Showing papers on "Inverse trigonometric functions published in 2016"

CRC Standard Curves and Surfaces with Mathematica

Abstract: Introduction Concept of a Curve Concept of a Surface Coordinate Systems Qualitative Properties of Curves and Surfaces Classification of Curves and Surfaces Basic Curve and Surface Operations Method of Presentation References Algebraic Functions Plotting Information for This Chapter Functions with xn/m Functions with xn and (a + bx)m Functions with (a2 + x 2) and xm Functions with (a2 x2) and xm Functions with (a3 + x3) and xm Functions with (a3 x3) and xm Functions with (a4 + x4) and xm Functions with (a4 x4) and xm Functions with a + bx and xm Functions with a2 x2 and xm Functions with x2 a2 and xm Functions with a2 + x2 and xm Miscellaneous Functions Functions Expressible in Polar Coordinates Functions Expressed Parametrically Transcendental Functions Plotting Information for This Chapter Functions with sinn (2 ax) and cosm(2 bx)(n,m integers) Functions with 1 +/- sinn (2 ax) and 1 +/-} cosm (2 bx) Functions with c sinn (ax) + d cosm (bx) Functions of More Complicated Arguments Inverse Trigonometric Functions Logarithmic Functions Exponential Functions Hyperbolic Functions Inverse Hyperbolic Functions Trigonometric Combined with Exponential Functions Trigonometric Functions Combined with Powers of x Logarithmic Functions Combined with Powers of x Exponential Functions Combined with Powers of x Hyperbolic Functions Combined with Powers of x Combined Trigonometric Functions, Exponential Functions, and Powers of x Miscellaneous Functions Functions Expressible in Polar Coordinates Functions Expressible Parametrically Polynomial Sets Plotting Information for This Chapter Orthogonal Polynomials Nonorthogonal Polynomials References Special Functions in Mathematical Physics Plotting Information for This Chapter Exponential and Related Integrals Sine and Cosine Integrals Gamma and Related Functions Error Functions Fresnel Integrals Legendre Functions Bessel Functions Modified Bessel Functions Kelvin Functions Spherical Bessel Functions Modified Spherical Bessel Functions Airy Functions Riemann Functions Parabolic Cylinder Functions Elliptic Integrals Jacobi Elliptic Functions References Green's Functions and Harmonic Functions Plotting Information for This Chapter Green's Function for the Poisson Equation Green's Function for the Wave Equation Green's Function for the Diffusion Equation Green's Function for the Helmholtz Equation Miscellaneous Green's Functions Harmonic Functions: Solutions to Laplace's Equation References Special Functions in Probability and Statistics Plotting Information for This Chapter Discrete Probability Densities Continuous Probability Densities Sampling Distributions Laplace Transforms Plotting Information for This Chapter Elementary Functions Algebraic Functions Exponential Functions Trigonometric Functions References Nondifferentiable and Discontinuous Functions Plotting Information for This Chapter Functions with a Finite Number of Discontinuities Functions with an Infinite Number of Discontinuities Functions with a Finite Number of Discontinuities in First Derivative Functions with an Infinite Number of Discontinuities in First Derivative Random Processes Plotting Information for This Chapter Elementary Random Processes General Linear Processes Integrated Processes Fractal Processes Poisson Processes References Polygons Plotting Information for This Chapter Polygons with Equal Sides Irregular Triangles Irregular Quadrilaterals Polyiamonds Polyominoes Polyhexes Miscellaneous Polygons Three-Dimensional Curves Plotting Information for This Chapter Helical Curves Sine Waves in Three Dimensions Miscellaneous 3-D Curves Knots Links References Algebraic Surfaces Plotting Information for This Chapter Functions with ax + by Functions with x2/a2 +/- y2/b2 Functions with x2/a2 + y2/b2 +/-c2)1/2 Functions with x3/a3 +/- y3/b3 Functions with x4/a4 +/- y4/b4 Miscellaneous Functions Miscellaneous Functions Expressed Parametrically Transcendental Surfaces Plotting Information for This Chapter Trigonometric Functions Logarithmic Functions Exponential Functions Trigonometric and Exponential Functions Combined Surface Spherical Harmonics Complex Variable Surfaces Plotting Information for This Chapter Algebraic Functions Transcendental Functions Minimal Surfaces Plotting Information for This Chapter Elementary Minimal Surfaces Complex Minimal Surfaces References Regular and Semi-Regular Solids with Edges Plotting Information for This Chapter Platonic Solids Archimedean Solids Duals of Platonic Solids Stellated (Star) Polyhedra References Irregular and Miscellaneous Solids Plotting Information for This Chapter Irregular Polyhedra Miscellaneous Closed Surfaces with Edges Index

Full-Color Computational Imaging with Single-Pixel Detectors Based on a 2D Discrete Cosine Transform

Abstract: We propose and demonstrate a computational imaging technique that uses structured illumination based on a two-dimensional discrete cosine transform to perform imaging with a single-pixel detector. A scene is illuminated by a projector with two sets of orthogonal patterns, then by applying an inverse cosine transform to the spectra obtained from the single-pixel detector a full-color image is retrieved. This technique can retrieve an image from sub-Nyquist measurements, and the background noise is easily canceled to give excellent image quality. Moreover, the experimental setup is very simple.

Breaking the Limits -- Redefining the Instantaneous Frequency

Abstract: The Carson and Fry (1937) introduced the concept of variable frequency as a generalization of the constant frequency. The instantaneous frequency (IF) is the time derivative of the instantaneous phase and it is well-defined only when this derivative is positive. If this derivative is negative, the IF creates problem because it does not provide any physical significance. This study proposes a mathematical solution and eliminate this problem by redefining the IF such that it is valid for all monocomponent and multicomponent signals which can be nonlinear and nonstationary in nature. This is achieved by using the property of the multivalued inverse tangent function. The efforts and understanding of all the methods based on the IF would improve significantly by using this proposed definition of the IF. We also demonstrate that the decomposition of a signal, using zero-phase filtering based on the well established Fourier and filter theory, into a set of desired frequency bands with proposed IF produces accurate time-frequency-energy (TFE) distribution that reveals true nature of signal. Simulation results demonstrate the efficacy of the proposed IF that makes zero-phase filter based decomposition most powerful, for the TFE analysis of a signal, as compared to other existing methods in the literature.

Best Possible Bounds for Yang Mean Using Generalized Logarithmic Mean

Abstract: We prove that the double inequality holds for all with if and only if and and find several sharp inequalities involving the trigonometric, hyperbolic, and inverse trigonometric functions, where is the unique solution of the equation on the interval , , and , and are the Yang, and th generalized logarithmic means of and , respectively.

Shafer-type inequalities for inverse trigonometric functions and Gauss lemniscate functions

Abstract: In this paper, we present Shafer-type inequalities for inverse trigonometric functions and Gauss lemniscate functions.

Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms

Abstract: Theoretical and computational aspects of matrix inverse trigonometric and inverse hyperbolic functions are studied. Conditions for existence are given, all possible values are characterized, and the principal values acos, asin, acosh, and asinh are defined and shown to be unique primary matrix functions. Various functional identities are derived, some of which are new even in the scalar case, with care taken to specify precisely the choices of signs and branches. New results include a “round trip” formula that relates $acos(\cos A)$ to $A$ and similar formulas for the other inverse functions. Key tools used in the derivations are the matrix unwinding function and the matrix sign function. A new inverse scaling and squaring type algorithm employing a Schur decomposition and variable-degree Pade approximation is derived for computing acos, and it is shown how it can also be used to compute asin, acosh, and asinh. In numerical experiments the algorithm is found to behave in a forward stable fashion and to be...

Approximation of Some Classes of Set-Valued Periodic Functions by Generalized Trigonometric Polynomials

Abstract: We generalize some known results on the best, best linear, and best one-sided approximations by trigonometric polynomials from the classes of 2π-periodic functions represented in the form of convolutions to the case of classes of set-valued functions.

Fixed point implementation of trigonometric function using Taylor's series and error characterization

Abstract: The aim of this paper is to analyze fixed point implementation of functions designed for signal processing algorithms. In this paper, basic building block function is analyzed for errors while implementing the algorithm using fixed point technique. To streamline the process of converting the floating point to fixed point, Model Based Design(MBD) concept is adopted where models of the basic functions are designed and tested in MATLAB (Matrix Laboratory) environment before converting to fixed point. Hardware Description Language (HDL) coder is used to convert the model in VHSIC Hardware Description Language (VHDL) code to implement in Field Programmable Gate Array (FPGA). The work proposed in this paper is abstract function which is implemented using Taylor's approximation series as some functions are not supported by HDL code generation tool for inverse trigonometric functions like inverse sine (ARCSINE), inverse tangent (ARCTANGENT). These functions are calculated with different number of iterations in MATLAB and using HDL coder. The script is successfully converted to fixed point VHDL code and the error between floating and fixed point is calculated which is presented in this paper.

Non-parametric smoothing for gradient methods in non-differentiable optimization problems

Abstract: In this article, a method has been established for optimizing multivariate nonlinear discontinuous cost functions having multiple simple kinks in their domains of definition, by applying simple non-parametric inverse trigonometric functions and combinations thereof as smoothing agents. The original function is locally replaced by these smoothing agents at the points of jump discontinuity, while retaining the global structure of the original objective function. The non-parametric smoothing function is exact and devoid of complicated special functions or integral approximations and, hence, the resulting optimization algorithm is relatively simpler and faster. The absence of the parameter ensures little ill-conditioning effects in subsequent calculations. Relevant properties of the smoothing function are developed analytically and the shape of the resulting modified objective functions are illustrated with appropriate numerical simulations. Conjugate gradient and Broyden family of methods (DFP/BFGS) were employed to minimize the modified functions; although Levenberg Marquardt and gradient descent methods were also used occasionally to confirm the universal applicability of the proposed method. Over fifty problems were successfully minimized (maximized) and results for some of them are tabulated before the concluding remarks of the article. Constrained optimization of discontinuous functions with such smoothing agents is an active research within the working group.

Definition of weber-ampere characteristics of electrical devices harmonic balance method in solving inverse problems of diagnostics

Abstract: The authors have developed a method of determining the weber-ampere characteristics of the electrical alternating current devices based on the method of harmonic balance. However, the use of the polynomial approximation as a power function requires a large number of coefficients, which complicates the task. An arc tangent function has been selected as an approximating weber-ampere characteristic expression in this article, which allows using less power compared with the approximate coefficients of the polynomial, which simplifies the procedure of calculation and reduces the time spent. It is found that the harmonic balance method may be used with this type of approximating expressions to determine the weber-ampere characteristic electrical AC devices with an accuracy of no more than 5%.

On a more accurate half-discrete Hardy-Hilbert-type inequality related to the kernel of arc tangent function.

Abstract: By means of weight functions and Hermite–Hadamard’s inequality, and introducing a discrete interval variable, a more accurate half-discrete Hardy–Hilbert-type inequality related to the kernel of arc tangent function and a best possible constant factor is given, which is an extension of a published result. The equivalent forms and the operator expressions are also considered.

The Ellis-Gohberg inverse problem for matrix-valued Wiener functions on the line

Abstract: This talk deals with the Ellis-Gohberg inverse problem for matrix-valued Wiener functions on the line, instead of on the circle, as was done in [1] for scalar functions and in [2] for matrixvalued functions. Using elements of mathematical system theory, the problem is reduced to a linear finite matrix equation of which the right hand side is described explicitly in terms of one of the given functions. Necessary and sufficient conditions will be given in order that the problem is solvable and the solution is unique. The results obtained parallel and extend those derived in [2] for Wiener functions on the circle. Special attention is paid to the scalar case and to the case when the given functions are Fourier transforms of functions of finite support. The talk is based on joint work with Freek van Schagen.

Graphs of q-exponentials and q-trigonometric functions

Abstract: Here we will show the behavior of some of q-functions. In particular we plot the q-exponential and the q-trigonometric functions. Since these functions are not generally included as software routines, a Fortran program was necessary to give them.

Calculating Some Integrals of Trigonometric Functions

Abstract: In t his article, we study three types of integrals related with the trigonometric functions . The infinite series forms of the three types of integrals can be obtained using binomial series and integration term by term theorem . On the other hand, we propose some example s to do calculation practically. The research methods adopted in this paper is to find solutions through manual calculations and verify the answers using Maple. Key Words: integrals; trigonometric functions; infinite series forms; binomial series; integration term by term theorem; Maple

Trig-Disc: An innovative device for determination of values of trigonometric functions with single measurement

Abstract: Angles and lengths of a triangle engaged the mathematicians from historical time. The branch of mathematics that resulted from these studies is now known as Trigonometry. The discovery that lengths of a right angled triangle and the angles between them have a definite relationship led to the invention of trigonometric functions. The determination of values of various trigonometric functions like sine, cosine, tangent, cotangent, cosecant and secant for various angles in four quarters of a circle was a challenging task. Various methods and tables were generated for determining these values. Efforts were made to invent various devices for these measurements. There exist various U.S. patents [1-7] based on devices developed to visualize, teach and calculate values of various trigonometric functions. In the present paper, we are reporting a very accurate, economic, simple, and portable device designed for measuring the values of Trigonometric functions. The patent application number for the reported device is 1068/DEL/2015. The device consists of (i) a unit radius circular disc (with 360o angle scale indication on its circumference of 1o accuracy, X- and Y-linear scales markings with accuracy of 0.01 unit and marking of four quadrants of the circle), (ii) a corresponding linear scale with positive marking, (iii) a corresponding linear scale with negative marking and (iv) a blank transparent strip. The device can directly measure all the six trigonometric identities (sine, cosine, tangent, cotangent, cosecant and secant) for any value of angle up to the accuracy of 0.01 units just by measurement of only one parameter on one of the provided scale by appropriately placing it on Trig Disc along with blank linear strip in respective quadrants. Single measurement on a linear scale and no division or calculations of values is main feature of the device.

Remarks on trigonometric functions after Eisenstein

Abstract: We modify the Whittaker-Watson account of the Eisenstein approach to the trigonometric functions, basing these functions independently on the Eisenstein function $\varepsilon_2$.

A Direct Proof of the Integral Formulae for the Inverse Hyperbolic Functions

Abstract: In the same vein as a Classroom Capsule of Arnold Insel on the arctangent, we present a direct geometric derivation of the integral formulae for the inverse hyperbolic functions. We then use these ...

The group inverse of a product

Abstract: In this paper, we characterize the existence and give an expression of the group inverse of a product of two regular elements by means of a ring unit.

High accuracy ESPRIT without left-right ambiguity using an acoustic vector sensor array

Abstract: Through in-pressure sensor array signal processing, the signal arrival angle can be estimated by the estimation of signal parameters via rotational invariance techniques (ESPRIT) if we regard the whole array as two sub-arrays with the same structure. The estimation angles arrived at by ESPRIT have a left-right ambiguity because of the inverse trigonometric function operation. An acoustic vector sensor array (AVSA), which measures particle velocity and pressure at the same time and point of the acoustic field, offers significant potential. The traditional ESPRIT algorithm can also be used for an AVSA with left-right ambiguity. A new algorithm with omnibearing and an high accuracy based on ESPRIT using an AVSA was proposed in this paper. An analytic particle velocity, which composes by acoustic vector sensor x-axis and y-axis velocities, embeds the signal angle in its phase. The signal broadside can be known by the ESPRIT based on analytic particle velocity and pressure sub-arrays. The estimation angle containing broadside information can be used as the driving angle for combined particle velocity. According to the signal broadside, an average cross-variance matrix of pressure and combined velocity was constructed. The signal angle can be estimated by ESPRIT based on this new combined cross-variance matrix. Theoretical analysis and computer simulations show that the proposed algorithm has no left-right ambiguity and also has higher accuracy than the analytic particle velocity algorithm in the case of low signal-to-noise ratio (SNR). The algorithm is based on pressure and particle velocity combined processing, which can suppress interference better than existing algorithms in isotropic noise. The algorithm is also based on the analytic particle velocity ESPRIT, which has no left-right ambiguity. So the new algorithm has better performance than the existing algorithms.

Best m-Term Trigonometric Approximation for Periodic Functions with Low Mixed Smoothness from the Nikol’skii–Besov-Type Classes

Abstract: We establish the exact-order estimates of the best m-term trigonometric approximation for periodic functions of many variables (with low mixed smoothness) from the Nikol’skii–Besov-type classes.

Inequalities on generalized trigonometric functions

Abstract: when 2, p = the p − functions sin p , cos p , tan p become our familiar trigonometric functions. Recently, the generalized trigonometric functions have been studied by many mathematicians from different viewpoints(see [2,4,5,6,7]). In [5,9], the authors gave basic properties of the generalized trigonometric functions. In [6], Klen, Vuorinen and Zhang generalized some classical inequalities for trigonometric functions, such as Mitrinovic-Adamovic’s inequality, Lazarevic’s inequality, Huygens-type inequalities, and Wilker-type inequalities, to the case of generalized functions. 3rd International Conference on Mechatronics and Information Technology (ICMIT 2016)

A Multi-mode Coordinate Rotation Digital Computer (CORDIC)

Abstract: This paper presents a 24-bit fixed-point multi-mode Coordinate Rotation Digital Computer (CORDIC) engine for VLSI implementation of Independent Component Analysis (ICA). Three different modes are integrated for computing sine/cosine, arc tangent and square root to save system resource. We describe the design method for deciding iteration time and fixed-point bits, and present the architecture of a pipelined VLSI implementation. An approximation method is proposed to decrease the data to be pre-stored. The CORDIC engine is designed and implemented with SMIC 65 nm CMOS technology. The performance and computation results of this engine are shown to be very high-accurate and area-efficient.

Interpolation methods for multivalued functions

Abstract: The notion of multivalued functions appeared in the first half of the twentieth century. A multivalued function also known as multi-function, multimap, set-valued function. This is a ”function” that assume two or more values for each point from the domain. These functions are not functions in the classical way because for each point assign a set of points, so there is not a one-to-one correspondence. The term of ”multivalued function” is not correct, but became very popular. Multivalued functions often arise as inverse of functions which are not-injective. For example the inverse of the trigonometric, exponential, power or hyperbolic functions are multivalued functions. Also the indefinite integral can be considered as a multivalued function. These functions appears in many areas, for example in physics in the theory of defects of crystals, for vortices in superfluids and superconductors but also in optimal control theory or game theory in mathematics.

On the basis property of a two-part trigonometric series

Abstract: Conditions are presented under which two-part trigonometric systems arising in mixed type equations form a Riesz basis in the space of Lebesgue square integrable functions. For such systems, biorthogonal systems can be obtained in explicit form. As a result, an integral representation of the solution to the Frankl problem in a special domain can be found. The results are extended to two-part systems of broader functions.

Displacement measurement device, signal processing device, and signal processing method

Abstract: PROBLEM TO BE SOLVED: To provide a displacement measurement device capable of generating as much accurate signal as possible and heightening measurement accuracy, and a signal processing device and a signal processing method that are used therein.SOLUTION: A displacement measurement device 100 includes an encoder unit 10 and a signal processing device 30. The signal processing device 30 is provided with an AD conversion unit 31, a standardization unit 33, an inverse trigonometric function calculation unit 35, a change amount calculation unit 37, and a displacement calculation unit 39. A sinusoidal signal generated by the encoder unit 10 is standardized by the standardization unit 33, and calculation pertaining to the amplitude value of the signal can thereby be simplified. Also, a triangular wave signal including a linear area can be obtained by the inverse trigonometric function calculation of the inverse trigonometric function calculation unit 35. Due to the fact that the change amount calculation unit 37 calculates a change amount per prescribed time of the signal in the linear area, the displacement measurement device 100 can generate an accurate signal and perform high-accuracy measurement.SELECTED DRAWING: Figure 1

Precise digital demodulation for fiber optic interferometer sensors

Abstract: Different methods are used in the interferometer sensors for target signal extraction. Digital technologies provide new opportunities for precise signal detection. We have developed the principle of signal demodulation using an additional harmonic phase modulation and digital signal processing. The principle allows implementation of processing algorithms using different ratios between modulation and discretization frequencies. The expressions allowing calculation of the phase difference using the inverse trigonometric functions were derived. The method was realized in LabVIEW programming environment and was demonstrated for various signal shapes.