Asymptotic normality of the Parzen-Rosenblatt density estimator for strongly mixing random fields

We prove the asymptotic normality of the kernel density estimator (introduced by Rosenblatt (1956) and Parzen (1962)) in the context of stationary strongly mixing random fields. Our approach is based on the Lindeberg's method rather than on Bernstein's small-block-large-block technique and coupling arguments widely used in previous works on nonparametric estimation for spatial processes. Our method allows us to consider only minimal conditions on the bandwidth parameter and provides a simple criterion on the (non-uniform) strong mixing coefficients which do not depend on the bandwith.Comment: 16 page

Probing the fuzzy sphere regularisation in simulations of the 3d \lambda \phi^4 model

We regularise the 3d \lambda \phi^4 model by discretising the Euclidean time and representing the spatial part on a fuzzy sphere. The latter involves a truncated expansion of the field in spherical harmonics. This yields a numerically tractable formulation, which constitutes an unconventional alternative to the lattice. In contrast to the 2d version, the radius R plays an independent r\^{o}le. We explore the phase diagram in terms of R and the cutoff, as well as the parameters m^2 and \lambda. Thus we identify the phases of disorder, uniform order and non-uniform order. We compare the result to the phase diagrams of the 3d model on a non-commutative torus, and of the 2d model on a fuzzy sphere. Our data at strong coupling reproduce accurately the behaviour of a matrix chain, which corresponds to the c=1-model in string theory. This observation enables a conjecture about the thermodynamic limit.Comment: 31 pages, 15 figure

A spatially uniform illumination source for widefield multi-spectral optical microscopy

Illumination uniformity is a critical parameter for excitation and data extraction quality in widefield biological imaging applications. However, typical imaging systems suffer from spatial and spectral non-uniformity due to non-ideal optical elements, thus require complex solutions for illumination corrections. We present Effective Uniform Color-Light Integration Device (EUCLID), a simple and cost-effective illumination source for uniformity corrections. EUCLID employs a diffuse-reflective, adjustable hollow cavity that allows for uniform mixing of light from discrete light sources and modifies the source field distribution to compensate for spatial non-uniformity introduced by optical components in the imaging system. In this study, we characterize the light coupling efficiency of the proposed design and compare the uniformity performance with the conventional method. EUCLID demonstrates a remarkable illumination improvement for multi-spectral imaging in both Nelsonian and Koehler alignment with a maximum spatial deviation of ~1% across a wide field-of-view

Neel to spin-Peierls transition in a quasi-1D Heisenberg model coupled to bond phonons

The zero and finite temperature spin-Peierls transitions in a quasi-one-dimensional spin-1/2 Heisenberg model coupled to adiabatic bond phonons is investigated using the Stochastic Series Expansion (SSE) Quantum Monte Carlo (QMC) method. The quantum phase transition from a gapless Neel state to a spin-gapped Peierls state is studied in the parameter space spanned by spatial anisotropy, inter-chain coupling strength and spin-lattice coupling strength. It is found that for any finite inter-chain coupling, the transition to a dimerized Peierls ground state only occurs when the spin-lattice coupling exceeds a finite, non-zero critical value. This is in contrast to the pure 1D model (zero inter-chain coupling), where adiabatic/classical phonons lead to a dimerized ground state for any non-zero spin-phonon interaction. The phase diagram in the parameter space shows that for a strong inter-chain coupling, the relation between the inter-chain coupling and the critical value of the spin-phonon interaction is linear whereas for weak inter-chain coupling, this behavior is found to have a natural logarithm-like relation. No region was found to have a long range magnetic order and dimerization occurring simultaneously. Instead, the Neel state order vanishes simultaneously with the setting in of the spin-Peierls state. For the thermal phase transition, a continuous heat capacity with a peak at the critical temperature, $T_{c}$, shows a second order phase transition. The variation of the equilibrium bond length distortion, $\delta_{eq}$, with temperature showed a power law relation which decayed to zero as the temperature was increased to $T_{c}$, indicating a continuous transition from the dimerized phase to a paramagnetic phase with uniform bond length and zero antiferromagnetic susceptibility.Comment: 7 pages, 5 figures, updated and extended version of arXiv:cond-mat/030774

Simulation of macrosegregation benchmark on a non-uniform computational node arrangement with a meshless method

An application of a meshless numerical method on a macrosegregation benchmark case is developed in the present paper. The test case is solidification in 2D rectangular cavity, filled with liquid metal and chilled from both sides. This is a highly non-linear problem due to a strong coupling of the macroscopic transport equations with the microsegregation model. The main result is the macrosegregation pattern of the solidified metal Al4.5wt%Cu alloy is used for evaluation of the problem. The model uses diffuse approximate meshless method with the second-order polynomial basis for spatial integration and explicit time-stepping. Simulations are performed on uniform and non-uniform computational node arrangements and compared to each other. The results on uniform and non-uniform node arrangements show a very good matching with the finite volume method results and results based on radial basis function collocation method. This shows that diffuse approximate method based on non-uniform node arrangements can be used for solving macrosegregation problems

Simulating generic spin-boson models with matrix product states

The global coupling of few-level quantum systems ("spins") to a discrete set of bosonic modes is a key ingredient for many applications in quantum science, including large-scale entanglement generation, quantum simulation of the dynamics of long-range interacting spin models, and hybrid platforms for force and spin sensing. We present a general numerical framework for treating the out-of-equilibrium dynamics of such models based on matrix product states. Our approach applies for generic spin-boson systems: it treats any spatial and operator dependence of the two-body spin-boson coupling and places no restrictions on relative energy scales. We show that the full counting statistics of collective spin measurements and infidelity of quantum simulation due to spin-boson entanglement, both of which are difficult to obtain by other techniques, are readily calculable in our approach. We benchmark our method using a recently developed exact solution for a particular spin-boson coupling relevant to trapped ion quantum simulators. Finally, we show how decoherence can be incorporated within our framework using the method of quantum trajectories, and study the dynamics of an open-system spin-boson model with spatially non-uniform spin-boson coupling relevant for trapped atomic ion crystals in the presence of molecular ion impurities.Comment: 13 pages+refs. 13 figure

Weak localization and conductance fluctuations of a chaotic quantum dot with tunable spin-orbit coupling

In a two-dimensional quantum dot in a GaAs heterostructure, the spin-orbit scattering rate is substantially reduced below the rate in a bulk two-dimensional electron gas [B.I. Halperin et al, Phys. Rev. Lett. 86, 2106 (2001)]. Such a reduction can be undone if the spin-orbit coupling parameters acquire a spatial dependence, which can be achieved, e.g., by a metal gate covering only a part of the quantum dot. We calculate the effect of such spatially non-uniform spin-orbit scattering on the weak localization correction and the universal conductance fluctuations of a chaotic quantum dot coupled to electron reservoirs by ballistic point contacts, in the presence of a magnetic field parallel to the plane of the quantum dot.Comment: 4 pages, RevTeX; 2 figures. Substantial revision

Spin-polarized electric currents in quantum transport through tubular two-dimensional electron gases

Scattering theory is employed to derive a Landauer-type formula for the spin and the charge currents, through a finite region where spin-orbit interactions are effective. It is shown that the transmission matrix yields the spatial direction and the magnitude of the spin polarization. This formula is used to study the currents through a tubular two-dimensional electron gas. In this cylindrical geometry, which may be realized in experiment, the transverse conduction channels are not mixed (provided that the spin-orbit coupling is uniform). It is then found that for modest boundary scattering, each step in the quantized conductance is split into two, and the new steps have a non-zero spin conductance, with the spin polarization perpendicular to the direction of the current.Comment: 6 pages, 5 figure

A BIOLOGICAL BASED MODEL OF THE HUMAN VISUAL SYSTEM INCORPORATING LATERAL SUBTRACTIVE INHIBITION WITH NON-UNIFORM SAMPLING AND MULTIPLE SPATIAL FREQUENCY FILTERS

The human visual system has been an interesting topic of scientific research for decades. It is known that the cone photo-receptors are arrayed in a non-linear fashion and that a lateral subtractive inhibitory process is occurring in the visual pathway. This thesis outlines for the first time how lateral subtractive inhibition manifests itself in the context of a non-uniform sensor distribution where the distance between cone photo-receptors, and size of the receptors, are varying in a log manner when moving radially away from the foveal area. Range limits on the parameters that control the non-uniform sampling and coupling coefficients are presented and optimal values are identified for specific image resolutions. The results of this analysis are then coupled to a proposed model of spatial frequency filtering to assist in subsequent studies of feature extraction and pattern analysis. The filters generated are based on three spatial-frequency channels that are designed to model the human eye contrast sensitivity curve. Simulated results are presented