Archiv der Mathematik最新文献

The characterization of dual Steffensen–Popoviciu measures has so far been an open problem. As the main contribution of this paper, we give a complete characterization of dual Steffensen–Popoviciu measures on compact intervals.

The aim of this study is twofold. Initially, by employing a perturbation semigroup approach and admissible observation operators, a novel variation of constants formula is presented for the mild solutions of a specific set of integrodifferential equations in Banach spaces. Subsequently, utilizing this formula, an examination of the maximal (L^p)-regularity for such equations is conducted through the application of the sum operator method established by Da Prato and Grisvard. Importantly, it is demonstrated that the maximal (L^p)-regularity of an integrodifferential equation is equivalent to that of the same equation when the integral term is omitted. Furthermore, a finding concerning the strong solution of an initial value integrodifferential equation is provided when the initial condition pertains to the trace space.

In this article, we investigate some isoperimetric-type inequalities related to the first eigenvalue of the fractional composite membrane problem. First, we establish an analogue of the renowned Faber–Krahn inequality for the fractional composite membrane problem. Next, we investigate an isoperimetric inequality for the first eigenvalue of the fractional composite membrane problem on the intersection of two domains - a problem that was first studied by Lieb (Invent Math 74(3):441–448, 1983) for the Laplacian. Similar results in the local case were previously obtained by Cupini–Vecchi (Commun Pure Appl Anal 18(5):2679–2691, 2019) for the composite membrane problem. Our findings provide further insights into the fractional setting, offering a new perspective on these classical inequalities.

We determine the average size of the eigenvalues of the Hecke operators acting on the cuspidal modular forms space (S_k(Gamma _0(N))) in both the vertical and the horizontal perspective. The “average size” is measured via the quadratic mean.

Let (C_{n}) be the n-th Catalan number. In this note, we prove that the product of two different Catalan numbers cannot be a square of an integer. On the other hand, for each (kge 3), there are infinitely many k-tuples of pairwise different Catalan numbers with product being squares. We also obtain a characterization of (xin mathbb {N}_{+}) such that (C_{x}C_{x+1}) is a power-full number and prove that there are infinitely many such x. Moreover we present some numerical results which motivate further problems.

Hou, Krattenthaler, and Sun have introduced two q-analogues of a remarkable series for (pi ^2) due to Guillera, and these q-identities were, respectively, proved with the use of a q-analogue of a Wilf–Zeilberger pair provided by Guillera and with the use of ( _{3}phi _{2})-transforms. We prove a q-analogue of Guillera’s formula for (pi ^2) that is inequivalent to previously known q-analogues of the same formula due to Guillera, including the Hou–Krattenthaler–Sun q-identities and a subsequent q-identity due to Wei. In contrast to previously known q-analogues of Guillera’s formula, our new q-analogue involves another free parameter apart from the q-parameter. Our derivation of this new result relies on the q-analogue of Zeilberger’s algorithm.

Let P be a nonzero projective module over an integral group ring. We consider the question of whether the rank of P is necessarily positive.

In this short note, we provide a quantitative global Poincaré inequality for one-forms on a closed Riemannian four manifold, in terms of an upper bound on the diameter, a positive lower bound on the volume, and a two-sided bound on the Ricci curvature. This seems to be the first non-trivial result giving such an inequality without any higher curvature assumptions. The proof is based on a Hodge theoretic result on orbifolds, a comparison for fundamental groups, and a spectral convergence with respect to Gromov–Hausdorff convergence, via a degeneration result to orbifolds by Anderson.

In a recent paper, Hou et al. conjectured that there exist no regular maps of order (2^n) and of type ({2^k,2^s}), where n, k, and s are positive integers satisfying (2le s<k<n-1) and (s+k>n). In this paper, we give an affirmative answer to this conjecture.