Improved calculation of the γ*γ → π process at low Q2 using LCSR’s and renormalization-group summation

We study two versions of lightcone sum rules to calculate the γ*γ → π0 transition form factor (TFF) within QCD. While the standard version is based on fixed-order perturbation theory by means of a power-series expansion in the strong coupling, the new method incorporates radiative corrections by renormalization-group summation and generates an expansion within a generalized fractional analytic perturbation theory involving only analytic couplings. Using this scheme, we determine the relative nonperturbative parameters and the first two Gegenbauer coefficients of the pion distribution amplitude (DA) to obtain TFF predictions in good agreement with the preliminary BESIII data, while the best-fit pion DA satisfies the most recent lattice constraints on the second moment of the pion DA at the three-loop level.

Symmetry Methods and Conservation Laws for the Nonlinear Generalized 2D Equal-Width Partial Differential Equation of Engineering

In this work, we study the generalized 2D equal-width equation which arises in various fields of science. With the aid of numerous methods which includes Lie symmetry analysis, power series expansion and Weierstrass method, we produce closed-form solutions of this model. The exact solutions obtained are the snoidal wave, cnoidal wave, Weierstrass elliptic function, Jacobi elliptic cosine function, solitary wave and exponential function solutions. Moreover, we give a graphical representation of the obtained solutions using certain parametric values. Furthermore, the conserved vectors of the underlying equation are constructed by utilizing two approaches: the multiplier method and Noether’s theorem. The multiplier method provided us with four local conservation laws, whereas Noether’s theorem yielded five nonlocal conservation laws. The conservation laws that are constructed contain the conservation of energy and momentum.

A vortex sheet based analytical model of the curled wake behind yawed wind turbines

Motivated by the need for compact descriptions of the evolution of non-classical wakes behind yawed wind turbines, we develop an analytical model to predict the shape of curled wakes. Interest in such modelling arises due to the potential of wake steering as a strategy for mitigating power reduction and unsteady loading of downstream turbines in wind farms. We first estimate the distribution of the shed vorticity at the wake edge due to both yaw offset and rotating blades. By considering the wake edge as an ideally thin vortex sheet, we describe its evolution in time moving with the flow. Vortex sheet equations are solved using a power series expansion method, and an approximate solution for the wake shape is obtained. The vortex sheet time evolution is then mapped into a spatial evolution by using a convection velocity. Apart from the wake shape, the lateral deflection of the wake including ground effects is modelled. Our results show that there exists a universal solution for the shape of curled wakes if suitable dimensionless variables are employed. For the case of turbulent boundary layer inflow, the decay of vortex sheet circulation due to turbulent diffusion is included. Finally, we modify the Gaussian wake model by incorporating the predicted shape and deflection of the curled wake, so that we can calculate the wake profiles behind yawed turbines. Model predictions are validated against large-eddy simulations and laboratory experiments for turbines with various operating conditions.

Computational Methods for Light Scattering by Metallic Nanoparticles

<p>The unifying theme of this thesis is that of light scattering by particles, using computational approaches. This contributions here are separated into two main areas. The first consists of examining the behaviour of the extended boundary-condition method and T-matrix method, and providing a modified set of equations to use to calculate the relevant integrals. From this, some linear relations between integrals were found, which hint at the possibility of a more efficient means of performing these calculations. As well as this, the severe numerical problems associated with this method were investigated, and the primary source of these problems was identified in the case of two commonly-used shapes, spheroids and offset spheres. The cause of these numerical problems is that dominant, leading terms in the power series expansion of the integrands integrate identically to zero, but in practice, numerical calculations have insufficient precision to compute this exactly, and the overwhelming errors from this lead to drastically incorrect results. Following this identification, a new formulation of the integrals for spheroids is presented, which allows the much easier treatment of spheroids, approaching the level of ease of calculations for spheres in Mie theory. This formulation replaces some terms in the integrands with modified terms, that do not contain the parts of the power series that cause problems. As these should integrate to zero, we are able to remove them from the integrand without affecting the correct result. The second area of this thesis is concerned with calculations of the near-field for systems of interest in plasmonics, and specifically in surface-enhanced Raman spectroscopy. Here, the enhancement of the electric field in the vicinity of a metallic surface has a large effect on measured signals. The contribution of this thesis is to study the geometric parameters that influence the distribution of the field enhancement at the particle's resonance, specifically focusing on different effects caused by the overall shape of the particle, as opposed to those effects due to the local shape of the particle in regions of high enhancement. It is shown that the overall shape determines the location of the resonance, while the local shape determines how strongly the enhancement is localised. Understanding the factors that influence the enhancement localisation will help in guiding the design of suitable plasmonic substrates.</p>

Computational Methods for Light Scattering by Metallic Nanoparticles

<p>The unifying theme of this thesis is that of light scattering by particles, using computational approaches. This contributions here are separated into two main areas. The first consists of examining the behaviour of the extended boundary-condition method and T-matrix method, and providing a modified set of equations to use to calculate the relevant integrals. From this, some linear relations between integrals were found, which hint at the possibility of a more efficient means of performing these calculations. As well as this, the severe numerical problems associated with this method were investigated, and the primary source of these problems was identified in the case of two commonly-used shapes, spheroids and offset spheres. The cause of these numerical problems is that dominant, leading terms in the power series expansion of the integrands integrate identically to zero, but in practice, numerical calculations have insufficient precision to compute this exactly, and the overwhelming errors from this lead to drastically incorrect results. Following this identification, a new formulation of the integrals for spheroids is presented, which allows the much easier treatment of spheroids, approaching the level of ease of calculations for spheres in Mie theory. This formulation replaces some terms in the integrands with modified terms, that do not contain the parts of the power series that cause problems. As these should integrate to zero, we are able to remove them from the integrand without affecting the correct result. The second area of this thesis is concerned with calculations of the near-field for systems of interest in plasmonics, and specifically in surface-enhanced Raman spectroscopy. Here, the enhancement of the electric field in the vicinity of a metallic surface has a large effect on measured signals. The contribution of this thesis is to study the geometric parameters that influence the distribution of the field enhancement at the particle's resonance, specifically focusing on different effects caused by the overall shape of the particle, as opposed to those effects due to the local shape of the particle in regions of high enhancement. It is shown that the overall shape determines the location of the resonance, while the local shape determines how strongly the enhancement is localised. Understanding the factors that influence the enhancement localisation will help in guiding the design of suitable plasmonic substrates.</p>

Semi-analytic treatment of mixed hyperbolic–elliptic Cauchy problem modeling three-phase flow in porous media

This work provides a technical applied description of the residual power series method (RPSM) to develop a fast and accurate algorithm for mixed hyperbolic–elliptic systems of conservation laws with Riemann initial datum. The RPSM does not require discretization, reduces the system to an explicit system of algebraic equations and consequently of massive and complex computations, and provides the solution in a form of Taylor power series expansion of closed-form exact solution (if exists). Theoretically, convergence hypotheses are discussed, and error bounds of exponential rates are derived. Numerically, the convergence and stability of approximate solutions are achieved for systems of mixed type. The reported results, with application to general Cauchy problems, which rise in diverse branches of physics and engineering, reveal the reliability, efficiency, and economical implementation of the proposed algorithm for handling nonlinear partial differential equations in applied mathematics.

Optimization of a Range-Separated UW12 Hybrid Functional

In a previous paper we presented a new hybrid functional B-LYP-osUW12-D3(BJ) containing the Unsöld-w12 (UW12) hybrid correlation model. In this paper we present a new 15-parameter range-separated hybrid density functional using a power series expansion together with UW12 correlation. This functional is optimised using the survival of the fittest strategy developed for the ωB97X-V functional, fitted to data from the Main Group Chemistry Database (MGCDB84). In addition we optimize a standard hybrid and double hybrid using the same method. We show that our fully self-consistent UW12 hybrid functional WM21-D3(BJ) outperforms both of these functionals and other range-separated hybrid functionals.

New Bounds for the Sine Function and Tangent Function

Using the power series expansion technique, this paper established two new inequalities for the sine function and tangent function bounded by the functions x2sin(λx)/(λx)α and x2tan(μx)/(μx)β. These results are better than the ones in the previous literature.

The HVT Technique and the "Uncertainty" Relation for Central

The quantum mechanical hypervirial theorems (HVT) technique is used to treat the so-called "uncertainty" relation for quite a wide class of central potential wells, including the (reduced) Poeschl-Teller and the Gaussian one.It is shown that this technique is quite suitable in deriving an approximate analytic expression in the form of a truncated power series expansion for the dimensionless product $P_{nl}\equiv <r^2>_{nl}<p^2>_{nl}/\hbar^2$, for every (deeply) bound state of a particle moving non-relativistically in the well, provided that a (dimensionless) parameter s is sufficiently small. Numerical results are also given and discussed.