0}^n$. The Gröbner fan of *J* is a complete fan in R*n* whose elements represent weight vectors on the variables *x*1, …, *x**n*. Two weight vectors lie in the same open cone if and only if they give rise to the same initial ideal of *J*, see . The ideal *J* defines a weighted projective variety *V* inside the weighted projective space $\\mathbb P({\\bf d})$. Every open cone in the Gröbner fan gives rise to a one-parameter flat family degenerating *V* to the variety defined by the associated initial ideal of *J*. This construction is realized by choosing a weight in the relative interior of the cone.\nWe modify the classical one-parameter construction as follows: for a maximal cone *C* in the Gröbner fan of *J* we choose integral generators of its rays *r*1, …, *r**m* and denote by ${\\bf r}$ the (*m* × *n*)-matrix whose rows are *r*1, …, *r**m*. For an element $f=\\sum\\_{\\alpha\\in \\mathbb Z\\_{\\ge 0}^n} c\\_\\alpha{\\bf x}^\\alpha$ of *J*, we define $\\mu(f)\\in \\mathbb Z^m$ as the vector whose *i*-th entry is $\\min\\_{c\\_\\alpha\\not=0}\\{r\\_i\\cdot\\alpha\\}$. Then the *lift* of *f* is\n \n$$\\tilde f:=f\\big({\\bf t}^{{\\bf r}\\cdot e\\_1}x\\_1,\\dots, {\\bf t}^{{\\bf r}\\cdot e\\_n}x\\_n\\big) {\\bf t}^{-\\mu(f)},$$ \nwhere ${\\bf t}^{\\bf a}$ for ${\\bf a}\\in \\mathbb Z^m$ denotes the monomial ∏*i* = 1*m**t**i**a**i*. The *lifted ideal* $\\tilde J \\subseteq \\mathbb K[t\\_1,\\dots,t\\_m][x\\_1,\\dots,x\\_n]$ is the ideal generated by the lifts of all polynomials in *J*. We prove that $\\tilde J$ is generated by the lifts of elements of the reduced Gröbner basis for *J* and *C*, see . The lifts of these elements are independent of the choice of ${\\bf r}$ and homogeneous with respect to the ${\\bf d}$-grading on *x**i*’s, see . Consequently, $\\tilde J$ is independent of the choice of ${\\bf r}$ and it defines a variety inside $\\mathbb P({\\bf d})\\times \\mathbb A^m$. Our first main result is the following.\n[] Let *J* be a weighted homogeneous ideal, *C* a maximal cone in the Gröbner fan of *J* and ${\\bf r}$ an (*m* × *n*)-matrix whose rows are integral ray generators of *C*. Then the algebra $\\tilde A:=\\mathbb K[t\\_1,\\dots,t\\_m][x\\_1,\\dots,x\\_n]/\\tilde J$ is a free $\\mathbb K[t\\_1,\\dots,t\\_m]$-algebra. It defines a flat family\n \n$$\\begin{tikzcd}[ampersand replacement=\\&,cramped]\n {\\rm Proj}\\big(\\tilde A\\big) \\ar[r,hook]\\ar[d,\"\\pi\"] \\& \\mathbb P({\\bf d})\\times \\mathbb A^m \\ar[ld,twoheadrightarrow]\\\\\n \\mathbb A^m \\&\n \\end{tikzcd}$$ \nsuch that for every face *τ* of *C* there exists ${\\bf a}\\_\\tau\\in \\mathbb A^m$ with fiber $\\pi^{-1}({\\bf a}\\_\\tau)$ isomorphic to the variety defined by the initial ideal associated to *τ*. In particular, generic fibers are isomorphic to ${\\rm Proj}(A)$, where *A* = K[*x*1, …, *x**n*]/*J*, and there exist special fibers for every proper face *τ* ⊂ *C*.\nNext, we explain how the algebra *Ã* arises in Kaveh–Manon’s recent work on the classification of affine toric flat families of finite type over toric varieties . Consider a fan Σ defining a toric variety *X*Σ which contains a dense torus *T*. Then a *toric family* is a *T*-equivariant flat sheaf $\\mathcal A$ of positively graded algebras of finite type over *X*Σ such that: (*i*) the relative spectrum of $\\mathcal A$ has reduced fibers and (*i**i*) its generic fibers are isomorphic to the spectrum of some positively graded $\\mathbb K$-algebra *A*. Such families are classified by so-called PL-quasivaluations on *A* whose codomain is the semifield of piecewise linear functions on the intersection of Σ with the cocharacter lattice of *T* (see or Section [sec:toric families] below). Given a PL-quasivaluation, Kaveh–Manon construct a sheaf of Rees algebras on *X*Σ that is a toric family. In the special case when Σ is a cone, their construction yields a single Rees algebra rather than a sheaf. We prove the following result:\n[] Let $J\\subseteq\\mathbb K[{x\\_1,\\dots,x\\_n}]$ be a weighted homogeneous ideal and *C* a maximal cone in the Gröbner fan of *J*. Let $\\mathcal R\\_C(A)$ be the Rees algebra associated to the PL-quasivaluation on $A=\\mathbb K[{x\\_1,\\dots,x\\_n}]/J$ defined by *C* and let $\\psi\\colon \\operatorname{Spec}(\\mathcal R\\_C(A))\\to X\\_C$ be the corresponding toric family. The morphism $\\pi\\colon \\operatorname{Spec}\\big(\\tilde A\\big)\\to \\mathbb A^m$ fits into a pull-back diagram as follows:\n \n$$\\begin{tikzcd}[ampersand replacement=\\&,cramped]\n \\operatorname{Spec}(\\mathcal R\\_C (A))\\ar[d,\"\\psi\"'] \\&\n \\operatorname{Spec}\\big(\\tilde A\\big)\\ar[d,\"\\pi\"] \\ar[l]\\\\\n X\\_{C}\\& \\mathbb A^m. \\ar[l,\"p\\_C\"]\n \\end{tikzcd}$$ \nHere $p\\_C\\colon \\mathbb A^m\\to X\\_C$ is the universal torsor of *X**C* (see for details) and *m* is the number of rays of *C*. Both flat families, the one defined by *ψ* and the one defined by *π*, are direct sums of line bundles indexed by the standard monomial basis associated to the maximal cone *C*.\nToric degenerations are a particular class of flat families with a single special fiber that is a toric variety. They are of significant interest and have been widely studied in recent years. Among the faces of the maximal cone *C* one may find binomial prime initial ideals that hence define toric Gröbner degenerations of *V*. Such faces lie in the tropicalization of *J* denoted Trop(*J*), which by definition is the subfan of the Gröbner fan consisting of cones whose associated initial ideal does not contain monomials. Let Σ be the intersection of *C* with the tropicalization of *J*. Then the affine space $\\mathbb A^m$ contains the universal torsor $\\mathcal T\\_\\Sigma$ of the toric variety *X*Σ and all fibers of $\\pi^{-1}(\\mathcal T\\_\\Sigma)\\to \\mathcal T\\_\\Sigma$ correspond to initial ideals of cones in Σ. In we show how the family defined by $\\pi\\vert\\_{\\pi^{-1}(\\mathcal T\\_\\Sigma)}$ also arises as a pull-back from a toric family defined by a sheaf of Rees algebras on *X*Σ.\nThe interest of studying toric degenerations in the context of cluster algebras has grown in the last years (see for example ). Therefore, we would like to understand the framework introduced above from the perspective of cluster algebras. As a first step in this direction, we analyze in depth the situation for the Grassmannians $\\operatorname{Gr}(2,\\mathbb C^n)$ and $\\operatorname{Gr}\\big(3,\\mathbb C^6\\big)$ whose coordinate rings are cluster algebras.\n=-1 For $\\operatorname{Gr}(2,\\mathbb C^n)$ we choose its Plücker embedding and obtain the homogeneous coordinate ring *A*2, *n* as a quotient of the polynomial ring in Plücker variables $\\mathbb C[p\\_{ij}\\colon 1\\le i<j\\le n]$ by the Plücker ideal *I*2, *n*. It was shown in that *A*2, *n* is a cluster algebra in which the cluster variables are in one-to-one correspondence with Plücker coordinates $\\bar p\\_{ij}\\in A\\_{2,n}$ and the seeds are in one-to-one correspondence with triangulations of the *n*-gon. Every triangulation of the *n*-gon gives rise to a toric degeneration of $\\operatorname{Gr}(2,\\mathbb C^n)$ obtained by adding *principal coefficients* to *A*2, *n* at the corresponding seed . Principal coefficients were introduced by Fomin and Zelevinsky in and are a core concept in the theory of cluster algebras. A central piece of the construction is endowing every cluster variable (i.e., every Plücker coordinate) with a so-called ${\\bf g}$-vector depending on the fixed seed. We want to understand the toric degenerations coming from principal coefficients in the context of Gröbner theory: to achieve this first fix a triangulation *T*. We use ${\\bf g}$-vectors for Plücker coordinates to construct a weight vector ${\\bf w}\\_T$, see . In we prove that the initial ideal with respect to ${\\bf w}\\_T$ is binomial and prime, hence ${\\bf w}\\_T$ lies in the tropicalization of *I*2, *n*. Moreover, the central fiber of the Gröbner degeneration induced by ${\\bf w}\\_T$ is isomorphic to the central fiber of the toric degeneration induced by endowing *A*2, *n* with principal coefficients at the seed determined by *T*, see .\nThere is a single object in cluster theory that simultaneously encodes principal coefficients at all seeds and thus the ${\\bf g}$-vectors of all Plücker coordinates with respect to all triangulations. Namely, the cluster algebra with *universal coefficients* $A^{\\rm univ}\\_{2,n}$ associated to *A*2, *n*, see . In spirit, this algebra is very similar to the algebra defining the flat family associated to a maximal cone in the Gröbner fan of *I*2, *n* as it encodes various toric degenerations of $\\operatorname{Gr}(2,\\mathbb C^n)$ at the same time. It is therefore natural to ask if the cluster algebra with universal coefficients fits into the above framework. The following result answers this question for $\\operatorname{Gr}(2,\\mathbb C^n)$:\nThere exists a maximal cone *C* in the Gröbner fan of *I*2, *n* whose rays are in bijection with diagonals of the *n*-gon. Moreover, the unique cone *C* has the following properties:\n* The standard monomial basis for *A*2, *n* associated with *C* coincides with the basis of cluster monomials ().\n* For every triangulation *T* of the *n*-gon the weight vector ${\\bf w}\\_T$ lies in the boundary of *C* and the intersection of *C* with the tropicalization of *I*2, *n* is the totally positive part of Trop(*I*2, *n*) ( and ).\n* Let $\\tilde I\\_{2,n}$ be the lift of *I*2, *n* with respect to *C* and denote by $\\tilde A\\_{2,n}$ the quotient $\\mathbb C[t\\_{ij}\\colon 2\\le i+1<j\\le n][p\\_{ij}\\colon 1\\le i<j\\le n]/\\tilde I\\_{2,n}$. Then there exists a canonical isomorphism between the cluster algebra with universal coefficients $A\\_{2,n}^{\\rm univ}$ and $\\tilde A\\_{2,n}$ ().\n* The monomial initial ideal of *I*2, *n* with respect to *C* is squarefree and coincides with the Stanley–Reisner ideal of the cluster complex ().\nSimilarly, we describe the cluster algebra with universal coefficients for $\\operatorname{Gr}\\big(3,\\mathbb C^6\\big)$ from the viewpoint of Gröbner theory. In this case, we fix its *cluster embedding* obtained as follows: consider the Plücker embedding and let *A*3, 6 be the homogeneous coordinate ring. As a cluster algebra of type $\\mathtt D\\_4$, *A*3, 6 has 22 cluster variables, 20 of which are the Plücker coordinates. The additional two cluster variables are homogeneous binomials in Plücker coordinates of degree 2. Hence, we can present *A*3, 6 as the quotient of a polynomial ring in 22 variables by a weighted homogeneous ideal denoted by $I^{\\rm ex}$. This yields an embedding of $\\operatorname{Gr}\\big(3,\\mathbb C^6\\big)$ in the weighted projective space $\\mathbb P(1,\\dots,1,2,2)$, where the Plücker coordinates correspond to the coordinates of weight one, and the additional cluster variables to those of weight two, see Section [sec:Gr(3,6)]. Our main result is the following:\nThere exists a unique maximal cone *C* in the Gröbner fan of $I^{\\rm ex}$ such that\n* The algebra $\\tilde A\\_{3,6}$ is canonically isomorphic to the cluster algebra with universal coefficients $A^{\\rm univ}\\_{3,6}$, where rays of *C* are identified with mutable cluster variables of *A*3, 6 (Theorem $\\ref{thm:Gr36}(ii))$.\n* For every seed of *A*3, 6 there exists a face of *C* whose associated initial ideal is a totally positive binomial prime ideal. More precisely, the intersection of *C* with $\\operatorname{Trop}(I^{\\rm ex})$ is the totally positive part of $\\operatorname{Trop}(I^{\\rm ex})$ ((*i**i**i*)).\n* The monomial initial ideal $\\operatorname{in}\\_C(I^{\\rm ex})$ associated to *C* is squarefree and coincides with the Stanley–Reisner ideal of the cluster complex. In particular, the basis of standard monomials associated to *C* coincides with the basis of cluster monomials ().\nAs mentioned before, the toric fibers of the above families are of particular interest. They arise from maximal cones in the tropicalization whose initial ideal is binomial and prime; such cones are called *maximal prime cones*. The corresponding projective toric varieties (respectively, their normalizations) have associated polytopes. Following these polytopes can be realized as Newton–Okounkov bodies of full rank valuations constructed from maximal prime cones. In the recent preprint Escobar and Harada study wall-crossing phenomena of Newton–Okounkov bodies associated to maximal prime cones that intersect in a facet. They give piecewise linear maps called *flip* and *shift* that relate the Newton–Okounkov bodies. For cluster algebras like *A*3, 6 and *A*2, *n* it has been shown in several cases that Newton–Okounkov bodies can equivalently be obtained from the cluster structure (see, e.g., ). Hence, one might wonder how Escobar–Harada’s wall-crossing formulas can be understood in the context of cluster algebras. In the particular case of *A*2, *n* each Newton–Okounkov body arising from Trop(*I*2, *n*) is unimodularly equivalent to the convex hull of **g**-vectors of Plücker coordinates for some triangulation. More precisely, we obtain the following result:\n[] Let *σ*1 and *σ*2 be two maximal prime cones in Trop(*I*2, *n*) that intersect in a facet. Then their associated Newton–Okounkov bodies are (up to unimodular equivalence) related by a shear map obtained from tropicalizing a cluster mutation.\nIn combination with Escobar–Harada’s results about the flip map for $\\operatorname{Gr}(2,\\mathbb C^n)$ this corollary implies that it is of cluster nature (for details see Section [sec:NO]).\nWe would like to remark that this paper is not the first to make a connection between cluster algebras and Gröbner theory. In Muller, Rajchgot and Zykoski obtained presentations for lower bounds of cluster algebras using Gröbner theory. Further, it is worth noticing that the theory of universal coefficients for cluster algebras is particularly well-developed for finite and surface type cluster algebras, see . We believe that the above results can be extended to projective varieties containing a cluster variety of finite type. It is further an interesting and challenging problem to extend the results of this paper to cluster algebras of (infinite) surface type as it would involve Gröbner theory of non-Noetherian algebras.\nIn Section [sec:prep] we recall the background on weighted projective varieties and Gröbner basis theory. In Section [sec:flat families] we introduce the construction of the flat families and prove the main theorem. We relate to Kaveh–Manon’s work in Section [sec:toric families]. In Section [sec:Gr] we turn to the Grassmannians $\\operatorname{Gr}(2,\\mathbb C^n)$ and $\\operatorname{Gr}\\big(3,\\mathbb C^6\\big)$. We recall the background on cluster algebras and universal coefficients in Section [sec:cluster]. We explain in detail how toric degenerations from cluster algebras arise as Gröbner degenerations for $\\operatorname{Gr}(2,\\mathbb C^n)$ in Section [sec: toric deg]. Then we apply the main construction to $\\operatorname{Gr}(2,\\mathbb C^n)$ in Section [sec:monomialdegeneration] and afterwards to $\\operatorname{Gr}\\big(3,\\mathbb C^6\\big)$ in Section [sec:Gr(3,6)]. We explore further connections to Escobar–Harada’s work in Section [sec:NO]. Finally, in the Appendix [sec:data Gr36] we present computational results used for the application to $\\operatorname{Gr}\\big(3,\\mathbb C^6\\big)$.\nPreliminaries\n=============\nWe first fix our notation throughout the note. Let K be an algebraically closed field. We are mainly interested in the case when K = C. In the polynomial ring $\\mathbb K[x\\_1, \\dots, x\\_n]$, we fix the notation ${\\bf x}^\\alpha$ with $\\alpha=(a\\_1,\\ldots,a\\_n)\\in \\mathbb Z\\_{\\ge 0}^n$ denoting the monomial *x*1*a*1…*x**n**a**n*. Throughout when we write $f = \\sum\\_{\\alpha\\in \\mathbb{Z}^n\\_{\\geq 0}} c\\_{\\alpha} {\\bf x}^{\\alpha}$ for $f \\in \\mathbb K[x\\_1, \\dots, x\\_n]$ we refer to the expression of *f* in the basis of monomials.\nWeighted projective varieties\n-----------------------------\nFix a vector $ {\\bf d}=(d\\_1, \\dots, d\\_n)\\in \\mathbb R^n$. Let $\\mathbb K\\_{\\bf d}[x\\_1, \\dots, x\\_n]$ be the polynomial ring $\\mathbb K [x\\_1, \\dots, x\\_n]$ endowed with the $\\mathbb R$-grading determined by setting deg(*x**i*) = *d**i* for *i* = 1, …, *n*.\nFor $c \\in \\mathbb K^{\\ast}$ and $ \\alpha=(a\\_1,\\dots, a\\_n) \\in \\mathbb Z\\_{\\geq 0}^n$ the *weight* of the monomial $ c {\\bf x}^{\\alpha}\\in \\mathbb K\\_{\\bf d}[x\\_1, \\dots, x\\_n]$ is ${\\bf d}\\cdot \\alpha = \\sum\\_{i=1}^nd\\_ia\\_i$. We extend this definition and say $f \\in \\mathbb K\\_{\\bf d}[x\\_1, \\dots, x\\_n]$ is **d**-*homogeneous* if *f* can be expressed as a sum of monomials of the same weight. Similarly, an ideal $J\\subseteq \\mathbb K\\_{\\bf d}[x\\_1, \\dots, x\\_n]$ is **d**-*homogeneous* if it is generated by ${\\bf d}$-homogeneous elements.\nAssume ${\\bf d}\\in \\mathbb Z^n\\_{> 0}$. In this case $\\mathbb K\\_{\\bf d}[x\\_1, \\dots, x\\_n]$ is $\\mathbb Z$-graded and we can define the *weighted projective space* $\\mathbb P (d\\_1, \\dots, d\\_n)$ as the quotient of $\\mathbb K^n$ under the equivalence relation\n \n$$(x\\_1,\\dots, x\\_n) \\sim \\big(\\lambda^{d\\_1}x\\_1, \\dots,\\lambda^{d\\_n} x\\_n\\big) \\qquad\\text{for}\\quad\\lambda \\in \\mathbb K^{\\ast}.$$ \nFor $\\mathbb K = \\mathbb C$ it is well-known that $\\mathbb P (d\\_1, \\dots, d\\_n)$ is a projective toric variety . Moreover, as a scheme $\\mathbb P(d\\_1,\\dots,d\\_n)$ is the homogeneous spectrum (or Proj) of $\\mathbb K\\_{\\bf d}[x\\_1,\\dots,x\\_n]$ . For simplicity, we denote $\\mathbb P (d\\_1, \\dots, d\\_n)$ by $\\mathbb P ({\\bf d})$ and let [*x*1 : … : *x**n*] denote the class of (*x*1, …, *x**n*) in $\\mathbb P ({\\bf d})$. Given a **d**-homogeneous ideal $J \\subseteq \\mathbb K\\_{\\bf d}[x\\_1, \\dots, x\\_n]$ its set of zeros is\n \n$$V(J)=\\big\\{ [x\\_1:\\dots : x\\_n]\\in \\mathbb P ({\\bf d}) \\colon f(x\\_1,\\dots, x\\_n)=0\\text{ for all }f\\in J\\big\\}.$$ \nConversely, given a subset $X \\subseteq \\mathbb P (d\\_1,\\dots,d\\_n)$, its associated ideal *I*(*X*) is generated by polynomials:\n \n$$\\big\\{ f \\in \\mathbb K\\_{\\bf d}[x\\_1,\\dots, x\\_n] \\colon f \\text{ is }{\\bf d}\\text{-homogeneous and }f(p)=0\\text{ for all }p\\in X \\big\\}.$$ \nSubsets of $\\mathbb P({\\bf d})$ of the form *V*(*J*) for some weighted projective ideal *J* are called weighted projective varieties. These sets are the closed sets of a Zariski-type topology on $ \\mathbb P({\\bf d})$. The theory of weighted projective varieties is very similar to the theory of usual projective varieties and background on this theory can be found for example in . In particular, the projective space P*n* − 1 can be realized as $\\mathbb{P}\n^{n-1}=\\mathbb P ({\\bf 1})$, where ${\\bf 1}=(1, \\dots, 1)\\in \\mathbb Z^n$. So every projective variety can be considered as a weighted projective variety of $\\mathbb P ({\\bf 1})$.\nThe *weighted homogeneous coordinate ring* of a weighted projective variety $X\\subseteq \\mathbb P ({\\bf d})$ is defined as\n \n$$S(X):= \\mathbb{K}\\_{\\bf d}[x\\_1,\\dots, x\\_n]/ I(X).$$ \nBy construction, *S*(*X*) is a positively graded ring. Moreover, the weighted homogeneous coordinate ring of a projective variety (considered as a weighted projective variety) coincides with its homogeneous coordinate ring. There is a natural notion of morphism between weighted projective varieties which we will not need in this work. It will be sufficient to recall that if $X\\subseteq \\mathbb P(d\\_1, \\dots, d\\_n)$ and $Y\\subseteq \\mathbb P(d'\\_1, \\dots, d'\\_m)$ are weighted projective varieties, then an isomorphism between the graded rings *S*(*X*) and *S*(*Y*) induces an isomorphism of the weighted projective varieties *X* and *Y*.\nGröbner basis theory\n--------------------\nWe first review some results in Gröbner basis theory in order to fix our notation and keep the paper self-contained. Most of the material here is well-known and we refer to Buchberger’s thesis and standard books (e.g., ) for proofs and more details.\nLet $f = \\sum\\_{\\alpha\\in \\mathbb{Z}^n\\_{\\geq 0}} c\\_{\\alpha} {\\bf x}^{\\alpha} \\in \\mathbb K[x\\_1, \\dots, x\\_n]$. Given a *weight vector* $w\\in \\mathbb R^n$ the **initial form** of *f* with respect to *w* is defined as\n \n$$\\operatorname{in}\\_w(f):=\\sum\\_{\\alpha\\colon w\\cdot \\alpha=a} c\\_{\\alpha}{\\bf x}^{\\alpha},$$ \nwhere $a= \\min \\{w\\cdot \\alpha \\colon c\\_{\\alpha}\\not =0\\}$. For an ideal $J\\subseteq \\mathbb K[x\\_1, \\dots, x\\_n]$ its *initial ideal* with respect to *w* is defined as in*w*(*J*): = ⟨in*w*(*f*): *f* ∈ *J*⟩. A finite set G = {*g*1, …, *g**r*} ⊆ *J* is called a *Gröbner basis* for *J* with respect to *w* if in*w*(*J*) = ⟨in*w*(*g*1), …, in*w*(*g**r*)⟩.\nA *monomial term order* on $\\mathbb K[x\\_1, \\dots, x\\_n]$ is a total order < on the set of monic monomials in $\\mathbb K[x\\_1, \\dots, x\\_n]$ such that for every *α*, *β*, *γ* in $\\mathbb Z\\_{\\ge 0}^n$ we have that \n$$(i)\\quad 1\\le {\\bf x}^\\alpha,\\qquad \\text{and}\\qquad\n(ii)\\quad \\text{if}\\quad {\\bf x}^\\alpha<{\\bf x}^\\beta, \\quad \\text{then}\\quad {\\bf x}^{\\alpha+\\gamma}<{\\bf x}^{\\beta+\\gamma}.$$ \n The *initial monomial* of an element $f=\\sum\\_{\\alpha\\in \\mathbb{Z}^n\\_{\\geq 0}} c\\_{\\alpha} {\\bf x}^{\\alpha}\\in \\mathbb K[x\\_1, \\dots, x\\_n]$ with respect to < is $\\operatorname{in}\\_<(f):=c\\_\\beta{\\bf x}^{\\beta}$, where ${\\bf x}^\\beta=\\max\\_<\\{{\\bf x}^\\alpha\\colon c\\_\\alpha\\not=0\\}$. The *initial ideal* of an ideal $J\\subseteq \\mathbb K[x\\_1, \\dots, x\\_n]$ with respect to < is defined as in<(*J*) : = ⟨in<(*f*): *f* ∈ *J*⟩.\nRecall that given an ideal $J\\subseteq \\mathbb K[x\\_1, \\dots, x\\_n]$ and a monomial term order < there always exists a weight vector − *w* ∈ Z≥ 0*n* such that in*w*(*J*) = in<(*J*), see, e.g., (Note the sign switch here, this is due to our min-convention for initial ideals with respect to weight vectors and max-convention for initial ideals with respect to monomial term orders.) On the other hand, if in*w*(*J*) is generated by monomials and there exists a monomial term order < such that in*w*(*J*) = in<(*J*), then we say that *w* is *compatible* with < .\nLet in<(*J*) be a monomial initial ideal of the ideal *J* for some monomial term order < on K[*x*1, …, *x**n*]. Then the set $\\mathbb B\\_{<}:=\\{\\bar{\\bf x}^\\alpha \\colon {\\bf x}^\\alpha\\not \\in \\operatorname{in}\\_<(J)\\}$ is a vector space basis of $\\mathbb K[x\\_1, \\dots, x\\_n]/J$ (and $\\mathbb K[x\\_1, \\dots, x\\_n]/\\operatorname{in}\\_<(J)$) called *standard monomial basis*.\n[the division algorithm] Fix a monomial term order < on $\\mathbb K[x\\_1,\\dots,x\\_n]$ and let *g*1, …, *g**s* be non-zero polynomials in $\\mathbb K[x\\_1,\\dots,x\\_n]$. Given $0\\not=f\\in\\mathbb K[x\\_1,\\dots,x\\_n]$ there exist $f\\_1,\\dots,f\\_s,f'\\in \\mathbb K[x\\_1,\\dots,x\\_n]$ with *f* = *f*1*g*1 + … + *f**s**g**s* + *f*ʹ, such that\n* if *f*ʹ ≠ 0 then no monomial of *f*ʹ is divisible by any of the monomials in<(*g*1), …, in<(*g**s*);\n* if $f\\_i\\not=0$ then in<(*f*) ≥ in<(*f**i**g**i*) for all *i*.\nWe say that *f* reduces to *f*ʹ with respect to {*g*1, …, *g**s*}.\nLet < be a monomial term order on $\\mathbb K[x\\_1, \\dots, x\\_n]$ and G = {*g*1, …, *g**s*} a finite generating set of an ideal *J*. Then the **S*-polynomial* of *g**i* and *g**j* is defined as\n \n$$S(g\\_i,g\\_j):=\\frac{{\\rm lcm}\\big(\\operatorname{in}\\_{<}(g\\_i),\\operatorname{in}\\_{<}(g\\_j)\\big)}{\\operatorname{in}\\_{<}(g\\_i)}g\\_i-\\frac{{\\rm lcm}\\big(\\operatorname{in}\\_{<}(g\\_i),\\operatorname{in}\\_{<}(g\\_j)\\big)}{\\operatorname{in}\\_{<}(g\\_j)}g\\_j.$$ \nWe say that *Buchberger’s criterion* holds if for all 1 ≤ *i* < *j* ≤ *s*, the *S*-pairs reduce to zero with respect to {*g*1, …, *g**s*}. If Buchberger’s criterion holds, then G forms a Gröbner basis for *J* with respect to < . Moreover, a Gröbner basis G for *J* with respect to < is *reduced* if\n* in<(*g**i*) is monic for all 1 ≤ *i* ≤ *s*, and\n* for *i* ≠ *j* no monomial in *g**i* is divisible by in<(*g**j*).\nA reduced Gröbner basis for *J* with respect to < always exists and is unique, see, e.g., . We let G<(*J*) denote the reduced Gröbner basis for *J* with respect to < .\nStudying all possible initial ideals of a given ideal leads to the notion of Gröbner fan (see, e.g., ):\nLet $J\\subseteq \\mathbb K[x\\_1, \\dots, x\\_n]$ be an ideal. The *Gröbner region* of *J* denoted by GR(*J*) is the set of $w\\in \\mathbb R^n$ such that there exists a monomial term order < with in<(in*w*(*J*)) = in<(*J*). The *Gröbner fan* of *J* denoted by GF(*J*) is a fan with support GR(*J*) in which a pair of elements $u,w\\in \\mathbb R^n$ lie in the relative interior of the same cone $C\\subset \\mathbb R^n$ (denoted by *C*∘) if and only if in*u*(*J*) = in*w*(*J*). We introduce the notation in*C*(*J*) : = in*w*(*J*) for any *w* ∈ *C*∘. By definition of GR(*J*), for every full-dimensional cone *C*, there exists a monomial term order < such that *C* is the topological closure of {*w* ∈ R*n*: in*w*(*J*) = in<(*J*)}. Moreover, we define the *lineality space* $\\mathcal L(J)$ as the linear subspace of GF(*J*) that contains all elements *u* for which in*u*(*J*) = *J*.\nIntegral weight vectors in GF(*J*) can be seen as points in a lattice $N=\\mathbb Z^n$ whose dual lattice $M=(\\mathbb Z^n)^\\*$ contains exponent vectors of monomials in $\\mathbb K[x\\_1,\\dots,x\\_n]$. Consequently, for *w* ∈ *N* and *α* ∈ *M* we denote by *w* ⋅ *α* the pairing between the two lattices.\n[rem:Eisenbud] Proposition 15.16 in gives a criterion for whether a weight vector is compatible with a monomial term order < for *J*, or not. Namely, a weight *w* is compatible with a monomial term order < if and only if in*w*(*g*) = in<(*g*) for every element of G<(*J*). Let *C* be the topological closure of the corresponding Gröbner cone of < , then *w* ∈ *C* if and only if in<(*g*) = in<(in*w*(*g*)) for every *g* ∈ G<(*J*).\nLet ${\\bf d}\\in \\mathbb Z\\_{>0}^n$ and $J\\subseteq \\mathbb K\\_{\\bf d}[x\\_1, \\dots, x\\_n]$ be a ${\\bf d}$-homogeneous ideal. Then ${\\bf d} \\in \\mathcal{L}(J)$. More generally, $\\mathcal L(J)$ contains all elements ${\\bf u}\\in \\mathbb R^n$ such that *J* is ${\\bf u}$-homogeneous.\nObserve that for every ${\\bf d}$-homogeneous polynomial $f\\in \\mathbb K\\_{\\bf d}[x\\_1, \\dots, x\\_n]$ the equality $\\operatorname{in}\\_{\\bf d}(f)=f$ holds. Next, every *f* ∈ *J* can be uniquely written as a finite sum *f* = ∑*i* = 0*d**f**i* for some $d\\in \\mathbb Z\\_{\\geq 0}$ and *f**i* ∈ *J* of **d**-degree *i* (see ). Then $\\operatorname{in}\\_{\\bf d}(f)=f\\_j$, where *j* = min{*i*: *f**i* ≠ 0}. In particular, for any *f* ∈ *J* we have that $\\operatorname{in}\\_{\\bf d}(f) \\in J$. Therefore, $\\operatorname{in}\\_{\\bf d}(J )= J$.\nFix an arbitrary monomial term order < . Then for every $\\ell \\in \\mathcal L(J)$ and every $g\\in \\mathcal G\\_<(J)$ we have inℓ(*g*) = *g*. In particular, if *J* is ${\\bf d}$-homogeneous then so is every $g\\in \\mathcal G\\_<(J)$.\nConsider a Gröbner basis *F* : = {*f*1, …, *f**t*} of *J* with respect to a fixed element. Then *J* = inℓ(*J*) is generated by in(*F*) : = {inℓ(*f*1), …, inℓ(*f**t*)}. In particular, we have inℓ(inℓ(*f*)) = inℓ(*f*) for all *f* ∈ *F*. From the set *F* we can construct the reduced Gröbner basis $\\mathcal G\\_<(J)$ by doing the following three steps: First, extend *F* to a Gröbner basis *H* with respect to < by adding *S*-pairs using Buchberger’s criterion. Then if necessary eliminate elements from *H* until the outcome is a minimal Gröbner basis *G* (see ). Finally, reduce all elements in *G* to obtains $\\mathcal G\\_<(J)$ (see ). We invite the reader to verify that the property inℓ(*g*) = *g* holds for all *g* ∈ *H*, hence for all *g* ∈ *G* and finally for all $g\\in \\mathcal G\\_<(J)$.\nRecall from that for a ${\\bf d}$-homogeneous ideal $J\\subseteq \\mathbb K\\_{\\bf d}[x\\_1, \\dots, x\\_n]$ the support of its Gröbner fan, i.e., GR(*J*), is $\\mathbb R^n$. In other words, there exists a compatible monomial term order for every maximal cone *C* ∈ GF(*J*).\nLet $J\\subseteq \\mathbb K\\_{\\bf d}[x\\_1, \\dots, x\\_n]$ be a ${\\bf d}$-homogeneous ideal, *C* ∈ GF(*J*) a maximal cone, and < a compatible monomial term order of *C*. Consider *v* ∈ *C* \\ *C*∘ and *u* ∈ R*n* such that *w* : = *u* + *v* ∈ *C*∘. Then for every *g* ∈ G<(*J*), we have in<(*g*) = in*w*(*g*) = in*u*(in*v*(*g*)).\nConsider an element *g* in G<(*J*). Since *w* ∈ *C*∘ we have in<(*g*) = in*w*(*g*). On the other hand, since *v* ∈ *C*, by Remark [rem:Eisenbud] we have that in<(*g*) = in<(in*v*(*g*)). This implies that in*w*(*g*) is a refinement of in*v*(*g*). In other words, in*v*(*g*) contains the monomial in*w*(*g*) with possibly some extra terms which will disappear after taking its initial form with respect to *u*.\nWe note that does not hold for arbitrary elements of *J*. See .\nFor every cone $C\\in\\operatorname{GF}(J) \\subseteq \\mathbb R^n$, the cone $\\overline C :=C / \\mathcal{L}(J)\\subseteq \\mathbb R^n / \\mathcal{L}(J)$ is strongly convex, that is $\\overline C\\cap (- \\overline C)=\\{0\\}$.\nAs we have chosen *J* to be ${\\bf d}$-homogeneous every cone in GF(*J*) is a face of a full-dimensional cone. Hence, we may assume without loss of generality that *C* is a cone of dimension *n*. Let < be the corresponding monomial term order and *g* ∈ G<(*J*). Let *v* and − *v* be in *C* \\ *C*∘ and choose *u*, *u*ʹ ∈ R*n* such that *u* + *v*, *u*ʹ − *v* ∈ *C*∘. By we have that in<(*g*) = in*u*(in*v*(*g*)) = in*u*ʹ(in− *v*(*g*)). However, in<(*g*) cannot simultaneously be a monomial in in*v*(*g*) and in− *v*(*g*) unless $v\\in \\mathcal L(J)$.\nFor a ${\\bf d}$-homogeneous ideal $J\\subseteq \\mathbb K\\_{\\bf d}[x\\_1, \\dots, x\\_n]$ we define its *tropicalization*\n \n$$\\operatorname{Trop}(J):=\\big\\{w\\in \\mathbb R^n \\colon \\operatorname{in}\\_w(J) \\text{ does not contain any monomial}\\big\\}.$$ \nIn fact, Trop(*J*) is a *d*-dimensional subfan of GF(*J*), where *d* is the Krull-dimension of $\\mathbb K\\_{\\bf d}[x\\_1,\\dots,\\linebreak x\\_n]/J$.\nAn ideal $J\\subseteq \\mathbb R[x\\_1,\\dots,x\\_n]$ is called *totally positive* if it does not contain any non-zero element of $\\mathbb R\\_{\\ge 0}[x\\_1,\\dots,x\\_n]$. For a ${\\bf d}$-homogeneous ideal $J\\subseteq \\mathbb R[x\\_1,\\dots,x\\_n]$ the *totally positive part* of its tropicalization is defined as\n \nTrop+(*J*) : = {*w* ∈ Trop(*J*): in*w*(*J*)is totally positive}. \nDue to Trop+(*J*) is a closed subfan of Trop(*J*) (that may be empty).\nFamilies of Gröbner degenerations\n=================================\nIn this section, we introduce the main construction of the paper. Let $J \\subseteq \\mathbb K\\_{\\bf d}[x\\_1, \\dots, x\\_n]$ be a **d**-homogeneous ideal, *C* a maximal cone in GF(*J*) and *A* the quotient $\\mathbb K\\_{\\bf d}[x\\_1, \\dots, x\\_n]/J$. Our aim is to construct a flat family of degenerations of ${\\rm Proj}(A)$ that contains as fibers the varieties corresponding to the various initial ideals associated to the interior of the faces of *C*. To achieve this we generalize the following classical construction: take *w* ∈ *C*∘ and consider the ideal\n \n$$\\label{eq:flat}\n\\hat J\\_w:=\\bigg\\langle \\sum\\_{\\alpha \\in \\mathbb Z^n\\_{\\ge 0}} c\\_\\alpha {\\bf x}^\\alpha t^{w\\cdot \\alpha -\\min\\{w\\cdot \\beta \\colon c\\_\\beta \\not =0\\}} \\colon \\sum\\_{\\alpha \\in \\mathbb Z^n\\_{\\ge 0}} c\\_\\alpha {\\bf x}^\\alpha\\in J \\bigg\\rangle \\subseteq \\mathbb K[t][x\\_1,\\dots,x\\_n].$$ \n[rem:flat] The ideal $\\hat J\\_w$ induces a flat family $\\operatorname{Spec} \\big(\\mathbb K[t][x\\_1,\\dots,x\\_n]/\\hat J\\_w\\big) \\to \\operatorname{Spec}(\\mathbb{K}[t])$ whose fiber over the closed point (*t*) is isomorphic to $\\operatorname{Spec}(\\mathbb K[x\\_1,\\dots,x\\_n]/\\operatorname{in}\\_w(J))$ and the fiber over any non-zero closed point (*t* − *a*) is isomorphic to Spec(*A*), see . In fact, and hold more generally for arbitrary cones in GF(*J*). However, for the following generalization we focus on maximal cones for simplicity.\nTo generalize the construction we fix vectors *r*1, …, *r**m* ∈ *C* such that $\\{ \\overline{r}\\_1,\\dots, \\overline{r}\\_m \\}$ is the set of primitive ray generators for $\\overline{C}$, which is possible due to . Using we may assume if necessary that *r*1, …, *r**m* are non-negative or positive vectors. We denote by ${\\bf r}$ the (*m* × *n*)-matrix whose rows are *r*1, …, *r**m*. Additionally, we write < for a monomial term order compatible with *C* and denote by $\\mathcal G$ the associated reduced Gröbner basis.\nFor $f=\\sum\\_{\\alpha \\in \\mathbb Z^n\\_{\\ge 0}} c\\_\\alpha {\\bf x}^{\\alpha} \\in J$ set ${\\mu}\\_{\\bf r}(f):=\\big(\\min\\_{c\\_\\alpha\\not =0}\\{r\\_i\\cdot \\alpha\\}\\big)\\_{i=1,\\dots,m}\\in \\mathbb Z^{m\\times 1}$, that is we think of $\\mu\\_{\\bf r}(f)$ as a column vector with *m* entries. We define the *lift of *f** to be the polynomial $ \\tilde f\\_{\\bf r} \\in \\mathbb K[t\\_1,\\dots,t\\_m][x\\_1,\\dots,x\\_n]$ given by the following formula\n \n$$\\tilde f\\_{\\bf r} := \\tilde f\\_{\\bf r} ({\\bf t},{\\bf x})\n := f({\\bf t}^{{\\bf r}\\cdot e\\_1}x\\_1,\\dots, {\\bf t}^{{\\bf r}\\cdot e\\_n}x\\_n) {\\bf t}^{-\\mu\\_{\\bf r}(f)}\n =\\sum\\_{\\alpha \\in \\mathbb Z^n\\_{\\ge 0}} c\\_\\alpha {\\bf x}^{\\alpha} {\\bf t}^{{\\bf r}\\cdot \\alpha -\\mu\\_{\\bf r}(f)}.$$ \nSimilarly, we define the *lifted ideal* as\n \n$$\\tilde J\\_{\\bf r}:=\\big\\langle \\tilde f\\_{\\bf r} \\colon f \\in J \\big\\rangle \\subseteq \\mathbb K[t\\_1,\\dots,t\\_m][x\\_1,\\dots,x\\_n],$$ \nand the *lifted algebra* as the quotient\n \n$$\\label{eq:lifted algebra}\n\\tilde A\\_{\\bf r}:=\\mathbb K[t\\_1,\\dots,t\\_m][x\\_1,\\dots,x\\_n]/\\tilde J\\_{\\bf r}.$$ \nWe proceed by showing that the lifted algebra $\\tilde A\\_{\\bf r}$ is independent of the choice of vectors *r*1, …, *r**m* ∈ *C* that represent the primitive ray generators $\\overline{r}\\_1,\\dots, \\overline{r}\\_m$ of $\\overline{C}$. For this, we identify generators of the ideal $\\tilde J\\_{\\bf r}$ which is a crucial matter for applications.\nSuppose *r*ʹ1, …, *r*ʹ*m* ∈ *C* are such that $\\overline{r}'\\_i= \\overline{r}\\_i$ in $ \\mathbb R^n / \\mathcal{L}(J)$. Let ${\\bf r'}$ be the (*m* × *n*)-matrix whose rows are *r*ʹ1, …, *r*ʹ*m*. Then for *g* ∈ G we have that $\\tilde{g}\\_{\\bf r}=\\tilde{g}\\_{\\bf r'}$.\nWe have that *r*ʹ*i* = *r**i* + *l**i* for some *l**i* ∈ L(*J*). Let *L* be the (*m* × *n*)-matrix whose rows are *l*1, …, *l**m*. In particular, ${\\bf r'}={\\bf r}+L$. Write $g=\\sum\\_{\\alpha \\in \\mathbb Z^n\\_{\\geq 0}}c\\_{\\alpha} {\\bf x}^{\\alpha}$. Since *g* ∈ G we have by that the value *l**i* ⋅ *α* is the same for all *α* with *c**α* ≠ 0. Let *a**i* be this common value. Observe that min*c**α* ≠ 0{*r*ʹ*i* ⋅ *α*} = min*c**α* ≠ 0{*r**i* ⋅ *α*} + *a**i*. In particular, we have that the column vector (*a*1, …, *a**m*) is equal to *L* ⋅ *α* for all *α* with *c**α* ≠ 0 and therefore $\\mu\\_{\\bf r'}(f)= \\mu\\_{\\bf r}(f)+L\\cdot \\alpha$. Finally, we compute\n \n$$\\tilde{g}\\_{\\bf r'}=\\sum\\_{\\alpha \\in \\mathbb Z^n\\_{\\geq 0}}c\\_{\\alpha} {\\bf x}^{\\alpha}{\\bf t}^{{\\bf r}'\\cdot \\alpha-\\mu\\_{\\bf r'}(g)}=\n\\sum\\_{\\alpha \\in \\mathbb Z^n\\_{\\geq 0}}c\\_{\\alpha} {\\bf x}^{\\alpha}{\\bf t}^{{\\bf r}\\cdot \\alpha+ L\\cdot \\alpha -(\\mu\\_{\\bf r}(g)+L\\cdot \\alpha)}=\n\\tilde{g}\\_{\\bf r}.\\tag\\*{\\qed}$$ \nIn the following, when there is no risk of confusion, we write $\\tilde J$ for $\\tilde J\\_{\\bf r},\\tilde f$ for $\\tilde f\\_{\\bf r}$ and $\\tilde A$ for $\\tilde A\\_{\\bf r}$.\nFor ${\\bf a}=(a\\_1, \\dots, a\\_m)\\in \\mathbb K^m$ denote by $({\\bf t}-{\\bf a})$ the maximal ideal ⟨*t*1 − *a*1, …, of $\\mathbb K[t\\_1,\\dots,t\\_m]$. Then we denote by $\\tilde A\\vert\\_{{\\bf t}={\\bf a}}$ the tensor product $\\tilde A\\otimes\\_{\\mathbb K[t\\_1,\\dots,t\\_m]} \\mathbb K[t\\_1,\\dots,t\\_m]$ $/({\\bf t}-{\\bf a})$. Additionally, for $f\\in \\mathbb K[t\\_1,\\dots,t\\_m][x\\_1,\\dots,x\\_n]$ let $f\\vert\\_{{\\bf t}={\\bf a}} \\in \\mathbb K[x\\_1,\\dots,x\\_n]$ be the evaluation of *f* at *t**i* = *a**i* for *i* ∈ [*m*]. Similarly, denote by $\\tilde J\\vert\\_{{\\bf t}={\\bf a}}$ the ideal of $\\mathbb K[x\\_1,\\dots,x\\_n]$ generated by the set $\\lbrace \\tilde{f}\\vert\\_{{\\bf t}={\\bf a}} \\colon f\\in J \\rbrace$. For a set $B\\subset \\mathbb K[t\\_1,\\dots,t\\_m][x\\_1,\\dots,x\\_n]$ let $B\\vert\\_{{\\bf t}={\\bf a}}:=\\{b\\vert\\_{{\\bf t}={\\bf a}}\\colon b\\in B\\}$. Recall the isomorphism $\\mathbb K[t\\_1,\\dots,t\\_m]/({\\bf t}-{\\bf a})\\to \\mathbb K$ that sends an element $\\bar f$ to $f\\vert\\_{{\\bf t}={\\bf a}}$. It extends to an isomorphism $\\tilde A\\vert\\_{{\\bf t}={\\bf a}}\\cong \\mathbb K[x\\_1,\\dots,x\\_n]/\\tilde J\\vert\\_{{\\bf t}={\\bf a}}$, which explains our choice of notation.\nDenote ${\\bf 1}:=(1,\\dots,1)\\in \\mathbb Z^m$.\nLet *w* = (*w*1, …, *w**n*) = *r*1 + ⋯ + *r**m*, and *w*ʹ = ( − 1, …, − 1, *w*1, …, *w**n*) ∈ Z*m* + *n*, where *w**i* is the weight of *x**i* and − 1 is the weight of *t**j* for *i* = 1, …, *n* and *j* = 1, …, *m*. Then for every *f* ∈ *J* its lift $\\tilde f$ is *w*ʹ-homogeneous. In particular, the ideal *J̃* is *w*ʹ-homogeneous.\nLet $f=\\sum\\_{\\alpha\\in \\mathbb Z\\_{\\ge 0}^n} c\\_\\alpha{\\bf x}^{\\alpha}$. Then the lift of *f* is by definition $\\tilde f=\\sum c\\_\\alpha{\\bf x}^\\alpha{\\bf t}^{{\\bf r}\\cdot \\alpha-\\mu(f)}$. Notice that the *w*ʹ-weight of a monomial $c\\_\\alpha{\\bf x}^\\alpha{\\bf t}^{{\\bf r}\\cdot \\alpha-\\mu(f)}$ in $\\tilde f$ is\n \n$$w\\cdot \\alpha -{\\bf 1}\\cdot ({\\bf r}\\cdot \\alpha-\\mu(f)) = \\sum\\_{i=1}^m r\\_i\\cdot \\alpha - \\sum\\_{i=1}^m r\\_i\\cdot \\alpha + \\sum\\_{i=1}^m \\min\\_{c\\_\\beta\\not=0}(r\\_i\\cdot \\beta) = \\sum\\_{i=1}^m \\min\\_{c\\_\\beta\\not=0}(r\\_i\\cdot \\beta).$$ \nThis implies the first claim. The second claim is immediate as $\\tilde J=\\langle \\tilde f\\colon f\\in J\\rangle$.\nIn more generally we can choose *v* = *c*1*r*1 + … + *c**m**r**m* ∈ *C*∘ with $c\\_i\\in \\mathbb R\\_{\\ge 0}$ and *v*ʹ = ( − *c*1, …, − *c**m*, *v*1, …, *v**n*). Then $\\tilde J$ is *v*ʹ-homogeneous.\nLet $h\\in \\tilde J$ be *w*ʹ-homogeneous with *w*ʹ as in . Then $h={\\bf t}^{\\bf u}\\tilde{f}$ for some *f* ∈ *J* and ${\\bf u}\\in \\mathbb{Z}\\_{\\geq 0}^m$.\nLet *h* = ∑*i* = 1*s**p**i**f̃**i* with *f**i* ∈ *J* and *p**i* ∈ K[*t*1, …, *t**m*][*x*1, …, *x**n*] *w*ʹ-homogeneous. Then $h\\vert\\_{{\\bf t}={\\bf 1}}=\\sum\\_{i=1}^s p\\_i\\vert\\_{{\\bf t}={\\bf 1}}f\\_i\\in J$. Furthermore, since *h* is *w*ʹ-homogeneous, $h={\\bf t}^{\\bf u}\\widetilde{h\\vert\\_{{\\bf t}={\\bf 1}}}$ for some ${\\bf u}\\in \\mathbb{Z}\\_{\\geq 0}^m$. Hence, we can take $h\\vert\\_{{\\bf t}={\\bf 1}}$ as *f*.\nConsider $f=\\sum\\_{\\alpha \\in \\mathbb Z^n\\_{\\ge 0}} c\\_\\alpha{\\bf x}^\\alpha\\in J$ with a unique monomial $c\\_\\beta{\\bf x}^\\beta\\in \\operatorname{in}\\_<(J)$. That is, for every *α* ≠ *β* with *c**α* ≠ 0, there exists no $g'\\in\\mathcal G$ such that ${\\bf x}^\\alpha$ is divisible by in<(*g*ʹ). Then\n \n$$\\label{eq:lifted element}\n\\tilde f = \\sum\\_{\\alpha \\in \\mathbb Z^n\\_{\\ge 0}} c\\_\\alpha {\\bf x}^{\\alpha} {\\bf t}^{{\\bf r}\\cdot(\\alpha -\\beta)} = c\\_\\beta {\\bf x}^{\\beta} + \\sum\\_{\\alpha\\not=\\beta} c\\_\\alpha{\\bf x}^{\\alpha}{\\bf t}^{{\\bf r}\\cdot (\\alpha-\\beta)},$$ \ncontains a unique monomial with coefficient in $\\mathbb K$. In particular, this is the case for elements of the reduced Gröbner basis $\\mathcal G$.\nThe assumption that only the monomial $c\\_\\beta{\\bf x}^\\beta$ of *f* is contained in in<(*J*) ensures that $c\\_\\beta{\\bf x}^\\beta$ is also a monomial appearing in in*r**i*(*f*) for all *i*. To see this, assume there is a ray *r**i* such that $c\\_\\beta{\\bf x}^\\beta$ is not a monomial in in*r**i*(*f*) and let $w'=\\sum\\_{j\\not=i} r\\_j$. By we can find an *ε* so that in*ε**w*ʹ + *r**i*(*f*) = in*w*ʹ(in*r**i*(*f*)). As we assumed $c\\_\\beta{\\bf x}^\\beta$ is not a monomial in in*r**i*(*f*), it is also not a monomial in in*ε**w*ʹ + *r**i*(*f*). But *ε**w*ʹ + *r**i* ∈ *C*∘ and so in*ε**w*ʹ + *r**i*(*f*) ∈ in*ε**w*ʹ + *r**i*(*J*) = in<(*J*). This however is a contradiction as we assumed that $c\\_\\beta{\\bf x}^\\beta$ is the only monomial in *f* that is contained in in<(*J*). Therefore, $r\\_i\\cdot\\beta=\\min\\_{c\\_\\alpha\\not =0}\\{r\\_i\\cdot \\alpha\\}$ for all *i* and ${\\bf r}\\cdot \\beta=\\mu\\_{\\bf r}(f)$. Assume on the contrary that there exists another monomial $c\\_\\gamma{\\bf x}^\\gamma$ in *f* with ${\\bf r}\\cdot\\gamma={\\bf r}\\cdot\\beta$. Then $c\\_\\gamma{\\bf x}^\\gamma$ is also a monomial in in*r**i*(*f*) for all *i*. Moreover, for *w* = *r*1 + … + *r**m* ∈ *C*∘ the initial form in*w*(*f*) contains both monomials $c\\_\\beta{\\bf x}^\\beta$ and $c\\_\\gamma{\\bf x}^\\gamma$. But as in*w*(*J*) = in<(*J*) this implies that $c\\_\\gamma{\\bf x}^\\gamma\\in \\operatorname{in}\\_<(J)$, a contradiction. Hence, $c\\_\\beta{\\bf x}^\\beta$ is the unique monomial in *f* with $\\mu\\_{\\bf r}(f)={\\bf r}\\cdot \\beta$ and we obtain as ${\\bf r}\\cdot(\\alpha-\\beta)\\not=0$ for all $\\alpha\\not=\\beta$. This completes the proof of the first claim.\nTo prove the second part, assume that *g* ∈ G has a monomial term $c\\_\\gamma{\\bf x}^{\\gamma}\\in \\operatorname{in}\\_<(J)$ with $c\\_\\gamma{\\bf x}^{\\gamma}\\neq \\operatorname{in}\\_<(g)$. Then by the definition of the reduced Gröbner basis, $c\\_\\gamma{\\bf x}^{\\gamma}$ is not divisible by in<(*g*ʹ) for any $g'\\in \\mathcal G\\setminus \\{g\\}$. Hence, it should be divisible by in<(*g*), but this cannot happen as *g* is ${\\bf d}$-homogeneous by .\nWe extend the monomial term order < on $\\mathbb K[x\\_1,\\dots,x\\_n]$ to a monomial term order ≪ on the polynomial ring $\\mathbb K[t\\_1,\\dots,t\\_m,x\\_1,\\dots,x\\_n]$ in such a way that *t**i* ≪ *x**j* for all 1 ≤ *i* ≤ *m* and 1 ≤ *j* ≤ *n*, and\n \n$$\\begin{gathered}\n\\label{eq:<<}\n{\\bf x}^\\alpha{\\bf t}^{\\lambda}\\ll{\\bf x}^\\beta{\\bf t}^{\\mu}\\quad\\text{if and only if}\\quad\n(i)\\quad {\\bf x}^\\alpha<{\\bf x}^\\beta \\quad \\text{or}\\quad\n(ii)\\quad {\\bf x}^\\alpha={\\bf x}^\\beta\\quad \\text{and}\\quad {\\bf t}^{\\lambda}<\\_{\\rm lex}{\\bf t}^{\\mu}.\\end{gathered}$$ \nThe set $\\tilde{\\mathcal{G}} =\\{\\tilde g\\colon g\\in \\mathcal G\\}$ is a Gröbner basis for *J̃* with respect to ≪ . In particular, $\\tilde{\\mathcal{G}} $ is a generating set for *J̃*.\nSince *J̃* is *w*ʹ-homogeneous by Lemma [lem:w’homogeneous], it is enough to show that for every *w*ʹ-homogeneous polynomial *h* ∈ *J̃* there exists some $\\tilde g\\_i\\in\\tilde{\\mathcal{G}} $ whose initial term divides in≪(*h*). Let *h* ∈ *J̃* be *w*ʹ-homogeneous. By Lemma [lem:homogeneous] there exist *f* ∈ *J* and ${\\bf u}\\in \\mathbb{Z}\\_{\\geq 0}^m$ such that $h={\\bf t}^{\\bf u}\\tilde{f}$. We compute\n \n$$\\begin{gathered}\n\\operatorname{in}\\_{\\ll}(h) = {\\bf t}^{\\bf u}\\operatorname{in}\\_{\\ll}(\\tilde f) \\overset{\\text{Def. }\\ll}{=} {\\bf t}^{{\\bf u}+{\\bf v}}\\operatorname{in}\\_<(f)\n\\overset{\\exists \\ g\\in \\mathcal G}{=} {\\bf t}^{{\\bf u}+{\\bf v}}c\\_\\alpha{\\bf x}^{\\alpha}\\operatorname{in}\\_{<}(g) \\overset{\\text{\\thref{lem:lifts}}}{=} {\\bf t}^{{\\bf u}+{\\bf v}}c\\_\\alpha{\\bf x}^{\\alpha}\\operatorname{in}\\_{\\ll}(\\tilde g),\\end{gathered}$$ \nwhere *g* is a suitable element of the Gröbner basis $\\mathcal G$ for *J* and ${\\bf v}\\in \\mathbb Z\\_{\\ge 0}^m$. As $\\tilde g\\in \\tilde{\\mathcal{G}}$, this shows that $\\tilde{\\mathcal{G}}$ is a Gröbner basis for *J̃*. Hence, by $\\tilde{\\mathcal{G}}$ is a generating set for *J̃*.\nAs a direct consequence of and we obtain that the lifted algebra is independent of the choice of vectors *r*1, …, *r**m*.\nFor ${\\bf r}$ and ${\\bf r}'$ as in we have that $\\tilde{A}\\_{\\bf r} = \\tilde{A}\\_{\\bf r '}$.\nThe algebras $\\tilde A\\vert\\_{{\\bf t}={\\bf a}}$ constitute the fibers of a flat family introduced below. The following result leads to fibers over different points being isomorphic if they have zero entries in the same positions.\nConsider ${\\bf a}=(a\\_1,\\dots,a\\_m)$ and ${\\bf b}=(b\\_1,\\dots,b\\_m)$ in $\\mathbb K^m$ with the property that *a**i* = 0 if and only if *b**i* = 0. Then the algebras $\\tilde A\\vert\\_{{\\bf t}={\\bf a}}$ and $\\tilde A\\vert\\_{{\\bf t}={\\bf b}}$ are isomorphic.\nLet *φ* be the $\\mathbb K$-algebra automorphism of $\\mathbb K[t\\_1,\\dots,t\\_m]$ defined by $\\varphi(t\\_i)=\\frac{a\\_i}{b\\_i}t\\_i$ if $b\\_i\\not =0$ and *φ*(*t**i*) = *t**i* if *b**i* = 0. Then $\\varphi({\\bf t}-{\\bf a})=({\\bf t}-{\\bf b})$, so the claim follows by .\nFor computational reasons it might be desirable to work with ray generators for *C* that are not representatives of primitive ray generators for $\\overline{C}$. Let ${\\bf r}$ and ${\\bf r'}$ be two choices of ray matrices whose rows satisfy $\\overline{r'}\\_j=q\\_j\\overline{r}\\_j$ with $q\\_j\\in \\mathbb Q$ for all *j*. Then we still have an isomorphism between $\\tilde A\\_{\\bf r}\\vert\\_{{\\bf t}={\\bf a}}$ and $\\tilde A\\_{\\bf r'}\\vert\\_{{\\bf t}={\\bf a}}$ for all ${\\bf a}\\in \\mathbb K^m$. For the proof it is necessary to extend the polynomial ring $\\mathbb K[t\\_1,\\dots,t\\_m]$ to a ring $\\mathbb K\\big[t\\_1^{\\mathbb Q},\\dots,t\\_m^{\\mathbb Q}\\big]$, where the *t**i*’s are allowed to have rational exponents. The automorphism of $\\mathbb K\\big[t\\_1^{\\mathbb Q},\\dots,t\\_m^{\\mathbb Q}\\big]$ defined by *h*(*t**i*) = *t**i**q**i* extends to an automorphism of $\\mathbb K\\big[t\\_1^{\\mathbb Q},\\dots,t\\_m^{\\mathbb Q}\\big][x\\_1,\\dots,x\\_n]$ with the property $h\\big(\\tilde f\\_{\\bf r}\\big)=\\tilde f\\_{\\bf r'}$ for all ${\\bf d}$-homogeneous polynomials *f* ∈ *J*. The rest follows from .\nBefore presenting our main result we explain how the ideal $\\tilde J\\_{\\bf r}$ is related to the ideal $\\hat J\\_w$ in .\nConsider a face *τ* of *C* spanned by a subset {*r**i*1, …, *r**i**s*} of {*r*1, …, *r**m*} and the lineality space $\\mathcal L(J)$. We define ${\\bf t}\\_\\tau\\in \\mathbb K[t]^{m}$ by\n \n$$({\\bf t}\\_\\tau)\\_k:= \\begin{cases}t& \\text{if}\\quad k\\in\\{i\\_1,\\ldots,i\\_s\\},\n\\\\\n1 &\\text{otherwise,}\n\\end{cases}\n\\qquad\n\\text{for}\n\\quad k=1,\\ldots,m.$$ \nThen for *w**τ* = *r**i*1 + … + *r**i**s* we have $\\tilde J\\vert\\_{{\\bf t}={\\bf t}\\_\\tau}=\\hat J\\_{w\\_\\tau}$.\nLet < be the monomial term order compatible with *C*. Consider an element $g=\\sum\\_{\\alpha \\in \\mathbb Z^n\\_{\\ge 0}}c\\_{\\alpha}{\\bf x}^{\\alpha}$ in G with $\\operatorname{in}\\_<(g)=c\\_\\beta {\\bf x}^\\beta$. Since the vectors *w**τ*, *w* : = *w**τ* − *r**i**s* and *r**i**s* are all in *C*, by Remark [rem:Eisenbud]:\n \n$$\\operatorname{in}\\_<(\\operatorname{in}\\_{w\\_\\tau}(g))=\\operatorname{in}\\_<(\\operatorname{in}\\_{w}(g))=\\operatorname{in}\\_{<}(\\operatorname{in}\\_{r\\_{i\\_s}}(g))=\\operatorname{in}\\_<(g)=c\\_\\beta {\\bf x}^\\beta.$$ \nThis implies that the initial forms of in*w**τ*(*g*), in*w*(*g*) and in*r**i**s*(*g*) contain $c\\_\\beta {\\bf x}^\\beta$. In other words,\n \n$$\\min\\_{c\\_{\\alpha}\\not =0}\\{w\\_\\tau\\cdot \\alpha\\}=w\\_\\tau\\cdot\\beta=(w+r\\_{i\\_s})\\cdot \\beta, \\qquad \\min\\_{c\\_{\\alpha}\\not =0}\\{w\\cdot \\alpha\\}=w\\cdot\\beta,\\qquad \\min\\_{c\\_{\\alpha}\\not =0}\\{r\\_{i\\_s}\\cdot \\alpha\\}=r\\_{i\\_s}\\cdot\\beta.$$ \nTherefore,\n \n$$(w+r\\_{i\\_s})\\cdot\\beta=\\min\\_{c\\_{\\alpha}\\not =0}\\{(w+r\\_{i\\_s})\\cdot \\alpha\\} =\\min\\_{ c\\_{\\alpha}\\not =0}\\{w\\cdot \\alpha\\} + \\min\\_{c\\_{\\alpha}\\not =0} \\{r\\_{i\\_s}\\cdot \\alpha\\}.$$ \nUsing the same argument multiple times we obtain that\n \n$$\\min\\_{c\\_{\\alpha}\\not =0}\\Bigg\\{ \\bigg(\\sum\\_{j=1}^s r\\_{i\\_j}\\bigg)\\cdot \\alpha \\Bigg\\} = \\sum\\_{j=1}^s \\min\\_{c\\_{\\alpha}\\not =0}\\{r\\_{i\\_j}\\cdot \\alpha\\}.$$ \nNow the claim follows by comparing the generating sets of $\\hat J\\_{w\\_\\tau}$ and $\\tilde J\\vert\\_{{\\bf t}={\\bf t}\\_{\\tau}}$.\nWe are now prepared to present our main theorem:\nLet *J* be a ${\\bf d}$-homogeneous ideal in $\\mathbb K\\_{\\bf d}[x\\_1,\\dots,x\\_n]$, $A=\\mathbb K\\_{\\bf d}[x\\_1,\\ldots,x\\_n]/J$, *C* a maximal cone in GF(*J*) with compatible monomial term order < and ${\\bf r}$ an (*m* × *n*)-matrix whose rows are representatives of the primitive ray generators of $\\overline{C}\\subset \\mathbb R^n/\\mathcal L(J)$. Then:\n* The algebra $\\tilde A\\_{\\bf r}$ is a free $\\mathbb K[t\\_1,\\dots,t\\_m]$-module with basis $\\mathbb B\\_<$, the standard monomial basis of *A* with respect to in<(*J*). In particular, we have a flat family\n \n$$\\begin{tikzcd}[ampersand replacement=\\&,cramped]\n {\\rm Proj}(\\tilde A) \\ar[r,hook]\\ar[d,\"\\pi\"] \\& \\mathbb P({\\bf d})\\times \\mathbb A^m \\ar[ld,twoheadrightarrow]\\\\\n \\mathbb A^m. \\&\n \\end{tikzcd}$$\n* For every face *τ* of *C* there exists ${\\bf a}\\_\\tau\\in \\mathbb A^m$ such that $\\pi^{-1}({\\bf a}\\_\\tau) = {\\rm Proj}(\\mathbb K\\_{\\bf d}[x\\_1,\\dots,x\\_n]/$ in*τ*(*J*)). In particular, generic fibers are isomorphic to ${\\rm Proj}(A)$ and there exist special fibers for every proper face *τ* ⊂ *C*.\n(*i*) We extend < to a monomial term order ≪ on $\\mathbb K[t\\_1,\\dots,t\\_m,x\\_1,\\dots,x\\_n]$ as in . Let $\\mathbb B\\_{\\ll}$ be the standard monomial basis for $\\tilde A\\_{\\bf r}$ induced by $\\operatorname{in}\\_{\\ll}\\big(\\tilde J\\_{\\bf r}\\big)$. Then $\\mathbb B\\_{\\ll}$ is a basis for $\\tilde A\\_{\\bf r}$ as a K-vector space and a generating set for $\\tilde A\\_{\\bf r}$ as a $\\mathbb K[t\\_1,\\dots,t\\_m]$-module. By and we have\n \n$$\\begin{gathered}\n \\mathbb B\\_{\\ll} = \\big\\{\\bar {\\bf t}^\\beta\\bar {\\bf x}^\\alpha \\colon \\operatorname{in}\\_{\\ll}(\\tilde g)\\not\\vert \\ {\\bf t}^\\beta{\\bf x}^\\alpha \\text{ for all }g\\in \\mathcal G\\big\\},\\qquad\n \\mathbb B\\_{<} = \\big\\{\\bar {\\bf x}^\\alpha \\colon \\operatorname{in}\\_{<}(g)\\not\\vert \\ {\\bf x}^\\alpha \\text{ for all }g\\in \\mathcal G\\big\\}.\n \\end{gathered}$$ \nObserve that $\\mathbb B\\_{\\ll}$ can be reduced to a $\\mathbb K[t\\_1,\\dots,t\\_m]$-basis of $\\tilde A\\_{\\bf r}$ by defining\n \n$$\\mathbb B'\\_{\\ll}:= \\mathbb B\\_{\\ll} \\setminus \\big\\{\\bar{\\bf t}^\\beta\\bar{\\bf x}^\\alpha \\colon \\exists \\ \\bar{\\bf t}^{\\gamma}\\bar{\\bf x}^\\alpha\\in \\mathbb B\\_{\\ll} \\text{ with } \\gamma < \\beta\\big\\},$$ \nwhere *γ* < *β* means that *γ**i* ≤ *β**i* for all *i* and *γ**j* < *β**j* for some *j*. Note that $\\mathbb B\\_{<}\\subseteq \\mathbb B'\\_{\\ll}$. Now assume there is a monomial $\\bar{\\bf t}^\\beta\\bar{\\bf x}^\\alpha \\in \\mathbb B'\\_{\\ll}\\setminus \\mathbb B\\_<$. By we have that $\\operatorname{in}\\_{\\ll}\\big(\\tilde J\\big)\\vert\\_{{\\bf t}={\\bf 1}}=\\operatorname{in}\\_<(J)$. Hence, $\\mathbb B'\\_{\\ll}\\vert\\_{{\\bf t}={\\bf 1}}=\\mathbb B\\_{\\ll}\\vert\\_{{\\bf t}={\\bf 1}}=\\mathbb B\\_<$. In particular, this implies that $\\bar{\\bf x}^\\alpha\\in \\mathbb B\\_<$. But as $\\mathbb B\\_<\\subset \\mathbb B\\_{\\ll}$ this is a contradiction to the definition of $\\mathbb B'\\_{\\ll}$.\nFor the second claim, let $\\mathbb K[t\\_1,\\dots,t\\_m]\\_{\\bf d}[x\\_1,\\dots,x\\_n]$ denote the polynomial ring in *x*1, …, *x**n* with coefficients in $\\mathbb K[t\\_1,\\dots,t\\_m]$ and grading induced by ${\\bf d}$. Then by and $\\tilde J\\subseteq \\mathbb K[t\\_1,\\dots,t\\_m]\\_{\\bf d}[x\\_1,\\dots,x\\_n]$ is ${\\bf d}$-homogeneous. This yields the embedding ${\\rm Proj}\\big(\\tilde A\\big)\\hookrightarrow \\mathbb P({\\bf d})\\times \\mathbb A^m$. The projection $\\mathbb P({\\bf d})\\times \\mathbb A^m\\twoheadrightarrow \\mathbb A^m$ induces the flat morphism $\\pi\\colon {\\rm Proj}(\\tilde A)\\to \\mathbb A^m$ as $\\tilde A$ has a $\\mathbb K[t\\_1,\\dots,t\\_m]$-basis by the first claim.\n(*i**i*) Every face *τ* of *C* induces a face $\\overline{\\tau}$ of $\\overline{C}$ with primitive ray generators $\\overline{r}\\_{i\\_1},\\dots, \\overline{r}\\_{i\\_{s}}$ for some *i*1, …, *i**s* ∈ [*m*]. We define\n \n$$({\\bf a}\\_{\\tau})\\_k:=\n \\begin{cases}\n 0 & \\text{if}\\quad k\\in\\{i\\_1,\\dots,i\\_{s}\\},\\\\\n 1 & \\text{otherwise},\n \\end{cases}\n \\qquad\n\\text{for}\n\\quad k=1,\\ldots,m.$$ \nThen by we have $\\tilde J\\vert\\_{{\\bf t}={\\bf a}\\_\\tau}=\\hat J\\_{w\\_\\tau}\\vert\\_{t=0}$ which is equal to in*τ*(*J*) by Remark [rem:flat].\nTorus equivariant families\n--------------------------\nWe explain how the above results are related to Kaveh–Manon’s recent work on the classification of torus equivariant families.\nConsider the lattice $N=\\mathbb Z^n$, its dual lattice *M* = *N*\\* and a fan $\\Sigma\\subset N\\otimes\\_{\\mathbb Z}\\mathbb R$. We write *X*Σ for the toric variety associated to Σ. Furthermore, we define $\\mathcal O\\_N$ to be the semifield of piecewise linear functions on *N* and $\\mathcal O\\_{\\Sigma}$ the semifield of piecewise linear functions on |Σ| ∩ *N*. For $a,b\\in\\mathcal O\\_{\\Sigma}$ we have *a* ⊗ *b* : = *a* + *b* and *a* ⊕ *b* : = min{*a*, *b*}, where the minimum is taken pointwise. In particular, $\\mathcal O\\_{\\Sigma}$ is partially ordered: *a* ≥ *b* if *a* ⊕ *b* = *b*. A *PL-quasivaluation* on an algebra *A* is a map $\\operatorname{\\mathfrak v}\\colon A\\to \\mathcal O\\_{\\Sigma}$ that satisfies: (*i*) $\\operatorname{\\mathfrak v}(fg)\\ge \\operatorname{\\mathfrak v}(f)+\\operatorname{\\mathfrak v}(g)$ and (*i**i*) $\\operatorname{\\mathfrak v}(f+g)\\ge \\min\\{\\operatorname{\\mathfrak v}(f),\\operatorname{\\mathfrak v}(g)\\}$, where the minimum is taken pointwise in $\\mathcal O\\_{\\Sigma}$. If (*i*) is an equality, $\\operatorname{\\mathfrak v}$ is called a *PL-valuation*. If *A* is a graded algebra $A=\\bigoplus\\_{n\\in \\mathbb Z} A\\_n$ then $\\operatorname{\\mathfrak v}$ is called *homogeneous* if it is compatible with the grading, i.e., for $f=\\sum\\_{n\\in \\mathbb Z} c\\_n g\\_n$ with *g**n* ∈ *A**n* and coefficients *c**n* we have $\\operatorname{\\mathfrak v}(f)=\\operatorname{\\mathfrak v}(g\\_k)$ for the smallest *k* with $c\\_k\\not =0$.\nGiven a sheaf of algebras $\\mathcal A$ on *X*Σ its *relative spectrum* denoted by $\\mathbf{Spec}(\\mathcal A)$ is the scheme obtained from gluing affine schemes $\\operatorname{Spec}(\\mathcal A(U\\_i))$, where ⋃*i**U**i* is an open cover of *X*Σ and $\\mathcal A(U\\_i)$ is the corresponding section of $\\mathcal A$.\n[] Let *A* = ⨁*n* ≥ 0*A**n* be a positively graded algebra and a domain over an algebraically closed field of characteristic zero and Σ a fan in $N\\otimes\\_{\\mathbb Z}\\mathbb R$. Then the following are equivalent information:\n* a *T**N*-equivariant flat sheaf $\\mathcal A$ of positively graded algebras of finite type over *X*Σ such that $\\mathbf{Spec}(\\mathcal A)$ has reduced and irreducible fibers and its generic fibers are isomorphic to Spec(*A*);\n* a homogeneous PL-valuation $\\operatorname{\\mathfrak v}\\colon A\\to \\mathcal O\\_{\\Sigma}$ with finite Khovanskii basis. (Given $\\operatorname{\\mathfrak v}\\colon A\\to \\mathcal O\\_{\\Sigma}$ for every ray *ρ* ∈ Σ one constructs a quasivaluation $\\operatorname{\\mathfrak v}\\_\\rho\\colon A\\to \\mathbb Z\\cup \\{\\infty\\}$ with associated graded algebra ${\\rm gr}\\_{\\rho}(A)$, see . A *Khovanskii basis* for $\\operatorname{\\mathfrak v}$ is a subset $\\mathcal B\\subset A$ such that for every ray *ρ* ∈ Σ the image of $\\mathcal B$ in ${\\rm gr}\\_{\\rho}(A)$ is a set of algebra generators.)\nConsider a presentation of *A*, i.e., a surjection ${\\rm pr}\\colon \\mathbb K[x\\_1,\\dots,x\\_n]\\to A$ whose kernel is a weighted homogeneous prime ideal *J* and so it induces an isomorphism of graded algebras $A\\cong \\mathbb K[x\\_1,\\dots,x\\_n]/J$. Take a cone *σ* in GF(*J*). Then *σ* defines PL-quasivaluation $\\operatorname{\\mathfrak v}\\_{\\sigma}\\colon A\\to \\mathcal O\\_\\sigma$ as follows: let *C* be a maximal cone in GF(*J*) with face *σ* and denote by $\\mathbb B\\_<$ the standard monomial basis associated to *C* (here < is the compatible monomial term order). If in*σ*(*J*) is prime, then $\\operatorname{\\mathfrak v}\\_{\\sigma}$ is a PL-valuation. Then for $\\bar{\\bf x}^\\alpha\\in \\mathbb B\\_<$ we have $\\operatorname{\\mathfrak v}\\_{\\sigma}(\\bar{\\bf x}^\\alpha):= -\\cdot \\alpha\\colon \\sigma\\to \\mathbb Z$, where − ⋅ − is the pairing between the lattices *N* and *M*, as above. For an arbitrary element *f* ∈ *A* write $f=\\sum\\_{\\bar{\\bf x}^\\alpha\\in \\mathbb B\\_<} c\\_\\alpha \\bar{\\bf x}^\\alpha$. Then $\\operatorname{\\mathfrak v}\\_\\sigma(f):=\\min\\_{c\\_\\alpha\\not =0}\\{\\operatorname{\\mathfrak v}\\_\\sigma(\\bar{\\bf x}^\\alpha)\\}$.\n[] The PL-quasivaluation $\\operatorname{\\mathfrak v}\\_\\sigma\\colon A\\to\\mathcal O\\_\\sigma$ defines a *filtration* on *A* with filtered pieces $F\\_m(\\operatorname{\\mathfrak v}\\_\\sigma):=\\{f\\in A\\colon \\operatorname{\\mathfrak v}\\_\\sigma(f)\\ge -\\cdot m \\}$ for *m* ∈ *M*. The *Rees algebra* of the PL-valuation $\\operatorname{\\mathfrak v}\\_\\sigma$ is $\\mathcal R\\_\\sigma(A):=\\bigoplus\\_{m\\in M}F\\_m(\\operatorname{\\mathfrak v}\\_\\sigma)t^m$, where we think of *t**m* as a character of the torus $T\\_N=N\\otimes\\_{\\mathbb Z}\\mathbb C^\\*$.\nMore generally, a PL-quasivaluation can be obtained from a collection of equidimensional cones *σ*1, …, *σ**k* in Trop(*J*) which are faces of the same maximal cone *C* in GF(*J*). Let Σ ⊂ Trop(*J*) be the subfan whose maximal cones are *σ*1, …, *σ**k* and $\\operatorname{\\mathfrak v}\\_{\\Sigma}\\colon A \\to \\mathcal O\\_N$ be the corresponding PL-valuation. By Kaveh–Manon’s classification of toric families this yields a flat family\n \n$$\\label{eq:KM family}\n\\psi\\colon\\ \\mathbf{Spec}(\\mathcal R\\_\\Sigma(A))\\to X\\_\\Sigma.$$ \nHere $\\mathcal R\\_\\Sigma(A)$ is the *T**N*-equivariant flat sheaf of Rees algebras on *X*Σ. The scheme ${\\bf Spec}(\\mathcal{R}\\_{\\Sigma}(A))$ is glued from Spec(R*σ*) for *σ* ∈ Σ. For *p* ∈ *T**N* ⊂ *X*Σ we have *ψ*− 1(*p*) ≅ Spec(*A*). Moreover, *ψ*Σ has a special fiber over every torus fixed point of *X*. More precisely, let *p**σ**i* be the torus fixed point corresponding to the (maximal) cone *σ**i* ∈ Σ. Then\n \n$$\\label{eq:fibers psi}\n\\psi^{-1}(p\\_{\\sigma\\_i})\\cong \\operatorname{Spec}\\big(\\mathbb K[x\\_1,\\dots,x\\_n]/\\operatorname{in}\\_{\\sigma\\_i}(J)\\big).$$ \nNote that we can apply the construction of to the ideal *J* and the maximal cone *C* ∈ GF(*J*). In what follows we explore the relation between the flat families from and . Before stating our results ( and ), we recall necessary background from . We fix a maximal cone *C* in GF(*J*), denote by $\\mathcal G$ the associated reduced Gröbner basis and let $\\mathbb B\\_<$ be the standard monomial basis for the compatible monomial term order < .\n[] The PL-quasivaluation $\\mathfrak{w}\\_C\\colon \\mathbb K[x\\_1,\\dots,x\\_n]\\to \\mathcal O\\_C$ associated to *C* is defined by $\\mathfrak{w}\\_C({\\bf x}^\\alpha)= -\\cdot \\alpha\\colon C\\to \\mathbb Z$. For a polynomial $f=\\sum c\\_\\alpha {\\bf x}^\\alpha$ we have $\\mathfrak{w}\\_C(f):=\\min\\_{c\\_\\alpha\\not =0}\\{ -\\cdot \\alpha\\}$. The associated Rees algebra is $\\mathcal R\\_C(\\mathbb K[x\\_1,\\dots,x\\_n])=\\bigoplus\\_{m\\in M} F\\_m(\\mathfrak{w}\\_C)t^m$, where *F**m*(w*C*) is defined analogous to $F\\_m(\\operatorname{\\mathfrak v}\\_\\sigma)$ in .\nThe PL-quasivaluation $\\operatorname{\\mathfrak v}\\_C\\colon A\\to \\mathcal O\\_C$ is obtained as a pushforward of the PL-quasivaluation $\\mathfrak{w}\\_C\\colon \\mathbb K[x\\_1,\\dots,x\\_n]\\to \\mathcal O\\_C$ along the morphism ${\\rm pr}\\colon \\mathbb K[x\\_1,\\dots,x\\_n]\\to A$. An *adapted basis* for $\\operatorname{\\mathfrak v}\\_C$ is a vector space basis $\\mathbb B$ for *A* such that $\\mathbb B\\cap F\\_m(\\operatorname{\\mathfrak v}\\_\\sigma)$ is a vector space basis of $F\\_m(\\operatorname{\\mathfrak v}\\_\\sigma)$ for every *m* ∈ *M*. By the standard monomial basis $\\mathbb B\\_<$ is an adapted basis for $\\operatorname{\\mathfrak v}\\_C$. Similarly, the monomial basis of $\\mathbb K[x\\_1,\\dots,x\\_n]$ is adapted to w*C*. Hence, $\\mathcal R\\_C(A)$ is a free $\\mathbb K[S\\_C]$-algebra with basis $\\mathbb B\\_<$ and $\\mathcal R\\_C(\\mathbb K[x\\_1,\\dots,x\\_n])$ is a free $\\mathbb K[S\\_C]$-algebra whose basis is the monomial basis, where $S\\_C:=-C^\\vee \\cap \\mathbb Z^n$ due to our min-convention. In particular:\n \n$$\\label{eq:Rees poly}\n\\mathcal R\\_C\\big(\\mathbb K[x\\_1,\\dots,x\\_n]\\big)\\cong \\mathbb K[S\\_C][x\\_1,\\dots,x\\_n].$$ \nManon explained the proof of the following proposition to us. It follows from results in .\nFor any maximal cone *C* ∈ GF(*J*), the Rees algebra $\\mathcal R\\_C(A)$ has an explicit presentation of form $\\mathbb K[S\\_C][x\\_1,\\dots,x\\_n]/\\hat J$. More precisely, the ideal $\\hat J$ is generated by homogenizations of elements in the reduced Gröbner basis $\\mathcal G$: let $g={\\bf x}^\\gamma+\\sum c\\_\\alpha{\\bf x}^\\alpha\\in \\mathcal G$ and ${\\bf x}^\\gamma=\\operatorname{in}\\_C(g)$, then the *homogenization* of *g* is\n \n$$\\label{eq:homog}\n\\hat g= {\\bf x}^\\gamma+\\sum c\\_\\alpha {\\bf x}^\\alpha t^\\beta\\qquad \\text{with}\\quad \\gamma-\\alpha=\\beta\\in S\\_C.$$ \nConsider $f\\in \\mathbb K[x\\_1,\\dots,x\\_n]$. It can be written as $f=\\sum\\_{{\\bf x}^\\alpha\\in \\mathbb B\\_<} c\\_\\alpha {\\bf x}^\\alpha+\\sum\\_{{\\bf x}^\\beta\\not\\in \\mathbb B\\_<} c\\_\\beta {\\bf x}^\\beta$. In particular, if $\\mathfrak w\\_C(f)\\ge -\\cdot m$ in $\\mathcal O\\_C$, then $\\operatorname{\\mathfrak v}\\_C(\\text{pr}(f))\\ge -\\cdot m$. So the morphism $\\text{pr}\\colon \\mathbb K[x\\_1,\\dots,x\\_n]$ → *A* induces a map $\\text{pr}\\_m\\colon F\\_m(\\mathfrak w\\_C)\\to F\\_m(\\operatorname{\\mathfrak v}\\_C)$ for all *m*. Further, every element in $\\mathbb B\\_<\\cap F\\_m(\\operatorname{\\mathfrak v}\\_C)$ is the image of a monomial in $F\\_m(\\mathfrak w\\_C)$. Hence, pr*m* is surjective and we obtain:\n \n$$\\widehat{\\text{pr}}:=\\bigoplus\\_{m\\in (\\mathbb Z^n)^\\*} \\text{pr}\\_m\\colon\\\n\\mathcal R\\_C(\\mathbb K[x\\_1,\\dots,x\\_n])\\to \\mathcal R\\_C(A).$$ \nBy this implies the first claim. Now let $\\hat J$ be the kernel of $\\widehat{\\text{pr}}$. Then $\\hat J=\\bigoplus\\_{m\\in M} \\hat J\\_m $, where $\\hat J\\_m=\\ker(\\text{pr}\\_m)\\subset F\\_m(\\mathfrak w\\_C)$. Now consider an element $g\\in \\mathcal G$ of form ${\\bf x}^\\gamma+\\sum c\\_\\alpha{\\bf x}^\\alpha$ with ${\\bf x}^\\gamma=\\operatorname{in}\\_C(g)$. We want to show that the element $\\hat g$ defined in lies in $\\hat J$ for all $g\\in \\mathcal G$. Recall that by the proof of , *γ* is such that *w* ⋅ *γ* ≤ *w* ⋅ *α* for all *w* ∈ *C* and all *α* with $c\\_\\alpha\\not =0$ in *g*. Hence, *w* ⋅ (*γ* − *α*) ≤ 0 and so *β* = *γ* − *α* ∈ − *C*∨. In particular, this implies that $\\hat g\\in \\hat J\\_\\gamma\\subset F\\_\\gamma(\\mathfrak w\\_\\sigma)$.\nLastly, we need to show that $\\hat J$ is generated by $\\{\\hat g\\colon g\\in \\mathcal G\\}$. Consider $h\\in \\hat J$. As $\\hat J=\\bigoplus\\_{m\\in M} \\hat J\\_m$, we may assume that $h\\in \\hat J\\_m$. Further, as $\\hat J\\_m\\subset F\\_m(\\mathfrak w\\_C)\\subset \\mathbb K[x\\_1,\\dots,x\\_n]$ we can think of *h* as an element of $\\mathbb K[x\\_1,\\dots,x\\_n]$ as well as an element of $\\mathcal R\\_C(\\mathbb K[x\\_1,\\dots,x\\_n])$. Then $\\operatorname{in}\\_<(h)={\\bf x}^\\mu \\operatorname{in}\\_<(g)$ for some $g\\in \\mathcal G$ of form $g={\\bf x}^\\gamma+\\sum c\\_\\alpha{\\bf x}^\\alpha$ with ${\\bf x}^\\gamma=\\operatorname{in}\\_<(g)$. In particular, similar to the above the exponent of $\\operatorname{in}\\_<({\\bf x}^\\mu g)={\\bf x}^{\\mu+\\gamma}$ has the property that *w* ⋅ (*μ* + *γ*) ≤ *w* ⋅ (*μ* + *α*) for all *w* ∈ *C* and all *α* with $c\\_\\alpha\\not =0$ in *g*. So, $h-{\\bf x}^{\\mu} g\\in F\\_m(\\mathfrak w\\_C)$. In particular, the division algorithm with respect to $\\mathcal G$ takes place inside $F\\_m(\\mathfrak w\\_C)$ and yields an expression of *h* in terms of $\\{\\hat g\\colon g\\in \\mathcal G \\}$.\nWe now recall the Cox construction for toric varieties and refer to for more details. Let Σ = *C* ∩ Trop(*J*) and *l* denote the dimension of $\\mathcal L(J)$. Fix an (*m* × *n*)-matrix ${\\bf r}$ whose rows *r*1, …, *r**m* are representatives of the primitive ray generators of $\\overline{C}\\subset \\mathbb R^n/\\mathcal L(J)$. The rays of Σ, denoted by Σ(1), form a subset of {*r*1, …, *r**m*}. Recall the definition of the lifted algebra $\\tilde A\\_{\\bf r}$ from . We can apply the *Cox construction* in two ways, to *X**C* and to *X*Σ: First, let *X**C* be the affine toric variety associated to *C*. Recall that a *quasitorus* is a product of a torus with a finite abelian group. Then *X**C* is isomorphic to the almost geometric quotient $\\mathbb A^m/\\!/G$ (see, e.g., ), where the group *G* by is the quasitorus\n \n$$\\label{eq:quasitorus}\nG=\\{(t\\_1,\\dots,t\\_m)\\in(\\mathbb C^\\*)^m\\colon t\\_1^{r\\_1\\cdot e\\_i}\\cdots t\\_m^{r\\_m\\cdot e\\_i}=1\\, \\forall \\, i\\in [n]\\}.$$ \nOne can easily check that if $p,q\\in \\mathbb A^m$ lie in the same *G*-orbit, then *π*− 1(*p*) ≅ *π*− 1(*q*) by . Hence, the flat family $\\pi\\colon \\operatorname{Spec}(\\tilde A\\_{\\bf r})\\to \\mathbb A^m$ induces the commutative diagram\n \n$$\\label{eq:Cox C}\n \\begin{tikzcd}[ampersand replacement=\\&,cramped]\n \\& \\operatorname{Spec}(\\tilde A\\_{\\bf r})\\ar[d,\"\\pi\"] \\ar[dl,\"p\\_{C}\\circ \\pi\"'] \\\\\n X\\_{C} \\& \\mathbb A^m. \\ar[l,\"p\\_{C}\"]\n \\end{tikzcd}$$ \nIf *p*, *q* ∈ *X**C* lie in the same torus orbit, then (*p**C* ∘ *π*)− 1(*p*) ≅ (*p**C* ∘ *π*)− 1(*q*) by . Note that $p\\_C\\colon \\mathbb A^m\\to X\\_C$ is indeed a morphism. Namely, it is the *universal torsor* for *X**C*.\nSimilarly, we can apply the Cox construction to the toric variety *X*Σ. For simplicity we assume that Σ(1) = {*r*1, …, *r**m*}. In this case the construction has two steps: first we remove the locus of $\\mathbb A^m$ that does not correspond to torus orbits in *X*Σ, denoted by *Z*(Σ). Recall that $\\mathcal C\\subset \\{{r}\\_1,\\dots,{r}\\_m\\}$ is a *primitive collection* for Σ, if (*i*) $\\mathcal C\\not\\subset \\sigma(1)$ for all *σ* ∈ Σ, and (*i**i*) for every $\\mathcal C'\\subsetneq \\mathcal C$ there exists *σ* ∈ Σ with $\\mathcal C'\\subset \\sigma(1)$. Then by \n \n$$\\label{eq:Z(Sigma)}\n Z(\\Sigma)=\\bigcup\\_{\\mathcal C \\text{ primitive collection }} V(t\\_i\\colon r\\_i\\in \\mathcal C) \\subset \\mathbb A^m.$$ \nNow *X*Σ is isomorphic to the almost geometric quotient $(\\mathbb A^m-Z(\\Sigma))/\\!/G$, where *G* is as in (see, e.g., ). In contrast to the morphism $p\\_C\\colon \\mathbb A^m\\to X\\_C$ from , we obtain a rational map $p\\_\\Sigma\\colon \\mathbb A^m\\dashrightarrow X\\_\\Sigma$ as *p*Σ is only defined on $\\mathcal T\\_\\Sigma:=\\mathbb A^m-Z(\\Sigma)$. Moreover, $p\\_\\Sigma\\colon \\mathcal T\\_\\Sigma\\to X\\_\\Sigma$ is the universal torsor of *X*Σ. Similarly to the first case, this gives the diagram:\n \n$$\\label{eq:Cox Sigma}\n \\begin{tikzcd}[ampersand replacement=\\&,cramped]\n \\& \\pi^{-1}(\\mathcal T\\_\\Sigma)\\ar[d,\"\\pi\\vert\\_{\\pi^{-1}(\\mathcal T\\_\\Sigma)}\"] \\ar[dl,\"p\\_\\Sigma\\circ \\pi\"']\\ar[r,hook] \\& \\operatorname{Spec}(\\tilde A\\_{\\bf r})\\ar[d,\"\\pi\"] \\\\\n X\\_{\\Sigma} \\& \\mathcal T\\_\\Sigma \\ar[l,\"p\\_\\Sigma\"]\\ar[r,hook]\\& \\mathbb A^m\n \\end{tikzcd}$$ \nwhich is commutative. The fibers of the family defined by $p\\_\\Sigma\\circ \\pi\\vert\\_{\\pi^{-1}(\\mathcal T\\_\\Sigma)}$ coincide with the fibers of *ψ* defined in : (*p*Σ ∘ *π*)− 1(*x*) ≅ *ψ*− 1(*x*) for every *x* ∈ *X*Σ, see .\nIn the following, we explain how the flat family of is related to Kaveh–Manon’s flat family associated to a maximal cone in the Gröbner fan.\nLet $J\\subseteq\\mathbb K\\_{\\bf d}[{x\\_1,\\dots,x\\_n}]$ be a **d**-homogeneous ideal and *C* a maximal cone in GF(*J*). Let ${\\bf r}$ be an (*m* × *n*)-matrix whose rows are representatives of primitive ray generators for $\\overline C$. Then the morphism $\\pi\\colon \\operatorname{Spec}\\big(\\tilde A\\_{\\bf r}\\big)\\to \\mathbb A^m$ fits into a pull-back diagram\n \n$$\\begin{tikzcd}[ampersand replacement=\\&,cramped]\n \\operatorname{Spec}(\\mathcal R\\_C)\\ar[d,\"\\psi\"'] \\&\n \\operatorname{Spec}\\big(\\tilde A\\_{\\bf r}\\big)\\ar[d,\"\\pi\"] \\ar[l]\\\\\n X\\_{C}\\& \\mathbb A^m. \\ar[l,\"p\\_C\"]\n \\end{tikzcd}$$ \nHere $p\\_C\\colon \\mathbb A^m\\to X\\_C$ is the universal torsor of *X**C* obtained from the Cox construction as in .\nWe prove equivalently that the corresponding diagram between the algebras is a push-out. Hence, it suffices to show that\n \n$$\\mathcal R\\_C\\otimes\\_{\\mathbb K[S\\_C]} \\mathbb K[t\\_1,\\dots,t\\_m]\\cong \\tilde A.$$ \nNote that the map $p\\_C\\colon \\mathbb A^m \\to X\\_C$ corresponds to $p\\_C^\\#\\colon \\mathbb K[S\\_C] {\\hookrightarrow} \\mathbb K[t\\_1,\\dots,t\\_m]$. In particular, for a character $t^{-r\\_i^\\*}\\in \\mathbb K[S\\_C]$ with *r**i*\\* ∈ *C*∨ ⊂ *M* the dual of *r**i* ∈ *C* ⊂ *N* we have *p**C*#(*t*− *r**i*\\*) $=t\\_i\\in \\mathbb K[t\\_1,\\dots,t\\_m]$. Then under the extension of scalars (i.e., applying the functor $-\\otimes\\_{\\mathbb K[S\\_C]}\\mathbb K[t\\_1,\\dots,t\\_m]$) the homogenization $\\hat g$ of an element $g\\in \\mathcal G$ is sent to the lift $\\tilde g$. This can be seen as follows: by we have $\\tilde g={\\bf x}^\\gamma + \\sum c\\_\\alpha{\\bf x}^\\alpha{\\bf t}^{{\\bf r}\\cdot(\\alpha-\\gamma)}$. By for every $\\ell\\in \\mathcal L(J)$ we have inℓ(*g*) = *g*. In particular, ℓ ⋅ *α* = ℓ ⋅ *γ* and so $\\alpha-\\gamma\\in (\\mathcal L(J)\\cap N)^\\perp$. Let {ℓ1, …, ℓ*k*} be a basis for $\\mathcal L(J)\\cap N$. Then {*r*1, …, *r**m*, ℓ1, …, ℓ*k*} spans *N* and {*r*1\\*, …, *r**m*\\*} spans $(\\mathcal L(J)\\cap N)^\\perp\\subset M$. So every *α* − *γ* has an expression of form *α* − *γ* = ∑*c**i**r**i*\\*. We compute\n \n$$\\label{eq:t's}\nt^{-(\\alpha-\\gamma)}=t^{-c\\_1r\\_1^\\*}\\cdots t^{-c\\_mr\\_m^\\*} \\overset{p\\_C^\\#}{\\longmapsto} t\\_1^{c\\_1}\\cdots t\\_m^{c\\_m}={\\bf t}^{{\\bf r}\\cdot(\\alpha-\\gamma)}.$$ \nNote that *γ* has the property that $r\\_i\\cdot\\gamma\n=\\min\\_{c\\_\\alpha \\not=0}\\{r\\_i\\cdot\\alpha\\}$. So, *r**i* ⋅ (*α* − *γ*) ≥ 0 and therefore *α* − *γ* ∈ *C*∨. Hence, we have\n \n$$\\begin{gathered}\n\\mathcal R\\_C\\otimes\\_{\\mathbb K[S\\_C]} \\mathbb K[t\\_1,\\dots,t\\_m]\n\\overset{\\text{{Proposition}~\\ref{prop:Rees quotient}}}{\\cong}\n\\big(\\mathbb K[S\\_C][x\\_1,\\dots,x\\_n]/\\hat J \\big)\\otimes\\_{\\mathbb K[S\\_C]} \\mathbb K[t\\_1,\\dots,t\\_m]\n\\\\ \\hphantom{\\mathcal R\\_C\\otimes\\_{\\mathbb K[S\\_C]} \\mathbb K[t\\_1,\\dots,t\\_m]Pro}\n{}\\overset{\\eqref{eq:t's}}{\\cong}\n\\mathbb K[t\\_1,\\dots,t\\_m][x\\_1,\\dots,x\\_n]/\\tilde J=\\tilde A.\n\\tag\\*{{}}\\end{gathered}$$ \nLet $J\\subseteq\\mathbb K[{x\\_1,\\dots,x\\_n}]$ be a weighted homogeneous ideal as in Definition $\\ref{def:d homog}$ and *C* a maximal cone in GF(*J*). Let ${\\bf r}$ be the matrix whose rows are representatives of primitive ray generators for $\\overline C$ and denote $\\tilde A\\_{\\bf r}$ by $\\tilde A$. Let Σ be the intersection of *C* with Trop(*J*). Then the restriction of morphism $\\pi\\colon \\operatorname{Spec}\\big(\\tilde A\\big)\\to \\mathbb A^m$ to $\\pi^{-1}(\\mathcal T\\_\\Sigma)=\\operatorname{Spec}\\big(\\tilde A\\big)-\\pi^{-1}(Z(\\Sigma))$ fits into a pull-back diagram\n \n$$\\begin{tikzcd}[ampersand replacement=\\&,cramped]\n \\mathbf{Spec}(\\mathcal R\\_\\Sigma)\\ar[d,\"\\psi\"'] \\&\n \\pi^{-1}(\\mathcal T\\_\\Sigma)\\ar[d,\"\\pi\\vert\\_{\\pi^{-1}(\\mathcal T\\_\\Sigma)}\"] \\ar[l]\\\\\n X\\_{\\Sigma}\\& \\mathcal T\\_\\Sigma. \\ar[l,\"p\\_\\Sigma\"]\n \\end{tikzcd}$$ \nHere $p\\_\\Sigma\\colon \\mathcal T\\_\\Sigma\\to X\\_\\Sigma$ is the universal torsor of *X*Σ obtained from the Cox construction as in .\nThe scheme *X*Σ and the sheaf $\\mathcal R\\_\\Sigma$ are defined locally for affine pieces $U\\_\\sigma=\\operatorname{Spec}(\\mathbb K[S\\_\\sigma])$ ⊂ *X*Σ for *σ* ∈ Σ. Algebraically, for every *σ* ∈ Σ the pull-back of *ψ* and *p*Σ corresponds to the following push-out diagram:\n \n$$\\begin{tikzcd}[ampersand replacement=\\&,cramped]\n \\mathcal R\\_\\sigma\\ar[r] \\&\n \\mathcal R\\_\\sigma\\otimes\\_{\\mathbb K[S\\_\\sigma]}\\mathbb K[t\\_1,\\dots,t\\_m]\\_{\\langle t\\_i\\colon r\\_i\\not\\in \\sigma\\rangle } \\\\\n \\mathbb K[S\\_\\sigma] \\ar[r]\\ar[u]\\& \\mathbb K[t\\_1,\\dots,t\\_m]\\_{\\langle t\\_i\\colon r\\_i\\not\\in \\sigma\\rangle. } \\ar[u]\n \\end{tikzcd}$$ \nObserve that by the localization $\\mathbb K[t\\_1,\\dots,t\\_m]\\_{\\langle t\\_i\\colon r\\_i\\not\\in \\sigma\\rangle }$ corresponds to an affine piece in $\\mathbb A^m$ that does not intersect *Z*(Σ). So it is in fact an affine piece of $\\mathcal T\\_\\Sigma$. We have to show that\n \n$$\\mathcal R\\_\\sigma\\otimes\\_{\\mathbb K[S\\_\\sigma]}\\mathbb K[t\\_1,\\dots,t\\_m]\\_{\\langle t\\_i\\colon r\\_i\\not\\in \\sigma\\rangle } \\cong \\tilde A\\_{\\langle t\\_i\\colon r\\_i\\not\\in \\sigma\\rangle}.$$ \nAs *σ* is a face of the maximal cone *C* by we have $\\mathcal R\\_\\sigma\\cong \\mathcal R\\_C\\otimes\\_{\\mathbb K[S\\_C]}\\mathbb K[S\\_\\sigma]$. Using we compute\n \n$$\\begin{gathered}\n\\mathcal R\\_\\sigma\\otimes\\_{\\mathbb K[S\\_\\sigma]}\\mathbb K[t\\_1,\\dots,t\\_m]\\_{\\langle t\\_i\\colon r\\_i\\not\\in \\sigma\\rangle}\n\\cong\n\\big(\\mathcal R\\_C\\otimes\\_{\\mathbb K[S\\_C]}\\mathbb K[S\\_\\sigma]\\big)\\otimes\\_{\\mathbb K[S\\_\\sigma]}\\mathbb K[t\\_1,\\dots,t\\_m]\\_{\\langle t\\_i\\colon r\\_i\\not\\in \\sigma\\rangle}\n\\\\ \\hphantom{\\mathcal R\\_\\sigma\\otimes\\_{\\mathbb K[S\\_\\sigma]}\\mathbb K[t\\_1,\\dots,t\\_m]\\_{\\langle t\\_i\\colon r\\_i\\not\\in \\sigma\\rangle}}\n{}\\cong\\mathcal R\\_C\\otimes\\_{\\mathbb K[S\\_C]}\\mathbb K[t\\_1,\\dots,t\\_m]\\_{\\langle t\\_i\\colon r\\_i\\not\\in \\sigma\\rangle}\\cong \\tilde A\\_{\\langle t\\_i\\colon r\\_i\\not\\in \\sigma\\rangle}.\\tag\\*{{}}\\end{gathered}$$ \nGrassmannians and cluster algebras\n==================================\nWe now apply the machinery developed in Section [sec:flat families] to the Grassmannians $\\operatorname{Gr}(2,\\mathbb C^n)$ and $\\operatorname{Gr}\\big(3,\\mathbb C^6\\big)$. Our aim is to make this paper as self-contained as possible and accessible to readers of broad mathematical background. Therefore, we recall background on tropical Grassmannians below, on cluster algebras and the cluster structure of $\\operatorname{Gr}(2,\\mathbb C^n)$ in Section [sec:cluster] and then in Section [sec: toric deg] we translate between toric degenerations obtained from the tropicalizations and from the cluster structure. In Section [sec:monomialdegeneration], we turn to the application of the flat families introduced in Section [sec:flat families] and show how to recover the cluster algebra with universal coefficients in this case. Similarly, in Section [sec:Gr(3,6)] we treat the case of $\\operatorname{Gr}\\big(3,\\mathbb C^6\\big)$.\nWe first fix our notation for $\\operatorname{Gr}(2,\\mathbb C^n)$. Denote by *S* the polynomial ring in Plücker variables $\\mathbb C[p\\_{ij}\\colon 1\\leq i<j\\le n]$. Here, Plücker variables are associated with arcs in an *n*-gon whose vertices are labeled by [*n*] in clockwise order. Hence, we think of [*n*] : = {1, …, *n*} as a cyclically ordered set. In particular, *i* < *j* < *k* < *l* indicates that in clockwise order starting at the vertex *i* the other vertices appear in order *j*, *k*, *l*. Note that in general we might not have 1 ≤ *i* < *j* < *k* < *l* ≤ *n*. We set the convention *p**i**j* : = − *p**j**i*. In *S* we define for every *i* < *j* < *k* < *l* the *Plücker relation*:\n \n*R**i**j**k**l* : = *p**i**j**p**k**l* − *p**i**k**p**j**l* + *p**i**l**p**j**k* ∈ *S*. \nThe ideal *I*2, *n* ⊂ *S* generated by all Plücker relations *R**i**j**k**l* is called the *Plücker ideal*. The quotient *A*2, *n* : = *S*/*I*2, *n* is the *Plücker algebra* which is the homogeneous coordinate ring of the Grassmannian $\\operatorname{Gr}(2,\\mathbb C^n)$ with respect to its Plücker embedding into $\\mathbb P^{\\binom{n}{2}-1}$. The cosets of Plücker variables in the quotient are denoted by $\\bar p\\_{ij}\\in A\\_{2,n}$ and are called *Plücker coordinates*. Denote by *d* : = 2(*n* − 2) the dimension of $\\operatorname{Gr}(2,\\mathbb C^n)$, so *d* + 1 is the Krull-dimension of *A*2, *n*.\nIn *S*, we fix the notation ${\\bf p}^m$ with $m=(m\\_{ij})\\_{ij}\\in \\mathbb Z\\_{\\ge 0}^{\\binom{n}{2}}$ denoting the monomial ∏1 ≤ *i* < *j* ≤ *n**p**i**j**m**i**j*. The maximal cones in the Gröbner fan GF(*I*2, *n*) are those of full dimension in the ambient space $\\mathbb R^{\\binom{n}{2}}$ with associated monomial initial ideals. The lineality space $\\mathcal L\\_{2,n}:=\\mathcal L(I\\_{2,n})\\subset \\operatorname{GF}(I\\_{2,n})$ is *n*-dimensional and spanned by the elements ℓ*s* for *s* ∈ [*n*] which are defined by\n \n$$(\\ell\\_s)\\_{ij}:=\\begin{cases}\n1& \\text{if}\\quad s=i\\quad \\text{or}\\quad s=j, \\\\ 0& \\text{otherwise}.\n\\end{cases}$$ \nNote that ${\\bf 1}:=(1,\\dots,1)\\in \\mathbb R^{\\binom{n}{2}}$ lies in $\\mathcal L\\_{2,n}$. The Gröbner fan of the Plücker ideal has a subfan that is particularly interesting when studying toric degenerations of $\\operatorname{Gr}(2,\\mathbb C^n)$.\nThe *tropical Grassmannian* is a simplicial (*d* + 1)-dimensional subfan of GF(*I*2, *n*) defined as Trop(*I*2, *n*).\nMaximal cones in Trop(*I*2, *n*) are called *tropical maximal cones*. They are in correspondence with trivalent trees with *n* leaves as shown by Speyer and Sturmfels in . To a trivalent tree *Υ* one associates a weight vector in the relative interior of the corresponding cone of Trop(*I*2, *n*) by\n \n$$\\begin{gathered}\n\\label{eq:tree wt}\n \\underline{w}^{\\Upsilon}(p\\_{ij}):=-\\#\\{\\text{edges on the unique path } i\\to j \\text{ in }\\Upsilon\\}.\\end{gathered}$$ \nTo simplify notation we denote $\\operatorname{in}\\_{\\Upsilon}(I):=\\operatorname{in}\\_{\\underline{w}^{\\Upsilon}}(I)$.\n[] For every trivalent tree *Υ* with *n* leaves the ideal in*Υ*(*I*) is *toric*, i.e., binomial and prime. Moreover, it is generated by in*Υ*(*R**i**j**k**l*) for all 1 ≤ *i* < *j* < *k* < *l* ≤ *n*.\nThe *totally positive tropical* Grassmannian is Trop+(*I*2, *n*).\nIn , it was shown that maximal cones in Trop+(*I*2, *n*) are in bijection with *planar trivalent trees*, i.e., trivalent trees whose vertices are labeled 1, …, *n* in a clockwise manner.\nPreliminaries on cluster algebras\n---------------------------------\nThe Plücker algebra *A*2, *n* is in fact a cluster algebra (see Definition [def:cluster algebra]) whose cluster structure is closely related to the combinatorics governing the tropical Grassmannian. To exhibit this connection, we recall some facts about skew-symmetric cluster algebras following Fomin and Zelevinsky . Afterwards, we define finite type cluster algebras with frozen variables and universal coefficients.\nLet $m,f\\in \\mathbb N$ be positive integers and F = C(*u*1, …, *u**m* + *f*) be the field of rational functions on *m* + *f* variables. Here *m* stands for mutable and *f* for frozen. A *seed* in F is a pair $\\big(\\widetilde{{\\bf x}},\\widetilde{B}\\big)$, where $\\widetilde{{\\bf x}}=\\{x\\_1,\\dots, x\\_{m+f}\\}$ is a free generating set of $\\mathcal F$ and $\\widetilde{B}=(b\\_{ij})$ is an (*m* + *f*) × *m* rectangular matrix with the following property: the top square submatrix of $\\widetilde{B}$ (that is the *m* × *m* submatrix of $\\widetilde{B}$ formed by its first *m* rows) is skew-symmetric. We call $\\widetilde{B} $ an *extended exchange matrix* and refer to its top square submatrix *B* as an *exchange matrix*. We call ${\\bf x}=\\{x\\_1, \\dots, x\\_m \\}$ a *cluster*, {*x**m* + 1, …, *x**m* + *f*} the set of *frozen variables* and $\\widetilde{{\\bf x}}$ an *extended cluster*.\nGiven *k* ∈ {1, …, *m*}, the *mutation in direction *k** of a seed $\\big(\\widetilde{{\\bf x}}, \\widetilde{B}\\big)$ is the new seed $\\mu\\_k\\big(\\widetilde{{\\bf x}}, \\widetilde{B}\\big)=\\big(\\widetilde{{\\bf x}}', \\widetilde{B}'\\big)$, defined as follows:\n* The extended cluster $\\widetilde{{\\bf x}}'=\\widetilde{{\\bf x}}\\setminus\\{ x\\_k \\} \\cup \\{ x'\\_k \\} $, where\n \n*x**k**x*ʹ*k* = ∏*i*: *b**i**k* > 0*x**i**b**i**k* + ∏*i*: *b**i**k* < 0*x**i*− *b**i**k*. \nWe call an expression of the form an *exchange relation* and the monomial *x**k**x**k*ʹ an *exchange monomial*.\n* The extended exchange matrix $\\widetilde{B}'=(b'\\_{ij})$ is defined by the following formula\n \n$$b'\\_{ij}:= \\begin{cases} -b\\_{ij} \\qquad\\text{if}\\quad i=k\\quad \\text{or}\\quad j=k,\\\\\n b\\_{ij} +\\mathop{\\rm sgn}(b\\_{ik})\\mathop{\\rm max}(b\\_{ik}b\\_{kj},0) \\qquad\\text{else},\n \\end{cases}$$ \nwhere \n$$\\mathop{\\rm sgn}(b\\_{ik}):= \\begin{cases}\n-1 &\\text{if}\\quad b\\_{ik}<0,\\\\\n0 &\\text{if}\\quad b\\_{ik}=0,\\\\\n1 &\\text{if}\\quad b\\_{ik}>0.\\end{cases}$$\n[def:cluster algebra] The *cluster algebra* $A\\big(\\widetilde{{\\bf x}},\\widetilde{B}\\big)$ associated to a seed $\\big(\\widetilde{{\\bf x}},\\widetilde{B}\\big) $ is the $\\mathbb C$-subalgebra of $\\mathcal F$ generated by the elements of *all* the extended clusters that can be obtained from the *initial seed* $\\big(\\widetilde{{\\bf x}},\\widetilde{B}\\big)$ by a finite sequence of mutations. The elements of the clusters constructed in this way are called the *cluster variables*. If *f* ≠ 0 we say that the cluster algebra $A\\big(\\widetilde{{\\bf x}},\\widetilde{B}\\big)$ has frozen variables, namely, *x**m* + 1, …, *x**m* + *f*.\nIt can be easily verified that, up to a canonical isomorphism, $A\\big(\\widetilde{{\\bf x}},\\widetilde{B}\\big)$ is independent of $\\widetilde{{\\bf x}}$. Therefore, we write $A\\big(\\widetilde{B}\\big)$ instead of $A\\big(\\widetilde{{\\bf x}},\\widetilde{B}\\big)$.\nFomin and Zelevinsky usually define cluster algebras over $\\mathbb Q$. We recover the construction described above applying tensor product with $\\mathbb C$ to Fomin and Zelevinsky’s construction.\n[] The cluster algebra $A\\big(\\widetilde{B}\\big)$ is a $\\mathbb C[x\\_{m+1},\\dots, x\\_{m+f}]$-subalgebra of $\\mathbb C\\big[x\\_1^{\\pm 1}, \\dots, x\\_m^{\\pm 1},x\\_{m+1},\\dots, x\\_{m+f}\\big]$.\nThe cluster algebra with *principal coefficients* associated to an *m* × *m* skew-symmetric matrix *B* is the cluster algebra $A(B^{\\rm{prin}})$, where $B^{\\rm{prin}}$ is the 2*m* × *m* matrix whose top *m* × *m* square submatrix is *B* and whose lower *m* × *m* square submatrix is the identity matrix $\\mathbbm{1}\\_m$.\nThe notion of cluster algebras with principal coefficients is central in cluster theory. Its relevance in the framework of toric degenerations of cluster varieties was highlighted in . In particular, in loc. cit. the authors work implicitly with cluster algebras with frozen variables and principal coefficients. This is precisely the notion we need here.\nLet $ \\widetilde{B}$ be an extended exchange matrix. Let *B̂* be the (*m* + *f*) × (*m* + *f*) skew-symmetric square matrix whose left (*m* + *f*) × *m* submatrix is $\\widetilde{B}$ and whose bottom right *f* × *f* square submatrix is the zero matrix (this information fully determines *B̂* since it is skew-symmetric). Denote by *t*1, …, *t**m* + *f* the variables of $A\\big(\\widehat{B}^{\\rm prin}\\big) $ associated to the last *m* + *f* directions, which in this case are the frozen directions. Then the *cluster algebra with principal coefficients and frozen variables* $A\\big(\\widetilde{B}^{\\rm prin}\\big) $ associated to $\\widetilde{B}$ is the C[*t*1, …, *t**m* + *f*]-subalgebra of $A\\big(\\widehat{B}^{\\rm prin}\\big)$ spanned by the cluster variables that can be obtained from the initial extended cluster by iterated mutation in the first *m* mutable directions.\n[] Let *X* be a cluster variable associated to $A\\big(\\widetilde{B}^{\\rm{prin}}\\big)$. By Theorem [thm:SLaurent] we know that $X\\in \\mathbb C [t\\_1,\\dots,t\\_{m+f}]\\big[x\\_1^{\\pm 1}, \\dots, x\\_m^{\\pm 1}, x\\_{m+1}, \\dots, x\\_{m+f}\\big]$. Let $X|\\_{{\\bf t}=0}$ be the Laurent polynomial in $\\mathbb C \\big[x\\_1^{\\pm 1}, \\dots, x\\_m^{\\pm 1},x\\_{m+1}, \\dots, x\\_{m+f}\\big]$ obtained after the evaluation of *X* at *t*1 = 0, …, *t**m* + *f* = 0. Then $X|\\_{{\\bf t} = 0}$ is a non-constant Laurent monomial with coefficient 1. In other words, for a non-zero vector ${\\bf g}(X)=\\big(g\\_1^X,\\dots, g\\_{m+f}^X\\big)\\in \\mathbb Z^{m+f}$ we have that\n \n$$X|\\_{{\\bf t}=0}=\\prod\\_{i=1}^{m+f} x\\_i^{g^X\\_i}.$$ \nMoreover, if *X* and *X*ʹ are different cluster variables then ${\\bf g }(X)\\neq {\\bf g}(X')$.\nLet *X* be a cluster variable associated to $A\\big(\\widetilde{B}^{\\rm{prin}}\\big)$. The vector ${\\bf g}(X)\\in \\mathbb{Z}^{m+f}$ introduced in is the **g**-*vector* associated to *X*. We refer to the vector $\\big(g\\_1^X,\\dots, g\\_m^X\\big)\\in \\mathbb{Z}^m$ as the *truncated* **g**-*vector* of *X*. The set of (truncated) **g**-vectors associated to $\\widetilde{B}$ is the set of (truncated) **g**-vectors of all the cluster variables in $A\\big(\\widetilde{B}^{\\rm{prin}}\\big)$.\nLet $X|\\_{{\\bf t}=1}$ be the Laurent polynomial in $\\mathbb C \\big[x\\_1^{\\pm 1}, \\dots, x\\_m^{\\pm 1},x\\_{m+1}, \\dots, x\\_{m+f}\\big]$ obtained after the evaluation of *X* at *t*1 = 1, …, *t**m* + *f* = 1. Then $X|\\_{{\\bf t}=1}$ is a cluster variable of $A\\big(\\widetilde{B}\\big)$. By a slight abuse of terminology we say ${\\bf g}(X)$ is the ${\\bf g}$-vector of the cluster variable $X|\\_{{\\bf t}=1}$.\nWe now turn to the definition of finite type cluster algebras *with* frozen variables *and* universal coefficients. This notion is a natural (and slight) generalization of the usual notion of universal coefficients for cluster algebras *without* frozen variables introduced by Fomin and Zelevinsky in . The difference between these notions arises from the distinction we make between coefficients and frozen variables, notions that are usually identified. This distinction was first suggested in and is of particular importance in the study of toric degenerations of cluster varieties. It was shown in that finite type cluster algebras can be endowed with universal coefficients. This construction was categorified in and the categorification was used to perform various computations in the following sections. To further clarify the preceding discussion let us first recall the notion of finite type cluster algebras with universal coefficients.\nAn *ice quiver* is a pair (*Q*, *F*), where *Q* is a finite quiver with the vertex set *V*(*Q*) and *F* ⊂ *V*(*Q*) is a set of *frozen vertices*. The vertices in *V*(*Q*) ∖ *F* are called *mutable*. If (*Q*, *F*) is an ice quiver then the corresponding ∣*V*(*Q*)∣ × ∣*V*(*Q*) \\ *F*∣ matrix is *B*(*Q*, *F*) : = (*b**i**j*), where\n \n*b**i**j* : = #{arrows *i* → *j*in *Q*} − #{arrows *j* → *i*in *Q*} \nfor *i* ∈ *V*(*Q*) and *j* ∈ *V*(*Q*) \\ *F*. Note that $(\\ref{eq:matrix\\_quiver})$ can be similarly used to recover (*Q*, *F*) from *B*(*Q*, *F*). The *quiver mutation* is the operation induced by matrix mutation at the level of quivers. If *F* = ∅ we set *B**Q* : = *B*(*Q*, ∅).\n[] Let *Q* be a quiver of finite cluster type, that is *Q* is mutation equivalent to an orientation of a Dynkin diagram. The *cluster algebra with universal coefficients* associated to *Q* is $A\\big(B\\_Q^{{\\rm univ}}\\big)$, where\n \n$$B^{{\\rm univ}}\\_Q=\n\\left(\n\\begin{array}{c}\n B\\_Q \\\\ \\hdashline\n U\\_Q\n\\end{array}\n\\right)$$ \nand *U**Q* is the rectangular matrix whose rows are the **g**-vectors of the opposite quiver $Q^{\\rm op}$.\nObserve that we have described the rows of *U**Q* but did not specify in which order they appear. Any order we choose provides a realization of $A\\big(B\\_Q^{{\\rm univ}}\\big)$ since a rearrangement of the rows of *U**Q* amounts to reindexing the corresponding coefficients. So any choice gives rise to an isomorphic algebra. It can be verified that the cluster algebra with universal coefficients associated to *Q* is an invariant of the mutation class of *Q*. In other words, if *Q*ʹ is mutation equivalent to *Q* then $A\\big(B\\_Q^{{\\rm univ}}\\big)$ is canonically isomorphic to $A\\big(B\\_{Q'}^{{\\rm univ}}\\big)$. The canonical isomorphism is completely determined by sending the elements of the initial cluster of $B\\_{Q}^{{\\rm univ}}$ to those of $B\\_{Q'}^{{\\rm univ}}$.\nCluster algebras with universal coefficients as treated by Fomin and Zelevinsky are defined by a universal property . We have chosen Reading’s equivalent definition since we will not use the universal property here.\nIf *Q* is a bipartite orientation of a Dynkin diagram Δ then the matrix $B^{\\rm univ}\\_Q$ has a particularly nice and explicit description. More precisely, let Φ∨ be the root system dual to the root system associated to Δ. Fix Φ+∨ ⊂ Φ∨ a subset of positive coroots with {*α*1, …, *α**n*} ⊂ Φ+∨ the set of simple coroots. The almost positive coroots are the coroots in the set Φ≥ − 1∨ : = Φ+∨ ∪ { − *α*1, …, − *α**n*}. The cardinality of Φ≥ − 1∨ coincides with the number of cluster variables in $A\\big(B^{\\rm univ}\\_{Q}\\big)$ which is the number of coefficients for $A\\big(B^{\\rm univ}\\_{Q}\\big)$. Hence, we may consider $\\big\\{ y\\_{\\alpha} \\colon \\alpha \\in \\Phi^{\\vee}\\_{\\geq -1} \\big\\} $ as the set of coefficients for $A\\big(B^{\\rm univ}\\_{Q}\\big)$. Next, for a coroot *β* = ∑*i* = 1*n**c**i**α**i* we define [*β* : *α**i*] : = *c**i*. With this notation we have the following description of $ B^{\\rm univ}\\_Q $ which is a reformulation of .\nThe entry in $B\\_Q^{\\rm univ}$ in the column *i* and the row corresponding to *y**β* is [*β* : *α**i*] if *i* is a source and − [*β* : *α**i*] if *i* is a sink.\nLet (*Q*, *F*) be an ice quiver whose mutable part is a quiver $Q^{\\rm mut}$ of finite cluster type. The *associated cluster algebra with universal coefficients* is $A\\big(B\\_{(Q,F)}^{{\\rm univ}}\\big)$, where\n \n$$B^{{\\rm univ}}\\_{(Q,F)}=\n\\left(\n\\begin{array}{c}\n B\\_{(Q,F)} \\\\ \\hdashline\n U\\_{Q^{\\rm mut}}\n\\end{array}\n\\right)$$ \nand $U\\_{Q^{\\rm mut}}$ is the matrix whose rows are the truncated **g**-vectors of the opposite quiver $(Q^{\\rm mut})^{\\rm op}$.\nThe algebra $A\\big(B\\_{(Q,F)}^{{\\rm univ}}\\big)$ satisfies a universal property"},"data_id":{"kind":"string","value":"arxiv_0004101"}}},{"rowIdx":993,"cells":{"content":{"kind":"string","value":"Conductance of closed and open long Aharonov-Bohm-Kondo rings\n=============================================================\nWe calculate the finite temperature linear DC conductance of a generic single-impurity Anderson model containing an arbitrary number of Fermi liquid leads, and apply the formalism to closed and open long Aharonov-Bohm-Kondo (ABK) rings. We show that, as with the short ABK ring, there is a contribution to the conductance from the connected 4-point Green’s function of the conduction electrons. At sufficiently low temperatures this contribution can be eliminated, and the conductance can be expressed as a linear function of the T-matrix of the screening channel. For closed rings we show that at temperatures high compared to the Kondo temperature, the conductance behaves differently for temperatures above and below *v**F*/*L* where *v**F* is the Fermi velocity and *L* is the circumference of the ring. For open rings, when the ring arms have both a small transmission and a small reflection, we show from the microscopic model that the ring behaves like a two-path interferometer, and that the Kondo temperature is unaffected by details of the ring. Our findings confirm that ABK rings are potentially useful in the detection of the size of the Kondo screening cloud, the *π*/2 scattering phase shift from the Kondo singlet, and the suppression of Aharonov-Bohm oscillations due to inelastic scattering.\nIntroduction[sec:intro]\n=======================\nThe Kondo problem, one of the most influential problems in condensed matter physics, emerges from a deceptively unembellished model: a localized impurity spin coupled with a Fermi sea of conduction electrons. Perturbation theory in the coupling constant is plagued by infrared divergence, but after much theoretical endeavor it has been recognized that the model has a relatively simple low energy behavior. For the single-channel spin-1/2 model, at temperatures well below the Kondo temperature *T**K*, the impurity spin is screened by the conduction electrons, forming a local singlet state. The spatial extent of this singlet state, commonly termed the Kondo screening cloud, is expected to be *L**K* = *v**F*/*T**K* where *v**F* is the Fermi velocity. The remaining conduction electrons are well described by a Fermi liquid theory at zero temperature, and acquire a phase shift (*π*/2 in the presence of particle-hole symmetry) upon being elastically scattered by the Kondo singlet. Moreover, at finite temperatures, scattering by the Kondo impurity can have both elastic and inelastic contributions, and it has been suggested that the inelastic scattering can be the origin of decoherence in mesoscopic structure, as measured for example by weak localization.\nAdvances in semiconductor technology have made it possible to imitate the impurity spin with a quantum dot (QD): when the ground state number of electrons in the QD is odd, the QD hosts a nonzero spin at temperatures lower than the charging energy. This possibility has triggered renewed experimental and theoretical interest in mesoscopic manifestations of Kondo physics in QD devices, such as the observation of the length scale *L**K*, the *π*/2 phase shift, and also decoherence effects of inelastic Kondo scattering.\nMany mesoscopic configurations have been proposed in order to observe *L**K*. These include QD-terminated finite quantum wires, and also various geometries with an embedded QD, including finite quantum wires, small metallic grains/larger QDs, and in particular, closed long Aharonov-Bohm (AB) rings with and without external electrodes. (A closed ring conserves the electric current and there is no leakage current.) Another motivation for quantum rings is that they may be used to answer the question of whether or not the inelastic scattering from the Kondo QD can cause decoherence by suppressing the amplitude of AB oscillations. A common feature of all these configurations is that they introduce at least one additional mesoscopic length scale *L*. When the bare Kondo coupling strength is adjusted so that *L**K* crosses the scale *L*, the dependence of observables on other control parameters changes qualitatively. In the closed long AB ring with an embedded QD [also known as the Aharonov-Bohm-Kondo (ABK) ring], for instance, *L* is the circumference of the ring: it is known that both *L**K* itself and the conductance through the ring can have drastically different AB phase dependences for *L**K* ≫ *L* and $L\\_{K}\\lesssim L$. In the large Kondo cloud regime *L**K* ≫ *L*, corresponding to a relatively small bare Kondo coupling, the Kondo cloud leaks out of the ring and the size of the cloud becomes strongly influenced by the ring size and other mesoscopic details of the system. For a given bare Kondo coupling, *L**K* can be extremely sensitive to the AB phase at certain values of Fermi energy, varying by many orders of magnitude. This sensitivity is completely lost in the opposite small Kondo cloud regime $L\\_{K}\\lesssim L$, where the bare Kondo coupling is relatively large.\nThe conductance calculation of ABK rings, however, involves an additional layer of complication that went neglected in a number of early works. In mesoscopic Kondo problems with Fermi liquid electrodes, it is usually convenient to work with the scattering states and rotate to the basis of the so-called screening and non-screening channels: the screening channel *ψ* is coupled to the QD and therefore has a nonzero T-matrix, while the non-screening channel *ϕ* is described by a decoupled non-interacting theory. A careful evaluation by Kubo formula at finite temperatures reveals that, unlike a QD directly coupled to external leads, the interaction effects on the linear DC conductance of short ABK rings are generally *not* fully encoded by the screening channel T-matrix in the single-particle sector, or equivalently the two-point function. Instead, there exists a contribution from connected four-point diagrams, corresponding to two-particle scattering processes in the screening channel, which cannot be interpreted as resulting from a single-particle scattering amplitude. This is not in contradiction with the famous Meir-Wingreen formula due to the violation of the proportionate coupling condition. For the short ABK ring, the four-point contribution becomes comparable to the two-point contribution well above the Kondo temperature *T* ≫ *T**K*, but can be approximately eliminated at temperatures low compared to the bandwidth and the on-site repulsion of the QD, *T* ≪ min{*t*, *U*}, by applying the bias voltage and probing the current in a particular fashion. (This does not mean the four-point contribution is negligible for *T* ≪ min{*t*, *U*}, however.) One naturally wonders how this result generalizes to the closed long ring at high and low temperatures, and how it possibly modifies early predictions on conductance, which is again expected to display qualitatively different behaviors for *L**K* ≫ *L* and $L\\_{K}\\lesssim L$.\nOn the other hand, efforts to measure the *π*/2 phase shift are mainly concentrated on two-path AB interferometer devices. In these devices, electrons from the source lead propagate through two possible paths (QD path and reference path) to the drain lead; the two paths enclose a tunable AB phase *φ*, and a QD tuned into the Kondo regime is embedded in the QD path. Most importantly, the complex transmission amplitudes through the two paths *t**d**e**i**φ* and *t**r**e**f* should be independent of each other, and the total coherent transmission amplitude at zero temperature *t**s**d* = *t**r**e**f* + *t**d**e**i**φ* is the sum of the individual amplitudes (the two-slit condition), meaning multiple traversals of the ring are negligible. Using a multi-particle scattering formalism, and assuming that only single-particle scattering processes are coherent, Ref. calculates the conductance of such an interferometer with an embedded Kondo QD in terms of the single-particle T-matrix through the QD, and concludes that the AB oscillations are suppressed by inelastic multi-particle scattering processes due to the Kondo QD.\nThe two-path interferometer can in principle be realized through open AB rings, where in contrast to closed rings, the propagating electrons may leak into side leads attached to the ring. For a non-interacting QD, Ref. presents the criteria for an open long ring to yield the intrinsic transmission phase through the QD: all lossy arms with side leads should have a small transmission and a small reflection. A small transmission suppresses multiple traversals of the ring and guarantees the validity of the two-slit assumption, while a small reflection prevents electrons from rattling (tunneling back and forth) across the QD. However, when the QD is in the Kondo regime, as with the previously discussed closed AB rings, the transmission probability through the QD and even the Kondo temperature may be sensitive to the AB phase and other details of the geometry, hampering the detection of the intrinsic phase shift across the QD. In addition, since the screening channels in the open ABK ring and in the simple embedded QD geometry are usually not the same, it is not obvious that the single-particle sector T-matrices coincide in the two geometries. These issues are not addressed in Ref. , which simply assumes that the two-slit condition is obeyed by the coherent processes, and that the T-matrix of the open ABK ring is identical to that of the QD embedded between source and drain leads. To our knowledge, it has been a mystery whether in certain parameter regimes the open long ABK ring realizes the two-path interferometer with a Kondo QD, where the Kondo temperature and the transmission probability through the QD are independent of the details of other parts of the ring, and the T-matrix of the ring truthfully reflects the T-matrix of the QD.\nThe aforementioned problems in closed and open ABK rings prompt a unified treatment of linear DC conductance in different mesoscopic geometries containing an interacting QD. Much work has been done on generic mesoscopic geometries, but in our formalism presented in this paper we aim to take the connected contribution into account expressly, and refrain from making assumptions about the geometry in question (such as parity symmetry).\nWe study a QD represented by an Anderson impurity, which is embedded in a junction connecting an arbitrary number of Fermi liquid leads. The junction is regarded as a black box characterized only by its scattering S-matrix and its coupling with the QD, and all leads (including source, drain and possibly side leads) are treated on equal footing. In parallel with Ref. we find that the linear DC conductance is given by the sum of a disconnected part and a connected part. The disconnected part has the appearance of a linear response Landauer formula, where the transmission amplitude is linear in the T-matrix of the screening channel in the single-particle sector, and indeed reduces at zero temperature to a non-interacting transmission amplitude appropriate for the local Fermi liquid theory. The connected part is again a Fermi surface property, can be eliminated by proper application of bias voltages, and is calculated perturbatively at weak coupling *T* ≫ *T**K*, as well as at strong coupling *T* ≪ *T**K* provided the local Fermi liquid theory applies.\nOur formalism is subsequently applied to long ABK rings. In the case of closed rings, we show that for *T* ≫ *T**K*, the high-temperature conductance does exhibit qualitatively different behaviors as a function of the AB phase for *T* ≫ *v**F*/*L* and *T* ≪ *v**F*/*L*. In the case of open rings, when the small transmission condition is met, we find the mesoscopic fluctuations are suppressed, and the two-path interferometer behavior is indeed recovered at low temperatures. If in addition the small reflection condition is satisfied, the Kondo temperature of the QD and the complex transmission amplitude through the QD are both unaffected by the details of the ring. We then find the conductance at *T* ≫ *T**K* and *T* ≪ *T**K*, and in particular rigorously calculate the normalized visibility of the AB oscillations in the Fermi liquid regime. We show that while the deviation of normalized visibility from unity is indeed proportional to inelastic scattering as predicted by Ref. , the constant of proportionality depends on non-universal particle-hole symmetry breaking potential scattering. Our findings also suggest that the *π*/2 phase shift across the QD is measurable in our two-path interferometer when the criteria of small transmission and small reflection are fulfilled. We stress again that, while we focus on long ABK rings in this paper, our general formalism is applicable to a Kondo impurity embedded in an arbitrary non-interacting multi-terminal mesoscopic structure.\nThe rest of this paper is outlined below. In Sec. [sec:model] we provide a formulation of our generalized Anderson model with an interacting QD, separate the screening channel from the non-screening ones, and discuss the effective Kondo model in the local moment regime. In Sec. [sec:conductance] the linear DC conductance is calculated using Kubo formula. Disconnected and connected contributions are examined separately, along with the approximate elimination of the latter. Perturbation theories in the weak-coupling and Fermi liquid regimes are employed in Sec. [sec:pert]; weak-coupling results applicable at high temperatures formally resemble the short ring case. Sec. [sec:closed] applies the abstract formalism to the closed long ring, and Sec. [sec:open] studies open long rings and their potential utilization as two-path interferometers. Conclusions and open questions are presented in Sec. [sec:outlook]. In Appendix [sec:appcomp], we make contact with early results by applying our formalism in a few other mesoscopic systems. Appendix [sec:appdisc] consists of details related to the calculation of disconnected contributions. Appendix [sec:appNI] is a check of our formalism in the case of a non-interacting QD. Appendix [sec:appFermi] focuses on the Fermi liquid regime: we derive the T-matrix for the screening channel, and perform another consistency check on our formalism by calculating the connected contribution explicitly. Finally, Appendix [sec:appopen] presents the non-interacting Schroedinger equations for the open long ring, whose solutions are used in Sec. [sec:open].\nModel[sec:model]\n================\nOur generalized tight-binding Anderson model describes *N* Fermi liquid leads meeting at a junction containing a QD with an on-site Coulomb repulsion. In addition to the QD, the junction comprises an arbitrary configuration of non-interacting tight-binding sites. The full Hamiltonian contains a non-interacting part, a QD part, and a coupling term between the two:\n \n*H* = *H*0 + *H**T* + *H**d*. \nThe non-interacting part is made up of two terms,\n \n*H*0 = *H*0,leads + *H*0,junction; \nthe lead term\n \n*H*0,leads = − *t*∑*j* = 1*N*∑*n* = 0∞(*c**j*, *n*†*c**j*, *n* + 1 + h.c.) \nmodels the Fermi liquid leads as semi-infinite nearest-neighbor tight-binding chains with hopping *t*, where *j* is the lead index and *n* is the site index. For simplicity all leads are assumed to be identical, and we temporarily suppress the spin index. *H*0,junction is the non-interacting part of the junction; it glues all leads together and often includes additional sites (e.g. representing the arms of an ABK ring), but does not include coupling to the interacting QD. In a typical open ABK ring with electron leakage, two of the leads serve as source and drain electrodes, while the remaining *N* − 2 leads mimic the base contacts thorough which electrons escape the junction. In experiments usually the current flowing through the source or the drain is monitored, but the leakage current can also be measured.\nAssume that there are *M* sites in the junction to which the QD is directly coupled; hereafter we refer to these sites as the coupling sites. The coupling to the QD can be written as\n \n*H**T* = − ∑*r* = 1*M*(*t**r**c**C*, *r*†*d* + h.c.) \nwhere *d* annihilates an electron on the QD, and *c**C*, *r*† creates an electron on the *r*th coupling site. *c**C*, *r* may coincide with *c**j*, 0. In the simplest AB ring, there is only one physical AB phase, which may be incorporated in either *H*0,junction or *H**T*. In more complicated models both *H*0,junction and *H**T* can depend on AB phases.\nFinally, the Hamiltonian of the interacting QD is given by\n \n*H**d* = *ε**d**d*†*d* + *U**n**d* ↑*n**d* ↓ \nwhere *n**d**σ* = *d**σ*†*d**σ*. We assume *S**U*(2) spin symmetry throughout the paper.\nA generic system with *N* = 5 and *M* = 3 is sketched in Fig. [fig:sketch], with details of the mesoscopic junction hidden. We will analyze more concrete examples in Secs. [sec:closed] and [sec:open]; additional examples, previously studied in Refs. , are provided in Appendix [sec:appcomp].\nSketch of a generic system which allows the application of our formalism. Here *N* = 5 and *M* = 3.[fig:sketch]\nScreening and non-screening channels\n------------------------------------\nWhile it is *H*0,junction that ultimately determines the properties of the junction, its details are actually not important in our formalism. Instead, in the following we characterize the model by its background scattering S-matrix and coupling site wave functions. Both quantities are easily obtained from a given *H*0,junction, and as we show in Sec. [sec:conductance], they play a central role in our quest for the linear DC conductance.\nTo recast our model into the standard form of an interacting QD coupled to a continuum of states, it is convenient to first diagonalize the non-interacting part of the Hamiltonian *H*0 by introducing the scattering basis *q**j*, *k*:\n \n$$H\\_{0}=\\int\\_{0}^{\\pi }\\frac{dk}{2\\pi }\\sum\\_{j=1}^{N}\\epsilon\n\\_{k}q\\_{j,k}^{\\dag }q\\_{j,k}\\text{,}$$ \nwhere *ε**k* = − 2*t*cos*k* is the dispersion relation in the leads, and for simplicity we let the lattice constant *a* = 1. In addition to the scattering states *q**j*, *k*, there may exist a number of bound states with their energies outside of the continuum, but since their wave functions decay exponentially away from the junction region, they do not affect linear DC transport properties.\nThe scattering basis operator *q**j*, *k* annihilates a scattering state electron incident from lead *j* with momentum *k*, and obeys the anti-commutation relation {*q**j*, *k*, *q**j*′, *k*′†} = 2*π**δ**j**j*′*δ*(*k* − *k*′). The corresponding wave function has the following form on site *n* in lead *j*′,\n \n*χ**j*, *k*(*j*′, *n*) = *δ**j**j*′*e*− *i**k**n* + *S**j*′*j*(*k*)*e**i**k**n*; \nand on coupling site *r*,\n \n*χ**j*, *k*(*r*) = Γ*r**j*(*k*). \nIn other words, for an electron incident from lead *j*, *S**j*′*j* is the background reflection or transmission amplitude in lead *j*′, and Γ*r**j* is the wave function on coupling site *r*. The scattering S-matrix *S* is unitary: *S*†*S* = 1.\nFrom its wave function, *q**j*, *k* can be related to *c**j*, *n* and *c**C*, *r*:\n \n$$c\\_{j,n}=\\int\\_{0}^{\\pi }\\frac{dk}{2\\pi }\\sum\\_{j^{\\prime }=1}^{N}\\left[ \\delta\n\\_{jj^{\\prime }}e^{-ikn}+S\\_{jj^{\\prime }}\\left( k\\right) e^{ikn}\\right]\nq\\_{j^{\\prime },k}\\text{,} \\label{leadsb}$$ \nand\n \n$$c\\_{C,r}=\\int\\_{0}^{\\pi }\\frac{dk}{2\\pi }\\sum\\_{j^{\\prime }=1}^{N}\\Gamma\n\\_{rj^{\\prime }}\\left( k\\right) q\\_{j^{\\prime },k}\\text{.} \\label{cplsitesb}$$ \nWe now express *H**T* in the scattering basis. Inserting Eq. ([cplsitesb]) into Eq. ([HT0]), we find the QD is only coupled to one channel in the continuum, i.e. the screening channel:\n \n$$H\\_{T}=-\\int\\_{0}^{\\pi }\\frac{dk}{2\\pi }V\\_{k}\\left( \\psi \\_{k}^{\\dag }d+\\text{h.c.}\\right) \\text{,} \\label{Andersonmodel}$$ \nwhere the screening channel operator *ψ**k*† is defined by\n \n$$\\psi \\_{k}^{\\dag }=\\frac{1}{V\\_{k}}\\sum\\_{j=1}^{N}\\sum\\_{r=1}^{M}t\\_{r}\\Gamma\n\\_{r,j}^{\\ast }\\left( k\\right) q\\_{j,k}^{\\dag }\\text{,}$$ \nand the normalization factor *V**k* > 0 is defined by\n \n*V**k*2 = ∑*j* = 1*N*∑*r*, *r*′ = 1*M**t**r**t**r*′\\*Γ*r**j*\\*(*k*)Γ*r*′*j*(*k*) = tr[Γ†(*k*)*λ*Γ(*k*)]. \nThis ensures {*ψ**k*, *ψ**k*′†} = 2*π**δ*(*k* − *k*′). Here we also introduce the *M* × *M* Hermitian QD coupling matrix *λ*,\n \n*λ**r**r*′ = *t**r**t**r*′\\*. \nIt will be useful to define a series of non-screening channels *ϕ**l*, *k* orthogonal to *ψ*, where *l* = 1, ⋯, *N* − 1. The *ϕ* channels are decoupled from the QD. In a compact notation we can write the transformation from the scattering basis to the screening–non-screening basis as\n \n$$\\Psi \\_{k}\\equiv \\left( \n\\begin{array}{c}\n\\psi \\_{k} \\\\ \n\\phi \\_{1,k} \\\\ \n\\cdots \\\\ \n\\phi \\_{N-1,k}\\end{array}\\right) =\\mathbb{U}\\_{k}\\left( \n\\begin{array}{c}\nq\\_{1,k} \\\\ \nq\\_{2,k} \\\\ \n\\cdots \\\\ \nq\\_{N,k}\\end{array}\\right) \\text{,}$$ \nwhere U is a unitary matrix. The first row of U is known:\n \n$$\\mathbb{U}\\_{1,j,k}=\\frac{1}{V\\_{k}}\\sum\\_{r=1}^{M}t\\_{r}^{\\ast }\\Gamma\n\\_{rj}\\left( k\\right) \\text{.} \\label{bbUmatrow1}$$ \nAs long as U stays unitary, its remaining matrix elements can be chosen freely without affecting physical observables. Ψ*k* now also diagonalizes *H*0,\n \n$$H\\_{0}=\\int\\_{0}^{\\pi }\\frac{dk}{2\\pi }\\epsilon \\_{k}\\left( \\psi \\_{k}^{\\dag\n}\\psi \\_{k}+\\sum\\_{l=1}^{N-1}\\phi \\_{l,k}^{\\dag }\\phi \\_{l,k}\\right) \\text{;}$$ \nwe shall also need the inverse transformation,\n \n*q**j*, *k* = U1, *j*, *k*\\**ψ**k* + ∑*l* = 1*N* − 1U*l* + 1, *j*, *k*\\**ϕ**l*, *k*. \nKondo model\n-----------\nIn the local moment regime of the Anderson model, for *T* ≪ *U* we can perform the Schrieffer-Wolff transformation on *ψ* to obtain an effective Kondo model with a reduced bandwidth and a momentum-dependent coupling:\n \n$$H=H\\_{0}+\\int \\frac{dkdk^{\\prime }}{\\left( 2\\pi \\right) ^{2}}\\left(\nJ\\_{kk^{\\prime }}\\psi \\_{k}^{\\dag }\\frac{\\vec{\\sigma}}{2}\\psi \\_{k^{\\prime\n}}\\cdot \\vec{S}\\_{d}+K\\_{kk^{\\prime }}\\psi \\_{k}^{\\dag }\\psi \\_{k^{\\prime\n}}\\right) \\text{,} \\label{Kondomodel}$$ \nwhere all momenta are between *k**F* − Λ0 and *k**F* + Λ0 with Fermi wave vector *k**F*, 0 < Λ0 ≪ *k**F* is the initial momentum cutoff, and the dispersion is linearized near the Fermi energy as *ε**k* = *v**F*(*k* − *k**F*). For the tight-binding model *v**F* = 2*t*sin*k**F*.\nThe interaction consists of a spin-flip term *J*,\n \n*J**k**k*′ = *V**k**V**k*′(*j**k* + *j**k*′), \n \n$$j\\_{k}=\\frac{1}{\\epsilon \\_{k}-\\epsilon \\_{d}}+\\frac{1}{U+\\epsilon\n\\_{d}-\\epsilon \\_{k}}\\approx j\\text{,}$$ \nand a particle-hole symmetry breaking potential scattering term *K*,\n \n$$K\\_{kk^{\\prime}}=\\frac{1}{4}V\\_{k}V\\_{k^{\\prime}}\\left(\n\\kappa\\_{k}+\\kappa\\_{k^{\\prime}}\\right) \\text{,} \\label{KondoK}$$ \n \n$$\\kappa \\_{k}=\\frac{1}{\\epsilon \\_{k}-\\epsilon \\_{d}}-\\frac{1}{U+\\epsilon\n\\_{d}-\\epsilon \\_{k}}\\approx \\kappa \\text{.}$$ \nThe energy cutoff is initially *D*0 ≡ *v**F*Λ0. When we reduce the running energy cutoff from *D* to *D* + *d**D* (0 < − *d**D* ≪ *D*) to integrate out the high-energy degrees of freedom in the narrow strips of energy ( − *D*, − *D* − *d**D*) and (*D* + *d**D*, *D*), *K* is exactly marginal in the renormalization group (RG) sense, whereas *J* is marginally relevant and obeys the following RG equation:\n \n$$-\\frac{d\\left( \\nu J\\_{kk^{\\prime }}\\right) }{d\\ln D}=\\frac{1}{2}\\left( \\nu\nJ\\_{k,k\\_{F}+\\Lambda }\\nu J\\_{k\\_{F}+\\Lambda,k^{\\prime }}+\\nu\nJ\\_{k,k\\_{F}-\\Lambda }\\nu J\\_{k\\_{F}-\\Lambda,k^{\\prime }}\\right) \\text{,}\n\\label{KondoJkkprRG}$$ \nor equivalently\n \n$$-\\frac{dj}{d\\ln D}=\\nu j^{2}\\left( V\\_{k\\_{F}+\\Lambda }^{2}+V\\_{k\\_{F}-\\Lambda\n}^{2}\\right) \\text{,} \\label{KondojRG}$$ \nwhere *ν* is the density of states per channel per spin, *ν* = 1/(2*π**v**F*). Therefore, renormalization of the Kondo coupling is controlled by the momentum-dependent normalization factor *V**k*2, defined in Eq. ([Vksquared]). *j* is the only truly independently renormalized coupling constant despite the appearance of Eq. ([KondoJkkprRG]); this follows from the fact that the screening channel is the only channel coupled to the QD.\nThe prototype Kondo model possesses a momentum-independent coupling function, *J**k**k*′ ≈ 2*j**V**k**F*2. As a result, spin-charge separation occurs and the Kondo interaction is found to be exclusively in the spin sector. The charge sector is nothing but a non-interacting theory with a particle-hole symmetry breaking phase shift due to the potential scattering term *K*, while at very low energy scales the spin sector renormalizes to a local Fermi liquid theory with *π*/2 phase shift.\nOn the other hand, in a mesoscopic geometry *V**k*2 often exhibit fluctuations on a mesoscopic energy scale *E**V*. (More precisely, *E**V* can be defined as the energy corresponding to the largest Fourier component in the spectrum of *V**k*2, but for both specific models discussed in this paper we can simply read it off the analytic expression.) In the presence of a characteristic length scale *L*, *E**V* may be of the order of the Thouless energy *v**F*/*L*, as is the case for the closed long ABK ring in Sec. [sec:closed]; however this is not always true, with a counterexample provided by the open long ABK ring in Sec. [sec:open] where *E**V* is of the order of the bandwidth 4*t*. Well above *E**V*, *V**k*2 appears featureless and can be approximated by its mean value $\\overline{V\\_{k}^{2}}$ with respect to *k*. The Kondo temperature *T**K* can be loosely defined as the energy cutoff at which the dimensionless coupling $2\\nu j\\overline{V\\_{k}^{2}}$ becomes *O*(1). As briefly sketched in Sec. [sec:intro], there are two very different parameter regimes of the Kondo temperature:\na) The small Kondo cloud regime *T**K* ≫ *E**V*. For *E**V* ∼ *v**F*/*L*, the size of the Kondo screening cloud *L**K* ≡ *v**F*/*T**K* ≪ *L*; hence the name. In this regime, the bare Kondo coupling is sufficiently large, so that $2\\nu j\\overline{V\\_{k}^{2}}$ renormalizes to *O*(1) before it senses any mesoscopic fluctuation. By approximating $V\\_{k}^{2}\\approx \\overline{V\\_{k}^{2}}$, Eq. ([KondojRG]) has a solution\n \n$$j\\left( D\\right) \\approx \\frac{j\\_{0}}{1+2\\nu j\\_{0}\\overline{V\\_{k}^{2}}\\ln \n\\frac{D}{D\\_{0}}}\\text{,} \\label{KondojRGsolsc}$$ \nwhere *j*0 is the bare Kondo coupling constant at the initial energy cutoff *D*0. Eq. ([KondojRGsolsc]) gives the background Kondo temperature\n \n$$T\\_{K}\\approx T\\_{K}^{0}\\equiv D\\_{0}\\exp \\left( -\\frac{1}{2\\nu j\\_{0}\\overline{V\\_{k}^{2}}}\\right) \\text{,} \\label{TKKondobulk}$$ \nindependent of the mesoscopic details of the geometry. For *T* ≪ *T**K*, the low-energy effective theory is also conjectured to be a Fermi liquid, but the T-matrix (or the phase shift) of the screening channel is not yet known with certainty.\nb) The very large Kondo cloud regime *T**K* ≪ *E**V*. For *E**V* ∼ *v**F*/*L*, *L**K* ≫ *L*. In this regime, the bare Kondo coupling is very small, and *j* does not begin to renormalize significantly until the energy cutoff is well below *E**V*. The variation of *V**k*2 is hence negligible in the resulting low energy theory, *V**k*2 ≈ *V**k**F*2, but *V**k**F*2 may be significantly different from $\\overline{V\\_{k}^{2}}$, which means Kondo temperature is thus highly sensitive to the mesoscopic details of *V**k**F*2. Because *V**k*2 is almost independent of *k*, we may map the low-energy theory in question onto the conventional Kondo model, where conduction electrons are scattered by a point-like spin in real space. (We stress that this mapping would not be possible for a strongly *k*-dependent *V**k*2, which is the case for the small cloud regime.) Following well-known results in the conventional Kondo model, we see that the low-energy effective theory is a local Fermi liquid theory, with parameters also sensitive to mesoscopic details.\nLinear DC conductance[sec:conductance]\n======================================\nIn this section we calculate the DC conductance tensor of the system in linear response theory, generalizing the results in Ref. to our multi-terminal setup. The result is presented as the sum of a disconnected contribution and a connected one (Fig. [fig:discconn]). By disconnected and connected, we are referring to the topology of the corresponding Feynman diagrams: a disconnected contribution originates from a Feynman diagram without any cross-links, and can always be written as the product of two two-point functions. The disconnected contribution has a simple Landauer form, and is quadratic in the T-matrix of the screening channel *ψ*. The connected contribution is also shown to depend on properties near the Fermi surface only, but it is usually difficult to evaluate analytically except at high temperatures, or at low temperatures if the Fermi-liquid perturbation theory is applicable. Nevertheless, just as with the short ABK ring, the connected contribution can be approximately eliminated at temperatures low compared to another mesoscopic energy scale *T* ≪ *E*conn.\nDisconnected (self-energy) and connected (vertex correction) contributions to the density-density correlation function, which is directly related to the conductance through the Kubo formula Eq. ([Kubo]). The dashed lines represent external legs at times *t̄* and 0, the solid lines represent fully dressed Ψ fermion propagators, and the hatched circle represents all connected 4-point vertices of the screening channel.[fig:discconn]\nKubo formula in terms of screening and non-screening channels\n-------------------------------------------------------------\nThe linear DC conductance tensor *G**j**j*′ is defined through ⟨*I**j*⟩ = ∑*j*′*G**j**j*′*V**j*′, where *I**j* is the current operator in lead *j*, and *V**j*′ is the applied bias voltage on lead *j*′. *I**j*is given by *I**j* = − *e**d**N**j*/*d**t̄*, where *t̄* is the real time variable, and\n \n*N**j* ≡ ∑*n* = 0∞*c**j*, *n*†*c**j*, *n* \nis the density operator in lead *j*. *G**j**j*′ is then given by the Kubo formula\n \n$$G\\_{jj^{\\prime }}=\\frac{e^{2}}{h}\\lim\\_{\\Omega \\rightarrow 0}\\left( 2\\pi\ni\\Omega \\right) G\\_{jj^{\\prime }}^{\\prime }\\left( \\Omega \\right) \\text{,}\n\\label{Kubo}$$ \nwhere the retarded density-density correlation function is\n \n*G**j**j*′′(Ω) ≡ − *i*∫0∞*d**t̄**e**i*Ω+*t̄*⟨[*N**j*(*t̄*), *N**j*′(0)]⟩ \nand Ω+ ≡ Ω + *i*0+. The retarded correlation function can be obtained by means of analytic continuation *i**ω**p* → Ω+ from its imaginary time counterpart,\n \nG*j**j*′′(*i**ω**p*) = − ∫0*β**d**τ**e**i**ω**p**τ*⟨*T**τ**N**j*(*τ*)*N**j*′(0)⟩, \nwhere *ω**p* = 2*p**π*/*β* is a bosonic Matsubara frequency, *β* = 1/*T* is the inverse temperature, and *T**τ* is the imaginary time-ordering operator.\nTo calculate the correlation function we need the density operator *N**j* written as bilinears of Ψ. This is achieved by the insertion of Eq. ([leadsb]) and then Eq. ([scattoscrn]) into Eq. ([leaddensity0]). We find\n \n$$N\\_{j}=\\int\\_{0}^{\\pi }\\frac{dk\\_{1}dk\\_{2}}{\\left( 2\\pi \\right) ^{2}}\\Psi\n\\_{k\\_{1}}^{\\dag }\\mathbb{M}\\_{k\\_{1}k\\_{2}}^{j}\\Psi \\_{k\\_{2}}\\text{,}\n\\label{leaddensity}$$ \nwhere for *l*1, *l*2 = 1,..., *N*,\n \n$$\\begin{aligned}\n\\left( \\mathbb{M}\\_{k\\_{1}k\\_{2}}^{j}\\right) \\_{l\\_{1}l\\_{2}}& =\\sum\\_{j\\_{1}j\\_{2}}\\mathbb{U}\\_{l\\_{1},j\\_{1},k\\_{1}}\\mathbb{U}\\_{l\\_{2},j\\_{2},k\\_{2}}^{\\ast }\\left[\n\\delta \\_{jj\\_{1}}\\delta \\_{jj\\_{2}}\\frac{1}{1-e^{i\\left( k\\_{1}-k\\_{2}+i0\\right) }}+S\\_{jj\\_{1}}^{\\ast }\\left( k\\_{1}\\right) \\delta \\_{jj\\_{2}}\\frac{1}{1-e^{-i\\left( k\\_{1}+k\\_{2}-i0\\right) }}\\right. \\notag \\\\\n& \\left. +\\delta \\_{jj\\_{1}}S\\_{jj\\_{2}}\\left( k\\_{2}\\right) \\frac{1}{1-e^{i\\left( k\\_{1}+k\\_{2}+i0\\right) }}+S\\_{jj\\_{1}}^{\\ast }\\left( k\\_{1}\\right)\nS\\_{jj\\_{2}}\\left( k\\_{2}\\right) \\frac{1}{1-e^{i\\left( k\\_{2}-k\\_{1}+i0\\right) }}\\right] \\text{.} \\label{bbMmat}\\end{aligned}$$ \nThe matrix M obeys M*k*1*k*2*j* = (M*k*2*k*1*j*)† which ensures *N**j* is Hermitian.\nDisconnected part\n-----------------\nWe substitute Eq. ([leaddensity]) into Eq. ([GjjprOmega]). The disconnected part of the conductance is obtained by pairing up Ψ and Ψ† operators to form two two-point Green’s functions:\n \n$$G\\_{jj^{\\prime }}^{\\prime D}\\left( \\Omega \\right) =-2i\\int\\_{0}^{\\infty }d\\bar{t}e^{i\\Omega ^{+}\\bar{t}}\\int\\_{0}^{\\pi }\\frac{dk\\_{1}dk\\_{2}}{\\left( 2\\pi\n\\right) ^{2}}\\frac{dq\\_{1}dq\\_{2}}{\\left( 2\\pi \\right) ^{2}}\\operatorname{tr}\\left[ \\mathbb{M}\\_{k\\_{1}k\\_{2}}^{j}\\mathbb{G}\\_{k\\_{2}q\\_{1}}^{>}\\left( \\bar{t}\\right) \\mathbb{M}\\_{q\\_{1}q\\_{2}}^{j^{\\prime }}\\mathbb{G}\\_{q\\_{2}k\\_{1}}^{<}\\left( -\\bar{t}\\right) -\\mathbb{M}\\_{q\\_{1}q\\_{2}}^{j^{\\prime }}\\mathbb{G}\\_{q\\_{2}k\\_{1}}^{>}\\left( -\\bar{t}\\right) \\mathbb{M}\\_{k\\_{1}k\\_{2}}^{j}\\mathbb{G}\\_{k\\_{2}q\\_{1}}^{<}\\left( \\bar{t}\\right) \\right] \\text{.} \\label{GjjprDgl}$$ \nHere the factor of 2 in the second line is due to the spin degeneracy. The greater and lesser Green’s functions in the screening–non-screening basis are defined as\n \n$$\\mathbb{G}\\_{kq}^{>}\\left( \\bar{t}\\right) \\equiv-i\\left( \n\\begin{array}{cccc}\n\\left\\langle \\psi\\_{k}\\left( \\bar{t}\\right) \\psi\\_{q}^{\\dag}\\left( 0\\right)\n\\right\\rangle & & & \\\\ \n& \\left\\langle \\phi\\_{1,k}\\left( \\bar{t}\\right) \\phi\\_{1,q}^{\\dag}\\left(\n0\\right) \\right\\rangle & & \\\\ \n& & \\cdots & \\\\ \n& & & \\left\\langle \\phi\\_{N-1,k}\\left( \\bar{t}\\right) \\phi\\_{N-1,q}^{\\dag\n}\\left( 0\\right) \\right\\rangle\\end{array}\n\\right) \\text{,}$$ \nand\n \n$$\\mathbb{G}\\_{kq}^{<}\\left( \\bar{t}\\right) \\equiv +i\\left( \n\\begin{array}{cccc}\n\\left\\langle \\psi \\_{q}^{\\dag }\\left( 0\\right) \\psi \\_{k}\\left( \\bar{t}\\right)\n\\right\\rangle & & & \\\\ \n& \\left\\langle \\phi \\_{1,q}^{\\dag }\\left( 0\\right) \\phi \\_{1,k}\\left( \\bar{t}\\right) \\right\\rangle & & \\\\ \n& & \\cdots & \\\\ \n& & & \\left\\langle \\phi \\_{N-1,q}^{\\dag }\\left( 0\\right) \\phi\n\\_{N-1,k}\\left( \\bar{t}\\right) \\right\\rangle\\end{array}\\right) \\text{.}$$ \nIn equilibrium, fluctuation-dissipation theorem requires that G*k**q*>(*ω*) = 2*i*[1 − *f*(*ω*)]ImG*k**q**R*(*ω*), and G*k**q*<(*ω*) = − 2*i**f*(*ω*)ImG*k**q**R*(*ω*), where *f*(*ω*) = 1/(*e**β**ω* + 1) is the Fermi function. These equilibrium relations result from the fact that G*k**q**R*(*ω*) = G*q**k**R*(*ω*) for the Anderson model [see Eq. ([Dyson]) below]. With these relations Eq. ([GjjprDgl]) becomes\n \n$$G\\_{jj^{\\prime }}^{\\prime D}\\left( \\Omega \\right) =8\\int \\frac{d\\omega\nd\\omega ^{\\prime }}{\\left( 2\\pi \\right) ^{2}}\\frac{f\\left( \\omega \\right)\n-f\\left( \\omega ^{\\prime }\\right) }{\\omega -\\omega ^{\\prime }+\\Omega ^{+}}\\int\\_{0}^{\\pi }\\frac{dk\\_{1}dk\\_{2}}{\\left( 2\\pi \\right) ^{2}}\\frac{dq\\_{1}dq\\_{2}}{\\left( 2\\pi \\right) ^{2}}\\operatorname{tr}\\left[ \\mathbb{M}\\_{k\\_{1}k\\_{2}}^{j}\\operatorname{Im} \\mathbb{G}\\_{k\\_{2}q\\_{1}}^{R}\\left( \\omega ^{\\prime }\\right) \\mathbb{M}\\_{q\\_{1}q\\_{2}}^{j^{\\prime }}\\operatorname{Im} \\mathbb{G}\\_{q\\_{2}k\\_{1}}^{R}\\left(\n\\omega \\right) \\right] \\text{.} \\label{GjjprDImIm}$$ \nWe note that, in contrast to the case of Ref. , the momentum integral here is not necessarily real. Instead, its complex conjugate takes the same form but with *ω* and *ω*′ interchanged:\n \n$$\\begin{aligned}\n& \\int\\_{0}^{\\pi }\\frac{dk\\_{1}dk\\_{2}}{\\left( 2\\pi \\right) ^{2}}\\frac{dq\\_{1}dq\\_{2}}{\\left( 2\\pi \\right) ^{2}}\\operatorname{tr}\\left[ \\mathbb{M}\\_{k\\_{1}k\\_{2}}^{j}\\operatorname{Im} \\mathbb{G}\\_{k\\_{2}q\\_{1}}^{R}\\left( \\omega ^{\\prime }\\right) \\mathbb{M}\\_{q\\_{1}q\\_{2}}^{j^{\\prime }}\\operatorname{Im} \\mathbb{G}\\_{q\\_{2}k\\_{1}}^{R}\\left(\n\\omega \\right) \\right] ^{\\ast } \\notag \\\\\n& =\\int\\_{0}^{\\pi }\\frac{dk\\_{1}dk\\_{2}}{\\left( 2\\pi \\right) ^{2}}\\frac{dq\\_{1}dq\\_{2}}{\\left( 2\\pi \\right) ^{2}}\\operatorname{tr}\\left[ \\mathbb{M}\\_{k\\_{1}k\\_{2}}^{j}\\operatorname{Im} \\mathbb{G}\\_{k\\_{2}q\\_{1}}^{R}\\left( \\omega \\right) \\mathbb{M}\\_{q\\_{1}q\\_{2}}^{j^{\\prime }}\\operatorname{Im} \\mathbb{G}\\_{q\\_{2}k\\_{1}}^{R}\\left(\n\\omega ^{\\prime }\\right) \\right] \\text{.}\\end{aligned}$$ \nMaking use of this property, we can show that [*G**j**j*′′*D*( − Ω)]\\* = *G**j**j*′′*D*(Ω). Thus the disconnected contribution to the DC conductance can be written as\n \n$$G\\_{jj^{\\prime }}^{D}=\\frac{e^{2}}{h}\\lim\\_{\\Omega \\rightarrow 0}\\left( -2\\pi\n\\Omega \\right) \\operatorname{Im}G\\_{jj^{\\prime }}^{\\prime D}\\left( \\Omega \\right) \n\\text{.}$$ \nWe should realize, however, that taking the imaginary part of *G**j**j*′′*D* is generally not equivalent to taking the *δ*-function part of 1/(*ω* − *ω*′ + Ω+) in Eq. ([GjjprDImIm]).\nFor the Anderson model, it is not difficult to find the Dyson’s equation for the retarded Green’s function by the equation-of-motion technique:\n \nG*k*2*q*1*R*(*ω*) = 2*π**δ*(*k*2 − *q*1)*g**k*2*R*(*ω*) + *τ**ψ**g**k*2*R*(*ω*)T*k*2*q*1(*ω*)*g**q*1*R*(*ω*), \nwhere the free retarded Green’s function for *ψ* and *ϕ* is\n \n$$g\\_{k}^{R}\\left( \\omega \\right) =\\frac{1}{\\omega ^{+}-\\epsilon \\_{k}}\\text{,}$$ \nand *τ**ψ* is the projection operator onto the screening channel subspace. Again, only the Green’s function of the screening channel is modified by coupling to the QD. The retarded T-matrix of the screening channel in the single-particle sector is related to the retarded two-point function of the QD by\n \nT*k*2*q*1(*ω*) = *V**k*2*G**d**d**R*(*ω*)*V**q*1, \nwhere *G**d**d**R*(*ω*) ≡ − *i*∫0∞*d**t̄**e**i**ω*+*t̄*⟨{*d*(*t̄*), *d*†(0)}⟩.\nFrom Eqs. ([GjjprDImIm]) and ([Dyson]) we may express the disconnected contribution to the linear DC conductance in the Landauer form:\n \n$$G\\_{jj^{\\prime }}^{D}=\\frac{2e^{2}}{h}\\int\\_{-2t}^{2t}d\\epsilon \\_{p}\\left[\n-f^{\\prime }\\left( \\epsilon \\_{p}\\right) \\right] \\mathcal{T}\\_{jj^{\\prime\n}}^{D}\\left( \\epsilon \\_{p}\\right) \\text{,} \\label{GDTD}$$ \nwhere the disconnected transmission probability T*j**j*′*D* is written in terms of the absolute square of a transmission amplitude,\n \n$$\\begin{aligned}\n\\mathcal{T}\\_{jj^{\\prime }}^{D}\\left( \\epsilon \\_{p}\\right) &=&\\delta\n\\_{jj^{\\prime }}-\\left\\vert \\left\\{ S\\left( p\\right) \\left[ 1-2i\\pi \\nu \\_{p}\\text{T}\\_{pp}\\left( \\epsilon \\_{p}\\right) \\mathbb{U}\\_{p}^{\\dag }\\tau \\_{\\psi }\\mathbb{U}\\_{p}\\right] \\right\\} \\_{jj^{\\prime }}\\right\\vert ^{2} \\notag \\\\\n&=&\\delta \\_{jj^{\\prime }}-\\left\\vert S\\_{jj^{\\prime }}\\left( p\\right) +\\frac{2i}{V\\_{p}^{2}}\\left[ S\\left( p\\right) \\Gamma ^{\\dag }\\left( p\\right) \\lambda\n\\Gamma \\left( p\\right) \\right] \\_{jj^{\\prime }}\\left[ -\\pi \\nu \\_{p}\\text{T}\\_{pp}\\left( \\epsilon \\_{p}\\right) \\right] \\right\\vert ^{2}\\text{.}\n\\label{discprob}\\end{aligned}$$ \nAgain *λ* is the QD coupling matrix defined in Eq. ([couplingmatlambda]) and *ν**p* is the density of states per channel per spin for the tight-binding model\n \n$$\\nu \\_{p}=\\frac{1}{4\\pi t\\sin p}\\text{.}$$ \nThe detailed derivation of Eq. ([discprob]) by contour methods is left for Appendix [sec:appdisc]. As a consistency check, we show in Appendix [sec:appNI] that for a non-interacting QD, *U* = 0, Eq. ([discprob]) and solving the Schroedinger’s equation yield the same transmission probability.\nAt zero temperature, when the single-particle sector of the screening channel T-matrix obeys the optical theorem and the inelastic part of the T-matrix vanishes, there is no connected contribution and Eq. ([GDTD]) yields the full linear DC conductance. In this case a clear picture emerges from Eq. ([discprob]): the conductance is given by the Landauer formula with an effective single-particle S-matrix, which is obtained from the original S-matrix simply by imposing a phase shift on the screening channel, corresponding to the particle-hole symmetry breaking potential scattering and the elastic scattering by the Kondo singlet.\nAnother useful representation of the disconnected probability, similar to that in Ref. , is obtained by expanding Eq. ([discprob]):\n \n$$\\begin{aligned}\n\\mathcal{T}\\_{jj^{\\prime }}^{D}\\left( \\epsilon \\_{p}\\right) &=&\\mathcal{T}\\_{0,jj^{\\prime }}\\left( \\epsilon \\_{p}\\right) +\\mathcal{Z}\\_{R,jj^{\\prime\n}}\\left( \\epsilon \\_{p}\\right) \\operatorname{Re} \\left[ -\\pi \\nu \\_{p}\\text{T}\\_{pp}\\left( \\epsilon \\_{p}\\right) \\right] \\notag \\\\\n&&+\\mathcal{Z}\\_{I,jj^{\\prime }}\\left( \\epsilon \\_{p}\\right) \\operatorname{Im} \\left[\n-\\pi \\nu \\_{p}\\text{T}\\_{pp}\\left( \\epsilon \\_{p}\\right) \\right] +\\mathcal{Z}\\_{2,jj^{\\prime }}\\left( \\epsilon \\_{p}\\right) \\left\\vert -\\pi \\nu \\_{p}\\text{T}\\_{pp}\\left( \\epsilon \\_{p}\\right) \\right\\vert ^{2}\\text{,}\n\\label{discprobKYA}\\end{aligned}$$ \nwith a background transmission term\n \n$$\\mathcal{T}\\_{0,jj^{\\prime }}\\left( \\epsilon \\_{p}\\right) =\\delta \\_{jj^{\\prime\n}}-\\left\\vert S\\_{jj^{\\prime }}\\left( p\\right) \\right\\vert ^{2}\\text{,}\n\\label{discprobKYAbgt}$$ \na term linear in the real part of the T-matrix, proportional to\n \n$$\\mathcal{Z}\\_{R,jj^{\\prime }}\\left( \\epsilon \\_{p}\\right) =\\frac{4}{V\\_{p}^{2}}\\operatorname{Im}\\left\\{ \\left[ S\\left( p\\right) \\Gamma ^{\\dag }\\left( p\\right)\n\\lambda \\Gamma \\left( p\\right) \\right] \\_{jj^{\\prime }}S\\_{jj^{\\prime }}^{\\ast\n}\\left( p\\right) \\right\\} \\text{,} \\label{discprobKYAZR}$$ \na term linear in the imaginary part, proportional to\n \n$$\\mathcal{Z}\\_{I,jj^{\\prime }}\\left( \\epsilon \\_{p}\\right) =\\frac{4}{V\\_{p}^{2}}\\operatorname{Re}\\left\\{ \\left[ S\\left( p\\right) \\Gamma ^{\\dag }\\left( p\\right)\n\\lambda \\Gamma \\left( p\\right) \\right] \\_{jj^{\\prime }}S\\_{jj^{\\prime }}^{\\ast\n}\\left( p\\right) \\right\\} \\text{,} \\label{discprobKYAZI}$$ \nand a term quadratic in the T-matrix, proportional to\n \n$$\\mathcal{Z}\\_{2,jj^{\\prime }}\\left( \\epsilon \\_{p}\\right) =-\\frac{4}{V\\_{p}^{4}}\\left\\vert \\left[ S\\left( p\\right) \\Gamma ^{\\dag }\\left( p\\right) \\lambda\n\\Gamma \\left( p\\right) \\right] \\_{jj^{\\prime }}\\right\\vert ^{2}\\text{.}\n\\label{discprobKYAZ2}$$ \nIn the DC limit, the total current flowing out of the junction is zero, and a uniform voltage applied to all leads does not result in any current; hence the linear DC conductance satisfies current and voltage Kirchhoff’s laws ∑*j**G**j**j*′ = ∑*j*′*G**j**j*′ = 0. As a comparison it is interesting to consider the sum of the disconnected transmission probability, Eq. ([discprobKYA]), over *j* or *j*′. Using the unitarity of *S* and Eq. ([couplingmatlambda]) it is not difficult to find that\n \n$$\\sum\\_{j}\\mathcal{T}\\_{jj^{\\prime }}^{D}\\left( \\epsilon \\_{p}\\right) =\\left\\{ \n\\operatorname{Im} \\left[ -\\pi \\nu \\_{p}\\text{T}\\_{pp}\\left( \\epsilon \\_{p}\\right) \\right] -\\left\\vert -\\pi \\nu \\_{p}\\text{T}\\_{pp}\\left( \\epsilon \\_{p}\\right)\n\\right\\vert ^{2}\\right\\} \\frac{4}{V\\_{p}^{2}}\\left[ \\Gamma ^{\\dag }\\left(\np\\right) \\lambda \\Gamma \\left( p\\right) \\right] \\_{j^{\\prime }j^{\\prime }}\\text{,} \\label{TDrowsum}$$ \nand\n \n$$\\sum\\_{j^{\\prime }}\\mathcal{T}\\_{jj^{\\prime }}^{D}\\left( \\epsilon \\_{p}\\right)\n=\\left\\{ \\operatorname{Im}\\left[ -\\pi \\nu \\_{p}\\text{T}\\_{pp}\\left( \\epsilon\n\\_{p}\\right) \\right] -\\left\\vert -\\pi \\nu \\_{p}\\text{T}\\_{pp}\\left( \\epsilon\n\\_{p}\\right) \\right\\vert ^{2}\\right\\} \\frac{4}{V\\_{p}^{2}}\\left[ S\\left(\np\\right) \\Gamma ^{\\dag }\\left( p\\right) \\lambda \\Gamma \\left( p\\right)\nS^{\\dag }\\left( p\\right) \\right] \\_{jj}\\text{.} \\label{TDcolsum}$$ \nAs mentioned in Ref. , the quantity in curly brackets in Eqs. ([TDrowsum]) and ([TDcolsum]) measures the deviation of the single-particle sector of the T-matrix from the optical theorem. In the case of a non-interacting QD or the *T* = 0 Fermi liquid theory of the Kondo limit, where the connected contribution to the conductance vanishes, these row/column sum formulas conform to our expectations: the T-matrix obeys the optical theorem, leading to ∑*j*T*j**j*′*D* = ∑*j*′T*j**j*′*D* = 0, so that ∑*j**G**j**j*′ = ∑*j*′*G**j**j*′ = 0 is ensured.\nConnected part and its low-temperature elimination\n--------------------------------------------------\nIn this subsection we show that the connected contribution to the conductance is again a Fermi surface contribution, and discuss how it can be approximately eliminated at low temperatures. Following Ref. we construct a transmission probability for this contribution. After a partial insertion of Eq. ([leaddensity]) into Eq. ([Gjjpriomegap]), the connected part of the density-density correlation function can be written as\n \nG*j**j*′′*C*(*i**ω**p*) = ∫0*β**d**τ**e**i**ω**p**τ**P**j**j*′(*τ*, *τ*), \nwhere the connected four-point function *P**j**j*′ with two temporal arguments is\n \n$$P\\_{jj^{\\prime }}\\left( \\tau \\_{1},\\tau \\_{2}\\right) \\equiv -\\int\\_{0}^{\\pi }\\frac{dk\\_{1}dk\\_{2}}{\\left( 2\\pi \\right) ^{2}}\\left( \\mathbb{M}\\_{k\\_{1}k\\_{2}}^{j}\\right) \\_{11}\\sum\\_{\\sigma }\\left\\langle T\\_{\\tau }\\psi\n\\_{k\\_{1}\\sigma }^{\\dag }\\left( \\tau \\_{1}\\right) \\psi \\_{k\\_{2}\\sigma }\\left(\n\\tau \\_{2}\\right) N\\_{j^{\\prime }}\\left( 0\\right) \\right\\rangle \\_{C}\\text{;}$$ \nthe subscript *C* denotes connected diagrams. Note that only the screening channel contributes to the connected part, as the non-screening channels are free fermions. Using the equation-of-motion technique, it is easy to relate *P**j**j*′ to a partially amputated quantity:\n \n$$P\\_{jj^{\\prime}}\\left( \\tau,\\tau\\right) =\\int\\_{0}^{\\pi}\\frac{dk\\_{1}dk\\_{2}}{\\left( 2\\pi\\right) ^{2}}\\left( \\mathbb{M}\\_{k\\_{1}k\\_{2}}^{j}\\right)\n\\_{11}V\\_{k\\_{1}}V\\_{k\\_{2}}\\int d\\tau\\_{1}d\\tau\\_{2}g\\_{k\\_{1}}\\left(\n\\tau\\_{1}-\\tau\\right) g\\_{k\\_{2}}\\left( \\tau-\\tau\\_{2}\\right)\n\\sum\\_{\\sigma}\\left\\langle T\\_{\\tau}d\\_{\\sigma}^{\\dag}\\left( \\tau\\_{1}\\right)\nd\\_{\\sigma}\\left( \\tau \\_{2}\\right) N\\_{j^{\\prime}}\\left( 0\\right)\n\\right\\rangle \\_{C}\\text{,}$$ \nwhere\n \n*g**k*(*τ*) ≡ [*f*(*τ*) − *θ*(*τ*)]*e*− *ε**k**τ* \nis the imaginary time free Green’s function and *θ*(*τ*) is the Heaviside unit-step function. With *τ* only appearing in free propagators, we can perform the Fourier transform explicitly,\n \n$$\\begin{aligned}\n\\mathcal{G}\\_{jj^{\\prime }}^{\\prime C}\\left( i\\omega \\_{p}\\right) & =\\frac{1}{\\beta }\\sum\\_{\\omega \\_{m}}P\\_{jj^{\\prime }}\\left( i\\omega \\_{m},i\\omega\n\\_{m}+i\\omega \\_{p}\\right) \\notag \\\\\n& =\\frac{1}{\\beta }\\sum\\_{\\omega \\_{m}}\\int\\_{0}^{\\pi }\\frac{dk\\_{1}dk\\_{2}}{\\left( 2\\pi \\right) ^{2}}\\left( \\mathbb{M}\\_{k\\_{1}k\\_{2}}^{j}\\right)\n\\_{11}g\\_{k\\_{1}}\\left( i\\omega \\_{m}\\right) g\\_{k\\_{2}}\\left( i\\omega\n\\_{m}+i\\omega \\_{p}\\right) V\\_{k\\_{1}}V\\_{k\\_{2}} \\notag \\\\\n& \\times \\int d\\tau \\_{1}d\\tau \\_{2}e^{-i\\omega \\_{m}\\tau \\_{1}}e^{i\\left(\n\\omega \\_{m}+\\omega \\_{p}\\right) \\tau \\_{2}}\\sum\\_{\\sigma }\\left\\langle T\\_{\\tau\n}d\\_{\\sigma }^{\\dag }\\left( \\tau \\_{1}\\right) d\\_{\\sigma }\\left( \\tau\n\\_{2}\\right) N\\_{j^{\\prime }}\\left( 0\\right) \\right\\rangle \\_{C}\\text{.}\\end{aligned}$$ \nOne may now use the contour integration argument in Ref. . The final result is that the connected contribution to the DC conductance is expressed in terms of a transmission probability T*C* related to *P**j**j*′:\n \n$$G\\_{jj^{\\prime}}^{C}=\\frac{2e^{2}}{h}\\int\\_{-2t}^{2t}d\\omega\\left[ -f^{\\prime\n}\\left( \\omega\\right) \\right] \\mathcal{T}\\_{jj^{\\prime}}^{C}\\left(\n\\omega\\right) \\text{,}$$ \nwhere\n \n$$\\mathcal{T}\\_{jj^{\\prime }}^{C}\\left( \\omega \\right) =\\lim\\_{\\Omega\n\\rightarrow 0}\\frac{\\Omega ^{2}}{8}P\\_{jj^{\\prime }}\\left( \\omega -i\\eta\n\\_{1},\\omega +\\Omega +i\\eta \\_{2}\\right) +\\text{c.c.}$$ \nand\n \n$$\\begin{aligned}\n& P\\_{jj^{\\prime }}\\left( \\omega -i\\eta \\_{1},\\omega +\\Omega +i\\eta\n\\_{2}\\right) \\notag \\\\\n& =\\int\\_{0}^{\\pi }\\frac{dk\\_{1}dk\\_{2}}{\\left( 2\\pi \\right) ^{2}}\\left( \n\\mathbb{M}\\_{k\\_{1}k\\_{2}}^{j}\\right) \\_{11}g\\_{k\\_{1}}^{A}\\left( \\omega \\right)\ng\\_{k\\_{2}}^{R}\\left( \\omega +\\Omega \\right) V\\_{k\\_{1}}V\\_{k\\_{2}}\\int d\\tau\n\\_{1}d\\tau \\_{2}e^{-i\\omega ^{-}\\tau \\_{1}}e^{i\\left( \\omega ^{+}+\\Omega\n\\right) \\tau \\_{2}}\\sum\\_{\\sigma }\\left\\langle T\\_{\\tau }d\\_{\\sigma }^{\\dag\n}\\left( \\tau \\_{1}\\right) d\\_{\\sigma }\\left( \\tau \\_{2}\\right) N\\_{j^{\\prime\n}}\\left( 0\\right) \\right\\rangle \\_{C}\\text{.}\\end{aligned}$$ \nHere *η*1, *η*2 → 0+ are positive infinitesimal numbers.\nIt is in fact possible to do the *k*1 and *k*2 integrals. Using Eqs. ([bbUmatrow1]), ([bbMmat]) and finally ([Gammaminusk]), we obtain\n \n$$\\begin{aligned}\n& \\int\\_{0}^{\\pi }\\frac{dk\\_{1}dk\\_{2}}{\\left( 2\\pi \\right) ^{2}}\\left( \\mathbb{M}\\_{k\\_{1}k\\_{2}}^{j}\\right) \\_{11}g\\_{k\\_{1}}^{A}\\left( \\epsilon \\_{p}\\right)\ng\\_{k\\_{2}}^{R}\\left( \\epsilon \\_{p}+\\Omega \\right) V\\_{k\\_{1}}V\\_{k\\_{2}} \\notag\n\\\\\n& =\\sum\\_{r\\_{1}r\\_{2}}t\\_{r\\_{1}}^{\\ast }t\\_{r\\_{2}}\\int\\_{-\\pi }^{\\pi }\\frac{dk\\_{1}dk\\_{2}}{\\left( 2\\pi \\right) ^{2}}g\\_{k\\_{1}}^{A}\\left( \\epsilon\n\\_{p}\\right) g\\_{k\\_{2}}^{R}\\left( \\epsilon \\_{p}+\\Omega \\right) \\Gamma\n\\_{r\\_{1}j}\\left( k\\_{1}\\right) \\Gamma \\_{r\\_{2}j}^{\\ast }\\left( k\\_{2}\\right) \n\\frac{1}{1-e^{i\\left( k\\_{1}-k\\_{2}+i0\\right) }}\\text{.}\\end{aligned}$$ \nHere domains of the momentum integrals are extended to ( − *π*, *π*) according to Eq. ([Gammaminusk]), which facilitates the application of the residue method. As explained in Appendix [sec:appdisc], the poles of Γ(*k*1) and Γ\\*(*k*2) are not important in the DC limit Ω → 0. Therefore, the *O*(1/Ω) contribution is dominated by the poles of the free Green’s functions, and is given by\n \n$$\\int\\_{0}^{\\pi }\\frac{dk\\_{1}dk\\_{2}}{\\left( 2\\pi \\right) ^{2}}\\left( \\mathbb{M}\\_{k\\_{1}k\\_{2}}^{j}\\right) \\_{11}g\\_{k\\_{1}}^{A}\\left( \\epsilon \\_{p}\\right)\ng\\_{k\\_{2}}^{R}\\left( \\epsilon \\_{p}+\\Omega \\right) V\\_{k\\_{1}}V\\_{k\\_{2}}=\\frac{2\\pi i}{\\Omega }\\nu \\_{p}\\left[ S\\left( p^{\\prime }\\right) \\Gamma ^{\\dag\n}\\left( p^{\\prime }\\right) \\lambda \\Gamma \\left( p\\right) S^{\\dag }\\left(\np\\right) \\right] \\_{jj}+O\\left( 1\\right) \\text{.}$$ \nwhere *ε**p* + Ω ≡ *ε**p*′, 0 ≤ *p*, *p*′ ≤ *π*. This leads to\n \n$$\\mathcal{T}\\_{jj^{\\prime }}^{C}\\left( \\epsilon \\_{p}\\right) =\\nu \\_{p}\\left[\nS\\left( p\\right) \\Gamma ^{\\dag }\\left( p\\right) \\lambda \\Gamma \\left(\np\\right) S^{\\dag }\\left( p\\right) \\right] \\_{jj}\\left[ \\frac{i\\pi }{4}\\lim\\_{\\Omega \\rightarrow 0}\\Omega \\int d\\tau \\_{1}d\\tau \\_{2}e^{-i\\omega\n^{-}\\tau \\_{1}}e^{i\\left( \\omega ^{+}+\\Omega \\right) \\tau \\_{2}}\\sum\\_{\\sigma\n}\\left\\langle T\\_{\\tau }d\\_{\\sigma }^{\\dag }\\left( \\tau \\_{1}\\right) d\\_{\\sigma\n}\\left( \\tau \\_{2}\\right) N\\_{j^{\\prime }}\\left( 0\\right) \\right\\rangle \\_{C}+\\text{c.c.}\\right] \\text{.}$$ \nA similar manipulation can be done for the *N**j*′ part of the correlation function.\nOne can again consider the row and column sums of the tensor T*C*. Tracing over *j* immediately yields\n \n$$\\sum\\_{j}\\mathcal{T}\\_{jj^{\\prime }}^{C}\\left( \\epsilon \\_{p}\\right) =\\nu\n\\_{p}V\\_{p}^{2}\\left[ \\frac{i\\pi }{4}\\lim\\_{\\Omega \\rightarrow 0}\\Omega \\int\nd\\tau \\_{1}d\\tau \\_{2}e^{-i\\omega ^{-}\\tau \\_{1}}e^{i\\left( \\omega ^{+}+\\Omega\n\\right) \\tau \\_{2}}\\sum\\_{\\sigma }\\left\\langle T\\_{\\tau }d\\_{\\sigma }^{\\dag\n}\\left( \\tau \\_{1}\\right) d\\_{\\sigma }\\left( \\tau \\_{2}\\right) N\\_{j^{\\prime\n}}\\left( 0\\right) \\right\\rangle \\_{C}+\\text{c.c.}\\right] \\text{;}$$ \ncombining the last two equations, we have\n \n$$\\mathcal{T}\\_{jj^{\\prime }}^{C}\\left( \\epsilon \\_{p}\\right) -\\frac{1}{V\\_{p}^{2}}\\left[ S\\left( p\\right) \\Gamma ^{\\dag }\\left( p\\right) \\lambda \\Gamma\n\\left( p\\right) S^{\\dag }\\left( p\\right) \\right] \\_{jj}\\sum\\_{j^{\\prime \\prime\n}}\\mathcal{T}\\_{j^{\\prime \\prime }j^{\\prime }}^{C}\\left( \\epsilon \\_{p}\\right)\n=0\\text{.} \\label{connelimT}$$ \nLet us now define *E*conn as the characteristic energy scale below which both *S*(*p*) and Γ(*p*) vary slowly. By definition $E\\_{\\text{conn}}\\lesssim E\\_{V}$; while *E*conn is not necessarily the same as *E**V*, for the two ABK ring geometries considered in this paper *E**V* ∼ *E*conn. For a mesoscopic structure with characteristic length scale *L*, *E*conn is usually the Thouless energy, *E*conn ∼ *v**F*/*L*; however, this is again not always the case, and the open long ABK ring in Sec. [sec:open] provides a counterexample where *E*conn is comparable to the bandwidth. Below *E*conn, the function [*S*(*p*)Γ†(*p*)*λ*Γ(*p*)*S*†(*p*)]*j**j*/*V**p*2 is only weakly dependent on *p*.\nEq. ([connelimT]) suggests that we can approximately eliminate the connected part of *G**j**j*′, provided the temperature is low compared to *E*conn. Consider the linear combination\n \n$$G\\_{jj^{\\prime }}-\\frac{1}{V\\_{k\\_{F}}^{2}}\\left[ S\\left( k\\_{F}\\right) \\Gamma\n^{\\dag }\\left( k\\_{F}\\right) \\lambda \\Gamma \\left( k\\_{F}\\right) S^{\\dag\n}\\left( k\\_{F}\\right) \\right] \\_{jj}\\sum\\_{j^{\\prime \\prime }}G\\_{j^{\\prime\n\\prime }j^{\\prime }}\\equiv G\\_{jj^{\\prime }}\\text{;}$$ \nthis corresponds to measuring the conductance by measuring the current in lead *j*, plus a constant times the total current in all leads. (Note that here we include both disconnected and connected contributions.) By Kirchhoff’s law, this linear combination must equal *G**j**j*′ itself. We write it as a sum of disconnected and connected contributions:\n \n$$\\begin{aligned}\nG\\_{jj^{\\prime }} &=&\\left( G\\_{jj^{\\prime }}^{D}-\\frac{1}{V\\_{k\\_{F}}^{2}}\\left[\nS\\left( k\\_{F}\\right) \\Gamma ^{\\dag }\\left( k\\_{F}\\right) \\lambda \\Gamma\n\\left( k\\_{F}\\right) S^{\\dag }\\left( k\\_{F}\\right) \\right] \\_{jj}\\sum\\_{j^{\\prime \\prime }}G\\_{j^{\\prime \\prime }j^{\\prime }}^{D}\\right) \\notag \\\\\n&&+\\int d\\epsilon \\_{p}\\left[ -f^{\\prime }\\left( \\epsilon \\_{p}\\right) \\right]\n\\left\\{ \\mathcal{T}\\_{jj^{\\prime }}^{C}\\left( \\epsilon \\_{p}\\right) -\\frac{1}{V\\_{k\\_{F}}^{2}}\\left[ S\\left( k\\_{F}\\right) \\Gamma ^{\\dag }\\left( k\\_{F}\\right)\n\\lambda \\Gamma \\left( k\\_{F}\\right) S^{\\dag }\\left( k\\_{F}\\right) \\right]\n\\_{jj}\\sum\\_{j^{\\prime \\prime }}\\mathcal{T}\\_{j^{\\prime \\prime }j^{\\prime\n}}^{C}\\left( \\epsilon \\_{p}\\right) \\right\\} \\text{.}\\end{aligned}$$ \nFor *T* ≪ *E*conn, by Eq. ([connelimT]), the quantity in curly brackets approximately vanishes for $\\left\\vert \\epsilon \\_{p}-\\epsilon\n\\_{k\\_{F}}\\right\\vert \\lesssim T$, whereas the Fermi factor approximately vanishes for $\\left\\vert \\epsilon \\_{p}-\\epsilon \\_{k\\_{F}}\\right\\vert \\gg T$. Therefore\n \n$$G\\_{jj^{\\prime }}\\approx G\\_{jj^{\\prime }}^{D}-\\frac{1}{V\\_{k\\_{F}}^{2}}\\left[\nS\\left( k\\_{F}\\right) \\Gamma ^{\\dag }\\left( k\\_{F}\\right) \\lambda \\Gamma\n\\left( k\\_{F}\\right) S^{\\dag }\\left( k\\_{F}\\right) \\right] \\_{jj}\\sum\\_{j^{\\prime \\prime }}G\\_{j^{\\prime \\prime }j^{\\prime }}^{D}\\text{;}\n\\label{connelimG}$$ \nin other words, at *T* ≪ *E*conn it is possible to write the conductance in terms of disconnected contributions alone.\nSince Eq. ([connelimG]) contains only the disconnected contribution, we may calculate it explicitly using Eqs. ([discprobKYA]) and ([TDrowsum]). Since both *S*(*p*) and Γ(*p*) are slowly varying below the energy scale *E*conn, we find the conductance is approximately linear in the T-matrix,\n \n$$\\begin{aligned}\nG\\_{jj^{\\prime }} &\\approx &\\frac{2e^{2}}{h}\\int d\\epsilon \\_{p}\\left[\n-f^{\\prime }\\left( \\epsilon \\_{p}\\right) \\right] \\left\\{ \\mathcal{T}\\_{0,jj^{\\prime }}\\left( \\epsilon \\_{k\\_{F}}\\right) +\\mathcal{Z}\\_{R,jj^{\\prime\n}}\\left( \\epsilon \\_{k\\_{F}}\\right) \\operatorname{Re}\\left[ -\\pi \\nu \\_{p}\\text{T}\\_{pp}\\left( \\epsilon \\_{p}\\right) \\right] \\right. \\notag \\\\\n&&\\left. +\\left[ \\mathcal{Z}\\_{I,jj^{\\prime }}\\left( \\epsilon \\_{k\\_{F}}\\right)\n+\\mathcal{Z}\\_{2,jj^{\\prime }}\\left( \\epsilon \\_{k\\_{F}}\\right) \\right] \\operatorname{Im}\\left[ -\\pi \\nu \\_{p}\\text{T}\\_{pp}\\left( \\epsilon \\_{p}\\right) \\right]\n\\right\\} \\text{,} \\label{connelimG1}\\end{aligned}$$ \nprovided *T* ≪ *E*conn. Eq. ([connelimG1]) can also be obtained by eliminating the connected part with the column sum Eq. ([TDcolsum]) instead of the row sum,\n \n$$G\\_{jj^{\\prime }}-\\frac{1}{V\\_{k\\_{F}}^{2}}\\left[ \\Gamma ^{\\dag }\\left(\nk\\_{F}\\right) \\lambda \\Gamma \\left( k\\_{F}\\right) \\right] \\_{j^{\\prime\n}j^{\\prime }}\\sum\\_{j^{\\prime \\prime }}G\\_{jj^{\\prime \\prime }}\\text{,}$$ \nwhich corresponds to measuring the conductance by applying a small uniform bias voltage in all leads, in addition to the small bias voltage in lead *j*′.\nEq. ([connelimG1]) is the first central result of this paper. It generalizes the result of the two-lead short ABK ring in Ref. to an arbitrary ABK ring, and expresses the linear DC conductance as a linear function of the scattering channel T-matrix, as long as the temperature is low compared to the mesoscopic energy scale *E*conn at which *S*(*p*) and Γ(*p*) varies significantly.\nPerturbation theories[sec:pert]\n===============================\nWeak-coupling perturbation theory\n---------------------------------\nAlthough we now understand that the connected part of the conductance can be eliminated at low temperatures, this procedure may not be applicable in the weak-coupling regime *T* ≫ *T**K*. In this subsection we calculate the linear DC conductance perturbatively in powers of *V**k*2/*U*, again generalizing the short ring results of Ref. ; we expect the result to be valid in both small and large Kondo cloud regimes as long as *T* ≫ *T**K* and the renormalized Kondo coupling constant remains weak.\n### Disconnected part\nWe first find the disconnected part; the result is already given in Ref. , but for completeness we reproduce it here. As implied by Eq. ([discprob]), our task amounts to calculating the retarded T-matrix of the screening channel in the single-particle sector, which is in turn achieved by calculating the two-point Green’s function − ⟨*T**τ**ψ**k*(*τ*)*ψ**k*′†(0)⟩. The pertinent Feynman diagrams to *O*(*J*2) and *O*(*K*2) are depicted in Fig. [fig:KondoFD], and we find\n \n$$\\nu \\text{T}\\_{kk^{\\prime }}\\left( \\Omega \\right) =\\nu K\\_{kk^{\\prime }}+\\nu\n^{2}\\int d\\epsilon \\_{q}\\frac{1}{\\Omega ^{+}-\\epsilon \\_{q}}\\left(\nK\\_{kq}K\\_{qk^{\\prime }}+\\frac{3}{16}J\\_{kq}J\\_{qk^{\\prime }}\\right) \\text{,}\n\\label{TmatKondo}$$ \nwhere again *ν* = 1/(2*π**v**F*) for the model with a reduced band. The factor of 3/16 results from time-ordering and tracing over the impurity spin, where we have used the following identity\n \n$$\\left\\langle T\\_{\\tau }S\\_{d}^{a}\\left( \\tau \\_{1}\\right) S\\_{d}^{b}\\left( \\tau\n\\_{2}\\right) \\right\\rangle =\\frac{1}{4}\\delta ^{ab}\\text{.}\n\\label{timeorderSd2}$$ \nDiagrammatics of weak-coupling perturbation theory. a) The vertices corresponding to the Kondo coupling and the potential scattering in Eq. ([Kondomodel]). b) Diagrams contributing to the T-matrix of the screening channel *ψ* electrons up to *O*(*J*2) ∼ *O*(*K*2). We have traced over the impurity spin so that the double dashed lines (impurity spin propagators) form loops, and arranged the internal time variables from left to right in increasing order. c) Connected diagrams contributing to the linear DC conductance up to *O*(*J*2).[fig:KondoFD]\nThe *O*(*K*) and *O*(*K*2) terms, accounting for the particle-hole symmetry breaking potential scattering due to the QD, clearly obey the optical theorem $\\operatorname{Im}\\left[ -\\pi \\nu \\_{p}\\text{T}\\_{pp}\\left( \\epsilon \\_{p}\\right) \\right] =\\left\\vert -\\pi \\nu \\_{p}\\text{T}\\_{pp}\\left( \\epsilon \\_{p}\\right) \\right\\vert ^{2}$. If *κ* is comparable to the renormalized value of *j* then the *O*(*K*) term dominates the T-matrix.\nOn the other hand, if we tune the QD to be particle-hole symmetric, *ε**d* = − *U*/2 and *κ* = 0, both terms containing *K* will vanish, and the *O*(*J*2) term becomes the lowest order contribution to the T-matrix. For this term, one should also make a distinction between the real principal value part and the imaginary *δ*-function part. As noticed in Ref. and reiterated in Ref. , the principal value part introduces non-universalities due to its dependence on all energies in the reduced band ( − *D*, *D*); nevertheless it is merely an elastic potential scattering term, and we neglect it in the following. Meanwhile, the *δ*-function part is an inelastic effect stemming from the Kondo physics, as can be seen from its violation of the optical theorem. (The T-matrix apparently disobeys the optical theorem because it is restricted to the single-particle sector, and the sum over intermediate states excludes many-particle states.) Therefore, for a particle-hole symmetric QD, to *O*(*J*2) we have\n \n$$-\\pi \\nu \\text{T}\\_{pp}\\left( \\epsilon \\_{p}\\right) =i\\frac{3\\pi ^{2}}{16}\\nu\n^{2}J\\_{pp}^{2}\\text{.} \\label{TmatKondosym}$$ \nThe weak-coupling perturbation theory is famous for being infrared divergent at *O*(*J*3), but as long as *T* ≫ *T**K*, to logarithmic accuracy we can verify that the *O*(*J*3) corrections to the T-matrix can be absorbed into our *O*(*J*2) result by reinterpreting the bare Kondo coupling constant *J**p**p* as a renormalized one. The renormalization is governed by Eq. ([KondoJkkprRG]), and cut off at either the electron energy $\\left\\vert \\epsilon \\_{p}\\right\\vert $ or the temperature *T*, whichever is larger. In other words, the Kondo coupling *J**p**p* in Eq. ([TmatKondosym]) should be replaced by $J\\_{pp}\\left(\n\\max \\left\\{ \\left\\vert \\epsilon \\_{p}\\right\\vert,T\\right\\} \\right) $, where the argument in round brackets stands for the energy cutoff *D* in Eq. ([KondoJkkprRG]) where the running coupling constant is evaluated.\n### Connected part\nWe now calculate the connected part to *O*(*J*2); the calculation follows Ref. closely. Inserting Eq. ([leaddensity]) into Eq. ([Gjjpriomegap]), we write the connected part of the density-density correlation function in terms of a four-point correlation function of *ψ*:\n \n$$\\mathcal{G}\\_{jj^{\\prime }}^{\\prime C}\\left( i\\omega \\_{p}\\right)\n=\\int\\_{0}^{\\pi }\\frac{dk\\_{1}dk\\_{2}dq\\_{1}dq\\_{2}}{\\left( 2\\pi \\right) ^{4}}\\left( \\mathbb{M}\\_{k\\_{1}k\\_{2}}^{j}\\right) \\_{11}\\left( \\mathbb{M}\\_{q\\_{1}q\\_{2}}^{j^{\\prime }}\\right) \\_{11}\\mathcal{G}\\_{k\\_{1}k\\_{2}q\\_{1}q\\_{2}}^{C}\\left( i\\omega \\_{p}\\right) \\text{,}\n\\label{GprCimag}$$ \nwhere\n \nG*k*1*k*2*q*1*q*2*C*(*i**ω**p*) ≡ − ∫0*β**d**τ**e**i**ω**p**τ*∑*σ**σ*′⟨*T**τ**ψ**k*1*σ*†(*τ*)*ψ**k*2*σ*(*τ*)*ψ**q*1*σ*′†(0)*ψ**q*2*σ*′(0)⟩*C*. \nWe insert Eqs. ([bbUmatrow1]) and ([bbMmat]) into Eq. ([GprCimag]), and take the continuum limit, which is appropriate for the Kondo model. Because in the wide band limit the most divergent contribution to *G**j**j*′′*C*(Ω) is from *k*1 ≈ *k*2 and *q*1 ≈ *q*2, we can expand the integrand around these points,\n \n$$\\begin{aligned}\n& \\mathcal{G}\\_{jj^{\\prime }}^{\\prime C}\\left( i\\omega \\_{p}\\right) \\notag \\\\\n& =O\\left( 1\\right) +\\int \\frac{dk\\_{1}dk\\_{2}dq\\_{1}dq\\_{2}}{\\left( 2\\pi\n\\right) ^{4}}\\frac{1}{V\\_{k\\_{1}}V\\_{k\\_{2}}V\\_{q\\_{1}}V\\_{q\\_{2}}}\\mathcal{G}\\_{k\\_{1}k\\_{2}q\\_{1}q\\_{2}}^{C}\\left( i\\omega \\_{p}\\right)\n\\sum\\_{r\\_{1}r\\_{2}r\\_{1}^{\\prime }r\\_{2}^{\\prime }}t\\_{r\\_{1}}^{\\ast\n}t\\_{r\\_{2}}t\\_{r\\_{1}^{\\prime }}^{\\ast }t\\_{r\\_{2}^{\\prime }} \\notag \\\\\n& \\times \\left\\{ \\Gamma \\_{r\\_{1}j}\\left( k\\_{1}\\right) \\Gamma \\_{r\\_{2}j}^{\\ast\n}\\left( k\\_{2}\\right) \\frac{1}{-i\\left( k\\_{1}-k\\_{2}+i0\\right) }+\\left[ \\Gamma\nS^{\\dag }\\left( k\\_{1}\\right) \\right] \\_{r\\_{1}j}\\left[ \\Gamma S^{\\dag }\\left(\nk\\_{2}\\right) \\right] \\_{r\\_{2}j}^{\\ast }\\frac{1}{-i\\left(\nk\\_{2}-k\\_{1}+i0\\right) }\\right\\} \\notag \\\\\n& \\times \\left\\{ \\Gamma \\_{r\\_{1}^{\\prime }j^{\\prime }}\\left( q\\_{1}\\right)\n\\Gamma \\_{r\\_{2}^{\\prime }j^{\\prime }}^{\\ast }\\left( q\\_{2}\\right) \\frac{1}{-i\\left( q\\_{1}-q\\_{2}+i0\\right) }+\\left[ \\Gamma S^{\\dag }\\left( q\\_{1}\\right) \\right] \\_{r\\_{1}^{\\prime }j^{\\prime }}\\left[ \\Gamma S^{\\dag }\\left(\nq\\_{2}\\right) \\right] \\_{r\\_{2}^{\\prime }j^{\\prime }}^{\\ast }\\frac{1}{-i\\left(\nq\\_{2}-q\\_{1}+i0\\right) }\\right\\} \\text{.} \\label{GprCimag1}\\end{aligned}$$ \nThe only non-vanishing diagrams at *O*(*J*2) are shown in panel c) of Fig. [fig:KondoFD]:\n \n$$\\begin{aligned}\n\\mathcal{G}\\_{k\\_{1}k\\_{2}q\\_{1}q\\_{2}}^{C}\\left( i\\omega \\_{p}\\right) &=&\\frac{3}{8}J\\_{q\\_{2}k\\_{1}}J\\_{k\\_{2}q\\_{1}}\\int\\_{0}^{\\beta }d\\tau e^{i\\omega \\_{p}\\tau\n}\\int\\_{0}^{\\beta }d\\tau \\_{1}d\\tau \\_{2}g\\_{q\\_{2}}\\left( -\\tau \\_{1}\\right)\ng\\_{k\\_{1}}\\left( \\tau \\_{1}-\\tau \\right) g\\_{k\\_{2}}\\left( \\tau -\\tau\n\\_{2}\\right) g\\_{q\\_{1}}\\left( \\tau \\_{2}\\right) \\notag \\\\\n&=&\\frac{3}{8}J\\_{q\\_{2}k\\_{1}}J\\_{k\\_{2}q\\_{1}}\\frac{1}{\\beta }\\sum\\_{\\omega\n\\_{n\\_{1}}}g\\_{q\\_{2}}\\left( i\\omega \\_{n\\_{1}}\\right) g\\_{k\\_{1}}\\left( i\\omega\n\\_{n\\_{1}}\\right) g\\_{k\\_{2}}\\left( i\\omega \\_{n\\_{1}}+i\\omega \\_{p}\\right)\ng\\_{q\\_{1}}\\left( i\\omega \\_{n\\_{1}}+i\\omega \\_{p}\\right) \\text{;}\\end{aligned}$$ \nwe have again used Eq. ([timeorderSd2]). The frequency summation is performed by deforming the complex plane contour and wrapping it around the lines Im*z* = 0 and Im*z* = − *ω**p*. Analytic continuation yields\n \n$$\\begin{aligned}\nG\\_{k\\_{1}k\\_{2}q\\_{1}q\\_{2}}^{C}\\left( \\Omega \\right) & =-\\frac{3J\\_{q\\_{2}k\\_{1}}J\\_{k\\_{2}q\\_{1}}}{16\\pi i}\\int d\\omega f\\left( \\omega \\right)\n\\left\\{ \\left[ g\\_{q\\_{2}}^{R}\\left( \\omega \\right) g\\_{k\\_{1}}^{R}\\left( \\omega\n\\right) -g\\_{q\\_{2}}^{A}\\left( \\omega \\right) g\\_{k\\_{1}}^{A}\\left( \\omega\n\\right) \\right] g\\_{k\\_{2}}^{R}\\left( \\omega +\\Omega \\right)\ng\\_{q\\_{1}}^{R}\\left( \\omega +\\Omega \\right) \\right. \\notag \\\\\n& \\left. +g\\_{q\\_{2}}^{A}\\left( \\omega -\\Omega \\right) g\\_{k\\_{1}}^{A}\\left(\n\\omega -\\Omega \\right) \\left[ g\\_{k\\_{2}}^{R}\\left( \\omega \\right)\ng\\_{q\\_{1}}^{R}\\left( \\omega \\right) -g\\_{k\\_{2}}^{A}\\left( \\omega \\right)\ng\\_{q\\_{1}}^{A}\\left( \\omega \\right) \\right] \\right\\} \\text{.}\n\\label{GCkkqqretKondo2}\\end{aligned}$$ \nSubstituting Eqs. ([GCkkqqretKondo2]) and ([KondoJ]) into Eq. ([GprCimag1]), we are able to evaluate the momentum integrals in the Λ → ∞ limit by contour methods. The *R**R**R**R* and *A**A**A**A* terms vanish, and the *A**A**R**R* terms combine to produce a Fermi surface factor *f*′(*ω*):\n \n$$\\begin{aligned}\n& G\\_{jj^{\\prime }}^{\\prime C}\\left( \\Omega \\right) \\notag \\\\\n& =O\\left( 1\\right) -\\frac{1}{\\Omega }\\int d\\omega \\left[ -f^{\\prime }\\left(\n\\omega \\right) \\right] \\frac{3\\left( 2j\\right) ^{2}}{16\\pi i}\\sum\\_{r\\_{1}r\\_{2}r\\_{1}^{\\prime }r\\_{2}^{\\prime }}t\\_{r\\_{1}}^{\\ast\n}t\\_{r\\_{2}}t\\_{r\\_{1}^{\\prime }}^{\\ast }t\\_{r\\_{2}^{\\prime }}\\left[ \\Gamma\nS^{\\dag }\\left( k\\_{F}+\\frac{\\omega }{v\\_{F}}\\right) \\right] \\_{r\\_{1}j} \\notag\n\\\\\n& \\times \\left[ \\Gamma S^{\\dag }\\left( k\\_{F}+\\frac{\\omega +\\Omega }{v\\_{F}}\\right) \\right] \\_{r\\_{2}j}^{\\ast }\\Gamma \\_{r\\_{1}^{\\prime }j^{\\prime }}\\left(\nk\\_{F}+\\frac{\\omega +\\Omega }{v\\_{F}}\\right) \\Gamma \\_{r\\_{2}^{\\prime }j^{\\prime\n}}^{\\ast }\\left( k\\_{F}+\\frac{\\omega }{v\\_{F}}\\right) \\frac{1}{v\\_{F}^{2}} \n\\notag \\\\\n& =O\\left( 1\\right) +\\frac{1}{i\\pi \\Omega }\\int d\\epsilon \\_{p}\\left[\n-f^{\\prime }\\left( \\epsilon \\_{p}\\right) \\right] \\frac{3\\pi ^{2}\\nu\n^{2}J\\_{pp}^{2}}{16}\\mathcal{Z}\\_{2,jj^{\\prime }}\\left( \\epsilon \\_{p}\\right) \n\\text{,}\\end{aligned}$$ \nwhere we used Eq. ([discprobKYAZ2]). The connected contribution to the conductance is now clearly a Fermi surface property:\n \n$$G\\_{jj^{\\prime }}^{C}=\\frac{2e^{2}}{h}\\int d\\epsilon \\_{p}\\left[ -f^{\\prime\n}\\left( \\epsilon \\_{p}\\right) \\right] \\mathcal{T}\\_{jj^{\\prime }}^{C\\left(\n2\\right) }\\left( \\epsilon \\_{p}\\right) \\text{,}$$ \nwhere the connected transmission probability is\n \n$$\\mathcal{T}\\_{jj^{\\prime }}^{C}\\left( \\epsilon \\_{p}\\right) =\\mathcal{Z}\\_{2,jj^{\\prime }}\\left( \\epsilon \\_{p}\\right) \\frac{3\\pi ^{2}}{16}\\nu\n^{2}J\\_{pp}^{2}\\text{.} \\label{TC2Kondo}$$ \nThis is formally identical to the short ring result, and is of the same order of magnitude [*O*(*J*2)] as the disconnected contribution for a particle-hole symmetric QD. In fact, if leads *j* and *j*′ are not directly coupled to each other (i.e. they become decoupled when their couplings with the QD are turned off; the simplest example is a QD embedded between source and drain leads), we have *S**j**j*′ = 0, and the disconnected contribution for a particle-hole symmetric QD is *O*(*J*4). In this case the *O*(*J*2) connected contribution dominates.\nJust as with the T-matrix, when we calculate the connected part further to *O*(*J*3) to logarithmic accuracy, the result can be absorbed into Eq. ([TC2Kondo]) if the coupling constant *J* is understood as fully renormalized according to Eq. ([KondoJkkprRG]), with its renormalization cut off by $\\left\\vert \\epsilon \\_{p}\\right\\vert $ or *T*.\n### Total conductance\nWe write the total conductance at *T* ≫ *T**K* as a background term and a correction due to the QD:\n \n$$G\\_{jj^{\\prime }}=\\frac{2e^{2}}{h}\\int d\\epsilon \\_{p}\\left[ -f^{\\prime\n}\\left( \\epsilon \\_{p}\\right) \\right] \\left[ \\mathcal{T}\\_{0,jj^{\\prime\n}}\\left( \\epsilon \\_{p}\\right) +\\delta \\mathcal{T}\\_{jj^{\\prime }}\\left(\n\\epsilon \\_{p}\\right) \\right] \\text{.} \\label{GjjprKondo}$$ \nIf the QD is well away from particle-hole symmetry, *κ* can be of the same order of magnitude as *j*(*T*) even when the latter is fully renormalized to the given temperature. In this case, the *T* ≫ *T**K*correction to the background conductance will be dominated by the potential scattering term; the connected contribution is negligible. The expression for *δ*T is\n \n*δ*T*j**j*′(*ε**p*) ≈ − Z*R*, *j**j*′(*ε**p*)*π**ν**K**p**p*. \nIf, however, the QD is particle-hole symmetric, the connected contribution becomes important. Inserting Eq. ([TmatKondosym]) into Eq. ([discprobKYA]) and combining with ([TC2Kondo]), we find the Kondo-type correction to *O*(*J*2) at *T* ≫ *T**K*\n \n$$\\delta \\mathcal{T}\\_{jj^{\\prime }}\\left( \\epsilon \\_{p}\\right) =\\left[ \n\\mathcal{Z}\\_{I,jj^{\\prime }}\\left( \\epsilon \\_{p}\\right) +\\mathcal{Z}\\_{2,jj^{\\prime }}\\left( \\epsilon \\_{p}\\right) \\right] \\frac{3\\pi ^{2}}{16}\\nu\n^{2}J\\_{pp}^{2}\\text{.} \\label{deltaTjjprKondo}$$ \nAgain, in the RG improved perturbation theory, *J**p**p* in Eq. ([deltaTjjprKondo]) should be replaced by $J\\_{pp}\\left( \\max \\left(\n\\left\\vert \\epsilon \\_{p}\\right\\vert,T\\right) \\right) $, indicating that *J**p**p* is the renormalized value at the running energy cutoff $\\max \\left(\n\\left\\vert \\epsilon \\_{p}\\right\\vert,T\\right) $. This expression is valid as long as *T* ≫ *T**K*, irrespective of whether the system is in small or large Kondo cloud regime.\nEq. ([deltaTjjprKondo]) is formally similar to the previously obtained short ring result. It should be noted, however, that the energy dependence of Z*I*, Z2 and *J*2 is possibly much stronger than the short ring case, and the thermal averaging in Eq. ([GjjprKondo]) can lead to very different results in small and large Kondo cloud regimes. For instance, if *E*conn ≪ *T**K* ≪ *T* (which may happen in the small cloud regime), the Fermi factor in Eq. ([GjjprKondo]) averages over many peaks in T, Z*I*, Z2 and *V*2 that are associated with the underlying mesoscopic structure. In this case connected part elimination is not applicable. On the other hand, if *T**K* ≪ *T* ≪ *E*conn, the variation of T, Z*I*, Z2 or *V*2 is negligible on the scale of *T*, and the Fermi factor in Eq. ([GjjprKondo]) may be approximated by a *δ* function. This leads to\n \n$$G\\_{jj^{\\prime }}=\\frac{2e^{2}}{h}\\left\\{ \\mathcal{T}\\_{0,jj^{\\prime }}\\left(\n\\epsilon \\_{k\\_{F}}\\right) +\\left[ \\mathcal{Z}\\_{I,jj^{\\prime }}\\left( \\epsilon\n\\_{k\\_{F}}\\right) +\\mathcal{Z}\\_{2,jj^{\\prime }}\\left( \\epsilon \\_{k\\_{F}}\\right) \\right] \\frac{3\\pi ^{2}}{16}\\nu ^{2}J\\_{k\\_{F}k\\_{F}}^{2}\\right\\} \\text{,}$$ \nwhich agrees with our prescription of eliminating the connected part, Eq. ([connelimG1]).\nFermi liquid perturbation theory\n--------------------------------\nIt is also interesting to consider temperatures low compared to the Kondo temperature *T* ≪ *T**K*. Since our formalism does not by itself provide a low-energy effective theory of the small cloud regime for *T**K* ≫ *E**V*, we focus on the very large Kondo cloud regime *T**K* ≪ *E**V*, where as explained in Sec. [sec:model] the low-energy effective theory is simply a Fermi liquid theory. If we further assume *T* ≪ *E*conn, then we can simply eliminate the connected contribution to the conductance with Eq. ([connelimG1]).\nTo use Eq. ([connelimG1]) we need the low-energy T-matrix for the screening channel in the single-particle sector in the Fermi liquid regime, which is again well known. As mentioned in Sec. [sec:intro], the strong-coupling single particle wave function at zero temperature is obtained by imposing a phase shift on the weak-coupling wave function. This phase shift *δ**ψ**ψ* results from both elastic scattering off the Kondo singlet and particle-hole symmetry breaking potential scattering:\n \n$$\\delta \\_{\\psi \\psi,\\sigma }=\\sigma \\frac{\\pi }{2}+\\delta \\_{P}\\text{,}\n\\label{deltapsipsi}$$ \nwhere *σ* = ± 1 for spin-up/spin-down electrons. To the lowest order in potential scattering *O*(*K*), we have\n \ntan*δ**P* = − *π**ν**K**k**F**k**F*. \nLet us introduce the phase-shifted screening channel *ψ̃*, which is then related to the original screening channel *ψ* via a scattering basis transformation:\n \n$$\\psi \\_{k,\\sigma }=\\int\\_{k\\_{F}-\\Lambda }^{k\\_{F}+\\Lambda }\\frac{dp}{2\\pi }\\left( \\frac{i}{k-p+i0}-e^{2i\\delta \\_{\\psi \\psi,\\sigma }}\\frac{i}{k-p-i0}\\right) \\tilde{\\psi}\\_{p,\\sigma }\\text{.} \\label{psitildescatt}$$ \nUsing the definition of the T-matrix in the single-particle sector Eq. ([Dyson]) and the transformation Eq. ([psitildescatt]) one can show that retarded T-matrices for *ψ* and *ψ̃* are related by\n \n$$\\text{T}\\_{kk^{\\prime },\\sigma }\\left( \\omega \\right) =\\frac{i}{2\\pi \\nu }\\left( e^{2i\\delta \\_{\\psi \\psi,\\sigma }}-1\\right) +e^{2i\\delta \\_{\\psi \\psi\n,\\sigma }}\\text{\\~{T}}\\_{kk^{\\prime },\\sigma }\\left( \\omega \\right) \\text{.}\n\\label{TvsTtilde}$$ \nSince *ψ̃* diagonalizes the strong-coupling fixed point Hamiltonian, by definition T*k**F**k**F*, *σ*(*ω* = 0) = 0 at zero temperature.\nThe leading irrelevant operator perturbing the strong-coupling fixed point is localized at the QD (with a spatial extent of *v**F*/*T**K*), and quadratic in spin current. It is most conveniently written in terms of *ψ̃*:\n \n$$\\begin{aligned}\nH\\_{int} &=&\\frac{2\\pi v\\_{F}^{2}}{T\\_{K}}\\colon \\tilde{\\psi}\\_{\\alpha }^{\\dag }\\tilde{\\psi}\\_{\\alpha }\\tilde{\\psi}\\_{\\beta }^{\\dag }\\tilde{\\psi}\\_{\\beta\n}\\colon \\left( x=0\\right) \\notag \\\\\n&&-\\frac{v\\_{F}^{2}}{T\\_{K}}\\left[ \\colon \\tilde{\\psi}\\_{\\alpha }^{\\dag }\\left(\ni\\frac{d}{dx}-k\\_{F}\\right) \\tilde{\\psi}\\_{\\alpha }+\\left( -i\\frac{d}{dx}-k\\_{F}\\right) \\tilde{\\psi}\\_{\\alpha }^{\\dag }\\tilde{\\psi}\\_{\\alpha }\\colon \\right] \\left( x=0\\right) \\text{.} \\label{FLLIO}\\end{aligned}$$ \nHere : : denotes normal ordering, and sums over repeated spin indices *α* and *β* are implied. The *ψ̃* operators has been unfolded, so that their wave functions are now defined on the entire real axis instead of the positive real axis. The two terms in *H**i**n**t* are illustrated in panel a) of Fig. [fig:FermiFD1] as a four-point vertex and a two-point one. Both terms share a single coupling constant of *O*(1/*T**K*), because the leading irrelevant operator written in the *ψ̃* basis must be particle-hole symmetric by definition. The on-shell retarded T-matrix for *ψ̃* in the single-particle sector is calculated to *O*(1/*T**K*2) in Ref. :\n \n$$-\\pi \\nu \\text{\\~{T}}\\_{pp}\\left( \\epsilon \\_{p}\\right) =\\frac{\\epsilon \\_{p}}{T\\_{K}}+i\\frac{3\\epsilon \\_{p}^{2}+\\pi ^{2}T^{2}}{2T\\_{K}^{2}}\\text{.}\n\\label{TtildeAL}$$ \nFor completeness we give a derivation of this result in Appendix [sec:appFermi]. It is diagramatically represented by Fig. [fig:FermiFD1] panel b).\nDiagrammatics of Fermi liquid perturbation theory. a) The two vertices given by the leading irrelevant operator, Eq. ([FLLIO]). b) Diagrams contributing to the T-matrix of $\\tilde\\psi$ electrons up to *O*(1/*T**K*2). The propagators are those of the phase-shifted screening channel operators $\\tilde\\psi$.[fig:FermiFD1]\nSubstituting Eqs. ([deltapsipsi]), ([TvsTtilde]) and ([TtildeAL]) into Eq. ([connelimG1]), we eliminate the connected contribution, and obtain the *T* ≪ *T**K* conductance in the very large Kondo cloud regime:\n \n$$\\begin{aligned}\nG\\_{jj^{\\prime }} &\\approx &\\frac{2e^{2}}{h}\\left\\{ \\mathcal{T}\\_{0,jj^{\\prime\n}}\\left( \\epsilon \\_{k\\_{F}}\\right) -\\mathcal{Z}\\_{R,jj^{\\prime }}\\left(\n\\epsilon \\_{k\\_{F}}\\right) \\left( \\frac{1}{2}-\\frac{\\pi ^{2}T^{2}}{T\\_{K}^{2}}\\right) \\sin 2\\delta \\_{P}\\right. \\notag \\\\\n&&\\left. +\\left[ \\mathcal{Z}\\_{I,jj^{\\prime }}\\left( \\epsilon \\_{k\\_{F}}\\right)\n+\\mathcal{Z}\\_{2,jj^{\\prime }}\\left( \\epsilon \\_{k\\_{F}}\\right) \\right] \\left(\n\\cos ^{2}\\delta \\_{P}-\\frac{\\pi ^{2}T^{2}}{T\\_{K}^{2}}\\cos 2\\delta \\_{P}\\right)\n\\right\\} \\text{.} \\label{connelimG2}\\end{aligned}$$ \nWe note that the connected contribution can in fact be evaluated directly in the Fermi liquid theory. This is also done in Appendix [sec:appFermi], and provides further verification of our scheme of eliminating the connected contribution.\nPredictions of the conductance at high temperatures [Eq. ([GjjprKondo])] and at low temperatures [Eq. ([connelimG2])] together constitute the second main result of this paper. We emphasize once more that, while Eq. ([GjjprKondo]) is valid as long as *T* ≫ *T**K*, Eq. ([connelimG2]) is expected to be justified provided *T* ≪ *T**K* ≪ *E**V*, so that the Fermi liquid theory applies, and also *T* ≪ *E*conn, so that the connected contribution can be eliminated.\nFor clarity we tabulate various regimes of energy scales discussed so far (Table [tab:table1]). Note again that the connected contribution to conductance can be eliminated when *T* ≪ *E*conn. In general we have $E\\_{\\text{conn}}\\lesssim E\\_{V}$, but we assume in this table that *E*conn ∼ *E**V*, which is the case with the systems to be discussed in this paper.\n[b]\ncp4cmp4cmp4cm & & *T**K*& \n*T* ≫ *T**K* ≫ *E**V* & & & \n*T* ≫ *E**V* ≫ *T**K* & & & \n*E**V* ≫ *T* ≫ *T**K* & & & \ncp4cmp4cmp4cm & & *T**K*& \n*T* ≪ *T**K* ≪ *E**V* & & & \n*T* ≪ *E**V* ≪ *T**K* & & & \n*E**V* ≪ *T* ≪ *T**K* & & & \nClosed long ABK rings[sec:closed]\n=================================\nIn this section we apply our general formalism to the simplest model of a closed long ABK ring, studied in Ref. (Fig. [fig:longring1]): the QD is coupled directly to the source and drain leads, and a long reference arm connects the two leads smoothly. A weak link with hopping *t*′ splits the reference arm into two halves of equal length *d**r**e**f*/2 where *d**r**e**f* is an even integer. As opposed to Ref. , however, we use gauge invariance to assign the AB phase to the QD tunnel couplings rather than the weak link: *t*1 ≡ *t**L**e**i**φ*/2 and *t*2 ≡ *t**R**e*− *i**φ*/2. We assume no additional non-interacting long arms connecting the QD with the source and drain leads, because multiple traversal processes in such long QD arms will lead to interference effects independent of the AB phase, complicating the problem. The Hamiltonian representing this model takes the form\n \n$$\\begin{aligned}\nH\\_{0\\text{,junction}} &=&-t\\left[ \\left(\n\\sum\\_{n=1}^{d\\_{ref}/2-1}+\\sum\\_{n=d\\_{ref}/2+1}^{d\\_{ref}-1}\\right)\nc\\_{ref,n}^{\\dag }c\\_{ref,n+1}+\\text{h.c.}\\right] \\notag \\\\\n&&-t\\left( c\\_{ref,1}^{\\dag }c\\_{1,0}+c\\_{ref,d\\_{ref}}^{\\dag }c\\_{2,0}+\\text{h.c.}\\right) \\notag \\\\\n&&-t^{\\prime }\\left( c\\_{ref,d\\_{ref}/2}^{\\dag }c\\_{ref,d\\_{ref}/2+1}+\\text{h.c.}\\right) \\text{;}\\end{aligned}$$ \nthe coupling sites are *c**C*, *r* = 1 ≡ *c*1, 0 and *c**C*, *r* = 2 ≡ *c*2, 0.\nGeometry of the long ABK ring with short upper arms and a pinched reference arm.[fig:longring1]\nWe first repeat the Kondo temperature analysis in Ref. in order to distinguish between small and large Kondo cloud regimes, then carefully study the conductance at high and low temperatures, taking into account the previously neglected connected contribution.\nKondo temperature\n-----------------\nThe background S-matrix for this model is identical to the short ABK ring up to overall phases, due to the smooth connection between reference arm and leads:\n \n$$S\\left( k\\right) =e^{ikd\\_{ref}}\\left( \n\\begin{array}{cc}\n\\tilde{r}\\left( k\\right) & \\tilde{t}\\left( k\\right) \\\\ \n\\tilde{t}\\left( k\\right) & \\tilde{r}\\left( k\\right)\\end{array}\\right) \\text{;} \\label{SMatkYE}$$ \nwhere the S-matrix elements *r̃* and *t̃* for the weak link are\n \n$$\\tilde{r}\\left( k\\right) =-\\frac{1-\\tilde{\\tau}^{2}}{e^{-2ik}-\\tilde{\\tau}^{2}}\\text{, }\\tilde{t}\\left( k\\right) =-\\frac{2i\\tilde{\\tau}\\sin k}{e^{-2ik}-\\tilde{\\tau}^{2}}\\text{,}$$ \nand we introduce the shorthand *τ̃* = *t*′/*t*. The wave function is also straightforward to find:\n \nΓ*j**j*′(*k*) = *δ**j**j*′ + *S**j**j*′(*k*). \nIn the wide band limit, *r̃* and *t̃* are approximately independent of *k* in the reduced band *k**F* − Λ0 < *k* < *k**F* + Λ0 where the momentum cutoff Λ0 ≪ 1. This allows us to approximate them by their Fermi surface values, $\\tilde{r}\\left( k\\right)\n\\approx \\tilde{r}=\\left\\vert \\tilde{r}\\right\\vert e^{i\\theta }$ and $\\tilde{t}\\left( k\\right) \\approx \\tilde{t}=\\pm i\\left\\vert \\tilde{t}\\right\\vert\ne^{i\\theta }$ (the ± *π*/2 phase difference is required by unitarity of *S*); without loss of generality we focus on the $\\tilde{t}=i\\left\\vert \n\\tilde{t}\\right\\vert e^{i\\theta }$ case.\nFrom Eq. ([GammakYE]) one conveniently obtains the normalization factor\n \n$$\\begin{aligned}\nV\\_{k}^{2} &=&2\\left( t\\_{L}^{2}+t\\_{R}^{2}\\right) \\left[ 1+\\left\\vert \\tilde{r}\\right\\vert \\cos \\left( kd\\_{ref}+\\theta \\right) -\\gamma \\left\\vert \\tilde{t}\\right\\vert \\cos \\varphi \\sin \\left( kd\\_{ref}+\\theta \\right) \\right] \\notag\n\\\\\n&=&2\\left( t\\_{L}^{2}+t\\_{R}^{2}\\right) \\left[ 1+\\sqrt{1-\\left\\vert \\tilde{t}\\right\\vert ^{2}\\left( 1-\\gamma ^{2}\\cos ^{2}\\varphi \\right) }\\cos \\left(\nkd\\_{ref}+\\theta ^{\\prime }\\right) \\right] \\text{,} \\label{VksqYE}\\end{aligned}$$ \nwhere *γ* = 2*t**L**t**R*/(*t**L*2 + *t**R*2) measures the degree of symmetry of coupling to the QD. In the second line we have used $\\left\\vert \\tilde{t}\\right\\vert ^{2}+\\left\\vert \\tilde{r}\\right\\vert ^{2}=1$ and introduced another phase *θ*′, where *θ*′ − *θ* is a function of *γ*, $\\left\\vert \\tilde{t}\\right\\vert $ and *φ* but independent of *k*. We note that this expression is also applicable in the continuum limit, where the lattice constant *a* → 0 (we have previously set *a* = 1) but the arm length *d**r**e**f**a* is fixed. In that case *d**r**e**f* should be understood as the arm length *d**r**e**f**a*.\nFor long rings and filling factors not too small *k**F**d**r**e**f* ≫ 1, *V**k*2 oscillates around 2(*t**L*2 + *t**R*2) as a function of *k*, and has its extrema at *k**n* = (*n**π* − *θ*′)/*d**r**e**f* where *n* takes integer values. The only characteristic energy scale for *V**k*2 is therefore the peak/valley spacing Δ = *v**F**π*/*d**r**e**f*, and *E**V* ∼ *E*conn ∼ Δ. As in Ref. we define the reduced band such that Δ ≪ *D*0 ≡ *v**F*Λ0, and the reduced band initially contains many oscillations.\nIn the small Kondo cloud regime *T**K* ≫ Δ, one may assume the oscillations of *V**k*2 are smeared out when the energy cutoff is being reduced from *D*0, which is still well above *T**K*: $V\\_{k}^{2}\\simeq \n\\overline{V\\_{k}^{2}}=2\\left( t\\_{L}^{2}+t\\_{R}^{2}\\right) $. This means *T**K* in this regime is approximately the background Kondo temperature *T**K*0 defined in Eq. ([TKKondobulk]), independent of the position of the Fermi level at the energy scale Δ, and also independent of the magnetic flux.\nOn the other hand, in the large cloud regime $T\\_{K}\\lesssim \\Delta $, now that the Kondo temperature is largely determined by the value of *V**k*2 in a very narrow range of energies around the Fermi level, the mesoscopic *k* oscillations become much more important. When the running energy cutoff *D* is above the peak/valley spacing Δ, the renormalization of *j* is controlled by $\\overline{V\\_{k}^{2}}$ as in Eq. ([KondojRGsolsc]). Once *D* is reduced below Δ, we may approximate the renormalization of *j* as being dominated by *V**k**F*2. This leads to the following estimation of the Kondo temperature:\n \n$$T\\_{K}\\simeq \\Delta \\exp \\left[ -\\frac{1}{2V\\_{k\\_{F}}^{2}\\nu j\\left( \\Delta\n\\right) }\\right] =\\Delta \\left( \\frac{T\\_{K}^{0}}{\\Delta }\\right) ^{\\left[ 1+\\sqrt{1-\\left\\vert \\tilde{t}\\right\\vert ^{2}\\left( 1-\\gamma ^{2}\\cos\n^{2}\\varphi \\right) }\\cos \\left( k\\_{F}d\\_{ref}+\\theta ^{\\prime }\\right) \\right] ^{-1}}\\text{.}$$ \nIt is clear that *T**K* can be significantly dependent on the AB phase *φ* in this regime. In particular, *T**K* varies from $\\sim \\sqrt{T\\_{K}^{0}\\Delta }$ (on resonance) to practically 0 (off resonance) as *φ* is tuned between 0 and *π*, when the Fermi energy is located on a peak or in a valley *k**F* = *k**n*, the background transmission is perfect $\\left\\vert \\tilde{t}\\right\\vert =1$, and coupling to the QD is symmetric *γ* = 1; see Fig. [fig:longringTK]. (The special case *V**k**F*2 = 0 corresponds to a pseudogap problem *ν**V**k*2 ∝ (*k* − *k**F*)2, and the stable RG fixed point can be the local moment fixed point or the asymmetric strong coupling fixed point, depending on the degree of particle-hole symmetry.) As a general rule, stronger transmission through the pinch $\\left\\vert \\tilde{t}\\right\\vert $ and greater symmetry of coupling *γ* result in stronger interference between the two tunneling paths through the device, and hence increases the tunability of the Kondo temperature by the magnetic flux.\nKondo temperature *T**K* for the closed long ABK ring, calculated by numerical integration of the weak coupling RG equation Eq. ([KondojRG]), plotted against the AB phase *φ*. *T**K*(*φ*) is an even function of *φ* and has a period of 2*π*, so only 0 ≤ *φ* ≤ *π* is shown. System parameters are: *d**r**e**f* = 60, *θ* = *π*/2, ∣*r̃*∣ = 0, *t**L* = *t**R*, *D*0 = 10. The curves with *T**K* ≫ Δ (small Kondo cloud regime) have a large bare Kondo coupling (*t**L*2 + *t**R*2)*j*0/*π* = 0.15, whereas the curves with *T**K* ≪ Δ (large Kondo cloud regime) have a much smaller bare Kondo coupling (*t**L*2 + *t**R*2)*j*0/*π* = 0.02. In the small cloud regime *T**K* is almost independent of *φ* and *k**F*, as the curves are flat and overlapping with each other. In the large cloud regime, however, *T**K* highly sensitive to both *φ* and *k**F*.[fig:longringTK]\nHigh-temperature conductance\n----------------------------\nWe now calculate the conductance at *T* ≫ *T**K* by perturbation theory. Following the discussion in Ref. , we consider the case of a particle-hole symmetric QD *κ* = 0 and *K**k**k*′ = 0, and also ignore the elastic real part of the potential scattering generated at *O*(*J*2). These assumptions allow us to adopt Eq. ([deltaTjjprKondo]) for the *O*(*J*2) correction to the transmission probability:\n \n$$\\delta \\mathcal{T}\\_{jj^{\\prime }}\\left( \\epsilon \\_{k}\\right) =3\\pi ^{2}\\nu\n^{2}j^{2}\\left( V\\_{k}^{2}\\operatorname{Re}\\left\\{ \\left[ S\\left( k\\right) \\Gamma\n^{\\dag }\\left( k\\right) \\lambda \\Gamma \\left( k\\right) \\right] \\_{jj^{\\prime\n}}S\\_{jj^{\\prime }}^{\\ast }\\left( k\\right) \\right\\} -\\left\\vert \\left[\nS\\left( k\\right) \\Gamma ^{\\dag }\\left( k\\right) \\lambda \\Gamma \\left(\nk\\right) \\right] \\_{jj^{\\prime }}\\right\\vert ^{2}\\right) \\text{,}\n\\label{deltaTht}$$ \nwhere we have used Eqs. ([discprobKYAZI]) and ([discprobKYAZ2]).\nNote that Eq. ([deltaTht]) does not depend on details of the non-interacting part of the ring Hamiltonian *H*0, junction. For a parity-symmetric geometry with two leads and two coupling sites (*N* = *M* = 2), when coupling to the QD is also symmetric (*t**L* = *t**R*) and time-reversal symmetry is present (*φ* = 0 or *π*), we can further show that the sign of the *O*(*J*2) transmission probability correction is determined by the sign of $1-2\\left\\vert \\mathcal{T}\\_{0,12}\\right\\vert $, a property discussed in Ref. at the end of Sec. IV C. Indeed, parity symmetry implies that *S*11 = *S*22, *S*12 = *S*21, Γ11 = Γ22, Γ12 = Γ21; hence it is not difficult to find from Eq. ([deltaTht]) that\n \n$$\\frac{1}{4}\\left[ \\mathcal{\\delta T}\\_{11}\\left( \\epsilon \\_{k}\\right) +\\mathcal{\\delta T}\\_{22}\\left( \\epsilon \\_{k}\\right) -\\mathcal{\\delta T}\\_{12}\\left( \\epsilon \\_{k}\\right) -\\mathcal{\\delta T}\\_{21}\\left( \\epsilon\n\\_{k}\\right) \\right] =\\frac{3}{8}\\pi ^{2}\\nu ^{2}J\\_{kk}^{2}\\left[\n1-2\\left\\vert S\\_{12}\\left( k\\right) \\right\\vert ^{2}\\right] \\text{.}$$ \nThe left-hand side correspond to a particular way to measure the conductance, namely parity-symmetric bias voltage and parity-symmetric current probes, or *y* = 1/2 in Sec. V of Ref. .\nWe now return to the long ring geometry without assumptions about *t**L*, *t**R* and *φ*. Plugging Eqs. ([SMatkYE]) and ([GammakYE]) into Eq. ([deltaTht]) we find\n \n\\delta T11(*ε**k*) = − 3*π*2*ν*2*j*2[*C*0(*k*) + *C*1(*k*)cos*φ* + *C*2(*k*)cos2*φ*], \nwhere the coefficients *C*0(*k*), *C*1(*k*) and *C*2(*k*) are independent of *φ* but are usually complicated functions of *k*:\n \n$$\\begin{aligned}\nC\\_{0}\\left( k\\right) & =\\left( t\\_{L}^{4}+t\\_{R}^{4}\\right) \\left\\vert \\tilde{t}\\right\\vert ^{2}\\left[ 1+2\\left\\vert \\tilde{r}\\right\\vert \\cos \\left(\nkd\\_{ref}+\\theta \\right) +\\left\\vert \\tilde{r}\\right\\vert ^{2}\\cos 2\\left(\nkd\\_{ref}+\\theta \\right) \\right] \\notag \\\\\n& -2t\\_{L}^{2}t\\_{R}^{2}\\left[ 3-4\\left\\vert \\tilde{t}\\right\\vert\n^{2}+4\\left\\vert \\tilde{r}\\right\\vert ^{3}\\cos \\left( kd\\_{ref}+\\theta\n\\right) +\\left( \\left\\vert \\tilde{r}\\right\\vert ^{4}+\\left\\vert \\tilde{t}\\right\\vert ^{4}\\right) \\cos 2\\left( kd\\_{ref}+\\theta \\right) \\right] \\text{,}\\end{aligned}$$ \n \n$$\\begin{aligned}\nC\\_{1}\\left( k\\right) & =4\\left\\vert \\tilde{t}\\right\\vert t\\_{L}t\\_{R}\\left(\nt\\_{L}^{2}+t\\_{R}^{2}\\right) \\sin \\left( kd\\_{ref}+\\theta \\right) \\notag \\\\\n& \\times \\left[ \\left\\vert \\tilde{r}\\right\\vert ^{2}\\cos 2\\left(\nkd\\_{ref}+\\theta \\right) +\\left\\vert \\tilde{r}\\right\\vert \\left( \\left\\vert \n\\tilde{r}\\right\\vert ^{2}-\\left\\vert \\tilde{t}\\right\\vert ^{2}\\right) \\cos\n\\left( kd\\_{ref}+\\theta \\right) -\\left\\vert \\tilde{t}\\right\\vert ^{2}\\right] \n\\text{,}\\end{aligned}$$ \n \n$$C\\_{2}\\left( k\\right) =2\\left\\vert \\tilde{t}\\right\\vert\n^{2}t\\_{L}^{2}t\\_{R}^{2}\\left\\{ 1+2\\left\\vert \\tilde{r}\\right\\vert \\cos \\left(\nkd\\_{ref}+\\theta \\right) +\\left\\vert \\tilde{r}\\right\\vert ^{2}\\cos 2\\left(\nkd\\_{ref}+\\theta \\right) \\right\\} \\text{.}$$ \nIn the special case of a smooth reference arm $\\left\\vert \\tilde{r}\\right\\vert =0$ and $\\left\\vert \\tilde{t}\\right\\vert =1$, the Kondo-type correction becomes especially simple:\n \n$$\\begin{aligned}\n\\mathcal{\\delta T}\\_{11}\\left( \\epsilon \\_{k}\\right) & =-3\\pi ^{2}\\nu ^{2}j^{2}\\left[ t\\_{L}^{2}+t\\_{R}^{2}-2t\\_{L}t\\_{R}\\sin \\left( kd\\_{ref}+\\theta +\\varphi\n\\right) \\right] \\notag \\\\\n& \\times \\left[ t\\_{L}^{2}+t\\_{R}^{2}-2t\\_{L}t\\_{R}\\sin \\left( kd\\_{ref}+\\theta\n-\\varphi \\right) \\right] \\text{.}\\end{aligned}$$ \nAs in Refs. , only the first and the second harmonics of the AB phase *φ* appear in the correction to the transmission probability \\delta T11.\nWe may perform the thermal averaging in Eq. ([GjjprKondo]) at this stage. The Fermi factor − *f*′(*ε**k*) ensures only the energy range $\\left\\vert \\epsilon \\_{k}\\right\\vert \\lesssim T$ contributes significantly to the conductance; in this energy range the renormalization of *j* is cut off by *T*.\nIn the small Kondo cloud regime, *T* ≫ *T**K* means *T* ≫ Δ so thatwe can average over many peaks of \\delta T*j**j*′(*ε**k*), so we may drop all rapidly oscillating Fourier components in Eq. ([deltaTYE]). This leads to\n \n$$\\delta G\\approx -\\frac{2e^{2}}{h}3\\pi ^{2}\\nu ^{2}\\left[ j\\left( T\\right) \\right] ^{2}\\left[ \\left( t\\_{L}^{4}+t\\_{R}^{4}\\right) \\left\\vert \\tilde{t}\\right\\vert ^{2}-2t\\_{L}^{2}t\\_{R}^{2}\\left( 3-4\\left\\vert \\tilde{t}\\right\\vert ^{2}\\right) +2\\left\\vert \\tilde{t}\\right\\vert\n^{2}t\\_{L}^{2}t\\_{R}^{2}\\cos 2\\varphi \\right] \\text{.} \\label{deltaGYEsc}$$ \nWe see that the first harmonic in *φ* approximately drops out upon thermal averaging.\nOn the other hand, in the large Kondo cloud regime, for *T* ≫ *T**K* it is possible to have either *T* ≫ Δ or *T* ≪ Δ. In the former case Eq. ([deltaGYEsc]) continues to hold. In the latter case \\delta T*j**j*′(*ε**k*) has little variation in the energy range $\\left\\vert \\epsilon \\_{k}\\right\\vert \\lesssim T$, so it is appropriate to replace − *f*′(*ε**k*) with a *δ* function at the Fermi level; thus\n \n$$\\delta G\\approx -\\frac{2e^{2}}{h}3\\pi ^{2}\\nu ^{2}\\left[ j\\left( T\\right) \\right] ^{2}\\left[ C\\_{0}\\left( k\\_{F}\\right) +C\\_{1}\\left( k\\_{F}\\right) \\cos\n\\varphi +C\\_{2}\\left( k\\_{F}\\right) \\cos 2\\varphi \\right] \\text{.}\n\\label{deltaGYElc}$$ \nFig. [fig:longringht] illustrates these two different cases for the large Kondo cloud regime. We note that our *T* ≫ *T**K* results, Eq. ([deltaGYEsc]) for *T* ≫ Δ and Eq. ([deltaGYElc]) for *T* ≪ Δ, are different from those of Ref. . We believe the discrepancy is due to the fact that only single-particle scattering processes are taken into consideration by Ref. ; the connected contribution to the conductance is omitted, despite being of comparable magnitude with the disconnected contribution.\nKondo-type correction to the conductance *δ**G* at *T* ≫ *T**K* for the closed long ABK ring with a particle-hole symmetric QD, calculated by RG improved perturbation theory Eq. ([deltaTYE]), plotted against the AB phase *φ*. Again only 0 ≤ *φ* ≤ *π* is shown. System parameters are: *d**r**e**f* = 60, *θ* = *π*/2, ∣*r̃*∣ = 0, *t**L* = *t**R*, (*t**L*2 + *t**R*2)*j*0/*π* = 0.02 at *D*0 = 10 (i.e. the system is in the large cloud regime). *T*/Δ = 0.0955 in panel a) and *T*/Δ = 19.1 in panel b). For *T* ≪ Δ the conductance shows considerable *k**F* dependence, while for *T* ≫ Δ such dependence essentially vanishes and curves at different *k**F* overlap. Also, for *T* ≫ Δ the first harmonic cos*φ* drops out as predicted by Eq. ([deltaGYEsc]), and *δ**G*(*φ*) has a period of *π*. [fig:longringht]\nFermi liquid conductance\n------------------------\nIt is also interesting to calculate the conductance in the *T* ≪ *T**K* limit in the very large Kondo cloud regime, starting from Eq. ([connelimG2]). We make the assumption that the particle-hole symmetry breaking potential scattering is negligible, *δ**P* = 0, as in Ref. . Inserting Eqs. ([SMatkYE]) and ([GammakYE]) into Eqs. ([discprobKYAZI]) and ([discprobKYAZ2]), we find the total conductance has the form\n \n$$G=\\frac{2e^{2}}{h}\\left[ T\\_{s}+\\left( \\"},"data_id":{"kind":"string","value":"arxiv_0004110"}}},{"rowIdx":994,"cells":{"content":{"kind":"string","value":" a solution of the *n*-SE, then a map, defined by a transposed matrix *M**T* is also a solution of the *n*-SE.\nFortunately in some cases we already know solutions of *n*-SE. For example, all linear bijective solutions for the 2-SE, 3-SE and 4-SE are listed in. Thus we can list all verbal universal abelian solutions in these cases.\nLet *G* be an abelian group, *α*, *β* ∈ Z. Then all verbal solutions to the YBE are given by following mappings: \n$$\\begin{aligned}\n (a, b) & \\mapsto (a^{{\\alpha}}, b^{{\\beta}});\n \\\\\n (a, b) & \\mapsto (b, a);\n \\\\\n (a, b) & \\mapsto (a^{{\\alpha}}b^{1 - {\\alpha}{\\beta}}, b^{{\\beta}});\n \\\\\n (a, b) & \\mapsto (a^{{\\alpha}}, a^{1 - {\\alpha}{\\beta}} b^{{\\beta}}),\n \\end{aligned}$$ \n where *a*, *b* ∈ *G*.\nLet *G* be an abelian group, *α*, *β*, *γ* ∈ Z. Then all verbal solutions to the TE come either from *k*-amalgams of solutions of the YBE and 1-simplex equations or from the list of mappings below by conjugation from Proposition [invsymm] or transposition from Proposition [transposition]. \n$$\\begin{aligned}\n (a, b, c) & \\mapsto (a^{{\\alpha}}, a, a^{-{\\beta}} b c^{{\\beta}});\n \\\\\n (a, b, c) & \\mapsto (b, a, a^{-{\\alpha}} b c^{{\\alpha}});\n \\\\\n (a, b, c) & \\mapsto (a^{{\\alpha}} b^{1 - {\\alpha}{\\beta}} c^{{\\alpha}({\\beta}{\\gamma}- 1)}, b^{{\\beta}} c^{1 - {\\beta}{\\gamma}}, c^{{\\gamma}});\n \\\\\n (a, b, c) & \\mapsto (a^{{\\alpha}} b^{1 - {\\alpha}{\\beta}} c^{{\\gamma}({\\alpha}{\\beta}- 1)}, b^{{\\beta}} c^{1 - {\\beta}{\\gamma}}, c^{{\\gamma}});\n \\end{aligned}$$ \n where *a*, *b*, *c* ∈ *G*.\nLet *G* be an abelian group, *α*, *β*, *γ*, *δ* ∈ Z. Then all verbal solutions to the 4-SE are obtained either from *k*-amalgams of solutions of m-SE where *m* < 4 or from the mappings below by conjugation from Proposition [invsymm] and matrix transposition from Proposition [transposition]. \n$$\\begin{aligned}\n (a, b, c, d) & \\mapsto (b d^{-1}, a c, d, c);\n \\\\\n (a, b, c, d) & \\mapsto (b c^{-{\\alpha}} d^{{\\alpha}{\\beta}-1}, a c, c^{{\\alpha}} d^{1 - {\\alpha}{\\beta}}, d^{{\\beta}});\n \\\\\n (a, b, c, d) & \\mapsto (b, a, a^{-{\\alpha}} b c^{{\\alpha}}, a^{{\\alpha}{\\beta}-1}c^{1-{\\alpha}{\\beta}}d^{\\beta});\n \\\\\n (a, b, c, d) & \\mapsto (b c^{-{\\alpha}} d^{{\\alpha}{\\beta}}, a c d^{-{\\beta}}, c^{{\\alpha}} d^{1-{\\alpha}{\\beta}}, d^{{\\beta}});\n \\\\\n (a, b, c, d) & \\mapsto (b, a, a^{-{\\alpha}}bc^{{\\alpha}},a^{{\\alpha}{\\beta}}b^{-{\\beta}} c^{1-{\\alpha}{\\beta}} d^{{\\beta}});\n \\\\\n (a, b, c, d) & \\mapsto (a^{{\\alpha}} b c^{-{\\alpha}}, c, b, b^{-{\\beta}} c d^{{\\beta}});\n \\\\\n (a, b, c, d) & \\mapsto (a^{{\\alpha}}, b c d^{-{\\beta}}, a^{-{\\alpha}} b d, d^{{\\beta}});\n \\\\\n (a, b, c, d) & \\mapsto (a^{{\\alpha}} b c^{-{\\alpha}} d^{{\\alpha}{\\beta}}, c d^{-{\\beta}}, b d, d^{{\\beta}});\n \\\\\n (a, b, c, d) & \\mapsto (a^{{\\alpha}}b^{1-{\\alpha}{\\beta}}c^{{\\gamma}({\\alpha}{\\beta}-1)}d^{{\\gamma}{\\delta}(1-{\\alpha}{\\beta})}, b^{{\\beta}}c^{1-{\\beta}{\\gamma}}d^{{\\delta}({\\beta}{\\gamma}-1)}, c^{{\\gamma}}d^{1-{\\gamma}{\\delta}}, d^{{\\delta}});\n \\\\\n (a, b, c, d) & \\mapsto (a^{{\\alpha}}b^{1-{\\alpha}{\\beta}}c^{{\\alpha}({\\beta}{\\gamma}-1)}d^{{\\alpha}{\\delta}(1-{\\beta}{\\gamma})}, b^{{\\beta}}c^{1-{\\beta}{\\gamma}}d^{{\\delta}({\\beta}{\\gamma}-1)}, c^{{\\gamma}}d^{1-{\\gamma}{\\delta}}, d^{{\\delta}});\n \\\\\n (a, b, c, d) & \\mapsto (a^{{\\alpha}}b^{1-{\\alpha}{\\beta}}c^{{\\alpha}({\\beta}{\\gamma}-1)}d^{{\\alpha}{\\beta}(1-{\\gamma}{\\delta})}, b^{{\\beta}}c^{1-{\\beta}{\\gamma}}d^{{\\beta}({\\gamma}{\\delta}-1)}, c^{{\\gamma}}d^{1-{\\gamma}{\\delta}}, d^{{\\delta}}),\n \\\\\n \\end{aligned}$$ \n where *a*, *b*, *c*, *d* ∈ *G*.\nResearch problems\n=================\nAt the end, we would like to suggest some problems for further research.\n1. 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Rump, *Modules over braces*, Algebra Discrete Math., 2 (2006), 127–137.\nW. Rump, *Braces, radical rings, and the quantum Yang-Baxter equation*, J. Algebra, 307, no. 1 (2007), 153–170.\nA. Soloviev, *Non-unitary set-theoretical solutions to the quantum Yang–Baxter equation*, Math. Res. Lett. 7 (2000), 577–596.\nS. M. Sergeev, *Solutions of the Functional Tetrahedron Equation Connected with the Local Yang-Baxter Equation for the Ferro-Electric Condition*, Lett. Math. Phys., 45, 1998, 113–119.\nE. K. Sklyanin, L. A. Takhtadzhyan, and L. D. Faddeev, *The quantum inverse problem method*. I, Teor. Mat. Fiz. 40 (1979), no. 2, 194–220; English transl., Theor. and Math. Phys. 40 (1979), no. 2.\nL. A. Takhtadzhyan and L. D. Faddeev, *The quantum method for the inverse problem and the XYZ Heisenberg model*, Uspekhi Mat. Nauk 34 (1979), no. 5, 13–63; English transl. in Russian Math. Surveys 34 (1979), no. 5.\nC. N. Yang, *Some exact results for the many-body problem in one dimension with repulsive delta-function interaction.* Phys. Rev. Lett., 19 (1967), 1312–1315.\nA. B. Zamolodchikov, *Tetrahedra equations and integrable systems in three dimensional space*, Zh. Eksp. Teor. Fiz. 79 (1980) 641664. [English translation: Soviet Phys. JETP 52 (1980) 325–326].\nA. B. Zamolodchikov, *Tetrahedron equations and the relativistic S-matrix of straight-strings in 2+1 dimensions*, Commun. Math. Phys., 79 (1981), 489–505.\nValeriy Bardakov† , ‡ , \\* \\* (bardakov@math.nsc.ru), \nBogdan Chuzinov\\* , † (b.chuzhinov@g.nsu.ru), \nIvan Emel’yanenkov\\* (i.emelianenkov@g.nsu.ru), \nMaxim Ivanov\\* (m.ivanov2@g.nsu.ru), \nTatyana Kozlovskaya‡ (t.kozlovskaya@math.tsu.ru), \nVadim Leshkov\\* (v.leshkov@g.nsu.ru)\n \n\\* Novosibirsk State University, Pirogova 1, 630090 Novosibirsk, Russia, \n† Tomsk State University, pr. Lenina 36, 634050 Tomsk, Russia, \n‡ Sobolev Institute of Mathematics, Acad. Koptyug avenue 4, 630090 Novosibirsk, Russia, \n\\* \\* Novosibirsk State Agrarian University, Dobrolyubova 160, 630039 Novosibirsk, Russia. \n\nSet-theoretical solutions of simplex equations\n==============================================\nThe *n*-simplex equation (*n*-SE) was introduced by A. B. Zamolodchikov as a generalization of the Yang–Baxter equation, which is the 2-simplex equation in these terms. In the present paper we suggest some general approaches to constructing solutions of *n*-simplex equations, describe some types of solutions, introduce an operation which under some conditions allows us to construct a solution of (*n* + *m* + *k*)-SE from solution of (*n* + *k*)-SE and (*m* + *k*)-SE. We consider the tropicalization of rational solutions and discuss a way to generalize it. We prove that if a group *G* is an extension of a group *H* by a group *K*, then we can find a solution of the *n*-SE on *G* from solutions of this equation on *H* and on *K*. Also, we find solutions of the parametric Yang-Baxter equation on *H* with parameters in *K*. For studying solutions of the 3-simplex equations we introduce algebraic systems with ternary operations and give examples of these systems which gives solutions of the 3-SE. We find all elementary verbal solutions of the 3-SE on free groups.\n*Keywords:* Yang-Baxter equation, tetrahedral equation, *n*-simplex equation, set theoretical solution, groupoid, 2-groupoid, ternar, ternoid, group extension, virtual braid group.\n*Mathematics Subject Classification 2010:* 08A02, 16T25, 17A42, 20F36, 20F29.\nIntroduction\n============\nA solution of the quantum Yang–Baxter equation (YBE) is a linear map *R* : *V* ⊗ *V* → *V* ⊗ *V* satisfying\n \n*R*12*R*13*R*23 = *R*23*R*13*R*12, \nwhere *V* is a vector space and *R**i**j* : *V* ⊗ *V* ⊗ *V* → *V* ⊗ *V* ⊗ *V* acts as *R* on the *i*-th and *j*-th tensor factors and as the identity on the remaining factor. V. M. Bukhshtaber called the map *R*, which satisfies the YBE, a Yang-Baxter map.\nThe Yang-Baxter equation, or the 2-simplex equation, or the triangle equation, is one of the basic equations in mathematical physics and in low dimension topology. It lies in the foundation of the theory of quantum groups, solvable models of statistical mechanics, knot theory and braid theory. At first, the YBE appears in the paper of C. N. Yang on the many-body problem. Later R. J. Baxter introduced this equation for the study of solvable vertex models in statistical mechanics as the condition of commuting transfer matrices. Another derivation of the YBE follows from the factorization of the *S*-matrix in a 1 + 1 dimensional Quantum Field Theory (see papers of A. B. Zamolodchikov ). The YBE is also essential in Quantum Inverse Scattering Method for integrable systems.\nThe Yang-Baxter equation is equivalent to a system of *n*6 algebraic cubic equations for *n*4 variables, where *n* = dim*V*. So even in the two-dimensional case, one has 64 equations for 16 variables. In (see also ) one can find a complete description of solutions for the YBE in the case *n* = 2.\nV. G. Drinfeld suggested studying set-theoretical solutions, i.e. solutions for which the vector space *V* is spanned by a set *X*, and *R* is the linear operator induced by a permutation *R* : *X* × *X* → *X* × *X*. In this case, we say that (*X*, *R*) is a set-theoretical solution of the Yang–Baxter equation or simply a solution of the YBE. Set-theoretical solutions have connections for example with groups of I-type, Bieberbach groups, bijective 1-cocycles, Garside theory, etc. Involutive set-theoretical solutions for the YBE were studied in.\nIt is easy to see that for any *X* the map *P*(*x*, *y*) = (*y*, *x*) gives a set-theoretical solution of the YBE. On the other side, if *R* is a solution of the YBE, then the map *S* = *P**R* satisfies the braid relation \n(*S* × id)(id × *S*)(*S* × id) = (id × *S*)(*S* × id)(id × *S*). \n The braid relation is a relation in the braid group *B**n*. Topologically, the braid relation is simply the third Reidemeister move of planar diagrams of links. In 1980s, D. Joyce and S. V. Matveev introduced quandles as invariants of knots and links, and proved that any quandle gives an elementary set-theoretical solution to the braid equation. A lot of papers is dedicated to set-theoretical solution of the Yang-Baxter equation and the braid equation (see, for example ).\nIn solutions of a parametric YBE were constructed. In there was suggested a method to construct linear parametric solutions to the YBE from non-linear Darboux transformations of some Lax operators.\nIn 2006, W. Rump introduced braces to study involutive set-theoretical solutions to the Yang–Baxter equation, although the same algebraic objects were already considered by A. G. Kurosh in 1970s. In 2014, this notion was reformulated by F. Cedó, E. Jespers, and J. Okniński in . In 2017, L. Guarnieri and L. Vendramin defined skew left braces which give non-involutive solutions to the Yang–Baxter equation.\nIf *R*(*x*, *y*) = (*σ**y*(*x*), *τ**x*(*y*)) for *x*, *y* ∈ *X*, is a solution of the YBE, then it is said to be non-degenerate if *σ**x* and *τ**x* are invertible for all *x* ∈ *X*. It is said to be square-free if *R*(*x*, *x*) = (*x*, *x*) for all *x* ∈ *X*. The solution *R*(*x*, *y*) = (*σ**y*(*x*), *τ**x*(*y*)) defines a 2-groupoid (*X*; ⋅ , \\* ) that is a set with two binary algebraic operations ⋅ , \\* : *X* × *X* → *X* that are *x* ⋅ *y* = *σ**y*(*x*) and *y* \\* *x* = *τ**x*(*y*). If *σ**y* = id for all *y* ∈ *X* or *τ**x* = id for all *x* ∈ *X*, then the solution (*X*, *R*) is called by elementary solution. Any elementary solution defines a self-distributive groupoid on *X*, if the elementary solution is non-degenerate, then this groupoid is a rack. On the other side, any self-distributive groupoid gives an elementary solution for the YBE and any rack gives a non-degenerate solution for the YBE. Any quandle defines a non-degenerate square-free solution of the YBE.\nThe 3-simplex equation or the tetrahedron equation (TE) has been introduced by A. B. Zamolodchikov as a 3-dimensional generalization of the YBE. The TE is a non-linear relation in *V*⊗ 6 on an endomorphism *R* : *V*⊗ 3 → *V*⊗ 3: \n*R*123*R*145*R*246*R*356 = *R*356*R*246*R*145*R*123, \n where indexes denote the copies of *V* in the tensor product on which the operator acts as *R* on all other copies on acts as identity map. The tetrahedron equation is popular in electric nets. In particular, the famous transformation ‘star-triangle’ gives a rational solution to the tetrahedron equation. Also, the tetrahedron equation is used in the theory of 2-dimensional knots in a 4-sphere. It is some analogue of the third Reidemeister move in classical knot theory. The TE arrives in a problem of coloring 2-faces of a 4-dimensional cube by elements of some set *X* (see ).\nMany solutions of the TE have been already found. For example, J. Hietarinta considered the set of integers Z*d* modulo *d* as *X* and studied linear and affine maps *X**n* → *X**n* which give solutions to the *n*-simplex equation for *n* = 2, 3, 4. In linear solutions for the TE are described.\nThe *n*-simplex equation is a generalization of YBE and TE. It was introduced in and studied in,,,,,. In the present paper we suggest some general approaches to constructing solutions of *n*-simplex equations, describe some types of solutions, introduce an operation which under some conditions allows us to construct a solution of (*n* + *m* + *k*)-SE from solution of (*n* + *k*)-SE and (*m* + *k*)-SE. We consider the tropicalization of rational solutions and discuss a way to generalize it. We prove that if a group *G* is an extension of a group *H* by a group *K*, then we can find a solution of the *n*-SE on *G* from solutions of this equation on *H* and on *K*. Also, we find solutions of the parametric Yang-Baxter equation on *H* with parameters in *K*. For studying solutions of the 3-simplex equation we introduce algebraic systems with ternary operations and give examples of these systems which gives solutions of the 3-SE. We find all elementary verbal solutions of the 3-SE on free groups. \nThe paper is organized as follows. In Section [prelim], we recall connection between set-theoretical solutions of the Yang–Baxter equation and some algebraic systems (self-distributive groupoid, rack, quandle). We formulate conditions under which an algebraic system with two binary operations gives a set-theoretical solution of the YBE. We recall definitions and some properties of braid groups and virtual braid groups.\nIn Section [NSim] we introduce a general form of the *n*-simplex equation, describe an algorithm that gives a possibility to write the *n*-simplex equation for arbitrary *n* ≥ 2 and formulate results on general solutions. We also give a definition of the classical *n*-SE that is an analogous to the classical YBE.\nIt is known that if (*X*, *R*) is a solution of the YBE and *P*12(*x*, *y*) = (*y*, *x*) is the transposition, then the map *P*12*R**P*12 satisfies the YBE. We prove in Proposition [invsymm] a similar fact for any *n*-SE, constructing some permutation *P* such that the map *P**R**P* satisfies the *n*-SE.\nFor non-negative integer *k* we introduce *k*-amalgam of solutions of the (*n* + *k*)-SE and (*m* + *k*)-SE and find conditions (see Theorem [const]) under which *k*-amalgam is a solution of the (*n* + *m* + *k*)-SE. As a corollary of this general construction we study composition of linear solutions. In Proposition [sumlin] we prove that 0-amalgam of linear solutions of *n*-SE and *m*-SE gives a solution of the (*n* + *m*)-SE. If for these linear solutions 1-amalgam is defined, then it is a linear solution of the (*n* + *m* − 1)-SE. We also prove that if a solution of the *n*-SE has a fixed point, then a specialization of this solution gives a solution of the (*n* − 1)-SE.\nIn Section [Trop] we consider some class of rational solutions of the *n*-SE and give a definition of their tropicalization. We show that the tropicalization of such rational solutions gives piecewise linear solutions of the *n*-SE. In particular, we construct a linear solution of the TE using the famous electric solution. Further we generalize the tropicalization on other algebraic systems. As an example, we show that using the homomorphism *h* : R → Dʹ[R*m*] from the field R of real numbers with standard operations of addition and multiplication to the linear space Dʹ[R*m*] of generalized functions with compact support on the space R*m* with the standard operation of addition and the convolution \\* as the multiplication, one can construct a solution (Dʹ[R*m*], *R**h*) of the *n*-SE from a rational solution (R, *R*) of the *n*-SE.\nIn Section [GrExt] we consider a group *G* with a normal subgroup *H* and the quotient *G*/*H* ≅ *K*. We prove that if a structure of a self-distributive groupoid on *G* is defined, then, under some conditions on *K*, there is a solution of the YBE on *H* with parameters in *K*. In particular, it is true (see Corollary [trext]) if *K* is the trivial self-distributive groupoid. Further these results are extended to general *n*-SE. In Subsection [lin] using the known representations of the virtual braid group we construct (see Theorem [psol]) two types solutions of the Yang–Baxter equations on a group with parameters in an abelian group.\nFurther we consider a group *G* that is an extension of a group *H* by a group *K*. In Proposition [trext] we prove that by the solutions (*H*, *R*) and (*K*, *T*) of the *n*-SE it is possible to construct a solution of the *n*-SE on *G*.\nIn Section [Inlim] we define a category of solutions of the *n*-SE and prove in Proposition [lim] that the limit of solutions of the *n*-SE is also a solution of the *n*-SE. As a consequence we get (see Corollary [adic]) that if we have solutions (Z*p**k*, *R**k*), *k* = 1, 2, …, of the *n*-SE, then under some conditions we can define a solution of the *n*-SE on the set of the *p*-adic integer numbers.\nIn Section [algsust] by analogy with racks, bi-racks, skew braces, which were introduced for the studying of the Yang-Baxter equation, we introduce ternar and 3-ternoid, that are algebraic systems with one and three ternary operations, respectively. We show how they are connected with solutions of the tetrahedral equation (see Proposition [p7.1] and its corollaries). Also, we introduce some algebraic systems with four binary operations and show in Propositions [2-sol] and [1-sol] how it is possible to find elementary solutions for the tetrahedron equation.\nIn Section [verb] we defined verbal solution. These are some generalizations of verbal solutions of the YBE, which one can define on a group with structure of conjugacy quandle or core quandles. We describe all elementary verbal solutions of the TE, which are possible to define on free non-abelian groups.\nIn the paper we formulate some open questions for further research.\nThroughout the paper we use standard notations. If *G* is a group, and *a*, *b* ∈ *G*, then we denote by *a**b* = *b*− 1*a**b* the conjugate of *a* by *b*, and by [*a*, *b*] = *a*− 1*b*− 1*a**b* the commutator of *a* and *b*. Also, we shall assume that compositions acting from the right to the left, i.e. if *f*, *g* : *X* → *X*, then their composition *f**g* acts by the rule (*f**g*)(*x*) = *f*(*g*(*x*)) for *x* ∈ *X* in which it is defined.\nPreliminaries and some basic results\n====================================\nIn this section we give necessary preliminaries. We recall examples of solutions to the YBE, describe a connection of solutions with some groupoids (racks, quandles) and recall definitions of the braid group and the virtual braid group.\nYang–Baxter equations and racks\n-------------------------------\nThe following lemma is evident.\n[YBEF] Let *X* be a non-empty set and *f*, *g* : *X*2 → *X* are two maps. The map *R* : *X*2 → *X*2, *R*(*x*, *y*) = (*f*(*x*, *y*), *g*(*x*, *y*)) is a solution of the YBE if and only if for any *x*, *y*, *z* ∈ *X* the following identities hold\n \n$$\\begin{aligned}\nf\\left( f(x, y), z \\right) &=& f\\left( f(x, g(y, z)), f(y, z) \\right),\\\\\nf\\left( g(x,y), g(f(x,y), z) \\right) &=& g\\left( f(x, g(y, z)), f(y, z) \\right),\\\\\ng\\left( g(x, y), g(f(x, y), z) \\right) &=& g\\left( x, g(y, z) \\right).\\end{aligned}$$ \n1. Let *A* be an abelian group and *a*, *b* ∈ *A*, then \n*R*(*x*, *y*) = (*x* + *a*, *y* + *b*), *x*, *y* ∈ *A*, \n is a solution of the YBE on *A*.\n2. For arbitrary group *G* the map \n*R*(*x*, *y*) = (*x**y*2, *x**y*− 1*x*− 1), *x*, *y* ∈ *G*, \n is a solution of the YBE on *G*.\n3. Let *G* be a group and *φ* ∈ Aut(*G*) be its automorphism, then the map \n*R*(*x*, *y*) = (*φ*(*x*), *x**y**φ*(*x*)− 1), *x*, *y* ∈ *G*, \n is a solution of the YBE on *G*. It is easy to see that if the map *φ*(*x*) = *x**y*2 is an automorphism of *G*, we get the solution from the previous example.\nA groupoid is a non-empty set with one binary algebraic operation, if *k* is a natural number, *k* > 1, then a *k*-groupoid is a non-empty set with *k* binary algebraic operations. Suppose that (*X*, *R*) is a set-theoretical solution of the YBE. Writing *R*(*x*, *y*) = (*σ**y*(*x*), *τ**x*(*y*)) for *x*, *y* ∈ *X*, we say that the solution *R* is *non-degenerate* if *σ**x* and *τ**x* are invertible for all *x* ∈ *X*, we says that the solution *R* is *square-free* if *R*(*x*, *x*) = (*x*, *x*) for all *x* ∈ *X*.\nA *quandle* is a groupoid *Q* with a binary operation (*x*, *y*) ↦ *x* \\* *y* satisfying the following axioms:\n1. *x* \\* *x* = *x* for all *x* ∈ *Q*,\n2. for any *x*, *y* ∈ *Q* there exists a unique *z* ∈ *Q* such that *x* = *z* \\* *y*,\n3. (*x* \\* *y*) \\* *z* = (*x* \\* *z*) \\* (*y* \\* *z*) for all *x*, *y*, *z* ∈ *Q*.\nIf a groupoid satisfies (Q3) it is called a self-distributive groupoid, if it satisfies (Q2) and (Q3) it is called a *rack*. Many interesting examples of quandles come from groups showing deep connection with group theory.\n1. If *G* is a group, then the binary operation *a* \\* *b* = *b*− 1*a**b* turns *G* into the quandle Conj(*G*) called the *conjugation quandle* of *G*.\n2. A group *G* with the binary operation *a* \\* *b* = *b**a*− 1*b* turns the set *G* into the quandle Core(*G*) called the *core quandle* of *G*. In particular, if *G* = Z*n*, the cyclic group of order *n*, then it is called the *dihedral quandle* and denoted by R*n*.\n3. Let *G* be a group and *ϕ* ∈ Aut(*G*). Then the set *G* with binary operation *a* \\* *b* = *ϕ*(*a**b*− 1)*b* forms a quandle Alex(*G*, *ϕ*) referred as the *generalized Alexander quandle* of *G* with respect to *ϕ*.\nA quandle *Q* is called *trivial* if *x* \\* *y* = *x* for all *x*, *y* ∈ *Q*. Unlike groups, a trivial quandle can have arbitrary number of elements. We denote the *n*-element trivial quandle by T*n* and an arbitrary trivial quandle by T.\nNotice that the axioms (Q2) and (Q3) are equivalent to the map *S**x* : *Q* → *Q* given by \n*S**x*(*y*) = *y* \\* *x* \n being an automorphism of *Q* for each *x* ∈ *Q*. These automorphisms are called *inner automorphisms*, and the group generated by all such automorphisms is denoted by Inn(*X*). A quandle is said to be *connected* if it admits a transitive action by its group of inner automorphisms. For example, dihedral quandles of odd order are connected, whereas that of even order are disconnected. A quandle *X* is called *involutive* if *S**x*2 = id*Q* for each *x* ∈ *Q*. For example, all core quandles are involutive.\nFor a solution (*X*, *R*), *R*(*x*, *y*) = (*σ**y*(*x*), *τ**x*(*y*)), we can define a 2-groupoid (*X*; ⋅ , \\* ) with the operations *x* ⋅ *y* = *σ**y*(*x*) and *y* \\* *x* = *τ**x*(*y*). If *σ**y* = id for all *y* ∈ *X* or *τ**x* = id for all *x* ∈ *X*, then the solution (*X*, *R*) is called by *elementary solution*. Any elementary solution defines a structure of self-distributive groupoid on *X*, if the elementary solution is non-degenerate, then this groupoid is a rack. On the other side, any self-distributive groupoid gives an elementary solution to the YBE and any rack gives a non-degenerate solution to the YBE, any quandle defines a non-degenerate square-free solution of the YBE.\nConnections of solutions with 2-groupoids gives the next lemma, which follows from Lemma [YBEF].\n[YBT] Let (*X*, ⋅ , \\* ) be a 2-groupoid and *R* : *X* × *X* → *X* × *X* given by *R*(*x*, *y*) = (*x* ⋅ *y*, *y* \\* *x*) for *x*, *y* ∈ *X*. Then the following hold:\n1. The pair (*X*, *R*) is a set-theoretical solution of the Yang–Baxter equation if and only if the equalities\n \n$$\\begin{aligned}\n(x \\cdot y) \\cdot z &=& (x \\cdot (z \\* y)) \\cdot (y \\cdot z),\\\\\n(y \\* x) \\cdot (z \\* (x \\cdot y)) &=& (y \\cdot z) \\*(x \\cdot (z \\* y)),\\\\\n(z\\*(x \\cdot y)) \\* (y \\* x) &=& (z\\* y ) \\* x,\\end{aligned}$$ \nhold for all *x*, *y*, *z* ∈ *X*.\n2. If *x* ⋅ *y* = *y* for all *x*, *y* ∈ *X*, then the pair (*X*, *R*) is a set-theoretic solution of the Yang–Baxter equation if and only if the operation \\* is (right) self-distributive, i.e.\n \n(*z* \\* *x*) \\* (*y* \\* *x*) = (*z* \\* *y*) \\* *x* \nfor all *x*, *y*, *z* ∈ *X*.\n3. If *y* \\* *x* = *y* for all *x*, *y* ∈ *X*, then the pair (*X*, *R*) is a set-theoretic solution of the Yang–Baxter equation if and only if the operation ⋅ is (right) self-distributive, i.e.\n \n(*z* ⋅ *y*) ⋅ *x* = (*z* ⋅ *x*) ⋅ (*y* ⋅ *x*) \nfor all *x*, *y*, *z* ∈ *X*.\nLet us define on the set of real numbers R a binary operation \n*a* \\* *b* = max{*a*, *b*}, *a*, *b* ∈ R. \n Then (R, \\* ) is a self-distributive groupoid, which is not a rack, but any element *a* ∈ R is idempotent, that is *a* \\* *a* = *a*, and it is commutative groupoid. The map *R*(*x*, *y*) = (*x*, *y* \\* *x*) gives a degenerate solution of the YBE on R. If we take min instead of max we get a similar result.\nA. Soloviev (see also L. Guarnieri, L. Vendramin and D. Bachiller ) proves that any non-degenerate solution of the YBE conjugates to an elementary solution.\nIf *R*(*x*, *y*) = (*σ**y*(*x*), *τ**x*(*y*)), *x*, *y* ∈ *X*, gives a left non-degenerate (i.e. *σ**y* is a bijection for any *y* ∈ *X*) solution to the YBE, then it conjugates to an elementary solution: \n*R*ʹ(*x*, *y*) = (*σ**x*(*τ**σ**y*− 1(*x*)(*y*)), *y*). \n If for all *a*, *b* ∈ *X* there exists a unique *x* ∈ *X* such that\n \n*τ**σ**x*− 1(*a*)(*x*) = *σ**a*− 1(*b*), \nthen this solution is non-degenerate.\nBraid group and virtual braid group\n-----------------------------------\nFor *n* ≥ 2 the braid group *B**n* on *n* strands is defined as the group generated by (*n* − 1) elements *σ*1, *σ*2, …, *σ**n* − 1 with defining relations\n \n$$\\begin{aligned}\n \\label{eq1}\\sigma\\_i\\sigma\\_{i+1}\\sigma\\_i& = \\sigma\\_{i+1}\\sigma\\_i\\sigma\\_{i+1},&&i=1,2,\\ldots,n-2,\\\\\n\\label{eq2}\\sigma\\_i\\sigma\\_j &= \\sigma\\_j\\sigma\\_i,&& |i-j|\\geq 2. \\end{aligned}$$ \nThe *virtual braid group* *V**B**n* on *n* strands, introduced in, is the group with the generators *σ*1, *σ*2, …, *σ**n* − 1, *ρ*1, *ρ*2, …, *ρ**n* − 1, which is defined by the relations ([eq1])-([eq2]) and additional relations\n \n$$\\begin{aligned}\n\\label{eq12} \\rho\\_{i} \\rho\\_{i+1} \\rho\\_{i} &= \\rho\\_{i+1} \\rho\\_{i} \\rho\\_{i+1},&& i=1,2,\\ldots,n-2,\\\\\n\\label{eq13}\\rho\\_{i} \\rho\\_{j} &= \\rho\\_{j} \\rho\\_{i},&& |i-j| \\geq 2, \\\\\n\\label{eq14}\\rho\\_{i}^2 &= 1,&& i=1,2,\\ldots,n-1,\\\\ \n\\label{eq15}\\sigma\\_{i} \\rho\\_{j} &= \\rho\\_{j} \\sigma\\_{i},&& |i-j| \\geq 1,\\\\\n\\label{eq16}\\rho\\_{i} \\rho\\_{i+1} \\sigma\\_{i} &= \\sigma\\_{i+1} \\rho\\_{i} \\rho\\_{i+1},&& i=1,2,\\ldots,n-2. \\end{aligned}$$ \nThe elements *σ*1, *σ*2, …, *σ**n* − 1 generate a subgroup that is isomorphic to the braid group *B**n*. Relations ([eq12])-([eq14]) are defining relations of the symmetric group *S**n*, there, the elements *ρ*1, *ρ*2, …, *ρ**n* − 1 generate *S**n* in *V**B**n*, hence the virtual braid group *V**B**n* is generated by the braid group *B**n* and the symmetric group *S**n*. Relations ([eq15])-([eq16]) are called the mixed relations of the virtual braid group.\nQuantum and classical *n*-simplex equations\n===========================================\nConstruction of quantum *n*-simplex equations\n---------------------------------------------\nTo write a quantum *n*-simplex equation (*n*-SE) for any natural *n*, we give a geometric interpretation of the YBE, which is the 2-simplex equation. For simplicity, we will call a quantum *n*-simplex equation by a *n*-simplex equation. Suppose that we have three straight lines *l*1, *l*2, and *l*3 on the plane R2. The line *l*1 intersects with *l*2 in the point *R*12, with the line *l*3 in the point *R*13, and the line *l*2 intersects with *l*3 in the point *R*23. We assume that all points *R*12, *R*13, and *R*23 are different and are vertices of a non-degenerate triangle (2-simplex). See Figure 1.\nUsing the lexicographical order, we introduce an order on the vertices, \n*R*12 < *R*13 < *R*23. \n Then the YBE is the equality of words, where the first one is a word which we get by going around the vertices in the increasing order and the second word is a word which we get by going around the vertices in the decreasing order, i.e. \n*R*12*R*13*R*23 = *R*23*R*13*R*12. \n To get the tetrahedron equation (3-SE) we increment the indices of all lines by 3 and get the triangle with the vertices *R*45, *R*46, and *R*56. Further, embed our plane R2 into a 3-space R3 = R × R2, take a vertex *R*123, which does not lie in R2 and construct a straight line *l*1, which connects *R*123 with the first vertex *R*45 of the original triangle. Analogously, construct a straight line *l*2, which connects *R*123 with the second vertex, *R*46 and construct a straight line *l*3, which connects *R*123 with the third vertex, *R*56. We construct a tetrahedron with the vertices *R*123, *R*145, *R*246, and *R*356. See Figure 2.\nThen the tetrahedron equation (TE) is the equality of words, where the first one is a word which we get by going around the vertices of the tetrahedron in the increasing order and the second word is a word which we get by going around the vertices in the decreasing order, i.e. \n*R*123*R*145*R*246*R*356 = *R*356*R*246*R*145*R*123. \nUsing the same approach one can write the 4-simplex equation, \n*R*1234*R*1567*R*2589*R*368, 10*R*479, 10 = *R*479, 10*R*368, 10*R*2589*R*1567*R*1234. \nIn general case, the left side and the right side of the *n*-SE are words of length *n* + 1. Notice that *n* + 1 is the number of vertices of the *n*-simplex. Suppose that we have the *n*-SE for some *n* ≥ 3. We will write this equation in the form \n$$R\\_{\\overline{1}}R\\_{\\overline{2}} \\cdots R\\_{\\overline{n+1}}\n =\n R\\_{\\overline{n+1}} \\cdots R\\_{\\overline{2}} R\\_{\\overline{1}},$$ \n where $ \\overline{k} = (k\\_1, k\\_2, \\ldots, k\\_{n+1}) \\in \\mathbb{N}^{n+1} $ is a multi-index. We want to construct the (*n* + 1)-SE. To do it, define an operation *s**n* : N → N, *s**n*(*k*) = *k* + *n* + 1, and extend it to the multi-indices by the rule \n$$s\\_n(\\overline{k}) = (s\\_n(k\\_1), s\\_n(k\\_2), \\ldots, s\\_n(k\\_{n+1})) \\in \\mathbb{N}^{n+1}.$$ \n Using these notations we can write the (*n* + 1)-SE as \n$$R\\_{1,2,\\ldots,n+1} R\\_{1, s\\_n(\\overline{1})} R\\_{2, s\\_n(\\overline{2})} \\cdots R\\_{n+1, s\\_n(\\overline{n+1})}\n =\n R\\_{n+1, s\\_n(\\overline{n+1})} \\cdots R\\_{2,s\\_n(\\overline{2})} R\\_{1,s\\_n(\\overline{1})} R\\_{1,2,\\ldots,n+1}.$$ \nThe multi-indices in the *n*-SE can be regarded as rows of the *multi-indices matrix* *M**I**n* that is a (*n* + 1) × *n* matrix satisfying the recurrence relation \n$$MI\\_{n} = \n\\begin{pmatrix}\n 1 & & 2 & 3 & \\cdots & n \\\\\n \\hline\n1 &\\vline & & & & \\\\\n2 &\\vline & & & &\\\\\n\\vdots &\\vline & & MI\\_{n - 1} + (n) & & \\\\\nn-1 &\\vline & & & &\\\\\nn &\\vline & & & & \\\\\n\\end{pmatrix}$$ \n where (*n*) is the *n* by *n* − 1 matrix in which all elements are equal to *n*. The matrix *M**I**n* could be written explicitly as \n$$\\begin{pmatrix}\n 1 & 2 & 3 & \\cdots & n \\\\\n 1 & n + 1 & n + 2 & \\cdots & 2n - 1 \\\\\n 2 & n + 1 & 2n & \\cdots & 3n - 3 \\\\\n \\vdots & \\vdots & \\vdots & \\ddots &\\vdots \\\\\n n-1 & 2n - 2 & 3n - 4 & \\cdots & \\frac{n(n+1)}{2}\\\\\n n & 2n - 1 & 3n - 3 & \\cdots & \\frac{n(n+1)}{2}\n \\end{pmatrix}$$ \nA linear solution of the *n*-SE is a linear map *R* : *V*⊗ *n* → *V*⊗ *n* such that the next equality \n$$R\\_{\\overline{1}}R\\_{\\overline{2}} \\cdots R\\_{\\overline{n+1}}\n =\n R\\_{\\overline{n+1}} \\cdots R\\_{\\overline{2}} R\\_{\\overline{1}},$$ \n of two linear maps *V*⊗ *N* → *V*⊗ *N*, where *N* = *n*(*n* + 1)/2, holds. In this equality the map $R\\_{\\overline{k}} : V^{\\otimes N} \\to V^{\\otimes N}$ acts as *R* on the copies of *V* with the numbers $\\overline{k}$ and as identity map on copies with other numbers. Notice that *n* + 1 is the number of vertices of the *n*-simplex, and *N* is the number of its edges.\nA set-theoretic solution of the *n*-SE on a set *X* is a map *R* : *X**n* → *X**n* that is satisfies the *n*-SE.\nFor completeness, we define the 1-simplex equation *R*1*R*2 = *R*2*R*1 as equality of two maps *X* × *X* → *X* × *X*. In this case *R* is a map *R* : *X* → *X*. Since, \n*R*1(*x*, *y*) = (*R*(*x*), *y*), *R*2(*x*, *y*) = (*x*, *R*(*y*)), *R*1*R*2(*x*, *y*) = *R*2*R*1(*x*, *y*) = (*R*(*x*), *R*(*y*)), \n any map *R* : *X* → *X* gives a solution to the 1-simplex equation.\nClassical *n*-simplex equation\n------------------------------\nThe equation \n*R*12*R*13*R*23 = *R*23*R*13*R*12 \n also is called the Quantum Yang-Baxter Equation. If (*V*, *R*) is its linear solution and the linear map *R* can be presented in the form \n*R* = 1 + ℏ*r* + *O*(ℏ2), \n then *r*: *V* → *V* gives a solution to the Classical Yang-Baxter Equation, \n[*r*12, *r*13] + [*r*12, *r*23] + [*r*13, *r*23] = 0, \n where [*a*, *b*] = *a**b* − *b**a*. The *n*-SE which we considered before are quantum *n*-SE. It is interesting to understand what is a classical version of the *n*-SE? The proof of the following proposition is strait calculations.\nIf *V* is a vector space and *R*: *V**n* → *V**n* is a linear map that is a solution of the quantum *n*-SE, and there is a map *r*: *V**n* → *V**n* such that \n*R* = 1 + ℏ*r* + *O*(ℏ2), \n then *r* satisfied the equations \n$$\\sum\\_{\\overline{1} \\leq \\overline{i} < \\overline{j} \\leq \\overline{n+1}}\n[ r\\_{\\overline{i}}, r\\_{\\overline{j}}] = 0,$$ \n where we use the lexicographic order on the multi-indices. This equation is said to be the classical *n*-SE.\nSolutions of arbitrary *n*-SE\n=============================\nFurther we will consider only set-theoretic solutions and call them simply by solutions. We also will work with partial functions *R* : *X**n* → *X**n* defined only on some subset *D* ⊂ *X**n*. In these cases we will say that (*X*, *R*) is a solution of the *n*-SE when the *n*-SE holds for every point *x̄* ∈ *X**N* where it has sense, i.e. where left and right parts of the equation are defined.\nThe next proposition is a generalization of well-known result for the YBE.\nLet *R* : *X**n* → *X**n* be a solution of the *n*-SE.\n1. If *R* is invertible, then its inverse *R*− 1 is also a solution of the *n*-SE.\n2. If *φ*1, ... , *φ**n* are pairwise commuting maps from *X* to *X*, then the map *R* : *X**n* → *X**n* defined as \n*R*(*x*1, ... , *x**n*) = (*φ*1(*x*1), ... , *φ**n*(*x**n*)), \n is a solution of the *n*-SE.\n3. If *φ* ∈ *S**y**m*(*X*) is an arbitrary bijection of the set *X* onto itself, then the composition \n*φ*× *n* *R* (*φ*− 1)× *n* \n is a solution of the *n*-SE.\n(1) If we take inverse on both sides of *n*-SE, we get that *R*− 1 gives a solution.\n(2) It is easy to see that all functions $ R\\_{\\overline{\\imath}} $ derived from such *R* are pairwise commutative. Thus the left side of the *n*-SE could be reordered to obtain the right side of the equation.\n(3) Put $\\widetilde R = \\varphi^{\\times n} \\, R \\, (\\varphi^{-1})^{\\times n}$. Then \n$$\\widetilde{R}\\_{\\bar k} = \\varphi^{\\times n}\\_{\\bar k} \\, R\\_{\\bar k} \\, ({\\varphi^{-1}})^{\\times n}\\_{\\bar k} = \\varphi^{\\times N} \\, R\\_{\\bar k} \\, ({\\varphi^{-1}})^{\\times N}$$ \n The equation \n$$\\widetilde{R}\\_{\\overline{1}}\\widetilde R\\_{\\overline{2}} \\cdots \\widetilde R\\_{\\overline{n+1}}\n =\n \\widetilde R\\_{\\overline{n+1}} \\cdots \\widetilde R\\_{\\overline{2}} \\widetilde R\\_{\\overline{1}},$$ \n is equivalent to the equation\n \n$$\\begin{gathered}\n \\left( \\varphi^{\\times N} \\, R\\_{\\bar 1} \\, ({\\varphi^{-1}})^{\\times N} \\right) \\, \\left( \\varphi^{\\times N} \\, R\\_{\\bar 2} \\, ({\\varphi^{-1}})^{\\times N}\\right) \\cdots \\left( \\varphi^{\\times N} \\, R\\_{\\overline {n+1}} \\, ({\\varphi^{-1}})^{\\times N}\\right) \n =\\\\\n = \\left( \\varphi^{\\times N} \\, R\\_{\\overline{n+1}} \\, ({\\varphi^{-1}})^{\\times N} \\right) \\cdots \\left( \\varphi^{\\times N} \\, R\\_{\\bar 2} \\, ({\\varphi^{-1}})^{\\times N} \\right) \\left( \\varphi^{\\times N} \\, R\\_{\\bar 1} \\, ({\\varphi^{-1}})^{\\times N} \\right).\n \\end{gathered}$$ \n After cancellations we obtain an equation \n$$\\varphi^{\\times N} {R}\\_{\\overline{1}} R\\_{\\overline{2}} \\cdots R\\_{\\overline{n+1}} ({\\varphi^{-1}})^{\\times N} \n =\n \\varphi^{\\times N} R\\_{\\overline{n+1}} \\cdots R\\_{\\overline{2}} R\\_{\\overline{1}}\\ ({\\varphi^{-1}})^{\\times N}.$$ \n Since *φ**N* is a bijection, the obtained equation holds if and only if the initial equation holds.\nThe following proposition is a generalization of Proposition 2.2 from.\n[twist] If *R* is a solution of a *n*-SE and *φ* ∈ *S**y**m*(*X*) is such that \n(*φ*− 1)× *n* *R* *φ*× *n* = *R* \n then the map \n$$\\tilde R = (\\varphi \\times {\\mathrm{id}}\\times \\varphi \\times \\ldots) \\, R \\, ({\\mathrm{id}}\\times \\varphi^{-1} \\times {\\mathrm{id}}\\times \\ldots),$$ \n where (*φ* × id × *φ* × …) acts as *φ* on odd components and as id on even, (id × *φ*− 1 × id × …) acts as *φ*− 1 on even components and as id on odd is also a solution of the *n*-SE.\nNote that \n(*φ* × id × *φ* × …) = (id × *φ*− 1 × id × …) *φ*× *n*, \n \n(id × *φ*− 1 × id × …) = (*φ*− 1)× *n* (*φ* × id × *φ* × …) \n and \n$$\\begin{gathered}\nR\\_A := \\tilde R =\n(\\varphi \\times {\\mathrm{id}}\\times \\varphi \\times \\ldots) R ({\\mathrm{id}}\\times \\varphi^{-1} \\times {\\mathrm{id}}\\times \\ldots) = \\\\ =\n(\\varphi \\times {\\mathrm{id}}\\times \\varphi \\times \\ldots) \\left ( (\\varphi^{-1})^{\\times n} R \\varphi^{\\times n} \\right ) ({\\mathrm{id}}\\times \\varphi^{-1} \\times {\\mathrm{id}}\\times \\ldots) = \\\\ =\n({\\mathrm{id}}\\times \\varphi^{-1} \\times {\\mathrm{id}}\\times \\ldots) R (\\varphi^{} \\times {\\mathrm{id}}\\times \\varphi^{} \\times \\ldots) =: R\\_B.\n\\end{gathered}$$ \nWriting the *n*-SE one can observe that every index which first occurrence was in multi-index $\\overline{i}$ on the position *j* will appear again only in multi-index $\\overline{j+1}$ on the position *i*. It means that if we replace every $R\\_{\\overline{i}}$ where *i* is even with $(R\\_{A})\\_{\\overline{i}}$ and every $R\\_{\\overline{i}}$ where *i* is odd with $(R\\_{B})\\_{\\overline{i}}$ then for every *i* and every *j* ≥ *i* the component with index *i**j* would be trivially acted upon between $R\\_{\\overline{i}}$ and $R\\_{\\overline{j+1}}$. Indeed if *i* and *j* are even or odd simultaneously then *i**j* between $R\\_{\\overline{i}}$ and $R\\_{\\overline{j+1}}$ is acted upon by *i**d* ∘ *i**d*, and if they have distinct parity then it is acted upon by either *φ* ∘ *φ*− 1 or *φ*− 1 ∘ *φ*. The final observation of the proof is that since every index appears in precisely two different multi-indices the equation \n$$(R\\_B)\\_{\\overline{1}}(R\\_A)\\_{\\overline{2}} \\ldots (\\tilde R)\\_{\\overline{n+1}} =\n (\\tilde R)\\_{\\overline{n+1}}\\ldots (R\\_A)\\_{\\overline{2}}(R\\_B)\\_{\\overline{1}}$$ \n takes the form \n$$\\Phi R\\_{\\overline{1}}R\\_{\\overline{2}}...R\\_{\\overline{n+1}} \\Phi =\n \\Phi R\\_{\\overline{n+1}}...R\\_{\\overline{2}}R\\_{\\overline{1}} \\Phi,$$ \n where Φ is componentwise action of either *φ*, *φ*− 1 or id.\nIt is easy to check that if (*X*, *R*), *R*(*x*, *y*) = (*x*, *y* \\* *x*), is an elementary solution of the YBE and *P*(*x*, *y*) = (*y*, *x*) for all *x*, *y* ∈ *X*, then \n*P**R**P*(*x*, *y*) = (*x* \\* *y*, *y*) \n is also an elementary solution of the YBE.\nThe proof of the next lemma is elementary, but we give it to further generalize on arbitrary *n*-simplex equations.\nSuppose that *R*(*x*, *y*) = (*x* ⋅ *y*, *y* \\* *x*) is a solution of the YBE. Then *P**R**P*(*x*, *y*) = (*x* \\* *y*, *y* ⋅ *x*) is also a solution of the YBE.\nIt is easy to see that \n*R*13 = *P*23*R*12*P*23, *R*23 = *P*12*P*23*R*12*P*23*P*12, \n where *P**i**j* : *X*3 → *X*3 permutes the *i*-th and *j*-th components. Then the YBE has the form \n*R*12 ⋅ *R*12*P*23 ⋅ *R*12*P*23*P*12 = *R*12*P*23*P*12 ⋅ *R*12*P*23 ⋅ *R*12. \nDenote by $\\widetilde{R} = P R P$. Then $R = P \\widetilde{R} P$ and from the YBE \n$$\\widetilde{R}\\_{12}^{P\\_{12}} \\cdot \\widetilde{R}\\_{12}^{P\\_{12} P\\_{23}} \\cdot \\widetilde{R}\\_{12}^{P\\_{12}P\\_{23}P\\_{12}} = \n\\widetilde{R}\\_{12}^{P\\_{12}P\\_{23}P\\_{12}} \\cdot \\widetilde{R}\\_{12}^{P\\_{12} P\\_{23}} \\cdot \\widetilde{R}\\_{12}^{P\\_{12}}.$$ \n Conjugating both sides of this equation by *P*12*P*23*P*12 = *P*13 we get \n$$\\widetilde{R}\\_{12}^{P\\_{23} P\\_{12}} \\cdot \\widetilde{R}\\_{12}^{P\\_{23}} \\cdot \\widetilde{R}\\_{12}= \n\\widetilde{R}\\_{12} \\cdot \\widetilde{R}\\_{12}^{P\\_{23}} \\cdot \\widetilde{R}\\_{12}^{P\\_{23} P\\_{12}}.$$ \n That is equivalent to \n$$\\widetilde{R}\\_{12} \\widetilde{R}\\_{12} \\widetilde{R}\\_{12}= \n\\widetilde{R}\\_{12} \\widetilde{R}\\_{12} \\widetilde{R}\\_{12}.$$ \n It means that $\\widetilde{R} = P R P$ satisfied the YBE.\nTo generalize this result, let us consider a solution (*X*, *R*) of the *n*-SE and define a permutation *P* ∈ *S**y**m*(*X**n*) by \n*P* = *P*1, *n**P*2, *n* − 1…*P*[*n*/2] + 1, *n* − [*n*/2], \n where [*n*/2] is the integer part of *n*/2 and for *n* = 2*m* + 1, we assume that the last permutation *P*[*n*/2] + 1, *n* − [*n*/2] = *P**m* + 1, *m* + 1 is the identity. It is easy to see that *P*− 1 = *P*. In these notations we can formulate the following proposition.\n[invsymm] Let *R* : *X**n* → *X**n* be a solution of the *n*-SE. Then $ \\widetilde{R} = P R P $ is also a solution of the *n*-SE.\nThe *n*-SE \n$$R\\_{\\overline{1}}R\\_{\\overline{2}} \\cdots R\\_{\\overline{n+1}} = R\\_{\\overline{n+1}} \\cdots R\\_{\\overline{2}} R\\_{\\overline{1}}$$ \n can be presented in the form \n$$R\\_{\\overline{1}} R\\_{\\overline{1}}^{Q\\_1} \\cdots R\\_{\\overline{1}}^{Q\\_n} = R\\_{\\overline{1}}^{Q\\_n} \\cdots R\\_{\\overline{1}}^{Q\\_1} R\\_{\\overline{1}},$$ \n where *Q*1, …, *Q**n* ∈ *S**y**m*(*X**n*) are some permutations. Since $R = P \\widetilde{R} P$, we have \n$$\\widetilde{R}\\_{\\overline{1}}^{P} \\widetilde{R}\\_{\\overline{1}}^{P Q\\_1} \\cdots \\widetilde{R}\\_{\\overline{1}}^{P Q\\_n} = \\widetilde{R}\\_{\\overline{1}}^{P Q\\_n} \\cdots \\widetilde{R}\\_{\\overline{1}}^{P Q\\_1} \\widetilde{R}\\_{\\overline{1}}^{P}.$$ \n After conjugating both sides by *Q**n*− 1*P* we get \n$$\\widetilde{R}\\_{\\overline{1}}^{P Q\\_n^{-1} P} \\widetilde{R}\\_{\\overline{1}}^{P Q\\_1Q\\_n^{-1} P} \\cdots \\widetilde{R}\\_{\\overline{1}}^{P Q\\_{n-1} Q\\_n^{-1} P} \\widetilde{R}\\_{\\overline{1}} = \\widetilde{R}\\_{\\overline{1}} \\widetilde{R}\\_{\\overline{1}}^{P Q\\_{n-1} Q\\_n^{-1} P} \\cdots \\widetilde{R}\\_{\\overline{1}}^{P Q\\_1Q\\_n^{-1} P} \\ \\widetilde{R}\\_{\\overline{1}}^{P Q\\_n^{-1} P}.$$ \n Using the equalities \n*P**Q**n*− 1*P* = *Q**n*, *P**Q*1*Q**n*− 1*P* = *Q**n* − 1, …, *P**Q**n* − 1*Q**n*− 1*P* = *Q*1, \n we have \n$$\\widetilde{R}\\_{\\overline{1}}^{Q\\_n} \\widetilde{R}\\_{\\overline{1}}^{Q\\_{n-1}} \\cdots \\widetilde{R}\\_{\\overline{1}}^{Q\\_1} \\widetilde{R}\\_{\\overline{1}} = \n\\widetilde{R}\\_{\\overline{1}} \\widetilde{R}\\_{\\overline{1}}^{Q\\_1} \\cdots \\widetilde{R}\\_{\\overline{1}}^{Q\\_{n-1}} \\ \\widetilde{R}\\_{\\overline{1}}^{Q\\_n}.$$ \n Hence, $ \\widetilde{R}$ gives a solution of the *n*-SE.\nWe know that if *R* is a solution of the YBE, then *S* = *P**R* is a solution of the braid equation *S*12*S*23*S*12 = *S*23*S*12*S*23. What is an analogous of the braid equation for the *n*-SE when *n* ≥ 3?\nComposition of solutions\n------------------------\nFor the linear solutions of the YBE we know two operations on solutions (see, for example, ): tensor product and direct sum. If (*V*1, *R*(1)) and (*V*2, *R*(2)) are two solutions. Then their tensor product is the solution (*V*1 ⊗ *V*2, *R*(1) ⊗ *R*(2)) and the direct sum of solutions is the solution (*V*1 × *V*2, *R*(1) + *R*(2)), where the map *R*(1) + *R*(2) is defined on the basis of (*V*1 × *V*2) ⊗ (*V*1 × *V*2) by the rules\n \n$$\\begin{aligned}\n(R^{(1)} + R^{(2)}) (e\\_i \\otimes e\\_j) &=& R^{(1)}(e\\_i \\otimes e\\_j),\\\\\n(R^{(1)} + R^{(2)}) (e\\_i \\otimes f\\_q) &=& e\\_i \\otimes f\\_q,\\\\\n(R^{(1)} + R^{(2)}) (f\\_p \\otimes e\\_j) &=& f\\_p \\otimes e\\_j,\\\\\n(R^{(1)} + R^{(2)}) (f\\_p\\otimes f\\_q) &=& R^{(2)}(f\\_p\\otimes f\\_q),\\end{aligned}$$ \nwhere {*e**α*} is a basis of *V*1 and {*f**β*} is a basis of *V*2.\nIn this subsection we are considering the following question: Let (*X*, *A*) and (*Y*, *B*) be set-theoretic solutions of the *n*-SE and the *m*-SE, correspondingly. What new solutions can one construct?\nIf *m* = *n*, then one can define the direct product of solutions, \n*A* × *B* : (*X* × *Y*)*n* → (*X* × *Y*)*n*, \n \n*A* × *B*((*x*1, *y*1), (*x*2, *y*2), …, (*x**n*, *y**n*)) = ((*f*1(*x̄*), *g*1(*ȳ*)), (*f*2(*x̄*), *g*2(*ȳ*)), …, (*f**n*(*x̄*), *g**n*(*ȳ*))), \n where \n*A*(*x̄*) = (*f*1(*x̄*), *f*2(*x̄*), …, *f**n*(*x̄*)), *B*(*ȳ*) = (*g*1(*ȳ*), *g*2(*ȳ*), …, *g**n*(*ȳ*)), \n and we denote \n*x̄* = (*x*1, *x*2, …, *x**n*), *ȳ* = (*y*1, *y*2, …, *y**n*). \n The map *A* × *B* gives a solution of the *n*-SE.\n[com] Let *A*: *X**n* + *k* → *X**n* + *k* and *B*: *X**k* + *m* → *X**k* + *m* be two maps which have the form \n*A*(*x̄*, *ȳ*) = (*f*1(*x̄*, *ȳ*), ... , *f**n*(*x̄*, *ȳ*), *h*1(*ȳ*), ... , *h**k*(*ȳ*)), *x̄* ∈ *X**n*, *ȳ* ∈ *X**k*, \n \n*B*(*ȳ*, *z̄*) = (*h*1(*ȳ*), ... , *h**k*(*ȳ*), *g*1(*ȳ*, *z̄*), ... , *g**m*(*ȳ*, *z̄*)), *z̄* ∈ *X**m*, \n then a function *A* #*k**B*: *X**n* + *k* + *m* → *X**n* + *k* + *m*, defined by \n*A* #*k**B*(*x̄*, *ȳ*, *z̄*) : = (*f*1(*x̄*, *ȳ*), ... , *f**n*(*x̄*, *ȳ*), *h*1(*ȳ*), ... , *h**k*(*ȳ*), *g*1(*ȳ*, *z̄*), ... , *g**m*(*ȳ*, *z̄*)) \n is called a *k*-amalgam of *A* and *B*.\n[const] Let *n* > 0, *m* > 0, *k* ≥ 0, be integers, *A*: *X**n* + *k* → *X**n* + *k* and *B*: *X**k* + *m* → *X**k* + *m* be solutions of the (*n* + *k*)-SE and the (*m* + *k*)-SE, correspondingly, \n*A*(*x̄*, *ȳ*) = (*f*1(*x̄*, *ȳ*), ... , *f**n*(*x̄*, *ȳ*), *h*1(*ȳ*), ... , *h**k*(*ȳ*)), *x̄* ∈ *X**n*, *ȳ* ∈ *X**k*, \n \n*B*(*ȳ*, *z̄*) = (*h*1(*ȳ*), ... , *h**k*(*ȳ*), *g*1(*ȳ*, *z̄*), ... , *g**m*(*ȳ*, *z̄*)), *z̄* ∈ *X**m*, \n Then the *k*-amalgam *A* #*k**B* : *X**n* + *k* + *m* → *X**n* + *k* + *m* of *A* and *B*, \n*A* #*k**B*(*x̄*, *ȳ*, *z̄*) = (*f*1(*x̄*, *ȳ*), ... , *f**n*(*x̄*, *ȳ*), *h*1(*ȳ*), ... , *h**k*(*ȳ*), *g*1(*ȳ*, *z̄*), ... , *g**m*(*ȳ*, *z̄*)) \n is a solution of the (*n* + *k* + *m*)-SE if and only if for any pair of indices 1 ≤ *i* ≤ *n* and 1 ≤ *j* ≤ *m* and for any collection of elements *a**α*, *β* ∈ *X*, *α* ∈ {1, …, *n* + *k*}, *β* ∈ {1, …, *k* + *m*} the following equality holds: \n$$\\begin{gathered}\n f\\_i\\left(\\begin{matrix}\ng\\_j( & a\\_{1,1}, & a\\_{1,2}, & \\cdots & a\\_{1, k-1}, & a\\_{1, k}, & \\cdots & a\\_{1, k+m} & ),\\\\\ng\\_j( & a\\_{2,1}, & a\\_{2,2}, & \\cdots & a\\_{2, k-1}, & a\\_{2, k}, & \\cdots & a\\_{2, k+m} & ),\\\\\n & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\ddots & \\vdots & \\\\\ng\\_j( & a\\_{n+1,1}, & a\\_{n+1,2}, & \\cdots & a\\_{n+1, k-1}, & a\\_{n+1, k}, & \\cdots & a\\_{n+1, k+m} & ),\\\\\ng\\_j( & b^{n+1,1}, & a\\_{n+2,2}, & \\cdots & a\\_{n+2, k-1}, & a\\_{n+2, k}, & \\cdots & a\\_{n+2, k+m} & ),\\\\\ng\\_j( & b^{n+1,2}, & b^{n+2,2}, & \\cdots & a\\_{n+3, k-1}, & a\\_{n+3, k}, & \\cdots & a\\_{n+3, k+m} & ),\\\\\n & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\ddots & \\vdots & \\\\\ng\\_j( & b^{n+1,k-1}, & b^{n+2,k-1}, & \\cdots & b^{n+k-1,k-1} & a\\_{n+k, k}, & \\cdots & a\\_{n+k, k+m} & )\n \\end{matrix}\\right)\n \\\\\n {\\mathrel{\\rotatebox{90}{$=$}}}\\\\\n g\\_j\\left(\\begin{matrix}\nf\\_i( & a\\_{1,1}, & a\\_{2,1}, & \\cdots & a\\_{n+1, 1}, & b\\_{2, 2}, & b\\_{3, 2}, & \\cdots & b\\_{k, 2} & ),\\\\\nf\\_i( & a\\_{1,2}, & a\\_{2,2}, & \\cdots & a\\_{n+1, 2}, & a\\_{n+2, 2}, & b\\_{3, 3}, & \\cdots & b\\_{k, 3} & ),\\\\\n & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\\\\nf\\_i( & a\\_{1,k-1}, & a\\_{2,k-1}, & \\cdots & a\\_{n+1, k-1}, & a\\_{n+2, k-1}, & a\\_{n+3, k-1}, & \\cdots & b\\_{k, k} & ),\\\\\nf\\_i( & a\\_{1,k}, & a\\_{2,k}, & \\cdots & a\\_{n+1, k}, & a\\_{n+2, k}, & a\\_{n+3, k}, & \\cdots & a\\_{n+k, k} & ),\\\\\n & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\\\\nf\\_i( & a\\_{1,k+m}, & a\\_{2,k+m}, & \\cdots & b^{n+1,k+m} & a\\_{n+2, k+m}, & a\\_{n+3, k+m}, & \\cdots & a\\_{n+k, k+m} & )\n \\end{matrix}\\right),\n \\end{gathered}$$ \n where \n$$\\begin{aligned}\n b^{i, j} & := h\\_j(b^{n+1, i-n-1}, {...\\, }, b^{i-1, i-n-1}, a\\_{i, j}, {...\\, }, a\\_{i, k}),\n \\\\\n b\\_{i, j} & := h\\_j(a\\_{n+1, i}, {...\\, }, a\\_{n+i, i}, b^{i+1, i}, {...\\, }, b^{k, i}).\n\\end{aligned}$$ \nThis statement becomes obvious if one looks at the matrix *M**I**n* + *k* + *m* for the (*n* + *k* + *m*)-SE. Its (*n* + *k*) × (*n* + *k* + 1) top-left and (*k* + *m*) × (*k* + *m* + 1) bottom-right submatrices are independent of the rest of the matrix and represent the (*n* + *k*)-SE and the (*k* + *m*)-SE respectively.\nDue to functions *h**i* overlapping parts of those submatrices are independent of the rest of the matrix and thus the (*n* + *k* + *m*)-SE recedes to (*n* + *k*)-SE and (*k* + *m*)-SE for indices in corresponding submatrices. Remaining bottom-left and top-right submatrices consist of *n**m* different indices and are transposes of each other.\nExplicit form of the (*n* + *k* + *m*)-SE for indices in those matrices gives us the condition above.\nTo illustrate the construction of *k*-amalgam we consider some examples.\nLet *X* be a set and *A*, *B* : *X*2 → *X*2 are solutions of the YBE, which have the forms \n*A*(*x*, *y*) = (*f*1(*x*, *y*), *f*2(*x*, *y*)), *B*(*z*, *t*) = (*g*1(*z*, *t*), *g*2(*z*, *t*)), *x*, *y*, *z*, *t* ∈ *X*. \n By Lemma [YBEF] the maps *f**i* satisfy the identities\n \n$$\\begin{aligned}\nf\\_1\\left( f\\_1(x, y), z \\right) &=& f\\_1\\left( f\\_1(x, f\\_2(y, z)), f\\_1(y, z) \\right),\\\\\nf\\_1\\left( f\\_2(x,y), f\\_2(f\\_1(x,y), z) \\right) &=& f\\_2\\left( f\\_1(x, f\\_2(y, z)), f\\_1(y, z) \\right),\\\\\nf\\_2\\left( f\\_2(x, y), f\\_2(f\\_1(x, y), z) \\right) &=& f\\_2\\left( x, f\\_2(y, z) \\right),\\end{aligned}$$ \nand the maps *g**i* satisfy the identities\n \n$$\\begin{aligned}\ng\\_1\\left( g\\_1(x, y), z \\right) &=& g\\_1\\left( g\\_1(x, g\\_2(y, z)), g\\_1(y, z) \\right),\\\\\ng\\_1\\left( g\\_2(x,y), g\\_2(g\\_1(x,y), z) \\right) &=& g\\_2\\left( g\\_1(x, g\\_2(y, z)), g\\_1(y, z) \\right),\\\\\ng\\_2\\left( g\\_2(x, y), g\\_2(g\\_1(x, y), z) \\right) &=& g\\_2\\left( x, g\\_2(y, z) \\right),\\end{aligned}$$ \nfor all *x*, *y*, *z*, *t* ∈ *X*. Set \n*R*(*x*, *y*, *z*, *t*) = (*f*1(*x*, *y*), *f*2(*x*, *y*), *g*1(*z*, *t*), *g*2(*z*, *t*)), *x*, *y*, *z*, *t* ∈ *X*. \n Let us find the conditions under which this map *R* satisfies the 4-SE, \n*R*1234*R*1567*R*2589*R*368, 10*R*479, 10 = *R*479, 10*R*368, 10*R*2589*R*1567*R*1234. \n The straightforward calculations give \n*R*479, 10*R*368, 10*R*2589*R*1567*R*1234(*x*, *y*, *z*, *t*, *p*, *q*, *r*, *s*, *u*, *v*) = \n \n = (*f*1(*f*1(*x*, *y*), *p*), *f*1(*f*2(*x*, *y*), *f*2(*f*1(*x*, *y*), *p*)), *f*1(*g*1(*z*, *t*), *g*1(*q*, *r*)), *f*1(*g*2(*z*, *t*), *g*2(*q*, *r*)), \n \n*f*2(*f*2(*x*, *y*), *f*2(*f*1(*x*, *y*), *p*)), *f*2(*g*1(*z*, *t*), *g*1(*q*, *r*)), *f*2(*g*2(*z*, *t*), *g*2(*q*, *r*)), *g*1(*g*1(*s*, *u*), *v*), \n \n*g*1(*g*2(*s*, *u*), *g*2(*g*1(*s*, *u*), *v*)), *g*2(*g*2(*s*, *u*), *g*2(*g*1(*s*, *u*), *v*))), \n and \n*R*1234*R*1567*R*2589*R*368, 10*R*479, 10(*x*, *y*, *z*, *t*, *p*, *q*, *r*, *s*, *u*, *v*) = \n \n = (*f*1(*f*1(*x*, *f*2(*y*, *p*)), *f*1(*y*, *p*)), *f*2(*f*1(*x*, *f*2(*y*, *p*)), *f*1(*y*, *p*)), *g*1(*f*1(*z*, *q*), *f*1(*t*, *r*)), *g*2(*f*1(*z*, *q*), *f*1(*t*, *r*)), \n \n*f*2(*x*, *f*2(*y*, *p*)), *g*1(*f*2(*z*, *q*), *f*2(*t*, *r*)), *g*2(*f*2(*z*, *q*), *f*2(*t*, *r*)), *g*1(*g*1(*s*, *g*2(*u*, *v*)), *g*1(*u*, *v*)), \n \n*g*2(*g*1(*s*, *g*2(*u*, *v*)), *g*1(*u*, *v*)), *g*2(*s*, *g*2(*u*, *v*))). \n Equating the left-hand side with the right-hand side gives a system of 10 equalities. The first, second, and fifth equalities mean that *A* is a solution of the YBE. The last three equalities mean that *B* is a solution of the YBE. Hence,*R* is a solution of the 4-SE if and only if\n \n$$\\begin{aligned}\nf\\_1(g\\_1(z,t), g\\_1(q,r)) &=& g\\_1(f\\_1(z,q), f\\_1(t,r)),\\\\\nf\\_1(g\\_2(z,t), g\\_2(q, r)) &=& g\\_2(f\\_1(z,q), f\\_1(t, r)),\\\\\nf\\_2(g\\_1(z,t), g\\_1(q,r)) &=& g\\_2(f\\_2(z,q), f\\_2(t,r)),\\\\\nf\\_2(g\\_2(z,t), g\\_2(q,r)) &=& g\\_2(f\\_2(z,q), f\\_2(t,r)).\\end{aligned}$$ \nfor all *z*, *t*, *q*, *r* ∈ *X*. We see that these conditions are conditions on *f**i* and *g**j* from Theorem [const].\nLet *X* be a set and *A*, *B* : *X*2 → *X*2 are solutions of the YBE, which have the forms \n*A*(*x*, *y*) = (*f*(*x*, *y*), *h*(*y*)), *B*(*y*, *z*) = (*h*(*y*), *g*(*y*, *z*)), *x*, *y*, *z* ∈ *X*. \n By Lemma [YBEF] the maps *f*, *g* and *h* satisfy the identities \n*f*(*f*(*x*, *y*), *z*) = *f*(*f*(*x*, *h*(*z*)), *f*(*y*, *z*)), *f*(*h*(*y*), *h*(*z*)) = *h*(*f*(*y*, *z*)), \n \n*g*(*y*, *g*(*z*, *t*)) = *g*(*g*(*y*, *z*), *g*(*h*(*y*), *t*)), *g*(*h*(*y*), *h*(*z*)) = *h*(*g*(*y*, *z*)), \n for all *x*, *y*, *z*, *t* ∈ *X*. Set \n*R*(*x*, *y*, *z*) = (*f*(*x*, *y*), *h*(*y*), *g*(*y*, *z*)) \n is a map from *X*3 to *X*3. Then \n*R*123*R*145*R*246*R*356(*x*, *y*, *z*, *t*, *p*, *q*) = \n \n = (*f*(*f*(*x*, *h*(*t*)), *f*(*y*, *t*)), *h*(*f*(*y*, *t*)), *g*(*f*(*y*, *t*), *f*(*z*, *p*)), *h*2(*t*), *g*(*h*(*t*), *h*(*p*)), *g*(*t*, *g*(*p*, *q*))), \n and \n*R*356*R*246*R*145*R*123(*x*, *y*, *z*, *t*, *p*, *q*) = \n \n = (*f*(*f*(*x*, *y*), *t*), *f*(*h*(*y*), *h*(*t*)), *f*(*g*(*y*, *z*), *g*(*t*, *p*)), *h*2(*t*), *h*(*g*(*t*, *p*)), *g*(*g*(*t*, *p*), *g*(*h*(*t*), *q*))). \n Hence the map *R* is a solution of the TE if and only if for all *x*, *y*, *z*, *t*, *p*, *q* ∈ *X* the following equalities are true\n \n$$\\begin{aligned}\n f\\left( f(x, h(t)), f(y, t)\\right) &=& f\\left( f(x, y), t \\right), \\\\\n h (f(y, t)) &=& f(h(y), h(t)), \\\\\ng\\left( f(y, t), f(z, p) \\right) &=& f \\left( g(y, z), g(t, p) \\right),\\\\\ng(h(t), h(p)) &=& h (g(t, p)), \\\\\ng\\left( t, g(p, q) \\right) &=& g\\left( g(t, p), g(h(t), q) \\right).\\end{aligned}$$ \nSince *A* and *B* are solution of the YBE the first, second, forth and fifth equalities are satisfied. Hence, *R* is a solution of the TE if and only if for all *y*, *z*, *t*, *p* ∈ *X* holds \n*g*(*f*(*y*, *t*), *f*(*z*, *p*)) = *f*(*g*(*y*, *z*), *g*(*t*, *p*)). \nOn the other side, in this situation *n* = *m* = *k* = 1 and by Theorem [const] the map *A* #1*B* : *X*3 → *X*3, \n*A* #1*B*(*x*, *y*, *z*) = (*f*(*x*, *y*), *h*(*y*), *g*(*y*, *z*)) \n gives a solution for the TE if and only if the following equality true \n*g*(*f*(*a*11, *a*21), *f*(*a*12, *a*22)) = *f*(*g*(*a*11, *a*12), *g*(*a*21, *a*22)). \nNote that for *k* ∈ {0, 1} the conditions of Theorem [const] does not contain elements *b**i*, *j* and *b**i*, *j*. Moreover for *k* = 0 *A* #0*B* = *A* × *B*. It is easy to see that in this case if the function *A* × *B* is a solution of the *n*-SE then the function *B* × *A* is a solution as well.\nSimple solutions\n----------------\nIn this subsection we define some class of solutions to *n*-simplex equations.\nA solution *T* : *X**n* → *X**n* of the *n*-SE is said to be simple if \n*T*(*x*1, ... , *x**n*) = (*x**s*(1), ... , *x**s*(*n*)), where *s* : {1, ... , *n*} → {1, ... , *n*}is a map (not necessary injective). \n[simple] Let *X* be a set.\n1) The identity map id : *X* → *X* is a simple solution to the 1-SE.\n2) The maps *P*, *P**r*12, *P**r*22 : *X*2 → *X*2 that are defined by the rules \n$$\\begin{aligned}\n P : & & (x, y) & \\mapsto (y, x),\n \\\\\n Pr^2\\_1 : & & (x, y) & \\mapsto (x, x),\n \\\\\n Pr^2\\_2 : & & (x, y) & \\mapsto (y, y),\n \\end{aligned}$$ \n for all *x*, *y*, *z* ∈ *X*, are simple solutions to the 2-SE.\n3) The map *P**r*23 : *X*3 → *X*3, *P**r*23 : (*x*, *y*, *z*) ↦ (*y*, *y*, *y*),*x*, *y*, *z* ∈ *X*, is a simple solutions to the 3-SE.\nIt is evident that the direct product of simple solutions is a simple solution.\nFrom Theorem [const] follows\nLet *R* : *X**n* → *X**n* be a solution of the *n*-SE and *A* be any map from the list {id*X*, *P*, *P**r*12, *P**r*22, *P**r*23}. Then *R* × *A* and *A* × *R* are solutions of the (*n* + *m*)-SE.\nIt seems that any simple solution can be construct from the list {id*X*, *P*, *P**r*12, *P**r*22, *P**r*23} by a sequence of direct products.\nA solution *R* is called indecomposable if there are no solutions *A* and *B*, such that *R* = *A* × *B*.\n1) Is there a indecomposable simple solution of the *n*-SE for some *n* different from five solutions {id*X*, *P*, *P**r*12, *P**r*22, *P**r*23}?\n2) We know that the permutation *P*12 is a solution of the YBE. For which *n* > 2 there are non-identity permutations without fixed points that gives solutions of the *n*-SE?\nOf course, using Proposition [invsymm] it is not difficult to find some transpositions which are solutions of *n*-SE.\nLinear and affine solutions\n---------------------------\nLinear and affine solutions of the YBE and the TE are studied in.\nUsing Theorem [const] we are proving\n[sumlin] Let *A* and *B* be linear solutions of the *n*-SE and the *m*-SE, respectively. Then their 0-amalgam *A* × *B* is a linear solution of the (*n* + *m*)-SE. If *A* and *B* such that the 1-amalgam *A*#1*B* is defined, then it is a solution of the (*n* + *m* − 1)-SE.\nLet *f*: *X**n* → *X* be a linear map. Then it can be presented in terms of matrix multiplication \n*f*(*x̄*) = *a*1*x*1 + *a*2*x*2 + … + *a**n**x**n* = *x̄**T*[*f*] = [*f*]*T**x̄*, \n where [*f*]*T* = (*a*1, *a*2, …, *a**n*) is a vector of coefficients, *x̄**T* = (*x*1, *x*2, …, *x**n*) is a vector of variables and ⋅ *T* is the transposition. It is easy to see that the condition of Theorem [const] for *k* ∈ {0, 1} can be rewritten as \n[*f*]*T**M*[*g*] = [*g*]*T**M**T*[*f*], \n where *f* is a component of the map *A*, *g* is a component of the map *B* and *M* ia an *n* × *m* matrix and so the condition holds.\nThis means that for any linear solutions *A* and *B* of some *n*-SEs the map *A* #*k**B* for *k* ∈ {0, 1} will be a solution of some *n*-SE.\n1) What can we say on *k*-amalgam of affine solutions? Is it true that for *k* = 0 and/or *k* = 1 we get a solution?\n2) Under which conditions *k*-amalgam of rational solutions gives a rational solution?\nConstructing rational solutions from the linear solutions\n---------------------------------------------------------\nTake a linear solution (R, *R*) of the YBE: \n*R*(*x*, *y*) = (*α*1*x*, (1 − *α*1*α*2)*x* + *α*2*y*), where *α*1, *α*2 ∈ R. \n By proposition [sumlin] we have a solution *R* : R*n* → R*n* of an *n*-SE, defined by \n$$x\\_i \\mapsto \\begin{cases}\n \\alpha\\_i x\\_i & \\text{ if } i \\text{ is odd,}\\\\\n (1 - \\alpha\\_{i-1}\\alpha\\_i)x\\_{i-1} + \\alpha\\_i x\\_i + (1 - \\alpha\\_i \\alpha\\_{i+1})x\\_{i+1} & \\text{ if } i \\text{ is even,}\\\\\n(1 - \\alpha\\_{i-1}\\alpha\\_i)x\\_{n-1} + \\alpha\\_i x\\_n & \\text{ if } i=n \\text{ is even,}\\\\\n \\end{cases}$$ \n where $\\alpha\\_i \\in \\mathbb R$.\nTake any rational function *f* : R → R having a rational inverse, for example a linear fractional transformation \n$$f(x) = \\dfrac{ax + b}{cx + d}, \\quad f^{-1}(x) = \\dfrac{dx - b}{a - cx}.$$ \n We can conjugate our solution by *f*. This yields a rational solution (*f*− 1)*n**R**f**n*. In the case of a linear rational transformation, this conjugation takes the form \n$$x\\_i \\mapsto \\begin{cases} \\dfrac{(\\alpha\\_i ad - bc)x\\_i + (\\alpha\\_i - 1)bd}{ca(1-\\alpha\\_i) x\\_i + (ad - \\alpha\\_i bc)}& \\text{ if } i \\text{ is odd,}\\\\\n & \\\\\n \\dfrac{d \\left(\\beta\\_i \\tfrac{a x\\_{i-1} + b}{c x\\_{i-1} + d}+ \\alpha\\_{i} \\tfrac{a x\\_{i} + b}{c x\\_{i} + d} + \\beta\\_{i+1}\\frac{a x\\_{i+1} + b}{c x\\_{i+1} + d}\\right) - b}{a - c \\left(\\beta\\_i\\tfrac{a x\\_{i-1} + b}{c x\\_{i-1} + d}+ \\alpha\\_{i} \\tfrac{a x\\_{i} + b}{c x\\_{i} + d} + \\beta\\_{i+1} \\tfrac{a x\\_{i+1} + b}{c x\\_{i+1} + d}\\right)} & \\text{ if } i \\text{ is even, }\\\\\n& \\\\\n \\dfrac{d \\left(\\beta\\_n \\tfrac{a x\\_{n-1} + b}{c x\\_{n-1} + d}+ \\alpha\\_{n} \\tfrac{a x\\_{n} + b}{c x\\_{n} + d} \\right) - b}{a - c \\left( \\beta\\_n\\tfrac{a x\\_{n-1} + b}{c x\\_{n-1} + d}+ \\alpha\\_{n} \\tfrac{a x\\_{n} + b}{c x\\_{n} + d} \\right) } & \\text{ if } i=n \\text{ is even, } \n \\end{cases}$$ \n where *β**i* = 1 − *α**i* − 1*α**i* for *i* ∈ {2, 3, …, *n*}. This construction provides lots of examples of new rational solutions of arbitrary *n*-SE and one can wonder whether or not some solution is a conjugate to a linear one.\nIf we put \n*α*1 = *α*3 = 1, *α*2 = *α*4 = 0 \n then \n*β*2 = *β*3 = *β*4 = 1. \n Also let \n*a* = *c* = 1, *b* = 0, *d* = − 1, \n then we get a rational solution of the 4-SE \n$$(x\\_1, x\\_2, x\\_3, x\\_4) \\mapsto \\left(x\\_1, \\dfrac{x\\_1 + x\\_3 - 2 x\\_1 x\\_3}{1 - x\\_1 x\\_3}, x\\_3, x\\_3\\right).$$ \nLet *R* be a linear solution of the 4-SE defined by \n*R*(*x*1, *x*2, *x*3, *x*4) = (*x*2 − *x*4, *x*1 + *x*3, *x*3, *x*4). \n By taking *f* equal to $ \\dfrac{x}{x-1}$ we get a rational solution of the 4-SE \n$$(f^{-1})^n R f^n (x\\_1, x\\_2, x\\_3, x\\_4) = \\left(\\dfrac{x\\_2 - x\\_4}{1 + x\\_2 x\\_4}, \\dfrac{x\\_1 + x\\_3 - 2x\\_1x\\_3}{1 - x\\_1 x\\_3}, x\\_3, x\\_4\\right).$$ \nFrom solutions of the *n*-SE to solutions of the (*n* − 1)-SE\n-------------------------------------------------------------\nWe defined a *k*-amalgam of solutions, which gives a possibility to construct solutions of higher dimensions. Now we are going to the opposite direction: If (*X*, *R*) is a solution of the *n*-SE is it possible to construct a solution of the (*n* − 1)-SE?\nLet *R* : *X**n* → *X**n*, *n* ≥ 3, be a solution of the *n*-SE, and there exists *x*0 ∈ *X* such that *R*(*x*0, ... , *x*0) = (*x*0, ... , *x*0). Then \n*R**r*(*x*1, ... , *x**n* − 1) : = *R*(*x*0, *x*1, ... , *x**n* − 1) \n and \n*R**l*(*x*1, ... , *x**n* − 1) : = *R*(*x*1, ... , *x**n* − 1, *x*0) \n are solutions of the (*n* − 1)-SE.\nIf *R* is a solution of the *n*-SE, then the maps *R**ī*, 1 ≤ *i* ≤ *n* + 1, act on *X**N*, where *N* = *n*(*n* + 1)/2. Let us consider a set *X**x*0 ⊂ *X**N* with first *n* coordinates equal to *x*0, \n$$X\\_{x\\_0} = \\{ (\\underbrace{x\\_0, \\ldots, x\\_0}\\_n, x\\_{n+1}, x\\_{n+2}, \\ldots, x\\_N) \\in X^N \\}.$$ \n We restrict the *n*-SE to the set *X**x*0. In the right hand side of this equation we apply at first ${R}\\_{\\bar{1}}$ \n$${R}\\_{\\bar{1}} (\\underbrace{x\\_0, \\ldots, x\\_0}\\_n, x\\_{n+1}, x\\_{n+2}, \\ldots, x\\_N) = R(x\\_0, \\ldots, x\\_0) \\times {\\mathrm{id}}^{N-n}( x\\_{n+1}, x\\_{n+2}, \\ldots, x\\_N) =$$ \n \n$$=(\\underbrace{x\\_0, \\ldots, x\\_0}\\_n, x\\_{n+1}, x\\_{n+2}, \\ldots, x\\_N).$$ \n Hence, ${R}\\_{\\bar{1}} $ acts identically on *X**x*0 and on this set *X**x*0 the *n*-SE has the form \n$${R}\\_{\\bar{1}} {R}\\_{\\bar{2}} \\ldots {R}\\_{\\overline{n+1}} = {R}\\_{\\overline{n+1}} \\ldots {R}\\_{\\bar{2}}.$$ \n Both sides of the (*n* − 1)-SE act on *X**N*ʹ where *N*ʹ = *N* − *n*. If we remove the first *n* coordinates and consider the equation above restricted to the last *N*ʹ coordinate, we get a solution of the (*n* − 1)-SE after shifting all indices by − *n*.\nTo prove that *R**l* is a solution of the (*n* − 1)-SE it is need to take the subset of *X**N* in which the last *n* coordinates are equal to *x*0 and remark that the map $R\\_{\\overline{n+1}}$ acts identically on this subset.\nTropicalization\n===============\nTropicalization of rational solutions\n-------------------------------------\nBy tropicalization we mean a map from the set of rational function to the set of piecewise linear functions. I. Dynnikov used tropicalization to study representations of the braid group *B**n* into the permutation group *S**y**m*(Z2*n*). In particular, he finds a non-degenerate solution of the YBE over Z2.\nIn the present section we find a connection between rational solutions of *n*-SEs and piecewise linear solutions of *n*-SEs. To formulate the main result, we introduce some definitions and notations. Let R(*x*1, *x*2, …, *x**n*) be the field of rational fractions with coefficients in the field of real numbers R. Any solution (R, *R*) of the *n*-SE, where \n*R*(*x*1, *x*2, …, *x**n*) = (*r*1, *r*2, …, *r**n*), *r**i* ∈ R(*x*1, *x*2, …, *x**n*), \n is said to be a *rational solution*.\nLet us denote by *I**n* a subset of non-zero fractions *r* = *f*/*g* ∈ R(*x*1, *x*2, …, *x**n*) such that all coefficients of the numerator *f* are equal to 1 and the free term is equal to zero, the denominator *g* is equal to 1 or all its coefficients are equal to 1 and the free term is equal to zero. A rational solution \n*R*(*x*1, *x*2, …, *x**n*) = (*r*1, *r*2, …, *r**n*), *r**i* ∈ R(*x*1, *x*2, …, *x**n*), \n of the *n*-SE is said to be an *I**–rational solution* if all components *r**i* lie in *I**n*.\n[ex1] It is easy to see that the so called *electric solution* of the TE,\n \n$$\\begin{aligned}\n R\\_E(x,y,z) &= \\left(\\frac{xy}{x+z+xyz},\\ x+z+xyz,\\ \\frac{yz}{x+z+xyz}\\right)\\\\\\end{aligned}$$ \nthat appears in the theory of electric circuits as the famous Y-Δ transform of an electric network is an *I*–rational solution.\nThe map\n \n$$\\begin{aligned}\n R\\_e(x,y,z) &= \\left(\\frac{xy}{x+z},\\ x+z,\\ \\frac{yz}{x+z}\\right)\\end{aligned}$$ \nwhich is obtained from *R**E* by removing terms of degree three is also an *I*–rational solution.\nLet us denote by *P**L**n* the set of piecewise linear functions R*n* → R.\nThe *tropicalization* is a map ( ⋅ )*t* : *I**n* → *P**L**n* that is defined on *r* = *f*/*g* ∈ *I**n*, where \n*f* = ∑*i*1 + … + *i**n* > 0*α**i*1…*i**n**x*1*i*1…*x**n**i**n*, *g* = ∑*j*1 + … + *j**n* ≥ 0*β**j*1…*j**n**x*1*j*1…*x**n**j**n* \n by the rule rt =\n{ i1 x1 + …+ in xn } - { j1 x1 + …+jn xn }, & \n { i1 x1 + …+ in xn }, &\nNote that *r**t* can be obtained from *r* using the following recursive procedure.\n[rec] Let *r* = *r*(*x*1, …, *x**n*), *r*1 = *r*1(*x*1, …, *x**n*), *r*2 = *r*2(*x*1, …, *x**n*) be rational functions from *I**n*. Then\n1. if *r* = *x**i*, then *r**t* = *x**i*, for *i* = 1, …, *n*;\n2. (*r*1 + *r*2)*t* = max{*r*1*t*, *r*2*t*};\n3. (*r*1*r*2)*t* = *r*1*t* + *r*2*t*;\n4. $\\left(\\dfrac{r\\_1}{r\\_2}\\right)^t = r\\_1^t - r\\_2^t$.\nIn particular, tropicalization is correctly defined i.e. if *r*1 and *r*2 are two equal rational functions, then their tropicalizations are also equal functions.\nLet *R*(*x*1, …, *x**n*) = (*r*1, *r*2, …, *r**n*) ∈ (*I**n*)*n* be a rational vector-valued map of *n* variables. Define the *tropicalization* of the rational map *R* componentwise: \n*R**t*(*x*1, …, *x**n*) : = (*r*1*t*(*x*1, …, *x**n*), …, *r**n**t*(*x*1, …, *x**n*)). \nThe tropicalization of the electric solution *R**E* from Example [ex1] is \n*R**E**t*(*x*, *y*, *z*) = (*x* + *y* − max{*x*, *z*, *x* + *y* + *z*}, max{*x*, *z*, *x* + *y* + *z*}, *y* + *z* − max{*x*, *z*, *x* + *y* + *z*}). \n The tropicalization of *R**e* from Example [ex1] is \n*R**e**t*(*x*, *y*, *z*) = (*x* + *y* − max{*x*, *z*}, max{*x*, *z*}, *y* + *z* − max{*x*, *z*}). \n It is easy to see that *R**E**t* and *R**e**t* are piecewise linear solutions of the TE.\nThe solution *R**E**t* consists of 3 linear pieces: \n*R*1(*x*, *y*, *z*) = (*y*, *x*, *y* + *z* − *x*), \n \n*R*2(*x*, *y*, *z*) = (*x* + *y* − *z*, *z*, *y*), \n \n*R*3(*x*, *y*, *z*) = ( − *z*, *x* + *y* + *z*, − *x*). \n We can consider these maps as maps on R3. One can check that (R, *R*1) and (R, *R*2) are solutions of the TE, but (R, *R*3) is not.\nLet *r*1 and *r*2 be two rational functions in R(*x*1, …, *x**n*) and 1 ≤ *k* ≤ *n*. A *k*-*composition* of *r*1 and *r*2 is a rational function *r*1 ∘ *k**r*2 ∈ R(*x*1, …, *x**n*) defined as \n(*r*1 ∘ *k**r*2)(*x*1, …, *x**n*) : = *r*1(*x*1, …, *x**k* − 1, *r*2(*x*1, …, *x**n*), *x**k* + 1, …, *x**n*). \nIf *r*1, *r*2 ∈ *I**n*, then (*r*1 ∘ *k**r*2)*t* = *r*1*t* ∘ *k**r*2*t*.\nThe proof uses the recursive procedure given in Proposition [rec]. If *r*1(*x*1, …, *x**n*) = *x**i* is a variable, then the statement is obvious. If \n*r*1(*x*1, …, *x**n*) = *f*(*x*1, …, *x**n*) \\* *g*(*x*1, …, *x**n*), \n where *f* and *g* are rational functions in *I**n* and \\* ∈ { + , ⋅ , /} is a binary operation in rational function *r*1, then the tropicalization \n*r*1*t*(*x*1, …, *x**n*) = *f**t*(*x*1, …, *x**n*) \\* *t**g**t*(*x*1, …, *x**n*), \n where \\* *t* ∈ {max, + , − }, and \n(*r*1 ∘ *k**r*2)(*x*1, …, *x**n*) = (*f* ∘ *k**r*2)(*x*1, …, *x**n*) \\* (*g* ∘ *k**r*2)(*x*1, …, *x**n*). \n Hence, the tropicalization \n(*r*1 ∘ *k**r*2)*t*(*x*1, …, *x**n*) = (*f* ∘ *k**r*2)*t*(*x*1, …, *x**n*) \\* *t*(*g* ∘ *k**r*2)*t*(*x*1, …, *x**n*). \n The needed statement could be obtained by induction on a recursion depth.\n[cortr] Tropicalization preserves a composition of rational vector-valued maps of the form described above.\nIf (R> 0, *R*), *R* ∈ (*I**n*)*n* is a solution of the *n*-simplex equation on the set of positive real numbers, then its tropicalization (R> 0, *R**t*) is also a solution of the *n*-simplex equation.\nFor any *n* ≥ 2, the *n*-SE has the following form \n$$R\\_{\\overline{1}}R\\_{\\overline{2}} \\cdots R\\_{\\overline{n+1}}\n =\n R\\_{\\overline{n+1}} \\cdots R\\_{\\overline{2}} R\\_{\\overline{1}}.$$ \n Apply the tropicalization procedure to both sides: \n$$(R\\_{\\overline{1}}R\\_{\\overline{2}} \\cdots R\\_{\\overline{n+1}})^t = (R\\_{\\overline{n+1}} \\cdots R\\_{\\overline{2}} R\\_{\\overline{1}})^t.$$ \n Hence, according to Corollary [cortr], the following relation holds: \n$$R\\_{\\overline{1}}^t R\\_{\\overline{2}}^t \\cdots R\\_{\\overline{n+1}}^t\n =\n R\\_{\\overline{n+1}}^t \\cdots R\\_{\\overline{2}}^t R\\_{\\overline{1}}^t.$$ \n It means that *R**t* is a solution of the *n*-simplex equation.\nFormal tropicalization of solutions on algebraic systems\n--------------------------------------------------------\nThe reasonable question that arises is whether we can make a replacement, analogous to a tropicalization, in solutions of the *n*-SE such that we again obtain a solution. Obviously this question has a positive answer. Such substitutions of functions in formulas of solutions that transform solutions into solutions we will call formal tropicalizations. In order to give more formal definition we shall recall the notion of a partial algebra. A partial algebra is a generalization of universal algebra that allows partial operations. A natural example of partial algebra is a field, a multiplicative inverse is not defined for element 0.\nLet A be a partial algebra of signature Σ and *R* : *A**n* → *A**n* be a partial function that can be expressed as a realization of a collection (*f*1, ... , *f**n*) of *n*-ary functions–terms of a signature Σʹ ⊂ Σ such that (*A*, *R*) is a solution of the *n*-SE. Let B be a partial algebra of a signature Ξ, and *g* is a mapping that match every *n*-ary functional symbol from Σʹ with *n*-ary function of the signature Ξ. Then for every *f**i* we can define *g*(*f**i*) as *n*-ary function of the signature Ξ where every functional symbol from Σʹ is replaced with corresponding *n*-ary function of the signature Ξ. The function *R**g* : *B**n* → *B**n* that is realization of a collection (*g*(*f*1), ... , *g*(*f**n*)) will be called a *formal tropicalization* of the solution *R* if (*B*, *R**g*) is a solution of the *n*-SE.\nThe sentence “(*A*, *R*) is a solution of the *n*-SE” could be understood as a system of *N* equations Φ*R* with *N* variables such that for any interpretation of variables every equation is either true or at least one side is undefined. This is just an explicit form of the *n*-SE.\nThose equations can be viewed as formulas of equational logic. For some survey of equational logic in the context of algebras one can see.\nFollowing the notation above, let *T* be a set of equational formulas of signature Σʹ such that A satisfies *T* and every formula Φ*R* has finite proof starting from *T*. These formulae will be called *axioms* of the solutions (*A*, *R*).\nLet (*A*, *R*) be a solution of the *n*-SE and *T* its set of axioms. If *g* is a mapping such that B satisfies *g*(*T*) : = {*g*(*ϕ*) : *ϕ* ∈ *T*} then (*B*,"},"data_id":{"kind":"string","value":"arxiv_0004112"}}},{"rowIdx":995,"cells":{"content":{"kind":"string","value":"2, 0, 1, 1, 0]],\n[[2, 0, 0, 1, 0],[2, 0, 0, 1, 1],[2, 0, 1, 0, 0],[3, 0, 0, 0, 0],[3, 0, 1, 0, 1]],\n[[1, 0, 1, 0, 1],[2, 1, 0, 0, 0],[3, 0, 0, 0, 1],[3, 0, 0, 1, 0],[3, 1, 1, 1, 0]],\n[[2, 1, 1, 0, 0],[2, 1, 1, 0, 1],[2, 1, 1, 1, 0],[3, 0, 1, 0, 0],[3, 1, 0, 1, 1]],\n[[2, 1, 0, 0, 1],[2, 1, 0, 1, 0],[2, 1, 1, 1, 1],[3, 0, 0, 1, 1],[3, 1, 1, 1, 1]],\n[[0, 0, 1, 0, 1],[0, 0, 1, 1, 0],[0, 1, 1, 1, 1],[1, 1, 0, 1, 1],[3, 0, 1, 1, 1]],\n[[1, 1, 0, 0, 0],[2, 1, 0, 1, 1],[3, 0, 1, 1, 0],[3, 1, 0, 0, 0],[3, 1, 1, 0, 1]]\n]\nA partition for sets of sizes: 8*3 1*4 7*5\n[4, 2, 2, 2, 2]\n[6, 0, 9]\n[\n[[0, 0, 0, 0, 1],[0, 0, 0, 1, 0],[0, 0, 0, 1, 1]],\n[[0, 0, 1, 0, 0],[0, 1, 0, 0, 0],[0, 1, 1, 0, 0]],\n[[0, 0, 1, 1, 1],[0, 1, 0, 0, 1],[0, 1, 1, 1, 0]],\n[[0, 1, 0, 1, 0],[1, 0, 0, 0, 0],[3, 1, 0, 1, 0]],\n[[0, 1, 1, 0, 1],[1, 0, 0, 0, 1],[3, 1, 1, 0, 0]],\n[[0, 1, 0, 1, 1],[1, 0, 0, 1, 0],[3, 1, 0, 0, 1]],\n[[1, 0, 0, 1, 1],[1, 0, 1, 0, 0],[1, 0, 1, 0, 1],[2, 0, 0, 0, 0],[3, 0, 0, 1, 0]],\n[[1, 1, 0, 0, 0],[1, 1, 0, 0, 1],[1, 1, 0, 1, 0],[2, 0, 0, 1, 1],[3, 1, 0, 0, 0]],\n[[1, 1, 1, 0, 1],[1, 1, 1, 1, 0],[1, 1, 1, 1, 1],[2, 0, 0, 0, 1],[3, 1, 1, 0, 1]],\n[[2, 0, 0, 1, 0],[2, 0, 1, 0, 0],[2, 0, 1, 0, 1],[3, 0, 0, 0, 0],[3, 0, 0, 1, 1]],\n[[2, 0, 1, 1, 1],[2, 1, 0, 0, 0],[2, 1, 0, 0, 1],[3, 0, 0, 0, 1],[3, 0, 1, 1, 1]],\n[[2, 1, 1, 0, 0],[2, 1, 1, 0, 1],[2, 1, 1, 1, 0],[3, 0, 1, 0, 0],[3, 1, 0, 1, 1]],\n[[1, 0, 1, 1, 1],[2, 1, 0, 1, 1],[3, 0, 1, 0, 1],[3, 0, 1, 1, 0],[3, 1, 1, 1, 1]],\n[[0, 0, 1, 0, 1],[0, 0, 1, 1, 0],[0, 1, 1, 1, 1],[2, 0, 1, 1, 0],[2, 1, 0, 1, 0]],\n[[1, 0, 1, 1, 0],[1, 1, 0, 1, 1],[1, 1, 1, 0, 0],[2, 1, 1, 1, 1],[3, 1, 1, 1, 0]]\n]\nA partition for sets of sizes: 6*3 0*4 9*5\n[4, 2, 2, 2, 2]\n[3, 1, 10]\n[\n[[0, 0, 0, 0, 1],[0, 0, 0, 1, 0],[0, 0, 0, 1, 1]],\n[[0, 0, 1, 0, 0],[0, 1, 0, 0, 0],[0, 1, 1, 0, 0]],\n[[0, 0, 1, 1, 1],[0, 1, 0, 0, 1],[0, 1, 1, 1, 0]],\n[[0, 1, 0, 1, 0],[0, 1, 0, 1, 1],[1, 0, 0, 0, 0],[3, 0, 0, 0, 1]],\n[[0, 0, 1, 0, 1],[0, 1, 1, 1, 1],[1, 0, 0, 0, 1],[1, 0, 0, 1, 0],[2, 1, 0, 0, 1]],\n[[1, 0, 0, 1, 1],[1, 0, 1, 0, 0],[1, 0, 1, 0, 1],[2, 0, 0, 0, 0],[3, 0, 0, 1, 0]],\n[[1, 1, 0, 0, 0],[1, 1, 0, 0, 1],[1, 1, 0, 1, 0],[2, 0, 0, 0, 1],[3, 1, 0, 1, 0]],\n[[1, 1, 1, 0, 1],[1, 1, 1, 1, 0],[1, 1, 1, 1, 1],[2, 0, 0, 1, 0],[3, 1, 1, 1, 0]],\n[[1, 0, 1, 1, 0],[2, 0, 0, 1, 1],[3, 0, 0, 0, 0],[3, 0, 0, 1, 1],[3, 0, 1, 1, 0]],\n[[2, 0, 1, 1, 1],[2, 1, 0, 0, 0],[2, 1, 0, 1, 0],[3, 1, 0, 0, 0],[3, 1, 1, 0, 1]],\n[[2, 1, 1, 0, 0],[2, 1, 1, 0, 1],[2, 1, 1, 1, 0],[3, 0, 1, 0, 0],[3, 1, 0, 1, 1]],\n[[0, 1, 1, 0, 1],[1, 0, 1, 1, 1],[2, 0, 1, 0, 1],[2, 0, 1, 1, 0],[3, 1, 0, 0, 1]],\n[[0, 0, 1, 1, 0],[1, 1, 1, 0, 0],[2, 0, 1, 0, 0],[2, 1, 0, 1, 1],[3, 0, 1, 0, 1]],\n[[1, 1, 0, 1, 1],[2, 1, 1, 1, 1],[3, 0, 1, 1, 1],[3, 1, 1, 0, 0],[3, 1, 1, 1, 1]]\n]\nA partition for sets of sizes: 3*3 1*4 10*5\n[4, 2, 2, 2, 2]\n[1, 0, 12]\n[\n[[0, 0, 0, 0, 1],[0, 0, 0, 1, 0],[0, 0, 0, 1, 1]],\n[[0, 0, 1, 0, 0],[0, 0, 1, 0, 1],[0, 0, 1, 1, 0],[0, 1, 0, 0, 0],[0, 1, 1, 1, 1]],\n[[0, 1, 0, 0, 1],[0, 1, 0, 1, 0],[0, 1, 0, 1, 1],[1, 0, 0, 0, 0],[3, 1, 0, 0, 0]],\n[[0, 1, 1, 1, 0],[1, 0, 0, 0, 1],[1, 0, 0, 1, 0],[1, 0, 0, 1, 1],[1, 1, 1, 1, 0]],\n[[0, 0, 1, 1, 1],[1, 0, 1, 0, 0],[1, 0, 1, 0, 1],[1, 1, 0, 0, 1],[1, 1, 1, 1, 1]],\n[[1, 1, 0, 0, 0],[1, 1, 0, 1, 0],[1, 1, 0, 1, 1],[2, 0, 0, 0, 0],[3, 1, 0, 0, 1]],\n[[1, 1, 1, 0, 1],[2, 0, 0, 0, 1],[3, 0, 0, 0, 0],[3, 0, 0, 0, 1],[3, 1, 1, 0, 1]],\n[[2, 0, 0, 1, 0],[2, 0, 0, 1, 1],[2, 0, 1, 0, 0],[3, 0, 0, 1, 0],[3, 0, 1, 1, 1]],\n[[2, 0, 1, 1, 1],[2, 1, 0, 0, 0],[2, 1, 0, 0, 1],[3, 0, 0, 1, 1],[3, 0, 1, 0, 1]],\n[[2, 1, 1, 0, 0],[2, 1, 1, 0, 1],[2, 1, 1, 1, 0],[3, 0, 1, 0, 0],[3, 1, 0, 1, 1]],\n[[1, 0, 1, 1, 0],[2, 0, 1, 0, 1],[3, 0, 1, 1, 0],[3, 1, 0, 1, 0],[3, 1, 1, 1, 1]],\n[[2, 0, 1, 1, 0],[2, 1, 0, 1, 1],[2, 1, 1, 1, 1],[3, 1, 1, 0, 0],[3, 1, 1, 1, 0]],\n[[0, 1, 1, 0, 0],[0, 1, 1, 0, 1],[1, 0, 1, 1, 1],[1, 1, 1, 0, 0],[2, 1, 0, 1, 0]]\n]\nA partition for sets of sizes: 1*3 0*4 12*5\n```\n```\n[4, 2, 2, 2, 2, 2]\n[41, 1, 0]\n[\n[[0, 0, 0, 0, 0, 1],[0, 0, 0, 0, 1, 0],[0, 0, 0, 0, 1, 1]],\n[[0, 0, 0, 1, 0, 0],[0, 0, 1, 0, 0, 0],[0, 0, 1, 1, 0, 0]],\n[[0, 0, 0, 1, 1, 1],[0, 0, 1, 0, 0, 1],[0, 0, 1, 1, 1, 0]],\n[[0, 0, 1, 0, 1, 0],[0, 1, 0, 0, 0, 0],[0, 1, 1, 0, 1, 0]],\n[[0, 0, 1, 1, 0, 1],[0, 1, 0, 0, 0, 1],[0, 1, 1, 1, 0, 0]],\n[[0, 0, 1, 0, 1, 1],[0, 1, 0, 0, 1, 0],[0, 1, 1, 0, 0, 1]],\n[[0, 1, 0, 0, 1, 1],[1, 0, 0, 0, 0, 0],[3, 1, 0, 0, 1, 1]],\n[[0, 1, 0, 1, 1, 0],[1, 0, 0, 0, 0, 1],[3, 1, 0, 1, 1, 1]],\n[[0, 0, 0, 1, 0, 1],[0, 1, 1, 0, 1, 1],[0, 1, 1, 1, 1, 0]],\n[[0, 0, 1, 1, 1, 1],[0, 1, 0, 1, 1, 1],[0, 1, 1, 0, 0, 0]],\n[[0, 1, 1, 1, 1, 1],[1, 0, 0, 0, 1, 0],[3, 1, 1, 1, 0, 1]],\n[[0, 1, 0, 1, 0, 0],[1, 0, 0, 1, 0, 0],[3, 1, 0, 0, 0, 0]],\n[[1, 0, 0, 1, 0, 1],[1, 0, 0, 1, 1, 0],[2, 0, 0, 0, 1, 1]],\n[[1, 0, 1, 0, 0, 0],[1, 0, 1, 0, 0, 1],[2, 0, 0, 0, 0, 1]],\n[[1, 0, 1, 0, 1, 1],[1, 0, 1, 1, 0, 0],[2, 0, 0, 1, 1, 1]],\n[[1, 0, 1, 1, 1, 0],[1, 1, 0, 0, 0, 0],[2, 1, 1, 1, 1, 0]],\n[[1, 1, 0, 0, 0, 1],[1, 1, 0, 0, 1, 1],[2, 0, 0, 0, 1, 0]],\n[[1, 1, 0, 1, 0, 0],[1, 1, 1, 0, 0, 0],[2, 0, 1, 1, 0, 0]],\n[[1, 1, 0, 1, 1, 1],[1, 1, 1, 0, 0, 1],[2, 0, 1, 1, 1, 0]],\n[[1, 1, 1, 0, 1, 0],[1, 1, 1, 1, 0, 0],[2, 0, 0, 1, 1, 0]],\n[[1, 0, 1, 0, 1, 0],[1, 1, 1, 1, 1, 0],[2, 1, 0, 1, 0, 0]],\n[[1, 1, 0, 0, 1, 0],[1, 1, 1, 1, 0, 1],[2, 0, 1, 1, 1, 1]],\n[[1, 0, 0, 1, 1, 1],[1, 1, 1, 0, 1, 1],[2, 1, 1, 1, 0, 0]],\n[[2, 0, 0, 1, 0, 0],[3, 0, 0, 0, 0, 0],[3, 0, 0, 1, 0, 0]],\n[[2, 0, 1, 0, 0, 1],[3, 0, 0, 0, 0, 1],[3, 0, 1, 0, 0, 0]],\n[[2, 0, 1, 1, 0, 1],[3, 0, 0, 0, 1, 0],[3, 0, 1, 1, 1, 1]],\n[[2, 1, 0, 0, 0, 0],[3, 0, 0, 1, 0, 1],[3, 1, 0, 1, 0, 1]],\n[[2, 1, 0, 0, 1, 0],[3, 0, 0, 0, 1, 1],[3, 1, 0, 0, 0, 1]],\n[[2, 1, 0, 1, 0, 1],[3, 0, 0, 1, 1, 1],[3, 1, 0, 0, 1, 0]],\n[[2, 1, 1, 0, 0, 0],[3, 0, 0, 1, 1, 0],[3, 1, 1, 1, 1, 0]],\n[[2, 1, 1, 0, 1, 1],[3, 0, 1, 1, 0, 1],[3, 1, 0, 1, 1, 0]],\n[[1, 0, 1, 1, 1, 1],[1, 1, 0, 1, 1, 0],[2, 1, 1, 0, 0, 1]],\n[[2, 1, 0, 0, 1, 1],[3, 0, 1, 0, 1, 0],[3, 1, 1, 0, 0, 1]],\n[[2, 0, 1, 0, 0, 0],[3, 1, 0, 1, 0, 0],[3, 1, 1, 1, 0, 0]],\n[[2, 0, 0, 1, 0, 1],[3, 1, 1, 0, 1, 0],[3, 1, 1, 1, 1, 1]],\n[[1, 0, 0, 0, 1, 1],[1, 1, 0, 1, 0, 1],[2, 1, 0, 1, 1, 0]],\n[[2, 1, 0, 1, 1, 1],[3, 0, 1, 1, 0, 0],[3, 1, 1, 0, 1, 1]],\n[[0, 1, 0, 1, 0, 1],[2, 0, 1, 0, 1, 0],[2, 1, 1, 1, 1, 1]],\n[[2, 1, 0, 0, 0, 1],[3, 0, 1, 0, 0, 1],[3, 1, 1, 0, 0, 0]],\n[[0, 0, 0, 1, 1, 0],[1, 0, 1, 1, 0, 1],[3, 0, 1, 0, 1, 1]],\n[[0, 1, 1, 1, 0, 1],[2, 0, 0, 0, 0, 0],[2, 1, 1, 1, 0, 1]],\n[[1, 1, 1, 1, 1, 1],[2, 0, 1, 0, 1, 1],[2, 1, 1, 0, 1, 0],[3, 0, 1, 1, 1, 0]]\n]\nA partition for sets of sizes: 41*3 1*4 0*5\n```\n```\n[4, 2, 2, 2, 2, 2]\n[38, 2, 1]\n[\n[[0, 0, 0, 0, 0, 1],[0, 0, 0, 0, 1, 0],[0, 0, 0, 0, 1, 1]],\n[[0, 0, 0, 1, 0, 0],[0, 0, 1, 0, 0, 0],[0, 0, 1, 1, 0, 0]],\n[[0, 0, 0, 1, 1, 1],[0, 0, 1, 0, 0, 1],[0, 0, 1, 1, 1, 0]],\n[[0, 0, 1, 0, 1, 0],[0, 1, 0, 0, 0, 0],[0, 1, 1, 0, 1, 0]],\n[[0, 0, 1, 1, 0, 1],[0, 1, 0, 0, 0, 1],[0, 1, 1, 1, 0, 0]],\n[[0, 0, 1, 0, 1, 1],[0, 1, 0, 0, 1, 0],[0, 1, 1, 0, 0, 1]],\n[[0, 1, 0, 0, 1, 1],[1, 0, 0, 0, 0, 0],[3, 1, 0, 0, 1, 1]],\n[[0, 1, 0, 1, 1, 0],[1, 0, 0, 0, 0, 1],[3, 1, 0, 1, 1, 1]],\n[[0, 0, 0, 1, 0, 1],[0, 1, 1, 0, 1, 1],[0, 1, 1, 1, 1, 0]],\n[[0, 0, 1, 1, 1, 1],[0, 1, 0, 1, 1, 1],[0, 1, 1, 0, 0, 0]],\n[[0, 1, 1, 1, 1, 1],[1, 0, 0, 0, 1, 0],[3, 1, 1, 1, 0, 1]],\n[[0, 1, 0, 1, 0, 0],[1, 0, 0, 1, 0, 0],[3, 1, 0, 0, 0, 0]],\n[[1, 0, 0, 1, 0, 1],[1, 0, 0, 1, 1, 0],[2, 0, 0, 0, 1, 1]],\n[[1, 0, 1, 0, 0, 0],[1, 0, 1, 0, 0, 1],[2, 0, 0, 0, 0, 1]],\n[[1, 0, 1, 0, 1, 1],[1, 0, 1, 1, 0, 0],[2, 0, 0, 1, 1, 1]],\n[[1, 0, 1, 1, 1, 0],[1, 1, 0, 0, 0, 0],[2, 1, 1, 1, 1, 0]],\n[[1, 1, 0, 0, 0, 1],[1, 1, 0, 0, 1, 1],[2, 0, 0, 0, 1, 0]],\n[[1, 1, 0, 1, 0, 0],[1, 1, 1, 0, 0, 0],[2, 0, 1, 1, 0, 0]],\n[[1, 1, 0, 1, 1, 1],[1, 1, 1, 0, 0, 1],[2, 0, 1, 1, 1, 0]],\n[[1, 1, 1, 0, 1, 0],[1, 1, 1, 1, 0, 0],[2, 0, 0, 1, 1, 0]],\n[[1, 0, 1, 0, 1, 0],[1, 1, 1, 1, 1, 0],[2, 1, 0, 1, 0, 0]],\n[[1, 1, 0, 0, 1, 0],[1, 1, 1, 1, 0, 1],[2, 0, 1, 1, 1, 1]],\n[[1, 0, 0, 1, 1, 1],[1, 1, 1, 0, 1, 1],[2, 1, 1, 1, 0, 0]],\n[[2, 0, 0, 1, 0, 0],[3, 0, 0, 0, 0, 0],[3, 0, 0, 1, 0, 0]],\n[[2, 0, 1, 0, 0, 1],[3, 0, 0, 0, 0, 1],[3, 0, 1, 0, 0, 0]],\n[[2, 0, 1, 1, 0, 1],[3, 0, 0, 0, 1, 0],[3, 0, 1, 1, 1, 1]],\n[[2, 1, 0, 0, 0, 0],[3, 0, 0, 1, 0, 1],[3, 1, 0, 1, 0, 1]],\n[[2, 1, 0, 0, 1, 0],[3, 0, 0, 0, 1, 1],[3, 1, 0, 0, 0, 1]],\n[[2, 1, 0, 1, 0, 1],[3, 0, 0, 1, 1, 1],[3, 1, 0, 0, 1, 0]],\n[[2, 1, 1, 0, 0, 0],[3, 0, 0, 1, 1, 0],[3, 1, 1, 1, 1, 0]],\n[[2, 1, 1, 0, 1, 1],[3, 0, 1, 1, 0, 1],[3, 1, 0, 1, 1, 0]],\n[[1, 0, 1, 1, 1, 1],[1, 1, 0, 1, 1, 0],[2, 1, 1, 0, 0, 1]],\n[[2, 1, 1, 1, 1, 1],[3, 0, 1, 0, 1, 1],[3, 1, 0, 1, 0, 0]],\n[[2, 1, 0, 1, 1, 0],[3, 0, 1, 0, 0, 1],[3, 1, 1, 1, 1, 1]],\n[[2, 0, 0, 1, 0, 1],[3, 1, 1, 0, 0, 1],[3, 1, 1, 1, 0, 0]],\n[[1, 1, 0, 1, 0, 1],[1, 1, 1, 1, 1, 1],[2, 0, 1, 0, 1, 0]],\n[[2, 1, 0, 1, 1, 1],[3, 0, 1, 1, 0, 0],[3, 1, 1, 0, 1, 1]],\n[[0, 1, 0, 1, 0, 1],[2, 0, 1, 0, 0, 0],[2, 1, 1, 1, 0, 1]],\n[[1, 0, 0, 0, 1, 1],[2, 1, 0, 0, 1, 1],[2, 1, 1, 0, 1, 0],[3, 0, 1, 0, 1, 0]],\n[[0, 1, 1, 1, 0, 1],[2, 0, 1, 0, 1, 1],[3, 0, 1, 1, 1, 0],[3, 1, 1, 0, 0, 0]],\n[[0, 0, 0, 1, 1, 0],[1, 0, 1, 1, 0, 1],[2, 0, 0, 0, 0, 0],[2, 1, 0, 0, 0, 1],[3, 1, 1, 0, 1, 0]]\n]\nA partition for sets of sizes: 38*3 2*4 1*5\n```\n```\n[4, 2, 2, 2, 2, 2]\n[39, 0, 2]\n[\n[[0, 0, 0, 0, 0, 1],[0, 0, 0, 0, 1, 0],[0, 0, 0, 0, 1, 1]],\n[[0, 0, 0, 1, 0, 0],[0, 0, 1, 0, 0, 0],[0, 0, 1, 1, 0, 0]],\n[[0, 0, 0, 1, 1, 1],[0, 0, 1, 0, 0, 1],[0, 0, 1, 1, 1, 0]],\n[[0, 0, 1, 0, 1, 0],[0, 1, 0, 0, 0, 0],[0, 1, 1, 0, 1, 0]],\n[[0, 0, 1, 1, 0, 1],[0, 1, 0, 0, 0, 1],[0, 1, 1, 1, 0, 0]],\n[[0, 0, 1, 0, 1, 1],[0, 1, 0, 0, 1, 0],[0, 1, 1, 0, 0, 1]],\n[[0, 1, 0, 0, 1, 1],[1, 0, 0, 0, 0, 0],[3, 1, 0, 0, 1, 1]],\n[[0, 1, 0, 1, 1, 0],[1, 0, 0, 0, 0, 1],[3, 1, 0, 1, 1, 1]],\n[[0, 0, 0, 1, 0, 1],[0, 1, 1, 0, 1, 1],[0, 1, 1, 1, 1, 0]],\n[[0, 0, 1, 1, 1, 1],[0, 1, 0, 1, 1, 1],[0, 1, 1, 0, 0, 0]],\n[[0, 1, 1, 1, 1, 1],[1, 0, 0, 0, 1, 0],[3, 1, 1, 1, 0, 1]],\n[[0, 1, 0, 1, 0, 0],[1, 0, 0, 1, 0, 0],[3, 1, 0, 0, 0, 0]],\n[[1, 0, 0, 1, 0, 1],[1, 0, 0, 1, 1, 0],[2, 0, 0, 0, 1, 1]],\n[[1, 0, 1, 0, 0, 0],[1, 0, 1, 0, 0, 1],[2, 0, 0, 0, 0, 1]],\n[[1, 0, 1, 0, 1, 1],[1, 0, 1, 1, 0, 0],[2, 0, 0, 1, 1, 1]],\n[[1, 0, 1, 1, 1, 0],[1, 1, 0, 0, 0, 0],[2, 1, 1, 1, 1, 0]],\n[[1, 1, 0, 0, 0, 1],[1, 1, 0, 0, 1, 1],[2, 0, 0, 0, 1, 0]],\n[[1, 1, 0, 1, 0, 0],[1, 1, 1, 0, 0, 0],[2, 0, 1, 1, 0, 0]],\n[[1, 1, 0, 1, 1, 1],[1, 1, 1, 0, 0, 1],[2, 0, 1, 1, 1, 0]],\n[[1, 1, 1, 0, 1, 0],[1, 1, 1, 1, 0, 0],[2, 0, 0, 1, 1, 0]],\n[[1, 0, 1, 0, 1, 0],[1, 1, 1, 1, 1, 0],[2, 1, 0, 1, 0, 0]],\n[[1, 1, 0, 0, 1, 0],[1, 1, 1, 1, 0, 1],[2, 0, 1, 1, 1, 1]],\n[[1, 0, 0, 1, 1, 1],[1, 1, 1, 0, 1, 1],[2, 1, 1, 1, 0, 0]],\n[[2, 0, 0, 1, 0, 0],[3, 0, 0, 0, 0, 0],[3, 0, 0, 1, 0, 0]],\n[[2, 0, 1, 0, 0, 1],[3, 0, 0, 0, 0, 1],[3, 0, 1, 0, 0, 0]],\n[[2, 0, 1, 1, 0, 1],[3, 0, 0, 0, 1, 0],[3, 0, 1, 1, 1, 1]],\n[[2, 1, 0, 0, 0, 0],[3, 0, 0, 1, 0, 1],[3, 1, 0, 1, 0, 1]],\n[[2, 1, 0, 0, 1, 0],[3, 0, 0, 0, 1, 1],[3, 1, 0, 0, 0, 1]],\n[[2, 1, 0, 1, 0, 1],[3, 0, 0, 1, 1, 1],[3, 1, 0, 0, 1, 0]],\n[[2, 1, 1, 0, 0, 0],[3, 0, 0, 1, 1, 0],[3, 1, 1, 1, 1, 0]],\n[[2, 1, 1, 0, 1, 1],[3, 0, 1, 1, 0, 1],[3, 1, 0, 1, 1, 0]],\n[[1, 0, 1, 1, 1, 1],[1, 1, 0, 1, 1, 0],[2, 1, 1, 0, 0, 1]],\n[[2, 1, 1, 1, 1, 1],[3, 0, 1, 0, 1, 1],[3, 1, 0, 1, 0, 0]],\n[[2, 1, 0, 1, 1, 0],[3, 0, 1, 0, 0, 1],[3, 1, 1, 1, 1, 1]],\n[[2, 0, 0, 1, 0, 1],[3, 1, 1, 0, 0, 1],[3, 1, 1, 1, 0, 0]],\n[[1, 1, 0, 1, 0, 1],[1, 1, 1, 1, 1, 1],[2, 0, 1, 0, 1, 0]],\n[[2, 1, 0, 1, 1, 1],[3, 0, 1, 1, 0, 0],[3, 1, 1, 0, 1, 1]],\n[[0, 1, 0, 1, 0, 1],[1, 0, 1, 1, 0, 1],[3, 1, 1, 0, 0, 0]],\n[[0, 1, 1, 1, 0, 1],[2, 0, 0, 0, 0, 0],[2, 1, 1, 1, 0, 1]],\n[[0, 0, 0, 1, 1, 0],[1, 0, 0, 0, 1, 1],[2, 1, 0, 0, 0, 1],[2, 1, 1, 0, 1, 0],[3, 0, 1, 1, 1, 0]],\n[[2, 0, 1, 0, 0, 0],[2, 0, 1, 0, 1, 1],[2, 1, 0, 0, 1, 1],[3, 0, 1, 0, 1, 0],[3, 1, 1, 0, 1, 0]]\n]\nA partition for sets of sizes: 39*3 0*4 2*5\n```\n```\n[4, 2, 2, 2, 2, 2]\n[35, 3, 2]\n[\n[[0, 0, 0, 0, 0, 1],[0, 0, 0, 0, 1, 0],[0, 0, 0, 0, 1, 1]],\n[[0, 0, 0, 1, 0, 0],[0, 0, 1, 0, 0, 0],[0, 0, 1, 1, 0, 0]],\n[[0, 0, 0, 1, 1, 1],[0, 0, 1, 0, 0, 1],[0, 0, 1, 1, 1, 0]],\n[[0, 0, 1, 0, 1, 0],[0, 1, 0, 0, 0, 0],[0, 1, 1, 0, 1, 0]],\n[[0, 0, 1, 1, 0, 1],[0, 1, 0, 0, 0, 1],[0, 1, 1, 1, 0, 0]],\n[[0, 0, 1, 0, 1, 1],[0, 1, 0, 0, 1, 0],[0, 1, 1, 0, 0, 1]],\n[[0, 1, 0, 0, 1, 1],[1, 0, 0, 0, 0, 0],[3, 1, 0, 0, 1, 1]],\n[[0, 1, 0, 1, 1, 0],[1, 0, 0, 0, 0, 1],[3, 1, 0, 1, 1, 1]],\n[[0, 0, 0, 1, 0, 1],[0, 1, 1, 0, 1, 1],[0, 1, 1, 1, 1, 0]],\n[[0, 0, 1, 1, 1, 1],[0, 1, 0, 1, 1, 1],[0, 1, 1, 0, 0, 0]],\n[[0, 1, 1, 1, 1, 1],[1, 0, 0, 0, 1, 0],[3, 1, 1, 1, 0, 1]],\n[[0, 1, 0, 1, 0, 0],[1, 0, 0, 1, 0, 0],[3, 1, 0, 0, 0, 0]],\n[[1, 0, 0, 1, 0, 1],[1, 0, 0, 1, 1, 0],[2, 0, 0, 0, 1, 1]],\n[[1, 0, 1, 0, 0, 0],[1, 0, 1, 0, 0, 1],[2, 0, 0, 0, 0, 1]],\n[[1, 0, 1, 0, 1, 1],[1, 0, 1, 1, 0, 0],[2, 0, 0, 1, 1, 1]],\n[[1, 0, 1, 1, 1, 0],[1, 1, 0, 0, 0, 0],[2, 1, 1, 1, 1, 0]],\n[[1, 1, 0, 0, 0, 1],[1, 1, 0, 0, 1, 1],[2, 0, 0, 0, 1, 0]],\n[[1, 1, 0, 1, 0, 0],[1, 1, 1, 0, 0, 0],[2, 0, 1, 1, 0, 0]],\n[[1, 1, 0, 1, 1, 1],[1, 1, 1, 0, 0, 1],[2, 0, 1, 1, 1, 0]],\n[[1, 1, 1, 0, 1, 0],[1, 1, 1, 1, 0, 0],[2, 0, 0, 1, 1, 0]],\n[[1, 0, 1, 0, 1, 0],[1, 1, 1, 1, 1, 0],[2, 1, 0, 1, 0, 0]],\n[[1, 1, 0, 0, 1, 0],[1, 1, 1, 1, 0, 1],[2, 0, 1, 1, 1, 1]],\n[[1, 0, 0, 1, 1, 1],[1, 1, 1, 0, 1, 1],[2, 1, 1, 1, 0, 0]],\n[[2, 0, 0, 1, 0, 0],[3, 0, 0, 0, 0, 0],[3, 0, 0, 1, 0, 0]],\n[[2, 0, 1, 0, 0, 1],[3, 0, 0, 0, 0, 1],[3, 0, 1, 0, 0, 0]],\n[[2, 0, 1, 1, 0, 1],[3, 0, 0, 0, 1, 0],[3, 0, 1, 1, 1, 1]],\n[[2, 1, 0, 0, 0, 0],[3, 0, 0, 1, 0, 1],[3, 1, 0, 1, 0, 1]],\n[[2, 1, 0, 0, 1, 0],[3, 0, 0, 0, 1, 1],[3, 1, 0, 0, 0, 1]],\n[[2, 1, 0, 1, 0, 1],[3, 0, 0, 1, 1, 1],[3, 1, 0, 0, 1, 0]],\n[[2, 1, 1, 0, 0, 0],[3, 0, 0, 1, 1, 0],[3, 1, 1, 1, 1, 0]],\n[[2, 1, 1, 0, 1, 1],[3, 0, 1, 1, 0, 1],[3, 1, 0, 1, 1, 0]],\n[[1, 0, 1, 1, 1, 1],[1, 1, 0, 1, 1, 0],[2, 1, 1, 0, 0, 1]],\n[[2, 1, 1, 1, 1, 1],[3, 0, 1, 0, 1, 1],[3, 1, 0, 1, 0, 0]],\n[[2, 1, 0, 1, 1, 0],[3, 0, 1, 0, 0, 1],[3, 1, 1, 1, 1, 1]],\n[[2, 0, 0, 1, 0, 1],[3, 1, 1, 0, 0, 1],[3, 1, 1, 1, 0, 0]],\n[[3, 0, 1, 1, 0, 0],[3, 0, 1, 1, 1, 0],[3, 1, 1, 0, 0, 0],[3, 1, 1, 0, 1, 0]],\n[[1, 1, 0, 1, 0, 1],[2, 1, 0, 0, 1, 1],[2, 1, 1, 1, 0, 1],[3, 1, 1, 0, 1, 1]],\n[[0, 1, 0, 1, 0, 1],[0, 1, 1, 1, 0, 1],[2, 0, 0, 0, 0, 0],[2, 0, 1, 0, 0, 0]],\n[[1, 0, 0, 0, 1, 1],[1, 1, 1, 1, 1, 1],[2, 1, 0, 0, 0, 1],[2, 1, 0, 1, 1, 1],[2, 1, 1, 0, 1, 0]],\n[[0, 0, 0, 1, 1, 0],[1, 0, 1, 1, 0, 1],[2, 0, 1, 0, 1, 0],[2, 0, 1, 0, 1, 1],[3, 0, 1, 0, 1, 0]]\n]\nA partition for sets of sizes: 35*3 3*4 2*5\n```\n```\n[4, 2, 2, 2, 2, 2]\n[36, 1, 3]\n[\n[[0, 0, 0, 0, 0, 1],[0, 0, 0, 0, 1, 0],[0, 0, 0, 0, 1, 1]],\n[[0, 0, 0, 1, 0, 0],[0, 0, 1, 0, 0, 0],[0, 0, 1, 1, 0, 0]],\n[[0, 0, 0, 1, 1, 1],[0, 0, 1, 0, 0, 1],[0, 0, 1, 1, 1, 0]],\n[[0, 0, 1, 0, 1, 0],[0, 1, 0, 0, 0, 0],[0, 1, 1, 0, 1, 0]],\n[[0, 0, 1, 1, 0, 1],[0, 1, 0, 0, 0, 1],[0, 1, 1, 1, 0, 0]],\n[[0, 0, 1, 0, 1, 1],[0, 1, 0, 0, 1, 0],[0, 1, 1, 0, 0, 1]],\n[[0, 1, 0, 0, 1, 1],[1, 0, 0, 0, 0, 0],[3, 1, 0, 0, 1, 1]],\n[[0, 1, 0, 1, 1, 0],[1, 0, 0, 0, 0, 1],[3, 1, 0, 1, 1, 1]],\n[[0, 0, 0, 1, 0, 1],[0, 1, 1, 0, 1, 1],[0, 1, 1, 1, 1, 0]],\n[[0, 0, 1, 1, 1, 1],[0, 1, 0, 1, 1, 1],[0, 1, 1, 0, 0, 0]],\n[[0, 1, 1, 1, 1, 1],[1, 0, 0, 0, 1, 0],[3, 1, 1, 1, 0, 1]],\n[[0, 1, 0, 1, 0, 0],[1, 0, 0, 1, 0, 0],[3, 1, 0, 0, 0, 0]],\n[[1, 0, 0, 1, 0, 1],[1, 0, 0, 1, 1, 0],[2, 0, 0, 0, 1, 1]],\n[[1, 0, 1, 0, 0, 0],[1, 0, 1, 0, 0, 1],[2, 0, 0, 0, 0, 1]],\n[[1, 0, 1, 0, 1, 1],[1, 0, 1, 1, 0, 0],[2, 0, 0, 1, 1, 1]],\n[[1, 0, 1, 1, 1, 0],[1, 1, 0, 0, 0, 0],[2, 1, 1, 1, 1, 0]],\n[[1, 1, 0, 0, 0, 1],[1, 1, 0, 0, 1, 1],[2, 0, 0, 0, 1, 0]],\n[[1, 1, 0, 1, 0, 0],[1, 1, 1, 0, 0, 0],[2, 0, 1, 1, 0, 0]],\n[[1, 1, 0, 1, 1, 1],[1, 1, 1, 0, 0, 1],[2, 0, 1, 1, 1, 0]],\n[[1, 1, 1, 0, 1, 0],[1, 1, 1, 1, 0, 0],[2, 0, 0, 1, 1, 0]],\n[[1, 0, 1, 0, 1, 0],[1, 1, 1, 1, 1, 0],[2, 1, 0, 1, 0, 0]],\n[[1, 1, 0, 0, 1, 0],[1, 1, 1, 1, 0, 1],[2, 0, 1, 1, 1, 1]],\n[[1, 0, 0, 1, 1, 1],[1, 1, 1, 0, 1, 1],[2, 1, 1, 1, 0, 0]],\n[[2, 0, 0, 1, 0, 0],[3, 0, 0, 0, 0, 0],[3, 0, 0, 1, 0, 0]],\n[[2, 0, 1, 0, 0, 1],[3, 0, 0, 0, 0, 1],[3, 0, 1, 0, 0, 0]],\n[[2, 0, 1, 1, 0, 1],[3, 0, 0, 0, 1, 0],[3, 0, 1, 1, 1, 1]],\n[[2, 1, 0, 0, 0, 0],[3, 0, 0, 1, 0, 1],[3, 1, 0, 1, 0, 1]],\n[[2, 1, 0, 0, 1, 0],[3, 0, 0, 0, 1, 1],[3, 1, 0, 0, 0, 1]],\n[[2, 1, 0, 1, 0, 1],[3, 0, 0, 1, 1, 1],[3, 1, 0, 0, 1, 0]],\n[[2, 1, 1, 0, 0, 0],[3, 0, 0, 1, 1, 0],[3, 1, 1, 1, 1, 0]],\n[[2, 1, 1, 0, 1, 1],[3, 0, 1, 1, 0, 1],[3, 1, 0, 1, 1, 0]],\n[[1, 0, 1, 1, 1, 1],[1, 1, 0, 1, 1, 0],[2, 1, 1, 0, 0, 1]],\n[[2, 1, 1, 1, 1, 1],[3, 0, 1, 0, 1, 1],[3, 1, 0, 1, 0, 0]],\n[[2, 1, 0, 1, 1, 0],[3, 0, 1, 0, 0, 1],[3, 1, 1, 1, 1, 1]],\n[[2, 0, 0, 1, 0, 1],[3, 1, 1, 0, 0, 1],[3, 1, 1, 1, 0, 0]],\n[[1, 1, 0, 1, 0, 1],[1, 1, 1, 1, 1, 1],[2, 0, 1, 0, 1, 0]],\n[[1, 0, 0, 0, 1, 1],[2, 0, 1, 0, 0, 0],[2, 1, 0, 0, 1, 1],[3, 1, 1, 0, 0, 0]],\n[[2, 1, 0, 1, 1, 1],[2, 1, 1, 0, 1, 0],[2, 1, 1, 1, 0, 1],[3, 0, 1, 0, 1, 0],[3, 1, 1, 0, 1, 0]],\n[[0, 0, 0, 1, 1, 0],[0, 1, 0, 1, 0, 1],[2, 1, 0, 0, 0, 1],[3, 0, 1, 1, 0, 0],[3, 0, 1, 1, 1, 0]],\n[[0, 1, 1, 1, 0, 1],[1, 0, 1, 1, 0, 1],[2, 0, 0, 0, 0, 0],[2, 0, 1, 0, 1, 1],[3, 1, 1, 0, 1, 1]]\n]\nA partition for sets of sizes: 36*3 1*4 3*5\n```\n```\n[4, 2, 2, 2, 2, 2]\n[33, 2, 4]\n[\n[[0, 0, 0, 0, 0, 1],[0, 0, 0, 0, 1, 0],[0, 0, 0, 0, 1, 1]],\n[[0, 0, 0, 1, 0, 0],[0, 0, 1, 0, 0, 0],[0, 0, 1, 1, 0, 0]],\n[[0, 0, 0, 1, 1, 1],[0, 0, 1, 0, 0, 1],[0, 0, 1, 1, 1, 0]],\n[[0, 0, 1, 0, 1, 0],[0, 1, 0, 0, 0, 0],[0, 1, 1, 0, 1, 0]],\n[[0, 0, 1, 1, 0, 1],[0, 1, 0, 0, 0, 1],[0, 1, 1, 1, 0, 0]],\n[[0, 0, 1, 0, 1, 1],[0, 1, 0, 0, 1, 0],[0, 1, 1, 0, 0, 1]],\n[[0, 1, 0, 0, 1, 1],[1, 0, 0, 0, 0, 0],[3, 1, 0, 0, 1, 1]],\n[[0, 1, 0, 1, 1, 0],[1, 0, 0, 0, 0, 1],[3, 1, 0, 1, 1, 1]],\n[[0, 0, 0, 1, 0, 1],[0, 1, 1, 0, 1, 1],[0, 1, 1, 1, 1, 0]],\n[[0, 0, 1, 1, 1, 1],[0, 1, 0, 1, 1, 1],[0, 1, 1, 0, 0, 0]],\n[[0, 1, 1, 1, 1, 1],[1, 0, 0, 0, 1, 0],[3, 1, 1, 1, 0, 1]],\n[[0, 1, 0, 1, 0, 0],[1, 0, 0, 1, 0, 0],[3, 1, 0, 0, 0, 0]],\n[[1, 0, 0, 1, 0, 1],[1, 0, 0, 1, 1, 0],[2, 0, 0, 0, 1, 1]],\n[[1, 0, 1, 0, 0, 0],[1, 0, 1, 0, 0, 1],[2, 0, 0, 0, 0, 1]],\n[[1, 0, 1, 0, 1, 1],[1, 0, 1, 1, 0, 0],[2, 0, 0, 1, 1, 1]],\n[[1, 0, 1, 1, 1, 0],[1, 1, 0, 0, 0, 0],[2, 1, 1, 1, 1, 0]],\n[[1, 1, 0, 0, 0, 1],[1, 1, 0, 0, 1, 1],[2, 0, 0, 0, 1, 0]],\n[[1, 1, 0, 1, 0, 0],[1, 1, 1, 0, 0, 0],[2, 0, 1, 1, 0, 0]],\n[[1, 1, 0, 1, 1, 1],[1, 1, 1, 0, 0, 1],[2, 0, 1, 1, 1, 0]],\n[[1, 1, 1, 0, 1, 0],[1, 1, 1, 1, 0, 0],[2, 0, 0, 1, 1, 0]],\n[[1, 0, 1, 0, 1, 0],[1, 1, 1, 1, 1, 0],[2, 1, 0, 1, 0, 0]],\n[[1, 1, 0, 0, 1, 0],[1, 1, 1, 1, 0, 1],[2, 0, 1, 1, 1, 1]],\n[[1, 0, 0, 1, 1, 1],[1, 1, 1, 0, 1, 1],[2, 1, 1, 1, 0, 0]],\n[[2, 0, 0, 1, 0, 0],[3, 0, 0, 0, 0, 0],[3, 0, 0, 1, 0, 0]],\n[[2, 0, 1, 0, 0, 1],[3, 0, 0, 0, 0, 1],[3, 0, 1, 0, 0, 0]],\n[[2, 0, 1, 1, 0, 1],[3, 0, 0, 0, 1, 0],[3, 0, 1, 1, 1, 1]],\n[[2, 1, 0, 0, 0, 0],[3, 0, 0, 1, 0, 1],[3, 1, 0, 1, 0, 1]],\n[[2, 1, 0, 0, 1, 0],[3, 0, 0, 0, 1, 1],[3, 1, 0, 0, 0, 1]],\n[[2, 1, 0, 1, 0, 1],[3, 0, 0, 1, 1, 1],[3, 1, 0, 0, 1, 0]],\n[[2, 1, 1, 0, 0, 0],[3, 0, 0, 1, 1, 0],[3, 1, 1, 1, 1, 0]],\n[[2, 1, 1, 0, 1, 1],[3, 0, 1, 1, 0, 1],[3, 1, 0, 1, 1, 0]],\n[[1, 0, 1, 1, 1, 1],[1, 1, 0, 1, 1, 0],[2, 1, 1, 0, 0, 1]],\n[[2, 1, 1, 1, 1, 1],[3, 0, 1, 0, 1, 1],[3, 1, 0, 1, 0, 0]],\n[[2, 0, 1, 0, 0, 0],[2, 0, 1, 0, 1, 0],[2, 1, 0, 0, 0, 1],[2, 1, 0, 0, 1, 1]],\n[[3, 0, 1, 0, 0, 1],[3, 0, 1, 0, 1, 0],[3, 1, 1, 0, 0, 0],[3, 1, 1, 0, 1, 1]],\n[[2, 0, 0, 1, 0, 1],[2, 0, 1, 0, 1, 1],[2, 1, 1, 1, 0, 1],[3, 0, 1, 1, 0, 0],[3, 1, 1, 1, 1, 1]],\n[[0, 1, 0, 1, 0, 1],[1, 0, 0, 0, 1, 1],[2, 1, 0, 1, 1, 0],[2, 1, 1, 0, 1, 0],[3, 1, 1, 0, 1, 0]],\n[[0, 0, 0, 1, 1, 0],[1, 0, 1, 1, 0, 1],[2, 0, 0, 0, 0, 0],[2, 1, 0, 1, 1, 1],[3, 1, 1, 1, 0, 0]],\n[[0, 1, 1, 1, 0, 1],[1, 1, 0, 1, 0, 1],[1, 1, 1, 1, 1, 1],[3, 0, 1, 1, 1, 0],[3, 1, 1, 0, 0, 1]]\n]\nA partition for sets of sizes: 33*3 2*4 4*5\n```\n```\n[4, 2, 2, 2, 2, 2]\n[34, 0, 5]\n[\n[[0, 0, 0, 0, 0, 1],[0, 0, 0, 0, 1, 0],[0, 0, 0, 0, 1, 1]],\n[[0, 0, 0, 1, 0, 0],[0, 0, 1, 0, 0, 0],[0, 0, 1, 1, 0, 0]],\n[[0, 0, 0, 1, 1, 1],[0, 0, 1, 0, 0, 1],[0, 0, 1, 1, 1, 0]],\n[[0, 0, 1, 0, 1, 0],[0, 1, 0, 0, 0, 0],[0, 1, 1, 0, 1, 0]],\n[[0, 0, 1, 1, 0, 1],[0, 1, 0, 0, 0, 1],[0, 1, 1, 1, 0, 0]],\n[[0, 0, 1, 0, 1, 1],[0, 1, 0, 0, 1, 0],[0, 1, 1, 0, 0, 1]],\n[[0, 1, 0, 0, 1, 1],[1, 0, 0, 0, 0, 0],[3, 1, 0, 0, 1, 1]],\n[[0, 1, 0, 1, 1, 0],[1, 0, 0, 0, 0, 1],[3, 1, 0, 1, 1, 1]],\n[[0, 0, 0, 1, 0, 1],[0, 1, 1, 0, 1, 1],[0, 1, 1, 1, 1, 0]],\n[[0, 0, 1, 1, 1, 1],[0, 1, 0, 1, 1, 1],[0, 1, 1, 0, 0, 0]],\n[[0, 1, 1, 1, 1, 1],[1, 0, 0, 0, 1, 0],[3, 1, 1, 1, 0, 1]],\n[[0, 1, 0, 1, 0, 0],[1, 0, 0, 1, 0, 0],[3, 1, 0, 0, 0, 0]],\n[[1, 0, 0, 1, 0, 1],[1, 0, 0, 1, 1, 0],[2, 0, 0, 0, 1, 1]],\n[[1, 0, 1, 0, 0, 0],[1, 0, 1, 0, 0, 1],[2, 0, 0, 0, 0, 1]],\n[[1, 0, 1, 0, 1, 1],[1, 0, 1, 1, 0, 0],[2, 0, 0, 1, 1, 1]],\n[[1, 0, 1, 1, 1, 0],[1, 1, 0, 0, 0, 0],[2, 1, 1, 1, 1, 0]],\n[[1, 1, 0, 0, 0, 1],[1, 1, 0, 0, 1, 1],[2, 0, 0, 0, 1, 0]],\n[[1, 1, 0, 1, 0, 0],[1, 1, 1, 0, 0, 0],[2, 0, 1, 1, 0, 0]],\n[[1, 1, 0, 1, 1, 1],[1, 1, 1, 0, 0, 1],[2, 0, 1, 1, 1, 0]],\n[[1, 1, 1, 0, 1, 0],[1, 1, 1, 1, 0, 0],[2, 0, 0, 1, 1, 0]],\n[[1, 0, 1, 0, 1, 0],[1, 1, 1, 1, 1, 0],[2, 1, 0, 1, 0, 0]],\n[[1, 1, 0, 0, 1, 0],[1, 1, 1, 1, 0, 1],[2, 0, 1, 1, 1, 1]],\n[[1, 0, 0, 1, 1, 1],[1, 1, 1, 0, 1, 1],[2, 1, 1, 1, 0, 0]],\n[[2, 0, 0, 1, 0, 0],[3, 0, 0, 0, 0, 0],[3, 0, 0, 1, 0, 0]],\n[[2, 0, 1, 0, 0, 1],[3, 0, 0, 0, 0, 1],[3, 0, 1, 0, 0, 0]],\n[[2, 0, 1, 1, 0, 1],[3, 0, 0, 0, 1, 0],[3, 0, 1, 1, 1, 1]],\n[[2, 1, 0, 0, 0, 0],[3, 0, 0, 1, 0, 1],[3, 1, 0, 1, 0, 1]],\n[[2, 1, 0, 0, 1, 0],[3, 0, 0, 0, 1, 1],[3, 1, 0, 0, 0, 1]],\n[[2, 1, 0, 1, 0, 1],[3, 0, 0, 1, 1, 1],[3, 1, 0, 0, 1, 0]],\n[[2, 1, 1, 0, 0, 0],[3, 0, 0, 1, 1, 0],[3, 1, 1, 1, 1, 0]],\n[[2, 1, 1, 0, 1, 1],[3, 0, 1, 1, 0, 1],[3, 1, 0, 1, 1, 0]],\n[[1, 0, 1, 1, 1, 1],[1, 1, 0, 1, 1, 0],[2, 1, 1, 0, 0, 1]],\n[[2, 1, 1, 1, 1, 1],[3, 0, 1, 0, 1, 1],[3, 1, 0, 1, 0, 0]],\n[[2, 1, 0, 1, 1, 0],[3, 0, 1, 0, 0, 1],[3, 1, 1, 1, 1, 1]],\n[[2, 0, 1, 0, 0, 0],[2, 0, 1, 0, 1, 1],[2, 1, 0, 1, 1, 1],[3, 0, 1, 1, 0, 0],[3, 1, 1, 0, 0, 0]],\n[[2, 0, 0, 1, 0, 1],[2, 1, 0, 0, 0, 1],[2, 1, 0, 0, 1, 1],[3, 1, 1, 0, 1, 1],[3, 1, 1, 1, 0, 0]],\n[[2, 0, 0, 0, 0, 0],[2, 0, 1, 0, 1, 0],[2, 1, 1, 1, 0, 1],[3, 0, 1, 1, 1, 0],[3, 1, 1, 0, 0, 1]],\n[[0, 1, 0, 1, 0, 1],[0, 1, 1, 1, 0, 1],[1, 0, 1, 1, 0, 1],[1, 1, 1, 1, 1, 1],[2, 1, 1, 0, 1, 0]],\n[[0, 0, 0, 1, 1, 0],[1, 0, 0, 0, 1, 1],[1, 1, 0, 1, 0, 1],[3, 0, 1, 0, 1, 0],[3, 1, 1, 0, 1, 0]]\n]\nA partition for sets of sizes: 34*3 0*4 5*5\n```\n```\n[4, 2, 2, 2, 2, 2]\n[30, 3, 5]\n[\n[[0, 0, 0, 0, 0, 1],[0, 0, 0, 0, 1, 0],[0, 0, 0, 0, 1, 1]],\n[[0, 0, 0, 1, 0, 0],[0, 0, 1, 0, 0, 0],[0, 0, 1, 1, 0, 0]],\n[[0, 0, 0, 1, 1, 1],[0, 0, 1, 0, 0, 1],[0, 0, 1, 1, 1, 0]],\n[[0, 0, 1, 0, 1, 0],[0, 1, 0, 0, 0, 0],[0, 1, 1, 0, 1, 0]],\n[[0, 0, 1, 1, 0, 1],[0, 1, 0, 0, 0, 1],[0, 1, 1, 1, 0, 0]],\n[[0, 0, 1, 0, 1, 1],[0, 1, 0, 0, 1, 0],[0, 1, 1, 0, 0, 1]],\n[[0, 1, 0, 0, 1, 1],[1, 0, 0, 0, 0, 0],[3, 1, 0, 0, 1, 1]],\n[[0, 1, 0, 1, 1, 0],[1, 0, 0, 0, 0, 1],[3, 1, 0, 1, 1, 1]],\n[[0, 0, 0, 1, 0, 1],[0, 1, 1, 0, 1, 1],[0, 1, 1, 1, 1, 0]],\n[[0, 0, 1, 1, 1, 1],[0, 1, 0, 1, 1, 1],[0, 1, 1, 0, 0, 0]],\n[[0, 1, 1, 1, 1, 1],[1, 0, 0, 0, 1, 0],[3, 1, 1, 1, 0, 1]],\n[[0, 1, 0, 1, 0, 0],[1, 0, 0, 1, 0, 0],[3, 1, 0, 0, 0, 0]],\n[[1, 0, 0, 1, 0, 1],[1, 0, 0, 1, 1, 0],[2, 0, 0, 0, 1, 1]],\n[[1, 0, 1, 0, 0, 0],[1, 0, 1, 0, 0, 1],[2, 0, 0, 0, 0, 1]],\n[[1, 0, 1, 0, 1, 1],[1, 0, 1, 1, 0, 0],[2, 0, 0, 1, 1, 1]],\n[[1, 0, 1, 1, 1, 0],[1, 1, 0, 0, 0, 0],[2, 1, 1, 1, 1, 0]],\n[[1, 1, 0, 0, 0, 1],[1, 1, 0, 0, 1, 1],[2, 0, 0, 0, 1, 0]],\n[[1, 1, 0, 1, 0, 0],[1, 1, 1, 0, 0, 0],[2, 0, 1, 1, 0, 0]],\n[[1, 1, 0, 1, 1, 1],[1, 1, 1, 0, 0, 1],[2, 0, 1, 1, 1, 0]],\n[[1, 1, 1, 0, 1, 0],[1, 1, 1, 1, 0, 0],[2, 0, 0, 1, 1, 0]],\n[[1, 0, 1, 0, 1, 0],[1, 1, 1, 1, 1, 0],[2, 1, 0, 1, 0, 0]],\n[[1, 1, 0, 0, 1, 0],[1, 1, 1, 1, 0, 1],[2, 0, 1, 1, 1, 1]],\n[[1, 0, 0, 1, 1, 1],[1, 1, 1, 0, 1, 1],[2, 1, 1, 1, 0, 0]],\n[[2, 0, 0, 1, 0, 0],[3, 0, 0, 0, 0, 0],[3, 0, 0, 1, 0, 0]],\n[[2, 0, 1, 0, 0, 1],[3, 0, 0, 0, 0, 1],[3, 0, 1, 0, 0, 0]],\n[[2, 0, 1, 1, 0, 1],[3, 0, 0, 0, 1, 0],[3, 0, 1, 1, 1, 1]],\n[[2, 1, 0, 0, 0, 0],[3, 0, 0, 1, 0, 1],[3, 1, 0, 1, 0, 1]],\n[[2, 1, 0, 0, 1, 0],[3, 0, 0, 0, 1, 1],[3, 1, 0, 0, 0, 1]],\n[[2, 1, 0, 1, 0, 1],[3, 0, 0, 1, 1, 1],[3, 1, 0, 0, 1, 0]],\n[[2, 1, 1, 0, 0, 0],[3, 0, 0, 1, 1, 0],[3, 1, 1, 1, 1, 0]],\n[[2, 0, 0, 1, 0, 1],[2, 0, 1, 0, 1, 0],[2, 1, 0, 1, 1, 0],[2, 1, 1, 0, 0, 1]],\n[[1, 0, 1, 1, 0, 1],[2, 1, 1, 0, 1, 1],[2, 1, 1, 1, 1, 1],[3, 0, 1, 0, 0, 1]],\n[[2, 0, 0, 0, 0, 0],[2, 0, 1, 0, 1, 1],[2, 1, 0, 0, 0, 1],[2, 1, 1, 0, 1, 0]],\n[[2, 0, 1, 0, 0, 0],[2, 1, 0, 1, 1, 1],[2, 1, 1, 1, 0, 1],[3, 0, 1, 1, 0, 0],[3, 0, 1, 1, 1, 0]],\n[[1, 1, 0, 1, 0, 1],[2, 1, 0, 0, 1, 1],[3, 0, 1, 0, 1, 1],[3, 1, 0, 1, 0, 0],[3, 1, 1, 0, 0, 1]],\n[[0, 1, 0, 1, 0, 1],[1, 1, 0, 1, 1, 0],[1, 1, 1, 1, 1, 1],[3, 0, 1, 0, 1, 0],[3, 1, 0, 1, 1, 0]],\n[[0, 0, 0, 1, 1, 0],[3, 1, 1, 0, 0, 0],[3, 1, 1, 0, 1, 0],[3, 1, 1, 0, 1, 1],[3, 1, 1, 1, 1, 1]],\n[[0, 1, 1, 1, 0, 1],[1, 0, 0, 0, 1, 1],[1, 0, 1, 1, 1, 1],[3, 0, 1, 1, 0, 1],[3, 1, 1, 1, 0, 0]]\n]\nA partition for sets of sizes: 30*3 3*4 5*5\n```\n```\n[4, 2, 2, 2, 2, 2]\n[31, 1, 6]\n[\n[[0, 0, 0, 0, 0, 1],[0, 0, 0, 0, 1, 0],[0, 0, 0, 0, 1, 1]],\n[[0, 0, 0, 1, 0, 0],[0, 0, 1, 0, 0, 0],[0, 0, 1, 1, 0, 0]],\n[[0, 0, 0, 1, 1, 1],[0, 0, 1, 0, 0, 1],[0, 0, 1, 1, 1, 0]],\n[[0, 0, 1, 0, 1, 0],[0, 1, 0, 0, 0, 0],[0, 1, 1, 0, 1, 0]],\n[[0, 0, 1, 1, 0, 1],[0, 1, 0, 0, 0, 1],[0, 1, 1, 1, 0, 0]],\n[[0, 0, 1, 0, 1, 1],[0, 1, 0, 0, 1, 0],[0, 1, 1, 0, 0, 1]],\n[[0, 1, 0, 0, 1, 1],[1, 0, 0, 0, 0, 0],[3, 1, 0, 0, 1, 1]],\n[[0, 1, 0, 1, 1, 0],[1, 0, 0, 0, 0, 1],[3, 1, 0, 1, 1, 1]],\n[[0, 0, 0, 1, 0, 1],[0, 1, 1, 0, 1, 1],[0, 1, 1, 1, 1, 0]],\n[[0, 0, 1, 1, 1, 1],[0, 1, 0, 1, 1, 1],[0, 1, 1, 0, 0, 0]],\n[[0, 1, 1, 1, 1, 1],[1, 0, 0, 0, 1, 0],[3, 1, 1, 1, 0, 1]],\n[[0, 1, 0, 1, 0, 0],[1, 0, 0, 1, 0, 0],[3, 1, 0, 0, 0, 0]],\n[[1, 0, 0, 1, 0, 1],[1, 0, 0, 1, 1, 0],[2, 0, 0, 0, 1, 1]],\n[[1, 0, 1, 0, 0, 0],[1, 0, 1, 0, 0, 1],[2, 0, 0, 0, 0, 1]],\n[[1, 0, 1, 0, 1, 1],[1, 0, 1, 1, 0, 0],[2, 0, 0, 1, 1, 1]],\n[[1, 0, 1, 1, 1, 0],[1, 1, 0, 0, 0, 0],[2, 1, 1, 1, 1, 0]],\n[[1, 1, 0, 0, 0, 1],[1, 1, 0, 0, 1, 1],[2, 0, 0, 0, 1, 0]],\n[[1, 1, 0, 1, 0, 0],[1, 1, 1, 0, 0, 0],[2, 0, 1, 1, 0, 0]],\n[[1, 1, 0, 1, 1, 1],[1, 1, 1, 0, 0, 1],[2, 0, 1, 1, 1, 0]],\n[[1, 1, 1, 0, 1, 0],[1, 1, 1, 1, 0, 0],[2, 0, 0, 1, 1, 0]],\n[[1, 0, 1, 0, 1, 0],[1, 1, 1, 1, 1, 0],[2, 1, 0, 1, 0, 0]],\n[[1, 1, 0, 0, 1, 0],[1, 1, 1, 1, 0, 1],[2, 0, 1, 1, 1, 1]],\n[[1, 0, 0, 1, 1, 1],[1, 1, 1, 0, 1, 1],[2, 1, 1, 1, 0, 0]],\n[[2, 0, 0, 1, 0, 0],[3, 0, 0, 0, 0, 0],[3, 0, 0, 1, 0, 0]],\n[[2, 0, 1, 0, 0, 1],[3, 0, 0, 0, 0, 1],[3, 0, 1, 0, 0, 0]],\n[[2, 0, 1, 1, 0, 1],[3, 0, 0, 0, 1, 0],[3, 0, 1, 1, 1, 1]],\n[[2, 1, 0, 0, 0, 0],[3, 0, 0, 1, 0, 1],[3, 1, 0, 1, 0, 1]],\n[[2, 1, 0, 0, 1, 0],[3, 0, 0, 0, 1, 1],[3, 1, 0, 0, 0, 1]],\n[[2, 1, 0, 1, 0, 1],[3, 0, 0, 1, 1, 1],[3, 1, 0, 0, 1, 0]],\n[[2, 1, 1, 0, 0, 0],[3, 0, 0, 1, 1, 0],[3, 1, 1, 1, 1, 0]],\n[[2, 1, 1, 0, 1, 1],[3, 0, 1, 1, 0, 1],[3, 1, 0, 1, 1, 0]],\n[[1, 0, 1, 1, 1, 1],[2, 0, 1, 0, 1, 0],[2, 1, 1, 1, 1, 1],[3, 1, 1, 0, 1, 0]],\n[[1, 1, 0, 1, 1, 0],[2, 1, 0, 0, 1, 1],[3, 0, 1, 0, 0, 1],[3, 1, 0, 1, 0, 0],[3, 1, 1, 0, 0, 0]],\n[[2, 1, 0, 1, 1, 0],[2, 1, 1, 0, 0, 1],[2, 1, 1, 1, 0, 1],[3, 0, 1, 0, 1, 1],[3, 1, 1, 0, 0, 1]],\n[[2, 0, 0, 1, 0, 1],[2, 1, 0, 0, 0, 1],[2, 1, 0, 1, 1, 1],[3, 1, 1, 1, 0, 0],[3, 1, 1, 1, 1, 1]],\n[[1, 1, 1, 1, 1, 1],[2, 0, 0, 0, 0, 0],[3, 0, 1, 0, 1, 0],[3, 0, 1, 1, 1, 0],[3, 1, 1, 0, 1, 1]],\n[[0, 0, 0, 1, 1, 0],[0, 1, 0, 1, 0, 1],[1, 0, 1, 1, 0, 1],[1, 1, 0, 1, 0, 1],[2, 0, 1, 0, 1, 1]],\n[[0, 1, 1, 1, 0, 1],[1, 0, 0, 0, 1, 1],[2, 0, 1, 0, 0, 0],[2, 1, 1, 0, 1, 0],[3, 0, 1, 1, 0, 0]]\n]\nA partition for sets of sizes: 31*3 1*4 6*5\n```\n```\n[4, 2, 2, 2, 2, 2]\n[28, 2, 7]\n[\n[[0, 0, 0, 0, 0, 1],[0, 0, 0, 0, 1, 0],[0, 0, 0, 0, 1, 1]],\n[[0, 0, 0, 1, 0, 0],[0, 0, 1, 0, 0, 0],[0, 0, 1, 1, 0, 0]],\n[[0, 0, 0, 1, 1, 1],[0, 0, 1, 0, 0, 1],[0, 0, 1, 1, 1, 0]],\n[[0, 0, 1, 0, 1, 0],[0, 1, 0, 0, 0, 0],[0, 1, 1, 0, 1, 0]],\n[[0, 0, 1, 1, 0, 1],[0, 1, 0, 0, 0, 1],[0, 1, 1, 1, 0, 0]],\n[[0, 0, 1, 0, 1, 1],[0, 1, 0, 0, 1, 0],[0, 1, 1, 0, 0, 1]],\n[[0, 1, 0, 0, 1, 1],[1, 0, 0, 0, 0, 0],[3, 1, 0, 0, 1, 1]],\n[[0, 1, 0, 1, 1, 0],[1, 0, 0, 0, 0, 1],[3, 1, 0, 1, 1, 1]],\n[[0, 0, 0, 1, 0, 1],[0, 1, 1, 0, 1, 1],[0, 1, 1, 1, 1, 0]],\n[[0, 0, 1, 1, 1, 1],[0, 1, 0, 1, 1, 1],[0, 1, 1, 0, 0, 0]],\n[[0, 1, 1, 1, 1, 1],[1, 0, 0, 0, 1, 0],[3, 1, 1, 1, 0, 1]],\n[[0, 1, 0, 1, 0, 0],[1, 0, 0, 1, 0, 0],[3, 1, 0, 0, 0, 0]],\n[[1, 0, 0, 1, 0, 1],[1, 0, 0, 1, 1, 0],[2, 0, 0, 0, 1, 1]],\n[[1, 0, 1, 0, 0, 0],[1, 0, 1, 0, 0, 1],[2, 0, 0, 0, 0, 1]],\n[[1, 0, 1, 0, 1, 1],[1, 0, 1, 1, 0, 0],[2, 0, 0, 1, 1, 1]],\n[[1, 0, 1, 1, 1, 0],[1, 1, 0, 0, 0, 0],[2, 1, 1, 1, 1, 0]],\n[[1, 1, 0, 0, 0, 1],[1, 1, 0, 0, 1, 1],[2, 0, 0, 0, 1, 0]],\n[[1, 1, 0, 1, 0, 0],[1, 1, 1, 0, 0, 0],[2, 0, 1, 1, 0, 0]],\n[[1, 1, 0, 1, 1, 1],[1, 1, 1, 0, 0, 1],[2, 0, 1, 1, 1, 0]],\n[[1, 1, 1, 0, 1, 0],[1, 1, 1, 1, 0, 0],[2, 0, 0, 1, 1, 0]],\n[[1, 0, 1, 0, 1, 0],[1, 1, 1, 1, 1, 0],[2, 1, 0, 1, 0, 0]],\n[[1, 1, 0, 0, 1, 0],[1, 1, 1, 1, 0, 1],[2, 0, 1, 1, 1, 1]],\n[[1, 0, 0, 1, 1, 1],[1, 1, 1, 0, 1, 1],[2, 1, 1, 1, 0, 0]],\n[[2, 0, 0, 1, 0, 0],[3, 0, 0, 0, 0, 0],[3, 0, 0, 1, 0, 0]],\n[[2, 0, 1, 0, 0, 1],[3, 0, 0, 0, 0, 1],[3, 0, 1, 0, 0, 0]],\n[[2, 0, 1, 1, 0, 1],[3, 0, 0, 0, 1, 0],[3, 0, 1, 1, 1, 1]],\n[[2, 1, 0, 0, 0, 0],[3, 0, 0, 1, 0, 1],[3, 1, 0, 1, 0, 1]],\n[[2, 1, 0, 0, 1, 0],[3, 0, 0, 0, 1, 1],[3, 1, 0, 0, 0, 1]],\n[[2, 1, 0, 1, 0, 1],[2, 1, 0, 1, 1, 0],[2, 1, 1, 0, 0, 0],[2, 1, 1, 0, 1, 1]],\n[[2, 0, 0, 1, 0, 1],[2, 0, 1, 0, 0, 0],[2, 1, 0, 1, 1, 1],[2, 1, 1, 0, 1, 0]],\n[[1, 0, 1, 1, 1, 1],[2, 1, 1, 1, 0, 1],[3, 0, 0, 1, 1, 0],[3, 0, 1, 0, 1, 0],[3, 1, 1, 1, 1, 0]],\n[[1, 1, 0, 1, 1, 0],[2, 1, 0, 0, 1, 1],[3, 0, 0, 1, 1, 1],[3, 0, 1, 0, 0, 1],[3, 0, 1, 0, 1, 1]],\n[[2, 0, 1, 0, 1, 1],[2, 1, 1, 0, 0, 1],[2, 1, 1, 1, 1, 1],[3, 1, 0, 0, 1, 0],[3, 1, 1, 1, 1, 1]],\n[[1, 1, 0, 1, 0, 1],[2, 1, 0, 0, 0, 1],[3, 0, 1, 1, 1, 0],[3, 1, 0, 1, 1, 0],[3, 1, 1, 1, 0, 0]],\n[[1, 1, 1, 1, 1, 1],[2, 0, 1, 0, 1, 0],[3, 1, 0, 1, 0, 0],[3, 1, 1, 0, 0, 0],[3, 1, 1, 0, 0, 1]],\n[[0, 0, 0, 1, 1, 0],[0, 1, 0, 1, 0, 1],[0, 1, 1, 1, 0, 1],[1, 0, 0, 0, 1, 1],[3, 0, 1, 1, 0, 1]],\n[[1, 0, 1, 1, 0, 1],[2, 0, 0, 0, 0, 0],[3, 0, 1, 1, 0, 0],[3, 1, 1, 0, 1, 0],[3, 1, 1, 0, 1, 1]]\n]\nA partition for sets of sizes: 28*3 2*4 7*5\n```\n```\n[4, 2, 2, 2, 2, 2]\n[29, 0, 8]\n[\n[[0, 0, 0, 0, 0, 1],[0, 0, 0, 0, 1, 0],[0, 0, 0, 0, 1, 1]],\n[[0, 0, 0, 1, 0, 0],[0, 0, 1, 0, 0, 0],[0, 0, 1, 1, 0, 0]],\n[[0, 0, 0, 1, 1, 1],[0, 0, 1, 0, 0, 1],[0, 0, 1, 1, 1, 0]],\n[[0, 0, 1, 0, 1, 0],[0, 1, 0, 0, 0, 0],[0, 1, 1, 0, 1, 0]],\n[[0, 0, 1, 1, 0, 1],[0, 1, 0, 0, 0, 1],[0, 1, 1, 1, 0, 0]],\n[[0, 0, 1, 0, 1, 1],[0, 1, 0, 0, 1, 0],[0, 1, 1, 0, 0, 1]],\n[[0, 1, 0, 0, 1, 1],[1, 0, 0, 0, 0, 0],[3, 1, 0, 0, 1, 1]],\n[[0, 1, 0, 1, 1, 0],[1, 0, 0, 0, 0, 1],[3, 1, 0, 1, 1, 1]],\n[[0, 0, 0, 1, 0, 1],[0, 1, 1, 0, 1, 1],[0, 1, 1, 1, 1, 0]],\n[[0, 0, 1, 1, 1, 1],[0, 1, 0, 1, 1, 1],[0, 1, 1, 0, 0, 0]],\n[[0, 1, 1, 1, 1, 1],[1, 0, 0, 0, 1, 0],[3, 1, 1, 1, 0, 1]],\n[[0, 1, 0, 1, 0, 0],[1, 0, 0, 1, 0, 0],[3, 1, 0, 0, 0, 0]],\n[[1, 0, 0, 1, 0, 1],[1, 0, 0, 1, 1, 0],[2, 0, 0, 0, 1, 1]],\n[[1, 0, 1, 0, 0, 0],[1, 0, 1, 0, 0, 1],[2, 0, 0, 0, 0, 1]],\n[[1, 0, 1, 0, 1, 1],[1, 0, 1, 1, 0, 0],[2, 0, 0, 1, 1, 1]],\n[[1, 0, 1, 1, 1, 0],[1, 1, 0, 0, 0, 0],[2, 1, 1, 1, 1, 0]],\n[[1, 1, 0, 0, 0, 1],[1, 1, 0, 0, 1, 1],[2, 0, 0, 0, 1, 0]],\n[[1, 1, 0, 1, 0, 0],[1, 1, 1, 0, 0, 0],[2, 0, 1, 1, 0, 0]],\n[[1, 1, 0, 1, 1, 1],[1, 1, 1, 0, 0, 1],[2, 0, 1, 1, 1, 0]],\n[[1, 1, 1, 0, 1, 0],[1, 1, 1, 1, 0, 0],[2, 0, 0, 1, 1, 0]],\n[[1, 0, 1, 0, 1, 0],[1, 1, 1, 1, 1, 0],[2, 1, 0, 1, 0, 0]],\n[[1, 1, 0, 0, 1, 0],[1, 1, 1, 1, 0, 1],[2, 0, 1, 1, 1, 1]],\n[[1, 0, 0, 1, 1, 1],[1, 1, 1, 0, 1, 1],[2, 1, 1, 1, 0, 0]],\n[[2, 0, 0, 1, 0, 0],[3, 0, 0, 0, 0, 0],[3, 0, 0, 1, 0, 0]],\n[[2, 0, 1, 0, 0, 1],[3, 0, 0, 0, 0, 1],[3, 0, 1, 0, 0, 0]],\n[[2, 0, 1, 1, 0, 1],[3, 0, 0, 0, 1, 0],[3, 0, 1, 1, 1, 1]],\n[[2, 1, 0, 0, 0, 0],[3, 0, 0, 1, 0, 1],[3, 1, 0, 1, 0, 1]],\n[[2, 1, 0, 0, 1, 0],[3, 0, 0, 0, 1, 1],[3, 1, 0, 0, 0, 1]],\n[[2, 1, 0, 1, 0, 1],[3, 0, 0, 1, 1, 1],[3, 1, 0, 0, 1, 0]],\n[[2, 1, 1, 0, 0, 0],[2, 1, 1, 0, 0, 1],[2, 1, 1, 0, 1, 0],[3, 0, 1, 1, 0, 1],[3, 1, 0, 1, 1, 0]],\n[[1, 0, 1, 1, 1, 1],[2, 1, 1, 1, 0, 1],[3, 0, 0, 1, 1, 0],[3, 0, 1, 0, 1, 0],[3, 1, 1, 1, 1, 0]],\n[[1, 1, 0, 1, 1, 0],[2, 1, 0, 0, 1, 1],[3, 0, 1, 0, 0, 1],[3, 1, 0, 1, 0, 0],[3, 1, 1, 0, 0, 0]],\n[[2, 1, 0, 1, 1, 0],[2, 1, 1, 0, 1, 1],[2, 1, 1, 1, 1, 1],[3, 0, 1, 0, 1, 1],[3, 1, 1, 0, 0, 1]],\n[[1, 1, 0, 1, 0, 1],[2, 0, 1, 0, 1, 1],[3, 0, 1, 1, 0, 0],[3, 0, 1, 1, 1, 0],[3, 1, 1, 1, 0, 0]],\n[[0, 1, 0, 1, 0, 1],[1, 1, 1, 1, 1, 1],[2, 0, 0, 0, 0, 0],[2, 1, 0, 0, 0, 1],[3, 1, 1, 0, 1, 1]],\n[[0, 0, 0, 1, 1, 0],[1, 0, 0, 0, 1, 1],[2, 0, 1, 0, 0, 0],[2, 1, 0, 1, 1, 1],[3, 1, 1, 0, 1, 0]],\n[[0, 1, 1, 1, 0, 1],[1, 0, 1, 1, 0, 1],[2, 0, 0, 1, 0, 1],[2, 0, 1, 0, 1, 0],[3, 1, 1, 1, 1, 1]]\n]\nA partition for sets of sizes: 29*3 0*4 8*5\n```\n```\n[4, 2, 2, 2, 2, 2]\n[25, 3, 8]\n[\n[[0, 0, 0, 0, 0, 1],[0, 0, 0, 0, 1, 0],[0, 0, 0, 0, 1, 1]],\n[[0, 0, 0, 1, 0, 0],[0, 0, 1, 0, 0, 0],[0, 0, 1, 1, 0, 0]],\n[[0, 0, 0, 1, 1, 1],[0, 0, 1, 0, 0, 1],[0, 0, 1, 1, 1, 0]],\n[[0, 0, 1, 0, 1, 0],[0, 1, 0, 0, 0, 0],[0, 1, 1, 0, 1, 0]],\n[[0, 0, 1, 1, 0, 1],[0, 1, 0, 0, 0, 1],[0, 1, 1, 1, 0, 0]],\n[[0, 0, 1, 0, 1, 1],[0, 1, 0, 0, 1, 0],[0, 1, 1, 0, 0, 1]],\n[[0, 1, 0, 0, 1, 1],[1, 0, 0, 0, 0, 0],[3, 1, 0, 0, 1, 1]],\n[[0, 1, 0, 1, 1, 0],[1, 0, 0, 0, 0, 1],[3, 1, 0, 1, 1, 1]],\n[[0, 0, 0, 1, 0, 1],[0, 1, 1, 0, 1, 1],[0, 1, 1, 1, 1, 0]],\n[[0, 0, 1, 1, 1, 1],[0, 1, 0, 1, 1, 1],[0, 1, 1, 0, 0, 0]],\n[[0, 1, 1, 1, 1, 1],[1, 0, 0, 0, 1, 0],[3, 1, 1, 1, 0, 1]],\n[[0, 1, 0, 1, 0, 0],[1, 0, 0, 1, 0, 0],[3, 1, 0, 0, 0, 0]],\n[[1, 0, 0, 1, 0, 1],[1, 0, 0, 1, 1, 0],[2, 0, 0, 0, 1, 1]],\n[[1, 0, 1, 0, 0, 0],[1, 0, 1, 0, 0, 1],[2, 0, 0, 0, 0, 1]],\n[[1, 0, 1, 0, 1, 1],[1, 0, 1, 1, 0, 0],[2, 0, 0, 1, 1, 1]],\n[[1, 0, 1, 1, 1, 0],[1, 1, 0, 0, 0, 0],[2, 1, 1, 1, 1, 0]],\n[[1, 1, 0, 0, 0, 1],[1, 1, 0, 0, 1, 1],[2, 0, 0, 0, 1, 0]],\n[[1, 1, 0, 1, 0, 0],[1, 1, 1, 0, 0, 0],[2, 0, 1, 1, 0, 0]],\n[[1, 1, 0, 1, 1, 1],[1, 1, 1, 0, 0, 1],[2, 0, 1, 1, 1, 0]],\n[[1, 1, 1, 0, 1, 0],[1, 1, 1, 1, 0, 0],[2, 0, 0, 1, 1, 0]],\n[[1, 0, 1, 0, 1, 0],[1, 1, 1, 1, 1, 0],[2, 1, 0, 1, 0, 0]],\n[[1, 1, 0, 0, 1, 0],[1, 1, 1, 1, 0, 1],[2, 0, 1, 1, 1, 1]],\n[[1, 0, 0, 1, 1, 1],[1, 1, 1, 0, 1, 1],[2, 1, 1, 1, 0, 0]],\n[[2, 0, 0, 1, 0, 0],[3, 0, 0, 0, 0, 0],[3, 0, 0, 1, 0, 0]],\n[[2, 0, 1, 0, 0, 1],[3, 0, 0, 0, 0, 1],[3, 0, 1, 0, 0, 0]],\n[[1, 1, 0, 1, 1, 0],[2, 0, 1, 1, 0, 1],[2, 1, 0, 0, 0, 0],[3, 0, 1, 0, 1, 1]],\n[[1, 1, 0, 1, 0, 1],[2, 1, 0, 0, 0, 1],[2, 1, 0, 0, 1, 0],[3, 1, 0, 1, 1, 0]],\n[[1, 1, 1, 1, 1, 1],[2, 1, 0, 1, 0, 1],[2, 1, 0, 1, 1, 0],[3, 1, 1, 1, 0, 0]],\n[[2, 1, 1, 0, 0, 0],[2, 1, 1, 0, 0, 1],[2, 1, 1, 0, 1, 0],[3, 0, 0, 0, 1, 0],[3, 1, 1, 0, 0, 1]],\n[[1, 0, 1, 1, 1, 1],[2, 1, 1, 1, 0, 1],[3, 0, 0, 0, 1, 1],[3, 0, 0, 1, 0, 1],[3, 1, 0, 1, 0, 0]],\n[[2, 0, 1, 0, 0, 0],[2, 1, 1, 0, 1, 1],[2, 1, 1, 1, 1, 1],[3, 0, 0, 1, 1, 0],[3, 0, 1, 0, 1, 0]],\n[[2, 0, 1, 0, 1, 1],[2, 1, 0, 0, 1, 1],[2, 1, 0, 1, 1, 1],[3, 1, 0, 0, 0, 1],[3, 1, 1, 1, 1, 0]],\n[[1, 0, 0, 0, 1, 1],[2, 0, 1, 0, 1, 0],[3, 0, 1, 1, 1, 0],[3, 1, 0, 0, 1, 0],[3, 1, 0, 1, 0, 1]],\n[[1, 0, 1, 1, 0, 1],[2, 0, 0, 1, 0, 1],[3, 0, 1, 1, 0, 1],[3, 1, 1, 0, 1, 0],[3, 1, 1, 1, 1, 1]],\n[[0, 0, 0, 1, 1, 0],[3, 0, 1, 0, 0, 1],[3, 0, 1, 1, 0, 0],[3, 1, 1, 0, 0, 0],[3, 1, 1, 0, 1, 1]],\n[[0, 1, 0, 1, 0, 1],[0, 1, 1, 1, 0, 1],[2, 0, 0, 0, 0, 0],[3, 0, 0, 1, 1, 1],[3, 0, 1, 1, 1, 1]]\n]\nA partition for sets of sizes: 25*3 3*4 8*5\n```\n```\n[4, 2, 2, 2, 2, 2]\n[26, 1, 9]\n[\n[[0, 0, 0, 0, 0, 1],[0, 0, 0, 0, 1, 0],[0, 0, 0, 0, 1, 1]],\n[[0, 0, 0, 1, 0, 0],[0, 0, 1, 0, 0, 0],[0, 0, 1, 1, 0, 0]],\n[[0, 0, 0, 1, 1, 1],[0, 0, 1, 0, 0, 1],[0, 0, 1, 1, 1, 0]],\n[[0, 0, 1, 0, 1, 0],[0, 1, 0, 0, 0, 0],[0, 1, 1, 0, 1, 0]],\n[[0, 0, 1, 1, 0, 1],[0, 1, 0, 0, 0, 1],[0, 1, 1, 1, 0, 0]],\n[[0, 0, 1, 0, 1, 1],[0, 1, 0, 0, 1, 0],[0, 1, 1, 0, 0, 1]],\n[[0, 1, 0, 0, 1, 1],[1, 0, 0, 0, 0, 0],[3, 1, 0, 0, 1, 1]],\n[[0, 1, 0, 1, 1, 0],[1, 0, 0, 0, 0, 1],[3, 1, 0, 1, 1, 1]],\n[[0, 0, 0, 1, 0, 1],[0, 1, 1, 0, 1, 1],[0, 1, 1, 1, 1, 0]],\n[[0, 0, 1, 1, 1, 1],[0, 1, 0, 1, 1, 1],[0, 1, 1, 0, 0, 0]],\n[[0, 1, 1, 1, 1, 1],[1, 0, 0, 0, 1, 0],[3, 1, 1, 1, 0, 1]],\n[[0, 1, 0, 1, 0, 0],[1, 0, 0, 1, 0, 0],[3, 1, 0, 0, 0, 0]],\n[[1, 0, 0, 1, 0, 1],[1, 0, 0, 1, 1, 0],[2, 0, 0, 0, 1, 1]],\n[[1, 0, 1, 0, 0, 0],[1, 0, 1, 0, 0, 1],[2, 0, 0, 0, 0, 1]],\n[[1, 0, 1, 0, 1, 1],[1, 0, 1, 1, 0, 0],[2, 0, 0, 1, 1, 1]],\n[[1, 0, 1, 1, 1, 0],[1, 1, 0, 0, 0, 0],[2, 1, 1, 1, 1, 0]],\n[[1, 1, 0, 0, 0, 1],[1, 1, 0, 0, 1, 1],[2, 0, 0, 0, 1, 0]],\n[[1, 1, 0, 1, 0, 0],[1, 1, 1, 0, 0, 0],[2, 0, 1, 1, 0, 0]],\n[[1, 1, 0, 1, 1, 1],[1, 1, 1, 0, 0, 1],[2, 0, 1, 1, 1, 0]],\n[[1, 1, 1, 0, 1, 0],[1, 1, 1, 1, 0, 0],[2, 0, 0, 1, 1, 0]],\n[[1, 0, 1, 0, 1, 0],[1, 1, 1, 1, 1, 0],[2, 1, 0, 1, 0, 0]],\n[[1, 1, 0, 0, 1, 0],[1, 1, 1, 1, 0, 1],[2, 0, 1, 1, 1, 1]],\n[[1, 0, 0, 1, 1, 1],[1, 1, 1, 0, 1, 1],[2, 1, 1, 1, 0, 0]],\n[[2, 0, 0, 1, 0, 0],[3, 0, 0, 0, 0, 0],[3, 0, 0, 1, 0, 0]],\n[[2, 0, 1, 0, 0, 1],[3, 0, 0, 0, 0, 1],[3, 0, 1, 0, 0, 0]],\n[[2, 0, 1, 1, 0, 1],[3, 0, 0, 0, 1, 0],[3, 0, 1, 1, 1, 1]],\n[[2, 0, 0, 0, 0, 0],[2, 0, 0, 1, 0, 1],[2, 1, 0, 0, 1, 0],[2, 1, 0, 1, 1, 1]],\n[[2, 1, 0, 0, 1, 1],[2, 1, 0, 1, 0, 1],[2, 1, 0, 1, 1, 0],[3, 0, 0, 1, 0, 1],[3, 1, 0, 1, 0, 1]],\n[[2, 1, 1, 0, 0, 0],[2, 1, 1, 0, 0, 1],[2, 1, 1, 0, 1, 0],[3, 0, 0, 0, 1, 1],[3, 1, 1, 0, 0, 0]],\n[[1, 0, 1, 1, 1, 1],[2, 1, 1, 1, 0, 1],[3, 0, 0, 1, 1, 0],[3, 0, 1, 0, 1, 0],[3, 1, 1, 1, 1, 0]],\n[[1, 1, 0, 1, 1, 0],[2, 1, 1, 0, 1, 1],[3, 0, 0, 1, 1, 1],[3, 1, 0, 0, 0, 1],[3, 1, 1, 0, 1, 1]],\n[[2, 0, 1, 0, 0, 0],[2, 0, 1, 0, 1, 1],[2, 1, 0, 0, 0, 1],[3, 0, 1, 0, 1, 1],[3, 1, 1, 0, 0, 1]],\n[[1, 1, 0, 1, 0, 1],[2, 1, 0, 0, 0, 0],[3, 0, 1, 1, 1, 0],[3, 1, 0, 1, 0, 0],[3, 1, 1, 1, 1, 1]],\n[[1, 1, 1, 1, 1, 1],[2, 1, 1, 1, 1, 1],[3, 0, 1, 1, 0, 0],[3, 1, 0, 1, 1, 0],[3, 1, 1, 0, 1, 0]],\n[[0, 1, 0, 1, 0, 1],[1, 0, 0, 0, 1, 1],[1, 0, 1, 1, 0, 1],[3, 0, 1, 0, 0, 1],[3, 1, 0, 0, 1, 0]],\n[[0, 0, 0, 1, 1, 0],[0, 1, 1, 1, 0, 1],[2, 0, 1, 0, 1, 0],[3, 0, 1, 1, 0, 1],[3, 1, 1, 1, 0, 0]]\n]\nA partition for sets of sizes: 26*3 1*4 9*5\n```\n```\n[4, 2, 2, 2, 2, 2]\n[23, 2, 10]\n[\n[[0, 0, 0, 0, 0, 1],[0, 0, 0, 0, 1, 0],[0, 0, 0, 0, 1, 1]],\n[[0, 0, 0, 1, 0, 0],[0, 0, 1, 0, 0, 0],[0, 0, 1, 1, 0, 0]],\n[[0, 0, 0, 1, 1, 1],[0, 0, 1, 0, 0, 1],[0, 0, 1, 1, 1"},"data_id":{"kind":"string","value":"arxiv_0004114"}}},{"rowIdx":996,"cells":{"content":{"kind":"string","value":"h(\\pi (u-n\\pi)/\\gamma)$ for *γ* ≪ 1.\nWe compute Eq. , merging together (1)-(3), to get Eq. in the main text,\n \n$$\\begin{aligned}\n \\mathcal{G}\\_{\\delta p =1}&\\simeq \n- g\\_0 \\tilde{\\gamma} L (k\\_B T ) ^3\n\\exp(-\\frac{L}{L\\_{T,c}}) \\exp(-\\frac{\\Delta{L}}{L\\_{T,n}}) \\nonumber\n\\\\ &\\times \\exp(-\\frac{L - \\Delta L}{L\\_{T,n}})\n\\nonumber \\\\ \n&= -g\\_0 \\tilde{\\gamma} L (k\\_B T ) ^3\n\\exp(-\\frac{L}{L\\_{T,c}}) \\exp(-\\frac{L}{L\\_{T,n}}).\\end{aligned}$$ \nHere $g\\_0 = 16 e^2 \\sqrt {2\\pi} \\pi (a/\\hbar v\\_c)^{13/4} (a/\\hbar v\\_n)^{3/4} (\\Gamma(3/4))^2 / (ah)$ is a constant, Γ is the Gamma function, and we have used the integral formula of\n \n$$\\label{integralFM\\_sup}\n\\int\\_{-\\infty}^{\\infty}dt\\bigg (\\frac{\\sinh(\\frac{i\\pi a k\\_B T}{\\hbar v\\_{c}})}{\\sinh \\big(\\frac{\\pi k\\_B T}{\\hbar v\\_{c}}(ia-v\\_{c}t)\\big)} \\bigg )^{\\frac{3}{2}} = \\frac{2a}{v\\_c}\\Big(\\frac{2a k\\_B T}{\\hbar v\\_c}\\Big )^{\\frac{1}{2}} \\Gamma \\big(\\frac{3}{4}\\big)^2.$$ \nNotice that the factor exp( − (*L* − Δ*L*)2/(*L**L**T*, *n*)) of Re[*G*0] (zero-mode part) exactly cancels out exp[(*L* − Δ*L*)2/(*L**L**T*, *n*)] from *G**n*(*t*) (the plasmonic part). This fact was found in a Luttinger liquid with finite size .\nWe next move to G*δ**p* = 1 in the *v**c* ≫ *v**n* case. The main contribution to G*δ**p* = 1 also occurs near *t* = ± *L*/*v**c*. We observe the followings. (1) Because the portion of (*δ**p*qm, *δ**q*qm) = (1, 0) in Re[*G*0] fully vanishes, the dominant contribution comes from the portion of (*δ**p*qm, *δ**q*qm) = (1, − 1), which leads to Re[*G*0] ≃ − cos(2*π*Φ/Φ0)exp( − (*L* − Δ*L*)2/(*L**L**T*, *n*)). (2) *F* ≃ 4*π**k**B**T**L*exp( − 2*π**k**B**T**L*/ℏ*v**c*)/(ℏ*v**c*) = (8*L*/3*L**T*, *c*)exp( − *L*/*L**T*, *c*)exp( − Δ*L*/*L**T*, *n*). (3) For the QD plasmon part, the same expression is obtained as Eq. , except that *L* − Δ*L* is replaced by Δ*L*. Merging (1) − (3) and using Eq. , we obtain Eq. in the main text,\n \n$$\\begin{aligned}\n \\label{smallvsFirstHarmonics\\_sup}\n \\mathcal{G}\\_{\\delta p =1}&\\simeq \n-g\\_0 \\tilde{\\gamma} L (k\\_B T)^3 \\exp(-\\frac{L}{L\\_{T,c}}) \\exp(-\\frac{\\Delta{L}}{L\\_{T,n}}) \\nonumber \\\\ &\\times \n \\exp(-\\frac{\\Delta L}{L\\_{T,n}})\n \\exp[-\\frac{(L -\\Delta L)^2-(\\Delta L)^2}{L L\\_{T,n}}]\n \\nonumber \\\\ \n &= -g\\_0 \\tilde{\\gamma} L (k\\_B T)^3 \\exp(-\\frac{L}{L\\_{T,n}})\\exp(-\\frac{L}{L\\_{T,c}}).\\end{aligned}$$ \nIn the same way, we obtain Eq. ,\n \n$$\\begin{aligned}\n \\label{smallvsSecondHarmonics\\_sup}\n\\mathcal{G}\\_{\\delta p =2}& \\simeq\n2g\\_0 \\tilde{\\gamma} L (k\\_B T)^3 \\exp(-\\frac{2L}{L\\_{T,c}}) \\exp(-\\frac{2\\Delta L}{L\\_{T,n}}) \\exp(-\\frac{2\\Delta L}{L\\_{T,n}}) \\nonumber \\\\ &= \n 2g\\_0 \\tilde{\\gamma} L (k\\_B T)^3 \\exp(-\\frac{10L}{3L\\_{T,c}}).\\end{aligned}$$ \nThe quasiparticle fluctuation in the bulk\n=========================================\nWe consider quasiparticle fluctuations in the bulk, differentiating the QD bulk from the edge. In the limit of bulk charging energy smaller than temperature, we show that the zero-mode part *G*0 has the same expression as Eq. in the presence of quasiparticle fluctuations in the bulk, hence, that the period halving (*h*/2*e* oscillation) takes place.\nWe argue that the zero-mode part of the QD states is characterized by four numbers (*N**e*, *N**n*, *N**q**p*, *N**q**p*, *n*), when we additionally consider the bulk degrees of freedom. *N**e* and *N**n* are necessary to describe excess electrons in the QD. For an additional excess electron, the number *N**e* of excess electrons increases by 1. This electron is decomposed into 3 additional charge components (*N**c* → *N**c* + 3; the charge of the charge component is *e*/3) and ± 1 additional neutral component (*N**n* → *N**n* ± 1), as discussed in the main text. *N**n* counts the number of the neutral components by excess electrons in the QD, as in the main text. On the other hand, *N**q**p* and *N**q**p*, *n* are introduced to describe quasiparticle excitations in the QD bulk. When an additional quasiparticle excites in the bulk, the number *N**q**p* of quasiparticles increases by 1. This quasiparticle is decomposed into 1 charge component and ± 1 neutral component (*N**q**p*, *n* → *N**q**p*, *n* ± 1). *N**q**p*, *n* counts the number of the neutral components by bulk quasiparticle excitations.\nTaking into account quasiparticle fluctuations in the bulk and electron fluctuations inside the QD, the zero-mode part of the QD Hamiltonian *H*D0 is expressed as\n \n$$\\begin{aligned}\n \\label{HamiltonianZeroBulk\\_sup}\n& H\\_{\\textrm{D}}^{0} (N\\_{e}, N\\_{qp}, N\\_{n}, N\\_{qp,n}) \\nonumber \\\\ = &\\, \\frac{E\\_c}{12}\\bigg(3N\\_{e}+1-\\frac{2\\Phi}{\\Phi\\_0} - N\\_{qp}\\bigg)^2 +\n\\frac{E\\_n}{4} (N\\_{n} - N\\_{qp,n})^2 \\nonumber \\\\ + & \\, E\\_{\\textrm{bc}} \\bigg(\\frac{2\\Phi}{\\Phi\\_0} + N\\_{qp}\\bigg)^2, \\end{aligned}$$ \nwhere *E*bc is the bulk charging energy. The first (second) term describes the interaction between the charge (neutral) components on the edge while the third term describes the interaction between the charge components in the bulk. The flux dependence in the bulk-charging energy term describes charge accumulation in the bulk as the magnetic flux increases. We ignore the interactions between the neutral components in the bulk because they are of dipole type hence weaker than the interaction terms of the total charge. We also assume that it is in the Aharonov-Bohm regime neglecting electrostatic coupling between quasiparticles in the bulk and on the edge.\nWe consider the case that the relaxation time from the QD edge to the bulk is much longer than the QD dwell time of an electron injected from a lead edge, hence, that the electron enters only into the QD edge. Then, the zero-mode part $\\exp({i\\sqrt{3/2}\\phi\\_c^{0}}\\pm{i\\sqrt{1/2}\\phi\\_n^{0}})$ of the electron operators Ψ±(*x*) is evolved by Eq. as\n \n$$\\begin{aligned}\n& e^{{i\\sqrt{\\frac{3}{2}}\\phi\\_c^{0}(x,t)}\\pm{i\\sqrt{\\frac{1}{2}}\\phi\\_n^{0}(x,t)}} \\nonumber \\\\ \n=& e^{iH\\_{\\textrm{D}}^{0}t}e^{{i\\sqrt{\\frac{3}{2}}\\phi\\_c^{0}(x)}\\pm{i\\sqrt{\\frac{1}{2}}\\phi\\_n^{0}(x)}}\n e^{-iH\\_{\\textrm{D}}^{0}t}\\nonumber \\\\\n=&\\exp\\Big(i\\sqrt{\\frac{3}{2}}\\phi\\_c^{0}(x)\\pm i\\sqrt{\\frac{1}{2}}\\phi\\_n^{0}(x) \\nonumber \\\\ +&\\frac{i \\pi v\\_c t}{L} (3N\\_{e}+1-\\frac{2\\Phi}{\\Phi\\_0} - N\\_{qp} )\\pm \\frac{i \\pi v\\_n t}{L} (N\\_{n} - N\\_{qp,n})\\Big). \\end{aligned}$$ \nAnd, the zero-mode part *G*0(*t*) of the Green’s function $\\big\\langle \\hat{\\Psi}\\_{\\pm}^{\\dagger}(x\\_{\\mathcal{L}},t) \\hat{\\Psi}\\_{\\pm}(x\\_{\\mathcal{L}},0)\\big\\rangle$ is written as\n \n$$\\begin{aligned}\n G^0&\\propto \\Big\\langle \\exp \\Big[\\frac{i \\pi v\\_{c}t}{L} \\Big(3N\\_{e}-\\frac{2\\Phi}{\\Phi\\_0}+1-N\\_{qp}\\Big) \\Big]\n\\exp(\\pm\\frac{ i\\pi v\\_{n}t }{L} (N\\_{n} - N\\_{qp,n}) ) \n\\Big\\rangle \\nonumber \\\\\n& = \\bigg \\{\n\\sum\\limits\\_{\\delta p\\_{\\textrm{qm}}, \\delta p', \\delta q\\_{\\textrm{qm}} = -\\infty}^{\\infty} e^{-i\\pi \\delta p' /3} (1+(-1)^{\\delta p\\_{\\textrm{qm}}+\\delta q\\_{\\textrm{qm}}})(1+(-1)^{\\delta p'+\\delta q\\_{\\textrm{qm}}})\\exp(2\\pi i \\delta p\\_{\\textrm{qm}} \\Phi/ \\Phi\\_0) \\nonumber \\\\& \\times\n\\exp \\big [-\\frac{3 \\pi^2 k\\_B T }{E\\_c}\\big(\\frac{v\\_{c} t}{L} + \\frac{\\delta p'}{3} \\big)^2 \\big] \\exp \\big[-\\frac{\\pi^2 k\\_B T}{E\\_n}\\big( \\delta q\\_{\\textrm{qm}} \\pm \\frac{v\\_{n} t}{L} \\big)^2 \\big] \\exp \\big[-\\frac{\\pi^2 k\\_B T}{4E\\_{\\textrm{bc}}}\\big( \\delta p\\_{\\textrm{qm}} - \\frac{\\delta p'}{3} \\big)^2 \\big] \\bigg \\} \\nonumber \\\\\n& \\bigg/ \\bigg \\{\n\\sum\\limits\\_{\\delta p\\_{\\textrm{qm}}, \\delta p', \\delta q\\_{\\textrm{qm}} = -\\infty}^{\\infty} e^{-i\\pi \\delta p' /3} (1+(-1)^{\\delta p\\_{\\textrm{qm}}+\\delta q\\_{\\textrm{qm}}})(1+(-1)^{\\delta p'+\\delta q\\_{\\textrm{qm}}})\\exp(2\\pi i \\delta p\\_{\\textrm{qm}} \\Phi/ \\Phi\\_0) \\nonumber \\\\& \\times\n\\exp \\big [-\\frac{3 \\pi^2 k\\_B T }{E\\_c}\\big(\\frac{\\delta p'}{3}\\big)^2 \\big] \\exp \\big[-\\frac{\\pi^2 k\\_B T\\delta q\\_{\\textrm{qm}}^2 }{E\\_n}\\big] \\exp \\big[-\\frac{\\pi^2 k\\_B T}{4E\\_{\\textrm{bc}}}\\big( \\delta p\\_{\\textrm{qm}} - \\frac{\\delta p'}{3} \\big)^2 \\big] \\bigg \\}.\n\\label{ZeromodeaverageBulk\\_sup} \\end{aligned}$$ \nWe applied the Poisson summation formula as in Eq. . At temperature much higher than the bulk-charging energy, the portion of *δ**p*ʹ = 3*δ**p*qm survives, resulting in the same expression as Eq. .\nConfining potential and Coulomb interaction\n===========================================\nWhen the edge potential is smooth enough , we below show that *v**c* is much larger than *v**n* and *v*; see Eq. in the main text. In the absense of Coulomb interaction, the edge Hamiltonian at *ν* = 2/3 has the form of $H\\_{\\text{con}}=\\frac{1}{4\\pi} \\int dx [v\\_{1,\\text{con}} (\\partial\\_x \\phi\\_1)^2 + 3v\\_{2,\\text{con}}(\\partial\\_x \\phi\\_2)^2]$ in terms of the velocities of the original edge modes, *v*1, con and *v*2, con, which are solely determined by the edge confining potential. When the Coulomb interaction is tuned on, it leads to an interaction Hamiltonian $H\\_{\\text{int}}=\\frac{v\\_{\\text{int}}}{4\\pi} \\int dx (\\partial\\_x (\\phi\\_1 + \\phi\\_2))^2 $, which counts the interaction by the total charge density ∂*x*(*ϕ*1 + *ϕ*2)/(2*π*); here, we assume that *v*int is indepedent of momentum, (which is valid when the Coulomb interaction is short ranged due to the screening by gates). Then, the total Hamiltonian is $H\\_{\\textrm{D}} = H\\_{\\text{con}}+H\\_{\\text{int}} = \\frac{1}{4\\pi} \\int dx [v\\_{c} (\\partial\\_x \\phi\\_c)^2 + v\\_{n}(\\partial\\_x \\phi\\_n)^2+ v\\partial\\_x \\phi\\_c \\partial\\_x \\phi\\_n]$, where *v**c* = 3*v*1, con/2 + *v*2, con/2 + 2*v*int/3, *v**n* = *v*1, con/2 + 3*v*2, con/2, and $v = -\\sqrt{3}(v\\_{1,\\text{con}}+ v\\_{2,\\text{con}})$. This shows that *v* and *v**n* are much smaller than *v**c*, if the edge potential is smooth enough such that *v*int ≫ *v*1, con, *v*2, con.\n00\nX.-G. Wen, Int. J. Mod. Phys. 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Present affiliation: Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel[↩](#fnref1)\n\nTopological dephasing in the *ν* = 2/3 fractional Quantum Hall regime\n=====================================================================\n[1](#fn1)\nWe study dephasing in electron transport through a large quantum dot (a Fabry-Perot interferometer) in the fractional quantum Hall regime with filling factor 2/3. In the regime of sequential tunneling, dephasing occurs due to electron fractionalization into counterpropagating charge and neutral edge modes on the dot. In particular, when the charge mode moves much faster than the neutral mode, and at temperatures higher than the level spacing of the dot, electron fractionalization combined with the fractional statistics of the charge mode leads to the dephasing selectively suppressing *h*/*e* Aharonov-Bohm oscillations but not *h*/(2*e*) oscillations, resulting in oscillation-period halving.\nIntroduction\n============\nA fractional quantum Hall (QH) system of filling fraction *ν* has edge channels that support fractional charges obeying fractional braiding statistics . At *ν* = 2/3, the edge states are decomposed into a *ν**c*edge = 2/3 charge mode and a counterpropagating neutral mode . They originate from renormalization of two counterpropagating charge modes , *ν*1edge = 1 and *ν*2edge = − 1/3, and stabilize at low temperature under strong disorder. Neutral modes have attracted much attention, as they are charge neutral and carry energy. They have been recently detected through shot noise measurements , and their properties such as energy and decay length have been extensively studied .\nElectron interaction is a dominant source of dephasing at low temperature . It leads to electron fractionalization in quantum wires; an electron, injected into a wire, splits into constituents (spin-charge separation, charge fractionalization), showing reduction of interference visibility or dephasing . Interestingly, when the wire is finite, the constituents recombine after bouncing at wire ends, resulting in coherence revival . Fractionalization was detected in a non-chiral wire, and studied in the integer QH edge .\nCoherent transport, as well as dephasing, can be tackled through the study of low energy dynamics at the edge. This is particularly important in the context of the fractional QH regime. The present study implies that the presence of neutral modes could be a dominant source of dephasing. Note that neutral modes have been observed in almost all fractional QH systems . At the same time there is no uncontested observation of anyonic interference oscillations in the pure Aharonov-Bohm regime of a fractional QH interferometer.\nThe present study of the *ν* = 2/3 QH regime emphasizes two dephasing mechanisms by fractionalization of an electron into charge and neutral components, *plasmonic dephasing* and *topological dephasing*. Concerning the plasmonic dephasing mechanism, the overlap between the plasmonic parts of the charge and neutral components decreases with time, as the two components propagate with different velocities in the opposite directions. The resulting dephasing is similar to the plasmonic dephasing that takes place in a quantum wire or in integer QH edges. On the other hand, the topological dephasing is a new mechanism unnoticed so far. It occurs because the zero-mode parts of the components, satisfying fractional statistics, may braid with thermally excited anyons. Thermal average of the resulting braiding phase leads to dephasing that occurs only in the interfering processes characterized by particular values of topological winding numbers.\n[Setup1]\nOur analysis addresses the AB oscillation of differential conductance G through a quantum dot (QD) in the *ν* = 2/3 QH regime. We focus on linear response of electron sequential tunneling into the QD. G is decomposed into the harmonics of the AB flux Φ in the QD,\n \n$$\\begin{aligned}\n\\mathcal{G} = \\frac{e^2}{h} \\sum\\_{\\delta p=0,1,2,\\cdots} \\mathcal{G}\\_{\\delta p} \\cos ( 2 \\pi \\delta p \\frac{\\Phi}{\\Phi\\_0}), \\label{g\\_decomp}\\end{aligned}$$ \nwhere Φ0 ≡ ℏ*c*/∣*e*∣ is a flux quantum; see Fig. [Setup1]. Semiclassically, *δ**p* represents the relative winding number of a fractionalized charge component, around the circumference *L* of the QD, between two interfering paths: an electron, after tunneling into the QD, fractionalizes into charge and neutral components; see Fig. [Setup1]. The charge (neutral) component has propagation velocity *v**c*(*n*), spatial width *L**T*, *c*(*n*) ≡ ℏ*v**c*(*n*)/(2*π**k**B**T**δ**c*(*n*)) at temperature *T*, level spacing *E**c*(*n*) ≡ 2*π*ℏ*v**c*(*n*)/*L*, and scaling dimension *δ**c* = 3/4 (*δ**n* = 1/4) in the electron tunneling operator at low temperatures. G*δ**p* is determined by the overlaps of the components of the same kind between two interfering paths of relative charge winding *δ**p*.\nWe find two mechanisms suppressing G*δ**p* ≠ 0, the plasmonic dephasing and the topological dephasing; the former (latter) involves plasmon (zero-mode) parts of the components. In the plasmonic dephasing, G*δ**p* is contributed from the two interfering paths whose charge components overlap maximally between the paths. But, their neutral components overlap only partially between the interfering paths, reducing G*δ**p*; similar dephasing occurs in other fractionalizations . The topological dephasing additionally occurs, but depending on *δ**p*, in contrast to the other known mechanisms. When *v**c* ≫ *v**n* , the first harmonics G*δ**p* = 1 is suppressed at *k**B**T* > *E**n*/(4*π*2*δ**n*) (namely, *L* > *L**T*, *n*). It is because the charge component gains thermally fluctuating fractional braiding phase of *π**N**c* (leading to *e**i**π**N**c* = ± 1), while it winds once (*δ**p* = 1) around *N**c* electronic or anyonic thermal excitations on the QD edge or in the bulk. By contrast, the second harmonics G*δ**p* = 2 is not affected by the topological dephasing (as braiding phase *π**N**c**δ**p* and ( ± 1)*δ**p* = 1 are trivial) and dominates G, resulting in *h*/(2*e*) AB oscillations. These above findings occur in both the regimes of strong disorder and weak disorder in the edge of the QD. Note that the topological dephasing does not occur in the Coulomb dominated regime where Coulomb interactions between the bulk and edge of the QD is strong, as discussed later.\nSetup and Hamiltonian\n=====================\nThe *ν* = 2/3 QD is coupled to two lead edges via quantum point contacts (QPCs) ; see Fig. [Setup1]. The Hamiltonian is *H* = *H*D + *H*L + *H*R + *H*T. *H*D describes the edge of the QD, while *H*L(R) the *ν* = 2/3 left (right) lead edge. Each edge consists of the bosonic mode *ϕ*1 (*ν*1edge = 1) and the counterpropagating *ϕ*2 (*ν*2edge = − 1/3). *ϕ**i* = 1, 2 supports charge *e**ν**i*edge and satisfies [*ϕ**i*(*x*), *ϕ**i*ʹ(*x*ʹ)] = *i**π**ν**i*edgesgn(*x* − *x*ʹ)*δ**i**i*ʹ at positions *x*, *x*ʹ. Introducing the charge mode $\\phi\\_c \\equiv \\sqrt{3/2}(\\phi\\_1+\\phi\\_2)$ (supporting charge 2*e*/3) and the neutral mode $\\phi\\_n \\equiv (\\phi\\_1 + 3\\phi\\_2)/\\sqrt{2}$, one writes \n \n$$\\begin{aligned}\n \\label{Hamiltoniandot}\nH\\_\\textrm{D} &=& \\frac{\\hbar}{4\\pi} \\int^L\\_0 dx [v\\_c (\\partial\\_x \\phi\\_c)^2 + v\\_n(\\partial\\_x \\phi\\_n)^2+v\\partial\\_x\\phi\\_c\\partial\\_x \\phi\\_n] \\nonumber \\\\\n&+&\\int^L\\_0 dx [\\xi (x) \\exp(i\\sqrt{2}\\phi\\_n) + \\text{H.c.}].\\end{aligned}$$ \nDisorder-induced tunneling amplitude *ξ*(*x*) between *ϕ*1 and *ϕ*2 is modeled by a Gaussian random variable with mean zero and variance $\\overline{\\xi^{\\*}(x)\\xi(x)}=W\\delta (x-x')$. For a finite range of bare parameters, *ϕ**c* and *ϕ**n* decouple at low temperatures, rendering *v* irrelevant. *H*L,R is written similarly to *H*D, except ∫0*L* → ∫− ∞∞ in Eq. . Note that we ignore the Coulomb interaction between the bulk and edge of the QD, considering that the QD size is large enough .\nThe QPCs are almost closed, so electron tunneling is facilitated. Renormalization group analysis indicates four equally most relevant electron tunneling operators between the electron field operators, $\\Psi\\_{\\pm} (x\\_\\alpha) = e^{i\\sqrt{3/2}\\phi\\_c (x\\_\\alpha)}e^{\\pm i \\phi\\_n (x\\_\\alpha)/ \\sqrt{2}} / \\sqrt{2\\pi a}$ at *x**α* = L,R on the QD, and Ψ*α*, ±(0) on lead edge *α*; *a* is an ultra-violet cutoff and Ψ*α*, ± has the same form as Ψ±. So the tunneling Hamiltonian is *H*T = ∑*α* = L, R∑*i*, *j* = ±[*t**α**i**j*Ψ*α*, *i*†(0)Ψ*j*(*x**α*) + H.c.], where *t**α**i**j* is the tunneling strength.\nTopological dephasing\n=====================\nWe show that at *ν* = 2/3, fractionalization and fractional statistics cause the topological dephasing. We address the number operator *N**c*(*n*) of charge (neutral) mode at the QD edge,\n \n$$\\begin{aligned}\n \\label{OldnumberNewnumber}\n\\frac{1}{3}N\\_c = N\\_1 - \\frac{1}{3} N\\_2, \\quad \\quad N\\_n = N\\_2-N\\_1,\\end{aligned}$$ \ndefined through the zero-mode parts of *ϕ*1, 2 (see Appendix [AppenNumber]). The number operator *N*1(2) of *ϕ*1(2) is an integer since *e* and − *e*/3 are the elementary charges of *ϕ*1, 2; *N**c* is an integer measuring charge excitations in the units of *e*/3 (*N**c* = 1 for a quasiparticle of charge *e*/3; *N**c* = 3 for an electron).\nA quasiparticle of charge *e*/3 at position *x* on the QD edge is written as $e^{i \\phi\\_c(x) / \\sqrt{6}}e^{\\pm i \\phi\\_n(x) / \\sqrt{2}}$ . Consider clockwise exchange of two such quasiparticles. Since [*ϕ**c*(*x*), *ϕ**c*(*x*ʹ)] = *i**π*sgn(*x* − *x*ʹ), the exchange of the two charge components results in statistical phase *π*/6,\n \n$$e^{\\frac{i}{\\sqrt{6}} \\phi\\_c(x)} e^{\\frac{i}{\\sqrt{6}} \\phi\\_c(x')} = e^{\\pm i \\frac{\\pi}{6} \\textrm{sgn}(x'-x)} e^{\\frac{i }{\\sqrt{6}}\\phi\\_c(x')} e^{\\frac{i }{\\sqrt{6}}\\phi\\_c(x)}.\n\\label{Braid}$$ \nSo, after the charge component of the electron operator Ψ± winds once clockwise around *N**c* charge-mode excitations on the edge, a phase 3 × *N**c* × 2 × *π*/6 = *π**N**c* is gained . Here, 3 means the number of charge components forming Ψ±(*x*), and 2 refers to braiding (double exchanges). Similarly, the exchange of the neutral components of the two quasiparticles leads to exchange phase − *π*/2. So the neutral component of Ψ±(*x*) gains ± 1 × *N**n* × 2 × ( − *π*/2) = ∓ *π**N**n*, after winding once around *N**n* neutral-mode excitations; the number of the neutral components of Ψ± is ± 1.\nThis has implications on the dynamics of an electron which enters into the QD and then fractionalizes. When *v**c* ≫ *v**n*, there is a process where the charge component of the electron winds once around the QD, while the neutral component moves very little. In terms of the winding numbers of the charge and neutral components, *p* and *q*, this process is denoted by (*p*, *q*) = (1, 0). This process interferes with that of no winding (*p*ʹ, *q*ʹ) = (0, 0), contributing to the *h*/*e* harmonics G*δ**p* = 1; see Fig. [Interferencepaths2]. The relative winding numbers between the two interfering paths are (*δ**p* = *p* − *p*ʹ = 1, *δ**q* = *q* − *q*ʹ = 0), and the net braiding phase gained from that winding around *N**c* charge and *N**n* neutral excitations on the edge is *π*(*N**c**δ**p* ∓ *N**n**δ**q*) = *π**N**c*. Since *N**c* is an integer, thermal fluctuations of quasiparticle (or electron) excitations on the edge give rise to fluctuations of the braiding phase factor *e**i**π**N**c* = ± 1 [ + ( − ) for even (odd) *N**c*], suppressing the *h*/*e* harmonics. This topological braiding-induced dephasing also occurs due to thermal quasiparticle or electron fluctuations in the bulk; see Appendix [AppendixQuasiflucbulk] for quasiparticle fluctuations in the bulk. Note that this topological dephasing is utterly different from a dephasing mechanism at zero temperature, arising when quasiparticles travelling along an edge change internal degrees of freedom within the bulk (e.g., Ref. ).\nBy contrast, the main contribution to the *h*/(2*e*) harmonics G*δ**p* = 2 comes from (*δ**p*, *δ**q*) = (2,0). In this case, the braiding phase factor *e**π**i*(*N**c**δ**p* ∓ *N**n**δ**q*) = 1, regardless of *N**c* being even or odd. Hence, G*δ**p* = 2 is immune to the topological dephasing. In general, such dephasing occurs only with odd *δ**p* + *δ**q*, since the fluctuating *N**c* ± *N**n* is always even; see Eq. .\nWhen *v**c* ≃ *v**n*, the topological dephasing does not occur, since *δ**p* = − *δ**q* and *N**c* ± *N**n* is even.\nThe above arguments hold for the pure AB regime (or for the intermediate regime between pure AB and Coulomb-dominated regimes). An apt question is to what extent this analysis holds for the Coulomb dominated regime. When an electron of a given energy enters the QD by the process of sequential tunneling, it occupies a certain orbital state of the QD edge, satisfying energy conservation. In the pure AB regime, the area enclosed by the orbit (hence, the AB phase assigned to the orbit) is not modified when the number of quasiparticles or electrons in the bulk of the QD fluctuates thermally: edge-bulk interactions are negligible. Hence, such thermal fluctuations affect only the braiding phase gained by the electron, leading to topological dephasing. By contrast, in the Coulomb-dominated regime, the fluctuations are fully screened by the edge, reflecting the effect of edge-bulk interaction. This screening leads to modification of the area of the orbit, hence it modifies the AB phase of the orbit. This change of the AB phase exactly cancels out the change of the braiding phase caused by the thermal fluctuations. It follows that topological dephasing disappears in the Coulomb-dominated regime.\n[Interferencepaths2]\nSequential tunneling\n====================\nWe compute G in Eq. to the order of sequential tunneling,\n \n$$\\mathcal{G} \\simeq \\frac{e^2}{\\hbar} c\\_g \\tilde{\\gamma} k\\_B T \\sum\\_{j=\\pm, \\alpha} \\int\\_{-\\infty}^{0} dt F (t)\\text{Im} \\, G\\_j(x\\_\\alpha, x\\_\\alpha; t), \\label{conductance11}$$ \nwhere *γ̃* = *γ*L*γ*R/(*γ*L + *γ*R), *γ**α* ∝ ∣*t**α**i**j*∣2 is the (renormalized) electron tunneling rate between the QD and lead edge *α* = L,R, $G\\_j(x\\_{\\alpha}, x\\_{\\alpha}; t) \\equiv \\big\\langle \\big [\\Psi\\_{j}^{\\dagger}(x\\_{\\alpha},t), \\Psi\\_{j}(x\\_{\\alpha},0) \\big ]\n\\big\\rangle$ is the Green function describing the time (*t*) evolution of the fractionalized components of an injected electron described by Ψ*j*(*x**α*, *t* = 0), and *c**g* is a constant. The start and end positions of the Green function *G**j*(*x**α*, *x**α*; *t*) coincide, since the Green function describes the sequential tunneling. The injection leaves a hole behind on the lead edge. *F*(*t*) = (*π**k**B**T**t*/ℏ)sinh− 2(*δ**c* + *δ**n*)(*π**k**B**T**t*/ℏ) accounts for the fractionalization of the hole. For the detailed derivation of Eq. , see Appendix [secAppendixseq].\nG*δ**p* comes from the interference between two processes of relative charge winding number *δ**p*. At *k**B**T* > *E**c*/(2*π*2), the charge component has spatial width *L**T*, *c* < *L*. Then, *G**j*(*x**α*, *x**α*; *t*) contributes to G*δ**p* mainly around the times *δ**p**L*/*v**c*, at which the charge component arrives at the initial injection point *x**α* after winding *δ**p* times around *L*; the neutral component winding times *δ**q**L*/*v**n* are much less important, because of the scaling dimensions *δ**c* = 3*δ**n* > *δ**n*. We focus on G*δ**p* = 1 and G*δ**p* = 2, as they involve the shorter times of *δ**p**L*/*v**c*, are more robust against the dephasing discussed below, hence are much larger than G*δ**p* ≥ 3 at *k**B**T* ≫ *E**c*. At *k**B**T* ≫ *E**c*, we compute G*δ**p* = 1 and G*δ**p* = 2 analytically in the absence of disorder and interaction (*W* = 0, *v* = 0), and also in the strong disorder regime, based on a finite-size bosonization and a semiclassical approximation (see Appendix [appendixsemi]).\nClean regime\n============\nWe first deal with the regime of *W* = 0 and *v* = 0 [see Eq. ] and then discuss the regime of weak disorder and weak intermode interaction. We treat various contributions to dephasing quantitatively for the two cases of $v\\_c \\gtrsim v\\_n$ and *v**c* ≫ *v**n*.\nWhen $v\\_c \\gtrsim v\\_n$ and *k**B**T* ≫ *E**c*, only the plasmonic dephasing is important. The dominant contribution to G comes from the *h*/*e* harmonics. With the additional condition of *k**B**T* ≫ ℏ*v**n*/(*L* − Δ*L*), we obtain\n \n$$\\begin{aligned}\n \\label{conductancevcsimvn}\n\\mathcal{G}\\_{\\delta p = 1} \\propto\n\\tilde{\\gamma} L (k\\_BT)^3 \\exp(-\\frac{L}{L\\_{T,c}}-\\frac{\\Delta L}{L\\_{T,n}}-\\frac{L - \\Delta L}{L\\_{T,n}}), \\end{aligned}$$ \nwhere Δ*L* = *L**v**n*/*v**c*. We explain two processes, whose interference dominates G*δ**p* = 1. In one process [Fig. [Interferencepaths2](a)], an electron tunnels from lead edge L into the QD and fractionalizes at *x*L at time *t*1ʹ = 0, while at *t*2ʹ = − *L*/*v**c* in the other [Fig. [Interferencepaths2](b)]. The charge components of the two processes interfere at *x*L at *t* = 0, contributing to G*δ**p* = 1, after respective windings *p* = 0 and 1. At that time, the distance between the neutral components is *L* − Δ*L*, leading to partial overlap, hence, to the third factor exp( − (*L* − Δ*L*)/*L**T*, *n*) of G*δ**p* = 1. The tunneling leaves a hole behind on L, which also fractionalizes into charge and neutral components (not shown in Fig. [Interferencepaths2]). The partial overlap at *t* = 0 between the two charge components from the holes created at *t*1ʹ and *t*2ʹ, and that between the two neutral components, lead to the first two exponential factors of G*δ**p* = 1 in Eq. , respectively.\nIn the other limit of *v**c* ≫ *v**n*, both plasmonic and topological dephasings are crucial. There are two interfering processes for G*δ**p* = 1 shown in Figs. [Interferencepaths2](a) and [Interferencepaths2](c), and for G*δ**p* = 2 in Figs. [Interferencepaths2](a) and [Interferencepaths2](d). When *k**B**T* ≫ *E**c*, we obtain\n \n$$\\begin{aligned}\n\\mathcal{G}\\_{\\delta p = 1} &\\propto& \\tilde{\\gamma} L (k\\_BT)^3 \\exp{(-\\frac{L}{ L\\_{T,c}}-\\frac{\\Delta L}{L\\_{T,n}}-\\frac{\\Delta L}{L\\_{T,n}} +\\frac{ (\\Delta L)^2}{L L\\_{T,n}})} \\nonumber\\\\ && \\times \\exp[-\\frac{( L - \\Delta L)^2}{ L L\\_{T,n}}] \\label{conductancevcggvn1} \\\\\n\\mathcal{G}\\_{\\delta p = 2} \n&\\propto& \\tilde{\\gamma} L (k\\_BT)^3 \\exp[-2(\\frac{L}{L\\_{T,c}}+\\frac{\\Delta L}{L\\_{T,n}}+\\frac{\\Delta L}{L\\_{T,n}})]. \\label{conductancevcggvn2}\\end{aligned}$$ \nThe first three exponential factors of G*δ**p* = 1 and G*δ**p* = 2 result from plasmonic dephasing, as those of Eq. . The third factor has a form different from that of Eq. of $v\\_c \\gtrsim v\\_n$, as the interfering neutral components in the QD are now Δ*L* apart in space. The factor 2 in the arguments of G*δ**p* = 2 arises from the double winding. Another exponential factor exp[(Δ*L*)2/(*L**L**T*, *n*)] of G*δ**p* = 1 comes from the plasmonic part of the neutral component; it is cancelled out with zero-mode contributions in Eqs. and and also in other cases .\nThe last suppression factor in Eq. , exp[ − (*L* − Δ*L*)2/(*L**L**T*, *n*)], represents the topological dephasing, arising from the zero-mode parts of *G**j*(*x**α*, *x**α*; *t*). The process in Fig. [Interferencepaths2](c) (where the center of the neutral component hardly moves, while the charge component winds once around *L*) interferes with that of Fig. [Interferencepaths2](a), contributing to (*δ**p*, *δ**q*) = (1, 0). As discussed around Eq. , this interference with *δ**p* + *δ**q* = 1 is suppressed by the thermally fluctuating braiding phase factor of *e**i**π*(*δ**p**N**c* + *δ**q**N**n*) = ± 1. The suppression factor is interestingly determined by the spatial tail (or finite *L**T*, *n*) of the zero-mode part of the neutral component. The tail indicates that the neutral component can quantum mechanically wind once more than the semiclassical number *q* of the center; the quantum mechanical winding is well defined by the Poisson formula (see Appendix [appendixtop]). Hence, from the processes in Figs. [Interferencepaths2](a) and [Interferencepaths2](c), interference with the total relative winding number of *δ**p* + (*δ**q* + 1) can occur. As *δ**p* + *δ**q* + 1 is even, this interference avoids the topological dephasing, dominantly contributing to G*δ**p* = 1, but it is reduced by the separation *L* − Δ*L* of the neutral components of *δ**q* + 1 relative windings. This explains the factor exp[ − (*L* − Δ*L*)2/(*L**L**T*, *n*)]; the exponent is quadratic in *L* − Δ*L*, since it originates from the zero-mode part .\nWe point out that the topological dephasing occurs when *L* > *L**T*, *n* (*k**B**T* > *E**n*/(4*π*2*δ**n*)), as seen in the last exponential factor in Eq. . In contrast, the plasmonic dephasing occurs when *k**B**T* ≫ *E**c*. Note that we choose the condition of *k**B**T* ≫ *E**c* for the derivation of Eq. , to show both of the plasmonic dephasing and the topological dephasing simultaneously.\nBecause of the topological dephasing, G*δ**p* = 2 is much larger than G*δ**p* = 1 when *v**c* ≫ *v**n*; exp( − (*L* − Δ*L*)2/*L**L**T*, *n*) is much smaller than the other factors. As a result, G shows *h*/(2*e*) AB oscillations. In Fig. [conductance3], we numerically compute G for both $v\\_c \\gtrsim v\\_n$ and *v**c* ≫ *v**n* without employing the semiclassical approximation. The result for *v**c* ≫ *v**n* demonstrates the topological dephasing and consequent period halving even at *k**B**T* < *E**c*.\n[conductance3]\nSo far, we have discussed the regime of no intermode interaction (*v* = 0) and no disorder (*W* = 0). The argument of the regime holds also in the regime of weak intermode interaction and weak disorder, with slight modifications. In this regime, the plasmonic part of the neutral component decays, together with the diffusive spreading of the plasmonic part of the charge component . These slightly modify the plasmonic dephasing (the first three dephasing factors in Eqs. − , but do not affect the topological dephasing. Note that the weak disorder regime is realized when the renormalization of *W* stops by temperature *T* or QD size *L* before going to the strong disorder regime, and a weak intermode interaction occurs in a dot, when the Coulomb interaction between the charge modes is larger than the confining potential (see Appendix [appenCoulomb] and Refs. ). In recent experiments , neutral modes are measured with QDs of size 4 *μ*m2, implying that the intermode interaction is sufficiently weak in the QDs.\nStrong disorder regime\n======================\nWe show that Eqs. − hold in the strong disorder regime of a QD edge without any modification. In this regime, the neutral component is totally decoupled with the charge component (*v* = 0 in Eq. ) .\nWe start with the diagonalized form of *H*D *H*D = ∫0*L**d**x*[*v**c*(∂*x**ϕ**c*)2/(4*π*) + *v**n**ψ̃*†*i*∂*x**ψ̃*]. This form is obtained from Eq. , where the effect of disorders is included. Here, $\\tilde{\\psi}(x) \\equiv (e^{i(\\tilde{\\chi}+\\tilde{\\phi}\\_{n})/\\sqrt{2}}, e^{i(\\tilde{\\chi}-\\tilde{\\phi}\\_{n})/\\sqrt{2}})^T= U(x) \\psi(x)$, the unitary matrix *U*(*x*) = *T**x*exp[ − *i*∫0*x**d**x*ʹ(*ξ*(*x*ʹ)*σ*+ + *ξ*\\*(*x*ʹ)*σ*−)/*v**n*] represents random disorder scattering, $\\psi\\equiv (e^{i(\\chi+\\phi\\_{n})/\\sqrt{2}}, e^{i(\\chi-\\phi\\_{n})/\\sqrt{2}})^T$ is a two-component fermionic operator, *χ* is an auxiliary bosonic field, and *σ*± = *σ**x* ± *i**σ**y*, *σ**x* and *σ**y* are the Pauli matrices. The equal-position correlator ⟨[Ψ±†(*x*L, *t*), Ψ±(*x*L, 0)]⟩ is replaced by $\\langle [\\tilde{\\Psi}\\_{\\pm}^{\\dagger}(x\\_{\\mathcal{L}},t), \\tilde{\\Psi}\\_{\\pm}(x\\_{\\mathcal{L}},0)] \\rangle$ when we choose the global gauge transformation making *U*(*x*L) = 1. Then, it is readily computed because the Hamiltonian is free in the basis of $\\tilde{\\Psi}\\_{\\pm}$, and is the same as *G**j*(*x**α*, *x**α*; *t*) in Eq. that is obtained in the absence of interaction (*v* = 0) between the charge and neutral components and disorder (*W* = 0). Hence, Eqs. − can be also applied to the strong disorder regime.\nDiscussion and conclusion\n=========================\nWe have studied electron dephasing at *ν* = 2/3. Electron fractionalization into charge and neutral components leads to plasmonic dephasing. When *v**c* ≫ *v**n* (which is likely ) and at *k**B**T* > *E**n*/(4*π*2*δ**n*), a new type of dephasing additionally arises. This dephasing is topological, resulting from the fractionalization and the fractional braiding statistics of the components, and occurs depending on the topological sectors characterized by the winding numbers (*δ**p*, *δ**q*) of the components; its dependence on the even-odd parity of *δ**p* + *δ**q* is mathematical reminiscent of the parity (integer versus half-integer spin) dependent role of the topological *θ* term in antiferromagnetic spin chains . It leads to period halving of the AB oscillations.\nWe emphasize that the topological dephasing occurs in both the regimes of strong and weak disorder, when bulk-edge interactions are not strong. In the case of weak disorder, which may be realized in high temperatures, weak intermode interaction causes the decay of the plasmonic part of the neutral component, accompanied by the diffusive spreading of the plasmonic part of the charge component . These do not affect the topological dephasing, hence the emergence of the *h*/2*e* oscillations. On the other hand, in the case of strong disorder at low temperatures, *v* renormalizes towards zero , and then does not change the physics of the topological dephasing. Note that bulk-edge Coulomb interactions become weaker in QDs of larger area; the pure AB regime (or the intermediate regime between the pure AB and the Coulomb-dominated regimes) could be achieved when the edge-to-bulk capacitance is smaller than other capacitances even when strong backscattering occurs at QPCs.\nWe also note that the QH edges at *ν* = 2/3 may undergo more complex edge reconstruction at about *T* > 50 mK . At lower temperature our analysis is applicable, while at higher temperature different topological dephasing may occur. Assuming *v**n* ∼ 5 × 104 m/s, *v**c* ∼ 5 × 105 m/s, and *L* = 10 *μ*m, we expect that the *h*/(2*e*) oscillation will appear at temperature *k**B**T* > ℏ*v**n*/(2*π**δ**n**L*) ∼ 20 mK. In this case, the oscillation will be suppressed at *k**B**T* > ℏ*v**c*/(2*π**δ**c**L*) ∼ 60 mK, due to the plasmonic dephasing.\nDetection of the period halving supports the topological dephasing, thus, the fractional statistics of the charge component at *ν* = 2/3. The plasmonic dephasing and the topological dephasing will occur, with modifications, in other anyon interferometers or at other *ν*’s.\nIt should be mentioned that the known mechanisms yielding *h*/(2*e*) oscillations in other mesoscopic systems do not apply to our setup. The Altshuler-Aronov-Spivak mechanism employs disorder averaging in multi-channel geometries, which is not present in our setup. Another mechanism for *h*/2*e* oscillations relies on integer QH edge modes in an antidot at temperatures much below the charging energy of the antidot. Moreover, our setup does not have superconducting fluctuations that support such periodicity.\nWe thank G. Rafael for a helpful discussion. HSS was supported by Korea NRF (Grant No. NRF-2010-00491 and Grant No. NRF-2013R1A2A2A01007327). YG was supported by DFG (Grant No. RO 2247/8-1).\nQuantum-Dot Hamiltonian\n=======================\nWe derive the Hamiltonian *H*D (cf. Eq. in the main text). *H*D is written in terms of the mode *ϕ*1 of filling factor *ν*1 = 1 and the counterpropagating mode *ϕ*2 of *ν*2 = − 1/3, as\n \n$$\\begin{aligned}\n \\label{HamiltonianDotOrg\\_sup} \nH\\_{\\textrm{D}}&=\\frac{\\hbar}{4\\pi}\\int\\_{0}^{L} dx [v\\_1 (\\partial \\_x \\phi\\_1)^2 + 3 v\\_2 (\\partial \\_x \\phi\\_2)^2 + 2v\\_{12}\\partial\\_x \\phi\\_1 \\partial\\_x \\phi\\_2] \\nonumber \\\\\n&+\\int\\_{0}^{L} dx [\\xi (x) \\exp(i\\phi\\_1+3i\\phi\\_2)+\\text{H.c.}].\\end{aligned}$$ \n*v*1(2) is the velocity of *ϕ*1(2) (renormalized by the intra-mode interactions) and *v*12 describes the inter-mode interaction. *ϕ*1, 2 satisfies [*ϕ**i*(*x*), *ϕ**i*ʹ(*x*ʹ)] = *i**π**ν**i*edgesgn(*x* − *x*ʹ)*δ**i**i*ʹ. Each field *ϕ**i*(*x*) is decomposed through *ϕ**i*(*x*) = *ϕ**i*pl(*x*) + *ϕ**i*0(*x*) into a plasmonic mode *ϕ**i*pl(*x*), satisfying the periodic boundary condition of *ϕ**i*pl(*x* + *L*) = *ϕ**i*pl(*x*), and a zero mode *ϕ**i*0(*x*),\n \n$$\\phi\\_i^{0}(x) = \\frac{2\\pi \\nu\\_i x}{L} (N\\_i + \\frac{1}{2} - \\frac{\\Phi}{\\Phi\\_0}) - \\lambda\\_i. \n\\label{zero\\_12}$$ \nThe number operator *N**i* counts the excess number of quasiparticles of charge *ν**i*edge*e*. Its canonical conjugate *λ**i* satisfies [*λ**i*, *N**i*ʹ] = *i**δ**i**i*ʹ, and *e*± *i**λ**i* changes *N**i* by ± 1, acting as a Klein factor. This ensures [*ϕ**i*0(*x*), *ϕ**i*ʹ0(*x*ʹ)] = 2*i**δ**i**i*ʹ*π**ν**i*edge(*x* − *x*ʹ)/*L*. Combined with the commutation rule of the plasmonic part [*ϕ**i*pl(*x*), *ϕ**i*ʹpl(*x*ʹ)] = *i**π**ν**i*edge(sgn(*x* − *x*ʹ) − 2(*x* − *x*ʹ)/*L*)*δ**i**i*ʹ, this leads to [*ϕ**i*(*x*), *ϕ**i*ʹ(*x*ʹ)] = *i**π**ν**i*edgesgn(*x* − *x*ʹ)*δ**i**i*ʹ. The term 1/2 in the bracket of Eq. is introduced, to impose the boundary condition of electron operators, exp(*i**ϕ**i*(*x* + *L*)/*ν**i*edge) = exp(*i**ϕ**i*(*x*)/*ν**i*edge)exp( − 2*π**i*Φ/Φ0). The magnetic flux Φ enclosed by the QD edge states causes the shift of *N**i* → *N**i* − Φ/Φ0 in Eq. ; as Φ increases (decreases) by Φ0, the edge state with filling factor *ν**i*edge is energetically stabilized by removing (adding) its own quasiparticle of charge *ν**i*edge*e*.\nCombining *ϕ**i*’s, one introduces the charge mode $\\phi\\_c = \\sqrt {3/2} (\\phi\\_1 + \\phi\\_2)$ and the neutral mode $\\phi\\_n = \\sqrt {1/2} (\\phi\\_1 + 3\\phi\\_2)$, satisfying [*ϕ**c*/*n*(*x*), *ϕ**c*/*n*(*x*ʹ)] = ± *i**π*sgn(*x* − *x*ʹ) and [*ϕ**c*(*x*), *ϕ**n*(*x*ʹ)] = 0. Putting this into Eq. , we derives Eq. (2).\nDerivation of *N**c* and *N**n*\n===============================\nWe derive Eq. in the absence of disorder (*W* = 0). The charge (neutral) mode is decomposed into the zero-mode part *ϕ**c*0(*x*) (*ϕ**n*0(*x*)) and the plasmonic part *ϕ**c*pl(*x*) (*ϕ**n*pl(*x*)). The latter describes edge plasmonic excitations, while the former anyon number excitations. The zero-mode parts are determined from Eq. as\n \n$$\\begin{aligned}\n \\label{zeromode\\_sup}\n\\phi\\_{c}^{0}(x)&=\\sqrt{\\frac{1}{6}}\\frac{2\\pi x}{L}\\bigg(N\\_{c}+1 - 2\\frac{\\Phi}{\\Phi\\_0} \\bigg)-\\sqrt{6}\\lambda\\_{c}, \\nonumber \\\\ \n\\phi\\_{n}^{0}(x)&=-\\sqrt{\\frac{1}{2}}\\frac{2\\pi N\\_{n} x}{L}-\\sqrt{2}\\lambda\\_{n}.\\end{aligned}$$ \nWe impose [*λ**c*(*n*), *N**c*(*n*)] = *i*, [*λ**c*(*n*), *N**n*(*c*)] = 0, and [*ϕ**c*(*n*)pl(*x*), *ϕ**c*(*n*)pl(*x*ʹ)] = ± *i**π*sgn(*x* − *x*ʹ) ∓ 2*π**i*(*x* − *x*ʹ)/*L*. These ensure [*ϕ**c*/*n*0(*x*), *ϕ**c*/*n*0(*x*ʹ)] = ± 2*π**i*(*x* − *x*ʹ)/*L* and [*ϕ**c*/*n*(*x*), *ϕ**c*/*n*(*x*ʹ)] = ± *i**π*sgn(*x* − *x*ʹ). Comparing Eqs. with, one gets Eq. .\nDerivation of differential conductance\n======================================\nIn this appendix, we derive G in Eq. . The electron current IR along the right lead edge is given by IR,\n \n$$\\begin{aligned}\n \\label{electroncurrent}\n\\langle \\mathcal{I}\\_{\\mathcal{R}} \\rangle &= e\\frac {d \\langle n\\_{\\mathcal{R}} \\rangle}{dt} = i\\frac{e}{\\hbar} \\langle [H\\_\\textrm{T}, n\\_\\mathcal{R}] \\rangle \n\\nonumber \\\\\n&=-\\frac{ie}{\\hbar} \\sum\\_{i,j=\\pm} \\bigg \\langle \\textrm{T}\\_t \\exp \\bigg(\\frac{i}{\\hbar} \\int\\_{-\\infty}^{0} dt\\hat{H}\\_{\\textrm{T}} (t)\\bigg) \\big [t\\_{\\mathcal{R}ij} \\hat{\\Psi}\\_{\\mathcal{R},i}^{\\dagger}(0,t'=0) \\hat{\\Psi}\\_{j}(x\\_{\\mathcal{R}},t'=0)-\\text{H.c.}\\big ] \\textrm{T}\\_t \\exp \\bigg(-\\frac{i}{\\hbar} \\int\\_{-\\infty}^{0} dt \\hat{H}\\_{\\textrm{T}} (t)\\bigg) \\bigg \\rangle, \\end{aligned}$$ \n*n*R counts electron number in the right lead edge, T*t* is the time ordering, the operators with (without) caret are in the interaction (Heisenberg) picture, and ⟨ ⋅ ⟩ is the thermal average. We set *t* = 0 at which IR is measured. To second order in the tunneling strengths, ⟨IR⟩ is calculated as\n \n$$\\begin{aligned}\n \\langle \\mathcal{I}\\_{\\mathcal{R}}^{(2)} \\rangle & = \\frac{e}{\\hbar^2} \\sum\\_{i,j=\\pm}\\int\\_{-\\infty}^{0} dt\\big\\langle \\big[\\hat{H}\\_\\textrm{T} (t), (t\\_{\\mathcal{R}ij} \\langle \\hat{\\Psi}\n\\_{\\mathcal{R},i}^{\\dagger}(0,0) \\hat{\\Psi}\\_{j}(x\\_{\\mathcal{R}},0)\\rangle - \\text{H.c.})\\big]\n\\big\\rangle \\nonumber \\\\ &= \\frac{e}{\\hbar^2} \\sum\\_{i,j=\\pm} |t\\_{\\mathcal{R}ij}|^2 \\int\\_{-\\infty}^{0} dt\\text{Re} \\Big [\\exp \\big(-\\frac{i}{\\hbar}(\\mu\\_\\mathcal{R}-\\mu\\_\\textrm{D})t\\big ) \\Big ( G\\_{\\textrm{D}, j}^K(-t) (G\\_{\\mathcal{R}, i}^R-G\\_{\\mathcal{R}, i}^A)(t)- \n (G\\_{\\textrm{D}, j}^R-G\\_{\\textrm{D}, j}^A)(-t) G\\_{\\mathcal{R}, i}^K(t)\\Big) \\Big ]. \\end{aligned}$$ \n*G**α*(D), ±*R*, *G**α*(D), ±*A* and *G**α*(D), ±*K* are the retarded, advanced, and Keldysh Green’s functions of lead edge *α* = L, R (QD),\n \n$$\\begin{aligned}\n \\label{green'sfunction}\n(G\\_{\\alpha, \\pm}^R-G\\_{\\alpha, \\pm}^A)(t) &\\equiv -i\\langle \\{ \\hat{\\Psi}\\_{\\alpha, \\pm}(0,t), \\hat{\\Psi}\\_{\\alpha, \\pm}^{\\dagger}(0,0)\\}\\rangle, \n\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\nG\\_{\\alpha, \\pm}^K(t)\\equiv -i\\langle [ \\hat{\\Psi}\\_{\\alpha, \\pm}(0,t), \\hat{\\Psi}\\_{\\alpha, \\pm}^{\\dagger}(0,0)]\\rangle,\\nonumber\\\\\n(G\\_{\\textrm{D}, \\pm}^R-G\\_{\\textrm{D}, \\pm}^A)(t)&\\equiv -i\\langle \\{ \\hat{\\Psi}\\_{\\pm}(x\\_{\\alpha},t), \\hat{\\Psi}\\_{\\pm}^{\\dagger}(x\\_{\\alpha},0)\\}\\rangle, \n\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\nG\\_{\\textrm{D}, \\pm}^K(t)\\equiv -i\\langle [ \\hat{\\Psi}\\_{\\pm}(x\\_{\\alpha},t), \\hat{\\Psi}\\_{\\pm}^{\\dagger}(x\\_{\\alpha},0)]\\rangle, \\end{aligned}$$ \nand *μ**α* (*μ*D) is the chemical potential for lead edge *α* (the QD). *μ*D is assumed to be uniform over the entire region of the QD, which is valid in the linear response regime. The expression ⟨IL(2)⟩ of electron current in the left lead edge is similar to that of ⟨IR(2)⟩. To second order in the tunneling strengths, the current I through the QD is written as I = − ⟨IL(2)⟩ = ⟨IR(2)⟩.\nApplying the current conservation condition of ⟨IR(2)⟩ + ⟨IL(2)⟩ = 0, we write the symmetrized form of I as\n \n$$\\begin{aligned}\n\\mathcal{I}&=\\frac{\\gamma\\_{\\mathcal{L} } \\langle \\mathcal{I}\\_{\\mathcal{R}}^{(2)} \\rangle- \\gamma\\_{\\mathcal{R}} \\langle \\mathcal{I}\\_{\\mathcal{L}}^{(2)} \\rangle}{ \\gamma\\_{\\mathcal{L}}+\\gamma\\_{\\mathcal{R}}}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\n\\big[\\gamma\\_{\\mathcal{L}/\\mathcal{R}} = \\sum\\_{i,j=\\pm}|t\\_{\\mathcal{R}/\\mathcal{L}ij}|^2/ (\\hbar a v\\_c^{3/4} v\\_n^{1/4})\\big]\\nonumber \\\\ \n &= \\frac{e}{4\\hbar} a v\\_c^{3/4} v\\_n^{1/4} \\frac{ \\gamma\\_{\\mathcal{L}}\\gamma\\_{\\mathcal{R}}}{ \\gamma\\_{\\mathcal{L} }+\\gamma\\_{\\mathcal{R}}} \\sum\\_{i,j=\\pm}\\int\\_{-\\infty}^{0} dt \\nonumber \\\\ \n & \\times \n \\text{Re} \\Big [ \\big (e^{-\\frac{i}{\\hbar}(\\mu\\_{\\mathcal{R}}-\\mu\\_{\\textrm{D}})t}-e^{-\\frac{i}{\\hbar}(\\mu\\_{\\mathcal{L}}-\\mu\\_{\\textrm{D}})t} \\big )\n\\Big ( G\\_{\\textrm{D}, j}^K(-t) (G\\_{\\mathcal{R}, i}^R-G\\_{\\mathcal{R}, i}^A)(t)- \n (G\\_{\\textrm{D}, j}^R-G\\_{\\textrm{D}, j}^A)(-t) G\\_{\\mathcal{R}, i}^K(t)\\Big) \\Big].\\end{aligned}$$ \nIn the second equality, we used the simplification that the left and right lead edges are symmetric (*H*L = *H*R), namely *G*L, ±*R* = *G*R, ±*R*, *G*L, ±*A* = *G*R, ±*A*, and *G*L, ±*K* = *G*R, ±*K*. We also used the fact that the Green’s functions are independent of the index *i* = ± of the electron field operators in *H*T. The differential conductance G = *d*I/*d**V*∣*V* → 0 is written as\n \n$$\\begin{aligned}\n \\label{currentAndconductance}\n\\mathcal{G} = -\\frac{e^2}{8\\hbar^2} a v\\_c^{3/4} v\\_n^{1/4} \\frac{ \\gamma\\_{\\mathcal{L}}\\gamma\\_{\\mathcal{R}}}{ \\gamma\\_{\\mathcal{L} }+\\gamma\\_{\\mathcal{R}}} \\sum\\_{i,j} \\sum\\_{\\alpha=\\mathcal{R},\\mathcal{L}} \\int\\_{-\\infty}^{0} dt \n\\text{Im} \\Big [ t \\Big ( G\\_{D, j}^K(-t) (G\\_{\\alpha, i}^R-G\\_{\\alpha, i}^A)(t)- (G\\_{D, j}^R-G\\_{D, j}^A)(-t) G\\_{\\alpha, i}^K(t)\\Big)\\Big ], \\nonumber\\end{aligned}$$ \nwhere *e**V* ≡ *μ*L − *μ*R. The Green’s functions for the lead edges are computed as\n \n$$\\begin{aligned}\n(G\\_{\\alpha, i}^R-G\\_{\\alpha, i}^A)(t) &= -\\frac{i}{\\pi a} \\text{Re} \\bigg[ \\bigg (\\frac{\\sinh(\\frac{i\\pi a k\\_B T}{\\hbar v\\_{c}})}{\\sinh \\big(\\frac{\\pi k\\_B T}{\\hbar v\\_{c}}(ia-v\\_{c}t)\\big)} \\bigg )^{\\frac{3}{2}}\n\\bigg (\\frac{\\sinh(\\frac{i\\pi a k\\_B T}{\\hbar v\\_{n}})}{\\sinh \\big(\\frac{\\pi k\\_B T}{\\hbar v\\_{n}}(ia-v\\_{n}t)\\big)} \\bigg )^{\\frac{1}{2}} \\bigg], \\nonumber \\\\\nG\\_{\\alpha, i}^K(t) &= \\frac{1}{\\pi a} \\text{Im} \\bigg[ \\bigg (\\frac{\\sinh(\\frac{i\\pi a k\\_B T}{\\hbar v\\_{c}})}{\\sinh \\big(\\frac{\\pi k\\_B T}{\\hbar v\\_{c}}(ia-v\\_{c}t)\\big)} \\bigg )^{\\frac{3}{2}}\n\\bigg (\\frac{\\sinh(\\frac{i\\pi a k\\_B T}{\\hbar v\\_{n}})}{\\sinh \\big(\\frac{\\pi k\\_B T}{\\hbar v\\_{n}}(ia-v\\_{n}t)\\big)} \\bigg )^{\\frac{1}{2}} \\bigg].\\end{aligned}$$ \nThe Green’s functions are independent of the index *α* = L, R, because of the imposed symmetry between the left and right lead edges. Since (*G**α*, *i**R* − *G**α*, *i**A*)(*t*) ≫ *G**α*, *i**K*(*t*) at ∣*t*∣ > *a*/*v**n*, and since the processes of ∣*t*∣ < *a*/*v**n* do not contribute to the interference in G, we can ignore *G**α*, *i**K*(*t*) in the expression of G. Then, a simplified form of G is obtained as\n \n$$\\mathcal{G} = \\frac{e^2}{4\\hbar^2} \\frac{\\gamma\\_{\\mathcal{L}}\\gamma\\_{\\mathcal{R}}}{ \\gamma\\_{\\mathcal{L}}+\\gamma\\_{\\mathcal{R}}} \\frac{a^2 k\\_B T}{\\hbar v\\_c^{3/4} v\\_n^{1/4}} \\sum\\_{j=\\pm} \\sum\\_{\\alpha=\\mathcal{R},\\mathcal{L}} \\int\\_{-\\infty}^{0} dt F(t)\\text{Im} G\\_j(x\\_{\\alpha}, x\\_{\\alpha}; t)=\\frac{e^2}{\\hbar} c\\_g \\tilde{\\gamma} k\\_B T \\sum\\_{j=\\pm} \\sum\\_{\\alpha=\\mathcal{R},\\mathcal{L}} \\int\\_{-\\infty}^{0} dt F(t)\\text{Im} \\, G\\_j(x\\_\\alpha, x\\_\\alpha; t). \\nonumber$$ \nHere, *γ̃* = *γ*L*γ*R/(*γ*L + *γ*R), *c**g* = *a*2/(4ℏ2*v**c*3/4*v**n*1/4), the weight factor *F*(*t*) = (*π**k**B**T**t*/ℏ)sinh− 2(*π**k**B**T**t*/ℏ), and $G\\_j(x\\_{\\alpha}, x\\_{\\alpha}; t) \\equiv \\big\\langle \\big [\\Psi\\_{j}^{\\dagger}(x\\_{\\alpha},t), \\Psi\\_{j}(x\\_{\\alpha},0) \\big ]\n\\big\\rangle$ represents a Green’s function of the QD; its starting position *x**α* is the same with the ending one in the sequential tunneling regime. The weight factor *F*(*t*) decays rapidly as *e*− 2*π**k**B**T**t*/ℏ for *t* ≫ ℏ/*k**B**T*, describing the plasmonic dephasing (by partial overlap due to different positions of the neutral components between the interfering paths) occuring at lead edge *α*. The above is the derivation of Eq. in the main text.\nBelow, we further compute G in the case of *v* = 0. We note that in the absence of disorders (*W* = 0) and interaction (*v* = 0) between the two modes, the Hamiltonian *H*D0 of the zero-mode parts is obtained\n \n$$\\begin{aligned}\n \\label{HamiltonianZero\\_sup}\nH\\_{\\textrm{D}}^{0} &= \\frac{\\pi \\hbar v\\_c}{6L}\\bigg(N\\_{c}+1-2\\frac{\\Phi}{\\Phi\\_0}\\bigg)^2 +\n\\frac{\\pi \\hbar v\\_n}{2L} N\\_{n}^2 \\nonumber \\\\\n&= \\frac{E\\_c}{12} \\bigg(N\\_{c}+1-2\\frac{\\Phi}{\\Phi\\_0}\\bigg)^2 + \\frac{E\\_n}{4} N\\_{n}^2.\\end{aligned}$$ \nHere we define energy scales *E**c*(*n*) ≡ 2*π*ℏ*v**c*(*n*)/*L*. Then, the Green’s function *G**j* is decomposed into the charge and neutral components. Then G is simplified as\n \n$$\\begin{aligned}\n \\label{Sup:conductance}\n\\mathcal{G}\n& =\\frac{e^2}{ h} \\frac{\\tilde{\\gamma} a k\\_B T}{\\hbar^2 v\\_c^{3/4} v\\_n^{1/4}} \\int\\_{-\\infty}^{\\infty} dt F\\_{\\alpha} (t) \\text{Re} G^{0}(t) \\nonumber \\\\ & \\times \\text{Im} \\big[G\\_c(t) \nG\\_n(t) e^{-3\\pi i v\\_c t/2L} e^{-\\pi i v\\_n t/2L} \\big], \\end{aligned}$$ \nwhere the plasmonic parts *G**c*(*t*) and *G**n*(*t*) of the charge and neutral modes and the zero-mode part *G*0(*t*) are\n \n$$\\begin{aligned}\nG\\_c(t) &= \\langle e^{i\\sqrt{\\frac{3}{2}} \\phi\\_c(t)} e^{-i\\sqrt{\\frac{3}{2}} \\phi\\_c(0)}\\rangle \\nonumber \\\\&= \\bigg(\\exp\\big({\\frac{i\\pi v\\_c t}{L}}\\big)\\frac{ \\theta\\_1 (-\\frac{i\\pi a}{L}\n, e^{-\\gamma\\_{c}})}{\\theta\\_1 (\\frac{\\pi (v\\_{c}t -ia)}{L}\n, e^{-\\gamma\\_{c}})} \\bigg )^{\\frac{3}{2}}, \\nonumber \\\\ \nG\\_n(t) &= \\langle e^{i\\sqrt{\\frac{1}{2}} \\phi\\_n(t)} e^{-i\\sqrt{\\frac{1}{2}} \\phi\\_n(0)}\\rangle \\nonumber \\\\& = \\bigg ( \\exp\\big({\\frac{i \\pi v\\_n t}{L}}\\big) \\frac{ \\theta\\_1 (-\\frac{i\\pi a}{L}\n, e^{-\\gamma\\_{n}})}{\\theta\\_1 (\\frac{\\pi (v\\_{n}t-ia)}{L}\n, e^{-\\gamma\\_{n}})} \\bigg )^{\\frac{1}{2}}, \\nonumber \\\\\nG^{0} (t) &=\\big \\langle e^{i \\pi v\\_{c} t (N\\_{c}+1-2\\Phi/\\Phi\\_0 )/L} e^{-i \\pi N\\_{n} v\\_{n} t /L} \\big \\rangle. \n\\label{Greenplasmonic\\_sup}\\end{aligned}$$ \nThe elliptic-theta function of the first kind is $\\theta\\_1 (z, q) = 2 q^{1/4} \\sin z \\prod\\limits\\_{n=1}^{\\infty} (1-2q^{2n} \\cos(2z) + q^{4n})(1-q^{2n})$, and *γ**c*(*n*) ≡ *π*ℏ*v**c*(*n*)/(*k**B**T**L*). In the derivation of Eq. , we used the relations of *G**j* = +(*x**α*, *x**α*, *t*) = *G**j* = −(*x**α*, *x**α*, *t*), *G**c*(*n*)( − *t*) = *G**c*(*n*)\\*(*t*), and *F*(*t*) = − *F*( − *t*). The zero-mode part *G*0(*t*) will be calculated in Appendix [appendixtop].\nTopological dephasing\n=====================\nWe first sketch the topological dephasing in the case of *v**c* ≫ *v**n* and no disorder, and derive it, by expanding the zero-mode contribution to G in harmonics of the winding numbers. The discussion is valid even with strong disorder.\nAs in the main text, we consider the interference between the processes in Figs. 2(a) and 2(c). In the semiclassical regime of *L* ≫ *L**T*, *c*/*n*, counting the winding numbers *p* and *q* of the *center* of the spatial distributions of the charge and neutral components, we find that this interference contributes mainly to (*δ**p*, *δ**q*) = (1, 0), and gains the net phase of *θ* = *π*(*N**c* − 2Φ/Φ0)*δ**p* from those windings that braid with *N**c* charge excitations and *N**n* neutral excitations. At lower temperature of $L \\gtrsim L\\_{T,c/n}$, the tails (namely the spatial width *L**T*, *c*/*n*) of the spatial distributions of the components are non-negligible, and imply that the two processes can also contribute to quantum mechanical net windings (*δ**p*qm, *δ**q*qm) that can differ from (*δ**p*, *δ**q*). To see the contribution to different windings, we expand the average of ⟨*e**i**θ*⟩*k**B**T* over the thermal fluctuations of *N**c* and *N**n* in the harmonics of (*δ**p*qm, *δ**q*qm),\n \n$$\\begin{aligned}\n\\langle e^{i\\theta} \\rangle\\_{k\\_B T} &= \\sum\\_{\\delta p\\_{\\textrm{qm}}, \\delta q\\_{\\textrm{qm}} \\in \\mathbb{Z}} f(\\delta p\\_{\\textrm{qm}}, \\delta q\\_{\\textrm{qm}}, k\\_B T) \n\\nonumber \\\\ &\\times \\exp (2\\pi i \\delta p\\_{\\textrm{qm}} \\Phi / \\Phi\\_0) \\exp (2\\pi i \\delta q\\_{\\textrm{qm}} \\varphi\\_n ), \\end{aligned}$$ \nwhere we introduce a fictitious “neutral flux” *φ**n* in the mathematical analogy of Φ/Φ0 in order to have the expansion; *θ* is now generalized to *θ* = *π*(*N**c* − 2Φ/Φ0)*δ**p* + *π*(*N**n* − 2*φ**n*)*δ**q*, and we put *φ**n* → 0 at the end. The thermal fluctuations of *N**c* and *N**n* are governed by the QD-energy *H*D0 = *E**c*(*N**c* − 2Φ/Φ0 + 1)2/12 + *E**n*(*N**n* − 2*φ**n*)2/4 (cf. Eq. ). Notice *H*D0(Φ/Φ0, *φ**n*)∣*N**c*, *N**n* = *H*D0(Φ/Φ0 + 1, *φ**n* + 1)∣*N**c* + 2, *N**n* + 2 and *θ*(Φ/Φ0)∣*N**c*, *N**n* = *θ*(Φ/Φ0 + 1)∣*N**c* + 2, *N**n* + 2, meaning that *θ* and *H*D0 are restored by changing *N**c* and *N**n* by 2, when each flux shifts by one as Φ/Φ0 → Φ/Φ0 + 1 and *φ**n* → *φ**n* + 1.\nWe decompose the amplitude *f*(*δ**p*qm, *δ**q*qm, *k**B**T*) = *f*e(*δ**p*qm, *δ**q*qm, *k**B**T*) + *f*o(*δ**p*qm, *δ**q*qm, *k**B**T*) into the average *f*e over (even *N**c*, even *N**n*) and that *f*o over (odd *N**c*, odd *N**n*); *N**c* and *N**n* should have the same parity, according to Eq. (3) in the main text. We find a useful relation of *f*o(*δ**p*qm, *δ**q*qm, *k**B**T*) = ( − 1)*δ**p*qm + *δ**q*qm*f*e(*δ**p*qm, *δ**q*qm, *k**B**T*), obtained from the fact that the thermal average of ⟨*e**i**θ*⟩*k**B**T* over odd *N**c* and odd *N**n* at (Φ/Φ0, *φ**n*) is identical to that over even *N**c* and even *N**n* at (Φ/Φ0 + 1/2, *φ**n* + 1/2), according to *H*D0. This relation leads to *f*(*δ**p*qm, *δ**q*qm, *k**B**T*) = 0 for odd *δ**p*qm + *δ**q*qm, describing the topological dephasing. For the interference between those in Figs. 2(a) and 2(c), the contribution from the semiclassical winding of (*δ**p*, *δ**q*) = (*δ**p*qm, *δ**q*qm) = (1, 0) vanishes (independent of temperature *T*!), while G*δ**p* = 1 is contributed dominantly from the quantum mechanical winding of (*δ**p*qm, *δ**q*qm) = (1, − 1) of the tail.\nWe confirm the above discussion mathematically. The flux dependence of G in Eq. comes from the zero-mode part of the electron field operator Ψ±, hence, from the Green’s function *G*0 in Eq. . Using Eq. , *G*0 = ⟨exp[*i**π**v**c**t*(*N**c* − 2Φ/Φ0 + 1)/*L*]exp( − *i**π**v**n**t**N**n*/*L*)⟩ is computed as\n \n$$\\begin{aligned}\nG^{0}(t)\n& = \n\\Big [ \\sum\\limits\\_{n\\_c, n\\_n =-\\infty}^{\\infty} \n\\Big \\{ \\exp \\Big (- \\frac{\\frac{E\\_c}{12} \\big(2n\\_c + 1 -\\frac{2\\Phi}{\\Phi\\_0} + 1\\big)^2 + \\frac{E\\_n}{4} \\big(2n\\_n + 1\\big)^2}{k\\_B T} \\Big)\ne^{i\\frac{\\pi v\\_c t}{L} (2 n\\_c + 1 -\\frac{ 2 \\Phi}{\\Phi\\_0} +1 )} \ne^{-i\\frac{\\pi v\\_n t}{L} (2 n\\_n + 1)}\\nonumber \\\\ &\n+\\exp \\Big (- \\frac{\\frac{E\\_c}{12} \\big(2n\\_c-\\frac{2\\Phi}{\\Phi\\_0}+1\\big)^2 + \\frac{E\\_n}{4} (2n\\_n)^2}{k\\_B T} \\Big)\ne^{i \\frac{\\pi v\\_c t}{L} (2n\\_c -\\frac{2\\Phi}{\\Phi\\_0}+1)}\ne^{-i \\frac{\\pi v\\_n t}{L} (2n\\_n)} \\Big \\} \\Big] \\nonumber \\\\ &\n\\Big /\n \\Big [ \\sum\\limits\\_{n\\_c, n\\_n =-\\infty}^{\\infty} \n\\Big \\{ \\exp \\Big (-\\frac{\\frac{E\\_c}{12} \\big(2n\\_c + 1 -\\frac{2\\Phi}{\\Phi\\_0} +1\\big)^2 + \\frac{E\\_n}{4} \\big(2n\\_n + 1\\big)^2}{k\\_B T} \\Big) +\\exp \\Big (- \\frac{\\frac{E\\_c}{12} \\big(2n\\_c -\\frac{2\\Phi}{\\Phi\\_0} +1\\big)^2 + \\frac{E\\_n}{4} (2n\\_n)^2}{k\\_B T} \\Big) \\Big \\} \\Big ].\n\\label{Zeromodeaverage\\_sup} \\end{aligned}$$ \nThe first term of Eq. comes from odd integers *N**c* = 2*n**c* + 1 and *N**n* = 2*n**n* + 1, and the second from even integers *N**c* = 2*n**c* and *N**n* = 2*n**n*. Utilizing the Poisson summation formula of ∑*n* = − ∞∞exp[ − *a*(*n* + *δ*)2]exp[2*b**i*(*n* + *δ*)] = ∑*p*ʹ = − ∞∞exp( − 2*π**i**p*ʹ*δ*)exp[ − (*π**p*ʹ + *b*)2/*a*] with real constants, *a*, *b* and *δ*, we obtain\n \n$$\\begin{aligned}\nG^{0}(t)\n& = \\frac{\\sum\\limits\\_{\\delta p', \\delta q' = -\\infty}^{\\infty} (-1)^{\\delta p'} (1+(-1)^{\\delta p'+\\delta q'})\\exp(2\\pi i \\delta p' \\Phi/ \\Phi\\_0)\n\\exp \\big [-\\frac{3 \\pi^2 k\\_B T}{E\\_c}\\big(\\frac{v\\_{c} t}{L} - \\delta p' \\big)^2 \\big] \\exp \\big[-\\frac{\\pi^2 k\\_B T}{E\\_n}\\big( \\frac{v\\_{n} t}{L} +\\delta q'\\big)^2 \\big]}\n{\\sum\\limits\\_{\\delta p'', \\delta q'' = -\\infty}^{\\infty}\n(-1)^{\\delta p''} (1+(-1)^{\\delta p''+\\delta q''})\\exp (2\\pi i \\delta p'' \\Phi/ \\Phi\\_0)\\exp \\big(-\\frac{3 \\pi^2 k\\_B T}{E\\_c}{\\delta p''}^2 \\big ) \\exp \\big (-\\frac{\\pi^2 k\\_B T}{E\\_n}{\\delta q''}^2 \\big )} \\nonumber \\\\ &=\n\\frac{\\sum\\limits\\_{\\delta p'+ \\delta q' \\in 2 \\mathbb{Z}} (-1)^{\\delta p'} \n\\exp(2\\pi i \\delta p'\\Phi/ \\Phi\\_0)\n\\exp \\big [-\\frac{3 \\pi^2 k\\_B T}{E\\_c}\\big(\\frac{v\\_{c} t}{L} - \\delta p' \\big)^2 \\big] \\exp \\big[-\\frac{\\pi^2 k\\_B T}{E\\_n}\\big( \\frac{v\\_{n} t}{L} +\\delta q'\\big)^2 \\big]}\n{\\sum\\limits\\_{\\delta p''+ \\delta q''\\in 2 \\mathbb{Z}}\n(-1)^{\\delta p''} \\exp (2\\pi i \\delta p'' \\Phi/ \\Phi\\_0)\\exp \\big(-\\frac{3 \\pi^2 k\\_B T}{E\\_c}{\\delta p''}^2 \\big ) \\exp \\big (-\\frac{\\pi^2 k\\_B T}{E\\_n}{\\delta q''}^2 \\big )}. \n\\label{zeromodeaverage2\\_sup}\n \\end{aligned}$$ \nHere 2Z is the set of even integers. The denominator is approximated as 1 at high temperature *k**B**T* ≫ ℏ*v**c*/*L*; this condition of *k**B**T* ≫ ℏ*v**c*/*L* is chosen for simplicity, and it is not a condition for the topological dephasing. Then *G*0(*t*) is written by the harmonics of winding numbers *δ**p*ʹ → *δ**p*qm and *δ**q*ʹ → *δ**q*qm,\n \n$$\\begin{aligned}\nG^{0}(t)&\\simeq\n\\sum\\limits\\_{\\delta p\\_{\\textrm{qm}}+ \\delta q\\_{\\textrm{qm}} \\in 2 \\mathbb{Z}} (-1)^{\\delta p\\_{\\textrm{qm}}} \n\\exp(2\\pi i \\delta p\\_{\\textrm{qm}}\\Phi/ \\Phi\\_0) \\nonumber \\\\ & \\times\n\\exp \\big [-\\frac{3 \\pi^2 k\\_B T}{E\\_c}\\big(\\frac{v\\_{c} t}{L} - \\delta p\\_{\\textrm{qm}} \\big)^2 \\big] \n\\nonumber \\\\ & \\times\n\\exp \\big[-\\frac{\\pi^2 k\\_B T}{E\\_n}\\big( \\frac{v\\_{n} t}{L} +\\delta q\\_{\\textrm{qm}}\\big)^2 \\big]. \\label{G0}\\end{aligned}$$ \nNotice that the windings of odd *δ**p*qm + *δ**q*qm do not contribute to *G*0(*t*), as mentioned in the main text.\nSemiclassical approximation: Derivation of Eqs. −\n==================================================\nIn this Appendix, we compute the analytic expression of G*δ**p* for the two cases $v\\_c \\gtrsim v\\_n$ (Eq. ) and *v**c* ≫ *v**n* (Eqs. and ), utilizing a semiclassical approximation such that the time *t* in the integrand of Eq. is replaced by *δ**p**L*/*v**c* except for the time argument *t* in *G**c*(*t*). The approximation is based on the fact that the dominant contribution to the integrand of Eq. comes from a peak structure of *G**c*(*t*) near *t* = *δ**p**L*/*v**c*; the other peaks from *G**n*(*t*) near *t* = *δ**q**L*/*v**n* is more monotonous and less important because the scaling dimension (*δ**n* = 1/4) of the neutral component in the electron tunneling operator is smaller than that (*δ**c* = 3/4) of the charge component. This approximation is applicable when (i) the spatial distance between two interfering neutral components in the QD at *t* = *δ**p**L*/*v**c* is much larger than the width *L**T*, *n* ∝ ℏ*v**n*/*k**B**T* of the neutral components (then *G**n*(*t*) is sufficiently monotonous), and (ii) *L* ≫ ℏ*v**c*/*k**B**T* (then *F*(*t*), coming from the lead edge, is sufficientlly monotonous). We focus on the contribution from near *t* = ± *L*/*v**c* and near ± 2*L*/*v**c*, since that from larger times *t* = *δ**p**L*/*v**c* of ∣*δ**p*∣ > 2 is much more smaller due to more dephasing.\nWe first compute G*δ**p* = 1 in the $v\\_c \\gtrsim v\\_n$ case with the semiclassical conditions of *k**B**T* ≫ ℏ*v**c*/*L* and *k**B**T* ≫ ℏ*v**n*/(*L* − Δ*L*), where Δ*L* ≡ *L**v**n*/*v**c*. In Eq. , the main contribution occurs at *t* = ± *L*/*v**c*. Near *t* = *L*/*v**c*, we use the following approximations: (1) For the portion of (*δ**p*qm, *δ**q*qm) = (1, − 1) in the zero-mode part, Re[*G*0] ≃ − cos(2*π*Φ/Φ0)exp( − (*L* − Δ*L*)2/(*L**L**T*, *n*)); cf. Eq. . (2) For the lead edge part, *F* ≃ 4*π**k**B**T**L*exp[ − 2*π**k**B**T**L*/ℏ*v**c*]/(ℏ*v**c*) = (8*L*/3*L**T*, *c*)exp( − *L*/*L**T*, *c*)exp( − Δ*L*/*L**T*, *n*). (3) For the QD plasmon part,\n \n$$\\begin{aligned}\n \\label{plasmonic\\_sup}\n& G\\_{c}(t) G\\_{n}(t) \\exp\\Big(-\\frac{3\\pi i v\\_c t }{2L}\\Big) \\exp\\Big(-\\frac{i \\pi v\\_n t }{2L}\\Big ) \\nonumber \\\\ \\simeq & 2 e^{i \\pi / 4} \\bigg (\\frac{ \\sinh (\\frac{i\\pi k\\_B T a}{\\hbar v\\_{c}}) }{\\sinh \\big(\\frac{\\pi k\\_B T (L-v\\_{c}t +ia)}{\\hbar v\\_{c}}\\big)} \\bigg )^{3/2}\\bigg (\\frac{a}{L\\_{T,n}} \\bigg )^{1/2} \\nonumber \\\\ \\times & \\exp\\big(-\\frac{L - \\Delta L}{L\\_{T,n}}\\big)\n \\exp \\big(\\frac{(L - \\Delta L)^2}{L L\\_{T,n}}\\big). \\end{aligned}$$ \nWe have used $\\theta\\_1 (u,\\exp(-\\gamma)) \\simeq 2 (-1)^{n}\\sqrt{\\pi /\\gamma}\\exp [-(u-n\\pi)^2 /\\gamma]\\exp(-\\pi^2/4\\gamma)\\sinh(\\pi (u-n\\pi)/\\gamma)$ for *γ* ≪ 1.\nWe compute Eq. , merging together (1)-(3), to get Eq. in the main text,\n \n$$\\begin{aligned}\n \\mathcal{G}\\_{\\delta p =1}&\\simeq \n- g\\_0 \\tilde{\\gamma} L (k\\_B T ) ^3\n\\exp(-\\frac{L}{L\\_{T,c}}) \\exp(-\\frac{\\Delta{L}}{L\\_{T,n}}) \\nonumber\n\\\\ &\\times \\exp(-\\frac{L - \\Delta L}{L\\_{T,n}})\n\\nonumber \\\\ \n&= -g\\_0 \\tilde{\\gamma} L (k\\_B T ) ^3\n\\exp(-\\frac{L}{L\\_{T,c}}) \\exp(-\\frac{L}{L\\_{T,n}}).\\end{aligned}$$ \nHere $g\\_0 = 16 e^2 \\sqrt {2\\pi} \\pi (a/\\hbar v\\_c)^{13/4} (a/\\hbar v\\_n)^{3/4} (\\Gamma(3/4))^2 / (ah)$ is a constant, Γ is the Gamma function, and we have used the integral formula of\n \n$$\\label{integralFM\\_sup}\n\\int\\_{-\\infty}^{\\infty}dt\\bigg (\\frac{\\sinh(\\frac{i\\pi a k\\_B T}{\\hbar v\\_{c}})}{\\sinh \\big(\\frac{\\pi k\\_B T}{\\hbar v\\_{c}}(ia-v\\_{c}t)\\big)} \\bigg )^{\\frac{3}{2}} = \\frac{2a}{v\\_c}\\Big(\\frac{2a k\\_B T}{\\hbar v\\_c}\\Big )^{\\frac{1}{2}} \\Gamma \\big(\\frac{3}{4}\\big)^2.$$ \nNotice that the factor exp( − (*L* − Δ*L*)2/(*L**L**T*, *n*)) of Re[*G*0] (zero-mode part) exactly cancels out exp[(*L* − Δ*L*)2/(*L**L**T*, *n*)] from *G**n*(*t*) (the plasmonic part). This fact was found in a Luttinger liquid with finite size .\nWe next move to G*δ**p* = 1 in the *v**c* ≫ *v**n* case. The main contribution to G*δ**p* = 1 also occurs near *t* = ± *L*/*v**c*. We observe the followings. (1) Because the portion of (*δ**p*qm, *δ**q*qm) = (1, 0) in Re[*G*0] fully vanishes, the dominant contribution comes from the portion of (*δ**p*qm, *δ**q*qm) = (1, − 1), which leads to Re[*G*0] ≃ − cos(2*π*Φ/Φ0)exp( − (*L* − Δ*L*)2/(*L**L**T*, *n*)). (2) *F* ≃ 4*π**k**B**T**L*exp( − 2*π**k**B**T**L*/ℏ*v**c*)/(ℏ*v**c*) = (8*L*/3*L**T*, *c*)exp( − *L*/*L**T*, *c*)exp( − Δ*L*/*L**T*, *n*). (3) For the QD plasmon part, the same expression is obtained as Eq. , except that *L* − Δ*L* is replaced by Δ*L*. Merging (1) − (3) and using Eq. , we obtain Eq. in the main text,\n \n$$\\begin{aligned}\n \\label{smallvsFirstHarmonics\\_sup}\n \\mathcal{G}\\_{\\delta p =1}&\\simeq \n-g\\_0 \\tilde{\\gamma} L (k\\_B T)^3 \\exp(-\\frac{L}{L\\_{T,c}}) \\exp(-\\frac{\\Delta{L}}{L\\_{T,n}}) \\nonumber \\\\ &\\times \n \\exp(-\\frac{\\Delta L}{L\\_{T,n}})\n \\exp[-\\frac{(L -\\Delta L)^2-(\\Delta L)^2}{L L\\_{T,n}}]\n \\nonumber \\\\ \n &= -g\\_0 \\tilde{\\gamma} L (k\\_B T)^3 \\exp(-\\frac{L}{L\\_{T,n}})\\exp(-\\frac{L}{L\\_{T,c}}).\\end{aligned}$$ \nIn the same way, we obtain Eq. ,\n \n$$\\begin{aligned}\n \\label{smallvsSecondHarmonics\\_sup}\n\\mathcal{G}\\_{\\delta p =2}& \\simeq\n2g\\_0 \\tilde{\\gamma} L (k\\_B T)^3 \\exp(-\\frac{2L}{L\\_{T,c}}) \\exp(-\\frac{2\\Delta L}{L\\_{T,n}}) \\exp(-\\frac{2\\Delta L}{L\\_{T,n}}) \\nonumber \\\\ &= \n 2g\\_0 \\tilde{\\gamma} L (k\\_B T)^3 \\exp(-\\frac{10L}{3L\\_{T,c}}).\\end{aligned}$$ \nThe quasiparticle fluctuation in the bulk\n=========================================\nWe consider quasiparticle fluctuations in the bulk, differentiating the QD bulk from the edge. In the limit of bulk charging energy smaller than temperature, we show that the zero-mode part *G*0 has the same expression as Eq. in the presence of quasiparticle fluctuations in the bulk, hence, that the period halving (*h*/2*e* oscillation) takes place.\nWe argue that the zero-mode part of the QD states is characterized by four numbers (*N**e*, *N**n*, *N**q**p*, *N**q**p*, *n*), when we additionally consider the bulk degrees of freedom. *N**e* and *N**n* are necessary to describe excess electrons in the QD. For an additional excess electron, the number *N**e* of excess electrons increases by 1. This electron is decomposed into 3 additional charge components (*N**c* → *N**c* + 3; the charge of the charge component is *e*/3) and ± 1 additional neutral component (*N**n* → *N**n* ± 1), as discussed in the main text. *N**n* counts the number of the neutral components by excess electrons in the QD, as in the main text. On the other hand, *N**q**p* and *N**q**p*, *n* are introduced to describe quasiparticle excitations in the QD bulk. When an additional quasiparticle excites in the bulk, the number *N**q**p* of quasiparticles increases by 1. This quasiparticle is decomposed into 1 charge component and ± 1 neutral component (*N**q**p*, *n* → *N**q**p*, *n* ± 1). *N**q**p*, *n* counts the number of the neutral components by bulk quasiparticle excitations.\nTaking into account quasiparticle fluctuations in the bulk and electron fluctuations inside the QD, the zero-mode part of the QD Hamiltonian *H*D0 is expressed as\n \n$$\\begin{aligned}\n \\label{HamiltonianZeroBulk\\_sup}\n& H\\_{\\textrm{D}}^{0} (N\\_{e}, N\\_{qp}, N\\_{n}, N\\_{qp,n}) \\nonumber \\\\ = &\\, \\frac{E\\_c}{12}\\bigg(3N\\_{e}+1-\\frac{2\\Phi}{\\Phi\\_0} - N\\_{qp}\\bigg)^2 +\n\\frac{E\\_n}{4} (N\\_{n} - N\\_{qp,n})^2 \\nonumber \\\\ + & \\, E\\_{\\textrm{bc}} \\bigg(\\frac{2\\Phi}{\\Phi\\_0} + N\\_{qp}\\bigg)^2, \\end{aligned}$$ \nwhere *E*bc is the bulk charging energy. The first (second) term describes the interaction between the charge (neutral) components on the edge while the third term describes the interaction between the charge components in the bulk. The flux dependence in the bulk-charging energy term describes charge accumulation in the bulk as the magnetic flux increases. We ignore the interactions between the neutral components in the bulk because they are of dipole type hence weaker than the interaction terms of the total charge. We also assume that it is in the Aharonov-Bohm regime neglecting electrostatic coupling between quasiparticles in the bulk and on the edge.\nWe consider the case that the relaxation time from the QD edge to the bulk is much longer than the QD dwell time of an electron injected from a lead edge, hence, that the electron enters only into the QD edge. Then, the zero-mode part $\\exp({i\\sqrt{3/2}\\phi\\_c^{0}}\\pm{i\\sqrt{1/2}\\phi\\_n^{0}})$ of the electron operators Ψ±(*x*) is evolved by Eq. as\n \n$$\\begin{aligned}\n& e^{{i\\sqrt{\\frac{3}{2}}\\phi\\_c^{0}(x,t)}\\pm{i\\sqrt{\\frac{1}{2}}\\phi\\_n^{0}(x,t)}} \\nonumber \\\\ \n=& e^{iH\\_{\\textrm{D}}^{0}t}e^{{i\\sqrt{\\frac{3}{2}}\\phi\\_c^{0}(x)}\\pm{i\\sqrt{\\frac{1}{2}}\\phi\\_n^{0}(x)}}\n e^{-iH\\_{\\textrm{D}}^{0}t}\\nonumber \\\\\n=&\\exp\\Big(i\\sqrt{\\frac{3}{2}}\\phi\\_c^{0}(x)\\pm i\\sqrt{\\frac{1}{2}}\\phi\\_n^{0}(x) \\nonumber \\\\ +&\\frac{i \\pi v\\_c t}{L} (3N\\_{e}+1-\\frac{2\\Phi}{\\Phi\\_0} - N\\_{qp} )\\pm \\frac{i \\pi v\\_n t}{L} (N\\_{n} - N\\_{qp,n})\\Big). \\end{aligned}$$ \nAnd, the zero-mode part *G*0(*t*) of the Green’s function $\\big\\langle \\hat{\\Psi}\\_{\\pm}^{\\dagger}(x\\_{\\mathcal{L}},t) \\hat{\\Psi}\\_{\\pm}(x\\_{\\mathcal{L}},0)\\big\\rangle$ is written as\n \n$$\\begin{aligned}\n G^0&\\propto \\Big\\langle \\exp \\Big[\\frac{i \\pi v\\_{c}t}{L} \\Big(3N\\_{e}-\\frac{2\\Phi}{\\Phi\\_0}+1-N\\_{qp}\\Big) \\Big]\n\\exp(\\pm\\frac{ i\\pi v\\_{n}t }{L} (N\\_{n} - N\\_{qp,n}) ) \n\\Big\\rangle \\nonumber \\\\\n& = \\bigg \\{\n\\sum\\limits\\_{\\delta p\\_{\\textrm{qm}}, \\delta p', \\delta q\\_{\\textrm{qm}} = -\\infty}^{\\infty} e^{-i\\pi \\delta p' /3} (1+(-1)^{\\delta p\\_{\\textrm{qm}}+\\delta q\\_{\\textrm{qm}}})(1+(-1)^{\\delta p'+\\delta q\\_{\\textrm{qm}}})\\exp(2\\pi i \\delta p\\_{\\textrm{qm}} \\Phi/ \\Phi\\_0) \\nonumber \\\\& \\times\n\\exp \\big [-\\frac{3 \\pi^2 k\\_B T }{E\\_c}\\big(\\frac{v\\_{c} t}{L} + \\frac{\\delta p'}{3} \\big)^2 \\big] \\exp \\big[-\\frac{\\pi^2 k\\_B T}{E\\_n}\\big( \\delta q\\_{\\textrm{qm}} \\pm \\frac{v\\_{n} t}{L} \\big)^2 \\big] \\exp \\big[-\\frac{\\pi^2 k\\_B T}{4E\\_{\\textrm{bc}}}\\big( \\delta p\\_{\\textrm{qm}} - \\frac{\\delta p'}{3} \\big)^2 \\big] \\bigg \\} \\nonumber \\\\\n& \\bigg/ \\bigg \\{\n\\sum\\limits\\_{\\delta p\\_{\\textrm{qm}}, \\delta p', \\delta q\\_{\\textrm{qm}} = -\\infty}^{\\infty} e^{-i\\pi \\delta p' /3} (1+(-1)^{\\delta p\\_{\\textrm{qm}}+\\delta q\\_{\\textrm{qm}}})(1+(-1)^{\\delta p'+\\delta q\\_{\\textrm{qm}}})\\exp(2\\pi i \\delta p\\_{\\textrm{qm}} \\Phi/ \\Phi\\_0) \\nonumber \\\\& \\times\n\\exp \\big [-\\frac{3 \\pi^2 k\\_B T }{E\\_c}\\big(\\frac{\\delta p'}{3}\\big)^2 \\big] \\exp \\big[-\\frac{\\pi^2 k\\_B T\\delta q\\_{\\textrm{qm}}^2 }{E\\_n}\\big] \\exp \\big[-\\frac{\\pi^2 k\\_B T}{4E\\_{\\textrm{bc}}}\\big( \\delta p\\_{\\textrm{qm}} - \\frac{\\delta p'}{3} \\big)^2 \\big] \\bigg \\}.\n\\label{ZeromodeaverageBulk\\_sup} \\end{aligned}$$ \nWe applied the Poisson summation formula as in Eq. . At temperature much higher than the bulk-charging energy, the portion of *δ**p*ʹ = 3*δ**p*qm survives, resulting in the same expression as Eq. .\nConfining potential and Coulomb interaction\n===========================================\nWhen the edge potential is smooth enough , we below show that *v**c* is much larger than *v**n* and *v*; see Eq. in the main text. In the absense of Coulomb interaction, the edge Hamiltonian at *ν* = 2/3 has the form of $H\\_{\\text{con}}=\\frac{1}{4\\pi} \\int dx [v\\_{1,\\text{con}} (\\partial\\_x \\phi\\_1)^2 + 3v\\_{2,\\text{con}}(\\partial\\_x \\phi\\_2)^2]$ in terms of the velocities of the original edge modes, *v*1, con and *v*2, con, which are solely determined by the edge confining potential. 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Present affiliation: Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100, Israel[↩](#fnref1)\n"},"data_id":{"kind":"string","value":"arxiv_0004119"}}},{"rowIdx":997,"cells":{"content":{"kind":"string","value":"}^{a+3} - D\\_{\\rm complete}^{a+3} \\right]\\;.$$ \nThe result is $D\\_{\\rm PB} = \\sqrt[3]{{6\\over\\pi} V\\_{\\rm PB}} \\doteq 130\\,{\\rm km}$, some sort of an upper limit. The volumetric ratio between the largest fragment and the parent body is then $V\\_{\\rm LF}/V\\_{\\rm PB} \\doteq 0.2$, a fairly typical value for asteroid families in the main asteroid belt. Obviously, the assumption of a single power-law extrapolation of the *N*( > *D*) at small sizes is only approximate and can lead to a result with a 10% uncertainty. However, if we use an entirely different geometric method, developed by Tanga et al. (1998), we obtain $D\\_{\\rm PB}\\simeq 120\\mbox{--}130$km, i.e., comparable to our previous estimate.\nWhat is an approximate size $d\\_{\\rm disrupt}$ of a projectile necessary to disrupt the parent body of the Schubart family? Using Eq. (1) from Bottke et al. (2005):\n \n$$d\\_{\\rm disrupt} = \\left(2 Q\\_D^\\* / V\\_{\\rm imp}^2\\right)^{1\\over 3}\n D\\_{\\rm target} \\;. \\label{d\\_disrupt}$$ \nand substituting $Q\\_D^\\* = 10^{5}\\,{\\rm J}/{\\rm kg}$ for the strength (somewhat lower than that of basaltic objects to accommodate the assumed C/X spectral type; e.g., Kenyon et al. 2008 and references therein), $V\\_{\\rm imp} = 4.78\\,{\\rm km}/{\\rm s}$ for the typical impact velocity (see Dahlgren 1998) and $D\\_{\\rm target} \\simeq 130\\,{\\rm km}$, we obtain $d\\_{\\rm disrupt} \\simeq 25\\,{\\rm km}$. At this size the projectile population is dominated by main belt bodies. Considering also different intrinsic collisional probabilities between Hilda–Hilda asteroids ($2.3\\times 10^{-18}\\,{\\rm km}^{-2}\\,{\\rm yr}^{-1}$; Dahlgren 1998) and Hilda–main belt asteroids ($0.6\\times 10^{-18}\\,{\\rm km}^{-2}\\,{\\rm yr}^{-1}$), we find it more likely the Schubart family parent body was hit by a projectile originating from the main belt.\nHilda family\n------------\nWe repeated the same analysis as in Section [schubartfamily] for the Hilda family. The family remains statistically distinct from the whole J3/2 population in the range of cutoff velocities $(130, 170)\\,\n{\\rm m}/{\\rm s}$; we choose $v\\_{\\rm cutoff} = 150\\,{\\rm m}/{\\rm s}$ as the nominal value.\nThe slope *γ* of the cumulative absolute magnitude distribution *N*( < *H*) is ( + 0.50 ± 0.02) (Fig. [r32absmagdistribution]), again steeper than for the total J3/2 population and comparable to that of the Schubart family. The spectral slopes (PC1) are somewhat spread from flat (C/X-compatible values; PC1 < 0.3) to redder (D-compatible values; PC1 > 0.3) — see Fig. [r32woschubart.astorb.sloanpc1lt0.1hist]. Overall, though, the C/X members prevail such that the D-type objects might be actually interlopers, at least according to a simple estimate based on the volume of the Hilda family in the (*a**p*, *e**p*, sin*i**p*) space, compared to the total volume of the J3/2 population.\nTedesco et al. (2002) determine albedos for six family members. They range from 0.037 to 0.087, but three values are close to the median albedo 0.044 of all J3/2 asteroids. We thus consider this value to be representative of the Hilda family. The corresponding cumulative size distribution is plotted in Fig. [r32sizedistribution]. Using the same method as in Section [schubartfamily] we estimate the size of the parent body $D\\_{\\rm PB} = 180\\mbox{--}190\\,{\\rm km}$, with $V\\_{\\rm LF}/V\\_{\\rm PB} \\simeq 0.8$. With the model of Tanga et al. (1999) we would obtain $D\\_{\\rm PB} \\simeq 210$km and thus $V\\_{\\rm LF}/V\\_{\\rm PB}\\simeq 0.5$. This family forming event seems to be thus characterized in between the catastrophic disruption and a huge cratering. The necessary projectile size is $d\\_{\\rm disrupt} = 50\\mbox{--}55$km.\nWhile not so prominent as the Schubart family, we consider the group of asteroids around Hilda a fairly robust case of a collisionally-born family too.\n [r32-impact2aeinicond]\n [r32-impact4f0f135ae]\nSimulated disruption events\n---------------------------\nIn order to asses some limits for the age of the Schubart and Hilda families, we perform a number of numerical tests. In particular, we simulate a disruption of a parent body inside the resonance and numerically determine the long-term orbital evolution of fragments. The evolved synthetic family at different time steps is then compared with the observed family. Ideally, this approach should allow to constrain the time elapsed since the family formed.\nAs a first step, we need to create a synthetic family inside the resonance. We use current orbital elements of the largest family member, (1911) Schubart in this case, as representative to the parent body and only allow changes in the true anomaly *f* and in the argument of pericentre *ω* at the break-up event. By changing these two geometric parameters we can produce different initial positions of the fragments in the orbital element space. For sake of our test, fragments are assumed to be dispersed isotropically with respect to the parent body, with a velocity distribution given by the model of Farinella et al. (1993, 1994). The number of fragments *d**N*(*v*) launched with relative velocities in the interval (*v*, *v* + *d**v*) is given by\n \n$$dN(v) = C v (v^2 + v\\_{\\rm esc}^2)^{-(\\kappa+1)/2}\\, dv\\;,$$ \nwith *C* a normalization constant, $v\\_{\\rm esc}$ the escape velocity from the parent body and *κ* = 3.25. To prevent excessive escape velocities we introduce a maximum allowed value $v\\_{\\rm max}$. Nominally, we set $v\\_{\\rm max}=200$ m/s, but in the next Section [yarkovskye] we also use restricted values of this parameter to test sensitivity of our results to initial conditions.\nTo simulate an impact that might have created the Schubart family, we generated velocities randomly for 139 fragments with $v\\_{\\rm esc} = 65\\,{\\rm m}/{\\rm s}$ (note the number of fragments in the synthetic family is equal to the number of the Schubart family members). The resulting swarm of fragments is shown in Fig. [r32-impact2aeinicond], for the impact geometry *f* = 0∘ and *ω* + *f* = 180∘. We show both the initial osculating orbital elements and the pseudo-proper elements.\nThe synthetic family extends over significantly larger range of the semimajor axis than the observed Schubart family, but all fragments still fall within the J3/2 resonance. The eccentricity distribution is, on the other hand, substantially more compact. Only the distribution of inclinations of the synthetic family roughly matches that of the observed family. We verified this holds also for other isotropic-impact geometries (such as *f* = 135∘ and *ω* + *f* = 180∘ shown in Fig. [r32-impact4f0f135ae]). The peculiar shape of the synthetic family in the pseudo-proper element space (*a**p*, *e**p*) is an outcome of the isotropic disruption, simply because some fragments fall to the left from the libration centre of the J3/2 resonance (at 3.97AU) and they are ‘mapped’ to the right. This is because the pseudo-proper elements are the maxima and minima of *a* and *e*, respectively, over their resonant oscillations.\n [r32-impact2ae4Gyr]\nThe initial configuration of the synthetic family was propagated for 4Gyr, using the integrator described in Section [popul]. At this stage, we use only the gravitational perturbations from the 4 exterior giant planets. We performed such simulation for several impact geometries, as determined by *f* and *ω*, with similar results.\nFigure [r32-impact2ae4Gyr] shows the long-term evolution of the synthetic family. Because the family resides mostly in the stable zone of the J3/2 resonance, only little evolution can be seen for most of the bodies. This is in accord with findings of Nesvorný & Ferraz-Mello (1997) who concluded that the stable region in this resonance shows little or no diffusion over time scales comparable to the age of the Solar system. Only about 10% of orbits that initially started at the outskirts of the stable zone (with large libration amplitudes) escaped from the resonance during the 4Gyr simulation.\nThe removal of orbits with large semimajor axis *a**p* helps in part to reconcile the mismatch with the distribution of the observed Schubart family. However, the dispersion in eccentricity *e**p* does not evolve much and it still shows large mismatch if compared to the observed family. Even in the case *f* = 135∘ (*ω* + *f* = 180∘; Fig. [r32-impact4f0f135ae]), which maximizes the initial eccentricity dispersion of the synthetic fragments, the final value at 4Gyr is about three times smaller than that of the Schubart family. Clearly, our model is missing a key element to reproduce the current orbital configuration of this family.\nOne possibility to resolve the problem could be to release the assumption of an isotropic impact and explore anisotropies in the initial velocity field. This is an obvious suspect in all attempts to reconstruct orbital configurations of the asteroid families, but we doubt it might help much in this case. Exceedingly large relative velocities, compared to the escape velocity of the estimated parent body, would be required. Recall, the fragments located in the stable region of the J3/2 resonance would hardly evolve over the age of the Solar system.\nA more radical solution is to complement the force model, used for the long-term propagation, by additional effects. The only viable mechanism for the size range we are dealing with is the Yarkovsky effect. This tiny force, due to anisotropic thermal emission, has been proved to have determining role in understanding fine structures of the asteroid families in the main belt (e.g., Bottke et al. 2001; Vokrouhlický et al. 2006a,b). In these applications the Yarkovsky effect produces a steady drift of the semimajor axis, leaving other orbital elements basically constant.\nHowever, the situation is different for resonant orbits. The semimajor axis evolution is locked by the strong gravitational influence of Jupiter. For that reason we first ran simplified simulations with the Yarkovsky forces — results of these tests are briefly described in the Appendix A. We next applied the model containing both gravitational and Yarkovsky perturbations to the evolution of the synthetic family. Results of these experiments are described in the next Section [yarkovskye].\nYarkovsky drift in eccentricity\n-------------------------------\nWe ran our previous simulation of the long-term evolution of the synthetic family with the Yarkovsky forces included. Our best guess of thermal parameters for bodies of the C/X type is: $\\rho\\_{\\rm s} = \\rho\\_{\\rm b} = 1300\\,{\\rm kg}/{\\rm m}^3$ for the surface and bulk densities, $K = 0.01\\,{\\rm W}/{\\rm m}/{\\rm K}$ for the surface thermal conductivity, $C = 680\\,{\\rm J}/{\\rm kg}$ for the heat capacity, *A* = 0.02 for the Bond albedo and *ε* = 0.95 for the thermal emissivity parameter. Rotation periods are bound in the 2–12hours range. Spin axes orientations are assumed isotropic in space. Finally, we assign sizes to our test particles equal to the estimate of sizes for Schubart family members, based on their reported absolute magnitudes and albedo *p**V* = 0.025. The dependence of the Yarkovsky force on these parameters is described, e.g., in Bottke et al. (2002, 2006). We note the uncertainties of the thermal parameters, assigned to individual bodies, do not affect our results significantly, mainly because we simulate a collective evolution of more than 100 bodies; we are not interested in evolution of individual orbits.\n [r32-impye2ae1Gyr]\nWe let the synthetic family evolve for 4Gyr and recorded its snapshots every 50 kyr. As discussed in the Appendix A, the *resonant Yarkovsky effect* produces mainly secular changes of eccentricity. We recall this systematic drift in *e* must not be confused with the chaotic diffusion in *e*. We also note that inclination of the orbits remains stable, in accord with a good match of the Schubart family by the initial inclination distribution.\nBecause the initial eccentricity dispersion of the synthetic family is much smaller than that of the observed one, its steady increase due to the combined effects of the Yarkovsky forces and the resonant lock gives us a possibility to date the origin of the family (see Vokrouhlický et al. 2006a for a similar method applied to families in the main belt). To proceed in a quantitative way, we use a 1-dimensional Kolmogorov–Smirnov (KS) test to compare cumulative distribution of pseudo-proper eccentricity values *e**p* of the observed and synthetic families (e.g., Press et al. 2007).\nFigure [schubartkstest1D] shows the KS distance *D**K**S* of the two eccentricity distributions as a function of time. For sake of a test, we also use smaller $v\\_{\\rm max}$ values of the initial velocity field (essentially, this is like to start with a more compact synthetic family). Regardless of the $v\\_{\\rm max}$ value, our model rejects Schubart family ages smaller than 1Gyr and larger than 2.4Gyr with a 99% confidence level. For ages in between 1.5Gyr and 1.7Gyr the KS-tested likelihood of a similarity of the synthetic-family and the observed-family *e**p* distributions can reach up to 50%. We thus conclude the most likely age of the Schubart cluster is $(1.7\\pm 0.7)\\,{\\rm Gyr}$.\n [schubartkstest1D]\n [hildakstest1D]\nWe repeated the same analysis for Hilda family by creating a synthetic family of 233 particles about (153) Hilda. In this case we used $v\\_{\\rm esc}=110\\,{\\rm m}/{\\rm s}$. The situation is actually very similar to the Schubart — there is again a problem with the small dispersion of eccentricities in case of a purely gravitational model. Using the model with the Yarkovsky effect, we can eventually fit the spread of eccentricities and according to the KS test (Fig. [hildakstest1D]) the age of the family might be $\\gtrsim 4$Gyr. The match is still not perfect, but this problem might be partly due to numerous interlopers in the family. We also note that a 10% relative uncertainty of the mean albedo of the Hilda family members would lead to a 5% uncertainty of their sizes and, consequently, to a 5% uncertainty of the family age.\nThe Hilda family seems to be dated back to the Late Heavy Bombardment era (Morbidelli et al. 2005; see also Sec. [migration]). We would find such solution satisfactory, because the population of putative projectiles was substantially more numerous than today (note that a disruption of the Hilda family parent body is a very unlikely event during the last 3.5Gyr).\nWe finally simulated a putative collision in the J2/1 resonance, around the asteroid (3789) Zhongguo (the largest asteroid in the stable island B). There are two major differences as compared to the J3/2 resonance: (i) the underlying chaotic diffusion due to the gravitational perturbations is larger in the J2/1 resonance (e.g., Nesvorný & Ferraz-Mello 1997), such that an initially compact cluster would fill the whole stable region in 1–1.5Gyr and consequently becomes unobservable; (ii) sizes of the observed asteroids are generally smaller, which together with a slightly smaller heliocentric distance, accelerates the Yarkovsky drift in *e**p*. The latter effect would likely shorten the time scale to 0.5–1Gyr. Thus the non-existence of any significant orbital clusters in the J2/1 resonance (Sec. [fams] and Fig. [r21-hcm2007stalactitediagrammin5]) does not exclude a collisional origin of the long-lived population by an event older than 1Gyr. This would also solve the apparent problem of the very steep size distribution of the stable J2/1 population (Brož et al. 2005 and Fig. [ressfd]). Note the expected collisional lifetime of the smallest observed J2/1 asteroids is several Gyr (e.g., Bottke et al. 2005).\nResonant population stability with respect to planetary migration and the Yarkovsky effect\n==========================================================================================\nWe finally pay a brief attention to the overall stability of asteroid populations in the first-order resonances with respect to different configurations of giant planets. We are focusing on the situations when the orbits of Jupiter and Saturn become resonant. This is motivated by currently adopted views about final stages of building planetary orbits architecture, namely planet migration in a diluted planetesimal disk (e.g., Malhotra 1995; Hahn & Malhotra 1999; Tsiganis et al. 2005; Morbidelli et al. 2005; Gomes et al. 2005 – these last three references are usually described as the Nice model). Morbidelli et al. (2005) proved that the primordial Trojan asteroids were destabilized when Jupiter and Saturn crossed their mutual 1:2 mean motion resonance and, at the same time, the Trojan region was re-populated by particles of the planetesimal disk. Since the mutual 1:2 resonance of Jupiter and Saturn plays a central role in the Nice model, and since these two planets had to cross other (weaker) mutual resonances such as 4:9 and 3:7 before they acquired today’s orbits, one can naturally pose a question about the stability of primordial populations in the first-order mean motion resonances with Jupiter. Ferraz-Mello et al. (1998a,b) demonstrated that even subtler effects can influence the J2/1 population, namely the resonances between the asteroid libration period in the J2/1 resonance and the period of the Great Inequality (GI) terms in planetary perturbations (i.e., those associated with Jupiter and Saturn proximity to their mutual 2:5 mean motion resonance; Fig. [shiftedSaturnperiodGI]). A first glimpse to the stability of the first-order resonance populations with respect to these effects is given in Section [migration].\nIn Section [yarkovsky] we also briefly estimate the change of dynamical lifetimes for small J2/1 and J3/2 bodies caused by the Yarkovsky effect.\n[shiftedSaturnperiodGI]\nPlanetary migration effects\n---------------------------\nIn what follows we use a simple approach by only moving Saturn’s orbit into different resonance configurations with Jupiter’s orbit. We do not let orbits of these planets migrate, but consider them static. With such a crude approach we can only get a first hint about a relative role of depletion of the asteroid populations in the first-order resonances (note in reality the planets undergo steady, but likely not smooth, migration and exhibit jumps over different mutual resonant states; e.g., supplementary materials of Tsiganis et al. 2005).\n \n \n [r213243-jsloglftime]\nThe results are summarised in Fig. [r213243-jsloglftime]. (i) The Hilda group in the J3/2 resonance is very unstable (on the time scale $\\sim\\!1\\,{\\rm Myr}$) with respect to the 1:2 Jupiter–Saturn resonance;[9](#fn9) on the contrary J2/1 asteroids may survive several 10Myr in this configuration of Jupiter and Saturn, so this population is not much affected by this phase by planetary evolution (note, moreover, that Jupiter and Saturn would likely cross the zone of other mutual 1:2 resonance in $\\sim\\!1\\,{\\rm Myr}$ only; e.g., Tsiganis et al. 2005 and Morbidelli et al. 2005). (ii) The 4:9 resonance has a larger influence on the J2/1 population than on Hildas. (iii) In the case of 3:7 resonance it is the opposite: the J3/2 is more unstable than the J2/1. (iv) The Great Inequality resonance does indeed destabilise the J2/1 on a time scale 50Myr. Provided the last phases of the migration proceed very slowly, it may cause a significant depletion of the primordial J2/1 population. In the exact 2:5 resonance, the J2/1 population would not be affected at all.\nAccording to our preliminary tests with a more complete *N*-body model for planetary migration which includes a disk of 103 planetesimals beyond Neptune, the strong instability of the J3/2 asteroids indeed occurs during the Jupiter–Saturn 1:2 resonance crossing (see Fig. [j32pdisc50MEcoldI00jatplanets]). A vast reservoir of planetesimals residing beyond the giant planets, and to some extent also nearby regions of the outer asteroid belt, are probably capable to re-populate the J3/2 resonance zone at the same time and form the currently observed populations. This is similar to the Trojan clouds of Jupiter (Morbidelli et al. 2005).\n \n[j32pdisc50MEcoldI00jatplanets]\nThe Yarkovsky effect\n--------------------\nIn Section [yarkovskye] we already discussed the influence of the Yarkovsky effect on the families located inside the J3/2 resonance. In course of time it modified eccentricities of their members, but did not cause a large-scale instability; the families remained inside the resonance all the time. Here we seek the size threshold for which the Yarkovsky would case overall instability by quickly removing the bodies from the resonance.\nWe perform the following numerical test: we multiply sizes of the long-lived J2/1 objects by fudge factors 0.2, 0.02 and 0.002, for which the Yarkovsky effect is stronger, and compare respective dynamical lifetimes with the original long-lived objects. Results are summarized in Fig. [r21zgyloglftimeradii]. We can conclude that a significant destabilisation of the J2/1 resonant population occurs for sizes ∼ 0.1km and smaller (provided the nominal population has sizes mostly 4–12km).\nWe do not include the YORP effect (i.e., the torque induced by the infrared thermal emission) at this stage. The YORP is nevertheless theoretically capable to significantly decelerate (or accelerate) the rotation rate, especially of the smallest asteroids, which can lead to random reorientations of the spin axes due to collisions, because the angular momentum is low in this spin-down state. These reorientations can be simulated by a Monte-Carlo model with a typical time scale (Čapek & Vokrouhlický 2004; *R* is radius in kilometres and *a* orbital semimajor axis in AU):\n \n$$\\tau\\_{\\rm YORP} \\simeq 25\\,{\\rm Myr}\\, (R/1\\,{\\rm km})^2\\,\n (2.5\\,{\\rm AU}/a)^2\\;. \\label{tau\\_YORP}$$ \nSince the Yarkovsky effect depends on the obliquity value, the systematic drift would be changed to a random walk for bodies whose spin axis undergo frequent re-orientations by the YORP effect. We can thus expect that the YORP effect might significantly prolong dynamical lifetimes of resonant objects with sizes ∼ 0.1km or smaller, because $\\tau\\_{\\rm YORP} < 0.25\\,{\\rm Myr}$ for them.\n [r21zgyloglftimeradii]\nWe can also check, which orientation of the spin axis makes the escape from the J2/1 resonance more likely to happen. We consider 0.08–0.24km bodies, clone them 5 times and assign them different values of the obliquity *γ* = 0∘, 45∘, 90∘, 135∘ and 180∘. Figure [r21zgyloglftimeobliq0.01] shows clearly that the retrograde rotation increases the probability of the escape. This is consistent with the structure of the J2/1 resonance, for which low-*a* separatrix does not continue to *e* = 0. We conclude the remaining very small (yet unobservable) asteroids inside the J2/1 may exhibit a preferential prograde rotation.\n [r21zgyloglftimeobliq0.01]\nWe perform a similar simulation for the J3/2 population (first 100 bodies with sizes 10–60km), but now with YORP reorientations (Eq. [tauYORP]) included. The results (Fig. [r32zgyloglftimeradii]) show the J3/2 population is much less affected than the J2/1 by the Yarkovsky/YORP perturbation.\nWe conclude the Yarkovsky/YORP effect may destabilize the J2/1 and J3/2 bodies only partially on the 100Myr time scale and only for sizes smaller than ∼ 0.1km. It is obviously a remote goal to verify this conclusion by observations (note the smallest bodies in these resonances have several kilometers size). Nevertheless, the dynamical lifetimes of small asteroids determined in this section, might be useful for collisional models of asteroid populations, which include also dynamical removal.\n [r32zgyloglftimeradii]\nConclusions and future work\n===========================\nThe main results of this paper can be summarised as follows: (i) we provided an update of the observed J2/1, J3/2 and J4/3 resonant populations; (ii) we discovered two new objects in the J4/3 resonance; (iii) we described two asteroid families located inside the J3/2 resonance (Schubart and Hilda) and provided an evidence that they are of a collisional origin; (iv) we reported a new mechanism how the Yarkovsky effect systematically changes *eccentricities* of resonant asteroids; we used this phenomenon to estimate the ages of the Schubart and Hilda families ((1.7 ± 0.7)Gyr and $\\gtrsim 4$Gyr respectively); (v) collisionally-born asteroid clusters in the stable region of J2/1 would disperse in about 1Gyr; (vi) 20% of Hildas may escape from the J3/2 resonance within 4Gyr in the current configuration of planets; (vii) Hildas are strongly unstable when Jupiter and Saturn cross their mutual 1:2 mean motion resonance.\nThe J3/2 resonance is a unique ‘laboratory’ — the chaotic diffusion is so weak, that families almost do not disperse in eccentricity and inclination due to this effect over the age of the Solar system. What is even more important, they almost do not disperse in semimajor axis, even thought the Yarkovsky effect operates. The drift in *a* is transformed to a drift in *e*, due to a strong gravitational coupling with Jupiter. We emphasize, this is is not a chaotic diffusion in *e*, but a *systematic drift in *e**.\nAnother piece of information about the families in J3/2 resonance is hidden in the eccentricity *e**p* vs absolute magnitude *H* plots (see Fig. [hildaeH] for the Hilda family). The triangular shape (larger eccentricity dispersion of the family members for larger *H*) is a well-known combination of two effects: (i) larger ejection speed, and (ii) faster dispersal by the Yarkovky forces for smaller fragments. Interestingly, there is also a noticeable depletion of small bodies in the centre of the family and their concentration at the outskirts — a phenomenon known from (*a*, *H*) plots of main-belt families, which was interpreted as an interplay between the Yarkovsky and YORP effects (Vokrouhlický et al. 2006).[10](#fn10) Indeed the estimated ∼ 4Gyr age for this family matches the time scale of a YORP cycle for *D* ≃ 10km asteroids in the Hilda region (e.g., Vokrouhlický & Čapek 2002; Čapek & Vokrouhlický 2004). Vokrouhlický et al. (2006a) pointed out that this circumstance makes the uneven distribution of family members most pronounced.\n[hildaeH]\nWe postpone the following topics for the future work: (i) a more precise age determination for the resonant asteroid families, based on the Yarkovsky/YORP evolution in the (*e*, *H*) space; (ii) a more detailed modelling of analytic or *N*-body migration of planets and its influence on the stability of resonant populations.\nAcknowledgements\n================\nWe thank David Nesvorný, Alessandro Morbidelli and William F. Bottke for valuable discussions on the subject and also Rodney Gomes for a constructive review. The work has been supported by the Grant Agency of the Czech Republic (grants 205/08/0064 and 205/08/P196) and the Research Program MSM0021620860 of the Czech Ministry of Education. 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S., 2006b, Icarus, 182, 92 Weissman P.R., Bottke W.F., Levison H.L., 2002, in Bottke W.F., Cellino A., Paolicchi P., Binzel R.P., eds., Asteroids III. University of Arizona Press, Tucson, p. 669 Zappalà V., Cellino A., Farinella P., Knežević Z., 1990, AJ, 100, 2030 Zappalà V., Cellino A., Farinella P., Milani A., 1994, AJ, 107, 772 Zappalà V., Cellino A., Dell’Oro A., Paolicchi P., 2002, in Bottke W.F., Cellino A., Paolicchi P., Binzel R.P., eds., Asteroids III. University of Arizona Press, Tucson, p. 619\nResonant Yarkovsky effect\n=========================\n| | | |\n| --- | --- | --- |\n| | | |\n[r32-rtbpfilterSsigma100Myr]\nThe effects of weak dissipative forces, such as the tidal force, gas-drag force and the Poynting-Robertson force, on both non-resonant and resonant orbits were extensively studied in the past (e.g., Murray & Dermott 1999 and references therein). Interaction of the Yarkovsky drifting orbits with high-order, weak resonances was also numerically studied to some extent (e.g., Vokrouhlický & Brož 2002) but no systematic effort was paid to study Yarkovsky evolving orbits in strong low-order resonances. Here we do not intend to develop a detailed theory, rather give a numerical example that can both help to explain results presented in the main text and motivate a more thorough analytical theory.\n*The Yarkovsky effect outside the resonance.* We constructed the following simple numerical experiment: we took the current orbit of (1911) Schubart as a starting condition and integrated the motion of two 0.1km size objects with extreme obliquity values 0∘ and 180∘. Their thermal parameters were the same as in Section [yarkovskye]. Because the diurnal variant of the Yarkovsky effect dominates the evolution, the extreme obliquities would mean the two test bodies would normally (outside any resonances) drift in semimajor axis in two opposite directions (e.g., Bottke et al. 2002, 2006). The two orbits would secularly acquire Δ*a* ≃ + 0.25AU or − 0.25AU in 100 My, about the extent shown by the arrow on top of the left panel of Fig. [r32-rtbpfilterSsigma100Myr]. Since the strength of the Yarkovsky forces is inversely proportional to the size, we can readily scale the results for larger bodies.\n*The resonance without the Yarkovsky effect.* If we include gravitational perturbations by Jupiter only, within a restricted circular three body problem ($e\\_{\\rm J} = 0$), and remove short-period oscillations by a digital filter, the parameter *N* from Eq. ([resint]) would stay constant. The orbit would be characterized by a stable libration in (Σ, *σ*) variables with about 30∘ amplitude in *σ* (see the curve labelled 0Myr in the middle panel of Fig. [r32-rtbpfilterSsigma100Myr]).\nWhile evolving, some parameters known as the adiabatic invariants are approximately conserved (see, e.g., Landau & Lifschitz 1976; Henrard 1982; Murray & Dermott 1999). One of the adiabatic invariants is *N* itself. Another, slightly more involved quantity, is the area *J* enclosed by the resonant path in the $\\sqrt{2\\Sigma}\\, (\\cos\\sigma,\\sin\\sigma)$ space:\n \n$$J = \\oint \\sqrt{2\\Sigma}\\,d\\sigma\\;. \\label{adiab}$$ \nWe would thus expect these parameters be constant, except for strong-enough perturbation or long-enough time scales (recall the adiabatic invariants are constant to the second order of the perturbing parameter only).\n*Resonant Yarkovsky effect.* Introducing the Yarkovsky forces makes the system to evolve slowly. The lock in the resonance prevents the orbits to steadily drift away in the semimajor axis and the perturbation by the Yarkovsky forces acts adiabatically. This is because (i) the time scale of the resonance oscillations is much shorter than the characteristic time scale of the orbital evolution driven by the Yarkovsky forces, and (ii) the strength of the resonant terms in the equations of motion are superior to the strength of the Yarkovsky accelerations.\nWe let the two J3/2 orbits evolve over 100 Myr (Fig. [r32-rtbpfilterSsigma100Myr]). At the end of our simulation the orbits moved from *N*0 ≃ 0.45 to *N*+ ≃ 0.46 (for the outward migrating orbit) or to *N*− ≃ 0.44 (for the inward migrating orbit), respectively. During this evolution, both orbits remained permanently locked in the J2/1 resonance, librating about the periodic orbit (dashed line in the left panel of Fig. [r32-rtbpfilterSsigma100Myr]). Because the position of this centre has a steep progression in the eccentricity and only small progression in the semimajor axis, the evolution across different *N*-planes makes the orbital eccentricity evolve significantly more than the semimajor axis. This is also seen in the middle panel of Fig. [r32-rtbpfilterSsigma100Myr], where the librating orbits significantly split farther/closer with respect to origin of coordinates (note the polar distance from the origin is basically a measure of the eccentricity). The shape of the librating orbit is modified such that the area *J* stays approximately constant. We have verified that the relative change in both adiabatic invariants, acquired during the 100Myr of evolution, is about the same: *δ**N*/*N* ∼ *δ**J*/*J* ∼ 5 × 10− 2. It is a direct expression of the strength of the perturbation by the Yarkovsky forces.\nWe can conclude the Yarkovsky effect results in a significantly different type of secular evolution for orbits initially inside strong first-order mean motion resonances with Jupiter. Instead of secularly pushing the orbital semimajor axis inward or outward from the Sun, it drives the orbital eccentricity to smaller or larger values, while leaving the semimajor axis to follow the resonance centre.\nIf we were to leave the orbital evolution continue in our simple model, the inward-migrating orbit would leave the resonance toward the zone of low-eccentricity apocentric librators. Such bodies are observed just below the J2/1 resonance. On the other hand, the outward-migrating orbit would finally increase the eccentricity to the value when the orbit starts to cross the Jupiters orbit. Obviously, in a more complete model, with all planets included, the orbits would first encounter the unstable region surrounding the stable resonant zone. Such marginally stable populations exist in both the J3/2 and J2/1 resonances.\n[lastpage]\n---\n1. E-mail: mira@sirrah.troja.mff.cuni.cz (MB); vokrouhl@cesnet.cz (DV).[↩](#fnref1)\n2. Interestingly, at their discovery (158) Hilda and (279) Thule, residing in the J3/2 and J4/3 resonances, immediately attracted attention of astronomers by vastly extending asteroid zone toward giant planets and by their ability to apparently approach Jupiter near aphelia of their orbits (e.g., Kühnert 1876; Krueger 1889).[↩](#fnref2)\n3. Note the former P-type objects were reclassified to X-type in a newer taxonomy by Bus and Binzel (e.g., Bus et al. 2002).[↩](#fnref3)\n4. Our results for both J2/1 and J3/2 resonances are summarised in tables available through a website http://sirrah.troja.mff.cuni.cz/yarko-site/ (those for J4/3 bodies are given in Table [tab1]). These contain listing of all resonant asteroids, their pseudo-proper orbital elements with standard deviations, their dynamical residence time and some additional information.[↩](#fnref4)\n5. This is equivalent to a cumulative size distribution law *N*( > *D*) ∝ *D**α* with *α* = − 5*γ* = ( − 3.5 ± 0.1), assuming all bodies have the same albedo.[↩](#fnref5)\n6. While our initial search used AstOrb catalogue from September 2007, we repeated it using the catalogue as of June 2008. No additional J4/3 objects were found.[↩](#fnref6)\n7. Schubart lists 11 additional asteroids in the group on his website http://www.rzuser.uni-heidelberg.de/ s24/hilda.htm, but again he does not go into details of their putative collisional origin.[↩](#fnref7)\n8. We also mention that so far asteroid disruption simulations did not explore cases of weak-strength materials appropriate for the suggested C/X spectral taxonomy of the Schubart family parent body.[↩](#fnref8)\n9. Jupiter Trojans, which are already known to be strongly unstable (Morbidelli et al. 2005), would have the dynamical lifetime of the order $0.1\\,{\\rm Myr}$ in this kind of simulation.[↩](#fnref9)\n10. The YORP effect tilts the spin axes of asteroids preferentially towards obliquity *γ* = 0∘ or 180∘ and this enhances the diurnal Yarkovsky drift due to its cos*γ* dependence.[↩](#fnref10)\n\nAsteroid families in the first order resonances with Jupiter\n============================================================\n### Accepted???. Received???; in original form???\n[firstpage]\nAsteroids residing in the first-order mean motion resonances with Jupiter hold important information about the processes that set the final architecture of giant planets. Here we revise current populations of objects in the J2/1 (Hecuba-gap group), J3/2 (Hilda group) and J4/3 (Thule group) resonances. The number of multi-opposition asteroids found is 274 for J2/1, 1197 for J3/2 and 3 for J4/3. By discovering a second and third object in the J4/3 resonance, (186024) 2001 QG207 and (185290) 2006 UB219, this population becomes a real group rather than a single object. Using both hierarchical clustering technique and colour identification we characterise a collisionally-born asteroid family around the largest object (1911) Schubart in the J3/2 resonance. There is also a looser cluster around the largest asteroid (153) Hilda. Using *N*-body numerical simulations we prove that the Yarkovsky effect (infrared thermal emission from the surface of asteroids) causes a systematic drift in eccentricity for resonant asteroids, while their semimajor axis is almost fixed due to the strong coupling with Jupiter. This is a different mechanism from main belt families, where the Yarkovsky drift affects basically the semimajor axis. We use the eccentricity evolution to determine the following ages: $(1.7\\pm0.7)\\,{\\rm Gyr}$ for the Schubart family and $\\gtrsim 4\\,{\\rm Gyr}$ for the Hilda family. We also find that collisionally-born clusters in the J2/1 resonance would efficiently dynamically disperse. The steep size distribution of the stable population inside this resonance could thus make sense if most of these bodies are fragments from an event older than ≃ 1Gyr. Finally, we test stability of resonant populations during Jupiter’s and Saturn’s crossing of their mutual mean motion resonances. In particular we find primordial objects in the J3/2 resonance were efficiently removed from their orbits when Jupiter and Saturn crossed their 1:2 mean motion resonance.\ncelestial mechanics – minor planets, asteroids – methods: *N*-body simulations.\nIntroduction\n============\nPopulations of asteroids in the Jovian first order mean motion resonances –J2/1, J3/2 and J4/3– are closely linked to the orbital evolution of the giant planets. This is because of their orbital proximity to Jupiter.[2](#fn2) Stability or instability of these asteroid populations directly derives from the orbital configuration of the giant planets. As such it is also sensitive on the nature and amount of Jupiter’s migration and other finer details of its dynamics. As a result, the currently observed asteroids in the Jovian first order resonances contain valuable information about the early evolution of planets and, if correctly understood and properly modelled, they may help to constrain it.\nApart from the Trojan clouds (not studied in this paper) the largest known population in the Jovian mean motion resonances occupies the J3/2 resonance, and is frequently called the Hilda group. It was carefully studied in a parallel series of papers by Schubart and Dahlgren and collaborators during the past few decades. Schubart (1982a,b, 1991) analysed short-term dynamics of Hilda-type orbits and introduced quasi-constant orbital parameters that allowed their first classification. While pioneering, Schubart’s work had the disadvantage of having much smaller sample of known asteroids and computer power than today. Dahlgren & Lagerkvist (1995) and Dahlgren et al. (1997, 1998, 1999) conducted the first systematic spectroscopic and rotation-rate investigation of Hildas. They found about equal abundance of D- and P-type asteroids[3](#fn3) and suggested spectral-size correlation such that P-types dominate large Hildas and D-type dominate smaller Hildas. They also suggested small Hildas have large lightcurve amplitudes, as an indication of elongated or irregular shape, and that the distribution of their rotation rates is non-Maxwellian. Further analysis using the Sloan Digital Sky Survey (SDSS) data however does not support significant dominance of either of the two spectral types for small sizes and indicates about equal mix of them (Gil-Hutton & Brunini 2008; see also below). Smaller populations of asteroids in the J2/1 and J4/3 received comparatively less observational effort.\nSince the late 1990s powerful-enough computers allowed a more systematic analysis of fine details of the longer-term dynamics in the Jovian first order resonances. Ferraz-Mello & Michtchenko (1996) and Ferraz-Mello et al. (1998a,b) determined that asteroids in the J2/1 resonance can be very long-lived, possibly primordial, yet their motion is comparatively more chaotic than those in the J3/2 resonance. The latter paper showed that commensurability between the libration period and the period of Jupiter’s and Saturn’s Great Inequality might have played an important role in depletion of the J2/1 resonance. This would have occurred when both giant planets were farther from their mutual 2:5 mean motion configuration in the past. A still more complete analysis was obtained by Nesvorný & Ferraz-Mello (1997) who also pointed out that the J4/3 resonance stable zone is surprisingly void of asteroids, containing only (279) Thule. Roig et al. (2001) and Brož et al. (2005) recently revised the population of asteroids in the J2/1 resonance and classified them into several groups according to their long-term orbital stability. While the origin of the unstable resonant population was successfully interpreted using a model of a steady-state flow of main belt objects driven by the Yarkovsky semimajor axis drift, the origin of the long-lived asteroids in the J2/1 remains elusive. Population of Hildas and Thule was assumed primordial or captured by an adiabatic migration of Jupiter (e.g., Franklin et al. 2004).\nIt has been known for some time that the current configuration of giant planets does not correspond to that at their birth. However, a new momentum to that hypothesis was given by the so called Nice model (Tsiganis et al. 2005; Morbidelli et al. 2005; Gomes et al. 2005). The Nice model postulates the initial configuration of the giant planets was such that Jupiter and Saturn were interior of their mutual 1:2 mean motion resonance (see also Morbidelli et al. 2007). The event of crossing this resonance had a major influence on the final architecture of giant planets and strongly influenced structure of small-bodies populations in the Solar system. Morbidelli et al. (2005) showed that the population of Jupiters Trojan asteroids was destabilised and re-populated during this phase. In what follows we show that, within the Nice model, the same most probably occurs for populations of asteroids in the J3/2 and J4/3 resonances.\nThe paper is organised as follows: in Section [popul] we revise information about the current populations of asteroids in the Jovian first order resonances. We use an up-to-date AstOrb database of asteroid orbits from the Lowell Observatory (ftp.lowell.edu) as of September 2007 and eliminate only single-opposition cases to assure accurate orbital information.\nIn Section [fams] we apply clustering techniques and extract two families of asteroids on similar orbits in the J3/2 resonance. We strengthen their case with an additional colour analysis using the SDSS broadband data. We model the long-term orbital evolution of these families and estimate their ages on the basis of Yarkovsky-driven dispersion in eccentricity.\nIn Section [stability] we determine an orbital stability of the putative primordial populations of planetesimals in the Jovian first-order resonances. We show that those in the J3/2 and J4/3 are very efficiently eliminated when Jupiter and Saturn cross their mutual 1:2 mean motion resonance. We also determine the removal rate of very small resonant asteroids due to the Yarkovsky/YORP effects.\nCurrent asteroid populations in the Jovian first order resonances\n=================================================================\nDynamics of asteroid motion in the Jovian first order resonances has been extensively studied by both analytical and numerical methods in the past few decades (e.g., Murray 1986; Sessin & Bressane 1988; Ferraz-Mello 1988; Lemaitre & Henrard 1990; Morbidelli & Moons 1993; Moons et al. 1998; Nesvorný & Ferraz-Mello 1997; Roig et al. 2002; Schubart 2007). In what follows we review a minimum information needed to understand our paper, referring an interested reader to the literature mentioned above for more insights.\nIn the simplest framework of a circular restricted planar three-body problem (Sun-Jupiter-asteroid) the fundamental effects of the resonant dynamics is reduced to a one-degree of freedom problem defined by a pair of variables (Σ, *σ*). For J(*p* + 1)/*p* resonance (*p* = 1, 2 and 3 in our cases) we have\n \n$$\\begin{aligned}\n \\Sigma &=& \\sqrt{a}\\,\\left(1-\\sqrt{1-e^2}\\right)\\;, \\label{res11} \\\\\n \\sigma &=& \\left(p+1\\right)\\lambda' - p\\, \\lambda - \\varpi \\;,\n \\label{res12}\\end{aligned}$$ \nwhere *a* is the semimajor axis, *e* the eccentricity, $\\varpi$ the longitude of pericentre and *λ* the mean longitude in orbit of the asteroid, and *λ*ʹ is the mean longitude in orbit of Jupiter.\nIf the asteroid motion is not confined into the orbital plane of the planet, we have an additional pair of resonant variables (Σ*z*, *σ**z*) such that\n \n$$\\begin{aligned}\n \\Sigma\\_z &=& 2\\sqrt{a\\left(1-e^2\\right)}\\sin^2 \\frac{i}{2}\\;,\n \\label{res21} \\\\\n \\sigma\\_z &=& \\left(p+1\\right)\\lambda' - p\\, \\lambda - \\Omega \\;,\n \\label{res22}\\end{aligned}$$ \nwhere *i* denotes the inclination of asteroids orbit and Ω the longitude of its node. Remaining still with the simple averaged model, orbital effects with shorter periods are neglected, the motion obeys an integral of motion *N* given by\n \n$$N=\\sqrt{a}\\,\\left(\\frac{p+1}{p}-\\sqrt{1-e^2}\\cos i\\right)\\;.\n \\label{resint}$$ \nBecause of this integral of motion, variations of Σ imply oscillations of both *a* and *e*.\nThe two-degree of freedom character of the resonant motion prevents integrability. However, as an approximation we may introduce a hierarchy by noting that perturbation described by the (Σ, *σ*) variables is larger than that described by the (Σ*z*, *σ**z*) terms (e.g., Moons et al. 1998). This is usually true for real resonant asteroids of interest. Only the angle *σ* librates and *σ**z* circulates with a very long period. The (Σ*z*, *σ**z*) dynamics thus produces a long-period perturbation of the (Σ, *σ*) motion.\nWithin this model the minimum value of Σ in one resonant cycle (typically several hundreds of years) implies *a* is minimum and *e* is maximum. These values do not conserve exactly from one cycle to another because the (Σ*z*, *σ**z*) motion produces small oscillations. Since $\\Sigma+\\Sigma\\_z-N=-\\sqrt{a}/p$ one needs to wait until Σ*z* reaches maximum over its cycle to attain ‘real’ minimum of *a* values and ‘real’ maximum of *e* values over a longer time interval. From ([res21]) we note the maximum of Σ*z* occurs for the maximum of *i* variations. This situations occurs typically once in a few thousands of years. In an ideal situation, these extremal values of (*a*, *e*, *i*) would be constant and may serve as a set of proper orbital elements.\nThe motion of real asteroids in the Solar system is further complicated by Jupiter having non-zero and oscillating value of eccentricity. This brings further perturbations (e.g., Ferraz-Mello 1988; Sessin & Bressane 1988 for a simple analytic description) and sources of instability inside the resonance. Despite the non-integrability, we follow Roig et al. (2001) and introduce *pseudo-proper orbital elements* (*a**p*, *e**p*, sin*i**p*) as the osculating elements (*a*, *e*, sin*i*) at the moment, when the orbit satisfies the condition:\n \n$$\\sigma=0 \\land {d\\sigma\\over dt}<0 \\land \\varpi-\\varpi'=0 \\land\n \\Omega-\\Omega'=0\\;, \\label{prop1}$$ \nwhere $\\varpi'$ and Ωʹ denote the longitude of pericentre and the longitude of node of Jupiter. As above, when ([prop1]) holds the osculating orbital elements are such that *a* attains minimum, *e* attains maximum and *i* attains maximum. Numerical experiments show that with a complete perturbation model and a finite time-step it is difficult to satisfy all conditions of ([prop1]) simultaneously. Following Roig et al. (2001) we thus relax ([prop1]) to a more practical condition\n \n$$|\\sigma|<5^\\circ \\land {\\Delta\\sigma\\over \\Delta t}<0 \\land\n |\\varpi-\\varpi'|<5^\\circ\\;. \\label{prop2}$$ \nBecause this condition is only approximate, we numerically integrate orbits of resonant asteroids for 1Myr, over which the pseudo-proper orbital elements are recorded. We then compute their mean value and standard deviation, which is an expression of the orbital stability over that interval of time.\nIn the case of the J3/2 and J4/3 resonances, we use the condition ([prop2]) with a different sign ${\\Delta\\sigma\\over \\Delta t}>0$ and, moreover, we apply a digital filter (denoted as A in Quinn et al. (1991), using 1yr sampling and a decimation factor of 10) to *σ*(*t*). This intermediate stage serves to suppress oscillations faster than the libration period. The different sign of ${\\Delta\\sigma\\over\n\\Delta t}$ just means our pseudo-proper orbital elements correspond to maximum value of *a* and minimum values of *e* and *i*, in order to allow more direct comparison with previous analyses.\nAside to this short-term integration we perform long-term runs to determine the stability of a particular resonant orbit. With this aim we conduct integrations spanning 4Gyr for all resonant asteroids. Because of the inherent uncertainty in the initial conditions (orbital elements at the current epoch), we perform such integration for the nominal orbit and 10 clones that randomly span the uncertainty ellipsoid. We then define *dynamical lifetime* of the orbit as the median of time intervals, for which the individual clones stayed in the resonance.\nAll integrations are performed using the Swift package (Levison & Duncan 1994), slightly modified to include necessary on-line digital filters and a second order symplectic integrator (Laskar & Robutel 2001). Most of numerical simulations take into account gravitational interactions only, but in specific cases – and when explicitly mentioned – we include also Yarkovsky (thermal) accelerations. In this case we use an implementation described in detail by Brož (2006). Our simulations include 4 outer planets. We modify the initial conditions of the planets and asteroids by a barycentric correction to partially account for the influence of the terrestrial planets. The absence of the terrestrial planets as perturbers is a reasonable approximation in the outer part of the main belt and for orbits with *e* < 0.8 in general. We nevertheless checked the short-term computations (determination of pseudo-proper resonant elements) using a complete planetary model and noticed no significant difference in results. The second order symplectic scheme allows us to use a time step of 91 days.\nHecuba-gap group\n----------------\nIn order to determine, which objects are located in the J2/1 mean motion resonance, we first extracted orbits from the AstOrb database with osculating orbital elements in a broad box around this resonance (see, e.g., Roig et al. 2001 for a similar procedure). We obtained 7139 orbits, which we numerically integrated for 10kyr. We recorded and analysed behaviour of the resonance angle $\\sigma=2\\lambda'-\\lambda-\\varpi$ from Eq. ([res12]). Pericentric librators, for which *σ* oscillates about 0∘, were searched. We found 274 such cases; this extends the previous catalogue of Brož et al. (2005) almost twice. The newly identified resonant objects are mainly asteroids discovered or recovered after 2005 with accurate enough orbits. We disregard from our analysis asteroids at the border of the resonance, for which *σ*(*t*) exhibits alternating periods of libration and circulation, and also those asteroids for which *σ* oscillates but are not resonant anyway (*N* ≤ 0.8 in Eq. ([resint]); see, e.g., Morbidelli & Moons 1993). The latter reside on low-eccentricity orbits in the main asteroid belt adjacent to the J2/1 resonance.\nWe conducted short- and long-term integrations of the resonant asteroids as described above. They allowed us to divide the population into 182 long-lived asteroids (with the median dynamical lifetime longer than 70Myr, as defined in Brož et al. 2005) and 92 short-lived asteroids (the lifetime shorter than 70Myr), see Fig. [r21obsloglftime4].[4](#fn4) Among the short-lived objects we found 14 have dynamical lifetimes even less than 2Myr and we call them extremely unstable. Brož et al. (2005) suggested the unstable orbits in the J2/1 resonance are resupplied from the adjacent main belt due to a permanent flux driven by the Yarkovsky force, the extremely unstable objects are most probably temporarily captured Jupiter-family comets. The origin of the long-lived population in this resonance is still not known.\n [r21obsloglftime4]\nFigure [r21proper] shows the pseudo-proper orbital elements of the J2/1 asteroids projected onto the (*a**p*, *e**p*) and (*a**p*, sin*i**p*) planes. Our data confirm that the unstable population of J2/1 asteroids populates the resonance outskirts near its separatrix, where several secular resonances overlap and trigger chaotic motion (e.g., Morbidelli & Moons 1993; Nesvorný & Ferraz-Mello 1997; Moons et al. 1998). At low-eccentricities the chaos is also caused by an overlap on numerous secondary resonances (e.g., Lemaitre & Henrard 1990). Two ‘islands’ of stability –A and B– harbour the long-lived population of bodies. The high-inclination island A, separated from the low-inclination island B by the *ν*16 secular resonance, is much less populated. Our current search identifies 9 asteroids in the island A. The origin of the asymmetry in A/B islands is not known, but since the work of Michtchenko & Ferraz-Mello (1997) and Ferraz-Mello et al. (1998a,b) it is suspected to be caused by instability due to the libration period commensurability with the forcing terms produced by the Great Inequality.\n84mm \n[r21proper]\nThe size-frequency distribution of objects of a population is an important property, complementing that of the orbital distribution. Figure [ressfd] shows cumulative distribution *N*( < *H*) of the absolute magnitudes *H* for bodies in the J2/1 (and other Jovian first order resonances as well). In between $H=12\\,{\\rm mag}$ and $14.5\\,{\\rm mag}$ (an approximate completeness limit; R. Jedicke, personal communication) it can be matched by a simple power-law *N*( < *H*) ∝ 10*γ**H*, with *γ* = ( + 0.70 ± 0.02).[5](#fn5) We thus confirm that the J2/1 population is steeper than it would correspond to a standard collisionally evolved system (e.g., Dohnanyi 1969; O’Brien & Greenberg 2003) with *γ* = + 0.5. The same result holds for both the short- and long-lived sub-populations in this resonance separately.\n[ressfd]\nAlbedos of J2/1 bodies are not known, except for (1362) Griqua for which Tedesco et al. (2002) give *p**V* = 0.067. The surroundings main-belt population has an average *p**V* = 0.05. For sake of simplicity we convert absolute magnitudes to sizes using this averaged value when needed. For instance in Fig. [r21arerirsizes] we show a zoom on the long-lived population of objects in the J2/1 resonance with symbol size weighted by the estimated size of the body. We note large objects are located far from each other and they are quite isolated — no small asteroids are in close surroundings. Both these observations suggest that the long-lived J2/1 population does not contain recently-born collisional clusters.\n \n [r21arerirsizes]\n84mm \n[r32proper]\n [r32aeescape]\n[resalb]\nHilda group\n-----------\nBecause asteroids in the J3/2 constitute a rather isolated group, it is easy to select their candidates: we simply extracted from the AstOrb database those asteroids with semimajor axis in between 3.8 AU and 4.1 AU. With that we obtained 1267 multi-opposition objects. We numerically integrated these orbits for 10kyr and analysed the behaviour of the resonance angle $\\sigma=3\\lambda'-2\\lambda-\\varpi$. We obtained 1197 cases for which *σ* librates about 0∘ and which have *N* ≥ 0.44, a threshold of the resonance zone (e.g., Morbidelli & Moons 1993); see Fig. [r32proper].\nThe long-term evolution of Hildas indicates that not all of them are stable over 4Gyr, but 20% escape earlier. A brief inspection of Fig. [r32aeescape] shows, that the escapees are essentially asteroids located closer to the outer separatrix and exhibiting large amplitudes of librations. If the Hilda group has been constituted during the planetary formation some 4 Gyr ago, some non-conservative process must have placed these objects onto their currently unstable orbits. We suspect mutual collisions or gravitational scattering on the largest Hilda members might be the corresponding diffusive mechanisms. Small enough members might be also susceptible to the resonant Yarkovsky effect (see Sec. [yarkovsky] and the Appendix A).\nData in Fig. [ressfd] confirm earlier findings that the Hilda group is characterized by an anomalously shallow size distribution. In between absolute magnitudes $H=10.5\\,{\\rm mag}$ and $14.5\\,{\\rm mag}$ the cumulative distribution can be well matched by *N*( < *H*) ∝ 10*γ**H* with *γ* = ( + 0.35 ± 0.02) only.\nSub-populations among Hilda asteroids, namely two collisional families, are studied in Section [fams].\nThule group\n-----------\nIn spite of a frequent terminology “Thule group”, asteroids in the J4/3 resonance consisted of a single object (279) Thule up to now. Nesvorný & Ferraz-Mello (1997) considered this situation anomalous because the extent of the stable zone of this resonance is not much smaller than that of the J3/2 resonance (see also Franklin et al. 2004). In the same way, our knowledge about the low-*e* and low-*i* Thule-type stable orbits (e.g., *a* = 4.27AU, *e* = 0.1 and *i* = 5∘) should be observationally complete at about magnitudes $H=12.5\\mbox{--}13\\,{\\rm mag}$ (R. Jedicke, personal communication). A rough estimate also shows that even one magnitude in *H* beyond this completeness limit the Thule population should be known at ∼ 10% completeness, leaving only about 90% undiscovered population. We thus conclude that the objects in the magnitude range $H=9\\mbox{--}13\\,{\\rm mag}$ are very likely missing in this resonance. Where does the existing population of small Thule-type asteroids begin?\nOur initial search in the broad box around the J4/3 resonance detected only 13 objects. Six of them, including the well-known extinct comet (3552) Don Quixote (e.g., Weissman et al. 2002), are on typical orbits of Jupiter-family comets that happen to reside near this resonance with very high eccentricity and moderately high inclination. Two more are single-opposition objects and one has only poorly constrained orbit, leaving us with (279) Thule and three additional candidate objects: (52007) 2002 EQ47, (186024) 2001 QG207 and (185290) 2006 UB219.\n140mm\n| | | | | | | | | | | |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n| No. | Name | *a**p* | *e**p* | sin*i**p* | *δ**a**p* | *δ**e**p* | *δ*sin*i**p* | *σ**p*, *m**a**x* | | |\n| [3pt] | | [AU] | | | [AU] | | | [deg] | | |\n| 279 | Thule | 4.2855 | 0.119 | 0.024 | 0.0005 | 0.012 | 0.003 | ∼ 50 | 8.57 | 126.6 |\n| 186024 | 2001 QG207 | 4.2965 | 0.244 | 0.042 | 0.0003 | 0.014 | 0.003 | $\\phantom{\\sim}$25 | 14.36 | 8.9 |\n| 185290 | 2006 UB219 | 4.2979 | 0.234 | 0.102 | 0.0003 | 0.014 | 0.004 | $\\phantom{\\sim}$25 | 13.75 | 11.8 |\n176mm\n| | | |\n| --- | --- | --- |\n| | | |\n| | | |\n[43reson]\nFigure [43reson] (top panels) shows short-term tracks of (279) Thule, (186024) 2001 QG207 and (185290) 2006 UB219 in resonant variables $\\sqrt{2\\Sigma}\\,(\\cos\\sigma,\\sin\\sigma) \\simeq (e\\cos\\sigma,e\\sin\\sigma)$ of the J4/3 resonance ($\\sigma=4\\lambda'-3\\lambda-\\varpi$ in this case). In all cases their orbits librate about the pericentric branch (*σ* = 0∘) of this resonance, although this is complicated –mainly in the low-eccentricity case of (279) Thule– by the forced terms due to Jupiter’s eccentricity (see, e.g., Ferraz-Mello 1988; Sessin & Bressane 1988; Tsuchida 1990). The leftmost panel recovers the 40∘--50∘ libration of (279) Thule, determined previously by Tsuchida (1990; Fig. 3). The other two smaller asteroids show librations with comparable amplitudes. The last object, (52007) 2002 EQ47, appears to reside on an unstable orbit outside the J4/3 resonance. Our search thus lead to the detection of two new asteroids in this resonance, increasing its population by a factor of three.[6](#fn6)\nResults of a long-term numerical integration of the nominal orbits plus 10 close clones, placed within an orbital uncertainty, reveal that the orbit of (279) Thule is stable over 4Gyr, but the orbits of (186024) 2001 QG207 and (185290) 2006 UB219 are partially unstable. They are not ‘short-lived’ but 45% and 60% of clones, respectively, escaped before 4Gyr. Figure [r43-lftat] shows pseudo-proper semimajor axis vs time for nominal orbits and their clones of all J4/3 objects; the escaping orbits leave the figure before the simulation was ended at 4Gyr. We suspect similar non-conservative effects as mentioned above for the 20% fraction of long-term-unstable Hildas to bring these two Thule members onto their marginally stable orbits.\n[r43-lftat]\nWe would like to point out that it took more than a century from the discovery of (279) Thule (Palisa 1888; Krueger 1889) until further objects in this resonance were finally discovered. This is because there is an anomalously large gap in size of these bodies: (279) Thule is 127 km in size with *p**V* = 0.04 AU (e.g., Tedesco et al. 2002), while the estimated sizes of (186024) 2001 QG207 and (185290) 2006 UB219 for the same value of albedo are 8.9km and 11.8km only. It will be interesting to learn as much as possible about the Thule population in the *H* = 13--15mag absolute magnitude range using future-generation survey projects such as Pan-STARRS (e.g., Jedicke et al. 2007). Such a completed population may present an interesting constraint on the planetesimal size distribution 4Gyr ago.\nCollisional families among Hilda asteroids\n==========================================\nCollisions and subsequent fragmentations are ubiquitous processes since planets formed in the Solar system. Because the characteristic dispersal velocities of the ejecta (as a rule of thumb equal to the escape velocity of the parent body) are usually smaller than the orbital velocity, the resulting fragments initially reside on nearby orbits. If the orbital chaoticity is not prohibitively large in the formation zone, we can recognise the outcome of such past fragmentations as distinct clusters in the space of sufficiently stable orbital elements. More than 30 collisional families are known and studied in the main asteroid belt (e.g., Zappalà et al. 2002) with important additions in the recent years (e.g., Nesvorný et al. 2002; Nesvorný, Vokrouhlický & Bottke 2006). Similarly, collisional families have been found among the Trojan clouds of Jupiter (e.g., Milani 1993; Beaugé & Roig 2001; Roig et al. 2008), irregular satellites of Jupiter (e.g., Nesvorný et al. 2003, 2004) and even trans-Neptunian objects (e.g., Brown et al. 2007). Mean motion resonances, other than the Trojan librators of Jupiter, are typically too chaotic to hold stable asteroid populations, or the populations were too small to enable search for families. The only remaining candidate populations are those in the Jovian first order resonances, with Hilda asteroids the most promising group. However, low expectations for an existence of collisional families likely de-motivated systematic search. Note, the estimated intrinsic collisional probability of Hilda asteroids is about a factor 3 smaller than in the main asteroid belt (e.g., Dahlgren 1998; Dell’Oro et al. 2001) and the population is more than two orders of magnitude smaller.\nIn spite of the situation outlined above, Schubart (1982a, 1991) repeatedly noticed groups of Hilda-type asteroids with very similar proper elements. For instance, in his 1991 paper he lists 5 members of what we call Schubart family below and pointed out their nearly identical values of the proper inclination. Already in his 1982 paper Schubart mentions a similarity of such clusters to Hirayama families, but later never got back to the topic to investigate this problem with sufficient amount of data provided by the growing knowledge about the J3/2 population.[7](#fn7) Even a zero order inspection of Fig. [r32proper], in particular the bottom panel, implies the existence of two large clusters among the J3/2 population. In what follows we pay a closer analysis to both of them.\nWe adopt an approach similar to the hierarchical-clustering method (HCM) frequently used for identification of the asteroid families in the main belt (e.g., Zappalà et al. 1990, 1994, 2002). In the first step of our analysis, we compute the number of bodies $N\\_{\\rm min}$ which is assumed to constitute a statistically significant cluster for a given value of the cutoff velocity $v\\_{\\rm cutoff}$. We use a similar approach to that of Beaugé & Roig (2001): for all asteroids in the J3/2 resonance we determine the number $N\\_i(v\\_{\\rm cutoff})$ of asteroids which are closer than $v\\_{\\rm cutoff}$. Then we compute the average value $N\\_0 = \\bar N\\_i$. According to Zappalà et al. (1994), a cluster may be considered significant if $N > N\\_{\\rm min}\n= N\\_0 + 2\\sqrt{N\\_0}$. The plots $N\\_0(v\\_{\\rm cutoff})$ and the corresponding $N\\_{\\rm min}(v\\_{\\rm cutoff})$ for Hilda population are shown in Fig. [Nmin]. We use a standard metric (*d*1 defined by Zappalà et al., 1994), namely:\n \n$$\\delta v = na\\_p \\sqrt{{5\\over 4} \\left({\\delta a\\_p\\over a\\_p}\n \\right)^2 + 2\\, (\\delta e\\_p)^2 + 2\\, (\\delta \\sin i\\_p)^2} \\;,\n \\label{metric}$$ \nwhere (*a**p*, *e**p*, sin*i**p*) are 10 Myr averaged values of the resonant pseudo-proper elements (we checked that our results practically do not depend on the width of this averaging interval).\n[Nmin]\nNext, we construct a stalactite diagram for Hildas in a traditional way (e.g., Zappalà et al. 1990): we start with (153) Hilda as the first central body and we find all bodies associated with it at $v\\_{\\rm cutoff} = 300\\,{\\rm m}/{\\rm s}$, using a hierarchical clustering method (HCM; Zappalà et al. 1990, 1994). Then we select the asteroid with the lowest number (catalogue designation) from remaining (not associated) asteroids and repeat the HCM association again and again, until no asteroids are left. Then we repeat the whole procedure recursively for all clusters detected at $v\\_{\\rm cutoff} = 300\\,\n{\\rm m}/{\\rm s}$, but now for a lower value, e.g., $v\\_{\\rm cutoff} = \n299\\,{\\rm m}/{\\rm s}$. We may continue until $v\\_{\\rm cutoff} = 0\\,\n{\\rm m}/{\\rm s}$, but of course, for too low values of the cutoff velocity, no clusters can be detected and all asteroids are single. The resulting stalactite diagram at Fig. [r32-hcmstalactitediagrammin5] is simply the asteroid number (designation) vs $v\\_{\\rm cutoff}$ plot: a dot at a given place is plotted only if the asteroids belongs to a cluster of at least ${\\rm max}(5, N\\_{\\rm min}(v\\_{\\rm cutoff}))$ bodies. We are not interested in clusters with less than 5 members; they are most probably random flukes.\nWe can see two prominent clusters among Hildas: the first one around the asteroid (153) Hilda itself, and the second one around (1911) Schubart. In the remaining part of this Section we discuss each of them separately.\n [r32-hcmstalactitediagrammin5]\nThe stalactite diagram constructed in the same way for Zhongguos and Griquas is shown in Fig. [r21-hcm2007stalactitediagrammin5]. No grouping seem to be significant enough to be considered an impact-generated cluster. This is consistent with the discussion of the (*a**p*, *e**p*, sin*i**p*) plots in Section [hecuba].\n [r21-hcm2007stalactitediagrammin5]\nSchubart family\n---------------\nThe Schubart family can be distinguished from the remaining population of Hildas on a large range of cutoff velocities: from 50m/s to more than 100m/s (Fig. [r32-hcmstalactitediagrammin5]). It merges with the Hilda family at 200m/s. For the purpose of our analysis we selected $v\\_{\\rm cutoff} = 60\\,{\\rm m}/{\\rm s}$ as the nominal value. While the total number of Schubart family members is not too sensitive to this cutoff value, we refrain from using too high $v\\_{\\rm cutoff}$, for which we would expect and increasing number of interlopers to be associated with the family, and the family would attain a rather peculiar shape in the (*a**p*, *e**p*, sin*i**p*) space.\nFigure [r32absmagdistribution] shows the cumulative distribution of the absolute magnitudes for the Schubart family members, compared to the rest of the J3/2 population. Importantly, the slope *γ* = ( + 0.48 ± 0.02) of the *N*( < *H*) ∝ 10*γ**H* fit is quite *steeper* for the Schubart family, which supports the hypothesis of its collisional origin.\n [r32absmagdistribution]\nWe also analysed the available SDSS catalogue of moving objects (ADR3; Ivezić et al. 2002). We searched for the J3/2 asteroids among the entries of this catalogue and computed the principal component PC1 of the spectrum in the visible band. Note the PC1 value is an indicator of the spectral slope and allows thus to broadly distinguish principal spectral classes of asteroids (e.g., Bus et al. 2002). Figure [r32woschubart.astorb.sloanpc1lt0.1hist] shows our results. The top panel confirms the bimodal character of the J3/2 population (see also Dahlgren et al. 1997, 1998, 1999 and Gil-Hutton & Brunini 2008). More importantly, though, the bottom panel indicates a spectral homogeneity of the Schubart family, placing all members within the C/X taxonomy class branch. This finding strongly supports collisional origin of the Schubart family.\n [r32woschubart.astorb.sloanpc1lt0.1hist]\nTedesco et al. (2002) derive *D* = 80km size for (1911) Schubart, corresponding to a very low albedo *p**V* = 0.025. The same authors determine *D* = 38km size of (4230) van den Bergh and exactly the same albedo; this asteroid is among the five largest in the family. Assuming the same albedo for all other family members, we can construct a size-frequency distribution (Fig. [r32sizedistribution]). The slope *α* ≃ ( − 2.7 ± 0.1) fitted to the small end of the distribution, where we still assume observational completeness, is rather shallow, but marginally within the limits of population slopes produced in the numerical simulations of disruptions (e.g., Durda et al. 2007).[8](#fn8)\n[r32sizedistribution]\nIf we sum the volumes of the observed members, we end up with a lower limit for the parent body size $D\\_{\\rm PB} = 110\\,{\\rm km}$, provided there are no interlopers. We can also estimate the contribution of small (unobserved) bodies using the following simple method: (i) we sum only the volumes of the observed bodies larger than an assumed completeness limit $D\\_{\\rm complete} = 10\\,{\\rm km}$ ($V\\_{\\rm complete} = \\sum\\_i {\\pi\\over 6} D\\_i^3$); (ii) we fit the cumulative size distribution by a power law ($\\log N({>}D) = \\alpha\\log [D]\\_{\\rm km} + \\beta$; *α* = − 2.68, *β* = 4.73 for the Schubart); (iii) we prolong this slope from $D\\_{\\rm complete}$ down to $D\\_{\\rm min} = 0$ and calculate the total volume of the parent body (provided *α* > − 3):\n \n$$V\\_{\\rm PB} = V\\_{\\rm complete} + {\\pi\\over 6} 10^\\beta {\\alpha\\over\\alpha+3}\n \\left[ D\\_{\\rm min}^{a+3} - D\\_{\\rm complete}^{a+3} \\right]\\;.$$ \nThe result is $D\\_{\\rm PB} = \\sqrt[3]{{6\\over\\pi} V\\_{\\rm PB}} \\doteq 130\\,{\\rm km}$, some sort of an upper limit. The volumetric ratio between the largest fragment and the parent body is then $V\\_{\\rm LF}/V\\_{\\rm PB} \\doteq 0.2$, a fairly typical value for asteroid families in the main asteroid belt. Obviously, the assumption of a single power-law extrapolation of the *N*( > *D*) at small sizes is only approximate and can lead to a result with a 10% uncertainty. However, if we use an entirely different geometric method, developed by Tanga et al. (1998), we obtain $D\\_{\\rm PB}\\simeq 120\\mbox{--}130$km, i.e., comparable to our previous estimate.\nWhat is an approximate size $d\\_{\\rm disrupt}$ of a projectile necessary to disrupt the parent body of the Schubart family? Using Eq. (1) from Bottke et al. (2005):\n \n$$d\\_{\\rm disrupt} = \\left(2 Q\\_D^\\* / V\\_{\\rm imp}^2\\right)^{1\\over 3}\n D\\_{\\rm target} \\;. \\label{d\\_disrupt}$$ \nand substituting $Q\\_D^\\* = 10^{5}\\,{\\rm J}/{\\rm kg}$ for the strength (somewhat lower than that of basaltic objects to accommodate the assumed C/X spectral type; e.g., Kenyon et al. 2008 and references therein), $V\\_{\\rm imp} = 4.78\\,{\\rm km}/{\\rm s}$ for the typical impact velocity (see Dahlgren 1998) and $D\\_{\\rm target} \\simeq 130\\,{\\rm km}$, we obtain $d\\_{\\rm disrupt} \\simeq 25\\,{\\rm km}$. At this size the projectile population is dominated by main belt bodies. Considering also different intrinsic collisional probabilities between Hilda–Hilda asteroids ($2.3\\times 10^{-18}\\,{\\rm km}^{-2}\\,{\\rm yr}^{-1}$; Dahlgren 1998) and Hilda–main belt asteroids ($0.6\\times 10^{-18}\\,{\\rm km}^{-2}\\,{\\rm yr}^{-1}$), we find it more likely the Schubart family parent body was hit by a projectile originating from the main belt.\nHilda family\n------------\nWe repeated the same analysis as in Section [schubartfamily] for the Hilda family. The family remains statistically distinct from the whole J3/2 population in the range of cutoff velocities $(130, 170)\\,\n{\\rm m}/{\\rm s}$; we choose $v\\_{\\rm cutoff} = 150\\,{\\rm m}/{\\rm s}$ as the nominal value.\nThe slope *γ* of the cumulative absolute magnitude distribution *N*( < *H*) is ( + 0.50 ± 0.02) (Fig. [r32absmagdistribution]), again steeper than for the total J3/2 population and comparable to that of the Schubart family. The spectral slopes (PC1) are somewhat spread from flat (C/X-compatible values; PC1 < 0.3) to redder (D-compatible values; PC1 > 0.3) — see Fig. [r32woschubart.astorb.sloanpc1lt0.1hist]. Overall, though, the C/X members prevail such that the D-type objects might be actually interlopers, at least according to a simple estimate based on the volume of the Hilda family in the (*a**p*, *e**p*, sin*i**p*) space, compared to the total volume of the J3/2 population.\nTedesco et al. (2002) determine albedos for six family members. They range from 0.037 to 0.087, but three values are close to the median albedo 0.044 of all J3/2 asteroids. We thus consider this value to be representative of the Hilda family. The corresponding cumulative size distribution is plotted in Fig. [r32sizedistribution]. Using the same method as in Section [schubartfamily] we estimate the size of the parent body $D\\_{\\rm PB} = 180\\mbox{--}190\\,{\\rm km}$, with $V\\_{\\rm LF}/V\\_{\\rm PB} \\simeq 0.8$. With the model of Tanga et al. (1999) we would obtain $D\\_{\\rm PB} \\simeq 210$km and thus $V\\_{\\rm LF}/V\\_{\\rm PB}\\simeq 0.5$. This family forming event seems to be thus characterized in between the catastrophic disruption and a huge cratering. The necessary projectile size is $d\\_{\\rm disrupt} = 50\\mbox{--}55$km.\nWhile not so prominent as the Schubart family, we consider the group of asteroids around Hilda a fairly robust case of a collisionally-born family too.\n [r32-impact2aeinicond]\n [r32-impact4f0f135ae]\nSimulated disruption events\n---------------------------\nIn order to asses some limits for the age of the Schubart and Hilda families, we perform a number of numerical tests. In particular, we simulate a disruption of a parent body inside the resonance and numerically determine the long-term orbital evolution of fragments. The evolved synthetic family at different time steps is then compared with the observed family. Ideally, this approach should allow to constrain the time elapsed since the family formed.\nAs a first step, we need to create a synthetic family inside the resonance. We use current orbital elements of the largest family member, (1911) Schubart in this case, as representative to the parent body and only allow changes in the true anomaly *f* and in the argument of pericentre *ω* at the break-up event. By changing these two geometric parameters we can produce different initial positions of the fragments in the orbital element space. For sake of our test, fragments are assumed to be dispersed isotropically with respect to the parent body, with a velocity distribution given by the model of Farinella et al. (1993, 1994). The number of fragments *d**N*(*v*) launched with relative velocities in the interval (*v*, *v* + *d**v*) is given by\n \n$$dN(v) = C v (v^2 + v\\_{\\rm esc}^2)^{-(\\kappa+1)/2}\\, dv\\;,$$ \nwith *C* a normalization constant, $v\\_{\\rm esc}$ the escape velocity from the parent body and *κ* = 3.25. To prevent excessive escape velocities we introduce a maximum allowed value $v\\_{\\rm max}$. Nominally, we set $v\\_{\\rm max}=200$ m/s, but in the next Section [yarkovskye] we also use restricted values of this parameter to test sensitivity of our results to initial conditions.\nTo simulate an impact that might have created the Schubart family, we generated velocities randomly for 139 fragments with $v\\_{\\rm esc} = 65\\,{\\rm m}/{\\rm s}$ (note the number of fragments in the synthetic family is equal to the number of the Schubart family members). The resulting swarm of fragments is shown in Fig. [r32-impact2aeinicond], for the impact geometry *f* = 0∘ and *ω* + *f* = 180∘. We show both the initial osculating orbital elements and the pseudo-proper elements.\nThe synthetic family extends over significantly larger range of the semimajor axis than the observed Schubart family, but all fragments still fall within the J3/2 resonance. The eccentricity distribution is, on the other hand, substantially more compact. Only the distribution of inclinations of the synthetic family roughly matches that of the observed family. We verified this holds also for other isotropic-impact geometries (such as *f* = 135∘ and *ω* + *f* = 180∘ shown in Fig. [r32-impact4f0f135ae]). The peculiar shape of the synthetic family in the pseudo-proper element space (*a**p*, *e**p*) is an outcome of the isotropic disruption, simply because some fragments fall to the left from the libration centre of the J3/2 resonance (at 3.97AU) and they are ‘mapped’ to the right. This is because the pseudo-proper elements are the maxima and minima of *a* and *e*, respectively, over their resonant oscillations.\n [r32-impact2ae4Gyr]\nThe initial configuration of the synthetic family was propagated for 4Gyr, using the integrator described in Section [popul]. At this stage, we use only the gravitational perturbations from the 4 exterior giant planets. We performed such simulation for several impact geometries, as determined by *f* and *ω*, with similar results.\nFigure [r32-impact2ae4Gyr] shows the long-term evolution of the synthetic family. Because the family resides mostly in the stable zone of the J3/2 resonance, only little evolution can be seen for most of the bodies. This is in accord with findings of Nesvorný & Ferraz-Mello (1997) who concluded that the stable region in this resonance shows little or no diffusion over time scales comparable to the age of the Solar system. Only about 10% of orbits that initially started at the outskirts of the stable zone (with large libration amplitudes) escaped from the resonance during the 4Gyr simulation.\nThe removal of orbits with large semimajor axis *a**p* helps in part to reconcile the mismatch with the distribution of the observed Schubart family. However, the dispersion in eccentricity *e**p* does not evolve much and it still shows large mismatch if compared to the observed family. Even in the case *f* = 135∘ (*ω* + *f* = 180∘; Fig. [r32-impact4f0f135ae]), which maximizes the initial eccentricity dispersion of the synthetic fragments, the final value at 4Gyr is about three times smaller than that of the Schubart family. Clearly, our model is missing a key element to reproduce the current orbital configuration of this family.\nOne possibility to resolve the problem could be to release the assumption of an isotropic impact and explore anisotropies in the initial velocity field. This is an obvious suspect in all attempts to reconstruct orbital configurations of the asteroid families, but we doubt it might help much in this case. Exceedingly large relative velocities, compared to the escape velocity of the estimated parent body, would be required. Recall, the fragments located in the stable region of the J3/2 resonance would hardly evolve over the age of the Solar system.\nA more radical solution is to complement the force model, used for the long-term propagation, by additional effects. The only viable mechanism for the size range we are dealing with is the Yarkovsky effect. This tiny force, due to anisotropic thermal emission, has been proved to have determining role in understanding fine structures of the asteroid families in the main belt (e.g., Bottke et al. 2001; Vokrouhlický et al. 2006a,b). In these applications the Yarkovsky effect produces a steady drift of the semimajor axis, leaving other orbital elements basically constant.\nHowever, the situation is different for resonant orbits. The semimajor axis evolution is locked by the strong gravitational influence of Jupiter. For that reason we first ran simplified simulations with the Yarkovsky forces — results of these tests are briefly described in the Appendix A. We next applied the model containing both gravitational and Yarkovsky perturbations to the evolution of the synthetic family. Results of these experiments are described in the next Section [yarkovskye].\nYarkovsky drift in eccentricity\n-------------------------------\nWe ran our previous simulation of the long-term evolution of the synthetic family with the Yarkovsky forces included. Our best guess of thermal parameters for bodies of the C/X type is: $\\rho\\_{\\rm s} = \\rho\\_{\\rm b} = 1300\\,{\\rm kg}/{\\rm m}^3$ for the surface and bulk densities, $K = 0.01\\,{\\rm W}/{\\rm m}/{\\rm K}$ for the surface thermal conductivity, $C = 680\\,{\\rm J}/{\\rm kg}$ for the heat capacity, *A* = 0.02 for the Bond albedo and *ε* = 0.95 for the thermal emissivity parameter. Rotation periods are bound in the 2–12hours range. Spin axes orientations are assumed isotropic in space. Finally, we assign sizes to our test particles equal to the estimate of sizes for Schubart family members, based on their reported absolute magnitudes and albedo *p**V* = 0.025. The dependence of the Yarkovsky force on these parameters is described, e.g., in Bottke et al. (2002, 2006). We note the uncertainties of the thermal parameters, assigned to individual bodies, do not affect our results significantly, mainly because we simulate a collective evolution of more than 100 bodies; we are not interested in evolution of individual orbits.\n [r32-impye2ae1Gyr]\nWe let the synthetic family evolve for 4Gyr and recorded its snapshots every 50 kyr. As discussed in the Appendix A, the *resonant Yarkovsky effect* produces mainly secular changes of eccentricity. We recall this systematic drift in *e* must not be confused with the chaotic diffusion in *e*. We also note that inclination of the orbits remains stable, in accord with a good match of the Schubart family by the initial inclination distribution.\nBecause the initial eccentricity dispersion of the synthetic family is much smaller than that of the observed one, its steady increase due to the combined effects of the Yarkovsky forces and the resonant lock gives us a possibility to date the origin of the family (see Vokrouhlický et al. 2006a for a similar method applied to families in the main belt). To proceed in a quantitative way, we use a 1-dimensional Kolmogorov–Smirnov (KS) test to compare cumulative distribution of pseudo-proper eccentricity values *e**p* of the observed and synthetic families (e.g., Press et al. 2007).\nFigure [schubartkstest1D] shows the KS distance *D**K**S* of the two eccentricity distributions as a function of time. For sake of a test, we also use smaller $v\\_{\\rm max}$ values of the initial velocity field (essentially, this is like to start with a more compact synthetic family). Regardless of the $v\\_{\\rm max}$ value, our model rejects Schubart family ages smaller than 1Gyr and larger than 2.4Gyr with a 99% confidence level. For ages in between 1.5Gyr and 1.7Gyr the KS-tested likelihood of a similarity of the synthetic-family and the observed-family *e**p* distributions can reach up to 50%. We thus conclude the most likely age of the Schubart cluster is $(1.7\\pm 0.7)\\,{\\rm Gyr}$.\n [schubartkstest1D]\n [hildakstest1D]\nWe repeated the same analysis for Hilda family by creating a synthetic family of 233 particles about (153) Hilda. In this case we used $v\\_{\\rm esc}=110\\,{\\rm m}/{\\rm s}$. The situation is actually very similar to the Schubart — there is again a problem with the small dispersion of eccentricities in case of a purely gravitational model. Using the model with the Yarkovsky effect, we can eventually fit the spread of eccentricities and according to the KS test (Fig. [hildakstest1D]) the age of the family might be $\\gtrsim 4$Gyr. The match is still not perfect, but this problem might be partly due to numerous interlopers in the family. We also note that a 10% relative uncertainty of the mean albedo of the Hilda family members would lead to a 5% uncertainty of their sizes and, consequently, to a 5% uncertainty of the family age.\nThe Hilda family seems to be dated back to the Late Heavy Bombardment era (Morbidelli et al. 2005; see also Sec. [migration]). We would find such solution satisfactory, because the population of putative projectiles was substantially more numerous than today (note that a disruption of the Hilda family parent body is a very unlikely event during the last 3.5Gyr).\nWe finally simulated a putative collision in the J2/1 resonance, around the asteroid (3789) Zhongguo (the largest asteroid in the stable island B). There are two major differences as compared to the J3/2 resonance: (i) the underlying chaotic diffusion due to the gravitational perturbations is larger in the J2/1 resonance (e.g., Nesvorný & Ferraz-Mello 1997), such that an initially compact cluster would fill the whole stable region in 1–1.5Gyr and consequently becomes unobservable; (ii) sizes of the observed asteroids are generally smaller, which together with a slightly smaller heliocentric distance, accelerates the Yarkovsky drift in *e**p*. The latter effect would likely shorten the time scale to 0.5–1Gyr. Thus the non-existence of any significant orbital clusters in the J2/1 resonance (Sec. [fams] and Fig. [r21-hcm2007stalactitediagrammin5]) does not exclude a collisional origin of the long-lived population by an event older than 1Gyr. This would also solve the apparent problem of the very steep size distribution of the stable J2/1 population (Brož et al. 2005 and Fig. [ressfd]). Note the expected collisional lifetime of the smallest observed J2/1 asteroids is several Gyr (e.g., Bottke et al. 2005).\nResonant population stability with respect to planetary migration and the Yarkovsky effect\n==========================================================================================\nWe finally pay a brief attention to the overall stability of asteroid populations in the first-order resonances with respect to different configurations of giant planets. We are focusing on the situations when the orbits of Jupiter and Saturn become resonant. This is motivated by currently adopted views about final stages of building planetary orbits architecture, namely planet migration in a diluted planetesimal disk (e.g., Malhotra 1995; Hahn & Malhotra 1999; Tsiganis et al. 2005; Morbidelli et al. 2005; Gomes et al. 2005 – these last three references are usually described as the Nice model). Morbidelli et al. (2005) proved that the primordial Trojan asteroids were destabilized when Jupiter and Saturn crossed their mutual 1:2 mean motion resonance and, at the same time, the Trojan region was re-populated by particles of the planetesimal disk. Since the mutual 1:2 resonance of Jupiter and Saturn plays a central role in the Nice model, and since these two planets had to cross other (weaker) mutual resonances such as 4:9 and 3:7 before they acquired today’s orbits, one can naturally pose a question about the stability of primordial populations in the first-order mean motion resonances with Jupiter. Ferraz-Mello et al. (1998a,b) demonstrated that even subtler effects can influence the J2/1 population, namely the resonances between the asteroid libration period in the J2/1 resonance and the period of the Great Inequality (GI) terms in planetary perturbations (i.e., those associated with Jupiter and Saturn proximity to their mutual 2:5 mean motion resonance; Fig. [shiftedSaturnperiodGI]). A first glimpse to the stability of the first-order resonance populations with respect to these effects is given in Section [migration].\nIn Section [yarkovsky] we also briefly estimate the change of dynamical lifetimes for small J2/1 and J3/2 bodies caused by the Yarkovsky effect.\n[shiftedSaturnperiodGI]\nPlanetary migration effects\n---------------------------\nIn what follows we use a simple approach by only moving Saturn’s orbit into different resonance configurations with Jupiter’s orbit. We do not let orbits of these planets migrate, but consider them static. With such a crude approach we can only get a first hint about a relative role of depletion of the asteroid populations in the first-order resonances (note in reality the planets undergo steady, but likely not smooth, migration and exhibit jumps over different mutual resonant states; e.g., supplementary materials of Tsiganis et al. 2005).\n \n \n [r213243-jsloglftime]\nThe results are summarised in Fig. [r213243-jsloglftime]. (i) The Hilda group in the J3/2 resonance is very unstable (on the time scale $\\sim\\!1\\,{\\rm Myr}$) with respect to the 1:2 Jupiter–Saturn resonance;[9](#fn9) on the contrary J2/1 asteroids may survive several 10Myr in this configuration of Jupiter and Saturn, so this population is not much affected by this phase by planetary evolution (note, moreover, that Jupiter and Saturn would likely cross the zone of other mutual 1:2 resonance in $\\sim\\!1\\,{\\rm Myr}$ only; e.g., Tsiganis et al. 2005 and Morbidelli et al. 2005). (ii) The 4:9 resonance has a larger influence on the J2/1 population than on Hildas. (iii) In the case of 3:7 resonance it is the opposite: the J3/2 is more unstable than the J2/1. (iv) The Great Inequality resonance does indeed destabilise the J2/1 on a time scale 50Myr. Provided the last phases of the migration proceed very slowly, it may cause a significant depletion of the primordial J2/1 population. In the exact 2:5 resonance, the J2/1 population would not be affected at all.\nAccording to our preliminary tests with a more complete *N*-body model for planetary migration which includes a disk of 103 planetesimals beyond Neptune, the strong instability of the J3/2 asteroids indeed occurs during the Jupiter–Saturn 1:2 resonance crossing (see Fig. [j32pdisc50MEcoldI00jatplanets]). A vast reservoir of planetesimals residing beyond the giant planets, and to some extent also nearby regions of the outer asteroid belt, are probably capable to re-populate the J3/2 resonance zone at the same time and form the currently observed populations. This is similar to the Trojan clouds of Jupiter (Morbidelli et al. 2005).\n \n[j32pdisc50MEcoldI00jatplanets]\nThe Yarkovsky effect\n--------------------\nIn Section [yarkovskye] we already discussed the influence of the Yarkovsky effect on the families located inside the J3/2 resonance. In course of time it modified eccentricities of their members, but did not cause a large-scale instability; the families remained inside the resonance all the time. Here we seek the size threshold for which the Yarkovsky would case overall instability by quickly removing the bodies from the resonance.\nWe perform the following numerical test: we multiply sizes of the long-lived J2/1 objects by fudge factors 0.2, 0.02 and 0.002, for which the Yarkovsky effect is stronger, and compare respective dynamical lifetimes with the original long-lived objects. Results are summarized in Fig. [r21zgyloglftimeradii]. We can conclude that a significant destabilisation of the J2/1 resonant population occurs for sizes ∼ 0.1km and smaller (provided the nominal population has sizes mostly 4–12km).\nWe do not include the YORP effect (i.e., the torque induced by the infrared thermal emission) at this stage. The YORP is nevertheless theoretically capable to significantly decelerate (or accelerate) the rotation rate, especially of the smallest asteroids, which can lead to random reorientations of the spin axes due to collisions, because the angular momentum is low in this spin-down state. These reorientations can be simulated by a Monte-Carlo model with a typical time scale (Čapek & Vokrouhlický 2004; *R* is radius in kilometres and *a* orbital semimajor axis in AU):\n \n$$\\tau\\_{\\rm YORP} \\simeq 25\\,{\\rm Myr}\\, (R/1\\,{\\rm km})^2\\,\n (2.5\\,{\\rm AU}/a)^2\\;. \\label{tau\\_YORP}$$ \nSince the Yarkovsky effect depends on the obliquity value, the systematic drift would be changed to a random walk for bodies whose spin axis undergo frequent re-orientations by the YORP effect. We can thus expect that the YORP effect might significantly prolong dynamical lifetimes of resonant objects with sizes ∼ 0.1km or smaller, because $\\tau\\_{\\rm YORP} < 0.25\\,{\\rm Myr}$ for them.\n [r21zgyloglftimeradii]\nWe can also check, which orientation of the spin axis makes the escape from the J2/1 resonance more likely to happen. We consider 0.08–0.24km bodies, clone them 5 times and assign them different values of the obliquity *γ* = 0∘, 45∘, 90∘, 135∘ and 180∘. Figure [r21zgyloglftimeobliq0.01] shows clearly that the retrograde rotation increases the probability of the escape. This is consistent with the structure of the J2/1 resonance, for which low-*a* separatrix does not continue to *e* = 0. We conclude the remaining very small (yet unobservable) asteroids inside the J2/1 may exhibit a preferential prograde rotation.\n [r21zgyloglftimeobliq0.01]\nWe perform a similar simulation for the J3/2 population (first 100 bodies with sizes 10–60km), but now with YORP reorientations (Eq. [tauYORP]) included. The results (Fig. [r32zgyloglftimeradii]) show the J3/2 population is much less affected than the J2/1 by the Yarkovsky/YORP perturbation.\nWe conclude the Yarkovsky/YORP effect may destabilize the J2/1 and J3/2 bodies only partially on the 100Myr time scale and only for sizes smaller than ∼ 0.1km. It is obviously a remote goal to verify this conclusion by observations (note the smallest bodies in these resonances have several kilometers size). Nevertheless, the dynamical lifetimes of small asteroids determined in this section, might be useful for collisional models of asteroid populations, which include also dynamical removal.\n [r32zgyloglftimeradii]\nConclusions and future work\n===========================\nThe main results of this paper can be summarised as follows: (i) we provided an update of the observed J2/1, J3/2 and J4/3 resonant populations; (ii) we discovered two new objects in the J4/3 resonance; (iii) we described two asteroid families located inside the J3/2 resonance (Schubart and Hilda) and provided an evidence that they are of a collisional origin; (iv) we reported a new mechanism how the Yarkovsky effect systematically changes *eccentricities* of resonant asteroids; we used this phenomenon to estimate the ages of the Schubart and Hilda families ((1.7 ± 0.7)Gyr and $\\gtrsim 4$Gyr respectively); (v) collisionally-born asteroid clusters in the stable region of J2/1 would disperse in about 1Gyr; (vi) 20% of Hildas may escape from the J3/2 resonance within 4Gyr in the current configuration of planets; (vii) Hildas are strongly unstable when Jupiter and Saturn cross their mutual 1:2 mean motion resonance.\nThe J3/2 resonance is a unique ‘laboratory’ — the chaotic diffusion is so weak, that families almost do not disperse in eccentricity and inclination due to this effect over the age of the Solar system. What is even more important, they almost do not disperse in semimajor axis, even thought the Yarkovsky effect operates. The drift in *a* is transformed to a drift in *e*, due to a strong gravitational coupling with Jupiter. We emphasize, this is is not a chaotic diffusion in *e*, but a *systematic drift in *e**.\nAnother piece of information about the families in J3/2 resonance is hidden in the eccentricity *e**p* vs absolute magnitude *H* plots (see Fig. [hildaeH] for the Hilda family). The triangular shape (larger eccentricity dispersion of the family members for larger *H*) is a well-known combination of two effects: (i) larger ejection speed, and (ii) faster dispersal by the Yarkovky forces for smaller fragments. Interestingly, there is also a noticeable depletion of small bodies in the centre of the family and their concentration at the outskirts — a phenomenon known from (*a*, *H*) plots of main-belt families, which was interpreted as an interplay between the Yarkovsky and YORP effects (Vokrouhlický et al. 2006).[10](#fn10) Indeed the estimated ∼ 4Gyr age for this family matches the time scale of a YORP cycle for *D* ≃ 10km asteroids in the Hilda region (e.g., Vokrouhlický & Čapek 2002; Čapek & Vokrouhlický 2004). Vokrouhlický et al. (2006a) pointed out that this circumstance makes the uneven distribution of family members most pronounced.\n[hildaeH]\nWe postpone the following topics for the future work: (i) a more precise age determination for the resonant asteroid families, based on the Yarkovsky/YORP evolution in the (*e*, *H*) space; (ii) a more detailed modelling of analytic or *N*-body migration of planets and its influence on the stability of resonant populations.\nAcknowledgements\n================\nWe thank David Nesvorný, Alessandro Morbidelli and William F. Bottke for valuable discussions on the subject and also Rodney Gomes for a constructive review. The work has been supported by the Grant Agency of the Czech Republic (grants 205/08/0064 and 205/08/P196) and the Research Program MSM0021620860 of the Czech Ministry of Education. 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S., 2006b, Icarus, 182, 92 Weissman P.R., Bottke W.F., Levison H.L., 2002, in Bottke W.F., Cellino A., Paolicchi P., Binzel R.P., eds., Asteroids III. University of Arizona Press, Tucson, p. 669 Zappalà V., Cellino A., Farinella P., Knežević Z., 1990, AJ, 100, 2030 Zappalà V., Cellino A., Farinella P., Milani A., 1994, AJ, 107, 772 Zappalà V., Cellino A., Dell’Oro A., Paolicchi P., 2002, in Bottke W.F., Cellino A., Paolicchi P., Binzel R.P., eds., Asteroids III. University of Arizona Press, Tucson, p. 619\nResonant Yarkovsky effect\n=========================\n| | | |\n| --- | --- | --- |\n| | | |\n[r32-rtbpfilterSsigma100Myr]\nThe effects of weak dissipative forces, such as the tidal force, gas-drag force and the Poynting-Robertson force, on both non-resonant and resonant orbits were extensively studied in the past (e.g., Murray & Dermott 1999 and references therein). Interaction of the Yarkovsky drifting orbits with high-order, weak resonances was also numerically studied to some extent (e.g., Vokrouhlický & Brož 2002) but no systematic effort was paid to study Yarkovsky evolving orbits in strong low-order resonances. Here we do not intend to develop a detailed theory, rather give a numerical example that can both help to explain results presented in the main text and motivate a more thorough analytical theory.\n*The Yarkovsky effect outside the resonance.* We constructed the following simple numerical experiment: we took the current orbit of (1911) Schubart as a starting condition and integrated the motion of two 0.1km size objects with extreme obliquity values 0∘ and 180∘. Their thermal parameters were the same as in Section [yarkovskye]. Because the diurnal variant of the Yarkovsky effect dominates the evolution, the extreme obliquities would mean the two test bodies would normally (outside any resonances) drift in semimajor axis in two opposite directions (e.g., Bottke et al. 2002, 2006). The two orbits would secularly acquire Δ*a* ≃ + 0.25AU or − 0.25AU in 100 My, about the extent shown by the arrow on top of the left panel of Fig. [r32-rtbpfilterSsigma100Myr]. Since the strength of the Yarkovsky forces is inversely proportional to the size, we can readily scale the results for larger bodies.\n*The resonance without the Yarkovsky effect.* If we include gravitational perturbations by Jupiter only, within a restricted circular three body problem ($e\\_{\\rm J} = 0$), and remove short-period oscillations by a digital filter, the parameter *N* from Eq. ([resint]) would stay constant. The orbit would be characterized by a stable libration in (Σ, *σ*) variables with about 30∘ amplitude in *σ* (see the curve labelled 0Myr in the middle panel of Fig. [r32-rtbpfilterSsigma100Myr]).\nWhile evolving, some parameters known as the adiabatic invariants are approximately conserved (see, e.g., Landau & Lifschitz 1976; Henrard 1982; Murray & Dermott 1999). One of the adiabatic invariants is *N* itself. Another, slightly more involved quantity, is the area *J* enclosed by the resonant path in the $\\sqrt{2\\Sigma}\\, (\\cos\\sigma,\\sin\\sigma)$ space:\n \n$$J = \\oint \\sqrt{2\\Sigma}\\,d\\sigma\\;. \\label{adiab}$$ \nWe would thus expect these parameters be constant, except for strong-enough perturbation or long-enough time scales (recall the adiabatic invariants are constant to the second order of the perturbing parameter only).\n*Resonant Yarkovsky effect.* Introducing the Yarkovsky forces makes the system to evolve slowly. The lock in the resonance prevents the orbits to steadily drift away in the semimajor axis and the perturbation by the Yarkovsky forces acts adiabatically. This is because (i) the time scale of the resonance oscillations is much shorter than the characteristic time scale of the orbital evolution driven by the Yarkovsky forces, and (ii) the strength of the resonant terms in the equations of motion are superior to the strength of the Yarkovsky accelerations.\nWe let the two J3/2 orbits evolve over 100 Myr (Fig. [r32-rtbpfilterSsigma100Myr]). At the end of our simulation the orbits moved from *N*0 ≃ 0.45 to *N*+ ≃ 0.46 (for the outward migrating orbit) or to *N*− ≃ 0.44 (for the inward migrating orbit), respectively. During this evolution, both orbits remained permanently locked in the J2/1 resonance, librating about the periodic orbit (dashed line in the left panel of Fig. [r32-rtbpfilterSsigma100Myr]). Because the position of this centre has a steep progression in the eccentricity and only small progression in the semimajor axis, the evolution across different *N*-planes makes the orbital eccentricity evolve significantly more than the semimajor axis. This is also seen in the middle panel of Fig. [r32-rtbpfilterSsigma100Myr], where the librating orbits significantly split farther/closer with respect to origin of coordinates (note the polar distance from the origin is basically a measure of the eccentricity). The shape of the librating orbit is modified such that the area *J* stays approximately constant. We have verified that the relative change in both adiabatic invariants, acquired during the 100Myr of evolution, is about the same: *δ**N*/*N* ∼ *δ**J*/*J* ∼ 5 × 10− 2. It is a direct expression of the strength of the perturbation by the Yarkovsky forces.\nWe can conclude the Yarkovsky effect results in a significantly different type of secular evolution for orbits initially inside strong first-order mean motion resonances with Jupiter. Instead of secularly pushing the orbital semimajor axis inward or outward from the Sun, it drives the orbital eccentricity to smaller or larger values, while leaving the semimajor axis to follow the resonance centre.\nIf we were to leave the orbital evolution continue in our simple model, the inward-migrating orbit would leave the resonance toward the zone of low-eccentricity apocentric librators. Such bodies are observed just below the J2/1 resonance. On the other hand, the outward-migrating orbit would finally increase the eccentricity to the value when the orbit starts to cross the Jupiters orbit. Obviously, in a more complete model, with all planets included, the orbits would first encounter the unstable region surrounding the stable resonant zone. Such marginally stable populations exist in both the J3/2 and J2/1 resonances.\n[lastpage]\n---\n1. E-mail: mira@sirrah.troja.mff.cuni.cz (MB); vokrouhl@cesnet.cz (DV).[↩](#fnref1)\n2. Interestingly, at their discovery (158) Hilda and (279) Thule, residing in the J3/2 and J4/3 resonances, immediately attracted attention of astronomers by vastly extending asteroid zone toward giant planets and by their ability to apparently approach Jupiter near aphelia of their orbits (e.g., Kühnert 1876; Krueger 1889).[↩](#fnref2)\n3. Note the former P-type objects were reclassified to X-type in a newer taxonomy by Bus and Binzel (e.g., Bus et al. 2002).[↩](#fnref3)\n4. Our results for both J2/1 and J3/2 resonances are summarised in tables available through a website http://sirrah.troja.mff.cuni.cz/yarko-site/ (those for J4/3 bodies are given in Table [tab1]). These contain listing of all resonant asteroids, their pseudo-proper orbital elements with standard deviations, their dynamical residence time and some additional information.[↩](#fnref4)\n5. This is equivalent to a cumulative size distribution law *N*( > *D*) ∝ *D**α* with *α* = − 5*γ* = ( − 3.5 ± 0.1), assuming all bodies have the same albedo.[↩](#fnref5)\n6. While our initial search used AstOrb catalogue from September 2007, we repeated it using the catalogue as of June 2008. No additional J4/3 objects were found.[↩](#fnref6)\n7. Schubart lists 11 additional asteroids in the group on his website http://www.rzuser.uni-heidelberg.de/ s24/hilda.htm, but again he does not go into details of their putative collisional origin.[↩](#fnref7)\n8. We also mention that so far asteroid disruption simulations did not explore cases of weak-strength materials appropriate for the suggested C/X spectral taxonomy of the Schubart family parent body.[↩](#fnref8)\n9. Jupiter Trojans, which are already known to be strongly unstable (Morbidelli et al. 2005), would have the dynamical lifetime of the order $0.1\\,{\\rm Myr}$ in this kind of simulation.[↩](#fnref9)\n10. The YORP effect tilts the spin axes of asteroids preferentially towards obliquity *γ* = 0∘ or 180∘ and this enhances the diurnal Yarkovsky drift due to its cos*γ* dependence.[↩](#fnref10)\n"},"data_id":{"kind":"string","value":"arxiv_0004120"}}},{"rowIdx":998,"cells":{"content":{"kind":"string","value":"rightness galaxies and dwarf spheroids, where DM dominates over the baryonic mass, the situation is much more complicated for their massive elliptical counterparts. Here, our ignorance about the stellar initial mass function (IMF) introduces a degeneracy between dark and luminous matter, which makes it hard to constrain the behaviour of the DM in the inner regions; equally, the task of probing the gravitational potential at large radii is made hard by the fact that their outskirts are notoriously faint. However, one way of significantly alleviating these degeneracies is to use multiple dynamical tracer populations, spanning a range of galactocentric radii. Indeed, massive elliptical galaxies are often home to large populations of planetary nebulae (PNe), globular clusters (GCs) and even, in the case of brightest cluster galaxies (BCGs), satellite galaxies, and each of these populations, with its own signature spatial distribution and kinematic profile, can be used as an independent probe of the gravitational potential. The pool of ETGs for which such an analysis has been carried out is currently too small for any meaningful conclusions to be extracted, though the suggestion also appears to be that the NFW profile may not provide a good universal fit: studies of BCGs by, and have found evidence for sub-NFW density profiles in clusters, while a handful of studies of field ellipticals, have found super-NFW densities. Clearly, a lot remains to be understood here, from both observational and theoretical perspectives.\nThe massive elliptical M87, located at the centre of the Virgo cluster, is an ideal subject for continuing such studies, as it has an enormous GC population, making it one of the richest GC hosts in the local Universe. Further, the sample of this population for which we have high-resolution kinematic data has been greatly expanded in recent years through the wide-field study of. It is also understood to be a slow rotator and nearly spherical, indicating that it is suitable for mass modelling under the assumption of spherical symmetry. However, two recent studies of M87’s mass distribution, both primarily based on Strader’s GC dataset, are markedly inconsistent. The first of these,, divided the GC sample into three independent populations and used a virial analysis to infer a very cuspy central density profile (*γ* = 1.57), while combined SAURON central stellar kinematics with the Strader GC data (along with an additional GC sample from Hectospec) and modelled the density profile using a logarithmic potential, thus imposing a core, which they infer to be *r**s* = 42 ± 10 kpc. Their total inferred stellar masses also differ by almost a factor of two. As M87 is one of the nearest and most well-observed BCGs, it seems unsatisfactory that its mass distribution should still be so poorly constrained, and clearly there remains work to be done. The aim of this paper is therefore to use a synthesis of GC, satellite and stellar kinematic data in conjunction with flexible mass models in order to infer a density profile which is free to be cuspy or cored as the dynamics dictate.\nThe paper is organised as follows: in Section [sec:data], we introduce the tracer populations used in our analysis and the associated datasets. In Section [sec:massmodel] we describe our mass model and Jeans analysis, the results of which are presented in Section [sec:results]. We discuss the implications of our findings in Section [sec:discussion], and use Section [sec:conclusions] to summarise our main conclusions. Throughout this work, we assume a distance to M87 *D**L* = 16.5 Mpc.\nDATA\n====\nTo constrain M87’s density profile across a wide radius range, we use multiple dynamical tracers, combining stellar kinematics in the central regions with GC dynamics at large radii and satellite galaxies on cluster scales. We can then solve the Jeans equation for each population separately, provided that the underlying density distribution is known. We therefore take data from a number of sources, as summarised in Table 1.\n| | | | | |\n| --- | --- | --- | --- | --- |\n| | radial coverage | data type | instruments | sources |\n| | 1.6 × 10− 3 - 210 kpc | photometry | multiple | |\n| | 0 - 2.5 kpc | kinematics | | |\n| | 0 - 0.17 kpc | kinematics | NIFS/Gemini Telescope | |\n| | 1.3 - 240 kpc | photometry | MegaPrime/CFHT | |\n| | 2 - 200 kpc | kinematics | multiple | |\n| | 35 - 1000 kpc | kinematics | multiple (mostly SDSS) | |\n| | | | | |\n[tab:sources]\n[fig:SBs]\nStars\n-----\nThe deprojected stellar surface brightness profile comes into our analysis at two points: first, the use of the stars as dynamical tracers in the Jeans equation requires us to know their 3D density distribution; second, our goal is to model M87’s mass as the sum of dark and luminous components, and the latter is simply the product of the integrated 3D luminosity density with some constant mass-to-light ratio, *Υ*⋆, which can be inferred from the data. We use the radial profile for M87 presented in, in which 20 sets of observations across a range of radii and filter systems were synthesised into a single profile in the V band. As M87 is known to have a very extended cD envelope in addition to a stellar core, its surface brightness profile cannot be accurately modelled by a Sérsic profile at both small and large radii. We therefore chose to model it using a more flexible Nuker profile, according to the following relation\n \n$$I(R) = I\\_0 \\Big(\\frac{r}{r\\_b}\\Big)^{-\\zeta} \\Big(1 + \\Big[\\frac{r}{r\\_b}\\Big]^{\\alpha}\\Big)^{\\frac{\\zeta-\\eta}{\\alpha}}$$ \nwith amplitude *I*0, break radius *r**b* = 1.05 kpc, inner slope *ζ* = 0.186, outer slope *η* = 1.88 and break softening *α* = 1.27. While this has the disadvantage of having no analytic deprojection or normalisation, it is much more flexible than a Sérsic profile as it allows both the inner and outer slopes greater freedom. Assuming spherical symmetry, we deproject this profile to give the 3D density shown in Figure [fig:SBs].\nThe kinematics of the inner 33ʺ × 41ʺ of M87 have been observed with the integral-field unit (IFU) SAURON, and a catalogue of the first four moments of the Gauss-Hermite expansion of the line-of-sight velocity distribution is available online[2](#fn2). We use the velocity dispersions, which were obtained from the spectra using a direct pixel fitting routine. The spectra were adaptively binned to ensure a signal-to-noise of at least 60 per spectral resolution element; uncertainties are generally less than ∼ 20 kms− 1, with a mean uncertainty of ∼ 9kms− 1.\nM87 is known to host a supermassive black hole (SMBH) of mass ∼ 6.6 × 109*M*⊙, and this should make a significant contribution to stellar velocity dispersions at the smallest radii. The SAURON dataset extends right down to the centre, though its resolution of 1ʺ is too low to be able to set constraints on the SMBH mass. As we are mainly interested in the behaviour of the DM and stellar components, one option to deal with this would be to simply exclude the apertures within the central ∼ 3ʺ from our analysis. However, though the contribution of the SMBH to the enclosed mass becomes sub-dominant beyond this approximate radius, it still contributes non-negligibly at larger radii and may be covariant with the stellar mass. We therefore include the SMBH in our mass model and constrain it using high-resolution kinematics of the central 2ʺ, as observed with the IFU NIFS on the Gemini North Telescope. As explained in that paper, these data were obtained from spectra which used laser adaptive optics corrections, and have a resolution of 0.08ʺ and a signal-to-noise generally greater than 50. The velocity moments are provided in radius and position angle bins, though our simplifying assumption of spherical symmetry allows us to combine bins azimuthally.\n[fig:GCmap]\nGlobular clusters\n-----------------\nWe use the colour and radial profiles for the GC populations of, as shown in Figure [fig:SBs]. The details of the inference are explained fully in that paper, but in brief, archival CFHT/MegaPrime images in the *ugriz* bands were used to compile a sample of 17620 GC candidates, selected according to their colours, magnitudes and apparent sizes, and the resulting catalogue was modelled to infer the radial and colour distributions and the relative fractions of the two GC populations. The surface density for each GC component was modelled as a Sérsic profile\n \n$$N(R) = N\\_0 \\exp{{\\Bigg[-k\\_n\\Big(\\frac{R}{R\\_e}\\Big)^{\\frac{1}{n}}\\Bigg]}}$$ \nwith *k**n* = 2*n* − 0.324, and with the profiles normalised such that ∫0*R**m**a**x**N*(*R*)*d**R* = 1, where *R**m**a**x* = 240 kpc is the radius of the outermost GC in the catalogue. The *g* − *r*, *g* − *i* colours and the GC luminosity function for each GC population were modelled as Gaussians, with radius-dependent colour gradients in all but the latter. The 3D deprojected profiles are also shown in Figure [fig:SBs].\n presents a spectroscopic catalogue of 737 GC candidates around M87 at radii from 2 kpc to 200 kpc (plus one object at 800 kpc, which we exclude because it lies outside the region over which our model from the photometry is strictly valid). The catalogue combines new measurements for 451 GCs – obtained using Keck/DEIMOS, Keck/LRIS and MMT/Hectospec – with literature data, and provides a ‘classification’ of objects as blue GCs, red GCs, ultra-compact dwarfs and transient/unknown along with SDSS-band *g* − *r* and *g* − *i* colours. We cross-correlate our photometric catalogue with this spectroscopic catalogue, selecting only the objects classified in as GCs, to obtain a sample of 612 GCs with complete luminosity, spatial, colour and kinematic information. In the analysis that follows, we choose to use the photometry over that provided in, for consistency with our GC colour distributions.\nSatellite galaxies\n------------------\nThe Extended Virgo Cluster Catalogue provides the redshifts and positions on the sky of 1589 galaxies in a footprint of 725 deg2 centred on M87, extending to 3.5 times the cluster virial radius. The redshifts are compiled from the SDSS DR7 release and the NASA Extragalactic Database (NED), and each object is classified as either a certain cluster member, a possible member or a background source based on morphological and spectroscopic criteria. As it is important that our sample only contains galaxies moving in M87’s halo potential, we selected only those objects classed as certain members according to both criteria, and we further cross-correlated these with the catalogue of, which used surface brightness fluctuations to calculate distance moduli. To avoid contamination from the W cloud, a slightly more distant component of the Virgo cluster at a characteristic distance of 23 Mpc, we imposed a distance cut of 20 Mpc; further, to separate the A cloud (centred on M87) from the B cloud (centred on M49) we also imposed a declination angle cut of 9.5 degrees and a radius cut of 1 Mpc. The spatial and velocity distributions of the resulting sample are shown in Figure [fig:GCmap]: it comprises 60 galaxies, with radii relative to M87 ranging from 35 kpc to 1 Mpc.\nWe could use this tracer population in the same way as the stars and the GCs and require its velocity dispersion profile to satisfy the Jeans equation so as to obtain a further probe of the mass at scales comparable to the virial radius. However, as noted earlier, the Jeans equation requires us to know the tracer density distribution and, as our satellite sample is most likely highly incomplete, we do not have access to this quantity. Most importantly, the sample of satellites we use has been selected spectroscopically, and this imposes a non-trivial selection function which may alter the spatial distribution from the true underlying one, leading us to draw incorrect conclusions from a Jeans analysis. Instead, we choose to use these satellites to give an estimate of the total mass at large radii and so give a further constraint in our inference on the mass profile. We do this using the virial mass estimator developed in and applied to a galaxy group in, which is designed to be robust against simple approximations to the true distributions. We use the mass estimator given by\n \n$$M(R<R\\_{out}) = \\frac{C}{G}\\big<v\\_{los}^2 r^{\\mu}\\big>,$$ \nwhere\n \n$$C = \\frac{\\mu + \\nu - 2\\beta}{I\\_{\\mu,\\nu}}r\\_{out}^{1-\\mu}$$ \nand\n \n$$I\\_{\\mu,\\nu} = \\frac{\\pi^{0.5}\\Gamma(\\frac{\\mu}{2} + 1)}{4\\Gamma(\\frac{\\mu}{2} + \\frac{5}{2})}[\\mu + 3 - \\beta(\\mu + 2)].$$ \nHere *β* is the anisotropy parameter\n \n$$\\beta = 1 - \\frac{\\sigma\\_t^2}{\\sigma\\_r^2}\n\\label{eq:binney}$$ \nfor radial and tangential velocity dispersions *σ**r*2 and *σ**t*2, and *μ* and *ν* are the slopes of the potential and tracer density respectively, both assumed to be scale-free. As mentioned previously, our satellite sample is likely to be incomplete, and this means we cannot infer *β*, *μ* and *ν* directly from the data; instead we calibrate their values using simulations. Following, we use the *z* = 0 halo catalogue of the first MultiDark simulation, desribed in detail in. This uses the WMAP5 cosmology and contains about 8.6 billion particles per Gpc/h3: the halo finder uses the bound density maximum technique described in. We identify all haloes with more than 30 subhaloes and treat each subhalo as a distinct satellite galaxy. We then use the subhalo velocities and positions and the parent halo mass profiles to infer the posterior distributions on *β*, *μ* and *ν* that best describe the global properties of the population, and use these to generate a posterior on the total mass *M*(*R* < *R**o**u**t*) of our cluster, whose median and standard deviation we use to re-select haloes from the simulation and iterate the procedure until our the inference on *μ*, *β* and *ν* converges. We report median values *β* = 0.30, *μ* = 0.15, *ν* = 1.9, and find a cluster mass\n \nlog(*M*(*R* < 985kpc)/*M*⊙) = 14.11 ± 0.19. \nTo test the calibration, we use the median values of *β*, *μ* and *ν* to apply the virial mass estimator to the simulated parent haloes, and the resulting comparison is shown in Figure [fig:VME]. Encouragingly, the masses are consistent, with a negligible median offset and a scatter of 0.1 dex, thus illustrating the robustness of the estimator. The applicability of this to M87 is then dependent on the assumption that the properties of the simulated galaxy satellite populations are representative of real galaxies.\n [fig:VME]\nModelling\n=========\nMass models\n-----------\nWe model the galaxy density profile as the sum of luminous and DM components plus a SMBH:\n \n*ρ*(*r*) = *ρ*⋆(*r*) + *ρ**D**M*(*r*) + *ρ**B**H*(*r*). \nThe stellar density profile is obtained by deprojecting the stellar surface brightness and scaling by the stellar mass-to-light ratio *Υ*⋆, which we assume to be constant with radius and include as a free parameter in the model. To allow flexibility in the DM density profile and to explore the impact of changing the halo model on the mass inference, we carry out our analysis using four different halo profiles. The first is a standard NFW profile,\n \n$$\\rho\\_{DM}(r) = \\frac{\\rho\\_0}{4\\pi} \\Big(\\frac{r}{r\\_s}\\Big)^{-1} \\Big(1 + \\frac{r}{r\\_s}\\Big)^{-2},$$ \nwith scale radius *r**s* and density scale *ρ*0, and cumulative mass\n \n$$M\\_{DM}(r) = \\rho\\_0 r\\_s^3\\Big(\\ln\\big(1+\\frac{r}{r\\_s}\\big) - \\frac{r}{r+r\\_s}\\Big).$$ \nFollowing, we also use a cored isothermal profile with a logarithmic potential (the LOG model)\n \n$$\\rho\\_{DM}(r) = \\frac{\\rho\\_{0}r\\_s^2}{4\\pi} \\frac{3r\\_s^2 + r^2}{(r\\_s^2 + r^2)^2}.$$ \nThis model is inherently cored, but in contrast to the NFW has a *ρ* ∼ *r*− 2 dependence at large radii: for a given core, then, this profile allows more dark matter to be placed at large distances from the galaxy centre. Its cumulative mass is given by\n \n$$M\\_{DM}(r) = \\frac{\\rho\\_0 r\\_s^2 r^3}{r\\_s^2+r^2}.$$ \nWe also use a generalised NFW (gNFW),\n \n$$\\rho\\_{DM}(r) = \\frac{\\rho\\_{0}}{4\\pi}\\Big(\\frac{r}{r\\_s}\\Big)^{-\\gamma} \\Big(1 + \\frac{r}{r\\_s}\\Big)^{\\gamma-3},$$ \nwith free parameters *γ*, *ρ*0 and *r**s*, where *γ* is the inner slope. This has the advantage of leaving the profile free to choose between cusps (*γ* ≥ 1) and cores (*γ* = 0) in the centre, while still becoming NFW-like at large radii in a way that is consistent with both simulations and observations. This gives a cumulative mass profile that can be related to the Gauss hypergeometric function 2*F*1 as\n \n$$M\\_{DM}(r) = \\frac{\\rho\\_{0}r\\_s^3}{\\omega}\\Big(\\frac{r}{r\\_s}\\Big)^{\\omega}~ \\_2F\\_1[\\omega,\\omega;1+\\omega;-\\frac{r}{r\\_s}]$$ \nwhere *ω* = 3 − *γ*. The final profile we use is a cored generalised NFW (cgNFW)\n \n$$\\rho\\_{DM}(r) = \\rho\\_{0}\\Big(\\frac{r+r\\_c}{r\\_s}\\Big)^{-\\gamma}~\\Big(1+\\frac{r}{r\\_s}\\Big)^{\\gamma-3}.$$ \nin which *r**c* is now the scale radius of the core. This is a more general case of the gNFW, and, in addition to allowing the data to choose between cusps (*r**c* = 0, *γ* = 1) and cores (*r**c* > 0, or *r**c* = 0 and *γ* = 0), it has additional flexibility at intermediate radii. We carry out the integration of the cumulative mass for this profile numerically.\nThe BH is simply a point mass at the origin,\n \n$$\\rho\\_{BH}(r) = \\frac{M\\_{BH}}{4\\pi r^2}\\delta(r),$$ \ngiving a constant term in the cumulative mass distribution.\nAs the SAURON dataset far outweighs the GC, NIFS and satellite datasets in terms of size, we regularise its contribution to the likelihood calculation by additionally fitting for a ‘noise’ parameter Δ*σ*⋆2. This can be interpreted as accounting for scatter in the data not included in the uncertainties or, alternatively, as modifying the relative weight given to these data, and is added in quadrature to the measured uncertainties on the velocity dispersion.\nOur overall model therefore has a number of free parameters dependent on the halo model in question and our assumptions about the anisotropy. In the isotropic case, which we treat first, the number of parameters varies between five and seven. In common for all halo models are the mass-to-light ratio *Υ*⋆, the normalisation of the DM halo log(*ρ*0), the scale radius *r**s*, the BH mass log(*M**B**H*) and the noise in the SAURON data, Δ*σ*⋆2. The gNFW and cgNFW halo models then add the free parameters (*γ*), (*γ*, *r**c*) respectively. When the orbital anisotropy of each tracer population is allowed to vary, this adds three parameters, as will be discussed in Section [sec:bayes]. In our notation in the following sections, we use the gNFW parameter set whenever we write out the model parameters explicitly, but this should be understood as standing in for any of the halo models.\nJeans modelling\n---------------\nGiven the stellar velocity dispersion and the GC and satellite velocities, we want to infer the posterior probability distribution on M87’s density profile. For the satellite galaxies, this involves a direct comparison of the mass calculated from our virial estimator and that obtained by integrating the proposed density profile. For the stars and GCs, on the other hand, we can relate the observed velocities and velocity dispersions to the density profile via the Jeans equation. Assuming spherical symmetry and dynamical equilibrium, the Jeans equation has the simple form\n \n$$\\frac{d}{dr}(l\\sigma\\_r^2) + 2\\frac{\\beta(r)}{r}l\\sigma\\_r^2 = l(r) \\frac{GM(r)}{r^2}$$ \nwhere *l*(*r*) is the luminosity density of the tracer, *σ**r*(*r*) the radial velocity dispersion and *β*(*r*) is the anisotropy parameter defined in Equation [eq:binney]. This is a first-order differential equation with general solution\n \n$$l(r)\\sigma\\_r^2(r) = \\frac{1}{f(r)} \\int\\_r^{\\infty} f(s) l(s) \\frac{G M(s)}{s^2}ds$$ \nwhere\n \n$$f(r) = f(r\\_i) \\exp\\Big[\\int\\_{r\\_i}^r 2 \\beta(s) \\frac{ds}{s}\\Big]\n\\label{eq:IF}$$ \n. Projecting this along the line of sight gives *σ**l**o**s*2(*R*) as\n \n$$\\frac{1}{2G} I(R) \\sigma\\_{los}(R)^2 = \\int\\_R^{\\infty} \\frac{l \\sigma\\_r^2 r dr}{\\sqrt{r^2 - R^2}} - R^2 \\int\\_R^{\\infty}\\frac{\\beta l \\sigma\\_r^2 dr}{r\\sqrt{r^2-R^2}}\n\\label{eq:jeans}$$ \nwhich, for certain choices of anisotropy parameterisations, can be written in the simple form\n \n$$I(R) \\sigma\\_{los}^2(R) = 2G \\int\\_R^{\\infty} l K\\_{\\beta} M\\frac{dr}{r}\n\\label{eq:kernel}$$ \nwith the kernel *K**β* dependent on the particular anisotropy model. Initially, we assume all orbits to be isotropic; later we consider the effect of anisotropy on our inference by modelling the stellar population as following a radially-dependent anistropy profile and each GC population with a constant, non-zero anisotropy. For the stars, we use a scaled Osipkov-Merritt profile\n \n$$\\beta(r) = \\beta\\_{\\infty} \\frac{r^2}{r^2 + r\\_a^2};\n\\label{eq:scaledOM}$$ \nwhich is centrally isotropic and becomes radially anisotropic at large radii, tending to the asymptotic anisotropy *β*∞. The kernel for the scaled Osipkov-Merritt profile is presented in the Appendix; the kernel for the constant-anisotropy case is given in, and we refer the reader there for further details.\nGiven a prescription for the anisotropy, the surface brightness and luminosity profiles of Figure [fig:SBs] and the mass model, then, we can calculate line-of-sight velocity dispersions *σ**l**o**s* from the Jeans equation and so infer the posterior probability distribution of the model parameters, based on the data in hand.\nStatistical analysis\n--------------------\nWe use the methods of Bayesian analysis to infer the posterior probability distribution of our density profile parameters, given the data. Bayes theorem states that the posterior distribution is proportional to the product of the likelihood function of the data given the model and the priors on the model, and so the task here is to construct sensible likelihood functions for each dataset. The total likelihood is then, in turn, the product of these, as each consitutes an independent set of measurements.\nFirst, as we have velocity dispersion measurements for the stars, the likelihood of observing a particular velocity dispersion at radius *R* is assumed to be Gaussian, with a standard deviation equal to the uncertainty. Thus the *k**t**h* stellar velocity dispersion measurement gives a contribution to the likelihood:\n \n$$\\ln L\\_{\\star,k}(\\sigma\\_k,R\\_k|\\vec{M}) = -0.5\\Bigg(\\frac{\\sigma\\_k^2 - \\sigma\\_m^2}{\\delta\\_{\\sigma\\_k^2}}\\Bigg)^2 - 0.5\\ln(2\\pi\\delta\\_{\\sigma\\_k^2}^2)$$ \nfor uncertainty *δ**σ**k*2 and model prediction *σ**m*2, and model parameters *M⃗* = (*Υ*⋆, *M**B**H*, Δ*σ*⋆2, *ρ*0, *γ*, *r**s*). As the observations in each aperture are independent, the total log likelihood of observing the ensemble is just the sum:\n \n$$\\begin{split}\n\\ln L\\_{\\star} &= \\sum\\_k \\ln L\\_{\\star,k}.\n\\end{split}$$ \nNote that, for measurement uncertainty Δ*σ**k*22, the regularisation of the SAURON data gives a total uncertainty *δ**σ**k*22 = Δ*σ**k*22 + Δ*σ*⋆22. For the NIFS data, on the other hand, *δ**σ**k*22 = Δ*σ**k*22.\nThe virial mass estimate from the satellites can be treated in a similar way, though here we only have one measurement:\n \n$$\\ln L\\_{sat} = -0.5\\Bigg(\\frac{\\log(M\\_{sat})-\\log(M\\_{mod})}{\\delta M\\_{sat}}\\Bigg)^2 - 0.5 \\ln\\Big(2\\pi\\delta M\\_{sat}^2\\Big)$$ \nfor virial mass estimate *M**s**a**t*, model mass *M**m**o**d* and the logarithm of the uncertainty from the mass estimator, *δ**M**s**a**t*. This is a direct comparison without the need for recourse to the Jeans equation.\nFor the GCs we can assign probabilities, based on the colour and position information provided in the photometric catalogue, of each GC belonging to either the red or the blue population, although we are not able to classify them with certainty. As the two populations are assumed to be dynamically decoupled, we do not expect their velocity dispersion profiles to be the same. In contrast to the other tracer populations, then, we calculate the likelihood of observing a GC with a particular velocity under the assumption that the velocity distribution of each GC population can be described by a Gaussian with a standard deviation given by the velocity dispersion, such that\n \n$$\\ln L\\_{GC,k} = -0.5\\Bigg(\\frac{(v\\_k-v\\_{gal})^2}{\\delta v\\_k^2 +\\sigma\\_{m}^2}\\Bigg) - 0.5\\ln\\Big(2\\pi(\\delta v\\_k^2 + \\sigma\\_{m}^2)\\Big)$$ \nwhere *v**k* and *δ**v**k* are the measured line-of-sight velocity and velocity uncertainty of the *k**t**h* GC, *v**g**a**l* = 1284 kms− 1 is the heliocentric velocity of M87 and the line-of-sight velocity dispersion at the location of each globular cluster, *σ**m*, is modelled separately for the red and the blue populations.\nTo account for the uncertainty in assigning each GC to either the red or the blue population, for each set of model parameters *M⃗* we draw 1000 Monte Carlo samples which stochastically explore the population distribution based on the colour, magnitude and position information for the individual GCs. We then marginalise over the samples to give a final contribution to the likelihood. While some GCs have either very high or very low probabilities of belonging to one of the GC populations, with small uncertainty, there also exists a significant fraction with comparable probabilities of belonging to either: for these objects, it would not be meaningful to simply assign them to one population or the other. A further advantage of this stochastic sampling is that it allows us to explore different combinations of red and blue GCs.\nAs each tracer population is independent, the final log-likelihood of any set of model parameters is the sum over all contributions:\n \nln*L* = ∑*k*ln*L*⋆ , *k* + ∑*k*ln*L**G**C*, *k* + ln*L**M**s**a**t*. \nWe explore the parameter space using the ensemble-sampling code `emcee`.\nResults\n=======\nC1cmC1.4cmC1.4cmcccccchalo model & *M*⋆/*L* & log(*ρ**D**M*) & *r**s* & *γ* & *δ**σ*⋆2 & *r**c* & log(*M**v**i**r*) & *R**v**i**r* \nNFW & 6.6 ± 0.1 & 6.6 ± 0.1 & 448 ± 75 & – & 12.1 ± 0.2 & – & 14.39− 0.53+ 1.11 & 1620− 360+ 460 \nLOG & 6.9 ± 0.1 & 7.7 ± 0.1 & 48 ± 5 & – & 12.0 ± 0.2 & – & 14.21− 0.37+ 0.60 & 1410− 200+ 240 \ngNFW & 6.9 ± 0.1 & 8.3 ± 0.1 & 79 ± 10 & < 0.14 & 12 ± 0.1 & – & 14.13− 0.44+ 0.76 & 1320− 240+ 270 \ncgNFW\\* & 6.9 ± 0.1 & 5.6− 1.2+ 1.4 & 273− 180+ 430 & 2.54− 1.18+ 0.33 & 12.0 ± 0.2 & 63− 14+ 11 & 14.16− 0.40+ 0.67 & 1350− 200+ 240 \n[tab:results]\n [fig:triangleNFW]\n [fig:triangleLOG]\n [fig:triangleBPL]\n [fig:triangleBBPL]\n[fig:MassProfiles]\n [fig:triangleNFWaniso]\n [fig:triangleLOGaniso]\n [fig:triangleBPLaniso]\n [fig:triangleBBPLaniso]\nInitially, we treat all orbits as isotropic and investigate the uncertainty introduced to the mass inference by the choice of halo model. The inference for the four different halo models are presented in Figures [fig:triangleNFW] - [fig:triangleBBPL], and are summarised in Table [tab:results] along with the associated virial parameters. We find the results of the gNFW, cgNFW and LOG models to be very similar: the gNFW and cgNFW haloes are both centrally cored with break radii ∼ 80 kpc, while the LOG model is inherently cored and has a slightly smaller scale radius *r**s* = 48 ± 5 kpc (note, however, that these are *not* core radii, and that the scale radius is defined differently in the latter case). All three models also agree closely on the high stellar mass-to-light ratio *Υ*⋆ , *v* ∼ 6.9 and the virial parameters, with log(*M**v**i**r*/*M*⊙) ∼ 14.2 and R*v**i**r* ∼ 1.3 Mpc. Indeed, the best cgNFW model approximately recovers the gNFW solution, with an intermediate slope *γ* ∼ 3 and both the core and break radii comparable to the gNFW scale radius. Like the gNFW, this represents a solution in which the halo is cored centrally and becomes NFW-like at large radii; unlike the gNFW, it allows more flexibility in the intermediate regions, and the fact that this intermediate slope is found to be close to the NFW value *γ* ∼ 3 suggests that the NFW profile is an adequate description at intermediate radii as well as large radii.\nOn the other hand, the NFW model predicts a lower mass-to-light ratio *Υ*⋆ , *V* = 6.60 ± 0.05, though it does match the other models in terms of its virial parameters. The difference at small radii arises because of the NFW profile’s hard-wired cusp, which places more DM in the central regions at the expense of stellar mass. In Figure [fig:MassProfiles] we plot the mass profiles inferred in each case, along with the associated rotation curves in order to better highlight the differences between the four models. As can be seen, deviations start to arise at larger radii, where the constraints from our data are weakest. These are mainly due to the different structural features of the models: for instance, the LOG profile decays as *r*− 2 at large radii whereas the other three go as *r*− 3, which allows the former to place more mass at large radii. Equally, the NFW profile goes as *r*− 1 at small radii whereas the others are either intrinsically cored or have flexible inner slopes (which are found to favour cores here), and this causes the NFW profile to start deviating from the others at smaller radii.\nFollowing this, we test the robustness of our inference on the halo structure against model assumptions by relaxing the isotropy condition and introducing some element of anisotropy into each of the tracer populations. Guided by cosmological N-body simulations,, which find orbits to change smoothly from being radially anisotropic in the galaxy outskirts to being isotropic in the centre – a result of hierarchical formation in which BCGs form by the accretion of infalling satellites which become phase-mixed by the time they reach the galaxy centre – we choose to model the stellar anisotropy using the scaled Osipkov-Merritt profile of Equation [eq:scaledOM]. As the GC data are comparatively sparse, we treat each GC population as having some constant, non-zero anisotropy. This increases our parameter space by four, as we are now also fitting for the scale radius of the stellar scaled Osipkov-Merrit profile *r**a*, the asymptotic stellar anisotropy *β*∞, and the anisotropies of the two GC populatons, *β**r**e**d* and *β**b**l**u**e*. Again, we carry out the inference for the four halo models, and present the posteriors in Figures [fig:triangleNFWaniso] - [fig:triangleBBPLaniso] and summarise the results in Table [tab:anisoresults].\nInterestingly, we now find the some significant deviation among the halo models. Firstly, the LOG profile recovers a model for the stars that is quite isotropic, with *β*∞ > 0.99 at the 95% confidence level and *r**a* = 9.29 ± 0.05 kpc; given that the SAURON data only extend out to ∼ 2 kpc, this implies anisotropies *β* < 0.05 where we have data (though our data are in projection). This is presumably a result of the hard-wired core. The remaining three models agree, within uncertainties, on the mass profile – though, as might be expected, not so closely as when isotropy is enforced. In these cases, we find that the mass-to-light ratio drops significantly and that the stars become radially anisotropic within scale radii *r**a* < 4 kpc. This covariance between the central mass and anisotropy is a direct manifestation of the mass-anisotropy degeneracy: in the isotropic models, the assumed lack of any radial anisotropy drove the central mass to high values, whereas here we are able to go some way towards breaking the degeneracy through the use of multiple populations tracing the same gravitational potential. Note also that the introduction of radial anisotropy, coupled with the decrease in stellar mass, now allows the DM halo to become less cored in the centre: we find a gNFW inner slope which is still distinctly sub-NFW, but less extremely so, with *γ* < 0.81 at the 95 % confidence level, and the cgNFW model finds a smaller central core (*r**c* = 19.00− 8.34+ 8.38 kpc) and a sub-NFW intermediate slope *γ* = 2.39− 0.43+ 0.39 before transitioning to the *r*− 3 regime beyond the scale radius *r**s* = 412.1− 143.0+ 123.0 kpc. For these three profiles, both GC populations are found to have a mild radial anisotropies. We find a slightly higher BH mass than previous estimates, though it is consistent within uncertainties. However, we note that this is the first time the BH mass has been inferred jointly with the halo and stellar mass parameters using the high-spatial-resolution NIFS dataset; as we might anticipate a strong covariance between these parameters – especially between the stellar and BH masses – a slight change is not too surprising. On the other hand, there are still assumptions in our model which could affect our inference on the BH mass. For instance, our anisotropy profile enforces isotropy in the centre, which, while a physically sound assumption, restricts the range of parameter space explored by the BH mass. We are also only making use of the second-order moments of the line-of-sight velocity profile: while this should not cause any systematic bias in the BH mass inference, we may be able to obtain tighter constraints by incorporating higher-order moments.\nWe compare the goodness of fit of the four models using a reduced chi-squared criterion, and find that the cgNFW profile provides the best description of the dynamics – noting that the cgNFW, gNFW and NFW are nested models. We therefore present the best cgNFW mass profile in Figure [fig:mass] and the residuals on the SAURON and NIFS data in Figure [fig:residuals]. The mass profile appears to be generally consistent with earlier work; the residuals are noticeably better than in the isotropic case, indicating the importance of the anisotropy in the stellar dynamics.\nWe also note that the assumption that the MultiDark simulations are able to accurately reproduce the dynamics of real galaxies may introduce some additional systematic uncertainty into the mass measurement made using the virial estimator in Section [sec:data:satellites]. We therefore test the sensitivity of our inference to this uncertainty by carrying out the analysis with twice the calculated uncertainty, and we find that this has a negligible effect on our inferred density model. Essentially, the likelihood is dominated by contributions from the stars and the GCs, as the datasets for these populations are far more extensive; the satellite mass estimate rejects models with discrepant extrapolations of the total mass to large radii, but most of the models consistent with the smaller-radius data are still well-behaved at 1 Mpc.\ncC0.8cmC1.0cmC1.0cmC1.0cmC1.0cmC1.0cmC1.0cmC1.0cmC1.0cmC1.0cmC1.0cmC1.0cmC1.0cm& *M*⋆/*L* & log(*ρ**D**M*) & *r**s* & *γ* & *M**B**H* × 10− 9 & *δ**σ*⋆2 & *r**a* & *β*∞ & *β**r**e**d* & *β**b**l**u**e* & *r**c* & log(*M**v**i**r*) & *R**v**i**r* \nNFW & 3.05− 0.24+ 0.24 & 7.51− 0.12+ 0.13 & 128.4− 24.1+ 27.8 & – & 7.04− 0.2+ 0.2 & 9.34− 0.19+ 0.20 & 2.78− 1.30+ 1.26 & 0.690.04+ 0.04 & 0.16− 0.21+ 0.21 & 0.34− 0.17+ 0.15 & – & 14.00− 0.32+ 0.37 & 1190− 120+ 240 \nLOG & 6.48− 0.02+ 0.02 & 8.25− 0.19+ 0.23 & 22.1− 5.7+ 6.7 & – & 9.5− 0.23+ 0.22 & 10.2− 0.2+ 0.2 & 9.29− 0.05+ 0.05 & > 0.99 & − 0.17− 0.21+ 0.23 & 0.08− 0.29+ 0.23 & – & 14.03− 0.29+ 0.44 & 1200− 190+ 250 \ngNFW & 3.33− 0.45+ 0.38 & 8.64− 0.35+ 0.24 & 40.3− 8.8+ 16.4 & < 0.81 & 7.2− 0.4+ 0.35 & 9.3− 0.2+ 0.2 & 1.06− 0.10+ 0.07 & 0.67− 0.05+ 0.05 & 0.13− 0.20+ 0.21 & 0.23− 0.17+ 0.17 & – & 13.74− 0.32+ 0.43 & 990− 140+ 270 \ncgNFW & 3.50− 0.36+ 0.32 & 4.76− 0.40+ 0.55 & 412.1− 143.0+ 123.0 & 2.39− 0.43+ 0.39 & 7.22− 0.40+ 0.34 & 9.33− 0.20+ 0.21 & 1.06− 0.05+ 0.08 & 0.67− 0.06+ 0.03 & 0.13− 0.20+ 0.21 & 0.23− 0.18+ 0.16 & 19.00− 8.34+ 8.38 & 13.87− 0.35+ 0.42 & 1090− 150+ 220 \n[tab:anisoresults]\n[fig:mass]\n[fig:residuals]\nDiscussion\n==========\nResolving previous discrepancies\n--------------------------------\nPart of the impetus for this study was to resolve the discrepancies between recent models of M87’s mass structure. The two recent analyses of and, both of which relied heavily on the GC kinematics, disagreed on both the structure of the DM halo and the total stellar mass of the system, with finding a stellar mass-to-light ratio *Υ*⋆ , *V* ∼ 4.5 and a super-NFW cusp *γ* = 1.6 and inferring *Υ*⋆ , *I* = 6.0 ± 0.3 (which converts to a *V*-band measurement of ∼ 7.5) under the assumption of a cored halo. There are a number of possible reasons for this: first, while both studies overlapped in the majority of their GC data, each used a different mass model, and it is possible that these may have unnecessarily constrained the inference on the mass. Specifically, chose to use a logarithmic potential model for the dark matter, so enforcing a core, whereas used a power law for the dark halo, thus requiring a constant slope at all radii. Neither of these allows the halo much flexibility in the central regions. Further, while relied solely on the GCs, separating them into three independent populations based on their velocities, positions and colours, treated all the GCs as a single tracer population, but used the same SAURON data as in this study to constrain the stellar mass-to-light ratio. Thus it is possible that lacked the data coverage in these central regions that would have permitted a reliable distinction between cusps and cores.\nBy exploring multiple mass models and data combinations, we are able to reproduce the results of both studies. First, excluding the stellar kinematics and carrying out the inference using just the satellites and GCs in a way more similar to that of, we infer an isotropic gNFW profile with *γ* = 1.48 ± 0.22, thus reproducing their finding of a cusp. The inference is summarised in Figure [fig:cusp]. This data combination also provides only a very weak constraint on the stellar mass-to-light ratio, as might be expected given the relatively small number of GCs in the centre as compared to the stars. When we compare the predictions of the best-fit model in this case, however, we find that it significantly overpredicts the stellar velocity dispersions in the centre. Adding in the stellar data, then, requires an excavation of the central regions, giving rise to a sub-NFW inner slope more easily reconciled with the assumption of, although we do still find a lower stellar mass, which could be a consequence of the fact that, unlike, we include the BH in our models. The solution of the discrepancy between the two studies is presumably then that it is simply not possible to constrain the whole mass profile using GC data alone. This really shows the importance of combining multiple tracer populations with different characteristic radii.\n [fig:cusp]\nWe also perform the first joint inference on the BH, stellar and halo mass parameters that has been carried out with the high-resolution NIFS kinematics of the stars in the central 2ʺ. When these data were presented in, the structure and scale of the halo and stellar distributions were fixed at ’s values, which in turn were inferred assuming the BH mass measured in, based on lower-resolution data. The BH mass that we infer is consistent with the value within uncertainties: for the best cgNFW model, we find 7.22− 0.40+ 0.34 × 109*M*⊙.\nIn previous models of M87, significant tension has also arisen between the ratio of the stellar and halo masses and expectations from abundance matching, and this has tended to be related to a consistently high value of the stellar mass-to-light ratio, *Υ*⋆. For instance, infer a very high mass-to-light ratio *Υ*⋆ , *V* ∼ 7.5, despite the circular velocity of their LOG halo profile being slightly lower than ours (at *v**c* = 591 ± 50 kms− 1 compared to our 610 kms− 1), and this places it above the abundance matching predictions of, even allowing for uncertainties. infers both a larger stellar mass (with *Υ*⋆ , *V* = 9.1 ± 0.2 at an assumed distance *D**L* = 17.9 Mpc, which translates to *Υ*⋆ , *V* = 9.9 at our assumed distance) and a larger halo mass (log(*M**v**i**r*/*M*⊙) = 14.37, *R**v**i**r* = 1.6 Mpc), also placing M87 above abundance matching expectations. While our Nuker profile for the projected light is not strictly normalisable, we compare our results using the absolute magnitude of M87 *M**v* = 8.30 reported in, corrected for dust reddening using the extinction given in that study. This gives a stellar mass log(*M**l**u**m*/*M*⊙) = 11.61 ± 0.10. Compared with the virial mass log(*M**v**i**r*/*M*⊙) = 13.87− 0.35+ 0.42, this puts M87 just within the abundance-matching curves of, allowing for uncertainties, and so alleviating this tension. We note, however, that all of our isotropic models – as well as our anisotropic LOG model – also yield anomalously high stellar masses relative to the halo, and furthermore that all these previous studies assumed a form for the halo given by the LOG profile, which we find to not perform as well as the more generalised NFW-like parameterisations. Seen together, then, this highlights the need for flexible density and anisotropy models in order to properly understand the astrophysics of these systems.\nThe case for a cored halo\n-------------------------\nIt is important to compare our findings for the halo structure with other ETGs; as explained in the Introduction, this is crucial for developing our understanding of the role of baryons in shaping the halo and the diversity of structures that can arise.\nA number of other studies have found BCGs to have DM profiles which are flatter than NFW:, for instance, combined stellar dynamics with weak and strong graviational lensing and X-ray data to infer a density slope *γ* < 1 for the BCG Abell 383 with 95 % confidence, while went on to fit gNFW halo models to a sample of seven massive BCGs and found a mean slope of *γ* = 0.5 ± 0.1, inconsistent with the NFW prediction – and with three of those galaxies hitting the *γ* = 0 prior. This is also in line with the earlier studies of ; more recently, have also modelled multiple image families lensed by the cluster Abell S1063 to infer a significant core radius *r**c* ∼ 100 kpc. The picture, however, is quite different for elliptical galaxies in the field, and a number of strong lensing studies of ETGs have found halo slopes consistent with NFW or even super-NFW DM profiles:, for instance, modelled the global properties of a sample of 81 lenses and found the inner slope to be consistent with an NFW profile, while performed a similar analysis on a smaller sample of lenses from the Sloan Lens ACS Survey (SLACS) and found *γ* = 1.7 ± 0.5, which is very inconsistent with what we find here. At first sight, this might seem to point tentatively to the existence of two very different evolutionary paths for ETGs in these two environments, and the implausibility of a one-size-fits-all DM halo profile: however, the sample size of ETGs for which this analysis has been carried out clearly remains too small for any meanungful conclusions on this issue.\nWe now consider the implications of M87’s central core in the context of the baryonic feedback processes that play a role in the centres of galaxies. While a cored halo is inconsistent with DM-only simulations, there have been a number of recent simulations in which baryonic effects have been included, and we note some similarities with these. Here, we focus on two of the most recent studies of this type in order to elucidate the physics, though we stress that this comparison is by no means exhaustive and a number of similar projects have been carried out. Firstly, we consider the zoom-in simulations of, which followed two BCGs from redshift *z* = 2 to *z* = 0 and demonstrated that the effect of repeated dissipationless mergers is to soften the otherwise-NFW-like cusp by Δ*γ* ∼ 0.3 − 0.4 on scales of the stellar half-light radius. In this paradigm, infalling satellite galaxies experience dynamical friction from DM as they move through the halo, and this transfers energy to the DM, causing it to expand and so become less dense in the central regions. While we do not measure a core radius explicitly, in Figure [fig:drhodr] we plot the density slope as a function of radius, dlog*ρ*/dlog*r*, and see that the slope becomes sub-NFW at radii *r* < 10 kpc and that *γ* ∼ 0.6 (equivalent to Δ*γ* ∼ 0.4) at r ∼ 5 kpc, which is comparable to the scale radius of the starlight measured by. However, we find a cored centre while these simulations find only 0.6 < *γ* < 1 down to the innermost resolvable radius. While they suggest that the merging of SMBHs could then give rise to cores in the central 3-4 kpc – with the binary BH system spiralling inwards and transferring energy to the surrounding matter – their simulations do not include the effects of SMBHs and so they are unable to test this. It therefore remains unclear as to whether merging events would be capable of totally erasing the cusp.\nOn the other hand, M87 has a large AGN at its centre and this may also contribute to core formation. The second study that we examine is that of, which specifically simulated Virgo-like ETGs using a recipe for AGN feedback. They found cores to develop within the inner 10 kpc. In this scheme, outflows of gas due to the AGN are able to irreversibly modify the gravitational potential of the halo and so cause expansion of both the dark and luminous matter. In that study, they suggest that a combination of AGN feedback and SMBH effects are the main contributing processes, though they also note that the large amount of gas expelled by the AGN increases the efficiency of the energy transfer due to dynamical friction. This role for AGN is analogous to the role of supernovae in dwarf spheroids. However, while the core radius of 10 kpc seems consistent with the density profile that we infer, predict cores in the 3D stellar density, whereas M87 is known to have a ‘cuspy core’ in its 2D surface brightness (*γ* = 0.18), making it considerably cuspy in 3D. Indeed, the finding of cores in the 3D density distribution of the stars seems at odds with observations of large numbers of ETGs. Again, this seems to be a sign of the complex nature of the processes governing the shape of the density profile, and that there still remain many questions to be answered on this subject.\n [fig:drhodr]\nImplications for the stellar initial mass function\n--------------------------------------------------\nOne of the initial motivations for this investigation was to resolve the striking discrepancy between the *Υ*⋆ , *V* inferred by and, particularly in light of recent evidence that massive ETGs may have Salpeter-like stellar mass-to-light ratios. In our anisotropic cgNFW model, we find *Υ*⋆ , *V* = 3.50− 0.36+ 0.32; however, infer a much higher value *Υ*⋆ , *V* ∼ 7.5, and previous studies have found mass-to-light ratios even higher than this. We use the single stellar population models from to understand the implications of the stellar mass-to-light ratio for the age and IMF of M87.\nWe measure colours for M87 from the CFHT photometry and compare these with the BC03 models. In particular, we determine the mean surface brightness within a 2′′ annulus at 30′′ (i.e., ∼ 2.5 kpc) from the centre of M87, masking globular clusters and other artefacts, for each of the *u**g**r**i**z* filters. We find extinction-corrected *u* − *g*, *g* − *r*, *g* − *i*, and *g* − *z* colours of 1.48, 0.67, 1.13, and 1.45, respectively, which agree well with the colours calculated over a much larger aperture from SDSS photometry. Then, assuming a Salpeter IMF and solar metallicity, the BC03 models give an age of 10 Gyr and *Υ*\\* , *V* ≈ 5.7, significantly lower than the or values and thus implying the need for an IMF even more extreme than that found by if these high mass-to-light ratios are to be explained. If we instead assume a Chabrier IMF and solar metallicity, the BC03 models still find a ∼ 10 Gyr age, but *Υ*\\* , *V* ≈ 3.7, which is in good agreement with our dynamical measurement when we allow for anisotropy. We note, however, that our dynamical inference on *Υ**V*, \\* is very sensitive to our assumptions about anisotropy. Furthermore, our assumption of solar metallicity – and, indeed, a constant *Υ*\\* , *V* with radius – may be inappropriate. For example, use UV to NIR photometry to demonstrate that M87 has colour gradients, and they interpret these as being the result of a metallicity gradient.\nImportance of correctly modelling the underlying tracer populations\n-------------------------------------------------------------------\nThe Jeans equation requires us to know the underlying tracer density for each population, and this, along with the mass profile and anisotropy, determines the velocity dispersion that is measured. Therefore any uncertainty or bias in the parameterisation of the tracer density might manifest itself as uncertainty or bias in the resulting velocity dispersions and, in a study such as this in which the velocity dispersions are the primary tool for the inference, also as uncertainty or bias in the mass profile. A particular worry here is the distributions of the GCs, as the spectroscopic catalogue of is only a subsample of the wider population and, by virtue of its being selected for spectroscopy, is subject to some non-trivial selection function that may change the apparent distribution from that of the true underlying one. It is therefore important to model the distribution using an independent photometric sample. Indeed, this was the motivation for the initial photometric study of.\nTo demonstrate the impact on the inference of using the density of the spectroscopic subsample in the Jeans analysis rather than that of the more repsresentative photometric sample, we repeat our analysis using the three-population decomposition given in. This decomposition was carried out based solely on the spectroscopic catalogue, and separates the GCs into a compact red component along with two much more extended blue and intermediate-colour components. Our anaylsis differs from ’s in that they focussed mainly on virial decompositions and used the GC data exclusively, whereas we use a spherical Jeans analysis and combine the GCs with the stellar kinematics and satellite galaxy datasets. This comparison is not intended as a comment on their work, but merely aims to show that, in this particular analysis, the use of this approximation to the true distribution is important. When we carry out the inference using the three populations, we find that we underpredict the halo mass in the region in which the GCs dominate the fit by ∼ 0.25 dex. *M**v**i**r* also decreases by ∼ 10%, though the effect is presumably smaller here because the GC data only extend out to a fraction of the virial radius. Unsurprisingly, our inference on the stellar mass-to-light ratio is robust against the GC Sérsic distributions, as this is mainly determined by the properties of the stellar population, which is fixed by our fit to the surface brightness profile.\nAnother possible bias in spherical Jeans modelling could result from any flattening in the potential: indeed, there have been recent suggestions that the shape of a galaxy’s DM halo may correlate with the shape of its luminous matter, and, while the modelling of assumed the GCs to be spherically distributed, they do exhibit some ellipticity at larger radii. However, the effect of this was analysed in using a series of axisymmetric Jeans models, and it was found that the impact of any realistic amount of flattening on the mass profile was small, with up to a 10 % decrease in the mass for GCs with elliptical profiles. This is therefore unlikely to be a major concern here.\nAnisotropy, and a pinch radius from multiple tracer populations\n---------------------------------------------------------------\nAs explained in Section [sec:results], we incorporated anisotropy in our models by assuming a scaled Osipkov-Merritt profile for the stars and constant anisotropies for the GC populations, and inferred mild radial anisotropies for the three populations, which in turn led to changes in our inference on the mass model relative to the isotropic case. As the inclusion of the anisotropy parameters is considerably more expensive from a computational point of view, it is of interest to investigate ways of obtaining robust mass estimates using models with fixed anisotropy. It is well known that, for the Jeans analysis of a single tracer population, there exists a pinch radius at which the dependence of the inferred mass profile on the anisotropy is minimal. This occurs at approximately the 3D effective radius of the tracer and can be quoted as a robust measure of the enclosed mass – instead of an entire radial profile – when it is not possible to infer the anisotropy from the data. However, our case is slightly more complicated than those studied by the previous authors as we are combining three completely independent tracer populations. It is therefore of great interest to test whether the pinch radius is still recovered in this case.\nFor simplicity, we run a series of models in which the stellar anisotropy is fixed at some non-zero constant value while the GCs remain isotropic. This is guided by our earlier results, which found the stars to be considerably more anisotropic than the GCs. To span a range of anisotropies, we let the anisotropy run from *β* = − 1 (corresponding to the case where the velocity dispersion in the radial direction is twice that in the radial direction) to *β* = 0.5 (at which point the tangential velocity dispersion is half the radial velocity dispersion). The results are shown in Figure [fig:pinch]. Indeed, we do find the three *β*-curves to intersect at a single radius, as in the case of single-tracer population models: however, this occurs at a radius = 111 ± 4 kpc (with mass log(*M*(*R* < *R**p*)/*M*⊙) = 13.09 ± 0.40), which is much too large to be associated with the effective radius of the starlight. This is presumably a consequence of our use of additional tracer populations, all at larger radii than the stars. This is an interesting finding which will be useful for further multiple-population studies of this kind.\n [fig:pinch]\nConclusions\n===========\nWe have modelled the mass profile of M87 using a combination of stellar, GC and satellite galaxy kinematics in a Jeans analysis, and our main conclusions are as follows:\n1. M87 is a massive BCG with a sub-NFW DM halo. It has a virial mass of log(*M**v**i**r*/*M*⊙) = 13.87− 0.35+ 0.42, a virial radius *R**v**i**r* = 1.1 ± 0.2 Mpc and a stellar mass log(*M**l**u**m*/*M*⊙) = 11.61 ± 0.10, placing it at the top end of the galaxy mass distribution and just within the stellar-to-halo mass relations expected from abundance matching.\n2. Under assumptions of isotropy, the scatter introduced by the use of different models to parameterise the halo is small. In this paradigm, we find M87 to have a high stellar mass-to-light ratio of *Υ*⋆ = 6.9 ± 0.1 in the *V*-band and a cored DM halo, with inner slope *γ* < 0.14 at the 95 % confidence level.\n3. When the model is modified to allow each tracer population some element of anisotropy, differences arise between the halo models, and a cored generalised NFW profile provides the best description of the dynamics. This reduces the stellar mass-to-light ratio to *Υ*⋆ = 3.50− 0.36+ 0.55 and slightly relaxes the constraint on the inner slope, with a core radius *r**c* = 19.0 ± 8.3 kpc. All three tracer populations are characterised by mildly radially anisotropic orbits.\n4. The *V*-band stellar mass-to-light ratio of M87 is consistent with a picture in which its stellar populations are old ( ∼ 10 Gyrs), with solar metallicity. In the isotropic case, this implies a Salpeter-like IMF, whereas when anisotropy is accounted for, a Chabrier-like IMF is preferred.\n5. The inclusion of tracers at a variety of spatial scales has a significant impact on the inference. Modelling the mass using the GCs and satellite galaxies alone - that is, without the stars - we infer a cuspy halo, with inner slope *γ* ∼ 1.5, but when the stars are included in the inference, the halo becomes sub-NFW. This shows the importance of consistently modelling the profile across a range of spatial scales.\n6. It is important to properly characterise the distributions of the underlying tracer populations as opposed to those of their kinematic subsamples. We have shown that the use of GC colour and spatial distributions based on the kinematic dataset alone leads to a systematic underprediction of the total mass of the system.\n7. When the inference is carried out with different (constant) values for the stellar anisotropy between -1 and 0.5, a pinch radius clearly emerges at which the dependence of the enclosed mass on the anisotropy is minimised. This gives an estimate of the mass enclosed at 110 kpc as log(*M*(*R* < 110)/*M*⊙) = 13.09 ± 0.40 which should be robust against assumptions about anisotropy.\nAcknowledgements\n================\nWe thank the referee for providing helpful suggestions and corrections. We would also like to thank Wyn Evans, Sergey Koposov, Vasily Belokurov, Jason Sanders and Angus Williams for useful discussions, and Debora Sijacki for helpful comments on the draft of this paper. LJO thanks the Science and Technology Facilities Council (STFC) for the award of a studentship; MWA also acknowledges support from the STFC in the form of an Ernest Rutherford Fellowship.\nAnisotropy kernel for the scaled Osipkov-Merritt profile\n========================================================\nFor the anisotropy profile of Equation [eq:scaledOM], the integrating factor defined by Equation [eq:IF] is\n \n$$f(r) = f(r\\_i)\\Bigg(\\frac{r^2 + r\\_a^2}{r\\_i^2 + r\\_a^2}\\Bigg)^{\\beta\\_{\\infty}}.$$ \nThe kernel of Equation [eq:kernel] is then\n \n$$K\\_{\\beta}(R,r) = \\frac{\\sqrt{r^2 - R^2}}{r} \\Bigg( 1 + \\beta\\_{\\infty} \\Big( \\frac{r^2 + r\\_a^2}{R^2 + r\\_a^2}\\Big)^{\\beta\\_{\\infty}} \\Big[2 \\_2F\\_1 (\\frac{1}{2},\\beta\\_{\\infty}; \\frac{3}{2}; z) - \\frac{3R^2 + 2r\\_a^2}{R^2 + r\\_a^2} ~\\_2F\\_1(\\frac{1}{2},\\beta\\_{\\infty}+1; \\frac{3}{2};z)\\Big]\\Big)\\Bigg)$$ \nwhere\n \n$$z = \\frac{R^2 - r^2}{R^2 + r\\_a^2}.$$ \n---\n1. E-mail: loldham@ast.cam.ac.uk[↩](#fnref1)\n2. http://www.strw.leidenuniv.nl/sauron/[↩](#fnref2)\n\nGalaxy structure from multiple tracers: II. M87 from parsec to megaparsec scales\n================================================================================\nFollowing a number of conflicting studies of M87’s mass profile, we undertake a dynamical analysis of multiple tracer populations to constrain its mass over a large radius range. We combine stellar kinematics in the central regions with the dynamics of 612 globular clusters out to 200 kpc and satellite galaxies extending to scales comparable with the virial radius. Using a spherical Jeans analysis, we are able to disentangle the mass contributions from the dark and baryonic components and set constraints on the structure of each. Assuming isotropy, we explore four different models for the dark matter halo and find that a centrally-cored dark matter distribution is preferred. We infer a stellar mass-to-light ratio *Υ*⋆ , *v* = 6.9 ± 0.1 – consistent with a Salpeter-like IMF – and a core radius *r**c* = 67 ± 20 kpc. We then introduce anisotropy and find that, while the halo remains clearly cored, the radial stellar anisotropy has a strong impact on both *Υ*⋆ , *v* and the core’s radius; here we find *Υ*⋆ , *v* = 3.50− 0.36+ 0.32 – consistent with a Chabrier-like IMF – and *r**c* = 19.00− 8.34+ 8.38 kpc. Thus the presence of a core at the centre of the dark halo is robust against anisotropy assumptions, while the stellar mass and core size are not. We are able to reconcile previously discrepant studies by showing that modelling the globular cluster data alone leads to the very different inference of a super-NFW cusp, thus highlighting the value of multiple-population modelling, and we point to the possible role of M87’s AGN and the cluster environment in forming the central dark matter core.\ngalaxies: elliptical and lenticular, cD – galaxies: kinematics and dynamics – galaxies: haloes – galaxies: individual: M87 – galaxies: structure\nIntroduction\n============\nThe ΛCDM paradigm of structure formation has been very successful in describing the Universe on large scales, but there remains some tension regarding its predictions about galaxy structure. For instance, one main prediction of cold, collisionless gravitational collapse is the formation of a central cusp in the density profile of the dark matter (DM) haloes that envelope galaxies, with *ρ**D**M* ∼ *r*− *γ* and *γ* = 1 at small radii. However, real haloes also contain baryons, and the imprint of baryonic physics on the DM distribution could be significant. For instance, feedback from supernovae and AGN, as well as dynamical friction from infalling satellites, could lead to some degree of heating and expansion, thus hollowing out the DM, while the cooling and condensation of baryons could increase the density in the central regions via adiabatic contraction. The current observational picture reflects this complexity, with the haloes of an increasing number of systems being found to favour cored or only weakly cuspy central profiles. For instance, the recent local surveys THINGS and LITTLE THINGS found a large fraction of dwarf field galaxies to have DM density profiles that go as *ρ* ∝ *r*− 0.4 within the central kiloparsec, and studies of low-surface-brightness galaxies also point to relatively flat central profiles with a large scatter.\nWhile a great deal of progress has been made in constraining the halo structure of low-surface-brightness galaxies and dwarf spheroids, where DM dominates over the baryonic mass, the situation is much more complicated for their massive elliptical counterparts. Here, our ignorance about the stellar initial mass function (IMF) introduces a degeneracy between dark and luminous matter, which makes it hard to constrain the behaviour of the DM in the inner regions; equally, the task of probing the gravitational potential at large radii is made hard by the fact that their outskirts are notoriously faint. However, one way of significantly alleviating these degeneracies is to use multiple dynamical tracer populations, spanning a range of galactocentric radii. Indeed, massive elliptical galaxies are often home to large populations of planetary nebulae (PNe), globular clusters (GCs) and even, in the case of brightest cluster galaxies (BCGs), satellite galaxies, and each of these populations, with its own signature spatial distribution and kinematic profile, can be used as an independent probe of the gravitational potential. The pool of ETGs for which such an analysis has been carried out is currently too small for any meaningful conclusions to be extracted, though the suggestion also appears to be that the NFW profile may not provide a good universal fit: studies of BCGs by, and have found evidence for sub-NFW density profiles in clusters, while a handful of studies of field ellipticals, have found super-NFW densities. Clearly, a lot remains to be understood here, from both observational and theoretical perspectives.\nThe massive elliptical M87, located at the centre of the Virgo cluster, is an ideal subject for continuing such studies, as it has an enormous GC population, making it one of the richest GC hosts in the local Universe. Further, the sample of this population for which we have high-resolution kinematic data has been greatly expanded in recent years through the wide-field study of. It is also understood to be a slow rotator and nearly spherical, indicating that it is suitable for mass modelling under the assumption of spherical symmetry. However, two recent studies of M87’s mass distribution, both primarily based on Strader’s GC dataset, are markedly inconsistent. The first of these,, divided the GC sample into three independent populations and used a virial analysis to infer a very cuspy central density profile (*γ* = 1.57), while combined SAURON central stellar kinematics with the Strader GC data (along with an additional GC sample from Hectospec) and modelled the density profile using a logarithmic potential, thus imposing a core, which they infer to be *r**s* = 42 ± 10 kpc. Their total inferred stellar masses also differ by almost a factor of two. As M87 is one of the nearest and most well-observed BCGs, it seems unsatisfactory that its mass distribution should still be so poorly constrained, and clearly there remains work to be done. The aim of this paper is therefore to use a synthesis of GC, satellite and stellar kinematic data in conjunction with flexible mass models in order to infer a density profile which is free to be cuspy or cored as the dynamics dictate.\nThe paper is organised as follows: in Section [sec:data], we introduce the tracer populations used in our analysis and the associated datasets. In Section [sec:massmodel] we describe our mass model and Jeans analysis, the results of which are presented in Section [sec:results]. We discuss the implications of our findings in Section [sec:discussion], and use Section [sec:conclusions] to summarise our main conclusions. Throughout this work, we assume a distance to M87 *D**L* = 16.5 Mpc.\nDATA\n====\nTo constrain M87’s density profile across a wide radius range, we use multiple dynamical tracers, combining stellar kinematics in the central regions with GC dynamics at large radii and satellite galaxies on cluster scales. We can then solve the Jeans equation for each population separately, provided that the underlying density distribution is known. We therefore take data from a number of sources, as summarised in Table 1.\n| | | | | |\n| --- | --- | --- | --- | --- |\n| | radial coverage | data type | instruments | sources |\n| | 1.6 × 10− 3 - 210 kpc | photometry | multiple | |\n| | 0 - 2.5 kpc | kinematics | | |\n| | 0 - 0.17 kpc | kinematics | NIFS/Gemini Telescope | |\n| | 1.3 - 240 kpc | photometry | MegaPrime/CFHT | |\n| | 2 - 200 kpc | kinematics | multiple | |\n| | 35 - 1000 kpc | kinematics | multiple (mostly SDSS) | |\n| | | | | |\n[tab:sources]\n[fig:SBs]\nStars\n-----\nThe deprojected stellar surface brightness profile comes into our analysis at two points: first, the use of the stars as dynamical tracers in the Jeans equation requires us to know their 3D density distribution; second, our goal is to model M87’s mass as the sum of dark and luminous components, and the latter is simply the product of the integrated 3D luminosity density with some constant mass-to-light ratio, *Υ*⋆, which can be inferred from the data. We use the radial profile for M87 presented in, in which 20 sets of observations across a range of radii and filter systems were synthesised into a single profile in the V band. As M87 is known to have a very extended cD envelope in addition to a stellar core, its surface brightness profile cannot be accurately modelled by a Sérsic profile at both small and large radii. We therefore chose to model it using a more flexible Nuker profile, according to the following relation\n \n$$I(R) = I\\_0 \\Big(\\frac{r}{r\\_b}\\Big)^{-\\zeta} \\Big(1 + \\Big[\\frac{r}{r\\_b}\\Big]^{\\alpha}\\Big)^{\\frac{\\zeta-\\eta}{\\alpha}}$$ \nwith amplitude *I*0, break radius *r**b* = 1.05 kpc, inner slope *ζ* = 0.186, outer slope *η* = 1.88 and break softening *α* = 1.27. While this has the disadvantage of having no analytic deprojection or normalisation, it is much more flexible than a Sérsic profile as it allows both the inner and outer slopes greater freedom. Assuming spherical symmetry, we deproject this profile to give the 3D density shown in Figure [fig:SBs].\nThe kinematics of the inner 33ʺ × 41ʺ of M87 have been observed with the integral-field unit (IFU) SAURON, and a catalogue of the first four moments of the Gauss-Hermite expansion of the line-of-sight velocity distribution is available online[2](#fn2). We use the velocity dispersions, which were obtained from the spectra using a direct pixel fitting routine. The spectra were adaptively binned to ensure a signal-to-noise of at least 60 per spectral resolution element; uncertainties are generally less than ∼ 20 kms− 1, with a mean uncertainty of ∼ 9kms− 1.\nM87 is known to host a supermassive black hole (SMBH) of mass ∼ 6.6 × 109*M*⊙, and this should make a significant contribution to stellar velocity dispersions at the smallest radii. The SAURON dataset extends right down to the centre, though its resolution of 1ʺ is too low to be able to set constraints on the SMBH mass. As we are mainly interested in the behaviour of the DM and stellar components, one option to deal with this would be to simply exclude the apertures within the central ∼ 3ʺ from our analysis. However, though the contribution of the SMBH to the enclosed mass becomes sub-dominant beyond this approximate radius, it still contributes non-negligibly at larger radii and may be covariant with the stellar mass. We therefore include the SMBH in our mass model and constrain it using high-resolution kinematics of the central 2ʺ, as observed with the IFU NIFS on the Gemini North Telescope. As explained in that paper, these data were obtained from spectra which used laser adaptive optics corrections, and have a resolution of 0.08ʺ and a signal-to-noise generally greater than 50. The velocity moments are provided in radius and position angle bins, though our simplifying assumption of spherical symmetry allows us to combine bins azimuthally.\n[fig:GCmap]\nGlobular clusters\n-----------------\nWe use the colour and radial profiles for the GC populations of, as shown in Figure [fig:SBs]. The details of the inference are explained fully in that paper, but in brief, archival CFHT/MegaPrime images in the *ugriz* bands were used to compile a sample of 17620 GC candidates, selected according to their colours, magnitudes and apparent sizes, and the resulting catalogue was modelled to infer the radial and colour distributions and the relative fractions of the two GC populations. The surface density for each GC component was modelled as a Sérsic profile\n \n$$N(R) = N\\_0 \\exp{{\\Bigg[-k\\_n\\Big(\\frac{R}{R\\_e}\\Big)^{\\frac{1}{n}}\\Bigg]}}$$ \nwith *k**n* = 2*n* − 0.324, and with the profiles normalised such that ∫0*R**m**a**x**N*(*R*)*d**R* = 1, where *R**m**a**x* = 240 kpc is the radius of the outermost GC in the catalogue. The *g* − *r*, *g* − *i* colours and the GC luminosity function for each GC population were modelled as Gaussians, with radius-dependent colour gradients in all but the latter. The 3D deprojected profiles are also shown in Figure [fig:SBs].\n presents a spectroscopic catalogue of 737 GC candidates around M87 at radii from 2 kpc to 200 kpc (plus one object at 800 kpc, which we exclude because it lies outside the region over which our model from the photometry is strictly valid). The catalogue combines new measurements for 451 GCs – obtained using Keck/DEIMOS, Keck/LRIS and MMT/Hectospec – with literature data, and provides a ‘classification’ of objects as blue GCs, red GCs, ultra-compact dwarfs and transient/unknown along with SDSS-band *g* − *r* and *g* − *i* colours. We cross-correlate our photometric catalogue with this spectroscopic catalogue, selecting only the objects classified in as GCs, to obtain a sample of 612 GCs with complete luminosity, spatial, colour and kinematic information. In the analysis that follows, we choose to use the photometry over that provided in, for consistency with our GC colour distributions.\nSatellite galaxies\n------------------\nThe Extended Virgo Cluster Catalogue provides the redshifts and positions on the sky of 1589 galaxies in a footprint of 725 deg2 centred on M87, extending to 3.5 times the cluster virial radius. The redshifts are compiled from the SDSS DR7 release and the NASA Extragalactic Database (NED), and each object is classified as either a certain cluster member, a possible member or a background source based on morphological and spectroscopic criteria. As it is important that our sample only contains galaxies moving in M87’s halo potential, we selected only those objects classed as certain members according to both criteria, and we further cross-correlated these with the catalogue of, which used surface brightness fluctuations to calculate distance moduli. To avoid contamination from the W cloud, a slightly more distant component of the Virgo cluster at a characteristic distance of 23 Mpc, we imposed a distance cut of 20 Mpc; further, to separate the A cloud (centred on M87) from the B cloud (centred on M49) we also imposed a declination angle cut of 9.5 degrees and a radius cut of 1 Mpc. The spatial and velocity distributions of the resulting sample are shown in Figure [fig:GCmap]: it comprises 60 galaxies, with radii relative to M87 ranging from 35 kpc to 1 Mpc.\nWe could use this tracer population in the same way as the stars and the GCs and require its velocity dispersion profile to satisfy the Jeans equation so as to obtain a further probe of the mass at scales comparable to the virial radius. However, as noted earlier, the Jeans equation requires us to know the tracer density distribution and, as our satellite sample is most likely highly incomplete, we do not have access to this quantity. Most importantly, the sample of satellites we use has been selected spectroscopically, and this imposes a non-trivial selection function which may alter the spatial distribution from the true underlying one, leading us to draw incorrect conclusions from a Jeans analysis. Instead, we choose to use these satellites to give an estimate of the total mass at large radii and so give a further constraint in our inference on the mass profile. We do this using the virial mass estimator developed in and applied to a galaxy group in, which is designed to be robust against simple approximations to the true distributions. We use the mass estimator given by\n \n$$M(R<R\\_{out}) = \\frac{C}{G}\\big<v\\_{los}^2 r^{\\mu}\\big>,$$ \nwhere\n \n$$C = \\frac{\\mu + \\nu - 2\\beta}{I\\_{\\mu,\\nu}}r\\_{out}^{1-\\mu}$$ \nand\n \n$$I\\_{\\mu,\\nu} = \\frac{\\pi^{0.5}\\Gamma(\\frac{\\mu}{2} + 1)}{4\\Gamma(\\frac{\\mu}{2} + \\frac{5}{2})}[\\mu + 3 - \\beta(\\mu + 2)].$$ \nHere *β* is the anisotropy parameter\n \n$$\\beta = 1 - \\frac{\\sigma\\_t^2}{\\sigma\\_r^2}\n\\label{eq:binney}$$ \nfor radial and tangential velocity dispersions *σ**r*2 and *σ**t*2, and *μ* and *ν* are the slopes of the potential and tracer density respectively, both assumed to be scale-free. As mentioned previously, our satellite sample is likely to be incomplete, and this means we cannot infer *β*, *μ* and *ν* directly from the data; instead we calibrate their values using simulations. Following, we use the *z* = 0 halo catalogue of the first MultiDark simulation, desribed in detail in. This uses the WMAP5 cosmology and contains about 8.6 billion particles per Gpc/h3: the halo finder uses the bound density maximum technique described in. We identify all haloes with more than 30 subhaloes and treat each subhalo as a distinct satellite galaxy. We then use the subhalo velocities and positions and the parent halo mass profiles to infer the posterior distributions on *β*, *μ* and *ν* that best describe the global properties of the population, and use these to generate a posterior on the total mass *M*(*R* < *R**o**u**t*) of our cluster, whose median and standard deviation we use to re-select haloes from the simulation and iterate the procedure until our the inference on *μ*, *β* and *ν* converges. We report median values *β* = 0.30, *μ* = 0.15, *ν* = 1.9, and find a cluster mass\n \nlog(*M*(*R* < 985kpc)/*M*⊙) = 14.11 ± 0.19. \nTo test the calibration, we use the median values of *β*, *μ* and *ν* to apply the virial mass estimator to the simulated parent haloes, and the resulting comparison is shown in Figure [fig:VME]. Encouragingly, the masses are consistent, with a negligible median offset and a scatter of 0.1 dex, thus illustrating the robustness of the estimator. The applicability of this to M87 is then dependent on the assumption that the properties of the simulated galaxy satellite populations are representative of real galaxies.\n [fig:VME]\nModelling\n=========\nMass models\n-----------\nWe model the galaxy density profile as the sum of luminous and DM components plus a SMBH:\n \n*ρ*(*r*) = *ρ*⋆(*r*) + *ρ**D**M*(*r*) + *ρ**B**H*(*r*). \nThe stellar density profile is obtained by deprojecting the stellar surface brightness and scaling by the stellar mass-to-light ratio *Υ*⋆, which we assume to be constant with radius and include as a free parameter in the model. To allow flexibility in the DM density profile and to explore the impact of changing the halo model on the mass inference, we carry out our analysis using four different halo profiles. The first is a standard NFW profile,\n \n$$\\rho\\_{DM}(r) = \\frac{\\rho\\_0}{4\\pi} \\Big(\\frac{r}{r\\_s}\\Big)^{-1} \\Big(1 + \\frac{r}{r\\_s}\\Big)^{-2},$$ \nwith scale radius *r**s* and density scale *ρ*0, and cumulative mass\n \n$$M\\_{DM}(r) = \\rho\\_0 r\\_s^3\\Big(\\ln\\big(1+\\frac{r}{r\\_s}\\big) - \\frac{r}{r+r\\_s}\\Big).$$ \nFollowing, we also use a cored isothermal profile with a logarithmic potential (the LOG model)\n \n$$\\rho\\_{DM}(r) = \\frac{\\rho\\_{0}r\\_s^2}{4\\pi} \\frac{3r\\_s^2 + r^2}{(r\\_s^2 + r^2)^2}.$$ \nThis model is inherently cored, but in contrast to the NFW has a *ρ* ∼ *r*− 2 dependence at large radii: for a given core, then, this profile allows more dark matter to be placed at large distances from the galaxy centre. Its cumulative mass is given by\n \n$$M\\_{DM}(r) = \\frac{\\rho\\_0 r\\_s^2 r^3}{r\\_s^2+r^2}.$$ \nWe also use a generalised NFW (gNFW),\n \n$$\\rho\\_{DM}(r) = \\frac{\\rho\\_{0}}{4\\pi}\\Big(\\frac{r}{r\\_s}\\Big)^{-\\gamma} \\Big(1 + \\frac{r}{r\\_s}\\Big)^{\\gamma-3},$$ \nwith free parameters *γ*, *ρ*0 and *r**s*, where *γ* is the inner slope. This has the advantage of leaving the profile free to choose between cusps (*γ* ≥ 1) and cores (*γ* = 0) in the centre, while still becoming NFW-like at large radii in a way that is consistent with both simulations and observations. This gives a cumulative mass profile that can be related to the Gauss hypergeometric function 2*F*1 as\n \n$$M\\_{DM}(r) = \\frac{\\rho\\_{0}r\\_s^3}{\\omega}\\Big(\\frac{r}{r\\_s}\\Big)^{\\omega}~ \\_2F\\_1[\\omega,\\omega;1+\\omega;-\\frac{r}{r\\_s}]$$ \nwhere *ω* = 3 − *γ*. The final profile we use is a cored generalised NFW (cgNFW)\n \n$$\\rho\\_{DM}(r) = \\rho\\_{0}\\Big(\\frac{r+r\\_c}{r\\_s}\\Big)^{-\\gamma}~\\Big(1+\\frac{r}{r\\_s}\\Big)^{\\gamma-3}.$$ \nin which *r**c* is now the scale radius of the core. This is a more general case of the gNFW, and, in addition to allowing the data to choose between cusps (*r**c* = 0, *γ* = 1) and cores (*r**c* > 0, or *r**c* = 0 and *γ* = 0), it has additional flexibility at intermediate radii. We carry out the integration of the cumulative mass for this profile numerically.\nThe BH is simply a point mass at the origin,\n \n$$\\rho\\_{BH}(r) = \\frac{M\\_{BH}}{4\\pi r^2}\\delta(r),$$ \ngiving a constant term in the cumulative mass distribution.\nAs the SAURON dataset far outweighs the GC, NIFS and satellite datasets in terms of size, we regularise its contribution to the likelihood calculation by additionally fitting for a ‘noise’ parameter Δ*σ*⋆2. This can be interpreted as accounting for scatter in the data not included in the uncertainties or, alternatively, as modifying the relative weight given to these data, and is added in quadrature to the measured uncertainties on the velocity dispersion.\nOur overall model therefore has a number of free parameters dependent on the halo model in question and our assumptions about the anisotropy. In the isotropic case, which we treat first, the number of parameters varies between five and seven. In common for all halo models are the mass-to-light ratio *Υ*⋆, the normalisation of the DM halo log(*ρ*0), the scale radius *r**s*, the BH mass log(*M**B**H*) and the noise in the SAURON data, Δ*σ*⋆2. The gNFW and cgNFW halo models then add the free parameters (*γ*), (*γ*, *r**c*) respectively. When the orbital anisotropy of each tracer population is allowed to vary, this adds three parameters, as will be discussed in Section [sec:bayes]. In our notation in the following sections, we use the gNFW parameter set whenever we write out the model parameters explicitly, but this should be understood as standing in for any of the halo models.\nJeans modelling\n---------------\nGiven the stellar velocity dispersion and the GC and satellite velocities, we want to infer the posterior probability distribution on M87’s density profile. For the satellite galaxies, this involves a direct comparison of the mass calculated from our virial estimator and that obtained by integrating the proposed density profile. For the stars and GCs, on the other hand, we can relate the observed velocities and velocity dispersions to the density profile via the Jeans equation. Assuming spherical symmetry and dynamical equilibrium, the Jeans equation has the simple form\n \n$$\\frac{d}{dr}(l\\sigma\\_r^2) + 2\\frac{\\beta(r)}{r}l\\sigma\\_r^2 = l(r) \\frac{GM(r)}{r^2}$$ \nwhere *l*(*r*) is the luminosity density of the tracer, *σ**r*(*r*) the radial velocity dispersion and *β*(*r*) is the anisotropy parameter defined in Equation [eq:binney]. This is a first-order differential equation with general solution\n \n$$l(r)\\sigma\\_r^2(r) = \\frac{1}{f(r)} \\int\\_r^{\\infty} f(s) l(s) \\frac{G M(s)}{s^2}ds$$ \nwhere\n \n$$f(r) = f(r\\_i) \\exp\\Big[\\int\\_{r\\_i}^r 2 \\beta(s) \\frac{ds}{s}\\Big]\n\\label{eq:IF}$$ \n. Projecting this along the line of sight gives *σ**l**o**s*2(*R*) as\n \n$$\\frac{1}{2G} I(R) \\sigma\\_{los}(R)^2 = \\int\\_R^{\\infty} \\frac{l \\sigma\\_r^2 r dr}{\\sqrt{r^2 - R^2}} - R^2 \\int\\_R^{\\infty}\\frac{\\beta l \\sigma\\_r^2 dr}{r\\sqrt{r^2-R^2}}\n\\label{eq:jeans}$$ \nwhich, for certain choices of anisotropy parameterisations, can be written in the simple form\n \n$$I(R) \\sigma\\_{los}^2(R) = 2G \\int\\_R^{\\infty} l K\\_{\\beta} M\\frac{dr}{r}\n\\label{eq:kernel}$$ \nwith the kernel *K**β* dependent on the particular anisotropy model. Initially, we assume all orbits to be isotropic; later we consider the effect of anisotropy on our inference by modelling the stellar population as following a radially-dependent anistropy profile and each GC population with a constant, non-zero anisotropy. For the stars, we use a scaled Osipkov-Merritt profile\n \n$$\\beta(r) = \\beta\\_{\\infty} \\frac{r^2}{r^2 + r\\_a^2};\n\\label{eq:scaledOM}$$ \nwhich is centrally isotropic and becomes radially anisotropic at large radii, tending to the asymptotic anisotropy *β*∞. The kernel for the scaled Osipkov-Merritt profile is presented in the Appendix; the kernel for the constant-anisotropy case is given in, and we refer the reader there for further details.\nGiven a prescription for the anisotropy, the surface brightness and luminosity profiles of Figure [fig:SBs] and the mass model, then, we can calculate line-of-sight velocity dispersions *σ**l**o**s* from the Jeans equation and so infer the posterior probability distribution of the model parameters, based on the data in hand.\nStatistical analysis\n--------------------\nWe use the methods of Bayesian analysis to infer the posterior probability distribution of our density profile parameters, given the data. Bayes theorem states that the posterior distribution is proportional to the product of the likelihood function of the data given the model and the priors on the model, and so the task here is to construct sensible likelihood functions for each dataset. The total likelihood is then, in turn, the product of these, as each consitutes an independent set of measurements.\nFirst, as we have velocity dispersion measurements for the stars, the likelihood of observing a particular velocity dispersion at radius *R* is assumed to be Gaussian, with a standard deviation equal to the uncertainty. Thus the *k**t**h* stellar velocity dispersion measurement gives a contribution to the likelihood:\n \n$$\\ln L\\_{\\star,k}(\\sigma\\_k,R\\_k|\\vec{M}) = -0.5\\Bigg(\\frac{\\sigma\\_k^2 - \\sigma\\_m^2}{\\delta\\_{\\sigma\\_k^2}}\\Bigg)^2 - 0.5\\ln(2\\pi\\delta\\_{\\sigma\\_k^2}^2)$$ \nfor uncertainty *δ**σ**k*2 and model prediction *σ**m*2, and model parameters *M⃗* = (*Υ*⋆, *M**B**H*, Δ*σ*⋆2, *ρ*0, *γ*, *r**s*). As the observations in each aperture are independent, the total log likelihood of observing the ensemble is just the sum:\n \n$$\\begin{split}\n\\ln L\\_{\\star} &= \\sum\\_k \\ln L\\_{\\star,k}.\n\\end{split}$$ \nNote that, for measurement uncertainty Δ*σ**k*22, the regularisation of the SAURON data gives a total uncertainty *δ**σ**k*22 = Δ*σ**k*22 + Δ*σ*⋆22. For the NIFS data, on the other hand, *δ**σ**k*22 = Δ*σ**k*22.\nThe virial mass estimate from the satellites can be treated in a similar way, though here we only have one measurement:\n \n$$\\ln L\\_{sat} = -0.5\\Bigg(\\frac{\\log(M\\_{sat})-\\log(M\\_{mod})}{\\delta M\\_{sat}}\\Bigg)^2 - 0.5 \\ln\\Big(2\\pi\\delta M\\_{sat}^2\\Big)$$ \nfor virial mass estimate *M**s**a**t*, model mass *M**m**o**d* and the logarithm of the uncertainty from the mass estimator, *δ**M**s**a**t*. This is a direct comparison without the need for recourse to the Jeans equation.\nFor the GCs we can assign probabilities, based on the colour and position information provided in the photometric catalogue, of each GC belonging to either the red or the blue population, although we are not able to classify them with certainty. As the two populations are assumed to be dynamically decoupled, we do not expect their velocity dispersion profiles to be the same. In contrast to the other tracer populations, then, we calculate the likelihood of observing a GC with a particular velocity under the assumption that the velocity distribution of each GC population can be described by a Gaussian with a standard deviation given by the velocity dispersion, such that\n \n$$\\ln L\\_{GC,k} = -0.5\\Bigg(\\frac{(v\\_k-v\\_{gal})^2}{\\delta v\\_k^2 +\\sigma\\_{m}^2}\\Bigg) - 0.5\\ln\\Big(2\\pi(\\delta v\\_k^2 + \\sigma\\_{m}^2)\\Big)$$ \nwhere *v**k* and *δ**v**k* are the measured line-of-sight velocity and velocity uncertainty of the *k**t**h* GC, *v**g**a**l* = 1284 kms− 1 is the heliocentric velocity of M87 and the line-of-sight velocity dispersion at the location of each globular cluster, *σ**m*, is modelled separately for the red and the blue populations.\nTo account for the uncertainty in assigning each GC to either the red or the blue population, for each set of model parameters *M⃗* we draw 1000 Monte Carlo samples which stochastically explore the population distribution based on the colour, magnitude and position information for the individual GCs. We then marginalise over the samples to give a final contribution to the likelihood. While some GCs have either very high or very low probabilities of belonging to one of the GC populations, with small uncertainty, there also exists a significant fraction with comparable probabilities of belonging to either: for these objects, it would not be meaningful to simply assign them to one population or the other. A further advantage of this stochastic sampling is that it allows us to explore different combinations of red and blue GCs.\nAs each tracer population is independent, the final log-likelihood of any set of model parameters is the sum over all contributions:\n \nln*L* = ∑*k*ln*L*⋆ , *k* + ∑*k*ln*L**G**C*, *k* + ln*L**M**s**a**t*. \nWe explore the parameter space using the ensemble-sampling code `emcee`.\nResults\n=======\nC1cmC1.4cmC1.4cmcccccchalo model & *M*⋆/*L* & log(*ρ**D**M*) & *r**s* & *γ* & *δ**σ*⋆2 & *r**c* & log(*M**v**i**r*) & *R**v**i**r* \nNFW & 6.6 ± 0.1 & 6.6 ± 0.1 & 448 ± 75 & – & 12.1 ± 0.2 & – & 14.39− 0.53+ 1.11 & 1620− 360+ 460 \nLOG & 6.9 ± 0.1 & 7.7 ± 0.1 & 48 ± 5 & – & 12.0 ± 0.2 & – & 14.21− 0.37+ 0.60 & 1410− 200+ 240 \ngNFW & 6.9 ± 0.1 & 8.3 ± 0.1 & 79 ± 10 & < 0.14 & 12 ± 0.1 & – & 14.13− 0.44+ 0.76 & 1320− 240+ 270 \ncgNFW\\* & 6.9 ± 0.1 & 5.6− 1.2+ 1.4 & 273− 180+ 430 & 2.54− 1.18+ 0.33 & 12.0 ± 0.2 & 63− 14+ 11 & 14.16− 0.40+ 0.67 & 1350− 200+ 240 \n[tab:results]\n [fig:triangleNFW]\n [fig:triangleLOG]\n [fig:triangleBPL]\n [fig:triangleBBPL]\n[fig:MassProfiles]\n [fig:triangleNFWaniso]\n [fig:triangleLOGaniso]\n [fig:triangleBPLaniso]\n [fig:triangleBBPLaniso]\nInitially, we treat all orbits as isotropic and investigate the uncertainty introduced to the mass inference by the choice of halo model. The inference for the four different halo models are presented in Figures [fig:triangleNFW] - [fig:triangleBBPL], and are summarised in Table [tab:results] along with the associated virial parameters. We find the results of the gNFW, cgNFW and LOG models to be very similar: the gNFW and cgNFW haloes are both centrally cored with break radii ∼ 80 kpc, while the LOG model is inherently cored and has a slightly smaller scale radius *r**s* = 48 ± 5 kpc (note, however, that these are *not* core radii, and that the scale radius is defined differently in the latter case). All three models also agree closely on the high stellar mass-to-light ratio *Υ*⋆ , *v* ∼ 6.9 and the virial parameters, with log(*M**v**i**r*/*M*⊙) ∼ 14.2 and R*v**i**r* ∼ 1.3 Mpc. Indeed, the best cgNFW model approximately recovers the gNFW solution, with an intermediate slope *γ* ∼ 3 and both the core and break radii comparable to the gNFW scale radius. Like the gNFW, this represents a solution in which the halo is cored centrally and becomes NFW-like at large radii; unlike the gNFW, it allows more flexibility in the intermediate regions, and the fact that this intermediate slope is found to be close to the NFW value *γ* ∼ 3 suggests that the NFW profile is an adequate description at intermediate radii as well as large radii.\nOn the other hand, the NFW model predicts a lower mass-to-light ratio *Υ*⋆ , *V* = 6.60 ± 0.05, though it does match the other models in terms of its virial parameters. The difference at small radii arises because of the NFW profile’s hard-wired cusp, which places more DM in the central regions at the expense of stellar mass. In Figure [fig:MassProfiles] we plot the mass profiles inferred in each case, along with the associated rotation curves in order to better highlight the differences between the four models. As can be seen, deviations start to arise at larger radii, where the constraints from our data are weakest. These are mainly due to the different structural features of the models: for instance, the LOG profile decays as *r*− 2 at large radii whereas the other three go as *r*− 3, which allows the former to place more mass at large radii. Equally, the NFW profile goes as *r*− 1 at small radii whereas the others are either intrinsically cored or have flexible inner slopes (which are found to favour cores here), and this causes the NFW profile to start deviating from the others at smaller radii.\nFollowing this, we test the robustness of our inference on the halo structure against model assumptions by relaxing the isotropy condition and introducing some element of anisotropy into each of the tracer populations. Guided by cosmological N-body simulations,, which find orbits to change smoothly from being radially anisotropic in the galaxy outskirts to being isotropic in the centre – a result of hierarchical formation in which BCGs form by the accretion of infalling satellites which become phase-mixed by the time they reach the galaxy centre – we choose to model the stellar anisotropy using the scaled Osipkov-Merritt profile of Equation [eq:scaledOM]. As the GC data are comparatively sparse, we treat each GC population as having some constant, non-zero anisotropy. This increases our parameter space by four, as we are now also fitting for the scale radius of the stellar scaled Osipkov-Merrit profile *r**a*, the asymptotic stellar anisotropy *β*∞, and the anisotropies of the two GC populatons, *β**r**e**d* and *β**b**l**u**e*. Again, we carry out the inference for the four halo models, and present the posteriors in Figures [fig:triangleNFWaniso] - [fig:triangleBBPLaniso] and summarise the results in Table [tab:anisoresults].\nInterestingly, we now find the some significant deviation among the halo models. Firstly, the LOG profile recovers a model for the stars that is quite isotropic, with *β*∞ > 0.99 at the 95% confidence level and *r**a* = 9.29 ± 0.05 kpc; given that the SAURON data only extend out to ∼ 2 kpc, this implies anisotropies *β* < 0.05 where we have data (though our data are in projection). This is presumably a result of the hard-wired core. The remaining three models agree, within uncertainties, on the mass profile – though, as might be expected, not so closely as when isotropy is enforced. In these cases, we find that the mass-to-light ratio drops significantly and that the stars become radially anisotropic within scale radii *r**a* < 4 kpc. This covariance between the central mass and anisotropy is a direct manifestation of the mass-anisotropy degeneracy: in the isotropic models, the assumed lack of any radial anisotropy drove the central mass to high values, whereas here we are able to go some way towards breaking the degeneracy through the use of multiple populations tracing the same gravitational potential. Note also that the introduction of radial anisotropy, coupled with the decrease in stellar mass, now allows the DM halo to become less cored in the centre: we find a gNFW inner slope which is still distinctly sub-NFW, but less extremely so, with *γ* < 0.81 at the 95 % confidence level, and the cgNFW model finds a smaller central core (*r**c* = 19.00− 8.34+ 8.38 kpc) and a sub-NFW intermediate slope *γ* = 2.39− 0.43+ 0.39 before transitioning to the *r*− 3 regime beyond the scale radius *r**s* = 412.1− 143.0+ 123.0 kpc. For these three profiles, both GC populations are found to have a mild radial anisotropies. We find a slightly higher BH mass than previous estimates, though it is consistent within uncertainties. However, we note that this is the first time the BH mass has been inferred jointly with the halo and stellar mass parameters using the high-spatial-resolution NIFS dataset; as we might anticipate a strong covariance between these parameters – especially between the stellar and BH masses – a slight change is not too surprising. On the other hand, there are still assumptions in our model which could affect our inference on the BH mass. For instance, our anisotropy profile enforces isotropy in the centre, which, while a physically sound assumption, restricts the range of parameter space explored by the BH mass. We are also only making use of the second-order moments of the line-of-sight velocity profile: while this should not cause any systematic bias in the BH mass inference, we may be able to obtain tighter constraints by incorporating higher-order moments.\nWe compare the goodness of fit of the four models using a reduced chi-squared criterion, and find that the cgNFW profile provides the best description of the dynamics – noting that the cgNFW, gNFW and NFW are nested models. We therefore present the best cgNFW mass profile in Figure [fig:mass] and the residuals on the SAURON and NIFS data in Figure [fig:residuals]. The mass profile appears to be generally consistent with earlier work; the residuals are noticeably better than in the isotropic case, indicating the importance of the anisotropy in the stellar dynamics.\nWe also note that the assumption that the MultiDark simulations are able to accurately reproduce the dynamics of real galaxies may introduce some additional systematic uncertainty into the mass measurement made using the virial estimator in Section [sec:data:satellites]. We therefore test the sensitivity of our inference to this uncertainty by carrying out the analysis with twice the calculated uncertainty, and we find that this has a negligible effect on our inferred density model. Essentially, the likelihood is dominated by contributions from the stars and the GCs, as the datasets for these populations are far more extensive; the satellite mass estimate rejects models with discrepant extrapolations of the total mass to large radii, but most of the models consistent with the smaller-radius data are still well-behaved at 1 Mpc.\ncC0.8cmC1.0cmC1.0cmC1.0cmC1.0cmC1.0cmC1.0cmC1.0cmC1.0cmC1.0cmC1.0cmC1.0cmC1.0cm& *M*⋆/*L* & log(*ρ**D**M*) & *r**s* & *γ* & *M**B**H* × 10− 9 & *δ**σ*⋆2 & *r**a* & *β*∞ & *β**r**e**d* & *β**b**l**u**e* & *r**c* & log(*M**v**i**r*) & *R**v**i**r* \nNFW & 3.05− 0.24+ 0.24 & 7.51− 0.12+ 0.13 & 128.4− 24.1+ 27.8 & – & 7.04− 0.2+ 0.2 & 9.34− 0.19+ 0.20 & 2.78− 1.30+ 1.26 & 0.690.04+ 0.04 & 0.16− 0.21+ 0.21 & 0.34− 0.17+ 0.15 & – & 14.00− 0.32+ 0.37 & 1190− 120+ 240 \nLOG & 6.48− 0.02+ 0.02 & 8.25− 0.19+ 0.23 & 22.1− 5.7+ 6.7 & – & 9.5− 0.23+ 0.22 & 10.2− 0.2+ 0.2 & 9.29− 0.05+ 0.05 & > 0.99 & − 0.17− 0.21+ 0.23 & 0.08− 0.29+ 0.23 & – & 14.03− 0.29+ 0.44 & 1200− 190+ 250 \ngNFW & 3.33− 0.45+ 0.38 & 8.64− 0.35+ 0.24 & 40.3− 8.8+ 16.4 & < 0.81 & 7.2− 0.4+ 0.35 & 9.3− 0.2+ 0.2 & 1.06− 0.10+ 0.07 & 0.67− 0.05+ 0.05 & 0.13− 0.20+ 0.21 & 0.23− 0.17+ 0.17 & – & 13.74− 0.32+ 0.43 & 990− 140+ 270 \ncgNFW & 3.50− 0.36+ 0.32 & 4.76− 0.40+ 0.55 & 412.1− 143.0+ 123.0 & 2.39− 0.43+ 0.39 & 7.22− 0.40+ 0.34 & 9.33− 0.20+ 0.21 & 1.06− 0.05+ 0.08 & 0.67− 0.06+ 0.03 & 0.13− 0.20+ 0.21 & 0.23− 0.18+ 0.16 & 19.00− 8.34+ 8.38 & 13.87− 0.35+ 0.42 & 1090− 150+ 220 \n[tab:anisoresults]\n[fig:mass]\n[fig:residuals]\nDiscussion\n==========\nResolving previous discrepancies\n--------------------------------\nPart of the impetus for this study was to resolve the discrepancies between recent models of M87’s mass structure. The two recent analyses of and, both of which relied heavily on the GC kinematics, disagreed on both the structure of the DM halo and the total stellar mass of the system, with finding a stellar mass-to-light ratio *Υ*⋆ , *V* ∼ 4.5 and a super-NFW cusp *γ* = 1.6 and inferring *Υ*⋆ , *I* = 6.0 ± 0.3 (which converts to a *V*-band measurement of ∼ 7.5) under the assumption of a cored halo. There are a number of possible reasons for this: first, while both studies overlapped in the majority of their GC data, each used a different mass model, and it is possible that these may have unnecessarily constrained the inference on the mass. Specifically, chose to use a logarithmic potential model for the dark matter, so enforcing a core, whereas used a power law for the dark halo, thus requiring a constant slope at all radii. Neither of these allows the halo much flexibility in the central regions. Further, while relied solely on the GCs, separating them into three independent populations based on their velocities, positions and colours, treated all the GCs as a single tracer population, but used the same SAURON data as in this study to constrain the stellar mass-to-light ratio. Thus it is possible that lacked the data coverage in these central regions that would have permitted a reliable distinction between cusps and cores.\nBy exploring multiple mass models and data combinations, we are able to reproduce the results of both studies. First, excluding the stellar kinematics and carrying out the inference using just the satellites and GCs in a way more similar to that of, we infer an isotropic gNFW profile with *γ* = 1.48 ± 0.22, thus reproducing their finding of a cusp. The inference is summarised in Figure [fig:cusp]. This data combination also provides only a very weak constraint on the stellar mass-to-light ratio, as might be expected given the relatively small number of GCs in the centre as compared to the stars. When we compare the predictions of the best-fit model in this case, however, we find that it significantly overpredicts the stellar velocity dispersions in the centre. Adding in the stellar data, then, requires an excavation of the central regions, giving rise to a sub-NFW inner slope more easily reconciled with the assumption of, although we do still find a lower stellar mass, which could be a consequence of the fact that, unlike, we include the BH in our models. The solution of the discrepancy between the two studies is presumably then that it is simply not possible to constrain the whole mass profile using GC data alone. This really shows the importance of combining multiple tracer populations with different characteristic radii.\n [fig:cusp]\nWe also perform the first joint inference on the BH, stellar and halo mass parameters that has been carried out with the high-resolution NIFS kinematics of the stars in the central 2ʺ. When these data were presented in, the structure and scale of the halo and stellar distributions were fixed at ’s values, which in turn were inferred assuming the BH mass measured in, based on lower-resolution data. The BH mass that we infer is consistent with the value within uncertainties: for the best cgNFW model, we find 7.22− 0.40+ 0.34 × 109*M*⊙.\nIn previous models of M87, significant tension has also arisen between the ratio of the stellar and halo masses and expectations from abundance matching, and this has tended to be related to a consistently high value of the stellar mass-to-light ratio, *Υ*⋆. For instance, infer a very high mass-to-light ratio *Υ*⋆ , *V* ∼ 7.5, despite the circular velocity of their LOG halo profile being slightly lower than ours (at *v**c* = 591 ± 50 kms− 1 compared to our 610 kms− 1), and this places it above the abundance matching predictions of, even allowing for uncertainties. infers both a larger stellar mass (with *Υ*⋆ , *V* = 9.1 ± 0.2 at an assumed distance *D**L* = 17.9 Mpc, which translates to *Υ*⋆ , *V* = 9.9 at our assumed distance) and a larger halo mass (log(*M**v**i**r*/*M*⊙) = 14.37, *R**v**i**r* = 1.6 Mpc), also placing M87 above abundance matching expectations. While our Nuker profile for the projected light is not strictly normalisable, we compare our results using the absolute magnitude of M87 *M**v* = 8.30 reported in, corrected for dust reddening using the extinction given in that study. This gives a stellar mass log(*M**l**u**m*/*M*⊙) = 11.61 ± 0.10. Compared with the virial mass log(*M**v**i**r*/*M*⊙) = 13.87− 0.35+ 0.42, this puts M87 just within the abundance-matching curves of, allowing for uncertainties, and so alleviating this tension. We note, however, that all of our isotropic models – as well as our anisotropic LOG model – also yield anomalously high stellar masses relative to the halo, and furthermore that all these previous studies assumed a form for the halo given by the LOG profile, which we find to not perform as well as the more generalised NFW-like parameterisations. Seen together, then, this highlights the need for flexible density and anisotropy models in order to properly understand the astrophysics of these systems.\nThe case for a cored halo\n-------------------------\nIt is important to compare our findings for the halo structure with other ETGs; as explained in the Introduction, this is crucial for developing our understanding of the role of baryons in shaping the halo and the diversity of structures that can arise.\nA number of other studies have found BCGs to have DM profiles which are flatter than NFW:, for instance, combined stellar dynamics with weak and strong graviational lensing and X-ray data to infer a density slope *γ* < 1 for the BCG Abell 383 with 95 % confidence, while went on to fit gNFW halo models to a sample of seven massive BCGs and found a mean slope of *γ* = 0.5 ± 0.1, inconsistent with the NFW prediction – and with three of those galaxies hitting the *γ* = 0 prior. This is also in line with the earlier studies of ; more recently, have also modelled multiple image families lensed by the cluster Abell S1063 to infer a significant core radius *r**c* ∼ 100 kpc. The picture, however, is quite different for elliptical galaxies in the field, and a number of strong lensing studies of ETGs have found halo slopes consistent with NFW or even super-NFW DM profiles:, for instance, modelled the global properties of a sample of 81 lenses and found the inner slope to be consistent with an NFW profile, while performed a similar analysis on a smaller sample of lenses from the Sloan Lens ACS Survey (SLACS) and found *γ* = 1.7 ± 0.5, which is very inconsistent with what we find here. At first sight, this might seem to point tentatively to the existence of two very different evolutionary paths for ETGs in these two environments, and the implausibility of a one-size-fits-all DM halo profile: however, the sample size of ETGs for which this analysis has been carried out clearly remains too small for any meanungful conclusions on this issue.\nWe now consider the implications of M87’s central core in the context of the baryonic feedback processes that play a role in the centres of galaxies. While a cored halo is inconsistent with DM-only simulations, there have been a number of recent simulations in which baryonic effects have been included, and we note some similarities with these. Here, we focus on two of the most recent studies of this type in order to elucidate the physics, though we stress that this comparison is by no means exhaustive and a number of similar projects have been carried out. Firstly, we consider the zoom-in simulations of, which followed two BCGs from redshift *z* = 2 to *z* = 0 and demonstrated that the effect of repeated dissipationless mergers is to soften the otherwise-NFW-like cusp by Δ*γ* ∼ 0.3 − 0.4 on scales of the stellar half-light radius. In this paradigm, infalling satellite galaxies experience dynamical friction from DM as they move through the halo, and this transfers energy to the DM, causing it to expand and so become less dense in the central regions. While we do not measure a core radius explicitly, in Figure [fig:drhodr] we plot the density slope as a function of radius, dlog*ρ*/dlog*r*, and see that the slope becomes sub-NFW at radii *r* < 10 kpc and that *γ* ∼ 0.6 (equivalent to Δ*γ* ∼ 0.4) at r ∼ 5 kpc, which is comparable to the scale radius of the starlight measured by. However, we find a cored centre while these simulations find only 0.6 < *γ* < 1 down to the innermost resolvable radius. While they suggest that the merging of SMBHs could then give rise to cores in the central 3-4 kpc – with the binary BH system spiralling inwards and transferring energy to the surrounding matter – their simulations do not include the effects of SMBHs and so they are unable to test this. It therefore remains unclear as to whether merging events would be capable of totally erasing the cusp.\nOn the other hand, M87 has a large AGN at its centre and this may also contribute to core formation. The second study that we examine is that of, which specifically simulated Virgo-like ETGs using a recipe for AGN feedback. They found cores to develop within the inner 10 kpc. In this scheme, outflows of gas due to the AGN are able to irreversibly modify the gravitational potential of the halo and so cause expansion of both the dark and luminous matter. In that study, they suggest that a combination of AGN feedback and SMBH effects are the main contributing processes, though they also note that the large amount of gas expelled by the AGN increases the efficiency of the energy transfer due to dynamical friction. This role for AGN is analogous to the role of supernovae in dwarf spheroids. However, while the core radius of 10 kpc seems consistent with the density profile that we infer, predict cores in the 3D stellar density, whereas M87 is known to have a ‘cuspy core’ in its 2D surface brightness (*γ* = 0.18), making it considerably cuspy in 3D. Indeed, the finding of cores in the 3D density distribution of the stars seems at odds with observations of large numbers of ETGs. Again, this seems to be a sign of the complex nature of the processes governing the shape of the density profile, and that there still remain many questions to be answered on this subject.\n [fig:drhodr]\nImplications for the stellar initial mass function\n--------------------------------------------------\nOne of the initial motivations for this investigation was to resolve the striking discrepancy between the *Υ*⋆ , *V* inferred by and, particularly in light of recent evidence that massive ETGs may have Salpeter-like stellar mass-to-light ratios. In our anisotropic cgNFW model, we find *Υ*⋆ , *V* = 3.50− 0.36+ 0.32; however, infer a much higher value *Υ*⋆ , *V* ∼ 7.5, and previous studies have found mass-to-light ratios even higher than this. We use the single stellar population models from to understand the implications of the stellar mass-to-light ratio for the age and IMF of M87.\nWe measure colours for M87 from the CFHT photometry and compare these with the BC03 models. In particular, we determine the mean surface brightness within a 2′′ annulus at 30′′ (i.e., ∼ 2.5 kpc) from the centre of M87, masking globular clusters and other artefacts, for each of the *u**g**r**i**z* filters. We find extinction-corrected *u* − *g*, *g* − *r*, *g* − *i*, and *g* − *z* colours of 1.48, 0.67, 1.13, and 1.45, respectively, which agree well with the colours calculated over a much larger aperture from SDSS photometry. Then, assuming a Salpeter IMF and solar metallicity, the BC03 models give an age of 10 Gyr and *Υ*\\* , *V* ≈ 5.7, significantly lower than the or values and thus implying the need for an IMF even more extreme than that found by if these high mass-to-light ratios are to be explained. If we instead assume a Chabrier IMF and solar metallicity, the BC03 models still find a ∼ 10 Gyr age, but *Υ*\\* , *V* ≈ 3.7, which is in good agreement with our dynamical measurement when we allow for anisotropy. We note, however, that our dynamical inference on *Υ**V*, \\* is very sensitive to our assumptions about anisotropy. Furthermore, our assumption of solar metallicity – and, indeed, a constant *Υ*\\* , *V* with radius – may be inappropriate. For example, use UV to NIR photometry to demonstrate that M87 has colour gradients, and they interpret these as being the result of a metallicity gradient.\nImportance of correctly modelling the underlying tracer populations\n-------------------------------------------------------------------\nThe Jeans equation requires us to know the underlying tracer density for each population, and this, along with the mass profile and anisotropy, determines the velocity dispersion that is measured. Therefore any uncertainty or bias in the parameterisation of the tracer density might manifest itself as uncertainty or bias in the resulting velocity dispersions and, in a study such as this in which the velocity dispersions are the primary tool for the inference, also as uncertainty or bias in the mass profile. A particular worry here is the distributions of the GCs, as the spectroscopic catalogue of is only a subsample of the wider population and, by virtue of its being selected for spectroscopy, is subject to some non-trivial selection function that may change the apparent distribution from that of the true underlying one. It is therefore important to model the distribution using an independent photometric sample. Indeed, this was the motivation for the initial photometric study of.\nTo demonstrate the impact on the inference of using the density of the spectroscopic subsample in the Jeans analysis rather than that of the more repsresentative photometric sample, we repeat our analysis using the three-population decomposition given in. This decomposition was carried out based solely on the spectroscopic catalogue, and separates the GCs into a compact red component along with two much more extended blue and intermediate-colour components. Our anaylsis differs from ’s in that they focussed mainly on virial decompositions and used the GC data exclusively, whereas we use a spherical Jeans analysis and combine the GCs with the stellar kinematics and satellite galaxy datasets. This comparison is not intended as a comment on their work, but merely aims to show that, in this particular analysis, the use of this approximation to the true distribution is important. When we carry out the inference using the three populations, we find that we underpredict the halo mass in the region in which the GCs dominate the fit by ∼ 0.25 dex. *M**v**i**r* also decreases by ∼ 10%, though the effect is presumably smaller here because the GC data only extend out to a fraction of the virial radius. Unsurprisingly, our inference on the stellar mass-to-light ratio is robust against the GC Sérsic distributions, as this is mainly determined by the properties of the stellar population, which is fixed by our fit to the surface brightness profile.\nAnother possible bias in spherical Jeans modelling could result from any flattening in the potential: indeed, there have been recent suggestions that the shape of a galaxy’s DM halo may correlate with the shape of its luminous matter, and, while the modelling of assumed the GCs to be spherically distributed, they do exhibit some ellipticity at larger radii. However, the effect of this was analysed in using a series of axisymmetric Jeans models, and it was found that the impact of any realistic amount of flattening on the mass profile was small, with up to a 10 % decrease in the mass for GCs with elliptical profiles. This is therefore unlikely to be a major concern here.\nAnisotropy, and a pinch radius from multiple tracer populations\n---------------------------------------------------------------\nAs explained in Section [sec:results], we incorporated anisotropy in our models by assuming a scaled Osipkov-Merritt profile for the stars and constant anisotropies for the GC populations, and inferred mild radial anisotropies for the three populations, which in turn led to changes in our inference on the mass model relative to the isotropic case. As the inclusion of the anisotropy parameters is considerably more expensive from a computational point of view, it is of interest to investigate ways of obtaining robust mass estimates using models with fixed anisotropy. It is well known that, for the Jeans analysis of a single tracer population, there exists a pinch radius at which the dependence of the inferred mass profile on the anisotropy is minimal. This occurs at approximately the 3D effective radius of the tracer and can be quoted as a robust measure of the enclosed mass – instead of an entire radial profile – when it is not possible to infer the anisotropy from the data. However, our case is slightly more complicated than those studied by the previous authors as we are combining three completely independent tracer populations. It is therefore of great interest to test whether the pinch radius is still recovered in this case.\nFor simplicity, we run a series of models in which the stellar anisotropy is fixed at some non-zero constant value while the GCs remain isotropic. This is guided by our earlier results, which found the stars to be considerably more anisotropic than the GCs. To span a range of anisotropies, we let the anisotropy run from *β* = − 1 (corresponding to the case where the velocity dispersion in the radial direction is twice that in the radial direction) to *β* = 0.5 (at which point the tangential velocity dispersion is half the radial velocity dispersion). The results are shown in Figure [fig:pinch]. Indeed, we do find the three *β*-curves to intersect at a single radius, as in the case of single-tracer population models: however, this occurs at a radius = 111 ± 4 kpc (with mass log(*M*(*R* < *R**p*)/*M*⊙) = 13.09 ± 0.40), which is much too large to be associated with the effective radius of the starlight. This is presumably a consequence of our use of additional tracer populations, all at larger radii than the stars. This is an interesting finding which will be useful for further multiple-population studies of this kind.\n [fig:pinch]\nConclusions\n===========\nWe have modelled the mass profile of M87 using a combination of stellar, GC and satellite galaxy kinematics in a Jeans analysis, and our main conclusions are as follows:\n1. M87 is a massive BCG with a sub-NFW DM halo. It has a virial mass of log(*M**v**i**r*/*M*⊙) = 13.87− 0.35+ 0.42, a virial radius *R**v**i**r* = 1.1 ± 0.2 Mpc and a stellar mass log(*M**l**u**m*/*M*⊙) = 11.61 ± 0.10, placing it at the top end of the galaxy mass distribution and just within the stellar-to-halo mass relations expected from abundance matching.\n2. Under assumptions of isotropy, the scatter introduced by the use of different models to parameterise the halo is small. In this paradigm, we find M87 to have a high stellar mass-to-light ratio of *Υ*⋆ = 6.9 ± 0.1 in the *V*-band and a cored DM halo, with inner slope *γ* < 0.14 at the 95 % confidence level.\n3. When the model is modified to allow each tracer population some element of anisotropy, differences arise between the halo models, and a cored generalised NFW profile provides the best description of the dynamics. This reduces the stellar mass-to-light ratio to *Υ*⋆ = 3.50− 0.36+ 0.55 and slightly relaxes the constraint on the inner slope, with a core radius *r**c* = 19.0 ± 8.3 kpc. All three tracer populations are characterised by mildly radially anisotropic orbits.\n4. The *V*-band stellar mass-to-light ratio of M87 is consistent with a picture in which its stellar populations are old ( ∼ 10 Gyrs), with solar metallicity. In the isotropic case, this implies a Salpeter-like IMF, whereas when anisotropy is accounted for, a Chabrier-like IMF is preferred.\n5. The inclusion of tracers at a variety of spatial scales has a significant impact on the inference. Modelling the mass using the GCs and satellite galaxies alone - that is, without the stars - we infer a cuspy halo, with inner slope *γ* ∼ 1.5, but when the stars are included in the inference, the halo becomes sub-NFW. This shows the importance of consistently modelling the profile across a range of spatial scales.\n6. It is important to properly characterise the distributions of the underlying tracer populations as opposed to those of their kinematic subsamples. We have shown that the use of GC colour and spatial distributions based on the kinematic dataset alone leads to a systematic underprediction of the total mass of the system.\n7. When the inference is carried out with different (constant) values for the stellar anisotropy between -1 and 0.5, a pinch radius clearly emerges at which the dependence of the enclosed mass on the anisotropy is minimised. This gives an estimate of the mass enclosed at 110 kpc as log(*M*(*R* < 110)/*M*⊙) = 13.09 ± 0.40 which should be robust against assumptions about anisotropy.\nAcknowledgements\n================\nWe thank the referee for providing helpful suggestions and corrections. We would also like to thank Wyn Evans, Sergey Koposov, Vasily Belokurov, Jason Sanders and Angus Williams for useful discussions, and Debora Sijacki for helpful comments on the draft of this paper. LJO thanks the Science and Technology Facilities Council (STFC) for the award of a studentship; MWA also acknowledges support from the STFC in the form of an Ernest Rutherford Fellowship.\nAnisotropy kernel for the scaled Osipkov-Merritt profile\n========================================================\nFor the anisotropy profile of Equation [eq:scaledOM], the integrating factor defined by Equation [eq:IF] is\n \n$$f(r) = f(r\\_i)\\Bigg(\\frac{r^2 + r\\_a^2}{r\\_i^2 + r\\_a^2}\\Bigg)^{\\beta\\_{\\infty}}.$$ \nThe kernel of Equation [eq:kernel] is then\n \n$$K\\_{\\beta}(R,r) = \\frac{\\sqrt{r^2 - R^2}}{r} \\Bigg( 1 + \\beta\\_{\\infty} \\Big( \\frac{r^2 + r\\_a^2}{R^2 + r\\_a^2}\\Big)^{\\beta\\_{\\infty}} \\Big[2 \\_2F\\_1 (\\frac{1}{2},\\beta\\_{\\infty}; \\frac{3}{2}; z) - \\frac{3R^2 + 2r\\_a^2}{R^2 + r\\_a^2} ~\\_2F\\_1(\\frac{1}{2},\\beta\\_{\\infty}+1; \\frac{3}{2};z)\\Big]\\Big)\\Bigg)$$ \nwhere\n \n$$z = \\frac{R^2 - r^2}{R^2 + r\\_a^2}.$$ \n---\n1. E-mail: loldham@ast.cam.ac.uk[↩](#fnref1)\n2. http://www.strw.leidenuniv.nl/sauron/[↩](#fnref2)\n"},"data_id":{"kind":"string","value":"arxiv_0004125"}}},{"rowIdx":999,"cells":{"content":{"kind":"string","value":"Supplementary material for Bayesian inference for diffusion-driven mixed-effects models\n=======================================================================================\n### 1School of Mathematics & Statistics, Newcastle University,Newcastle-upon-Tyne, NE1 7RU, UK2Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF\nStochastic differential equations (SDEs) provide a natural framework for modelling intrinsic stochasticity inherent in many continuous-time physical processes. When such processes are observed in multiple individuals or experimental units, SDE driven mixed-effects models allow the quantification of both between and within individual variation. Performing Bayesian inference for such models using discrete-time data that may be incomplete and subject to measurement error is a challenging problem and is the focus of this paper. We extend a recently proposed MCMC scheme to include the SDE driven mixed-effects framework. Fundamental to our approach is the development of a novel construct that allows for efficient sampling of conditioned SDEs that may exhibit nonlinear dynamics between observation times. We apply the resulting scheme to synthetic data generated from a simple SDE model of orange tree growth, and real data on aphid numbers recorded under a variety of different treatment regimes. In addition, we provide a systematic comparison of our approach with an inference scheme based on a tractable approximation of the SDE, that is, the linear noise approximation.\n**Keywords:** Stochastic differential equation; mixed-effects; Markov chain Monte Carlo; modified innovation scheme; linear noise approximation.\nIntroduction\n============\nDiffusion processes satisfying Itô stochastic differential equations (SDEs) are a class of continuous-time, continuous-valued Markov stochastic processes that can be used to model a wide range of phenomena. Examples include (but are not limited to) epidemics, financial time series, population dynamics (including predator-prey systems) and intra-cellular processes. When repeated measurements on a system of interest are made, differences between individuals or experimental units can be incorporated through random effects. Quantification of both system (intrinsic) variation and variation between units leads to a stochastic differential mixed-effects model (SDMEM).\nUnfortunately, analytic intractability of SDEs governing most nonlinear multivariate diffusions can make likelihood-based inference methods problematic. Methods to overcome this difficulty include closed-form expansion of the transition density, exact simulation approaches and use of the Euler-Maruyama approximation coupled with data augmentation. Difficulty in performing inference for SDEs has resulted in relatively little work on SDMEMs.\n propose a procedure for obtaining approximate maximum likelihood estimates for SDMEM parameters based on a two step approach; they use a closed-form Hermite expansion to approximate the transition density, before using Gaussian quadrature to numerically integrate the conditional likelihood with respect to the random parameters. As noted by, the approach is, in practice, limited to a scalar random effect parameter since Gaussian quadrature is computationally inefficient when the dimension of the random effect parameter grows. The methodology is extended in to deal with multiple random effects. A number of limitations remain however. In particular a reducible diffusion process is required, that is, one which can be transformed to give a unit diffusion coefficient. Another drawback is that the method cannot account for measurement error. A promising approach appears to be the use of the extended Kalman filter (EKF) to provide a tractable approximation to the SDMEM. This has been the focus of, and. The `R` package `PSM` uses the EKF to estimate SDMEMs. Unfortunately, a quantification of the effect of using these approximate inferential models appears to be missing from the literature. discuss inference for SDMEMs in a Bayesian framework, and implement a Gibbs sampler when the SDE (for each experimental unit) has an explicit solution. When no explicit solution exists they suggest that a solution might be found using the Euler-Maruyama discretisation.\nContributions and organisation of the paper\n-------------------------------------------\nIn this article we provide a method that permits (simulation-based) Bayesian inference for a large class of multivariate SDMEMs using discrete-time observations that may be incomplete (so that only a subset of model components are observed) and subject to measurement error. The method makes use of a novel scheme that allows for observations made sparsely in time, as the process of interest may exhibit nonlinear dynamics between measurement times.\nAs a starting point, we consider a data augmentation approach that adopts an Euler-Maruyama approximation of unavailable transition densities and augments low frequency data with additional time points over which the approximation is satisfactory. Although a discretisation bias is introduced, this can be made arbitrarily small (at greater computational expense). Moreover, the approach is flexible, and is not restricted to reducible diffusions. A Bayesian approach then aims to construct the joint posterior density for parameters and the components of the latent process. The intractability of the posterior density necessitates simulation techniques such as Markov chain Monte Carlo. As is well documented in, care must be taken in the design of the MCMC sampler due to dependence between the parameters entering the diffusion coefficient and the latent process. We therefore adapt the reparameterisation technique (known as the modified innovation scheme) of and (see also ) to the SDMEM framework. A key requirement of the scheme is the ability to sample the latent process between two fixed values. Previous approaches have typically focused on the modified diffusion bridge construct of. For the SDMEM considered in Section [aphid] we find that this construct fails to capture the nonlinear dynamics exhibited between observation times. We therefore develop a novel bridge construct that is simple to implement and can capture nonlinear behaviour.\nFinally, we provide a systematic comparison of our approach with an inference scheme based on a linear noise approximation (LNA) of the SDE. The LNA approximates transition densities as Gaussian, and when combined with Gaussian measurement error, allows the latent process to be integrated out analytically. Essentially a forward (Kalman) filter can be implemented to calculate the marginal likelihood of all parameters of interest, allowing a marginal Metropolis-Hastings scheme targeting their posterior distribution. It should be noted, however, that evaluation of the Gaussian transition densities under the LNA require the solution of an ordinary differential equation (ODE) system whose order grows quadratically with the number of components (say *d*) governed by the SDE. The computational efficiency of an LNA based inference scheme will therefore depend on *d*, and on whether or not the ODE system can be solved analytically.\nWe apply the methods to two examples. First, we consider a synthetic dataset generated from an SDMEM driven by the simple univariate model of orange tree growth presented in and. The ODE system governing the LNA solution is tractable in this example. Second, we fit a model of aphid growth to both real and synthetic data. The real data are taken from and consist of Cotton aphid (*Aphis gossypii*) counts in the Texas High Plains obtained for three different levels of irrigation water, nitrogen fertiliser and block. This application is particularly challenging, due to the nonlinear drift and diffusion coefficients governing the SDMEM and the ability to only observe one of the model components (with error). Moreover, the ODE system governing the LNA solution is intractable and a numerical solver must be used.\nThe remainder of the article is organised as follows. The SDMEM framework is introduced in Section [sdmem]. Section [bayes] provides MCMC methods for Bayesian inference, with a novel bridge construct outlined in Section [dgode]. The linear noise approximation and its application as an inferential model is discussed in Section [lna]. The methods are applied in Section [apps] before conclusions are drawn in Section [disc].\nStochastic Differential Mixed-effects models\n============================================\nConsider the case where we have *N* experimental units randomly chosen from a theoretical population, and associated with each unit *i* is a continuous-time *d*-dimensional Itô process {*X**t**i*, *t* ≥ 0} governed by the SDE\n \n$$dX\\_t^i=\\alpha(X\\_t^i,\\theta,b^i)\\,dt+\\sqrt{\\beta(X\\_t^i,\\theta,b^i)}\\,dW\\_t^i,\n\\quad X\\_0^i=x\\_0^i, \\quad i=1,\\ldots,N. \\label{eqn:sdmem}$$ \nHere, *α* is a *d*-vector of drift functions, the diffusion coefficient *β* is a *d* × *d* positive definite matrix with a square root representation $\\sqrt{\\beta}$ such that $\\sqrt{\\beta}\\sqrt{\\beta}^T=\\beta$ and *W**t**i* is a *d*-vector of (uncorrelated) standard Brownian motion processes. The *p*-vector parameter *θ* = (*θ*1, …, *θ**p*)*T* is common to all units whereas the *q*-vectors *b**i* = (*b*1*i*, …, *b**q**i*)*T*, *i* = 1, …, *N*, are unit-specific effects, which may be fixed or random. In the most general random effects scenario we let *π*(*b**i*∣*ψ*) denote the joint distribution of *b**i*, parameterised by the *r*-vector *ψ* = (*ψ*1, …, *ψ**r*)*T*. The model defined by ([eqn:sdmem]) allows for differences between experimental units through different realisations of the Brownian motion paths *W**t**i* and the random effects *b**i*, accounting for inherent stochasticity within a unit, and variation between experimental units respectively.\nWe assume that each experimental unit {*X**t**i*, *t* ≥ 0} cannot be observed exactly, but observations *y**i* = (*y**t*0*i*, *y**t*1*i*, …, *y**t**n**i*)*T* are available and these are conditionally independent (given the latent process). We link the observations to the latent process via\n \n$$\\label{eqn:obs}\nY\\_t^i = F^T X\\_t^i + \\epsilon\\_t, \\qquad \\epsilon\\_t|\\Sigma{\\overset{indep}{\\sim }}N(0,\\Sigma),$$ \nwhere *Y**t**i* is a *d**o*-vector, *F* is a constant *d* × *d**o* matrix and *ε**t* is a random *d**o*-vector. Note that this setup allows for only observing a subset of components (*d**o* < *d*) and this aspect is explored further in Section [aphid].\nTogether and completely specify the stochastic differential mixed-effects model. However, for most problems of interest the form of the SDE associated with each unit will not permit an analytic solution, precluding straightforward inference for the unknown parameters. We therefore work with the Euler-Maruyama approximation \n$$\\Delta X\\_t^i\\equiv X^i\\_{t+\\Delta t}-X^i\\_t=\\alpha(X\\_t^i,\\theta,b^i)\\,\\Delta t+\\sqrt{\\beta(X\\_t^i,\\theta,b^i)}\\,\\Delta W\\_t^i$$ \n where Δ*W**t**i* ∼ *N*(0, *I**d*Δ*t*) and Δ*t* is the length of time between observations, assumed equally spaced for notational simplicity. It is, of course, unlikely that this approximation will be sufficiently accurate over the intervals between observation times and so we adopt a data augmentation scheme. Partitioning [*t**j*, *t**j* + 1] as \n*t**j* = *τ**j*, 0 < *τ**j*, 1 < *τ**j*, 2 < … < *τ**j*, *m* − 1 < *τ**j*, *m* = *t**j* + 1 \n introduces *m* − 1 intermediate time points with interval widths of length\n \n$$\\label{eqn:part}\n\\Delta\\tau\\equiv\\tau\\_{j,k+1}-\\tau\\_{j,k}=\\frac{t\\_{j+1}-t\\_j}{m}.$$ \nThe Euler-Maruyama approximation can then be applied over each interval of width Δ*τ*, and the associated discretisation bias can be made arbitrarily small at the expense of having to impute {*X**t**i*} at the intermediate times. We adopt the shorthand notation \n*x*[*t**j*, *t**j* + 1]*i* ≡ *x*[*j*, *j* + 1]*i* = (*x**τ**j*, 0*i*, *x**τ**j*, 1*i*, …, *x**τ**j*, *m**i*)*T* \n for the latent process over the time interval [*t**j*, *t**j* + 1] for unit *i*. Hence, the complete latent trajectory associated with unit *i* is given by \n(*x**i*)*T* = ((*x*[0, 1]*i*)*T*, (*x*(1, 2]*i*)*T*…, (*x*(*n* − 1, *n*]*i*)*T*) \n and we stack all unit-specific trajectories into a matrix *x* = (*x*1, …, *x**N*). Likewise the matrix *y* = (*y*1, …, *y**N*) denotes the entire set of observations. Next we focus on how to perform Bayesian inference for the model quantities *x*, *θ*, *b* = (*b*1, …, *b**N*)*T*, *ψ* and Σ.\nBayesian inference\n==================\nThe joint posterior for the common parameters *θ*, fixed/random effects *b*, hyperparameters *ψ*, measurement error variance Σ and latent values *x* is given by\n \n*π*(*θ*, *ψ*, Σ, *b*, *x*∣*y*) ∝ *π*(*θ*)*π*(*ψ*)*π*(Σ)*π*(*b*∣*ψ*)*π*(*x*∣*θ*, *b*)*π*(*y*∣*x*, Σ) \nwhere *π*(*θ*)*π*(*ψ*)*π*(Σ) is the joint prior density ascribed to *θ*, *ψ* and Σ. In addition we have that\n \n$$\\label{eqn:xdens}\n\\pi(x|\\theta,b)=\\prod\\limits\\_{i=1}^{N}\\prod\\limits\\_{j=0}^{n-1}\\prod\\limits\\_{k=1}^{m}\\pi(x^i\\_{\\tau\\_{j,k}}|x^i\\_{\\tau\\_{j,k-1}},\\theta,b^i)$$ \nwhere \n*π*(*x**τ**j*, *k**i*∣*x**τ**j*, *k* − 1*i*, *θ*, *b**i*) = *N*(*x**τ**j*, *k**i* ; *x**τ**j*, *k* − 1*i* + *α*(*x**τ**j*, *k* − 1*i* *θ*, *b**i*)Δ*τ*, *β*(*x**τ**j*, *k* − 1*i*, *θ*, *b**i*)Δ*τ*) \n and *N*( ⋅ ; *m*, *V*) denotes the multivariate Gaussian density with mean *m* and variance *V*. Similarly \n$$\\pi(y|x,\\Sigma)= \\prod\\limits\\_{i=1}^{N}\\prod\\limits\\_{j=0}^{n}\\pi(y\\_{t\\_{j}}^i|x\\_{t\\_{j}}^i,\\Sigma)$$ \n where *π*(*y**t**j**i*∣*x**t**j**i*, Σ) = *N*(*y**t**j**i* ; *x**t**j**i*, Σ). Given the intractability of the joint posterior distribution in ([eqn:posterior]) we aim to construct a Markov chain Monte Carlo (MCMC) scheme which generates realisations from this posterior. The form of the SDMEM admits a Gibbs sampling strategy with blocking that sequentially takes draws from the full conditionals \n[b]0.5\n1. *π*(*x*∣*θ*, *ψ*, Σ, *b*, *y*) = *π*(*x*∣*θ*, Σ, *b*, *y*),\n2. *π*(*θ*∣*ψ*, Σ, *b*, *x*, *y*) = *π*(*θ*∣*b*, *x*),\n3. *π*(*ψ*∣*θ*, Σ, *b*, *x*, *y*) = *π*(*ψ*∣*b*).\n4. \n[b]0.5\n1. *π*(Σ∣*θ*, *ψ*, *b*, *x*, *y*) = *π*(Σ∣*x*, *y*),\n2. *π*(*b*∣*θ*, *ψ*, Σ, *x*, *y*) = *π*(*b*∣*θ*, *ψ*, *x*),\n3. \n4. \nFurther blocking strategies that exploit the conditional dependencies between the model parameters and latent trajectories can be used. For example, in step 1 the latent trajectories can be updated separately for each experimental unit. Likewise, the unit-specific random effects can be updated separately. Where necessary, Metropolis-within-Gibbs updates can be used. We note that as written, this scheme will mix intolerably poorly as the degree of augmentation *m* is increased due to dependence between the latent values *x* and the parameters entering the diffusion coefficient (namely *θ* and *b*). We refer the reader to for a detailed discussion of this problem. A simple mechanism for overcoming this issue is to update the parameters and latent trajectories jointly (and this has been considered for SDE models by and ). For SDMEMs a joint update of *θ*, *b* and *x* is likely to result in a sampler with low acceptance rates. We therefore wish to preserve the blocking structure described above and instead adapt the reparameterisation of to our problem. In what follows, we describe in detail each step of the Gibbs sampler.\nPath updates\n------------\nThe full conditional density of the latent paths for all experimental units is given by\n \n$$\\begin{aligned}\n\\pi(x|\\theta,\\Sigma,b,y)&\\propto \\pi(x|\\theta,b)\\pi(y|x,\\Sigma)\n=\\prod\\limits\\_{i=1}^{N}\\pi(x^i|\\theta,b^i)\\pi(y^i|x^i,\\Sigma)\\end{aligned}$$ \nwhich suggests a scheme where unit-specific paths are updated separately. We now focus on an updating scheme for a single path, and drop *i* from the notation, writing *x* in place of *x**i* and *x*[*j*, *j* + 1] in place of *x*[*j*, *j* + 1]*i*. Since the parameters are fixed throughout this updating step, we also drop them from the notation.\nFollowing we update *x* in overlapping blocks of size 2*m* + 1. Consider times *t**j* and *t**j* + 2 at which the current values of the latent process are *x**t**j* and *x**t**j* + 2. The full conditional density of the latent process over the interval (*t**j*, *t**j* + 2) is given by\n \n$$\\label{eqn:FCDpath}\n\\pi(x\\_{(j,j+2)}|x\\_{t\\_j},y\\_{t\\_{j+1}},x\\_{t\\_{j+2}})\n\\propto \\pi(y\\_{t\\_{j+1}}|x\\_{t\\_{j+1}})\\prod\\limits\\_{l=j}^{j+1}\\prod\\limits\\_{k=1}^{m}\\pi(x\\_{\\tau\\_{l,k}}|x\\_{\\tau\\_{l,k-1}}).$$ \nUnder the nonlinear structure of the diffusion process, this full conditional is intractable and so we use a Metropolis-Hastings step to generate draws from ([eqn:FCDpath]). We use an independence sampler with proposal density of the form\n \n*q*(*x*(*j*, *j* + 2)∣*x**t**j*, *y**t**j* + 1, *x**t**j* + 2) = *q*1(*x*(*j*, *j* + 1]∣*x**t**j*, *y**t**j* + 1) *q*2(*x*(*j* + 1, *j* + 2)∣*x**t**j* + 1, *x**t**j* + 2). \nFigure [pic-2blocknoise] gives an illustration of the updating procedure which can be applied over intervals (*t**j*, *t**j* + 2), *j* = 0, 1, …, *n* − 2, with two additional Metropolis-Hastings steps (such as those described in ) that allow for updating *x* at times *t*0 and *t**n*. Deriving appropriate forms for *q*1 and *q*2 requires the ability to (approximately) generate a discrete-time realisation of a diffusion process between two time points at which the process is either observed exactly or subject to Gaussian noise. The resulting trajectory is typically referred to as a diffusion bridge.\n[scale=6,>=latex] (xo0) at (0,0.5) ; (xo1) at (1,0.5) ; (xo2) at (2,0.5) ; (xu0) at (0,0) ; (xu1) at (1,0) ; (xu2) at (2,0) ;\n(0,0.1) – (0,0.4); (1,0.1) – (1,0.4); (2,0.1) – (2,0.4);\n(0.1,0.35) – (0.1,0.65) – (1.09,0.65) – (1.09,0.45) – (0.93,0.45) – (0.93,0.35) – cycle;\n(1.12,0.35) rectangle (1.9,0.65);\n(0.15,0.5) – (0.85,0.5) node[midway, below, black] *m* − 1 latent values;\n(1.15,0.5) – (1.85,0.5) node[midway, below, black] *m* − 1 latent values;\nat (0.5, 0.3) Propose using *q*1; at (1.5, 0.3) Propose using *q*2;\n[pic-2blocknoise]\n[scale=5,>=latex] (xo0) at (0,0.5) ; (xo1) at (1,0.5) ; (xo2) at (2,0.5) ; (xu0) at (0,0.15) ; (xu1) at (1,0.15) ; (xu2) at (2,0.15) ;\n(0,0.25) – (0,0.4); (1,0.25) – (1,0.4); (2,0.25) – (2,0.4);\n(0.1,0.35) – (0.1,0.65) – (1.10,0.65) – (1.10,0.45) – (0.93,0.45) – (0.93,0.35) – cycle;\n(1.12,0.35) rectangle (1.9,0.65);\n(0.15,0.5) – (0.85,0.5) node[midway, below, black] *m* − 1 latent values;\n(1.15,0.5) – (1.85,0.5) node[midway, below, black] *m* − 1 latent values;\nat (0.5, 0.3) Propose using *q*1; at (1.5, 0.3) Propose using *q*2;\n[pic-2blocknoise]\nSeveral strategies for constructing diffusion bridges have been proposed in the literature. For example, used the Euler-Maruyama scheme to generate bridges myopically of the end point. use a linear Gaussian approximation of the distribution of the process conditional on the value at a previous and future time point, giving a construct known as the modified diffusion bridge. Extensions of this construct to the case of partial observation with additive Gaussian noise can be found in. Whilst this construct can, in principle, be applied to arbitrary nonlinear multivariate diffusion processes, the effect of the Gaussian approximation is to guide the bridge towards the observation in a linear way, unless there is large uncertainty in the observation process. This effect is exacerbated in the case of no measurement error, in which case the resulting construct is independent of the drift of the target process. Consequently, use of the modified diffusion bridge as a proposal mechanism (in a Metropolis-Hastings independence sampler) is likely to result in low acceptance rates, unless the drift is of little importance in dictating the dynamics of the target process between observation times. Several attempts to overcome this issue have been proposed in the recent literature.\nA time-dependent combination of the Pedersen and modified diffusion bridge approaches was proposed by. However, the resulting construct requires a model specific tuning parameter governing the relative weight of each contribution (either Pedersen or modified diffusion bridge). Moreover, the optimal value (in terms of maximising acceptance rate) may vary between observation intervals. use Hybrid Monte Carlo (HMC) on pathspace to generate SDE sample paths under various observation regimes. For the applications considered, the authors found reasonable gains in overall efficiency (as measured by minimum effective sample size per CPU time) over an independence sampler with a Brownian bridge proposal. However, we note that HMC also requires careful choice of tuning parameters (namely the number of steps (and their size) in the leapfrog integrator) to maximise efficiency. (see also ) combine the ideas of and to obtain a bridge based on the addition of a guiding term to the drift of the target SDE. The guiding term requires a tractable approximation of the unavailable transition densities governing the target process over the length of the inter-observation interval. suggest using the transition densities associated with a class of linear processes, although we note that finding an approximation that is both accurate and computationally efficient may be difficult in practice. Moreover, such an approximation can suffer from computational efficiency due to the fact that it must be obtained at each intermediate time point.\nIn the next section we describe a novel bridge construct that requires no tuning parameters, is simple to implement (even when only a subset of components are observed with Gaussian noise), computationally efficient and explicitly allows for the effect of the drift governing the target SDE.\nAn improved bridge construct\n----------------------------\nConsider a typical interval [*t**j*, *t**j* + 1], partitioned into *m* sub-intervals as in ([eqn:part]), over which we wish to generate a realisation of {*X**t*, *t* ∈ [*t**j*, *t**j* + 1]} conditional on *x**t**j* ≡ *x**τ**j*, 0 and the noisy measurement *y**t**j* + 1 ≡ *y**τ**j*, *m*. Our approach builds on the modified diffusion bridge of, which we briefly review before describing our extension.\n### Modified diffusion bridge\nKey to constructing the modified diffusion bridge is an approximation of the joint distribution of *X**τ**j*, *k* and *Y**t**j* + 1 (conditional on *x**τ**j*, *k* − 1). Under the Euler-Maruyama approximation *X**τ**j*, *k*∣*X**τ**j*, *k* − 1 and *Y**t**j* + 1∣*X**τ**j*, *k* are Gaussian, with the expressions for the mean and variance of the latter evaluated at *X**τ**j*, *k* − 1 to give a linear Gaussian structure. This leads to the approximation\n \n$$\\begin{aligned}\n\\label{eqn:mdbJ}\n\\begin{pmatrix}\nX\\_{\\tau\\_{j,k}} \\\\\nY\\_{t\\_{j+1}} \\end{pmatrix}\\bigg| x\\_{\\tau\\_{j,k-1}}&\\sim N\\left\\{\\begin{pmatrix}\nx\\_{\\tau\\_{j,k-1}}+\\alpha\\_{j,k-1}\\Delta\\tau \\\\[0.2em]\nF^T [x\\_{\\tau\\_{j,k-1}}+\\alpha\\_{j,k-1}\\Delta^-] \\end{pmatrix},\\right. \\nonumber\\\\\n& \\quad \\qquad \\qquad \\qquad \\qquad \\left. \\begin{pmatrix}\n\\beta\\_{j,k-1} \\Delta\\tau & \\beta\\_{j,k-1} F\\Delta\\tau \\\\\nF^T\\beta\\_{j,k-1}\\Delta\\tau & F^T\\beta\\_{j,k-1} F\\Delta^- + \\Sigma \\end{pmatrix}\\right\\}\\end{aligned}$$ \nwhere Δ− = *t**j* + 1 − *τ**j*, *k* − 1 and we have used the shorthand notation *α*(*x**τ**j*, *k* − 1) = *α**j*, *k* − 1 and *β*(*x**τ**j*, *k* − 1) = *β**j*, *k* − 1. Conditioning further on *y**t**j* + 1 gives a Gaussian approximation of *π*(*x**τ**j*, *k*∣*x**τ**j*, *k* − 1, *y**t**j* + 1), denoted *π̂*(*x**τ**j*, *k*∣*x**τ**j*, *k* − 1, *y**t**j* + 1), which can be sampled recursively to give a bridge *x**τ**j*, 0, …, *x**τ**j*, *m*. In the case of no measurement error and observation of all components (so that *y**t**j* + 1 = *x**t**j* + 1 and *F* = *I**d*, the *d* × *d* identity matrix), we obtain \n$$\\widehat{\\pi}(x\\_{\\tau\\_{j,k}}| x\\_{\\tau\\_{j,k-1}},x\\_{t\\_{j+1}})=\nN\\left(x\\_{\\tau\\_{j,k}}\\,;\\, x\\_{\\tau\\_{j,k-1}}+\\frac{x\\_{t\\_{j+1}}-x\\_{\\tau\\_{j,k-1}}}{t\\_{j+1}-\\tau\\_{j,k-1}}\\Delta\\tau\n\\,,\\,\\frac{t\\_{j+1}-\\tau\\_{j,k}}{t\\_{j+1}-\\tau\\_{j,k-1}}\\beta\\_{j,k-1}\\Delta\\tau\\right)$$ \n which is the form of the modified diffusion bridge first described by. In this case, *π̂*(*x**τ**j*, *k*∣*x**τ**j*, *k* − 1, *y**t**j* + 1) can be seen as a linear approximation of the Brownian bridge SDE\n \n$$\\label{eqn:SDEbridge0}\ndX\\_t=\\frac{X\\_{t\\_{j+1}}-X\\_t}{t\\_{j+1}-t}\\,dt+\\sqrt{\\beta(X\\_t)}\\,dW\\_t.$$ \nUse of ([eqn:SDEbridge0]) has been justified by, who show that the distribution of the target process (conditional on *x**t**j* + 1) is absolutely continuous with respect to the distribution of the solution to ([eqn:SDEbridge0]). We may therefore expect that a Metropolis-Hastings scheme that uses a proposal based on a discretisation of ([eqn:SDEbridge0]) will yield a non-zero acceptance rate as Δ*τ* → 0 (for a rigorous treatment of the limiting forms, we refer the reader to, and to for a recent discussion). However, it should also be noted that the linear drift function governing ([eqn:SDEbridge0]) is independent of the the drift function *α*( ⋅ ) governing the target process. Consequently, in situations where realisations of the target SDE (with the same initial condition) exhibit strong and similar nonlinearity over the inter-observation time, the modified diffusion bridge is likely to be unsatisfactory.\n### Residual bridge\nTo allow explicitly for dynamics based on the drift, we partition *X**t* into two parts, one that accounts for the drift in a deterministic way, and another as a residual stochastic process. The modified diffusion bridge is then applied to the *residual stochastic process* rather than the target process itself. The partition we require is\n \n*X**t* = *η**t* + *R**t*, \nwhere {*η**t*, *t* ∈ [*t**j*, *t**j* + 1]} is a deterministic process satisfying the ODE\n \n$$\\frac{d\\eta\\_t}{dt} = \\alpha(\\eta\\_t),\\quad \\eta\\_{t\\_{j}}=x\\_{t\\_{j}}, \\label{eqn:dgode\\_eta}$$ \nand {*R**t*, *t* ∈ [*t**j*, *t**j* + 1]} is a residual stochastic process satisfying\n \n$$\\begin{aligned}\ndR\\_t&\\equiv dX\\_t-d\\eta\\_t=\\{\\alpha(X\\_t)-\\alpha(\\eta\\_t)\\}dt+\\sqrt{\\beta(X\\_t)}\\,dW\\_t. \\label{eqn:dgode\\_r}\\end{aligned}$$ \nWe note that the partition in ([eqn:partx]) is used by (see also Section [lna]) to derive a tractable approximation to the intractable transition densities governing *X**t*, whereas our primary motivation for ([eqn:partx]) is the application of the modified diffusion bridge construct to the residual process, thus giving a proposal that is likely to perform well for arbitrarily fine discretisations and explicitly incorporates the drift of the target SDE. Therefore, we aim to derive an approximation *π̂*(*r**τ**j*, *k*∣*r**τ**j*, *k* − 1, *y**t**j* + 1), that can be sampled recursively for *k* = 1, …, *m* and combined with the deterministic process (through numerical solution of ([eqn:dgodeeta])) via ([eqn:partx]) to give a bridge *x**τ**j*, 0, …, *x**τ**j*, *m*. The scheme is illustrated in Figure [fig:dgodeexplain].\n [fig:dgodeexplain]\nThe initial condition *η**t**j* = *x**t**j* together with the Gaussian measurement error process imply that *r**t**j* = 0 and \n$$Y\\_{t\\_{j+1}}-F^T\\eta\\_{t\\_{j+1}}=F^T R\\_{t\\_{j+1}} + \\epsilon\\_{t\\_{j+1}},\\qquad \\epsilon\\_{t\\_{j+1}}|\\Sigma{\\overset{indep}{\\sim }}N(0,\\Sigma).$$ \n Hence, it should be clear that the joint distribution of *R**τ**j*, *k* and *Y**t**j* + 1 − *F**T**η**t**j* + 1 conditional on *R**τ**j*, *k* − 1 = *r**τ**j*, *k* − 1 can be approximated as\n \n$$\\begin{aligned}\n\\begin{pmatrix}\nR\\_{\\tau\\_{j,k}} \\\\\nY\\_{t\\_{j+1}}-F^T\\eta\\_{t\\_{j+1}} \\end{pmatrix}\\bigg| r\\_{\\tau\\_{j,k-1}}&\\sim N\\left\\{\\begin{pmatrix}\nr\\_{\\tau\\_{j,k-1}}+(\\alpha\\_{j,k-1}-\\alpha^{\\eta}\\_{j,k-1})\\Delta\\tau \\\\[0.2em]\nF^T r\\_{\\tau\\_{j,k-1}}+F^T (\\alpha\\_{j,k-1}-\\alpha^\\eta\\_{j,k-1})\\Delta^- \\end{pmatrix},\\right.\\\\\n& \\quad \\qquad \\qquad \\qquad \\left. \\begin{pmatrix}\n\\beta\\_{j,k-1} \\Delta\\tau & \\beta\\_{j,k-1} F\\Delta\\tau \\\\\nF^T\\beta\\_{j,k-1}\\Delta\\tau & F^T\\beta\\_{j,k-1} F\\Delta^- + \\Sigma \\end{pmatrix}\\right\\}\\end{aligned}$$ \nwhere *α*(*η**τ**j*, *k* − 1) = *α**j*, *k* − 1*η*. Conditioning further on *y**t**j* + 1 − *F**T**η**t**j* + 1 gives\n \n*π̂*(*r**τ**j*, *k*∣*r**τ**j*, *k* − 1, *y**t**j* + 1) = *N*(*r**τ**j*, *k*; *μ**j*, *k* , Ψ*j*, *k*), \nwhere\n \n$$\\begin{aligned}\n\\begin{split}\n\\mu\\_{j,k}&=r\\_{\\tau\\_{j,k-1}}+(\\alpha^x\\_{j,k-1}-\\alpha^\\eta\\_{j,k-1})\\Delta\\tau+\n\\beta\\_{j,k-1} F\\Delta\\tau(F^T\\beta\\_{j,k-1} F\\Delta^- + \\Sigma)^{-1} \\\\\n&\\qquad\\qquad\\qquad\\times(y\\_{t\\_{j+1}}-F^T\\eta\\_{t\\_{j+1}}-\n\\{F^T r\\_{\\tau\\_{j,k-1}}+F^T (\\alpha^x\\_{j,k-1}-\\alpha^\\eta\\_{j,k-1})\\Delta^-\\})\n\\end{split} \n\\label{eqn:dgodebridge2}\\\\[0.5em]\n\\intertext{and}\n\\Psi\\_{j,k}&=\\beta\\_{j,k-1}\\Delta\\tau-\\beta\\_{j,k-1} F\\Delta\\tau(F^T\\beta\\_{j,k-1} F\\Delta^- + \\Sigma)^{-1}F^T\\beta\\_{j,k-1}\\Delta\\tau. \\label{eqn:dgodebridge3}\\end{aligned}$$ \nTogether - define our bridge construct. These can be used to define the proposal mechanism in for generating {*X**t*, *t* ∈ [*t**j*, *t**j* + 2]} by taking\n \n$$\\begin{aligned}\nq\\_1(x\\_{(j,j+1]}|x\\_{t\\_j},y\\_{t\\_{j+1}})&=\n\\prod\\limits\\_{k=1}^m\\widehat{\\pi}(x\\_{\\tau\\_{j,k}}-\\eta\\_{\\tau\\_{j,k}}| r\\_{\\tau\\_{j,k-1}},y\\_{t\\_{j+1}})\\intertext{and}\nq\\_2(x\\_{(j+1,j+2)}| x\\_{t\\_{j+1}},x\\_{t\\_{j+2}})&=\n\\prod\\limits\\_{k=1}^{m-1}\\widehat{\\pi}(x\\_{\\tau\\_{j+1,k}}-\\eta\\_{\\tau\\_{j+1,k}}| r\\_{\\tau\\_{j+1,k-1}},x\\_{t\\_{j+2}}), \\end{aligned}$$ \nwhere *π̂*(*x**τ**j* + 1, *k* − *η**τ**j* + 1, *k*∣*r**τ**j* + 1, *k* − 1, *x**t**j* + 2) can be sampled using ([eqn:partx]) and ([eqn:dgodebridge1])-([eqn:dgodebridge3]) with *j* replaced by *j* + 1, Σ = 0 and *F* = *I**d*.\nIn the special case of no measurement error and observation of all components we have that \n$$\\widehat{\\pi}(r\\_{\\tau\\_{j,k}}| r\\_{\\tau\\_{j,k-1}},x\\_{t\\_{j+1}}) = N\\left(r\\_{\\tau\\_{j,k}}\\,;\\,r\\_{\\tau\\_{j,k-1}}+\\frac{r\\_{t\\_{j+1}}-r\\_{\\tau\\_{j,k-1}}}{t\\_{j+1}-\\tau\\_{j,k-1}}\\Delta\\tau\\,,\\,\n\\frac{t\\_{j+1}-\\tau\\_{j,k}}{t\\_{j+1}-\\tau\\_{j,k-1}}\\beta\\_{j,k-1}\\Delta\\tau\\right),$$ \n which can be seen as a linear approximation of the Brownian bridge SDE\n \n$$\\label{SDEbridge}\ndR\\_t=\\frac{R\\_{t\\_{j+1}}-R\\_t}{t\\_{j+1}-t}+\\sqrt{\\beta(X\\_t)}\\,dW\\_t.$$ \nWe also note that ([SDEbridge]) has the same diffusion coefficient as the target process and appeal again to, to deduce that the distribution of the residual process governed by ([eqn:dgoder]) (conditional on *r**t**j* + 1) is absolutely continuous with respect to the distribution of the solution to ([eqn:SDEbridge0]).\nParameter updates\n-----------------\nThe full conditional densities of Σ and *ψ* are \n*π*(Σ∣*x*, *y*) ∝ *π*(Σ)*π*(*y*∣Σ) and *π*(*ψ*∣*b*) ∝ *π*(*ψ*)*π*(*b*∣*ψ*). \n Often, semi-conjugate priors can be specified for Σ and *ψ* negating the need for Metropolis-within-Gibbs steps. For the remaining parameters *θ* and *b* = (*b*1, …, *b**N*)*T* we have \n$$\\pi(\\theta| b,x)\\propto \\pi(\\theta)\\pi(x|\\theta,b)\n\\qquad\\text{and}\\qquad\n\\pi(b|\\theta,\\psi,x)\\propto \\pi(b|\\psi)\\pi(x|\\theta,b)\n=\\prod\\limits\\_{i=1}^{N}\\pi(b^i|\\psi)\\pi(x^i|\\theta,b^i)$$ \n where the last expression suggests unit-specific updates of the components of *b*.\nAs discussed earlier, since *θ* and the components of *b* enter into the diffusion coefficient of ([eqn:sdmem]), sampling the full conditionals of *θ*∣*b*, *x* and *b*∣*θ*, *ψ*, *x* as part of a Gibbs sampler will result in a reducible Markov chain as *m* → ∞ (or Δ*τ* → 0). To overcome this problem we use a reparameterisation which is outlined in the next section.\n### Modified Innovation scheme\nThe innovation scheme was first outlined in and exploits the fact that, given *θ* and *b*, under the Euler-Maruyama approximation there is a one-to-one relationship between the increments of the process (Δ*X**t*) and the increments of the driving Brownian motion (Δ*W**t*). Moreover, whilst the quadratic variation of *X* determines *θ* and *b* (as *m* → ∞), the quadratic variation of the Brownian process is independent of *θ* and *b* *a priori*. Conditioning on the Brownian increment innovations in a Gibbs update should therefore be effective in overcoming the dependence problem. The resulting algorithm is known as the innovation scheme. Unfortunately, combining an updated parameter value with the Brownian increments will not necessarily give an imputed path that is consistent with the observations. Therefore, suggest that a diffusion bridge (such as the modified diffusion bridge of ) be used to determine the innovation process, leading to a modified innovation scheme.\n considers the modified innovation scheme in a continuous-time framework. Adapting their innovation process to an SDMEM, we have, for an interval [*t**j*, *t**j* + 1], an innovation process {*Z**t**i*, *t* ∈ [*t**j*, *t**j* + 1]} satisfying\n \n$$\\begin{aligned}\ndZ\\_t^i&=\\beta(X\\_t^i,\\theta,b^i)^{-{{1/2}}}\n\\left(dX\\_t^i-\\frac{x\\_{t\\_{j+1}}^i-X\\_t^i}{t\\_{j+1}-t}\\,dt\\right), \\label{eqn:innovation} \\\\\n&= \\beta(X\\_t^i,\\theta,b^i)^{-{{1/2}}}\n\\left\\{\\alpha(X\\_t^i,\\theta,b^i)-\\frac{x\\_{t\\_{j+1}}^i-X\\_t^i}{t\\_{j+1}-t}\\right\\}dt+dW\\_t^i \\nonumber\\end{aligned}$$ \nwith *Z**t**j**i* = 0. Clearly, each process *Z**i* has unit diffusion coefficient and whilst not Brownian motion processes, the probability measures induced by each *Z**i* are absolutely continuous with respect to Wiener measure. A proof of this result can be found in as well as a justification for using this form of innovation process as the effective component in a Gibbs sampler.\nThe aim is to apply a discretisation of ([eqn:innovation]) between observation times. We therefore define *x**o**i* = (*x**t*0*i*, …, *x**t**n**i*)*T* to be the current values of the (unit-specific) latent process at the observation times, and stack all *x**o**i* values into the matrix *x**o*. We have for *k* = 1, …, *m* \n$$Z\\_{\\tau\\_{j,k}}^i-Z\\_{\\tau\\_{j,k-1}}^i=\n\\beta^\\*(X\\_{\\tau\\_{j,k-1}}^i,\\theta,b^i)^{-{{1/2}}}\n\\left(X\\_{\\tau\\_{j,k}}^i-X\\_{\\tau\\_{j,k-1}}^i-\\frac{x\\_{t\\_{j+1}}^i-X\\_{\\tau\\_{j,k-1}}^i}{t\\_{j+1}-\\tau\\_{j,k-1}}\\Delta\\tau\\right),$$ \n where *Z**τ**j*, 0 = 0 and \n$$\\beta^\\*(X\\_{\\tau\\_{j,k-1}}^i,\\theta,b^i)=\n\\frac{t\\_{j+1}-\\tau\\_{j,k}}{t\\_{j+1}-\\tau\\_{j,k-1}}\\beta(X\\_{\\tau\\_{j,k-1}}^i,\\theta,b^i).$$ \n Note that our discretisation of ([eqn:innovation]) follows by using the modified diffusion bridge to construct the innovation process. Now define a function *f* so that and *Z**τ**j*, *k**i* = *f*− 1(*X**τ**j*, *k**i*, *θ*, *b**i*). Let *z**i**m**p**i* denote the (unit-specific) innovation values over [*t*0, *t**n*] and stack all *z**i**m**p**i* values into the matrix *z**i**m**p*. Define *x**i**m**p**i* and *x**i**m**p* similarly. The modified innovation scheme samples *θ*∣*b*, *z**i**m**p*, *x**o* and *b**i*∣*θ*, *ψ*, *z**i**m**p**i*, *x**o**i*, *i* = 1, …, *N*. Note that for an updated value of *b**i*, say *b**i* \\*, a new *x**i**m**p**i* \\* is updated deterministically through *x**i**m**p**i* \\* = *f*(*z**i**m**p**i* \\*, *θ*, *b**i* \\*). Likewise, for a new *θ*\\*, a new *x**i**m**p*\\* is updated deterministically through *x**i**m**p**i* \\* = *f*(*z**i**m**p**i* \\*, *θ*\\*, *b**i*), *i* = 1, …, *N*. The full conditional density of *θ* is\n \n$$\\begin{aligned}\n& \\pi(\\theta| b,z\\_{imp},x\\_o) \\propto \\pi(\\theta)\\prod\\limits\\_{i=1}^{N}\\prod\\limits\\_{j=1}^{n-1}\n\\left[\\prod\\limits\\_{k=1}^{m}\\pi(x^i\\_{\\tau\\_{j,k}}| x^i\\_{\\tau\\_{j,k-1}},\\theta,b^i)\n\\prod\\limits\\_{k=1}^{m-1}J\\{f(z\\_{\\tau\\_{j,k}}^i,\\theta,b^i)\\} \\right], \\label{eqn:innovationFC1}\\end{aligned}$$ \nwhere \n*J*{*f*(*z**τ**j*, *k**i*, *θ*, *b**i*)} = ∣*β*\\*(*x**τ**j*, *k* − 1*i*, *θ*, *b**i*)∣− 1/2 \n is the Jacobian determinant of *f*. Similarly, the full conditional density of *b**i*, *i* = 1, …, *N* is\n \n$$\\begin{aligned}\n&\\pi(b^i|\\theta,\\psi,z\\_{imp}^i,x\\_o^i)\\propto \\pi(b^i|\\psi)\\prod\\limits\\_{j=1}^{n-1}\\left[\\prod\\limits\\_{k=1}^{m}\n\\pi(x^i\\_{\\tau\\_{j,k}}|x^i\\_{\\tau\\_{j,k-1}},\\theta,b^i)\n\\prod\\limits\\_{k=1}^{m-1}J\\{f(z\\_{\\tau\\_{j,k}}^i,\\theta,b^i)\\} \\right]. \\label{eqn:innovationFC2}\\end{aligned}$$ \nNaturally, the full conditionals in ([eqn:innovationFC1]) and ([eqn:innovationFC2]) will typically be intractable, requiring the use of Metropolis-within-Gibbs updates.\nLinear noise approximation\n==========================\nIn this section we outline a competing solution which uses an inference scheme based on a linear noise approximation (LNA) to the SDMEM. The LNA typically refers to an approximation to the solution of the forward Kolmogorov equation governing the transition probability of a Markov jump process. Specifically, the forward Kolmogorov equation is approximated by a Fokker-Planck equation with linear coefficients. Equivalently, a general Fokker-Planck equation can be deduced and then linearised. In this context, therefore, the LNA aims to replace intractable transition densities with Gaussian approximations. In what follows, we give a brief informal derivation of the LNA and refer the reader to and the references therein for further details.\nSetup\n-----\nFor notational simplicity and clarity of exposition, we suppress parameter dependence and the unit-specific *i* for the remainder of this sub-section.\nWithout loss of generality, consider a time *t* ∈ [*t**j*, *t**j* + 1] at which we wish to approximate the intractable transition density associated with *X**t*∣*X**t**j* = *x**t**j*. The LNA uses the same partition of *X**t* given in ([eqn:partx]), that is *X**t* = *η**t* + *R**t* where the deterministic process *η**t* satisfies ([eqn:dgodeeta]) and the residual stochastic process satisfies ([eqn:dgoder]). The key assumption underpinning the LNA is that the residual stochastic perturbation is “small” relative to the deterministic process, allowing suitable truncation of a Taylor series expansion of *α*(*X**t*) and *β*(*X**t*) about *η**t*. Taking the first two terms in the expansion of *α*(*X**t*), and the first term in the expansion of *β*(*X**t*) gives an SDE satisfied by an approximate residual process {*R̃**t*, *t* ∈ [*t**j*, *t**j* + 1]} of the form\n \n$$\\label{eqn:tilder}\nd\\tilde{R}\\_t = H\\_t\\tilde{R}\\_t\\,dt + \\sqrt{\\beta(\\eta\\_t)}\\,dW\\_t,$$ \nwhere *H**t* is the Jacobian matrix with (*i*, *j*)th element (*H**t*)*i*, *j* = ∂*α**i*(*η**t*)/∂*η**j*, *t*.\nAssuming fixed or Gaussian initial conditions *R̃**t**j* ∼ *N*(*m**t**j*, *V**t**j*) gives *R̃**t* ∼ *N*(*m**t*, *V**t*), where *m**t* and *V**t* satisfy the ODE system\n \n$$\\begin{aligned}\n\\dfrac{dm\\_t}{dt} &= H\\_tm\\_t, \\label{eqn:lna\\_m} \\\\\n\\dfrac{dV\\_t}{dt} &= H\\_tV\\_t + \\beta(\\eta\\_t,\\theta,b) + V\\_tH\\_t^T. \\label{eqn:lna\\_v}\\end{aligned}$$ \nIn the absence of an analytic solution, the system of coupled ODEs and which characterise the LNA, must be solved numerically. For initial conditions *η**t**j* = *x**t**j*, we have *m**t**j* = 0 and *V**t**j* = 0 so that does not need to be solved, and the approximating transition distribution is *X**t*∣*X**t**j* = *x**t**j* ∼ *N*(*η**t*, *V**t*).\nIt is worth noting here that the linear form of the SDE ([eqn:tilder]) satisfied by the approximate residual process coupled with the additive Gaussian observation regime admits a closed form expression for densities of the form *π̂*(*r̃**τ**j*, *k*∣*r̃**τ**j*, *k* − 1, *y**t**j* + 1), suggesting use of the LNA as a proposal mechanism inside the Bayesian imputation approach of Section [bayes]. Whilst the LNA could in principle be used to directly approximate the conditioned residual process governed by the SDE in ([eqn:dgoder]) we note that the SDEs in ([eqn:dgoder]) and ([eqn:tilder]) have different diffusion coefficients. Consequently, the probability law governing *R̃**t* is not absolutely continuous with respect to the law of *R**t*. We therefore do not advocate use of the LNA in this way.\nIn the next section we outline an inference scheme for SDMEMs of the form based on the LNA. It exploits the computational efficiency of a filtering algorithm proposed by that allows closed-form calculation of the marginal likelihood *π*(*y*∣*θ*, *b*, Σ) under our Gaussian observation regime ; see the supplementary material for further details.\nApplication to SDMEMs\n---------------------\nUnder the linear noise approximation of the marginal posterior for all parameters is given by\n \n$$\\begin{aligned}\n\\pi(\\theta,\\psi,\\Sigma,b| y)&\\propto\\pi(\\theta)\\pi(\\psi)\\pi(\\Sigma)\\pi(b|\\psi)\\pi(y|\\theta,\\Sigma,b)\\\\\n&\\propto \\pi(\\theta)\\pi(\\psi)\\pi(\\Sigma)\\prod\\limits\\_{i=1}^N\\pi(b^i|\\psi)\\pi(y^i|\\theta,\\Sigma,b^i). \\end{aligned}$$ \nThis factorisation suggests a Gibbs sampler with blocking that sequentially takes draws from the full conditionals *π*(Σ∣*θ*, *ψ*, *b*, *y*) = *π*(Σ∣*y*), *π*(*θ*∣*ψ*, Σ, *b*, *y*) = *π*(*θ*∣*b*, *y*),*π*(*b*∣*θ*, *ψ*, Σ, *y*) = *π*(*b*∣*θ*, *ψ*, *y*) and *π*(*ψ*∣*θ*, Σ, *b*, *y*) = *π*(*ψ*∣*b*). A Metropolis-Hastings step can be used when a full conditional density is intractable. An algorithm for computing the marginal likelihood *π*(*y**i*∣*θ*, Σ, *b**i*) for each experimental unit is given in the supplementary material. Interest may also lie in the joint posterior *π*(*θ*, *ψ*, Σ, *b*, *x*∣*y*) where, since no imputation is required for the LNA, *x**i* = (*x**t*0, …, *x**t**n*)*T* and *x* = (*x*1, …, *x**N*). Realisations from this posterior can be obtained using the above Gibbs sampler with an extra step that draws from *π*(*x**i*∣*θ*, *ψ*, Σ, *b**i*, *y**i*) = *π*(*x**i*∣*θ*, Σ, *b**i*, *y**i*) for *i* = 1, …, *N*. An efficient mechanism for making such draws can also be found in the supplementary material. The method uses a forward filter, backward sampling (FFBS) algorithm.\nNote that the backward sweep requires\n \n$$\\begin{aligned}\nCov(X\\_{t\\_{j+1}},X\\_{t\\_j})&=Cov(\\tilde{R}\\_{t\\_{j+1}},\\tilde{R}\\_{t\\_j})\n=Cov(P\\_{t\\_{j+1}}\\tilde{R}\\_{t\\_j},\\tilde{R}\\_{t\\_j})\n= P\\_{t\\_{j+1}}Var(\\tilde{R}\\_{t\\_j}).\\end{aligned}$$ \nHere *P**t* is a *d* × *d* matrix that can be shown to satisfy the ODE\n \n$$\\dfrac{dP\\_t}{dt} = H\\_tP\\_t, \\label{eqn:lna\\_p}$$ \nwith initial condition *P*0 = *I**d*, the *d* × *d* identity matrix.\nApplications\n============\nWe now compare the accuracy and efficiency of our Bayesian imputation approach (coupled with the modified innovation scheme) with an LNA-based solution. We consider two scenarios: one in which the ODEs governing the LNA are tractable and one in which numerical solvers are required. In the first we use synthetic data generated from a simple univariate SDE description of orange tree growth. The second example uses real data taken from to fit an SDMEM driven by the bivariate diffusion approximation of a stochastic kinetic model of aphid dynamics. The resulting SDMEM is particularly challenging to fit as both the drift and diffusion functions are nonlinear and also only one component of the model is observed (with error). We also include (in the supplementary material) a simulation study based on synthetic data generated from the model of aphid dynamics, to explore further any differences between the Bayesian imputation and LNA-based approaches.\nOrange tree growth\n------------------\nThe SDMEM developed by and to model orange tree growth describes the dynamics of the circumference (*X**t**i*) of individual trees (mm) by \n$$dX\\_t^i = \\dfrac{1}{\\phi\\_1^i\\phi\\_2^i}X\\_t^i(\\phi\\_1^i-X\\_t^i)\\,dt \n+ \\sigma\\sqrt{X\\_t^i}\\,dW\\_t^i,\\quad X\\_0^i=x\\_0^i, \\quad i=1,\\ldots,N$$ \n with *ϕ*1*i* ∼ *N*(*ϕ*1, *σ**ϕ*12) and *ϕ*2*i* ∼ *N*(*ϕ*2, *σ**ϕ*22) independently. Here *θ* = *σ* is common to all trees, the random effects are *b**i* = (*ϕ*1*i*, *ϕ*2*i*)*T*, *i* = 1, …, *N* and the parameter vector governing the random effects distributions is *ψ* = (*ϕ*1, *ϕ*2, *σ**ϕ*1, *σ**ϕ*2)*T*. Note that the *ϕ*1*i* can be interpreted as asymptotic circumferences and the *ϕ*2*i* as the time-distance between the inflection point of the model obtained by ignoring stochasticity and the point where *X**t**i* = *ϕ*1*i*/(1 + *e*− 1).\nTo allow identifiability of all model parameters we generated 16 observations for the circumference of *N* = 100 trees at intervals of 100 days. Following we gave each tree the same initial condition (*x*0*i* = 30) and took (*ϕ*1, *ϕ*2, *σ**ϕ*1, *σ**ϕ*2, *σ*) = (195, 350, 25, 52.5, 0.08), which gives random effects distributions *ϕ*1*i* ∼ *N*(195, 252) and *ϕ*2*i* ∼ *N*(350, 52.52). For our analysis of these data we assumed the parameters to be independent *a priori* with *ϕ*1 and *ϕ*2 having weak *N*(0, 1002) priors and 1/*σ**ϕ*12, 1/*σ**ϕ*22 and 1/*σ*2 having weak gamma *G**a*(1, 0.01) priors. In this example we assume there is no measurement error and therefore the target posterior is given by \n*π*(*θ*, *ψ*, *b*∣*x*) ∝ *π*(*θ*)*π*(*ψ*)*π*(*b*∣*ψ*)*π*(*x*∣*θ*, *b*). \n In the Bayesian imputation approach, *π*(*x*∣*θ*, *b*) is as in whereas for the LNA–based solution \n$$\\pi(x|\\theta,b)=\n\\prod\\limits\\_{i=1}^{N}\\prod\\limits\\_{j=0}^{n-1}N(x^{i}\\_{t\\_{j+1}}\\,;\\, \\eta^{i}\\_{t\\_{j+1}},V^{i}\\_{t\\_{j+1}}),$$ \n where, for each interval [*t**j*, *t**j* + 1] and each tree *i*, the *η**t**i* and *V**t**i* satisfy the ODE system\n \n$$\\begin{aligned}\n\\dfrac{d\\eta\\_t^i}{dt} &= \\dfrac{1}{\\phi\\_1^i\\phi\\_2^i}\\eta\\_t^i(\\phi\\_1^i-\\eta\\_t^i),\\qquad \\eta\\_{t\\_j}^i=x\\_{t\\_j}^i, \\\\\n\\dfrac{dV\\_t^i}{dt} &= \\dfrac{2}{\\phi\\_1^i\\phi\\_2^i}(\\phi\\_1^i-2\\eta\\_t^i)V\\_t^i + \\sigma^2\\eta\\_t^i,\\qquad V\\_{t\\_j}^i=0.\\end{aligned}$$ \nFortunately this ODE system can be solved analytically giving *η**t**i* = *A**ϕ*1*i**e**t*/*ϕ*2*i*/(1 + *A**e**t*/*ϕ*2*i*) and \n$$V\\_t^i=B\\left(\\frac{1}{2}A^3\\phi\\_2^i e^{2t/\\phi\\_2^i} + 3A^2\\phi\\_2^ie^{t/\\phi\\_2^i}-\\phi\\_2^ie^{-t/\\phi\\_2^i}\n+3At - \\dfrac{1}{2}A^3\\phi\\_2^i - 3A^2\\phi\\_2^i +\\phi\\_2^i \\right)$$ \n where *A* = *x**t*0*i*/(*ϕ*1*i* − *x**t*0*i*) and *B* = *σ*2*A**ϕ*1*i**e*2*t*/*ϕ*2*i*/(1 + *A**e**t*/*ϕ*2*i*)4.\nThe MCMC scheme can make use of simple semi-conjugate updates for *ϕ*1, *ϕ*2, *σ**ϕ*1 and *σ**ϕ*2. However the remaining parameters (*σ* and the *b**i*) require Metropolis-within-Gibbs updates and we have found that componentwise normal random walk updates (so-called random walk Metropolis) on the log scale work particularly well. Also, for the modified innovation scheme, the dynamics of the SDMEM permit the use of the modified diffusion bridge construct to update the latent trajectories between observation times: the improved bridge construct of Section [dgode] is not needed.\nThe modified innovation scheme requires specification of the level of discretisation *m*. We performed several short pilot runs of the scheme with *m* ∈ {5, 10, 20, 40} and found no discernible difference in posterior output for *m* ≥ 10. We therefore took *m* = 10. The sample output was also used to estimate the marginal posterior variances of *σ* and the *b**i*, to provide sensible innovation variances in the random walk Metropolis updates. Both the modified innovation scheme and the LNA–based scheme required a burn in of 500 iterations, a thin of 100 iterates and were run long enough to yield a sample of approximately 10K independent posterior draws. Figure [orange figparams] shows the marginal posterior densities and autocorrelations for the common parameter *σ* and the parameters governing the random effects distributions. The marginal posterior means and standard deviations of (*ϕ*1, *ϕ*2, *σ**ϕ*1, *σ**ϕ*2, *σ*) are given in Table [tab orangeparam]. The figures and table show that for these parameters both the imputation approach and LNA–based approach generally give similar output and are consistent with the true values from which the data were simulated. Similar results are obtained for the random effects parameters (see the supplementary material).\nBoth schemes were coded in C and run on an Intel Xeon 3.0GHz processor; the modified innovation scheme took 43504 seconds to run whilst the LNA inference scheme took 2483 seconds. We use the minimum (over each parameter chain) effective sample size (minESS) to measure the statistical efficiency of each scheme. The modified innovation scheme produced a minESS of 7949 and the LNA–based approach gave 7821. Therefore, in terms of minESS/sec, using the LNA outperforms the imputation approach in this example by a factor of approximately 17. It should be noted, however, that for most nonlinear SDMEMs the ODEs governing the LNA solution will rarely be tractable and the consequent use of numerical schemes will degrade its performance.\nIn the next section we consider an example in which the LNA ODEs are intractable.\n[b]0.28 \n[b]0.28 \n[b]0.28 \n \n[b]0.28 \n[b]0.28 \n[b]0.28 \n[orange figparams]\n[h]\nc|ccccc & *ϕ*1 & *ϕ*2 & *σ**ϕ*1 & *σ**ϕ*2 & *σ* \n & 194.229 & 344.799 & 24.316 & 53.219 & 0.079 \n& (3.509) & (10.098) & (3.149) & (10.410) & (0.002) \n---\n& 194.634 & 347.631 & 24.207 & 53.960 & 0.079 \n& (4.025) & (10.844) & (3.154) & (10.193) & (0.002) \n[tab orangeparam]\n[b]0.28 \n[b]0.28 \n[b]0.28 \n \n[b]0.28 \n[b]0.28 \n[b]0.28 \n \n[b]0.28 \n[b]0.28 \n[b]0.28 \n \n[b]0.28 \n[b]0.28 \n[b]0.28 \n[orange figphii]\nCotton aphid dynamics\n---------------------\n### Model and data\nAphids (also known as plant lice or greenfly) are small sap sucking insects which live on the leaves of plants. As they suck the sap they also secrete honey-dew which forms a protective cover over the leaf, ultimately resulting in aphid starvation. describe a model for aphid dynamics in terms of population size (*N**t*) and cumulative population size (*C**t*). The model is a stochastic birth-death model with linear birth rate *λ**N**t* and death rate *μ**N**t**C**t*. The key probabilistic laws governing the time-evolution of the process over a small interval (*t*, *t* + *d**t*] are\n \n$$\\begin{split}\n\\Pr(N\\_{t+dt}=n\\_t +1,\\,C\\_{t+dt}=c\\_t +1\\,|\\,n\\_t,c\\_t)&=\\lambda n\\_t\\,dt+o(dt), \\label{eqn:aphid1and2} \\\\\n\\Pr(N\\_{t+dt}=n\\_t -1,\\,C\\_{t+dt}=c\\_t\\,|\\,n\\_t,c\\_t)&=\\mu n\\_t c\\_t\\,dt+o(dt).\n\\end{split}$$ \nThe diffusion approximation of the Markov jump process defined by is\n \n$$\\label{eqn:aphidSDE}\n\\begin{pmatrix}\ndN\\_t\\\\\ndC\\_t\\\\ \\end{pmatrix}= \\begin{pmatrix}\n\\lambda N\\_t-\\mu N\\_tC\\_t \\\\\n\\lambda N\\_t \\\\ \\end{pmatrix}dt+ \\begin{pmatrix}\n\\lambda N\\_t+\\mu N\\_tC\\_t & \\lambda N\\_t \\\\\n\\lambda N\\_t & \\lambda N\\_t \\\\ \\end{pmatrix}^{{1/2}}dW\\_t.$$ \n also provide a dataset of cotton aphid counts collected from three blocks (1/2/3) and using treatments constructed from two factors: water irrigation (low/medium/high) and nitrogen (blanket/variable/none). The data were collected in July 2004 in Lamesa, Texas and consist of five observations of aphid counts aggregated over twenty randomly chosen leaves in each plot for the twenty-seven treatment-block combinations. The data were recorded at times *t* = 0, 1.14, 2.29, 3.57 and 4.57 weeks (approximately every 7/8 days).\nWe now formulate an appropriate SDMEM model driven by for these data and then fit the model. For notational simplicity, let *i*, *j*, *k* denote the level of water, nitrogen and block number respectively with *i*, *j*, *k* ∈ {1, 2, 3}, where 1 represents low water/blanket nitrogen, 2 represents medium water/variable nitrogen and 3 represents high water/zero nitrogen. Let *N**t**i**j**k* denote the number of aphids at time *t* for combination *i**j**k* and *C**t**i**j**k* the corresponding cumulative population size. We write *X**t**i**j**k* = (*N**t**i**j**k*, *C**t**i**j**k*)*T* and consider the SDMEM \n$$dX\\_t^{ijk}=\\alpha(X\\_t^{ijk},b^{ijk})\\,dt+\\sqrt{\\beta(X\\_t^{ijk},b^{ijk})}\\,dW\\_t^{ijk}, \\quad i,j,k\\in\\{1,2,3\\},$$ \n where\n \n$$\\begin{aligned}\n\\alpha(X\\_t^{ijk},b^{ijk})&=\\begin{pmatrix}\n\\lambda^{ijk} N\\_t^{ijk}-\\mu^{ijk} N\\_t^{ijk}C\\_t^{ijk} \\\\\n\\lambda^{ijk} N\\_t^{ijk} \\\\ \\end{pmatrix},\\\\\n\\beta(X\\_t^{ijk},b^{ijk})&=\\begin{pmatrix}\n\\lambda^{ijk} N\\_t^{ijk}+\\mu^{ijk} N\\_t^{ijk}C\\_t^{ijk} & \\lambda^{ijk} N\\_t^{ijk} \\\\\n\\lambda^{ijk} N\\_t^{ijk} & \\lambda^{ijk} N\\_t^{ijk} \\\\ \\end{pmatrix}.\\end{aligned}$$ \nThe fixed effects *b**i**j**k* = (*λ**i**j**k*, *μ**i**j**k*)*T* have a standard structure which allows for main factor and block effects and single factor-block interactions, with\n \n$$\\begin{split}\n\\lambda^{ijk}&=\\lambda + \\lambda\\_{W\\_i} + \\lambda\\_{N\\_j} + \\lambda\\_{B\\_k} + \\lambda\\_{WN\\_{ij}} + \\lambda\\_{WB\\_{ik}} + \\lambda\\_{NB\\_{jk}} \\label{aphid\\_lambdamu}\\\\\n\\mu^{ijk}&=\\mu + \\mu\\_{W\\_i} + \\mu\\_{N\\_j} + \\mu\\_{B\\_k} + \\mu\\_{WN\\_{ij}} + \\mu\\_{WB\\_{ik}} + \\mu\\_{NB\\_{jk}}. \n\\end{split}$$ \nAlso for identifiability we use the corner constraints *λ**W*1 = *λ**N*1 = *λ**B*1 = 0, *λ**W**N**i**j* = *λ**W**N**i**j*(1 − *κ**i**j*), *λ**W**B**i**k* = *λ**W**B**i**k*(1 − *κ**i**k*) and *λ**N**B**j**k* = *λ**N**B**j**k*(1 − *κ**j**k*), where *κ**r**s* = max(*δ*1*r*, *δ*1*s*) and *δ*1 ⋅ is the Kronecker delta, with equivalent constraints on the death rates. The interpretation of is straightforward. For example, *λ*111 = *λ* and *μ*111 = *μ* are the baseline birth and death rates inferred using all 5 × 33 = 135 observations, and correspond to the treatment combination low water, blanket nitrogen and block 1. Likewise, all 5 × 32 = 45 observations taken from block 2 inform the main effects of block 2 (*λ**B*2 and *μ**B*2) relative to the baseline.\nA related approach can be found in, where the diffusion approximation is eschewed in favour of a further approximation via moment closure. Our approach further differs from theirs by allowing for measurement error and leads to a much improved predictive fit. The measurement error model is in part motivated by an over-dispersed Poisson error structure which we then approximate by a Gaussian distribution. Specifically, we assume that aphid population size *N**t* is observed with Gaussian error and that the error variance is proportional to the latent aphid numbers, giving\n \n$$\\label{eqn:obsAphid}\nY\\_t^{ijk}|N\\_t^{ijk},\\sigma {\\overset{indep}{\\sim }}N(N\\_t^{ijk},\\sigma^{2}N\\_t^{ijk}),\\quad t=0, 1.14, 2.29, 3.57, 4.57.$$ \n### Implementation\nOur prior beliefs for 1/*σ*2 are described by a *G**a*(*a*, *a*) distribution. We found little difference in results for *a* ∈ {0.01, 0.1, 1} and so here we report results for *a* = 1. The prior for the elements in consists of independent components subject to the birth and death rates for each treatment combination (*λ**i**j**k*, *μ**i**j**k*) being positive. The baseline rates *λ* and *μ* must be positive and so, following, we assign weak *U*( − 10, 10) priors to log*λ* and log*μ* and also to the remaining parameters. We also take a fairly weak *N*(24, 90) prior for each *N**t*0*i**j**k* and use a proposal of the form *N*(*N**t*0,*σ*2*N**t*0) for updates. The cumulative population sizes must be at least as large as their equivalent population size. However, we do not expect them to be greatly different *a priori*. We investigated using a truncated distribution of the form *C**t*0∣*N**t*0 ∼ *N*(*N**t*0, *d**c*2), *C**t*0 > *N**t*0 as the prior and found that this led to little difference in posterior output for *d**c* ∈ {1, 10, 100}. We have, therefore, chosen to fix *C**t*0*i**j**k* = *N**t*0*i**j**k* in our analysis. Note that the form of the prior for *σ* gives a semi-conjugate update. The remaining parameters in are updated using random walk Metropolis on the pairwise *λ*, *μ* component blocks (*λ*, *μ*), (*λ**W*2, *μ**W*2), (*λ**W*3, *μ**W*3), …, (*λ**N**B*33, *μ**N**B*33).\nThe nonlinear form of the observation model can be problematic for the modified innovation scheme. In particular, the proposal mechanism for the path update requires an observation model that is linear in *N**t*. Therefore, when proposing from the bridge construct in Section [dgode], we replace Σ in and with *σ*2*η**N*, *t**j* + 1, where *η**t**j* + 1 = (*η**N*, *t**j* + 1, *η**C*, *t**j* + 1)*T* is the solution of. Since the proposal mechanism is corrected for via the Metropolis-Hastings step, no additional approximations to the target distribution are needed.\nIn order to obtain a statistically efficient implementation of the modified innovation scheme, we investigate the performance of the modified diffusion bridge construct of and our improved bridge construct of Section [dgode] in a scenario typical of the real dataset. Using the simulation study of, we take (*λ*, *μ*)*T* = (1.75, 0.00095)*T*, and recursively apply the Euler-Maruyama approximation to give *x*3.57 = (829.08, 1406.07)*T*. We then compare the performance of each bridge construct over the final observation interval [3.57, 4.57] by taking *y*4.57 as the median of ([eqn:obsAphid]) with *σ* = 1. Figure [aphid figbridges] shows 95% credible regions of the true conditioned process *N**t*∣*x*3.57, *y*4.57 (found via Monte Carlo simulation) with 95% credible regions obtained by repeatedly simulating from the modified diffusion bridge and our improved construct. It is clear that the modified diffusion bridge fails to adequately account for the nonlinear behaviour of the conditioned process. Use of each construct as a proposal mechanism inside a Metropolis-Hastings independence sampler (100K iterations) results in an acceptance rate of around 58% for the improved bridge construct and just 1% for the modified diffusion bridge. It is for these reasons that the modified diffusion bridge is eschewed in favour of our improved bridge construct when applying the Bayesian imputation approach.\n[b]0.49\n[b]0.49\n[aphid figbridges]\nFinally, fitting the LNA requires the solution of an ODE system given by and where the Jacobian matrix is \n$$H\\_t= \\begin{pmatrix}\n\\lambda - \\mu\\eta\\_{C,t} & -\\mu\\eta\\_{N,t} \\\\\n\\lambda & 0 \\end{pmatrix}.$$ \n This ODE system is intractable and so our C implementation uses a standard ODE solver from the GNU scientific library, namely the explicit embedded Runge-Kutta-Fehlberg (4, 5) method. Note that the tractability of the marginal likelihood under the LNA requires a linear Gaussian observation model. Therefore, when applying the FFBS algorithm in the supplementary material, we make an approximation to the marginal likelihood calculation by replacing Σ with *σ*2*η**N*, *t**j* + 1.\n### Results\nThe time between observations is almost but not quite constant and so we have allowed each interval to have its own discretisation level, *m*. That said, the interval-specific values vary very little, and by at most two for the larger *m* values. Several short pilot runs of the modified innovation scheme were performed with typical *m* ∈ {5, 10, 20, 40, 50}. These gave no discernible difference in posterior output for *m* ≥ 20 and so we took *m* = 20. The sample output was also used to estimate the marginal posterior variances of the *λ*, *μ* component blocks of the parameters in, to be used in the random walk Metropolis updates. Both the modified innovation scheme and MCMC scheme under the LNA were run for 40M iterations with the output thinned by taking every 4Kth iterate to give a final sample of size 10K.\n[b]0.28 \n[b]0.28 \n[b]0.28 \n \n[b]0.28 \n[b]0.28 \n[b]0.28 \n \n[b]0.28 \n[b]0.28 \n[b]0.28 \n[aphid figbaselines]\nFigure [aphid figbaselines] shows the marginal posterior densities of the baseline parameters, the parameter *σ* controlling the observation error variance and a selection of the remaining parameters. As in we find that block 2 plays an important role. The 95% credible regions for *μ**B*2, the main block 2 death rate, and *λ**N**B*22, the birth rate characterising the interaction with nitrogen, are plausibly non-zero. Whilst the imputation approach and LNA generally give consistent output, there are some notable differences. For example, we find, in general, that the LNA tends to underestimate parameter values (and slightly exaggerates the confidence in these estimates) compared to those obtained under the modified innovation scheme.\nWe also compared the predictive distributions obtained under each inferential model. The within-sample predictive distribution for the observation process {*Y**t*, *t* = 0, …, 4.57} can be obtained by integrating over the posterior uncertainty of the latent process and parameter values in the observation model. Specifically, given samples {(*n**t**i**j**k*(*l*), *σ*(*l*)), *l* = 1, …, *L*} from the marginal posterior *π*(*n**t**i**j**k*, *σ*∣*y*), the predictive density at time *t* can be estimated by \n$$\\frac{1}{L}\\sum\\_{l=1}^{L}N\\left(y\\_t\\,;\\,n\\_t^{ijk(l)},(\\sigma^{(l)})^2 n\\_t^{ijk(l)}\\right).$$ \n Likewise, for a new experiment repeated under the same conditions, the out-of-sample predictive distribution for the aphid population size can be determined for each treatment combination. This is estimated by averaging realisations of *N**t* (obtained by applying the Euler-Maruyama approximation to ([eqn:aphidSDE])) over draws from the marginal posterior *π*(*n*0*i**j**k*, *b**i**j**k*∣*y*) obtained using either Bayesian imputation or the LNA. Figures [aphid figfit] and [aphid figpreds] summarise these predictive distributions for a random selection of treatment combinations. Both the SDMEM and LNA give a satisfactory fit to the observed data, with all observations within or close to the central 50% of the distribution, and no observation outside the equi-tailed 95% credible intervals. As expected, the SDMEM gives a better fit over the LNA, although there is little difference between the two. There are however noticeable differences in the out-of-sample predictives, especially in the lower credible bound (in Figure [aphid figpreds]) suggesting that in some situations, using the inferences made under the LNA to predict the outcome of future experiments can give misleading results. These differences lead us to examine the marginal posterior densities of the treatment-block specific birth and death rates, *λ**i**j**k* and *μ**i**j**k*, over whose uncertainty we average. Samples from these posteriors are straightforward to obtain, using the posterior samples of the constituent parameters in ([aphidlambdamu]). Figure [aphid figre] shows marginal posterior densities of the overall birth rates (*λ**i**j**k*) associated with the six treatment-block combinations for which predictives are presented in Figure [aphid figpreds]. We see distinct differences between posteriors obtained under the Bayesian imputation approach and the LNA approach. The posteriors displayed are indicative of those obtained for all treatment combinations. Moreover, similar patterns are evident in the overall death rates (*μ**i**j**k*).\n[b]0.28\n[b]0.28\n[b]0.28\n \n[b]0.28\n[b]0.28\n[b]0.28\n \n[b]0.28\n[b]0.28\n[b]0.28\n \n[b]0.28\n[b]0.28\n[b]0.28\n[aphid figfit]\n[b]0.28\n[b]0.28\n[b]0.28\n \n[b]0.28\n[b]0.28\n[b]0.28\n[aphid figpreds]\n[b]0.28 \n[b]0.28 \n[b]0.28 \n \n[b]0.28 \n[b]0.28 \n[b]0.28 \n[aphid figre]\nWe obtained a minESS of 1039 under the modified innovation scheme. The LNA, however, clearly benefits from analytically integrating out the latent process and gave a minESS of 8908. For this example, we found that significant gains in computational efficiency were possible by performing the parameter updates and, for the modified innovation scheme, the path updates, in parallel. For example, updating *λ**B*2 and *μ**B*2 involves calculating a product of likelihoods (or marginal likelihoods for the LNA) over all 32 = 9 treatment combinations that include block 2. These constituent likelihoods can be calculated in parallel. Similarly, for the modified innovation scheme, the treatment specific path updates can be performed in parallel. Both the modified innovation scheme and the LNA–based scheme were again coded in C and run on a high performance computing cluster with 14 cores (made up of Intel Xeon 3.0GHz processors). The modified innovation scheme took approximately 18 days to run whereas the LNA–based scheme required only approximately 4.3 days. Note that here the speed advantage of the LNA–based scheme has reduced, now being roughly 4 times faster than the modified innovation scheme, whereas in Section [orange], the LNA was approximately 20 times faster. The intractability of the ODEs driving the LNA clearly plays a significant role in computational efficiency. In terms of overall efficiency (as measured by minESS/sec) the LNA–based scheme outperforms the Bayesian imputation approach by a factor of around 36. These computational advantages of the LNA must be weighted against the inaccuracies of the resulting posterior and predictive distributions, inaccuracies which can at times be substantial, as demonstrated by the simulation study in the supplementary material.\n### Simulation study\nIn this section, we investigate further differences between the Bayesian imputation approach and an inference scheme based on the LNA using synthetic data generated from ([eqn:aphidSDE]). For simplicity, we consider a fixed treatment (low water, blanket nitrogen) and three blocks. We therefore write *X**t*11*k* = (*N**t*11*k*, *C**t*11*k*)*T* and consider the \n$$dX\\_t^{11k}=\\alpha(X\\_t^{11k},b^{11k})\\,dt+\\sqrt{\\beta(X\\_t^{11k},b^{11k})}\\,dW\\_t^{11k}, \\quad k\\in\\{1,2,3\\},$$ \n where\n \n$$\\begin{aligned}\n\\alpha(X\\_t^{11k},b^{11k})&=\\begin{pmatrix}\n\\lambda^{11k} N\\_t^{11k}-\\mu^{11k} N\\_t^{11k}C\\_t^{11k} \\\\\n\\lambda^{11k} N\\_t^{11k} \\\\ \\end{pmatrix},\\\\\n\\beta(X\\_t^{11k},b^{11k})&=\\begin{pmatrix}\n\\lambda^{11k} N\\_t^{11k}+\\mu^{11k} N\\_t^{11k}C\\_t^{11k} & \\lambda^{11k} N\\_t^{11k} \\\\\n\\lambda^{11k} N\\_t^{11k} & \\lambda^{11k} N\\_t^{11k} \\\\ \\end{pmatrix}.\\end{aligned}$$ \nThe fixed effects *b*11*k* = (*λ*11*k*, *μ*11*k*)*T* have a standard structure to incorporate block effects, with \n*λ*11*k* = *λ* + *λ**B**k* and *μ*11*k* = *μ* + *μ**B**k*, \n where we again impose the corner constraints *λ**B*1 = *μ**B*1 = 0 to allow for identifiability.\nTo mimic the real dataset, we took *λ* = 1.75, *μ* = 0.00095, *λ**B*2 = − 0.1154, *λ**B*3 = − 0.0225, *μ**B*2 = − 0.0004 and *μ**B*3 = 0.0002. For each block, we generated five observations (on a regular grid) by using the Euler-Maruyama approximation with a small time-step (Δ*t* = 0.001) and an initial condition of *x*0 = (5, 5)*T*. To assess the impact of measurement error on the quality of inferences that can be made about each parameter, we corrupted our data via the observation model \n$$Y\\_t^{11k}|N\\_t^{11k},\\sigma {\\overset{indep}{\\sim }}N(N\\_t^{11k},\\sigma^{2}N\\_t^{11k}),\\quad t=0, 1, 2, 3, 4,$$ \n and took *σ* ∈ {0, 0.5, 1, 5} to give four synthetic datasets. We adopt the same prior specification for the unknown parameters as used in the real data application.\nBoth the modified innovation scheme (again incorporating the improved bridge construct) and the LNA-based inference scheme were run long enough to yield a sample of approximately 10K independent posterior draws. For the former, we fixed the discretisation level by taking *m* = 20 and note that *m* > 20 gave little difference in posterior output. Figure [aphid figsimbaselines] shows the marginal posterior densities of the baseline parameters (*λ* and *μ*) and the measurement error variance (*σ*). The joint posterior densities of (*μ*, *λ*)*T* are shown in Figure [aphid figsimjoint]. It is clear that when fitting the SDMEM using the Bayesian imputation approach, the posterior samples obtained are consistent with the ground truth. This is true to a lesser extent when using the LNA, with the ground truth found in the tail of the posterior distribution in three out of the four scenarios. In fact, when using synthetic data with *σ* < 5, we see substantive differences in posterior output. As was observed when using real data, the LNA underestimates parameter values compared to those obtained under the Bayesian imputation scheme. In this case, the LNA provides a relatively poor approximation to the true posterior distribution.\nIncreasing *σ* to 5 (and beyond) gives output from both schemes which is largely in agreement. This is intuitively reasonable, since, as the variance of the measurement process is increased, the ability of both inference schemes to accurately infer the underlying dynamics is diminished. Essentially, the relative difference between the LNA and SDE is reduced.\n[b]0.28 \n[b]0.28 \n[b]0.28\n \n[b]0.28 \n[b]0.28 \n[b]0.28 \n \n[b]0.28 \n[b]0.28 \n[b]0.28 \n \n[b]0.28 \n[b]0.28 \n[b]0.28 \n[aphid figsimbaselines]\n[b]0.49\n[b]0.49\n \n[b]0.49\n[b]0.49\n[aphid figsimjoint]\nDiscussion\n==========\nWe have provided a framework that permits (simulation-based) Bayesian inference for a large class of multivariate SDMEMs using discrete-time observations that may be incomplete and subject to measurement error. By adopting a Bayesian imputation approach, we have shown how the modified innovation scheme of, which is necessary for overcoming the problematic dependence between the latent process and any parameters that enter the diffusion coefficient, can be applied to SDMEMs. Fundamental to our approach is the development of a novel bridge construct that can be used to sample a discretisation of a conditioned diffusion process, and does not break down when the process exhibits strong nonlinearity over inter-observation times of interest. The computational cost of the Bayesian imputation scheme is dictated by the number of imputed points (characterised by *m*) between observation times. In the examples considered here we see little difference in posterior output under the Bayesian imputation scheme for *m* ≥ 20.\nWe also considered a tractable approximation to the SDMEM, the linear noise approximation, and provided a systematic comparison using two applications. The computational efficiency of the LNA depends on the dimension of the SDE driving the SDMEM. For a *d*-dimensional SDE system, the LNA requires the solution of a system of order *d*2 coupled ODEs. In our first application, the resulting ODE system can be solved analytically, leading to increases in both computational and overall efficiency (as measured by minimum ESS per second) of around a factor of 20. Moreover, we found little difference in the accuracy of inferences made under the LNA and imputation approaches. In our second application, we fitted the diffusion approximation of a Markov jump process description of aphid dynamics using data from. In this example, the ODE system governing the LNA is intractable and the computational advantage of using the LNA over an imputation approach reduced to around a factor of 4. However, the benefit of using the LNA to analytically integrate over the latent process is clear, giving an overall increase in efficiency of around a factor of 36. It is important to note that whilst the LNA is preferred in terms of overall efficiency for the examples considered here, as the dimension *d* of the SDE is increased, the LNA is likely to become infeasible. Moreover, whilst both the imputation and LNA approaches provided a reasonable fit to the aphid data, differences were found between the parameter posteriors, leading to differences in the out-of-sample predictive distributions. A simulation study (given in the supplementary material) highlighted further differences between the LNA and Bayesian imputation approaches. Care must therefore be taken in trying to fit the SDMEM by using an LNA–based inference approach.\nLNA FFBS\n========\nTo ease the notation, consider a single experimental unit and drop *i* from the notation. Since the parameters *θ*, *ψ*, *b* and Σ remain fixed throughout this section, we also drop them from the notation where possible. Define *y*0 : *j* = (*y**t*0, …, *y**t**j*)*T*. Now suppose that *X*0 ∼ *N*(*a*, *C*) *a priori*. The marginal likelihood under the LNA, *π*(*y*∣*θ*, Σ, *b*), can be obtained from the following forward filter.\n1. Forward filter. Initialisation. Compute *π*(*y**t*0) = *N*(*y**t*0 ; *F**T**a* , *F**T**C**F* + Σ). The posterior at time *t*0 = 0 is therefore *X**t*0∣*y**t*0 ∼ *N*(*a*0, *C*0), where\n \n$$\\begin{aligned}\na\\_0 &= a+CF(F^TCF+\\Sigma)^{-1}(y\\_{t\\_{0}}-F^Ta) \\\\\nC\\_0 &= C-CF(F^TCF+\\Sigma)^{-1}F^TC\\,.\\end{aligned}$$ \nStore the values of *a*0 and *C*0.\n2. For *j* = 0, 1, …, *n* − 1,\n\t* Prior at *t**j* + 1. Initialise the LNA with *η**t**j* = *a**t**j*, *V**t**j* = *C**t**j* and *P**t**j* = *I**d*. Integrate the ODEs ([eqn:dgodeeta]), ([eqn:lnav]) and ([eqn:lnap]) forward to *t**j* + 1 to obtain *η**t**j* + 1, *V**t**j* + 1 and *P**t**j* + 1. Hence *X**t**j* + 1∣*y*0 : *j* + 1 ∼ *N*(*η**t**j* + 1, *V**t**j* + 1).\n\t* One step forecast. Using the observation equation, we have that \n\t*Y**t**j* + 1∣*y*0 : *j* ∼ *N*(*F**T**η**t**j* + 1, *F**T**V**t**j* + 1*F* + Σ). \n\t Compute the updated marginal likelihood\n\t\n\t\n\t \n\t$$\\begin{aligned}\n\t\\pi(y\\_{0:j+1})&=\\pi(y\\_{0:j})\\pi(y\\_{t\\_{j+1}}|y\\_{0:j})\n\t=\\pi(y\\_{0:j})N(y\\_{t\\_{j+1}}\\,;\\, F^T\\eta\\_{t\\_{j+1}}\\,,\\,F^TV\\_{t\\_{j+1}}F+\\Sigma).\\end{aligned}$$\n\t* Posterior at *t**j* + 1. Combining the distributions in (a) and (b) gives the joint distribution of *X**t**j* + 1 and *Y**t**j* + 1 (conditional on *y*0 : *j*) as \n\t$$\\begin{pmatrix}\n\t X\\_{t\\_{j+1}} \\\\ \n\t Y\\_{t\\_{j+1}}\n\t \\end{pmatrix}\\sim N\\left\\{\\begin{pmatrix}\n\t \\eta\\_{t\\_{j+1}} \\\\\n\t F^T\\eta\\_{t\\_{j+1}} \n\t \\end{pmatrix}\\,,\\, \\begin{pmatrix}\n\t V\\_{t\\_{j+1}} & V\\_{t\\_{j+1}}F \\\\\n\t F^TV\\_{t\\_{j+1}} & F^TV\\_{t\\_{j+1}}F+\\Sigma \n\t \\end{pmatrix} \\right \\}$$ \n\t and therefore *X**t**j* + 1∣*y*0 : *j* + 1 ∼ *N*(*a**t**j* + 1, *C**t**j* + 1), where\n\t\n\t\n\t \n\t$$\\begin{aligned}\n\ta\\_{t\\_{j+1}} &= \\eta\\_{t\\_{j+1}}+V\\_{t\\_{j+1}}F(F^TV\\_{t\\_{j+1}}F+\\Sigma)^{-1}(y\\_{t\\_{j+1}}-F^T\\eta\\_{t\\_{j+1}}) \\\\\n\tC\\_{t\\_{j+1}} &= V\\_{t\\_{j+1}}-V\\_{t\\_{j+1}}F(F^TV\\_{t\\_{j+1}}F+\\Sigma)^{-1}F^TV\\_{t\\_{j+1}}\\,.\\end{aligned}$$ \n\t\n\t\n\t\n\tStore the values of *a**t**j* + 1, *C**t**j* + 1, *η**t**j* + 1, *V**t**j* + 1 and *P**t**j* + 1.\nSample *π*(*x*∣*y*) using the following backward sampler.\n1. Backward sampler. First draw *x**t**n* from *X**t**n*∣*y* ∼ *N*(*a**t**n*, *C**t**n*).\n2. For *j* = *n* − 1, *n* − 2, …, 0,\n\t* Joint distribution of *X**t**j* and *X**t**j* + 1. Note that *X**t**j*∣*y*0 : *j* ∼ *N*(*a**t**j*, *C**t**j*). The joint distribution of *X**t**j* and *X**t**j* + 1 (conditional on *y*0 : *j*) is \n\t$$\\begin{pmatrix}\n\t X\\_{t\\_{j}} \\\\ \n\t X\\_{t\\_{j+1}}\n\t \\end{pmatrix}\\sim N\\left\\{\\begin{pmatrix}\n\t a\\_{t\\_{j}} \\\\\n\t \\eta\\_{t\\_{j+1}} \n\t \\end{pmatrix}\\,,\\, \\begin{pmatrix}\n\t C\\_{t\\_{j}} & C\\_{t\\_{j}}P\\_{t\\_{j+1}}^T \\\\\n\t P\\_{t\\_{j+1}}C\\_{t\\_{j}} & V\\_{t\\_{j+1}} \n\t \\end{pmatrix} \\right \\}.$$\n\t* Backward distribution. The distribution of *X**t**j*∣*X**t**j* + 1, *y*0 : *j* is *N*(*ã**t**j*, *C̃**t**j*), where\n\t\n\t\n\t \n\t$$\\begin{aligned}\n\t\\tilde{a}\\_{t\\_j} &= a\\_{t\\_j}+C\\_{t\\_j}P\\_{t\\_{j+1}}^TV\\_{t\\_{j+1}}^{-1}(x\\_{t\\_{j+1}}-\\eta\\_{t\\_{j+1}}), \\\\\n\t\\tilde{C}\\_{t\\_j} &= C\\_{t\\_j}-C\\_{t\\_j}P\\_{t\\_{j+1}}^TV\\_{t\\_{j+1}}^{-1}P\\_{t\\_{j+1}}C\\_{t\\_{j}}.\\end{aligned}$$ \n\t\n\t\n\t\n\tDraw *x**t**j* from *X**t**j*∣*X**t**j* + 1, *y*0 : *j* ∼ *N*(*ã**t**j*, *C̃**t**j*).\nAcknowledgements\n----------------\nThe authors would like to thank the associate editor and two anonymous referees for their suggestions which improved the paper.\nLNA FFBS algorithm\n==================\nTo ease the notation, consider a single experimental unit and drop *i* from the notation. Since the parameters *θ*, *ψ*, *b* and Σ remain fixed throughout this section, we also drop them from the notation where possible. Define *y*0 : *j* = (*y**t*0, …, *y**t**j*)*T*. Now suppose that *X*0 ∼ *N*(*a*, *C*) *a priori*. The marginal likelihood under the LNA, *π*(*y*∣*θ*, Σ, *b*), can be obtained from the forward filter described below. After execution of the forward filter, realisations of *π*(*x*∣*y*, *θ*, Σ, *b*) can be generated using a backward sampler. Note that the backward sweep requires\n \n$$\\begin{aligned}\nCov(X\\_{t\\_{j+1}},X\\_{t\\_j})&=Cov(\\tilde{R}\\_{t\\_{j+1}},\\tilde{R}\\_{t\\_j})\n=Cov(P\\_{t\\_{j+1}}\\tilde{R}\\_{t\\_j},\\tilde{R}\\_{t\\_j})\n= P\\_{t\\_{j+1}}Var(\\tilde{R}\\_{t\\_j}).\\end{aligned}$$ \nHere *P**t* is a *d* × *d* matrix that can be shown to satisfy the ODE\n \n$$\\dfrac{dP\\_t}{dt} = H\\_tP\\_t, \\label{eqn:lna\\_p}$$ \nwith initial condition *P*0 = *I**d*, the *d* × *d* identity matrix.\n1. Forward filter. Initialisation. Compute *π*(*y**t*0) = *N*(*y**t*0 ; *F**T**a* , *F**T**C**F* + Σ). The posterior at time *t*0 = 0 is therefore *X**t*0∣*y**t*0 ∼ *N*(*a*0, *C*0), where\n \n$$\\begin{aligned}\na\\_0 &= a+CF(F^TCF+\\Sigma)^{-1}(y\\_{t\\_{0}}-F^Ta) \\\\\nC\\_0 &= C-CF(F^TCF+\\Sigma)^{-1}F^TC\\,.\\end{aligned}$$ \nStore the values of *a*0 and *C*0.\n2. For *j* = 0, 1, …, *n* − 1,\n\t* Prior at *t**j* + 1. Initialise the LNA with *η**t**j* = *a**t**j*, *V**t**j* = *C**t**j* and *P**t**j* = *I**d*. Integrate the ODEs (11), (22) and forward to *t**j* + 1 to obtain *η**t**j* + 1, *V**t**j* + 1 and *P**t**j* + 1. Hence *X**t**j* + 1∣*y*0 : *j* + 1 ∼ *N*(*η**t**j* + 1, *V**t**j* + 1).\n\t* One step forecast. Using the observation equation, we have that \n\t*Y**t**j* + 1∣*y*0 : *j* ∼ *N*(*F**T**η**t**j* + 1, *F**T**V**t**j* + 1*F* + Σ). \n\t Compute the updated marginal likelihood\n\t\n\t\n\t \n\t$$\\begin{aligned}\n\t\\pi(y\\_{0:j+1})&=\\pi(y\\_{0:j})\\pi(y\\_{t\\_{j+1}}|y\\_{0:j})\n\t=\\pi(y\\_{0:j})N(y\\_{t\\_{j+1}}\\,;\\, F^T\\eta\\_{t\\_{j+1}}\\,,\\,F^TV\\_{t\\_{j+1}}F+\\Sigma).\\end{aligned}$$\n\t* Posterior at *t**j* + 1. Combining the distributions in (a) and (b) gives the joint distribution of *X**t**j* + 1 and *Y**t**j* + 1 (conditional on *y*0 : *j*) as \n\t$$\\begin{pmatrix}\n\t X\\_{t\\_{j+1}} \\\\ \n\t Y\\_{t\\_{j+1}}\n\t \\end{pmatrix}\\sim N\\left\\{\\begin{pmatrix}\n\t \\eta\\_{t\\_{j+1}} \\\\\n\t F^T\\eta\\_{t\\_{j+1}} \n\t \\end{pmatrix}\\,,\\, \\begin{pmatrix}\n\t V\\_{t\\_{j+1}} & V\\_{t\\_{j+1}}F \\\\\n\t F^TV\\_{t\\_{j+1}} & F^TV\\_{t\\_{j+1}}F+\\Sigma \n\t \\end{pmatrix} \\right \\}$$ \n\t and therefore *X**t**j* + 1∣*y*0 : *j* + 1 ∼ *N*(*a**t**j* + 1, *C**t**j* + 1), where\n\t\n\t\n\t \n\t$$\\begin{aligned}\n\ta\\_{t\\_{j+1}} &= \\eta\\_{t\\_{j+1}}+V\\_{t\\_{j+1}}F(F^TV\\_{t\\_{j+1}}F+\\Sigma)^{-1}(y\\_{t\\_{j+1}}-F^T\\eta\\_{t\\_{j+1}}) \\\\\n\tC\\_{t\\_{j+1}} &= V\\_{t\\_{j+1}}-V\\_{t\\_{j+1}}F(F^TV\\_{t\\_{j+1}}F+\\Sigma)^{-1}F^TV\\_{t\\_{j+1}}\\,.\\end{aligned}$$ \n\t\n\t\n\t\n\tStore the values of *a**t**j* + 1, *C**t**j* + 1, *η**t**j* + 1, *V**t**j* + 1 and *P**t**j* + 1.\nWe sample *π*(*x*∣*y*) using a backward sampler. The algorithm is as follows.\n1. Backward sampler. First draw *x**t**n* from *X**t**n*∣*y* ∼ *N*(*a**t**n*, *C**t**n*).\n2. For *j* = *n* − 1, *n* − 2, …, 0,\n\t* Joint distribution of *X**t**j* and *X**t**j* + 1. Note that *X**t**j*∣*y*0 : *j* ∼ *N*(*a**t**j*, *C**t**j*). The joint distribution of *X**t**j* and *X**t**j* + 1 (conditional on *y*0 : *j*) is \n\t$$\\begin{pmatrix}\n\t X\\_{t\\_{j}} \\\\ \n\t X\\_{t\\_{j+1}}\n\t \\end{pmatrix}\\sim N\\left\\{\\begin{pmatrix}\n\t a\\_{t\\_{j}} \\\\\n\t \\eta\\_{t\\_{j+1}} \n\t \\end{pmatrix}\\,,\\, \\begin{pmatrix}\n\t C\\_{t\\_{j}} & C\\_{t\\_{j}}P\\_{t\\_{j+1}}^T \\\\\n\t P\\_{t\\_{j+1}}C\\_{t\\_{j}} & V\\_{t\\_{j+1}} \n\t \\end{pmatrix} \\right \\}.$$\n\t* Backward distribution. The distribution of *X**t**j*∣*X**t**j* + 1, *y*0 : *j* is *N*(*ã**t**j*, *C̃**t**j*), where\n\t\n\t\n\t \n\t$$\\begin{aligned}\n\t\\tilde{a}\\_{t\\_j} &= a\\_{t\\_j}+C\\_{t\\_j}P\\_{t\\_{j+1}}^TV\\_{t\\_{j+1}}^{-1}(x\\_{t\\_{j+1}}-\\eta\\_{t\\_{j+1}}), \\\\\n\t\\tilde{C}\\_{t\\_j} &= C\\_{t\\_j}-C\\_{t\\_j}P\\_{t\\_{j+1}}^TV\\_{t\\_{j+1}}^{-1}P\\_{t\\_{j+1}}C\\_{t\\_{j}}.\\end{aligned}$$ \n\t\n\t\n\t\n\tDraw *x**t**j* from *X**t**j*∣*X**t**j* + 1, *y*0 : *j* ∼ *N*(*ã**t**j*, *C̃**t**j*).\nAdditional graphics from the orange tree growth example\n=======================================================\nFigure [orange figphii] shows the marginal posterior densities of five randomly chosen random effects, based on synthetic data generated from the SDMEM of orange tree growth given in Section 5.1 of the main article.\n[b]0.28 \n[b]0.28 \n[b]0.28 \n \n[b]0.28 \n[b]0.28 \n[b]0.28 \n \n[b]0.28 \n[b]0.28 \n[b]0.28 \n \n[b]0.28 \n[b]0.28 \n[b]0.28 \n[orange figphii]\nSimulation study\n================\nIn this section, we investigate further differences between the Bayesian imputation approach and an inference scheme based on the LNA using synthetic data generated from (24) in the main paper. For simplicity, we consider a fixed treatment (low water, blanket nitrogen) and three blocks. We therefore write *X**t*11*k* = (*N**t*11*k*, *C**t*11*k*)*T* and consider the \n$$dX\\_t^{11k}=\\alpha(X\\_t^{11k},b^{11k})\\,dt+\\sqrt{\\beta(X\\_t^{11k},b^{11k})}\\,dW\\_t^{11k}, \\quad k\\in\\{1,2,3\\},$$ \n where\n \n$$\\begin{aligned}\n\\alpha(X\\_t^{11k},b^{11k})&=\\begin{pmatrix}\n\\lambda^{11k} N\\_t^{11k}-\\mu^{11k} N\\_t^{11k}C\\_t^{11k} \\\\\n\\lambda^{11k} N\\_t^{11k} \\\\ \\end{pmatrix},\\\\\n\\beta(X\\_t^{11k},b^{11k})&=\\begin{pmatrix}\n\\lambda^{11k} N\\_t^{11k}+\\mu^{11k} N\\_t^{11k}C\\_t^{11k} & \\lambda^{11k} N\\_t^{11k} \\\\\n\\lambda^{11k} N\\_t^{11k} & \\lambda^{11k} N\\_t^{11k} \\\\ \\end{pmatrix}.\\end{aligned}$$ \nThe fixed effects *b*11*k* = (*λ*11*k*, *μ*11*k*)*T* have a standard structure to incorporate block effects, with \n*λ*11*k* = *λ* + *λ**B**k* and *μ*11*k* = *μ* + *μ**B**k*, \n where we again impose the corner constraints *λ**B*1 = *μ**B*1 = 0 to allow for identifiability.\nTo mimic the real dataset, we took *λ* = 1.75, *μ* = 0.00095, *λ**B*2 = − 0.1154, *λ**B*3 = − 0.0225, *μ**B*2 = − 0.0004 and *μ**B*3 = 0.0002. For each block, we generated five observations (on a regular grid) by using the Euler-Maruyama approximation with a small time-step (Δ*t* = 0.001) and an initial condition of *x*0 = (5, 5)*T*. To assess the impact of measurement error on the quality of inferences that can be made about each parameter, we corrupted our data via the observation model \n$$Y\\_t^{11k}|N\\_t^{11k},\\sigma {\\overset{indep}{\\sim }}N(N\\_t^{11k},\\sigma^{2}N\\_t^{11k}),\\quad t=0, 1, 2, 3, 4,$$ \n and took *σ* ∈ {0, 0.5, 1, 5} to give four synthetic datasets. We adopt the same prior specification for the unknown parameters as used in the real data application.\nBoth the modified innovation scheme (again incorporating the improved bridge construct) and the LNA-based inference scheme were run long enough to yield a sample of approximately 10K independent posterior draws. For the former, we fixed the discretisation level by taking *m* = 20 and note that *m* > 20 gave little difference in posterior output. Figure [aphid figsimbaselines] shows the marginal posterior densities of the baseline parameters (*λ* and *μ*) and the measurement error variance (*σ*). The joint posterior densities of (*μ*, *λ*)*T* are shown in Figure [aphid figsimjoint]. It is clear that when fitting the SDMEM using the Bayesian imputation approach, the posterior samples obtained are consistent with the ground truth. This is true to a lesser extent when using the LNA, with the ground truth found in the tail of the posterior distribution in three out of the four scenarios. In fact, when using synthetic data with *σ* < 5, we see substantive differences in posterior output. As was observed when using real data, the LNA underestimates parameter values compared to those obtained under the Bayesian imputation scheme. In this case, the LNA provides a relatively poor approximation to the true posterior distribution.\nIncreasing *σ* to 5 (and beyond) gives output from both schemes which is largely in agreement. This is intuitively reasonable, since, as the variance of the measurement process is increased, the ability of both inference schemes to accurately infer the underlying dynamics is diminished. Essentially, the relative difference between the LNA and SDE is reduced.\n[b]0.28 \n[b]0.28 \n[b]0.28\n \n[b]0.28 \n[b]0.28 \n[b]0.28 \n \n[b]0.28 \n[b]0.28 \n[b]0.28 \n \n[b]0.28 \n[b]0.28 \n[b]0.28 \n[aphid figsimbaselines]\n[b]0.49\n[b]0.49\n \n[b]0.49\n[b]0.49\n[aphid figsimjoint]\n\nSupplementary material for Bayesian inference for diffusion-driven mixed-effects models\n=======================================================================================\n### 1School of Mathematics & Statistics, Newcastle University,Newcastle-upon-Tyne, NE1 7RU, UK2Department of Mathematics and Statistics, Lancaster University, Lancaster, LA1 4YF\nStochastic differential equations (SDEs) provide a natural framework for modelling intrinsic stochasticity inherent in many continuous-time physical processes. When such processes are observed in multiple individuals or experimental units, SDE driven mixed-effects models allow the quantification of both between and within individual variation. Performing Bayesian inference for such models using discrete-time data that may be incomplete and subject to measurement error is a challenging problem and is the focus of this paper. We extend a recently proposed MCMC scheme to include the SDE driven mixed-effects framework. Fundamental to our approach is the development of a novel construct that allows for efficient sampling of conditioned SDEs that may exhibit nonlinear dynamics between observation times. We apply the resulting scheme to synthetic data generated from a simple SDE model of orange tree growth, and real data on aphid numbers recorded under a variety of different treatment regimes. In addition, we provide a systematic comparison of our approach with an inference scheme based on a tractable approximation of the SDE, that is, the linear noise approximation.\n**Keywords:** Stochastic differential equation; mixed-effects; Markov chain Monte Carlo; modified innovation scheme; linear noise approximation.\nIntroduction\n============\nDiffusion processes satisfying Itô stochastic differential equations (SDEs) are a class of continuous-time, continuous-valued Markov stochastic processes that can be used to model a wide range of phenomena. Examples include (but are not limited to) epidemics, financial time series, population dynamics (including predator-prey systems) and intra-cellular processes. When repeated measurements on a system of interest are made, differences between individuals or experimental units can be incorporated through random effects. Quantification of both system (intrinsic) variation and variation between units leads to a stochastic differential mixed-effects model (SDMEM).\nUnfortunately, analytic intractability of SDEs governing most nonlinear multivariate diffusions can make likelihood-based inference methods problematic. Methods to overcome this difficulty include closed-form expansion of the transition density, exact simulation approaches and use of the Euler-Maruyama approximation coupled with data augmentation. Difficulty in performing inference for SDEs has resulted in relatively little work on SDMEMs.\n propose a procedure for obtaining approximate maximum likelihood estimates for SDMEM parameters based on a two step approach; they use a closed-form Hermite expansion to approximate the transition density, before using Gaussian quadrature to numerically integrate the conditional likelihood with respect to the random parameters. As noted by, the approach is, in practice, limited to a scalar random effect parameter since Gaussian quadrature is computationally inefficient when the dimension of the random effect parameter grows. The methodology is extended in to deal with multiple random effects. A number of limitations remain however. In particular a reducible diffusion process is required, that is, one which can be transformed to give a unit diffusion coefficient. Another drawback is that the method cannot account for measurement error. A promising approach appears to be the use of the extended Kalman filter (EKF) to provide a tractable approximation to the SDMEM. This has been the focus of, and. The `R` package `PSM` uses the EKF to estimate SDMEMs. Unfortunately, a quantification of the effect of using these approximate inferential models appears to be missing from the literature. discuss inference for SDMEMs in a Bayesian framework, and implement a Gibbs sampler when the SDE (for each experimental unit) has an explicit solution. When no explicit solution exists they suggest that a solution might be found using the Euler-Maruyama discretisation.\nContributions and organisation of the paper\n-------------------------------------------\nIn this article we provide a method that permits (simulation-based) Bayesian inference for a large class of multivariate SDMEMs using discrete-time observations that may be incomplete (so that only a subset of model components are observed) and subject to measurement error. The method makes use of a novel scheme that allows for observations made sparsely in time, as the process of interest may exhibit nonlinear dynamics between measurement times.\nAs a starting point, we consider a data augmentation approach that adopts an Euler-Maruyama approximation of unavailable transition densities and augments low frequency data with additional time points over which the approximation is satisfactory. Although a discretisation bias is introduced, this can be made arbitrarily small (at greater computational expense). Moreover, the approach is flexible, and is not restricted to reducible diffusions. A Bayesian approach then aims to construct the joint posterior density for parameters and the components of the latent process. The intractability of the posterior density necessitates simulation techniques such as Markov chain Monte Carlo. As is well documented in, care must be taken in the design of the MCMC sampler due to dependence between the parameters entering the diffusion coefficient and the latent process. We therefore adapt the reparameterisation technique (known as the modified innovation scheme) of and (see also ) to the SDMEM framework. A key requirement of the scheme is the ability to sample the latent process between two fixed values. Previous approaches have typically focused on the modified diffusion bridge construct of. For the SDMEM considered in Section [aphid] we find that this construct fails to capture the nonlinear dynamics exhibited between observation times. We therefore develop a novel bridge construct that is simple to implement and can capture nonlinear behaviour.\nFinally, we provide a systematic comparison of our approach with an inference scheme based on a linear noise approximation (LNA) of the SDE. The LNA approximates transition densities as Gaussian, and when combined with Gaussian measurement error, allows the latent process to be integrated out analytically. Essentially a forward (Kalman) filter can be implemented to calculate the marginal likelihood of all parameters of interest, allowing a marginal Metropolis-Hastings scheme targeting their posterior distribution. It should be noted, however, that evaluation of the Gaussian transition densities under the LNA require the solution of an ordinary differential equation (ODE) system whose order grows quadratically with the number of components (say *d*) governed by the SDE. The computational efficiency of an LNA based inference scheme will therefore depend on *d*, and on whether or not the ODE system can be solved analytically.\nWe apply the methods to two examples. First, we consider a synthetic dataset generated from an SDMEM driven by the simple univariate model of orange tree growth presented in and. The ODE system governing the LNA solution is tractable in this example. Second, we fit a model of aphid growth to both real and synthetic data. The real data are taken from and consist of Cotton aphid (*Aphis gossypii*) counts in the Texas High Plains obtained for three different levels of irrigation water, nitrogen fertiliser and block. This application is particularly challenging, due to the nonlinear drift and diffusion coefficients governing the SDMEM and the ability to only observe one of the model components (with error). Moreover, the ODE system governing the LNA solution is intractable and a numerical solver must be used.\nThe remainder of the article is organised as follows. The SDMEM framework is introduced in Section [sdmem]. Section [bayes] provides MCMC methods for Bayesian inference, with a novel bridge construct outlined in Section [dgode]. The linear noise approximation and its application as an inferential model is discussed in Section [lna]. The methods are applied in Section [apps] before conclusions are drawn in Section [disc].\nStochastic Differential Mixed-effects models\n============================================\nConsider the case where we have *N* experimental units randomly chosen from a theoretical population, and associated with each unit *i* is a continuous-time *d*-dimensional Itô process {*X**t**i*, *t* ≥ 0} governed by the SDE\n \n$$dX\\_t^i=\\alpha(X\\_t^i,\\theta,b^i)\\,dt+\\sqrt{\\beta(X\\_t^i,\\theta,b^i)}\\,dW\\_t^i,\n\\quad X\\_0^i=x\\_0^i, \\quad i=1,\\ldots,N. \\label{eqn:sdmem}$$ \nHere, *α* is a *d*-vector of drift functions, the diffusion coefficient *β* is a *d* × *d* positive definite matrix with a square root representation $\\sqrt{\\beta}$ such that $\\sqrt{\\beta}\\sqrt{\\beta}^T=\\beta$ and *W**t**i* is a *d*-vector of (uncorrelated) standard Brownian motion processes. The *p*-vector parameter *θ* = (*θ*1, …, *θ**p*)*T* is common to all units whereas the *q*-vectors *b**i* = (*b*1*i*, …, *b**q**i*)*T*, *i* = 1, …, *N*, are unit-specific effects, which may be fixed or random. In the most general random effects scenario we let *π*(*b**i*∣*ψ*) denote the joint distribution of *b**i*, parameterised by the *r*-vector *ψ* = (*ψ*1, …, *ψ**r*)*T*. The model defined by ([eqn:sdmem]) allows for differences between experimental units through different realisations of the Brownian motion paths *W**t**i* and the random effects *b**i*, accounting for inherent stochasticity within a unit, and variation between experimental units respectively.\nWe assume that each experimental unit {*X**t**i*, *t* ≥ 0} cannot be observed exactly, but observations *y**i* = (*y**t*0*i*, *y**t*1*i*, …, *y**t**n**i*)*T* are available and these are conditionally independent (given the latent process). We link the observations to the latent process via\n \n$$\\label{eqn:obs}\nY\\_t^i = F^T X\\_t^i + \\epsilon\\_t, \\qquad \\epsilon\\_t|\\Sigma{\\overset{indep}{\\sim }}N(0,\\Sigma),$$ \nwhere *Y**t**i* is a *d**o*-vector, *F* is a constant *d* × *d**o* matrix and *ε**t* is a random *d**o*-vector. Note that this setup allows for only observing a subset of components (*d**o* < *d*) and this aspect is explored further in Section [aphid].\nTogether and completely specify the stochastic differential mixed-effects model. However, for most problems of interest the form of the SDE associated with each unit will not permit an analytic solution, precluding straightforward inference for the unknown parameters. We therefore work with the Euler-Maruyama approximation \n$$\\Delta X\\_t^i\\equiv X^i\\_{t+\\Delta t}-X^i\\_t=\\alpha(X\\_t^i,\\theta,b^i)\\,\\Delta t+\\sqrt{\\beta(X\\_t^i,\\theta,b^i)}\\,\\Delta W\\_t^i$$ \n where Δ*W**t**i* ∼ *N*(0, *I**d*Δ*t*) and Δ*t* is the length of time between observations, assumed equally spaced for notational simplicity. It is, of course, unlikely that this approximation will be sufficiently accurate over the intervals between observation times and so we adopt a data augmentation scheme. Partitioning [*t**j*, *t**j* + 1] as \n*t**j* = *τ**j*, 0 < *τ**j*, 1 < *τ**j*, 2 < … < *τ**j*, *m* − 1 < *τ**j*, *m* = *t**j* + 1 \n introduces *m* − 1 intermediate time points with interval widths of length\n \n$$\\label{eqn:part}\n\\Delta\\tau\\equiv\\tau\\_{j,k+1}-\\tau\\_{j,k}=\\frac{t\\_{j+1}-t\\_j}{m}.$$ \nThe Euler-Maruyama approximation can then be applied over each interval of width Δ*τ*, and the associated discretisation bias can be made arbitrarily small at the expense of having to impute {*X**t**i*} at the intermediate times. We adopt the shorthand notation \n*x*[*t**j*, *t**j* + 1]*i* ≡ *x*[*j*, *j* + 1]*i* = (*x**τ**j*, 0*i*, *x**τ**j*, 1*i*, …, *x**τ**j*, *m**i*)*T* \n for the latent process over the time interval [*t**j*, *t**j* + 1] for unit *i*. Hence, the complete latent trajectory associated with unit *i* is given by \n(*x**i*)*T* = ((*x*[0, 1]*i*)*T*, (*x*(1, 2]*i*)*T*…, (*x*(*n* − 1, *n*]*i*)*T*) \n and we stack all unit-specific trajectories into a matrix *x* = (*x*1, …, *x**N*). Likewise the matrix *y* = (*y*1, …, *y**N*) denotes the entire set of observations. Next we focus on how to perform Bayesian inference for the model quantities *x*, *θ*, *b* = (*b*1, …, *b**N*)*T*, *ψ* and Σ.\nBayesian inference\n==================\nThe joint posterior for the common parameters *θ*, fixed/random effects *b*, hyperparameters *ψ*, measurement error variance Σ and latent values *x* is given by\n \n*π*(*θ*, *ψ*, Σ, *b*, *x*∣*y*) ∝ *π*(*θ*)*π*(*ψ*)*π*(Σ)*π*(*b*∣*ψ*)*π*(*x*∣*θ*, *b*)*π*(*y*∣*x*, Σ) \nwhere *π*(*θ*)*π*(*ψ*)*π*(Σ) is the joint prior density ascribed to *θ*, *ψ* and Σ. In addition we have that\n \n$$\\label{eqn:xdens}\n\\pi(x|\\theta,b)=\\prod\\limits\\_{i=1}^{N}\\prod\\limits\\_{j=0}^{n-1}\\prod\\limits\\_{k=1}^{m}\\pi(x^i\\_{\\tau\\_{j,k}}|x^i\\_{\\tau\\_{j,k-1}},\\theta,b^i)$$ \nwhere \n*π*(*x**τ**j*, *k**i*∣*x**τ**j*, *k* − 1*i*, *θ*, *b**i*) = *N*(*x**τ**j*, *k**i* ; *x**τ**j*, *k* − 1*i* + *α*(*x**τ**j*, *k* − 1*i* *θ*, *b**i*)Δ*τ*, *β*(*x**τ**j*, *k* − 1*i*, *θ*, *b**i*)Δ*τ*) \n and *N*( ⋅ ; *m*, *V*) denotes the multivariate Gaussian density with mean *m* and variance *V*. Similarly \n$$\\pi(y|x,\\Sigma)= \\prod\\limits\\_{i=1}^{N}\\prod\\limits\\_{j=0}^{n}\\pi(y\\_{t\\_{j}}^i|x\\_{t\\_{j}}^i,\\Sigma)$$ \n where *π*(*y**t**j**i*∣*x**t**j**i*, Σ) = *N*(*y**t**j**i* ; *x**t**j**i*, Σ). Given the intractability of the joint posterior distribution in ([eqn:posterior]) we aim to construct a Markov chain Monte Carlo (MCMC) scheme which generates realisations from this posterior. The form of the SDMEM admits a Gibbs sampling strategy with blocking that sequentially takes draws from the full conditionals \n[b]0.5\n1. *π*(*x*∣*θ*, *ψ*, Σ, *b*, *y*) = *π*(*x*∣*θ*, Σ, *b*, *y*),\n2. *π*(*θ*∣*ψ*, Σ, *b*, *x*, *y*) = *π*(*θ*∣*b*, *x*),\n3. *π*(*ψ*∣*θ*, Σ, *b*, *x*, *y*) = *π*(*ψ*∣*b*).\n4. \n[b]0.5\n1. *π*(Σ∣*θ*, *ψ*, *b*, *x*, *y*) = *π*(Σ∣*x*, *y*),\n2. *π*(*b*∣*θ*, *ψ*, Σ, *x*, *y*) = *π*(*b*∣*θ*, *ψ*, *x*),\n3. \n4. \nFurther blocking strategies that exploit the conditional"},"data_id":{"kind":"string","value":"arxiv_0004126"}}}],"truncated":false,"partial":true},"paginationData":{"pageIndex":9,"numItemsPerPage":100,"numTotalItems":46503,"offset":900,"length":100}},"jwt":"eyJhbGciOiJFZERTQSIsImtpZCI6Im1sR1pFYktIVHRVSGRhX1RRdEczQ0N1S3k5ME9Qd25DNHY5elRjM3FVYzgifQ.eyJyZWFkIjp0cnVlLCJwZXJtaXNzaW9ucyI6eyJyZXBvLmNvbnRlbnQucmVhZCI6dHJ1ZX0sImlhdCI6MTc4Mzg1MjY4OCwianRpIjoiNzVlOTE2MTgtZjk1MS00OTNlLTlmODctMzkwZGI3NzY4ZWI2Iiwic3ViIjoiL2RhdGFzZXRzL0x5dW4wOTEyL0xvbmdBQkMiLCJleHAiOjE3ODM4NTYyODgsImlzcyI6Imh0dHBzOi8vaHVnZ2luZ2ZhY2UuY28ifQ.fseTKSit62PO8Rh8KxtTTKqN9G5oyVwjN0V3Sm-xjmWoM6h47h16gBW8hiICA5_rXiqkBKxwdtaeQfbjDfT8CA","displayUrls":true,"splitSizeSummaries":[{"config":"default","split":"train","numRows":46503,"numBytesParquet":2327876865}]},"discussionsStats":{"closed":0,"open":2,"total":2},"fullWidth":true,"hasGatedAccess":true,"hasFullAccess":true,"isEmbedded":false,"savedQueries":{"community":[],"user":[]}}">
content stringlengths 34.5k 188k | data_id stringlengths 13 13 |
|---|---|
Angular, spectral, and time distributions of highest energy protons andassociated secondary gamma-rays and neutrinos propagating throughextragalactic magnetic and radiation fields
============================================================================================================================================... | arxiv_0003692 |
Approximating the Revenue Maximization Problem with Sharp Demands
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We consider the revenue maximization problem with sharp multi-demand, in which *m* indivisible items have to be sold to *n* potential buyers. Each buyer *i* is interested in getting exactl... | arxiv_0003698 |
A Unified Framework For Quantum Unforgeability
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In this paper, we continue the line of work initiated by Boneh and Zhandry at CRYPTO 2013 and EUROCRYPT 2013 in which they formally define the notion of unforgeability against quantum adversaries. We develop a general and para... | arxiv_0003702 |
On uniqueness of *p*-adic period morphisms, II
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[1](#fn1)
We prove equality of the various rational *p*-adic period morphisms for smooth, not necessarily proper, schemes. We start with showing that the *K*-theoretical uniqueness criterium we had found earlier for proper smo... | arxiv_0003703 |
**Interpolating Population Distributions using Public-use Data: An Application to Income Segregation using American Community Survey Data**
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#1
Income segregation measures the exten... | arxiv_0003705 |
A Grid of 3D Stellar Atmosphere Models of Solar Metallicity:I. General Properties, Granulation and Atmospheric Expansion
========================================================================================================================
Present grids of stellar atmosphere models are the workhorses in interpreting ... | arxiv_0003706 |
Second-order perturbations of cosmological fluids:Relativistic effects of pressure, multi-component, curvature, and rotation
============================================================================================================================
We present general relativistic correction terms appearing in Newton’s... | arxiv_0003707 |
) & $ \leq 0.040(15)\footnotemark[3] $
66As & 0.020(20) & 0.110 & 0.087 & 0.050(30) &
70Br & 0.070(20) & 0.226 & 0.235 & 0.150(80) &
74Rb & 0.050(30) & 0.045& 0.047 & 0.050(30)& $ \leq 0.075\footnotemark[5] $
For the heavier nuclei there are experimental data on Fermi transitions to the non-analog excited 0+ st... | arxiv_0003710 |
Embedded complex curves in the affine plane
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This paper brings several contributions to the classical Forster–Bell–Narasimhan conjecture and the Yang problem concerning the existence of proper, almost proper, and complete injective holomorphic immersions of open Riemann surfac... | arxiv_0003711 |
Fiber-Base Duality and Global Symmetry Enhancement
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Introduction
============
Gauge theories in five dimensional (5D) spacetime are perturbatively non-renormalizable, infrared (IR) free and may have a Landau pole singularity in the ultraviolet (UV) region. Thus, usually... | arxiv_0003712 |
*v* do not depend on the initial conditions and tend to the behaviour of Eqs. ([1Dresult-1]) and ([1Dresult-2]) at later-times.
Spherically symmetric 3D case
-----------------------------
I have shown that the solution for a 1D cloud has distinct characteristic features. In homogeneous steady initial conditions, the d... | arxiv_0003720 |
1); (1,1)–(0,0); at (2.5,1) = ; (3,2) [rounded corners=0.2cm] – (4,1) – (3,0); (5,2) [rounded corners=0.2cm] – (4,1) – (5,0); at (5.5,1) + ; at (6,1) *v*; (6.25,2) – (8.25,0); (8.25,2) – (6.25,0);
[yshift=-6cm] (2,2)–(1,1); (1,1)–(2,0); (0,0)–(1,1); (1,1)–(0,2); at (2.5,1) = ; (3,0) [rounded corners=0.2cm] – (4,1) –... | arxiv_0003726 |
Spectroscopy of states in $^{136}\rm{Ba}$ using the $^{138}\rm{Ba}(p,t)$ reaction
=================================================================================
Background
The 136Ba isotope is the daughter nucleus in 136Xe *β**β* decay. It also lies in a shape transitional region of the nuclear chart, making it a su... | arxiv_0003727 |
Joint IRS Location and Size Optimization in Multi-IRS Aided Two-Way Full-Duplex Communication Systems
=====================================================================================================
Intelligent reflecting surfaces (IRSs) have emerged as a promising wireless technology for the dynamic configuration... | arxiv_0003730 |
about two dozens of unsaturated isolated astrometric standards from the USNO-B1 catalogue. Their catalogue and image positional uncertainties for both coordinates were $\la$ 02 and $\la$ 05, respectively. IRAF *ccmap/cctran* tasks were used to find plate solutions. Formal rms uncertainties of the astrometric fit were ... | arxiv_0003734 |
Quasar clustering at redshift 6
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### Received 12 March 2021 / Accepted 20 July 2021
We aim at quantifying quasar pairs at high redshift for a large sample of objects. This provides a new key constraint on a combination of parameters related to the origin and assembly for the most massive... | arxiv_0003745 |
A Fundamental Class for Intersection Spaces of Depth One Witt Spaces
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By a theorem of Banagl-Chriestenson, intersection spaces of depth one pseudomanifolds exhibit generalized Poincaré duality of Betti numbers, provided that certain characteristic clas... | arxiv_0003746 |
A comparison of shock-cloud and wind-cloud interactions: The longer survival of clouds in winds
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### Accepted... Received...; in original form...
[section] [firstpage]
The interaction of a hot, high-velocity wind with a cold,... | arxiv_0003748 |
Wisdom of the Contexts: Active Ensemble Learning for Contextual Anomaly Detection
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### Received: date / Accepted: date
example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave.4 setgr... | arxiv_0003751 |
Minimum Vector Rank and Complement Critical Graphs [4](#fn4)
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Given a graph *G*, a real orthogonal representation of *G* is a function from its set of vertices to R*d* such that two vertices are mapped to orthogonal vectors if and only if they are not neighbor... | arxiv_0003753 |
Witnessing quantum memory in non-Markovian processes
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[apsrev4-1modified]
**We present a method to detect quantum memory in a non-Markovian process. We call a process Markovian when the environment does not provide a memory that retains correlations across different s... | arxiv_0003755 |
*s*2*s*2*s*2, *s*1*s*1*s*2*s*1, *s*1*s*2*s*1*s*1, *s*1*s*2*s*1*s*2*s*1},
all of which are resolvable.
The Bergman diamond lemma states that, if the set of monomials possesses a sufficiently nice partial order compatible with our relations, then the irreducible monomials form a basis if and only if every minimal amb... | arxiv_0003756 |
Robustness to Unbounded Smoothness of Generalized SignSGD
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Traditional analyses in non-convex optimization typically rely on the smoothness assumption, namely requiring the gradients to be Lipschitz. However, recent evidence shows that this smoothness condition d... | arxiv_0003758 |
Production rate and ellipticity of lepton pairs from a rotating hot and dense QCD medium
========================================================================================
[1](#fn1)
[2](#fn2)
[3](#fn3)
Using a current-current correlation function (CF), the photon polarization tensor is calculated for a rotating h... | arxiv_0003762 |
Sparse Matrix Multiplication and Triangle Listingin the Congested Clique Model[4](#fn4)
=======================================================================================
We show how to multiply two *n* × *n* matrices *S* and *T* over semirings in the Congested Clique model, where *n* nodes communicate in a fully ... | arxiv_0003766 |
Introduction
============
Regression model
----------------
We consider the following regression model
$$\begin{aligned}
\label{modele\_regression}
Y & = & X\theta + \epsilon\,\end{aligned}$$
where *θ* is an unknown vector of R*p*. The row vector *X* : = (*X**i*)1 ≤ *i* ≤ *p* follows a real zero mean Gaussian dis... | arxiv_0003770 |
Non-pluripolar products on vector bundles and Chern–Weil formulae
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In this paper, we develop several pluripotential-theoretic techniques for singular metrics on vector bundles. We first introduce the theory of non-pluripolar products on holomorphic vector... | arxiv_0003773 |
Hybrid Conditional Planningusing Answer Set Programming
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[firstpage]
We introduce a parallel offline algorithm for computing hybrid conditional plans, called HCP-ASP, oriented towards robotics applications. HCP-ASP relies on modeling actuation actions and sensing a... | arxiv_0003782 |
On the Complexity of the Correctness Problem for Non-Zeroness Test Instruction Sequences
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This paper concerns the question to what extent it can be efficiently determined whether an arbitrary program correctly solves a given problem... | arxiv_0003784 |
DNA-Based Data Storage Systems: A Review of Implementations and Code Constructions
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Coded trace reconstruction, coding for unique reconstruction, molecular data storage, synthetic biology.
Introduction and Motivation
=====================... | arxiv_0003792 |
**Generically split octonion algebrasand ${{\mathbb A}^1}$-homotopy theory**
============================================================================
We study generically split octonion algebras over schemes using techniques of ${\mathbb A}^1$-homotopy theory. By combining affine representability results with techn... | arxiv_0003800 |
cal{I}}, Pre, Post\rangle$$
where:
* *A* is a non-empty finite set of deontic actions.
* $\leq^{\A\_i}\_{j \to k}$ is a reflexive and transitive relation on *A*
* *P**r**e* : *A* → L is a precondition function.
* *P**o**s**t* : *A* → (*P**r**o**p* → L) is a post-condition function that assigns to each action and eac... | arxiv_0003805 |
| p30pt | p30pt | p30pt | Factors & Dataset Name & Top-k & SAPNSP & k-means & k-SDPP & EINSP
& **C6**\_T4\_S8\_I8\_DB10k\_N0.1k & 14.86% & 15.92% & 16.90% & 23.53% & 32.58%
& **C8**\_T4\_S8\_I8\_DB10k\_N0.1k & 15.99% & 16.65% & 17.88% & 24.84% & 33.67%
& **C10**\_T4\_S8\_I8\_DB10k\_N0.1k & 18.52% & 19.25% & 20... | arxiv_0003816 |
**Living with ghosts in Hořava–Lifshitz gravity**
=================================================
We consider the branch of the projectable Hořava-Lifshitz model which exhibits ghost instabilities in the low energy limit. It turns out that, due to the Lorentz violating structure of the model and to the presence of a ... | arxiv_0003819 |
The solution space geometry of random linear equations
=======================================================
c ł ¶ §
[lemma]Corollary [lemma]Claim [lemma]Remark [lemma]Proposition [lemma]Definition [lemma]Observation
We consider random systems of linear equations over GF(2) in which every equation binds *k* variable... | arxiv_0003828 |
Human Hands as Probes for Interactive Object Understanding
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Interactive object understanding, or what we can do to objects and how is a long-standing goal of computer vision. In this paper, we tackle this problem through observation of human hands in in-the-wild... | arxiv_0003830 |
respectively). Next, the polarization and amplitude of the wave are fixed by setting either the electric or magnetic field and then deriving the remaining one via Maxwell’s equations. For L- and R-mode waves, the fields are arranged as transverse, circularly polarized, plane waves with frequency *ω* and wave number *k... | arxiv_0003839 |
ologically similar to the MR and RR states. These states not only provide candidate wave functions for several FQH plateaus but can also be further associated with frustration-free positive semi-definite two-body parent Hamiltonians in flat bands with higher Chern numbers. Our specific analysis focused on degenerate LL... | arxiv_0003841 |
Correlation, hierarchies, and networks in financial markets
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We discuss some methods to quantitatively investigate the properties of correlation matrices. Correlation matrices play an important role in portfolio optimization and in several other quantitative de... | arxiv_0003844 |
Ultra-slow dynamics of free-running ring lasers:towards a minimal model
=======================================================================
The dynamics of a resonant, free-running ring laser, in the common case of a fast relaxation of the atomic polarization, is unexpectedly highly singular. As shown in [Phys. Rev... | arxiv_0003848 |
Different Types of Isomorphisms of Drawings of Complete Multipartite Graphs[1](#fn1)
====================================================================================
Simple drawings are drawings of graphs in which any two edges intersect at most once (either at a common endpoint or a proper crossing), and no edge i... | arxiv_0003855 |
User Selection and Power Allocation inFull Duplex Multi-Cell Networks
=====================================================================
Full duplex (FD) communications has the potential to double the capacity of a half duplex (HD) system at the link level. However, in a cellular network, FD operation is not a strai... | arxiv_0003856 |
metries.
In this work, we use the hot-spot renormalization group developed in Ref. as well as other subsequent works to study the infrared phases of hole-doped RTG at doping close to the VHSs under different types of electron interactions. While previous studies often assumed a hierarchy of interaction scales, our RG... | arxiv_0003857 |
**Symmetry Improved CJT Effective Action**
==========================================
MAN/HEP/2013/07
May 2013
The formalism introduced by Cornwall, Jackiw and Tomboulis (CJT) provides a systematic approach to consistently resumming non-perturbative effects in Quantum Thermal Field Theory. One major limitation of the... | arxiv_0003858 |
Rotational component spaces for infinite-type translation surfaces
==================================================================
Finite translation surfaces can be classified by the order of their singularities. When generalizing to infinite translation surfaces, however, the notion of order of a singularity is no... | arxiv_0003859 |
monopole operators leading to the *U*(1)*m**i* → *U*(*m**i*) flavor enhancement (again, this statement holds assuming generic *m**i* such that *m*1 + *m*2 + *m*3 = 2*n* and *m**i* < *n*).
.5cm
[place/.style=circle,draw=blue!50,fill=blue!20,thick,inner sep=0pt,minimum size=10mm,transition2/.style=rectangle,draw=black!5... | arxiv_0003860 |
The importance of Fe fragmentation for LiBeB analyses
=====================================================
Is a Li primary source needed to explain AMS-02 data?
=====================================================
### Received / Accepted
epsf
We re-evaluate the impact of Fe fragmentation on the Li, Be, and B modellin... | arxiv_0003867 |
${\cal
N}\_y$ represents the only source maintaining the shearwise velocity and hence the stress.
The linear nonmodal dynamics for the non-symmetric harmonics is described by the terms of linear origin related to shear in the above dynamical equations in Fourier space. First consider the non-symmetric mode with ${\bf ... | arxiv_0003885 |
1 while also serving the users in other sectors as much as it could. With *β*\* = 1 − *ρ*/*R* in or the blue solid dot in Fig. 3(b), the UAV maximizes its coverage area in *S*1 and avoids wasting any coverage outside the cell.
* If $\rho\in[\sqrt{2}R/2,R)$, the coverage region of the UAV is larger than *S*1 as shown in... | arxiv_0003886 |
Tidally Forced Planetary Waves in the Tachocline of Solar-like Stars
====================================================================
Can atmospheric waves in planet-hosting solar-like stars substantially resonate to tidal forcing? Substantially at a level of impacting the space weather or even of being dynamo-rele... | arxiv_0003888 |
Tight Algorithm for Connected Odd Cycle Transversal Parameterized by Clique-width
=================================================================================
**Abstract**
Recently, Bojikian and Kratsch [2023] have presented a novel approach to tackle connectivity problems parameterized by clique-width ($\cw$), ba... | arxiv_0003896 |
based techniques. For example, vector *e*1 = (1, 0, …, 0) satisfies *e*1 ≥ lex*γ*⋆(*e*1), but violates . We therefore also discuss a second approach: a generalization of lexicographic reduction to reflection symmetries.
The lexicographic reduction algorithm of receives a permutation symmetry *π* of as well as upper ... | arxiv_0003898 |
to certain criteria”. Since focus groups are explicitly mentioned by Mayring as a conceivable source of data and additionally, the data are available in the form of written transcripts, this method seems appropriate. The desired content is extracted by means of a category system which is not only the core instrument ... | arxiv_0003908 |
�∣) = 0,
by virtue of the fact that this is satisfied by the tree-level numerators, and that these are all multiplied by a common factor. Although we have been somewhat schematic in our notation, eq. ([BCJ2loopex1]) is indeed the relevant limit of the BCJ relation with all basis momenta properly accounted for, anala... | arxiv_0003910 |
Quantum Mechanics and the Principle of Least Radix Economy
==========================================================
### Received: date / Accepted: date
example.eps gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave.4 setgray fill grestore stroke grestore
A new variati... | arxiv_0003911 |
The number of *C*2ℓ-free graphs
===============================
[1](#fn1)
One of the most basic questions one can ask about a graph *H* is: how many *H*-free graphs on *n* vertices are there? For non-bipartite *H*, the answer to this question has been well-understood since 1986, when Erds, Frankl and Rödl proved that t... | arxiv_0003913 |
, J. L. 2005,, 618, 259 Meier, D. S., Turner, J. L. & Hurt, R. L. 2000,, 531, 200 Mezger, P. G. & Henderson, A. P. 1967,, 147, 471 Milam, S. N., Savage, C., Brewster, M. A., Ziurys, L. M., & Wyckoff, S. 2005,, 634, 1126 Nagai, M., Tanaka, K., Kamegai, K., & Oka, T. 2007,, 59, 25 Padin, S., Scott, S. L., Woody, D. P., S... | arxiv_0003921 |
*m* ≥ 0*π*\*L⊗ *m* as a graded OP1-algebra. This completes the proof of Theorem [fg].
[Proof of Corollary [sph.YTD]] The result of combined with Theorem [fg] readily imply the existence part of Corollary [sph.YTD]. The uniqueness part is due to for general cscK metrics.
Note that in the toric case, *i.e.*, when *G* i... | arxiv_0003924 |
A new catalogue of polar-ring galaxies selected from the SDSS. [2](#fn2)
========================================================================
### Accepted 9 Jule 2011 Received 21 Apr 2011
[firstpage]
Galaxies with polar rings (PRGs) are a unique class of extragalactic objects allowing to investigate a wide range of... | arxiv_0003929 |
Solar wind driving of magnetospheric ULF waves: Field line resonances driven by dynamic pressure fluctuations
=============================================================================================================
#1Equation ([eq:#1]) #1Figure [fig:#1] #1Table [tbl:#1] #1Chapter [ch:#1] #1Section [sec:#1]
Several... | arxiv_0003933 |
\int\_0^T\int\_{C\_{\varepsilon}}\big| f\_i({\bf R}^\varepsilon) - f\_i({\bf u}^\varepsilon) + \Psi^\varepsilon\_i \big|\,
\big| R^\varepsilon\_i - u^\varepsilon\_i \big| \, dx d\tau \notag
\\+\,
\varepsilon \int\_0^T\int\_{\Upsilon\_1}\big| g\_i({\bf R}^\varepsilon) - g\_i({\bf u}^\varepsilon) + \Phi^\varepsilon\_... | arxiv_0003939 |
of this term has no effect on the estimates in the existence proof of Chang-Yang.
We are now ready to prove Theorem [LDThm]:
[The proof of Theorem [LDThm]] We first remark that if (*M*4, *g*0) is conformally equivalent to the round sphere (suitably normalized), then *g* = *g**c* satisfies the conclusions of the Theore... | arxiv_0003944 |
; (n9) at (5,2) ; (n10) at (5,1) ; (n11) at (6,2) ; (n12) at (5,4) ; (n13) at (5,5) ; (n14) at (6,4) ;
(n15) at (6.2,4) ; (n16) at (8.8,4) ;
(n29) at (6.2,2) ; (n30) at (8.8,2) ;
(n17) at (9,2) ; (n18) at (10,1) ; (n19) at (10,2) ; (n20) at (10,4) ; (n21) at (9,4) ; (n22) at (10,5) ; (n23) at (13,2) ; (n24) at (13,1) ... | arxiv_0003948 |
a measure of convective efficiency, and it is most commonly expressed in a non-dimensional form as $\alpha\_{\rm MLT}=\ell/H\_P$, i.e., as the mixing length ℓ divided by the local pressure scale height *H**P*. Although the MLT contains several other free parameters (see, e.g., the discussions of the MLT formalism by )... | arxiv_0003953 |
Image Cartoon-Texture Decomposition Using Isotropic Patch Recurrence
====================================================================
Aiming at separating the cartoon and texture layers from an image, cartoon-texture decomposition approaches resort to image priors to model cartoon and texture respectively. In recen... | arxiv_0003957 |
Age Optimal Information Gathering and Dissemination on Graphs
=============================================================
=1
We consider the problem of timely exchange of updates between a central station and a set of ground terminals *V*, via a mobile agent that traverses across the ground terminals along a mobility... | arxiv_0003966 |
Asteroseismic and orbital analysis of the triple star system HD 188753 observed by *Kepler*
===========================================================================================
### Version 1.0 – 24 April 2018.
In this work, we aim to derive the fundamental parameters of a close triple star system, HD 188753, for... | arxiv_0003973 |
*, [Sci. Rep. **5**, 9408 ( 2015)](\doibase 10.1038/srep09408) , , , , , , and , *Scanning tunneling spectroscopy and vortex imaging in the iron pnictide superconductor BaFe1.8Co0.2As2*, [Phys. Rev. Lett. **102**, 097002 ( 2009)](\doibase 10.1103/PhysRevLett.102.097002) , , , , , , , , , , ... | arxiv_0003974 |
Confidence bounds for the sensitivity lack of a less specific diagnostic test, without gold standard
====================================================================================================
.tex
We consider the problem of comparing two diagnostic tests based on a sample of paired test results without tru... | arxiv_0003976 |
delta\_m^{\prime\prime}+\mathcal{H}\delta\_m^\prime- 4\pi {G\_{\rm eff}}(\bar{\varphi}) a^2\,\bar{\rho}\_m\delta\_m=0\,.$$
The quantity
$$\label{eq:MainGeffective}
{G\_{\rm eff}}(\bar{\varphi})=\frac{G\_N}{\bar{\varphi}}\left(\frac{4+2\omega\_{BD}}{3+2\omega\_{BD}}\right)=\frac{G\_N}{\bar{\varphi}}\left(\frac{2+4... | arxiv_0003977 |
Upper bounds for Newton’s method onmonotone polynomial systems,and P-time model checking ofprobabilistic one-counter automata[1](#fn1)
======================================================================================================================================
A central computational problem for analyzing and ... | arxiv_0003988 |
Analysis of evanescently-coupled pairs of spin-polarised vertical-cavity surface-emitting lasers
================================================================================================
A general model for the dynamics of arrays of coupled, spin-polarised lasers is derived, which is shown to reduce to both the ... | arxiv_0003990 |
Funnel MPC for nonlinear systems with relative degree one [3](#fn3)
===================================================================
[1]
@twocolumn
#1
==
.2in **#1:**
@twocolumn
We show that Funnel MPC, a novel Model Predictive Control (MPC) scheme, allows tracking of smooth reference signals with prescribed perfor... | arxiv_0003997 |
H2∣2-model and Vertex-Reinforced Jump Process on Regular Trees: Infinite-Order Transition and an Intermediate Phase
===================================================================================================================
**Abstract:** We explore the supercritical phase of the vertex-reinforced jump process (... | arxiv_0004005 |
Corollary 4 of .[↩](#fnref5)
6. As shown in , a tight *a priori* bound for all *s* is usually far from necessary. We remark that the algorithm of Theorem [thm:main1] can be straightforwardly adapted to the framework of ; this would exponentially improve the error dependence of the algorithms of , but is beyond the scop... | arxiv_0004012 |
A Bayesian linear model for the high-dimensional inverse problem of seismic tomography
======================================================================================
,
We apply a linear Bayesian model to seismic tomography, a high-dimensional inverse problem in geophysics. The objective is to estimate the three... | arxiv_0004013 |
Comonotonicity and Choquet integrals of Hermitian operators and their applications
==================================================================================
In a quantum system with *d*-dimensional Hilbert space, the *Q*-function of a Hermitian positive semidefinite operator *θ*, is defined in terms of the *d*... | arxiv_0004023 |
given and *σ* > 0 is a small parameter characterizing the level of the error in the data *ψ*(**x**, *s*). The connection between *σ* and the error in *ψ*(**x**, *s*) is clear from comparison of ([3.9]), ([3.11]) with ([4.17]). Condition ([4.15]) is a direct analog of ([3.103]). Each function *p**n*\* is the weak solut... | arxiv_0004053 |
hat{A}-\hat{B})r\_{o}+\hat{r}\_{2} \hat{B}-\hat{r}\_{1} A}{(\hat{A}+\hat{B})r\_{o}-\hat{r}\_{2} \hat{B}-\hat{r}\_{1}\hat{A}} \right]=am\left[\tau \sqrt{\hat{A}\hat{B}}+\nu\_{r}F(\hat{\varphi}\_{P,s} \mid \hat{k}\_{P}) \right].\end{aligned}$$
### Motion in *t* direction
The motion in *t* direction also entails both th... | arxiv_0004054 |
in Algorithm [fig:stopwithinmcmc1] is a log normal random walk, i.e. we use *ζ*ʹ = *ζ*(*i* − 1) + 0.4N(0, 1) with *ζ* = log(*μ*).
We will compare PMMH when implemented with a simple adaptive scheme for *p* and when *p* is fixed. In the latter case we set *p* = 14. When an adaptive strategy is employed we will sample e... | arxiv_0004058 |
Magnetic Field Properties in High Mass Star Formation from Large to Small Scales - A Statistical Analysis from Polarization Data
================================================================================================================================
Polarization data from high mass star formation regions (W51 e... | arxiv_0004060 |
Resurgent Analysis of Localizable Observables in Supersymmetric Gauge Theories
==============================================================================
Introduction and Summary
========================
Understanding the strong–coupling dynamics of gauge theories requires detailed knowledge of their nonperturbativ... | arxiv_0004071 |
to contradict DD search limits from Xenon-10 and CDMS2, assuming a standard local DM density, the data can be accommodated by models with non-universal Higgs mass parameters. However, it recently appears that the latest FGST data are *in accord* with background expectations, which may end up ruling out this galactic E... | arxiv_0004074 |
be the left vertical edge of the unit torus, and this completes the proof.
Modification of polysquare translation surfaces
===============================================
We next consider modification of a polysquare translation surface by the inclusion of a one-sided vertical barrier on part of a vertical edge. We lo... | arxiv_0004077 |
An intrinsic hyperboloid approach forEinstein Klein-Gordon equations
====================================================================
In Klainerman introduced the hyperboloidal method to prove the global existence results for nonlinear Klein-Gordon equations by using commuting vector fields. In this paper, we exte... | arxiv_0004079 |
Composition Theorems for Interactive Differential Privacy
=========================================================
An interactive mechanism is an algorithm that stores a data set and answers adaptively chosen queries to it. The mechanism is called differentially private, if any adversary cannot distinguish whether a s... | arxiv_0004081 |
On the Fourier analysis of the Einstein-Klein-Gordon system: Growth and Decay of the Fourier constants
======================================================================================================
We consider the (1 + 3)-dimensional Einstein equations with negative cosmological constant coupled to a sphericall... | arxiv_0004083 |
while making decisions based on individual confidence levels of 96% or greater.
As shown in Figure [fig:MonotonicityDirectionsAndCIExample1], it is not easy to classify cases like *s* = 3 where all the parameter estimates are close to zero and their confidence intervals are big enough to make the monotonicity directio... | arxiv_0004089 |
M* domain walls are stable at scale *M*, and which domain walls in an optimized segmentation containing *M*ʹ > *M* domain walls are stable at scale *M*ʹ. However, these two sets of stable domain walls can disagree significantly because of the recursive segmentation scheme’s sensitivity to global contexts. Again, we can... | arxiv_0004090 |
C} = \{\bold{x} \in \mathbb{R}^{n\times 1}~\vert~ \bold{m}\_{{l}} \leq \bold{x} \leq \bold{m}\_{{u}}\}$ is the set of all feasible models bounded by the lower bound $\bold{m}\_{{l}}$ and the upper bound $\bold{m}\_{{u}}$ and $\|\bold{m} \|\_{\text{TT}}$ is the TT regularization functional as defined by
$$\label{TT\... | arxiv_0004093 |
N**c* = 3, while one has only real eigenvalues. We notice that the normalized eigenvectors corresponding to the complex eigenvalues are all directed along the four-fermi couplings, and have very small contributions (∣*η**α**g*∣ < 0.007) along the *e**α**g* direction, meaning that this complex scaling behavior involves ... | arxiv_0004095 |
the best-fit model with two photoionization zones. The spectra are shown as a ratio to the best-fit continuum, so that the wind absorption lines can be examined in detail. The ratio is particularly useful in that it is directly related to the equivalent width of the wind lines, and therefore to the column density in t... | arxiv_0004098 |
=1
Introduction
============
The theory of Gröbner fans, introduced by Mora and Robbiano , is a popular tool in commutative algebra, and one of its modern applications is degenerating ideals into simpler ones such as monomial, binomial or toric ideals. More precisely, let $\mathbb K$ be an algebraically closed field an... | arxiv_0004101 |
Conductance of closed and open long Aharonov-Bohm-Kondo rings
=============================================================
We calculate the finite temperature linear DC conductance of a generic single-impurity Anderson model containing an arbitrary number of Fermi liquid leads, and apply the formalism to closed and op... | arxiv_0004110 |
a solution of the *n*-SE, then a map, defined by a transposed matrix *M**T* is also a solution of the *n*-SE.
Fortunately in some cases we already know solutions of *n*-SE. For example, all linear bijective solutions for the 2-SE, 3-SE and 4-SE are listed in. Thus we can list all verbal universal abelian solutions in ... | arxiv_0004112 |
2, 0, 1, 1, 0]],
[[2, 0, 0, 1, 0],[2, 0, 0, 1, 1],[2, 0, 1, 0, 0],[3, 0, 0, 0, 0],[3, 0, 1, 0, 1]],
[[1, 0, 1, 0, 1],[2, 1, 0, 0, 0],[3, 0, 0, 0, 1],[3, 0, 0, 1, 0],[3, 1, 1, 1, 0]],
[[2, 1, 1, 0, 0],[2, 1, 1, 0, 1],[2, 1, 1, 1, 0],[3, 0, 1, 0, 0],[3, 1, 0, 1, 1]],
[[2, 1, 0, 0, 1],[2, 1, 0, 1, 0],[2, 1, 1, 1, 1],[3, 0... | arxiv_0004114 |
h(\pi (u-n\pi)/\gamma)$ for *γ* ≪ 1.
We compute Eq. , merging together (1)-(3), to get Eq. in the main text,
$$\begin{aligned}
\mathcal{G}\_{\delta p =1}&\simeq
- g\_0 \tilde{\gamma} L (k\_B T ) ^3
\exp(-\frac{L}{L\_{T,c}}) \exp(-\frac{\Delta{L}}{L\_{T,n}}) \nonumber
\\ &\times \exp(-\frac{L - \Delta L}{L\_{T,n}}... | arxiv_0004119 |
}^{a+3} - D\_{\rm complete}^{a+3} \right]\;.$$
The result is $D\_{\rm PB} = \sqrt[3]{{6\over\pi} V\_{\rm PB}} \doteq 130\,{\rm km}$, some sort of an upper limit. The volumetric ratio between the largest fragment and the parent body is then $V\_{\rm LF}/V\_{\rm PB} \doteq 0.2$, a fairly typical value for asteroid fami... | arxiv_0004120 |
rightness galaxies and dwarf spheroids, where DM dominates over the baryonic mass, the situation is much more complicated for their massive elliptical counterparts. Here, our ignorance about the stellar initial mass function (IMF) introduces a degeneracy between dark and luminous matter, which makes it hard to constrai... | arxiv_0004125 |
Supplementary material for Bayesian inference for diffusion-driven mixed-effects models
=======================================================================================
### 1School of Mathematics & Statistics, Newcastle University,Newcastle-upon-Tyne, NE1 7RU, UK2Department of Mathematics and Statistics, Lancast... | arxiv_0004126 |
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