Coloring random graphs online without creating monochromatic subgraphs

Consider the following random process: The vertices of a binomial random graph $G_{n,p}$ are revealed one by one, and at each step only the edges induced by the already revealed vertices are visible. Our goal is to assign to each vertex one from a fixed number $r$ of available colors immediately and irrevocably without creating a monochromatic copy of some fixed graph $F$ in the process. Our first main result is that for any $F$ and $r$, the threshold function for this problem is given by $p_0(F,r,n)=n^{-1/m_1^*(F,r)}$, where $m_1^*(F,r)$ denotes the so-called \emph{online vertex-Ramsey density} of $F$ and $r$. This parameter is defined via a purely deterministic two-player game, in which the random process is replaced by an adversary that is subject to certain restrictions inherited from the random setting. Our second main result states that for any $F$ and $r$, the online vertex-Ramsey density $m_1^*(F,r)$ is a computable rational number. Our lower bound proof is algorithmic, i.e., we obtain polynomial-time online algorithms that succeed in coloring $G_{n,p}$ as desired with probability $1-o(1)$ for any $p(n) = o(n^{-1/m_1^*(F,r)})$.Comment: some minor addition

Investigation of Electrochemistry of High Energy Compounds in Organic Electrolytes. Bibliography on Propylene Carbonate, Gamma- Butyrolactone, and Related Subjects Supplement to Third Progress Report

Bibliography on electrochemistry of high energy compounds in organic electrolyte

An Exploration of the Effects of Collectivism and Individualism on Maintaining Societal Cohesiveness and Encouraging Cross-Societal International Relations by the Review of a Sample of Societial Mechanics

A functional society is a society that has found its balance between the extremes of 100% Individualism (Chaos) and 100% Collectivism (Stagnation). Thorough exploration of the influences that the institutions of Society, Family, Education, Work Environment and Religion have on the members of a particular society allows for an understanding of the mechanics which work to shape and influence societies from generation to generation. Following a review of multiple pieces of literature and personal interactions and observations in select societies (China, Czech Republic, Greece and Sweden) the extrapolation can be made that there are two distinct societal types, collectivist and individualist. Although it is often speculated that a society must utilize one extreme or the other, in reality societies try to balance between the extremes of stagnation and chaos as a means of self preservation. The institutions used by societies to enforce compliance among its membership are essentially universal; however, the methods by which societies choose to maintain cohesiveness differ from society to society and are influenced by changes in technology and information sharing on a global scale. These factors serve to explain why degrees of both individualism and collectivism can be found in each functional society currently in existence. The balance of these two societal extremes allows a society to optimally function and maintain harmony. This balance is by no means stagnant. Societies constantly struggle toward one extreme or the other. Functional societies find themselves being drawn back toward the middle over time, with no society every truly obtaining a perfect equilibrium. An understanding of the dynamics at work in this cycle enhances our ability to function in our own society and interact with other societies on an international scale

What Has Happened Since Chakrabarty

It is conventional wisdom that the patent system is designed to undergrid the investment in pushing technology forward. The patent system is innovation-oriented. And (sic) it functions most effectively in the expensive, breakthrough technologies, where uncertainties of success or payback abound. If, in assessing the risk of commitment, the penalties of failure outweigh the prizes of success, the prudent money will go elsewhere. The patent system moves the equation to the right, not by better assuring success (for only public needs and market values can do that), but by aiding success through offering the innovator a temporary respite from non-innovative copying. However faulted and flawed our system might be, only the most compelling legal logic should deny this country its benefits in a nascent technology full of promise for so much good

Online vertex-coloring games in random graphs

Consider the following one-player game. The vertices of a random graph on n vertices are revealed to the player one by one. In each step, also all edges connecting the newly revealed vertex to preceding vertices are revealed. The player has a fixed number of colors at her disposal, and has to assign one of these to each vertex immediately. However, she is not allowed to create any monochromatic copy of some fixed graph F in the process. For n → ∞, we study how the limiting probability that the player can color all n vertices in this online fashion depends on the edge density of the underlying random graph. For a large family of graphs F, including cliques and cycles of arbitrary size, and any fixed number of colors, we establish explicit threshold functions for this edge density. In particular, we show that the order of magnitude of these threshold functions depends on the number of colors, which is in contrast to the corresponding offline coloring proble

Upper Bounds for Online Ramsey Games in Random Graphs

Consider the following one-player game. Starting with the empty graph on n vertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one of r available colours. The player's goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. We prove an upper bound on the typical duration of this game if F is from a large class of graphs including cliques and cycles of arbitrary size. Together with lower bounds published elsewhere, explicit threshold functions follo

Online Ramsey Games in Random Graphs

Consider the following one-player game. Starting with the empty graph on n vertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one of r available colours. The player's goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. We prove a lower bound of nβ(F,r) on the typical duration of this game, where β(F,r) is a function that is strictly increasing in r and satisfies limr→∞ β(F,r) = 2 − 1/m2(F), where n2−1/m2(F) is the threshold of the corresponding offline colouring proble

Candida albicans Possesses Sap7 as a Pepstatin A-Insensitive Secreted Aspartic Protease

BACKGROUND: Candida albicans, a commensal organism, is a part of the normal flora of healthy individuals. However, once the host immunity is compromised, C. albicans opportunistically causes recurrent superficial or fatal systemic candidiasis. Secreted aspartic proteases (Sap), encoded by 10 types of SAP genes, have been suggested to contribute to various virulence processes. Thus, it is important to elucidate their biochemical properties for better understanding of the molecular mechanisms that how Sap isozymes damage host tissues. METHODOLOGY/PRINCIPAL FINDINGS: The SAP7 gene was cloned from C. albicans SC5314 and heterogeneously produced by Pichia pastoris. Measurement of Sap7 proteolytic activity using the FRETS-25Ala library showed that Sap7 was a pepstatin A-insensitive protease. To understand why Sap7 was insensitive to pepstatin A, alanine substitution mutants of Sap7 were constructed. We found that M242A and T467A mutants had normal proteolytic activity and sensitivity to pepstatin A. M242 and T467 were located in close proximity to the entrance to an active site, and alanine substitution at these positions widened the entrance. Our results suggest that this alteration might allow increased accessibility of pepstatin A to the active site. This inference was supported by the observation that the T467A mutant has stronger proteolytic activity than the wild type. CONCLUSIONS/SIGNIFICANCE: We found that Sap7 was a pepstatin A-insensitive protease, and that M242 and T467 restricted the accessibility of pepstatin A to the active site. This finding will lead to the development of a novel protease inhibitor beyond pepstatin A. Such a novel inhibitor will be an important research tool as well as pharmaceutical agent for patients suffering from candidiasis