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General and refined Montgomery Lemmata

MATHEMATISCHE ANNALEN 2019年第3-4期373卷 1283-1297页

作者： Bilyk, Dmitriy Dai, Feng Steinerberger, Stefan Univ Minnesota Sch Math Minneapolis MN 55408 USA Univ Alberta Dept Math & Stat Sci Edmonton AB T6G 2G1 Canada Yale Univ Dept Math New Haven CT 06510 USA

Montgomery's Lemma on the torus Td states that a sum of N Dirac masses cannot be orthogonal to many low-frequency trigonometric functions in a quantified way. We provide an extension to general manifolds that also allows for positive weights: let (M, g) be a smooth compact d-dimensional manifold without boundary, let (fk) 8 k= 0 denote the Laplacian eigenfunctions, let {x1,..., xN}. M be a set of points and {a1,..., aN}. R= 0 be a sequence of nonnegative weights. Then, for all X = 0, Sigma(X) (k= 0) N n= 1 anfk (xn 2 (M, g) N i= 1 a2 i X (log X) d 2. This result is sharp up to the logarithmic factor. Furthermore, we prove a refined spherical version of Montgomery's Lemma, and provide applications to estimates of discrepancy and discrete energies of N points on the sphere S-d.

关键词： Lemma Orthogonal Manifolds Chpher(0) Commissioned Officers Eigenfunctions Trigonometry circular functions foreign key Set of points MOMENT MAGNITUDE

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EXTREME STATISTICS FROM THE SINE DISTRIBUTION

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AMERICAN STATISTICIAN 1986年第3期40卷 216-217页

作者： BURROWS, PM College of Agricultural Sciences Clemson University Clemson SC 29634 United States

The sine distribution can be added to the short list of teaching examples for which exact moment properties of extreme order statistics can be expressed in terms of elementary functions and for which the limiting joint distribution of extreme order statistics is easy to derive. Asymptotic efficiency of the midrange relative to the sample average, as unbiased estimators of the population mean, is very different from that attained for the rectangular distribution.

关键词： circular functions Order statistics Limiting distribution

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An Ionosphere Specification Technique Based on Data Ingestion Algorithm and Empirical Orthogonal Function Analysis Method

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SPACE WEATHER-THE INTERNATIONAL JOURNAL OF RESEARCH AND APPLICATIONS 2018年第9期16卷 1410-1423页

作者： Aa, Ercha Ridley, Aaron Huang, Wengeng Zou, Shasha Liu, Siqing Coster, Anthea J. Zhang, Shunrong Chinese Acad Sci Natl Space Sci Ctr Beijing Peoples R China Univ Michigan Dept Climate & Space Sci & Engn Ann Arbor MI 48109 USA Univ Chinese Acar ly Sci Coll Earth & Planetary Sci Beijing Peoples R China MIT Haystack Observ 77 Massachusetts Ave Cambridge MA 02139 USA

A data ingestion method in reproducing ionospheric electron density and total electron content (TEC) was developed to incorporate TEC products from the Madrigal Database into the NeQuick 2 model. The method is based on retrieving an appropriate global distribution of effective ionization parameter (Az) to drive the NeQuick 2 model, which can be implemented through minimizing the difference between the measured and modeled TEC at each grid in the local time-modified dip latitude coordinates. The performance of this Madrigal TEC-driven-NeQuick 2 result is validated through the comparison with various International Global Navigation Satellite Systems Services global ionospheric maps and ionosonde data. The validation results show that a general accuracy improvement of 30-50% can be achieved after data ingestion. In addition, the empirical orthogonal function (EOF) analysis technique is used to construct a parameterized time-varying global Az model. The quick convergence of EOF decomposition makes it possible to use the first six EOF series to represent over 90% of the total variances. The intrinsic diurnal variation and spatial distribution in the original data set can be well reflected by the constructed EOF base functions. The associated EOF coefficients can be expressed as a set of linear functions of F-10.7 and Ap indices, combined with a series of trigonometric functions with annual/seasonal variation components. The NeQuick TEC driven by EOF-modeled Az shows 10-15% improvement in accuracy over the standard ionosphere correction algorithm in the Galileo navigation system. These preliminary results demonstrate the effectiveness of the combined data ingestion and EOF modeling technique in improving the specifications of ionospheric density variations.

关键词： Ionosphere Specification Technique Data Ingestion Algorithm ionospheres correction algorithm Dataset TEC gene Ingestion EOF Orthogonal functions linear function circular functions

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Vieta-like products of nested radicals with Fibonacci and Lucas numbers

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FIBONACCI QUARTERLY 2007年第3期45卷 202-204页

作者： Osler, Thomas J. Rowan Univ Dept Math Glassboro NJ 08028 USA

We present two infinite products of nested radicals involving Fibonacci and Lucas numbers. These products resemble Vieta's classical product of nested radicals for 2/pi. A modern derivation of Vieta's product involves trigonometric functions, while our product involves similar manipulations involving hyperbolic functions.

关键词： Fibonacci Radicals Hyperbolic functions Infinite product Trigonometry circular functions

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Blending two major techniques in order to compute pi

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INTERNATIONAL JOURNAL OF MATHEMATICAL EDUCATION IN SCIENCE AND TECHNOLOGY 2005年第1期36卷 85-92页

作者： Guasti, M. Fernandez Univ Metropolitana Unidad Iztapalapa Dept Fis Lab Optica Cuantica Mexico City 55534 DF Mexico

Three major techniques are employed to calculate pi. Namely, (i) the perimeter of polygons inscribed or circumscribed in a circle, (ii) calculus based methods using integral representations of inverse trigonometric functions, and (iii) modular identities derived from the transformation theory of elliptic integrals. This note presents a combination of the first two procedures, which allows the derivation of a family of series that may exhibit very fast convergence rates. The geometrical interpretation gives good insight into the acceleration method that is being implemented.

关键词： Major Trigonometry Blending Elliptic integral circular functions Circumscribe derivation Polygon perimeter order

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Squigonometry

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Mathematics Magazine 2011年第4期84卷 257-265页

作者： Wood, William E. University of Northern Iowa Cedar Falls IA 50614 United States

The article focuses on the development of the theory of squigonometry to define the classical trigonometric functions of sine and cosine that parameterize the unit circle. It notes the acquisition of ideas in elliptic integrals, non-euclidean geometry, number theory and complex analysis with the application of the said theory using elementary calculus. The author also adapts this approach to develop analogous functions that parameterize the unit squircle.

关键词： Squigonometry equations offers trigonometric functions sine catch Unit circle Non-Euclidean geometry Trigonometry Number theory Elliptic integral circular functions Differential equations parameterization Complex Analysis sin Cosine

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Application of the Rayleigh-Ritz method to solve a class of fractional variational problem 20

Application of the Rayleigh-Ritz method to solve a class of ...

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20th International Conference on Methods and Models in Automation and Robotics (MMAR)

作者： Blaszczyk, Tomasz Czestochowa Tech Univ Inst Math Al Armii Krajowej 21 PL-42201 Czestochowa Poland

ISBN: (纸本)9781479987016

The aim of this paper is to investigate a direct numerical method for solving a class of a fractional variational problem (FVP). The fractional derivative in the FVP is in the Caputo sense. The Rayleigh-Ritz method is introduced for the numerical solution of the FVP. The corresponding Euler-Lagrange equation contain the left and right Caputo fractional derivatives. We approximate the solution of the FVP by using the trigonometric function. Finally, the illustrative example is presented.

关键词： Rayleigh-Ritz methods Numerical methods circular functions Euler-Lagrange equation Solve Lagrangian equations

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A New Duty Cycles Calculation Method in Pulse Width Modulation for Direct Matrix Converters Based on The Shape Function Concept

A New Duty Cycles Calculation Method in Pulse Width Modulati...

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8th International Conference on Compatibility and Power Electronics (CPE)

作者： Szczepankowski, Pawel Gdansk Univ Technol Gdansk Poland

ISBN: (纸本)9781467349130;9781467349116

This paper presents a new approach to duty cycle calculation in pulse width modulation for direct matrix converters. The essence of the proposed method is the use of elementary shape function of finite element. The algorithm for calculating the duty cycles was reduced to simple operations on the coordinates of input and output voltage vectors without trigonometric functions.

关键词： Pulse width modulation matrix converters shape functions computational methods DUTY CYCLE circular functions Essences Coordinates

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(M, p, k)-friendly points: a table-based method for trigonometric function evaluation

(M, p, k)-friendly points: a table-base...

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23rd IEEE International Conference on Application-Specific Systems, Architectures and Processors (ASAP)

作者： Brisebarre, Nicolas Ercegovac, Milos D. Muller, Jean-Michel Ecole Normale Super Lyon CNRS ENS Lyon Lab LIPInriaUCBL 46 Allee Italie F-69364 Lyon 07 France Univ Calif Los Angeles Dept Comp Sci Los Angeles CA 90024 USA

ISBN: (纸本)9780769547688

We present a new way of approximating the sine and cosine functions by a few table look-ups and additions. It consists in first reducing the input range to a very small interval by using rotations with "(M, p, k) friendly angles", proposed in this work, and then by using a bipartite table method in a small interval. An implementation of the method for 24-bit case is described and compared with CORDIC. Roughly, the proposed scheme offers a speedup of 2 compared with an unfolded double-rotation radix-2 CORDIC.

关键词： circular functions Speeding Digital arithmetic crop rotation 24-bit Uninterruptible power supply Tables

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A generalization of wilf's formula

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Kodai Mathematical Journal 2003年第1期26卷 44-48页

作者： Choi, Junesang Lee, Jungseob Srivastava, H.M. Department of Mathematics College of Natural Sciences Dongguk University Kyongju 780-714 South Korea Department of Mathematics Ajou University Suwon 442-749 South Korea Department of Mathematics and Statistics University of Victoria Victoria British Columbia V8W 3P4 Canada

A remarkable infinite product formula was recently posed as a problem by Wilf [6]. Subsequently, Choi and Seo [2] proved Wilf's formula as well as three other analogous product formulas. The main object of this sequel to these earlier works is to present several general infinite product formulas which include, as their special cases, the aforementioned product formulas of Wilf [6] and Choi and Seo [2]. Some other related results, involving the Riemann Zeta function, are also considered briefly. © 2003, Department of Mathematics, Tokyo Institute of Technology. All rights reserved.

关键词： circular functions Euler-Mascheorni constant hyperbolic functions infinite product formulas Riemann Zeta function Stirling's formula

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