Search results for "Mathematical analysis"

A simple method for limiting the quadrature oscillator amplitude

1981

Exact solutions for the mutual inductance of circular coils and elliptic coils

2012

An exact solution is presented for the mutual inductance between general noncoaxial thin circular and elliptic coils with parallel axes. The thin coil solution is given as an angular integral of an elliptic integral expression. In addition, for the coaxial case, an exact solution is given for the mutual inductance of a thick circular coil and a thick elliptic coil. The elliptic coil is such that the coil thickness is the same along both elliptic semi-axes. The thick coil solution is given as an integral of an expression involving Bessel and Struve functions. Extensive numerical results for sample geometries are given for both solutions, which are cross checked against each other in the limi…

Time-optimal selective pulses of two uncoupled spin-1/2 particles

2018

We investigate the time-optimal solution of the selective control of two uncoupled spin 1/2 particles. Using the Pontryagin Maximum Principle, we derive the global time-optimal pulses for two spins with different offsets. We show that the Pontryagin Hamiltonian can be written as a one-dimensional effective Hamiltonian. The optimal fields can be expressed analytically in terms of elliptic integrals. The time-optimal control problem is solved for the selective inversion and excitation processes. A bifurcation in the structure of the control fields occurs for a specific offset threshold. In particular, we show that for small offsets, the optimal solution is the concatenation of regular and sin…

Nonlocally-induced (fractional) bound states: Shape analysis in the infinite Cauchy well

2015

Fractional (L\'{e}vy-type) operators are known to be spatially nonlocal. This becomes an issue if confronted with a priori imposed exterior Dirichlet boundary data. We address spectral properties of the prototype example of the Cauchy operator $(-\Delta )^{1/2}$ in the interval $D=(-1,1) \subset R$, with a focus on functional shapes of lowest eigenfunctions and their fall-off at the boundaries of $D$. New high accuracy formulas are deduced for approximate eigenfunctions. We analyze how their shape reproduction fidelity is correlated with the evaluation finesse of the corresponding eigenvalues.

Comparative study of monotonically convergent optimization algorithms for the control of molecular rotation

2013

We apply two different monotonically convergent optimization algorithms to the control of molecular rotational dynamics by laser pulses. This example represents a quantum control problem where the interaction of the system with the external field is non-linear. We test the validity and accuracy of the two methods on the key control targets of producing molecular orientation and planar delocalization at zero temperature, and maximizing permanent alignment at non-zero temperature.

Distillation by repeated measurements: Continuous spectrum case

2010

Repeated measurements on a part of a bipartite system strongly affect the other part not measured, whose dynamics is regulated by an effective contracted evolution operator. When the spectrum of this operator is discrete, the latter system is driven into a pure state irrespective of the initial state, provided the spectrum satisfies certain conditions. We here show that even in the case of continuous spectrum an effective distillation can occur under rather general conditions. We confirm it by applying our formalism to a simple model.

Killing (absorption) versus survival in random motion

2017

We address diffusion processes in a bounded domain, while focusing on somewhat unexplored affinities between the presence of absorbing and/or inaccessible boundaries. For the Brownian motion (L\'{e}vy-stable cases are briefly mentioned) model-independent features are established, of the dynamical law that underlies the short time behavior of these random paths, whose overall life-time is predefined to be long. As a by-product, the limiting regime of a permanent trapping in a domain is obtained. We demonstrate that the adopted conditioning method, involving the so-called Bernstein transition function, works properly also in an unbounded domain, for stochastic processes with killing (Feynman-…

QCD condensates from tau-decay data: A functional approach

2004

We study a functional method to extract the V − A condensate of dimension 6 from a comparison of τ -decay data with the asymptotic space-like QCD prediction. Our result is in agreement within errors with that from conventional analyses based on finite energy sum rules.

Teaching stable two-mirror resonators through the fractional Fourier transform

2009

We analyse two-mirror resonators in terms of their fractional Fourier transform (FRFT) properties. We use the basic ABCD ray transfer matrix method to show how the resonator can be regarded as the cascade of two propagation–lens–propagation FRFT systems. Then, we present a connection between the geometric properties of the resonator (the g parameters) and those of the equivalent FRFT systems (the FRFT order and scaling parameters). Expressions connecting Gaussian beam q-transformation with FRFT parameters are derived. In particular, we show that the beam waist of the resonator's mode is located at the plane leading to two FRFT subsystems with equal scaling parameter which, moreover, coincid…

On the regularity of critical and minimal sets of a free interface problem

2015

We study a free interface problem of finding the optimal energy configuration for mixtures of two conducting materials with an additional perimeter penalization of the interface. We employ the regularity theory of linear elliptic equations to study the possible opening angles of Taylor cones and to give a different proof of a partial regularity result by Fan Hua Lin [Calc Var. Partial Differential Equations, 1993].