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Hypercontractive Inequality for Pseudo-Boolean Functions of Bounded Fourier Width  [PDF]
Gregory Gutin,Anders Yeo
Computer Science , 2011,
Abstract: A function $f:\ \{-1,1\}^n\rightarrow \mathbb{R}$ is called pseudo-Boolean. It is well-known that each pseudo-Boolean function $f$ can be written as $f(x)=\sum_{I\in {\cal F}}\hat{f}(I)\chi_I(x),$ where ${\cal F}\subseteq \{I:\ I\subseteq [n]\}$, $[n]=\{1,2,...,n\}$, and $\chi_I(x)=\prod_{i\in I}x_i$ and $\hat{f}(I)$ are non-zero reals. The degree of $f$ is $\max \{|I|:\ I\in {\cal F}\}$ and the width of $f$ is the minimum integer $\rho$ such that every $i\in [n]$ appears in at most $\rho$ sets in $\cal F$. For $i\in [n]$, let $\mathbf{x}_i$ be a random variable taking values 1 or -1 uniformly and independently from all other variables $\mathbf{x}_j$, $j\neq i.$ Let $\mathbf{x}=(\mathbf{x}_1,...,\mathbf{x}_n)$. The $p$-norm of $f$ is $||f||_p=(\mathbb E[|f(\mathbf{x})|^p])^{1/p}$ for any $p\ge 1$. It is well-known that $||f||_q\ge ||f||_p$ whenever $q> p\ge 1$. However, the higher norm can be bounded by the lower norm times a coefficient not directly depending on $f$: if $f$ is of degree $d$ and $q> p>1$ then $ ||f||_q\le (\frac{q-1}{p-1})^{d/2}||f||_p.$ This inequality is called the Hypercontractive Inequality. We show that one can replace $d$ by $\rho$ in the Hypercontractive Inequality for each $q> p\ge 2$ as follows: $ ||f||_q\le ((2r)!\rho^{r-1})^{1/(2r)}||f||_p,$ where $r=\lceil q/2\rceil$. For the case $q=4$ and $p=2$, which is important in many applications, we prove a stronger inequality: $ ||f||_4\le (2\rho+1)^{1/4}||f||_2.$
Numerical Solution of the Modified Equal Width Wave Equation  [PDF]
Seydi Battal Gazi Karako?,Turabi Geyikli
International Journal of Differential Equations , 2012, DOI: 10.1155/2012/587208
Abstract: Numerical solution of the modified equal width wave equation is obtained by using lumped Galerkin method based on cubic B-spline finite element method. Solitary wave motion and interaction of two solitary waves are studied using the proposed method. Accuracy of the proposed method is discussed by computing the numerical conserved laws and error norms. The numerical results are found in good agreement with exact solution. A linear stability analysis of the scheme is also investigated. 1. Introduction The modified equal width wave equation (MEW) based upon the equal width wave (EW) equation [1, 2] which was suggested by Morrison et al. [3] is used as a model partial differential equation for the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes. This equation is related with the modified regularized long wave (MRLW) equation [4] and modified Korteweg-de Vries (MKdV) equation [5]. All the modified equations are nonlinear wave equations with cubic nonlinearities and all of them have solitary wave solutions, which are wave packets or pulses. These waves propagate in non-linear media by keeping wave forms and velocity even after interaction occurs. Few analytical solutions of the MEW equation are known. Thus numerical solutions of the MEW equation can be important and comparison between analytic solution can be made. Geyikli and Battal Gazi Karako? [6, 7] solved the MEW equation by a collocation method using septic B-spline finite elements and using a Petrov-Galerkin finite element method with weight functions quadratic and element shape functions which are cubic B-splines. Esen applied a lumped Galerkin method based on quadratic B-spline finite elements which have been used for solving the EW and MEW equation [8, 9]. Saka proposed algorithms for the numerical solution of the MEW equation using quintic B-spline collocation method [10]. Zaki considered the solitary wave interactions for the MEW equation by collocation method using quintic B-spline finite elements [11] and obtained the numerical solution of the EW equation by using least-squares method [12]. Wazwaz investigated the MEW equation and two of its variants by the tanh and the sine-cosine methods [13]. A solution based on a collocation method incorporated cubic B-splines is investigated by and Saka and Da? [14]. Variational iteration method is introduced to solve the MEW equation by Lu [15]. Evans and Raslan [16] studied the generalized EW equation by using collocation method based on quadratic B-splines to obtain the numerical solutions of a single solitary
A More Accurate Fourier Transform  [PDF]
Elya Courtney,Michael Courtney
Physics , 2015,
Abstract: Fourier transform methods are used to analyze functions and data sets to provide frequencies, amplitudes, and phases of underlying oscillatory components. Fast Fourier transform (FFT) methods offer speed advantages over evaluation of explicit integrals (EI) that define Fourier transforms. This paper compares frequency, amplitude, and phase accuracy of the two methods for well resolved peaks over a wide array of data sets including cosine series with and without random noise and a variety of physical data sets, including atmospheric $\mathrm{CO_2}$ concentrations, tides, temperatures, sound waveforms, and atomic spectra. The FFT uses MIT's FFTW3 library. The EI method uses the rectangle method to compute the areas under the curve via complex math. Results support the hypothesis that EI methods are more accurate than FFT methods. Errors range from 5 to 10 times higher when determining peak frequency by FFT, 1.4 to 60 times higher for peak amplitude, and 6 to 10 times higher for phase under a peak. The ability to compute more accurate Fourier transforms has promise for improved data analysis in many fields, including more sensitive assessment of hypotheses in the environmental sciences related to $\mathrm{CO_2}$ concentrations and temperature. Other methods are available to address different weaknesses in FFTs; however, the EI method always produces the most accurate output possible for a given data set. On the 2011 Lenovo ThinkPad used in this study, an EI transform on a 10,000 point data set took 31 seconds to complete. Source code (C) and Windows executable for the EI method are available at https://sourceforge.net/projects/amoreaccuratefouriertransform/.
Multigrid Method for the Numerical Solution of the Modified Equal Width Wave Equation  [PDF]
Yasser M. Abo Essa
Applied Mathematics (AM) , 2016, DOI: 10.4236/am.2016.710102
Abstract: Numerical solutions of the modified equal width wave equation are obtained by using the multigrid method and finite difference method. The motion of a single solitary wave, interaction of two solitary waves and development of the Maxwellian initial condition into solitary waves are studied using the proposed method. The numerical solutions are compared with the known analytical solutions. Using error norms and conservative properties of mass, momentum and energy, accuracy and efficiency of the mentioned method will be established through comparison with other methods.
Accurate Interconnection Length and Routing Channel Width Estimates for FPGAs
Gao Haixi,Ma Xiaohua,Yang Yintang,
Gao Haixi
,Ma Xiaohu,Yang Yintang

半导体学报 , 2006,
Abstract: We study the problem of the prediction of interconnection dimensions for FPGAs,including estimating interconnection length and channel width.Experimental results show that our estimates are more accurate than those of existing methods.
Direct Inversion of the 3D Pseudo-polar Fourier Transform  [PDF]
Amir Averbuch,Gil Shabat,Yoel Shkolnisky
Computer Science , 2015,
Abstract: The pseudo-polar Fourier transform is a specialized non-equally spaced Fourier transform, which evaluates the Fourier transform on a near-polar grid, known as the pseudo-polar grid. The advantage of the pseudo-polar grid over other non-uniform sampling geometries is that the transformation, which samples the Fourier transform on the pseudo-polar grid, can be inverted using a fast and stable algorithm. For other sampling geometries, even if the non-equally spaced Fourier transform can be inverted, the only known algorithms are iterative. The convergence speed of these algorithms as well as their accuracy are difficult to control, as they depend both on the sampling geometry as well as on the unknown reconstructed object. In this paper, we present a direct inversion algorithm for the three-dimensional pseudo-polar Fourier transform. The algorithm is based only on one-dimensional resampling operations, and is shown to be significantly faster than existing iterative inversion algorithms.
An Accurate Conformal Fourier Transform Method for 2D Discontinuous Functions
Chun-Hui Zhu;Qing Huo Liu;Yanhui Liu;Yi Shen;Li Jun Liu
PIER , 2011, DOI: 10.2528/PIER11072306
Abstract: Fourier transform of discontinuous functions are often encountered in computational electromagnetics. A highly accurate, fast conformal Fourier transform (CFT) algorithm is proposed to evaluate the finite Fourier transform of 2D discontinuous functions. A curved triangular mesh combined with curvilinear coordinate transformation is adopted to flexibly model an arbitrary shape of the discontinuity boundary. This enables us to take full advantages of high order interpolation and Gaussian quadrature methods to achieve highly accurate Fourier integration results with a low sampling density and small computation time. The complexity of the proposed algorithm is similar to the traditional 2D fast Fourier transform algorithm, but with orders of magnitude higher accuracy. Numerical examples illustrate the excellent performance of the proposed CFT method.
Pseudorandomness and Fourier Growth Bounds for Width 3 Branching Programs  [PDF]
Thomas Steinke,Salil Vadhan,Andrew Wan
Computer Science , 2014,
Abstract: We present an explicit pseudorandom generator for oblivious, read-once, width-$3$ branching programs, which can read their input bits in any order. The generator has seed length $\tilde{O}( \log^3 n ).$ The previously best known seed length for this model is $n^{1/2+o(1)}$ due to Impagliazzo, Meka, and Zuckerman (FOCS '12). Our work generalizes a recent result of Reingold, Steinke, and Vadhan (RANDOM '13) for \textit{permutation} branching programs. The main technical novelty underlying our generator is a new bound on the Fourier growth of width-3, oblivious, read-once branching programs. Specifically, we show that for any $f:\{0,1\}^n\rightarrow \{0,1\}$ computed by such a branching program, and $k\in [n],$ $$\sum_{s\subseteq [n]: |s|=k} \left| \hat{f}[s] \right| \leq n^2 \cdot (O(\log n))^k,$$ where $\widehat{f}[s] = \mathbb{E}\left[f[U] \cdot (-1)^{s \cdot U}\right]$ is the standard Fourier transform over $\mathbb{Z}_2^n$. The base $O(\log n)$ of the Fourier growth is tight up to a factor of $\log \log n$.
An Accurate Approximation to the Distribution of the Sum of Equally Correlated Nakagami-m Envelopes and its Application in Equal Gain Diversity Receivers  [PDF]
Zoran Hadzi-Velkov,Nikola Zlatanov,George K. Karagiannidis
Mathematics , 2009, DOI: 10.1109/ICC.2009.5198714
Abstract: We present a novel and accurate approximation for the distribution of the sum of equally correlated Nakagami-m variates. Ascertaining on this result we study the performance of Equal Gain Combining (EGC) receivers, operating over equally correlating fading channels. Numerical results and simulations show the accuracy of the proposed approximation and the validity of the mathematical analysis.
Septic B-Spline Collocation Method for the Numerical Solution of the Modified Equal Width Wave Equation  [PDF]
Turabi Geyikli, Seydi Battal Gazi Karakoc
Applied Mathematics (AM) , 2011, DOI: 10.4236/am.2011.26098
Abstract: Numerical solutions of the modified equal width wave equation are obtained by using collocation method with septic B-spline finite elements with three different linearization techniques. The motion of a single solitary wave, interaction of two solitary waves and birth of solitons are studied using the proposed method. Accuracy of the method is discussed by computing the numerical conserved laws error norms L2 and L∞. The numerical results show that the present method is a remarkably successful numerical technique for solving the MEW equation. A linear stability analysis shows that this numerical scheme, based on a Crank Nicolson approximation in time, is unconditionally stable.
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