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Raj Chandra Bose
Raj Chandra Bose (19 June 1901 – 31 October 1987) was an Indian American mathematician and statistician best known for his work in design theory, finite geometry and the theory of error-correcting codes in which the class of BCH codes is partly named after him. He also invented the notions of partial geometry, association scheme, and strongly regular graph and started a systematic study of difference sets to construct symmetric block designs. He was notable for his work along with S. S. Shrikhande and E. T. Parker in their disproof of the famous conjecture made by Leonhard Euler dated 1782 that there do not exist two mutually orthogonal Latin squares of order 4''n'' + 2 for every ''n''. Early life Bose was born in Hoshangabad, India; he was the first of five children. His father was a physician and life was good until 1918 when his mother died in the influenza pandemic. His father died of a stroke the following year. Despite difficult circumstances, Bose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hoshangabad
Hoshangabad, officially known as Narmadapuram is a city and municipality in the Indian state of Madhya Pradesh. It serves as the headquarters of both Hoshangabad district and Narmadapuram division. It is located in central India, on the south bank of the Narmada River. Narmapuram is from the state capital and the nearest airport Bhopal. History The city was earlier called Narmadapur after the Narmada river. Later the name was changed to Hoshangabad after Hoshang Shah Gori, the first ruler of Malwa Sultanate. Hoshangabad district was part of the Nerbudda (Narmada) Division of the Central Provinces and Berar, which became the state of Madhya Bharat (later Madhya Pradesh) after India's independence in 1947. The city is famous for its beautiful ghats along the banks of the Narmada river, Sethani ghat is a major attraction. There are colourful celebrations in the city on Narmada Jayanti. During celebrations this year the CM announced the efforts to rename the town. A Satsan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indian Statistical Institute
Indian Statistical Institute (ISI) is a higher education and research institute which is recognized as an Institute of National Importance by the 1959 act of the Indian parliament. It grew out of the Statistical Laboratory set up by Prasanta Chandra Mahalanobis in Presidency College, Kolkata. Established in 1931, this unique institution of India is one of the oldest institutions focused on statistics, and its early reputation led it to being adopted as a model for the first US institute of statistics set up at the Research Triangle, North Carolina by Gertrude Mary Cox. Mahalanobis, the founder of ISI, was deeply influenced by the wisdom and guidance of Rabindranath Tagore and Brajendranath Seal. Under his leadership, the institute initiated and promoted the interaction of statistics with natural and social sciences to advance the role of statistics as a key technology by explicating the twin aspectsits general applicability and its dependence on other disciplines for its own ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of '' Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Block Design
In combinatorial mathematics, a block design is an incidence structure consisting of a set together with a family of subsets known as ''blocks'', chosen such that frequency of the elements satisfies certain conditions making the collection of blocks exhibit symmetry (balance). They have applications in many areas, including experimental design, finite geometry, physical chemistry, software testing, cryptography, and algebraic geometry. Without further specifications the term ''block design'' usually refers to a balanced incomplete block design (BIBD), specifically (and also synonymously) a 2-design, which has been the most intensely studied type historically due to its application in the design of experiments. Its generalization is known as a t-design. Overview A design is said to be ''balanced'' (up to ''t'') if all ''t''-subsets of the original set occur in equally many (i.e., ''λ'') blocks. When ''t'' is unspecified, it can usually be assumed to be 2, which means that e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Difference Set
In combinatorics, a (v,k,\lambda) difference set is a subset D of size k of a group G of order v such that every nonidentity element of G can be expressed as a product d_1d_2^ of elements of D in exactly \lambda ways. A difference set D is said to be ''cyclic'', ''abelian'', ''non-abelian'', etc., if the group G has the corresponding property. A difference set with \lambda = 1 is sometimes called ''planar'' or ''simple''. If G is an abelian group written in additive notation, the defining condition is that every nonzero element of G can be written as a ''difference'' of elements of D in exactly \lambda ways. The term "difference set" arises in this way. Basic facts * A simple counting argument shows that there are exactly k^2-k pairs of elements from D that will yield nonidentity elements, so every difference set must satisfy the equation k^2-k=(v-1)\lambda. * If D is a difference set, and g\in G, then gD=\ is also a difference set, and is called a translate of D (D + g in additi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strongly Regular Graph
In graph theory, a strongly regular graph (SRG) is defined as follows. Let be a regular graph with vertices and degree . is said to be strongly regular if there are also integers and such that: * Every two adjacent vertices have common neighbours. * Every two non-adjacent vertices have common neighbours. The complement of an is also strongly regular. It is a . A strongly regular graph is a distance-regular graph with diameter 2 whenever μ is non-zero. It is a locally linear graph whenever . Etymology A strongly regular graph is denoted an srg(''v'', ''k'', λ, μ) in the literature. By convention, graphs which satisfy the definition trivially are excluded from detailed studies and lists of strongly regular graphs. These include the disjoint union of one or more equal-sized complete graphs, and their complements, the complete multipartite graphs with equal-sized independent sets. Andries Brouwer and Hendrik van Maldeghem (see #References) use an alternate but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Association Scheme
The theory of association schemes arose in statistics, in the theory of experimental design for the analysis of variance. In mathematics, association schemes belong to both algebra and combinatorics. In algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory. In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups. Definition An ''n''-class association scheme consists of a set ''X'' together with a partition ''S'' of ''X'' × ''X'' into ''n'' + 1 binary relations, ''R''0, ''R''1, ..., ''R''''n'' which satisfy: *R_ = \ and is called the identity relation. *Defining R^* := \, if ''R'' in ''S'', then ''R*'' in ''S'' *If (x,y) \in R_, the number of z \in X such that (x,z) \in R_ and (z,y) \in R_ is a constant p^k_ depending on i, j, k but not on the particular choice of x and y. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Geometry
An incidence structure C=(P,L,I) consists of points P, lines L, and flags I \subseteq P \times L where a point p is said to be incident with a line l if (p,l) \in I. It is a (finite) partial geometry if there are integers s,t,\alpha\geq 1 such that: * For any pair of distinct points p and q, there is at most one line incident with both of them. * Each line is incident with s+1 points. * Each point is incident with t+1 lines. * If a point p and a line l are not incident, there are exactly \alpha pairs (q,m)\in I, such that p is incident with m and q is incident with l. A partial geometry with these parameters is denoted by pg(s,t,\alpha). Properties * The number of points is given by \frac and the number of lines by \frac. * The point graph (also known as the collinearity graph) of a pg(s,t,\alpha) is a strongly regular graph: srg((s+1)\frac,s(t+1),s-1+t(\alpha-1),\alpha(t+1)). * Partial geometries are dual structures: the dual of a pg(s,t,\alpha) is simply a pg(t,s,\alpha). Speci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
BCH Code
In coding theory, the Bose–Chaudhuri–Hocquenghem codes (BCH codes) form a class of cyclic error-correcting codes that are constructed using polynomials over a finite field (also called ''Galois field''). BCH codes were invented in 1959 by French mathematician Alexis Hocquenghem, and independently in 1960 by Raj Chandra Bose and D.K. Ray-Chaudhuri. The name ''Bose–Chaudhuri–Hocquenghem'' (and the acronym ''BCH'') arises from the initials of the inventors' surnames (mistakenly, in the case of Ray-Chaudhuri). One of the key features of BCH codes is that during code design, there is a precise control over the number of symbol errors correctable by the code. In particular, it is possible to design binary BCH codes that can correct multiple bit errors. Another advantage of BCH codes is the ease with which they can be decoded, namely, via an algebraic method known as syndrome decoding. This simplifies the design of the decoder for these codes, using small low-pow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Error-correcting Code
In computing, telecommunication, information theory, and coding theory, an error correction code, sometimes error correcting code, (ECC) is used for controlling errors in data over unreliable or noisy communication channels. The central idea is the sender encodes the message with redundant information in the form of an ECC. The redundancy allows the receiver to detect a limited number of errors that may occur anywhere in the message, and often to correct these errors without retransmission. The American mathematician Richard Hamming pioneered this field in the 1940s and invented the first error-correcting code in 1950: the Hamming (7,4) code. ECC contrasts with error detection in that errors that are encountered can be corrected, not simply detected. The advantage is that a system using ECC does not require a reverse channel to request retransmission of data when an error occurs. The downside is that there is a fixed overhead that is added to the message, thereby requiring a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Geometry
Finite is the opposite of infinite. It may refer to: * Finite number (other) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Groves from the album ''Invisible Empires'' See also * * Nonfinite (other) Nonfinite is the opposite of finite * a nonfinite verb is a verb that is not capable of serving as the main verb in an independent clause * a non-finite clause In linguistics, a non-finite clause is a dependent or embedded clause that represen ... {{disambiguation fr:Fini it:Finito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Design Of Experiments
The design of experiments (DOE, DOX, or experimental design) is the design of any task that aims to describe and explain the variation of information under conditions that are hypothesized to reflect the variation. The term is generally associated with experiments in which the design introduces conditions that directly affect the variation, but may also refer to the design of quasi-experiments, in which natural conditions that influence the variation are selected for observation. In its simplest form, an experiment aims at predicting the outcome by introducing a change of the preconditions, which is represented by one or more independent variables, also referred to as "input variables" or "predictor variables." The change in one or more independent variables is generally hypothesized to result in a change in one or more dependent variables, also referred to as "output variables" or "response variables." The experimental design may also identify control variables that must b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
