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We prove that a function $f(x,y)$ of real variables defined on a rectangle, having square integrable partial derivatives $f"_{xx}$ and $f"_{yy}$, has almost everywhere mixed derivatives $f"_{xy}$ and $f"_{yx}$.

A new integral identity for functions with continuous second partial derivatives is derived. It is shown that the value of any function f(r,t) at position r and time t is completely determined by its previous values at all other locations…

We study partial derivatives on the product of two metric measure structures, in particular in connection with calculus via modules as proposed by the first named author. Our main results are 1) The extension to this non-smooth framework of…

It is shown that for a function $f:\mathbb R^2\to \mathbb R$ which is measurable with respect to the first variable and upper semicontinuous quasicontinuous and increasing with respect to the second variable there exists a Caratheodory's…

It is proved that a differentiable with respect to each variable function $f:\mathbb R^2\to\mathbb R$ is a solution of the equation $ \frac{\partial u}{\partial x} + \frac{\partial u}{\partial y}=0$ if and only if there exists a function…

The Cauchy problem for fractional derivatives linear systems of ordinary differential equations with constant coefficients is considered, where at first the analytic expressions are given through the matrix exponent of its corresponding…

In order to describe more complex problem using the concept of fractional derivatives, we introduce in this paper the concept of fractional derivatives with orders. The new definitions are based upon the concept of power law together with…

We demonstrate that discrete m-functions with eventually periodic continued fraction coefficients have an algebraic relationship to their second solution if and only if the periodic part of the sequence of continued fraction coefficients is…

We investigate a collection of orthonormal functions that encodes information about the continued fraction expansion of real numbers. When suitably ordered these functions form a complete system of martingale differences and are a special…

It is a well-known result by Saks \cite{Saks1934} that there exists a function $f \in L^1(\mathbb{R}^2)$ so that for almost every $(x,y)\in \mathbb{R}^2$ \[ \lim_{\substack{\mathrm{diam} R\rightarrow 0, \\ (x,y) \in R \in…

We consider a class of doubly intermittent maps with critical points, unbounded derivative and regularly varying tails. Under some mild assumptions we prove the existence of a unique mixing absolutely continuous invariant measure and give…

A result is proved concerning meromorphic functions of finite order in the plane such that all but finitely many zeros of the second derivative are zeros of the first derivative.

Let $A$ be a pseudo-differential operator with symbol $q(x,\xi)$. In this paper we derive sufficient conditions which ensure the existence of a solution to the $(A,C_c^{\infty}(\mathbb{R}^d))$-martingale problem. If the symbol $q$ depends…

We study a second order scheme for spatial fractional differential equations with variable coefficients. Previous results mainly concentrate on equations with diffusion coefficients that are proportional to each other. In this paper, by…

This paper is an introduction to the theory of multivector functions of a real variable. The notions of limit, continuity and derivative for these objects are given. The theory of multivector functions of a real variable, even being similar…

The main purpose of this work is to characterize derivations through functional equations. This work consists of five chapters. In the first one, we summarize the most important notions and results from the theory of functional equations.…

We generalize some classical results about quasicontinuous and separately continuous functions with values in metrizable spaces to functions with values in certain generalized metric spaces, called Maslyuchenko spaces. We establish…

A new derivative, called deformable derivative, is introduced here which is equivalent to ordinary derivative in the sense that one implies other. The deformable derivative is defined using limit approach like that of ordinary one but with…

Some new inequalities of Ostrowski type for twice differentiable mappings whose derivatives in absolute value are s-convex in the second sense are given.Applications for special means are also provided.

Finite differences have been widely used in mathematical theory as well as in scientific and engineering computations. These concepts are constantly mentioned in calculus. Most frequently-used difference formulas provide excellent…