Evolution of curvatures on a surface with boundary to prescribed functions

Let (M, g ₀) be a compact Riemann surface with boundary and with negative Euler characteristic. Let f(x) be a strictly negative smooth function on \({\bar{M}}\) and denote by \({\sigma(x)}\) the value of f in the interior and \({\zeta(x)}\) the value of f on the boundary. By studying the evolution of curvatures on M, we prove that there exist a constant \({\lambda_\infty}\) and a conformal metric \({g_\infty}\) such that \({\lambda_\infty\sigma(x)}\) and \({\lambda_\infty\zeta(x)}\) can be realized as the Gaussian curvature and boundary geodesic curvature of \({g_\infty}\) respectively.     

New conjectured inequalities for zeros of Jacobi polynomials

Inequalities recently conjectured for all zeros of Jacobi polynomials \(P_n^{(\alpha,\beta)}\) of all degrees n are modified and conjectured to hold (in reverse direction) in considerably larger domains of the (α,β)-plane.     

Maximal Operator in Variable Exponent Generalized Morrey Spaces on Quasi-metric Measure Space

We consider generalized Morrey spaces \({\mathcal{L}^{p(\cdot),\varphi(\cdot)}( X )}\) on quasi-metric measure spaces \({X,d,\mu}\), in general unbounded, with variable exponent p(x) and a general function \({\varphi(x,r)}\) defining the Morrey-type norm. No linear structure of the underlying space X is assumed. The admission of unbounded X generates problems known in variable exponent analysis. We prove the boundedness results for maximal operator known earlier only for the case of bounded sets X. The conditions for the boundedness are given in terms of the so called supremal inequalities imposed on the function \({\varphi(x,r)}\), which are weaker than Zygmund-type integral inequalities often used for characterization of admissible functions \({\varphi}\). Our conditions do not suppose any assumption on monotonicity of \({\varphi(x,r)}\) in r.     

Curvature inequalities for operators in the Cowen-Douglas class and localization of the Wallach set

For any bounded domain Ω in ?^m, let B₁(Ω) denote the Cowen-Douglas class of commuting m-tuples of bounded linear operators. For an m-tuple T in the Cowen-Douglas class B₁(Ω), let N_T (w) denote the restriction of T to the subspace \(\cap^m_{i,j=1}{\rm{ker}}(T_i-w_iI)(T_j-w_jI)\). This commuting m-tuple N_T (w) of m + 1 dimensional operators induces a homomorphism \({\rho _{{N_T}\left( w \right)}}\) of the polynomial ring P[z₁, · · ·, z_m], namely, \({\rho _{{N_T}\left( w \right)}}\) (p) = p(N_T (w)), p ∈ P[z₁, · · ·, z_m]. We study the contractivity and complete contractivity of the homomorphism \({\rho _{{N_T}\left( w \right)}}\). Starting from the homomorphism \({\rho _{{N_T}\left( w \right)}}\), we construct a natural class of homomorphisms \(\rho_{N^{(\lambda)}(w)}\), λ > 0, and relate the properties of \(\rho_{N^{(\lambda)}(w)}\) to those of \({\rho _{{N_T}\left( w \right)}}\). Explicit examples arising from the multiplication operators on the Bergman space of Ω are investigated in detail. Finally, it is shown that contractive properties of \({\rho _{{N_T}\left( w \right)}}\) are equivalent to an inequality for the curvature of the Cowen-Douglas bundle E_T. However, we construct examples to show that the contractivity of the homomorphism ρ_T does not follow, even if \({\rho _{{N_T}\left( w \right)}}\) is contractive for all w in Ω.     

Infinite dimensional multiplicity free spaces III: matrix coefficients and regular functions

In earlier papers we studied direct limits \({(G,\,K) = \varinjlim\, (G_n,K_n)}\) of two types of Gelfand pairs. The first type was that in which the G _n /K _n are compact Riemannian symmetric spaces. The second type was that in which \({G_n = N_n\rtimes K_n}\) with N _n nilpotent, in other words pairs (G _n , K _n ) for which G _n /K _n is a commutative nilmanifold. In each we worked out a method inspired by the Frobenius–Schur Orthogonality Relations to define isometric injections \({\zeta_{m,n}: L^2(G_n/K_n) \hookrightarrow L^2(G_m/K_m)}\) for m ≧ n and prove that the left regular representation of G on the Hilbert space direct limit \({L^2(G/K) := \varinjlim L^2(G_n/K_n)}\) is multiplicity-free. This left open questions concerning the nature of the elements of L ²(G/K). Here we define spaces \({\mathcal{A}(G_n/K_n)}\) of regular functions on G _n /K _n and injections \({\nu_{m,n} : \mathcal{A}(G_n/K_n) \to \mathcal{A}(G_m/K_m)}\) for m ≧ n related to restriction by \({\nu_{m,n}(f)|_{G_n/K_n} = f}\). Thus the direct limit \({\mathcal{A}(G/K) := \varinjlim \{\mathcal{A}(G_n/K_n), \nu_{m,n}\}}\) sits as a particular G-submodule of the much larger inverse limit \({\varprojlim \{\mathcal{A}(G_n/K_n), {\rm restriction}\}}\). Further, we define a pre Hilbert space structure on \({\mathcal{A}(G/K)}\) derived from that of L ²(G/K). This allows an interpretation of L ²(G/K) as the Hilbert space completion of the concretely defined function space \({\mathcal{A}(G/K)}\), and also defines a G-invariant inner product on \({\mathcal{A}(G/K)}\) for which the left regular representation of G is multiplicity-free.     

Zigzag Strip Bundle Realization of B(Λ0) over $U_{q}(C_{n}^{(1)})$

Zigzag strip bundles are new combinatorial models realizing the crystals B(∞) for the quantum affine algebras \(U_{q}(\mathfrak {g})\), where \(\mathfrak {g}=B_{n}^{(1)},D_{n}^{(1)}\), \(D_{n+1}^{(2)}\), \(C_{n}^{(1)}\), \(A_{2n-1}^{(2)}\), \(A_{2n}^{(2)}\). Recently, these models were used to the realization of highest weight crystals except for the highest weight crystal B(Λ₀) over the quantum affine algebra \(U_{q}(C_{n}^{(1)})\). In this paper, we construct the highest weight crystal B(Λ₀) over the quantum affine algebra \(U_{q}(C_{n}^{(1)})\) using zigzag strip bundles, which completes the realizations of all highest weight crystals over \(U_{q}(\mathfrak {g})\).     

<Emphasis Type="Italic">k</Emphasis>-Extreme Points in Symmetric Spaces of Measurable Operators

Let \({\mathcal{M}}\) be a semifinite von Neumann algebra with a faithful, normal, semifinite trace \({\tau}\) and E be a strongly symmetric Banach function space on \({[0,\tau({\bf 1}))}\) . We show that an operator x in the unit sphere of \({E(\mathcal{M}, \tau)}\) is k-extreme, \({k \in {\mathbb{N}}}\) , whenever its singular value function \({\mu(x)}\) is k-extreme and one of the following conditions hold (i) \({\mu(\infty, x) = \lim_{t\to\infty}\mu(t, x) = 0}\) or (ii) \({n(x)\mathcal{M}n(x^*) = 0}\) and \({|x| \geq \mu(\infty, x)s(x)}\) , where n(x) and s(x) are null and support projections of x, respectively. The converse is true whenever \({\mathcal{M}}\) is non-atomic. The global k-rotundity property follows, that is if \({\mathcal{M}}\) is non-atomic then E is k-rotund if and only if \(E(\mathcal{M}, \tau)\) is k-rotund. As a consequence of the noncommutative results we obtain that f is a k-extreme point of the unit ball of the strongly symmetric function space E if and only if its decreasing rearrangement \({\mu(f)}\) is k-extreme and \({|f| \geq \mu(\infty,f)}\) . We conclude with the corollary on orbits Ω(g) and Ω′(g). We get that f is a k-extreme point of the orbit \({\Omega(g),\,g \in L_1 + L_{\infty}}\) , or \({\Omega'(g),\,g \in L_1[0, \alpha),\,\alpha < \infty}\) , if and only if \({\mu(f) = \mu(g)}\) and \({|f| \geq \mu(\infty, f)}\) . From this we obtain a characterization of k-extreme points in Marcinkiewicz spaces.     

Uniform asymptotic expansions for Laguerre polynomials and related confluent hypergeometric functions

Uniform asymptotic expansions involving exponential and Airy functions are obtained for Laguerre polynomials \(L_{n}^{(\alpha )}(x)\), as well as complementary confluent hypergeometric functions. The expansions are valid for n large and α small or large, uniformly for unbounded real and complex values of x. The new expansions extend the range of computability of \(L_{n}^{(\alpha )}(x)\) compared to previous expansions, in particular with respect to higher terms and large values of α. Numerical evidence of their accuracy for real and complex values of x is provided.     

The rate of the smallest value of the weighted measure of the nonnegativity set for polynomials with zero mean value on a closed interval

Let P _n (α) be the set of algebraic polynomials p _n of order n with real coefficients and zero weighted mean value with ultraspherical weight \(\phi ^{(\alpha )} (t) = (1 - t^2 )^\alpha \) on the interval \([ - 1,1]:\int_{ - 1}^1 {\phi ^{(\alpha )} (t)p_n (t)dx = 0} \). We study the problem on the smallest value µ _n = inf{m(p _n ): p _n ∈ P _n (α)} of the weighted measure \(m(p_n ) = \int_{\chi (p_n )} {\phi ^{(\alpha )} (t)dt} \) of the set where p _n is nonnegative. The order of µ _n with respect to n is found: it is proved that \(\mu _n (\alpha ) \asymp n^{ - 2(\alpha + 1)} \) as n→∞.     

Integral means of weighted Hardy–Bloch functions

Let \({\mathcal{B}^\omega(p, q, B_d)}\) denote the \({\omega}\)-weighted Hardy–Bloch space on the unit ball B _d of \({\mathbb{C}^d}\), \({d\ge 1}\). For \({2< p,q < \infty}\) and \({f\in \mathcal{B}^\omega(p, q, B_d)}\), we obtain sharp estimates on the growth of the p-integral means M _p (f, r) as \({r\to 1-}\).     

Depth,Stanley depth,and regularity of ideals associated to graphs

Let \({\mathbb{K}}\) be a field and \({S=\mathbb{K}[x_1,\dots,x_n]}\) be the polynomial ring in n variables over \({\mathbb{K}}\). Let G be a graph with n vertices. Assume that \({I=I(G)}\) is the edge ideal of G and \({J=J(G)}\) is its cover ideal. We prove that \({{\rm sdepth}(J)\geq n-\nu_{o}(G)}\) and \({{\rm sdepth}(S/J)\geq n-\nu_{o}(G)-1}\), where \({\nu_{o}(G)}\) is the ordered matching number of G. We also prove the inequalities \({{\rmsdepth}(J^k)\geq {\rm depth}(J^k)}\) and \({{\rm sdepth}(S/J^k)\geq {\rmdepth}(S/J^k)}\), for every integer \({k\gg 0}\), when G is a bipartite graph. Moreover, we provide an elementary proof for the known inequality reg\({(S/I)\leq \nu_{o}(G)}\).     

Isoperimetric inequality for the polydisk

Let \({n\in\mathbb{N}}\). For \({k\in\{1,\dots,n\}}\) let \({\Omega_k\subset \mathbb{C}}\) be a simply connected domain with a rectifiable boundary. Let \({\Omega^n=\prod_{k=1}^n\Omega_k\subset \mathbb{C}^n}\) be a generalized polydisk with distinguished boundary \({\partial\Omega^n=\prod_{k=1}^n\partial\Omega_k}\). Let E ^r (Ω ⁿ ) be the holomorphic Smirnov class on Ω ⁿ with index r. We show that the generalized isoperimetric inequality

$ \int\limits_{\Omega^n} |f_1|^p|f_2|^qdV\le \frac{1}{(4\pi)^n}\int\limits_{\partial \Omega^n}|f_1|^pdS \int\limits_{\partial \Omega^n} |f_2|^qdS, $

holds for arbitrary \({f_1\in E^p(\Omega^n)}\) and \({f_2\in E^q(\Omega^n)}\), where 0 < p, q < ∞. We also determine necessary and sufficient conditions for equality.

     

An inequality between Willmore functional and Weyl functional for submanifolds in space forms

Let \({\phi : M \to R^{n+p}(c)}\) be an n-dimensional submanifold in an (n + p)-dimensional space form R ^n+p(c) with the induced metric g. Willmore functional of \({\phi}\) is \({W(\phi) = \int_{M}(S - nH^{2})^{n/2}dv}\) , where \({S = \sum_{\alpha,i, j}(h^{\alpha}_{ij} )^2}\) is the square of the length of the second fundamental form, H is the mean curvature of M. The Weyl functional of (M, g) is \({\nu(g) = \int_{M}|W_{g}|^{n/2}dv}\) , where \({|W_{g}|^{2} = \sum_{i, j,k,l} W^{2}_{ijkl}}\) and W _ijkl are the components of the Weyl curvature tensor W _g of (M, g). In this paper, we discover an inequality relation between Willmore functional \({W(\phi)}\) and Weyl funtional ν(g).     

On the Local Time of Random Walks Associated with Gegenbauer Polynomials

The local time of random walks associated with Gegenbauer polynomials \(P_{n}^{(\alpha)}(x)\), x∈[?1,1], is studied in the recurrent case: \(\alpha\in [-\frac{1}{2},0]\). When α is nonzero, the limit distribution is given in terms of a Mittag-Leffler distribution. The proof is based on a local limit theorem for the random walk associated with Gegenbauer polynomials. As a by-product, we derive the limit distribution of the local time of some particular birth-and-death Markov chains on ?.     

Intersection properties of radial solutions and global bifurcation diagrams for supercritical quasilinear elliptic equations

We study the positive solution \({u(r,\rho)}\) of the quasilinear elliptic equation

$$\begin{cases}r^{-(\gamma-1)}(r^{\alpha}|u^{\prime}|^{\beta-1}u^{\prime})^{\prime}+|u|^{p-1}u=0, & 0 < r < \infty,\\ u(0) = \rho > 0,\ u^{\prime}(0)=0.\end{cases}$$

This class of differential operators includes the usual Laplace, m-Laplace, and k-Hessian operators in the space of radial functions. The equation has a singular positive solution u ^*(r) under certain conditions on \({\alpha}\), \({\beta}\), \({\gamma}\), and p. A generalized Joseph–Lundgren exponent, which we denote by \({p^*_{JL}}\), is obtained. We study the intersection numbers between \({u(r,\rho)}\) and u ^*(r) and between \({u(r,\rho_0)}\) and \({u(r,\rho_1)}\), and see that \({p^*_{JL}}\) plays an important role. We also determine the bifurcation diagram of the problem

$$\begin{cases}r^{-(\gamma-1)}(r^{\alpha}|u^{\prime}|^{\beta-1}u^{\prime})^{\prime} + \lambda(u+1)^p=0, & 0 < r < 1,\\ u(r) > 0, & 0 \le r < 1,\\ u^{\prime}(0)=0,\ u(1)=0.\end{cases}$$

The main technique used in the proofs is a phase plane analysis.

     

On a Composition Preserving Inequalities between Polynomials

The Schur-Szegö composition of two polynomials \(f\left( z \right) = \sum\nolimits_{j = 0}^n {{A_j}{z^j}} \) and \(g\left( z \right) = \sum\nolimits_{j = 0}^n {{B_j}{z^j}} \), both of degree n, is defined by \(f * g\left( z \right) = \sum\nolimits_{j = 0}^n {{A_j}{B_j}{{\left( {\begin{array}{*{20}{c}}n \\ j \end{array}} \right)}^{ - 1}}{z^j}} \). In this paper, we estimate the minimum and the maximum of the modulus of f * g(z) on z = 1 and thereby obtain results analogues to Bernstein type inequalities for polynomials.     

The dual space of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> of a vector measure

For a vector measure ν having values in a real or complex Banach space and \({p \in}\) [1, ∞), we consider L ^p (ν) and \({L_{w}^{p}(\nu)}\), the corresponding spaces of p-integrable and scalarly p-integrable functions. Given μ, a Rybakov measure for ν, and taking q to be the conjugate exponent of p, we construct a μ-Köthe function space E _q (μ) and show it is σ-order continuous when p > 1. In this case, for the associate spaces we prove that L ^p (ν) ^×  = E _q (μ) and \({E_q(\mu)^\times = L_w^p(\nu)}\). It follows that \({L_p (\nu) ^{**} = L_w^p (\nu)}\). We also show that L ¹ (ν) ^×  may be equal or not to E _∞ (μ).     

Uniform in bandwidth exact rates for a class of kernel estimators

Given an i.i.d sample (Y _i , Z _i ), taking values in \({\mathbb{R}^{d'}\times\mathbb{R}^d}\), we consider a collection Nadarya–Watson kernel estimators of the conditional expectations \({\mathbb{E}( <\,c_g(z),g(Y)>+d_g(z)\mid Z=z)}\), where z belongs to a compact set \({H\subset \mathbb{R}^d}\), g a Borel function on \({\mathbb{R}^{d'}}\) and c _g (·), d _g (·) are continuous functions on \({\mathbb{R}^d}\). Given two bandwidth sequences \({h_n<\mathfrak{h}_n}\) fulfilling mild conditions, we obtain an exact and explicit almost sure limit bounds for the deviations of these estimators around their expectations, uniformly in \({g\in\mathcal{G},\;z\in H}\) and \({h_n\le h\le \mathfrak{h}_n}\) under mild conditions on the density f _Z , the class \({\mathcal{G}}\), the kernel K and the functions c _g (·), d _g (·). We apply this result to prove that smoothed empirical likelihood can be used to build confidence intervals for conditional probabilities \({\mathbb{P}( Y\in C\mid Z=z)}\), that hold uniformly in \({z\in H,\; C\in \mathcal{C},\; h\in [h_n,\mathfrak{h}_n]}\). Here \({\mathcal{C}}\) is a Vapnik–Chervonenkis class of sets.     

Some properties of autocentral automorphisms of a group

Let G be a group, Aut(G) and L(G) denote the full automorphisms group and absolute centre of G, respectively. The automorphism \({\alpha\in Aut(G)}\) is called autocentral if \({g^{-1}\alpha(g)\in L(G)}\), for all \({g\in G}\). In the present paper, we investigate the properties of such automorphisms.     

Diameter-Preserving Spanning Trees in Sparse Weighted Graphs

We give a simple sufficient condition for a weighted graph to have a diameter-preserving spanning tree. More precisely, let G = (V, E, f _E ) be a connected edge weighted graph with f _E being the edge weight function. Let f _V be the vertex weight function of G induced by f _E as follows: f _V (v) = max{f _E (e) : e is incident with v} for all \({v \in V}\) . We show that G contains a diameter-preserving spanning tree if \({d(G)\ge \frac{2}{3} \sum_{v\in V} f_V(v)}\) where d(G) is the diameter of G. The condition is sharp in the sense that for any \({\epsilon >0 }\) , there exist weighted graphs G satisfying \({d(G) > (\frac{2}{3}-\epsilon)\sum_{v\in V} f_V(v)}\) and not containing a diameter-preserving spanning tree.