Some properties of third order differential operators

Consider the third order differential operator L given by and the related linear differential equation L(x)(t) + x(t) = 0. We study the relations between L, its adjoint operator, the canonical representation of L, the operator obtained by a cyclic permutation of coefficients a _i , i = 1,2,3, in L and the relations between the corresponding equations.We give the commutative diagrams for such equations and show some applications (oscillation, property A).     

On a cyclic disconjugate operator associated to linear differential equations

In this paper the disconjugate linear differential operator of n-th order D^1/(n) given by $$D_1^{(n)} (x)(t) = \frac{1}{{a_n (t)}}\frac{d}{{dt}}\frac{1}{{a_{n - 1} (t)}} \ldots \frac{1}{{a_1 (t)}}\frac{d}{{dt}}x(t)$$ is considered together with other n?1 operators, which are obtained from D ₁ ⁽ⁿ⁾ by an ordered cyclic permutation of the functions a_i. Such operators play an important role in the study of oscillation of the associated linear differential equation (*) $$D_1^{(n)} (x)(t) \pm x(t) = 0.$$ Some properties of these operators suggest the new idea of «isomorphism of oscillation». The existence of an isomorphism of oscillation allows to describe the oscillatory or nonoscillatory behavior of solutions of (*) by the oscillatory or nonoscillatory behavior of solutions of other n ?1 suitable linear differential equations. From this fact one can easily obtain new results about oscillation or nonoscillation of (*) that might be hard to prove directly. Several interesting consequences concerning the classification of solutions of (*) are also presented together with some new applications to the structure of the set of nonoscillatory solutions of (*).     

On nonoscillation of canonical or noncanonical disconjugate functional equations

Qualitative comparison of the nonoscillatory behavior of the equations

The number of partial orders of fixed width

We investigate the approximate number of n-element partial orders of width k, for each fixed k. We show that the number of width 2 partial orders with vertex set {1, 2, ..., n} is

Existence and uniqueness of limit cycles on a cubic kolmogorov differential system in the Predator-Prey relation

In this paper, we analyse qualitatively a cubic Kolmogorov system: which is one of the mathematical models in ecology describing the interaction between Predator-Prey of two populations; and give the conditions of nonexistence, existence and uniqueness of limit cycles for three different cases.Fulfilled during engagement in advanced studies at the Institute of Mathematics, Academia Sinica.     

Existence of Solutions for the Dirichlet Problem with Superlinear Nonlinearities

In this paper we establish the existence of nontrivial solutions to

When all trajectories in the Liénard plane cross the vertical isocline?

In this paper we study the problem whether all trajectories of the system =y–F(x) and =–g(x) cross the vertical isocline which is very important for the existence of periodic solutions and oscillation theory. The problem has not been solved for the critical case:

Construction of a solution of a mixed problem for the generalized heat conduction equation by the fourier method

By the Fourier method a solution of the equation

Estimates for Multiplicative Functions on the Set of Shifted Primes

Let fi, i = 1, ... k, be complex-valued multiplicative functions satisfying the conditions

Accuracy of the normal approximation

We denote by _n(x) the difference between the distribution fonctions of the sum of n independent random variables and the normal random variable with mean a and variance ²0. In the note one obtains lower estimates for

Behavior of Automorphic L-Functions at the Center of the Critical Strip

Let be the Hecke eigenbasis of the space of -cusp forms of weight 2. Let p be a prime. Let be the Hecke L-series of form . The following statements are proved:

Criteria for the coincidence of the kernel of a function with the kernels of its Riesz and Abel integral means

We indicate criteria for the coincidence of the Knopp kernels K(f) K(A _f), and K (R _f) of bounded functions f(t); here,

A Nonlocal Diffusion Equation whose Solutions Develop a Free Boundary

Let J:\mathbbR ? \mathbbRJ:\mathbb{R} \to \mathbb{R} be a nonnegative, smooth compactly supported function such that ò_\mathbbR J(r)dr = 1. \int_\mathbb{R} {J(r)dr = 1.} We consider the nonlocal diffusion problem

Lyapunov exponent and rotation number for stochastic Dirac operators

In this paper, we consider the stochastic Dirac operatoron a polish space (Ω,β, P). The relation between the Lyapunov index, rotation number andthe spectrum of L_ω is discussed. The existence of the Lyapunov index and rotation number isshown. By using the W-T functions and W-function we prove the theorems for L_ω as in Kotani[1], [2] for Schrodinger operatorB, and in Johnson [5] for Dirac operators on compact space.     

Minimal coupled diffusion process

An attempt is made to classify the boundary of the coupled operator and to construct the corresponding minimal semigroup and minimal coupled diffusion process. The sample properties and the conservative conditions of the process are discussed also.     

Die Ungleichungen von Vietoris

The inequalities $$P_{k,l} = \frac{1}{{B(l + 1,k)}}\int\limits_0^{l/(k + l)} {x^l (1 - x)^{k - 1} dx = I_{l/(k + l)} (l + 1,k)< \frac{1}{2}}$$ and $$\Phi _{k,l,\mu } = \frac{1}{{B(k,k\mu + 1)}}\int\limits_0^1 {x^{k - 1} (1 - x)^{k\mu } I_x (l + 1,l\mu )dx< \frac{1}{2}}$$ are shown to be valid for any positive real numbersk, l, μ.     

On the Lipschitz Constant for a Nonisometric Bi-Lipschitz Transformation of a Cantor Set

A bi-Lipschitz continuous mapping of a space X is a bijection such that , where . We write if f is a Lipschitz (bi-Lipschitz) mapping of X into itself and denote by the set of all bi-Lipschitz mappings of X that are not isometry. Thus, if and blip . For X we consider a standard Cantor set K on the real line (with standard metric). The main result of this paper is formulated as follows: where Bibliography: 2 titles.     

On the almost sure central limit theorem and domains of attraction

We give necessary and sufficient criteria for a sequence (X _n) of i.i.d. r.v.'s to satisfy the a.s. central limit theorem, i.e.,

Aubry–Mather Sets for Relativistic Oscillators with Anharmonic Potentials

In this paper, by the Aubry–Mather theory, it is proved that there are many periodic solutions and usual or generalized quasiperiodic solutions for relativistic oscillator with anharmonic potentials models d/dt(x/(1-|x|~2~(1/2))+ |x|~(α-1)x=p(t),where p(t) ∈ C~0(R~1) is 1-periodic and α 0.