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Model Theory
In mathematics, model theory is the study of classes of mathematical structures (e.g. groups, fields, graphs, universes of set theory) from the perspective of mathematical logic. The objects of study are models of theories in a formal language. A set of sentences in a formal language is called a theory; a model of a theory is a structure (e.g. an interpretation) that satisfies the sentences of that theory. Model theory recognises and is intimately concerned with a duality: it examines semantical elements (meaning and truth) by means of syntactical elements (formulas and proofs) of a corresponding language
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Computer Science
Computer science
Computer science
is the study of the theory, experimentation, and engineering that form the basis for the design and use of computers. It is the scientific and practical approach to computation and its applications and the systematic study of the feasibility, structure, expression, and mechanization of the methodical procedures (or algorithms) that underlie the acquisition, representation, processing, storage, communication of, and access to, information. An alternate, more succinct definition of computer science is the study of automating algorithmic processes that scale. A computer scientist specializes in the theory of computation and the design of computational systems.[1] Its fields can be divided into a variety of theoretical and practical disciplines
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Interdisciplinarity
Interdisciplinarity involves the combining of two or more academic disciplines into one activity (e.g., a research project). It draws knowledge from several other fields like sociology, anthropology, psychology, economics etc. It is about creating something new by thinking across boundaries. It is related to an interdiscipline or an interdisciplinary field, which is an organizational unit that crosses traditional boundaries between academic disciplines or schools of thought, as new needs and professions emerge. Large engineering teams are usually interdisciplinary, as a power station or mobile phone or other project requires the melding of several specialties
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Combinatorics
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, from evolutionary biology to computer science, etc. To fully understand the scope of combinatorics requires a great deal of further amplification, the details of which are not universally agreed upon.[1] According to H. J
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Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but the abstract formalization of the group axioms, detached as it is from the concrete nature of any particular group and its operation, applies much more widely. It allows entities with highly diverse mathematical origins in abstract algebra and beyond to be handled in a flexible way while retaining their essential structural aspects. The ubiquity of groups in numerous areas within and outside mathematics makes them a central organizing principle of contemporary mathematics.[1][2] Groups share a fundamental kinship with the notion of symmetry
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Infinitary Logic
An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. Some infinitary logics may have different properties from those of standard first-order logic. In particular, infinitary logics may fail to be compact or complete. Notions of compactness and completeness that are equivalent in finitary logic sometimes are not so in infinitary logics. Therefore for infinitary logics, notions of strong compactness and strong completeness are defined
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Algebraic Geometry
Algebraic geometry
Algebraic geometry
is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity
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Analytic Functions
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not hold generally for real analytic functions
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Finitary
In mathematics or logic, a finitary operation is an operation of finite arity, that is an operation that takes a finite number of input values. By contrast, an operation that may take an infinite number of input values is said to be infinitary. In standard mathematics, an operation is, by definition, finitary. Therefore these terms are used only in the context of infinitary logic.Contents1 Finitary argument 2 History 3 See also 4 Notes 5 External links Finitary argument[edit] A finitary argument is one which can be translated into a finite set of symbolic propositions starting from a finite[1] set of axioms. In other words, it is a proof (including all assumptions) that can be written on a large enough sheet of paper. By contrast, infinitary logic studies logics that allow infinitely long statements and proofs
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Association For Symbolic Logic
The Association for Symbolic Logic
Logic
(ASL) is an international organization of specialists in mathematical logic and philosophical logic. The ASL was founded in 1936, and its first president was Alonzo Church. The current president of the ASL is Ulrich Kohlenbach.[1]Contents1 Publications 2 Meetings 3 Awards3.1 Karp Prize 3.2 Sacks Prize 3.3 Shoenfield Prize4 References 5 External linksPublications[edit] The ASL publishes books and academic journals. Its three official journals are: Journal of Symbolic Logic (website) – publishes research in all areas of mathematical logic. Founded in 1936, ISSN 0022-4812. Bulletin of Symbolic Logic
Logic
(website) – publishes primarily expository articles and reviews. Founded in 1995, ISSN 1079-8986. Review of Symbolic Logic
Logic
(website) – publishes research relating to logic, philosophy, science, and their interactions
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Sentence (mathematical Logic)
In mathematical logic, a sentence of a predicate logic is a boolean-valued well-formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that must be true or false. The restriction of having no free variables is needed to make sure that sentences can have concrete, fixed truth values: As the free variables of a (general) formula can range over several values, the truth value of such a formula may vary. Sentences without any logical connectives or quantifiers in them are known as atomic sentences; by analogy to atomic formula. Sentences are then built up out of atomic formulas by applying connectives and quantifiers. A set of sentences is called a theory; thus, individual sentences may be called theorems. To properly evaluate the truth (or falsehood) of a sentence, one must make reference to an interpretation of the theory. For first-order theories, interpretations are commonly called structures
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Arithmetic Dynamics
Arithmetic dynamics[1] is a field that amalgamates two areas of mathematics, dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic properties of integer, rational, p-adic, and/or algebraic points under repeated application of a polynomial or rational function
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Philosophy
Philosophy
Philosophy
(from Greek φιλοσοφία, philosophia, literally "love of wisdom"[1][2][3][4]) is the study of general and fundamental problems concerning matters such as existence, knowledge, values, reason, mind, and language.[5][6] The term was probably coined by Pythagoras
Pythagoras
(c. 570–495 BCE)
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Syntax
In linguistics, syntax (/ˈsɪntæks/[1][2]) is the set of rules, principles, and processes that govern the structure of sentences in a given language, usually including word order. The term syntax is also used to refer to the study of such principles and processes.[3] The goal of many syntacticians is to discover the syntactic rules common to all languages. In mathematics, syntax refers to the rules governing the behavior of mathematical systems, such as formal languages used in logic
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Mathematics
Mathematics
Mathematics
(from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity,[1] structure,[2] space,[1] and change.[3][4][5] It has no generally accepted definition.[6][7] Mathematicians seek out patterns[8][9] and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back as written records exist
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Number Theory
Number
Number
theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called "The Queen of Mathematics" because of its foundational place in the discipline.[1] Number
Number
theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of analytical objects (e.g., the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory)
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