Polynomial Decomposition Research Articles

Relationship Between Aberration Coefficients of an Optical Device and Its Focusing Property

The best focus point of a focused Gaussian beam subject to a phase aberration is generally shifted with respect to the focal plane of the focusing lens. This focus shift is attributed to a lensing effect that belongs to the phase aberration, which mean focal length can be determined from the aberration coefficients determined in the framework of a Zernike polynomial decomposition. In this paper, we have checked the validity of this procedure, already available in literature, applied to three aberration types: a pure primary spherical aberration, the Kerr effect induced by a Gaussian beam, and an axicon illuminated by a Gaussian beam. Note that usually, the mean focal length of an aberrated lens is based on the relation between the effective radius of curvature of the wavefront before and after the lens. However, in this paper, the focal length associated with the phase aberration under study is defined from the point of the best focus, where the diffracted intensity on the axis is the maximum.

Planar decomposition of the HOMFLY polynomial for bipartite knots and links

AbstractThe theory of the Kauffman bracket, which describes the Jones polynomial as a sum over closed circles formed by the planar resolution of vertices in a knot diagram, can be straightforwardly lifted from $$\mathfrak {sl}(2)$$ sl ( 2 ) to $$\mathfrak {sl}(N)$$ sl ( N ) at arbitrary N – but for a special class of bipartite diagrams made entirely from the antiparallel lock tangle. Many amusing and important knots and links can be described in this way, from twist and double braid knots to the celebrated Kanenobu knots for even parameters – and for all of them the entire HOMFLY polynomials possess planar decomposition. This provides an approach to evaluation of HOMFLY polynomials, which is complementary to the arborescent calculus, and this opens a new direction to homological techniques, parallel to Khovanov–Rozansky generalizations of the Kauffman calculus. Moreover, this planar calculus is also applicable to other symmetric representations beyond the fundamental one, and to links which are not fully bipartite what is illustrated by examples of Kanenobu-like links.

The Mueller pupils Reduction-Zernike polynomial decomposition and polarization design criteria for the astronomical telescope system

This study introduces a novel method for decomposing polarization aberrations. It uses two sets of reduction matrices to decompose the diattenuation and retardance Mueller matrix into five independent reduction coefficients. And then, it uses Zernike polynomials to decompose these coefficients at the exit pupil, thereby providing a method for quantitatively analyzing the polarization aberrations. This method also gives the constraint values of the Zernike term of polarization aberrations during astronomical telescope design. Finally, the effectiveness and correctness of the process are verified by simulation of the actual optical system, pointing out that the D1, D2, D3, and R5 terms need to be calibrated and optimized to ensure polarization accuracy. This method offers a valuable tool and theoretical guidance for the polarization design of high-precision optical systems.

Representation of coxeter group and orthogonal group

The paper is primarily divided into two parts. The main focus of the first part is the construction of a representation of Coxeter groups. This begins with the definition of the Coxeter system and connected components, followed by the introduction of the length function and subsequent theorems. The faithfulness of this representation is then proven, allowing for the identification of isomorphisms that enable the final classification of finite Coxeter groups. This classification is achieved by leveraging the established relationship between irreducible representations of Coxeter groups and positive definite quadratic forms. Given the strong connection between Coxeter groups and orthogonal groups, the primary objective of the second part is to create a specific representation of orthogonal groups. This is accomplished through an examination of the decomposition of harmonic polynomials into subspaces of homogeneous harmonic polynomials, using the action of O(2) on these subspaces. The paper concludes by drawing connections to results in Invariant Theory, demonstrating the applicability of the presented concepts in a more general duality context.

Polynomial decompositions with invariance and positivity inspired by tensors

We present a framework to decompose real multivariate polynomials while preserving invariance and positivity. This framework has been recently introduced for tensor decompositions, in particular for quantum many-body systems. Here we transfer results about decomposition structures, invariance under permutations of variables, positivity, rank inequalities and separations, approximations, and undecidability to real polynomials. Specifically, we define invariant decompositions of polynomials, and characterize which polynomials admit such decompositions. We then include positivity: We define invariant separable and sum-of-squares decompositions, and characterize the polynomials similarly. We provide inequalities and separations between the ranks of the decompositions, and show that the separations are not robust with respect to approximations. For cyclically invariant decompositions, we show that it is undecidable whether the polynomial is nonnegative or sum-of-squares for all system sizes. Our framework is different from existing approaches for polynomial decompositions, since it covers symmetry and positivity combined, in a clean and uniform way. Also, our work sheds new light on polynomials by putting them on an equal footing with tensors, and opens the door to extending this framework to other tensor product structures.

Fractional Fischer decompositions by inframonogenic functions

Inframonogenic functions can be viewed as a non-commutative version of the more traditional harmonic functions. In this paper we obtain a new Fischer decomposition for homogeneous polynomials in Rm in terms of (φ,ψ)-inframonogenic homogeneous polynomials. The latter being a natural generalization arising when structural sets φ, ψ are considered instead of the standard orthonormal basis of Rm. Moreover, we extend our results to the fractional context by means of the Caputo derivative and Weyl relations.

On the zero set of the holomorphic sectional curvature

AbstractA notable example due to Heier, Lu, Wong, and Zheng shows that there exist compact complex Kähler manifolds with ample canonical line bundle such that the holomorphic sectional curvature is negative semi‐definite and vanishes along high‐dimensional linear subspaces in every tangent space. The main result of this note is an upper bound for the dimensions of these subspaces. Due to the holomorphic sectional curvature being a real‐valued bihomogeneous polynomial of bidegree (2,2) on every tangent space, the proof is based on making a connection with the work of D'Angelo on complex subvarieties of real algebraic varieties and the decomposition of polynomials into differences of squares. Our bound involves an invariant that we call the holomorphic sectional curvature square decomposition length, and our arguments work as long as the holomorphic sectional curvature is semi‐definite, be it negative or positive.

Sum of squares decomposition of positive polynomials with rational coefficients

We present an example of a strictly positive polynomial with rational coefficients that can be decomposed as a sum of squares of polynomials over R but not over Q. This answers an open question by C. Scheiderer posed as the second question in [14, Section 5.1]. We verify that the example we construct defines a nonsingular projective hypersurface, giving also a positive answer to the third question posed in that section.

The Properties of Topological Manifolds of Simplicial Polynomials

The formulations of polynomials over a topological simplex combine the elements of topology and algebraic geometry. This paper proposes the formulation of simplicial polynomials and the properties of resulting topological manifolds in two classes, non-degenerate forms and degenerate forms, without imposing the conditions of affine topological spaces. The non-degenerate class maintains the degree preservation principle of the atoms of the polynomials of a topological simplex, which is relaxed in the degenerate class. The concept of hybrid decomposition of a simplicial polynomial in the non-degenerate class is introduced. The decompositions of simplicial polynomial for a large set of simplex vertices generate ideal components from the radical, and the components preserve the topologically isolated origin in all cases within the topological manifolds. Interestingly, the topological manifolds generated by a non-degenerate class of simplicial polynomials do not retain the homeomorphism property under polynomial extension by atom addition if the simplicial condition is violated. However, the topological manifolds generated by the degenerate class always preserve isomorphism with varying rotational orientations. The hybrid decompositions of the non-degenerate class of simplicial polynomials give rise to the formation of simplicial chains. The proposed formulations do not impose strict positivity on simplicial polynomials as a precondition.

FACTORIZATION OF DEGREE k-ALMOST SIMPLE POLYNOMIALS

In this work, we introduce the notion of k-almost prime polynomials formed by the product over a Galois field of an arbitrary characteristic p of exactly k irreducible polynomials. The study aims to develop an effective algorithm for factorizing the degree of such composite polynomials providing a minimum of computational complexity. The proposed algorithm is based on the solution of a system of two equations functionally related to the a priori unknown degree of the components of the compound polynomial. One of these equations reflects that the degree of the composite polynomial equals the sum of degrees of elements of this polynomial. The second equation is based on the so-called cycle period of the compound polynomial, which coincides with the least common multiple of the unknown degrees of the components of the compound polynomial. It is found that the expansion of the cycle period contains all the degrees of the factorizable compound polynomial. The computational volume reduction is achieved by switching from the linear scale of the multiplicative group of subtractions generated by the cycle period of the compound polynomial to the logarithmic scale.

The hybrid method for finding the roots of a polynomial over finite fields based on affine expansion

This paper proposes a hybrid method to find the roots of a polynomial in a finite field based on combining the polynomial decomposition into the sum of multiples of the affine polynomial with the analytical method to find the roots of low-order polynomials. The proposed method allows a significant reduction in complexity and is applicable in the design of low-latency BCH and Reed-Solomon decoders.

Angle-dependent light scattering in tissue phantoms for the case of thin bone layers with predominant forward scattering.

The cochlea forms a key element of the human auditory system in the temporal bone. Damage to the cochlea continues to produce significant impairment for sensory reception of environmental stimuli. To improve this impairment, the optical cochlear implant forms a new research approach. A prerequisite for this method is to understand how light propagation, as well as scattering, reflection, and absorption, takes place within the cochlea. We offer a method to study the light distribution in the human cochlea through phantom materials which have the objective to mimic the optical behavior of bone and Monte-Carlo simulations. The calculation of an angular distribution after scattering requires a phase function. Often approximate functions like Henyey-Greenstein, two-term Henyey-Greenstein or Legendre polynomial decompositions are used as phase function. An alternative is to exactly calculate a Mie distribution for each scattering event. This method provides a better fit to the data measured in this work.

Almansi decomposition of polynomials of quaternionic Dirac operators

This article presents an innovative extension of the classical Almansi decomposition. Traditionally, the Almansi decomposition is employed to decompose polymonogenic functions into their monogenic counterparts, thereby elucidating the relationship between the Dirac operator and its successive iterations. Diverging from this conventional method, the new extension delves into the kernel of a polynomial related to the quaternionic Dirac operator, as opposed to focusing on the iterated operator. This is achieved by employing a polynomial-based partition of unity. The outcome of this approach is a distinctive decomposition that leverages the kernel of the Dirac–Helmholtz operators.

Generating probability distributions on intervals and spheres: Convex decomposition

This work studies the convexity of a class of positive definite probability measures with density functionsρ=cλ∑n=0∞g(n)Cnλwλ, that are generated through the interval characteristic sequences. We discuss when such measures supported in an interval are convex and demonstrate how convexity is preserved under multiplication by certain multipliers and under the intertwining product. A key message to be delivered from this work is: any integrable function f on the interval with a polynomial expansion that has relatively fast absolute convergence, can be decomposed into a non-unique pair of positive convex interval probabilities (f+,f−)∈Eλ′×Eλ′ up to normalization, in the sense that f=f+−f−, where Eλ′ is a class of convex functions. Consequently not only the study of an interval distribution is reduced to the study of the properties of class Eλ′, but also the discontinuous probabilistic Galerkin schemes can be simplified. As concrete examples the decomposition of polynomials and truncated normal distribution are computed and graphically illustrated. Such interval measures lead to localized spherical probability measures, and as a convex expression of probability measure is a Jensen type inequality for spherical random variables. Based on the spherical information entropy models, one dependent on the domain partition and the other not, we propose three minimization formulations for the probabilistic finite element method, including one that mixes the least-squares and entropy models.

Terracini Loci: Dimension and Description of Its Components

We study the Terracini loci of an irreducible variety X embedded in a projective space: non-emptiness, dimensions and the geometry of their maximal dimension’s irreducible components. These loci were studied because they describe where the differential of an important geometric map drops rank. Our best results are if X is either a Veronese embedding of a projective space of arbitrary dimension (the set-up for the additive decomposition of homogeneous polynomials) or a Segre–Veronese embedding of a multiprojective space (the set-up for partially symmetric tensors). For an arbitrary X, we give several examples in which all Terracini loci are empty, several criteria for non-emptiness and examples with the maximal defect possible a priori of an element of a minimal Terracini locus. We raise a few open questions.

Применение модального метода синтеза для дискретных систем

Works considering modal synthesis and using polynomial decomposition of transfer functions of object and controller models are considered mainly for continuous automatic control systems. At the same time, in some cases, it is necessary to consider systems of a discrete type. An example of the synthesis of an automatic control system for an accurate positioning of a table driven by a direct-drive motor is given. Such systems provide accurate positioning in the processes of manufacturing and processing of parts. They are used in various fields of industry: medical, aerospace, automotive and others. In particular, the above model of the precise positioning table is used for packaging semiconductor elements. A feature of this example is the representation of the transfer function of the object model in a discrete form and the presence of four-cycle delay links in it. In addition, this model is unstable due to the presence of poles lying on the stability boundary in it. As a synthesis method, a modal method is used, using a polynomial decomposition of the transfer function of the model of the control object and controller. An approach is also demonstrated that makes it possible to reduce the number of calculations in the synthesis of regulators by reducing the order of the transfer function of the considered object model and reducing the number of unknown parameters of the regulator. At the same time, a regulator is obtained that differs from the original one. Various options for choosing the desired poles of the transfer function of a closed system are analyzed. The choice of the desired poles is carried out in favor of reducing the amount of overshoot of the system relative to the transients of the system with the initial choice of poles.

Performance analysis of Petri net based on moment generating function

The Petri net structure of workflow is used to model, and the moment generating function is used to analyze the time performance of workflow, and the complexity of analysis is given. It provides basic theory and basis for analysis and verification. The calculation of time complexity is given for sequence, concurrency, cycle, conflict (selection) and mutual exclusion. The performance analysis method based on moment generating function can be used to analyze the performance of arbitrarily distributed bounded or unbounded random Petri nets. Establish a broad-random Petri net model that conforms to the concept of workflow. Then, based on statistical analysis and experience estimation of relevant data in the actual system, analyze the time nature of the on-demand service based on the analysis method based on behavioral expression, and obtain some valuable performance and index information. A necessary and sufficient condition for maintaining reliability of a workflow network model is given; A polynomial decomposition algorithm for P-invariants is proposed; Combining the moment function, a performance analysis method for workflow systems is established. An example is given to verify the effectiveness of the algorithm.

On Explicit Formulas of Hyperbolic Matrix Functions

Hyperbolic matrix functions are essential for solving hyperbolic coupled partial differential equations. In fact the best analytic-numerical approximations for resolving these equations come from the use of hyperbolic matrix functions. The hyperbolic matrix sine and cosine sh(A), ch(A) (A∈Mr(C)) can be calculated using numerous different techniques. In this article we derive some explicit formulas of sh(tA) and ch(tA) (t∈R) using the Fibonacci-H\"{o}rner and the polynomial decomposition, these decompositions are calculated using the generalized Fibonacci sequences combinatorial properties in the algebra of square matrices. Finally we introduce a third approach based on the homogeneous linear differential equations. And we provide some examples to illustrate your methods.

Watermarking approach based on Hermite transform and a sliding window algorithm.

Nowadays, the distribution of huge amounts of medical images through open networks in telemedicine applications has become increasingly faster and easier. Therefore, a number of considerations are introduced related to the risks of the illegal use of these images, as total diagnosis depends on them. Indeed, the patient's data management, storage, and transmission require a technique for boosting security, integrity and privacy measures in telehealthcare services. In fact, in our previous works, we used polynomial decompositions such as Chebychev orthogonal polynomial transform in medical image watermarking. We then customise our tools for finding the best candidate area for embedding the watermark, always seeking to provide the best solution to this issue. In this research, a variation to medical image watermarking approach based on Hermite transform (HT) is suggested for reliable management of medical data. In this approach, the HT is employed as a preprocessing step so as to extract the texture component of the host medical image. Afterward, a sliding window technique is done to select the most suitable regions for watermark embedding. Lastly, the Arnold transform is utilized for encrypting the watermark to strengthen the security of our scheme. Experiments were conducted on various modalities of medical images. Results indicate that the proposed scheme is robust when subjected to various attacks while preserving a high level of security and invisibility factors. Also, our method preserves the quality of medical images with a good embedding capacity. The obtained results support the use of Hermite Polynomials for the implementation of watermarking in the medical imaging context.

On polyharmonic polynomials

We study the orthogonal projection of homogeneous polynomials onto the space of homogeneous polyharmonic polynomials. To do this we derive the decomposition of homogeneous polynomials in terms of the Kelvin transform of derivatives of the fundamental solution $|x|^{2-n}$ or $\log |x|$. We consider also the vector bases of the space of homogeneous polyharmonic polynomials and study the problem of convergence of orthogonal series.