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Perrin Number
In mathematics, the Perrin numbers are a doubly infinite constant-recursive sequence, constant-recursive integer sequence with Characteristic equation (calculus), characteristic equation . The Perrin numbers, named after the French engineer , bear the same relationship to the Padovan sequence as the Lucas numbers do to the Fibonacci sequence. Definition The Perrin numbers are defined by the recurrence relation :\begin P(0)&=3, \\ P(1)&=0, \\ P(2)&=2, \\ P(n)&=P(n-2) +P(n-3) \mboxn>2, \end and the reverse :P(n) =P(n+3) -P(n+1) \mboxn<0. The first few terms in both directions are Perrin numbers can be expressed as sums of the three initial terms : |
Perrin Triangles
Perrin may refer to: Places in the United States *Perrin, Missouri, an unincorporated community *Perrin, Texas, an unincorporated community in southeastern Jack County, Texas Other *Famille Perrin, French winery owners *Perrin friction factors, in hydrodynamics *Perrin number, in mathematics *Éditions Perrin, a publishing house (est. 1827) *Perrin's beaked whale, a recently described species of whale *Perrin's cave beetle, an extinct freshwater beetle from France *Towers Perrin, a global professional services firm People Surname *Abner Monroe Perrin (1827–1864), Confederate States Army general *Alain Perrin (born 1956), French association football coach, former manager of China national team *Ami Perrin (died 1561), Swiss opponent of Calvinism reform *Benjamin Perrin, Canadian professor *Benny Perrin (1959–2017), American football safety *Bernadette Perrin-Riou (born 1955), French number theorist *Carmen Perrin (born 1953), Bolivian-born Swiss artist and educator *Cédric Pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maximal Independent Set
In graph theory, a maximal independent set (MIS) or maximal stable set is an Independent set (graph theory), independent set that is not a subset of any other independent set. In other words, there is no Vertex (graph theory), vertex outside the independent set that may join it because it is maximal with respect to the independent set property. For example, in the graph , a Path graph, path with three vertices , , and , and two edges and , the sets and are both maximal independent. The set is independent, but is not maximal independent, because it is a subset of the larger independent set In this same graph, the maximal cliques are the sets and A MIS is also a dominating set in the graph, and every dominating set that is independent must be maximal independent, so MISs are also called independent dominating sets. A graph may have many MISs of widely varying sizes; the largest, or possibly several equally large, MISs of a graph is called a maximum independent set, ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucas's Theorem
In number theory, Lucas's theorem expresses the remainder of division of the binomial coefficient \tbinom by a prime number ''p'' in terms of the base ''p'' expansions of the integers ''m'' and ''n''. Lucas's theorem first appeared in 1878 in papers by Édouard Lucas. Statement For non-negative integers ''m'' and ''n'' and a prime ''p'', the following congruence relation holds: :\binom\equiv\prod_^k\binom\pmod p, where :m=m_kp^k+m_p^+\cdots +m_1p+m_0, and :n=n_kp^k+n_p^+\cdots +n_1p+n_0 are the base ''p'' expansions of ''m'' and ''n'' respectively. This uses the convention that \tbinom = 0 if ''m'' < ''n''. Proofs There are several ways to prove Lucas's theorem.Consequences * A binomial coefficient is divisible by a prime ''p'' if and only if at least one of the digits of the base-''p'' representation of ''n'' is greater than the corresponding digit of ''m''. * In particular, |
Symmetric Polynomial
In mathematics, a symmetric polynomial is a polynomial in variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, is a ''symmetric polynomial'' if for any permutation of the subscripts one has . Symmetric polynomials arise naturally in the study of the relation between the roots of a polynomial in one variable and its coefficients, since the coefficients can be given by polynomial expressions in the roots, and all roots play a similar role in this setting. From this point of view the elementary symmetric polynomials are the most fundamental symmetric polynomials. Indeed, a theorem called the fundamental theorem of symmetric polynomials states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials. This implies that every ''symmetric'' polynomial expression in the roots of a monic polynomial can alternatively be given as a polynomial expression in the coefficients of the polynomial. Symme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Symmetric Polynomials
Elementary may refer to: Arts, entertainment, and media Music * Elementary (Cindy Morgan album), ''Elementary'' (Cindy Morgan album), 2001 * Elementary (The End album), ''Elementary'' (The End album), 2007 * ''Elementary'', a Melvin "Wah-Wah Watson" Ragin album, 1977 Other uses in arts, entertainment, and media * Elementary (TV series), ''Elementary'' (TV series), a 2012 American drama television series * "Elementary, my dear Watson", a Sherlock Holmes#"Elementary, my dear Watson", catchphrase of Sherlock Holmes Education * Elementary and Secondary Education Act, US * Elementary education, or primary education, the first years of formal, structured education * Elementary Education Act 1870, England and Wales * Elementary school, a school providing elementary or primary education Science and technology * ELEMENTARY, a class of objects in computational complexity theory * Elementary, a widget set based on the Enlightenment Foundation Libraries#Elementary, Enlightenment Foundation L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vieta's Formulas
In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (1540-1603), more commonly referred to by the Latinised form of his name, "Franciscus Vieta." Basic formulas Any general polynomial of degree ''n'' P(x) = a_n x^n + a_x^ + \cdots + a_1 x + a_0 (with the coefficients being real or complex numbers and ) has (not necessarily distinct) complex roots by the fundamental theorem of algebra. Vieta's formulas relate the polynomial coefficients to signed sums of products of the roots as follows: Vieta's formulas can equivalently be written as \sum_ \left(\prod_^k r_\right)=(-1)^k\frac for (the indices are sorted in increasing order to ensure each product of roots is used exactly once). The left-hand sides of Vieta's formulas are the elementary symmetric polynomials of the roots. Vieta's system can be solved by Newton's method through an explicit simple iterative formula, the Dura ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Coordinate System
In mathematics, the polar coordinate system specifies a given point in a plane by using a distance and an angle as its two coordinates. These are *the point's distance from a reference point called the ''pole'', and *the point's direction from the pole relative to the direction of the ''polar axis'', a ray drawn from the pole. The distance from the pole is called the ''radial coordinate'', ''radial distance'' or simply ''radius'', and the angle is called the ''angular coordinate'', ''polar angle'', or ''azimuth''. The pole is analogous to the origin in a Cartesian coordinate system. Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as spirals. Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more intuitive to model using polar coordinates. The polar coordinate system i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lucas Sequence
In mathematics, the Lucas sequences U_n(P,Q) and V_n(P, Q) are certain constant-recursive integer sequences that satisfy the recurrence relation : x_n = P \cdot x_ - Q \cdot x_ where P and Q are fixed integers. Any sequence satisfying this recurrence relation can be represented as a linear combination of the Lucas sequences U_n(P, Q) and V_n(P, Q). More generally, Lucas sequences U_n(P, Q) and V_n(P, Q) represent sequences of polynomials in P and Q with integer coefficients. Famous examples of Lucas sequences include the Fibonacci numbers, Mersenne numbers, Pell numbers, Lucas numbers, Jacobsthal numbers, and a superset of Fermat numbers (see below). Lucas sequences are named after the French mathematician Édouard Lucas. Recurrence relations Given two integer parameters P and Q, the Lucas sequences of the first kind U_n(P,Q) and of the second kind V_n(P,Q) are defined by the recurrence relations: :\begin U_0(P,Q)&=0, \\ U_1(P,Q)&=1, \\ U_n(P,Q)&=P\cdot U_(P,Q)-Q\cdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a and b are real numbers, then the complex conjugate of a + bi is a - bi. The complex conjugate of z is often denoted as \overline or z^*. In polar form, if r and \varphi are real numbers then the conjugate of r e^ is r e^. This can be shown using Euler's formula. The product of a complex number and its conjugate is a real number: a^2 + b^2 (or r^2 in polar coordinates). If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root. Notation The complex conjugate of a complex number z is written as \overline z or z^*. The first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate. The second is preferred in physics, where ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plastic Ratio
In mathematics, the plastic ratio is a geometrical aspect ratio, proportion, given by the unique real polynomial root, solution of the equation Its decimal expansion begins as . The adjective ''plastic'' does not refer to Plastic, the artificial material, but to the formative and sculptural qualities of this ratio, as in ''plastic arts''. Definition Three quantities are in the plastic ratio if \frac =\frac =\frac The ratio is commonly denoted Substituting b=\rho c \, and a=\rho b =\rho^2 c \, in the middle fraction, \rho =\frac. It follows that the plastic ratio is the unique real solution of the cubic equation \rho^3 -\rho -1 =0. Solving with Cubic equation#Cardano's formula, Cardano's formula, \begin w_ &=\frac12 \left( 1 \pm \frac13 \sqrt \right) \\ \rho &=\sqrt[3] +\sqrt[3] \end or, using the Cubic equation#Trigonometric and hyperbolic solutions, hyperbolic cosine, :\rho =\frac \cosh \left( \frac \operatorname \left( \frac \right) \right). is the superstabl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubic Equation
In algebra, a cubic equation in one variable is an equation of the form ax^3+bx^2+cx+d=0 in which is not zero. The solutions of this equation are called roots of the cubic function defined by the left-hand side of the equation. If all of the coefficients , , , and of the cubic equation are real numbers, then it has at least one real root (this is true for all odd-degree polynomial functions). All of the roots of the cubic equation can be found by the following means: * algebraically: more precisely, they can be expressed by a ''cubic formula'' involving the four coefficients, the four basic arithmetic operations, square roots, and cube roots. (This is also true of quadratic (second-degree) and quartic (fourth-degree) equations, but not for higher-degree equations, by the Abel–Ruffini theorem.) * trigonometrically * numerical approximations of the roots can be found using root-finding algorithms such as Newton's method. The coefficients do not need to be real ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
