Set Of Polynomials Research Articles

Translation invariant linear spaces of polynomials

A set of polynomials is called a <em>submodule</em> of $\mathbb C [x_1 ,\ldots , x_n ]$ if it is a translation invariant linear subspace of $\mathbb C [x_1 ,\ldots , x_n ]$. We present a description of the submodules of $\mathbb C [x,y]$ in terms of a

INFLUENCE OF DIMENSIONLESS WELLBORE RADIUS ON DIMENSIONLESS PRESSURE AND PRESSURE DERIVATIVES

Effects of wellbore radius on pressure derivatives of a vertical oil well is a major concern to oil and gas industry operators who intend to execute fluid production from a vertical oil well. Reservoir deliverability depends on several factors: the dimensionless wellbore radius. This study examines the influence of the dimensionless wellbore radius on the pressure and pressure derivative. A set of polynomials was implemented to calculate the dimensionless pressure variable and the pressure derivative. The polynomials were put into a computer program developed in the study to create a fast means through which the effect of wellbore radii can be studied easily. However, when several dimensionless radii were used as input parameters for the developed computer program, it was observed that the lower the dimensionless wellbore pressure, the higher the pressure derivative and vice versa. Also, lower dimensionless time yielded higher values of the dimensionless pressure variable. Results from the developed computer model were validated by comparing results from the model to those obtained from a published article while using the same input parameters (tD and rWD). Moreover, the percentage error was estimated to be less than 0.02%.

Краткие неинтерактивные аргументы с нулевым разглашением на основе наборов полиномов

The paper discusses the basic principles of construction and the main types of zeroknowledge succinct non-interactive argument of knowledge (zk-SNARK) which is used in the model of a three-way insecure computing environment and based on sets of polynomials. A number of zk-SNARK cryptographic protocols with different algorithms for generating public parameters (Trusted Setup) are given, constructing succinct proofs of reliability calculations (Prover) and public/designated verification of proofs (Verifier). The cases of satisfying the feasibility of discrete functions (arithmetic/ Boolean circuits) using different polynomial sets are presented in quadratic arithmetic programs (QAP), square arithmetic programs (SAP), quadratic span programs (QSP), square span programs (SSP), quadratic polynomial programs (QPP), etc., also the use of authenticated data are described. The cryptographic transformations needed to build zk-SNARKs based on symmetric and asymmetric hash functions, exponential knowledge problems, digital signatures, homomorphic encryption, bilinear pairings based on elliptic curves, etc. are presented. Examples of multilateral verifiable calculations based on zk-SNARK are given.

Decision making in railway interlocking systems based on calculating the remainder of dividing a polynomial by a set of polynomials

&lt;abstract&gt;&lt;p&gt;Decision-making in a railway station regarding the compatibility of the positions of the switches of the turnouts and the indications (proceed/stop) of the railway colour light signals is a safety-critical issue that is considered very labor-intensive. Different authors have proposed alternative solutions to automate its supervision, which is performed by the so-called railway interlocking systems. The classic railway interlocking systems are route-based and their compatibility is predetermined (usually by human experts): only some chosen routes are simultaneously allowed. Some modern railway interlocking systems are geographical and make decisions on the fly, but are unsuitable if the station is very large and the number of trains is high. In this paper, we present a completely new algebraic model for decision-making in railway interlocking systems, based on other computer algebra techniques, that bypasses the disadvantages of the approaches mentioned above (its performance does not depend on the number of trains in the railway station and can be used in large railway stations). The main goal of this work is to provide a mathematical solution to the interlocking problems. We prove that our approach solves it in linear time. Although our approach is interesting from a theoretical perspective, it has a significant limitation: it can hardly be adopted in an actual interlocking implementation, mainly due to the heavy certification requirements for this kind of safety-critical application. Nevertheless, the results may be useful for simulations that do not require certification credit.&lt;/p&gt;&lt;/abstract&gt;

НОВЫЙ ПОДХОД К РАЗРАБОТКЕ АЛГОРИТМОВ МНОГОМЕРНОЙ КРИПТОГРАФИИ

Purpose of work is the reduction in the size of the public key of public-key algorithms of multivariate cryptography based on the computational difficulty of solving systems of many power equations with many unknowns. Research method is use of non-linear mappings defined as exponentiation operations in finite extended fields GF(qm) represented in the form of finite algebras. The latter makes it possible to perform the exponentiation operation in the field GF(qm) by calculating the values of power polynomials over the field GF(q), which define a hardly reversible nonlinear mapping of the vector space over GF(q) with a secret trapdoor. Due to the use of nonlinear mappings of this type, it is possible to specify a public key in multidimensional cryptography algorithms in the form of a nonlinear mapping implemented as a calculation of the values of a set of polynomials of the third and sixth degree. At the same time, due to the use of masking linear mappings that do not lead to an increase in the number of terms in polynomials, the size of the public key is reduced in comparison with known analogue algorithms, in which the public key is represented by a set of polynomials of the second and third degrees. The proposed approach potentially expands the areas of practical application of post-quantum algorithms for public encryption and electronic digital signature, related to multidimensional cryptography, by significantly reducing the size of the public key. Results of the study are the main provisions of a new approach to the development of algorithms of multidimensional cryptography are formulated. Hardly invertible nonlinear mappings with a secret trapdoor are proposed in the form of exponentiation operations to the second and third powers in finite extended fields GF(qm), represented in a form of a finite algebra. A rationale is given for specifying a public key in a form that includes a superposition of two non-linear mappings performed as a calculation of a set of second and third degree polynomials defined over GF(q). Techniques for implementing mappings of this type are proposed and specific options for specifying the fields GF(qm) in the form of finite algebras are considered. An estimate of the size of the public key in the algorithms developed within the framework of the new approach is made. at a given security level.. Practical relevance includes the developed main provisions of a new method for constructing multidimensional cryptography algorithms based on the computational difficulty of solving systems of many power equations with many unknowns and related to post-quantum cryptoschemes. The proposed approach expands the areas of practical application of post-quantum algorithms of this type by significantly reducing the size of the public key, which provides the prerequisites for improving performance and reducing technical resources for their implementation

Irreducible Radical Field Extensions

This paper is devoted to irreducible radical field extensions, i.e., extensions that can be obtained by adjunction of roots of irreducible binomials. We find a criterion for cyclotomic extensions to be irreducible radical. This criterion developes the Gauss–Wantzel theorem about constructible polygons. We also prove that any normal solvable extension of some field K is irreducible radical until K has all roots of unity. This generalizes Abel’s theorem, which fills the gap in the uncomplete Ruffini proof of the impossibility theorem for the general equation of degree five or higher. Finally, we prove that the root set of any irreducible solvable polynomial coincides with the value set of some radical formula using irreducible radical extensions.

The algebra of difference operators associated to Meixner type polynomials

Meixner type polynomials are defined from the Meixner polynomials by using Casoratian determinants whose entries belong to two given finite sets of polynomials and . They are eigenfunctions of higher-order difference operators but only for a careful choice of the polynomials and , the sequence is orthogonal with respect to a measure. Under mild assumptions, we characterize in this paper the algebra formed by all difference operators with respect to which the family of Meixner type polynomials are eigenfunctions.

The smooth extension embedding method with Chebyshev polynomials

AbstractWe propose an implementation of the smooth extension embedding method (SEEM), first described by Agress and Guidotti's study, in the setting of Chebyshev polynomials. SEEM is a hybrid fictitious domain/collocation method which solves general boundary value problems in complex domains by recasting them as constrained optimization problems in a simple encompassing set. Previously, SEEM was introduced and implemented using a periodic box (read a torus) using Fourier series; here, it is implemented on a (non‐periodic) rectangle using Chebyshev polynomial expansions. This implementation has faster convergence on smaller grids. Numerical experiments will demonstrate that the method provides a simple, robust, efficient, and high order fictitious domain method which can solve elliptic and parabolic problems in complex geometries, with non‐constant coefficients, and for general boundary conditions. We consider applications to two and three dimensional boundary value problems as well as an initial boundary value problem via a genuinely space–time discretization.

The separating variety for 2 × 2 matrix invariants

Let G be a linear algebraic group acting linearly on a vector space (or more generally, an affine variety) V , and let k [ V ] G be the corresponding algebra of invariant polynomial functions. A separating set S ⊆ k [ V ] G is a set of polynomials with the property that for all v , w ∈ V , if there exists f ∈ k [ V ] G separating $v$ and $w$, then there exists f ∈ S separating $v$ and $w$. In this article, we consider the action of G = GL 2 ⁡ ( C ) on the C -vector space M 2 n of n-tuples of 2 × 2 matrices by simultaneous conjugation. Minimal generating sets S n of C [ M 2 n ] G are well known and | S n | = 1 6 ( n 3 + 11 n ) . In recent work, Kaygorodov et al. [Kaygorodov I, Lopatin A, Popov Y. Separating invariants for 2 × 2 matrices. Linear Algebra Appl. 2018;559:114-124.] showed that for all n ≥ 1 , S n is a minimal separating set by inclusion, i.e. that no proper subset of S n is a separating set. This does not necessarily mean that S n has minimum cardinality among all separating sets for C [ M 2 n ] G . Our main result shows that any separating set for C [ M 2 n ] G has cardinality ≥ 5 n − 5 . In particular, there is no separating set of size dim ⁡ ( C [ M 2 n ] G ) = 4 n − 3 for n ≥ 3 . Further, S 3 has indeed minimum cardinality as a separating set, but for n ≥ 4 there may exist a smaller separating set than S n . We show that a smaller separating set does in fact exist for all n ≥ 5 . We also prove similar results for the left–right action of SL 2 ⁡ ( C ) × SL 2 ⁡ ( C ) on M 2 n .

Stokes, Gibbs, and Volume Computation of Semi-Algebraic Sets

We consider the problem of computing the Lebesgue volume of compact basic semi-algebraic sets. In full generality, it can be approximated as closely as desired by a converging hierarchy of upper bounds obtained by applying the Moment-SOS (sums of squares) methodology to a certain infinite-dimensional linear program (LP). At each step one solves a semidefinite relaxation of the LP which involves pseudo-moments up to a certain degree. Its dual computes a polynomial of same degree which approximates from above the discontinuous indicator function of the set, hence with a typical Gibbs phenomenon which results in a slow convergence of the associated numerical scheme. Drastic improvements have been observed by introducing in the initial LP additional linear moment constraints obtained from a certain application of Stokes’ theorem for integration on the set. However and so far there was no rationale to explain this behavior. We provide a refined version of this extended LP formulation. When the set is the smooth super-level set of a single polynomial, we show that the dual of this refined LP has an optimal solution which is a continuous function. Therefore in this dual one now approximates a continuous function by a polynomial, hence with no Gibbs phenomenon, which explains and improves the already observed drastic acceleration of the convergence of the hierarchy. Interestingly, the technique of proof involves recent results on Poisson’s partial differential equation (PDE).

Mandelbrot and Julia Sets of Complex Polynomials Involving Sine and Cosine Functions via Picard–Mann Orbit

Mandelbrot and Julia Sets of Complex Polynomials Involving Sine and Cosine Functions via Picard–Mann Orbit

Algebraic properties of HTC-identifiable graphs

In this paper, we explore some algebraic properties of linear structural equation modelsthat can be represented by an HTC-identifiable graph. In particular, we prove that all mixedgraphs are HTC-identifiable if and only if all the regression coefficients can be recovered fromthe covariance matrix using straightforward linear algebra operations. We also find a set ofpolynomials that generates the ideal that encompasses all the equality constraints of the modelon the cone of positive definite matrices. We further prove that this set of polynomials are theminimal generators of said ideal for a subset of HTC-identifiable graphs.

A new algorithm for computing staggered linear bases

Considering a multivariate polynomial ideal over a given field as a vector space, we investigate for such an ideal a particular linear basis, a so-called staggered linear basis, which contains a Gröbner basis as well. In this paper, we present a simple and efficient algorithm to compute a staggered linear basis. The new framework is equipped with some novel criteria (including both Buchberger's criteria) to detect superfluous reductions. The proposed algorithm has been implemented in Maple, and we provide an illustrative example to show how it works. Finally, the efficiency of this algorithm compared to the existing methods is discussed via a set of benchmark polynomials.

Vibration analysis of thick-section sandwich structures in thermal environments

An ultra-light sandwich structure with a thin ceramic-based composite face sheet bonded to a thick thermal insulator is developed as an integrated thermal protection system (ITPS) structure to protect the substructure of a hypersonic flight vehicle. For vibration fatigue assessment and failure prediction, the global dynamic response of the three-dimensional ITPS structure when subjected to a combination of thermal and harmonic excitation is desired. In this study, a novel unified model for vibration analysis of a thick-section sandwich structure is proposed based on the variational asymptotic method. The original 3D sandwich structure problem is reduced to one-dimensional through-the-thickness analysis and two-dimensional (2D) reference plane analysis. The stiffness matrices, including an equivalent transverse shear matrix, are obtained; hence, a shear correction factor is not required. The mechanical response of the 2D reduced-order plate under general boundary conditions in thermal environments is investigated. The static thermal deformation and forced vibration of the sandwich structure with the temperature gradient in the thickness direction are determined using a set of algebraic orthogonal polynomials through the Gram–Schmidt procedure. The thermal deflection is first introduced into the kinematic relationship to modify the total strain energy for the sandwich plate with temperature-independent material properties. To demonstrate the validity and applicability of the model, some numerical results relating to displacements and fundamental frequencies of simply supported sandwich plates are presented and compared to those obtained by other models. The effects of temperature gradients in the thickness direction, boundary conditions, core thickness, and temperature-dependent material properties on the dynamic characteristics and forced vibration of the sandwich structure are discussed by a detailed parametric study. The global dynamic strain response is first obtained using the plate model, which is desirable to determine the 3D local stress and strain field and predict vibration fatigue failure for the ITPS structure.

On Hadamard's three‐hyperballs theorem and its applications to Whittaker‐Cannon hypercomplex theory

This paper shows a hypercomplex function theory emerging in the representation of paravector‐valued monogenic functions over the ‐dimensional Euclidean space through a basic set (or basis) of hypercomplex monogenic polynomials. We derive the properties of the arising hypercomplex Cannon function and present an extension of the well‐known Whittaker‐Cannon theorem to special monogenic functions defined in an open hyperball in . More precisely, we determine what conditions should be applied to a basic set of special monogenic polynomials to attain the effectiveness property in an open hyperball employing Hadamard's three‐hyperballs theorem. We also provide a necessary and sufficient condition for a special monogenic Cannon series to represent every function near the origin that is special monogenic there. Additionally, we investigate the effectiveness of a non‐Cannon basis and show that the underlying hypercomplex Cannon function maintains similar properties in both cases, the Cannon basis and the non‐Cannon basis.

On computing root polynomials and minimal bases of matrix pencils

We revisit the notion of root polynomials, thoroughly studied in (Dopico and Noferini, 2020 [9]) for general polynomial matrices, and show how they can efficiently be computed in the case of a matrix pencil λE+A. The method we propose makes extensive use of the staircase algorithm, which is known to compute the left and right minimal indices of the Kronecker structure of the pencil. In addition, we show here that the staircase algorithm, applied to the expansion (λ−λ0)E+(A−λ0E), constructs a block triangular pencil from which a minimal basis and a maximal set of root polynomials at the eigenvalue λ0, can be computed in an efficient manner.

Choosing better variable orderings for cylindrical algebraic decomposition via exploiting chordal structure

As is well-known, the choice of variable ordering while computing cylindrical algebraic decomposition (CAD) has a great effect on the time and memory use of the computation as well as the number of sample points computed. In this paper, we indicate that typical CAD algorithms, if executed with respect to a special kind of variable orderings (called “the perfect elimination orderings”, PEO), naturally preserve chordality, which is well compatible with an important (variable) sparsity pattern called “the correlative sparsity”. If the associated graph of the polynomial system in question is chordal (resp., is nearly chordal), then a PEO of the associated graph (resp., of a minimal chordal completion of the associated graph) can be a better variable ordering for the CAD computation than other naive variable orderings in the sense that it results in a much smaller full set of projection polynomials and thus more efficient computation. A new (m,d)-property of the full set of CAD projection polynomials obtained via a PEO is given, which indicates that when the corresponding perfect elimination tree has a lower height, the full set of projection polynomials also tends to have a smaller “size”. Furthermore, combining the lower-tree-height rule with Brown's heuristics, a new procedure is proposed to choose better PEOs for CAD computation.

Thermal buckling and post-buckling analyses of rotating Timoshenko microbeams reinforced with graphene platelet

A comprehensive study is established to investigate on the thermal buckling instability and the subsequent post-critical deflection of a rotating nanocomposite microbeam reinforced with graphene platelet. The main scope of the current research is shedding light on the effectiveness of reinforcing a rotating microbeam with graphene platelet subjected to a constant temperature development. The Timoshenko beam theory comes along the von-Karman strain displacement relations to extract the governing equations of motion in accordance with the modified couple stress theory. The modified Halpin–Tsai micromechanical model is implemented to determine the practical elasticity modulus of the nanocomposite beam. The Ritz technique is accompanied by the Chebyshev orthonormal polynomial set to discretize the weak form of the nonlinear equations of motion. The Newton–Raphson technique is applied to the discretized static nonlinear equations of motion to compute the static deformation due to the centrifugal force. On the basis of two effective algorithms the nonlinear equations of motion are attacked to find out the critical buckling temperature change as well as the succeeding post-critical deflection. It is observed that adjoining the graphene phase in the form of an X-pattern promotes the static strength of a rotating microbeam against the buckling instability. However, in contrast, for stationary microbeams and microbeams that rotate with a velocity below a threshold speed adjoining the graphene platelet reinforcement in the form of an O-Pattern not only does not strengthen the microbeam but also weakens it against the buckling instability. Moreover, the buckling modeshape of rotating simply-clamped microbeams is more impressed by the graphene phase.

KOLMOGOROV-WIENER FILTER FOR CONTINUOUS TRAFFIC PREDICTION IN THE GFSD MODEL

Context. We investigate the Kolmogorov-Wiener filter weight function for the prediction of continuous stationary telecommunication traffic in the GFSD (Gaussian fractional sum-difference) model.&#x0D; Objective. The aim of the work is to obtain an approximate solution for the corresponding weight function and to illustrate the convergence of the truncated polynomial expansion method used in this paper.&#x0D; Method. The truncated polynomial expansion method is used for the obtaining of an approximate solution for the KolmogorovWiener weight function under consideration. In this paper we used the corresponding method on the basis of the Chebyshev polynomials of the first kind orthogonal on the time interval on which the filter input data are given. It is expected that the results based on other polynomial sets will be similar to the results obtained in this paper.&#x0D; Results. The weight function is investigated in the approximations up to the eighteen-polynomial one. It is shown that approximations of rather large numbers of polynomials lead to a good coincidence of the left-hand side and the right-hand side of the Wiener-Hopf integral equation. The quality of the coincidence is illustrated by the calculation of the corresponding MAPE errors.&#x0D; Conclusions. The paper is devoted to the theoretical construction of the Kolmogorov-Wiener filter for the prediction of continuous stationary telecommunication traffic in the GFSD model. The traffic correlation function in the framework of the GFSD model is a positively defined one, which guarantees the convergence of the truncated polynomial expansion method. The corresponding weight function is obtained in the approximations up to the eighteen-polynomial one. The convergence of the method is illustrated by the calculation of the MAPE errors of misalignment of the left-hand side and the right-hand side of the Wiener-Hopf integral equation under consideration. The results of the paper may be applied to practical traffic prediction in telecommunication systems with data packet transfer.

Polynomial superlevel set representation of the multistationarity region of chemical reaction networks

In this paper we introduce a new representation for the multistationarity region of a reaction network, using polynomial superlevel sets. The advantages of using this polynomial superlevel set representation over the already existing representations (cylindrical algebraic decompositions, numeric sampling, rectangular divisions) is discussed, and algorithms to compute this new representation are provided. The results are given for the general mathematical formalism of a parametric system of equations and so may be applied to other application domains.