Quasi-exactly-solvable confining solutions for spin-1 and spin-0 bosons in (1+1)-dimensions with a scalar linear potential

We point out a misleading treatment in the recent literature regarding confining solutions for a scalar potential in the context of the Duffin-Kemmer-Petiau theory. We further present the proper bound-state solutions in terms of the generalized Laguerre polynomials and show that the eigenvalues and eigenfunctions depend on the solutions of algebraic equations involving the potential parameter and the quantum number. (C) 2014 Elsevier Inc. All rights reserved.

On the maximum and minimum values of a linear function of the radial co-ordinates of a point with respect to simplicissimum in space of n dimensions.

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Linear and nonlinear propagation of localised plasmon in metallic nanostructures

A major challenge in modern photonics and nano-optics is the diffraction limit of light which does not allow field localisation into regions with dimensions smaller than half the wavelength. Localisation of light into nanoscale regions (beyond its diffraction limit) has applications ranging from the design of optical sensors and measurement techniques with resolutions as high as a few nanometres, to the effective delivery of optical energy into targeted nanoscale regions such as quantum dots, nano-electronic and nano-optical devices. This field has become a major research direction over the last decade. The use of strongly localised surface plasmons in metallic nanostructures is one of the most promising approaches to overcome this problem. Therefore, the aim of this thesis is to investigate the linear and non-linear propagation of surface plasmons in metallic nanostructures. This thesis will focus on two main areas of plasmonic research –– plasmon nanofocusing and plasmon nanoguiding. Plasmon nanofocusing – The main aim of plasmon nanofocusing research is to focus plasmon energy into nanoscale regions using metallic nanostructures and at the same time achieve strong local field enhancement. Various structures for nanofocusing purposes have been proposed and analysed such as sharp metal wedges, tapered metal films on dielectric substrates, tapered metal rods, and dielectric V-grooves in metals. However, a number of important practical issues related to nanofocusing in these structures still remain unclear. Therefore, one of the main aims of this thesis is to address two of the most important of issues which are the coupling efficiency and heating effects of surface plasmons in metallic nanostructures. The method of analysis developed throughout this thesis is a general treatment that can be applied to a diversity of nanofocusing structures, with results shown here for the specific case of sharp metal wedges. Based on the geometrical optics approximation, it is demonstrated that the coupling efficiency from plasmons generated with a metal grating into the nanofocused symmetric or quasi-symmetric modes may vary between ~50% to ~100% depending on the structural parameters. Optimal conditions for nanofocusing with the view to minimise coupling and dissipative losses are also determined and discussed. It is shown that the temperature near the tip of a metal wedge heated by nanosecond plasmonic pulses can increase by several hundred degrees Celsius. This temperature increase is expected to lead to nonlinear effects, self-influence of the focused plasmon, and ultimately self-destruction of the metal tip. This thesis also investigates a different type of nanofocusing structure which consists of a tapered high-index dielectric layer resting on a metal surface. It is shown that the nanofocusing mechanism that occurs in this structure is somewhat different from other structures that have been considered thus far. For example, the surface plasmon experiences significant backreflection and mode transformation at a cut-off thickness. In addition, the reflected plasmon shows negative refraction properties that have not been observed in other nanofocusing structures considered to date. Plasmon nanoguiding – Guiding surface plasmons using metallic nanostructures is important for the development of highly integrated optical components and circuits which are expected to have a superior performance compared to their electronicbased counterparts. A number of different plasmonic waveguides have been considered over the last decade including the recently considered gap and trench plasmon waveguides. The gap and trench plasmon waveguides have proven to be difficult to fabricate. Therefore, this thesis will propose and analyse four different modified gap and trench plasmon waveguides that are expected to be easier to fabricate, and at the same time acquire improved propagation characteristics of the guided mode. In particular, it is demonstrated that the guided modes are significantly screened by the extended metal at the bottom of the structure. This is important for the design of highly integrated optics as it provides the opportunity to place two waveguides close together without significant cross-talk. This thesis also investigates the use of plasmonic nanowires to construct a Fabry-Pérot resonator/interferometer. It is shown that the resonance effect can be achieved with the appropriate resonator length and gap width. Typical quality factors of the Fabry- Pérot cavity are determined and explained in terms of radiative and dissipative losses. The possibility of using a nanowire resonator for the design of plasmonic filters with close to ~100% transmission is also demonstrated. It is expected that the results obtained in this thesis will play a vital role in the development of high resolution near field microscopy and spectroscopy, new measurement techniques and devices for single molecule detection, highly integrated optical devices, and nanobiotechnology devices for diagnostics of living cells.

Fast exact nearest neighbour matching in high dimensions using d-D Sort

Data structures such as k-D trees and hierarchical k-means trees perform very well in approximate k nearest neighbour matching, but are only marginally more effective than linear search when performing exact matching in high-dimensional image descriptor data. This paper presents several improvements to linear search that allows it to outperform existing methods and recommends two approaches to exact matching. The first method reduces the number of operations by evaluating the distance measure in order of significance of the query dimensions and terminating when the partial distance exceeds the search threshold. This method does not require preprocessing and significantly outperforms existing methods. The second method improves query speed further by presorting the data using a data structure called d-D sort. The order information is used as a priority queue to reduce the time taken to find the exact match and to restrict the range of data searched. Construction of the d-D sort structure is very simple to implement, does not require any parameter tuning, and requires significantly less time than the best-performing tree structure, and data can be added to the structure relatively efficiently.

Non-linear model predictive control for UAVs with slung/swung load

Suspended loads on UAVs can provide significant benefits to several applications in agriculture, law enforcement and construction. The load impact on the underlying system dynamics should not be neglected as significant feedback forces may be induced on the vehicle during certain flight manoeuvres. Much research has focused on standard multi-rotor position and attitude control with and without a slung load. However, predictive control schemes, such as Nonlinear Model Predictive Control (NMPC), have not yet been fully explored. To this end, we present software and flight system architecture to test controller for safe and precise operation of multi-rotors with heavy slung load in three dimensions.

Carbon nanotube-based super nanotubes: Tunable thermal conductivity in three dimensions

Advances in nanomaterials/nanostructures offer the possibility of fabricating multifunctional materials for use in engineering applications. Carbon nanotube (CNT)-based nanostructures are a representative building block for these multifunctional materials. Based on a series of in silico studies, we investigated the possibility of tuning the thermal conductivity of a three-dimensional CNT-based nanostructure: a single-walled CNT-based super-nanotube. The thermal conductivity of the super-nanotubes was shown to vary with different connecting carbon rings and super-nanotubes with longer constituent single-walled CNTs and larger diameters had a smaller thermal conductivity. The inverse of the thermal conductivity of the super-nanotubes showed a good linear relationship with the inverse of the length. The thermal conductivity was approximately proportional to the inverse of the temperature, but was insensitive to the axial strain as a result of the Poisson ratio. These results provide a fundamental understanding of the thermal conductivity of the super-nanotubes and will guide their future design/fabrication and engineering applications.

Refractive indices used by the Haag-Streit Lenstar to calculate axial biometric dimensions

PURPOSE To estimate refractive indices used by the Lenstar biometer to translate measured optical path lengths into geometrical path lengths within the eye. METHODS Axial lengths of model eyes were determined using the IOLMaster and Lenstar biometers; comparing those lengths gave an overall eye refractive index estimate for the Lenstar. Using the Lenstar Graphical User Interface, we noticed that boundaries between media could be manipulated and opposite changes in optical path lengths on either side of the boundary could be introduced. Those ratios were combined with the overall eye refractive index to estimate separate refractive indices. Furthermore, Haag-Streit provided us with a template to obtain 'air thicknesses' to compare with geometrical distances. RESULTS The axial length estimates obtained using the IOLMaster and the Lenstar agreed to within 0.01 mm. Estimates of group refractive indices used in the Lenstar were 1.340, 1.341, 1.415, and 1.354 for cornea, aqueous, lens, and overall eye, respectively. Those refractive indices did not match those of schematic eyes, but were close in the cases of aqueous and lens. Linear equations relating air thicknesses to geometrical thicknesses were consistent with our findings. CONCLUSION The Lenstar uses different refractive indices for different ocular media. Some of the refractive indices, such as that for the cornea, are not physiological; therefore, it is likely that the calibrations in the instrument correspond to instrument-specific corrections and are not the real optical path lengths.

Crystal engineering in two dimensions: An approach to molecular nanopatterning

We describe a surprising cooperative adsorption process observed by scanning tunneling microscopy (STM) at the liquid−solid interface. The process involves the association of a threefold hydrogen-bonding unit, trimesic acid (TMA), with straight-chain aliphatic alcohols of varying length (from C7 to C30), which coadsorb on highly oriented pyrolytic graphite (HOPG) to form linear patterns. In certain cases, the known TMA “flower pattern” can coexist temporarily with the linear TMA−alcohol patterns, but it eventually disappears. Time-lapsed STM imaging shows that the evolution of the flower pattern is a classical ripening phenomenon. The periodicity of the linear TMA−alcohol patterns can be modulated by choosing alcohols with appropriate chain lengths, and the precise structure of the patterns depends on the parity of the carbon count in the alkyl chain. Interactions that lead to this odd−even effect are analyzed in detail. The molecular components of the patterns are achiral, yet their association by hydrogen bonding leads to the formation of enantiomeric domains on the surface. The interrelation of these domains and the observation of superperiodic structures (moiré patterns) are rationalized by considering interactions with the underlying graphite surface and within the two-dimensional crystal of the adsorbed molecules. Comparison of the observed two-dimensional structures with the three-dimensional crystal structures of TMA−alcohol complexes determined by X-ray crystallography helps reveal the mechanism of molecular association in these two-component systems.

Fluorescent resonant excitation energy transfer in linear polyenes

We have studied the dynamics of excitation transfer between two conjugated polyene molecules whose intermolecular separation is comparable to the molecular dimensions. We have employed a correlated electron model that includes both the charge-charge, charge-bond, and bond-bond intermolecular electron repulsion integrals. We have shown that the excitation transfer rate varies as inverse square of donor-acceptor separation R-2 rather than as R-6, suggested by the Foumlrster type of dipolar approximation. Our time-evolution study alsom shows that the orientational dependence on excitation transfer at a fixed short donor-acceptor separation cannot be explained by Foumlrster type of dipolar approximation beyond a certain orientational angle of rotation of an acceptor polyene with respect to the donor polyene. The actual excitation transfer rate beyond a certain orientational angle is faster than the Foumlrster type of dipolar approximation rate. We have also studied the excitation transfer process in a pair of push-pull polyenes for different push-pull strengths. We have seen that, depending on the push-pull strength, excitation transfer could occur to other dipole coupled states. Our study also allows for the excitation energy transfer to optically dark states which are excluded by Foumlrster theory since the one-photon transition intensity to these states (from the ground state) is zero.

Colossal nonsaturating linear magnetoresistance in two-dimensional electron systems at a GaAs/(Al,Ga)As heterointerface

Engineering devices with a large electrical response to magnetic field is of fundamental importance for a range of applications such as magnetic field sensing and magnetic read heads. We show that a colossal nonsaturating linear magnetoresistance (NLMR) arises in two-dimensional electron systems hosted in a GaAs/AlGaAs heterostructure in the strongly insulating regime. When operated at high source-drain bias, the magnetoresistance of our devices increases almost linearly with magnetic field, reaching nearly 10 000% at 8 T, thus surpassing many known nonmagnetic materials that exhibit giant NLMR. The temperature dependence and mobility analysis indicate that the NLMR has a purely classical origin, driven by nanoscale inhomogeneities. A large NLMR combined with small device dimensions makes these systems an attractive candidate for on-chip magnetic field sensing.

Representing a cubic graph as the intersection graph of axis-parallel boxes in three dimensions

We show that every graph of maximum degree 3 can be represented as the intersection graph of axis parallel boxes in three dimensions, that is, every vertex can be mapped to an axis parallel box such that two boxes intersect if and only if their corresponding vertices are adjacent. In fact, we construct a representation in which any two intersecting boxes just touch at their boundaries. Further, this construction can be realized in linear time.

Analysis of instability of a cylindrical soap film of finite dimensions

By means of experiments of instability of a uniform cylindrical soap film, Boys had showed that the bubble molded by the film is unstable when its length is greater than its circumference. Recently that is generally called the Rayleigh Criterion. In this paper, a linear theory in hydrodynamics is applied to analyze the stability of the cylindrical soap film supported by two equal size disks; all conditions of the stationary wave on the end plates of two disks are given. From here we get that the Rayleigh Criterion on the stability of the cylindrical soap film is proved.

Dynamics of cell–matrix mechanical interactions in three dimensions

The forces cells apply to their surroundings control biological processes such as growth, adhesion, development, and migration. In the past 20 years, a number of experimental techniques have been developed to measure such cell tractions. These approaches have primarily measured the tractions applied by cells to synthetic two-dimensional substrates, which do not mimic in vivo conditions for most cell types. Many cell types live in a fibrous three-dimensional (3D) matrix environment. While studying cell behavior in such 3D matrices will provide valuable insights for the mechanobiology and tissue engineering communities, no experimental approaches have yet measured cell tractions in a fibrous 3D matrix.

This thesis describes the development and application of an experimental technique for quantifying cellular forces in a natural 3D matrix. Cells and their surrounding matrix are imaged in three dimensions with high speed confocal microscopy. The cell-induced matrix displacements are computed from the 3D image volumes using digital volume correlation. The strain tensor in the 3D matrix is computed by differentiating the displacements, and the stress tensor is computed by applying a constitutive law. Finally, tractions applied by the cells to the matrix are computed directly from the stress tensor.

The 3D traction measurement approach is used to investigate how cells mechanically interact with the matrix in biologically relevant processes such as division and invasion. During division, a single mother cell undergoes a drastic morphological change to split into two daughter cells. In a 3D matrix, dividing cells apply tensile force to the matrix through thin, persistent extensions that in turn direct the orientation and location of the daughter cells. Cell invasion into a 3D matrix is the first step required for cell migration in three dimensions. During invasion, cells initially apply minimal tractions to the matrix as they extend thin protrusions into the matrix fiber network. The invading cells anchor themselves to the matrix using these protrusions, and subsequently pull on the matrix to propel themselves forward.

Lastly, this thesis describes a constitutive model for the 3D fibrous matrix that uses a finite element (FE) approach. The FE model simulates the fibrous microstructure of the matrix and matches the cell-induced matrix displacements observed experimentally using digital volume correlation. The model is applied to predict how cells mechanically sense one another in a 3D matrix. It is found that cell-induced matrix displacements localize along linear paths. These linear paths propagate over a long range through the fibrous matrix, and provide a mechanism for cell-cell signaling and mechanosensing. The FE model developed here has the potential to reveal the effects of matrix density, inhomogeneity, and anisotropy in signaling cell behavior through mechanotransduction.

Some generalizations of commutativity for linear transformations on a finite dimensional vector space

Let L be the algebra of all linear transformations on an n-dimensional vector space V over a field F and let A, B, ƐL. Let A_i+1 = A_iB - BA_i, i = 0, 1, 2,…, with A = A_o. Let f_k (A, B; σ) = A_2K+1 - ^σ1^A2K-1 ^{+ σ}2^A2K-3 -… +(-1)^Kσ_KA₁ where σ = (σ₁, σ₂,…, σ_K), σ_i belong to F and K = k(k-1)/2. Taussky and Wielandt [Proc. Amer. Math. Soc., 13(1962), 732-735] showed that f_n(A, B; σ) = 0 if σ_i is the i^th elementary symmetric function of (β₄- β_s)², 1 ≤ r ˂ s ≤ n, i = 1, 2, …, N, with N = n(n-1)/2, where β₄ are the characteristic roots of B. In this thesis we discuss relations involving f_k(X, Y; σ) where X, Y Ɛ L and 1 ≤ k ˂ n. We show: 1. If F is infinite and if for each X Ɛ L there exists σ so that f_k(A, X; σ) = 0 where 1 ≤ k ˂ n, then A is a scalar transformation. 2. If F is algebraically closed, a necessary and sufficient condition that there exists a basis of V with respect to which the matrices of A and B are both in block upper triangular form, where the blocks on the diagonals are either one- or two-dimensional, is that certain products X₁, X₂…X_r belong to the radical of the algebra generated by A and B over F, where X_i has the form f₂(A, P(A,B); σ), for all polynomials P(x, y). We partially generalize this to the case where the blocks have dimensions ≤ k. 3. If A and B generate L, if the characteristic of F does not divide n and if there exists σ so that f_k(A, B; σ) = 0, for some k with 1 ≤ k ˂ n, then the characteristic roots of B belong to the splitting field of g_k(w; σ) = w^2K+1 - σ₁w^2K-1 + σ₂w^2K-3 - …. +(-1)^K σ_Kw over F. We use this result to prove a theorem involving a generalized form of property L [cf. Motzkin and Taussky, Trans. Amer. Math. Soc., 73(1952), 108-114]. 4. Also we give mild generalizations of results of McCoy [Amer. Math. Soc. Bull., 42(1936), 592-600] and Drazin [Proc. London Math. Soc., 1(1951), 222-231].