We consider a version of height on polynomial spaces defined by the integral over the normalized area measure on the unit disk. This natural analog of Mahler’s measure arises in connection with extremal problems for Bergman spaces. It inherits many nice properties such as the multiplicative one. However, this height is a lower bound for Mahler’s measure, and it can be substantially lower. We discuss some similarities and differences between the two.

This paper is concerned with derivation of the global or local in time Strichartz estimates for radially symmetric solutions of the free wave equation from some Morawetz-type estimates via weighted Hardy–Littlewood–Sobolev (HLS) inequalities. In the same way, we also derive the weighted end-point Strichartz estimates with gain of derivatives for radially symmetric solutions of the free Schrödinger equation.

The proof of the weighted HLS inequality for radially symmetric functions involves an application of the weighted inequality due to Stein and Weiss and the Hardy–Littlewood maximal inequality in the weighted Lebesgue space due to Muckenhoupt. Under radial symmetry, we get significant gains over the usual HLS inequality and Strichartz estimate.

We compute a basis for the $p$-adic Dwork cohomology of a smooth complete intersection in projective space over a finite field and use it to give $p$-adic estimates for the action of Frobenius on this cohomology. In particular, we prove that the Newton polygon of the characteristic polynomial of Frobenius lies on or above the associated Hodge polygon. This result was first proved by B. Mazur using crystalline cohomology.

This paper focuses on finite dimensional point derivations for the non-self-adjoint operator algebras corresponding to directed graphs. We begin by analyzing the derivations corresponding to full matrix representations of the tensor algebra of a directed graph. We determine when such a derivation is inner, and describe situations that give rise to noninner derivations. We also analyze the situation when the derivation corresponds to a multiplicative linear functional.

This paper is concerned with the problem of existence of invariant sublattices for a positive matrix or a positive operator on $L_p(\mu)$. Common invariant sublattices for certain semigroups of positive operators are constructed. The paper also provides extensions of Perron-Frobenius theorem.

We study the invertibility of the operator $ \beta I - K^*$ in $H^{-\alpha} (\partial\Omega),\ 0\leq\alpha\leq1$ for $\beta\in \mathbf{C} \setminus(-\frac12 , \frac12]$ where $K^*$ is a adjoint operator of the double layer potential $K$ related to the Laplace equation and $\Omega$ is a bounded Lipschitz domain in $\mathbf{R}^n$. Consequently, the spectrum on the real line lies in $(-\frac12 , \frac12]$.

We settle the problem of finding an entire function with three singular values whose Nevanlinna characteristic dominates an arbitrarily prescribed function.

We introduce a new quotient construction of toric Deligne–Mumford stacks. We use this new construction to define toric stack bundles which generalize the construction of toric bundles by Sankaran and Uma [Comment. Math. Helv. 78 (2003) 540–554]. The orbifold Chow ring of such toric stack bundles is computed. We show that the orbifold Chow ring of the toric stack bundle and the Chow ring of its crepant resolution are fibres of a flat family, generalizing a result of Borisov–Chen–Smith.

We give sufficient conditions on an asymptotic $\ell_p$ (for $1 < p < \infty$) Banach space to ensure the space admits an operator, which is not a compact perturbation of a multiple of the identity. These conditions imply the existence of strictly singular noncompact operators on the HI spaces constructed by G. Androulakis and the author and by Deliyanni and Manoussakis. Additionally, we show that under these same conditions on the space $X$, $\ell_\infty$ embeds isomorphically into the space of bounded linear operators on $X$.

For analytic self-maps $\varphi$ of the unit disk, we develop a necessary and sufficient condition for the composition operator $C_{\varphi}$ to be closed-range on the classical Bergman space $\mathbb{A}^2$. This condition is relatively easy to apply. Particular attention is given to the case that $\varphi$ is an inner function. Included are observations concerning angular derivatives of Blaschke products. In the case that $\varphi$ is univalent, it is shown that $C_{\varphi}$ is closed-range on $\mathbb{A}^2$ only if $\varphi$ is an automorphism of the disk.

An Artin algebra is by definition virtually Gorenstein if the class of modules which are right orthogonal (with respect to $\operatorname {Ext}^*(-,-)$) to all Gorenstein projective modules coincides with the class of modules which are left orthogonal to all Gorenstein injective modules. We provide a new characterization in terms of finitely generated modules. In addition, an example of an algebra is presented which is not virtually Gorenstein.

We develop a unified approach to characterize approximation properties defined by spaces of operators. Our main result describes them in terms of the approximability of weak*-weak continuous operators. In particular, we prove that if $\mathcal{A}$ and~$\mathcal{B}$ are operator ideals satisfying $\mathcal{A}\circ\mathcal{B}^{*}\subset\mathcal{K}$, then the $\mathcal{A}(X,X)$-approximation property of a Banach space $X$ is equivalent to the following “metric” condition: for every Banach space $Y$ and for every operator $T\in\mathcal{B}^{*}(Y,X)$, there exists a net $(S_{\alpha})\subset\mathcal{A}(X,X)$ such that $\sup_\alpha\|S_{\alpha}T\| \leq\|T\|$ and $T^{*}S_{\alpha}^{*}\rightarrow T^{*}$ in the strong operator topology on $\mathcal{L}(X^{\ast},Y^{\ast})$. As application, approximation properties of dual spaces and weak metric approximation properties are studied.

Let $\mathcal M$ be an $o$-minimal expansion of a real closed field. We prove the definable smoothing of definable Lipschitz continuous functions. In the case of Lipschitz functions of one variable, we are even able to preserve the Lipschitz constant.

Given a three-dimensional complex algebraic variety with isolated singular point and a sufficiently fine complete resolution of the singularity, we can make a careful choice of hyperplane that allows us to construct an exact sequence of weighted Nash complexes.

We study the restriction of the Fourier transform to quadratic surfaces in vector spaces over finite fields. In two dimensions, we obtain the sharp result by considering the sums of arbitrary two elements in the subset of quadratic surfaces on two dimensional vector spaces over finite fields. For higher dimensions, we estimate the decay of the Fourier transform of the characteristic functions on quadratic surfaces so that we obtain the Tomas–Stein exponent. Using incidence theorems, we also study the extension theorems in the restricted settings to sizes of sets in quadratic surfaces. Estimates for Gauss and Kloosterman sums and their variants play an important role.

In this paper, we construct some invariants of spatial graphs by disk-summing the constituent knots and show the delta edge-homotopy invariance of them. As an application, we show that there exist infinitely many slice spatial embeddings of a planar graph up to delta edge-homotopy, and there exist infinitely many boundary spatial embeddings of a planar graph up to delta edge-homotopy.

Let $R$ be an integral domain. We study the structure of $R$ under the condition that the orbit space $R/\mathit{Aut}(R)$ is finite. It is proved that if $R$ is Noetherian, then $ |R/\mathit{Aut}(R)|= \infty$ unless $R$ is a finite field (Theorem 15 and Corollary 16). Furthermore, we give an example of an infinite integral domain with $|R/\mathit{Aut}(R)|< \infty$.

It is shown that the $L^p_w,1<p<\infty$, operator norms of Littlewood--Paley operators are bounded by a multiple of $\|w\|_{A_p}^{\gamma_p}$, where $\gamma_p=\max\{1,p/2\}\frac {1}{p-1}$. This improves previously known bounds for all $p>2$. As a corollary, a new estimate in terms of $\|w\|_{A_p}$ is obtained for the class of Calderón-Zygmund singular integrals commuting with dilations.

Let $0<\alpha<2$ and $X_t$ be the isotropic $\alpha$-stable Lévy process. We define scattering length $\Gamma(v)$ of a positive potential $v$. We use the scattering length to find estimates for the first eigenvalue of the Schrödinger operator of the “Neumann” fractional Laplacian in a cube with a potential $v$.

In this paper, we show how to compute the $\Lambda_{\alpha}$ norm, $\alpha\ge0$, using the dyadic grid. This result is a consequence of the description of the Hardy spaces $H^p(\mathbb{R}^N)$ in terms of dyadic and special atoms.

We show that if a class of modules is closed under pure quotients, then it is precovering if and only if it is covering, and this happens if and only if it is closed under direct sums. This is inspired by a dual result by Rada and Saorín.

We also show that if a class of modules contains the ground ring and is closed under extensions, direct sums, pure submodules, and pure quotients, then it forms the first half of a so-called perfect cotorsion pair as introduced by Salce; this is stronger than being covering.

Some applications are given to concrete classes of modules such as kernels of homological functors and torsion free modules in a torsion pair.