Related papers: Differentiability of the $n$-Variable Function Ded…
Within the framework of mappings between affine spaces, the notion of $n$-th polarization of a function will lead to an intrinsic characterization of polynomial functions. We prove that the characteristic features of derivations, such as…
Suppose that we are given a formal power series of many variables with coefficients in $\mathbb{R}$ (or $\mathbb{C}$) and we want to compute its $n$-th (multiplicative) root. As can be expected coefficients of the root have to satisfy a…
A mixed type dual to a nondifferentiable variational problem involving higher order derivative is formulated and duality results are proved under generalized invexity conditions. Special cases are generated from our results.
We compute the nth derivative of a function given parametrically, and of one given implicitly, and some history for both problems. I am posting this version of the paper at the request of Shaul Zemel, whose forthcoming paper The…
We consider some possible approaches to the fractional-order generalization of definition of variation (functional) derivative. Some problems of formulation of a fractional-order variational derivative are discussed. To give a consistent…
We consider approximation problems for a special space of d variate functions. We show that the problems have small number of active variables, as it has been postulated in the past using concentration of measure arguments. We also show…
In classical complex analysis analyticity of a complex function $f$ is equivalent to differentiability of its real and imaginary parts $u$ and $v$, respectively, together with the Cauchy-Riemann equations for the partial derivatives of $u$…
The level of a function f on an n-dimensional space encloses a region. The volume of a region between two such levels depends on both levels. Fixing one of them the volume becomes a function of the remaining level. We show that if the…
In this paper we study differentiability properties of the map $T\mapsto\phi(T)$, where $\phi$ is a given function in the disk-algebra and $T$ ranges over the set of contractions on Hilbert space. We obtain sharp conditions (in terms of…
The concept of the derivative-dependent functional separable solution, as a generalization to the functional separable solution, is proposed. As an application, it is used to discuss the generalized nonlinear diffusion equations based on…
We prove a analogous of Stein theorem for rational functions in several variables: we bound the number of reducible fibers by a formula depending on the degree of the fraction.
The classical decision problem, as it is understood today, is the quest for a delineation between the decidable and the undecidable parts of first-order logic based on elegant syntactic criteria. In this paper, we treat the concept of…
The fractional calculus of variations is now a subject under strong research. Different definitions for fractional derivatives and integrals are used, depending on the purpose under study. In this paper the fractional operators are defined…
We consider a length functional for $C^1$ curves of fixed degree in graded manifolds equipped with a Riemannian metric. The first variation of this length functional can be computed only if the curve can be deformed in a suitable sense, and…
The first aim of this work is to establish a Peano type existence theorem for an initial value problem involving complex fractional derivative and the second is, as a consequence of this theorem, to give a partial answer to the local…
Let $f$ be a measurable function defined on $\mathbb{R}$. For each $n\in\mathbb{Z}$ define the operator $A_n$ by $$A_nf(x)=\frac{1}{2^n}\int_x^{x+2^n}f(y)\, dy.$$ Consider the variation operator…
We consider an equation with drift and either critical or supercritical fractional diffusion. Under a regularity assumption for the vector field that is marginally stronger than what is required for Holder continuity of the solutions, we…
We extend to several variables an earlier result of ours, according to which an entire function of one variable of sufficiently small exponential type, having all derivatives of even order taking integer values at two points, is a…
We provide a simple proof of a result, due to G. Alberti, concerning a rank-one property for the singular part of the derivative of vector-valued functions of bounded variation.
We give a proper fractional extension of the classical calculus of variations. Necessary optimality conditions of Euler-Lagrange type for variational problems containing both classical and fractional derivatives are proved. The fundamental…