formulasearchengine - User contributions [en]

Hammett acidity function

2014-01-15T19:09:44Z

182.186.219.90: /* Typical values */

{{for|the violinist|Louis Kaufman}}
[[File:Louis Kauffman.jpg|right|thumb|200px|Louis Kauffman]]
'''Louis Hirsch Kauffman''' (born February 3, 1945) is an American [[mathematician]], [[topology|topologist]], and professor of [[Mathematics]] in the Department of Mathematics, Statistics, and Computer science at the [[University of Illinois at Chicago]]. He is known for the introduction and development of the [[bracket polynomial]] and the [[Kauffman polynomial]].

== Biography ==
Kauffman was [[wiktionary:valedictorian|valedictorian]] of his graduating class at Norwood Norfolk Central High School in 1962. He received his [[B.S.]] at [[MIT]] in 1966 and his [[Ph.D.]] in [[mathematics]] from [[Princeton University]] in 1972.

Kauffman has worked at many places as a visiting professor and researcher, including the University of Zaragoza in Spain, the University of Iowa in Iowa City, the Institute Hautes Etudes Scientifiques in Bures Sur Yevette, France, the Institute Henri Poincaré in Paris, France, the Università di Bologna, Italy, the Universidade Federal de Pernambuco in Recife, Brasil, and the Newton Institute in Cambridge England.<ref name = "Knot Information"> http://www.math.uic.edu/~kauffman/569.html</ref>

He is the founding editor and one of the managing editors of the ''[[Journal of Knot Theory and Its Ramifications]]'', and editor of the ''World Scientific Book Series On Knots and Everything''. He writes a column entitled Virtual Logic for the journal ''Cybernetics and Human Knowing''

From 2005 to 2008 he was president of the [[American Society for Cybernetics]]. He plays
clarinet in the ChickenFat Klezmer Orchestra in Chicago.

== Work ==
Kauffman's research interests are in the fields of cybernetics, topology and foundations of mathematics and physics. His work is primarily in the topics of [[knot theory]] and connections with [[statistical mechanics]], [[Quantum field theory|quantum theory]], [[algebra]], [[combinatorics]] and foundations. <ref>[http://www.asci.org/artsci2002/artworks/Sunday/explorations.htm Presentation]</ref> In [[topology]] he introduced and developed the [[bracket polynomial]] and [[Kauffman polynomial]].

=== Bracket polynomial ===
{{main|Bracket polynomial}}
In the mathematical field of [[knot theory]], the [[bracket polynomial]], also known as the ''Kauffman bracket'', is a [[polynomial]] invariant of [[framed link]]s. Although it is not an invariant of knots or links (as it is not invariant under type I [[Reidemeister move]]s), a suitably "normalized" version yields the famous [[knot invariant]] called the [[Jones polynomial]]. The bracket polynomial plays an important role in unifying the Jones polynomial with other [[quantum invariant]]s. In particular, Kauffman's interpretation of the Jones polynomial allows generalization to state sum invariants of [[3-manifold]]s. Recently the bracket polynomial formed the basis for Mikhail Khovanov's construction of a homology for knots and links, creating
a stronger invariant than the Jones polynomial and such that the graded Euler chacteristic of the [[Khovanov homology]] is equal to the original
Jones polynomial. The generators for the chain complex of the Khovanov homology are states of the bracket polynomial decorated with elements
of a [[Frobenius algebra]].

=== Kauffman polynomial ===
{{main|Kauffman polynomial}}
The [[Kauffman polynomial]] is a 2-variable [[knot polynomial]] due to Louis Kauffman. It is defined as

:<math>F(K)(a,z)=a^{-w(K)}L(K)\,</math>

where <math>w(K)</math> is the [[writhe]] and <math>L(K)</math> is a [[regular isotopy]] invariant which generalizes the bracket polynomial.

=== Discrete ordered calculus ===
In 1994, Kauffman and Tom Etter wrote a draft proposal for a non-commutative ''discrete ordered calculus'' (DOC), which they presented in revised form in 1996.<ref>T. Etter, L.H. Kauffman, ANPA West Journal, vol. 6, no. 1, pp. 3–5</ref> In the mean time, the theory was presented in a modified form by Kauffman and [[H. Pierre Noyes]] together with a presentation of a derivation of free space Maxwell equations on this basis.<ref>Louis H. Kauffman, H. Pierre Noyes, Discrete physics and the derivation of electromagnetism from the formalism of quantum mechanics, Proceedings of the Royal Society London A (1996), vol. 452, pp. 81–95</ref>

==Awards and honors==
He is the 1993 recipient of the Warren McCulloch award of the American Society for Cybernetics and the 1996 award of the Alternative Natural Philosophy Association for his work in discrete physics.

In 2012 he became a fellow of the [[American Mathematical Society]].<ref>[http://www.ams.org/profession/fellows-list List of Fellows of the American Mathematical Society], retrieved 2013-01-27.</ref>

== Publications ==
[[Louis H. Kauffman]] is author of several monographs on knot theory and mathematical physics. His publication list numbers over 170.<ref name = "Knot Information"/> Books:
* 1987, ''On Knots'', Princeton University Press 498 pp.
* 1993, ''Quantum Topology (Series on Knots & Everything)'', with Randy A. Baadhio, World Scientific Pub Co Inc, 394 pp.
* 1994, ''Temperley-Lieb Recoupling Theory and Invariants of 3-Manifolds'', with [[Sostenes Lins]], Princeton University Press, 312 pp.
* 1995, ''Knots and Applications (Series on Knots and Everything, Vol 6)''
* 1995, ''The Interface of Knots and Physics: American Mathematical Society Short Course January 2-3, 1995 San Francisco, California (Proceedings of Symposia in Applied Mathematics)'', with the American Mathematical Society.
* 1998, ''Knots at Hellas 98: Proceedings of the International Conference on Knot Theory and Its Ramifications'', with [[Cameron Gordon (mathematician)|Cameron McA. Gordon]], [[Vaughan F. R. Jones]] and [[Sofia Lambropoulou]],
* 1999, ''Ideal Knots'', with Andrzej Stasiak and Vsevolod Katritch, World Scientific Publishing Company, 414 pp.
* 2001, ''Knots and Physics (Series on Knots and Everything, Vol. 1)'', World Scientific Publishing Company, 788 pp.
* 2002, ''Hypercomplex Iterations: Distance Estimation and Higher Dimensional Fractals (Series on Knots and Everything , Vol 17)'', with Yumei Dang and Daniel Sandin.
* 2006, ''Formal Knot Theory'', Dover Publications, 272 pp.
* 2007, ''Intelligence of Low Dimensional Topology 2006'', with J. Scott Carter and Seiichi Kamada.
* 2012, ''Knots and Physics (Fourth Edition)'', World Scientific Publishing Company, ISBN 978-981-4383-00-4
Articles and papers, a selection:
* 2001, [http://www2.math.uic.edu/~kauffman/CHK.pdf The Mathematics of Charles Sanders Peirce], in: ''Cybernetics & Human Knowing'', Vol.8, no.1–2, 2001, pp. 79–110

== References ==
<references/>

== External links ==
* [http://www2.math.uic.edu/~kauffman/ Louis Kauffman] Homepage at uic.edu.
* [http://www.evl.uic.edu/hypercomplex/] Hypercomplex Fractals.
* [http://front.math.ucdavis.edu/search?a=louis+kauffman&t=&q=&c=&n=25&s=Listings] Arxiv Papers.
* {{MathGenealogy|id=1306}}
* [http://chickenfat.bandcamp.com/album/chickenfat-demos] ChickenFat Klezmer Orchestra.

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| SHORT DESCRIPTION = American mathematician
| DATE OF BIRTH = 1945
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[[Category:1945 births]]
[[Category:Cyberneticists]]
[[Category:Living people]]
[[Category:Topologists]]
[[Category:University of Illinois at Chicago faculty]]
[[Category:Fellows of the American Mathematical Society]]

Invested capital

2013-12-25T15:30:17Z

182.186.83.165: /* References */

In [[machine learning]], '''Weighted Majority Algorithm (WMA)''' is a meta-learning algorithm used to construct a compound algorithm from a pool of prediction algorithms, which could be any type of learning algorithms, classifiers, or even real human experts. The algorithm assumes that we have no prior knowledge about the accuracy of the algorithms in the pool, but there are sufficient reasons to believe that one or more will perform well.

Assume that the problem is a binary decision problem. To construct the compound algorithm, a positive weight is given to each of the algorithms in the pool. The compound algorithm then collects weighted votes from all the algorithms in the pool, and gives the prediction that has a higher vote. If the compound algorithm makes a mistake, the algorithms in the pool that contributed to the wrong predicting will be discounted by a certain ratio β where 0<β<1.

It can be shown that the upper bounds on the number of mistakes made in a given sequence of predictions from a pool of algorithms <math> \mathbf{A} </math> is

:<math>\mathbf{O(log|A|+m)}</math>

if one algorithm in <math> \mathbf{x}_i </math> makes at most <math> \mathbf{m} </math> mistakes.

There are many variations of the Weighted Majority Algorithm to handle different situations, like shifting targets, infinite pools, or randomized predictions. The core mechanism remain similar, with the final performances of the compound algorithm bounded by a function of the performance of the '''specialist''' (best performing algorithm) in the pool.

== See also ==
* [[randomized weighted majority algorithm]]

==References==

* Littlestone,N. & [[Manfred K. Warmuth|Warmuth,M.]] (1989). ''Weighted Majority Algorithm.'' IEEE Symposium on Foundations of Computer Science.

[[Category:Machine learning algorithms]]

Income elasticity of demand

2013-12-17T05:18:15Z

182.186.34.62: /* Mathematical definition */

{{Noref|date=November 2009}}
In [[axiomatic set theory]], the '''axiom schema of predicative separation''', or of '''restricted''', or '''Δ<sub>0</sub>''' separation, is a [[schema (logic)|schema]] of [[axiom]]s which is a restriction of the usual [[axiom schema of separation]] in [[Zermelo–Fraenkel set theory]]. It only asserts the existence of a [[subset]] of a set if that subset can be defined without reference to the entire [[Von Neumann universe|universe]] of sets. The axiom appears in the systems of [[constructive set theory]] CST and CZF, as well as in the system of [[Kripke–Platek set theory]]. The name Δ<sub>0</sub> comes from the [[Levy hierarchy]] (in analogy with the [[arithmetic hierarchy]]).

The formal statement of this is the same as full separation schema, but with a restriction on the formulas that may be used. For any formula φ:

:<math>\forall x \; \exist y \; \forall z \; (z \in y \leftrightarrow z \in x \wedge \phi(z))</math>

provided, as usual, that the variable ''y'' is not free in φ; but also provided that φ contains only [[bounded quantifiers]]. That is, all quantifiers in φ (if there are any) must appear in the form <math>\exist x \in y \; \psi(x)</math> or <math>\forall x \in y \; \psi(x)</math> for some sub-formula ψ.

The meaning of this is that, given any set ''x'', and any predicate φ there is a set ''y'' whose elements are the elements of ''x'' which satisfy φ, provided φ only quantifies over existing sets, and never quantifies over all sets. This restriction is necessary from a [[impredicativity|predicative]] point of view, since the universe of all sets contains the set being defined. If it were referenced in the definition of the set, the definition would be circular.

Although the schema contains one axiom for each restricted formula φ, it is possible in CZF to replace this schema with a finite number of axioms.{{Citation needed|date=March 2012}}

{{settheory-stub}}

[[Category:Constructivism (mathematics)]]
[[Category:Axioms of set theory]]