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+{"text":"\\section{Introduction}\n\nThe study of birationality of algebraic varieties is a classical \nand well studied subject, with many open problems.\nIn some cases, it is interesting to study birational maps\npreserving additional structure, for example\ngroup actions, symplectic forms, or volume forms. \nSuch a study is already implicit in many questions of birational geometry,\n\\eg,  in the notion of crepant resolution of singularities.\n\nIn this paper, we consider the case of varieties\nendowed with volume forms with logarithmic poles\nand develop  a formalism of Burnside rings \nalong the lines of their counterpart introduced\nby~\\citet{KontsevichTschinkel-2019} to establish\nthe specialization of rationality, and \nits equivariant version by~\\citet{KreschTschinkel-2022}.\n\nLet $k$ be a field of characteristic zero.\nFor each integer~$n$,\nwe define \n\\[\u00a0\\Burn_n(k) \\]\nas the free abelian group\non birational equivalence classses of pairs $(X,\\omega)$\nconsisting of an integral smooth  proper $k$-variety~$X$ of dimension~$n$\nequipped with an $n$-form~$\\omega$ with at most logarithmic poles.\n\nThe graded abelian group\n\\[\u00a0\\Burn(k) = \\bigoplus_{n\\in\\N} \\Burn_n(k) \\]\ncarries a ring structure, induced by taking products of varieties,\ndecomposed into irreducible components,\nand equipped with the external product of the volume forms. \n\nIn section~\\ref{sec.residues},\nwe define morphisms of abelian groups\n\\[\u00a0\\partial \\colon \\Burn_n(k) \\to \\Burn_{n-1}(k) .\\]\nWhen $X$ is smooth and the polar divisor of~$\\omega$ has strict\nnormal crossings, the image of the class $[X,\\omega]$ is given\nby the following formula. Let $(D_\\alpha)_{\\alpha\\in\\mathscr A}$\nbe the family of irreducible components of the polar divisor of~$\\omega$.\nFor each subset~$A$ of~$\\mathscr A$, the intersection $D_A=\\bigcap_{\\alpha\\in A} D_\\alpha$ is a union of integral smooth varieties of codimension~$\\Card(A)$;\ntaking iterated residues, we may equip it \nwith a volume form with logarithmic poles~$\\omega_A$.\nThen\n\\[\u00a0\\partial ([X,\\omega]) = \\sum_{\\emptyset\\neq A \\subset\\mathscr A}\n (-1)^{\\Card(A)-1} [D_A, \\omega_A] \\cdot \\mathbf T^{\\Card(A)-1}, \\]\nwhere \n\\[\u00a0\\mathbf T = [\\P^1, \\mathrm dt\/t]. \\]\nIn particular, the existence of the map~$\\partial$ relies on the \nbirational invariance of this expression,\nsee theorem~\\ref{theo.residue-bir}.\n\nThis construction is reminiscent of the boundary map in polar homology\n\\citep{KhesinRosly-2003,KhesinRoslyThomas-2004,GorchinskiyRosly-2015}.\nHowever, apart from the obvious difference that we only record birational types\nof strata, rather than the strata themselves,\nour formula takes into account strata of all codimensions, rather\nthan those of codimension one.\n\nThe map~$\\partial$  is additive.\nFurthermore, we prove in theorem~\\ref{theo.partial-deriv} that\n\\[\u00a0\\partial(a \\cdot b) = \\eps^n \\cdot \\partial (a)\\cdot  b\n+ a \\cdot \\partial(b) - \\mathbf T \\cdot  \\partial(a) \\cdot \\partial(b), \\]\nwhen $a\\in\\Burn_m(k)$ and $b\\in\\Burn_n(k)$.\nHere $\\eps$ is the class of the point~$\\Spec(k)$\nequipped with the volume form equal to~$-1$.\n\nIn theorem~\\ref{theo.dd=0}, we show that \n\\[\u00a0\\partial \\circ \\partial  = 0 . \\]\n\nThese formulas may look complicated.\nHowever, as  we explain in \\S\\ref{sec.alg},\nthey simplify significantly after inverting~$2$.\n\nInspired by the constructions of~\\cite{LinShinderZimmermann-2020,LinShinder-2022,KreschTschinkel-2022a}, we define in~\\S\\ref{sec.biraut}\na homomorphism \n\\[\u00a0\\mathbf c \\colon \\Bir(X,\\omega) \\to \\Burn_{n-1}(k), \\]\nfrom the group of birational automorphisms of the pair\n$(X,\\omega)$, where $X$ is an $n$-dimensional integral proper smooth variety\nequipped with a logarithmic volume form~$\\omega$.\nAs in the above references, our map~$\\mathbf c$ is defined\nat the groupoid level of birational maps preserving logarithmic volume forms.\n\nWhen the birational isomorphism~$\\phi\u00a0\\colon (X,\\omega) \\dashrightarrow (Y,\\eta)$ is described\nby a diagram \n\\[\u00a0\\begin{tikzcd}[column sep = small]\n & W \\ar{dl}[swap]{p} \\ar{dr}{q} \\\\\nX \\ar[dashrightarrow]{rr}{\\phi} && Y \\end{tikzcd}\u00a0\\]\nof smooth proper integral $k$-varieties,\nwith birational morphisms~$p$ and~$q$, the two logarithmic\nvolume forms~$p^*\\omega$ and~$q^*\\eta$ on~$W$ are equal,\nand the element~$\\mathbf c(\\phi)\\in\\Burn_{n-1}(k)$ is given by\n\\[\u00a0\\mathbf c(\\phi) = \n\\sum_{E \\in\\operatorname{Exc}(q)} [E, p^*\\omega_E] - \n\\sum_{D \\in\\operatorname{Exc}(p)} [D, q^*\\eta_D] \\]\nwhere $\\operatorname{Exc}(p)$ is the set of irreducible components\nof the exceptional divisor of~$p$, and where, for each such component~$D$,\nthe logarithmic volume form~$p^*\\omega_D$ on~$D$\nis obtained by taking the residue of~$p^*\\omega$ along~$D$\n(and similarly for~$q$).\n\nFinally, consider a discrete valuation ring\nwith residue field~$k$ and field of fractions~$K$\nand let $t$ be a uniformizing element.\nIn this context, we define a specialization map\n\\[\u00a0\\rho_t \\colon \\Burn_n(K)\\to \\Burn_n(k). \\]\nThe image of the class $[X,\\omega]$  involves the combinatorics of a \ngood model $(\\mathscr X,\\omega)$ over the valuation ring, \nand a certain subcomplex of the Clemens complex of the special fiber.\nIn the particular case where $\\mathscr X$ is smooth,\nthe polar divisor of~$\\omega$ is a relative divisor with normal crossings,\nand denoting by~$\\omega_o$ the restriction of~$\\omega$ \nto the special fiber~$\\mathscr X_o$,\none has\n\\[\u00a0\\rho_t ([X,\\omega])  = [\\mathscr X_o,\\omega_o]. \\]\nNote that \nthe existence of such a specialization map implies,\nas in theorem~1 of~\\cite{KontsevichTschinkel-2019},\nor as in \\citep{NicaiseShinder-2019},\nthat birational equivalence of varieties with\nlogarithmic volume forms  is preserved under \"good specializations\".\n\nIn the geometric case, where the valuation is the local ring of a curve~$C$\nat point~$o$, the construction of the specialization map\ncan be viewed as a restriction to the special fiber\nof a normalization of a global residue map~$\\partial$\nthat takes place on a proper model whose special fiber is\n a divisor with normal crossings.\nThe normalization procedure extracts a subcomplex\nof the Clemens complex of the special fiber.\nA similar situation appeared in the study of Tamagawa\nmeasures on analytic manifolds \\citep{Chambert-LoirTschinkel-2010}.\n\nRelated constructions\nemerged from the seminal work of~\\citet{KontsevichSoibelman-2006}\ninspired by mirror symmetry,\nand its subsequent developments,\n\\eg, by \\citet{MustataNicaise-2015a, NicaiseXu-2016a, BoucksomJonsson-2017, JonssonNicaise-2020}.\n\nOur constructions use essentially only formal properties of the residue maps.\nConsequently, one can envision analogous theories \nfor logarithmic forms of smaller degree, Milnor $K$-theory,\nor even for the cycle modules of \\citet{Rost-1996}.\n\n\\subsection*{Acknowledgments}\nWe are grateful to Hsueh-Yung Lin and Evgeny Shinder for having pointed out a lapsus \nin~\\S\\ref{sec.biraut}.\nThe third author was partially supported by NSF grant 2000099.\n\n\\section{Logarithmic differential forms}\n\n\\subsection{K\u00e4hler differentials}\nLet $k$ be a field of characteristic zero\nand let $K$ be a finitely generated extension of~$k$;\nlet $n$ be its transcendence degree.\nThe space of K\u00e4hler differentials $\\Omega_{K\/k}$\nis the $K$-vector space generated\nby symbols $\\mathrm da$, for $a\\in K$, subject to the relations:\n\\begin{enumerate}\n\\item For $a\\in k$, one has $\\mathrm da = 0$;\n\\item For $a,b\\in K$, one has $\\mathrm d(a+b)=\\mathrm da+\\mathrm db$\nand $\\mathrm d(ab)=a\\mathrm db + b(\\mathrm da)$.\n\\end{enumerate}\n\nFor any integer~$m\\geq0$, we may consider its $m$th exterior\npower $\\Omega^{m}_{K\/k}$, which is a $K$-vector space\nof dimension~$\\binom nm$; in particular, it vanishes if $m>n$,\n$\\Omega^1_{K\/k}$ has dimension~$n$,\nand $\\Omega^n_{K\/k}$ has dimension~one.\nOne has $\\Omega^0_{K\/k}=k$, canonically.\n\nElements of $\\Omega^n_{K\/k}$, for $n=\\trdeg_k(K)$,\nare also called \\emph{volume forms}.\n\nFor $a\\in K^\\times$, we also write $\\dlog a=\\mathrm da\/a\\in\\Omega_{K\/k}$.\n\n\n\\subsection{Models}\nLet $m$ be an integer and \nlet $\\omega\\in\\Omega^m_{K\/k}$.\nA \\emph{model} of~$K$ is an integral $k$-scheme~$X$\ntogether with a $k$-isomorphism $K\\simeq k(X)$; we say\nthat this model is proper, resp.\\ smooth if $X$ is proper,\nresp.\\ smooth over~$k$. \nGiven such a model,\n $\\omega$ induces a meromorphic global section~$\\omega_X$\nof~$\\Omega^m_{X\/k}$.\nThe polar ideal of~$\\omega_X$ \nis the subsheaf of~$\\mathscr O_X$ whose local sections \nare the $a\\in\\mathscr O_X$\nsuch that $a\\omega_X$ is induced by a regular $m$-form. \nLet $D$ be the zero-locus of this ideal. Its complement~$U$\nis the largest open subscheme of~$X$ \nsuch that $\\omega_X$ is induced by a regular $m$-form on~$U$.\nIf $X$ is smooth, then $\\omega_X$ is locally free,\nhence the scheme~$D$ is an effective divisor (Hartogs's principle);\nwe call it the \\emph{polar divisor} of~$\\omega$ on~$X$.\n\n\\subsection{Logarithmic forms}\nBy Hironaka's theorem on embedded resolution of singularities,\nthere exist smooth projective  models $(X,\\omega_X)$ of $(K,\\omega)$\nsuch that the polar divisor~$D$ of~$\\omega_X$ has normal crossings.\n\nFollowing \\cite[chap.~2, \\S3]{Deligne-1970}, we then say that  $\\omega_X$ \nhas at most logarithmic poles, or that $\\omega$\nhas at most logarithmic poles on~$X$, \nif both $\\omega_X$ and $\\mathrm d\\omega_X$ have at most simple poles along~$D$. \n\nThe following lemma implies that this condition \nis essentially independent of the choice of~$X$\nsuch that the polar divisor of~$\\omega_X$ has normal crossings.\n\n\\begin{lemm}\\label{lemm.logarithmic-functoriality}\nLet $g\\colon X'\\to X$ be a morphism of smooth $k$-varieties,\nlet $D$ be a divisor with normal crossings in~$X$ and\nlet $D'$ be a divisor with normal crossings in~$X'$\nsuch that $D'=g^{-1}(D)$.\nLet $\\omega$ be a regular $m$-form on $X\\setminus D$\nand let $\\omega'=g^*\\omega$.\n\n\\begin{enumerate}\n\\item If $\\omega$ has at most logarithmic poles along~$D$, \nthen $\\omega'$ has at most logarithmic poles along~$D'$.\n\n\\item The converse holds if $g$ is proper and surjective.\n\\end{enumerate}\n\\end{lemm}\n\\begin{proof}\nThe first assertion is \\citep[chap.~II, prop.~3.2, (iv)]{Deligne-1970}.\nLet us prove the second one. \n\nConsider the\ngeneric point~$\\eta$ of~$X$ and a point $\\eta'\\in X'\\setminus D'$\nwhich is algebraic over~$k(\\eta)$.\nThe Zariski closure~$X'_1$ of~$\\eta'$ \nis proper and generically finite over~$X$,\nand $D'_1=D'\\cap X'_1$ is a divisor.\nThere is a proper modification $h\\colon X'_2\\to X'_1$ \nsuch that $D'_2=h^{-1}(D'_1)$ has normal crossings.\nBy the first part, the form $h^*\\omega'|_{X'_1}$ has at most\nlogarithmic poles along~$D'_2$. \nReplacing~$g$ by $g\\circ h$, we may assume \nthat $g$ is generically finite. \n\nSince the sheaf of forms with at most logarithmic poles along~$D$\nis locally free and $X$ is smooth, we can \ndelete from~$X$ a subset of codimension at least~2.\nThus, we may assume that $g$ is flat,\n$D$ is smooth and irreducible,\nand $g$ is \u00e9tale outside of $D$. \nIt suffices to argue \u00e9tale locally at the generic point of~$D$.\nBy the local description of ramified morphisms,\nthere are \u00e9tale local coordinates $(z_1,\\dots,z_n)$\non~$X$ such that $D_\\red=V(z_1)$, \nlocal coordinates $(z'_1,\\dots,z'_n)$ on~$X'$\nsuch that $g^*z_1=(z'_1)^e$, $g^*z_2=z'_2$, etc.,\nwhere $e$ is the ramification index of~$g$ along~$D$.\nLet $d$ be the order of the pole of~$\\omega$ along~$D$;\nwrite $\\omega = \\alpha\/z_1^d + \\beta \\wedge \\mathrm dz_1\/z_1^d$,\nwhere $\\alpha,\\beta$ are regular forms which do not involve~$\\mathrm dz_1$.\nThen \n\\[\u00a0\\omega'=g^*\\omega=g^*\\alpha\/(z'_1)^{de} \n+ e\\, g^*\\beta \\wedge \\mathrm dz'_1\/(z'_1)^{1+(d-1)e}. \\]\nAssume, by contradiction, that $d\\geq 2$, so that $de\\geq 2$\nand $1+(d-1)e\\geq 2$.\nSince $\\omega'$ has at most logarithmic poles along~$D$,\nwe get $g^*\\alpha=0$ and $g^*\\beta=0$.\nThis implies that both $\\alpha$ and $\\beta$\nare multiples of~$z_1$, contradicting the hypothesis\nthat $d$ was the order of the pole of~$\\omega$ along~$D$.\nTherefore, $d\\leq 1$. This concludes the proof.\n\\end{proof}\n\n\\subsection{}\nWe say that an $m$-form $\\omega\\in\\Omega^m_{K\/k}$ is  \\emph{logarithmic}\nif for all proper smooth models~$X$ of~$K$\nsuch that the polar divisor of~$\\omega_X$ has normal crossings,\nthe meromorphic differential form $\\omega_X$ \nhas at most logarithmic poles.\n\nBy resolution of singularities \\citep{Hironaka-1964},\ntwo models are dominated by a third one, \nhence lemma~\\ref{lemm.logarithmic-functoriality},\nimplies that it suffices that this condition \nis satisfied on some proper smooth model for which\nthe polar divisor of~$\\omega_X$ has normal crossings.\n\nAnalogously, if $X$ is a reduced $k$-variety, then we \nsay that a meromorphic $m$-form~$\\omega$ on~$X$ is logarithmic \"everywhere\"\nif for all proper birational models $(X',\\omega')$ of~$(X,\\omega)$,\nthe meromorphic $m$-form~$\\omega'$ on~$X'$ has at most logarithmic poles.\nIt suffices that this holds on one such model.\n\n\\section{Burnside rings for logarithmic forms}\n\n\\subsection{Burnside rings}\\label{ssec.burn}\nLet $k$ be a field of characteristic zero and $n$\nan integer such that $n\\geq0$.\n\\citet{KontsevichTschinkel-2019}\ndefined the Burnside group $\\burn_n(k)$ as the free abelian group\non isomorphism classes of \nfinitely generated extensions of~$k$\nof transcendence degree~$n$.\n\nAny integral $k$-variety~$X$ of dimension~$n$\nhas a class $[X]$ in $\\burn_n(k)$.\nThis gives rise to alternative useful presentations of~$\\burn_n(k)$,\nfor example involving only classes of integral projective smooth varieties.\n\nThe group\n\\[\n\\burn(k)=\\bigoplus_{n\\ge 0}  \\burn_n(k)\n\\]\ncarries a natural commutative ring structure, \nwith multiplication defined by taking products of (smooth projective) $k$-varieties:\n\\[\n[X]\\cdot [X'] = [X\\times X'].\n\\]\n\n\n\\subsection{Definition of a Burnside group for volume forms}\nLet $k$ be a field of characteristic zero and let $n$ be an integer\n$\\geq0$.\nWe define $\\Burn_n(k)$ to be the free abelian group\non isomorphisms classes of pairs $(K,\\omega)$,\nwhere \n\\begin{itemize}\n\\item\n$K$ is a finitely generated extension of $k$ of transcendence degree~$n$ and \n\\item\n$\\omega\\in\\Omega^n_{K\/k}$ is a logarithmic volume form.\n\\end{itemize}\nWe write \n\\[\n[K,\\omega] \\in \\Burn_n(k)\n\\]\nfor the class of a pair $(K,\\omega)$.\n\n\\begin{rema}\nThis definition has obvious more geometric formulations.\nFor example, we can take for generators equivalence classes\nof pairs $(X,\\omega)$, where \n\\begin{itemize}\n\\item\n$X$ is a smooth integral $k$-scheme of dimension~$n$,\nand \n\\item\n$\\omega$ a regular volume form on~$X$ which is logarithmic \n\"everywhere\",\n\\end{itemize}\nmodulo the smallest equivalence relation\nthat identifies $(X,\\omega)$ and $(X',\\omega')$\nif there exists an open immersion $f\\colon X'\\to X$\nsuch that $\\omega'=f^*\\omega$. \n\nAlternatively, we can assume that $X$ is proper, smooth and integral, \nthe form $\\omega$ is a logarithmic volume form on~$X$,\nand consider the smallest equivalence relation\nthat identifies $(X,\\omega)$ and $(X',\\omega')$\nif there exists a proper birational morphism $f\\colon X'\\to X$\nsuch that $\\omega'=f^*\\omega$.\nBy the weak factorization theorem of~\\citep{AbramovichKaruMatsukiEtAl-2002},\nthis equivalence relation is generated by such morphisms~$f$\nwhich are blowing-ups along smooth centers in good\nposition with respect to the polar divisor of~$X$.\n\nIn both contexts, if $X$ is an $n$-dimensional $k$-variety\nand $\\omega$ is a meromorphic $n$-form on~$X$ \nwhich is logarithmic \"everywhere\",\nthen we\ndefine $[X,\\omega]$ to be the sum, over all irreducible components~$Y$\nof~$X$ which have dimension~$n$,\nof the classes $[Y,\\omega|_Y]$.\n\\end{rema}\n\n\\begin{exem}\\label{exem.eps}\nFinitely generated extensions of~$k$ of transcendence degree~$0$\nare finite extensions of~$k$. Let $K$ be such an extension.\nSince $k$ has characteristic zero, \none has $\\Omega^1_{K\/k}=0$. However, $\\Omega^0_{K\/k}$,\nwhich is its $0$th exterior power, is canonically isomorphic to~$K$.\nConsequently, \n$\u00a0\\Burn_0(k) $ is the free  abelian group on isomorphism\nclasses of pairs $(K,\\lambda)$, where $K$ is a finite extension of~$k$\nand $\\lambda\\in K$.\n\nWe will let $\\mathbf 1=[\\Spec(k),1]$\nand $\\eps = [\\Spec(k),-1]$.\n\\end{exem}\n\n\\begin{exem}\\label{exem.T}\nLet $K=k(t)$. The differential form $\\mathrm dt\/t$ \nis a logarithmic volume form; indeed $X=\\P^1_k$ is a model\nof~$K$ and this form has poles of order~$1$ at~$0$ and~$\\infty$,\nand no other poles. We write $\\mathbf T$ for the class of\n$(k(t),\\mathrm dt\/t)$. \n\nNote that the $k$-isomorphism of $K$ that maps~$t$ to~$1\/t$\nmaps $\\mathrm dt\/t$ to its opposite;  consequently,\nwe also have $\\mathbf T=[k(t),-\\mathrm dt\/t]=\\eps\\cdot\\mathbf T$.\n\nIn the context of birational\ngeometry in presence of logarithmic volume forms,\n\"rational varieties\" would have class in~$\\mathbf T^n$,\nand similarly for stable birationality.\n\\end{exem}\n\n \n\\subsection{Multiplicative structure}\\label{ss.commut}\nWe view the direct sum \n\\[\u00a0\\Burn(k)=\\bigoplus_{n\\in\\N}\\Burn_n(k) \\]\nas a graded abelian group.\nIt is endowed with a multiplication such that\n\\[\u00a0[X,\\omega] \\cdot [X',\\omega'] =  [X\\times X', \\omega\\wedge \\omega'] \\]\nwhen $X,X'$ are proper, smooth and integral\nand $\\omega$, resp.~$\\omega'$ are logarithmic volume forms on~$X$, resp.~$X'$,\nand $Y$ ranges over the set of irreducible components of $X\\times X'$. \n\nLet $s\\colon X'\\times X\\to X\\times X'$ be the isomorphism\nexchanging the two factors. One has\n\\[\u00a0s^*(\\omega \\wedge \\omega') = (-1)^{nn'} \\omega'\\wedge\\omega,  \\]\nif $n=\\dim(X)$, $n'=\\dim(X')$, $\\omega$ is a volume form on~$X$ \nand $\\omega'$ is a  volume form on~$X'$.\nConsequently,\n\\[ a \\cdot b = \\eps^{nn'} \\cdot b \\cdot a \\]\nfor $a\\in\\Burn_n(k)$ and $b\\in \\Burn_{n'}(k)$.\nIn particular, classes in $\\Burn_n(k)$, for even~$n$, are central in~$\\Burn(k)$.\n\nWe remark that \nthe element~$\\mathbf T\\in\\Burn_1(k)$ is central as well.\nLet indeed $a\\in\\Burn_n(k)$. If $n$ is even, then $a\\cdot\\mathbf T=\\mathbf T\\cdot a$.\nOtherwise, we have $a\\cdot \\mathbf T=\\eps \\cdot \\mathbf T\\cdot a$,\nbut we have seen in example~\\ref{exem.T}\nthat \n$\\mathbf T=\\eps\\cdot\\mathbf T$. \nAs a consequence, $a\\cdot \\mathbf T=\\mathbf T\\cdot a$.\n\nHowever, the ring~$\\Burn(k)$ is \\emph{not} commutative.\nIndeed, consider curves $X$, $X'$ \nwithout automorphisms\nand no nonconstant morphism between them.\nThen the switch is the only isomorphism\nfrom $X'\\times X$ to~$X\\times X'$. \nTake nonzero logarithmic $1$-forms $\\omega,\\omega'$ on~$X,X'$ respectively. \nThe classes $[X\\times X',\\omega\\wedge\\omega']$\nand $[X'\\times X,\\omega'\\wedge\\omega]$ are then distinct.\n\n\n\\subsection{Functoriality}\nLet $k'$ be an extension of~$k$. Then there is a natural\nring homomorphism \n\\[\u00a0\\Burn(k)\\to\\Burn(k') \\]\ndescribed as follows.\nLet $(X,\\omega)$ be an integral $k$-variety\nof dimension~$n$ equiped with a logarithmic $q$-form. \nLet $X'=X\\otimes_k k'$\nbe its base change to~$k'$, and let $\\omega'$\nbe the volume form on~$X'$ deduced from~$\\omega$\nby base change. \nThen the class of $(X,\\omega)$ maps to \nthe sum of classes $(Y,\\omega'|_Y)$, where $Y$ runs\nthe (finite) set of irreducible components of~$X'$.\n\nIf $k'$ is a finite extension of~$k$,\nwe also have a trace map\n\\[\u00a0\\Tr_{k'\/k} \\colon \\Burn(k')\\to \\Burn(k) \\]\nobtained by averaging over a set of representatives\nof automorphisms of the Galois closure of~$k'$ over~$k$ modulo \nthose preserving~$k'$.\n\n\\subsection{Relation with the classical Burnside group}\nForgetting the form~$\\omega$ gives a ring morphism\n\\[\u00a0\\pi\\colon \\Burn(k) \\to \\burn(k). \\]\nOn the other hand, if $K$ is a finitely generated\nextension of~$k$ of transcendence degree~$n$, \nwe can endow it with the zero $n$-form.\nThe resulting map\n\\[\u00a0\\varpi\\colon \\burn(k) \\to \\Burn(k) \\]\nidentifies $\\burn(k)$ with an ideal of $\\Burn(k)$.\nOne has $\\pi\\circ\\varpi=\\id$.\n\n\\subsection{Variations on the theme}\nThe construction of the Burnside ring~$\\Burn(k)$ admits\nseveral natural variants that are relevant in more specific contexts.\nSome of them will be used in later sections.\n\n\\subsubsection{A relative ring}\nLet $n$ be an integer.\nFor any $k$-scheme~$S$, we define $\\Burn_n(S\/k)$\nas the free abelian group on triples $(X,\\omega,u)$\nwhere $X$ is an integral smooth $n$-dimensional $k$-scheme,\n$\\omega\\in\\Omega^n_{X\/k}$ is a regular volume form\nwhich is logarithmic \"everywhere\",\nand $u\\colon X\\to S$ is a morphism, \nmodulo the smallest equivalence relation \nthat identifies $(X,\\omega,u)$ and $(X',\\omega',u')$\nif there exists an open immersion $f\\colon X'\\to X$\nsuch that $\\omega'=f^*\\omega$ and $u'=u\\circ f$.\n\nLet $h\\colon S\\to T$ be a morphism of $k$-schemes.\nIt induces a morphism of abelian groups \n\\[\nh_*\\colon \\Burn_n(S\/k)\\to \\Burn_n(T\/k)\n\\]\nsuch that $h_*([X,\\omega,u])=[X,\\omega,h\\circ u]$\nfor any triple $(X,\\omega,u)$ as above.\n\n\\subsubsection{Pluriforms}\nOne can replace volume forms with volume $r$-pluriforms,\nthat is, elements of~$(\\Omega^n_{K\/k})^{\\otimes r}$,\nfor some given integer~$r$. \nThe corresponding logarithmic condition requires\nthat the pluriform has poles of order at most~$r$ on \nan adequate model.\nNote that when $r$ is even, the obtained ring is commmutative.\n\n\\subsubsection{Forms up to scalars}\nIn the construction, we may wish to identify\n$(K,\\omega)$ and $(K',\\omega')$\nif there exists $\\lambda\\in k^\\times$, resp. $\\lambda \\in\\{\\pm1\\}$,\nand a $k$-isomorphism $f\\colon K\\to K'$ such that $f^*\\omega'=\\lambda\\omega$.\nThese variants also give rise to a commutative ring.\n\n\\subsubsection{Group actions}\\label{sss.group}\nLet $G$ be a profinite group scheme over~$k$.\nOne can also consider pairs $(K,\\omega)$,\nwhere the field~$K$ is endowed with an action of~$G$\nleaving the form~$\\omega$ invariant.\nThe obtained ring will be denoted by~\n$\u00a0\\Burn^G(k)$.\n\n\n\\section{Residues}\\label{sec.residues}\n\n\\subsection{Residue of a volume form}\nLet $X$ be an equidimensional smooth $k$-variety of dimension~$n$.\n\nLet $D$ be a smooth divisor on~$X$.\nWe denote by $\\Omega^m_{X\/k}(\\log D)$\nthe sheaf of $m$-forms on~$X$ with logarithmic poles along~$D$, \nlocally of the form $\\eta \\wedge \\mathrm d\\log f + \\eta' $,\nwhere $\\eta$ and $\\eta'$ are regular and $f$ is a local equation of~$D$.\nThe residue map is the homomorphism  of $\\mathscr O_X$-modules\n\\[\n\\rho_D\\colon \\Omega^{m}_{X\/k}(\\log D)\\to \\Omega^{m-1}_{D\/k},\n\\]\ncharacterized by the relation\n\\[\n\\rho_D(\\eta \\wedge \\mathrm d \\log f + \\eta' )\n=\\eta|_D\n\\] \nfor every local sections $\\eta\\in\\Omega^{m-1}_{X\/k}$\nand $\\eta'\\in\\Omega^m_{X\/k}$, \nand any local generator~$f$ of the ideal of~$D$.\n\nIf $\\omega$ is a logarithmic $m$-form on~$X$,\nthere is an open subset~$U$ of~$X$ such that $U\\cap D\\neq\\emptyset$\nand such that $\\omega|_U$ belongs to $\\Omega^m_{X\/k}(\\log D)$.\nIts residue $\\rho_D(\\omega|_U)$ is then\na meromorphic section of~$\\Omega^{m-1}_{D\/k}$.\n\n\\begin{lemm}\\label{lemm.residue-log}\nLet $\\omega$ be a logarithmic differential form of degree~$m$ on~$X$.\nThen $\\rho_D(\\omega)$ is a logarithmic $(m-1)$-form on~$D$.\n\\end{lemm}\n\\begin{proof}\nWe may assume that the sum of~$D$ and of the polar divisor of~$\\omega$\nhas strict normal crossings. The assertion \nis then evident in  local  coordinates.\n\\end{proof}\n\n\n\n\n\\subsection{Blowing-ups and normal bundles}\n \nLet $Y$ be a smooth closed subscheme of~$X$.\nThe blow-up $\\Bl_Y(X)$ of~$X$ along~$Y$ is a smooth $k$-variety.\nThe blowing-up morphism $b_Y\\colon \\Bl_Y(X)\\to X$ is an isomorphism\nover the complement of~$Y$.\nIf $Y$ is nowhere dense and nonempty,\nthen $E_Y=b_Y^{-1}(Y)$ is a smooth divisor in~$\\Bl_Y(X)$. \n\nIn general, $E_Y=b_Y^{-1}(Y)$ identifies, as an $Y$-scheme, with\nthe projectivization of the normal bundle $\\mathscr N_Y(X)$\nof~$Y$ in~$X$. \n\nLet $W$ be a closed smooth subscheme of~$X$. Assume that $W$ and~$Y$ \nare transversal. Then the Zariski closure of $b_Y^{-1}(W\\setminus (Y\\cap W))$\nis called the strict transform of~$W$ in~$\\Bl_Y(X)$.\nIt identifies with $\\Bl_{Y\\cap W}(W)$.\n\nLet now $\\omega$ be a logarithmic $m$-form on~$X$.\nThen the form $b_Y^*\\omega$ on~$\\Bl_Y(X)$ is logarithmic;\nassuming that $Y$ is nonempty and nowhere dense, we can consider\nthe residue $\\rho_Y(\\omega)$\nof~$b_Y^*\\omega$ along~$E_Y$. It is a logarithmic $(n-1)$-form\non $\\P(\\mathscr N_Y(\\mathscr X))$.\n\n\\begin{defi}\\label{defi.rho}\nLet $X$ be an irreducible proper smooth $k$-variety,\nlet $n$ be its dimension and\nlet $\\omega\\in\\Omega^n_{X\/k}$ be  logarithmic volume form\nwhose polar divisor~$D$ has strict normal crossings.\nLet $(D_\\alpha)_{\\alpha\\in\\mathscr A}$ be the family of its\nirreducible components; for $A\\subset\\mathscr A$,\nwe let $D_A=\\bigcap_{\\alpha\\in A}D_\\alpha$.\nWe then define an element $\\rho(X,\\omega)$ in $\\Burn_{n-1}(X\/k)$\nby the formula:\n\\[\u00a0\\rho(X,\\omega) = \\sum_{\\emptyset\\neq A\\subset \\mathscr A}\n (-1)^{\\Card(A)-1}  \\rho_{D_A}(X,\\omega) .\\]\n\\end{defi}\n(In this formula and all similar ones below, it is always\nimplicit that the terms where $D_A=\\emptyset$ are omitted.)\n\n\\subsection{Iterated residues}\\label{ss.iter}\nWe retain the notation of definition~\\ref{defi.rho}\n\nFix \na logarithmic volume form~$\\omega$ on~$X$\nand \na nonempty subset~$A$ of~$\\mathscr A$\nsuch that $D_A\\neq\\emptyset$.\nIt will be useful to compute  inductively\nthe logarithmic volume form $\\rho_{D_A}(\\omega)$\nthat appears in definition~\\ref{defi.rho}.\n\nLet $b_A\\colon \\tilde X\\to X$ be the blowing-up of~$X$\nalong~$D_A$ and let $E$ be its exceptional divisor.\n\nWhen $A=\\{\\alpha\\}$ has a single element, \n$D_A$ is the divisor~$D_\\alpha$,\nthe blowing-up morphism~$b_A$ is an isomorphism\nand the exceptional divisor identifies with~$D_A$.\nThen \n\\[\u00a0\\rho_{D_A}(X,\\omega)=[D_\\alpha,\\rho_{D_\\alpha}(\\omega),j_\\alpha], \\]\nwhere  $j_\\alpha$ is the immersion of~$D_\\alpha$ into~$X$.\n\nThis construction can be pursued in higher codimension,\nusing iterated residues.\nFix a total order on~$\\mathscr A$.\nThere is a unique, strictly increasing sequence\n$(\\alpha_1,\\dots,\\alpha_m)$ in~$\\mathscr A$ such that \n$A=\\{\\alpha_1,\\dots,\\alpha_m\\}$.\nGiven the chosen order on~$\\mathscr A$,\nwe may apply the iterated residues construction \nand obtain a logarithmic form of degree~$n-m$\n\\[\u00a0\\rho_{D_A} (\\omega) \n= \\rho_{D_{\\alpha_1}} \\circ \\dots \\circ \\rho_{D_{\\alpha_m}}(\\omega). \\]\nOn a nonempty open subset~$U$ of~$X$ that meets~$D_A$, we may write\n\\[\u00a0\\omega=\\eta \\wedge \\dlog(f_{\\alpha_1})\\wedge \\dots\n\\dlog (f_{\\alpha_m}), \\]\nfor a regular form~$\\eta$, \nand then one has $\\rho_{D_A}(\\omega)=\\eta|_{U\\cap D_A}$.\n\nDenote by~$b_A$ the blowing-up of~$X$ along~$D_A$\nand by~$E_A$ its exceptional divisor;\nrecall that $E_A$ identifies with the projectivized normal\nbundle~$\\mathscr N_{D_A}(X)$ of~$D_A$ in~$X$.\nUsing local equations for the divisors~$D_\\alpha$, for $\\alpha\\in A$,\nwe trivialize $\\mathscr N_{D_A}(X)$ on a dense open subscheme of~$D_A$;\nthis gives a birational isomorphism of~$E_A$ with~$D_A \\times \\P^{m-1}$\n(with $m=\\Card(A)$), and a local computation gives the formula\n\\[\u00a0\\rho_{D_A}(X,\\omega) = \n[D_A,\\rho_{D_A}(\\omega)]\\cdot \\mathbf T^{m-1} \\]\n in $\\Burn_{n-1}(D_A\/k)$.\n\nWhen $m\\geq 2$, \nthe definition of~$\\rho_{D_A}$ actually depends \non the chosen order of~$\\mathscr A$, but only \nup to a sign, so that  the class $[D_A,\\rho_{D_A}(\\omega)]$\nis well defined up to multiplication\nby the class~$\\eps\\in\\Burn_0(k)$. \nOn the other hand,\nit is multiplied by $\\mathbf T^{m-1}$\nand we have observed that $\\eps \\cdot \\mathbf T=\\mathbf T$.\n\n\n\\begin{prop}\\label{prop.good-blow-up}\nLet $(X,\\omega)$, $D$, and $(D_\\alpha)_{\\alpha\\in \\mathscr A}$\nbe as in definition~\\ref{defi.rho}.\nLet $Y$ be a strict irreducible subvariety of~$X$;\nlet $\\mathscr A_Y$ be the set of all $\\alpha\\in \\mathscr A$\nsuch that $Y\\not\\subset D_\\alpha$;\nwe assume that $\\sum_{\\alpha\\in \\mathscr A_Y} D_\\alpha$\nmeets~$Y$ transversally.\n\nLet $g\\colon X'\\to X$ be the blowing-up of~$X$ along~$Y$\nand let $\\omega'=g^*\\omega$; it is a logarithmic\nform, its polar divisor\nhas strict normal crossings, and we have \n\\[\u00a0g_* \\rho (X',\\omega') = \\rho(X,\\omega) \\]\nin $\\Burn(X\/k)$.\n\\end{prop}\n\\begin{proof}\nLet $E=g^{-1}(Y)$ be the exceptional divisor;\nfor each $\\alpha\\in\\mathscr A$, let $D'_\\alpha$\nbe the strict transform of~$D_\\alpha$.\nThe blow-up~$X'$\nis smooth;\nthe  divisor $E+\\sum_{\\alpha\\in\\mathscr A}D'_\\alpha$\nhas strict normal crossings and\ncontains the polar divisor of~$\\omega'$.\n\nLet $B$ be the set of all~$\\beta\\in\\mathscr A$ such that $Y\\subset D_\\beta$,\nso that $D_B$ is the minimal stratum containing~$Y$.\n\nWe now split the discussion into two cases.\n\n\\begin{enumerate}\n\\item\n\\emph{Assume that $\\dim(Y)< \\dim(D_B)$.} \nSince $g$ is ramified along~$E$, its Jacobian vanishes along~$E$.\nSince $\\omega$ has poles of order at most~one, \nthe form $\\omega'=g^*\\omega$ is regular at the generic point of~$E$.\nConsequently, the polar divisor of~$\\omega'$ does not contain~$E$\nand we have to compare\n\\[\u00a0\\sum_{\\emptyset\\neq A\\subset \\mathscr A} (-1)^{\\Card(A)-1}\n  \\rho_{D'_A} (\\omega') \n\\]\nwith\n\\[\u00a0\\sum_{\\emptyset\\neq A\\subset \\mathscr A} (-1)^{\\Card(A)-1}\n  \\rho_{D_A} (\\omega) .\\]\n\nSince $g$ is a local isomorphism around\nthe generic points of~$D_\\alpha$, for $\\alpha\\in\\mathscr A$,\nwe see that the polar divisor of~$\\omega'$ \nis equal to $\\sum_{\\alpha\\in{\\mathscr A}}D'_\\alpha$.\nFor every nonempty subset~$A$ of~$\\mathscr A$,\none has\n\\[\u00a0g_* \\rho_{D_A}(X',\\omega')  = \\rho_{D_A}(X,\\omega) \\]\nfor every nonempty subset~$A$ of~$\\mathscr A$,\nwhich implies the desired formula in this case.\n\n\\item\n\\emph{Assume that $\\dim(Y)=\\dim(D_B)$.}\nIn this case, $Y$ is an irreducible component of~$D_B$. \nSince $D_{\\emptyset}=X$ and $Y\\neq X$, we have $B\\neq\\emptyset$.\nWe have to compare the expression\n\\[\u00a0\\sum_{\\emptyset\\neq A\\subset \\mathscr A} (-1)^{\\Card(A)-1}\n  \\rho_{D'_A} (\\omega') \n+ \\sum_{A\\subset\\mathscr A} (-1)^{\\Card(A)}  \\rho_{E\\cap D'_A}(\\omega') \\]\nwith\n\\[\u00a0\\sum_{\\emptyset\\neq A\\subset \\mathscr A} (-1)^{\\Card(A)-1}\n  \\rho_{D_A} (\\omega) .\\]\nThe argument takes place in a neighborhood of~$Y$, \nwhich allows us to assume that $Y=D_B$.\n\nLet $A$ be a nonempty subset of~$\\mathscr A$.\nOne has $D'_A=\\emptyset$ whenever $B\\subset A$, and the corresponding\nterms are absent from the second expression.\nOn the other hand, if $B\\not\\subset A$, \nthe morphism~$g$  identifies~$D'_A$ with the blow-up\nof~$D_A$ along~$D_{A}\\cap Y=D_{A\\cup B}$. \nIn particular, $g$ induces a birational isomorphism from~$D'_A$ to~$D_A$,\nso that $g_* \\rho_{D'_A}(X',\\omega')=\\rho_{D_A}(X,\\omega)$.\nMoreover, $E\\cap D'_A$ is the projectivized\nnormal bundle $\\mathbf P\\mathscr N_{D_{A\\cup B}}(D_A)$,\nand \n\\[\u00a0g_* \\rho_{E\\cap D'_A} (X',\\omega')= \\rho_{D_{A\\cup B}}(X,\\omega). \\]\nSimilarly, one has \n\\[\u00a0g_*\u00a0\\rho_E(X',\\omega') = \\rho_{D_B}(X,\\omega). \\]\nThis gives a formula of the form \n\\begin{align*}\n\u00a0g_*\\rho(X',\\omega') & = \\sum_{\\substack{\\emptyset \\neq A \\subset \\mathscr A \\\\ B \\not\\subset A}}  (-1)^{\\Card(A)-1} \\rho_{D_A}(X,\\omega)\n+ \\sum_{\\substack{ A \\subset \\mathscr A \\\\ B \\not\\subset A}} (-1)^{\\Card(A)} \\rho_{D_{A\\cup B}}(X,\\omega)  \\\\\n& = \\sum_{\\emptyset \\neq A \\subset\\mathscr A} n'_A \\rho_{D_A}(X,\\omega), \n\\end{align*}\nwhere\n\\[\n\u00a0n'_A = \\begin{cases} (-1)^{\\Card(A)-1} & \\text{if $B\\not\\subset A$,} \\\\\n \\sum_{\\substack{ C \\subset \\mathscr A \\\\ B \\not\\subset C \n \\\\ C \\cup B = A }} (-1)^{\\Card(C)} & \\text{if $B\\subset A$.} \\end{cases} \\]\nIt suffices to prove that $n'_A=n_A$ \nfor any nonempty subset~$A$ of~$\\mathscr A$.\nThis is obvious when $B\\not\\subset A$, so let us assume that $B\\subset A$.\nIn the sum that defines~$n'_A$,  we write $C=(C\\setminus B)\\cup C'$,\nwhere $C'=C\\cap B$ is a subset of~$B$;\nthe condition $C\\cup B=A$ means $C\\setminus B=A\\setminus B$;\nthe condition $B\\not\\subset C$ means $C'\\neq B$.\nConsequently, we have\n\\begin{align*}\n\u00a0n'_A & = (-1)^{\\Card(A\\setminus B)}\n\\sum_{\\substack{C' \\subset B \\\\\u00a0C' \\neq B}} (-1)^{\\Card(C')} \\\\\n& = (-1)^{\\Card(A\\setminus B)}\n\\left( \\sum_{C' \\subset B } (-1)^{\\Card(C')} \n- (-1)^{\\Card(B)}\\right) \\\\\n& = (-1)^{\\Card(A\\setminus B)}\n\\left( (1-1)^{\\Card(B)} - (-1)^{\\Card(B)}\\right) \\\\\n& = (-1)^{\\Card(A)-1}, \n\\end{align*}\nsince $\\Card(B)\\geq1$.\nThis concludes the proof of the proposition.\n\\qedhere\n\\end{enumerate}\n\\end{proof}\n\n\\begin{theo}\\label{theo.residue-bir}\nLet $(X,\\omega)$ be as in definition~\\ref{defi.rho}.\nIf $X$ is proper, then the image of $\\rho(X,\\omega)$\nin~$\\Burn_{n-1}(k)$ only depends on the class $[X,\\omega]\\in\n\\Burn_n(k)$.\nIt gives rise to a morphism of abelian groups\n\\[\u00a0\\partial_n\\colon \\Burn_n(k)\\to \\Burn_{n-1}(k). \\]\n\\end{theo}\n\\begin{proof}\nBy the definition of $\\Burn_n(k)$ involving\npairs $(X,\\omega)$ where $X$ is proper,\nit suffices to consider two pairs $(X,\\omega)$\nand $(X',\\omega')$ as in definition~\\ref{defi.rho}\nwhich are related by a proper birational morphism\n$g\\colon X'\\to X$ such that $g^*\\omega=\\omega'$.\nBy the weak factorization theorem of~\\cite{AbramovichKaruMatsukiEtAl-2002},\nin order to prove the theorem,\nwe may assume that $g$ is a blowing-up\nof~$X$ along a smooth subvariety which is transversal\nto the polar divisor of~$\\omega$. In\nthis case, proposition~\\ref{prop.good-blow-up} \nasserts that $g_*\\rho(X',\\omega')=\\rho(X,\\omega)$\nin $\\Burn(X\/k)$. In particular,\nthe images in~$\\Burn(k)$\nof $\\rho(X',\\omega')$ and $\\rho(X,\\omega)$\nare equal.\n\\end{proof}\n\n\\begin{exem}\\label{exem.residue-torus}\nThe meromorphic differential form $\\mathrm dt\/t$ on~$\\P^1_k$\nhas residues~$1$ and~$-1$ at~$0$ and~$\\infty$ respectively.\nBy construction, we thus have \n\\[\u00a0\\partial_1(\\mathbf T) = [\\Spec(k), 1] + [\\Spec(k),-1]\n = \\mathbf 1 + \\eps . \\]\n\nLet $n$ be an integer such that $n\\geq 2$ \nand let us compute $\\partial_n(\\mathbf T^n)$.\nWe view $\\mathbf T^n$ as the class of~$\\P^n$, with homogeneous\ncoordinates $[1:x_1:\\dots:x_n]$, and with the toric differential\nform\n\\[\n\\omega_n=(\\mathrm dx_1\/x_1)\\wedge \\dots (\\mathrm dx_n\/x_n).\n\\]\nIts divisor is the sum of the toric hyperplanes $D_0,\\dots,D_n$.\nEach of these hyperplanes identifies with $\\P^{n-1}$, \nand $\\rho_{D_j}(\\omega_n)$ is $(-1)^{n-j}\\omega_{n-1}$.\nLet $\\mathscr A=\\{0,\\dots,n\\}$.\nIf $A=\\mathscr A$, then $D_A=\\emptyset$.\nOtherwise, we see by induction\nthat  $D_A$ is isomorphic to~$\\P^{n-\\Card(A)}$\nand $\\rho_{D_A}(\\omega_n)$ identifies with $\\pm \\omega_{n-\\Card(A)}$,\nso that \n\\[\u00a0[D_A,\\rho_{D_A}(\\omega_n)]\\cdot \\mathbf T^{\\Card(A)-1} = \n[\\gm^{n-1},\\pm \\omega_{n-1}] = \\mathbf T^{n-1}, \\]\nsince $n-1\\geq 1$.\nThen,\n\\begin{align*}\n\\partial_n(\\mathbf T^n) \n& = \\sum_{\\emptyset \\neq A\\subset \\mathscr A}\n (-1)^{\\Card(A)-1} [D_A, \\rho_{D_A}(\\omega_n)]\u00a0\\cdot \\mathbf T^{\\Card(A)-1} \\\\\n& = \\sum_{\\emptyset \\neq A\\subsetneq \\mathscr A} (-1)^{\\Card(A)-1} \\mathbf T^{n-1}.\n\\end{align*}\nNow,\n\\[\u00a0\\sum_{\\emptyset \\neq A\\subsetneq\\mathscr A} (-1)^{\\Card(A)-1}\n= 1 - (1-1)^{n+1}+ (-1)^{n+1} = \n\\begin{cases} \n 2 & \\text{if $n$ is odd;} \\\\\n0 & \\text{if $n$ is even}. \n\\end{cases} \\]\nWe get \n$\u00a0\\partial_n(\\mathbf T^n) = 2 \\mathbf T^{n-1}$ \nif $n$ is odd and $\\partial_n(\\mathbf T^n)=0$ if $n$ is even.\n(Remind that $n\\geq 2$.)\nSince $\\mathbf T=\\eps\\cdot\\mathbf T$, the following formula unifies \nthe various cases: for $n\\geq 1$, we have\n\\[\u00a0\\partial_n(\\mathbf T^n) = (1+(-1)^{n-1}\\eps)\\cdot \\mathbf T^{n-1}. \\]\n\\end{exem}\n\n\\begin{prop}\\label{prop.residue-times-T}\nFor every class $b\\in\\Burn_n(k)$, we have\n\\[\u00a0\\partial_{n+1} (b\\cdot\\mathbf T)\n = - \\partial_n(b) \\cdot\\mathbf T+ b\\cdot \\partial_1\\mathbf T. \\]\n\\end{prop}\n\\begin{proof}\nWe may assume that $b=[X,\\omega]$, where \n$X$ is a proper integral smooth variety of dimension~$n$,\nand $\\omega$ is a logarithmic volume form on~$X$\nwhose polar divisor has strict normal crossings.\nLet $(D_\\alpha)_{\\alpha\\in\\mathscr A}$ be the family of its irreducible\ncomponents.\nWe view $b\\cdot\\mathbf T$\n as the class of $[X\\times \\mathbf P^1,\\omega \\wedge \\mathrm dt\/t]$.\nThe polar divisor of~$\\omega \\wedge\\mathrm dt\/t$ is equal to\n\\[\u00a0\\sum_{\\alpha\\in\\mathscr A} D_\\alpha \\times \\P^1 +\n X \\times \\{0\\} + X \\times \\{\\infty\\}. \\]\nIt has strict normal crossings, and its strata are \nof the form $D_A \\times \\P^1$, for nonempty $A\\subset\\mathscr A$,\nor $D_A \\times\\{0\\}$, or $D_A \\times\\{\\infty\\}$, for $A\\subset\\mathscr A$.\nThis decomposes $\\partial_{n+1}(b\\times\\mathbf T)$ as \nthe sum of three terms.\n\nThe first one is\n\\[\u00a0\\sum_{\\emptyset\\neq  A\\subset\\mathscr A} [D_A\\times\\P^1,\\rho_{D_A\\times\\P^1}(\\omega\\wedge\\mathrm dt\/t)]\\cdot \\mathbf T^{\\Card(A)-1}. \\]\nFor any nonempty subset~$A$ of~$\\mathscr A$, one has \n\\[\u00a0\\rho_{D_A\\times\\P^1}(\\omega\\wedge \\mathrm dt\/t) = \\pm \\rho_{D_A}(\\omega)\\wedge \\mathrm dt\/t, \\]\nso that\n\\[\u00a0 [D_A\\times\\P^1,\\rho_{D_A\\times\\P^1}(\\omega\\wedge\\mathrm dt\/t)]\\cdot \\mathbf T^{\\Card(A)-1}\n= [D_A,\\rho_{D_A}(\\omega)]\u00a0\\cdot \\mathbf T\\cdot \\mathbf T^{\\Card(A)-1}.  \\]\nConsequently, the first term equals\n$\u00a0\\partial_n(b)\\times\\mathbf T$.\n\nWrite $D_0=X\\times\\{0\\}$ and $D_\\infty=X\\times\\{\\infty\\}$, and\nidentify both divisors to~$X$.\nFor a subset~$A$ of~$\\mathscr A$, \nwe have\n\\[\u00a0\\rho_{D_{A\\cup\\{0\\}}} (\\omega\\wedge\\mathrm dt\/t)\n= \\rho_{D_A} \\circ \\rho_{D_0} (\\omega\\wedge\\mathrm dt\/t)\n = \\rho_{D_A} (\\omega). \\]\nConsequently, the second term is equal to\n\\[\u00a0\\sum_{A\\subset \\mathscr A}\n (-1)^{\\Card(A)} [D_A,\\rho_{D_A}(\\omega)] \\cdot \\mathbf T^{\\Card(A)}\n= [X,\\omega]\u00a0 - \\partial_n(b) \\cdot \\mathbf T. \\]\n\nSimilarly, the third term is equal to\n\\[\u00a0[X,-\\omega] - \\partial_n(b)\\cdot \\mathbf T. \\]\n\nSumming up these three terms, we get\n\\[\u00a0\\partial_{n+1}(b\\times\\mathbf T)\n= - \\partial_n(b) \\cdot \\mathbf T + [X,\\omega] + [X,-\\omega]. \\]\nWe now recall that $\\partial_1(\\mathbf T)=[\\Spec(k),1]+[\\Spec(k),-1]$,\nso that\n\\[\u00a0[X,\\omega] + [X,-\\omega] = [X,\\omega]\u00a0\\cdot \\partial_1(\\mathbf T)\n= b \\cdot \\partial_1(\\mathbf T). \\]\nThis concludes the proof.\n\\end{proof}\n\n\n\\begin{theo}\\label{theo.partial-deriv}\nLet $a\\in\\Burn_m(k)$ and $b\\in\\Burn_n(k)$;\nwe have\n\\[\u00a0\\partial_{m+n}(a\\cdot b)\n= \\eps^n\\cdot   \\partial_m(a) \\cdot b + a \\cdot \\partial_n(b) - \\mathbf T \\cdot \\partial_m(a)\\cdot \\partial_n(b) \\]\nin $\\Burn_{m+n-1}(k)$.\n\\end{theo}\n\\begin{proof}\nIt suffices to treat the case where $a$ and $b$ are classes \nof proper integral smooth varieties $(X,\\omega)$,\n$(Y,\\eta)$, endowed with meromorphic volume forms\nwhose polar divisors have strict normal crossings and no multiplicities.\nLet $(D_\\alpha)_{\\alpha\\in\\mathscr A}$ be the irreducible\ncomponents of the polar divisor of~$\\omega$,\nlet $(E_\\beta)_{\\beta\\in\\mathscr B}$ be the irreducible\ncomponents of the polar divisor of~$\\eta$.\nThen $[X,\\omega]\\cdot [Y,\\eta]$\nis the class of $[X\\times Y, \\omega\\wedge\\eta]$;\nthe polar divisor of~$\\omega\\wedge\\eta$ is equal to\n\\[\u00a0\\sum_{\\alpha\\in\\mathscr A} D_\\alpha \\times Y + \\sum_{\\beta\\in \\mathscr B} X \\times E_\\beta. \\]\nWe fix a total order on the disjoint union of~$\\mathscr A$\nand~$\\mathscr B$ such that the elements of~$\\mathscr A$ \nare smaller than those of~$\\mathscr B$.\nFor any subsets $A,B$ of~$\\mathscr A$ and~$\\mathscr B$,\nobserve that we have\n\\[\u00a0\\rho_{D_{A\\cup B}} (\\omega\\wedge\\eta) \n = \\pm \\rho_{D_A}(\\omega) \\wedge \\rho_{E_B}(\\eta), \\]\nwhere $\\rho_{D_A}$ has to be understood as\nthe identity when $A$ is empty, and similarly for~$\\rho_{E_B}$.\nThe sign is~$1$ when $A=\\emptyset$;\nwhen $B=\\emptyset$, it is equal to~$(-1)^{\\Card(A)n}$;\nwe won't need to use its explicit value in the other cases.\nThen we can write\n$\u00a0\\partial ([X,\\omega]\\cdot [Y,\\eta]) $ as\n\\[\n \\sum_{\\substack{A\\subset \\mathscr A \\\\ B\\subset \\mathscr B \\\\\n A\\cup B\\neq\\emptyset}}\n(-1)^{\\Card(A)+\\Card(B)-1}\n[\u00a0D_A\\times E_B, \\pm \\rho_{A}(\\omega)\\wedge\\rho_{E_B}(\\eta)]\n \\cdot \\mathbf T^{\\Card(A\\cup B)-1} \\]\nand we split it into the sum of three terms,\naccording to which $B=\\emptyset$, or $A=\\emptyset$, \nor none of them is empty.\nThe first two terms are respectively equal to \n\\[\u00a0\\sum_{\\emptyset\\neq A\\subset\\mathscr A}\n(-1)^{\\Card(A)-1} [D_A\\times Y,(-1)^{n\\Card(A)}\\rho_{D_A}(\\omega)\\wedge\\eta]\n\\cdot \\mathbf T^{\\Card(A)-1}\n= \\partial([X,(-1)^n \\omega])\\cdot [Y,\\eta] \\]\nand\n\\[\u00a0\\sum_{\\emptyset\\neq B\\subset\\mathscr B}\n(-1)^{\\Card(B)-1} [X\\times E_B, \\omega\\wedge \\rho_{E_B}(\\eta)]\n\\cdot \\mathbf T^{\\Card(B)-1}\n= [X,\\omega]\\cdot \\partial([Y,\\eta]), \\]\nsince $\\mathbf T$ belongs to the center of~$\\Burn(k)$.\nAs for the third one, \nwe obtain\n\\[ - \\sum_{\\emptyset \\neq B\\subset \\mathscr B} \n (-1)^{\\Card(B)-1} \\sum_{\\emptyset \\neq A\\subset \\mathscr A}\n (-1)^{\\Card(A)-1}\u00a0[D_A, \\rho_{D_A}(\\omega)]\u00a0\\cdot [E_B,\\rho_{E_B}(\\eta)]\n \\cdot \\mathbf T^{\\Card(A)+\\Card(B)-2}  \\]\nwhich equals\n\\[\u00a0- \\partial ([X,\\omega]) \\cdot \\partial([Y,\\eta])\\cdot \\mathbf T. \\]\nFinally, we get\n\\begin{align*}\n\u00a0\\partial_{m+n}(a\\cdot b)  & = \n\u00a0\\partial_{m+n}([X,\\omega]\\cdot [Y,\\eta])  \\\\\n& = \\partial_m([X,(-1)^n\\omega) \\cdot [Y,\\eta]\n+ [X,\\omega] \\cdot \\partial_n([Y,\\eta]) \\\\\n& \\qquad{}  - \\mathbf T \\cdot \\partial_m([X,\\omega]) \\cdot \\partial_n([Y,\\eta]) \\\\\n& = \\eps^n \\cdot \\partial_m(a) \\cdot b + a \\cdot \\partial_n(b) \n  - \\mathbf T \\cdot \\partial_m(a) \\cdot \\partial_n(b) \n\\end{align*}\nas was to be shown. \n\\end{proof}\nIn particular, using the computation of example~\\ref{exem.residue-torus},\nwe obtain the following generalization of proposition~\\ref{prop.residue-times-T}.\n\\begin{coro} \nFor any $a\\in\\Burn_m(k)$ and any integer~$n$, we have\n\\[\u00a0\\partial_{m+n}(a\\cdot \\mathbf T^n) \n= \\begin{cases}\n\\partial_m(a) \\cdot \\mathbf T^n  & \\text{if $n$ is even;} \\\\ \n-\\partial_m(a) \\cdot \\mathbf T^n +  a \\cdot\\partial_n(\\mathbf T^{n}) \n& \\text{if $n$ is odd.} \\end{cases}\n\\]\n\\end{coro}\n\n\\begin{rema}\nFor the variant of $\\Burn(k)$ where we consider forms up to sign,\nthe formula of theorem~\\ref{theo.partial-deriv}\nsimplifies to \n\\[\u00a0\\partial_{m+n}(a\\cdot b)\n=  \\partial_m(a) \\cdot b + a \\cdot \\partial_n(b) - \\mathbf T \\cdot \\partial_m(a)\\cdot \\partial_n(b). \\]\n\\end{rema}\n\n\\section{A complex of Burnside rings}\n\n\\begin{theo}\\label{theo.dd=0}\nFor any integer $n\\geq 2$, we have\n\\[\u00a0\\partial_{n-1} \\circ \\partial_n = 0. \\]\n\\end{theo}\nIn other words,\nthe residue morphisms of Burnside groups give rise to a complex\n\\[\u00a0\\dots \\to \\Burn_n(k) \\to \\Burn_{n-1}(k)\u00a0\\to \\dots \\to \\Burn_1(k) \\to\\Burn_0(k) \\]\n\n\\begin{proof}\nIt suffices to prove the following result:\nLet $(X,\\omega)$ be an integral proper smooth variety \nof dimension~$n$\nequipped with a meromorphic volume form~$\\omega$\nwhose polar divisor has strict normal crossings and \nno multiplicities;\nthen $\\partial_{n-1}(\\partial_n ([X,\\omega]))=0$.\n\nLet $(D_\\alpha)_{\\alpha\\in\\mathscr A}$ be the family\nof irreducible components of the polar divisor of~$\\omega$ in~$X$.\nBy definition, one has\n\\[\u00a0\\partial_n ([X,\\omega]) = \n\\sum_{\\emptyset\\neq A\\subset \\mathscr A}\n(-1)^{\\Card(A)-1}  \\rho_{D_A}(X,\\omega). \\]\nFix a total order on~$\\mathscr A$.\nLet $(\\alpha_1,\\dots,\\alpha_m)$ be a strictly increasing sequence\nin~$\\mathscr A$ and let $A=\\{\\alpha_1,\\dots,\\alpha_m\\}$.\nWe have seen in~\\S\\ref{ss.iter}\nthat $\\rho_{D_A}(X,\\omega)$ can be defined \nvia iterated residue maps: \n\\[\u00a0\\rho_{D_A}([X,\\omega])=\n[D_A, \\rho_{D_{\\alpha_1}}\\circ \\dots \\circ \\rho_{D_{\\alpha_m}} (\\omega)]\n\\cdot \\mathbf T^{\\Card(A)-1}\n= [D_A,\\omega_A]\\cdot \\mathbf T^{\\Card(A)-1} \n \\]\nwhere we wrote $\\omega_A$ for\nthe composition \n$\\rho_{D_{\\alpha_1}}\\circ \\dots \\circ \\rho_{D_{\\alpha_m}}(\\omega)$.\nWhen $\\Card(A)$ is odd, we have \n\\[\u00a0\\partial (\\rho_{D_A}([X,\\omega])) = \\partial([D_A,\\omega_A])\n\\cdot \\mathbf T^{\\Card(A)-1}, \\]\nwhile when $\\Card(A)$ is even, we have \n\\[\u00a0\\partial (\\rho_{D_A}([X,\\omega])) \n= - \\partial([D_A,\\omega_A])\\cdot \\mathbf T^{a-1}\n+ [D_A,\\omega_A] \\cdot \\partial(\\mathbf T^{\\Card(A)-1}) . \\]\nConsequently, we have\n\\[\u00a0\\partial \\circ\\partial ([X,\\omega])\n= \\sum_{\\emptyset\\neq A\\subset \\mathscr A}\n \\partial ([D_A,\\omega_A])\\cdot \\mathbf T^{\\Card(A)-1}\n- \\sum_{\\substack{\\emptyset\\neq A\\subset \\mathscr A \\\\\u00a0\\text{$\\Card(A) $ even}}}\n[D_A,\\omega_A]\\cdot \\partial(\\mathbf T^{\\Card(A)-1}). \n\\]\n\nThe polar divisor of the form\n$\\omega_A $ on~$D_A$\nis equal to $\\sum_{\\beta\\not\\in A} D_\\beta \\cap D_A$, \nso that, by definition (and computation of~$\\partial$ via iterated residues), \n\\[\u00a0\\partial([D_A,\\omega_A])\n= \\sum_{\\emptyset \\neq B \\subset \\complement A}\n(-1)^{\\Card(B)-1} [D_{AB},\\omega_{A\\cup B}]\u00a0\\cdot \\mathbf T^{\\Card(B)-1}.\\]\nAlso, when $A$ is nonempty and of even cardinality, \n$\\partial(\\mathbf T^{\\Card(A)-1})=2\\mathbf T^{\\Card(A)-2}$.\nWhen we put these two formulas into the antepenultimate one\nand collect the various terms, we obtain\n\\[\u00a0\\partial \\circ\\partial ([X,\\omega])\n= \\sum_{\\substack{ C\\subset \\mathscr A \\\\ 2\\leq \\Card(C)}} n_C [D_C,\\omega_C]\u00a0 \\cdot \\mathbf T^{\\Card(C)-2},\n\\]\nwhere\n\\[\u00a0n_C = - \\sum_{\\substack{\\emptyset \\neq A, B  \\\\\n A \\cup B = C, A \\cap B = \\emptyset}} (-1)^{\\Card(B)}\n - \u00a02 \\delta_{\\text{$\\Card(C)$ is even}}. \\]\nIn the first sum, the terms $A=\\emptyset $ or $B=\\emptyset$ are omitted,\nwhile  if we put them in, we obtain\n\\[ \\sum_{\\substack{A \\cup B=C \\\\ A \\cap B=\\emptyset}}\n(-1)^{\\Card(B)} = \\sum_{b=0}^{\\Card(C)} \\binom {\\Card(C)}b (-1)^b\n= (1-1)^{\\Card(C)}=0 \\]\nsince $\\Card(C)\\geq 1$.\nConsequently, \n\\[\u00a0n_C = 1 + (-1)^{\\Card(C)} - 2 \\delta_{\\text{$\\Card(C)$  is even}}\n= 0. \\]\nThis concludes the proof.\n\\end{proof}\n\n\\section{Algebraic structure of \\(\\Burn(k)\\) after localization at~\\(2\\)}\n\\label{sec.alg}\n\nIn this section, we study the algebraic structure of\nthe Burnside ring $\\Burn(k)$, endowed with its elements~$\\eps$, $\\mathbf T$\nand the operator~$\\partial$. \n\n\\subsection{}\nBy construction, $\\Burn(k)=\\bigoplus_{n\\geq 0}\\Burn_n(k)$\nis an associative unital $\\Z_{\\geq0}$-graded ring, $\\eps\\in\\Burn_0(k)$,\n$\\mathbf T\\in\\Burn_1(k)$ and $\\partial$ is a homogeneous\nadditive map of degree~$-1$. They satisfy the following relations,\nfor homogeneous elements $a,b\\in\\Burn(k)$:\n\\begin{align*}\n\\tag{1}\n & b\\cdot a = \\eps^{\\abs a\\abs b} \\cdot a\\cdot b \n\t&& \\text{(\\S\\ref{ss.commut});} \\\\\n\\tag{2}\n & \\eps^2 = 1 && \\text{(example~\\ref{exem.eps});}  \\\\\n\\tag{3}\n & \\mathbf T = \\eps\\cdot \\mathbf T && \\text{(example~\\ref{exem.T});} \\\\\n\\tag{4}\n & \\partial(\\mathbf T) = 1+\\eps && \\text{(example~\\ref{exem.residue-torus});} \\\\\n\\tag{5}\n & \\partial(a\\cdot b) = \\eps^{\\abs b}\\cdot\\partial(a)\\cdot b+ a\\cdot \\partial(b) -\\mathbf T\\cdot\\partial(a)\\cdot\\partial(b) && \\text{(theorem~\\ref{theo.partial-deriv});} \\\\\n\\tag{6}\n & \\partial(\\partial(a)) = 0 &&  \\text{(theorem~\\ref{theo.dd=0}).} \n\\end{align*}\nBy~(1), the element~$\\eps$ is central, and by~(2), we may view~$\\Burn(k)$\nas an algebra over $\\Z[\\eps]\/(\\eps^2-1)$. After inverting~2, \nthe algebra $\\Burn(k)$ splits into two components\n$\\Burn^{\\eps=1}(k)$ and $\\Burn^{\\eps=-1}(k)$, one over which $\\eps=1$,\nand the other over which $\\eps=-1$.\n\n\\emph{In the rest of this section, we implicitly assume that $2$ is inverted,\nwithout changing the notation.}\n\n\n\\subsection{Sector $\\eps=-1$}\\label{sect.sectorminus}\nHere, we have $\\mathbf T=-\\mathbf T$, \nhence $\\mathbf T=0$ since $2$ is invertible.\nAs a consequence, after replacing~$\\partial$ with\n$\\partial'\\colon a\\mapsto (-1)^{1+\\abs a}\\partial(a)$,\none gets from~(5) the usual graded Leibniz rule \n\\[\u00a0\\partial'(a\\cdot b)=\\partial'(a)\\cdot b+(-1)^{\\abs a}a\\cdot\\partial'(b) \\]\nand therefore\n$\\Burn^{\\eps=-1}(k)$ \nis a classical differential graded (super\\nobreak-)commutative algebra,\nsimilar to, \\eg, the de Rham complex.\n\n\\subsection{Sector $\\eps=1$}\nThe algebra $\\Burn^{\\eps=1}$ is now commutative (and not graded commutative).\nThis reflects the intuition in our constructions that they\nspeak about volume forms (as opposed to top-degree differential forms) \nfor which we have commutativity (as reflected by the change of order\nof integration in multiple integrals). \n\n\\begin{lemm}\nThe map $F\\colon a \\mapsto  a- \\mathbf T\\cdot \\partial(a)$\nis a ring endomorphism of $\\Burn^{\\eps=1}(k)$, and $F^2=\\id$.\nMoreover, one has $F \\circ \\partial =\\partial = - \\partial \\circ F$.\n\\end{lemm}\n\\begin{proof}\nThis map is additive. One has \n$\u00a0F(1) = 1 - \\mathbf T\\cdot \\partial(1)=1$. Let us show multiplicativity.\nIndeed, for $a,b\\in \\Burn^{\\eps=1}(k)$, one has\n\\begin{align*}\nF(a)\\cdot F(b) & = (a-\\mathbf T\\cdot \\partial(a))\\cdot (b-\\mathbf T\\cdot\\partial(b)) \\\\\n& = a\\cdot b - \\mathbf T\\cdot \\partial(a)\\cdot b-\\mathbf T\\cdot a\\cdot\\partial(b)\n+\\mathbf T^2 \\cdot\\partial(a)\\cdot\\partial(b)\\\\\n& = a\\cdot b - \\mathbf T\\cdot (\\partial(a)\\cdot b+ a\\cdot\\partial(b)\n-\\mathbf T \\cdot\\partial(a)\\cdot\\partial(b)) \\\\\n& = a\\cdot b - \\mathbf T\\cdot \\partial(a\\cdot b) & \\text{(using~(5))} \\\\\n& = F(a\\cdot b). \n\\end{align*}\n\nSince $\\partial^2=0$, one has \n\\[\u00a0F (\\partial(a))= \\partial(a)-\\mathbf T\\cdot \\partial(\\partial(a))=\\partial(a). \\]\nOn the other hand,\n\\begin{align*}\n\\partial(F(a))& = \\partial (a-\\mathbf T\\cdot \\partial(a)) \u00a0\\\\\n& = \\partial(a) - \\partial(\\mathbf T\\cdot \\partial(a)) \\\\\n& = \\partial(a) - \\partial(\\mathbf T) \\cdot \\partial(a)\n- \\mathbf T \\cdot \\partial(\\partial(a))+\\mathbf T\\cdot \\partial(\\mathbf T)\n\\cdot \\partial(\\partial(a)) \\\\\n& = -\\partial(a)\n\\end{align*}\nusing that $\\partial(\\mathbf T)=2$ and $\\partial^2=0$.\n\nConsequently, for $a\\in\\Burn^{\\eps=1}(k)$, we have\n\\[\nF^2(a)  = F(a) - \\mathbf T\\cdot \\partial (F(a)) \n = a - \\mathbf T\\cdot \\partial(a) + \\mathbf T \\cdot \\partial (a) \n = a \\]\nsince $\\partial\\circ F=-\\partial$.\n\\end{proof}\n\n\\subsection{}\nTo simplify the notation, write $\\mathscr B=\\Burn^{\\eps=1}(k)$.\nSince $F^2=\\id$ and $2$ is invertible, the algebra $\\mathscr B$\nsplits as a direct sum \n\\[\u00a0\\mathscr B= \\mathscr B_+ \\oplus \\mathscr B_-, \\]\nsuch that $F$ acts as~$\\id$ on $\\mathscr B_+$\nand as $-\\id$ on~$\\mathscr B_-$. Moreover, $\\mathscr B_+$ is a subalgebra.\n\nSince the operator~$\\partial$ anticommutes with~$F$,\nit induces maps\n\\[\u00a0\\partial_{\\pm}\\colon \\mathscr B_+\u00a0\\to \\mathscr B_-,\n\\qquad \\partial_{\\mp}\\colon \\mathscr B_-\u00a0\\to \\mathscr B_+. \\]\n\nNote that \n\\[\u00a0F(\\mathbf T) = \\mathbf T- \\mathbf T \\cdot \\partial(\\mathbf T)\n= - \\mathbf T , \\]\nso that $\\mathbf T\\in\\mathscr B_-$.\nConsequently, the multiplication by~$\\mathbf T$ map induces\ntwo maps\n\\[\u00a0t_{\\pm}\\colon \\mathscr B_+\\to\\mathscr B_-, \\qquad \nt_{\\mp}\\colon \\mathscr B_-\\to\\mathscr B_+. \\]\n\n\\begin{lemm}\nThe map~$\\partial$ vanishes on~$\\mathscr B_+$.\nEquivalently, $\\partial_{\\pm}=0$. \n\nThe maps $\\frac12 \\partial_{\\mp}$ and~$t_{\\pm}$ are inverses\nthe one of the other.\n\\end{lemm}\n\\begin{proof}\nFor $a\\in\\mathscr B_+$, one has $\\partial(a)=-\\partial(F(a))=-\\partial(a)$,\nsince $\\partial\\circ F=-\\partial$, hence $\\partial(a)=0$.\n\nOn the other hand, for $a\\in\\mathscr B_+$, one has\n\\[\u00a0\\partial (\\mathbf T\\cdot a)\n= 2 \\cdot a+\\mathbf T\\cdot \\partial(a)- 2\\mathbf T\\cdot \\partial(a)\n= 2a - \\mathbf T\\cdot\\partial(a)=a + F(a) = 2a \\]\nwhile for $a\\in\\mathscr B_-$, we have \n\\[\u00a0\\mathbf T\\cdot \\partial(a) = a-F(a) = 2a. \\]\nThis concludes the proof of the lemma.\n\\end{proof}\nIn particular, we see that the cohomology of the differential $\\partial$ vanishes in the sector $ \\Burn^{\\eps=1}(k) =\\mathscr B$.\n\n\\subsection{}\nIt follows from the lemma that we have a ring isomorphism\n\\[\u00a0\\mathscr B = \\mathscr B_+[t](t^2-\\mathbf T^2), \\]\nfrom which we see that all the algebraic structure of~$\\mathscr B_+$\n(namely $\\delta$, $\\mathbf T$, $F$) can be canonically reconstructed from a unital commutative associative $\\Z_{\\geq0}$-graded ring~$\\mathscr B_+$ endowed with an element in degree $+2$ (namely, the element $\\mathbf T^2$).\n\\begin{rema}\nThe situation clarifies even more if we invert the class~$\\mathbf T$.\nThen we can write $\\partial(a)=(a-F(a))\/\\mathbf T$,\nand all relations happen to follow from the fact that $F$ is an involution\nsuch that $F(\\mathbf T)=-1$.\nIndeed,\n\\[\n\\partial^2(a) = \\frac{\\partial(a)-F(\\partial(a))}{\\mathbf T}\n= \\frac1{\\mathbf T} \\left( \\frac{a-\\partial(a)}{\\mathbf T}\n- F( \\frac{a-\\partial(a)}{\\mathbf T} \\right) = 0 \\]\nexplains that $\\partial^2=0$.\nMoreover, for $a,b\\in\\mathscr B$, we  have\n\\begin{align*}\n\\partial(a\\cdot b) & = \\frac{a\\cdot b - F(a\\cdot b)}{\\mathbf T}\n = \\frac{a\\cdot b - F(a)\\cdot F(b)}{\\mathbf T} \\\\\n& = \\frac{a-F(a)}{\\mathbf T} \\cdot b + a \\cdot \\frac{b-F(b)}{\\mathbf T}\u00a0\n - \\mathbf T \\cdot \\frac{a- F(a)}{\\mathbf T} \\cdot \\frac{b-F(b)}{\\mathbf T}\u00a0\\\\\n & =\\partial(a)\\cdot b+a\\cdot\\partial(b)-{\\mathbf T}\\cdot\\partial(a)\\cdot\\partial(b).\n\\end{align*}\n\\end{rema}\n\n\n\n\\section{Birational morphisms preserving volume forms}\n\\label{sec.biraut}\n\n\\subsection{}\nLet $(X,\\omega_X)$ be a smooth integral $k$-variety of dimension~$n$\nequipped with a logarithmic volume form, \nand let $f\\colon Y\\to X$ be a proper birational morphism.\n\nLet $E$ be an exceptional divisor in~$Y$,\nthat is, such that $\\dim(f(E))<\\dim(E)$. \nBy lemma~\\ref{lemm.logarithmic-functoriality},\nand lemma~\\ref{lemm.residue-log}, \nthe residue $\\rho_E(f^*\\omega_X)$ along~$E$ \nof the meromorphic form~$f^*\\omega$ is a logarithmic volume\nform on~$E$.\n\nWe define $\\mathbf c(f;X,\\omega)$ to be the sum\nof all such classes $[E,\\rho_E(f^*\\omega)]$\nin the free abelian group $\\Burn_{n-1}(k)$.\n\n\\begin{lemm}\\label{lemm.c-additivity}\nLet $g\\colon Z\\to Y$ be a proper birational morphism\nof smooth integral varieties  of dimension~$n$.\nThen $g\\circ f$ is a proper birational morphism\nand one has\n\\[\u00a0\\mathbf c(g\\circ f;X,\\omega) = \\mathbf c(g;Y,f^*\\omega) + \\mathbf c(f;X,\\omega) \\]\nin $\\Burn_{n-1}(k)$.\n\\end{lemm}\n\\begin{proof}\nAn integral divisor~$F$ in~$Z$ is exceptional for\n$g\\circ f$ if and only if one of the two \nmutually excluding situations happen:\n\\begin{itemize}\n\\item The divisor~$F$ is exceptional for~$g$;\n\\item Or $g(F)$ is a divisor in~$Y$ which is exceptional for~$f$.\n\\end{itemize}\nMoreover, any divisor~$E$ in~$Y$ which is exceptional\nfor~$f$ appears once and only as  a divisor of the form~$g(F)$.\nThe contribution of~$F$ to~$\\mathbf c(g\\circ f;X,\\omega)$\nis given by the volume form~$\\rho_F((g\\circ f)^*\\omega)$.\nIn the first case, we write\n$\\rho_F((g\\circ f)^*\\omega)=\\rho_F(g^*(f^*\\omega))$,\nso that the contribution of~$F$ coincides with its contribution\nto the term $\\mathbf c(g;Y,f^*\\omega)$.\nIn the second case, $g$ induces a birational isomorphism\nfrom~$F$ to~$E=g(F)$;\nwriting $\\rho_F((g\\circ f)^*\\omega)=g^*(\\rho_F(f^*\\omega))$,\nwe see that the contribution of~$F$ coincides with the\ncontribution of~$E$ to $\\mathbf c(f;X,\\omega)$.\nThis concludes the proof.\n\\end{proof}\n\n\\subsection{}\nLet $(X,\\omega_X)$ and $(Y,\\omega_Y)$ be proper smooth $k$-varieties\nequipped with logarithmic volume forms and \nlet \n\\[\n\\phi\\colon (X,\\omega_X) \\dashrightarrow (Y,\\omega_Y)\n\\]\nbe a birational map\npreserving the volume forms.\nBy definition, this means that there exists a diagram\n\\[\u00a0\\begin{tikzcd}[column sep = small]\n & W \\ar{dl}[swap]{p} \\ar{dr}{q} \\\\\nX \\ar[dashrightarrow]{rr}{\\phi} && Y \\end{tikzcd}\u00a0\\]\nof integral $k$-varieties\nsuch that that $p$ and~$q$ are proper and birational,\nand such that $p^*\\omega=q^*\\omega'$ on~$W$.\nIn this situation, we may assume that $W$ is smooth.\n\n\\begin{lemm}\\label{lemm.c-independence}\nWith this notation, the element  \n\\[\n\\mathbf c(\\phi)=\\mathbf c(q)-\\mathbf c(p) \\in \\Burn_{n-1}(k)\n\\]\nonly depends on the birational map~$\\phi$,\nand not on the choice of the triple $(W,p,q)$.\n\\end{lemm}\n\n\\begin{proof}\nConsider two possible diagrams \n$X\\xleftarrow p V\\xrightarrow q Y$ \nand $X\\xleftarrow r W\\xrightarrow s Y$\ndescribing~$\\phi$. \nConsidering for example a resolution\nof singularities~$U$ of $V\\times_X W$,\nwe can fit these two diagrams in a common commutative diagram\nof the following form:\n\\[\u00a0\\begin{tikzcd}[column sep = small]\n && U \\ar{dl}[swap]{u} \\ar{dr}{v} \\\\\n & V \\ar{dl}[swap]{p} \\arrow{drrr}[very near start]{q}\n \t&& W \\ar[crossing over]{dlll}[swap,very near start]{r\\vphantom q} \\ar{dr}{s} \\\\\nX \\ar[dashrightarrow]{rrrr}{\\phi} &&&& \nY \n\\end{tikzcd}\u00a0\\]\nThe equalities $p^*\\omega_X=q^*\\omega_Y$\nand $r^*\\omega_X=s^*\\omega_Y$ imply\nthat \n\\[\u00a0(p\\circ u)^*\\omega_X = \nu^*p^*\\omega_X = u^* q^*\\omega_Y\n=(q\\circ u)^*\\omega_Y = (s\\circ v)^*\\omega_Y. \\]\nBy lemma~\\ref{lemm.c-additivity}, we then have\n\\[\n\\mathbf c(p)-\\mathbf c(q) \n = \\mathbf c(p\\circ u)-\\mathbf c(q\\circ u)  \n =  \\mathbf c(r\\circ v)-\\mathbf c(s\\circ v)  \n = \\mathbf c(r)-\\mathbf c(s) . \n\\]\nThis concludes the proof.\n\\end{proof}\n\n\\begin{theo}\\label{theo.additivity}\nIf $\\psi\\colon (Y,\\omega_Y)\\dashrightarrow (Z,\\omega_Z)$\nis another birational map preserving volume forms,\nthen one has\n\\[\u00a0\\mathbf c(\\psi\\circ\\phi) = \\mathbf c(\\psi)+\\mathbf c(\\psi). \\]\n\\end{theo}\n\\begin{proof}\nConsider two diagrams \n$X\\xleftarrow p V\\xrightarrow q Y$ \nand $Y\\xleftarrow r W\\xrightarrow s Y$\ndescribing~$\\phi$ and~$\\phi$. \nConsidering for example a resolution\nof singularities~$U$ of $V\\times_Y W$,\nwe can fit these two diagrams in a common commutative diagram\nof the following form:\n\\[\u00a0\\begin{tikzcd}[column sep = small]\n && U \\ar{dl}[swap]{u} \\ar{dr}{v} \\\\\n & V \\ar{dl}[swap]{p} \\arrow{dr}{q}\n \t&& W \\ar{dl}[swap]{r\\vphantom q} \\ar{dr}{s} \\\\\nX \\ar[dashrightarrow]{rr}{\\phi} && Y \\ar[dashrightarrow]{rr}{\\psi} && Z \n\\end{tikzcd}\u00a0\\]\nand the diagram $X\\xleftarrow{p\\circ u} U\\xrightarrow{s\\circ v}$\ndescribes the birational map $\\psi\\circ \\phi$.\nSince $q\\circ u=r\\circ v$, we then have\n\\begin{align*}\n\\mathbf c(\\psi\\circ \\phi) & = \n\\mathbf c(p\\circ u)-\\mathbf c(s\\circ v) \\\\\n& = \\mathbf c(p\\circ u)-\\mathbf c(q\\circ u)\n+ \\mathbf c(r\\circ v)- \\mathbf c(s\\circ v) \\\\\n& = \\mathbf c(p)-\\mathbf c(q) + \n+ \\mathbf c(r)- \\mathbf c(s) \\\\\n& = \\mathbf c(\\phi) + \\mathbf c(\\psi), \n\\end{align*}\nas was to be shown.\n\\end{proof}\n\n\\begin{coro}\nLet $\\Bir(X,\\omega)$ be the set of birational automorphisms\nof~$X$ preserving~$\\omega$. The map~$\\mathbf c$\ninduces a homomorphism of abelian groups\n\\[\n\\Bir(X,\\omega)\\to \\Burn_{n-1}(k).\n\\]\nIts kernel contains the group of automorphisms of~$X$\nthat preserve~$\\omega$.\n\\end{coro}\n\n\n\n\\section{Specialization}\n\nLet $K$ be the field of fractions of a discrete valuation ring~$R$\nwith residue field~$k$. \nFix a uniformizer~$t\\in R$.\n\nIn this context, \n\\citet{KontsevichTschinkel-2019} have defined\ntwo (distinct) \\emph{specialization morphisms}\n\\[\u00a0\\rho_t \\colon \\burn_n(K)\\to\\burn_n(k), \\]\nrelating the Burnside groups of~$K$ and~$k$ (see \\ref{ssec.burn}),\none of which is a ring homomorphism.\n(The latter homomorphism actually\ndepends on the choice of~$t$, \nsee example~6.2 of~\\citep{KreschTschinkel-2022}.)\n\nThe goal of this section is to define a similar homomorphism\n\\[\u00a0\\mathbf\\rho_t \\colon \\Burn(K) \\to \\Burn(k)  \\]\nfor varieties with logarithmic volume forms.\n\n\\subsection{}\\label{sec.kt}\nLet $\\mathscr X$ be an integral proper scheme over~$R$,\nof relative dimension~$n$,\nwhose special fiber~$\\Delta$ is a divisor with strict normal crossings.\n\nLet $(\\Delta_\\alpha)_{\\alpha\\in\\mathscr A}$  \nbe the family of irreducible components of the special fiber~$\\Delta$;\nfor $\\alpha\\in\\mathscr A$, let $e_\\alpha$ be the multiplicity\nof~$\\Delta_\\alpha$ in~$\\Delta$.\nFor every nonempty subset~$A$ of~$\\mathscr A$,\nlet $\\Delta_A$ be the intersection of all divisors~$\\Delta_\\alpha$,\nfor $\\alpha\\in\\mathscr A$\nand $e_A$ be the greatest common divisor of\nthe~$e_\\alpha$, for $\\alpha\\in A$; let also $\\Delta_A^\\circ$\nbe the complement $\\Delta_A\\setminus \\bigcup_{\\alpha\\not\\in A}\\Delta_\\alpha$.\n\nThe first specialization morphism of~\\citep{KontsevichTschinkel-2019}\nis defined by\n\\begin{equation}\n\u00a0\\rho_t( [\\mathscr X_K]) \n= \\sum_{\\emptyset\\neq A\\subset\\mathscr A} (-1)^{\\Card(A)-1} [\\Delta_A] \n\\Lef^{\\Card(A)-1},\n\\end{equation}\nwhere $\\Lef\\in\\burn(k)$ is the class of the affine line.\n\nAlthough this map is not multiplicative, it proved sufficient for\nmany applications to rationality problems.\n\nTo ensure multiplicativity, a more delicate construction\nwas necessary, valued in the Burnside ring \n\\[\u00a0\\burn^{\\widehat\\mu}(k) \\]\nof varieties endowed with an action of the profinite group~$\\widehat\\mu$,\nlimit of finite groups of roots of unity.\n\nFix a nonempty subset~$A$ of~$\\mathscr A$.\nWe identify the normal bundle of~$\\Delta_A$ in~$\\mathscr X$\nas a direct sum of line bundles:\n\\[\u00a0\\mathscr N_{\\Delta_A}(\\mathscr X) \\simeq \\bigoplus_{\\alpha\\in A}\n  \\mathscr N_{\\Delta_\\alpha}(\\mathscr X)|_{\\Delta_A} . \\]\nLet us consider its open subscheme~$\\mathscr N_{\\Delta_A}^\\circ(\\mathscr X)$\nobtained by restricting to~$\\Delta_A^\\circ$\nand taking out all \"coordinate\" hyperplanes.\nThis furnishes a morphism\n\\[\u00a0\\nu_A \\colon \\mathscr N_{\\Delta_A}^\\circ(\\mathscr X) \\to\n \\bigotimes_{\\alpha\\in A} \\mathscr N_{\\Delta_\\alpha}(\\mathscr X)^{\\otimes e_\\alpha}|_{\\Delta_A^\\circ} . \\]\nSince the uniformizer~$t$ has divisor \n$-\\sum_{\\alpha\\in\\mathscr A} e_\\alpha \\Delta_\\alpha$ on~$\\mathscr X$,\nit trivializes the line bundle on the target of~$\\nu_A$.\nWe set $\\Delta'_A=\\nu_A^{-1}(t)$.\nBy construction, the projection $\\Delta'_A\\to \\Delta_A$ \nis a torsor with group~$\\mu_{e_A}$.\n\nWith this notation, the correct, multiplicative, specialization\nmap of~\\citep{KontsevichTschinkel-2019} is given by the formula\n\\[\u00a0\\widehat\\rho_t(X)  = \\sum_{\\emptyset\\neq A\\subset \\mathscr A}\n (-1)^{\\Card(A)-1} [\\Delta'_A]\u00a0\\Lef ^{\\Card(A)-1} \\]\nin $\\burn^{\\widehat\\mu}(k)$.\n\n\\begin{rema}\nThe relation between the two specialization morphisms\nis as follows. Fix a nonempty subset~$A$ of~$\\mathscr A$.\nThe group~$\\gm$ acts diagonally on~$\\mathscr N_{\\Delta_A}^\\circ(\\mathscr X)$\n(the factors of index $\\alpha\\notin A$ don't act),\nand this induces an action \nof the finite group of roots of unity of order~$e_A$\non~$\\Delta'_A$,\nhence an action of~$\\widehat\\mu$,\nso that $\\widehat\\rho_t(X)$ naturally lives\nin the equivariant Burnside ring $\\burn^{\\widehat\\mu}(k)$.\nMoreover, taking the $\\widehat\\mu$-invariants of~$\\Delta'_A$,\nwe get $\\Delta_A^\\circ$, so that the specialization map~$\\rho_t$\nis the composition of~$\\widehat\\rho_t$ with the map\n\\[\u00a0\\burn^{\\widehat\\mu}(k) \\to \\burn(k) \\]\nobtained by taking~$\\widehat\\mu$-invariants.\n\nTaking invariants does not commute with taking products, in general,\nso that $\\rho_t$ is not multiplicative.\n\\end{rema}\n\n\n\\subsection{}\nLet us explain how to define analogous specialization\nhomomorphisms in our context\nof Burnside groups with volume forms.\n\nFor simplicity, we only consider\nthe case where $K$ has transcendence degree~$1$ over~$k$,\nin which case the idea can be explained geometrically as follows.\nWe assume that there exists an smooth integral curve~$C$\ntogether with a $k$-point $o\\in C(k)$\nsuch that $K=k(C)$ and $R=\\mathscr O_{C,o}$.\nWe fix a local parameter $t\\in R$ such that $V(t)=o$.\n\nLet us consider a pair $(X,\\omega)$ consisting of an integral\nproper $K$-variety~$X$ of dimension~$n$\nand a logarithmic $n$-form~$\\omega$ on~$X$.\n\n\\subsection{}\nConsider a regular flat proper model~$\\mathscr X$ is of~$X$ over~$C$,\nlet $\\Delta=(\\mathscr X_o)_\\red$ be its reduced special fiber, and \nconsider a divisor~$\\mathscr D$ with relative normal crossings on~$\\mathscr X$.\nWe assume that the divisor $\\mathscr D+\\Delta$ has normal crossings.\nIn this situation, \n\\citet[\\S3.3.2]{Deligne-1970} says that \na meromorphic relative differential $m$-form on~$\\mathscr X\/C$\nis logarithmic with respect to~$\\mathscr D+\\Delta$ \nif it is (locally) \nthe image of a logarithmic $m$-form~$\\tilde\\omega$\nin~$\\Omega^m_{\\mathscr X\/k}$ with poles $\\mathscr D+\\Delta$\nunder the  natural\nmorphism $\\Omega^m_{\\mathscr X\/k}\\to\\Omega^m_{\\mathscr X\/C}$.\n\nConsider a logarithmic relative $n$-form~$\\omega$ on~$\\mathscr X\/C$.\nWe consider an associated volume form~$\\omega'$ on~$\\mathscr X$, \ndefined locally by\n\\[\u00a0\\omega' = \\tilde \\omega \\wedge \\mathrm dt\/t, \\]\nwhere $\\tilde\\omega$ is any local lift of~$\\omega$.\nThis form $\\omega'$ is logarithmic\nand we can compute its \"residue along~$\\Delta$\" as\nin~\\S\\ref{sec.residues}, only taking into account\nthe strata of the polar divisor of~$\\omega'$\nwhich are contained in the special fiber~$\\Delta$.\n\nThere exists a subset $\\mathscr A_o$ of~$\\mathscr A$\nand a subset~$\\mathscr B_o$ of~$\\mathscr B$\nsuch that \nthe polar divisor of~$\\omega'$ is given by \n\\[\u00a0\\sum_{\\alpha\\in\\mathscr A_o}  \\Delta_\\alpha + \\sum_{\\beta\\in\\mathscr B_o} \\mathscr D_\\beta. \\]\nWe thus set\n\\[\u00a0\\rho_t(\\mathscr X,\\omega)\n = \\sum_{\\substack{\\emptyset \\neq A \\subset \\mathscr A_o \\\\\u00a0B \\subset\\mathscr B_o} }(-1)^{\\Card(A)+\\Card(B)-1}\n\\rho_{\\Delta_A\\cap \\mathscr D_B}(\\mathscr X,\\omega). \\]\nThis is an element of $\\Burn_n(\\mathscr X_o\/k)$.\n\\begin{prop}\nLet $\\mathscr Y$ be an irreducible closed subscheme of~$\\mathscr X$\nwhich is transverse to $\\mathscr D+\\Delta$\nand let $g\\colon \\mathscr X'\\to\\mathscr X$ \nbe the blowing-up of~$\\mathscr X$ along~$\\mathscr Y$.\nThe form $g^*\\omega$ on~$\\mathscr X'$ is logarithmic and \nwe have\n\\[\u00a0g_* \\rho_t (\\mathscr X',g^*\\omega) = \\rho_t (\\mathscr X,\\omega) \\]\nin $\\Burn_n(\\mathscr X_o\/k)$.\n\\end{prop}\n\\begin{proof}\nWith the notation of~\\S\\ref{sec.residues}, the difference\n\\[\u00a0\\rho (\\mathscr X,\\tilde\\omega)  - \\rho_t(\\mathscr X,\\omega) \\]\nis exactly the  part of $\\rho(\\mathscr X,\\tilde\\omega)$ \nwhich lies over the complement of the special fiber~$\\mathscr X_o$\nin~$\\mathscr X$.\nWe have seen in theorem~\\ref{theo.residue-bir} that\n\\[\u00a0g_* \\rho(\\mathscr X',\\tilde\\omega') = \\rho(\\mathscr X,\\omega), \\]\nand a similar formula holds over $\\mathscr X\\setminus\\mathscr X_o$.\nThis implies the proposition.\n\\end{proof}\n\n\\subsection{}\nStarting from a smooth proper $K$-variety~$X$\nand a logarithmic volume form~$\\omega$ on~$X$,\nwe can define a model~$\\mathscr X\/C$, with $\\mathscr D$ and~$\\Delta$\nas above, but the form~$\\omega$\nwill not necessarily extend to a logarithmic relative form with respect\nto $\\mathscr D+\\Delta$,\nnor does \nthe volume form~$\\tilde\\omega$ on~$\\mathscr X$.\nHowever, this can be achieved by multiplying~$\\omega$\nby a suitable power of the uniformizing element.\n\nLet us write the polar divisor of~$\\tilde\\omega$ on~$\\mathscr X$ as\n\\[\u00a0\\div_{\\mathscr X}(\\tilde\\omega) = D+\\Delta\n= \\sum_{\\alpha\\in\\mathscr A} d_\\alpha \\Delta_\\alpha\n + \\sum_{\\beta\\in\\mathscr B} d_\\beta \\mathscr D_\\beta. \\]\nWith this notation, the condition for~$\\tilde\\omega$ to be logarithmic\non~$\\mathscr X$ is just that \n\\[\u00a0d_\\alpha \\geq -1, \\qquad d_\\beta\\geq -1. \\]\nIn particular, while the conditions at the horizontal components follow \nfrom their counterparts on the generic fiber, \nthose for the vertical components are not automatic.\nOn the other hand, for any $\\kappa\\in\\Z$, the form $t^\\kappa \\tilde\\omega$\nis logarithmic if and only if \n\\[\u00a0\\kappa e_\\alpha + d_\\alpha \\geq -1 \\]\nfor all $\\alpha\\in\\mathscr A$, that is, if and only if\n$\u00a0\\kappa \\geq \\kappa(\\omega)$, \nwhere $\\kappa(\\omega)$ is defined by \n\\[\u00a0\\kappa(\\omega) = \\inf_{\\alpha\\in\\mathscr A} \\frac{1-d_\\alpha}{e_\\alpha}. \\]\n\nSince the rational number\n$\\kappa(\\omega)$ is defined in terms of logarithmic forms,\nit only depends on the class of $(X,\\omega)$ in~$\\Burn_n(K)$,\nand not on the actual model which is chosen to compute it.\n\n\\subsection{}\nWe assume for the moment that $\\kappa(\\omega) \\in\\Z$. \nThis holds in particular if the special fiber~$\\mathscr X_o$ is reduced.\nLet then $\\mathscr A_o$ be the subset of~$\\mathscr A$\nconsisting of all $\\alpha$ such that \n\\[\u00a0\\kappa (\\omega) e_\\alpha+d_\\alpha = -1,\\]\nand let $\\mathscr B_o$ be the subset of~$\\mathscr B$\nconsisting of all $\\beta$ such that $d_\\beta=-1$.\nThe polar divisor of $t^\\kappa\\tilde\\omega$ is equal to\n\\[\u00a0\\sum_{\\alpha\\in\\mathscr A_o} \\Delta_\\alpha\n + \\sum_{\\beta\\in\\mathscr B_o} \\mathscr D_\\beta, \\]\nand we set\n\\[\u00a0\\rho_t(\\mathscr X,\\omega) = \\rho_t(\\mathscr X,t^{\\kappa(\\omega)}\\omega) \\]\nin $\\Burn_n(\\mathscr X_o\/k)$.\n\nIn the particular case where $\\mathscr D$ is empty, \nthe strata of the Clemens complex of the special fiber \nthat actually appear in the definition of this class\nare those defined by~\\cite{KontsevichSoibelman-2006},\nmore precisely, by the adjustment provided by~\\cite{MustataNicaise-2015a}.\n\n\n\\subsection{}\nIn the general case, the rational number~$\\kappa(\\omega)$\nis not an integer.  \nLet us consider the finite ramified extension $K_d=K(t^{1\/d})$\nof~$K$, \nwhose ramification index~$d$\nis a multiple of the denominator of~$\\kappa(\\omega)$,\nbut which induces an isomorphism on the residue field.\nGeometrically, this furnishes a morphism $\\pi\\colon C_d\\to C$\nwhich is ramified at the point~$o$, together\nwith a lift of~$o$ in~$C_d(k)$ (still denoted by~$o$),\nand a distinguished uniformizing element~$t^{1\/d}$.\n\nWe consider the extension of $(X,\\omega)$ to~$K_d$ and\nintroduce a model $(\\mathscr X_d,\\omega_d)$ as above, over~$C_d$. \nNow, the corresponding $\\kappa$-parameter is integral, so that \nany choice of a uniformizing element~$t^{1\/d}$ in~$R_d$\ninduces a class \n$\\rho_{t^{1\/d}}(\\mathscr X_d,\\omega_d)$\nin~$\\Burn(k)$.\nIn fact, we can assume\nthat the scheme~$\\mathscr X_d$ carries an action\nof the group scheme~$\\mu_d$ of $d$th roots of unity\ninduced by its action on~$\\Spec(R_d)$, leaving the\nlogarithmic form~$\\omega_d$ invariant.\nIn other words, we obtain a class in the group $\\Burn^{\\widehat\\mu}(k)$.\n\nCombinining these classes, we obtain the desired\ngroup homomorphism\n\\[\u00a0\\widehat\\rho_t \\colon \\Burn(K)  \\to \\Burn^{\\widehat\\mu}(k). \\]\n\nIn fact, as explained in \\cite[\\S2.3]{Nicaise-2013},\nespecially proposition~2.3.2,\none can compute the normalisation of~$\\mathscr X\\otimes R_d$\nin terms of the given model~$\\mathscr X$.\nThis gives an explicit decomposition of~$\\widehat\\rho_t(X,\\omega)$\nas a sum\n\\[\u00a0\\sum_{\\emptyset \\neq A\\subset \\mathscr A_o}\n (-1)^{\\Card(A)-1} [D'_A,\\nu_A^*\\omega_A] \\cdot \\mathbf T^{\\Card(A)-1}, \\]\nwhere $\\nu_A\\colon D'_A\\to D_A$ is the $\\mu_{d_A}$-torsor\nintroduced in~\\S\\ref{sec.kt} for the definition of the classical\nspecialization map.\n\n\\begin{rema}\nIn the case  of specialization of rationality, it has proved\nfruitful to consider models with singularities on the special fiber,\nmild enough so that \nthe special fiber computes the specialization of\nthe birational type of the generic fiber.\nThis is in particular the case for rational double points.\n\nA parallel study can be developped in the context\nof varieties with logarithmic forms.\n\\end{rema}\n\nFollowing \\citep{KontsevichTschinkel-2019} and \nkeeping track of the various logarithmic volume forms on the strata,\nwe have:\n\\begin{theo}\nThe morphism~$\\widehat\\rho_t$ is a ring  homomorphism.\n\\end{theo}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOne of the main efforts of modern cosmology is to determine what is responsible for the observed accelerated expansion of the universe \\cite{riess:1998a, perlmutter:1999a}. Alcock and Paczynski \\cite{alcock:1979a} proposed a cosmological test (hereafter denoted AP) based on the assumption of statistical isotropy around any comoving location. For regions of space-time that expand with the background, observed angles and redshifts can be translated into proper distances using the angular diameter distance $d_A(z)$ and the reciprocal of the Hubble parameter $H(z)$. Requiring isotropy in proper distance, after translating from angle and redshift measurements, leads to measurements of the product $d_A(z)H(z)$.\n\nBecause radial information comes from redshifts, AP measurements are traditionally limited by peculiar velocities, also known as comoving velocities \\cite{ballinger:1996a, simpson:2009a}. These add to expansion-driven redshifts, leading to apparent anisotropic clustering if redshifts are assumed to be completely cosmological in origin, even if the correct $d_A(z)$ and $H(z)$ are used to analyze redshifts. These redshift-space distortions (hereafter RSD) are degenerate with the AP effect, removing signal \\cite{ballinger:1996a, simpson:2009a}, unless assumptions are made such as the Universe following a FLRW metric \\cite{samushia:2011a}. In fact, it is simply standard convention that makes us split redshift into cosmological and peculiar velocity components: considering that pairs of galaxies move due to local space-time curvature shows that the expansion rate and the RSD component can be strongly correlated. In the extreme case of bound systems, for example, the combined pairwise velocity is not dependent on background evolution, i.e.\\ the expansion-driven redshift difference across a pair is exactly canceled by the RSD signal (see Appendix \\ref{app:bound_systems}).\n\nMarinoni and Buzzi \\cite{marinoni:2010a} recently proposed a method to derive cosmological constraints from pairs of galaxies for which peculiar velocities can be modeled. They provided a fitting formula for the observed distribution of velocities, which can then be used to help break the AP-RSD degeneracy. They assume the normalization of the galaxy velocity distribution to be redshift and cosmology independent, whereas a more recent work by \\citet{jennings:2012a} questioned this statement using N-body simulations. In the work presented here we investigate this further, considering how well pairs of galaxies, selected using different properties, trace the cosmological expansion. \n\nWe use the Millennium simulation \\cite{springel:2005a} to test how the pairwise velocity of galaxy pairs may contain information about the background expansion of the Universe. We argue that the local density in which the pairs are found may affect the amount of information these pairs carry on cosmology, because each patch of the Universe expands in a way that depends on the local density. Our analysis suggests that selecting isolated pairs, as considered by Marinoni and Buzzi, can result in average pairwise velocities more in line with the Hubble expansion, i.e.\\ they need smaller, less cosmology dependent peculiar velocity corrections. We also find a better match if low-mass tracers are used. \n\n\nThe layout of this paper is as follows: in \\sref{sec:AP} we briefly review the AP effect. In \\sref{sec:millennium} we describe the Millennium simulation and the two semi-analytic models used: \\citet{guo:2011a} and \\citet{font:2008a}. In \\sref{sec:all_pairs} we present and discuss the results we obtained by analyzing all galaxy pairs regardless of their local density, while in \\sref{sec:isolated_pairs} we consider only isolated pairs. The effects of varying galaxy properties are studied in \\sref{sec:galaxy_properties}. In \\sref{sec:comparison_catalogs} we compare the results from the two different semi-analytic galaxy formation models used. We then conclude in \\sref{sec:conclusions}. \n\n\n\n\\section{The Alcock Paczynski effect}  \\label{sec:AP}\n\nConsider a distribution of particles expanding with the\nHubble flow, in the redshift interval ($z-\\Delta z\/2,\nz+\\Delta z\/2$) and subtended by an angle $\\Delta\\theta$. Assuming a\nFLRW cosmology, the proper size of the object perpendicular to our line of\nsight is given by\n\\begin{equation}\n   d_1=d_A(z)\\Delta\\theta, \n\\end{equation} \nwhere $d_A(z)$ is the angular diameter distance to the object. The size of the object parallel to the line of sight is given by\n\\begin{equation} \\label{eq:d2}\n  d_2=\\frac{\\Delta z}{(1+z)H(z)},\n\\end{equation}\nwhere $\\Delta z$ is the difference in the redshift of objects closest\nand furthest away from the observer and $H(z)$ is the Hubble parameter\nat the central redshift of the distribution.\n\nAssuming that the collection of particles statistically does not have a preferred direction with respect to one line of sight, then\n$\\langle d_1\\rangle =\\langle d_2 \\rangle$, allowing a statistical cosmological measurement \\cite{alcock:1979a}, from a sufficient number of pairs, of\n\\begin{equation}\n  H(z)d_A(z) = \\frac{\\Delta z}{(1+z)\\Delta\\theta}.\n\\end{equation}\nNote that $\\Delta z$, $z$ and $\\Delta \\theta$ are all directly\nobservable quantities. The AP effect, as described above assumes that $\\Delta z$ as measured only depends on the cosmological expansion. In fact, the relative  velocity of pairs of particles depends on the local curvature of space, so this is not necessarily a good approximation.\n\n\n\n\\section{The Millennium simulation and semi-analytic galaxy models} \\label{sec:millennium}\n\nIn order to quantify how the dynamics of galaxy pairs may be affected by factors like redshift, isolation radius, mass of the halo etc., we have considered a population of galaxies from  the Millennium simulation \\cite{springel:2005a}. This traces the evolution of 2160$^3$ dark matter particles of mass $1.18\\times 10^9\\ \\text{M}_\\odot$ from redshift 127 to the present day inside a periodic box of side \\mpch{500}. The simulation assumes a \\lcdm cosmology with parameters based on a combined analysis of the 2dFGRS \\cite{colless:2001a} and the first-year WMAP data \\cite{spergel:2003a}. The parameters are $\\Omega_m=0.25,\\ \\Omega_b=0.045,\\ \\Omega_\\Lambda=0.75,\\ n=1,\\ \\sigma_8=0.9$ and $H_0=73$ km s$^{-1}$ Mpc$^{-1}$.\n\nData on dark matter particles were stored at 64 different times. At each output time, the post-processing pipeline produced a friends-of-friends (FOF) catalog by linking particles with a separation less than 0.2 of the mean value. The SUBFIND algorithm \\cite{springel:2001a} was then applied to each FOF group to identify all the substructures. The merger trees, vital for galaxy formation modeling, were then constructed by linking each subhalo found in a given ``snapshot\" to one and only one descendant in the subsequent output time-slice.\n\nWe used the data from two semi-analytic models of galaxy formation based on the Millennium simulation: \\citet{guo:2011a} and \\citet{font:2008a}. Most of our analysis uses the semi-analytic model developed by \\citet{guo:2011a} which is based on the growth of and merging of the population of subhaloes. Within this catalog, each FOF group hosts a central galaxy which sits in the minimum of the potential of the main subhalo. Other galaxies associated to the same FOF group may sit at the potential minima of smaller subhaloes or may no longer correspond to a resolved dark matter substructure, the latter being know as `orphans'. The last two collectives of galaxies are referred to as satellites, although in \\citet{guo:2011a}, the physical processes affecting satellite galaxies only begin to differ from those affecting central galaxies when a satellite first enters within the virial radius of the larger system. This is the radius of the largest sphere with its center at the center of the FOF group and a mean overdensity exceeding 200 times the critical value.\n\nFor our analysis, we have varied several parameters from the galaxy catalogs, namely redshift, mass of the subhalo that hosts the galaxy, stellar mass and \\ensuremath{r\\text{-band}}\\xspace rest-frame magnitude.  Unless differently stated, the redshift shown in our plots corresponds to $z=0.989$.\n\n\n\n\\section{All galaxy pairs} \\label{sec:all_pairs}\nIn this section we study the average pairwise velocity of galaxies regardless of local density and galaxy properties.  We shall compare our findings with predictions from linear theory to examine general trends, and test the possibility of using randomly selected galaxy pairs to trace cosmological expansion. \n\n\n\\subsection{Method}\\label{sec:method_allpairs}\n\nFor each galaxy pair, we compute the comoving separation $ d $, the pairwise velocity $ v_{12} $ and its square $ v_{12}^2 $.  We define the pairwise velocity as:\n\\begin{equation}\n\\label{eq:pairwise_velocity_definition}\n  v_{12} = \\frac{\\operatorname{d}d}{\\operatorname{d}t} = \\frac{(\\vec{v_1} - \\vec{v_2}) \\, \\cdot \\, (\\vec{x_1} - \\vec{x_2})}{d} \\;,\n\\end{equation}\nwhere $ t $ is cosmic time, and $\\vec{x}$ and $\\vec{v}$ are galaxy positions and velocities.  Note that, following the Millennium simulation, we work in coordinates that are comoving with the Hubble flow, hence, $ v_{12} $ represents the peculiar, non-Hubble component of the pairwise velocity.  In the plots that follow, we shall always show the $ - H(z)\\,d $ curve and denote it as ``static solution\".  We shall also highlight the zero line, which in these plots represents the Hubble flow, and denote it as ``comoving solution\".  Any data point above the comoving solution represents pairs where the two galaxies are receding from each other faster than the Hubble expansion, while below they are moving towards each other in comoving coordinates.\n\n\nWe group pairs in bins according to their separation $ d $ and, for each bin, compute the average $ \\avg{v_{12}} $ and the variance $ \\text{var}(v_{12}) = \\avg{v_{12}^2} - \\avg{v_{12}}^2 $ of the pairwise velocity.  Our definition of expectation value is:\n\\begin{equation}\n\\label{eq:average_definition}\n  \\avg{v^n} \\equiv \\frac{1}{N_\\text{pairs}} \\, \\sum\\limits_{i=1}^{N_\\text{pairs}} (v_i)^n \\;,\n\\end{equation}\nwhere $ N_\\text{pairs} $ is the number of pairs in the separation bin.  In all the plots in this paper, we shall always show $ \\avg{v_{12}} $ as a function of pair separation, with the error bars for each bin taken as $ \\sqrt{\\text{var}(v_{12}) \/ N_\\text{pairs}} $.\n\nBy default, we employ a logarithmic binning in galaxy separation.  In order to better visualize the data, however, in some plots we combine underpopulated bins together so that each bin represents at least a minimum number of galaxy pairs, $ N_\\text{min}$.  In this section we use $ N_\\text{min} \\geq 1000 $ while, due the poor statistics, we shall employ $ N_\\text{min} \\geq 2 $ for some of the ``isolated\" plots in Sections \\ref{sec:isolated_pairs} and \\ref{sec:galaxy_properties}.\n\n\n\\subsection{Results}\\label{sec:results_all_pairs}\nIn \\fref{fig:generic_all_pairs} we show the average pairwise velocity $ v_{12} $ of all the galaxies within the \\citet{guo:2011a} semi-analytic model at redshift $ z=0 $, as a function of separation $ d $.  The velocity curve is represented by the red dots with one-sigma error bars, while the blue and black lines are, respectively, the static and comoving solutions. %\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0_z0_noiso_theory.pdf}\n\\caption{Average relative velocity for all galaxy pairs at redshift $z=0$. The blue solid line represents the static solution, followed by pairs that have already virialized and do not feel the background Hubble flow. The green dashed line represents the linear theory model for pairwise velocities according to the prescription found in References \\cite{fisher:1995a, reid:2011a} with a bias of $b=1$. Error bars, which are too small to be clearly seen are plotted at the one sigma confidence limit, assuming Poisson statistics.}\n\\label{fig:generic_all_pairs}\n\\end{figure}\n\nWe also plot the prediction of linear perturbation theory for $ v_{12} $ as the green dashed line, obtained using the prescription from \\cite{fisher:1995a, reid:2011a}:\n\\begin{equation}\n  v_{12}(d) = -\\frac{fb}{\\pi^2}\\int dk\\ k\\ P_m(k)j_1(kd),\n\\end{equation}\nwhere we have chosen a bias of $b=1$, which is in agreement with that measured from clustering within the galaxy catalogue.  \n\nFor separations larger than  $d>\\mpch{10}$, we can see that the average peculiar velocity is correctly predicted by linear theory.  As we follow the velocity curve into the nonlinear regime, galaxy pairs approach the static line until they cross it at $d\\sim\\mpch{3}$.  This crossing marks the beginning of the infall regime, where the galaxies in the pairs get closer to each other, but with smaller velocities as their separation decreases.  On the smallest scales, the pairs asymptote to the static solution.\n\nTo use galaxy pairs as tracers of the cosmological expansion, we need their peculiar velocity to be small with respect to the Hubble flow or modellable.  A smaller correction is required if $ v_{12} $ is closer to zero comoving velocity than to the static solution. As we noted above, only for $d>\\mpch{10}$ are the velocities closer to the comoving solution than the static solution, which is the regime of linear perturbation theory.  On scales $d\\lesssim\\mpch{3}$, galaxy pairs follow closely the static solution on average.  Their peculiar velocity component is equal and opposite to the Hubble flow, therefore they do not carry any cosmological information (refer to Appendix \\ref{app:bound_systems}).\n\n\\section{Isolated pairs} \\label{sec:isolated_pairs}\nIn this section we investigate how the dynamics of galaxy pairs changes when an isolation criterion is imposed and how this depends on the isolation radius, the allowed number of galaxies within this radius, and redshift. We use the same galaxy sample as in \\sref{sec:all_pairs}.\n\n\n\\subsection{Method}\\label{sec:method_isolated_pairs}\nWe initially define a galaxy pair to be isolated within a radius \\ensuremath{r_\\text{iso}}\\xspace if each galaxy in the pair has exactly 1 neighbor within \\ensuremath{r_\\text{iso}}\\xspace, and that neighbor is the other galaxy in the pair.  This is equivalent to drawing two spheres of radius \\ensuremath{r_\\text{iso}}\\xspace centered on the galaxies, and imposing the absence of galaxies extraneous to the pair in each of the spheres.  We shall weaken this requirement, allowing for the maximum number of neighbors \\ensuremath{N_\\text{neigh}}\\xspace in each sphere to be larger than 1.  Thus we can interpolate between the dynamics of galaxy pairs in the fully isolated case ($ \\ensuremath{N_\\text{neigh}}\\xspace = 1 $) and in the unconstrained case ($ \\ensuremath{N_\\text{neigh}}\\xspace \\rightarrow \\infty $). We implement such isolation criterion with arbitrary number of neighbors in a two-step process.  We first determine the number of neighbors for each galaxy in the simulation, and later use this information to select only those pairs where each galaxy has less than \\ensuremath{N_\\text{neigh}}\\xspace neighbors.\n\nWe fix the isolation radius to $ \\ensuremath{r_\\text{iso}}\\xspace = \\mpch{4} $, which matches the definition of isolation adopted by \\citet{marinoni:2010a}.  Note that in the plots that follow, we only look at separations less than the isolation radius to ensure that the pair is truly isolated.\n\n\n\n\\subsection{Results}\nIn the upper panel of \\fref{fig:generic_isolated} we present the relative motion of galaxy pairs as a function of their separation for galaxies that are isolated within a \\mpch{4} radius. The galaxies are taken from the \\citet{guo:2011a} semi-analytic model applied to the Millennium simulation at redshift $ z = 0 $. The data points are plotted in red with one-sigma error bars. The blue line is the static solution and the black line the comoving solution -- see \\sref{sec:method_allpairs} for their definition.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0_z0_isolation4_ratio.pdf}\n\\caption{Upper panel: average pairwise velocity for isolated galaxy pairs at redshift $z=0$. The isolation radius is taken to be \\mpch{4} for each member of the pair. The blue solid line represents the static solution, showing the virialization of pairs. The three shaded regions represent three different regimes. The left blue area represents the virialization regime: galaxies within these separations have virialized and do not experience the background expansion. The middle red area shows the infall regime, where galaxies start to collapse to form bound systems. The right green region shows what we denote as the `void effect': on average, the isolated pair feels a stronger gravitational pull separating the pair rather than making it closer. The error bars shown are the one sigma confidence limit, assuming Poisson statistics. Lower panel: ratio of average pairwise velocities to the static solution $Hd$. Note that the y axis of this panel is in logarithmic scale. The error bars shown are the propagated one-sigma errors from the upper panel.}\n\\label{fig:generic_isolated}\n\\end{figure}\n\nOur first comment on \\fref{fig:generic_isolated} regards the error bars, which are much larger than those in the non-isolated case of \\fref{fig:generic_all_pairs}.  The reason is that the imposition of an isolation criterion results in a drastic reduction of the galaxy pairs found, which in the case of \\fref{fig:generic_isolated} are only $ 694 $\n\\footnote{Equivalent to one isolated pair every $10^6$ other pairs for $ d = \\mpch{1} $}.\nThis number is in line with \\citet{marinoni:2010a}, who find $ 721 $ pairs for their low-redshift SDSS sample, and $ 509 $ for their DEEP2 sample. \n\nThe most striking feature about the dynamics of isolated pairs is the roughly logarithmic growth of the peculiar velocity $ v_{12} $ for scales larger than $ \\sim \\mpch{0.2} $.\nThe behavior of $ v_{12} $ can be explained when we recognize that the dynamics is determined by the combined effect of two competing forces: the mutual attraction between the two galaxies, dominant for small separations, and the disrupting outflow from the void -- the \\emph{void effect}, dominant for separations close to the isolation radius.\nFor small separations, the mutual attraction of the members of the pair overcomes the void effect and we see an infall regime.  As we study objects with larger separations, the void effect becomes dominant and we see a logarithmically growing pairwise velocity $ v_{12} $.\n\nIn the lower panel of \\fref{fig:generic_isolated} we plot $ v_{12}\/H d $, that is the ratio of peculiar velocity to Hubble flow. For separations $ 0.4 < d < \\mpch{4} $ we find an almost comoving regime where the peculiar velocities are less than $ 20\\% $ of the Hubble flow.  In such a regime,  the RSD corrections are small, and one could hope that they are easier to model so that $ v_{12} $ becomes a proxy of the expansion rate.  In the following subsections we shall investigate how the comoving regime depends on the isolation radius, the local density and redshift.\n\n\n\\subsubsection{Varying the isolation radius}\\label{sec:shrinking}\nIt is interesting to investigate whether the void effect, giving the approximately logarithmic growth of $ v_{12}(d) $ for isolated pairs, is still present when we allow the size of the isolation radius to vary.  We demonstrate this in \\fref{fig:isolation_radii}, where we show the peculiar velocities of galaxy pairs with $ \\ensuremath{r_\\text{iso}}\\xspace $ varying from \\mpch{0.1} to \\mpch{4}, at redshift $ z = 0.989 $.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{guo2010_isolation_radii01to4_z0989_np0.pdf}\n\\caption{Average pairwise velocity for isolated galaxy pairs at redshift $z=0.989$ for different isolation radii ranging from \\mpch{0.1} to \\mpch{4}.}\n\\label{fig:isolation_radii}\n\\end{figure}\n\nFor $ \\ensuremath{r_\\text{iso}}\\xspace \\geqslant \\mpch{0.6} $ the presence of the void severely affects the dynamics of galaxy pairs.  The logarithmic growth of the peculiar velocity is visible, even though it is just a hint for the $ \\ensuremath{r_\\text{iso}}\\xspace = \\mpch{0.6} $ data points.  The cosmological scale $ d_0 $, defined as the separation where the peculiar velocity $ v_{12} $ vanishes, occurs at $ d_0 \\simeq \\mpch{0.8} $ and appears to be independent of the isolation radius. \n\nAt small separations, we cannot see any noticeable differences between the various $ \\ensuremath{r_\\text{iso}}\\xspace $ datasets.  They all follow the static solution line for $d<\\mpch{0.05}$, suggesting that isolated galaxy pairs tend to virialize on the smallest scales just as non-isolated ones do.\n\nThe number of isolated galaxy pairs at $ z=0.989 $ increases from $ 435 $ to $ 71,201 $ when reducing the isolation radius from $ 4 $ to \\mpch{2}, a factor of roughly $ 160 $.  As we have noted above, the $ \\ensuremath{r_\\text{iso}}\\xspace = \\mpch{2} $ pairs still experience a regime where peculiar velocities are negligible with respect to the Hubble flow.  Hence their velocity difference still traces the cosmological expansion, although the maximum separation for which this is the case is halved with respect to the $ \\ensuremath{r_\\text{iso}}\\xspace = \\mpch{4} $ case.  We conclude that using pairs isolated within a \\mpch{2} radius as cosmological tracers would drastically reduce the statistical error with respect to the \\mpch{4} case, while still needing minimal corrections for RSD, provided that the cosmological dependence of the RSD could be modeled. We shall discuss this in more detail later.\n\n\n\\subsubsection{Varying the isolation density criterion}\\label{sec:populating}\nHere, we investigate how the dynamics of galaxy pairs changes if we relax the isolation criterion by increasing the allowed number of galaxies $ \\ensuremath{N_\\text{neigh}}\\xspace $ within the isolation sphere of \\mpch{4}.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{guo2010_neighbours1to5000_z0989_isolaton4_np0_lin.pdf}\n\\caption{Average pairwise velocity for isolated galaxy pairs at redshift $z=0.989$ for different number of neighbors within the isolation sphere of \\mpch{4}. The blue solid line represents the static solution, showing the virialization of pairs. The green dashed line represents the average pairwise velocity for non-isolated pairs. The error bars shown are the one sigma confidence limit, assuming Poisson statistics.}\n\\label{fig:neighbours}\n\\end{figure}\n\nIn the linear plot of \\fref{fig:neighbours}, we present the average peculiar velocity $ v_{12} $ at $ z=0.989 $ for 7 values of $ \\ensuremath{N_\\text{neigh}}\\xspace $ ranging from $ \\ensuremath{N_\\text{neigh}}\\xspace = 1 $ (equivalent to the pure isolated case of \\fref{fig:generic_isolated}) to $ \\ensuremath{N_\\text{neigh}}\\xspace = 5000 $.  We also plot $ v_{12} $ for the non-isolated galaxy pairs, as already shown in \\fref{fig:generic_all_pairs}, as a dashed green curve.  As $ \\ensuremath{N_\\text{neigh}}\\xspace $ increases, the different $ v_{12} $ curves monotonically fill the gap between the pure isolated case and the non-isolated case.  A good agreement between the dynamics of pairs with and without isolation criterion is reached once we allow each galaxy in the pair to have $ 5000 $ other neighbors.\n\nIn the $ \\ensuremath{N_\\text{neigh}}\\xspace = 10 $ case, we found $ 250,670 $ pairs, roughly a factor $ 600 $ more pairs than in the fully isolated case of $ \\ensuremath{N_\\text{neigh}}\\xspace = 1 $.  Nonetheless, the $ v_{12} $ curve for $ \\ensuremath{N_\\text{neigh}}\\xspace = 10 $ is strikingly similar to the $ \\ensuremath{N_\\text{neigh}}\\xspace = 1 $ one.  In particular, the void effect still seems to trigger the logarithmic growth of the peculiar velocity, with $ v_{12} $ crossing the zero line at $ d_0 \\simeq \\mpch{1} $ (in the fully isolated case, we have $ d_0 \\simeq \\mpch{0.8}) $. Hence, we suggest that pairs that are not completely isolated trace the cosmological expansion almost as well as the fully isolated ones, with the added benefit of a much better statistics.\n\n\n\\subsubsection{Varying redshift} \\label{sec:varying_redshift}\n\n\nWe illustrate the redshift dependence of the peculiar velocity for isolated pairs in the top panel of \\fref{fig:iso_redshift_variation}, where we plot $ v_{12}(d) $ for the four redshifts $ z = 0, 0.5085, 0.989, 1.504 $.  It is remarkable that for separations $ d \\gtrsim \\mpch{0.2} $ the peculiar velocity depends only slightly on redshift.  The scale $ d_0 $, defined as the separation where $ v_{12} $ vanishes, ranges from \\mpch{0.6} at $ z=0 $ to \\mpch{0.9} at $ z=1.504 $.  This is a small variation if we consider that at $ z=1.504 $ the Universe was at one third of its current age.\n\n\\begin{figure}[ht]\n\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0_z0to1504_isolation4_ratio_errorbars.pdf}\n\\caption{Upper panel: variation of the average pairwise velocity of isolated galaxy pairs with redshift as a function of separation. The isolation radius is taken to be \\mpch{4} for each member of the pair. Lower panel: range in pairwise velocity for each separation bin over range in the static solutions at each separation bin. Note that the y axis of this panel is in logarithmic scale. The error bars shown are the propagated one-sigma errors from the upper panel.}\n\\label{fig:iso_redshift_variation}\n\\end{figure}\n\nIn the bottom panel of \\fref{fig:iso_redshift_variation} we plot the ratio between the range in $ v_{12} $ and the range in $ H d $ at a given separation.  For separations $ 1 < d < \\mpch{4} $, the change of peculiar velocity with redshift is only $ 10 \\% $ of the change in Hubble flow.  This implies a weak dependence of $ v_{12} $ on cosmological expansion on those scales.\n\n\\subsection{Cosmological implications}\\label{sec:cosmological_implications}\n\\fref{fig:generic_isolated} shows that for $ 0.4 < d < \\mpch{4} $ isolated galaxy pairs at $ z=0 $ are nearly comoving with the Hubble flow.  Thus, they move with the cosmological expansion and, for this cosmology and epoch, only need a small RSD correction. Using different redshift slices as a way to test different cosmological expansion rates, in the lower panel of \\fref{fig:iso_redshift_variation}, we identify a second regime for $ 1 < d < \\mpch{4} $ where the peculiar velocity $ v_{12}(z) $ depends only slightly on the cosmological expansion. This suggests the intriguing possibility of isolated pairs behaving in the same way on those scales regardless of the assumed cosmology. In particular, the lower panel of \\fref{fig:iso_redshift_variation} shows that the variation in RSD model is less than the variation in expansion rates. Thus we conclude that there is cosmological signal to be extracted here.\n\nWe refer to the intersection of these regimes, where we have almost comoving pairs with small redshift evolution, as a cosmological regime, since we might be able to use these pairs as cosmological tracers. Measuring galaxy pairs in the cosmological regime would still induce a systematic error due to the fact that peculiar velocities are non-zero.  As the correction is of the $10\\%$ level -- see lower panel of \\fref{fig:generic_isolated} -- and we might suppose to be able to model this at the same level, we would have a $1\\%$ systematic correction to contend with.  This claim is little more than a speculation at this stage; in order to falsify or confirm it, one needs to model isolated pairs in detail and to analyze N-body simulations with different underlying cosmologies. \n\n\nPairs isolated within a radius of \\mpch{4} are rare objects -- see \\fref{fig:number_pairs} -- and this might result in significant statistical error when dealing with observations.  In Sections \\ref{sec:shrinking} and \\ref{sec:populating} we found that one can drastically increase the number of pairs while keeping the RSD correction small by either reducing the isolation radius to \\mpch{2} or allowing up to $ 10 $ galaxies to be neighbors of the pair.\n\n\n\n\\section{Varying galaxy properties} \\label{sec:galaxy_properties}\nThe main result of the previous section, illustrated in Fig.\\ \\ref{fig:generic_isolated} and \\ref{fig:iso_redshift_variation}, is that there is a regime where isolated galaxy pairs may be used as tracers of expansion with correction for RSD that is weaker than the signal to be measured.  Such a finding relies on the ability of measuring the redshift of galaxies regardless of their mass or luminosity.  This is clearly not the case when dealing with actual galaxy surveys, whose flux sensitivity is limited.   To model such a selection bias, we need to investigate ways of selecting galaxies from simulations which mimic the selection process of surveys.   In this section we address this issue by forming subsamples where galaxies are selected according to subhalo mass, stellar mass and magnitude.  We also study the redshift dependence of our results by analyzing 4 different redshifts: $ z = 0,\\,0.5085,\\,0.989,\\,1.504 $.\n\n\n\\subsection{Method} \\label{sec:method_varying}\n\\begin{table*}[t]\n    \\begin{tabular}{ r*{8}r }\n        $\\ensuremath{n_\\text{p}}\\xspace$  &\\:\\:\\:  $m(\\msun\/h)$  &\\:\\:\\: $\\ensuremath{r\\text{-mag}}\\xspace$  &\\:\\:\\:  $\\ensuremath{m_{\\star}}\\xspace(\\msun\/h)$  &\\:\\:\\:  n(z=0) (Mpc$^{-3}$)  &\\:\\:  \\%(z=0)  &\\:\\:\\:  \\%(z=0.5085)  &\\:\\:\\:  \\%(z=0.989)  &\\:\\:\\:  \\%(z=1.504)  \\\\\n        \\hline \\hline\n        20   & \\sci{1.72}{10} &        &                & \\sci{8.1}{-2} &   100  &   100  &   100  &   100 \\\\\n        100  & \\sci{8.6}{10}  & -19.27 & \\sci{1.40}{9}  & \\sci{1.4}{-2} & 17.10  & 17.81  & 17.77  & 17.49 \\\\\n        500  & \\sci{4.3}{11}  & -21.72 & \\sci{1.59}{10} & \\sci{3.1}{-3} &  3.85  &  3.73  &  3.45  &  3.09 \\\\\n        1000 & \\sci{8.6}{11}  & -22.17 & \\sci{2.60}{10} & \\sci{1.6}{-3} &  2.01  &  1.88  &  1.67  &  1.41 \\\\\n        5000 & \\sci{4.3}{12}  & -22.86 & \\sci{5.36}{10} & \\sci{3.4}{-4} &  0.42  &  0.35  &  0.27  &  0.19\n    \\end{tabular}\n    \\caption{Cuts imposed on our galaxy sample from the \\citet{guo:2011a} semianalytic model. The first line corresponds to no cut at all; in that case, for the particle number column, we report the \\citet{guo:2011a} resolution limit of $ \\ensuremath{n_\\text{p}}\\xspace = 20 $.  The columns with a percentage sign denote the percentage of galaxies surviving the cut at a given redshift.  The \\ensuremath{r\\text{-mag}}\\xspace and \\ensuremath{m_{\\star}}\\xspace columns refer to the cuts performed at $ z = 0.989 $, as it is the only redshift we plot for these quantities (see Figures \\ref{fig:rmag_variation} and \\ref{fig:stellar_mass_variation}).  Note that the \\ensuremath{r\\text{-mag}}\\xspace cuts are intended to be upper limits, while the mass cuts are lower limits.}\n\\label{tab:cuts}\n\\end{table*}\n\n\n\\begin{figure*}[t]\n\\subfloat[No isolation]{\\label{fig:npairs_noiso}\\includegraphics[width=0.5\\textwidth]{guo2010_npairs_np10to5000_z0989_noiso}}\n\\subfloat[Isolation of \\mpch{4}]{\\label{fig:npairs_isolation4}\\includegraphics[width=0.5\\textwidth]{guo2010_npairs_np10to5000_z0989_isolation4}}\n\\caption{Number of galaxy pairs for different mass cuts as a function of separation at redshift $z=0.989$.  In the right panel we omitted the $ m > \\msunh{8.6}{11} $ curve for the sake of readability.}\n\\label{fig:number_pairs}\n\\end{figure*}\n\nWe select galaxy subsamples from the semi-analytic models by applying cuts on galaxy properties.  We then study the dynamics of each subsample by applying the same analyses of Sections \\ref{sec:all_pairs} and \\ref{sec:isolated_pairs}.  Initially we look at the number \\ensuremath{n_\\text{p}}\\xspace of dark matter particles of the subhalo the galaxy is in%\n\\footnote{For reference, this is the \\emph{np} field of the \\emph{Guo2010a} database in the Millennium simulation servers.}.\nWe consider $ \\ensuremath{n_\\text{p}}\\xspace = 100,\\,500,\\,1000,\\,5000 $ corresponding to masses $ m = \\sci{8.6}{10},\\,\\sci{4.3}{11},\\,\\sci{8.6}{11},\\,\\msunh{4.3}{12} $.  Note that, where no cut is made, the number of particles in each subhalo is always $ \\ensuremath{n_\\text{p}}\\xspace > 20 $, corresponding to \\msunh{1.72}{10}, since this is the threshold that defines a bound subhalo according to \\citet{guo:2011a}. \n\nWe choose the dark matter mass as our main cut because it is directly related to the pairwise velocity dynamics, which is the main subject of this paper.  In order to make a more direct link with observations, we also study the pairwise velocity statistics when varying the rest-frame $ r $-band magnitude%\n\\footnote{More precisely, this is the rest-frame total absolute magnitude in the SDSS r-band, corresponding to the \\emph{r\\_mag} field in the \\emph{Guo2010a} database of the Garching mirror and to the \\emph{r\\_SDSS} field in the \\emph{Font2008a} database of the Durham mirror.}\nand stellar mass of galaxies.\n\nThe limits on $ r $-band magnitude (hereafter \\ensuremath{r\\text{-mag}}\\xspace) and stellar mass (hereafter \\ensuremath{m_{\\star}}\\xspace) are chosen such that, for a given \\ensuremath{n_\\text{p}}\\xspace cut, the corresponding \\ensuremath{r\\text{-mag}}\\xspace and \\ensuremath{m_{\\star}}\\xspace cuts yield the same number of surviving galaxies. \\tref{tab:cuts} reports the values of the limits used, together with the resulting fraction of surviving galaxies at each redshift. We apply these cuts to the data sets before running the pair-finder algorithms.  Thus, a pair that is isolated within its subsample may not be isolated when considering the full catalog, \\ie our isolation criterion is sample dependent.  This implementation is in line with an analysis of an actual galaxy survey, limited by these cuts.\n\n\nIn \\fref{fig:number_pairs} we show how many galaxy pairs we find after imposing the cuts given in \\tref{tab:cuts}.  The number of unconstrained pairs (left panel) decreases monotonically with increasing mass cut.  Note that the drop-off in the number of pairs at large separations is due to the size of the individual boxes we consider and has no physical meaning.  For isolated pairs (right panel), the number density increases as we increase the mass cut, as a higher mass cut results in a sparser distribution of galaxies where it is easier to find isolated pairs.  Only for our most stringent mass cut, that is for $ m>\\msunh{4.3}{12} $, do we see a slight decrease in the density of isolated pairs due to the small number of high mass galaxies.\n\n\\begin{figure*}[ht]\n\\subfloat[Redshift z=0]{\\label{fig:z_0}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0to5000_z0_noiso.pdf}}\n\\subfloat[Redshift z=0.5085]{\\label{fig:z_0.5}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0to5000_z05085_noiso.pdf}}\\\\\n\\subfloat[Redshift z=0.989]{\\label{fig:z_1}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0to5000_z0989_noiso.pdf}}\n\\subfloat[Redshift z=1.504]{\\label{fig:z_1.5}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0to5000_z1504_noiso.pdf}}\n\\caption{Variation of the average pairwise velocity of all galaxy pairs with redshift for different mass cuts (number of dark matter particles) as a function of separation.}\n\\label{fig:redshift_variation_noiso}\n\\end{figure*}\n\n\n\\subsection{Results}\n\n\\subsubsection{All galaxy pairs}\nIn \\fref{fig:redshift_variation_noiso} we present the average pairwise velocity $ v_{12} $ for non-isolated pairs above different subhalo mass thresholds.  The four panels show the same selection procedure at different redshifts. The mass cuts range from \\msunh{1.72}{10} up to \\msunh{4.3}{12} as tabulated in \\tref{tab:cuts}.  Note that the lowest mass cut corresponds to the smallest subhalo in the \\citet{guo:2011a} semianalytic model, consisting of $ 20 $ dark matter particles.\n\n\n\n\nThe imposition of a mass cut has a significant impact on non-linear scales. Independent of redshift, \\fref{fig:redshift_variation_noiso} shows that massive galaxy pairs experience an infall regime for separations smaller than $ d \\lesssim \\mpch{3} $.  While in such a regime, peculiar velocity increases with mass, with the most massive galaxies ranging from $ v_{12} \\simeq \\kms{330} $ at $ z=0 $ and $ v_{12} \\simeq \\kms{600} $ at $ z=1.504 $.  Lower mass galaxies, on the other hand, seem to follow the static solution up to higher separations, especially at low redshift. Our interpretation for such behavior is that galaxies in high mass pairs are more affected at small separations by their mutual attraction than the underlying density field.\n\n\\begin{figure*}[ht]\n\\subfloat[Redshift z=0]{\\label{fig:z_0_iso}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0to5000_z0_isolation4.pdf}}\n\\subfloat[Redshift z=0.5085]{\\label{fig:z_0.5_iso}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0to5000_z05085_isolation4.pdf}}\\\\\n\\subfloat[Redshift z=0.989]{\\label{fig:z_1_iso}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0to5000_z0989_isolation4.pdf}}\n\\subfloat[Redshift z=1.504]{\\label{fig:z_1.5_iso}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0to5000_z1504_isolation4.pdf}}\n\\caption{Variation of the average pairwise velocity of isolated galaxy pairs with redshift for different mass cuts (number of dark matter particles) as a function of separation. The isolation radius is taken to be \\mpch{4} for each member of the pair.}\n\\label{fig:redshift_variation_isolation4}\n\\end{figure*}\n\n\\begin{figure}[h]\n\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np10to5000_z0989_isolation4_scaling.pdf}\n\\caption{Same plot as in \\fref{fig:z_1_iso} where both the separation and the average pairwise velocity for each curve have been scaled by a factor $(M_{\\text{ref}}\/M)^{1\/3}$ where $M_{\\text{ref}}$ is the minimum mass of a subhalo: $M_{\\text{ref}}=\\sci{1.72}{10} \\msun\/h$. Note that the errors for each curve have been scaled accordingly.}\n\\label{fig:scaling_z1}\n\\end{figure}\n\nFor separations $ d > \\mpch{10} $, the velocity curves at each redshift seem to converge to a common asymptote.  In \\sref{sec:all_pairs}, we have shown that this limit is correctly predicted by linear theory -- see the agreement between the green curve and $ v_{12} $ in \\fref{fig:generic_all_pairs}.  This means that, even though the non-linear dynamics of the different mass limit pairs differs, their behavior at large separations seems to be predicted by the same linear theory.\n\nA remarkable feature of \\fref{fig:redshift_variation_noiso} is the different redshift dependence of the various $ v_{12} $ curves.  The pairwise velocity of the heaviest galaxies (cyan triangles) greatly increases with redshift, while in the uncut case (red circles) it decreases.  The intermediate curves seem to experience smaller variations. \n\n\n\\subsubsection{Isolated pairs}\n\n\\begin{figure*}[ht]\n\\subfloat[\\;No mass cut\\;\\;(\\:$ \\ensuremath{n_\\text{p}}\\xspace \\geqslant 20 \\:)$]\n {\\label{fig:np_0_iso}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np0_z0to1504_isolation4.pdf}}\n\\subfloat[Mass $ \\geqslant 4.3\\times10^{11}$ M$_\\odot\/h$\\, \\;\\;(\\:$ \\ensuremath{n_\\text{p}}\\xspace \\geqslant 500 \\:)$] {\\label{fig:np_100_iso}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np500_z0to1504_isolation4.pdf}}\\\\\n\\subfloat[Mass $ \\geqslant 8.6\\times10^{11}$ M$_\\odot\/h$\\, \\;\\;(\\:$ \\ensuremath{n_\\text{p}}\\xspace \\geqslant 1000 \\:)$] {\\label{fig:np_1000_iso}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np1000_z0to1504_isolation4.pdf}}\n\\subfloat[Mass $ \\geqslant 4.3\\times10^{12}$ M$_\\odot\/h$\\, \\;\\;(\\:$ \\ensuremath{n_\\text{p}}\\xspace \\geqslant 5000 \\:)$] {\\label{fig:np_5000_iso}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_np5000_z0to1504_isolation4.pdf}}\n\\caption{Variation of the average pairwise velocity of isolated galaxy pairs with redshift for different mass cuts (number of dark matter particles) as a function of separation. The isolation radius is taken to be \\mpch{4} for each member of the pair. Note that panel \\fref{fig:np_0_iso} is equivalent to the upper panel of \\fref{fig:iso_redshift_variation}.}\n\\label{fig:masscut_variation_isolation4}\n\\end{figure*}\n\n\nHaving analyzed the dynamics of non-isolated pairs with varying subhalo mass, we now do the same for pairs isolated within a \\mpch{4} radius. In \\fref{fig:redshift_variation_isolation4} we show the average pairwise velocity $ v_{12} $ for different mass cuts, with the redshift varying form panel to panel. This is the same setup as in \\fref{fig:redshift_variation_noiso}; note, however, that here we only plot separations up to \\mpch{4}.\n\n\nAll curves in \\fref{fig:redshift_variation_isolation4}, regardless of redshift, present the same features found in the uncut sample shown in \\fref{fig:generic_isolated} and explained in \\sref{sec:isolated_pairs}.  Namely, we see a virialized region on the smallest scales, an infalling regime on intermediate scales, and a roughly logarithmic growth due to the void effect on the largest separations analyzed.  The main difference from the uncut case is the separations at which these different regimes hold.  Most importantly, we can see that the logarithmic growth of $ v_{12} $ begins at larger separations for higher masses.  This is intuitive, since we expect the mutual attraction to be stronger in heavier pairs, thus overcoming the void effect even when the galaxies are closer to the edge of the void.\n\nAs a result of this stronger mutual attraction, the peculiar velocity contribution increases as we consider heavier pairs.  This means that when it comes to isolated galaxy pairs, low-mass pairs trace the cosmological expansion better than high-mass ones.  More quantitatively, the scale $ d_0 $ where the peculiar velocity vanishes is reached at larger separations for massive pairs.  At $ z=0.989 $, $ d_0 $ ranges from \\mpch{0.8} in the uncut case to almost \\mpch{4} for pairs with $ m>\\msunh{4.3}{11} $.   For higher masses, $ v_{12} $ does not even cross the zero line.\n\n\\begin{figure*}[ht]\n\\subfloat[No isolation]{\\label{fig:noiso_rmag}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_rmag-22to99_z0989_noiso}}\n\\subfloat[Isolation of \\mpch{4}]{\\label{fig:iso_rmag}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_rmag-22to99_z0989_isolation4}}\n\\caption{Average pairwise velocity of galaxy pairs at $z=0.989$ varying r-band magnitude.}\n\\label{fig:rmag_variation}\n\\end{figure*}\n\n\n\\begin{figure*}[ht]\n\\subfloat[No isolation]{\\label{fig:noiso_stellarm}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_stellarm0to536_z0989_noiso}}\n\\subfloat[Isolation of \\mpch{4}]{\\label{fig:iso_stellarm}\\includegraphics[width=0.5\\textwidth]{guo2010_vinfall_stellarm0to536_z0989_isolation4}}\n\\caption{Average pairwise velocity of galaxy pairs  at $z=0.989$ varying stellar mass.}\n\\label{fig:stellar_mass_variation}\n\\end{figure*}\n\n\nTo illustrate the redshift dependence of the peculiar velocity in more detail, in \\fref{fig:masscut_variation_isolation4} we plot $ v_{12} $ for a given mass-cut at four different redshifts, with the mass-cuts varying across the panels.  Increasing the mass-cut makes the redshift evolution of $ v_{12} $ more evident.  As a result, for $ m > \\msunh{4.3}{11} $, we cannot identify a cosmological regime where the pairs are comoving and have a redshift independent peculiar velocity. Where ``independence\" here means that the evolution is significantly less than the change in expansion rate.\n\nA closer look at \\fref{fig:redshift_variation_isolation4} shows a pattern in the different mass cuts. We notice that, as we increase the mass cut, the absolute value of the pairwise velocity increases and the minimum shifts to the right. This suggests that applying a mass dependent scaling to both separation and velocity may stack the curves. This is shown in \\fref{fig:scaling_z1}, where we have have taken as an example the plot at redshift $z=0.989$ (\\fref{fig:z_1_iso}) and scaled both the x and y axis by a factor $(M_{\\text{ref}}\/M)^{1\/3}$ where $M_{\\text{ref}}$ is the minimum mass of a subhalo: $M_{\\text{ref}}=\\sci{1.72}{10} \\msun\/h$. The physical motivation for this scaling is to have the same orbital period for all curves in the Keplerian regime \\ie on small scales. Indeed we see on this plot that such scaling collapses the curves specially in the infalling and virialized regions.\n\n\\subsubsection{Magnitude \\& stellar mass}\n\n\\begin{figure*}[ht]\n\\subfloat[No isolation]{\\label{fig:noiso_font}\\includegraphics[width=0.5\\textwidth]{font2008_vinfall_rmag-22to99_z0989_noiso}}\n\\subfloat[Isolation of \\mpch{4}]{\\label{fig:iso_font}\\includegraphics[width=0.5\\textwidth]{font2008_vinfall_rmag-22to99_z0989_isolation4}}\n\\caption{Average pairwise velocity of galaxy pairs for the \\citet{font:2008a} semi-analytic model at $ z=0.989 $.  This figure should be compared with \\fref{fig:rmag_variation}, where we plotted the same curves obtained for the \\citet{guo:2011a} model.}\n\\label{fig:rmag_font}\n\\end{figure*}\n\n\nWe now make a more direct link with observations and study the dependence of $ v_{12} $ on $ r $-band absolute magnitude and stellar mass.\n\nIn the left panel of \\fref{fig:rmag_variation}, we show the average pairwise velocity $ v_{12} $ of all galaxy pairs at $ z = 0.989 $ for the \\ensuremath{r\\text{-mag}}\\xspace cuts given in \\tref{tab:cuts}.  Although the curves retain their qualitative shape, there are two major differences with respect to the mass-cut sample in \\fref{fig:z_1}.  Firstly, the \\ensuremath{r\\text{-mag}}\\xspace selected pairs have smaller average velocities.  Secondly, the velocity minima are all approximately aligned at the same scale of $r_\\text{min}\\sim$ \\mpch{2}, while for the subhalo mass cuts the different velocity curves have their minima at different separations. These two differences are also seen when we apply the cuts in stellar mass (\\fref{fig:noiso_stellarm}).\n\nThe velocity differences can be explained by the fact that, although the subsamples chosen based on limits in $ r $-band magnitude and stellar mass preserve the number density of galaxies selected, these are not the same galaxies as the ones selected by the subhalo mass cuts. In particular, most massive galaxies do not necessarily coincide with the most luminous ones. In general, dark matter haloes trace the velocity of galaxies more directly than stellar mass or \\ensuremath{r\\text{-band}}\\xspace magnitude. Cuts based on stellar mass or luminosity add an additional dispersion, affecting the position of the minima with respect to subhalo mass cuts.\n\nThe right panels of \\fref{fig:rmag_variation} and \\fref{fig:stellar_mass_variation} show the average pairwise velocity $ v_{12} $ for pairs with an isolation radius of \\mpch{4} for cuts in \\ensuremath{r\\text{-band}}\\xspace magnitude and stellar mass respectively. These plots should be compared with the corresponding cuts in subhalo mass at redshift $z=0.989$ (\\fref{fig:z_1_iso}). Even though we again appreciate that the pairwise velocities in the \\ensuremath{r\\text{-mag}}\\xspace and stellar mass cut plots are smaller, the general dynamics shown on the plots are the same. It is worth noting that the almost comoving regime mentioned in \\sref{sec:cosmological_implications} for each curve remains unchanged both for the \\ensuremath{r\\text{-band}}\\xspace magnitude and the stellar mass cuts. It is clear that the effects of galaxy selection (be it subhalo mass, stellar mass or \\ensuremath{r\\text{-band}}\\xspace magnitude) play an important role in the behavior of pairwise velocities for isolated galaxy pairs.\n\n\n\\section{Comparison of two catalogs} \\label{sec:comparison_catalogs}\nThe results presented in the previous sections were based on the semi-analytic model of \\citet{guo:2011a}. To check the robustness of these results, we also compute the average peculiar velocities $ v_{12} $ for the semi-analytic model in \\citet{font:2008a}. This catalog is an improvement over the one presented in \\citet{bower:2006a} to better match the colors of satellite galaxies observed in the SDSS sample. In order to do this, the main modification introduced in \\citet{font:2008a} is the stripping of hot gaseous haloes of satellite galaxies into the GALFORM semi-analytic model for galaxy formation, while \\citet{guo:2011a} concentrate more on the independence of satellite galaxies from the FOF group. Both semi-analytic models have similar galaxy luminosity functions that fit the data well.\n\nIn \\fref{fig:rmag_font} we show the average pairwise velocity $ v_{12} $ as a function of separation for all the galaxy pairs (left panel) and for isolated pairs with an isolation radius of \\mpch{4} (right panel).  Each curve corresponds to one of the \\ensuremath{r\\text{-mag}}\\xspace cuts in \\tref{tab:cuts}, and should be compared with the matching curve for the \\citet{guo:2011a} catalog in \\fref{fig:rmag_variation}.   Note that we omitted to plot $ v_{12} $ for our most stringent cut of $ \\ensuremath{r\\text{-mag}}\\xspace < -22.86 $ because of poor statistics.  In general, we found that \\citet{font:2008a} has significantly less bright galaxies with $ \\ensuremath{r\\text{-mag}}\\xspace < -22.17 $ than \\citet{guo:2011a}, as can be seen by the large error bars in \\fref{fig:rmag_font}. \n\nA direct comparison between Figures \\ref{fig:rmag_font} and \\ref{fig:rmag_variation} shows that galaxy pairs have very similar dynamics regardless of the semi-analytic model used.   Not only do we see almost the same $ v_{12} $ range, but also the almost comoving regime introduced in \\sref{sec:cosmological_implications} is found approximately in the same range.  Such findings suggest that the parameters of the semi-analytic model used do not significantly affect the average pairwise velocities of galaxies. \n\n\n\\section{Conclusions}  \\label{sec:conclusions}\nWe have investigated the pairwise velocities of galaxies with a view towards using them as cosmological tracers by means of an AP style test, as recently proposed by \\citet{marinoni:2010a}.  We have analyzed the dynamics of such objects within the semi-analytic models of \\citet{guo:2011a} and \\citet{font:2008a} applied to the Millennium simulation \\cite{springel:2005a}, and studied the dependence of their relative velocity on local density, redshift, mass of the hosting subhaloes, \\ensuremath{r\\text{-band}}\\xspace magnitude and stellar mass.\n\nWe have first analyzed the dynamics of all galaxy pairs at redshift $ z=0 $ (see \\fref{fig:generic_all_pairs}).  We have found that, on scales $ d>\\mpch{10} $, the peculiar velocity is correctly predicted by linear theory \\cite{fisher:1995a, reid:2011a}.  On the other hand, for separations $ d<\\mpch{3} $, the pairs are decoupled from the Hubble flow and close to the static solution.  We argue that pairs in this regime cannot be used as cosmological tracers (see Appendix \\ref{app:bound_systems}).  \n\nBeing interested in investigating the claims by \\citet{marinoni:2010a}, we have studied the dynamics of galaxy pairs that are isolated within a radius of $ \\mpch{4} $.  At $ z=0 $, isolated galaxy pairs are almost comoving already for separations of $ 0.4<d<\\mpch{4} $ and only need up to 20\\% RSD correction (see \\fref{fig:generic_isolated}).   By analyzing redshift slices up to $ z=1.504 $, we have found that the peculiar velocities are only weakly dependent on the cosmological expansion ($ < 10\\% $ variation) for separations of $ 1<d<\\mpch{4} $ (see \\fref{fig:iso_redshift_variation}).  Since expansion is the main property characterizing a cosmological model, we might assume that in this regime the dynamics of isolated pairs are independent of the underlying cosmology.  Hence, we argue that isolated pairs in this regime could possibly be used as cosmological tracers with minimal RSD corrections.  \n\nImposing an isolation criterion of \\mpch{4}, as done in Ref.\\ \\cite{marinoni:2010a}, greatly reduces the number of pairs (see \\fref{fig:number_pairs}).  We have found that one can drastically increase the statistics while keeping the RSD corrections small by either reducing the isolation radius to \\mpch{2} or allowing up to $ 10 $ galaxies to be neighbors of the pair.  When dealing with observations, these adjustments may be helpful to reduce possibly large statistical uncertainties.\n\nAs galaxy surveys are flux limited, we have studied the feasibility of a measurement by varying the following properties of galaxy pairs: mass of the subhaloes that host the galaxies, \\ensuremath{r\\text{-band}}\\xspace absolute magnitude in the rest-frame, and stellar mass.   Low-mass pairs appear to be the best cosmological tracers, as RSD corrections increase with mass.  More precisely, a nearly comoving regime is reached in our analysis only for subhalo masses of $ m \\lesssim \\msunh{4.3}{11} $, corresponding to \\ensuremath{r\\text{-band}}\\xspace magnitudes of $ \\ensuremath{r\\text{-mag}}\\xspace \\gtrsim -21.27 $ and stellar masses of $ \\ensuremath{m_{\\star}}\\xspace \\lesssim \\msunh{1.59}{10} $ (see, respectively, Figures \\ref{fig:redshift_variation_isolation4}, \\ref{fig:rmag_variation} and \\ref{fig:stellar_mass_variation}).  We have also found that the peculiar velocities of galaxy pairs becomes more redshift-dependent as we increase the subhalo mass (see \\fref{fig:iso_redshift_variation}).  Therefore, we suggest that isolated pairs may not be adequate as cosmological tracers if their mass or luminosity is above the given thresholds.\n\nMarinoni and Buzzi \\cite{marinoni:2010a} selected isolated galaxy pairs from the DEEP2 galaxy survey \\cite{davis:2003a, newman:2012a} with comoving transverse separation $ \\ensuremath{r_\\perp}\\xspace $ in the range $ \\unit[20]{kpc\/h} $ -- $ \\mpch{0.7} $.  DEEP2 galaxies are known to reside in dark matter haloes of approximately $ \\unit[10^{12}]{\\msun\/h} $ \\cite{newman:2012a, conroy:2005a}.  The results of our analysis imply that such galaxy pairs do require corrections for evolution and cosmology dependent RSD component, which is significant with respect to the evolution being measured. \n\nIndeed, the primary concern for observational studies is the extent to which RSD ``corrections\" need to be modeled. Cosmological measurements from Baryon Acoustic Oscillation and RSD measurements on large-scales are reaching a precision at the 2--5\\% level \\cite{anderson:2012a,reid:2012a,padmanabhan:2012a}, and it is therefore reasonable to suppose that this is also the level at which we need to understand RSD corrections in order to make a useful contribution to the field from small-scale measurements. We have investigated whether selection based on local density can reduce the modeling burden, and find that low-mass, isolated galaxy pairs are preferred. However, even for these galaxies, the corrections depend on sample properties, and would need to be recalculated for each cosmological model to be tested: the only currently available way to do this is via numerical simulations. We conclude that observations of close-pairs of galaxies do show promise for AP-style cosmological measurements, particularly for low mass, isolated galaxies. However, it is likely that modeling limitations will continue to be the limiting factor for the foreseeable future.\n\n\\section{Acknowledgments}\nABB acknowledges financial support from the Spanish Ministry \nof Education (ME) within the FPU grant program with ref.\\ number AP2008-02679. \nShe also thanks the Institute of Cosmology and Gravitation for hospitality \nduring her stay from September 2011 to January 2012. WJP thanks the European Research Council and the UK Science and Technology Facilities Research Council for financial support. The authors would like to thank David Bacon, Marco Bruni, Phil Bull, Diego Capozzi, Chris Clarkson, Timothy Clifton, Robert Crittenden, Chris D'Andrea, Sean February, Pedro G.\\ Ferreira, Juan Garc\\'ia-Bellido, Alan Heavens, Elise Jennings, Marc Manera, Francesco Pace, John A.\\ Peacock, Mat Pieri and Lado Samushia for useful discussions. We would also like to thank the referee for useful suggestions.\n\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Credits}\n\n\n\\section{Introduction}\n\nKnowledge graph(KG) has received a lot of traction in the recent past and led to much research in this area. Most of the research is focused on the generation of Knowledge graphs and consumption of the information enshrined in the Knowledge Graph. Some of the earlier works to create KG are YAGO\\citep{Suchanek2007}, Freebase\\citep{Bollacker2008}, DBpedia\\citep{Lehmann2012} and WikiData\\citep{Vrandecic2014}. The evolution of the Knowledge Graph starts with the seminal paper from Berners-Lee \\citep{Berners-Lee2001}. The knowledge graph has evolved in three phases. In the first phase, the Knowledge representation was brought to the level of Web standard. The core focus shifted to Data management, linked data, and its application in the second phase. In the third phase, the focus shifted on the real-world application \\citep{Bonatti2018KnowledgeGN}. The real-world application ranges from Semantic parsing\\citep{Berant2013, Heck2014}, recommender system\\citep{Sun2020, Wang}, question answering\\citep{ Saxena}, named entity disambiguation\\citep{Lin}, information extraction \\citep{Xiong, Xiong, Liu2018a, Dietz2019} etc. The knowledge graph is a representation of structured relational information in the form of Entities and relations between them. It is a multi-relational graph where nodes are entities and edges are relations. Entities are real-world objects or abstract information. The representation entities and relations between them are represented as triple. For e.g. \\textit{(New Delhi, IsCapitalOf, India)} is an example of a triple. \\textit{New Delhi} and \\textit{Delhi} are entities and \\textit{IsCapitalOf} is relation. Though the representation looks scientific but consuming these in the real-world application is not an easy task. The consumption of information enshrined in the knowledge graph will be very easy if it can be converted to numerical representation. \\textit{Knowledge graph embedding} is a solution to incorporate the knowledge from the knowledge graph in a real-world application. \\\\\nThe motivation behind Knowledge graph embedding \\citep{Bordes} is to preserve the structural information, i.e., the relation between entities, and represent it in some vector space. This makes it easier to manipulate the information. Most of the work in \\textit{Knowledge graph embedding \\textbf{(KGE)}} is focused on generating a continuous vector representation for entities and relations and apply a relationship reasoning on the embedding. The relationship reasoning is supposed to optimize some scoring function to learn the embedding. Researchers have used different approaches to learn the embedding Path-based learning\\citep{Toutanova2016}, Entity-based learning, textual-based learning, etc. A lot of work was focused were translation model \\citep{Bordes} and semantic-based model\\citep{Bordes2014}. The representation of the triple results in a lot of information lost because it fails to take \"textual information\" into account. With the proposal of Graph attention network \\citep{Velickovic2017a} the representation of the entities has become more contextualized. In recent years, the proposal of multi-modal graph has extended the spectrum to a new level. In multi-modal knowledge graph, Knowledge graph can have multi-modal information like image and text etc. \\citep{Sun2020, Wei2019}. \\\\\nPrevious work on survey has focused on the KG Embedding \\citep{Wang}, KG Embedding and application\\citep{Ji2020b} , KG embedding with textual data \\citep{Lu2020a}, KG embedding based on the deep-learning \\citep{Wang2020a}. This work shall focus on the KG embedding from translation-based model, semantic-based model, embedding with enriched representation from textual data and multi-modal data and their application. In section 2, we shall provide the details of KGE; in section 3, we shall present the application area. In the summary section, we shall try to put emerging areas of research in KGE. \n\n\\section{Knowledge Graph embedding}\n\\label{sec:length}\n\nKnowledge Graph embedding is an approach to transform Knowledge Graphs (nodes, edges, and their feature vectors) into a low dimensional continuous vector space that preserves various graph structure, information, etc. These approaches are broadly classified into two groups: $\\textit{translation models}$ and $\\textit{semantic matching models}$.\n\n\\subsection{Translation Models}\nThe translation-based model uses distance-based measures to generate the similarity score for a pair of entities and their relationships. The translation-based model aims to find a vector representation of entities with relation to the translation of the entities. It maps entities to a low-dimensional vector space. \n\\subsubsection{TransE \\citep{Bordes2014}}\nThe first model proposed was TransE. It is an energy based model. If a triple \\textit{h, r, t} holds then the vector representation \\textit{h} and \\textit{r} should be as close as possible. It can be graphically represented as Figure-\\ref{fig:my_label}. Mathematically, it can be stated like $\\textit{h} + \\textit{r} \\approx \\textit{t}$. The energy of a triple is  \\textit{d(h + r,t)} for some similarity measure d. To learn the embedding, minimization of ranking based loss function over the training set.\n\\begin{equation}\n    \\mathcal{L= \\sum_{\\textit{h,r,t} \\in S}  \\sum_{\\textit{\\^{h},r,\\^{t}} \\in \\^{S} } [\\gamma +\\textit{d(h + r,t)} - \\textit{d(\\^{h},r,\\^{t})}] }\n\\end{equation}\n\\textit{d(\\^{h},r,\\^{t})} represents the set of the corrupt triples. This loss function will be optimized so that the valid triples are ranked above the corrupt triples. \nThis model fails in case of the one to many relation and many to many relation. To overcome this deficit new model TransH is proposed.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.9\\linewidth]{Trans E.png}\n    \\caption{The intuition of TransE model. Image credit \\cite{Wang}}\n    \\label{fig:my_label}\n\\end{figure}\n\n\\subsubsection{TransH \\cite{Wang2014KnowledgeGE}}\nIt was proposed to address the limitations of TransE. This model enables an entity to\nhave distributed representations based on their involvement in the relation. The representation of \\textit{$h_\\bot$}, \\textit{$t_\\bot$} are projected in a relation specific hyperplane. The relation between then is \\textit{$d_r$}. As in the Figure-\\ref{fig:my_label_2}, the vectors \\textit{h} and \\textit{t} are projected in the relation hyper-plane. The loss function and intuition remain similar to TransE. \n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.9\\linewidth]{TransH.png}\n    \\caption{TransH model. Image taken from \\cite{Wang}}\n    \\label{fig:my_label_2}\n\\end{figure}\n\n\\subsubsection{TransR \\cite{10.5555\/2886521.2886624}}\nIt proposes that an entity may have multiple attributes and various relations. Each relation may focus on different attributes of entities. TransR models entities  and relation in different embedding space. It means that two different spaces: entity space and relation space are modelled. Each entity is mapped into relation space. The translation construct is applied on the projected representation in the relationship space. The Figure-\\ref{fig:my_label_3} presents the intuition behind TransR model.\\\\ \n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.9\\linewidth]{TransR.png}\n    \\caption{TransR model. Image taken from \\cite{10.5555\/2886521.2886624}}\n    \\label{fig:my_label_3}\n\\end{figure}\nTo further refine the representation, a new model TransD was proposed by \\cite{ji-etal-2015-knowledge}. TransR captures the possibility of relations and their embedding from the relation space. But the TransD, extends it to the entity space as well. Here Entity-relation pair is considered as the first-class object. \n\\subsubsection{RotatE \\cite{DBLP:journals\/corr\/abs-1902-10197}}\nIn knowledge graphs, often, the relation is symmetric\/anti-symmetric, inversion and composition. e.g., \"Marriage\" is a symmetric relation, \"My niece is my sister's daughter\" is a composition, etc. The models discussed above are not capable of predicting these relations. The model proposed here is based on the intuition that the relation from head to tail is modeled as rotation in a complex plane. It is motivated by Euler's identity. \n    \\begin{equation}\n        e^{\\iota\\theta} = \\sin \\theta + \\iota \\cos \\theta\n    \\end{equation}\n\n\\noindent\nFor a triplet \\textit{(h,r,t)}, the relation among them can be represented as \\textit{t = h $\\circ$ r}. Where \\textit{h, r, t} is the k-dimension embedding of the head, relation and tail, restricting the $|{r_i}|$ = 1. It means we are in the unit circle and $\\circ$ represents element wise product. For each dimension $t_i = r_i h_i$ subjected to constraint $|{r_i}|$ = 1.\nUnder these condition the relationship is symmetric $\\iff$  for all the values of i, $e^{0\/\\iota\\theta} = \\pm 1$, The relationship is inverse $\\iff$ $r_j = \\bar{r}_i$, i.e. both are the complex conjugate. Two relations are composite $\\iff$ $r_j = r_i \\circ r_k$. It means the relation $r_j$ can be obtained by a combined rotation of $r_i$ and $r_k$, $\\theta_j = \\theta_i + \\theta_k$. The scoring function measures the angular distance.\n\\begin{equation}\n    d_r(h,t) = \\left\\| h \\circ r - t \\right\\|\n\\end{equation}\nThe Figure-\\ref{fig:my_label_4} shows the comparison of RotaE and TransE. \n\\begin{figure}\n    \\centering\n    \\includegraphics[width=.9\\linewidth]{rotaE.png}\n    \\caption{Representation of RotaE in comparison to TransE. In RotatE, the relationship between \\textit{h} and \\textit{t} is represented as angle of rotation. This image depicts the relationship in 1-dimension space. \\citep{DBLP:journals\/corr\/abs-1902-10197} }\n    \\label{fig:my_label_4}\n\\end{figure}\n\\subsubsection{HakE \\cite{DBLP:journals\/corr\/abs-1911-09419}}\nThe approaches we have discussed so far fails to capture the semantic hierarchies. In HakE, the authors have proposed to model the hierarchy in the entities as concentric circles in polar coordinate. The entity with smaller radius belongs to higher up in the hierarchy. The angle between them represents the variation in the meaning. \nTo represent a point on the circle, we should have $(r, \\theta)$. Similarly, this model has two components, one to map the modulus and the other one to map the angle. The modulus part considers the depth of the tree as moduli. \nLet $h_m, t_m$ and $r_m$ are the representation in modulus space then $h_m \\circ r_m = t_m$, where $h_m, t_m$ is a k-dim vector. \nThe distance function is similar to RotatE with modification to consider only modulo part. \n\\begin{equation}\n    d_{r,m}(h_m,t_m) = \\left\\| h_m \\circ r_m - t_m \\right\\|_2\n\\end{equation}\n    \nSimilarly, the phase part can be formulated as \n$(h_p + r_p)mod 2\\pi = t_p$ where $h_p, r_p, t_p \\in [0,2\\pi)^k$. The distance function is \n\\begin{equation}\n    d_{r,p}(h_p,t_p) = \\left\\|\\sin ( (h_p \\circ r_p - t_p)\/2) \\right\\|_1\n\\end{equation}\nBy combining both the part, an entity can be mapped in the polar coordinate space. \n\n\n\\subsection{Semantic Matching Models}\nSemantic Matching is one of the core tasks in Natural Language Processing. As we have seen that Translational Distance Models use distance-based scoring functions to calculate the similarity between the different entities and relations thus build the embedding accordingly. On the other hand, Semantic Matching Models use similarity-based scoring function. There are several Knowledge Graph Embedding algorithms comes under this model. Some of the algorithms are described below. \n\n\\subsubsection{RESCAL \\cite{nickel2011three}}\nRESCAL follows Statistical Relational Learning Approach which is based on a Tensor Factorization model that takes the inherent structure of relational data into account. A Tensor\\cite{kolda2009tensor} is a multidimensional array. More formally we can say that a first order Tensor is a vector, second order Tensor is a matrix and, Tensor with more than two order is called as higher order Tensor. Tensor Factorization is expressing a Tensor as a sequence of elementary operations acting on other, often a simpler Tensors.  Statistical Relational Learning inherits from Probability Theory and Statistics to address uncertainty and complexity of relational structures. \\\\\n\\cite{nickel2011three} models the Knowledge Graph triplet of the form (head, relation, tail) into three-way tensor, $\\mathcal{X}$ as shown in  Figure 5.\n\n\\begin{figure}[htb]\n    \\includegraphics[width=.9\\linewidth]{Fig_1.png}\n    \\caption{Tensor model for relational data. $E_{1} ... E_{n}$ denote the entities and $R_{1} ... R_{m}$ denote the relation in the domain\\cite{nickel2011three} }\n    \\label{fig: Menu}\n\\end{figure}\n\n\\noindent\nIn  $\\mathcal{X}$, two modes holds the concatenated entities (head and tail), and the third mode holds relation (relation). A Tensor entity $\\mathcal{X}_{ijk}$ = 1  denotes that there exist a relation and if $\\mathcal{X}_{ijk}$ = 0 denotes that there is unknown relation. It is assumed that data is given as n * n * m Tensor. Where n is the number of entities and m is the number of relations. RESCAL \"explains triples via pairwise interaction of latent features\". It performs the rank-r factorization on each slice of $\\mathcal{X}$ (relational data) and the score of a fact (head, relation, tail ) is given by the following bi-linear function.\n\\[f_{r}(h,t) = \\textbf{h}^T\\textbf{M}_r\\textbf{t} = \\sum_{i=0}^{d-1}\\sum_{j=0}^{d-1} [\\textbf{M}_r]_{ij} .[\\textbf{h}]_i . [\\textbf{t}]_j\\]\nwhere h,t $\\in$ $\\mathbb{R}^d$ are vector representation of entities, and $M_{r}$ $\\in$ $\\mathbb{R}^{d*d}$ is a matrix representation of $r^{th}$ relation. Thus from this equation we are able to calculating the score of the triple using the weighted sum of all the pairwise interactions between the latent features of the entities $h$ and $t$ as shown in the Figure 6.\n\n\\begin{figure}[htb]\n    \\includegraphics[width=.9\\linewidth]{Fig_2.png}\n    \\caption{ RESCAL Representation. Here number of latent features for entities are 3 and number of latent features for relations are 3. \\cite{Nickel_2016}  }\n    \\label{fig: Graph}\n\\end{figure}\n\n\\noindent\nThis method require $O(d^{2})$ parameters per relation and the space complexity of $O(nd + md^{2})$ where n is the number of entities and m is the number of relations.\n\n\\subsubsection{TATEC \\cite{10.1007\/978-3-662-44848-9_28}}\nTATEC stands for \\textit{Two And Three-way Embeddings Combination}. The main disadvantage of RESCAL is that it is a Three-way model which performs fairly good for relationships which occur frequently but it performs poor for the rare relationships and leads to major over-fitting. The issue of major over-fitting for rare relationships can be controlled by regularizing or reducing the expressivity of the model and former method is not feasible. The second method of reducing the expressivity is Two-way interaction which is implemented in TransE and SME. Two-way interaction approaches overperform the Three-way approaches on many datasets from which we can conclude that Two-way interactions are more efficient for the datasets and specially for those datasets which have more rare relationships. But the problem with the two-way interaction is that they are limited and are not able to represent all kind of relations with entities.\\\\\nTATEC is a latent factor model which is capable of incorporating the high capacity Three-way model with well-controlled two-way interactions and take  the advantage of both of them. Since two-way and three-way models do not use the the same kind of data pattern and do not encode the same kind of information in the embedding. So, in TATEC during first stage they used two different embeddings and then combined and fine-tuned them in the later stage. The scoring function of TATEC is given by $f_{r}(h,t) = f_{r}^1(h,t) + f_{r}^2(h,t) $ which is a linear combination of bi-gram and tri-gram terms, where $f_{r}^1(h,t)$ is a two-way interaction score and $f_{r}^2(h,t)$ is a three-way interaction score. These can be calculates as follows:\\\\\n1) Two-way interactions terms  can be given by:\n\\[  f_{r}^1(h,t) = \\textbf{h}^T\\textbf{r} + \\textbf{t}^T\\textbf{r} + \\textbf{h}^T\\textbf{D}\\textbf{t} \\]\n\\noindent\nwhere D is the diagonal matrix shared across all the different relations and does not depend on input triple and r $\\in$ $\\mathbb{R}^d$ is a vector  that depends on relationships.\\\\\n2) Three-way interactions terms  can be given by:\n\\[f_{r}^2(h,t) = \\textbf{h}^T\\textbf{M}_r\\textbf{t} \\]\n\n\\noindent\nThe final scoring function of TATEC is given by \n\\[f_{r}(h,t) = \\textbf{h}^T\\textbf{M}_r\\textbf{t} + \\textbf{h}^T\\textbf{r} + \\textbf{t}^T\\textbf{r} + \\textbf{h}^T\\textbf{D}\\textbf{t}\\]\n\n\\begin{figure}[htb]\n    \\includegraphics[width=.9\\linewidth]{Fig_3.png}\n    \\caption{ Link prediction results \\cite{10.1007\/978-3-662-44848-9_28} }\n    \\label{fig: Graph}\n\\end{figure}\n\n\\noindent\nAuthors of \\cite{10.1007\/978-3-662-44848-9_28} compared this model to the other existing models such as RESCAL \\cite{nickel2011three}, TransE \\citep{Bordes}, LFM, SE and, SME for link prediction on FB15k dataset and as a result TATEC performs better than all other available models as shown in  Figure 7.\n\n\\noindent\nTime complexity and the space complexity of TATEC is same as RESCAL as TATEC extends RESCAL. The time complexity of TATEC is $O(d^{2})$ parameters per relation and the space complexity of $O(nd + md^{2})$ where n is the number of entities and m is the number of relations.\n\n\\subsubsection{DistMult \\cite{yang2014embedding}}\nThis model compares with the NTN neural model, TransE and  bi-linear models like RESCAL. The problem with NTN is that it is the most expensive model it incorporate both linear and bilinear relation operations. Similarly TransE parameterizes  the linear operations with one dimensional vectors. DistMult is a simplified RESCAL, which uses the basic bilinear scoring function.\n\\[f_{r}(h,t) = \\textbf{h}^T\\textbf{M}_r\\textbf{t} \\]\n\n\\noindent\nThese bilinear formulations are combined with different forms of regularization to make different models. In DistMult authors considered a simpler approach where they reduced the number of parameters by imposing restrictions on $M_{r}$ to be a diagonal matrix. This results in a simpler model and this model enjoys the same scalable properties of TransE as well as it achieves better performance over TransE. Thus the final scoring function is given as \n\\[ f_{r}(h,t) = \\textbf{h}^T diag(\\textbf{r}) \\textbf{t} = \\sum_{i=0}^{d-1} [\\textbf{r}]_{i} .[\\textbf{h}]_i . [\\textbf{t}]_i \\]\n\n\\noindent\nwhere for each relation r, r $\\in \\mathbb{R}^d$ is a vector that depends on relationships. \n\n\\begin{figure}[htb]\n    \\includegraphics[width=.9\\linewidth]{DistMult.png}\n    \\caption{ DistMult simple Illustration }\n    \\label{fig: Graph}\n\\end{figure}\n\n\\noindent\nTime complexity and the space complexity of DistMult is more efficient as compared to RESCAL or TATEC. The time complexity of DistMult is $O(d)$ parameters per relation and the space complexity of $O(nd + md)$ where n is the number of entities and m is the number of relations. Due to its over-simplified mature of the model, this model is not enough powerful for the use in case of general Knowledge Graphs because it is only able to work efficiently  with symmetric relations.\n\n\\subsubsection{HolE \\cite{nickel2016holographic}}\nHolE stands for Holographic Embedding. HolE tried to overcome the problem of Tensor Product used in RESCAL by using circular correlation. Tensor product uses pairwise multiplicative interactions between feature vectors which results in increase in dimensionality of the representation i.e., $\\mathbb{R}^{d^2}$ thus increase the computational demand.\n\\[ \\textbf{h} \\circ \\textbf{t} = \\textbf{h} \\otimes \\textbf{t} \\in \\mathbb{R}^{d^2} \\]\nWhere \\textbf{a},\\textbf{b} $\\in \\mathbb{R}^d$ are entity embeddings. Tensor products are very rich in capturing the interactions but are computational intensive. On the other hand HolE  use Circular Correlation which can be seen as compression of the Tensor Product. The main advantage of Circular Correlation over Tensor Product is that it won't increase the dimensionality of the representation.\n\\[ \\textbf{h} \\circ \\textbf{t} = \\textbf{h} * \\textbf{t} \\in \\mathbb{R}^d \\]\nWhere * : $ \\mathbb{R}^d \\times \\mathbb{R}^d \\rightarrow \\mathbb{R}^d$ denotes the circular correlation. \n\\[ [\\textbf{h} * \\textbf{t}]_{i} = \\sum_{k=0}^{d-1} [\\textbf{h}_{k}] . \\textbf{t}_{(k+i)mod(d)}\\]\nThe final score of the fact in  HolE is given by matching the compositional vector ($\\textbf{h} * \\textbf{t}$) with the relational representation,i.e.,\n\n\\[ f_{r}(h,t) = \\textbf{r}^T  (\\textbf{h} * \\textbf{t} ) = \\sum_{i=0}^{d-1} [\\textbf{r}]_{i} \\sum_{k=0}^{d-1}[\\textbf{h}]_k . [\\textbf{t}]_{(k+i)mod(d)} \\]\n\n\\begin{figure}[htb]\n    \\includegraphics[width=.9\\linewidth]{HolE.png}\n    \\caption{ HolE simple Illustration (It requires only d components) \\cite{nickel2016holographic} }\n    \\label{fig: Graph}\n\\end{figure}\n\n\\noindent\nSo HolE is more efficient as compared to RESCAL or TransE. HolE take $O(d log d)$ parameters per relation and the space complexity of $O(nd + md)$ where n is the number of entities and m is the number of relations. Another advantage of HolE is that Circular Correlation is not commutative (  $\\textbf{h} * \\textbf{t} \\neq \\textbf{t} * \\textbf{h}$ ) thus HolE is able to model asymmetric relations (directed graphs) with compositional representations which is not possible in RESCAL.\n\n\\subsubsection{ComplEx\\cite{Trouillon2016}}\nKnowledge graphs represent the relation between entities. The entities may be termed as subjects and objects respectively of a given relation. However not all relations may be present in a given KG. One of the application of KG is the ability to predict missing relations or entities. \\\\\nDot product of vector embedding of KG triplets is being successfully used for symmetric, reflexive, anti-reflexive and even transitive relations \\citep{bouchard2015} however it can't be used for anti-symmetric relations. For example the relation capitalOf(New Delhi, India) is not symmetric since we cannot interchange subject and object entity in this relation therefore we need to have different embedding for an entity as subject and as object which increases the number of parameters.\\\\\nComplex embedding facilitates joint learning of subject and object entities while preserving the asymmetry of the relation. It uses Hermitian dot product of embedding of subject entities and object entities. The Eigen Vector decomposition is used to identify a low rank diagonal matrix W such that there exists X = Re($EWE^{T}$) such that X has same sign pattern as Y. The low rank diagonal matrix W is then used to predict missing relations by applying Re($e_{s}^{T}W\\bar{e}_{o}$).\n\\subsubsection{ANALOGY\\cite{Liu2017}}\nANALOGY is based on a multiplicative model where the relation (s,r,o) is scored by multiplying the vector representations of of subject (s), relation (r) and object (o). The relation present in the knowledge graph are expected to have higher score i.e. $\\phi(s,r,0)$ = $e_{s}^{T}W_{r}e_{o}$ will be high if the triplet (s,r,o) exists. \\\\\nAn example of analogy may be branch:tree::petal:leaf, in such an analogy, the relation $W_{r}$ \"is part of\" may be used to predict the missing entity from the analogy. The foundation of ANALOGY model is the linear maps of matrix representation of relations from the triplets present in the KG. Basically, the model uses the fact that there can be multiple paths to arrive at the entity from a given entity through a linear map of relations $W_{r1}, W_{r2}, W_{r3}$ and so on, and application of such relations in any sequence will give the same result. Hence such linear maps may be used to predict entities missing from the knowledge graph.\\\\\nFor example Let's consider the entities teacher (t), school (s), professor (p) and college (c). We have two relations in this set up: teachesAt (t,s) \\& teachesAt(p,c) and juniorOf(t,p) \\& juniorOf(s,c) therefore the relation between teacher and college is teachesAt*juniorOf = juniorOf*teachesAt. However such linear maps are feasible only if the commutative property holds for such relations. ANALOGY uses such linear maps to predict entities.\n\n\\subsection{Enrichment based embedding}\nIn the recent times, new emerging research areas are focusing on contextualized embedding. Under this, the entity under the consideration is enriched information from the neighbourhood information. A few notable approaches are Graph attention network (GAT) \\citep{Velickovic2017a} based information enrichment. Two methods based on the KGAT \\citep{Wei2019}, MMGAT \\citep{Sun2020} has proposed models to embed contextual information for an entity. Under MMGAT, they proposed model to combine the embedding from multi-modal data with an attention framework, as adopted from GAT's attention framework. Both the frameworks, were using translation model to learn the representation after the enrichment. The new emerging research areas are try to learn the structural information as well as path based information, multi-modal data. There are other research areas in the embedding are: \\textit{Text-enhanced embedding}, \\textit{Logic-enhanced embedding}, \\textit{Image-enhanced embedding} \\citep{Bianchi2020KnowledgeGE} etc. \n\n\\section{Applications of Knowledge graph embedding}\n\nThere are many applications of KG embedding learning methods. This section explores three of them namely- link prediction, triplet classification and recommender systems. \n\n\\noindent\nThe first two are In-KG applications, which are conducted within the scope of the KG. The last is an example of Out-of-KG applications that scale to broader domains \\citep{Wang2017}.\n\n\\subsection{Link Prediction}\n\nThe set of edges in a knowledge graph is a subset of \\textit{Entities$\\times$Relations$\\times$Entities}. The link prediction task focuses on finding an entity that can be represented as a fact (edge) together with a given relation and entity i.e., \\textit{(entity, relation, ?)} or \\textit{(?, relation, entity)} where ? refers to the missing entity. For e.g. \\textit{(New Delhi, isCapitalOf, ?)} or \\textit{(?, isCapitalOf, India)}. Link prediction is a way of Knowledge graph augmentation \\citep{paulheim2017knowledge}. It deduces missing information from the knowledge graph itself. \n\n\\noindent\nThe datasets for LP are constructed by sampling from the original knowledge graph. Then, the links removed can be used in validation set or the test set \\citep{bordes2013translating, dettmers2018convolutional}. The structure of such graphs play a vital role \nfor improving the results, multiple source entities making learning effective and multiple destination entities making learning difficult \\citep{Rossi_2021}.\n\n\\noindent\nThe LP models assigns a score to the triplet corresponding to each possible entity to fill the question mark (?). The triplets are then ranked by a function and entity corresponding to the lowest rank is predicted. If the predicted facts in the ranked predictions are already present in the Knowledge graph, they may or may not be excluded while calculating the ranks called raw and filtered rankings respectively \\citep{bordes2013translating}. For e.g. if the training knowledge graph contains the fact that \\textit{(Arjuna, isSonOf, Kunti)}, and the test query is \\textit{(?, isSonOf, Kunti)}. The target answer is \\textit{(Yudhishtra, isSonOf, Kunti)} and the system ranks  \\textit{(Arjuna, isSonOf, Kunti)} and then \\textit{(Yudhishtra, isSonOf, Kunti)}. The raw ranking of the triplet \\textit{(Yudhishtra, isSonOf, Kunti)} will be two and filtered ranking will be one. \n\n\\noindent\nThere are several tie breaking policies that are used by the ranking system. Assigning the \\textit{minimum} or the \\textit{\nmaximum} rank, or a \\textit{random} or the \\textit{average} rank to the targeted entity \\citep{Rossi_2021}.\n\n\\noindent\nThe ranks obtained are used to compute metrics such as \\textit{Mean Rank} (average of all the ranks), \\textit{Mean Reciprocal Rank} (average of the inverse of ranks), or \\textit{Hits@M} (proportion of ranks \\textit{$\\leq$ M}).\n\n\n\\subsection{Triple Classification}\nTriple Classification is the problem of identifying whether a given triple is correct. It aims to give a yes or no answer to questions such as \\textit{is New Delhi capital of India?} which can be written in the form of a triple \\textit{(New Delhi, isCapitalOf, India)} \\citep{socher2013reasoning}.\n\n\\noindent\nA scoring function is used to calculate score of a triple similar to the link prediction. If the score is greater than a certain threshold, then it is considered a fact else a wrong triple \\citep{Wang2017}. \n\n\\noindent\nBoth the classical methods, such as \\textit{micro} and \\textit{macro averaging}, and ranking methods such as \\textit{Mean Rank} are used as evaluation metrics \\citep{guo2016jointly}. \n\n\\subsection{Recommender Systems}\nRecommender system (RS) assists the user in an environment where multiple options are available by providing a certain ordering of choices that the recommendation algorithm infers. This inference can be based on the similarity of the choices and behaviour pattern of different users. This type of recommendation methods falls into the domain of collaborative filtering methods \\citep{adomavicius2005toward}. \n\n\\noindent\nThe CF methods suffer from problems of Data sparsity and cold start. Data sparsity arises from the fact that only a small proportion of items are rated by the users and most options have only limited feedback from the users. Cold start problem is the problem of having no historical data about the new users and options. To deal with these problems different types of side information about a user and item are utilized by the RS \\citep{Sun_2019}.\n\n\\noindent\nKG is utilised for side information in CF. It acts as a heterogeneous graph that represent entities as nodes and relation as edges. The KG connects various entities via latent relationships and also provide explainability in recommendations \\citep{Wang_2018}.\n\n\\noindent\nThe KG embedding based methods for RS use two modules - Graph embedding and Recommendation Module. The way that these modules are coupled lead to categorization of embedding based methods in a). two stage learning methods, b). joint learning method and c). multi task learning method\\citep{Guo_2020}. \n\n\\noindent\nTwo stage learning methods first uses graph embedding module to obtain the embeddings using various KG algorithms and then use recommendation module to infer. The advantages of this method lies in its simplicity and scalability but since the two modules are loosely coupled the embeddings might not be suitable for recommendation tasks.\n\n\\noindent\nJoint learning methods train both the modules in an end to end fashion. Thus, recommendation module guides the training in graph embedding layer. \n\n\\noindent\nMulti task learning method train the recommendation module with the guidance of KG related task such as KG completion. The primary intuition behind this method is that the bipartite graph of user and item in recommendation task share structures with the corresponding KG entities.\n\n\\section{Summary}\nKnowledge graphs provide an effective way of presenting real-world relationships. As a result Knowledge graphs have an inherent advantage w.r.t serving the information need. KG in itself is a growing area of research. KG embedding is a technique to represent all the components of the KG in vector form. These vectors represent the latent properties of the components of the graph. Various models for embedding methods are based on different combinations of vector algebra which present an interesting area of research. In this work, we have surveyed the embedding methods that started this active area of research, state-of-art models and the new frontiers which are being explored in the KG embedding. \\\\\nThe KG embedding methods progressed from translation-based models which are based on vector addition. In this work, we have presented how translation-based models improved over time to overcome shortcomings of the earlier models. While translation-based models used vector addition, semantic models can be clubbed together as multiplicative models. We have included the transition from basic semantic models to the more advanced semantic models which may be used to explain different types of real world relationships such as symmetric, anti-symmetric,inverse or composition.\\\\\nNew research areas have broaden the scope from structural embedding to more contextual embedding by encoding additional information in the learned representation. The latest area of research in this field is enrichment based embedding models. In this work, we have introduced those briefly. \\\\\nVector space representation has paved a way to use the information from Knowledge graph directly into the real world application. In this work, we have described a few real world applications of KG embedding such as link prediction, triple classification and recommender systems.\n\n\n\n\\chapter*{}\n\\setcounter{page}{1}%\n\\vspace{-4.65cm}\n\\noindent\n\\begin{tabular}[H]{lr}\\toprule [1pt] \\vspace{-0.33cm}\\\\\n\\hspace{-2mm}\\Journal{Journal Name} & \\hspace{49mm}\\multirow{5}*{\\includegraphics[height=22mm,width=58mm]{SciencePGLOGO.pdf}}\\\\\n\\hspace{-2mm}\\Address{2021; X(X): XX-XX} &\\\\\n\\hspace{-2mm}\\Address{http:\/\/www.sciencepublishinggroup.com\/x\/xxx} &\\\\\n\\hspace{-2mm}\\Address{doi: 10.11648\/j.XXXX.2021XXXX.XX} &\\\\\n\\hspace{-2mm}\\Address{ISSN: xxxx-xxxx (Print); ISSN: xxxx-xxxx (Online)} \\\\\\bottomrule [2.5pt]\n\\end{tabular}\n\\parskip=12pt\n\\begin{flushleft}\n\\noindent \\Head{A Survey of Knowledge Graph Embedding and Their Applications}\n\\end{flushleft}\n\\parskip=8pt\n\\noindent \\Name{Shivani Choudhary^{1}$, Tarun Luthra$^{2}$, Ashima Mittal$^{2}$, Rajat Singh$^{2} $}\n\\parskip=8pt\n\n\n\\noindent \\Address{$^1$School of Interdisciplinary Research, Indian Institute of Technology Delhi, Delhi, India\\\\$^2$Department of Computer Science and Engineering, Indian Institute of Technology Delhi, Delhi, India}\n\\vspace{5pt}\n\n\\parskip=4pt\n\\noindent \\Emailaddress{Email address:}\\\\\n\\noindent \\Emailaddresscontent{shivani@sire.iitd.ac.in (Shivani Choudhary), mcs202475@cse.iitd.ac.in (Tarun Luthra), anz208486@cse.iitd.ac.in (Ashima Mittal), csz208507@cse.iitd.ac.in (Rajat Singh)}\n\n\\parskip=8pt\n\\noindent \\Citation{To cite this article:}\n\\begin{flushleft}\n\\vspace{-0.56cm}\n\\noindent \\Citationcontent{Authors Name. Paper Title. {\\it International Journal of XXXXXX}. Vol. x, No. x, 2021, pp. x-x. doi: 10.11648\/j.xxx.xxxxxxxx.xx.}\n\\vspace{-0.3cm}\n\\end{flushleft}\n\\parskip=10pt\n\n\\noindent \\Date{{\\bf Received:} MM DD, 2021; {\\bf Accepted:} MM DD, 2021; {\\bf Published:} MM DD, 2021}\n\n\\parskip=5pt\n\\noindent \\rule{\\textwidth}{1pt}\n\n\\parskip=6pt\n\\noindent \\Abstract{Abstract: }\\Abstractcontent{\\baselineskip=12pt Knowledge Graph represents a network of real-world entities. The information embedded in the Knowledge graph, though structured, is challenging to consume in a real-world application. Consuming this information for real-world task necessitate a numerical representation of entities. Knowledge graph embedding focuses on approaches to generate a dense embedding. These techniques find their application in various tasks such as completion of knowledge graph to predict missing information, recommender systems, question answering, query expansion, etc. Knowledge graph embedding enables the real-world application to consume information to improve performance. Knowledge graph embedding is an active research area. Most of the embedding methods focus on structure-based information. Recent research has extended the boundary to include multimodal text-based information and image-based information to generate improved entity embedding. Efforts have been made to enhance the representation with context information. This paper introduces the growth in the field of KG embedding from simple translation-based models to enrichment-based models. Further, This paper covers the areas where Knowledge Graph finds its application. }\n\n\\parskip=8pt\n\\hangafter=1\n\\setlength {\\hangindent}{6.2em}\n\\noindent \\Abstract{Keywords: }\\Abstractcontent{knowledge Graph, Recommender Systems, Question Answering, Query Expansion}\n\\parskip=0pt\n\n\\noindent \\rule{\\textwidth}{1pt}\n\\parskip=0pt\n\\columnseprule=0pt\n\\setlength{\\columnsep}{1.5em}\n\\vspace{-12pt}\n\\begin{multicols}{2}\n\\section{Introduction}\n\\vspace{-6pt}\n\nKnowledge graph(KG) has received a lot of traction in the recent past and led to much research in this area. Most of the research is focused on the generation of Knowledge graphs and consumption of the information enshrined in the Knowledge Graph. Some of the earlier works to create KG are YAGO\\citep{Suchanek2007}, Freebase\\citep{Bollacker2008}, DBpedia\\citep{Lehmann2012} and WikiData\\citep{Vrandecic2014}. The evolution of the Knowledge Graph starts with the seminal paper from Berners-Lee \\citep{Berners-Lee2001}. The knowledge graph has evolved in three phases. In the first phase, the Knowledge representation was brought to the level of Web standard. The core focus shifted to Data management, linked data, and its application in the second phase.\n In the third phase, the focus shifted on the real-world application \\citep{Bonatti2018KnowledgeGN}. The real-world application ranges from Semantic parsing\\citep{Berant2013, Heck2014}, recommender system\\citep{Sun2020, Wang}, question answering\\citep{ Saxena}, named entity disambiguation\\citep{Lin}, information extraction \\citep{Xiong, Xiong, Liu2018a, Dietz2019} etc. The knowledge graph is a representation of structured relational information in the form of Entities and relations between them. It is a multi-relational graph where nodes are entities and edges are relations. Entities are real-world objects or abstract information. The representation entities and relations between them are represented as triple. For e.g. \\textit{(New Delhi, IsCapitalOf, India)} is an example of a triple. \\textit{New Delhi} and \\textit{Delhi} are entities and \\textit{IsCapitalOf} is relation. Though the representation looks scientific but consuming these in the real-world application is not an easy task. The consumption of information enshrined in the knowledge graph will be very easy if it can be converted to numerical representation. \\textit{Knowledge graph embedding} is a solution to incorporate the knowledge from the knowledge graph in a real-world application. \\\\\nThe motivation behind Knowledge graph embedding \\citep{Bordes} is to preserve the structural information, i.e., the relation between entities, and represent it in some vector space. This makes it easier to manipulate the information. Most of the work in \\textit{Knowledge graph embedding \\textbf{(KGE)}} is focused on generating a continuous vector representation for entities and relations and applying a relationship reasoning on the embedding. The relationship reasoning is supposed to optimize some scoring functions to learn the embedding. Researchers have used different approaches to learn the embedding of Path-based learning\\citep{Toutanova2016}, Entity-based learning, textual-based learning, etc. MuchA lot of work was focused onwere translation model \\citep{Bordes} and semantic-based model\\citep{Bordes2014}. The representation of the triple results in a lot of information lost because it fails to take \"textual information\" into account. With the proposal of Graph attention network \\citep{Velickovic2017a} the representation of the entities haves become more contextualized. In recent years, the proposal of a multi-modal graph has extended the spectrum to a new level. In the multi-modal knowledge graph, Knowledge graph can have multi-modal information like image and text, etc. \\citep{Sun2020, Wei2019}. \\\\\nPrevious work on survey has focused on the KG Embedding \\citep{Wang}, KG Embedding and application\\citep{Ji2020b} , KG embedding with textual data \\citep{Lu2020a}, KG embedding based on the deep-learning \\citep{Wang2020a}. This work shall focus on the KG embedding from translation-based model, semantic-based model, embedding with enriched representation from textual data, and multi-modal data and their application. In section 2, we shall provide the details of KGE; in section 3, we shall present the application area. In the summary section, we shall try to put emerging areas of research areas in KGE. \n\n\\section{Knowledge Graph embedding}\n\\label{sec:length}\n\nKnowledge Graph embedding is an approach to transform Knowledge Graphs (nodes, edges, and their feature vectors) into a low dimensional continuous vector space that preserves various graph structures, information, etc. These approaches are broadly classified into two groups: $\\textit{translation models}$ and $\\textit{semantic matching models}$.\n\n\\subsection{Translation Models}\nThe translation-based model uses distance-based measures to generate the similarity score for a pair of entities and their relationships. The translation-based model aims to find a vector representation of entities with relation to the translation of the entities. It maps entities to a low-dimensional vector space. \n\\subsubsection{TransE \\citep{Bordes2014}}\nThe first model proposed was TransE. It is an energy based model. If a triple \\textit{h, r, t} holds then the vector representation \\textit{h} and \\textit{r} should be as close as possible. It can be graphically represented as Figure-\\ref{fig:my_label}. Mathematically, it can be stated like $\\textit{h} + \\textit{r} \\approx \\textit{t}$. The energy of a triple is  \\textit{d(h + r,t)} for some similarity measure d. To learn the embedding, minimization of ranking based loss function over the training set.\n\\begin{equation}\n    \\mathcal{L= \\sum_{\\textit{h,r,t} \\in S}  \\sum_{\\textit{$\\hat{h}$,r,$\\hat{t}$} \\in \\hat{S} } [\\gamma+\\textit{d(h+r),t}-\\textit{d($\\hat{h}$,r,$\\hat{t}$)}]}\n\\end{equation}\n\\textit{d(\\^{h},r,\\^{t})} represents the set of the corrupt triples. This loss function will be optimized so that the valid triples are ranked above the corrupt triples. \nThis model fails in the case of the one- to- many relation and many- to- many relation. To overcome this deficit new model, TransH is proposed.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=.9\\linewidth]{TransE.png}\n\\caption{The intuition of TransE model. Image credit \\cite{Wang}}\n\\label{fig:my_label}\n\\end{figure}\n\n\\subsubsection{TransH \\cite{Wang2014KnowledgeGE}}\nIt was proposed to address the limitations of TransE. This model enables an entity to\nhave distributed representations based on their involvement in the relation. The representation of \\textit{$h_\\bot$}, \\textit{$t_\\bot$} are projected in a relation specific hyperplane. The relation between then is \\textit{$d_r$}. As in the Figure-\\ref{fig:my_label_2}, the vectors \\textit{h} and \\textit{t} are projected in the relation hyper-plane. The loss function and intuition remain similar to TransE. \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=.9\\linewidth]{TransH.png}\n\\caption{TransH model. Image taken from \\cite{Wang}}\n\\label{fig:my_label_2}\n\\end{figure}\n\n\\subsubsection{TransR \\cite{10.5555\/2886521.2886624}}\nIt proposes that an entity may have multiple attributes and various relations. Each relation may focus on different attributes of entities. TransR models entities  and relation in different embedding space. It means that two different spaces: entity space and relation space are modelled. Each entity is mapped into relation space. The translation construct is applied on the projected representation in the relationship space. The Figure-\\ref{fig:my_label_3} presents the intuition behind TransR model.\\\\ \n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=.9\\linewidth]{TransR.png}\n\\caption{TransR model. Image taken from \\cite{10.5555\/2886521.2886624}}\n\\label{fig:my_label_3}\n\\end{figure}\nTo further refine the representation, a new model TransD was proposed by \\cite{ji-etal-2015-knowledge}. TransR captures the possibility of relations and their embedding from the relation space. But the TransD, extends it to the entity space as well. Here Entity-relation pair is considered as the first-class object. \n\\subsubsection{RotatE \\cite{DBLP:journals\/corr\/abs-1902-10197}}\nIn knowledge graphs, often, the relation is symmetric\/anti-symmetric, inversion and composition. e.g., \"Marriage\" is a symmetric relation, \"My niece is my sister's daughter\" is a composition, etc. The models discussed above are not capable of predicting these relations. The model proposed here is based on the intuition that the relation from head to tail is modeled as rotation in a complex plane. It is motivated by Euler's identity. \n    \\begin{equation}\n        e^{\\iota\\theta} = \\sin \\theta + \\iota \\cos \\theta\n    \\end{equation}\n\n\\noindent\nFor a triplet \\textit{(h,r,t)}, the relation among them can be represented as \\textit{t = h $\\circ$ r}. Where \\textit{h, r, t} is the k-dimension embedding of the head, relation and tail, restricting the $|{r_i}|$ = 1. It means we are in the unit circle and $\\circ$ represents element wise product. For each dimension $t_i = r_i h_i$ subjected to constraint $|{r_i}|$ = 1.\nUnder these condition the relationship is symmetric $\\iff$  for all the values of i, $e^{0\/\\iota\\theta} = \\pm 1$, The relationship is inverse $\\iff$ $r_j = \\bar{r}_i$, i.e. both are the complex conjugate. Two relations are composite $\\iff$ $r_j = r_i \\circ r_k$. It means the relation $r_j$ can be obtained by a combined rotation of $r_i$ and $r_k$, $\\theta_j = \\theta_i + \\theta_k$. The scoring function measures the angular distance.\n\\begin{equation}\n    d_r(h,t) = \\left\\| h \\circ r - t \\right\\|\n\\end{equation}\nThe Figure-\\ref{fig:my_label_4} shows the comparison of RotaE and TransE. \n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=.9\\linewidth]{rotaE.png}\n    \\caption{Representation of RotaE in comparison to TransE. In RotatE, the relationship between \\textit{h} and \\textit{t} is represented as angle of rotation. This image depicts the relationship in 1-dimension space. \\citep{DBLP:journals\/corr\/abs-1902-10197} }\n    \\label{fig:my_label_4}\n\\end{figure}\n\\subsubsection{HakE \\cite{DBLP:journals\/corr\/abs-1911-09419}}\nThe approaches we have discussed so far fails to capture the semantic hierarchies. In HakE, the authors have proposed to model the hierarchy in the entities as concentric circles in polar coordinate. The entity with smaller radius belongs to higher up in the hierarchy. The angle between them represents the variation in the meaning. \nTo represent a point on the circle, we should have $(r, \\theta)$. Similarly, this model has two components, one to map the modulus and the other one to map the angle. The modulus part considers the depth of the tree as moduli. \nLet $h_m, t_m$ and $r_m$ are the representation in modulus space then $h_m \\circ r_m = t_m$, where $h_m, t_m$ is a k-dim vector. \nThe distance function is similar to RotatE with modification to consider only modulo part. \n\\begin{equation}\n    d_{r,m}(h_m,t_m) = \\left\\| h_m \\circ r_m - t_m \\right\\|_2\n\\end{equation}\n    \nSimilarly, the phase part can be formulated as \n$(h_p + r_p)mod 2\\pi = t_p$ where $h_p, r_p, t_p \\in [0,2\\pi)^k$. The distance function is \n\\begin{equation}\n    d_{r,p}(h_p,t_p) = \\left\\|\\sin ( (h_p \\circ r_p - t_p)\/2) \\right\\|_1\n\\end{equation}\nBy combining both the part, an entity can be mapped in the polar coordinate space. \n\n\n\\subsection{Semantic Matching Models}\nSemantic Matching is one of the core tasks in Natural Language Processing. As we have seen that Translational Distance Models use distance-based scoring functions to calculate the similarity between the different entities and relations thus build the embedding accordingly. On the other hand, Semantic Matching Models use similarity-based scoring function. There are several Knowledge Graph Embedding algorithms comes under this model. Some of the algorithms are described below. \n\n\\subsubsection{RESCAL \\cite{nickel2011three}}\nRESCAL follows Statistical Relational Learning Approach which is based on a Tensor Factorization model that takes the inherent structure of relational data into account. A Tensor\\cite{kolda2009tensor} is a multidimensional array. More formally we can say that a first order Tensor is a vector, second order Tensor is a matrix and, Tensor with more than two order is called as higher order Tensor. Tensor Factorization is expressing a Tensor as a sequence of elementary operations acting on other, often a simpler Tensors.  Statistical Relational Learning inherits from Probability Theory and Statistics to address uncertainty and complexity of relational structures. \\\\\n\\cite{nickel2011three} models the Knowledge Graph triplet of the form (head, relation, tail) into three-way tensor, $\\mathcal{X}$ as shown in  Figure 5.\n\n\\begin{figure}[H]\n    \\includegraphics[width=.9\\linewidth]{Fig_1.png}\n    \\caption{Tensor model for relational data. $E_{1} ... E_{n}$ denote the entities and $R_{1} ... R_{m}$ denote the relation in the domain\\cite{nickel2011three} }\n    \\label{fig: Menu}\n\\end{figure}\n\n\\noindent\nIn  $\\mathcal{X}$, two modes holds the concatenated entities (head and tail), and the third mode holds relation (relation). A Tensor entity $\\mathcal{X}_{ijk}$ = 1  denotes that there exist a relation and if $\\mathcal{X}_{ijk}$ = 0 denotes that there is unknown relation. It is assumed that data is given as n * n * m Tensor. Where n is the number of entities and m is the number of relations. RESCAL \"explains triples via pairwise interaction of latent features\". It performs the rank-r factorization on each slice of $\\mathcal{X}$ (relational data) and the score of a fact (head, relation, tail ) is given by the following bi-linear function.\n\\[f_{r}(h,t) = \\textbf{h}^T\\textbf{M}_r\\textbf{t} = \\sum_{i=0}^{d-1}\\sum_{j=0}^{d-1} [\\textbf{M}_r]_{ij} .[\\textbf{h}]_i . [\\textbf{t}]_j\\]\nwhere h,t $\\in$ $\\mathbb{R}^d$ are vector representation of entities, and $M_{r}$ $\\in$ $\\mathbb{R}^{d*d}$ is a matrix representation of $r^{th}$ relation. Thus from this equation we are able to calculating the score of the triple using the weighted sum of all the pairwise interactions between the latent features of the entities $h$ and $t$ as shown in the Figure 6.\n\n\\begin{figure}[H]\n    \\includegraphics[width=.9\\linewidth]{Fig_2.png}\n    \\caption{ RESCAL Representation. Here number of latent features for entities are 3 and number of latent features for relations are 3. \\cite{Nickel_2016}  }\n    \\label{fig: Graph}\n\\end{figure}\n\n\\noindent\nThis method require $O(d^{2})$ parameters per relation and the space complexity of $O(nd + md^{2})$ where n is the number of entities and m is the number of relations.\n\n\\subsubsection{TATEC \\cite{10.1007\/978-3-662-44848-9_28}}\nTATEC stands for \\textit{Two And Three-way Embeddings Combination}. The main disadvantage of RESCAL is that it is a Three-way model which performs fairly good for relationships which occur frequently but it performs poor for the rare relationships and leads to major over-fitting. The issue of major over-fitting for rare relationships can be controlled by regularizing or reducing the expressivity of the model and former method is not feasible. The second method of reducing the expressivity is Two-way interaction which is implemented in TransE and SME. Two-way interaction approaches overperform the Three-way approaches on many datasets from which we can conclude that Two-way interactions are more efficient for the datasets and specially for those datasets which have more rare relationships. But the problem with the two-way interaction is that they are limited and are not able to represent all kind of relations with entities.\\\\\nTATEC is a latent factor model which is capable of incorporating the high capacity Three-way model with well-controlled two-way interactions and take  the advantage of both of them. Since two-way and three-way models do not use the the same kind of data pattern and do not encode the same kind of information in the embedding. So, in TATEC during first stage they used two different embeddings and then combined and fine-tuned them in the later stage. The scoring function of TATEC is given by $f_{r}(h,t) = f_{r}^1(h,t) + f_{r}^2(h,t) $ which is a linear combination of bi-gram and tri-gram terms, where $f_{r}^1(h,t)$ is a two-way interaction score and $f_{r}^2(h,t)$ is a three-way interaction score. These can be calculates as follows:\\\\\n1) Two-way interactions terms  can be given by:\n\\[  f_{r}^1(h,t) = \\textbf{h}^T\\textbf{r} + \\textbf{t}^T\\textbf{r} + \\textbf{h}^T\\textbf{D}\\textbf{t} \\]\n\\noindent\nwhere D is the diagonal matrix shared across all the different relations and does not depend on input triple and r $\\in$ $\\mathbb{R}^d$ is a vector  that depends on relationships.\\\\\n2) Three-way interactions terms  can be given by:\n\\[f_{r}^2(h,t) = \\textbf{h}^T\\textbf{M}_r\\textbf{t} \\]\n\n\\noindent\nThe final scoring function of TATEC is given by \n\\[f_{r}(h,t) = \\textbf{h}^T\\textbf{M}_r\\textbf{t} + \\textbf{h}^T\\textbf{r} + \\textbf{t}^T\\textbf{r} + \\textbf{h}^T\\textbf{D}\\textbf{t}\\]\n\n\\begin{figure}[H]\n\\includegraphics[width=.9\\linewidth]{Fig_3.png}\n\\caption{ Link prediction results \\cite{10.1007\/978-3-662-44848-9_28} }\n\\label{fig: Graph}\n\\end{figure}\n\n\\noindent\nAuthors of \\cite{10.1007\/978-3-662-44848-9_28} compared this model to the other existing models such as RESCAL \\cite{nickel2011three}, TransE \\citep{Bordes}, LFM, SE and, SME for link prediction on FB15k dataset and as a result TATEC performs better than all other available models as shown in  Figure 7.\n\n\\noindent\nTime complexity and the space complexity of TATEC is same as RESCAL as TATEC extends RESCAL. The time complexity of TATEC is $O(d^{2})$ parameters per relation and the space complexity of $O(nd + md^{2})$ where n is the number of entities and m is the number of relations.\n\n\\subsubsection{DistMult \\cite{yang2014embedding}}\nThis model compares with the NTN neural model, TransE and  bi-linear models like RESCAL. The problem with NTN is that it is the most expensive model it incorporate both linear and bilinear relation operations. Similarly TransE parameterizes  the linear operations with one dimensional vectors. DistMult is a simplified RESCAL, which uses the basic bilinear scoring function.\n\\[f_{r}(h,t) = \\textbf{h}^T\\textbf{M}_r\\textbf{t} \\]\n\n\\noindent\nThese bilinear formulations are combined with different forms of regularization to make different models. In DistMult authors considered a simpler approach where they reduced the number of parameters by imposing restrictions on $M_{r}$ to be a diagonal matrix. This results in a simpler model and this model enjoys the same scalable properties of TransE as well as it achieves better performance over TransE. Thus the final scoring function is given as \n\\[ f_{r}(h,t) = \\textbf{h}^T diag(\\textbf{r}) \\textbf{t} = \\sum_{i=0}^{d-1} [\\textbf{r}]_{i} .[\\textbf{h}]_i . [\\textbf{t}]_i \\]\n\n\\noindent\nwhere for each relation r, r $\\in \\mathbb{R}^d$ is a vector that depends on relationships. \n\n\\begin{figure}[H]\n\\includegraphics[width=.9\\linewidth]{DistMult.png}\n\\caption{ DistMult simple Illustration }\n\\label{fig: Graph}\n\\end{figure}\n\n\\noindent\nTime complexity and the space complexity of DistMult is more efficient as compared to RESCAL or TATEC. The time complexity of DistMult is $O(d)$ parameters per relation and the space complexity of $O(nd + md)$ where n is the number of entities and m is the number of relations. Due to its over-simplified mature of the model, this model is not enough powerful for the use in case of general Knowledge Graphs because it is only able to work efficiently  with symmetric relations.\n\n\\subsubsection{HolE \\cite{nickel2016holographic}}\nHolE stands for Holographic Embedding. HolE tried to overcome the problem of Tensor Product used in RESCAL by using circular correlation. Tensor product uses pairwise multiplicative interactions between feature vectors which results in increase in dimensionality of the representation i.e., $\\mathbb{R}^{d^2}$ thus increase the computational demand.\n\\[ \\textbf{h} \\circ \\textbf{t} = \\textbf{h} \\otimes \\textbf{t} \\in \\mathbb{R}^{d^2} \\]\nWhere \\textbf{a},\\textbf{b} $\\in \\mathbb{R}^d$ are entity embeddings. Tensor products are very rich in capturing the interactions but are computational intensive. On the other hand HolE  use Circular Correlation which can be seen as compression of the Tensor Product. The main advantage of Circular Correlation over Tensor Product is that it won't increase the dimensionality of the representation.\n\\[ \\textbf{h} \\circ \\textbf{t} = \\textbf{h} * \\textbf{t} \\in \\mathbb{R}^d \\]\nWhere * : $ \\mathbb{R}^d \\times \\mathbb{R}^d \\rightarrow \\mathbb{R}^d$ denotes the circular correlation. \n\\[ [\\textbf{h} * \\textbf{t}]_{i} = \\sum_{k=0}^{d-1} [\\textbf{h}_{k}] . \\textbf{t}_{(k+i)mod(d)}\\]\nThe final score of the fact in  HolE is given by matching the compositional vector ($\\textbf{h} * \\textbf{t}$) with the relational representation,i.e.,\n\n\\[ f_{r}(h,t) = \\textbf{r}^T  (\\textbf{h} * \\textbf{t} ) = \\sum_{i=0}^{d-1} [\\textbf{r}]_{i} \\sum_{k=0}^{d-1}[\\textbf{h}]_k . [\\textbf{t}]_{(k+i)mod(d)} \\]\n\n\\begin{figure}[H]\n\\includegraphics[width=.9\\linewidth]{HolE.png}\n\\caption{ HolE simple Illustration (It requires only d components) \\cite{nickel2016holographic} }\n\\label{fig: Graph}\n\\end{figure}\n\n\\noindent\nSo HolE is more efficient as compared to RESCAL or TransE. HolE take $O(d log d)$ parameters per relation and the space complexity of $O(nd + md)$ where n is the number of entities and m is the number of relations. Another advantage of HolE is that Circular Correlation is not commutative (  $\\textbf{h} * \\textbf{t} \\neq \\textbf{t} * \\textbf{h}$ ) thus HolE is able to model asymmetric relations (directed graphs) with compositional representations which is not possible in RESCAL.\n\n\\subsubsection{ComplEx\\cite{Trouillon2016}}\nKnowledge graphs represent the relation between entities. The entities may be termed as subjects and objects respectively of a given relation. However not all relations may be present in a given KG. One of the application of KG is the ability to predict missing relations or entities. \\\\\nDot product of vector embedding of KG triplets is being successfully used for symmetric, reflexive, anti-reflexive and even transitive relations \\citep{bouchard2015} however it can't be used for anti-symmetric relations. For example the relation capitalOf(New Delhi, India) is not symmetric since we cannot interchange subject and object entity in this relation therefore we need to have different embedding for an entity as subject and as object which increases the number of parameters.\\\\\nComplex embedding facilitates joint learning of subject and object entities while preserving the asymmetry of the relation. It uses Hermitian dot product of embedding of subject entities and object entities. The Eigen Vector decomposition is used to identify a low rank diagonal matrix W such that there exists X = Re($EWE^{T}$) such that X has same sign pattern as Y. The low rank diagonal matrix W is then used to predict missing relations by applying Re($e_{s}^{T}W\\bar{e}_{o}$).\n\\subsubsection{ANALOGY\\cite{Liu2017}}\nANALOGY is based on a multiplicative model where the relation (s,r,o) is scored by multiplying the vector representations of of subject (s), relation (r) and object (o). The relation present in the knowledge graph are expected to have higher score i.e. $\\phi(s,r,0)$ = $e_{s}^{T}W_{r}e_{o}$ will be high if the triplet (s,r,o) exists. \\\\\nAn example of analogy may be branch:tree::petal:leaf, in such an analogy, the relation $W_{r}$ \"is part of\" may be used to predict the missing entity from the analogy. The foundation of ANALOGY model is the linear maps of matrix representation of relations from the triplets present in the KG. Basically, the model uses the fact that there can be multiple paths to arrive at the entity from a given entity through a linear map of relations $W_{r1}, W_{r2}, W_{r3}$ and so on, and application of such relations in any sequence will give the same result. Hence such linear maps may be used to predict entities missing from the knowledge graph.\\\\\nFor example Let's consider the entities teacher (t), school (s), professor (p) and college (c). We have two relations in this set up: teachesAt (t,s) \\& teachesAt(p,c) and juniorOf(t,p) \\& juniorOf(s,c) therefore the relation between teacher and college is teachesAt*juniorOf = juniorOf*teachesAt. However such linear maps are feasible only if the commutative property holds for such relations. ANALOGY uses such linear maps to predict entities.\n\n\\subsection{Enrichment based embedding}\nIn the recent times, new emerging research areas are focusing on contextualized embedding. Under this, the entity under the consideration is enriched information from the neighbourhood information. A few notable approaches are Graph attention network (GAT) \\citep{Velickovic2017a} based information enrichment. Two methods based on the KGAT \\citep{Wei2019}, MMGAT \\citep{Sun2020} has proposed models to embed contextual information for an entity. Under MMGAT, they proposed model to combine the embedding from multi-modal data with an attention framework, as adopted from GAT's attention framework. Both the frameworks, were using translation model to learn the representation after the enrichment. The new emerging research areas are try to learn the structural information as well as path based information, multi-modal data. There are other research areas in the embedding are: \\textit{Text-enhanced embedding}, \\textit{Logic-enhanced embedding}, \\textit{Image-enhanced embedding} \\citep{Bianchi2020KnowledgeGE} etc. \n\n\\section{Applications of Knowledge graph embedding}\n\nThere are many applications of KG embedding learning methods. This section explores three of them namely- link prediction, triplet classification and recommender systems. \n\n\\noindent\nThe first two are In-KG applications, which are conducted within the scope of the KG. The last is an example of Out-of-KG applications that scale to broader domains \\citep{Wang2017}.\n\n\\subsection{Link Prediction}\n\nThe set of edges in a knowledge graph is a subset of \\textit{Entities$\\times$Relations$\\times$Entities}. The link prediction task focuses on finding an entity that can be represented as a fact (edge) together with a given relation and entity i.e., \\textit{(entity, relation, ?)} or \\textit{(?, relation, entity)} where ? refers to the missing entity. For e.g. \\textit{(New Delhi, isCapitalOf, ?)} or \\textit{(?, isCapitalOf, India)}. Link prediction is a way of Knowledge graph augmentation \\citep{paulheim2017knowledge}. It deduces missing information from the knowledge graph itself. \n\n\\noindent\nThe datasets for LP are constructed by sampling from the original knowledge graph. Then, the links removed can be used in validation set or the test set \\citep{bordes2013translating, dettmers2018convolutional}. The structure of such graphs play a vital role \nfor improving the results, multiple source entities making learning effective and multiple destination entities making learning difficult \\citep{Rossi_2021}.\n\n\\noindent\nThe LP models assigns a score to the triplet corresponding to each possible entity to fill the question mark (?). The triplets are then ranked by a function and entity corresponding to the lowest rank is predicted. If the predicted facts in the ranked predictions are already present in the Knowledge graph, they may or may not be excluded while calculating the ranks called raw and filtered rankings respectively \\citep{bordes2013translating}. For e.g. if the training knowledge graph contains the fact that \\textit{(Arjuna, isSonOf, Kunti)}, and the test query is \\textit{(?, isSonOf, Kunti)}. The target answer is \\textit{(Yudhishtra, isSonOf, Kunti)} and the system ranks  \\textit{(Arjuna, isSonOf, Kunti)} and then \\textit{(Yudhishtra, isSonOf, Kunti)}. The raw ranking of the triplet \\textit{(Yudhishtra, isSonOf, Kunti)} will be two and filtered ranking will be one. \n\n\\noindent\nThere are several tie breaking policies that are used by the ranking system. Assigning the \\textit{minimum} or the \\textit{\nmaximum} rank, or a \\textit{random} or the \\textit{average} rank to the targeted entity \\citep{Rossi_2021}.\n\n\\noindent\nThe ranks obtained are used to compute metrics such as \\textit{Mean Rank} (average of all the ranks), \\textit{Mean Reciprocal Rank} (average of the inverse of ranks), or \\textit{Hits@M} (proportion of ranks \\textit{$\\leq$ M}).\n\n\n\\subsection{Triple Classification}\nTriple Classification is the problem of identifying whether a given triple is correct. It aims to give a yes or no answer to questions such as \\textit{is New Delhi capital of India?} which can be written in the form of a triple \\textit{(New Delhi, isCapitalOf, India)} \\citep{socher2013reasoning}.\n\n\\noindent\nA scoring function is used to calculate score of a triple similar to the link prediction. If the score is greater than a certain threshold, then it is considered a fact else a wrong triple \\citep{Wang2017}. \n\n\\noindent\nBoth the classical methods, such as \\textit{micro} and \\textit{macro averaging}, and ranking methods such as \\textit{Mean Rank} are used as evaluation metrics \\citep{guo2016jointly}. \n\n\\subsection{Recommender Systems}\nRecommender system (RS) assists the user in an environment where multiple options are available by providing a certain ordering of choices that the recommendation algorithm infers. This inference can be based on the similarity of the choices and behaviour pattern of different users. This type of recommendation methods falls into the domain of collaborative filtering methods \\citep{adomavicius2005toward}. \n\n\\noindent\nThe CF methods suffer from problems of Data sparsity and cold start. Data sparsity arises from the fact that only a small proportion of items are rated by the users and most options have only limited feedback from the users. Cold start problem is the problem of having no historical data about the new users and options. To deal with these problems different types of side information about a user and item are utilized by the RS \\citep{Sun_2019}.\n\n\\noindent\nKG is utilised for side information in CF. It acts as a heterogeneous graph that represent entities as nodes and relation as edges. The KG connects various entities via latent relationships and also provide explainability in recommendations \\citep{Wang_2018}.\n\n\\noindent\nThe KG embedding based methods for RS use two modules - Graph embedding and Recommendation Module. The way that these modules are coupled lead to categorization of embedding based methods in a). two stage learning methods, b). joint learning method and c). multi task learning method\\citep{Guo_2020}. \n\n\\noindent\nTwo stage learning methods first uses graph embedding module to obtain the embeddings using various KG algorithms and then use recommendation module to infer. The advantages of this method lies in its simplicity and scalability but since the two modules are loosely coupled the embeddings might not be suitable for recommendation tasks.\n\n\\noindent\nJoint learning methods train both the modules in an end to end fashion. Thus, recommendation module guides the training in graph embedding layer. \n\n\\noindent\nMulti task learning method train the recommendation module with the guidance of KG related task such as KG completion. The primary intuition behind this method is that the bipartite graph of user and item in recommendation task share structures with the corresponding KG entities.\n\n\\section{Summary}\nKnowledge graphs provide an effective way of presenting real-world relationships. As a result Knowledge graphs have an inherent advantage w.r.t serving the information need. KG in itself is a growing area of research. KG embedding is a technique to represent all the components of the KG in vector form. These vectors represent the latent properties of the components of the graph. Various models for embedding methods are based on different combinations of vector algebra which present an interesting area of research. In this work, we have surveyed the embedding methods that started this active area of research, state-of-art models and the new frontiers which are being explored in the KG embedding. \\\\\nThe KG embedding methods progressed from translation-based models which are based on vector addition. In this work, we have presented how translation-based models improved over time to overcome shortcomings of the earlier models. While translation-based models used vector addition, semantic models can be clubbed together as multiplicative models. We have included the transition from basic semantic models to the more advanced semantic models which may be used to explain different types of real world relationships such as symmetric, anti-symmetric,inverse or composition.\\\\\nNew research areas have broaden the scope from structural embedding to more contextual embedding by encoding additional information in the learned representation. The latest area of research in this field is enrichment based embedding models. In this work, we have introduced those briefly. \\\\\nVector space representation has paved a way to use the information from Knowledge graph directly into the real world application. In this work, we have described a few real world applications of KG embedding such as link prediction, triple classification and recommender systems.\n\n\n\n\\section*{Acknowledgements}\nThe preferred spelling of the word `acknowledgment\" in America is without an ``e\" after the ``g\". Avoid the stilted expression, ``One of us (R. B. G.) thanks . . .\" Instead, try ``R. B. G. thanks\". Put sponsor acknowledgments in the unnum-bered footnote on the first page.\n\n\n\n\\section{Introduction}\nConversational systems have acquired the center stage in NLP research. Compared to the conventional information retrieval task where we have to extract the passage or document from a vast collection of documents, the Conversational system requires extracting related information to respond to a series of questions. The turns in the conversation may follow the previous question. Complexity in this task arises due to the way we form the queries, which often have a reference to previous information using pronouns, co-reference. The presence of pronouns and unresolved co-references induces ambiguity in the query. Resolving the contextual dependency is one of the most challenging tasks in the Conversational system. \n\nThe Conversational assistance track (CAsT) has started in 2019. The long-term vision of the CAsT is ``to support natural conversations between a person and a search engine to satisfy information needs and support complex information tasks''. The task in both years of CAsT remained the same, to retrieve relevant passages depending upon the context evolution from the subsequent queries. \nraw\\_utterance\nCAsT-2019 had two tacks viz: automatic and manual track. Under Automatic track, response extraction was based on the raw utterances while the manual track has queries rewritten by humans. Co-reference and pronouns were resolved in the manually rewritten queries based on the historical context. Manually rewritten queries contain all of the information required to represent the single turn of the underlying information need. In CAsT-2019, additional information like description and title for the session was also provided \\citep{Dalton2020}. Three corpora, namely MSMARCO, TREC CAR (Wikipedia) paragraph Corpus V2.0\\footnote{http:\/\/trec-car.cs.unh.edu\/datareleases\/} were used. Initially, it had also considered WaPo, but it was later dropped. CAsT-2020 had some changes, title and description was \\textbf{removed from the query} info, and document id as a canonical response to the query is added in the task. CAsT-2020 had three tracks - automatic, automatic-canonical, and manual. Under automatic, only raw utterance is to be used, while automatic-canonical can use the provided canonical response. The manual track was similar to last year based on the manually rewritten queries \\citep{Dalton2021}. The dataset was MSMARCO and CAR \\citep{Dietz2018TRECCA}. CAsT-2021 has three tracks similar to CAsT-2020, but with a modification that we need to extract a specific passage out of the extracted document. The idea of specific passage to be extracted from the document centered around the idea that responses should be crisp and could be used by automatic voice assistants like Alexa, Google, etc. CAsT-2021 used MSMARCO (2019\/20 dump), WAPO-2020 and KILT \\citep{petroni-etal-2021-kilt}.  \n\n\\begin{table}[]\n    \\centering\n    \\begin{tabular}{|l|p{0.8\\linewidth}|}\n       \n       \n       \n       \n         \\hline\n         Turn & Queries  \\\\\n         \\hline\n         1 & I just had a breast biopsy for cancer. What are the most common types? \\\\\n         \\hline\n         2 & Once it breaks out, how likely is it to spread?\\\\\n         \\hline\n        3 &\tHow deadly is it?\\\\\n        \\hline\n        4 &\tWhat? No, I want to know about the deadliness of lobular carcinoma in situ. \\\\\n        \\hline\n        5 & Wow, that's better than I thought.  What are common treatments? \\\\\n        \\hline\n        6 & ... \\\\\n        \\hline\n       \n       \n       \n       \n       \n       \n       \n       \n       \n       \n    \\end{tabular}\n    \\caption{A sample query from CAsT-2021 Automatic Evaluation Topics }\n    \\label{tab:my_label}\n\\end{table}\n\n\\section{Problem Description}\nIn CAsT track, A conversation session $S$ has a series of utterances $\\{u_1, u_2, u_3, u_4\\dots\\}$ called as turns. Our task is to predict a set of top-K passages from the collection for each turn. \n\n\\textbf{Tracks:} There are three tracks of submission for CAsT-2021 - Automatic, Automatic-canonical, and Manual. Manual track has rewritten queries as input.\n\n\\textbf{Dataset}: CAsT-2021 uses two dataset corpus for the task - MSMARCO, KILT \\citep{petroni-etal-2021-kilt}  and WAPO. \n\nWe have attempted the Automatic track. Each turn (Queries) in this session is raw in nature. The onus to resolve references and pronouns lies on the model itself. \n\n\\section{Motivation and Method}\nWe have gone through the CAsT-2020 submissions, and most of the models were multistage models. In the initial stage, a set of documents were retrieved from the collection, followed by re-ranking using neural model. Most of the models used query rewriting using generative model based on Transformer architecture \\citep{DBLP:journals\/corr\/VaswaniSPUJGKP17}. For query rewriting, T5, BART, GPT-2 was used, and the re-ranker engine was based on BERT, ALBERT, and T5.  \n\nOur retrieval framework has three components: Query rewrite, document retrieval with pseudo-relevance feedback, and neural engine-based re-ranker. The Query rewrite framework is based on chatty-goose \\footnote{https:\/\/github.com\/castorini\/chatty-goose} and re-ranker is based on pygaggle \\footnote{https:\/\/github.com\/castorini\/pygaggle}. Core retrieval engine is based on the HQE \\citep{Yang2019QueryAA} and PQE  \\citep{Al-Thani2021}\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=\\linewidth] {trec.jpg}\n    \\caption{Multi-stage retrieval pipeline}\n    \\label{fig:my_label}\n\\end{figure}\n\n\\subsection{Historical Query expansion (\\textbf{HQE})}\nThis step is based on the submission from CAsT-2019 \\citep{Yang2019QueryAA}. It is a three-stage algorithm. In the first step, it extracts the keyword for the session and query level followed by measurement of ambiguity in query in second step, and in the last stage, query expansion with the session keyword and query level keyword is done. \n\nA keyword is deemed important if it strongly relates with the documents in the index. HQE uses BM25 score between a keyword and its highest-ranked document in the index as its measure of importance. HQE computes this measure for each token in every utterance when presented with a topic. The most informative ones are then selected for query expansion.\nFor a given topic, keywords can be locally important (i.e., it strongly relates to the topic being discussed in the current utterance) or globally important (i.e., it relates to the overall conversation theme). HQE thus creates two sets of expansion keywords, session and query, during the extraction phase.\nWhether the keyword is considered a session keyword or a query keyword is decided by two separate cutoffs, $Q_s$, and $Q_t$. If the importance score of a keyword is greater than either of the cutoff scores, it is added to corresponding expansion sets. Note that session keywords are always used for expansion, while query keywords are only used when current utterance is identified as ambiguous. An utterance is deemed ambiguous when BM25 score between it and its highest ranked document in index is low (i.e. by itself utterance is not important). If the ambiguity score is less than a certain threshold $\\theta$, then the query expansion will take place with query keywords.\n\nThe cutoff values for $Q_s$, $Q_t$, and $\\theta$ determine the performance of HQE. We follow a greedy approach to find the optimal values of these cutoffs. Note that a query is always expanded with current session keywords and occasionally with preceding query keywords. Thus, we start by tuning $Q_s$ session cutoff. We set $Q_t$ and $\\theta$ to arbitrarily high values allowing query expansion only via session keywords. Then we do a line search for $Q_s$ over the training set. Second, we set $Q_t$ to some fixed value ($> Q_s$) and in similar fashion tune $\\theta$. Finally, we tune $Q_s$. \n\n\\begin{table*}[]\n    \\centering\n    \\begin{tabular}{p{0.4\\linewidth}|c|c|c|c|c|c}\n        \\hline\n        Run Name & $K1$ & $b$ & NDCG@3 & NDCG@5 & P@5 & AP@500\\\\\n        \\hline\n        IITD-RAW\\_U\\_T5\\_1 & $0.9$ & $0.4$ & 0.3712 & 0.3631 & 0.5025 & 0.1759\\\\\n        \\hline\n        IITD-RAW\\_U\\_T5\\_2 & $1.2$ & $0.75$ & 0.3801 & 0.3731 & 0.5203  & 0.1874 \\\\\n         \\hline\n    \\end{tabular}\n    \\caption{Evaluation score of Submitted manual runs}\n    \\label{tab:my_label_2}\n\\end{table*}\n\n\n\\subsection{Passage Query Expansion (PQE)}\nPseudo-relevance feedback enriches the query by incorporating features from top-k relevant documents. PQE uses a pseudo-relevance feedback mechanism as follows:\n\\begin{itemize}\n\n\\item The expanded query is from HQE is used to fetch an initial set of top-k documents. These documents combined together form the corpus of responses for the query.\n\\item Individual tokens in the corpus are scored using TF-IDF. An IDF vector is pre-computed on complete MSMARCO documents.\n\\item The topmost unique tokens are then considered for query expansion.\n\\end{itemize}\nNote that PQE can be computationally prohibitive since it involves complete document retrieval. In order to avoid this, \\cite{Al-Thani2021} proposes to use a simple rule to decide whether a query is to be expanded using PQE or not. \\cite{Al-Thani2021} only perform PQE expansion when the query has at least one pronoun.\n\nThis process has two HyperParameters, top-k documents, and top-k tokens from the document based on the TF-IDF score. The TF-IDF score helps the system to select the important terms. While selecting the term, we have also placed a criterion that it should have a DF between 0.001 and 0.2. In the PQE step, we noticed that some digits also appeared in top-k terms. Later, we filter out those instances. \n\n\n\n\n\n\\subsection{T5 query rewriter and reranker}\nIn automatic track, query are presented in the bare format. In order to preserve the context of the query, we have used query rewriter. It uses T5 based model \\footnote{https:\/\/huggingface.co\/castorini\/t5-base-canard}. The model is trained on CANARD dataset \\citep{Elgohary:Peskov:Boyd-Graber-2019} that contains a set of rewritten queries based upon the history. To produce a $n^{th}$ rewitten utterance, we have supplied history of turns $\\{u_1, u_2, u_3, u_4\\dots, u_{n-1}\\}$ concatenated with $u_n$. Reformulated queries were used for the final stage ranking of passages. \n\nOn the other hand T5 based ranking model \\citep{DBLP:journals\/corr\/NguyenRSGTMD16} is trained on the MSMARCO, Robust04, Core17 and Core18. This model takes query $Q$ and a list of passages $\\{p_1, p_2, p_3, .. p_n\\}$. It returns submitted passages according to the descending order of their relevance.  \n\n\\subsection{Document Indexing}\nAll three datasets were pre-processed and converted into jsonl format. We have use Pyserini\\footnote{https:\/\/github.com\/castorini\/pyserini} to generate an index for faster retrieval of documents. Index was generated with an option to keep a copy of raw documents. We choose this option because the raw content was required at HQE and PQE stage. \n\n\n\\section{Evaluation Matrix}\nCAsT-2021 used following matrix to present the results of the participants NDCG@3, NDCG@5, NDCG@500 and AP@500.\n\n\\section{Results and Discussion}\n\nIndex for the cleaned document was generated using Pyserini\\footnote{https:\/\/github.com\/castorini\/pyserini} with the default setting. Default setting did not restrict us to keeping $K1$ and $b$ fixed. It can be changed during the retrieval of the documents. First, we performed our extraction and tuning on the 2019 training set. The best performance for HQE was obtained with $Q_s$ = 4, $Q_t$ = 4, $\\theta$ = 10 and PQE with top-k = 5, top-k token = 3.  \n\nWe have participated in automatic track and submitted two runs in the CAsT-2021. Two different set of parameters for BM25, $K1 = 0.9$, $b = 0.4$ and $K1 = 1.2$, $b = 0.75$ was selected for the retrieval of the document. The MAP is very low in both the results, and the main issue lies with the recall. Our retrieval engine extracted the top 100 documents from the corpus. Extracted documents were later chunked and re-ranked. In this process, lower retrieval numbers left the ranker with fewer relevant chunks, resulting in a lower than expected performance. \n\nIITD-RAW\\_U\\_T5\\_2 has produced a mean NDCG@3 performance better than the median model. Model's performance on evaluation query has a noticeable variation in scores compared with median scores. Results for queries like -- 115 and 119 were better than the median; however, results on the evaluation queries like -- 111 and 117 were worse than the median benchmark for NDCG@3. Analysis of query expansion term for the first stage retrieval suggests that query expansion term has carried the intent of the conversation to higher depths. While the poor performing queries have expansion terms with less relevant keywords, generic keywords. Query expansion terms for query 119 have terms like \\textit{swelling, shaking, infection, ear} that carried the conversation context to higher depth led to better performance. \n\nWe have also analyzed the hqe and pqe keywords for query expansion. We can take evaluation number 106; the first few queries are listed in Table-\\ref{tab:my_label}. For turn 3, query expansion terms were \\textit{biopsy, breast, deadly, cancer, comment and cell}. This turn has only one term, ``deadly'', that can explain the intent of the query. But, it is too generic in nature which has led to poor recall. On the flip side, turn five is expanded with terms \\textit{carcinoma, wow, deadly, situ, lobular, deadliness, biopsy, breast} which has specific terms related to cancer like ``lobular, situ, carcinoma'' has a good recall. Here the word ``specific'' means relevance to the topic. \n\nOur submission to CAsT-2021 aimed to preserve the key terms and the context in all subsequent turns and use classical Information retrieval methods. It was aimed to pull as relevant documents as possible from the corpus. It appears that it can retain some of the keywords in subsequent turns. But, it fails when the key term itself is generic. The performance of this model can be improved further by including the context word vectors in the HQE and PQE stages. Context vector-based selection may help to keep top-k terms that are relevant to the conversation theme. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWe consider a polynomial \nof complex variables $\\zz = (z_1, \\dots, z_n)$\nwhich is given by \n\\[\nP(\\zz, \\bar{\\zz}) := \\sum_{i = 1}^{m} c_{i}\\zz^{\\nu_{i}}\\bar{\\zz}^{\\mu_{i}}, \n\\]\nwhere $c_{i} \\in \\Bbb{C}^*$, \n$\\zz^{\\nu_{i}} = z^{\\nu_{i,1}}_1 \\cdots z^{\\nu_{i,n}}_n$ and \n$\\bar{\\zz}^{\\mu_{i}} = \\bar{z}_{1}^{\\mu_{i,1}} \\cdots \\bar{z}_{n}^{\\mu_{i,n}}$ \nfor $\\nu_{i} = (\\nu_{i,1}, \\dots, \\nu_{i,n})$ and \n$\\mu_{i} = (\\mu_{i,1}, \\dots, \\mu_{i,n})$ respectively.  \nHere $\\bar{\\zz}^{\\mu_{i}}$ \nrepresents the complex conjugate of $\\zz^{\\mu_{i}} = z_{1}^{\\mu_{i,1}} \\cdots z_{n}^{\\mu_{i,n}}$. \nWe call $P(\\zz, \\bar{\\zz})$ \\textit{a mixed polynomial} of complex variables $\\zz = (z_1, \\dots, z_n)$. \nA point $\\ww \\in \\Bbb{C}^{n}$ is called \\textit{a mixed singular point of $P(\\zz, \\bar{\\zz})$} \nif~the gradient vectors of $\\Re P$ and $\\Im P$ are linearly dependent at~$\\ww$ \\cite{O1, O2}. \nSuppose that $P(0,\\dots, 0) = 0$ and $P$ has an isolated singularity at the origin. \nThere exist positive real numbers $\\varepsilon$ and $\\delta$ with $\\delta \\ll \\varepsilon \\ll 1$ such that \nthe map \n\\[\nP : D^{2n}_{\\varepsilon} \\cap P^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta} \n\\]\nis a locally trivial fibration over $\\partial D^{2}_{\\delta}$, where \n$D^{2n}_{\\varepsilon} = \\{ \\zz \\in \\Bbb{C}^{n} \\mid \\| \\zz \\| \\leq \\varepsilon \\}$ and \n$D^{2}_{\\delta} = \\{ \\eta \\in \\Bbb{C} \\mid \\lvert \\eta \\rvert \\leq \\delta \\}$.  \nThis map is called \\textit{the Milnor fibration of $P$ at the origin}. \n\nRuas, Seade and Verjovsky \\cite{RSV} and Cisneros-Molina \\cite{C} introduced \nthe following classes of mixed polynomials. \nLet $p_{1}, \\dots, p_{n}$ and $q_{1}, \\dots, q_{n}$ be integers such that \n$\\gcd(p_1, \\dots, p_n) = \\gcd(q_1,\\dots,q_n) = 1$. \nWe define the~$S^1$-action and the~$\\Bbb{R}^{*}$-action on~$\\Bbb{C}^{n}$ as follows: \n\\begin{equation*}\n\\begin{split}\ns \\circ \\zz = (s^{p_{1}}z_{1}, \\dots, s^{p_{n}}z_{n}), \\ \\ \\ \nr \\circ \\zz = (r^{q_1}z_{1}, \\dots, r^{q_n}z_{n}), \n\\end{split}\n\\end{equation*}\nwhere $s \\in S^{1}$ and $r \\in \\Bbb{R}^{*}$. \nA mixed polynomial $P(\\zz, \\bar{\\zz})$ is called \\textit{a polar weighted homogeneous polynomial} \nif there exists a positive integer $d_p$ such that $P(\\zz, \\bar{\\zz})$ satisfies \n\\[\nP(s^{p_{1}}z_{1}, \\dots, s^{p_{n}}z_{n}, \\bar{s}^{p_1}\\bar{z}_{1}, \\dots, \\bar{s}^{p_1}\\bar{z}_{n})\n = s^{d_p}P(\\zz, \\bar{\\zz}), \\ \\ s \\in S^{1}.  \\notag \\\\\n\\]\nThe weight vector $(p_{1}, \\dots, p_{n})$ is called \\textit{the polar weight} and $d_p$ is called \n\\textit{the polar degree}. \nSimilarly $P(\\zz, \\bar{\\zz})$ is called \\textit{a~radial weighted homogeneous polynomial} \nif there exists a positive integer $d_{r}$ such that \n\\[\nP(r^{q_1}z_{1}, \\dots, r^{q_n}z_{n}, r^{q_1}\\bar{z}_{1}, \\dots, r^{q_n}\\bar{z}_{n}) \n= r^{d_{r}}P(\\zz, \\bar{\\zz}), \\ \\ r \\in \\Bbb{R}^{*}. \n\\]\nThe integer $d_r$ is called \\textit{the radial degree}. \nIf $P$ is  polar and radial weighted homogeneous,  \n$P$ admits the global Milnor fibration $P : \\Bbb{C}^{n} \\setminus P^{-1}(0) \\rightarrow \\Bbb{C}^{*}$ \nand the monodromy of the Milnor fibration is given by the $S^1$-action, see for instance \\cite{RSV, C, O1, O2}. \n\n\nWe study the topology of the Milnor fibration of a mixed polynomial $P$ by using a deformation of $P$. \nHere \\textit{a deformation of $P$} is a~polynomial map \n$F : \\Bbb{C}^{n} \\times \\Bbb{R} \\rightarrow \\Bbb{C}, (\\zz, t) \\mapsto F_{t}(\\zz)$, with $F_{0}(\\zz) = P(\\zz, \\bar{\\zz})$. \nA deformation of $P$ is useful to study the Milnor fibration of $P$. \nFor complex isolated singularities, \nit is known that there exist a~neighborhood $U$ of the origin of $\\Bbb{C}^{n}$ and \na~deformation $F_{t}$ of a complex polynomial $P(\\zz)$ \nsuch that $F_{t}(\\zz)$ is also a~complex polynomial and any singularity of $F_{t}(\\zz)$ in $U$ \nis a Morse singularity for any $0 < t \\ll 1$, see for instance \\cite[Chapter 4]{Lg}. \nA sufficiently small compact neighborhood of each Morse singularity can be regarded as a $2n$-dimensional $n$-handle and \nthus we have a decomposition \n\\[\nD^{2n}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta}) \\cong \n\\bigr(D^{2n}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}})\\bigl) \\cup_{\\varphi} \\bigr(\\sqcup_{i=1}^{\\ell} (\\text{$n$-handle})_{i}\\bigl), \n\\]\nwhere $\\ell$ is the Milnor number of the singularity of $P$ at $(0, \\dots, 0)$, \n$\\varphi = (\\varphi_{1}, \\dots \\varphi_{\\ell})$ is the $\\ell$-tuple of the attaching map $\\varphi_{i}$ of \n$(n\\text{-handle})_i$ \nand $D^{2}_{\\delta_{t}}$ is a $2$-disk centered at $0$ with radius $\\delta_{t}$ \nsuch that $\\delta_{t} < \\delta$ and $F_{t} \\mid_{F_{t}^{-1}(D^{2}_{\\delta_{t}})}$ has no singularities. \nNote that the framing of each handle attaching is determined by the vanishing cycle of the corresponding Morse singularity \\cite{K}. \nSuch a decomposition induces a decomposition of the monodromy of the Milnor fibration into those of \nthe Morse singularities. \n\nIn this paper, we observe analogous deformations for mixed singularities. \nLet $P$ be a $2$-variable polar and radial weighted homogeneous \npolynomial which has an isolated singularity at the origin of $\\Bbb{C}^{2}$ \nand let $F_t$ be a deformation of $P$. \nSet $S_{k}(F_{t}) = \\{ \\zz \\in U \\mid \\text{rank}\\>dF_{t}(\\zz) = 2-k \\}$ for $k = 1, 2$. \nWe assume that $F_t$ satisfies the following properties: \n\\begin{enumerate}\n\\item\n$F_t$ is polar weighted homogeneous for any $0 \\leq t \\ll 1$, which implies that, for each $0 < t \\ll 1$, \nthe singular set $S_{1}(F_{t}) \\cup S_{2}(F_{t})$ \nconsists of the union of a finite number of orbits of the $S^1$-action \\cite[Proposition 2]{In3} and \n$F_{t}(S_{1}(F_{t}))$ consists of circles centered at $0$ except the origin; \n\\item\nFor each point $\\ww \\in S_{1}(F_{t})$, \nthere exist local coordinates $(x_{1}, x_{2}, x_{3}, x_{4})$ centered at $\\ww$  \nsuch that $F_{t}$ is given by \n\\[\n(F_{t}\/\\lvert F_{t} \\rvert, \\lvert F_{t} \\rvert) = \\bigr(x_{1}, -x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+c_{t, \\ww}\\bigl), \n\\]\nwhere $c_{t,\\ww} = \\lvert F_{t}(\\ww)\\rvert$ for $\\ww \\in S_{1}(F_{t})$ and $0 < t \\ll 1$. \nIn particular, $S_{1}(F_{t})$ consists of indefinite fold singularities; \n\\item\n$S_{2}(F_{t}) = \\{ \\mathbf{o}\\}$ or $\\emptyset$. \n\\end{enumerate}\nIn \\cite{In3}, we focused on the mixed singularity of $f\\bar{g}$, \nwhere $f$ and $g$ are $2$-variable weighted homogeneous complex polynomials which have no common branches, \nand showed that there exists a deformation $F_t$ of $f\\bar{g}$ \nsuch that $F_t$ satisfies the above conditions. \nIt is important to notice that there exist polar weighted homogeneous polynomials which do not admit \ndeformations into smooth maps with only Morse singularities, see \\cite[Theorem 1]{In1}, \\cite[Corollary 1 and Corollary 2]{In2}. \n\n\nBy the condition $(2)$, the absolute value $\\lvert F_{t}\\rvert$ of $F_t$ defines a Morse function on \nthe orbit space $(D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta}))\/S^{1}$ of the $S^1$-action for $t>0$ and \noutside the image of the origin, and the indices of the Morse singularities are always $1$. \nThen the handle decomposition of the orbit space according to this Morse function induces a decomposition \nof $D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta})$ into a tubular neighborhood of a singular fiber over \nthe origin and a finite number of round $1$-handles, that is, we have \n\\begin{equation}\n\\label{0.1}\n\\tag{0.1}\nD^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta}) \\cong \n\\bigr(D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}})\\bigl) \\cup_{\\varphi} \n\\bigr(\\sqcup_{i=1}^{\\ell} (\\text{round $1$-handle})_{i}\\bigl) \n\\end{equation}\nwhere $\\ell$ is the number of the singularities of the Morse function on the orbit space outside the origin \nand $\\varphi = (\\varphi_{1}, \\dots \\varphi_{\\ell})$ is the attaching map of $\\ell$ copies of a \nround $1$-handle. \nHere we may assume that the images of $\\varphi_{1}, \\dots, \\varphi_{\\ell}$ in \n$\\partial(D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}}))$ are disjoint. \n\nIn this paper, we describe the structure of this decomposition more precisely. To state our theorem, \nwe introduce the notion of negative link components. \nLet $P$ be a polar weighted homogeneous polynomial. Then the link of $P$ at the origin is a Seifert link, \ndenoted by $L(P, \\mathbf{o})$. \nA fiber surface of the Seifert link induces an orientation to each link component canonically. \nA connected component of $L(P, \\mathbf{o})$ is called a \\textit{positive component} if the orientation of the link \ncomponent coincides with that of the $S^1$-action, and otherwise it is called a \\textit{negative component}. \nLet $\\lvert L^{+}(P, \\mathbf{o}) \\rvert$ and $\\lvert L^{-}(P, \\mathbf{o}) \\rvert$ \ndenote the number of positive components of $L(P, \\mathbf{o})$ and \nthe number of negative components of $L(P, \\mathbf{o})$, respectively. \nThen the decomposition is given as follows: \n\n\n\n\n\\begin{theorem}\\label{rthm}\nLet $P$ be a $2$-variable polar and radial weighted homogeneous \npolynomial which has an isolated singularity at the origin and \nlet $F_t$ be a deformation of $P$ satisfying the conditions $(1), (2)$ and $(3)$. \nThen \n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\begin{enumerate}\n\\item\n$D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}})$ is diffeomorphic to \nthe disjoint union of a $4$-ball and $\\ell$ copies of $S^{1} \\times B^{3}$, \nwhere $B^3$ is a $3$-ball, and each $\\varphi_{i}$ of the attaching map \n$\\varphi = (\\varphi_{1}, \\dots, \\varphi_{\\ell})$ \nmaps the two attaching regions of the $i$-th round $1$-handle to \nboth of the boundary of the $4$-ball and that of the $i$-th $S^1 \\times B^3$, \nand these $\\ell+1$ connected components are connected by $\\ell$ round $1$-handles attached by the map $\\varphi$; and\n\\item\nthe number $\\ell$ of round $1$-handles in the decomposition $\\eqref{0.1}$ is given as \n\\[\n\\ell = \\lvert L^{+}(P, \\mathbf{o})\\rvert - \\lvert L^{+}(F_{t}, \\mathbf{o})\\rvert = \\lvert L^{-}(P, \\mathbf{o})\\rvert - \\lvert L^{-}(F_{t}, \\mathbf{o})\\rvert \n\\]\nfor $0 < t \\ll 1$. \n\\end{enumerate}\n\\end{theorem}\nAs we mentioned, in \\cite{In3}, a deformation of a mixed singularity of type $(f\\bar{g}, \\mathbf{o})$ is given explicitly. \nIn that case, the number $\\ell$ is determined by the polar degree and the radial degree of $P$ as \n$\\ell = \\frac{d_{r} - d_{p}}{2pq}$. \nFrom the decomposition in $\\eqref{0.1}$, we can observe that the Milnor fiber of $(P, \\mathbf{o})$ is obtained from \nthe Milnor fiber of $(F_{t}, \\mathbf{o})$ by removing $2d_{p}$ disks from \ntwo connected components of $D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}})$ \nand gluing these boundary circles by $d_p$ annuli for each $i = 1, \\dots \\ell$. \nMoreover, we see that monodromy exchanges these $\\ell$ copies of the union of $d_p$ annuli \nand $2d_{p}$ disks cyclically. \n\nThis paper is organized as follows. In Section $2$ \nwe give the definitions of fold singularities and round handles and \nintroduce deformations of polar weighted homogeneous polynomials. \nIn Section $3$ we prove Theorem $1$. In Section $4$ we make a few comments on the monodromy of the Milnor fibration of $F_t$ \nand a specific deformation of $f\\bar{g}$. \n\nThe author would like to thank Professor Masaharu Ishikawa for precious comments. \n\n\n\\section{Preliminaries}\n\\subsection{Fold singularities}\nLet $X$ be a $n$-dimensional manifold and $W$ be a $2$-dimensional manifold. \nWe denote $C^{\\infty}(X, W)$ the set of smooth maps from $X$ to $W$. \nIt is known that the subset of smooth maps from $X$ to $W$ whose singularities are \nonly definite fold singularities, indefinite fold singularities or cusps \nis open and dense in $C^{\\infty}(X, W)$ topologized with \nthe $C^{\\infty}$-topology \\cite{L1}. \nHere \\textit{a fold singularity} is the singularity of the following map \n\\[\n(x_{1}, \\dots, x_{n}) \\mapsto (x_{1}, \\sum_{j=2}^{n}\\pm x_{j}^{2}), \n\\] \nwhere $(x_{1}, \\dots, x_{n})$ are coordinates of a small neighborhood of the singularity. \nIf the coefficients of $x_j$ for $j = 2, \\dots, n$ is \neither all positive or all negative, we say that \n$x$ is a \\textit{definite fold} singularity, \notherwise it is an \\textit{indefinite fold} singularity. \n\n\n\n\n\\subsection{Deformations of polar weighted homogeneous polynomials}\nLet $P : \\Bbb{C}^{2} \\rightarrow \\Bbb{C}$ be a polar weighted homogeneous polynomial map which has an isolated singularity at the origin of $\\Bbb{C}^2$. \nThen $P$ admits a Milnor fibration, i.e., there exist positive real numbers $\\varepsilon$ and $\\delta$ such that \nthe map \n\\[\nP : D^{4}_{\\varepsilon} \\cap P^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta} \n\\]\nis a locally trivial fibration over $\\partial D^{2}_{\\delta}$, where \n$D^{4}_{\\varepsilon} = \\{ \\zz \\in \\Bbb{C}^{2} \\mid \\| \\zz \\| \\leq \\varepsilon \\}$ and \n$D^{2}_{\\delta} = \\{ \\eta \\in \\Bbb{C} \\mid \\lvert \\eta \\rvert \\leq \\delta \\}$. \nWe fix such  positive real numbers $\\varepsilon$ and $\\delta$. \nLet $F$ be the fiber surface of a polar weighted homogeneous polynomial $P$. \nIn \\cite{O1, O2}, the monodromy $h : F \\rightarrow F$ of the Milnor fibration of $P$ is given by \n\\[\n(z_{1}, z_{2}) \\mapsto \\Bigr(\\exp\\Bigr(\\frac{2p\\pi i}{d_p}\\Bigl)z_{1}, \\exp\\Bigr(\\frac{2q\\pi i}{d_p}\\Bigl)z_{2}\\Bigl), \n\\] \nwhere $(p,q)$ is the polar weight of $P$. \nNote that the link $K_{P} = \\partial D^{4}_{\\varepsilon} \\cap P^{-1}(0)$ is an invariant set of the $S^1$-action. \nSo the link $K_{P}$ is a Seifert link in the $3$-sphere \\cite{EN}. \n\nLet $F_t$ be a deformation of $P$ which satisfies the conditions $(1), (2)$ and $(3)$. \nSince the fiber surface $F_{0}^{-1}(c)$ intersects $\\partial D^{4}_{\\varepsilon}$ transversely, \n$F_{t}^{-1}(c)$ intersects $\\partial D^{4}_{\\varepsilon}$ transversely for each $c \\in D^{2}_{\\delta}$ \nand $0 \\leq t \\ll 1$. \nBy the Ehresmann's fibration theorem \\cite{W}, the map \n\\[\nF_{t} : D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}\n\\] \nis a locally trivial fibration for $0 \\leq t \\ll 1$. \nThe polar weight of $F_t$ coincides with that of $F_{0}$ for $0 < t \\ll 1$. \nThus the monodromy of $F_{t} : D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$ \nis given by the same $S^1$-action on $\\Bbb{C}^{2}$ for each $0 \\leq t \\ll 1$. \n\n\\begin{lemma}\nThe Milnor fibration \n$F_{t} : D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$ \nis isomorphic to the fibration \n$F_{0}\/\\lvert F_{0} \\rvert : \\partial D^{4}_{\\varepsilon} \\setminus \\emph{Int}N(K_{F_{0}}) \\rightarrow S^{1}$ \nfor $0 \\leq t \\ll 1$, \nwhere $N(K_{F_{0}}) = \\{\\zz \\in \\partial D^{4}_{\\varepsilon} \\mid \\lvert F_{0}(\\zz)\\rvert \\leq \\delta\\}$. \n\\end{lemma}\n\\begin{proof}\nSince the fiber surface $F_{0}^{-1}(c)$ is transversal to $\\partial D^{4}_{\\varepsilon}$, \n$F_{t}^{-1}(c)$ is transversal to $\\partial D^{4}_{\\varepsilon}$ for any $c \\in D^{2}_{\\delta}$ \nand $0 \\leq t \\ll 1$. \nFix such a positive real number $t$. \nWe set \n\\[\\begin{split}\n\\partial \\mathcal{E}(\\delta, \\varepsilon)& := \\{(\\zz, t') \\in \\Bbb{C}^{2} \\times [0, t] \\mid \\lvert F_{t'}(\\zz)\\rvert = \\delta, \\| \\zz \\| \\leq \\varepsilon \\}\\\\\n\\partial ^2\\mathcal{E}(\\delta, \\varepsilon)& := \\{(\\zz, t') \\in \\Bbb{C}^{2}\\times [0, t] \\mid \\lvert F_{t'}(\\zz)\\rvert = \\delta, \\| \\zz \\| = \\varepsilon \\}.\n\\end{split}\n\\]\nThen the projection \n\\[\n\\pi' :( \\partial \\mathcal{E}(\\delta, \\varepsilon),\\partial^2\\mathcal{E}(\\delta, \\varepsilon)) \\rightarrow [0, t], \\ \\ \n(\\zz, t') \\mapsto t'\n\\]\nis a~proper submersion. \nBy the Ehresmann's fibration theorem \\cite{W}, \n$\\pi'$ is a~locally trivial fibration over $[0, t]$. \nThus the projection \n$\\pi'$ induces \na~family of isomorphisms $\\psi_{t'} : \\partial E_{0}(\\delta, \\varepsilon) \\rightarrow \\partial E_{t'}(\\delta, \\varepsilon)$ \nof fibrations \nsuch that the following diagram is commutative for $0 \\leq t' \\leq t$: \n\\def\\mapright#1{\\smash{\\mathop{\\longrightarrow}\\limits^{{#1}}}}\n\\def\\mapdown#1{\\Big\\downarrow\\rlap{$\\vcenter{\\hbox{$#1$}}$}}\n\\[\\begin{matrix}\n\\partial E_{0}(\\delta, \\varepsilon)&\\mapright{\\psi_{t'}}& \\partial E_{t'}(\\delta, \\varepsilon) \\\\\n\\mapdown{F_{0}}&&\\mapdown{F_{t'}}\\\\\nS^{1}_{\\delta}&=& S^{1}_{\\delta} \n\\end{matrix},\\]\nwhere $\\partial E_{t'}(\\delta, \\varepsilon) = \n\\{\\zz \\in \\Bbb{C}^{2} \\mid \\lvert F_{t'}(\\zz)\\rvert = \\delta, \\| \\zz \\| \\leq \\varepsilon \\}$. \nThus the two fibrations \n$F_{t'} : D^{4}_{\\varepsilon} \\cap F_{t'}^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$ and \n$F_{0} : D^{4}_{\\varepsilon} \\cap F_{0}^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$ \nare isomorphic for $0 \\leq t' \\leq t$. \nBy \\cite[Theorem 37]{O2}, the two fibrations \n\\[\nF_{0} : \\partial E_{0}(\\delta, \\varepsilon) \\rightarrow S^{1}_{\\delta}, \\ \\ \\\nF_{0}\/\\lvert F_{0}\\rvert : S^{3}_{\\varepsilon} \\setminus \\text{Int}N(K_{F_{0}}) \\rightarrow S^{1} \n\\]\nare isomorphic for $\\varepsilon > 0, \\delta > 0$ and $\\delta \\ll \\varepsilon$. \nThus $F_{t} : D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$ \nis isomorphic to \n$F_{0}\/\\lvert F_{0} \\rvert : \\partial D^{4}_{\\varepsilon} \\setminus \\text{Int}N(K_{F_{0}}) \\rightarrow S^{1}$ \nfor $0 \\leq t \\ll 1$. \n\\end{proof}\n\n\n\\begin{lemma}\\label{c1}\nThe orbit space of  \n$D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta})$ \nof the $S^1$-action is homeomorphic to \na holed $2$-sphere for $0 \\leq t \\ll 1$. \n\\end{lemma}\n\\begin{proof}\nThe monodromy of $F_{t} : D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$ \nis given by the same $S^1$-action on $\\Bbb{C}^{2}$ for each $0 \\leq t \\ll 1$. \nBy Lemma $1$, $(D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta}))\/S^{1}$ is homeomorphic to \n$(\\partial D^{4}_{\\varepsilon} \\setminus \\text{Int}N(K_{F_{0}}))\/S^{1}$. \n\n\nSince the orbit space $\\partial D^{4}_{\\varepsilon}\/S^{1}$ is homeomorphic to a $2$-sphere and \n$K_{F_{0}}$ is an invariant set of the $S^1$-action, \nthe orbit space $(D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta}))\/S^{1}$ is a holed $2$-sphere.\n\\end{proof}\n\n\n\n\n\n\n\\subsection{Round handles}\nLet $X$ and $Y$ be $n$-dimensional smooth manifolds. \nAccording to \\cite{As, M}, we say that \\textit{$X$ is obtained from $Y$ by attaching a round $k$-handle} if \n\\begin{enumerate}\n\\item there are disk bundles over $S^{1}$, $E_{s}^{k}$ and $E_{u}^{n-k-1}$, \\\\\n\\item there exists an embedding $\\varphi : \\partial E_{s}^{k} \\times_{S^{1}} E_{u}^{n-k-1} \\rightarrow \\partial Y$ \nsuch that $X \\cong Y \\cup_{\\phi} E_{s}^{k} \\oplus E_{u}^{n-k-1}$, \n\\end{enumerate}\nwhere $E_{s}^{k} \\oplus E_{u}^{n-k-1}$ is the Whitney sum of $E_{s}^{k}$ and $E_{u}^{n-k-1}$ over $S^1$. \nThe bundle $E_{s}^{k} \\oplus E_{u}^{n-k-1}$ over $S^1$ is called \n\\textit{an $n$-dimensional round $k$-handle} and $\\varphi$ is called \n\\textit{the attaching map of $E_{s}^{k} \\oplus E_{u}^{n-k-1}$}. \nNote that a sufficiently small compact neighborhood of a connected component of the set of fold singularities \ncan be regarded as an $n$-dimensional round handle. \nIn our case, a sufficiently small compact neighborhood of each connected component of $S_{1}(F_{t})$ \nis regarded as a $4$-dimensional round $1$-handle. \n\n\n\n\n\n\n\\section{Proof of Theorem \\ref{rthm}}\n\\subsection{Round $1$-handles determined by $S_{1}(F_{t})$}\nBy the condition $(3)$, the origin $\\mathbf{o}$ is an isolated singularity of $F_t$. \nThere exist positive real numbers \n$\\varepsilon_{t}$ and $\\delta_{t}$ such that $\\delta_{t} \\ll \\varepsilon_{t}$ and the map \n\\[\nF_{t} : D^{4}_{\\varepsilon'_{t}} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta'_{t}}) \\rightarrow \\partial D^{2}_{\\delta'_{t}} \n\\]\nis a locally trivial fibration over $\\partial D^{2}_{\\delta'_{t}}$, where \n$D^{4}_{\\varepsilon'_{t}} = \\{ \\zz \\in \\Bbb{C}^{2} \\mid \\| \\zz \\| \\leq \\varepsilon'_{t} \\}$ and \n$D^{2}_{\\delta'_{t}} = \\{ \\eta \\in \\Bbb{C} \\mid \\lvert \\eta \\rvert \\leq \\delta'_{t} \\}$ \nfor $0 < \\varepsilon'_{t} \\leq \\varepsilon_{t}, 0 < \\delta'_{t} \\leq \\delta_{t}$ and \n$\\delta'_{t} \\ll \\varepsilon'_{t}$, see \\cite{W} . \nThus $F_{t}^{-1}(c)$ intersects $\\partial D^{4}_{\\varepsilon_{t}}$ transversely for any $c \\in D^{2}_{\\delta_{t}}$ \nand $0 \\leq t \\ll 1$. \nWe assume that $\\varepsilon_{t}$ and $\\delta_{t}$ satisfy the following properties: \n\\[\nD^{4}_{\\varepsilon_{t}} \\cap S_{1}(F_{t}) = \\emptyset, \\ \\ \\ \nD^{2}_{\\delta_{t}} \\cap F_{t}(S_{1}(F_{t})) = \\emptyset. \n\\]\nSee Figure~\\ref{fig1}.\n\\begin{figure}[h]\n\\centering\n  \\includegraphics[scale=1.1]{F_t.eps}\n  \\caption{$D^{2}_{\\delta_{t}}$ and $F_{t}(S_{1}(F_{t}))$}\n  \\label{fig1}\n\\end{figure\nPut $M_{0} = D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}})$. \n\n\nFix $t$ with $0 < t \\ll 1$ and \nlet $\\ell$ be the number of singularities of $\\lvert F_{t} \\rvert$. \nNote that $\\lvert F_{t} \\rvert$ is a Morse function by the conditions $(1)$ and $(2)$ except the origin. \nLet $C_{1}, \\dots, C_{\\ell}$ be the connected components of $S_{1}(F_{t})$, where \nthe number of the connected components of $S_{1}(F_{t})$ is $\\ell$ because of the conditions $(1), (2)$ and $(3)$. \nWe may assume that $\\lvert F_{t} \\rvert$ satisfies \n\\[\n\\lvert F_{t}(c_{1}) \\rvert \\leq \\lvert F_{t}(c_{2}) \\rvert \\dots \\leq \\lvert F_{t}(c_{\\ell}) \\rvert\n\\]\nfor $c_{i} \\in C_{i}$ and $i = 1, \\dots, \\ell$. \nLet $c'_{i}$ be the singularity of $\\lvert F_{t} \\rvert$ corresponding to $C_{i}$ and \n$N'_{i}$ be a sufficiently small compact neighborhood of $c'_{i}$ for $i = 1, \\dots, \\ell$. \nSince $\\lvert F_{t} \\rvert$ is a Morse function, each $N'_{i}, i = 1,\\dots, \\ell$, \ncan be regarded as a $3$-dimensional $1$-handle $[-1, 1] \\times D^{2}_{i}$, \nwhere $D^{2}_{i}$ is a $2$-disk. \nWe set $M'_{0} = M_{0}\/S^{1}$ and $M'_{i} := M'_{i-1} \\cup_{\\varphi'_{i}} N'_{i}$, \nwhere $\\varphi'_{i} : \\{\\pm 1\\} \\times D^{2}_{i} \\rightarrow \\partial M'_{i-1}$ is the attaching map of $N'_{i}$ \nfor $i = 1,\\dots, \\ell$. \nWe may assume that $\\varphi'_{i}(\\{\\pm 1\\} \\times D^{2}_{i}) \\subset \\partial M'_{0}$ for $i = 1,\\dots, \\ell$. \nThen the orbit space $D^{4}_{\\varepsilon}\/S^{1}$ is a topological manifold obtained from \n$M_{0}'$ by attaching $3$-dimensional $1$-handles $N'_{1}, \\dots, N'_{\\ell}$. \nNote that $D^{4}_{\\varepsilon}\/S^{1}$ is homeomorphic to a $3$-ball. \n\n\n\\begin{lemma}\\label{l2}\nLet $M_{0}^{*}$ be a connected component of $M'_{0}$. \nThen $\\varphi'_{i}(\\{\\pm 1\\} \\times D^{2}_{i}) \\not\\subset \\partial M_{0}^{*}$ for $i = 1,\\dots, \\ell$. \n\\end{lemma}\n\\begin{proof}\nAssume that there exist \n$i \\in \\{ 1, \\dots, \\ell \\}$ and \na connected component $M_{0}^{*}$ of $M'_{0}$ such that \n$\\varphi'_{i}(\\{\\pm 1\\} \\times D^{2}_{i})$ is contained in $\\partial M_{0}^{*}$. \nThen the genus of $\\partial M'_{i}$ is greater than $0$. \nAfter attaching $1$-handles, the genus of the boundary of the orbit space does not decrease. \nThus the genus of $\\partial M'_{\\ell}$ is greater than $0$. \nAs Lemma \\ref{c1}, the genus of $\\partial M'_{\\ell}$\nis equal to $0$. This is a~contradiction. \n\\end{proof}\nLet $M_{i}$ and $N_i$ be $4$-dimensional manifolds such that \n$M_{i}\/S^{1} = M'_{i}$ and $N_{i}\/S^{1} = N'_{i}$ respectively for $i = 1, \\dots, \\ell$. \nThen $N_i$ can be regarded as a $4$-dimensional round $1$-handle and \n$M_{i}$ is a manifold obtained from $M_{i-1}$ by attaching $N_{i}$ for $i = 1, \\dots, \\ell$. \nBy Lemma \\ref{l2}, $N_{i}$ connects two connected components of $M_{0}$. \nNote that $M_{\\ell}$ is diffeomorphic to \n$D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta})$. \n\n\\subsection{The fiber surface of $F_{t} : D^{4}_{\\varepsilon_{t}} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta_{t}}) \\rightarrow \\partial D^{2}_{\\delta_{t}}$} \nWe consider the restricted Milnor fibration  \n$F_{t} : D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta_{t}}) \\rightarrow \\partial D^{2}_{\\delta_{t}}$ and \nconnected components of $M_0$. \n\n\\begin{lemma}\\label{l3} \nLet $S_{0}$ be the fiber surface of \n$F_{t} : D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta_{t}}) \\rightarrow \\partial D^{2}_{\\delta_{t}}$. \nThen $S_{0}$ \nis diffeomorphic to the disjoint union of the fiber surface of \n$F_{t} \\mid_{D^{4}_{\\varepsilon_{t}} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta_{t}})}$ and  \n$\\ell$ copies of an annulus, where $\\ell$ is the number of connected components of $S_{1}(F_{t})$. \n\\end{lemma}\n\\begin{proof}\nLet $M_{0}^{0}, M_{0}^{1}, \\dots, M_{0}^{k}$ denote the connected components of $M_{0}$ such that \n$\\mathbf{o} \\in M_{0}^{0}$. \nThen $M_{0}^{0} \\cap D^{4}_{\\varepsilon_{t}} \\neq \\emptyset$. \nThe restricted map $F_{t} : D^{4}_{\\varepsilon_{t}} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}}) \\rightarrow D^{2}_{\\delta_{t}}$ \nhas a~unique singularity at the origin $\\mathbf{o}$ of $\\Bbb{C}^{2}$.  \nBy \\cite[Lemma 11.3]{Mi}, \n$D^{4}_{\\varepsilon_{t}} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta_{t}})$ \nis homeomorphic to $\\partial D^{4}_{\\varepsilon_{t}} \\setminus \\text{Int}N(K_{F_{t}})$, where \n$N(K_{F_{t}}) = \\{\\zz \\in \\partial D^{4}_{\\varepsilon_{t}} \\mid \\lvert F_{t}(\\zz)\\rvert \\leq \\delta_{t} \\}$. \nSo any fiber surface of \n$F_{t}: D^{4}_{\\varepsilon_{t}} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta_{t}}) \\rightarrow \\partial D^{2}_{\\delta_{t}}$ \nis connected. \nThe boundary of the orbit space \n$(D^{4}_{\\varepsilon_{t}} \\cap M_{0}^{0})\/S^{1}$ is homeomorphic to a~$2$-sphere and  \n\\[\nM_{0}^{j} \\cap D^{4}_{\\varepsilon_{t}} = \\emptyset \n\\]\nfor $j = 1, \\dots, k$. \n\nLet $S_{0}^{0}$ be a fiber surface of $F_{t}\\mid_{M_{0}^{0} \\cap (D^{4}_{\\varepsilon_{t}} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta_{t}}))}$. \nWe divide the surface $S_{0}^{0}$ as follows: \n\\[\nS_{0}^{0} = (S_{0}^{0} \\cap D^{4}_{\\varepsilon_{t}}) \\cup (S_{0}^{0} \\cap (D^{4}_{\\varepsilon} \\setminus \\text{Int$D^{4}_{\\varepsilon_{t}}$})). \n\\]\nSince $F_{t} : (D^{4}_{\\varepsilon} \\setminus \n\\text{Int$D^{4}_{\\varepsilon_{t}}$}) \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}}) \\rightarrow D^{2}_{\\delta_{t}}$ \nhas no singularities and  \n$F_{t}^{-1}(c)$ intersects $\\partial D^{4}_{\\varepsilon} \\sqcup \\partial D^{4}_{\\varepsilon_{t}}$ \ntransversely for any $c \\in D^{2}_{\\delta_{t}}$, \n$F_{t}^{-1}(c) \\cap (D^{4}_{\\varepsilon} \\setminus \\text{Int$D^{4}_{\\varepsilon_{t}}$})$ \nis diffeomorphic to $F_{t}^{-1}(0) \\cap (D^{4}_{\\varepsilon} \\setminus \\text{Int$D^{4}_{\\varepsilon_{t}}$})$. \nNote that $F_{t}^{-1}(0)$ is an invariant set of the $S^{1}$-action and \n$F_{t}^{-1}(0)\/S^{1}$ is a $1$-dimensional algebraic set. \nThe orbit space $F_{t}^{-1}(0)\/S^{1}$ is diffeomorphic to $[0, 1]$. \nThus the connected component of \n$F_{t}^{-1}(c) \\cap (D^{4}_{\\varepsilon} \\setminus \\text{Int$D^{4}_{\\varepsilon_{t}}$})$ is diffeomorphic to an annulus. \nSo any connected component of \n$S_{0}^{0} \\cap (D^{4}_{\\varepsilon} \\setminus \\text{Int$D^{4}_{\\varepsilon_{t}}$})$ is an annulus. \nSince any fiber of $F_{t}$ intersects $\\partial D^{4}_{\\varepsilon_{t}}$ transversely, \n$S^{0}_{0} \\cap \\partial D^{4}_{\\varepsilon_{t}}$ consists of circles and \n$S_{0}^{0} \\cap (D^{4}_{\\varepsilon} \\setminus \\text{Int$D^{4}_{\\varepsilon_{t}}$})$ is diffeomorphic to \n$(S^{0}_{0} \\cap \\partial D^{4}_{\\varepsilon_{t}}) \\times [0, 1]$. \nSo we have \n\\begin{equation*}\n\\begin{split}\nS_{0}^{0} &= (S_{0}^{0} \\cap D^{4}_{\\varepsilon_{t}}) \\cup (S_{0}^{0} \\cap (D^{4}_{\\varepsilon} \\setminus \\text{Int$D^{4}_{\\varepsilon_{t}}$})) \\\\ \n&\\cong (S_{0}^{0} \\cap D^{4}_{\\varepsilon_{t}}) \\cup ((S^{0}_{0} \\cap \\partial D^{4}_{\\varepsilon_{t}}) \\times [0, 1]) \\\\\n&\\cong S_{0}^{0} \\cap D^{4}_{\\varepsilon_{t}}.\n\\end{split} \n\\end{equation*}\n\n\nWe consider $M_{0}^{j}$ for $j = 1, \\dots, k$. \nThe restricted map $F_{t} : M_{0}^{j} \\rightarrow D^{2}_{\\delta_{t}}$ has no singularities. \nFor any $c \\in D^{2}_{\\delta_{t}} \\setminus \\{0\\}$ \nand $j = 1, \\dots, k$, \n$F_{t}^{-1}(c) \\cap M_{0}^{j}$ is diffeomorphic to $F_{t}^{-1}(0) \\cap M_{0}^{j}$. \nSince $F_{t}^{-1}(0)$ is an invariant set of the~$S^{1}$-action, \nthe orbit space $F_{t}^{-1}(0)\/S^{1}$ is a $1$-dimensional algebraic set. \nSo $F_{t}^{-1}(0)\/S^{1}$ is diffeomorphic to $[0, 1]$ or $S^{1}$. \nAssume that $F_{t}^{-1}(0)\/S^{1} = S^{1}$. \nThen $F_{t}^{-1}(c)$ is a torus and \nthe orbit space $F_{t}^{-1}(c)\/S^{1}$ is also a torus. \nSince the boundary of $M_{\\ell}'$ is a~$2$-sphere, \nthis is a contradiction. \nLet $S_{0}^{j}$ denote the fiber surface of $F_{t}\\mid_{M_{0}^{j}}$. \nThen $S_{0}^{j}\/S^{1}$ is diffeomorphic to $[0, 1]$ and \nthe fiber surface $S_{0}^{j}$ is diffeomorphic to an~annulus for $j = 1, \\dots, k$. \n\nBy Lemma \\ref{l2}, each $N_{i}$ connects two connected components of $M_0$. \nSince $M_{\\ell}$ is connected, we have $k+1-\\ell = 1$. \nThus the number of connected components of $S_0$ other than that of \n$F_{t} : D^{4}_{\\varepsilon_{t}} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta_{t}})$ \nis equal to $\\ell$. \n\\end{proof}\n\n\\begin{lemma}\\label{c2}\nThe connected component $M_{0}^{0}$ of $M_{0}$ is diffeomorphic to a $4$-ball and \n$M_{0}^{j}$ is diffeomorphic to $S^1 \\times B^{3}$, where $B^3$ is a $3$-ball, for $j =1, \\dots, \\ell$. \n\\end{lemma}\n\\begin{proof}\nThe two fibrations $F_{t}: D^{4}_{\\varepsilon_{t}} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta'}) \\rightarrow \\partial D^{2}_{\\delta'}$ \nand $F_{t}\/\\lvert F_{t}\\rvert : \\partial D^{4}_{\\varepsilon_{t}} \\setminus (\\partial D^{4}_{\\varepsilon_{t}} \\cap F_{t}^{-1}(0)) \\rightarrow S^{1}$  \nare isomorphic for any $0 < \\delta' \\leq \\delta_{t}$. \nThus $M_{0}^{0}$ is diffeomorphic to a $4$-ball. \n\nThe map $F_{t}\\mid_{M_{0}^{j}}$ has no singularities for $j \\neq 0$. \nThen the fiber surface of \n$F_{t} : M_{0}^{j} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta'}) \\rightarrow \\partial D^{2}_{\\delta'}$\nis diffeomorphic to $S_{0}^{j}$ for $0 < \\delta' \\leq \\delta_{t}$. \nSince the monodromy of $F_{t} : M_{0}^{j} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta'}) \\rightarrow \\partial D^{2}_{\\delta'}$ \nis given by the $S^1$-action, \nthe orbit space $(M_{0}^{j} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}} \\setminus \\{0\\}))\/S^{1}$ \nis homeomorphic to $(S_{0}^{j}\/S^{1}) \\times (0, 1]$. \nBy Lemma \\ref{l3}, $S_{0}^{j}\/S^{1}$ is also an annulus. We identify $S_{0}^{j}\/S^{1}$ with $S^{1} \\times [0,1]$. \nSince $S^{1} \\times (0, 1]$ is diffeomorphic to $D^{2}\\setminus \\{0\\}$, \n$(M_{0}^{j} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}} \\setminus \\{0\\}))\/S^{1}$ is homeomorphic to \n$D^{2} \\times [0,1] \\setminus (\\{0\\}\\times [0,1])$, where $D^2$ is a $2$-disk centered at $0$. \nSince $F_{t}^{-1}(0)$ is an invariant set of the $S^1$-action, \nthe orbit space of $F_{t}^{-1}(0)$ is homeomorphic to $\\{0\\} \\times [0, 1]$.\nThus the orbit space of $M_{0}^{j}$ is homeomorphic to $D^{2} \\times [0,1]$. \nThe manifold $M_{0}^{j}$ is diffeomorphic to $S^{1} \\times B^{3}$. \n\\end{proof}\n\n\n\n\n\\subsection{The number of connected components of $S_{1}(F_{t})$}\nTo complete the proof of Theorem~\\ref{rthm}, \nit is enough to show the equality in Theorem \\ref{rthm} $(\\text{ii})$. \nWe set $\\tilde{M}_{0} = D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta_{t}})$ and \n$\\tilde{M}_{i} = \\tilde{M}_{i-1} \\cup_{\\varphi_{i}} \\partial N_{i}$ for $i = 1, \\dots, \\ell$. \nSince $F_t$ is polar weighted homogeneous, \nthe monodromy of $F_{t}\\mid_{\\tilde{M}_{i}}$ is given by the $S^{1}$-action on~$\\Bbb{C}^{2}$. \nBy the condition $(2)$, a fiber of $\\lvert F_{t}\\rvert : N'_{i} \\rightarrow \\Bbb{R}$ is as follows: \n\\begin{equation*}\n\\lvert F_{t}\\rvert^{-1}(\\lvert u\\rvert) \\cap N'_{i} \\cong \\begin{cases}                                             \n          \\text{two $2$-disks} & 0 < c_{t,\\ww} - \\lvert u\\rvert \\ll 1 \\\\                                                         \n          \\text{an annulus} &  0 < \\lvert u\\rvert - c_{t,\\ww} \\ll 1           \n          \\end{cases}. \n\\end{equation*}\nSince the polar degree of $F_t$ is equal to $d_p$ and $N_i$ is a neighborhood of an orbit of the $S^1$-action, \na fiber of $F_{t}: N_i \\rightarrow D^{2}_{\\delta}$ is a $d_p$-fold cover over a fiber of $\\lvert F_{t}\\rvert$. \nThus we have \n\\begin{equation*}\nF_{t}^{-1}(u) \\cap N_{i} \\cong \\begin{cases}                                             \n          (\\sqcup_{j=1}^{d_{p}} D^{2}_{1,j}) \\sqcup (\\sqcup_{j=1}^{d_{p}} D^{2}_{2,j}) & 0 < c_{t,\\ww} - \\lvert u\\rvert \\ll 1 \\\\                                                         \n          \\sqcup_{j=1}^{d_{p}} A_{j} &  0 < \\lvert u\\rvert - c_{t,\\ww} \\ll 1           \n          \\end{cases}, \n\\end{equation*}\nwhere $D^{2}_{k, j}$ is a $2$-disk and $A_j$ is an annulus for $k=1, 2$ and $j = 1, \\dots, d_{p}$. \nBy Lemma \\ref{l2}, we may assume that $\\sqcup_{j=1}^{d_{p}} D^{2}_{1,j}$ is contained in a connected component of $M_0$ \nwhich does not contain $\\sqcup_{j=1}^{d_{p}} D^{2}_{2,j}$. \nLet $S_{i}$ be the fiber surface of $F_{t}\\mid_{\\tilde{M}_{i}}$ for $i = 0, 1, \\dots, \\ell$. \nSet $A' = \\sqcup_{j=1}^{d_{p}}A_{j}, \\partial A' = \\sqcup_{j=1}^{d_{p}}\\partial A_{j}$ and \n$D'_{k} = \\sqcup_{j=1}^{d_{p}}D^{2}_{k,j}$ for $k = 1, 2$. \nThen $S_i$ is the surface obtained from $S_{i-1}$ by replacing $D'_{1} \\sqcup D'_{2}$ by $A'$. \nThe Euler characteristic $\\chi(S_{i})$ of $S_i$ is equal to $\\chi(S_{i-1}) - 2d_{p}$, where \n$d_p$ is the polar degree of $F_t$ for $i = 1, \\dots, \\ell$. \nThus we have \n\\[\n\\chi(S_{\\ell}) - \\chi(S_{0}) = - 2\\ell d_{p}. \n\\]\n\n\\begin{lemma}\\label{c3} \nLet $\\ell$ be the number of connected components of $S_{1}(F_{t})$. \nThen $\\ell$ is equal to $\\lvert L^{+}(P, \\mathbf{o})\\rvert - \\lvert L^{+}(F_{t}, \\mathbf{o})\\rvert$ and also to \n$\\lvert L^{-}(P, \\mathbf{o})\\rvert - \\lvert L^{-}(F_{t}, \\mathbf{o})\\rvert$. \n\\end{lemma}\n\\begin{proof}\nSince the fibration $F_{t}\\mid_{\\tilde{M}_{\\ell}}$ is isomorphic to \n$P : D^{4}_{\\varepsilon} \\cap P^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$ and \n$L(P, \\mathbf{o})$ is the Seifert link in $\\partial D^{4}_{\\varepsilon}$, \nwe have \n\\[\n\\chi(S_{\\ell}) = 1 - \\{pq(m+n) - p - q\\}(m-n), \n\\]\nwhere $m = \\lvert L^{+}(P, \\mathbf{o})\\rvert$ and $n = \\lvert L^{-}(P, \\mathbf{o})\\rvert$, see \\cite[Theorem 11.1]{EN}. \nBy Lemma \\ref{l3}, the fiber surface $S_{0}$ of $F_{t}\\mid_{\\tilde{M}_{0}}$ \nis diffeomorphic to $S^{0}_{0} \\sqcup S^{1}_{0} \\sqcup \\dots \\sqcup S^{k}_{0}$ \nand $S^{j}_{0}$ is an annulus for $j \\neq 0$. \nThe Euler characteristic $\\chi(S_{0})$ of $S_0$ is equal to $\\chi(S^{0}_{0})$. \nSince the link $L(F_{t}, \\mathbf{o})$ is also the Seifert link in a $3$-sphere, \n$\\chi(S^{0}_{0})$ is given by \n\\[\n\\chi(S^{0}_{0}) = 1 - \\{pq(m'+n') - p -q\\}(m'-n'), \n\\] \nwhere $m' = \\lvert L^{+}(F_{t}, \\mathbf{o})\\rvert$ and $n' = \\lvert L^{-}(F_{t}, \\mathbf{o})\\rvert$.\nOn the other hand, $\\chi(S_{\\ell})$ is equal to $\\chi(S_{0}) - 2\\ell d_{p}$. \nThe polar degree $d_p$ is equal to $pq(m-n)$ and also to $pq(m'-n')$ \\cite{EN}. \nThen we have \n\\begin{equation*}\n\\begin{split}\n\\chi(S_{\\ell}) - \\chi(S_{0}) &= - \\{pq(m+n) - p - q\\}(m-n) + \\{pq(m'+n') - p -q\\}(m'-n') \\\\\n                             &= -pq(m-n)(m+n) + pq(m'-n')(m'+n') \\\\\n                             &= -d_{p}\\{(m+n)-(m'+n')\\} \\\\\n                             &= - 2\\ell d_{p}. \n\\end{split}\n\\end{equation*}\nSo $2\\ell$ is equal to $m+n - (m'+n')$. Since $m-m'$ is equal to $n-n'$, \n$\\ell$ is equal to $m - m'$ and also to $n-n'$. \n\\end{proof}\n\nWe give an example of Lemma $\\ref{c3}$ which is considered in \\cite{In3}. \n\\begin{Example}\nSet $f(\\zz) = z_{1}^{m} + z_{2}^{m}$ and $g(\\zz) = z_{1} + 2z_{2}$ where $m \\geq 3$. \nWe consider a deformation $F_{t} = f(\\zz)\\overline{g(\\zz)} + t(z_{1}^{m}\\bar{z}_{1} + z_{1}^{m-1} + \\gamma z_{2}^{m-1})$ \nof $f(\\zz)\\overline{g(\\zz)}$ where $\\gamma \\in \\Bbb{C}$. \nIn \\cite{In3}, we take a~coefficient $\\gamma$ of $h(\\zz)$ such that \n\\[\n\\overline{\\gamma} \\neq \n\\frac{-(2\\alpha'f(z, 1) - mg(z, 1))(mz\\bar{z}^{m-1}r^{2} + (m-1)\\bar{z}^{m-2} - \\alpha' z^{m}r^{2})}\n{(m-1)(m\\bar{z}^{m-1}g(z, 1) - \\alpha' f(z, 1))} \n\\]\nwhere $(z^{m}+1)\\overline{(z+2)} = \\alpha'\\overline{(z^{m}+1)}(z+2), \\alpha' \\in S^1$. \nThen $S_{1}(F_{t})$ is the set of indefinite fold singularities and \nthe link $S^{3}_{\\varepsilon} \\cap F_{t}^{-1}(0)$ is a $(m-1, m-1)$-torus link, \nwhere \n$S^{3}_{\\varepsilon} = \\{ (z_{1}, z_{2}) \\in \\Bbb{C}^{2} \\mid  \\lvert z_{1} \\rvert^{2} + \\lvert z_{2} \\rvert^{2} = \\varepsilon \\}$, \n$\\varepsilon \\ll 1$. \nBy Lemma $\\ref{c3}$, the number of connected components of $S_{1}(F_{t})$ is equal to~$1$. \n\\end{Example}\n\n\n\\begin{proof}[Proof of Theorem 1]\nBy Lemma $\\ref{l2}$ and Lemma $\\ref{c2}$, \n$D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(D^{2}_{\\delta_{t}})$ consists of a $4$-ball and $\\ell$-copies of \n$S^{1} \\times B^{3}$ \nand the image of the attaching map $\\varphi_{i}$ of $i$-th round $1$-handle is \ncontained in both of the boundary of a $4$-ball and that of $S^{1} \\times B^{3}$ for $i=1, \\dots, \\ell$. \nBy Lemma $\\ref{c3}$, the number of connected components of $S_{1}(F_{t})$ is equal to \n$\\lvert L^{-}(P, \\mathbf{o})\\rvert - \\lvert L^{-}(F_{t}, \\mathbf{o})\\rvert$. \n\\end{proof}\n\n\n\n\n\\section{Remarks}\n\\subsection{Monodromy and characteristic polynomials}\nLet $h : S \\rightarrow S$ be a homeomorphism of a surface $S$. We define \n\\[\n\\Delta_{*}(h) = \\frac{\\Delta_{1}(h)}{\\Delta_{0}(h)}, \n\\]\nwhere $\\Delta_{i}(h)$ is the characteristic polynomial of the homological map from \n$H_{i}(S, \\Bbb{Z})$ to itself induced by $h$ for $i = 0, 1$. \n\nLet $h_{i} : S_{i} \\rightarrow S_{i}$ be the monodromy of $F_{t}\\mid_{\\tilde{M}_{i}}$ for $i=1, \\dots, \\ell$. \nSince $h_{i}$ is given by the $S^1$-action on $\\Bbb{C}^{2}$, $h_{i} : S_{i} \\rightarrow S_{i}$ satisfies the following conditions:  \n\\renewcommand{\\theenumi}{\\Roman{enumi}}\n\\begin{enumerate}\n\\item\n$h_{i}(S_{i-1} \\setminus (D'_{1} \\sqcup D'_{2})) = S_{i-1} \\setminus (D'_{1} \\sqcup D'_{2})$ and \n$h_{i}\\mid_{S_{i-1}\\setminus (D'_{1} \\sqcup D'_{2})} = h_{i-1}\\mid_{S_{i-1}\\setminus (D'_{1} \\sqcup D'_{2})}$, \n\\item\n$h_{i}\\mid_{D'_{k}}$ and $h_{i}\\mid_{A'}$ are periodic maps which satisfy \n$D^{2}_{k,j} \\rightarrow D^{2}_{k,j+1}$ and \n$A_{j} \\rightarrow A_{j+1}$ \n\\end{enumerate}\nfor $i = 1, \\dots, \\ell, j = 1,\\dots, d_{p}$ and $k = 1, 2$. Here $D^{2}_{k,d_{p}+1} = D^{2}_{k,1}$ and $A_{d_{p}+1} = A_{1}$. \nWe calculate $\\Delta_{*}(h_{i})$ from $\\Delta_{*}(h_{i-1})$ by using a round $1$-handle $N_i$. \n\n\n\n\\begin{lemma}\\label{l4} \nLet $S_{i}$ be the fiber surface of $F_{t}\\mid_{\\tilde{M}_{i}}$ and \n$h_{i} : S_{i} \\rightarrow S_{i}$ be the monodromy of $F_{t}\\mid_{\\tilde{M}_{i}}$ for $i = 1, \\dots, \\ell$. \nThen the characteristic polynomial of $h_i$ satisfies \n\\[\n\\Delta_{*}(h_{i}) = \\Delta_{*}(h_{i-1})(t^{d_{p}}-1)^{2}. \n\\]\n\\end{lemma}\n\\begin{proof}\nSince $S_i$ is the surface obtained from $S_{i-1}$ by replacing $D'_{1} \\sqcup D'_{2}$ by $A'$ \nand $h_i$ satisfies the above properties, \nwe have \n\\[\n\\Delta_{*}(h_{i}) = \\frac{\\Delta_{*}\\bigr(h_{i}\\mid_{S_{i-1} \\setminus (D'_{1} \\sqcup D'_{2})}\\bigl)\n\\Delta_{*}\\bigr(h_{i}\\mid_{A'}\\bigl)}\n{\\Delta_{*}\\bigr(h_{i}\\mid_{\\partial A'}\\bigl)}. \n\\]\nBy the condition $(\\text{II})$, the monodromy matrices of $H_{0}(D'_{k}, \\Bbb{Z}), H_{i}(A', \\Bbb{Z})$ and \n$H_{i}(\\partial A', \\Bbb{Z})$ are equal to \n\\begin{eqnarray*}\n \\left(\n        \\begin{array}{@{\\,}cccccccc@{\\,}}\n0 & 1 & 0 & \\ldots & 0 \\\\\n0 & 0 & 1 & \\ddots & \\vdots \\\\ \n\\vdots & \\ddots & \\ddots & \\ddots & \\vdots \\\\\n0 & \\ldots & 0 & 0 & 1 \\\\\n1 & 0 & \\ldots & 0 & 0\n\\end{array}\n       \\right)\n\\end{eqnarray*}\nfor $k = 1, 2$ and $i = 0, 1$. \nThe characteristic polynomial of the above matrix is equal to $t^{d_{p}} - 1$. \nSo $\\Delta_{*}\\bigr(h_{i}\\mid_{A'}\\bigl)$ and $\\Delta_{*}\\bigr(h_{i}\\mid_{\\partial A'}\\bigl)$\nare equal to $1$. \nAs the condition $(\\text{I})$, we have\n\\[\n\\Delta_{*}\\bigr(h_{i}\\mid_{S_{i-1} \\setminus (D'_{1} \\sqcup D'_{2})}\\bigl)\n\\Delta_{*}\\bigr(h_{i-1}\\mid_{D'_{1}}\\bigl)\\Delta_{*}\\bigr(h_{i-1}\\mid_{D'_{2}}\\bigl) = \n\\Delta_{*}\\bigr(h_{i-1}\\bigl).\n\\] \nThus the characteristic polynomial satisfies  \n\\begin{equation*}\n\\begin{split}\n\\Delta_{*}(h_{i}) \n&= \\Delta_{*}\\bigr(h_{i}\\mid_{S_{i-1} \\setminus (D'_{1} \\sqcup D'_{2})}\\bigl) \\\\\n&= \\Delta_{*}(h_{i-1})(t^{d_{p}} - 1)^{2}. \n\\end{split}\n\\end{equation*}\n\\end{proof}\n\nSince the two fibrations \n$P : D^{4}_{\\varepsilon} \\cap F_{t}^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$ and \n$F_{t} : D^{4}_{\\varepsilon} \\cap F_{0}^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$ \nare isomorphic, we have the following theorem. \n\n\\begin{theorem}\nLet $h$ be the monodromy of $P : D^{4}_{\\varepsilon} \\cap P^{-1}(\\partial D^{2}_{\\delta}) \\rightarrow \\partial D^{2}_{\\delta}$. \nThen $\\Delta_{*}(h)$ is equal to $\\Delta_{*}(h_{0})(t^{d_{p}}-1)^{2\\ell}$, \nwhere $\\ell$ is the number of connected components of $S_{1}(F_{t})$. \n\\end{theorem}\n\\begin{remark}\nThe algebraic set $P^{-1}(0) \\cap \\partial D^{4}_{\\varepsilon}$ is a fibered Seifert link in the $3$-sphere. \nThus the characteristic polynomial of the monodromy of the Milnor fibration of $P$ at the origin \ncan also be calculated from the splice diagram \\cite{EN}. \n\\end{remark}\n\n\\subsection{A specific deformation of $f\\bar{g}$}\nWe introduce a deformation of $f\\bar{g}$ given in \\cite{In3}, where $f$ and $g$ are $2$-variables convenient \ncomplex polynomials and $f\\overline{g}$ has an isolated singularity at the origin $\\mathbf{o}$. \nWe define the $\\Bbb{C}^{*}$-action on $\\Bbb{C}^{2}$ as follows: \n\\[\nc \\circ (z_{1}, z_{2}) := (c^{q}z_{1}, c^{p}z_{2}), \\ \\ c \\in \\Bbb{C}^{*}. \n\\]\nAssume that $f(\\zz)$ and $g(\\zz)$ satisfy\n\\[ \nf(c \\circ \\zz) = c^{pqm}f(\\zz), \\ \\ g(c \\circ \\zz) = c^{pqn}g(\\zz), \\ \\ m > n. \n\\]\nThen $f(\\zz)$ and $g(\\zz)$ are weighted homogeneous polynomials. \nTwo complex polynomials $f(\\zz)$ and $g(\\zz)$ can be written as \n\\[\n f(\\zz) =  \\textstyle\\prod\\limits_{j=1}^{m} (z^{p}_{1} + \\alpha_{j}z^{q}_{2}), \\ \\  \n g(\\zz) = \\textstyle\\prod\\limits_{j=1}^{n}(z^{p}_{1} + \\beta_{j}z^{q}_{2}), \\ \\  \\gcd(p, q) = 1, \n\\]\nwhere $\\alpha_{j} \\neq \\alpha_{j'}, \\beta_{j} \\neq \\beta_{j'} \\ (j \\neq j')$ and \n$\\alpha_{k} \\neq \\beta_{k'}$ for $1 \\leq k \\leq m$ and $1 \\leq k' \\leq n$. \nThe mixed polynomial $f(\\zz)\\overline{g(\\zz)}$ is a polar and radial weighted homogeneous polynomial, \ni.e., $f(\\zz)\\overline{g(\\zz)}$ satisfies that \n$f(s \\circ \\zz)\\overline{g(s \\circ \\zz)} = s^{pq(m - n)}f(\\zz)\\overline{g(\\zz)}$ and \n$f(r \\circ \\zz)\\overline{g(r \\circ \\zz)} = r^{pq(m + n)}f(\\zz)\\overline{g(\\zz)}$, \nwhere $s \\in S^{1}$ and $r \\in \\Bbb{R}^{*}$ \\cite{O1}. \nWe define a deformation of $f(\\zz)\\overline{g(\\zz)}$ as follows: \n\\[\nF_{t}(\\zz) := f(\\zz)\\overline{g(\\zz)} + th(\\zz),\n\\]\nwhere $0 < t \\ll 1$ and  \n\\begin{equation*}\nh(\\zz) = \\begin{cases}\n            \\gamma_{1}z_{1}^{p(m-n)} + \\gamma_{2}z_{2}^{q(m-n)}  & \\text{$(g(\\zz) \\neq z_{1} + \\beta z_{2})$}, \\\\\n            z_{1}^{m}\\overline{z}_{1} + z_{1}^{m-1} + \\gamma z_{2}^{m-1} & \\text{$(g(\\zz) = z_{1} + \\beta z_{2})$}. \n         \\end{cases}\n\\end{equation*} \nThen $F_{t}(\\zz)$ is also a polar weighted homogeneous polynomial with the polar degree $pq(m-n)$. \nBy \\cite[Theorem 1]{In3}, there exists $h(\\zz)$ such that $F_{t}(\\zz)$ satisfies the conditions $(1), (2)$ and $(3)$ \nfor $0 < t \\ll 1$. \nThe above deformation $F_t$ of $f\\bar{g}$ satisfies that $\\lvert L^{-}(F_{t}, \\mathbf{o})\\rvert = 0$. \nBy Lemma $\\ref{c3}$, the number $\\ell$ of connected components of $S_{1}(F_{t})$ is equal to $n$. \nSince the radial degree $d_r$ and the polar degree $d_p$ are equal to $pq(m+n)$ and $pq(m-n)$ respectively, \nwe have the following proposition. \n\\begin{proposition}\nLet $F_t$ be the above deformation of $f\\bar{g}$. \nThen the number $\\ell$ of connected components of $S_{1}(F_{t})$ is equal to $\\frac{d_{r} - d_{p}}{2pq}$, \nwhere $d_r$ is the radial degree of $f\\bar{g}$ and $d_p$ is the polar degree of $f\\bar{g}$. \n\\end{proposition}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Abstract}\n\nHadron therapy is a novel treatment against cancer. The main advantage\nof this therapy causes less side effect in comparison to X-ray\nirradiation methods. Hadron therapy is just ahead of a significant\nbreakthrough since this technique can be more precise, applying proton\ncomputer tomograph (pCT) to map the stopping power in the tissues.\n\nThe research and development of a pCT require a fast detector to\nmeasure the energy of hadrons behind the patient. The best detector\noption is called hadron-tracking calorimeter, which consists of sandwich\nlayers of silicon tracking detectors and absorber layers. The\ncombination of measuring the trajectory (tracking process), and, in parallel, the\nenergy of relativistic particles, can provide high-resolution hadron\nimaging. This semiconductor-based technology requires stable\ntemperature and homogeneous cooling.\n\nI have worked in the development of this detector in the Bergen pCT Collaboration for two years. Last year my work was to investigate the temperature distribution in the\ncalorimeter and examine two cooling concepts in detail. I performed\nboth analytical and numerical calculations to analyze the temperature\ndistribution of the calorimeter. The final decision about the design takes into account\nmany engineering aspects, such as reliability, flexibility, and\nperformance. \n\\end{titlepage}\n\n\\tableofcontents\n\n\\newpage\n\\section{Introduction}\n\\subsection{Cancer}\n\nCancer has become one of the leading reasons for death in the developed world. It is responsible for 25\\% of all death in Hungary \\cite{Cancerstatistics}. It affects all age groups, but the risk of cancer is increasing with age, as one can see in Figure \\ref{fig:procure}.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{16cm}\n\\centering\n\\includegraphics[width=16cm]{Pictures\/Rak_statisztika2.png}\n\\caption{Cancer affects all age groups, but the risk of cancer is increasing with age. This statistic was made in Scotland in 2016 \\cite{cancerage}.}\n\\label{fig:cancerstatistics}\n\\end{minipage}\n\\end{figure}\n\nThis section is about the nature of cancer, briefly summarized based on the literature \\cite{oncology1, oncology2, oncology3}.\n\nThe cell cycle is an important period of life of the cells because they divide and replace dead cells. In healthy tissues, there are stem cells, which can divide at any time. However, for most of the cells, the number of divisions and their times are limited. This is not true for the cancer cells because a DNA mutation turns off this limitation.\n\nUsually, a cell dies after a lifetime, or in case of serious injury in its DNA. This function is called apoptosis, which, in general, does not work in cancer cells.\n\nThere are two types of cancer. The first is called bengilus. In the case of bengilus cancer, the tumor does not reduce the life expectancy and quality of life of the patient. The second one is called malignus, which means risk in life or the life quality of the patient. It is hard to determinate whether a tumor is bengilus or malignus. Usually, bengilus tumors expand slower, and they have a tegument, but the lack of these effects does not evidence that the tumor is malignus.\n\nUnfortunately, tumors can expand in the surrounding tissues. This phenomenon is called an invasion. Cells of the tumor can spread far from the original one, this is called metastasis. After the treatment of cancer, it is still possible to find other cancer cells in the surrounding tissues that can cause a new, so-called residual tumor.\n\nThere are status groups, called stages, to categorize the tumors. The traditional staging has five stages:\n\\\\0. stage: there is no invasion,\n\\\\I. stage: small, local tumor, with small invasion (without metastasis),\n\\\\II. stage: local tumor with more significant invasion, probably with metastasis in the local  lymph nodes,\n\\\\III. stage: local tumor with significant invasion with high probability to has big metastasis in the local  lymph nodes,\n\\\\IV. stage: local tumor with huge invasion and with far-reaching metastasis.\n\n\\subsection{Mapping methodes in the detection and treatment of cancer}\n\nThere are lots of methods to measure or find a tumor in the human body. \nThis subsection presents them briefly based on \\cite{oncology4}.\n\nUltrasound method uses sound waves to take a picture from the inside of the body. Applying this method, the sound wave penetrates into the body of the patient and is scattered on the boundaries of the organs with different speed of sound. A microphone records that reflection and a computer reconstructs the map.  A usual ultrasound machine takes a 2D picture from a segment of the human body, but nowadays it is also possible to obtain a 3D picture. The ultrasound technology does not have any side effect.  \n\nAn X-ray computer tomograph uses X-ray beams to fluoroscope through the human body. It creates a 3D picture from the nucleon density of the body. This type of imaging causes radiation in the body, which can damage healthy tissues.\n\nThe MR technology is based on the measurement of the magnetic field caused by the water molecules in the human body. The water molecules are excited by a strong magnetic field generated by superconductor magnets. The detector measures the changes of the magnetic field after the magnets are turned off. For instance, it makes visible the blood flow in the veins. MR technology does not cause any side effect, based on the current knowledge of science. The mapping process takes a long time, and the patient have to lie in a narrow, and closed space that causes feeling of being locked up in $4-6\\%$ of the patients.\n\nMapping with radiative isotopes can give information about the life cycle of a tumor. This technology is based on the density of the chemical material which is absorbed by the tissues and the tumor. Usually, the tumor absorbes different amount of chemical element than the surrounding healty tissues. Thus, the concentration difference of the isotopes is measured here. The directional dependence of the radiation makes possible to construct the 3D map. Moreover, the amount of the absorbed radiation offers information about the type, size and the current state of the tumor. It is also useful to find metastasis; however, the irradiation of healthy tissues is unavoidable.\n\n\\subsection{Treatment of cancer}\n\nThe treatment aims to ensure high life expectancy (or as high as possible) and life quality (or as high as possible). In order to reach this goal, modern health care uses surgery, radiotherapy, chemotherapy, and hormone therapy. These treatments can be applied either standalone or combined as a complex therapy to improve the effectiveness.\nThat subsection is summarized based on \\cite{oncology5, oncology6, oncology7}.\n\nEvery treatment starts with canceling the tumor.  It is crucial not to leave any cancer cell in the surrounding tissues because it can cause a residual tumor later. If it is not possible due to some reason, then it becomes necessary to kill these cells in the second part of the treatment. That part is about to kill all the remaining cancer cells in order to avoid metastasis. These cells are probably to be a metastasis, or cells in the vascular and lymphatic system, which try to find a new living space and create a new metastasis.\n\nSurgery is the leading treatment of cancer. It is the oldest treatment of tumor diseases. In general, it is used to remove the original tumor, but it is useful in removing the metastasis, too. Its one of the most significant advantage is that the cancer cells cannot become resistive to the surgery. During this process, doctors cut out the tumor and usually remove the local lymph nodes also from the body to reduce the risk of metastasis. The surgery burdens the body of the patient. If the patient is not in adequate conditions, the operation becomes impossible. There are other situations in which the outcome is the same: either the tumor is in a hardly available location, or too big to cut out. In these cases, doctors have to find another type of treatment or use complex methods.\n\nRadiotherapy uses the effects of the ionization of X-ray or ion beams. The beam exerts its effect in two ways, directly and indirectly. The direct way is when the particles of the beam damage the DNA and the critical parts of the cell. The indirect way is when the particles of the beam ionize the water molecules, and these molecules damage the DNA of the cancer cells. In both cases, the damage induces a self-destruct function of the cancer cell, thus the cell destroys itself. This self-destruction process is called apoptosis. Unfortunately, there are such cancer cells which have resistance to the radiation, the apoptosis process is prevented. \n\nUsually, chemotherapy uses apoptosis to destroy cancer cells. In the case of chemotherapy, doctors give medicine to the patient, which consists of chemical elements, inducing the apoptosis process. The most common chemicals are called cytostatics. \nThe cytostatics affect the cancer cells more seriously than healthy cells. Besides, there are other chemicals to reduce the effect of cytostatics in the healthy regions or increase their exterminating effect in the cancer cells.\n\nSurgery as a standalone treatment is usual for a tumor in the I.~or II.~stages. In the III.~stage, radiotherapy is recommended. Chemotherapy is typical for IV.~stage. \n\nBeyond these possibilities, sometimes the surgeon must remove the entire organ, this is called `organ removal surgery'. However, this is not preferred and tried to be avoided. For instance, using radiotherapy first reduces the size of the tumor, hence it is possible to preserve the organ as much as possible.\nMoreover, chemotherapy helps to reduce the risk of metastasis with canceling the remaining cells in the vascular and lymphatic system.\n\n\\subsection{Radiotherapy}\n\nRadiotherapy uses the ionizing effect of radiation to destroy cancer cells. There are different ways to transfer the radiation to the tumor \\cite{OrvosiBiofizika}. As a first option, one can use beams, which penetrates the tissues and the tumor, and in the meantime, it damages them. This treatment is called external beam radiation therapy. Its advantage is that it is usable in many segments of the body, without using surgery. Secondly, one can situate a radioactive source in the vicinity of the tumor, for example, into a body cavity. This technology is called brachytherapy. The advantage is the same, no operation is needed. As a third option, it is possible to ingest certain radioactive isotopes into the body. In fact, a tumor absorbs more isotopes than any of the other healthy organs. This treatment is called systemic radiation therapy. The disadvantage of this technology is that the patient becomes radioactive for a short time, so he or she has to stay in the hospital for some days.\n\nOne of the primary development in radiotherapy is to reduce the side effects caused by the radiation in healthy tissues. If one improves the ratio of the ionization in the tumor, it reduces the amount of ionization in the healthy tissues, thus it reduces the side effects.\n\nX-ray photon beams are the most common in external beam radiation therapy. The usage of protons or heavier ions instead of X-ray photons results in less ionization in the healthy tissues \\cite{OrvosiBiofizika, protonTherapy0}. Its advantage is based on the radiation concentration distribution. The X-ray photons generate the highest radiation after the entry point in the body, then the radiation is monotonously decreasing. Using ions instead of X-ray photons, it is possible to concentrate the radiation onto the tumor, since the ions stop in the tumor mostly. Ions create the highest radiation right before they stop.\n One can see the resulted radiation by X-ray photons and protons in Figure \\ref{fig:EnergiaLeadas}. One can also see the radiation of each treatment in a cross-section of a head in Figure \\ref{fig:procure}.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/EnergialeadasFotonproton.jpg}\n\\caption{The energy-loss of X-ray photons and protons in function of the amount of material in their previous path \\cite{ProCure}.}\n\\label{fig:EnergiaLeadas}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/treatement_more_angles.png}\n\\caption{Radiation in a head in case of proton and X-ray therapy \\cite{ProCure}.}\n\\label{fig:procure}\n\\end{minipage}\n\\end{figure}\n\nAs it is apparent, the application of ions in the external beam radiation therapy is beneficial. Nowadays, the number of hadron therapy centers is increasing. One can see the treatment room of a hadron therapy facility in Figure \\ref{fig:ProtonKezeles}. However, there is a design barrier in the spread of hadron therapy centers. This barrier is the lack of an accurate imaging tool for  dose planning. \nDoctors use X-ray computer tomographs (CTs) to take a three-dimensional map from the nucleon density of the body. However, the energy loss of ions depends on the electron density of the body. It is possible to calculate the electron density using the measured nucleon density, but there is no simple relation between them, because of the different ratio between proton and nucleon number of atoms of the human body. As a result, one can calculate the electron density with $1.7\\%$ statistical error from nucleon density \\cite{CTcomparison}. If one could measure the electron density map of the human body directly with the usage of ions, the statistical error could be reduced to $0.5\\%$ without significant error \\cite{CTcomparison}. Overall, the development of a hadron computer tomography (usually called proton computer tomography or pCT) is beneficial.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/Proton_therapy2.jpg}\n\\caption{The patient is fixed into a bed during the radiotherapy. The patient is moved with the bed by a robot arm that can be moved in all directions. First of all they take a 3D map of the tissues nearby the tumor of the patient, after they treat the tumor by a proton beam \\cite{pKezeles}.}\n\\label{fig:ProtonKezeles}\n\\end{minipage}\n\\end{figure}\n\n\\subsection{Proton computer tomography}\n\nThe original idea of computer tomography (CT) comes from Allan M. Cormack (1963) \\cite{Cormack}. He won the Nobel Prize with this idea in 1979.\n\nA CT equipment moves around the body and takes pictures from the inside in several directions, and a computer calculates the 3D map using these pictures. In case of proton CT, one can use a beam to fluoroscope through the body. One scans the body with this beam in a particular direction, so one obtains a two-dimensional picture about the body. After, the detector is turned into a different direction, and the scanning process starts again. One can see the scanning and rotating process in Figure \\ref{fig:workingofapCT}. Finally, a computer algorithm calculates the three-dimensional map of the body from the two-dimensional pictures.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/Working_of_proton_CT.pdf}\n\\caption{Left: it is visible how the proton beam scans the body. Right: it is visible how the detector and the beam rotate around the patient. }\n\\label{fig:workingofapCT}\n\\end{minipage}\n\\end{figure}\n\nDuring the scanning process, one has to use another detector at the other side of the patient in order to measure the energy and the path of particles coming from the beam. Such a detector is called a tracking calorimeter since it measures both the energy and the tracks of hadrons. $10^7-10^9~\\frac{particle}{second}$ particle rate is necessary to obtain a three-dimensional picture within a reasonable time for clinical use \\cite{Dieter}. To measure the path of the particles with such a rate was impossible five years ago. However, thanks to the development of tracking silicon pixel detectors in the last years in CERN LHC, there are available tracking detectors that can meet with this particle rate now \\cite{ALICETEC}. These tracking detectors can measure the path of the particles, but there is not any available detector that can measure the energy of each particle with this particle rate.\n\nThe Bergen pCT Collaboration aims to develop a calorimeter (calorimeter is a detector, which measures the energy of hadrons) build of alternating tracking detector and aluminum absorber layers, as a sandwich structure \\cite{Dieter}. One can see the concept of this detector in Figure \\ref{fig:Concept1}. As a tracking calorimeter, we are going to use a tracking detector, which was developed in ALICE, CERN LHC, and which is called ALPIDE.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/Patient_and_detector.pdf}\n\\caption{Concept of the proton CT and its tracking calorimeter.}\n\\label{fig:Concept1}\n\\end{minipage}\n\\end{figure}\n\nMy role in the Bergen pCT Collaboration is the investigation of the temperature distribution in the calorimeter and compare two cooling system concept. That work is crucial to ensure the accuracy of the detector. That part of the detector development is clearly separated from the other tasks related to the design. Thus the present thesis reflects my work.\n\n\n\n\\subsection{ALPIDE tracking detector}\n\nALPIDE was developed to upgrade the Inner Tracking System (ITS) of `A Large Ion Collider Experiment' (ALICE) in 2019-2020, which is one of the four large experiment programs of the Large Hadron Collider of the European Organization for Nuclear Research (CERN) \t\\cite{ALICE_ALL}. One can see the ALPIDE detector in Figure \\ref{fig:ALPIDE}.\n\n\\begin{figure}[!h]\n\\centering\n\\begin{minipage}{10cm}\n\\centering\n\\includegraphics[width=10cm]{Pictures\/pALPIDE.png}\n\\caption{ALPIDE detector \\cite{MoniKonyve}.}\n\\label{fig:ALPIDE}\n\\end{minipage}\n\\end{figure}\n\nALPIDE is a monolithic active pixel sensor (MAPS) type silicon detector \\cite{ALICETEC}, which contains the sensitive sensors together with the readout electronics on the same silicon layer, see Figure \\ref{fig:MAPS} for details. \n When a charged particle goes through the sensitive (light blue area, called Epitaxial Layer P-) area of the detector, it generates electron-hole pairs. The electrons and the holes are transferred in the sensitive area by diffusion. The holes are absorbed by the Substrate P++ layer (dark blue). There is an area (white), which is evacuated by electric voltage. When an electron reaches this zone, the electric field of the evacuated zone transfers it into the NWELL diode (green). This diode collects the electrons, then a microelectronic amplifier amplifies the signal. Another microelectronics decides whether the signal is higher than a threshold or not. If so, the microerectronic device sends a signal to the readout electronics with the coordinates of the pixel. This is a state of the art solution because the sensitive silicon layer contains the complete amplifier electronics. It is possible, thanks to the DEEP PWELL (red), which separates the NWELL parts of the microelectronics, thus the NWELL parts cannot operate as an undesirable diode.\n\n\\begin{figure}[!h]\n\\centering\n\\begin{minipage}{10cm}\n\\centering\n\\includegraphics[width=10cm]{Pictures\/technology.pdf}\n\\caption{Main parts of a monolithic active pixel sensor in a section view of one pixel \\cite{ALICETEC}.}\n\\label{fig:MAPS}\n\\end{minipage}\n\\end{figure}\n\nThe electrons, generated by the same particle, can be transfered to more than one pixels. This effect causes that more than one pixel sends signals to the readout electronics. We call as cluster the pixels send signals after a particle go through the detector. The size of the cluster depends on the absorbed energy in the detector, so it is possible to use this information to increase the accuracy of the calorimeter. However the cluster size also depends on the temperature of the detector, so it is very important to know the accurate temperature of the detector, because a small difference can cause error in the energy measurement.\n\n\\newpage\n\\section{Detector}\n\\subsection{Concept of the detector}\n\nLet us recall the concept of the detector in Figure \\ref{fig:Concept}. The detector is built of two parts. The first part is a tracking detector, made of two layers of ALPIDE sensors. This part measures the direction of incoming particles. The second part of the detector is a calorimeter. It has a sandwich structure, and it is made of 35 ALPIDE sensor layers, separated by 4 mm thick aluminum absorber layers. This calorimeter measures the energy of the particles.\n\n\\begin{figure}[!h]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/Patient_and_detector.pdf}\n\\caption{Concept of the proton CT and its tracking calorimeter.}\n\\label{fig:Concept}\n\\end{minipage}\n\\end{figure}\n\nOne can see the structure of a detector layer in Figure \\ref{fig:Layers}. The ALPIDEs are glued on both sides of the aluminum absorber layer, alternately with a small overlap, hence the absorber functions as a support structure also. Here, two definitions are introduced for the layers, depicted in Fig.~\\ref{fig:Layers}. Its first meaning is about the engineering aspects, and consists of the aluminium absorber with APLIDE sensors on its both sides. Its second meaning is about the data analysis aspects, which considers the ALPIDE sensors between two aluminium absorbers. \n\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/What_is_a_LAYER.pdf}\n\\caption{This figure shows the structure of a layer. Gray: absorber, red: ALPIDE, blue: chip cable and orange: glue.}\n\\label{fig:Layers}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/Stack_on_a_Layer.pdf}\n\\caption{A stack is a group of ALPIDEs, which is connected to the readout electronics with the same chip cable.}\n\\label{fig:Stacks}\n\\end{minipage}\n\\end{figure}\n\nThe ALPIDEs are grouped into stacks that form a readout unit. One can see a stack on one side of the absorber in Figure \\ref{fig:Stacks}. The stacks are placed in this way, but they are mirrored.\nThe thickness of the glue, ALPIDE, and chip cable are small (Figure \\ref{fig:Layers}) but not necessarily negligible. \nThe resulted temperature difference $\\Delta T$ in the perpendicular direction of the layer can be calculated with the following equation in steady-state \\cite{hokozles}:\n\n\\begin{equation}\n\\Delta T=\\frac{q \\cdot h}{\\lambda},\n\\end{equation}\nwhere $q$ is the heat flux, $\\lambda$ stands for the thermal conductivity, and $h$ being the thickness. One can find the constants and the results in Table \\ref{tab:smalllayers}. Indeed, these $\\Delta T$ values are small, thus in later calculations their thermal resistances are neglected. \nA layer model is depicted in Figure \\ref{fig:Layer_Mosell}. The volumetric heat generation occuring in the ALPIDE detectors is modelled  as an equivalent volumetric heat generation in the absorber layer.\n\n\\begin{table}[h!]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|l|r|r|r|r|r|}\n      \\hline\n      \\textbf{Layer} & \\textbf{Material} & \\textbf{Thermal} & \\textbf{Thickness} &  \\textbf{Typical} & \\textbf{Temperature}\\\\\n      \\textbf{} & \\textbf{} & \\textbf{conductivity} & \\textbf{}  & \\textbf{heat flux} & \\textbf{difference}\\\\\n      \\hline\n       &  & $\\lambda~\\big [\\frac{W}{mK}\\big]$ & $h~[\\mu m]$ & $q~\\big[\\frac{W}{m^2}\\big]$ & $\\Delta T~[K]$\\\\\n      \\hline\n      Glue & Glue & 0.22 & 10 & 410 & 0.0186\\\\\n      ALPIDE & Silicone & 149 & 50 & 410 & $1.38\\times10^{-4}$\\\\\n      Chip cable & Polyamide & 0.24 & 50 & 410& 0.0854\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Thermal resistance parameters of ALPIDE, chip cable and glue.}\n  \\label{tab:smalllayers}\n\\end{table}\n\n\\begin{figure}[!h]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/Detector_and_absolber_connection.pdf}\n\\caption{Comparing the real detector to the model. $Q_{v_1}$ is the real volumetric heat generation in ALPIDEs, and $Q_{v_2}$ is the equivalent heat generation in absorber layer.}\n\\label{fig:Layer_Mosell}\n\\end{minipage}\n\\end{figure}\n\nOne can see the essential geometrical parameters and the thermal load of a real layer in Table \\ref{tab:layerinreality}. It is not worth to model it in its original arrangement. Consequently, a simplified model is prepared using the parameters in Table \\ref{tab:layermodell}. There are several layers side by side in which almost equal heat generation occurs. Thus, applying a periodic boundary condition, one can define a `unit' in which half of the air gap appears on both sides together with one layer. Between the layer and the air heat convection occurs. \n\n\\begin{table}[h!]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|l|r|r|r|r|r|}\n      \\hline\n      \\textbf{Part} & \\textbf{Material} & \\textbf{Height} & \\textbf{Width} &  \\textbf{Thickness} & \\textbf{Heat generation}\\\\\n      \\hline\n       &  & [mm] & $[mm]$ & $[mm]$ & $\\big [\\frac{W}{m^3}\\big ]$\\\\\n      \\hline\n      Absorber & Pure aluminum & 200 & 300 & 4 & -\\\\\n      ALPIDE & Silicone & 15 & 30 & 0.05 & $8.2\\cdot 10^6$\\\\\n      Chip cable & Polyamide & 27 & 285 & 0.05& -\\\\\n      Glue & Glue & 27 & 270 & 0.01 & -\\\\\n      Air gap & Air & 200 & 300 & 1 & -\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{The essential parameters of the real layer.}\n  \\label{tab:layerinreality}\n\\end{table}\n\n\\begin{table}[H]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|l|r|r|r|r|r|}\n      \\hline\n      \\textbf{Part} & \\textbf{Material} & \\textbf{Height} & \\textbf{Width} &  \\textbf{Thickness} & \\textbf{Heat generation}\\\\\n      \\hline\n       &  & $[mm]$ & $ [mm]$ & $[mm]$ & $\\big [\\frac{W}{m^3}\\big]$\\\\\n      \\hline\n      Absorber & Pure aluminum & 200 & 300 & 4 & $8.3\\cdot 10^4$\\\\\n      Air gap & Air & 200 & 300 & 1 & -\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Overall dimensions and the heat load in the layer model.}\n  \\label{tab:layermodell}\n\\end{table}\n\n\\subsection{Without any active cooling system}\n\nHere, I consider the detector that is covered by a casing, it protects the detector from external effects. As a first approximation, I neglect the inner temperature difference, and the entire detector is modeled as a lumped capacitance. Despite this simplification, it offers quantitative information about the characteristic heating of the device, without any active cooling system.\nOne can see the material data in Table \\ref{tab:materialmodell}, which are used to characterize the detector. In Table \\ref{tab:freeairmodell}, further parameters are summarized for modeling. Table \\ref{tab:generatedheat} contains the number of ALPIDEs and the heat generation is specified for each part. The amount of heat generation is determined based on the description of ALPIDE elements, provided by the research group. One can see the heat generated by the particles of the beam in Table \\ref{tab:particleheat}. The heat generated by the beam is several orders of magnitude smaller than the heat generated by the electronics, so I neglected it. Regarding the heat transfer coefficient, I assumed air at rest around the detector casing.\n\n\\begin{table}[H]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|r|r|r|r|}\n      \\hline\n      \\textbf{Part} & \\textbf{Density} & \\textbf{Mass} & \\textbf{Specific heat } &  \\textbf{Heat capacity}\\\\\n      \\hline\n       & $\\big [\\frac{kg}{m^3}\\big ]$ & $[kg]$ & $\\big [\\frac{J}{kg\\cdot K}\\big ]$ & $\\big [\\frac{J}{K}\\big ]$\\\\\n      \\hline\n      One layer & 2850 & 0.684 & 434 & 297\\\\\n      35 layers & 2850 & 23.94 & 434 & 10395\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Thermal parameters for the lumped capacitance model.}\n  \\label{tab:materialmodell}\n\\end{table}\n\n\\begin{table}[H]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|r|r|r|r|r|}\n      \\hline\n      \\textbf{Height} & \\textbf{Width} &  \\textbf{Thickness} & \\textbf{Surface area} & \\textbf{Heat transfer coefficient}\\\\\n      \\hline\n      $[mm]$ & $[mm]$ & $[ mm]$ & $[m^2]$ & $\\big[\\frac{W}{m^2\\cdot K}\\big ]$\\\\\n      \\hline\n      200 & 300 & 175 & $0.295$ & $ 5 $\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{The dimensions of the detectors, and the applied heat transfer coefficient. }\n  \\label{tab:freeairmodell}\n\\end{table}\n\n\\begin{table}[H]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|r|r|r|r|}\n      \\hline\n      & \\textbf{ALPIDE} & \\textbf{Stack} &  \\textbf{Layer} & \\textbf{Detector}\\\\\n      \\hline\n      Number of ALPIDEs & 1 & 9 & 108 & 3780\\\\\n      \\hline\n      Generated power & 185 mW & 1.66 W & 19.9 W & 697 W\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{The generated heat and the number of ALPIDEs.}\n  \\label{tab:generatedheat}\n\\end{table}\n\n\\begin{table}[H]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|r|r|r|}\n      \\hline\n      \\textbf{Energy of a particles} & \\textbf{Particle rate} &  \\textbf{Generated power}\\\\\n      \\hline\n      $[MeV]$ & $[\\frac{particle}{s}]$ & $[mW]$\\\\\n      \\hline\n      200 & $10^9$ &  35.2\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{The generated heat by the particles of the beam.}\n  \\label{tab:particleheat}\n\\end{table}\n\nThe lumped capacitance method is the mathematical formulation of the I.~law of thermodynamics, i.e., one has to formulate the balance of internal energy. On the left hand side, there is the time evolution of the internal energy, considering constant specific heat $c$ and mass density $\\rho$. On the right hand side, I consider the thermal load and the heat transfer by convection \\cite{hokozles}:\n\\begin{equation}\n\\label{eqn:powerballance}\n\\frac{dT}{dt}\\cdot m\\cdot c=Q-(T-T_{\\infty})\\cdot \\alpha \\cdot A~.\n\\end{equation}\nThe solution of the differential equation (\\ref{eqn:powerballance}) reads:\n\\begin{equation}\n\\label{eqn:powerballancesolution}\nT(t)=C_{const}\\cdot e^{-\\frac{ \\alpha \\cdot A}{m \\cdot c}t}+T_{\\infty}+\\frac{Q}{\\alpha \\cdot A},\n\\end{equation}\nin which I can define the time constant:\n\\begin{equation}\n\\label{eqn:timeconstant}\n\\tau=\\frac{m \\cdot c}{\\alpha \\cdot A}~,\n\\end{equation}\nand the $C_{const}$ is about to consider the initial condition\n\\begin{equation}\nT(0)=T_0.\n\\end{equation}\nOne obtains the following form of the $T(t)$ function  \\eqref{eqn:powerballancesolution}:\n\\begin{equation}\nT(t)=\\frac{Q}{\\alpha \\cdot A} + T_{\\infty}+(T_0 - T_{\\infty} - \\frac{Q}{\\alpha \\cdot A}) \\cdot e^{\\frac{-t}{\\tau}}~.\n\\end{equation}\nThe maximum allowable temperature for the detector is $40~^\\circ C$, which can be exceeded easily in continous operation, even for a one hour interval (see  Figure \\ref{fig:T_t_freeair}), and Table \\ref{tab:calsulationdata} summarizes the constants used in the calculation. The allowable $40~^\\circ C$ exceeded under $230$ s, which time interval is too short for the detection. Moreover, one should wait too much time to use the detector again. Consequently, an active cooling system is desired.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Python_code_analitic\/Angular_gap\/T_without_active_cooling_t18000.png}\n\\caption{The temperature history of the detector without active cooling.}\n\\label{fig:T_t_freeair}\n\\end{minipage}\n\\end{figure}\n\n\\begin{table}[h!]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|r|r|r|r|r|r|}\n      \\hline\n      \\textbf{$T_0$} & \\textbf{$T_{\\infty}$} &  \\textbf{$Q$} & \\textbf{$\\alpha \\cdot A$} & \\textbf{$m \\cdot c$} & \\textbf{$\\tau$}\\\\\n      \\hline\n      [$^\\circ C$] & [$^\\circ C$] & $W$ & $\\big [\\frac{W}{K}\\big ]$ & $\\big[\\frac{kJ}{K}\\big]$ & $[s]$\\\\\n      \\hline\n      25 & 25 & 697 & 1.48 & 10.4 & 7030\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{The constants used to determine the temperature history.}\n  \\label{tab:calsulationdata}\n\\end{table}\n\n\\subsection{Cooling concepts}\nHere, I compare three significantly different cooling concepts:\n\\begin{itemize}\n\\item water cooling at the edge of aluminum layers,\n\\item air cooling between the layers,\n\\item water cooling inside the aluminum layers.\n\\end{itemize}\nTwo of them use water as coolant and one concept uses air. Let me review them in the following.\n\n\\textbf{A) Water cooling at the edge of aluminum layers.}\nIn concept A, the detector layers are cooled at their top and bottom edges. The scheme of this concept is depicted in Figure \\ref{fig:Water_Geom}. The benefit, compared with concept C, is that it does not need any extra material between the layers, thus it does not cause any extra scattering, consequently, it does not have any negative effect in the data reconstruction.\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/WatercoolingGeometry.pdf}\n\\caption{Schematic arrangement of concept A.}\n\\label{fig:Water_Geom}\n\\end{minipage}\n\\end{figure}\n\nOn one hand, this is a compact and quiet arrangement, so it is perfect to use in treatment as well. It does not need a high coolant volume flow because of the high density and heat capacity of water. One needs a heat pump to cool the coolant that can be set far from the detector, and it makes possible to change the mean temperature of the detector. On the other hand, we cannot influence the temperature distribution. That could be a problem when a significant temperature difference occurs in a layer. In this case, one cannot use this concept of detector cooling. It reveals that the knowledge about the temperature distribution could be essential.\n\n\\textbf{B) Air cooling between layers.} In concept B, there is forced airflow between the detector layers, which cools the detectors directly. One can see this cooling concept in Figure \\ref{fig:Air_Geom}. Using this cooling method, almost the entire surface of the detector is exposed to forced convection. Here, I expect a more advantageous temperature distribution with less significant temperature gradient within a layer. \n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/AircoolingGeometry.pdf}\n\\caption{Schematic arrangement of concept B.}\n\\label{fig:Air_Geom}\n\\end{minipage}\n\\end{figure}\n\nIt could be easier to realize this concept since one can use a simple fan instead of a water circuit system, thus the engineering design is easier than the others. Moreover, it becomes completely irrelevant about how to protect the electronics from the coolant. However, it uses the air of the room as a coolant which restricts the minimum temperature and the cooling rate. It may result in high airspeed and disadvantageous vibrations in the structure. It is also an interesting question to investigate the temperature distribution in this case. \n\n\\textbf{C) Water cooling inside aluminum layers.}\nThis concept can produce the most homogenous temperature distribution. However, it is disadvantageous for the data analysis due to the flowing medium among the layer. Moreover, it would be significantly harder to protect the electronics in case of any failure. Later, this concept is excluded from further investigations.\n\n\n\n\\newpage\n\\section{Comparing the concepts}\nNow, my aim is to present the basic differences between concepts A and B by comparing the resulting temperature distributions to each other. In the following, I consider only one-dimensional heat conduction along the layer (direction of x axis). In case of concept A I also used symmetry boundary condition at the middle ($x=0$) of the layer.\n\n\\subsection{Concept A}\n\nIn general, the heat transfer coefficient can be high enough to replace the convection boundary condition with a so-called first-type boundary in which the temperature is prescribed directly. This is what we apply here. \nIntegrating the Fourier heat equation in Cartesian coordinate system, one obtains \\cite{hokozles}\n\\begin{equation}\nT(x)= - \\frac{q_v}{2 \\lambda} x^2 + \\frac{q_v}{2 \\lambda} \\left(\\frac{L_1}{2}\\right)^2 + T_0,\n\\end{equation}\nin which $T_0$ is the coolant temperature that used as a boundary condition, and $L_1$ stands for the height of the detector.\nDue to the symmetrical arrangement, the maximum temperature occurs at the middle, i.e., considering $x=0$,\n\\begin{equation}\nT_{max}=\\frac{q_v}{2 \\lambda} \\left(\\frac{L_1}{2}\\right)^2 + T_0.\n\\end{equation}\nThe desired operation condition requires as small as possible temperature difference inside a layer, thus the difference between $T_{max}$ and $T_0$ has the importance:\n\\begin{equation}\nT_{diff}=T_{max}-T_0=\\frac{q_v}{2 \\lambda} \\left(\\frac{L_1}{2}\\right)^2.\n\\end{equation}\nOne can see the temperature distribution in Figure \\ref{fig:Tdist_water}, and the parameters can be found in Table \\ref{tab:constantsofwatercooling}.\n\n\\begin{figure}[!h]\n\\centering\n\\centering\n\\includegraphics[width=10.5cm]{Python_code_analitic\/Angular_gap\/Tdistribution_waterCooling.png}\n\\caption{Temperature distribution in the layer, where the direction of x axis is visible in Figure \\ref{fig:Water_Geom}.}\n\\label{fig:Tdist_water}\n\\end{figure}\n\n\\begin{table}[!h]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|r|r|}\n      \\hline\n      \\textbf{Constant} & \\textbf{Notation} &  \\textbf{Value}\\\\\n      \\hline\n      Coolant temperature & $T_0$ & $25~^\\circ C$\\\\\n      Thermal conductivity of the absorber & $\\lambda$ & $237~\\frac{W}{m \\cdot K}$\\\\\n      Height of the detector & $L_1$ & $0.2~m$\\\\\n      Equivalent volumetric heat generation in the absorber & $q_v$ & $8.3 \\cdot 10^4~\\frac{W}{m^3}$\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Constants used to the calculation of the temperature distribution in concept A.}\n  \\label{tab:constantsofwatercooling}\n\\end{table}\n\nIn the current state of the detector design, the research group did not decide the material of the absorber. Most probably, it will be made from an aluminium alloy. Previously, the material properties of pure aluminium is used. However, I have found important to investigate the possible temperature difference $T_{diff}$ when the thermal conductivity is changed, accounting the fact that the material might be different in the real design. Thus thermal conductivity from $100 \\frac{W}{mK}$ to $237 \\frac{W}{mK}$ is considered, which are the realistic values for an aluminum alloy. The maximum temperature has a strong dependence on this material property as one can see in Figure \\ref{fig:Tmax_TC_Absorber_Water}. Most likely, the material will be Al1050, which has a thermal conductivity of $222 \\frac{W}{mK}$. The difference in maximum temperature between pure aluminum and Al1050 is $0.12~^\\circ C$, which is small, and the calculations with pure aluminum are also a good approximation for Al1050 in steady-state situations.\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10.5cm]{Python_code_analitic\/Angular_gap\/Tmax_TCofAbsolber_WaterCooling.png}\n\\caption{Maximum layer temperature as a function of thermal conductivity of the absorber layer. The temperature of the edge of the detector is set to 25 $^\\circ C$.}\n\\label{fig:Tmax_TC_Absorber_Water}\n\\end{figure}\n\n\\subsection{Concept B}\n\n\\subsubsection{Heat transfer coefficient}\n\nThe calculation of the heat transfer coefficient is not evident in this case. There is a wide and long, but very narrow gap between the layers. Considering it as a tube with an equivalent tube diameter does not work evidently. The heat transfer coefficient in narrow annular gaps was measured previously by Tachibana and Fukui \\cite{HTC_article1}. \nNow, I am considering the equivalent dimension between a narrow annular gap and a gap with rectangular cross-section by wrapping the rectangular one around a cylinder (see Figure \\ref{fig:Equ_Geom}). One can find the geometrical results in Table \\ref{tab:equvalentgeom}. The equivalent tube diameter is necessary for the calculation of Reynolds (Re) number. It is calculated based on the dimensions of the annular gap, but one would obtain almost the same results using the rectangular cross-section directly.\n\n\\begin{figure}[!h]\n\\centering\n\\begin{minipage}{12cm}\n\\centering\n\\includegraphics[width=12cm]{Pictures\/Equivalent_geometry.pdf}\n\\caption{Transformation between a rectangular cross-section gap and a narrow annular gap.}\n\\label{fig:Equ_Geom}\n\\end{minipage}\n\\end{figure}\n\n\\begin{table}[h!]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|r|r|r|}\n      \\hline\n      \\textbf{Dimension} & \\textbf{Notation} &  \\textbf{Calculation} & \\textbf{Value [mm]}\\\\\n      \\hline\n      Hight of the air gap & $L_1$ & - & 200\\\\\n      Width of the air gap & $L_2$ & - & 300\\\\\n      Thickness of the air gap & $L_3$ & - & 1\\\\\n      Equivalent outer diameter & $D_1$ & $\\frac{L_2}{\\pi} + L_3$ & 96,5\\\\\n      Equivalent inner diameter & $D_2$ & $\\frac{L_2}{\\pi} - L_3$ & 94,5\\\\\n      Equivalent tube diameter & $D_e$ & $\\frac{(D_1^2 - D_2^2)\\pi}{(D_1 + D_2)\\pi}=D_1 - D_2=2L_3$ & 2\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Geometry of the air gap between two layers and the equivalent annular air gap.}\n  \\label{tab:equvalentgeom}\n\\end{table}\n\nThe general strategy to determines the heat transfer coefficient requires an empirical (or semi-empirical) relation between the Nusselt (Nu) number (the dimensionless heat transfer coefficient) and some other dimensionless numbers such as the Reynolds (Re), Prandtl (Pr), Grashof (Gr), depending on the particular situation.\n\nHere, I use the following relation to determine the Nusselt number \\cite{HTC_article1}:\n\\begin{equation}\nNu=0.017\\left(1+2.3\\frac{D_e}{L}\\right)\\left(\\frac{D_2}{D_1}\\right)^{0.45} Re^{0.8} Pr^{1\/3},\n\\end{equation}\nwhere the Reynolds number and the Prandtl number are calculated using the equations \\cite{hokozles}:\n\n\\begin{figure}[H]\n        \\centering\n        \\begin{minipage}{6cm}\n        \\begin{equation}\n        Re=\\frac{wD_e}{\\nu}\n        \\end{equation}\n        \\end{minipage}\n        \\qquad\n        \\begin{minipage}{2cm}\n        ~~~~~~~~~~~~~~and\n        \\end{minipage}\n        \\qquad\n        \\begin{minipage}{6cm}\n        \\begin{equation}\n        Pr=\\frac{\\nu}{a}~.\n        \\end{equation}\n        \\end{minipage}\n\\end{figure}\n\nUsing the Nu number, I can determine the heat transfer coefficient as\n\\begin{equation}\n{\\alpha}=\\frac{Nu \\lambda}{D_e}.\n\\end{equation}\nFor this particular situation, one can see the heat transfer coefficient as a function of airspeed in Figure \\ref{fig:HTC_w}, and the constants are summarized in Table \\ref{tab:constantsofair}.\n\n\\begin{figure}[h!]\n\\centering\n\\centering\n\\includegraphics[width=12cm]{Python_code_analitic\/Angular_gap\/HTCoefficient_Airspeed.png}\n\\caption{The heat transfer coefficient as a function of airspeed.}\n\\label{fig:HTC_w}\n\\end{figure}\n\n\\begin{table}[!h]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|r|r|r|}\n      \\hline\n      \\textbf{Constant} & \\textbf{Notation} &  \\textbf{Value} & \\textbf{Unit}\\\\\n      \\hline\n      Air temperature & $T$ & $25$ & [$^\\circ C$]\\\\\n      Density & $\\rho$ & $1.19$ & $\\big[\\frac{kg}{m^3}\\big]$\\\\\n      Kinematic viscosity & $\\nu$ & $1.58 \\cdot 10^{-5}$ & \\big[$\\frac{m^2}{s}\\big]$\\\\\n      Specific heat  & $c$ & $1013$ & $\\big[\\frac{J}{kg \\cdot K}\\big]$\\\\\n      Thermal conductivity & $\\lambda$ & $0.0261$ & $\\big[\\frac{W}{m \\cdot K}\\big]$\\\\\n      Thermal diffusivity & $a$ & $2.17 \\cdot 10^{-5}$ & $\\big[\\frac{m^2}{s}\\big]$\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Parameters of the air.}\n  \\label{tab:constantsofair}\n\\end{table}\n\n\\subsubsection{Temperature distribution}\n\nIn order to obtain the temperature distribution, I have applied the following simple numerical approximation and discretization. \nThe solution is based on the internal energy balance, accounting the heat transfer among each cell. The aluminium plate and the air gap are splitted into $n$ parts, see Figure \\ref{fig:Al_and_Air} for details. Every cell of the aluminum (`w-type' cell) are connected by heat conduction and connected by heat convection to the air (`a-type' cell). Heat conduction among the a-type cells is not considered. It is possible to determine the steady-state temperature distribution by iteration among the cells.\n\n\\begin{figure}[h!]\n\\centering\n\\centering\n\\includegraphics[width=6cm]{Python_code_analitic\/Drawing_from_the_aircooling_numerical_solution.png}\n\\caption{The red cells denote the aluminum layer, the blue cells denote the air.}\n\\label{fig:Al_and_Air}\n\\end{figure}\n\nThe results are presented in Figure \\ref{fig:T_air_w5}. It makes apparent how the temperature of air increases between the entry and the exit points. More importantly, it influences the temperature distribution of a layer as well. It makes the heat transfer less intensive at the end, thus higher temperature difference can occur together with less homogeneous distribution. Consequently, the highest temperature can be found at the uttermost point from the entry of the air.\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{Python_code_analitic\/Angular_gap\/AirandWalltemperature_w10.png}\n\\caption{The air and the layer temperature distribution in a streamline. The direction of x axis is visible in Figure \\ref{fig:Air_Geom}. Average airspeed is set to 10 $\\frac{m}{s}$, and the heat transfer coefficient is 62 $\\frac{W}{m^2 \\cdot K}$.}\n\\label{fig:T_air_w5}\n\\end{figure}\n\n\\subsubsection{Maximum temperature}\n\nOne can see the maximum temperature as a function of airspeed in the Figure \\ref{fig:Tmax_airspeed}. The maximum temperature is lower than the allowed $40~^\\circ C$ if the airspeed is higher than $5 \\frac{m}{s}$. However, it is safer to ensure $15 \\frac{m}{s}$ airspeed, but it is not worth to increase it. The optimum interval could be between $10$ to $15$ m\/s. In the further calculations, $10$ m\/s is used. Unfortunately, that speed could be too noisy and too high for clinical applications and may cause other problems as well.\n\n\\begin{figure}[!h]\n\\centering\n\\centering\n\\includegraphics[width=11cm]{Python_code_analitic\/Angular_gap\/MaxWalltemperature_Airspeed2.png}\n\\caption{Maximum temperature as a function of airspeed. The room temperature is 25 $^\\circ C$.}\n\\label{fig:Tmax_airspeed}\n\\end{figure}\n\nOne significant uncertainty is the determination of the heat transfer coefficient. The heat convection between the wall and airflow is a very complicated process because of the irregularity and uncertainty in the shape of the surface. Hence it is important to investigate the effect of changing the heat transfer coefficient. One can see the maximum temperature as a function of heat transfer coefficient in the Figure \\ref{fig:Tmax_HTC}. The domain of interest starts at $40~\\frac{W}{m^2 \\cdot K}$ to $80~\\frac{W}{m^2 \\cdot K}$. Within this domain, the maximum temperature is safely lower than the allowed $40~^\\circ C$.\n\n\\begin{figure}[!h]\n\\centering\n\\centering\n\\includegraphics[width=11cm]{Python_code_analitic\/Angular_gap\/MaxT_HTCoefficient_w10.png}\n\\caption{The maximum temperature as a function of the heat transfer coefficient. The room temperature is set to 25 $^\\circ C$.}\n\\label{fig:Tmax_HTC}\n\\end{figure}\n\nPreviously, I mentioned the uncertainty due to the material change that should be investigated. Now, I handle the thermal conductivity of the absorber layer as a parameter, see Fig.~\\ref{fig:Tmax_TC} for details. The domain of interest is $100 \\frac{W}{mK}$ to $237 \\frac{W}{mK}$ which covers all the possibilities for aluminium alloys. \n The maximum temperature variation is found to be $0.3~^\\circ C$ in this interval, hence the thermal conductivity of the absorber does not have a big impact on the maximum. One of the most likely aluminum alloy for the material of the detector is Al1050, which has the thermal conductivity of $222 \\frac{W}{mK}$. The difference between the maximum temperature in case of pure aluminum and Al1050 is $0.02~^\\circ C$, which is negligible.\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{Python_code_analitic\/Angular_gap\/MaxTemperature_tC_of_Al_w10.png}\n\\caption{The maximum temperature as a function of thermal conductivity.}\n\\label{fig:Tmax_TC}\n\\end{figure}\n\n\\subsection{Comparison}\n\nThe maximum temperature difference in a layer is very important, since the cluster size and the noise rate of the ALPIDE are temperature dependent. I compared the maximum temperature differences for concepts A and B as a function of the airspeed, thermal conductivity and heat transfer coefficient. \n\nOne can see the maximum temperature difference as a function of airspeed in case of air and water cooling in the Figure \\ref{fig:Tdiff_a_w}. As one can see, there is no significant difference in this value. If the airspeed is higher than $8 \\frac{m}{s}$ the temperature difference is slightly lower for concept B. Otherwise, the temperature difference is higher than for water cooling. The temperature difference in case of concept B, considering $10 \\frac{m}{s}$ airseed, is $1.6~^\\circ C$. The maximum temperature difference in case of concept A is given as a benchmark, and it is constantly $1.8~^\\circ C$. In both cases, the temperature difference is lower than the allowed 5 $^\\circ C$.\n\n\\begin{figure}[h!]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{Python_code_analitic\/Angular_gap\/temperatureDifference_Air_vs_Water.png}\n\\caption{The temperature difference in a  layer in the case of air and water cooling. The temperature difference in case of concept A is given as benchmark as it is independent of the airspeed.}\n\\label{fig:Tdiff_a_w}\n\\end{figure}\n\nThe maximum temperature difference depends on the thermal conductivity of the layer, as one can see in Figure \\ref{fig:Tdiff_a_w_TC}, using $10 \\frac{m}{s}$ airspeed. One can see that the temperature difference is increasing in both cases with decreasing the thermal conductivity. The concept A, using water cooling, seems to be more sensitive for this parameter. Considering again the Al1050 alloy, the temperature difference in the layer is $1.9~^\\circ C$ for concept A and $1.7~^\\circ C$ in case of concept B.\n\n\\begin{figure}[!h]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{Python_code_analitic\/Angular_gap\/temperatureDifference_thermalConductivityofAir_W10.png}\n\\caption{Temperature difference in layer in case of air- and watercooling in function of the thermal conductivity of the layer. The average airspeed is set to 10 $\\frac{m}{s}$ in case of concept B.}\n\\label{fig:Tdiff_a_w_TC}\n\\end{figure}\n\nChanges in the heat transfer coefficient between the layer and the air also has an effect in the maximum temperature difference. One can see the outcome in the Figure \\ref{fig:Tdiff_a_w_HTC}. If the heat transfer coefficient is increasing the temperature difference is also increasing. The reason behind this effect is the changes in the temperature distribution in the streamline. In Figure \\ref{fig:T_line_lowHTC}, one can see the temperature distribution using unrealistically low heat transfer coefficient, and in Figure \\ref{fig:T_line_highHTC}, with unrealistically high  heat transfer coefficient. The realistic zone of the heat transfer coefficient is between $40~\\frac{W}{m^2 \\cdot K}$ and $80~\\frac{W}{m^2 \\cdot K}$.\nIn this domain, the maximum temperature difference starts from $1.2~^\\circ C$ to $1.8~^\\circ C$ in case of concept B. In case of concept A, the maximum temperature difference is not affected by the heat transfer coefficient, so it is only given as a benchmark, it is constantly $1.8~^\\circ C$.\n\n\\begin{figure}[!h]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{Python_code_analitic\/Angular_gap\/DeltaT_HTCoefficient_w10.png}\n\\caption{The temperature difference in a layer in case of air and water cooling as a function of the heat transfer coefficient. The temperature difference in case of concept A is given as a benchmark.}\n\\label{fig:Tdiff_a_w_HTC}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{Python_code_analitic\/Angular_gap\/AirandWalltemperature_w10_alpha15.png}\n\\caption{The layer and the air temperature with unrealistically low (15 $\\frac{W}{m^2 \\cdot K}$) heat transfer coefficient, in stream direction. The direction of x axis is visible in Figure \\ref{fig:Air_Geom}. The average air speed is set to 10 $\\frac{m}{s}$.}\n\\label{fig:T_line_lowHTC}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{Python_code_analitic\/Angular_gap\/AirandWalltemperature_w10_alpha200.png}\n\\caption{The layer and the air temperature with unrealistically high (200 $\\frac{W}{m^2 \\cdot K}$) heat transfer coefficient, in stream direction. The direction of x axis is visible in Figure \\ref{fig:Air_Geom}. The average air speed is set to 10 $\\frac{m}{s}$.}\n\\label{fig:T_line_highHTC}\n\\end{figure}\n\n\\newpage\n\\section{Cost estimation and environmental effects}\n\nAs a result of the previous calculations, I considered that both concept A and concept B meets with the requirements of the hadron-tracking calorimeter, so the decision between these concepts can be made based on other aspects. One of these aspects is the cost of the cooling system, so I compare the possible cost of concept A and concept B in this chapter. This comparison is a approximation of the costs, so it contains only the cost of the main elements of the cooling system. The prices in Table \\ref{tab:costofA} and Table \\ref{tab:costofB} are given as an example for the price of the parts.\n\n\\subsection{Concept A}\n\nIn case of concept \"A\" I use a water cooling circle. To protect the hadron-tracking calorimeter from a possible leak, I need to use water cooling circle, which operates under room pressure. If the cooling system operates under room pressure, in case of a leak the air of the room move into the tube, not the water move out. To operate under room pressure I need to use a vacuum pump. In case of a leak the vacuum pump has to be able to remove all water from the cooling circle, so I need the insert a vacuum tank between the cooling circle and the vacuum pump. In normal working this tank mainly filled with air. If there is a leak in the cooling circle this tank will be filled with the water of the cooling circle. The cooling circle contains a cooler, to cool the water under the temperature of the room. The cooler contains a pump, which circulates the water in the cooling circle, so I did not calculate the price of the pump separately.  The cooling circle also contains cooling plates to cool the detector. I only find smaller cooling plates then the top or the bottom of the hadron-tracking calorimeter, so I calculated with two cooling plate in the top and two cooling plate in the bottom of the detector. The cooling circle also contains tubes and fittings. I just estimated price of them, because without an exact plan of the cooling system I could not find price for them. The elements of concept A is visible in Figure \\ref{fig:A_concept_gazd}. One can see the estimated cost of this solution in Table \\ref{tab:costofA}.\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{Pictures\/Aconcept_gazdasagielemzes.pdf}\n\\caption{The water cooling circuit of concept A.}\n\\label{fig:A_concept_gazd}\n\\end{figure}\n\n\\begin{table}[!h]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|r|r|r|}\n      \\hline\n      \\textbf{Component} & \\textbf{Main operation data} &  \\textbf{Number of components} & \\textbf{Price}\\\\\n      \\hline\n\tWater cooler \\cite{cooler} & $2400~W,~16.3~\\frac{l}{min},~2.1~bar$ & 1 & 4090 \\texteuro \\\\\n\tVacuum pump \\cite{vacuumpump} & 100 mbar & 1 & 640 \\texteuro \\\\\n\tCooling plate \\cite{coolingplate} & 304.8 mm $\\cdot$ 88.9 mm& 4 & 110 \\texteuro \\\\\n\tVacuum tank \\cite{vacuumtank}  & 25 liter & 1 & 50 \\texteuro \\\\\n\tTubes and fittings & - & - & 300 \\texteuro \\\\\n\tOverall & - & - & 5520 \\texteuro \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Cost of concept \"A\".}\n  \\label{tab:costofA}\n\\end{table}\n\n\\subsection{Concept B}\n\nI need to calculate the volume flow through the detector and the pressure drop in the detector to find a corresponding fan to concept B.\n\nPressure drop in case of concept B, if the temperature of the room is $T_{room}=25~^\\circ C$. The equivalent diameter is:\n\\begin{equation}\nd_{eq}=4 \\cdot \\frac{L_2 \\cdot L_3}{2 \\cdot L_2 + 2 \\cdot L_3}=1.99~mm,\n\\end{equation}\n where $L_2=300~mm$ and $L_3=1~mm$. The airspeed is $w_{air}=10~\\frac{m}{s}$, and the kinematic viscosity is $\\nu=1.58 \\cdot 10^{-5}$ $\\frac{m^2}{s}$. The Reynolds-number of the airflow is:\n\\begin{equation}\nRe=\\frac{w_{air} \\cdot d_{eq}}{\\nu}=1250,\n\\end{equation}\nwhich means the airflow is laminar between the detector layers. The pressure drop can be calculated with this equation\n\\begin{equation}\n\\Delta p=\\frac{3 \\cdot L_1 \\nu \\cdot \\rho w_{air}}{\\left(\\frac{L_3}{2}\\right)^2}=438~Pa\n\\end{equation}\nbased on Newton's law of viscosity, where $\\rho=1.19~\\frac{kg}{m^3}$ and $L_1=200~mm$. The volume flow between the layers is\n\\begin{equation}\nQ=n_{layers} \\cdot L_3 \\cdot L_2 \\cdot w_{air}=0.105~\\frac{m^3}{s}=378~\\frac{m^3}{h},\n\\end{equation}\nwhere $n_{layers}=35$.\n\nI need to have a fun in the cooling system to generate airflow in the detector. The fan is as big as the hadron-tracking calorimeter and it is possible the fan generate vibrations, so it has to operate separately from the detector connected with a tube. I also need an air filter before the detector, to protect the hadron-tracking calorimeter from the dust in the air. The elements of the cooling system is visible in Figure \\ref{fig:B_concept_gazd}.\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=3.5cm]{Pictures\/Bconcept_gazdasagielemzes.pdf}\n\\caption{The elements of B cooling system concept.}\n\\label{fig:B_concept_gazd}\n\\end{figure}\n\n\\begin{table}[!h]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|l|r|r|r|}\n      \\hline\n      \\textbf{Component} & \\textbf{Main operation data} &  \\textbf{Number of components} & \\textbf{Price}\\\\\n      \\hline\n\tFan \\cite{fan} & 950 Pa, $400~\\frac{m^3}{h}, 485 W$ & 1 & 890 \\texteuro \\\\\n\tTube \\cite{tube} & l=10 m, d=250 mm & 1 & 50 \\texteuro \\\\\n\tFittings and filter & - & - & 300 \\texteuro \\\\\n\tOverall & - & - & 1240 \\texteuro \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Cost of concept \"B\".}\n  \\label{tab:costofB}\n\\end{table}\n\n\\subsection{Comparison}\n\nAs a result of the cost estimation one can conclude that the concept B is cheaper than concept A.  As environmental effect one can consider that concept B use less energy to operate than concept A, because the water cooler needs more energy to cool the water under room temperature, than the fan needs to transfer air through the detector.\n\nIt is possible to use a concept like A, but use thermoelectric cooler and heat sinks with heat pipe and funs to cool the top and the bottom of the detector. This solution results same temperature distribution than concept A, but it can be cheaper, smaller than concept A. In case of this concept it is also possible to cool the detector under room temperature and the maximum temperature also adjustable. This concept can be more advantageous than concept A in case of beam test, but concept A is more suitable for clinical use, because in this case the generated heat is transported out of the treatment room.\n\n\\newpage\n\\section{Finite element simulation of concept A}\n\nMy next work in the research team was the more accurate calculation of the temperature distribution in case of concept A. I decided to check the effect of contact thermal resistances and inhomogeneous load on the temperature distribution. I took into account the heat generation of the ALPIDEs as a heat flux in the contact surfaces of them. I calculated with a two dimensional model (Figure \\ref{fig:Tresistance_layer_structure}) when I calculate the effect of the contact resistances, and I calculate with a three dimensional model (Figure \\ref{fig:Stacks}) when I calculated the effect of the inhomogeneous load. I used $\\lambda_{Al1050}=222~\\frac{W}{m \\cdot K}$ as the thermal conductivity of the absorber layers in both calculation.\n\n\\subsection{The effect of contact thermal resistances}\n\nFirstly I calculated with 4 mm thick absorber layer. In reality the layer will be build from 2 mm thick absorber layer in the middle, and two 1 mm thick layer in both side. The ALPIDEs will be glued into the 1 mm thick plates, as one can see in \\ref{fig:Tresistance_layer_structure}. There will be contact resistances between this plates and between the cooling plate and this plates. I calculate the contact resistance between the plates as the resistance of the average air gap between the layers, based on \\cite{contactRes}. I calculate the average air gap between two plate with $\\delta_1=\\frac{2 \\cdot R_z}{2}=R_z$ equation. The aluminum plate is cold rolled aluminum, so the mean roughness depth is $R_z=12.5~\\mu m$. The average distance is $\\delta_1=12.5~\\mu m$. Between the cooling plate and the edge of the absolber plates I calculated with $\\delta_2=200~\\mu m$, because of the uncertainty of the positioning of the plates. It is necessary to use thermal conductive grease between the cooling plate and the edge of the absolber plates. I considered $\\lambda_{air}=0.0261~\\frac{W}{m \\cdot K}$ as the thermal conductivity of air and $\\lambda_{conductive}=8.5~\\frac{W}{m \\cdot K}$ \\cite{thermalpaste} as the thermal conductivity of a typical thermal conductive paste. The thermal resistance between two layer is $R_1=\\frac{\\delta_1}{\\lambda_{air}}=4.79 \\cdot 10^{-4}~\\frac{m^2 \\cdot K}{W}$ and between the cooling plate and the edge of the absorber layers is $R_2=\\frac{\\delta_2}{\\lambda_{conductive}}=2.35 \\cdot 10^{-5}~\\frac{m^2 \\cdot K}{W}$.\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{Pictures\/ContactResistances.pdf}\n\\caption{The model of the layer with thermal resistances.}\n\\label{fig:Tresistance_layer_structure}\n\\end{figure}\n\nOne can see the model of the layer and the contact resistances in Figure \\ref{fig:Tresistance_layer_structure} and the result of the two dimensional simulation in Figure \\ref{fig:Tresistance_results1} and Figure \\ref{fig:Tresistance_results2}. I calculated the maximum temperature difference in that part of the surface of the absorber layer, where ALPIDEs are glued.\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{SzakdogaAnsys\/Thermal_Resistance_Water_Cooling\/Results\/Delta_T_Thermal_Insulance.png}\n\\caption{The effect of the thermal resistance between layers in the maximum temperature difference.}\n\\label{fig:Tresistance_results1}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{SzakdogaAnsys\/Thermal_Resistance_Water_Cooling\/Results\/Delta_T_Thermal_Insulance_Cooling_Plate.png}\n\\caption{The effect of the thermal resistance between the cooling plate and absorber the layers in the maximum temperature difference.}\n\\label{fig:Tresistance_results2}\n\\end{figure}\n\nI have checked the mesh dependence of the solution in case when  $R_1$ is $4.79 \\cdot 10^{-4}~\\frac{m^2 \\cdot K}{W}$ and $R_2$ is $2.35 \\cdot 10^{-5}~\\frac{m^2 \\cdot K}{W}$. I checked the maximum temperature difference ($\\Delta T_{max}$) in y=2 mm line, which is visible in Table \\ref{tab:Tresistance_mesh}. I found that the mesh with 0.5 mm edge length seems to be the most accurate. The maximum temperature difference between 1 mm edge length and 0.5 mm edge length is small compared with the measured temperature differences, so the solution is mesh independent.\n\n\\begin{table}[H]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|r|r|r|r|}\n      \\hline\n      $L_{max}$ in x direction (mm) & $L_{max}$ in y direction (mm) & Number of elements & $\\Delta T_{max}~(^\\circ C)$\\\\\n      \\hline\n\t1 & 1 & 800 & - \\\\\n\t0.5 & 0.5 & 3200 & 0.017\\\\\n\t2 & 0.5 & 824 & - \\\\\n\t1 & 0.25 & 3216 & 0.104\\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Outcome of the mesh dependence test, where $\\Delta T_{max}$ is the maximum difference of temperature compared with the previous mesh. $L_{max}$ is the maximum edge length in the given direction.}\n  \\label{tab:Tresistance_mesh}\n\\end{table}\n\nI considered based on Figure \\ref{fig:Tresistance_results1} and Figure \\ref{fig:Tresistance_results2} that the contact thermal resistance between the layers has minimal effect on the temperature difference, and the contact thermal resistance between the cooling plate and the layers has only marginal effect on the temperature distribution.\n\n\\subsection{The effect of inhomogeneous load}\n\nI obtained the temperature distribution in the case, if only the middle of the detector is used. I expect the maximum temperature difference will be smaller than in case of full load, but it is interesting how much it will be, because the partial load will be a typical usage, if we have to take image from a smaller part of the human body then the size of the full detector area.\n\nI used $P_{standby}=60~mW$ heat generation in every ALPIDE detector, which was inactive in the measurement, which is the power consumption of waiting for hits. I used $P_{active}=185~mW$ heat generation in every active ALPIDE, which is the heat generation working with maximum readout capacity.\n\nI obtained the temperature distribution in an x direction path, defined by z=0 mm and y=0.002 mm. The temperature distribution in this path is visible compared with the temperature distribution of full load in Figure \\ref{fig:T_x_partial_load1}.\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{SzakdogaAnsys\/Inhomogenity_Water_Cooling\/Results\/Full_and_partial_load_x.png}\n\\caption{The temperature distribution in case of full and partial load in path z=0 mm and y=0.002 mm.}\n\\label{fig:T_x_partial_load1}\n\\end{figure}\n\nOne can see the temperature in a path, defined by x=0 mm and y=0.002 mm, in Figure \\ref{fig:T_x_partial_load2}.\n\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{SzakdogaAnsys\/Inhomogenity_Water_Cooling\/Results\/Full_and_partial_load_z.png}\n\\caption{The temperature distribution in case of full and partial load in path x=0 mm and y=0.002 mm.}\n\\label{fig:T_x_partial_load2}\n\\end{figure}\n\nThe result of the mesh dependence test is visible in Table \\ref{tab:T_inhomogen_mesh}.\n\n\\begin{table}[H]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|r|r|r|r|}\n      \\hline\n      Maximum edge size (mm) & Number of elements & $\\Delta T_{max}~(^\\circ C)$ full load & $\\Delta T_{max}~(^\\circ C)$ full load \\\\\n      \\hline\n\t7.5 & 1234 & - & - \\\\\n\t5 & 2400 & 0.171 & 0.077 \\\\\n\t2.5 & 19200 & 0.086 & 0.038 \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Outcome of the mesh dependence test in case of full and partial load, where the $\\Delta T_{max}$ is the maximum temperature difference compared with the previous mesh.}\n  \\label{tab:T_inhomogen_mesh}\n\\end{table}\n\nThe temperature difference was $2.1~^\\circ C$ in case of full load and $1.1~^\\circ C$ in case of partial load.\n\n\\newpage\n\\section{Calculation of transient behavior}\n\nIn this subsection I focus on how much time does the heat up takes, because it effects both the clinical use, both the measurements during the development of the detector. I used the same geometry than I used in Chapter 3.1, so I used a one dimensional model. I calculated with the thermal conductivity of Al1050, which is $222~\\frac{W}{m \\cdot K}$. The heat transfer is described by the following equation:\n\\begin{equation}\n\\label{mainEq}\n\\frac{\\partial T}{\\partial t}=\\alpha \\cdot \\frac{\\partial^2 T}{\\partial x^2} + \\frac{\\dot q_v}{\\rho \\cdot c},\n\\end{equation}\nwhere $\\alpha=\\frac{\\lambda}{\\rho \\cdot c}$.\n\nInitial condition:\n\\begin{equation}\n\\label{eqn:init1}\nT(x,0)=T_0=25~^\\circ C.\n\\end{equation}\nBoundary conditions:\n\\begin{figure}[H]\n        \\centering\n        \\begin{minipage}{6cm}\n        \\begin{equation}\n\tT(0,t)=T_1=25~^\\circ C\n        \\end{equation}\n        \\end{minipage}\n        \\qquad\n        \\begin{minipage}{2cm}\n        ~~~~~~~~~~~~~~and\n        \\end{minipage}\n        \\qquad\n        \\begin{minipage}{6cm}\n        \\begin{equation}\n\tT(0,t)=T_2=25~^\\circ C.\n        \\end{equation}\n        \\end{minipage}\n\\end{figure}\n\nThe temperature can be calculated as the sum of steady state and the transient temperature:\n\\begin{equation}\nT(x,t)=T_s(x) + T_h(x,t).\n\\end{equation}\nThe steady state temperature is \\cite{hokozles}:\n\\begin{equation}\nT_s(x)=-\\frac{\\dot q_v}{\\lambda} \\cdot \\frac{x^2}{2} + C_1 \\cdot x + C_2,\n\\end{equation}\nwhere $C_2=T_1$ and $C_1=\\frac{T_2 - T_1}{L} + \\frac{\\dot q_v}{\\lambda} \\cdot \\frac{L}{2}$. The transient temperature is \\cite{hokozles}:\n\\begin{equation}\nT_h(x,t)=\\sum_{n=1}^{n_{max}} D_n \\cdot e^{-\\Lambda_n \\cdot t} \\cdot sin \\left(\\frac{n \\cdot \\pi}{L} \\cdot x \\right),\n\\end{equation}\nwhere $\\Lambda_n=\\alpha \\cdot \\left( \\frac{n \\cdot \\pi}{L} \\right)^2$ and $D_n=\\frac{\\int_{0}^{L} (T(x,0)-T_s(x)) \\cdot sin \\left( \\frac{n \\cdot \\pi}{L} \\cdot x \\right) dx}{\\int_{0}^{L} sin^2 \\left( \\frac{n \\cdot \\pi}{L} \\cdot x \\right) dx}$.\n\nOne can see the temperature of the middle of the detector in function of time in Figure \\ref{fig:Tmiddle_t}, and the parameters, which was used to the calculation, in Table \\ref{tab:consHeatup}.\n\\begin{figure}[H]\n\\centering\n\\centering\n\\includegraphics[width=10cm]{SzakdogaAnsys\/Heatup_T_t.png}\n\\caption{The temperature of the middle point of the detector in function of time.}\n\\label{fig:Tmiddle_t}\n\\end{figure}\n\n\\begin{table}[H]\n  \\begin{center}\n   \n   \n    \\begin{tabular}{|r|r|r|}\n      \\hline\n      Parameter & Value\\\\\n      \\hline\n\tInitial temperature of the layer ($T_1$) & $25~^\\circ C$\\\\\n\tTemperature of the edge of the layer ($T_1$ and $T_2$) & $25~^\\circ C$\\\\\n\tThermal conductivity of the layer ($\\lambda$) & $222~\\frac{W}{m \\cdot K}$ \\\\\n\tMass density of the layer ($\\rho$) & $2710~\\frac{kg}{m^3}$ \\\\\n\tSpecific heat of the layer (c) & $434~\\frac{J}{kg \\cdot K}$ \\\\\n\tEquivalent volumetric heat generation ($q_v$) & $8.3 \\cdot 10^4~\\frac{W}{m^3}$ \\\\\n\tMaximal item of Fourier series ($n_{max}$) & 50 \\\\\n      \\hline\n    \\end{tabular}\n  \\end{center}\n  \\caption{Parameters used in the heat up calculation.}\n  \\label{tab:consHeatup}\n\\end{table}\n\nIt takes 134 s, while the temperature of the middle point of the layer reach $\\pm 0.1~^\\circ C$ zone around the steady state temperature of the middle point. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\newpage\n\\section{Looking Ahead}\n\nIn this work I analysed the temperature distribution of a layer of the hadron-tracking calorimeter. The detector is going to contain two layer before the calorimeter, which measure the path of the particles. This layers are going to contains a carbon composite layer in the middle instead of an aluminum absorber layer. This carbon composite layer is going to work as a support structure, and it is thinner than aluminum absorber layer. The thermal conductivity of carbon composite is anisotropic, it is reasonable in longitudinal direction, but low in perpendicular direction. It is essential to analyze the temperature distribution in this layers, and design new cooling concepts for these layers, if it is necessary.\n\nAnother important engineering task in the development of  proton computer tomograph is the comparison of the temperature distribution between layers. This task requires more information about the power consumption of the tracking detectors and the number of particles, which goes through them. It is possible to reach this information with test beam measurements, so one of the next engineering task should be the measurement of the electric power consumption in case of different particle fluxes.\n\n\\newpage\n\\section{Summary}\n\nCancer started to become the leading cause of death in the developed world, as it is responsible for 25\\% of all death in Hungary. Mainly, there are three ways of treatment: surgery, radiotherapy, and chemotherapy. Hadron therapy is a novel treatment in radiotherapy. It is beneficial because it generates less radiation in healthy tissues than X-ray radiotherapy, thus it leads to fewer side effects, and it allows a higher daily dose, which increases the effectiveness of treatment.\n\nHadron therapy requires an accurate three-dimensional map about the electron density of the body. The most accurate solution to obtain the map uses hadrons, however, it requires the development of a proton computer tomograph (pCT). Our group, the Bergen pCT Collaboration, aims to develop a pCT that can meet the requirements of the clinical use. The detector of the pCT is built of silicon pixel tracking detectors and aluminum absorber layers.\n\nIn the collaboration, my role was to investigate the temperature distribution of the detector, which is essential because of the accuracy of the energy measurement based on the temperature of the detector. It is indispensable in the development of the pCT detector.\n\nI applied the lumped capacitance method to estimate the thermal behaviour of the detector, containing 35 absorber layers. This simple analysis shows the necessity for cooling. Hence, two cooling concepts are elaborated in this work. In concept A, water cools the edges of the aluminum absorber layers. In concept B, the air is circulated between the detector layers. My goal was to quantify the differences in the steady-state temperature distribution and compare the concepts to each other.\n\nFirstly I calculated with analytic methods the steady state temperature distribution in case of both concepts. Both of the them satisfies the first requirement, which is to keep the maximum temperature lower than 40  $^\\circ C$. The allowed temperature difference is 5 $^\\circ C$. Both of the concepts meets this requirement also, but concept B has a bit better performance as the temperature difference is 1.6 $^\\circ C$ in this case, compared with 1.8 $^\\circ C$ in case of concept A.\n\nSecondly I estimated the cost of both concept. I considered concept A is more expensive than concept B. It is possible to use a third solution, which cools the top and the bottom of the hadron-tracking calorimeter, like concept A, but contains thermoelectric coolers, heat pipes, heat sinks and funs instead of a water cooling circuit. This concept can be smaller and cheaper than concept A, but it has the advantageous properties of concept A.\n\nThirdly I used finite element simulations to determinate the effect of the contact thermal resistances and partial load in the temperature distribution, in case of concept A. I obtained that a reasonable contact thermal resistance has small effect on the temperature distribution, as it cause maximum $0.2~^\\circ C$ increase in the temperature difference. In case of partial load the temperature difference is decrease, as it was expected. I calculated the time, which is required to reach steady state temperature distribution. It takes 134 s, which is suitable for clinical use and test beam measurements.\n\n\\newpage\n\\section{\u00d6sszefoglal\u00f3}\n\nNapjainkban a r\u00e1k az egyik vezet\u0151 hal\u00e1lokk\u00e1 kezd v\u00e1llni a fejledt vil\u00e1gban, Magyarorsz\u00e1gon a teljes elhal\u00e1loz\u00e1s 25\\%-\u00e1\u00e9rt felel\u0151s. A r\u00e1knak h\u00e1rom f\u00e9le f\u0151 gy\u00f3gy\\-m\u00f3d\\-ja van: az els\u0151 a seb\u00e9szet, a m\u00e1sodik a sug\u00e1rter\u00e1pia \u00e9s a harmadik a kemoter\u00e1pia. A hadronter\u00e1pia egy \u00faj sug\u00e1rter\u00e1pi\u00e1s kezel\u00e9s. El\u0151ny\u00f6s, mert kevesebb ioniz\u00e1ci\u00f3t okoz az eg\u00e9szs\u00e9ges sz\u00f6vetekben mint a hagyom\u00e1nyos, r\u00f6ntgen sug\u00e1rz\u00e1ssal v\u00e9gzet sug\u00e1rter\u00e1pia. Ennek k\u00f6sz\u00f6nhet\u0151en kevesebb mell\u00e9khat\u00e1ssal j\u00e1r, \u00e9s nagyobb napi d\u00f3zist tesz lehet\u0151v\u00e9, ezzel n\u00f6velve a kezel\u00e9s hat\u00e1soss\u00e1g\u00e1t.\n\nA hadronter\u00e1pi\u00e1hoz sz\u00fcks\u00e9ges egy pontos t\u00e9rbeli k\u00e9pet k\u00e9sz\u00edteni a p\u00e1ciens test\u00e9nek elektrons\u0171r\u0171s\u00e9g\u00e9r\u0151l. A legpontosabb megold\u00e1s, ha ezt a k\u00e9pet hadronok haszn\u00e1lat\u00e1val k\u00e9sz\u00edtj\u00fck el, de ez egy proton computer tomogr\u00e1f (pCT) kifejleszt\u00e9s\u00e9t ig\u00e9nyli. A Bergen pCT egy\u00fcttm\u0171k\u00f6d\u00e9s c\u00e9lja, hogy kifejlesszen egy ilyen eszk\u00f6zt, ami a klinikai felhaszn\u00e1l\u00e1si szempontoknak eleget tesz. A pCT detektora szil\u00edciumpixel-nyomk\u00f6vet\u0151 detektorokb\u00f3l \u00e9s alum\u00ednum abszorber r\u00e9tegekb\u0151l \u00e9p\u00fcl fel.\n\nAz egy\u00fcttm\u0171k\u00f6d\u00e9sben bet\u00f6lt\u00f6tt szerepem a detektor h\u0151m\u00e9rs\u00e9klet-eloszl\u00e1s\u00e1nak meg\\-ha\\-t\u00e1\\-ro\\-z\u00e1\\-s\u00e1\\-ra ir\u00e1nyult. Ennek vizsg\u00e1lata elengedhetetlen a pontos m\u00e9r\u00e9sek elv\u00e9gz\u00e9s\u00e9hez. El\u0151sz\u00f6r, egy egy\\-sze\\-r\u0171, koncentr\u00e1lt param\u00e9ter\u0171 modellel becs\u00fcltem a 35 r\u00e9tegb\u0151l \u00e1ll\u00f3 detektor termikus viselked\u00e9s\u00e9t. Ebb\u0151l egy\u00e9rtelm\u0171en l\u00e1tni, hogy a h\u0171t\u00e9s elengedhetetlen.\nK\u00e9t f\u00e9le h\u0171t\u00e9si koncepci\u00f3t dolgoztam ki. Az ``A'' koncepci\u00f3 eset\u00e9n v\u00edzzel h\u0171tj\u00f6m az abszorber r\u00e9tegek sz\u00e9leit, m\u00edg a ``B'' koncepci\u00f3 eset\u00e9n leveg\u0151t keringtetek a r\u00e9tegek k\u00f6z\u00f6tt. C\u00e9lom a kialakul\u00f3, \u00e1lland\u00f3sult \u00e1llapot\u00fa h\u0151m\u00e9rs\u00e9klet-eloszl\u00e1sok meghat\u00e1roz\u00e1sa \u00e9s a k\u00e9t koncepci\u00f3 \u00f6sszehasonl\u00edt\u00e1sa.\n\nEl\u0151sz\u00f6r analitikusan meghat\u00e1roztam az \u00e1lland\u00f3sult h\u0151m\u00e9rs\u00e9kleteloszl\u00e1st mindk\u00e9t koncepci\u00f3 eset\u00e9ben. A megengedett maximum h\u0151m\u00e9rs\u00e9klet a detekteorban $40~^\\circ C$, amit mindk\u00e9t koncepci\u00f3 teljes\u00edt. A megengedett h\u0151m\u00e9rs\u00e9klet-k\u00fcl\u00f6nbs\u00e9g $5~^\\circ C$, amit \u00fagyszint\u00e9n mindk\u00e9t koncepci\u00f3 teljes\u00edt, de a B koncepci\u00f3 ilyen szempontb\u00f3l el\u0151ny\u00f6sebb, mivel ebben az esetben a h\u0151m\u00e9rs\u00e9klet-k\u00fcl\u00f6nbs\u00e9g csak $1.6~^\\circ C$, am\u00edg az A koncepci\u00f3 eset\u00e9ben $1.8~^\\circ C$.\n\nM\u00e1sodszor megbecs\u00fcltem a k\u00e9t koncepci\u00f3 lehets\u00e9ges k\u00f6lts\u00e9geit. Eredm\u00e9ny\u00fcl azt kaptam, hogy az A koncepci\u00f3 l\u00e9nyegesen dr\u00e1g\u00e1bb. Lehets\u00e9ges egy harmadik k\u00f6ncepci\u00f3 alkalmaz\u00e1s is, amely eset\u00e9ben ugyan\u00fagy a detektor als\u00f3 \u00e9s fels\u0151 fel\u00fclet\u00e9t h\u0171tj\u00fak, mint az A koncepci\u00f3 eset\u00e9ben, azonban v\u00edzh\u0171t\u00e9s helyett Peltier-elemek, h\u0151cs\u00f6vek, bord\u00e1k \u00e9s ventill\u00e1torok seg\u00edts\u00e9g\u00e9vel val\u00f3s\u00edtjuk meg a h\u0171t\u00e9st. Ez a koncepci\u00f3 kisebb \u00e9s olcs\u00f3bb megold\u00e1st jelenthet az A koncepci\u00f3n\u00e1l, azonban rendelkezik annak az el\u0151ny\u00f6s tulajdons\u00e1gaival.\n\nHarmadszor v\u00e9geselemes-szimul\u00e1ci\u00f3k seg\u00edts\u00e9g\u00e9vel vizsg\u00e1ltam a kontakt-h\u0151ellen\u00e1ll\u00e1sok \u00e9s az r\u00e9szleges terhel\u00e9s hat\u00e1s\u00e1t a h\u0151m\u00e9rs\u00e9klet eloszl\u00e1sra az A koncepci\u00f3 eset\u00e9ben. Eredm\u00e9ny\u00fcl azt kaptam, hogy a re\u00e1lis kontakt-h\u0151ellen\u00e1ll\u00e1sok maximum $0.2~^\\circ C$ h\u0151m\u00e9rs\u00e9kletk\u00fcl\u00f6nbs\u00e9g-n\u00f6veked\u00e9st okoz. R\u00e9szleges terhel\u00e9s eset\u00e9n a maxim\u00e1lis h\u0151m\u00e9rs\u00e9klet-k\u00fcl\u00f6nbs\u00e9g  kisebb, mint teljes terhel\u00e9s eset\u00e9n, ahogyan ezt el\u0151re v\u00e1rtuk. Analitikus sz\u00e1m\u00edt\u00e1ssal meghat\u00e1roztam az \u00e1lland\u00f3sult h\u0151m\u00e9rs\u00e9kleteloszl\u00e1s kialakul\u00e1s\u00e1hoz sz\u00fcks\u00e9ges id\u0151t. Ez 134 m\u00e1sodpercnek ad\u00f3dott, ami megfelel\u0151, mind az orvosi felhaszn\u00e1l\u00e1s, mind a tesztm\u00e9r\u00e9sek sz\u00e1m\u00e1ra.\n\n\\newpage\n\\addcontentsline{toc}{section}{Acknowledgements}\n\\section*{Acknowledgements}\n\nI would like to thank for the work of my supervisor, R\u00f3bert Kov\u00e1cs, and my consultant M\u00f3nika Varga-K\u0151farag\u00f3. Both of them helped me a lot. First, I would like to thank to M\u00f3nika, who found this topic for me, and helped a lot in the initial difficulties. I studied a lot from her about data analysis, programming, and how to behave as a scientist while we worked together. I would like to thank R\u00f3bert for his help that I got from him during the creation of this thesis. I studied a lot from him about thermodynamics, and about how to work as an engineer. I hope we will continue the work together, and I can learn a lot more from him.\n\nI would like to thank Gergely G\u00e1bor Barnaf\u00f6ldi, who helped me a lot with his advice and encouragement. I started to learn from him how to write not just grammatically correct, but readable sentences, that will be useful in my entire career to become a scientist.\nI would like to thank Dieter R\u00f6rich, who gave me the possibility to spend three months in Bergen and study a lot from him and his colleagues. I also would like to thank Shruti Vineet Mehendale, who helped me a lot in my work during the last half-year. I owe a lot of gratitude for the Wigner \\-ALICE Group and the Bergen pCT Group for the possibility of joining to their project. Finally, this work would not be possible without the support of NKFIH\/OTKA K 120660, Hungarian Scientific Research Fund \u2013 OTKA.\n\n\\newpage\n\\addcontentsline{toc}{section}{Bibliography}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}