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Mathematics of the Quantum Zeno Effect

Schmidt, Andreas U.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
35.41%
We present an overview of the mathematics underlying the quantum Zeno effect. Classical, functional analytic results are put into perspective and compared with more recent ones. This yields some new insights into mathematical preconditions entailing the Zeno paradox, in particular a simplified proof of Misra's and Sudarshan's theorem. We empahsise the complex-analytic structures associated to the issue of existence of the Zeno dynamics. On grounds of the assembled material, we reason about possible future mathematical developments pertaining to the Zeno paradox and its counterpart, the anti-Zeno paradox, both of which seem to be close to complete characterisations.; Comment: 32 pages, 1 figure, AMSLaTeX. In: Mathematical Physics Research at the Leading Edge, Charles V. Benton ed. Nova Science Publishers, Hauppauge NY, pp. 111-141, ISBN 1-59033-905-3, 2003; revision contains corrections from the published corrigenda to Reference [64]

Distinguished graduates in mathematics of the Jagiellonian University in the interwar period. Part I: 1918-1926

Domoradzki, Stanisław; Stawiska, Małgorzata
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
35.41%
In this study, we present profiles of some distinguished graduates in mathematics of the Jagiellonian University from the years 1918-1939. We discuss their professional paths and scholarly achievements, instances of scientific collaboration, connections with other academic centers in Poland and worldwide, involvement in mathematical education and teacher training, as well as their later roles in Polish scientific and academic life. We also try to understand in what way they were shaped by their studies and how much of Krak\'ow scientific traditions they continued. We find strong support for the claim that there was a distinct, diverse and deep mathematical stream in Krak\'ow between the wars, rooted in classical disciplines like differential equations and geometry, but also open to new trends in mathematics. Part I concerns the graduates before the university reform, in 1918-1926.; Comment: 5 photos; minor corrections to the first version

Four Departures in Mathematics and Physics

Rosinger, Elemer E
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 01/03/2010 Português
Relevância na Pesquisa
35.41%
Much of Mathematics, and therefore Physics as well, have been limited by four rather consequential restrictions. Two of them are ancient taboos, one is an ancient and no longer felt as such bondage, and the fourth is a surprising omission in Algebra. The paper brings to the attention of those interested these four restrictions, as well as the fact that each of them has by now ways, even if hardly yet known ones, to overcome them.

Transfinite Recursion in Higher Reverse Mathematics

Schweber, Noah
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 21/10/2013 Português
Relevância na Pesquisa
35.41%
In this paper we investigate the reverse mathematics of higher-order analogues of the theory \ATRz{} within the framework of higher order reverse mathematics developed by Kohlenbach \cite{Koh01}. We define a theory \RCAzthr, a close higher-type analogue of the classical base theory \RCAz, and show that it is essentially a conservative subtheory of Kohlenbach's base theory \RCAzo. Working over \RCAzthr, we study higher-type analogues of statements classically equivalent to \ATRz, including open and clopen determinacy, as well as two choice principles, and prove several equivalences and separations. Our main result is the separation of open and clopen determinacy for reals, using a variant of Steel forcing; in the presentation of this result, we develop a new, more flexible framework for Steel-type forcing.; Comment: 39 pages, 1 figure

Liber Mathematicae: A Web-Based Documentation and Collaboration Project for Mathematics

Pflaum, Markus J.; Tuley, John
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 28/02/2011 Português
Relevância na Pesquisa
35.41%
Traditionally, mathematical knowledge is published in printed media such as books or journals. With the advent of the Internet, a new method of publication became available. To date, however, most online mathematical publications do not employ the full capabilities of the medium. We describe a project to modernize online mathematics presentation and build a community-focused environment in which the lines between "author" and "reader" are blurred, enhancing collaboration and improving publication quality.; Comment: 1 page. Presented at the "Workshop on the Future of Mathematics Journals," MSRI, Berkeley, CA (USA) on February 15, 2011

Electronic Mathematics Journals

Steinberger, Mark
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 25/08/1997 Português
Relevância na Pesquisa
35.41%
In the Forum section of the November, 1993 Notices of the American Mathematical Society, John Franks discussed the electronic journal of the future. Since then, the New York Journal of Mathematics, the first electronic general mathematics journal, has begun publication. In this article, we explore the issues of electronic journal publishing in the context of this new project. We also discuss future developments.; Comment: 7 pages, latex, nyj class

Some Thoughts on the Teaching of Mathematics -- ten years later

Rivin, Igor
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 04/01/2014 Português
Relevância na Pesquisa
35.41%
I describe some deep-seated problems in higher mathematical education, and give some ideas for their solution -- I advocate a move away from the traditional introduction of mathematics through calculus, and towards computation and discrete mathematics.; Comment: 10 pages, to appear in Notices of the AMS

On the Mathematics of Music: From Chords to Fourier Analysis

Lenssen, Nathan; Needell, Deanna
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 11/06/2013 Português
Relevância na Pesquisa
35.41%
Mathematics is a far reaching discipline and its tools appear in many applications. In this paper we discuss its role in music and signal processing by revisiting the use of mathematics in algorithms that can extract chord information from recorded music. We begin with a light introduction to the theory of music and motivate the use of Fourier analysis in audio processing. We introduce the discrete and continuous Fourier transforms and investigate their use in extracting important information from audio data.

Dialectics of Counting and the Mathematics of Vagueness

Mani, A.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 05/08/2014 Português
Relevância na Pesquisa
35.41%
New concepts of rough natural number systems are introduced in this research paper from both formal and less formal perspectives. These are used to improve most rough set-theoretical measures in general Rough Set theory (\textsf{RST}) and to represent rough semantics. The foundations of the theory also rely upon the axiomatic approach to granularity for all types of general \textsf{RST} recently developed by the present author. The latter theory is expanded upon in this paper. It is also shown that algebraic semantics of classical \textsf{RST} can be obtained from the developed dialectical counting procedures. Fuzzy set theory is also shown to be representable in purely granule-theoretic terms in the general perspective of solving the contamination problem that pervades this research paper. All this constitutes a radically different approach to the mathematics of vague phenomena and suggests new directions for a more realistic extension of the foundations of mathematics of vagueness from both foundational and application points of view. Algebras corresponding to a concept of \emph{rough naturals} are also studied and variants are characterised in the penultimate section.; Comment: This paper includes my axiomatic approach to granules. arXiv admin note: substantial text overlap with arXiv:1102.2558

A minimalist two-level foundation for constructive mathematics

Maietti, Maria Emilia
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
35.41%
We present a two-level theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin. One level is given by an intensional type theory, called Minimal type theory. This theory extends the set-theoretic version introduced in the mentioned paper with collections. The other level is given by an extensional set theory which is interpreted in the first one by means of a quotient model. This two-level theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; it is constructive thanks to the way the extensional level is linked to the intensional one which fulfills the "proofs-as-programs" paradigm and acts as a programming language.; Comment: 46 pages, revised version (I corrected typos)

A topological approach to non-Archimedean Mathematics

Benci, Vieri; Baglini, Lorenzo Luperi
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
35.41%
Non-Archimedean mathematics (in particular, nonstandard analysis) allows to construct some useful models to study certain phenomena arising in PDE's; for example, it allows to construct generalized solutions of differential equations and variational problems that have no classical solution. In this paper we introduce certain notions of non-Archimedean mathematics (in particular, of nonstandard analysis) by means of an elementary topological approach; in particular, we construct non-Archimedean extensions of the reals as appropriate topological completions of $\mathbb{R}$. Our approach is based on the notion of $\Lambda $-limit for real functions, and it is called $\Lambda $-theory. It can be seen as a topological generalization of the $\alpha $-theory presented in \cite{BDN2003}, and as an alternative topological presentation of the ultrapower construction of nonstandard extensions (in the sense of \cite{keisler}). To motivate the use of $\Lambda $-theory for applications we show how to use it to solve a minimization problem of calculus of variations (that does not have classical solutions) by means of a particular family of generalized functions, called ultrafunctions.; Comment: 22 pages

``Theoretical mathematics'': Toward a cultural synthesis of mathematics and theoretical physics

Jaffe, Arthur; Quinn, Frank
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 30/06/1993 Português
Relevância na Pesquisa
35.41%
Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and destructive consequences. Serious caution is required, and the issue should be considered before, rather than after, obvious damage occurs. With the hazards carefully in mind, we propose a framework that should allow a healthy and positive role for speculation.; Comment: 13 pages

Role of skew-symmetric differential forms in mathematics

Petrova, L. I.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 05/04/2010 Português
Relevância na Pesquisa
35.41%
Skew-symmetric forms possess unique capabilities. The properties of closed exterior and dual forms, namely, invariance, covariance, conjugacy and duality, either explicitly or implicitly appear in all invariant mathematical formalisms. This enables one to see an internal connection between various branches of mathematics. However, the theory of closed exterior forms cannot be completed without an answer to a question of how the closed exterior forms emerge. In the present paper we discus essentially new skew-symmetric forms, which generate closed exterior forms. Such skew-symmetric forms, which are evolutionary ones, are derived from differential equations, and, in contrast to exterior forms, they are defined on nonintegrable manifolds.; Comment: 37 pages

Expander Graphs in Pure and Applied Mathematics

Lubotzky, Alexander
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 12/05/2011 Português
Relevância na Pesquisa
35.41%
Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number theory, group theory, geometry and more. This expository article describes their constructions and various applications in pure and applied mathematics.; Comment: This paper is based on notes prepared for the Colloquium Lectures at the Joint Annual Meeting of the American Mathematical Society (AMS) and the Mathematical Association of America (MAA). New Orleans, January 6-9, 2011

The exterior algebra and central notions in mathematics

Fløystad, Gunnar
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 27/04/2015 Português
Relevância na Pesquisa
35.41%
This note surveys how the exterior algebra and deformations or quotients of it, gives rise to centrally important notions in five domains of mathematics: Combinatorics, Topology, Lie theory, Mathematical physics, and Algebraic geometry.; Comment: 16 pages

Popularizing mathematics: from eight to infinity

Hansen, Vagn Lundsgaard
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 30/04/2003 Português
Relevância na Pesquisa
35.41%
It is rare to succeed in getting mathematics into ordinary conversation without meeting all kinds of reservations. In order to raise public awareness of mathematics effectively, it is necessary to modify such attitudes. In this paper, we point to some possible topics for general mathematical conversation.

Reverse mathematics, Young diagrams, and the ascending chain condition

Hatzikiriakou, Kostas; Simpson, Stephen G.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 11/10/2015 Português
Relevância na Pesquisa
35.41%
Let $S$ be the group of finitely supported permutations of a countably infinite set. Let $K[S]$ be the group algebra of $S$ over a field $K$ of characteristic $0$. According to a theorem of Formanek and Lawrence, $K[S]$ satisfies the ascending chain condition for two-sided ideals. We study the reverse mathematics of this theorem, proving its equivalence over RCA$_0$ (or even over RCA$_0^*$) to the statement that $\omega^\omega$ is well ordered. Our equivalence proof proceeds via the statement that the Young diagrams form a well partial ordering.; Comment: 14 pages

The Roles Leon Henkin Played in Mathematics Education

Manzano, María
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 16/07/2015 Português
Relevância na Pesquisa
35.41%
This paper is divided into two sections. In the first I give reasons for strongly recommending reading some of Henkin's expository papers. In the second I describe Leon Henkin's work as a social activists in the field of mathematics education, as he labored in much of his career to boost the number of women and underrepresented minorities in the upper echelons of mathematics.; Comment: Proceedings of the Fourth International Conference on Tools for Teaching Logic (TTL2015), Rennes, France, June 9-12, 2015. Editors: M. Antonia Huertas, Jo\~ao Marcos, Mar\'ia Manzano, Sophie Pinchinat, Fran\c{c}ois Schwarzentruber

The Kadison-Singer Problem in Mathematics and Engineering

Casazza, Peter G.; Fickus, Matt; Tremain, Janet C.; Weber, Eric
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
35.41%
We will show that the famous, intractible 1959 Kadison-Singer problem in $C^{*}$-algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. This gives all these areas common ground on which to interact as well as explaining why each of these areas has volumes of literature on their respective problems without a satisfactory resolution. In each of these areas we will reduce the problem to the minimum which needs to be proved to solve their version of Kadison-Singer. In some areas we will prove what we believe will be the strongest results ever available in the case that Kadison-Singer fails. Finally, we will give some directions for constructing a counter-example to Kadison-Singer.

Experimental Mathematics and Mathematical Physics

Bailey, David H.; Borwein, Jonathan M.; Broadhurst, David; Zudilin, Wadim
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 03/05/2010 Português
Relevância na Pesquisa
35.41%
One of the most effective techniques of experimental mathematics is to compute mathematical entities such as integrals, series or limits to high precision, then attempt to recognize the resulting numerical values. Recently these techniques have been applied with great success to problems in mathematical physics. Notable among these applications are the identification of some key multi-dimensional integrals that arise in Ising theory, quantum field theory and in magnetic spin theory.; Comment: 18 pages, 2 figures