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sense that it is a function of the coordinate system transforming functorially under coordinate changes. Examples of objects obeying more general kinds of transformation laws are jets and, more generally still, natural bundles. === Spinors === When changing from one orthonormal basis (called a frame) to another by a ro... |
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while a... |
An on which the functions in S simultaneously vanish, that is to say Z ( S ) = { x ∈ A n ∣ f ( x ) = 0 for all f ∈ S } . {\displaystyle Z(S)=\left\{x\in \mathbf {A} ^{n}\mid f(x)=0{\text{ for all }}f\in S\right\}.} A subset V of An is called an affine algebraic set if V = Z(S) for some S.: 2 A nonempty affine algebraic... |
by this ideal.: 10 A quasi-projective variety is a Zariski open subset of a projective variety. Notice that every affine variety is quasi-projective. Notice also that the complement of an algebraic set in an affine variety is a quasi-projective variety; in the context of affine varieties, such a quasi-projective variet... |
is straightforward to construct toric varieties that are not quasi-projective but complete. == Examples == === Subvariety === A subvariety is a subset of a variety that is itself a variety (with respect to the topological structure induced by the ambient variety). For example, every open subset of a variety is a variet... |
a single point. Let A3 be the three-dimensional affine space over C. The set of points (x, x2, x3) for x in C is an algebraic variety, and more precisely an algebraic curve that is not contained in any plane. It is the twisted cubic shown in the above figure. It may be defined by the equations y − x 2 = 0 z − x 3 = 0 {... |
) {\displaystyle \operatorname {GL} _{n}(k)} is the localization k [ x i j ∣ 0 ≤ i , j ≤ n ] [ det − 1 ] {\displaystyle k[x_{ij}\mid 0\leq i,j\leq n][{\det }^{-1}]} , which can be identified with k [ x i j , t ∣ 0 ≤ i , j ≤ n ] / ( t det − 1 ) {\displaystyle k[x_{ij},t\mid 0\leq i,j\leq n]/(t\det -1)} . The multiplicat... |
defined by x = 0. For another example, first consider the affine cubic curve y 2 = x 3 − x . {\displaystyle y^{2}=x^{3}-x.} in the 2-dimensional affine space (over a field of characteristic not two). It has the associated cubic homogeneous polynomial equation: y 2 z = x 3 − x z 2 , {\displaystyle y^{2}z=x^{3}-xz^{2},} ... |
{\displaystyle \operatorname {Jac} (C)} is a projective variety. The tangent space to Jac ( C ) {\displaystyle \operatorname {Jac} (C)} at the identity element is naturally isomorphic to H 1 ( C , O C ) ; {\displaystyle \operatorname {H} ^{1}(C,{\mathcal {O}}_{C});} hence, the dimension of Jac ( C ) {\displaystyl... |
curves exemplifies a typical situation: a moduli of nice objects tend not to be projective but only quasi-projective. Another case is a moduli of vector bundles on a curve. Here, there are the notions of stable and semistable vector bundles on a smooth complete curve C {\displaystyle C} . The moduli of semistable vecto... |
nor projective. To give an example, let X = P1 × A1 and p: X → A1 the projection. Here X is an algebraic variety since it is a product of varieties. It is not affine since P1 is a closed subvariety of X (as the zero locus of p), but an affine variety cannot contain a projective variety of positive dimension as a closed... |
projective varieties is projective. == Isomorphism of algebraic varieties == Let V1, V2 be algebraic varieties. We say V1 and V2 are isomorphic, and write V1 ≅ V2, if there are regular maps φ : V1 → V2 and ψ : V2 → V1 such that the compositions ψ ∘ φ and φ ∘ ψ are the identity maps on V1 and V2 respectively. == Discuss... |
aren't algebraically closed), so the rings R may not be integral domains. A more significant modification is to allow nilpotents in the sheaf of rings, that is, rings which are not reduced. This is one of several generalizations of classical algebraic geometry that are built into Grothendieck's theory of schemes. Allow... |
In abstract algebra, a magma, binar, or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation that must be closed by definition. No other properties are imposed. == History and terminology == The term groupoid was introduced in 1927 by ... |
of the magma (M, •) to (N, ∗). proof: log x y = log x + log y 2 {\displaystyle \log {\sqrt {xy}}\ =\ {\frac {\log x+\log y}{2}}} Note that these commutative magmas are not associative; nor do they have an identity element. This morphism of magmas has been used in economics since 1863 when W. Stanley Jevons calcul... |
(a • a) • b = ((a)(a))(b). MX can be described as the set of non-associative words on X with parentheses retained. It can also be viewed, in terms familiar in computer science, as the magma of full binary trees with leaves labelled by elements of X. The operation is that of joining trees at the root. A free magma has t... |
z, relation xy = xz implies y = z Right-cancellative If, for all x, y, z, relation yx = zx implies y = z Cancellative If it is both right-cancellative and left-cancellative A semigroup with left zeros If it is a semigroup and it satisfies the identity xy ≡ x A semigroup with right zeros If it is a semigroup and it sati... |
In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mat... |
spin-off domains. The classification of finite simple groups is a vast body of work from the mid 20th century, classifying all the finite simple groups. == Main classes of groups == The range of groups being considered has gradually expanded from finite permutation groups and special examples of matrix groups to abstra... |
numbers, permutations, or matrices. It was not until the late nineteenth century that the idea of an abstract group began to take hold, where "abstract" means that the nature of the elements are ignored in such a way that two isomorphic groups are considered as the same group. A typical way of specifying an abstract gr... |
Γ. A comparatively recent trend in the theory of finite groups exploits their connections with compact topological groups (profinite groups): for example, a single p-adic analytic group G has a family of quotients which are finite p-groups of various orders, and properties of G translate into the properties of its fini... |
easily manageable than the whole V (via Schur's lemma). Given a group G, representation theory then asks what representations of G exist. There are several settings, and the employed methods and obtained results are rather different in every case: representation theory of finite groups and representations of Lie groups... |
connection of graphs via their fundamental groups. A fundamental theorem of this area is that every subgroup of a free group is free. There are several natural questions arising from giving a group by its presentation. The word problem asks whether two words are effectively the same group element. By relating the probl... |
that exchanges the two roots is the Galois group belonging to the equation. Every polynomial equation in one variable has a Galois group, that is a certain permutation group on its roots. The axioms of a group formalize the essential aspects of symmetry. Symmetries form a group: they are closed because if you take a sy... |
are spaces with prescribed homotopy groups. Similarly algebraic K-theory relies in a way on classifying spaces of groups. Finally, the name of the torsion subgroup of an infinite group shows the legacy of topology in group theory. === Algebraic geometry === Algebraic geometry likewise uses group theory in many ways. Ab... |
classify regular polyhedra, and the symmetries of molecules, and space groups to classify crystal structures. The assigned groups can then be used to determine physical properties (such as chemical polarity and chirality), spectroscopic properties (particularly useful for Raman spectroscopy, infrared spectroscopy, circ... |
principal axis of rotation, it is called σh (horizontal). Other planes, which contain the principal axis of rotation, are labeled vertical (σv) or dihedral (σd). Inversion (i ) is a more complex operation. Each point moves through the center of the molecule to a position opposite the original position and as far from t... |
and Galois theory. Free downloadable PDF with open-source GFDL license. Kleiner, Israel (1986), "The evolution of group theory: a brief survey", Mathematics Magazine, 59 (4): 195–215, doi:10.2307/2690312, ISSN 0025-570X, JSTOR 2690312, MR 0863090 La Harpe, Pierre de (2000), Topics in geometric group theory, University ... |
Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into two or three major branches: the natural sciences (e.g., physics, chemistry, and biology), which study the physical world; and the social... |
health care, public infrastructure, and environmental protection. == Etymology == The word science has been used in Middle English since the 14th century in the sense of "the state of knowing". The word was borrowed from the Anglo-Norman language as the suffix -cience, which was borrowed from the Latin word scientia, m... |
behaviour, and astrology for divinatory purposes. The Mesopotamians had an intense interest in medicine and the earliest medical prescriptions appeared in Sumerian during the Third Dynasty of Ur. They seem to have studied scientific subjects which had practical or religious applications and had little interest in satis... |
the beginnings of calculus. Pliny the Elder was a Roman writer and polymath, who wrote the seminal encyclopaedia Natural History. Positional notation for representing numbers likely emerged between the 3rd and 5th centuries CE along Indian trade routes. This numeral system made efficient arithmetic operations more acce... |
as by contributing to the improvement and development of technology such as the camera obscura and the telescope. At the start of the Renaissance, Roger Bacon, Vitello, and John Peckham each built up a scholastic ontology upon a causal chain beginning with sensation, perception, and finally apperception of the individu... |
other things. In Bacon's words, "the real and legitimate goal of sciences is the endowment of human life with new inventions and riches", and he discouraged scientists from pursuing intangible philosophical or spiritual ideas, which he believed contributed little to human happiness beyond "the fume of subtle, sublime o... |
of conservation of energy, conservation of momentum and conservation of mass suggested a highly stable universe where there could be little loss of resources. However, with the advent of the steam engine and the Industrial Revolution there was an increased understanding that not all forms of energy have the same energy... |
The Human Genome Project was completed in 2003 by identifying and mapping all of the genes of the human genome. The first induced pluripotent human stem cells were made in 2006, allowing adult cells to be transformed into stem cells and turn into any cell type found in the body. With the affirmation of the Higgs boson ... |
according to a set of rules. It includes mathematics, systems theory, and theoretical computer science. The formal sciences share similarities with the other two branches by relying on objective, careful, and systematic study of an area of knowledge. They are, however, different from the empirical sciences as they rely... |
basic assumptions that are needed to justify the scientific method: there is an objective reality shared by all rational observers; this objective reality is governed by natural laws; these laws were discovered by means of systematic observation and experimentation. Mathematics is essential in the formation of hypothes... |
archival record of science. The first scientific journals, Journal des sçavans followed by Philosophical Transactions, began publication in 1665. Since that time the total number of active periodicals has steadily increased. In 1981, one estimate for the number of scientific and technical journals in publication was 11... |
to science, first defined by Austrian-British philosopher Karl Popper. Popper rejected the way that empiricism describes the connection between theory and observation. He claimed that theories are not generated by observation, but that observation is made in the light of theories, and that the only way theory A can be ... |
maintain the quality of research methodology and objectivity when interpreting results. === Scientists === Scientists are individuals who conduct scientific research to advance knowledge in an area of interest. Scientists may exhibit a strong curiosity about reality and a desire to apply scientific knowledge for the be... |
projects are evaluated and only the most promising receive funding. Such processes, which are run by government, corporations, or foundations, allocate scarce funds. Total research funding in most developed countries is between 1.5% and 3% of GDP. In the OECD, around two-thirds of research and development in scientific... |
scientific claims in terms of their credibility within the scientific community as a whole. Determining how much weight to give different sides in a scientific debate may require considerable expertise regarding the matter. Few journalists have real scientific knowledge, and even beat reporters who are knowledgeable ab... |
scientific evidence. Tactics such as shifting conversation, failing to acknowledge facts, and capitalising on doubt of scientific consensus have been used to gain more attention for views that have been undermined by scientific evidence. Examples of issues that have involved the politicisation of science include the gl... |
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, in universal algebra one takes the class of groups as an object of stud... |
used to talk about these structures uses equations only. Not all algebraic structures in a wider sense fall into this scope. For example, ordered groups involve an ordering relation, so would not fall within this scope. The class of fields is not an equational class because there is no type (or "signature") in which al... |
and one nullary (signature (2, 1, 0)) 3 equational laws (associativity, identity, and inverse) no quantified laws (except outermost universal quantifiers, which are allowed in varieties) A key point is that the extra operations do not add information, but follow uniquely from the usual definition of a group. Although t... |
isomorphism theorems, which encompass the isomorphism theorems of groups, rings, modules, etc. Birkhoff's HSP Theorem, which states that a class of algebras is a variety if and only if it is closed under homomorphic images, subalgebras, and arbitrary direct products. == Motivations and applications == In addition to it... |
categories of a special sort, known as Lawvere theories or more generally algebraic theories. Alternatively, one can describe algebraic structures using monads. The two approaches are closely related, with each having their own advantages. In particular, every Lawvere theory gives a monad on the category of sets, while... |
light thereby thrown on the general theory of symbolic reasoning, and on algebraic symbolism in particular. The comparative study necessarily presupposes some previous separate study, comparison being impossible without knowledge." Whitehead, however, had no results of a general nature. Work on the subject was minimal ... |
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of single variable calculus, vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry ... |
geometry, in the time of the ancient Greek mathematicians. Famously, Eratosthenes calculated the circumference of the Earth around 200 BC, and around 150 AD Ptolemy in his Geography introduced the stereographic projection for the purposes of mapping the shape of the Earth. Implicitly throughout this time principles tha... |
Leibniz began the study of plane curves and the investigation of concepts such as points of inflection and circles of osculation, which aid in the measurement of curvature. Indeed, already in his first paper on the foundations of calculus, Leibniz notes that the infinitesimal condition d 2 y = 0 {\displaystyle d^{2}y=0... |
of a space curve on a surface in terms of the principal curvatures, known as Euler's theorem. Later in the 1700s, the new French school led by Gaspard Monge began to make contributions to differential geometry. Monge made important contributions to the theory of plane curves, surfaces, and studied surfaces of revolutio... |
Riemannian metric, denoted by d s 2 {\displaystyle ds^{2}} by Riemann, was the development of an idea of Gauss's about the linear element d s {\displaystyle ds} of a surface. At this time Riemann began to introduce the systematic use of linear algebra and multilinear algebra into the subject, making great use of the th... |
importance of the Einstein Field equations. Einstein's theory popularised the tensor calculus of Ricci and Levi-Civita and introduced the notation g {\displaystyle g} for a Riemannian metric, and Γ {\displaystyle \Gamma } for the Christoffel symbols, both coming from G in Gravitation. Élie Cartan helped reformulate the... |
necessarily flat, though they still resemble Euclidean space at each point infinitesimally, i.e. in the first order of approximation. Various concepts based on length, such as the arc length of curves, area of plane regions, and volume of solids all possess natural analogues in Riemannian geometry. The notion of a dire... |
symplectic manifolds which preserves the symplectic form is called a symplectomorphism. Non-degenerate skew-symmetric bilinear forms can only exist on even-dimensional vector spaces, so symplectic manifolds necessarily have even dimension. In dimension 2, a symplectic manifold is just a surface endowed with an area for... |
, endowed with a tensor of type (1, 1), i.e. a vector bundle endomorphism (called an almost complex structure) J : T M → T M {\displaystyle J:TM\rightarrow TM} , such that J 2 = − 1. {\displaystyle J^{2}=-1.\,} It follows from this definition that an almost complex manifold is even-dimensional. An almost complex manifo... |
results in differential geometry and differential topology. === Gauge theory === Gauge theory is the study of connections on vector bundles and principal bundles, and arises out of problems in mathematical physics and physical gauge theories which underpin the standard model of particle physics. Gauge theory is concern... |
structure additional to the intrinsic one. (See the Nash embedding theorem.) In the formalism of geometric calculus both extrinsic and intrinsic geometry of a manifold can be characterized by a single bivector-valued one-form called the shape operator. == Applications == Below are some examples of how differential geom... |
geometry of curves and surfaces. Englewood Cliffs, N.J.: Prentice-Hall. ISBN 978-0-13-212589-5. OCLC 1529515. Frankel, Theodore (2004). The geometry of physics : an introduction (2nd ed.). New York: Cambridge University Press. ISBN 978-0-521-53927-2. OCLC 51855212. Elsa Abbena; Simon Salamon; Alfred Gray (2017). Modern... |
In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a comm... |
of planar subalgebras as they are all isomorphic to C. == Subalgebras in universal algebra == In universal algebra, a subalgebra of an algebra A is a subset S of A that also has the structure of an algebra of the same type when the algebraic operations are restricted to S. If the axioms of a kind of algebraic structure... |
In mathematics, a linear equation is an equation that may be put in the form a 1 x 1 + … + a n x n + b = 0 , {\displaystyle a_{1}x_{1}+\ldots +a_{n}x_{n}+b=0,} where x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} are the variables (or unknowns), and b , a 1 , … , a n {\displaystyle b,a_{1},\ldots ,a_{n}} are the coe... |
not both 0. If a and b are real numbers, it has infinitely many solutions. === Linear function === If b ≠ 0, the equation a x + b y + c = 0 {\displaystyle ax+by+c=0} is a linear equation in the single variable y for every value of x. It therefore has a unique solution for y, which is given by y = − a b x − c b . {\disp... |
y = m x + y 0 . {\displaystyle y=mx+y_{0}.} If, moreover, the line is not horizontal, it can be defined by its slope and its x-intercept x0. In this case, its equation can be written y = m ( x − x 0 ) , {\displaystyle y=m(x-x_{0}),} or, equivalently, y = m x − m x 0 . {\displaystyle y=mx-mx_{0}.} These forms rely on th... |
verify that the two given points satisfy the equation). This form is not symmetric in the two given points, but a symmetric form can be obtained by regrouping the constant terms: ( y 1 − y 2 ) x + ( x 2 − x 1 ) y + ( x 1 y 2 − x 2 y 1 ) = 0 {\displaystyle (y_{1}-y_{2})x+(x_{2}-x_{1})y+(x_{1}y_{2}-x_{2}y_{1})=0} (exchan... |
is either inconsistent (for b ≠ 0) as having no solution, or all n-tuples are solutions. The n-tuples that are solutions of a linear equation in n variables are the Cartesian coordinates of the points of an (n − 1)-dimensional hyperplane in an n-dimensional Euclidean space (or affine space if the coefficients are compl... |
Algebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra. == History == The term "algebraic combinatorics" was introduced in the l... |
by Alfred Young, a mathematician at Cambridge University, in 1900. They were then applied to the study of the symmetric group by Georg Frobenius in 1903. Their theory was further developed by many mathematicians, including Percy MacMahon, W. V. D. Hodge, G. de B. Robinson, Gian-Carlo Rota, Alain Lascoux, Marcel-Paul Sc... |
In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0 {\displaystyle P=0} , where P is a polynomial with coefficients in some field, often the field of the rational numbers. For example, x 5 − 3 x + 1 = 0 {\displaystyle x^{5}-3x+1=0} is an algebraic equation with integer coeffici... |
AD) explicitly described the quadratic formula in his treatise Brāhmasphuṭasiddhānta published in 628 AD, but written in words instead of symbols. In the 9th century Muhammad ibn Musa al-Khwarizmi and other Islamic mathematicians derived the quadratic formula, the general solution of equations of degree 2, and recogniz... |
a polynomial equation in the four variables x, y, z, and T over the rational numbers. However, it is a polynomial equation in the three variables x, y, and z over the field of the elementary functions in the variable T. == Theory == === Polynomials === Given an equation in unknown x ( E ) a n x n + a n − 1 x n − 1 + ⋯ ... |
There exist formulas giving the solutions of real or complex polynomials of degree less than or equal to four as a function of their coefficients. Abel showed that it is not possible to find such a formula in general (using only the four arithmetic operations and taking roots) for equations of degree five or higher. Ga... |
. If the polynomial has real coefficients, it has: two distinct real roots if Δ > 0 {\displaystyle \Delta >0} ; one real double root if Δ = 0 {\displaystyle \Delta =0} ; no real root if Δ < 0 {\displaystyle \Delta <0} , but two complex conjugate roots. === Cubic equations === The best-known method for solving cubic equ... |
In algebra, a septic equation is an equation of the form a x 7 + b x 6 + c x 5 + d x 4 + e x 3 + f x 2 + g x + h = 0 , {\displaystyle ax^{7}+bx^{6}+cx^{5}+dx^{4}+ex^{3}+fx^{2}+gx+h=0,\,} where a ≠ 0. A septic function is a function of the form f ( x ) = a x 7 + b x 6 + c x 5 + d x 4 + e x 3 + f x 2 + g x + h {\displays... |
x − 4 = 0 {\displaystyle x^{7}-2x^{6}+(\alpha +1)x^{5}+(\alpha -1)x^{4}-\alpha x^{3}-(\alpha +5)x^{2}-6x-4=0\,} whose members appear in Kluner's Database of Number Fields. Its discriminant is Δ = − 4 4 ( 4 α 3 + 99 α 2 − 34 α + 467 ) 3 {\displaystyle \Delta =-4^{4}\left(4\alpha ^{3}+99\alpha ^{2}-34\alpha +467\right)^{... |
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and explores the relationships between these classifications. A computational problem is a task solved by a computer. A computation problem is solvable by mec... |
the following instance of the decision version of the travelling salesman problem: Is there a route of at most 2000 kilometres passing through all of Germany's 14 largest cities? The quantitative answer to this particular problem instance is of little use for solving other instances of the problem, such as asking for a... |
notion of function problems is much richer than the notion of decision problems. However, this is not really the case, since function problems can be recast as decision problems. For example, the multiplication of two integers can be expressed as the set of triples ( a , b , c ) {\displaystyle (a,b,c)} such that the re... |
such as a RAM machine, Conway's Game of Life, cellular automata, lambda calculus or any programming language can be computed on a Turing machine. Since Turing machines are easy to analyze mathematically, and are believed to be as powerful as any other model of computation, the Turing machine is the most commonly used m... |
by a deterministic Turing machine M {\displaystyle M} on input x {\displaystyle x} is the total number of state transitions, or steps, the machine makes before it halts and outputs the answer ("yes" or "no"). A Turing machine M {\displaystyle M} is said to operate within time f ( n ) {\displaystyle f(n)} if the time re... |
or smallest value in the list (so the list is never divided). In this case, the algorithm takes time O( n 2 {\displaystyle n^{2}} ). If we assume that all possible permutations of the input list are equally likely, the average time taken for sorting is O ( n log n ) {\displaystyle O(n\log n)} . The best case occurs w... |
such as "polynomial time", "logarithmic space", "constant depth", etc. Some complexity classes have complicated definitions that do not fit into this framework. Thus, a typical complexity class has a definition like the following: The set of decision problems solvable by a deterministic Turing machine within time f ( n... |
about how much more additional time or space is needed in order to increase the number of problems that can be solved. More precisely, the time hierarchy theorem states that D T I M E ( o ( f ( n ) ) ) ⊊ D T I M E ( f ( n ) ⋅ log ( f ( n ) ) ) {\displaystyle {\mathsf {DTIME}}{\big (}o(f(n)){\big )}\subsetneq {\mathsf... |
For complexity classes larger than P, polynomial-time reductions are commonly used. In particular, the set of problems that are hard for NP is the set of NP-hard problems. If a problem X {\displaystyle X} is in C {\displaystyle C} and hard for C {\displaystyle C} , then X {\displaystyle X} is said to be complete for C ... |
then there exist problems in NP {\displaystyle {\textsf {NP}}} that are neither in P {\displaystyle {\textsf {P}}} nor NP {\displaystyle {\textsf {NP}}} -complete. Such problems are called NP-intermediate problems. The graph isomorphism problem, the discrete logarithm problem and the integer factorization problem are e... |
{P}}\subseteq {\textsf {NP}}\subseteq {\textsf {PP}}\subseteq {\textsf {PSPACE}}} , but it is possible that P = PSPACE {\displaystyle {\textsf {P}}={\textsf {PSPACE}}} . If P {\displaystyle {\textsf {P}}} is not equal to NP {\displaystyle {\textsf {NP}}} , then P {\displaystyle {\textsf {P}}} is not equal to PSPACE {\d... |
that are EXPTIME-hard. If NP {\displaystyle {\textsf {NP}}} is not the same as P {\displaystyle {\textsf {P}}} , then NP-hard problems are also intractable in this sense. However, this identification is inexact: a polynomial-time solution with large degree or large leading coefficient grows quickly, and may be impracti... |
systems. == History == An early example of algorithm complexity analysis is the running time analysis of the Euclidean algorithm done by Gabriel Lamé in 1844. Before the actual research explicitly devoted to the complexity of algorithmic problems started off, numerous foundations were laid out by various researchers. M... |
A Guide to the Theory of NP-Completeness. Series of Books in the Mathematical Sciences (1st ed.). New York: W. H. Freeman and Company. ISBN 9780716710455. MR 0519066. OCLC 247570676. Goldreich, Oded (2008), Computational Complexity: A Conceptual Perspective, Cambridge University Press van Leeuwen, Jan, ed. (1990), Hand... |
In mathematics, a quadratic equation (from Latin quadratus 'square') is an equation that can be rearranged in standard form as a x 2 + b x + c = 0 , {\displaystyle ax^{2}+bx+c=0\,,} where the variable x represents an unknown number, and a, b, and c represent known numbers, where a ≠ 0. (If a = 0 and b ≠ 0 then the equa... |
can be found by several alternative methods. === Factoring by inspection === It may be possible to express a quadratic equation ax2 + bx + c = 0 as a product (px + q)(rx + s) = 0. In some cases, it is possible, by simple inspection, to determine values of p, q, r, and s that make the two forms equivalent to one another... |
4 = 0 {\displaystyle 2x^{2}+4x-4=0} x 2 + 2 x − 2 = 0 {\displaystyle \ x^{2}+2x-2=0} x 2 + 2 x = 2 {\displaystyle \ x^{2}+2x=2} x 2 + 2 x + 1 = 2 + 1 {\displaystyle \ x^{2}+2x+1=2+1} ( x + 1 ) 2 = 3 {\displaystyle \left(x+1\right)^{2}=3} x + 1 = ± 3 {\displaystyle \ x+1=\pm {\sqrt {3}}} x = − 1 ± 3 {\displaystyle \ x=-... |
by zero for one root and an indeterminate form 0/0 for the other root. On the other hand, when c = 0, the more common formula yields two correct roots whereas this form yields the zero root and an indeterminate form 0/0. When neither a nor c is zero, the equality between the standard quadratic formula and Muller's meth... |
expressions i is the imaginary unit. Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative. === Geometric interpretation === The function f(x) = ax2 + bx + c is a quadratic function. The graph of any quadratic function has the sam... |
quadratic function f ( x ) = a x 2 + b x + c , {\displaystyle f(x)=ax^{2}+bx+c,} which is a parabola. If the parabola intersects the x-axis in two points, there are two real roots, which are the x-coordinates of these two points (also called x-intercept). If the parabola is tangent to the x-axis, there is a double root... |
lead to loss of up to half of correct significant figures in the roots. == Examples and applications == The golden ratio is found as the positive solution of the quadratic equation x 2 − x − 1 = 0. {\displaystyle x^{2}-x-1=0.} The equations of the circle and the other conic sections—ellipses, parabolas, and hyperbolas—... |
terms of x and y, were as follows: Compute half of p. Square the result. Subtract q. Find the (positive) square root using a table of squares. Add together the results of steps (1) and (4) to give x. In modern notation this means calculating x = p 2 + ( p 2 ) 2 − q {\displaystyle x={\frac {p}{2}}+{\sqrt {\left({\frac {... |
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