In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f of some function f. An important class of pointwise concepts are the pointwise operations, operations defined on functions by applying the operations to function values separately for each point in the domain of definition. Important relations can be defined pointwise. A binary operation o: Y × Y → Y on a set Y can be lifted pointwise to an operation O: × → on the set X→Y of all functions from X to Y as follows: Given two functions f1: X → Y and f2: X → Y, define the function O: X → Y by = o for all x∈X. O and O are denoted by the same symbol. A similar definition is used for unary operations o, for operations of other arity. = f + g = f ⋅ g = λ ⋅ f where f, g: X → R. See pointwise product, scalar. An example of an operation on functions, not pointwise is convolution. Pointwise operations inherit such properties as associativity and distributivity from corresponding operations on the codomain.
If A is some algebraic structure, the set of all functions X to the carrier set of A can be turned into an algebraic structure of the same type in an analogous way. Componentwise operations are defined on vectors, where vectors are elements of the set K n for some natural number n and some field K. If we denote the i -th component of any vector v as v i componentwise addition is i = u i + v i. Componentwise operations can be defined on matrices. Matrix addition, where i j = A i j + B i j is a componentwise operation while matrix multiplication is not. A tuple can be regarded as a function, a vector is a tuple. Therefore, any vector v corresponds to the function f: n → K such that f = v i, any componentwise operation on vectors is the pointwise operation on functions corresponding to those vectors. In order theory it is common to define a pointwise partial order on functions. With A, B posets, the set of functions A → B can be ordered by only if f ≤ g. Pointwise orders inherit some properties of the underlying posets.
For instance if A and B are continuous lattices so is the set of functions A → B with pointwise order. Using the pointwise order on functions one can concisely define other important notions, for instance: A closure operator c on a poset P is a monotone and idempotent self-map on P with the additional property that idA ≤ c, where id is the identity function. A projection operator k is called a kernel operator if and only if k ≤ idA. An example of infinitary pointwise relation is pointwise convergence of functions — a sequence of functions n = 1 ∞ with f n: X ⟶ Y converges pointwise to a function f if for each x in X lim n → ∞ f n = f. For order theory examples: T. S. Blyth and Ordered Algebraic Structures, Springer
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Free abelian group
In abstract algebra, a free abelian group or free Z-module is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation, associative and invertible. A basis is a subset such that every element of the group can be found by adding or subtracting basis elements, such that every element's expression as a linear combination of basis elements is unique. For instance, the integers under addition form a free abelian group with basis. Addition of integers is commutative and has subtraction as its inverse operation, each integer is the sum or difference of some number of copies of the number 1, each integer has a unique representation as an integer multiple of the number 1. Free abelian groups have properties, they have applications in algebraic topology, where they are used to define chain groups, in algebraic geometry, where they are used to define divisors. Integer lattices form examples of free abelian groups, lattice theory studies free abelian subgroups of real vector spaces.
The elements of a free abelian group with basis B may be described in several equivalent ways. These include formal sums over B, which are expressions of the form ∑ a i b i where each coefficient ai is a nonzero integer, each factor bi is a distinct basis element, the sum has finitely many terms. Alternatively, the elements of a free abelian group may be thought of as signed multisets containing finitely many elements of B, with the multiplicity of an element in the multiset equal to its coefficient in the formal sum. Another way to represent an element of a free abelian group is as a function from B to the integers with finitely many nonzero values; every set B has a free abelian group with B as its basis. This group is unique in the sense that every two free abelian groups with the same basis are isomorphic. Instead of constructing it by describing its individual elements, a free group with basis B may be constructed as a direct sum of copies of the additive group of the integers, with one copy per member of B.
Alternatively, the free abelian group with basis B may be described by a presentation with the elements of B as its generators and with the commutators of pairs of members as its relators. The rank of a free abelian group is the cardinality of a basis; every subgroup of a free abelian group is itself free abelian. The integers, under the addition operation, form a free abelian group with the basis; every integer n is a linear combination of basis elements with integer coefficients: namely, n = n × 1, with the coefficient n. The two-dimensional integer lattice, consisting of the points in the plane with integer Cartesian coordinates, forms a free abelian group under vector addition with the basis. Letting these basis vectors be denoted e 1 = and e 2 =, the element can be written = 4 e 1 + 3 e 2 where'multiplication' is defined so that 4 e 1:= e 1 + e 1 + e 1 + e 1. In this basis, there is no other way to write. However, with a different basis such as, where f 1 = and f 2 =, it can be written as = f 1 + 3 f 2.
More every lattice forms a finitely-generated free abelian group. The d-dimensional integer lattice has a natural basis consisting of the positive integer unit vectors, but it has many other bases as well: if M is a d × d integer matrix with determinant ±1 the rows of M form a basis, conversely every basis of the integer lattice has this form. For more on the two-dimensional case, see fundamental pair of periods; the direct product of two free abelian groups is itself free abelian, with basis the disjoint union of the bases of the two groups. More the direct product of any finite number of free abelian groups is free abelian; the d-dimensional integer lattice, for instance, is isomorphic to the direct product of d copies of the integer group Z. The trivial group is considered to be free abelian, with basis the empty set, it may be interpreted as a direct product of zero copies of Z. For infinite families of free abelian groups, the direct product is not free abelian. For instance the Baer–Specker group Z N
International Standard Serial Number
An International Standard Serial Number is an eight-digit serial number used to uniquely identify a serial publication, such as a magazine. The ISSN is helpful in distinguishing between serials with the same title. ISSN are used in ordering, interlibrary loans, other practices in connection with serial literature; the ISSN system was first drafted as an International Organization for Standardization international standard in 1971 and published as ISO 3297 in 1975. ISO subcommittee TC 46/SC 9 is responsible for maintaining the standard; when a serial with the same content is published in more than one media type, a different ISSN is assigned to each media type. For example, many serials are published both in electronic media; the ISSN system refers to these types as electronic ISSN, respectively. Conversely, as defined in ISO 3297:2007, every serial in the ISSN system is assigned a linking ISSN the same as the ISSN assigned to the serial in its first published medium, which links together all ISSNs assigned to the serial in every medium.
The format of the ISSN is an eight digit code, divided by a hyphen into two four-digit numbers. As an integer number, it can be represented by the first seven digits; the last code digit, which may be 0-9 or an X, is a check digit. Formally, the general form of the ISSN code can be expressed as follows: NNNN-NNNC where N is in the set, a digit character, C is in; the ISSN of the journal Hearing Research, for example, is 0378-5955, where the final 5 is the check digit, C=5. To calculate the check digit, the following algorithm may be used: Calculate the sum of the first seven digits of the ISSN multiplied by its position in the number, counting from the right—that is, 8, 7, 6, 5, 4, 3, 2, respectively: 0 ⋅ 8 + 3 ⋅ 7 + 7 ⋅ 6 + 8 ⋅ 5 + 5 ⋅ 4 + 9 ⋅ 3 + 5 ⋅ 2 = 0 + 21 + 42 + 40 + 20 + 27 + 10 = 160 The modulus 11 of this sum is calculated. For calculations, an upper case X in the check digit position indicates a check digit of 10. To confirm the check digit, calculate the sum of all eight digits of the ISSN multiplied by its position in the number, counting from the right.
The modulus 11 of the sum must be 0. There is an online ISSN checker. ISSN codes are assigned by a network of ISSN National Centres located at national libraries and coordinated by the ISSN International Centre based in Paris; the International Centre is an intergovernmental organization created in 1974 through an agreement between UNESCO and the French government. The International Centre maintains a database of all ISSNs assigned worldwide, the ISDS Register otherwise known as the ISSN Register. At the end of 2016, the ISSN Register contained records for 1,943,572 items. ISSN and ISBN codes are similar in concept. An ISBN might be assigned for particular issues of a serial, in addition to the ISSN code for the serial as a whole. An ISSN, unlike the ISBN code, is an anonymous identifier associated with a serial title, containing no information as to the publisher or its location. For this reason a new ISSN is assigned to a serial each time it undergoes a major title change. Since the ISSN applies to an entire serial a new identifier, the Serial Item and Contribution Identifier, was built on top of it to allow references to specific volumes, articles, or other identifiable components.
Separate ISSNs are needed for serials in different media. Thus, the print and electronic media versions of a serial need separate ISSNs. A CD-ROM version and a web version of a serial require different ISSNs since two different media are involved. However, the same ISSN can be used for different file formats of the same online serial; this "media-oriented identification" of serials made sense in the 1970s. In the 1990s and onward, with personal computers, better screens, the Web, it makes sense to consider only content, independent of media; this "content-oriented identification" of serials was a repressed demand during a decade, but no ISSN update or initiative occurred. A natural extension for ISSN, the unique-identification of the articles in the serials, was the main demand application. An alternative serials' contents model arrived with the indecs Content Model and its application, the digital object identifier, as ISSN-independent initiative, consolidated in the 2000s. Only in 2007, ISSN-L was defined in the
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series and functions. A ring is an abelian group with a second binary operation, associative, is distributive over the abelian group operation, has an identity element. By extension from the integers, the abelian group operation is called addition and the second binary operation is called multiplication. Whether a ring is commutative or not has profound implications on its behavior as an abstract object; as a result, commutative ring theory known as commutative algebra, is a key topic in ring theory. Its development has been influenced by problems and ideas occurring in algebraic number theory and algebraic geometry. Examples of commutative rings include the set of integers equipped with the addition and multiplication operations, the set of polynomials equipped with their addition and multiplication, the coordinate ring of an affine algebraic variety, the ring of integers of a number field.
Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators in the theory of differential operators, the cohomology ring of a topological space in topology. The conceptualization of rings was completed in the 1920s. Key contributors include Dedekind, Hilbert and Noether. Rings were first formalized as a generalization of Dedekind domains that occur in number theory, of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. Afterward, they proved to be useful in other branches of mathematics such as geometry and mathematical analysis; the most familiar example of a ring is the set of all integers, Z, consisting of the numbers …, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, …The familiar properties for addition and multiplication of integers serve as a model for the axioms for rings. A ring is a set R equipped with two binary operations + and · satisfying the following three sets of axioms, called the ring axioms R is an abelian group under addition, meaning that: + c = a + for all a, b, c in R. a + b = b + a for all a, b in R.
There is an element 0 in R such that a + 0 = a for all a in R. For each a in R there exists −a in R such that a + = 0. R is a monoid under multiplication, meaning that: · c = a · for all a, b, c in R. There is an element 1 in R such that a · 1 = a and 1 · a = a for all a in R. Multiplication is distributive with respect to addition, meaning that: a ⋅ = + for all a, b, c in R. · a = + for all a, b, c in R. As explained in § History below, many authors follow an alternative convention in which a ring is not defined to have a multiplicative identity; this article adopts the convention that, unless otherwise stated, a ring is assumed to have such an identity. A structure satisfying all the axioms except the requirement that there exists a multiplicative identity element is called a rng. For example, the set of integers with the usual + and ⋅ is a rng, but not a ring; the operations + and ⋅ are called multiplication, respectively. The multiplication symbol ⋅ is omitted, so the juxtaposition of ring elements is interpreted as multiplication.
For example, xy means x ⋅ y. Although ring addition is commutative, ring multiplication is not required to be commutative: ab need not equal ba. Rings that satisfy commutativity for multiplication are called commutative rings. Books on commutative algebra or algebraic geometry adopt the convention that ring means commutative ring, to simplify terminology. In a ring, multiplication does not have to have an inverse. A commutative ring such; the additive group of a ring is the ring equipped just with the structure of addition. Although the definition assumes that the additive group is abelian, this can be inferred from the other ring axioms; some basic properties of a ring follow from the axioms: The additive identity, the additive inverse of each element, the multiplicative identity are unique. For any element x in a ring R, one has x0 = 0 = 0x and x = –x. If 0 = 1 in a ring R R has only one element, is called the zero ring; the binomial formula holds for any commuting pair of elements. Equip the set Z 4 = with the following operat
In mathematics, the natural numbers are those used for counting and ordering. In common mathematical terminology, words colloquially used for counting are "cardinal numbers" and words connected to ordering represent "ordinal numbers"; the natural numbers can, at times, appear as a convenient set of codes. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, …, whereas others start with 1, corresponding to the positive integers 1, 2, 3, …. Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers; the natural numbers are a basis from which many other number sets may be built by extension: the integers, by including the neutral element 0 and an additive inverse for each nonzero natural number n. These chains of extensions make the natural numbers canonically embedded in the other number systems.
Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics. In common language, for example in primary school, natural numbers may be called counting numbers both to intuitively exclude the negative integers and zero, to contrast the discreteness of counting to the continuity of measurement, established by the real numbers; the most primitive method of representing a natural number is to put down a mark for each object. A set of objects could be tested for equality, excess or shortage, by striking out a mark and removing an object from the set; the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers; the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1, 10, all the powers of 10 up to over 1 million.
A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds, 7 tens, 6 ones. The Babylonians had a place-value system based on the numerals for 1 and 10, using base sixty, so that the symbol for sixty was the same as the symbol for one, its value being determined from context. A much advance was the development of the idea that 0 can be considered as a number, with its own numeral; the use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, but they omitted such a digit when it would have been the last symbol in the number. The Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica; the use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628. However, 0 had been used as a number in the medieval computus, beginning with Dionysius Exiguus in 525, without being denoted by a numeral; the first systematic study of numbers as abstractions is credited to the Greek philosophers Pythagoras and Archimedes.
Some Greek mathematicians treated the number 1 differently than larger numbers, sometimes not as a number at all. Independent studies occurred at around the same time in India and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the exact nature of the natural numbers. A school of Naturalism stated that the natural numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized "God made the integers, all else is the work of man". In opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not natural but a consequence of definitions. Two classes of such formal definitions were constructed. Set-theoretical definitions of natural numbers were initiated by Frege and he defined a natural number as the class of all sets that are in one-to-one correspondence with a particular set, but this definition turned out to lead to paradoxes including Russell's paradox.
Therefore, this formalism was modified so that a natural number is defined as a particular set, any set that can be put into one-to-one correspondence with that set is said to have that number of elements. The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, further explored by Giuseppe Peano, it is based on an axiomatization of the properties of ordinal numbers: each natural number has a