scholarly journals Colorings and Nowhere-Zero Flows of Graphs in Terms of Berlekamp's Switching Game

10.37236/552 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Normal Forms ◽  
Oriented Graph ◽  
Vertex Coloring ◽  
Cycle Space ◽  
The Core ◽  
Light Pattern ◽  
Special Cases ◽  
Axis Parallel ◽  
Switching Game ◽  
Zero Points

We work with a unifying linear algebra formulation for nowhere-zero flows and colorings of graphs and matrices. Given a subspace (code) $U\leq{\mathbb{Z}_k^n}$ – e.g. the bond or the cycle space over ${\mathbb{Z}}_k$ of an oriented graph – we call a nowhere-zero tuple $f\in{\mathbb{Z}_k^n}$ a flow of $U$ if $f$ is orthogonal to $U$. In order to detect flows, we view the subspace $U$ as a light pattern on the $n$-dimensional Berlekamp Board ${\mathbb{Z}_k^n}$ with $k^n$ light bulbs. The lights corresponding to elements of $U$ are ON, the others are OFF. Then we allow axis-parallel switches of complete rows, columns, etc. The core result of this paper is that the subspace $U$ has a flow if and only if the light pattern $U$ cannot be switched off. In particular, a graph $G$ has a nowhere-zero $k$-flow if and only if the ${\mathbb{Z}}_k$-bond space of $G$ cannot be switched off. It has a vertex coloring with $k$ colors if and only if a certain corresponding code over ${\mathbb{Z}}_k$ cannot be switched off. Similar statements hold for Tait colorings, and for nowhere-zero points of matrices. Studying different normal forms to equivalence classes of light patterns, we find various new equivalents, e.g., for the Four Color Problem, Tutte's Flow Conjectures and Jaeger's Conjecture. Two of our equivalents for colorability and existence of nowhere zero flows of graphs include as special cases results by Matiyasevich, by Balázs Szegedy, and by Onn. Alon and Tarsi's sufficient condition for $k$-colorability also arrives, remarkably, as a generalized full equivalent.

scholarly journals Anti-Codes in Terms of Berlekamp's Switching Game

10.37236/17 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Projective Space ◽  
Linear Code ◽  
Coding Theory ◽  
Normal Forms ◽  
Light Pattern ◽  
Maximal Weight ◽  
Axis Parallel ◽  
Generalized Games ◽  
Switching Game ◽  
Zero Points

We view a linear code (subspace) $C\leq\mathbb{F}_{q}^n$ as a light pattern on the \(n\)-dimensional Berlekamp Board $\mathbb{F}_{q}^n$ with $q^n$ light bulbs. The lights corresponding to elements of $C$ are ON, the others are OFF. Then we allow axis-parallel switches of complete rows, columns, etc. We show that the dual code $C^\perp$ contains a vector $v$ of full weight, i.e. $v_1,v_2,\dots,v_n\neq0$, if and only if the light pattern $C$ cannot be switched off. Generalizations of this allow us to describe anti-codes with maximal weight $\delta$ in a similar way, or, alternatively, in terms of a switching game in projective space. We provide convenient bases and normal forms to the modules of all light patterns of the generalized games. All our proofs are purely combinatorial and simpler than the algebraic ones used for similar results about anti-codes in $\mathbb{Z}_k^n$.  Aside from coding theory, the game is also of interest in the study of nowhere-zero points of matrices and nowhere-zero flows and colorings of graphs.

Computing Cores for Existential Rules with the Standard Chase and ASP

2020 ◽  
Author(s):  
Markus Krötzsch
Keyword(s):  
Large Class ◽  
Data Exchange ◽  
Answer Set Programming ◽  
Query Answering ◽  
Universal Model ◽  
The Core ◽  
Universal Models ◽  
General Terms ◽  
Special Cases ◽  
The Many

To reason with existential rules (a.k.a. tuple-generating dependencies), one often computes universal models. Among the many such models of different structure and cardinality, the core is arguably the “best”. Especially for finitely satisfiable theories, where the core is the unique smallest universal model, it has advantages in query answering, non-monotonic reasoning, and data exchange. Unfortunately, computing cores is difficult and not supported by most reasoners. We therefore propose ways of computing cores using practically implemented methods from rule reasoning and answer set programming. Our focus is on cases where the standard chase algorithm produces a core. We characterise this desirable situation in general terms that apply to a large class of cores, derive concrete approaches for decidable special cases, and generalise these approaches to non-monotonic extensions of existential rules.

ON TWO THEOREMS OF QUINZII AND RENT CONTROLLED HOUSING ALLOCATION IN SWEDEN

2007 ◽  
Vol 09 (03) ◽  
pp. 515-525
Author(s):  
KIMMO ERIKSSON ◽  
JONAS SJÖSTRAND
Keyword(s):  
Control System ◽  
Empirical Data ◽  
General Theorem ◽  
Black Market ◽  
Utility Functions ◽  
Rent Control ◽  
Black And White ◽  
The Core ◽  
Special Cases ◽  
Housing Allocation

The Swedish rent control system creates a white market for swapping rental contracts and a black market for selling rental contracts. Empirical data suggests that in this black-and-white market some people act according to utility functions that are both discontinuous and locally decreasing in money. We discuss Quinzii's theorem for the nonemptiness of the core of generalized house-swapping games, and show how it can be extended to cover the Swedish game. In a second part, we show how this theorem of Quinzii and her second theorem on nonemptiness of the core in two-sided models are both special cases of a more general theorem.

scholarly journals Probability of infectious disease in humans during epidemic

2021 ◽  
Author(s):  
AM Karmishin ◽  
IV Borisevich ◽  
VI Skvortsova ◽  
AA Goryaev ◽  
SM Yudin
Keyword(s):  
Infectious Disease ◽  
Biological Object ◽  
Epidemiological Modeling ◽  
Pathogen Virulence ◽  
The Core ◽  
Special Cases ◽  
Quantitative Characteristics ◽  
Infective Dose ◽  
The Law ◽  
Generalized Time

Popular SIR models and their modifications used to generate predictions about epidemics and, specifically, the COVID-19 pandemic, are inadequate. The aim of this study was to find the laws describing the probability of infection in a biological object. Using theoretical methods of research based on the probability theory, we constructed the laws describing the probability of infection in a human depending on the infective dose and considering the temporal characteristics of a given infection. The so-called generalized time-factor law, which factors in the time of onset and the duration of an infectious disease, was found to be the most general. Among its special cases are the law describing the probability of infection developing by some point in time t, depending on the infective dose, and the law that does not factor in the time of onset. The study produced a full list of quantitative characteristics of pathogen virulence. The laws described in the study help to solve practical tasks and should lie at the core of mathematical epidemiological modeling.

Effect of Cell Geometry on the Shear Modulus and on Density of Sandwich Panel Cores

1961 ◽  
Vol 83 (4) ◽  
pp. 513-517 ◽  
Author(s):  
Chieh C. Chang ◽  
Ibrahim K. Ebcioglu
Keyword(s):  
Aspect Ratio ◽  
Shear Modulus ◽  
Sandwich Panel ◽  
Analytic Theory ◽  
Cell Geometry ◽  
Cell Weight ◽  
The Core ◽  
Special Cases ◽  
Square Cells ◽  
Sandwich Core

A simple analytic theory for the effect of cell geometry on both the shear modulus and the density of sandwich panel core is presented. The core shear modulus in different directions is analyzed to include the effects of the angle α and the aspect ratio b/a of the cell. It is also found that the minimum cell weight of the sandwich core depends both on the cell angle α and the cell aspect ratio b/a. The theory compares fairly well with some available experiments. The cell geometry chosen is so general that the regular hexagonal and square cells of commercial sandwich cores are special cases.

Singularities of dispersion relations

1983 ◽  
Vol 95 (3-4) ◽  
pp. 301-316 ◽  
Author(s):  
G. Dangelmayr
Keyword(s):  
Catastrophe Theory ◽  
Bifurcation Theory ◽  
Normal Forms ◽  
Dispersion Relations ◽  
Bifurcation Parameter ◽  
Special Cases ◽  
Wave Numbers ◽  
Imperfect Bifurcation ◽  
Generic Singularities ◽  
Zero Frequency

SynopsisGeneric singularities occurring in dispersion relations are discussed within the framework of imperfect bifurcation theory and classified up to codimension four. Wave numbers are considered as bifurcation variables x =(x1,…, xn) and the frequency is regarded as a distinguished bifurcation parameter λ. The list of normal forms contains, as special cases, germs of the form ±λ +f(x), where f is a standard singularity in the sense of catastrophe theory. Since many dispersion relations are ℤ(2)-equivariant with respect to the frequency, bifurcation equations which are ℤ(2)-equivariant with respect to the bifurcation parameter are introduced and classified up to codimension four in order to describe generic singularities which occur at zero frequency. Physical implications of the theory are outlined.

THE CORE AND CONSISTENCY PROPERTIES: A GENERAL CHARACTERISATION

2001 ◽  
Vol 03 (02n03) ◽  
pp. 175-187 ◽  
Author(s):  
YUKIHIKO FUNAKI ◽  
TAKEHIKO YAMATO
Keyword(s):  
General Theorem ◽  
General Definition ◽  
Tu Games ◽  
The Core ◽  
Special Cases ◽  
Reduced Games ◽  
Definition Of ◽  
Consistency Properties

In this paper, we unify various axiomatisations of the core of TU games by means of consistency with respect to different definitions of reduced games. First, we introduce a general definition of reduced games including the reduced games due to Davis and Maschler (1965), Moulin (1985), and Funaki (1995) as special cases. Then, we provide a general theorem from which the characterisations due to Peleg (1986), Tadenuma (1992), and Funaki (1995) can be obtained. Our general theorem clarifies how the three characterisations of the core differ and are related.

Indexed containers

2015 ◽  
Vol 25 ◽  
Author(s):  
THORSTEN ALTENKIRCH ◽  
NEIL GHANI ◽  
PETER HANCOCK ◽  
CONOR MCBRIDE ◽  
PETER MORRIS
Keyword(s):  
Type Theory ◽  
Normal Forms ◽  
Fixed Number ◽  
The Novel ◽  
The Core

AbstractWe show that the syntactically rich notion of strictly positive families can be reduced to a core type theory with a fixed number of type constructors exploiting the novel notion of indexed containers. As a result, we show indexed containers provide normal forms for strictly positive families in much the same way that containers provide normal forms for strictly positive types. Interestingly, this step from containers to indexed containers is achieved without having to extend the core type theory. Most of the construction presented here has been formalized using the Agda system.

scholarly journals Some New Characterizations of Graph Colorability and of Blocking Sets of Projective Spaces

10.37236/3767 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Uwe Schauz
Keyword(s):  
Prime Power ◽  
Projective Spaces ◽  
Vertex Coloring ◽  
General Level ◽  
Blocking Sets ◽  
Proper Vertex ◽  
Switching Game

Let $G=(V,E)$ be a graph and $q$ be an odd prime power. We prove that $G$ possess a proper vertex coloring with $q$ colors if and only if there exists an odd vertex labeling $x\in F_q^V$ of $G$. Here, $x$ is called odd if there is an odd number of partitions $\pi=\{V_1,V_2,\dotsc,V_t\}$ of $V$ whose blocks $V_i$ are \(G\)-bipartite and \(x\)-balanced, i.e., such that $G|_{V_i}$ is connected and bipartite, and $\sum_{v\in V_i}x_v=0$. Other new characterizations concern edge colorability of graphs and, on a more general level, blocking sets of projective spaces. Some of these characterizations are formulated in terms of a new switching game.

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