Related papers: On perturbations of continuous maps
We prove several results regarding the homology and homotopy type of images of real maps and their complexification. In particular, we study the local behavior of singular points after deformations. In this context, we prove a restrictive…
The perturbation theory based on typicality introduced in Ref. [1] and further refined in Refs. [2, 3] provides a powerful tool since it is intended to be applicable to a wide range of scenarios while relying only on a few parameters. Even…
We present an analysis of two different approximations to the scalar field theory on the fuzzy sphere, a nonperturbative and a perturbative one, which are both multitrace matrix models. We show that the former reproduces a phase diagram…
We develop the general theory of \emph{topometric spaces}, i.e., topological spaces equipped with a well-behaved lower semi-continuous metric function. Spaces of global and local types in continuous logic are the motivating examples for the…
Is a given map between compact topological manifolds homotopic to the projection map of a fiber bundle? In this paper obstructions to this question are introduced with values in higher algebraic K-theory. Their vanishing implies that the…
We study the topology associated with physical vector and scalar fields. A mathematical object, e.g., a ball, can be continuously deformed, without tearing or gluing, to make other topologically equivalent objects, e.g., a cube or a solid…
In the following text we show if $X$ is an Alexandroff space, then $f:X\to Y$ has closed graph if and only if it has constant closed value on each connected component of $X$. Moreover, if $X$ an Alexandroff space and $f:X\to Y$ has closed…
We describe energy--momentum conservation in relativistic perturbation theory in general FRW backgrounds with causal source terms, such as the presence of cosmic defect networks. We provide a prescription for a linear energy--momentum…
We study composed map germs with respect to their local fibrations. Under most general conditions, inspired by the tameness condition that was introduced recently, we prove the existence of singular tube fibrations, and we determine the…
The reductions of a square complex matrix A to its canonical forms under transformations of similarity, congruence, or *congruence are unstable operations: these canonical forms and reduction transformations depend discontinuously on the…
This paper studies the obstructions to deforming a map from a complex variety to another variety which is an immersion of codimension one. We extend the classical notion of semiregularity of subvarieties to maps between varieties, and show…
We derive the matching conditions for cosmological perturbations in a Friedmann Universe where the equation of state undergoes a sharp jump, for instance as a result of a phase transition. The physics of the transition which is needed to…
We consider several ways to test for topology directly in harmonic space by comparing the measured a_lm with the expected correlation matrices. Two tests are of a frequentist nature while we compute the Bayesian evidence as the third test.…
For a continuous self-map $f$ on a compact interval $I$ and the induced map $\hat f$ on the space $\mathcal{M}(I)$ of probability measures, we obtain a sharp condition to guarantee that $(I,f)$ is transitive if and only if…
A convex subset X of a linear topological space is called compactly convex if there is a continuous compact-valued map $\Phi:X\to exp(X)$ such that $[x,y]\subset\Phi(x)\cup \Phi(y)$ for all $x,y\in X$. We prove that each convex subset of…
We introduce a fairly general concept of functional equation for $k$-tuples of functions $f_1,\dots,f_k\colon X \to Y$ between arbitrary sets. The homomorphy equations for mappings between groups and other algebraic systems, as well as…
We prove a Milnor-L\^e type fibration theorem for a subanalytic map $f: X \to Y$ between subanalytic sets $X \subset \mathbb{R}^m$ and $Y \subset \mathbb{R}^n$. Moreover, if $f$ extends to an analytic map $\mathbb{R}^m \to \mathbb{R}^n$, we…
We prove that the inclusion of map(X,Y) into map(K(X),K(Y)) is continuous, where K(X) is the space of non-empty compact subsets of X (also known as the hyperspace of compact subsets of X), and both spaces of maps are endowed with the…
We consider two natural topologies on the space $S(X\times Y,Z)$ of all separately continuous functions defined on the product of two topological spaces $X$ and $Y$ and ranged into a topological or metric space $X$. These topologies are the…
This paper discusses a more general contractive condition for a class of extended cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same…