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Differential operator in mathematics
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean
Laplace_operator
Operator generalizing the Laplacian in differential geometry
In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean
Laplace–Beltrami_operator
Analog of the continuous Laplace operator
In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a graph or a discrete
Discrete_Laplace_operator
Elliptic differential operators in geometry mathematics
tensors of rank 0), the connection Laplacian is often called the Laplace–Beltrami operator. It is defined as the trace of the second covariant derivative:
Laplace operators in differential geometry
Laplace_operators_in_differential_geometry
Inequality relating to the Laplace operator
Kato's inequality is a distributional inequality for the Laplace operator or certain elliptic operators. It was proven in 1972 by the Japanese mathematician
Kato's_inequality
Type of differential operator
partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that
Elliptic_operator
Class of numerical techniques
The discrete Laplace operator Δ h u {\displaystyle \Delta _{h}u} depends on the dimension n {\displaystyle n} . In 1D the Laplace operator is approximated
Finite_difference_method
Fourth-order PDE in continuum mechanics
\nabla ^{4}} , which is the fourth power of the del operator and the square of the Laplacian operator ∇ 2 {\displaystyle \nabla ^{2}} (or Δ {\displaystyle
Biharmonic_equation
Eigenvalue problem for the Laplace operator
mathematics, the Helmholtz equation is the eigenvalue problem for the Laplace operator. It corresponds to the elliptic partial differential equation: ∇ 2
Helmholtz_equation
Type of discrete calculus
discrete operators on graphs which are analogous to differential operators in calculus, such as graph Laplacians (or discrete Laplace operators) as discrete
Calculus on finite weighted graphs
Calculus_on_finite_weighted_graphs
Nonlocal mathematical operator
mathematical analysis, the fractional Laplacian is an operator that generalizes the notion of the Laplace operator to fractional powers of spatial derivatives.
Fractional_Laplacian
Inequality. (See talk on the article). As a simple example, consider the Laplace operator Δ. More specifically, suppose that one wishes to solve, for f ∈ L2(Ω)
Gårding's_inequality
Exterior algebraic map taking tensors from p forms to n-p forms
be realized as the codifferential opposite to the gradient operator, and the Laplace operator on a function is the divergence of its gradient. An important
Hodge_star_operator
Second-order partial differential equation
=\nabla \cdot \nabla =\nabla ^{2}} is the Laplace operator, ∇ ⋅ {\displaystyle \nabla \cdot } is the divergence operator (also symbolized "div"), ∇ {\displaystyle
Laplace's_equation
Matrix representation of a graph
Named after Pierre-Simon Laplace, the graph Laplacian matrix can be viewed as a matrix form of the negative discrete Laplace operator on a graph approximating
Laplacian_matrix
Partial differential equations
the Green's function (or fundamental solution) for the Laplacian (or Laplace operator) in three variables is used to describe the response of a particular
Green's function for the three-variable Laplace equation
Green's_function_for_the_three-variable_Laplace_equation
Elliptic partial differential operator
p-Laplace operator, is a quasilinear elliptic partial differential operator of 2nd order. It is a nonlinear generalization of the Laplace operator, where
P-Laplacian
First-order differential linear operator on spinor bundle, whose square is the Laplacian
{\displaystyle D} is called a Dirac operator. Note that there are two different conventions as to how the Laplace operator is defined: the "analytic" Laplacian
Dirac_operator
Metric tensor describing constant negative (hyperbolic) curvature
{z}}}}\Phi (z,{\overline {z}})=\lambda ^{2}(z,{\overline {z}}).} The Laplace–Beltrami operator is given by Δ = 4 λ 2 ∂ ∂ z ∂ ∂ z ¯ = 1 λ 2 ( ∂ 2 ∂ x 2 + ∂ 2
Poincaré_metric
Integral transform useful in probability theory, physics, and engineering
In mathematics, the Laplace transform, named after Pierre-Simon Laplace (/ləˈplɑːs/), is an integral transform that converts a function of a real variable
Laplace_transform
French polymath (1749–1827)
Pierre-Simon, Marquis de Laplace (/ləˈplɑːs/; French: [pjɛʁ simɔ̃ laplas]; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has
Pierre-Simon_Laplace
Image edge detection algorithm
Feature extraction Discrete Laplace operator Prewitt operator Irwin Sobel, 2014, History and Definition of the Sobel Operator K. Engel (2006). Real-time
Sobel_operator
Typically linear operator defined in terms of differentiation of functions
differential equation. In applications to the physical sciences, operators such as the Laplace operator play a major role in setting up and solving partial differential
Differential_operator
Topics referred to by the same term
Del squared may refer to: Laplace operator, a differential operator often denoted by the symbol ∇2 Hessian matrix, sometimes denoted by ∇2 Aitken's delta-squared
Del_squared
Second-order differential operator
sometimes quabla operator (cf. nabla symbol) is the Laplace operator of Minkowski space. The operator is named after French mathematician and physicist
D'Alembert_operator
Discrete (i.e., incremental) version of infinitesimal calculus
to define discrete Lie derivative on general polygonal meshes. The Laplace operator Δ f {\displaystyle \Delta f} of a function f {\displaystyle f} at a
Discrete_calculus
Topics referred to by the same term
Search for "laplace" or "la-place" on Wikipedia. Laplace distribution, a probability distribution Laplace operator, a differential operator equal to the
Laplace_(disambiguation)
Vector differential operator
operator component-wise to each component of the vector. The Laplace operator is a scalar operator that can be applied to either vector or scalar fields; for
Del
mathematics, the infinity Laplace (or L ∞ {\displaystyle L^{\infty }} -Laplace) operator is a 2nd-order partial differential operator, commonly abbreviated
Infinity_Laplacian
Complex-valued smooth functions of the upper half plane (harmonic analysis topic)
_{2}(\mathbb {R} )} as modular forms. They are eigenforms of the hyperbolic Laplace operator Δ {\displaystyle \Delta } defined on the upper half plane and satisfy
Maass_wave_form
Equation in differential geometry
^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}} is the flat Laplace operator in two dimensions. This equation governs how to deform flat space to
Liouville's_equation
Quantum operator for the sum of energies of a system
\nabla ^{2}} . In three dimensions using Cartesian coordinates the Laplace operator is ∇ 2 = ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 {\displaystyle \nabla ^{2}={\frac
Hamiltonian (quantum mechanics)
Hamiltonian_(quantum_mechanics)
Laplacian vector field Laplace's equation Laplace operator Discrete Laplace operator Laplace–Beltrami operator Laplacian, see Laplace operator Infinity Laplacian
List of things named after Pierre-Simon Laplace
List_of_things_named_after_Pierre-Simon_Laplace
Finite difference equation
analog of the Poisson equation. In it, the discrete Laplace operator takes the place of the Laplace operator. The discrete Poisson equation is frequently used
Discrete_Poisson_equation
Elliptic partial differential equation
Euclidean space, the Laplace operator is often denoted as ∇2 to denote its second-order nature relative to the classical gradient operator, and so Poisson's
Poisson's_equation
Fundamental solution to the heat equation, given boundary values
is also one of the main tools in the study of the spectrum of the Laplace operator, and is thus of some auxiliary importance throughout mathematical physics
Heat_kernel
Ornstein–Uhlenbeck operator is a generalization of the Laplace operator to an infinite-dimensional setting. The Ornstein–Uhlenbeck operator plays a significant
Ornstein–Uhlenbeck_operator
Mathematical model of wave dynamics
_{\perp }=\partial _{yy}^{2}+\partial _{zz}^{2}} is the transverse Laplace operator, λ 1 {\displaystyle \lambda _{1}} and γ 1 {\displaystyle \gamma _{1}}
Shrira-Pesenson_equation
Method of solution to differential equations
separation of variables, and Laplace transforms. Let L {\displaystyle L} be the general linear second-order differential operator defined on [ a , b ] ∈ R
Green's_function
Discrete differentiation operator used in image processing
figure, imshow(output_image); title('Edge Detected Image'); Sobel operator Laplace operator Roberts Cross Edge detection Feature detection (computer vision)
Prewitt_operator
Stochastic process generalizing Brownian motion
X {\displaystyle X} is one-half the Laplace–Beltrami operator, which is a generalization of the Laplace operator for Riemannian manifolds. The special
Wiener_process
eigenfunctions) of the Laplace–Beltrami operator to compare and analyze geometric shapes. Since the spectrum of the Laplace–Beltrami operator is invariant under
Spectral_shape_analysis
Mathematical potential
potential of order s is the operator ( I − Δ ) − s / 2 {\displaystyle (I-\Delta )^{-s/2}} where Δ is the Laplace operator and the fractional power is
Bessel_potential
Special mathematical functions defined on the surface of a sphere
functions on the sphere which are harmonic with respect to the Laplace-Beltrami operator for the standard round metric on the sphere: the only harmonic
Spherical_harmonics
is Shizuo Kakutani's 1944 solution of the Dirichlet problem for the Laplace operator using Brownian motion. However, it turns out that for a large class
Stochastic processes and boundary value problems
Stochastic_processes_and_boundary_value_problems
Functions in mathematics
functions on U {\displaystyle U} . The Laplace operator Δ {\displaystyle \Delta } and the partial derivative operator will commute on this class of functions
Harmonic_function
Fundamental study of potential theory
continuous mass distribution ρ(r), then ρ can be recovered using the Laplace operator, Δ: ρ ( x ) = 1 4 π G Δ V ( x ) . {\displaystyle \rho (\mathbf {x}
Gravitational_potential
Type of constraint on solutions to differential equations
\nabla ^{2}y+y=0,} where ∇ 2 {\displaystyle \nabla ^{2}} denotes the Laplace operator, the Dirichlet boundary conditions on a domain Ω ⊂ Rn take the form
Dirichlet_boundary_condition
Field in mathematics
differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian manifold has been most intensively studied, although other Laplace operators
Spectral_geometry
Potential in mathematics
power ( − Δ ) − α / 2 {\displaystyle (-\Delta )^{-\alpha /2}} of the Laplace operator on Euclidean space. It generalizes to several variables the Riemann–Liouville
Riesz_potential
Mathematical problem in spectral theory
Ja. (1980), "The second term of the spectral asymptotics for a Laplace–Beltrami operator on manifolds with boundary", Funktsional. Anal. I Prilozhen, 14
Hearing_the_shape_of_a_drum
Type of graph in mathematics and physics
of an open edge. The simplest example of an operator on a metric graph is the Laplace operator. The operator on an edge is − d 2 d x e 2 {\displaystyle
Quantum_graph
Green's function for Laplacian
the Newtonian potential of a function is a partial inverse to the Laplace operator. Then w {\displaystyle w} will be a classical solution, that is twice
Newtonian_potential
Indian mathematician (1913–1968)
paper together called, Some properties of the eigenfunctions of the Laplace-operator on Riemannian manifolds, in which they introduced the Minakshisundaram-Pleijel
Subbaramiah_Minakshisundaram
Nonlinear partial differential equation
\left(u^{m}\right),\quad m>1} where Δ {\displaystyle \Delta } is the Laplace operator. It may also be put into its equivalent divergence form: ∂ u ∂ t =
Porous_medium_equation
grad, 2. A connection, such as the covariant derivative. ∇2 or ∇⋅∇ Laplace operator or Laplacian: ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 + ∂ 2 ∂ z 2 {\displaystyle \textstyle
Glossary of mathematical symbols
Glossary_of_mathematical_symbols
Quantum mechanical operator related to rotational symmetry
}}.\end{aligned}}} In spherical coordinates the angular part of the Laplace operator can be expressed by the angular momentum. This leads to the relation
Angular_momentum_operator
Linear operators with a common spectrum
studied isospectral problem in infinite dimensions is that of the Laplace operator on a domain in R2. Two such domains are called isospectral if their
Isospectral
Mathematical conjecture about eigenvalues
conjectured by Selberg (1965, p. 13), states that the eigenvalues of the Laplace operator on Maass wave forms of congruence subgroups are at least 1/4. Selberg
Selberg's_1/4_conjecture
Type of differential equation
commutativity of partial derivatives. The Greek letter Δ denotes the Laplace operator; if u is a function of n variables, then Δ u = u 11 + u 22 + ⋯ + u
Partial_differential_equation
Partial differential equation describing the evolution of temperature in a region
prototypical example of a parabolic partial differential equation. Using the Laplace operator, the heat equation can be simplified, and generalized to similar equations
Heat_equation
Millennium Prize Problem
_{j}u_{j}} (using the Einstein summation convention) with the inverted Laplace operator Δ − 1 {\displaystyle \Delta ^{-1}} being defined using the Fourier
Navier–Stokes existence and smoothness
Navier–Stokes_existence_and_smoothness
Approach used in computer vision systems
the Laplacian/difference of Gaussian operator, the determinant of the Hessian and the Hessian–Laplace operator. The Wang and Brady detector considers
Corner_detection
Partial differential equation used in physics
in a medium with permeability μ, and permittivity ε, and ∇2 is the Laplace operator. In a vacuum, vph = c0 = 299792458 m/s, a fundamental physical constant
Electromagnetic_wave_equation
{L}}\{f(t)\}+b{\mathcal {L}}\{g(t)\}} and is, therefore, regarded as a linear operator. The Laplace transform of f ( t − a ) u ( t − a ) {\displaystyle f(t-a)u(t-a)}
List_of_Laplace_transforms
Kind of linear transformation
bounded. This operator is in fact a compact operator. The compact operators form an important class of bounded operators. The Laplace operator Δ : H 2 ( R
Bounded_operator
process is a second-order elliptic operator. The infinitesimal generator of Brownian motion is the Laplace operator and the transition probability density
Stochastic analysis on manifolds
Stochastic_analysis_on_manifolds
Topics referred to by the same term
Difference operator (Δ) Dirac delta function (δ function) Increment operator (∆) Kronecker delta ( δ i j {\displaystyle \delta _{ij}} ) Laplace operator (Δ)
Delta
Equation describing the evolution of the vorticity of a fluid particle as it flows
the kinematic viscosity and ∇ 2 {\displaystyle \nabla ^{2}} is the Laplace operator. Under the further assumption of two-dimensional flow, the equation
Vorticity_equation
Equations of motion for viscous fluids
relationship can be expanded upon due to its uniqueness with the vector Laplace operator ∇ 2 u = ∇ ( ∇ ⋅ u ) − ∇ × ( ∇ × u ) {\displaystyle \nabla ^{2}\mathbf
Navier–Stokes_equations
Fourth letter in the Greek alphabet
differentials, which also describe change by infinitesimal amounts.) The Laplace operator: Δ f = ∑ i = 1 n ∂ 2 f ∂ x i 2 {\displaystyle \Delta f=\sum _{i=1}^{n}{\frac
Delta_(letter)
Partial differential equation
{\displaystyle a,b} are constants, Δ {\displaystyle \Delta } is the Laplace operator, ∇ {\displaystyle \nabla } is the gradient, and ‖ ⋅ ‖ {\displaystyle
Cole–Hopf_transformation
Broad concept generalizing scalars in mathematics and physics
vector differential operator represented by the nabla symbol ∇ {\displaystyle \nabla } Vector Laplacian, the vector Laplace operator, denoted by ∇ 2 {\displaystyle
Vector (mathematics and physics)
Vector_(mathematics_and_physics)
Mathematics
y + y = 0 , {\displaystyle \nabla ^{2}y+y=0,} where ∇2 denotes the Laplace operator, the Neumann boundary conditions on a domain Ω ⊂ Rn take the form ∂
Neumann_boundary_condition
Velocity field as the gradient of a scalar function
\varphi } satisfies the Laplace equation ∇ 2 φ = 0 , {\displaystyle \nabla ^{2}\varphi =0\,,} where ∇2 = ∇ ⋅ ∇ is the Laplace operator (sometimes also written
Potential_flow
Function used in quantum chemistry
r}\left(r^{2}{\partial \over \partial r}\right)} The first differential operator of the Laplace operator yields ( r 2 ∂ ∂ r ) R ( r ) = [ ( n − 1 ) r − ζ r 2 ] R (
Slater-type_orbital
Equation known for chaotic behavior
where Δ {\displaystyle \Delta } is the Laplace operator, and Δ 2 {\displaystyle \Delta ^{2}} is the biharmonic operator. The Cauchy problem for the 1d Kuramoto–Sivashinsky
Kuramoto–Sivashinsky_equation
Spectral Geometry Phenomenon
is an inequality concerning the lowest Dirichlet eigenvalue of the Laplace operator on a bounded domain in R n {\displaystyle \mathbb {R} ^{n}} , n ≥ 2
Rayleigh–Faber–Krahn inequality
Rayleigh–Faber–Krahn_inequality
Research topic in computational geometry
using the Laplace operator, geometric smoothing might be achieved by convolving a surface geometry with a blur kernel formed using the Laplace-Beltrami
Geometry_processing
Type of vector space in math
physically meaningful eigenfunctions of a differential operator (typically the Laplace operator): this forms the foundation for the spectral study of functions
Hilbert_space
Swiss mathematician
Laplaceoperators auf kompakten Flächen (Studies on the first eigenvalue of the Laplace operator on compact surfaces). As a post-doctoral student he was at the University
Jürg_Peter_Buser
Differential equation for the description of waves or standing wave
to time, Δ {\displaystyle \Delta } is the Laplace operator and ◻ {\displaystyle \Box } the d'Alembert operator, defined as: u t t = ∂ 2 u ∂ t 2 , Δ = ∂
Wave_equation
invariant of a compact Riemannian manifold related to the spectrum of the Laplace operator acting on square-integrable differential forms on its universal cover
Novikov–Shubin_invariant
Property of functions which is weaker than continuity
used in the Perron method for solving the Dirichlet problem for the Laplace operator in a domain. The key condition for the class of subharmonic solutions
Semi-continuity
Mathematical function of a linear operator
In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f {\displaystyle f} in that space that
Eigenfunction
approximations to derivatives Discrete Laplace operator — finite-difference approximation of the Laplace operator Eigenvalues and eigenvectors of the second
List of numerical analysis topics
List_of_numerical_analysis_topics
Concepts from linear algebra
(increasingly) of the graph's Laplacian matrix due to its discrete Laplace operator, which is either D − A (sometimes called the combinatorial Laplacian)
Eigenvalues_and_eigenvectors
space) for an open and connected domain Ω ⊆ Rn and let A = Δ, the Laplace operator, defined on the dense subspace of compactly supported smooth functions
Dissipative_operator
Mathematical equation
\mathbb {R} ^{n}} , and let Δ {\displaystyle \Delta } denote the usual Laplace operator. Weyl's lemma states that if a locally integrable function u ∈ L l
Weyl's lemma (Laplace equation)
Weyl's_lemma_(Laplace_equation)
Mathematical gradient operator in certain coordinate systems
Academic Press. p. 192. ISBN 9789381269558. Weisstein, Eric W. "Convective Operator". Mathworld. Retrieved 23 March 2011. Fernández-Guasti, M. (2012). "Green's
Del in cylindrical and spherical coordinates
Del_in_cylindrical_and_spherical_coordinates
Function defined on an inner product space
in X. This also applies for an operator that acts on such functions. An example is the two-dimensional Laplace operator ∇ 2 = ∂ 2 ∂ x 2 + ∂ 2 ∂ y 2 , {\displaystyle
Rotational_invariance
Branch of mathematical analysis
={\frac {\partial ^{2}}{\partial \mathbf {r} ^{2}}}} is the Laplace operator and the operator (−ħ2Δ)β(t)/2 is the variable-order fractional quantum Riesz
Fractional_calculus
Functional analysis theorem
{\displaystyle \partial _{t}u(t,x)=\Delta u(t,x),} where Δ denotes the Laplace operator. Two questions arise immediately: on what domain in spacetime is the
Lions–Lax–Milgram_theorem
Mathematical function
modular group, being an eigenfunction of the corresponding hyperbolic Laplace operator, and having at most linear exponential growth at the cusps. If the
Harmonic_Maass_form
Vector calculus formulas relating the bulk with the boundary of a region
{\displaystyle \Delta f=\nabla ^{2}f=\nabla \cdot \nabla f} is the Laplace operator, ∂U is the boundary of region U, n is the outward pointing unit normal
Green's_identities
Doughnut-shaped surface of revolution
operators of vector calculus can be calculated using the same parametrization to obtain their ring toroidal forms. For example, the Laplace operator for
Torus
Particular task in computer vision
the Laplacian/Difference of Gaussian operator, the determinant of the Hessian and the Hessian-Laplace operator (see also Harris-Affine and Hessian-Affine)
Blob_detection
Equation in electrochemistry
cylindrical, and rectangular geometries by using the corresponding Laplace operator and boundary conditions in conjunction with Fick's second law of diffusion
Cottrell_equation
Random motion of particles suspended in a fluid
generator (and hence characteristic operator) of Brownian motion on Euclidean Rn is 1/2Δ, where Δ denotes the Laplace operator. Brownian motion on an m-dimensional
Brownian_motion
American mathematician (1953–2022)
He has done research on the spectral and scattering theory of the Laplace operator on Riemannian manifolds and especially the asymptotic and distribution
Steven_Zelditch
travel, tourism, insurance
LAPLACE OPERATOR
LAPLACE OPERATOR
LAPLACE OPERATOR
LAPLACE OPERATOR
LAPLACE OPERATOR
LAPLACE OPERATOR
LAPLACE OPERATOR
LAPLACE OPERATOR
LAPLACE OPERATOR
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