Abstract. Let p > 3 be a prime. A p-adic congruence is called a super congruence if it happens to hold modulo some higher power of p. The topic of super congruences is related to many fields including Gauss and Jacobi sums and hypergeometric series. We prove that
Abstract. In this paper we first investigate for what positive integers a, b, c every nonnegative integer n can be written as x(ax + 1) + y(by + 1) + z(cz + 1) with x, y, z integers. We show that (a, b, c) can be either of the following seven triples
Let $p$ be a prime, and let $d\in\{0,...,p^a\}$ with $a\in\Z^+$. In this paper we determine $\sum_{k=0}^{p^a-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=1}^{p-1}\binom{2k}{k+d}/(km^{k-1})$ modulo $p$ where $m$ is an integer not divisible by $p$. For example, we show that if $p\not=2,5$ then $$\sum_{k=1}^{p-1}(-1)^k\frac{\binom{2k}k}k=-5\frac{F_{p-(\frac p5)}}p (mod p),$$ where $F_n$ denotes the $n$th Fibonacci number. We also prove that if $p>3$ then $$\sum_{k=1}^{p-1}\frac{\binom{2k}k}k={8/9} p^2B_{p-3} (mod p^3),$$ where $B_n$ is the $n$th Bernoulli number
We consider the problem of minimizing a continuous function f over a compact set K. We analyze a hierarchy of upper bounds proposed by Lasserre (SIAM J Optim 21(3):864-885, 2011), obtained by searching for an optimal probability density function h on K which is a sum of squares of polynomials, so that the expectationWe show that the rate of convergence is no worse than O(1/ √ r ), where 2r is the degree bound on the density function. This analysis applies to the case when f is Lipschitz continuous and K is a full-dimensional compact set satisfying some boundary condition (which is satisfied, e.g., for convex bodies). The r th upper bound in the hierarchy may be computed using semidefinite programming if f is a polynomial of degree d, and if all moments of order up to 2r +d of the Lebesgue measure on K are known, which holds, for example, if K is a simplex, hypercube, or a Euclidean ball.
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