Exceptions and counterexamples: Understanding Abel's comment on Cauchy's Theorem

It may seem odd that Abel, a protagonist of Cauchy's new rigor, spoke of “exceptions” when he criticized Cauchy's theorem on the continuity of sums of continuous functions. However, when interpreted contextually, exceptions appear as both valid and viable entities in the early 19th century. First, Abel's use of the term “exception” and the role of the exception in his binomial paper is documented and analyzed. Second, it is suggested how Abel may have acquainted himself with the exception and his use of it in a process denoted critical revision is discussed. Finally, an interpretation of Abel's exception is given that identifies it as a representative example of a more general transition in the understanding of mathematical objects that took place during the period. With this interpretation, exceptions find their place in a fundamental transition during the early 19th century from a formal approach to analysis toward a more conceptual one.     

The introduction of logarithms into Spain

During the first half of the 17th century, logarithms were taught by some professors in Spain, but knowledge of this subject remained scanty until the publication of Architectura civil by Juan Caramuel (1678) and especially of Trigonometria española by José Zaragoza (1672). Logarithms were considered only as an aid for computation up to the second half of the 18th century. Only when the infinitesimal calculus became more widely spread in Spanish mathematics, analytical interpretations of logarithms were also taken into account in books such as Elementos de matemáticas by Benito Bails (1776).     

Thomas Harriot on continuous compounding

In the unpublished papers of Thomas Harriot (BL Add Ms 6782 f. 67) there is a remarkable calculation. It concerns what is now known as the continuous compounding of interest, and was probably written before 1620. This article describes the background to Harriot's calculation, and its significance. The general solution of Harriot's problem depended on some of the important mathematical developments in seventeenth century, such as logarithms and infinite series. By 1727 John Arbuthnot was able to give a recognizably modern account of it, but the final step was the work of Leonhard Euler.     

Maximum thick paths in static and dynamic environments

We consider the problem of finding a large number of disjoint paths for unit disks moving amidst static or dynamic obstacles. The problem is motivated by the capacity estimation problem in air traffic management, in which one must determine how many aircraft can safely move through a domain while avoiding each other and avoiding “no-fly zones” and predicted weather hazards. For the static case we give efficient exact algorithms, based on adapting the “continuous uppermost path” paradigm. As a by-product, we establish a continuous analogue of Menger's Theorem.Next we study the dynamic problem in which the obstacles may move, appear and disappear, and otherwise change with time in a known manner; in addition, the disks are required to enter/exit the domain during prescribed time intervals. Deciding the existence of just one path, even for a 0-radius disk, moving with bounded speed is NP-hard, as shown by Canny and Reif [J. Canny, J.H. Reif, New lower bound techniques for robot motion planning problems, in: Proc. 28th Annu. IEEE Sympos. Found. Comput. Sci., 1987, pp. 49–60]. Moreover, we observe that determining the existence of a given number of paths is hard even if the obstacles are static, and only the entry/exit time intervals are specified for the disks. This motivates studying “dual” approximations, compromising on the radius of the disks and on the maximum speed of motion.Our main result is a pseudopolynomial-time dual-approximation algorithm. If K unit disks, each moving with speed at most 1, can be routed through an environment, our algorithm finds (at least) K paths for disks of radius somewhat smaller than 1 moving with speed somewhat larger than 1.     

The Ladies' Diary or Woman's Almanack, 1704–1841

The existence of the Ladies' Diary or the Woman's Almanack, an 18th century English magazine devoted largely to problems and puzzles in mathematics, indicates that stereotypes about the inability of women to understand and enjoy mathematics were less strongly believed in the 18th century than they are today. The beginning of the Ladies' Diary coincides with the popularization of mathematics and the growth of mathematical literacy. However, as mathematical literacy spread in response to developing technology's requirements for more mathematically sophisticated workers, women, not part of this need, were left behind. This effect is reflected in the decline in the number of women contributors over the life of the publication.     

Methods of evaluation and scale of functional spaces

Linear equations of mathematical physics with constant coefficients have fed calculational mathematics since the 18th century. The area of nonlinear equations with variable coefficients arose due to gas-hydrodynamic problems in the 20th century. Now, one of the methods of research for properties of solutions of such equations and, accordingly, applied problems is the use of calculations on modern computers. The capacities of computers and their efficiency have increased in the 21st century and allowed progress to be made in solving applied problems, except for cases of methodical errors in calculations. One of the basic sources of such methodical errors is the “uncontrollable” machine accuracy of calculations. One of the methods of solving such numerical problems is a suitable localization of the problem and the choice of an adequate basis of the necessary functional space. Below, we state new results in this area of mathematical and applied research.     

The difference in a problem of rations from the Rhind mathematical papyrus

In an ancient Egyptian problem of bread distribution from the Rhind mathematical papyrus (dated between 1794 and 1550 B.C.), a procedure of “false position” is used in the calculation of a series of five rations. The algorithm is only partially illustrated in the problem text, and last century's prevailing interpretations suggested a determination of the series by trial and error. The missing part of the computational procedure is reconstructed in this article as an application of the algorithm, exemplified in the preceding section of the papyrus, to calculate an unknown quantity by means of the method of “false position.”     

Changing canons of mathematical and physical intelligibility in the later 17th century

Learning to use the new calculus in the late 17th century meant looking at quantities and configurations, and the relationships among them, in fundamentally new ways. In part, as Leibniz argued implicitly in his articles, the new concepts lay along lines established by Viète, Fermat, Descartes, and other “analysts” in their development of algebraic geometry and the theory of equations. But in part too, those concepts drew intuitive support from the new mechanics that they were being used to explicate and that was rapidly becoming the primary area of their application. So it was that the world machine that emerged from the Scientific Revolution could be both mechanically intelligible and mathematically transcendental.     

Cartan,Schouten and the search for connection

In this paper we provide an analysis, both historical and mathematical, of two joint papers on the theory of connections by Élie Cartan and Jan Arnoldus Schouten that were published in 1926. These papers were the result of a fertile collaboration between the two eminent geometers that flourished in the two-year period 1925–1926. We describe the birth and the development of their scientific relationship especially in the light of unpublished sources that, on the one hand, offer valuable insight into their common research interests and, on the other hand, provide a vivid picture of Cartan's and Schouten's different technical choices. While the first part of this work is preeminently of a historical character, the second part offers a modern mathematical treatment of some contents of the two contributions.     

Mathematical foundations of a cultural project or Ramchandra's treatise “through the unsentimentalised light of mathematics”

The nineteenth century witnessed a number of projects of cultural rapprochement between the knowledge traditions of the East and West. This paper discusses the attempt to render elementary calculus amenable to an Indian audience in the indigenous mathematical idiom, undertaken by an Indian polymath, Ramchandra. The exercise is specifically located in his book A Treatise on the Problems of Maxima and Minima. The paper goes on to discuss the “vocation of failure” of the book within the context of encounter and the pedagogy of mathematics.     

Geometric separation and exact solutions for the parameterized independent set problem on disk graphs

We consider the parameterized problem, whether for a given set  of n disks (of bounded radius ratio) in the Euclidean plane there exists a set of k non-intersecting disks. For this problem, we expose an algorithm running in time that is—to our knowledge—the first algorithm with running time bounded by an exponential with a sublinear exponent. For λ-precision disk graphs of bounded radius ratio, we show that the problem is fixed parameter tractable with a running time  . The results are based on problem kernelization and a new “geometric ( -separator) theorem” which holds for all disk graphs of bounded radius ratio. The presented algorithm then performs, in a first step, a “geometric problem kernelization” and, in a second step, uses divide-and-conquer based on our new “geometric separator theorem.”     

Passage to the limit in Proposition I, Book I of Newton's Principia

The purpose of this paper is to analyze the way in which Newton uses his polygon model and passes to the limit in Proposition I, Book I of his Principia. It will be evident from his method that the limit of the polygon is indeed the orbital arc of the body and that his approximation of the actual continuous force situation by a series of impulses passes correctly in the limit into the continuous centripetal force situation. The analysis of the polygon model is done in two ways: (1) using the modern concepts of force, linear momentum, linear impulse, and velocity, and (2) using Newton's concepts of motive force and quantity of motion. It should be clearly understood that the term “force” without the adjective “motive,” is used in the modern sense, which is that force is a vector which is the time rate of change of the linear momentum. Newton did not use the word “force” in this modern sense. The symbol F denotes modern force. For Newton “force” was “motive force,” which is measured by the change in the quantity of motion of a body. Newton's “quantity of motion” is proportional to the magnitude of the modern vector momentum. Motive force is a scalar and the symbol F_m is used for motive force.     

Māl, enunciations, and the prehistory of Arabic algebra

Medieval Arabic algebra books intended for practical training generally have in common a first “book” which is divided into two sections: one on the methods of solving simplified equations and manipulating expressions, followed by one consisting of worked-out problems. By paying close attention to the wording of the problems in the books of al-Khwārizmī, Abū Kāmil, and Ibn Badr, we reveal the different ways the word māl was used. In the enunciation of a problem it is a common noun meaning “quantity,” while in the solution it is the proper noun naming the square of “thing” (shay '). We then look into the differences between the wording of enunciations and equations, which clarify certain problems solved without “thing,” and help explain the development of algebra before the time of al-Khwārizmī.     

Proposal and implementation of the “science SQC” quality control principle

In forecasting the operation of the manufacturing industry in the 21^st century, the authors recently proposed “science SQC” as a demonstrative-scientific methodology and discussed its effectiveness on the basis of verification studies conducted by Toyota Motor Corporation. This study outlines a new SQC principle “science SQC”, as a demonstrative-scientific methodology, which enables the principle of TQM to be improved systematically.     

On a decision rule using dichotomies for identifying the nonnegligible parameter in certain linear models

Consider the class of linear models (with uncorrelated observation, each having variance σ²), in which it is known that at most k (location) parameters are negligible, but it is not known which are negligible. The problem is to identify the nonnegligible parameters. In this paper, for k = 1, and under certain restrictions on the model, a technique is developed for solving this problem, which has the feature of requiring (in an information theoretic sense) the minimum amount of computation. (It can “search through” 2^m objects, using m “steps.”) The technique consists of dichotomizing the set of parameters (one known subset possibly containing the nonnegligible element, and the other not), using chi-square variables. A method for computing the probability that the correct parameter is identified, is presented, and an important application to factorial search designs is established.     

Explosive instability in nonlinear waves in media with negative viscosity : PMM vol. 38, n 6, 1974, pp. 991–995

The propagation of a wave of a finite amplitude in a medium with a nonlinearity of the second degree and negative viscosity, is examined. It is shown that in a finite time singularities appear in the solution. The exact solution of the Cauchy problem is given for a specific case. Recently the effects of negative viscosity which cause an increase in the energy of the wave motion have been studied intensively in electrodynamics, plasma physics, the Earth's atmosphere, in the theory of the circulation of the oceans and of flow in open channels [1–4], Wave amplification caused by an energy transfer from turbulent to regular motions, is possible in any medium having space-time fluctuations, provided the correlation time is sufficiently small [5, 6]. As the wave amplitude increases, nonlinear effects become important; they have been taken into account in cases where the interaction of a finite number of harmonics [2, 4] and the structure of steady motions have been examined [3].It is shown in this paper that in a medium with negative viscosity and a second degree dynamic nonlinearity, a solution of the Cauchy problem for an arbitrary “good” form of the initial perturbation, exists over a finite time interval. An example of such a solution is given.     

Macrocosmos/microcosmos: celestial mechanics and old quantum theory

The Bohr atom was a solar system in miniature. Despite many deep foundational questions related to the origin of quantized motion, rapid progress was made in its mathematical development and its apparently successful application to spectral line series. In United States, where celestial mechanics flourished throughout the 19th and well into the 20th century, mathematicians and physicists were well prepared for just this sort of problem and made it their own far faster than many areas of the new physics. This paper examines the link between classical problems of perturbation theory, three-body and N-body orbital trajectories, the Hamilton–Jacobi equation, and the old quantum theory. I discuss why it was comparatively easy for American applied mathematicians, astronomers, and mathematical physicists to make significant contributions quickly to quantum theory and why further progress toward quantum mechanics by the same cohort was, in contrast, so slow.     

Bier Spheres and Posets

In 1992 Thomas Bier presented a strikingly simple method to produce a huge number of simplicial (n – 2)-spheres on 2n vertices, as deleted joins of a simplicial complex on n vertices with its combinatorial Alexander dual. Here we interpret his construction as giving the poset of all the intervals in a boolean algebra that “cut across an ideal.” Thus we arrive at a substantial generalization of Bier’s construction: the Bier posets Bier(P, I) of an arbitrary bounded poset P of finite length. In the case of face posets of PL spheres this yields cellular “generalized Bier spheres.” In the case of Eulerian or Cohen–Macaulay posets P we show that the Bier posets Bier(P, I) inherit these properties. In the boolean case originally considered by Bier, we show that all the spheres produced by his construction are shellable, which yields “many shellable spheres,” most of which lack convex realization. Finally, we present simple explicit formulas for the g-vectors of these simplicial spheres and verify that they satisfy a strong form of the g-conjecture for spheres.     

Detecting simple dynamics in Cournot-like models

We introduced the so-called Cournot-like models, i.e. n-dimensional discrete dynamical systems which constitute the mathematical environment for modeling some economic and biological processes. The main aim of this work is to present a characterization of the dynamical simplicity for these types of systems through the property “to have zero topological entropy”. Cournot-like systems generalize the well-known economic situation of competition in a duopolistic market introduces by Cournot in 1838.     

Elements in accepting an explanation

This paper addresses the question of what criteria influenced the acceptance of two “explanations” by grade 5 students. The students accepted the use of deductive reasoning as explanatory, as well as using reasoning by analogy in their own explanations. The “explanations” can be interpreted as proofs by mathematical induction. The main weakness of mathematical induction as a form of explanation was the arbitrariness of the initial step. The induction step did not seem to trouble these students. Other elements in their acceptance of explanations were concreteness, familiarity, and opportunities for multiple interpretations.