|
Harmonic Measure
In mathematics, especially potential theory, harmonic measure is a concept related to the theory of harmonic functions that arises from the solution of the classical Dirichlet problem. In probability theory, the harmonic measure of a subset of the boundary of a bounded domain in Euclidean space R^n, n\geq 2 is the probability that a Brownian motion started inside a domain hits that subset of the boundary. More generally, harmonic measure of an Itō diffusion ''X'' describes the distribution of ''X'' as it hits the boundary of ''D''. In the complex plane, harmonic measure can be used to estimate the modulus of an analytic function inside a domain ''D'' given bounds on the modulus on the boundary of the domain; a special case of this principle is Hadamard's three-circle theorem. On simply connected planar domains, there is a close connection between harmonic measure and the theory of conformal maps. The term ''harmonic measure'' was introduced by Rolf Nevanlinna in 1928 for plan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Open Set
In mathematics, an open set is a generalization of an Interval (mathematics)#Definitions_and_terminology, open interval in the real line. In a metric space (a Set (mathematics), set with a metric (mathematics), distance defined between every two points), an open set is a set that, with every point in it, contains all points of the metric space that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, an open set is a member of a given Set (mathematics), collection of Subset, subsets of a given set, a collection that has the property of containing every union (set theory), union of its members, every finite intersection (set theory), intersection of its members, the empty set, and the whole set itself. A set in which such a collection is given is called a topological space, and the collection is called a topology (structure), topology. These conditions are very loose, and allow enormous flexibility in the choice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff Measure
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in \R^n or, more generally, in any metric space. The zero-dimensional Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite. Likewise, the one-dimensional Hausdorff measure of a simple curve in \R^n is equal to the length of the curve, and the two-dimensional Hausdorff measure of a Lebesgue measure#Construction of the Lebesgue measure, Lebesgue-measurable subset of \R^2 is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes the Lebesgue measure and its notions of counting, length, and area. It also generalizes volume. In fact, there are ''d''-dimensional Hausdorff measures for any ''d'' ≥ 0, which is n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Kernel
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson. Poisson kernels commonly find applications in control theory and two-dimensional problems in electrostatics. In practice, the definition of Poisson kernels are often extended to ''n''-dimensional problems. Two-dimensional Poisson kernels On the unit disc In the complex plane, the Poisson kernel for the unit disc is given by P_r(\theta) = \sum_^\infty r^e^ = \frac = \operatorname\left(\frac\right), \ \ \ 0 \le r < 1. This can be thought of in two ways: either as a function of ''r'' and ''θ'', or as a family of functions of ''θ'' indexed by ''r''. If is the open unit disc in C, T is the boundary of the disc, and ''f'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radon–Nikodym Theorem
In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A ''measure'' is a set function that assigns a consistent magnitude to the measurable subsets of a measurable space. Examples of a measure include area and volume, where the subsets are sets of points; or the probability of an event, which is a subset of possible outcomes within a wider probability space. One way to derive a new measure from one already given is to assign a density to each point of the space, then Lebesgue integration, integrate over the measurable subset of interest. This can be expressed as :\nu(A) = \int_A f \, d\mu, where is the new measure being defined for any measurable subset and the function is the density at a given point. The integral is with respect to an existing measure , which may often be the canonical Lebesgue measure on the real line or the ''n''-dimensional Euclidean space (corr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lipschitz Domain
In mathematics, a Lipschitz domain (or domain with Lipschitz boundary) is a domain in Euclidean space whose boundary is "sufficiently regular" in the sense that it can be thought of as locally being the graph of a Lipschitz continuous function. The term is named after the German mathematician Rudolf Lipschitz. Definition Let n \in \mathbb N. Let \Omega be a domain of \mathbb R^n and let \partial\Omega denote the boundary of \Omega. Then \Omega is called a Lipschitz domain if for every point p \in \partial\Omega there exists a hyperplane H of dimension n-1 through p, a Lipschitz-continuous function g : H \rightarrow \mathbb R over that hyperplane, and reals r > 0 and h > 0 such that * \Omega \cap C = \left\ * (\partial\Omega) \cap C = \left\ where :\vec is one of the two unit vectors that are normal to H, :B_ (p) := \ is the open ball of radius r, :C := \left\. In other words, at each point of its boundary, \Omega is locally the set of points located above the graph of so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Measure
In mathematics, two positive (or signed or complex) measures \mu and \nu defined on a measurable space (\Omega, \Sigma) are called singular if there exist two disjoint measurable sets A, B \in \Sigma whose union is \Omega such that \mu is zero on all measurable subsets of B while \nu is zero on all measurable subsets of A. This is denoted by \mu \perp \nu. A refined form of Lebesgue's decomposition theorem decomposes a singular measure into a singular continuous measure and a discrete measure. See below for examples. Examples on R''n'' As a particular case, a measure defined on the Euclidean space \R^n is called ''singular'', if it is singular with respect to the Lebesgue measure on this space. For example, the Dirac delta function is a singular measure. Example. A discrete measure. The Heaviside step function on the real line, H(x) \ \stackrel \begin 0, & x 0 but \delta_0(U) = 0. Example. A singular continuous measure. The Cantor distribution has a cumulative dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arc Length
Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the most basic formulation of arc length for a vector valued curve (thought of as the trajectory of a particle), the arc length is obtained by integrating speed, the magnitude of the velocity vector over the curve with respect to time. Thus the length of a continuously differentiable curve (x(t),y(t)), for a\le t\le b, in the Euclidean plane is given as the integral L = \int_a^b \sqrt\,dt, (because \sqrt is the magnitude of the velocity vector (x'(t),y'(t)), i.e., the particle's speed). The defining integral of arc length does not always have a closed-form expression, and numerical integration may be used instead to obtain numerical values of arc length. Determining the length of an irregular arc segment by approximating the arc segment as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harnack's Inequality
In mathematics, Harnack's inequality is an inequality relating the values of a positive harmonic function at two points, introduced by . Harnack's inequality is used to prove Harnack's theorem about the convergence of sequences of harmonic functions. , and generalized Harnack's inequality to solutions of elliptic or parabolic partial differential equations. Such results can be used to show the interior Hölder condition, regularity of weak solutions. Grigori Perelman, Perelman's solution of the Poincaré conjecture uses a version of the Harnack inequality, found by , for the Ricci flow. The statement Harnack's inequality applies to a non-negative function ''f'' defined on a closed ball in R''n'' with radius ''R'' and centre ''x''0. It states that, if ''f'' is continuous on the closed ball and harmonic function, harmonic on its interior, then for every point ''x'' with , ''x'' − ''x''0, = ''r'' 0 (depending only on ''K'', \tau, t-\tau, and the c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indicator Function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator function of is the function \mathbf_A defined by \mathbf_\!(x) = 1 if x \in A, and \mathbf_\!(x) = 0 otherwise. Other common notations are and \chi_A. The indicator function of is the Iverson bracket of the property of belonging to ; that is, \mathbf_(x) = \left x\in A\ \right For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers. Definition Given an arbitrary set , the indicator function of a subset of is the function \mathbf_A \colon X \mapsto \ defined by \operatorname\mathbf_A\!( x ) = \begin 1 & \text x \in A \\ 0 & \text x \notin A \,. \end The Iverson bracket provides the equivalent notation \left x\in A\ \right/math> or that can be used instead of \mathbf_\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a σ-algebra that satisfies Measure (mathematics), measure properties such as ''countable additivity''. The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint (mutually exclusive) events by the measure should be the sum of the probabilities of the events; for example, the value assigned to the outcome "1 or 2" in a throw of a dice should be the sum of the values assigned to the outcomes "1" and "2". Probability measures have applications in diverse fields, from physics to finance and biology. Definition The requirements for a set function \mu to be a probability measure on a σ-algebra are that: * \mu must return results in the unit interval ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximum Principle
In the mathematical fields of differential equations and geometric analysis, the maximum principle is one of the most useful and best known tools of study. Solutions of a differential inequality in a domain ''D'' satisfy the maximum principle if they achieve their maxima at the boundary of ''D''. The maximum principle enables one to obtain information about solutions of differential equations without any explicit knowledge of the solutions themselves. In particular, the maximum principle is a useful tool in the numerical approximation of solutions of ordinary and partial differential equations and in the determination of bounds for the errors in such approximations. In a simple two-dimensional case, consider a function of two variables such that :\frac+\frac=0. The weak maximum principle, in this setting, says that for any open precompact subset of the domain of , the maximum of on the closure of is achieved on the boundary of . The strong maximum principle says that, unle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
