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Determinant in functional analysis
In functional analysis, a branch of mathematics, it is sometimes possible to generalize the notion of the determinant of a square matrix of finite order
Functional_determinant
In mathematics, invariant of square matrices
In mathematics, the determinant is a scalar-valued function of the entries of a square matrix. The determinant of a matrix A is commonly denoted det(A)
Determinant
Matrix of second derivatives
Hesse and later named after him. Hesse originally used the term "functional determinants". The Hessian is sometimes denoted by H or ∇ ∇ {\displaystyle \nabla
Hessian_matrix
Summability method in physics
divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators. The technique is now commonly
Zeta_function_regularization
Number, approximately 3.14
is that π can be obtained from the functional determinant of the harmonic oscillator. This functional determinant can be computed via a product expansion
Pi
Conjecture on zeros of the zeta function
square of the imaginary part of the Riemann zeros, and also that the functional determinant of this Hamiltonian operator is just the Riemann Xi function. In
Riemann_hypothesis
Set of functions between two fixed sets
mathematical functions Clifford algebra Tensor field Spectral theory Functional determinant Equicontinuity Ascoli's theorem Fulton, William; Harris, Joe (1991)
Function_space
Variable associated with an increased risk of disease or infection
determinant is a variable associated with an increased risk of disease or infection. Due to a lack of harmonization across disciplines, determinant,
Risk_factor
In functional analysis, the Hilbert–Carleman determinant is an operator determinant for certain integral operators on Banach spaces, whose kernels are
Hilbert–Carleman_determinant
Method of calculating chiral anomalies
anomaly in quantum field theory. It uses the correspondence between functional determinants and the partition function, effectively making use of the Atiyah–Singer
Fujikawa_method
Type of unique key in a database
relational model of data, a natural key is a superkey and is therefore a functional determinant for all attributes in a relation. A natural key serves two complementary
Natural_key
Polyakov formula expresses the conformal variation of the zeta functional determinant of a Riemannian manifold. Proposed by Alexander Markovich Polyakov
Polyakov_formula
Quillen metric Quillen, Daniel (1985), "Determinants of Cauchy-Riemann operators over a Riemann surface", Functional Analysis and Its Applications, 19, Springer
Quillen determinant line bundle
Quillen_determinant_line_bundle
Computational quantum mechanical modelling method to investigate electronic structure
wavefunction can be represented as a Slater determinant of orbitals. Further, the kinetic energy functional of such a system is known exactly. The exchange–correlation
Density_functional_theory
Array of numbers
symmetric matrices are real. Jacobi studied "functional determinants"—later called Jacobi determinants by Sylvester—which can be used to describe geometric
Matrix_(mathematics)
Relational database theory concept
database theory, a functional dependency (FD) is constraint between two attribute sets, whereby values in one set (the determinant set) determine the
Functional_dependency
Chemical entity which can be bound by an antibody
An epitope, also known as antigenic determinant, is the part of an antigen that is recognized by the immune system, specifically by antibodies, B cells
Epitope
Integral of the Gaussian function, equal to sqrt(π)
) ∞ {\displaystyle (2\pi )^{\infty }} is infinite and also, the functional determinant would also be infinite in general. This can be taken care of if
Gaussian_integral
Italian-French scientist (1736–1813)
is also treated in these papers. In 1773, Lagrange considered a functional determinant of order 3, a special case of a Jacobian. He also proved the expression
Joseph-Louis_Lagrange
_{i}^{-s}} . It is used in giving a rigorous definition to the functional determinant of an operator, which is given by det O := e − ζ O ′ ( 0 ) . {\displaystyle
Zeta_function_(operator)
Stochastic differential equation
(A){\big )}\det(\mathrm {d} \eta /\mathrm {d} A)\,\mathrm {d} A} . The functional determinant and associated mathematical subtleties drop out if the Langevin
Langevin_equation
Branch of mathematics
since Gaussian elimination is a faster algorithm. The determinant of an endomorphism is the determinant of the matrix representing the endomorphism in terms
Linear_algebra
Formulation to quantize gauge field theories in physics
perturbation takes place. The ghost term in the Lagrangian represents the functional determinant of the Jacobian of this embedding, and the properties of the ghost
BRST_quantization
Scale of cardiac health
classifying patients in class II or III, because functional capacity is such a powerful determinant of outcome, it remains arguably the most important
New York Heart Association Functional Classification
New_York_Heart_Association_Functional_Classification
Schrödinger equation of a fictitious system of non-interacting particles
fermions (non-fermion density functional theory has been studied), the Kohn–Sham wavefunction is a single Slater determinant constructed from a set of orbitals
Kohn–Sham_equations
For a square matrix, the transpose of the cofactor matrix
(entries not on the main diagonal are zero) whose diagonal entries are the determinant of the original matrix: A adj ( A ) = det ( A ) I , {\displaystyle
Adjugate_matrix
Sum of elements on the main diagonal
to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). The trace of an n × n square matrix A is defined
Trace_(linear_algebra)
Sequences. OEIS Foundation. Sloane, N. J. A. (ed.). "Sequence A001970 (Functional determinants; partitions of partitions; Euler transform applied twice to all
1000_(number)
Mathematical operation in calculus
ISBN 978-0-201-00288-1. Goursat, Édouard (1904). "II. Implicit Functions; Functional Determinants; Change of Variable". A Course in Mathematical Analysis. Vol. 1
Implicit_differentiation
Extension of Lidskii's theorem
In functional analysis, the Grothendieck trace theorem is an extension of Lidskii's theorem about the trace and the determinant of a certain class of nuclear
Grothendieck_trace_theorem
Relation between determinant and trace via eigenvalues). The number Δ ( X ) {\displaystyle \Delta (X)} is well-defined by continuous functional calculus. Δ
Fuglede−Kadison_determinant
homogeneous polynomial in several variables, also called a quantic. functional determinant An archaic name for Jacobians fundamental 1. The first fundamental
Glossary_of_invariant_theory
Function of the coefficients of a polynomial that gives information on its roots
characteristic different from 2, the discriminant or determinant of Q is the determinant of A. The Hessian determinant of Q is 2 n {\displaystyle 2^{n}} times its
Discriminant
Swiss mathematician and physicist (1939–2015)
; Schrader, R.; Seiler, R. (1978). "A conjecture on the spinor functional determinant". Nuclear Physics B. 142 (4): 525–534. Bibcode:1978NuPhB.142..525H
Robert_Schrader
Group of 𝑛 × 𝑛 invertible matrices
{\displaystyle R} is invertible if and only if its determinant is a unit in R {\displaystyle R} , that is, if its determinant is invertible in R {\displaystyle R}
General_linear_group
Theory on customer satisfaction
for the functional and dysfunctional questions, one can infer the feature category. Illogical answers (e.g., "I like it" for both the functional and dysfunctional
Kano_model
Non-conservation of chiral current in physics
Fujikawa derived this anomaly using the correspondence between functional determinants and the partition function using the Atiyah–Singer index theorem
Chiral_anomaly
Technique in quantum field theory
(x)\delta \phi (y)}}[B]\right)^{-1/2}+\cdots } where "det" signifies a functional determinant and C is a constant. The power of minus one half will naturally
Background_field_method
Probability distribution
a random Hermitian matrix. The distribution is defined as a Fredholm determinant. In practical terms, Tracy–Widom is the crossover function between the
Tracy–Widom_distribution
Mathematical computing environment
data processing, visualization, and others. A toolbox, MapleSim, adds functionality for multidomain physical modeling and code generation. Maple's capacity
Maple_(software)
Fourth letter in the Greek alphabet
given quantity. Any of the delta particles in particle physics. The determinant of the matrix of coefficients of a set of linear equations (see Cramer's
Delta_(letter)
Pattern of limb movements made during locomotion
resulting in some energy conservation. But recent studies testing this third determinant of gait have reported varied results. It was found out that stance-phase
Gait_(human)
Approximation method in quantum physics
-body wave function of the system can be approximated by a single Slater determinant (in the case where the particles are fermions) or by a single permanent
Hartree–Fock_method
Orthonormalization of a set of vectors
≥ 1 {\displaystyle j\geq 1} , D j {\displaystyle D_{j}} is the Gram determinant D j = | ⟨ v 1 , v 1 ⟩ ⟨ v 2 , v 1 ⟩ ⋯ ⟨ v j , v 1 ⟩ ⟨ v 1 , v 2 ⟩ ⟨ v
Gram–Schmidt_process
Result about when a matrix can be diagonalized
In linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented
Spectral_theorem
Unscientific healthcare practices
Wake, Getu Engida; Fitie, Girma Wogie (29 April 2022). "Magnitude and Determinant Factors of Herbal Medicine Utilization Among Mothers Attending Their
Alternative_medicine
Matrix-variate probability distribution
measure. For example, this distribution arises in the study of the functional determinant under transformations involving orthogonal or semi-orthogonal matrices
Uniform distribution on a Stiefel manifold
Uniform_distribution_on_a_Stiefel_manifold
Societal problems that disrupt mental health
The social determinants of mental health (SDOMH) are societal problems that disrupt mental health, increase risk of mental illness among certain groups
Social determinants of mental health
Social_determinants_of_mental_health
Algebra describing 2D conformal symmetry
singular vectors. The determinant of the Shapovalov form at a given level N {\displaystyle N} is given by the Kac determinant formula, det ( S L , L
Virasoro_algebra
Theorem
Hadamard's theorem on determinants) is a result first published by Jacques Hadamard in 1893. It is a bound on the determinant of a matrix whose entries
Hadamard's_inequality
German mathematician (1868–1950)
relationship between the invariants of three quadratic forms and their functional determinants. Schilling's theory was presented by Felix Klein in his lectures
Friedrich_Schilling
Level of database normalization
Codd by performing the set difference between the dependent and the determinant. In short, Zaniolo points to a loophole more clearly than the transitive
Third_normal_form
Quantum-mechanical framework for simulating molecules and solids
Time-dependent density-functional theory (TDDFT) is a quantum mechanical theory used in physics and chemistry to investigate the properties and dynamics
Time-dependent density functional theory
Time-dependent_density_functional_theory
Property of operations
In a ring of quadratic matrices, the determinant of an idempotent matrix is either 0 or 1. If the determinant is 1, the matrix necessarily is the identity
Idempotence
appears naturally in connection with extremal problems for the functional determinant of the Laplacian (via the Polyakov formula; see Branson & Ørsted
Paneitz_operator
Metric on a determinant line bundle
especially differential geometry, the Quillen metric is a metric on the determinant line bundle of a family of operators. It was introduced by Daniel Quillen
Quillen_metric
Form of amenorrhea and chronic anovulation
Functional hypothalamic amenorrhea (FHA) is a form of amenorrhea and chronic anovulation and is one of the most common types of secondary amenorrhea. It
Functional hypothalamic amenorrhea
Functional_hypothalamic_amenorrhea
Alternate way to define a function in APL
Similarly, the determinant of a square complex matrix using Gaussian elimination can be computed with tail recursion: det←{ ⍝ determinant of a square complex
Direct_function
Matrix with shifting rows
well-conditioned linear systems). The algorithms can also be used to find the determinant of a Toeplitz matrix in O ( n 2 ) {\displaystyle O(n^{2})} time. A Toeplitz
Toeplitz_matrix
Concept in linear algebra
to some basis for V, a bilinear form is degenerate if and only if the determinant of the associated matrix is zero – if and only if the matrix is singular
Degenerate_bilinear_form
Condition of impaired digestion
Hiroto (November 2021). "Meta-Analysis: Placebo Response and Its Determinants in Functional Dyspepsia". American Journal of Gastroenterology. 116 (11): 2184–2196
Indigestion
(Cauchy–Kowalevski theorem) Cauchy–Lipschitz theorem Cauchy matrix (and Cauchy determinant) Cauchy net Cauchy–Peano theorem Cauchy point Cauchy principal value
List of things named after Augustin-Louis Cauchy
List_of_things_named_after_Augustin-Louis_Cauchy
Concept in mathematics
In mathematics, the quasideterminant is a replacement for the determinant for matrices with noncommutative entries. Example 2 × 2 quasideterminants are
Quasideterminant
Application of geometry in number theory
this lattice. The discriminant Δ K {\displaystyle \Delta _{K}} is the determinant of the Gram matrix of an integral basis of O K {\displaystyle {\mathcal
Geometry_of_numbers
Theory of evolution by changes at the molecular level
environmental conditions to be the major determinants of polymorphisms rather than structural and functional factors. According to the neutral theory
Neutral theory of molecular evolution
Neutral_theory_of_molecular_evolution
Factors determining maximum running speed in animals
while humans can train to achieve high sprint speeds. There is no single determinant of maximum running speed: however, certain factors stand out against
Limitations of animal running speed
Limitations_of_animal_running_speed
Concept in computational chemistry
Mathematically, configuration simply describes the linear combination of Slater determinants used for the wave function. In terms of a specification of orbital occupation
Configuration_interaction
In addition, there is an argument that functional diversity in teams would make the system “less determinant, less standardized, and therefore more fluid
Functional diversity (organizational)
Functional_diversity_(organizational)
Scientific theories
The functional matrix hypothesis was popularized by Melvin Moss in 1962. This theory said that neither bone nor cartilage is a major determinant of growth
Theories of craniofacial growth
Theories_of_craniofacial_growth
Application of Lagrangian mechanics to field theories
the metric determinant is equivalent to the Jacobian determinant. The minus sign is a consequence of the metric signature (the determinant by itself is
Lagrangian_(field_theory)
Method in quantum chemistry
expansion of the wave function is) and Hartree–Fock (where there is only one determinant, but the molecular orbitals are varied). MCSCF wave functions are often
Multi-configurational self-consistent field
Multi-configurational_self-consistent_field
Typographic symbol
string S" determinant: | A | {\displaystyle |A|} , read "the determinant of the matrix A". When the matrix entries are written out, the determinant is denoted
Vertical_bar
Neuroscientist and award-winning author
Today. Griffith T, Swanson G (2015). "Identification of critical functional determinants of kainate receptor modulation by auxiliary protein Neto2". Journal
Theanne_Griffith
In density functional theory (DFT), the Harris energy functional is a non-self-consistent approximation to the Kohn–Sham density functional theory. It
Harris_functional
Concept in general relativity
where g = det ( g μ ν ) {\displaystyle g=\det(g_{\mu \nu })} is the determinant of the metric tensor matrix, R {\displaystyle R} is the Ricci scalar
Einstein–Hilbert_action
Parts of plant enabling sexual reproduction
depends on the reproductive morphology, and is the single most important determinant of the genetic structure of nonclonal plant populations. Christian Konrad
Plant_reproductive_morphology
German mathematician (1804–1851)
contributions to elliptic functions, dynamics, differential equations, determinants and number theory. Jacobi was born of Ashkenazi Jewish parentage in Potsdam
Carl_Gustav_Jacob_Jacobi
theory) Spectral theorem (linear algebra, functional analysis) Sylvester's determinant theorem (determinants) Sylvester's law of inertia (quadratic forms)
List_of_theorems
Function acting on function spaces
operators between finite-dimensional vector spaces are the ones of rank, determinant, inverse operator, and eigenspace. Linear operators also play a great
Operator_(mathematics)
Complete set of nucleic acid sequences for humans
mitochondrial genome. Human genomes include both genes and various other types of functional DNA elements. The latter is a diverse category that includes regulatory
Human_genome
Continual use of drugs despite negative consequences
broadly fall into three domains—biological determinants, psychological determinants, and social determinants. These are collectively referred to as the
Substance_use_disorder
Process of achieving a goal by overcoming obstacles
Funke" [No system of systems! Reply to the paper 'System features as determinants of behavior in dynamic task environments' by Joachim Funke]. Sprache
Problem_solving
Concept in mathematical physics
mechanics. Wu and Sprung also showed that the zeta-regularized functional determinant is the Riemann Xi-function ξ ( s ) ξ ( 0 ) = det ( H − s ( 1 − s
Wu–Sprung_potential
Physical model of solid metals as electron gases
functional theory; the formalism itself provides the basis for the local-density approximation to the exchange-correlation energy density functional.
Jellium
Part of the human heart
73 (1): 92–105. doi:10.1016/0002-8703(67)90313-4. PMID 6016029. The Determinants of Cardiac Output (video produced by University of Arizona Biomedical
Atrium_(heart)
of the paper: "...the number of functionally different roles represented in an ecosystem may be a stronger determinant of ecosystem processes than the
Diversity–function_debate
Creation of a whole that is greater than the simple sum of its parts
social groups, are favored by natural selection because they provide functional advantages for survival and reproduction. This principle also applies
Synergy
Extension of superfactorials to the complex numbers
Theory of the Barnes function". arXiv:math/0308086. Vardi, I. (1988). "Determinants of Laplacians and multiple gamma functions". SIAM J. Math. Anal. 19 (2):
Barnes_G-function
Hormonal theory of sexuality
later in the individual. Prenatal hormones may be seen as the primary determinant of adult sexual orientation, or a co-factor. The hormonal theory of sexuality
Prenatal hormones and sexual orientation
Prenatal_hormones_and_sexual_orientation
Biological molecule
Rutherford, Steven T; Papenfort, Kai; Bassler, Bonnie L (2013-07-09). "Functional determinants of the quorum-sensing non-coding RNAs and their roles in target
Qrr_RNA
Mathematical theory of integral equations
{\displaystyle R(\lambda )={\frac {1}{I-\lambda K}}.} The Fredholm determinant is commonly defined as det ( I − λ K ) = exp [ − ∑ n λ n n Tr K n ]
Fredholm_theory
RNA family
A, Sriram SI (April 1996). "Functional analysis of the Enterococcus faecalis plasmid pAD1-encoded stability determinant par". Mol. Microbiol. 20 (1):
Par_stability_determinant
Neurodevelopmental disorder involving motor and vocal tics
Sul JH, Tsetsos F, et al. (March 2019). "Interrogating the genetic determinants of Tourette's syndrome and other tic disorders through genome-wide association
Tourette_syndrome
Branch of chemistry
cases, the Hartree–Fock wave function occupies a single configuration or determinant. In some cases, particularly for bond-breaking processes, this is inadequate
Computational_chemistry
Quantum-mechanical simulation framework
In density functional theory (DFT) the adiabatic-connection fluctuation-dissipation theorem (ACFD) is an exact formula for the Kohn–Sham correlation energy
Adiabatic connection fluctuation dissipation theorem
Adiabatic_connection_fluctuation_dissipation_theorem
Country in South Asia
"Pakistan's Foreign Policy: An Abridgement Of Internal And External Determinants". Journal of Positive School Psychology. 7 (4): 1089–1098. ISSN 2717-7564
Pakistan
Within Hartree–Fock theory, the wave function is approximated as a Slater determinant of spin-orbitals. For an open-shell system, the mean-field approach of
Spin_contamination
Active region of an enzyme
; Borhani, David W.; Dror, Ron O.; Shaw, David E. (2013). "Molecular determinants of drug–receptor binding kinetics". Drug Discovery Today. 18 (13–14):
Active_site
to evolutionary ideas of the Enlightenment. Lagrange considers a functional determinant of order 3, a special case of a Jacobian. He also proves the expression
1773_in_science
Deterioration of function with age
Senescence (/ˌsɪˈnɛsəns/) or biological aging is the gradual deterioration of functional characteristics in living organisms. Whole organism senescence involves
Senescence
FUNCTIONAL DETERMINANT
FUNCTIONAL DETERMINANT
Biblical
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Male
Egyptian
, the son of the functionary Heknofre.
Boy/Male
American, Australian, British, English, French
Mighty Spearman; The Fictional Character Jorel Father of Superman
Male
Egyptian
, Functionary of the Interior.
Boy/Male
Australian, French
Fictional Swordsman; Ambitious and Filled with Religious Aspirations; From Alexander Dumas's Three Musketeers
Boy/Male
French
Fictional swordsman: (ambitious and filled with religious aspirations) from Alexander Dumas's...
Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, a high Egyptian functionary.
Boy/Male
English
The fictional character Jorel father of Superman.
Boy/Male
American, British, English
Mighty Spearman; One who Saves; The Fictional Character Jorel Father of Superman
Boy/Male
English
Modern. The fictional character Jorel father of Superman.
Boy/Male
American, Australian, British, Danish, English, Finnish, French, German, Scandinavian
Farmer; The Fictional Character Jorel Father of Superman; Earth Worker
Male
Egyptian
, a great functionary.
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
Celtic
, great justiciary, or functionary.
Boy/Male
English
The fictional character Jorel father of Superman.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Boy/Male
English
The fictional character Jorel father of Superman.
Male
Egyptian
, an Egyptian functionary.
Boy/Male
American, British, English
Mighty Spearman; The Fictional Character Jorel Father of Superman
FUNCTIONAL DETERMINANT
FUNCTIONAL DETERMINANT
Female
Native American
Native American Dakota name WICAPI WAKAN means "holy star."
Boy/Male
Indian, Sanskrit
Divine Soul
Boy/Male
Muslim
Supporter, Helper, One who helps
Girl/Female
Assamese, Indian
River
Boy/Male
Tamil
Hrydayesh | ஹà¯à®°à¯à®¯à¯à®¤à®¯à¯‡à®·
King of heart, Lord of hearts
Girl/Female
African, Arabic, Muslim
Honour; Power; Fame
Biblical
the cloak, glory, grandeur or power of the king
Boy/Male
Indian
Girl/Female
Hindu, Indian, Kannada, Malayalam, Marathi, Tamil, Telugu
Seeta
Boy/Male
Arabic, Muslim
Sword of the Faith
FUNCTIONAL DETERMINANT
FUNCTIONAL DETERMINANT
FUNCTIONAL DETERMINANT
FUNCTIONAL DETERMINANT
FUNCTIONAL DETERMINANT
n.
The office, duties, or functions of a minister, servant, or agent; ecclesiastical, executive, or ambassadorial function or profession.
v. i.
Alt. of Functionate
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.
a.
Fractional.
a.
Of or pertaining to fractions or a fraction; constituting a fraction; as, fractional numbers.
a.
Pertaining to the function of an organ or part, or to the functions in general.
n.
Paper fractional currency.
a.
Pertaining to, or connected with, a function or duty; official.
a.
Relatively small; inconsiderable; insignificant; as, a fractional part of the population.
v. t.
To supply with an organ or organs having a special function or functions.
v. i.
To execute or perform a function; to transact one's regular or appointed business.
a.
Relating to friction; moved by friction; produced by friction; as, frictional electricity.
a.
Pertaining to, or characterized by, fiction; fictitious; romantic.
n.
One charged with the performance of a function or office; as, a public functionary; secular functionaries.
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.
a.
Capable of, or pertaining to, flection or inflection.
n.
A derived function; a function obtained from a given function by a certain algebraic process.
pl.
of Functionary
n.
An angle upon which the value of some function depends; -- a term used more especially in connection with elliptic functions.
adv.
In a functional manner; as regards normal or appropriate activity.