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This paper addresses a topic within university mathematics education which has been somewhat underexplored: the teaching practices actually used by university mathematics teachers when giving lectures. The study investigates the teaching practices of seven Swedish university teachers on the topic of functions using a discursive approach, the commognitive framework of Sfard. In the paper a categorization of the construction and substantiation routines used by the teachers is presented, for instance various routines for constructing definitions and examples, and for verifying whether an example satisfies a given definition. The findings show that although the overall form of the lectures is similar, with teachers using ‘chalk talk’, and overt student participation limited to asking and answering questions, there are in fact significant differences in the way the teachers present and do mathematics in their lectures. These differences present themselves both on the level of discursive routines and on a more general level in how the process of doing mathematics is made visible in the teachers’ teaching practices. Moreover, I believe that many of the results of the study could be relevant for investigating the teaching of other mathematical topics. [ABSTRACT FROM PUBLISHER]

For PM functions of height 1, the existence of continuous iterative roots of any order was obtained under the characteristic endpoints condition. A natural open question about iterative roots without that condition was raised. This question was answered partially in the case that the function is increasing on its characteristic interval. In this paper, to the opposite, we consider the decreasing case and give the existence and nonexistence results for their iterative roots. [ABSTRACT FROM AUTHOR]

In the present paper, we discuss the approximation properties of Durrmeyer-Stancu type variant of Jain operators with the modified forms of the Beta basis functions. We establish some direct results, which include the asymptotic formula, the error estimation in terms of the modulus of continuity and weighted approximation. Also, we construct a King modification of these operators which preserves the test functions e0 and e1. [ABSTRACT FROM AUTHOR]

The study on the existence of periodic solutions of higher order nonlinear functional difference equations with p-Laplacian is made. Sufficient conditions for the existence of at least one periodic solution of these equations are established, respectively. We give no restriction on the deviating function, which is the significance of the paper. Our result is based on Mawhin's continuation theorem. The methods we used to estimate a priori bound on periodic solutions are also new. [ABSTRACT FROM AUTHOR]

In this paper, we study the uniform approximation of functions by Meyer-König and Zeller operators of max-product type in some weighted spaces. We estimate the rate of approximation in terms of a suitable modulus of continuity. [ABSTRACT FROM AUTHOR]

This paper characterizes the existence and form of the possible limit error signals in typical parameter-optimal iterative learning control. The set of limit errors has attracting and repelling components and the behaviour of the algorithm in the vicinity of these sets can be associated with the undesirable properties of apparent (but in fact temporary) convergence or permanent slow convergence properties in practice. The avoidance of these behaviours in practice is investigated using novel switching strategies. Deterministic strategies are analysed to prove the feasibility of the concept by proving that each of a number of such strategies is guaranteed to produce global convergence of errors to zero independent of the details of plant dynamics. For practical applications a random switching strategy is proposed to replace these approaches and shown, by example, to produce substantial potential improvements when compared with the non-switching case. The work described in this paper is covered by pending patent applications in the UK and elsewhere. [ABSTRACT FROM AUTHOR]

The article explores the theory of dynamical systems in mathematics, examining the papers of A. N. Sharkovsky and T.-Y. Li and J. A. Yorke on interval dynamics, among others. It should be noted that neither Sharkovsky's theorem nor Li-Yorke's definition of chaos have a significant role in the core of modern one-dimensional dynamics. Meanwhile, J. Guckenheimer's paper established the foundations for the almost everywhere approach to dynamics, which states that if a map is sufficiently smooth, there may be a fair chance of offering an effective description of the asymptotic behavior of almost every point of the interval.

In this paper, robust regulation of stable infinite-dimensional plants with transfer functions in the H 8 -algebra is considered. The reference and disturbance signals are allowed to have an infinite number of poles on the imaginary axis, which makes it possible to consider e.g. arbitrary periodic signals. A controller depending on a positive scalar e and design parameters K k is proposed. The design parameters are matrices over H 8 , and by the main result of the paper, if the design parameters satisfy a certain condition, the proposed controller is a robust regulator for sufficiently small values of e. The only knowledge needed to verify this condition is the so-called -stability of the plant and values of the plant transfer function at the poles of the reference and disturbance signals. The analysis in this paper is in the frequency domain and it is based on the fractional representation approach. [ABSTRACT FROM AUTHOR]

Comparing the large number of research papers on the ordered weighted averaging (OWA) operator, the researches on relative quantifier are relatively rare so far. In the present paper, based on the quantifier guided aggregation method with OWA operator which was proposed by Yager [“Quantifier guided aggregation using OWA operators”, Int. J. Intell. Syst. , 11, pp. 49–73, 1996], a generating function representation method for regular increasing monotone (RIM) quantifiers is proposed. We extend the the properties of OWA operator to the RIM quantifier which is represented with a monotone function instead of the OWA weighting vector. A class of parameterized equidifferent RIM quantifier which has minimum variance generating function is proposed and its properties are also analyzed. The equidifferent RIM quantifier is consistent with its orness level for any aggregated elements, which can be used to represent the decision maker's preference. [ABSTRACT FROM AUTHOR]

M. Fait, J. Krzyz, and J. Zygmunt in their paper Explicit quasiconformal extensions for some classes of univalent functions, comment. Math. Helv. 51(1976), 279-285, proved that a strongly starlike function of order a on the unit disk can be extended to a k-quasiconformal mapping with k ≤ sin(απ/2) on the whole complex plane &Csline; which fixes the point at infinity. An important question is to determine whether the bound k ≤ sin(απ/2) can be improved so that k ≤ α. By the theory of extremal quasiconformal mappings, we can give a negative approach to the question. [ABSTRACT FROM AUTHOR]

This paper discusses the estimation of average treatment effects in observational causal inferences. By employing a working propensity score and two working regression models for treatment and control groups, Robins et al. (1994, 1995) introduced the augmented inverse probability weighting (AIPW) method for estimation of average treatment effects, which extends the inverse probability weighting (IPW) method of Horvitz and Thompson (1952); the AIPW estimators are locally efficient and doubly robust. In this paper, we study a hybrid of the empirical likelihood method and the method of moments by employing three estimating functions, which can generate estimators for average treatment effects that are locally efficient and doubly robust. The proposed estimators of average treatment effects are efficient for the given choice of three estimating functions when the working propensity score is correctly specified, and thus are more efficient than the AIPW estimators. In addition, we consider a regression method for estimation of the average treatment effects when working regression models for both the treatment and control groups are correctly specified; the asymptotic variance of the resulting estimator is no greater than the semiparametric variance bound characterized by the theory of Robins et al. (1994, 1995). Finally, we present a simulation study to compare the finite-sample performance of various methods with respect to bias, efficiency, and robustness to model misspecification. [ABSTRACT FROM AUTHOR]

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Let G be a finite group and σ = {σi|i∈I} some partition of the set of all primes. A subgroup A of G is said to be generalized σ-subnormal in G if A = ⟨L,T⟩, where L is a modular subgroup and T is a σ-subnormal subgroup of G. In this paper, we prove that if every Schmidt subgroup of G is generalized σ-subnormal in G, then the commutator subgroup G′ of G is σ-nilpotent. [ABSTRACT FROM AUTHOR]

This paper investigates the determinants of grades achieved in maths by first-year students in economics. We use individual administrative data from 1993 to 2005 to fit an educational production function. Our main findings suggest that good secondary school achievements and the type of school attended are significantly associated with maths grades. Ceteris paribus, females typically do better than males. Since students can postpone the exam or repeat it when they fail, we also analyze the determinants of the elapsed time to pass the exam using a survival analysis. Modeling simultaneously maths grades and the hazard of passing the exam, we find that the overall hazard rate of passing the exam is higher for those students who get the higher grades. The longer the students wait to take the exam, the less likely they are to obtain high grades. [ABSTRACT FROM AUTHOR]

In some recent investigations involving differential operators for generalized Lagrange polynomials, Chan et al. [W.-Ch.C. Chan et al. The Lagrange polynomials in several variables, Integral Transforms Spec. Funct. 12 (2001), pp. 139-148] encountered and proved a certain summation identity for the Lagrange polynomials in several variables. The present paper introduces some bilateral generating functions for the Lagrange polynomials with multiple variables and the Lauricella functions. [ABSTRACT FROM AUTHOR]

In this paper, the nonautonomous competing two-species Lotka-Volterra models with impulsive effect are considered, where all the parameters are time-dependent and asymptotically approach the corresponding periodic functions. Under some conditions, it is shown that the semi-trivial positive solutions of the models asymptotically approach the semi-trivial positive periodic solutions of the corresponding periodic system. It is also shown that the positive solution of the models asymptotically approach the positive periodic solution of the corresponding periodic system. [ABSTRACT FROM AUTHOR]

Let (tn(t)) be the sequence of trigonometric functions on [0, 1] with real scalars. We prove a generalization of the likely known result: - for every p ⩾ 1 the series Σ (∣tn(t)∣p)/n is unbounded on every set A with positive measure. This provides an alternate proof of a result in a recent paper of the author; the methods used are classical. [ABSTRACT FROM AUTHOR]

This paper considers the problem of approximating an integrable function by piecewise linear functions that keep the integral and positivity of the original function. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical program with complementarity constraints (MPCC) is reformulated as a nonsmooth constrained mathematical program via the Fischer–Burmeister function. Smooth penalty functions are used to treat this nonsmooth constrained program. Under linear independence constraint qualification, and upper level strict complementarity condition, together with some other mild conditions, we prove that the limit point of stationary points satisfying second-order necessary conditions of unconstrained penalized problems is a strongly stationary point, hence a B-stationary point of the original MPCC. Furthermore, this limit point also satisfies a second-order necessary condition of the original MPCC. Numerical results are presented to test the performance of this method. [ABSTRACT FROM AUTHOR]

Let G =( V , E ) be a graph with vertex set V of size n and edge set E of size m . A vertex v ? V is called a hinge vertex if there exist two vertices in V \{ v } such that their distance becomes longer when v is removed. In this paper, we present a distributed algorithm that finds all hinge vertices on an arbitrary graph. The proposed algorithm works for named static asynchronous networks and achieves O ( n 2 ) time complexity and O ( m ) message complexity. In particular, the total messages exchanged during the algorithm are at most 2 m (log n + n log n +1) bits. [ABSTRACT FROM AUTHOR]

Boolean functions are widely used because they can be used to precisely describe logical circuits. Properties of Boolean functions with respect to their applications to cryptography have been studied, but relationship between Boolean functions are rarely studied. This paper studies the independence of Boolean functions, gives necessary and sufficient conditions for judging whether two given Boolean functions are independent. Some enumeration formulae are given. [ABSTRACT FROM AUTHOR]

This paper studies the ring structure of the Hochschild cohomology ring of an algebra. The first main result gives a ring homomorphism from the Hochschild cohomology ring of an algebra A to the Ext-algebra of an A-module. Then, for a one point extension B of a finite dimensional algebra A, we relate the ring structures of the Hochschild cohomology rings of A and B thus giving a method of constructing algebras with non-trivial ring structure on the Hochschild cohomology ring. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generating function for a new generalization of Laguerre-based Apostol-Bernoulli polynomials, Apostol-Euler and Apostol-Genocchi polynomials. By making use of the generating function method and some functional equations mentioned in the paper, we conduct a further investigation in order to obtain symmetric identities of these polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we express the scattered Debye potentials via a new generalized convolution related to the Kontorovich-Lebedev integral. The uniform asymptotics of the scattered Debye potentials under very mild conditions on spectral functions are obtained, and an inverse problem of finding spectral functions from given Debye potentials is considered. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some new circular summation formulas of theta functions using the theory of elliptic functions. As applications we also show some interesting identities of theta functions. [ABSTRACT FROM AUTHOR]

In this paper, we give two generalized homogeneousq-operator identities, which are obtained byq-difference equations. In addition, we generalize certain generating functions from Chen et al. [W.Y.C. Chen, A. M. Fu, and B. Zhang,The homogeneousq-difference operator, Adv. Appl. Math. 31 (2003), pp. 659–668] and Saad and Sukhi [H. L. Saad and A.A. Sukhi,Another homogeneousq-difference operator, Appl. Math. Comput. 215 (2010), pp. 4332–4339] by the method ofq-difference equations. Moreover, we gainU(n+1) generalizations of generating functions for Verma–Jain polynomials byq-difference equations. Finally, we obtain two transformational identities byq-difference equations. [ABSTRACT FROM AUTHOR]

This paper is devoted to derive some properties and expansions associated with the q-digamma function. The Newton series which is consisting of terms of forward difference operator, is established for the q-digamma function. The maltiplication formula of the q-gamma function is used to present some recurrence relations for the q-digamma function. The q-analogue of well-known results in the theory of the digamma function are investigated. One of the most important identities of q-beta integrals is used to introduce some integral representations of the q-digamma function. [ABSTRACT FROM AUTHOR]

Let , where is the -th cyclotomic polynomial. Let or , depending if the leading coefficient of the polynomial is ‘+1’ or ‘ − 1’, respectively. The rational function can be written as , where , the 's are positive integers, 's are integers and is a positive integer depending on . In the present paper, we study the set where the intersection is considered over all the possible decompositions of of the type mentioned above. Here, we describe the set in terms of the arithmetic properties of the integers . We also study the question: given S a finite subset of the natural numbers, does exists a , such that ? The set is called the minimal set of Lefschetz periods associated with q(t). The motivation of these problems comes from differentiable dynamics, when we are interested in describing the minimal set of periods for a class of differentiable maps on orientable surfaces. In this class of maps, the Morse–Smale diffeomorphisms are included (cf. Llibre and Sirvent, Houston J. Math. 35 (2009), pp. 835–855). [ABSTRACT FROM AUTHOR]

In this paper we shall define the special-valued multiple Hurwitz zeta functions, namely the multiple t-values t(α) and define similarly the multiple star t-values as t⋆(α). Then we consider the sum of all such multiple (star) t-values of fixed depth and weight with even argument and prove that such a sum can be evaluated when the evaluations of t({2m}n) and t*({2m}n) are clear. We give the evaluations of them in terms of the classical Euler numbers through their generating functions. [ABSTRACT FROM AUTHOR]

In this paper, some new inequalities for power series via Buzano's result are obtained. Applications for some fundamental complex functions are also provided. [ABSTRACT FROM AUTHOR]

In this paper, we consider kth order difference equations of the form[image omitted] where every function [image omitted] is continuous and monotonic in each of its arguments. We generalize and improve upon some previous techniques to prove convergence using invariant intervals as the common theme. We apply our techniques to several different special cases of the equation above. [ABSTRACT FROM AUTHOR]

The paper presents theorems on the global asymptotic behaviour of non-autonomous population models which for the special case of periodically forced populations generalize known results. It is shown that a non-autonomous discrete dynamical system on an interval possesses the property of path stability, if the system is given by cave functions. From this it follows for a periodic system with strictly cave functions that there exists a globally asymptotically stable cycle. Old and new examples are discussed. [ABSTRACT FROM AUTHOR]

In this paper the authors study the hyperbolic geometric flow on Riemann surfaces. This new nonlinear geometric evolution equation was recently introduced by the first two authors, motivated by Einstein equation and Hamilton's Ricci flow. We prove that, for any given initial metric on 2 in certain class of metrics, one can always choose suitable initial velocity symmetric tensor such that the solution exists for all time, and the scalar curvature corresponding to the solution metric gij keeps uniformly bounded for all time; moreover, if the initial velocity tensor is suitably “large", then the solution metric gij converges to the flat metric at an algebraic rate. If the initial velocity tensor does not satisfy the condition, then the solution blows up at a finite time, and the scalar curvature R(t, x) goes to positive infinity as (t, x) tends to the blowup points, and a flow with surgery has to be considered. The authors attempt to show that, comparing to Ricci flow, the hyperbolic geometric flow has the following advantage: the surgery technique may be replaced by choosing suitable initial velocity tensor. Some geometric properties of hyperbolic geometric flow on general open and closed Riemann surfaces are also discussed. [ABSTRACT FROM AUTHOR]

This paper is an attempt to stress the usefulness of multi-variable special functions. We derive generating relations involving multi-variable Hermite-Tricomi functions by using the Lie-algebraic method. Further, we derive certain new and known generating relations involving other forms of Hermite-Tricomi functions as applications. [ABSTRACT FROM AUTHOR]

In this paper, new classes of generalized invex functions called B-(p, r)-pseudo-invex and B-(p, r)-quasi-invex (with respect to η and b) are introduced. These classes of functions are defined by relaxing the definitions of the classes of quasi-invex and pseudo-invex functions earlier defined by Hanson (J. Math. Anal. Appl. 1981, 80:545-550), and also the concept of B-(p, r)-invexity (with respect to η and b) earlier introduced by Antczak (J. Math. Anal. Appl. 2003, 286:187-206). Further, mathematical programming problems involving generalized B-(p, r)-invex functions with respect to η and b are considered. The sufficient optimality conditions and various duality results are established for such optimization problems. [ABSTRACT FROM AUTHOR]

In this paper, we consider a perturbation of a Laguerre-Hahn functional by adding a derivative of a Dirac delta. This transformation leaves invariant the family of Laguerre-Hahn linear functionals. We shall also analyse the class of the perturbed linear functional. Finally, we illustrate these perturbations by adding the derivative of a Dirac delta to the first kind associated functional of the classical Laguerre functional. The expression of the new orthogonal polynomials is obtained. [ABSTRACT FROM AUTHOR]

The basic rules of elementary algebra can often appear beyond the grasp of many students. Even though most subjects, including calculus, prove to be more difficult, it is the simple rules of algebra that continue to be the “thorn in the side” of many mathematics students. In this paper we present a result intended to help students achieve a greater understanding of a familiar algebraic error. [ABSTRACT FROM AUTHOR]

The Reissner-Mindlin plate model for calculation of functionally graded materials has been proposed in literature by using shear correction coefficient of homogeneous model. However, this use is a priori not appropriate for the gradient material. Identification of the transverse shear factors is thus investigated in this paper. The transverse shear stresses are derived by using energy considerations from the expression of membrane stresses. Using the obtained transverse shear factor, a numerical analysis is performed on a simply supported FG square plate whose elastic properties are isotropic at each point and vary through the thickness according to a power law distribution. The numerical results of a static analysis are compared with available solutions from previous studies. [ABSTRACT FROM AUTHOR]

The focus of this paper is on student interpretation and usage of the existence and uniqueness theorems for first-order ordinary differential equations. The inherent structure of the theorems is made explicit by the introduction of a framework of layers concepts-conditions-connectives-conclusions, and we discuss the manners in which students' understanding of these layers construct their perception of the theorems as a whole. [ABSTRACT FROM AUTHOR]

Let X, Y be Banach modules over a [image omitted] -algebra and let [image omitted] be given. We prove the generalized Hyers-Ulam stability of the following functional equation in Banach modules over a unital [image omitted] -algebra:[image omitted] We show that if [image omitted] and an odd mapping [image omitted] satisfies the functional equation (0.1) then the odd mapping [image omitted] is Cauchy additive. As an application, we show that every almost linear bijection [image omitted] of a unital [image omitted] -algebra A onto a unital [image omitted] -algebra B is a [image omitted] -algebra isomorphism when [image omitted] for all unitaries [image omitted] , all [image omitted] , and all [image omitted] . The concept of generalized Hyers-Ulam stability originated from Th.M. Rassias' stability Theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. [ABSTRACT FROM AUTHOR]

The paper details a comprehensive system for the treatment of the topic of limits—conceptually, computationally, and formally. The system addresses fundamental linguistic flaws in the standard presentation of limits, which attempts to force limit discussion into the language of individual real numbers and equality. The system of near-numbers properly fits the context of limits, allowing precise and unambiguous notation in limit computation. More importantly, the near-numbers are a set of concrete mathematical objects that provide a scaffold both for building an accurate intuitive understanding of the limit concept and for connecting that understanding to the formal definition of the limit. [ABSTRACT FROM AUTHOR]

This paper introduces a new enhancement to the set-theoretic reconstructability analysis (RA) decomposition of logic functions called multiple-level modified reconstructability analysis (MRA). The new variation of MRA decomposition is achieved by implementing a map superposition method called Iterative Error Maps (IEM) as a pre-processor to the standard MRA. The introduced multiple-level MRA decomposition solves successfully for the decomposition of 3-variable NPN-classified Boolean functions in classes 2, 6, and 9 that were previously non-decomposable using the standard 2-level MRA decomposition. An extension to the case of many-valued multiple-level multiple-output MRA is also introduced. [ABSTRACT FROM AUTHOR]

The article explores the so-called λ-permutations shown by Steven Krantz and Jeffery McNeal in their paper "Creating more convergent series." They specified that a permutation σ of the positive integers is a λ-permutation if it has two characteristics. The first property is that for every convergent series, the summation of ak from k =1 to infinity, the summation of aσ(k) from k =1 to infinity also converges. The second property is that there is at least one divergent series which is the summation of ak from k =1 to infinity such that the summation of aσ(k) from k =1 to infinity converges.

The Goulden-Jackson cluster method is a powerful method to find generating functions of pattern occurrences in random sequences [1]. The method is clearly explained, extended and implemented by Noonan and Zeilberger [2]. In this paper, we elaborate on one of the several extensions in [2], namely the extension from symmetrical Bernoulli sequences where the occurrences of each symbol have equal probability, to asymmetrical Bernoulli sequences with different probabilities of symbol generations. An explicit formula is derived for the extension, which is implicitly embedded in the treatment of [2]. The extended result is then compared with the method of Régnier-Szpankowski [3], a method which was developed independently to tackle the same problem. By manipulating some matrix inversions, we show that the Régnier-Szpankowski method can be simplified to the extended Goulden-Jackson method. [ABSTRACT FROM AUTHOR]

In this paper, we consider a nonlinear difference equation of the form xn+1 =f(pn,xn-2s,xn-2t-1), n = 0,1,…, under some certain assumptions, where s,t ∈ {0,1,2,…} with s ⩽ t and the initial values x-2t-1 ,x-2t,ߪ, x0 ∈ (0, + ∞) and pn is the period-two sequence. We give sufficient conditions under which every positive solution of this equation converges to the period-two solution. [ABSTRACT FROM AUTHOR]

The article describes a method for allowing students to manipulate mathematical objects. It involves computer activities that can draw a student's attention to the big picture in a way that paper pencil textbook exercises cannot. The main feature is a window that displays the graph of a function. The expression that defines the function appears in a text box. Throughout the activity, student interaction with the computer consists mainly of clicking push buttons and watching the results. There is a bit of typing when a new function is to be defined. It is significant that the interaction is completely orchestrated. Teachers should sympathize with this perspective, because most of us have had the experience many times of completely losing the thread of meaning while listening to a lecture or colloquium. The carefully prepared explanations of the speaker cease to have any meaning, because they describe interconnections between mathematical constructs that has not been internalized. Students should be able to conduct experiments to answer questions.

In this paper, we study global attractivity of solutions of the delay difference equation with a forcing term where {p(n)} is a nonnegative sequence, f:R?R is a continuous function with xf(x)>0 for x?0, {r(n)} is a real sequence, and k is a nonnegative integer. Several global attractivity results are obtained. [ABSTRACT FROM AUTHOR]

This paper gives a broad descriptive account of some activities that the author has designed using Sketchpad to develop teachers' understanding of other functions of proof than just the traditional function of 'verification'. These other functions of proof illustrated here are those of explanation, discovery and systematization (in the context of defining and classifying some quadrilaterals). A solid theoretical rationale is provided for dealing with these other functions in teaching by analysing actual mathematical practice where verification is not always the most important function. The activities are designed according to the so-called 'reconstructive' approach, and are structured more or less in accordance with the Van Hiele theory of learning geometry. [ABSTRACT FROM AUTHOR]

In this paper there is proved a generalization of a theorem of Stein and Weiss concerning metric properties of the conjugate characteristic functions of given sets on the interval [0, 2π]. As an application for the Hilbert transform of a bounded function, the optimal correlation where the Favard's constant : is also established. [ABSTRACT FROM AUTHOR]

In this paper, we prove, under natural assumptions, that the semilinear elliptic system -Δu+u=g(x,v), x ∈ RN, -Δv+v=f(x,u), x∈RN (0.1) possesses at least one positive solution pair (u, v) ∈ H¹(RN) × H¹(RN), where f(x, t) and g(x, t) are continuous functions on RN × R and asymptotically linear as t → + ∞. [ABSTRACT FROM AUTHOR]