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We revisit the question of regularity for minimizers of scalar autonomous integral functionals with so-called $(p,q)$-growth. In particular, we establish Lipschitz regularity under the condition $\frac{q}p<1+\frac{2}{n-1}$ for $n\geq3$…

We prove some abstract multiplicity theorems that can be used to obtain multiple nontrivial solutions of critical growth $p$-Laplacian and $(p,q)$-Laplacian type problems. We show that the problems considered here have arbitrarily many…

Let S be an abelian semigroup, and A a finite subset of S. The sumset hA consists of all sums of h elements of A, with repetitions allowed. Let |hA| denote the cardinality of hA. Elementary lattice point arguments are used to prove that an…

In this paper, the $(p,q)$-derivative and the $(p,q)$-integration are investigated. Two suitable polynomials bases for the $(p,q)$-derivative are provided and various properties of these bases are given. As application, two $(p,q)$-Taylor…

Let $\|A\|_{p,q}$ be the norm induced on the matrix $A$ with $n$ rows and $m$ columns by the H\"older $\ell_p$ and $\ell_q$ norms on $R^n$ and $R^m$ (or $C^n$ and $C^m$), respectively. It is easy to find an upper bound for the ratio…

We consider the problem of computing the q->p norm of a matrix A, which is defined for p,q \ge 1, as |A|_{q->p} = max_{x !=0 } |Ax|_p / |x|_q. This is in general a non-convex optimization problem, and is a natural generalization of the…

We determine the minimal bi-degree(s) of an irreducible filling curve over $\mathbb{F}_q$ for $\mathbb{P}^1\times \mathbb{P}^1$. It is $(q+1, q+1)$ if $q\neq 2$, and they are $(4,3)$ and $(3,4)$ if $q=2$.

In this paper we aim to establish some results depending on the comparative growth properties of composite transcendental entire or meromorphic functions and some special type of differential polynomials generated by one of the factors on…

Let G be a planar graph with polynomial growth and isoperimetric dimension bigger than 1. Then the critical p for Bernoulli percolation on G satisfies p<1.

For every nonnegative integer $n$, let $r_F(n)$ be the number of ways to write $n$ as a sum of Fibonacci numbers, where the order of the summands does not matter. Moreover, for all positive integers $p$ and $N$, let \begin{equation*}…

We show that for any relatively prime integers $1\leq p<q$ and for any finite $A \subset \mathbb{Z}$ one has $$|p \cdot A + q \cdot A | \geq (p + q) |A| - (pq)^{(p+q-3)(p+q) + 1}.$$

For prime powers q, let s(q) denote the probability that a randomly-chosen principally-polarized abelian surface over the finite field F_q is not simple. We show that there are positive constants B and C such that for all q, B (log…

Let $|| \cdot ||$ denote the distance to the nearest integer and, for a prime number $p$, let $| \cdot |_p$ denote the $p$-adic absolute value. In 2004, de Mathan and Teuli\'e asked whether $\inf_{q \ge 1} \, q \cdot || q \alpha || \cdot |…

We show that near a point where the equilibrium density of eigenvalues of a matrix model behaves like y ~ x^{p/q}, the correlation functions of a random matrix, are, to leading order in the appropriate scaling, given by determinants of the…

We consider the maximal p-norm associated with a completely positive map and the question of its multiplicativity under tensor products. We give a condition under which this multiplicativity holds when p = 2, and we describe some maps which…

We obtain a set of necessary and sufficient conditions for $| \bar{N}, p_{n} |_{k} $ to imply $|\bar{N}, q_{n} |_{s}$ for $1 < k \leq s < \infty$. Using this result we establish several inclusion theorems as well as conditions for the…

We consider the weight w: 1<w<T on the unit circle and prove that the corresponding orthonormal polynomials can grow.

Any finite word $w$ of length $n$ contains at most $n+1$ distinct palindromic factors. If the bound $n+1$ is reached, the word $w$ is called rich. The number of rich words of length $n$ over an alphabet of cardinality $q$ is denoted…

We prove that for $1<p\le q<\infty$, $qp\geq {p'}^2$ or $p'q'\geq q^2$, $\frac{1}{p}+\frac{1}{p'}=\frac{1}{q}+\frac{1}{q'}=1$, $$\|\omega P_\alpha(f)\|_{L^p(\mathcal{H},y^{\alpha+(2+\alpha)(\frac{q}{p}-1)}dxdy)}\le…

Let $P$ be a polynomial that depends on two variables $X$ and $Y$ and has algebraic coefficients. If $x$ and $y$ are algebraic numbers with $P(x,y)=0$, then by work of N\'eron $h(x)/q$ is asymptotically equal to $h(y)/p$ where $p$ and $q$…