Existence of Tchebycheff extensions

Given a Tchebycheff System {y_o ··· y_n} defined on an interval I, it is proved that there exists a function y_n+1, such that the system {y_o ··· y_n, y_n+1} is a Tchebycheff System on I. A function such as y_n+1 is called a Tchebycheff extension of the system {y_i}ⁿ_{(i = 0)}.     

The Jacobian problem as a system of ordinary differential equations

LetP=x ⁿ +P _n?1(y)x ^n?1+…+P ₀(y),Q=x ^m +Q _m?2(y)x ^m?2+…+Q ₀(y) belong toK[x, y], whereK is a field of characteristic zero. The main result of this paper is the following: Assume thatP _x Q _y ?P _y Q _x =1. Then:*

1. K[Q _m?2(y), …,Q ₀(y)]=K[y],
2. K[P, Q]=K[x, y] ifQ=x ^m +Q _k (y)x ^k +Q _r (y)x ^r

     

On the construction of cyclic quasigroups

Let F(x,y) be the free groupoid on two generators x and y. Define an infinite class of words in F(x,y) by w₀(x,y) = x,w₁(x,y) = y and w_i+2(x,y) = w_i(x,y)w_i+1(x,y). An identity of the form w_3n(x,y) = x is called a cyclic identity and a quasigroup satisfying a cyclic identity is called a cyclic quasigroup. The most extensively studied cyclic quasigroups have been models of the identity y(xy) = x. The more general notion of cyclic quasigroups was introduced by N.S. Mendelsohn. In this paper a new construction for cyclic quasigroups is given. This construction is useful in constructing large numbers of nonisomorphic quasigroups satisfying a given cyclic identity or a consequence of a cyclic identity. The construction is based on a generalization of A. Sade's singular direct product of quasigroups.     

Asymptotic properties of matrix differential operators

For given matrices A(s) and B(s) whose entries are polynomials in s, the validity of the following implication is investigated: ?_ylim_{t → ∞}A(D) y(t) = 0 ? lim_{t → ∞}B(D) y(t) = 0. Here D denotes the differentiation operator and y stands for a sufficiently smooth vector valued function. Necessary and sufficient conditions on A(s) and B(s) for this implication to be true are given. A similar result is obtained in connection with an implication of the form ?_yA(D) y(t) = 0, lim_{t → ∞}B(D) y(t) = 0, C(D) y(t) is bounded ? lim_{t → ∞}E(D) y(t) = 0.     

A Dini type test on the pointwise convergence of double Fourier integrals

We give sufficient conditions for the convergence of the double Fourier integral of a complex-valued function f ∈ L ¹(?²) with bounded support at a given point (x ₀,y ₀) ∈ ?². It turns out that this convergence essentially depends on the convergence of the single Fourier integrals of the marginal functions f(x,y ₀), x ∈ ?, and f(x ₀,y), y ∈ ?, at the points x:= x ₀ and y:= y ₀, respectively. Our theorem applies to functions in the multiplicative Zygmund classes of functions in two variables.     

On an application of convexity to discrete systems

We prove the following result: Let A be a symmetric matrix, f be gradient (or certain subgradient) of a convex function, and {y_i} be a sequence defined by y_{i + 1} = f(Ay_i), y₀ arbitrary. Then the only possible periods of {y_i} are 1 or 2.     

Extension of periodic Tchebycheff Systems

Given a periodic Tchebycheff System {y_i}_{i = 0}²ⁿ it is proved that there exist two functions y_{2n + 1}, y_{2n + 2}, such that also the system {y_i}_{i = 0}^{2n + 2} is a periodic T-System.     

Optimal interval lengths for nonlocal boundary value problems associated with third order Lipschitz equations

For the third order differential equation, y^?=f(x,y,y^′,y^″), where f(x,y₁,y₂,y₃) is Lipschitz continuous in terms of yi, i=1,2,3, we obtain optimal bounds on the length of intervals on which there exist unique solutions of certain nonlocal three and four point boundary value problems. These bounds are obtained through an application of the Pontryagin Maximum Principle from the theory of optimal control.     

High-order one-step P-stable methods for the numerical integration of periodic initial value problems

Using Lobatto nodes, one-step methods of order six and eight have been obtained for the second-order differential equation y″ = f(x, y), y(x₀) = y₀, y′(x₀) = y′₀. The methods are shown to be P-stable. If

, then at each integration step a system of dimension 3s, 4s, respectively, has to be solved. The numerical results, for two problems, obtained by using these methods are given in the end.     

The behavior of solutions of second order delay differential equations

In this paper, we study the behavior of solutions of second order delay differential equation

y^″(t)=p₁y^′(t)+p₂y^′(t−τ)+q₁y(t)+q₂y(t−τ),     

Boundary data smoothness for solutions of nonlocal boundary value problems for second order differential equations

Under certain conditions, solutions of the boundary value problem, y^″=f(x,y,y^′), y(x₁)=y₁, and , are differentiated with respect to boundary conditions, where a<x₁<η₁<?<ηm<x₂<b, r₁,…,rm∈R, and y₁,y₂∈R.     

Some theorems on the existence,uniqueness, and nonexistence of limit cycles for quadratic systems

Given a quadratic system (QS) with a focus or a center at the origin we write it in the form ẋ = y + P₂(x, y), ẏ = −x + dy + Q₂(x, y) where P₂ and Q₂ are homogeneous polynomials of degree 2. If we define F(x, y) = (x − dy) P₂(x, y) + yQ₂(x, y) and g(x, y) = xQ₂(x, y) − yP₂(x, y) we give results of existence, nonexistence, and uniqueness of limit cycles if F(x, y) g(x, y) does not change of sign. Then, by using these results plus the properties on the evolution of the limit cycles of the semicomplete families of rotated vector fields we can study some particular families of QS, i.e., the QS with a unique finite singularity and the bounded QS with either one or two finite singularities.     

Numerical solution of a class of random boundary value problems

This paper deals with the nonlinear two point boundary value problem y″ = f(x, y, y′, R₁,…, R_n), x₀ < x < x_fS₁y(x₀) + S₂y′(x₀) = S₃, S₄y(x_f) + S₅y′(x_f) = S₆ where R₁,…, R_n, S₁,…, S₆ are bounded continuous random variables. An approximate probability distribution function for y(x) is constructed by numerical integration of a set of related deterministic problems. Two distinct methods are described, and in each case convergence of the approximate distribution function to the actual distribution function is established. Primary attention is placed on problems with two random variables, but various generalizations are noted. As an example, a nonlinear one-dimensional heat conduction problem containing one or two random variables is studied in some detail.     

A note on the time series which is the product of two stationary time series

The time series […,x_-1y_-1,x₀y₀,x₁y₁,…]> which is the product of two stationary time series x_t and y_t is studied. Such sequences arise in the study of nonlinear time series, censored time series, amplitude modulated time series, time series with random parameters, and time series with missing observations. The mean and autocovariance function of the product sequence are derived.     

The asymptotic estimates of solutions of difference equations

The asymptotic estimate of the solution y(t) of linear difference equations with almost constant coefficients and the condition of asymptotic equivalence between y(t) and y₀(t) are given, where y₀(t) is the corresponding solution of linear equations with constant coefficients. A brief and elementary proof of the results is presented. The method of proof carries over to differential equations in which some similar results are stated.     

Exact number of local extreme points of curvature function of solutions of second-order linear differential equations

We study local properties of the curvature ?? _y (x) of every nontrivial solution y=y(x) of the second-order linear differential equation?(P): (p(x)y??)??+q(x)y=0, x??(a,b)=I, where p(x) and q(x) are smooth enough functions. It especially includes the Euler, Bessel and other important types of second-order linear differential equations. Some sufficient conditions on the coefficients p(x) and q(x) are given such that the curvature ?? _y (x) of every nontrivial solution y of (P) has exactly one extreme point between each two its consecutive simple zeros. The problem of three local extreme points of ?? _y (x) is also considered but only as an open problem. It seems it is the first paper dealing with this kind of problems. Finally in Appendix, we pay attention to an application of the main results to a study of non-regular points (the cusps) of the ??-parallels of graph ??(y) of?y (the offset curves of???(y)).     

The semicycles of solutions of neutral difference equations

In this paper, we study the semicycles of solutions of neutral delay difference equation Δ(y_n + p_ny_n−τ) + q_ny_n−σ = 0, where {p_n} and {q_n} are sequences of nonnegative real numbers, τ and σ are positive integers. Upper bound of numbers of terms of semicycles are determined.     

On the Baire Classification of Positive Characteristic Exponents in the Perron Effect of Change of Their Values

In the complete Perron effect of change of values of characteristic exponents, where all nontrivial solutions y(t, y₀) of the perturbed two-dimensional differential system are infinitely extendible and have finite positive exponents (the exponents of the linear approximation system being negative), we prove that the Lyapunov exponent λ[y(·, y₀)] of these solutions is a function of the second Baire class of their initial vectors y₀ ∈ ?ⁿ {0}.     

Minkowski-Bouligand dimension of solutions of the one-dimensional p-Laplacian

The upper Minkowski-Bouligand dimension of the graph of continuous solutions, denoted by dim_My, is studied for a class of nonlinear one-dimensional p-Laplacian. Some sufficient conditions on the nonlinearities are given such that dim_My takes the prescribed fractional values. Next, a relation between dim_My and the order of growth for singular behaviour of L^p norm of derivatives of solutions is given. Finally, the change and stability of dim_My are considered by means of a double-parametric problem.     

Pringsheim type tests on the pointwise convergence of integrals conjugate to double fourier integrals

We prove sufficient conditions for the convergence of the integrals conjugate to the double Fourier integral of a complex-valued function f ∈ L ¹ (?²) with bounded support at a given point (x ₀, g ₀) ∈ ?². It turns out that this convergence essentially depends on the convergence of the integral conjugate to the single Fourier integral of the marginal functions f(x, y ₀), x ∈ ?, and f(x ₀, y), y ∈ ?, at x:= x ₀ and y:= y ₀, respectively. Our theorems apply to functions in the multiplicative Lipschitz and Zygmund classes introduced in this paper.