In set theory, a Cartesian product is a mathematical operation that returns a set from multiple sets. That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs where a ∈ A and b ∈ B. Products can be specified using e.g.. A × B =. A table can be created by taking the Cartesian product of a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form. More a Cartesian product of n sets known as an n-fold Cartesian product, can be represented by an array of n dimensions, where each element is an n-tuple. An ordered pair is a couple; the Cartesian product is named after René Descartes, whose formulation of analytic geometry gave rise to the concept, further generalized in terms of direct product. An illustrative example is the standard 52-card deck; the standard playing card ranks form a 13-element set. The card suits form a four-element set; the Cartesian product of these sets returns a 52-element set consisting of 52 ordered pairs, which correspond to all 52 possible playing cards.
Ranks × Suits returns a set of the form. Suits × Ranks returns a set of the form. Both sets are distinct disjoint; the main historical example is the Cartesian plane in analytic geometry. In order to represent geometrical shapes in a numerical way and extract numerical information from shapes' numerical representations, René Descartes assigned to each point in the plane a pair of real numbers, called its coordinates; such a pair's first and second components are called its x and y coordinates, respectively. The set of all such pairs is thus assigned to the set of all points in the plane. A formal definition of the Cartesian product from set-theoretical principles follows from a definition of ordered pair; the most common definition of ordered pairs, the Kuratowski definition, is =. Under this definition, is an element of P, X × Y is a subset of that set, where P represents the power set operator. Therefore, the existence of the Cartesian product of any two sets in ZFC follows from the axioms of pairing, power set, specification.
Since functions are defined as a special case of relations, relations are defined as subsets of the Cartesian product, the definition of the two-set Cartesian product is prior to most other definitions. Let A, B, C, D be sets; the Cartesian product A × B is not commutative, A × B ≠ B × A, because the ordered pairs are reversed unless at least one of the following conditions is satisfied: A is equal to B, or A or B is the empty set. For example: A =. × C ≠ A × If for example A = × A = ≠ = A ×. The Cartesian product behaves nicely with respect to intersections. × = ∩. × ≠ ∪ In fact, we have that: ∪ = ∪ ∪ [ ( B
In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length and volume. A important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length and volume of Euclidean geometry to suitable subsets of the n-dimensional Euclidean space Rn. For instance, the Lebesgue measure of the interval in the real numbers is its length in the everyday sense of the word 1. Technically, a measure is a function that assigns a non-negative real number or +∞ to subsets of a set X, it must further be countably additive: the measure of a'large' subset that can be decomposed into a finite number of'smaller' disjoint subsets is equal to the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure.
This problem was resolved by defining measure only on a sub-collection of all subsets. This means that countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined are complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of the axiom of choice. Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel, Henri Lebesgue, Johann Radon, Maurice Fréchet, among others; the main applications of measures are in the foundations of the Lebesgue integral, in Andrey Kolmogorov's axiomatisation of probability theory and in ergodic theory. In integration theory, specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space. Probability theory considers measures that assign to the whole set the size 1, considers measurable subsets to be events whose probability is given by the measure.
Ergodic theory considers measures that are invariant under, or arise from, a dynamical system. Let X be a set and Σ a σ-algebra over X. A function μ from Σ to the extended real number line is called a measure if it satisfies the following properties: Non-negativity: For all E in Σ: μ ≥ 0. Null empty set: μ = 0. Countable additivity: For all countable collections i = 1 ∞ of pairwise disjoint sets in Σ: μ = ∑ k = 1 ∞ μ One may require that at least one set E has finite measure; the empty set automatically has measure zero because of countable additivity, because μ = μ = μ + μ + μ + …, which implies that μ = 0. If only the second and third conditions of the definition of measure above are met, μ takes on at most one of the values ±∞ μ is called a signed measure; the pair is called a measurable space, the members of Σ are called measurable sets. If and are two measurable spaces a function f: X → Y is called measurable if for every Y-measurable set B ∈ Σ Y, the inverse image is X-measurable – i.e.: f ∈ Σ X.
In this setup, the composition of measurable functions is measurable, making the measurable spaces and measurable functions a category, with the measurable spaces as objects and the set of measurable functions as arrows. See Measurable function#Term usage variations about another setup. A triple is called a measure space. A probability measure is a measure with total measure one – i.e. Μ = 1. A probability space is a measure space with a probability measure. For measure spaces that are topological spaces various compatibility conditions can be
Paul Richard Halmos was a Hungarian-born American mathematician who made fundamental advances in the areas of mathematical logic, probability theory, operator theory, ergodic theory, functional analysis. He was recognized as a great mathematical expositor, he has been described as one of The Martians. Born in Hungary into a Jewish family, Halmos arrived in the U. S. at 13 years of age. He obtained his B. A. from the University of Illinois, majoring in mathematics, but fulfilling the requirements for both a math and philosophy degree. He took only three years to obtain the degree, was only 19 when he graduated, he began a Ph. D. in philosophy, still at the Champaign-Urbana campus. Joseph L. Doob supervised his dissertation, titled Invariants of Certain Stochastic Transformations: The Mathematical Theory of Gambling Systems. Shortly after his graduation, Halmos left for the Institute for Advanced Study, lacking both job and grant money. Six months he was working under John von Neumann, which proved a decisive experience.
While at the Institute, Halmos wrote his first book, Finite Dimensional Vector Spaces, which established his reputation as a fine expositor of mathematics. Halmos taught at Syracuse University, the University of Chicago, the University of Michigan, the University of California at Santa Barbara, the University of Hawaii, Indiana University. From his 1985 retirement from Indiana until his death, he was affiliated with the Mathematics department at Santa Clara University. In a series of papers reprinted in his 1962 Algebraic Logic, Halmos devised polyadic algebras, an algebraic version of first-order logic differing from the better known cylindric algebras of Alfred Tarski and his students. An elementary version of polyadic algebra is described in monadic Boolean algebra. In addition to his original contributions to mathematics, Halmos was an unusually clear and engaging expositor of university mathematics, he won the Lester R. Ford Award in 1971 and again in 1977. Halmos chaired the American Mathematical Society committee that wrote the AMS style guide for academic mathematics, published in 1973.
In 1983, he received the AMS's Steele Prize for exposition. In the American Scientist 56: 375–389, Halmos argued that mathematics is a creative art, that mathematicians should be seen as artists, not number crunchers, he discussed the division of the field into mathology and mathophysics, further arguing that mathematicians and painters think and work in related ways. Halmos's 1985 "automathography" I Want to Be a Mathematician is an account of what it was like to be an academic mathematician in 20th century America, he called the book "automathography" rather than "autobiography", because its focus is entirely on his life as a mathematician, not his personal life. The book contains the following quote on Halmos' view of what doing mathematics means: Don't just read it. Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?
In these memoirs, Halmos claims to have invented the "iff" notation for the words "if and only if" and to have been the first to use the "tombstone" notation to signify the end of a proof, this is agreed to be the case. The tombstone symbol ∎ is sometimes called a halmos. In 2005, Halmos and his wife Virginia funded the Euler Book Prize, an annual award given by the Mathematical Association of America for a book, to improve the view of mathematics among the public; the first prize was given in 2007, the 300th anniversary of Leonhard Euler's birth, to John Derbyshire for his book about Bernhard Riemann and the Riemann hypothesis: Prime Obsession. 1942. Finite-Dimensional Vector Spaces. Springer-Verlag. 1950. Measure Theory. Springer Verlag. 1951. Introduction to Hilbert Space and the Theory of Spectral Multiplicity. Chelsea. 1956. Lectures on Ergodic Theory. Chelsea. 1960. Naive Set Theory. Springer Verlag. 1962. Algebraic Logic. Chelsea. 1963. Lectures on Boolean Algebras. Van Nostrand. 1967. A Hilbert Space Problem Book.
Springer-Verlag. 1973.. How to Write Mathematics. American Mathematical Society. 1978.. Bounded Integral Operators on L² Spaces. Springer Verlag 1985. I Want to Be a Mathematician. Springer-Verlag. 1987. I Have a Photographic Memory. Mathematical Association of America. 1991. Problems for Mathematicians and Old, Dolciani Mathematical Expositions, Mathematical Association of America. 1996. Linear Algebra Problem Book, Dolciani Mathematical Expositions, Mathematical Association of America. 1998.. Logic as Algebra, Dolciani Mathematical Expositions No. 21, Mathematical Association of America. 2009. Introduction to Boolean Algebras, Springer. Criticism of non-standard analysis The Martians J. H. Ewing. Paul Halmos: Celebrating 50 Years of Mathematics. Springer-Verlag. ISBN 0-387-97509-8. OCLC 22859036. Includes a bibliography of Halmos's writings through 1991. John Ewing. "Paul Halmos: In His Own Words". Notices of the American Mathematical Society. 54: 1136–1144. Retrieved 2008-01-15. Paul Halmos. I want to be a Mathematician: An Automathography.
Springer-Verlag. ISBN 0-387-96470-3. OCLC 230812318. O'Connor, John J..
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
Michel Loève was a French-American probabilist and mathematical statistician, of Jewish origin. He is known in mathematical statistics and probability theory for the Karhunen–Loève theorem and Karhunen–Loève transform. Michel Loève was born in Jaffa to a Jewish family, he passed most of his childhood years in Egypt and received his primary and secondary education there in French schools. After achieving the grades of B. L. in 1931 and A. B. in 1936, he studied mathematics at the Université de Paris under Paul Lévy, received his Doctorat ès Sciences in 1941. In 1936 was employed as actuaire of the University of Lyon; because of his Jewish origin, he was arrested during the German occupation of France and sent to Drancy internment camp. One of his books is dedicated "To Line and To the students and teachers of the School in the Camp de Drancy". Having survived the Holocaust, after the liberation he became between 1944 and 1946 chief of research at the Institut Henri Poincaré at Paris University until 1948 worked at the University of London.
After one term as a visiting professor at Columbia University he accepted the position of professor of Mathematics at Berkeley, in 1955 adding the title professor of Statistics. He is the author of one of the earliest books on measure-theoretic probability theory and one of the best known textbooks, he is memorialized via the Loève Prize created by his widow Line. Paul Lévy Kari Karhunen Harold Hotelling University of California in Memoriam Photographs Photograph from Portraits of Statisticians Michel Loève at the Mathematics Genealogy Project