Some inequalities concerning the characteristic frequencies of elastic continuum

Summary We consider the boundary value problem αz″(x)+m(x)y(x)=0, αy″(x)+p(x)z(x)=0, xε[0, 1], y(0)=y(1)=z(0)=0, where the functions m(x) and p(x) are assumed integrable and positive everywhere in [0, 1]. As the main result we obtain the inequalities for n=1, 2, ... where δ_n(m, p) stands for the product of the first n eigenvalues α_i(m, p) of the above system and where δ_n(m) abbreviates δ_n(m, m). Entrata in Redazione il 6 febbraio 1976.     

Convolution properties of some classes of analytic functions

Let A denote the class of functions which are analytic in |z|<1 and normalized so that f(0)=0 and f′(0)=1, and let R(α, β)⊂A be the class of functions f such thatRe[f′(z)+αzf″(z)]>β,Re α>0, β<1. We determine conditions under which (i) f ∈ R(α₁, β₁), g ∈ R(α₂, β₂) implies that the convolution f×g of f and g is convex; (ii) f ∈ R(0, β₁), g ∈ R(0, β₂) implies that f×g is starlike; (iii) f≠A such that f′(z)[f(z)/z]^μ-1 ≺ 1 + λz, μ>0, 0<λ<1, is starlike, and (iv) f≠A such that f′(z)+αzf″(z) ≺ 1 + λz, α>0, δ>0, is convex or starlike. Bibliography: 16 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 138–154.     

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.     

Weakly metric spaces

The paper contains some initial results of the theory of weakly metric spaces. The weak triangle axiom: for any ε > 0 there exists δ > 0 such that for any points x, y, and z with d(y, z) ≤ δ, the inequality d(x, z) ≤ d(x, y) + ε holds. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 352, 2008, pp. 94–105.     

A unified approach for univalent functions with negative coefficients using the Hadamard product

For given analytic functions ϕ(z) = z + Σ _n=2^∞ λ _n z ⁿ , Ψ(z) = z + Σ _n=2^∞ μ with λ _n ≥ 0, μ _n ≥ 0, and λ _n ≥ μ _n and for α, β (0≤α<1, 0<β≤1), let E(φ,ψ; α, β) be of analytic functions ƒ(z) = z + Σ _n=2^∞ a _n z ⁿ in U such that f(z)*ψ(z)≠0 and

A Cauchy problem for the heat equation

Summary Let u(x, t) satisfy the heat equation in 0<x<1, 0<t≤T. Let u(x, 0)=0 for 0<x<1 and let |u(0, t)|<ε, | u_x(0, t) |<ε, and | u(1, t) |<M for 0≤t≤T. Then, , where M₁ and β(x) are given explicitly by simple formulas. The application of the a priori bound to obtain error estimates for a numerical solution of the Cauchy problem for the heat equation with u(x, 0)=h(x), u(0, t)=f(t), and u_x(0, t)=g(t) is discussed. Work performed under the auspices of the U. S. Atomic Energy Commission.     

A collocation by spline in tension

The collocation method by spline in tension for the problem: −εy"+p(x)y=f(x), y(0)=α₀,y(1)=α₁, p(x)>0, 0<ε<<1, is derived. The method has the second order of the global uniform convergence. For the corresponding difference scheme the optimal estimate: O (h_imin(h_i, ε) is obtained. This research was supported partly by NSF and SIZ for Science of SAP Vojvodina through funds made available to the U.S.—Yugoalav Joint Board on Scientific and Tchnological Cooperation (grants JF554, JF799).     

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.     

On Translation and Affine Systems Spanning L1(ℝ)

We show that the discrete translation parameter sets Λ ⊂ ℝ for which some φ ∈ L¹(ℝ) exists such that the translates φ(x − λ), λ ∈ Λ, span L¹(ℝ) are exactly the uniqueness sets for certain quasianalytic classes, and give explicit constructions of such generators φ. We also consider a similar situation for affine systems of the type φ(μx − λ), μ ∈ Γ, λ ∈ Λ.     

The region of values of a system of functionals in the class of typically real functions

Let T_R be the class of functions that are regular and typically real in the disk E={z:⋱z⋱<1}. For this class, the region of values of the system {f(z₀), f(r)} for z₀ ∈ ℝ, r∈(-1,1) is studied. The sets D_r={f(z₀):f∈T_R, f(r)=a} for −1≤r≤1 and Δ_r={(c₂, c₃): f ∈ T_R, −f(−r)=a} for 0<r≤1 are found, where aε(r(1+r)⁻², r(1−r)⁻²) is an arbitrary fixed number. Bibliography: 11 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 226, 1996, pp. 69–79.     

Abel-Lidskii bases in the non-self-adjoint inverse boundary problem

Let M be a manifold with boundary δM≠Ф. Let A be a second-order elliptic partial differential operator given on M. Denote by R_λ(x, y), x, y∈M, λε \σ(A) the Schwartz kernel of (A−λI)⁻¹. Consider the Gel'fand inverse boundary problem of the reconstruction of (M, A) via a given R_λ(x, y), x, y∈δM,λε . We prove that if the principal symbol of A satisfies some geometric condition (the Bardos-Lebeau-Rauch condition), then these data determine M uniquely, and they determine A to within the group of generalized gauge transformation on M. The above-mentioned geometric condition means, roughly speaking, that any geodesic (in the metric generated by A) leaves M. Bibliography: 29 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 161–190. Translated by Ya. Kurylev.     

非对称Cantor集的Hausdorff中心测度

§ 1　 IntroductionThe class of Cantor sets is a typical one of sets in fractal geometry.Mathematicianshave paid their attentions to such sets for a long time.Itis well known that the Hausdorffmeasure of the Cantor middle- third set is1(see[1]) .Recently,Feng[3] obtained the exactvalues of the packing measure for a class of linear Cantor sets.Using Feng s method,Zhuand Zhou[5] obtained the exactvalue of Hausdorff centred measure of the symmetry Cantorsets.In this papar,we consider the Ha…     

Differential subordinations concerning starlike functions

Denote byS ^* (⌕), (0≤⌕<1), the family consisting of functionsf(z)=z+a ₂z²+...+a_nzⁿ+... that are analytic and starlike of order ⌕, in the unit disc ⋎z⋎<1. In the present article among other things, with very simple conditions on μ, ⌕ andh(z) we prove the f’(z) (f(z)/z)^μ−1<h(z) implies f∈S^*(⌕). Our results in this direction then admit new applications in the study of univalent functions. In many cases these results considerably extend the earlier works of Miller and Mocanu [6] and others.     

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.     

Compact composition operators on the Bloch space in polydiscs

Let Uⁿ be the unit polydisc of ℂⁿ and φ=(φ₁, ⋯, φ _n ) a holomorphic self-map of Uⁿ. As the main result of the paper, it shows that the composition operator C is compact on the Bloch space β(Uⁿ) if and only if for every ε > 0, there exists a δ > 0, such that

Value distribution of Painlevé transcendents of the first and the second kind

We examine value distribution properties of the first and the second Painlevé transcendents. For every transcendental meromorphic solution ϕ(z) (resp. ψ(z)) of the first (resp. second) Painlevé equation, the deficiency δ(g,ϕ) (resp. δ(g, ψ)) of a small functiong(z) does not exceed 1/2. Furthermore, for ϕ(z), the ramification index satisfies ϑ(gφ)≤5/12.     

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     

High-Dimensional Centrally Symmetric Polytopes with Neighborliness Proportional to Dimension

Let A be a d by n matrix, d < n. Let C be the regular cross polytope (octahedron) in Rⁿ. It has recently been shown that properties of the centrosymmetric polytope P = AC are of interest for finding sparse solutions to the underdetermined system of equations y = Ax [9]. In particular, it is valuable to know that P is centrally k-neighborly. We study the face numbers of randomly projected cross polytopes in the proportional-dimensional case where d ∼ δn, where the projector A is chosen uniformly at random from the Grassmann manifold of d-dimensional orthoprojectors of Rⁿ. We derive ρ_N(δ) > 0 with the property that, for any ρ < ρ_N(δ), with overwhelming probability for large d, the number of k-dimensional faces of P = AC is the same as for C, for 0 ≤ k ≤ ρd. This implies that P is centrally ⌊ ρ d ⌋-neighborly, and its skeleton Skel_{⌊ ρ d ⌋}(P) is combinatorially equivalent to Skel_{⌊ ρ d⌋}(C). We display graphs of ρ_N. Two weaker notions of neighborliness are also important for understanding sparse solutions of linear equations: weak neighborliness and sectional neighborliness [9]; we study both. Weak (k,ε)-neighborliness asks if the k-faces are all simplicial and if the number of k-dimensional faces f_k(P) ≥ f_k(C)(1 – ε). We characterize and compute the critical proportion ρ_W(δ) > 0 such that weak (k,ε) neighborliness holds at k significantly smaller than ρ_W · d and fails for k significantly larger than ρ_W · d. Sectional (k,ε)-neighborliness asks whether all, except for a small fraction ε, of the k-dimensional intrinsic sections of P are k-dimensional cross polytopes. (Intrinsic sections intersect P with k-dimensional subspaces spanned by vertices of P.) We characterize and compute a proportion ρ_S(δ) > 0 guaranteeing this property for k/d ∼ ρ < ρ_S(δ). We display graphs of ρ_S and ρ_W.     

Averaging operators on normed köthe spaces

Summary Under study is the existence of averaging operators determined by measurable maps φ from a measure space (S, Σ, μ) into an arbitrary Hausdorff topological space T. The map φ induces a continuous map φ^e from the space C_b(T) into the normed (Banach) function space L_ϱ = L_ϱ(S, Σ, μ) defined by φ^e(f)=foφ for all f ε C_b(T). An integral representation for such operators is first studied. The existence is then determined by the existence of an averaging operator U₁ for the restriction of φ to a certain measurable subset B₁ of S. Utilizing a representation of L_ϱ(S, Σ, μ) as a Banach function space over a compact extremally disconnected Hausdorff space Ŝ, we are able to give a definition for the concept of plural points and irreducible map. A significant upper bound is given for the operator U₁. Finally conditions are considered under which no bounded projection from L_ϱ onto the range of φ^e may exist. From a topological point of view the development is pursued in a general setting. Averaging operators have recently been used for the study of injective Banach spaces of the type C_b(T) and in non-linear prediction and approximation theory relative to Tshebyshev subspaces of L_ϱ. Entrata in Redazione l’ll settembre 1975.     

Some Cubic Birth and Death Processes and their Related Orthogonal Polynomials

The orthogonal polynomials with recurrence relation (λ,_n +μ_n-z)F_n(z) = μ_n+1F_n+1(z)+λ_n-1F_n-1(z) with two kinds of cubic transition rates λ_n and μ_n, corresponding to indeterminate Stieltjes moment problems, are analyzed. We derive generating functions for these two classes of polynomials, which enable us to compute their Nevanlinna matrices. We discuss the asymptotics of the Nevanlinna matrices in the complex plane.