An algebroid function and its derivative

In this paper, we investigate the value distribution of an algebroid function and its derivative, and obtain two inequations between Nevanlinna characteristic function of an algebroid function and that of its derivative. We extend Chuang Chitai's theorem of meromorphic functions to algebroid functions.     

Nonparametric tests for bounds on the derivative of a regression function

We consider two tests of the null hypothesis that the k-th derivative of a regression function is uniformly bounded by a specified constant. These tests can be used to study the shape of the regression function. For instance, we can test for convexity of the regression function by setting k=2 and the constant equal to zero. Our tests are based on k-th order divided difference of the observations. The asymptotic distribution and efficacies of these tests are computed and simulation results presented.Research supported by Natural Sciences and Engineering Research Council of Canada Grant OGP0007969.Research supported by National Science Foundation Grant DMS-9306738.     

Characterizing the derivative and the entropy function by the Leibniz rule

Consider an operator T:C¹(R)→C(R) satisfying the Leibniz rule functional equation

     

Detecting derivative discontinuity locations in piecewise continuous functions from Fourier spectral data

We present a method that uses Fourier spectral data to locate jump discontinuities in the first derivatives of functions that are continuous with piecewise smooth derivatives. Since Fourier spectral methods yield strong oscillations near jump discontinuities, it is often difficult to distinguish true discontinuities from artificial oscillations. In this paper we show that by incorporating a local difference method into the global derivative jump function approximation, we can reduce oscillations near the derivative jump discontinuities without losing the ability to locate them. We also present an algorithm that successfully locates both simple and derivative jump discontinuities. This work was partially supported by NSF grants CNS 0324957 and DMS 0510813, and NIH grant EB 02553301 (AG).     

Inequalities for a polynomial and its derivative

Let , 1?μ?n, be a polynomial of degree n such that p(z)≠0 in |z|<k, k>0, then for 0<r?R?k, Dewan, Yadav and Pukhta [K.K. Dewan, R.S. Yadav, M.S. Pukhta, Inequalities for a polynomial and its derivative, Math. Inequal. Appl. 2 (2) (1999) 203-205] proved

     

The quasiderivative method for derivative estimates of solutions to degenerate elliptic equations

We give an example of quasiderivatives constructed by random time change, Girsanov’s Theorem and Levy’s Theorem. As an application, we investigate the smoothness and estimate the derivatives up to second order for the probabilistic solution to the Dirichlet problem for the linear degenerate elliptic partial differential equation of second order, under the assumption of non-degeneracy with respect to the normal to the boundary and an interior condition to control the moments of quasiderivatives, which is weaker than non-degeneracy.     

The logarithmic derivative of a meromorphic function

A well-known lemma on the logarithmic derivative for a function f(z), f(0) = 1 (0 < r="> m( r,\fracf￠f ) < ln+ { \fracT(r,f)r\fracrr- r } + 5.8501.m\left( {r,\frac{{f'}}{f}} \right)< \ln + \left\{ {\frac{{T(\rho ,f)}}{r}\frac{\rho }{{\rho - r}}} \right\} + 5.8501.     

An example of a nonconstant bianalytic function vanishing everywhere on a nowhere analytic boundary

A Jordan domain with a smooth nowhere analytic boundary and a function that is bianalytic in this domain, belongs to the class Lip₁ in its closure, vanishes on the boundary, but is not identically zero are constructed. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 629–632, October, 1997. Translated by O. V. Sipacheva     

Remarks on a Theorem of Amitsur

Amitsur proved the following theorem for an uncountable field k: Let A be a k-algebra, then any finite-dimensional nil subspace of A has bounded index, and any finite dimensional algebraic subspace of A has bounded degree. Here we give a new proof of Amitsur's theorem, a proof which is more elementary and direct, than Amitsur's original proof.     

On the interior derivative blow-up for the curvature evolution of capillary surfaces

We give examples of the interior derivative blow-up solutions for the curvature evolution of capillary surfaces over a bounded domain in .

     

Geometry of interpolation sets in derivative free optimization

We consider derivative free methods based on sampling approaches for nonlinear optimization problems where derivatives of the objective function are not available and cannot be directly approximated. We show how the bounds on the error between an interpolating polynomial and the true function can be used in the convergence theory of derivative free sampling methods. These bounds involve a constant that reflects the quality of the interpolation set. The main task of such a derivative free algorithm is to maintain an interpolation sampling set so that this constant remains small, and at least uniformly bounded. This constant is often described through the basis of Lagrange polynomials associated with the interpolation set. We provide an alternative, more intuitive, definition for this concept and show how this constant is related to the condition number of a certain matrix. This relation enables us to provide a range of algorithms whilst maintaining the interpolation set so that this condition number or the geometry constant remain uniformly bounded. We also derive bounds on the error between the model and the function and between their derivatives, directly in terms of this condition number and of this geometry constant.     

The local Hölder function of a continuous function

This work focuses on the local Hölder exponent as a measure of the regularity of a function around a given point. We investigate in detail the structure and the main properties of the local Hölder function (i.e., the function that associates to each point its local Hölder exponent). We prove that it is possible to construct a continuous function with prescribed local and pointwise Hölder functions outside a set of Hausdorff dimension 0.     

Exploring the role of metonymy in mathematical understanding and reasoning: The concept of derivative as an example

In this paper we examine the roles that metonymy may play in student reasoning. To organize this discussion we use the lens of a structured derivative framework. The derivative framework consists of three layers of process-object pairs, one each for ratio, limit, and function. Each of the layers can then be illustrated in any appropriate context, for example graphically as slope, verbally as rate of change, or kinesthetically as velocity. We will illustrate three main cases of metonymy using data from student reasoning with derivative-paradigmatic metonymy, individual metonymy and part-part metonymy. Metonymies may function both in powerful and problematic ways for students as they come to understand and work with a complicated concept such as the derivative. It is the goal of this paper to illuminate and clarify how metonymy may function in student reasoning in the hopes of providing insights to teachers and researchers.     

Sixth order derivative free family of iterative methods

In this study, we develop a four-parameter family of sixth order convergent iterative methods for solving nonlinear scalar equations. Methods of the family require evaluation of four functions per iteration. These methods are totally free of derivatives. Convergence analysis shows that the family is sixth order convergent, which is also verified through the numerical work. Though the methods are independent of derivatives, computational results demonstrate that family of methods are efficient and demonstrate equal or better performance as compared with other six order methods, and the classical Newton method.     

Thematization of derivative schema in university students: nuances in constructing relations between a function's successive derivatives

This study is part of a more extensive research project that addresses the understanding of the derivative concept in university students with prior instruction in differential calculus. In particular, we focus on the analysis of students’ responses to a sequence of tasks that require a high level of understanding of the concept, and complement this information with clinical interviews. APOS (Action-Process-Object-Schema) theory and the configuration of the derivative concept that is characterized by: mathematical elements, logical relations and the representation modes that students use to solve a task were used in the analysis of students’ responses. The results obtained suggest that thematizing the derivative schema is difficult to achieve. In addition, nuances were observed in responses given by those students who succeeded, indicating differences in the construction of relations between the successive derivatives of a function.