Poincaré disk model Metric[edit] If u and v are two vectors in real n-dimensional vector space Rn with the usual Euclidean norm, both of which have norm less than 1, then we may define an isometric invariant by where denotes the usual Euclidean norm. Then the distance function is Such a distance function is defined for any two vectors of norm less than one, and makes the set of such vectors into a metric space which is a model of hyperbolic space of constant curvature −1. The associated metric tensor of the Poincaré disk model is given by where the xi are the Cartesian coordinates of the ambient Euclidean space. Relation to the hyperboloid model[edit] The hyperboloid model can be seen as the equation of t2=x2+y2+1 can can be used to construct a Poincaré disk model as a perspective projection viewed from (t=-1,x=0,y=0), projecting the upper half hyperboloid onto an (x,y) unit disk at t=0. The Poincaré disk model, as well as the Klein model, are related to the hyperboloid model projectively. Angles[edit] M.
Challenge Math Online: An Online Program for Gifted, Talented and Promising Math Students | Noetic Learning I am very excited about Challenge Math ... The fact that the problems are chosen carefully to work on developing the skills of the kids that excel in math and to prepare them for national and international competitions is invaluable. There is little of this in school, so we intend to take full advantage of the program in the future. - Dr. Stanislavova Associate Professor, MathematicsUniversity of Kansas Welcome to Challenge Math Online Challenge Math is designed to hone young students' mathematical problem-solving skills and logical reasoning skills. Challenge Math gives out challenging word problem assignments weekly. Challenge Math is primarily designed for gifted elementary students. provides challenge beyond regular school curriculum, strengthens creative problem solving, and logical reasoning skills, further develops gifted students' intellect in math, and helps students excel in national math competitions. Problem Solving Strategies Covered Model with Diagrams | Draw a Picture |
Cayley graph The Cayley graph of the free group on two generators a and b Definition[edit] Suppose that is a generating set. The Cayley graph is a colored directed graph constructed as follows: [2] Each element of is assigned a vertex: the vertex set of is identified with Each generator of is assigned a color .For any the vertices corresponding to the elements and are joined by a directed edge of colour Thus the edge set consists of pairs of the form with providing the color. In geometric group theory, the set is usually assumed to be finite, symmetric (i.e. Examples[edit] Cayley graph of the dihedral group Dih4 on two generators a and b On two generators of Dih4, which are both self-inverse A Cayley graph of the dihedral group D4 on two generators a and b is depicted to the left. A different Cayley graph of Dih4 is shown on the right. b is still the horizontal reflection and represented by blue lines; c is a diagonal reflection and represented by green lines. is depicted to the right. The group
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Calabi–Yau manifold A 2D slice of the 6D Calabi-Yau quintic manifold. Calabi–Yau manifolds are complex manifolds that are higher-dimensional analogues of K3 surfaces. They are sometimes defined as compact Kähler manifolds whose canonical bundle is trivial, though many other similar but inequivalent definitions are sometimes used. They were named "Calabi–Yau spaces" by Candelas et al. (1985) after E. Definitions[edit] There are many different inequivalent definitions of a Calabi–Yau manifold used by different authors. A Calabi–Yau n-fold or Calabi–Yau manifold of (complex) dimension n is sometimes defined as a compact n-dimensional Kähler manifold M satisfying one of the following equivalent conditions: These conditions imply that the first integral Chern class c1(M) of M vanishes, but the converse is not true. In particular if a compact Kähler manifold is simply connected then the weak definition above is equivalent to the stronger definition. Examples[edit] All hyper-Kähler manifolds are Calabi–Yau.
AMS :: Who Wants to Be a Mathematician Hopf fibration The Hopf fibration can be visualized using a stereographic projection of S3 to R3 and then compressing R3 to a ball. This image shows points on S2 and their corresponding fibers with the same color. Pairwise linked keyrings mimic part of the Hopf fibration. In the mathematical field of topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. This fiber bundle structure is denoted meaning that the fiber space S1 (a circle) is embedded in the total space S3 (the 3-sphere), and p : S3 → S2 (Hopf's map) projects S3 onto the base space S2 (the ordinary 2-sphere). This has many implications: for example the existence of this bundle shows that the higher homotopy groups of spheres are not trivial in general. Stereographic projection of the Hopf fibration induces a remarkable structure on R3, in which space is filled with nested tori made of linking Villarceau circles.
PRIMES | MIT Mathematics In mathematics you don't understand things, you just get used to them. - John von Neumann PRIMES includes three sections: MIT PRIMES is a free year-long after-school program that offers research projects and guided reading to high school students living within driving distance from Boston. Program participants work with MIT researchers on exciting unsolved problems in mathematics, computer science, and computational biology. PRIMES-USA is a free year-long distance mentoring math research section for high school juniors from across the United States (outside of Greater Boston). PRIMES Circle is a free spring-term math enrichment section for high school students from underrepresented groups living within commuting distance from Boston. PRIMES runs two collaborative initiatives: CrowdMath (a joint program with the Art of Problem Solving) is a massive online collaborative year-long research project open to all high school and college students around the world. Contact