Search

A classical result of K. L. Chung and W. Feller deals with the partial sums S k S_k arising in a fair coin-tossing game. If N n N_n is the number of “positive” terms among S 1 S_1 , S 2 S_2 , …, S n S_n then the quantity P ( N 2 n = 2 r ) P(N_{2n} = 2r) takes an elegant form. We lift the restriction on an even number of tosses and give a simple expression for P ( N 2 n + 1 = r ) P(N_{2n+1} = r) , r = 0 r = 0 , 1 1 , 2 2 , …, 2 n + 1 2n+1 . We get to this ansatz by adaptating the Feynman–Kac methodology.

We show that a relative entropy condition recently shown by Leger and Vasseur to imply uniqueness and stable $L^2$ dependence on initial data of Lax 1- or $n$-shock solutions of an $n\times n$ system of hyperbolic conservation laws with convex entropy implies Lopatinski stability in the sense of Majda. This means in particular that Leger and Vasseur's relative entropy condition represents a considerable improvement over the standard entropy condition of decreasing shock strength and increasing entropy along forward Hugoniot curves, which, in a recent example exhibited by Barker, Freist\"uhler and Zumbrun, was shown to fail to imply Lopatinski stability, even for systems with convex entropy. This observation bears also on the parallel question of existence, at least for small $BV$ or $H^s$ perturbations, Comment: to appear in Proceedings of the AMS

In this paper we give an elementary proof of the proper homotopy invariance of the equivariant stable homotopy type of the configuration space $F(M,k)$ for a topological manifold $M$. Our technique is to compute the Spanier-Whitehead dual of $\Sigma^\infty_+ F(M,k)$ and use the results of Spivak and Wall on normal spherical fibrations to deduce that the Spanier-Whitehead dual is a proper homotopy invariant. This stable invariance was recently proved by Knudsen using factorization homology. Aside from being elementary, our proof has the advantage that it readily extends to ``generalized configuration spaces'' which have recently undergone study., Comment: Minor changes to structure of paper; to appear in "Proceedings of the American Mathematical Society"

The main result of this paper is that, under PFA, for every regular space X X with F ( X ) = ω F(X) = \omega we have | X | ≤ w ( X ) ω |X| \le w(X)^\omega ; in particular, w ( X ) ≤ c w(X) \le \mathfrak {c} implies | X | ≤ c |X| \le \mathfrak {c} . This complements numerous prior results that yield consistent examples of even compact Hausdorff spaces X X with F ( X ) = ω F(X) = \omega such that w ( X ) = c w(X) = \mathfrak {c} and | X | = 2 c |X| = 2^\mathfrak {c} . We also show that regularity cannot be weakened to the Hausdorff property in this result because we can find in ZFC a Hausdorff space X X with F ( X ) = ω F(X) = \omega such that w ( X ) = c w(X) = \mathfrak {c} and | X | = 2 c |X| = 2^\mathfrak {c} . In fact, this space X X has the strongly anti-Urysohn (SAU) property that any two infinite closed sets in X X intersect, which is much stronger than F ( X ) = ω F(X) = \omega . Moreover, any non-empty open set in X X also has size 2 c 2^\mathfrak {c} , and thus our example answers one of the main problems of both Juhász, Soukup, and Szentmiklóssy [Topology Appl. 213 (2016), pp. 8–23] and Juhász, Shelah, Soukup, and Szentmiklóssy [Topology Appl. 323 (2023), Paper No. 108288, 15 pp.] by providing in ZFC a SAU space with no isolated points.

The paper is devoted to coverings by translative homothets and illuminations of convex bodies. For a given positive number $\alpha$ and a convex body $B$, $g_{\alpha}(B)$ is the infimum of $\alpha$-powers of finitely many homothety coefficients less than 1 such that there is a covering of $B$ by translative homothets with these coefficients. $h_{\alpha}(B)$ is the minimal number of directions such that the boundary of $B$ can be illuminated by this number of directions except for a subset whose Hausdorff dimension is less than $\alpha$. In this paper, we prove that $g_{\alpha}(B)\leq h_{\alpha}(B)$, find upper and lower bounds for both numbers, and discuss several general conjectures. In particular, we show that $h_{\alpha} (B) > 2^{d-\alpha}$ for almost all $\alpha$ and $d$ when $B$ is the $d$-dimensional cube, thus disproving the conjecture from Research Problems in Discrete Geometry by Brass, Moser, and Pach.

We give a decomposition of the equivariant Kasparov category for discrete quantum group with torsions. As an outcome, we show that the crossed product by a discrete quantum group in a certain class preserves the UCT. We then show that quasidiagonality of a reduced C*-algebra of a countable discrete quantum group $\Gamma$ implies that $\Gamma$ is amenable, and deduce from the work of Tikuisis, White and Winter, and the results in the first part of the paper, the converse (i.e. the quantum Rosenberg Conjecture) for a large class of countable discrete unimodular quantum groups. We also note that the unimodularity is a necessary condition., Comment: 15 pages, v2 corrects a few minor points. The final version of the paper will appear in the Proceedings of the American Mathematical Society

In this paper the authors introduce an analogue of the nilpotent cone, N {\mathcal N} , for a classical Lie superalgebra, g {\mathfrak g} , that generalizes the definition for the nilpotent cone for semisimple Lie algebras. For a classical simple Lie superalgebra, g = g 0 ¯ ⊕ g 1 ¯ {\mathfrak g}={\mathfrak g}_{\bar 0}\oplus {\mathfrak g}_{\bar 1} with Lie G 0 ¯ = g 0 ¯ \text {Lie }G_{\bar 0}={\mathfrak g}_{\bar 0} , it is shown that there are finitely many G 0 ¯ G_{\bar 0} -orbits on N {\mathcal N} . Later the authors prove that the Duflo-Serganova commuting variety, X {\mathcal X} , is contained in N {\mathcal N} for any classical simple Lie superalgebra. Consequently, our finiteness result generalizes and extends the work of Duflo-Serganova on the commuting variety. Further applications are given at the end of the paper.

This paper is devoted to the isochronicity of two kinds of planar piecewise polynomial potential systems separated by x x -axis and y y -axis, respectively. By means of the expansion of the period function near the infinity, we obtain in this paper that for these two cases, the piecewise potential system has an isochronous center at the origin if and only if the subsystems are linear. It is however not true in the analytic scenario; one can find in the paper by F. Mañosas and P. J. Torres [Proc. Amer. Math. Soc. 133 (2005), pp. 3027–3035] that the isochronicity of the center of the piecewise analytic potential system cannot imply the linearity of the two subsystems.

We are concerned with the study of the Lane-Emden equation with variable exponent and Dirichlet boundary condition. The feature of this paper is that the analysis that we develop does not assume any subcritical hypotheses and the reaction can fulfill a mixed regime (subcritical, critical and supercritical). We consider the radial and the nonradial cases, as well as a singular setting. The proofs combine variational and analytic methods with a version of the Palais principle of symmetric criticality., The final version this paper will be published in Proc. AMS

In this paper we are concerned with the study of a long-standing open problem posed by Lau in 1976. This problem is about whether the left amenability property of the space of left uniformly continuous functions of a semitopological semigroup is equivalent to the existence of a common fixed point for every jointly weak* continuous norm nonexpansive action on a nonempty weak* compact convex subset of a dual Banach space. We establish in this paper a positive answer.

In this paper we study the b b -ary expansions of the square roots of the function defined by the recurrence f b ( n ) = b f b ( n − 1 ) + n f_b(n)=b f_b(n-1)+n with initial value f ( 0 ) = 0 f(0)=0 taken at odd positive integers n n , of which the special case b = 10 b=10 is often referred to as the “schizophrenic” or “mock-rational” numbers. Defined by Darling in 2004 2004 and studied in more detail by Brown in 2009 2009 , these irrational numbers have the peculiarity of containing long strings of repeating digits within their decimal expansion. The main contribution of this paper is the extension of schizophrenic numbers to all integer bases b ≥ 2 b\geq 2 by formally defining the schizophrenic pattern present in the b b -ary expansion of these numbers and the study of the lengths of the non-repeating and repeating digit sequences that appear within.

Let S n S_n be partial sums of an i.i.d. sequence { X i } \{X_i\} . We assume that E X 1 > 0 \mathbb {E} X_1 >0 and P [ X 1 > 0 ] > 0 \mathbb {P}[X_1>0]>0 . In this paper we study the first passage time τ u = inf { n : S n > u } . \begin{equation*} \tau _u = \inf \{n:\; S_n > u\}. \end{equation*} The classical Cramér’s estimate of the ruin probability says that P [ τ u > ∞ ] ∼ C e − α 0 u as u → ∞ , \begin{equation*} \mathbb {P}[\tau _u>\infty ] \sim C e^{-\alpha _0 u}\qquad \text {as } u\to \infty , \end{equation*} for some parameter α 0 \alpha _0 . The aim of the paper is to describe precise large deviations of the first crossing by S n S_n a linear boundary. More precisely for a fixed parameter ρ \rho we study asymptotic behavior of P [ τ u = ⌊ u / ρ ⌋ ] \mathbb {P}\big [\tau _u = \lfloor u/\rho \rfloor \big ] as u u tends to infinity.

In this note, we give a generalization of Gossez's example of a maximally monotone multifunction such that the closure of its range is not convex, using more elementary techniques than in Gossez's original papers. We also discuss some new properties of Gossez's skew linear operator and its adjoint. While most of this paper uses elementary functional analysis, we correlate our results with those obtained by using the Stone-Cech compactification of the integers.

In his celebrated paper "Generic projections", John Mather has shown that almost all linear projections from a submanifold of a vector space into a subspace are transverse with respect to a given modular submanifold. In this paper, an improvement of Mather's result is stated. Namely, we show that almost all linear perturbations of a smooth mapping from a submanifold of $\mathbb{R}^m$ into $\mathbb{R}^\ell$ yield a transverse mapping with respect to a given modular submanifold. Moreover, applications of this result are given., Comment: 11 pages, to appear in Proceedings of the American Mathematical Society

In this paper we study properties of functions of triples of not necessarily commuting self-adjoint operators. The main result of the paper shows that unlike in the case of functions of pairs of self-adjoint operators there is no Lipschitz type estimates in any Schatten--von Neumann norm $\boldsymbol S_p$, $1\le p\le\infty$, for arbitrary functions in the Besov class $B_{\infty,1}^1({\Bbb R}^3)$. In other words, we prove that for $p\in[1,\infty]$, there is no constant $K>0$ such that the inequality \begin{align*} \|f(A_1,B_1,C_1)&-f(A_2,B_2,C_2)\|_{\boldsymbol S_p}\\[.1cm] &\le K\|f\|_{B_{\infty,1}^1} \max\big\{\|A_1-A_2\|_{\boldsymbol S_p},\|B_1-B_2\|_{\boldsymbol S_p},\|C_1-C_2\|_{\boldsymbol S_p}\big\} \end{align*} holds for an arbitrary function $f$ in $B_{\infty,1}^1({\Bbb R}^3)$ and for arbitrary finite rank self-adjoint operators $A_1,\,B_1,\,C_1,\,A_2,\,B_2$ and $C_2$., 14 pages. arXiv admin note: substantial text overlap with arXiv:1606.08961

In this paper we show that any Bohr-zero non-periodic set B B of traceless integer-valued matrices, denoted by Λ \Lambda , intersects non-trivially the conjugacy class of any matrix from Λ \Lambda . As a corollary, we obtain that the family of characteristic polynomials of B B contains all characteristic polynomials of matrices from Λ \Lambda . The main ingredient used in this paper is an equidistribution result for an S L d ( Z ) SL_d(\mathbb {Z}) random walk on a finite-dimensional torus deduced from Bourgain-Furman-Lindenstrauss-Mozes work [J. Amer. Math. Soc. 24 (2011), 231–280].

This paper studies vanishing of Ext modules over Cohen-Macaulay local rings. The main result of this paper implies that the Auslander-Reiten conjecture holds for maximal Cohen-Macaulay modules of rank one over Cohen-Macaulay normal local rings. It also recovers a theorem of Avramov-Buchweitz-Şega and Hanes-Huneke, which shows that the Tachikawa conjecture holds for Cohen-Macaulay generically Gorenstein local rings.

This paper studies vanishing of Ext modules over Cohen–Macaulay local rings. The main result of this paper implies that the Auslander–Reiten conjecture holds for maximal Cohen–Macaulay modules of rank one over Cohen–Macaulay normal local rings. It also recovers a theorem of Avramov–Buchweitz–Şega and Hanes–Huneke, which shows that the Tachikawa conjecture holds for Cohen–Macaulay generically Gorenstein local rings.

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.

Given a multi-variable polynomial, there is an associated divided symmetrization (in particular turning it into a symmetric function). Postinkov has found the volume of a permutohedron as a divided symmetrization (DS) of the power of a certain linear form. The main task in this paper is to exhibit and prove closed form DS-formulas for a variety of polynomials. We hope the results to be valuable and available to the research practitioner in these areas. Also, the methods of proof utilized here are simple and amenable to many more analogous computations. We conclude the paper with a list of such formulas., Comment: 13 pages, no figures

We force the existence of a non-trivial κ \kappa -complete ultrafilter over κ \kappa which fails to satisfy the Galvin property. This answers a question asked by Benhamou and Gitik [Ann. Pure Appl. Logic 173 (2022), Paper No. 103107].

A well-known result in [Hsu-Smith-Waltman, Trans. Amer. Math. Soc. (1996)] states that in a competitive semiflow defined on X + = X 1 + × X 2 + X^+ = X_1^+ \times X_2^+ , the product of two cones in respective Banach spaces, if ( u ∗ , 0 ) (u^*,0) and ( 0 , v ∗ ) (0,v^*) are the global attractors in X 1 + × { 0 } X_1^+ \times \{0\} and { 0 } × X 2 + \{0\}\times X_2^+ respectively, then one of the following three outcomes is possible for the two competitors: either there is at least one coexistence steady state, or one of ( u ∗ , 0 ) , ( 0 , v ∗ ) (u^*,0), (0,v^*) attracts all trajectories initiating in the order interval I = [ 0 , u ∗ ] × [ 0 , v ∗ ] I = [0,u^*] \times [0,v^*] . However, it was demonstrated by an example that in some cases neither ( u ∗ , 0 ) (u^*,0) nor ( 0 , v ∗ ) (0,v^*) is globally asymptotically stable if we broaden our scope to all of X + X^+ . In this paper, we give two sufficient conditions that guarantee, in the absence of coexistence steady states, the global asymptotic stability of one of ( u ∗ , 0 ) (u^*,0) or ( 0 , v ∗ ) (0,v^*) among all trajectories in X + X^+ . Namely, one of ( u ∗ , 0 ) (u^*,0) or ( 0 , v ∗ ) (0,v^*) is (i) linearly unstable, or (ii) linearly neutrally stable but zero is a simple eigenvalue. Our results complement the counterexample mentioned in the above paper as well as applications that frequently arise in practice.

In his paper "A Construction for Coisotropic Subalgebras of Lie Bialgebras", Marco Zambon gave a way to use a long root of a complex semisimple Lie biaglebra $\mathfrak{g}$ to construct a coisotropic subalgebra of $\mathfrak{g}$. In this paper, we generalize Zambon's construction. Our construction is based on the theory of Lagrangian subalgebras of the double $\mathfrak{g}\oplus\mathfrak{g}$ of $\mathfrak{g}$, and our coisotropic subalgebras correspond to torus fixed points in the variety $\mathcal{L}(\mathfrak{g}\oplus\mathfrak{g})$ of Lagrangian subalgebras of $\mathfrak{g}\oplus\mathfrak{g}$.

In this paper we disprove a conjecture stated in (4) on the equality of two notions of dimension for closed cones. Moreover, we answer in the negative to the following question, raised in the same paper. Given a compact family C of closed cones and a set S such that every blow-up of S at every point x 2 S is contained in some element of C, is it true that the dimension of S is smaller than or equal to the largest dimension of a vector space contained is some element of C?

In this paper we consider Artin local K-algebras with maximal length in the class of Artin algebras with given embedding dimension and socle type. They have been widely studied by several authors, among others by Iarrobino, Froberg and Laksov. If the local K-algebra is Gorenstein of socle degree 3, then the authors proved that it is canonically graded, i.e. analytically isomorphic to its associated graded ring, see (6). This unexpected result has been extended to compressed level K-algebras of socle degree 3 in (4). In this paper we end the investigation proving that extremal Artin Gorenstein local K-algebras of socle degree s � 4 are canonically graded, but the result does not extend to extremal Artin Gorenstein local rings of socle degree 5 or to compressed level local rings of socle degree 4 and type > 1. As a consequence we present results on Artin compressed local K-algebras having a specified socle type.

We continue to study the rank functions of tropical matrices. In this paper, we explain how to reduce the computation of ranks for matrices over the `supertropical semifield' to the standard tropical case. Using a counting approach, we prove the existence of a $01$-matrix with many ones and without large all-one submatrices, and we put our results together and construct an $n\times n$ matrix with tropical rank $o(n^{0.5+\varepsilon})$ and Kapranov rank $n-o(n)$., Comment: 9 pages

In this paper we study the motion of an internal water wave and an internal wave in a porous medium. For these problems we establish that, if the free boundary and, in the case of the Euler equations, also the tangential velocity at the interface are sufficiently smooth, the depth cannot vanish in finite time. This results holds regardless of gravity and surface tension effects or, if applicable, the stratification in multiphase flows. The first author was supported by the Basque Government through the BERC 2022-2025 program and by the Spanish State Research Agency through BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and through project PID2020-114189RB-I00 funded by Agencia Estatal de Investigaci´on (PID2020-14189RB-I00 / AEI / 10.13039/501100011033). The second author was supported by the project “Mathematical Analysis of Fluids and Applications” Grant PID2019-109348GA-I00 funded by MCIN/AEI/ 10.13039/501100011033 and acronym “MAFyA”. This publication is part of the project PID2019-109348GA-I00 funded by MCIN/ AEI /10.13039/501100011033. The second author was also supported by a 2021 Leonardo Grant for Researchers and Cultural Creators, BBVA Foundation.

This paper considers the problems of finite determinacy and approximation of flat analytic maps from germs of real or complex analytic spaces. It is shown that the flatness of analytic maps from germs of real or complex analytic spaces whose local rings are Cohen-Macaulay is finitely determined. Further, it is shown that flat maps from complete intersection and Cohen-Macaulay analytic germs can be arbitrarily closely approximated by algebraic and Nash maps respectively in such a way that the Hilbert-Samuel function of the special fibre is preserved. It is also proved that in the complex case the preservation of the Hilbert-Samuel function implies the preservation of Whitney's tangent cone., Comment: 13 pages. Substantial improvements in writing, existing theorems strengthened, several results added. arXiv admin note: text overlap with arXiv:1910.11498

In this paper, we prove that certain contravariant endofunctors of singularity categories characterize Gorenstein rings. Let Λ be a noetherian ring. Denote by Dsg(Λ) the singularity category of Λ, that is, the Verdier quotient of the bounded derived category D(Λ) of finitely generated (right) Λ-modules by the full subcategory consisting of bounded complexes of finitely generated projective Λ-modules. We are interested in the following question. Question 1. What contravariant endofunctor of Dsg(Λ) characterizes the Iwanaga– Gorenstein property of Λ? In this paper we shall consider this question in the case where Λ is commutative and Cohen–Macaulay. Let R be a commutative Cohen–Macaulay local ring of Krull dimension d. Denote by CM(R) the category of (maximal) Cohen–Macaulay R-modules and by CM(R) its stable category : the objects of CM(R) are the Cohen–Macaulay R-modules, and the hom-set HomCM(R)(M,N) is defined as HomR(M,N), the quotient module of HomR(M,N) by the submodule consisting homomorphisms factoring through finitely generated projective (or equivalently, free) R-modules. The natural full embedding functor CM(R) → D(R) induces an additive covariant functor η : CM(R) → Dsg(R). Furthermore, the assignment M 7→ ΩTrM , where Ω and Tr stand for the syzygy and transpose functors respectively (see [1, Chapter 2, §1] for details of the functors Ω and Tr), makes an additive contravariant functor

In this paper, we examine linear conditions on finite sets of points in projective space implied by the Cayley–Bacharach condition. In particular, by bounding the number of points satisfying the Cayley–Bacharach condition, we force them to lie on unions of low-dimensional linear spaces. These results are motivated by investigations into degrees of irrationality of complete intersections, which are controlled by minimum-degree rational maps to projective space. As an application of our main theorem, we describe the fibers of such maps for certain complete intersections of codimension two.

In this paper, we establish two gap theorems for ends of smooth metric measure space $(M^n, g,e^{-f}dv)$ with the Bakry-\'Emery Ricci tensor $\mathrm{Ric}_f\ge-(n-1)$ in a geodesic ball $B_o(R)$ with radius $R$ and center $o\in M^n$. When $\mathrm{Ric}_f\ge 0$ and $f$ has some degeneration outside $B_o(R)$, we show that there exists an $\epsilon=\epsilon(n,\sup_{B_o(1)}|f|)$ such that such a space has at most two ends if $R\le\epsilon$. When $\mathrm{Ric}_f\ge\frac 12$ and $f(x)\le\frac 14d^2(x,B_o(R))+c$ for some constant $c>0$ outside $B_o(R)$, we can also get the same gap conclusion., Comment: Proc AMS, final version

In this paper, we consider the entire solution to the parabolic 2 2 -Hessian equation − u t σ 2 ( D 2 u ) = 1 -u_t\sigma _2(D^2 u)=1 in R n × ( − ∞ , 0 ] \mathbb {R}^n\times (-\infty ,0] . We prove a rigidity theorem for the parabolic 2 2 -Hessian equation in R n × ( − ∞ , 0 ] \mathbb {R}^n\times (-\infty ,0] by establishing Pogorelov type estimates for 2 2 -convex-monotone solutions.

This paper extends the results from a work of Hytönen, Petermichl, and Volberg about sharp A 2 A_2 - A ∞ A_\infty estimates with matrix weights to the non-homogeneous situation.

Given a fixed hypergraph H H , let wsat ⁡ ( n , H ) \operatorname {wsat}(n,H) denote the smallest number of edges in an n n -vertex hypergraph G G , with the property that one can sequentially add the edges missing from G G , so that whenever an edge is added, a new copy of H H is created. The study of wsat ⁡ ( n , H ) \operatorname {wsat}(n,H) was introduced by Bollobás in 1968, and turned out to be one of the most influential topics in extremal combinatorics. While for most H H very little is known regarding wsat ⁡ ( n , H ) \operatorname {wsat}(n,H) , Alon proved in 1985 that for every graph H H there is a limiting constant C H C_H so that wsat ⁡ ( n , H ) = ( C H + o ( 1 ) ) n \operatorname {wsat}(n,H)=(C_H+o(1))n . Tuza conjectured in 1992 that Alon’s theorem can be (appropriately) extended to arbitrary r r -uniform hypergraphs. In this paper we prove this conjecture.

All knots are known to be hyperbolic, satellite, or torus knots, and one important family is Lorenz links, or T-links, which arise from dynamics. However, it remains difficult to determine the geometric type of a Lorenz link from a description via dynamics or as a T-link. In this paper, we consider those T-links that are torus links. We show that T-links obtained by full twists along torus links can never be torus links, aside from a family of cases. This addresses a question of Birman and Kofman., Comment: 8 pages, 1 figure. V2 contains minor changes. To appear in Proceedings of the AMS

This paper presents a curvature-free version of the Log ( 2 k − 1 ) \text {Log}(2k-1) Theorem of Anderson, Canary, Culler, and Shalen [J. Differential Geometry 44 (1996), pp. 738–782]. It generalizes a result by Hou [J. Differential Geometry 57 (2001), no. 1, pp. 173–193] and its proof is rather straightforward once we know the work by Lim [Trans. Amer. Math. Soc. 360 (2008), no. 10, pp. 5089–5100] on volume entropy for graphs. As a byproduct we obtain a curvature-free version of the Collar Lemma in all dimensions.

For any $n\in\mathbb{N}=\{0,1,2,\ldots\}$ and $b,c\in\mathbb{Z}$, the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Let $p$ be an odd prime. In this paper, we determine the summation $\sum_{k=0}^{p-1}T_k(b,c)^2/m^k$ modulo $p^2$ for integers $m$ with certain restrictions. As applications, we confirm some conjectural congruences of Sun [Sci. China Math. 57 (2014), 1375--1400]., 13 pages. This is a preliminary manuscript

The paper presents conditions on entry permutations that induce asymptotic freeness when acting on Gaussian random matrices. The class of permutations described includes the matrix transpose, as well as entry permutations relevant in Quantum Information Theory and Quantum Physics., Comment: Second version, some errors corrected, less pages, more results

The aim of this paper is to prove that the $p$-Wasserstein space $\mathcal{W}_p(X)$ is isometrically rigid for all $p\geq 1$ whenever $X$ is a countable graph metric space. As a consequence, we obtain that for every countable group $H$ and any $p\geq 1$ there exists a $p$-Wasserstein space whose isometry group is isomorphic to $H$., 14 pages with 3 figures

The Frobenius characteristic of $Lie_n,$ the representation of the symmetric group $S_n$ afforded by the multilinear component of the free Lie algebra, is known to satisfy many interesting plethystic identities. In this paper we prove a conjecture of Richard Stanley establishing the plethystic inverse of the sum $\sum_{n\geq 0} Lie_{2n+1}$ of the odd Lie characteristics. We obtain an apparently new plethystic decomposition of the regular representation of $S_n$ in terms of irreducibles indexed by hooks, and the Lie representations. We determine the plethystic inverse of the alternating sum of the odd Lie characteristics., Comment: 12 pages; Section 3 reorganised per referee comments; added Lemma 3.2 and Proposition 3.9. To appear in Proceedings of the American Mathematical Society

In this paper, it is proved that the group generated by Bass units contains a subgroup of finite index in the group of central units Z ( U ( Z G ) ) \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) of the integral group ring Z G \mathbb {Z}G for a subgroup closed monomial group G G with the property that every cyclic subgroup of order not a divisor of 4 4 or 6 6 is subnormal in G G . If G G is a generalized strongly monomial group, then it is also shown that the group generated by generalized Bass units contains a subgroup of finite index in Z ( U ( Z G ) ) \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) . Furthermore, for a generalized strongly monomial group G G , the rank of Z ( U ( Z G ) ) \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) is determined. The formula so obtained is in terms of generalized strong Shoda pairs of G G .

A Π 1 0 \Pi ^{0}_{1} class P P is thin if every Π 1 0 \Pi ^{0}_{1} subclass Q Q of P P is the intersection of P P with some clopen set. In 1993, Cenzer, Downey, Jockusch and Shore initiated the study of Turing degrees of members of thin Π 1 0 \Pi ^{0}_{1} classes, and proved that degrees containing no members of thin Π 1 0 \Pi ^{0}_{1} classes can be recursively enumerable, and can be minimal degree below 0 ′ \mathbf {0}’ . In this paper, we work on this topic in terms of genericity, and prove that all 2-generic degrees contain no members of thin Π 1 0 \Pi ^{0}_{1} classes. In contrast to this, we show that all 1-generic degrees below 0 ′ \mathbf {0}’ contain members of thin Π 1 0 \Pi ^{0}_{1} classes.

In this paper, we consider the initial and boundary value problem of Ericksen-Leslie system modeling nematic liquid crystal flows in dimension three. Two examples of singularity at finite time are constructed. The first example is constructed in a special axisymmetric class with suitable axisymmetric initial and boundary data, while the second example is constructed for an initial data with small energy but nontrivial topology. A counter example of maximum principle to the system is constructed by utilizing the Poiseuille flow in dimension one., arXiv admin note: text overlap with arXiv:1504.01080

This paper extends two recent improvements in the Hilbert space setting of the well-known Katznelson-Tzafriri theorem by establishing both a version of the result valid for bounded representations of a large class of abelian semigroups and a quantified version for contractive representations. The paper concludes with an outline of an improved version of the Katznelson-Tzafriri theorem for individual orbits, whose validity extends even to certain unbounded representations., 13 pages, to appear in Proceedings of the American Mathematical Society

In this paper, on Riemannian manifolds with boundary, we establish a Yau type gradient estimate and Liouville theorem for harmonic functions under Dirichlet boundary condition. Under a similar setting, we also formulate a Souplet-Zhang type gradient estimate and Liouville theorem for ancient solutions to the heat equation., 10 pages

In this paper, for a locally compact commutative hypergroup K K and for a pair ( Φ 1 , Φ 2 ) (\Phi _1, \Phi _2) of Young functions satisfying sequence condition, we give a necessary condition in terms of aperiodic elements of the center of K , K, for the convolution f ∗ g f\ast g to exist a.e., where f f and g g are arbitrary elements of Orlicz spaces L Φ 1 ( K ) L^{\Phi _1}(K) and L Φ 2 ( K ) L^{\Phi _2}(K) , respectively. As an application, we present some equivalent conditions for compactness of a compactly generated locally compact abelian group. Moreover, we also characterize compact convolution operators from L w 1 ( K ) L^1_w(K) into L w Φ ( K ) L^\Phi _w(K) for a weight w w on a locally compact hypergroup K K .

In this paper we classify varieties of Picard number two having two projective bundle structures of any relative dimension, under the assumption that these structures are mutually uniform. As an application we prove the Campana--Peternell conjecture for varieties of Picard number one admitting $\mathbb C^*$-actions of a certain kind., Comment: 15 pages

In this paper, we use Rouché’s theorem and the pleasant properties of the arithmetic of the logarithmic derivative to establish several new results and bounds regarding the geometry of the zeros, poles, and critical points of a rational function. Included is an improvement on a result by Alexander and Walsh regarding the “exclusion region” around a given zero or pole of a rational function in which no critical point may lie.

The primary aim of this paper is to present the complete description of the isomorphisms between positive definite cones of C ∗ C^* -algebras with respect to the recently introduced Wasserstein mean and to show the nonexistence of nonconstant such morphisms into the positive reals in the case of von Neumann algebras without type I 2 _2 , I 1 _1 direct summands. A comment on the algebraic properties of the Wasserstein mean relating associativity is also made.