A hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac delta source

In this paper, we investigate a hybridizable discontinuous Galerkin method for second order elliptic equations with Dirac measures. Under assumption that the domain is convex and the mesh is quasi-uniform, a priori error estimate for the error in $L^2$-norm is proved. By duality argument and Oswald interpolation, a posteriori error estimates for the errors in $L^2$-norm and $W^{1,p}$-seminorm are also obtained. Finally, numerical examples are provided to validate the theoretical analysis.

Singularly perturbed rank one linear operators

The basic principles of the theory of singularly perturbed self-adjoint operatorsare generalized to the case of closed linear operators with non-symmetric perturbation of rank one.Namely, firstly linear closed operators are considered that coincide with each other on a dense set in a Hilbert space.The theory of singularly perturbed self-adjoint operators arose from the need to consider differential expressions in such terms as the Dirac $\delta$-function.Since it is important to consider expressions given not only by symmetric operators, the generalization (transfer) of the basic principles of the theory of singularly perturbed self-adjoint operators in the case of non-symmetric ones is important problem. The main facts of the theory include the definition of a singularly perturbed linear operator and the resolvent formula in the cases of ${\mathcal H}_{-1}$-class and ${\mathcal H}_{-2}$-class.The paper additionally describes the possibility of the appearance a point of the point spectrum and the construction of a perturbation with a predetermined point.In comparison with self-adjoint perturbations, the description of perturbations by non-symmetric terms is unexpected.Namely, in some cases, when the perturbed by a vectors from ${\mathcal H}_{-2}$ operator can be conveniently described by methods of class ${\mathcal H}_{-1}$, that is impossible in the case of symmetric perturbations of a self-adjoint operator. The perturbation of self-adjoint operators in a non-symmetric manner fully fits into the proposed studies.Such operators, for example, generalize models with nonlocal interactions, perturbations of the harmonic oscillator by the $\delta$-potentials, and can be used to study perturbations generated by a delay or an anticipation.

Global propagation of massive quantum fields in the plane gravitational waves and electromagnetic backgrounds

The behavior of massive quantum fields in the general plane wave spacetime and external, non-plane, electromagnetic waves is studied. The asymptotic conditions, the ``in" (``out") states and the cross sections are analysed. It is observed that, despite of the singularities encountered, the global form of these states can be obtained: at the singular points the Dirac delta-like behavior emerges and there is a discrete change of phase of the wave function after passing through each singular point. The relations between these phase corrections and local charts are discussed. Some examples of waves of infinite range (including the circularly polarized ones) are presented for which the explicit form of solutions can be obtained. All these results concern both the scalar as well as spin one-half fields; in latter case the change of the spin polarization after the general sandwich wave has passed is studied.

Impact action of a projectiles on special protective structures and ways to increase their protective capacity

A method for studying the reaction of elastic elements of protective structures to a series of impact actions of shells has been developed. In the work, the elastic elements of the protective structure are modeled by homogeneous beams, and the dynamic action of the shells is simulated by instantaneous point-applied forces. A mathematical model of this dynamic process is constructed, which is a boundary value problem for a hyperbolic equation with an irregular right-hand side. The latter is described using Dirac delta functions. Cases of both fixed and free ends of protective elements are considered. The main ideas of perturbation methods are used for the researches carried out in the work. Analytical dependences for the description of elastic deformations of a protective element which are basic for definition of its strength characteristics are received. They and the graphical dependences built on their basis for specific cases show that the dynamic deformations of the protective element for the fixed ends are greater in the case of the projectile closer to its middle, at the same time for the free ends – closer to the end. With regard to the modernization of protective structures, the dynamic effect on their elements can be reduced by using elastic reinforcement or changing the method of fixing the ends of the protective element: elastic or with a certain angle of inclination of the bearing surfaces. It is proposed to use special plastics, soil layer, flexible wood flooring, etc. as elastic reinforcement. The technique used in the work is the basis for determining the strength characteristics of protective elements, and from so – to check the reliability of the protective structure; study of the dynamics of protective and similar types of structures, taking into account the nonlinear characteristics of the elastic elements of protective structures; study of more complex oscillations of elements of protective structures. In the case of a series of impacts, it is obvious that the amplitude of deflection of the protective element after each impact will increase over time, because the model does not take into account the force of viscoelastic friction. These tasks will be the subject of further research.

DETERMINATION OF FIBER ORIENTATION IN HIGH ANGULAR RESOLUTION DIFFUSION IMAGING USING SPHERICAL HARMONICS AND PARTICLE SWARM OPTIMIZATION

The fiber directions in High Angular Resolution Diffusion Imaging (HARDI) with low fractional anisotropy or low Signal to Noise Ratio (SNR) cannot be estimated accurately. In this paper, the fiber directions are estimated using Particle Swarm Optimization and Spherical Deconvolution (PSO-SD). Fiber orientation is modeled as a Dirac delta function in [Formula: see text]. The Spherical Harmonic Coefficients (SHC) of the Dirac delta function in the [Formula: see text] direction are obtained using the rotational harmonic matrix and the SHC of the Dirac delta function in the [Formula: see text]-axis. The PSO-SD method is used to determine ([Formula: see text]). We generated noise-free synthetic data for isotropic regions (FA varied from 0.1 to 0.8) and synthetic data with two crossing fibers for anisotropic regions with SNRs of 20, 15, 10 and 5 (FA [Formula: see text] 0.78). In the noise-free signal (FA [Formula: see text] 0.3), the Success Ratio (SR) and Mean Difference Angle (MDA) of the PSO-SD method were 1∘ and 9.48∘, respectively. In the noisy signal (FA [Formula: see text] 0.78, SNR [Formula: see text] 10, crossing angle [Formula: see text] 40), the SR and MDA of PSO-SD (with [Formula: see text]) were 0.46∘ and 12.3∘, respectively. The PSO-SD method can estimate fiber directions in HARDI with low fractional anisotropy and low SNR. Moreover, it has a higher SR and lower MDA in comparison with those of the super-CSD method.

Variational approach to the Schrödinger equation with a delta-function potential

We obtain accurate eigenvalues of the one-dimensional Schr\"{o}dinger equation with a Hamiltonian of the form $H_{g}=H+g\delta (x)$, where $\delta (x)$ is the Dirac delta function. We show that the well known Rayleigh-Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction. Present analysis may be suitable for an introductory course on quantum mechanics to illustrate the application of the Rayleigh-Ritz variational method to a problem where the boundary conditions play a relevant role and have to be introduced carefully into the trial function. Besides, the examples are suitable for motivating the students to resort to any computer-algebra software in order to calculate the required integrals and solve the secular equations.

Statistical description of coalescing magnetic islands via magnetic reconnection

The physical picture of interacting magnetic islands provides a useful paradigm for certain plasma dynamics in a variety of physical environments, such as the solar corona, the heliosheath and the Earth's magnetosphere. In this work, we derive an island kinetic equation to describe the evolution of the island distribution function (in area and in flux of islands) subject to a collisional integral designed to account for the role of magnetic reconnection during island mergers. This equation is used to study the inverse transfer of magnetic energy through the coalescence of magnetic islands in two dimensions. We solve our island kinetic equation numerically for three different types of initial distribution: Dirac delta, Gaussian and power-law distributions. The time evolution of several key quantities is found to agree well with our analytical predictions: magnetic energy decays as $\tilde {t}^{-1}$ , the number of islands decreases as $\tilde {t}^{-1}$ and the averaged area of islands grows as $\tilde {t}$ , where $\tilde {t}$ is the time normalised to the characteristic reconnection time scale of islands. General properties of the distribution function and the magnetic energy spectrum are also studied. Finally, we discuss the underlying connection of our island-merger models to the (self-similar) decay of magnetohydrodynamic turbulence.