Showing papers in "Mathematica Slovaca in 1993"

Criteria of property $A$ for third order superlinear differential equations

Abstract: Some new criteria of property A for superlinear differential equations y'" + p(t)y' + q(t)\y\ sgny = 0 and y'" + p(t)y' + q(t)f(y) = 0 are established. The obtained results extend and improve a sufficient condition for the equation y"' +p(t)y' + q(t)y -= 0 , where a > 1 is a quotient of odd positive integers. 1» Introduction This paper is concerned with the criteria of property A for the third order superlinear differential equations of the form y'" + p(t)y'+ q(t)\y\ sgny = 0 , (A) y"'+p(t)y' + q(t)f(y) = 0, (F) where p: I —• ( —oo,0], q: I —• (0,co), / : R —> R = ( —oo,oo), / = (a,oo) C (0,oo) are continuous, 0 0 for x ^ 0 . In general, we assume that p(t) ^ 0 on I. We restrict our attention to those solutions of equations (A) and (F) which exist on some ray (t*,oo) C I and which are nontrivial in any neighbourhood of infinity. Such a solution is called oscillatory if it has arbitrarily large zeros, otherwise it is called nonoscillatory. A M S S u b j e c t C l a s s i f i c a t i o n (1991): Primary 34C11. Secondary 34C10. K e y w o r d s : Superlinear differential equations, Nonoscillatory solution.

Weak solutions of a boundary value problem for nonlinear ordinary differential equation of second order in banach spaces

Abstract: Using the measure of weak noncompactness we give sufficient con­ ditions for the existence of a weak solution of a boundary value problem for the equation x" = f(t,x,xf) in Banach space.

On distribution of sequences of integers

Abstract: In the paper we introduce on sequences of integers a notion ana­ logical to t h a t of distribution function. Due to some topological peculiarities of the set of integers we shall s tudy more general notions of distribution measure and distribution density. 1. In t roduc t ion In 1916 H e r m a n W e y l in his famous paper [12] introduced the notion of uniformly distributed sequences of real numbers modulo 1. This notion was subsequently generalized in various ways. One of them stems from I . N i v e n [6] who in 1961 introduced the notion of uniform distribution of integers. Another generalization can be done via the notion of asymptotic distribution function mod 1 which was initiated b y S c h o e n b e r g [11]. The aim of this paper is to join this two approaches. However the structure of positive integers gives us small space for the study of \"distribution functions\". Therefore we shall use a more general notion of distribution measure and we shall investigate the sequences of integers from the point of view of the uni­ form distribution in compact spaces of so called polyadic numbers. This is a generalization of M e i j e r 's method [5], [4] in the space of g-adic numbers. Nevertheless, instead of \"distribution function\" we shall use \" T -distribution\". In the first chapter we prove an existence theorem (Theorem 1), which is an analog of the existence theorem known for the distribution function. Then we shall focus on the properties of T -distributed sequences and the distribution measures. The principal result is given in chapter 3 which establishes (Theo­ rem 3) that an integer sequence is T -distributed if and only if it is uniformly distributed in the space of polyadic numbers with respect to distribution mea­ sures; (Corollary 3 of Th. 6) any integer polynomial sequence is distributed. In chapter 4 we collected technical results. A M S S u b j e c t C l a s s i f i c a t i o n (1991): Pr imary 11B05. Secondary 28E99. K e y w o r d s : Density, Uniform distribution, Measure sequence. x ) Supported by grants 363 & 622/03. 521 MILAN PASTEKA — STEFAN PORUBSKY In the sequel we will denote by N the set of all nonnegative integers and by R the set of all real numbers. 2. T -distr ibu t ion The following sequences T will be of fundamental importance for us. Let F = {h(j,m): 0)= ^2 h(j + rm,m1). (7) г = 0 Thus (5) and (6) are necessary conditions for the existence of a T -distributed sequence. A system T satisfying conditions (5) and (6) will be called a distribution. The fact that the distribution properties are also sufficient for the existence of a T-distributed sequences will follow from Theorem 1. However before stating this theorem, we have to recall some preliminaries from the theory of the uniform distribution in compact spaces. Various parts presented here can be found in monograph [3, Chapter 3]. Let X be a compact separable Hausdorff space with a probability measure P which is a regular normed Borel measure in X . A sequence {xn} of elements in X will be called P-uniformly distributed in X (or shortly P-u.d. in X) if IV lim ^è / (*n)= ífdP IV—>oo П — J 7 1 = 1 for every continuous functions / : X —> K (cf. [3, p. 171]). The next three results form a springboard for us: THEOREM A. ([3, p. 175]) A sequence {xn} is P-u.d. in X if and only if lim j V 1 = P(M) iV-oo N ^ n

Some bases of the Stickelberger ideal

Abstract: Various bases of the Stickelberger Ideal are introduced.Some bases are used for expressions of the Kummer system of congruences in another way. The p-adic situation is also investigated.

Topological entropy and variation for transitive maps

Abstract: We s tudy the continuous functions which m a p a compact real inter­ val back into itself. We investigate the relations between two important concepts of the dynamical systems and real analysis for transitive functions, topological entropy and variation. 0. Introduct ion This paper is concerned with investigation of relations between the topolog­ ical entropy and variation for transitive maps. Topological entropy, denoted ent(-), is a numerical conjugacy invariant of continuous maps. Variation of a function / on the interval 7, denoted Var(/, 7) , is a length of a way of a point f(x) if a point x goes through the interval 7. A continuous map is transitive if some point has a dense orbit. Let 7 = [0,1] be the closed unit interval and C(I, I) be the set of all con­ tinuous functions which map the interval 7 back into itself. MAIN THEOREM. Let (x, y) be a pair of numbers. Then there exists a tran­ sitive function f E (7(7,7) such that (x,y) = (Var(/, 7), ent(/)) if and only if (x,y) e ((l,oo] x (logx/2,oo]) U ((1,2) x { l o g v ^ } ) . In section 1 we give the definitions and some known results. In section 2 we define three transitive maps with prescribed topological entropy and in section 3 we show that all of the cases for a pair of numbers (Var(/, 7), ent(/)) from Main Theorem are possible. In section 4 we prove that, up to conjugacy, there A M S S u b j e c t C l a s s i f i c a t i o n (1991): Pr imary 58F03, 58F13. K e y w o r d s : Topological entropy, Variation, Topological conjugacy.

On the family of functions with a closure of its graph of measure zero

Abstract: In this paper I consider the family S of all functions / : R —> R with a closure of its graph of Lebesgue measure zero. In the first part I make known a full characterization of the family S. In the second part of this paper I observe that in like manner it is possible to characterize the family Si of functions / : R —> R with a closure of its graph of the first category. Let R be a real line. We will consider functions / : R —> R. Let G(f) be the graph of / and C1(H) be the closure of a set H. Symbols m and m 2 denote Lebesgue measure on R and R 2 , respectively. Denote by S the family {/: R -> R : m 2 ( C l (G(f))) = 0} . Let K+(f, x) and K~(f, x) be the right and the left cluster sets respectively for each x of the domain of / . Moreover, let K~*~(f, c o ) = {y G R; there exists xn —> —oo such that lim f(xn) = #} n—•oo J and K~(f,oo) = { t / G R ; there exists xn —» oo such that lim f(xn) = y\\ . ^ n-->oo ' Put K(f,x) = K+(f,x)UK~(f,x) and S(f,x) = K(f,x) U {/(*)} . R e m a r k 1.1. S ( / , x ) \\ { o o ; + o c } = [CI {G(f))]x for each x € R, where [H]x denotes the section of a set ffcR , i.e. the set { y £ R : (x,y)eH}. AMS S u b j e c t C l a s s i f i c a t i o n (1991): Primary 26A30.

Some remarks on pseudorandom sequences

Abstract: The paper presents some results illustrating the difficulty of giving a general definition of a pseudorandom sequence. It seems to indicate that rather technical distribution properties studied in the theory of uniform distribution may be the best one can expect as criterions. The method is to study different notions of tests defining sets of acceptance that are "natural" from a probabilistic point of view. Some results on Baire properties are added.

Baire functions and their restrictions to special sets

Abstract: In this paper I compare some families of functions whose restrictions to special sets have continuity points or intervals of continuity points. Moreover, I investigate the Darboux property in some of these families. Notat ion s Let R denote the set of all reals. A function / : X —• R (0 ?-X C R) is said to be quasicontinuous (cliquish) at a point x E X ([6] and respectively ([1])) if for every open neighbourhood U of x and for every positive number r there is an open set V dU such that V n X 7-0 and \\f(t) — f(x)\\ < r for every point teVnX (and osc /