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optimization, the perturbation function is any function which relates to primal and dual problems. The name comes from the fact that any such function defines a
Perturbation_function
Topics referred to by the same term
one Perturbation (biology), an alteration of the function of a biological system, induced by external or internal mechanisms Perturbation function, mathematical
Perturbation
Mathematical approach to quantum physics
In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated
Perturbation theory (quantum mechanics)
Perturbation_theory_(quantum_mechanics)
Methods of mathematical approximation
In mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related
Perturbation_theory
Principle in mathematical optimization
} otherwise). Then extend f ~ {\displaystyle {\tilde {f}}} to a perturbation function F : X × Y → R ∪ { + ∞ } {\displaystyle F:X\times Y\to \mathbb {R}
Duality_(optimization)
Generalized function whose value is zero everywhere except at zero
Dirac delta function (or δ {\displaystyle {\boldsymbol {\delta }}} distribution), also known as the unit impulse, is a generalized function on the real
Dirac_delta_function
Method in ab initio Quantum Chemistry
real parameter that controls the size of the perturbation. In MP theory the zeroth-order wave function is an exact eigenfunction of the Fock operator
Møller–Plesset perturbation theory
Møller–Plesset_perturbation_theory
Mathematical theorem in convex analysis
(via the perturbation function). Let ( X , τ ) {\displaystyle (X,\tau )} be a Hausdorff locally convex space, for any extended real valued function f : X
Fenchel–Moreau_theorem
Mathematical description of quantum state
element of L2. For instance, in perturbation theory one may construct a sequence of functions approximating the true wave function. This sequence will be guaranteed
Wave_function
Concept in mathematics
In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system A x = λ x {\displaystyle Ax=\lambda
Eigenvalue_perturbation
Solid-state physics model
In solid-state physics, the k·p perturbation theory is an approximated semi-empirical approach for calculating the band structure (particularly effective
K·p_perturbation_theory
Mathematics of convex functions and sets
of the perturbation function F {\displaystyle F} can lead to different dual problems for the same primal optimization problem. Convex functions need not
Convex_analysis
Optimization algorithm
Simultaneous perturbation stochastic approximation (SPSA) is an algorithmic method for optimizing systems with multiple unknown parameters. It is a type
Simultaneous perturbation stochastic approximation
Simultaneous_perturbation_stochastic_approximation
Transition rate formula
initial state. If H' is oscillating sinusoidally as a function of time (i.e. it is a harmonic perturbation) with an angular frequency ω, the transition is into
Fermi's_golden_rule
Condition in mathematical optimization
F ∗ ∗ {\displaystyle F=F^{**}} where F {\displaystyle F} is the perturbation function relating the primal and dual problems and F ∗ ∗ {\displaystyle F^{**}}
Strong_duality
Relationship of a signal transducer
formula, which considers the general case that the "force" h(t) is a perturbation of the basic operator of the system, the Hamiltonian, H ^ 0 → H ^ 0 −
Linear_response_function
Mathematical result in convex functions theory
respectively, and A ∗ {\displaystyle A^{*}} is the adjoint operator. The perturbation function for this dual problem is given by F ( x , y ) = f ( x ) + g ( A
Fenchel's_duality_theorem
Classical approach to the many-body problem of astronomy
In astronomy, perturbation is the complex motion of a massive body subjected to forces other than the gravitational attraction of a single other massive
Perturbation_(astronomy)
Functions that can't be described by perturbation theory
physics, a non-perturbative function or process is one that cannot be described by perturbation theory. An example is the function f ( x ) = e − 1 / x 2 ,
Non-perturbative
Type of differential equation
an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves
Partial_differential_equation
indicator function. Then let F : X × Y → R ∪ { + ∞ } {\displaystyle F:X\times Y\to \mathbb {R} \cup \{+\infty \}} be a perturbation function such that
Duality_gap
Testing how computer systems behave under unusual stresses
pFunc(aFunction(atoi(argv[1]))); if (a > 20) { /* do something */ } else { /* do something else */ } } In this case, pFunc is the perturbation function and
Fault_injection
Method in physics used to deal with infinities
applicability, though more limited but rigorous approaches like causal perturbation theory are also used. For example, an electron theory may begin by postulating
Renormalization
Metaheuristic
instances passed. Since there is no function a priori that tells which one is the most suitable value for a given perturbation, the best criterion is to get
Iterated_local_search
Function that encodes the dependence of a coupling parameter on the energy scale
graphs). Here are some examples of beta functions computed in perturbation theory: The one-loop beta function in quantum electrodynamics (QED) is β (
Beta_function_(physics)
Fractal sets in complex dynamics of mathematics
under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the
Julia_set
problems with maximization problems of convex conjugates Perturbation function — any function which relates to primal and dual problems Slater's condition
List of numerical analysis topics
List_of_numerical_analysis_topics
Summability method in physics
In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent
Zeta_function_regularization
Analyzes the topology of a manifold by studying differentiable functions on that manifold
by a slight perturbation of f . {\displaystyle f.} In the case of a landscape or a manifold embedded in Euclidean space, this perturbation might simply
Morse_theory
Mathematical approximation of a function
the sine function with its linear approximation. Such approximations are used throughout mathematics, physics, and engineering. In perturbation theory,
Taylor_series
Concept in mathematics
In mathematics, a singular perturbation problem is a problem containing a small parameter that cannot be approximated by setting the parameter value to
Singular_perturbation
Physical model of solid metals as electron gases
modification of the free-electron gas model which includes a weak periodic perturbation meant to model the interaction between the conduction electrons and the
Nearly_free_electron_model
Special function occurring in problems possessing elliptic symmetry
In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation d 2 y d x 2 + ( a − 2
Mathieu_function
Complete active space perturbation theory (CASPTn) is a multireference electron correlation method for computational investigation of molecular systems
Complete active space perturbation theory
Complete_active_space_perturbation_theory
Theoretical framework in physics
through an infinite perturbation series of the free two-point function. In canonical quantization, the two-point correlation function can be written as:
Quantum_field_theory
Differential equation that is linear with respect to the unknown function
differential equation is a differential equation that is linear in the unknown function and its derivatives, so it can be written in the form a 0 ( x ) y + a 1
Linear_differential_equation
Area of mathematics
points can be unfolded by expanding the potential function as a Taylor series in small perturbations of the parameters. When the degenerate points are
Catastrophe_theory
Solution method for linear differential equations
calculation in quantum mechanics in which the wave function is recast as an exponential function, semiclassically expanded, and then either the amplitude
WKB_approximation
Determinant of the matrix of first derivatives of a set of functions
Wronskian of n {\displaystyle n} differentiable functions is the determinant of a matrix formed by the functions and their derivatives up to order n − 1 {\displaystyle
Wronskian
Property of functions which is weaker than continuity
dimension if the perturbation is small enough. Another example of a similar character is that matrix rank is a lower semicontinuous function on the space
Semi-continuity
Single cell RNA sequencing method
gene expression phenotypes for each perturbation. Inferring a gene’s function by applying genetic perturbations to knock down or knock out a gene and
Perturb-seq
Auxiliary functions used to probe equations, distributions, and weak formulations
relevant Sobolev norm. Test functions also appear in variational arguments. If a functional is differentiated along perturbations of the form u + ε φ {\displaystyle
Test_function
Method in computational chemistry
Free-energy perturbation (FEP) is an alchemical method based on statistical mechanics that is used in computational chemistry for computing free-energy
Free-energy_perturbation
Collection of random variables
a sample function of a stochastic process X {\displaystyle X} is a continuous function of t ∈ T {\displaystyle t\in T} ; a sample function of a stochastic
Stochastic_process
Special function defined by an integral
is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument
Exponential_integral
Parameter describing the strength of a force
quantum electrodynamics (QED), where one finds by using perturbation theory that the beta function is positive. In particular, at low energies, α ≈ 1/137
Coupling_constant
Effective particle coupling beyond tree level
electrodynamics, the vertex function describes the coupling between a photon and an electron beyond the leading order of perturbation theory. In particular
Vertex_function
Effective field theory of quantum chromodynamics
Chiral perturbation theory (ChPT) is an effective field theory constructed with a Lagrangian consistent with the (approximate) chiral symmetry of quantum
Chiral_perturbation_theory
Methods of calculating definite integrals
\int _{a}^{b}f(x)\,dx} to a given degree of accuracy. If f(x) is a smooth function integrated over a small number of dimensions, and the domain of integration
Numerical_integration
Function that maps matrices to matrices
In mathematics, every analytic function can be used for defining a matrix function that maps square matrices with complex entries to square matrices of
Analytic_function_of_a_matrix
Spectral line splitting in electrical field
of the Wigner D-matrix. The first-order perturbation matrix on basis of the unperturbed rigid rotor function is non-zero and can be diagonalized. This
Stark_effect
exponential functions cancel each other out and, in addition, the different detector properties shorten themselves. The pure perturbation function remains
Perturbed_angular_correlation
Theorem in mathematics
mathematical analysis, the inverse function theorem gives sufficient conditions for a function to have an inverse function. The essential idea is that if
Inverse_function_theorem
Linear operator used in quantum mechanics
In physics, particularly in quantum perturbation theory, the matrix element refers to the linear operator of a modified Hamiltonian using Dirac notation
Matrix_element_(physics)
Spectral density of light emitted by a black body
coefficients can be calculated using dipole approximation in time dependent perturbation theory in quantum mechanics. Calculation of A also requires second quantization
Planck's_law
Theory of rapid universe expansion
of perturbations that were formed as quantum mechanical fluctuations in the inflationary epoch. The detailed form of the spectrum of perturbations, called
Cosmic_inflation
Second-order partial differential equation describing motion of mechanical system
a stationary point of S {\displaystyle S} with respect to any small perturbation in q {\displaystyle {\boldsymbol {q}}} . See proofs below for more rigorous
Euler–Lagrange_equation
Mathematically rigorous approach to renormalization theory
Causal perturbation theory is a mathematically rigorous approach to renormalization theory, which makes it possible to put the theoretical setup of perturbative
Causal_perturbation_theory
Astrophysical models for the formation of galaxies and clusters of galaxies
metric includes four scalar perturbations, two vector perturbations, and one tensor perturbation. Only the scalar perturbations are significant: the vectors
Structure_formation
Energy level of a quantum system
to the application of a small perturbation potential can be calculated using time-independent degenerate perturbation theory. This is an approximation
Degenerate_energy_levels
Type of boundary condition in mathematics
Robin boundary condition specifies a linear combination of the value of a function and the value of its derivative at the boundary of a given domain. It is
Robin_boundary_condition
Description of a quantum-mechanical system
Schrödinger equation is a partial differential equation that governs the wave function of a non-relativistic quantum-mechanical system. Its discovery was a significant
Schrödinger_equation
Expressions for approximation accuracy
effect that we only worry about at the annual calibration." Linearization Perturbation theory Chapman–Enskog method Big O notation Order of accuracy "Approximation
Order_of_approximation
Trace radiation from the early universe
primordial density perturbation spectrum predict different mixtures. Adiabatic density perturbations In an adiabatic density perturbation, the fractional
Cosmic_microwave_background
Computational quantum mechanical modelling method to investigate electronic structure
a term in the gradient of the density. In a perturbation theory approach the direct correlation function is given by the sum of the direct correlation
Density_functional_theory
Set of methods in computational chemistry
Configuration interaction (CI) Coupled cluster (CC) Møller–Plesset perturbation theory (MP2, MP3, MP4, etc.) Quadratic configuration interaction (QCI)
Post–Hartree–Fock
Quantum mechanics mathematical equation
function. Suppose now that just after some time t = t 0 {\displaystyle t=t_{0}} an external perturbation is applied to the system. The perturbation is
Kubo_formula
Theory of the evolution of cosmological structure
cosmological perturbation theory is the theory by which the evolution of structure is understood in the Big Bang model. Cosmological perturbation theory may
Cosmological perturbation theory
Cosmological_perturbation_theory
Measure of a system's order
correlation function's initial value and allowed to evolve. Equilibrium fluctuations of the system can be related to its response to external perturbations via
Correlation function (statistical mechanics)
Correlation_function_(statistical_mechanics)
Generating function for quantum correlation functions
not. Instead the partition function can be evaluated at weak coupling perturbatively, which amounts to regular perturbation theory using Feynman diagrams
Partition function (quantum field theory)
Partition_function_(quantum_field_theory)
In mathematics, variational perturbation theory (VPT) is a mathematical method to convert divergent power series in a small expansion parameter, say s
Variational perturbation theory
Variational_perturbation_theory
Differential equations involving stochastic processes
quantum effects are either unimportant or can be taken into account as perturbations. SDEs can be viewed as a generalization of the dynamical systems theory
Stochastic differential equation
Stochastic_differential_equation
Property of differential equations describing physical phenomena
generally not a continuous function of the parameters specifying the objective, even when the objective itself is a smooth function of those parameters. Inverse
Well-posed_problem
Screening phenomenon in metals
decay in the fermionic density near the perturbation followed by an ongoing sinusoidal decay resembling sinc function. The phenomenon is named after Jacques
Friedel_oscillations
Type of constraint on solutions to differential equations
differential equation. The dependent unknown u in the same form as the weight function w appearing in the boundary expression is termed a primary variable, and
Dirichlet_boundary_condition
Approximation method in quantum physics
multi-electron wave function. One of these approaches, Møller–Plesset perturbation theory, treats correlation as a perturbation of the Fock operator
Hartree–Fock_method
Pictorial representation of the behavior of subatomic particles
delta-function repulsion. Two such particles have an aversion to occupying the same point at the same time. Thinking of Feynman diagrams as a perturbation series
Feynman_diagram
Method in evaluating divergent integrals
large, and can be analytically continued from this region to a meromorphic function defined for all complex d. In general, there will be a pole at the physical
Dimensional_regularization
Divergence in perturbative quantum field theory
\left(\Lambda /Q\right)} . The name was suggested by Gerard 't Hooft in 1977. Perturbation series in quantum field theory are usually divergent as was firstly indicated
Renormalon
Class of numerical techniques
of PDE, along with finite element methods. For a n-times differentiable function, by Taylor's theorem the Taylor series expansion is given as f ( x 0 +
Finite_difference_method
Fundamental theorem in condensed matter physics
{\hat {H}}_{\mathbf {k} +\mathbf {q} }} We can consider the following perturbation problem in q: H ^ k + q = H ^ k + ℏ 2 m q ⋅ ( − i ∇ + k ) + ℏ 2 2 m q
Bloch's_theorem
Part of mathematics that addresses the stability of solutions
small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial
Stability_theory
Describes the range of energies of an electron within the solid
of a region of free space that has been divided into a lattice. k·p perturbation theory is a technique that allows a band structure to be approximately
Electronic_band_structure
Type of ordinary differential equation
differential equation is homogeneous if it is a homogeneous function of the unknown function and its derivatives. In the case of linear differential equations
Homogeneous differential equation
Homogeneous_differential_equation
Equation describing the universe's density contrast
the matter power spectrum is best understood in terms of the linear perturbation theory analysis of the growth of structure, which predicts to first order
Matter_power_spectrum
Materials simulation software
written in Fortran. It uses density functional theory and many-body perturbation theory to simulate chemical and physical properties of atoms, molecules
FHI-aims
Study of mathematical algorithms for optimization problems
solutions. The function f is variously called an objective function, criterion function, loss function, cost function (minimization), utility function or fitness
Mathematical_optimization
Quantum-mechanical framework for simulating molecules and solids
ground-state wave-function so that we can simply use all the properties of DFT. Consider a small time-dependent external perturbation δ V e x t ( t ) {\displaystyle
Time-dependent density functional theory
Time-dependent_density_functional_theory
Possible outcome of renormalization in physics
interpretation. The beta function β ( g ) {\displaystyle \beta (g)} was recently studied by different methods: (1) by summation the usual perturbation series in powers
Quantum_triviality
Process of modeling orbits
precision due to the need to model small perturbations to very large orbits. Because of this, perturbation methods are often used to model the orbit
Orbit_modeling
Integral used in the theory of vibrations
external perturbation. It is named after Jean-Marie Duhamel. It is the convolution of a linear system's impulse response function with the input function. The
Duhamel's_integral
Quartic potential in quantum mechanics
in this reference the perturbation method is developed for the cosine potential, i.e. the Mathieu equation; see Mathieu function. Harald J.W. Müller-Kirsten
Double-well_potential
Method for solving continuous operator problems (such as differential equations)
problem by applying linear constraints determined by finite sets of basis functions. They are named after the Soviet mathematician Boris Galerkin. Often when
Galerkin_method
Type of problem involving ODEs or PDEs
problems to be studied is the Dirichlet problem, of finding the harmonic functions (solutions to Laplace's equation); the solution was given by the Dirichlet's
Boundary_value_problem
Generating function in integrable systems
occupied and unoccupied sites on the integer lattice that are a finite perturbation of the Dirac sea are referred to as a Maya diagram. The case of the null
Tau function (integrable systems)
Tau_function_(integrable_systems)
Optimization method
descent finite-difference SA by Kiefer and Wolfowitz (1952) simultaneous perturbation SA by Spall (1992) scenario optimization On the other hand, even when
Stochastic_optimization
Type of calculus problem
together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or other sciences
Initial_value_problem
from the perturbation growth rate completely. The growth rate thus becomes only a function of the initial radius of the thread, the perturbation wavelength
Fluid_thread_breakup
American mathematician
analysis, perturbation theory, and their applications in aerodynamics and fluid dynamics. Kevorkian co-authored textbooks on multiple scale perturbation methods
Jerry_Kevorkian
Initial estimate or framework to the solution of a mathematical problem
been established, the equations are solved more precisely for the general function of interest, which then constitutes a confirmation of the assumption. In
Ansatz
PERTURBATION FUNCTION
PERTURBATION 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
Celtic
, great justiciary, or functionary.
Male
Egyptian
, an Egyptian functionary.
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.
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.
Biblical
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Male
Egyptian
, a high Egyptian functionary.
Male
Egyptian
, Functionary of the Interior.
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.
Boy/Male
Buddhist, Indian, Japanese
Mysterious Function
Male
Egyptian
, the son of the functionary Heknofre.
Male
Egyptian
, an Egyptian functionary.
Male
Egyptian
, a great functionary.
PERTURBATION FUNCTION
PERTURBATION FUNCTION
Girl/Female
Hindu
Pleasure of the earth, Flower
Female
Spanish
Variant spelling of Spanish Esmeralda, ESMERELDA means "emerald."
Girl/Female
Hindu, Indian
Soft Devotee Stone
Boy/Male
English
Happy; merry.
Girl/Female
Arabic, Australian, Muslim
Guiding Light; Shinning Light
Boy/Male
Tamil
Rich Man
Boy/Male
Assamese, Bengali, Hindu, Indian
Lord of the Earth
Girl/Female
Muslim
Loving nature
Boy/Male
French, German, Italian, Latin
Fidelity; Faithful
Girl/Female
Indian, Punjabi, Sikh
Supreme Longing of Beloved
PERTURBATION FUNCTION
PERTURBATION FUNCTION
PERTURBATION FUNCTION
PERTURBATION FUNCTION
PERTURBATION FUNCTION
n.
A stirring up or arousing; disturbance of tranquillity; disturbance of mind which shows itself by physical excitement; perturbation; as, to cause any one agitation.
n.
The act of disconcerting, or state of being disconcerted; discomposure; perturbation.
a.
Destitute of function, or of an appropriate organ. Darwin.
a.
Tending to cause perturbation; disturbing.
n.
Alt. of Perduration
n.
The act of deturbating.
n.
Disturbance; perturbation.
n.
A perturber.
n.
One charged with the performance of a function or office; as, a public functionary; secular functionaries.
n.
Agitation, perturbation, or disorder, of mind; heat; excitement.
n.
The state of being abashed or disconcerted; loss self-possession; perturbation; shame.
n.
Long continuance.
n.
One who, or that which, perturbs, or cause perturbation.
n.
A disturbance in the regular elliptic or other motion of a heavenly body, produced by some force additional to that which causes its regular motion; as, the perturbations of the planets are caused by their attraction on each other.
n.
Perturbation of mind; mental uneasiness.
a.
Of or pertaining to perturbation, esp. to the perturbations of the planets.
n.
Freedom from agitation of mind; calmness; quietude.
n.
The state of being discomposed; disturbance; disorder; agitation; perturbation.
n.
The act of perturbing, or the state of being perturbed; esp., agitation of mind.
n.
Agitation from violent emotions; perturbation of mind; despair.