The Chu construction started out as a way of constricting easy examples of *-autonomous categories. It first became more widely known as a way of providing easy models of linear logic and then as a way of simplifying the construction of *-autonomous categories. It actually originates in the construction of ``dual pairs'' in the topological vector spaces, which goes back ultimately in George Mackey's dissertation.
Although the standard biographical sources are silent on Abu al-Jud, Omar Khayyam tells us he solved an algebra problem, leading to a cubic equation, that none of his contemporaries had been able to solve. Our survey of his works, which we report in this talk, suggests that he was indeed a talented mathematician. But he was also one who occasionally `rushed into print' he had checked all the details of his argument and who was embroiled in a nasty controversy with a contemporary on the solution of a major unsolved problem.
THE DIGITAL MATHEMATICS LIBRARY. The `DML' project proposes over the next decade to put on line (scanned images) the entire printed corpus of Mathematics and to make it generally available. It is estimated that between five and ten percent is already available, though hard to find or access! A good idea of some of the progress already made can be gathered at the European Math Society's website ( http://elib.uni-osnabrueck.de/EMIS/). As was clear from a meeting I attended at NSF in late July, the project has significant support from NSF and from its German counterpart. NRC-CISTI was also present, and seems likely to assist in digitizing our own Canadian content.
It is generally agreed that the greatest obstacle to success is neither financial1 nor technical but lies in the incredibly complicated intellectual property and rights management issues that will have to be addressed. For example, in some settings one may have to request permission from the estate of authors deceased as much as 70 years ago, as they certainly never anticipated such a use of their work.2 More surely, while Springer-Verlag is already `on-board', we shall have to come to some `modus vivendi' with other large publishers such as Elsvier.
That said, success would represent an epochal event in cultural history. The material will, with caveats, be assured for posterity, it will be searchable (eventually the mathematics as well as the text), and we (mathematicians and others) will discover many things we do not know that we know.
The Magic Square has been a wonderful mathematics puzzle for years. It has shown up in many ancient architectural sites and in many mathematics writings. This discussion will trace the roots of the magic square and go through the steps of creating different dimensions of the magic square. There will also be visuals of where the magic square appears throughout history.
We discuss the contributions of Spiru Haretu to the problem of the solar system's stability and show their importance relative to the mathematics research of the late 19th century. We also give a brief survey of the subsequent developments and the consequences of Haretu's results.
The famous 15th-century cardinal appears on at least one list of ``great'' mathematicians; on the other hand his contemporary Regiomontanus dismissed his efforts in mathematics as ``ridiculous''. But whatever his technical competence, it is quite certain that Cusanus's perception of mathematics coloured deeply his influential views on such issues as the limits of human knowledge and the relation of man to God. I shall try to sketch from both perspectives-the technical and the philosophical-the place of mathematics in the world-view of this fascinating figure.
Between 1929 and 1932, Einstein and Elie Cartan carried an intense scientific correspondence on the geometric and analytic aspects of a unified field theory of gravitation and electromagnetism which had been proposed by Einstein in 1929. This correspondence was edited by Robert Debever, and published by Princeton University Press in 1979 on the occasion of Einstein's centenary. The framework of this theory is that of a differentiable manifold endowed with a connection for which all frames are parallel. Such a connection has necessarily zero curvature, but it will in general have non-zero torsion. The issues that Einstein and Cartan discussed in great detail dealt mostly with the local existence of analytic solutions to the field equations, and their degree of generality in the sense of Cartan-Kaehler theory. We will present some of the mathematical and historical highlights of this fascinating (and sometimes frustrating) correspondence.
I will introduce the structural graph we have been using for a representation of ribonucleic acid (RNA) structure since the late 80's. Then, I will formally introduce RNA motifs, that is functional or structural significant patterns, and present three approaches to discover them. The first one, from biologists, is subjective and uses visual examination of RNA structures. The second uses a greedy and incremental algorithm that is costly and uses a subjective definition of significance. Finally, the third one was discovered from taking a natural step in the graph representation, that is by dividing the structrual graph in a minimal cycle basis. We found redundant cycles that correspond directly to and others that compose acknowledged structural and functional motifs. The new approach has also allowed us to discover new instances of the classical GNRA motif that do not fit the GNRA sequence definition. We are now building a theory of RNA cycles that we see as an expression of fundamental thermodynamics rules at a higher than the atomic level.
I shall follow the development of ideas originating from elliptic integrals to the j-function and its natural generalization in recent years to the class of replicable functions.
We give a brief preliminary survey of the life and times of Roland George Dwight Richardson, Canadian born mathematician of the last century who, among his many contributions, served as Dean at Brown University and was ultimately responsible for attracting John D. Tamarkin there.
Divergent series have been used successfully in mathematics for centuries and have occupied an important place in mathematics until the middle of the 19th century. During this period mathematicians could not explain their success. In the 20th century mathematicians have justified rigorously the use of divergent series and also explained why they are so powerful. However divergent series remain a relatively marginal subject in contemporary mathematics. In this lecture I will present some history of divergent series related to differential equations and explain why they are not so marginal in the subject. This will bring me to the future.
This talk will overview the past history and current situation of the use of mathematics in the financial industry.
Life amidst politics, religion and mathematics. Deposed by the emperor Franz I. His contribution in philosophy, theology, logic, and mainly his example of continous nowhere differentiable function 40 years before Weierstrass.
In order to envision the future of network computing it is important to understand its past. How did we get from a world dominated by large-scale monolithic mainframe computers with few, if any, interconnections to a world where the personal computer and the Internet reign supreme? Understanding key evolutionary events can provide insight into the future of network computing and its impact on our futures.
Looking back in time, the invention and development of the telegraph, telephone, radio and computer formed the building blocks for unprecedented integration of capabilities. In its short history, the Internet has revolutionized the computer and communications world like nothing before. A scant 10 years ago, the World Wide Web didn't exist. The idea of doing business over the Internet was ludicrous. There was no Windows operating system, no Fast Ethernet or Gigabit Ethernet, no laptops, no PDA's no e-mail, not even shopping at home.
This session reviews some of the significant events in the origins, history, and evolution of the Internet. In addition, this session reviews some major trends identified by leading futurists and pundits, that may shape the industry over the next decade.
Olinde Rodrigues lived in the first half of the 19th century, and was an outstanding mathematician whose achievements are only being fully recognized today. He was Portugese by birth but French by training and education. Even his most famous contribution, the Rodrigues formula for the Legendre polynomials (which appears nowadays in most texts on differential equations, advanced calculus, etc.), was not attributed to him until about 50 years after his death. In this talk we will discuss his mathematical contributions which are in diverse fields such as analysis, rotation groups (in particular what we now call SO(3) and Spin(3)), group theory, and combinatorics. We will also discuss his contributions to an amazingly broad collection of other disciplines: philosophy (in particular socialism), music, women's rights, and racial equality. In the latter half of the 19th century France named a ship in his honour. We will conclude by showing that recently Rodrigues is finally getting some of the recognition he deserves, in both mathematics and theoretical physics.
1Though the cost is likely to be somewhere between $100 and $200 million US.
2A recent US Supreme Court ruling told the New York Times that it had to pay free-lancers again when it put pre-digital material on its website.