Sum Of Cosines Research Articles

Inequalities for trigonometric sums and applications

We present various new inequalities for cosine and sine sums. Among others, we prove that 0.1$$\begin{aligned} 0\le \sum _{k=0}^n \frac{ (a)_{2k}}{(2k)!} \frac{\cos ((2k+1)x)}{2k+1} \quad {(a\in \mathbb {R})} \end{aligned}$$is valid for all $$n\ge 0$$ and $$x\in [0,\pi /2]$$ if and only if $$a\in [-2,1]$$, and that 0.2$$\begin{aligned} 0\le \sum _{k=0}^n \frac{ (b)_{2k}}{(2k)!} \frac{\sin ((2k+1)x)}{2k+1} \quad {(b\in \mathbb {R})} \end{aligned}$$holds for all $$n\ge 0$$ and $$x\in [0,\pi ]$$ if and only if $$b\in [-3,2]$$. Here, $$(a)_n=\prod _{j=0}^{n-1} (a+j)$$ denotes the Pochhammer symbol. Inequality (0.1) with $$a=1$$ is due to Gasper. We use it to obtain an integral inequality in the complex domain and to provide a one-parameter class of absolutely monotonic functions. An application of (0.2) leads to a new extension of the classical Fejer–Jackson inequality.

Obtaining the Optimal Shape of Planar Antenna Arrays for Direction-of-Arrival Joint Estimation of One and Two Coordinates of Signal Sources on Azimuth

In this paper, an approach for obtaining optimal planar antenna arrays consisting of omnidirectional sensors for the direction-of-arrival estimation is proposed. Exact expressions of the Cramer-Rao lower bound are used. The obtained formulas describe the influence of the location of antenna elements on the accuracy of the direction-of-arrival non-joint estimation for one arriving signal and joint estimation with two signal sources. It has been shown that the accuracy of the direction-of-arrival non-joint estimation is determined as the sum of the squares of the differences between all the coordinates of omnidirectional elements along x- and y-axis if one signal arrives. If two signals arrive, the accuracy of the direction-of-arrival joint estimation depends on the sum of cosines having the argument with the difference between sensor coordinates and signals radius-vectors. The optimal locations of the antenna elements using the obtained expressions can be calculated very easily in order to reduce the direction-finding errors near particular sectors. In order to confirm the proposed methodology the optimal antenna arrays constructed after minimization of the new formulas are researched. It is found out that the new shapes of antenna arrays based on the analytical expressions have better direction-of-arrival accuracy in comparison with the circular ones.

Navier-Stokes Three Dimensional Equations Solutions Volume Three

Solution of Navier-Stokes equation is found by introducing new method for solving differential equations. This new method is writing periodic scalar function in any dimensions and any dimensional vector fields as the sum of sine and cosine series with proper coefficients. The method is extension of Fourier series representation for one variable function to multi-variable functions and vector fields.Before solving Navier-Stokes equations we introduce a new technique for writing periodic scalar functions or vector fields as the sum of cosine and sine series with proper coefficients. Fourier series representation is background for our new technique.Periodic nature of initial velocity for Navier-Stokes problem helps us write the vector field in the form of cosine and sine series sum which simplify the problem.

Klaus Friedrich Roth, 1925-2015

Klaus Friedrich Roth, 1925-2015

Geometric Properties of Cesàro Averaging Operators

Using positivity of trigonometric cosine and sine sums whose coefficients are generalization of Vietoris numbers, we find the conditions on coefficient {ak} to characterize the geometric properties of the corresponding analytic function f(z)=z+∑k=2∞akzk in the unit disc D. As an application, we also find geometric properties of generalized Cesàro-type polynomials.

A scientific report on heat transfer analysis in mixed convection flow of Maxwell fluid over an oscillating vertical plate

This scientific report investigates the heat transfer analysis in mixed convection flow of Maxwell fluid over an oscillating vertical plate with constant wall temperature. The problem is modelled in terms of coupled partial differential equations with initial and boundary conditions. Some suitable non-dimensional variables are introduced in order to transform the governing problem into dimensionless form. The resulting problem is solved via Laplace transform method and exact solutions for velocity, shear stress and temperature are obtained. These solutions are greatly influenced with the variation of embedded parameters which include the Prandtl number and Grashof number for various times. In the absence of free convection, the corresponding solutions representing the mechanical part of velocity reduced to the well known solutions in the literature. The total velocity is presented as a sum of both cosine and sine velocities. The unsteady velocity in each case is arranged in the form of transient and post transient parts. It is found that the post transient parts are independent of time. The solutions corresponding to Newtonian fluids are recovered as a special case and comparison between Newtonian fluid and Maxwell fluid is shown graphically.

The Green’s function for the Hückel (tight binding) model

Applications of the Hückel (tight binding) model are ubiquitous in quantum chemistry and solid state physics. The matrix representation of this model is isomorphic to an unoriented vertex adjacency matrix of a bipartite graph, which is also the Laplacian matrix plus twice the identity. In this paper, we analytically calculate the determinant and, when it exists, the inverse of this matrix in connection with the Green’s function, G, of the N×N Hückel matrix. A corollary is a closed form expression for a Harmonic sum (Eq. (12)). We then extend the results to d− dimensional lattices, whose linear size is N. The existence of the inverse becomes a question of number theory. We prove a new theorem in number theory pertaining to vanishing sums of cosines and use it to prove that the inverse exists if and only if N + 1 and d are odd and d is smaller than the smallest divisor of N + 1. We corroborate our results by demonstrating the entry patterns of the Green’s function and discuss applications related to transport and conductivity.

A new series representation derived for periodic functions that can be represented by the Fourier series

The Fourier series are, as we know, widely used in a number of mathematical problems arising in various topics of Communication systems, Signal Processing, Image processing and many more. The main contribution through these series is to analyze and build periodic functions in the form of infinite or finite sums of sines and cosines. In this paper, a new series representation like the Fourier series has been derived for every periodic function which can be represented by the Fourier series, thereby increasing the extent of applications of Fourier analysis in the related fields.

Local random-phase noise for procedural texturing

Local random-phase noise is a noise model for procedural texturing. It is defined on a regular spatial grid by local noises, which are sums of cosines with random phase. Our model is versatile thanks to separate sampling in the spatial and spectral domains. Therefore, it encompasses Gabor noise and noise by Fourier series. A stratified spectral sampling allows for a faithful yet compact and efficient reproduction of an arbitrary power spectrum. Noise by example is therefore obtained faster than state-of-the-art techniques. As a second contribution we address texture by example and generate not only Gaussian patterns but also structured features present in the input. This is achieved by fixing the phase on some part of the spectrum. Generated textures are continuous and non-repetitive. Results show unprecedented framerates and a flexible visual result: users can control with one parameter the blending between noise by example and structured texture synthesis.

Integrability and L1-convergence of Certain Cosine Sums with Quasi Hyper Convex Coefficients

In this paper criterion for L1- convergence of a certain cosine sums with third quasi hyper-convex coecients is obtained.

Neighbors’ distribution property and sample reduction for support vector machines

For data pre-processing of SVMs, many scholars tried to find those samples, which would become support vectors. Generally, support vectors locate in the overlap regions, which are between different classes. But overlap region does not always exist. In this paper, a new method is proposed to find the boundary regions of each class instead of overlap regions. This method could deal with the dataset without overlap regions. Summing the cosine of the sample-neighbor angle, the sum ranges from 0 to k. When the sample locates in the boundary region of data distribution, the sum would be close to k; when the sample locates in the interior of the data distribution, the sum would be close to 0. Using cosine sum, the samples locating in the interior of each class can be disposed before SVMs training. Experimental results show that the proposed method can solve the problem, which the methods based on finding overlap regions cannot deal with.

Boundary detection and sample reduction for one-class Support Vector Machines

There are a large number of target data and fewer outlier data in the problem of one-class classification. The sample reduction methods used for two-class or multi-class classification cannot be applied to this problem. This paper presents a novel method to reduce the scale of training set for one-class classification. This method only preserves the sample locating near the data distribution which may become support vector. If summing the cosine value of the angle which is between the difference between the sample and the neighbor, and the difference between the sample and mean of neighbors, the cosine sum will be close to k (k is the number of the neighbors), while the sample locating near the boundary; close to 0, while the sample locating within the data distribution. Experimental results demonstrate that the proposed method can reduce the scale of training set to get faster training speed and use less support vectors in the model of one-class SVMs, and the performance does not deteriorate.

Harmonic analysis: from Fourier to wavelets

In the last 200 years, harmonic analysis has been one of the most influential bodies of mathematical ideas, having been exceptionally significant both in its theoretical implications and in its enormous range of applicability throughout mathematics, science, and engineering. In this book, the authors convey the remarkable beauty and applicability of the ideas that have grown from Fourier theory. They present for an advanced undergraduate and beginning graduate student audience the basics of harmonic analysis, from Fourier's study of the heat equation, and the decomposition of functions into sums of cosines and sines (frequency analysis), to dyadic harmonic analysis, and the decomposition of functions into a Haar basis (time localization). While concentrating on the Fourier and Haar cases, the book touches on aspects of the world that lies between these two different ways of decomposing functions: time-frequency analysis (wavelets). Both finite and continuous perspectives are presented, allowing for the introduction of discrete Fourier and Haar transforms and fast algorithms, such as the Fast Fourier Transform (FFT) and its wavelet analogues. The approach combines rigorous proof, inviting motivation, and numerous applications. Over 250 exercises are included in the text. Each chapter ends with ideas for projects in harmonic analysis that students can work on independently. This book is published in cooperation with IAS/Park City Mathematics Institute.

A Generalization of the Identity

SummaryUsing Euler's Theorem and the Geometric Sum Formula, we prove trigonometric identities for alternating sums of sines and cosines.

Stable Functions and Extension of Vietoris’ Theorem

In this work, using subordination results for the Cesaro sum of certain analytic functions, a concept called Cesaro stable is given and two conjectures in this direction are also proposed. Using an extension of Vietoris’ theorem on positivity of cosine sums, some results on positivity of cosine and sine sums are obtained. These results are useful in discussing some particular cases of the proposed conjectures.

Simulation study of beam extraction from a synchrotron using colored noise with digital filter

A simulation program is developed for a slow-extraction method using a fast Q magnet (FQ) and an RF-knockout. In this extraction method, after the separatrix is produced with excitation of sextupole magnets, a required quantity of circulating beam is extracted by shrinking the separatrix with excitation of the FQ. Then the emittance of circulating beam is diffused to an original size with the RF-knockout. This process is repeated with required timing until the entire circulating beam is completely extracted. An algorithm using a digital filter and white noise is proposed for a colored noise as a signal source for the RF-knockout. Spill structures with the present computing method were similar to the results obtained using a conventional algorithm with the sum of cosines of many frequency components. The results are also in agreement with the experimental results using the HIMAC synchrotron. The computing time of colored noise for simulation of 10 6 turns was 0.5 h for the filter method and 5.0 h for the conventional one. It is indicated that a colored noise with a wider frequency bandwidth gives a better spill structure that smoothly increases with time.

Design of a ridge filter structure based on the analysis of dose distributions

Dose distributions distorted by a periodic structure, such as a ridge filter, are analytically investigated. Based on the beam optics, the fluence distributions of scanned beams passing through the ridge filter are traced. It is shown that the periodic lateral dose distribution blurred by multiple Coulomb scattering can be expressed by a sum of cosine functions through Fourier transform. The result shows that the dose homogeneity decreases exponentially as the period of the structure becomes longer. This analysis is applied to the example case of a mini-ridge filter. The mini-ridge filter is designed to broaden sharp Bragg peaks for an energy-stacking irradiation method. The dose distributions depend on the period of the ridge filter structure and the angular straggling at the ridge filter position. Several cases are prepared where the period and angular straggling are supposed to be probable values. In these cases, the lateral distributions obtained by the analytical method are compared to Monte Carlo simulation results. Both distributions show good agreement with each other within 1%, which means that this analysis allows estimation of the dose distribution downstream of the ridge filter quantitatively. The appropriate period of grooves and scatterer width can be determined which ensures sufficient homogeneity.

Math Bite: Sums of Sines and Cosines

Math Bite: Sums of Sines and Cosines

Math Bite: Sums of Sines and Cosines

Math Bite: Sums of Sines and Cosines

On the 𝐿1-Convergence of Modified Cosine Sums

Abstract Concerning the 𝐿1-convergence of modified cosine sums introduced by Kumari and Ram [Indian J. Pure Appl. Math. 19: 1101–1104, 1988], we generalise the results of Kumari, Ram [Indian J. Pure Appl. Math. 19: 1101–1104, 1988] and Bor [Proc. Indian Acad. Sci. Math. Sci. 102: 235–238, 1992]. Also a necessary and sufficient condition for the 𝐿1-convergence of cosine series has been deduced as a corollary for the said class.