Related papers: The Burnside problem for odd exponents

Let $m\leq n$ be positive integers and $\mathfrak X$ a class of groups which is closed for subgroups, quotient groups and extensions. Suppose that a finite group $G$ satisfies the condition that for every two subsets $M$ and $N$ of…

We show, by explicit computation, that bare lattice perturbation theory in the two-dimensional O(n) nonlinear $\sigma$ models with superinstanton boundary conditions is divergent in the limit of an infinite number of points $|\Lambda|$.…

We establish some upper bounds for the number of integer solutions to the Thue inequality $|F(x , y)| \leq m$, where $F$ is a binary form of degree $n \geq 3$ and with non-zero discriminant $D$, and $m$ is an integer. Our upper bounds are…

We show that the number of non-isotopic commutative semifields of odd order $p^{n}$ is exponential in $n$ when $n = 4t$ and $t$ is not a power of $2$. We introduce a new family of commutative semifields and a method for proving isotopy…

In this paper, we provide an method to obtain the lower bound on the number of the distinct maximum genus embedding of the complete bipartite graph Kn;n (n be an odd number), which, in some sense, improves the results of S. Stahl and H.…

We prove a broad generalization of a theorem of W. Burnside on real characters using permutation characters. Under a necessary hypothesis, We can give some control on multiplicities (a result that needs the Classification of Finite Simple…

We prove a lower bound of exp(-C (log(2/alpha))^7)N^{k-1} to the number of solutions of an invariant equation in k variables, contained in a set of density alpha. Moreover, we give a Behrend-type construction for the same problem with the…

In this paper we study geometric versions of Burnside's Problem and the von Neumann Conjecture. This is done by considering the notion of a translation-like action. Translation-like actions were introduced by Kevin Whyte as a geometric…

The induced Ramsey number $R_{\mathrm{ind}}(H; r)$ of a graph $H$ is the minimum number $N$ such that there exists a graph with $N$ vertices for which all $r$-colourings of its edges contain a monochromatic induced copy of $H$. Our main…

This paper extends the notion of $B$-group to a relative context. For a finite group $K$ and a field $\mathbb{F}$ of characteristic 0, the lattice of ideals of the Green biset functor $\mathbb{F}B_K$ obtained by shifting the Burnside…

In this paper we rectify two previous results found in the literature. Our work leads to a new upper bound for the largest sum-free subset of $[1,n]$ with lowest value in $\left [\frac{n}{3},\frac{n}{2}\right ]$, and the identification of…

The Burning Number Conjecture, that a graph on $n$ vertices can be burned in at most $\lceil \sqrt{n} \ \rceil$ rounds, has been of central interest for the past several years. Much of the literature toward its resolution focuses on two…

The best known method to give a lower bound for the Noether number of a given finite group is to use the fact that it is greater than or equal to the Noether number of any of the subgroups or factor groups. The results of the present paper…

The Monotone Upper Bound Problem asks for the maximal number M(d,n) of vertices on a strictly-increasing edge-path on a simple d-polytope with n facets. More specifically, it asks whether the upper bound M(d,n)<=M_{ubt}(d,n) provided by…

We prove that Knapsack problem (KP) is undecidable for any group of nilpotency class two if the number of generators (without torsion) of the derived subgroup is at least 322. This result together with the fact that if KP is undecidable for…

In this paper, we alternately obtain the almost sure uniqueness of the infinite open cluster in bond percolation in $\mathbb{Z}^d, d \geq 2,$ without requiring the use of ergodicity of translation action. If $N$ denotes the number of…

Let n(2,k) denote the largest integer n for which there exists a set A of k nonnegative integers such that the sumset 2A contains {0,1,2,...,n-1}. A classical problem in additive number theory is to find an upper bound for n(2,k). In this…

We prove that an additive form of degree $d=2m$, $m$ odd, $m\ge3$, over the unramified quadratic extension $\mathbb{Q}_2(\sqrt{5})$ has a nontrivial zero if the number of variables $s$ satisifies $s \ge 4d+1$. If $3 \nmid d$, then there…

We prove new lower bounds on the maximum size of sets $A\subseteq \mathbb{F}_p^n$ or $A\subseteq \mathbb{Z}_m^n$ not containing three-term arithmetic progressions (consisting of three distinct points). More specifically, we prove that for…

Let $s(n)$ be the number of nonzero bits in the binary digital expansion of the integer $n$. We study, for fixed $k,\ell,m$, the Diophantine system $$ s(ab)=k, \quad s(a)=\ell,\quad \mbox{and }\quad s(b)=m, $$ in odd integer variables…