Congruence Restricted Modular Forms

Given f(z), a modular form on a congruence subgroup (of the full modular group), we construct the function f(z;r,t) by summing over the terms of the Fourier expansion of f(z) with index congruent to r modulo t. In this paper, we determine a condition on the multiplier system of f(z) which guarantees that f(z;r,t) is itself a modular form on a (smaller) congruence subgroup.2000 Mathematics Subject Classification: Primary—11F11; Secondary—11F30     

On the Maximal Multiplicity of Parts in a Random Integer Partition

We study the asymptotic behavior of the maximal multiplicity μ_n = μ_n(λ) of the parts in a partition λ of the positive integer n, assuming that λ is chosen uniformly at random from the set of all such partitions. We prove that πμ_n/(6n)^1/2 converges weakly to max _jX_j/j as n→∞, where X₁, X₂, … are independent and exponentially distributed random variables with common mean equal to 1.2000 Mathematics Subject Classification: Primary—05A17; Secondary—11P82, 60C05, 60F05     

The estimation of the hessian matrix in nonlinear least squares problems with non-zero residuals

Computational experiments by McKeown [11] have shown that specialised methods, based on the Gauss—Newton iteration, are not necessarily the best choice for minimising functions that are sums of squared terms. Difficulties arise when the Gauss—Newton approach does not yield a good approximation to the second derivative matrix of the function: and this is more likely to happen when the function value at the optimum is not near zero and the terms in the sum of squares are significantly nonlinear. This paper considers some specialised methods for the nonlinear least squares problem which seek to improve the Gauss—Newton estimate of the Hessian matrix without explicitly calculating second derivatives.     

The Multiple Gamma Function and Its Application to Computation of Series

The multiple gamma function Γ_n, defined by a recurrence-functional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma function has been revived. This paper discusses some theoretical aspects of the Γ_n function and their applications to summation of series and infinite products.This work was supported by NFS grant CCR-0204003.2000 Mathematics Subject Classification: Primary—33E20, 33F99, 11M35, 11B73     

Mean square values of Hecke <Emphasis Type="Italic">L</Emphasis>–series formed with <Emphasis Type="Italic">r</Emphasis>–th order characters

Let L be a number field containing the r–th roots of unity. Starting with the Rankin-Selberg convolution of a metaplectic Eisenstein series on the r–fold cover of GL(2) with itself, we construct a Dirichlet series defined over L whose coefficients involve the r–th order twists of a fixed Hecke L–function. We then observe that a group of functional equations can be naturally associated with this construction. Combining this with the convexity theorem for holomorphic functions of several complex variables, we show that this object, as a function of two complex variables, admits meromorphic continuation to ². As an application, we obtain asymptotic formulae for mean square values of the r–th order twists of an arbitrary Hecke L–function defined over L. Mathematics Subject Classification (1991) 11R42, 11F66, 11F67, 11F70, 11M41, 11R47     

The first cohomology group of leaves and local stability of compact foliations

A foliation with all leaves compact (compact foliation) is called locally stable if every leaf has a basis of neighborhoods which are unions of leaves. We study the relationship between the first real cohomology group of leaves and the local stability of compact foliations. We show by example that the topology of the typical leaves (i.e. leaves with zero holonomy) has no influence on the local stability of the foliation while — at least for small codimensions — (less than 4 in general or less than 5 for foliations on compact minifolds) — a locally unstable foliation has a leaf F with infinite holonomy and a finite covering F' of F such that H¹(F'; IR) O. We also prove a related structural stability result for fibre bundles.     

Liftings and Mean Value Theorems for Automorphic L-Functions

A general mean value theorem for Dirichlet series, with a sharperror estimate near the boundary of the critical strip, is proved.Applications of this theorem to various automorphic L-functionsare discussed. Moreover, sharp upper bounds of mean square valuesof L-functions are obtained when they are attached to liftedforms, such as Doi–Naganuma and Ikeda lifts in the theoryof Siegel modular forms. 2000 Mathematics Subject Classification11F66, 11M41.     

Fastalgebren mit zweiseitigem bzw. kommutativem Inzidenzgruppoid

In this paper it is shown that a nearalgebra (F,H) — in the sense of [4] — is an algebra (a commutative algebra), if its incidencegroupoid F^*/H^* is twosided (commutative).

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Herrn Professor Martin Barner zum 65. Geburtstag     

Study of the prefracture states of rigid polymers 1. Forced high-elastic deformation

It is shown that in the process of extension specimens of a rigid unoriented polymer — polyethylene terephthalate — go over into the oriented state before failure. Various cases of transition to the oriented state are considered: with the formation of a neck and deformation bands, with and without loss of continuity. The degree of molecular orientation of the specimens and their fracture conditions are estimated.A. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 1, pp. 8–14, January–February, 1976.     

The method of Iterated Defect-Correction and its application to two-point boundary value problems

Summary In Part II of this paper we present a rigorous analysis of the Iterated Defect-Correction — applied to two-point boundary value problems — which was introduced in Part I of this paper [1]. A complete proof of Theorem 5. 1 of [1] is given.     

Equivalence of the derived chains corresponding to a boundary-value problem on a finite interval,for polynomial operator pencils

Under investigation is the equivalence of derived chains constructed from root vectors of polynomial pencils of operators acting in a Hilbert space. These derived chains correspond to various boundary—value problems on a finite interval for an operator—differential equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 1, pp. 83–95, January, 1990.     

Deformation and fracture of polymers

On the basis of experimental research it is suggested that in polymers under load two physically different kinetic processes — deformation and fracture — develop simultaneously. Their rates depend on different components of the stress tensor; accordingly, the relation between these rates can be experimentally controlled. It is suggested that these processes involve the overcoming of physically different potential barriers and that their elementary events embrace essentially different activation volumes. Deformation involves the overcoming of the forces of intermolecular interaction, and fracture, with the rupture of the main chains.Review of work carried out in the Materials Laboratory of the A. F. Ioffe Physico-Technical Institute, Academy of Sciences of the USSR.A. F. Ioffe Physico-Technical Institute, Leningrad. Translated from Mekhanika Polimerov, No. 1, pp. 95–106, January–February, 1975.     

Products of Ratios of Consecutive Integers

Take the product of the numbers (n/(n+1))^∊_n for 1≤ n < N, where each ∊_n is ± 1. Express the product as a/b in lowest terms. Evidently the minimal possible value for a over all choices for ∊_n is 1; just take each ∊_n = 1, or each ∊_n = 0. Denote the maximal possible value of a by A(N). It is known from work of Nicolas and Langevin that (log 4+o(1))N≤ log A(N)≤(2/3+o(1))Nlog N. Using the Rosse–Iwaniec sieve, we improve the lower bound to the same order of magnitude as the upper bound.For Jean-Louis Nicolas, on his sixtieth birthday2000 Mathematics Subject Classification: Primary—11N56; Secondary—11N36     

Certain class of nonuniform estimates in multidimensional limit theorems

Quantities of the form | F(X) — G(X) | are estimated, where F and G are the convolutions of certain k-dimensional probability distributions, while X is a convex polyhedron in R^k. Estimates of the form | F(X) — G(X) | c(k)(F, G, X) are proved, differing from the known ones by the presence of the factor (F, G, X) in the right-hand side, which may turn out to be small if the polyhedron X is small in a definite sense.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 92–105, 1990.     

Representing the mean residual life in terms of the failure rate

In survival or reliability studies, the mean residual life or life expectancy is an important characteristic of the model. Whereas the failure rate can be expressed quite simply in terms of the mean residual life and its derivative, the inverse problem—namely that of expressing the mean residual life in terms of the failure rate—typically involves an integral of a complicated expression. In this paper, we obtain simple expressions for the mean residual life in terms of the failure rate for certain classes of distributions which subsume many of the standard cases. Several results in the literature can be obtained using our approach. Additionally, we develop an expansion for the mean residual life in terms of Gaussian probability functions for a broad class of ultimately increasing failure rate distributions. Some examples are provided to illustrate the procedure.     

Dependence Between Extreme Values of Discrete and Continuous Time Locally Stationary Gaussian Processes

The maximum of a continuous, locally stationary Gaussian process which satisfies Bermans condition on the long range dependence is compared with the maximum of this process sampled at discrete time points. These two extreme values are asymptotically totally dependent if the grid of the discrete time points is sufficiently dense, and asymptotically independent if the the grid points are sparse.AMS 2000 Subject Classification. Primary—60F05, Secondary—60G15     

Eine äquiaffine Charakterisierung der Cardan-Bewegung mittels der höheren Wendepole

Characterizations for subfamilies of an underlying one parametric transformation familyT are or natural interest. The paper gives an affine characterization — taking forT the area preserving plane affine motions with one finite and two infinite instantaneous centers — of the Cardan motions, by means of the higher inflection poles.

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Herrn Professor Dr. F. Hohenberg zum 70. Geburtstag gewidmet     

Stability of Maxima of Random Variables with Multidimensional Indices

The paper discusses the stability of suitably-defined maxima of a set of i.i.d. random variables with multidimensional indices.It is shown that theorems of Gnedenko (1943) and Tomkins (1986) concerning relative stability and complete relative stability of maxima remain valid in the new setting.Moreover, a criterion for almost sure relative stability for maxima with multidimensional indices is presented, extending a result of Barndorff-Nielsen (1963).AMS 2000 Subject Classification. Primary—60F15, 60G60, 62G30, Secondary—10A25, 60G99     

On quantum multi-Hamiltonian systems

Laboratoire de Physique Théorique et Mathématique, Université Paris VII—Denis Diderot, F. 75251 Paris Cedex 05, France. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 99, No. 3, pp. 377–381, June, 1994.     

Discrete symmetries in quantum scattering

Using the discrete symmetries of the Klein—Gordon, Dirac, and Schrödinger wave equations, we obtain from one solution, considered as a function of the quantum numbers and the parameters of the potentials, three other solution. Taken together, these solutions form two complete sets of solutions of the wave equation. The coefficients of the linear relations between the functions of these sets — the connection coefficients — are related in a simple manner to the wave transmission and reflection amplitudes. By virtue of the discrete symmetries of the wave equation, the connection coefficients satisfy certain symmetry relations. We show that in a number of simple cases, the behavior of the wave function near the center of formation of an additional wave determines the amplitude of the wave that is formed at infinity.P. N. Lebedev Physics Institute, Russian Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98, No. 1, pp. 60–79, January, 1994.