Derandomizing Restricted Isometries via the Legendre Symbol

The restricted isometry property (RIP) is an important matrix condition in compressed sensing, but the best matrix constructions to date use randomness. This paper leverages pseudorandom properties of the Legendre symbol to reduce the number of random bits in an RIP matrix with Bernoulli entries. In this regard, the Legendre symbol is not special—our main result naturally generalizes to any small-bias sample space. We also conjecture that no random bits are necessary for our Legendre symbol-based construction.     

Construction of large families of pseudorandom binary sequences

In a series of papers Mauduit and Sárközy (partly with coauthors) studied finite pseudorandom binary sequences. They showed that the Legendre symbol forms a “good” pseudorandom sequence, and they also tested other sequences for pseudorandomness, however, no large family of “good” pseudorandom sequences has been found yet.In this paper, a large family of this type is constructed by extending the earlier Legendre symbol construction.     

Construction of large families of pseudorandom binary sequences

Oon constructed large families of finite binary sequences with strong pseudorandom properties by using Dirichlet characters of large order. In this paper Oon’s construction is generalized and extended. We prove that in our construction the well-distribution and correlation measures are as “small” as in the case of the Legendre symbol.      

The symbol of type and the wilson theorem

We introduce the notion of an element type for the field of residues modulo a prime number and study its role in factorial computation. We obtain an expression for its quantitative characteristic (the symbol of type) in terms of continued fractions and establish its connection with the Legendre symbol.     

On certain determinants involving Legendre symbols

The Ramanujan Journal - The evaluation of determinants with Legendre symbol entries is a classical topic both in number theory and in linear algebra. Recently Sun posed some conjectures on this...     

On the complexity of a family related to the Legendre symbol

Ahlswede, Khachatrian, Mauduit and A. Sárközy introduced the notion of family-complexity of families of binary sequences. They estimated the family-complexity of a large family related to Legendre symbol introduced by Goubin, Mauduit and Sárközy. Here their result is improved, and apart from the constant factor the best lower bound is given for the family-complexity.     

Toward an arithmetic of polynomials

Summary ForR a commutative ring, which may have divisors of zero but which has no idempotents other than zero and one, we consider the problem of unique factorization of a polynomial with coefficients inR. We prove that, if the polynomial is separable, then such a unique factorization exists. We also define a Legendre symbol for a separable polynomial and a prime of commutative ring with exactly two idempotents in such a way that the symbols of classical number theory are subsumed. We calculate this symbol forR = Q in two cases where it has classically been of interest, namely quadratic extensions and cyclotomic extensions. We then calculate it in a situation which is new, namely the so called generalized cyclotomic extensions from a paper by S. Beale and D. K. Harrison. We study the Galois theory in the general ring situation and in particular define a category of separable polynomials (this is an extension of a paper by D. K. Harrison and M. Vitulli) and a cohomology theory of separable polynomials.     

Existence and uniqueness for Legendre curves

We give a moving frame of a Legendre curve (or, a frontal) in the unit tangent bundle and define a pair of smooth functions of a Legendre curve like as the curvature of a regular plane curve. It is quite useful to analyse the Legendre curves. The existence and uniqueness for Legendre curves hold similarly to the case of regular plane curves. As an application, we consider contact between Legendre curves and the arc-length parameter of Legendre immersions in the unit tangent bundle.     

Recursive computation of Padé–Legendre approximants and some acceleration properties

Summary. In this paper, after recalling the two definitions of the generalizations of the Padé approximants to orthogonal series, we will define the Padé–Legendre approximants of a Legendre series. We will propose two algorithms for the recursive computation of some sequences of these approximants. We will also estimate the speed of convergence of the columns of the Padé–Legendre table from the asymptotic behaviour of the coefficients of the Legendre series. Finally we will illustrate these results with some numerical examples. Received June 20, 1998 / Published online March 20, 2001     

Elliptic curve analogue of Legendre sequences

The Legendre symbol is applied to the rational points over an elliptic curve to output a family of binary sequences with strong pseudorandom properties. That is, both the well-distribution measure and the correlation measure of order k, which are evaluated by using estimation of certain character sums along elliptic curves, of the resulting binary sequences are “small”. A lower bound on the linear complexity profile of these sequences is also presented. Our results indicate that the behavior of such sequences is very similar to that of the Legendre sequences. Research partially supported by the Science and Technology Foundation of Putian City (No. 2005S04), the Open Funds of Key Lab of Fujian Province University Network Security and Cryptology (No. 07B005) and the Foundation of the Education Department of Fujian Province (No. JA07164). Author’s addresses: Department of Mathematics, Putian University, Putian, Fujian 351100, China; and Key Lab of Network Security and Cryptology, Fujian Normal University, Fuzhou, Fujian 350007, China     

On the Galois groups of Legendre polynomials

Ever since Legendre introduced the polynomials that bear his name in 1785, they have played an important role in analysis, physics and number theory, yet their algebraic properties are not well-understood. Stieltjes conjectured in 1890 how they factor over the rational numbers. In this paper, assuming Stieltjes’ conjecture, we formulate a conjecture about the Galois groups of Legendre polynomials, to the effect that they are “as large as possible,” and give theoretical and computational evidence for it.     

The comparison of two reliable methods for accurate solution of time‐fractional Kaup–Kupershmidt equation arising in capillary gravity waves

In this paper, we compared two different methods, one numerical technique, viz Legendre multiwavelet method, and the other analytical technique, viz optimal homotopy asymptotic method (OHAM), for solving fractional‐order Kaup–Kupershmidt (KK) equation. Two‐dimensional Legendre multiwavelet expansion together with operational matrices of fractional integration and derivative of wavelet functions is used to compute the numerical solution of nonlinear time‐fractional KK equation. The approximate solutions of time fractional Kaup–Kupershmidt equation thus obtained by Legendre multiwavelet method are compared with the exact solutions as well as with OHAM. The present numerical scheme is quite simple, effective, and expedient for obtaining numerical solution of fractional KK equation in comparison to analytical approach of OHAM. Copyright © 2016 John Wiley & Sons, Ltd.     

A generalization of the Legendre symbol for finite Abelian groups

Let G be a finite Abelian group. Given an integer a relatively prime to the order of G, let σ_a be the permutation on G defined by σ_a (x) = x^a. The problem studied in the paper is to determine the parity of σ_a. Generalizations of well-known theorems on the Legendre symbol are obtained in this context.     

The study of collision and avalanche effect in a family of pseudorandom binary sequences

In an earlier paper we studied collisions and avalanche effect in two of the most important constructions given for large families of binary sequences possessing strong pseudorandom properties. It turned out that one of the two constructions (which is based on the use of the Legendre symbol) is ideal from this point of view, while the other construction (which is based on the size of the modulo p residue of f(n) for some polynomial f(x) ∈ $ \mathbb{F}_p $ \mathbb{F}_p [x]) is not satisfactory since there are “many” collisions in it. Here it is shown that this weakness of the second construction can be corrected: one can take a subfamily of the given family which is just slightly smaller and collision free.     

Comparison of two orthogonal series methods of estimating a density and its derivatives on an interval

We compare the merits of two orthogonal series methods of estimating a density and its derivatives on a compact interval—those based on Legendre polynomials, and on trigonometric functions. By examining the rates of convergence of their mean square errors we show that the Legendre polynomial estimators are superior in many respects. However, Legendre polynomial series can be more difficult to construct than trigonometric series, and to overcome this difficulty we show how to modify trigonometric series estimators to make them more competitive.     

A quadrature tau method for fractional differential equations with variable coefficients

In this article, we develop a direct solution technique for solving multi-order fractional differential equations (FDEs) with variable coefficients using a quadrature shifted Legendre tau (Q-SLT) method. The spatial approximation is based on shifted Legendre polynomials. A new formula expressing explicitly any fractional-order derivatives of shifted Legendre polynomials of any degree in terms of shifted Legendre polynomials themselves is proved. Extension of the tau method for FDEs with variable coefficients is treated using the shifted Legendre–Gauss–Lobatto quadrature. Numerical results are given to confirm the reliability of the proposed method for some FDEs with variable coefficients.     

On error estimates for Galerkin spectral discretizations of parabolic problems with nonsmooth initial data

We analyze the Legendre and Chebyshev spectral Galerkin semidiscretizations of a one dimensional homogeneous parabolic problem with nonconstant coefficients. We present error estimates for both smooth and nonsmooth data. In the Chebyshev case a limit in the order of approximation is established. On the contrary, in the Legendre case we find an arbitrary high order of convegence.

     

The Lebesgue constants of the biorthonormal system {Pn,Qn} and “Jackson” type theorem for a kind of new approximation problem

In this paper first we discuss the Lebesgue constants of the biorthonormal system {P_n,Q_n} on an ellipse, where P_n, Q_n are the Legendre polynomials and the second kind Legendre functions respectively. Secondly, for a kind of new approximation problem we give a corresponding “Jackson” type theorem of continuous functions on an ellipse.     

Hilbert scales and Sobolev spaces defined by associated Legendre functions

In this paper we study the Hilbert scales defined by the associated Legendre functions for arbitrary integer values of the parameter. This problem is equivalent to studying the left-definite spectral theory associated to the modified Legendre equation. We give several characterizations of the spaces as weighted Sobolev spaces and prove identities among the spaces corresponding to the lower regularity index.