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Deep Result
The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal shorthand for rigorous arguments or precise ideas. Much of this is common English, but with a specific non-obvious meaning when used in a mathematical sense. Some phrases, like "in general", appear below in more than one section. Philosophy of mathematics ; abstract nonsense:A tongue-in-cheek reference to category theory, using which one can employ arguments that establish a (possibly concrete) result without reference to any specifics of the present problem. For that reason, it's also known as ''general abstract nonsense'' or ''generalized abstract nonsense''. ; canonical:A reference to a standard or choice-free presentation of some mathematical object (e.g., canonical map, canonical form, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Language Of Mathematics
The language of mathematics or mathematical language is an extension of the natural language (for example English) that is used in mathematics and in science for expressing results (scientific laws, theorems, proofs, logical deductions, etc) with concision, precision and unambiguity. Features The main features of the mathematical language are the following. * Use of common words with a derived meaning, generally more specific and more precise. For example, " or" means "one, the other or both", while, in common language, "both" is sometimes included and sometimes not. Also, a " line" is straight and has zero width. * Use of common words with a meaning that is completely different from their common meaning. For example, a mathematical ring is not related to any other meaning of "ring". Real numbers and imaginary numbers are two sorts of numbers, none being more real or more imaginary than the others. * Use of neologisms. For example polynomial, homomorphism. * Use of symbols as word ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irrational Number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being '' incommensurable'', meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio of a circle's circumference to its diameter, Euler's number ''e'', the golden ratio ''φ'', and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational. Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the cas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Folklore
In common mathematical parlance, a mathematical result is called folklore if it is an unpublished result with no clear originator, but which is well-circulated and believed to be true among the specialists. More specifically, folk mathematics, or mathematical folklore, is the body of theorems, definitions, proofs, facts or techniques that circulate among mathematicians by word of mouth, but have not yet appeared in print, either in books or in scholarly journals. Quite important at times for researchers are folk theorems, which are results known, at least to experts in a field, and are considered to have established status, though not published in complete form. Sometimes, these are only alluded to in the public literature. An example is a book of exercises, described on the back cover: Another distinct category is well-knowable mathematics, a term introduced by John Conway. These mathematical matters are known and factual, but not in active circulation in relation with curren ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deep
Deep or The Deep may refer to: Places United States * Deep Creek (Appomattox River tributary), Virginia * Deep Creek (Great Salt Lake), Idaho and Utah * Deep Creek (Mahantango Creek tributary), Pennsylvania * Deep Creek (Mojave River tributary), California * Deep Creek (Pine Creek tributary), Pennsylvania * Deep Creek (Soque River tributary), Georgia * Deep Creek (Texas), a tributary of the Colorado River * Deep Creek (Washington), a tributary of the Spokane River * Deep River (Indiana), a tributary of the Little Calumet River * Deep River (Iowa), a minor tributary of the English River * Deep River (North Carolina) * Deep River (Washington), a minor tributary of the Columbia River * Deep Voll Brook, New Jersey, also known as Deep Brook Elsewhere * Deep Creek (Bahamas) * Deep Creek (Melbourne, Victoria), Australia, a tributary of the Maribyrnong River * Deep River (Western Australia) People * Deep (given name) * Deep (rapper), Punjabi rapper from Houston, Texas * Ravi Deep (bor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts. The basic intuitions are that projective space has more points than Euclidean space, for a given dimension, and that geometric transformations are permitted that transform the extra points (called "points at infinity") to Euclidean points, and vice-versa. Properties meaningful for projective geometry are respected by this new idea of transformation, which is more radical in its effects than can be expressed by a transformation matrix and translations (the affine transformations). The first issue for geometers is what kind of geometry is adequate for a novel situation. It is not possible to refer to angles in projective geometry as it is in Euclidean geometry, because angle is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gian-Carlo Rota
Gian-Carlo Rota (April 27, 1932 – April 18, 1999) was an Italian-American mathematician and philosopher. He spent most of his career at the Massachusetts Institute of Technology, where he worked in combinatorics, functional analysis, probability theory, and phenomenology. Early life and education Rota was born in Vigevano, Italy. His father, Giovanni, an architect and prominent antifascist, was the brother of the mathematician Rosetta, who was the wife of the writer Ennio Flaiano. Gian-Carlo's family left Italy when he was 13 years old, initially going to Switzerland. Rota attended the Colegio Americano de Quito in Ecuador, and graduated with an A.B. in mathematics from Princeton University in 1953 after completing a senior thesis, titled "On the solubility of linear equations in topological vector spaces", under the supervision of William Feller. He then pursued graduate studies at Yale University, where he received a Ph.D. in mathematics in 1956 after completing a do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Beauty
Mathematical beauty is the aesthetic pleasure derived from the abstractness, purity, simplicity, depth or orderliness of mathematics. Mathematicians may express this pleasure by describing mathematics (or, at least, some aspect of mathematics) as beautiful or describe mathematics as an art form, e.g., a position taken by G. H. Hardy) or, at a minimum, as a creative activity. Comparisons are made with music and poetry. In method Mathematicians describe an especially pleasing method of proof as ''elegant''. Depending on context, this may mean: * A proof that uses a minimum of additional assumptions or previous results. * A proof that is unusually succinct. * A proof that derives a result in a surprising way (e.g., from an apparently unrelated theorem or a collection of theorems). * A proof that is based on new and original insights. * A method of proof that can be easily generalized to solve a family of similar problems. In the search for an elegant proof, mathematicians oft ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Analysis
In mathematics, the branch of real analysis studies the behavior of real numbers, sequences and series of real numbers, and real functions. Some particular properties of real-valued sequences and functions that real analysis studies include convergence, limits, continuity, smoothness, differentiability and integrability. Real analysis is distinguished from complex analysis, which deals with the study of complex numbers and their functions. Scope Construction of the real numbers The theorems of real analysis rely on the properties of the real number system, which must be established. The real number system consists of an uncountable set (\mathbb), together with two binary operations denoted and , and an order denoted . The operations make the real numbers a field, and, along with the order, an ordered field. The real number system is the unique ''complete ordered field'', in the sense that any other complete ordered field is isomorphic to it. Intuitively, completeness means ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Proof
In mathematics, an elementary proof is a mathematical proof that only uses basic techniques. More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis. Historically, it was once thought that certain theorems, like the prime number theorem, could only be proved by invoking "higher" mathematical theorems or techniques. However, as time progresses, many of these results have also been subsequently reproven using only elementary techniques. While there is generally no consensus as to what counts as elementary, the term is nevertheless a common part of the mathematical jargon. An elementary proof is not necessarily simple, in the sense of being easy to understand or trivial. In fact, some elementary proofs can be quite complicated — and this is especially true when a statement of notable importance is involved.. Prime number theorem The distinction between elementary and non-elementary proofs has been considered especially important ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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