370 research outputs found
The Algebra of Logic Tradition
The algebra of logic, as an explicit algebraic system showing the underlying mathematical structure of logic, was introduced by George Boole (1815-1864) in his book The Mathematical Analysis of Logic (1847). The methodology initiated by Boole was successfully continued in the 19th century in the work of William Stanley Jevons (1835-1882), Charles Sanders Peirce (1839-1914), Ernst Schröder (1841-1902), among many others, thereby establishing a tradition in (mathematical) logic. From Boole's first book until the influence after WWI of the monumental work Principia Mathematica (1910 1913) by Alfred North Whitehead (1861-1947) and Bertrand Russell (1872-1970), versions of thealgebra of logic were the most developed form of mathematical above allthrough Schröder's three volumes Vorlesungen über die Algebra der Logik(1890-1905). Furthermore, this tradition motivated the investigations of Leopold Löwenheim (1878-1957) that eventually gave rise to model theory. Inaddition, in 1941, Alfred Tarski (1901-1983) in his paper On the calculus of relations returned to Peirce's relation algebra as presented in Schröder's Algebra der Logik. The tradition of the algebra of logic played a key role in thenotion of Logic as Calculus as opposed to the notion of Logic as Universal Language . Beyond Tarski's algebra of relations, the influence of the algebraic tradition in logic can be found in other mathematical theories, such as category theory. However this influence lies outside the scope of this entry, which is divided into 10 sections.Fil: Burris, Stanley. University of Waterloo; CanadáFil: Legris, Javier. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Interdisciplinario de Economía Politica de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Económicas. Instituto Interdisciplinario de Economía Politica de Buenos Aires; Argentin
Parametric Toricity of Steady State Varieties of Reaction Networks
We study real steady state varieties of the dynamics of chemical reaction
networks. The dynamics are derived using mass action kinetics with parametric
reaction rates. The models studied are not inherently parametric in nature.
Rather, our interest in parameters is motivated by parameter uncertainty, as
reaction rates are typically either measured with limited precision or
estimated. We aim at detecting toricity and shifted toricity, using a framework
that has been recently introduced and studied for the non-parametric case over
both the real and the complex numbers. While toricity requires that the variety
specifies a subgroup of the direct power of the multiplicative group of the
underlying field, shifted toricity requires only a coset. In the non-parametric
case these requirements establish real decision problems. In the presence of
parameters we must go further and derive necessary and sufficient conditions in
the parameters for toricity or shifted toricity to hold. Technically, we use
real quantifier elimination methods. Our computations on biological networks
here once more confirm shifted toricity as a relevant concept, while toricity
holds only for degenerate parameter choices.Comment: Computations available as ancillary file
Workshop on Verification and Theorem Proving for Continuous Systems (NetCA Workshop 2005)
Oxford, UK, 26 August 200
Special Algorithm for Stability Analysis of Multistable Biological Regulatory Systems
We consider the problem of counting (stable) equilibriums of an important
family of algebraic differential equations modeling multistable biological
regulatory systems. The problem can be solved, in principle, using real
quantifier elimination algorithms, in particular real root classification
algorithms. However, it is well known that they can handle only very small
cases due to the enormous computing time requirements. In this paper, we
present a special algorithm which is much more efficient than the general
methods. Its efficiency comes from the exploitation of certain interesting
structures of the family of differential equations.Comment: 24 pages, 5 algorithms, 10 figure
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
The domination monoid in henselian valued fields
We study the domination monoid in various classes of structures arising from
the model theory of henselian valuations, including RV-expansions of henselian
valued fields of residue characteristic 0 (and, more generally, of benign
valued fields), p-adically closed fields, monotone D-henselian differential
valued fields with many constants, regular ordered abelian groups, and pure
short exact sequences of abelian structures. We obtain Ax-Kochen-Ershov type
reductions to suitable fully embedded families of sorts in quite general
settings, and full computations in concrete ones.Comment: 35 pages. Minor revisio
- …