Sine-cosine Function Research Articles

Traveling Wave Solutions for Generalized Bretherton Equation

This paper studies the Generalized Bretherton equation using trigonometric function method including the sech-function method, the sine-cosine function method, and the tanh-function method, and He's semi-inverse method (He's variational method). Various traveling wave solutions are obtained, revealing an intrinsic relationship among the amplitude, frequency, and wave speed.

Exact solitary solutions of an inhomogeneous modified nonlinear Schrödinger equation with competing nonlinearities

We employed both tanh and sine–cosine function methods to study the inhomogeneous modified nonlinear Schrödinger (IMNLS) equation and constructed its exact solitary solutions by using symbolic computation software. We compared the solutions obtained by both the methods.

The reported incidence of campylobacteriosis modelled as a function of earlier temperatures and numbers of cases, Montreal, Canada, 1990–2006

Previous studies have detected an effect of earlier temperatures on the incidence of campylobacteriosis in humans, but without adjustment for earlier numbers of cases of the disease. We estimated the effect of temperature on the number of cases notified by week in Montreal, Canada, from 1 January 1990 to 26 March 2006, simultaneously with the effect of the numbers of cases notified in the preceding weeks. The current campylobacteriosis count (week 0) was modelled by negative binomial regression, with earlier weekly average temperatures and earlier counts as predictors. Secular trends were accounted for by cubic spline functions and seasonal variations by sine-cosine functions. Indicator variables identified weeks with fewer than 5 working days. In the final statistical model, a 1°C increase in temperature above 10°C during any of weeks -1 to -6 was associated with a 0.8% (95% CI: 0.3% to 1.3%) increase in the current count. For each additional notified case during any of weeks -1 to -5 or -9 to -12, the increase in the current count was approximately 0.5% (95% CI: 0.2% to 1.0%). Thus, earlier temperatures and earlier counts have independent effects, that of temperatures being the larger one. The temperature effect is too small to require short term public health planning. However, in Montreal, an increase in average temperature of the order of 4.5°C, forecast by some for 2055, could produce a 23% increase in incidence, resulting in about 4,000 excess cases per year.

Creation and annihilation of solitons in a ferromagnet with competing nonlinear inhomogeneities

In this paper, we consider the nonlinear dynamics of an inhomogeneous (site-dependent) bilinear and biquadratic isotropic Heisenberg ferromagnetic spin chain in the classical continuum limit. We establish the generalized solutions to the inhomogeneous nonlinear Schrödinger equation through the sine–cosine function method with minimal algebra. Finally, we demonstrate the creation and annihilation of solitons under the influence of nonlinear inhomogeneities on the spin chain.

Study on Strain Solitary Waves in the Nonlinear Elastic Circular Rod

Taking into account the nonlinear constitutive relationship and transverse Poisson effects, the propagation characteristic of nonlinear wave for one-dimension elastic thin rod is studied. With the help of Mathematic, two traveling wave solutions for this nonlinear wave equation are obtained by sine-cosine function method, which include the shock wave solution and the solitary wave solution. The necessary condition of these solutions is given also.

A Cross-Type Thermal Wind Sensor With Self-Testing Function

A 2-D smart flow sensor with self-testing function was designed, fabricated and tested in this paper. It is composed of cross-structure heaters and four symmetrically located sensing parts. When the flow angle changes in the clockwise direction, the temperature differences among the four parts, namely Mode 1 given by T13(T24) and Model 2 given by T12(T14) will be Sine(Cosine) functions of wind direction. Further study shows that the magnitude of mode 1 is 1.414 times that of Mode 2, and the phase of Mode 2 leads Mode 1 by 45°. By using the fixed phase gap between the two modes, the self-test function can be realized without additional components. In order to achieve a high sensitivity and robust sensor, thermal insulated substrate was introduced to fabricate calorimetric flow sensor instead of silicon wafers. The proposed sensor was fabricated on the Pyrex7740 substrate using single liftoff process. Finally, a wind tunnel test was carried out in constant power (CP) mode, and the test results show reasonable agreement with the simulation curves.

Superstability of Some Pexider-Type Functional Equation

We will investigate the superstability of the sine functional equation from the following Pexider-type functional equation (), which can be considered the mixed functional equation of the sine and cosine functions, the mixed functional equation of the hyperbolic sine and hyperbolic cosine functions, and the exponential-type functional equations.

On the Superstability of the Pexider Type Trigonometric Functional Equation

We will investigate the superstability of the (hyperbolic) trigonometric functional equation from the following functional equations: and, which can be considered the mixed functional equations of the sine function and cosine function, the hyperbolic sine function and hyperbolic cosine function, and the exponential functions, respectively.

Design of Hybrid Fuzzy Neural Network for Function Approximation

In this paper, a hybrid Fuzzy Neural Network (FNN) system for function approximation is presented. The proposed FNN can handle numeric and fuzzy inputs simultaneously. The numeric inputs are fuzzified by input nodes upon presentation to the network while the Fuzzy rule based knowledge is translated directly into network architecture. The connections between input to hidden nodes represent rule antecedents and hidden to output nodes represent rule consequents. All the connections are represented by Gaussian fuzzy sets. The method of activation spread in the network is based on a fuzzy mutual subsethood measure. Rule (hidden) node activations are computed as a fuzzy inner product. For a given numeric o fuzzy input, numeric outputs are computed using volume based defuzzification. A supervised learning procedure based on gradient descent is employed to train the network. The model has been tested on two different approximation problems: sine-cosine function approximation and Narazaki-Ralescu function and shows its natural capability of inference, function approximation, and classification.

On the Superstability Related with the Trigonometric Functional Equation

We will investigate the superstability of the (hyperbolic) trigonometric functional equation from the following functional equations: , , , , which can be considered the mixed functional equations of the sine function and cosine function, of the hyperbolic sine function and hyperbolic cosine function, and of the exponential functions, respectively.

A new method for fast transforms in parity-mixed PDEs: Part I. Numerical techniques and analysis

We address the problem of parity mixing, where the projection of a variable expressed as a finite series of half-period cosine (sine) functions onto a half-period sine (cosine) function basis is not finite. We propose new fast methods for computing these complicated projections exactly up to some arbitrary degree using fast Fourier transforms. This method has immediate applications for pseudospectral solutions of many systems of partial differential equations.

Alternative proofs for inequalities of some trigonometric functions

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.

Season of birth and childhood intelligence: Findings from the Aberdeen Children of the 1950s cohort study

In this study, 2 main hypotheses have been put forward to explain the variation in childhood intelligence or school performance by season of birth. In the first hypothesis, it is suggested that it is due to school policy concerning school entry, whereas the second suggests that a seasonally patterned exposure such as temperature, maternal nutrition, or infection during critical periods of brain development have a lasting effect on intelligence. To determine whether childhood performance on tests of different domains of intelligence is patterned by season of birth and to examine possible mechanisms for any associations. 12,150 individuals born in Aberdeen, Scotland between 1950 and 1956. Birth cohort study in which the variation in different domains of childhood intelligence measured at ages 7, 9, and 11 by season of birth were examined. Reading ability at age 9 and arithmetic ability at age 11 varied by season of birth, with lowest scores among those born in autumn or early winter (September-December) and highest scores among those born in later winter or spring (February-April); p=.002 for joint sine-cosine functions for reading ability at age 9 and p=.05 for sine-cosine function for arithmetic ability at age 11. The child's perception and understanding of pictorial differences at age 7, verbal reasoning at 11, and English language ability at 11 did not vary by season of birth. Age at starting primary school and age relative to class peers were both associated with the different measurements of childhood intelligence and both attenuated the association between month of birth and reading ability at age 9 and arithmetic ability at age 11 towards the null. Both adjusted and unadjusted differences in reading ability at age 9 and arithmetic ability at age 11 between those born from September to December compared with other times of the year were less than 0.1 of a standard deviation of the test scores. Ambient temperature around the time of conception, during gestation, and around the time of birth did not affect intelligence. Any variation in mean childhood intelligence by season of birth is weak and largely explained by age at school entry and age relative to class peers.

Comments on "Optimal control via Fourier series of operational matrix of integration"

In the last four decades, the orthogonal functions approach has been extensively applied to study problems like analysis, identification, and control of continuous-time dynamical systems. The success of this approach lies in the fact that it converts calculus into algebra approximately in the sense of least squares. Sine-cosine functions are a class of orthogonal functions, and they were applied to study the optimal control problem with some success. In order to have a computationally attractive method for solving the optimal control problem via sine-cosine functions, an attempt was made. In this note, we show that the Fourier series approach to the optimal control problem in a paper by Endow (Optimal control via Fourier series of operational matrix of integration, IEEE Trans. Autom. Control, vol. 34, no. 7, p. 770-3, Jul. 1989) is incorrect.

The association of ambient outdoor temperature throughout pregnancy and offspring birthweight: findings from the Aberdeen Children of the 1950s cohort

We assessed the effect of mean ambient outdoor temperature during gestation on birthweight. To assess the effect of mean ambient outdoor temperature during gestation on birth weight. Birth cohort study with record linkage to climate databases. Aberdeen, Scotland. A total of 12,150 individuals born in Aberdeen, Scotland between 1950 and 1956. Perinatal data from a cohort of 12,150 individuals born in Aberdeen, Scotland between 1950 and 1956 were linked to daily outdoor temperature data. Birthweight was seasonally patterned, with lowest birthweights among those born in the winter months (December-February) and highest birthweights among those born in the autumn months (September-November); P= 0.01 for joint sine-cosine functions. Mean ambient outdoor temperature during the first trimester of pregnancy was inversely associated with birthweight and mean ambient outdoor temperature during the third trimester of pregnancy was positively associated with birthweight. In fully adjusted (for sex, maternal age, birth year, birth order and social class) models a 1 degrees C increase in mean ambient outdoor temperature in the mid 10-day period of the first trimester was associated with a 5.4-g (95% confidence interval [CI] 2.9, 7.9 g) decrease in birthweight, whereas a 1 degrees C increase in the mid 10-day period of the third trimester was associated with a 1.3-g (95% CI 0.50, 2.1 g) increase in birthweight. Ambient outdoor temperature in the first trimester of pregnancy explained the seasonal patterning of birthweight. Birthweight. Birthweight was seasonally patterned, with lowest birthweights among those born in the winter months (December-February) and highest birthweights among those born in the autumn months (September-November); P= 0.01 for joint sine-cosine functions. Mean ambient outdoor temperature during the first trimester of pregnancy was inversely associated with birthweight and mean ambient outdoor temperature during the third trimester of pregnancy was positively associated with birthweight. In fully adjusted (for sex, maternal age, birth year, birth order and social class) models a 1 degrees C increase in mean ambient outdoor temperature in the mid 10-day period of the first trimester was associated with a 5.4 g (95% confidence interval 2.9, 7.9 g) decrease in birthweight, whereas a 1 degrees C increase in the mid 10-day period of the third trimester was associated with a 1.3 g (95% confidence interval 0.50, 2.1 g) increase in birthweight. Ambient outdoor temperature in the first trimester of pregnancy explained the seasonal patterning of birthweight. Higher ambient outdoor temperature in the first trimester of pregnancy and/or lower ambient outdoor temperature in the third trimester are associated with reduced offspring birthweight. With the increasing occurrence of temperature extremes, in particular, heat waves, these findings, if replicated in other studies, have important public health implications.

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.

Dirichlet multiplication and easily-generated function analysis

Following sine-cosine functions, sawtooth wave, square wave, triangular wave, and trapezoidal wave become new easily-generated periodic functions. Can a signal be considered to be a superposition of easily-generated functions with different frequencies? In order to answer this question, we generalize Fourier analysis to easily-generated function analysis including easily-generated functions series, easily-generated function transformation, and discrete transformation for easily-generated functions. The results in this paper make it possible to represent a signal by use of easily-generated functions. A lot of techniques based on sine-cosine functions can be translated into techniques based on easily-generated functions. Because Dirichlet multiplication in number theory plays a basic role in easily-generated function analysis, we briefly introduce this concept and present a related formula. The main contents of this paper include Dirichlet multiplication and a related formula, relations between basic waveforms in electronics, easily-generated function series, easily-generated function transformation, discrete transformation for easily-generated functions, and techniques of easily-generated function analysis.

New methods for Fourier analysis and Fourier synthesis

A basic and important relation between square wave and sine-cosine function is presented. This relation leads to completely new methods for Fourier analysis and Fourier synthesis.

Spatial and Temporal Characteristics of Saline Springs: Sea of Galilee, Israel

Spatial and temporal characteristics of the saline springs that emerge along the western shore of the Sea of Galilee (Lake Kinneret) are analyzed. Three groups of onshore springs (Tiberias, Fuliya, and Tabgha) and two groups of offshore springs (Barbutim and Maagan), contribute saline water to the lake with concentrations in the range of 300 to 18,000 mg Cl/L, depending on location and season. It is well accepted that water emerging from these springs is a mixture of two endmembers: deep-seated saline ground water and shallow, fresh circulating ground water. Temporal trends of discharge rates and of chloride (representing the deep saline aquifer) and nitrate (representing the shallow fresh water aquifer) concentrations within each group of springs are presented. Results show the proportions of the two water bodies while mixing are time dependent. Discharge and concentration peaks in Tabgha springs precede those in Fuliya and Tiberias springs by approximately two months. An analytical solution shows that in Tabgha, variations of these parameters are mainly controlled by recharge variations in the Galilee, and follow an exponential function. In Fuliya and Tiberias, variations of these parameters are mainly dependent on lake level, and follow a sine-cosine function. The different patterns are attributed to different hydraulic properties of the discharge area.

Frequency analysis based on general periodic functions

Following the sine–cosine function, the sawtooth wave, square wave, triangular wave, trapezoidal wave, and so on become new easily generated periodic functions in modern electronics. Similar to Fourier’s idea, a natural question is whether a signal can be considered as a superposition of easily generated functions with different frequencies. Therefore it is necessary to generalize Fourier analysis based on sine–cosine functions into frequency analysis based on general periodic functions. In this paper, we introduce the frequency series and frequency transformation based on general periodic functions. We discuss when a frequency system is a complete system or an unconditional basis in L2[−π,π], and when a frequency transformation can be carried out in L2(−∞,+∞). For practical convenience almost all easily generated functions in electronics are considered carefully as examples. As a new and practical generalization of classical Fourier analysis, these results will become a theoretical foundation for the technique of easily generated function analysis in signal processing.