Related papers: The Derivative of a Constructible Function is Cons…
A full subcategory of a Grothendieck category is called deconstructible if it consists of all transfinite extensions of some set of objects. This concept provides a handy framework for structure theory and construction of approximations for…
A proof is reconstructed for a useful theorem on the zeros of derivatives of analytic functions due to H. M. Macdonald, which appears to be now little known. The Theorem states that, if a function $f(z)$ is analytic inside a bounded region…
For a subfield K of C, we denote by C^K the category of algebras of functions defined on the globally subanalytic sets that are generated by all K-powers and logarithms of positively-valued globally subanalytic functions. For any function f…
We prove a closed formula for the derivative, of any order, of a implicit function, in terms of some binomial building blocks, and explain the combinatorics behind the coefficients appearing in the formula.
We prove two theorems about Goodwillie calculus and use those theorems to describe new models for Goodwillie derivatives of functors between pointed compactly-generated infinity-categories. The first theorem say that the construction of…
We develop synthetic notions of oracle computability and Turing reducibility in the Calculus of Inductive Constructions (CIC), the constructive type theory underlying the Coq proof assistant. As usual in synthetic approaches, we employ a…
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…
Given an integer $a\ge 1$, a function $f: \mathbb{R}\to \mathbb{R}$ is said to be $a$-subadditive if $$ f(ax+y) \le af(x)+f(y) \,\,\,\text{ for all }x,y \in \mathbb{R}. $$ Of course, $1$-subadditive functions (which correspond to ordinary…
We present a detailed introduction of the theory of constructible sheaf complexes in the complex algebraic and analytic setting. All concepts are illustrated by many interesting examples and relevant applications, while some important…
As established by R T. Rockafellar, real valued convex-concave functions are generically differentiable. It this paper we shall show that for a convex-concave function defined on an open convex set $C \times D,$ there exist dense subsets…
One proves that any everywhere defined constructive mapping from a complete metric space into a complete metric space which preserves the property of precompacity of subsets is locally uniformly continuous. This fact can be viewed as…
In this paper, we prove that slice polyanalytic functions on quaternions can be considered as solutions of a power of some special global operator with nonconstant coefficients as it happens in the case of slice hyperholomorphic functions.…
This short article contains the construction of a construction that generalizes the concept of the derivative of a function of one variable, using the theory of filters. The paper presents a new concept, demonstrates that it really…
We formulate a theory of shape valid for objects of arbitrary dimension whose contours are path connected. We apply this theory to the design and modeling of viable trajectories of complex dynamical systems. Infinite families of…
An algebraic version of Kashiwara and Schapira's calculus of constructible functions is used to describe local topological properties of real algebraic sets, including Akbulut and King's numerical conditions for a stratified set of…
For families of smooth complex projective varieties we show that normal functions arising from algebraically trivial cycle classes are algebraic, and defined over the field of definition of the family. In particular, the zero loci of those…
We prove that continuous reducibility is a well-quasi-order on the class of continuous functions between separable metrizable spaces with analytic zero-dimensional domain. To achieve this, we define scattered functions, which generalize…
We view a conic optimization problem that has a unique solution as a map from its data to its solution. If sufficient regularity conditions hold at a solution point, namely that the implicit function theorem applies to the normalized…
A classification is provided of functors, in particular polynomial ones, from a category with a zero object in which every object is a finite sum of copies of a generating object, into an abelian category. This classification is extended to…
We study a new notion of reduction between structures called enumerable functors related to the recently investigated notion of computable functors. Our main result shows that enumerable functors and effective interpretability with the…