EXACT PERIODIC SOLUTIONS OF SECOND-ORDER DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENTS

In the paper is given a method of finding periodical solutions of the differential equation of the form $x''(t)+p(t)x''(t-1)=q(t)x([t])+f(t),$ where $[\cdot]$ denotes the greatest integer function, $p(t)$,$q(t)$ and $f(t)$ are continuous periodic functions of $t$. This reduces $n$-periodic soluble problem to a system of $n+1$ linear equations, where $n=2,3$. Furthermore, by using the well known properties of linear system in the algebra, all existence conditions for $2$ and $3$-periodical solutions are described, and the explicit formula for these solutions are obtained.

On the alternating sums of reciprocal generalized Fibonacci numbers

In this paper, we consider finite alternating sums derived from the generalized Fibonacci numbers [Formula: see text] [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are positive integers with [Formula: see text], [Formula: see text]. Applying the greatest integer function to these sums, we obtain some equalities involving the generalized Fibonacci numbers.

Joint ergodicity along generalized linear functions

A criterion of joint ergodicity of several sequences of transformations of a probability measure space $X$ of the form $T_{i}^{\unicode[STIX]{x1D711}_{i}(n)}$ is given for the case where $T_{i}$ are commuting measure-preserving transformations of $X$ and $\unicode[STIX]{x1D711}_{i}$ are integer-valued generalized linear functions, that is, the functions formed from conventional linear functions by an iterated use of addition, multiplication by constants, and the greatest integer function. We also establish a similar criterion for joint ergodicity of families of transformations depending on a continuous parameter, as well as a condition of joint ergodicity of sequences $T_{i}^{\unicode[STIX]{x1D711}_{i}(n)}$ along primes.

Congruences for q[p/8] (mod p) under the condition 4n2p = x2 + qy2

Let ℤ be the set of integers, and let p be a prime of the form 4k + 1 and so p = c2 + d2 with c, d ∈ ℤ. Let q be an integer of the form 4k + 3. Assume that 4n2p = x2 + qy2 with c, d, n, x, y ∈ ℤ and (q, n) = (x, y) = 1, where (a, b) is the greatest common divisor of integers a and b. In this paper, we establish congruences for (-q)[p/8] ( mod p) in terms of c, d, n, x and y, where [⋅] is the greatest integer function. In particular, we establish a reciprocity law and give an explicit criterion for (-11)[p/8] ( mod p).

ALMOST AUTOMORPHIC SOLUTIONS FOR DIFFERENTIAL EQUATIONS WITH PIECEWISE CONSTANT ARGUMENT

Using the method of exponential dichotomies, we establish a new existence and uniqueness theorem for almost automorphic solutions of differential equations with piecewise constant argument of the form $$\begin{eqnarray*}{x}^{\prime } (t)= A(t)x(t)+ B(t)x(\lfloor t\rfloor )+ f(t), \quad t\in \mathbb{R} ,\end{eqnarray*}$$ where $\lfloor \cdot \rfloor $ denotes the greatest integer function, and $A(t), B(t): \mathbb{R} \rightarrow { \mathbb{R} }^{q\times q} $, $f(t): \mathbb{R} \rightarrow { \mathbb{R} }^{q} $ are all almost automorphic.

An Analytical Model to Determine Fundamental Frequency of Free Vibration of Perforated Plate by Using Greatest Integer Functions to Express Non Homogeneity

This paper aims at determining the fundamental frequency of square perforated plate with square perforation pattern of square holes. Rayleigh’s method is used for the solution of this problem. Non homogeneity in Young’s modulus and density at the perforation is expressed by using greatest integer function i.e. floor function. Boundary condition considered is clamped on all edges. Perforated plate is considered as plate with uniformly distributed mass and holes are considered as non homogeneous patches. The deflected surface of the plate is approximated by a function which satisfies the boundary conditions. Finite Element Method (FEM) modal analysis is carried out to validate the results of the proposed approach.