Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a relatively slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day. History Evolution Etymology Definitions of mathematics
List of theoremsThis is a list of theorems, by Wikipedia page. See also Most of the results below come from pure mathematics, but some are from theoretical physics, economics, and other applied fields. 0–9 A B C D E F G H I J K L M N O P Q R S T U V W ZFractalsRepresenting geometry in SecondLife Some comments on the suitability of SecondLife as a way of presenting fractal geometry Fractal forms found by using Google Earth Readers are encouraged to contribute their own findings. POVRay Fractal Raytracing Contest Rendering Wada-type basins of attraction Here's a party trick for over Christmas. Fractal Dimension and Self Similarity A particular box counting software package, Ruler or Compass Dimension, Lacunarity, Multifractal spectrum, Recurrence plots, Self Similarity. Time exists so that not everything happens at once. Natural Fractals in Grand Canyon National Park Fractals: A Symmetry Approach Gasket Photography Fractals and Computer Graphics Interface Magazine Article I wonder whether fractal images are not touching the very structure of our brains. Diffusion Limited Aggregation DLA in 3D Platonic solids Tropical leaf Random Attractors Marketing idea: A potato chip in the shape of the Sierpinski gasket! It would infinitely crispy but have zero calories. Lemon
Metric spaceThe most familiar metric space is 3-dimensional Euclidean space. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance. The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them. Other metric spaces occur for example in elliptic geometry and hyperbolic geometry, where distance on a sphere measured by angle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities. History Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, Rendic. Definition A metric space is an ordered pair where is a set and such that for any , the following holds: The first condition follows from the other three. The function is also called distance function or simply distance. is omitted and one just writes Examples of metric spaces About any point (where of
TheoremMany mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from the hypotheses. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol. Although they can be written in a completely symbolic form, for example, within the propositional calculus, theorems are often expressed in a natural language such as English. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed.
Math O' Man : Le Blog des MathsBoundary (topology)A set (in light blue) and its boundary (in dark blue). A connected component of the boundary of S is called a boundary component of S. If the set consists of discrete points only, then the set has only a boundary and no interior. There are several common (and equivalent) definitions to the boundary of a subset S of a topological space X: These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure. In the space of rational numbers with the usual topology (the subspace topology of R), the boundary of , where a is irrational, is empty. Hence: p is a boundary point of a set if and only if every neighborhood of p contains at least one point in the set and at least one point not in the set.A set is closed if and only if it contains its boundary, and open if and only if it is disjoint from its boundary.The closure of a set equals the union of the set with its boundary. Jump up ^ Mendelson, Bert (1990) . Munkres, J.
DualityFrom Wikipedia, the free encyclopedia Duality may refer to: Mathematics Philosophy, logic, and psychology Science Electrical and mechanical Physics Titles Film Music Other See alsomathématiques : cours et exercices de mathématiques, forums...