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Menger Curvature
In mathematics, the Menger curvature of a triple of points in ''n''-dimensional Euclidean space R''n'' is the reciprocal of the radius of the circle that passes through the three points. It is named after the Austrian-American mathematician Karl Menger. Definition Let ''x'', ''y'' and ''z'' be three points in R''n''; for simplicity, assume for the moment that all three points are distinct and do not lie on a single straight line. Let Π ⊆ R''n'' be the Euclidean plane spanned by ''x'', ''y'' and ''z'' and let ''C'' ⊆ Π be the unique Euclidean circle in Π that passes through ''x'', ''y'' and ''z'' (the circumcircle of ''x'', ''y'' and ''z''). Let ''R'' be the radius of ''C''. Then the Menger curvature ''c''(''x'', ''y'', ''z'') of ''x'', ''y'' and ''z'' is defined by :c (x, y, z) = \frac1. If the three points are collinear, ''R'' can be informally considered to be +∞, and it makes rigorous sense to define ''c''(''x'', ''y'', ''z'') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non-Collinearity, collinear, determine a unique triangle and simultaneously, a unique Plane (mathematics), plane (i.e. a two-dimensional Euclidean space). In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted. Types of triangle The terminology for categorizing triangles is more than two thousand years old, having been defined on the very first page of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The American Mathematical Society
''Proceedings of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. As a requirement, all articles must be at most 15 printed pages. According to the ''Journal Citation Reports'', the journal has a 2018 impact factor of 0.813. Scope ''Proceedings of the American Mathematical Society'' publishes articles from all areas of pure and applied mathematics, including topology, geometry, analysis, algebra, number theory, combinatorics, logic, probability and statistics. Abstracting and indexing This journal is indexed in the following databases: 2011. American Mathematical Society. * |
Curvature Of A Measure
In mathematics, the curvature of a measure defined on the Euclidean plane R2 is a quantification of how much the measure's "distribution of mass" is "curved". It is related to notions of curvature in geometry. In the form presented below, the concept was introduced in 1995 by the mathematician Mark S. Melnikov; accordingly, it may be referred to as the Melnikov curvature or Menger-Melnikov curvature. Melnikov and Verdera (1995) established a powerful connection between the curvature of measures and the Cauchy kernel. Definition Let ''μ'' be a Borel measure on the Euclidean plane R2. Given three (distinct) points ''x'', ''y'' and ''z'' in R2, let ''R''(''x'', ''y'', ''z'') be the radius of the Euclidean circle that joins all three of them, or +∞ if they are collinear. The Menger curvature ''c''(''x'', ''y'', ''z'') is defined to be :c(x, y, z) = \frac, with the natural convention that ''c''(''x'', ''y'', ''z'') = 0 if ''x'', ''y'' a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analyst's Traveling Salesman Theorem
The analyst's traveling salesman problem is an analog of the traveling salesman problem in combinatorial optimization. In its simplest and original form, it asks which plane sets are subsets of curve, rectifiable curves of finite length. Whereas the original traveling salesman problem asks for the shortest way to visit every vertex in a finite set with a discrete path, this analytical version may require the curve to visit infinitely many points. β-numbers A rectifiable curve has tangents at almost all of its points, where in this case "almost all" means all but a subset whose one-dimensional Hausdorff measure is zero. Accordingly, if a set is contained in a rectifiable curve, the set must look ''flat'' when zooming in on almost all of its points. This suggests that testing us whether a set could be contained in a rectifiable curve must somehow incorporate information about how flat it is when one zooms in on its points at different scales. This discussion motivates the definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frostman's Lemma
In mathematics, and more specifically, in the theory of fractal dimensions, Frostman's lemma provides a convenient tool for estimating the Hausdorff dimension of sets. Lemma: Let ''A'' be a Borel subset of R''n'', and let ''s'' > 0. Then the following are equivalent: *''H''''s''(''A'') > 0, where ''H''''s'' denotes the ''s''-dimensional Hausdorff measure. *There is an (unsigned) Borel measure ''μ'' on R''n'' satisfying ''μ''(''A'') > 0, and such that ::\mu(B(x,r))\le r^s :holds for all ''x'' ∈ R''n'' and ''r''>0. Otto Frostman proved this lemma for closed sets ''A'' as part of his PhD dissertation at Lund University , motto = Ad utrumque , mottoeng = Prepared for both , established = , type = Public research university , budget = SEK 9 billion [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pertti Mattila
Pertti Esko Juhani Mattila (born 28 March 1948) is a Finnish mathematician working in geometric measure theory, complex analysis and harmonic analysis. He is Professor of Mathematics in the Department of Mathematics and Statistics at the University of Helsinki, Finland. He is known for his work on geometric measure theory and in particular applications to complex analysis and harmonic analysis. His work include a counterexample to the general Vitushkin's conjecture and with Mark Melnikov and Joan Verdera he introduced new techniques to understand the geometric structure of removable sets for complex analytic functions which together with other works in the field eventually led to the solution of Painlevé's problem by Xavier Tolsa. His book ''Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability'' is now a widely cited and a standard textbook in this field. Mattila has been the leading figure on creating the geometric measure theory school in Finland a ... [...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 ,∞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-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 not necessarily an integer. These measures are fundamenta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Borel Measure
In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. Formal definition Let X be a locally compact Hausdorff space, and let \mathfrak(X) be the smallest σ-algebra that contains the open sets of X; this is known as the σ-algebra of Borel sets. A Borel measure is any measure \mu defined on the σ-algebra of Borel sets. A few authors require in addition that \mu is locally finite, meaning that \mu(C) 0 and μ(''B''(''x'', ''r'')) ≤ ''rs'' holds for some constant ''s'' > 0 and for every ball ''B''(''x'', ''r'') in ''X'', then the Hausdorff dimension dimHaus(''X'') ≥ ''s''. A partial converse is provided by the Frostman lemma: Lemma: Let ''A'' be a Borel subset of R''n'', and let ''s'' > 0. Then the following are equivalent: *''H''''s''(''A'') > 0, where ''H''''s'' den ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectifiable Set
In mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense. It is an extension of the idea of a rectifiable curve to higher dimensions; loosely speaking, a rectifiable set is a rigorous formulation of a piece-wise smooth set. As such, it has many of the desirable properties of smooth manifolds, including tangent spaces that are defined almost everywhere. Rectifiable sets are the underlying object of study in geometric measure theory. Definition A Borel subset E of Euclidean space \mathbb^n is said to be m-rectifiable set if E is of Hausdorff dimension m, and there exist a countable collection \ of continuously differentiable maps :f_i:\mathbb^m \to \mathbb^n such that the m-Hausdorff measure \mathcal^m of :E\setminus \bigcup_^\infty f_i\left(\mathbb^m\right) is zero. The backslash here denotes the set difference. Equivalently, the f_i may be taken to be Lipschitz continuous without altering the definition. Other authors have different defini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' meaning "equal", and μέτρον ''metron'' meaning "measure". Introduction Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional Euclidean space, two geometric figures are congruent if they are related by an isometry; the isometry that relates them is either a rigid motion (translation or rotation), or a composition of a rigid motion and a reflection. Isometries are often used in constructions where one space i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
