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The continuous -Laplace operator is a second-order differential operator that can be well-translated to finite weighted graphs. It allows the translation of various partial differential equations, e.g., the heat equation, to the graph setting.
The Hodge Laplacian, also known as the Laplace–de Rham operator, is a differential operator acting on differential forms. (Abstractly, it is a second order operator on each exterior power of the cotangent bundle.) This operator is defined on any manifold equipped with a Riemannian- or pseudo-Riemannian metric.
As a second-order differential operator, the Laplace operator maps C k functions to C k−2 functions for k ≥ 2.It is a linear operator Δ : C k (R n) → C k−2 (R n), or more generally, an operator Δ : C k (Ω) → C k−2 (Ω) for any open set Ω ⊆ R n.
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties.This is often written as = or =, where = = is the Laplace operator, [note 1] is the divergence operator (also symbolized "div"), is the gradient operator (also symbolized "grad"), and (,,) is a twice-differentiable real-valued function.
Discrete Laplace operator is often used in image processing e.g. in edge detection and motion estimation applications. [4] The discrete Laplacian is defined as the sum of the second derivatives and calculated as sum of differences over the nearest neighbours of the central pixel. Since derivative filters are often sensitive to noise in an image ...
The second smallest eigenvalue of L (could be zero) is the algebraic connectivity (or Fiedler value) of G and approximates the sparsest cut of a graph. The Laplacian is an operator on the n-dimensional vector space of functions :, where is the vertex set of G, and = | |.
The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
Verbally, the second version is the second derivative in the direction of the gradient. In the case of the infinity Laplace equation Δ ∞ u = 0 {\displaystyle \Delta _{\infty }u=0} , the two definitions are equivalent.