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We analyze matrix convex functions of a fixed order defined on a real interval by differential methods as opposed to the characterization in terms of divided differences given by Kraus. We obtain for each order conditions for matrix…
This article shows a very elementary and straightforward proof of the Implicit Function Theorem for differentiable maps $F(x,y)$ defined on a finite-dimensional Euclidean space. There are no hypothesis on the continuity of the partial…
Cartesian differential categories come equipped with a differential operator which formalises the total derivative from multivariable calculus. Cofree Cartesian differential categories always exist over a specified base category, where the…
Exponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, familiar from toric varieties and their moment maps. Among them are varieties of inverses…
We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…
The entropic discriminant is a non-negative polynomial associated to a matrix. It arises in contexts ranging from statistics and linear programming to singularity theory and algebraic geometry. It describes the complex branch locus of the…
In regular dynamics, discrete maps are model presentations of discrete dynamical systems, and they may approximate continuous dynamical systems. Maps are used to investigate general properties of dynamical systems and to model various…
In this paper we extend the notion of an $\alpha$-family of maps to discrete systems defined by simple difference equations with the fractional Caputo difference operator. The equations considered are equivalent to maps with falling…
Sufficient conditions are determined on the parameters such that the generalized and normalized Bessel function of the first kind and other related functions belong to subclasses of starlike and convex functions defined in the unit disk…
We introduce a sheaf theoretic viewpoint on functional analysis designed for infinite dimensional Lie group actions. We develop functional calculus for Banach valued functors and, in particular, prove the existence of an exponential map for…
Functional equations satisfied by additive functions have a special interest not only in the theory of functional equations, but also in the theory of (commutative) algebra because the fundamental notions such as derivations and…
We define and study symmetrized and antisymmetrized multivariate exponential functions. They are defined as determinants and antideterminants of matrices whose entries are exponential functions of one variable. These functions are…
Using the theory of group action, we first introduce the concept of the automorphism group of an exponential family or a graphical model, thus formalizing the general notion of symmetry of a probabilistic model. This automorphism group…
We propose a quadrature-based formula for computing the exponential function of matrices with a non-oscillatory integral on an infinite interval and an oscillatory integral on a finite interval. In the literature, existing quadrature-based…
We give a combinatorial classification of postsingularly finite exponential maps in terms of external addresses starting with the entry 0. This is an extension of the classification results for critically preperiodic polynomials \cite{BFH}…
We define and study analogues of exponentials for functions on noncommutative two-tori that depend on a choice of a complex structure. The major difference with the commutative case is that our noncommutative exponentials can be defined…
We survey recent classification theorems for expansive matrices that generate the same anisotropic homogeneous Triebel-Lizorkin function space or sequence space. The function spaces are classified precisely by those matrices for which their…
A differential module is a module equipped with a square-zero endomorphism. This structure underpins complexes of modules over rings, as well as differential graded modules over graded rings. We establish lower bounds on the class--a…
We introduce a class of diffeological spaces, called elastic, on which the left Kan extension of the tangent functor of smooth manifolds defines an abstract tangent functor in the sense of Rosicky. On elastic spaces there is a natural…
A procedure to obtain differentiation matrices is extended straightforwardly to yield new differentiation matrices useful to obtain derivatives of complex rational functions. Such matrices can be used to obtain numerical solutions of some…