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This article investigates certain q-analogue of the fractional Agarwal integral operator and its application to a class of polynomials and a series of functions. By utilizing various types of q-Bessel functions, the fractional q-Agarwal integral has been discussed and formulated in a series expression form involving q-shifted factorials and gamma functions. Moreover, certain results and applications of the q-Bessel theory are reported by establishing suitable forms of the fractional integral. Furthermore, the fractional Agarwal integral has been evaluated for some multiple power series formulas. Meanwhile, some desirable results involving q-generating Heine's series of the first type are provided. Over and above, certain conclusions associated with various exponential, hyperbolic sine and cosine functions are analysed. [ABSTRACT FROM AUTHOR]

In the filed of remote sensing, arbitrary-oriented object detection methods has gained great attention, benefiting from the accurate detection ability of dense objects. However, the existing methods, which are designed based on ResNet, are not fast enough for real-time application. To solve this problem, our paper proposes a new fast arbitrary-oriented object detection methods based on YOLOX. First, a new head for rotational box prediction is proposed, in which a new branch is designed to extract the angle information through weighted averaging from different angles. Then, a new loss function with sine function is designed to avoid the boundary problem for rotational box prediction. The advantage of this loss is that the value of loss is also periodic which corresponds exactly to the periodicity of rotational box. Experiment results verify that the detection speed of the proposed method is fastest in comparison with the state-of-the-art methods, while maintaining the competitive detection accuracy. Code is available at . [ABSTRACT FROM AUTHOR]

The paper has two main objectives. Firstly, it explores the properties of hyperbolic cosine and hyperbolic sine functions by using Volkenborn and the fermionic p-adic integrals, respectively. It also investigates interesting identities involving Bernoulli and degenerate Bernoulli numbers from differential equations with their solutions. Secondly, it introduces a random variable whose mass function is given in terms of Euler polynomials and find some expressions for the expectation of the random variable. [ABSTRACT FROM AUTHOR]

Generalized sine functions, primarily viewed as real-valued functions, are studied herein as complex analytic functions. Their natural domains of analyticity are identified and their mapping properties are detailed. Some implications of this theory for calculus and geometry are explored. [ABSTRACT FROM AUTHOR]

We discuss connectivity properties of Julia sets of the parameterized Dixon elliptic functions. Our main result is that the connectivity locus of the parameterized Dixon sine function is the exterior of the open unit disc, and the Julia set is Cantor in the open unit disc minus the origin. We prove the parameterized Dixon cosine function also exhibits a fundamental dichotomy in the connectivity of the Julia set. The Julia and Fatou sets exhibit a variety of rotational symmetries, and no Herman rings exist for any function in either family. [ABSTRACT FROM AUTHOR]

In this paper a practical power system planning and control strategy based on a new adaptive sine cosine arithmetic optimization algorithm (ASC_AOA) is proposed to enhance the technical performances of radial distribution (RD) systems. The main objective considered in this study is to optimize the location and the size of shunt compensators based STATCOM devices and multi distributed generation to reduce the total power losses, and to maximize the loading margin stability of practical RD systems. As well confirmed in many recent researches, the success of metaheuristic algorithms is related to the interactivity between intensification and diversification stages. In this study, an adaptive process based on sine and cosine functions is incorporated within the standard AOA to guide the search process toward the best solution. The particularity of the proposed variant (ASC_AOA) has been validated on many benchmark functions, and also applied for optimal automation control of multi DGs and shunt compensators based STATCOM Controllers to enhance the performances of various types of RD systems, such as the 33-bus, the 69-bus, and 85-bus. Obtained results are compared to many recent optimization methods. It is confirmed that the proposed variant achieves the best solution in all of the cases studies elaborated. The proposed variant seems to be a competitive technique and an alternative tool for solving various combinatorial planning and control problems of modern RD systems. [ABSTRACT FROM AUTHOR]

In the textile industry, selecting the optimal fabric is crucial in the design and production process. The multiple-criteria decision-making (MCDM) technique has been applied to address fabric selection problems. Because the sine trigonometry function maintains the periodicity and symmetry of the origin in nature, it satisfies the preferences of experts over multiple parameters. Considering the advantages of the sine trigonometry function, we introduce sine trigonometry Pythagorean fuzzy sets (ST-PFSs) and weighted averaging (ST-PFWA) and geometric (ST-PFWG) aggregation operators (AOs). Additionally, we propose a novel method based on ST-PFWA and ST-PFWG AOs to solve two different types of fabric selection MCDM problems. The first problem is provided by Pythagorean fuzzy numbers (PFNs), and we apply the proposed method to address the issue and conduct a comparative analysis with other PFS AOs. The second problem is offered with crisp values, and we apply a PFS linguistic term transform system to convert the values into PFNs. Then, we apply ST-PFWA and ST-PFWG AOs to settle the problem and perform a comparative analysis among different PFS linguistic terms. The complete ranking results illustrate that our proposed methodology is more efficient and reliable than previous approaches and can be used in other textile fields. [ABSTRACT FROM AUTHOR]

In this paper, a novel technique to design Multiband Pass Integrator (MPI) will be presented. This novel technique involves the implementation of the MPI using the approximation of equations abstracted from Z-Transform using sine and cosine function, F n ω and H n ω . The weights of the integrator are made dependable of the same sine and cosine function. After that, a model is predicted from the integrator weights having minimum error-related ideal integrator. Then, using this model, future values of the weight coefficients of integrator are forecasted for higher frequencies. The relative error for F n ω and H n ω are found to be in the range of 0.09 ≤ E R ≤ 0.08 and 0.082 ≤ E R ≤ 0.078. [ABSTRACT FROM AUTHOR]

In the medical industry, digital images are frequently used to diagnose a variety of diseases. These images comprise of lot of secret information regarding health of the patient. Image encryption is used to protect this secret information by changing the image into unrecognizable form. This research paper proposes a novel chaos map and a pseudo random key. The method comprises of three phases to encrypt the medical images. Firstly, a novel pseudorandom key is generated using the XOR operations and left rotate operations. Secondly, a novel chaos map is created using the cosine and sine function. Finally, pseudorandom key and chaos map are used to construct the cipher image. The novel chaotic map's robustness has been assessed according to the Shannon Entropy, Lyapunov Exponent, and Bifurcation Diagram. The proposed mathematically transformed hybrid chaos map provides a wider chaotic range and shows more complex behavior than the various existing chaotic maps. [ABSTRACT FROM AUTHOR]

We prove that, for p ∈ (1 , ∞) and β ∈ R , the function x ↦ β − log ⁡ 1 − x p K p (x p) is strictly concave on (0 , 1) if and only if β ≥ λ (p) := 2 p (p 2 − 2 p + 2) (p − 1) (2 p 2 − 3 p + 3) , where K p represents the generalized complete p-elliptic integrals of the first kind defined by K p (r) := ∫ 0 π p / 2 d θ (1 − r p sin p p ⁡ θ) 1 − 1 / p , where π p := 2 p B (1 / p , 1 − 1 / p) , π 2 = π , and sin p is the generalized sine function, with sin 2 = sin. This extends the recently obtained corresponding result for the case that p = 2. We then apply this concavity property to obtain the following functional inequality (likewise extending the previously established result for the case that p = 2): For all r ∈ (0 , 1) , we have 2 β π p + 1 < β − log ⁡ (r ′) K p (r) + β − log ⁡ (r) K p (r ′) ≤ 2 β + 2 log ⁡ (2 p) K p (1 / 2 p) , where r ′ = 1 − r p p , p ∈ (1 , ∞) , and β ≥ λ (p). Both bounds are sharp. The sign of equality holds if and only if r = 1 / 2 p . [ABSTRACT FROM AUTHOR]

This paper proposes a nonlinear robust track keeping control algorithm for energy efficiency from the perspective of ship motion control, based on the great significance of the twenty-first Century Maritime Silk Road. Replacing the course error e with the sin ⁡ (ω e) driven by the sine function in course keeping loop to serve as the input of the controller, the simulation results tested on the ship autopilot simulation platform show that the controller output (maximum rudder angle) decreases, and the control effect of track keeping is better than the previous one in aspects of energy efficiency and rudder shake inhibition. Otherwise, the proposed nonlinear track keeping control algorithm still works in relatively severe sea states. Substituting the nonlinear feedback for the linear feedback has the generality of control effect improving. [ABSTRACT FROM AUTHOR]

Hyperspectral image (HSI) has become one of the most important remote sensing techniques for object interpretation by its abundant band information. As the data dimension increases, band selection technique is utilized to achieve the highest possible classification accuracy with fewer bands. Essentially, it is considered as an NP-hard problem, which is a non-deterministic problem within polynomial complexity and difficult to achieve a satisfactory solution using traditional search methods. Sine cosine algorithm (SCA) is a recently developed swarm intelligence algorithm based on the calculation of sine and cosine functions. To determine the parameters setting in SCA, Lévy flight technique is employed to improve the exploitation phase of the algorithm. In the paper, a new band selection method on the basis of SCA with a Lévy flight is proposed, and an alternative distribution is utilized to decrease the band dimension of HSI. In addition, crossover operation is conducted to enhance the optimal bits of each individual, and an evaluation criterion for assessing the performance of band selection is defined, allowing the classification accuracy and selected number of bands to be as balanced as possible. Experimental results demonstrate that the proposed band selection method is superior to other state-of-the-art approaches in terms of band subsets that achieve higher classification accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of (probability) distributions, based on a cosine-sine transformation, obtained by compounding a baseline distribution with cosine and sine functions. Some of its properties are explored. A special focus is given to a particular cosine-sine transformation using the exponential distribution as baseline. Estimations of parameters of a particular cosine-sine exponential distribution are performed via the maximum likelihood estimation method. A simulation study investigates the performances of these estimates. Applications are given for four real data sets, showing a better fit in comparison to some existing distributions based on some goodness-of-fit tests. [ABSTRACT FROM AUTHOR]

This paper focuses on chaotification problems for first-order partial difference equations. Two chaotification schemes of the difference equations via sine functions are established, and all the controlled systems are proved to be chaotic in the sense of both Devaney and Li-Yorke by applying the coupled-expansion theory of general discrete dynamical systems. At the end, one illustrative example is provided. [ABSTRACT FROM AUTHOR]

A high-accuracy resolver-to-linear signal converter for the measurement of angular displacement is proposed in this paper. The proposed converter comprises two sections: a demodulator and linear shaper. In the first section, the demodulator makes use of the sample-and-hold circuit (SHC) to sample the peak amplitude of the resolver signal. The control signal of the SHC is provided from the resolver signals instead of the excitation signal used in traditional approaches. The proposed demodulator requires no analogue multiplier and low-pass filter. Therefore, the fast response time of the proposed demodulator is achieved. In the second section, the linear shaper consists of the inverse-sine function scheme together with a switched-gain amplifier to produce the linear signal proportional to the shaft angle. The hyperbolic tangent characteristic of the operational transconductance amplifier is utilised to realise the inverse-sine function scheme. The proposed technique requires one phase of the resolver signal to obtain the linear signal. Therefore, the position error caused by amplitude imbalance between the two resolver signals is avoided. The performances of the proposed converter are discussed in detail and demonstrated by an experimental implementation using commercial devices. The experimental results show that the maximum relative error and response time for the excitation frequency of 3 kHz are measured as 0.06% and 0.11 ms, respectively. [ABSTRACT FROM AUTHOR]

In this note, we present an elementary proof of the classical formula for the zeta function at the positive even integers. This proof could also have been given by Euler in the 18th century since it uses many of his (sneaky) analytic techniques, namely the infinite product representation of the sine function. [ABSTRACT FROM AUTHOR]

In this paper, a scale-corrected minimal skew simplex sampling unscented Kalman filter (UKF) algorithm for the permanent magnet (PM) brushless DC motors (BLDCM) sensorless control has been studied to cancel the position sensor by the use of a systematical and analytical approach. Compared with the general UKF, the sampling method with the least Sigma points called minimal skew simplex sampling is adopted to reduce amount of computation and increase the estimation precision. Moreover, the scale-corrected strategy is introduced into the minimal skew simplex sampling UKF to overcome the nonlocal effects. On the other hand, for more easily calculating the value of back-EMF, the shape function of counter electromotive force is approximated by a series of sine and cosine functions based on the law of Fourier series. The purpose of the problem addressed is, by the method of scale-corrected minimal skew simplex sampling UKF, to properly estimate the rotor speed and position without installing encoders. Finally, the effectiveness of the proposed sensorless control technique is verified by simulation in MATLAB/Simulink. [ABSTRACT FROM AUTHOR]

We give expressions for products of sinusoids as sums of sinusoids. [ABSTRACT FROM AUTHOR]

Daily college drinking data often have highly skewed distributions with many zeroes and a rising and falling pattern of use across the week. Alcohol researchers have typically relied on statistical models with dummy variables for either the weekend or all days of the week to handle weekly patterns of use. However, weekend versus weekday categorizations may be too simplistic and saturated dummy variable models too unwieldy, particularly when covariates of weekly patterns are included. In the present study we evaluate the feasibility of cyclical (sine and cosine) covariates in a multilevel hurdle count model for evaluating daily college alcohol use data. Results showed that the cyclical parameterization provided a more parsimonious approach than multiple dummy variables. The number of drinks when drinking had a smoothly rising and falling pattern that was reasonably approximated by cyclical terms, but a saturated set of dummy variables was a better model for the probability of any drinking. Combining cyclical terms and multilevel hurdle models is a useful addition to the data analyst toolkit when modeling longitudinal drinking with high zero counts. However, drinking patterns were not perfectly sinusoidal in the current application, highlighting the need to consider multiple models and carefully evaluate model fit. [ABSTRACT FROM PUBLISHER]

The student who has learned the binomial expansion and the formula for exp (ix), is shown how to obtain simple expansions for sin nx, cos nx, sinh nx, cosh nx, sin αx and cos αx for n any positive integer and α any real number. [ABSTRACT FROM AUTHOR]

In this paper, we consider the problem of estimating the number of components of a superimposed nonlinear sinusoids model of a signal in the presence of additive noise. We propose and provide a detailed empirical comparison of robust methods for estimation of the number of components. The proposed methods, which are robust modifications of the commonly used information theoretic criteria, are based on various M-estimator approaches and are robust with respect to outliers present in the data and heavy-tailed noise. The proposed methods are compared with the usual non-robust methods through extensive simulations under varied model scenarios. We also present real signal analysis of two speech signals to show the usefulness of the proposed methodology. [ABSTRACT FROM AUTHOR]

The article offers information on sine formula and its use in a complex simple Lie algebra. Topics include introduction to the meaning of the symbols appearing in the formula such as the Cartan matrix, the rank, the exponent and the Coxeter number of a complex simple Lie algebra, using the root system such that each vector in the set of vectors used is a linear combination of these three vectors with integer coefficient, and an example that considers a Coxeter system with a graph.

A two-stage autocorrelation method is proposed for frequency estimation of a complex sinusoidal signal in complex additive white Gaussian noise. In the first stage, a function about signal frequency is constructed via an autocorrelation procedure. By the Least-Square (LS) principle, the signal frequency is estimated from the function. In the second stage, the estimated error of signal frequency in the first stage is defined. Applying the Taylor series, a function about the estimated error of signal frequency is conducted from the autocorrelation function. Applying the LS estimation once more, the estimate of estimated error of frequency in the first stage is obtained. Then the fine estimate of signal frequency is obtained. Simulation results show that the performance of proposed method can approach the Cramer–Rao lower bound in the whole frequency range. [ABSTRACT FROM AUTHOR]

The article explores the use of differential equations to define exponential and trigonometric functions. It suggests that some functions can be defined through their governing differential equations. Other topics discussed are the basic theorems for differential equations and the definition and properties of the exponential functions, sines and cosines.

The article focuses on the ways of defining the sine function in trigonometry, which finds applications in mathematics and physics. It notes the geometrical and differential forms in which sine and cosine functions appear as solutions of harmonic differential equations. It puts forth the use of variables to represent arc's length in a unit circle by considering movement of object around the circle having constant speed.

An application of multipole moments for the determination of toroidal plasma column shift is presented. First, we present analytical details for using this technique. Then, the principle of different models based on this technique for design and fabrication of a six coils will be presented: four modified Rogowski coils (two cosine coils and two sine coils) and two saddle coils (saddle sine coil and saddle cosine coil). As continuous measurements of magnetic field distribution around the toroidal plasma can be made using these coils, this technique is a good method for the determination of toroidal plasma column shift. [ABSTRACT FROM PUBLISHER]

This article studies the iterative behaviour of a quasiregular mappingthat is an analogue of a sine function. We prove that the periodic points ofSform a dense subset of. We also show that the Julia set of this map isin the sense that the forward orbit underSof any non-empty open set is the whole space. The mapSwas constructed by Bergweiler and Eremenko (Ann. Acad. Sci. Fenn. Math. 36 (2011), pp. 165–175) who proved that the escaping setis also dense in. [ABSTRACT FROM PUBLISHER]

As students explore the technological tools available to them for learning mathematics, some will eventually discover what happens when a function button is repeatedly pressed on a calculator. We explore several examples of this, presenting tabular and graphical results for the square root, natural logarithm and sine and cosine functions. Observed behaviour is proven and then discussed in the context of fixed points. [ABSTRACT FROM PUBLISHER]

Two new sine models including the transverse normal deformation are presented for the analysis of laminated plates with a cross-ply layout in cylindrical bending. A layer refinement for each component of the displacements is considered. The transverse shear strain is obtained by using a cosine function avoiding the use of shear correction factors. This kinematics accounts for the interlaminar continuity conditions on the interfaces between the layers for the transverse normal and shear stresses, and the boundary conditions on the upper and lower surfaces of the plate. Note that the number of unknowns is independent of the number of layers. The main topics are focused on the evaluation of the transverse normal and shear stresses by the constitutive relation. Mechanical and thermomechanical results within a closed-form solution technique are compared to elasticity reference solutions for thin and very thick plates. [ABSTRACT FROM AUTHOR]

We construct the exact finite difference equation discretizations for the nonlinear differential equations whose solutions are the Jacobi cosine and sine functions. Our derivations clarify and extend previous work done on this topic. [ABSTRACT FROM AUTHOR]

The article focuses on the curvature functions, both power and sine functions, resulting in integrals not being assessed in terms of elementary functions. It refers to curves such as Euler spiral and clothoid out of which clothoid is useful for building railroads and freeways. It mentions that each spiral comprises two parts which are centrally symmetric in terms of origin for odd values and vertical axis for even values.

Motivated by examples in protein bioinformatics, we study a mixture model of multivariate angular distributions. The distribution treated here (multivariate sine distribution) is a multivariate extension of the well-known von Mises distribution on the circle. The density of the sine distribution has an intractable normalizing constant and here we propose to replace it in the concentrated case by a simple approximation. We study the EM algorithm for this distribution and apply it to a practical example from protein bioinformatics. [ABSTRACT FROM PUBLISHER]

The article discusses sinc integral evaluation. It illustrates mathematical solutions and computations of the three sinc functions such as sine, cosine, and tangent. It identifies the solution that involves the principal value (PV) of the integral using the formula for same partial fraction expansion.

An L 2-optimal identification method is extended to cope with MIMO errors-in-variables (EIV) model estimation based on a geometrical interpretation for the v-gap metric. The L 2-optimal approximate models are composed of system and noise models and characterised by a normalised right graph symbol (NRGS) and its complementary inner factor (CIF), respectively. This metric can be evaluated as the supreme of sine values of the maximal principal angles between NRGS frequency responses of two concerned models. In order to make full use of the angular cosine formula for complex vectors to reduce computational loads, a CIF of the NRGS of the perturbed model is introduced and thus, the system parameter optimisation can be efficiently solved by sequential quadratic programming methods. With the estimated system model, the associated noise model can be built by right multiplication of an inner matrix. Finally, a simulation example demonstrates the effectiveness of the proposed identification method. [ABSTRACT FROM AUTHOR]

The article focuses on a mathematical proof of mathematician Leonhard Euler's infinite product for sine product formula. It is stated that Gamma function and elementary probability theory prove product formula for the sine function. Some deductions or inferences of sine formula are also presented in the article. Validity of sine product formula is for the whole complex plane.

The article discusses the theorems on Rolling Sinusoidal Spirals given by mathematicians Tom A. Apostol and Mamikon A. Mnatsakanian. The theorems describe the relationship between the areas of the cycloidal sector with that of corresponding segment cut by a circle rolling on a line. The result is generalized by comparing the ratios of the areas of the sinusoidal and logarithmic spirals.

By using an identity relating to Bernoulli's numbers and power series expansions of cotangent function and logarithms of functions involving sine function, cosine function and tangent function, four inequalities involving cotangent function, sine function, secant function and tangent function are established. [ABSTRACT FROM AUTHOR]

Proposes an approach to closed form expressions for the birth trajectory produced by a regime of changing vital rates. Consideration for the exponentiated sinusoidal net maternity function; Relevance of the analytical approach for the analyses of interacting populations.

The article offers information on the differentiation formulae for sine, tangent, and inverse tangent. Topics discussed include promoting the Hartig's approach for differentiation formulae, using the symmetric derivative to make Hartig's argument correspond to an actual triangle and proving derivative formula for sine using a geometric argument.

A diagram proving a sine result for the angles of a triangle is presented.

The article presents a mathematical equation for the visual proof of sine sum identity using sine and cos entities.