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The density of polynomials in a weighted space of infinitely differentiable functions in a multidimensional real space is proved under minimal conditions on weight functions and on differences between weight functions. We apply this result…
From a suitable integral representation of the Laplace transform of a positive semi-definite quadratic form of independent real random variables with not necessarily identical densities a univariate integral representation is derived for…
The Laplace transform in Komatsu ultradistributions is considered. Also, conditions are given under which an analytic function is a Laplace transformation of an ultradistribution.
In engineering practice one often encounters planar problems, where the corresponding vector space of forces, velocities or (infinitesimal) displacements is three dimensional. This paper shows how these spaces can be factorized, such that…
Space-like and time-like invariant space-time intervals are used to analyse measurements of spatial and temporal distances. The former are found to be Lorentz invariant --there is no `relativistic length contraction', whereas the latter…
The apparent times and positions of moving clocks as predicted by both `non-local' and `local' Lorentz Transformations are considered. Only local transformations respect translational invariance. Such transformations change temporal but not…
Translational invariance requires that physical predictions are independent of the choice of spatial coordinate system used. The time dilatation effect of special relativity is shown to manifestly respect this invariance. Consideration of…
The Special Affine Fourier Transformation(SAFT), which generalizes several well-known unitary transformations, has been demonstrated as a valuable tool in signal processing and optics. In this paper, we explore the multivariate dynamical…
We prove that any given function can be smoothly approximated by functions lying in the kernel of a linear operator involving at least one fractional component. The setting in which we work is very general, since it takes into account…
The transformation property of the Caputo fractional derivative operator of a scalar function under rotation in two dimensional space is derived. The study of the transformation property is essential for the formulation of fractional…
In this paper, we introduce and analyze multidimensional vector-valued Laplace transform of functions with values in sequentially complete locally convex spaces. A great number of our results seem to be new even for the functions with…
Two-dimensional systems with time-dependent controls admit a quadratic Hamiltonian modelling near potential minima. Independent, dynamical normal modes facilitate inverse Hamiltonian engineering to control the system dynamics, but some…
Integral transformations of the QCD invariant (running) coupling and of some related objects are discussed. Special attention is paid to the Fourier transformation, that is to transition from the space-time to the energy--momentum…
It is demonstrated that the measured spatial separation of two objects, at rest in some inertial frame, is invariant under space-time transformations. This result holds in both Galilean and Special Relativity. A corollary is that there are…
A calculus based on pointer-mark coincidences is proposed to define, in a mathematically rigorous way, measurements of space and time intervals. The connection between such measurements in different inertial frames according to the Galilean…
We find a formula that relates the Fourier transform of a radial function on $\mathbf{R}^n$ with the Fourier transform of the same function defined on $\mathbf{R}^{n+2}$. This formula enables one to explicitly calculate the Fourier…
In this paper path integration in two- and three-dimensional spaces of constant curvature is discussed: i.e.\ the flat spaces $\bbbr^2$ and $\bbbr^3$, the two- and three-dimensional sphere and the two- and three dimensional pseudosphere.…
Time-like and space-like invariant space-time intervals are used to analyse measurements of spatial and temporal distances defined by two spatially-separated clocks. The time dilatation effect is confirmed, but not `relativity of…
We introduce a fractional calculus on time scales using the theory of delta (or nabla) dynamic equations. The basic notions of fractional order integral and fractional order derivative on an arbitrary time scale are proposed, using the…
Description of linear continuous functionals on a space of rapidly decreasing infinitely differentiable functions on an unbounded closed convex set in $\mathbb R^n$ in terms of their Fourier-Laplace transform is obtained.