66 research outputs found
Factorising non-monic quadratic equations
No full textIn the November 2014 issue of At Right Angles, author Shashidhar Jagadeeshan, in the article “Completing the Square . . . A powerful technique, not a feared enemy!” talked about completing the square, quadratics having the shape of a parabola . . In this article, student Anushka Tonapi explains a few methods of solving non-monic quadratic equations
[[alternative]]The Number of Steps in the Polynomial Euclidean Algorithm
No full text[[abstract]]Let M be a monic polynomial over some finite fields. For polynomials a with deg a<deg M and (a,M)=1, we estimate the average value of the Euclidean Algorithm.
On an inverse problem for a linear combination of orthogonal polynomials
No full textThis paper deals with the analysis of the orthogonality of a monic polynomial sequence defined as a linear combination of a sequence of monic orthogonal polynomials withwhere for . Moreover, we obtain the relation between the corresponding linear functionals as well as an explicit expression for the sequence of monic orthogonal polynomials . We obtain the connection between the Jacobi matrices associated with and , respectively, by using an LU factorization. Some special cases of the above type relation are analysed.The work of the first author (FM) has been supported by Ministerio de Economía y Competitividad of Spain,
grant MTM 2012 36732 C03 01. This paper was completed during a stay of the second author (SV)
in the Department of Mathematics of Universidad Carlos III de Madrid in the spring semester of the
academic year 2012 2013. He acknowledges the kind reception there
Recommended from our members
Paying the price for black masculinity: Investigating sexual behaviors contributing to HIV/AIDS infection and death of black women
No full textAccording to the Center for Disease Control (2005), African American women account for 64% of the nearly 127,000 women living with HIV/AIDS in the United States. The author conducted qualitative interviews with 30 adult Black males with a history of practicing Down Low (DL) sexual behavior-- engaging in homoerotic activities with their female partners. The purpose of this study is to provide insight into the DL subculture and to obtain useful information that is crucial to understanding how the DL male\u27s personal struggles with sexuality and masculinity contribute to the disporportionate amount of Black women contracting HIV/AIDS
On the characterizations of third-degree semiclassical forms via polynomial mappings
No full textThe aim of this contribution is the study of orthogonal polynomials via polynomial mappings in the framework of the third-degree semiclassical linear forms. Let u and v be two regular forms and let denote by {pn}n≥0 and {qn}n≥0 the corresponding sequences of monic orthogonal polynomials such that there exists a monic polynomial θm of degree m, with 0 ≤ m ≤ k − 1 and r ∈ C, in such a way (...)The work of the first author (Francisco Marcellán) has been supported by [grant number PGC2018-096504-B-C31] by FEDER/ Spanish Ministry of Science and Innovation – Agencia Estatal de Investigación (AEI) of Spain
A Personal Remembrance of Warren Bennis
No full textIn January 2010, I fulfilled a decades-long dream of driving Route 66 from Chicago to the Santa Monica Pier and back. On Jan. 21, 2010 I celebrated making it to the Pacific Ocean by 41 I visiting with a dear friend, the noted leadership author Warren Bennis, who lived in Santa Monic
A note on the factorization of iterated quadratics over finite fields
No full textLet be a monic quadratic polynomial over a finite field of odd
characteristic. In 2012, Boston and Jones constructed a Markov process based on
the post-critical orbit of , and conjectured that its limiting distribution
explains the factorization of large iterates of . Later on, Xia, Boston, and
the author did extensive Magma computations and found some exceptional families
of quadratics that do not seem to follow the original Markov model conjectured
by Boston and Jones. They did this by empirically observing that certain
factorization patterns predicted by the Boston-Jones model never seem to occur
for these polynomials, and suggested a multi-step Markov model which takes
these missing factorization patterns into account. In this note, we provide
proofs for all these missing factorization patterns. These are the first
provable results that explain why the original conjecture of Boston and Jones
does not hold for all monic quadratic polynomials.Comment: 13 page
Continuous symmetric Sobolev inner products
No full text24 pages, no figures.-- MSC2000 codes: 42C05.MR#: MR1953647 (2003j:42029)Zbl#: Zbl pre05368623In this paper we consider the sequence of monic polynomials (Qn) orthogonal with respect to a symmetric Sobolev inner product. If Q_2n(x)=Pn(x^2) and Q_2n+1(x)=xRn(x^2), then we deduce the integral representation of the inner products such that (Pn) and (Rn) are, respectively, the corresponding sequences of monic orthogonal polynomials. In the semiclassical case, algebraic relations between such sequences are deduced. Finally, an application of the above results to Freud-Sobolev polynomials is given.The work of the second author was partially supported by Dirección General de Investigación (Ministerio de Ciencia y Tecnología) of Spain under grant BFM 2000-0206-C04-01 and INTAS project INTAS 2000-272.Publicad
Spherical Harmonic Solution of the Robin Problem for the Helmholtz Equation in a Supershaped Shell
No full textThe Robin problem for the Helmholtz equation in normal-polar shells is addressed by using a suitable spherical har-monic expansion technique. Attention is in particular focused on the wide class of domains whose boundaries are de-fined by a generalized version of the so-called “superformula” introduced by Gielis. A dedicated numerical procedure based on the computer algebra system Mathematica© is developed in order to validate the proposed methodology. In this way, highly accurate approximations of the solution, featuring properties similar to the classical ones, are obtainedMicroelectronicsElectrical Engineering, Mathematics and Computer Scienc
Orthogonal polynomials for the weakly equilibrium cantor sets
No full textLet K(γ) be the weakly equilibrium Cantor-type set introduced by the second author in an earlier work. It is proven that the monic orthogonal polynomials Q2 s with respect to the equilibrium measure of K(γ) coincide with the Chebyshev polynomials of the set. Procedures are suggested to find Qn of all degrees and the corresponding Jacobi parameters. It is shown that the sequence of the Widom factors is bounded below. © 2016 American Mathematical Society
- …
