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https://en.wikipedia.org/wiki/Basis%20function | In mathematics, a basis function is an element of a particular basis for a function space. Every function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors.
In numerical analysis and approximation theory, basis functions are also called blending functions, because of their use in interpolation: In this application, a mixture of the basis functions provides an interpolating function (with the "blend" depending on the evaluation of the basis functions at the data points).
Examples
Monomial basis for Cω
The monomial basis for the vector space of analytic functions is given by
This basis is used in Taylor series, amongst others.
Monomial basis for polynomials
The monomial basis also forms a basis for the vector space of polynomials. After all, every polynomial can be written as for some , which is a linear combination of monomials.
Fourier basis for L2[0,1]
Sines and cosines form an (orthonormal) Schauder basis for square-integrable functions on a bounded domain. As a particular example, the collection
forms a basis for L2[0,1].
See also
Basis (linear algebra) (Hamel basis)
Schauder basis (in a Banach space)
Dual basis
Biorthogonal system (Markushevich basis)
Orthonormal basis in an inner-product space
Orthogonal polynomials
Fourier analysis and Fourier series
Harmonic analysis
Orthogonal wavelet
Biorthogonal wavelet
Radial basis function
Finite-elements (bases)
Functional analysis
Approximation theory
Numerical analysis
References
Numerical analysis
Fourier analysis
Linear algebra
Numerical linear algebra
Types of functions |
https://en.wikipedia.org/wiki/Liouville%20number | In number theory, a Liouville number is a real number with the property that, for every positive integer , there exists a pair of integers with such that
Liouville numbers are "almost rational", and can thus be approximated "quite closely" by sequences of rational numbers. Precisely, these are transcendental numbers that can be more closely approximated by rational numbers than any algebraic irrational number can be. In 1844, Joseph Liouville showed that all Liouville numbers are transcendental, thus establishing the existence of transcendental numbers for the first time.
It is known that and are not Liouville numbers.
The existence of Liouville numbers (Liouville's constant)
Liouville numbers can be shown to exist by an explicit construction.
For any integer and any sequence of integers such that for all and for infinitely many , define the number
In the special case when , and for all , the resulting number is called Liouville's constant:
L = 0.11000100000000000000000100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001...
It follows from the definition of that its base- representation is
where the th term is in the th place.
Since this base- representation is non-repeating it follows that is not a rational number. Therefore, for any rational number , .
Now, for any integer , and can be defined as follows:
Then,
Therefore, any such is a Liouville number.
Notes on the proof
The inequality follows since ak ∈ {0, 1, 2, …, b−1} for all k, so at most ak = b−1. The largest possible sum would occur if the sequence of integers (a1, a2, …) were (b−1, b−1, ...), i.e. ak = b−1, for all k. will thus be less than or equal to this largest possible sum.
The strong inequality follows from the motivation to eliminate the series by way of reducing it to a series for which a formula is known. In the proof so far, the purpose for introducing the inequality in #1 comes from intuition that (the geometric series formula); therefore, if an inequality can be found from that introduces a series with (b−1) in the numerator, and if the denominator term can be further reduced from to , as well as shifting the series indices from 0 to , then both series and (b−1) terms will be eliminated, getting closer to a fraction of the form , which is the end-goal of the proof. This motivation is increased here by selecting now from the sum a partial sum. Observe that, for any term in , since b ≥ 2, then , for all k (except for when n=1). Therefore, (since, even if n=1, all subsequent terms are smaller). In order to manipulate the indices so that k starts at 0, partial sum will be selected from within (also less than the total value since it's a partial sum from a series whose terms are all positive). Choose the partial sum formed by starting at k = (n+1)! which follows from the motivation to write a new series with k=0, namely by noticing that .
For the final inequality , this particular inequality ha |
https://en.wikipedia.org/wiki/Natural%20deduction | In logic and proof theory, natural deduction is a kind of proof calculus in which logical reasoning is expressed by inference rules closely related to the "natural" way of reasoning. This contrasts with Hilbert-style systems, which instead use axioms as much as possible to express the logical laws of deductive reasoning.
Motivation
Natural deduction grew out of a context of dissatisfaction with the axiomatizations of deductive reasoning common to the systems of Hilbert, Frege, and Russell (see, e.g., Hilbert system). Such axiomatizations were most famously used by Russell and Whitehead in their mathematical treatise Principia Mathematica. Spurred on by a series of seminars in Poland in 1926 by Łukasiewicz that advocated a more natural treatment of logic, Jaśkowski made the earliest attempts at defining a more natural deduction, first in 1929 using a diagrammatic notation, and later updating his proposal in a sequence of papers in 1934 and 1935. His proposals led to different notations
such as Fitch-style calculus (or Fitch's diagrams) or Suppes' method for which Lemmon gave a variant called system L.
Natural deduction in its modern form was independently proposed by the German mathematician Gerhard Gentzen in 1933, in a dissertation delivered to the faculty of mathematical sciences of the University of Göttingen. The term natural deduction (or rather, its German equivalent natürliches Schließen) was coined in that paper:
Gentzen was motivated by a desire to establish the consistency of number theory. He was unable to prove the main result required for the consistency result, the cut elimination theorem—the Hauptsatz—directly for natural deduction. For this reason he introduced his alternative system, the sequent calculus, for which he proved the Hauptsatz both for classical and intuitionistic logic. In a series of seminars in 1961 and 1962 Prawitz gave a comprehensive summary of natural deduction calculi, and transported much of Gentzen's work with sequent calculi into the natural deduction framework. His 1965 monograph Natural deduction: a proof-theoretical study was to become a reference work on natural deduction, and included applications for modal and second-order logic.
In natural deduction, a proposition is deduced from a collection of premises by applying inference rules repeatedly. The system presented in this article is a minor variation of Gentzen's or Prawitz's formulation, but with a closer adherence to Martin-Löf's description of logical judgments and connectives.
Judgments and propositions
A judgment is something that is knowable, that is, an object of knowledge. It is evident if one in fact knows it. Thus "it is raining" is a judgment, which is evident for the one who knows that it is actually raining; in this case one may readily find evidence for the judgment by looking outside the window or stepping out of the house. In mathematical logic however, evidence is often not as directly observable, but rather deduced from more b |
https://en.wikipedia.org/wiki/Coefficient | In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or an expression. It may be a number (dimensionless), in which case it is known as a numerical factor. It may also be a constant with units of measurement, in which it is known as a constant multiplier. In general, coefficients may be any expression (including variables such as , and ). When the combination of variables and constants is not necessarily involved in a product, it may be called a parameter.
For example, the polynomial has coefficients 2, −1, and 3, and the powers of the variable in the polynomial have coefficient parameters , , and .
The , also known as constant term or simply constant is the quantity not attached to variables in an expression. For example, the constant coefficients of the expressions above are the number 3 and the parameter c, respectively.
The coefficient attached to the highest degree of the variable in a polynomial is referred to as the leading coefficient. For example, in the expressions above, the leading coefficients are 2 and a, respectively.
In the context of differential equations, an equation can often be written as equating to zero a polynomial in the unknown functions and their derivatives. In this case, the coefficients of the differential equation are the coefficients of this polynomial, and are generally non-constant functions. A coefficient is a constant coefficient when it is a constant function. For avoiding confusion, the coefficient that is not attached to unknown functions and their derivative is generally called the constant term rather the constant coefficient. In particular, in a linear differential equation with constant coefficient, the constant term is generally not supposed to be a constant function.
Terminology and definition
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or any expression. For example, in the polynomial
with variables and , the first two terms have the coefficients 7 and −3. The third term 1.5 is the constant coefficient. In the final term, the coefficient is 1 and is not explicitly written.
In many scenarios, coefficients are numbers (as is the case for each term of the previous example), although they could be parameters of the problem—or any expression in these parameters. In such a case, one must clearly distinguish between symbols representing variables and symbols representing parameters. Following René Descartes, the variables are often denoted by , , ..., and the parameters by , , , ..., but this is not always the case. For example, if is considered a parameter in the above expression, then the coefficient of would be , and the constant coefficient (with respect to ) would be .
When one writes
it is generally assumed that is the only variable, and that , and are parameters; thus the constant coefficient is in this case.
Any polynomial in a single variable can be written as
for som |
https://en.wikipedia.org/wiki/Buckingham%20%CF%80%20theorem | In engineering, applied mathematics, and physics, the Buckingham theorem is a key theorem in dimensional analysis. It is a formalization of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically meaningful equation involving a certain number n of physical variables, then the original equation can be rewritten in terms of a set of p = n − k dimensionless parameters 1, 2, ..., p constructed from the original variables, where k is the number of physical dimensions involved; it is obtained as the rank of a particular matrix.
The theorem provides a method for computing sets of dimensionless parameters from the given variables, or nondimensionalization, even if the form of the equation is still unknown.
The Buckingham theorem indicates that validity of the laws of physics does not depend on a specific unit system. A statement of this theorem is that any physical law can be expressed as an identity involving only dimensionless combinations (ratios or products) of the variables linked by the law (for example, pressure and volume are linked by Boyle's law – they are inversely proportional). If the dimensionless combinations' values changed with the systems of units, then the equation would not be an identity, and the theorem would not hold.
History
Although named for Edgar Buckingham, the theorem was first proved by the French mathematician Joseph Bertrand in 1878. Bertrand considered only special cases of problems from electrodynamics and heat conduction, but his article contains, in distinct terms, all the basic ideas of the modern proof of the theorem and clearly indicates the theorem's utility for modelling physical phenomena. The technique of using the theorem ("the method of dimensions") became widely known due to the works of Rayleigh. The first application of the theorem in the general case to the dependence of pressure drop in a pipe upon governing parameters probably dates back to 1892, a heuristic proof with the use of series expansions, to 1894.
Formal generalization of the theorem for the case of arbitrarily many quantities was given first by in 1892, then in 1911—apparently independently—by both A. Federman and D. Riabouchinsky, and again in 1914 by Buckingham. It was Buckingham's article that introduced the use of the symbol "" for the dimensionless variables (or parameters), and this is the source of the theorem's name.
Statement
More formally, the number of dimensionless terms that can be formed is equal to the nullity of the dimensional matrix, and is the rank. For experimental purposes, different systems that share the same description in terms of these dimensionless numbers are equivalent.
In mathematical terms, if we have a physically meaningful equation such as
where are any physical variables, and there is a maximal dimensionally independent subset of size , then the above equation can be restated as
where are dimensionless parameters constructed from the by dime |
https://en.wikipedia.org/wiki/Fundamental%20theorem%20of%20algebra | The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero.
Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed.
The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots. The equivalence of the two statements can be proven through the use of successive polynomial division.
Despite its name, there is no purely algebraic proof of the theorem, since any proof must use some form of the analytic completeness of the real numbers, which is not an algebraic concept. Additionally, it is not fundamental for modern algebra; its name was given at a time when algebra was synonymous with theory of equations.
History
Peter Roth, in his book Arithmetica Philosophica (published in 1608, at Nürnberg, by Johann Lantzenberger), wrote that a polynomial equation of degree n (with real coefficients) may have n solutions. Albert Girard, in his book L'invention nouvelle en l'Algèbre (published in 1629), asserted that a polynomial equation of degree n has n solutions, but he did not state that they had to be real numbers. Furthermore, he added that his assertion holds "unless the equation is incomplete", by which he meant that no coefficient is equal to 0. However, when he explains in detail what he means, it is clear that he actually believes that his assertion is always true; for instance, he shows that the equation although incomplete, has four solutions (counting multiplicities): 1 (twice), and
As will be mentioned again below, it follows from the fundamental theorem of algebra that every non-constant polynomial with real coefficients can be written as a product of polynomials with real coefficients whose degrees are either 1 or 2. However, in 1702 Leibniz erroneously said that no polynomial of the type (with real and distinct from 0) can be written in such a way. Later, Nikolaus Bernoulli made the same assertion concerning the polynomial , but he got a letter from Euler in 1742 in which it was shown that this polynomial is equal to
with
Also, Euler pointed out that
A first attempt at proving the theorem was made by d'Alembert in 1746, but his proof was incomplete. Among other problems, it assumed implicitly a theorem (now known as Puiseux's theorem), which would not be proved until more than a century later and using the fundamental theorem of algebra. Other attempts were made by Euler (1749), de Foncenex (1759), Lagrange (1772), and Laplace (1795). These last four attempts assumed implicitly Girard's assertion; to be more precise, the existence of solutions was assumed and all that remained to be |
https://en.wikipedia.org/wiki/Integer%20sequence | In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers.
An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description . The sequence 0, 3, 8, 15, ... is formed according to the formula n2 − 1 for the nth term: an explicit definition.
Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, , even though we do not have a formula for the nth perfect number.
Computable and definable sequences
An integer sequence is a computable sequence if there exists an algorithm which, given n, calculates an, for all n > 0. The set of computable integer sequences is countable. The set of all integer sequences is uncountable (with cardinality equal to that of the continuum), and so not all integer sequences are computable.
Although some integer sequences have definitions, there is no systematic way to define what it means for an integer sequence to be definable in the universe or in any absolute (model independent) sense.
Suppose the set M is a transitive model of ZFC set theory. The transitivity of M implies that the integers and integer sequences inside M are actually integers and sequences of integers. An integer sequence is a definable sequence relative to M if there exists some formula P(x) in the language of set theory, with one free variable and no parameters, which is true in M for that integer sequence and false in M for all other integer sequences. In each such M, there are definable integer sequences that are not computable, such as sequences that encode the Turing jumps of computable sets.
For some transitive models M of ZFC, every sequence of integers in M is definable relative to M; for others, only some integer sequences are (Hamkins et al. 2013). There is no systematic way to define in M itself the set of sequences definable relative to M and that set may not even exist in some such M. Similarly, the map from the set of formulas that define integer sequences in M to the integer sequences they define is not definable in M and may not exist in M. However, in any model that does possess such a definability map, some integer sequences in the model will not be definable relative to the model (Hamkins et al. 2013).
If M contains all integer sequences, then the set of integer sequences definable in M will exist in M and be countable and countable in M.
Complete sequences
A sequence of positive integers is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once.
Examples
Integer sequences that have th |
https://en.wikipedia.org/wiki/P-adic%20number | In number theory, given a prime number , the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending (possibly infinitely) to the left rather than to the right. Formally, given a prime number , a -adic number can be defined as a series
where is an integer (possibly negative), and each is an integer such that A -adic integer is a -adic number such that
In general the series that represents a -adic number is not convergent in the usual sense, but it is convergent for the -adic absolute value where is the least integer such that (if all are zero, one has the zero -adic number, which has as its -adic absolute value).
Every rational number can be uniquely expressed as the sum of a series as above, with respect to the -adic absolute value. This allows considering rational numbers as special -adic numbers, and alternatively defining the -adic numbers as the completion of the rational numbers for the -adic absolute value, exactly as the real numbers are the completion of the rational numbers for the usual absolute value.
-adic numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.
Motivation
Roughly speaking, modular arithmetic modulo a positive integer consists of "approximating" every integer by the remainder of its division by , called its residue modulo . The main property of modular arithmetic is that the residue modulo of the result of a succession of operations on integers is the same as the result of the same succession of operations on residues modulo . If one knows that the absolute value of the result is less than , this allows a computation of the result which does not involve any integer larger than .
For larger results, an old method, still in common use, consists of using several small moduli that are pairwise coprime, and applying the Chinese remainder theorem for recovering the result modulo the product of the moduli.
Another method discovered by Kurt Hensel consists of using a prime modulus , and applying Hensel's lemma for recovering iteratively the result modulo If the process is continued infinitely, this provides eventually a result which is a -adic number.
Basic lemmas
The theory of -adic numbers is fundamentally based on the two following lemmas
Every nonzero rational number can be written where , , and are integers and neither nor is divisible by . The exponent is uniquely determined by the rational number and is called its -adic valuation (this definition is a particular case of a more general definition, given below). The proof of the lemma results directly from the fundamental theorem of arithmetic.
Every nonzero rational number of valuation can be unique |
https://en.wikipedia.org/wiki/Cantor%27s%20diagonal%20argument | In set theory, Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument, the anti-diagonal argument, the diagonal method, and Cantor's diagonalization proof, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers. Such sets are now known as uncountable sets, and the size of infinite sets is now treated by the theory of cardinal numbers which Cantor began.
The diagonal argument was not Cantor's first proof of the uncountability of the real numbers, which appeared in 1874.
However, it demonstrates a general technique that has since been used in a wide range of proofs, including the first of Gödel's incompleteness theorems and Turing's answer to the Entscheidungsproblem. Diagonalization arguments are often also the source of contradictions like Russell's paradox and Richard's paradox.
Uncountable set
Cantor considered the set T of all infinite sequences of binary digits (i.e. each digit is zero or one).
He begins with a constructive proof of the following lemma:
If s1, s2, ... , sn, ... is any enumeration of elements from T, then an element s of T can be constructed that doesn't correspond to any sn in the enumeration.
The proof starts with an enumeration of elements from T, for example
{|
|-
| s1 = || (0, || 0, || 0, || 0, || 0, || 0, || 0, || ...)
|-
| s2 = || (1, || 1, || 1, || 1, || 1, || 1, || 1, || ...)
|-
| s3 = || (0, || 1, || 0, || 1, || 0, || 1, || 0, || ...)
|-
| s4 = || (1, || 0, || 1, || 0, || 1, || 0, || 1, || ...)
|-
| s5 = || (1, || 1, || 0, || 1, || 0, || 1, || 1, || ...)
|-
| s6 = || (0, || 0, || 1, || 1, || 0, || 1, || 1, || ...)
|-
| s7 = || (1, || 0, || 0, || 0, || 1, || 0, || 0, || ...)
|-
| ...
|}
Next, a sequence s is constructed by choosing the 1st digit as complementary to the 1st digit of s1 (swapping 0s for 1s and vice versa), the 2nd digit as complementary to the 2nd digit of s2, the 3rd digit as complementary to the 3rd digit of s3, and generally for every n, the nth digit as complementary to the nth digit of sn. For the example above, this yields
{|
|-
| s1 || = || (0, || 0, || 0, || 0, || 0, || 0, || 0, || ...)
|-
| s2 || = || (1, || 1, || 1, || 1, || 1, || 1, || 1, || ...)
|-
| s3 || = || (0, || 1, || 0, || 1, || 0, || 1, || 0, || ...)
|-
| s4 || = || (1, || 0, || 1, || 0, || 1, || 0, || 1, || ...)
|-
| s5 || = || (1, || 1, || 0, || 1, || 0, || 1, || 1, || ...)
|-
| s6 || = || (0, || 0, || 1, || 1, || 0, || 1, || 1, || ...)
|-
| s7 || = || (1, || 0, || 0, || 0, || 1, || 0, || 0, || ...)
|-
| ...
|-
|
|-
| s || = || (1, || 0, || 1, || 1, || 1, || 0, || 1, || ...)
|}
By construction, s is a member of T that differs from each sn, since their nth digits differ (highlighted in the example).
Hence, s cannot occur in the enumeration.
Based on this lemma, Cantor then uses a proof by contradiction to show that:
The set T is uncountable.
The proof starts b |
https://en.wikipedia.org/wiki/Hyperreal%20number | In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form
(for any finite number of terms).
Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948.
The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic law of continuity. The transfer principle states that true first-order statements about R are also valid in *R. For example, the commutative law of addition, , holds for the hyperreals just as it does for the reals; since R is a real closed field, so is *R. Since for all integers n, one also has for all hyperintegers . The transfer principle for ultrapowers is a consequence of Łoś's theorem of 1955.
Concerns about the soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Archimedes replacing such proofs with ones using other techniques such as the method of exhaustion. In the 1960s, Abraham Robinson proved that the hyperreals were logically consistent if and only if the reals were. This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules that Robinson delineated.
The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called nonstandard analysis. One immediate application is the definition of the basic concepts of analysis such as the derivative and integral in a direct fashion, without passing via logical complications of multiple quantifiers. Thus, the derivative of f(x) becomes for an infinitesimal , where st(·) denotes the standard part function, which "rounds off" each finite hyperreal to the nearest real. Similarly, the integral is defined as the standard part of a suitable infinite sum.
The transfer principle
The idea of the hyperreal system is to extend the real numbers R to form a system *R that includes infinitesimal and infinite numbers, but without changing any of the elementary axioms of algebra. Any statement of the form "for any number x ..." that is true for the reals is also true for the hyperreals. For example, the axiom that states "for any number x, x + 0 = x" still applies. The same is true for quantification over several numbers, e.g., "for any numbers x and y, xy = yx." This ability to carry over statements from the reals to the hyperreals is called the transfer principle. However, statements of the form "for any set of numbers S ..." may not carry over. The only properties that differ between the reals and the hyperreals are those that rely on quantification over sets, or other higher-level structures such as functions and relations, which are typically constructed out of sets. Each real set, functi |
https://en.wikipedia.org/wiki/Surreal%20number | In mathematics, the surreal number system is a totally ordered proper class containing not only the real numbers but also infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. Research on the Go endgame by John Horton Conway led to the original definition and construction of surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness.
The surreals share many properties with the reals, including the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers (including the hyperreal numbers) can be realized as subfields of the surreals. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations. It has also been shown (in von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorphic to the maximal class surreal field.
History of the concept
Research on the Go endgame by John Horton Conway led to the original definition and construction of the surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned On to Pure Mathematics and Found Total Happiness. In his book, which takes the form of a dialogue, Knuth coined the term surreal numbers for what Conway had called simply numbers. Conway later adopted Knuth's term, and used surreals for analyzing games in his 1976 book On Numbers and Games.
A separate route to defining the surreals began in 1907, when Hans Hahn introduced Hahn series as a generalization of formal power series, and Hausdorff introduced certain ordered sets called ηα-sets for ordinals α and asked if it was possible to find a compatible ordered group or field structure. In 1962, Norman Alling used a modified form of Hahn series to construct such ordered fields associated to certain ordinals α and, in 1987, he showed that taking α to be the class of all ordinals in his construction gives a class that is an ordered field isomorphic to the surreal numbers.
If the surreals are considered as 'just' a proper-class-sized real closed field, Alling's 1962 paper handles the case of strongly inaccessible cardinals which can naturally be considered as proper classes by cutting off the cumulative hierarchy of the universe one stage above the cardinal, and Alling accordingly deserves much credit for the discovery/invention of the surreals in this sense. There is an important additional field structure on the surreals that isn't visible through this lens however, namely the notion of a 'birthday' and the correspon |
https://en.wikipedia.org/wiki/Sedenion | In abstract algebra, the sedenions form a 16-dimensional noncommutative and nonassociative algebra over the real numbers, usually represented by the capital letter S, boldface or blackboard bold . They are obtained by applying the Cayley–Dickson construction to the octonions, and as such the octonions are isomorphic to a subalgebra of the sedenions. Unlike the octonions, the sedenions are not an alternative algebra. Applying the Cayley–Dickson construction to the sedenions yields a 32-dimensional algebra, sometimes called the 32-ions or trigintaduonions. It is possible to continue applying the Cayley–Dickson construction arbitrarily many times.
The term sedenion is also used for other 16-dimensional algebraic structures, such as a tensor product of two copies of the biquaternions, or the algebra of 4 × 4 matrices over the real numbers, or that studied by .
Arithmetic
Like octonions, multiplication of sedenions is neither commutative nor associative.
But in contrast to the octonions, the sedenions do not even have the property of being alternative.
They do, however, have the property of power associativity, which can be stated as that, for any element x of , the power is well defined. They are also flexible.
Every sedenion is a linear combination of the unit sedenions , , , , ..., ,
which form a basis of the vector space of sedenions. Every sedenion can be represented in the form
Addition and subtraction are defined by the addition and subtraction of corresponding coefficients and multiplication is distributive over addition.
Like other algebras based on the Cayley–Dickson construction, the sedenions contain the algebra they were constructed from. So, they contain the octonions (generated by to in the table below), and therefore also the quaternions (generated by to ), complex numbers (generated by and ) and real numbers (generated by ).
The sedenions have a multiplicative identity element and multiplicative inverses, but they are not a division algebra because they have zero divisors. This means that two nonzero sedenions can be multiplied to obtain zero: an example is . All hypercomplex number systems after sedenions that are based on the Cayley–Dickson construction also contain zero divisors.
A sedenion multiplication table is shown below:
Sedenion properties
From the above table, we can see that:
and
Anti-associative
The sedenions are not fully anti-associative. Choose any four generators, and . The following 5-cycle shows that these five relations cannot all be anti-associative.
In particular, in the table above, using and the last expression associates.
Quaternionic subalgebras
The 35 triads that make up this specific sedenion multiplication table with the 7 triads of the octonions used in creating the sedenion through the Cayley–Dickson construction shown in bold:
The binary representations of the indices of these triples bitwise XOR to 0.
{ {1, 2, 3}, {1, 4, 5}, {1, 7, 6}, {1, 8, 9}, {1, 11, 10}, {1, 13, 12}, |
https://en.wikipedia.org/wiki/Octonion | In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold . Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative. They are also power associative.
Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. Octonions are related to exceptional structures in mathematics, among them the exceptional Lie groups. Octonions have applications in fields such as string theory, special relativity and quantum logic. Applying the Cayley–Dickson construction to the octonions produces the sedenions.
History
The octonions were discovered in 1843 by John T. Graves, inspired by his friend William Rowan Hamilton's discovery of quaternions. Graves called his discovery "octaves", and mentioned them in a letter to Hamilton dated 26 December 1843. He first published his result slightly later than Arthur Cayley's article. The octonions were discovered independently by Cayley and are sometimes referred to as "Cayley numbers" or the "Cayley algebra". Hamilton described the early history of Graves's discovery.
Definition
The octonions can be thought of as octets (or 8-tuples) of real numbers. Every octonion is a real linear combination of the unit octonions:
where is the scalar or real element; it may be identified with the real number 1. That is, every octonion can be written in the form
with real coefficients .
Addition and subtraction of octonions is done by adding and subtracting corresponding terms and hence their coefficients, like quaternions. Multiplication is more complex. Multiplication is distributive over addition, so the product of two octonions can be calculated by summing the products of all the terms, again like quaternions. The product of each pair of terms can be given by multiplication of the coefficients and a multiplication table of the unit octonions, like this one (due to Cayley, 1845, and Graves, 1843):
Most off-diagonal elements of the table are antisymmetric, making it almost a skew-symmetric matrix except for the elements on the main diagonal, as well as the row and column for which is an operand.
The table can be summarized as follows:
where is the Kronecker delta (equal to 1 if and only if ), and is a completely antisymmetric tensor with value 1 when .
The above definition is not unique, however; it is only one of 480 possible definitions for octonion multiplication with . The others can be obtained by permuting and changing the signs of the non-scalar basis elements . The 480 different algebras are isomorphic, and there is rarely a need to consider which particular multiplication rule is used.
Each of these 480 definitions is invari |
https://en.wikipedia.org/wiki/Hypercomplex%20number | In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers.
The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory.
History
In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established concepts in mathematical literature, added to the real and complex numbers. The concept of a hypercomplex number covered them all, and called for a discipline to explain and classify them.
The cataloguing project began in 1872 when Benjamin Peirce first published his Linear Associative Algebra, and was carried forward by his son Charles Sanders Peirce. Most significantly, they identified the nilpotent and the idempotent elements as useful hypercomplex numbers for classifications. The Cayley–Dickson construction used involutions to generate complex numbers, quaternions, and octonions out of the real number system. Hurwitz and Frobenius proved theorems that put limits on hypercomplexity: Hurwitz's theorem says finite-dimensional real composition algebras are the reals , the complexes , the quaternions , and the octonions , and the Frobenius theorem says the only real associative division algebras are , , and . In 1958 J. Frank Adams published a further generalization in terms of Hopf invariants on H-spaces which still limits the dimension to 1, 2, 4, or 8.
It was matrix algebra that harnessed the hypercomplex systems. First, matrices contributed new hypercomplex numbers like 2 × 2 real matrices (see Split-quaternion). Soon the matrix paradigm began to explain the others as they became represented by matrices and their operations. In 1907 Joseph Wedderburn showed that associative hypercomplex systems could be represented by square matrices, or direct product of algebras of square matrices. From that date the preferred term for a hypercomplex system became associative algebra as seen in the title of Wedderburn's thesis at University of Edinburgh. Note however, that non-associative systems like octonions and hyperbolic quaternions represent another type of hypercomplex number.
As Hawkins explains, the hypercomplex numbers are stepping stones to learning about Lie groups and group representation theory. For instance, in 1929 Emmy Noether wrote on "hypercomplex quantities and representation theory". In 1973 Kantor and Solodovnikov published a textbook on hypercomplex numbers which was translated in 1989.
Karen Parshall has written a detailed exposition of the heyday of hypercomplex numbers, including the role of mathematicians including Theodor Molien and Eduard Study.<ref>{{citation |author-link=Eduard Study |first=Eduard |last=Study |year=1898 |chapter=Theorie der gemeinen und höhern komplexen Grössen |title=[[Klein's encyclopedia|Encyclopädie der mathematischen Wissenschaften]] |volume=I A |issue=4 |pages=147–183}}</ref> For the transition to mode |
https://en.wikipedia.org/wiki/Quaternion | In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space, or, equivalently, as the quotient of two vectors. Multiplication of quaternions is noncommutative.
Quaternions are generally represented in the form
where , , , and are real numbers; and , and are the basis vectors or basis elements.
Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, and crystallographic texture analysis. They can be used alongside other methods of rotation, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application.
In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore a ring, being both a division ring and a domain. The algebra of quaternions is often denoted by (for Hamilton), or in blackboard bold by It can also be given by the Clifford algebra classifications In fact, it was the first noncommutative division algebra to be discovered.
According to the Frobenius theorem, the algebra is one of only two finite-dimensional division rings containing a proper subring isomorphic to the real numbers; the other being the complex numbers. These rings are also Euclidean Hurwitz algebras, of which the quaternions are the largest associative algebra (and hence the largest ring). Further extending the quaternions yields the non-associative octonions, which is the last normed division algebra over the real numbers. (The sedenions, the extension of the octonions, have zero divisors and so cannot be a normed division algebra.)
The unit quaternions can be thought of as a choice of a group structure on the 3-sphere that gives the group Spin(3), which is isomorphic to SU(2) and also to the universal cover of SO(3).
History
Quaternions were introduced by Hamilton in 1843. Important precursors to this work included Euler's four-square identity (1748) and Olinde Rodrigues' parameterization of general rotations by four parameters (1840), but neither of these writers treated the four-parameter rotations as an algebra. Carl Friedrich Gauss had also discovered quaternions in 1819, but this work was not published until 1900.
Hamilton knew that the complex numbers could be interpreted as points in a plane, and he was looking for a way to do the same for points in three-dimensional space. Points in space can be represented by their coordinates, which are triples of numbers, and for many years he had known how to add and subtract triples of numbers. However, for a long time, he had been stuck on the problem of multiplication |
https://en.wikipedia.org/wiki/Zero%20divisor | In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero divisor if there exists a nonzero in such that . This is a partial case of divisibility in rings. An element that is a left or a right zero divisor is simply called a zero divisor. An element that is both a left and a right zero divisor is called a two-sided zero divisor (the nonzero such that may be different from the nonzero such that ). If the ring is commutative, then the left and right zero divisors are the same.
An element of a ring that is not a left zero divisor (respectively, not a right zero divisor) is called left regular or left cancellable (respectively, right regular or right cancellable).
An element of a ring that is left and right cancellable, and is hence not a zero divisor, is called regular or cancellable, or a non-zero-divisor. A zero divisor that is nonzero is called a nonzero zero divisor or a nontrivial zero divisor. A non-zero ring with no nontrivial zero divisors is called a domain.
Examples
In the ring , the residue class is a zero divisor since .
The only zero divisor of the ring of integers is .
A nilpotent element of a nonzero ring is always a two-sided zero divisor.
An idempotent element of a ring is always a two-sided zero divisor, since .
The ring of n × n matrices over a field has nonzero zero divisors if n ≥ 2. Examples of zero divisors in the ring of 2 × 2 matrices (over any nonzero ring) are shown here:
A direct product of two or more nonzero rings always has nonzero zero divisors. For example, in with each nonzero, , so is a zero divisor.
Let be a field and be a group. Suppose that has an element of finite order . Then in the group ring one has , with neither factor being zero, so is a nonzero zero divisor in .
One-sided zero-divisor
Consider the ring of (formal) matrices with and . Then and . If , then is a left zero divisor if and only if is even, since , and it is a right zero divisor if and only if is even for similar reasons. If either of is , then it is a two-sided zero-divisor.
Here is another example of a ring with an element that is a zero divisor on one side only. Let be the set of all sequences of integers . Take for the ring all additive maps from to , with pointwise addition and composition as the ring operations. (That is, our ring is , the endomorphism ring of the additive group .) Three examples of elements of this ring are the right shift , the left shift , and the projection map onto the first factor . All three of these additive maps are not zero, and the composites and are both zero, so is a left zero divisor and is a right zero divisor in the ring of additive maps from to . However, is not a right zero divisor and is not a left zero divisor: the composite is the identity. is a two-sided z |
https://en.wikipedia.org/wiki/Zorn%27s%20lemma | Zorn's lemma, also known as the Kuratowski–Zorn lemma, is a proposition of set theory. It states that a partially ordered set containing upper bounds for every chain (that is, every totally ordered subset) necessarily contains at least one maximal element.
The lemma was proved (assuming the axiom of choice) by Kazimierz Kuratowski in 1922 and independently by Max Zorn in 1935. It occurs in the proofs of several theorems of crucial importance, for instance the Hahn–Banach theorem in functional analysis, the theorem that every vector space has a basis, Tychonoff's theorem in topology stating that every product of compact spaces is compact, and the theorems in abstract algebra that in a ring with identity every proper ideal is contained in a maximal ideal and that every field has an algebraic closure.
Zorn's lemma is equivalent to the well-ordering theorem and also to the axiom of choice, in the sense that within ZF (Zermelo–Fraenkel set theory without the axiom of choice) any one of the three is sufficient to prove the other two. An earlier formulation of Zorn's lemma is Hausdorff's maximum principle which states that every totally ordered subset of a given partially ordered set is contained in a maximal totally ordered subset of that partially ordered set.
Motivation
To prove the existence of a mathematical object that can be viewed as a maximal element in some partially ordered set in some way, one can try proving the existence of such an object by assuming there is no maximal element and using transfinite induction and the assumptions of the situation to get a contradiction. Zorn's lemma tidies up the conditions a situation needs to satisfy in order for such an argument to work and enables mathematicians to not have to repeat the transfinite induction argument by hand each time, but just check the conditions of Zorn's lemma.
Statement of the lemma
Preliminary notions:
A set P equipped with a binary relation ≤ that is reflexive ( for every x), antisymmetric (if both and hold, then ), and transitive (if and then ) is said to be (partially) ordered by ≤. Given two elements x and y of P with x ≤ y, y is said to be greater than or equal to x. The word "partial" is meant to indicate that not every pair of elements of a partially ordered set is required to be comparable under the order relation, that is, in a partially ordered set P with order relation ≤ there may be elements x and y with neither x ≤ y nor y ≤ x. An ordered set in which every pair of elements is comparable is called totally ordered.
Every subset S of a partially ordered set P can itself be seen as partially ordered by restricting the order relation inherited from P to S. A subset S of a partially ordered set P is called a chain (in P) if it is totally ordered in the inherited order.
An element m of a partially ordered set P with order relation ≤ is maximal (with respect to ≤) if there is no other element of P greater than m, that is, if there is no s in P with s ≠ m and m ≤ |
https://en.wikipedia.org/wiki/Singular%20function | In mathematics, a real-valued function f on the interval [a, b] is said to be singular if it has the following properties:
f is continuous on [a, b]. (**)
there exists a set N of measure 0 such that for all x outside of N the derivative f (x) exists and is zero, that is, the derivative of f vanishes almost everywhere.
f is non-constant on [a, b].
A standard example of a singular function is the Cantor function, which is sometimes called the devil's staircase (a term also used for singular functions in general). There are, however, other functions that have been given that name. One is defined in terms of the circle map.
If f(x) = 0 for all x ≤ a and f(x) = 1 for all x ≥ b, then the function can be taken to represent a cumulative distribution function for a random variable which is neither a discrete random variable (since the probability is zero for each point) nor an absolutely continuous random variable (since the probability density is zero everywhere it exists).
Singular functions occur, for instance, as sequences of spatially modulated phases or structures in solids and magnets, described in a prototypical fashion by the Frenkel–Kontorova model and by the ANNNI model, as well as in some dynamical systems. Most famously, perhaps, they lie at the center of the fractional quantum Hall effect.
When referring to functions with a singularity
When discussing mathematical analysis in general, or more specifically real analysis or complex analysis or differential equations, it is common for a function which contains a mathematical singularity to be referred to as a 'singular function'. This is especially true when referring to functions which diverge to infinity at a point or on a boundary. For example, one might say, "1/x becomes singular at the origin, so 1/x is a singular function."
Advanced techniques for working with functions that contain singularities have been developed in the subject called distributional or generalized function analysis. A weak derivative is defined that allows singular functions to be used in partial differential equations, etc.
See also
Absolute continuity
Mathematical singularity
Generalized function
Distribution
Minkowski's question-mark function
References
(**) This condition depends on the references
Fractal curves
Types of functions |
https://en.wikipedia.org/wiki/Geostatistics | Geostatistics is a branch of statistics focusing on spatial or spatiotemporal datasets. Developed originally to predict probability distributions of ore grades for mining operations, it is currently applied in diverse disciplines including petroleum geology, hydrogeology, hydrology, meteorology, oceanography, geochemistry, geometallurgy, geography, forestry, environmental control, landscape ecology, soil science, and agriculture (esp. in precision farming). Geostatistics is applied in varied branches of geography, particularly those involving the spread of diseases (epidemiology), the practice of commerce and military planning (logistics), and the development of efficient spatial networks. Geostatistical algorithms are incorporated in many places, including geographic information systems (GIS).
Background
Geostatistics is intimately related to interpolation methods, but extends far beyond simple interpolation problems. Geostatistical techniques rely on statistical models that are based on random function (or random variable) theory to model the uncertainty associated with spatial estimation and simulation.
A number of simpler interpolation methods/algorithms, such as inverse distance weighting, bilinear interpolation and nearest-neighbor interpolation, were already well known before geostatistics. Geostatistics goes beyond the interpolation problem by considering the studied phenomenon at unknown locations as a set of correlated random variables.
Let be the value of the variable of interest at a certain location . This value is unknown (e.g. temperature, rainfall, piezometric level, geological facies, etc.). Although there exists a value at location that could be measured, geostatistics considers this value as random since it was not measured, or has not been measured yet. However, the randomness of is not complete, but defined by a cumulative distribution function (CDF) that depends on certain information that is known about the value :
Typically, if the value of is known at locations close to (or in the neighborhood of ) one can constrain the CDF of by this neighborhood: if a high spatial continuity is assumed, can only have values similar to the ones found in the neighborhood. Conversely, in the absence of spatial continuity can take any value. The spatial continuity of the random variables is described by a model of spatial continuity that can be either a parametric function in the case of variogram-based geostatistics, or have a non-parametric form when using other methods such as multiple-point simulation or pseudo-genetic techniques.
By applying a single spatial model on an entire domain, one makes the assumption that is a stationary process. It means that the same statistical properties are applicable on the entire domain. Several geostatistical methods provide ways of relaxing this stationarity assumption.
In this framework, one can distinguish two modeling goals:
Estimating the value for , typically by the expectation, |
https://en.wikipedia.org/wiki/Burali-Forti%20paradox | In set theory, a field of mathematics, the Burali-Forti paradox demonstrates that constructing "the set of all ordinal numbers" leads to a contradiction and therefore shows an antinomy in a system that allows its construction. It is named after Cesare Burali-Forti, who, in 1897, published a paper proving a theorem which, unknown to him, contradicted a previously proved result by Cantor. Bertrand Russell subsequently noticed the contradiction, and when he published it in his 1903 book Principles of Mathematics, he stated that it had been suggested to him by Burali-Forti's paper, with the result that it came to be known by Burali-Forti's name.
Stated in terms of von Neumann ordinals
We will prove this by reductio ad absurdum.
Let be a set consisting of all ordinal numbers.
is transitive because for every element of (which is an ordinal number and can be any ordinal number) and every element of (i.e. under the definition of Von Neumann ordinals, for every ordinal number ), we have that is an element of because any ordinal number contains only ordinal numbers, by the definition of this ordinal construction.
is well ordered by the membership relation because all its elements are also well ordered by this relation.
So, by steps 2 and 3, we have that is an ordinal class and also, by step 1, an ordinal number, because all ordinal classes that are sets are also ordinal numbers.
This implies that is an element of .
Under the definition of Von Neumann ordinals, is the same as being an element of . This latter statement is proven by step 5.
But no ordinal class is less than itself, including because of step 4 ( is an ordinal class), i.e. .
We have deduced two contradictory propositions ( and ) from the sethood of and, therefore, disproved that is a set.
Stated more generally
The version of the paradox above is anachronistic, because it presupposes the definition of the ordinals due to John von Neumann, under which each ordinal is the set of all preceding ordinals, which was not known at the time the paradox was framed by Burali-Forti.
Here is an account with fewer presuppositions: suppose that we associate with each well-ordering
an object called its order type in an unspecified way (the order types are the ordinal numbers). The order types (ordinal numbers) themselves are well-ordered in a natural way,
and this well-ordering must have an order type . It is easily shown in
naïve set theory (and remains true in ZFC but not in New Foundations) that the order
type of all ordinal numbers less than a fixed is itself.
So the order
type of all ordinal numbers less than is itself. But
this means that , being the order type of a proper initial segment of the ordinals, is strictly less than the order type of all the ordinals,
but the latter is itself by definition. This is a contradiction.
If we use the von Neumann definition, under which each ordinal is identified as the set of all preceding ordinals, the paradox is unavoidab |
https://en.wikipedia.org/wiki/Soliton | In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is strongly stable, in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such localized wave packets. Its remarkable stability can be traced to a balanced cancellation of nonlinear and dispersive effects in the medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons were subsequently found to provide stable solutions of a wide class of weakly nonlinear dispersive partial differential equations describing physical systems.
The soliton phenomenon was first described in 1834 by John Scott Russell (1808–1882) who observed a solitary wave in the Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the "Wave of Translation". The term soliton was coined by Zabusky and Kruskal to describe localized, strongly stable propagating solutions to the Korteweg–de Vries equation, which models waves of the type seen by Russell. The name was meant to characterize the solitary nature of the waves, with the 'on' suffix recalling the usage for particles such as electrons, baryons or hadrons, reflecting their observed particle-like behaviour.
Definition
A single, consensus definition of a soliton is difficult to find. ascribe three properties to solitons:
They are of permanent form;
They are localized within a region;
They can interact with other solitons, and emerge from the collision unchanged, except for a phase shift.
More formal definitions exist, but they require substantial mathematics. Moreover, some scientists use the term soliton for phenomena that do not quite have these three properties (for instance, the 'light bullets' of nonlinear optics are often called solitons despite losing energy during interaction).
Explanation
Dispersion and nonlinearity can interact to produce permanent and localized wave forms. Consider a pulse of light traveling in glass. This pulse can be thought of as consisting of light of several different frequencies. Since glass shows dispersion, these different frequencies travel at different speeds and the shape of the pulse therefore changes over time. However, also the nonlinear Kerr effect occurs; the refractive index of a material at a given frequency depends on the light's amplitude or strength. If the pulse has just the right shape, the Kerr effect exactly cancels the dispersion effect and the pulse's shape does not change over time. Thus, the pulse is a soliton. See soliton (optics) for a more detailed description.
Many exactly solvable models have soliton solutions, including the Korteweg–de Vries equation, the nonlinear Schrödinger equation, the coupled nonlinear Schrödinger equation, and the sine-Gordon equation. The soliton solutions are typically obtained by means of the inverse scattering transform, and owe their stability to the integrability o |
https://en.wikipedia.org/wiki/Vacuous%20truth | In mathematics and logic, a vacuous truth is a conditional or universal statement (a universal statement that can be converted to a conditional statement) that is true because the antecedent cannot be satisfied.
It is sometimes said that a statement is vacuously true because it does not really say anything. For example, the statement "all cell phones in the room are turned off" will be true when no cell phones are in the room. In this case, the statement "all cell phones in the room are turned on" would also be vacuously true, as would the conjunction of the two: "all cell phones in the room are turned on and turned off", which would otherwise be incoherent and false.
More formally, a relatively well-defined usage refers to a conditional statement (or a universal conditional statement) with a false antecedent. One example of such a statement is "if Tokyo is in France, then the Eiffel Tower is in Bolivia".
Such statements are considered vacuous truths, because the fact that the antecedent is false prevents using the statement to infer anything about the truth value of the consequent. In essence, a conditional statement, that is based on the material conditional, is true when the antecedent ("Tokyo is in France" in the example) is false regardless of whether the conclusion or consequent ("the Eiffel Tower is in Bolivia" in the example) is true or false because the material conditional is defined in that way.
Examples common to everyday speech include conditional phrases used as idioms of improbability like "when hell freezes over..." and "when pigs can fly...", indicating that not before the given (impossible) condition is met will the speaker accept some respective (typically false or absurd) proposition.
In pure mathematics, vacuously true statements are not generally of interest by themselves, but they frequently arise as the base case of proofs by mathematical induction. This notion has relevance in pure mathematics, as well as in any other field that uses classical logic.
Outside of mathematics, statements which can be characterized informally as vacuously true can be misleading. Such statements make reasonable assertions about qualified objects which do not actually exist. For example, a child might truthfully tell their parent "I ate every vegetable on my plate", when there were no vegetables on the child's plate to begin with. In this case, the parent can believe that the child has actually eaten some vegetables, even though that is not true. In addition, a vacuous truth is often used colloquially with absurd statements, either to confidently assert something (e.g. "the dog was red, or I'm a monkey's uncle" to strongly claim that the dog was red), or to express doubt, sarcasm, disbelief, incredulity or indignation (e.g. "yes, and I'm the King of England" to disagree a previously made statement).
Scope of the concept
A statement is "vacuously true" if it resembles a material conditional statement , where the antecedent is known to |
https://en.wikipedia.org/wiki/Extended%20real%20number%20line | In mathematics, the extended real number system is obtained from the real number system by adding two infinity elements: and where the infinities are treated as actual numbers. It is useful in describing the algebra on infinities and the various limiting behaviors in calculus and mathematical analysis, especially in the theory of measure and integration. The extended real number system is denoted or or It is the Dedekind–MacNeille completion of the real numbers.
When the meaning is clear from context, the symbol is often written simply as
There is also the projectively extended real line where and are not distinguished so the infinity is denoted by only .
Motivation
Limits
It is often useful to describe the behavior of a function as either the argument or the function value gets "infinitely large" in some sense. For example, consider the function defined by
The graph of this function has a horizontal asymptote at Geometrically, when moving increasingly farther to the right along the -axis, the value of approaches . This limiting behavior is similar to the limit of a function in which the real number approaches except that there is no real number to which approaches.
By adjoining the elements and to it enables a formulation of a "limit at infinity", with topological properties similar to those for
To make things completely formal, the Cauchy sequences definition of allows defining as the set of all sequences of rational numbers such that every is associated with a corresponding for which for all The definition of can be constructed similarly.
Measure and integration
In measure theory, it is often useful to allow sets that have infinite measure and integrals whose value may be infinite.
Such measures arise naturally out of calculus. For example, in assigning a measure to that agrees with the usual length of intervals, this measure must be larger than any finite real number. Also, when considering improper integrals, such as
the value "infinity" arises. Finally, it is often useful to consider the limit of a sequence of functions, such as
Without allowing functions to take on infinite values, such essential results as the monotone convergence theorem and the dominated convergence theorem would not make sense.
Order and topological properties
The extended real number system , defined as or , can be turned into a totally ordered set by defining for all With this order topology, has the desirable property of compactness: Every subset of has a supremum and an infimum (the infimum of the empty set is , and its supremum is ). Moreover, with this topology, is homeomorphic to the unit interval Thus the topology is metrizable, corresponding (for a given homeomorphism) to the ordinary metric on this interval. There is no metric, however, that is an extension of the ordinary metric on
In this topology, a set is a neighborhood of if and only if it contains a set for some real number The notion of |
https://en.wikipedia.org/wiki/Taylor%27s%20theorem | In calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial.
Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1715, although an earlier version of the result was already mentioned in 1671 by James Gregory.
Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathematical analysis. It gives simple arithmetic formulas to accurately compute values of many transcendental functions such as the exponential function and trigonometric functions.
It is the starting point of the study of analytic functions, and is fundamental in various areas of mathematics, as well as in numerical analysis and mathematical physics. Taylor's theorem also generalizes to multivariate and vector valued functions.
Motivation
If a real-valued function is differentiable at the point , then it has a linear approximation near this point. This means that there exists a function h1(x) such that
Here
is the linear approximation of for x near the point a, whose graph is the tangent line to the graph at . The error in the approximation is:
As x tends to a, this error goes to zero much faster than , making a useful approximation.
For a better approximation to , we can fit a quadratic polynomial instead of a linear function:
Instead of just matching one derivative of at , this polynomial has the same first and second derivatives, as is evident upon differentiation.
Taylor's theorem ensures that the quadratic approximation is, in a sufficiently small neighborhood of , more accurate than the linear approximation. Specifically,
Here the error in the approximation is
which, given the limiting behavior of , goes to zero faster than as x tends to a.
Similarly, we might get still better approximations to f if we use polynomials of higher degree, since then we can match even more derivatives with f at the selected base point.
In general, the error in approximating a function by a polynomial of degree k will go to zero much faster than as x tends to a. However, there are functions, even infinitely differentiable ones, for which increasing the degree of the approximating polynomial does not increase the accuracy of approximation: we say such a function fails to be analytic at x = a: it is not (locally) determined by its derivatives at this point.
Taylor's theorem is of asymptotic nature: it only tells us that the error in an a |
https://en.wikipedia.org/wiki/Martin%20Dunwoody | Martin John Dunwoody (born 3 November 1938) is an emeritus professor of Mathematics at the University of Southampton, England.
He earned his PhD in 1964 from the Australian National University. He held positions at the University of Sussex before becoming a professor at the University of Southampton in 1992. He has been emeritus professor since 2003.
Dunwoody works on geometric group theory and low-dimensional topology. He is a leading expert in splittings and accessibility of discrete groups, groups acting on graphs and trees, JSJ-decompositions, the topology of 3-manifolds and the structure of their fundamental groups.
Since 1971 several mathematicians have been working on Wall's conjecture, posed by Wall in a 1971 paper, which said that all finitely generated groups are accessible. Roughly, this means that every finitely generated group can be constructed from finite and one-ended groups via a finite number of amalgamated free products and HNN extensions over finite subgroups. In view of the Stallings theorem about ends of groups, one-ended groups are precisely those finitely generated infinite groups that cannot be decomposed nontrivially as amalgamated products or HNN-extensions over finite subgroups.
Dunwoody proved the Wall conjecture for finitely presented groups in 1985. In 1991 he finally disproved Wall's conjecture by finding a finitely generated group that is not accessible.
Dunwoody found a graph-theoretic proof of Stallings' theorem about ends of groups in 1982, by constructing certain tree-like automorphism invariant graph decompositions. This work has been developed to an important theory in the book Groups acting on graphs, Cambridge University Press, 1989, with Warren Dicks. In 2002 Dunwoody put forward a proposed proof of the Poincaré conjecture. The proof generated considerable interest among mathematicians, but a mistake was quickly discovered and the proof was withdrawn. The conjecture was later proven by Grigori Perelman, following the program of Richard S. Hamilton.
References
External links
home page of Martin Dunwoody.
1938 births
Living people
20th-century British mathematicians
21st-century British mathematicians
Australian National University alumni
Academics of the University of Sussex
Academics of the University of Southampton |
https://en.wikipedia.org/wiki/Distribution%20%28mathematics%29 | Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative.
Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions (weak solutions) than classical solutions, or where appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the Dirac delta function.
A function is normally thought of as on the in the function domain by "sending" a point in the domain to the point Instead of acting on points, distribution theory reinterprets functions such as as acting on in a certain way. In applications to physics and engineering, are usually infinitely differentiable complex-valued (or real-valued) functions with compact support that are defined on some given non-empty open subset . (Bump functions are examples of test functions.) The set of all such test functions forms a vector space that is denoted by or
Most commonly encountered functions, including all continuous maps if using can be canonically reinterpreted as acting via "integration against a test function." Explicitly, this means that such a function "acts on" a test function by "sending" it to the number which is often denoted by This new action of defines a scalar-valued map whose domain is the space of test functions This functional turns out to have the two defining properties of what is known as a : it is linear, and it is also continuous when is given a certain topology called . The action (the integration ) of this distribution on a test function can be interpreted as a weighted average of the distribution on the support of the test function, even if the values of the distribution at a single point are not well-defined. Distributions like that arise from functions in this way are prototypical examples of distributions, but there exist many distributions that cannot be defined by integration against any function. Examples of the latter include the Dirac delta function and distributions defined to act by integration of test functions against certain measures on Nonetheless, it is still always possible to reduce any arbitrary distribution down to a simpler of related distributions that do arise via such actions of integration.
More generally, a is by definition a linear functional on that is continuous when is given a topology called the . This leads to space of (all) distributions on , usually denoted by (note the prime), which by definition is the space of all distributions on (that is, it is the c |
https://en.wikipedia.org/wiki/Cylinder%20%28disambiguation%29 | A cylinder is a basic curvilinear geometric shape.
Cylinder may also refer to:
Cylinder (algebra), the Cartesian product of a set with its superset
Cylinder (disk drive), a division of data in a disk drive
Cylinder (engine), the space in which a piston travels in an engine
Cylinder (firearms), the rotating part of a revolver containing multiple chambers
Cylinder (gastropod), a subgenus of sea snails
Cylinder (locomotive), the components that convert steam power into motion
Cylinder (optometry)
Cylinder, Iowa, a city in Palo Alto County, Iowa, United States
Cylinder set, a natural basic set in product spaces
Cylinder set measure, a way to generate a measure over product spaces
Gas cylinder, a high-strength container for storing gases at high pressure
Phonograph cylinder, the earliest commercial medium for recording and reproducing sound
See also
Cylindera, a genus of ground beetles |
https://en.wikipedia.org/wiki/Embedded | Embedded or embedding (alternatively imbedded or imbedding) may refer to:
Science
Embedding, in mathematics, one instance of some mathematical object contained within another instance
Graph embedding
Embedded generation, a distributed generation of energy, also known as decentralized generation
Self-embedding, in psychology, an activity in which one pushes items into one's own flesh in order to feel pain
Embedding, in biology, a part of sample preparation for microscopes
Embeddedness, in economics and economic sociology, refers to the degree to which economic activity is constrained by non-economic institutions.
Computing
Embedded system, a special-purpose system in which the computer is completely encapsulated by the device it controls
Embedding, installing media into a text document to form a compound document
, a HyperText Markup Language (HTML) element that inserts a non-standard object into the HTML document
Web embed, an element of a host web page that is substantially independent of the host page
Font embedding, inclusion of font files inside an electronic document
Word embedding, a text representation technique used in natural language processing
Data representations generated through feature learning
Vector embedding, representing concepts and information by reference to similar concepts, in a multi-dimensional space. Similar to word embeddings but with wide applicability.
Linguistics
Embedded clause or dependent clause: one that provides a sentence element with additional information, but which cannot stand alone as a sentence
Center embedding, recursive nesting of an element in the middle of a similar element
Art
Embedded journalism, under the control of one side's army in a military conflict
Embedded (play), a 2003 play by Tim Robbins about embedded journalists covering military conflict in the US-Iraq war
Embedded (Mark Seymour album), 2004
Embedded (Meathook Seed album), 1993
"Embedded", a song by Job for a Cowboy from the 2007 album Genesis
Embed Series, works by artist Mark Jenkins
Embedded, a novel by Dan Abnett
Embedded (film), a 2016 erotic political thriller film |
https://en.wikipedia.org/wiki/Embedding | In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object is said to be embedded in another object , the embedding is given by some injective and structure-preserving map . The precise meaning of "structure-preserving" depends on the kind of mathematical structure of which and are instances. In the terminology of category theory, a structure-preserving map is called a morphism.
The fact that a map is an embedding is often indicated by the use of a "hooked arrow" (); thus: (On the other hand, this notation is sometimes reserved for inclusion maps.)
Given and , several different embeddings of in may be possible. In many cases of interest there is a standard (or "canonical") embedding, like those of the natural numbers in the integers, the integers in the rational numbers, the rational numbers in the real numbers, and the real numbers in the complex numbers. In such cases it is common to identify the domain with its image contained in , so that .
Topology and geometry
General topology
In general topology, an embedding is a homeomorphism onto its image. More explicitly, an injective continuous map between topological spaces and is a topological embedding if yields a homeomorphism between and (where carries the subspace topology inherited from ). Intuitively then, the embedding lets us treat as a subspace of . Every embedding is injective and continuous. Every map that is injective, continuous and either open or closed is an embedding; however there are also embeddings which are neither open nor closed. The latter happens if the image is neither an open set nor a closed set in .
For a given space , the existence of an embedding is a topological invariant of . This allows two spaces to be distinguished if one is able to be embedded in a space while the other is not.
Related definitions
If the domain of a function is a topological space then the function is said to be if there exists some neighborhood of this point such that the restriction is injective. It is called if it is locally injective around every point of its domain. Similarly, a is a function for which every point in its domain has some neighborhood to which its restriction is a (topological, resp. smooth) embedding.
Every injective function is locally injective but not conversely. Local diffeomorphisms, local homeomorphisms, and smooth immersions are all locally injective functions that are not necessarily injective. The inverse function theorem gives a sufficient condition for a continuously differentiable function to be (among other things) locally injective. Every fiber of a locally injective function is necessarily a discrete subspace of its domain
Differential topology
In differential topology:
Let and be smooth manifolds and be a smooth map. Then is called an immersion if its derivative is everywhere injective. An embeddin |
https://en.wikipedia.org/wiki/Julius%20Pl%C3%BCcker | Julius Plücker (16 June 1801 – 22 May 1868) was a German mathematician and physicist. He made fundamental contributions to the field of analytical geometry and was a pioneer in the investigations of cathode rays that led eventually to the discovery of the electron. He also vastly extended the study of Lamé curves.
Biography
Early years
Plücker was born at Elberfeld (now part of Wuppertal). After being educated at Düsseldorf and at the universities of Bonn, Heidelberg and Berlin he went to Paris in 1823, where he came under the influence of the great school of French geometers, whose founder, Gaspard Monge, had only recently died.
In 1825 he returned to Bonn, and in 1828 was made professor of mathematics.
In the same year he published the first volume of his Analytisch-geometrische Entwicklungen, which introduced the method of "abridged notation".
In 1831 he published the second volume, in which he clearly established on a firm and independent basis projective duality.
Career
In 1836, Plücker was made professor of physics at University of Bonn. In 1858, after a year of working with vacuum tubes of his Bonn colleague Heinrich Geißler, he published his first classical researches on the action of the magnet on the electric discharge in rarefied gases. He found that the discharge caused a fluorescent glow to form on the glass walls of the vacuum tube, and that the glow could be made to shift by applying an electromagnet to the tube, thus creating a magnetic field. It was later shown that the glow was produced by cathode rays.
Plücker, first by himself and afterwards in conjunction with Johann Hittorf, made many important discoveries in the spectroscopy of gases. He was the first to use the vacuum tube with the capillary part now called a Geissler tube, by means of which the luminous intensity of feeble electric discharges was raised sufficiently to allow of spectroscopic investigation. He anticipated Robert Wilhelm Bunsen and Gustav Kirchhoff in announcing that the lines of the spectrum were characteristic of the chemical substance which emitted them, and in indicating the value of this discovery in chemical analysis. According to Hittorf, he was the first who saw the three lines of the hydrogen spectrum, which a few months after his death, were recognized in the spectrum of the solar protuberances.
In 1865, Plücker returned to the field of geometry and invented what was known as line geometry in the nineteenth century. In projective geometry, Plücker coordinates refer to a set of homogeneous co-ordinates introduced initially to embed the space of lines in projective space as a quadric in . The construction uses 2×2 minor determinants, or equivalently the second exterior power of the underlying vector space of dimension 4. It is now part of the theory of Grassmannians
(-dimensional subspaces of an -dimensional vector space ), to which the generalization of these co-ordinates to minors of the matrix of homogeneous coordinates, also known |
https://en.wikipedia.org/wiki/Perfect%20square | A perfect square is an element of algebraic structure that is equal to the square of another element.
Square number, a perfect square integer
Entertainment
Perfect Square, a live recording by the band R.E.M.
Perfect Square (publisher), a children's imprint label by Viz Media.
See also
Perfect square dissection, a dissection of a geometric square into smaller squares, all of different sizes
Perfect square trinomials, a method of factoring polynomials |
https://en.wikipedia.org/wiki/Magic%20square | In recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diagonals are the same. The 'order' of the magic square is the number of integers along one side (n), and the constant sum is called the 'magic constant'. If the array includes just the positive integers , the magic square is said to be 'normal'. Some authors take magic square to mean normal magic square.
Magic squares that include repeated entries do not fall under this definition and are referred to as 'trivial'. Some well-known examples, including the Sagrada Família magic square and the Parker square are trivial in this sense. When all the rows and columns but not both diagonals sum to the magic constant this gives a semimagic square (sometimes called orthomagic square).
The mathematical study of magic squares typically deals with their construction, classification, and enumeration. Although completely general methods for producing all the magic squares of all orders do not exist, historically three general techniques have been discovered: by bordering method, by making composite magic squares, and by adding two preliminary squares. There are also more specific strategies like the continuous enumeration method that reproduces specific patterns. Magic squares are generally classified according to their order n as: odd if n is odd, evenly even (also referred to as "doubly even") if n is a multiple of 4, oddly even (also known as "singly even") if n is any other even number. This classification is based on different techniques required to construct odd, evenly even, and oddly even squares. Beside this, depending on further properties, magic squares are also classified as associative magic squares, pandiagonal magic squares, most-perfect magic squares, and so on. More challengingly, attempts have also been made to classify all the magic squares of a given order as transformations of a smaller set of squares. Except for n ≤ 5, the enumeration of higher order magic squares is still an open challenge. The enumeration of most-perfect magic squares of any order was only accomplished in the late 20th century.
Magic squares have a long history, dating back to at least 190 BCE in China. At various times they have acquired occult or mythical significance, and have appeared as symbols in works of art. In modern times they have been generalized a number of ways, including using extra or different constraints, multiplying instead of adding cells, using alternate shapes or more than two dimensions, and replacing numbers with shapes and addition with geometric operations.
History
The third-order magic square was known to Chinese mathematicians as early as 190 BCE, and explicitly given by the first century of the common era. The first dateable instance of the fourth-order magic square occurred in 587 CE in India. Specimens of magic squares of order 3 to 9 appear in an encyclopedia fro |
https://en.wikipedia.org/wiki/Fourier%20transform | In physics, engineering and mathematics, the Fourier transform (FT) is an integral transform that converts a function into a form that describes the frequencies present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches.
Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.
The Fourier transform can be formally defined as an improper Riemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory. For example, many relatively simple applications use the Dirac delta function, which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint.
The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of 'position space' to a function of momentum (or a function of space and time to a function of 4-momentum). This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanics, where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both. In general, functions to which Fourier methods are applicable are complex-valued, and possibly vector-valued. Still further generalization is possible to functions on groups, which, besides the original Fourier transform on or , notably includes the discrete-time Fourier transform (DTFT, group = ), the discrete Fourier transform (DFT, group = ) and the Fourier series or circular Fourier transform (group = , the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle periodic functions. The fast Fourier transform (FFT) is an algorithm for computing the DFT.
Definitions
The Fourier transform on R
The Fourier transform is an extension of the Fourier series from bounded real interval of width to the infinite domain . The coefficients of Fourier series of a periodic |
https://en.wikipedia.org/wiki/Cyclic%20group | In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted Cn, that is generated by a single element. That is, it is a set of invertible elements with a single associative binary operation, and it contains an element g such that every other element of the group may be obtained by repeatedly applying the group operation to g or its inverse. Each element can be written as an integer power of g in multiplicative notation, or as an integer multiple of g in additive notation. This element g is called a generator of the group.
Every infinite cyclic group is isomorphic to the additive group of Z, the integers. Every finite cyclic group of order n is isomorphic to the additive group of Z/nZ, the integers modulo n. Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of cyclic groups.
Every cyclic group of prime order is a simple group, which cannot be broken down into smaller groups. In the classification of finite simple groups, one of the three infinite classes consists of the cyclic groups of prime order. The cyclic groups of prime order are thus among the building blocks from which all groups can be built.
Definition and notation
For any element g in any group G, one can form the subgroup that consists of all its integer powers: , called the cyclic subgroup generated by g. The order of g is |⟨g⟩|, the number of elements in ⟨g⟩, conventionally abbreviated as |g|, as ord(g), or as o(g). That is, the order of an element is equal to the order of the cyclic subgroup that it generates,
A cyclic group is a group which is equal to one of its cyclic subgroups: for some element g, called a generator of G.
For a finite cyclic group G of order n we have , where e is the identity element and whenever (mod n); in particular , and . An abstract group defined by this multiplication is often denoted Cn, and we say that G is isomorphic to the standard cyclic group Cn. Such a group is also isomorphic to Z/nZ, the group of integers modulo n with the addition operation, which is the standard cyclic group in additive notation. Under the isomorphism χ defined by the identity element e corresponds to 0, products correspond to sums, and powers correspond to multiples.
For example, the set of complex 6th roots of unity forms a group under multiplication. It is cyclic, since it is generated by the primitive root that is, G = ⟨z⟩ = { 1, z, z2, z3, z4, z5 } with z6 = 1. Under a change of letters, this is isomorphic to (structurally the same as) the standard cyclic group of order 6, defined as C6 = ⟨g⟩ = with multiplication gj · gk = gj+k (mod 6), so that g6 = g0 = e. These groups are also isomorphic to Z/6Z = with the operation of addition modulo 6, with zk and gk corresponding to k. For example, corresponds to , and corresponds to , and so on. Any element generates its own cyclic subgroup, such as ⟨ |
https://en.wikipedia.org/wiki/Axiom%20of%20extensionality | In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of extensionality, or axiom of extension, is one of the axioms of Zermelo–Fraenkel set theory. It says that sets having the same elements are the same set.
Formal statement
In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:
or in words:
Given any set A and any set B, if for every set X, X is a member of A if and only if X is a member of B, then A is equal to B.
(It is not really essential that X here be a set — but in ZF, everything is. See Ur-elements below for when this is violated.)
The converse, of this axiom follows from the substitution property of equality.
Interpretation
To understand this axiom, note that the clause in parentheses in the symbolic statement above simply states that A and B have precisely the same members.
Thus, what the axiom is really saying is that two sets are equal if and only if they have precisely the same members.
The essence of this is:
A set is determined uniquely by its members.
The axiom of extensionality can be used with any statement of the form
,
where P is any unary predicate that does not mention A, to define a unique set whose members are precisely the sets satisfying the predicate .
We can then introduce a new symbol for ; it's in this way that definitions in ordinary mathematics ultimately work when their statements are reduced to purely set-theoretic terms.
The axiom of extensionality is generally uncontroversial in set-theoretical foundations of mathematics, and it or an equivalent appears in just about any alternative axiomatisation of set theory.
However, it may require modifications for some purposes, as below.
In predicate logic without equality
The axiom given above assumes that equality is a primitive symbol in predicate logic.
Some treatments of axiomatic set theory prefer to do without this, and instead treat the above statement not as an axiom but as a definition of equality.
Then it is necessary to include the usual axioms of equality from predicate logic as axioms about this defined symbol. Most of the axioms of equality still follow from the definition; the remaining one is the substitution property,
and it becomes this axiom that is referred to as the axiom of extensionality in this context.
In set theory with ur-elements
An ur-element is a member of a set that is not itself a set.
In the Zermelo–Fraenkel axioms, there are no ur-elements, but they are included in some alternative axiomatisations of set theory.
Ur-elements can be treated as a different logical type from sets; in this case, makes no sense if is an ur-element, so the axiom of extensionality simply applies only to sets.
Alternatively, in untyped logic, we can require to be false whenever is an ur-element.
In this case, the usual axiom of extensionality would then imply that every ur-element is equal to the empty set.
To avoid this consequence, we can modify the axiom of extensio |
https://en.wikipedia.org/wiki/Axiom%20of%20pairing | In axiomatic set theory and the branches of logic, mathematics, and computer science that use it, the axiom of pairing is one of the axioms of Zermelo–Fraenkel set theory. It was introduced by as a special case of his axiom of elementary sets.
Formal statement
In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:
In words:
Given any object A and any object B, there is a set C such that, given any object D, D is a member of C if and only if D is equal to A or D is equal to B.
Or in simpler words:
Given two objects, there is a set whose members are exactly the two given objects.
Consequences
As noted, what the axiom is saying is that, given two objects A and B, we can find a set C whose members are exactly A and B.
We can use the axiom of extensionality to show that this set C is unique.
We call the set C the pair of A and B, and denote it {A,B}.
Thus the essence of the axiom is:
Any two objects have a pair.
The set {A,A} is abbreviated {A}, called the singleton containing A.
Note that a singleton is a special case of a pair. Being able to construct a singleton is necessary, for example, to show the non-existence of the infinitely descending chains from the Axiom of regularity.
The axiom of pairing also allows for the definition of ordered pairs. For any objects and , the ordered pair is defined by the following:
Note that this definition satisfies the condition
Ordered n-tuples can be defined recursively as follows:
Alternatives
Non-independence
The axiom of pairing is generally considered uncontroversial, and it or an equivalent appears in just about any axiomatization of set theory. Nevertheless, in the standard formulation of the Zermelo–Fraenkel set theory, the axiom of pairing follows from the axiom schema of replacement applied to any given set with two or more elements, and thus it is sometimes omitted. The existence of such a set with two elements, such as { {}, { {} } }, can be deduced either from the axiom of empty set and the axiom of power set or from the axiom of infinity.
In the absence of some of the stronger ZFC axioms, the axiom of pairing can still, without loss, be introduced in weaker forms.
Weaker
In the presence of standard forms of the axiom schema of separation we can replace the axiom of pairing by its weaker version:
.
This weak axiom of pairing implies that any given objects and are members of some set . Using the axiom schema of separation we can construct the set whose members are exactly and .
Another axiom which implies the axiom of pairing in the presence of the axiom of empty set is the axiom of adjunction
.
It differs from the standard one by use of instead of .
Using {} for A and x for B, we get {x} for C. Then use {x} for A and y for B, getting {x,y} for C. One may continue in this fashion to build up any finite set. And this could be used to generate all hereditarily finite sets without using the axiom of union.
Stronger
Together with the axiom of empty set and the |
https://en.wikipedia.org/wiki/Axiom%20schema%20of%20specification | In many popular versions of axiomatic set theory, the axiom schema of specification, also known as the axiom schema of separation, subset axiom scheme or axiom schema of restricted comprehension is an axiom schema. Essentially, it says that any definable subclass of a set is a set.
Some mathematicians call it the axiom schema of comprehension, although others use that term for unrestricted comprehension, discussed below.
Because restricting comprehension avoided Russell's paradox, several mathematicians including Zermelo, Fraenkel, and Gödel considered it the most important axiom of set theory.
Statement
One instance of the schema is included for each formula φ in the language of set theory with free variables among x, w1, ..., wn, A. So B does not occur free in φ. In the formal language of set theory, the axiom schema is:
or in words:
Given any set A, there is a set B (a subset of A) such that, given any set x, x is a member of B if and only if x is a member of A and φ holds for x.
Note that there is one axiom for every such predicate φ; thus, this is an axiom schema.
To understand this axiom schema, note that the set B must be a subset of A. Thus, what the axiom schema is really saying is that, given a set A and a predicate , we can find a subset B of A whose members are precisely the members of A that satisfy . By the axiom of extensionality this set is unique. We usually denote this set using set-builder notation as . Thus the essence of the axiom is:
Every subclass of a set that is defined by a predicate is itself a set.
The preceding form of separation was introduced in 1930 by Thoralf Skolem as a refinement of a previous form by Zermelo. The axiom schema of specification is characteristic of systems of axiomatic set theory related to the usual set theory ZFC, but does not usually appear in radically different systems of alternative set theory. For example, New Foundations and positive set theory use different restrictions of the axiom of comprehension of naive set theory. The Alternative Set Theory of Vopenka makes a specific point of allowing proper subclasses of sets, called semisets. Even in systems related to ZFC, this scheme is sometimes restricted to formulas with bounded quantifiers, as in Kripke–Platek set theory with urelements.
Relation to the axiom schema of replacement
The axiom schema of specification can almost be derived from the axiom schema of replacement.
First, recall this axiom schema:
for any functional predicate F in one variable that doesn't use the symbols A, B, C or D.
Given a suitable predicate P for the axiom of specification, define the mapping F by F(D) = D if P(D) is true and F(D) = E if P(D) is false, where E is any member of A such that P(E) is true.
Then the set B guaranteed by the axiom of replacement is precisely the set B required for the axiom of specification. The only problem is if no such E exists. But in this case, the set B required for the axiom of separation is the empty set, |
https://en.wikipedia.org/wiki/Axiom%20schema%20of%20replacement | In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite sets in ZF.
The axiom schema is motivated by the idea that whether a class is a set depends only on the cardinality of the class, not on the rank of its elements. Thus, if one class is "small enough" to be a set, and there is a surjection from that class to a second class, the axiom states that the second class is also a set. However, because ZFC only speaks of sets, not proper classes, the schema is stated only for definable surjections, which are identified with their defining formulas.
Statement
Suppose is a definable binary relation (which may be a proper class) such that for every set there is a unique set such that holds. There is a corresponding definable function , where if and only if . Consider the (possibly proper) class defined such that for every set , if and only if there is an with . is called the image of under , and denoted or (using set-builder notation) .
The axiom schema of replacement states that if is a definable class function, as above, and is any set, then the image is also a set. This can be seen as a principle of smallness: the axiom states that if is small enough to be a set, then is also small enough to be a set. It is implied by the stronger axiom of limitation of size.
Because it is impossible to quantify over definable functions in first-order logic, one instance of the schema is included for each formula in the language of set theory with free variables among ; but is not free in . In the formal language of set theory, the axiom schema is:
For the meaning of , see uniqueness quantification.
For clarity, in the case of no variables , this simplifies to:
So whenever specifies a unique -to- correspondence, akin to a function on , then all reached this way can be collected into a set , akin to .
Applications
The axiom schema of replacement is not necessary for the proofs of most theorems of ordinary mathematics. Indeed, Zermelo set theory (Z) already can interpret second-order arithmetic and much of type theory in finite types, which in turn are sufficient to formalize the bulk of mathematics. Although the axiom schema of replacement is a standard axiom in set theory today, it is often omitted from systems of type theory and foundation systems in topos theory.
At any rate, the axiom schema drastically increases the strength of ZF, both in terms of the theorems it can prove - for example the sets shown to exist - and also in terms of its proof-theoretic consistency strength, compared to Z. Some important examples follow:
Using the modern definition due to von Neumann, proving the existence of any limit ordinal greater than ω requires the replacement axiom. The ordinal number ω·2 = ω + ω is the first such ordinal. The axiom of infinity asserts th |
https://en.wikipedia.org/wiki/Tait%27s%20conjecture | In mathematics, Tait's conjecture states that "Every 3-connected planar cubic graph has a Hamiltonian cycle (along the edges) through all its vertices". It was proposed by and disproved by , who constructed a counterexample with 25 faces, 69 edges and 46 vertices. Several smaller counterexamples, with 21 faces, 57 edges and 38 vertices, were later proved minimal by .
The condition that the graph be 3-regular is necessary due to polyhedra such as the rhombic dodecahedron, which forms a bipartite graph with six degree-four vertices on one side and eight degree-three vertices on the other side; because any Hamiltonian cycle would have to alternate between the two sides of the bipartition, but they have unequal numbers of vertices, the rhombic dodecahedron is not Hamiltonian.
The conjecture was significant, because if true, it would have implied the four color theorem: as Tait described, the four-color problem is equivalent to the problem of finding 3-edge-colorings of bridgeless cubic planar graphs. In a Hamiltonian cubic planar graph, such an edge coloring is easy to find: use two colors alternately on the cycle, and a third color for all remaining edges. Alternatively, a 4-coloring of the faces of a Hamiltonian cubic planar graph may be constructed directly, using two colors for the faces inside the cycle and two more colors for the faces outside.
Tutte's counterexample
Tutte's fragment
The key to this counter-example is what is now known as Tutte's fragment, shown on the right.
If this fragment is part of a larger graph, then any Hamiltonian cycle through the graph must go in or out of the top vertex (and either one of the lower ones). It cannot go in one lower vertex and out the other.
The counterexample
The fragment can then be used to construct the non-Hamiltonian Tutte graph, by putting
together three such fragments as shown on the picture. The "compulsory" edges of the fragments, that must be part of any Hamiltonian path through the fragment, are connected at the central vertex; because any cycle can use only two of these three edges, there can be no Hamiltonian cycle.
The resulting Tutte graph is 3-connected and planar, so by Steinitz' theorem it is the graph of a polyhedron. In total it has 25 faces, 69 edges and 46 vertices.
It can be realized geometrically from a tetrahedron (the faces of which correspond to the four large faces in the drawing, three of which are between pairs of fragments and the fourth of which forms the exterior) by multiply truncating three of its vertices.
Smaller counterexamples
As show, there are exactly six 38-vertex non-Hamiltonian polyhedra that have nontrivial three-edge cuts. They are formed by replacing two of the vertices of a pentagonal prism by the same fragment used in Tutte's example.
See also
Grinberg's theorem, a necessary condition on the existence of a Hamiltonian cycle that can be used to show that a graph forms a counterexample to Tait's conjecture
Barnette's conjecture, a still-open |
https://en.wikipedia.org/wiki/Stephen%20Wolfram | Stephen Wolfram ( ; born 29 August 1959) is a British-American computer scientist, physicist, and businessman. He is known for his work in computer science, mathematics, and theoretical physics. In 2012, he was named a fellow of the American Mathematical Society.
As a businessman, he is the founder and CEO of the software company Wolfram Research where he works as chief designer of Mathematica and the Wolfram Alpha answer engine.
Early life
Family
Stephen Wolfram was born in London in 1959 to Hugo and Sybil Wolfram, both German Jewish refugees to the United Kingdom. His maternal grandmother was British psychoanalyst Kate Friedlander.
Wolfram's father, Hugo Wolfram, was a textile manufacturer and served as managing director of the Lurex Company—makers of the fabric Lurex. Wolfram's mother, Sybil Wolfram, was a Fellow and Tutor in Philosophy at Lady Margaret Hall at University of Oxford from 1964 to 1993.
Stephen Wolfram is married to a mathematician. They have four children together.
Education
Wolfram was educated at Eton College, but left prematurely in 1976. As a young child, Wolfram had difficulties learning arithmetic. He entered St. John's College, Oxford, at age 17 and left in 1978 without graduating to attend the California Institute of Technology the following year, where he received a PhD in particle physics in 1980. Wolfram's thesis committee was composed of Richard Feynman, Peter Goldreich, Frank J. Sciulli and Steven Frautschi, and chaired by Richard D. Field.
Early career
Wolfram, at the age of 15, began research in applied quantum field theory and particle physics and published scientific papers in peer-reviewed scientific journals including Nuclear Physics B, Australian Journal of Physics, Nuovo Cimento, and Physical Review D. Working independently, Wolfram published a widely cited paper on heavy quark production at age 18 and nine other papers. Wolfram's work with Geoffrey C. Fox on the theory of the strong interaction is still used in experimental particle physics.
Following his PhD, Wolfram joined the faculty at Caltech and became the youngest recipient of a MacArthur Fellowship in 1981, at age 21.
Later career
Complex systems and cellular automata
In 1983, Wolfram left for the School of Natural Sciences of the Institute for Advanced Study in Princeton. By that time, he was no longer interested in particle physics. Instead, he began pursuing investigations into cellular automata, mainly with computer simulations. He produced a series of papers systematically investigating the class of elementary cellular automata, conceiving the Wolfram code, a naming system for one-dimensional cellular automata, and a classification scheme for the complexity of their behaviour. He conjectured that the Rule 110 cellular automaton might be Turing complete, which a research assistant to Wolfram, Matthew Cook, later proved correct. Wolfram sued Cook and temporarily blocked publication of the work on Rule 110 for allegedly violating a no |
https://en.wikipedia.org/wiki/Axiom%20of%20empty%20set | In axiomatic set theory, the axiom of empty set is a statement that asserts the existence of a set with no elements. It is an axiom of Kripke–Platek set theory and the variant of general set theory that Burgess (2005) calls "ST," and a demonstrable truth in Zermelo set theory and Zermelo–Fraenkel set theory, with or without the axiom of choice.
Formal statement
In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:
or in words:
There is a set such that no element is a member of it.
Interpretation
We can use the axiom of extensionality to show that there is only one empty set. Since it is unique we can name it. It is called the empty set (denoted by { } or ∅). The axiom, stated in natural language, is in essence:
An empty set exists.
This formula is a theorem and considered true in every version of set theory. The only controversy is over how it should be justified: by making it an axiom; by deriving it from a set-existence axiom (or logic) and the axiom of separation; by deriving it from the axiom of infinity; or some other method.
In some formulations of ZF, the axiom of empty set is actually repeated in the axiom of infinity. However, there are other formulations of that axiom that do not presuppose the existence of an empty set. The ZF axioms can also be written using a constant symbol representing the empty set; then the axiom of infinity uses this symbol without requiring it to be empty, while the axiom of empty set is needed to state that it is in fact empty.
Furthermore, one sometimes considers set theories in which there are no infinite sets, and then the axiom of empty set may still be required. However, any axiom of set theory or logic that implies the existence of any set will imply the existence of the empty set, if one has the axiom schema of separation. This is true, since the empty set is a subset of any set consisting of those elements that satisfy a contradictory formula.
In many formulations of first-order predicate logic, the existence of at least one object is always guaranteed. If the axiomatization of set theory is formulated in such a logical system with the axiom schema of separation as axioms, and if the theory makes no distinction between sets and other kinds of objects (which holds for ZF, KP, and similar theories), then the existence of the empty set is a theorem.
If separation is not postulated as an axiom schema, but derived as a theorem schema from the schema of replacement (as is sometimes done), the situation is more complicated, and depends on the exact formulation of the replacement schema. The formulation used in the axiom schema of replacement article only allows to construct the image F[a] when a is contained in the domain of the class function F; then the derivation of separation requires the axiom of empty set. On the other hand, the constraint of totality of F is often dropped from the replacement schema, in which case it implies the separation schema without using the axiom of e |
https://en.wikipedia.org/wiki/Axiom%20of%20power%20set | In mathematics, the axiom of power set is one of the Zermelo–Fraenkel axioms of axiomatic set theory.
In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:
where y is the power set of x, .
In English, this says:
Given any set x, there is a set such that, given any set z, this set z is a member of if and only if every element of z is also an element of x.
More succinctly: for every set , there is a set consisting precisely of the subsets of .
Note the subset relation is not used in the formal definition as subset is not a primitive relation in formal set theory; rather, subset is defined in terms of set membership, . By the axiom of extensionality, the set is unique.
The axiom of power set appears in most axiomatizations of set theory. It is generally considered uncontroversial, although constructive set theory prefers a weaker version to resolve concerns about predicativity.
Consequences
The power set axiom allows a simple definition of the Cartesian product of two sets and :
Notice that
and, for example, considering a model using the Kuratowski ordered pair,
and thus the Cartesian product is a set since
One may define the Cartesian product of any finite collection of sets recursively:
Note that the existence of the Cartesian product can be proved without using the power set axiom, as in the case of the Kripke–Platek set theory.
Limitations
The power set axiom does not specify what subsets of a set exist, only that there is a set containing all those that do. Not all conceivable subsets are guaranteed to exist. In particular, the power set of an infinite set would contain only "constructible sets" if the universe is the constructible universe but in other models of ZF set theory could contain sets that are not constructible.
References
Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition).
Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. .
Kunen, Kenneth, 1980. Set Theory: An Introduction to Independence Proofs. Elsevier. .
Axioms of set theory
de:Zermelo-Fraenkel-Mengenlehre#Die Axiome von ZF und ZFC |
https://en.wikipedia.org/wiki/Axiom%20of%20union | In axiomatic set theory, the axiom of union is one of the axioms of Zermelo–Fraenkel set theory. This axiom was introduced by Ernst Zermelo.
The axiom states that for each set x there is a set y whose elements are precisely the elements of the elements of x.
Formal statement
In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:
or in words:
Given any set A, there is a set B such that, for any element c, c is a member of B if and only if there is a set D such that c is a member of D and D is a member of A.
or, more simply:
For any set , there is a set which consists of just the elements of the elements of that set .
Relation to Pairing
The axiom of union allows one to unpack a set of sets and thus create a flatter set.
Together with the axiom of pairing, this implies that for any two sets, there is a set (called their union) that contains exactly the elements of the two sets.
Relation to Replacement
The axiom of replacement allows one to form many unions, such as the union of two sets.
However, in its full generality, the axiom of union is independent from the rest of the ZFC-axioms:
Replacement does not prove the existence of the union of a set of sets if the result contains an unbounded number of cardinalities.
Together with the axiom schema of replacement, the axiom of union implies that one can form the union of a family of sets indexed by a set.
Relation to Separation
In the context of set theories which include the axiom of separation, the axiom of union is sometimes stated in a weaker form which only produces a superset of the union of a set. For example, Kunen states the axiom as
which is equivalent to
Compared to the axiom stated at the top of this section, this variation asserts only one direction of the implication, rather than both directions.
Relation to Intersection
There is no corresponding axiom of intersection. If is a nonempty set containing , it is possible to form the intersection using the axiom schema of specification as
,
so no separate axiom of intersection is necessary. (If A is the empty set, then trying to form the intersection of A as
{c: for all D in A, c is in D}
is not permitted by the axioms. Moreover, if such a set existed, then it would contain every set in the "universe", but the notion of a universal set is antithetical to Zermelo–Fraenkel set theory.)
References
{{reflist|refs=
<ref name=Zermelo1908>Ernst Zermelo, 1908, "Untersuchungen über die Grundlagen der Mengenlehre I", Mathematische Annalen 65(2), pp. 261–281.English translation: Jean van Heijenoort, 1967, From Frege to Gödel: A Source Book in Mathematical Logic, pp. 199–215 </ref>
}}
Further reading
Paul Halmos, Naive set theory. Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974. (Springer-Verlag edition).
Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. .
Axioms of set theory
de:Zermelo-Fraenkel-Mengenlehre#Die Axiom |
https://en.wikipedia.org/wiki/Partial%20differential%20equation | In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similar to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniquity, regularity and stability. Among the many open questions are the existence and smoothness of solutions to the Navier–Stokes equations, named as one of the Millennium Prize Problems in 2000.
Partial differential equations are ubiquitous in mathematically oriented scientific fields, such as physics and engineering. For instance, they are foundational in the modern scientific understanding of sound, heat, diffusion, electrostatics, electrodynamics, thermodynamics, fluid dynamics, elasticity, general relativity, and quantum mechanics (Schrödinger equation, Pauli equation etc.). They also arise from many purely mathematical considerations, such as differential geometry and the calculus of variations; among other notable applications, they are the fundamental tool in the proof of the Poincaré conjecture from geometric topology.
Partly due to this variety of sources, there is a wide spectrum of different types of partial differential equations, and methods have been developed for dealing with many of the individual equations which arise. As such, it is usually acknowledged that there is no "general theory" of partial differential equations, with specialist knowledge being somewhat divided between several essentially distinct subfields.
Ordinary differential equations form a subclass of partial differential equations, corresponding to functions of a single variable. Stochastic partial differential equations and nonlocal equations are, as of 2020, particularly widely studied extensions of the "PDE" notion. More classical topics, on which there is still much active research, include elliptic and parabolic partial differential equations, fluid mechanics, Boltzmann equations, and dispersive partial differential equations.
Introduction
A function of three variables is "harmonic" or "a solution of the Laplace equation" if it satisfies the condition
Such functions were widely studied in the nineteenth century due to their relevance for classical mechanics, for example the equilibrium temperature distribution of a homogeneous solid is a harmonic function. If |
https://en.wikipedia.org/wiki/Partial%20derivative | In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.
The partial derivative of a function with respect to the variable is variously denoted by
It can be thought of as the rate of change of the function in the -direction.
Sometimes, for the partial derivative of with respect to is denoted as Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in:
The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differences. The modern partial derivative notation was created by Adrien-Marie Legendre (1786), although he later abandoned it; Carl Gustav Jacob Jacobi reintroduced the symbol in 1841.
Definition
Like ordinary derivatives, the partial derivative is defined as a limit. Let be an open subset of and a function. The partial derivative of at the point with respect to the -th variable is defined as
Even if all partial derivatives exist at a given point , the function need not be continuous there. However, if all partial derivatives exist in a neighborhood of and are continuous there, then is totally differentiable in that neighborhood and the total derivative is continuous. In this case, it is said that is a function. This can be used to generalize for vector valued functions, by carefully using a componentwise argument.
The partial derivative can be seen as another function defined on and can again be partially differentiated. If the direction of derivative is repeated, it is called a mixed partial derivative. If all mixed second order partial derivatives are continuous at a point (or on a set), is termed a function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem:
Notation
For the following examples, let be a function in , , and .
First-order partial derivatives:
Second-order partial derivatives:
Second-order mixed derivatives:
Higher-order partial and mixed derivatives:
When dealing with functions of multiple variables, some of these variables may be related to each other, thus it may be necessary to specify explicitly which variables are being held constant to avoid ambiguity. In fields such as statistical mechanics, the partial derivative of with respect to , holding and constant, is often expressed as
Conventionally, for clarity and simplicity of notation, the partial derivative function and the value of the function at a specific point are conflated by including the function arguments when the partial derivative symbol (Leibniz notation) is used. Thus, an ex |
https://en.wikipedia.org/wiki/Alice%27s%20Adventures%20in%20Wonderland | Alice's Adventures in Wonderland (commonly Alice in Wonderland) is an 1865 English children's novel by Lewis Carroll, a mathematics don at Oxford University. It details the story of a young girl named Alice who falls through a rabbit hole into a fantasy world of anthropomorphic creatures. It is seen as an example of the literary nonsense genre. The artist John Tenniel provided 42 wood-engraved illustrations for the book.
It received positive reviews upon release and is now one of the best-known works of Victorian literature; its narrative, structure, characters and imagery have had a widespread influence on popular culture and literature, especially in the fantasy genre. It is credited as helping end an era of didacticism in children's literature, inaugurating an era in which writing for children aimed to "delight or entertain". The tale plays with logic, giving the story lasting popularity with adults as well as with children. The titular character Alice shares her name with Alice Liddell, a girl Carroll knew.
The book has never been out of print and has been translated into 174 languages. Its legacy covers adaptations for screen, radio, art, ballet, opera, musicals, theme parks, board games and video games. Carroll published a sequel in 1871 entitled Through the Looking-Glass and a shortened version for young children, The Nursery "Alice" in 1890.
Background
"All in the golden afternoon..."
Alice's Adventures in Wonderland was inspired when, on 4 July 1862, Lewis Carroll and Reverend Robinson Duckworth rowed up The Isis with the three young daughters of Carroll's friend Henry Liddell: Lorina Charlotte (aged 13; "Prima" in the book's prefatory verse); Alice Pleasance (aged 10; "Secunda" in the verse); and Edith Mary (aged 8; "Tertia" in the verse).
The journey began at Folly Bridge, Oxford, and ended away in Godstow, Oxfordshire. During the trip Carroll told the girls a story that he described in his diary as "Alice's Adventures Under Ground" and which his journal says he "undertook to write out for Alice". Alice Liddell recalled that she asked Carroll to write it down: unlike other stories he had told her, this one she wanted to preserve. She finally received the manuscript more than two years later.
4 July was known as the "golden afternoon", prefaced in the novel as a poem. In fact, the weather around Oxford on 4 July was "cool and rather wet," although at least one scholar has disputed this claim. Scholars debate whether Carroll in fact came up with Alice during the "golden afternoon" or whether the story was developed over a longer period.
Carroll had known the Liddell children since around March 1856, when he befriended Harry Liddell. He met Lorina by early March as well. In June 1856, he took the children out on the river. Robert Douglas-Fairhurst, who wrote a literary biography of Carroll, suggests that Carroll favoured Alice Pleasance Liddell in particular because her name was ripe for allusion. "Pleasance" means pleasure and |
https://en.wikipedia.org/wiki/Stone%E2%80%93%C4%8Cech%20compactification | In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX. The Stone–Čech compactification βX of a topological space X is the largest, most general compact Hausdorff space "generated" by X, in the sense that any continuous map from X to a compact Hausdorff space factors through βX (in a unique way). If X is a Tychonoff space then the map from X to its image in βX is a homeomorphism, so X can be thought of as a (dense) subspace of βX; every other compact Hausdorff space that densely contains X is a quotient of βX. For general topological spaces X, the map from X to βX need not be injective.
A form of the axiom of choice is required to prove that every topological space has a Stone–Čech compactification. Even for quite simple spaces X, an accessible concrete description of βX often remains elusive. In particular, proofs that βX \ X is nonempty do not give an explicit description of any particular point in βX \ X.
The Stone–Čech compactification occurs implicitly in a paper by and was given explicitly by and .
History
Andrey Nikolayevich Tikhonov introduced completely regular spaces in 1930 in order to avoid the pathological situation of Hausdorff spaces whose only continuous real-valued functions are constant maps.
In the same 1930 article where Tychonoff defined completely regular spaces, he also proved that every Tychonoff space (i.e. Hausdorff completely regular space) has a Hausdorff compactification (in this same article, he also proved Tychonoff's theorem).
In 1937, Čech extended Tychonoff's technique and introduced the notation βX for this compactification.
Stone also constructed βX in a 1937 article, although using a very different method.
Despite Tychonoff's article being the first work on the subject of the Stone–Čech compactification and despite Tychonoff's article being referenced by both Stone and Čech, Tychonoff's name is rarely associated with βX.
Universal property and functoriality
The Stone–Čech compactification of the topological space X is a compact Hausdorff space βX together with a continuous map iX : X → βX that has the following universal property: any continuous map f : X → K, where K is a compact Hausdorff space, extends uniquely to a continuous map βf : βX → K, i.e. (βf)iX = f .
As is usual for universal properties, this universal property characterizes βX up to homeomorphism.
As is outlined in , below, one can prove (using the axiom of choice) that such a Stone–Čech compactification iX : X → βX exists for every topological space X. Furthermore, the image iX(X) is dense in βX.
Some authors add the assumption that the starting space X be Tychonoff (or even locally compact Hausdorff), for the following reasons:
The map from X to its image in βX is a homeomorphism if and only if X is Tychonoff.
The map from X to its image in βX is a homeomo |
https://en.wikipedia.org/wiki/East%20of%20England | The East of England is one of the nine official regions of England in the United Kingdom. This region was created in 1994 and was adopted for statistics purposes from 1999. It includes the ceremonial counties of Bedfordshire, Cambridgeshire, Essex, Hertfordshire, Norfolk and Suffolk. Essex has the highest population in the region.
The population of the East of England region in 2018 was 6.24 million. Bedford, Luton, Basildon, Peterborough, Southend-on-Sea, Norwich, Ipswich, Colchester, Chelmsford and Cambridge are the region's most populous settlements. The southern part of the region lies in the London commuter belt.
Geography
The East of England region has the lowest elevation range in the UK. Twenty percent of the region is below mean sea level, most of this in North Cambridgeshire, Norfolk and on the Essex Coast. Most of the remaining area is of low elevation, with extensive glacial deposits. The Fens, a large area of reclaimed marshland, are mostly in North Cambridgeshire. The Fens include the lowest point in the country in the village of Holme: 2.75 metres (9.0 ft) below mean sea level. This area formerly included the body of open water known as Whittlesey Mere. The highest point in the region is at Clipper Down at 817 ft (249 m) above mean sea level, in the far southwestern corner of the region in the Ivinghoe Hills.
Communities known as New Towns, responses to urban congestion and World War II destruction, appeared in Basildon and Harlow (Essex), as well as in Stevenage and Hemel Hempstead (Hertfordshire), in the 1950s and 1960s. In the late 1960s, the Roskill Commission considered Cublington in Buckinghamshire, Thurleigh in Bedfordshire, Nuthampstead in Hertfordshire and Foulness in Essex as locations for a possible third airport for London. A new airport was not built, but a former Royal Air Force base at Stansted, which had previously been converted to civilian use redeveloped and expanded in the following decades.
Historical use
The East of England succeeded the standard statistical region East Anglia (which excluded Essex, Hertfordshire and Bedfordshire, then in the South East). The East of England civil defence region was identical to today's region.
East Anglia with Home Counties
Essex, despite meaning East-Saxons, previously formed part of South East England, along with Bedfordshire and Hertfordshire, a mixture of definite and debatable Home counties. The earliest use of the term is from 1695. Charles Davenant, in An essay upon ways and means of supplying the war, wrote, "The Eleven Home Counties, which are thought in Land Taxes to pay more than their proportion..." then cited a list including these four. The term does not appear to have been used in taxation since the 18th century.
Historic counties
The historic counties ceased to be used for any administrative purpose in 1899 but remain important to some people, notably for county cricket.
Climate
East Anglia is one of the driest parts of the United Kingdom, with averag |
https://en.wikipedia.org/wiki/Pareto%20distribution | The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto, is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actuarial, and many other types of observable phenomena; the principle originally applied to describing the distribution of wealth in a society, fitting the trend that a large portion of wealth is held by a small fraction of the population. The Pareto principle or "80-20 rule" stating that 80% of outcomes are due to 20% of causes was named in honour of Pareto, but the concepts are distinct, and only Pareto distributions with shape value () of log45 ≈ 1.16 precisely reflect it. Empirical observation has shown that this 80-20 distribution fits a wide range of cases, including natural phenomena and human activities.
Definitions
If X is a random variable with a Pareto (Type I) distribution, then the probability that X is greater than some number x, i.e., the survival function (also called tail function), is given by
where xm is the (necessarily positive) minimum possible value of X, and α is a positive parameter. The type I Pareto distribution is characterized by a scale parameter xm and a shape parameter α, which is known as the tail index. If this distribution is used to model the distribution of wealth, then the parameter α is called the Pareto index.
Cumulative distribution function
From the definition, the cumulative distribution function of a Pareto random variable with parameters α and xm is
Probability density function
It follows (by differentiation) that the probability density function is
When plotted on linear axes, the distribution assumes the familiar J-shaped curve which approaches each of the orthogonal axes asymptotically. All segments of the curve are self-similar (subject to appropriate scaling factors). When plotted in a log-log plot, the distribution is represented by a straight line.
Properties
Moments and characteristic function
The expected value of a random variable following a Pareto distribution is
The variance of a random variable following a Pareto distribution is
(If α ≤ 1, the variance does not exist.)
The raw moments are
The moment generating function is only defined for non-positive values t ≤ 0 as
Thus, since the expectation does not converge on an open interval containing we say that the moment generating function does not exist.
The characteristic function is given by
where Γ(a, x) is the incomplete gamma function.
The parameters may be solved for using the method of moments.
Conditional distributions
The conditional probability distribution of a Pareto-distributed random variable, given the event that it is greater than or equal to a particular number exceeding , is a Pareto distribution with the same Pareto index but with minimum instead of . This implies that the conditional expected value (if it is finite, i.e. ) is proportional to . In case of random |
https://en.wikipedia.org/wiki/Regular%20space | In topology and related fields of mathematics, a topological space X is called a regular space if every closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. Thus p and C can be separated by neighborhoods. This condition is known as Axiom T3. The term "T3 space" usually means "a regular Hausdorff space". These conditions are examples of separation axioms.
Definitions
A topological space X is a regular space if, given any closed set F and any point x that does not belong to F, there exists a neighbourhood U of x and a neighbourhood V of F that are disjoint. Concisely put, it must be possible to separate x and F with disjoint neighborhoods.
A or is a topological space that is both regular and a Hausdorff space. (A Hausdorff space or T2 space is a topological space in which any two distinct points are separated by neighbourhoods.) It turns out that a space is T3 if and only if it is both regular and T0. (A T0 or Kolmogorov space is a topological space in which any two distinct points are topologically distinguishable, i.e., for every pair of distinct points, at least one of them has an open neighborhood not containing the other.) Indeed, if a space is Hausdorff then it is T0, and each T0 regular space is Hausdorff: given two distinct points, at least one of them misses the closure of the other one, so (by regularity) there exist disjoint neighborhoods separating one point from (the closure of) the other.
Although the definitions presented here for "regular" and "T3" are not uncommon, there is significant variation in the literature: some authors switch the definitions of "regular" and "T3" as they are used here, or use both terms interchangeably. This article uses the term "regular" freely, but will usually say "regular Hausdorff", which is unambiguous, instead of the less precise "T3". For more on this issue, see History of the separation axioms.
A is a topological space where every point has an open neighbourhood that is regular. Every regular space is locally regular, but the converse is not true. A classical example of a locally regular space that is not regular is the bug-eyed line.
Relationships to other separation axioms
A regular space is necessarily also preregular, i.e., any two topologically distinguishable points can be separated by neighbourhoods.
Since a Hausdorff space is the same as a preregular T0 space, a regular space which is also T0 must be Hausdorff (and thus T3).
In fact, a regular Hausdorff space satisfies the slightly stronger condition T2½.
(However, such a space need not be completely Hausdorff.)
Thus, the definition of T3 may cite T0, T1, or T2½ instead of T2 (Hausdorffness); all are equivalent in the context of regular spaces.
Speaking more theoretically, the conditions of regularity and T3-ness are related by Kolmogorov quotients.
A space is regular if and only if its Kolmogorov quotient is T3; and, as mentioned, a space is T3 if and only if it's both regular and T0.
|
https://en.wikipedia.org/wiki/Scalar | Scalar may refer to:
Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers
Scalar (physics), a physical quantity that can be described by a single element of a number field such as a real number
Lorentz scalar, a quantity in the theory of relativity which is invariant under a Lorentz transformation
Pseudoscalar, a quantity that behaves like a scalar, except that it changes sign under a parity inversion
Scalar (computing), any non-composite value
Scalar boson, in physics, a boson subatomic particle whose spin equals zero
See also
dot product, also known as scalar product
dimensionless quantity, also known as scalar quantity
Inner product space
Scalar field
Scale (music)
Scaler (disambiguation)
Pterophyllum scalare (Lichtenstein, 1823), a species of freshwater angelfish
Scala (disambiguation) |
https://en.wikipedia.org/wiki/Convolution%20theorem | In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Other versions of the convolution theorem are applicable to various Fourier-related transforms.
Functions of a continuous variable
Consider two functions and with Fourier transforms and :
where denotes the Fourier transform operator. The transform may be normalized in other ways, in which case constant scaling factors (typically or ) will appear in the convolution theorem below. The convolution of and is defined by:
In this context the asterisk denotes convolution, instead of standard multiplication. The tensor product symbol is sometimes used instead.
The convolution theorem states that:
Applying the inverse Fourier transform , produces the corollary:
The theorem also generally applies to multi-dimensional functions.
This theorem also holds for the Laplace transform, the two-sided Laplace transform and, when suitably modified, for the Mellin transform and Hartley transform (see Mellin inversion theorem). It can be extended to the Fourier transform of abstract harmonic analysis defined over locally compact abelian groups.
Periodic convolution (Fourier series coefficients)
Consider -periodic functions and which can be expressed as periodic summations:
and
In practice the non-zero portion of components and are often limited to duration but nothing in the theorem requires that. The Fourier series coefficients are:
where denotes the Fourier series integral.
The pointwise product: is also -periodic, and its Fourier series coefficients are given by the discrete convolution of the and sequences:
The convolution: is also -periodic, and is called a periodic convolution. The corresponding convolution theorem is:
Functions of a discrete variable (sequences)
By a derivation similar to Eq.1, there is an analogous theorem for sequences, such as samples of two continuous functions, where now denotes the discrete-time Fourier transform (DTFT) operator. Consider two sequences and with transforms and :
The of and is defined by:
The convolution theorem for discrete sequences is:
Periodic convolution
and as defined above, are periodic, with a period of 1. Consider -periodic sequences and :
and
These functions occur as the result of sampling and at intervals of and performing an inverse discrete Fourier transform (DFT) on samples (see ). The discrete convolution:
is also -periodic, and is called a periodic convolution. Redefining the operator as the -length DFT, the corresponding theorem is:
And therefore:
Under the right conditions, it is possible for this N-length sequence to contain a distortion-free segment of a convolution. But when the non-zero por |
https://en.wikipedia.org/wiki/Tangloids | Tangloids is a mathematical game for two players created by Piet Hein to model the calculus of spinors.
A description of the game appeared in the book "Martin Gardner's New Mathematical Diversions from Scientific American" by Martin Gardner from 1996 in a section on the mathematics of braiding.
Two flat blocks of wood each pierced with three small holes are joined with three parallel strings. Each player holds one of the blocks of wood. The first player holds one block of wood still, while the other player rotates the other block of wood for two full revolutions. The plane of rotation is perpendicular to the strings when not tangled. The strings now overlap each other. Then the first player tries to untangle the strings without rotating either piece of wood. Only translations (moving the pieces without rotating) are allowed. Afterwards, the players reverse roles; whoever can untangle the strings fastest is the winner. Try it with only one revolution. The strings are of course overlapping again but they can not be untangled without rotating one of the two wooden blocks.
The Balinese cup trick, appearing in the Balinese candle dance, is a different illustration of the same mathematical idea. The anti-twister mechanism is a device intended to avoid such orientation entanglements. A mathematical interpretation of these ideas can be found in the article on quaternions and spatial rotation.
Mathematical articulation
This game serves to clarify the notion that rotations in space have properties that cannot be intuitively explained by considering only the rotation of a single rigid object in space. The rotation of vectors does not encompass all of the properties of the abstract model of rotations given by the rotation group. The property being illustrated in this game is formally referred to in mathematics as the "double covering of SO(3) by SU(2)". This abstract concept can be roughly sketched as follows.
Rotations in three dimensions can be expressed as 3x3 matrices, a block of numbers, one each for x,y,z. If one considers arbitrarily tiny rotations, one is led to the conclusion that rotations form a space, in that if each rotation is thought of as a point, then there are always other nearby points, other nearby rotations that differ by only a small amount. In small neighborhoods, this collection of nearby points resembles Euclidean space. In fact, it resembles three-dimensional Euclidean space, as there are three different possible directions for infinitesimal rotations: x, y and z. This properly describes the structure of the rotation group in small neighborhoods. For sequences of large rotations, however, this model breaks down; for example, turning right and then lying down is not the same as lying down first and then turning right. Although the rotation group has the structure of 3D space on the small scale, that is not its structure on the large scale. Systems that behave like Euclidean space on the small scale, but possibly have a more |
https://en.wikipedia.org/wiki/Euler%27s%20totient%20function | In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as or , and may also be called Euler's phi function. In other words, it is the number of integers in the range for which the greatest common divisor is equal to 1. The integers of this form are sometimes referred to as totatives of .
For example, the totatives of are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since and . Therefore, . As another example, since for the only integer in the range from 1 to is 1 itself, and .
Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then .
This function gives the order of the multiplicative group of integers modulo (the group of units of the ring ). It is also used for defining the RSA encryption system.
History, terminology, and notation
Leonhard Euler introduced the function in 1763. However, he did not at that time choose any specific symbol to denote it. In a 1784 publication, Euler studied the function further, choosing the Greek letter to denote it: he wrote for "the multitude of numbers less than , and which have no common divisor with it". This definition varies from the current definition for the totient function at but is otherwise the same. The now-standard notation comes from Gauss's 1801 treatise Disquisitiones Arithmeticae, although Gauss did not use parentheses around the argument and wrote . Thus, it is often called Euler's phi function or simply the phi function.
In 1879, J. J. Sylvester coined the term totient for this function, so it is also referred to as Euler's totient function, the Euler totient, or Euler's totient. Jordan's totient is a generalization of Euler's.
The cototient of is defined as . It counts the number of positive integers less than or equal to that have at least one prime factor in common with .
Computing Euler's totient function
There are several formulae for computing .
Euler's product formula
It states
where the product is over the distinct prime numbers dividing . (For notation, see Arithmetical function.)
An equivalent formulation is
where is the prime factorization of (that is, are distinct prime numbers).
The proof of these formulae depends on two important facts.
Phi is a multiplicative function
This means that if , then . Proof outline: Let , , be the sets of positive integers which are coprime to and less than , , , respectively, so that , etc. Then there is a bijection between and by the Chinese remainder theorem.
Value of phi for a prime power argument
If is prime and , then
Proof: Since is a prime number, the only possible values of are , and the only way to have is if is a multiple of , that is, , and there are such multiples not greater than . Therefore, the other numbers are all relative |
https://en.wikipedia.org/wiki/Fermat%27s%20little%20theorem | In number theory, Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . In the notation of modular arithmetic, this is expressed as
For example, if and , then , and is an integer multiple of .
If is not divisible by ; that is, if is coprime to , then Fermat's little theorem is equivalent to the statement that is an integer multiple of , or in symbols:
For example, if and , then , and is thus a multiple of .
Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory. The theorem is named after Pierre de Fermat, who stated it in 1640. It is called the "little theorem" to distinguish it from Fermat's Last Theorem.
History
Pierre de Fermat first stated the theorem in a letter dated October 18, 1640, to his friend and confidant Frénicle de Bessy. His formulation is equivalent to the following:
If is a prime and is any integer not divisible by , then is divisible by .
Fermat's original statement was
This may be translated, with explanations and formulas added in brackets for easier understanding, as:
Every prime number [] divides necessarily one of the powers minus one of any [geometric] progression [] [that is, there exists such that divides ], and the exponent of this power [] divides the given prime minus one [divides ]. After one has found the first power [] that satisfies the question, all those whose exponents are multiples of the exponent of the first one satisfy similarly the question [that is, all multiples of the first have the same property].
Fermat did not consider the case where is a multiple of nor prove his assertion, only stating:
(And this proposition is generally true for all series [sic] and for all prime numbers; I would send you a demonstration of it, if I did not fear going on for too long.)
Euler provided the first published proof in 1736, in a paper titled "Theorematum Quorundam ad Numeros Primos Spectantium Demonstratio" in the Proceedings of the St. Petersburg Academy, but Leibniz had given virtually the same proof in an unpublished manuscript from sometime before 1683.
The term "Fermat's little theorem" was probably first used in print in 1913 in Zahlentheorie by Kurt Hensel:
(There is a fundamental theorem holding in every finite group, usually called Fermat's little theorem because Fermat was the first to have proved a very special part of it.)
An early use in English occurs in A.A. Albert's Modern Higher Algebra (1937), which refers to "the so-called 'little' Fermat theorem" on page 206.
Further history
Some mathematicians independently made the related hypothesis (sometimes incorrectly called the Chinese hypothesis) that if and only if is prime. Indeed, the "if" part is true, and it is a special case of Fermat's little theorem. However, the "only if" part is false: For example, , but 341 = 11 × 31 is a pseudoprime to base 2. See below.
Proof |
https://en.wikipedia.org/wiki/Minkowski%27s%20theorem | In mathematics, Minkowski's theorem is the statement that every convex set in which is symmetric with respect to the origin and which has volume greater than contains a non-zero integer point (meaning a point in that is not the origin). The theorem was proved by Hermann Minkowski in 1889 and became the foundation of the branch of number theory called the geometry of numbers. It can be extended from the integers to any lattice and to any symmetric convex set with volume greater than , where denotes the covolume of the lattice (the absolute value of the determinant of any of its bases).
Formulation
Suppose that is a lattice of determinant in the -dimensional real vector space and is a convex subset of that is symmetric with respect to the origin, meaning that if is in then is also in . Minkowski's theorem states that if the volume of is strictly greater than , then must contain at least one lattice point other than the origin. (Since the set is symmetric, it would then contain at least three lattice points: the origin 0 and a pair of points , where .)
Example
The simplest example of a lattice is the integer lattice of all points with integer coefficients; its determinant is 1. For , the theorem claims that a convex figure in the Euclidean plane symmetric about the origin and with area greater than 4 encloses at least one lattice point in addition to the origin. The area bound is sharp: if is the interior of the square with vertices then is symmetric and convex, and has area 4, but the only lattice point it contains is the origin. This example, showing that the bound of the theorem is sharp, generalizes to hypercubes in every dimension .
Proof
The following argument proves Minkowski's theorem for the specific case of .
Proof of the case: Consider the map
Intuitively, this map cuts the plane into 2 by 2 squares, then stacks the squares on top of each other. Clearly has area less than or equal to 4, because this set lies within a 2 by 2 square. Assume for a contradiction that could be injective, which means the pieces of cut out by the squares stack up in a non-overlapping way. Because is locally area-preserving, this non-overlapping property would make it area-preserving for all of , so the area of would be the same as that of , which is greater than 4. That is not the case, so the assumption must be false: is not injective, meaning that there exist at least two distinct points in that are mapped by to the same point: .
Because of the way was defined, the only way that can equal is for
to equal for some integers and , not both zero.
That is, the coordinates of the two points differ by two even integers.
Since is symmetric about the origin, is also a point in . Since is convex, the line segment between and lies entirely in , and in particular the midpoint of that segment lies in . In other words,
is a point in . But this point is an integer point, and is not the origin since and are not both zero.
Th |
https://en.wikipedia.org/wiki/Group%20object | In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets. A typical example of a group object is a topological group, a group whose underlying set is a topological space such that the group operations are continuous.
Definition
Formally, we start with a category C with finite products (i.e. C has a terminal object 1 and any two objects of C have a product). A group object in C is an object G of C together with morphisms
m : G × G → G (thought of as the "group multiplication")
e : 1 → G (thought of as the "inclusion of the identity element")
inv : G → G (thought of as the "inversion operation")
such that the following properties (modeled on the group axioms – more precisely, on the definition of a group used in universal algebra) are satisfied
m is associative, i.e. m (m × idG) = m (idG × m) as morphisms G × G × G → G, and where e.g. m × idG : G × G × G → G × G; here we identify G × (G × G) in a canonical manner with (G × G) × G.
e is a two-sided unit of m, i.e. m (idG × e) = p1, where p1 : G × 1 → G is the canonical projection, and m (e × idG) = p2, where p2 : 1 × G → G is the canonical projection
inv is a two-sided inverse for m, i.e. if d : G → G × G is the diagonal map, and eG : G → G is the composition of the unique morphism G → 1 (also called the counit) with e, then m (idG × inv) d = eG and m (inv × idG) d = eG.
Note that this is stated in terms of maps – product and inverse must be maps in the category – and without any reference to underlying "elements" of the group object – categories in general do not have elements of their objects.
Another way to state the above is to say G is a group object in a category C if for every object X in C, there is a group structure on the morphisms Hom(X, G) from X to G such that the association of X to Hom(X, G) is a (contravariant) functor from C to the category of groups.
Examples
Each set G for which a group structure (G, m, u, −1) can be defined can be considered a group object in the category of sets. The map m is the group operation, the map e (whose domain is a singleton) picks out the identity element u of G, and the map inv assigns to every group element its inverse. eG : G → G is the map that sends every element of G to the identity element.
A topological group is a group object in the category of topological spaces with continuous functions.
A Lie group is a group object in the category of smooth manifolds with smooth maps.
A Lie supergroup is a group object in the category of supermanifolds.
An algebraic group is a group object in the category of algebraic varieties. In modern algebraic geometry, one considers the more general group schemes, group objects in the category of schemes.
A localic group is a group object in the category of locales.
The group objects in the category of groups (or monoids) are the abelian groups. The reason for this is that, if inv is assumed to be a hom |
https://en.wikipedia.org/wiki/Physical%20Quality%20of%20Life%20Index | The Physical Quality of Life Index (PQLI) is an attempt to measure the quality of life or well-being of a country. The value is the average of three statistics: basic literacy rate at the age of 15 years , infant mortality, and life expectancy at age one, all equally weighted on a 1 to 100 scale.
It was developed for the Overseas Development Council in the mid-1970s by M.D Morris, as one of a number of measures created due to dissatisfaction with the use of GNP as an indicator of development. He thought that they would cover a wide range of indicators like health, sanitation, drinking water, nutrition, education etc. PQLI might be regarded as an improvement but shares the general problems of measuring quality of life in a quantitative way. It has also been criticized because there is a considerable overlap between infant mortality and life expectancy.
The UN Human Development Index is a more widely used means of measuring well-being.
Steps to Calculate Physical Quality of Life:
1) Find percentage of the population that is literate (literacy rate).
2) Find the infant mortality rate. (out of 1000 births)
INDEXED Infant Mortality Rate = (166 - infant mortality) × 0.625
3) Find the Life Expectancy.
INDEXED Life Expectancy = (Life expectancy - 42) × 2.7
4) Physical Quality of Life =
(Literacy Rate + INDEXED Infant Mortality Rate + INDEXED Life Expectancy)
_
3
- ABOUT PHYSICAL QUALITY OF LIFE INDEX= PQLI : Increase in national income and per capita income are not the real indicators of economic development, as it has a number of limitations. Increasing incomes of the country are concentrated in the hands of a few people, which is not development. The development of a country should be such that the living standards of the poor rises, and the basic requirements of the citizens are fulfilled. Keeping this in mind, Morris Davis Morris presented the physical quality of life index, in short known as the PQLI. In this index, betterment of physical quality of life of human beings is considered economic development. The level of physical quality of life determines the level of economic development. If any country's physical quality of life is higher than that of the other country, then that country is considered as more developed. There are three standards to measure the physical quality, which are depicted here:
1)- Extent of Education,
2)- Life Expectancy &
3)- Infant Mortality Rate
See also
Basic Well-being Index (BWI)
Human Poverty Index
Quality-of-life Index, a different index calculated in 2005
Quality of well-being scale
Gross National Happiness
Bhutan GNH Index
Happiness economics
References
Quality of life |
https://en.wikipedia.org/wiki/Division%20%28mathematics%29 | Division is one of the four basic operations of arithmetic. The other operations are addition, subtraction, and multiplication. What is being divided is called the dividend, which is divided by the divisor, and the result is called the quotient.
At an elementary level the division of two natural numbers is, among other possible interpretations, the process of calculating the number of times one number is contained within another. For example, if 20 apples are divided evenly between 4 people, everyone receives 5 apples (see picture). However, this number of times or the number contained (divisor) need not be integers.
The division with remainder or Euclidean division of two natural numbers provides an integer quotient, which is the number of times the second number is completely contained in the first number, and a remainder, which is the part of the first number that remains, when in the course of computing the quotient, no further full chunk of the size of the second number can be allocated. For example, if 21 apples are divided between 4 people, everyone receives 5 apples again, and 1 apple remains.
For division to always yield one number rather than an integer quotient plus a remainder, the natural numbers must be extended to rational numbers or real numbers. In these enlarged number systems, division is the inverse operation to multiplication, that is means , as long as is not zero. If , then this is a division by zero, which is not defined. In the 21-apples example, everyone would receive 5 apple and a quarter of an apple, thus avoiding any leftover.
Both forms of division appear in various algebraic structures, different ways of defining mathematical structure. Those in which a Euclidean division (with remainder) is defined are called Euclidean domains and include polynomial rings in one indeterminate (which define multiplication and addition over single-variabled formulas). Those in which a division (with a single result) by all nonzero elements is defined are called fields and division rings. In a ring the elements by which division is always possible are called the units (for example, 1 and −1 in the ring of integers). Another generalization of division to algebraic structures is the quotient group, in which the result of "division" is a group rather than a number.
Introduction
The simplest way of viewing division is in terms of quotition and partition: from the quotition perspective, means the number of 5s that must be added to get 20. In terms of partition, means the size of each of 5 parts into which a set of size 20 is divided. For example, 20 apples divide into five groups of four apples, meaning that "twenty divided by five is equal to four". This is denoted as , or . In the example, 20 is the dividend, 5 is the divisor, and 4 is the quotient.
Unlike the other basic operations, when dividing natural numbers there is sometimes a remainder that will not go evenly into the dividend; for example, leaves a remainder of 1, a |
https://en.wikipedia.org/wiki/Symmetry | Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations, such as translation, reflection, rotation, or scaling. Although these two meanings of the word can sometimes be told apart, they are intricately related, and hence are discussed together in this article.
Mathematical symmetry may be observed with respect to the passage of time; as a spatial relationship; through geometric transformations; through other kinds of functional transformations; and as an aspect of abstract objects, including theoretic models, language, and music.
This article describes symmetry from three perspectives: in mathematics, including geometry, the most familiar type of symmetry for many people; in science and nature; and in the arts, covering architecture, art, and music.
The opposite of symmetry is asymmetry, which refers to the absence of symmetry.
In mathematics
In geometry
A geometric shape or object is symmetric if it can be divided into two or more identical pieces that are arranged in an organized fashion. This means that an object is symmetric if there is a transformation that moves individual pieces of the object, but doesn't change the overall shape. The type of symmetry is determined by the way the pieces are organized, or by the type of transformation:
An object has reflectional symmetry (line or mirror symmetry) if there is a line (or in 3D a plane) going through it which divides it into two pieces that are mirror images of each other.
An object has rotational symmetry if the object can be rotated about a fixed point (or in 3D about a line) without changing the overall shape.
An object has translational symmetry if it can be translated (moving every point of the object by the same distance) without changing its overall shape.
An object has helical symmetry if it can be simultaneously translated and rotated in three-dimensional space along a line known as a screw axis.
An object has scale symmetry if it does not change shape when it is expanded or contracted. Fractals also exhibit a form of scale symmetry, where smaller portions of the fractal are similar in shape to larger portions.
Other symmetries include glide reflection symmetry (a reflection followed by a translation) and rotoreflection symmetry (a combination of a rotation and a reflection).
In logic
A dyadic relation R = S × S is symmetric if for all elements a, b in S, whenever it is true that Rab, it is also true that Rba. Thus, the relation "is the same age as" is symmetric, for if Paul is the same age as Mary, then Mary is the same age as Paul.
In propositional logic, symmetric binary logical connectives include and (∧, or &), or (∨, or |) and if and only if (↔), while the connective if (→) is not symmetric. Other symmetric logical connectives include nand (not-and, or ⊼), xor (not-biconditional, or ⊻), |
https://en.wikipedia.org/wiki/Category%20%28mathematics%29 | In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose objects are sets and whose arrows are functions.
Category theory is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent. Virtually every branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. As such, category theory provides an alternative foundation for mathematics to set theory and other proposed axiomatic foundations. In general, the objects and arrows may be abstract entities of any kind, and the notion of category provides a fundamental and abstract way to describe mathematical entities and their relationships.
In addition to formalizing mathematics, category theory is also used to formalize many other systems in computer science, such as the semantics of programming languages.
Two categories are the same if they have the same collection of objects, the same collection of arrows, and the same associative method of composing any pair of arrows. Two different categories may also be considered "equivalent" for purposes of category theory, even if they do not have precisely the same structure.
Well-known categories are denoted by a short capitalized word or abbreviation in bold or italics: examples include Set, the category of sets and set functions; Ring, the category of rings and ring homomorphisms; and Top, the category of topological spaces and continuous maps. All of the preceding categories have the identity map as identity arrows and composition as the associative operation on arrows.
The classic and still much used text on category theory is Categories for the Working Mathematician by Saunders Mac Lane. Other references are given in the References below. The basic definitions in this article are contained within the first few chapters of any of these books.
Any monoid can be understood as a special sort of category (with a single object whose self-morphisms are represented by the elements of the monoid), and so can any preorder.
Definition
There are many equivalent definitions of a category. One commonly used definition is as follows. A category C consists of
a class ob(C) of objects,
a class mor(C) of morphisms or arrows,
a domain or source class function dom: mor(C) → ob(C),
a codomain or target class function cod: mor(C) → ob(C),
for every three objects a, b and c, a binary operation hom(a, b) × hom(b, c) → hom(a, c) called composition of morphisms. Here hom(a, b) denotes the subclass of morphisms f in mor(C) such that dom(f) = a and cod(f) = b. Morphi |
https://en.wikipedia.org/wiki/Euclidean%20distance | In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occasionally being called the Pythagorean distance. These names come from the ancient Greek mathematicians Euclid and Pythagoras, although Euclid did not represent distances as numbers, and the connection from the Pythagorean theorem to distance calculation was not made until the 18th century.
The distance between two objects that are not points is usually defined to be the smallest distance among pairs of points from the two objects. Formulas are known for computing distances between different types of objects, such as the distance from a point to a line. In advanced mathematics, the concept of distance has been generalized to abstract metric spaces, and other distances than Euclidean have been studied. In some applications in statistics and optimization, the square of the Euclidean distance is used instead of the distance itself.
Distance formulas
One dimension
The distance between any two points on the real line is the absolute value of the numerical difference of their coordinates, their absolute difference. Thus if and are two points on the real line, then the distance between them is given by:
A more complicated formula, giving the same value, but generalizing more readily to higher dimensions, is:
In this formula, squaring and then taking the square root leaves any positive number unchanged, but replaces any negative number by its absolute value.
Two dimensions
In the Euclidean plane, let point have Cartesian coordinates and let point have coordinates . Then the distance between and is given by:
This can be seen by applying the Pythagorean theorem to a right triangle with horizontal and vertical sides, having the line segment from to as its hypotenuse. The two squared formulas inside the square root give the areas of squares on the horizontal and vertical sides, and the outer square root converts the area of the square on the hypotenuse into the length of the hypotenuse.
It is also possible to compute the distance for points given by polar coordinates. If the polar coordinates of are and the polar coordinates of are , then their distance is given by the law of cosines:
When and are expressed as complex numbers in the complex plane, the same formula for one-dimensional points expressed as real numbers can be used, although here the absolute value sign indicates the complex norm:
Higher dimensions
In three dimensions, for points given by their Cartesian coordinates, the distance is
In general, for points given by Cartesian coordinates in -dimensional Euclidean space, the distance is
The Euclidean distance may also be expressed more compactly in terms of the Euclidean norm of the Euclidean vector difference:
Objects other than points
For pairs of objects that are not both poi |
https://en.wikipedia.org/wiki/Triangle%20inequality | In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but some authors, especially those writing about elementary geometry, will exclude this possibility, thus leaving out the possibility of equality. If , , and are the lengths of the sides of the triangle, with no side being greater than , then the triangle inequality states that
with equality only in the degenerate case of a triangle with zero area.
In Euclidean geometry and some other geometries, the triangle inequality is a theorem about distances, and it is written using vectors and vector lengths (norms):
where the length of the third side has been replaced by the vector sum . When and are real numbers, they can be viewed as vectors in , and the triangle inequality expresses a relationship between absolute values.
In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles, a consequence of the law of cosines, although it may be proved without these theorems. The inequality can be viewed intuitively in either or . The figure at the right shows three examples beginning with clear inequality (top) and approaching equality (bottom). In the Euclidean case, equality occurs only if the triangle has a angle and two angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.
In spherical geometry, the shortest distance between two points is an arc of a great circle, but the triangle inequality holds provided the restriction is made that the distance between two points on a sphere is the length of a minor spherical line segment (that is, one with central angle in ) with those endpoints.
The triangle inequality is a defining property of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the Lp spaces (), and inner product spaces.
Euclidean geometry
Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure. Beginning with triangle , an isosceles triangle is constructed with one side taken as and the other equal leg along the extension of side . It then is argued that angle has larger measure than angle , so side is longer than side . However:
so the sum of the lengths of sides and is larger than the length of . This proof appears in Euclid's Elements, Book 1, Proposition 20.
Mathematical expression of the constraint on the sides of a triangle
For a proper triangle, the triangle inequality, as stated in words, literally translates into three inequalities (given that a proper triangle has side lengths that |
https://en.wikipedia.org/wiki/Adjoint%20functors | In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology.
By definition, an adjunction between categories and is a pair of functors (assumed to be covariant)
and
and, for all objects in and in , a bijection between the respective morphism sets
such that this family of bijections is natural in and . Naturality here means that there are natural isomorphisms between the pair of functors and for a fixed in , and also the pair of functors and for a fixed in .
The functor is called a left adjoint functor or left adjoint to , while is called a right adjoint functor or right adjoint to . We write .
An adjunction between categories and is somewhat akin to a "weak form" of an equivalence between and , and indeed every equivalence is an adjunction. In many situations, an adjunction can be "upgraded" to an equivalence, by a suitable natural modification of the involved categories and functors.
Terminology and notation
The terms adjoint and adjunct are both used, and are cognates: one is taken directly from Latin, the other from Latin via French. In the classic text Categories for the working mathematician, Mac Lane makes a distinction between the two. Given a family
of hom-set bijections, we call an adjunction or an adjunction between and . If is an arrow in , is the right adjunct of (p. 81). The functor is left adjoint to , and is right adjoint to . (Note that may have itself a right adjoint that is quite different from ; see below for an example.)
In general, the phrases " is a left adjoint" and " has a right adjoint" are equivalent. We call a left adjoint because it is applied to the left argument of , and a right adjoint because it is applied to the right argument of .
If F is left adjoint to G, we also write
The terminology comes from the Hilbert space idea of adjoint operators , with , which is formally similar to the above relation between hom-sets. The analogy to adjoint maps of Hilbert spaces can be made precise in certain contexts.
Introduction and Motivation
Common mathematical constructions are very often adjoint functors. Consequently, general theorems about left/right adjoint functors encode the details of many useful and otherwise non-trivial results. Such general theorems include the equivalence of the various definitions of adjoint functors, the uniqueness of a right ad |
https://en.wikipedia.org/wiki/Sylow%20theorems | In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundamental part of finite group theory and have very important applications in the classification of finite simple groups.
For a prime number , a Sylow p-subgroup (sometimes p-Sylow subgroup) of a group is a maximal -subgroup of , i.e., a subgroup of that is a p-group (meaning its cardinality is a power of or equivalently, the order of every group element is a power of ) that is not a proper subgroup of any other -subgroup of . The set of all Sylow -subgroups for a given prime is sometimes written .
The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for any finite group the order (number of elements) of every subgroup of divides the order of . The Sylow theorems state that for every prime factor of the order of a finite group , there exists a Sylow -subgroup of of order , the highest power of that divides the order of . Moreover, every subgroup of order is a Sylow -subgroup of , and the Sylow -subgroups of a group (for a given prime ) are conjugate to each other. Furthermore, the number of Sylow -subgroups of a group for a given prime is congruent to 1 (mod ).
Theorems
Motivation
The Sylow theorems are a powerful statement about the structure of groups in general, but are also powerful in applications of finite group theory. This is because they give a method for using the prime decomposition of the cardinality of a finite group to give statements about the structure of its subgroups: essentially, it gives a technique to transport basic number-theoretic information about a group to its group structure. From this observation, classifying finite groups becomes a game of finding which combinations/constructions of groups of smaller order can be applied to construct a group. For example, a typical application of these theorems is in the classification of finite groups of some fixed cardinality, e.g. .
Statement
Collections of subgroups that are each maximal in one sense or another are common in group theory. The surprising result here is that in the case of , all members are actually isomorphic to each other and have the largest possible order: if with where does not divide , then every Sylow -subgroup has order . That is, is a -group and . These properties can be exploited to further analyze the structure of .
The following theorems were first proposed and proven by Ludwig Sylow in 1872, and published in Mathematische Annalen.
The following weaker version of theorem 1 was first proved by Augustin-Louis Cauchy, and is known as Cauchy's theorem.
Consequences
The Sylow theorems imply that for a prime number every Sylow -subgroup is of the same order, . Conversely, if a s |
https://en.wikipedia.org/wiki/Sophus%20Lie | Marius Sophus Lie ( ; ; 17 December 1842 – 18 February 1899) was a Norwegian mathematician. He largely created the theory of continuous symmetry and applied it to the study of geometry and differential equations. He also made substantial contributions to the development of algebra.
Life and career
Marius Sophus Lie was born on 17 December 1842 in the small town of Nordfjordeid. He was the youngest of six children born to Lutheran pastor Johann Herman Lie and his wife, who came from a well-known Trondheim family.
He had his primary education in the south-eastern coast of Moss, before attending high school in Oslo (known then as Christiania). After graduating from high school, his ambition towards a military career was dashed when the army rejected him due to poor eyesight. He then enrolled at the University of Christiania.
Sophus Lie's first mathematical work, Repräsentation der Imaginären der Plangeometrie, was published in 1869 by the Academy of Sciences in Christiania and also by Crelle's Journal. That same year he received a scholarship and travelled to Berlin, where he stayed from September to February 1870. There, he met Felix Klein and they became close friends. When he left Berlin, Lie travelled to Paris, where he was joined by Klein two months later. There, they met Camille Jordan and Gaston Darboux. But on 19 July 1870 the Franco-Prussian War began and Klein (who was Prussian) had to leave France very quickly. Lie left for Fontainebleau where he was arrested, suspected of being a German spy, garnering him fame in Norway. He was released from prison after a month, thanks to the intervention of Darboux.
Lie obtained his PhD at the University of Christiania (in present-day Oslo) in 1871 with a thesis entitled Over en Classe geometriske Transformationer (On a Class of Geometric Transformations). It would be described by Darboux as "one of the most handsome discoveries of modern Geometry". The next year, the Norwegian Parliament established an extraordinary professorship for him. That same year, Lie visited Klein, who was then at Erlangen and working on the Erlangen program.
In 1872, Lie spent eight years together with Peter Ludwig Mejdell Sylow, editing and publishing the mathematical works of their countryman, Niels Henrik Abel.
At the end of 1872, Sophus Lie proposed to Anna Birch, then eighteen years old, and they were married in 1874. The couple had three children: Marie (b. 1877), Dagny (b. 1880) and Herman (b. 1884).
From 1876, he co-edited the journal Archiv for Mathematik og Naturvidenskab, together with the physician Jacob Worm-Müller, and the biologist Georg Ossian Sars.
In 1884, Friedrich Engel arrived at Christiania to help him, with the support of Klein and Adolph Mayer (who were both professors at Leipzig by then). Engel would help Lie to write his most important treatise, Theorie der Transformationsgruppen, published in Leipzig in three volumes from 1888 to 1893. Decades later, Engel would also be one of the two edito |
https://en.wikipedia.org/wiki/Reciprocal | Reciprocal may refer to:
In mathematics
Multiplicative inverse, in mathematics, the number 1/x, which multiplied by x gives the product 1, also known as a reciprocal
Reciprocal polynomial, a polynomial obtained from another polynomial by reversing its coefficients
Reciprocal rule, a technique in calculus for calculating derivatives of reciprocal functions
Reciprocal spiral, a plane curve
Reciprocal averaging, a statistical technique for aggregating categorical data
In science and technology
Reciprocal aircraft heading, 180 degrees (the opposite direction) from a stated heading
Reciprocal lattice, a basis for the dual space of covectors, in crystallography
Reciprocal length, a measurement used in science
Reciprocating engine or piston engine
Reciprocating oscillation in physical wave theory
Life sciences and medicine
Hybrid (biology), in genetics, the result of a reciprocal pair of crossings, forming reciprocal hybrids
Reciprocal altruism, a form of symbiotic relationship in evolutionary biology
Reciprocal cross, a breeding experiment in genetics
Sherrington's law of reciprocal innervation in the theory of muscle activation
Social sciences
Reciprocal determinism, a theory in psychology
Linguistics
Reciprocal construction, a construction in which agent and patient are in a mutual relationship
In reciprocal relationships
Reciprocal license, a type of software licenses also known as copyleft
Reciprocal link between two web pages
Reciprocal Public License (RPL), a software license
See also
Reciprocation (disambiguation)
Reciprocity (disambiguation) |
https://en.wikipedia.org/wiki/Pigeonhole%20principle | In mathematics, the pigeonhole principle states that if items are put into containers, with , then at least one container must contain more than one item. For example, of three gloves (none of which is ambidextrous/reversible), at least two must be right-handed or at least two must be left-handed, because there are three objects but only two categories of handedness to put them into. This seemingly obvious statement, a type of counting argument, can be used to demonstrate possibly unexpected results. For example, given that the population of London is greater than the maximum number of hairs that can be on a human's head, the principle requires that there must be at least two people in London who have the same number of hairs on their heads.
Although the pigeonhole principle appears as early as 1624 in a book attributed to Jean Leurechon, it is commonly called Dirichlet's box principle or Dirichlet's drawer principle after an 1834 treatment of the principle by Peter Gustav Lejeune Dirichlet under the name ("drawer principle" or "shelf principle").
The principle has several generalizations and can be stated in various ways. In a more quantified version: for natural numbers and , if objects are distributed among sets, the pigeonhole principle asserts that at least one of the sets will contain at least objects. For arbitrary and , this generalizes to , where and denote the floor and ceiling functions, respectively.
Though the principle's most straightforward application is to finite sets (such as pigeons and boxes), it is also used with infinite sets that cannot be put into one-to-one correspondence. To do so requires the formal statement of the pigeonhole principle: "there does not exist an injective function whose codomain is smaller than its domain". Advanced mathematical proofs like Siegel's lemma build upon this more general concept.
Etymology
Dirichlet published his works in both French and German, using either the German or the French . The strict original meaning of these terms corresponds to the English drawer, that is, an open-topped box that can be slid in and out of the cabinet that contains it. (Dirichlet wrote about distributing pearls among drawers.) These terms morphed to pigeonhole in the sense of a small open space in a desk, cabinet, or wall for keeping letters or papers, metaphorically rooted in structures that house pigeons.
Because furniture with pigeonholes is commonly used for storing or sorting things into many categories (such as letters in a post office or room keys in a hotel), the translation pigeonhole may be a better rendering of Dirichlet's original "drawer". That understanding of the term pigeonhole, referring to some furniture features, is fading—especially among those who do not speak English natively but as a lingua franca in the scientific world—in favor of the more pictorial interpretation, literally involving pigeons and holes. The suggestive (though not misleading) interpretation of "pigeonhol |
https://en.wikipedia.org/wiki/Aleph%20%28disambiguation%29 | Aleph is the first letter of many Semitic abjads (alphabets).
Aleph may also refer to:
Science, technology and mathematics
ALEPH experiment (Apparatus for LEP Physics at CERN), detector of the Large Electron-Positron Collider
Aleph kernel, a computer operating system kernel
Aleph number, in mathematics set theory
Aleph, an advanced system for inductive logic programming
Aleph (Automated Library Expandable Program), software by Ex Libris Group
Aleph, an investigative data platform containing an archive of government records and open databases, maintained by OCCRP
Aleph (TeX), a TeX engine extension consolidating Unicode features from Omega and directional features from ε-TeX
Aleph (psychedelic), a psychoactive drug
Literature
Aleph (novel), by Brazilian author Paulo Coelho
The Aleph and Other Stories, short story collection by Argentine author Jorge Luis Borges
"The Aleph" (short story), title work of the collection
Aleph, a character in the Warren Ellis comic series Global Frequency
Aleph, a plot element in the novel Mona Lisa Overdrive by William Gibson
Aleph (), a shorthand designation for Codex Sinaiticus, a 4th-century manuscript of the Bible
Music
Aleph (band), a 1980s Italo disco band
Aleph (pianist), stage name of Fady Abi Saad, born 1980)
Aleph, a 2013 album by Gesaffelstein
"The Aleph", a song on the album Saints by Destroy the Runner
"Aleph", a song by Anahí
Organizations
Aleph Institute, a Jewish humanitarian organization for both prisoners and military personnel
ALEPH: Alliance for Jewish Renewal
Aleph Zadik Aleph, international youth-led fraternal organization for Jewish teenagers
Aleph Melbourne, an LGBT Jewish organization
Aleph, the current name of the Japanese cult and terrorist group Aum Shinrikyo
People with the surname
Patrick Aleph (born 1983), American writer and musician
Other
Aleph (film), a silent film by Wallace Berman
Aleph (journal), an academic journal on Jewish history and the history of science
Aleph One game engine for the Marathon trilogy
Aleph Sailing Team, a French America's Cup syndicate
Mount Aleph, a location in the video game Golden Sun
Aleph, the protagonist of the video game Shin Megami Tensei II
See also
Alef (disambiguation) |
https://en.wikipedia.org/wiki/Singularity%20%28mathematics%29 | In mathematics, a singularity is a point at which a given mathematical object is not defined, or a point where the mathematical object ceases to be well-behaved in some particular way, such as by lacking differentiability or analyticity.
For example, the reciprocal function has a singularity at , where the value of the function is not defined, as involving a division by zero. The absolute value function also has a singularity at , since it is not differentiable there.
The algebraic curve defined by in the coordinate system has a singularity (called a cusp) at . For singularities in algebraic geometry, see singular point of an algebraic variety. For singularities in differential geometry, see singularity theory.
Real analysis
In real analysis, singularities are either discontinuities, or discontinuities of the derivative (sometimes also discontinuities of higher order derivatives). There are four kinds of discontinuities: type I, which has two subtypes, and type II, which can also be divided into two subtypes (though usually is not).
To describe the way these two types of limits are being used, suppose that is a function of a real argument , and for any value of its argument, say , then the left-handed limit, , and the right-handed limit, , are defined by:
, constrained by and
, constrained by .
The value is the value that the function tends towards as the value approaches from below, and the value is the value that the function tends towards as the value approaches from above, regardless of the actual value the function has at the point where .
There are some functions for which these limits do not exist at all. For example, the function
does not tend towards anything as approaches . The limits in this case are not infinite, but rather undefined: there is no value that settles in on. Borrowing from complex analysis, this is sometimes called an essential singularity.
The possible cases at a given value for the argument are as follows.
A point of continuity is a value of for which , as one expects for a smooth function. All the values must be finite. If is not a point of continuity, then a discontinuity occurs at .
A type I discontinuity occurs when both and exist and are finite, but at least one of the following three conditions also applies:
;
is not defined for the case of ; or
has a defined value, which, however, does not match the value of the two limits.
Type I discontinuities can be further distinguished as being one of the following subtypes:
A jump discontinuity occurs when , regardless of whether is defined, and regardless of its value if it is defined.
A removable discontinuity occurs when , also regardless of whether is defined, and regardless of its value if it is defined (but which does not match that of the two limits).
A type II discontinuity occurs when either or does not exist (possibly both). This has two subtypes, which are usually not considered separately:
An infinite discontinuit |
https://en.wikipedia.org/wiki/Partition%20function | Partition function may refer to:
Rotational partition function
Vibrational partition function
Partition function (number theory)
Partition function (mathematics), which generalizes its use in statistical mechanics and quantum field theory:
Partition function (statistical mechanics)
Partition function (quantum field theory) |
https://en.wikipedia.org/wiki/Floor%20and%20ceiling%20functions | In mathematics and computer science, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or equal to , denoted or .
For example, for floor: , , and for ceiling: , and .
Historically, the floor of has been–and still is–called the integral part or integer part of , often denoted (as well as a variety of other notations). However, the same term, integer part, is also used for truncation towards zero, which differs from the floor function for negative numbers.
For an integer, .
Although and produce graphs that appear exactly alike, they are not the same when the value of x is an exact integer. For example, when =2.0001; . However, if =2, then , while .
Notation
The integral part or integer part of a number ( in the original) was first defined in 1798 by Adrien-Marie Legendre in his proof of the Legendre's formula.
Carl Friedrich Gauss introduced the square bracket notation in his third proof of quadratic reciprocity (1808). This remained the standard in mathematics until Kenneth E. Iverson introduced, in his 1962 book A Programming Language, the names "floor" and "ceiling" and the corresponding notations and . (Iverson used square brackets for a different purpose, the Iverson bracket notation.) Both notations are now used in mathematics, although Iverson's notation will be followed in this article.
In some sources, boldface or double brackets are used for floor, and reversed brackets or for ceiling.
The fractional part is the sawtooth function, denoted by for real and defined by the formula
For all x,
.
These characters are provided in Unicode:
In the LaTeX typesetting system, these symbols can be specified with the and commands in math mode, and extended in size using and as needed.
Some authors define as the round-toward-zero function, so and , and call it the "integer part". This is truncation to zero decimal digits.
Definition and properties
Given real numbers x and y, integers m and n and the set of integers , floor and ceiling may be defined by the equations
Since there is exactly one integer in a half-open interval of length one, for any real number x, there are unique integers m and n satisfying the equation
where and may also be taken as the definition of floor and ceiling.
Equivalences
These formulas can be used to simplify expressions involving floors and ceilings.
In the language of order theory, the floor function is a residuated mapping, that is, part of a Galois connection: it is the upper adjoint of the function that embeds the integers into the reals.
These formulas show how adding an integer to the arguments affects the functions:
The above are never true if is not an integer; however, for every and , the following inequalities hold:
Monotonicity
Both floor and ceiling functions are the monotonically non-d |
https://en.wikipedia.org/wiki/List%20of%20mathematical%20functions | In mathematics, some functions or groups of functions are important enough to deserve their own names. This is a listing of articles which explain some of these functions in more detail. There is a large theory of special functions which developed out of statistics and mathematical physics. A modern, abstract point of view contrasts large function spaces, which are infinite-dimensional and within which most functions are 'anonymous', with special functions picked out by properties such as symmetry, or relationship to harmonic analysis and group representations.
See also List of types of functions
Elementary functions
Elementary functions are functions built from basic operations (e.g. addition, exponentials, logarithms...)
Algebraic functions
Algebraic functions are functions that can be expressed as the solution of a polynomial equation with integer coefficients.
Polynomials: Can be generated solely by addition, multiplication, and raising to the power of a positive integer.
Constant function: polynomial of degree zero, graph is a horizontal straight line
Linear function: First degree polynomial, graph is a straight line.
Quadratic function: Second degree polynomial, graph is a parabola.
Cubic function: Third degree polynomial.
Quartic function: Fourth degree polynomial.
Quintic function: Fifth degree polynomial.
Sextic function: Sixth degree polynomial.
Rational functions: A ratio of two polynomials.
nth root
Square root: Yields a number whose square is the given one.
Cube root: Yields a number whose cube is the given one.
Elementary transcendental functions
Transcendental functions are functions that are not algebraic.
Exponential function: raises a fixed number to a variable power.
Hyperbolic functions: formally similar to the trigonometric functions.
Logarithms: the inverses of exponential functions; useful to solve equations involving exponentials.
Natural logarithm
Common logarithm
Binary logarithm
Power functions: raise a variable number to a fixed power; also known as Allometric functions; note: if the power is a rational number it is not strictly a transcendental function.
Periodic functions
Trigonometric functions: sine, cosine, tangent, cotangent, secant, cosecant, exsecant, excosecant, versine, coversine, vercosine, covercosine, haversine, hacoversine, havercosine, hacovercosine, etc.; used in geometry and to describe periodic phenomena. See also Gudermannian function.
Special functions
Piecewise special functions
Arithmetic functions
Sigma function: Sums of powers of divisors of a given natural number.
Euler's totient function: Number of numbers coprime to (and not bigger than) a given one.
Prime-counting function: Number of primes less than or equal to a given number.
Partition function: Order-independent count of ways to write a given positive integer as a sum of positive integers.
Möbius μ function: Sum of the nth primitive roots of unity, it depends on the prime factorization of n.
Prime omega |
https://en.wikipedia.org/wiki/Complement%20%28set%20theory%29 | In set theory, the complement of a set , often denoted by (or ), is the set of elements not in .
When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is the set of elements in that are not in .
The relative complement of with respect to a set , also termed the set difference of and , written is the set of elements in that are not in .
Absolute complement
Definition
If is a set, then the absolute complement of (or simply the complement of ) is the set of elements not in (within a larger set that is implicitly defined). In other words, let be a set that contains all the elements under study; if there is no need to mention , either because it has been previously specified, or it is obvious and unique, then the absolute complement of is the relative complement of in :
Or formally:
The absolute complement of is usually denoted by Other notations include
Examples
Assume that the universe is the set of integers. If is the set of odd numbers, then the complement of is the set of even numbers. If is the set of multiples of 3, then the complement of is the set of numbers congruent to 1 or 2 modulo 3 (or, in simpler terms, the integers that are not multiples of 3).
Assume that the universe is the standard 52-card deck. If the set is the suit of spades, then the complement of is the union of the suits of clubs, diamonds, and hearts. If the set is the union of the suits of clubs and diamonds, then the complement of is the union of the suits of hearts and spades.
When the universe is the universe of sets described in formalized set theory, the absolute complement of a set is generally not itself a set, but rather a proper class. For more info, see universal set.
Properties
Let and be two sets in a universe . The following identities capture important properties of absolute complements:
De Morgan's laws:
Complement laws:
(this follows from the equivalence of a conditional with its contrapositive).
Involution or double complement law:
Relationships between relative and absolute complements:
Relationship with a set difference:
The first two complement laws above show that if is a non-empty, proper subset of , then is a partition of .
Relative complement
Definition
If and are sets, then the relative complement of in , also termed the set difference of and , is the set of elements in but not in .
The relative complement of in is denoted according to the ISO 31-11 standard. It is sometimes written but this notation is ambiguous, as in some contexts (for example, Minkowski set operations in functional analysis) it can be interpreted as the set of all elements where is taken from and from .
Formally:
Examples
If is the set of real numbers and is the set of rational numbers, then is the set of irrational numbers.
Properties
Let , , and be three sets. The following identities capture n |
https://en.wikipedia.org/wiki/Boolean%20ring | In mathematics, a Boolean ring is a ring for which for all in , that is, a ring that consists of only idempotent elements. An example is the ring of integers modulo 2.
Every Boolean ring gives rise to a Boolean algebra, with ring multiplication corresponding to conjunction or meet , and ring addition to exclusive disjunction or symmetric difference (not disjunction , which would constitute a semiring). Conversely, every Boolean algebra gives rise to a Boolean ring. Boolean rings are named after the founder of Boolean algebra, George Boole.
Notation
There are at least four different and incompatible systems of notation for Boolean rings and algebras:
In commutative algebra the standard notation is to use for the ring sum of and , and use for their product.
In logic, a common notation is to use for the meet (same as the ring product) and use for the join, given in terms of ring notation (given just above) by .
In set theory and logic it is also common to use for the meet, and for the join . This use of is different from the use in ring theory.
A rare convention is to use for the product and for the ring sum, in an effort to avoid the ambiguity of .
Historically, the term "Boolean ring" has been used to mean a "Boolean ring possibly without an identity", and "Boolean algebra" has been used to mean a Boolean ring with an identity. The existence of the identity is necessary to consider the ring as an algebra over the field of two elements: otherwise there cannot be a (unital) ring homomorphism of the field of two elements into the Boolean ring. (This is the same as the old use of the terms "ring" and "algebra" in measure theory.)
Examples
One example of a Boolean ring is the power set of any set , where the addition in the ring is symmetric difference, and the multiplication is intersection. As another example, we can also consider the set of all finite or cofinite subsets of , again with symmetric difference and intersection as operations. More generally with these operations any field of sets is a Boolean ring. By Stone's representation theorem every Boolean ring is isomorphic to a field of sets (treated as a ring with these operations).
Relation to Boolean algebras
Since the join operation in a Boolean algebra is often written additively, it makes sense in this context to denote ring addition by , a symbol that is often used to denote exclusive or.
Given a Boolean ring , for and in we can define
,
,
.
These operations then satisfy all of the axioms for meets, joins, and complements in a Boolean algebra. Thus every Boolean ring becomes a Boolean algebra. Similarly, every Boolean algebra becomes a Boolean ring thus:
If a Boolean ring is translated into a Boolean algebra in this way, and then the Boolean algebra is translated into a ring, the result is the original ring. The analogous result holds beginning with a Boolean algebra.
A map between two Boolean rings is a ring homomorphism if and only if it is |
https://en.wikipedia.org/wiki/TI-89%20series | The TI-89 and the TI-89 Titanium are graphing calculators developed by Texas Instruments (TI). They are differentiated from most other TI graphing calculators by their computer algebra system, which allows symbolic manipulation of algebraic expressions—equations can be solved in terms of variables, whereas the TI-83/84 series can only give a numeric result.
TI-89
The TI-89 is a graphing calculator developed by Texas Instruments in 1998. The unit features a 160×100 pixel resolution LCD and a large amount of flash memory, and includes TI's Advanced Mathematics Software. The TI-89 is one of the highest model lines in TI's calculator products, along with the TI-Nspire. In the summer of 2004, the standard TI-89 was replaced by the TI-89 Titanium.
The TI-89 runs on a 32-bit microprocessor, the Motorola 68000, which nominally runs at 10 or 12 MHz, depending on the calculator's hardware version. The calculator has 256 kB of RAM, (190 kB of which are available to the user) and 2 MB of flash memory (700 kB of which is available to the user). The RAM and Flash ROM are used to store expressions, variables, programs, text files, and lists.
The TI-89 is essentially a TI-92 Plus with a limited keyboard and smaller screen. It was created partially in response to the fact that while calculators are allowed on many standardized tests, the TI-92 was not due to the QWERTY layout of its keyboard. Additionally, some people found the TI-92 unwieldy and overly large. The TI-89 is significantly smaller—about the same size as most other graphing calculators. It has a flash ROM, a feature present on the TI-92 Plus but not on the original TI-92.
User features
The major advantage of the TI-89 over other TI calculators is its built-in computer algebra system, or CAS. The calculator can evaluate and simplify algebraic expressions symbolically. For example, entering x^2-4x+4 returns . The answer is "prettyprinted" by default; that is, displayed as it would be written by hand (e.g. the aforementioned rather than x^2-4x+4). The TI-89's abilities include:
Algebraic factoring of expressions, including partial fraction decomposition.
Algebraic simplification; for example, the CAS can combine multiple terms into one fraction by finding a common denominator.
Evaluation of trigonometric expressions to exact values. For example, sin(60°) returns instead of 0.86603.
Solving equations for a certain variable. The CAS can solve for one variable in terms of others; it can also solve systems of equations. For equations such as quadratics where there are multiple solutions, it returns all of them. Equations with infinitely many solutions are solved by introducing arbitrary constants: solve(tan(x+2)=0,x) returns x=2.(90.@n1-1), with the @n1 representing any integer.
Symbolic and numeric differentiation and integration. Derivatives and definite integrals are evaluated exactly when possible, and approximately otherwise.
Calculate greatest common divisor (gcd) and least commo |
https://en.wikipedia.org/wiki/Computer%20algebra%20system | A computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The development of the computer algebra systems in the second half of the 20th century is part of the discipline of "computer algebra" or "symbolic computation", which has spurred work in algorithms over mathematical objects such as polynomials.
Computer algebra systems may be divided into two classes: specialized and general-purpose. The specialized ones are devoted to a specific part of mathematics, such as number theory, group theory, or teaching of elementary mathematics.
General-purpose computer algebra systems aim to be useful to a user working in any scientific field that requires manipulation of mathematical expressions. To be useful, a general-purpose computer algebra system must include various features such as:
a user interface allowing a user to enter and display mathematical formulas, typically from a keyboard, menu selections, mouse or stylus.
a programming language and an interpreter (the result of a computation commonly has an unpredictable form and an unpredictable size; therefore user intervention is frequently needed),
a simplifier, which is a rewrite system for simplifying mathematics formulas,
a memory manager, including a garbage collector, needed by the huge size of the intermediate data, which may appear during a computation,
an arbitrary-precision arithmetic, needed by the huge size of the integers that may occur,
a large library of mathematical algorithms and special functions.
The library must not only provide for the needs of the users, but also the needs of the simplifier. For example, the computation of polynomial greatest common divisors is systematically used for the simplification of expressions involving fractions.
This large amount of required computer capabilities explains the small number of general-purpose computer algebra systems. Significant systems include Axiom, GAP, Maxima, Magma, Maple, Mathematica, and SageMath.
History
Computer algebra systems began to appear in the 1960s and evolved out of two quite different sources—the requirements of theoretical physicists and research into artificial intelligence.
A prime example for the first development was the pioneering work conducted by the later Nobel Prize laureate in physics Martinus Veltman, who designed a program for symbolic mathematics, especially high-energy physics, called Schoonschip (Dutch for "clean ship") in 1963. Another early system was FORMAC.
Using Lisp as the programming basis, Carl Engelman created MATHLAB in 1964 at MITRE within an artificial-intelligence research environment. Later MATHLAB was made available to users on PDP-6 and PDP-10 systems running TOPS-10 or TENEX in universities. Today it can still be used on SIMH emulations of the PDP-10. MATHLAB ("mathematical laboratory") should not b |
https://en.wikipedia.org/wiki/Vertex | Vertex, vertices or vertexes may refer to:
Science and technology
Mathematics and computer science
Vertex (geometry), a point where two or more curves, lines, or edges meet
Vertex (computer graphics), a data structure that describes the position of a point
Vertex (curve), a point of a plane curve where the first derivative of curvature is zero
Vertex (graph theory), the fundamental unit of which graphs are formed
Vertex (topography), in a triangulated irregular network
Vertex of a representation, in finite group theory
Physics
Vertex (physics), the reconstructed location of an individual particle collision
Vertex (optics), a point where the optical axis crosses an optical surface
Vertex function, describing the interaction between a photon and an electron
Biology and anatomy
Vertex (anatomy), the highest point of the head
Vertex (urinary bladder), alternative name of the apex of urinary bladder
Vertex distance, the distance between the surface of the cornea of the eye and a lens situated in front of it
Vertex presentation, a head-first presentation at childbirth
Businesses
Vertex (company), an American business services provider
Vertex Holdings, an investment holding company in Singapore
Vertex Inc, an American tax compliance software and services company
Vertex Pharmaceuticals, an American biotech company
Vertex Railcar, a Chinese-American manufacturer of railroad rolling stock 2014–2018
Vertex Resource Group, a Canadian environmental services company
Other uses
Vertex (album), by Buck 65, 1997
Vertex (band), formed in 1996
Vertex (astrology), the point where the prime vertical intersects the ecliptic
See also
Virtex (disambiguation)
Vortex (disambiguation)
Vertex model, a type of statistical mechanics model
Vertex operator algebra in conformal field theory
External links |
https://en.wikipedia.org/wiki/Arc | Arc may refer to:
Mathematics
Arc (geometry), a segment of a differentiable curve
Circular arc, a segment of a circle
Arc (topology), a segment of a path
Arc length, the distance between two points along a section of a curve
Arc (projective geometry), a particular type of set of points of a projective plane
arc (function prefix) (arcus), a prefix for inverse trigonometric functions
Directed arc, a directed edge in graph theory
Minute and second of arc, a unit of angular measurement equal to 1/60 of one degree.
Wild arc, a concept from geometric topology
Science and technology
Geology
Arc, in geology a mountain chain configured as an arc due to a common orogeny along a plate margin or the effect of back-arc extension
Hellenic arc, the arc of islands positioned over the Hellenic Trench in the Aegean Sea off Greece
Back-arc basin, a subsided region caused by back-arc extension
Back-arc region, the region created by back-arc extension, containing all the basins, faults, and volcanoes generated by the extension
Island arc, an arc-shaped archipelago, usually so configured for geologic causes, such as sea-floor spreading, common orogeny on the margin of the same plate, or back-arc extension
Northeastern Japan Arc, an island arc
Banda Arc, a set of island arcs in Indonesia
Continental arc, in geology a continental mountain chain or parallel alignment of chains (as opposed to island arcs), configured in an arc
Eastern Arc Mountains, a continental arc of Africa
Volcanic arc, a chain of volcanoes positioned in an arc shape as seen from above
Aleutian Arc, a large volcanic arc in the U.S. state of Alaska
Nastapoka arc, a circular coastline in Hudson Bay
Technology
arc, the command-line interface for ArcInfo
ARC (file format), a file name extension for archive files
ARC (processor), 32-bit RISC architecture
ARC (adaptive replacement cache), a page replacement algorithm for high-performance filesystems
Arc (browser), a freeware browser developed by The Browser Company
Arc (programming language), a Lisp dialect designed by Paul Graham
Sony Ericsson Xperia Arc, a cellphone
Audio Return Channel, an audio technology working over HDMI
Authenticated Received Chain, an email authentication system
Arc lamp, a lamp that produces light by an electric arc
Xenon arc lamp, a highly specialized type of gas discharge lamp
Deuterium arc lamp, a low-pressure gas-discharge light source
Hydrargyrum medium-arc iodide lamp, the trademark name of Osram's brand of metal-halide gas discharge medium arc-length lamp
Electric arc furnace, a furnace that heats charged material by means of an electric arc
Arc welding, a welding process that is used to join metal to metal
Arc-fault circuit interrupter, a specialized circuit breaker
Arc converter, a spark transmitter
Intel Arc, brand of graphics processing units designed by Intel
Other science
Electric arc, an ongoing plasma discharge (an electric current through a gas), producing light and hea |
https://en.wikipedia.org/wiki/Recursively%20enumerable%20language | In mathematics, logic and computer science, a formal language is called recursively enumerable (also recognizable, partially decidable, semidecidable, Turing-acceptable or Turing-recognizable) if it is a recursively enumerable subset in the set of all possible words over the alphabet of the language, i.e., if there exists a Turing machine which will enumerate all valid strings of the language.
Recursively enumerable languages are known as type-0 languages in the Chomsky hierarchy of formal languages. All regular, context-free, context-sensitive and recursive languages are recursively enumerable.
The class of all recursively enumerable languages is called RE.
Definitions
There are three equivalent definitions of a recursively enumerable language:
A recursively enumerable language is a recursively enumerable subset in the set of all possible words over the alphabet of the language.
A recursively enumerable language is a formal language for which there exists a Turing machine (or other computable function) which will enumerate all valid strings of the language. Note that if the language is infinite, the enumerating algorithm provided can be chosen so that it avoids repetitions, since we can test whether the string produced for number n is "already" produced for a number which is less than n. If it already is produced, use the output for input n+1 instead (recursively), but again, test whether it is "new".
A recursively enumerable language is a formal language for which there exists a Turing machine (or other computable function) that will halt and accept when presented with any string in the language as input but may either halt and reject or loop forever when presented with a string not in the language. Contrast this to recursive languages, which require that the Turing machine halts in all cases.
All regular, context-free, context-sensitive and recursive languages are recursively enumerable.
Post's theorem shows that RE, together with its complement co-RE, correspond to the first level of the arithmetical hierarchy.
Example
The set of halting Turing machines is recursively enumerable but not recursive. Indeed, one can run the Turing machine and accept if the machine halts, hence it is recursively enumerable. On the other hand, the problem is undecidable.
Some other recursively enumerable languages that are not recursive include:
Post correspondence problem
Mortality (computability theory)
Entscheidungsproblem
Closure properties
Recursively enumerable languages (REL) are closed under the following operations. That is, if L and P are two recursively enumerable languages, then the following languages are recursively enumerable as well:
the Kleene star of L
the concatenation of L and P
the union
the intersection .
Recursively enumerable languages are not closed under set difference or complementation. The set difference is recursively enumerable if is recursive. If is recursively enumerable, then the complement of is rec |
https://en.wikipedia.org/wiki/Transition | Transition or transitional may refer to:
Mathematics, science, and technology
Biology
Transition (genetics), a point mutation that changes a purine nucleotide to another purine (A ↔ G) or a pyrimidine nucleotide to another pyrimidine (C ↔ T)
Transitional fossil, any fossilized remains of a lifeform that exhibits the characteristics of two distinct taxonomic groups
A phase during childbirth contractions during which the cervix completes its dilation
Gender and sex
Gender transitioning, the process of changing one's gender presentation to accord with one's internal sense of one's gender – the idea of what it means to be a man or woman
Sex reassignment therapy, the physical aspect of a gender transition
Physics
Phase transition, a transformation of the state of matter; for example, the change between a solid and a liquid, between liquid and gas or between gas and plasma
Quantum phase transition, a phase transformation between different quantum phases
Quantum Hall transitions, a quantum phase transition that occurred because of the Quantum Hall Effect
Transition radiation, contrasts to the Cherenkov radiation
Atomic electron transition, the transition of an electron from one quantum state to another within an atom
Beta decay transition, nuclear beta decay determined by changes in spin
Laminar–turbulent transition, the process of a laminar fluid flow becoming turbulent
Glass transition, the reversible transition in amorphous materials
Lambda transition, universality class in condensed matter physics
Chemistry
Transition metal, either an element whose atom has an incomplete d sub-shell, or any element in the d-block of the periodic table
Transition state, of a chemical reaction is a particular configuration along the reaction coordinate
SRM transition, the precursor and product ion pair in Selected reaction monitoring (SRM) in analytical chemistry
Computing
A movement between states of an abstract computer, described by a transition system
A phase of the project lifecycle in the Rational Unified Process
A paradigm describing changes of communication mechanisms, see Transition (computer science)
Other uses in technology
Transitions, a brand of photochromic eyeglass lens and sponsor of the PGA Tour Transitions Championship
Transition (roadable aircraft), a flying car (or drivable airplane) made by Terrafugia
"Shifting gears" on a railroad locomotive; see Diesel locomotive#Propulsion system operation
Government and politics
Democratic transition
Transition management (governance)
Social change, often synonymous with social transition
Arts and entertainment
Literature
Transition (fiction), a narrative element or general aspects of writing style that signal changes in a story
Transition (linguistics), a certain word, expression, or other device that gives text or speech greater cohesion by making it more explicit
Works
Transition (literary journal), an experimental literary journal that featured surrealist, expression |
https://en.wikipedia.org/wiki/Baire%20category%20theorem | The Baire category theorem (BCT) is an important result in general topology and functional analysis. The theorem has two forms, each of which gives sufficient conditions for a topological space to be a Baire space (a topological space such that the intersection of countably many dense open sets is still dense). It is used in the proof of results in many areas of analysis and geometry, including some of the fundamental theorems of functional analysis.
Versions of the Baire category theorem were first proved independently in 1897 by Osgood for the real line and in 1899 by Baire for Euclidean space . The more general statement for completetely metrizable spaces was first shown by Hausdorff in 1914.
Statement
A Baire space is a topological space in which every countable intersection of open dense sets is dense in See the corresponding article for a list of equivalent characterizations, as some are more useful than others depending on the application.
(BCT1) Every complete pseudometric space is a Baire space. In particular, every completely metrizable topological space is a Baire space.
(BCT2) Every locally compact regular space is a Baire space. In particular, every locally compact Hausdorff space is a Baire space.
Neither of these statements directly implies the other, since there are complete metric spaces that are not locally compact (the irrational numbers with the metric defined below; also, any Banach space of infinite dimension), and there are locally compact Hausdorff spaces that are not metrizable (for instance, any uncountable product of non-trivial compact Hausdorff spaces is such; also, several function spaces used in functional analysis; the uncountable Fort space).
See Steen and Seebach in the references below.
Relation to the axiom of choice
The proof of BCT1 for arbitrary complete metric spaces requires some form of the axiom of choice; and in fact BCT1 is equivalent over ZF to the axiom of dependent choice, a weak form of the axiom of choice.
A restricted form of the Baire category theorem, in which the complete metric space is also assumed to be separable, is provable in ZF with no additional choice principles.
This restricted form applies in particular to the real line, the Baire space the Cantor space and a separable Hilbert space such as the -space .
Uses
BCT1 is used in functional analysis to prove the open mapping theorem, the closed graph theorem and the uniform boundedness principle.
BCT1 also shows that every nonempty complete metric space with no isolated point is uncountable. (If is a nonempty countable metric space with no isolated point, then each singleton in is nowhere dense, and is meagre in itself.) In particular, this proves that the set of all real numbers is uncountable.
BCT1 shows that each of the following is a Baire space:
The space of real numbers
The irrational numbers, with the metric defined by where is the first index for which the continued fraction expansions of and differ |
https://en.wikipedia.org/wiki/Sector | Sector may refer to:
Places
Sector, West Virginia, U.S.
Geometry
Circular sector, the portion of a disc enclosed by two radii and a circular arc
Hyperbolic sector, a region enclosed by two radii and a hyperbolic arc
Spherical sector, a portion of a sphere enclosed by a cone of radii from the center of the sphere
Social and economic
Business sector, part of the economy which involves the trading and sale of products by companies
Economic sector, the manufacturing, finance and production of goods for consumers
Private sector, business activity created by private enterprise for profit
Public sector, delivers social services, infrastructure and institutions administered by government
Voluntary sector, a non-profit and voluntary part of an economy provided by organisations
The sector of the sector directive in government procurement in the European Union
Computing
Cylinder-head-sector, an early method for giving addresses to blocks of data on a hard drive
Disk sector, a subdivision of a track on a disk
Sector, or zone, in portal rendering
Sector/Sphere, an open source software suite
Sector (instrument), a historic calculating instrument
science fiction
Sector general
Other uses
Sector (country subdivision)
Sector clock
Sector light
Sector commander
United States Coast Guard Sector, a shore-based unit of the U.S. Coast Guard
Sector, a fictional area of space, e.g. in the Foundation series, Star Wars, StarCraft, Warhammer 40,000, Lollipop Chainsaw
Sector, or galactic quadrant, regions of space in Star Trek
Sector, and Sector No Limits, watch brands
See also
Area (disambiguation)
Region (disambiguation)
Zone (disambiguation)
Sectoria, a small genus of stone loaches |
https://en.wikipedia.org/wiki/Discriminant | In mathematics, the discriminant of a polynomial is a quantity that depends on the coefficients and allows deducing some properties of the roots without computing them. More precisely, it is a polynomial function of the coefficients of the original polynomial. The discriminant is widely used in polynomial factoring, number theory, and algebraic geometry.
The discriminant of the quadratic polynomial is
the quantity which appears under the square root in the quadratic formula. If this discriminant is zero if and only if the polynomial has a double root. In the case of real coefficients, it is positive if the polynomial has two distinct real roots, and negative if it has two distinct complex conjugate roots. Similarly, the discriminant of a cubic polynomial is zero if and only if the polynomial has a multiple root. In the case of a cubic with real coefficients, the discriminant is positive if the polynomial has three distinct real roots, and negative if it has one real root and two distinct complex conjugate roots.
More generally, the discriminant of a univariate polynomial of positive degree is zero if and only if the polynomial has a multiple root. For real coefficients and no multiple roots, the discriminant is positive if the number of non-real roots is a multiple of 4 (including none), and negative otherwise.
Several generalizations are also called discriminant: the discriminant of an algebraic number field; the discriminant of a quadratic form; and more generally, the discriminant of a form, of a homogeneous polynomial, or of a projective hypersurface (these three concepts are essentially equivalent).
Origin
The term "discriminant" was coined in 1851 by the British mathematician James Joseph Sylvester.
Definition
Let
be a polynomial of degree (this means ), such that the coefficients belong to a field, or, more generally, to a commutative ring. The resultant of and its derivative,
is a polynomial in with integer coefficients, which is the determinant of the Sylvester matrix of and . The nonzero entries of the first column of the Sylvester matrix are and and the resultant is thus a multiple of Hence the discriminant—up to its sign—is defined as the quotient of the resultant of and by :
Historically, this sign has been chosen such that, over the reals, the discriminant will be positive when all the roots of the polynomial are real. The division by may not be well defined if the ring of the coefficients contains zero divisors. Such a problem may be avoided by replacing by 1 in the first column of the Sylvester matrix—before computing the determinant. In any case, the discriminant is a polynomial in with integer coefficients.
Expression in terms of the roots
When the above polynomial is defined over a field, it has roots, , not necessarily all distinct, in any algebraically closed extension of the field. (If the coefficients are real numbers, the roots may be taken in the field of complex numbers, where the fundament |
https://en.wikipedia.org/wiki/Interior%20%28topology%29 | In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of the closure of the complement of .
In this sense interior and closure are dual notions.
The exterior of a set is the complement of the closure of ; it consists of the points that are in neither the set nor its boundary.
The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty).
The interior and exterior of a closed curve are a slightly different concept; see the Jordan curve theorem.
Definitions
Interior point
If is a subset of a Euclidean space, then is an interior point of if there exists an open ball centered at which is completely contained in
(This is illustrated in the introductory section to this article.)
This definition generalizes to any subset of a metric space with metric : is an interior point of if there exists a real number such that is in whenever the distance
This definition generalizes to topological spaces by replacing "open ball" with "open set".
If is a subset of a topological space then is an of in if is contained in an open subset of that is completely contained in
(Equivalently, is an interior point of if is a neighbourhood of )
Interior of a set
The interior of a subset of a topological space denoted by or or can be defined in any of the following equivalent ways:
is the largest open subset of contained in
is the union of all open sets of contained in
is the set of all interior points of
If the space is understood from context then the shorter notation is usually preferred to
Examples
In any space, the interior of the empty set is the empty set.
In any space if then
If is the real line (with the standard topology), then whereas the interior of the set of rational numbers is empty:
If is the complex plane then
In any Euclidean space, the interior of any finite set is the empty set.
On the set of real numbers, one can put other topologies rather than the standard one:
If is the real numbers with the lower limit topology, then
If one considers on the topology in which every set is open, then
If one considers on the topology in which the only open sets are the empty set and itself, then is the empty set.
These examples show that the interior of a set depends upon the topology of the underlying space.
The last two examples are special cases of the following.
In any discrete space, since every set is open, every set is equal to its interior.
In any indiscrete space since the only open sets are the empty set and itself, and for every proper subset of is the empty set.
Properties
Let be a topological space and let and be subsets of
is open in
If is open in then if and only if
is an open subset of whe |
https://en.wikipedia.org/wiki/Alexandroff%20extension | In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. It is named after the Russian mathematician Pavel Alexandroff.
More precisely, let X be a topological space. Then the Alexandroff extension of X is a certain compact space X* together with an open embedding c : X → X* such that the complement of X in X* consists of a single point, typically denoted ∞. The map c is a Hausdorff compactification if and only if X is a locally compact, noncompact Hausdorff space. For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification. The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications; the disadvantage lies in the fact that it only gives a Hausdorff compactification on the class of locally compact, noncompact Hausdorff spaces, unlike the Stone–Čech compactification which exists for any topological space (but provides an embedding exactly for Tychonoff spaces).
Example: inverse stereographic projection
A geometrically appealing example of one-point compactification is given by the inverse stereographic projection. Recall that the stereographic projection S gives an explicit homeomorphism from the unit sphere minus the north pole (0,0,1) to the Euclidean plane. The inverse stereographic projection is an open, dense embedding into a compact Hausdorff space obtained by adjoining the additional point . Under the stereographic projection latitudinal circles get mapped to planar circles . It follows that the deleted neighborhood basis of given by the punctured spherical caps corresponds to the complements of closed planar disks . More qualitatively, a neighborhood basis at is furnished by the sets as K ranges through the compact subsets of . This example already contains the key concepts of the general case.
Motivation
Let be an embedding from a topological space X to a compact Hausdorff topological space Y, with dense image and one-point remainder . Then c(X) is open in a compact Hausdorff space so is locally compact Hausdorff, hence its homeomorphic preimage X is also locally compact Hausdorff. Moreover, if X were compact then c(X) would be closed in Y and hence not dense. Thus a space can only admit a Hausdorff one-point compactification if it is locally compact, noncompact and Hausdorff. Moreover, in such a one-point compactification the image of a neighborhood basis for x in X gives a neighborhood basis for c(x) in c(X), and—because a subset of a compact Hausdorff space is compact if and only if it is closed—the open neighborhoods of must be all sets obtained by adjoining to the image under c of a subset of X with compact complement.
The Alexandroff extension
Let be a topological space. Put and topologi |
https://en.wikipedia.org/wiki/Linear%20combination | In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants). The concept of linear combinations is central to linear algebra and related fields of mathematics.
Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.
Definition
Let V be a vector space over the field K. As usual, we call elements of V vectors and call elements of K scalars.
If v1,...,vn are vectors and a1,...,an are scalars, then the linear combination of those vectors with those scalars as coefficients is
There is some ambiguity in the use of the term "linear combination" as to whether it refers to the expression or to its value. In most cases the value is emphasized, as in the assertion "the set of all linear combinations of v1,...,vn always forms a subspace". However, one could also say "two different linear combinations can have the same value" in which case the reference is to the expression. The subtle difference between these uses is the essence of the notion of linear dependence: a family F of vectors is linearly independent precisely if any linear combination of the vectors in F (as value) is uniquely so (as expression). In any case, even when viewed as expressions, all that matters about a linear combination is the coefficient of each vi; trivial modifications such as permuting the terms or adding terms with zero coefficient do not produce distinct linear combinations.
In a given situation, K and V may be specified explicitly, or they may be obvious from context. In that case, we often speak of a linear combination of the vectors v1,...,vn, with the coefficients unspecified (except that they must belong to K). Or, if S is a subset of V, we may speak of a linear combination of vectors in S, where both the coefficients and the vectors are unspecified, except that the vectors must belong to the set S (and the coefficients must belong to K). Finally, we may speak simply of a linear combination, where nothing is specified (except that the vectors must belong to V and the coefficients must belong to K); in this case one is probably referring to the expression, since every vector in V is certainly the value of some linear combination.
Note that by definition, a linear combination involves only finitely many vectors (except as described in Generalizations below).
However, the set S that the vectors are taken from (if one is mentioned) can still be infinite; each individual linear combination will only involve finitely many vectors.
Also, there is no reason that n cannot be zero; in that case, we declare by convention that the result of the linear combination is the zero vector in V.
Examples and counterexamples
Euclidean vectors
Let the field K be the set R of real numbers, an |
https://en.wikipedia.org/wiki/Harmonic%20function | In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function where is an open subset of that satisfies Laplace's equation, that is,
everywhere on . This is usually written as
or
Etymology of the term "harmonic"
The descriptor "harmonic" in the name harmonic function originates from a point on a taut string which is undergoing harmonic motion. The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as harmonics. Fourier analysis involves expanding functions on the unit circle in terms of a series of these harmonics. Considering higher dimensional analogues of the harmonics on the unit n-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and over time "harmonic" was used to refer to all functions satisfying Laplace's equation.
Examples
Examples of harmonic functions of two variables are:
The real or imaginary part of any holomorphic function.
The function this is a special case of the example above, as and is a holomorphic function. The second derivative with respect to x is while the second derivative with respect to y is
The function defined on This can describe the electric potential due to a line charge or the gravity potential due to a long cylindrical mass.
Examples of harmonic functions of three variables are given in the table below with
{| class="wikitable"
! Function !! Singularity
|-
|align=center|
|Unit point charge at origin
|-
|align=center|
|x-directed dipole at origin
|-
|align=center|
|Line of unit charge density on entire z-axis
|-
|align=center|
|Line of unit charge density on negative z-axis
|-
|align=center|
|Line of x-directed dipoles on entire z axis
|-
|align=center|
|Line of x-directed dipoles on negative z axis
|}
Harmonic functions that arise in physics are determined by their singularities and boundary conditions (such as Dirichlet boundary conditions or Neumann boundary conditions). On regions without boundaries, adding the real or imaginary part of any entire function will produce a harmonic function with the same singularity, so in this case the harmonic function is not determined by its singularities; however, we can make the solution unique in physical situations by requiring that the solution approaches 0 as r approaches infinity. In this case, uniqueness follows by Liouville's theorem.
The singular points of the harmonic functions above are expressed as "charges" and "charge densities" using the terminology of electrostatics, and so the corresponding harmonic function will be proportional to the electrostatic potential due to these charge distributions. Each function above will yield another harmonic function when multiplied by a constant, rotated, and/or has a constant added. The inversion of each function will yield another harmonic function which has singularities which are t |
https://en.wikipedia.org/wiki/Discrete%20space | In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are isolated from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology.
Definitions
Given a set :
A metric space is said to be uniformly discrete if there exists a such that, for any one has either or The topology underlying a metric space can be discrete, without the metric being uniformly discrete: for example the usual metric on the set
Properties
The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology.
Thus, the different notions of discrete space are compatible with one another.
On the other hand, the underlying topology of a non-discrete uniform or metric space can be discrete; an example is the metric space (with metric inherited from the real line and given by ).
This is not the discrete metric; also, this space is not complete and hence not discrete as a uniform space.
Nevertheless, it is discrete as a topological space.
We say that is topologically discrete but not uniformly discrete or metrically discrete.
Additionally:
The topological dimension of a discrete space is equal to 0.
A topological space is discrete if and only if its singletons are open, which is the case if and only if it doesn't contain any accumulation points.
The singletons form a basis for the discrete topology.
A uniform space is discrete if and only if the diagonal is an entourage.
Every discrete topological space satisfies each of the separation axioms; in particular, every discrete space is Hausdorff, that is, separated.
A discrete space is compact if and only if it is finite.
Every discrete uniform or metric space is complete.
Combining the above two facts, every discrete uniform or metric space is totally bounded if and only if it is finite.
Every discrete metric space is bounded.
Every discrete space is first-countable; it is moreover second-countable if and only if it is countable.
Every discrete space is totally disconnected.
Every non-empty discrete space is second category.
Any two discrete spaces with the same cardinality are homeomorphic.
Every discrete space is metrizable (by the discrete metric).
A finite space is metrizable only if it is discrete.
If is a topological space and is a set carrying the discrete topology, then is evenly covered by (the projection map is the desired covering)
The subspace topology on the integers as a subspace of the real line is the discrete topology.
A discrete space is separable if and only if it is countable.
Any topological subspace of (with its usual Euclidean topology) that is discrete is necessarily countable.
Any function from a discrete to |
https://en.wikipedia.org/wiki/Krull%20dimension | In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules.
The Krull dimension was introduced to provide an algebraic definition of the dimension of an algebraic variety: the dimension of the affine variety defined by an ideal I in a polynomial ring R is the Krull dimension of R/I.
A field k has Krull dimension 0; more generally, k[x1, ..., xn] has Krull dimension n. A principal ideal domain that is not a field has Krull dimension 1. A local ring has Krull dimension 0 if and only if every element of its maximal ideal is nilpotent.
There are several other ways that have been used to define the dimension of a ring. Most of them coincide with the Krull dimension for Noetherian rings, but can differ for non-Noetherian rings.
Explanation
We say that a chain of prime ideals of the form
has length n. That is, the length is the number of strict inclusions, not the number of primes; these differ by 1. We define the Krull dimension of to be the supremum of the lengths of all chains of prime ideals in .
Given a prime ideal in R, we define the of , written , to be the supremum of the lengths of all chains of prime ideals contained in , meaning that . In other words, the height of is the Krull dimension of the localization of R at . A prime ideal has height zero if and only if it is a minimal prime ideal. The Krull dimension of a ring is the supremum of the heights of all maximal ideals, or those of all prime ideals. The height is also sometimes called the codimension, rank, or altitude of a prime ideal.
In a Noetherian ring, every prime ideal has finite height. Nonetheless, Nagata gave an example of a Noetherian ring of infinite Krull dimension. A ring is called catenary if any inclusion of prime ideals can be extended to a maximal chain of prime ideals between and , and any two maximal chains between
and have the same length. A ring is called universally catenary if any finitely generated algebra over it is catenary. Nagata gave an example of a Noetherian ring which is not catenary.
In a Noetherian ring, a prime ideal has height at most n if and only if it is a minimal prime ideal over an ideal generated by n elements (Krull's height theorem and its converse). It implies that the descending chain condition holds for prime ideals in such a way the lengths of the chains descending from a prime ideal are bounded by the number of generators of the prime.
More generally, the height of an ideal I is the infimum of the heights of all prime ideals containing I. In the language of algebraic geometry, this is the codimension of the subvariety of Spec() corresponding to I.
Schemes
It follows readily from the definition of the spectrum of a r |
https://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings%20algorithm | In statistics and statistical physics, the Metropolis–Hastings algorithm is a Markov chain Monte Carlo (MCMC) method for obtaining a sequence of random samples from a probability distribution from which direct sampling is difficult. This sequence can be used to approximate the distribution (e.g. to generate a histogram) or to compute an integral (e.g. an expected value). Metropolis–Hastings and other MCMC algorithms are generally used for sampling from multi-dimensional distributions, especially when the number of dimensions is high. For single-dimensional distributions, there are usually other methods (e.g. adaptive rejection sampling) that can directly return independent samples from the distribution, and these are free from the problem of autocorrelated samples that is inherent in MCMC methods.
History
The algorithm is named in part for Nicholas Metropolis, the first coauthor of a 1953 paper, entitled Equation of State Calculations by Fast Computing Machines, with Arianna W. Rosenbluth, Marshall Rosenbluth, Augusta H. Teller and Edward Teller. For many years the algorithm was known simply as the Metropolis algorithm. The paper proposed the algorithm for the case of symmetrical proposal distributions, but in 1970, W.K. Hastings extended it to the more general case. The generalized method was eventually identified by both names, although the first use of the term "Metropolis-Hastings algorithm" is unclear.
Some controversy exists with regard to credit for development of the Metropolis algorithm. Metropolis, who was familiar with the computational aspects of the method, had coined the term "Monte Carlo" in an earlier article with Stanisław Ulam, and led the group in the Theoretical Division that designed and built the MANIAC I computer used in the experiments in 1952. However, prior to 2003 there was no detailed account of the algorithm's development. Shortly before his death, Marshall Rosenbluth attended a 2003 conference at LANL marking the 50th anniversary of the 1953 publication. At this conference, Rosenbluth described the algorithm and its development in a presentation titled "Genesis of the Monte Carlo Algorithm for Statistical Mechanics". Further historical clarification is made by Gubernatis in a 2005 journal article recounting the 50th anniversary conference. Rosenbluth makes it clear that he and his wife Arianna did the work, and that Metropolis played no role in the development other than providing computer time.
This contradicts an account by Edward Teller, who states in his memoirs that the five authors of the 1953 article worked together for "days (and nights)". In contrast, the detailed account by Rosenbluth credits Teller with a crucial but early suggestion to "take advantage of statistical mechanics and take ensemble averages instead of following detailed kinematics". This, says Rosenbluth, started him thinking about the generalized Monte Carlo approach – a topic which he says he had discussed often with John Von Neuman |
https://en.wikipedia.org/wiki/Noetherian%20ring | In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence of left (or right) ideals has a largest element; that is, there exists an such that:
Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. right-ideal) is finitely generated. A ring is Noetherian if it is both left- and right-Noetherian.
Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics are Noetherian (in particular the ring of integers, polynomial rings, and rings of algebraic integers in number fields), and many general theorems on rings rely heavily on Noetherian property (for example, the Lasker–Noether theorem and the Krull intersection theorem).
Noetherian rings are named after Emmy Noether, but the importance of the concept was recognized earlier by David Hilbert, with the proof of Hilbert's basis theorem (which asserts that polynomial rings are Noetherian) and Hilbert's syzygy theorem.
Characterizations
For noncommutative rings, it is necessary to distinguish between three very similar concepts:
A ring is left-Noetherian if it satisfies the ascending chain condition on left ideals.
A ring is right-Noetherian if it satisfies the ascending chain condition on right ideals.
A ring is Noetherian if it is both left- and right-Noetherian.
For commutative rings, all three concepts coincide, but in general they are different. There are rings that are left-Noetherian and not right-Noetherian, and vice versa.
There are other, equivalent, definitions for a ring R to be left-Noetherian:
Every left ideal I in R is finitely generated, i.e. there exist elements in I such that .
Every non-empty set of left ideals of R, partially ordered by inclusion, has a maximal element.
Similar results hold for right-Noetherian rings.
The following condition is also an equivalent condition for a ring R to be left-Noetherian and it is Hilbert's original formulation:
Given a sequence of elements in R, there exists an integer such that each is a finite linear combination with coefficients in R.
For a commutative ring to be Noetherian it suffices that every prime ideal of the ring is finitely generated. However, it is not enough to ask that all the maximal ideals are finitely generated, as there is a non-Noetherian local ring whose maximal ideal is principal (see a counterexample to Krull’s intersection theorem at Local ring#Commutative case.)
Properties
If R is a Noetherian ring, then the polynomial ring is Noetherian by the Hilbert's basis theorem. By induction, is a Noetherian ring. Also, , the power series ring, is a Noetherian ring.
If is a Noetherian ring and is a two-sided ideal, then the quotient ring is also Noet |
https://en.wikipedia.org/wiki/Artinian | Artinian may refer to:
Mathematics
Objects named for Austrian mathematician Emil Artin (1898–1962)
Artinian ideal, an ideal I in R for which the Krull dimension of the quotient ring R/I is 0
Artinian ring, a ring which satisfies the descending chain condition on (one-sided) ideals
Artinian module, a module which satisfies the descending chain condition on submodules
Artinian group, a group which satisfies the descending chain condition on subgroups
People
Araz Artinian, Armenian-Canadian filmmaker and photographer
Artine Artinian (1907–2005), French literature scholar
See also
Descending chain condition
List of things named after Emil Artin |
https://en.wikipedia.org/wiki/Dyadic%20rational | In mathematics, a dyadic rational or binary rational is a number that can be expressed as a fraction whose denominator is a power of two. For example, 1/2, 3/2, and 3/8 are dyadic rationals, but 1/3 is not. These numbers are important in computer science because they are the only ones with finite binary representations. Dyadic rationals also have applications in weights and measures, musical time signatures, and early mathematics education. They can accurately approximate any real number.
The sum, difference, or product of any two dyadic rational numbers is another dyadic rational number, given by a simple formula. However, division of one dyadic rational number by another does not always produce a dyadic rational result. Mathematically, this means that the dyadic rational numbers form a ring, lying between the ring of integers and the field of rational numbers. This ring may be denoted .
In advanced mathematics, the dyadic rational numbers are central to the constructions of the dyadic solenoid, Minkowski's question-mark function, Daubechies wavelets, Thompson's group, Prüfer 2-group, surreal numbers, and fusible numbers. These numbers are order-isomorphic to the rational numbers; they form a subsystem of the 2-adic numbers as well as of the reals, and can represent the fractional parts of 2-adic numbers. Functions from natural numbers to dyadic rationals have been used to formalize mathematical analysis in reverse mathematics.
Applications
In measurement
Many traditional systems of weights and measures are based on the idea of repeated halving, which produces dyadic rationals when measuring fractional amounts of units. The inch is customarily subdivided in dyadic rationals rather than using a decimal subdivision. The customary divisions of the gallon into half-gallons, quarts, pints, and cups are also dyadic. The ancient Egyptians used dyadic rationals in measurement, with denominators up to 64. Similarly, systems of weights from the Indus Valley civilisation are for the most part based on repeated halving; anthropologist Heather M.-L. Miller writes that "halving is a relatively simple operation with beam balances, which is likely why so many weight systems of this time period used binary systems".
In computing
Dyadic rationals are central to computer science as a type of fractional number that many computers can manipulate directly. In particular, as a data type used by computers, floating-point numbers are often defined as integers multiplied by positive or negative powers of two. The numbers that can be represented precisely in a floating-point format, such as the IEEE floating-point datatypes, are called its representable numbers. For most floating-point representations, the representable numbers are a subset of the dyadic rationals. The same is true for fixed-point datatypes, which also use powers of two implicitly in the majority of cases. Because of the simplicity of computing with dyadic rationals, they are also used for exact rea |
https://en.wikipedia.org/wiki/Linear%20span | In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , is defined as the set of all linear combinations of the vectors in .
For example, two linearly independent vectors span a plane.
The linear span can be characterized either as the intersection of all linear subspaces that contain , or as the smallest subspace containing . The linear span of a set of vectors is therefore a vector space itself. Spans can be generalized to matroids and modules.
To express that a vector space is a linear span of a subset , one commonly uses the following phrases—either: spans , is a spanning set of , is spanned/generated by , or is a generator or generator set of .
Definition
Given a vector space over a field , the span of a set of vectors (not necessarily finite) is defined to be the intersection of all subspaces of that contain . is referred to as the subspace spanned by , or by the vectors in . Conversely, is called a spanning set of , and we say that spans .
Alternatively, the span of may be defined as the set of all finite linear combinations of elements (vectors) of , which follows from the above definition.
In the case of infinite , infinite linear combinations (i.e. where a combination may involve an infinite sum, assuming that such sums are defined somehow as in, say, a Banach space) are excluded by the definition; a generalization that allows these is not equivalent.
Examples
The real vector space has {(−1, 0, 0), (0, 1, 0), (0, 0, 1)} as a spanning set. This particular spanning set is also a basis. If (−1, 0, 0) were replaced by (1, 0, 0), it would also form the canonical basis of .
Another spanning set for the same space is given by {(1, 2, 3), (0, 1, 2), (−1, , 3), (1, 1, 1)}, but this set is not a basis, because it is linearly dependent.
The set } is not a spanning set of , since its span is the space of all vectors in whose last component is zero. That space is also spanned by the set {(1, 0, 0), (0, 1, 0)}, as (1, 1, 0) is a linear combination of (1, 0, 0) and (0, 1, 0). Thus, the spanned space is not It can be identified with by removing the third components equal to zero.
The empty set is a spanning set of {(0, 0, 0)}, since the empty set is a subset of all possible vector spaces in , and {(0, 0, 0)} is the intersection of all of these vector spaces.
The set of monomials , where is a non-negative integer, spans the space of polynomials.
Theorems
Equivalence of definitions
The set of all linear combinations of a subset of , a vector space over , is the smallest linear subspace of containing .
Proof. We first prove that is a subspace of . Since is a subset of , we only need to prove the existence of a zero vector in , that is closed under addition, and that is closed under scalar multiplication. Letting , it is trivial that the zero vector of exists in , since . Adding together two linear combinations of also produces a linear combinat |
https://en.wikipedia.org/wiki/Linear%20subspace | In mathematics, and more specifically in linear algebra, a linear subspace or vector subspace is a vector space that is a subset of some larger vector space. A linear subspace is usually simply called a subspace when the context serves to distinguish it from other types of subspaces.
Definition
If V is a vector space over a field K and if W is a subset of V, then W is a linear subspace of V if under the operations of V, W is a vector space over K. Equivalently, a nonempty subset W is a linear subspace of V if, whenever are elements of W and are elements of K, it follows that is in W.
As a corollary, all vector spaces are equipped with at least two (possibly different) linear subspaces: the zero vector space consisting of the zero vector alone and the entire vector space itself. These are called the trivial subspaces of the vector space.
Examples
Example I
In the vector space V = R3 (the real coordinate space over the field R of real numbers), take W to be the set of all vectors in V whose last component is 0.
Then W is a subspace of V.
Proof:
Given u and v in W, then they can be expressed as and . Then . Thus, u + v is an element of W, too.
Given u in W and a scalar c in R, if again, then . Thus, cu is an element of W too.
Example II
Let the field be R again, but now let the vector space V be the Cartesian plane R2.
Take W to be the set of points (x, y) of R2 such that x = y.
Then W is a subspace of R2.
Proof:
Let and be elements of W, that is, points in the plane such that p1 = p2 and q1 = q2. Then ; since p1 = p2 and q1 = q2, then p1 + q1 = p2 + q2, so p + q is an element of W.
Let p = (p1, p2) be an element of W, that is, a point in the plane such that p1 = p2, and let c be a scalar in R. Then ; since p1 = p2, then cp1 = cp2, so cp is an element of W.
In general, any subset of the real coordinate space Rn that is defined by a system of homogeneous linear equations will yield a subspace.
(The equation in example I was z = 0, and the equation in example II was x = y.)
Example III
Again take the field to be R, but now let the vector space V be the set RR of all functions from R to R.
Let C(R) be the subset consisting of continuous functions.
Then C(R) is a subspace of RR.
Proof:
We know from calculus that .
We know from calculus that the sum of continuous functions is continuous.
Again, we know from calculus that the product of a continuous function and a number is continuous.
Example IV
Keep the same field and vector space as before, but now consider the set Diff(R) of all differentiable functions.
The same sort of argument as before shows that this is a subspace too.
Examples that extend these themes are common in functional analysis.
Properties of subspaces
From the definition of vector spaces, it follows that subspaces are nonempty, and are closed under sums and under scalar multiples. Equivalently, subspaces can be characterized by the property of being closed under linear combinations. That is, a nonempty set W is a |
https://en.wikipedia.org/wiki/Julia%20set | In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values.
Thus the behavior of the function on the Fatou set is "regular", while on the Julia set its behavior is "chaotic".
The Julia set of a function is commonly denoted and the Fatou set is denoted These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century.
Formal definition
Let be a non-constant holomorphic function from the Riemann sphere onto itself. Such functions are precisely the non-constant complex rational functions, that is, where and are complex polynomials. Assume that p and q have no common roots, and at least one has degree larger than 1. Then there is a finite number of open sets that are left invariant by and are such that:
The union of the sets is dense in the plane and
behaves in a regular and equal way on each of the sets .
The last statement means that the termini of the sequences of iterations generated by the points of are either precisely the same set, which is then a finite cycle, or they are finite cycles of circular or annular shaped sets that are lying concentrically. In the first case the cycle is attracting, in the second case it is neutral.
These sets are the Fatou domains of , and their union is the Fatou set of . Each of the Fatou domains contains at least one critical point of , that is, a (finite) point z satisfying , or if the degree of the numerator is at least two larger than the degree of the denominator , or if for some c and a rational function satisfying this condition.
The complement of is the Julia set of . If all the critical points are preperiodic, that is they are not periodic but eventually land on a periodic cycle, then is all the sphere. Otherwise, is a nowhere dense set (it is without interior points) and an uncountable set (of the same cardinality as the real numbers). Like , is left invariant by , and on this set the iteration is repelling, meaning that for all w in a neighbourhood of z (within ). This means that behaves chaotically on the Julia set. Although there are points in the Julia set whose sequence of iterations is finite, there are only a countable number of such points (and they make up an infinitesimal part of the Julia set). The sequences generated by points outside this set behave chaotically, a phenomenon called deterministic chaos.
There has been extensive research on the Fatou set and Julia set of iterated rational functions, known as rational maps. For example, it i |
https://en.wikipedia.org/wiki/Hyperbolic%20functions | In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the unit hyperbola. Also, similarly to how the derivatives of and are and respectively, the derivatives of and are and respectively.
Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity.
The basic hyperbolic functions are:
hyperbolic sine "" (),
hyperbolic cosine "" (),
from which are derived:
hyperbolic tangent "" (),
hyperbolic cosecant "" or "" ()
hyperbolic secant "" (),
hyperbolic cotangent "" (),
corresponding to the derived trigonometric functions.
The inverse hyperbolic functions are:
area hyperbolic sine "" (also denoted "", "" or sometimes "")
area hyperbolic cosine "" (also denoted "", "" or sometimes "")
and so on.
The hyperbolic functions take a real argument called a hyperbolic angle. The size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector.
In complex analysis, the hyperbolic functions arise when applying the ordinary sine and cosine functions to an imaginary angle. The hyperbolic sine and the hyperbolic cosine are entire functions. As a result, the other hyperbolic functions are meromorphic in the whole complex plane.
By Lindemann–Weierstrass theorem, the hyperbolic functions have a transcendental value for every non-zero algebraic value of the argument.
Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert. Riccati used and () to refer to circular functions and and () to refer to hyperbolic functions. Lambert adopted the names, but altered the abbreviations to those used today. The abbreviations , , , are also currently used, depending on personal preference.
Notation
Definitions
There are various equivalent ways to define the hyperbolic functions.
Exponential definitions
In terms of the exponential function:
Hyperbolic sine: the odd part of the exponential function, that is,
Hyperbolic cosine: the even part of the exponential function, that is,
Hyperbolic tangent:
Hyperbolic cotangent: for ,
Hyperbolic secant:
Hyperbolic cosecant: for ,
Differential equation definitions
The hyperbolic functions may be defined as solutions of differential equations: The hyperbolic sine and cosine are the solution of the system
with the initial conditions The initial conditions make the solution unique; without them any pair |
https://en.wikipedia.org/wiki/Fuzzy%20set | In mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set.
At the same time, defined a more general kind of structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics , decision-making , and clustering , are special cases of L-relations when L is the unit interval [0, 1].
In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition—an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1]. Fuzzy sets generalize classical sets, since the indicator functions (aka characteristic functions) of classical sets are special cases of the membership functions of fuzzy sets, if the latter only takes values 0 or 1. In fuzzy set theory, classical bivalent sets are usually called crisp sets. The fuzzy set theory can be used in a wide range of domains in which information is incomplete or imprecise, such as bioinformatics.
Definition
A fuzzy set is a pair where is a set (often required to be non-empty) and a membership function.
The reference set (sometimes denoted by or ) is called universe of discourse, and for each the value is called the grade of membership of in .
The function is called the membership function of the fuzzy set .
For a finite set the fuzzy set is often denoted by
Let . Then is called
not included in the fuzzy set if (no member),
fully included if (full member),
partially included if
The (crisp) set of all fuzzy sets on a universe is denoted with (or sometimes just ).
Crisp sets related to a fuzzy set
For any fuzzy set and the following crisp sets are defined:
is called its α-cut (aka α-level set)
is called its strong α-cut (aka strong α-level set)
is called its support
is called its core (or sometimes kernel ).
Note that some authors understand "kernel" in a different way; see below.
Other definitions
A fuzzy set is empty () iff (if and only if)
Two fuzzy sets and are equal () iff
A fuzzy set is included in a fuzzy set () iff
For any fuzzy set , any element that satisfies
is called a crossover point.
Given a fuzzy set , any , for which is not empty, is called a level of A.
The level set of A is the set of all levels representing distinct cuts. It is the image of :
For a fuzzy set , its height is given by
where denotes the supremum, which exists because is non-empty and bounded above by 1. If U is finite, we can simply replace the supremum by the maximum.
A fuzzy set is said to be normalized iff
In the finite case, wh |
https://en.wikipedia.org/wiki/Geography%20of%20Mauritius | Mauritius is an island of Africa's southeast coast located in the Indian Ocean, east of Madagascar. It is geologically located within the Somali plate.
Statistics
Area (includes Agaléga, Cargados Carajos (Saint Brandon), and Rodrigues):
total:
2,011 km2
land:
2,030 km2
water:
10 km2
note: includes Agalega Islands, Cargados Carajos Shoais (Saint Brandon), and Rodrigues.
Coastline:
177 km
Maritime claims:
territorial sea:
continental shelf:
or to the edge of the continental margin
exclusive economic zone:
Elevation extremes:
lowest point:
Indian Ocean 0 m
highest point:
Piton de la Petite Rivière Noire 828 m
Natural resources:
arable land, fish
Land use:
arable land:
38.24%
permanent crops:
1.96%
other:
59.80% (2011)
Irrigated land:
212.2 km2 (2003)
Total renewable water resources:
2.75 km3 (2011)
Environment - current issues:
water pollution, degradation of coral reefs
Environment - international agreements:
party to:
Antarctic-Marine Living Resources, Biodiversity, Climate Change, Climate Change-Kyoto Protocol, Desertification, Endangered Species, Environmental Modification, Hazardous Wastes, Law of the Sea, Marine Life Conservation, Ozone Layer Protection, Ship Pollution, Wetlands
Geography - note:
The main island is from which the country derives its name, former home of the dodo, a large flightless bird related to pigeons, driven to extinction by the end of the 17th century through a combination of hunting and the introduction of predatory species.
Table of Islands
notes: excludes Tromelin and other îles éparses
Climate
The local climate is tropical, modified by southeast trade winds; there is a warm, dry winter from May to November and a hot, wet, and humid summer from November to May. Anticyclones affect the country during May to September.
Cyclones affect Mauritius during November–April. Hollanda (1994) and Dina (2002) were the worst two of the more recent cyclones to have affected the island.
Terrain
The country's landscape consists of a small coastal plain rising to discontinuous mountains encircling a central plateau. Mauritius is almost completely surrounded by reefs that may pose maritime hazards. The main island is of volcanic origin.
The mountains with the greatest prominence include:
Piton de la Petite Rivière Noire, 828 m, the highest point of the island
Le Morne Brabant, 556 m
Tourelle de Tamarin, 563 m
Corps de Garde, 720 m, prominence 382 m
Le Pouce, 820 m, prominence 352 m
Pieter Both, 820 m, prominence 229 m
Montagne Cocotte, 780 m
Extreme points
This is a list of the extreme points of Mauritius, the points that are farther north, south, east or west than any other location.
Northernmost point – Tappe à Terre, North Island, Agaléga Islands
Easternmost point – Trou d’Argent, Rodrigues Island
Southernmost point - Le Gris Gris, Savanne District, Mauritius
Westernmost point - North West Point, North Island, Agaléga Islands
See also
Outer islands of Mauritius
References
External links
Ma |
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