Riemannian geometry Elliptic geometry is also sometimes called "Riemannian geometry". Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area, and volume. From those some other global quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugurational lecture Ueber die Hypothesen, welche der Geometrie zu Grunde liegen (On the Hypotheses which lie at the Bases of Geometry). It is a very broad and abstract generalization of the differential geometry of surfaces in R3. Introduction[edit] Riemannian geometry was first put forward in generality by Bernhard Riemann in the nineteenth century. The following articles provide some useful introductory material:
Monism Monism is the philosophical view that a variety of existing things can be explained in terms of a single reality or substance. The wide definition states that all existing things go back to a source which is distinct from them (e.g. in Neoplatonism everything is derived from The One). A commonly-used, restricted definition of monism asserts the presence of a unifying substance or essence. One must distinguish "stuff monism" from "thing monism".[3] According to stuff monism there is only one kind of stuff (e.g. matter or mind), although there may be many things made out of this stuff. The term monism originated from Western philosophy,[4] and has often been applied to various religions. History[edit] It was later also applied to the theory of absolute identity set forth by Hegel and Schelling. According to Jonathan Schaffer, monism lost popularity due to the emergence of Analytic philosophy in the early twentieth century, which revolted against the neo-Hegelians. Definitions[edit] a. b.
Transfinite number Transfinite numbers are numbers that are "infinite" in the sense that they are larger than all finite numbers, yet not necessarily absolutely infinite. The term transfinite was coined by Georg Cantor, who wished to avoid some of the implications of the word infinite in connection with these objects, which were nevertheless not finite. Few contemporary writers share these qualms; it is now accepted usage to refer to transfinite cardinals and ordinals as "infinite". Definition[edit] As with finite numbers, there are two ways of thinking of transfinite numbers, as ordinal and cardinal numbers. ω (omega) is defined as the lowest transfinite ordinal number and is the order type of the natural numbers under their usual linear ordering.Aleph-null, , is defined as the first transfinite cardinal number and is the cardinality of the infinite set of the natural numbers. Some authors, including P. m is a transfinite cardinal. See also[edit] References[edit]
Analytic geometry Cartesian coordinates Analytic geometry, or analytical geometry, has two different meanings in mathematics. The modern and advanced meaning refers to the geometry of analytic varieties. In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system. Analytic geometry is widely used in physics and engineering, and is the foundation of most modern fields of geometry, including algebraic, differential, discrete, and computational geometry. History[edit] The eleventh century Persian mathematician Omar Khayyám saw a strong relationship between geometry and algebra, and was moving in the right direction when he helped to close the gap between numerical and geometric algebra[4] with his geometric solution of the general cubic equations,[5] but the decisive step came later with Descartes.[4] Pierre Fermat also pioneered the development of analytic geometry. Basic principles[edit] Coordinates[edit] . -axis. and
Calculus ratiocinator The Calculus Ratiocinator is a theoretical universal logical calculation framework, a concept described in the writings of Gottfried Leibniz, usually paired with his more frequently mentioned characteristica universalis, a universal conceptual language. Two views[edit] There are two contrasting points of view on what Leibniz meant by calculus ratiocinator. The first is associated with computer software, the second is associated with computer hardware. The analytic view[edit] The received point of view in analytic philosophy and formal logic, is that the calculus ratiocinator anticipates mathematical logic — an "algebra of logic".[1] The analytic point of view understands that the calculus ratiocinator is a formal inference engine or computer program which can be designed so as to grant primacy to calculations. The synthetic view[edit] A contrasting point of view stems from synthetic philosophy and fields such as cybernetics, electronic engineering and general systems theory. Notes[edit]
Sample space For example, if the experiment is tossing a coin, the sample space is typically the set {head, tail}. For tossing two coins, the corresponding sample space would be {(head,head), (head,tail), (tail,head), (tail,tail)}. For tossing a single six-sided die, the typical sample space is {1, 2, 3, 4, 5, 6} (in which the result of interest is the number of pips facing up).[2] Multiple sample spaces[edit] Equally likely outcomes[edit] Up or down? In some sample spaces, it is reasonable to estimate or assume that all outcomes in the space are equally likely (that they occur with equal probability). Though most random phenomena do not have equally likely outcomes, it can be helpful to define a sample space in such a way that outcomes are at least approximately equally likely, since this condition significantly simplifies the computation of probabilities for events within the sample space. Simple random sample[edit] Infinitely large sample spaces[edit] See also[edit] References[edit]
Metric space The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space. In fact, the notion of "metric" is a 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. A metric space also induces topological properties like open and closed sets which leads to the study of even more abstract topological spaces. History[edit] Maurice Fréchet introduced metric spaces in his work Sur quelques points du calcul fonctionnel, Rendic. Definition[edit] A metric space is an ordered pair where is a set and such that for any about of in
Limit of a sequence As the positive integer n becomes larger and larger, the value n sin(1/n) becomes arbitrarily close to 1. We say that "the limit of the sequence n sin(1/n) equals 1." In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to".[1] If such a limit exists, the sequence is called convergent. A sequence which does not converge is said to be divergent.[2] The limit of a sequence is said to be the fundamental notion on which the whole of analysis ultimately rests.[1] Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers. Real numbers[edit] The plot of a convergent sequence {an} is shown in blue. Formal Definition[edit] We call the limit of the sequence if the following condition holds: For each real number , there exists a natural number such that, for every natural number , we have . In other words, for every measure of closeness , the sequence's terms are eventually that close to the limit. , written or and then
Real line The real line Just like the set of real numbers, the real line is usually denoted by the symbol R (or alternatively, , the letter “R” in blackboard bold). However, it is sometimes denoted R1 in order to emphasize its role as the first Euclidean space. As a linear continuum[edit] The real line is a linear continuum under the standard < ordering. In addition to the above properties, the real line has no maximum or minimum element. As a metric space[edit] The real line forms a metric space, with the distance function given by absolute difference: d(x, y) = | x − y | . The metric tensor is clearly the 1-dimensional Euclidean metric. This real line has several important properties as a metric space: The real line is a complete metric space, in the sense that any Cauchy sequence of points converges.The real line is path-connected, and is one of the simplest examples of a geodesic metric spaceThe Hausdorff dimension of the real line is equal to one. As a topological space[edit] As a vector space[edit]
Cardinality The cardinality of a set A is usually denoted | A |, with a vertical bar on each side; this is the same notation as absolute value and the meaning depends on context. Alternatively, the cardinality of a set A may be denoted by n(A), A, card(A), or # A. Comparing sets[edit] Definition 1: | A | = | B |[edit] For example, the set E = {0, 2, 4, 6, ...} of non-negative even numbers has the same cardinality as the set N = {0, 1, 2, 3, ...} of natural numbers, since the function f(n) = 2n is a bijection from N to E. Definition 2: | A | ≥ | B |[edit] A has cardinality greater than or equal to the cardinality of B if there exists an injective function from B into A. Definition 3: | A | > | B |[edit] A has cardinality strictly greater than the cardinality of B if there is an injective function, but no bijective function, from B to A. If | A | ≥ | B | and | B | ≥ | A | then | A | = | B | (Cantor–Bernstein–Schroeder theorem). Cardinal numbers[edit] Above, "cardinality" was defined functionally. For each .
Mathematical induction Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. Mathematical induction is a method of mathematical proof typically used to establish a given statement for all natural numbers. It is done in two steps. The first step, known as the base case, is to prove the given statement for the first natural number. Although its namesake may suggest otherwise, mathematical induction should not be misconstrued as a form of inductive reasoning (also see Problem of induction). History[edit] An implicit proof by mathematical induction for arithmetic sequences was introduced in the al-Fakhri written by al-Karaji around 1000 AD, who used it to prove the binomial theorem and properties of Pascal's triangle. None of these ancient mathematicians, however, explicitly stated the inductive hypothesis. Description[edit] The basis (base case): prove that the statement holds for the first natural number n. Example[edit] Algebraically: Variants[edit]