Mapping nn grid points onto a square forces an arbitrarily large Lipschitz constant

We prove that the regular $n\times n$ square grid of points in the integer lattice $\mathbb{Z}^{2}$ cannot be recovered from an arbitrary $n^{2}$-element subset of $\mathbb{Z}^{2}$ via a mapping with prescribed Lipschitz constant (independent of $n$). This answers negatively a question of Feige from 2002. Our resolution of Feige's question takes place largely in a continuous setting and is based on some new results for Lipschitz mappings falling into two broad areas of interest, which we study independently. Firstly the present work contains a detailed investigation of Lipschitz regular mappings on Euclidean spaces, with emphasis on their bilipschitz decomposability in a sense comparable to that of the well known result of Jones. Secondly, we build on work of Burago and Kleiner and McMullen on non-realisable densities. We verify the existence, and further prevalence, of strongly non-realisable densities inside spaces of continuous functions.Comment: 60 pages (43 pages of the main part, 13 pages of appendices), 10 figures. This is a revised version according to referees' comments. Our version of the proof of the theorem about bilipschitz decomposition of Lipschitz regular mappings was greatly simplified. To appear in GAF

Database Subsetting with Respect to the Workload

Tato bakalářská práce se zabývá návrhem a implementací nástroje pro hledání podmnožiny databáze s ohledem na vytížení. První část práce se zabývá problematikou subsettingu a rešerší existujících nástrojů s ohledem na naše požadavky. V druhé části je popis samotné implementace nástroje a řešení problémů, které nastaly. Součástí je také otestování nástroje a analýza získaných dat.This thesis describes the design and implementation of a tool for database subsetting with respect to workload. The first part deals with problematics of subsetting and the analysis of existing tools with regard to our requirements. In the second part is a description of~the~implementation itself and~the solution to problems that have occurred. It also includes testing of tools and~analysis of acquired data.460 - Katedra informatikyvelmi dobř

Shortest path embeddings of graphs on surfaces

The classical theorem of F\'{a}ry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of F\'{a}ry's theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest path between its two endpoints and we call an embedding with this property a shortest path embedding. The main question addressed in this paper is whether given a closed surface S, there exists a Riemannian metric for which every topologically embeddable graph admits a shortest path embedding. This question is also motivated by various problems regarding crossing numbers on surfaces. We observe that the round metrics on the sphere and the projective plane have this property. We provide flat metrics on the torus and the Klein bottle which also have this property. Then we show that for the unit square flat metric on the Klein bottle there exists a graph without shortest path embeddings. We show, moreover, that for large g, there exist graphs G embeddable into the orientable surface of genus g, such that with large probability a random hyperbolic metric does not admit a shortest path embedding of G, where the probability measure is proportional to the Weil-Petersson volume on moduli space. Finally, we construct a hyperbolic metric on every orientable surface S of genus g, such that every graph embeddable into S can be embedded so that every edge is a concatenation of at most O(g) shortest paths.Comment: 22 pages, 11 figures: Version 3 is updated after comments of reviewer

Even maps, the Colin de~Verdi\`ere number and representations of graphs

Van der Holst and Pendavingh introduced a graph parameter $\sigma$, which coincides with the more famous Colin de Verdi\`{e}re graph parameter $\mu$ for small values. However, the definition of $\sigma$ is much more geometric/topological directly reflecting embeddability properties of the graph. They proved $\mu(G) \leq \sigma(G) + 2$ and conjectured $\mu(G) \leq \sigma(G)$ for any graph $G$. We confirm this conjecture. As far as we know, this is the first topological upper bound on $\mu(G)$ which is, in general, tight. Equality between $\mu$ and $\sigma$ does not hold in general as van der Holst and Pendavingh showed that there is a graph $G$ with $\mu(G) \leq 18$ and $\sigma(G)\geq 20$. We show that the gap appears on much smaller values, namely, we exhibit a graph $H$ for which $\mu(H)\leq 7$ and $\sigma(H)\geq 8$. We also prove that, in general, the gap can be large: The incidence graphs $H_q$ of finite projective planes of order $q$ satisfy $\mu(H_q) \in O(q^{3/2})$ and $\sigma(H_q) \geq q^2$.Comment: 28 pages, 4 figures. In v2 we slightly changed one of the core definitions (previously "extended representation" now "semivalid representation"). We also use it to introduce a new graph parameter, denoted eta, which did not appear in v1. It allows us to establish an extended version of the main result showing that mu(G) is at most eta(G) which is at most sigma(G) for every graph

USPOREDBA FRONT END FRAMEWORKA ZA IZRADU WEB-APLIKACIJA

Modern web applications, due to the functionalities they provide in their user interfaces, have a complex program structure. Manually writing a program code, due to the complexity of the entire application, can result in uneven quality and content of individual application parts. Maintaining such developed applications is more difficult. Because of this, web applications are often developed by using different frameworks. A framework allows structuring, simpler and more uniform program script writing, and thus easier web application maintenance. There are various frameworks that can be used in the development of web applications, for different parts of the application. Those analyzed in this paper are used in the development of front end parts of web applications. According to their design, a web application can be developed as the Multi Page (MPA) or the Single Page (SPA). This paper explains the difference between MPA and SPA web applications. The advantages and disadvantages of MPA are demonstrated in relation to SPA web applications. Required characteristics that the framework should have in order to be optimized for creating MPA and SPA web applications are set. The hypothesis has been tested: There is a framework that is optimized for the development of both MPA and SPA applications. Possibilities, architecture and development techniques of a web application using front end frameworks, as well as the suitability of such frameworks for the development of MPA and SPA web applications have been analysed. Choosing a framework for the hypothesis testing has been performed based on the popularity of available frameworks. The required characteristics have been analyzed on the three most popular frameworks: Angular, Vue.js and React-js. It has been shown that the Vue.js framework is the most optimized framework for the development of both MPA and SPA applications.Moderne web-aplikacije, zbog funkcionalnosti koje omogućuju u korisničkom sučelju, imaju složenu programsku strukturu. Ručno pisanje programskog koda zbog složenosti cijele aplikacije može rezultirati neujednačenom kvalitetom i sadržajem pojedinih aplikacijskih djelova. Održavanje tako razvijanih aplikacija otežano je. Zbog toga se web-aplikacije često razvijaju korištenjem različitih frameworka. Framework omogućuje strukturiranje, jednostavnije i ujednačenije pisanje programskog koda, te time olakšava održavanje web-aplikacije. Postoji puno frameworka koji se mogu koristiti u razvoju web-aplikacija, i to za različite dijelove aplikacije, a oni analizirani u ovom radu koriste se u razvoju front end dijela web-aplikacije. Prema načinu izvođenja webaplikacije mogu biti Multi Page (MPA) ili Single Page (SPA). U radu je objašnjena različitost između MPA i SPA web-aplikacija. Pokazane su prednosti i nedostatci MPA u odnosu na SPA web-aplikacije. Postavljene su zahtijevane karakteristike frameworka koji je optimiziran za izradu MPA i SPA web-aplikacija. Testirana je hipoteza: Postoji framework koji je prilagođen za izradu i MPA i SPA aplikacija. Analizirane su mogućnosti, arhitektura i načini razvoja web-aplikacija pomoću front end frameworka te prilagođenost takvih frameworka za razvoj MPA i SPA web-aplikacija. Izvršen je odabir frameworka za testiranje hipoteze prema popularnosti na tržištu. Zahtijevane karakteristike analizirane su na 3 najpopularnija frameworka: Angular, Vue.js i React-js. Pokazano je da je Vue.js framework najoptimiziraniji framework za izradu i MPA i SPA aplikacija

The Black Death\u27s influence on the Iconographic Motif of the Death Personification in the Painting of the Tuscan trecento

Toskanska personifikacija smrti v trecentu v splošnem velja za prvi ikonografski motiv v upodabljajoči umetnosti zahodnega sveta, ki smrt občinstvu predstavi v oprijemljivi, človeku podobni obliki. Oborožena in predstavljena v različnih fazah razkroja, srh vzbujajoča prikazen tudi po skoraj sedmih stoletjih v gledalcu sproža čustven odziv. Njeno jasno izražana sporočila o minljivosti človeka in tesnobah ob čakanju na njen prihod, se, žal zgolj navidezno, dobro ujemajo z aktualnimi historiografskimi razpravami o črni smrti, ki s pomočjo demografskih in epidemioloških prijemov, v posameznih regijah kažejo na kar 50% smrtnost, kar je mnogo več od ustaljenih ocen (20-30%). Pojav makabristične ikonografije je v starejši literaturi složno pripisan črni smrti, omenjene raziskave pa utegnejo nekatera zakoreninjena prepričanja o izvorih personificirane smrti v prvi epidemiji kuge še okrepiti. S pomočjo ohranjenih primerov prvih upodobitev personificirane smrti bomo skušali pojasniti, da črna smrt zagotovo ni bila odločilni faktor pri nastanku obravnavanega ikonografskega motiva, saj se je ta pojavil že pred prihodom epidemije, sicer v izoliranem samostanskem okolju. Črna smrt pri proučevanju ikonografskega motiva personificirane smrti ni nepomembna, saj je omogočila nadvse gladek prehod motiva iz kroga samostanske askeze v širše, tudi laične množice, ki so se odtlej v njem prepoznale. Zanimalo nas bo, kako je motiv v desetletjih pred kugo nastal ter, kako je epidemija s posegom v ustaljene navade, povezane s smrtjo in umiranjem, vanj posegla in ga spreminjala. Pri tem nam bodo pomagale tako demografske in epidemiološke študije, kot tudi sočasni literarni in kronistični viri.The Tuscan death personification of the trecento is generally acknowledged as the very first iconographic motif in the pictorial arts of the West to present the audiences with a figure of Death in a more concrete, humanlike form. Armed and depicted in various stages of decomposition, the ghastly apparition triggers the viewer\u27s emotional response even after almost seven centuries. Her rather clearly expressed messages regarding the vanity of man and the anxieties upon awaiting her arrival are, unfortunately only at first glance, well paired with extant historiographical theses on the Black Death, which, with the help of demographical and epidemiological approaches, reveal certain regions\u27 striking mortality of over 50%, a much higher percentage than the usual assessments give (2030%). In older literature, the appearance of macabre iconography is consensually attributed to the Black Death, which is essentially one of many conservative beliefs about the roots of death personifications, the aforementioned theses tend to emphasize even more. With the help of the remaining examples depicting death personified, we will try to explain how the Black Death itself definitely was not the decisive factor in creating the iconographic motif, for it has appeared more than a decade before the arrival of the first epidemic, although isolated behind monastery walls. The Black Death is not unimportant when studying the iconographic motif of the death personification, for it was the very plague that made a smooth transition of the motif from ascetic ecclesiastic circles to wider, even lay audiences, possible, since they too have started identifying with it. We will focus on how the motif started appearing in the decades before the plague and the ways in which the epidemic had changed it, while at the same time changing core beliefs and rituals surrounding death and dying. Help will be drawn from both demographic or epidemiologic studies and contemporary literary texts and chronicles alike