Private versus public high schools in Spain: disentangling managerial and programme efficiencies

The purpose of this paper is to compare the efficiency of a set of Spanish public and private high schools using data envelopment analysis (hereafter DEA). In view of the usual difficulties of obtaining reliable budget figures on private schools, we have used a restrictive efficiency notion which focuses on the relation between the academic results obtained by each school and the socio-economic background and academic profile of its pupils. In this study, special emphasis is placed upon decomposing the overall inefficiencies of each school into managerial (due to individual performance) and programme (due to structural differences between management models) components. Our results reveal that although, in general, private schools obtain better academic results than public schools in absolute terms, this is not the consequence of comparatively more effective management but rather of having pupils with a more favourable background for the educational process.     

Approximation by Linear Fractional Transformations of Simple Partial Fractions and Their Differences

Let f be a function and ρ be a simple partial fraction of degree at most n. Under linear-fractional transformations, the difference f ? ρ becomes the difference of another function and a certain simple partial fraction of degree at most n with a quadratic weight. We study applications of this important property. We prove a theorem on uniqueness of interpolating simple partial fraction, generalizing known results, and obtain estimates for the best uniform approximation of certain functions on the real semi-axis ?⁺. For continuous functions of rather common type we first obtain estimates of the best approximation by differences of simple partial fractions on ?⁺. For odd functions we obtain such estimates on the whole axis ?.     

Partial spread and vectorial generalized bent functions

In 2007, Sun et al. (IEEE Trans Inf Theory 53(8):2922–2933, 2007) presented new variants of RSA, called Dual RSA, whose key generation algorithm outputs two distinct RSA moduli having the same public and private exponents, with an advantage of reducing storage requirements for keys. These variants can be used in some applications like blind signatures and authentication/secrecy. In this paper, we give an improved analysis on Dual RSA and obtain that when the private exponent is smaller than \(N^{0.368}\), the Dual RSA can be broken, where N is an integer with the same bitlength as the modulus of Dual RSA. The point of our work is based on the observation that we can split the private exponent into two much smaller unknown variables and solve a related modular equation on the two unknown variables and other auxiliary variables by making use of lattice based methods. Moreover, we extend this method to analyze the common private exponent RSA scheme, a variant of Dual RSA, and obtain a better bound than previous analyses. While our analyses cannot be proven to work in general, since we rely on some unproven assumptions, our experimental results have shown they work in practice.     

Rogue-wave solutions of the Zakharov equation

Using the bilinear transformation method, we derive general rogue-wave solutions of the Zakharov equation. We present these Nth-order rogue-wave solutions explicitly in terms of Nth-order determinants whose matrix elements have simple expressions. We show that the fundamental rogue wave is a line rogue wave with a line profile on the plane (x, y) arising from a constant background at t ? 0 and then gradually tending to the constant background for t ? 0. Higher-order rogue waves arising from a constant background and later disappearing into it describe the interaction of several fundamental line rogue waves. We also consider different structures of higher-order rogue waves. We present differences between rogue waves of the Zakharov equation and of the first type of the Davey–Stewartson equation analytically and graphically.     

Behavior of weak type bounds for high dimensional maximal operators defined by certain radial measures

As shown in Aldaz (Bull. Lond. Math. Soc. 39:203–208, 2007), the lowest constants appearing in the weak type (1, 1) inequalities satisfied by the centered Hardy-Littlewood maximal operator associated to certain finite radial measures, grow exponentially fast with the dimension. Here we extend this result to a wider class of radial measures and to some values of p > 1. Furthermore, we improve the previously known bounds for p = 1. Roughly speaking, whenever \({p\in (1, 1.03]}\), if μ is defined by a radial, radially decreasing density satisfying some mild growth conditions, then the best constants c _p,d,μ in the weak type (p, p) inequalities satisfy c _p,d,μ ≥ 1.005 ^d for all d sufficiently large. We also show that exponential increase of the best constants occurs for certain families of doubling measures, and for arbitrarily high values of p.     

Series in multiplicative systems in Lorentz spaces

Series in multiplicative systems χ with generalized monotone coefficients are studied. Necessary and sufficient Hardy–Littlewood type conditions for the sums of such series to belong to the Lorentz space are proved. As corollaries, we establish estimates of best approximation in the system χ and Konyushkov-type theorems on the equivalence of O- and ?-relations for the weighted sums of the Fourier coefficients in the system χ and for the best approximations.     

A comparative analysis of British and Taiwanese students’ conceptual and procedural knowledge of fraction addition

This study examines students’ procedural and conceptual achievement in fraction addition in England and Taiwan. A total of 1209 participants (561 British students and 648 Taiwanese students) at ages 12 and 13 were recruited from England and Taiwan to take part in the study. A quantitative design by means of a self-designed written test is adopted as central to the methodological considerations. The test has two major parts: the concept part and the skill part. The former is concerned with students’ conceptual knowledge of fraction addition and the latter is interested in students’ procedural competence when adding fractions.

There were statistically significant differences both in concept and skill parts between the British and Taiwanese groups with the latter having a higher score. The analysis of the students’ responses to the skill section indicates that the superiority of Taiwanese students’ procedural achievements over those of their British peers is because most of the former are able to apply algorithms to adding fractions far more successfully than the latter. Earlier, Hart [1 Hart KM. Children's understanding of mathematics: 11–16. Oxford: Alden Press; 1981. [Google Scholar]] reported that around 30% of the British students in their study used an erroneous strategy (adding tops and bottoms, for example, 2/3 + 1/7 = 3/10) while adding fractions. This study also finds that nearly the same percentage of the British group remained using this erroneous strategy to add fractions as Hart found in 1981.

The study also provides evidence to show that students’ understanding of fractions is confused and incomplete, even those who are successfully able to perform operations. More research is needed to be done to help students make sense of the operations and eventually attain computational competence with meaningful grounding in the domain of fractions.     

Design of balanced MBA student teams

In some schools and universities, students must sometimes be divided into several teams in such a way that each team provides a good representation of the classroom population. In this paper, two different ways of measuring the balance among teams are proposed: min-sum and min-max objective functions. For the first function and the L₁-norm used in the space of attributes, an exact solution method based on a set partitioning formulation and on the enumeration of all possible team patterns is presented. For the second objective function, a set partitioning formulation is also considered, but as an approximation. In order to solve large problem instances, we have also developed metaheuristics based on variable neighbourhood search. Models and methods are tested on data from an MBA programme.     

A linear set view on KM-arcs

In this paper, we study KM-arcs of type t, i.e., point sets of size \(q+t\) in \(\mathrm {PG}(2,q)\) such that every line contains 0, 2 or t of its points. We use field reduction to give a different point of view on the class of translation arcs. Starting from a particular \(\mathbb {F}_2\)-linear set, called an i -club, we reconstruct the projective triads, the translation hyperovals as well as the translation arcs constructed by Korchmáros-Mazzocca, Gács-Weiner and Limbupasiriporn. We show the KM-arcs of type \(q/4\) recently constructed by Vandendriessche are translation arcs and fit in this family. Finally, we construct a family of KM-arcs of type \(q/4\). We show that this family, apart from new examples that are not translation KM-arcs, contains all translation KM-arcs of type \(q/4\).     

On the sharpness of Jackson’s inequality in the spaces L p on the half-line with power weight

In the space L _p , 1 ≤ p < 2, on the half-line with power weight, Jackson’s inequality between the value of the best approximation of a function by even entire functions of exponential type and its modulus of continuity defined by means of a generalized shift operator is well known. The question of the sharpness of the inequality remained open. For the constant in Jackson’s inequality, we obtain a lower bound, which proves its sharpness.     

Multiplicative convolutions of functions from Lorentz spaces and convergence of series of Fourier–Vilenkin coefficients

Let f and g be functions from different Lorentz spaces L ^{p, q} [0, 1), h be theirmultiplicative convolution and xxxx be Fourier coefficients of h with respect to a multiplicative system with bounded generating sequence. We estimate the remainder of the series of xxxx with multiplicators of type k ^b in terms of the best approximations of f and g in the corresponding Lorentz spaces. We establish sharpness of this result and of its corollaries for the Lebesgue spaces.     

Generalized Absolute Convergence of Series of Fourier Coefficients With Respect to Haar Type Systems

For the orthogonal systems ofHaar type, introduced by Vilenkin in 1958, we study absolute convergence of series composed from positive powers of Fourier coefficients with multiplicators from the Gogoladze–Meskhia class. The conditions for convergence of the series are given in terms of either best approximations of functions in L ^p spaces by polynomials with respect to Haar type systems or fractional modulus of continuity for functions from the Wiener spaces V _p , p > 1. We establish the sharpness of the obtained results.     

$$L^{p}$$-Norms a nd Mahler’s Measure of Poly nomials o n the n-Dime nsio nal Torus

We prove Nikol’skii type inequalities that, for polynomials on the n-dimensional torus \(\mathbb {T}^n\), relate the \(L^p\)-norm with the \(L^q\)-norm (with respect to the normalized Lebesgue measure and \(0 <p <q < \infty \)). Among other things, we show that \(C=\sqrt{q/p}\) is the best constant such that \(\Vert P\Vert _{L^q}\le C^{\text {deg}(P)} \Vert P\Vert _{L^p}\) for all homogeneous polynomials P on \(\mathbb {T}^n\). We also prove an exact inequality between the \(L^p\)-norm of a polynomial P on \(\mathbb {T}^n\) and its Mahler measure M(P), which is the geometric mean of |P| with respect to the normalized Lebesgue measure on \(\mathbb {T}^n\). Using extrapolation, we transfer this estimate into a Khintchine–Kahane type inequality, which, for polynomials on \(\mathbb {T}^n\), relates a certain exponential Orlicz norm and Mahler’s measure. Applications are given, including some interpolation estimates.     

Identity-based non-interactive key distribution with forward security

Identity-based non-interactive key distribution (ID-NIKD) is a cryptographic primitive that enables two users to establish a common secret key without exchanging messages. All users of the system have access to public system parameters and a private key, obtained through the help of a trusted key generation center. In this contribution, we discuss how to capture an intuitive form of forward security for ID-NIKD schemes in a security model. Building on results of Sakai et?al. as well as of Paterson and Srinivasan, we discuss how the proposed notion of forward security can be achieved in the random oracle model, using a Bilinear Diffie-Hellman assumption in combination with a forward-secure pseudorandom bit generator. We also show how a forward-secure ID-NIKD scheme can be used to realize forward-secure identity-based encryption.     

The optimal solution set of the interval linear programming problems

Several methods exist for solving the interval linear programming (ILP) problem. In most of these methods, we can only obtain the optimal value of the objective function of the ILP problem. In this paper we determine the optimal solution set of the ILP as the intersection of some regions, by the best and the worst case (BWC) methods, when the feasible solution components of the best problem are positive. First, we convert the ILP problem to the convex combination problem by coefficients 0 ≤ λ _j , μ _ij , μ _i  ≤ 1, for i = 1, 2, . . . , m and j = 1, 2, . . . , n. If for each i, j, μ _ij  = μ _i  = λ _j  = 0, then the best problem has been obtained (in case of minimization problem). We move from the best problem towards the worst problem by tiny variations of λ _j , μ _ij and μ _i from 0 to 1. Then we solve each of the obtained problems. All of the optimal solutions form a region that we call the optimal solution set of the ILP. Our aim is to determine this optimal solution set by the best and the worst problem constraints. We show that some theorems to validity of this optimal solution set.     

Bounds for partial derivatives: necessity of UMD and sharp constants

We prove the necessity of the UMD condition, with a quantitative estimate of the UMD constant, for any inequality in a family of \(L^p\) bounds between different partial derivatives \(\partial ^\beta u\) of \(u \in C^\infty _c({\mathbb {R}}^n,X)\). In particular, we show that the estimate \(\Vert u_{xy}\Vert _p\le K(\Vert u_{xx}\Vert _p+\Vert u_{yy}\Vert _p)\) characterizes the UMD property, and the best constant K is equal to one half of the UMD constant. This precise value of K seems to be new even for scalar-valued functions.     

Data-Driven Model Reduction and Transfer Operator Approximation

In this review paper, we will present different data-driven dimension reduction techniques for dynamical systems that are based on transfer operator theory as well as methods to approximate transfer operators and their eigenvalues, eigenfunctions, and eigenmodes. The goal is to point out similarities and differences between methods developed independently by the dynamical systems, fluid dynamics, and molecular dynamics communities such as time-lagged independent component analysis, dynamic mode decomposition, and their respective generalizations. As a result, extensions and best practices developed for one particular method can be carried over to other related methods.     

Posets and VPG Graphs

We investigate the class of intersection graphs of paths on a grid (VPG graphs), and specifically the relationship between the bending number of a cocomparability graph and the poset dimension of its complement. We show that the bending number of a cocomparability graph G is at most the poset dimension of the complement of G minus one. Then, via Ramsey type arguments, we show our upper bound is best possible.     

Continuity of topological entropy for perturbation of time-one maps of hyperbolic flows

In this paper we derive necessary and sufficient homological and cohomological conditions for profinite groups and modules to be of type FP_n over a profinite ring R, analogous to the Bieri–Eckmann criteria for abstract groups. We use these to prove that the class of groups of type FP_n is closed under extensions, quotients by subgroups of type FP_n, proper amalgamated free products and proper HNN-extensions, for each n. We show, as a consequence of this, that elementary amenable profinite groups of finite rank are of type FP_∞ over all profinite R. For any class C of finite groups closed under subgroups, quotients and extensions, we also construct pro-C groups of type FP_n but not of type FP_n+1 over Z_? for each n. Finally, we show that the natural analogue of the usual condition measuring when pro-p groups are of type FP_n fails for general profinite groups, answering in the negative the profinite analogue of a question of Kropholler.