Mathematics

We consider the relation between Euler's trinomial problem and the problem of decomposition of tensor powers of adjoint representation of A 1 Lie algebra. By using this approach, some new results for both problems are obtained.

The exact solutions of both the cubic Duffing equation and cubic-quintic Duffing equation are presented by using only leaf functions. In previous studies, exact solutions of the cubic Duffing equation have been proposed using functions that integrate leaf functions in the phase of trigonometric functions. Because they are not simple, the procedures for transforming the exact solutions are complicated and not convenient. The first derivative of the leaf function can be derived as the root. This derivative can be factored. These factors or multiplications of factors are exact solutions to the Duffing equation. Some of these exact solutions are of the same type as the cubic Duffing equation reported in previously. Some of these exact solutions satisfy the exact solutions of the cubic--quintic Duffing equations with high nonlinearity. In this study, the relationship between the parameters of these exact solutions and the coefficients of the terms of the Duffing equation is clarified. We numerically analyze these exact solutions, plot the waveform, and discuss the periodicity and amplitude of the waveform.

In this paper we present an introduction to morphological calculus in which geometrical objects play the rule of generalised natural numbers.

A distance magic labeling of an n -dimensional hypercube is a labeling of its vertices by natural numbers from {0,?? 2 n ??} , such that for all vertices v the sum of the labels of the neighbors of v is the same. Such a labeling is called neighbor-balanced, if, moreover, for each vertex v and an index i=0,??n?? , exactly half of the neighbors of v have digit 1 at i -th position of the binary representation of their label. We demonstrate examples of non-neighbor-balanced distance magic labelings of 6 -dimensional hypercube obtained by a SAT solver.

A fundamental problem in analysis is to decide whether a smooth solution exists for the Navier-Stokes equations in three dimensions. In this paper we shall study this problem. The Navier-Stokes equations are given by: u it (x,t)−ρ△ u i (x,t)− u j (x,t) u i x j (x,t)+ p x i (x,t)= f i (x,t) , divu(x,t)=0 with initial conditions u | (t=0)⋃∂Ω =0 . We introduce the unknown vector-function: ( w i (x,t) ) i=1,2,3 : u it (x,t)−ρ△ u i (x,t)− dp(x,t) d x i = w i (x,t) with initial conditions: u i (x,0)=0, u i (x,t) ∣ ∂Ω =0 . The solution u i (x,t) of this problem is given by: u i (x,t)= ∫ t 0 ∫ Ω G(x,t;ξ,τ) ( w i (ξ,τ)+ dp(ξ,τ) d ξ i )dξdτ where G(x,t;ξ,τ) is the Green function. We consider the following N-Stokes-2 problem: find a solution w(x,t)∈ L 2 ( Q t ),p(x,t): p x i (x,t)∈ L 2 ( Q t ) of the system of equations: w i (x,t)−G( w j (x,t)+ dp(x,t) d x j )⋅ G x j ( w i (x,t)+ dp(x,t) d x i )= f i (x,t) satisfying almost everywhere on Q t . Where the v-function p x i (x,t) is defined by the v-function w i (x,t) . Using the following estimates for the Green function: |G(x,t;ξ,τ)|≤ c (t−τ ) μ ⋅|x−ξ | 3−2μ ;| G x (x,t;ξ,τ)|≤ c (t−τ ) μ ⋅|x−ξ | 3−(2μ−1) (1/2<μ<1), from this system of equations we obtain: w(t)<f(t)+b( ∫ t 0 w(τ) (t−τ ) μ dτ ) 2 ; w(t)=∥w(x,t) ∥ L 2 (Ω) ,f(t)=∥f(x,t) ∥ L 2 (Ω) . Further, using the replacement of the unknown function by \textbf{Riccati}, from this inequality we obtain the a priori estimate. By the Leray-Schauder's method and this a priori estimate the existence and uniqueness of the solution is proved.

Our aim in this paper is to prove, under some growth conditions on the datas, the solvability in a Gevrey class of a polynomially nonlinear functional differential equation.

In this article, we investigate global norm of potential vector field in Ricci soliton. In particular, we have deduced certain conditions so that the potential vector field has finite global norm in expanding Ricci soliton. We have also showed that if the potential vector field has finite global norm in complete non-compact Ricci soliton having finite volume, then the scalar curvature becomes constant.

Legendre's Conjecture is one of the most elegant open problems in Number Theory, which states that there is a prime between consecutive two perfect squares. In this note, we prove the conjecture holds true and also discuss the related results.

In this paper, we consider a class of nonlinear fractional differential equations involving Hilfer derivative with boundary conditions. First, we obtain an equivalent integral for the given boundary value problem in weighted space of continuous functions. Then we obtain the existence results for a given problem under a new approach and minimal assumptions on nonlinear function f. The technique used in the analysis relies on a variety of tools including Schauder, Schaefer and Krasnoselski fixed point theorems. We demonstrate our results through illustrative examples.

This paper is devoted to study the existence of a solution to Hilfer fractional differential equation with nonlocal boundary condition. We use the equivalent integral equation to study the considered Hilfer differential problem with nonlocal boundary condition. The Monch type fixed point theorem and the measure of the noncompactness technique are the main tools in this study. We demonstrate the existence of a solution with a suitable illustrative example.