Mathematics

We establish the existence of a uniformly bounded C ∞ solution of the Navier-Stokes equations on R 3 x [0,∞) without external forces or boundaries for a divergence free initial condition u o ∈ ∩ m H m when the viscosity ν is >0 .

The present article aims to introduce a unified family of the Apostol type-truncated exponential-Gould-Hopper polynomials and to characterize its properties via generating functions. A unified presentation of the generating function for the Apostol type-truncated exponential-Gould-Hopper polynomials is established and its applications are given. By the use of operational techniques, the quasi-monomial properties for the unified family are proved. Several explicit representations and multiplication formulas related to these polynomials are obtained. Some general symmetric identities involving multiple power sums and Hurwitz-Lerch zeta functions are established by applying different analytical means on generating functions.

In the paper the distribution of prime numbers and digits of π were presented as chaotic.

The main purpose of this paper is to present the spherical characterization of Legendre curves in 3 -dimensional quasi-Sasakian pseudo-metric manifolds. Furthermore, null Legendre curves are also characterized in this class of manifold.

In this paper, our aim is to give surfaces in the Galilean 3-space G3 with the property that there exist four geodesics through each point such that every surface built with the normal lines and the binormal lines along these geodesics is a surface with a minimal surface and a constant negative Gaussian curvature. We show that ψ should be an isoparametric surface in G3: A plane or a circular hyperboloid.

Nullnorms with a zero element being at any point of a bounded lattice are an important generalization of triangular norms and triangular conorms. This paper obtains an equivalent characterization for the existence of idempotent nullnorms with the zero element a on any bounded lattice containing two distinct elements incomparable with a . Furthermore, some basic properties for the bounded lattice containing two distinct element incomparable with a are presented.

The present article examines the Lie group invariants of the Navier-Stokes equation (NSE) for incompressible fluids. This is accomplished by applying the invariant theory of Charles Bouton which shows that the self-similar solutions of the NSE are relative invariants of the scaling group. The scaling transformation admitted by the NSE has been recently revisited and a general form of the transformation has been discovered from which it follows that Leray's self-similar solutions are an isolated case. The general form of such solutions is derived by the application of Bouton's first theorem and shows that the standard NSE system is not always supercritical, but can be critical or subcritical. Criticality criteria are derived. Using the criterion of Beale-Kato-Majda, we rule out blow-up for a subset of Bouton's self-similar solutions. For another subset, we show that the system exhibits a conserved quantity, the cavitation number of the fluid. It is coercive, scale- and rotationally invariant. By extending the analysis of Bouton to higher-dimensioned manifolds and by virtue of Bouton's theorems, additional conserved quantities are found, which could further elucidate the physics of fluid turbulence.

In this paper we consider a class of Burgers equation. We propose a new method of investigation for existence of classical solutions.

Presented is a two-tier analysis of the location of the real roots of the general quartic equation x 4 +a x 3 +b x 2 +cx+d=0 with real coefficients and the classification of the roots in terms of a , b , c , and d , without using any numerical approximations. Associated with the general quartic, there is a number of subsidiary quadratic equations (resolvent quadratic equations) whose roots allow this systematization as well as the determination of the bounds of the individual roots of the quartic. In many cases the root isolation intervals are found. The second tier of the analysis uses two subsidiary cubic equations (auxiliary cubic equations) and solving these, together with some of the resolvent quadratic equations, allows the full classification of the roots of the general quartic and also the determination of the isolation interval of each root. These isolation intervals involve the stationary points of the quartic (among others) and, by solving some of the resolvent quadratic equations, the isolation intervals of the stationary points of the quartic are also determined. Each possible case has been carefully studied and illustrated with a detailed figure containing a description of its specific characteristics, analysis based on solving cubic equations and analysis based on solving quadratic equations only. As the analysis of the roots of the quartic equation is done by studying the intersection points of the "sub-quartic" x 4 +a x 3 +b x 2 with a set of suitable parallel lines, a beautiful Pythagorean analogy can be found between these intersection points and the set of parallel lines on one hand and the musical notes and the staves representing different musical pitches on the other: each particular case of the quartic equation has its own short tune.

Triangular numbers that are multiple of other triangular numbers are investigated. It is known that for any positive non-square integer multiplier, there is an infinity of multiples of triangular numbers which are triangular numbers. If the multiplier is a squared integer, there is either one or no solution, depending on the multiplier value. Instead of recurrent relations, we develop in this paper closed form equations to calculate directly the values of triangular numbers and their indices without the need of knowing the previous solutions. We develop the theoretical equations for four cases of ranks from 1 to 4 and we give several examples for non-square multipliers 2, 3, 5 and 8.