Odd Polynomial Research Articles

Sharp estimates for coefficients of odd trigonometric polynomials under a one-sided constraint

We study the largest and the smallest values of coefficients of odd trigonometric polynomials upper bounded by the function ϕ(x) = x on the interval [0, 2π]. A similar problem for the first and second coefficients was studied by the author earlier.

Polynomial invariants via odd parities for virtual link diagrams

We introduce the odd index polynomial and the odd arrow polynomial for virtual links which are different from the original index polynomial and arrow polynomial.

Partial Interior Stabilization of a Coupled Wave Equations on an Exterior Bounded Obstacle

We consider a stabilization problem for a coupled wave equations on an exterior of bounded domain with interior stabilization. Under a geometrical control condition (BLR condition ), for any initial data in the energy space, we show a result of exponential stability in odd dimensional case and polynomial stability in the case of even dimension.

Optimizing Shrinkage Curves and Application in Image Denoising

A shrinkage curve optimization is proposed for weighted nuclear norm minimization and is adapted to image denoising. The proposed optimization method employs a penalty function utilizing the difference between a latent matrix and its observation and uses odd polynomials to shrink the singular values of the observation matrix. As a result, the coefficients of polynomial characterize the shrinkage operator fully. Furthermore, the Frobenius norm of the penalty function is converted into the corresponding spectral norm, and thus the parameter optimization problem can be easily solved by using off-and-shelf plain least-squares. In the practical application, the proposed denoising method does not work on the whole image at once, but rather a series of matrix termed Rank-Ordered Similar Matrix (ROSM). Simulation results on 256 noisy images demonstrate the effectiveness of the proposed algorithms.

On convergence of odd trigonometric interpolation polynomials with equidistant points in the metric L

Odd, 2π-periodical, continuous functions represented by Fourier series are considered. The question of the convergence in the metric L of trigonometric interpolation sine polinomials of functions with monotone and quasimonotone Fourier coefficients is considered.

On the sign characteristics of Hermitian matrix polynomials

The sign characteristics of Hermitian matrix polynomials are discussed, and in particular an appropriate definition of the sign characteristics associated with the eigenvalue infinity. The concept of sign characteristic arises in different forms in many scientific fields, and is essential for the stability analysis in Hamiltonian systems or the perturbation behavior of eigenvalues under structured perturbations. We extend classical results by Gohberg, Lancaster, and Rodman to the case of infinite eigenvalues. We derive a systematic approach, studying how sign characteristics behave after an analytic change of variables, including the important special case of Möbius transformations, and we prove a signature constraint theorem. We also show that the sign characteristic at infinity stays invariant in a neighborhood under perturbations for even degree Hermitian matrix polynomials, while it may change for odd degree matrix polynomials. We argue that the non-uniformity can be resolved by introducing an extra zero leading matrix coefficient.

Characterizing regularity of domains via the Riesz transforms on their boundaries

Given a domain D in R^d with mild geometric measure theoretic assumptions on its boundary, we show that boundedness of the principal value Riesz tranforms (witn kernel of homogeneity -(d-1)) on H\"older spaces of order alpha on the boundary of D is equivalent to D being a Lyapunov domain of order alpha (i.e., the boundary of D is an hypersurface of class 1+alpha). Another equivalent condition involving Riesz transforms on D is discussed. We also prove that on Lyapunov domains of order alpha the higher order Riesz transforms associated with an odd polynomial are bounded on the H\"older space of order alpha on the boundary of D. Finally, a limiting case of the above results dealing with VMO and Semmes-Kenig-Toro domains is considered.

Multifield direct design method for ultrashort throw ratio projection optics with two tailored mirrors.

In this work, we present a multifield direct design method for ultrashort throw ratio projection optics. The multifield design method allows us to directly calculate two freeform mirror profiles, which are fitted by odd polynomials and imported into an optical design program as an excellent starting point. Afterward, these two mirrors are represented by XY polynomial freeform surfaces for further optimization. The final configuration consists of an off-the-shelf projection lens and two XY polynomial freeform mirrors to greatly shorten the regular projection distance from 2 m to 48 cm for a 78.3 inch diagonal screen. The values of the modulation transfer function for the optimized freeform mirror system are improved to over 0.6 at 0.5 lp/mm, in comparison with its rotationally symmetric counterpart's 0.43, and the final distortion is less than 1.5%, showing a very good and well-tailored imaging performance over the entire field of view.

An invariant subspace problem for multilinear operators on finite dimensional spaces

We introduce the notion of invariant subspaces for multilinear operators from which the invariant subspace problems for multilinear and polynomial operators arise. We prove that polynomial operators acting in a finite dimensional complex space and even polynomial operators acting in a finite dimensional real space have eigenvalues. These results enable us to prove that polynomial and multilinear operators acting in a finite dimensional complex space, even polynomial and even multilinear operators acting in a finite dimensional real space have nontrivial invariant subspaces. Furthermore, we prove that odd polynomial operators acting in a finite dimensional real space either have eigenvalues or are homotopic to scalar operators; we then use this result to prove that odd polynomial and odd multilinear operators acting in a finite dimensional real space may or may not have invariant subspaces.

$L^2$ stable discontinuous Galerkin methods for one-dimensional two-way wave equations

Simulating wave propagation is one of the fundamental problems in scientific computing. In this paper, we consider one-dimensional two-way wave equations, and investigate a family of L stable high order discontinuous Galerkin methods defined through a general form of numerical fluxes. For these L stable methods, we systematically establish stability (hence energy conservation), error estimates (in both L and negative-order norms), and dispersion analysis. One novelty of this work is to identify a sub-family of the numerical fluxes, termed αβ-fluxes. Discontinuous Galerkin methods with αβ-fluxes are proven to have optimal L error estimates and superconvergence properties. Moreover, both the upwind and alternating fluxes belong to this sub-family. Dispersion analysis, which examines both the physical and spurious modes, provides insights into the sub-optimal accuracy of the methods using the central flux and the odd degree polynomials, and demonstrates the importance of numerical initialization for the proposed non-dissipative schemes. Numerical examples are presented to illustrate the accuracy and the long-term behavior of the methods under consideration.

Photoelectron circular dichroism observed in the above-threshold ionization signal from chiral molecules with femtosecond laser pulses

Photoelectron circular dichroism is investigated experimentally as a function of the number of absorbed circularly polarized photons. Three structurally different chiral molecules yet showing similar absorption spectra are studied. They are isotropically distributed in the gas phase and ionized with femtosecond laser pulses. We measure and analyze the photoelectron angular distribution of threshold electrons ionized with three photons and compare them to those of above-threshold (ATI) electrons ionized with four photons. Additionally to an increase in high even order Legendre polynomials the coefficients of the high odd order Legendre polynomials rise with increasing photon number. Consequently, the ATI electrons also carry the chirality signature. All investigated chiral molecules reveal an individual set of coefficients for the threshold and ATI signatures despite their similarities in chemical structure. The presented data set can serve as a guideline for theoretical modeling of the interaction of circularly polarized light with chiral molecules in the multiphoton regime.

Explicit Form of the Faulhaber Polynomials

SummaryHersh recently showed that the Faulhaber polynomials, related to sums of consecutive powers, can be expressed as even or odd polynomials in the variable offset by one-half. In this article, we use generating functions to give explicit formulas for all terms in these polynomials.

Characteristics Description of Electromagnetic Stiffness Compensator

This research presents the traction characteristic of the stiffness compensator using the odd polynomial. Due to the various components of the polynomial we can get any stiffness characteristics. The calculation of the polynomial coefficients is presented. For the representation of the characteristic in automatic control systems the harmonic linearization coefficients are obtained for the polynomial n-th degree, which corresponds accurately to the description of the precise method.

35.2: Design of Hybrid Refractive/Reflective Projection Optics for Family Theatre

An ultra‐short focal projector equipped with a set of refractive lenses and an odd polynomial mirror is developed in this paper. Currently, short focal projectors generally employ a fisheye lens that produces wide field of view (WFOV). However, the lens distortion is inevitable so the design a projection optics with WFOV, small distortion and low f‐number is still the challenge for an ultra‐short focal projector system. The projector described in this paper overcomes these limitations by using a set of refractive lenses and an odd polynomial mirror. This enables projection optics with a focal length of −4.6 mm, an f‐number of 2.0, a WFOV of about 140° and a distortion of less than < 1 % to be presented. Results of the proposed projection optics will be described. Moreover, a portable projector based on light‐emitting diodes (LED) is proposed.

The role of the Bézier extraction operator for T‐splines of arbitrary degree: linear dependencies, partition of unity property, nesting behaviour and local refinement

SummaryWe determine linear dependencies and the partition of unity property of T‐spline meshes of arbitrary degree using the Bézier extraction operator. Local refinement strategies for standard, semi‐standard and non‐standard T‐splines – also by making use of the Bézier extraction operator – are presented for meshes of even and odd polynomial degrees. A technique is presented to determine the nesting between two T‐spline meshes, again exploiting the Bézier extraction operator. Finally, the hierarchical refinement of standard, semi‐standard and non‐standard T‐spline meshes is discussed. This technique utilises the reconstruction operator, which is the inverse of the Bézier extraction operator. Copyright © 2015 John Wiley & Sons, Ltd.

New Even and ODD Coherent States Attached to the Hermite Polynomials

Following systematic strategies, which were introduced by M. M. Nieto in 1995, coherent states (CSs) may be derived from their generating functions. In the present paper we generalize the latter procedure to new types of generating functions of even and odd Hermite polynomials. In this case, new CSs are obtained, as superposition of even and Fock states, and the generating functions are proportional to the corresponding even (odd) CSs. It is shown that they realize resolution of the identity conditions and can be considered as the nonlinear CSs. In addition, in each case, one can identify some of nonclassical features, such as sub-Poissonian statistics and quadrature squeezing effects, which occur simultaneously.

Lumping Techniques for DFEM SN Transport in Slab Geometry

We examine several mass matrix lumping techniques for the discrete ordinates (SN) particle transport equations spatially discretized with arbitrary order discontinuous finite elements in one-dimensional (1-D) slab geometry. Though positive outflow angular flux is guaranteed with traditional mass matrix lumping for linear solution representations in source-free, purely absorbing 1-D slab geometry, we show that when used with higher-degree polynomial trial spaces, traditional lumping does not yield strictly positive outflows and does not increase the solution accuracy with increase in the polynomial degree of the trial space. As an alternative, we examine quadrature-based lumping strategies, which we term “self-lumping” (SL). Self-lumping creates diagonal mass matrices by using a numerical quadrature restricted to the Lagrange interpolatory points. When choosing equally spaced interpolatory points, SL is achieved through the use of closed Newton-Cotes formulas, resulting in strictly positive outflows for odd degree polynomial trial spaces in 1-D slab geometry. When selecting the interpolatory points to be the abscissas of a Gauss-Legendre or a Lobatto-Gauss-Legendre quadrature, it is possible to obtain solution representations with a strictly positive outflow in source-free pure absorber problems for any degree polynomial trial space in 1-D slab geometry. Furthermore, there is no inherent limit to local truncation error order of accuracy when using interpolatory points that correspond to Gauss-Legendre or Lobatto-Gauss-Legendre quadrature points.A single-cell analysis is performed to investigate outflow positivity and truncation error as a function of the trial space polynomial degree, the choice of interpolatory points, and the numerical integration strategy. We also verify that the single-cell local truncation error analysis translates into the expected global spatial convergence rates in multiple-cell problems.

An isogeometric collocation method using superconvergent points

We develop an IGA collocation method modified by collocating at points other than the standard Greville abscissae. The method is related to orthogonal collocation used for solving differential equations and to the superconvergence theory, therefore we refer to this method as “super-collocation” (IGA-SC). By carefully choosing the collocation points, it can be seen that the IGA-SC converges in the first derivative (energy) norms at rates similar to that of the Galerkin solution. This is different from the collocation at Greville abscissae (IGA-C), where the convergence in energy norm for odd polynomial degrees is typically suboptimal. The method is tested on 1D, 2D and 3D numerical examples, in which it is compared to IGA-C and Galerkin’s method (IGA-G). The comparison includes a detailed cost vs. accuracy analysis, which shows an improved efficiency of the proposed method in particular for odd polynomial degrees.

Parities and polynomial invariants for virtual links

We introduce the odd Jones–Kauffman polynomial and odd Miyazawa polynomials of virtual link diagrams by using the parity of virtual link diagrams given in [Y. H. Im and K. I. Park, A parity and a multi-variable polynomial invariant for virtual links, J. Knot Theory Ramifications22(13) (2013), Article ID: 1350073, 18pp.], which are different from the original Jones–Kauffman and Miyazawa polynomials. Also, we give a family of parities and odd polynomials for virtual knots so that many virtual knots can be distinguished.

Field curvature correction method for ultrashort throw ratio projection optics design using an odd polynomial mirror surface.

This paper presents a field curvature correction method of designing an ultrashort throw ratio (TR) projection lens for an imaging system. The projection lens is composed of several refractive optical elements and an odd polynomial mirror surface. A curved image is formed in a direction away from the odd polynomial mirror surface by the refractive optical elements from the image formed on the digital micromirror device (DMD) panel, and the curved image formed is its virtual image. Then the odd polynomial mirror surface enlarges the curved image and a plane image is formed on the screen. Based on the relationship between the chief ray from the exit pupil of each field of view (FOV) and the corresponding predescribed position on the screen, the initial profile of the freeform mirror surface is calculated by using segments of the hyperbolic according to the laws of reflection. For further optimization, the value of the high-order odd polynomial surface is used to express the freeform mirror surface through a least-squares fitting method. As an example, an ultrashort TR projection lens that realizes projection onto a large 50 in. screen at a distance of only 510 mm is presented. The optical performance for the designed projection lens is analyzed by ray tracing method. Results show that an ultrashort TR projection lens modulation transfer function of over 60% at 0.5 cycles/mm for all optimization fields is achievable with f-number of 2.0, 126° full FOV, <1% distortion, and 0.46 TR. Moreover, in comparing the proposed projection lens' optical specifications to that of traditional projection lenses, aspheric mirror projection lenses, and conventional short TR projection lenses, results indicate that this projection lens has the advantages of ultrashort TR, low f-number, wide full FOV, and small distortion.