Existence and concentration of nontrivial solutions for an elastic beam equation with local nonlinearity

<abstract><p>In this paper, by using the mountain pass lemma and the skill of truncation function, we investigate the existence and concentration phenomenon of nontrivial weak solutions for a class of elastic beam differential equation with two parameters $ \lambda $ and $ \mu $ when the nonlinear term satisfies some growth conditions only near the origin. In particular, we obtain a concrete lower bound of the parameter $ \lambda $, and analyze the relationship between $ \lambda $ and $ \mu $. In the end, we investigate the concentration phenomenon of solutions when $ \mu\to 0 $, and obtain a specific lower bound of the parameter $ \lambda $ which is independent of $ \mu $.</p></abstract>

Global random attractor of stochastic hydrodynamical equation for the Heisenberg paramagnet with multiplicative noise on ℝd

The stochastic hydrodynamical equation for the Heisenberg paramagnet with multiplicative noise defined on the entire [Formula: see text] is mainly investigated. The global random attractor for the random dynamical system associated with the equation is obtained. The method is to transform the stochastic equation into the corresponding partial differential equations with random coefficients by Ornstein–Uhlenbeck process. The uniform priori estimates for far-field values of solutions have been studied via a truncation function, and then the asymptotic compactness of the random dynamical system is established.

Photometric Precision in Infrared Spectra Measured by Fourier Transform Spectroscopy

Digitally recorded spectra containing lines of width less than or equal to the resolution at which the measurement has been made are observed to show some photometric In spectra measured by Fourier transform spectroscopy, this error may be substantially reduced if the interferogram is zero-filled. To prevent the generation of artifacts in the spectrum (of a similar origin to the sine x side-lobes formed when a non-zero-filled spectrum is computed with a box-car truncation function), the zero-filled interferogram must be suitably apodized.

Analysis of High Angle Diffuse Scattering from Small Platelets

A detailed analysis of the high angle diffuse scattering from small platelets is given. A large number of statistically centrosymmetric platelets is considered, and it is shown that, in this cage, the positive square root of the diffuse intensity from the platelets is proportional to the amplitude of the scattered radiation over particular regions in reciprocal space. The measured amplitude distribution is truncated from the true amplitude distribution by the limits of measurement and the influence of Bragg scattering. A truncation function is introduced to describe this truncated amplitude distribution in terms of the true amplitude distribution. This truncation introduces modulations on the measured electron density distribution. The measured electron density distribution is described in terms of the convolution of the true electron density distribution and the transform of the truncation function. The transform of the truncation function is knowi analytically, so the true electron density distribution can be found by a relaxation method. The true electron density distribution is given in terms of composition and strain parameters which are independently adjusted during the relaxation procedure to fit the measured values. Examples of the influence of the truncation function arc given and the technique is applied to G–P 1 zones in an aluminum - 1.67 at % copper alloy.