# Volume 2 (2010), Number 1

## One Derivative of One Component Regularity Criterion for the Navier-Stokes Equations

### Author(s): YUAN-SHAN ZHAO AND YUE HU

**Abstract:** We study the incompressible Navier-Stokes equations in
the entire three-dimensional space and we prove that if there exists one
derivative of one component of the velocity ${\partial}_{3}{u}_{3}\in {L}_{t}^{s}{L}_{x}^{r},\text{\hspace{0.17em}}\text{where}\text{\hspace{0.17em}}\frac{2}{s}+\frac{3}{r}\le \frac{1}{4},\text{\hspace{0.17em}}12\le r\le \infty $
then the solution is regular. This extends one result of Patrick Penel, Toulon,
Milan Pokorý, Praha [Appl.Math.,49,483-493(2004)]

## On Some Multiplier Difference Sequence Spaces Defined over a 2-normed Linear Space

### Author(s): B.SURENDER REDDY AND HEMEN DUTTA

**Abstract:** In this paper, we introduce a new class of generalized
difference sequences with base space, a real linear 2-normed space and by
means of a fixed multiplier. We study the spaces of thus constructed classes
of sequences for relevant linear topological structures. Further we investigate
the spaces for solidity, monotonicity, symmetricity etc. We also obtain some
relations between these spaces as well as prove some inclusion results.

## On Some Inequalities of Simpson-type Via Quasi-Convex Functions and Applications

### Author(s): MOHAMMAD ALOMARI AND MASLINA DARUS

**Abstract:** Some inequalities of Simpson's type for quasi-convex
functions are introduced. In the literature the error estimates for the midpoint
rule is $\left|{E}_{mid}\left(f,d\right)\right|\le \frac{K}{24}{\displaystyle \sum _{i=0}^{n-1}{\left({x}_{i+1}-{x}_{i}\right)}^{3}}$ , in this
paper we restrict the conditions on f to get better error estimates than the original.

## Solving Cauchy Problem for a Class of Sixth-order Hyperbolic Equations with Triple Characteristics

### Author(s): LAZHAR BOUGOFFA AND HIND K. AL-JEAID

**Abstract:** In this paper, the Cauchy problem for a class of the homogeneous
hyperbolic equations for sixth-order with triple characteristics is considered and
can be solved analytically by direct integration techniques. Also, an efficient
modification of Adomian decomposition method is proposed for solving this type of
problems. We then conduct a comparative study between the ADM and direct method
with the help of several illustrative examples.

## New Explicit and Implicit Solutions to Elliptic Equations with Two Space Variables

### Author(s): MOHAMMED A. AL-KADHI

**Abstract:** We present a direct method for finding a new explicit
and implicit solutions of elliptic equations with hyperbolic, trigonometric and
exponential nonlinearities.

## BAICA-CARDU Paratrigonometry, a Generalization of the Classical and Some New Non-classical Trigonometries and Its Application in Mechanics and Wave Theory

### Author(s): M. BAICA AND M.CARDU

**Abstract:** In their previous papers the authors introduced some new
Trigonometries as: 1. Quadratic Trigonometry (QT), 2. Polygonal Trigonometry
(PT), 3. Trans Trigonometry (TT), 4. Infra Trigonometry (IT), 5. Ultra-Trigonometry
(UT), 6. Extra Trigonometry (ET), 7. Para-Trigonometry (PRT). This time in this
paper we perform a synthesis of all these Trigonometries and state some of their
applications.

## Oscillation of a Class of Two-variables Functional Equations with Variable Coefficients

### Author(s): WENGUI YANG

**Abstract:** In this paper we will establish some sufficient conditions
of oscillation of a class of two-variables functional equations with variable
coefficients. Our results extend Zhang and Zhou's results (B.G. Zhang and Y.
Zhou, Comput. Math. Appl. 42 (3-5) (2001) 369-378).

## On Algebraic Properties of the Generalized Chebyshev Polynomials

### Author(s): AHMET İPEK

**Abstract:** Chebyshev polynomials are of great importance in many areas
of mathematics, particularly approximation theory. Numerous articles and books
have been written about this topic. Analytical properties of Chebyshev polynomials
are well understood, but algebraic properties less so. In this paper, new generalized
Chebyshev polynomials of the first and second kinds have been introduced and studied.
Many of the properties of these polynomials are proved.

## A Note on Bounds for the Spectral Norms of Circulant-Cauchy-Toeplitz Matrices

### Author(s): AHMET İPEK

**Abstract:** In this paper, we established lower and upper bounds for
the spectral norms of some Circulant-Cauchy-Toeplitz matrices.

## A Nonlinear Mixed Type Volterra-Fredholm Functional Integral Equation Via Perov's Theorem

### Author(s): MARCEL-ADRIAN ȘERBAN

**Abstract:** In this paper we study the following mixed type Volterra-Fredholm
functional integral equation $x\left(t\right)=F\left(t,x(t),{\displaystyle {\int}_{{a}_{1}}^{{t}_{1}}\dots}{\displaystyle {\int}_{{a}_{m}}^{{t}_{m}}K}(t,s,x(s))ds,{\displaystyle {\int}_{{a}_{1}}^{{b}_{1}}\dots}{\displaystyle {\int}_{{a}_{m}}^{{b}_{m}}H}(t,s,x(s))ds\right)$.
Using the Perov's Theorem and the Picard operator technique we establish existence,
uniqueness, data dependence and Gronwall results for the solutions.

## The Solution of a System of Nonlinear Integral Equations from Physics

### Author(s): MARIA DOBRIŢOIU

**Abstract:** Using the Perov's Theorem and the General data dependence
Theorem, in this paper, we obtain some conditions concerning the existence and
uniqueness of the solution in the Banach space $C([a,b],{\mathbb{R}}^{m})$ and the continuous data dependence of the solution
of the following system of nonlinear integral equations from physics:
$x(t)={\displaystyle {\int}_{a}^{b}K}(t,s,x(s),x(a),x(b))ds+f(t),t\in [a,b]$. An example is also given here.

## Double Inequalities of Boole's Quadrature Rule

### Author(s): MARIUS HELJIU

**Abstract:** In this paper double inequalities of Boole's type quadrature
rule are given. These inequalities are sharp.

## The Green's Matrix and the Green's Type Integral Formula for an Elastic Strip

### Author(s): TATIANA SPEIANU

**Abstract:** An efficient unified method to derive Green's matrices,
called the incompressible influence elements method (IIEM), had been elaborated
and published earlier by V. D. Șeremet [Handbook of Green's Functions and Matrices,
WIT press, Southampton, Boston, 2003]. The main point of this method is general
integral representations for Green's matrices in terms of Green's functions for
Poisson's equation. This paper uses above mentioned representations to derive
the Green's matrix and the Green's type integral formula for a boundary value
problem (BVP) for an elastic strip. All results are obtained exactly and in
elementary functions. To obtain these results some Green's functions for
Posson's equation for a strip are derived. An exact solution for a particular
BVP for an elastic strip also is included.