Mathematics

We prove a~general form of Chebyshev type inequality for generalized upper Sugeno integral in the form of necessary and sufficient condition. A key role in our considerations is played by the~class of m -positively dependent functions which includes comonotone functions as a~proper subclass. As a~consequence, we state an equivalent condition for Chebyshev type inequality to be true for all comonotone functions and any monotone measure. Our results generalize many others obtained in the framework of q-integral, seminormed fuzzy integral and Sugeno integral on the real half-line. Some further consequences of these results are obtained, among others Chebyshev type inequality for any functions. We also point out some flaws in existing results and provide their improvements.

Motivated by the substantial development of the special functions, we contribute to establish some rigorous results on the general series identities with bounded sequences and hypergeometric functions with different arguments, which are generally applicable in nature. For the application purpose, we apply our results to some functions e.g. Trigonometric functions, Elliptic integrals, Dilogarithmic function, Error function, Incomplete gamma function, and many other special functions.

Significant research has been carried out in the past half-century on defining generalised determinants for transformations between (typically real) vector spaces of different dimensions. We review three different generalisations of the determinant to non-square matrices, that we term for convenience the determinant-like function, the vector determinant and the g-determinant. We introduce and motivate these generalisations, note certain formal similarities between them and discuss their known properties.

With a new proof approach we prove in a more general setting the classical convergence theorem that almost everywhere convergence of measurable functions on a finite measure space implies convergence in measure. Specifically, we generalize the theorem for the case where the codomain is a separable metric space and for the case where the limiting map is constant and the codomain is an arbitrary topological space.

The method of separation of variables is significant, it has been applied to physics, engineering , chemistry and other fields. It allows to reduce the diffculity of problems by separating the variables from partial differential equation system into ordinary differential equations system. However, this method has complexity in higher order partial differential equations. In this reserach, we generalize this method by using multinomial theorem of n-harmonic equation to solve n-harmonic equation with m dimension and then solving an important class of partial differential equations with unbounded boundary conditions. Additionaly, application of convolution.

In this paper we discuss generalized group, provides some interesting examples. Further we introduce a generalized module as a module like structure obtained from a generalized group and discuss some of its properties and we also describes generalized module groupoids.

The generalized number-theoretic transformation (NPT) is formulated on the basis of the exponential function theorem, which allows us to replace operations modulo the expression as a whole by modulo operations on the exponent of this function, which makes this theorem fundamental for NPT, since it is such a function used in NPT as a weight conversion function. On the basis of this theorem, all the main theorems of the generalized NPT, their duality, as well as the properties of the weight functions of this transformation are formulated and proved. The choice of the basis of this function, as which any number can be chosen, including a complex one, determines not only one or another type of transformation, but also the module of the transformation itself. This allows us to generalize a number of well-known NPTs, such as Mersen, Gauss, and even Fourier, in the form of a unified theory of discrete transformations.

We show that a Fuchsian differential equation having five regular singular points admits solutions in terms of a single generalized hypergeometric function for infinitely many particular choices of equation parameters. Each solution assumes four restrictions imposed on the parameters: two of the singularities should have non-zero integer characteristic exponents and the accessory parameters should obey polynomial equations.

Let p 1 , p 2 ,… denote the prime numbers 2,3,… numbered in increasing order. The following method is used to generate primes. Start with p 1 =2 , p 2 =3 , Ip P 1 =[2,3] , p P 1 = {2,3} , MIp P 1 =3=Mp P 1 , #p P 1 =2 and for j=1,2,… , MIp P j = max Ip P j , Mp P j = max p P j , #p P j = |p P j | , Ip P j+1 =[MIp P j +1, Mp P 2 j +4Mp P j +3] and p P j+1 =Φ(Ip P j+1 )= the set of all primes in Ip P j+1 =Φ([MIp P j +1 , Mp P 2 j +4Mp P j +2]) . We use elementary method to obtain p P j+1 , j=1,2,… . This algorithm generates primes in a faster way to any given limit and the width of interval Ip P j+1 is a tight bound, in general, in the sense that if we increase further, then the algorithm fails. Also, we restate the Twin prime conjecture in an easier way.

In this paper, we investigate the geometric invariant properties of a normal curve on a smooth immersed surface under conformal transformation. We obtain an invariant-sufficient condition for the conformal image of a normal curve. We also find the deviations of normal and tangential components of the normal curve under the same motion. The results in \cite{9} are claimed as special cases of this paper.