Easy knapsacks and the complexity of energy allocation problems in the smart grid

Motivated by the growing interest in the smart grid and intelligent energy management mechanisms we study two classes of domestic energy allocation problems. In the first case we work with a system that is tasked with scheduling the work on a number of appliances over a given time window. In the second one a collection of air conditioning appliances is used to control the temperature of a given domestic environment. Our framework for this case includes a simplified mechanism for modelling the heat exchange between the interior and the exterior of the given environment. We present various polynomial time algorithms and NP-hardness proofs. In particular the main result of the paper is a proof that although it is NP-hard to schedule the operation of a single air-conditioning unit, working at various temperature levels in a variable energy price regime, there is a polynomial time algorithm for controlling one such device working at a single temperature level, for houses with low thermal inertia. The algorithm analysis hinges on the properties of a polynomial time variant of the minimisation version of the knapsack problem which may be of independent interest.     

Note on roots location of a symmetric polynomial with respect to the imaginary axis

In the theory of the separation of roots of algebraic equations, the well-known Routh–Hurwitz–Fujiwara theorem enables us to separate the complex roots of a polynomial with complex coefficients in terms of the inertia of a related Hermitian matrix. Unfortunately, it fails if the polynomial has a nontrivial factor which is symmetric with respect to the imaginary axis. In this article, we present a method to overcome the fault and formulate the inertia of a scalar polynomial with complex coefficients in terms of the inertia of several Hermitian matrices based on a factorization of a monic symmetric polynomial into products of monic symmetric polynomials with only simple roots in the complex plane and on computing the inertia of each factor by means of a subtle perturbation.     

Topology optimization of space vehicle structures considering attitude control effort

The topology optimization of load-bearing structural components for reducing attitude control efforts of miniature space vehicles is investigated. Based on the derivation of the cold gas consumption rate of three-axis stabilization actuators, it is pointed out that the attitude control efforts associated with cold gas micro thrusters are closely related to the mass moment inertia of the system. Therefore, the need to restrict the mass moments of inertia of the structural components is highlighted in the design of the load-bearing structural components when the attitude control performance is concerned. The optimal layout design of the space vehicle structure considering attitude control effort is, thus, reformulated as a topology optimization problem for minimum compliance under constraints on mass moments of inertia. Numerical techniques for the optimization problem are discussed. For the case of a single constraint on the mass moment of inertia about a given axis, a design variable updating scheme based on the Karush–Kuhn–Tucker optimality criteria is used to solve the minimization problem. For the problem with multiple constraints, mathematical programming approach is employed to seek the optimum. Numerical examples will be given to demonstrate the validity and applicability of the present problem statement.     

Risk minimization for atmospheric data restoration

The paper deals with filling gaps in meteorological fields aimed at weather simulation. A universal local approximation approach is described which allows us to fill gaps in meteorological fields under various conditions, and at the same time, effect time-step weather forecasts. We present results of small sample gap filling for some meteorological elements. The results are compared against those obtained with local averaging and inertia methods. Some possibilities of data restoration and prediction based on available global meteorological data are discussed.     

Moment matching approximation of Asian basket option prices

In this paper we propose some moment matching pricing methods for European-style discrete arithmetic Asian basket options in a Black & Scholes framework. We generalize the approach of [M. Curran, Valuing Asian and portfolio by conditioning on the geometric mean price, Management Science 40 (1994) 1705-1711] and of [G. Deelstra, J. Liinev, M. Vanmaele, Pricing of arithmetic basket options by conditioning, Insurance: Mathematics & Economics 34 (2004) 55-57] in several ways. We create a framework that allows for a whole class of conditioning random variables which are normally distributed. We moment match not only with a lognormal random variable but also with a log-extended-skew-normal random variable. We also improve the bounds of [G. Deelstra, I. Diallo, M. Vanmaele, Bounds for Asian basket options, Journal of Computational and Applied Mathematics 218 (2008) 215-228]. Numerical results are included and on the basis of our numerical tests, we explain which method we recommend depending on moneyness and time-to-maturity.     

A new theory of regular functions of a quaternionic variable

In this paper we develop the fundamental elements and results of a new theory of regular functions of one quaternionic variable. The theory we describe follows a classical idea of Cullen, but we use a more geometric formulation to show that it is possible to build a rather complete theory. Our theory allows us to extend some important results for polynomials in the quaternionic variable to the case of power series.     

截面连续变化悬臂构件挠度计算的等效柱法

基于最大位移相等原理,应用辛普生积分公式导出了截面连续变化悬臂构件挠度计算的等效柱法,将变截面构件挠度计算的复杂积分运算转化为等截面等效柱的惯性矩计算,力学概念明确.算例表明,等效柱法简便实用,适用性广,精度较高,是截面连续变化构件挠度计算的有效方法.     

Causal compositional models in valuation-based systems with examples in specific theories

We show that Pearl's causal networks can be described using causal compositional models (CCMs) in the valuation-based systems (VBS) framework. One major advantage of using the VBS framework is that as VBS is a generalization of several uncertainty theories (e.g., probability theory, a version of possibility theory where combination is the product t-norm, Spohn's epistemic belief theory, and Dempster–Shafer belief function theory), CCMs, initially described in probability theory, are now described in all uncertainty calculi that fit in the VBS framework. We describe conditioning and interventions in CCMs. Another advantage of using CCMs in the VBS framework is that both conditioning and intervention can be easily described in an elegant and unifying algebraic way for the same CCM without having to do any graphical manipulations of the causal network. We describe how conditioning and intervention can be computed for a simple example with a hidden (unobservable) variable. Also, we illustrate the algebraic results using numerical examples in some of the specific uncertainty calculi mentioned above.     

The Dirichlet Energy Integral and Variable Exponent Sobolev Spaces with Zero Boundary Values

We define and study variable exponent Sobolev spaces with zero boundary values. This allows us to prove that the Dirichlet energy integral has a minimizer in the variable exponent case. Our results are based on a Poincaré-type inequality, which we prove under a certain local jump condition for the variable exponent.     

Quantile approximations in auto-regressive portfolio models

This paper develops an analytical approximation for the distribution function of a terminal value of a periodic series of buy-and-hold investments placed over a fixed time horizon for the case when log-returns of assets follow a p-th order vector auto-regressive process. The derivation is based on a first order Taylor conditioned approximation with a suitably chosen conditioning variable. The results indicate a remarkably good fit between the approximating procedure and simulations based on realistic parameters.     

A bending–stretching model in adhesive contact for elastic rods obtained by using asymptotic methods

This paper is devoted to the derivation and mathematical justification of models for the bending–stretching of an elastic rod in adhesive contact with a deformable foundation. The process is assumed to be quasistatic, and therefore the effects of inertia are neglected. Contact is modeled with normal compliance and the adhesion is modeled by introducing a surface internal variable, the bonding function, the evolution of which is described by an ordinary differential equation. To derive the models we consider the three-dimensional contact problem of an elastic body in adhesive contact with a foundation, introduce a change of variable together with the scaling of the unknowns and parameters of the problem, and we obtain a limit model under the assumption of suitable asymptotic expansions for the scaled unknowns. After that, we obtain error estimates and convergence results which legitimate the limit model. Finally we show that our limit model contains as particular cases models previously considered by other authors. To our knowledge it is for the first time that a rigorous justification and a generalization of those models is provided.     

A Hybrid Mesh Selection Strategy Based on Conditioning for Boundary Value ODE Problems

An appropriate mesh selection strategy is one of the fundamental tools in designing robust codes for differential problems, especially if the codes are required to work for difficult multi scale problems. Most of the existing codes base the mesh selection on an estimate of the error (or the residual). Our strategy, based on the estimation of two parameters characterising the conditioning of the continuous problem, as well as on an estimate of the error, not only permits us to obtain a well adapted mesh, thus reducing the cost of the code, but also provides a measure of the conditioning of both continuous and discrete problems.     

On conditional weak convergence

In this paper we discuss a number of technical issues associated with conditional weak convergence. The main modes of convergence of conditional probability distributions areuniform, probability, andalmost sure convergence in the conditioning variable. General results regarding conditional convergence are obtained, including details of sufficient conditions for each mode of convergence, and characterization theorems for uniform conditional convergence.     

惯性定律的新证法

本文从实对称矩阵的角度 ,给出了惯性定律的一种新证法 .这种证法可使我们从已给的一个实二次型 f( x1 ,x2 ,… ,xn) =∑ni,j=1aijxixj 的矩阵 A=( aij) n× n的诸元素 ,来直接研究这个二次型的性质的这一体系更加完整 .同时 ,本文还大大地改进了雅可比方法 ,使雅可比的方法更加完美 ,应用更加广泛 .     

On a conjecture of Dixon and other topics in variable metric methods

A conjecture of Dixon relating to the behaviour of variable metric methods on functions with special symmetry is validated under suitable onditions. The relation between Huang's class and Oren's class is explored. Then the equivalence of Davidon's and Oren and Spedicato's approaches to optimal conditioning is demonstrated.     

The Evolution of Internal Undular Bores over a Slope in the Presence of Rotation

The large‐amplitude internal waves commonly observed in the coastal ocean often take the form of unsteady undular bores. Hence, here, we examine the long‐time combined effect of variable topography and background rotation on the propagation of internal undular bores, using the framework of a variable‐coefficient Ostrovsky equation. Because the leading waves in an internal undular bore are close to solitary waves, we first examine the evolution of a single solitary wave. Then, we consider an internal undular bore, for which two methods of generation are used. One method is the matured undular bore developed from an initial shock box in the Korteweg–de Vries equation, that is the Ostrovsky equation with the rotational term omitted, and the other method is a modulated cnoidal wave solution of the same Korteweg–de Vries equation. It transpires that in the long‐time model simulations, the rotational effect disintegrates the nonlinear waves into inertia‐gravity waves, and then there emerge complicated interactions between these inertia‐gravity waves and the modulated periodic waves of the undular bore, especially at the rear part of the undular bore. However, near the front of the undular bore, nonlinear effects further modulate these waves, with the eventual emergence of nonlinear envelope wave packets.     

Numerical solution of vector Sturm–Liouville problems with Dirichlet conditions and nonlinear dependence on the spectral parameter

A numerical-analytical iterative method is proposed for solving generalized self-adjoint regular vector Sturm–Liouville problems with Dirichlet boundary conditions. The method is based on eigenvalue (spectral) correction. The matrix coefficients of the equations are assumed to be nonlinear functions of the spectral parameter. For a relatively close initial approximation, the method is shown to have second-order convergence with respect to a small parameter. Test examples are considered, and the model problem of transverse vibrations of a hinged rod with a variable cross section is solved taking into account its rotational inertia.     

Novel inertia weight strategies for particle swarm optimization

The performance of PSO algorithm depends greatly on the appropriate parameter selection strategies for fine tuning its parameters. This paper proposes three new nonlinear strategies for selecting inertia weight which plays a significant role in particle’s foraging behaviour. The PSO variants implying these strategies are named as: fine grained inertia weight PSO (FGIWPSO); Double Exponential Self Adaptive IWPSO (DESIWPSO) and Double Exponential Dynamic IWPSO (DEDIWPSO). In FGIWPSO, inertia weight is obtained adaptively, depending on particle’s iteration wise performance and decreases exponentially. DESIWPSO and DEDIWPSO employ Gompertz function, a double exponential function for selecting inertia weight. In DESIWPSO the particles’ iteration wise performance is fed as input to the Gompertz function. On the other hand DEDIWPSO evaluates the inertia weight for whole swarm iteratively using Gompertz function where relative iteration is fed as input. The efficacy and efficiency of proposed approaches is validated on a suite of benchmark functions. The proposed variants are compared with non linear inertia weight and exponential inertia weight strategies. Experimental results assert that the proposed modifications help in improving PSO performance in terms of solution quality as well as convergence rate.     

A numerical study on statistical temporal scales in inertia particle dispersion

The statistical temporal scales involved in inertia particle dispersion are analyzed numerically. The numerical method of large eddy simulation, solving a filtered Navier-Stokes equation, is utilized to calculate fully developed turbulent channel flows with Reynolds numbers of 180 and 640, and the particle Lagrangian trajectory method is employed to track inertia particles released into the flow fields. The Lagrangian and Eulerian temporal scales are obtained statistically for fluid tracer particles and three different inertia particles with Stokes numbers of 1, 10 and 100. The Eulerian temporal scales, decreasing with the velocity of advection from the wall to the channel central plane, are smaller than the Lagrangian ones. The Lagrangian temporal scales of inertia particles increase with the particle Stokes number. The Lagrangian temporal scales of the fluid phase ‘seen’ by inertia particles are separate from those of the fluid phase, where inertia particles travel in turbulent vortices, due to the particle inertia and particle trajectory crossing effects. The effects of the Reynolds number on the integral temporal scales are also discussed. The results are worthy of use in examining and developing engineering prediction models of particle dispersion.     

Improved penalty calculations for a mixed integer branch-and-bound algorithm

This paper presents an extension of Tomlin's penalties for the branch-and-bound linear mixed integer programming algorithm of Beale and Small. Penalties which are uniformly stronger are obtained by jointly conditioning on a basic variable and the non-basic variable yielding the Tomlin penalty. It is shown that this penalty can be computed with a little additional arithmetic and some extra bookkeeping. The improvement is easy to incorporate for the normal case as well as when the variables are grouped into ordered sets with generalized upper bounds. Computational experience bears out the usefulness of the extra effort for predominantly integer problems.