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## Abstract

The concept of inequalities in time scales has attracted the attention of mathematicians for a quarter century. And these studies have inspired the solution of many problems in the branches of physics, biology, mechanics and economics etc. In this article, new principles of non-linear integral inequalities are presented in time scales via diamond-α dynamic integral and the nabla integral.

## Abstract

We compute a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. In this special case, we require the given supercuspidal representation to contain a skew maximal simple stratum, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a canonical lifting of its underlying simple character, together with the base change of the level-zero component of its inducing cuspidal type, modified by a sign attached to a quadratic Gauss sum defined by the internal structure of the simple character. To obtain this result, we study the reducibility points of a parabolic induction and the corresponding module over the affine Hecke algebra, defined by the covering type over the product of types of the given supercuspidal representation and of a candidate of its base change.

## Abstract

We show that any Weyl group orbit of weights for the Tannakian group of semisimple holonomic $𝒟$-modules on an abelian variety is realized by a Lagrangian cycle on the cotangent bundle. As applications we discuss a weak solution to the Schottky problem in genus five, an obstruction for the existence of summands of subvarieties on abelian varieties, and a criterion for the simplicity of the arising Lie algebras.

OPEN ACCESS

## Abstract

The Plateau–Douglas problem asks to find an area minimizing surface of fixed or bounded genus spanning a given finite collection of Jordan curves in Euclidean space. In the present paper we solve this problem in the setting of proper metric spaces admitting a local quadratic isoperimetric inequality for curves. We moreover obtain continuity up to the boundary and interior Hölder regularity of solutions. Our results generalize corresponding results of Jost and Tomi-Tromba from the setting of Riemannian manifolds to that of proper metric spaces with a local quadratic isoperimetric inequality. The special case of a disc-type surface spanning a single Jordan curve corresponds to the classical problem of Plateau, in proper metric spaces recently solved by Lytchak and the second author.

## Abstract

We give an affirmative solution to a conjecture of Cheng proposed in 1979 which asserts that the Bergman metric of a smoothly bounded strongly pseudoconvex domain in $ℂn,n≥2$, is Kähler–Einstein if and only if the domain is biholomorphic to the ball. We establish a version of the classical Kerner theorem for Stein spaces with isolated singularities which has an immediate application to construct a hyperbolic metric over a Stein space with a spherical boundary.

## Abstract

In recent years, a number of papers have been devoted to the study of zeros of period polynomials of modular forms. In the present paper, we study cohomological analogues of the Eichler–Shimura period polynomials corresponding to higher L-derivatives. We state a general conjecture about the locations of the zeros of the full and odd parts of the polynomials, in analogy with the existing literature on period polynomials, and we also give numerical evidence that similar results hold for our higher derivative “period polynomials” in the case of cusp forms. The unimodularity of the roots seems to be a very subtle property which is special to our “period polynomials”. This is suggested by numerical experiments on families of perturbed “period polynomials” (Section ) suggested by Zagier. We prove a special case of our conjecture in the case of Eisenstein series.

Although not much is currently known about derivatives higher than first order ones for general modular forms, celebrated recent work of Yun and Zhang established the analogues of the Gross–Zagier formula for higher L-derivatives in the function field case. A critical role in their work was played by a notion of “super-positivity”, which, as recently shown by Goldfeld and Huang, holds in infinitely many cases for classical modular forms. As will be discussed, this is similar to properties which were required by Jin, Ma, Ono, and Soundararajan in their proof of the Riemann Hypothesis for Period Polynomials, thus suggesting a connection between the analytic nature of our conjectures here and the framework of Yun and Zhang.

FREE ACCESS

## Abstract

In this paper we obtain rigidity results for a non-constant entire solution u of the Allen–Cahn equation in $ℝn$, whose level set ${u=0}$ is contained in a half-space. If $n≤3$, we prove that the solution must be one-dimensional. In dimension $n≥4$, we prove that either the solution is one-dimensional or stays below a one-dimensional solution and converges to it after suitable translations. Some generalizations to one phase free boundary problems are also obtained.