Construction Of Polynomials Research Articles

Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras

We propose a systematic procedure to construct polynomial algebras from intermediate Casimir invariants arising from (semisimple or non-semisimple) Lie algebras . In this approach, we deal with explicit polynomials in the enveloping algebra of . We present explicit examples of low-dimensional Lie algebras (up to dimension six) to show how they can display different behaviours and can lead to abelian algebras, quadratic algebras or more complex structures involving higher order nested commutators. Within this framework, we also demonstrate how virtual copies of the Levi factor of a Levi decomposable Lie algebra can be used as a tool to construct ‘copies’ of polynomial algebras. Different schemes to obtain polynomial algebras associated to algebraic Hamiltonians have been proposed in the literature, among them the use of commutants of various type. The present approach is different and relies on the construction of intermediate Casimir invariants in the enveloping algebra .

ON J-UNITARY MATRIX POLYNOMIALS

An efficient method for construction of J-unitary matrix polynomials is proposed, associated with companion matrix functions the last row of which is a polynomial in 1/t. The method relies on Wiener-Hopf factorization theory and stems from recently developed J-spectral factorization algorithm for certain Hermitian matrix functions.

Constructions of irreducible polynomials over finite fields with even characteristic

Constructions of irreducible polynomials over finite fields with even characteristic

Construction of Permutation Polynomials Using Additive and Multiplicative Characters

Permutation is a natural phenomenon useful for understanding and explaining the structural and functional behavior of objects or concepts. The mathematical formulation of permutation behavior can be readily achieved by permutation polynomials. Permutation polynomials are constructed by suitably modifying linearized polynomials and associated affine polynomials with the help of additive characters, multiplicative characters, and special types of Trace functions for the polynomials in one and more than one indeterminates. The permutation properties of the obtained polynomials are verified using the AGW criterion.

On Sobolev orthogonal polynomials on a triangle

We use the invariance of the triangle T 2 = { ( x , y ) ∈ R 2 : 0 ⩽ x , y , 1 − x − y } \mathbf {T}^2=\{(x,y)\in \mathbb {R}^2:\, 0\leqslant x,y,\, 1-x-y\} under the permutations of { x , y , 1 − x − y } \{x,y,1-x-y\} to construct and study two-variable orthogonal polynomial systems with respect to several distinct Sobolev inner products defined on T 2 \mathbf {T}^2 . These orthogonal polynomials can be constructed from two sequences of univariate orthogonal polynomials. In particular, one of the two univariate sequences of polynomials is orthogonal with respect to a Sobolev inner product and the other is a sequence of classical Jacobi polynomials.

Solution of the linearly structured partial polynomial inverse eigenvalue problem

In this paper, we consider the linearly structured partial polynomial inverse eigenvalue problem (LPPIEP) of constructing the matrices Ai∈Rn×n for i=0,1,2,…,(k−1) of specified linear structure such that the matrix polynomial P(λ)=λkIn+∑i=0k−1λiAi has the m (1⩽m⩽kn) prescribed eigenpairs as its eigenvalues and eigenvectors. Many practical applications give rise to linearly structured matrix polynomials. Typical linearly structured matrices are symmetric, skew-symmetric, tridiagonal, diagonal, pentagonal, Hankel, Toeplitz, etc. Therefore, construction of the matrix polynomial with the aforementioned structures is an important but challenging aspect of the polynomial inverse eigenvalue problem (PIEP). In this paper, a necessary and sufficient condition for the existence of solution to this problem is derived. Additionally, we characterize the class of all solutions to this problem by giving the explicit expressions of the solutions. It should be emphasized that the results presented in this paper resolve some important open problems in the area of PIEP namely, the inverse eigenvalue problems for structured matrix polynomials such as symmetric, skew-symmetric, alternating matrix polynomials as pointed out by De Terán et al. (2015). Further, we study sensitivity of solution to the perturbation of the eigendata. An attractive feature of our solution approach is that it does not impose any restriction on the number of eigendata for computing the solution of LPPIEP. Towards the end, the proposed method is validated with various numerical examples on a spring mass problem.

A construction of swap or switch polynomials

We discuss several constructions of swap polynomials that is non commutative polynomials in matrix variables xi∈Md(Q) with values in Md(Q)⊗2 which are multiples of the transposition operator (1,2).

Security limitations of Shamir’s secret sharing

The security is so important for both storing and transmitting the digital data, the choice of parameters is critical for a security system, that is, a weak parameter will make the scheme very vulnerable to attacks, for example the use of supersingular curves or anomalous curves leads to weaknesses in elliptic curve cryptosystems, for RSA cryptosystem there are some attacks for low public exponent or small private exponent. In certain circumstances the secret sharing scheme is required to decentralize the risk. In the context of the security of secret sharing schemes, it is known that for the scheme of Shamir, an unqualified set of shares cannot leak any information about the secret. This paper aims to show that the well-known Shamir’s secret sharing is not always perfect and that the uniform randomization before sharing is insufficient to obtain a secure scheme. The second purpose of this paper is to give an explicit construction of weak polynomials for which the Shamir’s (k, n) threshold scheme is insecure in the sense that there exist a fewer than k shares which can reconstruct the secret. Particular attention is given to the scheme whose threshold is less than or equal to 6. It also showed that for certain threshold k, the secret can be calculated by a pair of shares with the probability of 1/2. Finally, in order to address the mentioned vulnerabilities, several classes of polynomials should be avoided.

Epidemiology of benign thyroid disorders in the adult population of the Republic of Belarus: analysis of nationwide statistics 2009 to 2019

Nowadays, the Republic of Belarus belongs to the countries with sufficient iodine supply, which made it possible to reduce the incidence of non-toxic goiter and congenital hypothyroidism. However, even a slight change in iodine consumption influences the pattern of thyroid diseases. In addition to iodine deficiency, other environmental conditions, aswell as genetic factors, play a significant role in the etiology of thyroid diseases. To analyze the dynamics of the main epidemiological indicators of benign thyroid diseases from 2009 to 2019 in theadult population of the Republic of Belarus, using the data of official state statistics. The indicators of the incidence and prevalence of benign thyroid diseases were studied on the basis of state statistics for 2009-2019. To analyze the dynamics of the studied indicators, regression analysis was used with the construction of linear and polynomial models. A decrease in the incidence and prevalence of diffuse euthyroid goiter and an increase in the incidence and prevalence of nodular euthyroid goiter, thyroiditis, acquired hypothyroidism, Graves' disease, as well as the incidence of nodular toxic goiter were revealed. Obtained data indicate, that there is an increase in the prevalence of most of the studied thyroid diseases, despite the adequate iodine supply. The above justifies the need for further study of the causes of the identified trends, as well as the necessity of developing new methods of diagnosis and treatment of thyroid diseases.

Arbitrary multi-resolution multi-wavelet-based polynomial chaos expansion for data-driven uncertainty quantification

Various real world problems deal with data-driven uncertainty. In particular, in geophysical applications the amount of available data is often limited, posing a challenge in the construction of an appropriate stochastic discretization. Arbitrary polynomial chaos is an alternative to tackle this challenge. Approximating the dependence of model output on the uncertain model parameters by expansion in an orthogonal polynomial basis using data-driven principles. This type of global polynomial representation suffers often from Gibbs’ phenomena, especially if applied in non-linear convection dominated problems that require to deal with discontinuities. The multi-resolution or multi-element framework has been successfully used for reducing Gibbs’ phenomena in intrusive stochastic discretizations. In the present work, we introduce a multi-resolution extension of the arbitrary polynomial chaos expansion which is based on the construction of piecewise polynomial. Gaussian quadrature nodes and weights that are computed using only stochastic (localized) moments provided by the underlying raw data. We enhance our approach by a multi-wavelet based stochastic adaptivity that assures a significant reduction of the computational costs. Numerical experiments of increasing complexity demonstrate the performance of the non-intrusive implementation of the introduced methods in relevant scenarios. The use of a carbon dioxide storage benchmark scenario allows one to compare the presented methodology with other stochastic discretization techniques applied to this benchmark.

Multivariate Orthogonal Laurent Polynomials and Integrable Systems

An ordering for Laurent polynomials in the algebraic torus (\mathbb{C}^*)^D , inspired by the Cantero–Moral–Vel´azquez approach to orthogonal Laurent polynomials in the unit circle, leads to the construction of a moment matrix for a given Borel measure in the unit torus \mathbb{T}^D . The Gauss–Borel factorization of this moment matrix allows for the construction of multivariate biorthogonal Laurent polynomials in the unit torus, which can be expressed as last quasi-determinants of bordered truncations of the moment matrix. The associated second-kind functions are expressed in terms of the Fourier series of the given measure. Persymmetries and partial persymmetries of the moment matrix are studied and Cauchy integral representations of the second-kind functions are found, as well as Plemelj-type formulae. Spectral matrices give string equations for the moment matrix, which model the three-term relations as well as the Christoffel–Darboux formulae. Christoffel-type perturbations of the measure given by the multiplication by Laurent polynomials are studied. Sample matrices on poised sets of nodes, which belong to the algebraic hypersurface of the perturbing Laurent polynomial, are used to find a Christoffel formula that expresses the perturbed orthogonal Laurent polynomials in terms of a last quasi-determinant of a bordered sample matrix constructed in terms of the original orthogonal Laurent polynomials. Poised sets exist only for prepared Laurent polynomials, which are analyzed from the perspective of Newton polytopes and tropical geometry. Then, an algebraic geometrical characterization of prepared Laurent polynomial perturbation and poised sets is given; full-column-rankness of the corresponding multivariate Laurent–Vandermonde matrices and a product of different prime prepared Laurent polynomials leads to such sets. Some examples are constructed in terms of perturbations of the Lebesgue–Haar measure. Discrete and continuous deformations of the measure lead to a Toda-type integrable hierarchy, being the corresponding flows described through Lax and Zakharov–Shabat equations; bilinear equations and vertex operators are found. Varying size matrix nonlinear partial difference and differential equations of two-dimensional Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows are connected with a Gauss–Borel factorization of the Jacobi-type matrices and its quasi-determinants allow for expressions for the multivariate orthogonal polynomials in terms of shifted quasi-tau matrices, which generalize those that relate the Baker functions with ratios of Miwa shifted \tau -functions in the one-dimensional scenario. It is shown that the discrete and continuous flows are deeply connected and determine nonlinear partial difference-differential equations that involve only one site in the integrable lattice behaving as a Kadomtsev–Petviashvili-type system.

EXPLICIT FIFTH-ORDER HYBRID METHODS FOR DYNAMICAL SYSTEMS WITH DELAY

In the paper we consider the systems of delay differential equations, which are mathematical models of many technical processes with delay in time. An explicit fifth-order hybrid method for systems of delay differential equations for the variable step of numerical integration is constructed. This algorithm is based on the most commonly used explicit fifth-order Runge-Kutta methods for ordinary systems of differential equations and the construction of Newton polynomials for the prehistory of the model and Taylor formula. The basic principles of construction of such explicit hybrid methods of higher orders of convergence are given. An accurate estimate of the local error of numerical integration by this method is obtained.
 This algorithm allows the use of numerical integration steps greater than the magnitude of the delay or procedures for adjusting the magnitude of the step depending on the calculation error. Such a problem of numerical integration of time-delay dynamical systems occurs when the time interval is large enough compared to the delay. Implicit continuous Runge-Kutta methods were previously used for numerical integration such problems, which complicated the numerical algorithm, because nonlinear systems of equations have to be solved at each step of numerical integration. The explicit hybrid method constructed by us for systems of delay differential equations is convenient for programming, has a high speed of calculation of numerical solution, in comparison with implicit methods for such models of problems. Also, this method has no restrictions on the magnitude of the delay, in contrast to the hybrid methods with decomposition in Taylor series for delay. The method obtained by us can be used to construct maps of dynamic modes in the study of regular and chaotic behavior of time-delay dynamical systems.

Explicit lower bounds on |L(1,χ)|

Let χ denote a primitive, non-quadratic Dirichlet character with conductor q, and let L(s,χ) denote its associated Dirichlet L-function. We show that |L(1,χ)|≥1/(9.12255log⁡(q/π)) for sufficiently large q, and that |L(1,χ)|≥1/(9.69030log⁡(q/π)) for all q≥2, improving some results of Louboutin. The improvements come from an averaging argument that simplifies Louboutin's approach, and the construction of certain trigonometric polynomials using simulated annealing, a technique often used in applied mathematics and computer science. We highlight the benefits of this approach to a pure mathematical problem.

Interactive Proofs for Rounding Arithmetic

Interactive proofs are a type of verifiable computing that secures the integrity of computations. The need is increasing as more computations are outsourced to untrusted parties, e.g., cloud computing platforms. Existing techniques, however, have mainly focused on exact computations, but not approximate arithmetic (e.g., floating-point or fixed-point arithmetic). This makes it hard to apply them to certain types of computations (e.g., machine learning or financial applications) that inherently require approximate arithmetic. In this paper, we present an efficient interactive proof system for arithmetic circuits with rounding gates that can represent approximate arithmetic. The main idea is to reduce the rounding gate into a small sub-circuit without rounding, and reuse the machinery of the Goldwasser, Kalai, and Rothblum’s protocol (also known as the GKR protocol) and its recent refinements. Specifically, we shift the algebraic structure from a field to a ring to better deal with the notion of “digits”, and generalize the original GKR protocol over a ring. Then, we reduce the rounding operation to a low-degree polynomial over a ring, and develop a novel, optimal circuit construction of an arbitrary polynomial to transform the rounding polynomial to an optimal circuit representation. Moreover, further optimization on the proof generation cost for rounding is presented employing a Galois ring. Our experimental results show the efficiency of our protocol for approximate arithmetic, e.g., the implementation performed two orders of magnitude better than the existing system for a nested 128 by 128 matrix multiplication of depth 12 on the 16-bit fixed-point arithmetic.

New examples of Krall–Meixner and Krall–Hahn polynomials, with applications to the construction of exceptional Meixner and Laguerre polynomials

We construct new examples of Krall discrete orthogonal polynomials, i.e., orthogonal polynomials with respect to a measure which are also eigenfunctions of a higher order difference operator. The new examples include the orthogonal polynomials with respect to the measures obtained from the Meixner measure ρa,c and Hahn measure ρa,b,N by removing a finite number of their mass points when the parameter c of the Meixner measure and a or b of the Hahn measure are positive integers. From the new Krall–Meixner families we construct new families of exceptional Meixner and Laguerre polynomials.

On the construction of irreducible and primitive polynomials from [formula omitted] to [formula omitted

In this paper, we present two constructions of irreducible (primitive) polynomials over Fq of degree rm from irreducible (primitive) polynomial over Fqm of degree r, and we show that these two constructions coincide. The first construction is based on the Frobenius automorphism of Fqm over Fq. The second one comes from a generalization of a construction of primitive polynomials over Fq which uses the companion matrix. From this generalization, given an irreducible (resp. primitive) polynomial over Fqm of degree r, we generate multiple (resp. all) irreducible polynomials over Fq of degree rm. As an application, a characterization of the generator polynomial of a BCH code over Fq is given. Then, we show how two BCH codes over Fq and Fqm, respectively, and their generator polynomials are related.

Polynomial Approximations for Several Neural Network Activation Functions

Active deployment of machine learning systems sets a task of their protection against different types of attacks that threaten confidentiality, integrity and accessibility of both processed data and trained models. One of the promising ways for such protection is the development of privacy-preserving machine learning systems, that use homomorphic encryption schemes to protect data and models. However, such schemes can only process polynomial functions, which means that we need to construct polynomial approximations for nonlinear functions used in neural models. The goal of this paper is the construction of precise approximations of several widely used neural network activation functions while limiting the degree of approximation polynomials as well as the evaluation of the impact of the approximation precision on the resulting value of the whole neural network. In contrast to the previous publications, in the current paper we study and compare different ways for polynomial approximation construction, introduce precision metrics, present exact formulas for approximation polynomials as well as exact values of corresponding precisions. We compare our results with the previously published ones. Finally, for a simple convolutional network we experimentally evaluate the impact of the approximation precision on the bias of the output neuron values of the network from the original ones. Our results show that the best approximation for ReLU could be obtained with the numeric method, and for the sigmoid and hyperbolic tangent – with Chebyshev polynomials. At the same time, the best approximation among the three functions could be obtained for ReLU. The results could be used for the construction of polynomial approximations of activation functions in privacy-preserving machine learning systems.

Skew-Orthogonal Polynomials in the Complex Plane and Their Bergman-Like Kernels

Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. These point processes are characterised by a matrix valued kernel of skew-orthogonal polynomials. We develop their theory in providing an explicit construction of skew-orthogonal polynomials in terms of orthogonal polynomials that satisfy a three-term recurrence relation, for general weight functions in the complex plane. New examples for symplectic ensembles are provided, based on recent developments in orthogonal polynomials on planar domains or curves in the complex plane. Furthermore, Bergman-like kernels of skew-orthogonal Hermite and Laguerre polynomials are derived, from which the conjectured universality of the elliptic symplectic Ginibre ensemble and its chiral partner follow in the limit of strong non-Hermiticity at the origin. A Christoffel perturbation of skew-orthogonal polynomials as it appears in applications to quantum field theory is provided.

Construction of irreducible polynomials over finite fields

Irreducible polynomials over finite fields and their applications have been quite well studied. Here, we discuss the construction of the irreducible polynomials of degree [Formula: see text] over the finite field [Formula: see text] for a given irreducible polynomial of degree [Formula: see text]. Furthermore, we construct the irreducible polynomials of degree [Formula: see text] over the finite field [Formula: see text] for a given irreducible polynomial of degree [Formula: see text] by using the method of composition of polynomials with some conditions on coefficients and degree of a given irreducible polynomial.

An Effective Algorithm for the Synthesis of Irreducible Polynomials over a Galois Fields of Arbitrary Characteristics

The known algorithms for synthesizing irreducible polynomials have a significant drawback: their computational complexity, as a rule, exceeds the quadratic one. Moreover, consequently, as a consequence, the construction of large-degree polynomials can be implemented only on computing systems with very high performance. The proposed algorithm is base on the use of so-called fiducial grids (ladders). At each rung of the ladder, simple recurrent modular computations are performers. The purpose of the calculations is to test the irreducibility of polynomials over Galois fields of arbitrary characteristics. The number of testing steps coincides with the degree of the synthesized polynomials. Upon completion of testing, the polynomial is classifieds as either irreducible or composite. If the degree of the synthesized polynomials is small (no more than two dozen), the formation of a set of tested polynomials is carried out using the exhaustive search method. For large values of the degree, the test polynomials are generating by statistical modeling. The developed algorithm allows one to synthesize binary irreducible polynomials up to 2Kbit on personal computers of average performance