A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area

Motivated by a problem from fluid mechanics, we consider a generalization of the standard curve shortening flow problem for a closed embedded plane curve such that the area enclosed by the curve is forced to decrease at a prescribed rate. Using formal asymptotic and numerical techniques, we derive possible extinction shapes as the curve contracts to a point, dependent on the rate of decreasing area; we find there is a wider class of extinction shapes than for standard curve shortening, for which initially simple closed curves are always asymptotically circular. We also provide numerical evidence that self-intersection is possible for non-convex initial conditions, distinguishing between pinch-off and coalescence of the curve interior.

An FETI-preconditioned conjuerate gradient method for large-scale stochastic finite element problems

In the spectral stochastic finite element method for analyzing an uncertain system. the uncertainty is represented by a set of random variables, and a quantity of Interest such as the system response is considered as a function of these random variables Consequently, the underlying Galerkin projection yields a block system of deterministic equations where the blocks are sparse but coupled. The solution of this algebraic system of equations becomes rapidly challenging when the size of the physical system and/or the level of uncertainty is increased This paper addresses this challenge by presenting a preconditioned conjugate gradient method for such block systems where the preconditioning step is based on the dual-primal finite element tearing and interconnecting method equipped with a Krylov subspace reusage technique for accelerating the iterative solution of systems with multiple and repeated right-hand sides. Preliminary performance results on a Linux Cluster suggest that the proposed Solution method is numerically scalable and demonstrate its potential for making the uncertainty quantification Of realistic systems tractable.

A kinematic theory for radially foldable planar linkages

By using the algebraic locus of the coupler curve of a PRRP planar linkage, in this paper, a kinematic theory is developed for planar, radially foldable closed-loop linkages. This theory helps derive the previously invented building blocks, which consist of only two inter-connected angulated elements, for planar foldable structures. Furthermore, a special case of a circumferentially actuatable foldable linkage (which is different from the previously known cases) is derived from the theory, A quantitative description of some known and some new properties of planar foldable linkages, including the extent of foldability, shape-preservation of the interior polygons, multi-segmented assemblies and heterogeneous circumferential arrangemants, is also presented. The design equations derived here make the conception of even complex planar radially foldable linkages systematic and straightforward. Representative examples are presented to illustrate the usage of the design equations and the construction of prototypes. The current limitations and some possible extensions of the theory are also noted. (c) 2007, Elsevier Ltd. All ri-hts reserved.

Bounding Run-Times of Local Adiabatic Algorithms

A common trick for designing faster quantum adiabatic algorithms is to apply the adiabaticity condition locally at every instant. However it is often difficult to determine the instantaneous gap between the lowest two eigenvalues, which is an essential ingredient in the adiabaticity condition. In this paper we present a simple linear algebraic technique for obtaining a lower bound on the instantaneous gap even in such a situation. As an illustration, we investigate the adiabatic un-ordered search of van Dam et al. [17] and Roland and Cerf [15] when the non-zero entries of the diagonal final Hamiltonian are perturbed by a polynomial (in log N, where N is the length of the unordered list) amount. We use our technique to derive a bound on the running time of a local adiabatic schedule in terms of the minimum gap between the lowest two eigenvalues.

Lifting the singular nature of a model for peeling of an adhesive tape

We investigate the dynamics of peeling of an adhesive tape subjected to a constant pull speed. Due to the constraint between the pull force, peel angle and the peel force, the equations of motion derived earlier fall into the category of differential-algebraic equations (DAE) requiring an appropriate algorithm for its numerical solution. By including the kinetic energy arising from the stretched part of the tape in the Lagrangian, we derive equations of motion that support stick-slip jumps as a natural consequence of the inherent dynamics itself, thus circumventing the need to use any special algorithm. In the low mass limit, these equations reproduce solutions obtained using a differential-algebraic algorithm introduced for the earlier singular equations. We find that mass has a strong influence on the dynamics of the model rendering periodic solutions to chaotic and vice versa. Apart from the rich dynamics, the model reproduces several qualitative features of the different waveforms of the peel force function as also the decreasing nature of force drop magnitudes.

Single-resolution and multiresolution extended-Kalman-filter-based reconstruction approaches to optical refraction tomography

The problem of reconstruction of a refractive-index distribution (RID) in optical refraction tomography (ORT) with optical path-length difference (OPD) data is solved using two adaptive-estimation-based extended-Kalman-filter (EKF) approaches. First, a basic single-resolution EKF (SR-EKF) is applied to a state variable model describing the tomographic process, to estimate the RID of an optically transparent refracting object from noisy OPD data. The initialization of the biases and covariances corresponding to the state and measurement noise is discussed. The state and measurement noise biases and covariances are adaptively estimated. An EKF is then applied to the wavelet-transformed state variable model to yield a wavelet-based multiresolution EKF (MR-EKF) solution approach. To numerically validate the adaptive EKF approaches, we evaluate them with benchmark studies of standard stationary cases, where comparative results with commonly used efficient deterministic approaches can be obtained. Detailed reconstruction studies for the SR-EKF and two versions of the MR-EKF (with Haar and Daubechies-4 wavelets) compare well with those obtained from a typically used variant of the (deterministic) algebraic reconstruction technique, the average correction per projection method, thus establishing the capability of the EKF for ORT. To the best of our knowledge, the present work contains unique reconstruction studies encompassing the use of EKF for ORT in single-resolution and multiresolution formulations, and also in the use of adaptive estimation of the EKF's noise covariances. (C) 2010 Optical Society of America

Conformal Field Theory Methods for Variants of Schramm-Loewner Evolutions

This thesis consists of an introduction, four research articles and an appendix. The thesis studies relations between two different approaches to continuum limit of models of two dimensional statistical mechanics at criticality. The approach of conformal field theory (CFT) could be thought of as the algebraic classification of some basic objects in these models. It has been succesfully used by physicists since 1980's. The other approach, Schramm-Loewner evolutions (SLEs), is a recently introduced set of mathematical methods to study random curves or interfaces occurring in the continuum limit of the models. The first and second included articles argue on basis of statistical mechanics what would be a plausible relation between SLEs and conformal field theory. The first article studies multiple SLEs, several random curves simultaneously in a domain. The proposed definition is compatible with a natural commutation requirement suggested by Dubédat. The curves of multiple SLE may form different topological configurations, ``pure geometries''. We conjecture a relation between the topological configurations and CFT concepts of conformal blocks and operator product expansions. Example applications of multiple SLEs include crossing probabilities for percolation and Ising model. The second article studies SLE variants that represent models with boundary conditions implemented by primary fields. The most well known of these, SLE(kappa, rho), is shown to be simple in terms of the Coulomb gas formalism of CFT. In the third article the space of local martingales for variants of SLE is shown to carry a representation of Virasoro algebra. Finding this structure is guided by the relation of SLEs and CFTs in general, but the result is established in a straightforward fashion. This article, too, emphasizes multiple SLEs and proposes a possible way of treating pure geometries in terms of Coulomb gas. The fourth article states results of applications of the Virasoro structure to the open questions of SLE reversibility and duality. Proofs of the stated results are provided in the appendix. The objective is an indirect computation of certain polynomial expected values. Provided that these expected values exist, in generic cases they are shown to possess the desired properties, thus giving support for both reversibility and duality.

STBC's from representation of extended Clifford Algebras

A set of sufficient conditions to construct lambda-real symbol Maximum Likelihood (ML) decodable STBCs have recently been provided by Karmakar et al. STBCs satisfying these sufficient conditions were named as Clifford Unitary Weight (CUW) codes. In this paper, the maximal rate (as measured in complex symbols per channel use) of CUW codes for lambda = 2(a), a is an element of N is obtained using tools from representation theory. Two algebraic constructions of codes achieving this maximal rate are also provided. One of the constructions is obtained using linear representation of finite groups whereas the other construction is based on the concept of right module algebra over non-commutative rings. To the knowledge of the authors, this is the first paper in which matrices over non-commutative rings is used to construct STBCs. An algebraic explanation is provided for the 'ABBA' construction first proposed by Tirkkonen et al and the tensor product construction proposed by Karmakar et al. Furthermore, it is established that the 4 transmit antenna STBC originally proposed by Tirkkonen et al based on the ABBA construction is actually a single complex symbol ML decodable code if the design variables are permuted and signal sets of appropriate dimensions are chosen.

Max-Coloring Paths: Tight Bounds and Extensions

The max-coloring problem is to compute a legal coloring of the vertices of a graph G = (V, E) with a non-negative weight function w on V such that Sigma(k)(i=1) max(v epsilon Ci) w(v(i)) is minimized, where C-1, ... , C-k are the various color classes. Max-coloring general graphs is as hard as the classical vertex coloring problem, a special case where vertices have unit weight. In fact, in some cases it can even be harder: for example, no polynomial time algorithm is known for max-coloring trees. In this paper we consider the problem of max-coloring paths and its generalization, max-coloring abroad class of trees and show it can be solved in time O(vertical bar V vertical bar+time for sorting the vertex weights). When vertex weights belong to R, we show a matching lower bound of Omega(vertical bar V vertical bar log vertical bar V vertical bar) in the algebraic computation tree model.

Ideal Structure of the Silver Code

The Silver code has captured a lot of attention in the recent past,because of its nice structure and fast decodability. In their recent paper, Hollanti et al. show that the Silver code forms a subset of the natural order of a particular cyclic division algebra (CDA). In this paper, the algebraic structure of this subset is characterized. It is shown that the Silver code is not an ideal in the natural order but a right ideal generated by two elements in a particular order of this CDA. The exact minimum determinant of the normalized Silver code is computed using the ideal structure of the code. The construction of Silver code is then extended to CDAs over other number fields.

Models of Dispersal and Diversification

Ecology and evolutionary biology is the study of life on this planet. One of the many methods applied to answering the great diversity of questions regarding the lives and characteristics of individual organisms, is the utilization of mathematical models. Such models are used in a wide variety of ways. Some help us to reason, functioning as aids to, or substitutes for, our own fallible logic, thus making argumentation and thinking clearer. Models which help our reasoning can lead to conceptual clarification; by expressing ideas in algebraic terms, the relationship between different concepts become clearer. Other mathematical models are used to better understand yet more complicated models, or to develop mathematical tools for their analysis. Though helping us to reason and being used as tools in the craftmanship of science, many models do not tell us much about the real biological phenomena we are, at least initially, interested in. The main reason for this is that any mathematical model is a simplification of the real world, reducing the complexity and variety of interactions and idiosynchracies of individual organisms. What such models can tell us, however, both is and has been very valuable throughout the history of ecology and evolution. Minimally, a model simplifying the complex world can tell us that in principle, the patterns produced in a model could also be produced in the real world. We can never know how different a simplified mathematical representation is from the real world, but the similarity models do strive for, gives us confidence that their results could apply. This thesis deals with a variety of different models, used for different purposes. One model deals with how one can measure and analyse invasions; the expanding phase of invasive species. Earlier analyses claims to have shown that such invasions can be a regulated phenomena, that higher invasion speeds at a given point in time will lead to a reduction in speed. Two simple mathematical models show that analysis on this particular measure of invasion speed need not be evidence of regulation. In the context of dispersal evolution, two models acting as proof-of-principle are presented. Parent-offspring conflict emerges when there are different evolutionary optima for adaptive behavior for parents and offspring. We show that the evolution of dispersal distances can entail such a conflict, and that under parental control of dispersal (as, for example, in higher plants) wider dispersal kernels are optimal. We also show that dispersal homeostasis can be optimal; in a setting where dispersal decisions (to leave or stay in a natal patch) are made, strategies that divide their seeds or eggs into fractions that disperse or not, as opposed to randomized for each seed, can prevail. We also present a model of the evolution of bet-hedging strategies; evolutionary adaptations that occur despite their fitness, on average, being lower than a competing strategy. Such strategies can win in the long run because they have a reduced variance in fitness coupled with a reduction in mean fitness, and fitness is of a multiplicative nature across generations, and therefore sensitive to variability. This model is used for conceptual clarification; by developing a population genetical model with uncertain fitness and expressing genotypic variance in fitness as a product between individual level variance and correlations between individuals of a genotype. We arrive at expressions that intuitively reflect two of the main categorizations of bet-hedging strategies; conservative vs diversifying and within- vs between-generation bet hedging. In addition, this model shows that these divisions in fact are false dichotomies.

Simplified Analysis of Steady-State Performance of a Voltage Controlled Induction Motor Drive

Utilizing a circuit model [1, 2] of an induction motor, a simplified analysis of steady state performance of a voltage controlled induction motor (VCIM) drive is described in this paper. By solving a set of nonlinear algebraic equations which describe the VCIM drive under steady operation, the operating variables such as constant components of torque, rotor flux linkages, fundamental components of stator voltage and current and phase angle are obtained for any given value of slip, triggering angle and supply voltage.

A Cellular Array for Multivalued Logic Functions

An algebraic generalization of the well-known binary q-function array to a multivalued q-function array is presented. It is possible to associate tree-structure realizations for binary q-functions and multivalued q-functions. Synthesis of multivalued functions using this array is very simple

Fractal boundaries of basin of attraction of Newton-Raphson method in helicopter trim

A key problem in helicopter aeroelastic analysis is the enormous computational time required for a numerical solution of the nonlinear system of algebraic equations required for trim, particularly when free wake models are used. Trim requires calculation of the main rotor and tail rotor controls and the vehicle attitude which leads to the six steady forces and moments about the helicopter center of gravity to be zero. An appropriate initial estimate of the trim state is needed for successful helicopter trim. This study aims to determine the control inputs that can have considerable effect on the convergence of trim solution in the aeroelastic analysis of helicopter rotors by investigating the basin of attraction of the nonlinear equations (set of initial guess points from which the nonlinear equations converge). It is illustrated that the three main rotor pitch controls of collective pitch, longitudinal cyclic pitch and lateral cyclic pitch have a significant contribution to the convergence of the trim solution. Trajectories of the Newton iterates are shown and some ideas for accelerating the convergence of a trim solution in the aeroelastic analysis of helicopters are proposed. It is found that the basins of attraction can have fractal boundaries. (C) 2010 Elsevier Ltd. All rights reserved.

Unified Galerkin- and DAE-Based Approximation of Fractional Order Systems

We consider numerical solutions of nonlinear multiterm fractional integrodifferential equations, where the order of the highest derivative is fractional and positive but is otherwise arbitrary. Here, we extend and unify our previous work, where a Galerkin method was developed for efficiently approximating fractional order operators and where elements of the present differential algebraic equation (DAE) formulation were introduced. The DAE system developed here for arbitrary orders of the fractional derivative includes an added block of equations for each fractional order operator, as well as forcing terms arising from nonzero initial conditions. We motivate and explain the structure of the DAE in detail. We explain how nonzero initial conditions should be incorporated within the approximation. We point out that our approach approximates the system and not a specific solution. Consequently, some questions not easily accessible to solvers of initial value problems, such as stability analyses, can be tackled using our approach. Numerical examples show excellent accuracy. DOI: 10.1115/1.4002516]