Archiv der Mathematik最新文献

Let A be a finite group (or a finite algebra), and (omega ) be a word map (resp. polynomial map) on n many generators. We define the quantity (|omega (A)|/|A|) as the image ratio of (omega ) on A and denote it by (mu (omega ,A)). In this article, we investigate the set (textrm{R}(omega )={mu (omega , A) : A {text { is a finite group}}}), and study the same for the case of rings. We demonstrate the existence of word maps whose set of image ratios is dense in [0, 1] for groups (and rings).

We investigate the inverse problem of identifying a space-depen- dent potential in a Caputo time-fractional diffusion equation from boundary observations. Our analysis establishes a maximum principle for subdiffusion equations with non-homogeneous Neumann boundary conditions, demonstrating the positivity of solutions under specific assumptions on the boundary data. Building upon this result, we leverage the Gâteaux differentiability of the forward map and the non-vanishing property of its derivative to derive a local Lipschitz stability estimate for the inverse potential problem. This provides a rigorous foundation for the stable reconstruction of the potential, highlighting the interplay between fractional dynamics and stability in inverse problems.

In this paper, by using a special Euler–Ramanujan identity and the idea of Wick rotation, we show that a one-parameter family of solutions to the zero mean curvature equation in the Lorentz–Minkowski 3-space (mathbb {E}_1^3), namely Scherk-type zero mean curvature surfaces, can be expressed as an infinite superposition of dilated helicoids. Further, we also obtain different finite decompositions for these surfaces. We end this paper with an application of these decompositions to formulate maximal codimension 2 surfaces into finite and infinite “sums” of weakly untrapped and (*)-surfaces in the Lorentz–Minkowski 4-space.

By virtue of the Nash–Moser iteration, we obtain a local gradient estimate of positive weak solutions to the weighted p-Laplacian equation

defined on a complete smooth metric measure space under the condition that the m-Bakry–Émery Ricci curvature has a lower bound, where (p>2) and the function (a(x)le 0.) As applications, Liouville-type theorems for positive solutions to the above equation are achieved.

We determine which simplicial complexes with a given number of vertices have the maximum sum of bigraded Betti numbers.

For a connected orientable hyperbolic surface S without boundary and of finite topological type, the Johnson kernel (mathcal {K}(S)) is the subgroup of the mapping class group of S generated by Dehn twists about separating simple closed curves on S. We prove that (mathcal {K}(S)) is generated by the Dehn twists about separating simple closed curves on S bounding either: a closed subsurface of genus 1 or 2; a closed subsurface of genus 1 minus one point; a closed disc minus two points.

For even (kge 6) and square free (D>1) with (Dequiv 1pmod 4), let (chi _D) be the primitive quadratic Dirichlet character mod D and (S_k(D,chi _D)) be the space of cusp forms of weight k, level D, and nebentypus (chi _D). We show that if (D>2^{k-2}), then the critical values of symmetric square L-functions on (S_k(D,chi _D)) are linearly independent.

In the situation where a finite group A acts coprimely by automorphisms on another finite group G, we characterize when the principal p-block of G contains a unique A-invariant irreducible character, in terms of the subgroup (textbf{C}_{G}(A)) of fixed points under such action.