On Computability and Triviality of Well Groups

The concept of well group in a special but important case captures homological properties of the zero set of a continuous map $f:K\to R^n$ on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within $L_\infty$ distance r from f for a given r>0. The main drawback of the approach is that the computability of well groups was shown only when dim K=n or n=1. Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K<2n-2, our approximation of the (dim K-n)th well group is exact. For the second part, we find examples of maps $f,f': K\to R^n$ with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status.Comment: 20 pages main paper including bibliography, followed by 22 pages of Appendi

Persistence of Zero Sets

We study robust properties of zero sets of continuous maps $f:X\to\mathbb{R}^n$. Formally, we analyze the family $Z_r(f)=\{g^{-1}(0):\,\,\|g-f\|<r\}$ of all zero sets of all continuous maps $g$ closer to $f$ than $r$ in the max-norm. The fundamental geometric property of $Z_r(f)$ is that all its zero sets lie outside of $A:=\{x:\,|f(x)|\ge r\}$. We claim that once the space $A$ is fixed, $Z_r(f)$ is \emph{fully} determined by an element of a so-called cohomotopy group which---by a recent result---is computable whenever the dimension of $X$ is at most $2n-3$. More explicitly, the element is a homotopy class of a map from $A$ or $X/A$ into a sphere. By considering all $r>0$ simultaneously, the pointed cohomotopy groups form a persistence module---a structure leading to the persistence diagrams as in the case of \emph{persistent homology} or \emph{well groups}. Eventually, we get a descriptor of persistent robust properties of zero sets that has better descriptive power (Theorem A) and better computability status (Theorem B) than the established well diagrams. Moreover, if we endow every point of each zero set with gradients of the perturbation, the robust description of the zero sets by elements of cohomotopy groups is in some sense the best possible (Theorem C)

Specifics of management of enterprise innovation activities in the Czech republic - the decision-making mechanism

The majority of Czech managers are aware that the long-term competitiveness of the company depends primarily on the use of innovative technical solutions and investments in new technologies. Despite awareness of the importance of innovation, many companies do not know how to manage, implement, and evaluate them. Empirical research showed that most innovation firms implement, but do not systematically manage the implementation of innovative projects and the allocation of funds. There is a contradiction between companies' ability to orientate themselves in the approaches available in the area of innovation management and the existence of a large number of approaches that can be used to address a particular type of innovation problem. A set of innovation concepts has been created to solve those challenges. Practical steps of the decision-making mechanism for selecting innovation concepts have been proposed. The decision-making mechanism is based on the analytic hierarchy process (AHP) and serves primarily for managers of medium and large enterprises.Web of Science26314213

How many double squares can a string contain?

Counting the types of squares rather than their occurrences, we consider the problem of bounding the number of distinct squares in a string. Fraenkel and Simpson showed in 1998 that a string of length n contains at most 2n distinct squares. Ilie presented in 2007 an asymptotic upper bound of 2n - Theta(log n). We show that a string of length n contains at most 5n/3 distinct squares. This new upper bound is obtained by investigating the combinatorial structure of double squares and showing that a string of length n contains at most 2n/3 double squares. In addition, the established structural properties provide a novel proof of Fraenkel and Simpson's result.Comment: 29 pages, 20 figure

Parametric families for the Lorenz curve: an analysis of income distribution in European countries

The European Union Survey on Income and Living Conditions (EU-SILC) is the main source of information about living standards and poverty in the EU member states. We compare different parametric models for the Lorenz curve (LC) with an empirical analysis of the income distributions of 26 European countries in the year 2017. The objective of our empirical study is to verify whether simple mono-parametric models for the LCs can represent similarities or differences between European income distributions in sufficient detail, or whether an alternative, more sophisticated multi-parametric model should be used instead. In particular, we consider the power LC, the Pareto LC, the Lamè LC, a generalised bi-parametric version of the Lamè LC, a bi-parametric mixture of power LCs and the recently introduced arctan family of LCs. Whilst the first three families are ordered, in that different parametric values correspond to a situation of Lorenz ordering, the latter three may also identify the ambiguous situation of intersecting LCs. Therefore, besides focusing on the goodness-of-fit of the models considered and their mathematical simplicity, we evaluate the effectiveness of multi-parametric models in identifying the non-dominated cases

Computing simplicial representatives of homotopy group elements

A central problem of algebraic topology is to understand the homotopy groups $\pi_d(X)$ of a topological space $X$. For the computational version of the problem, it is well known that there is no algorithm to decide whether the fundamental group $\pi_1(X)$ of a given finite simplicial complex $X$ is trivial. On the other hand, there are several algorithms that, given a finite simplicial complex $X$ that is simply connected (i.e., with $\pi_1(X)$ trivial), compute the higher homotopy group $\pi_d(X)$ for any given $d\geq 2$. %The first such algorithm was given by Brown, and more recently, \v{C}adek et al. However, these algorithms come with a caveat: They compute the isomorphism type of $\pi_d(X)$, $d\geq 2$ as an \emph{abstract} finitely generated abelian group given by generators and relations, but they work with very implicit representations of the elements of $\pi_d(X)$. Converting elements of this abstract group into explicit geometric maps from the $d$-dimensional sphere $S^d$ to $X$ has been one of the main unsolved problems in the emerging field of computational homotopy theory. Here we present an algorithm that, given a~simply connected space $X$, computes $\pi_d(X)$ and represents its elements as simplicial maps from a suitable triangulation of the $d$-sphere $S^d$ to $X$. For fixed $d$, the algorithm runs in time exponential in $size(X)$, the number of simplices of $X$. Moreover, we prove that this is optimal: For every fixed $d\geq 2$, we construct a family of simply connected spaces $X$ such that for any simplicial map representing a generator of $\pi_d(X)$, the size of the triangulation of $S^d$ on which the map is defined, is exponential in $size(X)$

Two Personalities of Comparative Literary Studies (Claudio Guillén and Dionýz Ďurišin)

The study aims to analyze two different conceptualizations of comparative literary studies – one by a Spanish scholar (Claudio Guillén) and the other by a Slovak one (Dionýz Ďurišin). It focuses on the circumstances of development of this discipline on the basis of particular sources and impulses with regard to the geographic limits of the study.In the first part, the notion of supra-nationality is characterized as an effort to free oneself from the narrow frame of national literary history. Guillén was very aware of the tension between the local and the universal, or the particular and the general, which, according to him, requires the scholar to transcend conventional approaches and respect the reader’s ordinary experience. Instead of a rigid critical frame, what is needed is a historical and critical horizon that does not exclude the individual dimension, nor a unifying perspective. At the same time, Guillén emphasizes the search for the universal dimension of literature. Guillén is sceptical about the focus on formal-linguistic approaches in studying literary development, which he observes at Spanish universities today.On the other hand, Slovak comparative literary studies had different points of departure. The Slovak comparatist Ďurišin took many impulses from the Russian formalists, who focused particularly on the issues of the national literary development. As an example, the study uses the term historical poetics, applied in the study of the development of Slovak verse (Mikuláš Bakoš). Instead of the prevalent genetic method used by Guillén, Ďurišin at the same time used a theoretical-developmental model in studying the relationship between national and world literature, with an emphasis on the role of the receiving literature