One-variable Polynomial Research Articles

Modeling Human Macular Cone Photoreceptor Spatial Distribution.

The purpose of this study was to investigate the spatial distribution of human cone photoreceptors and examine cone density differences between the retinal meridians and quadrants. Using adaptive optics scanning laser ophthalmoscopy, the maculae were imaged in 17 eyes of 11 subjects with normal chorioretinal health aged 54 to 72 years. We measured cone density at 325 points within the central 10degrees radius of the retina. Cone density spatial distributions along the primary retinal meridians and in four macular quadrants (superior-nasal, superior-temporal, inferior-temporal, and inferior-nasal) were analytically modeled using the polynomial function to assess the meridional and quadrantal difference. The mean and 95% confidence interval for the prediction of cone density along the primary retinal meridians was modeled with a 7-degree one-variable polynomial (R2 = 0.9761, root mean squared error [RMSE] = 0.0585). In the 4 retinal quadrants, cone density distribution was described by a 2-variable polynomial with X degree 3 and Y degree 4 (R² = 0.9834, RMSE = 0.0377). The models suggest no statistically significant difference between medians and between quadrants. However, cone density difference at corresponding spatial locations in different areas can be up to 25.6%. The superior-nasal region has more areas with high cone density, followed by quadrants of inferior-nasal, inferior-temporal, and superior-temporal. Analytical modeling provides comprehensive knowledge of cone distribution across the entire macula. Although modeling analysis suggests no statistically significant difference between medians and between quadrants, the remarkable cone density discrepancies in certain regions should be accounted for in applications requiring sensitive detection of cone variation.

A polynomial analogue of Berggren’s theorem on Pythagorean triples

Say that ( x , y , z ) (x, y, z) is a positive primitive integral Pythagorean triple if x , y , z x, y, z are positive integers without common factors satisfying x 2 + y 2 = z 2 x^2 + y^2 = z^2 . An old theorem of Berggren gives three integral invertible linear transformations whose semi-group actions on ( 3 , 4 , 5 ) (3, 4, 5) and ( 4 , 3 , 5 ) (4, 3, 5) generate all positive primitive Pythagorean triples in a unique manner. We establish an analogue of Berggren’s theorem in the context of a one-variable polynomial ring over a field of characteristic ≠ 2 \neq 2 . As its corollaries, we obtain some structure theorems regarding the orthogonal group with respect to the Pythagorean quadratic form over the polynomial ring.

Skew characteristic polynomial of graphs and embedded graphs

We introduce a new one-variable polynomial invariant of graphs, which we call the skew characteristic polynomial. For an oriented simple graph, this is just the characteristic polynomial of its anti-symmetric adjacency matrix. For nonoriented simple graphs the definition is different, but for a certain class of graphs (namely, for intersection graphs of chord diagrams), it gives the same answer if we endow such a graph with an orientation induced by the chord diagram. We prove that this invariant satisfies Vassiliev's $4$-term relations and determines therefore a finite type knot invariant. We investigate the behaviour of the polynomial with respect to the Hopf algebra structure on the space of graphs and show that it takes a constant value on any primitive element in this Hopf algebra. We also provide a two-variable extension of the skew characteristic polynomial to embedded graphs and delta-matroids. The $4$-term relations for the extended polynomial prove that it determines a finite type invariant of multicomponent links.

An Algorithm for Finding Self-Orthogonal and Self-Dual Codes Over Gaussian and Eisenstein Integer Residue Rings Via Chinese Remainder Theorem

A code over Gaussian or Eisenstein integer residue ring is an additive group of vectors with entries in this integer residue ring which is closed under the action of constant multiplication by the Gaussian or Eisenstein integers. In this paper, we define the dual codes for the codes over the Gaussian and Eisenstein integer residue rings, and consider the construction of the self-dual codes. Because, in the Gaussian and Eisenstein integer rings, the uniqueness of the prime element decomposition holds in the same way as the one-variable polynomial rings over finite fields and the rational integer ring, we provide an efficient construction method for self-dual code generator matrices using that of moduli. As numerical examples, for Gaussian and Eisenstein integer rings, we enumerate and construct the self-dual codes for the actual moduli when the matrix size of the generator matrices is two.

A Multi-Secret Reputation Adjustment Method in the Secret Sharing for Internet of Vehicles

Secure data sharing for the Internet of Vehicles (IoV) has drawn much attention in developing smart cities and smart transportation. One way to achieve that is based on the reputation of the participant vehicles and the interaction between them. Most of the reputation adjustment schemes are aimed at single-secret situations, and there are cases where shared information leaks after reconstruction; and the shared information of all participants need to be updated again. This paper uses binary asymmetric polynomials to realize the asynchronous reconstruction of secrets, that is, multiple secrets are distributed at once, and each secret is independent of each other when it is restored. Each participant holds the shared divided information part of a one-variable polynomial. When the secret is reconstructed, the shared information part is not directly shown to ensure that the shared information is not leaked. Furthermore, we provide another way to keep the secret unchanged and modify the secret threshold.

Generic Newton polygons for L-functions of (A,B)-exponential sums

In this paper, we consider the following (A,B)-polynomial f over finite field:f(x0,x1,⋯,xn)=x0Ah(x1,⋯,xn)+g(x1,⋯,xn)+PB(1/x0), where h is a Deligne polynomial of degree d, g is an arbitrary polynomial of degree less than dB/(A+B) and PB(y) is a one-variable polynomial of degree less than or equal to B. Let Δ be the Newton polyhedron of f at infinity. We show that Δ is generically ordinary if p≡1modD, where D is a constant determined by Δ. In other words, we prove that the Adolphson–Sperber conjecture is true for Δ.

On a generalization of Jacobi sums

We prove an estimate for multi-variable multiplicative character sums over affine subspaces of Akn, which generalizes the well known estimates for both classical Jacobi sums and one-variable polynomial multiplicative character sums.

Tschirnhaus transformations after Hilbert

In this paper, we use enumerative geometry to simplify the formula for the roots of the general one-variable polynomial of degree n , for all n . More precisely, let \mathrm{RD}(n) denote the minimum d for which there exists a formula for the roots of the general degree n polynomial using only algebraic functions of d or fewer variables. In 1927, Hilbert sketched how the 27 lines on a cubic surface could be used to construct a 4-variable formula for the general degree 9 polynomial (implying \mathrm{RD}(9)\le 4 ). In this paper, we turn Hilbert's sketch into a general method. We show this method produces best-to-date upper bounds on \mathrm{RD}(n) for all n , improving earlier results of Hamilton, Sylvester, Segre and Brauer.

Evaluating Thin Flat Surfaces

We consider recognizable evaluations for a suitable category of oriented two-dimensional cobordisms with corners between finite unions of intervals. We call such cobordisms thin flat surfaces. An evaluation is given by a power series in two variables. Recognizable evaluations correspond to series that are ratios of a two-variable polynomial by the product of two one-variable polynomials, one for each variable. They are also in a bijection with isomorphism classes of commutative Frobenius algebras on two generators with a nondegenerate trace fixed. The latter algebras of dimension n correspond to points on the dual tautological bundle on the Hilbert scheme of n points on the affine plane, with a certain divisor removed from the bundle. A recognizable evaluation gives rise to a functor from the above cobordism category of thin flat surfaces to the category of finite-dimensional vector spaces. These functors may be non-monoidal in interesting cases. To a recognizable evaluation we also assign an analogue of the Deligne category and of its quotient by the ideal of negligible morphisms.

Bounds for discrete moments of Weyl sums and applications

We prove two bounds for discrete moments of Weyl sums. The first one can be obtained using a standard approach. The second one involves an observation how this method can be improved, which leads to a sharper bound in certain ranges. The proofs both build on the recently proved main conjecture for Vinogradov's mean value theorem. We present two selected applications: First, we prove a new $k$- th derivative test for the number of integer points close to a curve by an exponential sum approach. This yields a stronger bound than existing results obtained via geometric methods, but it is only applicable for specific functions. As second application we prove a new improvement of the polynomial large sieve inequality for one-variable polynomials of degree $k \geq 4$.

A Noniterative Method for Solving Nonlinear Equations for Steady-State Regimes of Electrical Networks

Methods for solving linear and nonlinear equations for steady-state operation modes are important for the development and operation of electrical networks. Linear equations are solved analytically or with the use of iterative methods, while nonlinear equations are solved only with use of iterative methods, usually the Newton–Raphson method. Iterative methods have two significant drawbacks. First, the iterative process can diverge, with the divergence being dependent on both the form of the nonlinear equation and the choice of the initial approximation. Second, in the case of convergence, the iterative method allows finding only one solution for each initial approximation, whereas a nonlinear equation can have several solutions corresponding to different operation modes of the electric network. Development of methods that are free of these drawbacks is an important problem. This paper proposes a noniterative method for solving nonlinear equations describing the steady-state operation modes of electric networks; the equations are written in the complex form. The method uses the resultant to transform set of nonlinear equations to a set of polynomials. The polynomials are composed in such a way that their zeros determine one of the coordinates of the solution vector of the nonlinear equation system. Thus, the problem of solving of the nonlinear equations system is reduced to the well-studied problem of finding the zeros of one-variable polynomials. There are numerous methods of solving this problem that do not require specifying the initial approximation and allow finding all zeros. The effectiveness of the method is illustrated by solving the equations of node network voltages in the form of power balance. The equations were solved by the traditional iterative method and the proposed method using the MathCAD-15 program. The drawbacks of the iterative method are illustrated by its divergence at certain values of the initial approximation and by the failure of finding all the solutions. In contrast, the use of the proposed noniterative method allows one to find two solutions and to establish that there is no other solutions. Further development of this method will be connected with taking into account the sparseness of the matrix of conductivities that would allow reducing the degree used polynomials, as well as with the extension of this method to the domain of real values in which the equations can be written down in the algebraic and trigonometric form.

On Some Families of New Constructed Polynomials

The purpose of this paper is to introduce some new polynomials obtained from second- and third-order algebraic equations by using a simple iterative method. One-variable polynomials obtained in this study deal with special form of Poschl–Teller potential with constant energy, and the two-variable polynomials are related to time-dependent wave equations. We present some recurrence relations, Binet formula and get various families of linear, multilinear and multilateral generating functions for these polynomials. In addition, we derive some special cases. At the end of the paper we also give an extension to the multidimensional case of our results.

Multiplicative Structure and Hecke Rings of Generator Matrices for Codes over Quotient Rings of Euclidean Domains

In this study, we consider codes over Euclidean domains modulo their ideals. In the first half of the study, we deal with arbitrary Euclidean domains. We show that the product of generator matrices of codes over the rings mod a and mod b produces generator matrices of all codes over the ring mod a b , i.e., this correspondence is onto. Moreover, we show that if a and b are coprime, then this correspondence is one-to-one, i.e., there exist unique codes over the rings mod a and mod b that produce any given code over the ring mod a b through the product of their generator matrices. In the second half of the study, we focus on the typical Euclidean domains such as the rational integer ring, one-variable polynomial rings, rings of Gaussian and Eisenstein integers, p-adic integer rings and rings of one-variable formal power series. We define the reduced generator matrices of codes over Euclidean domains modulo their ideals and show their uniqueness. Finally, we apply our theory of reduced generator matrices to the Hecke rings of matrices over these Euclidean domains.

Cn-moves and the difference of Jones polynomials for links

The Jones polynomial [Formula: see text] for an oriented link [Formula: see text] is a one-variable Laurent polynomial link invariant discovered by Jones. For any integer [Formula: see text], we show that: (1) the difference of Jones polynomials for two oriented links which are [Formula: see text]-equivalent is divisible by [Formula: see text], and (2) there exists a pair of two oriented knots which are [Formula: see text]-equivalent such that the difference of the Jones polynomials for them equals [Formula: see text].

A Categorification of One-Variable Polynomials

We develop a diagrammatic categorification of the polynomial ring $\mathbb{Z} [x]$, based on a geometrically-defined graded algebra and show how to lift various operations on polynomials to the categorified setting. Our categorification satisfies a version of the Bernstein-Gelfand-Gelfand reciprocity property, with indecomposable projective modules corresponding to $x^n$ and standard modules to $(x -1)^n$ in the Grothendieck ring. This construction generalizes tocategorification of various orthogonal polynomials. Résumé. Catégorification de l’anneau des polynômes $\mathbb{Z} [x]$, Nous développons une catégorification diagrammatique de l’anneau des polynômes $\mathbb{Z} [x]$, s’appuyant sur une algèbre graduée définie de manière géométrique, et nous décrivons comment on peut relever certaines opérations sur les polynômes dans cette catégorification. Notre catégorification vérifie une version de la réciprocité de Bernstein-Gelfand-Gelfand, avec les modules projectifs indécomposables correspondants à $x^n$ et les modules standards correspondants à $(x - 1)^n$ dans l’anneau de Grothendieck. Cette construction se généralise à certains polynômes orthogonaux.

Turning Points

Traditionally in calculus, the maximum number of turning points in the graph of a onevariable polynomial function is presented as (n − 1), where n is the degree of the polynomial. However, by using Descartes' rule of signs and certain properties of polynomial functions, we can establish a better estimate than the (n − 1) rule provides.

A Non-Uniform Regional Partition Coding of One Dimension Signal and its Application

In this paper, a novel method for one-dimension-signal coding based on non-uniform regional partition is proposed. A given 1D signal can automatically be partitioned into different regions with different lengths and the one-variable polynomial is used to do the Least Square Approximation for signal values in each sub-region. When the approximation error and initial partition are specified, a specific signal partition result is obtained. Based on this algorithm, an effective denoising scheme of 1D signal is implemented and some of the experimental examples are illustrated to prove that the quality of the re-constructed signal, the denoising effect are all satisfactory and can be referenced by other researchers.

Assessing the quality of multilevel graph clustering

“Lifting up” a non-hierarchical approach to handle hierarchical clustering by iteratively applying the approach to hierarchically cluster a graph is a popular strategy. However, these lifted iterative strategies cannot reasonably guide the overall nesting process precisely because they fail to evaluate the very hierarchical character of the clustering they produce. In this study, we develop a criterion that can evaluate the quality of the subgraph hierarchy. The multilevel criterion we present and discuss in this paper generalizes a measure designed for a one-level (flat) graph clustering to take nesting of the clusters into account. We borrow ideas from standard techniques in algebraic combinatorics and exploit a variable \(q\) to keep track of the depth of clusters at which edges occur. Our multilevel measure relies on a recursive definition involving variable \(q\) outputting a one-variable polynomial. This paper examines archetypal examples as proofs-of-concept; these simple cases are useful in understanding how the multilevel measure actually works. We also apply this multilevel modularity to real world networks to demonstrate how it can be used to compare hierarchical clusterings of graphs.

A FOUR-VARIABLE INDEX POLYNOMIAL INVARIANT OF LONG VIRTUAL KNOTS

We introduce a four-variable index polynomial invariant of long virtual knots that is non-trivial for many long virtual knots, but is trivial for classical knots so that it is an extension of a one-variable index polynomial invariant introduced in [A polynomial invariant of long virtual knots, European J. Combin.30 (2009) 1289–1296]. We give various properties of this polynomial and examples. Also, we use this polynomial invariant for long virtual knots to distinguish virtual knots, and we obtain a polynomial invariant for long flat virtual knots which is very useful to determine whether long flat virtual knots are invertible or not.

A DIAGRAMMATIC ALEXANDER INVARIANT OF TANGLES

We give a new construction of the one-variable Alexander polynomial of an oriented knot or link, and show that it generalizes to a vector valued invariant of oriented tangles.