Odd Coefficients Research Articles

Heights of roots of polynomials with odd coefficients

Soit α un zero d'un polynome de degre n a coefficients impairs qui n'est pas une racine de l'unite. Nous montrons que la hauteur de α satisfait h (α,) ≥ 0.4278/n+1. Plus generalement, nous obtenons des bornes dans le cas ou chaque coefficient est congru a 1 modulo m, avec m > 2.

An efficient Chebyshev–Tau spectral method for Ginzburg–Landau–Schrödinger equations

We propose an efficient time-splitting Chebyshev–Tau spectral method for the Ginzburg–Landau–Schrödinger equation with zero/nonzero far-field boundary conditions. The key technique that we apply is splitting the Ginzburg–Landau–Schrödinger equation in time into two parts, a nonlinear equation and a linear equation. The nonlinear equation is solved exactly; while the linear equation in one dimension is solved with Chebyshev–Tau spectral discretization in space and Crank–Nicolson method in time. The associated discretized system can be solved very efficiently since they can be decoupled into two systems, one for the odd coefficients, the other for the even coefficients. The associated matrices have a quasi-tridiagonal structure which allows a direction solution to be obtained. The computation cost of the method in one dimension is O ( N log ( N ) ) compared with that of the non-optimized one, which is O ( N 2 ) . By applying the alternating direction implicit (ADI) technique, we extend this efficient method to solve the Ginzburg–Landau–Schrödinger equation both in two dimensions and in three dimensions, respectively. Numerical accuracy tests of the method in one dimension, two dimensions and three dimensions are presented. Application of the method to study the semi-classical limits of Ginzburg–Landau–Schrödinger equation in one dimension and the two-dimensional quantized vortex dynamics in the Ginzburg–Landau–Schrödinger equation are also presented.

Effects of higher-order dispersion on resonant dispersive waves emitted by solitons

Dispersive waves (DW) are generated owing to perturbation of solitons by higher-order dispersion (HOD) and nonlinearity during supercontinuum (SC) generation. The frequencies of these waves are governed by a phase-matching condition in the form of a polynomial whose coefficients depend on the numerical values of the properly normalized third- and HOD parameters. Our extensive numerical solutions show that all odd HOD terms generate a single peak on the blue or the red side of the carrier frequency, depending on the sign of the corresponding term. In contrast, positive even HOD terms create conjugate DW peaks, in both the blue and red sides. No radiation is observed for negative values of these parameters. The combination of all even and odd HOD coefficients may generate more than two DW peaks for some specific choice of parameters. The results predicted by the phase-matching condition agree well with extensive numerical simulations revealing interesting facts of SC generation.

Measurement of the Transverse Polarization of Electrons Emitted in Free-Neutron Decay

Both components of the transverse polarization of electrons (sigmaT1, sigmaT2) emitted in the beta-decay of polarized, free neutrons have been measured. The T-odd, P-odd correlation coefficient quantifying sigmaT2, perpendicular to the neutron polarization and electron momentum, was found to be R=0.008+/-0.015+/-0.005. This value is consistent with time reversal invariance and significantly improves limits on the relative strength of imaginary scalar couplings in the weak interaction. The value obtained for the correlation coefficient associated with sigmaT1, N=0.056+/-0.011+/-0.005, agrees with the Standard Model expectation, providing an important sensitivity test of the experimental setup.

Odd coefficients of weakly holomorphic modular forms

). We will consider the question ofestimating the number of integers n for which a(n) 6≡0 (mod v).For a well-studied example, let p(n) be the ordinary partition function. Manyauthors have considered the problem of estimating the number of odd values of p(n).Among other references, one may see [1], [5], [15], [16], [17], [18], [19], [22], or [24].To see the connection to the general situation above, we recall the definition of theDedekind eta-function:(1.2) η(z) := q

Pion-photon transition distribution amplitudes in the Nambu–Jona-Lasinio model

We define the pion-photon transition distribution amplitudes (TDA) in a field theoretic formalism from a covariant Bethe-Salpeter approach for the determination of the bound state. We apply our formalism to the Nambu-Jona-Lasinio model, as a realistic theory of the pion. The obtained vector and axial TDAs satisfy all features required by general considerations. In particular, sum rules and the polynomiality condition are explicitly verified. We have numerically proved that the odd coefficients in the polynomiality expansion of the vector TDA vanish in the chiral limit. The role of PCAC and the presence of a pion pole are explicitly shown.

Differences between charged-current coefficient functions

Second- and third-order results are presented for the structure functions of charged-current deep-inelastic scattering in the framework of massless perturbative QCD. We write down the two-loop differences between the corresponding crossing-even and -odd coefficient functions, including those for the longitudinal structure function not covered in the literature so far. At three loops we compute the lowest five moments of these differences for all three structure functions and provide approximate expressions in Bjorken- x space. Also calculated is the related third-order coefficient-function correction to the Gottfried sum rule. We confirm the conjectured suppression of these quantities if the number of colours is large. Finally we derive the second- and third-order QCD contributions to the Paschos–Wolfenstein ratio used for the determination of the weak mixing angle from neutrino–nucleon deep-inelastic scattering. These contributions are found to be small.

Lehmer’s problem for polynomials with odd coefficients

We prove that if f(x) = YlkZo akxk is a polynomial with no cyclotomic factors whose coefficients satisfy ak = 1 mod 2 for 0 2. We also characterize the polynomials that appear as the noncyclotomic part of a polynomial whose coefficients satisfy ak = 1 mod p for each ft, for a fixed prime p. Last, we prove that the smallest Pisot number whose minimal polynomial has odd coefficients is a limit point, from both sides, of Salem [19] numbers whose minimal polynomials have coef ficients in {?1,1}.

On the Littlewood cyclotomic polynomials

In this article, we study the cyclotomic polynomials of degree N − 1 with coefficients restricted to the set { + 1 , − 1 } . By a cyclotomic polynomial we mean any monic polynomial with integer coefficients and all roots of modulus 1. By a careful analysis of the effect of Graeffe's root squaring algorithm on cyclotomic polynomials, P. Borwein and K.K. Choi gave a complete characterization of all cyclotomic polynomials with odd coefficients. They also proved that a polynomial p ( x ) with coefficients ±1 of even degree N − 1 is cyclotomic if and only if p ( x ) = ± Φ p 1 ( ± x ) Φ p 2 ( ± x p 1 ) ⋯ Φ p r ( ± x p 1 p 2 ⋯ p r − 1 ) , where N = p 1 p 2 ⋯ p r and the p i are primes, not necessarily distinct. Here Φ p ( x ) : = x p − 1 x − 1 is the pth cyclotomic polynomial. Based on substantial computation, they also conjectured that this characterization also holds for polynomials of odd degree with ±1 coefficients. We consider the conjecture for odd degree here. Using Ramanujan's sums, we solve the problem for some special cases. We prove that the conjecture is true for polynomials of degree 2 α p β − 1 with odd prime p or separable polynomials of any odd degree.

Structure unanimity multiple description coding

A novel multiple description coding framework, called structure unanimity multiple description coding, is proposed in this paper. Each wavelet coefficient is first decomposed into an odd position coefficient made from the bits in the odd positions and an even position coefficient made from the bits in the even positions. These two types of coefficients form the odd image and the even image, respectively, which are next encoded, and then transmitted using two different channels. Since the two images share the same hierarchical tree structure, only coefficient values in the redundancy parts need to be coded, resulting in improved coding efficiency. To meet the requirements of the packet switching network, these two images are rearranged in tree blocks, and the tree blocks in the even image and odd image are grouped to form the row and column packets, respectively. These packets are next separately dispatched through diverse channels. Since the row and the column packets have limited intersections, even with simultaneous loss of row and column packets, the proposed system exhibits good error resilient ability. Experimental results have verified the improved performance of the proposed algorithm.

SCATTERING OF ATOMS BY LIGHT: MEASURING THE QUANTUM STATE OF THE FIELD IN A CAVITY

This work consists in a generalization of the developed method to measure the quantum state of the field in a cavity with all non null coefficients, in the number representation, through one photon interaction. We present a strategy to measure the quantum state of the field when it has all non-null coefficients or only even or odd coefficients through a two-photons atom-field interaction.

Momentum anisotropies in the quark coalescence model

Based on the quark coalescence model, we derive relations among the momentum anisotropies of mesons and baryons in relativistic heavy ion collisions from a given, but arbitrary azimuthal distribution for the partons. Besides the familiar even Fourier coefficients such as the elliptic flow, we also pay attention to odd Fourier coefficients such as the directed flow, which has been observed at finite rapidity even at RHIC energies.

New formulae for solutions of quantum Knizhnik–Zamolodchikov equations on level-4

We present a new form of solution to the quantum Knizhnik-Zamolodchikov equation on level -4 in a special case corresponding to the Heisenberg XXX spin chain. Our form is equivalent to the integral representation obtained by Jimbo and Miwa in 1996 . An advantage of our form is that it is reduced to the product of single integrals. This fact is deeply related to a cohomological nature of our formulae. Our approach is also based on the deformation of hyper-elliptic integrals and their main property -- deformed Riemann bilinear relation. Jimbo and Miwa also suggested a nice conjecture which relates solution of the qKZ on level -4 to any correlation function of the XXX model. This conjecture together with our form of solution to the qKZ makes it possible to prove a conjecture that any correlation function of the XXX model can be expressed in terms of the Riemann zeta-function at odd arguments and rational coefficients suggested in our previous papers. This issue will be discussed in our next publication.

87.73 The number of odd multinomial coefficients

87.73 The number of odd multinomial coefficients

A de-blocking algorithm and a blockiness metric for highly compressed images

Blockiness is a typical artifact in reconstructed images that have been coded by a block-based discrete cosine transform (BDCT). In highly compressed images and video, the blocking artifacts are easily noticeable as the discontinuities between relatively homogeneous regions. Many current noniterative de-blocking algorithms attempt to remove the blocking artifacts by smoothing a few pixels around the block boundaries; however, the results are not satisfactory, especially at very low bit rates. We propose a de-blocking algorithm based on the number of connected blocks in a relatively homogeneous region, the magnitude of abrupt changes between neighboring blocks, and the quantization step size of DCT coefficients. Due to its adaptability, the proposed algorithm can smooth out the blocking artifacts while keeping the strong edges and texture areas untouched. Since this algorithm is noniterative and only identifies those block pairs that actually need de-blocking, its computation cost is low. In addition, we have developed a new metric to measure the blocking artifacts in images. Through analyzing the 2N-point (N is the block size) one-dimensional DCT coefficients of the two neighboring blocks with blocking artifacts, we show that all of the even DCT coefficients of the combined 2N points are zeros (except frequency k=0). The odd DCT coefficients are proportional to the pixel value difference between these two blocks with their magnitudes almost inversely proportional to frequency k. We selected the first DCT coefficient (frequency k=1) as an indicator for the strength of blocking artifacts in the reconstructed images. For the postprocessed images, we used a weighted summation of the squared first DCT coefficient to measure their blocking artifacts. Experimental results demonstrate that our proposed de-blocking algorithm produces better results than other methods, both visually and quantitatively, while the proposed blocking artifact metric is more consistent with subjective evaluation than the peak signal-to-noise ratio.

Radix-2 decimation-in-frequency algorithm for the computation of the real-valued FFT

An efficient algorithm for computing the real-valued FFT (of length N) using radix-2 decimation-in-frequency (DIF) approach has been introduced. The fact that the odd coefficients are the DFT values of an N/2-length linear phase sequence introduces a redundancy in the form of the symmetry X(2k+1)=X/sup */(N-2k-1), which can be exploited to reduce the arithmetic complexity and memory requirements. The arithmetic complexity and, memory requirements of the algorithm presented are exactly the same as the most efficient decimation-in-time (DIT) algorithm for the real-valued FFT that exists to date. A C++ program that implements this algorithm has been included.

On Cyclotomic Polynomials with ± 1 Coefficients

We characterize all cyclotomic polynomials of even degree with coefficients restricted to the set {+1, −1}. In this context a cyclotomic polynomial is any monic polynomial with integer coefficients and all roots of modulus 1. Inter alia we characterize all cyclotomic polynomials with odd coefficients. The characterization is as follows. A polynomial P(x) with coefficients ±1 of even degree N–l is cyclotomic if and only if where N = P1P1 … Pr and the Pi are primes, not necessarily distinct, and where ϕp(x) := (xp – 1)/ (x – 1) isthe p-th cyclotomic polynomial. We conjecture that this characterization also holds for polynomials of odd degree with ±1 coefficients. This conjecture is based on substantial computation plus a number of special cases. Central to this paper is a careful analysis of the effect of Graeffe's root squaring algorithm on cyclotomic polynomials.

Survey Design Procedures and Data Processing Techniques Applied to the EM Azimuthal Resistivity Method

The electromagnetic (EM) azimuthal resistivity method is an alternative to the galvanic azimuthal resistivity survey. The advantages of the EM approach include (1) reduced acquisition time, (2) simple field procedure, and (3) a reduced data acquisition area. The primary disadvantage of the EM method is the magnitude of the data noise. Signal processing was applied to assist quantification of noise in EM azimuthal resistivity datasets and to enable noise reduction. Comparison of the energy in the even and odd coefficients of the power spectra allowed the signal‐to‐noise ratio to be identified. Linear phase filters were used to suppress high frequency noise. These techniques were applied to EM data collected at three study sites. Following data processing, the EM azimuthal resistivity datasets revealed apparent resistivity lobes consistent with the orientation of fracture strike mapped at two of these sites. At one site the processed EM dataset correlated closely with an azimuthal resistivity dataset collected at the same location. Scale modeling of the EM azimuthal resistivity response over anisotropic and heterogeneous features was performed. The EM response obtained from a symmetric and asymmetric array configuration was measured. The results illustrate that an asymmetric array must be used if the presence of heterogeneity is to be distinguished from anisotropy. Signal processing was applied to assist interpretation of anisotropy and heterogeneity. The magnitude of the energy in the odd coefficients of the power spectrum assists recognition of heterogeneity. For an anisotropic medium the odd coefficients were all zero whereas heterogeneity resulted in significant energy in these coefficients. The ability to (1) characterize and remove noise, and (2) differentiate heterogeneity from anisotropy, improves the value of the EM azimuthal resistivity technique in site characterization. The survey design procedures and data processing applied in this study are equally applicable to the azimuthal resistivity method.

On a construction of the fundamental solution for the free Weyl equation by Hamiltonian path-integral method---an exactly solvable case with ``odd variable coefficients''

On a construction of the fundamental solution for the free Weyl equation by Hamiltonian path-integral method---an exactly solvable case with ``odd variable coefficients''

On a Coefficient Conjecture of Brannan

In 1972. D.A. Branna conjectured that all of the odd coefficients, a 2n+1 of the power series (1+xz)α/(1-z) were dominated by those of the series (1 + z)α(1-z) for the parameter range 0 < α < 1 after having shown that this was not true for the even coefficients. He verified the case when 2n + 1 = 3 The case when 2n + 1 = 5 was verified in the mid-eighties by J. G. Milcetich. In this paper, we verify the case when 2n + l = 7 using classical Sturm sequence arguments and some computer algebra.