196 research outputs found

    Neighborhood Variants of the KKM Lemma, Lebesgue Covering Theorem, and Sperner's Lemma on the Cube

    Full text link
    We establish a "neighborhood" variant of the cubical KKM lemma and the Lebesgue covering theorem and deduce a discretized version which is a "neighborhood" variant of Sperner's lemma on the cube. The main result is the following: for any coloring of the unit dd-cube [0,1]d[0,1]^d in which points on opposite faces must be given different colors, and for any ε>0\varepsilon>0, there is an \ell_\infty ε\varepsilon-ball which contains points of at least (1+ε1+ε)d(1+\frac{\varepsilon}{1+\varepsilon})^d different colors, (so in particular, at least (1+23ε)d(1+\frac{2}{3}\varepsilon)^d different colors for all sensible ε(0,12]\varepsilon\in(0,\frac12]).Comment: 18 pages plus appendices (30 pages total), 3 figure

    Wait-free approximate agreement on graphs

    No full text
    Approximate agreement is one of the few variants of consensus that can be solved in a wait-free manner in asynchronous systems where processes communicate by reading and writing to shared memory. In this work, we consider a natural generalisation of approximate agreement on arbitrary undirected connected graphs. Each process is given a node of the graph as input and, if non-faulty, must output a node such that – all the outputs are within distance 1 of one another, and – each output value lies on a shortest path between two input values. From prior work, it is known that there is no wait-free algorithm among processes for this problem on any cycle of length , by reduction from 2-set agreement (Castañeda et al., 2018). In this work, we investigate the solvability of this task on general graphs. We give a new, direct proof of the impossibility of approximate agreement on cycles of length , via a generalisation of Sperner's Lemma to convex polygons. We also extend the reduction from 2-set agreement to a larger class of graphs, showing that approximate agreement on these graphs is unsolvable. On the positive side, we present a wait-free algorithm for a different class of graphs, which properly contains the class of chordal graphs

    MATHEMATICS OF HUNG-PING TSAO II: BUSINESS MATHEMATICS

    Get PDF
    Tsao, Hung-ping (2023). Mathematics of Hung-Ping Tsao II: Business Mathematics.In: "Evolutionary Progress in Science, Technology, Engineering, Arts, and Mathematics (STEAM)", Wang, Lawrence K. and Tsao, Hung-ping (editors). 5 (4), April ; 238 p. Lenox Institute Press, MA, USA ..... ABSTRACT: During my seventeen years (1985-2002) of teaching at College of Business, San Francisco State University, the tailor-made textbook “College Mathematics” for my classes has been out of print for over twenty years now. I would like to share its contents, except for problem sections, with readers who might benefit from quite a few innovative ideas in pedagogical perspectives. The purpose of this sequel of “MATHEMATICS OF HUNG-PING TSAO” (see the link h-tsao-2020-hcommons.org) is to share, retrospectively, with readers the stem of my teaching resources. I would especially like to refresh my Kung-fu analogy of those differentiation rules, new approach in solving optimization problems in calculus and the cross-multiplication method for solving linear programming problems. All in all, my doctoral dissertation “Some Extremal Problems in Ordered Structures” played an important role in my tortuous career, Part I and Part II of which were published more than ten years after their submissions to the Journal of Discrete Mathematics. I was mistreated by the Journal of Discrete Mathematics when Daniel J. Kleitman was the Chief Editor. To support my claim, I present in the end a rejection letter from Daniel J. Kleitman in 1975 with a referee’s comments, contrary to what the reviewer of Mathematical Review said about Part I of my doctoral dissertation. Partly because of my frustration, I pursued eight years of actuarial career, for which I have no remorse. As a matter of fact, I had benefited a lot from it. In “MATHEMATICS OF HUNG-PING TSAO” , I have included many tidbits in Actuarial Mathematics that I previously published in Transactions and ARCH of the Society of Actuaries

    Geometry of Rounding: Near Optimal Bounds and a New Neighborhood Sperner's Lemma

    Full text link
    A partition P\mathcal{P} of Rd\mathbb{R}^d is called a (k,ε)(k,\varepsilon)-secluded partition if, for every pRd\vec{p} \in \mathbb{R}^d, the ball B(ε,p)\overline{B}_{\infty}(\varepsilon, \vec{p}) intersects at most kk members of P\mathcal{P}. A goal in designing such secluded partitions is to minimize kk while making ε\varepsilon as large as possible. This partition problem has connections to a diverse range of topics, including deterministic rounding schemes, pseudodeterminism, replicability, as well as Sperner/KKM-type results. In this work, we establish near-optimal relationships between kk and ε\varepsilon. We show that, for any bounded measure partitions and for any d1d\geq 1, it must be that k(1+2ε)dk\geq(1+2\varepsilon)^d. Thus, when k=k(d)k=k(d) is restricted to poly(d){\rm poly}(d), it follows that ε=ε(d)O(lndd)\varepsilon=\varepsilon(d)\in O\left(\frac{\ln d}{d}\right). This bound is tight up to log factors, as it is known that there exist secluded partitions with k(d)=d+1k(d)=d+1 and ε(d)=12d\varepsilon(d)=\frac{1}{2d}. We also provide new constructions of secluded partitions that work for a broad spectrum of k(d)k(d) and ε(d)\varepsilon(d) parameters. Specifically, we prove that, for any f:NNf:\mathbb{N}\rightarrow\mathbb{N}, there is a secluded partition with k(d)=(f(d)+1)df(d)k(d)=(f(d)+1)^{\lceil\frac{d}{f(d)}\rceil} and ε(d)=12f(d)\varepsilon(d)=\frac{1}{2f(d)}. These new partitions are optimal up to O(logd)O(\log d) factors for various choices of k(d)k(d) and ε(d)\varepsilon(d). Based on the lower bound result, we establish a new neighborhood version of Sperner's lemma over hypercubes, which is of independent interest. In addition, we prove a no-free-lunch theorem about the limitations of rounding schemes in the context of pseudodeterministic/replicable algorithms

    Almost envy-free allocations with connected bundles

    No full text
    We study the existence of allocations of indivisible goods that are envy-free up to one good (EF1), under the additional constraint that each bundle needs to be connected in an underlying item graph. If the graph is a path and the utility functions are monotonic over bundles, we show the existence of EF1 allocations for at most four agents, and the existence of EF2 allocations for any number of agents; our proofs involve discrete analogues of the Stromquist's moving-knife protocol and the Su–Simmons argument based on Sperner's lemma. For identical utilities, we provide a polynomial-time algorithm that computes an EF1 allocation for any number of agents. For the case of two agents, we characterize the class of graphs that guarantee the existence of EF1 allocations as those whose biconnected components are arranged in a path; this property can be checked in linear time

    O grao topolóxico de Leray-Schauder e aplicacións ás ecuacións diferenciais

    Get PDF
    Traballo Fin de Grao en Matemáticas. Curso 2021-2022Na primeira década do século XX comezan os primeiros estudos sobre o grao topolóxico, unha ferramenta de utilidade na topoloxía alxébrica e na análise funcional non linear. No presente traballo, desenvolvemos unha pormenorizada introdución á teoría do grao. Partindo de espazos de dimensión finita, definimos o grao de Brouwer para funcións continuamente diferenciables e, posteriormente, para funcións continuas. Presentamos as propiedades máis importantes do grao, que nos permiten, entre outras aplicacións, garantir a existencia de solución dunha ecuación dada. Partindo do grao, probamos teoremas clásicos como o de punto fixo de Brouwer ou o da bóla peluda. En espazos de dimensión infinita, os resultados relativos ó grao de Brouwer non son certos, en xeral. Por este motivo, cómpre redefinir o grao, coa limitación de facelo para unha clase máis restritiva de funcións, as perturbacións compactas da identidade. Construímos, partindo do grao de Brouwer, o grao de Leray-Schauder. Destacamos algunhas das propiedades máis interesantes e xeneralizamos os teoremas de punto fixo para espazos de dimensión infinita. Existen aplicacións da teoría do grao en moitos eidos das matemáticas. Facendo uso do Teorema de punto fixo de Schauder e das propiedades do grao, probamos a existencia de solución local dun problema de valor inicial e a conexidade do seu espazo de soluciónsIn the first decade of the twentieth century began the first studies on the topological degree, a useful tool in algebraic topology and in nonlinear functional analysis. In the present project, we have developed a detailed introduction to the degree theory. Starting from finite dimensional spaces, we define the Brouwer degree for continuously diferentiable functions and subsequently for continuous functions. We present the most important properties of the degree, which allow us, among other applications, to ensure existence of solution of a given equation. Based on the degree, we prove several classic theorems such us the Brouwer fixed point or the one of the hairy ball. In infinite dimensional spaces, the results relating to the Brouwer degree are not true, in general. Therefore it is necessary to redefine the degree, with the limitation of doing so for a more restrictive class of functions, the compact perturbations of the identity. Using the Brouwer degree, we build the Leray-Schauder degree. We highlight some of the most interesting properties and we generalize the xed point theorems for in nite dimensional spaces. Degree theory can be applied in many mathematical elds. We prove, using the Schauder xed point theorem and the degree properties, the existence of local solutions of an initial value problem and the connection of its solution se

    From Points to Potlucks: An Exploration of Fixed Point Theorems with Applications to Game Theory Models of Successful Integration Practices

    Get PDF
    Potlucks have many names: shared community dinners, faith suppers, “bring-a-dish” dinners, etc. They represent the desire to share food with other people and make new friends, sometimes learning about other cultures in the process. Not only does one have to decide what dish to bring, but one must also decide how large of a dish, if there will be a theme, and what course it will fit. For instance, if everyone brings side dishes, there will not be enough food for everyone, and if someone brings food that most of the group cannot eat, then feelings will be hurt on all sides. And in a way, having a potluck is similar to creating integration policies. Successful integration policies are fair to all people and take a “two-street” approach, while simultaneously being a collaborative affair. This paper will first explore fixed point theory, including the Kakutani Fixed Point Theorem and Brouwer Fixed Point Theorem; fixed point theorems are a significant field of mathematics and have many well-known applications. One of these applications is game theory, which is the study of how rational actors make decisions in everyday situations. Building upon the mathematical aspects of the first few chapters and the basics of game theory, this paper aims to build its own game theory model called the “Potluck Metaphor” that will model several methods of integration in the European Union; context for the model will be provided by critiquing three primary integration models and a brief literature review of the related field. Starting off with a simple game theory model for a dinner party, this paper will then slowly expand these models to show their applicability to European integration policy on an organizational level and on a member-specific level

    Teoremas de Jordan y Brouwer: demostraciones elementales

    Full text link
    Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Ignasi Mundet i Riera[en] In mathematics, an elementary proof is one that uses only basic techniques. In this work we provide the elementary proof of the Jordan curve at R2\mathbb{R}^2 and the Brouwer fixed point theorems. The Jordan curve theorem on R2\mathbb{R}^2 tells us that every simple closed curve separates the plane into two connected components. The elementary proof uses four lemmas, whose proofs we also carry out, and consists of approximating the curve by means of polygons. Let BnRn\mathbb{B}^n \subset \mathbb{R}^n be the n-dimensional unit ball. Brouwer's fixed point theorem tells us that every continuous function f:BnBnf: \mathbb{B}^n \rightarrow \mathbb{B}^n has a fixed point. The elementary proof is based on the fact that the compact, convex and non-empty subsets of Rn\mathbb{R}^n are homeomorphic and on Sperner's Lemma, which we also state and prove. Sperner's Lemma is a combinatorial result of coloring n-dimensional simplices

    Kaksoislaskenta

    Get PDF
    Tiivistelmä. Tämän pro gradu -tutkielman tarkoituksena on esitellä kaksoislaskentaa diskreetin matematiikan näkökulmasta. Lyhyesti määriteltynä kaksoislaskenta tarkoittaa laskutapaa, jossa lasketaan samat joukot kahdella eri tavalla ja lopputulos on yhtäsuuri. Kaksoislaskennan perusajatus tiivistyy hyvin esimerkkiin, jossa äärellinen määrä ihmisiä tapaa juhlissa ja jotkut heistä kättelevät, mutta kukaan ei itseään eikä ketään kahdesti. Tällöin niiden ihmisten määrä, jotka kättelevät parittoman määrän muita ihmisiä, on aina parillinen. Tästä esimerkistä saadaan Kättelylemma, joka todistetaan kappaleessa 2.3. Tutkielmassa tarkastellaan kaksoislaskennasta kahta mielenkiintoista ongelmaa. Ensimmäisenä tutkitaan kiintopisteen olemassaoloa. Monista kiintopistelauseista paneudutaan Brouwerin kiintopistelauseeseen tasossa ja havannoidaan sen käyttöä muun muassa peliteoriassa. Toinen kiinnostava ongelma on, kuinka suuria tai pieniä joukot voivat olla jollain tietyillä ehdoilla. Tällaisten ongelmien ratkaiseminen antaa lopputuloksena hyödyllisiä ylä- tai alarajoja esimerkiksi erilaisten verkkojen koolle. Tutkielma pohjautuu teokseen Matousek, Nesetril: Invitation to discrete mathematics. Oxford OUP, Oxford, 2008. Tämä teos on lähteenä ellei toisin mainita. Tutkielman ymmärtämiseksi oletetaan lukijalta perustietoja yliopistotason matematiikasta. Verkkoteoriaa käsittelevä kappale on tiivistetty suoraan kandidaatin työstäni, jossa on sama lähdeteos
    corecore