Cosecant Function Research Articles

Construction of a class of half-discrete Hilbert-type inequalities in the whole plane

Abstract In this work, we first define two special sets of real numbers, and then, we construct a half-discrete kernel function where the variables are defined in the whole plane, and the parameters in the kernel function are limited to the newly constructed special sets. Estimate the kernel function in the whole plane by converting it to the first quadrant, and then, a class of new Hilbert-type inequality is established. Additionally, it is proved that the constant factor of the newly established inequality is the best possible. Furthermore, assigning special values to the parameters and using rational fraction expansion of cosecant function, some special results are presented at the end of this article.

A HILBERT-TYPE LOCAL FRACTIONAL INTEGRAL INEQUALITY WITH THE KERNEL OF A HYPERBOLIC COSECANT FUNCTION

By using Yang’s local fractional calculus theory, the method of weight function, and real-analysis techniques in the fractal set, a general Hilbert-type local fractional integral inequality with the kernel of a hyperbolic cosecant function is established. The necessary and sufficient condition for the constant factor of the general Hilbert-type local fractional integral inequality to be the best possible is discovered. Furthermore, two equivalent inequalities with the best constant factors were obtained.

An improved reptile search algorithm based on mathematical optimization accelerator and elementary functions

The reptile search algorithm (RSA) is a dynamic and effective meta-heuristic algorithm inspired by the behavior of crocodiles in nature and the way of hunting prey. Unlike other crawler search algorithms, it uses four novel mechanisms to update the location of the solutions, such as walking at high or on the belly, and hunting in a coordinated or cooperative manner. In this algorithm, the total number of iterations is divided into four intervals, and different position-updating strategies are used to make the algorithm easily fall into the local optimum. Therefore, an improved reptile search algorithm based on a mathematical optimization accelerator (MOA) and elementary functions is proposed to improve its search efficiency and make it not easily fall into local optimum. MOA was used to realize the switching of RSA’s four searching modes by introducing random perturbations of six elementary functions (sine function, cosine function, tangent function, arccosine function, hyperbolic secant function and hyperbolic cosecant function), four mechanisms are distinguished by random number instead of the original RSA algorithm’s inherent four mechanisms by iteration number, which increases the randomness of the algorithm and avoids falling into local optimum. The random perturbations generated by elementary functions are added to the variation trend of parameter MOA to improve the optimization accuracy of the algorithm. To verify the effectiveness of the proposed algorithm, 30 benchmark functions in CEC2017 were used for carrying out simulation experiments, and the optimization performance was compared with BAT, PSO, ChOA, MRA and SSA. Finally, two practical engineering design problems are optimized. Simulation results show that the proposed sechRSA has strong global optimization ability.

Finding on Convergence of the Flint Hills and Cookson Hills Series based on a Summation Formula of Adamchik and Srivastava involving the Riemann Zeta Function

This article showcases significant progress in solving two renowned problems in the calculus of series: the Flint Hills and Cookson Hills series. For almost twenty years, a long-standing question has remained unanswered in regard to their convergence. Mainly, proving the convergence of the Flint Hills series would significantly impact the redefinition of the upper bound for the irrationality measure of the number π. One of the results presented in this article is that the Flint Hills series converges to 30.3144... which leads to a redefinition of the upper bound for the irrationality measure of π, specifically μ(π)≤ 2.5. This work proposes a transformation that solves the mystery of the Flint Hills and Cookson Hills series. It is based on a summation formula developed by mathematicians Adamchik and Srivastava. By leveraging a specialized series supported by the Riemann zeta function, this approach successfully transforms the original Flint Hills and Cookson Hills series into novel convergent versions with unique significance. The resulting sequences linked to these series are positive and bounded and satisfy convergence. Moreover, this article extends the Flint Hills series when the cosecant function has an arbitrary complex argument n+iβ, with i=√(-1), establishing a new series representation based on the polylogarithm 〖Li〗_3 (e^i2k), with k=1,2,3,…, e the Euler’s number, which bears resemblance to the famous integral of the Bose-Einstein distribution as a relevant finding. This is a never-seen-before link between the Flint Hills series and polylogarithms. Furthermore, a relationship between the Apéry constant and the Flint Hills and Cookson Hills series has been established. This article presents a significant breakthrough in the calculus of series by introducing a new method based on the Riemann Zeta function and logarithmical expressions derived from the Adamchik and Srivastava summation formula. The novel approach extends the analysis of convergence criteria for series, addressing ambiguous cases characterized by abrupt jumps. Thus, the Flint Hills series converges to 30.3144... and the Cookson Hills series to 42.9949... as proved in this article.

INEQUALITIES INVOLVING GENERALIZED TRIGONOMETRIC AND HYPERBOLIC FUNCTIONS

Trigonometric and Hyperbolic inequalities, which have been obtained by C. Huygens[5], D. S. Mitrinovic[10] and many more, have attracted attention of several researchers. We oer several renements and generalization of few trigonometric and hyperbolic inequalities involving tangent function, cotangent function, sine function, secant function and cosecant function. The established results are obtained with the aid of the Schwab-Borchardt mean.

Recent and new results on octonionic Bergman and Szegö kernels

Very recently one has started to study Bergman and Szegö kernels in the setting of octonionic monogenic functions. In particular, explicit formulas for the Bergman kernel for the octonionic unit ball and for the octonionic right half‐space as well as a formula for the Szegö kernel for the octonionic unit ball have been established. In this paper, we extend this line of investigation by developing explicit formulas for the Szegö kernel of strip domains of the form from which we derive by a limit argument consideringd → ∞the Szegö kernel of the octonionic right half‐space. Additionally, we set up formulas for the Bergman kernel of such strip domains and relate both kernels with each other. In fact, these kernel functions can be expressed in terms of onefold periodic octonionic monogenic generalizations of the cosecant function and the cotangent function, respectively.

On a class of Hilbert-type inequalities in the whole plane related to exponent function

By introducing a kernel involving an exponent function with multiple parameters, we establish a new Hilbert-type inequality and its equivalent Hardy form. We also prove that the constant factors of the obtained inequalities are the best possible. Furthermore, by introducing the Bernoulli number, Euler number, and the partial fraction expansion of cotangent function and cosecant function, we get some special and interesting cases of the newly obtained inequality.

Fault-Tolerant Quantized Control for Flexible Air-Breathing Hypersonic Vehicles With Appointed-Time Tracking Performances

This article contrives a fault-tolerant quantized control for flexible air-breathing hypersonic vehicles (FAHVs) with appointed-time tracking performances. At first, to online identify the lumped effect of actuator faults, flexible modes, parameter uncertainties as well as external disturbances, a hysteresis quantizer based neural estimator (HQNE) using finite precision state information is proposed, enabling a reduced communication load and computational time with a competitive estimation capability. Utilizing the estimation of HQNE, fault-tolerant quantized control laws equipped with auxiliary systems are established for FAHVs to realize a stable reference tracking result using discrete-time control signals, where auxiliary systems are employed to automatically monitor the impact of input saturation for command regulation. Furthermore, an appointed-time prescribed performance control based on a hyperbolic cosecant function is developed to make the tracking errors of velocity and altitude reach to the pregiven residual sets with a prescribed time in the absence of exact initial system states. The presented controller achieves a preassigned-time tracking performance for FAHVs under actuator faults and input saturation maintaining a decreased communication burden. The ultimately uniformly bounded stability of closed-loop system is proved through Lyapunov stability analysis, while numerical simulations are designed to verify the effectiveness of presented controller.

Flexible satellite control via fixed-time prescribed performance control and fully adaptive component synthesis vibration suppression

In this work, the problems of active vibration suppression and high accuracy attitude control for a flexible satellite with piezoelectric actuators are studied. Firstly, the attitude error dynamic equation is developed. By utilizing a novel hyperbolic cosecant function and the error transformation equation, the attitude errors can be transferred into new prescribed performance state variables. In order to further ensure better convergence of the new variables, a fixed-time sliding mode control is proposed. Subsequently, a novel fully adaptive component synthesis vibration suppression method is presented to realize vibration suppression during the attitude maneuver by utilizing piezoelectric actuators. Stability analysis of the proposed prescribed performance control is given. Finally, abundant numerical simulation results demonstrate the excellent performances of the proposed control scheme.

Study on inertia weight decreasing strategy of particle swarm optimization based on inverse coseca

The standard particle swarm optimization (pso), which introduces inertia weight w, is an effective method to find the extreme value of the function. However, particle swarm optimization (pso) has some disadvantages. When dealing with optimization problems, pso lacks effective parameter control and is prone to fall into local optimization, which leads to low convergence accuracy. In, this paper, put forward a new improved particle swarm optimization (pso) algorithm, The nonlinear decreasing inertia weight by the CSC function strategy, at the same time to join the beta distribution on random Numbers, thus to balance the global search and local search ability of the algorithm. The learning factor is the changed asynchronously to improve the learning ability of the algorithm. By adopting Griewank, Rastrigrin, J.D. Schaffer three standard test functions to simulate experiment, at the same time and the basic particle swarm algorithm the inertia weight in a fixed value, the linear regressive LDIW and nonlinear regressive comparison. The experimental results show that the nonlinear decreasing strategy with dynamic adjustment of inverse cosecant function can improve the convergence speed and stability.

Heavy quarkonia properties from a hard-wall confinement potential model with conformal symmetry perturbing effects

Heavy [Formula: see text] and [Formula: see text] quarkonia are considered as systems confined within a hard-wall potential shaped after a linear combination of a cotangent — with a square co-secant function. Wave functions and energy spectra are then obtained in closed forms in solving by the Nikiforov–Uvarov method the associated radial Schrödinger equation in the presence of a centrifugal term. The interest in this potential is that in one parametrization, it can account for a conformal symmetry of the strong interaction, and in another for its perturbation, a reason for which we here employ it to study status of conformal symmetry in the heavy flavor sector. The resulting predictions on heavy quarkonia mass spectra and root mean square radii are compared with the available experimental data, as well as with predictions by other theoretical approaches. We observe that a relatively small conformal symmetry perturbing term in the potential suffices to achieve good agreement with data.

A more accurate half-discrete Hardy-Hilbert-type inequality with the best possible constant factor related to the extended Riemann-Zeta function

By the method of weight coefficients, techniques of real analysis and Hermite-Hadamard's inequality, a half-discrete Hardy-Hilbert-type inequality related to the kernel of the hyperbolic cosecant function with the best possible constant factor expressed in terms of the extended Riemann-zeta function is proved. The more accurate equivalent forms, the operator expressions with the norm, the reverses and some particular cases are also considered.

Localization for broadband source by a single vector sensor in shallow water

The physical meaning of array invariant is the absolute traveling time of different modal from a source to a receiver under the average sound speed. It can be gotten by the derivative of the arriving time against the corresponding cosecant function of elevation angle for each modal, which can be used for the passive source localization. A vector sensor can be seen as a small volumetric array. It has its capability of elevation angle determination to describe the multi-modal propagation and the single-modal dispersion in shallow water, which may imply a chance to get the array invariant for the source localization. Here, we take advantage of the dispersion based short time Fourier transform and warping transform techniques for the time-frequency analysis of vector sensor signals. This technique can improve acoustic normal mode identification and thereby extract the modal data more accurately for array invariant determination. The idea had been effectively validated during the experiment that took place in ...

A Kramers-Kronig integral relation turned into a simple convolution operation and its application to dielectric measurements

This paper proposes that the conduction-free imaginary-part of dielectric constant may be obtained from a convolution between its real-part on a logarithmic scale of frequency and the hyperbolic cosecant function. This equation is, in fact, one of the Kramers-Kronig relations and is deduced from the relationships that relate a distribution of relaxation frequencies both to the real-part and to the conduction-free imaginary-part of the dielectric constant. This simple result has a large practical interest as it allows, for instance, checking for the presence of conductivity on experimental data.

Delayed luminescence as an optical indicator of tobacco leaf quality

In this paper, we present our study on the spectral discrimination between high and low quality tobacco leaves using a time-resolved ultraweak luminescence detection system. Photoinduced delayed luminescence (DL) is employed as a nondestructive and objective indicator of tobacco leaf quality. DL decay kinetics of tobacco leaf samples is measured, and the data are fitted by a hyperbolic cosecant function. Results show that the function’s parameter A is significantly related to the quality grades of tobacco leaves—compared with the low quality tobacco leaves, an increase of the A value by a factor of 7 is obtained for the high quality tobacco leaves. Research from this work contributes to the development of a novel optical method applicable for the quality evaluation of agricultural crops and food products.

Closed-form evaluation of some families of cotangent and cosecant integrals

Several families of integrals involving the cotangent or cosecant function have been evaluated in closed form by making use of elementary arguments. Some of them are expressible in terms of the values of the (generalized) Clausen function and Hurwitz zeta function.

Inequalities of some trigonometric functions

By using two identities and two inequalities relating to Bernoulli’s and Euler’s numbers and power series expansions of cotangent function, secant function, cosecant function and logarithms of functions involving sine function, cosine function and tangent function, six inequalities involving tangent function, cotangent function, sine function, secant function and cosecant function are established.

Monogenic muitiperiodic functions in clifford analysis

In this paper we deal with generalizations of the classical Eisenstein series in the frame-work of Clifford analysis. They provide monogenic generalizations of the classical tangent, cotangent, secant and cosecant function and the elliptic functions to higher dimensions. We study properties of them and characterize them by certain duplication formulas and their principal parts.

A polynomial method for solving the problems for lateral instability of cantilever plates

The present article researches the problems of the lateral instability of cantilever rectangular plates under a concentrated force or a uniformly distributed load respectively. We select the polynomial (2.1) instead of the cosecant function in Ref. [1] as the flexural functions.

Accurate coupling constants from two-dimensional correlation spectra by “J deconvolution”

A practical procedure is described for the accurate determination of coupling constants from cross peaks in homonuclear correlation (COSY) spectra. It lends itself to computer automation and is applicable to cases where the splittings are not well resolved. In an extension of the method suggested by Bothner-By and Dadok ( J. Magn. Reson. 72, 540, 1987) the active splitting J a is deconvoluted by multiplication of the experimental time-domain signal S( t 2) by cosec( πJ ∗t ), where J ∗ is a trial coupling. The trick is to sample the cosecant function at appropriate symmetrically spaced intervals that avoid the singularities. Fourier transformation of S( t 2) Cosec ( πJ ∗t ) gives a “test spectrum” containing antiphase doublets, and the integral of its absolute magnitude goes through a pronounced minimum when J ∗ = J a . Accuracies of ±0.04 Flz can be achieved. Passive couplings can be measured by an analogous procedure. Once the couplings have been determined it is possible to reduce a two-dimensional cross peak to a singlet at the chemical-shift coordinates by deconvoluting the splittings one at a time.