Volume 71, Issue 3

Effect algebras form a formal algebraic description of the structure of the so-called effects in a Hilbert space which serve as an event-state space for effects in quantum mechanics. This is why effect algebras are considered as logics of quantum mechanics, more precisely as an algebraic semantics of these logics. Because every productive logic is equipped with implication, we introduce here such a concept and demonstrate its properties. In particular, we show that this implication is connected with conjunction via a certain “unsharp” residuation which is formulated on the basis of a strict unsharp residuated poset. Though this structure is rather complicated, it can be converted back into an effect algebra and hence it is sound. Further, we study the Modus Ponens rule for this implication by means of so-called deductive systems and finally we study the contraposition law.

In 1939, H. S. Zuckerman provided a Hardy-Ramanujan-Rademacher-type convergent series that can be used to compute an isolated value of the overpartition function p ( n ). Computing p ( n ) by this method requires arithmetic with very high-precision approximate real numbers and it is complicated. In this paper, we provide a formula to compute the values of p ( n ) that requires only the values of p ( k ) with k ≤ n /2. This formula is combined with a known linear homogeneous recurrence relation for the overpartition function p ( n ) to obtain a simple and fast computation of the value of p ( n ). This new method uses only (large) integer arithmetic and it is simpler to program.

In this article, we attempt to survey all that has been written about 6-cycle systems. In addition, several new results, including many enumeration results, are included here for the first time. Our survey concludes with a list of open problems.

In this paper, we propose the notion of BF-closure spaces as concrete representation of bifinite domains. We prove that every bifinite domain can be obtained as the set of F-closed sets of some BF-closure space under set inclusion. Furthermore, we obtain that the category of bifinite domains and Scott-continuous functions is equivalent to that of BF-closure spaces and F-morphisms.

In this paper, we introduce the notion of fuzzy costs in a more general context, in universal algebra by the use of coset terms. We study the structure of fuzzy cosets by applying the general theory of algebraic fuzzy systems. Fuzzy cosets generated by a fuzzy set are characterized in different ways. It is also proved that the class of fuzzy cosets determined by an element forms an algebraic closure fuzzy set system. Finally, we give a set of necessary and sufficient conditions for a given variety of algebras to be congruence permutable by applying the theory of fuzzy cosets.

For a given set M of positive integers, a well-known problem of Motzkin asks to determine the maximal asymptotic density of M -sets, denoted by μ ( M ), where an M -set is a set of non-negative integers in which no two elements differ by an element in M . In 1973, Cantor and Gordon find μ ( M ) for | M | ≤ 2. Partial results are known in the case | M | ≥ 3 including some results in the case when M is an infinite set. Motivated by some 3 and 4-element families already discussed by Liu and Zhu in 2004, we study μ ( M ) for two families namely, M = { a , b , a + b , n ( a + b )} and M = { a , b , b − a , n ( b − a )}. For both of these families, we find some exact values and some bounds on μ ( M ). This number theory problem is also related to various types of coloring problems of the distance graphs generated by M . So, as an application, we also study these coloring parameters associated with these families.

A new concept of independence of real numbers, called degree independence, which contains those of linear and algebraic independences, is introduced. A sufficient criterion for such independence is established based on a 1988 result of Bundschuh, which in turn makes use of a generalization of Liouville’s estimate due to Feldman in 1968. Applications to numbers represented by Cantor series and product expansions are derived.

Let G be a finite group and let ψ ( G ) denote the sum of element orders of G . It is well-known that the maximum value of ψ on the set of groups of order n , where n is a positive integer, will occur at the cyclic group C n . For nilpotent groups, we prove a natural generalization of this result, obtained by replacing the element orders of G with the element orders relative to a certain subgroup H of G .

We introduce the notion of the fuzzy McShane integral in the linear topology sense and we discuse its relation with the fuzzy Pettis integral introduced recently by Chun-Kee Park in [ On the Pettis integral of fuzzy mappings in Banach spaces , Commun. Korean Math. Soc. 22 (2007), 535–545].

In this paper we give the upper bounds of the Hankel determinants of the second and third order for the class 𝓢 of univalent functions in the unit disc.

In this work, we introduce the notion of Jacobi-Dunkl positive definite sequences using the properties of the discrete Jacobi-Dunkl translation. Then, we establish the analogous of Herglotz’s theorem to characterize these sequences.

In the article, we present several new functional inequalities for the Gaussian hypergeometric function and generalized elliptic integral of the first kind.

In this paper, we develop the theory of a class of casual differential equations on a time scale. An existence theorem for casual differential equations on a time scale is given under mixed Lipschitz and compactness conditions by the fixed point theorem in Banach algebra due to Dhage. Some fundamental differential inequalities on a time scale are also presented which are utilized to investigate the existence of extremal solutions. The comparison principle on casual differential equations on a time scale is established. Our results in this paper extend and improve some well-known results.

The aim of this paper is to study the following second order system of difference equations xn+1=f(yn,yn− 1),yn+1=g(xn,xn− 1)$$\begin{array}{} x_{n+1} = f(y_{n},y_{n-1}),\quad y_{n+1} = g(x_{n},x_{n-1}) \end{array}$$ where n ∈ ℕ 0 , the initial values x −1 , x 0 , y −1 and y 0 are positive real numbers, the functions f , g : (0, +∞) 2 → (0, +∞) are continuous and homogeneous of degree zero. In this study, we establish results on local stability of the unique equilibrium point and to deal with the global attractivity, and so the global stability, some general convergence theorems are provided. Necessary and sufficient conditions on existence of prime period two solutions of our system are given. Also, a result on oscillatory solutions is proved. As applications of the obtained results, concrete models of systems of difference equations defined by homogeneous functions of degree zero are investigated. Our system generalize some existing works in the literature and our results can be applied to study new models of systems of difference equations. For interested readers, we left in the conclusion as open problems two more general systems of higher order defined by homogenous functions of degree zero.

In this paper, we introduce and study the concepts of Jordan amenability and Jordan biflatness of Banach algebras and compare those with the classical notions of amenability and biflatness. We also find some relations between Jordan amenability and the existence of Jordan approximate and virtual diagonals. These could be considered as Jordan versions of the classical results due to Johnson and Helemskii. We show that, for all C * -algebras, the concepts of amenability and Jordan amenability coincide.

In this paper, a space called geometric space, involving both the notions of area and length, is introduced in general setting. The interplay, between these two ideas, is studied. As a result, a new notion of orthogonality, called area-length orthogonality or A-L orthogonality, is demonstrated. It is shown that A-L orthogonality coincides with the standard notion of orthogonality for inner product spaces. Finally, it is proved that A-L orthogonality implies Birkhoff orthogonality, but not conversely.

In this paper, we investigate the ∗-Ricci operators on trans-Sasakian three-manifolds. We find conditions at which ∗- Ricci tensor on trans-Sasakian three-manifolds is symmetric and under which the ∗-Ricci operators are Reeb flow invariant.

We prove that a finite graph (allowing loops and multiple edges) is homeomorphic (isomorphic up to vertices of degree two) to the Reeb graph of a Morse–Bott function on a smooth closed n -manifold, for any dimension n ≥ 2. The manifold can be chosen orientable or non-orientable; we estimate the co-rank of its fundamental group (or the genus in the case of surfaces) from below in terms of the cycle rank of the graph. The function can be chosen with any number k ≥ 3 of critical values, and in a few special cases with k < 3. In the case of surfaces, the function can be chosen, except for a few special cases, as the height function associated with an immersion ℝ 3 .

If q : X → Y is a quotient map, then, in general, q × q : X × X → Y × Y may fail to be a quotient map. In this paper, by reviewing the concept of homotopy groups and quotient maps, we find under which conditions the map q × q is a quotient map, where q : Ω n ( X , x 0 ) → π n ( X , x 0 ), is the natural quotient map from the n th loop space of ( X , x 0 ), Ω n ( X , x 0 ), with compact-open topology to the quasitopological n th homotopy group π n ( X , x 0 ). Ultimately, using these results, we found some properties of first countable homotopy groups.