Results in Mathematics最新文献

We examine the properties of achievement sets of series in (mathbb {R}^2). We show several examples of unusual sets of subsums on the plane. We prove that we can obtain any set of P-sums as a cut of an achievement set in (mathbb {R}^2.) We introduce a notion of the spectre of a set in an Abelian group, which is an algebraic version of the notion of the center of distances. We examine properties of the spectre and we use it, for example, to show that the Sierpiński carpet is not an achievement set of any series.

In this paper, we study four discrete Bessel functions which are solutions to the discretization of Bessel differential equations when the time derivative is replaced by the forward and the backward difference. We focus on discrete Bessel equations with the time derivative being the backward difference and derive their solutions: the discrete I-Bessel function (overline{I}_n^c(t)) and the discrete J-Bessel function (overline{J}_n^c(t)), (tin mathbb {Z}), (nin mathbb {N}_0). We then study transformation properties of those functions and describe their asymptotic behaviour as (trightarrow infty ) and as (nrightarrow infty ). Moreover, we prove that the (unilateral) Laplace transform of (overline{I}_n^c) and (overline{J}_n^c) in the timescale (T=mathbb {Z}) with the delta derivative being the backward difference equals the Laplace transform of classical I-Bessel and J-Bessel functions (mathcal {I}_n(cx)) and (mathcal {J}_n(cx)), respectively. As an application, we study the discrete wave equation on the integers in the timescale (T=mathbb {Z}) and express its fundamental and general solution in terms of (overline{J}_n^c(t)). Going further, we show that the first fundamental solution of this discrete wave equation oscillates with the exponentially decaying amplitude as time tends to infinity.

We introduce and describe relations between Sobolev, Besov and Paley–Wiener spaces associated with three representations of the Lie group G of affine transformations of the line, also known as the ( ax + b) group. These representations are: left and right regular representations and a representation in a space of functions defined on the half-line. The Besov spaces are described as interpolation spaces between respective Sobolev spaces in terms of the K-functional and in terms of a relevant moduli of continuity. By using a Laplace operators associated with these representations a scales of relevant Paley–Wiener spaces are developed and a corresponding (L_{2})-approximation theory is constructed in which our Besov spaces appear as approximation spaces. Another description of our Besov spaces is given in terms of a frequency-localized Hilbert frames. A Jackson-type inequalities are also proven.

This paper is concerned with the homogeneous Dirichlet problem for a sublinear elliptic equation with unbounded coefficients in a Lipschitz domain. Bilateral a priori estimates for positive solutions and a priori upper estimates for their gradients are presented as a byproduct of the boundary Harnack principle. These estimates allow us to show the uniqueness of a positive solution of the homogeneous Dirichlet problem under no information about normal derivatives unlike in smooth domains.

Let G and H be locally compact groups. We will show that each contractive Jordan isomorphism (Phi :L^1(G)rightarrow L^1(H)) is either an isometric isomorphism or an isometric anti-isomorphism. We will apply this result to study isometric two-sided zero product preservers on group algebras and, further, to study local and approximately local isometric automorphisms of group algebras.

In this work, we introduce the notion of local and 2-local (delta )-derivations and describe local and 2-local (frac{1}{2})-derivation of finite-dimensional solvable Lie algebras with filiform, Heisenberg, and abelian nilradicals. Moreover, we describe the local (frac{1}{2})-derivation of oscillator Lie algebras, Schrödinger algebras, and the Lie algebra with the three-dimensional simple part, whose radical is an irreducible module. We prove that an algebra with only trivial (frac{1}{2})-derivation does not admit local and 2-local (frac{1}{2})-derivation, which is not (frac{1}{2})-derivation.

We consider tilings ((mathcal {Q},Phi )) of (mathbb {R}^d) where (mathcal {Q}) is the d-dimensional unit cube and the set of translations (Phi ) is constrained to lie in a pre-determined lattice (A mathbb {Z}^d) in (mathbb {R}^d). We provide a full characterization of matrices A for which such cube tilings exist when (Phi ) is a sublattice of (Amathbb {Z}^d) with any (d in mathbb {N}) or a generic subset of (Amathbb {Z}^d) with (dle 7). As a direct consequence of our results, we obtain a criterion for the existence of linearly constrained frequency sets, that is, (Phi subseteq Amathbb {Z}^d), such that the respective set of complex exponential functions (mathcal {E} (Phi )) is an orthogonal Fourier basis for the space of square integrable functions supported on a parallelepiped (Bmathcal {Q}), where (A, B in mathbb {R}^{d times d}) are nonsingular matrices given a priori. Similarly constructed Riesz bases are considered in a companion paper (Lee et al., Exponential bases for parallelepipeds with frequencies lying in a prescribed lattice, 2024. arXiv:2401.08042).

Beck introduced two important partition statistics NT(r, m, n) and (M_{omega }(r,m,n)) which count the total number of parts in the partitions of n with rank congruent to r modulo m and the total number of ones in the partitions of n with crank congruent to r modulo m, respectively. Andrews confirmed two conjectures of Beck on congruences of NT(r, m, n). Inspired by Andrews’ work, Chern discovered a number of congruences modulo 5, 7, 11 and 13 of NT(r, m, n) and (M_{omega }(r,m,n) ). Recently, Mao, and Xia, Yan and Yao established several identities on NT(r, 7, n) and (M_{omega }(r,7,n)) which yield some congruences modulo 7 due to Chern. Unfortunately, there are six congruences modulo 7 of Chern which are not implied by the identities given by Mao, and Xia, Yan and Yao. In this paper, we establish several new identities on NT(r, 7, n) and (M_{omega }(r,7,n)). In particular, we prove six identities which are analogous to “Ramanujan’s most beautiful identity”and imply Chern’s six congruences.

In the recent paper (Wang et al. in Differ Geom Appl 80:101840, 2022), the authors and Xu have established a Simons-type integral inequality for holomorphic curves in a complex Grassmann manifold G(k, N). In this paper, we completely classify holomorphic immersions from the two-sphere of constant curvature into G(3, N) with the norm of the second fundamental form satisfying the equality case of the inequality and prove that any such immersion can be decomposed as the “direct sum” of some “foundation stones” up to congruence.

Recently, Guo and Schlosser (Results Math 78:105, 2023) gave two interesting q-supercongruences. With the help of the creative microscoping method introduced by Guo and Zudilin, and Jackson’s (_{6}phi _{5}) summation formula, we establish one-parameter generalizations of them in this paper.