Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg最新文献

Assuming that a plane partition of the positive integer n is chosen uniformly at random from the set of all such partitions, we propose a general asymptotic scheme for the computation of expectations of various plane partition statistics as n becomes large. The generating functions that arise in this study are of the form Q(x)F(x), where (Q(x)=prod _{j=1}^infty (1-x^j)^{-j}) is the generating function for the number of plane partitions. We show how asymptotics of such expectations can be obtained directly from the asymptotic expansion of the function F(x) around (x=1). The representation of a plane partition as a solid diagram of volume n allows interpretations of these statistics in terms of its dimensions and shape. As an application of our main result, we obtain the asymptotic behavior of the expected values of the largest part, the number of columns, the number of rows (that is, the three dimensions of the solid diagram) and the trace (the number of cubes in the wall on the main diagonal of the solid diagram). Our results are similar to those of Grabner et al. (Comb Probab Comput 23:1057–1086, 2014) related to linear integer partition statistics. We base our study on the Hayman’s method for admissible power series.

A symplectic polarity of a building (varDelta ) of type (mathsf {E_6}) is a polarity whose fixed point structure is a building of type (mathsf {F_4}) containing residues isomorphic to symplectic polar spaces (i.e., so-called split buildings of type (mathsf {F_4})). In this paper, we show in a geometric way that every building of type (mathsf {E_6}) contains, up to conjugacy, a unique class of symplectic polarities. We also show that the natural point-line geometry of each split building of type (mathsf {F_4}) fully embedded in the natural point-line geometry of (varDelta ) arises from a symplectic polarity.

We choose some special unit vectors ({mathbf {n}}_1,ldots ,{mathbf {n}}_5) in ({mathbb {R}}^3) and denote by ({mathscr {L}}subset {mathbb {R}}^5) the set of all points ((L_1,ldots ,L_5)in {mathbb {R}}^5) with the following property: there exists a compact convex polytope (Psubset {mathbb {R}}^3) such that the vectors ({mathbf {n}}_1,ldots ,{mathbf {n}}_5) (and no other vector) are unit outward normals to the faces of P and the perimeter of the face with the outward normal ({mathbf {n}}_k) is equal to (L_k) for all (k=1,ldots ,5). Our main result reads that ({mathscr {L}}) is not a locally-analytic set, i.e., we prove that, for some point ((L_1,ldots ,L_5)in {mathscr {L}}), it is not possible to find a neighborhood (Usubset {mathbb {R}}^5) and an analytic set (Asubset {mathbb {R}}^5) such that ({mathscr {L}}cap U=Acap U). We interpret this result as an obstacle for finding an existence theorem for a compact convex polytope with prescribed directions and perimeters of the faces.

We classify the Seifert fibrations of any given lens space L(p, q). Starting from any pair of coprime non-zero integers (alpha _1^0,alpha _2^0), we give an algorithmic construction of a Seifert fibration (L(p,q)rightarrow S^2(alpha |alpha _1^0|,alpha |alpha _2^0|)), where the natural number (alpha ) is determined by the algorithm. This algorithm produces all possible Seifert fibrations, and the isomorphisms between the resulting Seifert fibrations are described completely. Also, we show that all Seifert fibrations are isomorphic to certain standard models.

Motivated by analytic number theory, we explore remainder versions of Ikehara’s Tauberian theorem yielding power law remainder terms. More precisely, for (f:[1,infty )rightarrow {mathbb R}) non-negative and non-decreasing we prove (f(x)-x=O(x^gamma )) with (gamma <1) under certain assumptions on f. We state a conjecture concerning the weakest natural assumptions and show that we cannot hope for more.

Let p be an odd prime number and (ell ) an odd prime number dividing (p-1). We denote by (F=F_{p,ell }) the real abelian field of conductor p and degree (ell ), and by (h_F) the class number of F. For a prime number (r ne p,,ell ), let (F_{infty }) be the cyclotomic (mathbb {Z}_r)-extension over F, and (M_{infty }/F_{infty }) the maximal pro-r abelian extension unramified outside r. We prove that (M_{infty }) coincides with (F_{infty }) and consequently (h_F) is not divisible by r when r is a primitive root modulo (ell ) and r is smaller than an explicit constant depending on p.

We give a systematic way to construct almost conjugate pairs of finite subgroups of (mathrm {Spin}(2n+1)) and ({{mathrm{Pin}}}(n)) for (nin {mathbb {N}}) sufficiently large. As a geometric application, we give an infinite family of pairs (M_1^{d_n}) and (M_2^{d_n}) of nearly Kähler manifolds that are isospectral for the Dirac and Laplace operator with increasing dimensions (d_n>6). We provide additionally a computation of the volume of (locally) homogeneous six dimensional nearly Kähler manifolds and investigate the existence of Sunada pairs in this dimension.

We discuss generalizations of classical theta series, requiring only some basic properties of the classical setting. As it turns out, the existence of a generalized theta transformation formula implies that the series is defined over a quasi-symmetric Siegel domain. In particular the exceptional symmetric tube domain does not admit a theta function.

We observe that derived equivalent K3 surfaces have isomorphic Chow motives. The result holds more generally for arbitrary surfaces, as pointed out by Charles Vial.

In this note we consider a special case of the famous Coarea Formula whose initial proof (for functions from any Riemannian manifold of dimension 2 into ({mathbb {R}})) is due to Kronrod (Uspechi Matem Nauk 5(1):24–134, 1950) and whose general proof (for Lipschitz maps between two Riemannian manifolds of dimensions n and p) is due to Federer (Am Math Soc 93:418–491, 1959). See also Maly et al. (Trans Am Math Soc 355(2):477–492, 2002), Fleming and Rishel (Arch Math 11(1):218–222, 1960) and references therein for further generalizations to Sobolev mappings and BV functions respectively. We propose two counterexamples which prove that the coarea formula that we can find in many references (for example Bérard (Spectral geometry: direct and inverse problems, Springer, 1987), Berger et al. (Le Spectre d’une Variété Riemannienne, Springer, 1971) and Gallot (Astérisque 163(164):31–91, 1988), is not valid when applied to (C^infty ) functions. The gap appears only for the non generic set of non Morse functions.