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Raj Chandra Bose
Raj Chandra Bose (or Basu) (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 for no ''n'' do there exist two mutually orthogonal Latin squares of order 4''n'' + 2. Early life Bose was born in Hoshangabad, India into a Bengali family; 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 ci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Narmadapuram
Hoshangabad (Hindi: ), officially Narmadapuram (), is a city 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. Hoshangabad is from the capital of Madhya Pradesh 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 was renamed to Narmadapuram in March 2021. Geography Hoshangabad is located at . It has an average elevation of . Climate The climate of Hoshangabad district Hoshangabad district (), officially Narmadapuram district (), is one of the districts of Madhya Pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Colorado State University
Colorado State University (Colorado State or CSU) is a Public university, public Land-grant university, land-grant research university in Fort Collins, Colorado, United States. It is the flagship university of the Colorado State University System. It was founded in 1870 as Colorado Agricultural College and assumed its current name in 1957. In 2024, enrollment was approximately 34,000 students, including resident and non-resident instruction students. The university has approximately 1,500 faculty in 8 colleges and 55 academic departments. Bachelor's degrees are offered in 65 fields of study and master's degrees are offered in 55 fields. Colorado State confers doctoral degrees in 40 fields of study, in addition to a professional degree in veterinary medicine. In fiscal year 2023, CSU spent $498.1 million on research and development. It is Carnegie Classification of Institutions of Higher Education, classified among "R1: Doctoral Universities – Very high research activity". CS ... [...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 or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Resolution of conjectures Proof Formal mathematics is based on ''provable'' truth. In mathematics, any number of cases supporting a universally quantified conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 101 ... [...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 number of occurrences of each element satisfies certain conditions making the collection of blocks exhibit symmetry (balance). Block designs 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Difference Set
In combinatorics, a (v,k,\lambda) difference set is a subset D of cardinality, size k of a group (mathematics), group G of order of a group, order v such that every non-identity element, identity 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 non-zero 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 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strongly Regular Graph
In graph theory, a strongly regular graph (SRG) is a regular graph with vertices and degree such that for some given integers \lambda, \mu \ge 0 * every two adjacent vertices have common neighbours, and * every two non-adjacent vertices have common neighbours. Such a strongly regular graph is denoted by . Its complement graph is also strongly regular: it is an . 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 as 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 bu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Association Scheme
The theory of association schemes arose in statistics, in the theory of design of experiments, 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, the theory of error-correcting codes. In algebra, the theory of association schemes generalizes the group character, character theory of group representation, linear representations of groups. Definition An ''n''-class association scheme consists of a Set (mathematics), set ''X'' together with a partition of a set, partition ''S'' of ''X'' × ''X'' into ''n'' + 1 binary relations, ''R''0, ''R''1, ..., ''R''''n'' which satisfy: *R_ = \; it 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) \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Geometry
An incidence structure C=(P,L,I) consists of a set of points, a set of lines, and an incidence relation, or set of flags, I \subseteq P \times L; a point p is said to be ''incident'' with a line l if . 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 , 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 , such that p is incident with m and q is incident with . A partial geometry with these parameters is denoted by . Properties * The number of points is given by \frac and the number of lines by . * The point graph (also known as the collinearity graph) of a \mathrm(s,t,\alpha) is a strongly regular graph: . * Partial geometries are dualizable structures: the dual of a \mathrm(s,t,\alpha) is simply a . Special cases * The generalized quadrangl ... [...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 a '' 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Error-correcting Code
In computing, telecommunication, information theory, and coding theory, forward error correction (FEC) or channel coding is a technique used for controlling errors in data transmission over unreliable or noisy communication channels. The central idea is that the sender encodes the message in a redundant way, most often by using an error correction code, or error correcting code (ECC). The redundancy allows the receiver not only to detect errors that may occur anywhere in the message, but often to correct a limited number of errors. Therefore a reverse channel to request re-transmission may not be needed. The cost is a fixed, higher forward channel bandwidth. 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. FEC can be applied in situations where re-transmissions are costly or impossible, such as one-way communication links or when transmitting to multiple receivers in m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Geometry
A finite geometry is any geometry, geometric system that has only a finite set, finite number of point (geometry), points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite projective space, projective and affine spaces because of their regularity and simplicity. Other significant types of finite geometry are finite Möbius plane, Möbius or inversive planes and Laguerre planes, which are examples of a general type called Benz planes, and their higher-dimensional analogs such as higher finite inversive geometry, inversive geometries. Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field; the affine and projective planes so const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Design Of Experiments
The design of experiments (DOE), also known as experiment design 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
