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Shapiro Inequality
In mathematics, the Shapiro inequality is an inequality (mathematics), inequality proposed by Harold S. Shapiro in 1954. Statement of the inequality Suppose n is a natural number and x_1, x_2, \dots, x_n are positive numbers and: * n is even and less than or equal to 12, or * n is odd and less than or equal to 23. Then the Shapiro inequality states that :\sum_^ \frac \geq \frac where x_=x_1, x_=x_2. For greater values of n the inequality does not hold and the strict lower bound is \gamma \frac with \gamma \approx 0.9891\dots. The initial proofs of the inequality in the pivotal cases n=12 (Godunova and Levin, 1976) and n=23 (Troesch, 1989) rely on numerical computations. In 2002, P.J. Bushell and J.B. McLeod published an analytical proof for n=12. The value of \gamma was determined in 1971 by Vladimir Drinfeld. Specifically, he proved that the strict lower bound \gamma is given by \frac \psi(0), where the function \psi is the convex hull of f(x)=e^ and g(x) = \frac. (Tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequality (mathematics)
In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different notations used to represent different kinds of inequalities: * The notation ''a'' ''b'' means that ''a'' is greater than ''b''. In either case, ''a'' is not equal to ''b''. These relations are known as strict inequalities, meaning that ''a'' is strictly less than or strictly greater than ''b''. Equivalence is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: * The notation ''a'' ≤ ''b'' or ''a'' ⩽ ''b'' means that ''a'' is less than or equal to ''b'' (or, equivalently, at most ''b'', or not greater than ''b''). * The notation ''a'' ≥ ''b'' or ''a'' ⩾ ''b'' means that ''a'' is greater than or equal to ''b'' (or, equivalently, at least ''b'', or not less than ''b''). The re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harold S
Harold may refer to: People * Harold (given name), including a list of persons and fictional characters with the name * Harold (surname), surname in the English language * András Arató, known in meme culture as "Hide the Pain Harold" Arts and entertainment * Harold (film), ''Harold'' (film), a 2008 comedy film * ''Harold'', an 1876 poem by Alfred, Lord Tennyson * ''Harold, the Last of the Saxons'', an 1848 book by Edward Bulwer-Lytton, 1st Baron Lytton * ''Harold or the Norman Conquest'', an opera by Frederic Cowen * ''Harold'', an 1885 opera by Eduard Nápravník * Harold, a character from the cartoon List of The Grim Adventures of Billy & Mandy characters#Harold, ''The Grim Adventures of Billy & Mandy'' *Harold & Kumar, a US movie; Harold/Harry is the main actor in the show. Places ;In the United States * Alpine, Los Angeles County, California, an erstwhile settlement that was also known as Harold * Harold, Florida, an unincorporated community * Harold, Kentucky, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal number, cardinal numbers'', and numbers used for ordering are called ''Ordinal number, ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports Number (sports), jersey numbers). Some definitions, including the standard ISO/IEC 80000, ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Number
In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it may be considered both positive and negative (having both signs). Whenever not specifically mentioned, this article adheres to the first convention. In some contexts, it makes sense to consider a signed zero (such as floating-point representations of real numbers within computers). In mathematics and physics, the phrase "change of sign" is associated with the generation of the additive inverse (negation, or multiplication by −1) of any object that allows for this construction, and is not restricted to real numbers. It applies among other objects to vectors, matrices, and complex numbers, which are not prescribed to be only either positive, negative, or zero. The word "sign" is also often used to indicate other binary aspects of mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Drinfeld
Vladimir Gershonovich Drinfeld ( uk, Володи́мир Ге́ршонович Дрінфельд; russian: Влади́мир Ге́ршонович Дри́нфельд; born February 14, 1954), surname also romanized as Drinfel'd, is a renowned mathematician from the former USSR, who emigrated to the United States and is currently working at the University of Chicago. Drinfeld's work connected algebraic geometry over finite fields with number theory, especially the theory of automorphic forms, through the notions of elliptic module and the theory of the geometric Langlands correspondence. Drinfeld introduced the notion of a quantum group (independently discovered by Michio Jimbo at the same time) and made important contributions to mathematical physics, including the ADHM construction of instantons, algebraic formalism of the quantum inverse scattering method, and the Drinfeld–Sokolov reduction in the theory of solitons. He was awarded the Fields Medal in 1990. In 2016, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
