We present a smoothed analysis of Gaussian elimination, both with partial pivoting and without pivoting. Let A be any matrix and let A be a slight random perturbation of A. We prove that it is unlikely that A has large condition number. Using this result, we prove it is unlikely that A has large growth factor under Gaussian elimination without pivoting. By combining these results, we bound the smoothed precision needed to perform Gaussian elimination without pivoting. Our results improve the average-case analysis of Gaussian elimination without pivoting performed by Yeung and Chan (SIAM J. Matrix Anal. Appl., 1997). We then extend the result on the growth factor to the case of partial pivoting, and present the first analysis of partial pivoting that gives a sub-exponential bound on the growth factor. In particular, we show that if the random perturbation is Gaussian with variance [sigma]², then the growth factor is bounded by (n/[sigma])[to the power of] (o log n) with very high probability.; by Arvind Sankar.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.; Includes bibliographical references (p. 59-60).

In the study of electrochemical transport processes, experimental exploration currently outpaces theoretical understanding of new phenomena. Classical electrochemical transport theory is not equipped to explain the behavior of electrochemical systems in the extreme operating conditions required by modern devices. In this thesis, we extend the classical theory to examine the response of two electrochemical systems that form the basis for novel electrochemical devices. We first examine the DC response of an electrochemical thin film, such as the separator in a micro-battery, driven by current applied through reactive electrodes. The model system consists of a binary electrolyte between parallel-plate electrodes, each possessing a compact Stern layer which mediates Faradaic reactions with Butler-Volmer kinetics. Our analysis differs from previous studies in two significant ways. First, we impose the full nonlinear, reactive boundary conditions appropriate for electrolytic/galvanic cells.; (cont.) Since surface effects become important for physically small systems, the use of reactive boundary conditions is critical in order to gain insight into the behavior of actual electrochemical thin films that are sandwiched between reactive electrodes...

Students’ attitudes toward mathematics have been known to influence students’ participation, engagement, and achievement in mathematics. A variety of instruments have been developed to measure students’ attitudes toward mathematics for example Mathematics Attitude Scale (Aiken, 1974), Fennema-Sherman Mathematics Attitudes Scales (Fennema & Sherman, 1976), and Attitudes Toward Mathematics Inventory (ATMI) (Tapia & Marsh, 1996). The purpose of this paper is to report the validation of the ATMI instrument. It was administered to 699 Year 7 and 8 students in 14 schools in South Australia. The students responded on a five-point Likert scale. Confirmatory factor analysis (CFA) supported the original four-factor correlated structure based on several fit indices. The validation provided evidence that ATMI can be a viable scale to measure students’ attitudes toward mathematics in a South Australian context; Aysha Abdul Majeed, I Gusti Ngurah Darmawan, Peggy Lynch

peer-reviewed; This research study is aimed at improving the quality of the mathematics textbooks available for junior cycle students. It is widely agreed that there is room for improvement with regard to the quality of mathematics at both junior and senior cycle level in Ireland. One such area which can be improved is the e ectiveness of the resources available in both junior and senior cycle mathematics classrooms. While the TIMSS report (Valverde et al., 2002) has explored textbooks on an international scale, minimal research (minor role in TIMSS Report) has been carried out on Irish mathematics textbooks. Considering the level of responsibility shouldered by mathematics textbooks, there is an obvious gap in mathematics education research. The aim of this study is to investigate the quality of the mathematics textbooks currently in use at junior secondary school level in Ireland. This is achieved by investigating, extending and applying suitable methodological tools for textbook analysis. Ultimately the aim of this research is to improve the quality of teaching and learning of mathematics at junior cycle level which should feed directly into improving the quality of mathematics at senior cycle. This will be achieved by rst measuring the quality of the current junior cycle mathematics textbooks and then highlighting the role of improved textbooks in students' conceptual under- standing. At present in Ireland...

We examine the classical/intuitionist divide, and how it reflects on modern theories of infinitesimals. When leading intuitionist Heyting announced that "the creation of non-standard analysis is a standard model of important mathematical research", he was fully aware that he was breaking ranks with Brouwer. Was Errett Bishop faithful to either Kronecker or Brouwer? Through a comparative textual analysis of three of Bishop's texts, we analyze the ideological and/or pedagogical nature of his objections to infinitesimals a la Robinson. Bishop's famous "debasement" comment at the 1974 Boston workshop, published as part of his Crisis lecture, in reality was never uttered in front of an audience. We compare the realist and the anti-realist intuitionist narratives, and analyze the views of Dummett, Pourciau, Richman, Shapiro, and Tennant. Variational principles are important physical applications, currently lacking a constructive framework. We examine the case of the Hawking-Penrose singularity theorem, already analyzed by Hellman in the context of the Quine-Putnam indispensability thesis.; Comment: 88 pages, to appear in Intellectica 56 (2011), no. 2

In this paper we use theoretical frameworks from mathematics education and cognitive psychology to analyse Cauchy's ideas of function, continuity, limit and infinitesimal expressed in his Cours D'Analyse. Our analysis focuses on the development of mathematical thinking from human perception and action into more sophisticated forms of reasoning and proof, offering different insights from those afforded by historical or mathematical analyses. It highlights the conceptual power of Cauchy's vision and the fundamental change involved in passing from the dynamic variability of the calculus to the modern set-theoretic formulation of mathematical analysis. This offers a re-evaluation of the relationship between the natural geometry and algebra of elementary calculus that continues to be used in applied mathematics, and the formal set theory of mathematical analysis that develops in pure mathematics and evolves into the logical development of non-standard analysis using infinitesimal concepts. It suggests that educational theories developed to evaluate student learning are themselves based on the conceptions of the experts who formulate them. It encourages us to reflect on the principles that we use to analyse the developing mathematical thinking of students...

We study the adapted solution, numerical methods, and related convergence analysis for a unified backward stochastic partial differential equation (B-SPDE). The equation is vector-valued, whose drift and diffusion coefficients may involve nonlinear and high-order partial differential operators. Under certain generalized Lipschitz and linear growth conditions, the existence and uniqueness of adapted solution to the B-SPDE are justified. The methods are based on completely discrete schemes in terms of both time and space. The analysis concerning error estimation or rate of convergence of the methods is conducted. The key of the analysis is to develop new theory for random field based Malliavin calculus to prove the existence and uniqueness of adapted solutions to the first-order and second-order Malliavin derivative based B-SPDEs under random environments.; Comment: 39 pages

The infinite-dimensional unitary group U(infinity) is the inductive limit of growing compact unitary groups U(N). In this paper we solve a problem of harmonic analysis on U(infinity) stated in the previous paper math/0109193. The problem consists in computing spectral decomposition for a remarkable 4-parameter family of characters of U(infinity). These characters generate representations which should be viewed as analogs of nonexisting regular representation of U(infinity). The spectral decomposition of a character of U(infinity) is described by the spectral measure which lives on an infinite-dimensional space Omega of indecomposable characters. The key idea which allows us to solve the problem is to embed Omega into the space of point configurations on the real line without 2 points. This turns the spectral measure into a stochastic point process on the real line. The main result of the paper is a complete description of the processes corresponding to our concrete family of characters. We prove that each of the processes is a determinantal point process. That is, its correlation functions have determinantal form with a certain kernel. Our kernels have a special `integrable' form and are expressed through the Gauss hypergeometric function. In simpler situations of harmonic analysis on infinite symmetric group and harmonic analysis of unitarily invariant measures on infinite hermitian matrices similar results were obtained in our papers math/9810015...

The present work stemmed from the study of the problem of harmonic analysis on the infinite-dimensional unitary group U(\infty). That problem consisted in the decomposition of a certain 4-parameter family of unitary representations, which replace the nonexisting two-sided regular representation (Olshanski, J. Funct. Anal., 2003, arXiv:0109193). The required decomposition is governed by certain probability measures on an infinite-dimensional space \Omega, which is a dual object to U(\infty). A way to describe those measures is to convert them into determinantal point processes on the real line, it turned out that their correlation kernels are computable in explicit form --- they admit a closed expression in terms of the Gauss hypergeometric function 2-F-1 (Borodin and Olshanski, Ann. Math., 2005, arXiv:0109194). In the present work we describe a (nonevident) q-discretization of the whole construction. This leads us to a new family of determinantal point processes. We reveal its connection with an exotic finite system of q-discrete orthogonal polynomials --- the so-called pseudo big q-Jacobi polynomials. The new point processes live on a double q-lattice and we show that their correlation kernels are expressed through the basic hypergeometric function 2-\phi-1. A crucial novel ingredient of our approach is an extended version G of the Gelfand-Tsetlin graph (the conventional graph describes the Gelfand-Tsetlin branching rule for irreducible representations of unitary groups). We find the q-boundary of G...

The generalized Taylor expansion including a secret auxiliary parameter $h$ which can control and adjust the convergence region of the series is the foundation of the homotopy analysis method proposed by Liao. The secret of $h$ can't be understood in the frame of the homotopy analysis method. This is a serious shortcoming of Liao's method. We solve the problem. Through a detailed study of a simple example, we show that the generalized Taylor expansion is just the usual Taylor's expansion at different point $t_1$. We prove that there is a relationship between $h$ and $t_1$, which reveals the meaning of $h$ and the essence of the homotopy analysis method. As an important example, we study the series solution of the Blasius equation. Using the series expansion method at different points, we obtain the same result with liao's solution given by the homotopy analysis method.; Comment: 5 pages

A simple analytic tool namely the general series expansion method is proposed to find the solutions for nonlinear differential equations. By choosing a set of suitable basis functions $\{e_n(t,t_0)\}_{n=0}^{+\infty}$ such that the solution to the equation can be expressed by $u(t)=\sum_{n=0}^{+\infty}c_ne_n(t,t_0)$. In general, $t_0$ can control and adjust the convergence region of the series solution such that our method has the same effect as the homotopy analysis method proposed by Liao, but our method is more simple and clear. As a result, we show that the secret parameter $h$ in the homotopy analysis methods can be explained by using our parameter $t_0$. Therefore, our method reveals a key secret in the homotopy analysis method. For the purpose of comparison with the homotopy analysis method, a typical example is studied in detail.; Comment: 8 pages

We seek solutions $u\in\R^n$ to the semilinear elliptic partial difference equation $-Lu + f_s(u) = 0$, where $L$ is the matrix corresponding to the Laplacian operator on a graph $G$ and $f_s$ is a one-parameter family of nonlinear functions. This article combines the ideas introduced by the authors in two papers: a) {\it Nonlinear Elliptic Partial Difference Equations on Graphs} (J. Experimental Mathematics, 2006), which introduces analytical and numerical techniques for solving such equations, and b) {\it Symmetry and Automated Branch Following for a Semilinear Elliptic PDE on a Fractal Region} wherein we present some of our recent advances concerning symmetry, bifurcation, and automation fo We apply the symmetry analysis found in the SIAM paper to arbitrary graphs in order to obtain better initial guesses for Newton's method, create informative graphics, and be in the underlying variational structure. We use two modified implementations of the gradient Newton-Galerkin algorithm (GNGA, Neuberger and Swift) to follow bifurcation branches in a robust way. By handling difficulties that arise when encountering accidental degeneracies and higher-dimension we can find many solutions of many symmetry types to the discrete nonlinear system. We present a selection of experimental results which demonstrate our algorithm's capability to automatically generate bifurcation diagrams and solution graphics starting with only an edgelis of a graph. We highlight interesting symmetry and variational phenomena.

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's foundational work associated with the work of Boyer and Grabiner; and to Bishop's constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.; Comment: 57 pages; 3 figures. Corrected misprints

A brief introduction into Idempotent Mathematics and an idempotent version of Interval Analysis are presented. Some applications are discussed.; Comment: 22 pages, no figures; to be published in an abridged form in Reliable Computing (Kluwer); the bibliography is updated in this version

Stability analysis and control of linear impulsive systems is addressed in a hybrid framework, through the use of continuous-time time-varying discontinuous Lyapunov functions. Necessary and sufficient conditions for stability of impulsive systems with periodic impulses are first provided in order to set up the main ideas. Extensions to stability of aperiodic systems under minimum, maximum and ranged dwell-times are then derived. By exploiting further the particular structure of the stability conditions, the results are non-conservatively extended to quadratic stability analysis of linear uncertain impulsive systems. These stability criteria are, in turn, losslessly extended to stabilization using a particular, yet broad enough, class of state-feedback controllers, providing then a convex solution to the open problem of robust dwell-time stabilization of impulsive systems using hybrid stability criteria. Relying finally on the representability of sampled-data systems as impulsive systems, the problems of robust stability analysis and robust stabilization of periodic and aperiodic uncertain sampled-data systems are straightforwardly solved using the same ideas. Several examples are discussed in order to show the effectiveness and reduced complexity of the proposed approach.; Comment: 12 pages...

This is a survey paper on applications of mathematics of semirings to numerical analysis and computing. Concepts of universal algorithm and generic program are discussed. Relations between these concepts and mathematics of semirings are examined. A very brief introduction to mathematics of semirings (including idempotent and tropical mathematics) is presented. Concrete applications to optimization problems, idempotent linear algebra and interval analysis are indicated. It is known that some nonlinear problems (and especially optimization problems) become linear over appropriate semirings with idempotent addition (the so-called idempotent superposition principle). This linearity over semirings is convenient for parallel computations.; Comment: 36 pages, 1 figure. To appear in Springer Lecture Notes in Computational Science and Engineering.

In this paper, we describe a numerical continuation method that enables harmonic analysis of nonlinear periodic oscillators. This method is formulated as a boundary value problem that can be readily implemented by resorting to a standard continuation package - without modification - such as AUTO, which we used. Our technique works for any kind of oscillator, including electronic, mechanical and biochemical systems. We provide two case studies. The first study concerns itself with the autonomous electronic oscillator known as the Colpitts oscillator, and the second one with a nonlinear damped oscillator, a non-autonomous mechanical oscillator. As shown in the case studies, the proposed technique can aid both the analysis and the design of the oscillators, by following curves for which a certain constraint, related to harmonic analysis, is fulfilled.; Comment: 20 pages, 4 figures

A new iterative algorithm for solving initial data inverse problems from partial observations has been recently proposed in Ramdani, Tucsnak and Weiss [15]. Based on the concept of observers (also called Luenberger observers), this algorithm covers a large class of abstract evolution PDE's. In this paper, we are concerned with the convergence analysis of this algorithm. More precisely, we provide a complete numerical analysis for semi-discrete (in space) and fully discrete approximations derived using finite elements in space and finite differences in time. The analysis is carried out for abstract Schr\"odinger and wave conservative systems with bounded observation (locally distributed).; Comment: 38 pages, 1 figures

In the framework of quaternionic Clifford analysis in Euclidean space $\mathbb{R}^{4p}$, which constitutes a refinement of Euclidean and Hermitian Clifford analysis, the Fischer decomposition of the space of complex valued polynomials is obtained in terms of spaces of so--called (adjoint) symplectic spherical harmonics, which are irreducible modules for the symplectic group Sp$(p)$. Its Howe dual partner is determined to be $\mathfrak{sl}(2,\mathbb{C}) \oplus \mathfrak{sl}(2,\mathbb{C}) = \mathfrak{so}(4,\mathbb{C})$.

A very brief introduction to tropical and idempotent mathematics is presented. Tropical mathematics can be treated as a result of a dequantization of the traditional mathematics as the Planck constant tends to zero taking imaginary values. In the framework of idempotent mathematics usually constructions and algorithms are more simple with respect to their traditional analogs. We especially examine algorithms of tropical/idempotent mathematics generated by a collection of basic semiring (or semifield) operations and other "good" operations. Every algorithm of this type has an interval version. The complexity of this interval version coincides with the complexity of the initial algorithm. The interval version of an algorithm of this type gives exact interval estimates for the corresponding output data. Algorithms of linear algebra over idempotent and semirings are examined. In this case, basic algorithms are polynomial as well as their interval versions. This situation is very different from the traditional linear algebra, where basic algorithms are polynomial but the corresponding interval versions are NP-hard and interval estimates are not exact.; Comment: 26 pages, submitted to the journal "Computers and Mathematics with Applications". arXiv admin note: substantial text overlap with arXiv:1001.4247...