Research in the Mathematical Sciences最新文献

The q-color Ramsey number of a k-uniform hypergraph G, denoted r(G; q), is the minimum integer N such that any coloring of the edges of the complete k-uniform hypergraph on N vertices contains a monochromatic copy of G. The study of these numbers is one of the most central topics in combinatorics. One natural question, which for triangles goes back to the work of Schur in 1916, is to determine the behavior of r(G; q) for fixed G and q tending to infinity. In this paper, we study this problem for 3-uniform hypergraphs and determine the tower height of r(G; q) as a function of q. More precisely, given a hypergraph G, we determine when r(G; q) behaves polynomially, exponentially or double exponentially in q. This answers a question of Axenovich, Gyárfás, Liu and Mubayi.

Let (rge 1) be a positive integer, A a real positive definite symmetric (rtimes r) matrix, B a vector of length r, and C a scalar. Nahm’s problem is to describe all such A, B and C with rational entries for which a specific r-fold q-hypergeometric series (denoted by (f_{A,B,C}(q))) involving the parameters A, B, C is modular. When the rank (r=2), Zagier provided eleven sets of examples of (A, B, C) for which (f_{A,B,C}(q)) is likely to be modular. We present a number of Rogers–Ramanujan type identities involving double sums, which give modular representations for Zagier’s rank two examples. Together with several known cases in the literature, we verified ten of Zagier’s examples and give conjectural identities for the remaining example.

A specialization of a K3 surface with Picard rank one to a K3 with rank two defines a vanishing class of order two in the Brauer group of the general K3 surface. We give the B-field invariants of this class. We apply this to the K3 double plane defined by a cubic fourfold with a plane. The specialization of such a cubic fourfold whose group of codimension two cycles has rank two to one which has rank three induces such a specialization of the double planes. We determine the Picard lattice of the specialized double plane as well as the vanishing Brauer class and its relation to the natural ‘Clifford’ Brauer class. This provides more insight in the specializations. It allows us to explicitly determine the K3 surfaces associated with infinitely many of the conjecturally rational cubic fourfolds obtained as such specializations.

Given a germ of an analytic variety X and a germ of a holomorphic function f with a stratified isolated singularity with respect to the logarithmic stratification of X, we show that under certain conditions on the singularity type of the pair (f, X), the following relative analog of the well-known K. Saito’s theorem holds true: equality of the relative Milnor and Tjurina numbers of f with respect to X (also known as Bruce–Roberts numbers) is equivalent to the relative quasihomogeneity of the pair (f, X), i.e. to the existence of a coordinate system such that both f and X are quasihomogeneous with respect to the same positive rational weights.

In this article we study the delta invariant of reduced curve germs via topological techniques. We describe an explicit connection between the delta invariant of a curve embedded in a rational singularity and the topological Poincaré series of the ambient surface. This connection is established by using another formula expressing the delta invariant as ‘periodic constants’ of the Poincaré series associated with the abstract curve and a ‘twisted’ duality developed for the Poincaré series of the ambient space.

We apply the theory of operadic Koszul duality to provide a cofibrant resolution of the colored operad whose algebras are prefactorization algebras on a fixed space M. This allows us to describe a notion of prefactorization algebra up to homotopy as well as morphisms up to homotopy between such objects. We make explicit these notions for several special M, such as certain finite topological spaces, or the real line.

The parabolic subset of a 3-manifold generically immersed in (mathbb {R}^4) is a surface. We analyze in this study the generic geometrical behavior of such surface, considered as a submanifold of (mathbb {R}^4). Typical Singularity Theory techniques based on the analysis of the family of height functions are applied in order to describe the geometrical characterizations of different singularity types.

Our results investigate mock theta functions and quantum modular forms via quantum q-series identities. After Lovejoy, quantum q-series identities are such that they do not hold as an equality between power series inside the unit disc in the classical sense, but do hold at dense sets of roots of unity on the boundary. We establish several general (multivariable) quantum q-series identities and apply them to various settings involving (universal) mock theta functions. As a consequence, we surprisingly show that limiting, finite, universal mock theta functions at roots of unity for which their infinite counterparts do not converge are quantum modular. Moreover, we show that these finite limiting universal mock theta functions play key roles in (generalized) Ramanujan radial limits. A further corollary of our work reveals that the finite Kontsevich–Zagier series is a kind of “universal quantum mock theta function,” in that it may be used to evaluate odd-order Ramanujan mock theta functions at roots of unity. (We also offer a similar result for even-order mock theta functions.) Finally, to complement the notion of a quantum q-series identity and the results of this paper, we also define what we call an “antiquantum q-series identity’ and offer motivating general results with applications to third-order mock theta functions.

In this paper, we discuss the concept of (rho )-regularity of analytic map germs and its close relationship with the existence of locally trivial smooth fibrations, known as the Milnor tube fibrations. The presence of a Thom regular stratification or the Milnor condition (b) at the origin, indicates the transversality of the fibers of the map G with respect to the levels of a function (rho ), which guarantees (rho )-regularity. Consequently, both conditions are crucial for the presence of fibration structures. The work aims to provide a comprehensive overview of the main results concerning the existence of Thom regular stratifications and the Milnor condition (b) for germs of analytic maps. It presents strategies and criteria to identify and ensure these regularity conditions and discusses situations where they may not be satisfied. The goal is to understand the presence and limitations of these conditions in various contexts.

Symmetric powers of matroids were first introduced by Lovasz (Combinatorial surveys, in: Proceedings 6th British combinatorial conference, pp 45-86, 1977) and Mason (Algebr Methods Graph Theory 1:519-561, 1981) in the 1970s, where it was shown that not all matroids admit higher symmetric powers. Since these initial findings, the study of matroid symmetric powers has remained largely unexplored. In this paper, we establish an equivalence between valuated matroids with arbitrarily large symmetric powers and tropical linear spaces that appear as the variety of a tropical ideal. In establishing this equivalence, we additionally show that all tropical linear spaces are connected through codimension one. These results provide additional geometric and algebraic connections to the study of matroid symmetric powers, which we leverage to prove that the class of matroids with second symmetric power is minor-closed and has infinitely many forbidden minors.