Structural formulas and value regions of functionals in certain classes of regular functions

We study the structural properties of the class M_k,λ,b(k≥2, 0≤λ≤1, b∈ℂ\{0}) of functions f(z)=z+ ... which are regular in |z|<1 and satisfy the conditions f(z)f′(z)z⁻¹≠0 and , where J(z)=λ(1+b⁻¹zf″(z)/f′(z)+(1−λ)(b⁻¹zf′(z)/f(z)+1−b⁻¹). The value regions of some functionals on this class are found. The case λ=1 was considered in our previous paper. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 204, 1993, pp. 55–60. Translated by O. A. Ivanov.     

The spectrum of the averaging operator for a finite homogeneous graph

We give an estimate for the spectrum of the averaging operator T₁(Γ, 1) over the radius 1 for the finite (q+1)-homogeneous quotient graph Γ/X, where X is an infinite (q+1)-homogeneous tree associated with the free group G over a finite set of generators S={x₁ ..., x_p} (2p=q+1), and Γ, a subgroup of finite index in G. T₁(Γ, 1) is defined on the subspace L²(Γ/G, 1) ⊖ E_ex, where E_ex is the subspace of eigenfunctions of T₁(Γ, 1) with eigenvalue λ such that |λ|=q+1. We present a construction of some finite homogeneous graphs such that the spectrum of their adjacency matrices can be calculated explicitly. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 205, 1993, pp. 92–109. Translated by A. M. Nikitin.     

A sturm-liouville problem with physical and spectral parameters in boundary conditions

We study the spectral probleml(u)=−u″+q(x)u(x)=λu(x),u′(0)=0, u′(π)=mλu(π), where λ andm are a spectral and a physical parameter. Form<0, we associate with the problem a self-adjoint operator in Pontryagin space II₁. Using this fact and developing analytic methods of the theory of Sturm-Liouville operators, we study the dynamics of eigenvalues and eigenfunctions of the problems asm→−0. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 163–172, August, 1999.     

Nonlinear differential equations with asymptotically stable solutions

We establish conditions of asymptotic stability of all solutions of the equation , t≥0in a Banach space E in the case where σ(A(x)⊂{λ:Reλ<0}∀x∈E. We give an example of an equation with unstable solutions. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 2, pp. 264–273, February, 1998.     

The value regions of initial coefficients in a certain class of meromorphic functions

Let M_k,λ(0≤λ≤1, k≥2) be the class of functions f(z)=1/z+a_o+a₁z+... that are regular and locally univalent for 0<⩛z⩛<1 and satisfy the condition where J_λ(z)=λ(1+zf″(z)/f'(z))+(1-λ)zf'(z)/f(z). In the class M_k,λ we consider sorne coefficient problems and problems concerning distortion theorems. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 91–96. Translated by N. Yu. Netsvetaev.     

Relationship between the Jung constant and acertain projection constant in Banach spaces

Summary For a Banach spaceX the parameter λ₁(X) is the infimum of the numbers a with the following property: for everyZ )X with dimZ/X=1 there exists a projectionP:Z→X with |P|≤a. We give an upper bound for λ₁(X) in terms of the Jung constantJ(X). Some known and some new results are then deduced. A characteristic property ofJ ₁-spaces is also proved.

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Riassunto Scopo di questa nota è mostrare la stretta relazione che intercorre tra la costante di JungJ(X) di uno spazio di BanachX e la costante λ₁(X). Si stabilisce una limitazione superiore di λ₁, come funzione diJ, dalla quale si possono dedurre alcuni nuovi risultati e altri conosciuti. è inoltre dimostrata una caratterizzazione deiJ ₁-spazi.

     

Some infinite groups with strongly embedded subgroup

An involution i of a group G is said to be finite if |ii^g|<∞ for all g ∃ G. Suppose that G contains a finite involution and an infinite elementary Abelian 2-subgroup S and, moreover, the normalizer H=N_G(S)=SλT is strongly embedded in G and is a Frobenius group with locally cyclic complement T. It is proved that G is isomorphic to L₂(Q) over a locally finite field Q of characteristic 2. In particular, part (a) of Question 10.76 raised by Shunkkov in the Kourovka Notebook is answered in the affirmative. Supported by RFFR grant No. 99-01-00542. Translated fromAlgebra i Logika, Vol. 39, No. 5, pp. 602–617, September–October, 2000.     

Extremal probabilities for Gaussian quadratic forms

 Denote by Q an arbitrary positive semidefinite quadratic form in centered Gaussian random variables such that E(Q)=1. We prove that for an arbitrary x>0, inf _Q P(Q≤x)=P(χ² _n /n≤x), where χ _n ² is a chi-square distributed rv with n=n(x) degrees of freedom, n(x) is a non-increasing function of x, n=1 iff x>x(1)=1.5364…, n=2 iff x[x(2),x(1)], where x(2)=1.2989…, etc., n(x)≤rank(Q). A similar statement is not true for the supremum: if 1<x<2 and Z ₁ ,Z ₂ are independent standard Gaussian rv's, then sup_0≤λ≤1/2 P{λZ ₁ ² +(1−λ)Z ₂ ² ≤x} is taken not at λ=0 or at λ=1/2 but at 0<λ=λ(x)<1/2, where λ(x) is a continuous, increasing function from λ(1)=0 to λ(2)=1/2, e.g. λ(1.5)=.15…. Applications of our theorems include asymptotic quantiles of U and V-statistics, signal detection, and stochastic orderings of integrals of squared Gaussian processes. Received: 24 June 2002 / Revised version: 26 January 2003 Published online: 15 April 2003 Research supported by NSA Grant MDA904-02-1-0091 Mathematics Subject Classification (2000): Primary 60E15, 60G15; Secondary 62G10     

Implicit functions determined by equations with unbounded operators and the solvability of operator equations

For implicit functions determined by the equation T_x+F(λ,x)=0, where T is an unbounded operator, existence and differentiability conditions are established. As an application, the solvability conditions for equations of the form Tx+f(x)=0 are derived. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 248, 1998, pp. 216–224. Translated by L. Yu. Kolotilina.     

Asymptotics of the eigenvalues of the dirichlet and neumann problems for the abstract sturm-liouville operator

Let A>0 be an unbounded self-adjoint operator in a Hilbert space H. In the Hilbert space H₁=L₂ (0, π; H) we study the spectrum of the differential equations−y″(x)+Ay=λy, y (0)=y(π)=0,−y″(x)+Ay=λy, y′(0) =y′(π)=0. We find the principal terms of the asymptotics of the functions N(λ) for these problems and we ascertain the conditions under which they are asymptotically not equivalent. Translated from Matematicheskie Zametki, Vol. 21, No. 2, pp. 209–212, February, 1977.     

On semigroups generated by restrictions of elliptic operators to invariant subspaces

Let A be the closed unbounded operator inL _p(G) that is associated with an elliptic boundary value problem for a bounded domainG. We prove the existence of a spectral projectionE determined by the set Γ = {λ;θ ₁≦argλ≦θ ₂} and show thatAE is the infinitesimal generator of an analytic semigroup provided that the following conditions hold: 1<p<∞; the boundary ϖΓ of Γ is contained in the resolvent setp(A) ofA;π/2≦θ<θ ₂≦3π/2 ; and there exists a constantc such that (I)││(λ-A)^-1││≦c/│λ│ for λ∈ϖΓ. The following consequence is obtained: Suppose that there exist constantsM andc such that λ∈p(A) and estimate (I) holds provided that |λ|≧M and Re λ=0. Then there exist bounded projectionE ⁻ andE ⁺ such thatA is completely reduced by the direct sum decompositionL _p(G)=E⁻L_p (G) ⊕E⁺L_p (G) and each of the operatorsAE ⁻ and—AE ⁺ is the infinitestimal generator of an analytic semigroup.     

Transitive groups with irreducible representations of bounded degree

A well-known theorem of Jordan states that there exists a function J(d) of a positive integer d for which the following holds: if G is a finite group having a faithful linear representation over ℂ of degree d, then G has a normal Abelian subgroup A with [G:A]≤J(d). We show that if G is a transitive permutation group and d is the maximal degree of irreducible representations of G entering its permutation representation, then there exists a normal solvable subgroup A of G such that [G:A]≤J(d) ^log ₂ ^d. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1995, pp. 108–119. Translated by S. A. Evdokimov.     

On the influence of parameters determining anisotropic spaces of smooth functions on characteristics of the embedding theorems

An anisotropic Sobolev and Nikol'skii-Besov space on a domain G is determined by its integro-differential (shortly, ID) parameters. On the other hand, the geometry of G is characterized by the set Λ(G) of all vectors λ=(λ₁,..., λ_n) such that G satisfies the λ-horn condition. We study the dependence of the totality of possible embeddings upon the set Λ(G) and theID-parameters of the space. We consider only embeddings with q≥pⁱ, where pⁱ are the integral parameters of the space and q is the integral embedding parameter. For a given space, we introduce its initial matrix A₀ determined by theID-parameters. A₀ turns out to be a Z-matrix. On the basis of a natural classification of Z-matrices, a classification of anisotropic spaces is introduced. This classification allows one to restate the existence of an embedding with q≥pⁱ in terms of certain specific properties of A₀. Let A₀ be a nondegenerate M-matrix. Any vector λ∈Λ(G) gives rise to a certain set of admissible values of the embedding parameters. We call λ optimal if this set is the largest possible. It turns out that the optimal vector λ _G ^* is determined by Λ(G) and A₀, and may be found by a linear optimization procedure. The following cases are possible: a) , b) , c) λ _G ^* does not exist. In case a) the set of admissible values of the embedding parameters is the biggest, while in case c) no embeddings with q≥pⁱ exist. In case b) the so-called saturation phenomenon occurs, i.e., certain variations of some differential parameters of the space do not change the set of admissible values of the embedding parameters. The latter fact has some applications to the problem of extension of all functions belonging to the given space from G to Eⁿ. Bibliography: 20 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 201, 1992, pp. 22–94. Translated by A. A. Mekler.     

Positive versus free boundary solutions to a singular elliptic equation

The equation-δu = χ uo(-1/u^Β + λf (x, u)) in Ω with Dirichlet boundary condition on ∂Ω has a maximal solution uλ ≥0 for every λ 0. For λ less than a constant λ*, the solution vanishes inside the domain; and for λ λ*, the solution is positive. We obtain optimal regularity ofuλ even in the presence of the free boundary. Supported in part by H. J. Sussmann’s NSF Grant DMS01-03901. Supported by FAPESP. He also thanks Rutgers University.     

Differential sobordination and Bazilevič functions

LetM(z)=z ⁿ +…,N(z)=z ⁿ +… be analytic in the unit disc Δ and let λ(z)=N(z)/zN′(z). The classical result of Sakaguchi-Libera shows that Re(M′(z)/N′(z))<0 implies Re(M(z)/N(z))>0 in Δ whenever Re(λ(z))>0 in Δ. This can be expressed in terms of differential subordination as follows: for anyp analytic in Δ, withp(0)=1,p(z)+λ(z)zp′(z)<1+z/1−z impliesp(z)<1+z/1−z, for Reλ(z)>0,z∈Δ. In this paper we determine different type of general conditions on λ(z),h(z) and ϕ(z) for which one hasp(z)+λ(z)zp′(z)<h(z) impliesp(z)<ϕ(z)<h(z) z∈Δ. Then we apply the above implication to obtain new theorems for some classes of normalized analytic funotions. In particular we give a sufficient condition for an analytic function to be starlike in Δ.     

Normal Cayley graphs of finite groups

LetG be a finite group and let S be a nonempty subset of G not containing the identity element 1. The Cayley (di) graph X = Cay(G, S) of G with respect to S is defined byV (X)=G, E (X)={(g,sg)|g∈G, s∈S} A Cayley (di) graph X = Cay (G,S) is said to be normal ifR(G) ◃A = Aut (X). A group G is said to have a normal Cayley (di) graph if G has a subset S such that the Cayley (di) graph X = Cay (G, S) is normal. It is proved that every finite group G has a normal Cayley graph unlessG≅ℤ₄×ℤ₂ orG≅Q ₈×ℤ ₂ ^r (r⩾0) and that every finite group has a normal Cayley digraph, where Z_m is the cyclic group of orderm and Q₈ is the quaternion group of order 8. Project supported by the National Natural Science Foundation of China (Grant No. 10231060) and the Doctorial Program Foundation of Institutions of Higher Education of China.     

Discrete spectrum in the gaps of a perturbed pseudorelativistic hamiltonian

The pseudorelativistic Hamiltonian is considered under wide conditions on potentials A(x), W(x). It is assumed that a real point λ is regular for G_1/2. Let G_1/2(α)=G_1/2−αV, where α>0, V(x)≥0, and V ∈L _d(ℝ^d). Denote by N(λ, α) the number of eigenvalues of G_1/2(t) that cross the point λ as t increases from 0 to α. A Weyl-type asymptotics is obtained for N(λ, α) as α→∞. Bibliography: 5 titles. To O. A. Ladyzhenskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 249, 1997. pp. 102–117. Translated by A. B. Pushnitskii.     

Invariant factors of combinatorial matrices

The Smith normal forms of an Hadamard matrix of order 4m (m square-free), and of the incidence matrix of a (ν, k, λ) configuration (n=k−λ square-free (n, λ)=1), are determined.     

Constants of simultaneous extension of continuous functions

It is shown that ifB is the unit ball of a non-separable Hilbert space with its weak topology, then for every number λ≧1, there exists a spaceK _λ containingB, such that the constant of simultaneous extension fromC(B) toC(K _λ) is exactly λ. This gives a negative answer to the question whether the constants of simultaneous extension ought to be odd integers, as was suggested by examples of Corson-Lindenstrauss and Corson-Pelczynski. This is a part of the author’s Ph.D. thesis prepared at the Hebrew University of Jerusalem under the supervision of Professor J. Lindenstrauss. I wish to thank Professor Lindenstrauss for his interest and advice.     

Orthogonal almost locally minimal projections on ℓ 1 n

A projectionP on a Banach spaceX with ‖P‖≤λ₀ is called almost locally minimal if, for every α>0 small enough, the ballB(P,α) in the space of operatorsL(X) does not contain a projectionQ with ‖Q‖≤‖P‖(1–Dα²), whereD=D(λ₀) is a constant independent of ‖P‖. It is shown that, for everyp≥1 and every compact abelian groupG, every translation invariant projection onL _p(G) is almost locally minimal. Orthogonal projections on ℓ ₁ ⁿ are investigated with respect to some weaker local minimality properties. Participant in Workshop in Linear Analysis and Probability, Texas A&M University, College Station, Texas 1998. Partially supported by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany).