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Mutually Orthogonal Latin Squares
In combinatorics, two Latin squares of the same size (''order'') are said to be ''orthogonal'' if when superimposed the ordered paired entries in the positions are all distinct. A set of Latin squares, all of the same order, all pairs of which are orthogonal is called a set of mutually orthogonal Latin squares. This concept of orthogonality in combinatorics is strongly related to the concept of blocking in statistics, which ensures that independent variables are truly independent with no hidden confounding correlations. "Orthogonal" is thus synonymous with "independent" in that knowing one variable's value gives no further information about another variable's likely value. An outdated term for pair of orthogonal Latin squares is ''Graeco-Latin square'', found in older literature. Graeco-Latin squares A Graeco-Latin square or Euler square or pair of orthogonal Latin squares of order over two sets and (which may be the same), each consisting of symbols, is an arrangement of ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaston Tarry
Gaston Tarry (27 September 1843 – 21 June 1913) was a French people, French mathematician. Born in Villefranche de Rouergue, Aveyron, he studied mathematics at high school before joining the civil service in Algeria. He pursued mathematics as an amateur. In 1901 Tarry confirmed Leonhard Euler's conjecture that no 6×6 Graeco-Latin square#Thirty-six officers problem, Graeco-Latin square was possible (the Mutually_orthogonal_Latin_squares#Thirty-six_officers_problem, 36 officers problem). See also *List of amateur mathematicians *Prouhet-Tarry-Escott problem *Tarry point *Tetramagic square References External links * * * People from Villefranche-de-Rouergue 1843 births 1913 deaths Combinatorialists 19th-century French mathematicians 20th-century French mathematicians {{France-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kathleen Ollerenshaw
Dame Kathleen Mary Ollerenshaw, (''née'' Timpson; 1 October 1912 – 10 August 2014) was a British mathematician and politician who was Lord Mayor of Manchester from 1975 to 1976 and an advisor on educational matters to Margaret Thatcher's government in the 1980s. Early life and education She was born Kathleen Mary Timpson in Withington, Manchester, where she attended Lady Barn House School (1918–26). She was a grandchild of the founder of the Timpson shoe repair business, who had moved to Manchester from Kettering and established the business there by 1870. She became fascinated with mathematics, inspired by the Lady Barn headmistress, Miss Jenkin Jones. While at Lady Barn, she met her future husband, Robert Ollerenshaw. Ollerenshaw became completely deaf at age eight and was taught to lip read. She gravitated toward the study of mathematics as it is not dependent on hearing. She was further inspired by a headmistress at Lady Barn House School who studied mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
AN/USQ-20
The AN/USQ-20, or CP-642 or Naval Tactical Data System (NTDS), was designed as a more reliable replacement for the Seymour Cray-designed AN/USQ-17 with the same instruction set. The first batch of 17 computers were delivered to the Navy starting in early 1961. A version of the AN/USQ-20 for use by the other military services and NASA was designated the UNIVAC 1206. Another version, designated the G-40, replaced the vacuum tube UNIVAC 1104 in the BOMARC Missile Program. The machine was the size and shape of an old-fashioned double-door refrigerator, about six feet tall (roughly 1.80 meters). Instructions were represented as 30-bit words in the following format: f 6 bits function code j 3 bits jump condition designator k 3 bits partial word designator b 3 bits which index register to use y 15 bits operand address in memory Numbers were represented as 30-bit words. This allowed for five 6-bit alphanumeric characters per word. The main me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Proof By Exhaustion
Proof by exhaustion, also known as proof by cases, proof by case analysis, complete induction or the brute force method, is a method of mathematical proof in which the statement to be proved is split into a finite number of cases or sets of equivalent cases, and where each type of case is checked to see if the proposition in question holds. This is a method of direct proof. A proof by exhaustion typically contains two stages: # A proof that the set of cases is exhaustive; i.e., that each instance of the statement to be proved matches the conditions of (at least) one of the cases. # A proof of each of the cases. The prevalence of digital computers has greatly increased the convenience of using the method of exhaustion (e.g., the first computer-assisted proof of four color theorem in 1976), though such approaches can also be challenged on the basis of mathematical elegance. Expert systems can be used to arrive at answers to many of the questions posed to them. In theory, the proof ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book '' Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singly And Doubly Even
In mathematics an even integer, that is, a number that is divisible by 2, is called evenly even or doubly even if it is a multiple of 4, and oddly even or singly even if it is not. The former names are traditional ones, derived from ancient Greek mathematics; the latter have become common in recent decades. These names reflect a basic concept in number theory, the 2-order of an integer: how many times the integer can be divided by 2. This is equivalent to the multiplicity of 2 in the prime factorization. *A singly even number can be divided by 2 only once; it is even but its quotient by 2 is odd. *A doubly even number is an integer that is divisible more than once by 2; it is even and its quotient by 2 is also even. The separate consideration of oddly and evenly even numbers is useful in many parts of mathematics, especially in number theory, combinatorics, coding theory (see even codes), among others. Definitions The ancient Greek terms "even-times-even" ( grc, ἀρτι� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scientific American November 1959 Graeco Latin Square
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe. Science may be as old as the human species, and some of the earliest archeological evidence for scientific reasoning is tens of thousands of years old. The earliest written records in the history of science come from Ancient Egypt and Mesopotamia in around 3000 to 1200 BCE. Their contributions to mathematics, astronomy, and medicine entered and shaped Greek natural philosophy of classical antiquity, whereby formal attempts were made to provide explanations of events in the physical world based on natural causes. After the fall of the Western Roman Empire, knowledge of Greek conceptions of the world deteriorated in Western Europe during the early centuries (400 to 1000 CE) of the Middle Ages, but was preserved in the Muslim world during the Islamic Golden Age and later by the efforts of Byzantine Greek scholars who brought Greek man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catherine The Great
, en, Catherine Alexeievna Romanova, link=yes , house = , father = Christian August, Prince of Anhalt-Zerbst , mother = Joanna Elisabeth of Holstein-Gottorp , birth_date = , birth_name = Princess Sophie of Anhalt-Zerbst , birth_place = Stettin, Pomerania, Prussia, Holy Roman Empire(now Szczecin, Poland) , death_date = (aged 67) , death_place = Winter Palace, Saint Petersburg, Russian Empire , burial_date = , burial_place = Saints Peter and Paul Cathedral, Saint Petersburg , signature = Catherine The Great Signature.svg , religion = Catherine II (born Sophie of Anhalt-Zerbst; 2 May 172917 November 1796), most commonly known as Catherine the Great, was the reigning empress of Russia from 1762 to 1796. She came to power following the overthrow of her husband, Peter III. Under her long reign, inspired by the ideas of the Enlightenment, Russia experienced a renaissance of culture and sciences, which led to the founding o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
36 Officers Problem
In combinatorics, two Latin squares of the same size (''order'') are said to be ''orthogonal'' if when superimposed the ordered paired entries in the positions are all distinct. A set of Latin squares, all of the same order, all pairs of which are orthogonal is called a set of mutually orthogonal Latin squares. This concept of orthogonality in combinatorics is strongly related to the concept of blocking in statistics, which ensures that independent variables are truly independent with no hidden confounding correlations. "Orthogonal" is thus synonymous with "independent" in that knowing one variable's value gives no further information about another variable's likely value. An outdated term for pair of orthogonal Latin squares is ''Graeco-Latin square'', found in older literature. Graeco-Latin squares A Graeco-Latin square or Euler square or pair of orthogonal Latin squares of order over two sets and (which may be the same), each consisting of symbols, is an arrangement of ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Up To
Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' are equal. This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count. For example, ''x'' is unique up to ''R'' means that all objects ''x'' under consideration are in the same equivalence class with respect to the relation ''R''. Moreover, the equivalence relation ''R'' is often designated rather implicitly by a generating condition or transformation. For example, the statement "an integer's prime factorization is unique up to ordering" is a concise way to say that any two lists of prime factors of a given integer are equivalent with respect to the relation ''R'' that relates two lists if one can be obtained by reordering (permutation) from the other. As another example, the sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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