Monatshefte für Mathematik最新文献

A lattice (Lambda ) is said to be an extension of a sublattice L of smaller rank if L is equal to the intersection of (Lambda ) with the subspace spanned by L. The goal of this paper is to initiate a systematic study of the geometry of lattice extensions. We start by proving the existence of a small-determinant extension of a given lattice, and then look at successive minima and covering radius. To this end, we investigate extensions (within an ambient lattice) preserving the successive minima of the given lattice, as well as extensions preserving the covering radius. We also exhibit some interesting arithmetic properties of deep holes of planar lattices.

We consider the Cauchy problem of a Camassa–Holm type equation with quadratic and cubic nonlinearities. We establish a new sufficient condition on the initial data that leads to the wave-breaking for this equation. Moreover, we obtain the persistence results of solutions for the equation in weighted (L^p) spaces.

Let G be a group which is generated by the set of its involutions, and assume that the set of integers which occur as orders of products of two involutions in G is ({1,2,3,4}). It is shown that (Gsimeq textrm{PSL}(2,7)) or (Gsimeq textrm{PSU}(3,3)) or G is a ({2,3})-group and (G/O_{2}(G) simeq S_{3}).

Let ({mathcal {M}}_{mathcal {D}}) be the dyadic maximal operator on ({mathbb {R}}^n). The paper contains the identification of the best constant in the two-weight estimate

under the assumption that the pair ((sigma ,w)) of weights satisfies an appropriate bump condition. The result is shown to be true in the larger context of abstract probability spaces equipped with a tree-like structure.

We provide answers to open questions from Banerjee and Gopaulsingh (Bull Pol Acad Sci Math 71: 1–21, 2023) about the relationship between the Erdős–Dushnik–Miller theorem ((textsf{EDM})) and certain weaker forms of the Axiom of Choice ((textsf{AC})), and we properly strengthen some results from Banerjee and Gopaulsingh (2023). We also settle a part of an open question of Lajos Soukup (stated in Banerjee and Gopaulsingh (2023) [Question 6.1]) about the relationship between the following two order-theoretic principles, which [as shown in Banerjee and Gopaulsingh (2023)] are weaker than (textsf{EDM}): (a) “Every partially ordered set such that all of its antichains are finite and all of its chains are countable is countable” (this is known as Kurepa’s theorem), and (b) “Every partially ordered set such that all of its antichains are countable and all of its chains are finite is countable”. In particular, we prove that (b) does not imply (a) in (textsf{ZF}) (i.e., Zermelo–Fraenkel set theory without (textsf{AC})). Moreover, with respect to (b), we answer an open question from Banerjee and Gopaulsingh (2023) about its relationship with the following weak choice form: “Every set is either well orderable or has an amorphous subset”; in particular, we show that (b) follows from, but does not imply, the latter weak choice principle in (textsf{ZFA}) (i.e., Zermelo–Fraenkel set theory with atoms).

Abstract

Main results of the paper are: the Lebesgue Density Theorem does not hold for the uniform density points and the uniform density topology is completely regular.

Abstract

We study the best coapproximation problem in Banach spaces, by using Birkhoff–James orthogonality techniques. We introduce two special types of subspaces, christened the anti-coproximinal subspaces and the strongly anti-coproximinal subspaces. We obtain a necessary condition for the strongly anti-coproximinal subspaces in a reflexive Banach space whose dual space satisfies the Kadets–Klee Property. On the other hand, we provide a sufficient condition for the strongly anti-coproximinal subspaces in a general Banach space. We also characterize the anti-coproximinal subspaces of a smooth Banach space. Further, we study these special subspaces in a finite-dimensional polyhedral Banach space and find some interesting geometric structures associated with them.

Abstract

We show that the power-law decay exponents in von Neumann’s Ergodic Theorem (for discrete systems) are the pointwise scaling exponents of a spectral measure at the spectral value 1. In this work we also prove that, under an assumption of weak convergence, in the absence of a spectral gap, the convergence rates of the time-average in von Neumann’s Ergodic Theorem depend on sequences of time going to infinity.

In this short note, we prove that given initial data (u_0in H^s(mathbb {R})) with (s>frac{3}{2}) and for some (T>0), the solution of the Camassa-Holm equation does not converges uniformly with respect to the initial data in (L^infty (0,T;H^s(mathbb {R}))) to the inviscid Burgers equation as the filter parameter (alpha ) tends to zero. This is a complement of our recent result on the zero-filter limit.

The study of the well-known partition function p(n) counting the number of solutions to (n = a_{1} + dots + a_{ell }) with integers (1 le a_{1} le dots le a_{ell }) has a long history in number theory and combinatorics. In this paper, we study a variant, namely partitions of integers into

with (1le a_1< cdots < a_ell ) and some fixed (0< alpha < 1). In particular, we prove a central limit theorem for the number of summands in such partitions, using the saddle-point method.