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## Mathematics of the Quantum Zeno Effect

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematical Physics#Mathematics - Functional Analysis#Mathematics - Operator Algebras#Quantum Physics#46L60, 47D03, 81P15, 81R15, 82B10

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]

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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

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## Four Departures in Mathematics and Physics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/03/2010
Português

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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.

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## Transfinite Recursion in Higher Reverse Mathematics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/10/2013
Português

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/02/2011
Português

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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

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## Electronic Mathematics Journals

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/08/1997
Português

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/01/2014
Português

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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

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## On the Mathematics of Music: From Chords to Fourier Analysis

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/06/2013
Português

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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.

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## Dialectics of Counting and the Mathematics of Vagueness

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/08/2014
Português

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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

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## A minimalist two-level foundation for constructive mathematics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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)

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## A topological approach to non-Archimedean Mathematics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/06/1993
Português

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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

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## Role of skew-symmetric differential forms in mathematics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/04/2010
Português

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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

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## Expander Graphs in Pure and Applied Mathematics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/05/2011
Português

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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

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## The exterior algebra and central notions in mathematics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/04/2015
Português

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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

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## Popularizing mathematics: from eight to infinity

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 30/04/2003
Português

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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.

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## Reverse mathematics, Young diagrams, and the ascending chain condition

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/10/2015
Português

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#Mathematics - Logic#Mathematics - Rings and Algebras#Primary 03B30, Secondary 16P40, 16S34, 05A17, 03F15

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

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## The Roles Leon Henkin Played in Mathematics Education

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/07/2015
Português

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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

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## The Kadison-Singer Problem in Mathematics and Engineering

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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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.

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## Experimental Mathematics and Mathematical Physics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/05/2010
Português

Relevância na Pesquisa

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#Mathematical Physics#High Energy Physics - Phenomenology#Mathematics - Classical Analysis and ODEs#Mathematics - Number Theory

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

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