Related papers: Functions of a Single Variable
The concept of a variance on a category is introduced as a two-sided strict factorization system. By employing variances, we define functors of variance in a more general setting than is usually considered, thereby eliminating the need for…
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…
It is shown that some class of differential inclusions has solutions that are defined and bounded for all real values of independent variable. Applications to dynamics are considered.
This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits…
The aim of these notes is to give a introduction to the ideas and techniques of handling rational curves on varieties. The main emphasis is on varieties with many rational curves which are the higher dimensional analogs of rational curves…
These informal notes are concerned with spaces of functions in various situations, including continuous functions on topological spaces, holomorphic functions of one or more complex variables, and so on.
A complete classification of all continuous, epi-translation and rotation invariant valuations on the space of super-coercive convex functions on ${\mathbb R}^n$ is established. The valuations obtained are functional versions of the…
This book concentrates on functional analysis. The text is written so that it can be followed on the basis of high school mathematics. The book introduces the set theoretical foundations of mathematics, the basic theories of linear algebra…
Working with letters that represent unknown constants, i.e., parameters, has been historically challenging for students. This is an important skill for their success in many future quantitative settings, and yet it appears this topic is…
Extending the notion of bounded variation, a function $u \in L_c^1(\mathbb R^n)$ is of bounded fractional variation with respect to some exponent $\alpha$ if there is a finite constant $C \geq 0$ such that the estimate \[ \biggl|\int u(x)…
This is a thesis on some applications of regularly varying functions. Three problems are considered. The first problem is about the randomly weighted sums, the second is on the behavior of the product under conditional extreme value model…
We define a normal form (called the canonical image) of an arbitrary measurable function of several variables with respect to a natural group of transformations; describe a new complete system of invariants of such a function (the system of…
The main aim of this note is to provide characterization theorems concerning real derivations. Among others the following implication will be verified: Assume that $\xi\colon \mathbb{R}\to \mathbb{R}$ is a given differentiable function and…
This note studies, and partially solves, 3 elementary questions about continuous rational functions on real (and p-adic) algebraic varieties: Can one restrict such a function to a subvariety? Can one extend such a function from 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.
Fourier Transforms is a first in a series of monographs we present on harmonic analysis. Harmonic analysis is one of the most fascinating areas of research in mathematics. Its centrality in the development of many areas of mathematics such…
This study introduces a procedure to obtain general expressions, $y = f(x)$, subject to linear constraints on the function and its derivatives defined at specified values. These constrained expressions can be used describe functions with…
We study real-valued valuations on the space of Lipschitz functions over the Euclidean unit sphere $S^{n-1}$. After introducing an appropriate notion of convergence, we show that continuous valuations are bounded on sets which are bounded…
We discuss $p$-variation regularity of real-valued functions defined on $[0,T]^2$, based on rectangular increments. When $p>1$, there are two slightly different notions of $p$-variation; both of which are useful in the context of Gaussian…
Functions of several quaternion variables are investigated and integral representation theorems for them are proved. With the help of them solutions of the $\tilde \partial $-equations are studied. Moreover, quaternion Stein manifolds are…