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Numerical Algebra, Control & Optimization

2012 , Volume 2 , Issue 2

Special Issue dedicated to Professor Charles Pearce on the occasion of his 70th birthday

Special Issue Papers: 223-375; Regular Papers: 377-435

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Song Wang and  Yong Hong Wu
2012, 2(2): i-ii doi: 10.3934/naco.2012.2.2i +[Abstract](37) +[PDF](76.2KB)
This Special Issue was intended to be dedicated to Charles Pearce, a Distinguished Professor at The University of Adelaide, on the occasion of his 70th birthday and in recognition of his many fundamental contributions in Applied Mathematics, Pure Mathematics and Mathematical Statistics. However, as this Special Issue is sent to print, we received the shocking news that Professor Charles Pearce was killed in a car accident near Fox Glacier, New Zealand on Friday, 9 June, 2012. With great sadness, we wish to dedicate this Special Issue to the late Charles Pearce in memory of his achievements in the elds and contributions to the mathematical communities.
On a family of means generated by the Hardy-Littlewood maximal inequality
Aleksandra Čižmešija , Iva Franjić , Josip Pečarić and  Dora Pokaz
2012, 2(2): 223-231 doi: 10.3934/naco.2012.2.223 +[Abstract](33) +[PDF](150.4KB)
The functional defined as the difference between the right-hand and the left-hand side of the Hardy-Littlewood maximal inequality is studied and its properties, such as exponential and logarithmic convexity, are explored. Furthermore, related analogues of the Lagrange and Cauchy mean value theorems are derived. Finally, using this functional, a new family of the Cauchy-type means is generated. These means are shown to be monotone.
Maximum entropy methods for generating simulated rainfall
Julia Piantadosi , Phil Howlett , Jonathan Borwein and  John Henstridge
2012, 2(2): 233-256 doi: 10.3934/naco.2012.2.233 +[Abstract](44) +[PDF](655.1KB)
We desire to generate monthly rainfall totals for a particular location in such a way that the statistics for the simulated data match the statistics for the observed data. We are especially interested in the accumulated rainfall totals over several months. We propose two different ways to construct a joint rainfall probability distribution that matches the observed grade correlation coefficients and preserves the prescribed marginal distributions. Both methods use multi-dimensional checkerboard copulas. In the first case we use the theory of Fenchel duality to construct a copula of maximum entropy and in the second case we use a copula derived from a multi-variate normal distribution. Finally we simulate monthly rainfall totals at a particular location using each method and analyse the statistical behaviour of the corresponding quarterly accumulations.
On Markovian solutions to Markov Chain BSDEs
Samuel N. Cohen and  Lukasz Szpruch
2012, 2(2): 257-269 doi: 10.3934/naco.2012.2.257 +[Abstract](51) +[PDF](314.2KB)
We study (backward) stochastic differential equations with noise coming from a finite state Markov chain. We show that, for the solutions of these equations to be `Markovian', in the sense that they are deterministic functions of the state of the underlying chain, the integrand must be of a specific form. This allows us to connect these equations to coupled systems of ODEs, and hence to give fast numerical methods for the evaluation of Markov-Chain BSDEs.
Some new bounds for two mappings related to the Hermite-Hadamard inequality for convex functions
S. S. Dragomir and  I. Gomm
2012, 2(2): 271-278 doi: 10.3934/naco.2012.2.271 +[Abstract](36) +[PDF](125.9KB)
Some new results concerning two mappings associated to the celebrated Hermite-Hadamard integral inequality for convex function with applications for special means are given.
Jensen's inequality for quasiconvex functions
S. S. Dragomir and  C. E. M. Pearce
2012, 2(2): 279-291 doi: 10.3934/naco.2012.2.279 +[Abstract](46) +[PDF](179.1KB)
Some inequalities of Jensen type and connected results are given for quasiconvex functions on convex sets in real linear spaces.
How to transform matrices $U_1, \ldots, U_p$ to matrices $V_1, \ldots, V_p$ so that $V_i V_j^T= {\mathbb O} $ if $ i \neq j $?
Vladimir Ejov and  Anatoli Torokhti
2012, 2(2): 293-299 doi: 10.3934/naco.2012.2.293 +[Abstract](30) +[PDF](344.1KB)
Methods of a transformation of matrices $U_1, \ldots, U_p$ to matrices $V_1, \ldots, V_p$ are proposed so that $V_i V_j^T={\mathbb O} $ for $ i\neq j $ and $i,j =1,\ldots,p $. We consider unconstrained and constrained problems associated with such a transformation. Solutions of the both problems are provided.
A sufficient optimality condition for nonregular problems via a nonlinear Lagrangian
A. C. Eberhard and  C.E.M. Pearce
2012, 2(2): 301-331 doi: 10.3934/naco.2012.2.301 +[Abstract](28) +[PDF](378.3KB)
A reformulation of a standard smooth mathematical program in terms of a nonlinear Lagrangian is used in conjunction with the calculus of subhessians to derive a set of sufficient optimality conditions that are applicable to some nonregular problems. These conditions are cast solely in terms of the first-- and second--order derivatives of the constituent functions and generalize standard second--order sufficiency conditions to a wide class of potentially nonregular problems.
Duration problem with multiple exchanges
Charles E. M. Pearce , Krzysztof Szajowski and  Mitsushi Tamaki
2012, 2(2): 333-355 doi: 10.3934/naco.2012.2.333 +[Abstract](29) +[PDF](342.6KB)
We treat a version of the multiple-choice secretary problem called the multiple-choice duration problem, in which the objective is to maximize the time of possession of relatively best objects. It is shown that, for the $m$--choice duration problem, there exists a sequence $(s_1,s_2,\ldots,s_m)$ of critical numbers such that, whenever there remain $k$ choices yet to be made, then the optimal strategy immediately selects a relatively best object if it appears at or after time $s_k$ ($1\leq k\leq m$). We also exhibit an equivalence between the duration problem and the classical best-choice secretary problem. A simple recursive formula is given for calculating the critical numbers when the number of objects tends to infinity. Extensions are made to models involving an acquisition or replacement cost.
Performance analysis of transmission rate control algorithm from readers to a middleware in intelligent transportation systems
Sangkyu Baek , Jinsoo Park and  Bong Dae Choi
2012, 2(2): 357-375 doi: 10.3934/naco.2012.2.357 +[Abstract](41) +[PDF](605.4KB)
Radio Frequency Identification (RFID) systems consisting of tags, readers and a middleware are known as one of the promising technologies to be applied to diverse fields for realizing a ubiquitous society. A typical RFID system where RFID readers collect information from tags on vehicles and send the collected data to a middleware has been used primarily for various distribution and logistics applications. The collected information by the middleware is used for many intelligent transportation systems (ITS) applications such as traffic estimation and real-time navigation. We propose a dynamic transmission rate control algorithm between readers and a middleware with limited buffer capacity environment and present analytical performances. We construct 3-dimensional continuous time Markov chains whose Q-matrix has the form of a quasi-birth and death structure. From the analytical model, we obtain performance measures such as packet loss probability and system throughput. We find the maximum number of readers associated with a middleware while satisfying a constraint on packet loss probability.
Numerical methods for estimating effective diffusion coefficients of three-dimensional drug delivery systems
Shalela Mohd--Mahali , Song Wang , Xia Lou and  Sungging Pintowantoro
2012, 2(2): 377-393 doi: 10.3934/naco.2012.2.377 +[Abstract](64) +[PDF](413.8KB)
This paper presents a numerical technique in three dimensions for estimating effective diffusion coefficients of drug release devices in rotating and flow-through fluid systems. We first formulate the drug release problems as diffusion equation systems with unknown effective diffusion coefficients. We then develop a numerical technique for estimating the unknown coefficients based on a nonlinear least-squares method and a finite volume discretization scheme for the 3D diffusion equations. Numerical experiments have been performed using experimental data and the numerical results are presented to show that our methods give accurate diffusivity estimations for the test problems.
A nonmonotone spectral projected gradient method for large-scale topology optimization problems
Rouhollah Tavakoli and  Hongchao Zhang
2012, 2(2): 395-412 doi: 10.3934/naco.2012.2.395 +[Abstract](50) +[PDF](429.6KB)
An efficient gradient-based method to solve the volume constrained topology optimization problems is presented. Each iterate of this algorithm is obtained by the projection of a Barzilai-Borwein step onto the feasible set consisting of box and one linear constraints (volume constraint). To ensure the global convergence, an adaptive nonmonotone line search is performed along the direction that is given by the current and projection point. The adaptive cyclic reuse of the Barzilai-Borwein step is applied as the initial stepsize. The minimum memory requirement, the guaranteed convergence property, and almost only one function and gradient evaluations per iteration make this new method very attractive within common alternative methods to solve large-scale optimal design problems. Efficiency and feasibility of the presented method are supported by numerical experiments.
Control design of linear systems with saturating actuators: A survey
Magdi S. Mahmoud and  Mohammed M. Hussain
2012, 2(2): 413-435 doi: 10.3934/naco.2012.2.413 +[Abstract](46) +[PDF](479.5KB)
In this paper, a survey of the extensive research investigation performed on linear systems subject to saturation including actuator, output and state types is presented. The survey takes into consideration several technical views on the analysis and design procedures leading to global or semi-global stability results and outlines basic assumptions. Research works on the design of linear feedback laws, decentralized controllers are equally emphasized. Results related stability with enlarging the domain of attraction and systems subject to multi-layered nested saturations are provided. Some typical examples are given to illustrate relevant issues.




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