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Type of mathematical function
In mathematics, Pfaffian functions are a certain class of functions whose derivative can be written in terms of the original function. They were originally
Pfaffian_function
Square root of the determinant of a skew-symmetric square matrix
called the Pfaffian polynomial. The value of this polynomial, when applied to the entries of a skew-symmetric matrix, is called the Pfaffian of that matrix
Pfaffian
German mathematician (1765–1825)
significant work, on partial differential equations of the first order Pfaffian systems, as they are now called, which became part of the theory of differential
Johann_Friedrich_Pfaff
Type of infinite structure
with restricted analytic functions), one can define its Pfaffian closure, which is again an o-minimal structure. (The Pfaffian closure of a structure is
O-minimal_theory
Mathematical field with an extra operation
theorem. This result, when combined with Khovanskii's theorem on Pfaffian functions, proves that R exp {\displaystyle \mathbb {R} _{\text{exp}}} is also
Exponential_field
A Pfaffian system is specified by 1-forms alone, but the theory includes other types of example of differential system. To elaborate, a Pfaffian system
Integrability conditions for differential systems
Integrability_conditions_for_differential_systems
Counting polynomial real roots based on coefficients
but for algebraic combinations of many transcendental functions, the so-called Pfaffian functions. Sturm's theorem – Counting polynomial roots in an interval
Descartes'_rule_of_signs
In dynamics, a Pfaffian constraint is a way to describe a dynamical system in the form: ∑ s = 1 n A r s d u s + A r d t = 0 ; r = 1 , … , L {\displaystyle
Pfaffian_constraint
Physics of heat, work, and temperature
Investigations on the Foundations of Thermodynamics, which made use of Pfaffian systems and the concept of adiabatic accessibility, a notion that was introduced
Thermodynamics
Type of constraints for mechanical systems
This form is called the Pfaffian form or the differential form. If the differential form is integrable, i.e., if there is a function f i ( u 1 , u 2 ,
Holonomic_constraints
Surname list
Pfaff, German mathematician Concepts named after him include the Pfaffian, Pfaffian functions, and the Pfaff problem 29491 Pfaff, a main-belt asteroid named
Pfaff_(surname)
Generating function in integrable systems
specific polynomial or quasi-polynomial functions, or parametric integrals, and their derivatives; 5) the Pfaffian of a skew symmetric matrix (either finite
Tau function (integrable systems)
Tau_function_(integrable_systems)
Algorithmic runtime requirements for common math procedures
Rote, G. (2001). "Division-free algorithms for the determinant and the pfaffian: algebraic and combinatorial approaches" (PDF). Computational discrete
Computational complexity of mathematical operations
Computational_complexity_of_mathematical_operations
Differential form of degree one or section of a cotangent bundle
{x}{x^{2}+y^{2}}}dy\end{aligned}}} While the angle "function" cannot be continuously defined – the function atan2 is discontinuous along the negative y {\displaystyle
One-form
Function of a matrix
the Pfaffian, but differs in that the signatures of the permutations are not taken into account. Thus the relationship of the hafnian to the Pfaffian is
Hafnian
Ordered field with a function generalizing the exponential function
theorem. This result, when combined with Khovanskiĭ's theorem on pfaffian functions, proves that Rexp is also o-minimal. Alfred Tarski posed the question
Ordered_exponential_field
Algorithm for counting perfect matchings in planar graphs
convert the problem into a Pfaffian computation of a skew-symmetric matrix derived from a planar embedding of the graph. The Pfaffian of this matrix is then
FKT_algorithm
Notion in supervised machine learning
dimension. Karpinski–Macintyre theorem, a bound on the VC dimension of general Pfaffian formulas. Vapnik, V. N.; Chervonenkis, A. Ya. (1971). "On the Uniform Convergence
Vapnik–Chervonenkis_dimension
Partial quantifier elimination for ordered fields with exponentials
using the same analytic functions that described the original set. It turns out the required functions are the Pfaffian functions. In particular the theory
Wilkie's_theorem
In mathematics, invariant of square matrices
by Catalan, Spottiswoode, Glaisher, and Scott; skew determinants and Pfaffians, in connection with the theory of orthogonal transformation, by Cayley;
Determinant
On finding a maximal set of solutions of a system of first-order homogeneous linear PDEs
language. Frobenius' original version of the theorem was stated in terms of Pfaffian systems, which today can be translated into the language of differential
Frobenius theorem (differential topology)
Frobenius_theorem_(differential_topology)
Topics referred to by the same term
time and the coordinates but not on the momenta) Nonholonomic constraints Pfaffian constraint Scleronomic constraint (not depending on time) Rheonomic constraint
Constraint
classes of exponential–polynomial equations also appear in the study of Pfaffian functions and tame real geometry, where finiteness and structural results play
Exponential_polynomial
Topics referred to by the same term
Photon Factory, a synchrotron located at KEK in Tsukuba, Japan pf(A), the Pfaffian of a matrix A Phenylphosphine, an organophosphorus compound Plasmodium
PF
British mathematician
Finiteness Theorems for sets definable using the exponential function, and more general Pfaffian functions. The results, going far beyond those obtained by conventional
Alex_Wilkie
Latvian mathematician
differential equations. During his later years he became an authority in Pfaffian functions. Reiziņš died in 1991. Taimina, Daina; Henina, Ingrida. "Some Incomplete
Linards_Reiziņš
Antihermitian matrix Positive-definite, positive-semidefinite matrix Pfaffian Projection Spectral theorem Perron–Frobenius theorem List of matrices Diagonal
Outline_of_linear_algebra
Geometric construct
A004003 in the OEIS). These numbers can be found by writing them as the Pfaffian of an m n × m n {\displaystyle mn\times mn} skew-symmetric matrix whose
Domino_tiling
Pictorial representation of the behavior of subatomic particles
Grassmann integral of a free Fermi field is a high-dimensional determinant or Pfaffian, which defines the new type of Gaussian integration appropriate for Fermi
Feynman_diagram
Tool used in probabilistic polynomial identity testing
determinant of this skew-symmetric matrix which coincides with the square of the pfaffian of the matrix A and is non-zero (as polynomial) if and only if a perfect
Schwartz–Zippel_lemma
Systems, Springer Verlag, New York, 1991. Alekseevskii, D.V. (2001) [1994], "Pfaffian problem", Encyclopedia of Mathematics, EMS Press R. Bryant, "Nine Lectures
Cartan–Kähler_theorem
Form of a matrix
{\displaystyle \det(A)=\operatorname {Pf} (A)^{2}.} This polynomial is called the Pfaffian of A {\displaystyle A} and is denoted Pf ( A ) {\displaystyle \operatorname
Skew-symmetric_matrix
Integration for Grassmann variables
n\times n} matrix, and P f M {\displaystyle \mathrm {Pf} \,M} being the Pfaffian of M {\displaystyle M} , which fulfills ( P f M ) 2 = det M {\displaystyle
Berezin_integral
Construct allowing differentiation of tangent vector fields of manifolds
the Pfaffian system θ j = 0 (for all j) is integrable, and its integral manifolds are the fibres of the principal bundle Aff(n) → A. the Pfaffian system
Affine_connection
Algebraic encoding of graph connectivity
partition function of the Ising model, i.e., the Tutte polynomial at the hyperbola H 2 {\displaystyle H_{2}} , can be expressed as a Pfaffian and computed
Tutte_polynomial
Baik, Jinho; Barraquand, Guillaume; Corwin, Ivan; Suidan, Toufic (2018). "Pfaffian Schur processes and last passage percolation in a half-quadrant". The Annals
Language_model_benchmark
is rotation-freeness. An even more general notion, in the language of Pfaffian systems, is that of a completely integrable 1-form ω, which amounts to
Complex_lamellar_vector_field
Mathematical theory
{\mathfrak {S}}_{2k}} (see Lie algebra-valued forms#Operations as well as Pfaffian). If, moreover, f is invariant; i.e., f ( Ad g x ) = f ( x ) {\displaystyle
Chern–Weil_homomorphism
Generalization of affine connections
general idea much earlier. The high point of his remarkable 1910 paper on Pfaffian systems in five variables is the construction of a Cartan connection modelled
Cartan_connection
Algebraic structure designed for geometry
_{n}}} where Pf ( A ) {\displaystyle \operatorname {Pf} (A)} is the Pfaffian of A {\displaystyle A} and C = ( n 2 i ) {\textstyle {\mathcal {C}}={\binom
Geometric_algebra
Indian-American physicist (born 1960)
Barkeshli; M. S. Rudner (2018). "Parton construction of a wave function in the anti-Pfaffian phase". doi:10.1103/PhysRevB.98.035127.{{cite web}}: CS1 maint:
Jainendra_K._Jain
Function in mathematics
developed a new notion of connection. He sought to apply the techniques of Pfaffian systems to the geometries of Felix Klein's Erlangen program. In these investigations
Connection_(mathematics)
Class of spinors constructed using Clifford algebras
correspondence, these may be expressed as infinite dimensional Fredholm Pfaffians. Cartan, Élie (1981) [1938]. The theory of spinors. New York: Dover Publications
Pure_spinor
Local ring in commutative algebra
structure theorem for Gorenstein rings of codimension 3 in terms of the Pfaffians of a skew-symmetric matrix, by Buchsbaum and Eisenbud. In 2011, Miles
Gorenstein_ring
Method of computing determinants
Mathematical Society, 46 (1999), 637-646. Knuth, Donald, Overlapping Pfaffians, Electronic Journal of Combinatorics, 3 no. 2 (1996). Lotkin, Mark (1959)
Dodgson_condensation
Vector satisfying some of the criteria of an eigenvector
to A {\displaystyle A} , which is useful in computing certain matrix functions of A {\displaystyle A} . The matrix J {\displaystyle J} is also useful
Generalized_eigenvector
Mathematical concept
always +1 for any field. One way to see this is through the use of the Pfaffian and the identity Pf ( M T Ω M ) = det ( M ) Pf ( Ω ) . {\displaystyle
Symplectic_matrix
American mathematician
Complex Powers Associated with the Twisted Cases of the Determinant and the Pfaffian, was supervised by Jun-Ichi Igusa. She taught briefly at Hampshire College
Margaret_M._Robinson
American mathematician
Rains, Eric M. (2005). "Eynard–Mehta Theorem, Schur Process, and their Pfaffian Analogs". Journal of Statistical Physics. 121 (3–4): 291–317. arXiv:math-ph/0409059
Eric_M._Rains
Perceptual Robotics Perrone Robotics Personal Robot Pete (Disney) Peter Nordin Pfaffian constraint Pharmacy automation Phidget Phil Tippett Philosophy Philosophy
Index_of_robotics_articles
noted for his work on partial differential equations of the first order (Pfaffian systems as they are now called) which became part of the theory of differential
List of German inventors and discoverers
List_of_German_inventors_and_discoverers
is a Hankel matrix. See Sylvester (1853, Glossary p. 543–548). Archaic. Pfaffian A square root of the determinant of a skew-symmetric matrix. pippian An
Glossary_of_invariant_theory
{\displaystyle \mu } is then the Chern-Gauss-Bonnet integrand, which is the Pfaffian of the Riemannian curvature tensor. If X {\displaystyle X} is Riemannian
Valuation_(geometry)
PFAFFIAN FUNCTION
PFAFFIAN FUNCTION
Surname or Lastname
English
English : nickname from the animal, Middle English catte ‘cat’. The word is found in similar forms in most European languages from very early times (e.g. Gaelic cath, Slavic kotu). Domestic cats were unknown in Europe in classical times, when weasels fulfilled many of their functions, for example in hunting rodents. They seem to have come from Egypt, where they were regarded as sacred animals.English : from a medieval female personal name, a short form of Catherine.Variant spelling of German and Dutch Katt.
Male
Egyptian
, a great functionary.
Male
Celtic
, great justiciary, or functionary.
Surname or Lastname
English
English : topographic name for someone who lived by the gates of a medieval walled town. The Middle English singular gate is from the Old English plural, gatu, of geat ‘gate’ (see Yates). Since medieval gates were normally arranged in pairs, fastened in the center, the Old English plural came to function as a singular, and a new Middle English plural ending in -s was formed. In some cases the name may refer specifically to the Sussex place Eastergate (i.e. ‘eastern gate’), known also as Gates in the 13th and 14th centuries, when surnames were being acquired.Americanized spelling of German Götz (see Goetz).Translated form of French Barrière (see Barriere).In New England, Gates was the preferred English version of the name of an extensive French family, called Barrière dit Langevin.
Male
Egyptian
, a high Egyptian functionary.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Surname or Lastname
English (chiefly Kent and Sussex)
English (chiefly Kent and Sussex) : occupational name for a designer or engineer, from a Middle English reduced form of Old French engineor ‘contriver’ (a derivative of engaigne ‘cunning’, ‘ingenuity’, ‘stratagem’, ‘device’). Engineers in the Middle Ages were primarily designers and builders of military machines, although in peacetime they might turn their hands to architecture and other more pacific functions.German : from the Latin personal name Januarius (see January 1). Jänner is a South German word for ‘January’, and so it is possible that this is one of the surnames acquired from words denoting months of the year, for example by converts who had been baptized in that month, people who were born or baptized in that month, or people whose taxes were due in January.
Surname or Lastname
English
English : occupational name for a dresser of cloth, Old English fullere (from Latin fullo, with the addition of the English agent suffix). The Middle English successor of this word had also been reinforced by Old French fouleor, foleur, of similar origin. The work of the fuller was to scour and thicken the raw cloth by beating and trampling it in water. This surname is found mostly in southeast England and East Anglia. See also Tucker and Walker.In a few cases the name may be of German origin with the same form and meaning as 1 (from Latin fullare).Americanized version of French Fournier.Samuel Fuller (1589–1633), born in Redenhall, Norfolk, England, was among the Pilgrim Fathers who sailed on the Mayflower in 1620. He was a deacon of the church and until his death functioned as Plymouth Colony’s physician.
Male
Egyptian
, the son of the functionary Heknofre.
Biblical
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Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, Functionary of the Interior.
PFAFFIAN FUNCTION
PFAFFIAN FUNCTION
Girl/Female
Indian, Sikh
One who is God Gifted
Boy/Male
Arabic
Respondent
Boy/Male
Arabic, Muslim
Patient
Male
Scottish
Scottish unisex name derived from Gaelic inis, INNIS means "island."
Male
English
English surname transferred to forename use, probably of Norman French origin, DARDEN means "from Ardern."
Girl/Female
Afghan, American, Arabic, Australian, British, Chinese, Christian, Danish, English, Finnish, French, German, Hebrew, Irish, Italian, Latin, Muslim, Portuguese, Swedish
Flower; The Goddess of Flower; Form of Florence; Blooming
Girl/Female
Tamil
Mahakali | மஹாகாலீ
Goddess Durga
Boy/Male
Muslim
Elegance
Male
Czechoslovakian
, ruler of glory.
Boy/Male
Muslim
Ally, Confederate
PFAFFIAN FUNCTION
PFAFFIAN FUNCTION
PFAFFIAN FUNCTION
PFAFFIAN FUNCTION
PFAFFIAN FUNCTION
v. i.
Alt. of Functionate
n.
A certain function relating to a system of forces and their points of application, -- first used by Clausius in the investigation of problems in molecular physics.
a.
Pertaining to, or connected with, a function or duty; official.
v. t.
To assign to some function or office.
a.
Belonging or relating to life, either animal or vegetable; as, vital energies; vital functions; vital actions.
n.
One deputed or authorized to perform the functions of another; a substitute in office; a deputy.
a.
Destitute of function, or of an appropriate organ. Darwin.
a.
Of, pertaining to, or designating, certain secret tribunals which flourished in Germany from the end of the 12th century to the middle of the 16th, usurping many of the functions of the government which were too weak to maintain law and order, and inspiring dread in all who came within their jurisdiction.
n.
The appropriate action of any special organ or part of an animal or vegetable organism; as, the function of the heart or the limbs; the function of leaves, sap, roots, etc.; life is the sum of the functions of the various organs and parts of the body.
n.
A quantity so connected with another quantity, that if any alteration be made in the latter there will be a consequent alteration in the former. Each quantity is said to be a function of the other. Thus, the circumference of a circle is a function of the diameter. If x be a symbol to which different numerical values can be assigned, such expressions as x2, 3x, Log. x, and Sin. x, are all functions of x.
pl.
of Functionary
a.
Of or pertaining to the vessels of animal and vegetable bodies; as, the vascular functions.
n.
The doctrine that all the functions of a living organism are due to an unknown vital principle distinct from all chemical and physical forces.
n.
One charged with the performance of a function or office; as, a public functionary; secular functionaries.
prep.
Acting as a substitute; -- said of abnormal action which replaces a suppressed normal function; as, vicarious hemorrhage replacing menstruation.
n.
Fig.: Any cavity, or hollow place, in which any function may be conceived of as operating.
v. i.
To execute or perform a function; to transact one's regular or appointed business.
a.
Having relation to growth or nutrition; partaking of simple growth and enlargement of the systems of nutrition, apart from the sensorial or distinctively animal functions; vegetal.
adv.
In a functional manner; as regards normal or appropriate activity.
a.
Pertaining to the function of an organ or part, or to the functions in general.