1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns 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, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
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Nonempty set
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In mathematics, and more specifically set theory, the empty set is the unique set having no elements, its size or cardinality is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, in other theories, many possible properties of sets are vacuously true for the empty set. Null set was once a synonym for empty set, but is now a technical term in measure theory. The empty set may also be called the void set, common notations for the empty set include, ∅, and ∅. The latter two symbols were introduced by the Bourbaki group in 1939, inspired by the letter Ø in the Norwegian, although now considered an improper use of notation, in the past,0 was occasionally used as a symbol for the empty set. The empty-set symbol ∅ is found at Unicode point U+2205, in LaTeX, it is coded as \emptyset for ∅ or \varnothing for ∅. In standard axiomatic set theory, by the principle of extensionality, hence there is but one empty set, and we speak of the empty set rather than an empty set. The mathematical symbols employed below are explained here, in this context, zero is modelled by the empty set. For any property, For every element of ∅ the property holds, There is no element of ∅ for which the property holds. Conversely, if for some property and some set V, the two statements hold, For every element of V the property holds, There is no element of V for which the property holds. By the definition of subset, the empty set is a subset of any set A. That is, every element x of ∅ belongs to A. Indeed, since there are no elements of ∅ at all, there is no element of ∅ that is not in A. Any statement that begins for every element of ∅ is not making any substantive claim and this is often paraphrased as everything is true of the elements of the empty set. When speaking of the sum of the elements of a finite set, the reason for this is that zero is the identity element for addition. Similarly, the product of the elements of the empty set should be considered to be one, a disarrangement of a set is a permutation of the set that leaves no element in the same position. The empty set is a disarrangment of itself as no element can be found that retains its original position. Since the empty set has no members, when it is considered as a subset of any ordered set, then member of that set will be an upper bound. For example, when considered as a subset of the numbers, with its usual ordering, represented by the real number line
3.
Associative
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In mathematics, the associative property is a property of some binary operations. In propositional logic, associativity is a rule of replacement for expressions in logical proofs. That is, rearranging the parentheses in such an expression will not change its value, consider the following equations, +4 =2 + =92 × = ×4 =24. Even though the parentheses were rearranged on each line, the values of the expressions were not altered, since this holds true when performing addition and multiplication on any real numbers, it can be said that addition and multiplication of real numbers are associative operations. Associativity is not to be confused with commutativity, which addresses whether or not the order of two operands changes the result. For example, the order doesnt matter in the multiplication of numbers, that is. Associative operations are abundant in mathematics, in fact, many algebraic structures explicitly require their binary operations to be associative, however, many important and interesting operations are non-associative, some examples include subtraction, exponentiation and the vector cross product. Z = x = xyz for all x, y, z in S, the associative law can also be expressed in functional notation thus, f = f. If a binary operation is associative, repeated application of the produces the same result regardless how valid pairs of parenthesis are inserted in the expression. This is called the generalized associative law, thus the product can be written unambiguously as abcd. As the number of elements increases, the number of ways to insert parentheses grows quickly. Some examples of associative operations include the following, the two methods produce the same result, string concatenation is associative. In arithmetic, addition and multiplication of numbers are associative, i. e. + z = x + = x + y + z z = x = x y z } for all x, y, z ∈ R. x, y, z\in \mathbb. }Because of associativity. Addition and multiplication of numbers and quaternions are associative. Addition of octonions is also associative, but multiplication of octonions is non-associative, the greatest common divisor and least common multiple functions act associatively. Gcd = gcd = gcd lcm = lcm = lcm } for all x, y, z ∈ Z. x, y, z\in \mathbb. }Taking the intersection or the union of sets, ∩ C = A ∩ = A ∩ B ∩ C ∪ C = A ∪ = A ∪ B ∪ C } for all sets A, B, C. Slightly more generally, given four sets M, N, P and Q, with h, M to N, g, N to P, in short, composition of maps is always associative. Consider a set with three elements, A, B, and C, thus, for example, A=C = A
4.
Property (philosophy)
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In philosophy, mathematics, and logic, a property is a characteristic of an object, a red object is said to have the property of redness. The property may be considered a form of object in its own right, a property however differs from individual objects in that it may be instantiated, and often in more than one thing. Understanding how different individual entities can in some sense have some of the properties is the basis of the problem of universals. The terms attribute and quality have similar meanings, in classical Aristotelian terminology, a property is one of the predicables. It is a quality of a species, but a quality which is nevertheless characteristically present in members of that species. For example, ability to laugh may be considered a characteristic of human beings. However, laughter is not a quality of the species human. A determinable property is one that can get more specific, for example, color is a determinable property because it can be restricted to redness, blueness, etc. A determinate property is one that become more specific. This distinction may be useful in dealing with issues of identity, in other words, it is the view that non-physical, mental properties inhere in some physical substances. e. The set, p is its indicator function and it may be objected that this defines merely the extension of a property, and says nothing about what causes the property to hold for exactly those values. The ontological fact that something has a property is represented in language by applying a predicate to a subject. However, taking any grammatical predicate whatsoever to be a property, or to have a property, leads to certain difficulties, such as Russells paradox. Other predicates, such as is an individual, or has some properties are uninformative or vacuous, there is some resistance to regarding such so-called Cambridge properties as legitimate. An intrinsic property is a property that an object or a thing has of itself, independently of other things, an extrinsic property is a property that depends on a things relationship with other things. A relation is considered to be a more general case of a property. Relations are true of several particulars, or shared amongst them, holds between two individuals, who would occupy the two ellipses. Relations can be expressed by N-place predicates, where N is greater than 1 and it is widely accepted that there are at least some apparent relational properties which are merely derived from non-relational properties
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Commutative
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In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says 3 +4 =4 +3 or 2 ×5 =5 ×2, the property can also be used in more advanced settings. The name is needed there are operations, such as division and subtraction. The commutative property is a property associated with binary operations and functions. If the commutative property holds for a pair of elements under a binary operation then the two elements are said to commute under that operation. The term commutative is used in several related senses, putting on socks resembles a commutative operation since which sock is put on first is unimportant. Either way, the result, is the same, in contrast, putting on underwear and trousers is not commutative. The commutativity of addition is observed when paying for an item with cash, regardless of the order the bills are handed over in, they always give the same total. The multiplication of numbers is commutative, since y z = z y for all y, z ∈ R For example,3 ×5 =5 ×3. Some binary truth functions are also commutative, since the tables for the functions are the same when one changes the order of the operands. For example, the logical biconditional function p ↔ q is equivalent to q ↔ p and this function is also written as p IFF q, or as p ≡ q, or as Epq. Further examples of binary operations include addition and multiplication of complex numbers, addition and scalar multiplication of vectors. Concatenation, the act of joining character strings together, is a noncommutative operation, rotating a book 90° around a vertical axis then 90° around a horizontal axis produces a different orientation than when the rotations are performed in the opposite order. The twists of the Rubiks Cube are noncommutative and this can be studied using group theory. Some non-commutative binary operations, Records of the use of the commutative property go back to ancient times. The Egyptians used the property of multiplication to simplify computing products. Euclid is known to have assumed the property of multiplication in his book Elements
6.
Cardinality
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In mathematics, the cardinality of a set is a measure of the number of elements of the set. For example, the set A = contains 3 elements, there are two approaches to cardinality – one which compares sets directly using bijections and injections, and another which uses cardinal numbers. The cardinality of a set is called its size, when no confusion with other notions of size is possible. The cardinality of a set A is usually denoted | A |, with a bar on each side, this is the same notation as absolute value. Alternatively, the cardinality of a set A may be denoted by n, A, card, while the cardinality of a finite set is just the number of its elements, extending the notion to infinite sets usually starts with defining the notion of comparison of arbitrary sets. Two sets A and B have the same cardinality if there exists a bijection, that is, such sets are said to be equipotent, equipollent, or equinumerous. This relationship can also be denoted A≈B or A~B, for example, the set E = of non-negative even numbers has the same cardinality as the set N = of natural numbers, since the function f = 2n is a bijection from N to E. A has cardinality less than or equal to the cardinality of B if there exists a function from A into B. A has cardinality less than the cardinality of B if there is an injective function. If | A | ≤ | B | and | B | ≤ | A | then | A | = | B |, the axiom of choice is equivalent to the statement that | A | ≤ | B | or | B | ≤ | A | for every A, B. That is, the cardinality of a set was not defined as an object itself. However, such an object can be defined as follows, the relation of having the same cardinality is called equinumerosity, and this is an equivalence relation on the class of all sets. The equivalence class of a set A under this relation then consists of all sets which have the same cardinality as A. There are two ways to define the cardinality of a set, The cardinality of a set A is defined as its class under equinumerosity. A representative set is designated for each equivalence class, the most common choice is the initial ordinal in that class. This is usually taken as the definition of number in axiomatic set theory. Assuming AC, the cardinalities of the sets are denoted ℵ0 < ℵ1 < ℵ2 < …. For each ordinal α, ℵ α +1 is the least cardinal number greater than ℵ α
7.
Algebra
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Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols, as such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine, abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are unknown or allowed to take on many values. For example, in x +2 =5 the letter x is unknown, in E = mc2, the letters E and m are variables, and the letter c is a constant, the speed of light in a vacuum. Algebra gives methods for solving equations and expressing formulas that are easier than the older method of writing everything out in words. The word algebra is used in certain specialized ways. A special kind of object in abstract algebra is called an algebra. A mathematician who does research in algebra is called an algebraist, the word algebra comes from the Arabic الجبر from the title of the book Ilm al-jabr wal-muḳābala by Persian mathematician and astronomer al-Khwarizmi. The word entered the English language during the century, from either Spanish, Italian. It originally referred to the procedure of setting broken or dislocated bones. The mathematical meaning was first recorded in the sixteenth century, the word algebra has several related meanings in mathematics, as a single word or with qualifiers. As a single word without an article, algebra names a broad part of mathematics, as a single word with an article or in plural, an algebra or algebras denotes a specific mathematical structure, whose precise definition depends on the author. Usually the structure has an addition, multiplication, and a scalar multiplication, when some authors use the term algebra, they make a subset of the following additional assumptions, associative, commutative, unital, and/or finite-dimensional. In universal algebra, the word refers to a generalization of the above concept. With a qualifier, there is the distinction, Without an article, it means a part of algebra, such as linear algebra, elementary algebra. With an article, it means an instance of some abstract structure, like a Lie algebra, sometimes both meanings exist for the same qualifier, as in the sentence, Commutative algebra is the study of commutative rings, which are commutative algebras over the integers
8.
Subset
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In mathematics, especially in set theory, a set A is a subset of a set B, or equivalently B is a superset of A, if A is contained inside B, that is, all elements of A are also elements of B. The relationship of one set being a subset of another is called inclusion or sometimes containment, the subset relation defines a partial order on sets. The algebra of subsets forms a Boolean algebra in which the relation is called inclusion. For any set S, the inclusion relation ⊆ is an order on the set P of all subsets of S defined by A ≤ B ⟺ A ⊆ B. We may also partially order P by reverse set inclusion by defining A ≤ B ⟺ B ⊆ A, when quantified, A ⊆ B is represented as, ∀x. So for example, for authors, it is true of every set A that A ⊂ A. Other authors prefer to use the symbols ⊂ and ⊃ to indicate proper subset and superset, respectively and this usage makes ⊆ and ⊂ analogous to the inequality symbols ≤ and <. For example, if x ≤ y then x may or may not equal y, but if x < y, then x definitely does not equal y, and is less than y. Similarly, using the convention that ⊂ is proper subset, if A ⊆ B, then A may or may not equal B, the set A = is a proper subset of B =, thus both expressions A ⊆ B and A ⊊ B are true. The set D = is a subset of E =, thus D ⊆ E is true, any set is a subset of itself, but not a proper subset. The empty set, denoted by ∅, is also a subset of any given set X and it is also always a proper subset of any set except itself. These are two examples in both the subset and the whole set are infinite, and the subset has the same cardinality as the whole. The set of numbers is a proper subset of the set of real numbers. In this example, both sets are infinite but the set has a larger cardinality than the former set. Another example in an Euler diagram, Inclusion is the partial order in the sense that every partially ordered set is isomorphic to some collection of sets ordered by inclusion. The ordinal numbers are a simple example—if each ordinal n is identified with the set of all ordinals less than or equal to n, then a ≤ b if and only if ⊆. For the power set P of a set S, the partial order is the Cartesian product of k = |S| copies of the partial order on for which 0 <1. This can be illustrated by enumerating S = and associating with each subset T ⊆ S the k-tuple from k of which the ith coordinate is 1 if and only if si is a member of T
9.
Set (mathematics)
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In mathematics, a set is a well-defined collection of distinct objects, considered as an object in its own right. For example, the numbers 2,4, and 6 are distinct objects when considered separately, Sets are one of the most fundamental concepts in mathematics. Developed at the end of the 19th century, set theory is now a part of mathematics. In mathematics education, elementary topics such as Venn diagrams are taught at a young age, the German word Menge, rendered as set in English, was coined by Bernard Bolzano in his work The Paradoxes of the Infinite. A set is a collection of distinct objects. The objects that make up a set can be anything, numbers, people, letters of the alphabet, other sets, Sets are conventionally denoted with capital letters. Sets A and B are equal if and only if they have precisely the same elements. Cantors definition turned out to be inadequate, instead, the notion of a set is taken as a notion in axiomatic set theory. There are two ways of describing, or specifying the members of, a set, one way is by intensional definition, using a rule or semantic description, A is the set whose members are the first four positive integers. B is the set of colors of the French flag, the second way is by extension – that is, listing each member of the set. An extensional definition is denoted by enclosing the list of members in curly brackets, one often has the choice of specifying a set either intensionally or extensionally. In the examples above, for instance, A = C and B = D, there are two important points to note about sets. First, in a definition, a set member can be listed two or more times, for example. However, per extensionality, two definitions of sets which differ only in one of the definitions lists set members multiple times, define, in fact. Hence, the set is identical to the set. The second important point is that the order in which the elements of a set are listed is irrelevant and we can illustrate these two important points with an example, = =. For sets with many elements, the enumeration of members can be abbreviated, for instance, the set of the first thousand positive integers may be specified extensionally as, where the ellipsis indicates that the list continues in the obvious way. Ellipses may also be used where sets have infinitely many members, thus the set of positive even numbers can be written as
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Structure
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Structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as biological organisms, abstract structures include data structures in computer science and musical form. Types of structure include a hierarchy, a network featuring many-to-many links, buildings, aircraft, skeletons, anthills, beaver dams and salt domes are all examples of load-bearing structures. The results of construction are divided into buildings and non-building structures, the effects of loads on physical structures are determined through structural analysis, which is one of the tasks of structural engineering. The structural elements can be classified as one-dimensional, two-dimensional, or three-dimensional, the latter was the main option available to early structures such as Chichen Itza. Two-dimensional elements with a third dimension have little of either. The structure elements are combined in structural systems, the majority of everyday load-bearing structures are section-active structures like frames, which are primarily composed of one-dimensional structures. In biology, structures exist at all levels of organization, ranging hierarchically from the atomic and molecular to the cellular, tissue, organ, organismic, population, usually, a higher-level structure is composed of multiple copies of a lower-level structure. Structural biology is concerned with the structure of macromolecules, particularly proteins. The function of molecules is determined by their shape as well as their composition. Protein structure has a four-level hierarchy, the primary structure is the sequence of amino acids that make it up. It has a backbone made up of a repeated sequence of a nitrogen. The secondary structure consists of repeated patterns determined by hydrogen bonding, the two basic types are the α-helix and the β-pleated sheet. The tertiary structure is a back and forth bending of the chain. Chemical structure refers to both molecular geometry and electronic structure, the structure can be represented by a variety of diagrams called structural formulas. Lewis structures use a dot notation to represent the valence electrons for an atom, bonds between atoms can be represented by lines with one line for each pair of electrons that is shared. In a simplified version of such a diagram, called a skeletal formula, only carbon-carbon bonds, atoms in a crystal have a structure that involves repetition of a basic unit called a unit cell. The atoms can be modeled as points on a lattice, and one can explore the effect of symmetry operations that include rotations about a point, reflections about a symmetry planes, and translations
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Partial order
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In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for pairs of elements in the set. The word partial in the partial order or partially ordered set is used as an indication that not every pair of elements need be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset, Partial orders thus generalize total orders, in which every pair is comparable. To be an order, a binary relation must be reflexive, antisymmetric. One familiar example of an ordered set is a collection of people ordered by genealogical descendancy. Some pairs of people bear the descendant-ancestor relationship, but other pairs of people are incomparable, a poset can be visualized through its Hasse diagram, which depicts the ordering relation. A partial order is a binary relation ≤ over a set P satisfying particular axioms which are discussed below, when a ≤ b, we say that a is related to b. The axioms for a partial order state that the relation ≤ is reflexive, antisymmetric. That is, for all a, b, and c in P, it must satisfy, in other words, a partial order is an antisymmetric preorder. A set with an order is called a partially ordered set. The term ordered set is also used, as long as it is clear from the context that no other kind of order is meant. In particular, totally ordered sets can also be referred to as ordered sets, for a, b, elements of a partially ordered set P, if a ≤ b or b ≤ a, then a and b are comparable. In the figure on top-right, e. g. and are comparable, while and are not, a partial order under which every pair of elements is comparable is called a total order or linear order, a totally ordered set is also called a chain. A subset of a poset in which no two elements are comparable is called an antichain. A more concise definition will be given using the strict order corresponding to ≤. For example, is covered by in the figure. Standard examples of posets arising in mathematics include, The real numbers ordered by the standard less-than-or-equal relation ≤, the set of subsets of a given set ordered by inclusion
12.
Topology
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In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, important topological properties include connectedness and compactness. Topology developed as a field of study out of geometry and set theory, through analysis of such as space, dimension. Such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs, Leonhard Eulers Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the fields first theorems. The term topology was introduced by Johann Benedict Listing in the 19th century, by the middle of the 20th century, topology had become a major branch of mathematics. It defines the basic notions used in all branches of topology. Algebraic topology tries to measure degrees of connectivity using algebraic constructs such as homology, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to geometry and together they make up the geometric theory of differentiable manifolds. Geometric topology primarily studies manifolds and their embeddings in other manifolds, a particularly active area is low-dimensional topology, which studies manifolds of four or fewer dimensions. This includes knot theory, the study of mathematical knots, Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries. Among these are certain questions in geometry investigated by Leonhard Euler and his 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750 Euler wrote to a friend that he had realised the importance of the edges of a polyhedron and this led to his polyhedron formula, V − E + F =2. Some authorities regard this analysis as the first theorem, signalling the birth of topology, further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term Topologie in Vorstudien zur Topologie, written in his native German, in 1847, the term topologist in the sense of a specialist in topology was used in 1905 in the magazine Spectator. Their work was corrected, consolidated and greatly extended by Henri Poincaré, in 1895 he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a case of a general topological space. In 1914, Felix Hausdorff coined the term topological space and gave the definition for what is now called a Hausdorff space, currently, a topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski
13.
Group (mathematics)
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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. The operation satisfies four conditions called the group axioms, namely closure and 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 areas within and outside mathematics makes them a central organizing principle of contemporary mathematics. Groups share a kinship with the notion of symmetry. The concept of a group arose from the study of polynomial equations, after contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right, to explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. A theory has developed for finite groups, which culminated with the classification of finite simple groups. Since the mid-1980s, geometric group theory, which studies finitely generated groups as objects, has become a particularly active area in group theory. One of the most familiar groups is the set of integers Z which consists of the numbers, −4, −3, −2, −1,0,1,2,3,4. The following properties of integer addition serve as a model for the group axioms given in the definition below. For any two integers a and b, the sum a + b is also an integer and that is, addition of integers always yields an integer. This property is known as closure under addition, for all integers a, b and c, + c = a +. Expressed in words, adding a to b first, and then adding the result to c gives the final result as adding a to the sum of b and c. If a is any integer, then 0 + a = a +0 = a, zero is called the identity element of addition because adding it to any integer returns the same integer. For every integer a, there is a b such that a + b = b + a =0. The integer b is called the element of the integer a and is denoted −a. The integers, together with the operation +, form a mathematical object belonging to a class sharing similar structural aspects. To appropriately understand these structures as a collective, the abstract definition is developed
14.
Monoid
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In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are semigroups with identity, monoids occur in several branches of mathematics, for instance, they can be regarded as categories with a single object. Thus, they capture the idea of composition within a set. In fact, all functions from a set into itself form naturally a monoid with respect to function composition, monoids are also commonly used in computer science, both in its foundational aspects and in practical programming. The set of strings built from a set of characters is a free monoid. The transition monoid and syntactic monoid are used in describing finite state machines, whereas trace monoids and history provide a foundation for process calculi. Some of the more important results in the study of monoids are the Krohn–Rhodes theorem, the history of monoids, as well as a discussion of additional general properties, are found in the article on semigroups. Identity element There exists an element e in S such that for every element a in S, in other words, a monoid is a semigroup with an identity element. It can also be thought of as a magma with associativity and identity, the identity element of a monoid is unique. A monoid in which each element has an inverse is a group. Depending on the context, the symbol for the operation may be omitted, so that the operation is denoted by juxtaposition, for example. This notation does not imply that it is numbers being multiplied, N is thus a monoid under the binary operation inherited from M. If there is a generator of M that has finite cardinality, not every set S will generate a monoid, as the generated structure may lack an identity element. A monoid whose operation is commutative is called a commutative monoid, commutative monoids are often written additively. Any commutative monoid is endowed with its algebraic preordering ≤, defined by x ≤ y if there exists z such that x + z = y. An order-unit of a commutative monoid M is an element u of M such that for any element x of M, there exists a positive integer n such that x ≤ nu. This is often used in case M is the cone of a partially ordered abelian group G. A monoid for which the operation is commutative for some, but not all elements is a trace monoid, trace monoids commonly occur in the theory of concurrent computation
15.
Band (mathematics)
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In mathematics, a band is a semigroup in which every element is idempotent. Bands were first studied and named by A. H. Clifford, a class of bands forms a variety if it is closed under formation of subsemigroups, homomorphic images and direct product. Each variety of bands can be defined by a single defining identity, semilattices are exactly commutative bands, that is, they are the bands satisfying the equation xy = yx for all x and y. A left zero band is a band satisfying the equation xy = x, symmetrically, a right zero band is one satisfying xy = y, so that the Cayley table has constant columns. There is a classification of rectangular bands. Left zero and right zero bands are bands, and in fact every rectangular band is isomorphic to a direct product of a left zero band. All rectangular bands of prime order are zero bands, either left or right, a rectangular band is said to be purely rectangular if it is not a left or right zero band. Note that if the set I is empty in the result, the rectangular band I × J is independent of J. This is why the above result only gives an equivalence between nonempty rectangular bands and pairs of nonempty sets, a normal band is a band S satisfying zxyz = zyxz for all x, y, and z ∈ S. This is the equation used to define medial magmas, and so a normal band may also be called a medial band. Left-regular bands thus show up naturally in the study of posets, a right-regular band is a band S satisfying xyx = yx for all x, y ∈ S Any right-regular band becomes a left-regular band using the opposite product. Indeed, every variety of bands has a version, this gives rise to the reflection symmetry in the figure below. The complete structure of this lattice is known, in particular, it is countable, complete, the sublattice consisting of the 13 varieties of regular bands is shown in the figure. The varieties of bands, semilattices, and right-zero bands are the three atoms of this lattice. Note that each variety of bands shown in the figure is defined by just one identity and this is not a coincidence, in fact, every variety of bands can be defined by a single identity. P. Varieties of idempotent semigroups, Algebra and Logic,9, 153–164, brown, Ken, Semigroups, rings, and Markov chains, J. Theoret. Clifford, Alfred Hoblitzelle, Bands of semigroups, Proceedings of the American Mathematical Society,5, 499–504, doi,10. 1090/S0002-9939-1954-0062119-9, Clifford, Alfred Hoblitzelle, Preston, Gordon Bamford, The Algebraic Theory of Semigroups, Moscow, Mir. Fennemore, Charles, All varieties of bands, Semigroup Forum,1, 172–179, the lattice of equational classes of idempotent semigroups, Journal of Algebra,15, 195–224, doi,10. 1016/0021-869390073-6
16.
Archimedean property
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Roughly speaking, it is the property of having no infinitely large or infinitely small elements. It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes’ On the Sphere and Cylinder. An algebraic structure in any two non-zero elements are comparable, in the sense that neither of them is infinitesimal with respect to the other, is said to be Archimedean. A structure which has a pair of elements, one of which is infinitesimal with respect to the other, is said to be non-Archimedean. For example, an ordered group that is Archimedean is an Archimedean group. This can be made precise in various contexts with different formulations. The concept was named by Otto Stolz after the ancient Greek geometer, the Archimedean property appears in Book V of Euclids Elements as Definition 4, Magnitudes are said to have a ratio to one another which can, when multiplied, exceed one another. Because Archimedes credited it to Eudoxus of Cnidus it is known as the Theorem of Eudoxus or the Eudoxus axiom. Archimedes used infinitesimals in heuristic arguments, although he denied that those were finished mathematical proofs, Let x and y be positive elements of a linearly ordered group G. Then x is infinitesimal with respect to y if, for natural number n, the multiple nx is less than y, that is. The group G is Archimedean if there is no x, y such that x is infinitesimal with respect to y. Additionally, if K is a structure with a unit — for example. If x is infinitesimal with respect to 1, then x is an infinitesimal element, likewise, if y is infinite with respect to 1, then y is an infinite element. The algebraic structure K is Archimedean if it has no infinite elements, an ordered field has some additional properties. One may assume that the numbers are contained in the field. If x is infinitesimal, then 1/x is infinite, and vice versa, therefore, to verify that a field is Archimedean it is enough to check only that there are no infinitesimal elements, or to check that there are no infinite elements. If x is infinitesimal and r is a number, then r x is also infinitesimal. As a result, given an element c, the three numbers c/2, c, and 2c are either all infinitesimal or all non-infinitesimal
17.
Exponential function
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In mathematics, an exponential function is a function of the form in which the input variable x occurs as an exponent. A function of the form f = b x + c, as functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function is directly proportional to the value of the function. The constant of proportionality of this relationship is the logarithm of the base b. The argument of the function can be any real or complex number or even an entirely different kind of mathematical object. Its ubiquitous occurrence in pure and applied mathematics has led mathematician W. Rudin to opine that the function is the most important function in mathematics. In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same change in the dependent variable. The graph of y = e x is upward-sloping, and increases faster as x increases, the graph always lies above the x -axis but can get arbitrarily close to it for negative x, thus, the x -axis is a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its y -coordinate at that point, as implied by its derivative function. Its inverse function is the logarithm, denoted log, ln, or log e, because of this. The exponential function exp, C → C can be characterized in a variety of equivalent ways, the constant e is then defined as e = exp = ∑ k =0 ∞. The exponential function arises whenever a quantity grows or decays at a proportional to its current value. One such situation is continuously compounded interest, and in fact it was this observation that led Jacob Bernoulli in 1683 to the number lim n → ∞ n now known as e, later, in 1697, Johann Bernoulli studied the calculus of the exponential function. If instead interest is compounded daily, this becomes 365, letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function, exp = lim n → ∞ n first given by Euler. This is one of a number of characterizations of the exponential function, from any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity, exp = exp ⋅ exp which is why it can be written as ex. The derivative of the function is the exponential function itself. More generally, a function with a rate of change proportional to the function itself is expressible in terms of the exponential function and this function property leads to exponential growth and exponential decay. The exponential function extends to a function on the complex plane. Eulers formula relates its values at purely imaginary arguments to trigonometric functions, the exponential function also has analogues for which the argument is a matrix, or even an element of a Banach algebra or a Lie algebra
18.
Subgroup
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In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗. More precisely, H is a subgroup of G if the restriction of ∗ to H × H is an operation on H. This is usually denoted H ≤ G, read as H is a subgroup of G, the trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group G is a subgroup H which is a subset of G. This is usually represented notationally by H < G, read as H is a subgroup of G. Some authors also exclude the group from being proper. If H is a subgroup of G, then G is sometimes called an overgroup of H, the same definitions apply more generally when G is an arbitrary semigroup, but this article will only deal with subgroups of groups. The group G is sometimes denoted by the pair, usually to emphasize the operation ∗ when G carries multiple algebraic or other structures. This article will write ab for a ∗ b, as is usual, a subset H of the group G is a subgroup of G if and only if it is nonempty and closed under products and inverses. In the case that H is finite, then H is a subgroup if and only if H is closed under products. The above condition can be stated in terms of a homomorphism, the identity of a subgroup is the identity of the group, if G is a group with identity eG, and H is a subgroup of G with identity eH, then eH = eG. The intersection of subgroups A and B is again a subgroup. The union of subgroups A and B is a if and only if either A or B contains the other, since for example 2 and 3 are in the union of 2Z and 3Z. Another example is the union of the x-axis and the y-axis in the plane, each of these objects is a subgroup and this also serves as an example of two subgroups, whose intersection is precisely the identity. An element of G is in <S> if and only if it is a product of elements of S. Every element a of a group G generates the cyclic subgroup <a>, if <a> is isomorphic to Z/nZ for some positive integer n, then n is the smallest positive integer for which an = e, and n is called the order of a. If <a> is isomorphic to Z, then a is said to have infinite order, the subgroups of any given group form a complete lattice under inclusion, called the lattice of subgroups. If e is the identity of G, then the group is the minimum subgroup of G
19.
Map (mathematics)
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There are also a few, less common uses in logic and graph theory. In many branches of mathematics, the map is used to mean a function. For instance, a map is a function in topology. Some authors, such as Serge Lang, use only to refer to maps in which the codomain is a set of numbers. Sets of maps of special kinds are the subjects of many important theories, see for instance Lie group, mapping class group, in the theory of dynamical systems, a map denotes an evolution function used to create discrete dynamical systems. A partial map is a function, and a total map is a total function. Related terms like domain, codomain, injective, continuous, etc. can be applied equally to maps and functions, all these usages can be applied to maps as general functions or as functions with special properties. In category theory, map is used as a synonym for morphism or arrow. In formal logic, the map is sometimes used for a functional predicate. In graph theory, a map is a drawing of a graph on a surface without overlapping edges, if the surface is a plane then a map is a planar graph, similar to a political map
20.
Composition of functions
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In mathematics, function composition is the pointwise application of one function to the result of another to produce a third function. The resulting composite function is denoted g ∘ f, X → Z, the notation g ∘ f is read as g circle f, or g round f, or g composed with f, g after f, g following f, or g of f, or g on f. Intuitively, composing two functions is a process in which the output of the inner function becomes the input of the outer function. The composition of functions is a case of the composition of relations. The composition of functions has some additional properties, Composition of functions on a finite set, If f =, and g =, then g ∘ f =. The composition of functions is always associative—a property inherited from the composition of relations, since there is no distinction between the choices of placement of parentheses, they may be left off without causing any ambiguity. In a strict sense, the composition g ∘ f can be only if fs codomain equals gs domain, in a wider sense it is sufficient that the former is a subset of the latter. The functions g and f are said to commute with each other if g ∘ f = f ∘ g, commutativity is a special property, attained only by particular functions, and often in special circumstances. For example, | x | +3 = | x + 3 | only when x ≥0, the composition of one-to-one functions is always one-to-one. Similarly, the composition of two functions is always onto. It follows that composition of two bijections is also a bijection, the inverse function of a composition has the property that −1 =. Derivatives of compositions involving differentiable functions can be using the chain rule. Higher derivatives of functions are given by Faà di Brunos formula. Suppose one has two functions f, X → X, g, X → X having the domain and codomain. Then one can form chains of transformations composed together, such as f ∘ f ∘ g ∘ f, such chains have the algebraic structure of a monoid, called a transformation monoid or composition monoid. In general, transformation monoids can have remarkably complicated structure, one particular notable example is the de Rham curve. The set of all functions f, X → X is called the transformation semigroup or symmetric semigroup on X. If the transformation are bijective, then the set of all combinations of these functions forms a transformation group
21.
Bijection
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In mathematics, injections, surjections and bijections are classes of functions distinguished by the manner in which arguments and images are related or mapped to each other. A function maps elements from its domain to elements in its codomain, given a function f, X → Y The function is injective if every element of the codomain is mapped to by at most one element of the domain. An injective function is an injection, notationally, ∀ x, x ′ ∈ X, f = f ⇒ x = x ′. Or, equivalently, ∀ x, x ′ ∈ X, x ≠ x ′ ⇒ f ≠ f, the function is surjective if every element of the codomain is mapped to by at least one element of the domain. A surjective function is a surjection, notationally, ∀ y ∈ Y, ∃ x ∈ X such that y = f. The function is bijective if every element of the codomain is mapped to by one element of the domain. A bijective function is a bijection, an injective function need not be surjective, and a surjective function need not be injective. The four possible combinations of injective and surjective features are illustrated in the diagrams to the right, a function is injective if every possible element of the codomain is mapped to by at most one argument. Equivalently, a function is injective if it maps distinct arguments to distinct images, an injective function is an injection. The formal definition is the following, the function f, X → Y is injective iff for all x, x ′ ∈ X, we have f = f ⇒ x = x ′. A function f, X → Y is injective if and only if X is empty or f is left-invertible, here f is the image of f. Since every function is surjective when its codomain is restricted to its image, more precisely, every injection f, X → Y can be factored as a bijection followed by an inclusion as follows. Let fR, X → f be f with codomain restricted to its image, a dual factorisation is given for surjections below. The composition of two injections is again an injection, but if g o f is injective, then it can only be concluded that f is injective, a function is surjective if every possible image is mapped to by at least one argument. In other words, every element in the codomain has non-empty preimage, equivalently, a function is surjective if its image is equal to its codomain. A surjective function is a surjection, the formal definition is the following. The function f, X → Y is surjective iff for all y ∈ Y, there is x ∈ X such that f = y. A function f, X → Y is surjective if and only if it is right-invertible, by collapsing all arguments mapping to a given fixed image, every surjection induces a bijection defined on a quotient of its domain
22.
Partial function
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In mathematics, a partial function from X to Y is a function f, X ′ → Y, for some subset X ′ of X. It generalizes the concept of an f, X → Y by not forcing f to map every element of X to an element of Y. If X ′ = X, then f is called a function and is equivalent to a function. Partial functions are used when the exact domain, X, is not known. Specifically, we say that for any x ∈ X, either. For example, we can consider the square root function restricted to the g, Z → Z g = n. Thus g is defined for n that are perfect squares. So, g =5, but g is undefined, there are two distinct meanings in current mathematical usage for the notion of the domain of a partial function. Most mathematicians, including recursion theorists, use the domain of f for the set of all values x such that f is defined. But some, particularly category theorists, consider the domain of a function f, X → Y to be X. Similarly, the range can refer to either the codomain or the image of a function. Occasionally, a function with domain X and codomain Y is written as f, X ⇸ Y. A partial function is said to be injective or surjective when the function given by the restriction of the partial function to its domain of definition is. A partial function may be both injective and surjective, because a function is trivially surjective when restricted to its image, the term partial bijection denotes a partial function which is injective. An injective partial function may be inverted to a partial function. Furthermore, a function which is injective may be inverted to an injective partial function. The notion of transformation can be generalized to functions as well. A partial transformation is a function f, A → B, total function is a synonym for function
23.
Matrix (mathematics)
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In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. For example, the dimensions of the matrix below are 2 ×3, the individual items in an m × n matrix A, often denoted by ai, j, where max i = m and max j = n, are called its elements or entries. Provided that they have the size, two matrices can be added or subtracted element by element. The rule for multiplication, however, is that two matrices can be multiplied only when the number of columns in the first equals the number of rows in the second. Any matrix can be multiplied element-wise by a scalar from its associated field, a major application of matrices is to represent linear transformations, that is, generalizations of linear functions such as f = 4x. The product of two matrices is a matrix that represents the composition of two linear transformations. Another application of matrices is in the solution of systems of linear equations, if the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a matrix has an inverse if and only if its determinant is not zero. Insight into the geometry of a transformation is obtainable from the matrixs eigenvalues. Applications of matrices are found in most scientific fields, in computer graphics, they are used to manipulate 3D models and project them onto a 2-dimensional screen. Matrix calculus generalizes classical analytical notions such as derivatives and exponentials to higher dimensions, Matrices are used in economics to describe systems of economic relationships. A major branch of analysis is devoted to the development of efficient algorithms for matrix computations. Matrix decomposition methods simplify computations, both theoretically and practically, algorithms that are tailored to particular matrix structures, such as sparse matrices and near-diagonal matrices, expedite computations in finite element method and other computations. Infinite matrices occur in planetary theory and in atomic theory, a simple example of an infinite matrix is the matrix representing the derivative operator, which acts on the Taylor series of a function. A matrix is an array of numbers or other mathematical objects for which operations such as addition and multiplication are defined. Most commonly, a matrix over a field F is an array of scalars each of which is a member of F. Most of this focuses on real and complex matrices, that is, matrices whose elements are real numbers or complex numbers. More general types of entries are discussed below, for instance, this is a real matrix, A =
24.
Linear transformation
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In mathematics, a linear map is a mapping V → W between two modules that preserves the operations of addition and scalar multiplication. An important special case is when V = W, in case the map is called a linear operator, or an endomorphism of V. Sometimes the term linear function has the meaning as linear map. A linear map always maps linear subspaces onto linear subspaces, for instance it maps a plane through the origin to a plane, Linear maps can often be represented as matrices, and simple examples include rotation and reflection linear transformations. In the language of algebra, a linear map is a module homomorphism. In the language of category theory it is a morphism in the category of modules over a given ring, let V and W be vector spaces over the same field K. e. that for any vectors x1. Am ∈ K, the equality holds, f = a 1 f + ⋯ + a m f. It is then necessary to specify which of these fields is being used in the definition of linear. If V and W are considered as spaces over the field K as above, for example, the conjugation of complex numbers is an R-linear map C → C, but it is not C-linear. A linear map from V to K is called a linear functional and these statements generalize to any left-module RM over a ring R without modification, and to any right-module upon reversing of the scalar multiplication. The zero map between two left-modules over the ring is always linear. The identity map on any module is a linear operator, any homothecy centered in the origin of a vector space, v ↦ c v where c is a scalar, is a linear operator. This does not hold in general for modules, where such a map might only be semilinear, for real numbers, the map x ↦ x2 is not linear. Conversely, any map between finite-dimensional vector spaces can be represented in this manner, see the following section. Differentiation defines a map from the space of all differentiable functions to the space of all functions. It also defines an operator on the space of all smooth functions. If V and W are finite-dimensional vector spaces over a field F, then functions that send linear maps f, V → W to dimF × dimF matrices in the way described in the sequel are themselves linear maps. The expected value of a variable is linear, as for random variables X and Y we have E = E + E and E = aE
25.
Vector space
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A vector space is a collection of objects called vectors, which may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be numbers, but there are also vector spaces with scalar multiplication by complex numbers, rational numbers. The operations of addition and scalar multiplication must satisfy certain requirements, called axioms. Euclidean vectors are an example of a vector space and they represent physical quantities such as forces, any two forces can be added to yield a third, and the multiplication of a force vector by a real multiplier is another force vector. In the same vein, but in a more geometric sense, Vector spaces are the subject of linear algebra and are well characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. Infinite-dimensional vector spaces arise naturally in mathematical analysis, as function spaces and these vector spaces are generally endowed with additional structure, which may be a topology, allowing the consideration of issues of proximity and continuity. Among these topologies, those that are defined by a norm or inner product are commonly used. This is particularly the case of Banach spaces and Hilbert spaces, historically, the first ideas leading to vector spaces can be traced back as far as the 17th centurys analytic geometry, matrices, systems of linear equations, and Euclidean vectors. Today, vector spaces are applied throughout mathematics, science and engineering, furthermore, vector spaces furnish an abstract, coordinate-free way of dealing with geometrical and physical objects such as tensors. This in turn allows the examination of local properties of manifolds by linearization techniques, Vector spaces may be generalized in several ways, leading to more advanced notions in geometry and abstract algebra. The concept of space will first be explained by describing two particular examples, The first example of a vector space consists of arrows in a fixed plane. This is used in physics to describe forces or velocities, given any two such arrows, v and w, the parallelogram spanned by these two arrows contains one diagonal arrow that starts at the origin, too. This new arrow is called the sum of the two arrows and is denoted v + w, when a is negative, av is defined as the arrow pointing in the opposite direction, instead. Such a pair is written as, the sum of two such pairs and multiplication of a pair with a number is defined as follows, + = and a =. The first example above reduces to one if the arrows are represented by the pair of Cartesian coordinates of their end points. A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below, elements of V are commonly called vectors. Elements of F are commonly called scalars, the second operation, called scalar multiplication takes any scalar a and any vector v and gives another vector av. In this article, vectors are represented in boldface to distinguish them from scalars
26.
Field (mathematics)
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In mathematics, a field is a set on which are defined addition, subtraction, multiplication, and division, which behave as they do when applied to rational and real numbers. A field is thus an algebraic structure, which is widely used in algebra, number theory. The best known fields are the field of numbers. In addition, the field of numbers is widely used, not only in mathematics. Finite fields are used in most cryptographic protocols used for computer security, any field may be used as the scalars for a vector space, which is the standard general context for linear algebra. Associativity of addition and multiplication For all a, b, and c in F, the following hold, a + = + c. Commutativity of addition and multiplication For all a and b in F, the following hold, a + b = b + a. Existence of additive and multiplicative identity elements There exists an element of F, called the identity element and denoted by 0, such that for all a in F. Likewise, there is an element, called the identity element and denoted by 1, such that for all a in F. To exclude the trivial ring, the identity and the multiplicative identity are required to be distinct. Existence of additive inverses and multiplicative inverses For every a in F, there exists an element −a in F, similarly, for any a in F other than 0, there exists an element a−1 in F, such that a · a−1 =1. In other words, subtraction and division operations exist, distributivity of multiplication over addition For all a, b and c in F, the following equality holds, a · = +. A simple example of a field is the field of numbers, consisting of numbers which can be written as fractions a/b, where a and b are integers. The additive inverse of such a fraction is simply −a/b, to see the latter, note that b a ⋅ a b = b a a b =1. In addition to number systems such as the rationals, there are other. The following example is a field consisting of four elements called O, I, A and B, the notation is chosen such that O plays the role of the additive identity element, and I is the multiplicative identity. One can check that all field axioms are satisfied, for example, A · = A · I = A, which equals A · B + A · A = I + B = A, as required by the distributivity. This field is called a field with four elements
27.
American Mathematical Society
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The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. It was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, john Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, the result was the Bulletin of the New York Mathematical Society, with Fiske as editor-in-chief. The de facto journal, as intended, was influential in increasing membership, the popularity of the Bulletin soon led to Transactions of the American Mathematical Society and Proceedings of the American Mathematical Society, which were also de facto journals. In 1891 Charlotte Scott became the first woman to join the society, the society reorganized under its present name and became a national society in 1894, and that year Scott served as the first woman on the first Council of the American Mathematical Society. In 1951, the headquarters moved from New York City to Providence. The society later added an office in Ann Arbor, Michigan in 1984, in 1954 the society called for the creation of a new teaching degree, a Doctor of Arts in Mathematics, similar to a PhD but without a research thesis. Mary W. Gray challenged that situation by sitting in on the Council meeting in Atlantic City, when she was told she had to leave, she refused saying she would wait until the police came. After that time, Council meetings were open to observers and the process of democratization of the Society had begun, julia Robinson was the first female president of the American Mathematical Society but was unable to complete her term as she was suffering from leukemia. In 1988 the Journal of the American Mathematical Society was created, the 2013 Joint Mathematics Meeting in San Diego drew over 6,600 attendees. Each of the four sections of the AMS hold meetings in the spring. The society also co-sponsors meetings with other mathematical societies. The AMS selects a class of Fellows who have made outstanding contributions to the advancement of mathematics. The AMS publishes Mathematical Reviews, a database of reviews of mathematical publications, various journals, in 1997 the AMS acquired the Chelsea Publishing Company, which it continues to use as an imprint. Blogs, Blog on Blogs e-Mentoring Network in the Mathematical Sciences AMS Graduate Student Blog PhD + Epsilon On the Market Some prizes are awarded jointly with other mathematical organizations. The AMS is led by the President, who is elected for a two-year term, morrey, Jr. Oscar Zariski Nathan Jacobson Saunders Mac Lane Lipman Bers R. H. Andrews Eric M. Friedlander David Vogan Robert L
28.
International Standard Book Number
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The International Standard Book Number is a unique numeric commercial book identifier. An ISBN is assigned to each edition and variation of a book, for example, an e-book, a paperback and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, the method of assigning an ISBN is nation-based and varies from country to country, often depending on how large the publishing industry is within a country. The initial ISBN configuration of recognition was generated in 1967 based upon the 9-digit Standard Book Numbering created in 1966, the 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108. Occasionally, a book may appear without a printed ISBN if it is printed privately or the author does not follow the usual ISBN procedure, however, this can be rectified later. Another identifier, the International Standard Serial Number, identifies periodical publications such as magazines, the ISBN configuration of recognition was generated in 1967 in the United Kingdom by David Whitaker and in 1968 in the US by Emery Koltay. The 10-digit ISBN format was developed by the International Organization for Standardization and was published in 1970 as international standard ISO2108, the United Kingdom continued to use the 9-digit SBN code until 1974. The ISO on-line facility only refers back to 1978, an SBN may be converted to an ISBN by prefixing the digit 0. For example, the edition of Mr. J. G. Reeder Returns, published by Hodder in 1965, has SBN340013818 -340 indicating the publisher,01381 their serial number. This can be converted to ISBN 0-340-01381-8, the check digit does not need to be re-calculated, since 1 January 2007, ISBNs have contained 13 digits, a format that is compatible with Bookland European Article Number EAN-13s. An ISBN is assigned to each edition and variation of a book, for example, an ebook, a paperback, and a hardcover edition of the same book would each have a different ISBN. The ISBN is 13 digits long if assigned on or after 1 January 2007, a 13-digit ISBN can be separated into its parts, and when this is done it is customary to separate the parts with hyphens or spaces. Separating the parts of a 10-digit ISBN is also done with either hyphens or spaces, figuring out how to correctly separate a given ISBN number is complicated, because most of the parts do not use a fixed number of digits. ISBN issuance is country-specific, in that ISBNs are issued by the ISBN registration agency that is responsible for country or territory regardless of the publication language. Some ISBN registration agencies are based in national libraries or within ministries of culture, in other cases, the ISBN registration service is provided by organisations such as bibliographic data providers that are not government funded. In Canada, ISBNs are issued at no cost with the purpose of encouraging Canadian culture. In the United Kingdom, United States, and some countries, where the service is provided by non-government-funded organisations. Australia, ISBNs are issued by the library services agency Thorpe-Bowker
29.
CRC Press
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The CRC Press, LLC is a publishing group based in the United States that specializes in producing technical books. Many of their books relate to engineering, science and mathematics and their scope also includes books on business, forensics and information technology. CRC Press is now a division of Taylor & Francis, itself a subsidiary of Informa, the company gradually expanded to include sales of laboratory equipment to chemists. In 1913 the CRC offered a manual called the Rubber Handbook as an incentive for any purchase of a dozen aprons. Since then the Rubber Handbook has evolved into the CRCs flagship book, in 1986 CRC Press was bought by the Times Mirror Company. Times Mirror began exploring the possibility of a sale of CRC Press in 1996, in 2003, CRC became part of Taylor & Francis, which in 2004 became part of the UK publisher Informa. Chapman & Hall CRC Press website
30.
Oxford University Press
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Oxford University Press is the largest university press in the world, and the second oldest after Cambridge University Press. It is a department of the University of Oxford and is governed by a group of 15 academics appointed by the known as the delegates of the press. They are headed by the secretary to the delegates, who serves as OUPs chief executive, Oxford University has used a similar system to oversee OUP since the 17th century. The university became involved in the print trade around 1480, and grew into a printer of Bibles, prayer books. OUP took on the project became the Oxford English Dictionary in the late 19th century. Moves into international markets led to OUP opening its own offices outside the United Kingdom, by contracting out its printing and binding operations, the modern OUP publishes some 6,000 new titles around the world each year. OUP was first exempted from United States corporation tax in 1972, as a department of a charity, OUP is exempt from income tax and corporate tax in most countries, but may pay sales and other commercial taxes on its products. The OUP today transfers 30% of its surplus to the rest of the university. OUP is the largest university press in the world by the number of publications, publishing more than 6,000 new books every year, the Oxford University Press Museum is located on Great Clarendon Street, Oxford. Visits must be booked in advance and are led by a member of the archive staff, displays include a 19th-century printing press, the OUP buildings, and the printing and history of the Oxford Almanack, Alice in Wonderland and the Oxford English Dictionary. The first printer associated with Oxford University was Theoderic Rood, the first book printed in Oxford, in 1478, an edition of Rufinuss Expositio in symbolum apostolorum, was printed by another, anonymous, printer. Famously, this was mis-dated in Roman numerals as 1468, thus apparently pre-dating Caxton, roods printing included John Ankywylls Compendium totius grammaticae, which set new standards for teaching of Latin grammar. After Rood, printing connected with the university remained sporadic for over half a century, the chancellor, Robert Dudley, 1st Earl of Leicester, pleaded Oxfords case. Some royal assent was obtained, since the printer Joseph Barnes began work, Oxfords chancellor, Archbishop William Laud, consolidated the legal status of the universitys printing in the 1630s. Laud envisaged a unified press of world repute, Oxford would establish it on university property, govern its operations, employ its staff, determine its printed work, and benefit from its proceeds. To that end, he petitioned Charles I for rights that would enable Oxford to compete with the Stationers Company and the Kings Printer and these were brought together in Oxfords Great Charter in 1636, which gave the university the right to print all manner of books. Laud also obtained the privilege from the Crown of printing the King James or Authorized Version of Scripture at Oxford and this privilege created substantial returns in the next 250 years, although initially it was held in abeyance. The Stationers Company was deeply alarmed by the threat to its trade, under this, the Stationers paid an annual rent for the university not to exercise its full printing rights – money Oxford used to purchase new printing equipment for smaller purposes
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John Wiley & Sons
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Founded in 1807, Wiley is also known for publishing For Dummies. As of 2015, the company had 4,900 employees, Wiley was established in 1807 when Charles Wiley opened a print shop in Manhattan. Wiley later shifted its focus to scientific, technical, and engineering subject areas, Charles Wileys son John took over the business when his father died in 1826. The firm was successively named Wiley, Lane & Co. then Wiley & Putnam, the company acquired its present name in 1876, when Johns second son William H. Wiley joined his brother Charles in the business. Through the 20th century, the company expanded its activities, the sciences. Since the establishment of the Nobel Prize in 1901, Wiley and its companies have published the works of more than 450 Nobel Laureates. Wiley in December 2010 opened an office in Dubai, to build on its business in the Middle East more effectively, the company has had an office in Beijing, China, since 2001, and China is now its sixth-largest market for STEM content. Wiley established publishing operations in India in 2006, and has established a presence in North Africa through sales contracts with academic institutions in Tunisia, Libya, and Egypt. On April 16,2012, the announced the establishment of Wiley Brasil Editora LTDA in São Paulo, Brazil. Wileys scientific, technical, and medical business was expanded by the acquisition of Blackwell Publishing in February 2007. Through a backfile initiative completed in 2007,8.2 million pages of content have been made available online. Other major journals published include Angewandte Chemie, Advanced Materials, Hepatology, International Finance, launched commercially in 1999, Wiley InterScience provided online access to Wiley journals, major reference works, and books, including backfile content. Journals previously from Blackwell Publishing were available online from Blackwell Synergy until they were integrated into Wiley InterScience on June 30,2008, in December 2007, Wiley also began distributing its technical titles through the Safari Books Online e-reference service. On February 17,2012, Wiley announced the acquisition of Inscape Holdings Inc. which provides DISC assessments and training for interpersonal business skills. On August 13,2012, Wiley announced it entered into an agreement to sell all of its travel assets, including all of its interests in the Frommers brand. On October 2,2012, Wiley announced it would acquire Deltak edu, LLC, Deltak is expected to contribute solid growth to both Wileys Global Education business and Wiley overall. Seventh-generation members Jesse and Nate Wiley work in the companys Professional/Trade and Scientific, Technical, Medical, and Scholarly businesses, respectively. Wiley has been owned since 1962, and listed on the New York Stock Exchange since 1995, its stock is traded under the symbols NYSE, JW. A and NYSE
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World Scientific
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World Scientific Publishing is an academic publisher of scientific, technical, and medical books and journals headquartered in Singapore. The company was founded in 1981 and it publishes about 500 books annually as well as more than 150 journals in various fields. In 1995, World Scientific co-founded the London-based Imperial College Press together with the Imperial College of Science, Technology, the company head office is in Singapore. The Chairman and Editor-in-Chief is Phua Kok Khoo, while the Managing Director is Doreen Liu, the company was co-founded by them in 1981. In 1995 the company co-founded Imperial College Press, specializing in engineering, medicine and information technology, in 2006, World Scientific assumed full ownership of Imperial College Press, under a license granted by the university. Finally in August 2016, ICP was fully incorporated into World Scientific under the new imprint, in 2014, the Singapore Literature Prize was criticized for naming the non-fiction prizes after World Scientific, who sponsored the category. A book World Scientific published, Lim Siong Guans The Leader, The Teacher & You, journals published by World Scientific Official website
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University of Kerala
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The University of Kerala, formerly the University of Travancore, is an affiliating university located in Thiruvananthapuram, capital of the south Indian state of Kerala, India. C. P. Ramaswami Iyer, the then Diwan of Travancore, was the first Vice-Chancellor, during the 7 decades since the University of Kerala grew and shrunk physically and transformed itself in many ways. The earliest origins of the University may be traced back to two institutions of learning in Kerala, the University College, Thiruvananthapuram and the Trivandrum Observatory. The University College was initially founded as the Maharajas Free School by Maharaja Swathi Thirunal in 1834, a Christian Missionary as Headmaster, and soon grew into a college in 1866, affiliated to the Madras University. The University College still retains its connection with the University as an affiliated college, the Trivandrum Observatory was founded in 1838 and had an internationally reputed scientist, John Caldecott FRS as its first Director. It became a part of the Travancore University, but for time was administered as an independent government institution. It is now the oldest institution under the Kerala University, the University of Travancore was established in 1937 by a promulgation of the Maharajah of Travancore, Sri Chithira Thirunal Balarama Varma who was also the first Chancellor of the University. P Ramaswamy Ayyar, the then Diwan of the State was the first Vice-Chancellor and he was an eminent scholar and an able administrator. It is said the Government made an attempt to invite Albert Einstein to be the first Vice-Chancellor. The University was modelled after the best Universities of the United Kingdom, the affiliating system of the University however evolved to be different from the college system in British Universities. Only ten colleges within the State of Travancore, which were at that time affiliated to the Madras University, in 1954, the unified state of Kerala came into being with most of Travancore and whole of state of Cochin and Malabar area of Madras presidency becoming part of it. The Kerala University Act was brought into force and the University of Travancore was renamed University of Kerala, the University had three campuses located in three different parts of the State viz. Thiruvananthapuram, Ernakulam and Kozhikode. The number of affiliated colleges grew phenomenally and these developments have shrunk the jurisdiction of the University of Kerala to Thiruvananthapuram, Kollam, Alappuzha Districts and some parts of Pathanamthitta District. At present, the University has sixteen faculties and forty one departments of teaching and research in addition to study centres, teaching, Research and Knowledge extension are the mandate of the Departments. They primarily focus on post-graduate programmes, MPhil programmes and doctoral research, in 2007, the University awarded over 100 PhDs. The University has had in its faculties eminent scholars who were trained under legendary figures, some of the faculty members themselves were legends, such as famous poet K Ayyappa Panciker, Oriental scholar T Ganapathi Sasthri etc. The total number of students in the University Departments is above 2000 including research students. The Institute of Distance Education offers a number of under-graduate and post graduate programmes which cater to more than 7000 students, all over the country, some of these centres have taught programmes and many offer PhD programmes
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Thiruvananthapuram
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Thiruvananthapuram, formerly known as Trivandrum, is the capital and the largest city of the Indian state of Kerala. It is located on the west coast of India near the south of the mainland. Referred to by Mahatma Gandhi as the Evergreen city of India, Thiruvananthapuram was a trading post for spices, sandalwood and ivory. The city was ruled by the Ays and was captured by the rulers of Venad in tenth century A. D, in 1729, Marthanda Varma founded the princely state of Thiruvithamkoor and made Thiruvananthapuram the capital in 1745. It remained as a state ruled by Travancore under the loose governance of the British before joining the Indian Union in 1948. With nearly 80% of the software exports, Thiruvananthapuram is a major IT hub with the Technopark. Indias first and only magic academy, Magic Academy Research Centre run by Merlin award-winning magician Gopinath Muthukad is situated in Thiruvananthapuram, the city is home to animation companies like Toonz India Ltd and Tata Elxsi Ltd. The Kinfra Film and Video Park is one of the most advanced film and it is consistently ranked among the best cities to live in Kerala as well as India. The city was referred to as Trivandrum until 1991, when the government decided to reinstate the citys original name Thiruvananthapuram. Thiruvananthapuram is an ancient region with trading traditions dating back to 1000 BCE, the city was a trading post for spices, sandalwood and ivory. The early rulers of the city were the Ays and after their fall in the 10th century, in 1729, Marthanda Varma founded the princely state of Thiruvithamkoor and Thiruvananthapuram was made the capital in 1745 after shifting the capital from Padmanabhapuram in Tamil Nadu. In the mid-19th century, the city was under the reign of Swathi Thirunal, an observatory was established in 1837 with the Oriental Research Institute & Manuscripts Library and the University College established in 1873. Several colleges were established by Moolam Thirunal, sree Moolam Assembly, established in 1904, was the first democratically elected legislative council in any Indian state. After Indian Independence in 1947, Travancore chose to join the Indian union, in 1949, Thiruvananthapuram became the capital of Travancore-Cochin, the state formed by the integration of Travancore with the Kingdom of Cochin. The king of Travancore, Chitra Thirunal Bala Rama Varma, became the Rajpramukh of the Travancore-Cochin Union from 1 July 1949 until 31 October 1956, when the state of Kerala was formed on 1 November 1956, Thiruvananthapuram became its capital. The city has a population of 752,490 according to the 2011 census, the sex ratio is 1,032 females for every 1,000 males. In October 2010, the number of wards was increased from 86 to 100 post expansion of city limits by adding Sreekaryam, Vattiyoorkavu, Kudappanakunnu, hindus comprise 68. 5% of the population, Christians about 16. 7% and Muslims form 13. 7%. The major languages spoken are Malayalam and English, in Palayam in the city centre, there is a mosque, a temple and a Christian church next to each other as neighbours, establishing the communal harmony of Keralites
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India
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India, officially the Republic of India, is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and it is bounded by the Indian Ocean on the south, the Arabian Sea on the southwest, and the Bay of Bengal on the southeast. It shares land borders with Pakistan to the west, China, Nepal, and Bhutan to the northeast, in the Indian Ocean, India is in the vicinity of Sri Lanka and the Maldives. Indias Andaman and Nicobar Islands share a border with Thailand. The Indian subcontinent was home to the urban Indus Valley Civilisation of the 3rd millennium BCE, in the following millennium, the oldest scriptures associated with Hinduism began to be composed. Social stratification, based on caste, emerged in the first millennium BCE, early political consolidations took place under the Maurya and Gupta empires, the later peninsular Middle Kingdoms influenced cultures as far as southeast Asia. In the medieval era, Judaism, Zoroastrianism, Christianity, and Islam arrived, much of the north fell to the Delhi sultanate, the south was united under the Vijayanagara Empire. The economy expanded in the 17th century in the Mughal empire, in the mid-18th century, the subcontinent came under British East India Company rule, and in the mid-19th under British crown rule. A nationalist movement emerged in the late 19th century, which later, under Mahatma Gandhi, was noted for nonviolent resistance, in 2015, the Indian economy was the worlds seventh largest by nominal GDP and third largest by purchasing power parity. Following market-based economic reforms in 1991, India became one of the major economies and is considered a newly industrialised country. However, it continues to face the challenges of poverty, corruption, malnutrition, a nuclear weapons state and regional power, it has the third largest standing army in the world and ranks sixth in military expenditure among nations. India is a constitutional republic governed under a parliamentary system. It is a pluralistic, multilingual and multi-ethnic society and is home to a diversity of wildlife in a variety of protected habitats. The name India is derived from Indus, which originates from the Old Persian word Hindu, the latter term stems from the Sanskrit word Sindhu, which was the historical local appellation for the Indus River. The ancient Greeks referred to the Indians as Indoi, which translates as The people of the Indus, the geographical term Bharat, which is recognised by the Constitution of India as an official name for the country, is used by many Indian languages in its variations. Scholars believe it to be named after the Vedic tribe of Bharatas in the second millennium B. C. E and it is also traditionally associated with the rule of the legendary emperor Bharata. Gaṇarājya is the Sanskrit/Hindi term for republic dating back to the ancient times, hindustan is a Persian name for India dating back to the 3rd century B. C. E. It was introduced into India by the Mughals and widely used since then and its meaning varied, referring to a region that encompassed northern India and Pakistan or India in its entirety
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Semigroup Forum
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Semigroup Forum is a mathematics research journal published by Springer. The journal serves as a platform for the speedy and efficient transmission of information on current research in semigroup theory, semigroups of operators were initially considered off-topic, but began being included in the journal in 1985. Semigroup Forum features survey and research articles, the journal published its first issue in 1970. It is indexed in Science Citation Index Expanded, Journal Citation Reports/Science Edition, SCOPUS and this journal has been called in many ways a point of crystallization for semigroup theory and its community, and an indicator of a field which is mathematically active