Fourier Analysis Techniques Research Articles

Probabilistic Harmonic Resonance Assessment Considering Power System Uncertainties

The presence of power system uncertainties results in variations of the harmonic resonance behaviors. There is, therefore, a need to perform the probabilistic assessment in harmonic resonance study. In this paper, a systematic methodology for probabilistic harmonic resonance assessment considering power system uncertainties is presented. First, potential system uncertainties are analyzed and modeled. The stochastic behavior of harmonic resonance due to system uncertainties is then studied using both Monte Carlo approach and harmonic resonance mode analysis technique. A modified power iteration method is further used to efficiently reduce the calculation time. Three indices, including probabilistic expressions of 1) resonance frequency band, 2) modal impedances in the resonance band, and 3) sensitivity information at the bus-level and the element-level are used to represent the stochastic behaviors of harmonic resonance. In addition, the resonance mitigation scheme based on probabilistic resonance frequency band shift technique is described. The effectiveness of the proposed method is demonstrated through case studies in an uncertain power system. Its potential applications are also discussed.

Small Amplitude Traveling Waves in the Full-Dispersion Whitham Equation

In this article, we provide an alternative way to construct small amplitude traveling waves for general Whitham type equations, in both periodic and whole line contexts. More specifically, Fourier analysis techniques allow us to reformulate the problem to the study of waves that are small and regular perturbations of well-understood ODE’s. In addition, rigorous spectral stability of these waves is established.

Fundamental solutions and decay of fully non-local problems

In this paper, we study a fully non-local reaction-diffusion equation which is non-local both in time and space. We apply subordination principles to construct the fundamental solutions of this problem, which we use to find a representation of the mild solutions. Moreover, using techniques of Harmonic Analysis and Fourier Multipliers, we obtain the temporal decay rates for the mild solutions.

Proper orthogonal decomposition analysis of a circular cylinder undergoing vortex-induced vibrations

Proper orthogonal decomposition (POD) is utilized to analyze the wake-dynamics of a low-mass ratio circular cylinder undergoing vortex-induced vibrations in the initial and upper branches (U* = U∞/fND = 4.07, 5.32). POD allows for characterizing dynamics at frequencies which differ from the cylinder oscillation that cannot be captured with conventional phase-averaging. POD modes contributing to the dominant coherent motions are described in detail. Fourier analysis techniques are used to identify relationships between the POD modes describing non-periodic dynamics linked to the slow-varying base flow and result in a modulation in the strength of vortex shedding. Heuristic models based on mean-field theory are proposed for the POD temporal coefficients. The modelled wake dynamics are found to account for a significant contribution to the Reynolds stresses. In the initial branch, it is found that 6 POD modes are required to capture the salient aspects of the flow, while in the upper branch, 7 modes are required.

ON THE DISTRIBUTION OF THE MAXIMUM OF CUBIC EXPONENTIAL SUMS

In this paper, we investigate the distribution of the maximum of partial sums of certain cubic exponential sums, commonly known as ‘Birch sums’. Our main theorem gives upper and lower bounds (of nearly the same order of magnitude) for the distribution of large values of this maximum, that hold in a wide uniform range. This improves a recent result of Kowalski and Sawin. The proofs use a blend of probabilistic methods, harmonic analysis techniques, and deep tools from algebraic geometry. The results can also be generalized to other types of $\ell$-adic trace functions. In particular, the lower bound of our result also holds for partial sums of Kloosterman sums. As an application, we show that there exist $x\in [1,p]$ and $a\in \mathbb{F}_{p}^{\times }$ such that $|\sum _{n\leqslant x}\exp (2\unicode[STIX]{x1D70B}i(n^{3}+an)/p)|\geqslant (2/\unicode[STIX]{x1D70B}+o(1))\sqrt{p}\log \log p$. The uniformity of our results suggests that this bound is optimal, up to the value of the constant.

Solving the backward problem in Riesz–Feller fractional diffusion by a new nonlocal regularization method

In this paper, we consider the backward problem introduced in [48] for Riesz–Feller fractional diffusion. To begin with, some basic properties of solution of the corresponding forward problem, such as the Lp estimates, symmetry property and asymptotic estimates, are established by Fourier analysis technique. And then, under various a priori bound assumptions, we give the L2 conditional stability estimates for the solution of backward problem and also its symmetry property. Moreover, in order to overcome the ill-posedness of the backward problem, we propose a new nonlocal regularization method (NLRM) to solve it. That is, the following nonlocal variational functional is introducedJ(φ)=12‖u(φ(x);x,T)−fδ(x)‖2+β2‖[Pα1⁎φ](x)‖2, where β∈(0,1) is a regularization parameter, “⁎” denotes the convolution operation and Pα1(x) is called a convolution kernel with parameter α1, which will be selected properly later. The minimizer of above variational problem is defined as the regularization solution, and the L2 estimates, symmetry property of regularization solution are given. These results actually show the well-posedness of nonlocal variational problem. Our idea is essentially that using this well-posed problem to approximate the backward (ill-posed) problem. Thus, under an a posteriori parameter choice rule, we deduce various convergence rate estimates under different a-priori bound assumptions for the exact solution. Finally, several numerical examples are given to show that the proposed numerical methods are effective and adaptive for different a-priori information.

A look at representations of $$SL_{2}({\mathbb {F}}_{q})$$ S L 2 ( F q ) through the lens of size

How to study a nice function on the real line? The physically motivated Fourier theory technique of harmonic analysis is to expand the function in the basis of exponentials and study the meaningful terms in the expansion. Now, suppose the function lives on a finite non-commutative group G, and is invariant under conjugation. There is a well-known analog of Fourier analysis, using the irreducible characters of G. This can be applied to many functions that express interesting properties of G. To study these functions one wants to know how the different characters contribute to the sum? In this note we describe the $$G=SL_{2}({\mathbb {F}}_{q})$$ case of the theory we have been developing in recent years which attempts to give a fairly general answer to the above question for finite classical groups. The irreducible representations of $$SL_{2}({\mathbb {F}}_{q})$$ are “well known” for a very long time (Frobenius in Sitzber Preuss Akad Wiss 985–1021, 1896; Jordan in Am J Math 29:387–405, 1907; Schur in Journal fur die reine und angewandte Mathematik 132:85–137, 1907) and are a prototype example in many introductory courses on the subject. We are happy that we can say something new about them. In particular, it turns out that the representations that were considered as “anomalous” in the “old” point of view (known as the “philosophy of cusp forms”) are the building blocks of the current approach.

A maximal Function Approach to Two-Measure Poincaré Inequalities

This paper extends the self-improvement result of Keith and Zhong in Keith and Zhong (Ann. Math. 167(2):575–599, 2008) to the two-measure case. Our main result shows that a two-measure (p, p)-Poincare inequality for $$1<p<\infty $$ improves to a $$(p,p-{\varepsilon })$$ -Poincare inequality for some $${\varepsilon }>0$$ under a balance condition on the measures. The corresponding result for a maximal Poincare inequality is also considered. In this case the left-hand side in the Poincare inequality is replaced with an integral of a sharp maximal function and the results hold without a balance condition. Moreover, validity of maximal Poincare inequalities is used to characterize the self-improvement of two-measure Poincare inequalities. Examples are constructed to illustrate the role of the assumptions. Harmonic analysis and PDE techniques are used extensively in the arguments.

Large-time behavior of strong solutions to the compressible magnetohydrodynamic system in the critical framework

We study the compressible viscous magnetohydrodynamic (MHD) system and investigate the large-time behavior of strong solutions near constant equilibrium (away from vacuum). In the 80s, Umeda et al. considered the dissipative mechanisms for a rather general class of symmetric hyperbolic–parabolic systems, which is given by [Formula: see text] Here, [Formula: see text] denotes the characteristic root of linearized equations. From the point of view of dissipativity, Kawashima in his doctoral dissertation established the optimal time-decay estimates of [Formula: see text]-[Formula: see text]) type for solutions to the MHD system. Now, by using Fourier analysis techniques, we present more precise description for the large-time asymptotic behavior of solutions, not only in extra Lebesgue spaces but also in a full family of Besov norms with the negative regularity index. Precisely, we show that the [Formula: see text] norm (the slightly stronger [Formula: see text] norm in fact) of global solutions with the critical regularity, decays like [Formula: see text] as [Formula: see text]. Our decay results hold in case of large highly oscillating initial velocity and magnetic fields, which improve Kawashima’s classical efforts.

Cyclicity of the left regular representation of a locally compact group

We combine harmonic analysis and operator algebraic techniques to give a concise argument that the left regular representation of a locally compact group is cyclic if and only if the group is first countable, a result first proved by Greenleaf and Moskowitz.

Stretching Convex Domains to Capture Many Lattice Points

Abstract We consider an optimal stretching problem for strictly convex domains in $\mathbb{R}^d$ that are symmetric with respect to each coordinate hyperplane, where stretching refers to transformation by a diagonal matrix of determinant 1. Specifically, we prove that the stretched convex domain which captures the most positive lattice points in the large volume limit is balanced: the (d − 1)-dimensional measures of the intersections of the domain with each coordinate hyperplane are equal. Our results extend those of Antunes and Freitas, van den Berg, Bucur and Gittins, Ariturk and Laugesen, van den Berg and Gittins, and Gittins and Larson. The approach is motivated by the Fourier analysis techniques used to prove the classical $\#\{(i,j) \in \mathbb{Z}^2 : i^2 +j^2 \le r^2 \} =\pi r^2 + \mathcal{O}(r^{2/3})$ result for the Gauss circle problem.

Preliminary Analysis of Surface Circulation off the Minjiang River Estuary near the Southwestern East China Sea

Better understanding the surface circulation in the marine environment is pertinent to the preparedness of pollution response and maritime safety. In the spring of 2012, two GPS surface drifters were deployed off the Minjiang River Estuary to study the surface circulation near the southwestern East China Sea. One drifter followed the China Coastal Current and the other one joined the Taiwan Warm Current. Classical harmonic analysis technique was employed to analyse the current velocities derived from the trajectories of these two drifters. M2 tidal component is dominant in the tidal current spectrum. The residual currents obtained from these 2 drifters agree very well with the previous studies.

Restriction estimates of varepsilon -removal type for k-th powers and paraboloids

We obtain restriction estimates of varepsilon -removal type for the set of k-th powers of integers, and for discrete d-dimensional surfaces of the form {(n1,⋯,nd,n1k+⋯+ndk):|n1|,⋯,|nd|⩽N},\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\{ (n_1,\\dots ,n_d,n_1^k + \\cdots + n_d^k) \\,:\\, |n_1|,\\dots ,|n_d| \\leqslant N \\}, \\end{aligned}$$\\end{document}which we term ‘k-paraboloids’. For these surfaces, we obtain a satisfying range of exponents for large values of d, k. We also obtain estimates of varepsilon -removal type in the full supercritical range for k-th powers and for k-paraboloids of dimension d < k(k-2). We rely on a variety of techniques in discrete harmonic analysis originating in Bourgain’s works on the restriction theory of the squares and the discrete parabola.

Numerical solution of the generalized, dissipative KdV–RLW–Rosenau equation with a compact method

The nonlinear dynamics of the one–dimensional, generalized Korteweg–de Vries–regularized–long wave–Rosenau (KdV–RLW–Rosenau) equation with second– and fourth–order dissipative terms subject to initial Gaussian conditions is analyzed numerically by means of three–point, fourth–order accurate, compact finite differences for the discretization of the spatial derivatives and a trapezoidal method for time integration. By means of a Fourier analysis and global integration techniques, it is shown that the signs of both the fourth–order dissipative and the mixed fifth–order derivative terms must be negative. It is also shown that an increase of either the linear drift or the nonlinear convection coefficients results in an increase of the steepness, amplitude and speed of the right–propagating wave, whereas the speed and amplitude of the wave decrease as the power of the nonlinearity is increased, if the amplitude of the initial Gaussian condition is equal to or less than one. It is also shown that the wave amplitude and speed decrease and the curvature of the wave’s trajectory increases as the coefficients of the second– and fourth–order dissipative terms are increased, while an increase of the RLW coefficient was found to decrease both the damping and the phase velocity, and generate oscillations behind the wave. For some values of the coefficients of both the fourth–order dissipative and the Rosenau terms, it has been found that localized dispersion shock waves may form in the leading part of the right–propagating wave, and that the formation of a train of solitary waves that result from the breakup of the initial Gaussian conditions only occurs in the absence of both Rosenau’s, Kortweg–de Vries’s and second– and fourth–order dissipative terms, and for some values of the amplitude and width of the initial condition and the RLW coefficient. It is also shown that negative values of the KdV term result in steeper, larger amplitude and faster waves and a train of oscillations behind the wave, whereas positive values of that coefficient may result in negative phase and group velocities, no wave breakup and oscillations ahead of the right–propagating wave.

Comparison of AC and pulsed magnetization-based elasto-magnetic methods for tensile force measurement in steel strand

AC and pulsed magnetization-based elasto-magnetic measurement methods of tensile force in steel strand are compared in the study. Completely opposite conclusions on the dependency of maximum magnetic induction on the tensile force are drawn from the two modes. The opposite conclusions are consistent with the predicted results before and after the Villari reversal point. Harmonic and spectral analysis techniques are introduced to investigate new feature parameters for tensile force characterization. In AC mode, the distortion factor of the induced voltage signals demonstrates the better stability and force prediction accuracy than the conventional time-domain feature parameters of maximum magnetic induction and coercive force. In the pulsed mode, the linear correlation between the spectral amplitudes of the induced pulsed signals and the tensile force is analyzed. A new parameter, the summation of the spectrum amplitudes in a particular band, is introduced to characterize tensile force. The calibration curve for tensile force measurement with the new parameter shows a lower prediction error and hysteresis error compared to the time-domain analysis results. Conclusions obtained from the comparative study on the performance of the two magnetization modes and two types of parameters may provide new insights into the improvements in the tensile force detection accuracy of the elasto-magnetic methods.

Reduction of Total Harmonic Distortion in Cascaded H-Bridge Inverter by Pattern Search Technique

Pattern Search technique can be used to find the solution for the optimization problem. In this paper, pattern search algorithm has been utilized to calculate the switching angles for the cascaded H-bridge inverter with the consideration of minimizing total harmonic distortion. Mathematical equations for the optimization problem were formulated by fourier analysis technique. Lower order harmonics such as third, fifth, seventh, ninth and eleventh order harmonics were taken into account to mitigate the total harmonic distortion of the inverter. Simulations have been carried out for thirteen level, fifteen level and seventeen level cascaded H-bridge inverter using matlab software. Total harmonic distortion of voltage and current for resistive load, resistive-inductive load and motor load were analyzed.

Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of the metric

We prove that the Atiyah-Singer Dirac operator ${\mathrm D}_{\mathrm g}$ in ${\mathrm L}^2$ depends Riesz continuously on ${\mathrm L}^{\infty}$ perturbations of complete metrics ${\mathrm g}$ on a smooth manifold. The Lipschitz bound for the map ${\mathrm g} \to {\mathrm D}_{\mathrm g}(1 + {\mathrm D}_{\mathrm g}^2)^{-\frac{1}{2}}$ depends on bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Calder\'on's first commutator and the Kato square root problem. We also show perturbation results for more general functions of general Dirac-type operators on vector bundles.

Season of collection and sperm head shape impacts expression of CARHSP and FTL from motile-rich boar sperm

The objective of the current study was to evaluate transcript activity of motile-rich sperm collected from June (spring) or August (summer), stored as cooled-extended (ExT) or cryopreserved (FrZ), and selected for least or most sperm head shape change, using Fourier harmonic analysis techniques, between June and August. Even with the lack of an extended heat stress, motile-rich sperm transcripts were influenced by season and putatively by sperm head shape change. Transcripts that had previously been associated with seasonality of sperm collection and methylation pathway transcripts were evaluated among semen samples. Calcium-regulated heat-stable protein 1-like transcript from motile-rich sperm tended (P=0.0829) to be greater in samples collected in June in comparison to August samples. Ferritin light polypeptide transcript tended (P=0.0838) to be greater from motile-rich sperm with least head shape change from June collection in contrast to sperm collected in August. Both transcripts have a functional role in cytoprotection and may serve to improve boar semen activity and quality during thermal stress or seasonal changes.

VARIATION OF CHART DATUM TOWARDS MARITIME DELIMITATION DUE TO RISING SEA LEVEL

Abstract. The importance of Chart Datum in hydrographic surveying is inarguable because its determination is part of the process to obtain the actual depth of bathymetry. The Chart Datum has a relationship with the determination of base points because any uncertainty of the base points would definitely cause uncertainty to the determination of the maritime baseline. If there is any doubt on the baselines, it will then cause doubt on the maritime zones as well which includes the equidistant line that forms the border between the two countries. However, due to the ongoing rising sea level, there has been some variations of the Chart Datum in some areas in Malaysia. This research discusses about the variation of Mean Sea Level and Chart Datum for the tide gauge stations at Geting, Cendering, Sedili and Tioman at East Coast and Kukup, Langkawi, Lumut and Penang at the West Coast of Peninsular Malaysia. The tidal analysis was carried out by using the 23 years of data beginning at 1993 to 2015. The observed tidal data for 23 years were processed and analysed by using GeoTide software. In this research, the Harmonic Analysis technique was used in order to calculate the values of Mean Sea Level and the Chart Datum while the slope of the shoreline is modelled by using Global Mapper. The linear trend of the Mean Sea Level and the Chart Datum was analysed to determine the increase of the annual sea level in millimetres accuracy and also to determine the variation of the Chart Datum for each tidal station and its impact towards maritime baseline. The result has shown that the linear trend of sea level rise varies from 24 millimetres per year up to 168 millimetres per year at the East Coast and 24 millimetres per year up to 96 millimetres per year at the West Coast of Peninsular Malaysia. As for the maritime baseline, results has indicated that there exist shifting in the horizontal which are varies from 1.564 metres per year to 3.299 metres per year at the East Coast and from 1.331 metres per year up to 5.857 metres per year at the West Cost of Peninsular Malaysia. From the analysis, it can be stated that the horizontal shift occur greater at the East Coast rather than at the West Coast of Peninsular Malaysia. As a conclusion, the sea level rise does have significant impact towards maritime baseline. Furthermore, the determination of a stable Chart Datum is important to define the maritime baseline in other to avoid conflict with other neighbouring countries.

Global well-posedness and long time decay of the 3D Boussinesq equations

The incompressible 3D Boussinesq equations is considered in this paper. The global well-posedness of mild solutions is proved. By using Fourier analysis and standard techniques, the global solutions u,θ will decay to zero in sense of the norm of the space χ−1 as time goes to infinity and the stability of global solutions is proved.