Equilibria, Fixed Points, and Complexity Classes

Many models from a variety of areas involve the computation of an equilibrium or fixed point of some kind. Examples include Nash equilibria in games; market equilibria; computing optimal strategies and the values of competitive games (stochastic and other games); stable configurations of neural networks; analysing basic stochastic models for evolution like branching processes and for language like stochastic context-free grammars; and models that incorporate the basic primitives of probability and recursion like recursive Markov chains. It is not known whether these problems can be solved in polynomial time. There are certain common computational principles underlying different types of equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP. Representative complete problems for these classes are respectively, pure Nash equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria in 2-player normal form games, and (mixed) Nash equilibria in normal form games with 3 (or more) players. This paper reviews the underlying computational principles and the corresponding classes

Connectivity, Complexity and Catastrophe in Large-scale Systems

This book represents an approach to large-scale system modeling that is a challenging synthesis for the systems analyst, the operations research worker, the system theorist, the policy analyst, and the student of social systems. After pointing out that the mathematical form of a system description dictates the types of questions that can be asked and answered by the model, the author declares that "there is no such thing as a model system: there are many models, each with its own characteristic mathematical features and each capable of addressing a certain subset of important questions about the system and its operation". The book supports this point with examples from a wide spectrum of contexts (such as physics, economic activity, water-resource management, ecology, transportation, and physiology) viewed from the points of view of various models and theories (such as general system theory, control theory, graph theory, linear and nonlinear system theory, and catastrophy theory). Against this broad background, the book then considers in depth the relations to large-scale systems of the theories of connectivity, complexity, stability, catastrophy, and resilience

Crab and cockle shells as heterogeneous catalysts in the production of biodiesel

In the present study, the waste crab and cockle shells were utilized as source of calcium oxide to transesterify palm olein into methyl esters (biodiesel). Characterization results revealed that the main component of the shells are calcium carbonate which transformed into calcium oxide upon activated above 700 °C for 2 h. Parametric studies have been investigated and optimal conditions were found to be catalyst amount, 5 wt.% and methanol/oil mass ratio, 0.5:1. The waste catalysts perform equally well as laboratory CaO, thus creating another low-cost catalyst source for producing biodiesel. Reusability results confirmed that the prepared catalyst is able to be reemployed up to five times. Statistical analysis has been performed using a Central Composite Design to evaluate the contribution and performance of the parameters on biodiesel purity

Molecular phylogeny of horseshoe crab using mitochondrial Cox1 gene as a benchmark sequence

An effort to assess the utility of 650 bp Cytochrome C oxidase subunit I (DNA barcode) gene in delineating the members horseshoe crabs (Family: xiphosura) with closely related sister taxa was made. A total of 33 sequences were extracted from National Center for Biotechnological Information (NCBI) which include horseshoe crabs, beetles, common crabs and scorpion sequences. Constructed phylogram showed beetles are closely related with horseshoe crabs than common crabs. Scorpion spp were distantly related to xiphosurans. Phylogram and observed genetic distance (GD) date were also revealed that Limulus polyphemus was closely related with Tachypleus tridentatus than with T.gigas. Carcinoscorpius rotundicauda was distantly related with L.polyphemus. The observed mean Genetic Distance (GD) value was higher in 3rd codon position in all the selected group of organisms. Among the horseshoe crabs high GC content was observed in L.polyphemus (38.32%) and lowest was observed in T.tridentatus (32.35%). We conclude that COI sequencing (barcoding) could be used in identifying and delineating evolutionary relatedness with closely related specie