418 research outputs found
Stevin numbers and reality
We explore the potential of Simon Stevin's numbers, obscured by shifting
foundational biases and by 19th century developments in the arithmetisation of
analysis.Comment: 22 pages, 4 figures. arXiv admin note: text overlap with
arXiv:1104.0375, arXiv:1108.2885, arXiv:1108.420
A Cauchy-Dirac delta function
The Dirac delta function has solid roots in 19th century work in Fourier
analysis and singular integrals by Cauchy and others, anticipating Dirac's
discovery by over a century, and illuminating the nature of Cauchy's
infinitesimals and his infinitesimal definition of delta.Comment: 24 pages, 2 figures; Foundations of Science, 201
MicroRNAs in a Cardiac Loop: Progenitor or Myocyte?
Like transcription factors, microRNAs are emerging as regulators of cell fate decisions. In this issue, Wang et al. (2010) identify a critical microRNA pathway under the control of Bmp signaling that promotes outflow tract myocardial differentiation from cardiac progenitors in vivo
Probing the Y2 Receptor on Transmembrane, Intra- and Extra-Cellular Sites for EPR Measurements
The function of G protein-coupled receptors is intrinsically linked to their conformational dynamics. In conjugation with site-directed spin labeling, electron paramagnetic resonance (EPR) spectroscopy provides powerful tools to study the highly dynamic conformational states of these proteins. Here, we explored positions for nitroxide spin labeling coupled to single cysteines, introduced at transmembrane, intra- and extra-cellular sites of the human neuropeptide Y2 receptor. Receptor mutants were functionally analyzed in cell culture system, expressed in Escherichia coli fermentation with yields of up to 10 mg of purified protein per liter expression medium and functionally reconstituted into a lipid bicelle environment. Successful spin labeling was confirmed by a fluorescence assay and continuous wave EPR measurements. EPR spectra revealed mobile and immobile populations, indicating multiple dynamic conformational states of the receptor. We found that the singly mutated positions by MTSL ((1-oxyl-2,2,5,5-tetramethyl-2,5-dihydro-1H-pyrrol-3-yl) methyl methanesulfonothioate) have a water exposed immobilized conformation as their main conformation, while in case of the IDSL (bis(1-oxyl-2,2,5,5-tetramethyl-3-imidazolin-4-yl) disulfide) labeled positions, the main conformation are mainly of hydrophobic nature. Further, double cysteine mutants were generated and examined for potential applications of distance measurements by double electron–electron resonance (DEER) pulsed EPR technique on the receptor
A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography
We analyze the developments in mathematical rigor from the viewpoint of a
Burgessian critique of nominalistic reconstructions. We apply such a critique
to the reconstruction of infinitesimal analysis accomplished through the
efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's
foundational work associated with the work of Boyer and Grabiner; and to
Bishop's constructivist reconstruction of classical analysis. We examine the
effects of a nominalist disposition on historiography, teaching, and research.Comment: 57 pages; 3 figures. Corrected misprint
Pointed Hopf Algebras with Triangular Decomposition: A Characterization of Multiparameter Quantum Groups
© 2016, The Author(s). In this paper, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable type over a group, we obtain a class of pointed Hopf algebras which can be viewed as natural generalizations of multiparameter deformations of universal enveloping algebras of Lie algebras. These Hopf algebras are instances of a new version of braided Drinfeld doubles, which we call asymmetric braided Drinfeld doubles. This is a generalization of an earlier result by Benkart and Witherspoon (Algebr. Represent. Theory 7(3) ? BC) who showed that two-parameter quantum groups are Drinfeld doubles. It is possible to recover a Lie algebra from these doubles in the case where the group is free abelian and the parameters are generic. The Lie algebras arising are generated by Lie subalgebras isomorphic to sl2
Cauchy's infinitesimals, his sum theorem, and foundational paradigms
Cauchy's sum theorem is a prototype of what is today a basic result on the
convergence of a series of functions in undergraduate analysis. We seek to
interpret Cauchy's proof, and discuss the related epistemological questions
involved in comparing distinct interpretive paradigms. Cauchy's proof is often
interpreted in the modern framework of a Weierstrassian paradigm. We analyze
Cauchy's proof closely and show that it finds closer proxies in a different
modern framework.
Keywords: Cauchy's infinitesimal; sum theorem; quantifier alternation;
uniform convergence; foundational paradigms.Comment: 42 pages; to appear in Foundations of Scienc
Ten Misconceptions from the History of Analysis and Their Debunking
The widespread idea that infinitesimals were "eliminated" by the "great
triumvirate" of Cantor, Dedekind, and Weierstrass is refuted by an
uninterrupted chain of work on infinitesimal-enriched number systems. The
elimination claim is an oversimplification created by triumvirate followers,
who tend to view the history of analysis as a pre-ordained march toward the
radiant future of Weierstrassian epsilontics. In the present text, we document
distortions of the history of analysis stemming from the triumvirate ideology
of ontological minimalism, which identified the continuum with a single number
system. Such anachronistic distortions characterize the received interpretation
of Stevin, Leibniz, d'Alembert, Cauchy, and others.Comment: 46 pages, 4 figures; Foundations of Science (2012). arXiv admin note:
text overlap with arXiv:1108.2885 and arXiv:1110.545
Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond
Many historians of the calculus deny significant continuity between
infinitesimal calculus of the 17th century and 20th century developments such
as Robinson's theory. Robinson's hyperreals, while providing a consistent
theory of infinitesimals, require the resources of modern logic; thus many
commentators are comfortable denying a historical continuity. A notable
exception is Robinson himself, whose identification with the Leibnizian
tradition inspired Lakatos, Laugwitz, and others to consider the history of the
infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies,
Robinson regards Berkeley's criticisms of the infinitesimal calculus as aptly
demonstrating the inconsistency of reasoning with historical infinitesimal
magnitudes. We argue that Robinson, among others, overestimates the force of
Berkeley's criticisms, by underestimating the mathematical and philosophical
resources available to Leibniz. Leibniz's infinitesimals are fictions, not
logical fictions, as Ishiguro proposed, but rather pure fictions, like
imaginaries, which are not eliminable by some syncategorematic paraphrase. We
argue that Leibniz's defense of infinitesimals is more firmly grounded than
Berkeley's criticism thereof. We show, moreover, that Leibniz's system for
differential calculus was free of logical fallacies. Our argument strengthens
the conception of modern infinitesimals as a development of Leibniz's strategy
of relating inassignable to assignable quantities by means of his
transcendental law of homogeneity.Comment: 69 pages, 3 figure
Ω-Arithmetization of Ellipses
International audienceMulti-resolution analysis and numerical precision problems are very important subjects in fields like image analysis or geometrical modeling. In the continuation of our previous works, we propose to apply the method of Ω-arithmetization to ellipses. We obtain a discrete multi-resolution representation of arcs of ellipses. The corresponding algorithms are completely constructive and thus, can be exactly translated into functional computer programs. Moreover, we give a global condition for the connectivity of the discrete curves generated by the method at every scale
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