Hadamard’s variational formula in terms of stress and strain tensors

Starting from a Lagrangian action functional for two scalar fields we construct, by variational methods, the Laplacian Green function for a bounded domain and an appropriate stress tensor. By a further variation, imposed by a given vector field, we arrive at an interior version of the Hadamard variational formula, previously considered by P. Garabedian. It gives the variation of the Green function in terms of a pairing between the stress tensor and a strain tensor in the interior of the domain, this contrasting the classical Hadamard formula which is expressed as a pure boundary variation.

Harmonic extension through conical surfaces

This paper establishes extension results for harmonic functions which vanish on a conical surface. These are based on a detailed analysis of expansions for the Green function of an infinite cone.

Green function for the Grad-Shafranov operator

The Grad-Shafranov equation, often written in cylindrical coordinates, is an elliptic partial differential equation in two dimensions. It describes magnetohydrodynamic equilibria in axisymmetric toroidal plasmas, such as tokamaks, and yields the poloidal magnetic flux function, which is related to the azimuthal component of the vector potential for the magnetic field produced by a circular (toroidal) current density. The Green function for the differential operator can be obtained from the vector potential for the magnetic field of a circular current loop, which is a typical problem in magnetostatics. The purpose of the paper is to collect results scattered in electrodynamics and plasma physics textbooks for the benefit of students in the field, as well as attracting the attention of a wider audience, in the context of electrodynamics and partial differential equations.

Steklov problem of the first class for a fractional order delay differential equation

Методом функции Грина получено решение задачи Стеклова первого класса для линейного уравнения с дробной производной Герасимова-Капуто с запаздывающим аргументом. Доказана теорема существования и единственности задачи. The solution to the Steklov problem with conditions of the first class for a linear delay differential equation with a Gerasimov-Caputo fractional derivative is obtained by Green function method. The existence and uniqueness theorem to the problem is proved.

Non-Markovian transients in transport across chiral quantum wires using space-time non-equilibrium Green functions

We study a system of two non-interacting quantum wires with fermions of opposite chirality with a point contact junction at the origin across which tunneling can take place when an arbitrary time-dependent bias between the wires is applied. We obtain the exact dynamical non-equilibrium Green function by solving Dyson’s equation analytically. Both the space-time dependent two and four-point functions are written down in a closed form in terms of simple functions of position and time. This allows us to obtain, among other things, the I-V characteristics for an arbitrary time-dependent bias. Our method is a superior alternative to competing approaches to non-equilibrium as we are able to account for transient phenomena as well as the steady state. We study the approach to steady state by computing the time evolution of the equal-time one-particle Green function. Our method can be easily applied to the problem of a double barrier contact whose internal properties can be adjusted to induce resonant tunneling leading to a conductance maximum. We then consider the case of a finite bandwidth in the point contact and calculate the non-equilibrium transport properties which exhibit non-Markovian behaviour. When a subsequently constant bias is suddenly switched on, the current shows a transient build up before approaching its steady state value in contrast to the infinite bandwidth case. This transient property is consistent with numerical simulations of lattice systems using time-dependent DMRG (tDMRG) suggesting thereby that this transient build up is merely due to the presence of a short distance cutoff in the problem description and not on the other details.

Random fiber network loaded by a point force

This article presents the displacement field produced by a point force acting on an athermal random fiber network (the Green function for the network). The problem is defined within the limits of linear elasticity and the field is obtained numerically for nonaffine networks characterized by various parameter sets. The classical Green function solution applies at distances from the point force larger than a threshold which is independent of the network parameters in the range studied. At smaller distances, the nonlocal nature of fiber interactions modifies the solution.

SHIP MOTIONS DURING REPLENISHMENT AT SEA OPERATIONS IN HEAD SEAS

During replenishment at sea operations the interaction between the two vessels travelling side by side can cause significant motions in the smaller vessel and affect the relative separation between their replenishment points. A study into these motions has been conducted including theoretical predictions and model experiments. The model tests investigated the influence of supply ship displacement and longitudinal separation on the ships’ motions. The data obtained from the experimental study has been used to validate a theoretical ship motion prediction method based on a 3-D zero-speed Green function with a forward speed correction in the frequency domain. The results were also used to estimate the expected extreme roll angle of the receiving vessel, and the relative motion between the vessels, during replenishment at sea operations in a typical irregular seaway. A significant increase in the frigate’s roll response was found to occur with an increase of the supply ship displacement, whilst a reduction in motion for the receiving vessel resulted from an increase in longitudinal separation between the vessels. It is proposed that to determine the optimal vessel separation it is vital that the motions of the vessels are not considered in isolation and all motions need to be considered for both vessels simultaneously.

Positivity of sums and integrals for n-convex functions via the Fink identity and new Green functions

We consider positivity of sum $\sum_{i=1}^np_if(x_i)$ involving convex functions of higher order. Analogous for integral $\int_a^bp(x)f(g(x))dx$ is also given. Representation of a function $f$ via the Fink identity and the Green function leads us to identities for which we obtain conditions for positivity of the mentioned sum and integral. We obtain bounds for integral remainders which occur in those identities as well as corresponding mean value theorems.

Conductance peaks and gaps in single-electron device with the presence of electron-electron interaction - “Nonequilibrium green function approach”

Consider a single-electron transistor (SET) with a small size quantum dot (QD), where confined energy and the Coulomb interaction control the charges adding to QD. In this paper, a theoretical analysis of the relation between source-drain voltage and gate voltage has been done to define quantum-Coulomb blocked (and unblocked) diamonds for QD that has N electrons. An analytical equation for the conductance has been derived using the non-equilibrium Green function technique (NEGFT). Further, the effect of QD size and the tunnelling rate on conductance peaks and gaps have been investigated. Finally, the effect of gate voltage on conductance peaks and gaps with respect the quantum-Coulomb blocked regions has been analysed.

Integrals and series related to propagators of neural and haemodynamic waves

The propagator, or Green function, of a class of neural activity fields and of haemodynamic waves is evaluated exactly. The results enable a number of related integrals to be evaluated, along with series expansions of key results in terms of Bessel functions of the second kind. Connections to other related equations are also noted.