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
Kazimierz Kuratowski was a Polish mathematician and logician. He was one of the leading representatives of the Warsaw School of Mathematics. Kazimierz Kuratowski was born in Warsaw, Vistula Land, on 2 February 1896, into an assimilated Jewish family, he was a son of Marek Kuratow, a barrister, Róża Karzewska. He completed a Warsaw secondary school, named after general Paweł Chrzanowski. In 1913, he enrolled in an engineering course at the University of Glasgow in Scotland, in part because he did not wish to study in Russian, he completed only one year of study when the outbreak of World War I precluded any further enrollment. In 1915, Russian forces withdrew from Warsaw and Warsaw University was reopened with Polish as the language of instruction. Kuratowski restarted his university education there the same year, this time in mathematics, he obtained his Ph. D. in 1921, in newly independent Poland. In autumn 1921 Kuratowski was awarded the Ph. D. degree for his groundbreaking work. His thesis statement consisted of two parts.
One was devoted to an axiomatic construction of topology via the closure axioms. This first part has been cited in hundreds of scientific articles; the second part of Kuratowski's thesis was devoted to continua irreducible between two points. This was the subject of a French doctoral thesis written by Zygmunt Janiszewski. Since Janiszewski was deceased, Kuratowski's supervisor was Stefan Mazurkiewicz. Kuratowski's thesis solved certain problems in set theory raised by a Belgian mathematician, Charles-Jean Étienne Gustave Nicolas, Baron de la Vallée Poussin. Two years in 1923, Kuratowski was appointed deputy professor of mathematics at Warsaw University, he was appointed a full professor of mathematics at Lwów Polytechnic in Lwów, in 1927. He was the head of the Mathematics department there until 1933. Kuratowski was dean of the department twice. In 1929, Kuratowski became a member of the Warsaw Scientific Society While Kuratowski associated with many of the scholars of the Lwów School of Mathematics, such as Stefan Banach and Stanislaw Ulam, the circle of mathematicians based around the Scottish Café he kept close connections with Warsaw.
Kuratowski left Lwów for Warsaw in 1934, before the famous Scottish Book was begun, hence did not contribute any problems to it. He did however, collaborate with Banach in solving important problems in measure theory. In 1934 he was appointed the professor at Warsaw University. A year Kuratowski was nominated as the head of Mathematics Department there. From 1936 to 1939 he was secretary of the Mathematics Committee in The Council of Science and Applied Sciences. During World War II, he gave lectures at the underground university in Warsaw, since higher education for Poles was forbidden under German occupation. In February 1945, Kuratowski started to lecture at the reopened Warsaw University. In 1945, he became a member of the Polish Academy of Learning, in 1946 he was appointed vice-president of the Mathematics department at Warsaw University, from 1949 he was chosen to be the vice-president of Warsaw Scientific Society. In 1952 he became a member of the Polish Academy of Sciences, of which he was the vice-president from 1957 to 1968.
After World War II, Kuratowski was involved in the rebuilding of scientific life in Poland. He helped to establish the State Mathematical Institute, incorporated into the Polish Academy of Sciences in 1952. From 1948 until 1967 Kuratowski was director of the Institute of Mathematics of the Polish Academy of Sciences, was a long-time chairman of the Polish and International Mathematics Societies, he was president of the Scientific Council of the State Institute of Mathematics. From 1948 to 1980 he was the head of the topology section. One of his students was Andrzej Mostowski. Kazimierz Kuratowski was one of a celebrated group of Polish mathematicians who would meet at Lwów's Scottish Café, he was a member of the Warsaw Scientific Society. What is more, he was chief editor in "Fundamenta Mathematicae", a series of publications in "Polish Mathematical Society Annals". Furthermore, Kuratowski worked as an editor in the Polish Academy of Sciences Bulletin, he was one of the writers of the Mathematical monographs, which were created in cooperation with the Institute of Mathematics of the Polish Academy of Sciences.
High quality research monographs of the representatives of Warsaw's and Lwów’s School of Mathematics, which concerned all areas of pure and applied mathematics, were published in these volumes. Kazimierz Kuratowski was an active member of many scientific societies and foreign scientific academies, including the Royal Society of Edinburgh, Germany, Hungary and the Union of Soviet Socialist Republics. In 1981, IMPAN, the Polish Mathematical Society, Kuratowski's daughter Zofia Kuratowska established a prize in his name for achievements in mathematics to people under the age of 30 years; the prize is considered the most prestigious of awards for young Polish mathematicians. Kuratowski’s research focused on abstract topological and metric structures, he implemented the closure axioms. This was fundamental for the development of topological space theory and irreducible continuum theory between two points; the most valuable results, which were obtained by Kazimierz K
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, more in functional analysis, a Banach space is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well defined limit, within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Banach spaces grew out of the study of function spaces by Hilbert, Fréchet, Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are Banach spaces. A Banach space is a vector space X over the field R of real numbers, or over the field C of complex numbers, equipped with a norm ‖ ⋅ ‖ X and, complete with respect to the distance function induced by the norm, to say, for every Cauchy sequence in X, there exists an element x in X such that lim n → ∞ x n = x, or equivalently: lim n → ∞ ‖ x n − x ‖ X = 0.
The vector space structure allows one to relate the behavior of Cauchy sequences to that of converging series of vectors. A normed space X is a Banach space if and only if each convergent series in X converges, ∑ n = 1 ∞ ‖ v n ‖ X < ∞ implies that ∑ n = 1 ∞ v n converges in X. Completeness of a normed space is preserved if the given norm is replaced by an equivalent one. All norms on a finite-dimensional vector space are equivalent; every finite-dimensional normed space over R or C is a Banach space. If X and Y are normed spaces over the same ground field K, the set of all continuous K-linear maps T: X → Y is denoted by B. In infinite-dimensional spaces, not all linear maps are continuous. A linear mapping from a normed space X to another normed space is continuous if and only if it is bounded on the closed unit ball of X. Thus, the vector space B can be given the operator norm ‖ T ‖ = sup. For Y a Banach space, the space B is a Banach space with respect to this norm. If X is a Banach space, the space B = B forms a unital Banach algebra.
If X and Y are normed spaces, they are isomorphic normed spaces if there exists a linear bijection T: X → Y such that T and its inverse T −1 are continuous. If one of the two spaces X or Y is complete so is the other space. Two normed spaces X and Y are isometrically isomorphic if in addition, T is an isometry, i.e. ||T|| = ||x|| for every x in X. The Banach–Mazur distance d between two isomorphic but not isometric spaces X and Y gives a measure of how much the two spaces X and Y differ; every normed space X can be isometrically embedded in a Banach space. More for every normed space X, there exist a Banach space Y and a mapping T: X → Y such that T is an isometric mapping and T is dense in Y. If Z is another Banach space such that there is an isometric isomorphism from X onto a dense subset of Z Z is isometrically isomorphic to Y; this Banach space Y is the completion of the normed space X. The underlying metric space for Y is the same as the metric completion of X, with the vector space operations extended from X to Y.
The completion of X is denoted by X ^. The cartesian product X × Y of two normed spaces is not canonically equipped with a norm. However, several equivalent norms are used, such as ‖ ‖ 1 = ‖ x ‖ + ‖ y ‖, ‖ ‖ ∞ = max and give rise to isomorphic normed spaces. In this sense, the product X × Y is only if the two factors are complete. If M is a closed linear subspace of a normed space X, there is a natural norm on the quotient space X / M, ‖ x + M ‖ = inf m ∈ M ‖ x + m ‖; the quotient X / M is a Banach space
Inner product space
In linear algebra, an inner product space is a vector space with an additional structure called an inner product. This additional structure associates each pair of vectors in the space with a scalar quantity known as the inner product of the vectors. Inner products allow the rigorous introduction of intuitive geometrical notions such as the length of a vector or the angle between two vectors, they provide the means of defining orthogonality between vectors. Inner product spaces generalize Euclidean spaces to vector spaces of any dimension, are studied in functional analysis; the first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898. An inner product induces an associated norm, thus an inner product space is a normed vector space. A complete space with an inner product is called a Hilbert space. An space with an inner product is called a pre-Hilbert space, since its completion with respect to the norm induced by the inner product is a Hilbert space.
Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. In this article, the field of scalars denoted F is either the field of real numbers R or the field of complex numbers C. Formally, an inner product space is a vector space V over the field F together with an inner product, i.e. with a map ⟨ ⋅, ⋅ ⟩: V × V → F that satisfies the following three axioms for all vectors x, y, z ∈ V and all scalars a ∈ F: Conjugate symmetry: ⟨ x, y ⟩ = ⟨ y, x ⟩ ¯ Linearity in the first argument: ⟨ a x, y ⟩ = a ⟨ x, y ⟩ ⟨ x + y, z ⟩ = ⟨ x, z ⟩ + ⟨ y, z ⟩ Positive-definite: ⟨ x, x ⟩ > 0, x ∈ V ∖. Positive-definiteness and linearity ensure that: ⟨ x, x ⟩ = 0 ⇒ x = 0 ⟨ 0, 0 ⟩ = ⟨ 0 x, 0 x ⟩ = 0 ⟨ x, 0 x ⟩ = 0 Notice that conjugate symmetry implies that ⟨x, x⟩ is real for all x, since we have: ⟨ x, x ⟩ = ⟨ x, x ⟩ ¯. Conjugate symmetry and linearity in the first variable imply ⟨ x, a y ⟩ = ⟨ a y, x ⟩ ¯ = a ¯ ⟨ y, x ⟩ ¯ = a ¯ ⟨ x, y ⟩ ⟨ x, y + z ⟩ = ⟨ y + z, x ⟩ ¯ = ⟨ y, x ⟩ ¯ + ⟨ z, x ⟩ ¯ = ⟨ x, y ⟩ + ⟨ x, z ⟩.
Partition of a set
In mathematics, a partition of a set is a grouping of the set's elements into non-empty subsets, in such a way that every element is included in one and only one of the subsets. Every equivalence relation on a set defines a partition of this set, every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a setoid in type theory and proof theory. A partition of a set X is a set of nonempty subsets of X such that every element x in X is in one of these subsets. Equivalently, a family of sets P is a partition of X if and only if all of the following conditions hold: The family P does not contain the empty set; the union of the sets in P is equal to X. The sets in P are said to cover X; the intersection of any two distinct sets in P is empty. The elements of P are said to be pairwise disjoint; the sets in P are called the parts or cells of the partition. The rank of P is | X | − | P |; the empty set ∅ has one partition, namely ∅. For any nonempty set X, P = is a partition of X, called the trivial partition.
Every singleton set has one partition, namely. For any non-empty proper subset A of a set U, the set A together with its complement form a partition of U, namely; the set has these five partitions:, sometimes written 1|2|3. Or 12|3. Or 13|2. Or 1|23. Or 123; the following are not partitions of: is not a partition because one of its elements is the empty set. Is not a partition because the element 2 is contained in more than one block. Is not a partition of because none of its blocks contains 3. For any equivalence relation on a set X, the set of its equivalence classes is a partition of X. Conversely, from any partition P of X, we can define an equivalence relation on X by setting x ~ y when x and y are in the same part in P, thus the notions of equivalence relation and partition are equivalent. The axiom of choice guarantees for any partition of a set X the existence of a subset of X containing one element from each part of the partition; this implies that given an equivalence relation on a set one can select a canonical representative element from every equivalence class.
A partition α of a set X is a refinement of a partition ρ of X—and we say that α is finer than ρ and that ρ is coarser than α—if every element of α is a subset of some element of ρ. Informally, this means that α is a further fragmentation of ρ. In that case, it is written that α ≤ ρ; this finer-than relation on the set of partitions of X is a partial order. Each set of elements has a least upper bound and a greatest lower bound, so that it forms a lattice, more it is a geometric lattice; the partition lattice of a 4-element set has 15 elements and is depicted in the Hasse diagram on the left. Based on the cryptomorphism between geometric lattices and matroids, this lattice of partitions of a finite set corresponds to a matroid in which the base set of the matroid consists of the atoms of the lattice, the partitions with n − 2 singleton sets and one two-element set; these atomic partitions correspond one-for-one with the edges of a complete graph. The matroid closure of a set of atomic partitions is the finest common coarsening of them all.
In this way, the lattice of partitions corresponds to the lattice of flats of the graphic matroid of the complete graph. Another example illustrates the refining of partitions from the perspective of equivalence relations. If D is the set of cards in a standard 52-card deck, the same-color-as relation on D – which can be denoted ~C – has two equivalence classes: the sets and; the 2-part partition corresponding to ~C has a refinement that yields the same-suit-as relation ~S, which has the four equivalence classes, and. A partition of the set N = with corresponding equivalence relation ~ is noncrossing if it has the following property: If four elements a, b, c and d of N having a < b < c < d satisfy a ~ c and b ~ d a ~ b ~ c ~ d. The name comes from the following equivalent definition: Imagine the elements 1, 2... N of N drawn as the n vertices of a regular n-gon. A partition can be visualized by drawing each block as a polygon; the partition is noncrossing if and only if these polygons do not intersect.
The lattice of noncrossing partitions of a finite set has taken on importance because of its role in free probability theory. These form a subset of the lattice of
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions. A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured. Furthermore, Hilbert spaces are complete: there are enough limits in the space to allow the techniques of calculus to be used. Hilbert spaces arise and in mathematics and physics as infinite-dimensional function spaces; the earliest Hilbert spaces were studied from this point of view in the first decade of the 20th century by David Hilbert, Erhard Schmidt, Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, quantum mechanics, Fourier analysis, ergodic theory. John von Neumann coined the term Hilbert space for the abstract concept that underlies many of these diverse applications.
The success of Hilbert space methods ushered in a fruitful era for functional analysis. Apart from the classical Euclidean spaces, examples of Hilbert spaces include spaces of square-integrable functions, spaces of sequences, Sobolev spaces consisting of generalized functions, Hardy spaces of holomorphic functions. Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in a Hilbert space. At a deeper level, perpendicular projection onto a subspace plays a significant role in optimization problems and other aspects of the theory. An element of a Hilbert space can be uniquely specified by its coordinates with respect to a set of coordinate axes, in analogy with Cartesian coordinates in the plane; when that set of axes is countably infinite, the Hilbert space can be usefully thought of in terms of the space of infinite sequences that are square-summable. The latter space is in the older literature referred to as the Hilbert space.
Linear operators on a Hilbert space are fairly concrete objects: in good cases, they are transformations that stretch the space by different factors in mutually perpendicular directions in a sense, made precise by the study of their spectrum. One of the most familiar examples of a Hilbert space is the Euclidean space consisting of three-dimensional vectors, denoted by ℝ3, equipped with the dot product; the dot product takes two vectors x and y, produces a real number x · y. If x and y are represented in Cartesian coordinates the dot product is defined by ⋅ = x 1 y 1 + x 2 y 2 + x 3 y 3; the dot product satisfies the properties: It is symmetric in x and y: x · y = y · x. It is linear in its first argument: · y = ax1 · y + bx2 · y for any scalars a, b, vectors x1, x2, y, it is positive definite: for all vectors x, x · x ≥ 0, with equality if and only if x = 0. An operation on pairs of vectors that, like the dot product, satisfies these three properties is known as a inner product. A vector space equipped with such an inner product is known as a inner product space.
Every finite-dimensional inner product space is a Hilbert space. The basic feature of the dot product that connects it with Euclidean geometry is that it is related to both the length of a vector, denoted ||x||, to the angle θ between two vectors x and y by means of the formula x ⋅ y = ‖ x ‖ ‖ y ‖ cos θ. Multivariable calculus in Euclidean space relies on the ability to compute limits, to have useful criteria for concluding that limits exist. A mathematical series ∑ n = 0 ∞ x n consisting of vectors in ℝ3 is convergent provided that the sum of the lengths converges as an ordinary series of real numbers: ∑ k = 0 ∞ ‖ x k ‖ < ∞. Just as with a series of scalars, a series of vectors that converges also converges to some limit vector L in the Euclidean space, in the sense that ‖ L − ∑ k = 0 N x k ‖ → 0 as N → ∞; this property expresses the completeness of