Mathematics

Derivative of a function can be expressed in terms of integration over a small neighborhood of the point of differentiation, so-called differentiation by integration method. In this text a maximal generalization of existing results which use one-dimensional integrals is presented together with some interesting non-analytic weight functions.

In this paper, we construct metallic Kähler and nearly metallic Kähler structures on Riemanian manifolds. For such manifolds with these structures, we study curvature properties. Also we describe linear connections on the manifold, which preserve the associated fundamental 2-form and satisfy some additional conditions and present some results concerning them.

When D:ξ?��?is a linear differential operator, a "direct problem " is to find the generating compatibility conditions (CC) in the form of an operator D 1 :η?��?such that Dξ=η implies D 1 η=0 . When D is involutive, the procedure provides successive first order involutive operators D 1 ,..., D n when the ground manifold has dimension n . Conversely, when D 1 is given, a more difficult " inverse problem " is to look for an operator D:ξ?��?having the generating CC D 1 η=0 . If this is possible, that is when the differential module defined by D 1 is torsion-free, one shall say that the operator D 1 is parametrized by D and there is no relation in general between D and D 2 . The parametrization is said to be " minimum " if the differential module defined by D has a vanishing differential rank and is thus a torsion module. The parametrization of the Cauchy stress operator in arbitrary dimension n has attracted many famous scientists (G.B. Airy in 1863 for n=2 , J.C. Maxwell in 1863, G. Morera and E. Beltrami in 1892 for n=3 , A. Einstein in 1915 for n=4 ) . This paper proves that all these works are using the Einstein operator and not the Ricci operator. As a byproduct, they are all based on a confusion between the so-called div operator induced from the Bianchi operator D 2 and the Cauchy operator which is the formal adjoint of the Killing operator D parametrizing the Riemann operator D 1 for an arbitrary n . Like the Michelson and Morley experiment, it is an open historical problem to know whether Einstein was aware of these previous works or not, as the comparison needs no comment.

We introduce an algebraic system which can be used as a model for spaces with geodesic paths between any two of their points. This new algebraic structure is based on the notion of mobility algebra which has recently been introduced as a model for the unit interval of real numbers. Mobility algebras consist on a set A together with three constants and a ternary operation. In the case of the closed unit interval A=[0,1] , the three constants are 0, 1 and 1/2 while the ternary operation is p(x,y,z)=x−yx+yz . A mobility space is a set X together with a map q:X×A×X→X with the meaning that q(x,t,y) indicates the position of a particle moving from point x to point y at the instant t∈A , along a geodesic path within the space X . A mobility space is thus defined with respect to a mobility algebra, in the same way as a module is defined over a ring. We introduce the axioms for mobility spaces, investigate the main properties and give examples. We also establish the connection between the algebraic context and the one of spaces with geodesic paths. The connection with affine spaces is briefly mentioned.

Successful application of Adomian decomposition method (ADM) in solving problems in nonlinear ordinary and partial differential equations depend strictly on the Adomian polynomial. In this paper, we present a simple modified known Adomian polynomial for nonlinear polynomial functionals with index as integers. The simple modified Adomian polynomial was tested for nonlinear functional with index 3 and 4 respectively. The result shows remarkable exact results as that given by Adomian himself. Also, the modifed simple Adomian polynomial was further tested on concrete problems and the numerical results were exactly the same as the exact solution. The large scale computation and evaluation was made possible by Maple software package.

We introduce a novel set-intersection operator called `most-intersection' based on the logical quantifier `most', via natural density of countable sets, to be used in determining the majority characteristic of a given countable (possibly infinite) collection of systems. The new operator determines, based on the natural density, the elements which are in `most' sets in a given collection. This notion allows one to define a majority set-membership characteristic of an infinite/finite collection with minimal information loss, compared to the standard intersection operator, when used in statistical ensembles. We also give some applications of the most-intersection operator in formal language theory and hypergraphs. The introduction of the most-intersection operator leads to a large number of applications in pure and applied mathematics some of which we leave open for further study.

A nested coordinate system is a reassigning of independent variables to take advantage of geometric or symmetry properties of a particular application. Polar, cylindrical and spherical coordinate systems are primary examples of such a regrouping that have proved their importance in the separation of variables method for solving partial differential equations. Geometric algebra offers powerful complimentary algebraic tools that are unavailable in other treatments.

In this article, we develop nested representations for cosine and inverse cosine functions, which is a generalization of Viète's formula for π . We explore a natural inverse relationship between these representations and develop numerical algorithms to compute them. Throughout this article, we perform numerical computation for various test cases, and demonstrate that these nested formulas are valid for complex arguments and a k th branch. We further extend the presented results to hyperbolic cosine and logarithm functions, and using additional trigonometric identities, we explore the sine and tangent functions and their inverses.

In this paper, the concept of neutrosophic soft filter and its basic properties are introduced. Later, we set up a neutrosophic soft topology with the help of a neutrosophic soft filter. We also give the notions of the greatest lower bound and the least upper bound of the family of neutrosophic soft filters, neutrosophic soft filter subbase and neutrosophic soft filter base and explore some basic properties of them.

In present paper, the definition of new metric space with neutrosophic numbers is given. Several topological and structural properties have been investigated. The analogues of Baire Category Theorem and Uniform Convergence Theorem are given for Neutrosophic metric spaces.