In this paper we study the distributional divergence of vector fields U in R of the form U = wB, where w is scalar function and B is a weakly differentiable vector field (or more in general the divergence of tensor fields of the form w⊗B). In particular we are interested in a kind of chain rule property, relating the divergence of h(w)B to the divergence of wB. In some sense, if one replaces “d...

In the present paper, measure differential equations involving the distributional Henstock-Kurzweil integral are investigated. Theorems on the existence and structure of the set of solutions are established by using Schauder$^prime s$ fixed point theorem and Vidossich theorem. Two examples of the main results paper are presented. The new results are generalizations of some previous results in t...

In this work, we consider the nonlocal obstacle problem with a given $\psi$ in bounded Lipschitz domain $\Omega$ $\mathbb{R}^{d}$, such that $\mathbb{K}\_\psi^s={v\in H^s\_0(\Omega):v\geq\psi \text{ a.e. }\Omega}\neq\emptyset$, by $$ u\in\mathbb{K}\_\psi^s:\quad\langle\mathcal{L}au,v-u\rangle\geq\langle F,v-u\rangle\quad\forall v\in\mathbb{K}^s\psi, for $F$ $H^{-s}(\Omega)$, dual space of fract...

It is shown that the nonstandard representatives of Schwartz-distributions, as introduced by K. D. Stroyan and W. A. J. Luxemburg in their book Introduction to the theory of infinitesimals [5], are locally equal to a finite-order derivative of a finite-valued and S-continuous function. By ‘equality’, we mean a pointwise equality, not an equality in a distributional sense. This proves a conjectu...

Over the last decades the research on monetary policy has largely concentrated on the impact of monetary authorities’ decisions on inflation and the fine-tuning of the macroeconomic, so that distributional effects of monetary policy which are non-trivial has been ignored. A view that has become increasingly popular since the financial crisis 2008 is that expansionary monetary policy can exacerb...

We give a local characterization of the class of functions having positive distributional derivative with respect to z̄ that are almost everywhere equal to one of finitely many analytic functions and satisfy some mild nondegeneracy assumptions. As a consequence, we show that any subharmonic piecewise harmonic function satisfying these conditions locally coincides with the maximum of finitely man...

Using the historically general growth condition on scalar-valued analytic functions, which have tempered distributions as boundary values, we show that vector-valued functions in tubes TC=Rn+iC obtain values. In a certain case, study structure of this value, is shown to be Fourier transform distributional derivative continuous function polynomial growth. A set used value one–one and onto relati...

In this paper we propose a novel numerical scheme for the Canham-Helfrich-Evans bending energy based on three-field lifting procedure of distributional shape operator to an auxiliary mean curvature field. Together with its energetic conjugate scalar stress field as Lagrange multiplier resulting fourth order problem is circumvented and reduced mixed saddle point involving only second differentia...

For this paper, we proposed the fractional category representation vector (FV) based on calculus (FC), of which one-hot label is only special case when derivative order 0. FV can be considered as a distributional negative probability considered. FVs used either regularization method or distributed representation. They gain significantly in generalization classification models and representabili...