Related papers: The critical exponent of the Arshon words

The classical SU(3) gauge theory is shown to be deterministic chaotic. Its largest Lyapunov exponent is dertermined, from which a short time scale of thermalization of a pure gluon system is estimated. The connection to gluon damping rate…

In this paper, we consider the global well-posedness of the defocusing, $L^{2}$ - critical nonlinear Schr{\"o}dinger equation in dimensions $n \geq 3$. Using the I-method, we show the problem is globally well-posed in $n = 3$ when $s >…

Partially ordered sets of type (k, n) are the sets such that a) cardinality of each set is n, b) dimension of each set is two, c) length of the maximal antichain in each set is k. Let \alpha_k(n) be the number of partially ordered sets of…

We present a sharp upper bound for the number of generators of a finite group in terms of the ratio between the order and the exponent.

We study an upper bound of ranks of $n$-tensors with size $2\times\cdots\times2$ over the complex and real number field. We characterize a $2\times 2\times 2$ tensor with rank 3 by using the Cayley's hyperdeterminant and some function. Then…

Improving a 2003 result of Bohman and Holzman, we show that for $n \geq 1$, the Shannon capacity of the complement of the $2n+1$-cycle is at least $(2^{r_n} + 1)^{1/r_n} = 2 + \Omega(2^{-r_n}/r_n)$, where $r_n = \exp(O((\log n)^2))$ is the…

For a 2-graph $F$, let $H_F^{(r)}$ be the $r$-graph obtained from $F$ by enlarging each edge with a new set of $r-2$ vertices. We show that if $\chi(F)=\ell>r \geq 2$, then $ {\rm ex}(n,H_F^{(r)})= t_r (n,\ell-1)+ \Theta( {\rm…

We improve the necessary condition for Carleson's problem regarding convergence for the Schr\"odinger equation in dimensions $n\ge 3$. We prove that if the solution converges almost everywhere to its initial datum as time tends to zero, for…

We survey properties of the set of possible exponents of subsets of $\Z_n$ (equivalently, exponents of primitive circulant digraphs on $n$ vertices). Let $E_n$ denote this exponent set. We point out that $E_n$ contains the positive integers…

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erd\H{o}s proved in 1935 that the weighted sum $\sum1/(n \log n)$ for $n$ ranging over a primitive set $A$ is universally bounded over all choices for…

We show that the maximum clique problem (decision version) can be expressed in existential second order (ESO) logic, where the first order part is a Horn formula in second-order quantified predicates. Without ordering, the first order part…

Let (G, +) be an abelian group. A subset of G is sumfree if it contains no elements x, y, z such that x +y = z. We extend this concept by introducing the Schur degree of a subset of G, where Schur degree 1 corresponds to sumfree. The…

Let m be a positive integer and A an elementary abelian group of order q^r with r greater than or equal to 2 acting on a finite q'-group G. We show that if for some integer d such that 2^{d} is less than or equal to (r-1) the dth derived…

Let $G$ be an additive abelian group and let $A,B \subseteq G$ be finite and nonempty. The pair $(A,B)$ is called critical if the sumset $A+B = {a+b \mid $a \in A$ and $b\in B$}$ satisfies $|A+B| < |A| + |B|$. Vosper proved a theorem which…

In this paper we confirm that $2^*(\gamma)=\frac{2(N+\gamma)}{N-2}$ with $\gamma>0$ is exactly the critical exponent for the embedding from $H_r^1(\mathbb{R}^N)$ into $L^q(\mathbb{R}^N;|x|^\gamma)$($N\geqslant 3$) (see…

We analyze the scaling behavior of the higher Lyapunov exponents at the Anderson transition. We estimate the critical exponent and verify its universality and that of the critical conductance distribution for box, Gaussian and Lorentzian…

We prove, via 30 seconds of Maple computation, that there are 10^n words in the alphabet {3,-2} of length 5n, sum 0, and such that every factor that sums to 0 and that starts with a 3 may not be immediately followed by a -2.

Let A and B be subsets of Z/pZ such that |A+B| < |A|+|B|+2. We prove that, if |A|>3, |B|>4, |A+B|<p-4 and p > 52, then A and B are included in arithmetic progressions with the same difference and of size |A|+2 and |B|+2 respectively. This…

We prove that assuming the Generalized Riemann Hypothesis every even integer larger than $\exp(\exp(15.85))$ can be written as the sum of a prime number and a number that has at most two prime factors.

In this paper, we show that all the exponents of degree greater than 2 of spinor groups divide the Dynkin index 2.