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	PlanetMath: Pfaffian
		It is a polynomial of the polynomial ring in elements of the matrix, such that its square is equal to the determinant of the matrix.
		The Pfaffian is applied in the generalized Gauss-Bonnet theorem.
		This is version 22 of Pfaffian, born on 2004-05-14, modified 2006-10-02.
	planetmath.org /encyclopedia/Pfafian.html (156 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-06)
		In this case the integration of the Pfaffian equation reduces to the integration of a system of ordinary differential equations.
		and the integral surfaces of the Pfaffian equation (3) are given by the equations
		If the Pfaffian equation (3) is not completely integrable, then it does not have integral surfaces but can have integral curves.
	eom.springer.de /p/p072510.htm (573 words)

	PlanetMath: Pfaffian
		It is a polynomial of the polynomial ring in elements of the matrix, such that its square is equal to the determinant of the matrix.
		The Pfaffian is applied in the generalized Gauss-Bonnet theorem.
		This is version 22 of Pfaffian, born on 2004-05-14, modified 2006-10-02.
	www.planetmath.org /encyclopedia/Pfafian.html (156 words)

	Pfaffian - Definition, explanation
		The Pfaffian is nonvanishing only for 2n × 2n skew-symmetric matrices, in which case it is a polynomial of degree n.
		The Pfaffian is an invariant polynomial of a skew-symmetric matrix (Note that it is not invariant under a general change of basis but rather under a proper orthogonal transformation).
		The term Pfaffian was introduced by Arthur Cayley, who used the term in 1852: "The permutants of this class (from their connection with the researches of Pfaff on differential equations) I shall term Pfaffians." The term honors German mathematician Johann Friedrich Pfaff.
	www.calsky.com /lexikon/en/txt/p/pf/pfaffian.php (321 words)

	On the number of dissimilar pfaffian orientations of graphs (Site not responding. Last check: 2007-11-06)
		An orientation D of G is Pfaffian if, for every conformal even circuit C, the number of edges of C whose directions in D agree with any prescribed sense of orientation of C is odd.
		However, if G has a Pfaffian orientation D, then the determinant of the adjacency matrix of D is the square of the number of perfect matchings of G.
		We deduce that the problem of determining whether or not a graph is Pfaffian is as difficult as the problem of determining whether or not a given orientation is Pfaffian, a result first proved by Vazirani and Yanakakis [Pfaffian orientation of graphs, 0,1 permanents, and even cycles in digraphs.
	www.edpsciences.org /articles/ita/abs/2005/01/ita0425/ita0425.html (387 words)

	A Theory Of Pfaffian Orientations II: T-joins, Edge-Cuts, And A Duality of Enumeration. (ResearchIndex)
		A Theory Of Pfaffian Orientations II: T-joins, Edge-Cuts, And A Duality of Enumeration.
		A Theory Of Pfaffian Orientations II: T-joins, Edge-Cuts, And A Duality of Enumeration (1997)
		A Theory of Pfaffian Orientations I: Perfect Matchings and..
	citeseer.ist.psu.edu /419360.html (434 words)

	TMAT Revista Latinoamericana de Ciencias e Ingeniería (Site not responding. Last check: 2007-11-06)
		The Pfaffian dimension (or still, the Pfaffian topological dimension or class) of A, is defined by the maximal number, m, of non-zero elements of this sequence; see page 209 of Forsyth [40], Schouten and van der Kulk [33].
		Thus, the Pfaffian dimension of the Cartan-Hilbert one-form is 2n + 1, i.e.
		This reduction of Pfaffian dimension from 2n + 1 to 2n coincides precisely with the the situation defined by a Finsler geometry on a 2n + 1-manifold on which H º 0 and the momentum are canonical [5].
	www.tmat.cl /rapoport.html (7182 words)

	15.4.1 Control-Affine Systems
		Velocity constraints on the C-space frequently are of the Pfaffian form (13.5).
		The Pfaffian constraints on configuration are often called kinematic constraints because they arise due to the kinematics of bodies in contact, such as a wheel rolling.
		Thus, Pfaffian constraints provide a dual way of specifying driftless control-affine systems.
	planning.cs.uiuc.edu /node825.html (538 words)

	Users develop pfaffian approach to describing many-body quantum systems
		A 3D cut through the fermion node hypersurface of an oxygen atom obtained by scanning the wave function with a pair of spin-up and spin-down of electrons, both sitting at the scanning point.
		The team's new method is based on what is called Pfaffian, a mathematical approach to solving skew-symmetric matrices.
		Pfaffian wave functions capture the same effects as conventional methods of solving, which are an order of magnitude or more expensive to calculate.
	access.ncsa.uiuc.edu /Stories/nuggets/pfaffian.html (325 words)

	The Vault at Pfaff's - Works - Search
		He is listed as one of the Pfaffian writers that "have fallen into obscurity." Stansell wonders how much influence these writers weilded on Whitman's literary career (108).
		She is listed as one of the Pfaffian writers that "have fallen into obscurity." Stansell wonders how much influence these writers weilded on Whitman's literary career (108).
		Stansell describes Menken as a "successful actress" and writes that "in 1860 she would achieve international noteriety as the star of a melodrama in which in the last scene, clad in flesh-colored tights and a G-string, whe rode into the horizon lashed to the back of a 'fiery steed'" (112).
	digital.lib.lehigh.edu /pfaffs/w1481 (3762 words)

	[No title]
		Abstract: An orientation of a graph G is Pfaffian if every even cycle C such that G\V(C) has a perfect matching has an odd number of edges directed in either direction of the cycle.
		The significance of Pfaffian orientations is that if a graph has one, then the number of perfect matchings (a.k.a.
		The question of what bipartite graphs have Pfaffian orientations is equivalent to many other problems of interest, such as a permanent problem of Polya, the even directed cycle problem, or the sign-nonsingular matrix problem for square matrices.
	math.ucf.edu /cllqm/view.shtml?03082007 (149 words)

	Etheses Record Display \| Library \| University of Waterloo
		We prove that when a pair of regular matroids is non-Pfaffian, there is a set of common bases which certifies this, and that the number of bases in the certificate is linear in the size of the ground set of the matroids.
		When both matroids in a pair are series-parallel, we prove that determining if the pair is Pfaffian is equivalent to finding an edge signing in an associated graph, and in the case that the pair is non-Pfaffian, we obtain a characterization of this associated graph.
		Pfaffian bipartite graphs are a class of graphs for which the number of perfect matchings can be determined; we show that the class of series-parallel Pfaffian matroid pairs is an extension of the class of Pfaffian bipartite graphs.
	etheses.uwaterloo.ca /display.cfm?ethesis_id=446 (349 words)

	CiteULike: Particle-hole symmetry and the Pfaffian state (Site not responding. Last check: 2007-11-06)
		We consider the properties of the Moore-Read Pfaffian state under particle-hole conjugation.
		We show that the particle-hole conjugate of the Pfaffian state - or "anti-Pfaffian" state - is in a different universality class from the Pfaffian state, with different topological order.
		The two states can be distinguished by both their bulk and edge physics though the difference is most dramatic at the edge: the edge of the anti-Pfaffian state has a composite structure that leads to a different thermal Hall conductance and different tunneling exponents than the Pfaffian state.
	www.citeulike.org /user/misha/article/1438148 (288 words)

	Bath University - Department of Computer Science
		A formalization of this idea (by Khovanskii in 1970s) leads to definition of a large class of analytic multivariate functions, called Pfaffian functions, which includes basic transcendental functions, algebraic functions (in particular, polynomials), as well as their various compositions.
		Classes of sets defined by Pfaffian functions, and their subclasses, are the most important examples of o-minimal structures.
		There are some important applications of Pfaffian functions also in computational complexity, in Hilbert's 16th problem, and in many other areas.
	www.cs.bath.ac.uk /seminar/logic/20030130.shtml (204 words)

	Topological types of Pfaffian manifolds
		A Pfaffian manifold of $\Omega$ is by definition a maximal integral $k$-manifold of $\Omega$.
		It is shown that the number of homeomorphism classes of all Pfaffian manifolds of Rolle Type of $\Omega$ is finite and, moreover, bounded by a computable function in variables $n$, $k$ and the degree of $\omega_{1}, \dots, \omega_{n-k}$.
		J.-M. Lion and P. Speissegger, Analytic strafication in the pfaffian closure of an o-minimal structure, Duke Math.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.nmj/1114631981 (380 words)

	Thesis - Chapter 3
		His proof relied heavily on the idea of a Pfaffian orientation of the graph.
		To assign a Pfaffian orientation, first distinguish each cycle C in the graph G for which C has even length and G\C has a perfect matching.
		A Pfaffian orientation of G, if one exists, is an orientation such that when traversing each distinguished cycle, an odd number of edges are oriented in the direction of the traversal.
	spider.ipac.caltech.edu /staff/brundage/presents/thesis/pages/chap3.html (1728 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		The first is an integer which denotes the order of the Pfaffian to be expanded.
		In order to speed the evaluation, a hashed array called PFAFFM is created and added to from time to time, whenever a PFAFFIAN of a new and higher order is evaluated.
		If the length of the list argument is not equal to (N+1)*N/2, or if the integer argument is not a positive integer, or if the second argument is not a list, an error is detected and both arguments are put into a list, to which ERREXP is then bound.
	www.unf.edu /public/cap4630/kmartin/gradfall94/maxima/share2/pfaff.usg (173 words)

	Matches for: (Site not responding. Last check: 2007-11-06)
		Singularities and the classification of 1-forms and Pfaffian equations are interesting not only as classical problems, but also because of their applications in contact geometry, partial differential equations, control theory, nonholonomic dynamics, and variational problems.
		In addition to collecting results on the geometry of singularities and classification of differential forms and Pfaffian equations, this monograph discusses applications and closely related classification problems.
		Zhitomirskiipresents proofs with all results and ends each chapter with a summary of the main results, a tabulation of the singularities, and a list of the normal forms.
	www.mathaware.org /bookstore?fn=50&arg1=saleanalysis&item=MMONO-113 (163 words)

	Pfaffian
		A manifold of dimension and of class is called an integral manifold of the Pfaffian equation (1) if on.
		The Pfaffian equation is said to be completely integrable if there is one and only one
		To locate works by a specific Pfaffian, choose the name of that individual in the Author menu.
	pfaffian.95e.i13uo.info (270 words)

	Little- Towards a Characterisation of Pfaffian Graphs. (Site not responding. Last check: 2007-11-06)
		A bipartite graph G is known to be Pfaffian if and only if it does not contain an even subdivision H of K3, 3 such that G - VH contains a 1-factor.
		However a general characterisation of Pfaffian graphs in terms of forbidden subgraphs is currently not known.
		We describe a possible approach to the derivation of such a characterisation.
	www-leibniz.imag.fr /DMD/sem/1999/Little.html (95 words)

	Buckingham Lecture (Site not responding. Last check: 2007-11-06)
		An orientation of a graph G is Pfaffian if every even cycle C such that G\V(C) has a perfect matching has an odd number of edges directed in either direction of the cycle.
		The significance of Pfaffian orientations is that if a graph has one, then the number of perfect matchings can be computed in polynomial time.
		The speaker will present a structural characterization of bipartite graphs that have a Pfaffian orientation, obtained independently by W.
	www.users.muohio.edu /jiangt/fall-conference/pfaffian.html (194 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Some graphs, called Pfaffian, have a special type of orientation that is also called Pfaffian.
		Given a Pfaffian orientation of a graph G, the number of perfect matchings of G may be evaluated in polynomial time.
		Two (not necessarily Pfaffian) orientations are similar if they differ by a cut.
	www.dcc.unicamp.br /ic-tr-ftp/2004/04-03.bib (74 words)

	CiteULike: Tag pfaffian (Site not responding. Last check: 2007-11-06)
		posted to coupled_random_matrices orthogonal_polynomials pfaffian p-point_correlation_functions quaternion_determinant skew_orthogonal_polynomials spectrum by RMT as
		posted to bessel_law characteristic_polynomials chiral_symmetry dirac_operator non-gaussian_ensembles numerics orthogonal_polynomials pfaffian p-point_correlation_functions qcd quaternion_determinant rmt_applications spectrum by RMT as
		posted to characteristic_polynomials goe gse gue orthogonal_polynomials pfaffian quaternion_determinant skew_orthogonal_polynomials spectrum by RMT as
	www.citeulike.org /tag/pfaffian (131 words)

	Andreas Malmendier: Talks (Site not responding. Last check: 2007-11-06)
		Witten has shown that these instantons are in fact the vacua of a supersymmetric gauge theory obtained by twisting the N=2 supersymmetric Yang-Mills theory.
		The fermionic part of its Lagrangian defines a Pfaffian line bundle over the set of reducilbe connections modulo gauge transformations.
		However, this line bundle is singular, and deforming its singularity amounts to computing quantum corrections to the effective action.
	www-math.mit.edu /~malmendi/talks/pfaffian.html (91 words)

	DROPS - Upper and Lower Bounds on Sizes of Finite Bisimulations of Pfaffian Dynamical Systems
		In this paper we study a class of dynamical systems defined by Pfaffian maps.
		It is a sub-class of o-minimal dynamical systems which capture rich continuous dynamics and yet can be studied using finite bisimulations.
		The first step in this direction was done by Korovina et al (2004) where a double exponential upper bound was shown for Pfaffian dynamical and hybrid systems.
	drops.dagstuhl.de /opus/volltexte/2006/713 (211 words)

	Atlas: Pfaffian Systems' Characterizations. Consequences in Dimension Five by Maria A. Canadas-Pinedo (Site not responding. Last check: 2007-11-06)
		The technique we use is based in the study of the incidence relations between a Pfaffian system and the characteristic of its first derived system.
		In particular we characterize Pfaffian systems with derived lenght one, that is, Pfaffian systems whose first derived system is an integrable system, and we also give a characterization for flag systems in terms of the class.
		Those results are used to obtain the whole classification of Pfaffian systems up to dimension four and the first classification in dimension five in an intrinsic way (see[CP-R;4]).
	atlas-conferences.com /c/a/d/q/94.htm (302 words)

	book pfaffian systems, k-symplectic systems, undergraduate level (part ii) - engineering colleges, lavoisier publishers
		The geometrical view of mechanics is based on the study of certain exterior systems, the most classical of which are Pfaffian systems.
		This book presents the classification theorems (Frobenius, Darboux) and the local classification of Pfaffian systems of five variables, following Cartan.
		It also presents a new class of exterior systems, called k-symplectic systems, generalizing the notion of symplectic form.
	www.lavoisier.fr /notice/gb088063.html (123 words)

	APS - 2006 APS March Meeting - Event - Multi-pfaffian pairing wave functions for quantum Monte Carlo (Site not responding. Last check: 2007-11-06)
		We investigate the limits of accuracy of trial wave function for quantum Monte Carlo based on pfaffian functional form with singlet and triplet pairing.
		Using a set of first row atoms and molecules we find that this wave function provides very consistent and systematic behaviour in recovering the correlation energies on the level of 95\%.
		We show that small number of pfaffians recovers another large fraction of the missing correlation energy comparable to the larger-scale configuration iteraction wave functions.
	meetings.aps.org /Meeting/MAR06/Event/44449 (176 words)

	Copyright Permission - Linear Algebra and Its Applications - α-Pfaffian, pfaffian point process and shifted Schur ... (Site not responding. Last check: 2007-11-06)
		Copyright.com is the place to go for the right to use and share content from millions of print and online information sources.
		The α-pfaffian is a pfaffian analogue of the α-determinant studied in [T. Shirai and Y. Takahashi, J. Funct.
		Also we provide a linear algebraic proof of the explicit pfaffian expression obtained in [S. Matsumoto, Correlation functions of the shifted Schur measure, math.CO/0312373] for the correlation function of the shifted Schur measure.
	www.copyright.com /articles/sd/7c/00243795/article_00243795_963.html (253 words)

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