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Mohammad Ashraf and Bilal Ahmad Wani

Abstract

The purpose of this paper is to investigate identities with Jordan *-derivations in semiprime *-rings. Let ℛ be a 2-torsion free semiprime *-ring. In this paper it has been shown that, if admits an additive mapping D : ℛ→ℛsatisfying either D(xyx) = D(xy)x*+ xyD(x) for all x,y, or D(xyx) = D(x)y*x*+ xD(yx) for all pairs x, y, then D is a *-derivation. Moreover this result makes it possible to prove that if satis es 2D(xn) = D(xn− 1)x* + xn− 1 D(x) + D(x)(x*)n− 1 + xD(xn− 1) for all x and some xed integer n ≥ 2, then D is a Jordan *-derivation under some torsion restrictions. Finally, we apply these purely ring theoretic results to standard operator algebras 𝒜(). In particular, we prove that if be a real or complex Hilbert space, with dim() > 1, admitting a linear mapping D : 𝒜() → ℬ() (where () stands for the bounded linear operators) such that

2D(An)=D(An1)A*+An1D(A)+D(A)(A*)n1+AD(An1)

for all A𝒜(). Then D is of the form D(A) = AB−BA* for all A𝒜() and some fixed B(), which means that D is Jordan *-derivation.

Open access

Emanuele Munarini

Abstract

In 1997, Richard André-Jeannin obtained a symmetric identity involving the reciprocal of the Horadam numbers Wn, defined by a three-term recurrence Wn +2 = P Wn +1 − QWn with constant coefficients. In this paper, we extend this identity to sequences {an}n ∈ℕ satisfying a three-term recurrence an +2 = pn +1 an +1 + qn +1 an with arbitrary coefficients. Then, we specialize such an identity to several q-polynomials of combinatorial interest, such as the q-Fibonacci, q-Lucas, q-Pell, q-Jacobsthal, q-Chebyshev and q-Morgan-Voyce polynomials.

Open access

Jean-Luc Baril, Sergey Kirgizov and Armen Petrossian

Abstract

We investigate an equivalence relation on permutations based on the pure descent statistic. Generating functions are given for the number of equivalence classes for the set of all permutations, and the sets of permutations avoiding exactly one pattern of length three. As a byproduct, we exhibit a permutation set in one-to-one correspondence with forests of ordered binary trees, which provides a new combinatorial class enumerated by the single-source directed animals on the square lattice. Furthermore, bivariate generating functions for these sets are given according to various statistics.

Open access

David Callan and Toufik Mansour

Abstract

This paper is one of a series whose goal is to enumerate the avoiders, in the sense of classical pattern avoidance, for each triple of 4-letter patterns. There are 317 symmetry classes of triples of 4-letter patterns, avoiders of 267 of which have already been enumerated. Here we enumerate avoiders for all small Wilf classes that have a representative triple containing the pattern 1342, giving 40 new enumerations and leaving only 13 classes still to be enumerated. In all but one case, we obtain an explicit algebraic generating function that is rational or of degree 2. The remaining one is shown to be algebraic of degree 3. We use standard methods, usually involving detailed consideration of the left to right maxima, and sometimes the initial letters, to obtain an algebraic or functional equation for the generating function.

Open access

David Callan and Toufik Mansour

Abstract

Recently, it has been determined that there are 242 Wilf classes of triples of 4-letter permutation patterns by showing that there are 32 non-singleton Wilf classes. Moreover, the generating function for each triple lying in a non-singleton Wilf class has been explicitly determined. In this paper, toward the goal of enumerating avoiders for the singleton Wilf classes, we obtain the generating function for all but one of the triples containing 1324. (The exceptional triple is conjectured to be intractable.) Our methods are both combinatorial and analytic, including generating trees, recurrence relations, and decompositions by left-right maxima. Sometimes this leads to an algebraic equation for the generating function, sometimes to a functional equation or a multi-index recurrence amenable to the kernel method.

Open access

Massimo Angrisani, Giovanni di Nella, Cinzia di Palo and Augusto Pianese

Abstract

This paper deals with the problem of the optimal rate of return to be paid by a defined contribution pension system to its participants’ savings, namely the rate that achieves the goal of the most favorable returns on their contributions jointly with the sustainability of the pension system.

We consider defined contribution pension systems provided with a funded component, and for their study we use the “theory of the logical sustainability of pension systems” already developed in several previous works. In this paper, we focus on pension systems in a demographically stable state, whereas the productivity of the active participants and the financial rate of return on the pension system’s fund, rates that constitute the “ingredients” of the optimal rate of return on contributions, are modeled by two stochastic processes.

We show that the decisional rule defining the optimal rate of return on contributions is optimal in the sense that it is effective in terms of sustainability, and also efficient in the sense that if the system pays to its participants’ contributions a rate of return that is either higher or lower than the one provided by the rule, then the pension system becomes unsustainable or overcapitalized, respectively.

Open access

Dorah Riah Mabule

Abstract

The aim of this article is to explore the dynamic of language choice and language use as well as to examine the effect of language policy on language attitudes in the Department of Correctional Services (DCS) with special reference to Pretoria Central Prison, now called Kgoši Mampuru Correctional facility where there is some resistance to the use of English as the only official language of business.

A case study was conducted at this facility to find out the language attitudes of the participants towards English as the only official language of business. A questionnaire was used to evoke the participants’ attitudes and beliefs regarding the importance of the use of other official languages (indigenous languages which, like English, also have official status) in their daily lives. A total of 60 correctional services staff and 280 offenders took part in this research study. Interviews and observations were mainly carried out at the research site to triangulate the data. Only the findings from the 280 offenders will be reported in this article.

The findings of this research study show that the participants were keen to use their languages of choice and favoured the language functions of their indigenous languages. The findings suggest that the prevailing language attitudes were in contrast with the aims of language policy at DCS thus making the effect of language planning not to be realized.

Open access

Dennis Matotoka and Kola O. Odeku

Abstract

In South Africa, progressive laws, policies and institutions established since 1996 seek to proliferate the representation of black African women in the private sector. However, the sector remains stagnant in giving opportunities to black African women to attain and occupy managerial and leadership positions. Black African women are not yet accepted as an integral of part of decision-making in the private sector contrary to the public sector that has somewhat progressed to place black African women in key decision-making positions in government. Consequently, black African women in the private sector predominately dominate the unskilled labour positions. The underrepresentation of black African women essentially denies them of economic participation and equality in the workplace. It is against the backdrop of this underrepresentation that this article analyses salient transformative legislative interventions that have been put in place to foster ample representation of black African women into managerial positions in the private sector. However, the concern is that the current legislative framework in South Africa does not explicitly make it mandatory for the private sector to achieve a specific target of black African women representation at the top management positions. The article showcases that the glass ceiling in the private sector is real and is nurtured by the organizational culture, policies and strategies which promote exclusion. Therefore, effective implementation and enforcement of laws and policies fostering mainstreaming of black African women into top managerial positions will help in breaking down the glass-ceiling. This will become realizable with the cooperation of all stake holders and role players where there is deliberate effort to empower and enhance the skill and capacity of women through quality training and education that will drive and deliver robust career development.

Open access

Hrvoje Jošić and Matej Metelko

Abstract

This paper presents empirical evidence on the validity of the Linder hypothesis in the case of Croatia. According to the Linder hypothesis, one of the new theories of international trade, countries with a similar level of income per capita should trade more. In order to investigate the trade pattern of Croatia's international trade, a panel regression model is formulated including 184 Croatia's import partner countries in the period from 2000 to 2016. The Linder effect was displayed and calculated using the Linder variable expressed as an absolute difference between GDP per capita of the importing and the exporting country. The cross-country panel regression model is estimated using Pooled OLS, Fixed and Random effects models. Results of the analysis have shown that the validity of the Linder hypothesis for Croatia cannot be accepted. Instead, the structure of Croatia's trade is in line with the gravity model of international trade.

Open access

Darja Boršič and Jani Bekő

Abstract

The paper assesses the existence of purchasing power parity (PPP) in ASEAN+3 economies taking into account EUR and USD as reference currencies. The research refers to the period from January 2000 to June 2017 and there are three points of view: we tested the period as a whole and then the pre-crisis period and the postcrisis period regarding the structural break due to the economic crisis. The evaluated economies include Brunei, Cambodia, China, Indonesia, Japan, Korea, Laos, Malaysia, Myanmar, the Philippines, Singapore, Thailand and Vietnam. A range of panel unit root tests are applied, covering the Levin, Lin and Chu test, the Breitung test, the Im, Pesaran and Shin test, the ADF-Fisher test and the PP-Fisher test. In cases where the unit root is rejected, the validity of PPP is confirmed. However, our results are ambiguous and depend on the selection of the base currency, the time period observed and on the choice of the methodology.