P G Tait's statistical models

This paper explains how the discovery of a pocket notebook brings to light P G Tait's surprising involvement in statistics. Tait (1831–1901) was Professor of Mathematics at the Queen's College, Belfast and later of Natural Philosophy at the University of Edinburgh. He was a Fellow of the Royal Society of Edinburgh and a former Fellow of Peterhouse, Cambridge (senior wrangler and first Smith's prizeman in 1852).     

Arthur Cayley as Sadleirian Professor: A Glimpse of Mathematics Teaching at 19th-Century Cambridge

This article contains the hitherto unpublished text of Arthur Cayley's inaugural professorial lecture given at Cambridge University on 3 November 1863. Cayley chose a historical treatment to explain the prevalent basic notions of analytical geometry, concentrating his attention in the period (1638–1750). Topics Cayley discussed include the geometric interpretation of complex numbers, the theory of pole and polar, points and lines at infinity, plane curves, the projective definition of distance, and Pascal's and Maclaurin's geometrical theorems. The paper provides a commentary on this lecture with reference to Cayley's work in geometry. The ambience of Cambridge mathematics as it existed after 1863 is briefly discussed.Copyright 1999 Academic Press.Cet article contient le texte jusqu'ici inédit de la leçon inaugurale de Arthur Cayley donnée à l'Université de Cambridge le 3 novembre 1863. Cayley choisit une approche historique pour expliquer les notions fondamentales de la géométrie analytique, qui existaient alors, en concentrant son attention sur la période 1638–1750. Les sujets discutés incluent l'interpretation géométrique des nombres complexes, la théorie des pôles et des polaires, les points et les lignes à l'infini, les courbes planes, la définition projective de la distance, et les théorèmes géométriques de Pascal et de Maclaurin. L'article contient aussi un commentaire reliant cette leçon à l'oeuvre de Cayley en géométrie. L'atmosphère des mathématiques à Cambridge après 1863 est brièvement discutée.Copyright 1999 Academic Press.MSC Classification: 01A55, 01A72, 01A73.     

Ein starker Normalisationssatz für die intuitionistische Typentheorie

In this paper we give a proof of the strong normalisation theorem for intuitionistic type theory. Cut-elimination is an immediate consequence of it. The basic ideas of our proof are the use of wellfoundedness predicates, originally due to J. Y. Girard [1] and a mapping from the derivations in type theory into a type-free -calculus, which we learned in a lecture, given by W. Tait at Munich 1972.     

On the intellectual heritage of Henri Poincaré

Poincaré's understanding of how mathematics grows over time was informed by the theory of evolution by natural selection and Mach's economy of thought. According to Poincaré, mathematics is neither created nor discovered but cultivated as part of our intellectual inheritance. This paper is based on a lecture delivered at the joint meeting of the BSHM and CSHPM at Trinity College, Dublin, 2011. The title echoes The mathematical heritage of Henri Poincaré (1983) a collection of essays edited by Felix Browder.     

ON MAXWELL EQUATIONS WITH THE TRANSPARENT BOUNDARY CONDITION

In this paper we show the well-posedness and stability of the Maxwell scattering problem with the transparent boundary condition. The proof depends on the well-posedness of the time-harmonic Maxwell scattering problem with complex wave numbers which is also established.     

Better Than Survival: The Need for Research to Strengthen Planning in Industry and the Public Service

Editor's NoteThis is an invited contribution in our occasional series O.R. Reflections, in which are presented essentially personal views of O.R., based on contributors' own experience. What follows is a slightly revised version of the author's inaugural lecture as Professor of Systems and Operational Research at the University of Warwick on 10th May 1982.The author draws on his experience in industry and in an international research institute (IIASA) o restate the importance of strategic planning if organisations are to do better than mere survival. He argues the need for problem-oriented, inter-disciplinary research to support such activity and suggests that such research, directed towards investment and strategic policy, has a potential value which is quite out of proportion to the money currently invested in it.     

Numerical analysis of two partitioned methods for uncoupling evolutionary MHD flows

Magnetohydrodynamics (MHD) studies the dynamics of electrically conducting fluids, involving Navier–Stokes (NSE) equations in fluid dynamics and Maxwell equations in eletromagnetism. The physical processes of fluid flows and electricity and magnetism are quite different and numerical simulations of each subprocess can require different meshes, time steps, and methods. In most terrestrial applications, MHD flows occur at low‐magnetic Reynold numbers. We introduce two partitioned methods to solve evolutionary MHD equations in such cases. The methods we study allow us at each time step to call NSE and Maxwell codes separately, each possibly optimized for the subproblem's respective physics. Complete error analysis and computational tests supporting the theory are given.Copyright © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1083–1102, 2014     

Mathematical modeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section

In this work, we focus on the time-domain simulation of the propagation of electromagnetic waves in non-homogeneous lossy coaxial cables. The full 3D Maxwell equations, that described the propagation of current and electric potential in such cables, are classically not tackled directly, but instead a 1D scalar model known as the telegraphist's model is used. We aim at justifying, by means of asymptotic analysis, a time-domain “homogenized” telegraphist's model. This model, which includes a nonlocal in time operator, is obtained via asymptotic analysis, for a lossy coaxial cable whose cross section is not homogeneous.     

Viability of circle and sphere theorems in potential theory and hydrodynamics via Maxwell's conjecture

In A Treatise on Electricity and Magnetism, Maxwell determines the angles of intersection for which one may use Kelvin's inversion method to obtain the perturbed electric potential upon placing intersecting spherical conductors into a region with a known potential. There are numerous modern applications utilizing this geometric construction in potential theory and hydrodynamics, and generalized circle and sphere theorems play a foundational role in this area of mathematical physics. In his work, Maxwell gives an intuitive argument for obtaining the perturbed potential based on intersecting planar conductors and a spherical inversion, and in this paper we extend his ideas to a full proof using rotational transformations and reflections. In the process, we disprove results in [Proc Lond Math Soc., 1966:3(16)] and [Stud Appl Math., 2001:106(4); Z. Angew. Math. Mech., 2001:81(8)] on boundary value problems in hydrodynamics involving intersecting circles and spheres, and we detail the angles of intersection for which these theorems are viable. Moreover, our proof recovers a special case overlooked by Maxwell for which Kelvin's inversion method may be utilized to obtain full solutions.     

求解两类Maxwell方程组棱元离散系统的快速算法和自适应方法

Maxwell方程组棱元离散系统的快速算法和自适应方法是当前计算电磁场中的研究热点和难点. 首先, 针对H(curl)椭圆方程组的棱元离散系统, 通过建立棱元空间的稳定性分解, 设计了相应的快速迭代法和高效预条件子, 并且证明了迭代算法的收敛率和预条件子的条件数均不依赖于模型参数和网格规模. 其次, 针对时谐Maxwell方程组的棱有限元方法, 利用离散的Helmholtz分解, 连续散度为零函数对离散散度为零函数的逼近性和对偶论证, 获得了在L²和H(curl)范数下的拟最优误差估计. 进而设计和分析了相应的两网格法. 最后, 分别针对变系数H(curl)椭圆方程组和不定时谐Maxwell方程组, 考虑了一种不需要标记振荡项和加密单元不需要满足“内节点” 性质的自适应棱有限元法(AEFEM), 并证明了AEFEM的收敛性. 进一步, 当初始网格和Dörfler标记策略参数满足一定的假设条件时, 利用AEFEM的收敛性、误差的整体下界和局部上界估计, 证明了AEFEM的拟最优复杂性.     

Variational Integrators for Thermo-Viscoelastic Discrete Systems

Variational integrators are modern time-integration schemes based on a discretization of the underlying variational principle. In this paper, Hamilton's principle is approximated by an action sum, whose vanishing variation results in discrete Euler-Lagrange equations or, equivalently, in discrete evolution equations for the position and momentum. In order to include the viscous and thermal virtual work (mechanical and thermal virtual dissipation), Hamilton's principle is extended by D'Alembert terms, which account for the time evolution equation of the internal variable and Fourier's law. From this variational formulation, variational integrators using different orders of approximation of the state variables as well as of the quadrature of the action integral are constructed and compared. A thermo-viscoelastic double pendulum comprised of two discrete masses connected by generalized Maxwell elements, and subject to heat conduction between them serves as a discrete model problem. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

James Clerk Maxwell (BSHM–Gresham Lecture 2012)

The following article is the text of the 2012 BSHM–Gresham Lecture, delivered at Gresham College, London, on 31 October 2012. This and other BSHM–Gresham Lectures can be seen in full on the Gresham College website: http://www.gresham.ac.uk.

James Clerk Maxwell (1831–1879) was one of the most important mathematical physicists of all time, coming only after Newton and Einstein. Within a relatively short lifetime he made enormous contributions to science which this lecture will survey. Foremost among these was the formulation of the theory of electromagnetism with light, electricity, and magnetism all shown to be manifestations of the electromagnetic field. He also made major contributions to the theory of colour vision and optics, the kinetic theory of gases and thermodynamics, and the understanding of the dynamics and stability of Saturn’s rings.     

Primary school geometry: some unusual activities†

The lecture given in Lesotho in November 1984 outlines a series of activities which could enliven primary school geometry. It begins with a brief look at the meaning of geometry and how this is related to children's conceptions of space. It puts forward a curriculum which places a high value on perspective and other questions about children's visual acquisition of information. It continues with drawing activities and with exercises in paper folding. It suggests further ways in which geometry is entwined with geography and asks teachers to determine how their children make connections between the conventional primary school geometry they receive and their individual spatial perceptual structures.

     

A proof of the normal form theorem for the closed terms of Girard's system F by means of computability

In this paper a proof of the normal form theorem for the closed terms of Girard's system F is given by using a computability method à la Tait. It is worth noting that most of the standard consequences of the normal form theorem can be obtained using this version of the theorem as well. From the proof-theoretical point of view the interest of the proof is that the definition of computable derivation here used does not seem to be well founded. MSC: 03F05, 03B15.     

Heisenberg's critical mass calculations for an explosive nuclear reaction

Based on the declassified records of Heisenberg's Farm-Hall “lecture”, we are attempting in the present work to follow Heisenberg's thoughts and results and show in a more transparent way how he arrived at his surprisingly quite accurate estimation of the critical mass of a reflected U235 sphere undergoing explosive neutrons fission. We conclude with a discussion regarding the contraversy concerning the so-called German Uranium bomb and the role played by Heisenberg and C.F. von Weizsäcker in this connection during and after the war.     

Calculus and Analysis in Early 19th-Century Britain: The Work of William Wallace

Scottish-born William Wallace (1768–1843) was an early exponent of the differential calculus in Britain and translator of French mathematical works. Encyclopaedias published during the early 19th century provided a valuable educational resource, to which Wallace and his colleague, James Ivory, contributed. Wallace's encyclopaedia articles on “Fluxions” and his other analytical writings are examined here, as are his relations with James Ivory, John Herschel, and others. Wallace's long 1815 article on “Fluxions” in the Edinburgh Encyclopaedia was the first complete account of calculus, using differential notation, to be published in English. There, he attempted an original and rigorous “doctrine of limits,” which deserved more attention than it received. In 1832, while professor of mathematics in Edinburgh, he applied analysis to support the reform of taxation proposed in the Reform Bill. It is suggested that the later neglect of Wallace's achievements is attributable to a mix of personal, institutional, political, and national rivalries. Copyright 1999 Academic Press.William Wallace né en Écosse (1768–1843) fut un interprète précoce du calcul différentiel en Grande Bretagne et traduisit des ouvrages mathématiques de langue française. Les encyclopédies publiées au début du XIXe siècle fournirent une précieuse ressource pédagogique à laquelle contribuèrent Wallace et son collègue James Ivory. Les articles encyclopédiques de Wallace sur les “Fluxions” et d'autres écrits analytiques font l'objet du présent article; et aussi ses relations avec James Ivory, John Herschel et d'autres mathématiciens. Le long article de Wallace sur les “Fluxions” paru dans l'Edinburgh Encyclopaedia de 1815 fut le premier exposé complet du calcul, utilisant la notation différentielle, à être publié en anglais. Dans cet article, il mit en oeuvre une “doctrine des limites” originale et rigoureuse, qui méritait plus d'attention qu'elle ne reçut. En 1832, alors qu'il était professeur de mathématiques à Edimbourg, il utilisa l'analyse pour soutenir la réforme fiscale proposée dans le Projet de loi de Réforme. Cet article suggère que l'oubli postérieur de l'oeuvre accomplie par Wallace doit être attribué à un mélange de rivalités personnelles, institutionnelles, politiques et nationales.     

Value and information: A profit maximizing strategy for Maxwell's demon

Applications of Lagrange's method of undetermined multipliers in physics and economics suggest some analogies between the two disciplines. Applying Jaynes' approach to information theory reveals the long standing problem of Maxwell's demon in thermodynamics as a constrained optimization problem where information and energy are dual concepts. Temperature emerges as the energetic price of information. This approach highlights the economic features of the Maxwell demon discussed by Bridgman. It also suggests that an analog to temperature (or the price of information) should be present in any optimization problem constrained by costly information. A trivial economics example is presented where generalized temperature measures informational efficiency in the sense of Grossman and Stiglitz.