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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

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.

A misuse of terminology has occurred in the statement and proof of Theorem 4.1 in our paper [Proc. Amer. Math. Soc. 143 (2015), pp. 1981–1989] which caused some justifiable misinterpretation of this result. To recover this result we provide a new definition and give the statement of our result in terms of this definition. The proof of the new version is an improvement of the old proof. The effect of the new definition on further relevant results in our paper has been adopted in a remark.

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 investigate asymptotic behaviors of Racah polynomials with fixed parameters and scaled variable as the polynomial degree tends to infinity. We start from the difference equation satisfied by the polynomials and derive an asymptotic formula in the outer region via ratio asymptotics. Next, we find the asymptotic formulas in the oscillatory region via a simple matching principle. Unlike the varying parameter case considered in a previous paper, the zeros of Racah polynomials with fixed parameters may not always be real. For this unusual case, we also provide a standard method to determine the oscillatory curve which attracts the zeros of Racah polynomials when the degree becomes large.

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.

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.

Recently, the authors in a joint paper obtained the structure of all capable nilpotent Lie algebras with derived subalgebra of dimension at most 1 1 . This paper is devoted to characterizing all capable nilpotent Lie algebras of class two with derived subalgebra of dimension 2 2 . It develops and generalizes the result due to Heineken for the group case.

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

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?

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.

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.

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.

In this paper, q-Gaussian distribution, q-analogy of Gaussian distribution, is introduced to characterize the prior information of unknown parameters for inverse problems. Based on q-Hermite polynomials, we propose a spectral likelihood approximation (SLA) algorithm of Bayesian inversion. Convergence results of the approximated posterior distribution in the sense of Kullback–Leibler divergence are obtained when the likelihood function is replaced with the SLA and the prior density function is truncated to its partial sum. In the end, two numerical examples are displayed, which verify our results.

In this paper, we set up the complete bubble tree theory for approximate harmonic maps from a Riemann surface with tension fields bounded in Zygmund class L ln + ⁡ L L\ln ^+ L . Some special cases of this theory have previously been used in a number of papers. On the other hand, one can see that this bubble tree theory is not true for the general target manifold if we only assume that the tension fields are bounded in L 1 L^1 uniformly.

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 multivariate approximation by weighted polynomials of the form w γ n ( x ) p n ( x ) w^{\gamma _n}(\mathbf {x})p_n(\mathbf {x}) , where p n p_n is a multivariate polynomial of degree at most n n , w w is a given nonnegative weight with nonempty zero set, and γ n ↑ ∞ \gamma _n\uparrow \infty . We study the question if every continuous function vanishing on the zero set of w w is a uniform limit of weighted polynomials w γ n ( x ) p n ( x ) w^{\gamma _n}(\mathbf {x})p_n(\mathbf {x}) . It turns out that for various classes of weights in order for this approximation property to hold it is necessary and sufficient that γ n = o ( n ) . \gamma _n=o(n).

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 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.

In this paper, we study the rough singular integral operator T Ω f ( x ) = p.v. ∫ R n f ( x − y ) Ω ( y ′ ) | y | n d y , \begin{equation*} T_\Omega f(x)=\text {p.v.}\int _{\mathbb {R}^n}f(x-y)\frac {\Omega (y’)}{|y|^n}dy, \end{equation*} and the corresponding maximal singular integral operator T Ω ∗ f ( x ) = sup ε > 0 | ∫ | y | ≥ ε f ( x − y ) Ω ( y ′ ) | y | n d y | , \begin{equation*} T^*_\Omega f(x)=\sup _{\varepsilon >0}\Big |\int _{|y|\geq \varepsilon }f(x-y)\frac {\Omega (y’)}{|y|^n}dy\Big |, \end{equation*} where the kernel Ω ∈ H 1 ( S n − 1 ) \Omega \in H^1(\mathrm {S}^{n-1}) has zero mean value and n ≥ 2 n\geq 2 . We prove that T Ω T_\Omega and T Ω ∗ T^*_\Omega are bounded on the mixed radial-angular spaces L | x | p L θ p ~ ( R n ) L_{|x|}^pL_{\theta }^{\tilde {p}}(\mathbb {R}^n) for some suitable indexes 1 > p , p ~ > ∞ 1>p, \tilde {p}>\infty . The corresponding vector-valued versions are also established.

This paper deals with the long-time behavior of viscosity solutions of evolutionary contact Hamilton-Jacobi equations w t + H ( x , w , w x ) = 0 , \begin{equation*} w_t+H(x,w,w_x)=0, \end{equation*} where H ( x , u , p ) H(x,u,p) is strictly decreasing in u u and satisfies Tonelli conditions in p p . We show that each viscosity solution of the ergodic contact Hamilton-Jacobi equation H ( x , u , u x ) = 0 H(x,u,u_x)=0 can be reached by many different viscosity solutions of the above evolutionary equation in a finite time.

In this paper we introduce the notion of global ultradifferentiable functions with respect to weight functions and include a discussion of its functional analytic theory and prove a characterization in terms of certain exponential decay of the Fourier-Broz-Iagolnitzer transform—a Paley-Wiener type theorem. As an application we investigate the regularity of global ultradifferentiable vectors proving a version of the Kotake-Narasimhan theorem in this setting.

In this paper we show that evolution algebras over any given field $\Bbbk$ are universally finite. In other words, given any finite group $G$, there exist infinitely many regular evolution algebras $X$ such that $Aut(X)\cong G$. The proof is built upon the construction of a covariant faithful functor from the category of finite simple (non oriented) graphs to the category of (finite dimensional) regular evolution algebras. Finally, we show that any constant finite algebraic affine group scheme $\mathbf{G}$ over $\Bbbk$ is isomorphic to the algebraic affine group scheme of automorphisms of a regular evolution algebra., Comment: Minor corrections. Bibliography updated. To appear in Proc. Amer. Math. Soc

In this paper, we study the sign-changing Kazdan-Warner's problem on two dimensional closed Riemannian manifold with negative Euler number $\chi(M)

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

Let G G be a locally compact group and let K K be its closed subgroup. Write G ^ K \widehat {G}_{K} for the set of irreducible representations with non-zero K K -invariant vectors. We call a pair ( G , K ) (G,K) admissible if for each irreducible representation ( π , V π ) (\pi , V_{\pi }) in G ^ K \widehat {G}_{K} , its K K -invariant subspace V π K V_{\pi }^{K} is of finite dimension. For each π \pi in G ^ K \widehat {G}_{K} , let π v i , ξ ¯ j \pi _{v_{i}, \overline {\xi }_{j}} ’s ( π v i , ξ ¯ j ( g K ) ≔ ⟨ v i , π ( g ) ξ j ⟩ ) (\pi _{v_{i}, \overline {\xi }_{j}}(gK)≔\langle v_{i}, \pi (g)\xi _{j}\rangle ) be the matrix coefficeints on G / K G/K induced by fixed orthonormal bases { v i } \{v_{i}\} and { ξ j } \{\xi _{j}\} for V π V_{\pi } and V π K V_{\pi }^{K} respectively. A probability measure μ \mu on G / K G/K is called a spectral measure if there is a subset Γ \Gamma of G ^ K \widehat {G}_{K} such that the set of all such matrix coefficients π v i , ξ ¯ j , π ∈ Γ , \pi _{v_{i}, \overline {\xi }_{j}},\ \pi \in \Gamma , constitutes an orthonormal basis for L 2 ( G / K , μ ) L^{2}(G/K, \mu ) with some suitable normalization of these matrix coordinate functions. In this paper, we shall give a characterization of a spectral measure for an admissible pair ( G , K ) (G,K) by using the Fourier transform on G / K G/K . Also, from this we show that there is a “local translation” (we call it locally regular representation in the sequel) of G G on L 2 ( G / K , μ ) L^{2}(G/K, \mu ) under a mild condition. This will give us some necessary conditions for the existence of spectral measures. In particular, the atomic spectral measures of finite supports for Gelfand pairs are studied.

The Littlewood-Richardson coefficients $c^{\lambda}_{\mu\nu}$ give the multiplicity of an irreducible polynomial ${\rm GL}_n$-representation $F^{\lambda}_n$ in the tensor product of polynomial representations $F^{\mu}_n\otimes F^{\nu}_n$. In this paper, we generalize these coefficients to an $r$-fold tensor product of rational representations, and give a new method for computing them using an analogue of statistical contingency tables. We demonstrate special cases in which our method reduces to counting statistical contingency tables with prescribed margins. Finally, we extend our result from the general linear group to both the orthogonal and symplectic groups.

In this paper, we extend the classical Bohr's inequality to the setting of the non-commutative Hardy space $H^1$ associated with a semifinite von Neumann algebra. As a consequence, we obtain Bohr's inequality for operators in the von Neumann-Schatten class $\cl C_1$ and square matrices of any finite order. Interestingly, we establish that the optimal bound for $r$ in the above mentioned Bohr's inequality concerning von Neumann-Shcatten class is 1/3 whereas it is 1/2 in the case of $2\times 2$ matrices and reduces to $\sqrt{2}-1$ for the case of $3\times 3$ matrices. We also obtain a generalization of our above-mentioned Bohr's inequality for finite matrices where we show that the optimal bound for $r$, unlike above, remains 1/3 for every fixed order $n\times n,\ n\ge 2$.

In 2012, Chan proved a number of congruences for the coefficients of Ramanujan’s ϕ \phi function. In this paper, we prove some new congruences modulo powers of 2 and 3 for Ramanujan’s ϕ \phi function by employing Newman’s identities and theta function identities.

The λ \lambda -property and the uniform λ \lambda -property were first introduced by R. Aron and R. Lohman in 1987 as geometric properties of Banach spaces. In 1989, Th. Shura and D. Trautman showed that the Schreier space possesses the λ \lambda -property and asked if it has the uniform λ \lambda -property. In this paper, we show that Schreier space does not have the uniform λ \lambda -property. Furthermore, we show that the dual of the Schreier space does not have the uniform λ \lambda -property.

Let ω \omega be a weight on a right cancellative semigroup S S . Let ‖ ⋅ ‖ ω \|\cdot \|_{\omega } be the weighted norm on the weighted discrete semigroup algebra ℓ 1 ( S , ω ) \ell ^1(S, \omega ) . In this paper, we prove that the weight ω \omega satisfies F-property if and only if the operator norm ‖ ⋅ ‖ ω o p \| \cdot \|_{\omega op} of ‖ ⋅ ‖ ω \| \cdot \|_{\omega } is exactly equal to another weighted norm ‖ ⋅ ‖ ω ~ 1 \| \cdot \|_{\widetilde {\omega }_1} . Though its proof is elementary, the result is unexpectedly surprising. In particular, the operator norm ‖ ⋅ ‖ 1 o p \| \cdot \|_{1 op} is same as the ℓ 1 \ell ^1 - norm ‖ ⋅ ‖ 1 \| \cdot \|_1 on ℓ 1 ( S ) \ell ^1(S) . Moreover, various examples are discussed to understand the relation among ‖ ⋅ ‖ ω o p \| \cdot \|_{\omega op} , ‖ ⋅ ‖ ω \| \cdot \|_{\omega } , and ℓ 1 ( S , ω ) \ell ^1(S, \omega ) .

This paper is concerned with a parabolic-elliptic system, which was originally proposed to model the evolution of biological transport networks. Recent results show that the corresponding initial-boundary value problem possesses a global weak solution, which, in particular, is also classical in the one and two dimensional cases. In this work, we establish a Serrin-type blowup criterion for classical solutions in the three dimensional setting.

This paper focuses on the stationary problem of the diffusive logistic equation on a bounded interval. We consider the ratio of a population size of a species to a carrying capacity which denotes a spatial heterogeneity of an environment. In one-dimensional case, Wei-Ming Ni proposed a variational conjecture that the supremum of this ratio varying a diffusion coefficient and a carrying function is 3. Recently, Xueli Bai, Xiaoqing He, and Fang Li [Proc. Amer. Math. Soc. 144 (2016), pp. 2161–2170] settled the conjecture by finding a special sequence of diffusion coefficients and carrying functions. Our interest is to derive a profile of the solutions corresponding to this maximizing sequence. Among other things, we obtain the exact order of the maximum and the minimum of solutions of the sequence. The proof is based on separating the stationary problem into two ordinary differential equations and smoothly adjoining each solution.

In this paper, we study the population optimization problem in the logistic reaction-diffusion model. The issue of maximizing the total population in a heterogeneous environment has attracted the attention of many researchers. For the n n -dimensional box domain, it has recently been shown that resource fragmentation is better than concentration in order to maximize the total population when the diffusion rate is sufficiently small. As resource concentration is known to be beneficial for the survival of the species, this contrasting phenomenon is quite surprising. We proved that the fragmentation phenomenon occurs for any general bounded domains in R n \mathbb {R}^n if the diffusion rate is sufficiently small.

In this paper, we introduce sequentially defined Besov spaces adapted to the scaling invariant property of the 3D incompressible Navier-Stokes equations. We first show that these spaces are Banach spaces. We then establish regularity conditions in these spaces.

In this paper, Lie group representations on Hilbert spaces are studied in relation with operator ranges. Let R \mathcal {R} be an operator range of a Hilbert space H \mathcal {H} . Given the set Λ \Lambda of R \mathcal {R} -invariant operators, and given a Lie group representation ρ : G → GL ( H ) \rho :G\rightarrow \text {GL}(\mathcal {H}) , we discuss the induced semigroup homomorphism ρ ~ : ρ − 1 ( Λ ) → B ( R ) \widetilde {\rho }: \rho ^{-1}(\Lambda ) \rightarrow \mathcal {B(R)} for the operator range topology on R \mathcal {R} . In one direction, we work under the assumption ρ − 1 ( Λ ) = G \rho ^{-1} (\Lambda ) = G , so ρ ~ : G → B ( R ) \widetilde {\rho }:G\rightarrow \mathcal {B}(\mathcal {R}) is in fact a group representation. In this setting, we prove that ρ ~ \widetilde {\rho } is continuous (and smooth) if and only if the tangent map d ρ d\rho is R \mathcal {R} -invariant. In another direction, we prove that for the tautological representations of unitary or invertible operators of an arbitrary infinite-dimensional Hilbert space H \mathcal {H} , the set ρ − 1 ( Λ ) \rho ^{-1}(\Lambda ) is neither a group for a large set of nonclosed operator ranges R \mathcal {R} nor closed for all nonclosed operator ranges R \mathcal {R} . Both results are proved by means of explicit counterexamples.

In this paper we will give a simple stopping time proof in the $\mathbb{R}^d$ setting of the matrix weighted Triebel-Lizorkin bounds proved by F. Nazarov/S. Treil and A. Volberg, respectively. Furthermore, we provide explicit matrix A${}_p$ characteristic dependence and also discuss some interesting open questions., 13 pages, no figures, submitted