Monatshefte fur Mathematik最新文献

We consider 2D Maxwell-Lorentz equations with an extended charged rotating particle. The system admits solitons which are solutions corresponding to a particle moving with constant velocity and rotating with constant angular velocity. Our main result is asymptotic stability of moving solitons with zero angular velocity.

In this article we completely characterise constant length substitution shifts which have a proper almost automorphic factor, or which have a bijective substitution factor such that the factor map is injective on at least one point. Our approach is algebraic: we characterise these dynamical properties in terms of a finite semigroup defined by the substitution. We characterise the existence of almost automorphic factors in terms of Green's $R$ -relation and the existence of bijective factors in terms of Green's $L$ -relation. Our results are constructive.

We present in this paper a new way to define weighted Sobolev spaces when the weight functions are arbitrary small. This new approach can replace the old one consisting in modifying the domain by removing the set of points where at least one of the weight functions is very small. The basic idea is to replace the distributional derivative with a new notion of weak derivative. In this way, non-locally integrable functions can be considered in these spaces. Indeed, assumptions under which a degenerate elliptic partial differential equation has a unique non-locally integrable solution are given. Tools like a Poincaré inequality and a trace operator are developed, and density results of smooth functions are established.

H. Weyl proved in Weyl (Eins Math Ann 77(3):313-352, 1916) that integer evaluations of polynomials are equidistributed mod 1 whenever at least one of the non-free coefficients (namely a coefficient of a monomial of degree at least 1) is irrational. We use Weyl's result to prove a higher dimensional analogue of this fact. Namely, we prove that evaluations of polynomials on lattice points are equidistributed mod 1 whenever at least one of the non-free coefficients is irrational. This result improves the main result of Arhipov et al. (Mat Zametki 25(1):3-14, 157, 1979). We prove this analogue as a corollary of a theorem that guarantees equidistribution of grid evaluations mod 1 for all functions which satisfy some restraints on their derivatives. Another corollary we prove is that for $p\in (1,\infty )$ the ${\ell }^p$ norms of integer vectors are equidistributed mod 1.

We study the spectral bounds of self-adjoint operators on the Hilbert space of square-integrable functions, arising from the representation theory of the Heisenberg group. Interestingly, starting either with the von Neumann lattice or the hexagonal lattice of density 2, the spectral bounds obey well-known arithmetic-geometric mean iterations. This follows from connections to Jacobi theta functions and Ramanujan's corresponding theories. As a consequence, we rediscover that these operators resemble the identity operator as the density of the lattice grows. We also prove that the conjectural value of Landau's constant is obtained as half the cubic arithmetic-geometric mean of $\sqrt[3]{2}$ and 1, which we believe to be a new result.

Gödel, working within axiomatic logic, succeeded in 1933 in establishing a translation from theorems of intuitionistic propositional logic to ones of classical logic enriched with a modal provability operator. The converse correspondence was established by semantical means in 1948, and by Gödel through a syntactic translation in unpublished work of 1941. It is shown through proof analysis of formal derivations in natural deduction for modal logic that steps of indirect proof in normal derivations of translations of intuitionistic formulas are vacuous. This conservativity of classical over intuitionistic modal logic for translated formulas is the reason why Gödel's modal translation succeeds in singling out a "provability fragment" within classical modal logic that coincides with intuitionistic logic.

We study conservation laws with a discontinuous flux function $f(x,\lambda )$ . The flux function can be expressed as $g\left(\beta \right(x,\lambda \left)\right)$ , where g is locally Lipschitz, $\beta (x,\lambda )$ is an increasing function in $\lambda$ for each fixed $x$ , $\nabla \beta (·,0)$ is a finite measure, and $\beta (·,0)$ is bounded. We consider this problem under the Audusse-Perthame entropy condition and derive a kinetic formulation. Using the kinetic approach, we prove an existence result under the assumption that the initial function belongs to $L^1$ . Uniqueness results are also presented.

We study the generic behavior of the method of successive approximations for set-valued mappings in separable Banach spaces. We consider the case of nonexpansive mappings with convex and compact point images and show that for the typical such mapping and typical points of its domain the sequence of successive approximations is unique and converges to a fixed point of the mapping.

We investigate the existence of solutions to a recent model for large-scale equatorial waves, derived recently by an asymptotic method driven by the thin-shell approximation of the Earth's atmosphere in rotating spherical coordinates.

We prove that a family of Diophantine series satisfies an approximate functional equation. It generalizes a result by Rivoal and Roques and proves an extended version of a conjecture posed in their paper. We also characterize the convergence points.