A note on Sommerfeld's diffraction problem

A direct transform technique is found to be most suitable for attacking two-dimensional diffraction problems. As a first example of the application of the technique, the well-known Sommerfeld problem is reconsidered and the solution of the problem of diffraction, by a half-plane, of a cylindrical pulse is made use of in deducing the solution of the problem of diffraction of a plane wave by a soft half-plane. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.

Mathematical model and rule extraction for tool wear monitoring problem using nature inspired techniques

In this paper, pattern classification problem in tool wear monitoring is solved using nature inspired techniques such as Genetic Programming(GP) and Ant-Miner (AM). The main advantage of GP and AM is their ability to learn the underlying data relationships and express them in the form of mathematical equation or simple rules. The extraction of knowledge from the training data set using GP and AM are in the form of Genetic Programming Classifier Expression (GPCE) and rules respectively. The GPCE and AM extracted rules are then applied to set of data in the testing/validation set to obtain the classification accuracy. A major attraction in GP evolved GPCE and AM based classification is the possibility of obtaining an expert system like rules that can be directly applied subsequently by the user in his/her application. The performance of the data classification using GP and AM is as good as the classification accuracy obtained in the earlier study.

Accurate Parameter Estimation of Contaminant Transport Inverse Problem using Particle Swarm Optimization

Swarm Intelligence techniques such as particle swarm optimization (PSO) are shown to be incompetent for an accurate estimation of global solutions in several engineering applications. This problem is more severe in case of inverse optimization problems where fitness calculations are computationally expensive. In this work, a novel strategy is introduced to alleviate this problem. The proposed inverse model based on modified particle swarm optimization algorithm is applied for a contaminant transport inverse model. The inverse models based on standard-PSO and proposed-PSO are validated to estimate the accuracy of the models. The proposed model is shown to be out performing the standard one in terms of accuracy in parameter estimation. The preliminary results obtained using the proposed model is presented in this work.

A Combinatorial Optimization Problem for High Order PODs with Few Sensors

Experimental characterization of high dimensional dynamic systems sometimes uses the proper orthogonal decomposition (POD). If there are many measurement locations and relatively fewer sensors, then steady-state behavior can still be studied by sequentially taking several sets of simultaneous measurements. The number required of such sets of measurements can be minimized if we solve a combinatorial optimization problem. We aim to bring this problem to the attention of engineering audiences, summarize some known mathematical results about this problem, and present a heuristic (suboptimal) calculation that gives reasonable, if not stellar, results.

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.

Effect of boundary condition description on convergence of solution in a boundary-value problem

It is well known that the numerical accuracy of a series solution to a boundary-value problem by the direct method depends on the technique of approximate satisfaction of the boundary conditions and on the stage of truncation of the series. On the other hand, it does not appear to be generally recognized that, when the boundary conditions can be described in alternative equivalent forms, the convergence of the solution is significantly affected by the actual form in which they are stated. The importance of the last aspect is studied for three different techniques of computing the deflections of simply supported regular polygonal plates under uniform pressure. It is also shown that it is sometimes possible to modify the technique of analysis to make the accuracy independent of the description of the boundary conditions.

Numerical procedure for second order non-linear ordinary differential equations and application to heat transfer problem

In this paper, we have first given a numerical procedure for the solution of second order non-linear ordinary differential equations of the type y″ = f (x;y, y′) with given initial conditions. The method is based on geometrical interpretation of the equation, which suggests a simple geometrical construction of the integral curve. We then translate this geometrical method to the numerical procedure adaptable to desk calculators and digital computers. We have studied the efficacy of this method with the help of an illustrative example with known exact solution. We have also compared it with Runge-Kutta method. We have then applied this method to a physical problem, namely, the study of the temperature distribution in a semi-infinite solid homogeneous medium for temperature-dependent conductivity coefficient.

Quantum kinematics of Fermi-Dirac vacuum-plus-one-electron problem

Following Weisskopf, the kinematics of quantum mechanics is shown to lead to a modified charge distribution for a test electron embedded in the Fermi-Dirac vacuum with interesting consequences.

An approximate algorithm for the minimal vertex nested polygon problem

Given two simple polygons, the Minimal Vertex Nested Polygon Problem is one of finding a polygon nested between the given polygons having the minimum number of vertices. In this paper, we suggest efficient approximate algorithms for interesting special cases of the above using the shortest-path finding graph algorithms.