Archiv der Mathematik最新文献

We determine the values of several infinite series involving the Riemann zeta function. In particular, degree one rationally weighted summands involving the Riemann function evaluated at even numbers give a finite sum involving only the Riemann function evaluated at odd numbers.

A complete characterization of the similarity between two operator-valued multishifts with invertible operator weights is obtained purely in terms of operator weights. This generalizes several existing results of the unitary equivalence of two (multi)shifts. Further, we utilize the aforementioned similarity criteria to determine the similarity between two tuples of operators of multiplication by the coordinate functions on certain reproducing kernel Hilbert spaces determined by diagonal kernels.

Let (f = f(z,t)) be a function holomorphic in (z in O subseteq {mathbb {C}}^d) for fixed (tin Omega ) and measurable in t for fixed z and such that (z mapsto f(z,cdot )) is bounded with values in (E:= textrm{L}_{p}(Omega )), (1le p le infty ). It is proved (among other things) that

whenever (mu in E') and (varphi ) is a bp-continuous linear functional on (textrm{H}^infty (O)).

This note presents an extension of a result within the concept of (left[ mathcal {S}right] )-lineability, originally due to Bernal-González, Conejero, Murillo-Arcila, and Seoane-Sepúlveda. Additionally, we provide a characterization of lineability in the context of complements of unions of closed subspaces in F-spaces in terms of (left[ ell _{infty }right] )-lineability. We also present a negative result in both normed spaces and p-Banach spaces. These findings contribute to the understanding of linearity within exotic settings in vector spaces.

Let (A=begin{bmatrix} A_{ij} end{bmatrix}) be an (ntimes n) operator matrix, where each (A_{ij}) is a bounded linear operator on a complex Hilbert space. Among other numerical radius bounds, we show that (w(A)le w({hat{A}}),) where ({hat{A}}=begin{bmatrix} {hat{a}}_{ij} end{bmatrix}) is an (ntimes n) complex matrix, with

This is a considerable improvement of the existing bound (w(A)le w({tilde{A}}),) where ({tilde{A}}=begin{bmatrix} {tilde{a}}_{ij} end{bmatrix}) is an (ntimes n) complex matrix, with

Further, applying the bounds, we develop the numerical radius bounds for the product of two operators and the commutator of operators. Also, we develop an upper bound for the spectral radius of the sum of the product of n pairs of operators, which improves the existing bound.

Finite groups are ubiquitous in mathematics and often arise as symmetry groups of objects. Consequently, finite group structure is of great interest. The transfer is a classical homomorphism of any finite group G into certain commutative sections of G. It has several basic applications in and has inspired new developments in finite group structure. In this article, we present a new characterization of the image of the transfer. Then we obtain new consequences and immediate proofs of old transfer consequences in finite group structure.

In 2016, Dovgoshey et al. introduced a metric (zeta ) on proper subdomains G of Euclidean spaces (mathbb {R}^k) and studied its connection with several hyperbolic type metrics. In this paper, we consider the Gromov hyperbolicity of ((G,zeta )) and show that there is a natural quasisymmetric homeomorphism between the Euclidean boundary of G and the Gromov boundary of ((G,zeta )) equipped with a visual metric.

Given a finite set of points (C subseteq {mathbb {R}}^d), we say that an ordering of C is protrusive if every point lies outside the convex hull of the points preceding it. We give an example of a set C of 5 points in the Euclidean plane possessing a protrusive ordering that cannot be obtained by ranking the points of C according to the sum of their distances to a finite multiset of points. This answers a question of Alon, Defant, Kravitz, and Zhu.

In this paper, we present counterexamples to maximal (L^p)-regularity for a parabolic PDE. The example is a second-order operator in divergence form with space and time-dependent coefficients. It is well-known from Lions’ theory that such operators admit maximal (L^2)-regularity on (H^{-1}) under a coercivity condition on the coefficients, and without any regularity conditions in time and space. We show that in general one cannot expect maximal (L^p)-regularity on (H^{-1}(mathbb {R}^d)) or (L^2)-regularity on (L^2(mathbb {R}^d)).

Jeon, Kang, and Kim defined the Zagier lifts between harmonic weak Maass forms of negative integral weights and half integral weights. These lifts were defined by establishing that traces related to cycle integrals of harmonic weak Maass forms of integral weights appear as Fourier coefficients of harmonic weak Maass forms of half integral weights. For fundamental discriminants d and (delta ,) they studied (delta )-th Fourier coefficients of the d-th Zagier lift with respect to the condition that (ddelta ) is not a perfect square. For (ddelta ) being a perfect square, the interpretation of coefficients in terms of traces is not possible due to the divergence of cycle integrals. In this paper, we provide an alternate definition of traces called modified trace in the condition that (ddelta ) is a perfect square and interpret such coefficients in terms of the modified trace.