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In this paper we are interested in lifting a prescribed group of automorphisms of a finite graph via regular covering projections. Here we describe with an example the problems we address and refer to the introductory section for the correct statements of our results. Let $P$ be the Petersen graph, say, and let $\wp:\tilde{P}\to P$ be a regular covering projection. With the current covering machinery, it is straightforward to find $\wp$ with the property that every subgroup of $\Aut(P)$ lifts via $\wp$. However, for constructing peculiar examples and in applications, this is usually not enough. Sometimes it is important, given a subgroup $G$ of $\Aut(P)$, to find $\wp$ along which $G$ lifts but no further automorphism of $P$ does. For instance, in this concrete example, it is interesting to find a covering of the Petersen graph lifting the alternating group $A_5$ but not the whole symmetric group $S_5$. (Recall that $\Aut(P)\cong S_5$.) Some other time it is important, given a subgroup $G$ of $\Aut(P)$, to find $\wp$ with the property that $\Aut(\tilde{P})$ is the lift of $G$. Typically, it is desirable to find $\wp$ satisfying both conditions. In a very broad sense, this might remind wallpaper patterns on surfaces: the group of symmetries of the dodecahedron is $S_5$, and there is a nice colouring of the dodecahedron (found also by Escher) whose group of symmetries is just $A_5$. In this paper, we address this problem in full generality., Comment: 10 pages

In this paper, we address the minimum-cost node-capacitated multiflow problem in an undirected network. For this problem, Babenko and Karzanov (2012) showed strongly polynomial-time solvability via the ellipsoid method. Our result is the first combinatorial weakly polynomial-time algorithm for this problem. Our algorithm finds a half-integral minimum-cost maximum multiflow in $O(m \log(nCD)\mathrm{SF}(kn, m, k))$ time, where $n$ is the number of nodes, $m$ is the number of edges, $k$ is the number of terminals, $C$ is the maximum node capacity, $D$ is the maximum edge cost, and $\mathrm{SF}(n', m', \eta)$ is the time complexity of solving the submodular flow problem in a network of $n'$ nodes, $m'$ edges, and a submodular function with $\eta$-time-computable exchange capacity. Our algorithm is built on discrete convex analysis on graph structures and the concept of reducible bisubmodular flows., Comment: A preliminary version of this paper was presented in at the 11th Hungarian-Japanese Symposium on Discrete Mathematics and Its Applications, 2019

Let $$\sigma $$ be an operator mean in the sense of Kubo and Ando. If the representation function $$f_\sigma $$ of $$\sigma $$ satisfies $$\begin{aligned} f_\sigma (t)^p\le f_\sigma (t^p) \text { for all } p>1, \end{aligned}$$then $$\sigma $$ is called a pmi mean. Our main interest is the class of pmi means (denoted by PMI). To study PMI, the operator means $$\sigma $$ satisfying $$\begin{aligned} f_\sigma (\sqrt{xy})\le \sqrt{f_\sigma (x)f_\sigma (y)}\quad (x,y>0) \end{aligned}$$are considered in this paper. The set of such means (denoted by GCV) includes certain significant examples and is included in PMI. The main result presented in this paper is that GCV is a proper subset of PMI. In addition, we investigate certain operator-mean classes including PMI.

Nielsen realization problem for the mapping class group $\text{Mod}(S_g)$ asks whether the natural projection $p_g: \text{Homeo}_+(S_g)\to \text{Mod}(S_g)$ has a section. While all the previous results use torsion elements in an essential way, in this paper, we focus on the much more difficult problem of realization of torsion-free subgroups of $\text{Mod}(S_g)$. The main result of this paper is that the Torelli group has no realization inside the area-preserving homeomorphisms., Comment: 22 pages, 5 figures

It is known that it is a very restrictive condition for a frame $\{f_k\}_{k=1}^\infty$ to have a representation $ \{T^n \varphi\}_{n=0}^\infty$ as the orbit of a bounded operator $T$ under a single generator $\varphi\in\mathcal{H}.$ In this paper we prove that, on the other hand, any frame can be approximated arbitrarily well by a suborbit $\{T^{\alpha(k)} \varphi\}_{k=1}^\infty$ of a bounded operator $T$. An important new aspect is that for certain important classes of frames, e.g., frames consisting of finitely supported vectors in $\ell^2(\mathbb{N}),$ we can be completely explicit about possible choices of the operator $T$ and the powers $\alpha(k),k\in \mathbb{N}.$ A similar approach carried out in $L^2(\mathbb{R})$ leads to an approximation of a frame using suborbits of two bounded operators. The results are illustrated with an application to Gabor frames generated by a compactly supported function. The paper is concluded with an appendix which collects general results about frame representations using multiple orbits of bounded operators.

This paper counts the number of reduced quasigroup words of a particular length in a certain number of generators. Taking account of the relationship with the Catalan numbers, counting words in a free magma, we introduce the term peri-Catalan number for the free quasigroup word counts. The main result of this paper is an exact recursive formula for the peri-Catalan numbers, structured by the Euclidean Algorithm. The Euclidean Algorithm structure does not readily lend itself to standard techniques of asymptotic analysis. However, conjectures for the asymptotic behavior of the peri-Catalan numbers, substantiated by numerical data, are presented. A remarkable aspect of the observed asymptotic behavior is the so-called asymptotic irrelevance of quasigroup identities, whereby cancelation resulting from quasigroup identities has a negligible effect on the asymptotic behavior of the peri-Catalan numbers for long words in a large number of generators.

In this work we prove general bounds for the diameter of random graphs generated by a preferential attachment model whose parameter is a function $f:\mathbb{N}\to[0,1]$ that drives the asymptotic proportion between the numbers of vertices and edges. These results are sharp when $f$ is a \textit{regularly varying function at infinity} with strictly negative index of regular variation~$-\gamma$. For this particular class, we prove a characterization for the diameter that depends only on~$-\gamma$. More specifically, we prove that the diameter of such graphs is of order $1/\gamma$ with high probability, although its vertex set order goes to infinity polynomially. Sharp results for the diameter for a wide class of \textit{slowly varying functions} are also obtained., Comment: After referee recommendation, all results not concerning the diameter have been removed to obtain a more streamlined paper. The paper has been published at Random Structures and Algorithms under the new title

We prove that there are arbitrarily large values of $t$ such that $|\zeta(1+it)| \geq e^{\gamma} (\log_2 t + \log_3 t) + \mathcal{O}(1)$. This essentially matches the prediction for the optimal lower bound in a conjecture of Granville and Soundararajan. Our proof uses a new variant of the "long resonator" method. While earlier implementations of this method crucially relied on a "sparsification" technique to control the mean-square of the resonator function, in the present paper we exploit certain self-similarity properties of a specially designed resonator function., Comment: 7 pages. Versions 2: several minor changes. The paper will appear in IMRN

Let $\mathscr{C}$ be a 2-Calabi-Yau triangulated category, and let $\mathscr{T}$ be a cluster tilting subcategory of $\mathscr{C}$. An important result from Dehy and Keller tells us that a rigid object $c \in \mathscr{C}$ is uniquely defined by its index with respect to $\mathscr{T}$. The notion of triangulated categories extends to the notion of $(d+2)$-angulated categories. Thanks to a paper by Oppermann and Thomas, we now have a definition for cluster tilting subcategories in higher dimensions. This paper proves that under a technical assumption, an indecomposable object in a $(d+2)$-angulated category is uniquely defined by its index with respect to a higher dimensional cluster tilting subcategory. We also demonstrate an application of this result in higher dimensional cluster categories., Comment: Revising the article based on advice from a referee received upon submitting to a journal. This serves to make the results slightly more general

We study the question whether rational homogeneous spaces are rigid under Fano deformation. In other words, given any smooth connected family f:X -> Zof Fano manifolds, if one fiber is biholomorphic to a rational homogeneous space S, whether is f an S-fibration? The cases of Picard number one were studied in a series of papers by J.-M. Hwang and N. Mok. For higher Picard number cases, we notice that the Picard number of a rational homogeneous space G/P is less or equal to the rank of G. Recently A. Weber and J. A. Wisniewski proved that rational homogeneous spaces G/P with Picard numbers equal to the rank of G (i.e. complete flag manifolds) are rigid under Fano deformation. In this paper we show that the rational homogeneous space G/P is rigid under Fano deformation, providing that G is a simple algebraic group of type ADE, the Picard number equal to rank(G)-1 and G/P is not biholomorphic to F(1, 2, P^3) or F(1, 2, Q^6). The variety F(1, 2, P^3) is the set of flags of projective lines and planes in P^3, and F(1, 2, Q^6) is the set of flags of projective lines and planes in 6-dimensional smooth quadric hypersurface. We show that F(1, 2, P^3) have a unique Fano degeneration, which is explicitly constructed. The structure of possible Fano degeneration of F(1, 2, Q^6) is also described explicitly. To prove our rigidity result, we firstly show that the Fano deformation rigidity of a homogeneous space of type ADE can be implied by that property of suitable homogeneous submanifolds. Then we complete the proof via the study of Fano deformation rigidity of rational homogeneous spaces of small Picard numbers. As a byproduct, we also show the Fano deformation rigidity of other manifolds such as F(0, 1, ..., k_1, k_2, k_2+1, ..., n-1, P^n) and F(0, 1, ..., k_1, k_2, k_2+1, ..., n, Q^{2n+2}) with 0, 39 pages, Comments are welcome!

In this paper, we introduce the concept of the Tutte polynomials of genus $g$ and discuss some of its properties. We note that the Tutte polynomials of genus one are well-known Tutte polynomials. The Tutte polynomials are matroid invariants, and we claim that the Tutte polynomials of genus $g$ are also matroid invariants. The main result of this paper and the forthcoming paper declares that the Tutte polynomials of genus $g$ are complete matroid invariants., 7 pages

Staircase tableaux are combinatorial objects that were first introduced due to a connection with the asymmetric simple exclusion process (ASEP) and Askey-Wilson polynomials. Since their introduction, staircase tableaux have been the object of study in many recent papers. Relevant to this paper, Hitczenko and Janson proved that distribution of parameters on the first diagonal is asymptotically normal. In addition, they conjectured that other diagonals would be asymptotically Poisson. Since then, only the second and the third diagonal were proven to follow the conjecture. This paper builds upon those results to prove the conjecture for the kth diagonal where k is fixed. In particular, we prove that the distribution of the number of α's (β's) on the kth diagonal, k > 1, is asymptotically Poisson with parameter 1/2. In addition, we prove that symbols on the kth diagonal are asymptotically independent and thus, collectively follow the Poisson distribution with parameter 1. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 795–818, 2016

In this paper we will show that certain universal homology classes which are fundamental in topology are algebraic. To be specific, the products of Eilenberg–MacLane spaces \({\mathcal K}_{2q}\equiv K(\mathbf Z,{2}) \times K(\mathbf Z,{4}) \times \cdots \times K(\mathbf Z,{2q}) \) have models which are limits of complex projective varieties. Precisely, we have \({\mathcal K}_{2q}= \varinjlim \nolimits _{d\rightarrow \infty }\mathcal C^{q}_{d}(\mathbf P^{n})\) where \(\mathcal C^{q}_{d}(\mathbf P^{n})\) denotes the Chow variety of effective cycles of codimension q and degree d on \(\mathbf P_{\mathbf C}^{n}\). It is natural to ask which elements in the homology of \({\mathcal K}_{2q}\) are represented by algebraic cycles in these approximations. In this paper we find such representations for the even dimensional classes which are known as Steenrod squares (as well as their Pontrjagin and join products). These classes are dual to the cohomology classes which correspond to the basic cohomology operations also known as the Steenrod squares.

In a recent paper, Duncan, Griffin and Ono provide exact formulas for the coefficients of McKay-Thompson series and use them to find asymptotic expressions for the distribution of irreducible representations in the moonshine module V ♮ = ⨁ n V n ♮ V^\natural = \bigoplus _n V_n^\natural . Their results show that as n n tends to infinity, V n ♮ V_n^\natural is dominated by direct sums of copies of the regular representation. That is, if we view V n ♮ V_n^\natural as a module over the group ring Z [ M ] \mathbb {Z}[\mathbb {M}] , the free part dominates. A natural problem, posed at the end of the aforementioned paper, is to characterize the distribution of irreducible representations in the non-free part. Here, we study asymptotic formulas for the coefficients of McKay-Thompson series to answer this question. We arrive at an ordering of the series by the magnitude of their coefficients, which corresponds to various contributions to the distribution. In particular, we show how the asymptotic distribution of the non-free part is dictated by the column for conjugacy class 2A in the monster’s character table. We find analogous results for the other monster modules V ( − m ) V^{(-m)} and W ♮ W^\natural studied by Duncan, Griffin, and Ono.

In this paper, we obtain a class of irreducible Virasoro modules by taking tensor products of the irreducible Virasoro modules $\Omega(\lambda,b)$ defined in [LZ], with irreducible highest weight modules $V(\theta,h)$ or with irreducible Virasoro modules Ind$_{\theta}(N)$ defined in [MZ2]. We determine the necessary and sufficient conditions for two such irreducible tensor products to be isomorphic. Then we prove that the tensor product of $\Omega(\lambda,b)$ with a classical Whittaker module is isomorphic to the module $\mathrm{Ind}_{\theta,\lambda}(\mathbb{C_\mathbf{m}})$ defined in [MW]. As a by-product we obtain the necessary and sufficient conditions for the module $\mathrm{Ind}_{\theta, \lambda}(\mathbb{C_\mathbf{m}})$ to be irreducible. We also generalize the module $\mathrm{Ind}_{\theta, \lambda}(\mathbb{C_\mathbf{m}})$ to $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ for any non-negative integer $ n$ and use the above results to completely determine when the modules $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ are irreducible. The submodules of $\mathrm{Ind}_{\theta,\lambda}(\mathcal{B}^{(n)}_{\mathbf{s}})$ are studied and an open problem in [GLZ] is solved. Feigin-Fuchs' Theorem on singular vectors of Verma modules over the Virasoro algebra is crucial to our proofs in this paper., Comment: 17 Pages

In a series of papers recently "checkerboard discrepancy" has been introduced, where a black-and-white checkerboard background induces a coloring on any curve, and thus a discrepancy, i.e., the difference of the length of the curve colored white and the length colored black. Mainly straight lines and circles have been studied and the general situation is that, no matter what the background coloring, there is always a curve in the family studied whose discrepancy is at least of the order of the square root of the length of the curve. In this paper we generalize the shape of the background, keeping the lattice structure. Our background now consists of lattice copies of any bounded fundamental domain of the lattice, and not necessarily of squares, as was the case in the previous papers. As the decay properties of the Fourier Transform of the indicator function of the square were strongly used before, we now have to use a quite different proof, in which the tiling and spectral properties of the fundamental domain play a role., Comment: 10 pages, 3 figures

In our previous paper, given an integral log smooth morphism $X\to S$ of fine log schemes of characteristic $p>0$, we studied the Azumaya nature of the sheaf of log differential operators of higher level and constructed a splitting module of it under an existence of a certain lifting modulo $p^{2}$. In this paper, under a certain liftability assumption which is stronger than our previous paper, we construct another splitting module of our Azumaya algebra over a scalar extension, which is smaller than our previous paper. As an application, we construct an equivalence, which we call the log local Cartier transform of higher level, between certain $\cal D$-modules and certain Higgs modules. We also discuss about the compatibility of the log Frobenius descent and the log local Cartier transform and the relation between the splitting module constructed in this paper and that constructed in the previous paper. Our result can be considered as a generalization of the result of Ogus-Vologodsky, Gros-Le Stum-Quir��s to the case of log schemes and that of Schepler to the case of higher level.

In his seminal paper, Schoenberg (1942) characterized the class P(S^d) of continuous functions f:[-1,1] \to \R such that f(\cos \theta) is positive definite over the product space S^d \times S^d, with S^d being the unit sphere of \R^{d+1} and \theta being the great circle distance. In this paper, we consider the product space S^d \times G, for G a locally compact group, and define the class P(S^d, G) of continuous functions f:[-1,1]\times G \to \C such that f(\cos \theta, u^{-1}\cdot v) is positive definite on S^d \times S^d \times G \times G. This offers a natural extension of Schoenberg's Theorem. Schoenberg's second theorem corresponding to the Hilbert sphere S^\infty is also extended to this context. The case G=\R is of special importance for probability theory and stochastic processes, because it characterizes completely the class of space-time covariance functions where the space is the sphere, being an approximation of Planet Earth., Comment: 26 pages

For every genus $g$, we prove that $S^2 \times R$ contains complete, properly embedded, genus-$g$ minimal surfaces whose two ends are asymptotic to helicoids of any prescribed pitch. We also show that as the radius of the $S^2$ tends to infinity, these examples converge smoothly to complete, properly embedded minimal surfaces in $R^3$ that are helicoidal at infinity. We prove that helicoidal surfaces in $R^3$ of every prescribed genus occur as such limits of examples in $S^2\times R$., Comment: 87 pages. This paper combines and supersedes our earlier two papers, Helicoidal minimal surfaces of prescribed genus, I and II [arXiv:1304.5861 and arXiv:1304.6180]. This version (16 November, 2016) fixes a few typos

In this paper, lower bounds on the probability of a finite union of events are considered, i.e. $P\left(\bigcup_{i=1}^N A_i\right)$, in terms of the individual event probabilities $\{P(A_i), i=1,\ldots,N\}$ and the sums of the pairwise event probabilities, i.e., $\{\sum_{j:j\neq i} P(A_i\cap A_j), i=1,\ldots,N\}$. The contribution of this paper includes the following: (i) in the class of all lower bounds that are established in terms of only the $P(A_i)$'s and $\sum_{j:j\neq i} P(A_i\cap A_j)$'s, the optimal lower bound is given numerically by solving a linear programming (LP) problem with $N^2-N+1$ variables; (ii) a new analytical lower bound is proposed based on a relaxed LP problem, which is at least as good as the bound due to Kuai, et al.; (iii) numerical examples are provided to illustrate the performance of the bounds., Comment: SIAM Journal on Discrete Mathematics, 2016

To provide a solid analytic foundation for the module approach to conditional risk measures, our purpose is to establish a complete random convex analysis over random locally convex modules by simultaneously considering the two kinds of topologies (namely the $(\varepsilon,\lambda)$--topology and the locally $L^0$-- convex topology). This paper is focused on the part of separation and Fenchel-Moreau duality in random locally convex modules. The key point of this paper is to give the precise relation between random conjugate spaces of a random locally convex module under the two kinds of topologies, which enables us to not only give a thorough treatment of separation between a point and a closed $L^{0}$-convex subset but also establish the complete Fenchel-Moreau duality theorems in random locally convex modules under the two kinds of topologies., Comment: 26 pages; this article draws heavily from arXiv:1210.1848v6. arXiv admin note: text overlap with arXiv:1503.08637

A subgroup $H$ of a group $G$ is said to be {pronormal} in $G$ if $H$ and $H^g$ are conjugate in $\langle H, H^g \rangle$ for every $g \in G$. Some problems in finite group theory, combinatorics, and permutation group theory were solved in terms of pronormality. In 2012, E. Vdovin and the third author conjectured that the subgroups of odd index are pronormal in finite simple groups. In this paper we disprove their conjecture and discuss a recent progress in the classification of finite simple groups in which the subgroups of odd index are pronormal., Comment: This is a survey paper for Proceedings of Groups St Andrews 2017, 12 pages

Markov numbers are integers that appear in the solution triples of the Diophantine equation, $x^2+y^2+z^2=3xyz$, called the Markov equation. A classical topic in number theory, these numbers are related to many areas of mathematics such as combinatorics, hyperbolic geometry, approximation theory and cluster algebras. There is a natural map from the rational numbers between zero and one to the Markov numbers. In this paper, we prove two conjectures seen in Martin Aigner's book, Markov's theorem and 100 years of the uniqueness conjecture, that determine an ordering on subsets of the Markov numbers based on their corresponding rational. The proof relies on a relationship between Markov numbers and continuant polynomials which originates in Frobenius' 1913 paper., 16 pages

Let $X \subseteq \mathbb{P}^r$ be a non-degenerate smooth projective variety of dimension $n$, codimension $e$, and degree $d$ defined over an algebraically closed field of characteristic zero. In this paper, we first show that $\text{reg} (\mathcal{O}_X) \leq d-e$, and classify the extremal and the next to extremal cases. Our result reduces the Eisenbud-Goto regularity conjecture for the smooth case to the problem finding a Castelnuovo-type bound for normality. It is worth noting that McCullough-Peeva recently constructed counterexamples to the regularity conjecture by showing that $\text{reg} (\mathcal{O}_X)$ is not even bounded above by any polynomial function of $d$ when $X$ is not smooth. For a normality bound in the smooth case, we establish that $\text{reg}(X) \leq n(d-2)+1$, which improves previous results obtained by Mumford, Bertram-Ein-Lazarsfeld, and Noma. Finally, by generalizing Mumford's method on double point divisors, we prove that $\text{reg}(X) \leq d-1+m$, where $m$ is an invariant arising from double point divisors associated to outer general projections. Using double point divisors associated to inner projection, we also obtain a slightly better bound for $\text{reg}(X)$ under suitable assumptions., Comment: 23 pages. This paper has been largely rewritten after McCullough-Peeva's counterexamples to the Eisenbud-Goto regularity conjecture, which appeared in J. Amer. Math. Soc. in 2018. We also added new results on the regularity of smooth projective varieties of arbitrary dimension

A polynomial knot in $\mathbb{R}^n$ is a smooth embedding of $\mathbb{R}$ in $\mathbb{R}^n$ such that the component functions are real polynomials. In the earlier paper with Mishra, we have studied the space $\mathcal{P}$ of polynomial knots in $\mathbb{R}^3$ with the inductive limit topology coming from the spaces $\mathcal{O}_d$ for $d\geq3$, where $\mathcal{O}_d$ is the space of polynomial knots in $\mathbb{R}^3$ with degree $d$ and having some conditions on the degrees of the component polynomials. In the same paper, we have proved that the space of polynomial knots in $\mathbb{R}^3$ has the same homotopy type as $S^2$. The homotopy type of the space is the mere consequence of the topology chosen. If we have another topology on $\mathcal{P}$, the homotopy type may change. With this in mind, we consider in general the set $\mathcal{L}^n$ of polynomial knots in $\mathbb{R}^n$ with various topologies on it and study the homotopy type of the respective spaces. Let $\mathcal{L}$ be the union of the sets $\mathcal{L}^n$ for $n\geq1$. We also explore the homotopy type of the space $\mathcal{L}$ with some natural topologies on it., 15 pages

Fix an alphabet $A=\{0,1,\dots,M\}$ with $M\in\mathbb{N}$. The univoque set $\mathscr{U}$ of bases $q\in(1,M+1)$ in which the number $1$ has a unique expansion over the alphabet $A$ has been well studied. It has Lebesgue measure zero but Hausdorff dimension one. This paper investigates how the set $\mathscr{U}$ is distributed over the interval $(1,M+1)$ by determining the limit $$f(q):=\lim_{\delta\to 0}\dim_H\big(\mathscr{U}\cap(q-\delta,q+\delta)\big)$$ for all $q\in(1,M+1)$. We show in particular that $f(q)>0$ if and only if $q\in\overline{\mathscr{U}}\backslash\mathscr{C}$, where $\mathscr{C}$ is an uncountable set of Hausdorff dimension zero, and $f$ is continuous at those (and only those) points where it vanishes. Furthermore, we introduce a countable family of pairwise disjoint subsets of $\mathscr{U}$ called {\emph relative bifurcation sets}, and use them to give an explicit expression for the Hausdorff dimension of the intersection of $\mathscr{U}$ with any interval, answering a question of Kalle et al.~[{\emph arXiv:1612.07982; to appear in Acta Arithmetica}, 2018]. Finally, the methods developed in this paper are used to give a complete answer to a question of the first author [{\emph Adv. Math.}, 308:575--598, 2017] about strongly univoque sets., Comment: 31 pages and 1 figure

For a finite group $G$, let $\H_{g,G,\xi}$ be the stack of admissible $G$-covers $C\to D$ of stable curves with ramification data $\xi$, $g(C)=g$ and $g(D)=g'$. There are source and target morphisms $\phi\colon \H_{g,G,\xi}\to \M_{g,r}$ and $\delta\colon \H_{g,G,\xi}\to \M_{g',b}$, remembering the curves $C$ and $D$ together with the ramification or branch points of the cover respectively. In this paper we study admissible cover cycles, i.e. cycles of the form $\phi_* [\H_{g,G,\xi}]$. Examples include the fundamental classes of the loci of hyperelliptic or bielliptic curves $C$ with marked ramification points. The two main results of this paper are as follows: Firstly, for the gluing morphism $\xi_A\colon \M_A\to \M_{g,r}$ associated to to a stable graph $A$ we give a combinatorial formula for the pullback $\xi^*_A \phi_*[\H_{g,G,\xi}]$ in terms of spaces of admissible $G$-covers and $\psi$ classes. This allows us to describe the intersection of the cycles $\phi_*[\H_{g,G,\xi}]$ with tautological classes. Secondly, the pull-push $\delta_*\phi^*$ sends tautological classes to tautological classes and we also give a combinatorial description of this map in terms of standard generators of the tautological rings. We show how to use the pullbacks to algorithmically compute tautological expressions for cycles of the form $\phi_* [\H_{g,G,\xi}]$. In particular, we compute the classes $[\Hyp_5]$ and $[\Hyp_6]$ of the hyperelliptic loci in $\M_5$ and $\M_6$ and the class $[\B_4]$ of the bielliptic locus in $\M_4$., Comment: 69 pages, comments very welcome

In this paper, we will show that the sheaf of p-adic vanishing cycles of a log smooth family over a DVR of mixed characteristic is generated by Milnor symboles. A key ingredient is a computation (due to K. Kato) on the graded quotients of a multi-indexed filtration on the sheaf concerned, which has been used in several papers of the first author., 26 pages. Setions 5 and 6 are reorganized, and typos have been fixed

We study stabilization of finite-dimensional representations of the periplectic Lie superalgebras $\mathfrak{p}(n)$ as $n \to \infty$. The paper gives a construction of the tensor category $Rep(\underline{P})$, possessing nice universal properties among tensor categories over the category $\mathtt{sVect}$ of finite-dimensional complex vector superspaces. First, it is the "abelian envelope" of the Deligne category corresponding to the periplectic Lie superalgebra, in the sense of arXiv:1511.07699. Secondly, given a tensor category $\mathcal{C}$ over $\mathtt{sVect}$, exact tensor functors $Rep(\underline{P})\longrightarrow \mathcal{C}$ classify pairs $(X, \omega)$ in $\mathcal{C}$ where $\omega: X \otimes X \to \Pi \mathbf{1}$ is a non-degenerate symmetric form and $X$ not annihilated by any Schur functor. The category $Rep(\underline{P})$ is constructed in two ways. The first construction is through an explicit limit of the tensor categories $Rep(\mathfrak{p}(n))$ ($n\geq 1$) under Duflo-Serganova functors. The second construction (inspired by P. Etingof) describes $Rep(\underline{P})$ as the category of representations of a periplectic Lie supergroup in the Deligne category $\mathtt{sVect} \boxtimes Rep(\underline{GL}_t)$. An upcoming paper by the authors will give results on the abelian and tensor structure of $Rep(\underline{P})$.

Let $N$ (resp., $U$) be a manifold (resp., an open subset of $\mathbb{R}^m$). Let $f:N\to U$ and $F:U\to \mathbb{R}^\ell$ be an immersion and a $C^{\infty}$ mapping, respectively. Generally, the composition $F\circ f$ does not necessarily yield a mapping transverse to a given subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$. Nevertheless, in this paper, for any $\mathcal{A}^1$-invariant fiber, we show that composing generic linearly perturbed mappings of $F$ and the given immersion $f$ yields a mapping transverse to the subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$ with the given fiber. Moreover, we show a specialized transversality theorem on crossings of compositions of generic linearly perturbed mappings of a given mapping $F:U\to \mathbb{R}^\ell$ and a given injection $f:N\to U$. Furthermore, applications of the two main theorems are given., 19 pages. Accepted for publication in Journal of the Mathematical Society of Japan. The paper is a generalization of arXiv:1610.02880. As the appendix, the statement of the main theorem in arXiv:1607.03220 (the proof of it is in arXiv:1607.03220) is introduced in the final section of this paper

In his influential 1947 paper that inaugurated the probabilistic method, Erdős proved the existence of 2logn-Ramsey graphs on n vertices. Matching Erdős’ result with a constructive proof is considered a central problem in combinatorics, that has gained a significant attention in the literature. The state of the art result was obtained in the celebrated paper by Barak, Rao, Shaltiel, and Wigderson who constructed a 22(loglogn)1−α-Ramsey graph, for some small universal constant α > 0. In this work, we significantly improve this result and construct 2(loglogn)c-Ramsey graphs, for some universal constant c. In the language of theoretical computer science, this resolves the problem of explicitly constructing dispersers for two n-bit sources with entropy (n). In fact, our disperser is a zero-error disperser that outputs a constant fraction of the entropy. Prior to this work, such dispersers could only support entropy Ω(n).

This paper answers a basic question about the Birman exact sequence in the theory of mapping class groups. We prove that the Birman exact sequence does not admit a section over any subgroup $\Gamma$ contained in the Torelli group with finite index. A fortiori this proves that there is no section of the Birman exact sequence for any finite-index subgroup of the full mapping class group. This theorem was announced in a 1990 preprint of G. Mess, but an error was uncovered and described in a recent paper of the first author., Comment: 17 pages, one figure

In previous work, joint with Bux, Fluch, Marschler and Witzel, we proved that the braided Thompson groups are of type $\textrm{F}_\infty$. The proof utilized certain contractible cube complexes, which in this paper we prove are CAT(0). We then use this fact to compute the geometric invariants $\Sigma^m(F_{\textrm{br}})$ of the pure braided Thompson group $F_{\textrm{br}}$. Only the first invariant $\Sigma^1(F_{\textrm{br}})$ was previously known. A consequence of our computation is that as soon as a subgroup of $F_{\textrm{br}}$ containing the commutator subgroup $[F_{\textrm{br}},F_{\textrm{br}}]$ is finitely presented, it is automatically of type $\textrm{F}_\infty$., Comment: 22 pages, 3 figures. v2: Removed Subsection 4.3, after a mistake came to light. This subsection was unrelated to the rest of the paper, so nothing else changed in any important ways

In a recent paper, Oliver Riordan shows that for $r \ge 4$ and $p$ up to and slightly larger than the threshold for a $K_r$-factor, the hypergraph formed by the copies of $K_r$ in $G(n,p)$ contains a copy of the binomial random hypergraph $H=H_r(n,\pi)$ with $\pi \sim p^{r \choose 2}$. For $r=3$, he gives a slightly weaker result where the density in the random hypergraph is reduced by a constant factor. Recently, Jeff Kahn announced an asymptotically sharp bound for the threshold in Shamir's hypergraph matching problem for all $r \ge 3$. With Riordan's result, this immediately implies an asymptotically sharp bound for the threshold of a $K_r$-factor in $G(n,p)$ for $r \ge 4$. In this note, we resolve the missing case $r=3$ by modifying Riordan's argument. This means that Kahn's result also implies a sharp bound for triangle factors in $G(n,p)$., Comment: 4 pages; minor corrections, and updated references to new version of Oliver Riordan's paper

In a paper by E. Dieterich 2017, the category $\mathscr{C}(k)$ of four-dimensional unital division algebras, whose right nucleus is non-trivial and whose automorphism group contains Klein's four group $V$, is studied over a general ground field $k$ with $\mathrm{char}\,k\neq 2$. In particular, the objects in $\mathscr{C}(k)$ are exhaustively constructed from parameters in $k^3$ and explicit isomorphism conditions for the constructed objects are found in terms of these parameters. In this paper, we specialize to the case $k=\mathbb{Q}$ and present results towards a classification of $\mathscr{C}(\mathbb{Q})$. In particular, for each field $\ell$ with $[\ell:k]=2$ we present explicity a two-parameter family of pairwise non-isomorphic non-associative objects in $\mathscr{C}(\mathbb{Q})$ that admit $\ell$ as a subfield and we provide a method for classifying the full subcategory of central skew fields admitting $\ell$ as a subfield and $kV$-submodule. We also classify the subcategory of $\mathscr{C}(\mathbb{Q})$ of all four-dimensional Galois extensions of $\mathbb{Q}$ with Galois group $V$ that admit $\ell$ as a subfield.

Extensions of the notions of polynomially and rationally convex hulls are introduced. Using these notions, a generalization of a result of Duval and Levenberg on polynomially convex hulls containing no analytic discs is presented. As a consequence it is shown that there exists a Cantor set $X$ in ${\mathbb C}^3$ with a nontrivial polynomially convex hull that contains no analytic discs. Using this Cantor set, it is shown that there exist arcs and curves in ${\mathbb C}^4$ with nontrivial polynomially convex hulls that contain no analytic discs. This answers a question raised a few years ago by Bercovici and can be regarded as a partial answer to a question raised by Wermer over 60 years ago. More generally, it is shown that every uncountable, compact subspace of a Euclidean space can be embedded as a subspace $X$ of ${\mathbb C}^N$, for some N, in such a way as to have a nontrivial polynomially convex hull that contains no analytic discs. In the case when the topological dimension of the space is at most one, $X$ can be chosen so as to have the stronger property that $P(X)$ has a dense set of invertible elements., Several revisions have been made to make the paper more succinct and focused. In particular, an entire section has been removed and made into a separate paper with the title "A doubly generated uniform algebra with a one-point Gleason part off its Shilov boundary"

The standard quadratic optimization problem (StQP), i.e. the problem of minimizing a quadratic form $$\mathbf{x}^TQ\mathbf{x}$$ on the standard simplex $$\{\mathbf{x}\ge \mathbf{0}: \mathbf{x}^T\mathbf{e}=1\}$$ , is studied. The StQP arises in numerous applications, and it is known to be NP-hard. Chen, Peng and Zhang showed that almost certainly the StQP with a large random matrix $$Q=Q^T$$ , $$Q_{i,j},\, (i\le j)$$ being independent and identically concave-distributed, attains its minimum at a point $$\mathbf{x}$$ with support size $$|\{j: x_j>0\}|$$ bounded in probability. Later Chen and Peng proved that for $$Q=(M+M^T)/2$$ , with $$M_{i,j}$$ i.i.d. normal, the likely support size is at most 2. In this paper we show that the likely support size is poly-logarithmic in n, the problem size, for a considerably broader class of the distributions. Unlike the cited papers, the mild constraints are put on the asymptotic behavior of the distribution at a single left endpoint of its support, rather than on the distribution’s shape elsewhere. It also covers the distributions with the left endpoint $$-\infty $$ , provided that the distribution of $$Q_{i,j},\, (i\le j)$$ (of $$M_{i,j}$$ , if $$Q=(M+M^T)/2$$ resp.) has a (super/sub) exponentially narrow left tail.

In the present paper, which is a direct sequel of our paper [12] joint with Roozbeh Hazrat, we prove unrelativised version of the standard commutator formula in the setting of Chevalley groups. Namely, let $\Phi$ be a reduced irreducible root system of rank $\ge 2$, let $R$ be a commutative ring and let $I,J$ be two ideals of $R$. We consider subgroups of the Chevalley group $G(\Phi,R)$ of type $\Phi$ over $R$. The unrelativised elementary subgroup $E(\Phi,I)$ of level $I$ is generated (as a group) by the elementary unipotents $x_{\alpha}(\xi)$, $\alpha\in\Phi$, $\xi\in I$, of level $I$. Obviously, in general $E(\Phi,I)$ has no chances to be normal in $E(\Phi,R)$, its normal closure in the absolute elementary subgroup $E(\Phi,R)$ is denoted by $E(\Phi,R,I)$. The main results of [12] implied that the commutator $\big[E(\Phi,I),E(\Phi,J)]$ is in fact normal in $E(\Phi,R)$. In the present paper we prove an unexpected result that in fact $\big[E(\Phi,I),E(\Phi,J)]=\big[E(\Phi,R,I),E(\Phi,R,J)\big]$. It follows that the standard commutator formula also holds in the unrelativised form, namely $\big[E(\Phi,I),C(\Phi,R,J)]=\big[E(\Phi,I),E(\Phi,J)\big]$, where $C(\Phi,R,I)$ is the full congruence subgroup of level $I$. In particular, $E(\Phi,I)$ is normal in $C(\Phi,R,I)$., Comment: 14 Pages

In this paper, we study the derivatives of an integer-valued polynomial of a given degree. Denoting by $E_n$ the set of the integer-valued polynomials with degree $\leq n$, we show that the smallest positive integer $c_n$ satisfying the property: $\forall P \in E_n, c_n P' \in E_n$ is $c_n = \mathrm{lcm}(1 , 2 , \dots , n)$. As an application, we deduce an easy proof of the well-known inequality $\mathrm{lcm}(1 , 2 , \dots , n) \geq 2^{n - 1}$ ($\forall n \geq 1$). In the second part of the paper, we generalize our result for the derivative of a given order $k$ and then we give two divisibility properties for the obtained numbers $c_{n , k}$ (generalizing the $c_n$'s). Leaning on this study, we conclude the paper by determining, for a given natural number $n$, the smallest positive integer $\lambda_n$ satisfying the property: $\forall P \in E_n$, $\forall k \in \mathbb{N}$: $\lambda_n P^{(k)} \in E_n$. In particular, we show that: $\lambda_n = \prod_{p \text{ prime}} p^{\lfloor\frac{n}{p}\rfloor}$ ($\forall n \in \mathbb{N}$)., Comment: 17 pages

We investigate generalised polynomials (i.e. polynomial-like expressions involving the use of the floor function) which take the value $0$ on all integers except for a set of density $0$. Our main result is that the set of integers where a sparse generalised polynomial takes non-zero value cannot contain a translate of an IP set. We also study some explicit constructions, and show that the characteristic functions of the Fibonacci and Tribonacci numbers are given by generalised polynomails. Finally, we show that any sufficiently sparse $\{0,1\}$-valued sequence is given by a generalised polynomial. (This paper is essentially the first half of our earlier submission arXiv:1610.03900 [math.NT]. Because the material in arXiv:1610.03900 [math.NT] touches upon many different subjects, we believe it is preferable to split it into two independent papers.), 28 pages. arXiv admin note: substantial text overlap with arXiv:1610.03900

Let $n$ be an integer$\geqq0$. Let $S^{n+2}_1$ (respectively, $S^{n+2}_2$) be the $(n+2)$-sphere embedded in the $(n+4)$-sphere $S^{n+4}$. Let $S^{n+2}_1$ and $S^{n+2}_2$ intersect transversely. Suppose that the smooth submanifold, $S^{n+2}_1 \cap S^{n+2}_2$ in $S^{n+2}_i$ is PL homeomophic to the $n$-sphere. Then $S^{n+2}_1$ and $S^{n+2}_2$ in $S^{n+2}_i$ is an $n$-knot $K_i$. We say that the pair $(K_1,K_2)$ of n-knots is realizable. We consider the following problem in this paper. Let $A_1$ and $A_2$ be n-knots. Is the pair $(A_1,A_2)$ of $n$-knots realizable? We give a complete characterization., Comment: 22 pages, 1 figure,Chapter I: Mathematical Research Letters, 1998, 5, 577-582. Chapter II: University of Tokyo preprint series UTMS 95-50. This paper is beased on the author's master thesis 1994, and his PhD thesis 1996

Suppose that $n\geq 1$ and that, for all $i$ and $j$ with $1\leq i,j\leq n$ and $i\neq j$, $z_{ij}\in{\mathbb T}$ are given such that $z_{ji}=\overline{z}_{ij}$ for all $i\neq j$. If $V_1,\dotsc, V_n$ are isometries on a Hilbert space such that $V_i^\ast V_j^{\phantom{\ast}}\!=\overline{z}_{ij} V_j^{\phantom{\ast}}\!V_i^\ast$ for all $i\neq j$, then $(V_1,\dotsc,V_n)$ is called an $n$-tuple of doubly non-commuting isometries. The generators of non-commutative tori are well-known examples. In this paper, we establish a simultaneous Wold decomposition for $(V_1,\dotsc,V_n)$. This decomposition enables us to classify such $n$-tuples up to unitary equivalence. We show that the joint listing of a unitary equivalence class of a representation of each of the $2^n$ non-commutative tori that are naturally associated with the structure constants is a classifying invariant. A dilation theorem is also established, showing that an $n$-tuple of doubly non-commuting isometries can be extended to an $n$-tuple of doubly non-commuting unitary operators on an enveloping Hilbert space., Comment: A remark on the relation between the dilation theorem in this paper and other dilation theorems for multiple operators in the literature has been added. Otherwise, there are only a few minor editorial changes compared to the first version; some typos have also been corrected. Final version, to appear in Advances in Mathematics

In this paper, we enumerate two families of pattern-avoiding permutations: those avoiding the vincular pattern $2-41-3$, which we call semi-Baxter permutations, and those avoiding the vincular patterns $2-41-3$, $3-14-2$ and $3-41-2$, which we call strong-Baxter permutations. We call semi-Baxter numbers and strong-Baxter numbers the associated enumeration sequences. We prove that the semi-Baxter numbers enumerate in addition plane permutations (avoiding $2-14-3$). The problem of counting these permutations was open and has given rise to several conjectures, which we also prove in this paper. For each family (that of semi-Baxter -- or equivalently, plane -- and that of strong-Baxter permutations), we describe a generating tree, which translates into a functional equation for the generating function. For semi-Baxter permutations, it is solved using (a variant of) the kernel method: this gives an expression for the generating function while also proving its D-finiteness. From the obtained generating function, we derive closed formulas for the semi-Baxter numbers, a recurrence that they satisfy, as well as their asymptotic behavior. For strong-Baxter permutations, we show that their generating function is (a slight modification of) that of a family of walks in the quarter plane, which is known to be non D-finite., Comment: Version 3 incorporates changes suggested by a referee. Most important changes are that the paths sections have been removed and that the proof of the asymptotic equivalent has been simplified

Every graph $G$ can be represented by a collection of equi-radii spheres in a $d$-dimensional metric $\Delta$ such that there is an edge $uv$ in $G$ if and only if the spheres corresponding to $u$ and $v$ intersect. The smallest integer $d$ such that $G$ can be represented by a collection of spheres (all of the same radius) in $\Delta$ is called the sphericity of $G$, and if the collection of spheres are non-overlapping, then the value $d$ is called the contact-dimension of $G$. In this paper, we study the sphericity and contact dimension of the complete bipartite graph $K_{n,n}$ in various $L^p$-metrics and consequently connect the complexity of the monochromatic closest pair and bichromatic closest pair problems., Comment: The paper was previously titled, "The Curse of Medium Dimension for Geometric Problems in Almost Every Norm"

[EN] While the classic separable quotient problem remains open, we survey general results related to this problem and examine the existence of infinite-dimensional separable quotients in some Banach spaces of vector-valued functions, linear operators and vector measures. Most of the presented results are consequences of known facts, some of them relative to the presence of complemented copies of the classic sequence spaces c_0 and lp. Also recent results of Argyros, Dodos, Kanellopoulos [see 1 in the paper] and Sliwa [see 64 in the paper] are provided. This makes our presentation supplementary to a previous survey (1997) due to Mujica., The first three named authors were supported by Grant PROMETEO/2013/058 of the Conselleria d'Educacio, Investigacio, Cultura i Esport of Generalitat Valenciana. The second named author was also supported by GACR Project 16-34860L and RVO: 67985840.

The main object in this paper is a certain rational convex polytope whose lattice points give a polyhedral realization of a highest weight crystal basis. This is also identical to a Newton-Okounkov body of a flag variety, and it gives a toric degeneration. In this paper, we prove that a specific class of this polytope is given by Kiritchenko's Demazure operators on polytopes. This implies that polytopes in this class are all lattice polytopes. As an application, we give a sufficient condition for the corresponding toric variety to be Gorenstein Fano., Comment: 21 pages

In a number of papers by Edge, and in a related paper by Fano, several properties are discussed of the family of scrolls of degree 8, in the complex projective space, whose plane sections are projected bicanonical models of a genus 3 curve C. This beautiful subject is implicitely related to the moduli of semistable rank two vector bundles on C with bicanonical determinant. We revisit and reconstruct this matter in modern terms with a modular point of view. This paper is the refereed version of a contribution to a volume in honor of W.L Edge, to be published by European Journal of Mathematics., Comment: 13 pages

In \cite[Problem 72]{Fuchs60} Fuchs posed the problem of characterizing the groups which are the groups of units of commutative rings. In the following years, some partial answers have been given to this question in particular cases. In a previous paper \cite{DDcharp} we dealt with finite characteristic rings. In this paper we consider Fuchs' question for finite groups and we address this problem in two cases. Firstly, we study the case of torson-free rings and we obtain a complete classification of the finite groups of units which arise in this case. Secondly, we examine the case of characteristic zero rings obtaining, a pretty good description of the possible groups of units equipped with families of examples of both realizable and non-realizable groups. The main tools to deal with this general case are the Pearson and Schneider splitting of a ring \cite{PearsonSchneider70}, our previous results on finite characteristic rings \cite{DDcharp} and our classification of the groups of units of torsion-free rings. As a consequence of our results we completely answer Ditor's question \cite{ditor} on the possible cardinalities of the group of units of a ring., arXiv admin note: substantial text overlap with arXiv:1607.00965

The main inspiration for this paper is a paper by Elek where he introduces combinatorial cost for graph sequences. We show that having cost equal to 1 and hyperfiniteness are coarse invariants. We also show `cost-1' for box spaces behaves multiplicatively when taking subgroups. We show that graph sequences coming from Farber sequences of a group have property A if and only if the group is amenable. The same is true for hyperfiniteness. This generalises a theorem by Elek. Furthermore we optimise this result when Farber sequences are replaced by sofic approximations. In doing so we introduce a new concept: property almost-A., 20 pages. Comments welcome

Let R be an integral domain and I a nonzero ideal of R. A sub-ideal J of I is a t-reduction of I if (JI^{n})_{t}=(I^{n+1})_{t} for some positive integer n. An element x in R is t-integral over I if there is an equation x^{n} + a_{1}x^{n-1} +...+ a_{n-1}x + a_{n} = 0 with a_{i} in (I^{i})_{t} for I = 1,...,n. The set of all elements that are t-integral over I is called the t-integral closure of I. This paper investigates the t-reductions and t-integral closure of ideals. Our objective is to establish satisfactory t-analogues of well-known results, in the literature, on the integral closure of ideals and its correlation with reductions. Namely, Section 2 identifies basic properties of t-reductions of ideals and features explicit examples discriminating between the notions of reduction and t-reduction. Section 3 investigates the concept of t-integral closure of ideals, including its correlation with t-reductions. Section 4 studies the persistence and contraction of t-integral closure of ideals under ring homomorphisms. All along the paper, the main results are illustrated with original examples., 13 pages. Rocky Mountain Journal of Mathematics 2016