Narrow Results

We provide algorithms for performing computations in generalized numerical semigroups, that is, submonoids of ${\mathbb{N}}^d$ with finite complement in ${\mathbb{N}}^d$. These semigroups are affine semigroups, which in particular implies that they are finitely generated. For a given finite set of elements in ${\mathbb{N}}^d$ we show how to deduce if the monoid spanned by this set is a generalized numerical semigroup and, if so, we calculate its set of gaps. Also, given a finite set of elements in ${\mathbb{N}}^d$ we can determine if it is the set of gaps of a generalized numerical semigroup and, if so, compute the minimal generators of this monoid. We provide a new algorithm to compute the set of all generalized numerical semigroups with a prescribed genus (the cardinality of their sets of gaps). Its implementation allowed us to compute (for various dimensions) the number of numerical semigroups of higher genus than has previously been computed.

Based on a systematic review of the literature, this article seeks to analyse the main questions, interpretations, and typologies for minority entrepreneurship over recent decades. To this end, we made recourse to the Scopus database for our article collection process that returned 220 articles for analysis. The results enable the identification of seven congruent research units (categories), with their own respective approaches and contributions: i) attitudes and motivations; ii) barriers and challenges; iii) interventionist policies and cultures; iv) comparisons between minorities and non-minorities; v) networks and resources; vi) impact on the local economy; vii) autonomous employment or entrepreneurship through need. This study further contributes by enabling future researchers to target their efforts on the still poorly explored shortcomings in the literature and providing a temporal overview of this theme.

We survey various developments in Number Theory that were inspired by classical papers by Roth [On the gaps between squarefree numbers, J. London Math. Soc. 26 (1951) 263–268] and by Halberstam and Roth [On the gaps between consecutive k-free integres, J. London Math. Soc. 26 (1951) 268–273].

In multiband superconductivity interband interaction channels creating the pairs of intraband $(W_a)$ and interband $(W_b)$ compositions can appear. Simultaneous functioning of these channels is investigated here. A three-band model where two similar bands $(a;b)$ interact with the itinerant-band has been proposed. The mean field Hamiltonian incorporates three order parameters ${\mathrm{\Delta }}_a$; ${\mathrm{\Delta }}_{ad}$; ${\mathrm{\Delta }}_{b1,2}$ genetically associated with dispersive bands. Calculated quasiparticle energies and operator averages lead to a coupled nonlinear equation system for the order parameters. Illustrative calculations versus temperature have been made for overlapping bands. At fixed parameters, the basic system has two independent solutions. The free energy has a complicated structure of extremal points. The interaction of channels with intra- and interband points is seen. These channels compete in general. From the point where the gap, type parameters associated with one channel become zero, the other continues as the first channel was logged out. The general behaviour of order parameters distribution is very sensible to $W_b$. Events of critical nature appear. Weakening of $W_b$ stimulates the formation of closed bubbles built up by the same type parameters from different parallel solutions.

We give a description of a class of numerical semigroups with embedding dimension equal to 4, defined by four pairwise relatively prime nonnegative integers $n$, $x$, $y_1$ and $y_2$ such that $y_1+y_2=tn$, for $t\ge 2$ and $t\in \mathbb{N}$. Such description provides a mode to determine the characteristics of the corresponding numerical semigroups: the Frobenius number, gaps, genus, etc.

Islamic Banking and Finance (IBF) has had forty years of impressive growth, yet, within the Muslim world, there is a paradox of huge accumulated wealth and seriously underdeveloped financial markets. This paper examines whether over its 40-year existence, IBF has lived up to its anticipated goals and objectives. It argues that growth has been patchy with huge gaps, missing links and missed opportunities. IBF has progressed not through innovation but imitation. Having replicated conventional banking and capital market products, IBF products like sukuk have largely mimicked conventional coupon bonds. Thus, IBF has not been able to offer new asset classes that can add value. Its products have the same interest rate risks and contagion effects. Given the higher transaction costs involved in structuring shariah compliant products, it is hard to see how they can offer even similar, if not superior returns. Further, with lower liquidity from less active and incomplete secondary markets, IBF cannot continue to play the same game as conventional players and hope to come out ahead. Yet, the paper argues, that the shariah offers huge latitude for innovation, especially of risk sharing instruments. A funding transaction or a sukuk based on a risk-sharing structure like Mudarabah, would have an entirely different risk return profile, no interest rate risk, very low correlation and no contagion. Further. increased reliance on risk-sharing contracts would move the financial system away from the fractional reserve framework, and closer to a mutual fund model. Such a move minimises systemic risk through risk dissipation and reduces the liquidity mismatch inherent to banking. Where the macro economy is concerned, system stability is enhanced by reducing risk concentration within the banking system and minimizes the contingent liabilities of governments by minimizing the use of deposit insurance. If IBF is to move on to a more successful and effective next phase, the gaps and inadequacies identified here, will have to be addressed.

In this report we highlight significant advances in university mathematics education research as well as areas that are in need for additional research insights. We add here to the rich set of literature reviews within the last several years. A novel aspect of this literature review is the fact that the areas of accomplishment and areas for growth were identified based on thematic analysis of survey responses from 119 experts in the field. The review provides a useful overview for both seasoned scholars and those new to research in university mathematics education.