Mathematics

The aim of this paper is to study Hyers-Ulam-Rassias stability for a Volterra-Hammerstein functional integral equation in three variables via Picard operators.

We consider the Navier-Stokes equations in a bounded domain with periodic boundary conditions. Let V=V(x,t) be the velocity of the fluid. The aim of this paper is to prove the bound ∥V(t) ∥ H 1 ≤c for any t∈ R + , where c depends on data. The proof is divided into two steps. In the first step the Lamé system with a special version of the convective term is considered. The system has two viscosities. Assuming that the second viscosity (the bulk one) is sufficiently large we are able to prove the existence of global regular solutions to this system. The proof is divided into two steps. First the long time existence in interval (0,T) is proved, where T is proportional to the bulk viscosity. Having the bulk viscosity large we are able to show that data at time T are sufficiently small. Then by the small data arguments a global existence follows. In this paper we are restricted to derive appropriate estimates only. To prove the existence we should use the method of successive approximations and the continuation argument. Let v be a solution to it. In the second step we consider a problem for u=v−V . Assuming that ∥u ∥ H 1 at t=0 is sufficiently small we show that ∥u(t) ∥ H 1 is also sufficiently small for any t∈ R + . Estimates for v and u in H 1 imply estimate for ∥V(t) ∥ H 1 for any t∈ R + .

Mathematics is one of the ways our species makes sense of this world and I believe that it is inherent in our thinking machinery. The mathematics we do in turn is dependent on the way we view our universe and ourselves. Lakoff and Nunez [17] argue carefully and eloquently for a mathematics inherently based on human cognition. In this note I attempt to engage with the construct of mathematical cognition through the lens of humanistic mathematics.

In this paper, we establish some formulas for the noncentral Tanny-Dowling polynomials including sums of products and explicit formulas which are shown to be generalizations of known identities. Other important results and consequences are also discussed and presented.

The Poincaré lemma (or Volterra theorem) is of utmost importance both in theory and in practice. It tells us every differential form which is closed, is locally exact. In other words, on a contractible manifold all closed forms are exact. The aim of this paper is to present some direct proofs of this lemma and explore some of its numerous consequences. Some connections with Cech-De Rham-Dolbeault cohomologies, ∂ ¯ ¯ ¯ -Poincaré lemma or Dolbeault-Grothendieck lemma are given.

Four new relations have been found between the Stirling numbers of first and second kind. They are derived directly from recently published relations.

A novel kind of self-referential square matrix is introduced. A certain subset of the matrix entries record the frequencies of occurrence of each distinct number appearing within the entire matrix. Such squares are necessarily elusive. Our investigation brings to light interesting cases, such as 'generic' squares that include algebraic variables and self-descriptive squares that are also magic squares.

We consider slant normal magnetic curves in (2n+1) -dimensional S -manifolds. We prove that γ is a slant normal magnetic curve in an -manifold ( M 2m+s ,φ, ξ α , η α ,g) if and only if it belongs to a list of slant φ -curves satisfying some special curvature equations. This list consists of some specific geodesics, slant circles, Legendre and slant helices of order 3 . We construct slant normal magnetic curves in R 2n+s (−3s) and give the parametric equations of these curves.

This study handles spatial three-dimensional solution of the nonlinear diffusion equation without particular initial conditions. The functional behavior of the equation and the concentration have been studied in new ways. An auxiliary function for diffusion is given having an interesting relationship with the concentration. A set of new integro-differential equations is given for diffusion.

The goal of the work is to take on and study one of the fundamental tasks studying Bidiophantine polygons (let us call a polygon Diophantine, if the distance between each two vertex of those is expressed by a natural number and we say that a Diophantine polygon is Bidiophantine if the coordinates of its each vertex are integer numbers).