Ship Capsize Risk Assessment and Designing Against Capsize

A promissing approach for assessing ship safety against capsize is presented by Melnikov's method. The mean zero up‐crossing rate of the dynamical system's associated Melnikov process provides an upper bound for the mean capsizing rate in a stationary sea state. Due to its moderate computational effort if low dimensional systems are considered, this approach is particularly suited for applications in ship design.     

Vizing's coloring algorithm and the fan number

Most upper bounds for the chromatic index of a graph come from algorithms that produce edge colorings. One such algorithm was invented by Vizing [Diskret Analiz 3 (1964), 25–30] in 1964. Vizing's algorithm colors the edges of a graph one at a time and never uses more than Δ+µ colors, where Δ is the maximum degree and µ is the maximum multiplicity, respectively. In general, though, this upper bound of Δ+µ is rather generous. In this paper, we define a new parameter fan(G) in terms of the degrees and the multiplicities of G. We call fan(G) the fan number of G. First we show that the fan number can be computed by a polynomial‐time algorithm. Then we prove that the parameter Fan(G)=max{Δ(G), fan(G)} is an upper bound for the chromatic index that can be realized by Vizing's coloring algorithm. Many of the known upper bounds for the chromatic index are also upper bounds for the fan number. Furthermore, we discuss the following question. What is the best (efficiently realizable) upper bound for the chromatic index in terms of Δ and µ ? Goldberg's Conjecture supports the conjecture that χ′+1 is the best efficiently realizable upper bound for χ′ at all provided that P ≠ NP . © 2009 Wiley Periodicals, Inc. J Graph Theory 65: 115–138, 2010     

Fractal dimension of a random invariant set

In recent years many deterministic parabolic equations have been shown to possess global attractors which, despite being subsets of an infinite-dimensional phase space, are finite-dimensional objects. Debussche showed how to generalize the deterministic theory to show that the random attractors of the corresponding stochastic equations have finite Hausdorff dimension. However, to deduce a parametrization of a ‘finite-dimensional’ set by a finite number of coordinates a bound on the fractal (upper box-counting) dimension is required. There are non-trivial problems in extending Debussche's techniques to this case, which can be overcome by careful use of the Poincaré recurrence theorem. We prove that under the same conditions as in Debussche's paper and an additional concavity assumption, the fractal dimension enjoys the same bound as the Hausdorff dimension. We apply our theorem to the 2d Navier–Stokes equations with additive noise, and give two results that allow different long-time states to be distinguished by a finite number of observations.     

A posteriori boundary element error estimation

An a posteriori error estimator is presented for the boundary element method in a general framework. It is obtained by solving local residual problems for which a local concept is introduced to accommodate the fact that integral operators are nonlocal operators. The estimator is shown to have an upper and a lower bound by the constant multiples of the exact error in the energy norm for Symm's and hypersingular integral equations. Numerical results are also given to demonstrate the effectiveness of the estimator for these equations. It can be used for adaptive h,p, and hp methods.     

Intersections of shifts of multiplicative subgroups

Using Stepanov’s method, we obtain an upper bound for the cardinality of the intersection of additive shifts of several multiplicative subgroups of a finite field. The resulting inequality is applied to a question dealing with the additive decomposability of subgroups.     

Dualization of regular Boolean functions

A monotonic Boolean function is regular if its variables are naturally ordered by decreasing ‘strength’, so that shifting to the right the non-zero entries of any binary false point always yields another false point. Peled and Simeone recently published a polynomial algorithm to generate the maximal false points (MFP's) of a regular function from a list of its minimal true points (MTP's). Another efficient algorithm for this problem is presented here, based on characterization of the MFP's of a regular function in terms of its MTP's. This result is also used to derive a new upper bound on the number of MFP's of a regular function.     

The Klee–Minty random edge chain moves with linear speed

An infinite sequence of 0's and 1's evolves by flipping each 1 to a 0 exponentially at rate 1. When a 1 flips, all bits to its right also flip. Starting from any configuration with finitely many 1's to the left of the origin, we show that the leftmost 1 moves right with bounded speed. Upper and lower bounds are given on the speed. A consequence is that a lower bound for the run time of the random‐edge simplex algorithm on a Klee–Minty cube is improved so as to be quadratic, in agreement with the upper bound. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2007     

Bounds to optimal burn‐in and optimal work size

Burn‐in is a widely used method to improve the quality of products or systems after they have been produced. In this paper, we consider the problem of determining the optimal burn‐in time and optimal work size maximizing the long‐run average amount of work saved per time unit in the computer applications. Assuming that the underlying lifetime distribution of the computer has an initially decreasing or/and eventually increasing failure rate function, an upper bound for the optimal burn‐in time is derived for each fixed work size and a uniform (with respect to the burn‐in time) upper bound for the optimal work size is also obtained. Furthermore, it is shown that a non‐trivial lower bound for the optimal burn‐in time can be derived if the underlying lifetime distribution has a large initial failure rate. Copyright © 2005 John Wiley & Sons, Ltd.     

Stochastic piecewise affine control with application to pitch control of helicopter

In this paper, at first the stability condition which gives an upper stochastic bound for a class of Stochastic Hybrid Systems (SHS) with deterministic jumps is derived. Here, additive noise signals are considered that do not vanish at equilibrium points. The presented theorem gives an upper bound for the second stochastic moment or variance of the system trajectories. Then, the linear case of SHS is investigated to show the application of the theorem. For the linear case of such stochastic hybrid systems, the stability criterion is obtained in terms of Linear Matrix Inequality (LMI) and an upper bound on state covariance is obtained for them. Then utilizing the stability theorem, an output feedback controller design procedure is proposed which requires the Bilinear Matrix Inequalities (BMI) to be solved. Next, the pitch dynamics of a helicopter is approximated with a set of linear stochastic systems, and the proposed controller is designed for the approximated model and implemented on the main nonlinear system to demonstrate the effectiveness of the proposed theorem and the control design method.     

Efficient computation of sparse structures

Basic graph structures such as maximal independent sets (MIS's) have spurred much theoretical research in randomized and distributed algorithms, and have several applications in networking and distributed computing as well. However, the extant (distributed) algorithms for these problems do not necessarily guarantee fault‐tolerance or load‐balance properties. We propose and study “low‐average degree” or “sparse” versions of such structures. Interestingly, in sharp contrast to, say, MIS's, it can be shown that checking whether a structure is sparse, will take substantial time. Nevertheless, we are able to develop good sequential/distributed (randomized) algorithms for such sparse versions. We also complement our algorithms with several lower bounds. Randomization plays a key role in our upper and lower bound results. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 322–344, 2016     

A modified Chebyshev ϑ‐weighted Crank–Nicolson method for analyzing fractional sub‐diffusion equations

This study presents a robust modification of Chebyshev ? ‐weighted Crank–Nicolson method for analyzing the sub‐diffusion equations in the Caputo fractional sense. In order to solve the problem, by discretization of the sub‐fractional diffusion equations using Taylor's expansion a linear system of algebraic equations that can be analyzed by numerical methods is presented. Furthermore, consistency, convergence, and stability analysis of the suggested method are discussed. In this framework, compact structures of sub‐diffusion equations are considered as prototype examples. The main advantage of the proposed method is that, it is more efficient in terms of CPU time, computational cost and accuracy in comparing with the existing ones in open literature.     

Upper bounds for the first zeros of Bessel functions

An upper bound for the first positive zero of the Bessel functions of first kind J_μ(z) for μ > −1 is given. This upper bound is better than a number of upper bounds found recently by several authors. The upper bound given in this paper follows from a step of the Ritz's approximation method, applied to the eigenvalue problem of a compact self-adjoint operator, defined on an abstract separable Hilbert space. Some advantages of this method in comparison with other approximation methods are presented.     

Modeling potential flow using Laurent series expansions and boundary elements

The fundamental underpinnings of the well‐known Bernoulli's Equation, as used to describe steady state two‐dimensional flow of an ideal incompressible irrotational flow, are typically described in terms of partial differential equations. However, it has been shown that the Cauchy Integral Theorem of standard complex variables also explain the Bernoulli's Equation and, hence, can be directly used to model problems of ideal fluid flow (or other potential problems such as electrostatics among other topics) using approximation function techniques such as the complex variable boundary element method (CVBEM). In this article, the CVBEM is extended to include Laurent Series expansions about singular points located outside of the problem domain union boundary. It is shown that by including such negatively powered complex monomials in the CVBEM formulation, considerable power is introduced to model potential flow problems. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 573–586,2012     

On an Uzawa smoother in multigrid for poroelasticity equations

A poroelastic saturated medium can be modeled by means of Biot's theory of consolidation. It describes the time‐dependent interaction between the deformation of porous material and the fluid flow inside of it. Here, for the efficient solution of the poroelastic equations, a multigrid method is employed with an Uzawa‐type iteration as the smoother. The Uzawa smoother is an equation‐wise procedure. It shall be interpreted as a combination of the symmetric Gauss‐Seidel smoothing for displacements, together with a Richardson iteration for the Schur complement in the pressure field. The Richardson iteration involves a relaxation parameter which affects the convergence speed, and has to be carefully determined. The analysis of the smoother is based on the framework of local Fourier analysis (LFA) and it allows us to provide an analytic bound of the smoothing factor of the Uzawa smoother as well as an optimal value of the relaxation parameter. Numerical experiments show that our upper bound provides a satisfactory estimate of the exact smoothing factor, and the selected relaxation parameter is optimal. In order to improve the convergence performance, the acceleration of multigrid by iterant recombination is taken into account. Numerical results confirm the efficiency and robustness of the acceleration scheme.     

On the impulsive boundary value problems for nonlinear Hamiltonian systems

In this work, we deal with two‐point boundary value problems for nonlinear impulsive Hamiltonian systems with sub‐linear or linear growth. A theorem based on the Schauder fixed point theorem is established, which gives a result that yields existence of solutions without implications that solutions must be unique. An upper bound for the solution is also established. Examples are given to illustrate the main result. Copyright © 2016 John Wiley & Sons, Ltd.     

Demailly's Conjecture and the containment problem

We investigate Demailly's Conjecture for a general set of sufficiently many points. Demailly's Conjecture generalizes Chudnovsky's Conjecture in providing a lower bound for the Waldschmidt constant of a set of points in projective space. We also study a containment between symbolic and ordinary powers conjectured by Harbourne and Huneke that in particular implies Demailly's bound, and prove that a general version of this containment holds for generic determinantal ideals and defining ideals of star configurations.     

On Dirichlet's approximation theorem

Though we cannot improve on the upper bound in Dirichlet's approximation theorem,Kaindl has shown that the upper bound can be lowered fromt ⁿ tot ⁿ ?t ^n?1?t ^n?2?...?t?1, if we admit equality. We show thatKaindl's upper bound is lowest possible in this case. The result is then generalized to linear forms.     

Non‐fragile synchronization control for complex networks with additive time‐varying delays

In this article, a synchronization problem for complex dynamical networks with additive time‐varying coupling delays via non‐fragile control is investigated. A new class of Lyapunov–Krasovskii functional with triple integral terms is constructed and using reciprocally convex approach, some new delay‐dependent synchronization criteria are derived in terms of linear matrix inequalities (LMIs). When applying Jensen's inequality to partition double integral terms in the derivation of LMI conditions, a new kind of linear combination of positive functions weighted by the inverses of squared convex parameters appears. To handle such a combination, an effective method is introduced by extending the lower bound lemma. Then, a sufficient condition for designing the non‐fragile synchronization controller is introduced. Finally, a numerical example is given to show the advantages of the proposed techniques. © 2014 Wiley Periodicals, Inc. Complexity 21: 296–321, 2015     

Approximate Separability of the Green's Function of the Helmholtz Equation in the High Frequency Limit

The minimum number of terms that are needed in a separable approximation for a Green's function reveals the intrinsic complexity of the solution space of the underlying differential equation. It also has implications for whether low‐rank structures exist in the linear system after numerical discretization. The Green's function for a coercive elliptic differential operator in divergence form was shown to be highly separable [2], and efficient numerical algorithms exploiting low‐rank structures of the discretized systems were developed. In this work, a new approach to study the approximate separability of the Green's function of the Helmholtz equation in the high‐frequency limit is developed. We show (1) lower bounds based on an explicit characterization of the correlation between two Green's functions and a tight dimension estimate for the best linear subspace to approximate a set of decorrelated Green's functions, (2) upper bounds based on constructing specific separable approximations, and (3) sharpness of these bounds for a few case studies of practical interest. © 2018 Wiley Periodicals, Inc.     

聚类分析在分枝定界法中的应用

针对界约束二次规划的分枝定界法中出现的紧、松弛策略,结合聚类分析方法,给出了新的剖分边的选取原则,把球约束二次规划作为子问题,使得原问题整体最优值的上、下界能较快的达到.