Let S(f) denote the set of integral ideals / such that / is a permutation polynomial modulo i", where / is a polynomial over the ring of integers of an algebraic number field. We obtain a classification for the sets S which may be written in the form S(f). Introduction. A polynomial f(x) with coefficients in a commutative ring R is said to be a permutation polynomial modulo an ideal I oi R (abb...

A sequence a,< . . . of integers is called primitive if no a divides any other . (a 1 < . . . will always denote a primitive sequence .) It is easy to see that if a i < . . . < a,,, < n then max k = [(n + 1) /2] . The following question seems to be very much more difficult . Put f(n) =niax E 1) , a,, where the maximum is taken over all primitive sequences all of wliose terms are not exceeding n...

A new semantics is presented for the logic of proofs (LP), [1, 2], based on the intuition that it is a logic of explicit knowledge. This semantics is used to give new proofs of several basic results concerning LP. In particular, the realization of S4 into LP is established in a way that carefully examines and explicates the role of the + operator. Finally connections are made with the conventio...

8 0. Preliminaries In this note we comment on the relationships among local versions of the parameter free induction, collection and pigeonhole schemas. We start by defining the versions that will interest us. We work in the usual language of arithmetic to which a new constant symbol a has been added. P-denotes a finite set of axioms such that if M C P-, then M is the nonnegative part of a comm...

Let X be a finite set with n elements. A function f : X −→ R such that ∑ x∈X f (x) ≥ 0 is called a n-weight function. In 1988 Manickam and Singhi conjectured that, if d is a positive integer and f is a n-weight function with n ≥ 4d there exist at least (n−1 d−1 ) subsets Y of X with |Y | = d for which ∑ y∈Y f (y) ≥ 0. In this paper we study this conjecture and we show that it is true if f is a ...

This article is a continuation of [1]. It is divided into five sections. The first one contains a few useful lemmas. In the second part there is a definition of conjugate sequences and proofs of some basic properties of such sequences. The third segment treats of bounded complex sequences,next one contains description of convergent complex sequences. The last and the biggest part of the article...

A basic question concerning zero-knowledge proof systems is whether their (sequential and/or parallel) composition is zero-knowledge too. This question is not only of natural theoretical interest, but is also of great practical importance as it concerns the use of zero-knowledge proofs as subroutines in cryptographic protocols. We prove that the original formulation of zero-knowledge as appeari...

In this paper we deliver the proofs for the claims, made in a companion paper, concerning the avoidance of cosmological curvature singularities in in full Loop Quantum Gravity (LQG). ∗[email protected],[email protected] [email protected],[email protected]