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.     

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.     

Eigenfunctions and associated functions of an n-th-order linear differential operator

For n2 we consider a differential operatorL [y] z ⁿ y ⁽ⁿ⁾ +P ₁(z)z ^n–1 y ^(n–1) +P ₂ (z)z ^n–2 y ^n–2 + ...+P _n (z)y = y, p ₁ (z), ..., P _n (z) A _R : here a_r is the space of functions which are analytic in the disk ¦z¦ < R, equipped with the topology of compact convergence. We prove the existence of sequences {f_k(z)} _k =o, consisting of a finite number of associated functions of the operator L and an infinite number of its eigenfunctions; we show that the sequence forms a basis in A_r for an arbitrary r, 0 < r <- R; and we establish some additional properties of the sequence ₀ (z), ₁ (z),..., _d–1 (z), f _d (z), f _d+1 (z),... Translated from Matematicheskie Zametki, Vol. 20, No. 6, pp. 869–878, December, 1976.     

Two combinatorial problems on posets

Bialostocki proposed the following problem: Let nk2 be integers such that k|n. Let p(n, k) denote the least positive integer having the property that for every poset P, |P|p(n, k) and every Z _k -coloring f: P Z _k there exists either a chain or an antichain A, |A|=n and _aA f(a) 0 (modk). Estimate p(n, k). We prove that there exists a constant c(k), depends only on k, such that (n+k–2)²–c(k) p(n, k) (n+k–2)²+1. Another problem considered here is a 2-dimensional form of the monotone sequence theorem of Erdös and Szekeres. We prove that there exists a least positive integer f(n) such that every integral square matrix A of order f(n) contains a square submatrix B of order n, with all rows monotone sequences in the same direction and all columns monotone sequences in the same direction (direction means increasing or decreasing).     

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.     

Pointwise estimation of comonotone approximation

We prove that, for a continuous functionf(x) defined on the interval [–1,1] and having finitely many intervals where it is either nonincreasing or nondecreasing, one can always find a sequence of polynomialsP _n (x) with the same local properties of monotonicity as the functionf(x) and such that ¦f(x)–P _n (x) ¦C₂(f;n^–2+n ^–11–x ²), whereC is a constant that depends on the length of the smallest interval.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 11, pp. 1467–1472, November, 1994.The author is grateful to Prof. I. A. Shevchuk for his permanent attention to the work.     

Finite difference methods for two-point boundary value problems involving high order differential equations

We discuss the construction of finite difference schemes for the two-point nonlinear boundary value problem:y ⁽²ⁿ⁾+f(x,y)=0,y ^(2j)(a)=A _2j ,y ^(2j)(b)=B _2j ,j=0(1)n–1,n2. In the case of linear differential equations, these finite difference schemes lead to (2n+1)-diagonal linear systems. We consider in detail methods of orders two, four and six for two-point boundary value problems involving a fourth order differential equation; convergence of these methods is established and illustrated by numerical examples.     

A functional inequality for real-analytic functions

Summary Forf ( C ⁿ() and 0 t x letJ _n (f, t, x) = (–1)ⁿ f(–x)f ⁽ⁿ⁾(t) +f(x)f ⁽ⁿ⁾ (–t). We prove that the only real-analytic functions satisfyingJ _n (f, t, x) 0 for alln = 0, 1, 2, are the exponential functionsf(x) = c e ^x,c, . Further we present a nontrivial class of real-analytic functions satisfying the inequalitiesJ ₀ (f, x, x) 0 and ₀ ^x (x – t)^{n – 1}J_n(f, t, x)dt 0 (n 1).     

On a lower bound for the perron eigenvalue

A lower boundn ^–1 _i,k a_ik for the Perron eigenvalue of a symmetric non-negative irreducible matrixA=(a _ik) is studied and compared with certain other lower bounds.     

Return probabilities for random walk on a half-line

A random walk with reflecting zone on the nonnegative integers is a Markov chain whose transition probabilitiesq(x, y) are those of a random walk (i.e.,q(x, y)=p(y–x)) outside a finite set {0, 1, 2,...,K}, and such that the distributionq(x,·) stochastically dominatesp(·–x) for everyx{0, 1, 2,..., K}. Under mild hypotheses, it is proved that when xp _x>0, the transition probabilities satisfyq ⁿ(x, y)C_xyR^–nn^–3/2 asn, and when xp _x=0,q ⁿ(x, y)C_xyn^–1/2.Supported by National Science Foundation Grant DMS-9307855.     

Algebras almost commuting with Clifford algebras in a II∞ factor

We are concerned here with certain Banach algebras of operators contained within a fixed II factor N. These algebras may be thought of as noncommutative classifying spaces for the functor Ext _* ^N The basic objects of study are the algebras A _kN (for n=1, 2,...). Here, we are given an essentially unique representation of the complex Clifford algebra C _k N and the elements of A _k are those operators in N which exactly commute with the first k–1 generators of C _k and also commute with the kth generator modulo a symmetric ideal N. Up to isomorphism, these algebras are periodic with period 2.We determine completely the homotopy types of A ₁ ^–1 and A ₂ ^–1 Here, A ₁ ^–1 is homotopy equivalent to the space of (Breuer) Fredholm operators in N, while A ₂ ^–1 is homotopy equivalent to the group K _N ^–1 ={x N^–1¦ x=1+k, k K_N}. We use these results to compute the K-theory of A ₁ and A ₂.For a fixed C ^*-algebra A, we define abelian groups G _k,N(A) of equivalence classes of homomorphisms AA _k. Letting N = M (H) for a II₁ factor M we define similar abelian groups G _k(A, M) where we replace N by L(E) for countably generated right Hilbert M-modules E with (left) actions C _k L(E). Using ideas of Skandalis, we show that G _k,NG_k(A, M) so that the G _k,N are stable half-exact homotopy functors because the G _k(·, M) are such.In general, we show that G _k(A, M)KK ^k(A, M) and so our theory fits neatly into Kasparov KK-theory. We investigate many interesting examples from our point of view.     

Control of stochastic distributed-parameter systems

A scheme is proposed for the feedback control of distributed-parameter systems with unknown boundary and volume disturbances and observation errors. The scheme consists of employing a nonlinear filter in the control loop such that the controller uses the optimal estimates of the state of the system. A theoretical comparison of feedback proportional control of a styrene polymerization reactor with and without filtering is presented. It is indicated how an approximate filter can be constructed, greatly reducing the amount of computing required.Notation a(t) l-vector noisy dynamic input to system - A(t, a) l-vector function - A frequency factor for first-order rate law (5.68×10⁶ sec^–1) - b distance to the centerline between two coil banks in the reactor (4.7 cm) - B k-vector function defining the control action - c(, ) concentration of styrene monomer, molel ^–1 - C _p heat capacity (0.43 cal · g^–1 · K^–1) - C _ij constants in approximate filter, Eqs. (49)–(52) - E activation energy (20330 cal · mole^–1) - expectation operator - f(t,...) n-vector function - g ₀,g ₁(t,...) n-vector functions - h(t, u) m-vector function relating observations to states - H(t) function defined in Eq. (36) - k dimensionality of control vectorv(x, t) - k _i constants in approximate filter, Eqs. (49)–(52) - K dimensionless proportional gain - l dimensionality of dynamic inputa(t) - m dimensionality of observation vectory(t) - n dimensionality of state vectoru(x, t) - P ^(vv)(x, s, t) n×n matrix governed by Eq. (9) - P ^(va)(x, t) n×l matrix governed by Eq. (10) - P ^(aa)(t) l×l matrix governed by Eq. (11) - q _i (t) diagonal elements ofm×m matrixQ(x, s, t) - Q(x, s, t) m×m weighting matrix - R universal gas constant (1.987 cal · mole^–1 · K^–1) - R(x, s, t) n×n weighting matrix - R _i (t) n×n weighting matrix - s dimensionless spatial variable - S(x, s, t) matrix defined in Eq. (11) - t dimensionless time variable - T(, ) temperature, K - u(x, t) n-dimensional state vector - u _c (t) wall temperature - u _d desired value ofu ₁(1,t) - u _c * reference control value ofu _c - u _c ^max maximum value ofu _c - u _c ^min minimum value of _c - v(x, t) k-dimensional control vector - W(t) l×l weighting matrix - x dimensionless spatial variable - y(t) m-dimensional observation vector - _i constants in approximate filter, Eqs. (49)–(52) - dimensionless parameter defined in Eq. (29) - H heat of reaction (17500 cal · mole^–1) - dimensionless activation energy, defined in Eq. (29) - (x) Dirac delta function - (x, t) m-dimensional observation noise - thermal conductivity (0.43×10^–3 cal · cm^–1 · sec^–1 · K^–1) - density (1 g · cm^–3) - time, sec - dimensionless parameter defined in Eq. (29) - spatial variable, cm - * reference value - ^ estimated value     

Pointwise decomposable sets

We show that, under the conditionala<0, every recursively enumerable (r.e.) A bia has a pointwise decomposable complement. If A T^B, A and ¯B are r.e. co-retraceable sets, and f(x)=f^B(x), then there exists a r.e. co-retraceable C, such thatA(c),BT ^C , (A n) (f(n) <c _n), where ¯C=C ₀<C ₁<C ₂<....Translated from Matematicheskie Zametki, Vol. 13, No. 6, pp. 893–898, June, 1973.The author thanks A. N. Degtev for his interest in this work.     

On the periodic solutions of the second-order wave equations. V

It is established that the linear problemu _u –a ² u _xx =g(x,t),u(0,t) =u(x, t + T) =u(x,t) is always solvable in the function spaceA = {g:g(x,t) =g(x,t+T) =g( –x,t) = –g(–x,t)} provided thataTq = (2p – 1) and (2p – 1,q) = 1, wherepandq are integer numbers. To prove this statement, an exact solution is constructed in the form of an integral operator, which is used to prove the existence of a solution of a periodic boundary-value problem for a nonlinear second-order wave equation. The results obtained can be used when studying the solutions to nonlinear boundary-value problems by asymptotic methods.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 8, pp. 1115–1121, August, 1993.     

Formulas for best extrapolation

We considern-point Lagrange-Hermite extrapolation forf(x), x>1, based uponf(x _i ),i=1(1)n, –1x _i 1, including non-distinct pointsx _i in confluent formulas involving derivatives. The problem is to find the pointsx _i that minimize the factor in the remainderP _n (x)f ⁽ⁿ⁾()/n, –1<<x subject to the condition|P _n (x)|M, –1x1,2^–n+1M2 ⁿ . The solution is significant only when a single set of pointsx _i suffices for everyx>1. The problem is here completely solved forn=1(1)4. Forn>4 it may be conjectured that there is a single minimal , 0 rn, whererr(M) is a non-decreasing function ofM, P _n (–1)=(–1) ⁿ M, and for 0rn–2, thej-th extremumP _n (x _{e, j} )=(–1) ^n–j M,j=1(1)n–r–1 (except forM=M _r ,r=1(1)n–1, whenj=1(1)n–r).     

On a generalization of a theorem of Erdős and Fuchs

Let A={a₁,a₂,…}(a₁<a₂<) be an infinite sequence of nonnegative integers, let k≥2 be a fixed integer and denote by r_k(A,n) the number of solutions of a_i1+a_i2++a_ik≤n. Montgomery and Vaughan proved that r₂(A,n)=cn+o(n^1/4) cannot hold for any constant c>0. In this paper, we extend this result to k>2.     

Simple minimum coverings ofK n with copies ofK 4 −e

Summary AK ₄–e design of ordern is a pair (S, B), whereB is an edge-disjoint decomposition ofK _n (the complete undirected graph onn vertices) with vertex setS, into copies ofK ₄–e, the graph on four vertices with five edges. It is well-known [1] thatK ₄–e designs of ordern exist for alln 0 or 1 (mod 5),n 6, and that if (S, B) is aK ₄–e design of ordern then |B| =n(n – 1)/10.Asimple covering ofK _n with copies ofK ₄–e is a pair (S, C) whereS is the vertex set ofK _n andC is a collection of edge-disjoint copies ofK ₄–e which partitionE(K_n)P, for some . Asimple minimum covering ofK _n (SMCK _n) with copies ofK ₄–e is a simple covering whereP consists of as few edges as possible. The collection of edgesP is called thepadding. Thus aK ₄–e design of ordern isSMCK _n with empty padding.We show that forn 3 or 8 (mod 10),n 8, the padding ofSMCK _n consists of two edges and that forn 2, 4, 7 or 9 (mod 10),n 9, the padding consists of four edges. In each case, the padding may be any of the simple graphs with two or four edges respectively. The smaller cases need separate treatment:SMCK ₅ has four possible paddings of five edges each,SMCK ₄ has two possible paddings of four edges each andSMCK ₇ has eight possible paddings of four edges each.The recursive arguments depend on two essential ingredients. One is aK ₄–e design of ordern with ahole of sizek. This is a triple (S, H, B) whereB is an edge-disjoint collection of copies ofK ₄–e which partition the edge set ofK _n\K_k, whereS is the vertex set ofK _n, and is the vertex set ofK _k. The other essential is acommutative quasigroup with holes. Here we letX be a set of size 2n 6, and letX = {x ₁, x₂, ..., x_n} be a partition ofX into 2-element subsets, calledholes of size two. Then a commutative quasigroup with holesX is a commutative quasigroup (X, ) such that for each holex _i X, (x_i, ) is a subquasigroup. Such quasigroups exist for every even order 2n 6 [4].     

On the uniqueness of a Walsh series converging on subsequences of partial sum

We show that if a Walsh series whose coefficients tend towards zero is such that the subsequence of its partial sums indexed by n_k, where n_k satisfies the condition 2^k–1k2^k (k=0, 1, 2, ...), tends everywhere, except possibly for a denumerable set, towards a bounded functionf(x), then this series is the Fourier series of the functionf(x).Translated from Matematicheskie Zametki, Vol. 16, No. 1, pp. 27–32, July, 1974.     

Eigenvalue problem for an irregular λ-matrix

The solution of the eigenvalue problem is examined for the polyomial matrixD()=A_o^s+A₁^s–1+...+A_s when the matricesA ₀ andA ₂ (or one of them) are singular. A normalized process is used for solving the problem, permitting the determination of linearly independent eigenvectors corresponding to the zero eigenvalue of matrixD() and to the zero eigenvalue of matrixA ₀. The computation of the other eigenvalues ofD() is reduced to the same problem for a constant matrix of lower dimension. An ALGOL program and test examples are presented.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 58, pp. 80–92, 1976.     

Conditioned limit theorems for random walks with negative drift

Summary In this paper we will solve a problem posed by Iglehart. In (1975) he conjectured that if S _n is a random walk with negative mean and finite variance then there is a constant so that (S _[n.]/n ^1/2¦N>n) converges weakly to a process which he called the Brownian excursion. It will be shown that his conjecture is false or, more precisely, that if ES ₁=–a<0, ES ₁ ² <, and there is a slowly varying function L so that P(S ₁>x)x ^–q L(x) as x then (S _[n.]/n¦S _n >0) and (S _[n.]/n¦N>n) converge weakly to nondegenerate limits. The limit processes have sample paths which have a single jump (with d.f. (1–(x/a)^–q )⁺) and are otherwise linear with slope –a. The jump occurs at a uniformly distributed time in the first case and at t=0 in the second.The research for this paper was started while the author was visiting W. Vervaat at the Katholieke Universiteit in Nijmegen, Holland, and was completed while the author was at UCLA being supported by funds from NSF grant MCS 77-02121