General and almost general periodic functions
Difference equations and Fourier constants for solving general periodic functions problems
PERIODIC FUNCTION↗
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Exponentially accurate approximations to piece-wise smooth periodic functions
A family of simple, periodic basis functions with 'built-in' discontinuities are introduced, and their properties are analyzed and discussed. Some of their potential usefulness is illustrated in conjunction with the Fourier series representations of functions with discontinuities. In particular, it is demonstrated how they can be used to construct a sequence of approximations which converges exponentially in the maximum norm to a piece-wise smooth function. The theory is illustrated with several examples and the results are discussed in the context of other sequences of functions which can be used to approximate discontinuous functions.
Greer, James↗
Numerical quadrature methods for integrals of singular periodic functions and their application to singular and weakly singular integral equations
High accuracy numerical quadrature methods for integrals of singular periodic functions are proposed. These methods are based on the appropriate Euler-Maclaurin expansions of trapezoidal rule approximations and their extrapolations. They are used to obtain accurate quadrature methods for the solution of singular and weakly singular Fredholm integral equations. Such periodic equations are used in the solution of planar elliptic boundary value problems, elasticity, potential theory, conformal mapping, boundary element methods, free surface flows, etc. The use of the quadrature methods is demonstrated with numerical examples.
Sidi, A.↗
On the Gibbs phenomenon 1: Recovering exponential accuracy from the Fourier partial sum of a non-periodic analytic function
It is well known that the Fourier series of an analytic or periodic function, truncated after 2N+1 terms, converges exponentially with N, even in the maximum norm, although the function is still analytic. This is known as the Gibbs phenomenon. Here, we show that the first 2N+1 Fourier coefficients contain enough information about the function, so that an exponentially convergent approximation (in the maximum norm) can be constructed.
Gottlieb, David↗
A critical period for functional vestibular development in zebrafish
We have determined a critical period for vestibular development in zebrafish by using a bioreactor designed by NASA to simulate microgravity for cells in culture. A critical period is defined as the briefest period of time during development when stimulus deprivation results in long lasting or permanent sensory deficits. Zebrafish eggs were collected within 3 hours of being laid and fertilized. In experiment 1, eggs were placed in the bioreactor at 3, 24, 30, 36, 48, or 72 hours postfertilization (hPF) and maintained in the bioreactor until 96 hPF. In experiment 2, eggs were placed in the bioreactor immediately after they were collected and maintained in the bioreactor until 24, 36, 48, 60, 66, 72, or 96 hPF. Beginning at 96 hPF, all larvae had their vestibulo-ocular reflexes (VOR) evaluated once each day for 5 days. Only larvae that hatched from eggs that were placed in the bioreactor before 30 hPF in experiment 1 or removed from the bioreactor later than 66 hPF in experiment 2 had VOR deficits that persisted for at least 5 days. These data suggest a critical period for vestibular development in the zebrafish that begins before 30 hPF and ends after 66 hPF. To confirm this, zebrafish eggs were placed in the bioreactor at 24 hPF and removed at 72 hPF. VORs were evaluated in these larvae once each day for 5 days beginning at 96 hPF. These larvae had VOR deficits that persisted for at least 5 days. In addition, larvae that had been maintained in the bioreactor from 24 to 66 hPF or from 30 to 72 hPF, had only temporary VOR deficits. In a final experiment, zebrafish eggs were placed in the bioreactor at 3 hPF and removed at 96 hPF but the bioreactor was turned off from 24 hPF to 72 hPF. These larvae had normal VORs when they were removed from the bioreactor at 96 hPF. Taken as a whole, these data support the idea that there is a critical period for functional maturation of the zebrafish vestibular system. The developmental period identified includes the timeframe during which the vestibular primary afferent neurons are born, innervate their central and peripheral targets, and remodel their central projections. Copyright 2002 Wiley-Liss, Inc.
Non-NASA Center↗
Solutions near simple periodic orbits of functional differential equations
Periodic solution with nondegenerate orbit for functional differential equations taking into account small perturbation
Hale, J. K.↗
Constrained nuclear–electronic orbital method for periodic density functional theory: Application to H 2 chemisorption on Si(001) surfaces
The nuclear–electronic orbital (NEO) method provides a powerful computational framework for incorporating nuclear quantum effects (NQE) in electronic structure calculations beyond the Born–Oppenheimer approximation. By incorporating additional constraints to the position operator on quantum particles like protons, the NEO method enables calculation of effective potential that accounts for NQE. Here, in this work, we present a new constrained NEO (cNEO) formulation for density functional theory (cNEO-DFT) calculations in the context of extended periodic systems. Using the nudged elastic band method, we discuss an application of the cNEO-DFT approach to studying the adsorption of a hydrogen molecule on the Si(001) surfaces. The calculation shows how NQE impacts the reaction energetics. The proton density changes are computed along the reaction pathways. This work demonstrates the capability of the new cNEO-DFT method to study a wide range of chemical processes, such as surface reactions where the quantum nature of light atoms like protons is non-negligible.
Chemical processes↗
Growth functions of periodic space tessellations
This work analyzes the rules governing the growth of the numbers of vertices, edges and faces in all possible periodic tessellations of the 2D Euclidean space, and encodes those rules in several types of polynomial growth functions. These encodings map the geometric, combinatorial and topological properties of the tessellations into sets of integer coefficients. Several general statements about these encodings are given with rigorous mathematical proof. The variation of the growth functions is represented graphically and analyzed in orphic diagrams, so named because of their similarity to orphic art. Several examples of 3D space groups are included, to emphasize the complexity of the growth functions in higher dimensions. A freely available Python library is presented to facilitate the discovery of the growth functions and the generation of orphic diagrams.
Chemistry↗
Left almost periodicity does not imply right almost periodicity.
Left almost periodic function defined on topological group G is not necessarily right almost periodic even if g is Lie group
TOPOLOGY↗
Behavior near a periodic orbit of functional differential equations
Behavior near periodic orbit of functional differential equations
Hale, J. K.↗
Optimal Force Control of Vibro-Impact Systems for Autonomous Drilling Applications
The need to maintain optimal energy efficiency is critical during the drilling operations performed on future and current planetary rover missions (see figure). Specifically, this innovation seeks to solve the following problem. Given a spring-loaded percussive drill driven by a voice-coil motor, one needs to determine the optimal input voltage waveform (periodic function) and the optimal hammering period that minimizes the dissipated energy, while ensuring that the hammer-to-rock impacts are made with sufficient (user-defined) impact velocity (or impact energy). To solve this problem, it was first observed that when voice-coil-actuated percussive drills are driven at high power, it is of paramount importance to ensure that the electrical current of the device remains in phase with the velocity of the hammer. Otherwise, negative work is performed and the drill experiences a loss of performance (i.e., reduced impact energy) and an increase in Joule heating (i.e., reduction in energy efficiency). This observation has motivated many drilling products to incorporate the standard bang-bang control approach for driving their percussive drills. However, the bang-bang control approach is significantly less efficient than the optimal energy-efficient control approach solved herein. To obtain this solution, the standard tools of classical optimal control theory were applied. It is worth noting that these tools inherently require the solution of a two-point boundary value problem (TPBVP), i.e., a system of differential equations where half the equations have unknown boundary conditions. Typically, the TPBVP is impossible to solve analytically for high-dimensional dynamic systems. However, for the case of the spring-loaded vibro-impactor, this approach yields the exact optimal control solution as the sum of four analytic functions whose coefficients are determined using a simple, easy-to-implement algorithm. Once the optimal control waveform is determined, it can be used optimally in the context of both open-loop and closed-loop control modes (using standard realtime control hardware).
Aldrich, Jack B.↗
Optimized Auxiliary Functions for Robust Mitigation of Finite-Size Errors in Periodic Hybrid Density Functional Theory
When calculating properties of periodic systems at the thermodynamic limit (TDL), the dominant source of finite size error (FSE) arises from the long-range Coulomb interaction, and can manifest as a slowly converging quadrature error when approximating an integral in the reciprocal space by a finite sum. The singularity subtraction (SS) method offers a systematic approach for reducing this quadrature error and thus the FSE. Here, in this work, we first investigate the performance of the SS method in the simplest setting, aiming at reducing the FSE in exact exchange calculations by subtracting the Coulomb contribution with a single, adjustable Gaussian auxiliary function. We demonstrate that a simple fitting method can robustly estimate the optimal Gaussian width and leads to rapid convergence toward the TDL. Furthermore, we suggest new forms of the auxiliary function, whose optimal parameters could also be determined through least-squares fitting. For a range of semiconductors and insulators, the proposed auxiliary functions achieve robust, millihartree-level accuracy in hybrid density functional theory calculations, including cases with sparse k-meshes and large basis sets.
Quiton, Stephen Jon [University of California, Ber↗
Behavior near a periodic orbit of functional differential equations
Equilibrium points and periodic orbits, using qualitative theory of autonomous functional differential equations
Hale, J. K.↗
A Bayesian Framework for Spectral Reprojection
Abstract Fourier partial sum approximations yield exponential accuracy for smooth and periodic functions, but produce the infamous Gibbs phenomenon for non-periodic ones. Spectral reprojection resolves the Gibbs phenomenon by projecting the Fourier partial sum onto a Gibbs complementary basis, often prescribed as the Gegenbauer polynomials. Noise in the Fourier data and the Runge phenomenon both degrade the quality of the Gegenbauer reconstruction solution, however. Motivated by its theoretical convergence properties, this paper proposes a new Bayesian framework for spectral reprojection, which allows a greater understanding of the impact of noise on the reprojection method from a statistical point of view. We are also able to improve the robustness with respect to the Gegenbauer polynomials parameters. Finally, the framework provides a mechanism to quantify the uncertainty of the solution estimate.
Li, Tongtong (ORCID:0000000276644764)↗
Longitude distribution of proton flares as a function of recurrence period.
Longitude distribution of solar proton flares as function of recurrence period
SOLAR FLARE↗
A graphical test for checking the stability of a linear time-invariant feedback system.
A continuous-time scalar linear time-invariant feedback system is considered for the purpose of checking Willems' (1969, 1970) graphical test for a scalar linear time-invariant feedback system with constant feedback. Heavy reliance is placed on the theory of almost periodic functions. Following definition of the problem and layout of notation, attention is given to solution of the problem, considering only the almost periodic part of the open-loop transfer function.
Callier, F. M.↗
Periodicity of the earth's magnetic reversals
Results are presented from an attempt to perform a relatively comprehensive analysis of the evidence for a periodicity, with harmonics, of the observed regular reversals of the earth's magnetic field. The database considered covers 296 reversals over the past 165 Myr. Histograms with bins 1 Myr apart reveal only 30 Myr reversal patterns. The reversal dates are fitted to a linear periodic function and a spectrum is computed for the residuals at the adopted dates. The possible presence of multiple periodicities is evaluated and over various time intervals. The analysis shows that a recently observed 15 Myr periodicity is probably a harmonic of the 29.5-30.5 Myr period. The calculations do not confirm an inherent magnetic reversal property of the earth. The reversals may arise from tectonic events or from impacts from extraterrestrial objects.
Stothers, R. B.↗
Aeroelastic stability of periodic systems with application to rotor blade flutter
The dynamics of a helicopter blade in forward flight are described by a system of linear differential equations with periodic coefficients. The stability of this periodic aeroelastic system is determined, using multivariable Floquet-Liapunov theory. The transition matrix at the end of the period is evaluated by: (1) direct numerical integration, and (2) a new, approximate method, which consists in approximating a periodic function by a series of step functions. The numerical accuracy and efficiency of the methods is compared, and the second method is shown to be superior by far. Results illustrating the effect of the periodic coefficients and various blade parameters are presented.
Friedmann, P.↗