Congruences Concerning Values of the Modular Functions <Emphasis Type="Italic">j</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

The j-function j(z) = q⁻¹+ 744 + 196884q + ⋅s plays an important role in many problems. In [7], Zagier, presented an interesting series of functions obtained from the j-function: j_m(ζ) = (j(ζ) – 744)∨T₀(m), where T₀(m) is the usual m′th normalized weight 0 Hecke operator. In [3], Bruinier et al. show how this series of functions can be used to describe all meromorphic modular forms on SL₂(ℤ). In this note we use these functions and basic notions about modular forms to determine previously unidentified congruence relations between the coefficients of Eisenstein series and the j-function. 2000 Mathematics Subject Classification: Primary–11B50, 11F03, 11F30 The author thanks the National Science Foundation for their generous support.     

Microlocal Analysis of Generalized Pullbacks of?Colombeau Functions

In distribution theory the pullback of a general distribution by a C ^∞-function is well-defined whenever the normal bundle of the C ^∞-function does not intersect the wave front set of the distribution. However, the Colombeau theory of generalized functions allows for a pullback by an arbitrary c-bounded generalized function. It has been shown in previous work that in the case of multiplication of Colombeau functions (which is a special case of a C ^∞ pullback), the generalized wave front set of the product satisfies the same inclusion relation as in the distributional case, if the factors have their wave front sets in favorable position. We prove a microlocal inclusion relation for the generalized pullback (by a c-bounded generalized map) of Colombeau functions. The proof of this result relies on a stationary phase theorem for generalized phase functions, which is given in the Appendix. Furthermore we study an example (due to Hurd and Sattinger), where the pullback function stems from the generalized characteristic flow of a partial differential equation.      

Higher covariant derivative regularization for calculations in supersymmetric theories

A variant of the higher covariant derivative regularization is used for calculation of a two-loop β-function for the general renormalizable N = 1 supersymmetric theory. It is shown that the β-function is given by integrals of total derivatives. Partially this can be explained by substituting solutions of Slavnov-Taylor identities into the Schwinger-Dyson equations.     

Selberg＇s Normal Density Theorem for Automorphic L-Functions for GLm

Let π be an irreducible unitary cuspidal representation of GLm（AQ） with m ≥ 2, and L（s, Tr） the L-function attached to π. Under the Generalized Riemann Hypothesis for L（s,π）, we estimate the normal density of primes in short intervals for the automorphic L-function L（s, π）. Our result generalizes the corresponding theorem of Selberg for the Riemann zeta-function.     

The Dirichlet series for the exterior square <Emphasis Type="Italic">L</Emphasis>-function on GL(<Emphasis Type="Italic">n</Emphasis>)

We present an elementary derivation of the Jacquet–Shalika construction for the exterior square L-function on GL(n), as a classical Dirichlet series in the Fourier coefficients A(m ₁,…,m _n−1).     

Circularm-functions

Circularm-functions are introduced on smooth manifolds with boundary. We study the distribution of their critical circles and construct an example of a four-dimensional manifoldM ⁴ with boundary ∂M ⁴ that satisfies the condition ξ(∂M ⁴)=ξ(M ⁴,∂M ⁴)=0 but does not contain any circularm-function. We prove that a manifold with boundaryM ⁿ (n≥5) such that ξ(∂M ⁿ , ∂M ⁿ )=0 always contains a circularm-function without critical points in the interior manifold. Sukhumi Branch of the Tbilisi University, Sukhumi. Translated from Ukrainskii Matermaticheskii Zhurnal, Vol. 46, No. 6, pp. 776–781, June, 1994.     

Symmetric functions,codes of partitions and the KP hierarchy

We consider an operator of Bernstein for symmetric functions and give an explicit formula for its action on an arbitrary Schur function. This formula is given in a remarkably simple form when written in terms of some notation based on the code of a partition. As an application, we give a new and very simple proof of a classical result for the KP hierarchy, which involves the Plücker relations for Schur function coefficients in a τ-function for the hierarchy. This proof is especially compact because we are able to restate the Plücker relations in a form that is symmetrical in terms of partition code notation.     

A short proof of the levy continuity theorem in Hilbert space

A short proof of the Levy continuity theorem in Hilbert space. In the theory of the normal distribution on a real Hilbert spaceH, certain functionsφ have been shown by L. Gross to give rise to random variablesφ∼ in a natural way; in particular, this is the case for functions which are “uniformly τ-continuous near zero”. Among such functions are the characteristic functionsφ of probability distributionsm onH, given byφ(y)=∫e ^i(y,x)dm(x). The following analogue of the Levy continuity theorem has been proved by Gross: Letφ _j be the characteristic function of the probability measurem _j onH, Then necessary and sufficient that ∫f dm _j → ∫f dm for some probability measurem and all bounded continuousf, is that there exists a functionφ, uniformly τ-continuous near zero, withφ _j∼ →φ∼ in probability.φ turns out, of course, to be the characteristic function ofm. In the present paper we give a short proof of this theorem. Research supported by National Science Foundation Grant GP-3977.     

Overdetermined problems with possibly degenerate ellipticity, a geometric approach

Given an open bounded connected subset Ω of ℝⁿ, we consider the overdetermined boundary value problem obtained by adding both zero Dirichlet and constant Neumann boundary data to the elliptic equation −div(A(|∇u|)∇u)=1 in Ω. We prove that, if this problem admits a solution in a suitable weak sense, then Ω is a ball. This is obtained under fairly general assumptions on Ω and A. In particular, A may be degenerate and no growth condition is required. Our method of proof is quite simple. It relies on a maximum principle for a suitable P-function, combined with some geometric arguments involving the mean curvature of ∂Ω.     

Fractional Moments of Automorphic L-Functions on GL（m）

Let π be an irreducible unitary cuspidal representation of GLm(AQ), m ≥ 2. Assume that π is self-contragredient. The author gets upper and lower bounds of the same order for fractional moments of automorphic L-function L(s, π) on the critical line under Generalized Ramanujan Conjecture; the upper bound being conditionally subject to the truth of Generalized Riemann Hypothesis.     

Σ-Bounded algebraic systems and universal functions. I

We introduce the concept of a Σ-bounded algebraic system and prove that if a system is Σ- bounded with respect to a subset A then in a hereditarily finite admissible set over this system there exists a universal Σ-function for the family of functions definable by Σ-formulas with parameters in A. We obtain a necessary and sufficient condition for the existence of a universal Σ-function in a hereditarily finite admissible set over a Σ-bounded algebraic system. We prove that every linear order is a Σ-bounded system and in a hereditarily finite admissible set over it there exists a universal Σ-function.     

The period function of a delay differential equation and an application

We consider the delay differential equation [(x)\dot](t) = - mx(t) + f(x(t - t))\dot x(t) = - \mu x(t) + f(x(t - \tau )), where μ, τ are positive parameters and f is a strictly monotone, nonlinear C ¹-function satisfying f(0) = 0 and some convexity properties. It is well known that for prescribed oscillation frequencies (characterized by the values of a discrete Lyapunov functional) there exists τ* > 0 such that for every τ > τ* there is a unique periodic solution. The period function is the minimal period of the unique periodic solution as a function of τ > τ*. First we show that it is a monotone nondecreasing Lipschitz continuous function of τ with Lipschitz constant 2. As an application of our theorem we give a new proof of some recent results of Yi, Chen and Wu [14] about uniqueness and existence of periodic solutions of a system of delay differential equations.     

On the analogue of Weil’s converse theorem for Jacobi forms and their lift to half-integral weight modular forms

We generalize Weil’s converse theorem to Jacobi cusp forms of weight k, index m and Dirichlet character χ over the group Γ ₀(N)⋉ℤ². Then two applications of this result are given; we generalize a construction of Jacobi forms due to Skogman and present a new proof for several known lifts of such Jacobi forms to half-integral weight modular forms.     

Counting permutations with no long monotone subsequence via generating trees and the kernel method

We recover Gessel’s determinantal formula for the generating function of permutations with no ascending subsequence of length m+1. The starting point of our proof is the recursive construction of these permutations by insertion of the largest entry. This construction is of course extremely simple. The cost of this simplicity is that we need to take into account in the enumeration m−1 additional parameters—namely, the positions of the leftmost increasing subsequences of length i, for i=2,…,m. This yields for the generating function a functional equation with m−1 “catalytic” variables, and the heart of the paper is the solution of this equation.     

Modularity of a Certain Calabi-Yau Threefold

 The Langlands program predicts that certain Calabi-Yau threefolds are modular in the sense that their L-series correspond to the Mellin transforms of weight 4 newforms. Here we prove that the L-function of the threefold given by is , the unique normalized eigenform in .     

Ergodicity of some mappings of the circle and the line

Ifφ is inner and has a fixed point inD, thenφ as a mapping of the circle is exact. Ifφ has a “fixed” point onT, then the condition Σ(1−|φ _n(0)|)=∞ impliesφ _m is weak mixing for allm. These results when transferred to the line by a conformal mapping of the disc onto the upper half plane give a proof of the total weak mixing for the Boole transformation.     

Convexity and a certain propertyP m

The propertyP _m (directly analogous to Valentine’s propertyP ₃) is used to prove several curious results concerning subsets of a topological linear space, among them the following: (a) If a closed setS has propertyP _m and containsk points of local nonconvexity no distinct pair of which can see each other viaS, thenS is the union ofm − k − 1 or fewer starshaped sets. (b) Any closed connected set with propertyP _m is polygonally connected. (c) A closed connected setS with propertyP _m is anL _m−1 set (each pair of points may be joined by a polygonal arc ofm − 1 of fewer sides inS). (d) A finite-dimensional set with propertyP _m is anL _{2m − 3} set. A new proof of Tietze’s theorem on locally convex sets is given, and various examples refute certain plausible conjectures.     

Two remarks on the relationship between BMO-regularity and analytic stability of interpolation for lattices of measurable functions

We study in this paper Hardy-type spaces on a measure space ( \mathbbT \mathbb{T} , m) × (Ω, μ), where ( \mathbbT \mathbb{T} , m) is the unit circle with Lebesgue measure. There is a characterization of analytic stability for real interpolation of weighted Hardy spaces on \mathbbT \mathbb{T} × Ω, a complete proof of which was present in the literature only for the case where μ is a point mass. Here this gap is filled, and a proof of the general case is presented. In a previous work by Kislyakov, certain results concerning BMO-regular lattices on ( \mathbbT \mathbb{T} × Ω, m × μ) were proved under the assumption that the measure μ is discrete. Here this extraneous assumption is lifted. Bibliography: 9 titles.