Applications of Spreadsheets in Education : The Amazing Power of a Simple Tool

This e-book is devoted to the use of spreadsheets in the service of education in a broad spectrum of disciplines: science, mathematics, engineering, business, and general education. The effort is aimed at collecting the works of prominent researchers and educators that make use of spreadsheets as a means to communicate concepts with high educational value. The e-book brings some of the most recent applications of spreadsheets in education and research to the fore. To offer the reader a broad overview of the diversity of applications, carefully chosen articles from engineering (power systems and control), mathematics (calculus, differential equations, and probability), science (physics and chemistry), and education are provided. Some of these applications make use of Visual Basic for Applications (VBA), a versatile computer language that further expands the functionality of spreadsheets. The material included in this e-book should inspire readers to devise their own applications and enhance their teaching and/or learning experience.

The use of a Riesz fractional differential-based approach for texture enhancement in image processing

Texture enhancement is an important component of image processing that finds extensive application in science and engineering. The quality of medical images, quantified using the imaging texture, plays a significant role in the routine diagnosis performed by medical practitioners. Most image texture enhancement is performed using classical integral order differential mask operators. Recently, first order fractional differential operators were used to enhance images. Experimentation with these methods led to the conclusion that fractional differential operators not only maintain the low frequency contour features in the smooth areas of the image, but they also nonlinearly enhance edges and textures corresponding to high frequency image components. However, whilst these methods perform well in particular cases, they are not routinely useful across all applications. To this end, we apply the second order Riesz fractional differential operator to improve upon existing approaches of texture enhancement. Compared with the classical integral order differential mask operators and other first order fractional differential operators, we find that our new algorithms provide higher signal to noise values and superior image quality.

Estimating link-dependent origin-destination matrices from sample trajectories and traffic counts

In transport networks, Origin-Destination matrices (ODM) are classically estimated from road traffic counts whereas recent technologies grant also access to sample car trajectories. One example is the deployment in cities of Bluetooth scanners that measure the trajectories of Bluetooth equipped cars. Exploiting such sample trajectory information, the classical ODM estimation problem is here extended into a link-dependent ODM (LODM) one. This much larger size estimation problem is formulated here in a variational form as an inverse problem. We develop a convex optimization resolution algorithm that incorporates network constraints. We study the result of the proposed algorithm on simulated network traffic.

The tensor distribution function

Diffusion weighted magnetic resonance imaging is a powerful tool that can be employed to study white matter microstructure by examining the 3D displacement profile of water molecules in brain tissue. By applying diffusion-sensitized gradients along a minimum of six directions, second-order tensors (represented by three-by-three positive definite matrices) can be computed to model dominant diffusion processes. However, conventional DTI is not sufficient to resolve more complicated white matter configurations, e.g., crossing fiber tracts. Recently, a number of high-angular resolution schemes with more than six gradient directions have been employed to address this issue. In this article, we introduce the tensor distribution function (TDF), a probability function defined on the space of symmetric positive definite matrices. Using the calculus of variations, we solve the TDF that optimally describes the observed data. Here, fiber crossing is modeled as an ensemble of Gaussian diffusion processes with weights specified by the TDF. Once this optimal TDF is determined, the orientation distribution function (ODF) can easily be computed by analytic integration of the resulting displacement probability function. Moreover, a tensor orientation distribution function (TOD) may also be derived from the TDF, allowing for the estimation of principal fiber directions and their corresponding eigenvalues.

Fourier spectral methods for fractional-in-space reaction-diffusion equations

Fractional differential equations are becoming increasingly used as a powerful modelling approach for understanding the many aspects of nonlocality and spatial heterogeneity. However, the numerical approximation of these models is demanding and imposes a number of computational constraints. In this paper, we introduce Fourier spectral methods as an attractive and easy-to-code alternative for the integration of fractional-in-space reaction-diffusion equations described by the fractional Laplacian in bounded rectangular domains ofRn. The main advantages of the proposed schemes is that they yield a fully diagonal representation of the fractional operator, with increased accuracy and efficiency when compared to low-order counterparts, and a completely straightforward extension to two and three spatial dimensions. Our approach is illustrated by solving several problems of practical interest, including the fractional Allen–Cahn, FitzHugh–Nagumo and Gray–Scott models, together with an analysis of the properties of these systems in terms of the fractional power of the underlying Laplacian operator.

Numerical analysis of a new space-time variable fractional order advection-dispersion equation

Many physical processes appear to exhibit fractional order behavior that may vary with time and/or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider a new space–time variable fractional order advection–dispersion equation on a finite domain. The equation is obtained from the standard advection–dispersion equation by replacing the first-order time derivative by Coimbra’s variable fractional derivative of order α(x)∈(0,1]α(x)∈(0,1], and the first-order and second-order space derivatives by the Riemann–Liouville derivatives of order γ(x,t)∈(0,1]γ(x,t)∈(0,1] and β(x,t)∈(1,2]β(x,t)∈(1,2], respectively. We propose an implicit Euler approximation for the equation and investigate the stability and convergence of the approximation. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

Some remarks on the symplectic pairing on the moduli space of representations of the fundamental group of surfaces

This work grew out of an attempt to understand a conjectural remark made by Professor Kyoji Saito to the author about a possible link between the Fox-calculus description of the symplectic structure on the moduli space of representations of the fundamental group of surfaces into a Lie group and pairs of mutually dual sets of generators of the fundamental group. In fact in his paper [3] , Prof. Kyoji Saito gives an explicit description of the system of dual generators of the fundamental group.

Energy-momentum conserving algorithm for nonlinear transient analysis within the framework of hybrid elements

This work deals with the formulation and implementation of an energy-momentum conserving algorithm for conducting the nonlinear transient analysis of structures, within the framework of stress-based hybrid elements. Hybrid elements, which are based on a two-field variational formulation, are much less susceptible to locking than conventional displacement-based elements within the static framework. We show that this advantage carries over to the transient case, so that not only are the solutions obtained more accurate, but they are obtained in fewer iterations. We demonstrate the efficacy of the algorithm on a wide range of problems such as ones involving dynamic buckling, complicated three-dimensional motions, et cetera.

Differential Games of Mixed Type with Control and Stopping Times

We study a zero sum differential game of mixed type where each player uses both control and stopping times. Under certain conditions we show that the value function for this problem exists and is the unique viscosity solution of the corresponding variational inequalities. We also show the existence of saddle point equilibrium for a special case of differential game.

Logic as the Universal Science : Bertrand Russell's Early Conception of Logic and Its Philosophical Context

Bertrand Russell (1872 1970) introduced the English-speaking philosophical world to modern, mathematical logic and foundational study of mathematics. The present study concerns the conception of logic that underlies his early logicist philosophy of mathematics, formulated in The Principles of Mathematics (1903). In 1967, Jean van Heijenoort published a paper, Logic as Language and Logic as Calculus, in which he argued that the early development of modern logic (roughly the period 1879 1930) can be understood, when considered in the light of a distinction between two essentially different perspectives on logic. According to the view of logic as language, logic constitutes the general framework for all rational discourse, or meaningful use of language, whereas the conception of logic as calculus regards logic more as a symbolism which is subject to reinterpretation. The calculus-view paves the way for systematic metatheory, where logic itself becomes a subject of mathematical study (model-theory). Several scholars have interpreted Russell s views on logic with the help of the interpretative tool introduced by van Heijenoort,. They have commonly argued that Russell s is a clear-cut case of the view of logic as language. In the present study a detailed reconstruction of the view and its implications is provided, and it is argued that the interpretation is seriously misleading as to what he really thought about logic. I argue that Russell s conception is best understood by setting it in its proper philosophical context. This is constituted by Immanuel Kant s theory of mathematics. Kant had argued that purely conceptual thought basically, the logical forms recognised in Aristotelian logic cannot capture the content of mathematical judgments and reasonings. Mathematical cognition is not grounded in logic but in space and time as the pure forms of intuition. As against this view, Russell argued that once logic is developed into a proper tool which can be applied to mathematical theories, Kant s views turn out to be completely wrong. In the present work the view is defended that Russell s logicist philosophy of mathematics, or the view that mathematics is really only logic, is based on what I term the Bolzanian account of logic . According to this conception, (i) the distinction between form and content is not explanatory in logic; (ii) the propositions of logic have genuine content; (iii) this content is conferred upon them by special entities, logical constants . The Bolzanian account, it is argued, is both historically important and throws genuine light on Russell s conception of logic.

Leibniz on Rational Decision-Making

In this study I discuss G. W. Leibniz's (1646-1716) views on rational decision-making from the standpoint of both God and man. The Divine decision takes place within creation, as God freely chooses the best from an infinite number of possible worlds. While God's choice is based on absolutely certain knowledge, human decisions on practical matters are mostly based on uncertain knowledge. However, in many respects they could be regarded as analogous in more complicated situations. In addition to giving an overview of the divine decision-making and discussing critically the criteria God favours in his choice, I provide an account of Leibniz's views on human deliberation, which includes some new ideas. One of these concerns is the importance of estimating probabilities in making decisions one estimates both the goodness of the act itself and its consequences as far as the desired good is concerned. Another idea is related to the plurality of goods in complicated decisions and the competition this may provoke. Thirdly, heuristic models are used to sketch situations under deliberation in order to help in making the decision. Combining the views of Marcelo Dascal, Jaakko Hintikka and Simo Knuuttila, I argue that Leibniz applied two kinds of models of rational decision-making to practical controversies, often without explicating the details. The more simple, traditional pair of scales model is best suited to cases in which one has to decide for or against some option, or to distribute goods among parties and strive for a compromise. What may be of more help in more complicated deliberations is the novel vectorial model, which is an instance of the general mathematical doctrine of the calculus of variations. To illustrate this distinction, I discuss some cases in which he apparently applied these models in different kinds of situation. These examples support the view that the models had a systematic value in his theory of practical rationality.

An absolute method for the determination of surface tension of liquids using pendent drop profiles

Drop formation at conical tips which is of relevance to metallurgists is investigated based on the principle of minimization of free energy using the variational approach. The dimensionless governing equations for drop profiles are computer solved using the fourth order Runge-Kutta method. For different cone angles, the theoretical plots of XT and ZT vs their ratio, are statistically analyzed, where XT and ZT are the dimensionless x and z coordinates of the drop profile at a plane at the conical tip, perpendicular to the axis of symmetry. Based on the mathematical description of these curves, an absolute method has been proposed for the determination of surface tension of liquids, which is shown to be preferable in comparison with the earlier pendent-drop profile methods.