3,061 research outputs found
A nonstandard Euler-Maruyama scheme
We construct a nonstandard finite difference numerical scheme to approximate
stochastic differential equations (SDEs) using the idea of weighed step
introduced by R.E. Mickens. We prove the strong convergence of our scheme under
locally Lipschitz conditions of a SDE and linear growth condition. We prove the
preservation of domain invariance by our scheme under a minimal condition
depending on a discretization parameter and unconditionally for the expectation
of the approximate solution. The results are illustrated through the geometric
Brownian motion. The new scheme shows a greater behavior compared to the
Euler-Maruyama scheme and balanced implicit methods which are widely used in
the literature and applications.Comment: Accepted in "Journal of Difference Equations and Applications", to
appear, 201
A monad measure space for logarithmic density
We provide a framework for proofs of structural theorems about sets with
positive Banach logarithmic density. For example, we prove that if has positive Banach logarithmic density, then contains an
approximate geometric progression of any length. We also prove that if
have positive Banach logarithmic density, then there
are arbitrarily long intervals whose gaps on are multiplicatively
bounded, a multiplicative version Jin's sumset theorem. The main technical tool
is the use of a quotient of a Loeb measure space with respect to a
multiplicative cut.Comment: 26 page
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
Arithmetic, Set Theory, Reduction and Explanation
Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In defense of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences
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