Chinese remainder theorem
The Chinese remainder theorem is a theorem of number theory, which states that if one knows the remainders of the Euclidean division of an integer n by several integers one can determine uniquely the remainder of the division of n by the product of these integers, under the condition that the divisors are pairwise coprime. The earliest known statement of the theorem is by the Chinese mathematician Sunzi in Sunzi Suanjing in the 3rd century AD; the Chinese remainder theorem is used for computing with large integers, as it allows replacing a computation for which one knows a bound on the size of the result by several similar computations on small integers. The Chinese remainder theorem is true over every principal ideal domain, it has been generalized with a formulation involving ideals. The earliest known statement of the theorem, as a problem with specific numbers, appears in the 3rd-century book Sunzi Suanjing by the Chinese mathematician Sunzi: There are certain things whose number is unknown.
If we count them by threes, we have two left over. How many things are there? Sunzi's work contains neither a full algorithm. What amounts to an algorithm for solving this problem was described by Aryabhata. Special cases of the Chinese remainder theorem were known to Brahmagupta, appear in Fibonacci's Liber Abaci; the result was generalized with a complete solution called Dayanshu in Qin Jiushao's 1247 Mathematical Treatise in Nine Sections. The notion of congruences was first introduced and used by Gauss in his Disquisitiones Arithmeticae of 1801. Gauss illustrates the Chinese remainder theorem on a problem involving calendars, namely, "to find the years that have a certain period number with respect to the solar and lunar cycle and the Roman indiction." Gauss introduces a procedure for solving the problem, used by Euler but was in fact an ancient method that had appeared several times. Let n1... ni... nk be integers greater than 1, which are called moduli or divisors. Let us denote by N the product of the ni.
The Chinese remainder theorem asserts that if the ni are pairwise coprime, if a1... ak are integers such that 0 ≤ ai < ni for every i there is one and only one integer x, such that 0 ≤ x < N and the remainder of the Euclidean division of x by ni is ai for every i. This may be restated as follows in term of congruences: If the ni are pairwise coprime, if a1... ak are any integers there exists an integer x such that x ≡ a 1 ⋮ x ≡ a k, any two such x are congruent modulo N. In abstract algebra, the theorem is restated as: if the ni are pairwise coprime, the map x mod N ↦ defines a ring isomorphism Z / N Z ≅ Z / n 1 Z × ⋯ × Z / n k Z between the ring of integers modulo N and the direct product of the rings of integers modulo the ni; this means that for doing a sequence of arithmetic operations in Z / N Z, one may do the same computation independently in each Z / n i Z and get the result by applying the isomorphism. This may be much faster than the direct computation if the number of operations are large.
This is used, under the name multi-modular computation, for linear algebra over the integers or the rational numbers. The theorem can be restated in the language of combinatorics as the fact that the infinite arithmetic progressions of integers form a Helly family; the existence and the uniqueness of the solution may be proven independently. However, the first proof of existence, given below, uses this uniqueness. Suppose that x and y are both solutions to all the congruences; as x and y give the same remainder, when divided by ni, their difference x − y is a multiple of each ni. As the ni are pairwise coprime, their product N divides x − y, thus x and y are congruent modulo N. If x and y are supposed to be non negative and less than N their difference may be a multiple of N only if x = y; the map x ↦ maps congruence classes modulo N
Gopal Prasad is an Indian-American mathematician. His research interests span the fields of Lie groups, their discrete subgroups, algebraic groups, arithmetic groups, geometry of locally symmetric spaces, representation theory of reductive p-adic groups, he is the Raoul Bott Professor of Mathematics at the University of Michigan in Ann Arbor. Prasad earned his bachelor's degree with honors in Mathematics from Magadh University in 1963. Two years in 1965, he received his masters in Mathematics from Patna University. After a brief stay at the Indian Institute of Technology Kanpur in their Ph. D. program for Mathematics, Prasad entered the Ph. D. program at the Tata Institute of Fundamental Research in 1966. There he began a long and extensive collaboration with his advisor M. S. Raghunathan on several topics including the study of lattices in semi-simple Lie groups. In 1976, Prasad received his Ph. D. from the University of Mumbai. Prasad became an Associate Professor at TIFR in 1979, a Professor in 1984.
In 1992 he left TIFR to join the faculty at the University of Michigan in Ann Arbor, where he is the Raoul Bott Professor of Mathematics. In 1969, he married Indu Devi of Deoria. Gopal Prasad and Indu Devi have a daughter and five grand-children. Shrawan Kumar, a professor of mathematics at the University of North Carolina at Chapel Hill, Dipendra Prasad, a professor of mathematics at the Tata Institute of Fundamental Research, are his younger brothers. Prasad's early work was on discrete subgroups of p-adic semi-simple groups, he proved the "strong rigidity" of lattices in real semi-simple groups of rank 1 and of lattices in p-adic groups, see and. He tackled group-theoretic and arithmetic questions on semi-simple algebraic groups, he proved the "strong approximation" property for connected semi-simple groups over global function fields. In collaboration with M. S. Raghunathan, Prasad determined the topological central extensions of these groups, computed the "metaplectic kernel" for isotropic groups, and.
Together with Andrei Rapinchuk, Prasad gave a precise computation of the metaplectic kernel for all connected semi-simple groups, see. Prasad and Raghunathan have obtained results on the Kneser-Tits problem. In 1987, Prasad found a formula for the volume of S-arithmetic quotients of semi-simple groups. Using this formula and certain number theoretic and Galois-cohomological estimates, Armand Borel and Gopal Prasad proved several finiteness theorems about arithmetic groups; the volume formula, together with number-theoretic and Bruhat-Tits theoretic considerations led to a classification, by Gopal Prasad and Sai-Kee Yeung, of fake projective planes into 28 non-empty classes. This classification, together with computations by Donald Cartwright and Tim Steger, has led to a complete list of fake projective planes; this list consists of 50 fake projective planes, up to isometry. This work was the subject of a talk in the Bourbaki seminar. Prasad has worked on the representation theory of reductive p-adic groups with Allen Moy.
The filtrations of parahoric subgroups, referred to as the "Moy-Prasad filtration", is used in representation theory and harmonic analysis. Moy and Prasad used these filtrations and Bruhat-Tits theory to prove the existence of "unrefined minimal K-types", to define the notion of "depth" of an irreducible admissible representation and to give a classification of representations of depth zero, see and. In collaboration with Andrei Rapinchuk, Prasad has studied Zariski-dense subgroups of semi-simple groups and proved the existence in such a subgroup of regular semi-simple elements with many desirable properties; these elements have been used in the investigation of geometric and ergodic theoretic questions. Prasad and Rapinchuk introduced a new notion of "weak-commensurability" of arithmetic subgroups and determined "weak- commensurability classes" of arithmetic groups in a given semi-simple group, they used their results on weak-commensurability to obtain results on length-commensurable and isospectral arithmetic locally symmetric spaces, and.
Together with Jiu-Kang Yu, Prasad has studied the fixed point set under the action of a finite group of automorphisms of a reductive p-adic group G on the Bruhat-Building of G. In another joint work, Prasad and Yu determined all the quasi-reductive group schemes over a discrete valuation ring. In collaboration with Brian Conrad and Ofer Gabber, Prasad has studied the structure of pseudo-reductive groups, provided proofs of the conjugacy theorems for general smooth connected linear algebraic groups, announced without detailed proofs by Armand Borel and Jacques Tits; the monograph contains a complete classification of pseudo-reductive groups, including a Tits-style classification and many interesting examples. The classification of pseudo-reductive groups has many applications. There was a Bourbaki seminar in March 2010 on the work of Tits, Conrad-Gabber-Prasad on pseudo-reductive groups. Prasad has received the Guggenheim Fellowship, the Humboldt Senior Research Award, the Raoul Bott Professorship at the University of Michigan.
He was awarded the Shanti Swarup Bhatnagar prize. He has received Fellowships in the Indian National Science Academy, the Indian Academy of Sciences and the American Mathematical Society. Prasad gave an invited talk in the International Congress of Mathematicians held in Kyoto
Martin Kneser was a German mathematician. His father Hellmuth Kneser and grandfather Adolf Kneser were mathematicians, he obtained his PhD in 1950 from Humboldt University of Berlin with the dissertation: Über den Rand von Parallelkörpern. His advisor was Erhard Schmidt, his name has been given to Kneser graphs which he studied in 1955. He gave a simplified proof of the Fundamental theorem of algebra. Kneser was an Invited Speaker of the ICM in 1962 at Stockholm, his main publications were on algebraic groups. Approximation in algebraic groups Kneser–Tits conjecture Kneser's theorem Kneser graphs Martin Kneser at the Mathematics Genealogy Project Martin Kneser’s Work on Quadratic Forms and Algebraic Groups
International Standard Serial Number
An International Standard Serial Number is an eight-digit serial number used to uniquely identify a serial publication, such as a magazine. The ISSN is helpful in distinguishing between serials with the same title. ISSN are used in ordering, interlibrary loans, other practices in connection with serial literature; the ISSN system was first drafted as an International Organization for Standardization international standard in 1971 and published as ISO 3297 in 1975. ISO subcommittee TC 46/SC 9 is responsible for maintaining the standard; when a serial with the same content is published in more than one media type, a different ISSN is assigned to each media type. For example, many serials are published both in electronic media; the ISSN system refers to these types as electronic ISSN, respectively. Conversely, as defined in ISO 3297:2007, every serial in the ISSN system is assigned a linking ISSN the same as the ISSN assigned to the serial in its first published medium, which links together all ISSNs assigned to the serial in every medium.
The format of the ISSN is an eight digit code, divided by a hyphen into two four-digit numbers. As an integer number, it can be represented by the first seven digits; the last code digit, which may be 0-9 or an X, is a check digit. Formally, the general form of the ISSN code can be expressed as follows: NNNN-NNNC where N is in the set, a digit character, C is in; the ISSN of the journal Hearing Research, for example, is 0378-5955, where the final 5 is the check digit, C=5. To calculate the check digit, the following algorithm may be used: Calculate the sum of the first seven digits of the ISSN multiplied by its position in the number, counting from the right—that is, 8, 7, 6, 5, 4, 3, 2, respectively: 0 ⋅ 8 + 3 ⋅ 7 + 7 ⋅ 6 + 8 ⋅ 5 + 5 ⋅ 4 + 9 ⋅ 3 + 5 ⋅ 2 = 0 + 21 + 42 + 40 + 20 + 27 + 10 = 160 The modulus 11 of this sum is calculated. For calculations, an upper case X in the check digit position indicates a check digit of 10. To confirm the check digit, calculate the sum of all eight digits of the ISSN multiplied by its position in the number, counting from the right.
The modulus 11 of the sum must be 0. There is an online ISSN checker. ISSN codes are assigned by a network of ISSN National Centres located at national libraries and coordinated by the ISSN International Centre based in Paris; the International Centre is an intergovernmental organization created in 1974 through an agreement between UNESCO and the French government. The International Centre maintains a database of all ISSNs assigned worldwide, the ISDS Register otherwise known as the ISSN Register. At the end of 2016, the ISSN Register contained records for 1,943,572 items. ISSN and ISBN codes are similar in concept. An ISBN might be assigned for particular issues of a serial, in addition to the ISSN code for the serial as a whole. An ISSN, unlike the ISBN code, is an anonymous identifier associated with a serial title, containing no information as to the publisher or its location. For this reason a new ISSN is assigned to a serial each time it undergoes a major title change. Since the ISSN applies to an entire serial a new identifier, the Serial Item and Contribution Identifier, was built on top of it to allow references to specific volumes, articles, or other identifiable components.
Separate ISSNs are needed for serials in different media. Thus, the print and electronic media versions of a serial need separate ISSNs. A CD-ROM version and a web version of a serial require different ISSNs since two different media are involved. However, the same ISSN can be used for different file formats of the same online serial; this "media-oriented identification" of serials made sense in the 1970s. In the 1990s and onward, with personal computers, better screens, the Web, it makes sense to consider only content, independent of media; this "content-oriented identification" of serials was a repressed demand during a decade, but no ISSN update or initiative occurred. A natural extension for ISSN, the unique-identification of the articles in the serials, was the main demand application. An alternative serials' contents model arrived with the indecs Content Model and its application, the digital object identifier, as ISSN-independent initiative, consolidated in the 2000s. Only in 2007, ISSN-L was defined in the
Gregori Aleksandrovich Margulis is a Russian-American mathematician known for his work on lattices in Lie groups, the introduction of methods from ergodic theory into diophantine approximation. He was awarded a Fields Medal in 1978 and a Wolf Prize in Mathematics in 2005, becoming the seventh mathematician to receive both prizes. In 1991, he joined the faculty of Yale University, where he is the Erastus L. De Forest Professor of Mathematics. Margulis was born to a Russian Jewish family in Moscow, Soviet Union. At age 16 in 1962 he won the silver medal at the International Mathematical Olympiad, he received his PhD in 1970 from the Moscow State University, starting research in ergodic theory under the supervision of Yakov Sinai. Early work with David Kazhdan produced the Kazhdan–Margulis theorem, a basic result on discrete groups, his superrigidity theorem from 1975 clarified an area of classical conjectures about the characterisation of arithmetic groups amongst lattices in Lie groups. He was awarded the Fields Medal in 1978, but was not permitted to travel to Helsinki to accept it in person due to anti-semitism against Jewish mathematicians in the Soviet Union.
His position improved, in 1979 he visited Bonn, was able to travel though he still worked in the Institute of Problems of Information Transmission, a research institute rather than a university. In 1991, Margulis accepted a professorial position at Yale University. Margulis was elected a member of the U. S. National Academy of Sciences in 2001. In 2012 he became a fellow of the American Mathematical Society. In 2005, Margulis received the Wolf Prize for his contributions to theory of lattices and applications to ergodic theory, representation theory, number theory and measure theory. Margulis's early work dealt with Kazhdan's property and the questions of rigidity and arithmeticity of lattices in semisimple algebraic groups of higher rank over a local field, it had been known since the 1950s that a certain simple-minded way of constructing subgroups of semisimple Lie groups produces examples of lattices, called arithmetic lattices. It is analogous to considering the subgroup SL of the real special linear group SL that consists of matrices with integer entries.
Margulis proved that under suitable assumptions on G, any lattice Γ in it is arithmetic, i.e. can be obtained in this way. Thus Γ is commensurable with the subgroup G of G, i.e. they agree on subgroups of finite index in both. Unlike general lattices, which are defined by their properties, arithmetic lattices are defined by a construction. Therefore, these results of Margulis pave a way for classification of lattices. Arithmeticity turned out to be related to another remarkable property of lattices discovered by Margulis. Superrigidity for a lattice Γ in G means that any homomorphism of Γ into the group of real invertible n × n matrices extends to the whole G; the name derives from the following variant: If G and G' are semisimple algebraic groups over a local field without compact factors and whose split rank is at least two and Γ and Γ ′ are irreducible lattices in them any homomorphism f: Γ → Γ ′ between the lattices agrees on a finite index subgroup of Γ with a homomorphism between the algebraic groups themselves.
While certain rigidity phenomena had been known, the approach of Margulis was at the same time novel and elegant. Margulis solved the Banach–Ruziewicz problem that asks whether the Lebesgue measure is the only normalized rotationally invariant finitely additive measure on the n-dimensional sphere; the affirmative solution for n ≥ 4, independently and simultaneously obtained by Dennis Sullivan, follows from a construction of a certain dense subgroup of the orthogonal group that has property. Margulis gave the first construction of expander graphs, generalized in the theory of Ramanujan graphs. In 1986, Margulis gave a complete resolution of the Oppenheim conjecture on quadratic forms and diophantine approximation; this was a question, open for half a century, on which considerable progress had been made by the Hardy–Littlewood circle method. He has formulated a further program of research in the same direction, that includes the Littlewood conjecture. Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, 17.
Springer-Verlag, Berlin, 1991. X+388 pp. ISBN 3-540-12179-X MR1090825 On some aspects of the theory of Anosov systems. With a survey by Richard Sharp: Periodic orbits of hyperbolic flows. Translated from the Russian by Valentina Vladimirovna Szulikowska. Springer-Verlag, Berlin, 2004. Vi+139 pp. ISBN 3-540-40121-0 MR2035655 Oppenheim conjecture. Fields Medallists' lectures, 272–327, World Sci. Ser. 20th Century Math. 5, World Sci. Publ. River Edge, NJ, 1997 MR1622909 Dynamical and ergodic properties of subgroup actions on homogeneous spaces with applications to number theory. Proceedings of the International Congress of Mathematicians, Vol. I, II, 193–215, Math. Soc. Japan, Tokyo, 1991 MR1159213 Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators. Problemy Peredachi I