Volume 62, Issue 3

We give a characterization of subsets of effect algebras, that can be embedded into a range of an observable. To give this characterization, we introduce a new notion of compatibility support mappings.

We introduce the so-called tense operators in lattice effect algebras. Tense operators express the quantifiers “it is always going to be the case that” and “it has always been the case that” and hence enable us to express the dimension of time in the logic of quantum mechanics. We present an axiomatization of these tense operators and prove that every lattice effect algebra whose underlying lattice is complete can be equipped with tense operators. Such an effect algebra is called dynamic since it reflects changes of quantum events from past to future.

In this note we shall give some results on annihilators preserving congruence relations, or AP-congruences, in bounded distributive lattices. We shall give some new characterizations, and a topological interpretation of the notion of annihilator preserving congruences introduced in [JANOWITZ, M. F.: Annihilator preserving congruence relations of lattices, Algebra Universalis 5 (1975), 391–394]. As an application of these results, we shall prove that the quotient of a quasicomplemented lattice by means of a AP-congruence is a quasicomplemented lattice. Similarly, we will prove that the quotient of a normal latttice by means of a AP-congruence is also a normal lattice.

Torsion classes and radical classes of lattice ordered groups have been investigated in several papers. The notions of torsion class and of radical class of generalized Boolean algebras are defined analogously. We denote by T g and R g the collections of all torsion classes or of all radical classes of generalized Boolean algebras, respectively. Both T g and R g are partially ordered by the class-theoretical inclusion. We deal with the relation between these partially ordered collection; as a consequence, we obtain that T g is a Brouwerian lattice. W. C. Holland proved that each variety of lattice ordered groups is a torsion class. We show that an analogous result is valid for generalized Boolean algebras.

In this paper we add one more characterization of intra-regular ordered semigroups in the existing bibliography by proving that the ordered semigroups whose elements are separated by prime ideals are actually the intra-regular ordered semigroups, a result which generalizes the corresponding result of semigroups (without order).

Let α be a cardinal. The notion of α-complete retract of a Boolean algebra has been studied by Dwinger. Specker lattice ordered groups were investigated by Conrad and Darnel. Assume that G is a Specker lattice ordered group generated by a Boolean algebra B(G). The notion of α-complete retract of G can be defined analogously as in the case of Boolean algebras. In the present paper we deal with the relations between α-complete retracts of G and α-complete retracts of B(G).

The aim of this paper is to give new results about factorizations of the Fibonacci numbers F n and the Lucas numbers L n. These numbers are defined by the second order recurrence relation a n+2 = a n+1+a n with the initial terms F 0 = 0, F 1 = 1 and L 0 = 2, L 1 = 1, respectively. Proofs of theorems are done with the help of connections between determinants of tridiagonal matrices and the Fibonacci and the Lucas numbers using the Chebyshev polynomials. This method extends the approach used in [CAHILL, N. D.—D’ERRICO, J. R.—SPENCE, J. P.: Complex factorizations of the Fibonacci and Lucas numbers, Fibonacci Quart. 41 (2003), 13–19], and CAHILL, N. D.—NARAYAN, D. A.: Fibonacci and Lucas numbers as tridiagonal matrix determinants, Fibonacci Quart. 42 (2004), 216–221].

The main purpose of this paper is to prove the following result: Let R be a 2-torsion free semiprime *-ring. Suppose that θ, φ are endomorphisms of R such that θ is onto. If there exists an additive mapping F: R → R associated with a (θ, φ)-derivation d of R such that F(xx*) = F(x)θ(x*) + φ(x)d(x*) holds for all x ∈ R, then F is a generalized (θ, φ)-derivation. Further, some more related results are obtained.

We introduce a new subgroup embedding property in a finite group called weakly S-quasinormality. We say a subgroup H of a finite group G is weakly S-quasinormal in G if there exists a normal subgroup K such that HK ⊴ G and H ∩ K is S-quasinormally embedded in G. We use the new concept to investigate the properties of some finite groups. Some previously known results are generalized.

In this paper, we present some results concerning functions which are represented as the maximum of two extra strong Świątkowski functions.

Let $\mathfrak{A}$ be a normed algebra with identity, Ω be a locally compact Hausdorf space and λ be a positive Radon measure on Ω with supp(λ) = Ω. In this paper, we establish a necessary and sufficient condition for L 1(Ω, $\mathfrak{A}$) to be an algebra with pointwise multiplication. Under this condition, we then characterize compact and weakly compact left multipliers on L 1(Ω, $\mathfrak{A}$).

By making use of a general linear operator we introduce several new subclasses of multvalent meromorphic functions and investigate various inclusion relationships. Several interesting integral-preversing properties are also discussed.

This paper is concerned with oscillation of the second-order quasilinear functional dynamic equation $(r(t)(x^\Delta (t))^\gamma )^\Delta + p(t)x^\beta (\tau (t)) = 0,$ on a time scale $\mathbb{T}$ where γ and β are quotient of odd positive integers, r, p, and τ are positive rd-continuous functions defined on $\mathbb{T},\tau :\mathbb{T} \to \mathbb{T}$ and $\mathop {\lim }\limits_{t \to \infty } \tau (t) = \infty $. We establish some new sufficient conditions which ensure that every solution oscillates or converges to zero. Our results improve the oscillation results in the literature when γ = β, and τ(t) ≤ t and when τ(t) > t the results are essentially new. Some examples are considered to illustrate the main results.

Let F(A) denote the set of all bounded sequences summable by Abel’s method. It is known, that F(A) is a linear subspace of the linear metric space (S, ρ) of all bounded sequences endowed with the sup metric. It is shown in [KOSTYRKO, P.: Convergence fields of regular matrix transformations 2, Tatra Mt. Math. Publ. 40 (2008), 143–147] that the convergence field of a regular matrix transformation is a σ-porous set. We show that F(A) is very porous in S.

We show that LUB of the set of G-expansive constants for a G-expansive homeomorphism h on a compact metric G-space, G compact, is not a G-expansive constant for h. We obtain a result regarding projecting and lifting of G-expansive homeomorphisms having interesting applications. We also prove that the G-expansiveness is a dynamical property for homeomorphisms on compact metric G-spaces and study G-periodic points.

We construct smooth circle actions on symplectic manifolds with non-symplectic fixed point sets or cyclic isotropy sets. All such actions are not compatible with any symplectic form on the manifold in question. In order to cover the case of non-symplectic fixed point sets, we use two smooth 4-manifolds (one symplectic and one non-symplectic) which become diffeomorphic after taking the products with the 2-sphere. The second type of actions is obtained by constructing smooth circle actions on spheres with non-symplectic cyclic isotropy sets, which (by the equivariant connected sum construction) we carry over from the spheres on products of 2-spheres. Moreover, by using the mapping torus construction, we show that periodic diffeomorphisms (isotopic to symplectomorphisms) of symplectic manifolds can provide examples of smooth fixed point free circle actions on symplectic manifolds with non-symplectic cyclic isotropy sets.

Let M 2n be a closed smooth manifold homotopy equivalent to the complex projective space ℂP(n). It is known that the first Pontrjagin class p 1(M) of M 2n has the form (n+1+24α(M))u 2 for some integer α(M) where u is a generator of H 2(M; ℤ). We prove that α(M) is even when n is even but not divisible by 64.

In this paper, a two-species nonautonomous Lotka-Volterra model of population growth in a polluted environment is proposed. Global asymptotic behaviour of this model by constructing suitable bounded functions has been investigated. It is proved that each population for competition, predation and cooperation systems respectively is uniformly persistent (permanent) under appropriate conditions. Sufficient conditions are derived to confirm that if each of competition, predation and cooperation systems respectively admits a positive periodic solution, then it is globally asymptotically stable.

We define the crossed product of a C*-algebra by a hypergroup via a group coaction. We generalize the results on Hecke C*-algebra crossed products to our setting.