A generalization of Schmidt’s Theorem on groups with all subgroups nilpotent

According to Schmidt’s Theorem a finite group whose proper subgroups are all nilpotent (or a finite group without non-nilpotent proper subgroups) is solvable. In this paper we prove that every finite group with less than 22 non-nilpotent subgroups is solvable and that this estimate is sharp.     

An attrition game on a network ruled by Lanchester’s square law

We consider two-person zero-sum attrition games in which an attacker and a defender are in combat with each other on a network. The attacker marches from a starting node to a destination node, hoping that the initial members survive the march. The defender deploys his forces on arcs in order to intercept the attacker. If the attacker encounters the defender on an arc, the attacker incurs casualties according to Lanchester’s square law. We consider two models: a one-shot game in which the two players have no information about their opponents, and a two-stage game in which both players have some information about their opponents. For both games, the payoff is defined as the number of survivors for the attacker. The attacker’s strategy is to choose a path, and the defender’s is to deploy the defending forces on arcs. We propose a numerical algorithm, in which nonlinear programming is embedded, to derive the equilibrium of the game.     

An effective Schmidt’s subspace theorem over function fields

We establish an effective version of the Schmidts subspace theorem for higher dimensional function fields of characteristic zero.Received: 30 January 2001     

A simple proof of Schmidt–Summerer’s inequality

In this paper we give a simple proof of an inequality for intermediate Diophantine exponents obtained recently by W.M. Schmidt and L. Summerer.     

Impact of neighborhood separation on the spatial reciprocity in the prisoner’s dilemma game

The evolutionary game theory is a very powerful tool to understand the collective cooperation behavior in many real-world systems. In the spatial game model, the payoff is often first obtained within a specific neighborhood (i.e., interaction neighborhood) and then the focal player imitates or learns the behavior of a randomly selected one inside another neighborhood which is named after the learning neighborhood. However, most studies often assume that the interaction neighborhood is identical with the learning neighborhood. Beyond this assumption, we present a spatial prisoner’s dilemma game model to discuss the impact of separation between interaction neighborhood and learning neighborhood on the cooperative behaviors among players on the square lattice. Extensive numerical simulations demonstrate that separating the interaction neighborhood from the learning neighborhood can dramatically affect the density of cooperators (ρ_C) in the population at the stationary state. In particular, compared to the standard case, we find that the medium-sized learning (interaction) neighborhood allows the cooperators to thrive and substantially favors the evolution of cooperation and ρ_C can be greatly elevated when the interaction (learning) neighborhood is fixed, that is, too little or much information is not beneficial for players to make the contributions for the collective cooperation. Current results are conducive to further analyzing and understanding the emergence of cooperation in many natural, economic and social systems.     

A note on Haj?asz-Sobolev spaces on fractals

We introduce Haj?asz-Sobolev spaces involving walk dimensions on subsets of that will generally be fractals. The relationship between this kind of new space and the Besov space of Jonsson and Wallin is investigated. The (compact) embedding theorems for Haj?asz-Sobolev spaces are obtained.     

Staircase rook polynomials and Cayley’s game of Mousetrap

First introduced by Arthur Cayley in the 1850’s, the game of Mousetrap involves removing cards from a deck according to a certain rule. In this paper we find the rook polynomial for the number of Mousetrap decks in which at least two specified cards are removed. We also find a new expression for the rook polynomial for the number of decks in which exactly one specified card is removed and give expressions for counts of two kinds of Mousetrap decks in terms of other known combinatorial numbers.     

The degenerated second main theorem and Schmidt’s subspace theorem

In this paper, we establish a Second Main Theorem for an algebraically degenerate holomorphic curve f : C → Pn(C) intersecting hypersurfaces in general position. The related Diophantine problems are also considered.     

Effects of limited interactions between individuals on cooperation in spatial evolutionary prisoner’s dilemma game

We study the spatial evolutionary prisoner’s dilemma game with limited interactions by introducing two kinds of individuals, say type-A and type-B with a fraction of p and (1 − p), respectively, distributed randomly on a square lattice. Each kind of individuals can adopt two pure strategies: either to cooperate or to defect. During the evolution, the individuals can only interact with others belonging to the same kind, but they can learn from either kinds of individuals in the nearest neighborhood. Using Monte Carlo simulations, the average frequency of cooperators ρ_C is calculated as a function of p in the equilibrium state. It is shown that, compared with the case of p = 0 (only one kind of individuals existing in the system), cooperation can be evidently promoted. In particular, the cooperator density can reach a maximum level at some moderate values of p in a wide range of payoff parameters. The results imply that certain limited interactions between individuals plays an important and nontrivial role in the evolution of cooperation.     

von Neumann’s trace inequality for Hilbert–Schmidt operators

von Neumann’s inequality in matrix theory refers to the fact that the Frobenius scalar product of two matrices is less than or equal to the scalar product of the respective singular values. Moreover, equality can only happen if the two matrices share a joint set of singular vectors, and this latter part is hard to find in the literature. We extend these facts to the separable Hilbert space setting, and provide a self-contained proof of the “latter part”.     

An optical model for implementing Parrondo’s game and designing stochastic game with long-term memory

An optical model for a photon propagating through a designed array of beam splitters is developed to give a physical implementation of Parrondo’s game and Parrondo’s history-dependent game. The winner in this optical model is a photon passed the beam splitter. The loser is a photon being reflected by the beam splitter. The optical beam splitter is the coin-tosser. We designed new games with long-term memory by using this optical diagram method. The optical output of the combined game of two losing games could be a win, or a loss, or an oscillation between win and loss. The modern technology to implement this optical model is well developed. A circularly polarized photon is a possible candidate for this physical implementation in laboratory.     

Dedekind’s and Frege’s views on logic

A contextual and comparative analysis shows that Dedekind and Frege do not understand the terms “logic” and “arithmetic” in the same way. More specifically the meaning and the scope of the corresponding concepts are essentially different for them. Consequently Dedekind and Frege have different conceptions of the relationship between arithmetic and logic.     

Cooperation in spatial prisoner’s dilemma game with delayed decisions

Phenomena that time delays of information lead to delayed decisions are extensive in reality. The effect of delayed decisions on the evolution of cooperation in the spatial prisoner’s dilemma game is explored in this work. Players with memory are located on a two dimensional square lattice, and they can keep the payoff information of his neighbors and his own in every historic generation in memory. Every player uses the payoff information in some generation from his memory and the strategy information in current generation to determine which strategy to choose in next generation. The time interval between two generations is set by the parameter m. For the payoff information is used to determine the role model for the focal player when changing strategies, the focal player’s decision to learn from which neighbor is delayed by m generations. Simulations show that cooperation can be enhanced with the increase of m. In addition, just like the original evolutionary game model (m = 0), pretty dynamic fractal patterns featuring symmetry can be obtained when m > 0 if we simulate the invasion of a single defector in world of cooperators on square lattice.     

Arveson’s Work on Entanglement

We summarize William Arveson’s work on entanglement in quantum information theory.     

Remarks on Harada’s conjecture

An open conjecture by Harada from 1981 gives an easy characterization of the p-blocks of a finite group in terms of the ordinary character table. Kiyota and Okuyama have shown that the conjecture holds for p-solvable groups. In the present work we extend this result using a criterion on the decomposition matrix. In this way we prove Harada’s Conjecture for several new families of defect groups and for all blocks of sporadic simple groups. In the second part of the paper we present a dual approach to Harada’s Conjecture.     

Note on Halphen’s system

The Ramanujan Journal - In this paper, we construct extensions of the differential field obtained from Halphen’s system of ordinary differential equations (ODEs) using quasi-modular forms of...     

Robin’s criterion on divisibility

The Ramanujan Journal - Robin’s criterion states that the Riemann hypothesis is true if and only if the inequality $$\sigma (n) < e^{\gamma } \times n \times \log \log n$$ holds for...