Related papers: Allowable Changes of Variable for Recently Develop…
A generalization of the definition of a one-dimensional improper integral with an infinite limit is presented. The new definition extends the range of valid integrals to include integrals which were previously considered to not be…
A generalization of the definition of a one-dimensional improper integral with a finite limit is presented. The new definition extends the range of valid integrals to include integrals which were previously considered to not be integrable.…
Assuming that the two integrals in the Change of Variable Formula for the Riemann integral on the real line are finite, one can rightfully ask if they have equal value. We give a positive answer to this question. The proof is very easy to…
The Fourier transform is naturally defined for integrable functrions. Otherwise, it should be stipulated in which sense the Fourier transform is understood. We consider some class of radial and, generally saying, nonintegrable functions.…
The change of variable theorem is proved under the sole hypothesis of differentiability of the transformation. Specifically, it is shown under this hypothesis that the transformed integral equals the given one over every measurable subset…
Various new sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained in the paper. The results are given in terms of $L^p$ integrability of the function and its…
In this note we prove a necessary and sufficient condition for the change of variables formula for the HK integral, with implications for the change of variables formula for the Lebesgue integral. As a corollary, we obtain a necessary and…
We consider general formulations of the change of variable formula for the Riemann-Stieltjes integral, including the case when the substitution is not invertible.
The routine definitions of both entropy, and differential entropy show inconsistencies that make them not reciprocally coherent. We propose a few possible modifications of these quantities so that 1) they no longer show incongruities, 2)…
A non-classical formulation of the central limit theorem is given for sequences of independent random variables with finite second moments. Singular sequences whose members all have a degenerate or normal distribution are excluded from…
In Physics, we are generally interested in real solutions involving natural phenomena, where knowledge of real functions of real variables is sufficient to obtain physically relevant results. However, the complexity of phenomena associated…
In this article it is proven the existence of integration of indefinite integrals as infinite derivative's series expansion. This also opens a new way to integrate a definite integral.
Estimation of the population total of a variable can be improved by calibration on a set of auxiliary variables. It is difficult to establish that such a set of variables is sufficient, that estimation could not be improved by calibration…
By topological arguments, we prove new results on the existence, non-existence, localization and multiplicity of nontrivial solutions of a class of perturbed nonlinear integral equations. These type of integral equations arise, for example,…
In this short note we explain how to construct resolutions or regular alterations admitting an ample exceptional divisor, assuming the existence of projective resolutions or regular alterations. In particular, this implies the existence of…
The Sinc quadrature and the Sinc indefinite integration are approximation formulas for definite integration and indefinite integration, respectively, which can be applied on any interval by using an appropriate variable transformation.…
We offer a conjecture on sharp estimation of a definite improper integral depend on a parameter $\lambda \in (0,+\infty)$ by means of given estimate of other definite integral depend on parameters $t\in [0,+\infty)$ and $\lambda$. Such…
The necessary and sufficient conditions for differentiability of a function of several real variables stated and proved and its ramifications discussed.
Writing the boundary integral equation for an exterior problem of elasticity is subordinate so far to hypotheses on the asymptotical behaviour at infinity of solutions. The sufficient conditions met in the literature are too restrictive and…
New sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained in the paper.