Let p be a prime, and let f(x) be an integer-valued polynomial. By a combinatorial approach, we obtain a nontrivial lower bound of the p-adic order of the sum ∑ k≡r (mod pβ) (n k ) (−1)f (⌊ k − r pα ⌋) , where α > β > 0, n > pα−1 and r ∈ Z. This polynomial extension of Fleck’s congruence has various backgrounds and several consequences such as ∑ k≡r (mod pα) (n k ) a ≡ 0 ( mod p ⌊ n−pα−1 φ(pα) ...

Let p be a prime, and let f (x) be an integer-valued polynomial. By a combinatorial approach, we obtain a nontrivial lower bound of the p-adic order of the sum k≡r (mod p β) n k (−1) k f k − r p α , where α β 0, n p α−1 and r ∈ Z. This polynomial extension of Fleck's congruence has various backgrounds and several consequences such as k≡r (mod p α) n k a k ≡ 0 mod p n−p α−1 ϕ(p α) provided that ...

We give a classification of dual functions, they are m -al functions. We call a function m-al with respect to an operator if the operator lives any function unchanged after m times of using the operator. And 2 ≤ m ≤ k. Functions with different m have very different properties. We give theoretical results for clones of self-dual (m = 2) and selfk -al (m = k) functions in k -valued logic at k ≤ 3...

Computing certain answers is the preferred way of answering queries in scenarios involving incomplete data. This, however, is computationally expensive, so practical systems use efficient techniques based on a particular three-valued logic, although this often leads to

In the standard theorems on weak convergence of stochastic processes, it is invariably assumed that the parameter set is a bounded interval. The object of this paper is to indicate that analogues of these theorems for unbounded intervals are also valid. We shall confine our attention to the results of Skorohod [l], and in particular to those results concerning his Ji topology. Let £ be a comple...

A LDP is proved for the inviscid shell model of turbulence. As the viscosity coefficient ν converges to 0 and the noise intensity is multiplied by √ ν, we prove that some shell models of turbulence with a multiplicative stochastic perturbation driven by a H-valued Brownian motion satisfy a LDP in C([0, T ], V) for the topology of uniform convergence on [0, T ], but where V is endowed with a top...

Let Sf be the Schwartz space of all rapidly decreasing functions on R, Sf be the topological dual space of Sf and for each positive integer p,Sfp be the space of all elements of £f' which are continuous in the p-th norm defining the nuclear Frechet topology of Sf. The main purpose of the present paper is to show that if {Xtί t e [0, + oo)} is an ^'-valued Gaussian process and for any fixed φβ^ ...

A LDP is proved for the inviscid shell model of turbulence. As the viscosity coefficient ν converges to 0 and the noise intensity is multiplied by ν , we prove that some shell models of turbulence with a multiplicative stochastic perturbation driven by a H-valued Brownian motion satisfy a LDP in ([0, T],V ) for the topology of uniform convergence on [0, T], but where V is endowed with a topolog...

We study semicontinuity of multiple integrals ~ f~,u,Du)dx~ where the vector-valued function u is defined for xE ~C N with values in ~ . The function f(x,s,~) is assumed to be Carath@odory and quasiconvex in Morrey's sense. We give conditions on the growth of f that guarantee the sequential lower semicontinuity of the given integral in the weak topology 1 N of the Sobolev space H 'P(~;~ ). The ...

This paper generalizes the Max-Flow Min-Cut (MFMC) theorem from the setting of numerical capacities to cellular sheaves of semimodules on directed graphs. Motivating examples of semimodules include probability distributions, multicommodity capacity constraints, and logical propositions. Directed algebraic topology provides the tools necessary for capturing the salient information in such a gene...