We show that there is no continuous bijection from ${ℝ}^n$ onto ${ℝ}^2$ for $n\ne 2$ by an elementary method. This proof is based on showing that for any cardinal number $\beta \le 2^{{\aleph }_0}$, there is a partition of $R^n$ ($n\ge 3$) into $\beta$ arcwise connected dense subsets.

Evolutionary graph theory combines evolutionary games with population structure, induced by the graph. The games used are limited to pairwise games occurring on the edges of the graph. Multiplayer games can be important in biological modelling, however, and so recently a new framework for modelling games in structured populations allowing games with arbitrary numbers of players was introduced. In this paper we develop the model to investigate the effect of population structure on the level of aggression, as opposed to a well-mixed population for two specific types of graph, using a multiplayer hawk-dove game. We find that the graph structure can have a significant effect on the level of aggression, and that a key factor is the variability of the group sizes formed to play the games; the more variable the group size, the lower the level of aggression, in general.

This paper examines binary codes from a frame-theoretic viewpoint. Binary Parseval frames have convenient encoding and decoding maps. We characterize binary Parseval frames that are robust to one or two erasures. These characterizations are given in terms of the associated Gram matrix and with graph-theoretic conditions. We illustrate these results with frames in lowest dimensions that are robust to one or two erasures. In addition, we present necessary conditions for correcting a larger number of erasures. As in a previous paper, we emphasize in which ways the binary theory differs from the theory of frames for real and complex Hilbert spaces.

Four infinite families of 2-groups are presented, all of whose members possess an outer automorphism that preserves conjugacy classes. The groups in these families are central extensions of their predecessors by a cyclic group of order 2. For each integer $r>1$, there is precisely one 2-group of nilpotency class $r$ in each of the four families. All other known families of 2-groups possessing a class-preserving outer automorphism consist entirely of groups of nilpotency class 2.

We provide a proof of the sharp log-Sobolev inequality on a compact interval.

Sudoku is a popular game of logic, and there are many variations of the standard puzzle. We investigate a variation of Sudoku that uses inequalities between cells rather than numerical clues. We begin with an overview of the rules and strategies of the game. We then examine the solvability of an individual $m\times n$ block with the use of partially ordered sets, and combine $2\times 2$ blocks to form $4\times 4$ puzzles.

This paper is devoted to estimates of the spanning tree congestion for some planar graphs. We present three main results: (1) We almost determined (up to $\pm 1$) the maximal possible spanning tree congestion for planar graphs. (2) The value of congestion indicator introduced by Ostrovskii [Discrete Math. 310, 1204–1209] can be very far from the value of the spanning tree congestion. (3) We find some more examples in which the congestion indicator can be used to find the exact value of the spanning tree congestion.

We explore the relationship between convex and subharmonic functions on discrete sets. Our principal concern is to determine the setting in which a convex function is necessarily subharmonic. We initially consider the primary notions of convexity on graphs and show that more structure is needed to establish the desired result. To that end, we consider a notion of convexity defined on lattice-like graphs generated by normed abelian groups. For this class of graphs, we are able to prove that all convex functions are subharmonic.

The number $N\left(k,r\right)$ is defined to be the first integer such that it and every subsequent integer can be written as the sum of the $k$-th powers of $r$ or more distinct positive integers. For example, it is known that $N\left(2,1\right)=129$, and thus the last number that cannot be written as the sum of one or more distinct squares is 128. We give a proof of a theorem that states if certain conditions are met, a number can be verified to be $N\left(k,r\right)$. We then use that theorem to find $N\left(2,r\right)$ for $1\le r\le 50$ and $N\left(3,r\right)$ for $1\le r\le 30$.