Animated Trigonometry | Jason Davies | Web Design and Development. Non-Euclidean Geometry. In three dimensions, there are three classes of constant curvature geometries. All are based on the first four of Euclid's postulates, but each uses its own version of the parallel postulate. The "flat" geometry of everyday intuition is called Euclidean geometry (or parabolic geometry), and the non-Euclidean geometries are called hyperbolic geometry (or Lobachevsky-Bolyai-Gauss geometry) and elliptic geometry (or Riemannian geometry).
Spherical geometry is a non-Euclidean two-dimensional geometry. It was not until 1868 that Beltrami proved that non-Euclidean geometries were as logically consistent as Euclidean geometry. Www.math.ucla.edu/~tao/ Dyadic product. Dyadic notation was first established by Josiah Willard Gibbs in 1884. In this article, upper-case bold variables denote dyadics (including dyads) whereas lower-case bold variables denote vectors. An alternative notation uses respectively double and single over- or underbars. Definitions and terminology Dyadic, outer, and tensor products A dyad is a tensor of order two and rank one, and is the result of the dyadic product of two vectors (complex vectors in general), whereas a dyadic is a general tensor of order two. There are several equivalent terms and notations for this product: the dyadic product of two vectors a and b is denoted by the juxtaposition ab,the outer product of two column vectors a and b is denoted and defined as a ⊗ b or abT, where T means transpose,the tensor product of two vectors a and b is denoted a ⊗ b, Three-dimensional Euclidean space To illustrate the equivalent usage, consider three-dimensional Euclidean space, letting: Classification Letting P.
Arithmetization of analysis. The arithmetization of analysis was a research program in the foundations of mathematics carried out in the second half of the 19th century. Kronecker originally introduced the term arithmetization of analysis, by which he meant its constructivization in the context of the natural numbers (see quotation at bottom of page). The meaning of the term later shifted to signify the set-theoretic construction of the real line.
Its main proponent was Weierstrass, who argued the geometric foundations of calculus were not solid enough for rigorous work. The highlights of this research program are: An important spinoff of the arithmetization of analysis is set theory. Naive set theory was created by Cantor and others after arithmetization was completed as a way to study the singularities of functions appearing in calculus. The arithmetization of analysis had several important consequences: Quotations: "God created the natural numbers, all else is the work of man. " -- Kronecker. Grigori Perelman. Grigori Yakovlevich Perelman (Russian: Григорий Яковлевич Перельман, IPA: [ɡrʲɪˈɡorʲɪj ˈjɑkəvlʲɪvʲɪtʃ pʲɪrʲɪlʲˈman] ( )/ˈpɛrɨlmən/ PERR-il-mən[dubious ]; Russian: Григо́рий Я́ковлевич Перельма́н; born 13 June 1966) is a Russian mathematician who made landmark contributions to Riemannian geometry and geometric topology before his presumed withdrawal from mathematics.
In 1994, Perelman proved the soul conjecture. In 2003, he proved Thurston's geometrization conjecture. This consequently solved in the affirmative the Poincaré conjecture, posed in 1904, which before its solution was viewed as one of the most important and difficult open problems in topology. On 18 March 2010, it was announced that he had met the criteria to receive the first Clay Millennium Prize for resolution of the Poincaré conjecture. Early life and education His mathematical education continued at the Leningrad Secondary School #239, a specialized school with advanced mathematics and physics programs.
Divergence. In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point. For example, consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the moving air at a point. If air is heated in a region it will expand in all directions such that the velocity field points outward from that region. Therefore the divergence of the velocity field in that region would have a positive value, as the region is a source. If the air cools and contracts, the divergence has a negative value, as the region is a sink. Definition of divergence In physical terms, the divergence of a three-dimensional vector field is the extent to which the vector field flow behaves like a source or a sink at a given point.
With or and . Curl (mathematics) The alternative terminology rotor or rotational and alternative notations rot F and ∇ × F are often used (the former especially in many European countries, the latter, using the del operator and the cross product, is more used in other countries) for curl and curl F. The name "curl" was first suggested by James Clerk Maxwell in 1871. The components of F at position r, normal and tangent to a closed curve C in a plane, enclosing a planar vector areaA = An. The curl of a vector field F, denoted by curl F, or ∇ × F, or rot F, at a point is defined in terms of its projection onto various lines through the point. If is any unit vector, the projection of the curl of F onto is defined to be the limiting value of a closed line integral in a plane orthogonal to as the path used in the integral becomes infinitesimally close to the point, divided by the area enclosed. As such, the curl operator maps continuously differentiable functions f : R3 → R3 to continuous functions g : R3 → R3.
Where Here and. Riemann,zeta,function. Statistical mechanics: the Riemann zeta function interpreted as a partition function. The Riemann zeta function interpreted as a partition function lattice-related number theory (involving Ising models, percolation, etc.) integer partition problems and physics entropy and number theory number theory and statistical mechanics – general probabilistic number theory the Riemann zeta function interpreted as a partition function One of the earliest, and perhaps most significant, examples of number theory influencing the development of physics was the application of Pólya's work on the Riemann zeta function to the theory of phase transitions by Lee and Yang in the early 1950's.
In 1951-2, Lee and Yang were developing this theory, and Mark Kac became aware of their conjecture which was later to become the "Lee-Yang circle theorem". Lee and Yang were then able to adapt the reasoning and, within a couple of weeks, produce a proof of their general theorem. In equilibrium statistical mechanics, the fundamental object of study for a system is its partition function. B.L. B.L. B.L. G.W. D. Analytic continuation. In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which it is initially defined becomes divergent.
The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of mathematical singularities. The case of several complex variables is rather different, since singularities then cannot be isolated points, and its investigation was a major reason for the development of sheaf cohomology. Initial discussion Analytic continuation of natural logarithm (imaginary part) Suppose f is an analytic function defined on a non-empty open subset U of the complex plane C. Then Let. . , hence. Riemann zeta function. , which converges when the real part of s is greater than 1. More general representations of ζ(s) for all s are given below.
The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. This function, as a function of a real argument, was introduced and studied by Leonhard Euler in the first half of the eighteenth century without using complex analysis, which was not available at that time. Bernhard Riemann in his article "On the Number of Primes Less Than a Given Magnitude" published in 1859 extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation and established a relation between its zeros and the distribution of prime numbers. The values of the Riemann zeta function at even positive integers were computed by Euler. Definition Bernhard Riemann's article on the number of primes below a given magnitude. Specific values A058303). Hyperreal number. The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities.
The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form Such a number is infinite, and its reciprocal is infinitesimal. The term "hyper-real" was introduced by Edwin Hewitt in 1948. The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic Law of Continuity.
The transfer principle states that true first order statements about R are also valid in *R. For example, the commutative law of addition, x + y = y + x, holds for the hyperreals just as it does for the reals; since R is a real closed field, so is *R. For all integers n, one also has for all hyperintegers H. The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called non-standard analysis.
For an infinitesimal The transfer principle but there is no such number in R. The Tesseract. Wolfram|Alpha. Mathematics. 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 theorems. This 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 Z
Theorem. Many 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 resources. Duality. From Wikipedia, the free encyclopedia Duality may refer to: Mathematics Philosophy, logic, and psychology Science Electrical and mechanical Physics Titles Film Music Other See also Beauty of Mathematics « Crazy People With Crazy Things. Chronology of Pure and Applied Mathematics. §. Portal @ MathLinks. MathPages. Category theory « The Unapologetic Mathematician. Comma Categories Another useful example of a category is a comma category.
The term comes from the original notation which has since fallen out of favor because, as Saunders MacLane put it, “the comma is already overworked”. We start with three categories , and , and two functors and . Are triples where is an object of is an arrow in. . — with an arrow in — making the following square commute: So what? To be the functor sending the single object of to the object . Be the identity functor on. . , where can be any other object in . Work out for yourself the category Here’s another example: the category . And another: check that given objects , the category is the discrete category (set) Neat! Cardinals and Ordinals as Categories We can import all of what we’ve said about cardinal numbers and ordinal numbers into categories. For cardinals, it’s actually not that interesting. And turn it into a category .
And just give every object its identity morphism — there are no other morphisms at all in this category. To if then . Category Theory Awodey Course. Www.liegroups.org. ATLAS of Finite Group Representations - V3. Www.liegroups.org. Algebraic structure. Algebraic geometry. Algebraic topology. Cohomology. Abelian Group. Algebraic geometry. Riemannian geometry. Symplectic geometry. Differential geometry. Non-Euclidean geometry. Special values of L-functions. John Cremona's home page. Number Theory.
Algebraic Topology. Homotopy. Euler characteristic. Abstract Algebra. Mathematics. Monad (category theory) Orbifold construction of the modes of the Poincare dodecahedral space - Jeffrey Weeks. Logarithmic spiral. Harvard extension school courses. Abstract Algebra - Free Harvard Courses. Algebra. Ring (mathematics) Abstract Algebra - Free Harvard Courses - StumbleUpon. Virtual Training Suite - free Internet tutorials to develop Internet research skills. The Thirty Greatest Mathematicians. Srinivasa Ramanujan. Carl Friedrich Gauss. Quantum field theory. Adjoint functors. Monad (category theory) Subgroup. Symmetrization. ABSTRACT ALGEBRA: OnLine Study Guide, Table of Contents. Vladimir Voevodsky. Groupoidification. Topology/Homotopy. Topology. Type theory in nLab.
Homotopy Type Theory, I | The n. Coherence theorem.