An Algebraic Approach to Harmonic Polynomials on S3
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Ladder Operators for Lam\'E Spheroconal Harmonic Polynomials
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 8 (2012), 074, 16 pages Ladder Operators for Lam´eSpheroconal Harmonic Polynomials? Ricardo MENDEZ-FRAGOSO´ yz and Eugenio LEY-KOO z y Facultad de Ciencias, Universidad Nacional Aut´onomade M´exico, M´exico E-mail: [email protected] URL: http://sistemas.fciencias.unam.mx/rich/ z Instituto de F´ısica, Universidad Nacional Aut´onomade M´exico, M´exico E-mail: eleykoo@fisica.unam.mx Received July 31, 2012, in final form October 09, 2012; Published online October 17, 2012 http://dx.doi.org/10.3842/SIGMA.2012.074 Abstract. Three sets of ladder operators in spheroconal coordinates and their respective actions on Lam´espheroconal harmonic polynomials are presented in this article. The poly- nomials are common eigenfunctions of the square of the angular momentum operator and of the asymmetry distribution Hamiltonian for the rotations of asymmetric molecules, in the body-fixed frame with principal axes. The first set of operators for Lam´epolynomials of a given species and a fixed value of the square of the angular momentum raise and lower and lower and raise in complementary ways the quantum numbers n1 and n2 counting the respective nodal elliptical cones. The second set of operators consisting of the cartesian components L^x, L^y, L^z of the angular momentum connect pairs of the four species of poly- nomials of a chosen kind and angular momentum. The third set of operators, the cartesian componentsp ^x,p ^y,p ^z of the linear momentum, connect pairs of the polynomials differing in one unit in their angular momentum and in their parities. -
10 7 General Vector Spaces Proposition & Definition 7.14
10 7 General vector spaces Proposition & Definition 7.14. Monomial basis of n(R) P Letn N 0. The particular polynomialsm 0,m 1,...,m n n(R) defined by ∈ ∈P n 1 n m0(x) = 1,m 1(x) = x, . ,m n 1(x) =x − ,m n(x) =x for allx R − ∈ are called monomials. The family =(m 0,m 1,...,m n) forms a basis of n(R) B P and is called the monomial basis. Hence dim( n(R)) =n+1. P Corollary 7.15. The method of equating the coefficients Letp andq be two real polynomials with degreen N, which means ∈ n n p(x) =a nx +...+a 1x+a 0 andq(x) =b nx +...+b 1x+b 0 for some coefficientsa n, . , a1, a0, bn, . , b1, b0 R. ∈ If we have the equalityp=q, which means n n anx +...+a 1x+a 0 =b nx +...+b 1x+b 0, (7.3) for allx R, then we can concludea n =b n, . , a1 =b 1 anda 0 =b 0. ∈ VL18 ↓ Remark: Since dim( n(R)) =n+1 and we have the inclusions P 0(R) 1(R) 2(R) (R) (R), P ⊂P ⊂P ⊂···⊂P ⊂F we conclude that dim( (R)) and dim( (R)) cannot befinite natural numbers. Sym- P F bolically, we write dim( (R)) = in such a case. P ∞ 7.4 Coordinates with respect to a basis 11 7.4 Coordinates with respect to a basis 7.4.1 Basis implies coordinates Again, we deal with the caseF=R andF=C simultaneously. Therefore, letV be an F-vector space with the two operations+ and . -
Selecting a Monomial Basis for Sums of Squares Programming Over a Quotient Ring
Selecting a Monomial Basis for Sums of Squares Programming over a Quotient Ring Frank Permenter1 and Pablo A. Parrilo2 Abstract— In this paper we describe a method for choosing for λi(x) and s(x), this feasibility problem is equivalent to a “good” monomial basis for a sums of squares (SOS) program an SDP. This follows because the sum of squares constraint formulated over a quotient ring. It is known that the monomial is equivalent to a semidefinite constraint on the coefficients basis need only include standard monomials with respect to a Groebner basis. We show that in many cases it is possible of s(x), and the equality constraint is equivalent to linear to use a reduced subset of standard monomials by combining equations on the coefficients of s(x) and λi(x). Groebner basis techniques with the well-known Newton poly- Naturally, the complexity of this sums of squares program tope reduction. This reduced subset of standard monomials grows with the complexity of the underlying variety, as yields a smaller semidefinite program for obtaining a certificate does the size of the corresponding SDP. It is therefore of non-negativity of a polynomial on an algebraic variety. natural to explore how algebraic structure can be exploited I. INTRODUCTION to simplify this sums of squares program. Consider the Many practical engineering problems require demonstrat- following reformulation [10], which is feasible if and only ing non-negativity of a multivariate polynomial over an if (2) is feasible: algebraic variety, i.e. over the solution set of polynomial Find s(x) equations. -
1 Expressing Vectors in Coordinates
Math 416 - Abstract Linear Algebra Fall 2011, section E1 Working in coordinates In these notes, we explain the idea of working \in coordinates" or coordinate-free, and how the two are related. 1 Expressing vectors in coordinates Let V be an n-dimensional vector space. Recall that a choice of basis fv1; : : : ; vng of V is n the same data as an isomorphism ': V ' R , which sends the basis fv1; : : : ; vng of V to the n standard basis fe1; : : : ; eng of R . In other words, we have ' n ': V −! R vi 7! ei 2 3 c1 6 . 7 v = c1v1 + ::: + cnvn 7! 4 . 5 : cn This allows us to manipulate abstract vectors v = c v + ::: + c v simply as lists of numbers, 2 3 1 1 n n c1 6 . 7 n the coordinate vectors 4 . 5 2 R with respect to the basis fv1; : : : ; vng. Note that the cn coordinates of v 2 V depend on the choice of basis. 2 3 c1 Notation: Write [v] := 6 . 7 2 n for the coordinates of v 2 V with respect to the basis fvig 4 . 5 R cn fv1; : : : ; vng. For shorthand notation, let us name the basis A := fv1; : : : ; vng and then write [v]A for the coordinates of v with respect to the basis A. 2 2 Example: Using the monomial basis f1; x; x g of P2 = fa0 + a1x + a2x j ai 2 Rg, we obtain an isomorphism ' 3 ': P2 −! R 2 3 a0 2 a0 + a1x + a2x 7! 4a15 : a2 2 3 a0 2 In the notation above, we have [a0 + a1x + a2x ]fxig = 4a15. -
Tight Monomials and the Monomial Basis Property
PDF Manuscript TIGHT MONOMIALS AND MONOMIAL BASIS PROPERTY BANGMING DENG AND JIE DU Abstract. We generalize a criterion for tight monomials of quantum enveloping algebras associated with symmetric generalized Cartan matrices and a monomial basis property of those associated with symmetric (classical) Cartan matrices to their respective sym- metrizable case. We then link the two by establishing that a tight monomial is necessarily a monomial defined by a weakly distinguished word. As an application, we develop an algorithm to compute all tight monomials in the rank 2 Dynkin case. The existence of Hall polynomials for Dynkin or cyclic quivers not only gives rise to a simple realization of the ±-part of the corresponding quantum enveloping algebras, but also results in interesting applications. For example, by specializing q to 0, degenerate quantum enveloping algebras have been investigated in the context of generic extensions ([20], [8]), while through a certain non-triviality property of Hall polynomials, the authors [4, 5] have established a monomial basis property for quantum enveloping algebras associated with Dynkin and cyclic quivers. The monomial basis property describes a systematic construction of many monomial/integral monomial bases some of which have already been studied in the context of elementary algebraic constructions of canonical bases; see, e.g., [15, 27, 21, 5] in the simply-laced Dynkin case and [3, 18], [9, Ch. 11] in general. This property in the cyclic quiver case has also been used in [10] to obtain an elementary construction of PBW-type bases, and hence, of canonical bases for quantum affine sln. In this paper, we will complete this program by proving this property for all finite types. -
Polynomial Interpolation CPSC 303: Numerical Approximation and Discretization
Lecture Notes 2: Polynomial Interpolation CPSC 303: Numerical Approximation and Discretization Ian M. Mitchell [email protected] http://www.cs.ubc.ca/~mitchell University of British Columbia Department of Computer Science Winter Term Two 2012{2013 Copyright 2012{2013 by Ian M. Mitchell This work is made available under the terms of the Creative Commons Attribution 2.5 Canada license http://creativecommons.org/licenses/by/2.5/ca/ Outline • Background • Problem statement and motivation • Formulation: The linear system and its conditioning • Polynomial bases • Monomial • Lagrange • Newton • Uniqueness of polynomial interpolant • Divided differences • Divided difference tables and the Newton basis interpolant • Divided difference connection to derivatives • Osculating interpolation: interpolating derivatives • Error analysis for polynomial interpolation • Reducing the error using the Chebyshev points as abscissae CPSC 303 Notes 2 Ian M. Mitchell | UBC Computer Science 2/ 53 Interpolation Motivation n We are given a collection of data samples f(xi; yi)gi=0 n • The fxigi=0 are called the abscissae (singular: abscissa), n the fyigi=0 are called the data values • Want to find a function p(x) which can be used to estimate y(x) for x 6= xi • Why? We often get discrete data from sensors or computation, but we want information as if the function were not discretely sampled • If possible, p(x) should be inexpensive to evaluate for a given x CPSC 303 Notes 2 Ian M. Mitchell | UBC Computer Science 3/ 53 Interpolation Formulation There are lots of ways to define a function p(x) to approximate n f(xi; yi)gi=0 • Interpolation means p(xi) = yi (and we will only evaluate p(x) for mini xi ≤ x ≤ maxi xi) • Most interpolants (and even general data fitting) is done with a linear combination of (usually nonlinear) basis functions fφj(x)g n X p(x) = pn(x) = cjφj(x) j=0 where cj are the interpolation coefficients or interpolation weights CPSC 303 Notes 2 Ian M. -
Arxiv:1805.04488V5 [Math.NA]
GENERALIZED STANDARD TRIPLES FOR ALGEBRAIC LINEARIZATIONS OF MATRIX POLYNOMIALS∗ EUNICE Y. S. CHAN†, ROBERT M. CORLESS‡, AND LEILI RAFIEE SEVYERI§ Abstract. We define generalized standard triples X, Y , and L(z) = zC1 − C0, where L(z) is a linearization of a regular n×n −1 −1 matrix polynomial P (z) ∈ C [z], in order to use the representation X(zC1 − C0) Y = P (z) which holds except when z is an eigenvalue of P . This representation can be used in constructing so-called algebraic linearizations for matrix polynomials of the form H(z) = zA(z)B(z)+ C ∈ Cn×n[z] from generalized standard triples of A(z) and B(z). This can be done even if A(z) and B(z) are expressed in differing polynomial bases. Our main theorem is that X can be expressed using ℓ the coefficients of the expression 1 = Pk=0 ekφk(z) in terms of the relevant polynomial basis. For convenience, we tabulate generalized standard triples for orthogonal polynomial bases, the monomial basis, and Newton interpolational bases; for the Bernstein basis; for Lagrange interpolational bases; and for Hermite interpolational bases. We account for the possibility of common similarity transformations. Key words. Standard triple, regular matrix polynomial, polynomial bases, companion matrix, colleague matrix, comrade matrix, algebraic linearization, linearization of matrix polynomials. AMS subject classifications. 65F15, 15A22, 65D05 1. Introduction. A matrix polynomial P (z) ∈ Fm×n[z] is a polynomial in the variable z with coef- ficients that are m by n matrices with entries from the field F. We will use F = C, the field of complex k numbers, in this paper. -
Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Outline
Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Outline 1 Interpolation 2 Polynomial Interpolation 3 Piecewise Polynomial Interpolation Michael T. Heath Scientific Computing 2 / 56 Chapter 7: Interpolation ❑ Topics: ❑ Examples ❑ Polynomial Interpolation – bases, error, Chebyshev, piecewise ❑ Orthogonal Polynomials ❑ Splines – error, end conditions ❑ Parametric interpolation ❑ Multivariate interpolation: f(x,y) Interpolation Motivation Polynomial Interpolation Choosing Interpolant Piecewise Polynomial Interpolation Existence and Uniqueness Interpolation Basic interpolation problem: for given data (t ,y ), (t ,y ),...(t ,y ) with t <t < <t 1 1 2 2 m m 1 2 ··· m determine function f : R R such that ! f(ti)=yi,i=1,...,m f is interpolating function, or interpolant, for given data Additional data might be prescribed, such as slope of interpolant at given points Additional constraints might be imposed, such as smoothness, monotonicity, or convexity of interpolant f could be function of more than one variable, but we will consider only one-dimensional case Michael T. Heath Scientific Computing 3 / 56 Interpolation Motivation Polynomial Interpolation Choosing Interpolant Piecewise Polynomial Interpolation Existence and Uniqueness Purposes for Interpolation Plotting smooth curve through discrete data points Reading between lines of table Differentiating or integrating tabular data Quick and easy evaluation of mathematical function Replacing complicated function by simple one Michael T. Heath Scientific Computing 4 / 56 Interpolation Motivation Polynomial Interpolation Choosing Interpolant Piecewise Polynomial Interpolation Existence and Uniqueness Interpolation vs Approximation By definition, interpolating function fits given data points exactly Interpolation is inappropriate if data points subject to significant errors It is usually preferable to smooth noisy data, for example by least squares approximation Approximation is also more appropriate for special function libraries Michael T. -
Spherical Harmonics and Homogeneous Har- Monic Polynomials
SPHERICAL HARMONICS AND HOMOGENEOUS HAR- MONIC POLYNOMIALS 1. The spherical Laplacean. Denote by S ½ R3 the unit sphere. For a function f(!) de¯ned on S, let f~ denote its extension to an open neighborhood N of S, constant along normals to S (i.e., constant along rays from the origin). We say f 2 C2(S) if f~ is a C2 function in N , and for such functions de¯ne a di®erential operator ¢S by: ¢Sf := ¢f;~ where ¢ on the right-hand side is the usual Laplace operator in R3. With a little work (omitted here) one may derive the expression for ¢ in polar coordinates (r; !) in R2 (r > 0;! 2 S): 2 1 ¢u = u + u + ¢ u: rr r r r2 S (Here ¢Su(r; !) is the operator ¢S acting on the function u(r; :) in S, for each ¯xed r.) A homogeneous polynomial of degree n ¸ 0 in three variables (x; y; z) is a linear combination of `monomials of degree n': d1 d2 d3 x y z ; di ¸ 0; d1 + d2 + d3 = n: This de¯nes a vector space (over R) denoted Pn. A simple combinatorial argument (involving balls and separators, as most of them do), seen in class, yields the dimension: 1 d := dim(P ) = (n + 1)(n + 2): n n 2 Writing a polynomial p 2 Pn in polar coordinates, we necessarily have: n p(r; !) = r f(!); f = pjS; where f is the restriction of p to S. This is an injective linear map p 7! f, but the functions on S so obtained are rather special (a dn-dimensional subspace of the in¯nite-dimensional space C(S) of continuous functions-let's call it Pn(S)) We are interested in the subspace Hn ½ Pn of homogeneous harmonic polynomials of degree n (¢p = 0). -
Harmonic Analysis on the Proper Velocity Gyrogroup ∗ 1 Introduction
Harmonic Analysis on the Proper Velocity gyrogroup ∗ Milton Ferreira School of Technology and Management, Polytechnic Institute of Leiria, Portugal 2411-901 Leiria, Portugal. Email: [email protected] and Center for Research and Development in Mathematics and Applications (CIDMA), University of Aveiro, 3810-193 Aveiro, Portugal. Email [email protected] Abstract In this paper we study harmonic analysis on the Proper Velocity (PV) gyrogroup using the gyrolanguage of analytic hyperbolic geometry. PV addition is the relativis- tic addition of proper velocities in special relativity and it is related with the hy- perboloid model of hyperbolic geometry. The generalized harmonic analysis depends on a complex parameter z and on the radius t of the hyperboloid and comprises the study of the generalized translation operator, the associated convolution operator, the generalized Laplace-Beltrami operator and its eigenfunctions, the generalized Poisson transform and its inverse, the generalized Helgason-Fourier transform, its inverse and Plancherel's Theorem. In the limit of large t; t ! +1; the generalized harmonic analysis on the hyperboloid tends to the standard Euclidean harmonic analysis on Rn; thus unifying hyperbolic and Euclidean harmonic analysis. Keywords: PV gyrogroup, Laplace Beltrami operator, Eigenfunctions, Generalized Helgason- Fourier transform, Plancherel's Theorem. 1 Introduction Harmonic analysis is the branch of mathematics that studies the representation of functions or signals as the superposition of basic waves called harmonics. It investigates and gen- eralizes the notions of Fourier series and Fourier transforms. In the past two centuries, it has become a vast subject with applications in diverse areas as signal processing, quantum mechanics, and neuroscience (see [18] for an overview). -
Poincaré Series and Homotopy Lie Algebras of Monomial Rings
ISSN: 1401-5617 Poincar´e series and homotopy Lie algebras of monomial rings Alexander Berglund Research Reports in Mathematics Number 6, 2005 Department of Mathematics Stockholm University Electronic versions of this document are available at http://www.math.su.se/reports/2005/6 Date of publication: November 28, 2005 2000 Mathematics Subject Classification: Primary 13D40, Secondary 13D02, 13D07. Postal address: Department of Mathematics Stockholm University S-106 91 Stockholm Sweden Electronic addresses: http://www.math.su.se/ [email protected] POINCARE´ SERIES AND HOMOTOPY LIE ALGEBRAS OF MONOMIAL RINGS ALEXANDER BERGLUND Abstract. This thesis comprises an investigation of (co)homological invari- ants of monomial rings, by which is meant commutative algebras over a field whose minimal relations are monomials in a set of generators for the algebra, and of combinatorial aspects of these invariants. Examples of monomial rings include the `Stanley-Reisner rings' of simplicial complexes. Specifically, we study the homotopy Lie algebra π(R), whose universal enveloping algebra is the Yoneda algebra ExtR(k; k), and the multigraded Poincar´e series of R, x i α i PR( ; z) = dimk ExtR(k; k)αx z : n i≥0,α2 ¡ To a set of monomials M we introduce a finite lattice KM , and show how to compute the Poincar´e series of an algebra R, with minimal relations M, in terms of the homology groups of lower intervals in this lattice. We introduce a ≥2 finite dimensional L1-algebra ¢ 1(M), and compute the Lie algebra π (R) ∗ in terms of the cohomology Lie algebra H ( ¢ 1(M)). Applications of these results include a combinatorial criterion for when a monomial ring is Golod. -
Lecture 26: Blossoming Blossom Abundantly, and Rejoice Even With
Lecture 26: Blossoming blossom abundantly, and rejoice even with joy and singing: Isaiah 35:2 1. Introduction Linear functions are simple; polynomials are complicated. If L(t) = at , then clearly L(µs+ λt) = µL(s)+ λL(t) . More generally, if L(t) = at + b , then it is easy to verify that L((1− λ)s+ λt) = (1− λ)L(s) + λ L(t) . € € In the first case L(t) preserves linear combinations; in the second case L(t) preserves affine n combinations.€ But if P(t) = a t + + a is a polynomial of degree n > 1, then n L 0 € P(µ s+ λt) ≠ µP( s)+ λP( t) P((1− λ)s+ λt) ≠ (1− λ)P(s) + λ P(t) . Thus arbitrary polynomials preserve neither linear nor affine combinations. The key idea behind €blossoming is to replace a complicated polynomial function P(t) in one variable by a simple €polynomial function p(u , , u ) in many variables that is either linear or affine in each variable. 1 K n The function p(u , , u ) is called the blossom of P(t) , and converting from P(t) to p(u , , u ) 1 K n 1 K n is called blossoming. Blossoming is intimately linked to Bezier curves. We shall see in Section 3 that the Bezier control points of a polynomial curve are given by the blossom of the curve evaluated at the end points of the parameter interval. Moreover, there is an algorithm for evaluating the blossom recursively that closely mimics the de Casteljau evaluation algorithm for Bezier curves. In this lecture we shall apply the blossom to derive two standard algorithms for Bezier curves: the de Casteljau subdivision algorithm and the procedure for differentiating the de Casteljau evaluation algorithm.