|
Hadwiger
Hugo Hadwiger (23 December 1908 in Karlsruhe, Germany – 29 October 1981 in Bern, Switzerland) was a Swiss mathematician, known for his work in geometry, combinatorics, and cryptography. Biography Although born in Karlsruhe, Germany, Hadwiger grew up in Bern, Switzerland.. He did his undergraduate studies at the University of Bern, where he majored in mathematics but also studied physics and actuarial science. He continued at Bern for his graduate studies, and received his Ph.D. in 1936 under the supervision of Willy Scherrer. He was for more than forty years a professor of mathematics at Bern. Mathematical concepts named after Hadwiger Hadwiger's theorem in integral geometry classifies the isometry-invariant valuations on compact convex sets in ''d''-dimensional Euclidean space. According to this theorem, any such valuation can be expressed as a linear combination of the intrinsic volumes; for instance, in two dimensions, the intrinsic volumes are the area, the perimete ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hugo Hadwiger
Hugo Hadwiger (23 December 1908 in Karlsruhe, Germany – 29 October 1981 in Bern, Switzerland) was a Swiss mathematician, known for his work in geometry, combinatorics, and cryptography. Biography Although born in Karlsruhe, Germany, Hadwiger grew up in Bern, Switzerland.. He did his undergraduate studies at the University of Bern, where he majored in mathematics but also studied physics and actuarial science. He continued at Bern for his graduate studies, and received his Ph.D. in 1936 under the supervision of Willy Scherrer. He was for more than forty years a professor of mathematics at Bern. Mathematical concepts named after Hadwiger Hadwiger's theorem in integral geometry classifies the isometry-invariant valuations on compact convex sets in ''d''-dimensional Euclidean space. According to this theorem, any such valuation can be expressed as a linear combination of the intrinsic volumes; for instance, in two dimensions, the intrinsic volumes are the area, the perimeter, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadwiger Conjecture (graph Theory)
In graph theory, the Hadwiger conjecture states that if G is loopless and has no K_t minor then its chromatic number satisfies It is known to be true for The conjecture is a generalization of the four-color theorem and is considered to be one of the most important and challenging open problems in the field. In more detail, if all proper colorings of an undirected graph G use k or more colors, then one can find k disjoint connected subgraphs of G such that each subgraph is connected by an edge to each other subgraph. Contracting the edges within each of these subgraphs so that each subgraph collapses to a single vertex produces a complete graph K_k on k vertices as a minor This conjecture, a far-reaching generalization of the four-color problem, was made by Hugo Hadwiger in 1943 and is still unsolved. call it "one of the deepest unsolved problems in graph theory." Equivalent forms An equivalent form of the Hadwiger conjecture (the contrapositive of the form stated above ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finsler–Hadwiger Theorem
The Finsler–Hadwiger theorem is statement in Euclidean plane geometry that describes a third square derived from any two squares that share a vertex. The theorem is named after Paul Finsler and Hugo Hadwiger, who published it in 1937 as part of the same paper in which they published the Hadwiger–Finsler inequality In mathematics, the Hadwiger–Finsler inequality is a result on the geometry of triangles in the Euclidean plane. It states that if a triangle in the plane has side lengths ''a'', ''b'' and ''c'' and area ''T'', then :a^ + b^ + c^ \geq (a - b)^ ... relating the side lengths and area of a triangle. Statement To state the theorem, suppose that ABCD and AB'C'D' are two squares with common vertex A. Let E and G be the midpoints of B'D and D'B respectively, and let F and H be the centers of the two squares. Then the theorem states that the quadrilateral EFGH is a square as well. The square EFGH is called the Finsler–Hadwiger square of the two given squares. Appl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Minor
In graph theory, an undirected graph is called a minor of the graph if can be formed from by deleting edges and vertices and by contracting edges. The theory of graph minors began with Wagner's theorem that a graph is planar if and only if its minors include neither the complete graph nor the complete bipartite graph ., p. 77; . The Robertson–Seymour theorem implies that an analogous forbidden minor characterization exists for every property of graphs that is preserved by deletions and edge contractions., theorem 4, p. 78; . For every fixed graph , it is possible to test whether is a minor of an input graph in polynomial time; together with the forbidden minor characterization this implies that every graph property preserved by deletions and contractions may be recognized in polynomial time. Other results and conjectures involving graph minors include the graph structure theorem, according to which the graphs that do not have as a minor may be formed by glui ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadwiger–Finsler Inequality
In mathematics, the Hadwiger–Finsler inequality is a result on the geometry of triangles in the Euclidean plane. It states that if a triangle in the plane has side lengths ''a'', ''b'' and ''c'' and area ''T'', then :a^ + b^ + c^ \geq (a - b)^ + (b - c)^ + (c - a)^ + 4 \sqrt T \quad \mbox. Related inequalities * Weitzenböck's inequality is a straightforward corollary of the Hadwiger–Finsler inequality: if a triangle in the plane has side lengths ''a'', ''b'' and ''c'' and area ''T'', then :a^ + b^ + c^ \geq 4 \sqrt T \quad \mbox. Hadwiger–Finsler inequality is actually equivalent to Weitzenböck's inequality. Applying (W) to the circummidarc triangle gives (HF) Weitzenböck's inequality can also be proved using Heron's formula, by which route it can be seen that equality holds in (W) if and only if the triangle is an equilateral triangle, i.e. ''a'' = ''b'' = ''c''. * A version for quadrilateral: Let ''ABCD'' be a convex quadrilateral with the length ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Valuation (geometry)
In geometry, a valuation is a finitely additive function on a collection of admissible subsets of a fixed set X with values in an abelian semigroup. For example, the Lebesgue measure is a valuation on finite unions of convex bodies (that is, non-empty compact convex sets) of Euclidean space \R^n. Other examples of valuations on finite unions of convex bodies are the surface area, the mean width, and the Euler characteristic. In the geometric setting, often continuity (or smoothness) conditions are imposed on valuations, but there are also purely discrete facets of the theory. In fact, the concept of valuation has its origin in the dissection theory of polytopes and in particular Hilbert's third problem, which has grown into a rich theory, heavily reliant on advanced tools from abstract algebra. Definition Let X be a set and \mathcal S be a collection of admissible subsets of X. A function \phi on \mathcal S with values in an abelian semigroup R is called a valuation if it satisfie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weitzenböck's Inequality
In mathematics, Weitzenböck's inequality, named after Roland Weitzenböck, states that for a triangle of side lengths a, b, c, and area \Delta, the following inequality holds: : a^2 + b^2 + c^2 \geq 4\sqrt\, \Delta. Equality occurs if and only if the triangle is equilateral. Pedoe's inequality is a generalization of Weitzenböck's inequality. The Hadwiger–Finsler inequality is a strengthened version of Weitzenböck's inequality. Geometric interpretation and proof Rewriting the inequality above allows for a more concrete geometric interpretation, which in turn provides an immediate proof. : \fraca^2 + \fracb^2 + \fracc^2 \geq 3\, \Delta. Now the summands on the left side are the areas of equilateral triangles erected over the sides of the original triangle and hence the inequation states that the sum of areas of the equilateral triangles is always greater than or equal to threefold the area of the original triangle. : \Delta_a + \Delta_b + \Delta_c \geq 3\, \Delta. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Finsler
Paul Finsler (born 11 April 1894, in Heilbronn, Germany, died 29 April 1970 in Zurich, Switzerland) was a German and Swiss mathematician. Finsler did his undergraduate studies at the Technische Hochschule Stuttgart, and his graduate studies at the University of Göttingen, where he received his Ph.D. in 1919 under the supervision of Constantin Carathéodory. He studied for his habilitation at the University of Cologne, receiving it in 1922. He joined the faculty of the University of Zurich in 1927, and was promoted to ordinary professor there in 1944. Finsler's thesis work concerned differential geometry, and Finsler spaces were named after him by Élie Cartan in 1934. The Hadwiger–Finsler inequality, a relation between the side lengths and area of a triangle in the Euclidean plane, is named after Finsler and his co-author Hugo Hadwiger, as is the Finsler–Hadwiger theorem on a square derived from two other squares that share a vertex. Finsler is also known for his work ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadwiger's Theorem
In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem characterises the valuations on convex bodies in \R^n. It was proved by Hugo Hadwiger. Introduction Valuations Let \mathbb^n be the collection of all compact convex sets in \R^n. A valuation is a function v : \mathbb^n \to \R such that v(\varnothing) = 0 and for every S, T \in \mathbb^n that satisfy S \cup T \in \mathbb^n, v(S) + v(T) = v(S \cap T) + v(S \cup T)~. A valuation is called continuous if it is continuous with respect to the Hausdorff metric. A valuation is called invariant under rigid motions if v(\varphi(S)) = v(S) whenever S \in \mathbb^n and \varphi is either a translation or a rotation of \R^n. Quermassintegrals The quermassintegrals W_j : \mathbb^n \to \R are defined via Steiner's formula \mathrm_n(K + t B) = \sum_^n \binom W_j(K) t^j~, where B is the Euclidean ball. For example, W_o is the volume, W_1 is proportional to the surface measure, W_ is proportional to th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixed Volume
In mathematics, more specifically, in convex geometry, the mixed volume is a way to associate a non-negative number to an of convex bodies in space. This number depends on the size and shape of the bodies and on their relative orientation to each other. Definition Let K_1, K_2, \dots, K_r be convex bodies in \mathbb^n and consider the function : f(\lambda_1, \ldots, \lambda_r) = \mathrm_n (\lambda_1 K_1 + \cdots + \lambda_r K_r), \qquad \lambda_i \geq 0, where \text_n stands for the n-dimensional volume and its argument is the Minkowski sum of the scaled convex bodies K_i. One can show that f is a homogeneous polynomial of degree n, therefore it can be written as : f(\lambda_1, \ldots, \lambda_r) = \sum_^r V(K_, \ldots, K_) \lambda_ \cdots \lambda_, where the functions V are symmetric. For a particular index function j \in \^n , the coefficient V(K_, \dots, K_) is called the mixed volume of K_, \dots, K_. Properties * The mixed volume is uniquely determined by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Coloring
In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color. Vertex coloring is often used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graph, and a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is. This is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Geometry
In mathematics, integral geometry is the theory of measures on a geometrical space invariant under the symmetry group of that space. In more recent times, the meaning has been broadened to include a view of invariant (or equivariant) transformations from the space of functions on one geometrical space to the space of functions on another geometrical space. Such transformations often take the form of integral transforms such as the Radon transform and its generalizations. Classical context Integral geometry as such first emerged as an attempt to refine certain statements of geometric probability theory. The early work of Luis Santaló and Wilhelm Blaschke was in this connection. It follows from the classic theorem of Crofton expressing the length of a plane curve as an expectation of the number of intersections with a random line. Here the word 'random' must be interpreted as subject to correct symmetry considerations. There is a sample space of lines, one on which the affin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
