Volume 60, Issue 6

New results about some sums s n(k, l) of products of the Lucas numbers, which are of similar type as the sums in [SEIBERT, J.—TROJOVSK Ý, P.: On multiple sums of products of Lucas numbers, J. Integer Seq. 10 (2007), Article 07.4.5], and sums σ(k) = $$ \sum\limits_{l = 0}^{\tfrac{{k - 1}} {2}} {(_l^k )F_k - 2l^S n(k,l)} $$ are derived. These sums are related to the numerator of generating function for the kth powers of the Fibonacci numbers. s n(k, l) and σ(k) are expressed as the sum of the binomial and the Fibonomial coefficients. Proofs of these formulas are based on a special inverse formulas.

A classical problem in analytic number theory is to study the distribution of αp modulo 1, where α is irrational and p runs over the set of primes. We consider the subsequence generated by the primes p such that p+2 is an almost-prime (the existence of infinitely many such p is another topical result in prime number theory) and prove that its distribution has a similar property.

Let K be a tamely ramified cyclic algebraic number field of prime degree l. In the paper one-to-one correspondence between all orders of K with a normal basis and all ideals of K with a normal basis is given.

Congruences of Ankeny-Artin-Chowla type modulo p 2 for a cyclic subfield K of prime conductor p were derived by Jakubec and expressed in terms of a technically defined map Φ. Later, Jakubec and Lassak found a decomposition of the map Φ modulo p 2 and simplified the formulation of these congruences. A corresponding decomposition of the map Φ modulo p 3 was obtained in [MARKO, F.: Towards Ankeny-Artin-Chowla type congruence modulo p 3, Ann. Math. Sil. 20 (2006), 31–55]. That technical step was important for the formulation of congruences of Ankeny-Artin-Chowla type modulo p 3. This paper will show how to produce an analogous decomposition of the map Φ modulo an arbitrary power p n which would allow a description of analogous congruences modulo p n.

We present a generalization of a result due to Thuswaldner and Tichy to the ring of polynomials over a finite fields. In particular, we want to show that every polynomial of sufficiently large degree can be represented as sum of kth powers, where the bases evaluated on additive functions meet certain congruence restrictions.

It is shown, that in the ring F ℚ[I] of generalized polynomials with several indeterminates from the set I over the field F and with rational exponents, each two elements have a greatest common divisor. On the other hand, this ring is Bezout only if I = O/ or I is a singleton. The arithmetic of the ring F ℚ[I] is transferred to the ring (V, F)[z] of generalized polynomials with one indeterminate z over F with exponents from the vector space V over ℚ. It is proved that the rings F ℚ[I] and (V, F)[z] are isomorphic provided dimV = cardI. It follows, for example, that the rings (ℝ, F)[z] and (ℂ, F)[z] of generalized polynomials with one indeterminate with real and complex exponents are isomorphic.

Bounded Rℓ-monoids form a large subclass of the class of residuated lattices which contains certain of algebras of fuzzy and intuitionistic logics, such as GMV-algebras (= pseudo-MV-algebras), pseudo-BL-algebras and Heyting algebras. Moreover, GMV-algebras and pseudo-BL-algebras can be recognized as special kinds of pseudo-MV-effect algebras and pseudo-weak MV-effect algebras, i.e., as algebras of some quantum logics. In the paper, bipartite, local and perfect Rℓ-monoids are investigated and it is shown that every good perfect Rℓ-monoid has a state (= an analogue of probability measure).

We prove that the correspondence between MV-algebras and abelian ℓ-groups with the strong unit is preserved in the direct limit construction. Further, several classes of MV-algebras which are closed under formation of direct limits will be distinguished.

We consider a initial-boundary value problem for a sixth order degenerate parabolic equation. Under some assumptions on the initial value, we establish the existence of weak solutions by the time-discrete method. The uniqueness, asymptotic behavior and the finite speed of propagation of perturbations of solutions are also discussed.

This paper provides a Korovkin type approximation theorem for a class of positive linear operators including Bleimann-Butzer and Hahn operators via J-convergence.

Our primary interest in the present paper is to prove a Korovkintype approximation theorem for sequences of positive linear operators defined on the space of all real valued n-variate B-continuous functions on a compact subset of the real n-dimensional space via statistical convergence. Also, we display an example such that our method of convergence is stronger than the usual convergence.