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A minimal model for the description of cuprate superconductor characteristics on doping scale (hole and electron) is developed. The leading interband pairing channel couples an itinerant band and defect states created by doping. Bare gaps between them are supposed and become closed by extended doping. Band overlap conditions determine special points in the phase diagram. Nodal and antinodal momentum regions are distinguished. Illustrative calculations have been made using a mean-field pair-transfer multiband Hamiltonian and corresponding free-energy expansions. The results are self-consistent and demonstrate that the elaborated approach is able to reproduce characteristic features of cuprate superconductors as, e.g., the doping dependence of T_c, superconducting gaps and pseudogaps, supercarrier density and effective mass, coherence length and penetration depth, critical magnetic fields and some other properties. Interband pairing scheme is suggested to be an essential aspect of cuprate superconductivity.

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.

The electronic structure and its derived valence and conduction charge distributions along with the optical properties of zinc-blende GaAs${}_{1-x}$P${}_x$ ternary alloys have been studied. The calculations are performed using a pseudopotential approach under the virtual crystal approximation (VCA) which takes into account the compositional disorder effect. Our findings are found to be generally in good accord with experiment. The composition dependence of direct and indirect bandgaps showed a clear bandgap bowing. The nature of the gap is found to depend on phosphorous content. The bonding and ionicity of the material of interest have been examined in terms of the anti-symmetric gap and charge densities. The variation in the optical constants versus phosphorous concentration has been discussed. The present investigation may give a useful applications in infrared and visible spectrum light emitters.

In contrast to the classical situation, it is known that many Laplacian operators on fractals have gaps in their spectrum. This surprising fact means there can be no limit in the Weyl counting formula and it is a key ingredient in proving that the convergence of Fourier series on fractals can be better than in the classical setting. Recently, it was observed that the Laplacian on the Sierpinski gasket has the stronger property that there are intervals which contain no ratios of eigenvalues. In this paper we give general criteria for this phenomena and show that Laplacians on many interesting classes of fractals satisfy our criteria.

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 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.

Starting from a result of Stewart, Tijdeman and Ruzsa on iterated difference sequences, we introduce the notion of iterated compositions of linear operations. We prove a general result on the stability of such compositions (with bounded coefficients) on sets of integers having a positive upper density.

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].

For an odd prime p, let ${𝔽}_p$ be the finite field of p elements. The main purpose of this paper is to establish new results on gaps between the elements of multiplicative subgroups of finite fields. For any $a,b,c\in {𝔽}_p^{*}$, we also obtain new upper bounds of the following double character sum

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.

In this study, we tried to examine the pros and cons of the annular type of fuel concerning mainly with the temperatures and stresses of pellet and cladding. The inner and outer gaps between pellet and cladding may play an important role on the temperature distribution and stress distribution of fuel system. Thus, we tested several inner and outer gap cases, and we evaluated the effect of gaps on fuel systems. We conducted thermo-elastic-plastic-creep analyses using an in-house thermo-elastic-plastic-creep finite element program that adopted the 'effective-stress-function' algorithm. Most analyses were conducted until the gaps disappeared; however, certain analyses lasted for 1582 days, after which the fuels were replaced. Further study on the optimal gaps sizes for annular nuclear fuel systems is still required.

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.

The growth of a second derivative of the logarithm of the maximum modulus of an entire function provides information about the location of zeros of the function and its sections. We present a survey of work on this topic along with some recent sharp results and open questions.

Let Ω be a periodic unbounded domain in ℝⁿ. It is well-known that the spectrum of the Neumann Laplacian in L₂(Ω) is a locally finite union of compact intervals called bands. The neighbouring bands may overlap, otherwise we have a gap in the spectrum. Our goal is to construct such periodic domain Ω that the spectrum of the corresponding Neumann Laplacian has at least m gaps (m ∈ ℕ is a given number) and these gaps are close to predefined intervals.