Volume 58, Issue 1

Relationship between automorphisms and residuated bounded mappings in atomistic lattices is studied.

In this note the sums s(k, N) of reciprocals $$\sum\limits_{\tfrac{{kp}}{N} < x < \tfrac{{(k + 1)p}}{N}} {\tfrac{1}{x}(mod p)} $$ are investigated, where p is an odd prime, N, k are integers, p does not divide N, N ≥ 1 and 0 ≤ k ≤ N − 1. Some linear relations for these sums are derived using “logarithmic property” and Lerch’s Theorem on the Fermat quotient. Particularly in case N = 10 another linear relation is shown by means of Williams’ congruences for the Fibonacci numbers.

It is well-known that Morgan-Voyce polynomials B n(x) and b n(x) satisfy both a Sturm-Liouville equation of second order and a three-term recurrence equation ([SWAMY, M.: Further properties of Morgan-Voyce polynomials, Fibonacci Quart. 6 (1968), 167–175]). We study Diophantine equations involving these polynomials as well as other modified classical orthogonal polynomials with this property. Let A, B, C ∈ ℚ and {pk(x)} be a sequence of polynomials defined by $$\begin{gathered} p_0 (x) = 1 \hfill \\ p_1 (x) = x - c_0 \hfill \\ p_{n + 1} (x) = (x - c_n )p_n (x) - d_n p_{n - 1} (x), n = 1,2,..., \hfill \\ \end{gathered} $$ with $$(c_0 ,c_n ,d_n ) \in \{ (A,A,B),(A + B,A,B^2 ),(A,Bn + A,\tfrac{1}{4}B^2 n^2 + Cn)\} $$ with A ≠ 0, B > 0 in the first, B ≠ 0 in the second and C > −¼B 2 in the third case. We show that the Diophantine equation with m > n ≥ 4, ≠ 0 has at most finitely many solutions in rational integers x, y.

In the paper, we obtain a congruence modulo p 3 among Euler numbers E p−1, E 2p−2, and Fermat quotients Q 2, Q a where p = a 2 + 4b 2.

In this paper we introduce the concept of the generalized AP-integral and discuss the properties of this integral.

In [GOULD, G. G.: Integration over vector-valued measures, Proc. London Math. Soc. (3) 15, (1965), 193–205], G. G. Gould introduced a type of integral of a bounded, real valued function with respect to a finite additive set function taking values in a Banach space, integral which is more general than the Lebesgue one. Recently, A. Precupanu and A. Croitoru gave the generalization, defining a Gould type integral for multimeasures with values in kc(X), X being a Banach space ([PRECUPANU, A.—CROITORU, A.: A Gould type integral with respect to a multimeasure, I, An. Ştiinţ. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. 48 (2002), 165–200]). Taking as starting point this work and [PRECUPANU, A.—CROITORU, A.: A Gould type integral with respect to a multimeasure, II, An. Ştiinţ. Univ. “Al. I. Cuza” Iaşi Secţ. I a Mat. 49 (2003), 183–207], we define here the notion of a Gould type integral with respect to a bf(X)-valued multisubmeasure, pointing out important properties of it. We also establish that, even if we deal with multisubmeasures, the integral is still a multimeasure.

In this paper an outer measure on IF-sets is studied as a mapping to the set of all compact subintervals of the unit interval. We characterize the properties of the outer measure by the help of the properties of functions given by the edges of the intervals. Then there are defined measurable elements and there is proved that the family of measurable elements is a lattice. Finally the outer measure induced by a measure is constructed.

Sufficient conditions are obtained for oscillation of solutions of a class of neutral parabolic differential equations with oscillating coefficients.

We study here the G-shadowing property of the shift map σ on the inverse limit space X f, generated by an equivariant self-map f on a metric G-space X.

The general concept of boundedness in a topological space generalizes both metric boundedness and relative compactness. A one-point extension o(ℱ X) of the space X is naturally associated to each boundedness ℱ X and every Hausdorff one-point extensions of a space X can be obtained in this way. Imitating this construction, it is possible to define a much more general class of Hausdorff extensions of a locally bounded space with respect to a given boundedness, the so-called B-extensions. In this paper we study separation properties and metrizability of this kind of extension.

The optimum linear estimators of the useful mean value parameters within a linear regression model with the stable and variable parameters and with the nuisance parameters are derived including their characteristics of accuracy. Some verification of theoretical results is presented.

We obtain the span of the real flag manifolds ℝF(1, 1, n−2), n ≥ 3, for the cases n ≡ 2 (mod 4), n ≡ 4 (mod 8) and n ≡ 8 (mod 16) and use the results to deduce that certain Stiefel-Whitney classes of the manifold are zero.