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For an arbitrary semi-direct product, we give a general description of its lower central series and an estimation of its derived series. In the second part of the paper, we study these series for the full braid group B n ( M ) and pure braid group P n ( M ) of a compact surface M, orientable or non-orientable, the aim being to determine the values of n for which B n ( M ) and P n ( M ) are residually nilpotent or residually soluble. We first solve this problem in the case where M is the 2-torus. We then use the results of the first part of the paper to calculate explicitly the lower central series of P n ( K ) , where K is the Klein bottle. Finally, if M is a non-orientable, compact surface without boundary, we determine the values of n for which B n ( M ) is residually nilpotent or residually soluble in the cases that were not already known in the literature.

We consider random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. In a companion paper, we have shown that if the random walk is pulled to the right by a positive bias $$\uplambda > 0$$ , then its asymptotic linear speed $$\overline{\mathrm {v}}$$ is continuous in the variable $$\uplambda > 0$$ and differentiable for all sufficiently small $$\uplambda > 0$$ . In the paper at hand, we complement this result by proving that $$\overline{\mathrm {v}}$$ is differentiable at $$\uplambda = 0$$ . Further, we show the Einstein relation for the model, i.e., that the derivative of the speed at $$\uplambda = 0$$ equals the diffusivity of the unbiased walk.

If $(G,V)$ is a polar representation with Cartan subspace $\mathfrak c$ and Weyl group $W$, it is shown that there is a natural morphism of Poisson schemes $\mathfrak c \oplus {\mathfrak c}^*/W \to V\oplus V^*/\!\!/\!\!/ G$. This morphism is conjectured to be an isomorphism of the underlying reduced varieties if $(G,V)$ is visible. The conjecture is proved for visible stable locally free polar representations and certain further examples., Comment: 18 pages, final version

Let $\mathfrak g$ be a simple Lie algebra over an algebraically closed field $\bf k$ of characteristic zero and $\bf G$ its adjoint group. Let $\mathfrak q$ be a biparabolic subalgebra of $\mathfrak g$. The algebra $Sy(\mathfrak q)$ of semi-invariants on $\mathfrak q^*$ is polynomial in most cases, in particular when $\mathfrak g$ is simple of type $A$ or $C$. On the other hand $\mathfrak q$ admits a canonical truncation $\mathfrak q_{\Lambda}$ such that $Sy(\mathfrak q)=Sy(\mathfrak q_{\Lambda})=Y(\mathfrak q_{\Lambda})$ where $Y(\mathfrak q_{\Lambda})$ denotes the algebra of invariant functions on $\mathfrak q_{\Lambda}^*$. An adapted pair for $\mathfrak q_{\Lambda}$ is a pair $(h,\,\eta)\in \mathfrak q_{\Lambda}\times\mathfrak q_{\Lambda}^*$ such that $\eta$ is regular and $(ad\,h)\eta=-\eta$. In a previous paper of A. Joseph (2008) adapted pairs for every truncated biparabolic subalgebra $\mathfrak q_{\Lambda}$ of a simple Lie algebra $\mathfrak g$ of type $A$ were constructed and then provide Weierstrass sections for $Y(\mathfrak q_{\Lambda})$ in $\mathfrak q_{\Lambda}^*$. These latter are linear subvarieties $\eta+V$ of $\mathfrak q_{\Lambda}^*$ such that the restriction map induces an algebra isomorphism of $Y(\mathfrak q_{\Lambda})$ onto the algebra of regular functions on $\eta+V$. Here we show that for each of the adapted pairs $(h,\,\eta)$ constructed in the paper mentioned above one can express $\eta$ as the image of a regular nilpotent element $y$ of $\mathfrak g^*$ under the restriction to $\mathfrak q$. Since $y$ must be a $\bf G$ translate of the standard regular nilpotent element defined in terms of the already chosen set $\pi$ of simple roots, one may attach to $y$ a unique element of the Weyl group. Ultimately one can then hope to be able to describe adapted pairs (in general) through the Weyl group., Comment: This is a rewriting of the version submitted on the arXiv in June 2013

This paper represents a step in our program towards the proof of the Pierce–Birkhoff conjecture. In the nineteen eighties J. Madden proved that the Pierce–Birkhoff conjecture for a ring A is equivalent to a statement about an arbitrary pair of points α , β ∈ Sper A and their separating ideal α , β > ; we refer to this statement as the local Pierce–Birkhoff conjecture at α, β. In [8] we introduced a slightly stronger conjecture, also stated for a pair of points α , β ∈ Sper A and the separating ideal α , β > , called the Connectedness conjecture. In this paper, for each pair ( α , β ) with h t ( α , β > ) = dim ⁡ A , we define a natural number, called complexity of ( α , β ) . Complexity 0 corresponds to the case when one of the points α, β is monomial; this case was settled in all dimensions in [8] . In the present paper we introduce a new conjecture, called the Strong Connectedness conjecture, and prove that the strong connectedness conjecture in dimension n − 1 implies the connectedness conjecture in dimension n in the case when h t ( α , β > ) ≤ n − 1 . We prove the Strong Connectedness conjecture in dimension 2, which gives the Connectedness and the Pierce–Birkhoff conjectures in any dimension in the case when h t ( α , β > ) ≤ 2 . Finally, we prove the Connectedness (and hence also the Pierce–Birkhoff) conjecture in the case when dim ⁡ A = h t ( α , β > ) = 3 , the pair ( α , β ) is of complexity 1 and A is excellent with residue field R .

The paper concerns area preserving homeomorphisms of surfaces that are isotopic to the identity. The purpose of the paper is to find a maximal identity isotopy such that we can give a fine descriptions of the dynamics of its transverse foliation. We will define a kind of identity isotopies: torsion-low isotopies. In particular, when $f$ is a diffeomorphism with finitely many fixed points such that every fixed point is not degenerate, an identity isotopy $I$ of $f$ is torsion-low if and only if for every point $z$ fixed along the isotopy, the (real) rotation number $\rho(I,z)$, which is well defined when one blows-up $f$ at $z$, is contained in $(-1,1)$. We will prove the existence of torsion-low maximal identity isotopies, and we will deduce the local dynamics of the transverse foliations of any torsion-low maximal isotopy near any isolated singularity., Comment: arXiv admin note: text overlap with arXiv:1409.3752

Traditional evidential clustering tends to build clusters where the number of data for each cluster fairly close to each other. However, it may not be suitable for imbalanced data. This paper proposes a new method, called credal clustering (CClu), to deal with imbalanced data based on the theory of belief functions. Consider a dataset with \(\mathcal {C}\) wanted classes, the credal c-means (CCM) clustering method is employed at first to divide the dataset into some (i.e., \(\mathcal {S}~(\mathcal {S}>\mathcal {C})\)) clusters. Then these clusters are gradually merged following a given principle based on the density of meta-clusters and the associated singleton clusters. The merging is finished when \(\mathcal {C}\) singleton wanted classes are obtained. During this merging procedure, the objects in each singleton cluster will be assigned to one new singleton class. Moreover, a weighted mean vector rule is developed to classify the objects in the unmerged meta-cluster to the associated new classes using the K-Nearest neighbor technique. Two experiments show that CClu can handle imbalanced datasets with high accuracy, and the errors are reduced by properly modeling imprecision.

We explore the structural similarities in three different languages, first in the protein language whose primary letters are the amino acids, second in the musical language whose primary letters are the notes, and third in the poetry language whose primary letters are the alphabet. For proteins, the non local (secondary) letters are the types of foldings in space (α-helices, β-sheets, etc.), for music, one is dealing with clear-cut repetition units called musical forms and for poems the structure consists of grammatical forms (names, verbs, etc.). We show in this paper that the mathematics of such secondary structures relies on finitely presented groups fp on r letters, where r counts the number of types of such secondary non local segments. The number of conjugacy classes of a given index (also the number of graph coverings over a base graph) of a group fp is found to be close to the number of conjugacy classes of the same index in the free group Fr−1 on r−1 generators. In a concrete way, we explore the group structure of a variant of the SARS-Cov-2 spike protein and the group structure of apolipoprotein-H, passing from the primary code with amino acids to the secondary structure organizing the foldings. Then, we look at the musical forms employed in the classical and contemporary periods. Finally, we investigate in much detail the group structure of a small poem in prose by Charles Baudelaire and that of the Bateau Ivre by Arthur Rimbaud.

In this paper we address the following shape optimization problem: find the planar domain of least area, among the sets with prescribed constant width and inradius. In the literature, the problem is ascribed to Bonnesen, who proposed it in \cite{BF}. In the present work, we give a complete answer to the problem, providing an explicit characterization of optimal sets for every choice of width and inradius. These optimal sets are particular Reuleaux polygons., Comment: to appear in Journal Ecole Polytechnique

In arXiv:2011.08830 we established a series of correspondences relating five enumerative theories of log Calabi-Yau surfaces, i.e. pairs $(Y,D)$ with $Y$ a smooth projective complex surface and $D=D_1+\dots +D_l$ an anticanonical divisor on $Y$ with each $D_i$ smooth and nef. In this paper we explore the generalisation to $Y$ being a smooth Deligne-Mumford stack with projective coarse moduli space of dimension 2, and $D_i$ nef $\mathbb{Q}$-Cartier divisors. We consider in particular three infinite families of orbifold log Calabi-Yau surfaces, and for each of them we provide closed form solutions of the maximal contact log Gromov-Witten theory of the pair $(Y,D)$, the local Gromov-Witten theory of the total space of $\bigoplus_i \mathcal{O}_Y(-D_i)$, and the open Gromov-Witten theory of toric orbi-branes in a Calabi-Yau 3-orbifold associated to $(Y,D)$. We also consider new examples of BPS integral structures underlying these invariants, and relate them to the Donaldson-Thomas theory of a symmetric quiver specified by $(Y,D)$, and to a class of open/closed BPS invariants., Comment: 26 pages, 11 figures. v2: typos fixed, minor changes, version accepted for the Boris Dubrovin Memorial Issue of LMP

We construct a completely normal bounded distributive lattice D in which for every pair (a, b) of elements, the set {x $\in$ D | a $\le$ b $\lor$ x} has a countable coinitial subset, such that D does not carry any binary operation - satisfying the identities x $\le$ y $\lor$(x-y),(x-y)$\land$(y-x) = 0, and x-z $\le$ (x-y)$\lor$(y-z). In particular, D is not a homomorphic image of the lattice of all finitely generated convex {\ell}-subgroups of any (not necessarily Abelian) {\ell}-group. It has $\aleph 2 elements. This solves negatively a few problems stated by Iberkleid, Mart{\'i}nez, and McGovern in 2011 and recently by the author. This work also serves as preparation for a forthcoming paper in which we prove that for any infinite cardinal $\lambda$, the class of Stone duals of spectra of all Abelian {\ell}-groups with order-unit is not closed under L $\infty$$\lambda$-elementary equivalence., Comment: 23 pages. v2 removes a redundancy from the definition of a Cevian operation in v1.In Theorem 5.12, Idc should be replaced by Csc (especially on the G side)

This paper is dedicated to the spectral optimization problemmin{λ1(Ω)+⋯+λk(Ω)+Λ|Ω| : Ω⊂Dquasi-open}whereD⊂ ℝdis a bounded open set and 0 <λ1(Ω) ≤⋯ ≤λk(Ω) are the firstkeigenvalues onΩof an operator in divergence form with Dirichlet boundary condition and Hölder continuous coefficients. We prove that the firstkeigenfunctions on an optimal set for this problem are locally Lipschtiz continuous inDand, as a consequence, that the optimal sets are open sets. We also prove the Lipschitz continuity of vector-valued functions that are almost-minimizers of a two-phase functional with variable coefficients.

International audience; This paper explores the main differences between the Shapley values of a set of taxa introduced by Haake et al. (J Math Biol 56:479–497, 2007. https://doi-org.lama.univ-amu.fr/10.1007/s00285-007-0126-2) and Fuchs and Jin (J Math Biol 71:1133–1147, 2015. https://doi-org.lama.univ-amu.fr/10.1007/s00285-014-0853-0), the latter having been found identical to the Fair Proportion index (Redding and Mooers in Conserv Biol 20:1670–1678, 2006. https://doi-org.lama.univ-amu.fr/10.1111/j.1523-1739.2006.00555.x). In line with Shapley (in: Kuhn, Tucker (eds) Contributions to to the theory of games, volume II, annals of mathematics studies 28, Princeton University Press, Princeton, 1953), we identify the cooperative game basis for each of these two classes of phylogenetic games and use them (i) to construct simple formulas for these two Shapley values and (ii) to compare these different approaches. Using the set of weights of a phylogenetic tree as a parameter space, we then discuss the conditions under which these two values coincide and, if they are not the same, revisit Hartmann’s (J Math Biol 67:1163–1170, 2013. https://doi-org.lama.univ-amu.fr/10.1007/s00285-012-0585-y) convergence result. An example illustrates our main argument. Finally, we compare the species ranking induced by these two values. Considering the Kendall and the Spearman rank correlation coefficient, simulations show that these rankings are strongly correlated. These results are consistent with Wicke and Fischer (J Theor Biol 430:207–214, 2017. https://doi-org.lama.univ-amu.fr/10.1016/j.jtbi.2017.07.010), who reach similar conclusions with a different simulation method.

The work presented in this paper consists of searching for all the longest common subsequences among r sequences of equal length n which exclude a particular string. This problem is named the constrained-multiple longest common subsequence (constrained-MLCS) and is a general case of the constrained-LCS. Due to its importance, particularly in bioinformatics, the constrained-MLCS is widely studied. Thus, the solution proposed here is a coarse-grained multicomputer-based algorithm that uses an existing serial algorithm for local computation and a master-slave paradigm. This solution requires O D × Σ + C S S s × M L C S p local computation time on each processor , O M L C S communication rounds and O n r | Σ | p time for preliminary works. |Σ| is the alphabet’s size, p is the number of processors used, |D| is the number of dominants generated during the entire resolution process, C S S s and M L C S are the number of all common subsequences, and the length of the longest common subsequences respectively. The experiments performed indicate that our proposed algorithm is scalable both with the number of processors and the number of input sequences.

Let F be a field of characteristic 2, Ω F m the space of m-differential forms over F, d : Ω F m − 1 ⟶ Ω F m the differential operator, and H 2 m + 1 ( F ) the cokernel of the Artin-Schreier operator ℘ : Ω F m ⟶ Ω F m / d Ω F m − 1 given on generators by: ℘ ( x d x 1 x 1 ∧ ⋯ ∧ d x m x m ) = ( x 2 − x ) d x 1 x 1 ∧ ⋯ ∧ d x m x m + d Ω F m − 1 . It was shown in [8] that given a multiquadratic purely inseparable extension L / F and a separable quadratic extension K / F , then the kernel of the natural homomorphism H 2 m + 1 ( F ) ⟶ H 2 m + 1 ( K ⋅ L ) is equal to the sum of the kernels of the extensions L / F and K / F . In this paper we extend this result to the compositum of a finite purely inseparable multiquadratic extension with a separable biquadratic extension, and we prove that this splitting property of kernels is no longer true by computing the kernel of the compositum of a separable quadratic (or biquadratic) extension with a simple purely inseparable extension of arbitrary degree.

Let $${ \rm I\!K}$$ be an ultrametric complete valued field and $${ \rm I\!E}$$ be an ultrametric space. We examine some Banach algebras $$S$$ of bounded continuous functions from $${ \rm I\!E}$$ to $${ \rm I\!K}$$ with the use of ultrafilters, particularly the relation of stickness. We recall and deepen results obtained in a previous paper by N. Mainetti and the third author concerning the whole algebra $$\cal A$$ of all bounded continuous functions from $${ \rm I\!E}$$ to $${ \rm I\!K}$$ . Every maximal ideal of finite codimension of $$\cal A$$ is of codimension $$1$$ and we show that this property also holds for every algebra $$S$$ , provided $${ \rm I\!K}$$ is perfect. If $$S$$ admits the uniform norm on $${ \rm I\!E}$$ as its spectral norm, then every maximal ideal is the kernel of only one multiplicative semi-norm, the Shilov boundary is equal to the whole multiplicative spectrum and the Banaschewski compactification of $${ \rm I\!E}$$ is homeomorphic to the multiplicative spectrum of $$S$$ .

The goal of this paper is to study geometric and extremal properties of the convex body $B_{\mathcal F(M)}$, which is the unit ball of the Lipschitz-free Banach space associated with a finite metric space $M$. We investigate $\ell_1$ and $\ell_\infty$-sums, in particular we characterize the metric spaces such that $B_{\mathcal F(M)}$ is a Hanner polytope. We also characterize the finite metric spaces whose Lipschitz-free spaces are isometric. We discuss the extreme properties of the volume product $\mathcal{P}(M)=|B_{\mathcal F(M)}|\cdot|B_{\mathcal F(M)}^\circ|$, when the number of elements of $M$ is fixed. We show that if $\mathcal P(M)$ is maximal among all the metric spaces with the same number of points, then all triangle inequalities in $M$ are strict and $B_{\mathcal F(M)}$ is simplicial. We also focus on the metric spaces minimizing $\mathcal P(M)$, and in the Mahler's conjecture for this class of convex bodies., 21 pages

Let N be a closed oriented k-dimensional submanifold of the (k+2)-dimensional sphere; denote its complement by C(N). Denote by x the 1-dimensional cohomology class in C(N), dual to N. The Morse-Novikov number of C(N) is by definition the minimal possible number of critical points of a regular Morse map f from C(N) to a circle, such that f belongs to x. In the first part of this paper we study the case when N is the twist frame spun knot associated to an m-knot K. We obtain a formula which relates the Morse-Novikov numbers of N and K and generalizes the classical results of D. Roseman and E.C. Zeeman about fibrations of spun knots. In the second part we apply the obtained results to the computation of Morse-Novikov numbers of surface-links in 4-sphere., 13 pages

The transfer distance (TD) was introduced in the classification framework and studied in the context of phylogenetic tree matching. Recently, Lemoine et al. (Nature 556(7702):452–456, 2018. 10.1038/s41586-018-0043-0) showed that TD can be a powerful tool to assess the branch support on large phylogenies, thus providing a relevant alternative to Felsenstein’s bootstrap. This distance allows a reference branch\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}β in a reference tree \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {T}}$$\end{document}T to be compared to a branch b from another tree T (typically a bootstrap tree), both on the same set of n taxa. The TD between these branches is the number of taxa that must be transferred from one side of b to the other in order to obtain \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}β. By taking the minimum TD from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}β to all branches in T we define the transfer index, denoted by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (\beta ,T)$$\end{document}ϕ(β,T), measuring the degree of agreement of T with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}β. Let us consider a reference branch \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta $$\end{document}β having p tips on its light side and define the transfer support (TS) as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1 - \phi (\beta ,T)/(p-1)$$\end{document}1-ϕ(β,T)/(p-1). Lemoine et al. (2018) used computer simulations to show that the TS defined in this manner is close to 0 for random “bootstrap” trees. In this paper, we demonstrate that result mathematically: when T is randomly drawn, TS converges in probability to 0 when n tends to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\infty $$\end{document}∞. Moreover, we fully characterize the distribution of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (\beta ,T)$$\end{document}ϕ(β,T) on caterpillar trees, indicating that the convergence is fast, and that even when n is small, moderate levels of branch support cannot appear by chance.

Let P = { p n } n = 1 ∞ be a sequence of positive integers with p n > 1 and D = { D n } n = 1 ∞ be a sequence of subsets of N with Card D n = 2 or 3. The infinite convolution of probability measures μ P , D = δ p 1 − 1 D 1 ⁎ δ ( p 1 p 2 ) − 1 D 2 ⁎ ⋯ is a Borel probability measure which is singular with respect to the Lebesgue measure on R . In this paper, we prove the spectrality of such measure, i.e., to find a set Λ ⊂ R such that the set { e − 2 π i λ x | λ ∈ Λ } is an orthonormal basis of L 2 ( μ P , D ) .

In this paper, we establish the Hitchin--Kobayashi correspondence for the $I_\pm$-holomorphic quiver bundle $\mathcal{E}=(E,\phi)$ over a compact generalized K\"{a}hler manifold $(X, I_+,I_-,g, b)$ such that $g$ is Gauduchon with respect to both $I_+$ and $I_-$, namely $\mathcal{E}$ is $(\alpha,\sigma,\tau)$-polystable if and only if $\mathcal{E}$ admits an $(\alpha,\sigma,\tau)$-Hermitian--Einstein metric., Comment: To appear in The Journal of Geometric Analysis

In this paper, we study the behaviour of the shortest distance between orbits and show that under some rapidly mixing conditions, the decay of the shortest distance depends on the correlation dimension. For irrational rotations, we prove a different behaviour depending on the irrational exponent of the angle of the rotation. For random processes, this problem corresponds to the longest common substring problem. We extend the result of Arratia and Waterman on sequence matching to $\alpha$-mixing processes with exponential decay., Comment: Final version. Accepted for publication in Advances in Mathematics

International audience; Denote by [Formula: see text] (respectively, [Formula: see text]) the largest (respectively, the smallest) prime factor of the integer [Formula: see text]. In this paper, we prove a lower bound of almost-primes [Formula: see text] with [Formula: see text] such that [Formula: see text] for [Formula: see text]. As an application, we study two patterns on the largest prime factors of consecutive integers with one of which without small prime factor.

The infinite pigeonhole principle for 2-partitions asserts the existence, for every set $A$, of an infinite subset of $A$ or of its complement. In this paper, we develop a new notion of forcing enabling a fine analysis of the computability-theoretic features of the pigeonhole principle. We deduce various consequences, such as the existence, for every set $A$, of an infinite subset of it or its complement of non-high degree. We also prove that every $\Delta^0_3$ set has an infinite low${}_3$ solution and give a simpler proof of Liu's theorem that every set has an infinite subset in it or its complement of non-PA degree., Comment: 20 pages

The first two authors [Proc. Lond. Math. Soc. (3) {\bf 114}(1):1--34, 2017] classified the behaviour near zero for all positive solutions of the perturbed elliptic equation with a critical Hardy--Sobolev growth $$-\Delta u=|x|^{-s} u^{2^\star(s)-1} -\mu u^q \hbox{ in }B\setminus\{0\},$$ where $B$ denotes the open unit ball centred at $0$ in $\mathbb{R}^n$ for $n\geq 3$, $s\in (0,2)$, $2^\star(s):=2(n-s)/(n-2)$, $\mu>0$ and $q>1$. For $q\in (1,2^\star-1)$ with $2^\star=2n/(n-2)$, it was shown in the op. cit. that the positive solutions with a non-removable singularity at $0$ could exhibit up to three different singular profiles, although their existence was left open. In the present paper, we settle this question for all three singular profiles in the maximal possible range. As an important novelty for $\mu>0$, we prove that for every $q\in (2^\star(s) -1,2^\star-1)$ there exist infinitely many positive solutions satisfying $|x|^{s/(q-2^\star(s)+1)}u(x)\to \mu^{-1/(q-2^\star(s)+1)}$ as $|x|\to 0$, using a dynamical system approach. Moreover, we show that there exists a positive singular solution with $\liminf_{|x|\to 0} |x|^{(n-2)/2} u(x)=0$ and $\limsup_{|x|\to 0} |x|^{(n-2)/2} u(x)\in (0,\infty)$ if (and only if) $q\in (2^\star-2,2^\star-1)$., Comment: Mathematische Annalen, to appear

The Tutte polynomial T(G; x, y) of a graph G, or equivalently the q-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in combinatorics and statistical physics. Graph operations have been extensively applied to model complex networks recently. In this paper, we study the Tutte polynomials of the diamond hierarchical lattices and a class of self-similar fractal models which can be constructed through graph operations. Firstly, we find out the behavior of the Tutte polynomial under k-inflation and k-subdivision which are two graph operations. Secondly, we compute and gain the Tutte polynomials of this two self-similar fractal models by using their structure characteristic. Moreover, as an application of the obtained results, some evaluations of their Tutte polynomials are derived, such as the number of spanning trees and the number of spanning forests.

Denote by$\mathbb{P}$the set of all prime numbers and by$P(n)$the largest prime factor of positive integer$n\geq 1$with the convention$P(1)=1$. In this paper, we prove that, for each$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$, there is a constant$c(\unicode[STIX]{x1D702})>1$such that, for every fixed nonzero integer$a\in \mathbb{Z}^{\ast }$, the set$$\begin{eqnarray}\{p\in \mathbb{P}:p=P(q-a)\text{ for some prime }q\text{ with }p^{\unicode[STIX]{x1D702}}has relative asymptotic density one in$\mathbb{P}$. This improves a similar result due to Banks and Shparlinski [‘On values taken by the largest prime factor of shifted primes’,J. Aust. Math. Soc.82(2015), 133–147], Theorem 1.1, which requires$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.0606\cdots \,)$in place of$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$.

Let $R_s M$ denote the Singer construction on an unstable module $M$ over the Steenrod algebra $A$ at the prime two; $R_s M$ is canonically a subobject of $P_s\otimes M$, where $P_s$ is the polynomial algebra on s generators of degree one. Passage to $A$-indecomposables gives the natural transformation $R_s M \rightarrow F \otimes_A (P_s \otimes M)$, which identifies with the dual of the composition of the Singer transfer and the Lannes-Zarati homomorphism. The main result of the paper proves the weak generalized algebraic spherical class conjecture, which was proposed by the first named author. Namely, this morphism is trivial on elements of positive degree when s>2. The condition s>2 is necessary, as exhibited by the spherical classes of Hopf invariant one and those of Kervaire invariant one., v2 15 pages. Minor revision. v3 17 pages, revision following referee's recommendations. Accepted for publication J. Alg

In this paper, we provide refined sufficient conditions for the quadratic Chabauty method to produce a finite set of points, with the conditions on the rank of the Jacobian replaced by conditions on the rank of a quotient of the Jacobian plus an associated space of Chow-Heegner points. We then apply this condition to prove the finiteness of this set for any modular curves $X_{\mathrm{ns} }^+ (N)$ and $X_0 ^+ (N)$ of genus at least 2 with N prime. The proof relies on the existence of a quotient of their Jacobians whose Mordell-Weil rank is equal to its dimension (and at least 2), which is proven via analytic estimates for orders of vanishing of L-functions of modular forms, thanks to a Kolyvagin-Logachev type result., Comment: 51 pages, comments welcome

The operator Schmidt rank of an operator acting on the tensor product C n ⊗ C m is the number of terms in a decomposition of the operator as a sum of simple tensors with factors forming orthogonal families in their respective matrix algebras. It has been known that for unitary operators acting on two copies of C 2 , the operator Schmidt rank can only take the values 1, 2, and 4, the value 3 being forbidden. In this paper, we settle an open question, showing that the above obstruction is the only one occurring. We do so by constructing explicit examples of bipartite unitary operators of all possible operator Schmidt ranks, for arbitrary dimensions ( n , m ) ≠ ( 2 , 2 ) .

In this paper, we study rhombic alternative tableaux, whose weight generating functions provide combinatorial formulae to compute the steady state probabilities of the two-species ASEP. In the ASEP, there are two species of particles, one heavy and one light, hopping right and left on a one-dimensional finite lattice with open boundaries. Parameters α , β , and q describe the hopping probabilities. The rhombic alternative tableaux are enumerated by the Lah numbers, which also enumerate certain assemblees of permutations. We describe a bijection between the rhombic alternative tableaux and these assemblees. We also provide an insertion algorithm that gives a weight generating function for the assemblees. Combined, these results give a bijective proof for the weight generating function for the rhombic alternative tableaux, which is also the partition function of the two-species ASEP at q = 1 .

In this paper we study the smallest non-zero eigenvalue $\lambda_1$ of the Laplacian on toric K\"ahler manifolds. We find an explicit upper bound for $\lambda_1$ in terms of moment polytope data. We show that this bound can only be attained for $\mathbb{CP}^n$ endowed with the Fubini-Study metric and therefore $\mathbb{CP}^n$ endowed with the Fubini-Study metric is spectrally determined among all toric K\"ahler metrics. We also study the equivariant counterpart of $\lambda_1$ which we denote by $\lambda_1^T$. It is the the smallest non-zero eigenvalue of the Laplacian restricted to torus-invariant functions. We prove that $\lambda_1^T$ is not bounded among toric K\"ahler metrics thus generalizing a result of Abreu-Freitas on $S^2$. In particular, $\lambda_1^T$ and $\lambda_1$ do not coincide in general., Comment: 24 pages. Added some more details and corrected an estimate. Fixed some typos

Let M be a compact manifold of dimension n . In this paper, we introduce the Mass Function a ≥ 0 ↦ X + M ( a ) (resp. a ≥ 0 ↦ X − M ( a ) ) which is defined as the supremum (resp. infimum) of the masses of all metrics on M whose Yamabe constant is larger than a and which are flat on a ball of radius 1 and centered at a point p ∈ M . Here, the mass of a metric flat around p is the constant term in the expansion of the Green function of the conformal Laplacian at p . We show that these functions are well defined and have many properties which allow to obtain applications to the Yamabe invariant (i.e. the supremum of Yamabe constants over the set of all metrics on M ).

Given a graph G = ( V , E ) of order n and maximum degree Δ , the NP -complete S - labeling problem consists in finding a labeling of G , i.e. a bijective mapping ϕ : V → { 1 , 2 … n } , such that SL ϕ ( G ) = ∑ u v ∈ E min { ϕ ( u ) , ϕ ( v ) } is minimized. In this paper, we study the S - labeling problem, with a particular focus on algorithmic issues. We first give intrinsic properties of optimal labelings, which will prove useful for our algorithmic study. We then provide lower bounds on SL ϕ ( G ) , together with a generic greedy algorithm, which collectively allow us to approximate the problem in several classes of graphs—in particular, we obtain constant approximation ratios for regular graphs and bounded degree graphs. We also show that deciding whether there exists a labeling ϕ of G such that SL ϕ ( G ) ≤ | E | + k is solvable in O ∗ ( 2 2 k ( 2 k ) ! ) time, thus fixed-parameterized tractable in k . We finally show that the S - Labeling problem is polynomial-time solvable for two classes of graphs, namely split graphs and (sets of) caterpillars.

Normal graphs are defined in terms of cross-intersecting set families: a graph is normal if it admits a clique cover $$\mathcal {Q}$$ and a stable set cover $$\mathcal {S}$$ s.t. every clique in $$\mathcal {Q}$$ intersects every stable set in $$\mathcal {S}$$ . Normal graphs can be considered as closure of perfect graphs by means of co-normal products and graph entropy. Perfect graphs have been characterized as those graphs without odd holes and odd antiholes as induced subgraphs (Strong Perfect Graph Theorem, Chudnovsky et al.). Korner and de Simone observed that $$C_5$$ , $$C_7$$ , and $$\overline{C}_7$$ are minimal not normal and conjectured, in analogy to the Strong Perfect Graph Theorem, that every $$(C_5, C_7, \overline{C}_7)$$ -free graph is normal (Normal Graph Conjecture, Korner and de Simone). Recently, this conjecture has been disproved by Harutyunyan, Pastor and Thomasse. However, in this paper we verify it for two classes of sparse graphs, 1-trees and cacti. In addition, we provide both linear time recognition algorithms and characterizations for the normal graphs within these two classes.

As usual, denote by P(n) the largest prime factor of the integer n ⩾ 1 with the convention P(1) = 1. For 0 < θ < 1, define $${T_\theta }\left( x \right): = \left| {\left\{ {p \leqslant x:P\left( {p - 1} \right) \geqslant {p^\theta }} \right\}} \right|.$$ In this paper, we obtain a new lower bound for T θ (x) as x → ∞, which improves some recent results of Luca et al. (2015) and of Chen and Chen (2017). As a corollary, we partially prove a conjecture of Chen and Chen (2017) about the size of T θ (x).

International audience; This paper is concerned with the study of the Monge optimal transport problem in sub-Riemannian manifolds where the cost is given by the square of the sub-Riemannian distance. Our aim is to extend previous results on existence and uniqueness of optimal transport maps to cases of sub-Riemannian structures which admit many singular minimizing geodesics. We treat here the case of sub-Riemannian structures of rank two in dimension four.

International audience; Bipartite graph matching algorithms become more and more popular to solve error-correcting graph matching problems and to approximate the graph edit distance of two graphs. However, the memory requirements and execution times of this method are respectively proportional to (n + m) 2 and (n + m) 3 where n and m are the order of the graphs. Subsequent developments reduced these complexities. However , these improvements are valid only under some constraints on the parameters of the graph edit distance. We propose in this paper a new formulation of the bipartite graph matching algorithm designed to solve efficiently the associated graph edit distance problem. The resulting algorithm requires O(nm) memory space and O(min(n, m) 2 max(n, m)) execution times.; L'appariement de graphes biparti deviennent de plus en plus populaires pour résoudre des problèmes d'appariement de graphes avec correction d'erreurs et pour approximer la distance d'édition sur graphes. Cependant, les exigences en mémoire et temps de calcul de cette méthode sont respectivement proportionnels à (n + m)^2 et (n + m)^3 où n et m représentent la taille des deux graphes. Des développements ultérieurs ont réduit ces complexités. Cependant, ces améliorations ne sont valables que sous certaines contraintes sur les paramètres de la distance d'édition. Nous proposons dans cet article une nouvelle formulation de l'algorithme Hongrois conçu pour résoudre efficacement le problème de distance d'édition associé. L'algorithme résultat nécessite un espace mémoire O (nm) et des temps d'exécution O (min (n, m)^2 max (n, m)).

Dynamic complexity is concerned with updating the output of a problem when the input is slightly changed. We study the dynamic complexity of Dyck reachability problems in directed and undirected graphs, where updates may add or delete edges. We show a strong dichotomy between such problems, based on the size of the Dyck alphabet. Some of them are P-complete (under a strong notion of reduction) while the others lie either in DynFO or in NL., 16 pages, 2 figures - Full version of FoSSaCS 2017 paper

Let n ≥ 3 . In this paper, we study the quotient group B n / [ P n , P n ] of the Artin braid group B n by the commutator subgroup of its pure Artin braid group P n . We show that B n / [ P n , P n ] is a crystallographic group, and in the case n = 3 , we analyse explicitly some of its subgroups. We also prove that B n / [ P n , P n ] possesses torsion, and we show that there is a one-to-one correspondence between the conjugacy classes of the finite-order elements of B n / [ P n , P n ] with the conjugacy classes of the elements of odd order of the symmetric group S n , and that the isomorphism class of any Abelian subgroup of odd order of S n is realised by a subgroup of B n / [ P n , P n ] . Finally, we discuss the realisation of non-Abelian subgroups of S n of odd order as subgroups of B n / [ P n , P n ] , and we show that the Frobenius group of order 21, which is the smallest non-Abelian group of odd order, embeds in B n / [ P n , P n ] for all n ≥ 7 .

The present paper presents a counterexample to the sequentially weak density of smooth maps between two manifolds $M$ and $N$ in the Sobolev space $W^{1, p} (M, N)$, in the case $p$ is an integer. It has been shown that, if $p, Comment: 68 pages

This paper introduces an inductively defined tree notation for all the faces of polytopes arising from a simplex by truncations. This notation allows us to view inclusion of faces as the process of contracting tree edges. Our notation instantiates to the well-known notations for the faces of associahedra and permutohedra. Various authors have independently introduced combinatorial tools for describing such polytopes. We build on the particular approach developed by Dosen and Petric, who used the formalism of hypergraphs to describe the interval of polytopes from the simplex to the permutohedron. This interval was further stretched by Petric to allow truncations of faces that are themselves obtained by truncations, and iteratively so. Our notation applies to all these polytopes. We illustrate this by showing that it instantiates to a notation for the faces of the permutohedron-based associahedra, that consists of parenthesised words with holes. Dosen and Petric have exhibited some families of hypergraph polytopes (associahedra, permutohedra, and hemiassociahedra) describing the coherences, and the coherences between coherences etc., arising by weakening sequential and parallel associativity of operadic composition. We complement their work with a criterion allowing us to recover the information whether edges of these "operadic polytopes" come from sequential, or from parallel associativity. We also give alternative proofs for some of the original results of Dosen and Petric., Comment: 42 pages, 4 figures

A number is said to be [Formula: see text]-friable if it has no prime factor greater than [Formula: see text]. In this paper, we prove a central limit theorem on average for the distribution of divisors of [Formula: see text]-friable numbers less than [Formula: see text], for all [Formula: see text] satisfying [Formula: see text]. This was previously known under the additional constraint [Formula: see text], by work of Basquin. Our argument relies on the two-variable saddle-point method.

In a recent paper, Chapuy conjectured that, for any positive integer k, the law for the fractions of total area covered by the k Voronoi cells defined by k points picked uniformly at random in the Brownian map of any fixed genus is the same law as that of a uniform k-division of the unit interval. For k=2, i.e. with two points chosen uniformly at random, it means that the law for the ratio of the area of one of the two Voronoi cells by the total area of the map is uniform between 0 and 1. Here, by a direct computation of the desired law, we show that this latter conjecture for k=2 actually holds in the case of large planar (genus 0) quadrangulations as well as for large general planar maps (i.e. maps whose faces have arbitrary degrees). This corroborates Chapuy's conjecture in its simplest realizations., 27 pages, 8 figures

In this paper we prove existence of matings between a large class of renormalizable cubic polynomials with one fixed critical point and another cubic polynomial having two fixed critical points. The resulting mating is a Newton map. Our result is the first part towards a conjecture by Tan Lei, stating that all (cubic) Newton maps can be described as matings or captures., Comment: 15 figures

In this paper, we give an equivariant compactification of the space PFlat(S) of homothety classes of half-translation structures on a compact, connected, orientable surface S. We introduce the space PMix(S) of homothety classes of mixed structures on S, that are CAT(0) tree-graded spaces in the sense of Drutu and Sapir, with pieces which are R-trees and completions of surfaces endowed with half-translation structures. Endowing Mix(S) with the equivariant Gromov topology, and using asymptotic cone techniques, we prove that PMix(S) is an equivariant compactification of PFlat(S), thus allowing us to understand in a geometric way the degenerations of half-translation structures on S. We finally compare our compactification to the one of Duchin-Leininger-Rafi, based on geodesic currents on S, by the mean of the translation distances of the elements of the covering group of S., 48 pages, 12 figures

This paper gives a connection between well chosen reductions of the Links-Gould invariants of oriented links and powers of the Alexander-Conway polynomial. We prove these formulas by showing the representations of the braid groups we derive the specialized Links-Gould polynomials from can be seen as exterior powers of copies of Burau representations., 19 pages

Let $R$ be a smooth affine algebra over an infinite perfect field $k$. Let $I\subset R$ be an ideal, $\omega_I:(R/I)^n\to I/I^2$ a surjective homomorphism and $Q_{2n}\subset \mathbb{A}^{2n+1}$ be the smooth quadric defined by the equation $\sum x_iy_i=z(1-z)$. We associate with the pair $(I,\omega_I)$ an obstruction in the set of homomorphisms $\mathrm{Hom}_{\mathbb{A}^1}(\mathrm{Spec}(R),Q_{2n})$ up to naive homotopy whose vanishing is sufficient for $\omega_I$ to lift to a surjection $R^n\to I$. Subsequently, we prove that the obstruction vanishes in case $R=k[T_1,\ldots,T_m]$ for $m\in \mathbb{N}$ where $k$ is an infinite perfect field having characteristic different from $2$ thus resolving an old conjecture of M. P. Murthy., Comment: Paper withdrawn. Due to an error in Lemma 3.2.3, the proof of the main result collapses

We consider in this paper the family of exponential Lie groups $G_{n,\mu}$, whose Lie algebra is an extension of the Heisenberg Lie algebra by the reals and whose quotient group by the centre of the Heisenberg group is an $ax+b$-like group. The C*-algebras of the groups $G_{n,\mu}$ give new examples of almost $C_0({\cal K})$-C*-algebras.

In this paper we study the topology of the stack $\mathcal{T}_g$ of smooth trigonal curves of genus g, over the complex field. We make use of a construction by the first named author and Vistoli, that describes $\mathcal{T}_g$ as a quotient stack of the complement of the discriminant. This allows us to use techniques developed by the second named author to give presentations of the orbifold fundamental group of $\mathcal{T}_g$, of its substrata with prescribed Maroni invariant and describe their relation with the mapping class group $\mathcal{M}ap_g$ of Riemann surfaces of genus g., To appear on Selecta Math