Archiv der Mathematik最新文献

For each even integer (k ge 2), we construct an explicit real two-dimensional family (C^{(k)}_{r,theta }) of non-hyperelliptic pseudo-real Riemann surfaces of genus (g=1+(2k-3)k^{4}). For each of them, we compute its field of moduli and also a minimal field of definition.

The illumination conjecture is a classical open problem in convex and discrete geometry, asserting that every compact convex body K in (mathbb {R}^n) can be illuminated by a set of no more than (2^n) points. If K has smooth boundary, it is known that (n+1) points are necessary and sufficient. We consider an effective variant of the illumination problem for bodies with smooth boundary, where the illuminating set is restricted to points of a lattice and prove the existence of such a set close to K with an explicit bound on the maximal distance. We produce improved bounds on this distance for certain classes of lattices, exhibiting additional symmetry or near-orthogonality properties. Our approach is based on the geometry of numbers.

Let G be a group with undecidable domino problem, such as ({mathbb {Z}}^2). We prove that all nontrivial dynamical properties for sofic G-subshifts are undecidable, that this is not true for G-SFTs, and an undecidability result for dynamical properties of G-SFTs similar to the Adian–Rabin theorem. Furthermore, we prove that every computable real-valued dynamical invariant for G-SFTs that is monotone by disjoint unions and products is constant.

The aim of this note is to exhibit proper first Brill-Noether loci inside the moduli spaces (M_{Y,H}(2;c_1,c_2)) of H-stable rank 2 vector bundles with fixed Chern classes of a certain type on an Enriques surface Y which is covered by a Jacobian Kummer surface X.

Let (beta in (0,n)). In this paper, we study the boundedness of the Calderón–Zygmund type singular integral

on the space (textrm{BMO}(mathbb {R}^n)). Precisely, let (qin (1,infty )) and (beta in (0,frac{(q-1)n}{q})). We prove that, for any (fin textrm{BMO}(mathbb {R}^n)cap L^{q'}(mathbb {R}^n)), (Tfin textrm{BMO}(mathbb {R}^n)) and

where (q'in (1,infty )) is given by (1/q+1/q'=1) and C is a positive constant independent of (beta ) and f. This estimate can be seen as a further development for the corresponding results in the scale of Lebesgue spaces, established by Chen and Guo (J Funct Anal 281:Paper No. 109196, 2021), in the endpoint case.

Let K be a global field, that is, a number field or a global function field. It is known that the answer to the question in the title over K is “Yes” when K has no real embeddings. We show that otherwise the answer is “No”. Namely, we show that when K is a number field admitting a real embedding, it is impossible to define a group structure on the first Galois cohomology sets (textrm{H}^1hspace{-0.8pt}(K,G)) for all reductive K-groups G in a functorial way.

The limiting function f(s) of the pair correlation

for a sequence ((x_N)_{N in mathbb {N}}) on the torus (mathbb {T}^1) is said to be Poissonian if it exists and equals 2s for all (s ge 0). For instance, independent, uniformly distributed random variables generically have this property. Obviously f(s) is always a monotonic function if existent. There are only few examples of sequences where (f(s) ne 2s), but where the limit can still be explicitly calculated. Therefore, it is an open question which types of functions f(s) can or cannot appear here. In this note, we give a partial answer on this question by addressing the case that the number of different gap lengths in the sequence is finite and showing that f cannot be continuous then.

Burness and Scott (J Aust Math Soc 87:329-357, 2009) classified finite groups G such that the number of prime order subgroups of G is greater than (|G|/2-1). In this note, we study finite groups G whose subgroup graph contains a vertex of degree greater than (|G|/2-1). The classification given for finite solvable groups extends the work of Burness and Scott.

Recent work by Craig, van Ittersum, and Ono constructs explicit expressions in the partition functions of MacMahon that detect the prime numbers. Furthermore, they define generalizations, the MacMahonesque functions, and prove there are infinitely many such expressions in these functions. Here, we show how to modify and adapt their construction to detect cubes of primes as well as primes in arithmetic progressions.

Let I be an arbitrary nonzero squarefree monomial ideal of dimension d in a polynomial ring (S = textrm{k}[x_1,ldots ,x_n]). Let (mu ) be the number of associated primes of S/I of dimension d. We prove that the multiplicity of powers of I is given by

for all (s ge 1). Consequently, we compute the multiplicity of all powers of path ideals of cycles.