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 calculus of the trigonometric functions

The trigonometric functions entered “analysis” when Isaac Newton derived the power series for the sine in his De Analysi of 1669. On the other hand, no textbook until 1748 dealt with the calculus of these functions. That is, in none of the dozen or so calculus texts written in England and the continent during the first half of the 18th century was there a treatment of the derivative and integral of the sine or cosine or any discussion of the periodicity or addition properties of these functions. This contrasts sharply with what occurred in the case of the exponential and logarithmic functions. We attempt here to explain why the trigonometric functions did not enter calculus until about 1739. In that year, however, Leonhard Euler invented this calculus. He was led to this invention by the need for the trigonometric functions as solutions of linear differential equations. In addition, his discovery of a general method for solving linear differential equations with constant coefficients was influenced by his knowledge that these functions must provide part of that solution.     

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.     

AIC, Overfitting Principles, and the Boundedness of Moments of Inverse Matrices for Vector Autotregressions and Related Models

In his somewhat informal derivation, Akaike (in “Proceedings of the 2nd International Symposium Information Theory” (C. B. Petrov and F. Csaki, Eds.), pp. 610–624, Academici Kiado, Budapest, 1973) obtained AIC's parameter-count adjustment to the log-likelihood as a bias correction: it yields an asymptotically unbiased estimate of the quantity that measures the average fit of the estimated model to an independent replicate of the data used for estimation. We present the first mathematically complete derivation of an analogous property of AIC for comparing vector autoregressions fit to weakly stationary series. As a preparatory result, we derive a very general “overfitting principle,” first formulated in a more limited context in Findley (Ann. Inst. Statist. Math.43, 509–514, 1991), asserting that a natural measure of an estimated model's overfit due to parameter estimation is equal, asymptotically, to a measure of its accuracy loss with independent replicates. A formal principle of parsimony for fitted models is obtained from this, which for nested models, covers the situation in which all models considered are misspecified. To prove these results, we establish a set of general conditions under which, for each τ1, the absolute τth moments of the entries of the inverse matrices associated with least squares estimation are bounded for sufficiently large sample sizes.     

Newton's solution to the equiangular spiral problem and a new solution using only the equiangular property

This paper is a study of Proposition IX of Book I of Newton's Principia, the problem of determining the centripetal force for an equiangular spiral. In Newton's main proof of this proposition there is an error concerning his reason for the figure SPRQT being “given in kind,” and a very interesting technique of varying things in the neighborhood of a limit. This main proof utilized Newton's formula for the limit of SP²QT²/QR given in Corollary I to Proposition VI of the Principia. Newton also gave an alternate proof which utilized his formula for SY²PV given in Corollary III to Proposition VI. The “given” of Proposition IX was “a spiral PQS, cutting all the radii SP, SQ, &c., in a given angle.” Both the main proof and the alternate proof implicitly depend on the property of the equiangular spiral that the radius of curvature at any point is proportional to the pole distance SP. We here offer a new proof of Newton's proposition which does not depend on this implicit assumption.     

Alpha-theory: An elementary axiomatics for nonstandard analysis

The methods of nonstandard analysis are presented in elementary terms by postulating a few natural properties for an infinite “ideal” number . The resulting axiomatic system, including a formalization of an interpretation of Cauchy's idea of infinitesimals, is related to the existence of ultrafilters with special properties, and is independent of ZFC. The Alpha-Theory supports the feeling that technical notions such as superstructure, ultrapower and the transfer principle are definitely not needed in order to carry out calculus with actual infinitesimals.     

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.     

Polynomial identity rings as rings of functions

We generalize the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where “varieties” carry a PGL_n-action, regular and rational “functions” on them are matrix-valued, “coordinate rings” are prime polynomial identity algebras, and “function fields” are central simple algebras of degree n. In particular, a prime polynomial identity algebra of degree n is finitely generated if and only if it arises as the “coordinate ring” of a “variety” in this setting. For n=1 our definitions and results reduce to those of classical affine algebraic geometry.     

The inverse problem for certain tree parameters

Let p be a graph parameter that assigns a positive integer value to every graph. The inverse problem for p asks for a graph within a prescribed class (here, we will only be concerned with trees), given the value of p. In this context, it is of interest to know whether such a graph can be found for all or at least almost all integer values of p. We will provide a very general setting for this type of problem over the set of all trees, describe some simple examples and finally consider the interesting parameter “number of subtrees”, where the problem can be reduced to some number-theoretic considerations. Specifically, we will prove that every positive integer, with only 34 exceptions, is the number of subtrees of some tree.     

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.     

The Reception of Frege''sBegriffsschrift

The topic of the paper is the public reception of Gottlob Frege's (1848–1925)Begriffsschriftright after its publication in 1879. According to a widespread conception, the reception of the book was “unfavorable” and even “tragic.” The aim of the paper is to correct this exaggerated and even false view. The arguments are based, above all, on the six journal reviews of Frege's book in 1879 and 1880, and on Leonhard Rabus's critical comment on it in his bookDie neuesten Bestrebungen auf dem Gebiete der Logik bei den Deutschen und die logische Frage(1880). The conclusion is that it is misleading to regard the reception of Frege's first work as unfair and hostile even though it is apparent thatsomereviewers of the book were rather poorly motivated to comment on theBegriffsschrift.Copyright 1998 Academic Press.Der Gegenstand dieses Beitrags ist die öffentliche Rezeption von Gottlob Freges (1848–1925)Begriffsschriftnach ihrer Publikation 1879. Nach einer weitverbreiteten Auffassung war die Rezeption “ungünstig,” sogar “tragisch.” Ziel des Beitrags ist es zu zeigen, dass solche Interpretationen überspannt, teilweise sogar falsch sind. Der Verfasser gründet seine Behauptung vor allem auf die Rezensionen, die in verschiedenen Zeitschriften in den Jahren 1879 und 1880 erschienen sind. Er benutzt auch den Kommentar von Leonhard RabusDie neuesten Bestrebungen auf dem Gebiete der Logik bei den Deutschen und Die logische Frage(1880). Die Untersuchung kommt zu dem Schluss, dass, obwohl offensichtlicheinigeRezensenten dem Buch eher ablehnend gegenüberstanden, es irreführend ist, von einer ungerechtfertigten oder abweisenden Rezeption der Fregeschen Begriffsschrift zu sprechenCopyright 1998 Academic Press.AMS subject classification: 00A30, 01A55, 03A05     

Herbart's influence on bernhard Riemann

B. Riemann (1826–1866) knew a great deal of the thought of the German philosopher J. Fr. Herbart (1776–1841). During his studies of the philosopher's work he copied out numerous excerpts and made a few notes which are preserved (at least partially) in the Riemann Archiv at Göttingen. This material reveals that Herbart influenced Riemann much more in his epistemology and the comprehension of science than in his particular philosophy of space and spatial thinking. Thus the relationship between Herbart and Riemann has to be looked upon as an example of an influence of German Bildungsphilosophie on the mathematics of the 19th century.     

Stability properties of Cauchy's surface area formula

Cauchy's surface area formula expresses the surface area of ad-dimensional convex body in terms of the mean value of the volume of its orthogonal projections onto (d–1)-dimensional linear subspaces. We consider here averages of the same kind as those in Cauchy's formula but with respect to some direction dependent density function and investigate the stability problem whether the density must be close to 1 if the formula produces approximately the correct surface area. It will be shown that this relationship between surface area and density is, in general, unstable; but if the density function satisfies suitable regularity conditions, then explicit stability estimates can be obtained.Dedicated to Professor Leopold Vietoris on the occasion of his one hundredth birthdaySupported by National Science Foundation Research Grant DMS 8922399.     

Ansätze zur begründung theoretischer terme in der mathematik—Die theorie des unendlichen bei Johann Schultz

In dem Artikel wird der Beitrag des Königsberger Mathematikers Johann Schultz (1739–1805) zur Theorie des Unendlichen diskutiert. Im Gegensatz zur vorherrschenden Benutzung des Unendlichen als einer heuristischen Fiktion begründet Schultz die Realität des mathematischen Unendlichen und entwickelt, aus Anlaß seines Versuchs, das Parallelenpostulat zu beweisen, eine Theorie zum Operieren mit diesem Begriff. Trotz der Inkonsequenzen seiner Theorie, die sich vor allem in der Anwendung auf eine “Geometrie des Unendlichgroßen” auswirken, hat Schultz in seiner Theorie wichtige Fortschritte erzielt, so die Trennung der Begriffe von Zahl und Größe, die Einführung des Begriffs der Menge als Verallgemeinerung des Zahlbegriffs für das Unendliche und die Trennung des Unendlich-Kleinen vom Unendlich-Großen. Da die zeitgenössischen Mathematiker nicht die Inkonsequenzen von Schultz' Theorien und die fehlerhafte Anwendung in der Geometrie erkannten, vermuteten sie den Fehler im theoretischen Ansatz, der daher—bis auf Ausnahmen—nicht weiterentwickelt wurde.This article discusses a publication by the Königsberg mathematician Johann Schultz (1739–1805) on a theory of the infinite. Rather than develop the infinite as a useful heuristic fiction, Schultz establishes the reality of the mathematical infinite and develops this concept to prove the parallel postulate. Despite the inconsistencies of his theory, which appeared particularly in applications to a “Geometry of the Infinite,” Schultz achieved important advances in his theory, including the distinction between the concepts of number and magnitude, the introduction of the concept of set as a generalization of the number concept for the infinite, and the separation of the infinitely-small from the infinitely-large. Since contemporary mathematicians did not recognize the inconsistencies of Schultz' theories and their faulty application in geometry, they assumed the errors to lie in his theoretical foundations, which therefore, with but few exceptions, were not developed further.     

A general framework of compactly supported splines and wavelets

Let {V_k} be a nested sequence of closed subspaces that constitute a multiresolution analysis of L²( ). We characterize the family Φ = {φ} where each φ generates this multiresolution analysis such that the two-scale relation of φ is governed by a finite sequence. In particular, we identify the ε Φ that has minimum support. We also characterize the collection Ψ of functions η such that each η generates the orthogonal complementary subspaces W_k of V_k, . In particular, the minimally supported ψ ε Ψ is determined. Hence, the “B-spline” and “B-wavelet” pair (, ψ) provides the most economical and computational efficient “spline” representations and “wavelet” decompositions of L² functions from the “spline” spaces V_k and “wavelet” spaces W_k, k . A very general duality principle, which yields the dual bases of both {(·−j):j and {η(·−j):j } for any η ε Ψ by essentially interchanging the pair of two-scale sequences with the pair of decomposition sequences, is also established. For many filtering applications, it is very important to select a multiresolution for which both and ψ have linear phases. Hence, “non-symmetric” and ψ, such as the compactly supported orthogonal ones introduced by Daubechies, are sometimes undesirable for these applications. Conditions on linear-phase φ and ψ are established in this paper. In particular, even-order polynomial B-splines and B-wavelets φ_m and ψ_m have linear phases, but the odd-order B-wavelet only has generalized linear phases.     

The Enumeration of Permutations with a Prescribed Number of “Forbidden” Patterns

We initiate a general approach for the fast enumeration of permutations with a prescribed number of occurrences of “forbidden” patterns that seems to indicate that the enumerating sequence is always P-recursive. We illustrate the method completely in terms of the patterns “abc,” “cab,” and “abcd.”