Small oscillations of a viscous incompressible fluid with a large number of small interacting particles in the case of their surface distribution

We study the asymptotic behavior of solutions of the problem that describes small motions of a viscous incompressible fluid filling a domain Ω with a large number of suspended small solid interacting particles concentrated in a small neighborhood of a certain smooth surface Γ ⊂ Ω. We prove that, under certain conditions, the limit of these solutions satisfies the original equations in the domain Ω\Γ and some averaged boundary conditions (conjugation conditions) on Γ.     

Transformation formulas associated with integrals involving the Riemann Ξ-function

Using residue calculus and the theory of Mellin transforms, we evaluate integrals of a certain type involving the Riemann Ξ-function, which give transformation formulas of the form F(z, α) = F(z, β), where αβ = 1. This gives a unified approach for generating certain modular transformation formulas, including a famous formula of Ramanujan and Guinand.     

Groups of typeF a,b,−c

For integersa, b andc, the groupF ^a,b,−c is defined to be the group 〈R, S : R ²=RS ^aRS^bRS^−c=1〉. In this paper we identify certain subgroups of the group of affine linear transformations of finite fields of orderp ⁿ (for certainp andn) as groups of typeF ^a,b,−c for certain (not unique) choices ofa, b andc.     

Vector valued Riesz distributions on Euclidian Jordan algebras

Let V be a Euclidean Jordan algebra, Гthe associated symmetric cone and G be the identity component of the linear automorphism group of Г.In this paper we associate to a certain class of spherical representations (ρ, ɛ) of G certain ɛ-valued Riesz distributions generalizing the classical scalar valued Riesz distributions on V. Our construction is motivated by the analytic theory of unitary highest weight representations where it permits to study certain holomorphic families of operator valued Riesz distributions whose positive definiteness corresponds to the unitarity of a representation of the automorphism group of the associated tube domain Г +iV.     

Achieving high-efficiency second harmonic generation in a sequence of laser pulses with random peak intensity. Part III. Propagation of pulses in a bulk medium

Interpretation of geoelectric fields gives rise to multidimensional inverse problems, which include many problems for quasi-layered media. Such problems are solved by an iterative-asymptotic method. In the present article we consider a particular implementation of the method designed for a certain model of the medium, a certain method of solving the one-dimensional inverse problem, and a certain representation of magnetotelluric sounding data. For Parts I and II, see Prikladnaya Matematika i Informatika, No. 26, pp. 5–24, 2007 and No. 28, pp. 50–65, 2008; English translation: Computational Mathematics and Modeling, Vol. 19, No. 4, pp. 333–342, 2008 and Vol. 20, No. 1, pp. 1–25, 2009. Translated from Prikladnaya Matematika i Informatika, No. 29, pp. 5–18, 2008.     

On the number of primitive representations of integers as sums of squares

Formulas for the number of primitive representations of any integer n as a sum of k squares are given, for 2 ≤ k ≤ 8, and for certain values of n, for 9 ≤ k ≤ 12. The formulas have a similar structure and are striking for their simplicity. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—11E25; Secondary—05A15, 33E05.     

The Bellman function,the two-weight Hilbert transform,and embeddings of the model spaces<Emphasis Type="Italic">K</Emphasis><Subscript>θ</Subscript>

This paper is devoted to embedding theorems for the spaceK _θ , where θ is an inner function in the unit disc D. It turns out that the question of embedding ofK _θ into L²(Μ) is virtually equivalent to the boundedness of the two-weight Hilbert transform. This makes the embedding question quite difficult (general boundedness criteria of Hunt-Muckenhoupt-Wheeden type for the twoweight Hilbert transform have yet to be found). Here we are not interested in sufficient conditions for the embedding ofKg into L²(Μ) (equivalent to a certain two-weight problem for the Hilbert transform). Rather, we are interested in the fact that a certain natural set of conditions is not sufficient for the embedding ofK _θ intoL ² (Μ) (equivalently, a certain set of conditions is not sufficient for the boundedness in a two-weight problem for the Hilbert transform). In particular, we answer (negatively) certain questions of W. Cohn about the embedding ofK _θ into L²(Μ). Our technique leads naturally to the conclusion that there can be a uniform embedding of all the reproducing kernels ofK _θ but the embedding of the wholeK _θ intoL ²(Μ) may fail. Moreover, it may happen that the embedding into a potentially larger spaceL ²(μ) fails too. Both authors are supported by the NSF grant DMS 9970395 and joint American-Israeli grant BSF 00030.     

Fitting classes and injectors

Given a Fitting class ℱ associated to a homomorph we examine the existence of the ℱ-injectors for a finite groupG, provided thatG verifies certain ℱ-constraint conditions.     

Continuity of Lyapunov exponent for analytic quasi-periodic cocycles on higher-dimensional torus

It is known that the Lyapunov exponent is not continuous at certain points in the space of continuous quasi-periodic cocycles. We show that the Lyapunov exponent is continuous for a higher-dimensional analytic category in this paper. It has a modulus of continuity of the form exp(−∣logt∣ ^σ ) for some 0 < σ < 1.     

<Emphasis Type="Italic">p</Emphasis>-adic interpolation of Taylor coefficients of modular forms

In this paper we show that the Taylor coefficients of a Hecke eigenform at a CM-point, suitably modified, form a sequence of algebraic numbers that satisfy the Kubota–Leopoldt generalization of the Kummer congruences for primes that split in the imaginary quadratic field associated with a CM-point. More generally, we show that these numbers are moments of a certain p-adic measure. In addition, we write down explicitly the “Euler factor” at p in terms of the p th Hecke eigenvalue of the modular form in question and certain data of the CM-point. P. Guerzhoy is supported by NSF grant DMS-0700933.     

On the Iwasawa invariants of certain non-cyclotomic ℤ2-extensions

Let k be an imaginary quadratic field in which the prime 2 splits. We consider the Iwasawa invariants of a certain non-cyclotomic ℤ₂-extension of k and give some sufficient conditions for the vanishing of λ- and μ-invariants.     

K-area, Hofer metric and geometry of conjugacy classes in Lie groups

Given a closed symplectic manifold (M,ω) we introduce a certain quantity associated to a tuple of conjugacy classes in the universal cover of the group Ham (M,ω) by means of the Hofer metric on Ham (M,ω). We use pseudo-holomorphic curves involved in the definition of the multiplicative structure on the Floer cohomology of a symplectic manifold (M,ω) to estimate this quantity in terms of actions of some periodic orbits of related Hamiltonian flows. As a corollary we get a new way to obtain Agnihotri-Belkale-Woodward inequalities for eigenvalues of products of unitary matrices. As another corollary we get a new proof of the geodesic property (with respect to the Hofer metric) of Hamiltonian flows generated by certain autonomous Hamiltonians. Our main technical tool is K-area defined for Hamiltonian fibrations over a surface with boundary in the spirit of L. Polterovich’s work on Hamiltonian fibrations over S ². Oblatum 23-II-2001 & 9-V-2001?Published online: 20 July 2001     

Characterizations of life distributions from percentile residual lifetimes

Summary An α-percentile residual life function does not uniquely determine a life distribution; however, a continuous life distribution can be uniquely determined by its α-percentile and β-percentile residual life functions if α and β satisfy a certain condition. Two characterizations in terms of percentle residual lifetimes are given for the Beta (1, θ,K), Exponential (λ) and Pareto (θ,K) family of distributions.     

Size of matrix coefficients characterizes anisotropic principal series representations of the free group

Let Γ be a free nonabelian group and let Ω be its boundary. Let π _h be one of the unitary representations of Γ introduced in More (Duke Math J 82:381–436, 1996). By its definition π _h acts on L ²(Ω, d ν _h ) for a certain measure ν _h and satisfies certain genericity conditions. Those conditions guarantee that π _h is not equivalent to a principal isotropic/anisotropic series representation of Figà-Talamanca–Picardello and Figà-Talamanca–Steger (J Funct Anal 47:281–304, 1982/Mem Am Math Soc 531:1–68). In this paper we show the converse: if the genericity conditions are not satisfied then, up to a twist by a unitary character, π _h belongs to the isotropic/anisotropic series.     

On the generalized Hankel-Clifford transformation of arbitrary order

Two generalized Hankel-Clifford integral transformations verifying a mixed Parseval relation are investigated on certain spaces of generalized functions for any real value of their orders (α-β).     

Projective graph theory and configurations of lines

The spaces of disjoint configurations of k-dimensional subspaces in ℝP ^2k+1 (for example, lines in ℝP ³) are studied. These spaces are modeled by various simplicial schemes, and the homology groups of the latter are computed in certain cases. We use the fact that every configuration can be assigned a so-called projective graph, which is a class of graphs with respect to a certain equivalence relation. Bibliography: 5 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 231, 1995, pp. 309–322. Translated by N. Yu. Netsvetaev.     

Blow-up of sign-changing solutions of a quasilinear heat equation

We consider the problem of the blow-up of sign-changing solutions of the Cauchy problem for a quasilinear heat equation. The solutions are considered in a weighted function space that admits a certain growth of functions as |x| → ∞.     

Some characters of Kac and Wakimoto and nonholomorphic modular functions

Answering a question of Kac, we relate the character formulas for certain sℓ(m, 1)^ modules to automorphic forms. We show that these q-series are the “holomorphic parts” of nonholomorphic modular functions. The authors thank the NSF, and the second author thanks the Manasse family, and the Hilldale Foundation for their support.     

An invariant for certain smooth manifolds

Summary Starting with Hirzebruch's A-genus we define a numerical invariantμ for certain (4k-1)-manifolds. We produce examples to show howμ can distinguish differentiable structures on certain topological manifolds. To Enrico Bompiani on his scientific Jubilee Research partially supported by the Office of Naval Rescarch under Contract 266 (57). This author has a research grant from the National Science Foundation NSF-G-13989.