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27 relations: Boundary (topology), Compact space, Connected space, Constant function, Convex function, Convex optimization, Convex set, Critical point (mathematics), Domain (mathematical analysis), Elliptic partial differential equation, Euclidean space, Harmonic function, Hopf lemma, Hopf maximum principle, Laplace operator, Mathematics, Maxima and minima, Maximum modulus principle, Neighbourhood (mathematics), Open set, Parabolic partial differential equation, Partial differential equation, Pontryagin's maximum principle, R. Tyrrell Rockafellar, Saddle point, Subharmonic function, Subset.
Boundary (topology)
In topology and mathematics in general, the boundary of a subset S of a topological space X is the set of points which can be approached both from S and from the outside of S. More precisely, it is the set of points in the closure of S not belonging to the interior of S. An element of the boundary of S is called a boundary point of S. The term boundary operation refers to finding or taking the boundary of a set.
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Compact space
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
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Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.
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Constant function
In mathematics, a constant function is a function whose (output) value is the same for every input value.
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Convex function
In mathematics, a real-valued function defined on an ''n''-dimensional interval is called convex (or convex downward or concave upward) if the line segment between any two points on the graph of the function lies above or on the graph, in a Euclidean space (or more generally a vector space) of at least two dimensions.
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Convex optimization
Convex optimization is a subfield of optimization that studies the problem of minimizing convex functions over convex sets.
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Convex set
In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations.
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Critical point (mathematics)
In mathematics, a critical point or stationary point of a differentiable function of a real or complex variable is any value in its domain where its derivative is 0.
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Domain (mathematical analysis)
In mathematical analysis, a domain is any connected open subset of a finite-dimensional vector space.
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Elliptic partial differential equation
Second order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic.
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Euclidean space
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
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Harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: U → R where U is an open subset of Rn that satisfies Laplace's equation, i.e. everywhere on U. This is usually written as or.
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Hopf lemma
In mathematics, the Hopf lemma, named after Eberhard Hopf, states that if a continuous real-valued function in a domain in Euclidean space with sufficiently smooth boundary is harmonic in the interior and the value of the function at a point on the boundary is greater than the values at nearby points inside the domain, then the derivative of the function in the direction of the outward pointing normal is strictly positive.
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Hopf maximum principle
The Hopf maximum principle is a maximum principle in the theory of second order elliptic partial differential equations and has been described as the "classic and bedrock result" of that theory.
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Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Maxima and minima
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema).
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Maximum modulus principle
In mathematics, the maximum modulus principle in complex analysis states that if f is a holomorphic function, then the modulus |f | cannot exhibit a true local maximum that is properly within the domain of f. In other words, either f is a constant function, or, for any point z0 inside the domain of f there exist other points arbitrarily close to z0 at which |f | takes larger values.
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Neighbourhood (mathematics)
In topology and related areas of mathematics, a neighbourhood (or neighborhood) is one of the basic concepts in a topological space.
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Open set
In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.
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Parabolic partial differential equation
A parabolic partial differential equation is a type of partial differential equation (PDE).
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Partial differential equation
In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.
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Pontryagin's maximum principle
Pontryagin's maximum (or minimum) principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls.
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R. Tyrrell Rockafellar
Ralph Tyrrell Rockafellar (born February 10, 1935) is an American mathematician and one of the leading scholars in optimization theory and related fields of analysis and combinatorics.
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Saddle point
In mathematics, a saddle point or minimax point is a point on the surface of the graph of a function where the slopes (derivatives) of orthogonal function components defining the surface become zero (a stationary point) but are not a local extremum on both axes.
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Subharmonic function
In mathematics, subharmonic and superharmonic functions are important classes of functions used extensively in partial differential equations, complex analysis and potential theory.
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Subset
In mathematics, a set A is a subset of a set B, or equivalently B is a superset of A, if A is "contained" inside B, that is, all elements of A are also elements of B. A and B may coincide.
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Maximum Principle, Maximum principal, Maximum principle for harmonic functions, Strong maximum principle, Weak maximum principle.
References
[1] https://en.wikipedia.org/wiki/Maximum_principle
