Linear Combinations Of Chebyshev Polynomials Research Articles

Semi-Automated Creation of Density Functional Tight Binding Models through Leveraging Chebyshev Polynomial-Based Force Fields.

Density functional tight binding (DFTB) is an attractive method for accelerated quantum simulations of condensed matter due to its enhanced computational efficiency over standard density functional theory (DFT) approaches. However, DFTB models can be challenging to determine for individual systems of interest, especially for metallic and interfacial systems where different bonding arrangements can lead to significant changes in electronic states. In this regard, we have created a rapid-screening approach for determining systematically improvable DFTB interaction potentials that can yield transferable models for a variety of conditions. Our method leverages a recent reactive molecular dynamics force field where many-body interactions are represented by linear combinations of Chebyshev polynomials. This allows for the efficient creation of multi-center representations with relative ease, requiring only a small investment in initial DFT calculations. We have focused our workflow on TiH2 as a model system and show that a relatively small training set based on unit-cell-sized calculations yields a model accurate for both bulk and surface properties. Our approach is easy to implement and can yield reliable DFTB models over a broad range of thermodynamic conditions, where physical and chemical properties can be difficult to interrogate directly and there is historically a significant reliance on theoretical approaches for interpretation and validation of experimental results.

LS-based and GL-based thermoelasticity in two dimensional bounded media: A Chebyshev collocation analysis

In this paper, the Chebyshev collocation numerical method is developed for solving generalized thermoelasticity problems of the isotropic homogeneous two dimensional media. The coupled thermoelastic equations are derived based on Lord-Shulman (LS) and Green-Lindsay (GL) theories. The temperature and displacement fields are approximated in space domain by linear combinations of Chebyshev polynomials. Also, the direct collocation method is applied to governing differential equations to generate the system of differential equations with respect to time. The resulted set of differential equations are solved in time domain by Wilson method. Both temperature and traction loadings are considered to be applied at the left side of the media. The obtained results from the present paper for the classical coupled thermoelasticity of two dimensional finite domains are compared with the same results extracted analytically in the literature and a very close agreement is observed.

Sums of finite products of Chebyshev polynomials of two different types

<abstract><p>In this paper, we consider sums of finite products of the second and third type Chebyshev polynomials, those of the second and fourth type Chebyshev polynomials and those of the third and fourth type Chebyshev polynomials, and represent each of them as linear combinations of Chebyshev polynomials of all types. Here the coefficients involve some terminating hypergeometric functions $ {}_{2}F_{1} $. This problem can be viewed as a generalization of the classical linearization problems and is done by explicit computations.</p></abstract>

Active learning for robust, high-complexity reactive atomistic simulations

Machine learned reactive force fields based on polynomial expansions have been shown to be highly effective for describing simulations involving reactive materials. Nevertheless, the highly flexible nature of these models can give rise to a large number of candidate parameters for complicated systems. In these cases, reliable parameterization requires a well-formed training set, which can be difficult to achieve through standard iterative fitting methods. Here, we present an active learning approach based on cluster analysis and inspired by Shannon information theory to enable semi-automated generation of informative training sets and robust machine learned force fields. The use of this tool is demonstrated for development of a model based on linear combinations of Chebyshev polynomials explicitly describing up to four-body interactions, for a chemically and structurally diverse system of C/O under extreme conditions. We show that this flexible training database management approach enables development of models exhibiting excellent agreement with Kohn-Sham density functional theory in terms of structure, dynamics, and speciation.

Spectral method solutions of a variable aspect ratio duct flow in a uniform transverse magnetic field with two different thermal boundary conditions

Abstract A laminar electrically-conducting flow inside an electrically-insulated rectangular duct in a uniform transverse magnetic field with both uniform surface temperature and uniform heat flux boundary conditions are considered numerically. The problem with aspect ratios of the range from 1:10 to 10:1 and Hartmann number M up to 1000 is solved using a highly accurate technique which is spectral method. The flow variables are expanded in terms of linear combinations of Chebyshev polynomials chosen to satisfy the boundary conditions implicitly. The resulting equations are collocated using the Gauss points to produce a system of nonlinear algebraic equations which is solved iteratively using Gauss elimination. Convergence properties of the numerical method reveals that for aspect ratio of less than 4 to 1, the flow is well resolved with as small as 29 by 29 Chebyshev polynomials; however, as the aspect ratio increases to more than 4 to 1, the number of polynomials required for an adequate resolution can be as high as 49 by 49 polynomials. It is found when the magnetic field is turned on, the pressure drop, in general, increases with the field for different aspect ratios. However, the pressure drop increase will be slower near the aspect ratio of 10. Also, the heat transfer increases with the field for most of the cases, but for some cases the field will have adverse effect on the heat transfer, particularly, for constant surface temperature boundary condition at aspect ratio > 4 and M 4 and M

Amplitude and Frequency Variation in Nonlinear Glucose Dynamics with Multiple Delays via Periodic Perturbation

Characterising the glycemic response to a glucose stimulus is an essential tool for detecting deficiencies in humans such as diabetes. In the presence of a constant glucose infusion in healthy individuals, it is known that this control leads to slow oscillations as a result of feedback mechanisms at the organ and tissue level. In this paper, we provide a novel quantitative description of the dependence of this oscillatory response on the physiological functions. This is achieved through the study of a model of the ultradian oscillations in glucose-insulin regulation which takes the form of a nonlinear system of equations with two discrete delays. While studying the behaviour of solutions in such systems can be mathematically challenging due to their nonlinear structure and non-local nature, a particular attention is given to the periodic solutions of the model. These arise from a Hopf bifurcation which is induced by an external glucose stimulus and the joint contributions of delays in pancreatic insulin release and hepatic glycogenesis. The effect of each physiological subsystem on the amplitude and period of the oscillations is exhibited by performing a perturbative analysis of its periodic solutions. It is shown that assuming the commensurateness of delays enables the Hopf bifurcation curve to be characterised by studying roots of linear combinations of Chebyshev polynomials. The resulting expressions provide an invaluable tool for studying the interplay between physiological functions and delays in producing an oscillatory regime, as well as relevant information for glycemic control strategies.

Searching for Hyperbolic Polynomials with Span Less than 4

A monic, irreducible polynomial in one variable having integer coefficients and all real roots deserves particular interest if its roots lie in an interval of length 4 whose end-points are not integers. This follows by some pioneering studies by R. Robinson. Thanks to the crucial support of computers, a number of contributions over the decades settled the existence question for such polynomials up to degree 18. In this article, we find out that almost all of these polynomials can be recovered with algebraic operations from a few polynomials of small degree. Furthermore, a great number of the polynomials discovered by Robinson can be actually obtained as simple linear combinations of Chebyshev polynomials. As a byproduct, we found several families of hyperbolic polynomials related to Salem’s numbers.

Application of the ChIMES Force Field to Nonreactive Molecular Systems: Water at Ambient Conditions.

We demonstrate development of the Chebyshev Interaction Model for Efficient Simulation (ChIMES) for molecular systems through application to water under ambient conditions (298 K, 1 g/cm3). These models, which are comprised of linear combinations of Chebyshev polynomials explicitly describing two- and three-body interactions, are largely fit by force matching to Kohn-Sham Density Functional Theory (DFT). Protocols for selecting user-specified parameters and inclusion of stress tensor data are investigated, and structural and dynamical property prediction for resulting models is benchmarked against DFT. We show that the present ChIMES force fields yield excellent agreement with DFT without the need for additional terms such as those for Coulomb interactions. Overall, we show that tractable parametrization and subsequent accuracy of the present models make ChIMES an ideal candidate for extension of DFT dynamics to larger system sizes and longer time scales.

Development of a Multicenter Density Functional Tight Binding Model for Plutonium Surface Hydriding.

We detail the creation of a multicenter density functional tight binding (DFTB) model for hydrogen on δ-plutonium, using a framework of new Slater-Koster interaction parameters and a repulsive energy based on the Chebyshev Interaction Model for Efficient Simulation (ChIMES), where two- and three-center atomic interactions are represented by linear combinations of Chebyshev polynomials. We find that our DFTB/ChIMES model yields a total electron density of states for bulk δ-Pu that compares well to that from Density Functional Theory, as well as to a grid of energy calculations representing approximate H2 dissociation paths on the δ-Pu (100) surface. We then perform molecular dynamics simulations and minimum energy pathway calculations to determine the energetics of surface dissociation and subsurface diffusion on the (100) and (111) surfaces. Our approach allows for the efficient creation of multicenter repulsive energies with a relatively small investment in initial DFT calculations. Our efforts are particularly pertinent to studies that rely on quantum calculations for interpretation and validation, such as experimental determination of chemical reactivity both on surfaces and in condensed phases.

ChIMES: A Force Matched Potential with Explicit Three-Body Interactions for Molten Carbon.

We present a new force field and development scheme for atomistic simulations of materials under extreme conditions. These models, which explicitly include two- and three-body interactions, are generated by fitting linear combinations of Chebyshev polynomials through force matching to trajectories from Kohn-Sham density functional theory (DFT). We apply our method to liquid carbon near the diamond/graphite/liquid triple point and at higher densities and temperatures, where metallization and many-body effects may be substantial. We show that explicit inclusion of three-body interaction terms allows our model to yield improved descriptions of both dynamic and structural properties over previous empirical potential efforts, while exhibiting transferability to nearby state points. The simplicity of our functional form and subsequent efficiency of parameter determination allow for extension of DFT to experimental time and length scales while retaining most of its accuracy.

Solution of inverse heat conduction equation with the use of Chebyshev polynomials

AbstractA direct problem and an inverse problem for the Laplace’s equation was solved in this paper. Solution to the direct problem in a rectangle was sought in a form of finite linear combinations of Chebyshev polynomials. Calculations were made for a grid consisting of Chebyshev nodes, what allows us to use orthogonal properties of Chebyshev polynomials. Temperature distributions on the boundary for the inverse problem were determined using minimization of the functional being the measure of the difference between the measured and calculated values of temperature (boundary inverse problem). For the quasi-Cauchy problem, the distance between set values of temperature and heat flux on the boundary was minimized using the least square method. Influence of the value of random disturbance to the temperature measurement, of measurement points (distance from the boundary, where the temperature is not known) arrangement as well as of the thermocouple installation error on the stability of the inverse problem was analyzed.

Generalized Markoff equations and Chebyshev polynomials

The Markoff equation is x2+y2+z2=3xyz, and all of the positive integer solutions of this equation occur on one tree generated from (1,1,1), called the Markoff tree. In this paper, we consider trees of solutions to x2+y2+z2=xyz+A. We say a tree satisfies the unicity condition if the maximum element of an ordered triple in the tree uniquely determines the other two. The unicity conjecture says that the Markoff tree satisfies the unicity condition. In this paper, we show that there exists a sequence of real numbers {cn} such that each tree generated from (1,cn,cn) satisfies the unicity condition, and that these trees converge to the Markoff tree. We accomplish this by recasting polynomial solutions as linear combinations of Chebyshev polynomials, showing that these polynomials are distinct, and evaluating them at certain values.

On linear combinations of Chebyshev polynomials

We investigate an infinite sequence of polynomials of the form: a0Tn(x) + a1Tn?1(x) + ... + amTn?m(x) where (a0, a1,..., am) is a fixed m-tuple of real numbers, a0, am ? 0, Ti(x) are Chebyshev polynomials of the first kind, n = m, m+ 1, m+ 2,.... Here we analyze the structure of the set of zeros of such polynomial, depending on A and its limit points when n tends to infinity. Also the expression of envelope of the polynomial is given. An application in number theory, more precise, in the theory of Pisot and Salem numbers, is presented.

Resultants of Chebyshev Polynomials

Recently, K.\~Dillcher and K.\~B.\~Stolarsky \[{\it Trans. Amer. Math. Soc.} \~357 (2004), 965 -- 981] used algebraic methods to evaluate the resultant of two linear combinations of Chebyshev polynomials of the second kind. In this paper we give an alternative method of computing the same resultant and resultants of more general combinations of Chebyshev polynomials of the second kind. We also consider resultants of linear combinations of Chebyshev polynomials of the first kind.

On the zeros of certain linear combinations of Chebyshev polynomials

We consider a system of masses and springs, such that all springs have equal stiffness K, except those springs that possibly connect the boundary masses to earth, and such that all masses except the first and the last are equal to M. The eigenfrequencies of this four parameter system can be written in terms of the zeros of certain linear combinations of Chebyshev polynomials. The physical behaviour of the system suggests sufficient conditions for the parameters such that these zeros are all real. It will be shown by Sturm's theorem that under these conditions the eigenvalues of the matrix of the system are real and ⩾0, which implies the conjecture about the zeros. When some additional conditions for the parameters are satisfied, we even can conclude that all the zeros are located in the interval (−1, 1).

Unsteady Lifting-Surface Theory for Ship Screws: Derivation and Numerical Treatment of Integral Equation

In an early stage of studies concerned with lifting-surface theory for ship screws, the Netherlands Ship Model Basin initiated the development of a numerical solution of the mathematical model. To achieve this, a rigorous method was followed consisting of separation of regular and singular parts of the kernel of the integral equation, preparation of numerical procedures for integration of the regular parts, and analytical integration of the singular parts. The general solution includes the steady load due to blade geometry in uniform flow, the unsteady load in nonuniform flow, and the unsteady load due to propeller vibrations. The pressure distribution in the chordwise direction is represented by a series of linear combinations of Chebyshev polynomials. In this paper an account is given of this method of solution.