On Stein?s method for infinite-dimensional Gaussian approximation in abstract Wiener spaces

In this paper, we generalize Stein?s method to “infinite-variate” normal approximation that is an infinite-dimensional approximation by abstract Wiener measures on a real separable Banach space. We first establish a Stein?s identity for abstract Wiener measures and solve the corresponding Stein?s equation. Then we will present a Gaussian approximation theorem using exchangeable pairs in an infinite-variate context. As an application, we will derive an explicit error bound of Gaussian approximation to the distribution of a sum of independent and identically distributed Banach space-valued random variables based on a Lindeberg-Lévy type limit theorem. In addition, an analogous of Berry-Esséen type estimate for abstract Wiener measures will be obtained.     

A short WZ-proof of Euler?s fundamental sum identity and more

We prove two identities that imply WZ-proofs of the fundamental Euler?s sum identity for ζ(3) and Gosper?s sum identity for ζ(5). In addition, we present a new similar representation for ζ(7).     

Infinite class of new sign ambiguities

In 1934, two kinds of multiplicative relations, the norm and the Davenport-Hasse relations, between Gauss sums, were known. In 1964, H. Hasse conjectured that the norm and the Davenport-Hasse relations were the only multiplicative relations connecting Gauss sums over Fp. However, in 1966, K. Yamamoto provided a simple counterexample disproving the conjecture. This counterexample was a new type of multiplicative relation, called a sign ambiguity, involving a ± sign not connected to elementary properties of Gauss sums. In this paper, we give an explicit product formula involving Gauss sums which generates an infinite class of new sign ambiguities, and we resolve the ambiguous sign by using Stickelberger?s theorem.     

On generalized Erd?s spaces

During the last years both Erd?s space and complete Erd?s space were topologically characterized by Dijkstra and van Mill. Applications include results about Erd?s type spaces in ?p-spaces as well as results about Polishable ideals on ω. We present an unifying theorem in terms of sets with a reflexive relation that among other things contains these apparently dissimilar results as special cases.     

Hensley?s problem for complex and non-Archimedean meromorphic functions

Büchi?s problem asks if there exists a positive integer M such that all x₁,…,xM∈Z satisfying the equations for all 3?r?M must also satisfy for some integer x. Hensley?s problem asks if there exists a positive integer M such that, for any integers ν and a, if ²(ν+r)−a is a square for 1?r?M, then a=0. It is not difficult to see that a positive answer to Hensley?s problem implies a positive answer to Büchi?s problem. One can ask a more general version of the Hensley?s problem by replacing the square by n-th power for any integer n?2 which is called the Hensley?s n-th power problem. In this paper we will solve Hensley?s n-th power problem for complex meromorphic functions and non-Archimedean meromorphic functions.     

On asymptotics of the q-exponential and q-gamma functions

In this work we present a derivation for the complete asymptotic expansions of Euler?s q-exponential function and Jackson?s q-gamma function via Mellin transform. These formulas are valid everywhere, uniformly on any compact subset of the complex plane.     

An effective isomorphy criterion for mod ? Galois representations

In this paper, we consider mod ? Galois representations of Q. In particular, we develop an effective criterion to decide whether or not two mod ? Galois representations Q are isomorphic. The proof uses methods from Khare-Wintenberger?s recent theorem on Serre?s conjecture along with theorems by Sturm and Kohnen.     

The structure of graphs with given lengths of cycles

The cycle spectrum of a given graph G is the lengths of cycles in G. In this paper, we introduce the following problem: determining the maximum number of edges of an n-vertex graph with given cycle spectrum. In particular, we determine the maximum number of edges of an n-vertex graph without containing cycles of lengths three and at least k. This can be viewed as an extension of a well-known result of Erd?s and Gallai concerning the maximum number of edges of an n-vertex graph without containing cycles of lengths at least k. We also determine the maximum number of edges of an n-vertex graph whose cycle spectrum is a subset of two positive integers.     

A note on the number of different inner products generated by a finite set of vectors

This note aims to introduce a new problem in combinatorial geometry. What is the minimum number of distinct inner products determined by n distinct vectors in R^d? We use some elementary methods to derive upper and lower bounds giving a good impression of where the answer might lie and show how it is connected to various known problems.     

Entropy and the fourth moment phenomenon

We develop a new method for bounding the relative entropy of a random vector in terms of its Stein factors. Our approach is based on a novel representation for the score function of smoothly perturbed random variables, as well as on the de Bruijn?s formula of information theory. When applied to sequences of functionals of a general Gaussian field, our results can be combined with the Carbery–Wright inequality in order to yield multidimensional entropic rates of convergence that coincide, up to a logarithmic factor, with those achievable in smooth distances (such as the 1-Wasserstein distance). In particular, our findings settle the open problem of proving a quantitative version of the multidimensional fourth moment theorem for random vectors having chaotic components, with explicit rates of convergence in total variation that are independent of the order of the associated Wiener chaoses. The results proved in the present paper are outside the scope of other existing techniques, such as for instance the multidimensional Stein?s method for normal approximations.     

On local spectral properties of complex symmetric operators

In this paper we study properties of complex symmetric operators. In particular, we prove that every complex symmetric operator having property (β) or (δ) is decomposable. Moreover, we show that complex symmetric operator T has Dunford?s property (C) and it satisfies Weyl?s theorem if and only if its adjoint does.     

Hall’s and Kőnig’s theorem in graphs and hypergraphs

We investigate the relation between Hall’s theorem and K?nig’s theorem in graphs and hypergraphs. In particular, we characterize the graphs satisfying a deficiency version of Hall’s theorem, thereby showing that this class strictly contains all K?nig–Egerváry graphs. Furthermore, we give a generalization of Hall’s theorem to normal hypergraphs.     

Chessboard complexes indomitable

We give a simpler, degree-theoretic proof of the striking new Tverberg type theorem of Blagojevi?, Ziegler and Matschke. Our method also yields some new examples of “constrained Tverberg theorems” including a simple colored Radon?s theorem for d+3 points in Rd. This gives us an opportunity to review some of the highlights of this beautiful theory and reexamine the role of chessboard complexes in these and related problems of topological combinatorics.     

On the weak–strong uniqueness of Koch–Tataruʼs solution for the Navier–Stokes equations

We prove the weak–strong uniqueness between Koch–Tataru?s solution and Leray?s weak solution for three-dimensional incompressible Navier–Stokes equations.     

Counting subset sums of finite abelian groups

In this paper, we obtain an explicit formula for the number of zero-sum k-element subsets in any finite abelian group.     

The discriminant of an algebraic torus

For a torus T defined over a global field K, we revisit an analytic class number formula obtained by Shyr in the 1970s as a generalization of Dirichlet?s class number formula. We prove a local-global presentation of the quasi-discriminant of T, which enters into this formula, in terms of cocharacters of T. This presentation can serve as a more natural definition of this invariant.     

On non-strong jumping numbers and density structures of hypergraphs

Estimating Turán densities of hypergraphs is believed to be one of the most challenging problems in extremal set theory. The concept of ‘jump’ concerns the distribution of Turán densities. A number α∈[0,1) is a jump for r-uniform graphs if there exists a constant c>0 such that for any family F of r-uniform graphs, if the Turán density of F is greater than α, then the Turán density of F is at least α+c. A fundamental result in extremal graph theory due to Erd?s and Stone implies that every number in [0,1) is a jump for graphs. Erd?s also showed that every number in [0,r!/r^r) is a jump for r-uniform hypergraphs. Furthermore, Frankl and Rödl showed the existence of non-jumps for hypergraphs. Recently, more non-jumps were found in [r!/r^r,1) for r-uniform hypergraphs. But there are still a lot of unknowns regarding jumps for hypergraphs. In this paper, we propose a new but related concept-strong-jump and describe several sequences of non-strong-jumps. It might help us to understand the distribution of Turán densities for hypergraphs better by finding more non-strong-jumps.     

Construction of the asymptotics of the solutions of the one-dimensional Schrödinger equation with rapidly oscillating potential

We obtain asymptotic formulas for the solutions of the one-dimensional Schrödinger equation ? y″ +q(x)y = 0 with oscillating potential q(x)=x ^β P(x ^1+α)+cx ^?2 as x→ +∞. The real parameters α and β satisfy the inequalities β ? α ≥ ?1, 2α ? β > 0 and c is an arbitrary real constant. The real function P(x) is either periodic with period T, or a trigonometric polynomial. To construct the asymptotics, we apply the ideas of the averaging method and use Levinson’s fundamental theorem.     

On some new P–Q mixed modular equations

In this paper, we establish several new P–Q mixed modular equations or P–Q eta-function identities akin to those recorded by Ramanujan in his notebooks. We also establish several new general formulas by giving explicit values to Ramanujan’s remarkable product of theta-functions.