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Register a new userApproximate Analytical Solution for the Dynamic Model of Large Amplitude Non-Linear Oscillations Arising in Structural Engineering
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In this work we obtain an approximate solution of the strongly nonlinear second order differential equation $frac{d^{2}u}{dt^{2}}+omega ^{2}u+alpha u^{2}frac{d^{2}u}{dt^{2}}+alpha uleft( frac{du}{dt}right)^{2}+beta omega ^{2}u^{3}=0$, describing the large amplitude free vibrations of a uniform cantilever beam, by using a method based on the Laplace transform, and the convolution theorem. By reformulating the initial differential equation as an integral equation, with the use of an iterative procedure, an approximate solution of the nonlinear vibration equation can be obtained in any order of approximation. The iterative approximate solutions are compared with the exact numerical solution of the vibration equation.
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In this paper, we develop a method for obtaining the approximate solution for the evolution of single-step transformations under non-isothermal conditions. We have applied it to many reaction models and obtained very simple analytical expressions for the shape of the corresponding transformation rate peaks. These analytical solutions represent a significant simplification of the systems description allowing easy curve fitting to experiment. A remarkable property is that the evolutions of the transformed fraction obtained at different heating rates are identical when time is scaled by a time constant. The accuracy achieved with our method is checked against several reaction models and different temperature dependencies of the transformation rate constant. It is shown that its accuracy is closely related with that of the Kissinger method.
In this paper, we investigate tree-level scattering amplitude relations in $U(N)$ non-linear sigma model. We use Cayley parametrization. As was shown in the recent works [23,24] both on-shell amplitudes and off-shell currents with odd points have to vanish under Cayley parametrization. We prove the off-shell $U(1)$ identity and fundamental BCJ relation for even-point currents. By taking the on-shell limits of the off-shell relations, we show that the color-ordered tree amplitudes with even points satisfy $U(1)$-decoupling identity and fundamental BCJ relation, which have the same formations within Yang-Mills theory. We further state that all the on-shell general KK, BCJ relations as well as the minimal-basis expansion are also satisfied by color-ordered tree amplitudes. As a consequence of the relations among color-ordered amplitudes, the total $2m$-point tree amplitudes satisfy DDM form of color decomposition as well as KLT relation.
Since the first reports of oscillations in prominences in 1930s there have been major theoretical and observational advances to understand the nature of these oscillatory phenomena leading to a whole new field of so called prominence seismology. There are two types of oscillatory phenomena observed in prominences; small amplitude oscillations (~2-3 km s$^{-1}$) which are quite common and large amplitude oscillations ($>$20 km s$^{-1}$) for which observations are scarce. Large amplitude oscillations have been found as winking filament in H$alpha$ as well as motion in the sky plane in H$alpha$, EUV, micro-wave and He 10830 observations. Historically, it was suggested that the large amplitude oscillations in prominences were triggered by disturbances such as fast-mode MHD waves (Moreton wave) produced by remote flares. Recent observations show, in addition, that near-by flares or jets can also create such large amplitude oscillations in prominences. Large amplitude oscillations, which are observed both in transverse as well as longitudinal direction, have a range of periods varying from tens of minutes to a couple of hours. Using the observed period of oscillation and simple theoretical models, the obtained magnetic field in prominences has shown quite a good agreement with directly measured one and therefore, justifies prominences seismology as a powerful diagnostic tool. On rare occasions, when the large amplitude oscillations have been observed before or during the eruption, the oscillations may be applied to diagnose the stability and the eruption mechanism. Here we review the recent developments and understanding in the observational properties of large amplitude oscillations and their trigger mechanisms and stability in the context of prominence seismology.
We apply moment methods to obtaining an approximate analytical solution to Knudsen layers. Based on the hyperbolic regularized moment system for the Boltzmann equation with the Shakhov collision model, we derive a linearized hyperbolic moment system to model the scenario with the Knudsen layer vicinity to a solid wall with Maxwell boundary condition. We find that the reduced system is in an even-odd parity form that the reduced system proves to be well-posed under all accommodation coefficients. We show that the system may capture the temperature jump coefficient and the thermal Knudsen layer well with only a few moments. With the increasing number of moments used, qualitative convergence of the approximate solution is observed.
This analysis paper presents previously unknown properties of some special cases of the Wright function whose consideration is necessitated by our work on probability theory and the theory of stochastic processes. Specifically, we establish new asymptotic properties of the particular Wright function [{}_1Psi_1(rho,k; rho,0;x)= sum_{n=0}^inftyfrac{Gamma(k+rho n)}{Gamma(rho n)},frac{x^n}{n!}qquad (|x|<infty)] when the parameter $rhoin (-1,0)cup (0,infty)$ and the argument $x$ is real. In the probability theory applications, which are focused on studies of the Poisson-Tweedie mixtures, the parameter $k$ is a non-negative integer. Several representations involving well-known special functions are given for certain particular values of $rho$. The asymptotics of ${}_1Psi_1(rho,k;rho,0;x)$ are obtained under numerous assumptions on the behavior of the arguments $k$ and $x$ when the parameter $rho$ is both positive and negative. We also provide some integral representations and structural properties involving the `reduced Wright function ${}_0Psi_1(-!!!-; rho,0;x)$ with $rhoin (-1,0)cup (0,infty)$, which might be useful for the derivation of new properties of members of the power-variance family of distributions. Some of these imply a reflection principle that connects the functions ${}_0Psi_1(-!!!-; pmrho, 0;cdot)$ and certain Bessel functions. Several asymptotic relationships for both particular cases of this function are also given. A few of these follow under additional constraints from probability theory results which, although previously available, were unknown to analysts.
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