Sum Of Cosines Research Articles

Functions of two variables applied to indigenous graphics

The aim of this work is to mathematically represent geometric figures that refer to the indigenous graphics of the Kariri-Xocó people. To do this, we will use the concept of functions with two variables, studied in Calculus III of the Curso de Licenciatura Intercultural Indígena em Matemática at Universidade Estadual de Alagoas. This connection is made based on body paintings that have familiar geometric shapes, which can be seen through the level surfaces of functions with two variables. One of the functions used is the exponential, while the other is made up of the sum of sine and cosine. In addition to this aspect of calculus teaching applied to indigenous culture, the results of this study motivate the development of research on the subject, contributing to the production of works that disseminate indigenous culture in Brazil and around the world.

Image watermarking based on a ratio of DCT coefficient sums using a gradient-based optimizer

Social networks and various websites provide a lot of tools for publishing and sharing digital images. However, publishing digital content online makes it vulnerable to illegal copying and distribution. An effective method for protecting images is watermarking technology, which embeds a logo or some kind of identifier into an image in an invisible way. If necessary, a watermark can be extracted from an image and used to prove authorship and ownership. A major challenge in designing image watermarking schemes is balancing embedding imperceptibility and resistance to various attacks. In this study, we propose a new robust and blind image watermarking algorithm that set a specific ratio of sum of discrete cosine transform (DCT) coefficient absolute values for each image block. We provide a balance between imperceptibility and robustness using a modern metaheuristic optimization algorithm called gradient-based optimizer, which finds the best vector of changes for mid-frequency DCT coefficients. An objective function of optimization algorithm is designed in such a way as to control the ratio of the sums of DCT coefficients absolute values to increase robustness and reduce a distortion level of an image block during embedding. Experimental results and comparison with some other image watermarking schemes demonstrate the high efficiency of the proposed algorithm in terms of imperceptibility and resistance to typical image processing attacks, such as changing brightness and contrast, sharpening, cropping, and others.

Solving the Matrix Exponential Function for Special Orthogonal Groups SO(n) up to n = 9 and the Exceptional Lie Group G2

In this work the matrix exponential function is solved analytically for the special orthogonal groups SO(n) up to n=9. The number of occurring k-th matrix powers gets limited to 0≤k≤n−1 by exploiting the Cayley–Hamilton relation. The corresponding expansion coefficients can be expressed as cosine and sine functions of a vector-norm V and the roots of a polynomial equation that depends on a few specific invariants. Besides the well-known case of SO(3), a quadratic equation needs to be solved for n=4,5, a cubic equation for n=6,7, and a quartic equation for n=8,9. As an interesting subgroup of SO(7), the exceptional Lie group G2 of dimension 14 is constructed via the matrix exponential function through a remarkably simple constraint on an invariant, ξ=1. The traces of the SO(n)-matrices arising from the exponential function are sums of cosines of several angles. This feature confirms that the employed method is equivalent to exponentiation after diagonalization, but avoids complex eigenvalues and eigenvectors and operates only with real-valued quantities.

Review of Cosine Sum Window Functions and Recent Advances in FIR Filter Design Methods

Review of Cosine Sum Window Functions and Recent Advances in FIR Filter Design Methods

ECG Identification Based on the Gramian Angular Field and Tested with Individuals in Resting and Activity States

In the last decade, biosignals have attracted the attention of many researchers when designing novel biometrics systems. Many of these works use cardiac signals and their representation as electrocardiograms (ECGs). Nowadays, these solutions are even more realistic since we can acquire reliable ECG records by using wearable devices. This paper moves in that direction and proposes a novel approach for an ECG identification system. For that, we transform the ECG recordings into Gramian Angular Field (GAF) images, a time series encoding technique well-known in other domains but not very common with biosignals. Specifically, the time series is transformed using polar coordinates, and then, the cosine sum of the angles is computed for each pair of points. We present a proof-of-concept identification system built on a tuned VGG19 convolutional neural network using this approach. We confirm our proposal's feasibility through experimentation using two well-known public datasets: MIT-BIH Normal Sinus Rhythm Database (subjects at a resting state) and ECG-GUDB (individuals under four specific activities). In both scenarios, the identification system reaches an accuracy of 91%, and the False Acceptance Rate (FAR) is eight times higher than the False Rejection Rate (FRR).

ESPRIT versus ESPIRA for reconstruction of short cosine sums and its application

In this paper we introduce two algorithms for stable approximation with and recovery of short cosine sums. The used signal model contains cosine terms with arbitrary real positive frequency parameters and therefore strongly generalizes usual Fourier sums. The proposed methods both employ a set of equidistant signal values as input data. The ESPRIT method for cosine sums is a Prony-like method and applies matrix pencils of Toeplitz + Hankel matrices while the ESPIRA method is based on rational approximation of DCT data and can be understood as a matrix pencil method for special Loewner matrices. Compared to known numerical methods for recovery of exponential sums, the design of the considered new algorithms directly exploits the special real structure of the signal model and therefore usually provides real parameter estimates for noisy input data, while the known general recovery algorithms for complex exponential sums tend to yield complex parameters in this case.

Back-calculation of soil parameters from displacement-controlled cavity expansion under geostatic stress by FEM and machine learning

Estimating soil properties from the mechanical reaction to a displacement is a common strategy, used not only in in situ soil characterization (e.g., pressuremeter and dilatometer tests) but also by biological organisms (e.g., roots, earthworms, razor clams), which sense stresses to explore the subsurface. Still, the absence of analytical solutions to predict the stress and deformation fields around cavities subject to geostatic stress, has prevented the development of characterization methods that resemble the strategies adopted by nature. We use the finite element method (FEM) to model the displacement-controlled expansion of cavities under a wide range of stress conditions and soil properties. The radial stress distribution at the cavity wall during expansion is extracted. Then, methods are proposed to prepare, transform and use such stress distributions to back-calculate the far field stresses and the mechanical parameters of the material around the cavity (Mohr-Coulomb friction angle phi , Young’s modulus E). Results show that: (i) The initial stress distribution around the cavity can be fitted to a sum of cosines to estimate the far field stresses; (ii) By encoding the stress distribution as intensity images, in addition to certain scalar parameters, convolutional neural networks can consistently and accurately back-calculate the friction angle and Young’s modulus of the soil.

Совершенствование формы плоских антенных решеток для повышения точности радиолокации со сверхразрешением

A promising way to increase the bandwidth and noise immunity of modern wireless information transmission systems is the use of antenna arrays equipped with a digital signal processing unit, which include MIMO systems (Multiple Input Multiple Output), as well as adaptive (smart) antennas. The main advantage of this approach is the antenna’s spacing, which makes it possible to assess the angular coordinates of radio signals with further developing 
 a radiation pattern. Gaps in studying the influence of the lattice geometry together with various kinds of antenna elements are known to be one of the factors of inaccuracy of such systems. The work is aimed at obtaining the shape 
 of a planar antenna array with a higher direction finding accuracy. There is described an algorithm for calculating such an arrangement of antenna elements of flat antenna arrays, in which the standard deviation of the estimates of the angular coordinates of one and two radio signal sources is reduced. The proposed approach is based on the analysis 
 of the influence of antenna location on the variance of estimates described by the lower boundary of Kramer-Rao. This value shows the influence of the location of antenna elements on the accuracy of estimating the direction of arrival of a joint assessment 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 squared differences between all coordinates of omnidirectional elements along the 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 location of the antenna elements using the obtained expressions can be calculated very easily to reduce the direction-finding errors near particular sectors. In order to confirm the proposed method, there were studied the antenna arrays built after minimizing the boundary of Kramer-Rao, where the target functions are the new expressions. 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.

Quaternionic Transform: A new Light on the Solar Power Spectrum

Asteroseismology, that is, the use of the frequency content of a time series caused by variations in brightness or radial velocity of a stellar object, is based on the hypothesis that such a series is harmonic and therefore can be described by a sum of sines and cosines. If this were not the case (e.g., the oscillations of an ellipsoid of revolution) it cannot be guaranteed that the Discrete Fourier transform is the least squares approximation to the time series. This report studies the effect of extending the Fourier kernel to a particular quaternion and exploring the impact when it is applied to the best time series that we have (GOLF/SoHO) from the closest star, our Sun. The results are consistent with a notable improvement in the signal-to-noise ratio in the low frequency range. This opens the possibility of detecting the elusive g modes of the Sun in future works.

New Properties of Triangular Orbits in Elliptic Billiards

New invariants in the one-dimensional family of 3-periodic orbits in the elliptic billiard were introduced by the authors in “Can the elliptic billiard still surprise us?” (2020) Math. Intelligencer 42(1): 6–17, some of which were generalized to N > 3. Invariants mentioned there included ratios of radii and/or areas, sum of angle cosines, and a special stationary circle. Here we present some of the proofs omitted there as well as a few new related facts.

Poncelet plectra: harmonious curves in cosine space

It has been shown that the family of Poncelet N-gons in the confocal pair (elliptic billiard) conserves the sum of cosines of its internal angles. Curiously, this quantity is equal to the sum of cosines conserved by its affine image where the caustic is a circle. We show that furthermore, (i) when N = 3, the cosine triples of both families sweep the same planar curve: an equilateral cubic resembling a plectrum (guitar pick). We also show that (ii) the family of triangles excentral to the confocal family conserves the same product of cosines as the one conserved by its affine image inscribed in a circle; and that (iii) cosine triples of both families sweep the same spherical curve. When the triple of log-cosines is considered, this curve becomes a planar, plectrum-shaped curve, rounder than the one swept by its parent confocal family.

New Invariants of Poncelet–Jacobi Bicentric Polygons

The 1d family of Poncelet polygons interscribed between two circles is known as the Bicentric family. Using elliptic functions and Liouville's theorem, we show (i) that this family has invariant sum of internal angle cosines and (ii) that the pedal polygons with respect to the family's limiting points have invariant perimeter. Interestingly, both (i) and (ii) are also properties of elliptic billiard N-periodics. Furthermore, since the pedal polygons in (ii) are identical to inversions of elliptic billiard N-periodics with respect to a focus-centered circle, an important corollary is that (iii) elliptic billiard focus-inversive N-gons have constant perimeter. Interestingly, these also conserve their sum of cosines (except for the N=4 case).

The talented Mr. Inversive Triangle in the elliptic billiard

Inverting the vertices of elliptic billiard N-periodics with respect to a circle centered on one focus yields a new “focus-inversive” family inscribed in Pascal’s limaçon. The following are some of its surprising invariants: (i) perimeter, (ii) sum of cosines, and (iii) sum of distances from inversion center (the focus) to vertices. We prove these for the \(N=3\) case, showing that this family (a) has a stationary Gergonne point, (b) is a 3-periodic family of a second, rigidly moving elliptic billiard, and (c) the loci of incenter, barycenter, circumcenter, orthocenter, nine-point center, and a great many other triangle centers are circles.

Algorithm 1013

A continuous spectral algorithm and computer routines in the R programming environment that enable the simulation of second-order stationary and intrinsic (i.e., with second-order stationary increments or generalized increments) vector Gaussian random fields in Euclidean spaces are presented. The simulation is obtained by computing a weighted sum of cosine and sine waves, with weights that depend on the matrix-valued spectral density associated with the spatial correlation structure of the random field to simulate. The computational cost is proportional to the number of locations targeted for simulation, below that of sequential, matrix decomposition and discrete spectral algorithms. Also, the implementation is versatile, as there is no restriction on the number of vector components, workspace dimension, number and geometrical configuration of the target locations. The computer routines are illustrated with synthetic examples and statistical testing is proposed to check the normality of the distribution of the simulated random field or of its generalized increments. A by-product of this work is a spectral representation of spherical, cubic, penta, Askey, J-Bessel, Cauchy, Laguerre, hypergeometric, iterated exponential, gamma, and stable covariance models in the d -dimensional Euclidean space.

On Zeros of Sums of Cosines

It is shown that there exist arbitrarily large natural numbers $$N$$ and distinct nonnegative integers $$n_1,\dots,n_N$$ for which the number of zeros on $$[-\pi,\pi)$$ of the trigonometric polynomial $$\sum_{j=1}^N \cos(n_j t)$$ is $$O(N^{2/3}\log^{2/3} N)$$ .

Billiards in ellipses revisited

We prove some recent experimental observations of Dan Reznik concerning periodic billiard orbits in ellipses. For example, the sum of cosines of the angles of a periodic billiard polygon remains constant in the 1-parameter family of such polygons (that exist due to the Poncelet porism). In our proofs, we use geometric and complex analytic methods.

Reduction of a noise influence with sinusoidal signal meters

The purpose of the article is to suggest digital processing methods of sinusoidal signal measurements data as well as compensation of sinusoidal noises. This kind of task arises, for example, in implementation of feeder control of electrical power networks, in which an additional ELF voltage source is included between the controlled network and the ground. A leakage current measuring device for this frequency should be installed on each feeder. The suggestion is to use averaging of the sinusoidal signal over several periods to reduce the effect of random interference. For this purpose, several periods of the signal are stored in digital form, measured after equidistant moments of time. Time sampling frequency is selected at least an order of magnitude higher than the frequency of the signal being measured. Then the values “bound” to a particular signal phase are averaged. In addition, it was proposed to use the approximation of the obtained points by a trigonometric polynomial. A series of averaged values corresponding to one period are approximated for this purpose by the sum of the constant component as well as the cosine and sinusoidal components. For approximation, the least squares method is used, which provides the minimum discrepancy between the measured values and the analytical function. The measurement result is presented in a complex form, i.e. as a sum of cosine (actual) and sinusoidal (imaginary) components. As a result, it is possible to isolate oneself from noise and measure the sinusoidal signal with high precision.

Peningkatan Hasil Belajar Siswa SMAN 2 Palembang melalui Model Discovery Learning pada Materi Trigonometri

The research conducted was classroom action research to improve the learning outcomes of students of class XI MIPA 6 Senior High School 2 Palembang. Improved learning outcomes are carried out by using the discovery learning model during the learning process. As for this research, using a subject consisting of 35 students. The subject used by researchers in this study is the formula of the sum and difference of sine, cosine, and tangent of two angles. In the first cycle consisted of 3 meetings and Cycle II consisted of 2 meetings. This study uses instruments in the form of test results of learning and observation. Tests of learning outcomes are used by researchers as a way to find out the achievements of students from each cycle and observations are used by researchers to collect data during the learning process takes place. Quantitative and descriptive data analysis was used in this study. Based on the results of the study, it was found that the average value of students of class XI MIPA 6 on the first cycle test was 63.00 and increased on the second cycle test to 83.69. The results of this study indicate that learning using the discovery learning model can improve students' mathematics learning outcomes, i.e. 74.28% of students have achieved indicators of success in the second cycle of research.

Direct Visual Servoing in the Frequency Domain

In this letter, we propose an original approach to extend direct visual servoing to the frequency domain. Whereas most of visual servoing approaches relied on the geometric features, recent works have highlighted the importance of taking into account the photometric information of the entire images. This leads to direct visual servoing (DVS) approaches. In this letter we propose no longer to consider the image itself in the spatial domain but its transformation in the frequency domain. The idea is to consider the Discrete Cosine Transform (DCT) which allows to represent the image in the frequency domain in terms of a sum of cosine functions that oscillate at various frequencies. This leads to a new set of coordinates in a new precomputed orthogonal basis, the coefficients of the DCT. We propose to use these coefficients as the visual features that are then considered in a visual servoing control law. We then exhibit the analytical formulation of the interaction matrix related to these coefficients. Experimental results validate our approach.

On Zeros of Sums of Cosines

Показано, что найдутся сколь угодно большие натуральные числа $N$ и различные неотрицательные целые числа $n_1,…,n_N$, для которых число нулей на $[-\pi,\pi)$ тригонометрического полинома $\sum_{j=1}^N \cos(n_j t)$ есть $O(N^{2/3}\log^{2/3} N)$. Библиография: 6 названий.