Reduction

Reduction and reductionism have been central philosophical topics in analytic philosophy of science for more than six decades. Together they encompass a diversity of issues from metaphysics and epistemology. This article provides an introduction to the topic that illuminates how contemporary epistemological discussions took their shape historically and limns the contours of concrete cases of reduction in specific natural sciences. The unity of science and the impulse to accomplish compositional reduction in accord with a layer-cake vision of the sciences, the seminal contributions of Ernest Nagel on theory reduction and how they strongly conditioned subsequent philosophical discussions, and the detailed issues pertaining to different accounts of reduction that arise in both physical and biological science (e.g., limit-case and part-whole reduction in physics, the difference-making principle in genetics, and mechanisms in molecular biology) are explored. The conclusion argues that the epistemological heterogeneity and patchwork organization of the natural sciences encourages a pluralist stance about reduction

Reduction

This is a contribution to the encyclopedia of systems biology on reduction

Pohlmeyer reduction revisited

A systematic group theoretical formulation of the Pohlmeyer reduction is presented. It provides a map between the equations of motion of sigma models with target-space a symmetric space M=F/G and a class of integrable multi-component generalizations of the sine-Gordon equation. When M is of definite signature their solutions describe classical bosonic string configurations on the curved space-time R_t\times M. In contrast, if M is of indefinite signature the solutions to those equations can describe bosonic string configurations on R_t\times M, M\times S^1_\vartheta or simply M. The conditions required to enable the Lagrangian formulation of the resulting equations in terms of gauged WZW actions with a potential term are clarified, and it is shown that the corresponding Lagrangian action is not unique in general. The Pohlmeyer reductions of sigma models on CP^n and AdS_n are discussed as particular examples of symmetric spaces of definite and indefinite signature, respectively.Comment: 45 pages, LaTeX, more references added, accepted for publication in JHE

Chemical reduction of hexavalent chromium (VI) in soil slurry by nano zero valent iron

The increasing industrial development of recent decades has lead to the production of increasing quantities of waste containing heavy metals, elements often harmful to the environment, which in the past were not properly disposed of, thus inducing soil and groundwater pollution. In particular, chromium (Cr) and its derivatives are largely used in industries such as textiles, electronics, metallurgy, tanneries. Consequently, large quantities of this element were released into the environment due to leakage or incorrect disposal. Chromium is a transition element present in nature in three stable forms: metallic Cr, trivalent Cr(III) and hexavalent Cr(VI). Metallic Chromium is rarely found in nature, mainly as natural chrome metallic inclusions in diamonds, fragments of as meteorites and metal alloys in fluvial deposits. The trivalent form is characterized by a relatively low toxicity, while the hexavalent chromium present in different compounds of industrial origin, is considered highly toxic towards the respiratory system and carcinogenic. In the present work, lab experiments of Cr(VI) contaminated soil clean-up by chemical reduction with nanoparticles of zero valent iron (nZVI) are presented and discussed. The aim of the work was to optimize the main operative parameters of the reduction process (pH, nZVI concentration, liquid/solid ratio). Cr(VI) reduction using nZVI was found to obey a pseudo-first-order kinetic: the kinetic constant depended upon the nZVI: Cr(VI) ratio. The use of nZVI in combination with sodium dithionite was also studied, by performing tests in batch conditions at pH = 1.3, in order to assess the optimal ratio between nZVI and Cr(VI), and between dithionite and Cr(VI). The results obtained showed an increase of Cr(VI) reduction rate with respect to the tests carried only with nZVI: for long treatment times, up to 24 hours, an almost total removal of Cr(VI) was achieved when a large excess of reagents was used

Deficit Reduction Through Diversity: How Affirmative Action at the FCC Increased Auction Competition

In recent auctions for paging licenses, the Federal Communications Commission has granted businesses owned by minorities and women substantial bidding credits. In this article, Professors Ayres and Cramton analyze a particular auction and argue that the affirmative action bidding preferences, by increasing competition among auction participants, increased the government's revenue by $45 million. Subsidizing the participation of new bidders can induce established bidders to bid more aggressively. The authors conclude that this revenue- enhancing effect does not provide a sufficient constitutional justification for affirmative action-but when such justification is independently present, affirmative actions can cost the government much less than is currently thought.Auctions; Affirmative Action

Optimal reduction

We generalize various symplectic reduction techniques to the context of the optimal momentum map. Our approach allows the construction of symplectic point and orbit reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a momentum map. We construct an orbit reduction procedure for canonical actions on a Poisson manifold that exhibits an interesting interplay with the von Neumann condition previously introduced by the author in his study of singular dual pairs. This condition ensures that the orbits in the momentum space of the optimal momentum map (we call them polar reduced spaces) admit a presymplectic structure that generalizes the Kostant--Kirillov--Souriau symplectic structure of the coadjoint orbits in the dual of a Lie algebra. Using this presymplectic structure, the optimal orbit reduced spaces are symplectic with a form that satisfies a relation identical to the classical one obtained by Marle, Kazhdan, Kostant, and Sternberg for free Hamiltonian actions on a symplectic manifold. In the symplectic case we provide a necessary and sufficient condition for the polar reduced spaces to be symplectic. In general, the presymplectic polar reduced spaces are foliated by symplectic submanifolds that are obtained through a generalization to the optimal context of the so called Sjamaar Principle, already existing in the theory of Hamiltonian singular reduction. We use these ideas in the construction of a family of presymplectic homogeneous manifolds and of its symplectic foliation and we show that these reduction techniques can be implemented in stages in total analogy with the case of free globally Hamiltonian proper actions.Comment: 42 page

Un-reduction

This paper provides a full geometric development of a new technique called un-reduction, for dealing with dynamics and optimal control problems posed on spaces that are unwieldy for numerical implementation. The technique, which was originally concieved for an application to image dynamics, uses Lagrangian reduction by symmetry in reverse. A deeper understanding of un-reduction leads to new developments in image matching which serve to illustrate the mathematical power of the technique.Comment: 25 pages, revised versio

Recursions of Symmetry Orbits and Reduction without Reduction

We consider a four-dimensional PDE possessing partner symmetries mainly on the example of complex Monge-Amp\`ere equation (CMA). We use simultaneously two pairs of symmetries related by a recursion relation, which are mutually complex conjugate for CMA. For both pairs of partner symmetries, using Lie equations, we introduce explicitly group parameters as additional variables, replacing symmetry characteristics and their complex conjugates by derivatives of the unknown with respect to group parameters. We study the resulting system of six equations in the eight-dimensional space, that includes CMA, four equations of the recursion between partner symmetries and one integrability condition of this system. We use point symmetries of this extended system for performing its symmetry reduction with respect to group parameters that facilitates solving the extended system. This procedure does not imply a reduction in the number of physical variables and hence we end up with orbits of non-invariant solutions of CMA, generated by one partner symmetry, not used in the reduction. These solutions are determined by six linear equations with constant coefficients in the five-dimensional space which are obtained by a three-dimensional Legendre transformation of the reduced extended system. We present algebraic and exponential examples of such solutions that govern Legendre-transformed Ricci-flat K\"ahler metrics with no Killing vectors. A similar procedure is briefly outlined for Husain equation