2,250 research outputs found
Steps towards "Quantum Gravity" and the practice of science: will the merger of mathematics and physics work?
The author recalls general tendencies of the "mathematization" of the
sciences and derives challenges and tentative obstructions for a successful
merger of mathematics and physics on fancied steps towards "Quantum Gravity".
This is an edited version of the author's opening words to an international
workshop "Quantum Gravity: An Assessment", Denmark, May 17-18, 2008. It
followed immediately after the Quantum Gravity Summer School 2008, see
http://QuantumGravity.ruc.dk/Comment: To appear as part of a Springer Lecture Notes in Physics publication:
"Quantum Gravity - New Paths towards Unification" (B. Booss-Bavnbek, G.
Esposito, M. Lesch, Eds.
The Topology of Probability Distributions on Manifolds
Let be a set of random points in , generated from a probability
measure on a -dimensional manifold . In this paper we study
the homology of -- the union of -dimensional balls of radius
around , as , and . In addition we study the critical
points of -- the distance function from the set . These two objects
are known to be related via Morse theory. We present limit theorems for the
Betti numbers of , as well as for number of critical points of index
for . Depending on how fast decays to zero as grows, these two
objects exhibit different types of limiting behavior. In one particular case
(), we show that the Betti numbers of perfectly
recover the Betti numbers of the original manifold , a result which is of
significant interest in topological manifold learning
Computation in Economics
This is an attempt at a succinct survey, from methodological and epistemological perspectives, of the burgeoning, apparently unstructured, field of what is often – misleadingly – referred to as computational economics. We identify and characterise four frontier research fields, encompassing both micro and macro aspects of economic theory, where machine computation play crucial roles in formal modelling exercises: algorithmic behavioural economics, computable general equilibrium theory, agent based computational economics and computable economics. In some senses these four research frontiers raise, without resolving, many interesting methodological and epistemological issues in economic theorising in (alternative) mathematical modesClassical Behavioural Economics, Computable General Equilibrium theory, Agent Based Economics, Computable Economics, Computability, Constructivity, Numerical Analysis
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