Archiv der Mathematik最新文献

Using Fedder’s criterion, we classify all non-F-split del Pezzo surfaces of degree 1. We give a necessary and sufficient criterion for the F-splitting of such del Pezzo surfaces in terms of their anti-canonical system.

We study the Dirichlet dynamical zeta function (eta _D(s)) for billiard flow corresponding to several strictly convex disjoint obstacles. For large ({{,textrm{Re},}}s), we have (eta _D(s) =sum _{n= 1}^{infty } a_n e^{-lambda _n s}, , a_n in {mathbb {R}}), and (eta _D) admits a meromorphic continuation to ({mathbb {C}}). We obtain some conditions of the frequencies (lambda _n) and some sums of coefficients (a_n) which imply that (eta _D) cannot be prolonged as an entire function.

In his 1934 paper, G. Birkhoff poses the problem of classifying pairs (G, U) where G is an abelian group and (Usubset G) a subgroup, up to automorphisms of G. In general, Birkhoff’s problem is not considered feasible. In this note, we fix a prime number p and assume that G is a direct sum of cyclic p-groups and (Usubset G) is a subgroup. Under the assumption that the factor group G/U is an elementary abelian p-group, we show that the pair (G, U) always has a direct sum decomposition into pairs of type (({mathbb {Z}}/(p^n),{mathbb {Z}}/(p^n))) or ((mathbb {Z}/(p^n), (p))). Surprisingly, in the dual situation, we need an additional condition. If we assume that U itself is an elementary subgroup of G, then we show that the pair (G, U) has a direct sum decomposition into pairs of type (({mathbb {Z}}/(p^n),0)) or ((mathbb {Z}/(p^n), (p^{n-1}))) if and only if G/U is a direct sum of cyclic p-groups. We generalize the above results to modules over commutative discrete valuation rings.

Let K be an N-dimensional simplicial complex. We investigate the spectrum of the up Laplacian matrix of K. Let L be the ((N-1))th up Laplacian matrix of K. We show that the largest eigenvalues of L and |L| are equal if and only if K is disorientable. We also derive lower bounds for the sum of the first k largest eigenvalues of L.

Let d be a square-free integer such that (d equiv 15 pmod {60}) and Pell’s equation (x^2 - dy^2 = -6) is solvable in rational integers x and y. In this paper, we prove that there exist infinitely many Diophantine quadruples in (mathbb {Z}[sqrt{d}]) with the property D(n) for certain n’s. As an application of it, we ‘unconditionally’ prove the existence of infinitely many rings (mathbb {Z}[sqrt{d}]) for which the conjecture of Franušić and Jadrijević (Conjecture 1.1) does ‘not’ hold. This conjecture states a relationship between the existence of a Diophantine quadruple in (mathcal {R}) with the property D(n) and the representability of n as a difference of two squares in (mathcal {R}), where (mathcal {R}) is a commutative ring with unity.

Let (textrm{Sym}_q(m)) be the space of symmetric matrices in ({mathbb {F}}_q^{mtimes m}). A subspace of (textrm{Sym}_q(m)) equipped with the rank distance is called an ({{mathbb {F}}}_{q})-linear symmetric rank-metric code. In this paper, we study the covering properties of ({{mathbb {F}}}_{q})-linear symmetric rank-metric codes. First we characterize ({{mathbb {F}}}_{q})-linear symmetric rank-metric codes which are perfect, i.e., that satisfy the equality in the sphere-packing like bound. We show that, despite the rank-metric case, there are non-trivial perfect codes. Indeed, we prove that the only perfect non-trivial ({{mathbb {F}}}_{q})-linear symmetric rank-metric codes in (textrm{Sym}_q(m)) are the symmetric MRD codes with minimum distance 3 and m odd. Also, we characterize families of codes which are quasi-perfect.

In (Differential Geom. Appl. 92: Paper No. 102078, 12 pp., 2024), an analogue of Brylinski’s knot beta function was defined for a compactly supported (Schwartz) distribution T on Euclidean space. Here we consider the Brylinski beta function of the distribution defined by a coaxial layer on a submanifold of Euclidean space. We prove that it has an analytic continuation to the whole complex plane as a meromorphic function with only simple poles, and in the case of a coaxial layer on a space curve, we compute some of the residues in terms of the curvature and torsion.

Lehmer conjectured that Ramanujan’s tau-function never vanishes. As a variation of this conjecture, it is proved that

where (ell <100) is an odd prime, by Balakrishnan, Ono, Craig, Tsai, and many people. We prove that

for (ell in L), where L is an explicit finite subset of odd primes less than 1000.

An element in a topological group is called compact or periodic if it is contained in a compact subgroup. In a general locally compact group, the compact elements will not be closed under multiplication. We show that the set of all compact elements forms a subgroup if a more general periodicity property is satisfied.

A simple necessary and sufficient condition is given for exponential polynomials and absolutely convergent Dirichlet series with imaginary exponents and only real zeros to be a finite product of sines. The proof is based on Meyer’s theorem on quasicrystals.