Application of the random eigenvalue problem in forced response analysis of a linear stochastic structure

The random eigenvalue problem arises in frequency and mode shape determination for a linear system with uncertainties in structural properties. Among several methods of characterizing this random eigenvalue problem, one computationally fast method that gives good accuracy is a weak formulation using polynomial chaos expansion (PCE). In this method, the eigenvalues and eigenvectors are expanded in PCE, and the residual is minimized by a Galerkin projection. The goals of the current work are (i) to implement this PCE-characterized random eigenvalue problem in the dynamic response calculation under random loading and (ii) to explore the computational advantages and challenges. In the proposed method, the response quantities are also expressed in PCE followed by a Galerkin projection. A numerical comparison with a perturbation method and the Monte Carlo simulation shows that when the loading has a random amplitude but deterministic frequency content, the proposed method gives more accurate results than a first-order perturbation method and a comparable accuracy as the Monte Carlo simulation in a lower computational time. However, as the frequency content of the loading becomes random, or for general random process loadings, the method loses its accuracy and computational efficiency. Issues in implementation, limitations, and further challenges are also addressed.

On the Erdos-Szekeres n-interior-point problem

The n-interior-point variant of the Erdos Szekeres problem is the following: for every n, n >= 1, does there exist a g(n) such that every point set in the plane with at least g(n) interior points has a convex polygon containing exactly n interior points. The existence of g(n) has been proved only for n <= 3. In this paper, we show that for any fixed r >= 2, and for every n >= 5, every point set having sufficiently large number of interior points and at most r convex layers contains a subset with exactly n interior points. We also consider a relaxation of the notion of convex polygons and show that for every n, n >= 1, any point set with at least n interior points has an almost convex polygon (a simple polygon with at most one concave vertex) that contains exactly n interior points. (C) 2013 Elsevier Ltd. All rights reserved.

Higher-Dimensional Analogues of the Map Coloring Problem

After a brief discussion of the history of the problem, we propose a generalization of the map coloring problem to higher dimensions.

The problem of clear air turbulence: Changing perspectives in the understanding of the phenomenon

Due to rapid improvements in on-board instrumentation and atmospheric observation systems, in most cases, aircraft are able to steer clear of regions of adverse weather. However, they still encounter unexpected bumpy flight conditions in regions away from storms and clouds. This is the phenomenon of clear air turbulence (CAT), which has been a challenge to our understanding as well as efforts at prediction. While most of such cases result in mild discomfort, a few cases can be violent leading to serious injuries to passengers and damage to the aircraft. The underlying physical mechanisms have been sought to be explained in terms of fluid dynamic instabilities and waves in the atmosphere. The main mechanisms which have been proposed are: (i) Kelvin-Helmholtz instability of shear layers, (ii) waves generated from flow over mountains, (iii) inertia-gravity waves from clouds and other sources, (iv) spontaneous imbalance theory and (v) horizontal vortex tubes. This has also undergone a change over the years. We present an overview of the mechanisms proposed and their implications for prediction.

STOKES' SYSTEM IN A DOMAIN WITH OSCILLATING BOUNDARY: HOMOGENIZATION AND ERROR ANALYSIS OF AN INTERIOR OPTIMAL CONTROL PROBLEM

Homogenization and error analysis of an optimal interior control problem in the framework of Stokes' system, on a domain with rapidly oscillating boundary, are the subject matters of this article. We consider a three dimensional domain constituted of a parallelepiped with a large number of rectangular cylinders at the top of it. An interior control is applied in a proper subdomain of the parallelepiped, away from the oscillating volume. We consider two types of functionals, namely a functional involving the L-2-norm of the state variable and another one involving its H-1-norm. The asymptotic analysis of optimality systems for both cases, when the cross sectional area of the rectangular cylinders tends to zero, is done here. Our major contribution is to derive error estimates for the state, the co-state and the associated pressures, in appropriate functional spaces.

A Phase Transition for the Uniform Distribution in the Pattern Maximum Likelihood Problem

In this paper, we consider the setting of the pattern maximum likelihood (PML) problem studied by Orlitsky et al. We present a well-motivated heuristic algorithm for deciding the question of when the PML distribution of a given pattern is uniform. The algorithm is based on the concept of a ``uniform threshold''. This is a threshold at which the uniform distribution exhibits an interesting phase transition in the PML problem, going from being a local maximum to being a local minimum.

Free turbulent shear layer in a point vortex gas as a problem in nonequilibrium statistical mechanics

This paper attempts to unravel any relations that may exist between turbulent shear flows and statistical mechanics through a detailed numerical investigation in the simplest case where both can be well defined. The flow considered for the purpose is the two-dimensional (2D) temporal free shear layer with a velocity difference Delta U across it, statistically homogeneous in the streamwise direction (x) and evolving from a plane vortex sheet in the direction normal to it (y) in a periodic-in-x domain L x +/-infinity. Extensive computer simulations of the flow are carried out through appropriate initial-value problems for a ``vortex gas'' comprising N point vortices of the same strength (gamma = L Delta U/N) and sign. Such a vortex gas is known to provide weak solutions of the Euler equation. More than ten different initial-condition classes are investigated using simulations involving up to 32 000 vortices, with ensemble averages evaluated over up to 10(3) realizations and integration over 10(4)L/Delta U. The temporal evolution of such a system is found to exhibit three distinct regimes. In Regime I the evolution is strongly influenced by the initial condition, sometimes lasting a significant fraction of L/Delta U. Regime III is a long-time domain-dependent evolution towards a statistically stationary state, via ``violent'' and ``slow'' relaxations P.-H. Chavanis, Physica A 391, 3657 (2012)], over flow time scales of order 10(2) and 10(4)L/Delta U, respectively (for N = 400). The final state involves a single structure that stochastically samples the domain, possibly constituting a ``relative equilibrium.'' The vortex distribution within the structure follows a nonisotropic truncated form of the Lundgren-Pointin (L-P) equilibrium distribution (with negatively high temperatures; L-P parameter lambda close to -1). The central finding is that, in the intermediate Regime II, the spreading rate of the layer is universal over the wide range of cases considered here. The value (in terms of momentum thickness) is 0.0166 +/- 0.0002 times Delta U. Regime II, extensively studied in the turbulent shear flow literature as a self-similar ``equilibrium'' state, is, however, a part of the rapid nonequilibrium evolution of the vortex-gas system, which we term ``explosive'' as it lasts less than one L/Delta U. Regime II also exhibits significant values of N-independent two-vortex correlations, indicating that current kinetic theories that neglect correlations or consider them as O(1/N) cannot describe this regime. The evolution of the layer thickness in present simulations in Regimes I and II agree with the experimental observations of spatially evolving (3D Navier-Stokes) shear layers. Further, the vorticity-stream-function relations in Regime III are close to those computed in 2D Navier-Stokes temporal shear layers J. Sommeria, C. Staquet, and R. Robert, J. Fluid Mech. 233, 661 (1991)]. These findings suggest the dominance of what may be called the Kelvin-Biot-Savart mechanism in determining the growth of the free shear layer through large-scale momentum and vorticity dispersal.

Cubic Sieve Congruence of the Discrete Logarithm Problem, and fractional part sequences

The Cubic Sieve Method for solving the Discrete Logarithm Problem in prime fields requires a nontrivial solution to the Cubic Sieve Congruence (CSC) x(3) equivalent to y(2)z (mod p), where p is a given prime number. A nontrivial solution must also satisfy x(3) not equal y(2)z and 1 <= x, y, z < p(alpha), where alpha is a given real number such that 1/3 < alpha <= 1/2. The CSC problem is to find an efficient algorithm to obtain a nontrivial solution to CSC. CSC can be parametrized as x equivalent to v(2)z (mod p) and y equivalent to v(3)z (mod p). In this paper, we give a deterministic polynomial-time (O(ln(3) p) bit-operations) algorithm to determine, for a given v, a nontrivial solution to CSC, if one exists. Previously it took (O) over tilde (p(alpha)) time in the worst case to determine this. We relate the CSC problem to the gap problem of fractional part sequences, where we need to determine the non-negative integers N satisfying the fractional part inequality {theta N} < phi (theta and phi are given real numbers). The correspondence between the CSC problem and the gap problem is that determining the parameter z in the former problem corresponds to determining N in the latter problem. We also show in the alpha = 1/2 case of CSC that for a certain class of primes the CSC problem can be solved deterministically in <(O)over tilde>(p(1/3)) time compared to the previous best of (O) over tilde (p(1/2)). It is empirically observed that about one out of three primes is covered by the above class. (C) 2013 Elsevier B.V. All rights reserved.

Error analysis of discontinuous Galerkin methods for the Stokes problem under minimal regularity

In this article, we analyse several discontinuous Galerkin (DG) methods for the Stokes problem under minimal regularity on the solution. We assume that the velocity u belongs to H-0(1)(Omega)](d) and the pressure p is an element of L-0(2)(Omega). First, we analyse standard DG methods assuming that the right-hand side f belongs to H-1(Omega) boolean AND L-1(Omega)](d). A DG method that is well defined for f belonging to H-1(Omega)](d) is then investigated. The methods under study include stabilized DG methods using equal-order spaces and inf-sup stable ones where the pressure space is one polynomial degree less than the velocity space.

On the Parameterized Complexity of the Maximum Edge 2-Coloring Problem

We investigate the parameterized complexity of the following edge coloring problem motivated by the problem of channel assignment in wireless networks. For an integer q >= 2 and a graph G, the goal is to find a coloring of the edges of G with the maximum number of colors such that every vertex of the graph sees at most q colors. This problem is NP-hard for q >= 2, and has been well-studied from the point of view of approximation. Our main focus is the case when q = 2, which is already theoretically intricate and practically relevant. We show fixed-parameter tractable algorithms for both the standard and the dual parameter, and for the latter problem, the result is based on a linear vertex kernel.

Parameterized Algorithms for MAX COLORABLE INDUCED SUBGRAPH Problem on Perfect Graphs

We address the parameterized complexity ofMaxColorable Induced Subgraph on perfect graphs. The problem asks for a maximum sized q-colorable induced subgraph of an input graph G. Yannakakis and Gavril IPL 1987] showed that this problem is NP-complete even on split graphs if q is part of input, but gave a n(O(q)) algorithm on chordal graphs. We first observe that the problem is W2]-hard parameterized by q, even on split graphs. However, when parameterized by l, the number of vertices in the solution, we give two fixed-parameter tractable algorithms. The first algorithm runs in time 5.44(l) (n+#alpha(G))(O(1)) where #alpha(G) is the number of maximal independent sets of the input graph. The second algorithm runs in time q(l+o()l())n(O(1))T(alpha) where T-alpha is the time required to find a maximum independent set in any induced subgraph of G. The first algorithm is efficient when the input graph contains only polynomially many maximal independent sets; for example split graphs and co-chordal graphs. The running time of the second algorithm is FPT in l alone (whenever T-alpha is a polynomial in n), since q <= l for all non-trivial situations. Finally, we show that (under standard complexitytheoretic assumptions) the problem does not admit a polynomial kernel on split and perfect graphs in the following sense: (a) On split graphs, we do not expect a polynomial kernel if q is a part of the input. (b) On perfect graphs, we do not expect a polynomial kernel even for fixed values of q >= 2.

Ecology of acoustic signalling and the problem of masking interference in insects

The efficiency of long-distance acoustic signalling of insects in their natural habitat is constrained in several ways. Acoustic signals are not only subjected to changes imposed by the physical structure of the habitat such as attenuation and degradation but also to masking interference from co-occurring signals of other acoustically communicating species. Masking interference is likely to be a ubiquitous problem in multi-species assemblages, but successful communication in natural environments under noisy conditions suggests powerful strategies to deal with the detection and recognition of relevant signals. In this review we present recent work on the role of the habitat as a driving force in shaping insect signal structures. In the context of acoustic masking interference, we discuss the ecological niche concept and examine the role of acoustic resource partitioning in the temporal, spatial and spectral domains as sender strategies to counter masking. We then examine the efficacy of different receiver strategies: physiological mechanisms such as frequency tuning, spatial release from masking and gain control as useful strategies to counteract acoustic masking. We also review recent work on the effects of anthropogenic noise on insect acoustic communication and the importance of insect sounds as indicators of biodiversity and ecosystem health.

Optimality Conditions and Duality for Semi-Infinite Mathematical Programming Problem with Equilibrium Constraints

This article considers a semi-infinite mathematical programming problem with equilibrium constraints (SIMPEC) defined as a semi-infinite mathematical programming problem with complementarity constraints. We establish necessary and sufficient optimality conditions for the (SIMPEC). We also formulate Wolfe- and Mond-Weir-type dual models for (SIMPEC) and establish weak, strong and strict converse duality theorems for (SIMPEC) and the corresponding dual problems under invexity assumptions.