Related papers: The Derivative of a Constructible Function is Cons…

We show that if the Kobayashi--Royden metric of a complex manifold is continuous and positive at a given point and any non-zero tangent vector, then the "derivatives" of the higher order Lempert functions exist and equal the respective…

The Donald-Flanigan conjecture asserts that any group algebra of a finite group has a separable deformation. We apply an inductive method to deform group algebras from deformations of normal subgroup algebras, establishing an infinite…

Let $\lambda$ be an uncountable cardinal such that $2^{< \lambda } = \lambda$. Working in the setup of generalized descriptive set theory, we study the structure of $\lambda^+$-Borel measurable functions with respect to various kinds of…

Containers conveniently represent a wide class of inductive data types. Their derivatives compute representations of types of one-hole contexts, useful for implementing tree-traversal algorithms. In the category of containers and cartesian…

Higher order derivatives of functions are structured high dimensional objects which lend themselves to many alternative representations, with the most popular being multi-index, matrix and tensor representations. The choice between them…

On a smooth manifold M, the Kashiwara index formula expresses the weighted Euler characteristic of a constructible function in terms of its characteristic cycle. We generalize this formula to the case when M is a smooth orbifold, answering…

We propose a definition of computable manifold by introducing computability as a structure that we impose to a given topological manifold, just in the same way as differentiability or piecewise linearity are defined for smooth and PL…

Ultrafunctions are a particular class of functions defined on a Non Archimedean field R^{*}\supset R. They have been introduced and studied in some previous works ([1],[2],[3]). In this paper we introduce a modified notion of ultrafunction…

We characterize left and right amenable semigroups of polynomials of one complex variable with respect to the composition operation. We also prove a number of results about amenable semigroups of arbitrary rational functions. In particular,…

Inspired from a joint work by A. Beckmann, S. Buss and S. Friedman, we propose a class of set-theoretic functions, predicatively computable functions. Each function in this class is polynomial time computable when we restrict to finite…

We extend the theory of Euler integration from the class of constructible functions to that of "tame" real-valued functions (definable with respect to an o-minimal structure). The corresponding integral operator has some unusual defects (it…

In their work on differential operators in positive characteristic, Smith and Van den Bergh define and study the derived functors of differential operators; they arise naturally as obstructions to differential operators reducing to positive…

A fractal function is a function whose graph is the attractor of an iterated function system. This paper generalizes analytic continuation of an analytic function to continuation of a fractal function.

We examine various categorical structures that can and cannot be constructed. We show that total computable functions can be mimicked by constructible functors. More generally, whatever can be done by a Turing machine can be constructed by…

A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory…

We prove the following analogue of a Theorem of R.O. Davies: Every $\Sigma^1_2$ function $f:\R\times\R\to\R$ can be represented as a sum of rectangular $\Sigma^1_2$ functions if and only if all reals are constructible.

We employ the notions of `sequential function' and `interrogation' (dialogue) in order to define new partial combinatory algebra structures on sets of functions. These structures are analyzed using J. Longley's preorder-enriched category of…

The set of trigonometric functions in taxicab geometry is completed and derivatives of all of the taxicab trigonometric functions are explored.

Graphical functions are positive functions on the punctured complex plane $\mathbb{C}\setminus\{0,1\}$ which arise in quantum field theory. We generalize a parametric integral representation for graphical functions due to Lam, Lebrun and…

Let $T$ be an o-minimal theory expanding $\mathrm{RCF}$ and $T_\mathrm{convex}$ be the common theory of its models expanded by predicate for a non-trivial $T$-convex valuation ring. We call an elementary extension $(\mathbb{E}, \mathcal{O})…