Cotangent Function Research Articles

A numerical study on parasitic capillary waves using unsteady conformal mapping

This paper describes fully nonlinear computation of unsteady motion of parasitic capillary waves that appear on the front face of steep gravity waves progressing on water of infinite depth, within the framework of irrotational plane flow. As an alternative to the widely-used boundary integral method with mixed-Eulerian–Lagrangian (MEL) time updating, we focus on a numerical method based on unsteady conformal mapping, which will be hereafter referred to as the unsteady hodograph transformation (UHT) method. In this method, we solve the nonlinear evolution equations to find an unsteady conformal map in a complex plane with which the flow domain is mapped onto the unit disk while the free surface is fixed on the unit circle. The aim of this work is to compare the UHT method with the MEL method and find a more efficient method to compute parasitic capillary waves. From linear stability analysis, it is found that a critical difference between these two methods arises from the kernel of cotangent function in singular integrals, and the UHT method can avoid some numerical instability due to it. Numerical examples demonstrate that the UHT method is more suitable than the MEL method for not only parasitic capillary waves, but also capillary dominated waves. In particular, the UHT method requires no artificial techniques, such as filtering, to control numerical errors, in these examples. In addition, another major difference between the two methods is observed in terms of the clustering property of sample points on the free surface, depending on the restoring force of waves (gravity or surface tension).

Cotangent zeta functions in function fields

We introduce and study an analogue of the cotangent zeta function in the function fields setting. We establish a relation of our newly introduced zeta function to the Apostol–Dedekind sum, and then prove a functional equation for our function. Finally, we compute special values of the cotangent zeta function at quadratic irrationals.

A new DOA-based factor graph geolocation technique for detection of unknown radio wave emitter position using the first-order Taylor series approximation

This paper proposes a new geolocation technique to improve the accuracy of the position estimate of a single unknown (anonymous) radio wave emitter. We consider a factor graph (FG)-based geolocation technique, where the input are the samples of direction-of-arrival (DOA) measurement results sent from the sensors. It is shown that the accuracy of the DOA-based FG geolocation algorithm can be improved by introducing approximated expressions for the mean and variance of the tangent and cotangent functions based on the first-order Taylor series (TS) at the tangent factor nodes of the FG. This paper also derives a closed-form expression of the Cramer-Rao lower bound (CRLB) for DOA-based geolocation, where the number of samples is taken into account. The proposed technique does not require high computational complexity because only mean and variance are to be exchanged between the nodes in the FG. It is shown that the position estimation accuracy with the proposed technique outperforms the conventional DOA-based least square (LS) technique and that the achieved root mean square error (RMSE) is very close to the theoretical CRLB.

Extended Capability Models for Carbon Fiber Composite (CFC) Panels in the Unstructured Transmission Line Modeling (UTLM) Method

An effective model of single- and multilayered thin panels, including those formed using carbon fiber composite (CFC) materials, is incorporated into the transmission line modeling (TLM) method. The thin-panel model is a one-dimensional (1-D) one based on analytical expansions of cotangent and cosecant functions that are used to describe the admittance matrix in the frequency domain; these are then converted into the time domain by using digital filter theory and an inverse Z transform. The model, which is extended to allow for material anisotropy, is executed within 1-D TLM codes. And, for the first time, the two-dimensional (2-D) thin-surface model is embedded in unstructured three-dimensional (3-D) TLM codes. The approach is validated by using it to study some canonical structures with analytic solutions, and against results taken from the literature. It is then used to investigate shielding effectiveness of CFC materials in a practical curved aerospace-related structure.

A new proof of Euler's formula for ζ(2n)

ABSTRACTWe use the hyperbolic cotangent function to deduce another proof of Euler's formula for ζ(2n).

An inverse trigonometric shear deformation theory for supersonic flutter characteristics of multilayered composite plates

In this study, an inverse trigonometric shear deformation theory developed by the authors is extended to assess the flutter behavior of multilayered composite plates subjected to yawed supersonic flow. The shear deformation is considered in terms of an inverse cotangent function which yields non-linear distribution of shear stresses. A generalized finite element formulation is presented to consider the shear strain function based theories. The displacement field is modified by a precise involvement of additional field variables to ensure the implementation of C0 continuous finite element. First order piston theory is employed to consider the aerodynamic load. The applicability, validity and accuracy of the present mathematical treatment are ascertained by performing various numerical tests and comparing the present results with the existing results. The influences of various parameters such as lamination sequences, boundary conditions, material anisotropy, flow angles, etc. on the free vibration and flutter behavior are examined and significant conclusions are made. It is concluded that flow angles, lamination sequence and material anisotropy should be considered as essential design parameters for enhanced flutter boundary of supersonic vehicles.

Finding sums for an infinite class of convergent series

We use Leibniz’s rule and the cotangent function to evaluate the following infinite series .

Single-valued neutrosophic similarity measures based on cotangent function and their application in the fault diagnosis of steam turbine

Similarity measure is an important tool in pattern recognition and fault diagnosis. This paper proposes two cotangent similarity measures for single-valued neutrosophic sets (SVNSs) based on cotangent function. Then, the weighted cotangent similarity measures are introduced by considering the importance of each element. Moreover, by the comparison between the cotangent similarity measures of SVNSs and existing cosine similarity measure of SVNSs, the developed cotangent similarity measures demonstrate their advantages and rationality and in some cases can overcome some disadvantages of the cosine similarity measure defined in vector space. Finally, the cotangent similarity measures are applied to the fault diagnosis of steam turbine. The proposed fault diagnosis method demonstrates its effectiveness and rationality by the comparative analysis with the cosine similarity measure in the fault diagnosis of steam turbine.

Probabilistic 2D Point Interpolation and Extrapolation via Data Modeling

Mathematics and computer science are interested in methods of 2D curve interpolation and extrapolation using the set of key points (knots). A proposed method of Hurwitz- Radon Matrices (MHR) is such a method. This novel method is based on the family of Hurwitz-Radon (HR) matrices which possess columns composed of orthogonal vectors. Two-dimensional curve is interpolated via different functions as probability distribution functions: polynomial, sinus, cosine, tangent, cotangent, logarithm, exponent, arcsin, arccos, arctan, arcctg or power function, also inverse functions. It is shown how to build the orthogonal matrix operator and how to use it in a process of curve reconstruction.

Floquet-Bloch analysis of refracting Huygens metasurfaces

We investigate the response of passive lossless Huygens metasurfaces (HMS), designed to implement a prescribed refraction, to arbitrary plane-wave excitations. Applying a rigorous Floquet-Bloch (FB) analysis of the cotangent electric and magnetic surface reactance modulations, we discover that the scattered FB modes follow an extraordinary ray-optical model. According to this model, the HMS can be replaced by a virtual region of constant wave impedance; incident power is incoupled into this virtual Fabry-P\'erot etalon, gradually leaking to successively higher-order FB modes. Analytical closed-form expressions are derived for the FB coefficients, indicating that the first FB mode in transmission is always dominant; this clarifies previous experimental findings which implied that the HMS retained its functionality over a broad angular range. Furthermore, we point out the implementation problems inherent in the cotangent function and offer a simple, realizable form of the reactive modulation which does not significantly deteriorate the HMS performance. These results provide fundamental physical insight into the working mechanism of passive lossless Huygens metasurfaces in general, revealing a new degree of freedom which enables control of the transmitted field amplitude: the virtual region wave impedance. Importantly, the implied broad acceptance angle of the HMS may be harnessed for synthesis of low-profile multifunctional devices, designed to respond to several excitations with reduced reflections.

An Archimedean Copula Family with Hyperbolic Cotangent Generator

Hyperbolic cotangent function is proposed as a generator of new Archimedean copula family and several properties are revealed. To show performance in real data analysis, application to modeling dependence between monthly temperature extremes as well as between flood peak and volume is given.

On a multidimensional half-discrete Hilbert-type inequality related to the hyperbolic cotangent function

In this paper, by the application of methods of weight functions and the use of analytic techniques, a multidimensional more accurate half-discrete Hilbert-type inequality with the kernel of the hyperbolic cotangent function is proved. We show that the constant factor related to the Riemann zeta function is the best possible. Equivalent forms as well as operator expressions are also investigated.

Curve interpolation and shape modeling via probabilistic nodes combination

The proposed method, called probabilistic nodes combination (PNC), is the method of 2D curve interpolation and modeling using the set of key points (knots or nodes). Nodes can be treated as characteristic points of the object for modeling. The model of each individual symbol or data can be built by choice of probability distribution function and nodes combination. PNC modeling via nodes combination and parameter $$\gamma $$ ? as probability distribution function enables curve parameterization and interpolation for each specific data or handwritten symbol. Two-dimensional curve is modeled and interpolated via nodes combination and different functions as discrete or continuous probability distribution functions: polynomial, sine, cosine, tangent, cotangent, logarithm, exponent, arcsin, arccos, arctan, arccot or power function. The novelty of the paper consists of two generalizations: generalization of previous MHR method with various nodes combinations and generalization of linear interpolation with different (no basic) probability distribution functions and nodes combinations.

Solution of Dirac Equations for Cotangent Potential with Coulomb-type Tensor Interaction for Spin and Pseudospin Symmetries Using Romanovski Polynomials

Normal 0 false false false EN-US X-NONE X-NONE MicrosoftInternetExplorer4 Abstract   The bound state solutions of Dirac equations for cotangent function potential with the Coulomb-type tensor potential under spin and pseudospin symmetric limits are obtained using finite Romanovski polynomials. The approximate relativistic energy spectra are obtained for spin and pseudospin symmetries exactly. The radial wave functions are obtained in terms of Romanovski polynomials in the limit of spin and pseudospin symmetric conditions. The Coulomb type tensor potential removes the doublet degeneracies for pseudospin and spin symmetric cases. The relativistic energy spectrum for the exact spin symmetric case reduces to the non-relativistic energy spectrum in the non-relativistic limit.     Abstrak   Solusi Persamaan Dirac untuk Potensial K otangen dengan Tensor Interaksi Tipe-Coulomb untuk Simetri Spin dan Pseudospin Menggunakan Polinomial Romanovski. Solusi keadaan terikat dari persamaan Dirac untuk potensial fungsi kotangen dengan potensial tensor tipe-Coulomb untuk simetri spin dan pseudospin diperoleh menggunakan polinomial Romanovski terbatas. Aproksimasi spektrum energi relativistik diperoleh secara eksak untuk simetri spin dan pseudospin. Fungsi gelombang radial diperoleh dalam bentuk polinomial Romanovski untuk keadaan simetri spin dan pseudospin. Potensial tensor tipe-Coulomb menghilangkan degenerasi doublet untuk kedua kasus simetri spin dan pseudospin. Pada batas non-relativistik, spektrum energi relativistik untuk kasus simetri spin eksak tereduksi menjadi spektrum energi non-relativistik.   Keywords: cotangent potential, Coulomb-type tensor, Dirac equation, pseudospin symmetry, Romanovski polynomials /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-qformat:yes; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri","sans-serif"; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-fareast-font-family:"Times New Roman"; mso-fareast-theme-font:minor-fareast; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:"Times New Roman"; mso-bidi-theme-font:minor-bidi;}

A general assessment of a new inverse trigonometric shear deformation theory for laminated composite and sandwich plates using finite element method

In this study, a general assessment of inverse trigonometric shear deformation theory, recently developed by the authors, is performed and the structural responses (static, buckling, and free vibration) of laminated-composite and sandwich plates are investigated. The in-plane displacement components are expressed in terms of an inverse cotangent function, which yields the nonlinear shear deformation while the constant transverse displacement is assumed over the thickness of the plate. A computationally efficient finite element model for the implementation of above-mentioned theory is proposed. The continuity requirement of the finite element model is maintained as C0 by a suitable choice of nodal field variables. Numerous analysis problems are selected to study the effects of various parameters such as span-to-thickness ratio, lamination sequence, loading conditions, boundary conditions, etc. on the response characteristics of plates. Higher modes are also obtained for the buckling and vibration problems and the ability to investigate higher modes is ensured. The comparison of the present results with the established results in literature indicates the efficiency and range of applicability of the present formulation. Moreover, the formulation is presented in a generalized approach which enables the implementation of all existing seven degree-of-freedom theories in a single computer algorithm thereby making it practically more significant.

The Computation of Azimuth and the Shortest Path between Two Points on the Earth's Surface

In this paper, first, analytic geometry are used to calculate the normal transversal azimuth on the ellipsoid surface, then modification value is calculated to get a more accurate orientation angle. Second, based on geodesic differential equations, segmented cumulation for distance between two points on the ground is brought forward. Each section is a short distance, so the result is more accurate. Moreover, when applyed to the program, angle is all got by anti cotangent function, thus avoiding to cumbersomely judge the quadrant.

Asymptotic Relative Efficiency and Exact Variance Stabilizing Transformation for the Generalized Gaussian Distribution

It is demonstrated that the sampling distributions of the maximum likelihood (ML) estimator and its Studentized statistic for the generalized Gaussian distribution do not pass the most powerful normality tests even for fairly large sample sizes. This disagreement with what the standard large sample ML theory predicts and the computational burden of having to deal with its associated polygamma functions motivate the consideration of a competing convexity-based estimator. The asymptotic normality of this estimator is derived. It is shown that the competing estimator is almost as efficient as the ML estimator and its asymptotic relative efficiency to the ML estimator is equal to 1 in the limit as the shape parameter approaches zero. More important, its asymptotic distribution admits an exact variance stabilizing transformation, whereas the asymptotic variance function of the ML estimator does not have a closed-form variance stabilizing transformation. The exact transformation is a composition of the inverse hyperbolic cotangent and square root functions. Besides stabilizing the variance, the inverse hyperbolic cotangent and square root transformation is remarkably effective for symmetrizing and normalizing the sampling distribution of the estimator and hence improving the standard normal approximation. Furthermore, this simple transformation provides a quite accurate approximation to the non-closed-form variance stabilizing transformation of the ML estimator.

One Entropy Measure for Interval-valued Intuitionistic Fuzzy Sets Based on Cotangent Function And Its Application in Multiple Attribute Decision Making

One Entropy Measure for Interval-valued Intuitionistic Fuzzy Sets Based on Cotangent Function And Its Application in Multiple Attribute Decision Making

The Fourier expansion of the hypermonogenic generalized trigonometric and elliptic functions

In this paper we deal with paravector valued multiperiodic solutions to the Dirac–Hodge equation on the upper half-space of R n which linearizes the Laplace–Beltrami operator. These functions provide basic building blocks for classes of hypermonogenic Eisenstein series on arithmetic subgroups of the Ahlfors–Vahlen group. In turn they can be regarded as hypermonogenic generalizations of the classical cotangent function and the elliptic functions. The goal of this paper is to develop explicit representations of the Fourier coefficients of these functions. The Fourier coefficients are composed by sums over Bell polynomials evaluated in lattice points multiplied by binomial coefficients.

A Fuzzy Multicriteria Group Decision-Making Method with New Entropy of Interval-Valued Intuitionistic Fuzzy Sets

A new entropy measure of interval-valued intuitionistic fuzzy set (IVIFS) is proposed by using cotangent function, which overcomes several limitations in the existing methods for calculating entropy of IVIFS. The efficiency of the new entropy is demonstrated by comparing it with several classical entropies. Moreover, an entropy weight model is established to determine the entropy weights for fuzzy multicriteria group decision-making (FMCGDMs) problems, which depends on incomplete weight information of criteria in IVIFSs setting. Finally, an illustrative supplier selection problem is used to demonstrate the practicality and effectiveness of the proposed method. It is capable of the handling the FMCGDM problems with incomplete known weights for criteria.