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.

Generalization of the Gross–Pitaevskii equation for multi-particle cases

A new method is proposed to obtain Gross–Pitaevskii equation by the chain of Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) quantum kinetic equations. In that sense, we investigate the dynamics of a quantum system including infinite number of identical particles which interact via a (special) pair potential on the form of Dirac delta-function.

Convergence of eigenfunction expansions for flexural gravity waves in infinite water depth

In the present paper, point-wise convergence of the eigenfunction expansion to the velocity potential associated with the flexural gravity waves problem in water wave theory is established for infinite water depth case. To take into account the hydroelastic boundary condition at the free surface, a flexible membrane is assumed to float in water waves. In this context, firstly the eigenfunction expansion for the velocity potentials is obtained. Thereafter, an appropriate Green’s function is constructed for the associated boundary value problem. Using suitable properties of the Green’s functions, the vertical components of the eigenfunction expansion is written in terms of the Dirac delta function. Finally, using the property of the Dirac delta function, the convergence of the eigenfunction expansion to the velocity potential is shown.

On the calculus of Dirac delta function with some applications in classical electrodynamics

The Dirac delta function is a concept that is useful throughout physics as a standard mathematical tool that appears repeatedly in the undergraduate physics curriculum including electrodynamics, optics, and quantum mechanics. Our analysis was guided by an analytical framework focusing on how students activate, construct, execute, and reflect on the Dirac delta function in the context of classical electrodynamics problems solving. It’s applications in solving the charge density associated with a point charge as well as electrostatic point dipole field, for more advanced situations to describe the charge density of hydrogen atom were presented.

Penerapan Fungsi Green dari Persamaan Poisson pada Elektrostatika

The Dirac delta function is a function that mathematically does not meet the criteria as a function, this is because the function has an infinite value at a point. However, in physics the Dirac Delta function is an important construction, one of which is in constructing the Green function. This research constructs the Green function by utilizing the Dirac Delta function and Green identity. Furthermore, the construction is directed at the Green function of the Poisson's equation which is equipped with the Dirichlet boundary condition. After the form of the Green function solution from the Poisson's equation is obtained, the Green function is determined by means of the expansion of the eigen functions in the Poisson's equation. These results are used to analyze the application of the Poisson equation in electrostatic.

The Fractional Derivative of the Dirac Delta Function and Additional Results on the Inverse Laplace Transform of Irrational Functions

Motivated from studies on anomalous relaxation and diffusion, we show that the memory function M(t) of complex materials, that their creep compliance follows a power law, J(t)∼tq with q∈R+, is proportional to the fractional derivative of the Dirac delta function, dqδ(t−0)dtq with q∈R+. This leads to the finding that the inverse Laplace transform of sq for any q∈R+ is the fractional derivative of the Dirac delta function, dqδ(t−0)dtq. This result, in association with the convolution theorem, makes possible the calculation of the inverse Laplace transform of sqsα∓λ where α<q∈R+, which is the fractional derivative of order q of the Rabotnov function εα−1(±λ,t)=tα−1Eα,α(±λtα). The fractional derivative of order q∈R+ of the Rabotnov function, εα−1(±λ,t) produces singularities that are extracted with a finite number of fractional derivatives of the Dirac delta function depending on the strength of q in association with the recurrence formula of the two-parameter Mittag–Leffler function.

On the asymptotics of wright functions of the second kind

The asymptotic expansions of the Wright functions of the second kind, introduced by Mainardi [see Appendix F of his book Fractional Calculus and Waves in Linear Viscoelasticity (2010)], F σ ( x ) = ∑ n = 0 ∞ ( − x ) n n ! Γ ( − n σ ) , M σ ( x ) = ∑ n = 0 ∞ ( − x ) n n ! Γ ( − n σ + 1 − σ ) ( 0 < σ < 1 ) $$F_\sigma(x)=\sum\limits_{n=0}^\infty \frac{(-x)^n}{n! {\mathrm{\Gamma}}(-n\sigma)}~,\quad M_\sigma(x)=\sum\limits_{n=0}^\infty \frac{(-x)^n}{n! {\mathrm{\Gamma}}(-n\sigma+1-\sigma)}\quad(0 \lt \sigma \lt 1) $$ for x → ± ∞ are presented. The situation corresponding to the limit σ → 1− is considered, where M σ (x) approaches the Dirac delta function δ(x − 1). Numerical results are given to demonstrate the accuracy of the expansions derived in the paper, together with graphical illustrations that reveal the transition to a Dirac delta function as σ → 1−.