An extension of Mangler transformation to a 3-D problem

Considering the linearized boundary layer equations for three-dimensional disturbances, a Mangler type transformation is used to reduce this case to an equivalent two-dimensional one.

An inverse problem for Helmholtz equation

This article is concerned with subsurface material identification for the 2-D Helmholtz equation. The algorithm is iterative in nature. It assumes an initial guess for the unknown function and obtains corrections to the guessed value. It linearizes the otherwise nonlinear problem around the background field. The background field is the field variable generated using the guessed value of the unknown function at each iteration. Numerical results indicate that the algorithm can recover a close estimate of the unknown function based on the measurements collected at the boundary.

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.

Finite Element Method For A Nonlocal Problem Of Kirchhoff Type

The nonlocal term in the nonlinear equations of Kirchhoff type causes difficulties when the equation is solved numerically by using the Newton-Raphson method. This is because the Jacobian of the Newton-Raphson method is full. In this article, the finite element system is replaced by an equivalent system for which the Jacobian is sparse. We derive quasi-optimal error estimates for the finite element method and demonstrate the results with numerical experiments.

An invariant subspace-based approach to the random eigenvalue problem of systems with clustered spectrum

The repeated or closely spaced eigenvalues and corresponding eigenvectors of a matrix are usually very sensitive to a perturbation of the matrix, which makes capturing the behavior of these eigenpairs very difficult. Similar difficulty is encountered in solving the random eigenvalue problem when a matrix with random elements has a set of clustered eigenvalues in its mean. In addition, the methods to solve the random eigenvalue problem often differ in characterizing the problem, which leads to different interpretations of the solution. Thus, the solutions obtained from different methods become mathematically incomparable. These two issues, the difficulty of solving and the non-unique characterization, are addressed here. A different approach is used where instead of tracking a few individual eigenpairs, the corresponding invariant subspace is tracked. The spectral stochastic finite element method is used for analysis, where the polynomial chaos expansion is used to represent the random eigenvalues and eigenvectors. However, the main concept of tracking the invariant subspace remains mostly independent of any such representation. The approach is successfully implemented in response prediction of a system with repeated natural frequencies. It is found that tracking only an invariant subspace could be sufficient to build a modal-based reduced-order model of the system. Copyright (C) 2012 John Wiley & Sons, Ltd.

Complexity of distance paired-domination problem in graphs

Suppose G = (V, E) is a simple graph and k is a fixed positive integer. A subset D subset of V is a distance k-dominating set of G if for every u is an element of V. there exists a vertex v is an element of D such that d(G)(u, v) <= k, where d(G)(u, v) is the distance between u and v in G. A set D subset of V is a distance k-paired-dominating set of G if D is a distance k-dominating set and the induced subgraph GD] contains a perfect matching. Given a graph G = (V, E) and a fixed integer k > 0, the MIN DISTANCE k-PAIRED-DOM SET problem is to find a minimum cardinality distance k-paired-dominating set of G. In this paper, we show that the decision version of MIN DISTANCE k-PAIRED-DOM SET iS NP-complete for undirected path graphs. This strengthens the complexity of decision version Of MIN DISTANCE k-PAIRED-DOM SET problem in chordal graphs. We show that for a given graph G, unless NP subset of DTIME (n(0)((log) (log) (n)) MIN DISTANCE k-PAIRED-Dom SET problem cannot be approximated within a factor of (1 -epsilon ) In n for any epsilon > 0, where n is the number of vertices in G. We also show that MIN DISTANCE k-PAIRED-DOM SET problem is APX-complete for graphs with degree bounded by 3. On the positive side, we present a linear time algorithm to compute the minimum cardinality of a distance k-paired-dominating set of a strongly chordal graph G if a strong elimination ordering of G is provided. We show that for a given graph G, MIN DISTANCE k-PAIRED-DOM SET problem can be approximated with an approximation factor of 1 + In 2 + k . In(Delta(G)), where Delta(G) denotes the maximum degree of G. (C) 2012 Elsevier B.V All rights reserved.

A note on the inverse mode shape problem for bars, beams, and plates

Motivated by the idea of designing a structure for a desired mode shape, intended towards applications such as resonant sensors, actuators and vibration confinement, we present the inverse mode shape problem for bars, beams and plates in this work. The objective is to determine the cross-sectional profile of these structures, given a mode shape, boundary condition and the mass. The contribution of this article is twofold: (i) A numerical method to solve this problem when a valid mode shape is provided in the finite element framework for both linear and nonlinear versions of the problem. (ii) An analytical result to prove the uniqueness and existence of the solution in the case of bars. This article also highlights a very important question of the validity of a mode shape for any structure of given boundary conditions.

Active Sequential Hypothesis Testing with Application to a Visual Search Problem

We consider a visual search problem studied by Sripati and Olson where the objective is to identify an oddball image embedded among multiple distractor images as quickly as possible. We model this visual search task as an active sequential hypothesis testing problem (ASHT problem). Chernoff in 1959 proposed a policy in which the expected delay to decision is asymptotically optimal. The asymptotics is under vanishing error probabilities. We first prove a stronger property on the moments of the delay until a decision, under the same asymptotics. Applying the result to the visual search problem, we then propose a ``neuronal metric'' on the measured neuronal responses that captures the discriminability between images. From empirical study we obtain a remarkable correlation (r = 0.90) between the proposed neuronal metric and speed of discrimination between the images. Although this correlation is lower than with the L-1 metric used by Sripati and Olson, this metric has the advantage of being firmly grounded in formal decision theory.

Spectral moment sum rules for the retarded Green's function and self-energy of the inhomogeneous Bose-Hubbard model in equilibrium and nonequilibrium

We derive exact expressions for the zeroth and the first three spectral moment sum rules for the retarded Green's function and for the zeroth and the first spectral moment sum rules for the retarded self-energy of the inhomogeneous Bose-Hubbard model in nonequilibrium, when the local on-site repulsion and the chemical potential are time-dependent, and in the presence of an external time-dependent electromagnetic field. We also evaluate these expressions for the homogeneous case in equilibrium, where all time dependence and external fields vanish. Unlike similar sum rules for the Fermi-Hubbard model, in the Bose-Hubbard model case, the sum rules often depend on expectation values that cannot be determined simply from parameters in the Hamiltonian like the interaction strength and chemical potential but require knowledge of equal-time many-body expectation values from some other source. We show how one can approximately evaluate these expectation values for the Mott-insulating phase in a systematic strong-coupling expansion in powers of the hopping divided by the interaction. We compare the exact moment relations to the calculated moments of spectral functions determined from a variety of different numerical approximations and use them to benchmark their accuracy. DOI: 10.1103/PhysRevA.87.013628

Observation of the nuclear magnetic octupole moment of Yb-173 from precise measurements of the hyperfine structure in the P-3(2) state

We measure hyperfine structure in the metastable P-3(2) state of Yb-173 and extract the nuclear magnetic octupole moment. We populate the state using dipole-allowed transitions through the P-3(1) and S-3(1) states. We measure frequencies of hyperfine transitions of the P-3(2) -> S-3(1) line at 770 nm using a Rb-stabilized ring cavity resonator with a precision of 200 kHz. Second-order corrections due to perturbations from the nearby P-3(1) and P-1(1) states are below 30 kHz. We obtain the hyperfine coefficients as A = -742.11(2) MHz and B = 1339.2(2) MHz, which represent a two orders-of-magnitude improvement in precision, and C = 0.54(2) MHz. From atomic structure calculations, we obtain the nuclear moments quadrupole Q = 2.46(12) b and octupole Omega = -34.4(21) b x mu(N). DOI: 10.1103/PhysRevA.87.012512

On the Erdos-Szekeres n-interior point problem

The n-interior point variant of the Erdos-Szekeres problem is to show the following: For any n, n-1, every point set in the plane with sufficient number of interior points contains a convex polygon containing exactly n-interior points. This has been proved only for n-3. In this paper, we prove it for pointsets having atmost logarithmic number of convex layers. We also show that any pointset containing atleast n interior points, there exists a 2-convex polygon that contains exactly n-interior points.

Optimal control problem for the time-dependent Kirchhoff-Love plate in a domain with rough boundary

In this paper, we study the asymptotic behavior of an optimal control problem for the time-dependent Kirchhoff-Love plate whose middle surface has a very rough boundary. We identify the limit problem which is an optimal control problem for the limit equation with a different cost functional.

Evaluation of strength of component-laminates in strip-based mechanisms

This paper deals with the evaluation of the component-laminate load-carrying capacity, i.e., to calculate the loads that cause the failure of the individual layers and the component-laminate as a whole in four-bar mechanism. The component-laminate load-carrying capacity is evaluated using the Tsai-Wu-Hahn failure criterion for various lay-ups. The reserve factor of each ply in the component-laminate is calculated by using the maximum resultant force and the maximum resultant moment occurring at different time steps at the joints of the mechanism. Here, all component bars of the mechanism are made of fiber reinforced laminates and have thin rectangular cross-sections. They could, in general, be pre-twisted and/or possess initial curvature, either by design or by defect. They are linked to each other by means of revolute joints. We restrict ourselves to linear materials with small strains within each elastic body (strip-like beam). Each component of the mechanism is modeled as a beam based on geometrically non-linear 3-D elasticity theory. The component problems are thus split into 2-D analyses of reference beam cross-sections and non-linear 1-D analyses along the three beam reference curves. For the thin rectangular cross-sections considered here, the 2-D cross-sectional nonlinearity is also overwhelming. This can be perceived from the fact that such sections constitute a limiting case between thin-walled open and closed sections, thus inviting the non-linear phenomena observed in both. The strong elastic couplings of anisotropic composite laminates complicate the model further. However, a powerful mathematical tool called the Variational Asymptotic Method (VAM) not only enables such a dimensional reduction, but also provides asymptotically correct analytical solutions to the non-linear cross-sectional analysis. Such closed-form solutions are used here in conjunction with numerical techniques for the rest of the problem to predict more quickly and accurately than would otherwise be possible. Local 3-D stress, strain and displacement fields for representative sections in the component-bars are recovered, based on the stress resultants from the 1-D global beam analysis. A numerical example is presented which illustrates the failure of each component-laminate and the mechanism as a whole.