An Extension of the Rényi Formula

In this paper, as a natural extension of the Rényi formula which counts labeled connected unicyclic graphs, we present a formula for the number of labeled(k + 1)-uniform(p, q)-unicycles as follows:U(k+1)p, q={p!/2[(k-1)!]~q·∑qt=2(q~(q-t-1)· sgn(tk- 2))/(q- t)!, p = qk,0, p≠qk,where k, p, q are positive integers.     

An Extension of Bernstein Polynomials on the Semi–Axis

The paper deals with the approximation of functions f on (0,+), where f can be singular at the origin, by means of Bernstein-type sequences. Error estimates in weighted uniform spaces with some converse results are given.Research was supported in part by grant INDAM-GNIM Progetto Equazioni Integrali e Problemi di Algebra Lineare Connessi.     

An Extension of the Weak Law of Large Numbers for Exchangeable Sequences

We prove an extension of the Kolmogorov-Feller weak law of large numbers for exchangeable sequences, by relaxing the finite mean condition therein.     

Derivation of Orowan's Law from the Peierls–Nabarro Model

In this paper we consider the time dependent Peierls-Nabarro model in dimension one. This model is a semi-linear integro-differential equation associated to the half Laplacian. This model describes the evolution of phase transitions associated to dislocations. At large scale with well separated dislocations, we show that the dislocations move at a velocity proportional to the effective stress. This implies Orowan's law which claims that the plastic strain velocity is proportional to the product of the density of dislocations by the effective stress.     

Do market share and efficiency matter for each other? An application of the zero-sum gains data envelopment analysis

Current studies that use traditional data envelopment analysis (DEA) neglect the 100% market share restriction. This study adopts zero-sum gains data envelopment analysis to measure the efficiency scores of securities firms (SFs) and indicates that the traditional DEA model underestimates the efficiency scores of inefficient SFs. This research analyses 266 integrated securities firms in Taiwan from 2001 to 2005 and employs three inputs (fixed assets, financial capital, and general expenses) and a single output (market share). The foreign-affiliated ownership of SFs positively affects the efficiency scores. The two-stage least squares procedure confirms that the market share and efficiency score simultaneously reinforce each other.     

How relevant is the lack of reciprocity in pairwise comparisons? An experiment with AHP

We consider the 1-median problem with euclidean distances with uncertainty in the weights, expressed as possible changes within given bounds and a single budget constraint on the total cost of change. The upgrading (resp. downgrading) problem consists of minimizing (resp. maximizing) the optimal 1-median objective value over these weight changes. The upgrading problem is shown to belong to the family of continuous single facility location-allocation problems, whereas the downgrading problem reduces to a convex but highly non-differentiable optimization problem. Several structural properties of the optimal solution are proven for both problems, using novel planar partitions, the knapsack Voronoi diagrams, and lead to polynomial time solution algorithms.     

Correction and strengthening of “How large are the level sets of the Takagi function?”

The purpose of this note is to correct an error in an earlier paper by the author about the level sets of the Takagi function (Monatsh Math 167:311–331, 2012), and to prove a stronger form of one of the main results of that paper.     

How to use the method of the improvising differentiation?

<正>We know the Substitution Rule,which is based on the following identity in differentiation.∫f(g(x))g′(x)dx=∫f(u)du(where u=g(x)).Now,we'll talk about how to use the method of the improvising differentiation,and give some examples by using basic     

An Optimal Extension of the Polak–Ribière–Polyak Conjugate Gradient Method

Based on a singular value analysis on an extension of the Polak–Ribière–Polyak method, a nonlinear conjugate gradient method with the following two optimal features is proposed: the condition number of its search direction matrix is minimum and also, the distance of its search direction from the search direction of a descent nonlinear conjugate gradient method proposed by Zhang et al. is minimum. Under proper conditions, global convergence of the method can be achieved. To enhance e?ciency of the proposed method, Powell’s truncation of the conjugate gradient parameters is used. The method is computationally compared with the nonlinear conjugate gradient method proposed by Zhang et al. and a modified Polak–Ribière–Polyak method proposed by Yuan. Results of numerical comparisons show e?ciency of the proposed method in the sense of the Dolan–Moré performance profile.     

How well prepared are the teachers of tomorrow? An examination of prospective mathematics teachers' pedagogical content knowledge

There is a growing emphasis in the teaching profession on pedagogical content knowledge (PCK) as an important knowledge component. The study reported in this article investigates Turkish prospective mathematics teachers’ mathematics teaching knowledge in the numbers content domain. A series of 10 open-ended scenario-type questions were adopted to challenge 83 prospective mathematics teachers’ knowledge of the learner and presentation of content in the context of PCK. The participants’ responses were analysed by means of rubrics and scoring guides developed by the researchers. The results showed that many of the future teachers performed well in determining what misconceptions students might express in the given scenarios. However, a majority of the participants performed poorly on presentation of content in terms of instructional strategies. In line with these results, the authors offer some suggestions for teacher training programmes.     

Do mergers enhance the performance of hospital efficiency?

Due to the wave of mergers that have taken place in the USA, the early 1990s could be labelled as a restructuring era for health care systems. The question of whether mergers have an impact on organizational performance is still an area of interest for health services researchers. In this study, we examined the impacts of horizontal mergers of US hospital's technical efficiency before and after merger using longitudinal Data Envelopment Analysis (DEA). The findings of our study illustrate that mergers do increase a hospital's level of efficiency. Constant returns-to-scale model indicated an overall reduction in input utilisation after merger, compared to variable returns-to-scale model. This indicates the role of scale efficiency as a dominant source of improvement in inefficiency of hospitals involved in horizontal mergers, but not for technical efficiency. Suggestions for future study are provided.     

How Does Tempering Affect the Local and Global Properties of Fractional Brownian Motion?

Journal of Theoretical Probability - The present paper investigates the effects of tempering the power law kernel of the moving average representation of a fractional Brownian motion (fBm) on some...     

Extension of a Function from the Exterior of an Interval to a Positive-Definite Function on the Entire Axis and an Approximation Characteristic of the Class W M r, β

We establish sufficient conditions for the extension of a function defined on [a, +), where a > 0, to a positive-definite function on the entire axis.     

Did Lewis Carroll own a copy of George Boole's Laws of thought? An argument from the sale catalogues

Private libraries of scientists offer valuable information on their character, work, and acquaintances. Charles L Dodgson (alias Lewis Carroll) constructed an impressive library of several thousand volumes. The sale catalogue of Carroll's library reveals that it contained at his death most of the major logic works that would be expected for a British mathematical logician of the time. However, there is dispute as to the presence of the most important logic book of all: George Boole's Laws of thought (1854). The absence of this work would make both an unfortunate and an intriguing gap. This paper explains the source of this dispute and introduces a new argument from the sale catalogues centred on the dissemination of the books after the sale of the library.     

Do Log Factors Matter? On Optimal Wavelet Approximation and the Foundations of Compressed Sensing

Foundations of Computational Mathematics - A signature result in compressed sensing is that Gaussian random sampling achieves stable and robust recovery of sparse vectors under optimal conditions...     

How Common Is the Common Core? A Global Analysis of Math Teaching and Learning

The U.S. educational system is undergoing rapid and substantial changes with many states grappling with the adoption of the Common Core State Standards in Mathematics (CCSSM). Important research questions remain unanswered regarding the potential efficacy of the CCSSM to improve student math performance compared with students around the globe. This article utilized TIMSS 2007 8th‐grade math assessment results and curricular frameworks to (1) measure the degree of overlap between the CCSSM and TIMSS standards, and (2) use this finding to create a predictive model to determine the potential efficacy of the CCSSM in improving the U.S. 8th‐grade student math performance compared with six culturally matched, TIMSS‐assessed countries, provinces, and states. Comparisons of CCSSM and TIMSS‐assessed jurisdictions show that the CCSSM holds many items in common with TIMSS‐assessed jurisdictions, but lacks rigor in some key areas. The CCSSM deficiencies include algebraic knowledge and problem solving at the 8th‐grade level, and are a significant detractor from the CCSSM when compared with TIMSS.