Δελτίο της Ελληνικής Μαθηματικής Εταιρίας: Τεύχος 48
The following theorem is proved: In a unitary ring of characteristic 2 the identity x^n=x implies that the ring is Boolean if and only if n ? 1 is not divisible by 2^p-1 for any prime p.
We consider the dual of an ideal and the ideal transforms of Nagata and Kaplansky in the context of monoids. In particular, we consider Pr?fer monoids with respect to a finitary ideal system and generalize the known results for Pr?fer domains. These generalizations embrace abstract monoids (without an ideal system), Pr?fer v ? multiplication domains and Pr?fer * - multiplication domains for any star ? operation of finite type.
We derive an alternate formula for Ramanujan?s function, denoted τ(n), by two different methods. The first method uses properties of the Weierstrass phi function, as well as identities for sums of 8 squares and for sums of 8 triangular numbers. The second method makes use of theta function identities.
Let W be a finite Coxeter group of type A_n and let H be the associated Hecke algebra. We investigate the involutions contained in a two ? sided cell of W and various H- module homomorphisms given by left multiplication by elements of H that induce isomorphisms between cell representations corresponding to elements contained in the same two ? sided cell of W.
Recently, numerous techniques and methods have been proposed to address hard and complex algebraic and number theoretical problems related to cryptography. We review several interpolation and approximation techniques. In particular, we discuss techniques related to polynomial interpolation, discrete Fourier transforms, as well as, polynomial approximation. Subsequently, we focus on a recently proposed approach to tackle these problems based on computational intelligence methods. More specifically, a study of the neural network approach to address the discrete logarithm problem, the Diffie ? Hellman mapping problem, and the factorization problem related to the RSA cryptosystem, is attempted.
Let be a prime number with q5(mod8)q?. If ? is the fundamental unit of (2)q and (2qκ=? ), then exactly one of 44(2?),(8?)κκ is embedded in a dihedral extension of which is cyclic and unramified outside 2 over κ. In this paper we show how we can find which field has the embedding property using numerical procedures.
This paper is an explanation of the Kummer theory of ideal numbers in the light of category theory, particularly of the concept of the categorical reflection. The topology on a semigroup established by its v ? ideals as a subbasis for closed sets is used. The central notion is the theory of divisors of a semigroup which plays the same role among the semigroups with a divisor theory as the ?ech ? Stone β?compactification among the completely regular topological spaces. At the conclusion the maximal choices of (semigroup) homomorphisms are mentioned which preserve the reflection property of the theory of divisors. An attention is drawn to the fact that the cardinal of these choices is equal to exp exp N_0.
The hyperstructures are algebraic structures endowed with at least one hyperoperation. We focus on H_v - structures which are generalized algebraic hyperstructures where, in the axioms of the classical hyperstructures, the equality is replaced by the non ? empty intersection. These axioms are called weak. The hyperstructures are classified according to their properties and they correspond to the classical structures by using the fundamental quotients. In the procedure to describe the fundamental relations, special elements appear. These elements are the elements of the core and the so ? called single ones, which are used to define and to study the H_v - structures. In this paper we present some applications on the above argument.