Polynomials In Variables Research Articles

Constraint polynomial approach: an alternative to the functional Bethe Ansatz method?

Recently developed general constraint polynomial approach is shown to replace a set of algebraic equations of the functional Bethe Ansatz method by a single polynomial constraint. As the proof of principle, the usefulness of the method is demonstrated for a number of quasi-exactly solvable (QES) potentials of the Schr\"odinger equation, such as two different sets of modified Manning potentials with three parameters, an electron in Coulomb and magnetic fields and relative motion of two electrons in an external oscillator potential, the hyperbolic Razavy potential, and a (perturbed) double sinh-Gordon system. The approach enables one to straightforwardly determine eigenvalues and wave functions. Odd parity solutions for the modified Manning potentials are also determined. For the QES examples considered here, constraint polynomials terminate a finite chain of orthogonal polynomials in an independent variable that need not to be necessarily energy. In the majority of cases the finite chain of orthogonal polynomials is characterized by a positive-definite moment functional ${\cal L}$, implying that a corresponding constraint polynomial has only real and simple zeros. Constraint polynomials are shown to be different from the weak orthogonal Bender-Dunne polynomials. At the same time the QES examples considered elucidate essential difference with various generalizations of the Rabi model. Whereas in the former case there are $n+1$ polynomial solutions at each point of a $n$th baseline, in the latter case there are at most $n+1$ polynomial solutions on entire $n$th baseline.

A natural frequency sensitivity-based stabilization in spectral stochasticfinite element method for frequency response analysis

In applying the spectral stochastic finite element methods to the frequency response analysis, the conventional methods are known to give unstable and inaccurate results near the natural frequencies. To address this issue, a new sensitivity based stabilized formulation for stochastic frequency response analysis is proposed in this paper. The main difference over the conventional spectral methods is that the polynomials of random variables are applied to both numerator and denominator in approximating the harmonic response solution. In order to reflect the resonance behavior of the structure, the denominator polynomials is constructed by utilizing the natural frequency sensitivity and the random mode superposition. The numerator is approximated by applying a polynomial chaos expansion, and its coefficients are obtained through the Galerkin or the spectral projection method. Through various numerical studies, it is seen that the proposed method improves accuracy, especially in the vicinities of structural natural frequencies compared to conventional spectral methods.

Elements of the q-Askey Scheme in the Algebra of Symmetric Functions

The classical q-hypergeometric orthogonal polynomials are assembled into a hierarchy called the q-Askey scheme. At the top of the hierarchy, there are two closely related families, the Askey-Wilson and q-Racah polynomials. As it is well known, their construction admits a generalization leading to remarkable orthogonal symmetric polynomials in several variables. We construct an analogue of the multivariable q-Racah polynomials in the algebra of symmetric functions. Next, we show that our q-Racah symmetric functions can be degenerated into the big q-Jacobi symmetric functions, introduced in a recent paper by the second author. The latter symmetric functions admit further degenerations leading to new symmetric functions, which are analogues of q-Meixner and Al-Salam--Carlitz polynomials. Each of the four families of symmetric functions (q-Racah, big q-Jacobi, q-Meixner, and Al-Salam--Carlitz) forms an orthogonal system of functions with respect to certain measure living on a space of infinite point configurations. The orthogonality measures of the four families are of independent interest. We show that they are linked by limit transitions which are consistent with the degenerations of the corresponding symmetric functions.

On the generalized q-poly-Euler polynomials of the second kind

In this work, we define the generalized q-poly-Euler numbers of the second kind of order ? and the generalized q-poly-Euler polynomials of the second kind of order ?. We investigate some basic properties for these polynomials and numbers. In addition, we obtain many identities, relations including the Roger-Sz?go polynomials, the Al-Salam Carlitz polynomials, q-analogue Stirling numbers of the second kind and two variable Bernoulli polynomials.

Cubification of [formula omitted]-SDE and exact moment equations

For an Itô-like Stochastic Differential Equation (SDE) system, with drift and diffusion that are formal polynomials of the independent variables, we show that all moments satisfy an infinite, countable, set of linear ordinary differential equations. This result is achieved by means of the exact cubification of the SDE, which consists in a set of deterministic transformations of the state variables, giving place to a new SDE with further finitely many state variables. Exact cubification can be considered as an extension to the ‘stochastic case’ of the exact quadratization of deterministic nonlinear systems, available in the literature. An example is finally shown, taken from systems biology, in which, for a basic chemical reaction network the exact moment equation is written down, and an approximate solution is calculated through a moment closure method.

Nektar++: Enhancing the capability and application of high-fidelity spectral/[formula omitted] element methods

Nektar++ is an open-source framework that provides a flexible, high-performance and scalable platform for the development of solvers for partial differential equations using the high-order spectral/hp element method. In particular, Nektar++ aims to overcome the complex implementation challenges that are often associated with high-order methods, thereby allowing them to be more readily used in a wide range of application areas. In this paper, we present the algorithmic, implementation and application developments associated with our Nektar++ version 5.0 release. We describe some of the key software and performance developments, including our strategies on parallel I/O, on in situ processing, the use of collective operations for exploiting current and emerging hardware, and interfaces to enable multi-solver coupling. Furthermore, we provide details on a newly developed Python interface that enables a more rapid introduction for new users unfamiliar with spectral/hp element methods, C++ and/or Nektar++. This release also incorporates a number of numerical method developments – in particular: the method of moving frames (MMF), which provides an additional approach for the simulation of equations on embedded curvilinear manifolds and domains; a means of handling spatially variable polynomial order; and a novel technique for quasi-3D simulations (which combine a 2D spectral element and 1D Fourier spectral method) to permit spatially-varying perturbations to the geometry in the homogeneous direction. Finally, we demonstrate the new application-level features provided in this release, namely: a facility for generating high-order curvilinear meshes called NekMesh; a novel new AcousticSolver for aeroacoustic problems; our development of a ‘thick’ strip model for the modelling of fluid–structure interaction (FSI) problems in the context of vortex-induced vibrations (VIV). We conclude by commenting on some lessons learned and by discussing some directions for future code development and expansion. Program summaryProgram Title: Nektar++Program Files doi:http://dx.doi.org/10.17632/9drxd9d8nx.1Code Ocean Capsule:https://doi.org/10.24433/CO.9865757.v1Licensing provisions: MITProgramming language: C++External routines/libraries: Boost, METIS, FFTW, MPI, Scotch, PETSc, TinyXML, HDF5, OpenCASCADE, CWIPINature of problem: The Nektar++ framework is designed to enable the discretisation and solution of time-independent or time-dependent partial differential equations.Solution method: spectral/hp element method

Decomposing algebraic m-isometric tuples

We show that any m-isometric tuple of commuting algebraic operators on a Hilbert space can be decomposed as a sum of a spherical isometry and a commuting nilpotent tuple. Our approach applies as well to tuples of algebraic operators that are hereditary roots of polynomials in several variables.

Stability of the Cohomology of the Space of Complex Irreducible Polynomials in Several Variables

AbstractWe prove that the space of complex irreducible polynomials of degree $d$ in $n$ variables satisfies two forms of homological stability: first, its cohomology stabilizes as $d\to \infty $, and second, its compactly supported cohomology stabilizes as $n\to \infty $. Our topological results are inspired by counting results over finite fields due to Carlitz and Hyde.

Matrix elements of irreducible representations of SU(n + 1)×SU(n + 1) and multivariable matrix-valued orthogonal polynomials

In Part 1 we study the spherical functions on compact symmetric pairs of arbitrary rank under a suitable multiplicity freeness assumption and additional conditions on the branching rules. The spherical functions are taking values in the spaces of linear operators of a finite dimensional representation of the subgroup, so the spherical functions are matrix-valued. Under these assumptions these functions can be described in terms of matrix-valued orthogonal polynomials in several variables, where the number of variables is the rank of the compact symmetric pair. Moreover, these polynomials are uniquely determined as simultaneous eigenfunctions of a commutative algebra of differential operators.In Part 2 we verify that the group case SU(n+1) meets all the conditions that we impose in Part 1. For any k∈N0 we obtain families of orthogonal polynomials in n variables with values in the N×N-matrices, where N=(n+kk). The case k=0 leads to the classical Heckman-Opdam polynomials of type An with geometric parameter. For k=1 we obtain the most complete results. In this case we give an explicit expression of the matrix weight, which we show to be irreducible whenever n≥2. We also give explicit expressions of the spherical functions that determine the matrix weight for k=1. These expressions are used to calculate the spherical functions that determine the matrix weight for general k up to invertible upper-triangular matrices. This generalizes and gives a new proof of a formula originally obtained by Koornwinder for the case n=1. The commuting family of differential operators that have the matrix-valued polynomials as simultaneous eigenfunctions contains an element of order one. We give explicit formulas for differential operators of order one and two for (n,k) equal to (2,1) and (3,1).

The Eneström–Kakeya Theorem for polynomials of a quaternionic variable

The well-known Eneström–Kakeya Theorem states that a polynomial with real, nonnegative, monotone increasing coefficients has all its complex zeros in the closed unit disk in the complex plane. In this paper, we extend this result by showing that all quaternionic zeros of such a polynomial lie in the unit sphere in the quaternions. We also extend related results from the complex to quaternionic setting.

Measurement-Based Estimation of Inertia in AC Microgrids

This paper presents an improved method to estimate the available inertia in an islanded AC microgrid. Inertia estimation is carried out based on measured frequency response for any arbitrary disturbance that occurs in the system. Modifications are made to the conventional swing equation-based curve-fitting method to obtain an accurate estimate for a system with high penetration of renewable generations. A polynomial curve fit over the total power generation is introduced to estimate the size of the disturbance accurately. Additionally, a variable order polynomial fit is carried out over the measured frequency, which not only improves the estimate of inertia but also helps to refrain the influence of network topology and size/location of the disturbance. The test microgrid system considered is a modified Standard IEEE distribution network, which consists of radial feeders and distributed generations. Firstly, the proposed method is tested on a system with only synchronous generations to assess the accuracy of the estimate. This is followed by the integration of Type 3 and Type 4 wind turbines, and a PV array within the microgrid system. Virtual inertia control is then implemented in the wind turbines to obtain inertial support. Estimation study of the microgrid system with virtual inertia is then carried out. The developed estimation method can accurately estimate the inertia provided by the synchronous sources within the generation mix. Finally, from all the results and observations, the inertia estimation process in a microgrid system is segregated into synchronous and nonsynchronous inertia estimation.

A new design paradigm for provably secure keyless hash function with subsets and two variables polynomial function

Provably secure keyless hash function uses Random Oracle (RO) or Sponge principles for the design and construction of security-centric hash algorithms. It capitalizes the aforesaid principles to produce outcomes like MD2, MD5, SHA-160, SHA-224/256, SHA-256, SHA-224/512, SHA-256/512, SHA-384/512, SHA-512, and SHA-3. These functions use bitwise AND, OR, XOR, and MOD operators to foresee randomness in their hash outputs. However, the partial breaking of SHA2 and SHA3 families and the breaking of MD5 and SHA-160 algorithms raise concerns on the use of bitwise operators at the block level. The proposed design tries to address this structural flaw through a polynomial function. A polynomial function of degree 128 demands arduous effort to be decoded in the opposing direction. The application of a polynomial on the blocks produces an unpredictable random response. It is a fact that the new design exhibits the merits of the polynomial function on subsets to achieve the avalanche response to a significant level. The output from experiments with more than 24 Million hash searches proves the proposed system is a provably secure hash function. The experiments on avalanche response and confusion and diffusion analysis prove it is an apt choice for security-centric cryptographic applications.

On higher Mahler measures and zeta Mahler measures for one variable polynomials

First, we give a formula for the limiting value of the higher Mahler measures for general polynomials of one variable, which refines Biswas and Monico's result. Second, we give an analytic continuation of the zeta Mahler measures for polynomials of one variable to the whole complex plane as meromorphic functions, and we show that all of their poles are simple.

Asymptotic expansion of Warlimont functions on Wright semigroups

We calculate full asymptotic expansions of prime-independent multiplicative functions on additive arithmetic semigroups that satisfy a strong form of Knopfmacher's axioms. When applied to the semigroup of unlabeled graphs, our method yields detailed asymptotic information on how graphs decompose into connected components. As a second class of examples, we discuss polynomials in several variables over a finite field.

Variable mesh non polynomial spline method for singular perturbation problems exhibiting twin layers

In this paper, we descend a variable mesh finite difference scheme based on non polynomial spline approximation for the solution of singular perturbation problems with twin boundary layers. We develop the discretization equation for the problem using the condition of continuity for the first order derivatives of the variable mesh non polynomial spline at the interior nodes. The discrete invariant imbedding algorithm is utilized to solve the tridiagonal system obtained by the method. Endeavor examples are illustrated and maximum absolute errors in comparison to the other methods in the literature are shown to vindicate the method.

Nimble evolution for pretzel Khovanov polynomials

We conjecture explicit evolution formulas for Khovanov polynomials, which for any particular knot are Laurent polynomials of complex variables q and T, for pretzel knots of genus g in some regions in the space of winding parameters n_0, dots , n_g. Our description is exhaustive for genera 1 and 2. As previously observed Anokhina and Morozov (2018), Dunin-Barkowski et al. (2019), evolution at Tne -1 is not fully smooth: it switches abruptly at the boundaries between different regions. We reveal that this happens also at the boundary between thin and thick knots, moreover, the thick-knot domain is further stratified. For thin knots the two eigenvalues 1 and lambda = q^2 T, governing the evolution, are the standard T-deformation of the eigenvalues of the R-matrix 1 and -q^2. However, in thick knots’ regions extra eigenvalues emerge, and they are powers of the “naive” lambda , namely, they are equal to lambda ^2, dots , lambda ^g. From point of view of frequencies, i.e. logarithms of eigenvalues, this is frequency doubling (more precisely, frequency multiplication) – a phenomenon typical for non-linear dynamics. Hence, our observation can signal a hidden non-linearity of superpolynomial evolution. To give this newly observed evolution a short name, note that when lambda is pure phase the contributions of lambda ^2, dots , lambda ^g oscillate “faster” than the one of lambda . Hence, we call this type of evolution “nimble”.

LFSR based versatile divider architectures for BCH and RS error correction encoders

Error correction codes play a major role in real world data communication systems. This paper proposes linear feedback shift register (LFSR) based flexible and vector serial/parallel dividers for Bose–Chaudhuri–Hocquenghem (BCH) and Reed Solomon (RS) error correction encoders (ECEs). These can be used in variable length error correction codecs. Also, this paper elaborates the versatile hardware implementations of Turbo product encoders using the proposed flexible/vector serial dividers based BCH/RS encoders. The proposed flexible architectures are used to perform the division operation with variable length generator polynomials in BCH and RS based ECEs. In these flexible designs, the same hardware circuit is used to achieve the versatility without much compromise in throughput by allowing the trade-offs in area and power-delay-product (PDP). The proposed vector architectures are used to perform multiple division operations in parallel to improve the throughput with the trade-offs in area and PDP. The proposed and existing designs are implemented using 45 nm CMOS technology. The synthesis results show that the proposed vector designs achieve significant improvement in throughput over the existing designs. For example, the LFSR based proposed vector serial dividers achieve 47.1% and 83.3% of improvement in throughput as compared with the conventional designs for BCH encoder with generator polynomial length 65 and RS encoder with generator polynomial length 17 respectively.

On the Relationship Between Matiyasevich's and Smorynski's Theorems

Yuri Matiyasevich's theorem states that the set of all Diophantine equations which have a solution in non-negative integers is not recursive. Craig Smory\'nski's theorem states that the set of all Diophantine equations which have at most finitely many solutions in non-negative integers is not recursively enumerable. Let R be a subring of Q with or without 1. By H_{10}(R), we denote the problem of whether there exists an algorithm which for any given Diophantine equation with integer coefficients, can decide whether or not the equation has a solution in R. We prove that a positive solution to H_{10}(R) implies that the set of all Diophantine equations with a finite number of solutions in R is recursively enumerable. We show the converse implication for every infinite set R \subseteq Q such that there exist computable functions \tau_1,\tau_2:N \to Z which satisfy (\forall n \in N \tau_2(n) \neq 0) \wedge ({\frac{\tau_1(n)}{\tau_2(n)}: n \in N}=R). This implication for R=N guarantees that Smory\'nski's theorem follows from Matiyasevich's theorem. Harvey Friedman conjectures that the set of all polynomials of several variables with integer coefficients that have a rational solution is not recursive. Harvey Friedman conjectures that the set of all polynomials of several variables with integer coefficients that have only finitely many rational solutions is not recursively enumerable. These conjectures are equivalent by our results for R=Q.

On Differential Operators for Chebyshev Polynomials in Several Variables

We obtain differential operators for the bivariate Chebyshev polynomials of the first kind associated with the root systems of the simple Lie algebras C2 and G2.

Real-time displacement calculation and offline geometric calibration of the grating interferometer system for ultra-precision wafer stage measurement

For ultra-precision wafer stage measurement in a photolithography scanner, real-time six-degree-of-freedom displacement calculation and offline geometric calibration methods are proposed to solve the multiple nonlinearities caused by rotation-translation coupling, Abbe error, and cosine error of the grating interferometer system with non-ideal geometry. First, the interference signal phase-shift model related to the displacements of the center of mass of the wafer stage is established. Then, through double approximations, the displacement calculation process is efficiently reduced to polynomials of substitution variables of phase-shift to improve the real-time performance of computing. Finally, the non-ideal geometry of the nonlinear computing model caused by manufacturing and assembly errors is calibrated offline by solving overdetermined equations set up by redundant measurement to improve the calculation precision. The proposed methods are verified through ZEMAX simulation.