Ein Einschliessungsverfahren für Lösungen Fehlerbehafteter Linearer Gleichungen

In a partially ordered space, the method x_n+1 = L⁺x _n ⁺ – N⁺x _n ^- – L^–y⁺ + N^– y _n ^- + r, y_n+1 = N⁺y⁺ – L⁺y _n ^- – N^–x _n ⁺ + L^–x^– + t of successive approximation is developed in order to enclose the solutions of a set of linear fixed point equations monotonously. The method works if only the inequalities x₀ y₀, x₀ x₁, y₁ y₀ related to the starting elements are satisfied. In finite-dimensional spaces suitable starting vectors can be computed if a sufficiently good approximation for the fixed points is known.

     

Continuation of semicontinuous processes with independent increments

We consider a process X _t ¹ with independent increments without positive jumps in the state space (–; +) VarX _t ¹ =+. For a stopped process in the space E⁰ we construct its continuation in E⁰ U {0}.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1269–1272, September, 1991.     

A new graphic characterization of non singular quadrics of PG(r,q)

LetK be ak-set of class [0, 1,m,n]₁ of anr-dimensional projective Galois space PG(r, q) of orderq. We prove that: Ifr = 2s (s 2),k = _2s–1 and if through each point ofK there are exactlyq ^2(s–1) tangent lines and at most _2s–3 n-secant lines, thenK is a non singular quadric of PG(2s,q). Ifr = 2s–1 (s2),k=_2(s–1) +q ^s–1 and if at each point ofK there are exactlyq ^2s–3 –q ^s–2 tangents and at most _2(s–2)+q ^s–2 n-secant lines, thenK is a hyperbolic quadric of PG(2s–1,q).     

Uniqueness of solutions of boundary-value problems for operator-differential equations on a finite segment and on a semiaxis

For the equationL ₀ x(t)+L ₁x(t)+...+L _n x ⁽ⁿ⁾(t)=O, whereL _k,k=0,1,...,n, are operators acting in a Banach space, we establish criteria for an arbitrary solutionx(t) to be zero provided that the following conditions are satisfied:x ^(1–1) (a)=0, 1=1, ..., p, andx ^(1–1) (b)=0, 1=1,...,q, for - <a< b< (the case of a finite segment) orx ^(1–1) (a)=0, 1=1,...,p, under the assumption that a solutionx(t) is summable on the semiaxista with its firstn derivatives.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 279–292, March, 1994.This research was supported by the Ukrainian State Committee on Science and Technology.     

Strong convergence and local limit theorems for functionals of supremum type

Let p_n be distributions in the space D([0, 1]^d), the multiparameter analogue of the Skorokhod space, which correspond to stepwise processes constructed on the basis of sums of independent random variables. Let P be the distribution corresponding to the d-parameter Brownian motion. We set where xD([O,1]^d),Tc [0,1]^d, and h is continuous on [0, 1]^d. Some results are established regarding convergence in variation of the distributions P_nf _i ^–1 ,Tand Pf _i ^–1 ,T, i=1, 2.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 85, pp. 39–45, 1979.In conclusion, the author thanks V. V. Gorodetskii for useful discussions.     

The Whitehead Group of the Novikov Ring

The Bass–Heller–Swan–Farrell–Hsiang–Siebenmann decomposition of the Whitehead group K ₁(A[z,z^-1]) of a twisted Laurent polynomial extension A[z,z^-1] of a ring A is generalized to a decomposition of the Whitehead group K ₁(A((z))) of a twisted Novikov ring of power series A((z))=A[[z]][z^-1]. The decomposition involves a summand W₁(A, ) which is an Abelian quotient of the multiplicative group W(A,) of Witt vectors 1+a₁z+a₂z²+ ··· A[[z]]. An example is constructed to show that in general the natural surjection W(A, )^ab W₁(A, ) is not an isomorphism.     

Spectrum of the nonself-adjoint Schrödinger operator in unbounded regions

It is proved that the discrete spectrum of the operator – + q(x) in the space L₂(E₂k) (k 1) where q(x) is a measurable complex-valued function satisfying the condition ¦q(x) ¦ Ce^–¦x¦, having no finite limit points, and for k=1 the discrete spectrum consists of a finite number of points.Translated from MatematicheskieZametki, Vol. 9, No. 1, pp. 19–26, January, 1971.In conclusion the author wishes to thank M. A. Naimark for directing this work.     

Connecting two semicontinuous processes with independent increments

We consider a stopped process X_t ⁰ in the phase space E⁰=(–, +)/{0} such that X_t ⁰=X_t ¹ if X_t ⁰ > 0 and X_t ⁰=X_t ² if X_t ⁰ < 0, where X_t ^j, j=1,2, are nonstopped stochastically continuous Markov processes with independent increments and with only negative jumps. We prove that there exists an extension of X_t ⁰ into a homogeneous, stochastically continuous, and strong Markov Feller process X_t in the phase space (–; +) and that the extension can be characterized by a measure N(dy) and three constants b, c₁ c₂.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 5, pp. 596–600, May, 1991.     

Necessary conditions for the existence of divisible designs

BOSE and CONNOR [2] proved that a symmetric regular divisible design with w classes of sizes g and joining numbers ₁ and ₂ must satisfy for every prime p the arithmetic condition (d₁, (–1)^sw)_p(d₂,(–l)^tg^w)_p=1, where d₁=k²–v₂, d₂= k–₁ s=(w-1)(w-2)/2, t=(v-w)(v-w-1)/2 and (*,*) is the Hilbert symbol. We show that if in addition _{1 2} and the design is fully symmetric divisible then (d₁, (–1)^s w)_p=(d₂, (–1)^tg^w)=1. Our assumption is by a result of CONNOR [5] fulfilled, if d₁ and ₁–₂ are relatively prime. Thus, we can exclude parameters not accessible to the Bose-Connor-Theorem. Our result can be derived from a theorem of RAGHAVARAO [9], and we give the precise assumptions of this theorem. We also discuss arithmetic restrictions for divisible designs which satisfy diverse other rules for the intersection numbers and generalize a result of DEMBOWSKI [6; 2.1.11].Dedicated to Professor Benz on occasion of his sixtieth birthday     

Relation between the vectors Э and S for a plastic material with a sequence of ray-like loading paths

Two different complex loading paths are investigated in stress space — longitudinal tension with subsequent total unloading and reloading in transverse tension or compression. For these loading paths the local strains theory [4] is used to determine the values of the components of the strain vector {₁, ₂} in five-dimensional space for nonlinearities n=3 and n=5 together with the components of the stress vector S{S₁, S₂}. A relation between the vectors and S is established in terms of the given loading parameter k.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 241–245, March–April, 1970.     

On a continuous analog of the gradient method

In a real Hilbert space H we consider the nonlinear operator equation P(x)=0 and the continuous gradient methodx (t)= –P (x)^* P (x), x (0) = x₀. Two theorems on the convergence of the process (*) to the solution of the equation P(x)=0 are proved.Translated from Matematicheskie Zametki, Vol. 3, No. 4, pp. 421–426, April, 1968.     

Accurate estimates of deviations of spline approximations to classes of differentiable functions

We derive the approximation on [0, 1] of functionsf(x) by interpolating spline-functions s_r(f; x) of degree 2r+1 and defect r+1 (r=1, 2,...). Exact estimates for ¦f(x)–s_r(f; x) ¦ and f(x)–s_r(f; x)|_c on the class W^mH for m=1, r=1, 2, ..., and m=2, 3, r=1 for the case of convex (t),are derived.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 483–494, May, 1971.     

Probabilities of large deviations in a certain scheme of summation of dependent random variables

In the paper one investigates questions regarding the CLT for an Ising scheme. If , where S _n = ₌₁ ⁿ X _n, while (X_nk) is a triangular array of random variables, constituting an Ising scheme, then in the zone one proves the equalities lim(1 — F_n(x))(1 — (x))^–1=1, lim F_n(–x)(–1))^–1=1.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 177, pp. 138–144, 1989.     

The self-adjointness conditions for a higher order differential operator with an operator coefficient

Certain sufficient conditions are found for self-adjointness of the differential operator generated by the expressionl (y)=(–1) ⁿ y ²ⁿ +Q (x)y, – <x <, where Q(x) is for each fixed value of x a bounded self-adjoint operator acting from the Hilbert space H into H, and y(x) is a vector function of H₁ for which .Translated from Matematicheskie Zametki, Vol. 5, No. 6, pp. 697–707, June, 1969.     

Ebene algebraische Kurven vom Typ p,q

Let p,q be relatively prime integers with 2pr ^p,q be the numerical semigroup generated by p,q,{(p–1) (q–1)–1–(ip+jq)¦i+jr–2}. Then there exists a smooth projective curve X and a point x on X, such that H _r ^p,q is the set of orders of poles of the rational functions on X, which are regular on X\{x}; in other words: H _r ^p,q is a Weierstraß semigroup.     

The approximation of functions of many variables by their Féjér sums

We construct elliptic Féjér polynomials K_n(x) of m variables. We prove some of their properties: a) the Féjér polynomials are positive on the m-dimensional torus T^m, K_n(x)0, b) (x)=o(n^–1), as n, c) we calculate their norms in the spaces L[T^m] and C[T^m]. We estimate the deviation of the Féjér sum _n(x,f) from the functionf(x). For the space C[T^m]: where c_,m c_1,m are constants.Translated from Matematicheskie Zametki, Vol. 13, No. 6, pp. 817–828, June, 1973.In conclusion, the author wishes to express his gratitude to S. B. Stechkin for help with the paper.     

Measurable solutions of a (2, 2)-type sum form functional equation

Summary In this paper we find the general measurable solutions of the functional equationF(xy) + F(x(1 – y)) – F((1 – x)y) – F((1 – x)(1 – y)) = G(x)H(y) (x, y ]0, 1[) whereF, G, H:]0, 1[ C are unknown functions. The solution of this equation is part of our program to determine the measurable solutions of the functional equationF ₁₁ (xy) + F ₁₂ (x(1 – y)) + F ₂₁ ((1 – x)y) + F ₂₂ ((1 – x)(1 – y)) = G(x)H(y) (x, y ]0, 1[). Our method of solution is based on the structure theorem of sum form equations of (2, 2)-type and on a result of B. Ebanks and the author concerning the linear independence of certain functions.     

Knaster's problem on continuous maps of a sphere into Euclidean space

A survey of known results and additional new ones on Knaster's problem: on the standard sphere S^n–1Rⁿ find configurations of points A₁, , A_k, such that for any continuous map fS^n–1R^m one can find a rotation a of the sphere S^n–1 such that f(a(A₁)==f(a(A_k)) and some problems closely connected with it. We study the connection of Knaster's problem with equivariant mappings, with Dvoretsky's theorem on the existence of an almost spherical section of a multidimensional convex body, and we also study the set {a S0(n)f(a(A₁))==f(a(A_k))} of solutions of Knaster's problem for a fixed configuration of points A₁, , A_kS^n–1 and a map fS^n–1R^m in general position. Unsolved problems are posed.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 167, pp. 169–178, 1987.     

The generating elements of certain Volterra operators connected with third- and fourth-order differential operators

Sufficient conditions are established forf (x) to be the generating function for the Volterra operator which is inverse to the Cauchy operator:l [y]=y⁽ⁿ⁾+p₂(x)y^(n–2) + ... +pn(x)y, y(0)=y(0)=...=y^(n–1)(0)=0 (n=3, 4), when the coefficients pi(x) are not analytic.Translated from Matematicheskie Zametki, Vol. 3, No. 6, pp. 715–720, June, 1968.     

Finite extinction time for some perturbed Hamilton-Jacobi equations

In this paper we study initial value problems likeu _t–R¦u¦^m+u^q=0 in ⁿ× ₊, u(·,0⁺)=u_o(·) in ^N, whereR > 0, 0 <q < 1,m 1, andu _o is a positive uniformly continuous function verifying –R¦u _o¦^m+u ₀ ^q 0 in ^N . We show the existence of the minimum nonnegative continuous viscosity solutionu, as well as the existence of the function t(·) defined byu(x, t) > 0 if 0<t<t (x) andu(x, t)=0 ift t (x). Regularity, extinction rate, and asymptotic behavior of t(x) are also studied. Moreover, form=1 we obtain the representation formulau(x, t)=max{([(u _o(x – t))^1–q –(1–q)t]₊)^1/(1–q): ¦¦R}, (x, t) ₊ ^N+1 .Partially supported by the DGICYT No. 86/0405 project.