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Topics referred to by the same term
algebraic groups over global fields Weak weak approximation, a form of weak approximation for varieties Weak-field approximation, a solution in general relativity
Weak_approximation
Hilbertian. The WWA property (weak 'weak approximation', sic) for a variety V over a number field is weak approximation (cf. approximation in algebraic groups)
Thin_set_(Serre)
in weak approximation is whether the embedding of G(k) in G(AS) has dense image. If the group G is connected and k-rational, then it satisfies weak approximation
Approximation in algebraic groups
Approximation_in_algebraic_groups
Meteorological concept
In atmospheric science, the weak temperature gradient approximation (WTG) is a theoretical framework used to simplify the equations governing tropical
Weak temperature gradient approximation
Weak_temperature_gradient_approximation
Linear perturbations to solutions of nonlinear Einstein field equations
of these that preserve the small scale that is required by the weak-field approximation. One may thus define ϕ {\displaystyle \phi } to denote an arbitrary
Linearized_gravity
Approximation for factorials
mathematics, Stirling's approximation (or Stirling's formula) is an asymptotic approximation for factorials. It is a good approximation, leading to accurate
Stirling's_approximation
Problem of finding the longest simple path for a given graph
understanding its approximation hardness". The best polynomial time approximation algorithm known for this case achieves only a very weak approximation ratio, n
Longest_path_problem
Mathematical concept
The approximation error in a given data value represents the significant discrepancy that arises when an exact, true value is compared against some approximation
Approximation_error
Mathematical theorem in the study of analysis
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly
Stone–Weierstrass_theorem
Concept in number theory
provides a unified interpretation of approximation theorems and local-global questions. The weak approximation theorem says that, for finitely many inequivalent
Adele_ring
Problem in number theory
Heath-Brown, D. R. (1992), "The density of zeros of forms for which weak approximation fails", Mathematics of Computation, 59 (200): 613–623, Bibcode:1992MaCom
Sums_of_three_cubes
Concept in optimization
only holds in certain cases. Many primal-dual approximation algorithms are based on the principle of weak duality. Consider a linear programming problem
Weak_duality
1995 video game
gamer's shelf." They also criticized the Genesis version as being a weak approximation of the arcade version, particularly the character sprites and sound
Mortal_Kombat_3
Rational-number approximation of a real number
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus
Diophantine_approximation
Averages of repeated trials converge to the expected value
numerical results. The larger the number of repetitions, the better the approximation tends to be. The reason that this method is important is mainly that
Law_of_large_numbers
In algebraic topology, the cellular approximation theorem states that a map between CW-complexes can always be taken to be of a specific type. Concretely
Cellular approximation theorem
Cellular_approximation_theorem
Family of iterative methods
Stochastic approximation methods are a family of iterative methods typically used for root-finding problems or for optimization problems. The recursive
Stochastic_approximation
Russian mathematician (1937–2023)
method. In 1978, Milstein introduced weak-sense approximations of SDEs for the first time and proposed a number of weak schemes. These papers became classics
Grigori_Milstein
Mathematician
Brauer-Manin obstruction is the only obstruction to the Hasse principle and weak approximation for homogeneous spaces of connected linear algebraic groups over number
Mikhail_Borovoi
Computational property
polynomially bounded objective function cannot have a fully polynomial-time approximation scheme (or FPTAS) unless P = NP. However, the converse fails: e.g. if
Strong_NP-completeness
Commutative ring with a well behaved theory of prime factorization
pairwise independent. As a consequence, there holds the so-called weak approximation theorem, an homologue of the Chinese remainder theorem: if p 1 , …
Krull_ring
Approximation valid for weakly non-linear and fairly long waves
the Boussinesq approximation for water waves is an approximation valid for weakly non-linear and fairly long waves. The approximation is named after Joseph
Boussinesq approximation (water waves)
Boussinesq_approximation_(water_waves)
Model used in atom optics and magnetic resonance
The rotating-wave approximation is an approximation used in atom optics and magnetic resonance. In this approximation, terms in a Hamiltonian that oscillate
Rotating-wave_approximation
Exercise in general relativity
are fully determined by some variant of this approach; from the weak-field approximation one arrives at the result: g 44 = K ( 1 + 1 S r ) ≈ − c 2 + 2 G
Derivation of the Schwarzschild solution
Derivation_of_the_Schwarzschild_solution
Statistical estimation method
an order- β {\displaystyle \beta } weak approximation to x {\displaystyle \mathbf {x} } satisfying (10) and the weak convergence condition sup t k ≤ t
Innovation_method
finding a sequence of weak approximation of maps in W 1 , p ( M , N ) {\displaystyle W^{1,p}(M,N)} is equivalent to the strong approximation when p {\displaystyle
Sobolev_mapping
Method for solving continuous operator problems (such as differential equations)
and approximation methods used: Ritz–Galerkin method (after Walther Ritz) typically assumes symmetric and positive-definite bilinear form in the weak formulation
Galerkin_method
Traditional form of Malagasy oral literature and poetry
an exemplar of the new wave of Malagasy poetry. "Proverb" offers a weak approximation of the meaning of ohabolana, which constitute no less than concise
Hainteny
Estimate of time taken for running an algorithm
arise in approximation algorithms; a famous example is the directed Steiner tree problem, for which there is a quasi-polynomial time approximation algorithm
Time_complexity
Gravitational deflection of light
In astronomy, weak gravitational lensing is a technique to map the mass distribution of astronomical objects. As predicted by general relativity, the presence
Weak_gravitational_lensing
Unsolved problem in computational complexity theory
unique game. It has broad applications in the theory of hardness of approximation. If the UGC is true and P ≠ NP, then for many important problems there
Unique_games_conjecture
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)
Physical model of solid metals as electron gases
cores can be modeled through the use of a "weak" perturbing potential. This may seem like a severe approximation, for the Coulomb attraction between these
Nearly_free_electron_model
Computational quantum mechanical modelling method to investigate electronic structure
hard spheres and a term in a weak interaction such as the long range London dispersion force. In a local density approximation the local excess free energy
Density_functional_theory
Paradigm in machine learning
Weak supervision (also known as semi-supervised learning) is a paradigm in machine learning, the relevance and notability of which increased with the advent
Weak_supervision
Near-field diffraction
the Fresnel diffraction equation for near-field diffraction is an approximation of the Kirchhoff–Fresnel diffraction that can be applied to the propagation
Fresnel_diffraction
In probability theory, the Komlós–Major–Tusnády approximation (also known as the KMT approximation, the KMT embedding, or the Hungarian embedding) refers
Komlós–Major–Tusnády approximation
Komlós–Major–Tusnády_approximation
Concept in physics
image points of infinite amplification. In weak lensing by large-scale structure, the thin-lens approximation may break down, and low-density extended structures
Gravitational lensing formalism
Gravitational_lensing_formalism
Mathematical concept
sequence might represent an attempt to construct 'better and better' approximations to a desired measure μ that is difficult to obtain directly. The meaning
Convergence_of_measures
Normed vector space that is complete
space X satisfies the bounded approximation property. The first example by Enflo of a space failing the approximation property was at the same time the
Banach_space
Method of approximation in general relativity
law of universal gravitation. This allows approximations to Einstein's equations to be made in the case of weak fields. Higher-order terms can be added
Post-Newtonian_expansion
Theoretical electronic band structure model in which the potential is periodic and weak
The empty lattice approximation is a theoretical electronic band structure model in which the potential is periodic and weak (close to constant). One may
Empty_lattice_approximation
2009 studio album by Handsome Furs
sludgy rock and tinsel-thin beats of 'Evangeline' to the couple's weak approximation of The Knife on '(White City)' to the half-assed 'Dancing In The Dark'
Face_Control
NP-hard problem in combinatorial optimization
optimal. It was one of the first approximation algorithms, and was in part responsible for drawing attention to approximation algorithms as a practical approach
Travelling_salesman_problem
Machine learning technique
real-valued y. It seeks an approximation F̂(x) in the form of a weighted sum of M functions hm(x) from some class ℋ, called base (or weak) learners: F̂(x)=∑m=1Mγmhm(x)+const
Gradient_boosting
Indicator function of positive numbers
variance can serve as an approximation, in the limit as the variance approaches zero. For example, all three of the above approximations are cumulative distribution
Heaviside_step_function
Mathematical formalism used in quantum field theory
The random phase approximation (RPA) is an approximation method in condensed matter physics and nuclear physics. It was first introduced by David Bohm
Random_phase_approximation
Most basic type of physical force
four fundamental interactions known to exist: gravity, electromagnetism, weak interaction, and strong interaction. The gravitational and electromagnetic
Fundamental_interaction
Type of algorithm
A parameterized approximation algorithm is a type of algorithm that aims to find approximate solutions to NP-hard optimization problems in polynomial
Parameterized approximation algorithm
Parameterized_approximation_algorithm
Problem in combinatorial optimization
algorithm using dynamic programming. There is a fully polynomial-time approximation scheme, which uses the pseudo-polynomial time algorithm as a subroutine
Knapsack_problem
Consonant sound change
alters consonants, making them "weaker" in some way. The word lenition means 'softening' or 'weakening' (from Latin lēnis 'weak'). Lenition can happen both
Lenition
Swedish mathematician
Szepessy, Anders; Tempone, Raúl; Zouraris, Georgios E. (2001). "Adaptive weak approximation of stochastic differential equations". Communications on Pure and
Anders_Szepessy
Computer-aided geometric design
In mathematics, the progressive-iterative approximation method is an iterative method of data fitting with geometric meanings. Given a set of data points
Progressive-iterative approximation method
Progressive-iterative_approximation_method
NP-complete problem in computer science
are not sorted, then the runtime is O(n) and the approximation ratio is at most 3/2 ("approximation ratio" means the larger sum in the algorithm output
Partition_problem
Matrix decomposition
polar decomposition, and is a common means for implementing low-rank approximation for matrices. Specifically, the singular value decomposition of an m×n
Singular_value_decomposition
Theory of forces and subatomic particles
describing three of the four known fundamental forces (electromagnetic, weak and strong interactions – excluding gravity) in the universe and classifying
Standard_Model
Darker version of the flag using RGB approximations of semiofficial Pantone approximations, and also the official 1997–2011 color scheme.
Flag_of_South_Korea
Fundamental theorem in probability theory and statistics
version of this theorem, that the normal distribution may be used as an approximation to the binomial distribution, is the de Moivre–Laplace theorem. Let
Central_limit_theorem
Numerical integration method
left and right Riemann sums and is sometimes defined this way. The approximation becomes more accurate as the resolution of the partition increases (that
Trapezoidal_rule
computational complexity, an NP-complete (or NP-hard) problem is weakly NP-complete (or weakly NP-hard) if there is an algorithm for the problem whose running
Weak_NP-completeness
Base that does not associate completely with hydrogen ions
A weak base is a base that does not accept all the available H+ ions, and forms an equilibrium between the base and its conjugate acid. Upon dissolution
Weak_base
Branch of mathematics
mathematics that studies functions, spaces, and operators through methods of approximation and convergence. It grew out of calculus, especially the use of derivatives
Mathematical_analysis
Chemical compound involving ionic bonding
structural information, the Kapustinskii equation. Using an even simpler approximation of the ions as impenetrable hard spheres, the arrangement of anions
Salt_(chemistry)
Integration kernels for smoothing out sharp features
In mathematics, mollifiers (also known as approximations to the identity) are particular smooth functions, used for example in distribution theory to create
Mollifier
Computational geometry and optimization concept
between: Strong coresets: The approximation guarantee holds uniformly for all candidate solutions x {\displaystyle x} . Weak coresets: The guarantee holds
Coreset
Solution concept of a non-cooperative game
A Beautiful Mind, Simon & Schuster. Aviad Rubinstein: "Hardness of Approximation Between P and NP", ACM, ISBN 978-1-947487-23-9 (May 2019), DOI: https://doi
Nash_equilibrium
general the first few terms of the Born series are good approximation to the expanded quantity for "weak" interaction V {\displaystyle V} and large collision
Born_series
Decomposition of periodic functions
norm or weak convergence is usually studied. Four partial sums (Fourier series) of lengths 1, 2, 3, and 4 terms, showing how the approximation to a square
Fourier_series
Design technique
graphical perspective projections, but there is attendant distortion in the approximation. Because pictorial projections innately contain this distortion, in
3D_projection
Approximations in density functional theory
Local-density approximations (LDA) are a class of approximations to the exchange–correlation (XC) energy functional in density functional theory (DFT)
Local-density_approximation
Quantity in quantum mechanics
In quantum mechanics (and computation), a weak value is a quantity related to a shift of a measuring device's pointer when usually there is pre- and postselection
Weak_value
Mathematical problem
viz. monotone polygons and polygons weakly visible from an edge. Krohn & Nilsson (2013) presented an approximation algorithm that computes in polynomial
Art_gallery_problem
Partition of a graph's nodes into cliques
there can be no polynomial time approximation algorithm for any ε > 0 that, on n-vertex graphs, achieves an approximation ratio better than n1 − ε. In graphs
Clique_cover
Markovian master equation of a quantum system weakly coupled to its environment
connection to the Lindblad master equation. If a so-called secular approximation is performed, where only certain resonant interactions with the environment
Redfield_equation
complex numbers Gamma function: Lanczos approximation Spouge's approximation — modification of Stirling's approximation; easier to apply than Lanczos AGM method
List of numerical analysis topics
List_of_numerical_analysis_topics
Approximation in many-body systems
The GW approximation is a method used to calculate the self-energy of a many-body system of electrons. The approximation is that the expansion of the
GW_approximation
Classical problem in combinatorics
the development of fundamental techniques for the entire field" of approximation algorithms. In the weighted set cover problem, each set is assigned
Set_cover_problem
Species of elementary particle
non-dynamical, discrete quantum numbers. In particular, the action of the weak force is such that it allows the conversion of quantum numbers describing
Flavour_(particle_physics)
Type of weak interaction in nuclear and atomic physics
nuclear physics and atomic physics, weak charge, or rarely neutral weak charge, refers to the Standard Model weak interaction coupling of a particle to
Weak_charge
Method in Itô calculus
solve this SDE on some interval of time [0, T]. Then the Euler–Maruyama approximation to the true solution X is the Markov chain Y defined as follows: Partition
Euler–Maruyama_method
Numerical method for solving physical or engineering problems
equations are often partial differential equations (PDEs). To explain the approximation of this process, FEM is commonly introduced as a special case of the
Finite_element_method
Quantum chemistry rule regarding vibronic transitions
of a series of approximations, principally the electrical dipole transition assumption and the Born–Oppenheimer approximation. Weaker magnetic dipole
Franck–Condon_principle
Task of computing complete subgraphs
maximum. Although the approximation ratio of this algorithm is weak, it is the best known to date. The results on hardness of approximation described below
Clique_problem
Even integers as sums of two primes
20 June 2025. Pintz, J.; Ruzsa, I. Z. (1 August 2020). "On Linnik's approximation to Goldbach's problem. II". Acta Mathematica Hungarica. 161 (2): 569–582
Goldbach's_conjecture
Instantaneous rate of change (mathematics)
of the function at that point. The tangent line is the best linear approximation of the function near that input value. The derivative is often described
Derivative
Weakly bound complex of atoms or molecules
adiabatic approximation or Born–Oppenheimer approximation has been adapted to separate angular and radial motions. Despite the challenges posed by the weak interactions
Van_der_Waals_molecule
Concept in topology
weak or strong topology. A basic approximation theorem says that 𝒞Ws(M,N) is dense in 𝒞Sr(M,N) for 1≤s≤∞,0≤r<s. See also: Grauert's approximation theorem
Mapping_space
Function that quantifies how near a number is to being rational
q>0} that satisfy the inequality. For example, whenever a rational approximation p q ≈ x {\displaystyle {\frac {p}{q}}\approx x} with p , q ∈ N {\displaystyle
Irrationality_measure
Generalized function whose value is zero everywhere except at zero
functions f, and so ηε converges weakly to δ in the sense of measures. The ηε constructed in this way are known as an approximation to the identity. This terminology
Dirac_delta_function
Class of numerical techniques
n and the original function. Following is the process to derive an approximation for the first derivative of the function f by first truncating the Taylor
Finite_difference_method
Gradient boosting machine learning library
works as gradient descent in function space, a second order Taylor approximation is used in the loss function to make the connection to Newton–Raphson
XGBoost
Concept in complexity theory
pseudo-polynomial time algorithm. It also does not have a fully-polynomial time approximation scheme. An example is the 3-partition problem. Both strong NP-hardness
Pseudo-polynomial_time
Type of artificial neural network
training image depicts a ring-textured sea urchin, the model also develops a weak association between ringed texture and sea urchin. Subsequent run of the
Feedforward_neural_network
Type of equation with integer coefficients
integer r {\displaystyle r} on the right hand side of the equation. This is a weaker form of a conjecture of Stewart, and is a special case of the uniform boundedness
Thue_equation
reasonably good approximation since in normal gases, Pradntl number is approximately equal to 0.72 {\displaystyle 0.72} . With this approximation and integrating
Becker–Morduchow–Libby solution
Becker–Morduchow–Libby_solution
Electron microscopy technique
{\displaystyle \xi _{eff}={\tfrac {1}{s_{eff}}}} which comes from the weak beam approximation. When the extinction distance is larger, it increases the fringe
Weak-beam dark-field microscopy
Weak-beam_dark-field_microscopy
Unitary matrix containing information on the weak interaction
weak interaction. Technically, it specifies the mismatch of quantum states of quarks when they propagate freely and when they take part in the weak interactions
Cabibbo–Kobayashi–Maskawa matrix
Cabibbo–Kobayashi–Maskawa_matrix
Class of stochastic process
\end{aligned}}} So { z t } {\displaystyle \{z_{t}\}} is a white noise in the weak sense (the mean and cross-covariances are zero, and the variances are all
Stationary_process
Type of radioactive decay
consequence of the weak force, which is characterized by relatively long decay times. Nucleons are composed of up quarks and down quarks, and the weak force allows
Beta_decay
Model of electrically conducting fluids
closure approximation must be applied to the highest moment of the particle distribution equation. This is often accomplished through approximations to the
Magnetohydrodynamics
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