Mathematische Nachrichten最新文献

We prove a generalized version of the $��3�G�$ Principle for Green's functions on bounded inner uniform domains in a wide class of Dirichlet spaces. In particular, our results apply to higher-dimensional fractals such as Sierpinski carpets in $�R^n$, $��n�\ge �3�$, as well as generalized fractal-type spaces that do not have a well-defined Hausdorff dimension or walk dimension. This yields new instances of the $��3�G�$ principle for these spaces. We also discuss applications to Schrödinger operators.

In this paper, we develop a method of evaluating general exponential sums with rational amplitude functions for multiple variables, which complements work by T. Cochrane and Z. Zheng on the single variable case. As an application, for $��n�\ge �3�$, a fixed natural number, we obtain an asymptotic formula for the (weighted) number of solutions of general quadratic congruences modulo $�p^m$ in small boxes, thus establishing an equidistribution result for these solutions.

This paper deals with the qualitative properties of solutions to the Cauchy problem for a quasilinear convection–diffusion equation with volumetric moisture content, which involves the cases of critical and fast decaying volumetric moisture content. Based on the method of directly constructing Barenblatt-type super- and sub-solutions, we establish the new criteria of global existence and nonexistence and provide the properties of expansion and shrinking of the support of the weak solution.

In this paper, we consider the following Schrödinger–Poisson–Slater equation:

In this paper, we study the dynamics of the focusing nonlinear Hartree equation with a Kato potential

Assume that the section conjecture holds over number fields. We prove then that it holds for a broad class of curves defined over finitely generated extensions of $�Q$. This class contains every projective, hyperelliptic curve, every hyperbolic, affine curve of genus $��\le �2�$, and a basis of open subsets of any curve. If we furthermore assume the weak Bombieri–Lang conjecture, we prove that the section conjecture holds for every hyperbolic curve over every finitely generated extension of $�Q$.

We give two geometric interpretations for the local type of a line that is highly tangent to a hypersurface in a single point. One interpretation is phrased in terms of the Wronski map, while the other interpretation relates to the fundamental forms of the hypersurface. These local types are the local contributions of a quadratic form-valued Euler number that depends on a choice of orientation.

We prove the existence of a positive radial solution in a unit ball centered at the origin for some classes of Hénon equations involving supercritical nonlinearity. More precisely, we study how Hénon's weight impacts the variable supercritical exponent in the context of the work by do Ó, Ruf, and Ubilla. For this purpose, we combine variational methods with a new Sobolev–Hardy type embedding for radial functions into variable exponent Lebesgue spaces. The suitable use of the Radial Lemma allows us to arrive at the necessary estimate with fewer assumptions than those found in the existing literature.

Let $�E$ be a module of projective dimension 1 over a Noetherian ring $�R$ and consider its Rees algebra $��R�\left(�E�\right)�$. We study this ring as a quotient of the symmetric algebra $��S�\left(�E�\right)�$ and consider the ideal $�A$ defining this quotient. In the case that $��S�\left(�E�\right)�$ is a complete intersection ring, we employ a duality between $�A$ and $��S�\left(�E�\right)�$ in order to study the Rees ring $��R�\left(�E�\right)�$ in multiple settings. In particular, when $�R$ is a complete intersection ring defined by quadrics, we consider its module of Kähler differentials $�{\Omega }_{�R�/�k�}$