Landau-Zener problem with waiting at the minimum gap and related quench dynamics of a many-body system

We discuss a technique for solving the Landau-Zener (LZ) problem of finding the probability of excitation in a two-level system. The idea of time reversal for the Schrodinger equation is employed to obtain the state reached at the final time and hence the excitation probability. Using this method, which can reproduce the well-known expression for the LZ transition probability, we solve a variant of the LZ problem, which involves waiting at the minimum gap for a time t(w); we find an exact expression for the excitation probability as a function of t(w). We provide numerical results to support our analytical expressions. We then discuss the problem of waiting at the quantum critical point of a many-body system and calculate the residual energy generated by the time-dependent Hamiltonian. Finally, we discuss possible experimental realizations of this work.

Isoperimetric Problem and Meta-fibonacci Sequences

Let G = (V,E) be a simple, finite, undirected graph. For S ⊆ V, let $\delta(S,G) = \{ (u,v) \in E : u \in S \mbox { and } v \in V-S \}$ and $\phi(S,G) = \{ v \in V -S: \exists u \in S$ , such that (u,v) ∈ E} be the edge and vertex boundary of S, respectively. Given an integer i, 1 ≤ i ≤ ∣ V ∣, the edge and vertex isoperimetric value at i is defined as b e (i,G) =  min S ⊆ V; |S| = i |δ(S,G)| and b v (i,G) =  min S ⊆ V; |S| = i |φ(S,G)|, respectively. The edge (vertex) isoperimetric problem is to determine the value of b e (i, G) (b v (i, G)) for each i, 1 ≤ i ≤ |V|. If we have the further restriction that the set S should induce a connected subgraph of G, then the corresponding variation of the isoperimetric problem is known as the connected isoperimetric problem. The connected edge (vertex) isoperimetric values are defined in a corresponding way. It turns out that the connected edge isoperimetric and the connected vertex isoperimetric values are equal at each i, 1 ≤ i ≤ |V|, if G is a tree. Therefore we use the notation b c (i, T) to denote the connected edge (vertex) isoperimetric value of T at i. Hofstadter had introduced the interesting concept of meta-fibonacci sequences in his famous book “Gödel, Escher, Bach. An Eternal Golden Braid”. The sequence he introduced is known as the Hofstadter sequences and most of the problems he raised regarding this sequence is still open. Since then mathematicians studied many other closely related meta-fibonacci sequences such as Tanny sequences, Conway sequences, Conolly sequences etc. Let T 2 be an infinite complete binary tree. In this paper we related the connected isoperimetric problem on T 2 with the Tanny sequences which is defined by the recurrence relation a(i) = a(i − 1 − a(i − 1)) + a(i − 2 − a(i − 2)), a(0) = a(1) = a(2) = 1. In particular, we show that b c (i, T 2) = i + 2 − 2a(i), for each i ≥ 1. We also propose efficient polynomial time algorithms to find vertex isoperimetric values at i of bounded pathwidth and bounded treewidth graphs.

A FEM-Based Forward Solver for Studying the Forward Problem of Electrical Impedance Tomography (EIT) with A Practical Biological Phantom

A Finite Element Method based forward solver is developed for solving the forward problem of a 2D-Electrical Impedance Tomography. The Method of Weighted Residual technique with a Galerkin approach is used for the FEM formulation of EIT forward problem. The algorithm is written in MatLAB7.0 and the forward problem is studied with a practical biological phantom developed. EIT governing equation is numerically solved to calculate the surface potentials at the phantom boundary for a uniform conductivity. An EIT-phantom is developed with an array of 16 electrodes placed on the inner surface of the phantom tank filled with KCl solution. A sinusoidal current is injected through the current electrodes and the differential potentials across the voltage electrodes are measured. Measured data is compared with the differential potential calculated for known current and solution conductivity. Comparing measured voltage with the calculated data it is attempted to find the sources of errors to improve data quality for better image reconstruction.

Crystal structure of peanut lectin, a protein with an unusual quaternary structure

The x-ray crystal structure of the tetrameric T-antigen-binding lectin from peanut, M(r) 110,000, has been determined by using the multiple isomorphous replacement method and refined to an R value of 0.218 for 22,155 reflections within the 10- to 2.95-A resolution range. Each subunit has essentially the same characteristic tertiary fold that is found in other legume lectins. The structure, however, exhibits an unusual quaternary arrangement of subunits. Unlike other well-characterized tetrameric proteins with identical subunits, peanut lectin has neither 222 (D2) nor fourfold (C4) symmetry. A noncrystallographic twofold axis relates two halves of the molecule. The two monomers in each half are related by a local twofold axis. The mutual disposition of the axes is such that they do not lead to a closed point group. Furthermore, the structure of peanut lectin demonstrates that differences in subunit arrangement in legume lectins could be due to factors intrinsic to the protein molecule and, contrary to earlier suggestions, are not necessarily caused by interactions involving covalently linked sugar. The structure provides a useful framework for exploring the structural basis and the functional implications of the variability in the subunit arrangement in legume lectins despite all of them having nearly the same subunit structure, and also for investigating the general problem of "open" quaternary assembly in oligomeric proteins.

Impedance parameters and the state-of-charge. III. Zinc-manganese dioxide and magnesium-manganese dioxide dry batteries

The problem of non-destructive determination of the state-of-charge of zinc- and magnesium-manganese dioxide dry batteries is examined experimentally from the viewpoint of internal impedance and open-circuit voltage at equilibrium. It is shown that the impedance is mainly charge-transfer controlled at relatively high states-of-charge and progressively changes over to diffusion control as the state-of-charge decreases in the case of zinc-manganese dioxide dry batteries. On the other hand, the impedance is mainly diffusion controlled for undischarged batteries but becomes charge-transfer controlled as soon as there is some discharge in the case of magnesium-manganese dioxide batteries. It is concluded that the determination of state-of-charge is not possible for both types of batteries by the measurement of impedance parameters due to film-induced fluctuations of these parameters. The measurement of open-circuit voltage at equilibrium can be used as a state-of-charge indicator for Zn-MnO2 batteries but not for Mg-MnO2 batteries.

Wind loading over the top portions of tall stacks with and without external landing platforms

The variation of the drag force near the top portions of tall stacks with and without external landing platforms, and with the exit open and closed, has been examined by model studies in a wind tunnel at Reynolds numbers of about 10(5). Pressure measurements on three models of different height to diameter ratios have been supplemented by flow visualisation studies. Observations confirm that when there is no platform, significant load enhancement over the top three to four diameters occurs, due to the high suction caused by the sharp separation of the flow over the top from the rim, in the aft regions of the stack. The enhanced loading is found to be greater if the exit is closed. A platform at the top, of less than twice the exit diameter, further increases the drag force near the top, but a still larger platform at the top, of about three times the exit diameter, decreases the drag force to values less than those much further below, effectively nullifying the enhanced drag force. It was found that such a reduction of the enhanced drag force in the top regions can also be achieved by a smaller platform of 1.1 to 1.3 times the local diameter, located at about three to five diameters below the top.

Control of transient thermal response during sequential open-die forging: A trajectory optimization approach

A trajectory optimization approach is applied to the design of a sequence of open-die forging operations in order to control the transient thermal response of a large titanium alloy billet. The amount of time tire billet is soaked in furnace prior to each successive forging operation is optimized to minimize the total process time while simultaneously satisfying constraints on the maximum and minimum values of the billet's temperature distribution to avoid microstructural defects during forging. The results indicate that a "differential" heating profile is the most effective at meeting these design goals.

Tailoring of Structural Morphology of Silver Nanowires in Electrochemical Growth

Noble metal such as Ag normally exists in an fcc crystal structure. However as the size of the material is decreased to nanometer lengthscales, a structural transformation from that of its bulk state can be expected with new atomic arrangements due to competition between internal packing and minimization of surface energy. In many previous studies, it has been shown that silver nanowires (AGNWs) grown inside anodic alumina (AAO) templates by ac or dc electrochemical deposition from silver salts or complexes, adopt fcc structure and below some critical diameter ∼ 20 nm they may acquire hcp structure at low temperature. This is, however, critically dependant on the nature of confinement, as AgNWs grown inside nanotube confinement with subnanometer diameter have been reported to have fcc structure. Hence the question of the crystal structure of metal nanowires under combined influence of confinement, temperature and deposition condition remains open. In this abstract we show that the alternative crystal structures of AGNWs at room temperature can be achieved with electrochemical growth processes under specific conditions determined by the deposition parameters and nature of confinement. We fabricated AgNWs of 4H hexagonal structure with diameters 30 – 80 nm inside polycarbonate (PC) templates with a modified dc electrodeposition technique, where the nanowires were grown at deposition potentials as low as 10 mV in 2 M silver nitrate solution[1]. We call this low-potential electrodeposition (LPED) since the electrodeposition process occurs at potential much less than the standard Nernst potential (770 mV) of silver. Two types of electrodes were used – stainless steel and sputtered thin Pt film, neither of which had any influence on the crystal structure of the nanowires. EDS elemental analysis showed the nanowires to consist only of silver. Although the precise atomic dynamics during the LPED process is unclear at present, we investigated this with HRTEM (high-resolution transmission electron microscopy) characterization of nanowires grown over various deposition times, as well as electrical conductivity measurements. These experiments indicate that nanowire growth does not occur through a three-dimensional diffusion controlled process, as proposed for conventional over-potential deposition, but follow a novel instantaneous linear growth mechanism. Further experiments showed that, (a) conventional electrochemical growth at a small over-potential in a 2 mM AgNO3 solution yields nanowires with expected fcc structure inside the same PC templates, and (2) no nanowire was observed under the LPED conditions inside hard AAO templates, indicating that LPED-growth process, and hcp structure of the corresponding nanowires depend on deposition parameters, as well as nature of confinement.

Pickup and delivery problem using metaheuristics techniques

Dial-a-ride problem (DARP) is an optimization problem which deals with the minimization of the cost of the provided service where the customers are provided a door-to-door service based on their requests. This optimization model presented in earlier studies, is considered in this study. Due to the non-linear nature of the objective function the traditional optimization methods are plagued with the problem of converging to a local minima. To overcome this pitfall we use metaheuristics namely Simulated Annealing (SA), Particle Swarm Optimization (PSO), Genetic Algorithm (GA) and Artificial Immune System (AIS). From the results obtained, we conclude that Artificial Immune System method effectively tackles this optimization problem by providing us with optimal solutions. Crown Copyright (C) 2011 Published by Elsevier Ltd. All rights reserved.

Experimental implementation of quantumUlam’s problem in a nuclear magnetic resonance quantum information processor

The Ulam’s problem is a two person game in which one of the player tries to search, in minimum queries, a number thought by the other player. Classically the problem scales polynomially with the size of the number. The quantum version of the Ulam’s problem has a query complexity that is independent of the dimension of the search space. The experimental implementation of the quantum Ulam’s problem in a Nuclear Magnetic Resonance Information Processor with 3 quantum bits is reported here.

Comparison of Optimum Reactive Power Schedule with Different Objectives Using LP Technique

Three algorithms for reactive power optimization are proposed in this paper with three different objective functions. The objectives in the proposed algorithm are to minimize the sum of the squares of the voltage deviations of the load buses, minimization of sum of squares of voltage stability L-indices of load buses (:3L2) algorithm, and also the objective of system real power loss (Ploss) minimization. The approach adopted is an iterative scheme with successive power flow analysis using decoupled technique and solution of the linear programming problem using upper bound optimization technique. Results obtained with all these objectives are compared. The analysis of these objective functions are presented to illustrate their advantages. It is observed comparing different objective functions it is possible to identify critical On Load Tap Changers (OLTCs) that should be made manual to avoid possible voltage instability due to their operation based on voltage improvement criteria under heavy load conditions. These algorithms have been tested under simulated conditions on few test systems. The results obtained on practical systems of 24-node equivalent EHV Indian power network, and for a 205 bus EHV system are presented for illustration purposes.

A numerical study of variable coefficient elliptic Cauchy problem via projection method

In this paper, we investigate a numerical method for the solution of an inverse problem of recovering lacking data on some part of the boundary of a domain from the Cauchy data on other part for a variable coefficient elliptic Cauchy problem. In the process, the Cauchy problem is transformed into the problem of solving a compact linear operator equation. As a remedy to the ill-posedness of the problem, we use a projection method which allows regularization solely by discretization. The discretization level plays the role of regularization parameter in the case of projection method. The balancing principle is used for the choice of an appropriate discretization level. Several numerical examples show that the method produces a stable good approximate solution.

Linearization and Reconstruction of Non-linear Diffuse Optical Tomographic Image

Diffuse optical tomography (DOT) is one of the ways to probe highly scattering media such as tissue using low-energy near infra-red light (NIR) to reconstruct a map of the optical property distribution. The interaction of the photons in biological tissue is a non-linear process and the phton transport through the tissue is modelled using diffusion theory. The inversion problem is often solved through iterative methods based on nonlinear optimization for the minimization of a data-model misfit function. The solution of the non-linear problem can be improved by modeling and optimizing the cost functional. The cost functional is f(x) = x(T)Ax - b(T)x + c and after minimization, the cost functional reduces to Ax = b. The spatial distribution of optical parameter can be obtained by solving the above equation iteratively for x. As the problem is non-linear, ill-posed and ill-conditioned, there will be an error or correction term for x at each iteration. A linearization strategy is proposed for the solution of the nonlinear ill-posed inverse problem by linear combination of system matrix and error in solution. By propagating the error (e) information (obtained from previous iteration) to the minimization function f(x), we can rewrite the minimization function as f(x; e) = (x + e)(T) A(x + e) - b(T)(x + e) + c. The revised cost functional is f(x; e) = f(x) + e(T)Ae. The self guided spatial weighted prior (e(T)Ae) error (e, error in estimating x) information along the principal nodes facilitates a well resolved dominant solution over the region of interest. The local minimization reduces the spreading of inclusion and removes the side lobes, thereby improving the contrast, localization and resolution of reconstructed image which has not been possible with conventional linear and regularization algorithm.

Analysis of Transverse Shear Strains in Pre-Twisted Thick Beams using Variational Asymptotic Method

The cross-sectional stiffness matrix is derived for a pre-twisted, moderately thick beam made of transversely isotropic materials and having rectangular cross sections. An asymptotically-exact methodology is used to model the anisotropic beam from 3-D elasticity, without any further assumptions. The beam is allowed to have large displacements and rotations, but small strain is assumed. The strain energy is computed making use of the beam constitutive law and kinematical relations derived with the inclusion of geometrical nonlinearities and an initial twist. The energy functional is minimized making use of the Variational Asymptotic Method (VAM), thereby reducing the cross section to a point on the beam reference line with appropriate properties, forming a 1-D constitutive law. VAM is a mathematical technique employed in the current problem to rigorously split the 3-D analysis of beams into two: a 2-D analysis over the beam cross-sectional domain, which provides a compact semi-analytical form of the properties of the cross sections, and a nonlinear 1-D analysis of the beam reference curve. In this method, as applied herein, the cross-sectional analysis is performed asymptotically by taking advantage of a material small parameter and two geometric small parameters. 3-D strain components are derived using kinematics and arranged in orders of the small parameters. Closed-form expressions are derived for the 3-D non-linear warping and stress fields. Warping functions are obtained by the minimization of strain energy subject to certain set of constraints that render the 1-D strain measures well-defined. The zeroth-order 3-D warping field thus yielded is then used to integrate the 3-D strain energy density over the cross section, resulting in the 1-D strain energy density, which in turn helps identify the corresponding cross-sectional stiffness matrix. The model is capable of predicting interlaminar and transverse shear stresses accurately up to first order.

An out-of-plane linear motion measurement system based on optical beam deflection

Measurement of out-of-plane linear motion with high precision and bandwidth is indispensable for development of precision motion stages and for dynamic characterization of mechanical structures. This paper presents an optical beam deflection (OBD) based system for measurement of out-of-plane linear motion for fully reflective samples. The system also achieves nearly zero cross-sensitivity to angular motion, and a large working distance. The sensitivities to linear and angular motion are analytically obtained and employed to optimize the system design. The optimal shot-noise limited resolution is shown to be less than one angstrom over a bandwidth in excess of 1 kHz. Subsequently, the system is experimentally realized and the sensitivities to out-of-plane motions are calibrated using a novel strategy. The linear sensitivity is found to be in agreement with theory. The angular sensitivity is shown to be over 7.5-times smaller than that of conventional OBD. Finally, the measurement system is employed to measure the transient response of a piezo-positioner, and, with the aid of an open-loop controller, reduce the settling time by about 90%. It is also employed to operate the positioner in closed-loop and demonstrate significant minimization of hysteresis and positioning error.