We show that the median m(x) in the gamma distribution with parameter x is a strictly convex function on the positive half-line.

We characterize componentwise linear monomial ideals with minimal Taylor resolution and consider the lower bound for the Betti numbers of componentwise linear ideals.

Let Ω be an open subset of R^d, d≥2, and let x∈Ω. A Jensen measure for x on Ω is a Borel probability measure μ, supported on a compact subset of Ω, such that ∫u dμ≤u(x) for every superharmonic function u on Ω. Denote by J_x(Ω) the family of Jensen measures for x on Ω. We present two characterizations of ext(J_x(Ω)), the set of extreme elements of J_x(Ω). The first is in terms of finely harmonic measures, and the second as limits of harmonic measures on decreasing sequences of domains.

This allows us to relax the local boundedness condition in a previous result of B. Cole and T. Ransford, Jensen measures and harmonic measures, J. Reine Angew. Math. 541 (2001), 29–53.

As an application, we give an improvement of a result by Khabibullin on the question of whether, given a complex sequence {α_n}_n=1^∞ and a continuous function $M\colon\textbf{C}\rightarrow\textbf{R}^+$, there exists an entire function f≢0 satisfying f(α_n)=0 for all n, and |f(z)|≤M(z) for all z∈C.

Let X be a rationally convex compact subset of the unit sphere S in ℂ², of three-dimensional measure zero. Denote by R(X) the uniform closure on X of the space of functions P/Q, where P and Q are polynomials and Q≠0 on X. When does R(X)=C(X)?

Our work makes use of the kernel function for the $\bar{\delta}_{b}$ operator on S, introduced by Henkin in [5] and builds on results obtained in Anderson–Izzo–Wermer [3].

We define a real-valued function ε_X on the open unit ball int B, with ε_X(z, w) tending to 0 as (z, w) tends to X. We give a growth condition on ε_X(z, w) as (z, w) approaches X, and show that this condition is sufficient for R(X)=C(X) (Theorem 1.1).

In Section 4, we consider a class of sets X which are limits of a family of Levi-flat hypersurfaces in int B.

For each compact set Y in ℂ², we denote the rationally convex hull of Y by $\widehat{Y}$. A general reference is Rudin [8] or Aleksandrov [1].

We present a method of finding weighted Koppelman formulas for (p, q)-forms on n-dimensional complex manifolds X which admit a vector bundle of rank n over X×X, such that the diagonal of X×X has a defining section. We apply the method to ℙⁿ and find weighted Koppelman formulas for (p, q)-forms with values in a line bundle over ℙⁿ. As an application, we look at the cohomology groups of (p, q)-forms over ℙⁿ with values in various line bundles, and find explicit solutions to the $\overline{\partial}$-equation in some of the trivial groups. We also look at cohomology groups of (0, q)-forms over ℙⁿ×ℙ^m with values in various line bundles. Finally, we apply our method to developing weighted Koppelman formulas on Stein manifolds.

Let X be a smooth n-dimensional projective variety embedded in some projective space ℙ^N over the field ℂ of the complex numbers. Associated with the general projection of X to a space ℙ^N-m (N-m> n+1) one defines an extended Gauss map $\overline{\gamma}\colon\overline{X}\rightarrow\text{Gr}(n;N-m)$ (in case N-m>2n-1 this is the Gauss map of the image of X under the projection). We prove that $\overline{X}$ is smooth. In case any two different points of X do have disjoint tangent spaces then we prove that $\overline{\gamma}$ is injective.

Laplace operators on metric graphs are considered. It is proven that for compact graphs the spectrum of the Laplace operator determines the total length, the number of connected components, and the Euler characteristic. For a class of non-compact graphs the same characteristics are determined by the scattering data consisting of the scattering matrix and the discrete eigenvalues.

We introduce the notion of compactly locally reflexive Banach spaces and show that a Banach space X is compactly locally reflexive if and only if $\mathcal{K}(Y,X^{**})\subseteq\mathcal{K}(Y,X)^{**}$ for all reflexive Banach spaces Y. We show that X^* has the approximation property if and only if X has the approximation property and is compactly locally reflexive. The weak metric approximation property was recently introduced by Lima and Oja. We study two natural weak compact versions of this property. If X is compactly locally reflexive then these two properties coincide. We also show how these properties are related to the compact approximation property and the compact approximation property with conjugate operators for dual spaces.

Decomposable mappings from the space of symmetric k-fold tensors over E, $\bigotimes_{s,k}E$, to the space of k-fold tensors over F, $\bigotimes_{s,k}F$, are those linear operators which map nonzero decomposable elements to nonzero decomposable elements. We prove that any decomposable mapping is induced by an injective linear operator between the spaces on which the tensors are defined. Moreover, if the decomposable mapping belongs to a given operator ideal, then so does its inducing operator. This result allows us to classify injective linear operators between spaces of homogeneous approximable polynomials and between spaces of nuclear polynomials which map rank-1 polynomials to rank-1 polynomials.

The wave equation, ∂_ttu=Δu, in ℝⁿ⁺¹, considered with initial data u(x,0)=f∈H^s(ℝⁿ) and u’(x,0)=0, has a solution which we denote by $\frac{1}{2}(e^{it\sqrt{-\Delta}}f+e^{-it\sqrt{-\Delta}}f)$. We give almost sharp conditions under which $\sup_{0<t<1}|e^{\pm it\sqrt{-\Delta}}f|$ and $\sup_{t\in\mathbb{R}}|e^{\pm it\sqrt{-\Delta}}f|$ are bounded from H^s(ℝⁿ) to L^q(ℝⁿ).

In this note, we consider the regularity of solutions of the nonlinear elliptic systems of n-Laplacian type involving measures, and prove that the gradients of the solutions are in the weak Lebesgue space L^n,∞. We also obtain the a priori global and local estimates for the L^n,∞-norm of the gradients of the solutions without using BMO-estimates. The proofs are based on a new lemma on the higher integrability of functions.