Reasoning about variation in the intensity of change in covarying quantities involved in rate of change

This paper extends work in the area of quantitative reasoning related to rate of change by investigating numerical and nonnumerical reasoning about covarying quantities involved in rate of change via tasks involving multiple representations of covarying quantities. The findings suggest that by systematically varying one quantity, an individual could simultaneously attend to variation in the intensity of change in a quantity indicating a relationship between covarying quantities. The results document how a secondary student, prior to formal instruction in calculus, reasoned numerically and nonnumerically about covarying quantities involved in rate of change in a way that was mathematically powerful and yet not ratio-based. I discuss how coordinating covariational and transformational reasoning supports attending to variation in the intensity of change in quantities involved in rate of change.     

The inverse partial correlation function of a time series and its applications

The concept of the inverse correlation function of a stationary process was introduced by Cleveland (Technometrics14 (1972), 277–293). The inverse partial correlation function of a stationary process may intuitively be thought of as the corresponding extension of the concept of the partial correlation function. A precise mathematical definition of this function is given. Its importance in describing the structure of a moving average of finite order h is discussed. Having observed X₁,…,X_T, the autoregressive method of estimating the inverse correlations is employed for constructing sample estimates of the inverse partial correlations. For the hth-order moving average process, the estimates beyond h are, as T → ∞, asymptotically independent normally distributed with 0 mean and variance T^?1. Their use for estimating h and for testing hypotheses concerning h is examined.     

Use of 3D-transient analytical solution based on Green’s function to reduce computational time in inverse heat conduction problems

Inverse problems can be found in many areas of science and engineering and can be applied in different ways. Two examples can be cited: thermal properties estimation or heat flux function estimation in some engineering thermal process. Different techniques for the solution of inverse heat conduction problem (IHCP) can be found in literature. However, any inverse or optimization technique has a basic and common characteristic: the need to solve the direct problem solution several times. This characteristic is the cause of the great computational time consumed. In heat conduction problem, the time consumed is, usually, due to the use of numerical solutions of multidimensional models with refined mesh. In this case, if analytical solutions are available the computational time can be reduced drastically. This study presents the development and application of a 3D-transient analytical solution based on Green’s function. The inverse problem is due to the thermal properties estimation of conductors. The method is based on experimental determination of thermal conductivity and diffusivity using partially heated surface method without heat flux transducer. Originally developed to use numerical solution, this technique can, using analytical solution, estimate thermal properties faster and with better accuracy.     

The inverse gravimetry problem: An application to the northern San Francisco craton granite

From the knowledge of the anomalies of the gravitational field, we reconstruct the mass density distribution in a large region of the state of Bahia (Brazil). This inverse gravimetry problem has been translated into a linear programming problem and solved using the simplex method. Both two-dimensional and three-dimensional models have been considered.This research was supported by the European Research Office of the US Army under Contract No. DAJA 45-86-C-0028 with the University of Rome-La Sapienza and by the Ministero Pubblica Istruzione under Contract (60%)-1987 with the University of Camerino.     

Attractors and symmetries of vector fields: The inverse problem

The inverse problem of constructing dynamical systems X with a prescribed compact attracting set K⊂Rn and non-trivial dynamics on it is solved. The boundaries ∂ of the attracting basins corresponding to different connected components of K and the relationship between the symmetries of K, X and ∂ are also studied.     

Connecting the learning of advanced mathematics with the teaching of secondary mathematics: Inverse functions,domain restrictions,and the arcsine function

Prospective secondary mathematics teachers are typically required to take advanced university mathematics courses. However, many prospective teachers see little value in completing these courses. In this paper, we present the instantiation of an innovative model that we have previously developed on how to teach advanced mathematics to prospective teachers in a way that informs their future pedagogy. We illustrate this model with a particular module in real analysis in which theorems about continuity, injectivity, and monotonicity are used to inform teachers’ instruction on inverse trigonometric functions and solving trigonometric equations. We report data from a design research study illustrating how our activities helped prospective teachers develop a more productive understanding of inverse functions. We then present pre-test/post-test data illustrating that the prospective teachers were better able to respond to pedagogical situations around these concepts that they might encounter.     

Making sense of qualitative geometry: The case of Amanda

This article presents a case study of a seven-year-old girl named Amanda who participated in an eighteen-week teaching experiment I conducted in order to model the development of her intuitive and informal topological ideas. I designed a new dynamic geometry environment that I used in each of the episodes of the teaching experiment to elicit these conceptions and further support their development. As the study progressed, I found that Amanda developed significant and authentic forms of geometric reasoning. It is these newly identified forms of reasoning, which I refer to as “qualitative geometry,” that have implications for the teaching and learning of geometry and for research into students’ mathematical reasoning.     

整代数体函数与其导函数的增长级

对整代数体函数与其导函数的增长性进行了研究,得到了整代数体函数与其导函数的特征函数之间关系式,并证明了它们具有相同的级和下级,从而将亚纯函数的相应结果推广到了整代数体函数.     

Class groups of imaginary function fields: The inert case

Let be a finite field and a transcendental element over . An imaginary function field is defined to be a function field such that the prime at infinity is inert or totally ramified. For the totally imaginary case, in a recent paper the second author constructed infinitely many function fields of any fixed degree over in which the prime at infinity is totally ramified and with ideal class numbers divisible by any given positive integer greater than 1. In this paper, we complete the imaginary case by proving the corresponding result for function fields in which the prime at infinity is inert. Specifically, we show that for relatively prime integers and , there are infinitely many function fields of fixed degree such that the class group of contains a subgroup isomorphic to and the prime at infinity is inert.

     

On an inverse problem: Recovery of non-smooth solutions to backward heat equation

We have recently developed two quasi-reversibility techniques in combination with Euler and Crank–Nicolson schemes and applied successfully to solve for smooth solutions of backward heat equation. In this paper, we test the viability of using these techniques to recover non-smooth solutions of backward heat equation. In particular, we numerically integrate the backward heat equation with smooth initial data up to a time of singularity (corners and discontinuities) formation. Using three examples, it is shown that the numerical solutions are very good smooth approximations to these singular exact solutions. The errors are shown using pseudo-L- and U-curves and compared where available with existing works. The limitations of these methods in terms of time of simulation and accuracy with emphasis on the precise set of numerical parameters suitable for producing smooth approximations are discussed. This paper also provides an opportunity to gain some insight into developing more sophisticated filtering techniques that can produce the fine-scale features (singularities) of the final solutions. Techniques are general and can be applied to many problems of scientific and technological interests.     

Tor gives the inverse to the Hilbert function of a graded algebra

The Hilbert generating function of Tor^U(A, A) gives the inverse of the Hilbert generating function of U where U is a graded A algebra.     

Class group frequencies of real quadratic function fields: The degree 4 case

The distribution of ideal class groups of is examined for degree-four monic polynomials when is a finite field of characteristic greater than 3 with or and is irreducible or has an irreducible cubic factor. Particular attention is paid to the distribution of the -Sylow part of the class group, and these results agree with those predicted using the Cohen-Lenstra heuristics to within about 1 part in 10000. An alternative set of conjectures specific to the cases under investigation is in even sharper agreement.

     

Degradation modeling of 2 fatigue‐crack growth characteristics based on inverse Gaussian processes: A case study

Most modern products that are highly reliable are complex in their inner and outer structures. This situation indicates quality characterization by the interaction of multiple performance characteristics, which motivates the utilization of robust reliability models to obtain robust estimates. It is paramount to obtaining substantial information about a product's life cycle; therefore, when multiple performance characteristics are dependent, it is important to find models that address the joint distribution of performance degradation of such. In this paper, a reliability model for products with 2 fatigue‐crack growth characteristics related to 2 degradation processes is developed. The proposed model considers the dependence among degradation processes by using copula functions considering the marginal degradation processes as inverse Gaussian processes. The statistical inference is performed by using a Bayesian approach to estimate the parameters of the joint bivariate model. A time‐scale transformation is considered to assure monotone paths of the degradation trajectories. The comparison results of the reliability analysis, under both dependent and independent assumptions, are reported with the implementation of the proposed modeling in a case study, which consists of the crack propagation data of 2 terminals of an electronic device.     

The Laplace transform of the digamma function: An integral due to Glasser, Manna and Oloa

The definite integral

is related to the Laplace transform of the digamma function

by when . Certain analytic expressions for in the complementary range, , are also provided.

     

The Hilbert problem: The case of infinitely many discontinuity points of coefficients

We obtain a solution to the Hilbert boundary value problem in the theory of analytic functions on the half-plane in the case that the coefficients of the boundary condition have countably many discontinuity points of the first kind. We elaborate the two substantially different situations: the series consisting of the jumps of the argument of the coefficient function and the increments of its continuous part converges and this series diverges. Accordingly, Hilbert problems with finite and infinite indices result. We derive formulas for the general solution and investigate the pictures of solvability of these problems.     

The role of framing in productive classroom discussions: A case for teacher learning

In this paper, we contrast two mathematical arguments that occurred during an algebra lesson to illustrate the importance of relevant framings in the ensuing arguments. The lesson is taken from a graduate course for elementary teachers who are enrolled in a mathematics specialist program. We use constructs associated with enthnography of argumentation to characterize the framings for warrants and backings that supported the ensuing arguments. Our findings suggest that teachers fully participated in argumentations that were framed by problem situations that were familiar to them, ones that were couched in elementary, fundamental mathematical ideas, and that these types of argumentations were arguably more productive in terms of opportunities for learning.     

On the inverse problem relative to dynamics of the <Emphasis Type="Italic">w</Emphasis> function

Let P be the set of prime numbers and P(n) denote the largest prime factor of integer n > 1. Write

Reinventing solutions to systems of linear differential equations: A case of emergent models involving analytic expressions

An enduring challenge in mathematics education is to create learning environments in which students generate, refine, and extend their intuitive and informal ways of reasoning to more sophisticated and formal ways of reasoning. Pressing concerns for research, therefore, are to detail students’ progressively sophisticated ways of reasoning and instructional design heuristics that can facilitate this process. In this article we analyze the case of student reasoning with analytic expressions as they reinvent solutions to systems of two differential equations. The significance of this work is twofold: it includes an elaboration of the Realistic Mathematics Education instructional design heuristic of emergent models to the undergraduate setting in which symbolic expressions play a prominent role, and it offers teachers insight into student thinking by highlighting qualitatively different ways that students reason proportionally in relation to this instructional design heuristic.     

A simplification of Laplace’s method: Applications to the Gamma function and Gauss hypergeometric function

The main difficulties in the Laplace’s method of asymptotic expansions of integrals are originated by a change of variables. We propose a variant of the method which avoids that change of variables and simplifies the computations. On the one hand, the calculation of the coefficients of the asymptotic expansion is remarkably simpler. On the other hand, the asymptotic sequence is as simple as in the standard Laplace’s method: inverse powers of the asymptotic variable. New asymptotic expansions of the Gamma function Γ(z) for large z and the Gauss hypergeometric function ₂F₁(a,b,c;z) for large b and c are given as illustrations. An explicit formula for the coefficients of the classical Stirling expansion of Γ(z) is also given.