Sum Of Polynomials Research Articles

The Eccentric-Distance Sum Polynomials of Graphs by Using Graph Products

The correlations between the physico-chemical properties of a chemical structure and its molecular structure-properties are used in quantitative structure-activity and property relationship studies (QSAR/QSPR) by using graph-theoretical analysis and techniques. It is well known that some structure-activity and quantitative structure-property studies, using eccentric distance sum, are better than the corresponding values obtained by using the Wiener index. In this article, we give precise expressions for the eccentric distance sum polynomial of some graph products such as join, Cartesian, lexicographic, corona and generalized hierarchical products. We implement our outcomes to calculate this polynomial for some significant families of molecular graphs in the form of the above graph products.

The Root Distributions of Ehrhart Polynomials of Free Sums of Reflexive Polytopes

In this paper, we study the root distributions of Ehrhart polynomials of free sums of certain reflexive polytopes. We investigate cases where the roots of the Ehrhart polynomials of the free sums of $A_d^\vee$'s or $A_d$'s lie on the canonical line $\mathrm{Re}(z)=-\frac{1}{2}$ on the complex plane $\mathbb{C}$, where $A_d$ denotes the root polytope of type A of dimension $d$ and $A_d^\vee$ denotes its polar dual. For example, it is proved that $A_m^\vee \oplus A_n^\vee$ with $\min\{m,n\} \leq 1$ or $m+n \leq 7$, $A_2^\vee \oplus (A_1^\vee)^{\oplus n}$ and $A_3^\vee \oplus (A_1^\vee)^{\oplus n}$ for any $n$ satisfy this property. We also perform computational experiments for other types of free sums of $A_n^\vee$'s or $A_n$'s.

On divisors of sums of polynomials

Let A and B be sets of polynomials of degree n over a finite field. We show, that if A and B are large enough, then A+B has an irreducible divisor of large degree for some A∈A and B∈B.

AUTOMATIC QUADRATURE SCHEME FOR CAUCHY TYPE SINGULAR INTEGRAL ON THE VARIABLE INTERVAL

In this note, we consider the product indefinite integral of the form
 
 An automatic quadrature scheme (AQS) is constructed for evaluating Cauchy principal singular integrals in two cases. In the first case c∈ [y,z] ⊂ [-1,1] where -1 < y < z < 1, density function h(t) is approximated by the truncated sum of Chebyshev polynomials of the first kind. Direct substitution does not give solutions so we have used the AQS and reduced problems into algebraic equation with unknown parameters bk which can be found in terms of the singular point with some front conditions. In the second case c ∈ [-1,1], the application of the AQS reduced the number of calculations twice and accuracy is increased. As a theoretical result, the convergence theorem of the proposed method is proven in a Hilbert space. Numerical examples with exact solutions and comparisons with other methods are also given, and they are in the line with theoretical findings.

Duality of sum of nonnegative circuit polynomials and optimal SONC bounds

Circuit polynomials are polynomials with properties that make it easy to compute sharp and certifiable global lower bounds for them. Consequently, one may use them to find certifiable lower bounds for any polynomial by writing it as a sum of circuit polynomials with known lower bounds. Recent work has shown that sums of nonnegative circuit polynomials (or SONC polynomials for short) can be used to compute global lower bounds (called SONC bounds) for polynomials in this manner very efficiently both in theory and in practice, if the polynomial is bounded from below and its support satisfies a certain nondegeneracy assumption. The quality of the SONC bound depends on the circuits used in the computation but finding the set of circuits that yield the best attainable SONC bound among the astronomical number of candidate circuits is a non-trivial task that has not been addressed so far. We propose an efficient method to compute the optimal SONC lower bound by iteratively identifying the optimal circuits to use in the SONC bounding process. The method is derived from a new proof of the result that every SONC polynomial decomposes into SONC polynomials on the same support. This proof is based on convex programming duality and motivates a column generation approach that is particularly attractive for sparse polynomials of high degree and with many unknowns. The method is implemented and tested on a large set of sparse polynomial optimization problems with up to 40 unknowns, of degree up to 60, and up to 3000 monomials in the support. The results indicate that the method is efficient in practice and requires only a small number of iterations to identify the optimal circuits.

Smearing scale in Laguerre reconstructions of the correlation function

To a good approximation, on large cosmological scales the evolved two-point correlation function of biased tracers is related to the initial one by a convolution. For Gaussian initial conditions, the smearing kernel is Gaussian, so if the initial correlation function is parametrized using simple polynomials then the evolved correlation function is a sum of generalized Laguerre functions of half-integer order. This motivates an analytic Laguerre reconstruction algorithm which previous work has shown is fast and accurate. This reconstruction requires as input the width of the smearing kernel. We show that the method can be extended to estimate the width of the smearing kernel from the same dataset. This estimate, and associated uncertainties, can then be used to marginalize over the distribution of reconstructed shapes, and hence provide error estimates on the value of the distance scale which are not tied to a particular cosmological model. We also show that if, instead, we parametrize the evolved correlation function using simple polynomials, then the initial one is a sum of Hermite polynomials, again enabling fast and accurate deconvolution. If one is willing to use constraints on the smearing scale from other datasets, then marginalizing over its value is simpler for Hermite reconstruction, potentially providing further speed-up in cosmological analyses.

On the Permanental Polynomial and Permanental Sum of Signed Graphs

On the Permanental Polynomial and Permanental Sum of Signed Graphs

Finite free convolutions of polynomials

We study three convolutions of polynomials in the context of free probability theory. We prove that these convolutions can be written as the expected characteristic polynomials of sums and products of unitarily invariant random matrices. The symmetric additive and multiplicative convolutions were introduced by Walsh and Szegö in different contexts, and have been studied for a century. The asymmetric additive convolution, and the connection of all of them with random matrices, is new. By developing the analogy with free probability, we prove that these convolutions produce real rooted polynomials and provide strong bounds on the locations of the roots of these polynomials.

Further improvement on index bounds

In this paper we obtain further improvement of index bounds for character sums of polynomials over finite fields. We present some examples, which show that our new bound is an improved bound compared to both the Weil bound and the index bound given by Wan and Wang. As an application, we give an estimation of the number of all the solutions of some algebraic curves by using our result.

Modeling wind speed time series by Chebyshev polynomial expansion method

ABSTRACT This paper sets out to simulate non-Gaussian wind speed time series. Given a set of historical wind speed observations, if the cumulative distribution function is not explicitly known, a probability distribution based on Fourier series is developed to recover the analytical expression of the cumulative distribution function. The marginal transformation is then applied to map wind speed observations to the standard normal space, where the autocorrelation function is represented by a weighted sum of Chebyshev polynomials. Finally, an explicit formula is derived to generate samples of wind speed time series. Testing on three sets of historical wind speed observations, it shows that the proposed method can well match the cumulative distribution function and autocorrelation function of wind speed time series, the absolute errors between the fitted cumulative distribution function and empirical cumulative distribution function are less than ; the autocorrelation function of generated wind speed time series is in a good agreement with that of historical wind speed observations, the differences are less than .

Characterising the relationships between physiological indicators and all-cause mortality (NHANES): a population-based cohort study

SummaryBackgroundMortality risk stratification based on dichotomising a physiological indicator with a cutoff point might not adequately capture increased mortality risk and might not account for non-linear associations. We aimed to characterise the linear and non-linear relationships of 27 physiological indicators with all-cause mortality to evaluate whether the current clinical thresholds are suitable in distinguishing patients at high risk for mortality from those at low risk.MethodsFor this observational cohort study of the US non-institutionalised population, we used data from adults (≥18 years) included in the 1999–2014 National Health and Nutrition Examination Survey (NHANES) linked with National Death Index mortality data collected from Jan 1, 1999, up until Dec 31, 2015. We used Cox proportional hazards regression models adjusted for age, sex, and race or ethnicity to assess associations of physiological indicators with all-cause mortality. We assessed non-linear associations by discretising the physiological indicator into nine quantiles (termed novemtiles) and by using a weighted sum of cubic polynomials (spline). We used ten-fold cross validation to select the most appropriate model using the concordance index, Nagelkerke R2, and Akaike Information Criterion. We identified the level of each physiological indicator that led to a 10% increase in mortality risk to define our cutoffs used to compare with the current clinical thresholds.FindingsWe included 47 266 adults of 82 091 assessed for eligibility. 25 (93%) of 27 indicators showed non-linear associations with substantial increases compared with linear models in mortality risk (1·5–2·5-times increase). Height and 60 s pulse were the only physiological indicators to show linear associations. For example, participants with an estimated glomerular filtration rate (GFR) of less than 65 mL/min per 1·73 m2 or between 90–116 mL/min per 1·73 m2 are at moderate (hazard ratio 1–2) mortality risk. Those with a GFR greater than 117 mL/min per 1·73 m2 show substantial (hazard ratio ≥2) mortality risk. Both lower and higher values of cholesterol are associated with increased mortality risk. The current clinical thresholds do not align with our mortality-based cutoffs for fat deposition indices, 60 s pulse, triglycerides, cholesterol-related indicators, alkaline phosphatase, glycohaemoglobin, homoeostatic model assessment of insulin resistance, and GFR. For these indicators, the misalignment suggests the need to consider an additional bound when only one is provided.InterpretationMost clinical indicators were shown to have non-linear associations with all-cause mortality. Furthermore, considering these non-linear associations can help derive reliable cutoffs to complement risk stratification and help inform clinical care delivery. Given the poor alignment with our proposed cutoffs, the current clinical thresholds might not adequately capture mortality risk.

Irreducibility of a sum of polynomials depending on disjoint sets of variables

In this article, we give two different sufficient conditions for the irreducibility of a polynomial of more than one variable, over an algebraically closed field, that can be written as a sum of two polynomials which depend on mutually disjoint sets of variables. These conditions are derived from analyzing the Newton polytope of such a polynomial and then applying the ‘Irreducibility criterion’ introduced by Gao.

On the product of the singular values of a binary tensor

A real binary tensor consists of $2^d$ real entries arranged into hypercube format $2^{\times d}$. For $d=2$, a real binary tensor is a $2\times 2$ matrix with two singular values. Their product is the determinant. We generalize this formula for any $d\ge 2$. Given a partition $\mu\vdash d$ and a $\mu$-symmetric real binary tensor $t$, we study the distance function from $t$ to the variety $X_{\mu,\mathbb{R}}$ of $\mu$-symmetric real binary tensors of rank one. The study of the local minima of this function is related to the computation of the singular values of $t$. Denoting with $X_\mu$ the complexification of $X_{\mu,\mathbb{R}}$, the Euclidean Distance polynomial $\mathrm{EDpoly}_{X_\mu^\vee,t}(\epsilon^2)$ of the dual variety of $X_\mu$ at $t$ has among its roots the singular values of $t$. On one hand, the lowest coefficient of $\mathrm{EDpoly}_{X_\mu^\vee,t}(\epsilon^2)$ is the square of the $\mu$-discriminant of $t$ times a product of sum of squares polynomials. On the other hand, we describe the variety of $\mu$-symmetric binary tensors that do not admit the maximum number of singular values, counted with multiplicity. Finally, we compute symbolically all the coefficients of $\mathrm{EDpoly}_{X_\mu^\vee,t}(\epsilon^2)$ for tensors of format $2\times 2\times 2$.

The monodromy problem for hyperelliptic curves

We study the Dynkin diagrams associated to the monodromy of direct sums of polynomials. The monodromy problem asks under which conditions on a polynomial, the monodromy of a vanishing cycle generates the whole homology of a regular fiber. We consider the case y4+g(x), which is a generalization of the results of Christopher and Mardešić about the monodromy problem for hyperelliptic curves. Moreover, we solve the monodromy problem for direct sums of fourth degree polynomials.

Rota-Baxter operators and Bernoulli polynomials

Abstract We develop the connection between Rota-Baxter operators arisen from algebra and mathematical physics and Bernoulli polynomials. We state that a trivial property of Rota-Baxter operators implies the symmetry of the power sum polynomials and Bernoulli polynomials. We show how Rota-Baxter operators equalities rewritten in terms of Bernoulli polynomials generate identities for the latter.

The character sum of polynomials with k variables and two-term exponential sums

The main purpose of this paper is using the properties of the classical Gauss sums and the analytic methods to study the computational problem of one kind of hybrid power mean involving the character sums of polynomials with k variables and the two-term exponential sums, and give an identity and asymptotic formula for it.

The Mean Values of Character Sums and Their Applications

In this paper, we use the elementary methods and properties of classical Gauss sums to study the calculation problems of some mean values of character sums of special polynomials, and obtained several interesting calculation formulae for them. As an application, we give a criterion for determining that 2 is the cubic residue for any odd prime p.

The infinite sums of reciprocals and the partial sums of Chebyshev polynomials

<abstract><p>In this paper, the infinite sums of reciprocals and the partial sums derived from Chebyshev polynomials are studied. For the infinite sums of reciprocals, we apply the floor function to the reciprocals of these sums to obtain some new and interesting identities involving the Chebyshev polynomials. Simultaneously, we get several identities about the partial sums of Chebyshev polynomials by the relation of two types of Chebyshev polynomials.</p></abstract>

ЗАДАЧА АНИЗОТРОПИИ УПРУГОСТИ И ПРОЧНОСТИ ВОЛОКНИСТЫХ АНИЗОТРОПНЫХ МАТЕРИАЛОВ

Introduction: The paper presents new results of studies on the anisotropy of fiber materials with cylindrical anisotropy, which include filament-wound composite materials reinforced with various fibers. Methods: We suggest a mathematical solution to a fourth-order partial differential equation in polar coordinates with two variables for an orthotropic anisotropic body. To solve this equation, we converted it into Cartesian coordinates and presented the stress function as a sum of polynomials. Results and Discussion: As a result of the solution, we obtained two relationships between the elastic constants in the principal directions of anisotropy (so-called elasticity parameters). One of them was obtained for the first time, and the other results from the solution of the anisotropy problem for an orthotropic curved body, suggested by S. G. Lekhnitsky. The obtained solution does not contradict Lekhnitsky’s solution. Thus, in our opinion, orthotropic materials can be divided into two groups. In one group, when shifting from the radial to the tangential direction, the elastic constants take on extreme values when the layers are at angles of 0, 60, and 90°. In the other group, there is no intermediate extreme value and the elastic constants take on extreme values when the layers are at angles of 0 and 90°. The obtained results can be applied in the development of new high-strength composite materials and new technologies for the design and manufacture of building structures, as well as in the design of high-strength structures from synthetic composite materials.

О способах экономии компьютерной памяти при классификации данных с помощью полносвязных нейронных сетей

In solving the classification problem, a fully connected trainable neural network (with adjusting the parameters represented by double-precision real numbers) is used as a mathematical model. After the training is completed, the neural network parameters are rounded and represented as fixed-point numbers (integers). The aim of the study is to reduce the required amount of the computing system memory for storing the obtained integer parameters. To reduce the amount of memory, the following methods for storing integer parameters are developed, which are based on representing the linear polynomials included in a fully connected neural network using compositions of simpler functions: - a method based on representing the considered polynomial as a sum of simpler polynomials; - a method based on separately storing the information about additions and multiplications. In the experiment with the MNIST data set, it took 1.41 MB to store real parameters of a fully connected neural network, 0.7 MB to store integer parameters without using the proposed methods, 0.47 MB in the RAM and 0.3 MB in compressed form on the disk when using the first method, and 0.25 MB on the disk when using the second method. In the experiment with the USPS data set, it took 0.25 MB to store real parameters of a fully connected neural network, 0.1 MB to store integer parameters without using the proposed methods, 0.05 MB in the RAM and approximately the same amount in compressed form on the disk when using the first method, and 0.03 MB on the disk when using the second method. The study results can be applied in using fully connected neural networks to solve various recognition problems under the conditions of limited hardware capacities.