Olga Taussky Research Articles

The Murder of Professor Schlick: The Rise and Fall of the Vienna Circle

It is not unusual to speculate on the contrary-to-fact implications of political assassinations. Lincoln's is the classic case in point, but we need only think of Julius Caesar, Gandhi, or John Kennedy, if we require further examples. One totally neglected case in this context is that of Moritz Schlick. One of the remote consequences of his murder, on June 22, 1936, which was most definitely a political assassination, is that today's academic world may well have been an entirely different one if he had gone on to lead the Vienna Circle into exile. That group of young philosophers of science had formed as he assumed the University of Vienna's chair of natural philosophy in 1922. A quarter of a century later, the entire English-speaking world (and Scandinavia) was in thrall to “logical positivism,” a caricature of scientific philosophy as the circle conceived it but an incredibly powerful intellectual standpoint, for all that. The view that all genuine knowledge can be quantified not only dominated philosophy but transformed the nature of universities to the point of threatening the very existence of the humanities, which could scarcely meet this demand. The polemical and oversimplified view of Rudolf Carnap's strong program for scientific philosophy simply found no place for disciplines based on mere description (ethnology) or on understanding complex interlinked constellations of events as they develop over time (history and its myriad allied disciplines).The die was cast, and the humanities became saddled with an inferiority complex that, even today, they have only partially shaken off. Only disciplines that employed mathematical methods in aid of rigorous explanation, working collectively in networks like those of physics or chemistry, were worthy of being institutionalized at universities. Since the humanities were, for the most part, embodied in individual researchers burning the Midnight Oil as they pondered the Eternal Verities, they were disqualified as sciences. They would have to reform themselves radically or disappear. The battle commenced, and sensitive observers can attest that, administrative rhetoric to the contrary, it still goes on. Had Schlick lived on, the pluralism and tolerance that was part of his view of the world might well have carried more influence than the more radical elements in the group ultimately did, and so the humanities might well have fared better in a more secure environment than they have done.Edmonds's The Murder of Professor Schlick does not venture into such speculations. Instead, it offers us a vibrant, suspense-filled narrative replete with as much information as the reader can digest (and more). Briefly, Edmonds presents a fully detailed, complex version of the high drama of the rise and decline of logical positivism in both its philosophical and its political dimensions, complete with vivid portraits of the characters whose thought and personalities supplied that movement with its dazzling dynamism. They include such characters as Schlick himself; the irrepressible Otto Neurath, who had been a minister in the short-lived Bavarian Soviet Republic; Rudolf Carnap, the circle's standard-bearer; Ludwig Wittgenstein, its unwilling ideal; Bertrand Russell, its Promethean inspiration; the eccentric Kurt Gödel; the rabbi Josef Schächter; the clever Englishman A. J. Ayer; America's rising star, W. V. Quine; the perennial contrarian, Karl Popper; the unfortunate Friedrich Waismann, doomed to failure in all his efforts; the tragic Edgar Zilsel, who committed suicide in American exile; and several frequently neglected women, such as Rose Rand, Olga Hahn, Marie Reidemeister, and Olga Taussky. In addition, the strong Jewish presence in the circle and the surrounding anti-Semitism that was a continuing stumbling block to it are clearly and powerfully described.These are but highlights of Edmond's narrative, but they are proof positive that his study is where to begin an encounter with the Vienna Circle and thus with the philosophical view that, more than any other, has conferred its character on modern thought and deeply molded people who have never heard of it.

Integer circulant determinants of order 16

We solve Olga Taussky–Todd’s circulant problem in the case of order 16.

Skew-symmetric tridiagonal Bohemian matrices

Image at right: Olga Taussky−Todd in her Caltech office circa 1960, wearing the famous "numbers" dress Abstract: Skew-symmetric tridiagonal Bohemian matrices with population P = [1,i] have eigenvalues with some interesting properties. We explore some of these here, and I prove a theorem showing that the only possible dimensions where nilpotent matrices can occur are one less than a power of two. I explicitly give a set of matrices in this family at dimension m=2ᵏ−1 which are nilpotent, and recursively constructed from those at smaller dimension. I conjecture that these are the only matrices in this family which are nilpotent. This paper will chiefly be of interest to those readers of my prior paper on Bohemian matrices with this structure who want more mathematical details than was provided there, and who want details of what has been proved versus what has been conjectured by experiment. I also give a terrible pun. Don't say you weren't warned.

Olga Taussky, a torchbearer for mathematics and teacher of mathematicians

AbstractOlga Taussky‐Todd's mathematical and personal life (1906‐1995), her achievements and obstacles, her scientific reasoning and teaching all have served as inspiration to many mathematicians. We describe her role in the mathematics world of the previous century as a torchbearer for mathematics and mathematicians, bearing the “torch of scientific truth” that burns inside of mathematics and its applications. Besides her many deep math contributions – too many to elaborate – she excelled at distilling and presenting mathematical concepts and ideas in her work and gave us many visionary papers and math talks. By sharing her mathematical vision freely she has inspired many of us. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Gersgorin variations I: on a theme of Pupkov and Solov'ev

We unify and extend various “additive” sufficient conditions of Pupkov and of Solov'ev for the nonsingularity of a complex matrix. This paper is intended to be the first in a sequcnce of “variations” on theorems of the Gersgorin genre. The whole sequence is dedicated to the memory of Olga Taussky-Todd, whose lovely paper [O. Taussky, Amer. Math. Monthly 56 (1949) 672–676,] inspired an interest in matrix theory for a generation of mathematicians.

The Weyr Characteristic

Click to increase image sizeClick to decrease image size Additional informationNotes on contributorsHelene ShapiroHELENE SHAPIRO received her B.A. degree from Kenyon College in 1975 and her Ph.D. from the California Institute of Technology in 1979, where she was a student of Olga Taussky Todd. She spent one year at the University of Wisconsin and then joined the Department of Mathematics and Statistics at Swarthmore College, where she has been teaching since 1980.

Some personal reminiscences of Olga Taussky-Todd

The year was probably 1953 and I was living in Belfast. One day our doorbell rang and there stood Olga Taussky and Jack Todd who were visiting Jack's parents whose home was not far from ours. It turned out that Olga had heard of me when she visited my supervisor A.C. Aitken in Edinburgh. I remember little of my first meeting with Olga, but for years she would tell me I was surrounded by three small children and looked like a boy myself (at age 26). Jack has reminded me that we walked Olga and him back to his parents' house and that his father said Bring in the wains when he saw us. Occasionally I would wonder whether Olga had been the referee of my first paper [1] written two years earlier while I was a graduate student. About 25 years later I dared ask her whether this was so. She told me plainly that I should not have asked, but admitted that I was right. I got punished for my question. During her Torch talk [2] at a SIAM Linear Algebra meeting in N. Carolina I interrupted her with some small comment and she rebuked me publicly with Let me say it my way; I refereed your first paper!. In 1976 I was invited to speak at a meeting in honor of Olga Taussky at Caltech and I wished to illustrate the influence of her work on other mathematicians by describing the history of three classical topics within matrix theory which are well described in Johnson's article [3] in this issue: diagonal dominance, inertia theory and the Taussky unification problem. In particular, of course, I wanted to describe the effect on my work. I had used her famous

Some aspects of Olga Taussky's work in algebra

Olga Taussky's pivotal role as a ‘torch bearer’ for matrix theory is well understood and well-appreciated, as is her role in number theory and in particular there her key contribution to the development of matrix-theoretic techniques in algebraic number theory. Her contributions in these subjects have been highlighted in numerous tributes to her. In my own case, I can strongly affirm the positive impact that her work and her personal advice and interest has had in inspiring and encouraging my own research. However, in this note, I wish to draw attention to her contributions to other parts of algebra and the influence that the results themselves and the questions she posed have had in these areas. A number of those results arose in connection with problems in linear algebra or number theory, but are of interest in more general terms. I would also like to put the results in the context of current research. Of necessity, I have had to make a short selection from her numerous papers and my choice inevitably is a subjective one.

To the Latimer—Macduffee theorem and beyond!

In this paper we discuss some of the mathematical contributions made by Olga Taussky Todd. We begin with fairly general considerations, commenting on her role as a thesis advisor and giving an overview of her research contributions. Then we consider a specific work of hers, Another look at a matrix of Mark Kac, her last published work, which was written jointly with John Todd. We make a series of observations that explain certain numerical phenomena that they prove in this paper and end with a generalization of their work.

Olga Taussky-Todd’s work in class field theory

Olga Taussky’s interest in class field theory began when she was a student with Ph. Furtwangler at the University of Vienna (she received her doctorate under his supervision in 1930). At this time, E. Artin, using his general reciprocity law, reformulated the “Principal Ideal Theorem” as a purely grouptheoretic question regarding the triviality of the “Verlagerung” homomorphism from a finite group to its commutator subgroup. Since Furtwangler had been developing group theoretic methods in his own number theoretic investigations, Artin communicated this new idea to him. This was considered to be one of the central problems in the subject at the time, and so there was a good deal of excitement when Furtwangler eventually succeeded in proving the result using this new approach. Olga Taussky had been looking for a thesis topic and Furtwangler suggested several questions related to the finer structure of the Principal Ideal Theorem. The optimistic atmosphere, and the prospect of contributing at the frontiers of class field theory, sparked in Olga a deep interest in this problem which lasted her entire career. She returned to the questions of “capitulation”, a term coined by one of her co-authors Arnold Scholz, several times in her life, always with the sense that these were questions of deep arithmetic significance. I believe that it was the experience of her early contact with class field theory which dominated her research interests in number theory and which provided her a fertile source of inspiration for most of her career. I will describe some of her work in this area below. I would like to take this opportunity to thank the referee for many valuable suggestions. For a number field F , (i.e., an extension of finite degree over the rational field Q) its Hilbert class field H = H(F ) is the maximal Galois extension of F which is everywhere unramified and whose Galois group Gal(H/F ) is abelian. It is a consequence of Artin’s reciprocity law that Gal(H/F ) is isomorphic to the ideal class group C(F ) of F. The Principal Ideal Theorem is the statement that every ideal of F becomes a principal ideal when considered

Renewal of the doctorate of olga taussky todd

Renewal of the doctorate of olga taussky todd

Ternary quadratic forms and norms Koh, K.: edited by Olga Taussky, 57:291–293 (1984)

Ternary quadratic forms and norms Koh, K.: edited by Olga Taussky, 57:291–293 (1984)

The conjugacy problem of the modular group and the class number of real quadratic number fields

We shall discuss the conjugacy problem of the modular group, and show how its solution, in conjunction with a theorem of Olga Taussky can be used to compute the class number of certain real quadratic number fields.

Review of Ternary Quadratic Forms and Norms: edited by Olga Taussky

Review of Ternary Quadratic Forms and Norms: edited by Olga Taussky

On a problem suggested by Olga Taussky-Todd

The problem considered is to characterize those integers $m$ such that $m = \mathrm{det}(C)$, $C$ an integral $n \times n$ circulant. It is shown that if $(m,n) = 1$ then such circulants always exist, and if $(m,n)> 1$ and $p$ is a prime dividing $(m,n)$ such that $p^{t}||n$, then $p^{t+1}|m$. This implies for example, that $n$ never occurs as the determinant of an integral $n \times n$ circulant, if $n > 1$.

Olga Taussky-Todd's influence on matrix theory and matrix theorists

Olga Taussky-Todd's influence on matrix theory and matrix theorists

The April meeting in Santa Barbara

The seven hundred fourteenth meeting of the American Mathematical Society was held at the University of California, Santa Barbara, on Saturday, April 27, 1974. There were 94 registrants including 49 members of the Society. Two invited hour addresses were given by invitation of the Committee to Select Hour Speakers for Far Western Sectional Meetings. Professor Garth W. Warner of the University of Washington gave a lecture entitled L'functions on reductive groups. He was introduced by Professor V. S. Varadarajan. Professor Hung-Hsi Wu of the University of California, Berkeley, lectured on Applications of some theorems in partial differential equations to geometry. He was introduced by Professor John Douglas Moore. Professors Marvin Marcus, Henryk Mine, and Robert C. Thompson organized two special sessions of thirty-minute papers on Linear Algebra. The speakers were John DePillis, Eugene C. Johnsen, Herbert J. Ryser, Robert C. Thompson, Olga Taussky-Todd, and Stanley G. Williamson. There were two sessions of contributed ten-minute papers. They were chaired by Professors Jack G. Ceder and Max L. Weiss. Local arrangements were handled by Professor Stephen Simons. KENNETH A. Ross

Simultaneous block diagonalization of two real symmetric matrices

The simultaneous diagonalization of two real symmetric (r.s.) matrices has long been of interest. This subject is generalized here to the following problem (this question was raised by Dr. Olga Taussky-Todd, my thesis advisor at the California Institute of Technology): What is the first simultaneous block diagonal structure of a nonsingular pair of r.s. matrices ? For example, given a nonsingular pair of r.s. matrices S and T, which simultaneous block diagonalizations X′SX = diag( A 1, , A k ), X′TX = diag( B 1,, B k ) with dim A i = dim B i and X nonsingular are possible for 1 ⩽ k ⩽ n; and how well defined is a simultaneous block diagonalization for which k, the number of blocks, is maximal? Here a pair of r.s. matrices S and T is called nonsingular if S is nonsingular. If the number of blocks k is maximal, then one can speak of the finest simultaneous block diagonalization of S and T, since then the sizes of the blocks A i are uniquely determined (up to permutations) by any set of generators of the pencil P( S, T) = { aS + bT| a, tb ε R} via the real Jordan normal form of S −1 T. The proof uses the canonical pair form theorem for nonsingular pairs of r.s. matrices. The maximal number k and the block sizes dim A i are also determined by the factorization over C of ƒ (λ, μ) = det(λ S + μ T) for λ, μ ε R.

On certain discrete inequalities and their continuous analogs

In a 1955 paper, Ky Fan, Olga Taussky, and John Todd presented discrete analogues of inequalities of Wirtinger type, and by taking limits they were able to recover the continuous inequalities. We generalize their techniques to mixed and higher derivatives and inequalities with weight functions in the integrals. We have also considered analogues of inequalities of Muller and Redheffer and have used these inequalities to derive a necessary and sufficient condition on ordered pairs of numbers so that the first number is the square norm of the k th derivative of some periodic function and the second number is the square norm of the m th derivative of the same periodic function.

Bounds for the spectral radius of a matrix

obtain a sequence of bounds for p(A ) in terms of p(f A' I) (r = 1, 2, ) which are less than or equal to p( l A j) and converge to p(A ). In this manner we are partially accounting for the effect on p(A) of the phases of the ai,. In Section II we derive bounds for p(A) in terms of the Frobenius norm of A. These bounds always lie in the field of values of A, are computationally well suited to complex matrices and can be used in conjunction with the techniques of Section I. The authors are indebted to Olga Taussky and Alston Householder for suggestions.