVert\leq\sqrt{2\gamma p},\\}$. Clearly, $G$ is a closed, convex set. Furthermore, on the set $G$, we obtain from the gradient and ${\boldsymbol{\sigma}}^{\ell}({\mathbf{y}})$ estimates above that: $\displaystyle\left\lVert{{\nabla}_{{\mathbf{Z}}}\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle}\right\rVert^{2}$ $\displaystyle=\sum_{\ell\in E^{c}}\left\lVert{{\nabla}_{{\mathbf{\tilde{z}}}_{\ell}}\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle}\right\rVert^{2}$ $\displaystyle\leq\sum_{\ell\in E^{c}}\frac{4y_{\ell}^{2}}{n^{2}}\left\lVert{{\mathbf{Z}}}\right\rVert^{2}\left\lVert{{\boldsymbol{\sigma}}^{\ell}({\mathbf{y}})}\right\rVert^{2}$ $\displaystyle\leq\frac{4\left\lVert{{\mathbf{Z}}}\right\rVert^{2}}{n^{2}}\max_{\ell\in E^{c}}\left\lVert{{\boldsymbol{\sigma}}^{\ell}({\mathbf{y}})}\right\rVert^{2}$ $\displaystyle\leq\frac{4A\left\lVert{{\mathbf{Z}}}\right\rVert^{4}\max_{\ell\in E^{c}}\left\lVert{{\mathbf{\tilde{z}}}_{\ell}}\right\rVert^{2}}{n^{3}p\tau^{2}}$ (21) $\displaystyle\leq\frac{128A\gamma^{3}\alpha^{3}}{p\tau^{2}}.$ (22) Here we treat ${\nabla}_{{\mathbf{Z}}}\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle$ as a vector in $\mathbb{R}^{np}$, hence the norm above is the standard $\ell_{2}$ norm on vectors. We also write the gradient as ${\nabla}_{({\mathbf{\tilde{z}}}_{\ell})_{\ell\in[p]}}\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle$ to avoid ambiguity in specifying the norm. We now bound ${\mathbb{P}}\\{G^{c}\\}$ as follows. Lemma 4.6 implies that with probability at least $1-\exp(-\Gamma p/2)$: $\displaystyle\left\lVert{{\mathbf{Z}}}\right\rVert_{2}$ $\displaystyle\leq(1+\sqrt{\Gamma}+\alpha^{-1/2})\sqrt{p}$ $\displaystyle\leq 2\sqrt{\gamma p},$ (23) since $\gamma\geq(1+\alpha^{-1/2})^{2}$. Further, the standard Chernoff bound implies that, for a fixed $\ell$, $\left\lVert{{\mathbf{\tilde{z}}}_{\ell}}\right\rVert^{2}\leq 2\gamma\alpha n=2\gamma p$ with probability at least $1-\exp(-\gamma p/2)$. By the union bound, we then obtain that ${\mathbb{P}}\\{G^{c}\\}\leq p\exp(-\gamma p/2)+\exp(-\Gamma p/2)\leq(p+1)\exp(-\Gamma p/2)$. Define $K=\sqrt{128A\gamma^{3}\alpha^{3}/p\tau^{2}}$. Let $F_{K}({\mathbf{Z}})$ denote the $G,K$-Lipschitz extension of $F({\mathbf{Z}})=\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle$. We have the following remark for $F_{K}({\mathbf{Z}})$ which is proved in Appendix A. ###### Remark 6.9. We have ${\mathbb{E}}\\{\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle\\}=0$. Further, for every $p$ large enough, $\|{\mathbb{E}}\\{F_{K}({\mathbf{Z}})\\}\|\leq p^{-1}$. We can now use Lemma 4.4 for $F({\mathbf{Z}})$, Thus for any $\Delta_{2}\geq 2/p$: $\displaystyle{\mathbb{P}}\left\\{\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle\geq\Delta_{2}\right\\}$ $\displaystyle\leq\exp\left(-\frac{\Delta_{2}^{2}}{4K^{2}}\right)+2p\exp\left(-\frac{\Gamma p}{2}\right).$ (24) Using $\Delta_{2}=\sqrt{2\Gamma p}K=16\sqrt{A\Gamma\gamma^{3}\alpha^{3}}/\tau$ we obtain: $\displaystyle{\mathbb{P}}\left\\{\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle\geq 16\frac{\sqrt{A\Gamma\gamma^{3}\alpha^{3}}}{\tau}\right\\}$ $\displaystyle\leq(2p+2)\exp(-\Gamma p/2).$ (25) Now we can use the essentially same strategy on the term $\langle{\mathbf{y}}_{E},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle$. For $\ell\in E$ we have as before: $\displaystyle{\nabla}_{{\mathbf{\tilde{z}}}_{\ell}}\langle{\mathbf{y}}_{E},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle$ $\displaystyle=\frac{y_{\ell}}{n}\sum_{i\in E^{c}}\partial\eta\left(\frac{\langle\widetilde{z}_{i},\widetilde{z}_{\ell}\rangle}{n};\frac{\tau}{\sqrt{n}}\right)y_{i}{\mathbf{\tilde{z}}}_{i},$ $\displaystyle\left\lVert{{\nabla}_{{\mathbf{\tilde{z}}}_{\ell}}\langle{\mathbf{y}}_{E},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle}\right\rVert^{2}$ $\displaystyle\leq\frac{y_{\ell}^{2}A\left\lVert{{\mathbf{Z}}}\right\rVert^{4}\max_{i\in E^{c}}\left\lVert{{\mathbf{\tilde{z}}}_{i}}\right\rVert^{2}}{\tau^{2}pn^{3}}.$ Hence: $\displaystyle\sum_{\ell\in E}\left\lVert{{\nabla}_{{\mathbf{\tilde{z}}}_{\ell}}\langle{\mathbf{y}}_{E},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle}\right\rVert^{2}$ $\displaystyle\leq\frac{A\left\lVert{{\mathbf{Z}}}\right\rVert^{4}\max_{i}\left\lVert{{\mathbf{\tilde{z}}}_{i}}\right\rVert^{2}}{\tau^{2}pn^{3}}.$ (26) Analogously, for $\ell\in E^{c}$: $\displaystyle{\nabla}_{{\mathbf{\tilde{z}}}_{\ell}}\langle{\mathbf{y}}_{E},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle$ $\displaystyle=\frac{y_{\ell}}{n}\sum_{i\in E}\partial\eta\left(\frac{\langle{\mathbf{\tilde{z}}}_{i},{\mathbf{\tilde{z}}}_{\ell}\rangle}{n};\frac{\tau}{\sqrt{n}}\right)y_{i}{\mathbf{\tilde{z}}}_{i}$ $\displaystyle=\frac{y_{\ell}}{n}{\mathbf{Z}}_{E}{\boldsymbol{\sigma}}^{\ell}_{E}({\mathbf{y}}),$ where we define the vector ${\boldsymbol{\sigma}}^{\ell}_{E}({\mathbf{y}})\in\mathbb{R}^{E}$ as: $\displaystyle\forall i\in E,\quad\sigma^{\ell}_{E}({\mathbf{y}})_{i}$ $\displaystyle=y_{i}\partial\eta\left(\frac{\langle{\mathbf{\tilde{z}}}_{i},{\mathbf{\tilde{z}}}_{\ell}\rangle}{n};\frac{\tau}{\sqrt{n}}\right).$ By Cauchy-Schwarz: $\displaystyle\left\lVert{{\nabla}_{{\mathbf{\tilde{z}}}_{\ell}}\langle{\mathbf{y}}_{E},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle}\right\rVert^{2}$ $\displaystyle\leq\frac{y_{\ell}^{2}}{n^{2}}\left\lVert{{\mathbf{Z}}_{E}}\right\rVert^{2}\left\lVert{{\boldsymbol{\sigma}}^{\ell}_{E}({\mathbf{y}})}\right\rVert^{2}.$ Summing over $\ell\in E^{c}$: $\displaystyle\sum_{\ell\in E^{c}}\left\lVert{{\nabla}_{{\mathbf{\tilde{z}}}_{\ell}}\langle{\mathbf{y}}_{E},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle}\right\rVert^{2}$ $\displaystyle\leq\frac{\left\lVert{{\mathbf{Z}}_{E}}\right\rVert^{2}}{n^{2}}\sum_{\ell\in E^{c}}y_{\ell}^{2}\left\lVert{{\boldsymbol{\sigma}}^{\ell}_{E}({\mathbf{y}})^{2}}\right\rVert^{2}$ $\displaystyle\leq\frac{A\left\lVert{{\mathbf{Z}}}\right\rVert^{2}}{pn^{2}}\sum_{\ell\in E^{c}}\left\lVert{{\boldsymbol{\sigma}}^{\ell}_{E}({\mathbf{y}})}\right\rVert^{2}$ $\displaystyle=\frac{A\left\lVert{{\mathbf{Z}}}\right\rVert^{2}}{pn^{2}}\sum_{\ell\in E^{c}}\sum_{i\in E}y_{i}^{2}\partial\eta\left(\frac{\langle{\mathbf{\tilde{z}}}_{i},{\mathbf{\tilde{z}}}_{\ell}\rangle}{n};\frac{\tau}{\sqrt{n}}\right)^{2}$ $\displaystyle\leq\frac{A\left\lVert{{\mathbf{Z}}}\right\rVert^{2}}{pn^{2}}\sum_{i\in E}y_{i}^{2}\sum_{\ell\in E^{c}}\frac{\langle{\mathbf{\tilde{z}}}_{i},{\mathbf{\tilde{z}}}_{\ell}\rangle^{2}}{\tau^{2}n}$ $\displaystyle=\frac{A\left\lVert{{\mathbf{Z}}}\right\rVert^{2}}{\tau^{2}pn^{3}}\sum_{i\in E}y_{i}^{2}\langle{\mathbf{\tilde{z}}}_{i},{\mathbf{Z}}_{E^{c}}^{\sf T}{\mathbf{Z}}_{E^{c}}{\mathbf{\tilde{z}}}_{i}\rangle$ $\displaystyle\leq\frac{A\left\lVert{{\mathbf{Z}}}\right\rVert^{2}\left\lVert{{\mathbf{Z}}_{E^{c}}}\right\rVert^{2}\max_{i\in[E]}\left\lVert{{\mathbf{\tilde{z}}}_{i}}\right\rVert^{2}}{\tau^{2}np^{3}}$ $\displaystyle\leq\frac{A\left\lVert{{\mathbf{Z}}}\right\rVert^{4}\max_{i\in[p]}\left\lVert{{\mathbf{\tilde{z}}}_{i}}\right\rVert^{2}}{\tau^{2}pn^{3}}.$ This bound along with Eq. (26) gives: $\displaystyle\left\lVert{{\nabla}_{({\mathbf{\tilde{z}}}_{\ell})_{\ell\in[p]}}\langle{\mathbf{y}}_{E},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle}\right\rVert^{2}$ $\displaystyle\leq\frac{2A\left\lVert{{\mathbf{Z}}}\right\rVert^{4}\max_{i\in[p]}\left\lVert{{\mathbf{\tilde{z}}}_{i}}\right\rVert^{2}}{\tau^{2}np^{3}}.$ On the set $G$ defined before, we have that: $\displaystyle\left\lVert{{\nabla}_{({\mathbf{\tilde{z}}}_{\ell})_{\ell\in[p]}}\langle{\mathbf{y}}_{E},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle}\right\rVert^{2}$ $\displaystyle\leq\frac{64A\gamma^{3}\alpha^{3}}{p\tau^{2}}.$ Proceeding as before, applying Lemma 4.4 we have: $\displaystyle{\mathbb{P}}\left\\{\langle{\mathbf{y}}_{E},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle\geq 16\frac{\sqrt{A\Gamma\gamma^{2}\alpha^{3}}}{\tau}\right\\}$ $\displaystyle\leq 2p\exp\left(-\frac{\Gamma p}{2}\right).$ (27) We can now use Eqs.(25), (27) in Eq. (20): $\displaystyle{\mathbb{P}}\left\\{\left\lVert{{\mathbf{\widetilde{N}}}}\right\rVert_{2}\geq(1-2{\varepsilon})^{-1}\left(32\sqrt{\frac{\alpha\log A}{A}}+48\sqrt{\frac{{A\Gamma\gamma^{2}\alpha^{3}}}{\tau^{2}}}\right)\right\\}$ $\displaystyle\leq\exp\left(-\frac{p\log A}{4A}\right)$ $\displaystyle\quad+\|T^{\varepsilon}_{p}\|(4p+4)\exp\left(\frac{-\Gamma p}{2}\right)$ We first simplify the probability bound. Since $A=\tau\log\tau$, $\log A/A\geq 1/\tau$ when $\tau\geq\exp(1)$. Further, choosing ${\varepsilon}=1/4$, with Lemma 4.2 we get that $\|T^{{\varepsilon}}_{p}\|\leq(1+2/{\varepsilon})^{p}=9^{p}$. Since $\log 9=2.19\dots<\Gamma/2=5/2$, we have $(4p+4)\|T^{\varepsilon}_{p}\|\exp(-\Gamma p/2)\leq\exp(-p/20)$ for large enough $p$. Thus the right hand side is bounded above by $2\exp\left(-p/4\max(\tau,5)\right)$ for every $p$ large enough. Now we simplify the operator norm bound. As $A=\tau\log\tau$, $\log A/A\leq\log\tau/\tau$ since $\log z/z$ is decreasing. Further $\alpha\leq\max(1,\alpha^{3})$ and $\Gamma=5$ imply: $\displaystyle(1-2{\varepsilon})^{-1}\left(32\sqrt{\frac{\alpha\log A}{A}}+64\sqrt{\frac{{A\Gamma\gamma^{3}\alpha^{3}}}{\tau^{2}}}\right)$ $\displaystyle\leq 2(32+64\Gamma^{2})\sqrt{\frac{\max(1,\alpha^{3})\log\tau}{\tau}}$ $\displaystyle\leq 5000\sqrt{\frac{\max(1,\alpha^{3})\log\tau}{\tau}}.$ Our conditions on $\tau$ and $A$ were: $(i)$ $\tau\geq\max(4\sqrt{\Gamma\gamma\alpha},\exp(1))=20\max(1,\sqrt{\alpha})$ and $(ii)$ $A\geq 180\max(\sqrt{\alpha},1)$. Using $\tau\geq 100\max(1,\alpha^{2}\log\alpha)$ satisfies both conditions. #### 6.2.1 Proof of Lemma 6.8 This proof also follows an ${\varepsilon}$-net argument. Let $a$ denote the size of the set $E$. For notational simplicity, we will permute the rows and columns of ${\mathbf{\widetilde{N}}}$ to ensure $E=[a]$ (i.e. $E$ is the first $a$ entries of $[p]$). For a fixed ${\mathbf{y}}\in T^{{\varepsilon}}_{a}$, we bound the Rayleigh quotient $\langle{\mathbf{y}},{\widetilde{\cal P}}_{E,E}({\mathbf{\widetilde{N}}}){\mathbf{y}}\rangle$ with high probability. Note that $\langle{\mathbf{y}},{\widetilde{\cal P}}_{E,E}({\mathbf{\widetilde{N}}}){\mathbf{y}}\rangle$ is a function of ${\mathbf{\tilde{z}}}_{\ell},\ell\in E$. The gradient of this function with respect to ${\mathbf{\tilde{z}}}_{\ell}$ is: $\displaystyle{\nabla}_{{\mathbf{\tilde{z}}}_{\ell}}\langle{\mathbf{y}},{\widetilde{\cal P}}_{E,E}({\mathbf{\widetilde{N}}}){\mathbf{y}}\rangle$ $\displaystyle=\frac{2y_{\ell}}{n}\sum_{i\in E\backslash\ell}\partial\eta\left(\frac{\langle{\mathbf{\tilde{z}}}_{i},{\mathbf{\tilde{z}}}_{\ell}\rangle}{n};\frac{\tau}{\sqrt{n}}\right)y_{i}{\mathbf{\tilde{z}}}_{i},$ where ${\boldsymbol{\sigma}}^{\ell}({\mathbf{y}})\in\mathbb{R}^{p}$ is the vector defined as: $\displaystyle\sigma_{i}^{\ell}({\mathbf{y}})$ $\displaystyle=\begin{cases}y_{i}\partial\eta\left(\frac{\langle{\mathbf{\tilde{z}}}_{i},{\mathbf{\tilde{z}}}_{\ell}\rangle}{n};\frac{\tau}{\sqrt{n}}\right)&\text{ when }i\in E\backslash\ell\\\ 0&\text{ otherwise.}\end{cases}$ The (square of the) total gradient is thus given by: $\displaystyle\left\lVert{{\nabla}\langle{\mathbf{y}},{\widetilde{\cal P}}_{E,E}({\mathbf{\widetilde{N}}}){\mathbf{y}}\rangle}\right\rVert^{2}$ $\displaystyle=\frac{4}{n^{2}}\sum_{\ell\in E}\left\lVert{{\mathbf{Z}}_{E}{\boldsymbol{\sigma}}^{\ell}({\mathbf{y}})}\right\rVert_{2}^{2}y_{\ell}^{2}$ $\displaystyle\leq\frac{4}{n^{2}}\sum_{\ell\in E}\left\lVert{{\mathbf{Z}}_{E}}\right\rVert_{2}^{2}\left\lVert{{\boldsymbol{\sigma}}^{\ell}({\mathbf{y}})}\right\rVert^{2}y_{\ell}^{2}$ $\displaystyle\leq\left(\frac{2\left\lVert{{\mathbf{Z}}_{E}}\right\rVert_{2}}{n}\right)^{2}\sum_{\ell\in E\backslash\ell}\left\lVert{{\boldsymbol{\sigma}}^{\ell}({\mathbf{y}})}\right\rVert^{2}y_{\ell}^{2},$ Since $\|\partial\eta(\cdot;\tau/\sqrt{n})\|\leq 1$ we have that $\left\lVert{{\boldsymbol{\sigma}}^{\ell}({\mathbf{y}})}\right\rVert^{2}\leq\left\lVert{{\mathbf{y}}}\right\rVert^{2}\leq 1$. Consequently we obtain the bound: $\displaystyle\left\lVert{{\nabla}\langle{\mathbf{y}},{\widetilde{\cal P}}_{E,E}({\mathbf{\widetilde{N}}}){\mathbf{y}}\rangle}\right\rVert^{2}$ $\displaystyle\leq\left(\frac{2\left\lVert{{\mathbf{Z}}_{E}}\right\rVert_{2}}{n}\right)^{2}.$ From Lemma 4.6 we have that: $\displaystyle\left\lVert{{\mathbf{Z}}_{E}}\right\rVert_{2}$ $\displaystyle\leq\sqrt{n}+\sqrt{a}+t\sqrt{p},$ with probability at least $1-\exp(-pt^{2}/2)$. Let $G=\\{{\mathbf{Z}}_{E}:\left\lVert{{\mathbf{Z}}_{E}}\right\rVert_{2}\leq\sqrt{n}+\sqrt{a}+t\sqrt{p}\\}$. Then: $\displaystyle\left\lVert{{\nabla}\langle{\mathbf{y}},{\widetilde{\cal P}}_{E,E}({\mathbf{\widetilde{N}}}){\mathbf{y}}\rangle}\right\rVert^{2}$ $\displaystyle\leq\frac{4\alpha}{p}\Bigg{(}1+\sqrt{\frac{a\alpha}{p}}+t\sqrt{\alpha}\Bigg{)}^{2}\equiv L^{2}$ (28) $\displaystyle\text{and }{\mathbb{P}}(G^{c})$ $\displaystyle\leq e^{-pt^{2}/2}.$ (29) We let $F_{L}({\mathbf{Z}}_{E})$ denote the $G,L$-Lipschitz extension of ${\nabla}\langle{\mathbf{y}},{\widetilde{\cal P}}_{E,E}({\mathbf{\widetilde{N}}}){\mathbf{y}}\rangle$. The following remark is proved in Appendix A: ###### Remark 6.10. Firstly, ${\mathbb{E}}\\{\langle{\mathbf{y}},{\widetilde{\cal P}}_{E,E}({\mathbf{\widetilde{N}}}){\mathbf{y}}\rangle\\}=0$. Secondly, for every $p$ large enough: $\|{\mathbb{E}}(F_{L}({\mathbf{Z}}))\|\leq p^{-1}$. Let $\widetilde{\Delta}=\Delta(1-2{\varepsilon})$ and $\nu=1+\sqrt{\alpha a/p}$. We choose $t=\left(\sqrt{\nu^{2}+\widetilde{\Delta}/2\sqrt{\alpha}}-\nu\right)/2$ and apply Lemma 4.4 and Remark 6.10. This choice of $t$ ensures that the two unfavorable events of Lemma 4.4 are both bounded above by $\exp(-pt^{2}/2)$. Thus, $\displaystyle{\mathbb{P}}\\{\langle{\mathbf{y}},{\widetilde{\cal P}}_{E,E}({\mathbf{\widetilde{N}}}){\mathbf{y}}\rangle\geq\widetilde{\Delta}\\}$ $\displaystyle\leq 2e^{-pt^{2}/2},$ for $p$ large enough. Further, our choice of $t$ implies: $\displaystyle t^{2}$ $\displaystyle=\frac{1}{4}\left(\sqrt{\nu^{2}+\frac{\widetilde{\Delta}}{2\sqrt{\alpha}}}-\nu\right)^{2}$ $\displaystyle=\frac{\nu^{2}}{2}\left(1+\frac{\widetilde{\Delta}}{4\nu^{2}\sqrt{\alpha}}-\sqrt{1+\frac{\widetilde{\Delta}}{2\nu^{2}\sqrt{\alpha}}}\right)$ $\displaystyle\geq\frac{\widetilde{\Delta}^{2}}{128\nu^{2}\alpha},$ where the last inequality follows from the fact that $g(x)=1+x/2-\sqrt{1+x}\geq x^{2}/16$ when $x\leq 2$. This requires $\widetilde{\Delta}\leq 4\nu^{2}\sqrt{\alpha}$). Now, Lemma 4.2 and 4.3 imply: $\displaystyle{\mathbb{P}}\left\\{\left\lVert{{\widetilde{\cal P}}_{E,E}({\mathbf{\widetilde{N}}})}\right\rVert_{2}\geq\Delta\right\\}$ $\displaystyle\leq 2\left(1+\frac{2}{{\varepsilon}}\right)^{a}\exp\left(-\frac{p\widetilde{\Delta}^{2}}{256\nu^{2}\alpha}\right)$ $\displaystyle\leq\exp\left(-\frac{p\widetilde{\Delta}^{2}}{256\alpha\nu^{2}}+a\log\left(2+\frac{4}{{\varepsilon}}\right)\right).$ There are $\binom{p}{a}\leq(pe/a)^{a}$ possible choices for the set $E$. Using the union bound we have that: $\displaystyle{\mathbb{P}}\left\\{\sup_{E\subseteq[p],\|E\|=a}\left\lVert{{\widetilde{\cal P}}_{E,E}({\mathbf{\widetilde{N}}})}\right\rVert_{2}\geq\Delta\right\\}$ $\displaystyle\leq\exp\left\\{-\frac{p\widetilde{\Delta}^{2}}{256\alpha\nu^{2}}+a\log\left(2+\frac{4}{{\varepsilon}}\right)+a\log\left(\frac{pe}{a}\right)\right\\}.$ Since $a\leq p/A$, $\nu=1+\sqrt{a\alpha/p}\leq 2$ when $A\geq\max(\sqrt{\alpha},1)$. Using ${\varepsilon}=1/4$ we obtain that $\displaystyle{\mathbb{P}}\left\\{\sup_{E\subseteq[p],\|E\|=a}\left\lVert{{\widetilde{\cal P}}_{E,E}({\mathbf{\widetilde{N}}})}\right\rVert_{2}\geq\Delta\right\\}$ $\displaystyle\leq\exp\left(-p\left(\frac{\Delta^{2}}{1024\alpha}-\frac{\log\left(18eA\right)}{A}\right)\right).$ We required $\widetilde{\Delta}\leq 4\nu^{2}\sqrt{\alpha}$, and $\widetilde{\Delta}=\Delta/2$. Hence we require $\Delta\leq 8\sqrt{\alpha}\leq 8\nu^{2}\sqrt{\alpha}$. Choosing $\Delta=32\sqrt{\alpha\log A/A}$, where $A\geq 180\max(\sqrt{\alpha},1)$ satisfies this condition. Further, with this choice of $A$, $\log(18eA)\leq 1.75\log A$. Consequently: $\displaystyle P\left\\{\sup_{E\subseteq[p],\|E\|=a}\left\lVert{{\widetilde{\cal P}}_{E,E}(N)}\right\rVert_{2}\geq 32\sqrt{\frac{\alpha\log A}{A}}\right\\}$ $\displaystyle\leq\exp\left(-\frac{p\log A}{4A}\right).$ ### 6.3 Proof of Proposition 6.3 We explicitly write the $(i,j)^{\mathrm{th}}$ entry of ${\mathbf{R}}_{1}$ (when $(i,j)\in{\sf G}$) as: $\displaystyle(R_{1})_{ij}$ $\displaystyle=\eta\left(\frac{\langle\sum_{a}\sqrt{\beta_{q}}{\mathbf{u}}_{q}(v_{q})_{i}+{\mathbf{\tilde{z}}}_{i},\sum_{q}{\mathbf{u}}_{q}(v_{q})_{j}+{\mathbf{\tilde{z}}}_{j}\rangle}{n};\frac{\tau}{\sqrt{n}}\right)$ Since ${\sf G}$ is a symmetric set of entries excluding the diagonal, it suffices to consider the case $i<j$ above. Denote by ${\mathbf{R}}$ the upper triangle of ${\mathbf{R}}_{1}$. Let $g$ denote the number of nonzero rows in ${\mathbf{R}}$. By the definition of $g$, $g\leq\sum_{q}\left\lvert{{\sf Q}_{q}}\right\rvert=k$. We wish to bound (with slight abuse of notation) the quantity: $\sup_{{\mathbf{x}}\in{S}^{g-1}}\sup_{{\mathbf{y}}\in{S}^{p-1}}\langle{\mathbf{x}},{\mathbf{R}}{\mathbf{y}}\rangle$. The proof follows an epsilon net argument entirely analogous to the proof of Proposition 6.2. The only difference is the further dependence on the Gaussian random vectors ${\mathbf{u}}_{q}$. Hence we only give a proof sketch, highlighting the difference with the proof of Proposition 6.2. Fix a vector ${\mathbf{y}}\in T^{1/4}_{p}$ and ${\mathbf{x}}\in T^{1/4}_{g}$, and let $E$ be the subset of indices $E=\\{i\in[p]:\left\lvert{y_{i}}\right\rvert\geq\sqrt{A/p}\\}$ for some constant $A$ to be fixed later in the proof. As before, we split the Rayleigh quotient $\langle{\mathbf{x}},{\mathbf{R}}_{3}{\mathbf{y}}\rangle=\langle{\mathbf{x}},{\mathbf{R}}{\mathbf{y}}_{E}\rangle+\langle{\mathbf{x}},{\mathbf{R}}{\mathbf{y}}_{E^{c}}\rangle\leq\left\lVert{{\cal P}_{[p]\times E}({\mathbf{R}})}\right\rVert+\langle{\mathbf{x}},{\mathbf{R}}{\mathbf{y}}_{E^{c}}\rangle$. By the condition on $E$, we have that $\left\lvert{E}\right\rvert\leq p/A$. Consequently: $\displaystyle{\mathbb{P}}\left\\{\left\lVert{{\mathbf{R}}}\right\rVert\geq\Delta\right\\}$ $\displaystyle\leq\sup_{{\mathbf{x}}\in T^{1/4}_{g},{\mathbf{y}}\in T^{1/4}_{p}}{\mathbb{P}}\left\\{\langle{\mathbf{x}},{\mathbf{R}}{\mathbf{y}}\rangle\geq\Delta/2\right\\}$ $\displaystyle\leq{\mathbb{P}}\left\\{\max_{\left\lvert{E}\right\rvert\leq p/A}\left\lVert{{\cal P}_{[p]\times E}\left({\mathbf{R}}\right)}\right\rVert\geq\frac{\Delta}{4}\right\\}+\sup_{{\mathbf{x}}\in T^{1/4}_{g},{\mathbf{y}}\in T^{1/4}_{p}}{\mathbb{P}}\left\\{\langle{\mathbf{x}},{\mathbf{R}}{\mathbf{y}}_{E^{c}}\rangle\geq\frac{\Delta}{4}\right\\}.$ We first concentrate on the second term, whose gradient with respect to a fixed ${\mathbf{\tilde{z}}}_{i}$ is given by: $\displaystyle{\nabla}_{{\mathbf{\tilde{z}}}_{i}}\langle{\mathbf{x}},{\mathbf{R}}{\mathbf{y}}_{E^{c}}\rangle$ $\displaystyle=\frac{x_{i}}{n}\sum_{j>i,(i,j)\in{\sf G}}(y_{E^{c}})_{i}\partial\eta\left(\langle\sum_{q}\sqrt{\beta_{q}}{\mathbf{u}}_{q}(v_{q})_{i}+{\mathbf{\tilde{z}}}_{i},\sum_{q}\sqrt{\beta_{q}}{\mathbf{u}}_{q}(v_{q})_{j}+{\mathbf{\tilde{z}}}_{j}\rangle;\tau\sqrt{n}\right)\left(\sum_{q}\sqrt{\beta_{q}}{\mathbf{u}}_{q}(v_{q})_{j}+{\mathbf{\tilde{z}}}_{j}\right)$ $\displaystyle\quad+\frac{(y_{E^{c}})_{i}}{n}\sum_{j<i,(i,j)\in{\sf G}}x_{j}\partial\eta\left(\langle\sum_{q}\sqrt{\beta_{q}}{\mathbf{u}}_{q}(v_{q})_{i}+{\mathbf{\tilde{z}}}_{i},\sum_{q}\sqrt{\beta_{q}}{\mathbf{u}}_{q}(v_{q})_{j}+{\mathbf{\tilde{z}}}_{j}\rangle;\tau\sqrt{n}\right)\left(\sum_{q}\sqrt{\beta_{q}}{\mathbf{u}}_{q}(v_{q})_{j}+{\mathbf{\tilde{z}}}_{j}\right).$ Defining ${\boldsymbol{\sigma}}^{i}({\mathbf{y}})$ and ${\boldsymbol{\sigma}}^{i}({\mathbf{x}})$ similar to Proposition 6.2, we have by Cauchy Schwarz: $\displaystyle\left\lVert{{\nabla}_{{\mathbf{\tilde{z}}}_{i}}\langle{\mathbf{x}},{\mathbf{R}}{\mathbf{y}}_{E^{c}}\rangle}\right\rVert^{2}$ $\displaystyle\leq\frac{2\left\lVert{\sum_{q}\sqrt{\beta_{q}}{\mathbf{u}}_{q}{\mathbf{v}}_{q}^{\sf T}+{\mathbf{Z}}}\right\rVert^{2}}{n^{2}}\left(x_{i}^{2}\left\lVert{{\boldsymbol{\sigma}}^{i}({\mathbf{y}})}\right\rVert^{2}+(y_{E^{c}})^{2}_{i}\left\lVert{{\boldsymbol{\sigma}}^{i}({\mathbf{x}})}\right\rVert^{2}\right).$ Summing over $i$: $\displaystyle\sum_{i}\left\lVert{{\nabla}_{{\mathbf{\tilde{z}}}_{i}}\langle{\mathbf{x}},{\mathbf{R}}{\mathbf{y}}_{E^{c}}\rangle}\right\rVert^{2}$ $\displaystyle\leq\frac{2\left\lVert{{\mathbf{X}}}\right\rVert^{2}}{n^{2}}\sum_{i}\left(x_{i}^{2}\left\lVert{{\boldsymbol{\sigma}}^{i}({\mathbf{y}})^{2}}\right\rVert+(y_{E^{c}})^{2}_{i}\left\lVert{{\boldsymbol{\sigma}}^{i}({\mathbf{x}})}\right\rVert^{2}\right)$ $\displaystyle\leq\frac{2\left\lVert{{\mathbf{X}}}\right\rVert^{2}}{n^{2}}\sup_{i}\left\lVert{{\boldsymbol{\sigma}}^{i}({\mathbf{y}})}\right\rVert^{2}+\frac{2\left\lVert{{\mathbf{X}}}\right\rVert^{2}}{n}\sum_{i}(y_{E^{c}})_{i}^{2}\left\lVert{{\boldsymbol{\sigma}}^{i}({\mathbf{x}})}\right\rVert^{2}$ Let $G=\\{({\mathbf{u}})_{q\leq r},{\mathbf{Z}}:\forall q\left\lVert{{\mathbf{u}}_{q}}\right\rVert\leq C^{\prime}\sqrt{n},\left\lVert{{\mathbf{Z}}}\right\rVert\leq C^{\prime}(\sqrt{p}+\sqrt{n}),\forall i\left\lVert{{\mathbf{\tilde{z}}}_{i}}\right\rVert\leq C^{\prime}\sqrt{n}\\}$. It is clear that $G$ is convex, and that ${\mathbb{P}}\\{G^{c}\\}\leq p\exp(-C^{\prime\prime}p)$ for some $C^{\prime\prime}$ dependent on $C^{\prime}$. It is not hard to show that: $\displaystyle\sum_{i}\left\lVert{{\nabla}_{{\mathbf{\tilde{z}}}_{i}}\langle{\mathbf{x}},{\mathbf{R}}{\mathbf{y}}_{E^{c}}\rangle}\right\rVert^{2}$ $\displaystyle\leq\frac{AC(\alpha,(\beta)_{q\leq r},r)}{p\tau^{2}},$ (30) for some constant $C$, when $C^{\prime}$ is large enough. Similarly, taking derivatives with respect to ${\mathbf{u}}_{q}$ for a fixed $q$, we have: $\displaystyle{\nabla}_{{\mathbf{u}}_{q}}\langle{\mathbf{x}},{\mathbf{R}}{\mathbf{y}}_{E^{c}}\rangle$ $\displaystyle=\frac{1}{n}\sum_{(i,j)\in{\sf G}}\partial\eta\left(\langle\sum_{q}\sqrt{\beta_{q}}{\mathbf{u}}_{q}(v_{q})_{i}+{\mathbf{\tilde{z}}}_{i},\sum_{q}\sqrt{\beta_{q}}{\mathbf{u}}_{q}(v_{q})_{j}+{\mathbf{\tilde{z}}}_{j}\rangle;\tau\sqrt{n}\right)\quad\cdot\quad$ $\displaystyle\quad\left(x_{i}(y_{E^{c}})_{j}\sqrt{\beta_{q}}(v_{q})_{i}(\sum_{q^{\prime}}\sqrt{\beta_{q^{\prime}}}{\mathbf{u}}_{q^{\prime}}(v_{q^{\prime}})_{j}+{\mathbf{\tilde{z}}}_{j})+x_{j}(y_{E^{c}})_{i}\sqrt{\beta_{q}}(v_{q})_{j}(\sum_{q^{\prime}}\sqrt{\beta_{q^{\prime}}}{\mathbf{u}}_{q^{\prime}}(v_{q^{\prime}})_{i}+{\mathbf{\tilde{z}}}_{i})\right)$ $\displaystyle=\frac{{\mathbf{X}}({\boldsymbol{\sigma}}^{1}_{{\sf G}}({\mathbf{x}},{\mathbf{y}})+{\boldsymbol{\sigma}}^{2}_{{\sf G}}({\mathbf{x}},{\mathbf{y}}))}{n},$ where we define the vectors ${\boldsymbol{\sigma}}_{\sf G}^{1}({\mathbf{x}},{\mathbf{y}}),{\boldsymbol{\sigma}}_{\sf G}^{2}({\mathbf{x}},{\mathbf{y}})$ appropriately. By Cauchy Schwarz: $\displaystyle\left\lVert{{\nabla}_{{\mathbf{u}}_{q}}\langle{\mathbf{x}},{\mathbf{R}}{\mathbf{y}}_{E^{c}}\rangle}\right\rVert^{2}$ $\displaystyle\leq\frac{2\left\lVert{{\mathbf{X}}}\right\rVert^{2}}{n^{2}}(\left\lVert{{\boldsymbol{\sigma}}_{\sf G}^{1}({\mathbf{x}},{\mathbf{y}})}\right\rVert^{2}+\left\lVert{{\boldsymbol{\sigma}}_{\sf G}^{2}({\mathbf{x}},{\mathbf{y}})}\right\rVert^{2}).$ We now bound the first term above, and the second term follows from a similar argument. $\displaystyle\left\lVert{{\boldsymbol{\sigma}}^{1}({\mathbf{x}},{\mathbf{y}})}\right\rVert^{2}$ $\displaystyle=\sum_{j}(y_{E^{c}})_{j}^{2}\left(\sum_{i:(i,j)\in{\sf G}}\sqrt{\beta_{q}}x_{i}(v_{q})_{i}\partial\eta\left(\langle\sum_{q}\sqrt{\beta_{q}}{\mathbf{u}}_{q}(v_{q})_{i}+{\mathbf{\tilde{z}}}_{i},\sum_{q}\sqrt{\beta_{q}}{\mathbf{u}}_{q}(v_{q})_{j}+{\mathbf{\tilde{z}}}_{j}\rangle;\tau\sqrt{n}\right)\right)^{2}$ For simplicity of notation, define $D_{ij}=\partial\eta\left(\langle\sum_{q}\sqrt{\beta_{q}}{\mathbf{u}}_{q}(v_{q})_{i}+{\mathbf{\tilde{z}}}_{i},\sum_{q}\sqrt{\beta_{q}}{\mathbf{u}}_{q}(v_{q})_{j}+{\mathbf{\tilde{z}}}_{j}\rangle;\tau\sqrt{n}\right)$. The above sum then can be reduced to: $\displaystyle\left\lVert{{\boldsymbol{\sigma}}^{1}({\mathbf{x}},{\mathbf{y}})}\right\rVert^{2}$ $\displaystyle=\sum_{i_{1},i_{2}}\beta_{q}x_{i_{1}}x_{i_{2}}(v_{q})_{i_{1}}(v_{q})_{i_{2}}\sum_{j:(i_{1},j)\in{\sf G}\text{ or }(i_{2},j)\in{\sf G}}(y_{E^{c}})_{j}^{2}D_{i_{1}j}D_{i_{2}j}.$ We first bound the inner summation uniformly in $i_{1},i_{2}$ as follows: $\displaystyle\sum_{j:(i_{1},j)\in{\sf G}\text{ or }(i_{2},j)\in{\sf G}}(y_{E^{c}})_{j}^{2}D_{i_{1},j}D_{i_{2},j}$ $\displaystyle\leq\frac{A}{p}\sum_{j}\frac{\left\lvert{\langle{\mathbf{\tilde{x}}}_{i_{1}},{\mathbf{\tilde{x}}}_{j}\rangle\langle{\mathbf{\tilde{x}}}_{i_{2}},{\mathbf{\tilde{x}}}_{j}\rangle}\right\rvert}{n\tau^{2}}$ $\displaystyle\leq\frac{A}{p}\sum_{j}\frac{\langle{\mathbf{\tilde{x}}}_{i_{1}},{\mathbf{\tilde{x}}}_{j}\rangle^{2}+\langle{\mathbf{\tilde{x}}}_{i_{2}},{\mathbf{\tilde{x}}}_{j}\rangle^{2}}{2n\tau^{2}}$ $\displaystyle\leq\frac{A}{n\tau^{2}p}\left\lVert{{\mathbf{X}}}\right\rVert^{2}(\left\lVert{{\mathbf{\tilde{x}}}_{i_{1}}}\right\rVert^{2}+\left\lVert{{\mathbf{\tilde{x}}}_{i_{2}}}\right\rVert^{2})$ Employing a similar strategy for the other term, it is not hard to show that: $\displaystyle\left\lVert{{\boldsymbol{\sigma}}^{1}({\mathbf{x}},{\mathbf{y}})}\right\rVert^{2}$ $\displaystyle\leq\frac{A\beta_{q}\left\lVert{{\mathbf{X}}}\right\rVert^{2}\sup_{i}\left\lVert{{\mathbf{\tilde{x}}}_{i}}\right\rVert^{2}}{pn\tau^{2}}.$ Thus, on the set $G$, we obtain that: $\displaystyle\sum_{q}\left\lVert{{\nabla}_{{\mathbf{u}}_{q}}\langle{\mathbf{x}},{\mathbf{R}}{\mathbf{y}}_{E^{c}}\rangle}\right\rVert^{2}$ $\displaystyle\leq\frac{AC(\alpha,\beta_{1},r)}{p\tau^{2}},$ (31) for every $\tau$ sufficiently large. Indeed the same bound, with a modified value for $C$ holds for the gradient with respect to all the variables $(({\mathbf{u}}_{q})_{q\leq r},({\mathbf{\tilde{z}}}_{i})_{i\leq p})$ using Eqs.(30), (31). Lemma 4.4 then implies that $\displaystyle\sup_{{\mathbf{x}}\in T^{1/4}_{g},{\mathbf{y}}\in T^{1/4}_{p}}{\mathbb{P}}\left\\{\langle{\mathbf{x}},{\mathbf{R}}{\mathbf{y}}\rangle\geq\sqrt{AC(\alpha,\beta_{1},r)}{\tau^{2}}\right\\}\leq\exp(-cp),$ for an appropriate $c$. We omit the proof of the following remark that uses similar techniques as above, followed by a union bound. ###### Remark 6.11. For every $A\geq A_{0}(\alpha,\beta_{1},r)$ we have that: $\displaystyle{\mathbb{P}}\left\\{\sup_{\left\lvert{E}\right\rvert\leq p/A}{\cal P}_{[p]\times E}({\mathbf{R}})\geq C(\alpha,\beta_{1},r)\sqrt{\frac{\log A}{A}}\right\\}$ $\displaystyle\leq\exp(-c_{2}(\tau)p).$ Here $c_{2}(\tau)=1/4\tau$ suffices. Using $A=\tau\log\tau$ for $\tau$ large enough completes the proof. ### 6.4 Proof of Proposition 6.4 Since ${\mathbf{R}}_{2}$ is a diagonal matrix, its spectral norm is bounded by the maximum of its entries. This is easily done as, for every $i\in{\sf Q}^{c}$: $\displaystyle\left\lvert{({\mathbf{R}}_{2})_{ii}}\right\rvert$ $\displaystyle=\left\lvert{\eta\left(\frac{\left\lVert{{\mathbf{\tilde{z}}}_{i}}\right\rVert^{2}}{n}-1;\frac{\tau}{\sqrt{n}}\right)}\right\rvert$ $\displaystyle\leq\left\lvert{\frac{\left\lVert{{\mathbf{\tilde{z}}}_{i}}\right\rVert^{2}-n}{n}}\right\rvert.$ By the Chernoff bound for $\chi$-squared random variables followed by the union bound we have that: $\displaystyle\max_{i}\left\lvert{\frac{\left\lVert{{\mathbf{\tilde{z}}}_{i}}\right\rVert^{2}}{n}-1}\right\rvert\geq t,$ with probability at most $p(\exp(n(-t+\log(1+t))/2)+\exp(n(t-\log(1-t))/2))$. Setting $t=\sqrt{C\log n/n}$ and using $\log(1+t)\leq t-t^{2}/3,\log(1-t)\geq- t-t^{2}/3$ for every $t$ small enough we obtain the probability bound of $pn^{-C/6}=\alpha n^{-C/6+1}$. ## Acknowledgements We are grateful to David Donoho for his feedback on this manuscript. This work was partially supported by the NSF CAREER award CCF-0743978, the NSF grant CCF-1319979, and the grants AFOSR/DARPA FA9550-12-1-0411 and FA9550-13-1-0036. ## Appendix A Some technical proofs In this Appendix we prove Remarks 6.7, 6.9 and 6.10. We begin with a preliminary lemma bounding the tail of Gaussian random variables. ###### Lemma A.1. Let ${\mathbf{X}}=\sum_{q}\sqrt{\beta_{q}}{\mathbf{u}}_{q}{\mathbf{v}}_{q}^{\sf T}+{\mathbf{Z}}$ as according to Eq. (1). Further assume we are given an event $B$ such that ${\mathbb{P}}\left\\{B\right\\}\leq\exp(-bn)$ for some constant $b$. Then for every $n$ large enough: $\displaystyle{\mathbb{E}}\left\\{\left\lVert{{\mathbf{X}}}\right\rVert_{F}^{2}\mathbb{I}(B)\right\\}$ $\displaystyle\leq\min(n^{-4},p^{-4}).$ $\displaystyle{\mathbb{E}}\left\\{\left\lVert{{\mathbf{X}}}\right\rVert_{F}\mathbb{I}(B)\right\\}$ $\displaystyle\leq\min(n^{-2},p^{-2}).$ ###### Proof. Note that $\left\lVert{{\mathbf{X}}}\right\rVert_{F}^{2}\leq(r+1)(\sum_{q}\beta_{q}\left\lVert{{\mathbf{u}}_{q}{\mathbf{v}}_{q}}\right\rVert_{F}^{2}+\left\lVert{{\mathbf{Z}}}\right\rVert_{F}^{2})=(r+1)\beta\left\lVert{{\mathbf{u}}_{q}}\right\rVert^{2}+2\left\lVert{{\mathbf{Z}}}\right\rVert_{F}^{2}$. Consider the event $\widetilde{G}=\left\\{({\mathbf{u}},{\mathbf{Z}}):\left\lVert{{\mathbf{u}}_{q}}\right\rVert\leq 2\sqrt{n},\left\lVert{{\mathbf{Z}}}\right\rVert_{F}\leq 2\sqrt{np}\right\\}$. We write: $\displaystyle{\mathbb{E}}\left\\{\left\lVert{{\mathbf{X}}}\right\rVert_{F}^{2}\mathbb{I}(B)\right\\}$ $\displaystyle={\mathbb{E}}\left\\{\left\lVert{{\mathbf{X}}}\right\rVert_{F}^{2}\mathbb{I}(B)\mathbb{I}(\widetilde{G})\right\\}+{\mathbb{E}}\left\\{\left\lVert{{\mathbf{X}}}\right\rVert_{F}^{2}\mathbb{I}(B)\mathbb{I}(\widetilde{G}^{c})\right\\}$ $\displaystyle\leq{\mathbb{E}}\left\\{(r+1)(\beta\left\lVert{{\mathbf{u}}}\right\rVert^{2}+\left\lVert{{\mathbf{Z}}}\right\rVert_{F}^{2})\mathbb{I}(B)\mathbb{I}(\widetilde{G})\right\\}+{\mathbb{E}}\left\\{(r+1)(\beta\left\lVert{{\mathbf{u}}}\right\rVert^{2}+\left\lVert{{\mathbf{Z}}}\right\rVert_{F}^{2})\mathbb{I}(\widetilde{G}^{c})\right\\}$ $\displaystyle\leq 4(r+1)(r\beta_{1}n+np){\mathbb{P}}\left\\{B\right\\}+(r+1)\sum_{q}\beta_{q}{\mathbb{E}}\left\\{\left\lVert{{\mathbf{u}}_{q}}\right\rVert^{2}\mathbb{I}(\widetilde{G}^{c})\right\\}+2{\mathbb{E}}\left\\{\left\lVert{{\mathbf{Z}}}\right\rVert_{F}^{2}\mathbb{I}(\widetilde{G}^{c})\right\\}$ $\displaystyle\leq 8(\beta n+np)\exp(-bn)+(r+1)\sum_{q}\beta_{q}\int_{4n}^{\infty}{\mathbb{P}}\left\\{\left\lVert{{\mathbf{u}}_{q}}\right\rVert^{2}\geq t\right\\}\mathrm{d}t+2\int_{4np}^{\infty}{\mathbb{P}}\left\\{\left\lVert{{\mathbf{Z}}}\right\rVert_{F}^{2}\geq t\right\\}\mathrm{d}t.$ Here the last line follows from the standard formula ${\mathbb{E}}\\{X\\}=\int_{0}^{\infty}{\mathbb{P}}\\{X\geq t\\}\mathrm{d}t$ for positive random variables $X$. By the Chernoff bound on $\chi$-squared random variable, ${\mathbb{P}}\left\\{\left\lVert{{\mathbf{u}}_{q}}\right\rVert^{2}\geq t\right\\}\leq\exp(-t/8)$ and ${\mathbb{P}}\left\\{\left\lVert{{\mathbf{Z}}}\right\rVert_{F}^{2}\geq t\right\\}\leq\exp(-t/8)$. Using this we have: $\displaystyle{\mathbb{E}}\left\\{\left\lVert{{\mathbf{X}}}\right\rVert_{F}^{2}\mathbb{I}(B)\right\\}$ $\displaystyle\leq(r+1)(r\beta_{1}n+np)\exp(-bn)+\sum_{q}\beta_{q}\exp\left(-\frac{n}{2}\right)+16\exp\left(-\frac{n}{2}\right).$ This implies the first claim of the lemma. The second claim follows from Jensen’s inequality. ∎ We now use this lemma to prove Remarks 6.7, 6.9 and 6.10. ### Proof of Remark 6.7 Recall the definition of $F_{L}({\mathbf{Z}}_{\sf Q})$ as the $G,L$-Lipschitz extension of $\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle$ where $G$ is the set: $\displaystyle G$ $\displaystyle=\left\\{{\mathbf{Z}}:\left\lVert{{\nabla}\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle}\right\rVert\leq L^{2}\right\\},$ $\displaystyle\text{ where }L$ $\displaystyle=\frac{8(2+\sqrt{\alpha}+\sqrt{\beta})}{\sqrt{n}}.$ Further, we have already shown ${\mathbb{P}}\left\\{G^{c}\right\\}\leq 2\exp(-n/4)$. It suffices, hence to show that $\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle$ grows at most polynomially in $n$. We have: $\displaystyle\|{\mathbb{E}}\\{\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle\\}-{\mathbb{E}}\\{F_{L}({\mathbf{Z}}_{\sf Q})\\}\|$ $\displaystyle\leq{\mathbb{E}}\left\\{\|\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle- F_{L}({\mathbf{Z}}_{{\sf Q}})\|\right\\}$ $\displaystyle={\mathbb{E}}\left\\{\left\lvert{\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle- F_{L}({\mathbf{Z}}_{{\sf Q}})}\right\rvert\mathbb{I}({\mathbf{Z}}_{{\sf Q}}\in G^{c})\right\\}.$ Since $F_{L}({\mathbf{Z}}_{\sf Q})=\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{v}}\rangle$ whenever ${\mathbf{Z}}_{{\sf Q}}\in G$. We continue, employing the triangle inequality: $\displaystyle\|{\mathbb{E}}\\{\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle\\}-{\mathbb{E}}\\{F_{L}({\mathbf{Z}}_{\sf Q})\\}\|$ $\displaystyle\leq{\mathbb{E}}\left\\{(\|\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle\|+\|F_{L}({\mathbf{Z}}_{\sf Q})\|)\mathbb{I}({\mathbf{Z}}_{{\sf Q}}\in G^{c})\right\\}.$ First consider the term ${\mathbb{E}}\left\\{\left\lvert{F_{L}({\mathbf{Z}}_{\sf Q})\mathbb{I}({\mathbf{Z}}_{{\sf Q}}\in G^{c})}\right\rvert\right\\}$. Since $F_{L}({\mathbf{Z}}_{\sf Q})$ is $L$-Lipschitz, $\displaystyle\left\lvert{F_{L}({\mathbf{Z}}_{\sf Q})}\right\rvert$ $\displaystyle\leq\left\lvert{F_{L}(0)}\right\rvert+L\left\lVert{{\mathbf{Z}}_{\sf Q}}\right\rVert_{F}$ $\displaystyle\leq\left\lvert{F_{L}(0)}\right\rvert+L\left\lVert{{\mathbf{Z}}}\right\rVert_{F}$ $\displaystyle\leq\left\lvert{F_{L}(0)}\right\rvert+L\left\lVert{{\mathbf{X}}}\right\rVert_{F},$ since ${\mathbf{X}}=\sum_{q}\sqrt{\beta_{q}}{\mathbf{u}}_{q}{\mathbf{v}}_{q}^{\sf T}+{\mathbf{Z}}$. We bound $\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle$ in an analogous manner. As ${\mathbf{S}}={\cal P}_{{\sf E}}\\{\eta({\mathbf{\widehat{\Sigma}}})\\}$ and $\left\lvert{\eta(z;\tau/\sqrt{n})}\right\rvert\leq\left\lvert{z}\right\rvert$: $\displaystyle\left\lvert{\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle}\right\rvert$ $\displaystyle\leq\left\lVert{{\cal P}_{{\sf E}}\left\\{\eta\left({\mathbf{\widehat{\Sigma}}}\right)\right\\}}\right\rVert_{F}$ $\displaystyle\leq\left\lVert{{\mathbf{\widehat{\Sigma}}}}\right\rVert_{F}$ $\displaystyle\leq\frac{1}{n}\left\lVert{{\mathbf{X}}^{\sf T}{\mathbf{X}}-{\rm I}_{p}}\right\rVert_{F}$ $\displaystyle\leq\frac{1}{n}\left\lVert{{\mathbf{X}}}\right\rVert_{F}^{2}+p,$ Consequently: $\displaystyle\|{\mathbb{E}}\\{\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle\\}-{\mathbb{E}}\\{F_{L}({\mathbf{Z}}_{\sf Q})\\}\|$ $\displaystyle\leq{\mathbb{E}}\left\\{(\|\langle{\mathbf{y}},{\mathbf{S}}{\mathbf{y}}\rangle\|+\|F_{L}({\mathbf{Z}}_{\sf Q})\|)\mathbb{I}({\mathbf{Z}}_{{\sf Q}}\in G^{c})\right\\}$ $\displaystyle\leq{\mathbb{E}}\left\\{(\frac{1}{n}\left\lVert{{\mathbf{X}}}\right\rVert_{F}^{2}+p+\left\lvert{F_{L}(0)}\right\rvert+L\left\lVert{{\mathbf{X}}}\right\rVert_{F})\mathbb{I}({\mathbf{Z}}_{\sf Q}\in G^{c})\right\\}$ $\displaystyle\leq 3\min(n^{-2},p^{-2}),$ where the last line follows from an application of Lemma A.1. This completes the proof of the remark. ### Proof of Remarks 6.9 and 6.10 We only prove Remark 6.9. The proof of Remark 6.10 is similar. The first claim follows from the fact that ${\mathbb{E}}\\{{\mathbf{\widetilde{N}}}_{ij}\\}=0$, by symmetry of the distribution of $\langle{\mathbf{\tilde{z}}}_{i},{\mathbf{\tilde{z}}}_{j}\rangle/n$ and of the soft thresholding function. As for the second claim, we follow a line of argument similar to that of Remark 6.7. Recall that $F_{K}({\mathbf{Z}}_{E^{c}})$ is the $G,K$\- Lipschitz extension of $\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle$. Here: $\displaystyle G$ $\displaystyle=\left\\{{\mathbf{Z}}_{E^{c}}:\left\lVert{{\nabla}\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle}\right\rVert^{2}\leq K^{2}\right\\},$ $\displaystyle K$ $\displaystyle=\sqrt{\frac{256A\gamma^{2}\alpha^{3}}{p\tau^{2}}}.$ Further we have shown that ${\mathbb{P}}\left\\{G^{c}\right\\}\leq\exp(-p/2)$. Since $F_{K}({\mathbf{Z}}_{E^{c}})$ is $K$-Lipschitz: $\displaystyle\|F_{K}({\mathbf{Z}}_{E^{c}})\|$ $\displaystyle\leq\|F_{K}(0)\|+K\left\lVert{{\mathbf{Z}}_{E^{c}}}\right\rVert_{F}$ $\displaystyle\leq K\left\lVert{{\mathbf{Z}}_{E^{c}}}\right\rVert_{F}$ $\displaystyle\leq K\left\lVert{{\mathbf{Z}}}\right\rVert_{F},$ since $F_{K}(0)=0$. Further, using the fact that $\left\lvert{\eta(z;\tau/\sqrt{n})}\right\rvert\leq\left\lvert{z}\right\rvert$: $\displaystyle\|\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle\|$ $\displaystyle\leq\left\lVert{{\mathbf{\widetilde{N}}}}\right\rVert_{F}$ $\displaystyle\leq\left\lVert{\frac{1}{n}{\mathbf{Z}}^{\sf T}{\mathbf{Z}}}\right\rVert_{F}$ $\displaystyle\leq\frac{1}{n}\left\lVert{{\mathbf{Z}}}\right\rVert_{F}^{2}.$ Now we have $\displaystyle\|{\mathbb{E}}\\{\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle\\}-{\mathbb{E}}\\{F_{K}({\mathbf{Z}}_{E^{c}})\\}\|$ $\displaystyle\leq{\mathbb{E}}\left\\{\|\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle- F_{K}({\mathbf{Z}}_{E^{c}})\|\right\\}$ $\displaystyle={\mathbb{E}}\left\\{\left\lvert{\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle- F_{K}({\mathbf{Z}}_{E^{c}})}\right\rvert\mathbb{I}({\mathbf{Z}}_{E^{c}}\in G^{c})\right\\}$ $\displaystyle\leq{\mathbb{E}}\left\\{\left\lvert{\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle}\right\rvert+\left\lvert{F_{K}({\mathbf{Z}}_{E^{c}})}\right\rvert\mathbb{I}({\mathbf{Z}}_{E^{c}}\in G^{c})\right\\},$ where the penultimate equality follows from the definition of the Lipschitz extension. Using the above estimates: $\displaystyle\|{\mathbb{E}}\\{\langle{\mathbf{y}}_{E^{c}},{\mathbf{\widetilde{N}}}{\mathbf{y}}_{E^{c}}\rangle\\}-{\mathbb{E}}\\{F_{K}({\mathbf{Z}}_{E^{c}})\\}\|$ $\displaystyle\leq{\mathbb{E}}\left\\{\left(\frac{1}{n}\left\lVert{{\mathbf{Z}}}\right\rVert_{F}^{2}+K\left\lVert{{\mathbf{Z}}}\right\rVert_{F}\right)\mathbb{I}({\mathbf{Z}}_{E^{c}}\in G^{c})\right\\}$ $\displaystyle\leq{\mathbb{E}}\left\\{\left(\frac{1}{n}\left\lVert{{\mathbf{X}}}\right\rVert_{F}^{2}+K\left\lVert{{\mathbf{X}}}\right\rVert_{F}\right)\mathbb{I}({\mathbf{Z}}_{E^{c}}\in G^{c})\right\\}$ The remark then follows by an application of Lemma A.1. ## Appendix B Empirical Results Figure 3: The results of Simple PCA, Diagonal Thresholding and Covariance Thresholding (respectively) for a synthetic block-constant function (which is sparse in the Haar wavelet basis). We use $\beta=1.4,p=4096$, and the rows correspond to sample sizes $n=1024,1625,2580,4096$ respectively. Parameters for Covariance Thresholding are chosen as in Section 3, with $\nu^{\prime}=4.5$. Parameters for Diagonal Thresholding are from [JL09]. 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1311.5276	# Characterizing derivations for any nest algebras on Banach spaces by their behaviors at an injective operator Yanfang Zhang Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, P. R. China. [email protected] , Jinchuan Hou Department of Mathematics, Taiyuan University of Technology, Taiyuan 030024, P. R. China [email protected] and Xiaofei Qi Department of Mathematics, Shanxi University, Taiyuan 030006, P. R. China. [email protected] ###### Abstract. Let ${\mathcal{N}}$ be a nest on a complex Banach space $X$ and let $\mbox{\rm Alg}{\mathcal{N}}$ be the associated nest algebra. We say that an operator $Z\in{\rm Alg}{\mathcal{N}}$ is an all-derivable point of $\mbox{\rm Alg}{\mathcal{N}}$ if every linear map $\delta$ from $\mbox{\rm Alg}{\mathcal{N}}$ into itself derivable at $Z$ (i.e. $\delta$ satisfies $\delta(A)B+A\delta(B)=\delta(Z)$ for any $A,B\in{\rm Alg}{\mathcal{N}}$ with $AB=Z$) is a derivation. In this paper, it is shown that every injective operator and every operator with dense range in ${\rm Alg}{\mathcal{N}}$ are all-derivable points of ${\rm Alg}{\mathcal{N}}$ without any additional assumption on the nest. 2010 Mathematical Subject Classification. 47B47, 47L35 Key words and phrases. Banach space nest algebras, the unite operator, all- derivable point. This work is partially supported by National Natural Science Foundation of China (11171249,11101250, 11271217) and Youth Foundation of Shanxi Province (2012021004). ## 1\. Introduction Let $\mathcal{A}$ be an (operator) algebra. Recall that a linear map $\delta:{\mathcal{A}}\rightarrow{\mathcal{A}}$ is a derivation if $\delta(AB)=\delta(A)B+A\delta(B)$ for all $A$, $B\in\mathcal{A}$. As well known, the class of derivations is a very important class of linear maps both in theory and applications, and was studied intensively. The question of under what conditions that a linear (even additive) map becomes a derivation attracted much attention of authors (for instance, see [2, 5, 6, 10] and the references therein). One approach is to characterize derivations by their local behaviors. We say that a map $\varphi:{\mathcal{A}}\rightarrow{\mathcal{A}}$ is derivable at a point $Z\in{\mathcal{A}}$ if $\varphi(A)B+A\varphi(B)=\varphi(Z)$ for any $A,B\in{\mathcal{A}}$ with $AB=Z$, and we call such $Z$ a derivable point of $\varphi$. Obviously, a linear map is a derivation if and only if it is derivable at every point. It is natural and interesting to ask the question whether or not a linear map is a derivation if it is derivable only at one given point. As usual, we say that an element $Z\in{\mathcal{A}}$ is an all- derivable point of $\mathcal{A}$ if every linear map on ${\mathcal{A}}$ derivable at $Z$ is in fact a derivation. So far, we have known that there exist many all-derivable points (or full-drivable points) for certain (operator) algebras (see [1, 4, 7, 8, 12, 14] and the references therein). However, zero point $0$ is not an all-derivable point for any algebra because the generalized derivations are derivable at $0$ [8]. The unit $I$, and more generally, invertible elements, are all-derivable points for many algebras and is a start point to find other all-derivable points. For instance, $I$ is an all-derivable point of prime rings and triangular algebras, and every invertible element is an all-derivable point of $\mathcal{J}$-subspace lattice algebras (see [1, 4, 9]). As nest algebras are of an important class of operator algebras, there are many papers on finding all-derivable points of certain nest algebras. We mention some results related to this paper. Zhu and Xiong [13] proved that every strongly operator topology continuous linear map derivable at $I$ between nest algebras on complex separable Hilbert spaces is a derivation. There they said that $I$ is an all- derivable point related to strong operator topology. An and Hou in [1] generalized the above result and showed that every linear map derivable at $I$ between nest algebras on complex Banach spaces is a derivation, under the additional assumption of the existence of a complemented nontrivial element in the nest. In [8], Qi and Hou showed further that, if the nest ${\mathcal{N}}$ on a Banach space satisfies “$N\in{\mathcal{N}}$ is complemented in the given Banach space whenever $N_{-}=N$”, then the unit operator $I$ is an all- derivable point of the nest algebra Alg$\mathcal{N}$; furthermore, they show that every injective operator and every operator with dense range in the nest algebra are all-derivable points of the nest algebra. This additional assumption on the nest is quite weak: such nest concludes all nests on Hilbert spaces, all finite nest, all nest with order-type $\omega+1$ or $1+\omega^{}$ or $1+\omega^{}+\omega+1$, where $\omega$ is the order-type of natural numbers and $\omega^{}$ is its anti-order-type. But the problem whether the unit operator $I$, or every injective operator, or every operator with dense range, is an all-derivable point of any nest algebras on any Banach spaces remains open. The purpose of the present paper is to solve the above problem and show that every injective operator and every operator with dense range are all-derivable points for all nest algebras on complex Banach spaces without any additional assumptions on the nests. The following is the main result of this paper. Theorem 1.1. Let $\mathcal{N}$ be a nest on a complex Banach space $X$ with $\dim X\geq 2$ and $\delta:\mbox{\rm Alg}{\mathcal{N}}\rightarrow\mbox{\rm Alg}{\mathcal{N}}$ be a linear map. Let $Z\in{\rm Alg}{\mathcal{N}}$ be an injective operator or an operator with dense range in ${\rm Alg}{\mathcal{N}}$. Then $\delta$ is derivable at the operator $Z$ if and only if $\delta$ is a derivation. That is, every injective operator and every operator with dense range are all-derivable points of any nest algebras. Particularly, we have Corollary 1.2. Let $\mathcal{N}$ be a nest on a complex Banach space $X$ with $dimX\geq 2$. Then every invertible operator in ${\rm Alg}{\mathcal{N}}$ is an all-derivable point of ${\rm Alg}{\mathcal{N}}$. The paper is organized as follows. We fix some notations and preliminary lemmas in Section 2 and give a proof of Theorem 1.1 in Section 3. ## 2\. Preliminaries and lemmas In this section we fix some notations and give some lemmas. Assume that $X$ is a Banach space over the complex field $\mathbb{C}$. Denote by ${\mathcal{B}}(X)$ the algebra of all bounded linear operators on $X$. The topological dual space of $X$ (i.e. the set of all bounded linear functionals on $X$) is denoted by $X^{}$. Let $X^{}$ be the second dual space of $X$. The map $\kappa:x\mapsto x^{}$, defined by $x^{*}(f)=f(x)$ for all $f$ in $X^{}$, is the canonical embedding from $X$ into $X^{*}$. For any $T\in{\mathcal{B}}(X)$, its Banach adjoint operator $T^{}$ is the map from $X^{}$ into $X^{}$ defined by $(T^{}f)(x)=f(Tx)$ for any $f\in X^{}$ and $x\in X$. If $f\in X^{}$ and $x\in X$, the operator $x\otimes f$ on $X$ is defined by $(x\otimes f)(y)=f(y)x$ for any $y\in X$. $x\otimes f$ is rank one whenever both $x$ and $f$ are nonzero, and every rank one operator has this form. It is easily seen that $(x\otimes f)^{}=f\otimes x^{}$ and $x^{}T^{}=(Tx)^{}$. For any non-empty subset $N\subseteq X$, denote by $N^{\perp}$ the annihilator of $N$, that is, $N^{\perp}=\\{f\in X^{}:f(x)=0$ for every $x\in N\\}$. Some times we use $\langle x,f\rangle$ to present the value $f(x)$ of $f$ at $x$. In addition, we use the symbols ${\rm ran}(T)$ and ${\rm ker}(T)$ for the range and the kernel of operator $T$, respectively. Recall that a nest $\mathcal{N}$ in $X$ is a chain of closed (under norm topology) linear subspaces of $X$ containing the trivial subspaces $\\{0\\}$ and $X$, which is closed under the formation of arbitrary closed linear span (denoted by $\bigvee$ ) and intersection (denoted by $\bigwedge)$. ${\mbox{\rm Alg}}\mathcal{N}$ denotes the associated nest algebra, which is the algebra of all operators $T$ in ${\mathcal{B}}(X)$ such that $TN\subseteq N$ for every element $N\in\mathcal{N}$. When ${\mathcal{N}}\neq\\{\\{0\\},X\\}$, we say that $\mathcal{N}$ is nontrivial. It is clear that ${\mbox{\rm Alg}}\mathcal{N}=\mathcal{B}(X)$ if $\mathcal{N}$ is trivial. For $N\in\mathcal{N}$, let $N_{-}=\bigvee\\{M\in\mathcal{N}\mid M\subset N\\}$, $N_{+}=\bigwedge\\{M\in\mathcal{N}\mid N\subset M\\}$ and $N_{-}^{\perp}=(N_{-})^{\perp}$. Also, let $\\{0\\}_{-}=\\{0\\}$ and $X_{+}=X$. Denote $\mathcal{D}_{1}(\mathcal{N})=\bigcup\\{N\in\mathcal{N}\mid N_{-}\neq X\\}$ and $\mathcal{D}_{2}(\mathcal{N})=\bigcup\\{N_{-}^{\perp}\mid N\in\mathcal{N}\ \ {\mbox{\rm{and}}}\ \ N\neq\\{0\\}\\}$. Note that $\mathcal{D}_{1}(\mathcal{N})$ is dense in $X$ and $\mathcal{D}_{2}(\mathcal{N})$ is dense in $X^{}$. Clearly, $\mathcal{D}_{1}(\mathcal{N})=X$ if and only if $X_{-}\not=X$ and $\mathcal{D}_{2}(\mathcal{N})=X^{}$ if and only if $\\{0\\}\not=\\{0\\}_{+}$. For more informations on nest algebras, we refer to [3, 11]. The following lemmas are needed to prove the main result. Lemma 2.1. ([3, 11]) Let $\mathcal{N}$ be a nest on a real or complex Banach space $X$. The rank one operator $x\otimes f$ belongs to ${\mbox{\rm Alg}}\mathcal{N}$ if and only if there is some $N\in\mathcal{N}$ such that $x\in N$ and $f\in N_{-}^{\perp}$. Lemma 2.2. Let $\mathcal{N}$ be a nest on a real or complex Banach space $X$ with $\dim X\geq 2$. Assume that $\delta:\mbox{\rm Alg}{\mathcal{N}}\rightarrow\mbox{\rm Alg}{\mathcal{N}}$ is a linear map satisfying $\delta(P)=\delta(P)P+P\delta(P)$ for all idempotent operators $P\in\mbox{\rm Alg}{\mathcal{N}}$ and $\delta(N)N+N\delta(N)=0$ for all operators $N\in\mbox{\rm Alg}{\mathcal{N}}$ with $N^{2}=0$. If $X_{-}\not=X$, then, (1) for any $x\in X$ and $f\in X_{-}^{\perp}$, we have $\delta(x\otimes f)\ker(f)\subseteq\mbox{\rm span}\\{x\\}$; (2) there exist linear transformations $B:X\rightarrow X$ and $C:X_{-}^{\perp}\rightarrow X^{\ast}$ such that $\delta(x\otimes f)=Bx\otimes f+x\otimes Cf$ and $\langle Bx,f\rangle+\langle x,Cf\rangle=0$ for all $x\in X$ and $f\in X_{-}^{\perp}$. Proof. (1) For any nonzero $x\in X$ and $f\in X_{-}^{\perp}$, it follows from Lemma 2.1 that $x\otimes f\in{\rm Alg}\mathcal{N}$. If $\langle x,f\rangle\neq 0$, letting $\bar{x}=\langle x,f\rangle^{-1}x$, then $\bar{x}\otimes f$ is an idempotent operator in ${\rm Alg}\mathcal{N}$. By the assumption on $\delta$, we have $None$ $\delta(\bar{x}\otimes f)=\delta(\bar{x}\otimes f)(\bar{x}\otimes f)+(\bar{x}\otimes f)\delta(\bar{x}\otimes f).$ For any $y\in\ker(f)$, letting the operators in Eq.(2.1) act at $y$, we obtain $\delta(\bar{x}\otimes f)y=\langle\delta(\bar{x}\otimes f)y,f\rangle\bar{x}$, which implies $\delta(x\otimes f)y=\langle\delta(\bar{x}\otimes f)y,f\rangle x\in{\rm span}\\{x\\}.$ If $\langle x,f\rangle=0$, we can take $z\in X$ such that $\langle z,f\rangle=1$ as $X_{-}\not=X$. It is obvious that both $(x+z)\otimes f$ and $z\otimes f$ are idempotents in ${\rm Alg}\mathcal{N}$. By the assumption on $\delta$ again, one has $None$ $\begin{array}[]{rl}&\delta(x\otimes f)=\delta((x+z)\otimes f)-\delta(z\otimes f)\\\ =&\delta(x\otimes f)(z\otimes f)+(x\otimes f)\delta(x\otimes f)+\delta(z\otimes f)(x\otimes f)+(z\otimes f)\delta(x\otimes f)\\\ \end{array}$ and $None$ $0=\delta(x\otimes f)(x\otimes f)+(x\otimes f)\delta(x\otimes f).$ Then, for any $y\in\ker(f)$, Eqs.(2.2)-(2.3) become to $\delta(x\otimes f)y=\langle\delta(z\otimes f)y,f\rangle x+\langle\delta(x\otimes f)y,f\rangle z$ and $\langle\delta(x\otimes f)y,f\rangle=0$. So $\delta(x\otimes f)y=\langle\delta(z\otimes f)y,f\rangle x$ holds for every $y\in\ker(f)$, that is, (1) holds. (2) For any $x\in X$ and $f\in X_{-}^{\perp}$, by (1), there exists a functional $\varphi_{x,f}$ on $\ker(f)$ such that $\delta(x\otimes f)y=\varphi_{x,f}(y)x$ for all $y\in\ker(f)$. It is easy to see that $\varphi_{x,f}$ is linear. Take a non-zero vector $w_{f}\in X\setminus X_{-}$ such that $\langle w_{f},f\rangle=1$. Let $\bar{\varphi}_{x,f}$ be a linear extension of $\varphi_{x,f}$ to $X$. Then $\tilde{\varphi}_{x,f}=\bar{\varphi}_{x,f}-\bar{\varphi}_{x,f}(w_{f})f$ is also a linear extension of $\varphi_{x,f}$ which vanishes at $w_{f}$. Define a map $B_{f}:X\rightarrow X$ by $B_{f}x=\delta(x\otimes f)w_{f}$. Obviously, $B_{f}$ is linear by the linearity of $\delta$, and $B_{\lambda f}=B_{f}$ for any nonzero scalar $\lambda$. For any $\tilde{x}\in X$, as $z_{x}=\tilde{x}-f(\tilde{x})w_{f}\in\ker(f)$, we have $\begin{array}[]{rl}\delta(x\otimes f)\tilde{x}=&\delta(x\otimes f)(z_{x}+f(\tilde{x})w_{f})=\delta(x\otimes f)z_{x}+f(\tilde{x})\delta(x\otimes f)w_{f}\\\ =&\varphi_{x,f}(z_{x})x+f(\tilde{x})\delta(x\otimes f)w_{f}=\varphi_{x,f}(\tilde{x}-f(\tilde{x})w_{f})+f(\tilde{x})B_{f}x\\\ =&\tilde{\varphi}_{x,f}(\tilde{x})x+f(\tilde{x})B_{f}x.\end{array}$ So $None$ $\delta(x\otimes f)=x\otimes\tilde{\varphi}_{x,f}+B_{f}x\otimes f\quad{\rm for\ \ all}\quad x\in X,\ f\in X_{-}^{\perp}.$ As $\delta(x\otimes f)$ and $B_{f}x\otimes f$ are bounded linear operators on $X$, we see that $\tilde{\varphi}_{x,f}\in X^{}$. For simplicity seek, in the following we still denote ${\varphi}_{x,f}$ for $\tilde{\varphi}_{x,f}$. Claim 1. $\varphi_{x,f}$ only depends on $f$. Take any $x_{1},x_{2}\in X$ with $x_{1}\not=0$. For any $y\in X$, by Eq.(2.4), we have $\delta((x_{1}+x_{2})\otimes f)y=\langle y,\varphi_{x_{1}+x_{2},f}\rangle(x_{1}+x_{2})$ and $\delta((x_{1}+x_{2})\otimes f)y=\delta(x_{1}\otimes f)y+\delta(x_{1}\otimes f)y=\langle y,\varphi_{x_{1},f}\rangle x_{1}+\langle y,\varphi_{x_{2},f}\rangle x_{2}.$ So $0=(\langle y,\varphi_{x_{1}+x_{2},f}\rangle-\langle y,\varphi_{x_{1},f}\rangle)x_{1}+(\langle y,\varphi_{x_{1}+x_{2},f}\rangle-\langle y,\varphi_{x_{2},f}\rangle)x_{2}.$ If $x_{1}$ and $x_{2}$ are linearly independent, the above equation implies $\varphi_{x_{1},f}=\varphi_{x_{1}+x_{2},f}=\varphi_{x_{2},f}$. If $x_{1}$ and $x_{2}$ are linearly dependent, we take $x_{3}\in X$ such that it is independent of $x_{1}$. By the preceding proof, one gets $\varphi_{x_{1},f}=\varphi_{x_{3},f}=\varphi_{x_{2},f}$. The claim is true. Now we can write $\varphi_{x,f}=\varphi_{f}\in X^{}$. Then $None$ $\delta(x\otimes f)=x\otimes\varphi_{f}+B_{f}x\otimes f\quad{\rm for\ \ all}\quad x\in X,\ f\in X_{-}^{\perp}.$ Claim 2. There exist linear transformations $B:X\rightarrow X$ and $C:X_{-}^{\perp}\rightarrow X^{\ast}$ such that $\delta(x\otimes f)=Bx\otimes f+x\otimes Cf$ for all $x\in X$ and $f\in X_{-}^{\perp}$. Fix a non-zero $f_{0}\in X_{-}^{\perp}$ and put $B=B_{f_{0}}.$ For any nonzero $f_{1},\ f_{2}\in X_{-}^{\perp}$, we show that the difference $B_{f_{1}}-B_{f_{2}}$ is a scalar multiple of the identity $I$. If $f_{1}$ and $f_{2}$ are linearly independent, then there are $x_{1},x_{2}\in X$ such that $f_{i}(x_{i})=1$ and $f_{i}(x_{j})=0$ for $1\leq i\neq j\leq 2$. For any $x\in X$, we have $\delta(x\otimes(f_{1}+f_{2}))=x\otimes\varphi_{f_{1}+f_{2}}+B_{f_{1}+f_{2}}x\otimes(f_{1}+f_{2})$ and $\begin{array}[]{rl}\delta(x\otimes(f_{1}+f_{2}))=&\delta(x\otimes f_{1})+\delta(x\otimes f_{2})\\\ =&x\otimes(\varphi_{f_{1}}+\varphi_{f_{2}})+B_{f_{1}}x\otimes f_{1}+B_{f_{2}}x\otimes f_{2}.\end{array}$ So $None$ $0=x\otimes(\varphi_{f_{1}+f_{2}}-\varphi_{f_{1}}-\varphi_{f_{1}})+B_{f_{1}}x\otimes f_{2}+B_{f_{2}}x\otimes f_{1}.$ By acting at $x_{1}-x_{2}$, Eq.(2.6) gives $B_{f_{1}}x-B_{f_{2}}x=(\varphi_{f_{1}+f_{2}}(x_{1}-x_{2})-\varphi_{f_{1}}(x_{1}-x_{2})-\varphi_{f_{1}}(x_{1}-x_{2}))x=\lambda_{f_{1},f_{2}}x.$ Thus, $B_{f_{1}}-B_{f_{2}}=\lambda_{f_{1},f_{2}}I$ is a scalar multiple of the identity $I$. If $f_{1}$ and $f_{2}$ are linearly dependent, then $B_{f_{1}}-B_{f_{2}}=0$ as $B_{f_{1}}=B_{f_{2}}$. By now we have shown that, for any $f\in X_{-}^{\perp}$, $B_{f}=B_{f_{0}}+b_{f}I=B+b_{f}I$ for some scalar $b_{f}$. Then Eq.(2.5) becomes $\begin{array}[]{rl}\delta(x\otimes f)=&x\otimes\varphi_{f}+(B+b_{f}I)x\otimes f\\\ =&Bx\otimes f+x\otimes(b_{f}f+\varphi_{f})=Bx\otimes f+x\otimes Cf,\end{array}$ where $C:X_{-}^{\perp}\rightarrow X^{\ast}$ is defined by $Cf=b_{f}f+\varphi_{f}$ for all $f\in X_{-}^{\perp}$. By the linearity of $\delta$, we see that $C$ is linear. Claim 3. $\langle Bx,f\rangle+\langle x,Cf\rangle=0$ for all $x\in X$ and $f\in X_{-}^{\perp}$. For any $x\in X$ and $f\in X_{-}^{\perp}$, by the assumptions on $\delta$, one can get $(x\otimes f)\delta(x\otimes f)(x\otimes f)=0$. Then, by Claim 2, $(\langle Bx,f\rangle+\langle x,Cf\rangle)(x\otimes f)=0$, and so $\langle Bx,f\rangle+\langle x,Cf\rangle=0$. Combining Claims 1-3, (2) is true. $\Box$ Lemma 2.3. Let $\mathcal{N}$ be a nest on a real or complex Banach space $X$ with $\dim X\geq 2$. Assume that $\delta:\mbox{\rm Alg}{\mathcal{N}}\rightarrow\mbox{\rm Alg}{\mathcal{N}}$ is a linear map satisfying $\delta(P)=\delta(P)P+P\delta(P)$ for all idempotent operators $P\in\mbox{\rm Alg}{\mathcal{N}}$ and $\delta(N)N+N\delta(N)=0$ for all operators $N\in\mbox{\rm Alg}{\mathcal{N}}$ with $N^{2}=0$. If $\\{0\\}\not=\\{0\\}_{+}$, then (1) for any $f\in X^{\ast}$ and any $x\in\\{0\\}_{+}$, we have $\delta(x\otimes f)^{\ast}(\ker x^{\ast\ast})\subseteq{\rm span}\\{f\\}$; (2) there exist linear transformations $B:X^{}\rightarrow X^{}$ and $C:\kappa(\\{0\\}_{+})\rightarrow X^{\ast\ast}$ such that $\delta(x\otimes f)^{\ast}=Bf\otimes x^{\ast\ast}+f\otimes Cx^{\ast\ast}$ and $\langle Bf,x^{\ast\ast}\rangle+\langle f,Cx^{\ast\ast}\rangle=0$ holds for all $x\in\\{0\\}_{+}$ and $f\in X^{}$. Proof. This is the “dual” of Lemma 2.2. We give a proof of the conclusion (1) in detail as a sample, and give a proof of the conclusion (2) in sketch. (1) Let $f\in X^{\ast}$ and $x\in\\{0\\}_{+}$ be nonzero. If $\langle f,x^{\ast\ast}\rangle\neq 0$, it is enough to consider the case $\langle f,x^{\ast\ast}\rangle=\langle x,f\rangle=1$. In this case, $(x\otimes f)^{2}=x\otimes f$. By the assumptions on $\delta$, we get $None$ $\delta(x\otimes f)^{\ast}=(f\otimes x^{\ast\ast})\delta(x\otimes f)^{\ast}+\delta(x\otimes f)^{\ast}(f\otimes x^{\ast\ast}).$ Applying Eq.(2.7) to any $g\in\ker(x^{\ast\ast})$ gives $\delta(x\otimes f)^{\ast}g=\langle\delta(x\otimes f)^{\ast}g,x^{\ast\ast}\rangle f\in{\rm span}\\{f\\}$. If $\langle f,x^{\ast\ast}\rangle=\langle x,f\rangle=0$, we can find $f_{1}\in X^{}$ with $\langle x,f_{1}\rangle=1$ as $\\{0\\}\not=\\{0\\}_{+}$. Then both $x\otimes f_{1}$ and $x\otimes(f_{1}+f)$ are idempotents in $\mbox{\rm Alg}{\mathcal{N}}$ and $(x\otimes f)^{2}=0$. So $\delta(x\otimes(f+f_{1}))=\delta(x\otimes(f+f_{1}))(x\otimes(f+f_{1}))+(x\otimes(f+f_{1}))\delta(x\otimes(f+f_{1})),$ $None$ $0=\delta(x\otimes f)(x\otimes f)+(x\otimes f)\delta(x\otimes f)$ and $\delta(x\otimes f_{1})=\delta(x\otimes f_{1})(x\otimes f_{1})+(x\otimes f_{1})\delta(x\otimes f_{1}).$ The above two equations imply $None$ $\begin{array}[]{rl}\delta(x\otimes f)^{\ast}=&(f_{1}\otimes x^{\ast\ast})\delta(x\otimes f)^{\ast}+(f\otimes x^{\ast\ast})\delta(x\otimes f_{1})^{\ast}\\\ &+\delta(x\otimes f_{1})^{\ast}(f\otimes x^{\ast\ast})+\delta(x\otimes f)^{\ast}(f_{1}\otimes x^{\ast\ast}).\end{array}$ Note that Eq.(2.8) implies $0=(f\otimes x^{\ast\ast})\delta(x\otimes f)^{\ast}+\delta(x\otimes f)^{\ast}(f\otimes x^{\ast\ast}).$ For any $g\in\ker(x^{\ast\ast})$, letting the equation act at $g$, one gets $\langle\delta(x\otimes f)^{\ast}g,x^{\ast\ast}\rangle f=0$, and so $\langle\delta(x\otimes f)^{\ast}g,x^{\ast\ast}\rangle=0$. Thus, letting Eq.(2.9) act at any $g\in\ker(x^{\ast\ast})$ leads to $\delta(x\otimes f)^{\ast}g=\langle\delta(x\otimes f_{1})^{\ast}g,x^{\ast\ast}\rangle f\in{\rm span}\\{f\\}$. (2) For any nonzero $f\in X^{}$ and $x\in\\{0\\}_{+}$, we have $x\otimes f\in\mbox{\rm Alg}{\mathcal{N}}$. Take $h_{x}\in X^{\ast}$ such that $\langle x,h\rangle=1$. Define a map $B_{x}:X^{\ast}\rightarrow X^{\ast}$ by $B_{x}f=\delta(x\otimes f)^{\ast}h_{x}$. Let $\kappa:X\rightarrow X^{\ast\ast}$ be the canonical map from $X$ into $X^{\ast\ast}$. By (1) in the lemma, there exists a linear functional $\Phi_{f,x}$ on $\ker x^{\ast\ast}$ such that $\delta(x\otimes f)^{\ast}g=\Phi_{x,f}(g)f$ for any $g\in\ker(x^{\ast\ast})$. Still by $\Phi_{x,f}$ denotes such a special extension of $\Phi_{x,f}$ on $X^{\ast\ast}$ which vanishes at $h_{x}$. For any $\tilde{f}\in X^{\ast}$, let $Z_{f}=\tilde{f}-\langle f,x^{\ast\ast}\rangle h_{x}$. It is obvious that $Z_{f}\in\ker x^{\ast\ast}$, and so $\delta(x\otimes f)^{\ast}\tilde{f}=\delta(x\otimes f)^{\ast}(Z_{f}+\langle f,x^{\ast\ast}\rangle h_{x})=\Phi_{x,f}(\tilde{f})f+(B_{x}f\otimes x^{\ast\ast})\tilde{f}.$ Thus we can get $\delta(x\otimes f)^{\ast}=B_{x}f\otimes x^{\ast\ast}+f\otimes\Phi_{x,f}$ for $f\in X^{}$ and $x\in\\{0\\}_{+}$. Similarly to the proof of Lemma 2.2(2), the following Claims 1-3 hold. Claim 1. $\Phi_{x,f}$ depends only on $x$. Fix a non-zero vector $x_{0}\in\\{0\\}_{+}$ and put $B=B_{x_{0}}$. Claim 2. There exist linear transformations $B:X^{}\rightarrow X^{}$ and $C:\kappa(\\{0\\}_{+})\rightarrow X^{\ast\ast}$ such that $\delta(x\otimes f)^{\ast}=Bf\otimes x^{\ast\ast}+f\otimes Cx^{\ast\ast}$ holds for all $x\in\\{0\\}_{+}$ and $f\in X^{}$. Claim 3. $\langle Bf,x^{\ast\ast}\rangle+\langle f,Cx^{\ast\ast}\rangle=0$ holds for all $x\in\\{0\\}_{+}$ and $f\in X^{}$. Hence (2) holds, the proof of the lemma is finished. $\Box$ Lemma 2.4. Let $X$ be a Banach space over the real or complex field $\mathbb{F}$. Suppose that $\mathcal{N}$ is a nest on $X$ and $\delta:\mbox{\rm Alg}{\mathcal{N}}\rightarrow\mbox{\rm Alg}{\mathcal{N}}$ is a linear map satisfying $\delta(P)=\delta(P)P+P\delta(P)$ for all idempotent operators $P\in\mbox{\rm Alg}{\mathcal{N}}$ and $\delta(N)N+N\delta(N)=0$ for all operators $N\in\mbox{\rm Alg}{\mathcal{N}}$ with $N^{2}=0$. If $X_{-}=X$ and $\\{0\\}=\\{0\\}_{+}$, then (1) there exists a bilinear functional $\beta:({\mathcal{D}}_{1}({\mathcal{N}})\times{\mathcal{D}}_{2}({\mathcal{N}}))\cap{\rm Alg}{\mathcal{N}}\rightarrow\mathbb{F}$ such that $(\delta(x\otimes f)-\beta_{x,f}I)\ker(f)\subseteq\mbox{\rm span}\\{x\\}$ holds for all $x\otimes f\in{\rm Alg}\mathcal{N}$; (2) there exist linear transformations $B:\mathcal{D}_{1}(\mathcal{N})\rightarrow\mathcal{D}_{1}(\mathcal{N})$ and $C:\mathcal{D}_{2}(\mathcal{N})\rightarrow\mathcal{D}_{2}(\mathcal{N})$ such that $\delta(x\otimes f)-\beta_{x,f}I=x\otimes Cf+Bx\otimes f$ holds for all $x\otimes f\in{\mbox{\rm Alg}}\mathcal{N}$. Proof. Since $X_{-}=X$ and $\\{0\\}=\\{0\\}_{+}$, it is obvious that $\mathcal{N}$ is non-trivial, $\mathcal{D}_{1}(\mathcal{N})$ and $\mathcal{D}_{2}(\mathcal{N})$ are dense proper linear manifolds in $X$ and $X^{}$, respectively, and, for each nontrivial $N\in{\mathcal{N}}$, both $N$ and $N_{-}^{\perp}$ are infinite dimensional. For any $x\otimes f\in{\rm Alg}\mathcal{N}$, if $\langle x,f\rangle\neq 0$, by the assumption on $\delta$ for idempotents, it is easily seen that $\delta(x\otimes f)\ker(f)\in{\rm span}\\{x\\}$. In this case let $\beta_{x,f}=0$. Now assume that $x\otimes f\in$Alg$\mathcal{N}$ and $\langle x,f\rangle=0$. By Lemma 2.1, there exists some $N_{x}\in\mathcal{N}$ such that $x\in N_{x}$ and $f\in(N_{x})_{-}^{\perp}$. It is easy to check that $\vee\\{\ker(f)\cap N:N\in\mathcal{N},N_{x}\subseteq N,N_{-}\not=X\\}={\mathcal{D}}_{1}({\mathcal{N}})\cap\ker(f)$ is dense in $\ker(f)$. For any $N\in\mathcal{N}$ with $N_{x}\subseteq N$ and for every $y\in N\cap\ker(f)$, take $g\in N^{\perp}$ such that $g$ is linearly independent of $f$. Then $x\otimes f+y\otimes g\in\mbox{\rm Alg}{\mathcal{N}}$ with $(x\otimes f+y\otimes g)^{2}=0$. By the assumption on $\delta$, we have $0=\delta(x\otimes f+y\otimes g)(x\otimes f+y\otimes g)+(x\otimes f+y\otimes g)\delta(x\otimes f+y\otimes g).$ Also note that $(x\otimes f)^{2}=0$ and $(y\otimes g)^{2}=0$. The above equation can be reduced to $None$ $0=\delta(x\otimes f)(y\otimes g)+\delta(y\otimes g)(x\otimes f)+(y\otimes g)\delta(x\otimes f)+(x\otimes f)\delta(y\otimes g).$ Choose $z_{1}\in X$ such that $\langle z_{1},f\rangle=-1$ and $\langle z_{1},g\rangle=1$. Let Eq.(2.10) act at $z_{1}$, one gets $None$ $0=\delta(x\otimes f)y-\delta(y\otimes g)x+\langle\delta(x\otimes f)z_{1},g\rangle y+\langle\delta(y\otimes g)z_{1},f\rangle x.$ Since $f$ and $g$ are linearly independent, we can pick $w\in X$ such that $\langle w,f\rangle=2$ and $\langle w,g\rangle=0$. Let $z_{2}=z_{1}+w$; then $\langle z_{2},g\rangle=\langle z_{2},f\rangle=1$. Letting each operator in Eq.(2.10) act at $z_{2}$, we get $None$ $0=\delta(x\otimes f)y+\delta(y\otimes g)x+\langle\delta(x\otimes f)z_{2},g\rangle y+\langle\delta(y\otimes g)z_{2},f\rangle x.$ It follows from Eqs.(2.11)-(2.12) that $\delta(x\otimes f)y=-\frac{1}{2}(\langle\delta(x\otimes f)z_{1},g\rangle+\langle\delta(x\otimes f)z_{2},g\rangle)y-\frac{1}{2}(\langle\delta(y\otimes g)z_{1},f\rangle+\langle\delta(y\otimes g)z_{2},f\rangle)x,$ which implies that $\delta(x\otimes f)y\in{\rm span}\\{x,y\\},$ that is, there exist scalars $\alpha_{x,f}^{N}(y)$, $\beta_{x,f}^{N}(y)$ such that $None$ $\delta(x\otimes f)y=\alpha_{x,f}^{N}(y)x+\beta_{x,f}^{N}(y)y.$ Next we show that, for any $y_{1},y_{2}\in N\cap\ker(f)$, $\beta_{x,f}^{N}(y_{1})=\beta_{x,f}^{N}(y_{2})$. In fact, by Eq.(2.13), we have $\delta(x\otimes f)y_{i}=\alpha_{x,f}^{N}(y_{i})x+\beta_{x,f}^{N}(y_{i})y_{i},\quad\quad i=1,2$ and $\delta(x\otimes f)(y_{1}+y_{2})=\alpha_{x,f}^{N}(y_{1}+y_{2})x+\beta_{x,f}^{N}(y_{1}+y_{2})(y_{1}+y_{2}).$ So $\begin{array}[]{rl}0=&(\alpha_{x,f}^{N}(y_{1}+y_{2})-\alpha_{x,f}^{N}(y_{1})-\alpha_{x,f}^{N}(y_{2}))x\\\ &+(\beta_{x,f}^{N}(y_{1}+y_{2})-\beta_{x,f}^{N}(y_{1}))y_{1}+(\beta_{x,f}^{N}(y_{1}+y_{2})-\beta_{x,f}^{N}(y_{2}))y_{2}.\\\ \end{array}$ If $y_{1}$, $y_{2}$ and $x$ are linearly independent, obviously $\beta_{x,f}^{N}(y_{1})=\beta_{x,f}^{N}(y_{2});$ if $y_{1}$ and $y_{2}$ are linearly dependent and $x\not\in{\rm span}\\{y_{1},y_{2}\\}$, as ${\rm dim}(\ker f\cap N)=\infty$, there exists $y_{3}\in\ker f\cap N$ so that $y_{3}\not\in{\rm span}\\{y_{1},y_{2}\\}$. Then, by what just proved above, we have $\beta_{x,f}^{N}(y_{1})=\beta_{x,f}^{N}(y_{3})=\beta_{x,f}^{N}(y_{2}).$ For the case $\dim{\rm span}\\{x,y_{1},y_{2}\\}=1$, choosing $y_{4}\in\ker(f)\cap N$ so that $y_{4}$ is linearly independent of $x$, we see that $\beta_{x,f}^{N}(y_{1})=\beta_{x,f}^{N}(y_{2})=\beta_{x,f}^{N}(y_{4})$. Thus we have shown that $\beta_{x,f}^{N}(y)$ is independent of $y$. So, there exists a scalar $\beta_{x,f}^{N}$ such that $\beta_{x,f}^{N}(y)=\beta_{x,f}^{N}$ holds for all $y\in N\cap\ker(f)$. We claim that $\alpha_{x,f}^{N}(y)$ and $\beta_{x,f}^{N}$ are independent of $N$, and thus $\alpha_{x,f}^{N}(y)=\alpha_{x,f}(y)$, $\beta_{x,f}^{N}=\beta_{x,f}.$ Indeed, if ${N}^{{}^{\prime}},{N}^{{}^{\prime\prime}}\in\mathcal{N}$ with $N_{x}\subseteq N^{{}^{\prime}}\cap N^{\prime\prime}$ and $y\in{N}^{{}^{\prime}}\cap N^{{}^{\prime\prime}}\cap\ker(f)$, then by Eq.(2.13), we have $\delta(x\otimes f)y=\alpha_{x,f}^{{N}^{{}^{\prime}}}(y)x+\beta_{x,f}^{{N}^{{}^{\prime}}}y\quad{\rm and}\quad\delta(x\otimes f)y=\alpha_{x,f}^{{N}^{{}^{\prime\prime}}}(y)x+\beta_{x,f}^{{N}^{{}^{\prime\prime}}}y.$ The above two equations give $0=(\alpha_{x,f}^{{N}^{{}^{\prime}}}(y)-\alpha_{x,f}^{{N}^{{}^{\prime\prime}}}(y))x+(\beta_{x,f}^{{N}^{{}^{\prime}}}-\beta_{x,f}^{{N}^{{}^{\prime\prime}}})y.$ As $\dim N^{\prime}\cap N^{\prime\prime}\cap\ker(f)=\infty$, one can choose $y$ such that $y$ is linearly independent of $x$. Then we get $\alpha_{x,f}^{{N}^{{}^{\prime}}}(y)=\alpha_{x,f}^{{N}^{{}^{\prime\prime}}}(y)$ and $\beta_{x,f}^{{N}^{{}^{\prime}}}=\beta_{x,f}^{{N}^{{}^{\prime\prime}}}$. It follows that $\beta_{x,f}^{{N}}$ is independent of $N$ and there is a scalar $\beta_{x,f}$ such that $\beta_{x,f}^{{N}}=\beta_{x,f}$ holds for all $N\in{\mathcal{N}}$ with $N_{x}\subseteq N$. Now it is clear that $\alpha_{x,f}^{{N}^{{}^{\prime}}}(y)=\alpha_{x,f}^{{N}^{{}^{\prime\prime}}}(y)$ also holds for all $y\in N^{\prime}\cap N^{\prime\prime}\cap\ker(f)$. Hence there exists a scalar $\alpha_{x,f}(y)$ such that $\alpha_{x,f}^{{N}}(y)=\alpha_{x,f}(y)$ for all $y\in N\cap\ker(f)$ with $N_{x}\subseteq N$. Thus we have shown that $None$ $\delta(x\otimes f)y=\alpha_{x,f}(y)x+\beta_{x,f}y$ holds for all $y\in{\mathcal{D}}_{1}({\mathcal{N}})\cap\ker(f)$. Since ${\mathcal{D}}_{1}({\mathcal{N}})\cap\ker(f)$ is dense in $\ker(f)$, Eq.(2.14) holds for all $y\in\ker(f)$. Finally we will prove that $\beta_{x,f}$ is bilinear in $x\in{\mathcal{D}}_{1}({\mathcal{N}})$ and $f\in{\mathcal{D}}_{2}({\mathcal{N}})$ with $x\otimes f\in$Alg$\mathcal{N}$. Indeed, for any $x_{1},x_{2}\in{\mathcal{D}}_{1}({\mathcal{N}})$, there exists some $N\in{\mathcal{N}}$ such that $x_{1},\ x_{2}\in N$. Then, for any $f\in N_{-}^{\perp}$, $x_{1}\otimes f$, $x_{2}\otimes f\in{\rm Alg}\mathcal{N}$. By Eq.(2.14), for any $y\in\ker(f)$, one has $\delta((x_{1}+x_{2})\otimes f)y=\alpha_{x_{1}+x_{2},f}(y)(x_{1}+x_{2})+\beta_{x_{1}+x_{2},f}y,$ $\delta(x_{1}\otimes f)y=\alpha_{x_{1},f}(y)(x_{1})+\beta_{x_{1},f}y\quad{\rm and}\quad\delta(x_{2}\otimes f)y=\alpha_{x_{2},f}(y)(x_{2})+\beta_{x_{2},f}y.$ By the additivity of $\delta$, the above equations yield $\begin{array}[]{rl}0=&(\alpha_{x_{1}+x_{2},f}-\alpha_{x_{1},f})(y)x_{1}+(\alpha_{x_{1}+x_{2},f}-\alpha_{x_{1},f})(y)x_{2}\\\ &+(\beta_{x_{1}+x_{2},f}-\beta_{x_{1},f}-\beta_{x_{2},f})y.\end{array}$ Since $\ker(f)\cap N\supseteq N_{-}$ is infinite-dimensional, we can choose $y\in\ker f\cap N$ such that $y\not\in{\rm span}\\{x_{1},x_{2}\\}$. It follows that $\beta_{x_{1}+x_{2},f}=\beta_{x_{1},f}+\beta_{x_{2},f}$, that is, $\beta_{x,f}$ is additive for $x\in{\mathcal{D}}_{1}({\mathcal{N}})$. Now for any non-zero scalar $s$, by Eq.(2.14), we have $\delta(sx_{1}\otimes f)y=s\alpha_{sx_{1},f}(y)x_{1}+\beta_{sx_{1},f}y\quad{\rm and}\quad s\delta(x_{1}\otimes f)y=s\alpha_{x_{1},f}(y)x_{1}+s\beta_{x_{1},f}y,$ which imply $s(\alpha_{sx_{1},f}(y)-\alpha_{x_{1},f}(y))x_{1}+(\beta_{sx_{1},f}-s\beta_{x_{1},f})y=0.$ Still choosing $y$ linearly independent of $x_{1}$ gives $\beta_{sx_{1},f}=s\beta_{x_{1},f}$. Hence $\beta_{x,f}$ is linear in $x$. For any $f_{1},f_{2}\in{\mathcal{D}}_{2}({\mathcal{N}})$, there exists some $N\in{\mathcal{N}}$ such that $f_{1},\ f_{2}\in N_{-}^{\perp}$. Take any $x\in N$; and then $x\otimes f_{1}$, $x\otimes f_{2}\in{\rm Alg}\mathcal{N}$. Since $f_{1},f_{2}\in{\mathcal{D}}_{2}({\mathcal{N}})$, $f_{1},f_{2}\in M^{\perp}$ for some $M\in{\mathcal{N}}\setminus\\{\\{0\\},X\\}$. Thus $M\subset\ker(f_{1})\cap\ker(f_{2})$ and $\ker f_{1}\cap\ker f_{2}\cap N=M$ or $N$. So $\dim(\ker f_{1}\cap\ker f_{2}\cap N)=\infty$. For any $y\in\ker f_{1}\cap\ker f_{2}\cap N$, by Eq.(2.14), one has $\delta(x\otimes(f_{1}+f_{2}))y=\alpha_{x,f_{1}+f_{2}}(y)x+\beta_{x,f_{1}+f_{2}}y,$ $\delta(x\otimes f_{1})y=\alpha_{x,f_{1}}(y)x+\beta_{x,f_{1}}y\quad{\rm and}\quad\delta(x\otimes f_{2})y=\alpha_{x,f_{2}}(y)x+\beta_{x,f_{2}}y.$ Comparing the above three equations, we get $None$ $0=(\alpha_{x,f_{1}+f_{2}}(y)-\alpha_{x,f_{1}}(y)-\alpha_{x,f_{2}}(y))x+(\beta_{x,f_{1}+f_{2}}-\beta_{x,f_{1}}-\beta_{x,f_{2}})y.$ Choosing $y$ linearly independent of $x$ in Eq.(2.15) entails $\beta_{x,f_{1}+f_{2}}=\beta_{x,f_{1}}+\beta_{x,f_{2}}.$ For $x\otimes(sf_{1})$, by Eq.(2.14), we get $\delta(x\otimes sf_{1})y=\alpha_{x,sf_{1}}(y)x+\beta_{x,sf_{1}}y\quad{\rm and}\quad s\delta(x_{1}\otimes f)y=s\alpha_{x,f_{1}}(y)x+s\beta_{x,f_{1}}y.$ Thus $(\alpha_{x,sf_{1}}-s\alpha_{x,f_{1}})(y)x+(\beta_{x,sf_{1}}-s\beta_{x,f_{1}})y=0.$ Since, for $x\in\ker f_{1}\cap\ker f_{2}\cap N$, we can find $y\in\ker f_{1}\cap\ker f_{2}\cap N$ so that $y$ is linearly independent of $x$, it follows that $\beta_{x,sf_{1}}=s\beta_{x,f_{1}}$. Hence $\beta_{x,f}$ is linear in $f$, completing the proof of (1). Now let us prove the conclusion (2). It follows from (1) that, for any $x\otimes f\in{\rm Alg}{\mathcal{N}}$, there exists a functional $\alpha_{x,f}:\ker(f)\rightarrow{\mathbb{F}}$ such that $\delta(x\otimes f)y-\beta_{x,f}y=\alpha_{x,f}(y)x$ for any $y\in\ker(f)$. Similar to the proof of Lemma 2.2(2), we can find linear maps $B:\mathcal{D}_{1}(\mathcal{N})\rightarrow\mathcal{D}_{1}(\mathcal{N})$ and $C:\mathcal{D}_{2}(\mathcal{N})\rightarrow\mathcal{D}_{2}(\mathcal{N})$ such that $\delta(x\otimes f)-\beta_{x,f}I=x\otimes Cf+Bx\otimes f$ holds for all $x\otimes f\in{\mbox{\rm Alg}}\mathcal{N}$. We complete the proof of the lemma. $\Box$ The following Lemmas 2.5-2.6 come from [8]. For the completeness, we give their proofs here. Lemma 2.5. Let $\mathcal{N}$ be a nest on a real or complex Banach space $X$. Suppose that $\delta:\mbox{\rm Alg}{\mathcal{N}}\rightarrow\mbox{\rm Alg}{\mathcal{N}}$ is a linear map. If there exists an injective operator or an operator with dense range $Z\in{\rm Alg}{\mathcal{N}}$ such that $\delta$ is derivable at $Z$, then, $\delta(I)=0$. Proof. Since $\delta$ is derivable at $Z$ and $Z=IZ=ZI$, we have $\delta(Z)=\delta(I)Z+I\delta(Z)=\delta(Z)+Z\delta(I)$. So $\delta(I)Z=Z\delta(I)=0$. If $Z$ is injective, by $Z\delta(I)=0$, we get $\delta(I)=0$; if $Z$ is an operator with dense range, then, by $\delta(I)Z=0$, we get again $\delta(I)=0$. $\Box$ Lemma 2.6. Let $\mathcal{N}$ be a nest on a complex Banach space $X$ and $\delta:\mbox{\rm Alg}{\mathcal{N}}\rightarrow\mbox{\rm Alg}{\mathcal{N}}$ be a linear map derivable at $Z\in{\rm Alg}{\mathcal{N}}$. If $Z$ is an operator with dense range, then (1) for every idempotent operator $P\in\mbox{\rm Alg}{\mathcal{N}},$ we have $\delta(PZ)=\delta(P)Z+P\delta(Z)$, and moreover, $\delta(P)=\delta(P)P+P\delta(P)$; (2) for every operator $N\in\mbox{\rm Alg}{\mathcal{N}}$ with $N^{2}=0$, we have $\delta(NZ)=\delta(N)Z+N\delta(Z)$, and moreover, $\delta(N)N+N\delta(N)=0$. If $Z$ is an injective operator, then ($1^{\prime}$) for every idempotent operator $P\in\mbox{\rm Alg}{\mathcal{N}}$, we have $\delta(ZP)=\delta(Z)P+Z\delta(P)$, and moreover, $\delta(P)=\delta(P)P+P\delta(P)$; ($2^{\prime}$) for every operator $N\in\mbox{\rm Alg}{\mathcal{N}}$ with $N^{2}=0$, we have $\delta(ZN)=\delta(Z)N+Z\delta(N)$, and moreover, $\delta(N)N+N\delta(N)=0$. Proof. Let $P\in\mbox{Alg}{\mathcal{N}}$ be any idempotent operator. If $Z$ is an operator with dense range, then by Lemma 2.5, we have $\begin{array}[]{rl}\delta(Z)=&\delta(I-\frac{1-\sqrt{3}i}{2}P)(I-\frac{1+\sqrt{3}i}{2}P)Z\\\ &+(I-\frac{1-\sqrt{3}i}{2}P)\delta(Z-\frac{1+\sqrt{3}i}{2}PZ)\\\ =&-\frac{1-\sqrt{3}i}{2}\delta(P)Z+\delta(P)PZ+\delta(Z)\\\ &-\frac{1+\sqrt{3}i}{2}\delta(PZ)-\frac{1-\sqrt{3}i}{2}P\delta(Z)+P\delta(PZ)\end{array}$ since $Z=(I-\frac{1-\sqrt{3}i}{2}P)(I-\frac{1+\sqrt{3}i}{2}P)Z$. Thus we get $None$ $-\frac{1-\sqrt{3}i}{2}\delta(P)Z+\delta(P)PZ-\frac{1+\sqrt{3}i}{2}\delta(PZ)-\frac{1-\sqrt{3}i}{2}P\delta(Z)+P\delta(PZ)=0.$ On the other hand, $Z=(I-\frac{1+\sqrt{3}i}{2}P)(I-\frac{1-\sqrt{3}i}{2}P)Z$ gives $None$ $-\frac{1+\sqrt{3}i}{2}\delta(P)Z+\delta(P)PZ-\frac{1-\sqrt{3}i}{2}\delta(PZ)-\frac{1+\sqrt{3}i}{2}P\delta(Z)+P\delta(PZ)=0.$ Combining Eq.(2.16) with Eq.(2.17), we get $\delta(PZ)=\delta(P)Z+P\delta(Z).$ Replacing $\delta(PZ)$ by $\delta(P)Z+P\delta(Z)$ in Eq.(2.16), one obtains $\delta(P)Z=\delta(P)PZ+P\delta(P)Z$. Note that $Z$ is an operator with dense range. It follows that $\delta(P)=\delta(P)P+P\delta(P).$ This completes the proof of assertion (1). If $Z$ is an injective operator, then by the equation $Z=Z(I-\frac{1-\sqrt{3}i}{2}P)(I-\frac{1+\sqrt{3}i}{2}P)=Z(I-\frac{1+\sqrt{3}i}{2}P)(I-\frac{1-\sqrt{3}i}{2}P),$ using a similar argument as above, one can get that $\delta(ZP)=\delta(Z)P+Z\delta(P)$ and $\delta(P)=\delta(P)P+P\delta(P).$ Hence (1′) holds true. For every operator $N\in\mbox{\rm Alg}{\mathcal{N}}$ with $N^{2}=0$, if $Z$ is an operator with dense range, then, noting that $Z=(I-N)(I+N)Z=(I+N)(I-N)Z$, we have $None$ $\delta(N)Z-\delta(N)NZ-\delta(NZ)+N\delta(Z)-N\delta(NZ)=0$ and $-\delta(N)Z-\delta(N)NZ+\delta(NZ)-N\delta(Z)-N\delta(NZ)=0$ since $\delta$ is derivable at $Z$. Comparing the above two equations, one gets $\delta(NZ)=\delta(N)Z+N\delta(Z).$ Replacing $\delta(NZ)$ by $\delta(N)Z+N\delta(Z)$ in Eq.(2.18) and noting that the range of $Z$ is dense, it follows that $\delta(N)N+N\delta(N)=0$. If $Z$ is an injective operator, then by the equation $Z=Z(I-N)(I+N)=Z(I+N)(I-N)$, a similar argument shows that $\delta(ZN)=\delta(Z)N+Z\delta(N)$ and $\delta(P)=\delta(P)P+P\delta(P).$ Hence the assertions (2) and (2′) hold true, completing the proof. $\Box$ By Lemmas 2.5-2.6, Lemmas 2.2-2.4 hold for linear maps between nest algebras on complex Banach space derivable at an injective operator or an operator with dense range. ## 3\. Proof of Theorem 1.1 In this section we complete the proof of Theorem 1.1. Proof of Theorem 1.1. The “if ” part is obvious. We only need to prove the “only if” part. In the following, we always assume that $Z\in{\rm Alg}{\mathcal{N}}$ is an injective operator or an operator with dense range and $\delta:{\rm Alg}{\mathcal{N}}\rightarrow{\rm Alg}{\mathcal{N}}$ is a linear map derivable at $Z$. Then, for any invertible element $A\in{\rm Alg}\mathcal{N}$, since $Z=AA^{-1}Z=ZA^{-1}A$ and $\delta$ is derivable in $Z$, we have $\delta(Z)=\delta(A)A^{-1}Z+A\delta(A^{-1}Z)$ and $\delta(Z)=\delta(ZA^{-1})A+ZA^{-1}\delta(Z),$ which imply respectively that $None$ $\delta(A^{-1}Z)=A^{-1}\delta(Z)-A^{-1}\delta(A)A^{-1}Z$ and $None$ $\delta(ZA^{-1})=\delta(Z)A^{-1}-ZA^{-1}\delta(A)A^{-1}.$ In the sequel, we will prove that $\delta$ is a derivation by considering three cases. Case 1. $X_{-}\not=X.$ In this case, by Lemmas 2.2 and 2.6, there exist linear transformations $B:X\rightarrow X$ and $C:X_{-}^{\perp}\rightarrow X^{\ast}$ such that $\delta(x\otimes f)=Bx\otimes f+x\otimes Cf$ and $\langle Bx,f\rangle+\langle x,Cf\rangle=0$ for all $x\in X$ and $f\in X_{-}^{\perp}$. Thus, by Lemma 2.6 and the linearity of $\delta$, if $Z$ is an operator with dense range, we have $None$ $\delta(x\otimes fZ)=(Bx\otimes f)Z+(x\otimes Cf)Z+(x\otimes f)\delta(Z);$ if $Z$ is an injective operator, we have $None$ $\delta(Zx\otimes f)=\delta(Z)(x\otimes f)+ZBx\otimes f+Zx\otimes Cf.$ Note that, for any $T$ and any $x\otimes f\in{\rm Alg}{\mathcal{N}}$, there exists some $\lambda\in{\mathbb{C}}$ such that $\|\lambda\|>\\|T\\|$ and $\\|(\lambda I-T)^{-1}x\\|\\|f\\|<1$. Then both $\lambda I-T$ and $\lambda I-T-x\otimes f=(\lambda I-T)(I-(\lambda I-T)^{-1}x\otimes f)$ are invertible with their inverses are still in ${\rm Alg}{\mathcal{N}}$. It is obvious that $(I-(\lambda I-T)^{-1}x\otimes f)^{-1}=I+(1-\alpha)^{-1}(\lambda I-T)^{-1}x\otimes f$, where $\alpha=\langle(\lambda I-T)^{-1}x,f\rangle$. Claim 1.1. $\delta$ is a derivation if $Z$ is an operator with dense range. For any $T$ and any $x\otimes f\in{\rm Alg}{\mathcal{N}}$ with $x\in X$ and $f\in X_{-}^{\perp}$, take $\lambda\in{\mathbb{C}}$ such that $\|\lambda\|>\\|T\\|$ and $\\|(\lambda I-T)^{-1}x\\|\\|f\\|<1$. By Lemma 2.2, Eqs.(3.1), (3.3) and the fact $\delta(I)=0$ (Lemma 2.5), we have $\begin{array}[]{rl}\delta(Z)=&\delta(\lambda I-T-x\otimes f)(I+(1-\alpha)^{-1}(\lambda I-T)^{-1}x\otimes f)(\lambda I-T)^{-1}Z\\\ &+(\lambda I-T-x\otimes f)\delta((I+(1-\alpha)^{-1}(\lambda I-T)^{-1}x\otimes f)(\lambda I-T)^{-1}Z)\\\ =&[-\delta(T)-Bx\otimes f-x\otimes Cf][(\lambda I-T)^{-1}Z\\\ &+(1-\alpha)^{-1}(\lambda I-T)^{-1}(x\otimes f)(\lambda I-T)^{-1}Z]\\\ &+(\lambda I-T-x\otimes f)[(\lambda I-T)^{-1}\delta(Z)+(\lambda I-T)^{-1}\delta(T)(\lambda I-T)^{-1}Z\\\ &+(1-\alpha)^{-1}B(\lambda I-T)^{-1}(x\otimes f)(\lambda I-T)^{-1}Z\\\ &+(1-\alpha)^{-1}(\lambda I-T)^{-1}x\otimes C((\lambda I-T^{})^{-1}f)Z\\\ &+(1-\alpha)^{-1}(\lambda I-T)^{-1}x\otimes f(\lambda I-T)^{-1}\delta(Z)]\\\ =&\delta(Z)-(1-\alpha)^{-1}B(x\otimes(\lambda I-T^{})^{-1}f)Z\\\ &-(1-\alpha)^{-1}\delta(T)(\lambda I-T)^{-1}(x\otimes(\lambda I-T^{})^{-1}f)Z\\\ &-(x\otimes(\lambda I-T^{})^{-1}Cf)Z+(x\otimes C(\lambda I-T^{})^{-1}f)Z\\\ &+(1-\alpha)^{-1}(\lambda I-T)B(\lambda I-T)^{-1}(x\otimes(\lambda I-T^{})^{-1}f)Z\\\ &-(x\otimes(\lambda I-T^{})^{-1}\delta(T)^{}(\lambda I-T^{})^{-1}f)Z\\\ &-(1-\alpha)^{-1}(\langle(\lambda I-T)^{-1}x,Cf\rangle+\langle B(\lambda I-T)^{-1}x,f\rangle)(x\otimes(\lambda I-T^{})^{-1}f)Z.\end{array}$ As $\langle(\lambda I-T)^{-1}x,Cf\rangle+\langle B(\lambda I-T)^{-1}x,f\rangle=0$, the above equation becomes $\begin{array}[]{rl}0=&(1-\alpha)^{-1}B(x\otimes(\lambda I-T^{})^{-1}f)Z+(1-\alpha)^{-1}\delta(T)(\lambda I-T)^{-1}(x\otimes(\lambda I-T^{})^{-1}f)Z\\\ &+(x\otimes(\lambda I-T^{})^{-1}Cf)Z-(x\otimes C(\lambda I-T^{})^{-1}f)Z\\\ &-(1-\alpha)^{-1}(\lambda I-T)B(\lambda I-T)^{-1}(x\otimes(\lambda I-T^{})^{-1}f)Z\\\ &+(x\otimes(\lambda I-T^{})^{-1}\delta(T)^{}(\lambda I-T^{})^{-1}f)Z.\end{array}$ Since the range of $Z$ is dense, it follows that $\begin{array}[]{rl}0=&(1-\alpha)^{-1}Bx\otimes(\lambda I-T^{})^{-1}f+(1-\alpha)^{-1}\delta(T)(\lambda I-T)^{-1}x\otimes(\lambda I-T^{})^{-1}f\\\ &+x\otimes(\lambda I-T^{})^{-1}Cf-x\otimes C(\lambda I-T^{})^{-1}f\\\ &-(1-\alpha)^{-1}(\lambda I-T)B(\lambda I-T)^{-1}x\otimes(\lambda I-T^{})^{-1}f\\\ &+x\otimes(\lambda I-T^{})^{-1}\delta(T)^{}(\lambda I-T^{})^{-1}f,\end{array}$ and so $\begin{array}[]{rl}&[\delta(T)(\lambda I-T)^{-1}-(\lambda I-T)B(\lambda I-T)^{-1}+B]x\otimes(\lambda I-T^{})^{-1}f\\\ =&x\otimes(1-\alpha)[C(\lambda I-T^{})^{-1}-(\lambda I-T^{})^{-1}C-(\lambda I-T^{})^{-1}\delta(T)^{}(\lambda I-T^{})^{-1}]f.\end{array}$ Hence $[\delta(T)(\lambda I-T)^{-1}-(\lambda I-T)B(\lambda I-T)^{-1}+B]x$ is linearly dependent of $x$ for every $x\in X$. This entails that there is a scalar $\beta$ such that $\delta(T)(\lambda I-T)^{-1}-(\lambda I-T)B(\lambda I-T)^{-1}+B=\beta I$ on $X$. It follows that $\delta(T)=BT-TB+\beta(\lambda I-T).$ By taking different $\lambda$ in the equation, we see that $\beta=0$ and consequently $\delta(T)=BT-TB$ holds for all $T\in{\rm Alg}{\mathcal{N}}$, that is, $\delta$ is a derivation. Claim 1.2. $\delta$ is a derivation if $Z$ is an injective operator. For any $T$ and any $x\otimes f\in{\rm Alg}{\mathcal{N}}$ with $x\in X$ and $f\in X_{-}^{\perp}$, take $\lambda\in{\mathbb{C}}$ such that $\|\lambda\|>\\|T\\|$ and $\\|(\lambda I-T)^{-1}x\\|\\|f\\|<1$. Note that $(\lambda I-T^{})^{-1}f\in X_{-}^{\perp}$ and $Z=Z(I+(1-\alpha)^{-1}(\lambda I-T)^{-1}x\otimes f)(\lambda I-T)^{-1}(\lambda I-T-x\otimes f)$. Then, by Lemma 2.2, Eqs.(3.2), (3.4) and the fact $\delta(I)=0$, we have $\begin{array}[]{rl}\delta(Z)=&\delta(Z(\lambda I-T)^{-1}+(1-\alpha)^{-1}Z(\lambda I-T)^{-1}x\otimes f(\lambda I-T)^{-1})(\lambda I-T-x\otimes f)\\\ &+(Z(\lambda I-T)^{-1}+(1-\alpha)^{-1}Z(\lambda I-T)^{-1}x\otimes f(\lambda I-T)^{-1})\delta(\lambda I-T-x\otimes f)\\\ =&[\delta(Z)(\lambda I-T)^{-1}+Z(\lambda I-T)^{-1}\delta(T)(\lambda I-T)^{-1}\\\ &+(1-\alpha)^{-1}\delta(Z)(\lambda I-T)^{-1}(x\otimes f)(\lambda I-T)^{-1}\\\ &+(1-\alpha)^{-1}ZB(\lambda I-T)^{-1}(x\otimes f)(\lambda I-T)^{-1}\\\ &+(1-\alpha)^{-1}Z(\lambda I-T)^{-1}(x\otimes C(\lambda I-T^{})^{-1}f)][(\lambda I-T)-x\otimes f]\\\ &+[Z(\lambda I-T)^{-1}\\\ &+(1-\alpha)^{-1}Z(\lambda I-T)^{-1}x\otimes f(\lambda I-T)^{-1}][-\delta(T)-Bx\otimes f-x\otimes Cf]\\\ =&\delta(Z)-Z(\lambda I-T)^{-1}\delta(T)(\lambda I-T)^{-1}x\otimes f\\\ &+(1-\alpha)^{-1}Z(\lambda I-T)^{-1}(x\otimes(\lambda I-T^{})C(\lambda I-T^{})^{-1}f)\\\ &+ZB(\lambda I-T)^{-1}x\otimes f-Z(\lambda I-T)^{-1}Bx\otimes f\\\ &-(1-\alpha)^{-1}Z(\lambda I-T)^{-1}x\otimes\delta(T)^{}(\lambda I-T^{})^{-1}f-(1-\alpha)^{-1}Z(\lambda I-T)^{-1}x\otimes Cf.\end{array}$ It follows that $\begin{array}[]{rl}0=&Z(\lambda I-T)^{-1}\delta(T)(\lambda I-T)^{-1}x\otimes f\\\ &-(1-\alpha)^{-1}Z(\lambda I-T)^{-1}x\otimes(\lambda I-T^{})C(\lambda I-T^{})^{-1}f\\\ &-ZB(\lambda I-T)^{-1}x\otimes f+Z(\lambda I-T)^{-1}Bx\otimes f\\\ &+(1-\alpha)^{-1}Z(\lambda I-T)^{-1}x\otimes\delta(T)^{}(\lambda I-T^{})^{-1}f+(1-\alpha)^{-1}Z(\lambda I-T)^{-1}x\otimes Cf.\end{array}$ Since $\ker Z=\\{0\\}$, we get $\begin{array}[]{rl}0=&(\lambda I-T)^{-1}\delta(T)(\lambda I-T)^{-1}x\otimes f\\\ &-(1-\alpha)^{-1}(\lambda I-T)^{-1}x\otimes(\lambda I-T^{})C(\lambda I-T^{})^{-1}f\\\ &-B(\lambda I-T)^{-1}x\otimes f+(\lambda I-T)^{-1}Bx\otimes f\\\ &+(1-\alpha)^{-1}(\lambda I-T)^{-1}x\otimes\delta(T)^{}(\lambda I-T^{})^{-1}f+(1-\alpha)^{-1}(\lambda I-T)^{-1}x\otimes Cf.\end{array}$ Multiplying the above equation by $(\lambda I-T)$ from the left, one has $\begin{array}[]{rl}0=&\delta(T)(\lambda I-T)^{-1}x\otimes f-(1-\alpha)^{-1}x\otimes(\lambda I-T^{})C(\lambda I-T^{})^{-1}f\\\ &-(\lambda I-T)B(\lambda I-T)^{-1}x\otimes f+Bx\otimes f\\\ &+(1-\alpha)^{-1}x\otimes\delta(T)^{}(\lambda I-T^{})^{-1}f+(1-\alpha)^{-1}x\otimes Cf.\end{array}$ That is, $\begin{array}[]{rl}&[\delta(T)(\lambda I-T)^{-1}-(\lambda I-T)B(\lambda I-T)^{-1}+B]x\otimes f\\\ =&(1-\alpha)^{-1}x\otimes[(\lambda I-T^{})C(\lambda I-T^{})^{-1}-\delta(T)^{}(\lambda I-T^{})^{-1}-C]f.\end{array}$ Now using the same argument as in the proof of Claim 1.1, we see that $\delta$ is a derivation. Case 2. $\\{0\\}\not=\\{0\\}_{+}$. In this case, one can use Lemmas 2.3, 2.5-2.6 and a similar argument of Case 1 to check that $\delta$ is a derivation, and we omit the details here. We remark that, the trivial case, that is, Alg${\mathcal{N}}={\mathcal{B}}(X)$, is included in both Case 1 and Case 2. Finally, let us consider the case that both end points of the nest are limit points. Case 3. $\\{0\\}=\\{0\\}_{+}$ and $X_{-}=X$. In this case $X$ is infinite dimensional and every nonzero element $N$ in $\mathcal{N}$ is infinite dimensional. By Lemmas 2.4 and 2.6, there exists a bilinear functional $\beta:({\mathcal{D}}_{1}({\mathcal{N}})\times{\mathcal{D}}_{2}({\mathcal{N}}))\cap{\rm Alg}{\mathcal{N}}\rightarrow\mathbb{C}$, linear transformations $B:\mathcal{D}_{1}(\mathcal{N})\rightarrow\mathcal{D}_{1}(\mathcal{N})$ and $C:\mathcal{D}_{2}(\mathcal{N})\rightarrow\mathcal{D}_{2}(\mathcal{N})$ such that $(\delta(x\otimes f)-\beta_{x,f}I)\ker(f)\subseteq\mbox{\rm span}\\{x\\}$ and $None$ $\delta(x\otimes f)-\beta_{x,f}I=x\otimes Cf+Bx\otimes f$ hold for all $x\otimes f\in{\mbox{\rm Alg}}\mathcal{N}$. Then, by Lemma 2.6 and the linearity of $\delta$, if $Z$ is an operator with dense range, we have $None$ $\delta(x\otimes fZ)=(Bx\otimes f)Z+(x\otimes Cf)+\beta_{x,f}Z+(x\otimes f)\delta(Z);$ if $Z$ is an injective operator, we have $None$ $\delta(Zx\otimes f)=\delta(Z)(x\otimes f)+ZBx\otimes f+Zx\otimes Cf+\beta_{x,f}Z.$ Now let $\tilde{\beta}_{x,f}=\langle x,Cf\rangle+\langle Bx,f\rangle$. We claim that $None$ $\left\\{\begin{array}[]{ll}\tilde{\beta}_{x,f}=0,&\mbox{\rm if}\ \langle x,f\rangle\not=0;\\\ \tilde{\beta}_{x,f}=-2\beta_{x,f},&\mbox{\rm if}\ \langle x,f\rangle=0.\end{array}\right.$ In fact, if $\langle x,f\rangle\not=0$, then, by Lemma 2.6(1) and (1′), $(x\otimes f)\delta(x\otimes f)(x\otimes f)=0$, and hence $\langle x,Cf\rangle+\langle Bx,f\rangle=-\beta_{x,f}=0$; if $\langle x,f\rangle=0$, then, by Lemma 2.6(2) and (2′), $\delta(x\otimes f)(x\otimes f)+(x\otimes f)\delta(x\otimes f)=0$, which, together with Eq.(3.5), implies that $\langle x,Cf\rangle+\langle Bx,f\rangle=-2\beta_{x,f}$. For any $T$ and $x\otimes f\in{\rm Alg}{\mathcal{N}}$, take $\lambda$ such that $\|\lambda\|>\\|T\\|$ and $\\|(\lambda I-T)^{-1}x\\|\\|f\\|<1$. Note that $I=(\lambda I-T-x\otimes f)(I+(1-\alpha)^{-1}(\lambda I-T)^{-1}x\otimes f)(\lambda I-T)^{-1},$ where $\alpha=\langle(\lambda I-T)^{-1}x,f\rangle$. We prove that $\delta$ is a derivation respectively by assume that $Z$ is injective or of dense range. Claim 3.1. $\delta$ is a derivation if $Z$ has dense range. Since $\delta$ is derivable at $Z$ of dense range, by Lemma 2.5, Eqs.(3.1) and (3.3), we have $\begin{array}[]{rl}\delta(Z)=&\delta(\lambda I-T-x\otimes f)[I+(1-\alpha)^{-1}(\lambda I-T)^{-1}x\otimes f](\lambda I-T)^{-1}Z\\\ &+(\lambda I-T-x\otimes f)\delta((I+(1-\alpha)^{-1}(\lambda I-T)^{-1}x\otimes f)(\lambda I-T)^{-1}Z)\\\ =&[-\delta(T)-Bx\otimes f-x\otimes Cf-\beta_{x,f}I][(\lambda I-T)^{-1}Z\\\ &+(1-\alpha)^{-1}(\lambda I-T)^{-1}(x\otimes f)(\lambda I-T)^{-1}Z]\\\ &+[\lambda I-T-x\otimes f][(\lambda I-T)^{-1}\delta(Z)+(\lambda I-T)^{-1}\delta(T)(\lambda I-T)^{-1}Z\\\ &+(1-\alpha)^{-1}(B(\lambda I-T)^{-1}(x\otimes f)(\lambda I-T)^{-1}Z\\\ &+(\lambda I-T)^{-1}x\otimes((\lambda I-T)^{-1})^{}f\delta(Z)+\beta_{(\lambda I-T)^{-1}x,((\lambda I-T)^{-1})^{}f}Z)].\end{array}$ Note that $(1-\alpha)^{-1}\alpha=(1-\alpha)^{-1}-1$ and $\tilde{\beta}_{x,f}=\langle x,Cf\rangle+\langle Bx,f\rangle$. It follows that $\begin{array}[]{rl}&\beta_{x,f}(\lambda I-T)^{-1}Z-(1-\alpha)^{-1}\beta_{(\lambda I-T)^{-1}x,(\lambda I-T^{})^{-1}f}(\lambda I-T)Z\\\ =&(1-\alpha)^{-1}\delta(\lambda I-T)((\lambda I-T)^{-1}x\otimes(\lambda I-T^{})^{-1}f)Z\\\ &-(1-\alpha)^{-1}(Bx\otimes(\lambda I-T^{})^{-1}f)Z-(x\otimes(\lambda I-T^{})^{-1}Cf)Z\\\ &-(1-\alpha)^{-1}\tilde{\beta}_{(\lambda I-T)^{-1}x,f}(x\otimes(\lambda I-T^{})^{-1}f)Z\\\ &-(1-\alpha)^{-1}\beta_{x,f}((\lambda I-T)^{-1}x\otimes(\lambda I-T^{})^{-1}f)Z\\\ &+(x\otimes f(\lambda I-T)^{-1}\delta(\lambda I-T)(\lambda I-T)^{-1})Z+(x\otimes C(\lambda I-T^{})^{-1}f)Z\\\ &+(1-\alpha)^{-1}((\lambda I-T)B(\lambda I-T)^{-1}x\otimes(\lambda I-T^{})^{-1}f)Z\\\ &-(1-\alpha)^{-1}\beta_{(\lambda I-T)^{-1}x,(\lambda I-T^{})^{-1}f}(x\otimes f)Z.\end{array}$ Since the range of $Z$ is dense, the above equation implies $None$ $\begin{array}[]{rl}&\beta_{x,f}(\lambda I-T)^{-1}-(1-\alpha)^{-1}\beta_{(\lambda I-T)^{-1}x,(\lambda I-T^{})^{-1}f}(\lambda I-T)\\\ =&(1-\alpha)^{-1}\delta(\lambda I-T)(\lambda I-T)^{-1}x\otimes(\lambda I-T^{})^{-1}f\\\ &-(1-\alpha)^{-1}Bx\otimes(\lambda I-T^{})^{-1}f-x\otimes(\lambda I-T^{})^{-1}Cf\\\ &-(1-\alpha)^{-1}\tilde{\beta}_{(\lambda I-T)^{-1}x,f}x\otimes(\lambda I-T^{})^{-1}f\\\ &-(1-\alpha)^{-1}\beta_{x,f}(\lambda I-T)^{-1}x\otimes(\lambda I-T^{})^{-1}f\\\ &+x\otimes f(\lambda I-T)^{-1}\delta(\lambda I-T)(\lambda I-T)^{-1}+x\otimes C(\lambda I-T^{})^{-1}f\\\ &+(1-\alpha)^{-1}(\lambda I-T)B(\lambda I-T)^{-1}x\otimes(\lambda I-T^{})^{-1}f\\\ &-(1-\alpha)^{-1}\beta_{(\lambda I-T)^{-1}x,(\lambda I-T^{})^{-1}f}(x\otimes f).\end{array}$ Next we show that $\beta_{x,f}=0$ for any $x\otimes f\in{\rm Alg}{\mathcal{N}}$. If $\langle x,f\rangle\neq 0$, it is obvious that $\beta_{x,f}=0$. For the case $\langle x,f\rangle=0$, to prove $\beta_{x,f}=0$, we simplify Eq.(3.9) by letting $W=(\lambda I-T)^{-1}$. Then $\alpha=\langle Wx,f\rangle$ and Eq.(3.9) becomes $None$ $\begin{array}[]{rl}&(1-\langle Wx,f\rangle)(\beta_{x,f}W+x\otimes W^{}Cf-x\otimes fW\delta(W^{-1})W-x\otimes CW^{}f)\\\ =&\beta_{Wx,W^{}f}W^{-1}+\delta(W^{-1})Wx\otimes W^{}f-Bx\otimes W^{}f-\tilde{\beta}_{Wx,f}x\otimes W^{}f\\\ &-\beta_{x,f}Wx\otimes W^{}f+W^{-1}BW^{-1}x\otimes W^{}f-\beta_{Wx,W^{}f}x\otimes f.\end{array}$ Let $t$ be a real or complex number with $t\neq 0,1$. Replacing $x$ by $tx$ in Eq.(3.10) and using the bilinearity of $\beta$, we get $None$ $\begin{array}[]{rl}&t(1-t\langle Wx,f\rangle)(\beta_{x,f}W+x\otimes W^{}Cf-x\otimes fW\delta(W^{-1})W-x\otimes CW^{}f)\\\ =&t\beta_{Wx,W^{}f}W^{-1}+t\delta(W^{-1})Wx\otimes W^{}f-tBx\otimes W^{}f-t^{2}\tilde{\beta}_{Wx,f}x\otimes W^{}f\\\ &-t^{2}\beta_{x,f}Wx\otimes W^{}f+tW^{-1}BW^{-1}x\otimes W^{}f-t^{2}\beta_{Wx,W^{}f}x\otimes f.\end{array}$ Comparing Eq.(3.10) and Eq.(3.11) gives $None$ $\begin{array}[]{rl}\langle Wx,f\rangle\beta_{x,f}W=&-\langle Wx,f\rangle x\otimes W^{}Cf+\langle Wx,f\rangle x\otimes fW\delta(W^{-1})W\\\ &+\langle Wx,f\rangle x\otimes CW^{}f+\tilde{\beta}_{Wx,f}x\otimes W^{}f\\\ &+\beta_{x,f}Wx\otimes W^{}f-\beta_{Wx,W^{}f}x\otimes f.\end{array}$ Note that, the right side of Eq.(3.12) is a finite rank operator. Thus, we must have $\langle Wx,f\rangle\beta_{x,f}=0$. If $\langle Wx,f\rangle\neq 0$, then $\beta_{x,f}=0$; if $\langle Wx,f\rangle=0$, then Eq.(3.12) leads to $None$ $0=-2{\beta}_{Wx,f}x\otimes W^{}f+\beta_{x,f}Wx\otimes W^{}f-\beta_{Wx,W^{}f}x\otimes f.$ Assume on the contrary that $\beta_{x,f}\neq 0$. Then Eq.(3.13) gives $Wx\otimes W^{}f=x\otimes(\frac{2\beta_{Wx,f}}{\beta_{x,f}}W^{}f+\frac{\beta_{Wx,W^{}f}}{\beta_{x,f}}f),$ which implies that, for any $x\otimes f\in$Alg$\mathcal{N}$ with $x\in\ker f$, there exists a scalar $t_{x,f}\not=0$ such that $Wx=t_{x,f}x$. It follows that $Tx=\xi_{x,f,\lambda}x$ for some scalar $\xi_{x,f,\lambda}$. This implies that $x$ is an eigenvector for every $T\in$Alg$\mathcal{N}$, which is imposable. Hence we must have $\beta_{x,f}=0$ for all $x\otimes f\in{\rm Alg}{\mathcal{N}}.$ Then by Eq.(3.5), we have $\delta(x\otimes f)=x\otimes Cf+Bx\otimes f\quad{\rm holds\ for\ all}\quad x\otimes f\in{\rm Alg}{\mathcal{N}}.$ Now, for any $T,x\otimes f\in$Alg$\mathcal{N}$, by Eq.(3.9), we get $\begin{array}[]{rl}0=&(1-\alpha)^{-1}\delta(\lambda I-T)(\lambda I-T)^{-1}x\otimes(\lambda I-T^{})^{-1}f\\\ &-(1-\alpha)^{-1}Bx\otimes(\lambda I-T^{})^{-1}f-x\otimes(\lambda I-T^{})^{-1}Cf\\\ &+x\otimes f(\lambda I-T)^{-1}\delta(\lambda I-T)(\lambda I-T)^{-1}+x\otimes C(\lambda I-T^{})^{-1}f\\\ &+(1-\alpha)^{-1}(\lambda I-T)B(\lambda I-T)^{-1}x\otimes(\lambda I-T^{})^{-1}f,\end{array}$ and hence $\begin{array}[]{rl}0=&(1-\alpha)^{-1}(-\delta(T)(\lambda I-T)^{-1}-B+(\lambda I-T)B(\lambda I-T)^{-1})x\otimes(\lambda I-T^{})^{-1}f\\\ &+x\otimes((\lambda I-T^{})^{-1}C-(\lambda I-T^{})^{-1}\delta(T)^{}(\lambda I-T^{})^{-1}+C(\lambda I-T^{})^{-1})f.\end{array}$ This implies that $[-\delta(T)(\lambda I-T)^{-1}-B+(\lambda I-T)B(\lambda I-T)^{-1}]x\in{\rm span}\\{x\\}$ for each $x\in\mathcal{D}_{1}(\mathcal{N})$. So there is a scalar $p_{\lambda}$ such that $\delta(T)(\lambda I-T)^{-1}+B-(\lambda I-T)B(\lambda I-T)^{-1}=p_{\lambda}I$ on $\mathcal{D}_{1}(\mathcal{N})$. It follows that $\delta(T)=BT- TB+p_{\lambda}(\lambda I-T)$ on $\mathcal{D}_{1}(\mathcal{N})$. By taking different $\lambda$, we see that $p_{\lambda}=0$, and consequently, $None$ $\delta(T)\|_{\mathcal{D}_{1}(\mathcal{N})}=BT\|_{\mathcal{D}_{1}(\mathcal{N})}-TB\quad{\rm holds\ for\ all}\quad T\in{\rm Alg}{\mathcal{N}}.$ Now for any $T,S\in{\rm Alg}{\mathcal{N}}$, by Eq.(3.14), we have $\delta(TS)\|_{\mathcal{D}_{1}(\mathcal{N})}=BTS\|_{\mathcal{D}_{1}(\mathcal{N})}-TSB=BT\|_{\mathcal{D}_{1}(\mathcal{N})}S\|_{\mathcal{D}_{1}(\mathcal{N})}-TSB$ and $\begin{array}[]{rl}(\delta(T)S+T\delta(S))\|_{\mathcal{D}_{1}(\mathcal{N})}=&(BT\|_{\mathcal{D}_{1}(\mathcal{N})}-TB)S\|_{\mathcal{D}_{1}(\mathcal{N})}+T(BS\|_{\mathcal{D}_{1}(\mathcal{N})}-SB)\\\ =&BT\|_{\mathcal{D}_{1}(\mathcal{N})}S\|_{\mathcal{D}_{1}(\mathcal{N})}-TSB.\end{array}$ Comparing the above two equations gives $\delta(TS)\|_{\mathcal{D}_{1}(\mathcal{N})}=(\delta(T)S+T\delta(S))\|_{\mathcal{D}_{1}(\mathcal{N})}$ holds for all $T,S\in{\rm Alg}{\mathcal{N}}$. Thus $\delta(TS)=\delta(T)S+T\delta(S)$ holds for all $T,S\in{\rm Alg}{\mathcal{N}}$ since ${\mathcal{D}_{1}}$ is dense in $X$. Therefore, $\delta$ is a derivation. Claim 3.2. $\delta$ is a derivation if $Z$ is an injective operator. In this case, by Lemma 2.5, Eqs.(3.2), (3.4) and the equation $Z=[Z(I+(1-\alpha)^{-1}(\lambda I-T)^{-1}x\otimes f)(\lambda I-T)^{-1}](\lambda I-T-x\otimes f),$ using a similar argument to that of Claim 3.1, one can show that $\delta$ is a derivation. The proof of Theorem 1.1 is completed. $\Box$ ## References [1] R. An, J. Hou, Characterizations of derivations on triangular ring: Additive maps derivable at idempotents, Linear Algebra and Applications, 431 (2009), 1070-1080. * [2] R. L. Crist, Local derivations on operator algebras, J. Func. Anal., 135 (1996), 76-92. * [3] K. R. Davision, Nest Algebras, Pitman Research Notes in Mathematics Series, vol. 191, Longman Scientific and Technical, Burnt mill Harlow, Essex, UK. 1988. * [4] J. C. Hou, X. F. Qi, Additive maps derivable at some point on $\mathcal{J}$-subspace lattice algebras, Linear Algebra and Applications, 429 (2008), 1851-1863. * [5] R. V. Kadison, Local derivations, J. Algebra, 130 (1990), 494-509. * [6] D. R. Larson, A. R. Sourour, Local derivations and local automorphisms of ${\mathcal{B}}(X)$, Proceedings Symposia in Pure Mathematics, 51 (1990), 187-194. * [7] J. Li, Z. Pan, On derivable mappings, J. Math. Anal. Appl., 374 (2011), 311-322. * [8] X. F. Qi, J. C. Hou, Characterizations of derivations of Banach space nest algebras: All-derivable points, Linear Algebra and Applications, 432 (2010), 3183-3200. * [9] X. F. Qi, J. C. Hou, Full-derivable points of $\mathcal{J}$-subspace lattice algebras, Rocky Mountain J. Math., to appear. * [10] P. Šemrl, Local automorphisms and derivations on ${\mathcal{B}}(H)$, Proc. Amer. Math. Soc., 125 (1997), 2667-2680. * [11] M. Spivac, Derivations and nest algebras on Banach spaces, Israel. J. Math., 50(2) (1985), 193-200. * [12] J. Zhou, Linear mappings derivable at some nontrivial elements, Linear Algebra Appl., 435 (2011), 1972-1986. * [13] J. Zhu, C. Xiong, Derivable mappings at unit operator on nest algebras, Lin. Alg. Appl., 422 (2007), 721-735. * [14] J. Zhu, S. Zhao, Characterization of all-derivable points in nest algebras, Proc. Amer. Math. Soc., 141 (2013), 2343-2350.	arxiv-papers	2013-11-21T00:23:59	2024-09-04T02:49:54.044000	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Yanfang Zhang, Jinchuan Hou, Xiaofei Qi", "submitter": "Jinchuan Hou", "url": "https://arxiv.org/abs/1311.5276" }
1311.5329	Remarks on the energy release rate for an antiplane moving crack in couple stress elasticity L. Morini(1)**Corresponding author. Tel.: +39 0461 282583, email address: [email protected]., A. Piccolroaz(1) and G. Mishuris(2) (1)_Department of Civil, Environmental and Mechanical Engineering, University of Trento,_ _Via Mesiano 77, 38123, Trento, Italy._ (2)_Institute of Mathematical and Physical Sciences, Aberystwyth University,_ _Ceredigion SY23 3BZ, Wales, U.K._ ###### Abstract This paper is concerned with the steady-state propagation of an antiplane semi-infinite crack in couple stress elastic materials. A distributed loading applied at the crack faces and moving with the same velocity of the crack tip is considered, and the influence of the loading profile variations and microstructural effects on the dynamic energy release rate is investigated. The behaviour of both energy release rate and maximum total shear stress when the crack tip speed approaches the critical speed (either that of the shear waves or that of the localised surface waves) is studied. The limit case corresponding to vanishing characteristic scale lengths is addressed both numerically and analytically by means of a comparison with classical elasticity results. _Keywords:_ Couple stress elasticity, Energy release rate, Couple stress surface waves, Shielding effects, Weakening effects. ## 1 Introduction Influence of the microstructure on the mechanical behaviour of brittle materials such as ceramics, composites, cellular materials, foams, masonry, bones tissues, glassy and semicrystalline polymers, has been detected in many experimental analyses (Park and Lakes, 1986; Lakes, 1993; Waseem et al., 2013; Beverige et al., 2013). In particular, relevant size effects have been found when the representative scale of the deformation field becomes comparable to the length scale of the microstructure (Lakes, 1986, 1995). These size effects influence strongly the macroscopic fracture toughness of the materials (Rice et al., 1980, 1981), and cannot be predicted by classical elasticity theory. In order to describe accurately these phenomena, generalized theories of continuum mechanics involving characteristic lengths, such as micropolar elasticity (Cosserat and Cosserat, 1909), indeterminate couple stress elasticity (Koiter, 1964) and strain gradient theories (Mindlin and Eshel, 1968; Fleck and Hutchinson, 2001; Dal Corso and Willis, 2011), have been developed and used in many experimental and theoretical studies (Radi and Gei, 2004; Itou, 2013a, b). Indeterminate couple stress elasticity theory developed by Koiter (1964) provides two distinct characteristic length scales for bending and torsion. Moreover, it includes the effects of the rotational inertia, which can be considered as an additional dynamic length scale. Full-field solution for steady-state propagating semi-infinite Mode III crack under distributed loading has been obtained by means of Fourier transform and Wiener-Hopf analytic continuation technique by Mishuris et al. (2013). A general expression for the dynamic energy release rate (ERR) corresponding to the same steady-state antiplane problem has been derived in Morini et al. (2013), and the stability of the propagation has been analyzed by means of both maximum total shear stress (Georgiadis, 2003; Radi, 2008) and energy-based Griffith criterion (Willis, 1971). In order to investigate how the variation of the applied loading can affect both energy release rate and maximum total shear stress, in this paper the solution derived in Mishuris et al. (2013) is extended considering different distributions for the loading acting on the crack faces and moving with the same velocity as that of the crack tip. In particular, the behaviour of the energy release rate in the limiting cases when the crack tip speed approaches the shear waves speed or alternatively the Rayleigh-type surface waves speed and when the characteristic scale lengths of the material vanish is studied assuming various amplitudes for the loading profile. The paper starts with a short description of the problem of a semi-infinite Mode III crack steadily propagating in couple stress elastic materials in Section 2, followed by an overview of results concerning the dispersive propagation of antiplane surface waves. For both antiplane and in-plane problems, indeterminate couple stress theory predicts the existence of surface waves analogous to Rayleigh waves observed in plane classical elasticity (Ottosen et al., 2000). In the paper, these are referred to as couple stress surface waves, and it is demonstrated that the critical maximum value for the crack tip speed introduced in Mishuris et al. (2013) and Morini et al. (2013) coincides with the minimum velocity for couple stress surface waves propagation in the material. A velocity range for the crack propagation, denominated for brevity sub-Rayleigh regime, is introduced: in cases where subsonic couple stress surface waves propagation is detected, a maximum crack tip velocity smaller than shear waves speed in classical elastic materials $c_{s}$ is defined and explicitly evaluated as a function of the microstructural parameters, while in cases where the surface waves propagation can be only supersonic the limit value for the crack tip speed is given by $c_{s}$. The analytical full-field solution of the problem is then addressed in Section 3 using Wiener-Hopf technique (Noble, 1958). The crack is assumed to propagate in the sub-Rayleigh regime under generalized distributed loading conditions of variable amplitude. In Section 4, the dynamic energy release rate is evaluated explicitly by means of the method developed by Freund (1972) and extended by Georgiadis (2003), Morini et al. (2013) and Gourgiotis and Piccolroaz (2013) to static and dynamic problems in couple stress elasticity. The effects of the microstructure as well as the influence of the loading profile gradients on displacements, stress fields, maximum total shear stress and energy release rate are illustrated and discussed by means of several numerical examples in Section 5. A strong localization of the applied loading around a maximum near to the crack tip is not associated with to higher levels of the shear traction and to a larger crack opening. This behaviour, detected by maximum total shear stress analysis, means that in couple stress elastic materials the action of loading forces concentrated near to the crack tip is shielded by the microstructure. This shielding effect is confirmed also by the energy release rate analysis. It is shown indeed that the energy release rate decreases as the applied loading is more and more localized near the crack tip. The behaviour of the energy release rate shows that if the distance between the position of application of the maximum loading and the crack tip grows, in presence of couple stress more energy is provided for propagating the crack at constant speed with respect to the classical elastic case, and then the fracture propagation is favored. Also this weakening effect is due to the microstructural contributions, and it is in agreement with the results detected in Gourgiotis et al. (2011) for plane strain crack problems under concentrated shear loading. Numerical results illustrate also that, when the crack tip speed approaches the shear waves speed in classical elastic materials or alternatively the couple stress surface waves speed, the energy release rate assumes a finite limit value depending on the microstructural parameters. Conversely, if the characteristic lengths vanish, for any arbitrary loading profile the value of the energy release rate becomes identical to that of the classical elastic case. This is an important proof of the fact that, if the microstructural effects are negligible, the material behaviour is identical to that of a classical elastic body for what concerns crack propagation. This result, observed in all the proposed numerical examples, is validated by means of the analytical evaluation of the limit of the energy release rate for vanishing characteristic lengths reported in Section 6. In this Section, indeed, it is demonstrated that, if the characteristic lengths vanish, for any arbitrary applied loading the energy release rate for couple stress materials tends to the energy release rate associated to an antiplane steady-state crack in classical elasticity. ## 2 Problem formulation A Cartesian coordinate system $(0,x_{1},x_{2},x_{3})$ centered at the crack- tip at time $t=0$ is assumed. The micropolar behaviour of the material is described by the indeterminate theory of couple stress elasticity (Koiter, 1964). The non-symmetric Cauchy stress tensor $t$ can be decomposed into a symmetric part $\sigma$ and a skew-symmetric part $\tau$, namely $\mbox{\boldmath$t$} = \mbox{\boldmath$\sigma$} +\mbox{\boldmath$\tau$}$. The reduced tractions vector $p$ and couple stress tractions vector $q$ are defined as $\mbox{\boldmath$p$}=\mbox{\boldmath$t$}^{T}\mbox{\boldmath$n$}+\frac{1}{2}\nabla\mu_{nn}\times\mbox{\boldmath$n$},\quad\mbox{\boldmath$q$}=\mbox{\boldmath$\mu$}^{T}\mbox{\boldmath$n$}-\mu_{nn}\mbox{\boldmath$n$},$ (1) where $\mu$ is the couple stress tensor, $n$ denotes the outward unit normal and $\mu_{nn}=\mbox{\boldmath$n$}\cdot\mbox{\boldmath$\mu$}\mbox{\boldmath$n$}$. For the dynamic antiplane problem, stresses and couple stresses can be expressed in terms of the out-of plane displacement $u_{3}$: $\sigma_{13}=G\frac{\partial u_{3}}{\partial x_{1}},\quad\sigma_{23}=G\frac{\partial u_{3}}{\partial x_{2}},$ (2) $\tau_{13}=-\frac{G\ell^{2}}{2}\Delta\frac{\partial u_{3}}{\partial x_{1}}+\frac{J}{4}\frac{\partial\ddot{u}_{3}}{\partial x_{1}},\quad\tau_{23}=-\frac{G\ell^{2}}{2}\Delta\frac{\partial u_{3}}{\partial x_{2}}+\frac{J}{4}\frac{\partial\ddot{u}_{3}}{\partial x_{2}},$ (3) $\displaystyle\mu_{11}=-\mu_{22}=G\ell^{2}(1+\eta)\frac{\partial^{2}u_{3}}{\partial x_{1}\partial x_{2}},\quad\mu_{21}=G\ell^{2}\left(\frac{\partial^{2}u_{3}}{\partial x_{2}^{2}}-\eta\frac{\partial u_{3}}{\partial x_{1}^{2}}\right),$ $\displaystyle\mu_{12}=-G\ell^{2}\left(\frac{\partial^{2}u_{3}}{\partial x_{1}^{2}}-\eta\frac{\partial^{2}u_{3}}{\partial x_{2}^{2}}\right).$ (4) where $\Delta$ denotes the Laplace operator, $J$ is the rotational inertia, $G$ is the elastic shear modulus, $\ell$ and $\eta$ the couple stress parameters, with $-1<\eta<1$. Both material parameters $\ell$ and $\eta$ depend on the microstructure and can be connected to the material characteristic lengths in bending and in torsion (Radi, 2008), namely $\ell_{b}=\ell/\sqrt{2}$ and $\ell_{t}=\ell\sqrt{1+\eta}$. Typical values of $\ell_{b}$ and $\ell_{t}$ for some classes of materials with microstructure can be found in Lakes (1986, 1995). Substituting expressions (2), (3) and (2) in the dynamic equilibrium equations (Mishuris et al., 2013), the following equation of motion is derived: $G\Delta u_{3}-\frac{G\ell^{2}}{2}\Delta^{2}u_{3}+\frac{J}{4}\Delta\ddot{u}_{3}=\rho\ddot{u}_{3}.$ (5) ### 2.1 Steady-state crack propagation We assume that the crack propagates with a constant velocity $V$ straight along the $x_{1}$-axis and is subjected to reduced force traction $p_{3}$ applied on the crack faces, moving with the same velocity $V$, whereas reduced couple traction $q_{1}$ is assumed to be zero (Georgiadis, 2003), $p_{3}(x_{1},0^{\pm},t)=\mp\tau(x_{1}-Vt),\quad q_{1}(x_{1},0^{\pm},t)=0,\quad\text{for}\quad x_{1}-Vt<0.$ (6) We also assume that the function $\tau$ decays at infinity sufficiently fast and it is bounded at the crack tip. These requirements are the same requirements for tractions as in the classical theory of elasticity. It is convenient to introduce a moving framework $X=x_{1}-Vt$, $y=x_{2}$. By assuming that the out of plane displacement field has the form $u_{3}(x_{1},x_{2},t)=w(X,y)$, then the equation of motion (5) writes: $\left(1-m^{2}\right)\frac{\partial^{2}w}{\partial X^{2}}+\frac{\partial^{2}w}{\partial y^{2}}-\frac{\ell^{2}}{2}\left(1-2m^{2}h_{0}^{2}\right)\frac{\partial^{4}w}{\partial X^{4}}-\ell^{2}\left(1-m^{2}h_{0}^{2}\right)\frac{\partial^{4}w}{\partial X^{2}\partial y^{2}}-\frac{\ell^{2}}{2}\frac{\partial^{4}w}{\partial y^{4}}=0,$ (7) where $m=V/c_{s}$ is the crack velocity normalized to the shear waves speed $c_{s}$, and $h_{0}=\sqrt{J/4\rho}/\ell$ is the normalized rotational inertia defined in Mishuris et al. (2013). According to (1), the non-vanishing components of the reduced force traction and reduced couple traction vectors along the crack line $y=0$, where $\mbox{\boldmath$n$}=(0,\pm 1,0)$, can be written as $p_{3}=t_{23}+\frac{1}{2}\frac{\partial\mu_{22}}{\partial X},\quad q_{1}=\mu_{21},$ (8) respectively. By using (2)2, (2)1,2, (3)2, and (8), the loading conditions (6) on the upper crack surface require the following conditions for the function $w$: $\displaystyle\frac{\partial w}{\partial y}-\frac{\ell^{2}}{2}\frac{\partial}{\partial y}\left[(2+\eta-2m^{2}h_{0}^{2})\frac{\partial^{2}w}{\partial X^{2}}+\frac{\partial^{2}w}{\partial y^{2}}\right]=-\frac{1}{G}\tau(X),$ $\displaystyle\frac{\partial^{2}w}{\partial y^{2}}-\eta\frac{\partial^{2}w}{\partial X^{2}}=0,\quad\text{for}\quad X<0,\quad y=0^{+}.$ (9) Ahead of the crack tip, the skew-symmetry of the Mode III crack problem requires $w=0,\quad\frac{\partial^{2}w}{\partial y^{2}}-\eta\frac{\partial^{2}w}{\partial X^{2}}=0,\ \text{for}\ X>0,\ y=0^{+}.$ (10) Note that the ratio $\eta$ enters the boundary conditions (9)-(10), but it does not appear into the governing PDE (7). ### 2.2 Preliminary analysis on couple stress surface waves propagation In couple stress elastic materials the existence of surface waves has been demonstrated for both in-plane and antiplane problems (Ottosen et al., 2000). Considering a material occupying the upper half-plane under antiplane deformations, the solution of the governing equation (5) is assumed in the form: $u_{3}(x_{1},x_{2},t)=W(x_{2})e^{i(kx_{1}-\omega t)},\quad x_{1}\geq 0,$ (11) where $W$ is the amplitude, k is the wave number and $\omega$ the radian frequency. Substituting (11) into (5) the following ODE is obtained: $W^{{}^{\prime\prime\prime\prime}}-\frac{2}{\ell^{2}}\left[k^{2}\ell^{2}+\left(1-\frac{\omega^{2}}{\theta^{2}}\right)\right]W^{{}^{\prime\prime}}+\frac{2}{\ell^{2}}\left[\frac{k^{4}\ell^{2}}{2}+\left(1-\frac{\omega^{2}}{\theta^{2}}\right)k^{2}-\frac{\omega^{2}}{c_{s}^{2}}\right]W=0,$ (12) where $c_{s}=\sqrt{G/\rho}$ is the shear wave speed for classical elastic materials, $\theta=\sqrt{4G/J}$ and the superscript ′ indicates the derivative with respect to $x_{2}$ variable. Equation (12) can be rewritten in the form $W^{{}^{\prime\prime\prime\prime}}-\frac{2}{\ell^{2}}\left[1+\left(\frac{1}{m_{R}^{2}}-h_{0}^{2}\right)\frac{\omega^{2}\ell^{2}}{c_{s}^{2}}\right]W^{{}^{\prime\prime}}+\frac{1}{\ell^{4}}\left[\left(\frac{1}{m_{R}^{2}}-2h_{0}^{2}\right)\frac{\omega^{4}\ell^{4}}{m_{R}^{2}c_{s}^{4}}-2\left(1-\frac{1}{m_{R}^{2}}\right)\frac{\omega^{2}\ell^{2}}{c_{s}^{2}}\right]W=0,$ (13) where $m_{R}=v_{R}/c_{s}$, $v_{R}=\omega/k$ is the couple stress surface waves speed and $h_{0}=c_{s}/\theta\ell=\sqrt{J/4\rho}/\ell$ is the normalized rotational inertia introduced in the previous section. Equation (13) admits the following bounded solution in the upper half-plane, vanishing for $x_{2}\rightarrow+\infty$ $W(x_{2})=Ae^{-\alpha(\omega,m_{R})x_{2}/\ell}+Be^{-\beta(\omega,m_{R})x_{2}/\ell},\quad\mbox{for}\ x_{2}>0,$ (14) where $\displaystyle\alpha(\omega,m_{R})$ $\displaystyle=$ $\displaystyle\sqrt{1-\left(h_{0}^{2}-\frac{1}{m_{R}^{2}}\right)\frac{\omega^{2}\ell^{2}}{c_{s}^{2}}+\chi(\omega)}=\sqrt{1+\left(1-h_{0}^{2}m_{R}^{2}\right)k^{2}\ell^{2}+\chi(k,m_{R})},$ (15) $\displaystyle\beta(\omega,m_{R})$ $\displaystyle=$ $\displaystyle\sqrt{1-\left(h_{0}^{2}-\frac{1}{m_{R}^{2}}\right)\frac{\omega^{2}\ell^{2}}{c_{s}^{2}}-\chi(\omega)}=\sqrt{1+\left(1-h_{0}^{2}m_{R}^{2}\right)k^{2}\ell^{2}-\chi(k,m_{R})},$ (16) $\chi(\omega)=\sqrt{1+2(1-h_{0}^{2})\frac{\omega^{2}\ell^{2}}{c_{s}^{2}}+h_{0}^{4}\frac{\omega^{4}\ell^{4}}{c_{s}^{4}}}=\sqrt{1+2(1-h_{0}^{2})m_{R}^{2}k^{2}\ell^{2}+h_{0}^{4}m_{R}^{4}k^{4}\ell^{4}}.$ (17) Similarly to the procedure commonly carried out for studying Rayleigh waves in classical elasticity, traction-free boundary conditions are imposed at the free surface: $p_{2}(x_{1},0^{+},t)=0,\quad q_{1}(x_{1},0^{+},t)=0,\quad\mbox{for}\ -\infty<x_{1}<\infty,$ (18) by using relations (2), (3), (2) together with expression (11), equation (18) becomes $W^{{}^{\prime}}(0)-\frac{\ell^{2}}{2}\left[-\frac{\omega^{2}}{c_{s}^{2}m_{R}^{2}}(2+\eta-2h_{0}^{2}m_{R}^{2})W^{{}^{\prime}}(0)+W^{{}^{\prime\prime\prime}}(0)\right]=0,$ (19) $W^{{}^{\prime\prime}}(0)+\frac{\eta\omega^{2}}{c_{s}^{2}m_{R}^{2}}W(0)=0.$ (20) Substituting expression (14) into equations (19) and (20), the following system of two algebraic equations for the unknown constants $A$ and $B$ is derived $\mbox{\boldmath$D$}(m_{R},\omega)\mbox{\boldmath$c$}=0,$ (21) where $\mbox{\boldmath$c$}=(A,B)^{T}$ and the matrix $D$ is given by $\mbox{\boldmath$D$}(m_{R},\omega)=\left[\begin{array}[]{cc}\alpha^{3}-\alpha\left(2-\cfrac{\omega^{2}\ell^{2}}{c_{s}^{2}m_{R}^{2}}\left(2+\eta-2h_{0}^{2}m_{R}^{2}\right)\right)&\beta^{3}-\beta\left(2-\cfrac{\omega^{2}\ell^{2}}{c_{s}^{2}m_{R}^{2}}\left(2+\eta-2h_{0}^{2}m_{R}^{2}\right)\right)\\\ &\\\ \alpha^{2}+\eta\cfrac{\omega^{2}\ell^{2}}{m_{R}^{2}c_{s}^{2}}&\beta^{2}+\eta\cfrac{\omega^{2}\ell^{2}}{m_{R}^{2}c_{s}^{2}}\end{array}\right],$ the system (21) possesses non-trivial solutions only if $\mathcal{D}(m_{R},\omega)=\det\mbox{\boldmath$D$}(m_{R},\omega)=0.$ (22) Expression (22) is the dispersion relation for antiplane couple stress surface waves, and the propagation velocity corresponding to a given value of the frequency $\omega$ or alternatively of the wave vector $k$ can be evaluated by solving this equation. Figure 1: Variation of the normalized Rayleigh waves speed with the normalized frequency. Figure 2: Variation of the normalized Rayleigh waves speed with the normalized wave vector. The normalized wave speed $m_{R}=v_{R}/c_{s}$ is shown in Figs. 1 and 2 as a function of the normalized frequency $\omega\ell/c_{s}$ and the normalized wave number $k\ell$, respectively. Different values for the characteristic parameter $\eta$ and for the normalized rotational inertia $h_{0}$ have been considered. For small values of the rotational inertia, the value of the couple stress surface waves speed is always greater than the shear waves velocity in classical elastic materials, and then the couple stress surface waves propagation is supersonic for any value of the wave number and frequency. In particular, for the case of vanishing rotational inertia $h_{0}=0$, the wave propagation is dispersive and supersonic with monotonically increasing speed, as it as been detected in Ottosen et al. (2000) and Askes and Aifantis (2011). As the rotational inertia increases, the phase speed behaviour changes: the values of $v_{R}$ may become smaller then $c_{s}$, and it decreases with the frequency and the wave number until a limit value corresponding to $m_{R}<1$ and depending by $h_{0}$ and $\eta$ is reached. This means that for large values of the rotational inertia and high frequencies the couple stress surface waves propagation becomes subsonic, and a minimum value for the phase speed is individuated for $\omega\rightarrow\infty$. For $\omega\rightarrow\infty$, the dispersion relation (22) exhibits the following asymptotic behaviour $\mathcal{D}(m_{R},\omega)=\left[(1+\eta)\sqrt{1-2h_{0}^{2}m_{R}^{2}}-(1-2h_{0}^{2}m_{R}^{2}+\eta)^{2}\right]\frac{\omega^{5}\ell^{5}}{m_{R}^{5}c_{s}^{5}}+O(\omega^{4}).$ (23) The minimum value for the normalized surface waves speed, depending on $\eta$ and $h_{0}$, is given by the value of $m_{R}$ for which the coefficient of the leading order term of (23) vanishes, and then it can be evaluated by solving the equation: $\Lambda(\eta,h_{0},m_{R})=(1+\eta)\sqrt{1-2h_{0}^{2}m_{R}^{2}}-(1-2h_{0}^{2}m_{R}^{2}+\eta)^{2}=0.$ (24) By means of simple algebra, it can be verified that equation (24) is equivalent to $\Upsilon(\eta,h_{0},m_{R})=\frac{1-\eta^{2}-2h_{0}^{2}m_{R}^{2}+2\sqrt{1-2h_{0}^{2}m_{R}^{2}}(1+\eta- h_{0}^{2}m_{R}^{2})}{1+\sqrt{1-2h_{0}^{2}m_{R}^{2}}}=0.$ (25) The function $\Upsilon$ introduced in expression (25) is the same defined in the Wiener-Hopf factorization of steady-state crack propagation problem in Mishuris et al. (2013), where the regime $\Upsilon(\eta,h_{0},m)>0$ is studied and a critical limit value for the crack tip speed is individuated by relation (25). Consequently, the minimum couple stress surface waves propagation velocity coincides with the critical value for steady-state crack propagation, and the condition $\Upsilon(\eta,h_{0},m)>0$ introduced in Mishuris et al. (2013) defines the transition between two different ranges of velocities, which further in the text will be called sub-Rayleigh and super-Rayleigh propagation regimes. These regimes are reported in the $h_{0}-m$ plane in Fig. 3A. For the case $\eta=0$ the dispersion curves shown in Fig.1 are identical to that obtained in Mishuris et al. (2013) for the shear waves. Consequently, for $\eta=0$ the couple stress surface waves degenerate to shear waves and subsonic and sub-Rayleigh regimes are equivalent. This can be demonstrated by the fact that for $\eta=0$ the eigenvalue $\beta$ given by (16) vanishes, and only the term of the matrix (21) depending by $\alpha^{2}$ is non-zero: in that case the factor $A$ is also zero and the solution coincides with the planar shear waves solution. Figure 3: A): Sub-Rayleigh and super-Rayleigh regimes in the $m-h_{0}$ plane. The continuous line coincides with the transition between subsonic and supersonic ranges. B): Variation of $h_{0}^{}$ as a function of $\eta$. In Fig. 3A it can be observed that for small values of the rotational inertia the crack propagation is both subsonic and sub-Rayleigh, and the limit value for the normalized crack tip speed is $m=1$. As $h_{0}$ increases, the limit speed for sub-Rayleigh regime becomes smaller than for subsonic regime, and the critical velocity $m_{c}(h_{0},\eta)$ is determined by solving equation (24) or alternatively (25). The limit value $h_{0}^{}$ such that for $h_{0}>h_{0}^{}$ the maximum normalized velocity for sub-Rayleigh regime is given by $m_{c}(h_{0},\eta)<1$ is plotted in Fig. 3B as a function of the microstructural parameter $\eta$. ## 3 Full-field solution The following form for the loading applied on the crack faces is assumed $\tau(X)=\frac{(-1)^{p}}{\Gamma(1+p)}\frac{T_{0}}{L}\left(\frac{X}{L}\right)^{p}e^{X/L},\quad X<0,\quad p=0,1,2,\dots$ (26) where $\Gamma$ is the Gamma function. It is important to note that the resultant force applied to the upper crack face is $T_{0}$, indeed $\int_{-\infty}^{0}\tau(X)dX=\frac{(-1)^{p}}{\Gamma(1+p)}\frac{T_{0}}{L}\int_{-\infty}^{0}\left(\frac{X}{L}\right)^{p}e^{X/L}dX=T_{0}.$ (27) Moreover, the maximum of the distributed traction $\tau(X)$ is attained at $X_{\text{max}}=-pL$. The normalized loading profile $\tau\ell/T_{0}$ is reported in Fig. 4 as a function of $X/\ell$ for several values of the exponent $p$ and of the ratio $L/\ell$. Note that for $p=0$, the loading is bounded but different from zero at the crack tip, for $p>0$ the loading tends to zero at the crack tip. Moreover, as $L/\ell$ decreases, the loading is more and more concentrated around a peak close to the crack tip. Sub-Rayleigh regime of propagation defined in previous Section is considered, so that $0\ \leq\ m\ \leq\mbox{min}\bigg{\\{}1,m_{c}(h_{0},\eta)\bigg{\\}},$ (28) where the critical value $m_{c}(h_{0},\eta)$ is obtained by the solution of equation (24) or (25) for given values of $\eta$ and $h_{0}$. Figure 4: Distributed loading applied to the crack faces ### 3.1 Solution of the Wiener-Hopf equation Since the Mode III crack problem is skew-symmetric, only the upper half-plane ($y\geq 0$) is considered for deriving the solution. The direct and inverse Fourier transforms of the out-of-plane displacements $w(X,y)$ are $\overline{w}(s,y)=\int_{-\infty}^{\infty}w(X,y)e^{isX}dX,\quad w(X,y)=\frac{1}{2\pi}\int_{\mathcal{L}}\overline{w}(s,y)e^{-isX}ds,$ (29) respectively, where $s$ is a real variable and the line of integration $\mathcal{L}$ will be defined later. Applying the Fourier transform (29)(1) to equation (9)(1) and using the general factorization procedure illustrated in details in Mishuris et al. (2013), the following functional equation of the Wiener–Hopf type can be obtained $\overline{p}_{3}^{+}(s)+\frac{G\sqrt{s^{2}\ell^{2}}}{2\ell}\Psi(s\ell)k(s\ell)\overline{w}^{-}(s)=\overline{\tau}^{-}(s),$ (30) where $\overline{\tau}^{-}(s)$ is analytic in the lower half complex $s$-plane, $\mathop{\mathrm{Im}}s<0$ and it is given by $\overline{\tau}^{-}(s)=\frac{T_{0}}{(1+isL)^{1+p}},$ (31) where $k(s\ell)=\frac{1}{\sqrt{s\ell}\Psi(s\ell)(\alpha+\beta)}\Big{\\{}\alpha\beta(\alpha^{2}+\beta^{2}+2\eta s^{2}\ell^{2})+\alpha^{2}\beta^{2}-\eta^{2}s^{4}\ell^{4}\Big{\\}},$ (32) $\alpha(s\ell)=\sqrt{1+(1-h_{0}^{2}m^{2})s^{2}\ell^{2}+\chi(s\ell)},\quad\beta(s\ell)=\sqrt{1+(1-h_{0}^{2}m^{2})s^{2}\ell^{2}-\chi(s\ell)},$ (33) $\chi(s\ell)=\sqrt{1+2(1-h_{0}^{2})m^{2}s^{2}\ell^{2}+h_{0}^{4}m^{4}s^{4}\ell^{4}},$ (34) $\Psi(s\ell)=\Upsilon(\eta,h_{0},m)s^{2}\ell^{2}+2\sqrt{1-m^{2}},$ (35) and $\Upsilon(\eta,h_{0},m)$ is defined in (25). The function $k(s\ell)$ has been factorized in Mishuris et al. (2013) as $k(s\ell)=k^{-}(s\ell)/k^{+}(s\ell)$, where $s\ell\in\mathbb{R}$, and $k^{+}(s\ell)$ and $k^{-}(s\ell)$ are analytic in the upper and lower half- planes, respectively. Since sub-Rayleigh regime is investigated, $\Upsilon(\eta,h_{0},m)$ is positive for all values of crack tip speed and microstructural parameters considered. The Wiener-Hopf equation (30) can then be rewritten in the form: $\frac{k^{+}(s\ell)\overline{p}_{3}^{+}(s)}{(s\ell)_{+}^{1/2}}+\frac{G}{2\ell}(s\ell)_{-}^{1/2}\Psi(s\ell)k^{-}(s\ell)\overline{w}^{-}(s)=\frac{T_{0}k^{+}(s\ell)}{(s\ell)_{+}^{1/2}(1+isL)^{1+p}},$ (36) The right-hand side of (36) can be easily split in the sum of plus and minus functions. Indeed, we use the fact that the function $k^{+}(s\ell)/(s\ell)_{+}^{1/2}$ is analytical in the point $sL=+i$ and thus can be represented as $\frac{k^{+}(s\ell)}{(s\ell)_{+}^{1/2}}=\sum_{j=0}^{p}(1+isL)^{j}F_{j}+F_{p+1}^{+}(s)=\sum_{j=0}^{p}(1+isL)^{j}F_{j}+{\mathcal{G}}^{+}(s)(1+isL)^{p+1},$ (37) where ${\mathcal{G}}^{+}(s)\equiv\frac{F_{p+1}^{+}(s)}{(1+isL)^{p+1}}=\frac{1}{(1+isL)^{p+1}}\left(\frac{k^{+}(s\ell)}{(s\ell)_{+}^{1/2}}-\sum_{j=0}^{p}(1+isL)^{j}F_{j}\right)=O(1),\quad s\to+i/L.$ (38) Note that the function ${\mathcal{G}}^{+}(s\ell)$ exhibits the following asymptotic behaviour: ${\mathcal{G}}^{+}(s)=i\frac{F_{p}}{sL}+O(s^{-2}),\ \|s\|\to\infty;\quad{\mathcal{G}}^{+}(s)=\frac{k^{+}(0)}{(s\ell)_{+}^{1/2}}+O(1),\ \|s\|\to 0,\quad\mbox{with}\ \mathop{\mathrm{Im}}s>0.$ (39) Taking this fact into account, the right-hand side of the equation (36) can be written in the form $\frac{T_{0}k^{+}(s\ell)}{(s\ell)_{+}^{1/2}(1+isL)^{1+p}}=T_{0}{\mathcal{G}}^{-}(s)+T_{0}{\mathcal{G}}^{+}(s),$ (40) where ${\mathcal{G}}^{-}(s)=\sum_{j=0}^{p}\frac{F_{j}}{(1+isL)^{p+1-j}},$ (41) and ${\mathcal{G}}^{-}(s)=-i\frac{F_{p}}{sL}+O(s^{-2}),\ \|s\|\to\infty;\quad{\mathcal{G}}^{-}(s)=\sum_{j=0}^{p}F_{j}+O(s),\ \|s\|\to 0\quad\mbox{with}\ \mathop{\mathrm{Im}}s<0.$ (42) The unknown constants $F_{j}$ are computed by evaluating the integrals: $F_{j}=\frac{L}{2\pi}\oint_{\gamma}\left(\frac{1}{(1+isL)^{j+1}}\frac{k^{+}(s\ell)}{(s\ell)_{+}^{1/2}}\right)ds,$ (43) where $\gamma$ is an arbitrary contour centered at the point $s=i/L$ and lying in the analyticity domain. Substituting (40) in (36), we finally obtain: $\frac{k^{+}(s\ell)\overline{p}_{3}^{+}(s)}{(s\ell)_{+}^{1/2}}-T_{0}{\mathcal{G}}^{+}(s)=T_{0}{\mathcal{G}}^{-}(s)-\frac{G}{2\ell}(s\ell)_{-}^{1/2}\Psi(sl)k^{-}(s\ell)\overline{w}^{-}(s).$ (44) The left and right hand sides of (44) are analytic functions in the upper and lower half-planes, respectively, and thus define an entire function on the $s$-plane. The Fourier transform of the reduced force traction ahead of the crack tip and the crack opening gives $\overline{p}_{3}^{+}\sim s^{1/2}$ and $\overline{w}^{-}\sim s^{-5/2}$ as $\|s\|\to\infty$. Therefore, both sides of (44) are bounded as $\|s\|\to\infty$ and according to the Liouville’s theorem must be equal to a constant $F$ in the entire $s$-plane. As a result, we obtain $\overline{p}_{3}^{+}(s)=\frac{T_{0}(s\ell)_{+}^{1/2}}{k^{+}(s\ell)}[F+\mathcal{G}^{+}(s)],\qquad\overline{w}^{-}(s)=\frac{2T_{0}\ell}{G}\frac{\mathcal{G}^{-}(s)-F}{(s\ell)_{-}^{1/2}\Psi(s\ell)k^{-}(s\ell)}.$ (45) The constant $F$ is determined by the condition that the displacement $w(X)$ is zero at the crack tip $X=0$, that is $\int_{-\infty}^{\infty}\overline{w}^{-}(s)ds=0,$ (46) which leads to $F=\frac{\displaystyle\int_{-\infty}^{\infty}\frac{\mathcal{G}^{-}(s)ds}{(s\ell)_{-}^{1/2}\Psi(s\ell)k^{-}(s\ell)}}{\displaystyle\int_{-\infty}^{\infty}\frac{ds}{(s\ell)_{-}^{1/2}\Psi(s\ell)k^{-}(s\ell)}}={\mathcal{G}}^{-}(-i\zeta/\ell),$ (47) where $\zeta$ is given by $\zeta=\sqrt{\frac{2\sqrt{1-m^{2}}}{\Upsilon(\eta,h_{0},m)}}.$ (48) Note here that according to (39), $\overline{p}_{3}^{+}(0)=T_{0}$, that is the standard balance condition for this problem. The equivalence between the two alternative expressions for the constant $F$ reported in relation (47) can be easily demonstrated by applying the Cauchy integral theorem (Arfken and Weber, 2005). ### 3.2 Analytical representation of displacements, stresses and couple stresses The reduced force traction ahead of the crack tip $p_{3}(X)$ and the crack opening $w(X)$ can be obtained applying the inverse Fourier transform (29)2 to expressions (45). Since the integrand does not have branch cuts along the real line, the path of integration $\mathcal{L}$ coincides with the real $s$-axis. Further, we introduce the change of variable $\xi=s\ell$, thus obtaining $w(X)=\frac{T_{0}}{\pi G}\int_{-\infty}^{\infty}\frac{\mathcal{G}^{-}(\xi/\ell)-F}{\xi_{-}^{1/2}\psi(\xi)k(\xi)k^{+}(\xi)}e^{-iX\xi/\ell}d\xi,\quad X<0,$ (49) $p_{3}(X)=\frac{T_{0}}{2\pi\ell}\int_{-\infty}^{\infty}\frac{\xi_{+}^{1/2}k(\xi)}{k^{-}(\xi)}[F+\mathcal{G}^{+}(\xi/\ell)]e^{-iX\xi/\ell}d\xi,\quad X>0.$ (50) The Fourier transform of stress (symmetric and skew-symmetric) and couple stress fields can be derived from (2), (3) and (2) namely $\overline{\sigma}_{23}(s,0)=-\frac{G}{\ell}\frac{\alpha\beta-\eta s^{2}\ell^{2}}{\alpha+\beta}\overline{w}^{-}(s),$ (51) $\overline{\tau}_{23}(s,0)=-\frac{G}{2\ell}\frac{1}{\alpha+\beta}\Big{\\{}\alpha^{2}\beta^{2}+(\alpha^{2}+\beta^{2}+\alpha\beta)\eta s^{2}\ell^{2}-(1-2h_{0}^{2}m^{2})s^{2}\ell^{2}(\eta s^{2}\ell^{2}-\alpha\beta)\Big{\\}}\overline{w}^{-}(s),$ (52) $\overline{\mu}_{22}(s,0)=-G(1+\eta)(is\ell)\frac{\alpha\beta-\eta s^{2}\ell^{2}}{\alpha+\beta}\overline{w}^{-}(s).$ (53) The inverse Fourier transform can be performed as explained above, thus obtaining for $X>0$ $\sigma_{23}(X,0)=-\frac{T_{0}}{\pi\ell}\int_{-\infty}^{\infty}\frac{\alpha(\xi)\beta(\xi)-\eta\xi^{2}}{\alpha(\xi)+\beta(\xi)}\frac{\mathcal{G}^{-}(\xi/\ell)-F}{\xi_{-}^{1/2}\psi(\xi)k^{-}(\xi)}e^{-iX\xi/\ell}d\xi,$ (54) $\displaystyle\tau_{23}(X,0)$ $\displaystyle=-\frac{T_{0}}{2\pi\ell}\int_{-\infty}^{\infty}\frac{1}{\alpha(\xi)+\beta(\xi)}\Big{\\{}\alpha^{2}(\xi)\beta^{2}(\xi)+(\alpha^{2}(\xi)+\beta^{2}(\xi)+\alpha(\xi)\beta(\xi))\eta\xi^{2}-$ (55) $\displaystyle\hskip 56.9055pt{}-(1-2h_{0}^{2}m^{2})\xi^{2}(\eta\xi^{2}-\alpha(\xi)\beta(\xi))\Big{\\}}\frac{\mathcal{G}^{-}(\xi/\ell)-F}{\xi_{-}^{1/2}\psi(\xi)k^{-}(\xi)}e^{-iX\xi/\ell}d\xi,$ (56) $\mu_{22}(X,0)=-\frac{iT_{0}(1+\eta)}{\pi}\int_{-\infty}^{\infty}\xi\frac{\alpha(\xi)\beta(\xi)-\eta\xi^{2}}{\alpha(\xi)+\beta(\xi)}\frac{\mathcal{G}^{-}(\xi/\ell)-F}{\xi_{-}^{1/2}\psi(\xi)k^{-}(\xi)}e^{-iX\xi/\ell}d\xi.$ (57) ## 4 Dynamic energy release rate In this Section the dynamic energy release rate for a Mode III steady-state propagating crack in couple stress elastic materials under distributed loading conditions given by expression (26) is evaluated. ### 4.1 Explicit evaluation The general expression for the dynamic J-integral in couple stress elasticity, including also the rotational inertia contribution, has been derived and proved to be path-independent in the steady-state case assuming traction free crack faces by Morini et al. (2013). Considering the moving framework $OXy$ with the origin at the crack tip introduced in Section 2, the J-integral for a steady state crack propagating along the $X-$axis is given by: $\displaystyle\mathcal{J}$ $\displaystyle=$ $\displaystyle\int_{\Gamma}\left[(W+T)n_{X}-\mbox{\boldmath$p$}\cdot\frac{\partial\mbox{\boldmath$u$}}{\partial X}-\mbox{\boldmath$q$}\cdot\frac{\partial\mbox{\boldmath${\varphi}$}}{\partial X}\right]ds=$ (58) $\displaystyle=$ $\displaystyle\int_{\Gamma}\left\\{(W+T)dy-\left[\mbox{\boldmath$p$}\cdot\frac{\partial\mbox{\boldmath$u$}}{\partial X}+\mbox{\boldmath$q$}\cdot\frac{\partial\mbox{\boldmath${\varphi}$}}{\partial X}\right]ds\right\\},$ where $\Gamma$ is an arbitrary closed path surrounding the crack tip, and $n_{X}$ is the Cartesian component directed along the $X-$axis of the outward unit vector normal to $\Gamma$, defined by $\mbox{\boldmath$n$}=(n_{X},n_{Y},0)$. Since the distributed loading of profile (26) acting on the crack line is assumed, in our case the contribution of the crack faces must be taken into account, and then in principle the J-integral (58) is not path-independent. Nevertheless, in this Section the J-integral is used to determine the dynamic energy release rate evaluating the limit for $\Gamma\rightarrow 0$ in (58) (Freund, 1998). This means that the asymptotic expressions of displacement and stresses can be used for calculating the energy release rate. Remembering the asymptotics behaviour of displacement and stresses for antiplane cracks reported in Morini et al. (2013) and the loading function (26), it is easy to verify that in the limit $\Gamma\rightarrow 0$ the contribution of the crack faces to the J-integral (58) vanishes. We assume the rectangular-shaped integration contour $\Gamma$ considered in Morini et al. (2013), and in order to evaluate the energy release rate we allow the height of the path along the $y-$direction to vanish and we make the limit $\varepsilon\rightarrow 0$. Assuming this type of contour, first introduced by Freund (1972), solely asymptotic expressions of displacements and stress fields are required for evaluating the energy release rate. Moreover, upon this choice of path, allowing the height of the rectangle along the $y-$direction to vanish, the integral $\int_{\Gamma}(W+T)dy$ becomes zero and then the energy release rate is given by $\mathcal{E}=\lim_{\Gamma\rightarrow 0}\mathcal{J}=-2\lim_{\varepsilon\rightarrow 0}\int_{-\varepsilon}^{\varepsilon}\left[\mbox{\boldmath$p$}\cdot\frac{\partial\mbox{\boldmath$u$}}{\partial X}+\mbox{\boldmath$q$}\cdot\frac{\partial\mbox{\boldmath${\varphi}$}}{\partial X}\right]ds.$ (59) Since boundary conditions (9) together with anti-symmetry conditions (10) provide that the reduced traction $q_{1}=\mu_{21}$ is zero along the whole crack $y=0$, the dynamic energy release rate for a steady-state Mode III crack becomes: $\displaystyle\mathcal{E}$ $\displaystyle=-2\lim_{\varepsilon\rightarrow 0^{+}}\int_{-\varepsilon}^{+\varepsilon}\left\\{\left[t_{23}(X,0^{+})+\frac{1}{2}\mu_{22}(X,0^{+})\right]\frac{\partial w(X,0^{+})}{\partial X}+\mu_{21}(X,0^{+})\frac{\partial\varphi_{1}(X,0^{+})}{\partial X}\right\\}dX$ $\displaystyle=-2\lim_{\varepsilon\rightarrow 0^{+}}\int_{-\varepsilon}^{+\varepsilon}\left[t_{23}(X,0^{+})+\frac{1}{2}\mu_{22}(X,0^{+})\right]\frac{\partial w(X,0^{+})}{\partial X}dX.$ (60) In the limit $\|s\|\rightarrow\infty$, the Fourier transform of displacements, total shear stress and couple stress fields derived in Section 3 assume the following behaviour: $\displaystyle\overline{w}^{-}(s,0^{+})$ $\displaystyle=-\frac{2FT_{0}\ell}{G\Upsilon(h_{0},m,\eta)}(s\ell)_{-}^{-5/2}+O\left((s\ell)_{-}^{-7/2}\right),\quad\mathop{\mathrm{Im}}s<0.$ (61) $\displaystyle\overline{t}_{23}^{+}(s,0^{+})$ $\displaystyle=-\frac{FT_{0}(1+\eta-2h_{0}^{2}m^{2})}{\Upsilon(h_{0},m,\eta)}(s\ell)_{+}^{1/2}+O\left((s\ell)_{+}^{-1/2}\right),\quad\mathop{\mathrm{Im}}s>0,$ (62) $\displaystyle\overline{\mu}_{22}^{+}(s,0^{+})$ $\displaystyle=\frac{2iFT_{0}\ell\left(\sqrt{1-2h_{0}^{2}m^{2}}-\eta\right)(1+\eta)}{\Upsilon(h_{0},m,\eta)\left(1+\sqrt{1-2h_{0}^{2}m^{2}}\right)}(s\ell)_{+}^{-1/2}+O\left((s\ell)_{+}^{-1}\right),\quad\mathop{\mathrm{Im}}s>0,$ (63) further, we consider the following transformation formula (Roos, 1969): $x^{\kappa}\stackrel{{\scriptstyle ft}}{{\leftrightarrow}}i^{\kappa+1}\Gamma(\kappa+1)s^{-\kappa-1},\ \mbox{with}\ \kappa\neq-1,-2,-3\ldots,$ (64) where $\Gamma$ is the gamma function and the symbol $\stackrel{{\scriptstyle ft}}{{\leftrightarrow}}$ indicates that the quantities on the two sides of the (64) are connected by means of unilateral Fourier transform. Applying the formula (64) to expressions (61)-(63), we get: $\displaystyle w(X,0^{+})$ $\displaystyle=-\frac{8FT_{0}(i\ell)^{-3/2}}{3\sqrt{\pi}G\Upsilon(h_{0},m,\eta)}(-X)^{3/2},\quad X<0.$ (65) $\displaystyle t_{23}(X,0^{+})$ $\displaystyle=-\frac{FT_{0}(1+\eta-2h_{0}^{2}m^{2})(i\ell)^{1/2}}{2\sqrt{\pi}\Upsilon(h_{0},m,\eta)}X^{-3/2},\quad X>0,$ (66) $\displaystyle\mu_{22}(X,0^{+})$ $\displaystyle=\frac{2FT_{0}\left(\sqrt{1-2h_{0}^{2}m^{2}}-\eta\right)(1+\eta)(i\ell)^{1/2}}{\sqrt{\pi}\Upsilon(h_{0},m,\eta)\left(1+\sqrt{1-2h_{0}^{2}m^{2}}\right)}X^{-1/2},\quad X>0.$ (67) Then, by substituting expressions (65), (66), and (67) into equation (60), we obtain: $\displaystyle\mathcal{E}$ $\displaystyle=-\frac{4iF^{2}T_{0}^{2}\left[(1+\eta-2h_{0}^{2}m^{2})+\left(\sqrt{1-h_{0}^{2}m^{2}}-\eta\right)\left(1+\eta\right)\right]}{\pi G\ell\Upsilon^{2}(h_{0},m,\eta)\left(1+\sqrt{1-h_{0}^{2}m^{2}}\right)}\lim_{\varepsilon\rightarrow 0^{+}}\int_{-\varepsilon}^{+\varepsilon}X_{-}^{1/2}X_{+}^{-3/2}dX$ $\displaystyle=-\frac{4iF^{2}T_{0}^{2}}{\pi G\ell\Upsilon(h_{0},m,\eta)}\lim_{\varepsilon\rightarrow 0^{+}}\int_{-\varepsilon}^{+\varepsilon}X_{-}^{1/2}X_{+}^{-3/2}dX$ (68) where $X_{-}^{1/2}$ and $X_{+}^{-3/2}$ are distributions of the bisection type. For any real $\lambda$ with the exception of $\lambda=1,2,3,\dots$, this particular type of distribution is defined as follows: $X_{+}^{\lambda}=\left\\{\begin{array}[]{cc}\|X\|^{\lambda},&\text{for}\leavevmode\nobreak\ X>0,\\\ 0,&\mbox{for}\leavevmode\nobreak\ X<0.\end{array}\right.,\ X_{-}^{\lambda}=\left\\{\begin{array}[]{cc}0,&\mbox{for}\leavevmode\nobreak\ X>0,\\\ \|X\|^{\lambda},&\text{for}\leavevmode\nobreak\ X<0.\end{array}\right.$ The products of distributions inside the integrals in (68) is evaluated through the application of Fisher’s theorem (Fischer, 1971), that leads to the relation: $(X_{-})^{\lambda}(X_{+})^{-1-\lambda}=-\frac{\pi\delta(x)}{2\sin(\pi\lambda)},\ \mbox{with}\ \lambda\neq-1,-2,-3\dots,$ (69) where $\delta(x)$ is the Dirac delta distribution. Then, by using the relation (69) into (68) and considering the fundamental property of the Dirac delta distribution $\int_{-\varepsilon}^{+\varepsilon}\delta(x)dx=1$, we finally get: $\mathcal{E}=\frac{2iF^{2}T_{0}^{2}}{G\ell\Upsilon(h_{0},m,\eta)}.$ (70) A general explicit expression for the dynamic energy release rate associated to an antiplane steady state crack in couple stress elastic materials where a distributed loading of the form (26) is applied on the crack faces has been derived. Equation (70) can be compared with the energy release rate corresponding to a Mode III steady state crack in classical elastic materials under the same loading conditions: $\mathcal{E}^{cl}=\frac{T_{0}^{2}}{GL}\frac{K_{p}^{2}}{\sqrt{1-m^{2}}},\quad\mbox{with}\quad K_{p}=\frac{(-1)^{p}}{p!}\frac{\sqrt{\pi}}{\Gamma(\frac{1}{2}-p)},$ (71) the ratio between the two expressions (70) and (71) is given by $\frac{\mathcal{E}}{\mathcal{E}^{cl}}=\frac{2iF^{2}L}{\ell K_{p}^{2}\Upsilon(h_{0},m,\eta)}\sqrt{1-m^{2}}.$ (72) ## 5 Results and discussion In order to study the effects of loading variations and microstructures on crack propagation, several numerical computations have been performed assuming loading configurations of the form (26) with different values of the exponent $p$ and the ratio $L/\ell$. Total shear stress ahead of the crack tip and crack opening profiles are reported and analyzed in subsection 5.1. Effects of $p$ and $L/\ell$ variation on maximum total shear stress ahead of the crack tip and on dynamic energy release rate are discussed in subsections 5.2 and 5.3, respectively. The limit cases when the crack tip speed approaches shear waves and couple stress surface waves velocities and when the characteristic length $\ell$ vanishes are investigated. ### 5.1 Total shear stress and crack opening In Fig. 5 the normalized variation of the total shear stress is reported for the same values of the crack tip speed $m=0.3$ and of the normalized rotational inertia $h_{0}=0.707$, and assuming three different values of $\eta=\left\\{-0.9,0,0.9\right\\}$. Four different values of $p=\left\\{0,\ 1,\ 2,\ 3\right\\}$ and three different values of $L/\ell=\left\\{0.5,1,10\right\\}$ have been considered for the computations. It can be observed that, as $p$ decreases, and then the maximum of the loading function approaches the crack tip (see Fig. 4), the level of the shear stress increases. This behaviour is more pronounced for $\eta=-0.9$, whereas it becomes less evident for $\eta=0$ and $\eta=0.9$. Consequently, for large values of the parameter $\eta$, corresponding to relevant microstructural effects, the increasing of the shear stress associate to maximum loading level approaching the crack tip is shielded. As it is shown in Fig. 4, small values of the ratio $L/\ell$ correspond to a localization of the applied loading close to the crack tip. In classical elastic media, this implies an increasing of the stress level ahead of the crack tip. In presence of couple stress, this increasing is detected for $\eta=-0.9$. In this case, since $\eta$ is close to the limit value $\eta=-1$, the microstructural effects are not very pronounced and the behaviour of the material differs slightly from that of a classical elastic medium (Radi, 2008). In Fig. 5, the increasing of the total shear stress associate to the decreasing of the ratio $L/\ell$ is not observed in the cases $\eta=0$ and $\eta=0.9$. It means that in couple stress elastic materials, the increasing effect due to the localization of the applied loading is counterbalanced by relevant microstructural contributions, corresponding to large values of $\eta$. An analogous behaviour is detected for the crack opening in Fig. 6: the value of $w$ increases as the exponent $p$ decreases and then the maximum of the loading function approaches the crack tip, while for small values of $L/\ell$ such as for example $L/\ell=0.5$ the expected increasing of $w$ due to the major localization of the loading is not observed. Conversely, as the distance from the crack tip increases, the crack opening corresponding to small values of $L/\ell$ approaches a maximum and decreases becoming less than in cases where this ratio is greater. This confirms that, as it has been deduced observing total shear stress behaviour ahead of the crack tip, the effect of the applied loading localization is shielded by microstructures of the material. Figure 5: Variation of the total shear stress $t_{23}$ along the $X-$axis. Figure 6: Variation of the crack opening displacement $w$ along the crack faces. ### 5.2 Maximum total shear stress analysis The normalized profile of the maximum total shear stress, $t_{23}^{\text{max}}$, is plotted as a function of the crack tip speed $m$ for several values of the exponent $p$ and of the rotational inertia $h_{0}$ in Fig. 7. For all sets of parameters considered in the study, numerical results show that for the limit cases $m=1$ and $m=m_{c}$ the maximum total shear stress assumes a finite critical value. Observing Fig. 7, it can be noted that in the cases $\eta=-0.9$ and $\eta=0$ the level of $t_{23}^{\text{max}}$ is greater for small values of $p$, corresponding to a maximum of the applied loading localized near to the crack tip. Conversely, for $\eta=0.9$ the value of $t_{23}^{\text{max}}$ associated to $p=0$ (dotted lines) is greater than for $p=1$ (dashed lines). This is due to the fact that for large values of $\eta$ the presence of the microstructures counterbalances the force action near to the crack tip, where the maximum of the loading is applied for $p=0$. In Radi (2008) and Georgiadis (2003), a fracture criterion based on the achieving of a critical level of the maximum shear stress $t_{23}^{\text{max}}=\tau_{C}$ at which the crack starts propagating is defined. Fig. 7 shows that for $\eta=-0.9$ and $h_{0}=0.01$ the maximum shear stress decreases as the crack speed increases until $m\approx 0.9$, whereas for $m>0.9$ it starts to increase until it reaches the maximum value for $m=1$, when the crack speed approaches the shear waves speed $c_{s}$. Differently, for $h_{0}=0.707$, $t_{23}^{\text{max}}$ increases monotonically up to the maximum value corresponding to $m=m_{c}=0.441$, when the crack tip speed approaches the minimum velocity for couple stress surface waves propagation. Therefore, referring to the maximum shear stress criterion, for $\eta=-0.9$ and $h_{0}=0.01$ the crack propagation turns out to be initially stable at speed sufficiently lower than the shear wave velocity in classical elastic materials, whereas it becomes unstable when the velocity approaches $c_{s}$. Conversely, for $\eta=-0.9$ and $h_{0}=0.707$ the propagation is unstable for any $m$ such that $m<m_{c}$. It can be observed that for $\eta=0$ and $h_{0}=0.01$, $t_{23}^{\text{max}}$ decreases as the crack tip speed becomes faster and reaches a minimum at $m=1$, while for $h_{0}=0.707$ it grows as $m$ increases until the maximum value corresponding to $m=1$. Consequently, for $\eta=0$ and $h_{0}=0.01$ the crack propagation can be considered stable, whereas for $\eta=0$ and $h_{0}=0.707$ it turns out to be unstable. On the basis of the same criterion, the figures show that for $\eta=0.9$ the crack propagation is stable for both $h_{0}=0.01$ and $h_{0}=0.707$. The reported results confirm the analysis performed in Mishuris et al. (2013), which shows that relevant microstructural effects, associated to large values of $\eta$, provide a stabilizing effect of the crack propagation. Moreover, it is important to observe that the variation of the exponent $p$ influences the value of $t_{23}^{\text{max}}$ but not the qualitative behaviour of its profiles as a function of $m$. This means that if the position of application of the maximum loading is changed, it does not affect the stability of the propagation. In Fig. 7 it can also be noted that for large values of the normalized rotational inertia $h_{0}$, the level of maximum shear stress ahead of the crack tip becomes higher. As a consequence, if the contribution of the rotational inertia is not negligible (as for the case $h_{0}=0.707$), a major amount of energy must be provided by the loading in order to initiate the propagation and to allow the crack propagating at constant speed. In Fig. 8 the variation of $t_{23}^{\text{max}}$ is reported as a function of the ratio $L/\ell$ for $m=0.3,\ p=1$, assuming $\eta=\left\\{-0.9,\ 0,\ 0.9\right\\}$ and considering four different values for the normalized rotational inertia $h_{0}=\left\\{0.01,\ 0.6,\ 0.707,\ 0.8\right\\}$. The decreasing of $L/\ell$, is associated with a strong localization of the applied loading around a maximum close to the crack front. As just discussed, in classical elasticity this implies an increasing of the stress level ahead of the crack tip. Conversely, Fig. 8 shows that in couple stress materials the maximum shear stress is zero for $L/\ell=0$, then it increases and after reaching a peak it starts decreasing. This means that if the loading profile is localized around a maximum close to the crack tip, then its action is shielded by the effects of the microstructure. This phenomena is more pronounced for the case $\eta=0.9$, where the microstructural contributions are more relevant. Figure 7: Variation of the maximum total shear stress $t_{23}^{\text{max}}$ with the crack tip speed $m$. Figure 8: Variation of the maximum total shear stress $t_{23}^{\text{max}}$ with the ratio $L/\ell$. ### 5.3 Energy release rate The normalized variation of the energy release rate versus the crack tip speed $m$ is reported in Fig. 9 for the same value of the ratio $L/\ell=10$, three different values of $\eta=\left\\{0,\ 0.9,\ -0.9\right\\}$ and of the rotational inertia $h_{0}=\left\\{0.01,\ 0.707,\ 0.8\right\\}$. Four different values of $p=\left\\{0,\ 1,\ 2,\ 3\right\\}$ have been considered in the computations. The curves reported present the same qualitative behaviour for all values of the exponent $p$: the energy release rate is initially constant for $m\leq 0.3$, then it increases monotonically until its limiting value corresponding to $m=1$ or $m=m_{c}$. This means that, once the critical value $\mathcal{E}_{c}=2\gamma$ (depending on the material properties) is achieved (Freund, 1998), the energy release rate always increases as a function of the velocity, and then if the applied loading provides the energy necessary for starting the fracture process, the crack has enough energy to accelerate rapidly up to the limiting values of the speed (Willis, 1971; Obrezanova et al., 2002). It follows that, analysing these results by means of Griffith criterion as it has been done in Morini et al. (2013), the crack propagation turns out to be unstable for any value of the exponent $p$, of the rotational inertia $h_{0}$ and of $\eta$. Moreover, the variation of the loading profile (26) does not affect significantly the stability of the propagation, and the stabilizing effect observed for large values of $\eta$ applying the $t_{23}^{\text{max}}$ is not detected. As it has been explained and discussed in details by Morini et al. (2013), this discrepancy is due to the fact that the energy release rate is evaluated using the term of order $r^{3/2}$ of the asymptotic displacement field, corresponding to the singular shear stress term of order $r^{-3/2}$ (see expressions (65) and (66) in Section 4). This singular contribution dominates very near to the crack tip, but it is not sufficient to describe accurately the physical behaviour of the stresses at few characteristic lengths from the crack tip, where higher order terms of the expansions become important (Hancock and Du, 1991; Smith et al., 2006). In Fig. 9 it can be observed that in the cases where a rotational inertia greater than the reference value $h_{0}^{}$ defined in Section 2 is considered, the limit value for the energy release rate associated to $m=m_{c}$ is finite for any set of microstructural parameters. Numerical results show that, also in the cases where a small rotational inertia $h_{0}<h_{0}^{}$ is assumed, the limit maximum value $\mathcal{E}_{\text{max}}$ corresponding to $m=1$ is finite. The only exception is represented by the case $\eta=0$ and $h_{0}=1/\sqrt{2}\approx 0.707$: for these particular values of microstructural parameter $\eta$ and rotational inertia $h_{0}$, couple stress surface waves degenerate to non- dispersive shear waves (see dispersion curves in Fig. 1), and for $m=1$ the energy release rate becomes unbounded. The ratio between the energy release rate in couple stress materials and the energy release rate in classical elastic materials (72) is plotted in Fig. 10 as a function of the normalized crack tip speed $m$. These figures show that $\mathcal{E}/\mathcal{E}^{cl}$ is less than one for $p=0$, while it is greater than one for $p>0$. As a consequence, if the maximum of the loading is applied at the crack tip, in couple stress elastic materials a minor quantity of energy is provided for propagating cracks at a constant speed respect to classical elastic material. This means that for $p=0$ the action of the applied forces is shielded by the effects of the microstructures. Conversely, if the maximum loading is not applied at the crack tip, a major amount of energy is available in order to propagate the fracture at a given constant velocity respect to classical elastic media. It follows that for $p>0$ the presence of microstructures facilitate the propagation and a weakening effect is detected. The observed shielding effect is in agreement with what has been illustrated analyzing crack opening and maximum shear stress for $p=0$. Note that also in this case both shielding and weakening phenomena are more pronounced for great values of $\eta$, corresponding to relevant microstructural effects. Fig. 10 shows that for $h_{0}=0.01$, and in general for values of the rotational inertia such that $h_{0}<h_{0}^{}$, the ratio $\mathcal{E}/\mathcal{E}^{cl}$ tends to zero at $m=1$. This is due to the fact that while the energy release rate in couple stress materials reach a finite limit value for $m=1$, in classical elasticity it diverges (see expression (71)). The only case where $\mathcal{E}/\mathcal{E}^{cl}$ may reach a non-zero value for $m=1$ corresponds to $\eta=0$ and $h_{0}=1/\sqrt{2}\approx 0.707$. For this particular values of $\eta$ and $h_{0}$, both $\mathcal{E}$ and $\mathcal{E}^{cl}$ becomes unbounded as $m=1$, and then their ratio can be different from zero. Observing Fig. 10, it can also be noted that in all cases where a rotational inertia greater than $h_{0}^{}$ is considered, the ratio $\mathcal{E}/\mathcal{E}^{cl}$ assumes a finite non-zero limit value for $m=m_{c}$. In particular, due to the fact that for small values of $\eta$ the microstructural effects are negligible and the behaviour of the material is similar to that of a classical elastic body (Radi, 2008), in the case $\eta=-0.9$ the ratio $\mathcal{E}/\mathcal{E}^{cl}$ tends to one for $m=m_{c}$ independently of the value of the exponent $p$. As $\eta$ increases, and then the action of the microstructures becomes relevant, the difference between the limit values of the ratio associated to different values of $p$ grows. The limit values for the normalized energy release rate and for the ratio $\mathcal{E}/\mathcal{E}^{cl}$, denominated respectively as $\mathcal{E}_{\text{max}}$ and and $\mathcal{E}_{\text{max}}/\mathcal{E}^{cl}$, are reported in Fig. 11 as functions of $h_{0}$. As we can expect on the basis of previous considerations, the limit value for the ratio $\mathcal{E}_{\text{max}}/\mathcal{E}^{cl}$ is zero for $h_{0}<h_{0}^{}$, and it presents a constant non-zero value for $h_{0}>h_{0}^{}$. In agreement with Fig. 10, it can be noted that for $\eta=-0.9$ and $h_{0}>h_{0}^{}$ $\mathcal{E}_{\text{max}}/\mathcal{E}^{cl}\approx 1$. Figure 9: Variation of the energy release rate with the normalized crack tip speed. Figure 10: Variation of the ratio $\mathcal{E}/\mathcal{E}^{cl}$ with the normalized crack tip speed. Figure 11: Variation of the maximum value of the energy release rate and of the ratio $\mathcal{E}_{\text{max}}/\mathcal{E}^{cl}$ with $h_{0}$ plotted for $p=0$ and $L/\ell=10$. Figure 12: Variation of the normalized energy release rate and of the ratio $\mathcal{E}/\mathcal{E}^{cl}$ with $L/\ell$ plotted for $p=1,2$, $m=0.3$ and $h_{0}=0.6$. In Fig. 12 the variation of the normalized energy release rate and of the ratio $\mathcal{E}/\mathcal{E}^{cl}$ are plotted as functions of $L/\ell$ for $p=1$, $m=0.3$, $h_{0}=0.707$ and $\eta=\left\\{-0.9,\ 0,\ 0.9\right\\}$. The energy release rate tends to zero in the limit $L/\ell\rightarrow 0$, then it increases until it reaches a maximum for $L/\ell\approx 0.5$ and then it start decreasing. This behaviour means that, due to the shielding effect induced by microstructures, for small values of $L/\ell<0.5$, corresponding to a major localization of the applied loading, less energy is provided for propagating the crack at constant speed, and then fracture advancing is hindered. This shielding effect is also shown by profiles of $\mathcal{E}/\mathcal{E}^{cl}$. Indeed, for $L/\ell<0.5$, if a highly concentrated load is applied close to the crack tip in couple stress materials, then $\mathcal{E}/\mathcal{E}^{cl}<1$ and less energy is provided in order to propagate the crack with respect to classical elastic media. Differently, for $L/\ell>0.5$ a weakening effect analogous to that observed in Fig. 10 is detected: $\mathcal{E}/\mathcal{E}^{cl}>1$ and more energy is provided with respect to elastic materials in order to propagate the crack, such that crack propagation is favored. It is important to observe that, for all sets of microstructural parameters, as $\ell\rightarrow 0$ and then $L/\ell\rightarrow+\infty$ the ratio $\mathcal{E}/\mathcal{E}^{cl}$ tends to one, and the material assumes the classical elastic behaviour. This behaviour is in agreement with the effects observed for plane strain problems in Gourgiotis and Georgiadis (2008) and Gourgiotis et al. (2011). It means that as the characteristic scale lengths of the material decrease, couple stress effects becomes negligible, and then the material behaviour is identical to that of a classical elastic body for what concerns crack initiation and propagation. This result is validated by means of the analytical evaluation of the limit of the ratio $\mathcal{E}/\mathcal{E}^{cl}$ as $\ell\rightarrow 0$, reported in the next Section. ## 6 Limit of the energy release rate as $\ell\rightarrow 0$ for a general loading function $\tau(X)$ In order to validate the numerical results illustrated in the previous section, the asymptotic behaviour of the dynamic energy release rate (70) as $\ell\rightarrow 0$ is studied. For this purpose, the evaluation of the limit of the Liouville constant $F$ as $\ell\rightarrow 0$ is needed. Using explicit expression (47) together with relation (40) and Cauchy integral formula, this constant becomes $F={\mathcal{G}}^{-}(-i\zeta/\ell)=-\frac{1}{2\pi iT_{0}}\displaystyle\int_{-\infty}^{\infty}\frac{k^{+}(s\ell)\overline{\tau}^{+}(s)}{(s\ell)_{+}^{1/2}(s+i\zeta/\ell)}ds.$ (73) Introducing the definition of Fourier transform of the loading function, and remembering that $k^{+}(z)=1+O(z)$ for $\|z\|\rightarrow\infty$ (see Mishuris et al. (2013) for details), the limit of (73) can be written as $\displaystyle\lim_{\ell\rightarrow 0}F$ $\displaystyle=-\frac{1}{2\pi iT_{0}}\lim_{\ell\rightarrow 0}\left[\displaystyle\int_{-\infty}^{0}\tau(X)dX\displaystyle\int_{-\infty}^{\infty}\frac{e^{isX}}{(s\ell)_{+}^{1/2}(s+i\zeta/\ell)}ds\right]$ $\displaystyle=-\frac{1}{2\pi iT_{0}}\lim_{\ell\rightarrow 0}\left[\displaystyle\int_{-\infty}^{0}\tau(X)\frac{\|X\|^{1/2}}{\ell^{1/2}}dX\int_{-\infty}^{\infty}\frac{e^{-iy}}{y_{+}^{1/2}(y+i\|X\|\zeta/\ell)}dy\right]$ $\displaystyle=-\frac{1}{T_{0}}\lim_{\ell\rightarrow 0}\left[p\left(i\frac{\|X\|\zeta}{\ell}\right)\cdot\int_{-\infty}^{0}\tau(X)\frac{\|X\|^{1/2}}{\ell^{1/2}}dX\right],$ (74) where $y=s\|X\|$. Introducing $t=i\|X\|\zeta/\ell$, the integral function $p(i\|X\|\zeta/\ell)$ can be written as $p\left(t\right)=\frac{1}{2\pi i}\int_{-\infty}^{\infty}\frac{e^{-iy}}{y_{+}^{1/2}(y+t)}dy.$ (75) For $\ell\rightarrow 0$ and then $\|t\|\rightarrow\infty$, $p(t)$ exhibits the following asymptotic behaviour $p(t)=\frac{p_{1}}{t}+O\left(\frac{1}{t^{2}}\right)=\frac{1}{t\sqrt{\pi}(-i)^{1/2}_{+}}+O\left(\frac{1}{t^{2}}\right),\quad\mbox{for}\quad\|t\|\rightarrow+\infty,$ (76) where $p_{1}$ is given by the integral $p_{1}=\frac{1}{2\pi i}\int_{-\infty}^{\infty}\frac{e^{-iy}}{y_{+}^{1/2}}dy=\frac{1}{\sqrt{\pi}(-i)^{1/2}_{+}},$ (77) Substituting expression (75) into the limit (6), it finally becomes $\lim_{\ell\rightarrow 0}F=-\lim_{\ell\rightarrow 0}\left[\frac{\ell^{1/2}}{\sqrt{\pi}i(-i)^{1/2}_{+}\zeta T_{0}}\int_{-\infty}^{0}\tau(X)\|X\|^{-1/2}dX\right].$ (78) Using expression (78), the limit for $\ell\rightarrow 0$ of the energy release rate (70) can be evaluated: $\displaystyle\lim_{\ell\rightarrow 0}\mathcal{E}$ $\displaystyle=\lim_{\ell\rightarrow 0}\frac{2iF^{2}T_{0}^{2}}{G\ell\Upsilon(h_{0},m,\eta)}$ $\displaystyle=\frac{2}{G\Upsilon(h_{0},m,\eta)\pi\zeta^{2}}\left(\int_{-\infty}^{0}\tau(X)\|X\|^{-1/2}dX\right)^{2}$ $\displaystyle=\frac{1}{\pi G\sqrt{1-m^{2}}}\left(\int_{-\infty}^{0}\tau(X)\|X\|^{-1/2}dX\right)^{2}=\mathcal{E}_{cl}.$ (79) The final result of the limit (6) coincides with the definition of energy release rate for a steady-state crack propagating in classical elastic material. It is important to note that expression (6) is valid for any arbitrary loading acting on the crack faces. This is in perfect agreement with numerical examples presented in Section 5, which show that $\mathcal{E}/\mathcal{E}_{cl}\rightarrow 1$ for $\ell\rightarrow 0$ and then $L/\ell\rightarrow+\infty$. As a consequence, we can say that if $\ell$ and then both characteristic scale lengths $\ell_{t}$ and $\ell_{b}$ tend to zero, couple stress effects disappear regardless of the applied loading, and then the material behaviour is identical to that of a classical elastic body for what concerns crack initiation and propagation. ## 7 Conclusions The influence of size effects due to microstructures on antiplane dynamic crack propagation in elastic materials is investigated by means of indeterminate couple stress theory. Sub-Rayleigh regime for the crack propagation in couple stress media is defined, and the behaviour of the dynamic energy release rate and of the maximum total shear stress is studied considering several different loading distributions applied at the crack faces. In the cases where the crack tip speed approaches the shear waves velocity in classical elastic media or altenatively the minimum couple stress surface waves propagation velocity in the material, a finite limit value for the energy release rate is detected. The analysis shows that if the applied loading is localized around a maximum close to the crack tip, its action is shielded by the microstructural effects. Conversely, as the profile of the applied loading becomes more uniformly distributed away from the crack tip a greater amount of energy is provided for propagating the crack, and a weakening effect is observed. Since the predicted shielding and weakening phenomena can strongly influence the level of stress ahead of the crack tip, the analytical results derived in the present work can represent an important contribution for modelling the mechanical behaviour of microstructured materials. The asymptotic behaviour of the energy release rate in the limit of vanishing material characteristic lengths is studied: numerical examples show that as the microstructural lengths decrease the energy release rate approaches the classical elasticity result. These numerical findings are validated by means of a rigorous demonstration. We prove that, independently of the applied loading, in this limit the energy release rate for couple stress materials tends exactly to the energy release rate for classical elastic materials. This is an important proof of the fact that as the characteristic scale length becomes negligibly small, size effects vanish and then the material behaviour is identical to that of a classical elastic body for what concerns dynamic crack propagation. ## Acknowledgements L.M. gratefully thank financial support of the Italian Ministry of Education, University and Research in the framework of the FIRB project 2010 “Structural mechanics models for renewable energy applications”, A.P. and G.M. gratefully acknowledge the support from European Union FP7 projects under contract numbers PCIG13-GA-2013-618375-MeMic and PIAP-GA-2011-286110-INTERCER2, respectively. ## Appendix A In this Appendix the analytical expression for the Liouville constant (47) is derived. This constant is defined as follows $F=\frac{\displaystyle\int_{-\infty}^{\infty}\frac{\mathcal{G}^{-}(s)ds}{(s\ell)_{-}^{1/2}\Psi(s\ell)k^{-}(s\ell)}}{\displaystyle\int_{-\infty}^{\infty}\frac{ds}{(s\ell)_{-}^{1/2}\Psi(s\ell)k^{-}(s\ell)}}.$ (80) Commonly, this constant is computed by means of numerical integration procedures. In order to estimate it analytically, we need to calculate explicitly the following two integrals $\displaystyle I_{1}$ $\displaystyle=\displaystyle\int_{-\infty}^{\infty}\frac{\mathcal{G}^{-}(s)ds}{(s\ell)_{-}^{1/2}\Psi(s\ell)k^{-}(s\ell)},$ (81) $\displaystyle I_{2}$ $\displaystyle=\displaystyle\int_{-\infty}^{\infty}\frac{ds}{(s\ell)_{-}^{1/2}\Psi(s\ell)k^{-}(s\ell)}.$ (82) These integrals can be represented as limits for $r\rightarrow\infty$: $\displaystyle I_{1}$ $\displaystyle=\lim_{r\rightarrow\infty}\displaystyle\int_{-r}^{+r}\frac{\mathcal{G}^{-}(s)ds}{(s\ell)_{-}^{1/2}\Psi(s\ell)k^{-}(s\ell)}=\lim_{r\rightarrow\infty}I_{1}(r),$ (83) $\displaystyle I_{2}$ $\displaystyle=\lim_{r\rightarrow\infty}\displaystyle\int_{-r}^{+r}\frac{ds}{(s\ell)_{-}^{1/2}\Psi(s\ell)k^{-}(s\ell)}=\lim_{r\rightarrow\infty}I_{2}(r).$ (84) The definite integrals $I_{1}(r)$ and $I_{2}(r)$ can be evaluated considering the closed integration path in the complex plane illustrated in Fig. 13 $\displaystyle I_{1}(r)$ $\displaystyle=\frac{1}{\ell}\oint_{\Gamma_{r}}\frac{\mathcal{G}^{-}(z/\ell)dz}{z_{-}^{1/2}\Upsilon(z+i\zeta)(z-i\zeta)k^{-}(z)}-\frac{1}{\ell}\int_{C_{r}}\frac{\mathcal{G}^{-}(z/\ell)dz}{z_{-}^{1/2}\Upsilon(z+i\zeta)(z-i\zeta)k^{-}(z)},$ (85) $\displaystyle I_{2}(r)$ $\displaystyle=\frac{1}{\ell}\oint_{\Gamma_{r}}\frac{dz}{z_{-}^{1/2}\Upsilon(z+i\zeta)(z-i\zeta)k^{-}(z)}-\frac{1}{\ell}\int_{C_{r}}\frac{dz}{z_{-}^{1/2}\Upsilon(z+i\zeta)(z-i\zeta)k^{-}(z)},$ (86) where $z=s\ell$, and the function $\Psi(z)$ given by expression (35) has been decomposed as follows $\Psi(z)=\Upsilon z^{2}+2\sqrt{1-m^{2}}=\Upsilon(z+i\zeta)(z-i\zeta),$ (87) where $\zeta$ is given by $\zeta=\sqrt{\frac{2\sqrt{1-m^{2}}}{\Upsilon}}.$ (88) Figure 13: Integration path in the complex plane considered for the evaluation of $I_{1}$ and $I_{2}$. Remembering the asymptotic behaviour of the function $\mathcal{G}^{-}$ studied in Section 3 (see expression (42)) and of $k^{-}(z)$ reported in Mishuris et al. (2013), it can be easily verified that: $\displaystyle\lim_{\|z\|\rightarrow\infty}\frac{z\mathcal{G}^{-}(z/\ell)}{z_{-}^{1/2}\Upsilon(z+i\zeta)(z-i\zeta)k^{-}(z)}$ $\displaystyle=0$ (89) $\displaystyle\lim_{\|z\|\rightarrow\infty}\frac{z}{z_{-}^{1/2}\Upsilon(z+i\zeta)(z-i\zeta)k^{-}(z)}$ $\displaystyle=0.$ (90) Since the conditions (89) and (90) are satisfied, for the estimation lemma (Arfken and Weber, 2005), the integrals along $C_{r}$ vanish in the limit $r\rightarrow\infty$ $\displaystyle\lim_{r\rightarrow\infty}\int_{C_{r}}\frac{\mathcal{G}^{-}(z/\ell)dz}{z_{-}^{1/2}\Upsilon(z+i\zeta)(z-i\zeta)k^{-}(z)}$ $\displaystyle=0,$ (91) $\displaystyle\lim_{r\rightarrow\infty}\int_{C_{r}}\frac{dz}{z_{-}^{1/2}\Upsilon(z+i\zeta)(z-i\zeta)k^{-}(z)}$ $\displaystyle=0.$ (92) and then the integrals (83) and (83) can be evaluated using Cauchy integral formula (Roos, 1969). Since the only singularity contained in the integration contour is the one at $z=-i\zeta$, the final result is $\displaystyle I_{1}$ $\displaystyle=\frac{1}{\ell}\lim_{r\rightarrow\infty}\oint_{\Gamma_{r}}\frac{\mathcal{G}^{-}(z/\ell)dz}{z_{-}^{1/2}\Upsilon(z+i\zeta)(z-i\zeta)k^{-}(z)}=\frac{\pi}{\ell\Upsilon}\frac{\mathcal{G}^{-}(-i\zeta/\ell)}{(-i\zeta)_{-}^{1/2}\zeta k^{-}(-i\zeta)},$ (93) $\displaystyle I_{2}$ $\displaystyle=\frac{1}{\ell}\lim_{r\rightarrow\infty}\oint_{\Gamma_{r}}\frac{dz}{z_{-}^{1/2}\Upsilon(z+i\zeta)(z-i\zeta)k^{-}(z)}=\frac{\pi}{\ell\Upsilon}\frac{1}{(-i\zeta)_{-}^{1/2}\zeta k^{-}(-i\zeta)}.$ (94) The analytical expression for the constant $F$ is finally obtained by the ratio between $I_{1}$ and $I_{2}$: $F=\frac{I_{1}}{I_{2}}=\mathcal{G}^{-}(-i\zeta/\ell)$ (95) ## Appendix B In this Appendix we derive the expression (71) for the energy release rate corresponding to a Mode III steady state propagating crack in a classical isotropic elastic material. For antiplane dynamical problems in classical elasticity the equation of motion (5) becomes $G\Delta u_{3}=\rho\ddot{u}_{3}.$ (96) Since we are interested in studying steady state crack propagation along $x_{1}-$axis, we perform the trasformation $u_{3}(x_{1},x_{2},t)=w(X,y)$ where $X=x_{1}-Vt,y=x_{2}$, (it is the same substitution illustrated in Section 2), and the (96) then becomes: $(1-m^{2})\frac{\partial^{2}w}{\partial X^{2}}+\frac{\partial^{2}w}{\partial y^{2}}=0,$ (97) where $m=v/c_{s}$ and $c_{s}=\sqrt{G/\rho}$. The Cauchy stresses are given by $\sigma_{13}=G\frac{\partial w}{\partial X},\quad\sigma_{23}=G\frac{\partial w}{\partial y}.$ (98) The following conditions, equivalent to those imposed for couple stress materials (see equations (9) and (10)), are assumed on the crack surface, at $y=0$: $\displaystyle\sigma_{23}(y=0)$ $\displaystyle=-\tau(x),\quad-\infty<x<0,$ (99) $\displaystyle w(y=0)$ $\displaystyle=0,\quad 0<x<+\infty,$ (100) where the same distributed loading configuration (26) considered for couple stress materials is applied at the crack faces. An exact solution of the boundary value problem formulated can be obtained by means of Fourier transform and Wiener-Hopf technique. The direct and inverse Fourier transform of an arbitrary function $f(x)$ is defined as follows: $\overline{f}(s,y)=\int_{-\infty}^{+\infty}f(x,y)e^{isx}dx,\quad f(s,y)=\frac{1}{2\pi}\int_{L}\overline{f}(s,y)e^{-isx}ds,$ (101) where $L$ denotes the inversion path within the region of analyticity of the function $\overline{f}(s,y)$ in the complex $s-$plane. Transforming the evolution equation (97) we obtain the following ODE: $\overline{w}^{\prime\prime}-s^{2}(1-m^{2})\overline{w}=0,$ (102) where the prime symbol denotes the total derivative with respect to $y$. The equation (102) possesses the following general solution that is required to be bounded as $y\rightarrow+\infty$: $\overline{w}(s,y)=B(s)e^{-\alpha(s)y},$ (103) where $\alpha(s)=\sqrt{s^{2}(1-m^{2})}$. The transformed stresses are given by: $\overline{\sigma}_{13}=-isG\overline{w},\quad\overline{\sigma}_{23}=G\overline{w}^{\prime}.$ (104) The Fourier transforms of the unknown stress ahead of the crack tip $\sigma_{23}(x>0,y=0)$ and of the crack faces displacements $w(x<0,y=0)$ are defined as follows: $\Sigma_{23}^{+}(s)=\int_{0}^{+\infty}\sigma_{23}(x,y=0)e^{isx}dx,$ (105) $\sigma_{23}(x,y=0)=\frac{1}{2\pi}\int_{D}\Sigma_{23}^{+}(s)e^{-isx}ds,\quad x>0,$ (106) and $W^{-}(s)=\int_{-\infty}^{0}w(x,y=0)e^{isx}dx,$ (107) $w(x,y=0)=\frac{1}{2\pi}\int_{D}W^{-}(s)e^{-isx}ds,\quad x<0,$ (108) where the inversion path is assumed to lie inside the region of analyticity of each transformed function. The transformed stress $\Sigma_{23}^{+}(s)$ is analytic and defined in the lower half complex $s-$plane, $\mbox{Im}s<0$, whereas the transformed displacement $W^{-}(s)$ is analytic and defined in the upper half complex $s-$plane, $\mbox{Im}s>0$. Taking into account (103), and substituting this expression into the (104)(2), in the limit $y\rightarrow 0$ we obtain: $B(s)=W^{-}(s),\quad\Sigma_{23}^{+}(s)=-\alpha(s)GW^{-}(s).$ (109) As a consequence, equation (109) together with the condition (99) provides the following Wiener-Hopf equation connecting the two unknown functions $\Sigma_{23}^{+}(s)$ and $W^{-}(s)$: $\Sigma_{23}^{+}(s)-\overline{\tau}^{-}(s)=-s_{+}^{1/2}s_{-}^{1/2}\nu GW^{-}(s),$ (110) where $\nu=\sqrt{1-m^{2}}$, $\overline{\tau}^{-}(s)$ is the Fourier transform of the loading function (26), defined by expression (31), and the function $\sqrt{s^{2}}$ is factorized as follows (Mishuris et al., 2013): $\sqrt{s^{2}}=s_{+}^{1/2}s_{-}^{1/2},$ (111) where the functions $s_{+}$ and $s_{-}$ are analytic in the upper and in the lower half plane, respectively. Equation (110) can then be rewritten as $\frac{\Sigma_{23}^{+}(s)}{s_{+}^{1/2}}+s_{-}^{1/2}\nu GW^{-}(s)=\frac{T_{0}}{s_{+}^{1/2}(1+isL)^{1+p}}.$ (112) The right-hand side of the Wiener-Hopf equation (112) can be split in the sum of plus and minus functions. Indeed, since the function $s_{+}^{-1/2}$ is analytical in the point $s=i/L$, it can be represented as follows $\frac{1}{s_{+}^{1/2}}=\sum_{j=0}^{p}(1+isL)^{j}H_{j}+H_{p+1}^{+}(s)=\sum_{j=0}^{p}(1+isL)^{j}H_{j}+{\mathcal{I}}^{+}(s)(1+isL)^{p+1}$ (113) where ${\mathcal{I}}^{+}(s)\equiv\frac{H_{p+1}^{+}(s)}{(1+isL)^{p+1}}=\frac{1}{(1+isL)^{p+1}}\left(\frac{1}{s_{+}^{1/2}}-\sum_{j=0}^{p}(1+isL)^{j}H_{j}\right).$ (114) The function ${\mathcal{I}}^{+}(s)$ exhibits the following asymptotic behaviour: ${\mathcal{I}}^{+}(s)=i\frac{H_{p}}{sL}+O(s^{-2}),\ \|s\|\to\infty;\quad{\mathcal{I}}^{+}(s)=\frac{1}{s^{1/2}}+O(1),\ \|s\|\to 0,\quad\mbox{with}\ \mathop{\mathrm{Im}}s>0.$ (115) therefore, the right-hand side of the equation (112) can be written in the form $\frac{T_{0}}{s_{+}^{1/2}(1+isL)^{1+p}}=T_{0}{\mathcal{I}}^{-}(s)+T_{0}{\mathcal{I}}^{+}(s),$ (116) where ${\mathcal{I}}^{-}(s)=\sum_{j=0}^{p}\frac{H_{j}}{(1+isL)^{p+1-j}},$ (117) and ${\mathcal{I}}^{-}(s)=-i\frac{H_{p}}{sL}+O(s^{-2}),\ \|s\|\to\infty;\quad{\mathcal{I}}^{-}(s)=\sum_{j=0}^{p}H_{j}+O(s),\ \|s\|\to 0,\quad\mbox{with}\ \mathop{\mathrm{Im}}s<0.$ (118) The coefficients $H_{j}$ can be computed analitically applying the definition of generalized derivative of a function $s^{\alpha}$ to the case $\alpha=-1/2$: $H_{j}=\frac{(-1)^{j}}{j!}\frac{\sqrt{\pi}}{\Gamma\left(\frac{1}{2}-j\right)}\left(\frac{i}{L}\right)^{-1/2}.$ (119) It has been verified that for any $p$ expression (119) is equivalent to the following integral definition, analogous to the (43) introduced for solving the same crack problem in couple stress materials: $H_{j}=\frac{L}{2\pi}\oint_{\gamma}\left(\frac{1}{(1+isL)^{j+1}}\frac{1}{s_{+}^{1/2}}\right)ds,$ (120) where $\gamma$ is an arbitrary contour centered at the point $s=i/L$ and lying in the analyticity domain. Using decomposition (116), the Wiener-Hopf equation (112) becomes $\frac{\Sigma_{23}^{+}(s)}{s_{+}^{1/2}}-T_{0}\mathcal{I}^{+}(s)=-s_{-}^{1/2}\nu GW^{-}(s)+T_{0}\mathcal{I}^{-}(s)\equiv E(s).$ (121) The functional equation (121) defines the function $E(s)$ only on the real line. In order to evaluate this function, it is first necessary to examine the asymptotic behaviour of the functions $\Sigma_{23}^{+}(s)$ and $W^{-}(s)$. It has been demonstrated that for $X\rightarrow 0\pm$ the stress and the displacement along the crack faces exhibit the following behaviour: $\displaystyle\sigma_{23}(X,y=0)$ $\displaystyle=O(X^{-1/2})\ \mbox{as}\ X\rightarrow 0+,$ (122) $\displaystyle w(X,y=0)$ $\displaystyle=O(X^{1/2})\ \mbox{as}\ X\rightarrow 0-.$ (123) Following the same procedure illustrated for couple stress materials, expressions (122) and (123) can be transformed applying Abel-Tauper type theorems (Roos, 1969): $\displaystyle\Sigma_{23}^{+}(s)$ $\displaystyle=O(s^{-1/2})\ \text{as}\ \|s\|\rightarrow\infty\ \mbox{with}\ \mathop{\mathrm{Im}}s>0,$ (124) $\displaystyle W^{-}(s)$ $\displaystyle=O(s^{-3/2})\ \text{as}\ \|s\|\rightarrow\infty\ \mbox{with}\ \mathop{\mathrm{Im}}s<0.$ (125) Considering the asymptotic behaviour of $\Sigma_{23}$ and $W^{+}$ and observing expressions (114) and (117), we note that the first member of the Wiener-Hopf equation (121) is a bounded analytic function for $\mathop{\mathrm{Im}}s>0$ that is zero as $\|s\|\rightarrow\infty$, whereas the second member is a bounded analytic function for $\mathop{\mathrm{Im}}s<0$ that is also zero as $\|s\|\rightarrow\infty$. Then, for the theorem of analytic continuation, the two members define one and the same analytic function $E(s)$ over the entire complex $s-$plane. Moreover, Liouville’s theorem leads to the conclusion that $E(s)=0$. As a consequence, the transformed shear stress and displacement are given by: $\displaystyle\Sigma_{23}^{+}(s)$ $\displaystyle=T_{0}\mathcal{I}^{+}(s)s_{+}^{1/2},\ \mathop{\mathrm{Im}}s>0,$ (126) $\displaystyle W^{-}(s)$ $\displaystyle=\frac{T_{0}\mathcal{I}^{-}(s)}{\nu Gs_{-}^{1/2}},\ \mathop{\mathrm{Im}}s<0.$ (127) Evaluating the asymptotic leading term $\|s\|\rightarrow\infty$ of these expressions, we get: $\displaystyle\Sigma_{23}^{+}(s)$ $\displaystyle=\frac{iT_{0}H_{p}}{L}s_{-}^{-1/2}+O(s^{-1})\ \text{as}\ \|s\|\rightarrow\infty\ \text{with}\ \mathop{\mathrm{Im}}s>0,$ (128) $\displaystyle W^{-}(s)$ $\displaystyle=-\frac{iT_{0}H_{p}}{\nu GL}s_{-}^{-3/2}+O(s^{-2})\ \text{as}\ \|s\|\rightarrow\infty\ \text{with}\ \mathop{\mathrm{Im}}s<0,$ (129) applying the transformation formula (64) to the (128) and (129) we finally obtain: $\displaystyle\sigma_{23}(X,y=0)$ $\displaystyle=\frac{i^{1/2}T_{0}F_{p}}{L\sqrt{\pi}}X^{-1/2}=\frac{(-1)^{p}}{p!}\frac{T_{0}}{\sqrt{L}\Gamma\left(\frac{1}{2}-p\right)}X^{-1/2}\ \mbox{as}\ X\rightarrow 0+,$ (130) $\displaystyle w(X,y=0)$ $\displaystyle=-\frac{2i^{-3/2}T_{0}F_{p}}{\nu GL\sqrt{\pi}}(-X)^{1/2}=\frac{(-1)^{p}}{p!}\frac{2T_{0}}{\nu G\sqrt{L}\Gamma\left(\frac{1}{2}-p\right)}(-X)^{1/2}\ \mbox{as}\ X\rightarrow 0-.$ (131) The shear traction expression (130) can then be used for calculating the stress intensity factor: $K_{III}^{cl}=\lim_{x\rightarrow 0}\sqrt{2\pi X}\sigma_{23}(X,y=0)=\frac{(-1)^{p}}{p!\Gamma\left(\frac{1}{2}-p\right)}\sqrt{\frac{2\pi}{L}}T_{0}.$ (132) The dynamic J-integral for an antiplane steady state propagating crack is evaluated using the (130) and (131) and performing the same procedure illustrated for couple stress materials, choosing a rectangular shaped path surrounding the tip and applying the Fisher theorem: $\mathcal{E}^{cl}=\frac{iF_{p}^{2}T_{0}^{2}}{\nu GL^{2}}=\frac{T_{0}^{2}K_{p}^{2}}{GL}\frac{1}{\sqrt{1-m^{2}}},$ (133) where $K_{p}=\frac{(-1)^{p}}{p!}\frac{\sqrt{\pi}}{\Gamma\left(\frac{1}{2}-p\right)}.$ (134) ## References Arfken and Weber (2005) Arfken, G. 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Solids 19, 335–368.	arxiv-papers	2013-11-21T08:17:31	2024-09-04T02:49:54.056125	{ "license": "Public Domain", "authors": "L. Morini, A. Piccolroaz, G. Mishuris", "submitter": "Lorenzo Morini", "url": "https://arxiv.org/abs/1311.5329" }
1311.5348		arxiv-papers	2013-11-21T09:57:19	2024-09-04T02:49:54.067450	{ "license": "Public Domain", "authors": "Zehra Bozkurt, Ismail G\\\"ok, Yusuf Yayl{\\i} and Faik Nejat Ekmekci", "submitter": "Zehra Bozkurt", "url": "https://arxiv.org/abs/1311.5348" }
1311.5358	# On wavenumber spectra for sound within subsonic jets A. Agarwal S. Sinayoko R. D. Sandberg Lecturer, Department of Engineering, University of Cambridge ([email protected])Brunel Research Fellow, ISVR, University of Southampton ([email protected])Professor, AFM, University of Southampton ([email protected]) ###### Abstract This paper clarifies the nature of sound spectra within subsonic jets. Three problems, of increasing complexity, are presented. Firstly, a point source is placed in a two-dimensional plug flow and the sound field is obtained analytically. Secondly, a point source is embedded in a diverging axisymmetric jet and the sound field is obtained by solving the linearised Euler equations. Finally, an analysis of the acoustic waves propagating through a turbulent jet obtained by direct numerical simulation is presented. In each problem, the pressure or density field are analysed in the frequency-wavenumber domain. It is found that acoustic waves can be classified into three main frequency- dependent groups. A physical justification is provided for this classification. The main conclusion is that, at low Strouhal numbers, acoustic waves satisfy the d’Alembertian dispersion relation. ## 1 Introduction Our initial motivation for understanding the sound spectra in jets came from the article by Goldstein, (2005) in which he proposed that it may be possible to identify the “true” sources of noise in jets if the radiating and non- radiating components could be separated. It is possible to achieve this separation for the Euler equations linearised about either a steady uniform base flow (Chu and Kovasznay,, 1958) or a steady parallel flow (Agarwal et al.,, 2004). Unfortunately, the separation techniques presented in these papers cannot be applied to full nonlinear Navier-Stokes equations and hence are not useful for realistic jets. Sinayoko et al., (2011) showed that filtering in the frequency-wavenumber domain is an effective technique for separating radiating and non-radiating components in subsonic jets. Their filtering technique relied on the dispersion relation $k=\|\omega\|/c_{\infty}$ (where $k$ denotes the magnitude of the wavenumber, $\omega$ the angular frequency and $c_{\infty}$ the farfield speed of sound) satisfied by acoustic waves radiating to a quiescent farfield. But inside the jet we can have waves that travel supersonically relative to the ambient medium. In this paper, we define acoustic waves in jets as those satisfying the dispersion relation $k_{z}\leq\|\omega\|/c_{\infty}$, where $k_{z}$ denotes the axial wavenumber. In other words, in the axial direction, acoustic waves travel either upstream ($k_{z}\leq 0$) or downstream ($0\leq k_{z}\leq\|\omega\|/c_{\infty}$; in the downstream case, the axial phase speed is therefore sonic or supersonic. The characterization of acoustic waves by supersonic axial phase speed was used by Freund, (2001), Cabana et al., (2008), Tinney and Jordan, (2008) and Obrist, (2009). The results presented in this paper support this definition of acoustic waves. Figure 1: Algorithm for filtering out the radiating field. For numerical implementation, $W(\bm{k},\omega)$ has a finite width (see Sinayoko et al., (2011) for details) The filtering technique is represented diagrammatically in figure 1. The radiating part $q^{\prime}(\bm{x},t)$ of a flowfield variable $q(\bm{x},t)$ can be obtained by convolving $q$ with an appropriate filter function $w(\bm{x},t)$, which is defined in the frequency-wavenumber domain ($W(\bm{k},\omega)$). Sinayoko et al., (2011) considered a model problem in which the base flow corresponding to the experiment of the Mach 0.9, Re 3600 jet by Stromberg et al., (1980) was excited by two instability waves at nondimensional frequencies of 2.2 and 3.4. These waves interact nonlinearly to produce acoustic waves at the difference frequency of 1.2. The density field at frequency 1.2 is shown in figure 2 (a). In this problem, both acoustic and hydrodynamic waves are being generated. The Fourier transform of this field, $P(\bm{k},\omega)$, is shown in figure 2 (b). Multiplying $P(\bm{k},\omega)$ with $W(\bm{k},\omega)$ as defined in figure 1 gives $P^{\prime}(\bm{k},\omega)$, which is shown in figure 2 (d). The Fourier transform of the remaining field, $\bar{P}(\bm{k},\omega)=P-P^{\prime}$, is shown in figure 2 (f). The corresponding density fields in the space-time domain are obtained by applying the inverse Fourier transforms and are shown in figures 2 (c) and (e). Radiating components have captured all the acoustic waves. Clearly the acoustic waves have been separated from the hydrodynamic waves. However, the efficacy of the filter is puzzling as it is based on the dispersion relation for sound propagation in a uniform quiescent medium. Inside the jet we do not have a quiescent medium, so how can this dispersion relation separate acoustic waves both outside and inside the jet? \| ---\|--- \| \| (a) $\rho_{\omega=1.2}(\mathbf{x},t_{0})$ (b) $P(\mathbf{k},1.2)$ (c) $\rho^{\prime}_{\omega=1.2}(\mathbf{x},t_{0})$ (d) $P^{\prime}(\mathbf{k},1.2)$ (e) $\overline{\rho}_{\omega=1.2}(\mathbf{x},t_{0})$ (f) $\overline{P}(\mathbf{k},1.2)$ Figure 2: Density field in a turbulent jet of exit radius $a$ plotted in the physical domain (snapshot in left column) and wavenumber domain (magnitude of Fourier transform in right column), with colour range $[-5\times 10^{-5},5\times 10^{-5}]$ and $[0,0.5]$ respectively, for a given normalized frequency $\omega=1.2$ and time $t_{0}$ (c.f. equation (19)). The top row shows the density field $\rho$, the middle row the radiating field $\rho^{\prime}$ and the bottom row the non-radiating field $\overline{\rho}$. In order to answer this question, we have constructed a simple model problem for sound radiation from a point source in a two-dimensional plug flow (§2). We show that, for this problem, it is possible to obtain an analytical expression for the Fourier transform for both the axial and cross-stream directions. This is a crucial step in obtaining the spectral characteristics of sound propagation and it enables us to understand and explain the observed acoustic wavenumber spectra for different frequencies. The solution to this problem also indicates how to identify acoustic waves for more general (turbulent) jets. In §3 we consider a more general problem of sound radiation from a point source in a diverging cylindrical jet and in §4 we identify the acoustic waves using data obtained from a DNS of a Mach 0.84, Re 7200 turbulent jet. Even though our motivation for identifying acoustic waves in turbulent jets stems from a particular application as mentioned above, this work can be used in other ways. For example, the flow filtering technique defined here could be used to separate convecting and propagating components in a jet. This can help define various source models or correlate the nearfield hydrodynamic data to the acoustic farfield to identify the noise producing regions in the jet. The technique can also be used to correctly identify the radiating part of Lighthill’s source term (Freund, (2001), Cabana et al., (2008), Sinayoko and Agarwal, (2012)). ## 2 Model problem Figure 3: Schematic sketch of a point source located at the origin inside a two-dimensional plug flow of width $2a$. The flow Mach number is $M>0$ for $\|y\|<a$ (region II) and $M=0$ otherwise (region I). Consider the problem of a time-harmonic monopole point mass source, $\rho_{o}\delta(\bm{x})\exp(-i\omega t)$ ($\rho_{o}$ denotes ambient density), embedded in a plug flow. Several authors (e.g. Morgan, (1975), Mani, (1972)) have considered the problem of farfield sound radiation from a point source in axisymmetric jets. The main difference between their analysis and ours is that we seek the spectral content in the frequency-wavenumber domain instead of the farfield characteristics of sound in the physical domain. For simplicity, we consider a two-dimensional problem. The problem set up is described in figure 3. Assuming an $\exp(-i\omega t)$ response ($\phi(\bm{x},t)=\phi(\bm{x};\omega)\exp(-i\omega t)$), the linear velocity potential $\phi_{I}$ for small disturbances satisfies, in region I (outside the jet), $\nabla^{2}\phi_{I}+\kappa^{2}\phi_{I}=0,$ (1) and in region II, inside the jet, $\nabla^{2}\phi_{II}-\left(-i\kappa+M\frac{\partial}{\partial x}\right)^{2}\phi_{II}=\delta(x)\delta(y),$ (2) where $\nabla^{2}$ denotes the Laplacian operator, $\kappa=\omega/c$ is the acoustic wavenumber, and $c$ is the speed of sound, which is uniform for the present problem. Because the velocity potential and pressure are symmetric about the mid-plane axis of the jet ($y=0$), it is sufficient to solve the problem for $y\geq 0$. Continuity of pressure at $y=a$ requires that $p=-\rho_{o}D\phi/Dt$ be continuous (D/Dt denotes material derivative), therefore $\left(-i\kappa+M\frac{\partial}{\partial x}\right)\phi_{II}(x,a)=-i\kappa\phi_{I}(x,a).$ (3) The kinematic constraint requires that particle displacement $\eta$ at the interface be continuous. Therefore, $\frac{\partial}{\partial y}\phi_{I}(x,a)=\frac{\partial}{\partial t}\eta(x,a),$ (4) $\frac{\partial}{\partial y}\phi_{II}(x,a)=\frac{\partial}{\partial t}\eta(x,a)+Mc\frac{\partial}{\partial x}\eta(x,a).$ (5) Applying the Fourier transform in $x$, defined by ${\hat{\phi\mkern 3.0mu}\mkern-3.0mu}{}(k_{x})=\int_{-\infty}^{\infty}\phi(x)e^{-ik_{x}x}dx,$ (6) to equations (1) and (2), we get, $\frac{d^{2}{\hat{\phi\mkern 3.0mu}\mkern-3.0mu}{}_{I}}{dy^{2}}+(\kappa^{2}-k_{x}^{2}){\hat{\phi\mkern 3.0mu}\mkern-3.0mu}{}_{I}=0,$ (7) $\frac{d^{2}{\hat{\phi\mkern 3.0mu}\mkern-3.0mu}{}_{II}}{dy^{2}}+\left((\kappa- k_{x}M)^{2}-k_{x}^{2}\right){\hat{\phi\mkern 3.0mu}\mkern-3.0mu}{}_{II}=\delta(y).$ (8) Application of the Fourier transform in the axial direction to Eqs. (3) – (5) gives $(\kappa-k_{x}M){\hat{\phi\mkern 3.0mu}\mkern-3.0mu}{}_{II}(k_{x},a)=\kappa{\hat{\phi\mkern 3.0mu}\mkern-3.0mu}{}_{I}(k_{x},a),$ (9) $\kappa\frac{d{\hat{\phi\mkern 3.0mu}\mkern-3.0mu}{}_{II}}{dy}(k_{x},a)=(\kappa- k_{x}M)\frac{d{\hat{\phi\mkern 3.0mu}\mkern-3.0mu}{}_{I}}{dy}(k_{x},a).$ (10) Let $\beta^{2}=\kappa^{2}-k_{x}^{2}$ and $\gamma^{2}=(\kappa- k_{x}M)^{2}-k_{x}^{2}$. The locations of the branch cuts for $\beta$ and $\gamma$ are shown in figure 4. The branch of the square roots are chosen such that both $\beta$ and $\gamma$ are equal to $\kappa$ for $k_{x}=0$. Acoustic waves propagate to the farfield only when $\beta$ is real, i.e. when $\|k_{x}\|<\kappa$. Therefore, we will focus on this range of wavenumbers. In region I for outgoing waves to infinity ${\hat{\phi\mkern 3.0mu}\mkern-3.0mu}{}_{I}=Ae^{i\beta y}.$ (11) Taking into account the symmetry about the mid-plane axis, the solution in region II is given by ${\hat{\phi\mkern 3.0mu}\mkern-3.0mu}{}_{II}=2B\cos(\gamma y)-\frac{i}{2}\frac{e^{i\gamma y}}{\gamma}.$ (12) Application of conditions (9) and (10) yields $\displaystyle A$ $\displaystyle=$ $\displaystyle-\frac{i\kappa\exp(-ia\beta)(\kappa- Mk_{x})}{2\Delta(\kappa,k_{x},\gamma)},$ (13) $\displaystyle B$ $\displaystyle=$ $\displaystyle\frac{\exp(ia\gamma)\left(\kappa^{2}\gamma-\beta(\kappa- Mk_{x})^{2}\right)}{4\gamma i\Delta(\kappa,k_{x},\gamma)},$ (14) where $\Delta(\kappa,k_{x},k_{y})=\beta\cos(ak_{y})(\kappa- Mk_{x})^{2}-i\kappa^{2}k_{y}\sin(ak_{y})$. Note that for $M=0$ we recover the free-field Green’s function of the Helmholtz equation. Figure 4: Location of the branch cuts for $\beta$ and $\gamma$ (hatched lines) and the Kelvin-Helmholtz instability pole, $k_{x_{0}}$, in the complex $k_{x}$ domain. Equation $\Delta(\kappa,k_{x},\gamma)=0$ is the dispersion relation for the hydrodynamic wave. The roots of this equation represent poles of $\hat{\phi}$ in the complex $k_{x}$ domain. For the present problem there is one root, $k_{x_{0}}$, which is located in the lower-half $k_{x}$-plane and is associated with a Kelvin-Helmholtz instability wave; it is purely hydrodynamic (Agarwal et al.,, 2004) and does not affect our analysis. If our problem had a pipe (or two splitter plates in 2D) the Kelvin-Helmholtz instability wave would have an amplitude given by the edge condition, usually an unsteady Kutta condition (see for example, Crighton, (1985)). Again the Kelvin-Helmholtz wave would be subsonic and hence would not interfere with our analysis. Defining the Fourier transform in $y$ by $\hat{\hat{\phi}}(k_{x},k_{y})=\int_{-\infty}^{\infty}{\hat{\phi\mkern 3.0mu}\mkern-3.0mu}{}(k_{x},y)e^{-ik_{y}y}dy,$ (15) the Fourier transform of the pressure field can be written as $\displaystyle\hat{\hat{p}}(k_{x},k_{y})=-2i\rho_{o}\int_{0}^{\infty}\left[\kappa{\hat{\phi\mkern 3.0mu}\mkern-3.0mu}{}_{II}(k_{x},y)H(a-y)+\right.$ $\displaystyle\left.(\kappa-k_{x}M){\hat{\phi\mkern 3.0mu}\mkern-3.0mu}{}_{I}(k_{x},y)H(y-a)\right]\cos(k_{y}y)\,dy.$ (16) This integral can be evaluated analytically: $\displaystyle\hat{\hat{p}}(k_{x},k_{y})$ $\displaystyle=$ $\displaystyle\frac{\rho_{o}(\kappa- Mk_{x})}{\Delta(\kappa,k_{x},\gamma)}\left[\frac{i\left(\Delta(\kappa,k_{x},k_{y})-\Delta(\kappa,k_{x},\gamma)\right)}{k_{y}^{2}-\gamma^{2}}+\right.$ (17) $\displaystyle\left.\frac{\kappa^{2}\left(k_{y}\sin(ak_{y})+i\beta\cos(ak_{y})\right)}{k_{y}^{2}-\beta^{2}}\right].$ ### 2.1 Frequency-wavenumber spectra The acoustic-wave solution in the physical domain can be obtained by applying the inverse Fourier transforms to Eq. (17) (for details on the geometry of the Fourier integration contours $F_{x}$ and $F_{y}$ and the implications on causality, see Agarwal et al., (2004)) $p(x,y;\omega)=\int_{F_{x}}\frac{dk_{x}}{2\pi}e^{ik_{x}x}\int_{F_{y}}\frac{dk_{y}}{2\pi}e^{ik_{y}y}\hat{\hat{p}}(k_{x},k_{y}).$ (18) If we look at the $k_{y}$ integral, $\hat{\hat{p}}(k_{x},k_{y})$, from Eq. (17), has two terms. The second term has poles at $k_{y}=\pm\beta$. Using the method of residues, it can be shown that only these poles contribute to the integral. Therefore, regardless of the frequency, only the wavenumbers that satisfy the dispersion relation is $k_{x}^{2}+k_{y}^{2}=\kappa^{2}$ contribute to the integral. We refer to this as the radiation circle. The other term in the integrand has zeroes in the denominator at $k_{y}=\pm\gamma$, which corresponds to an ellipse in the wavenumber domain. Note that these zeroes do not represent poles as the numerator also goes to zero at $k_{y}=\pm\gamma$. Therefore, the contribution from the integrand is more complicated for this term. Further insight can be obtained by plotting the integrand as a function of frequency. Recall that $\kappa=\omega/c$ and from hereon, for brevity, the reduced frequency $\kappa a$ is referred to as frequency. Figure 5 shows the wavenumber spectra $\|\hat{\hat{p}}(k_{x},k_{y})\|$ for $M=0.9$ for four different frequencies. At low frequencies ($\kappa a\ll 1$) most of the energy is concentrated around the radiation circle (figure 5(a)). For higher frequencies ($\kappa a=O(1)$), the energy is concentrated along the radiation circle as well, but there is a small amount of energy around the vertical line $k_{x}=\kappa/(1+M)$ (figures 5(b) and 5(c)). For very high frequencies ($\kappa a\gg 1$) we see the radiation circle and a part of the ellipse $k_{y}^{2}=\gamma^{2}$ (figure 5(d)). We observe some ringing around the ellipse. For jet noise another useful non-dimensional frequency is the Strouhal number, $St$, based on the jet diameter and exit velocity. It can be shown that $St=\kappa a/(\pi M)$. High-speed jet noise peaks at $St\sim 0.2$ ($\kappa a=0.2M\pi$). This suggests that for filtering out acoustic waves in a jet, around the peak radiation frequency, one need not worry about the ellipse in figure 5 (d). For sound radiation near the peak frequency, the dispersion characteristics are very similar to that of the ordinary wave equation. This explains why Sinayoko et al., (2011) obtained a good separation of acoustic and hydrodynamic fields by using a filter based on the dispersion characteristics of the ordinary wave equation. #### Low frequencies A mathematical justification for this low-frequency result can be obtained as follows. Assume $\kappa a\ll 1$. For acoustic waves, the wavenumbers $k_{x}$ and $k_{y}$ that satisfy the dispersion relation are of the same order of magnitude as $\kappa$, so that $\gamma a\ll 1$ and $k_{y}a\ll 1$. In Eq. (17) if we expand the trigonometric functions in a power series up to the second order, it simplifies to $\displaystyle\hat{\hat{p}}(k_{x},k_{y})\approx\frac{\rho_{o}\beta(\kappa- Mk_{x})\left[\kappa^{2}(i-a\beta)+O\left((a\kappa)^{4}\right)\right]}{\Delta(\kappa,k_{x},\gamma)(k_{y}^{2}-\beta^{2})}$ From this equation it is clear that at low frequencies, the dispersion relation for the problem is $k_{y}^{2}-\beta^{2}=0$, i.e. $k_{x}^{2}+k_{y}^{2}=(\omega/c)^{2}$, which is the dispersion relation for sound propagation through a uniform medium at rest. This indicates that mean flow has a negligible effect on sound propagation at low frequencies. This has been observed experimentally by Cavalieri et al., (2012). Figure 5: Wavenumber spectra of the pressure field, $\|\hat{\hat{p}}\|$, from a point source in a jet (set-up of figure 3) at frequency, (a) $\kappa a=0.1\pi$, (b) $\kappa a=\pi$, (c) $\kappa a=2\pi$, (d) $\kappa a=10\pi$. The density plot uses a linear scale from 0 (white) to $10\pi/\kappa$ (black). The results can be interpreted by considering the potential field $\hat{\phi}(k_{x},y)$ given by Eqs. (11) and (12). Figure 6(a) shows the associated pressure field $\hat{p}(k_{x},y)$ as a function of $y$ for $k_{x}=\kappa/(1+M)$ for two different values of $\kappa a$, $0.1\pi$ and $\pi$. For $k_{x}=\kappa/(1+M)$, $\gamma=0$ and from Eq. (12) it is clear that $\hat{p}$ does not have a wave-like solution for $y\leq a$. For brevity, we consider only the real part of $\hat{p}$, which can be shown to be a constant with respect to $y$ in the limit $\gamma\to 0$. For $y>a$, the response is given by the harmonic function $\exp(i\beta y)$. At low frequencies (e.g., $\kappa a=0.1\pi$), the constant part (when $y<a$) is a very small part of a wavelength (figure 6(a)) and therefore, when we Fourier transform this field, most of the energy is concentrated around $k_{y}=\pm\beta$. A similar argument applies to other values of $k_{x}$. Therefore, the presence of the jet has a negligible effect on the wavenumber spectrum at low frequencies. #### Mid-range frequencies As the frequency increases, the region in the $y$-domain over which the pressure field is a constant (when $k_{x}=\kappa/(1+M)$) occupies a larger part of the wavelength (see the dashed line in figure 6 (a)). This has a significant impact on the Fourier transform (in $y$) of $\hat{p}$. Instead of being concentrated just around $k_{y}=\pm\beta$, the energy in the $k_{y}$ space gets distributed over a range of $k_{y}$ values. The same reasoning applies to other values of $k_{x}$ close to $\kappa/(1+M)$. This explains the vertical patch in the spectra around $k_{x}=\kappa/(1+M)$ in figures 5 (b) and (c). Away from these values, in the range $0<k_{x}<\kappa/(1+M)$ , unless the frequency is very high, the pressure field inside the jet is very similar to that outside the jet. This is illustrated in figure 6(b), which shows the pressure field $\hat{p}(k_{x},y)$ for $k_{x}=0.5\kappa/(1+M)$, for $\kappa a=0.1\pi$ (solid line) and $\kappa a=\pi$ (dashed line). Therefore, for $0<k_{x}<\kappa/(1+M)$, the Fourier transform of the pressure field is concentrated around the radiation circle $k_{y}=\pm\beta$. The energy in the $k_{y}$ direction appears to be contained mainly inside the radiation circle around $k_{x}=\kappa/(1+M)$. This can be explained as follows. The difference between the constant field for $y<a$ and the sinusoid $\exp(i\beta y)$, that would exist in the absence of a jet, is related to the Heaviside function $H(a-y)$. The Fourier transform of this function is given by the sinc function, $\mathrm{sinc}(k_{y}a)$. Most of the energy of this function is contained in the first lobe $k_{y}a<\pi$. This is why there is little energy outside the radiation circle for $\kappa a=\pi$ (figure 5(b)). Thus the energy content inside the radiation circle is a consequence of the pressure field inside the jet. Also the phase speed of the content inside the radiation circle is supersonic relative to a laboratory reference frame. This energy content inside the radiation circle represents modes trapped within the jet. The implication for flow decomposition into radiating and non-radiating components is that some acoustic components lie within the radiation circle as well($\kappa^{2}<(\omega/c)^{2}$). From Eq. (12) and the definition of $\gamma$, it can be seen that for $k_{x}>\kappa/(1+M)$, we get an exponentially decaying response inside the jet. Physically this represents a subsonically propagating wave inside the jet that leads to an evanescent wave. This explains why the spectrum decays for $k_{x}>\kappa/(1+M)$ for moderate to high frequencies. The decay can also be explained using ray theory, which predicts a shadow region in a wedge in the forward direction with half angle $\cos^{-1}[1/(1+M)]$ (Morse and Ingard, (1968)). A point on the radiation circle determines the direction of sound radiation to the far field. The angle to the jet axis is given by $\cos^{-1}(k_{x}/\kappa)$ (Goldstein, (2005)). Thus the shadow region predicted by ray theory is in agreement with the region of decay on the radiation circle in Figs. 6(b) – (d). #### High frequencies For very high frequencies, we see a combination of the radiation circle and an ellipse. At high frequencies we would get several wavelengths inside the jet ($y<a$, compare with figure 6(b)). If $a$ is large (say, infinite), then the spectra would correspond to that of a point source in a uniformly moving medium, which is an ellipse. This is what we see in figure 5(d). The ringing is because of the finiteness of $a$, which in the wavenumber domain, results in the convolution of the sinc function with the radiation ellipse. Figure 6: Comparison of the pressure field $\hat{p}(k_{x},y)$ at $\kappa a=0.1\pi$ (solid line) and $\kappa a=\pi$ (dashed line) for (a) $k_{x}=\kappa/(1+M)$, (b) $k_{x}=0.5\kappa/(1+M)$. ## 3 Diverging jet In order to consider the effect of three-dimensionality and divergence of a jet on the results presented in the preceding section, we seek the acoustic field radiating from a monopole source embedded in diverging axisymmetric mean flow at $(z/a,r/a)=(3,0)$. The mean flow was obtained from the DNS of a Mach 0.84 and Re 7200 turbulent jet embedded in a co-flow of Mach 0.2. The full details of the DNS are available in Sandberg et al., (2012), and its sound field is analysed in the next section. The acoustic field was obtained by solving the linearised Euler equations. Figures 7 (a), (c) and (e) show the density field at frequencies of $ka=0.1\pi$, $\pi$ and $2\pi$, respectively. The notation $\rho_{ka}(\mathbf{x},t_{0})$ describes a linear combination of the real and imaginary parts of the temporal Fourier transform of $\rho$ at frequency $ka$, defined as $\rho_{ka}(\mathbf{x},t_{0})=\frac{1}{\pi}\left(\Gamma_{r}(\mathbf{x},ka)\cos(\omega t_{0})-\Gamma_{i}(\mathbf{x},ka)\sin(\omega t_{0})\right),$ (19) where $\Gamma=\Gamma_{r}+i\Gamma_{i}$ is the temporal Fourier transform of $\rho(\mathbf{x},t)$. This allows to follow the evolution of the density field at frequency $ka$ with time. Figures 7 (b), (d) and (f) show the corresponding wavenumber spectra. Comparing these figures for the case of monopole in the plug flow at the same frequencies (figures 5(a), (b) and (c)), we can see that the diverging jet has not changed the nature of the spectra. They look very similar. We see the radiation circle, the shadow region and the vertical line around $k_{x}=\kappa/(1+M)$, which can again be identified as trapped waves. These waves are clearly visible around the centerline in the physical domain (figure 7 (c) and (e)). The trapped modes propagate to the farfield, as can be seen by looking at the density along the axis of the jet (figure 8). The decay rate is a polynomial of order less than 2.5 for all frequencies and not exponential and thus, these waves are not evanescent. The presence of trapped modes inside the radiation circle implies that the filter $\|\mathbf{k}\|=\|\omega\|/c_{\infty}$ (figure 1) does not capture all the acoustic waves within the jet. To do so, one may use the condition $\|\mathbf{k}\|\leq\|\omega\|/c_{\infty}$ instead. \| ---\|--- \| \| (a) $\rho_{ka=0.1\pi}(\mathbf{x},t_{0})$ (b) $P(\mathbf{k},0.1\pi)$ (c) $\rho_{ka=\pi}(\mathbf{x},t_{0})$ (d) $P(\mathbf{k},\pi)$ (e) $\rho_{ka=2\pi}(\mathbf{x},t_{0})$ (f) $P(\mathbf{k},2\pi)$ Figure 7: Density field radiating from a monopole in a diverging mean flow of a turbulent jet plotted in the physical domain (snapshot in left column) and wavenumber domain (magnitude of Fourier transform in right column), with colour range $[-5.10^{-5},5.10^{-5}]$ and $[0,0.15]$ respectively, for normalized frequencies $ka/\pi=0.1,1$ and $2$ and an arbitrary time $t_{0}$ (c.f. equation (19)). In the right column, the arc shows the radiation circle $\|\mathbf{k}\|=\omega/c_{\infty}$. The dashed arcs are at $\|\mathbf{k}\|=\omega/c_{\infty}\pm\Delta k/2$, where $\Delta k=\sqrt{\Delta k_{z}^{2}+\Delta k_{r}^{2}}$ is the grid step in the wavenumber domain. Figure 8: Profiles of the density field and the associated decay rate along $r=0$ for the data of figures 7(a, c, e). ## 4 Turbulent jet The density field from the DNS Sandberg et al., (2012) of a Mach 0.84 and Re 7200 turbulent jet embedded in a co-flow of Mach 0.2 is shown in figure 9 (a) at a Strouhal number (based on the jet exit diameter and velocity) of 1.1. This corresponds to $\kappa a=0.92\pi$. The usual Fast Fourier Transform algorithm is inconvenient for computing the temporal Fourier transform, since it requires storing the complete time history of the three dimensional density field. Here, Goertzel’s algorithm (1958) was used instead, as it consumes the time history one frame at a time. No special windowing, i.e. a rectangular window, was used to keep the main lobe as narrow as possible. However, a small amount of spectral leakage from low frequencies was observed, resulting in highly supersonic components ($\|\mathbf{k}\|\leq.6k_{\infty})$ appearing in the wavenumber plots. These supersonic components are un-physical and have been filtered out by means of a low pass filter of the form $W_{L}(\mathbf{k})=\frac{1}{2}\left(1+\tanh((\|\mathbf{k}\|-k_{0})/(\sigma k_{\infty})\right),$ (20) where $k_{\infty}=\omega/c_{\infty}$, $k_{0}/k_{\infty}=0.6$, $\sigma=0.1$. Figure 9 (b) shows the Fourier transform of the density field. If we filter only along the radiation circle as for the laminar jet, the density field in the Fourier domain and the corresponding field in the physical domain are shown in figures 9 (d) and 9 (c) respectively. A qualitative comparison between figures 9 (a) and 9 (c) shows that this filter based on the radiation circle captures most of the acoustic waves. But it can be seen from figure 9 (b) that there is a significant amount of energy inside the radiation circle. From the above analysis we expect that this region of the spectrum should contribute to trapped waves along the axis of the jet. Figure 9 (f) shows the filtered spectra when only the interior of the radiation circle is considered. The corresponding density field in the physical domain is shown in figure 9 (e). As expected, the waves propagate close to the axis of the jet. These trapped waves propagate to the far field, as can be seen in the inset of figure 9 (e), which shows the magnitude of the density field along the centerline of the jet. As in the preceding section, we get a polynomial decay rate for these trapped waves. \| ---\|--- \| \| (a) $\rho_{St=1.1}(\mathbf{x},t_{0})$ (b) $P(\mathbf{k},1.1)$ (c) $\rho^{\prime}_{St=1.1}(\mathbf{x},t_{0})$ (d) $P^{\prime}(\mathbf{k},1.1)$ (e) $\overline{\rho}_{St=1.1}(\mathbf{x},t_{0})$ (f) $\overline{P}(\mathbf{k},1.1)$ Figure 9: Density field in a turbulent jet plotted in the physical domain (snapshot in left column) and wavenumber domain (magnitude of Fourier transform in right column), with colour range $[-8\times 10^{-6},8\times 10^{-6}]$ and $[0,0.5]$ respectively, for a given normalized frequency $St=1.1$ and time $t_{0}$ (c.f. equation (19)). The top row shows the density field $\rho$, the middle row the radiating field $\rho^{\prime}$ and the bottom row the non-radiating field $\overline{\rho}$. In the right column, the arc shows the radiation circle $\|\mathbf{k}\|=\omega/c_{\infty}$. The dashed arcs are at $\|\mathbf{k}\|=\omega/c_{\infty}\pm\Delta k/2$, where $\Delta k=\sqrt{\Delta k_{z}^{2}+\Delta k_{r}^{2}}$ is the grid size in the wavenumber domain. Based on the above discussions, we can predict that if we are interested in the far field acoustic wave radiating at a particular angle $\theta$, then we can obtain this by filtering on the radiation circle in the frequency- wavenumber domain for the same polar angle in the $k_{r}-k_{z}$ plane. This is not surprising and was shown theoretically by Goldstein, (2005). However for mid to high frequencies, this procedure would miss the effect of trapped waves that propagate close to the axis of the jet. It is worth noting that the wavenumber-frequency make-up of the acoustic spectra is different than that of the turbulence. The turbulent kinetic energy spectrum is generally obtained from numerical data by the application of the three-dimensional Fourier transform in space. The resulting spectrum contains a broad range of wavenumbers and tends to be maximum for low wavenumbers. Since the radiation circle is also located over low wavenumbers, one may think that there is no separation between the turbulence and acoustic spectra. This is not the case because the radiation circle is defined for a particular frequency: a Fourier transform in time is required in addition to the Fourier transform in space. In that case, for a given frequency, only a narrow band of wavenumbers make up the turbulent spectrum. In general, that narrow band would lie outside the radiation circle (corresponding to subsonic propagation speeds). For example, to compute the turbulence spectrum from experimental data, it is common practise to measure at a single point in space and to Fourier transform the signal in time. The Fourier transform in space is then obtained by invoking Taylor’s hypothesis: assuming a frozen pattern of turbulence convected at a local flow speed $U$, the wavenumber and frequency are related by $\omega/k=U$. Thus, for a given frequency, we are picking out a single wavenumber of the turbulence spectrum that corresponds to this dispersion relation. For a subsonic jet this corresponds to a subsonic wave that lies outside the radiation circle. Thus, for the example problem considered here the turbulence spectrum corresponds to the energy content to the right of the radiation circle in 9 (b). ## 5 Conclusions Most of the acoustic waves radiating from a jet satisfy the d’Alembertian dispersion relation $k=\omega/c_{\infty}$, i.e. they lie on the radiation circle in the frequency-wavenumber domain. This validates the radiation criterion proposed by Goldstein, (2005) and used by Sinayoko et al., (2011). At low Strouhal numbers (e.g. $St\leq 0.5$), acoustic waves lie mainly on the radiation circle ($k=\omega/c_{\infty}$). This explains why the dispersion relation based on an ordinary wave equation was sufficient to filter out the acoustic waves even inside the jet, shown in figure 2(c). At mid-Strouhal numbers ($St\sim 1$), some acoustic waves are trapped in the jet. These trapped waves can be classified as acoustic based on the observation that: 1. 1. they propagate to the far field ; 2. 2. they have supersonic phase speed $k<\omega/c_{\infty}$ (in a subsonic jet the hydrodynamic waves and the energy associated with turbulent structures convect at subsonic speeds). These trapped acoustic waves can be identified by using the criterion $k\leq\omega/c_{\infty}$. Alternatively, one can use the axial wavenumber, $\|k_{x}\|\leq\omega/c_{\infty}$ since there are usually no hydrodynamic components such that $\|k_{x}\|<\omega/c_{\infty}$ and $\|k_{r}\|>\omega/c_{\infty}$. These would represent (unphysical) waves with axial wavelengths larger than acoustic waves travelling at subsonic speeds at high angles to the downstream jet axis. An advantage of this approach is that it does not require computing the radial Fourier transform. Finally, at high Strouhal numbers ($St\gg 1$), some acoustic waves lie around the radiation ellipse corresponding to the dispersion relation for waves propagating through a flow of Mach number equal to the average convection Mach number. These waves can therefore extend outside the radiation circle. Although the results presented in this paper are for high Mach number subsonic jets, the solution of the plug flow problem at low Mach numbers indicates that the conclusions are valid for low Mach number flows as well. The above conclusions were shown to hold for sound propagation through a time- averaged diverging jet and a turbulent jet. Similar results could likely be obtained for other turbulent flows, such as mixing layers and wakes. If the flow field in the far field is non-quiescent, which would be the case for a mixing layer, then the radiation circle turns into a radiation ellipse. ## References * Agarwal et al., (2004) Agarwal, A., Morris, P., and Mani, R. (2004). Calculation of sound propagation in nonuniform flows: suppression of instability waves. AIAA J., 42(1):80–88. * Cabana et al., (2008) Cabana, M., Fortuné, V., and Jordan, P. (2008). Identifying the radiating core of Lighthill’s source term. Theoretical and Computational Fluid Dynamics, 22(2):87–106. * Cavalieri et al., (2012) Cavalieri, A. V., Jordan, P., Colonius, T., and Gervais, Y. (2012). Axisymmetric superdirectivity in subsonic jets. Journal of Fluid Mechanics, 704:388–420. * Chu and Kovasznay, (1958) Chu, B. and Kovasznay, L. (1958). Non-Linear Interactions in a Viscous Heat-Conducting Compressible Gas. J. Fluid Mech., 3(5):494–514. * Crighton, (1985) Crighton, D. G. (1985). The kutta condition in unsteady flow. Annual Review of Fluid Mechanics, 17(1):411–445. * Freund, (2001) Freund, J. B. (2001). Noise sources in a low-Reynolds-number turbulent jet at Mach 0.9. Journal of Fluid Mechanics, 438:277–305. * Goertzel, (1958) Goertzel, G. (1958). An algorithm for the evaluation of finite trigonometric series. The American Mathematical Monthly, 65(1):34–35. * Goldstein, (2005) Goldstein, M. (2005). On identifying the true sources of aerodynamic sound. J. Fluid Mech., 526:337–347. * Mani, (1972) Mani, R. (1972). A moving source problem relevant to jet noise. J. Sound Vib., 25(2):337–347. * Morgan, (1975) Morgan, J. (1975). The interaction of sound with a subsonic cylindrical vortex layer. Proc. R. Soc. A, 344(1638):341–362. * Morse and Ingard, (1968) Morse, P. and Ingard, K. (1968). Theoretical acoustics. Princeton University Press. * Obrist, (2009) Obrist, D. (2009). Directivity of acoustic emissions from wave packets to the far field. Journal of Fluid Mechanics, 640:165–186. * Sandberg et al., (2012) Sandberg, R., Suponitsky, V., and Sandham, N. (2012). DNS of compressible pipe flow exiting into a coflow. Int. J. Heat Fluid Fl., 35:33–44. * Sinayoko and Agarwal, (2012) Sinayoko, S. and Agarwal, A. (2012). The silent base flow and the sound sources in a laminar jet. The Journal of the Acoustical Society of America, 131(3):1959. * Sinayoko et al., (2011) Sinayoko, S., Agarwal, A., and Hu, Z. (2011). Flow decomposition and aerodynamic sound generation. J. Fluid Mech., 668:335–350. * Stromberg et al., (1980) Stromberg, J., McLaughlin, D., and Troutt, T. (1980). Flow field and acoustic properties of a Mach number 0· 9 jet at a low Reynolds number. Journal of sound and vibration, 72(2):159–176. * Tinney and Jordan, (2008) Tinney, C. E. and Jordan, P. (2008). The near pressure field of co-axial subsonic jets. Journal of Fluid Mechanics, 611:175–204.	arxiv-papers	2013-11-21T10:40:08	2024-09-04T02:49:54.072763	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "S. Sinayoko, A. Agarwal, R. D. Sandberg", "submitter": "Samuel Sinayoko", "url": "https://arxiv.org/abs/1311.5358" }
1311.5523	# Long range order in a hard disk model in statistical mechanics Alexisz Tamás Gaál (October 2, 2013) ###### Abstract We model two-dimensional crystals by a configuration space in which every admissible configuration is a hard disk configuration and a perturbed version of some triangular lattice with side length one. In this model we show that, under the uniform distribution, expected configurations in a given box are arbitrarily close to some triangular lattice whenever the particle density is chosen sufficiently high. This choice can be made independent of the box size. Keywords: Spontaneous symmetry breaking, hard-core potential, rigidity estimate. ## 1 Introduction The breaking of rotational symmetry in two-dimensional models of crystals at low temperature has been indicated since long, see [Mer68] and [NH79]. F. Merkl and S. W.W. Rolles showed the breaking of rotation symmetry in [MR09] in a simple model without defects. In this model of crystals, atoms can be enumerated by a triangular lattice. In the very recent work [HMR13] by M. Heydenreich, F. Merkl and S. W.W. Rolles, defects were integrated into the model; defects are single, isolated, missing atoms. However, the results in [HMR13] can be generalized to larger bounded islands of missing atoms as also mentioned in [HMR13], but non-local defects are not included. The first model in [MR09] treated pair potentials with at least quadratic growth; the second one, [HMR13], tackled the case of strictly convex potentials. We are going to examine an analogue of the models in [MR09] and [HMR13] with a hard-core repulsion. For this potential we show the breaking of the rotational symmetry in a strong sense. Our model does not include defects, but the result extends to models with isolated defects as in [HMR13]. Uniformity in the box size ensures the existence of infinite volume measures with the analogous property. This work is motivated by the following open problem: is there a Gibbs measure on the set of locally finite point configurations in ${\mathbb{R}}^{2}$ which breaks the rotational symmetry of the hard-core potential? This question is analogous to the problem which was solved in [Geo99] and [Ric09] for translational symmetry. However, the outcome is different than what is expected in the case of rotational symmetry, as translational symmetry is preserved, see [Geo99] and [Ric09]. ## 2 Configuration space The _standard triangular lattice_ in ${\mathbb{R}}^{2}$ is the set $I={\mathbb{Z}}+\tau{\mathbb{Z}}$ with $\tau=e^{\frac{i\pi}{3}}$. We identify ${\mathbb{Z}}\subset{\mathbb{R}}\subset{\mathbb{R}}^{2}$ by ${\mathbb{R}}\ni x\ \hat{=}\ (x,0)\in{\mathbb{R}}^{2}$ and ${\mathbb{R}}^{2}\subset{\mathbb{C}}$ by $(x,y)\ \hat{=}\ x+iy$. The set $I$ is an index set, which is going to be used to parametrize countable point configurations in the real plane. Let us define the quotient space $I_{N}=I/(NI)$ for an $N\in{\mathbb{N}}:=\\{1,2,3,...\\}$. We identify $I_{N}$ with the following specific set of representatives: $I_{N}=\\{x+y\tau\ \|\ x,y\in\\{0,...,N-1\\}\\}.$ (2.1) A _parametrized point configuration_ in ${\mathbb{R}}^{2}$ is a function $\omega:I\rightarrow{\mathbb{R}}^{2}$, $x\mapsto\omega(x)$, which determines the point configuration $\\{\omega(x)\ \|\ x\in I\\}\subset{\mathbb{R}}^{2}$. For the set of all parametrized point configurations we introduce the character $\Omega=\\{\omega:I\rightarrow{\mathbb{R}}^{2}\\}$. Note that a single point configuration $\\{\omega(x)\ \|\ x\in I\\}\subset{\mathbb{R}}^{2}$ can be parametrized by many different $\omega\in\Omega$. Let $\epsilon\in(0,1]$. An _$N$ -periodic parametrized point configuration_ with side length $l\in(1,1+\epsilon)$ is a parametrized configuration $\omega$ which satisfies the _periodic boundary conditions_ : $\omega(x+Ny)=\omega(x)+lNy\quad\textrm{for all}\ x,y\in I.$ (2.2) The set of $N$-periodic parametrized configurations with side length $l$ is denoted by $\Omega^{per}_{N,l}\subset\Omega$. From now on we will omit the word parametrized because we are going to work solely with _point configurations_ which are parametrized by $I$. An $N$-periodic configuration is uniquely determined by its values on $I_{N}$. Therefore, we identify $N$-periodic configurations $\omega\in\Omega^{per}_{N,l}$ with functions $\omega:I_{N}\rightarrow{\mathbb{R}}^{2}$. The bond set $E\subset I\times I$ contains index-pairs with Euclidean distance one; this is $E=\\{(x,y)\in I\times I\ \|\ \|x-y\|=1\\}$. In order to transfer the definition to the quotient space $I_{N}$, we define an equivalence relation $\sim_{N}$ on $E$ by $(x,y)\ \sim_{N}\ (x^{\prime},y^{\prime})$ if and only if there is a $z\in NI$ such that $x=x^{\prime}+z$ and $y=y^{\prime}+z$. We set $E_{N}=E/\sim_{N}$. We can think of $E_{N}$ as a bond set $E_{N}\subset I_{N}\times I_{N}$. For $x\in I$ and $z\in\\{1,\tau\\}$, define the open _triangle_ $\triangle_{x,z}=\\{x+sz+t\tau z\ \|\ 0<s,t,\ s+t<1\\}$ with corner points $x,\ x+z$ and $x+\tau z$. For $\triangle_{x,z}$ denote the set of corner points by ${\mathcal{S}}(\triangle_{x,z})=\\{x,x+z,x+\tau z\\}$. On the set of all triangles ${\mathcal{T}}=\\{\triangle_{x,z}\ \|\ x\in I\ \textrm{and}\ z\in\\{1,\tau\\}\\},$ we define an equivalence relation: $\triangle_{x,z}\sim_{N}\triangle_{x^{\prime},z^{\prime}}$ if and only if $x-x^{\prime}\in NI$ and $z=z^{\prime}$. The set of equivalence classes is denoted by ${\mathcal{T}}_{N}={\mathcal{T}}/\sim_{N}$. We identify equivalence classes $\triangle\in{\mathcal{T}}_{N}$ with their unique representative with corners in the set $\\{x+\tau y\ \|\ x,y\in\\{0,...,N\\}\\}$. The closures of the triangles in ${\mathcal{T}}_{N}$ cover the convex hull of the above set, which is denoted by $U_{N}=\textrm{conv}(\\{x+\tau y\ \|\ x,y\in\\{0,...,N\\}\\})$. ## 3 Probability space By definition $\Omega=({\mathbb{R}}^{2})^{I}$, and we can identify $\Omega^{per}_{N,l}=({\mathbb{R}}^{2})^{I_{N}}$. Both sets are endowed with the corresponding product $\sigma$-fields ${\mathcal{F}}=\bigotimes_{x\in I}{\mathcal{B}}({\mathbb{R}}^{2})$ and ${\mathcal{F}}_{N}=\bigotimes_{x\in I_{N}}{\mathcal{B}}({\mathbb{R}}^{2})$ where ${\mathcal{B}}({\mathbb{R}}^{2})$ denotes the Borel $\sigma$-field on each factor. The event of admissible, N-periodic configurations $\Omega_{N,l}\subset\Omega^{per}_{N,l}$ is defined by the properties $(\Omega 1)-(\Omega 3)$: $(\Omega 1)\quad\|\omega(x)-\omega(y)\|\in(1,1+\epsilon)$ for all $(x,y)\in E$. For $\omega\in\Omega$ we define the extension $\hat{\omega}:{\mathbb{R}}^{2}\to{\mathbb{R}}^{2}$ such that $\hat{\omega}(x)=\omega(x)$ if $x\in I$, and on the closure of any triangle $\triangle\in{\mathcal{T}}$, the map $\hat{\omega}$ is defined to be the unique affine linear extension of the mapping defined on the corners of $\triangle$. $(\Omega 2)\quad$ The map $\hat{\omega}:{\mathbb{R}}^{2}\to{\mathbb{R}}^{2}$ is injective. $(\Omega 3)\quad$ The map $\hat{\omega}$ is orientation preserving, this is to say that $\det(\nabla\hat{\omega}(x))>0$ for all $\triangle\in{\mathcal{T}}$ and $x\in\triangle$ with the Jacobian $\nabla\hat{\omega}:\cup{\mathcal{T}}\to{\mathbb{R}}^{2\times 2}$. Define the set of _admissible, $N$-periodic configurations_ as $\Omega_{N,l}=\\{\omega\in\Omega_{N,l}^{per}\ \|\ \omega\ \textrm{satisfies}\ (\Omega 1)\textrm{--}(\Omega 3)\\}$ and the set of all _admissible configurations_ as $\Omega_{\infty}=\\{\omega\in\Omega\ \|\ \omega\ \textrm{satisfies}\ (\Omega 1)\textrm{--}(\Omega 3)\\}$. Note that for $\omega\in\Omega_{N,l}^{per}$, $(\Omega 2)$ is fulfilled if and only if $\hat{\omega}$ is a bijection. This observation is a consequence of the periodic boundary conditions (2.2) and the continuity of $\hat{\omega}$. Figure 1: A part of an admissible, $4$-periodic configuration. The set $\Omega_{N,l}$ is non-empty and open in $({\mathbb{R}}^{2})^{I_{N}}$. The scaled standard configuration $\omega_{l}(x)=lx$, for $x\in I$ and $1<l<1+\epsilon$, is an element both of $\Omega_{N,l}$ and $\Omega_{\infty}$. Figure 1 illustrates a part of an admissible, $4$-periodic configuration. The points of the configuration are illustrated by hard disks with radii 1/2. The image of $I_{4}$ and those of two equivalent triangles are shaded in the figure. Clearly, $0<\delta_{0}\otimes\lambda^{I_{N}\setminus\\{0\\}}(\Omega_{N,l})<\infty$ with the Lebesgue measure $\lambda$ on ${\mathbb{R}}^{2}$ and the Dirac measure $\delta_{0}$ in $0\in{\mathbb{R}}^{2}$. The lower bound holds because sections of $\Omega_{N,l}$ are non-empty and open in $({\mathbb{R}}^{2})^{I_{N}\setminus\\{0\\}}$ if $\omega(0)$ is fixed; the upper bound is a consequence of the parameter $\epsilon$ in $(\Omega 1)$. Let the probability measure $P_{N,l}$ be $P_{N,l}(A)=\frac{\delta_{0}\otimes\lambda^{I_{N}\setminus\\{0\\}}(\Omega_{N,l}\cap A)}{\delta_{0}\otimes\lambda^{I_{N}\setminus\\{0\\}}(\Omega_{N,l})}$ for any Borel measurable set $A\in{\mathcal{F}}_{N}$, thus $P_{N,l}$ is the uniform distribution on the set $\Omega_{N,l}$ with respect to the _reference measure_ $\delta_{0}\otimes\lambda^{I_{N}\setminus\\{0\\}}$. The first factor in this product refers to the component $\omega(0)$ of $\omega\in\Omega$. We call the measures $P_{N,l}$ _finite-volume Gibbs measures_ and the parameter $l$ in the definition of $\Omega_{N,l}$ and $P_{N,l}$ is the _pressure parameter_ of the system. In fact, the pressure parameter $l$ controls the density of periodic configurations, and therefore is inversely related to the physical pressure of the system. ## 4 Result We have the following finite-volume result: ###### Theorem 4.1. For $\epsilon$ sufficiently small $($such that equation (5.7) holds for all $1<a_{i}<1+\epsilon)$, one has $\lim_{l\downarrow 1}\sup_{N\in{\mathbb{N}}}\sup_{\triangle\in{\mathcal{T}}_{N}}E_{P_{N,l}}[\ \|\nabla\hat{\omega}(\triangle)-\textnormal{Id}\|^{2}\ ]=0$ (4.1) with the constant value of the Jacobian $\nabla\hat{\omega}(\triangle)$ on the set $\triangle\in{\mathcal{T}}_{N}$. Weak limits of $(P_{N,l})_{N\in{\mathbb{N}}}$ are called _infinite-volume Gibbs measures_. Since the convergence in Theorem 4.1 is uniform in $N$, there is an infinite-volume Gibbs measure $P$ such that $E_{P}[\ \|\nabla\hat{\omega}(\triangle)-\textrm{Id}\|^{2}\ ]$ is small on every triangle $\triangle\in{\mathcal{T}}$. This is actually a result about a spontaneous breaking of the rotational symmetry in a strong sense. The set $\Omega_{\infty}$ is rotational-invariant, and this symmetry is broken by some infinite-volume Gibbs measure as per (4.1). Spontaneous breaking of the rotational symmetry in the usual sense can be proved immediately. This observation is formulated and proved in the next proposition. A similar result and its proof is also mentioned in [HMR13, Section 1.3]. ###### Proposition 4.2. For all $l\in(1,1+\epsilon)$, $N\in{\mathbb{N}}$, $x\in I$ and $z\in I$ with $(0,z)\in E$, we have $\displaystyle E_{P_{N,l}}[\omega(x+z)-\omega(x)]=lz.$ (4.2) ###### Proof. We follow the ideas stated in [HMR13, Section 1.3]. The reference measure $\delta_{0}\otimes\lambda^{I_{N}\setminus\\{0\\}}$ is invariant under the bijective translations $\psi_{b}:\Omega^{per}_{N,l}\rightarrow\Omega^{per}_{N,l}\quad(\omega(x))_{x\in I}\mapsto(\omega(x+b)-\omega(b))_{x\in I}$ (4.3) for all $b\in I$. The set $\Omega_{N,l}$ is also invariant under $\psi^{-1}_{b}=\psi_{-b}$. As a consequence, the measures $P_{N,l}$ are invariant under $\psi_{b}$ for all $b\in I$, and the random vectors $\omega(x+z)-\omega(x)$ have the same distribution under $P_{N,l}$ for all $x\in I$ and a fixed $z$. Therefore, we obtain (4.2) from the periodic boundary conditions (2.2). ∎ The expression $\|\omega(x+z)-\omega(x)\|$ is $P_{N,l}$-almost surely uniformly bounded in $N$, hence (4.2) carries over to weak limits of $P_{N,l}$ as $N\to\infty$. Consequently, such weak limits are not rotational-invariant. However, in the next section, we show Theorem 4.1, which states symmetry breaking in a much stronger sense. ## 5 Proof As in [HMR13], the central argument is the following rigidity theorem from [FJM02, Theorem 3.1], which generalizes Liouville’s Theorem. ###### Theorem 5.1 (Friesecke, James and Müller). Let $U$ be a bounded Lipschitz domain in ${\mathbb{R}}^{n},\ n\geq 2$. There exists a constant $C(U)$ with the following property: For each $v\in W^{1,2}(U,{\mathbb{R}}^{n})$ there is an associated rotation $R\in\textnormal{SO($n$)}$ such that $\|\|\nabla v-R\|\|_{L^{2}(U)}\leq C(U)\|\|\textnormal{dist}(\nabla v,\textnormal{SO}(n))\|\|_{L^{2}(U)}.$ Liouville’s Theorem states that a function $v$, fulfilling $\nabla v(x)\in\mathrm{SO}(n)$ almost everywhere, is a rigid motion. Theorem 5.1 generalizes this result. We are going to set $v=\hat{\omega}\|_{U_{N}}$ and $U=U_{N}$, which is a bounded Lipschitz domain. The function $\hat{\omega}\|_{U_{N}}$ is affine linear on each triangle $\triangle\in{\mathcal{T}}_{N}$, thus piecewise affine linear on $U_{N}$. As a consequence, $\hat{\omega}\|_{U_{N}}$ belongs to the class $W^{1,2}(U_{N},{\mathbb{R}}^{n})$. The following remark, which also appears in [FJM02] at the end of Section 3, is essential to achieve uniformity in Theorem 4.1 in the parameter $N$. ###### Remark 5.2. The constant $C(U)$ in Theorem 5.1 is invariant under scaling of the domain: $C(\alpha U)=C(U)$ for all $\alpha>0$. By setting $v_{\alpha}(\alpha x)=\alpha v(x)$ for $x\in U$, we have $\nabla v_{\alpha}(\alpha x)=\nabla v(x)$, and therefore $\|\|\nabla v_{\alpha}-R\|\|_{L^{2}(\alpha U)}=\alpha^{n/2}\|\|\nabla v-R\|\|_{L^{2}(U)}$, and $\|\|\textnormal{dist}(\nabla v_{\alpha},\textnormal{SO}(n))\|\|_{L^{2}(\alpha U)}\ =\ \alpha^{n/2}\ \|\|\textnormal{dist}(\nabla v,\textnormal{SO}(n))\|\|_{L^{2}(U)}$. Consequently, the constants $C(U_{N})$ for the domains $U_{N}$ $(N\geq 1)$ can be chosen independently of $N$. We are going to show that for $\omega\in\Omega_{N,l}$, the $L^{2}$-distance on $U_{N}$ of the Jacobian matrix $\nabla\hat{\omega}$ from the scaled identity matrix $l\ \textrm{Id}$ can be controlled by the difference of the areas of $\hat{\omega}(U_{N})$ and $U_{N}$. Due to the periodic boundary conditions, $\lambda(\hat{\omega}(U_{N}))$ does not depend on configurations $\omega$ with $(\Omega 2)$, thus the mentioned area difference provides a suitable uniform control on the set $\Omega_{N,l}$. First, we show that the $L^{2}$-distance of $\nabla\hat{\omega}$ from the scaled identity $l\ \textrm{Id}$ can be controlled by the sum over the squared deviations of the triangles’ side lengths from one. The one should be associated with the side length of an equilateral triangle. To achieve this estimate, we will apply the rigidity theorem, Theorem 5.1, but first we cite an analogous result which holds locally on each triangle. The following lemma provides the desired estimate on each triangle. It states that the distance from $\textrm{SO}(2)$ of a linear map near $\textrm{SO}(2)$ can be controlled by terms which measure how the linear map deforms the side lengths of a standard equilateral triangle. ###### Lemma 5.3. There is a positive constant $C$ such that, for all linear maps $A:{\mathbb{R}}^{2}\rightarrow{\mathbb{R}}^{2}$ with $\textnormal{det}(A)>0$ and the property $\|\|Av_{i}\|-1\|\leq 1\quad\textrm{for all}\ i\in\\{1,2,3\\}$ (5.1) where $v_{1}=(1,0)$, $v_{2}=(\frac{1}{2},\frac{\sqrt{3}}{2})$, $v_{3}=v_{1}-v_{2}$, the following inequality holds: $\textnormal{dist}\left(A\ ,\ \textnormal{SO}(2)\right)^{2}:=\inf_{R\in\textnormal{SO}(2)}\left\|A-R\right\|^{2}\leq C\max_{i\in\\{1,2,3\\}}\|\|Av_{i}\|-1\|^{2}$ (5.2) where $\|M\|=\sqrt{\textnormal{tr}(M^{t}M)}$ is the Frobenius norm and $\|v\|$ is the Euclidean norm of $v$. A proof can be found in [Th06, Lemma 4.2. in the appendix]. In this proof the requirement (5.1) is formulated by means of a positive constant $\alpha_{0}$: $\|\|Av_{i}\|-1\|\leq\alpha_{0}\quad\textrm{for all}\ i\in\\{1,2,3\\}$, although the proof also applies to the special case $\alpha_{0}=1$ as stated in Lemma 5.3. Now, we prove the mentioned estimate, which provides control over the $L^{2}$-distance of $\nabla\hat{\omega}$ from the scaled identity matrix in terms of the side length deviations. ###### Lemma 5.4. There is a constant $c$ such that for all $N\geq 1$ and $1<l<1+\epsilon$ the inequality $\|\|\ \nabla\hat{\omega}-l\ \textnormal{Id}\ \|\|^{2}_{L^{2}(U_{N})}\leq c\sum_{(x,y)\in E_{N}}(\|\omega(x)-\omega(y)\|-1)^{2}$ (5.3) holds for all $\omega\in\Omega_{N,l}$, and hence $E_{P_{N,l}}[\ \|\|\ \nabla\hat{\omega}-l\ \textnormal{Id}\ \|\|^{2}_{L^{2}(U_{N})}\ ]\leq c\sum_{(x,y)\in E_{N}}E_{P_{N,l}}[\ (\|\omega(x)-\omega(y)\|-1)^{2}\ ]$ (5.4) where the $L^{2}$-norm is defined with respect to some scalar product on ${\mathbb{R}}^{2\times 2}$, and $\|\cdot\|$ denotes the Euclidean norm on ${\mathbb{R}}^{2}$. Note that the right side in equation (5.3) is strictly positive because of the boundary conditions (2.2) and because $l>1$, whereas the left is zero for $\omega=\omega_{l}\in\Omega^{per}_{N,l}$. Since the measure $P_{N,l}$ is supported on the set $\Omega_{N,l}$, (5.4) follows from (5.3). Also note that $c$ does not depend on $N$. ###### Proof. Let $\omega\in\Omega_{N,l}$. By Lemma 5.3 we conclude that on every triangle $\triangle\in{\mathcal{T}}_{N}$, we have $\textrm{dist}\left(\nabla\hat{\omega}(\triangle),\ \textrm{SO}(2)\right)^{2}\leq C\max_{x\not=y\in{S(\triangle)}}(\|\omega(x)-\omega(y)\|-1)^{2}\leq\frac{C}{2}\sum_{x\not=y\in{S(\triangle)}}(\|\omega(x)-\omega(y)\|-1)^{2}$ where we used the assumption $\epsilon\leq 1$ together with $(\Omega 1)$ and $(\Omega 3)$ to apply Lemma 5.3. The factor $1/2$ is a consequence of summing over all non-equal pairs $(x,y)$. Orthogonality of the functions which are non-zero on different triangles gives $\|\|\ \textrm{dist}(\nabla\hat{\omega},\textrm{SO}(2))\ \|\|^{2}_{L^{2}(U_{N})}\leq c_{1}\sum_{(x,y)\in E_{N}}(\|\omega(x)-\omega(y)\|-1)^{2}$ with $c_{1}=C\ \lambda(\triangle_{0,1})=C\sqrt{3}/4$ because we sum again over both pairs $(x,y)$ and $(y,x)$ on the right side. With application of Theorem 5.1 about geometric rigidity, we find an $R(\omega)\in\textrm{SO}(2)$ such that $\|\|\ \nabla\hat{\omega}-R(\omega)\ \|\|^{2}_{L^{2}(U_{N})}\leq c_{2}\ \|\|\ \textrm{dist}(\nabla\hat{\omega},\textrm{SO}(2))\ \|\|^{2}_{L^{2}(U_{N})},$ with a constant $c_{2}$, which does not depend on $N$ by Remark 5.2. Due to the periodic boundary conditions (2.2), the function $\hat{\omega}-l\ \textrm{Id}$ is $N$-periodic, this is to say $\hat{\omega}(x+Ny)-l(x+Ny)=\hat{\omega}(x)-lx\quad\textrm{for all }x\in{\mathbb{R}}^{2}\textrm{ and }\ y\in I.$ (5.5) Let $A\in{\mathbb{R}}^{2\times 2}$ be a constant matrix. Integrating the function $\langle\nabla\hat{\omega}-l\ \textrm{Id},A\rangle$ over the set $U_{N}$, the result equals zero since, by (5.5) and the fundamental theorem of calculus, $\int_{0}^{1}\langle\nabla\hat{\omega}-l\ \textrm{Id},A\rangle(x+tN)\textnormal{d}t=0\quad\textrm{for all }x\in{\mathbb{R}}^{2}$ where we used the embedding ${\mathbb{R}}\subset{\mathbb{R}}^{2}$. Consequently, we obtain the orthogonality property: $\nabla\hat{\omega}-l\ \textrm{Id}\perp_{L^{2}(U_{N})}A$, for any constant matrix $A\in{\mathbb{R}}^{2\times 2}$ and thus $\|\|\ \nabla\hat{\omega}-l\ \textrm{Id}\ \|\|^{2}_{L^{2}(U_{N})}+\|\|\ l\ \textrm{Id}-R(\omega)\ \|\|^{2}_{L^{2}(U_{N})}=\|\|\ \nabla\hat{\omega}-R(\omega)\ \|\|^{2}_{L^{2}(U_{N})}$ by Pythagoras. Since $\|\|\ l\ \textrm{Id}-R(\omega)\ \|\|^{2}_{L^{2}(U_{N})}\geq 0$ and because $P_{N,l}$ is supported on the set $\Omega_{N,l}$, the lemma is established with $c=c_{1}c_{2}$. ∎ With Lemma 5.4 we can now prove Theorem 4.1. ###### Proof of Theorem 4.1. Heron’s formula states that the area $\lambda(\triangle)$ of the triangle $\triangle$ with side lengths $a_{1},a_{2},a_{3}$ is given by $\lambda(\triangle)=\frac{1}{4}\sqrt{(a_{1}+a_{2}+a_{3})(-a_{1}+a_{2}+a_{3})(a_{1}-a_{2}+a_{3})(a_{1}+a_{2}-a_{3})}.$ (5.6) By first order Taylor approximation of (5.6) at the point $a_{i}=1$, $i\in\\{1,2,3\\}$ we obtain $\lambda(\triangle)-\lambda({\triangle_{0,1}})=\frac{1}{2\sqrt{3}}\sum_{i=1}^{3}(a_{i}-1)+o\left(\sum_{i=1}^{3}\|a_{i}-1\|\right)\quad\textrm{as}\ (a_{1},a_{2},a_{3})\to(1,1,1).$ Since the function $\lambda$ is smooth in a neighborhood of $(1,1,1)$, we could also express the remainder term as Big ${\mathcal{O}}$ of the sum of the squares. In the following we only need the weaker estimate on the remainder. We choose $\epsilon$ so small that the inequality $\frac{1}{4\sqrt{3}}\sum_{i=1}^{3}(a_{i}-1)\leq\lambda(\triangle)-\lambda({\triangle_{0,1}})$ (5.7) is satisfied whenever $1<a_{i}<1+\epsilon$. Note that we have divided the constant by two preceding the sum. Let us fix such an $\epsilon$ and assume that $\Omega^{per}_{N,l}$ is defined by means of this $\epsilon.$ Using (5.7) we can also estimate the squared side length deviations: $\sum_{i=1}^{3}(a_{i}-1)^{2}\leq 4\sqrt{3}\ \epsilon\ (\lambda(\triangle)-\lambda({\triangle_{0,1}})).$ (5.8) By equation (5.3) from Lemma 5.4 and (5.8), we get an upper bound on $\|\|\nabla\hat{\omega}-l\ \textnormal{Id}\|\|_{L^{2}(U_{N})}^{2}$ in terms of the area differences. By summing up the contributions (5.8) of the triangles $\triangle\in{\mathcal{T}}_{N}$, we conclude for all $\omega\in\Omega_{N,l}$ that $\|\|\ \nabla\hat{\omega}-l\ \textnormal{Id}\ \|\|_{L^{2}(U_{N})}^{2}\leq 4\sqrt{3}\ \epsilon\ c\sum_{\triangle\in{\mathcal{T}}_{N}}(\lambda(\hat{\omega}(\triangle))-\lambda({\triangle_{0,1}})).$ (5.9) As a consequence of $(\Omega 2)$ and the periodic boundary conditions (2.2), the right hand side in (5.9) does not depend on $\omega\in\Omega_{N,l}$. Hence, with $\omega_{l}\in\Omega_{N,l}$ we can compute $\sum_{\triangle\in{\mathcal{T}}_{N}}(\lambda(\hat{\omega}(\triangle))-\lambda({\triangle_{0,1}}))=\sum_{\triangle\in{\mathcal{T}}_{N}}(\lambda(\hat{\omega}_{l}(\triangle))-\lambda({\triangle_{0,1}}))=\|{\mathcal{T}}_{N}\|\ \lambda(\triangle_{0,1})(l^{2}-1).$ (5.10) The combination of the equations (5.9) and (5.10) gives $\|\|\ \nabla\hat{\omega}-l\ \textnormal{Id}\ \|\|_{L^{2}(U_{N})}^{2}\leq 4\sqrt{3}\ \epsilon\ c\ \|{\mathcal{T}}_{N}\|\ \lambda(\triangle_{0,1})(l^{2}-1).$ (5.11) The reference measure $\delta_{0}\otimes\lambda^{I_{N}\setminus\\{0\\}}$ and the set of allowed configurations $\Omega_{N,l}$ are invariant under the reflection $\phi:\omega\mapsto(-\omega(-x))_{x\in I}$ and the translations $\psi_{b}$ for $b\in I$, defined in (4.3). As a consequence, the measure $P_{N,l}$ is also invariant under these maps, and therefore the matrix valued random variables $\nabla(\hat{\omega}(\triangle))$ are identically distributed for all $\triangle\in{\mathcal{T}}_{N}$. Thus, for all $\triangle\in{\mathcal{T}}_{N}$, one has $E_{P_{N,l}}[\ \|\|\ \nabla\hat{\omega}-l\ \textnormal{Id}\ \|\|_{L^{2}(U_{N})}^{2}\ ]=\|{\mathcal{T}}_{N}\|\ \lambda(\triangle_{0,1})E_{P_{N,l}}[\ \|\nabla\hat{\omega}(\triangle)-l\ \textnormal{Id}\|^{2}\ ].$ This equation, together with (5.11), implies $\lim_{l\downarrow 1}\sup_{N\in{\mathbb{N}}}\sup_{\triangle\in{\mathcal{T}}_{N}}E_{P_{N,l}}[\ \|\nabla\hat{\omega}(\triangle)-l\ \textnormal{Id}\|^{2}\ ]=0.$ By means of the triangle inequality, we see that for all $\triangle\in{\mathcal{T}}_{N}$ and $\omega\in\Omega_{N,l}$ $\|\nabla\hat{\omega}(\triangle)-\textnormal{Id}\|^{2}\leq\|\nabla\hat{\omega}(\triangle)-l\ \textnormal{Id}\|^{2}+c_{3}^{2}(l-1)^{2}+2c_{3}\ \|l-1\|\ \|\nabla\hat{\omega}(\triangle)-l\ \textnormal{Id}\|$ with $c_{3}=\|\mathrm{Id}\|>0$. For $\omega\in\Omega_{N,l}$, the term $\|\nabla\hat{\omega}(\triangle)-l\ \textnormal{Id}\|$ is uniformly bounded for $l\in(1,\epsilon)$ and $N\in{\mathbb{N}}$, which proves the theorem. ∎ Acknowledgement I would like to thank Prof. Dr. Merkl for his useful comments and suggestions. Without his support, this work would have not been possible. ## References * [FJM02] G. Friesecke, R. D. James and S. Müller _A Theorem on Geometric Rigidity and the Derivation of Nonlinear Plate Theory from Three-Dimensional Elasticity_ , Comm. Pure Appl. Math., 55: pp 1461–1506, (2002). * [Geo99] H.-O. Georgii _Translation invariance and continuous symmetries in two-dimensional continuum systems_ , In Mathematical results in statistical mechanics (Marseilles, 1998): pp 53–69, Word Sci. Publ., River Edge, NJ (1999). * [HMR13] M. Heydenreich, F. Merkl and S. W.W. Rolles: _Spontaneous breaking of rotational symmetry in the presence of defects_ , arXiv:1308.3959 [math.PR], Submitted on 19 Aug 2013, http://arxiv.org/pdf/1308.3959v1.pdf (2013). * [MR09] F. Merkl and S. W.W. Rolles: _Spontaneous breaking of continuous rotational symmetry in two dimensions_ , Electron. J. of Probab. 14, Paper no. 57, pp 1705–1726, (2009). * [Mer68] N. D. Mermin _Crystalline Order in Two Dimensions_ , Phys. Rev., 176: pp 250–254, (1968). * [NH79] D. R. Nelson and B. I. Halperin _Dislocation-mediated melting in two dimensions_ , Phys. Rev. B 19: pp 2457–2484, (1979). * [Ric05] T. Richthammer _Two-dimensional Gibbsian point processes with continuous spin symmetries_ , Stochastic Process. Appl., 115: pp 827–848, (2005). * [Ric09] T. Richthammer _Translational invariance of two-dimensional Gibbsian systems of particles with internal degrees of freedom_ , Stochastic Process. Appl., 119: pp 700–736, (2009). * [Th06] F. Theil: _A proof of Crystallization in Two Dimensions_ , Commun. Math. Phys., 262: pp 209–236 (2006).	arxiv-papers	2013-11-21T19:23:33	2024-09-04T02:49:54.084453	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Alexisz Tam\\'as Ga\\'al", "submitter": "Alexisz Tam\\'as Ga\\'al", "url": "https://arxiv.org/abs/1311.5523" }
1311.5525	# SCANNING WIRE BEAM POSITION MONITOR FOR ALIGNMENT OF A HIGH BRIGHTNESS INVERSE-COMPTON X-RAY SOURCE††thanks: Work supported by the US Department of Homeland Security DNDO ARI program GRANT NO. 2010-DN-077-ARI045-02 M. R. Hadmack and E. B. Szarmes University of Hawai‘i Free-Electron Laser Laboratory [email protected] Honolulu HI 96822 USA ###### Abstract The Free-Electron Laser Laboratory at the University of Hawai‘i has constructed and tested a scanning wire beam position monitor to aid the alignment and optimization of a high spectral brightness inverse-Compton scattering x-ray source. X-rays are produced by colliding the $40\text{\,}\mathrm{MeV}$ electron beam from a pulsed S-band linac with infrared laser pulses from a mode-locked free-electron laser driven by the same electron beam. The electron and laser beams are focused to $60\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameters at the interaction point to achieve high scattering efficiency. This wire-scanner allows for high resolution measurements of the size and position of both the laser and electron beams at the interaction point to verify spatial coincidence. Time resolved measurements of secondary emission current allow us to monitor the transverse spatial evolution of the e-beam throughout the duration of a $4\text{\,}\mathrm{\SIUnitSymbolMicro s}$ macro-pulse while the laser is simultaneously profiled by pyrometer measurement of the occulted infrared beam. Using this apparatus we have demonstrated that the electron and laser beams can be co-aligned with a precision better than $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ as required to maximize x-ray yield. ## 1 INTRODUCTION A compact high brightness x-ray source is currently under development at the University of Hawai‘i Free-Electron Laser Laboratory, based on inverse-Compton scattering of $40\text{\,}\mathrm{MeV}$ electron bunches with synchronous laser pulses from an infrared free-electron laser (FEL)[1, 2]. One of the more challenging aspects of realizing a Compton backscatter x-ray source is co- alignment of the electron and laser beams. With high intensities, and spot sizes as small as $30\text{\,}\mathrm{\SIUnitSymbolMicro m}$, it is not possible to align these beams without special diagnostic tools. The resolution of available beam position monitors (BPMs) and optical transition radiation (OTR) screens is limited to about $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ by the sampling electronics and video cameras used. Wire scanners are commonly employed on accelerator beam-lines as alignment aides. The “flying wire” type scanners, such as those used at CERN, are too large for use in the space allocated on the Mk V beam-line at UH and are incompatible with the bunch structure of our accelerator. The x-ray interaction point is shared by two other insertable diagnostic devices in a crowded vacuum chamber, also housing the x-ray interaction point laser optics. The wire scanner described here is based on the designs used at NBS-LANL[3] and the SLC[4] and adapted to the constraints of our beamline configuration. This system also includes the capability to resolve the temporal evolution of the electron beam profile over the macropulse duration (approximately $4\text{\,}\mathrm{\SIUnitSymbolMicro s}$). ## 2 HARDWARE The wire scanner head shown in Fig. 1 consists of two $34\text{\,}\mathrm{\SIUnitSymbolMicro m}$-diameter graphite fibers stretched across the $12.3\text{\,}\mathrm{mm}$ gap in an aluminum fork. The wires are oriented such that when the scanner insertion axis is inclined $None$ above the beam plane, the two wires are oriented horizontally and vertically. In this way a single axis of motion allows the beam to be scanned in both axes. Figure 1: The wire scanner fork electrically isolates the carbon fiber from the grounded fork. Fibers are soldered to the signal lead and clamped at the other end. The secondary emission current from the wire is conducted via the scanner shaft to a vacuum feedthrough on the assembly shown in Fig. 2. The fork itself is grounded to avoid charge accumulation from the beam halo. Figure 2: Beam profiler drive assembly with motor and LVDT with an early prototype fork. The support rod conducts the signal to the vacuum feedthrough on the far end. Figure 3: The wire scanner installed with other diagnostic devices in the x-ray scattering chamber. Most wire scanners in operation today utilize bremsstrahlung radiation detectors to measure beam interception of the wire. However, on a linear machine it is more difficult to position a PIN diode detector close to the source without substantial background radiation. Moving the detector to a suitably shielded location results in a large bremsstrahlung beam diameter, making efficient detection difficult and introducing errors due to diffraction. Figure 3 shows the assembly integrated in the x-ray scattering chamber installed at the interaction point. The wire scanner assembly consists of a precision vacuum translation stage, a linear-variable-differential-transformer (LVDT) position sensor, and a DC motor drive. The motor speed is controlled by software to achieve high resolution at the $5\text{\,}\mathrm{Hz}$ beam repetition rate. Position is measured with an LVDT attached to the translation stage; its resolution is limited by 12 bit readout electronics to $7.2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ steps over a $14\text{\,}\mathrm{mm}$ range. The LVDT read-back is calibrated against the actual translation stage motion with calipers. Wire scans are typically performed with the actuator speed set to $100\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{s}}^{-1}$ so that the position changes by twice the LVDT limiting step size each macropulse event, thus ensuring monotonic position data. A full $14\text{\,}\mathrm{mm}$ scan using both the horizontal and vertical wire takes approximately $140\text{\,}\mathrm{s}$. For each accelerator macropulse trigger three quantities are measured: the current from the wire, the laser pulse transmission, and the position. The intercepted electron beam current is inferred by the current resulting from secondary electrons ejected from the wire. The wire current signal is terminated in $50\text{\,}\mathrm{\SIUnitSymbolOhm}$ and sampled with a $300\text{\,}\mathrm{MHz}$ digital oscilloscope. A pyroelectric detector viewing the transmitted beam measures the occlusion of the laser beam by the wire. The pyrometer’s response time is considerably slower than that of the wire current. The pulse peak voltage is sampled with a boxcar integrator and digitized with 12 bit precision. The data acquisition software is implemented in Python with a graphical user interface (GUI) built using wxPython. The software acquisition is triggered using control lines on an RS232 serial port to monitor the accelerator’s TTL trigger. When a trigger event is detected, data is read from the oscilloscope and boxcar integrator, both of which are synchronously triggered. Data is also acquired asynchronously from the LVDT controller via a serial connection. Figure 4 illustrates the data acquisition system. The GUI shown in Fig. 5 provides the operator with a live stripchart of both the laser and current measurements throughout a scan. The GUI allows for configuration and control of automated scans and data storage. Data is stored in a custom binary format and includes full oscilloscope waveforms for every position step. Figure 4: A PC acquires beam current, laser intensity, and scan position data and controls the drive motor. Figure 5: The GUI displays the electron beam (yellow) and laser (purple) beam profiles in real time during a scan. Figure shows background data, not actual scan results. The GUI stripchart is only used as a rough guide for scan operation while detailed data analysis is performed using an offline tool, also implemented in Python. Figure 6 shows a sample wire scan analysis. The graphic in the upper part of the figure is a representation of the evolution of the beam current spatial distribution over a $4\text{\,}\mathrm{\SIUnitSymbolMicro s}$ macropulse. The vertical columns in the image are individual oscilloscope waveforms for each position along the horizontal axis; interpolation is applied to account for non-uniformly spaced positions. The lower plot shows the transmitted laser pulse energy compared to the wire current integrated over a particular duration of interest within the pulse. The integration region is typically chosen to overlap the laser pulse in the last $\mathrm{\SIUnitSymbolMicro s}$ of the macropulse. Figure 6: Secondary emission current as a function of position and time within the macropulse. The lower plot compares the integrated current from $2\text{\,}\mathrm{\SIUnitSymbolMicro s}3\text{\,}\mathrm{\SIUnitSymbolMicro s}$ with the laser signal (axis inverted). This scan includes both the horizontal (right) and vertical (left) profiles. ## 3 RESULTS Preliminary experiments have been conducted to measure the sizes, positions, and stability of the laser and electron beams. The wire scanner was initially commissioned with a $200\text{\,}\mathrm{\SIUnitSymbolMicro m}$ tungsten wire. This wire was operated successfully for several months with large beam diameters and limited resolution; however, a microfocused (sub $100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ diameter) beam quickly severed the wire. Next, a $25\text{\,}\mathrm{\SIUnitSymbolMicro m}$ tungsten wire was chosen to reduce the absorption volume and to enhance the spatial resolution of the scans by a factor of eight. Again, however, a microfocused beam destroyed the wire on the first pass. Ultimately, $34\text{\,}\mathrm{\SIUnitSymbolMicro m}$-diameter carbon monofilament from Specialty Materials, Inc. has proven robust enough to endure the highly focused $150\text{\,}\mathrm{mA}$ electron beam and several $\mathrm{mJ}$ of infrared laser exposure. Since the carbon filament could not be wrapped in the same manner as the more flexible tungsten, it was necessary to modify the wire scanner fork. Figure 1 illustrates how the carbon fibers are clamped on one end between Vespel plastic discs while the other ends are soldered to a copper tab attached to the signal lead. The scan data in Fig. 6 shows an electron beam focused to $w_{x},w_{y}=$350\text{\,}\mathrm{\SIUnitSymbolMicro m}$,$115\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, where the left feature is the evolution of the vertical beam profile and the right represents the horizontal. The total charge intercepted on each wire is the same, so the area under the beam profile curves is constant, resulting in lower peak signal for the horizontal scan. In this scan the laser is well aligned to the electron beam resulting in suppression of laser operation while the beam is intercepted by the wire. Figure 7: Horizontal axis wire scan showing the laser displaced from the e-beam position. Interception of the e-beam by the wire inhibits lasing and results in a second co-aligned peak. Figure 8: The Mark V FEL beamline at the University of Hawai‘i Figure 7 shows an example where the laser is misaligned from the electron beam. This horizontal scan (vertical wire only) gives an electron beam width of $175\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and a laser beam width of $143\text{\,}\mathrm{\SIUnitSymbolMicro m}$ with a beam separation of $419\text{\,}\mathrm{\SIUnitSymbolMicro m}$, correctable with a motorized mirror. The laser was intentionally defocused at the interaction point for this scan to preclude wire damage during these early experiments. It is interesting to note that while the laser size and position are measurable by transmission to the pyroelectric detector (red trace), a laser signal also appears distinctly in the wire current. We believe that this laser induced wire current is the result of thermionic emission from the wire due to heating from laser exposure. This hypothesis is supported by the observation that the laser induced current extends several hundred $\mathrm{ns}$ beyond both the end of the electron beam and laser macropulses while in fact the laser beam exposure of the wire begins a microsecond earlier and is therefore presumed to be a thermal artifact. The photon energy of the $3000\text{\,}\mathrm{nm}$ laser is not sufficient to generate photoelectrons. This feature provides a useful way to measure both laser and electron beams with a single sensor signal. Figure 7 also illustrates the time resolved nature of this wire scanner system. The trajectory in this image indicates that the beam’s horizontal position slews nearly a $\mathrm{mm}$ over the $4\text{\,}\mathrm{\SIUnitSymbolMicro s}$ macropulse. The large slew in the first microsecond is an inevitable consequence of beam-loading in the linear accelerator and is typically ignored for experimental purposes. The remainder of the macropulse, however, also shows a position slew that is the consequence energy slew in the beam. The diagnostic chicane shown in Fig. 8 contains a number dipole bend magnets upstream of the interaction point that produce energy dependent deflections in the beam. Characterization and mitigation of this energy/position slew in the beam are of critical importance to operation of the free-electron laser and for beam alignment of the inverse-Compton scattering interaction. A naïve integration of the wire current or a transition radiation image would significantly overestimate the instantaneous beam size as well as produce an ambiguity in the centroid position during the time interval of interest for scattering. Even with a significant transverse evolution in the beam, the instantaneous size and position can be precisely measured. The “flying wire” type beam profilers employed on many large accelerators and storage rings employ a high velocity wire capable of scanning many stored bunches in a single sweep[5]. While this technique enables much faster profile acquisition, the position becomes correlated with a particular time within the bunch train. Thus, these systems are not capable of revealing the temporal structure of the macropulse in the manner described above and can overestimate the beam size. Typically this is not a problem for a storage ring that is filled with nearly identical bunches. However, the transient beam loading experienced in a linac with a thermionic gun results in beam evolution that must be considered. Wire scan repeatability was measured from the analysis of a scan sequence with the same e-beam configuration. The beam centroid can be measured with an uncertainty of $\sigma_{x,y}=$9\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and a beam width uncertainty of $\sigma_{w}=$4\text{\,}\mathrm{\SIUnitSymbolMicro m}$$. Scans are always performed in the same direction to eliminate hysteresis due to a $30\text{\,}\mathrm{\SIUnitSymbolMicro m}$ backlash in the translator lead-screw. ## 4 CONCLUSION A scanning wire beam position monitor has been successfully constructed and operated at the University of Hawaii Free-Electron Laser Laboratory. This custom design satisfies the tight space restrictions imposed by the need to share access to the interaction point of the inverse-Compton x-ray source with other diagnostic devices and laser optics. The use of a commercially available linear vacuum translator significantly reduces the engineering time and cost of the system. $34\text{\,}\mathrm{\SIUnitSymbolMicro m}$ carbon fiber has been selected as a suitable material for scans of a sub-$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ microfocused electron beam operated at $40\text{\,}\mathrm{MeV}$ with $150\text{\,}\mathrm{mA}$ average current over $4\text{\,}\mathrm{\SIUnitSymbolMicro s}$ macropulses. Further studies of the carbon filament damage threshold for both electron beam and laser exposure are necessary for effective optimization of the inverse- Compton scattering interaction point focus. The laser can easily be attenuated to avoid wire damage once this is known. However, the electron beam current cannot be varied and the damage threshold will impose a limit to the average current density allowed on the wire. In principle, the $7\text{\,}\mathrm{\SIUnitSymbolMicro m}$ width resolution is sufficient to achieve the $30\text{\,}\mathrm{\SIUnitSymbolMicro m}$ focal spot specification of the UH x-ray source. Alignment of the beams can be verified with a scan repeatability of better than $10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ when the $30\text{\,}\mathrm{\SIUnitSymbolMicro m}$ hysteresis is accounted for. Combining time-resolved wire scans with quadrupole magnet scans will give us the capability to perform time-resolved emittance measurements of the e-beam. This will be a vital capability for our continued efforts to improve extended pulse length thermionic electron gun technology[6]. ## 5 ACKNOWLEDGMENT We acknowledge Specialty Materials, Inc. for providing the carbon monofilament and John M. J. Madey for his advice and operational support on this project. ## References * [1] M.R. Hadmack. Ph.D. thesis, University of Hawai‘i, 2012. * [2] J. M. J. Madey et al. SPIE X-Ray Nanoimaging Conference, San Diego, CA, 2013. * [3] R.I. Cutler, J. Owen, and J. Whittaker. PAC’87, p. 625, 1987. * [4] M.C. Ross et al. PAC’91, San Francisco, CA, 1991. * [5] S. Igarashi et al. Nucl. Instrum. Meth. A, 482(1–2):32, 2002. * [6] J. M. D. Kowalczyk, M. R. Hadmack, and J. M. J. Madey. FEL’13, New York, NY, 2013.	arxiv-papers	2013-11-21T19:30:22	2024-09-04T02:49:54.092289	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Michael R. Hadmack, Eric B. Szarmes", "submitter": "Michael Hadmack", "url": "https://arxiv.org/abs/1311.5525" }
1311.5745	Measurement of production asymmetries Hamish Gordon, for the LHCb Collaboration CERN, Geneva, Switzerland > The knowledge of charm production asymmetries is an important prerequisite > for many of the possible searches for CP violation in charm. Measurements of > these asymmetries at hadron colliders can also help to improve our > understanding of QCD. These proceedings review existing measurements and > discuss some of the experimental challenges of determining charge > asymmetries at the per-mille level. > PRESENTED AT > > > > > The 6th International Workshop on Charm Physics > (Charm 2013) > 29th August-3rd September, Manchester, UK ## 1 Introduction The recent hints of CP violation (CPV) in singly Cabibbo suppressed $D^{0}$ decays to two-body final states from LHCb [1] and CDF [2] have heightened interest from theoreticians in charm physics. Despite the lack of confirmation of these hints by further studies [3], searches for direct CPV in charm remain well motivated. Measurements of charm production asymmetries have the potential to increase the number of possible techniques for CPV searches in charm, and also to make existing searches more precise. For example, the most powerful search technique is currently the measurement of the $\Delta A_{CP}$ observable, which is the difference in CP-violating asymmetries between $D^{0}\to K^{-}K^{+}$ and $D^{0}\to\pi^{-}\pi^{+}$. This quantity is equal to the difference between the measured raw asymmetries in these decay modes, where the raw asymmetry is defined for observed numbers of decays $N$ as $A_{raw}=\frac{N(D^{0})-N(\overline{D^{0}})}{N(D^{0})+N(\overline{D^{0}})}.$ (1) The largest useable samples of these decays are those that originate from $D^{+}$ decays to $D^{0}$ and a charged pion, which tags the flavour of the $D^{0}$. Unfortunately, the values of $\Delta A_{CP}$ expected in the Standard Model are difficult to calculate, partly due to the lack of a good understanding of the strong interaction effects and partly because the charge asymmetries in the individual decay modes are not known. Knowledge of the production asymmetry in $D^{+}$ decays would enable measurements of the charge asymmetries in $D^{0}\to K^{-}K^{+}$ and $D^{0}\to\pi^{-}\pi^{+}$ separately, solving the second of these problems. Furthermore, a precise production asymmetry measurement could in principle lead to a more precise measurement of CP violation in $D^{0}\to K^{-}K^{+}$ than that in $\Delta A_{CP}$, where the statistical uncertainty is limited by the $D^{0}\to\pi^{-}\pi^{+}$ decay channel. Measurements of production asymmetries in the $D^{+}$, $D_{s}^{+}$ and $\Lambda_{c}^{+}$ sectors are also worthy endeavours which will pave the way for more precise searches for CP violation in their Cabibbo-suppressed decay modes. Measurements of charm production asymmetries are also interesting in their own right. The huge samples of charm decays from proton-proton collisions available at the LHC experiments can be used to improve our knowledge of the structure of the proton. It is conceivable that the charm samples at the B-factories could also be used to make precise tests of QCD symmetries via an investigation of the foward-backward asymmetry in charm meson production. The forward-backward asymmetry in $D^{\pm}$ production has been measured at the Belle experiment, and the $D^{+}$ and $D_{s}^{+}$ production asymmetries in $pp$ collisions have been measured at LHCb. These measurements, discussed in the next sections, all use large Cabibbo-favoured charm samples, in which no CP violation is expected. To make full use of the high statistical precision possible with these samples, careful studies of the systematic effects intrinsic to charge asymmetry measurements in particle physics detectors are required, and these are also discussed here. ## 2 Production asymmetry measurements at $e^{+}e^{-}$ colliders In the search for CPV in $D^{+}\to K^{0}_{S}\pi^{+}$ at the Belle experiment [4], the CP, detector and production asymmetries are intertwined. The raw measured charge asymmetry is $\displaystyle A^{K^{0}_{S}\pi^{+}}_{\rm rec}$ $\displaystyle=$ $\displaystyle A^{K^{0}_{S}\pi^{+}}_{CP}~{}+~{}A^{D^{+}}_{FB}(\cos\theta^{\rm CMS}_{D^{+}})$ (2) $\displaystyle+$ $\displaystyle A^{\pi^{+}}(p^{\rm lab}_{T\pi^{+}},\cos\theta^{\rm lab}_{\pi^{+}})~{}+~{}A_{\mathcal{D}}(p^{\rm lab}_{K^{0}_{S}})$ where $A^{\pi^{+}}(p^{\rm lab}_{T\pi^{+}},\cos\theta^{\rm lab}_{\pi^{+}})$ and $A_{\mathcal{D}}(p^{\rm lab}_{K^{0}_{S}})$ are the detection asymmetries of charged pions and neutral kaons respectively. The quantities $p^{\rm lab}$ and $p^{\rm lab}_{\rm T}$ refer to momentum and transverse momentum in the laboratory frame. The angle $\theta$ is the angle of the pion with respect to the axis of the beam, in either the laboratory (lab) frame or the centre of mass (CMS) frame. $A^{D^{+}}_{FB}(\cos\theta^{\rm CMS}_{D^{+}})$ is the forward-backward production asymmetry. $A^{\pi^{+}}(p^{\rm lab}_{T\pi^{+}},\cos\theta^{\rm lab}_{\pi^{+}})$ is measured as the difference in raw charge asymmetries between $D^{+}\to K^{-}\pi^{+}\pi^{+}$ and $D^{0}\to K^{-}\pi^{+}\pi^{0}$ under the assumption that $D^{0}$ and $D^{+}$ have the same forward-backward asymmetry. $A_{\mathcal{D}}(p^{\rm lab}_{K^{0}_{S}})$ is calculated as discussed in Sect. 4, and subsequently subtracted. The asymmetries due to production and CP violation are then determined. In terms of the charge asymmetry after correction for detector effects, $A^{K^{0}_{S}\pi^{+}_{\rm corr}}_{\rm rec}$, they are $\displaystyle A^{K^{0}_{S}\pi^{+}}_{CP}$ $\displaystyle=$ $\displaystyle[A^{K^{0}_{S}\pi^{+}_{\rm corr}}_{\rm rec}(+\cos\theta^{\rm CMS}_{D^{+}})$ (3) $\displaystyle+$ $\displaystyle~{}A^{K^{0}_{S}\pi^{+}_{\rm corr}}_{\rm rec}(-\cos\theta^{\rm CMS}_{D^{+}})]/2,$ $\displaystyle A^{D^{+}}_{FB}$ $\displaystyle=$ $\displaystyle[A^{K^{0}_{S}\pi^{+}_{\rm corr}}_{\rm rec}(+\cos\theta^{\rm CMS}_{D^{+}})$ (4) $\displaystyle-$ $\displaystyle~{}A^{K^{0}_{S}\pi^{+}_{\rm corr}}_{\rm rec}(-\cos\theta^{\rm CMS}_{D^{+}})]/2$ respectively. The CP asymmetry is consistent with CPV in the neutral kaon system as expected: $D^{+}\to K^{0}_{S}\pi^{+}$ is a predominantly Cabibbo- favoured decay with no loop contribution at first order. Figure 1: CP (top) and production (bottom) asymmetries measured in the $D^{+}\to K^{0}_{S}\pi^{+}$ channel with a data sample corresponding to an integrated luminosity of 977 fb−1 at the Belle experiment [4]. The assumption that the forward-backward asymmetries in charm meson production at $e^{+}e^{-}$ colliders does not depend on the flavour of the other quark in the meson could be tested if the pion efficiency asymmetry could be determined using another method, for example the technique outlined in Sect. 4 that has been employed at LHCb. ## 3 Production asymmetries at hadron colliders When two protons collide, the baryon number conservation law implies that two more baryons than antibaryons will form in the final state. These will sometimes contain charm quarks, and thus one expects an excess of charmed baryons over charmed antibaryons, with the effect being more pronounced at high rapidity where the valence quarks tend to end up. The anticharm quark formed with the charm quark must form part of a meson, resulting in an excess of $\overline{D^{0}}$ over $D^{0}$ and of $D^{-}$ over $D^{+}$. It is helpful to define the Feynman momentum $x_{F}$ as the fraction of the longitudinal momentum $p$ carried by the relevant parton. This $x_{F}$ is related approximately to rapidity $\eta$, transverse mass $m_{T}$ and centre-of-mass energy $\sqrt{s}$ by $x_{F}\sim 2m_{T}e^{\eta}/\sqrt{s}$ (5) for $\eta>1$. Since the valence quarks are found at high rapidity, the production asymmetry is likely to increase with $x_{F}$. In perturbative quantum chromodynamics (pQCD), charm quarks are produced by processes such as $q+\overline{q}\to c+\overline{c}$ and $g+g\to c+\overline{c}$, with the second dominating at high energy. Neither of these yield an overall excess of one quark type over the other, however. Such net production asymmetries cannot be explained with pQCD, nor with the string fragmentation model contained in the PYTHIA framework typically used to simulate $pp$ interactions at collider experiments. More creative explanations must therefore be devised. Some models, for example the ‘meson cloud model’ [5], assume that the incoming proton fluctuates into a virtual charm meson - charm baryon pair which can sometimes escape and become real. Alternatively it has been proposed that $\overline{c}c$ pairs exist in the sea and have some probability to ‘recombine’ with valence quarks and hadronise [6]. These two models lead to different forecasts for the energy dependence of the production asymmetry, and Ref. [5] contains concrete predictions which are compared to LHCb measurements of the $D^{\pm}$ asymmetry. The LHCb collaboration has measured both the $D^{\pm}$ and the $D_{s}^{\pm}$ production asymmetries [7, 8]. The $D_{s}^{\pm}$ asymmetry was determined using $D_{s}^{+}\to\phi\pi^{+}$ decays. In this case, the raw measured asymmetry and the production asymmetry must be corrected by the detection asymmetry of the charged pion. This was determined using tagged $D^{0}$ decays to $K^{-}\pi^{+}\pi^{-}\pi^{+}$. Due to the large number of kinematic constraints provided by the four-body final state, it is possible to reconstruct $D^{}(2010)^{+}$-tagged decays of this type with one pion missing. The pion tracking efficiency is then the yield of fully reconstructed decays divided by the yield of decays partially reconstructed with a missing pion. The mass distributions for these two cases in the data recorded at LHCb in 2011 are shown in Fig. 2. Figure 2: Mass distributions for the tagging $D^{}(2010)^{+}$ particle in the partially (left) and fully (right) reconstructed $D^{0}$ decays to $K^{-}\pi^{+}\pi^{-}\pi^{+}$. The tracking efficiencies are determined for $D^{+}$ and $D^{-}$ separately, and thus the charge asymmetry in the pion detection efficiency is obtained. When averaged over the LHCb acceptance, the asymmetry is small, of order 0.1%. However, for a given polarity of the LHCb magnet, the asymmetry varies quite strongly according to where in the detector the pions end up, as shown in Fig. 3. This is discussed further in Sect. 4. Figure 3: The variation of the asymmetry in the pion tracking asymmetry with the azimuthal angle $\phi$ made by the pion with a horizontal plane defined across the centre of the detector, which is the bending plane of the magnet. The causes and ramifications of the variation in the data split according to the polarity of the magnet are discussed in Sec. 4. In a similar analysis, the $D^{+}$ asymmetry was measured using $D^{+}\to K^{0}_{S}\pi^{+}$ decays. Here $A_{prod}=A_{raw}-A_{\pi^{+}}-A_{K^{0}_{S}}$ (6) where the $K^{0}_{S}$ asymmetry $A_{K^{0}_{S}}$ is due to CP violation and material interactions in the neutral kaon system. There is assumed to be no CPV in the $D^{+}$ decay. Figure 4: Invariant mass distributions of the two final state particle combinations used to measure the $D^{+}_{s}$ (left) and $D^{+}$ (right) production asymmetries. In both cases, the $D^{+}_{s}$ and $D^{+}$ mass peaks are visible. To determine the production asymmetries, the yields of $D_{(s)}^{+}$ and $D_{(s)}^{-}$ decays, and the average pion efficiency asymmetries, are determined in $p_{\rm T}$ and $\eta$ bins. The overall yields are shown in Fig. 4. The raw asymmetries are thus corrected for the pion asymmetry on a per-bin basis. Measured raw asymmetries in bins of $p_{\rm T}$ and $\eta$ are weighted by the reconstruction efficiency in these bins to determine an average asymmetry, and finally the charge asymmetry due to the neutral kaon is subtracted in the case of the $D^{+}$ measurement. This last quantity is very small because only neutral kaons with very short lifetimes are selected for use in the analysis, and thus its variation with $p_{\rm T}$ and $\eta$ is negligible. The results are asymmetries for $D_{(s)}^{+}$ decays produced in $pp$ collisions in the LHCb acceptance. The average asymmetries are $A_{prod}(D_{(s)}^{+})=(-0.33\pm 0.13\pm 0.18\pm 0.10)\%$ (7) $A_{prod}(D^{+})=(-0.96\pm 0.19\pm 0.18\pm 0.18)\%$ (8) where the uncertainties are statistical on the $D_{(s)}^{+}$ decays, statistical on the pion efficiency asymmetry correction, and systematic. There are some hints of the expected dependence on $p_{\rm T}$ and $\eta$ in the $D^{+}$ case, as shown in Fig. 5. Figure 5: Dependence of the $D^{+}$ production asymmetry on $p_{\rm T}$ (left) and $\eta$ (right). The comparison of the results with the theoretical model of Ref. [5] is shown in Fig. 6. It is clear that the effect is relatively small and the dependence on kinematic variables relatively weak, so more precise data will be needed before a fully rigorous test of the theory can be performed. Figure 6: Comparison of production asymmetries measured at LHCb with the predictions of the meson cloud theory [5]. The parameter $\Lambda$ is a cut- off. Note that the opposite convention is used here to define asymmetry, with an excess of $D^{-}$ decays being defined as positive. ## 4 Experimental challenges As data samples get larger and larger, systematic uncertainties are becoming increasingly important. It is a generally held view that systematics can be controlled at the level of the statistical uncertainty, but to achieve this they must be studied in ever more detail. In charge asymmetry measurements, important systematic uncertainties arise from the fact that the magnetic field used to separate the charges bends oppositely-charged particles in opposite directions so they pass through different parts of the detector. The different detector elements could have different efficiencies. This is illustrated for the LHCb detector in Fig. 7. The different acceptance and efficiency in different radial directions is responsible for the large asymmetries seen in, for example, the pion detection efficiency as a function of azimuthal angle in Fig. 3 for data taken with one magnet polarity. This figure highlights the importance of taking data with both magnet polarities and averaging the results, as this leads to near- complete cancellation of the effects. $D^{0}$$K^{-}$$\pi^{-}$$\pi^{+}$$\pi^{+}$$x$$z$${\cal C}$$\overline{D^{0}}$$\pi^{-}$$\pi^{-}$$\pi^{+}$$K^{+}$ Figure 7: Schematic of the LHCb detector showing the path of charged particles from a $D^{0}\to K^{-}\pi^{+}\pi^{-}\pi^{+}$ decay and its charge conjugate. In this case, the raw asymmetry will be dominated by the material interaction effects of the charged kaon, but when the pion tracking efficiency is measured, this cancels between the numerator and the denominator. Other key systematic uncertainties in LHCb production and CP asymmetry measurements are associated with material interaction effects. The asymmetric interaction of positive and negative pions with detector material are responsible for most of the angle-independent asymmetry in Fig. 3. Charged kaon material interactions lead to still larger asymmetries. There are also nuisance effects from neutral kaon mixing and CP violation. Neutral kaons are particularly interesting because they violate CP and their mixing can be affected by material interactions. To parameterise neutral kaon material interactions, one usually defines a ‘regeneration parameter’ $r$ in terms of forward scattering amplitudes $f$ and $\overline{f}$ for $K^{0}$ and $\overline{K^{0}}$ respectively, $r=-\frac{\pi{\cal N}(f-\overline{f})}{\Delta m-\frac{i}{2}(\Gamma_{L}-\Gamma_{S})}$ (9) where ${\cal N}$ is the number density of atoms in the material, $\Delta m$ is the mass difference between $K^{0}$ and $\overline{K^{0}}$, and $\Gamma_{L,S}$ are their lifetimes. The imaginary part of $f$ is related to the cross section by the optical theorem and the real part of $f$ is related to the imaginary part by dispersion integrals [9]. The difference between $K^{0}$ and $\overline{K^{0}}$ scattering amplitudes follows the scaling law $f-\overline{f}\propto-\frac{23.2pA^{0.758}}{[p\;({\rm GeV}/c)]^{0.614}}\;{\rm mb}$ (10) where $A$ is the nucleon number of the material and $p$ is the momentum of the neutral kaon [10]. Ko _et al_ [11] model a detector as a series of layers of material, calculate $r$ for each layer using measured cross sections, and solve a set of recursive equations to determine the asymmetry as a function of the kaon decay time and momentum. The CPV in the neutral kaon system decouples from this regeneration at first order. Neglecting direct CPV, it is given by $A(t)=2{\rm Re}(\epsilon)-2e^{-\frac{1}{2}\Delta\Gamma t}\left({\rm Re}(\epsilon)\cos\Delta mt+{\rm Im}(\epsilon)\sin\Delta mt\right)$ (11) where the indirect CP violation parameter $\epsilon$ is approximately $2\times 10^{-3}$. The formalism has now been employed at LHCb, but to date both material interactions and CPV lead to small effects on the measured raw asymmetries in charm decays of a few times $10^{-4}$. This is because kaons used in current analyses are very short-lived compared to $K^{0}_{S}$ lifetime of 89 ps, due to peculiarities in the trigger and selection criteria. The decay time distribution of the kaons is shown in Fig. 8. Figure 8: The $K^{0}_{S}$ decay time distribution for neutral kaons selected for use in the production asymmetry analysis. ## 5 Perspective With production asymmetries under control, it is possible to search for CP- violation more precisely and in more different ways. Sometimes one can extract the production and CPV asymmetries together, as done in the analysis of $D^{+}\to K^{0}_{S}h^{+}$ by the Belle collaboration. Production asymmetries are also interesting for QCD and those measured in $pp$ collisions should help theorists to develop non-perturbative models of the proton. The prospects for the future include measurements of the $\Lambda_{c}^{+}$ and $D^{+}$ production asymmetries at LHCb. These are likely to be challenging but rewarding analyses and the results will be highly pertinent to our understanding of both particle production and CP violation in charm decays. ## References [1] R. Aaij et al. [LHCb Collaboration], Phys. Rev. Lett. 108, 111602 (2012). [arXiv:1112.0938 [hep-ex]]. * [2] T. Aaltonen _et al_ , [CDF collaboration], Phys. Rev. Lett. 109, 111801 [arXiv:1207.2158 [hep-ex]]. * [3] R. Aaij et al. [LHCb Collaboration], Phys. Lett. B 723 (2013) 33 [arXiv:1303.2614 [hep-ex]]. * [4] B. R. Ko et al. [Belle collaboration], Phys. Rev. Lett. 109 021601 (2012) [arXiv:1203.6409 [hep-ex]]. * [5] E. Cazaroto et al., Phys. Lett. B 724 108 (2013). [arXiv:1302.0035 [hep-ex]]. * [6] K.P. Das et al., Phys. Lett. B 68, 459 (1977). * [7] R. Aaij et al. [LHCb Collaboration], Phys. Lett. B 713 186 (2012). [arXiv:1205.0897 [hep-ex]]. * [8] R. Aaij et al. [LHCb Collaboration], Phys. Lett. B 718 902 (2013) [arXiv:1210.4112 [hep-ex]]. * [9] R. A. Briere and B. Winstein Phys. Rev. Lett. 75 402 (1995). * [10] A. Gsponer et al. Phys. Rev. Lett. 42 13 (1979). * [11] B. R. Ko, E. Won, B. Golob and P. Pakhlov, Phys. Rev. D. 84 (2011) 111501 [arXiv:1006.1938 [hep-ex]].	arxiv-papers	2013-11-22T13:18:15	2024-09-04T02:49:54.103205	{ "license": "Creative Commons - Attribution - https://creativecommons.org/licenses/by/3.0/", "authors": "Hamish Gordon (for the LHCb collaboration)", "submitter": "Hamish Gordon", "url": "https://arxiv.org/abs/1311.5745" }

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