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Anand Pillay
Anand Pillay (born 7 May 1951) is a British mathematician and logician working in model theory and its applications in algebra and number theory. Biography Pillay studied as an undergraduate at the University of Oxford, obtaining a Bachelor in Mathematics and Philosophy in 1973 at Balliol College. At the University of London, he received his master's degree in mathematics in 1974 and his PhD in 1978 with Wilfrid Hodges at Bedford College, titled ''Gaifman Operations, Minimal Models, and the Number of Countable Models''. In 1978, he was a Royal Society Fellow and visiting scientist at CNRS at Paris Diderot University. After teaching at the University of Manchester starting in 1981 and at McGill University in Canada, he joined the University of Notre Dame as an assistant professor in 1983, where he became an associate professor in 1986 and a full professor in 1988. From 1996 to 2006, he was Swanlund Professor at the University of Illinois Urbana-Champaign, where he is now P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Algebra
In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with finitely many derivations, which are unary functions that are linear and satisfy the Leibniz product rule. A natural example of a differential field is the field of rational functions in one variable over the complex numbers, \mathbb(t), where the derivation is differentiation with respect to t. Differential algebra refers also to the area of mathematics consisting in the study of these algebraic objects and their use in the algebraic study of differential equations. Differential algebra was introduced by Joseph Ritt in 1950. Open problems The biggest open problems in the field include the Kolchin Catenary Conjecture, the Ritt Problem, and The Jacobi Bound Problem. All of these deal with the structure of differential ideals in differential rings. Differential ring A ''differential ring'' is a ring R equipped with one or more ''derivations'', whi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
British People
British people or Britons, also known colloquially as Brits, are the citizens of the United Kingdom of Great Britain and Northern Ireland, the British Overseas Territories, and the Crown dependencies.: British nationality law governs modern British citizenship and nationality, which can be acquired, for instance, by descent from British nationals. When used in a historical context, "British" or "Britons" can refer to the Ancient Britons, the indigenous inhabitants of Great Britain and Brittany, whose surviving members are the modern Welsh people, Cornish people, and Bretons. It also refers to citizens of the former British Empire, who settled in the country prior to 1973, and hold neither UK citizenship nor nationality. Though early assertions of being British date from the Late Middle Ages, the Union of the Crowns in 1603 and the creation of the Kingdom of Great Britain in 1707 triggered a sense of British national identity.. The notion of Britishness and a shared Brit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Notre Dame
The University of Notre Dame du Lac, known simply as Notre Dame ( ) or ND, is a private Catholic research university in Notre Dame, Indiana, outside the city of South Bend. French priest Edward Sorin founded the school in 1842. The main campus covers 1,261 acres (510 ha) in a suburban setting and contains landmarks such as the Golden Dome, the ''Word of Life'' mural (commonly known as ''Touchdown Jesus''), Notre Dame Stadium, and the Basilica. Originally for men, although some women earned degrees in 1918, the university began formally accepting undergraduate female students in 1972. Notre Dame has been recognized as one of the top universities in the United States. The university is organized into seven schools and colleges. Notre Dame's graduate program includes more than 50 master, doctoral and professional degrees offered by the six schools, including the Notre Dame Law School and an MD–PhD program offered in combination with the Indiana University School of Medicine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. Such relations are common; therefore, differential equations play a prominent role in many disciplines including engineering, physics, economics, and biology. Mainly the study of differential equations consists of the study of their solutions (the set of functions that satisfy each equation), and of the properties of their solutions. Only the simplest differential equations are solvable by explicit formulas; however, many properties of solutions of a given differential equation may be determined without computing them exactly. Often when a closed-form expression for the solutions is not available, solutions may be approximated numerically using computers. The theory of d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set and an Binary operation, operation that combines any two Element (mathematics), elements of the set to produce a third element of the set, in such a way that the operation is Associative property, associative, an identity element exists and every element has an Inverse element, inverse. These three axioms hold for Number#Main classification, number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nash Manifold
In real algebraic geometry, a Nash function on an open semialgebraic subset ''U'' ⊂ R''n'' is an analytic function ''f'': ''U'' → R satisfying a nontrivial polynomial equation ''P''(''x'',''f''(''x'')) = 0 for all ''x'' in ''U'' (A semialgebraic subset of R''n'' is a subset obtained from subsets of the form or , where ''P'' is a polynomial, by taking finite unions, finite intersections and complements). Some examples of Nash functions: *Polynomial and regular rational functions are Nash functions. *x\mapsto \sqrt is Nash on R. *the function which associates to a real symmetric matrix its ''i''-th eigenvalue (in increasing order) is Nash on the open subset of symmetric matrices with no multiple eigenvalue. Nash functions are those functions needed in order to have an implicit function theorem in real algebraic geometry. Nash manifolds Along with Nash functions one defines Nash manifolds, which are semialgebraic analytic submanifolds of some R''n''. A Nash mapping between Na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stability Theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance. In dynamical systems, an orbit is called ''Lyapunov stable'' if the forward orbit of any point is in a small enough neighborhood or it stays in a small (but perhaps, larger) neighborhood. Various criteria have been developed to prove stability or instability of an orbit. Under favorable circumstances, the question may be reduced to a well-studied problem involvi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theory (mathematical Logic)
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios, a deductive system is first understood from context, after which an element \phi\in T of a deductively closed theory T is then called a theorem of the theory. In many deductive systems there is usually a subset \Sigma \subseteq T that is called "the set of axioms" of the theory T, in which case the deductive system is also called an "axiomatic system". By definition, every axiom is automatically a theorem. A first-order theory is a set of first-order sentences (theorems) recursively obtained by the inference rules of the system applied to the set of axioms. General theories (as expressed in formal language) When defining theories for foundational purposes, additional care must be taken, as normal set-theoretic language may not be appropriate. The construction of a theory begins by specifying a definite non-empty ''conceptual class'' \mathcal, the element ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model (logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations that are defined on it. Universal algebra studies structures that generalize the algebraic structures such as groups, rings, fields and vector spaces. The term universal algebra is used for structures with no relation symbols. Model theory has a different scope that encompasses more arbitrary theories, including foundational structures such as models of set theory. From the model-theoretic point of view, structures are the objects used to define the semantics of first-order logic. For a given theory in model theory, a structure is called a model if it satisfies the defining axioms of that theory, although it is sometimes disambiguated as a ''semantic model'' when one discusses the notion in the more general setting of mathematical models. Logicians sometimes refer to structures as " interpretations", whereas the term "interpretation" generally has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Countable
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements. In more technical terms, assuming the axiom of countable choice, a set is ''countable'' if its cardinality (its number of elements) is not greater than that of the natural numbers. A countable set that is not finite is said countably infinite. The concept is attributed to Georg Cantor, who proved the existence of uncountable sets, that is, sets that are not countable; for example the set of the real numbers. A note on terminology Although the terms "countable" and "countably infinite" as defined here are quite comm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isaac Newton Institute
The Isaac Newton Institute for Mathematical Sciences is an international research institute for mathematics and its many applications at the University of Cambridge. It is named after one of the university's most illustrious figures, the mathematician and natural philosopher Sir Isaac Newton and occupies one of the buildings in the Cambridge Centre for Mathematical Sciences. History After a national competition run by SERC, the Science and Engineering Research Council (now known as EPSRC Engineering and Physical Sciences Research Council), this institute was chosen to be the national research institute for mathematical sciences in the UK. It opened in 1992 with support from St John's College and Trinity College. St. John's provided the land and a purpose-built building, Trinity provided running costs for the first five years and the London Mathematical Society provided other support. Shortly afterwards at the institute, the British mathematician Andrew Wiles announced hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Sciences Research Institute
The Simons Laufer Mathematical Sciences Institute (SLMath), formerly the Mathematical Sciences Research Institute (MSRI), is an independent nonprofit mathematical research institution on the University of California campus in Berkeley, California. It is widely regarded as a world leading mathematical center for collaborative research, drawing thousands of leading researchers from around the world each year. The institute was founded in 1982, and its funding sources include the National Science Foundation, private foundations, corporations, and more than 90 universities and institutions. The institute is located at 17 Gauss Way on the Berkeley campus, close to Grizzly Peak in the Berkeley Hills. Because of its contribution to the nation's scientific potential, SLMath's activity is supported by the National Science Foundation and the National Security Agency. Private individuals, foundations, and nearly 100 Academic Sponsor Institutions, including the top mathematics departm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
