Polynomial Representation Research Articles

Extended Exton’s Triple and Horn’s Double Hypergeometric Functions and Associated Bounding Inequalities

This paper introduces extensions H4,p and X8,p of Horn’s double hypergeometric function H4 and Exton’s triple hypergeometric function X8, taking into account recent extensions of Euler’s beta function, hypergeometric function, and confluent hypergeometric function. Among the numerous extended hypergeometric functions, the primary rationale for choosing H4 and X8 is their comparable extension type. Next, we present various integral representations of Euler and Laplace types, Mellin and inverse Mellin transforms, Laguerre polynomial representations, transformation formulae, and a recurrence relation for the extended functions. In particular, we provide a generating function for X8,p and several bounding inequalities for H4,p and X8,p. We explore the utilization of the H4,p function within a probability distribution. Most special functions, such as the generalized hypergeometric function, the Beta function, and the p-extended Beta integral, exhibit natural symmetry.

Explicit Properties of Apostol-Type Frobenius–Euler Polynomials Involving q-Trigonometric Functions with Applications in Computer Modeling

In this article, we define q-cosine and q-sine Apostol-type Frobenius–Euler polynomials and derive interesting relations. We also obtain new properties by making use of power series expansions of q-trigonometric functions, properties of q-exponential functions, and q-analogues of the binomial theorem. By using the Mathematica program, the computational formulae and graphical representation for the aforementioned polynomials are obtained. By making use of a partial derivative operator, we derived some interesting finite combinatorial sums. Finally, we detail some special cases for these results.

The truncated moment problem on curves y = q(x) and yx ℓ = 1

In this paper, we study the bivariate truncated moment problem (TMP) on curves of the form y = q ( x ) , q ( x ) ∈ R [ x ] , deg ⁡ q ≥ 3 and y x ℓ = 1 , ℓ ∈ N ∖ { 1 } . For even degree sequences, the solution based on the size of moment matrix extensions was first given by Fialkow [Fialkow L. Solution of the truncated moment problem with variety y = x 3 . Trans Amer Math Soc. 2011;363:3133–3165.] using the truncated Riesz–Haviland theorem [Curto R, Fialkow L. An analogue of the Riesz–Haviland theorem for the truncated moment problem. J Funct Anal. 2008;255:2709–2731.] and a sum-of-squares representations for polynomials, strictly positive on such curves [Fialkow L. Solution of the truncated moment problem with variety y = x 3 . Trans Amer Math Soc. 2011;363:3133–3165.; Stochel J. Solving the truncated moment problem solves the moment problem. Glasgow J Math. 2001;43:335–341.]. Namely, the upper bound on this size is quadratic in the degrees of the sequence and the polynomial determining a curve. We use a reduction to the univariate setting technique, introduced in [Zalar A. The truncated Hamburger moment problem with gaps in the index set. Integral Equ Oper Theory. 2021;93:36.doi: 10.1007/s00020-021-02628-6.; Zalar A. The truncated moment problem on the union of parallel lines. Linear Algebra Appl. 2022;649:186–239. http://doi.org/10.1016/j.laa.2022.05.008.; Zalar A. The strong truncated Hamburger moment problem with and without gaps. J Math Anal Appl. 2022;516:126563. doi: 10.1016/j.jmaa.2022.126563.], and improve Fialkow's bound to deg ⁡ q − 1 (resp. ℓ + 1 ) for curves y = q ( x ) (resp. y x ℓ = 1 ). This in turn gives analogous improvements of the degrees in the sum-of-squares representations referred to above. Moreover, we get the upper bounds on the number of atoms in the minimal representing measure, which are kdeg ⁡ q (resp. k ( ℓ + 1 ) ) for curves y = q ( x ) (resp. y x ℓ = 1 ) for even degree sequences, while for odd ones they are kdeg ⁡ q − ⌈ deg ⁡ q 2 ⌉ (resp. k ( ℓ + 1 ) − ⌊ ℓ 2 ⌋ + 1 ) for curves y = q ( x ) (resp. y x ℓ = 1 ). In the even case, these are counterparts to the result by Riener and Schweighofer [Riener C, Schweighofer M. Optimization approaches to quadrature:a new characterization of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions. J Complex. 2018;45:22–54., Corollary 7.8], which gives the same bound for odd degree sequences on all plane curves. In the odd case, their bound is slightly improved on the curves we study. Further on, we give another solution to the TMP on the curves studied based on the feasibility of a linear matrix inequality, corresponding to the univariate sequence obtained, and finally we solve concretely odd degree cases to the TMP on curves y = x ℓ , ℓ = 2 , 3 , and add a new solvability condition to the even degree case on the curve y = x 2 .

A Note on Location of the Zeros of Quaternionic Polynomials

The purpose of this paper is to investigate the extensions of the classical Eneström-Kakeya theorem and its various generalizations concerning the distribution of zeros of polynomials from the complex to the quaternionic setting. Using a maximum modulus theorem and the zero set structure in the recently published theory of regular functions and polynomials of a quaternionic variable, we construct new bounds of the Eneström-Kakeya type for the zeros of these polynomials. The obtained results for this subclass of polynomials and slice regular functions give generalizations of a number of results previously reported in the relevant literature.

NASA Polynomial representation of molecular specific heats

So called NASA polynomials are widely used in plasma and combustion models to represent the specific heat of molecules as a function of temperature. In this work, we compute seven-term NASA polynomials for 464 molecules of which 44 are cations and 9 are anions; polynomials are not currently available for almost 200 of these species. Calculation of the NASA polynomials utilises data provided by the ExoMol database, the HITRAN database, the diatomic partition functions computed by Barklem and Collet, and the JANAF thermodynamic tables. Our results are compared against existing polynomial compilations where available, and for cases where there are multiple datasets the recommended polynomials are identified. As proposed in the original compilation, the seven-term polynomials are fitted separately for the temperature ranges 200 – 1000 K and 1000 – 6000 K. In general, different data sources give good agreement in the lower temperature range but there are significant discrepancies at higher temperatures, which can be attributed to the underlying assumptions made about highly excited rotation-vibration energy levels.

Holistic Optimal Design of Face-Milled Hypoid Gearsets

Abstract The aim of this paper is to present a holistic framework to design optimized spiral bevel and hypoid gearsets with accurate finite element simulations in the loop. Starting from the basic transmission data, we first size gear and pinion blanks, and then we synthesize the basic machine-tool settings required to generate the two toothed members. This first step represents the macro-geometry design phase and its outcome is a conjugate spiral bevel or hypoid gearset. The second design phase is represented by the definition of the optimal pinion micro-geometry. This is formulated as a multi-objective optimization problem (MOOP) where the obtained optimal ease-off is guaranteed to be manufacturable. To this end, an original strategy is proposed where the search for the pinion optimal tooth surface happens in the space of the coefficients of a polynomial representation of its micro-topography. However, thanks to a fast identification algorithm that can handle all the higher-order motions, the ideal ease-off is projected onto set of machine-tool settings, thus ensuring manufacturability from the outset. It is worth remarking that the objective functions in the MOOP are evaluated by calling as a back-end solver one of the most accurate loaded tooth contact analysis software available on the market. A dedicated parallel implementation of such MOOP allows to maintain computation times within very reasonable limits. A fully worked out numerical test case clearly demonstrates that the whole procedure far surpasses the current state of the art.

On classical orthogonal polynomials and the Cholesky factorization of a class of Hankel matrices

Classical moment functionals (Hermite, Laguerre, Jacobi, Bessel) can be characterized as those linear functionals whose moments satisfy a second-order linear recurrence relation. In this work, we use this characterization to link the theory of classical orthogonal polynomials and the study of Hankel matrices whose entries satisfy a second-order linear recurrence relation. Using the recurrent character of the entries of such Hankel matrices, we give several characterizations of the triangular and diagonal matrices involved in their Cholesky factorization and connect them with a corresponding characterization of classical orthogonal polynomials.

Direct Synthesis of Acoustic Wave Multiplexers Built on Fully Canonical Multiport Functions

This work proposes the first analytical method for the direct synthesis of multiplexers based on acoustic wave (AW) ladder filters connected in a star junction topology. It describes the computation of the polynomial representation of a multiport ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${K} +$ </tex-math></inline-formula> 1 ports) network with fully canonical (FC) channels and derives a set of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${K}$ </tex-math></inline-formula> distorted channel polynomials from which each of the AW ladder filters is synthesized. The channel polynomials are distorted in a way such that when connected together at the star junction, the desired multiplexer response is obtained avoiding degradation due to channel loading. The method allows to freely set the fractional bandwidth (FBW), return loss (RL), and transmission zeros (TZs) of each channel enabling the search for multiplexers that fulfill the technological requirements of AW technology in terms of electromechanical coupling coefficient and area occupancy among others. Moreover, since the multiport network polynomials incorporate the complete effects of the loading between channels as, for example, additional TZs, the synthesized multiplexers depict an improvement of isolation when compared with a design based on only controlling the reflection phase of each channel filter. The proposed method is rigorously demonstrated by means of a synthesized multiplexer example providing the Butterworth–Van Dyke (BVD) elements of each channel filter. This approach avoids intensive optimization procedures and represents a useful technique for the design of carrier aggregation (CA) filter modules in a direct and controlled manner.

A Survey of Wave Energy Converter Mechanisms Presented Under the Topological and Functional Viewpoints

Abstract Wave energy converter (WEC) mechanisms have been increasingly attracting the interest of institutions and companies, because of the energetic crisis and the energy demanded by small off-grid off-shore systems. Examples of WEC systems design date back to 1892 and, since then, different mechanisms have been proposed, based on different working principles. In this work, an attempt has been made to build an atlas of WEC devices, collecting them from the scientific and technical literature, as broadly as possible to the best of the authors’ abilities. The working principle, the wave propagation direction, and the coast proximity have been used to classify the mechanisms of the atlas that have been illustrated by means of standardized esthetics. The topological nature of each device has been also extracted by applying both the polynomial representation of its kinematic chain (KC) together with a planar representation of the corresponding graph. These representations gave rise to a further and more detailed classification of the collected WEC devices that have been gathered together in peculiar topological families. Finally, system power has been also included to complete the information related to the devices illustrated in the atlas.

Unisolvence of Symmetric Node Patterns for Polynomial Spaces on the Simplex

Finite elements with polynomial basis functions on the simplex with a symmetric distribution of nodes should have a unique polynomial representation. Unisolvence not only requires that the number of nodes equals the number of independent polynomials spanning a polynomial space of a given degree, but also that the Vandermonde matrix controlling their mapping to the Lagrange interpolating polynomials can be inverted. Here, a necessary condition for unisolvence is presented for polynomial spaces that have non-decreasing degrees when going from the edges and the various faces to the interior of the simplex. It leads to a proof of a conjecture on a necessary condition for unisolvence, requiring the node pattern to be the same as that of the regular simplex.

Affine Super Schur Duality

Schur duality is an equivalence, for d\leq n , between the category of finite-dimensional representations over \mathbb{C} of the symmetric group S_d on d letters, and the category of finite-dimensional representations over \mathbb{C} of \operatorname{GL}(n,\mathbb{C}) whose irreducible subquotients are subquotients of \smash{\overline{\mathbb{E}}^{\otimes d}} , \overline{\mathbb{E}}=\mathbb{C}^n . The latter are called polynomial representations homogeneous of degree d . It is based on decomposing \smash{\overline{\mathbb{E}}^{\otimes d}} as a \mathbb{C}[S_d]\times\operatorname{GL}(n,\mathbb{C}) -bimodule. It was used by Schur to conclude the semisimplicity of the category of finite-dimensional complex \operatorname{GL}(n,\mathbb{C}) -modules from the corresponding result for S_d that had been obtained by Young. Here we extend this duality to the affine super case by constructing a functor \mathcal{F}\colon M\mapsto M\otimes_{\mathbb{C}[S_d]}\mathbb{E}^{\otimes d} , \mathbb{E} now being the super vector space \mathbb{C}^{m|n} , from the category of finite-dimensional \mathbb{C}[S_d\ltimes\mathbb{Z}^d] -modules, or representations of the affine Weyl, or symmetric, group S_d^a=S_d\ltimes\mathbb{Z}^d , \emph{to} the category of finite-dimensional representations of the universal enveloping algebra of the affine Lie superalgebra \mathfrak{U}(\widehat{\operatorname{sl}}(m|n)) that are \mathbb{E}^{\otimes d} -compatible, namely the subquotients of whose restriction to \mathfrak{U}(\operatorname{sl}](m|n)) are constituents of \mathbb{E}^{\otimes d} . Both categories are not semisimple. When d&lt;m+n the functor defines an equivalence of categories. As an application we conclude that the irreducible finite-dimensional \mathbb{E}^{\otimes d} -compatible representations of the affine superalgebra \widehat{\operatorname{sl}}(m|n) are tensor products of evaluation representations at distinct points of \mathbb{C}^\times .

Power-product matrix: nonsingularity, sparsity and determinant

In this paper, we are interested in a special class of integer matrices, namely the power-product matrix, defined with two positive integers n and d. Each matrix element is computed by a power-product of two weak compositions of d into n parts. The power-product matrix has several interesting applications such as the power-sum representation of polynomials and the difference-of-convex-sums-of-squares decomposition of polynomials. We investigate some properties of this matrix including: nonsingularity, sparsity and determinant. Based on techniques in enumerative combinatorics, we prove that the power-product matrix is nonsingular and the number of nonzero entries can be computed exactly. This matrix shows sparse structure which is a good feature in numerical computation of its inverse required in some applications. Special attention is devoted to the computation of the determinant for n = 2 whose explicit formulation is obtained.

Controllability Properties of Solar Sails

Trajectory design and stationkeeping of solar sails about a celestial body can be formulated as control problems with positivity constraints. Specifically, when reemitted radiation is neglected and the sail is modeled as a flat surface, which are reasonable assumptions for control purposes, the force generated by the solar radiation pressure is contained in a pointed convex cone of revolution with the axis in the sun–satellite direction. Therefore, classical approaches to infer controllability based on the Lie algebra rank condition do not apply to these problems. This study offers a novel condition to decide on controllability of control systems with positivity constraints. This condition is effective because it can be verified by solving an auxiliary convex optimization problem for which reliable numerical methods are available. A crucial ingredient of this approach is the theory of positive trigonometric polynomials. The practical interest of this condition is the assessment of a minimum requirement on the optical properties of the sail, which may be of use for mission design purposes.

FROM GRAPHS TO POLYNOMIAL RINGS: STAR POLYNOMIAL REPRESENTATION OF GRAPHS

Let G be a simple connected graph. An i-subset of V(G) is a subset of V(G) of cardinality i. An induced i-star of G is a star in G induced by an i-subset of V(G). The star polynomial representation of G is the generating function of the sequence of the number of induced i-stars in G. In this paper, we establish the star polynomials of some special graphs such as the star graph, spider graph, complete bipartite graph, and the complete q-partite graph.

An SDP method for fractional semi-infinite programming problems with SOS-convex polynomials

In this paper, we study a class of fractional semi-infinite polynomial programming problems involving sos-convex polynomial functions. For such a problem, by a conic reformulation proposed in our previous work and the quadratic modules associated with the index set, a hierarchy of semidefinite programming (SDP) relaxations can be constructed and convergent upper bounds of the optimum can be obtained. In this paper, by introducing Lasserre’s measure-based representation of nonnegative polynomials on the index set to the conic reformulation, we present a new SDP relaxation method for the considered problem. This method enables us to compute convergent lower bounds of the optimum and extract approximate minimizers. Moreover, for a set defined by infinitely many sos-convex polynomial inequalities, we obtain a procedure to construct a convergent sequence of outer approximations which have semidefinite representations (SDr). The convergence rate of the lower bounds and outer SDr approximations are also discussed.

New presentations of real numbers with k-Lucas numbers

In this paper, we first obtain a series of k-Lucas numbers using k-Lucas numbers. We give new presentations of any real number u ≠ 0 using this obtained k-Lucas series and show polynominal repsesentations that every nonzero real number can be uniquely represented as the sum of the squares of consecutive k-Lucas numbers. To do this, we give new presentation theorems for any real number using k-Lucas series. Finally, to support these theorems, we give examples where we obtain the roots of the polynomial representations of a selected real number u ≠ 0, as well as the values representing the first ten prime numbers corresponding to a chosen k-Lucas polynomial.

Fast 3D “wave-consistent” ray tomography based on a polynomial representation of a model

In the paper, we describe an original 3D travel-time tomography approach. It is based on the new realization of the bending method which to some extent takes into account the band-limited nature of real seismic signals propagation. As a result, two-point ray tracing provides more reliable ray trajectories and travel times in complex media. Another original feature of the proposed tomography is that the model is represented using the Chebyshev polynomials. Such parameterization allows analytical calculation of travel times and their derivatives with respect to model parameters and significantly reduces the number of parameters to be recovered during inversion compared to more common grid tomography. In certain situations, the proposed approach provides significant computational advantages. Numerical examples prove its efficiency.

Systematically derived thermodynamic properties for alkane oxidation

Key combustion properties, such as ignition delay time, show strong sensitivity to the thermochemistry of the main species in the standard radical oxidation pathway at low temperatures (600–1000 K). Significant uncertainties persist in current estimates of thermodynamic properties, particularly for the larger species of relevance to common practical fuels. In this article, we use advanced computational schemes to evaluate thermodynamic properties for the fuel, fuel radical, peroxy, hydroperoxy-alkyl radical, and hydroperoxide species for a set of 17 fuels containing up to 9 carbon atoms, and with various degrees of branching in the alkane backbone. The procedure, termed STAR-1D, combines conformer sampling to find the minimum geometry, B2PLYP-D3/cc-pVTZ harmonic frequency evaluations, and ωB97X-D/cc-pVTZ one-dimensional torsional mapping. It includes two physically-based scaling routines: a frequency-dependent scaling to B2PLYP-D3/cc-pVTZ anharmonic frequencies and a scaling of one-dimensional ωB97X-D/cc-pVTZ torsional profiles to reproduce the product of the B2PLYP-D3/cc-pVTZ frequencies in the harmonic limit. Substantive comparisons with existing experimental databases, together with careful examinations of key theoretical assumptions, are used to explore the accuracy of the predictions. These computationally intensive explorations of 195 species were facilitated by automated thermochemistry software. High accuracy 0 K heats of formation from a separate study are used with the STAR-1D computations to generate NASA polynomial representations. In addition to their intrinsic value, the present results also provide a reliable database for the optimization of group additivity or machine learning schemes for scaling to larger combustion systems. Towards this end a complementary extensive conformational analysis is carried out for the medium sized species and the thermodynamic properties of the lowest energy hydrogen-bonded and non-hydrogen bonded conformers are contrasted for larger hydroperoxy-alkyl radical species.

Atomization Theorems in Mathematical Physics and General Relativity

Formulated Atomization Theorems extend the theory of Atomic AString Functions evolving since the 1970s allowing representation of polynomials, complex analytic functions, and solutions of linear and nonlinear differential equations via Atomic Series over smooth finite Atomic Splines. Noting the preservation of analyticity for Ricci and Einstein tensors, special new theorems are formulated for General Relativity representing spacetime field via superpositions of flexible finite “solitonic atoms” resembling quanta. The novel Atomic Spacetime model correlates with A. Einstein’s 1933 paper predicting a new “atomic theory”. The theorems can be applied to many theories of mathematical physics, elasticity, hydrodynamics, soliton, and field theories for unified representation of fields via series over finite Atomic AString Functions which may offer a unified theory under research where fields are connected with a common mathematical ancestor.

Development of a reliability model for crack growth occurrence for a secondary hull component

Ship hull vibration is a major contributor to fatigue crack growth and main engine excitation is identified as an important vibration source. A general method to solve any vibration problem arising onboard a ship does not exist, which encourages the use of a reliability-based framework for assessing ship vibration and its consequences. A stochastic model of vibration response is developed for the probabilistic formulation of the failure probability of the occurrence of crack propagation of a secondary structural hull component. The secondary structural component considered is a pipe stack support. The pipe stack support connects a cargo pump pipe stack to the wall inside the cargo tank, and the support is welded directly onto this wall. First, a generic cargo hold model is analysed with engine speed and the relative distance between the engine and the structural component under consideration as stochastic variables. Then, submodels are used to investigate the local vibration of the support and the stress response is evaluated for a combination of different engine speeds and relative distances. A surface is fitted to the vibration response and used for probabilistic analysis by Monte Carlo (MC/DSPS) and FORM/SORM reliability methods. The limit state is formulated as the possibility of fatigue crack growth based on a threshold stress intensity factor. This threshold factor depends on the initial crack size and different initial sizes are investigated. The adequacy of the functional representation for the stochastic model, which is fitted to discrete data points, is also assessed. It is seen that a functional representation using a sum of sine terms give an adequate fit for describing the stress response induced by engine speed, while a polynomial representation was adequate for the relative distance variable. The failure probability estimated by Monte Carlo simulations and SORM indicates that the pipe stack support is not critical for the occurrence of fatigue crack growth. A main observation from the analysis is that the reliability-based design of secondary structural components, also looking at the interaction with the global structure, may help to improve the vibration-induced stresses in local hull details by application of proper design measures.