In this paper, we present two new fuzzy inner product spaces and investigate some basic properties of these spaces. Specially, we prove parallelogram law for two and three vectors. Also, we introduce a suitable notion of orthogonality.
In the present paper, monotone relations and maximal monotone relations from an Hadamard space to its linear dual space are investigated. Fitzpatrick transform of monotone relations in Hadamard spaces is introduced. It is shown that Fitzpatrick transform of a special class of monotone relations is proper, convex and lower semi-continuous. Finally, a representation result for monotone relations is given.
In this paper, the notion of W-property for subsets of X × X ♦ is introduced and investigated, where X is an Hadamard space and X ♦ is its linear dual space. It is shown that an Hadamard space X is flat if and only if X × X ♦ has W-property. Moreover, the notion of monotone relation from an Hadamard space to its linear dual space is introduced. Finally, a characterization result for monotone relations with W-property (and hence in flat Hadamard spaces) is proved.
The Dirac Delta function is usually used to express the discrete distribution of electric charges in electrostatic problems. The integration of the product of the Dirac Delta function and the Green functions can calculate the electric potential and the electric field. Using fractal calculus, characteristic function, χ Cn (x), as an alternative for dirac delta function is used to describe Cantor set charge distribution which is typical example of a discrete set. In these cases we deal with F α -integration and F α -derivative of the product of characteristic function and function of staircase function, namely f (S α Cn (x)), which lead to calculation of electric potential and electric field. Recently, a calculus based fractals, called F α -calculus, has been developed which involve F α -integral and F α -derivative, of orders α, 0 < α < 1, where α is dimension of F . In F α -calculus the staircase function and characteristic function have special roles. Finally, using COMSOL Multiphysics software we solve ordinary Laplace's equation (not fractional) in the fractal region with Koch snowflake boundary which is non-differentiable fractal, and give their graphs for the three first iterations.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.