Article

Trigonometric polynomials with critical points of predetermined respective multiplicity

January 1994
Annales des Sciences Mathematiques du Quebec 18(1)

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Article

Equal-ripple rational functions with multiple critical pole pair

February 1976 · Electronics Letters

A new class of equal-ripple rational functions is considered here for the design of lowpass transfer functions of RC active filters. Using a critical pole pair with a multiplicity greater than unity (as in the MUCROMAF and MUCROER polynomials), we can obtain lower pole Q-factors with this new class than with the classical Cauer functions.

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On implementing the symbolic preprocessing function over Boolean polynomial rings in Gröbner basis a...

August 2015 · Journal of Systems Science and Complexity

Some techniques using linear algebra was introduced by Faugêre in F4 to speed up the reduction process during Gröbner basis computations. These techniques can also be used in fast implementations of F5 and some other signature-based Gröbner basis algorithms. When these techniques are applied, a very important step is constructing matrices from critical pairs and existing polynomials by the ... [Show full abstract] Symbolic Preprocessing function (given in F4). Since multiplications of monomials and polynomials are involved in the Symbolic Preprocessing function, this step can be very costly when the number of involved polynomials/monomials is huge. In this paper, multiplications of monomials and polynomials for a Boolean polynomial ring are investigated and a specific method of implementing the Symbolic Preprocessing function over Boolean polynomial rings is reported. Many examples have been tested by using this method, and the experimental data shows that the new method is very efficient.

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Branching rules for symmetric functions and basic hypergeometric series

January 2011 · Advances in Applied Mathematics

A one-parameter rational function generalisation Rλ(X;b)Rλ(X;b) of the symmetric Macdonald polynomial and interpolation Macdonald polynomial is studied from the point of view of branching rules. We establish a Pieri formula, evaluation symmetry, principal specialisation formula and q-difference equation for Rλ(X;b)Rλ(X;b). Our main motivation for studying Rλ(X;b)Rλ(X;b) is that it leads to a new ... [Show full abstract] class of slnsln basic hypergeometric series, generalising the well-known basic hypergeometric series with Macdonald polynomial argument. For these new series we prove slnsln analogues of the q-Gauss and q-Kummer–Thomae–Whipple formulas. In a special limit, one of our results implies an elegant binomial formula for Jack polynomials, different to that of Kaneko, Lassalle, Okounkov and Olshanski.

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Preprint

Semidefinite Relaxations of Products of Nonnegative Forms on the Sphere

February 2021

We study the problem of maximizing the geometric mean of d low-degree non-negative forms on the real or complex sphere in n variables. We show that this highly non-convex problem is NP-hard even when the forms are quadratic and is equivalent to optimizing a homogeneous polynomial of degree O(d) on the sphere. The standard Sum-of-Squares based convex relaxation for this polynomial ... [Show full abstract] optimization problem requires solving a semidefinite program (SDP) of size

n^{O(d)}

, with multiplicative approximation guarantees of

\Omega(\frac{1}{n})

. We exploit the compact representation of this polynomial to introduce a SDP relaxation of size polynomial in n and d, and prove that it achieves a constant factor multiplicative approximation when maximizing the geometric mean of non-negative quadratic forms. We also show that this analysis is asymptotically tight, with a sequence of instances where the gap between the relaxation and true optimum approaches this constant factor as

d \rightarrow \infty

. Next we propose a series of intermediate relaxations of increasing complexity that interpolate to the full Sum-of-Squares relaxation, as well as a rounding algorithm that finds an approximate solution from the solution of any intermediate relaxation. Finally we show that this approach can be generalized for relaxations of products of non-negative forms of any degree.

Conference Paper

Towards Optimal Toom-Cook Multiplication for Univariate and Multivariate Polynomials in Characterist...

June 2007 · Lecture Notes in Computer Science

Marco Bodrato

Toom-Cook strategy is a well-known method for building algorithms to efficiently multiply dense univariate polynomials. Efficiency of the algorithm depends on the choice of interpolation points and on the exact sequence of operations for evaluation and interpolation. If carefully tuned, it gives the fastest algorithm for a wide range of inputs. This work smoothly extends the Toom strategy to ... [Show full abstract] polynomial rings, with a focus on . Moreover a method is proposed to find the faster Toom multiplication algorithm for any given splitting order. New results found with it, for polynomials in characteristic 2, are presented. A new extension for multivariate polynomials is also introduced; through a new definition of density leading Toom strategy to be efficient.

Conference Paper

Worst-Case Software Safety Level for Braking Distance Algorithm of a Train

November 2007

Subhasis Patra

This paper presents a worst-case software safety level analysis of braking distance algorithm of a train. This analysis will prove that the safety critical software algorithm is valid. Sometimes algorithms cannot be shown to be valid through testing due to being too complex. The proposed approach should show the algorithm never goes outside a stated bound in worst case condition. The inputs are ... [Show full abstract] chosen to not only show the normal condition, but also include combinations of inputs at the worst case limits in order to show worst case performance of the algorithm. For braking distance algorithm the performance could vary depending on multiple varying inputs, such as the location, direction, speed or acceleration rate of a train. Only algorithm related errors are considered in the present analysis. Hardware related errors are not considered in this analysis. The analysis is done for different algorithms such as single order interpolation and 2^nd order interpolation equations. The analysis result shows that single order interpolation algorithm gives better results than 2^nd order interpolation algorithm. In the analysis report the safety level is shown for different train speeds for high speed passenger train and freight train. The analysis is carried out for rounding error, division-by-zero, overflow error and underflow error.

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A fast stationary iterative method for a partial integro-differential equation in pricing options

December 2013 · Calcolo

A jump-diffusion model for the pricing of options leads to a partial integro-differential equation (PIDE). Discretizing the PIDE by certain method, we get a sequence of systems of linear equations, where the coefficient matrices are Toeplitz matrices. In this paper, we decompose the coefficient matrix as the sum of a tridiagonal matrix and a near low-rank matrix, and approximate the near low-rank ... [Show full abstract] matrix by low-rank matrices. Then we introduce a stationary iterative method for the approximate systems of linear equations. Comparison of the performance of our algorithm to that proposed in Pang et al. (Linear Algebra Appl. 434:2325–2342, 2011) is presented.

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Siegel disks and wandering domains of transcendental entire functions

January 1970

Koh Katagata

We give a brief survey of results on the dynamics of transcendental entire functions with Siegel disks whose singular values are just two points. One of the two singular values is not only a superattracting fixed point with multiplic-ity more than two but also an asymptotic value. Another one is a critical value with ffee dynamics under iterations. If the multiplicity of the superattracting fixed ... [Show full abstract] point is large enough, then the restriction of the transcendental entire ffinction near the Siegel point is a quadratic-like map. As its applications, the logarithmic lift of the above transcendental entire ffinction has a wandering domain whose shape looks like a Siegel disk of a quadratic polynomial.

Article

On the two-variable Dirichlet q-L-series

December 2005 · Advanced Studies in Contemporary Mathematics (Kyungshang)

In this study, we construct the two-variable Dirichlet q-L-function and the two-variable multiple Dirichlet-type Changhee q-L-function. These functions interpolate the q-Bernoulli polynomials and generalized Changhee q-Bernoulli polynomials. By using the Mellin transformation, we give an integral representation for the two-variable multiple Dirichlet-type Changhee q-L-function. We also obtain ... [Show full abstract] relations between the Barnes-type q-zeta function and the two-variable multiple Dirichlet-type Changhee q-L-function. 1.

Conference Paper

Hermite Interpolation Using ERBS with Trigonometric Polynomial Local Functions

July 2011 · Lecture Notes in Computer Science

Given a sequence of knots

t_0<t_1<\cdots<t_{n+1}

, an expo-rational B-spline (ERBS) function f(t) is defined by

f(t)=\sum_{k=1}^n \ell_k(t)B_k(t), \quad t\in[t_1,t_n],

where

B_k(t)

are the ERBS and

ℓ_k(t)

are local functions defined on

(t_{k−1},t_{k+1})

. Consider the Hermite interpolation problem at the knots

0 ≤ t_1<t_2<\cdots<t_n<2\pi

of arbitrary multiplicities. In [3] a ... [Show full abstract] formula was suggested for Hermite-interpolating ERBS function with

ℓ_k(t)

being algebraic polynomials. Here we construct Hermite interpolation by an ERBS function with trigonometric polynomial local functions. We provide also numerical results for the performance of the new trigonometric ERBS (TERBS) interpolant in graphical comparison with the interpolant from [3]. Potential applications and some topics for further research on TERBS are briefly outlined.

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Asymptotically homogeneous solutions to differential equations with homogeneous polynomial symbols w...

August 2014 · Proceedings of the Steklov Institute of Mathematics

We study solutions to partial differential equations with homogeneous polynomial symbols with respect to a multiplicative one-parameter transformation group such that all eigenvalues of the infinitesimal matrix are positive. The infinitesimal matrix may contain a nilpotent part. In the asymptotic scale of regularly varying functions, we find conditions under which such differential equations have ... [Show full abstract] asymptotically homogeneous solutions in the critical case.

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New formulations of the multiple sequence alignment problem

February 2011 · Optimization Letters

A well known formulation of the multiple sequence alignment (MSA) problem is the maximum weight trace (MWT), a 0–1 linear programming problem. In this paper, we propose a new integer quadratic programming formulation of the MSA. The number of constraints and variables in the problem are only of the order of kL 2, where, k is the number of sequences and L is the total length of the sequences, that ... [Show full abstract] is,

{L= \sum_{i=1}^{k}l_{i}}

, where l i is the length of sequence i. Based on this formulation we introduce an equivalent linear constrained 0–1 quadratic programming problem. We also propose a 0–1 linear programming formulation of the MWT problem, with polynomially many constraints. Our formulation provides the first direct compact formulation that ensures that the critical circuit inequalities (which are exponentially many) are all met.

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Trigonometric polynomials with critical points of predetermined respective multiplicity

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