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Poincaré–Miranda Theorem
In mathematics, the Poincaré–Miranda theorem is a generalization of intermediate value theorem, from a single function in a single dimension, to functions in dimensions. It says as follows: ::Consider n continuous functions of n variables, f_1,\ldots, f_n. Assume that for each variable x_i, the function f_i is nonpositive when x_i=-1 and nonnegative when x_i=1. Then there is a point in the n-dimensional cube 1,1n in which all functions are ''simultaneously'' equal to 0. The theorem is named after Henri Poincaré - who conjectured it in 1883, and Carlo Miranda - who in 1940 showed that it is equivalent to the Brouwer fixed-point theorem. It is sometimes called the Miranda theorem or the Bolzano-Poincare-Miranda theorem. Intuitive description The picture on the right shows an illustration of the Poincaré–Miranda theorem for functions. Consider a couple of functions whose domain of definition is (i.e., the unit square). The function is negative on the left boundary and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intermediate Value Theorem
In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two important corollaries: # If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem). # The image of a continuous function over an interval is itself an interval. Motivation This captures an intuitive property of continuous functions over the real numbers: given ''f'' continuous on ,2/math> with the known values f(1) = 3 and f(2) = 5, then the graph of y = f(x) must pass through the horizontal line y = 4 while x moves from 1 to 2. It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting a pencil from the paper. Theorem The intermediate value theorem states the following: Consider an interval I = ,b/math> of real n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cube
In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only regular hexahedron and is one of the five Platonic solids. It has 6 faces, 12 edges, and 8 vertices. The cube is also a square parallelepiped, an equilateral cuboid and a right rhombohedron a 3-zonohedron. It is a regular square prism in three orientations, and a trigonal trapezohedron in four orientations. The cube is dual to the octahedron. It has cubical or octahedral symmetry. The cube is the only convex polyhedron whose faces are all squares. Orthogonal projections The ''cube'' has four special orthogonal projections, centered, on a vertex, edges, face and normal to its vertex figure. The first and third correspond to the A2 and B2 Coxeter planes. Spherical tiling The cube can also be represented as a spherical tiling, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Poincaré
Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The Last Universalist", since he excelled in all fields of the discipline as it existed during his lifetime. As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern chaos theory. He is also considered to be one of the founders of the field of topology. Poincaré made clear the importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the Lorentz transformations in their modern symmetrical form. Poincaré discove ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
