Related papers: On identity theorem for real functions

Entropy can signify different things: For instance, heat transfer in thermodynamics or a measure of information in data analysis. Many entropies have been introduced and it can be difficult to ascertain their different importance and…

In the theory of complex valued functions of a complex variable arguably the first striking theorem is that pointwise differentiability implies $C^{\infty}$ regularity. As mentioned in Ahlfors's standard textbook there have been a number of…

We find an arc-parameterization of the contour on which an given analytic function has constant modulus. This contour is seen to satisfy a differential equation which we explicitly give.

We characterize normal families in the unit ball as those families of analytic functions whose restrictions to each complex line through the origin are normal. We then generalize this result to a characterization of normal functions…

We generalize the classical Bernstein theorem concerning the constructive description of classes of functions uniformly continuous on the real line. The approximation of continuous bounded functions by entire functions of exponential type…

When developing and assessing density functional theory methods, a finite basis set is usually employed. In most cases, however, the issue of basis set dependency is neglected. Here, we assess several basis sets and functionals. In…

In graph theory and its practical networking applications, e.g., telecommunications and transportation, the problem of finding paths has particular importance. Selecting paths requires giving scores to the alternative solutions to drive a…

The theory of computational complexity focuses on functions and, hence, studies programs whose interactive behavior is reduced to a simple question/answer pattern. We propose a broader theory whose ultimate goal is expressing and analyzing…

Cartan's uniqueness theorem does not hold in general for CR mappings, but it does hold under certain conditions guaranteeing extendibility of CR functions to a fixed neighborhood. These conditions can be defined naturally for a wide class…

A formal definition of epsilon-complexity of an individual continuous function defined on a unit cube is proposed. This definition is consistent with the Kolmogorov's idea of the complexity of an object. A definition of epsilon-complexity…

We present a characterization of sets for which Cartwright's theorem holds true. The connection is discussed between these sets and sampling sets for entire functions of exponential type.

Analyzing in detail the analytic continuation of the Riemann zeta function we are able to generate several new identities which may be useful for application in physics and mathematics.

We prove that any complex or real analytic set or function germ is topologically equivalent to a germ defined by polynomial equations whose coefficients are algebraic numbers.

In the framework of superanalysis we get a functions theory close to complex analysis, under a suitable condition (A) on the real superalgebras in consideration. Under the condition (A), we get an integral representation formula for the…

We give a simple algebraic derivation of a useful determinantal identity for multilevel systems such as random matrix chains and finite determinantal point processes, with applications to the calculation of point correlators, gap…

We use type-theoretic techniques to present an algebraic theory of $\infty$-categories with strict units. Starting with a known type-theoretic presentation of fully weak $\infty$-categories, in which terms denote valid operations, we extend…

Feature Learning aims to extract relevant information contained in data sets in an automated fashion. It is driving force behind the current deep learning trend, a set of methods that have had widespread empirical success. What is lacking…

For integral representations of associated Legendre functions in terms of modified Bessel functions, we establish justification for differentiation under the integral sign with respect to parameters. With this justification, derivatives for…

We discuss the proof of a certain integral theorem obtained by C. G. Cullen, originally stated on the class of the analytic intrinsic functions on the quaternions. It is shown that this integral theorem is true for a larger class of…

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 $u$…