Optimal Control of Non-well-posed Heat Equations

This work is concerned with Pontryagin＇s maximum principle of optimal control problems governed by some non-well-posed semilinear heat equations. A type of approach to the non-well-posed optimal control problem is given.     

Optimal control and properties of nonlinear multistage dynamical system for planning horizontal well paths

The goal of planning a horizontal well path is to obtain a trajectory that arrives at a given target subject to various constraints. In this paper, the optimal control problem subject to a nonlinear multistage dynamical system (NMDS) for horizontal well paths is investigated. Some properties of the multistage system are proved. In order to derive the optimality conditions, we transform the optimal control problem into one with control constraints and inequality-constrained trajectories by defining some functions. The properties of these functions are then discussed and optimality conditions for optimal control problem are also given. Finally, an improved simplex method is developed and applied to the optimal design for well Ci-16-Cp146 in Oil Field of Liaohe, and the numerical results illustrate the validity of both the model and the algorithm.     

Valuation of soccer spread bets

Simple statistical and probabilistic arguments are used to value the most commonly traded online soccer spread bets. Such markets typically operate dynamically during the course of a match and accurate valuations must, therefore, reflect the changing state of the match. Both goals and corners are assumed to evolve as Poisson processes with constant means. Although many of the bets that are typically traded are relatively easy to value, some (including the ‘four flags’ market) require more detailed analysis. Examples are given of the evolution of the spread during typical matches and theoretical predictions are shown to compare closely to spreads quoted by online bookmakers during some of the important matches of the EURO2004 tournament.     

Multidimensional scaling analysis of soccer dynamics

This paper studies the behavior of teams competing within soccer national leagues. The dissimilarities between teams are measured using the match results at each round and that information feeds a multidimensional scaling (MDS) algorithm for visualizing teams’ performance. Data characterizing four European leagues during season 2014–2015 is adopted and processed using three distinct approaches. In the first, one dissimilarity matrix and one MDS map per round are generated. After, Procrustes analysis is applied to linearly transform the MDS charts for maximum superposition and to build one global MDS representation for the whole season. In the second approach, all data is combined into one dissimilarity matrix leading to a single global MDS chart. In the third approach, the results per round are used to generate time series for all teams. Then, the time series are compared, generating a dissimilarity matrix and the corresponding MDS map. In all cases, the points on the maps represent teams state up to a given round. The set of points corresponding to each team forms a locus representative of its performance versus time.     

The Cost of the Missing Bit: Communication Complexity with Help

Dedicated to the Memory of Paul Erdős We generalize the multiparty communication model of Chandra, Furst, and Lipton (1983) to functions with b-bit output (b = 1 in the CFL model). We allow the players to receive up to b - 1 bits of information from an all-powerful benevolent Helper who can see all the input. Extending results of Babai, Nisan, and Szegedy (1992) to this model, we construct families of explicit functions for which bits of communication are required to find the "missing bit", where n is the length of each player's input and k is the number of players. As a consequence we settle the problem of separating the one-way vs. multiround communication complexities (in the CFL sense) for players, extending a result of Nisan and Wigderson (1991) who demonstrated this separation for k = 3 players. As a by-product we obtain lower bounds for the multiparty complexity (in the CFL sense) of new families of explicit boolean functions (not derivable from BNS). The proofs exploit the interplay between two concepts of multicolor discrepancy; discrete Fourier analysis is the basic tool. We also include an unpublished lower bound by A. Wigderson regarding the one-way complexity of the 3-party pointer jumping function. Received November 12, 1998 RID="*" ID="*" Supported in part by NSA grant MSPR-96G-184. RID="†" ID="†" Supported in part by an NSF Graduate Fellowship.     

Missing Observations in Spatial Models: A Spectral EM Algorithm

This article presents a model-based thin-plate smoothing method for optimal signal extraction and interpolation of missing data in spatial datasets. The method is based on a spectral EM algorithm where the two steps can be carried out in the frequency domain. In essence, the approach allows both dimensions to be treated separately from each other effectively rendering a likelihood that is easy to evaluate. As a result the algorithm is computationally inexpensive, in terms of both memory size and computing time, while allowing us to obtain an analytic expression for the asymptotic variance of the signal-to-noise ratio with which to construct confidence intervals of the missing data. Some numerical Monte Carlo simulations and a real data example using remotely sensed global aerosol optical thickness data illustrate the results given. Supplemental materials (Matlab computer code and dataset) are available online.     

Short shots and industrial case studies: Understanding fluid flow and solidification in high pressure die casting

The geometric complexity and high fluid speeds involved in high pressure die casting (HPDC) combine to give strongly three dimensional fluid flow with significant free surface fragmentation and splashing. A simulation method that has proved particularly suited to modelling HPDC is Smoothed Particle Hydrodynamics (SPH). Materials are approximated by particles that are free to move around rather than by fixed grids, enabling more accurate prediction of fluid flows involving complex free surface motion. Three practical industrial case studies of SPH simulated HPDC flows are presented; aluminium casting of a differential cover (automotive), an electronic housing and zinc casting of a door lock plate. These show significant detail in the fragmented fluid free surfaces and allow us to understand the predisposition to create defects such as porosity in the castings. The validation of flow predictions coupled with heat transfer and solidification is an important area for such modelling. One powerful approach is to use short shots, where insufficient metal is used in the casting or the casting shot is halted part way through, to leave the die cavity only partially filled. The frozen partial castings capture significant detail about the order of fill and the flow structures occurring during different stages of filling. Validation can occur by matching experimental and simulated short shots. Here we explore the effect of die temperature, metal super-heat and volume fill on the short shots for the casting of a simple coaster. The bulk features of the final solid castings are found to be in good agreement with the predictions, but the fine details appear to depend on surface behaviour of the solidifying metals. This potentially has significant implications for modelling HPDC.     

Classroom note: A simple non-trivial student model of a bouncing soccer ball

When students have made some progress in studying ordinary differential equations (ODEs) during a semester, small groups are given a soccer ball and asked to determine the simplest model describing one bounce of the ball from shoulder height. To start with, they play with the ball. However, after a few weeks, they begin to ask penetrating questions and they are led to a simple model that has a non-trivial twist to the solution.     

Missing boundary data recovery using Nash games: the Stokes system

We consider the Cauchy-Stokes problem. We use a new method based on Nash game theory to recover the missing velocity and normal stress on some inaccessible part of the boundary. This method is used with two different approaches. The first one is compared to a control type one. The numerical study attests that both approaches give accurate results. We compare these results with those of the energy-like minimization method.     

Quality function deployment for the good of soccer

This article presents a model for prioritizing and designing rule changes for the game of soccer in order to make it more attractive to soccer enthusiasts. The model, which is loosely based on quality function deployment (QFD), incorporates market segments, sports enthusiast interests, soccer activities, and rules of the game as rows and columns of interconnected QFD matrices. In addition, analytic hierarchy process (AHP) is used to determine the intensity of the relationship between the row and column variables of each matrix, while analytic network process (ANP) is used to determine the intensity of synergy effects among column variables. Finally, a trend extrapolation forecasting technique is used to suggest rule change specifications. The model fine-tunes and adds precision to the otherwise ad hoc process concerning the improvement of the game of soccer, and can provide insight about improvements in other professional sports.     

A comparison of penalty shootout designs in soccer

4OR - Penalty shootout in soccer is recognized to be unfair because the team kicking first in all rounds enjoys a significant advantage. The so-called Catch-Up Rule has been suggested recently to...     

Solution Extensions: Optimal Parameter Method in Optimal Control Problems

Using the extension of solution by the optimal parameter method, we obtain a numerical solution for a certain class of optimal control problems.     

Erratum to: Exponentially confining potential well

Theoretical and Mathematical Physics - An Erratum to this paper has been published: https://doi.org/10.1134/S0040577921010050     

Missing notions of equilibria for general game theory

Doklady Mathematics -     

关于数据缺失机制的检验方法探讨

在调查研究中,缺失数据是一个非常普遍的问题,各种处理缺失数据的方法都是建立在数据缺失机制的某种假定上.在总结他人研究成果的基础上,分别给出了MCAR、MAR和NMAR机制的检验识别方法,MCAR机制的检验从分布特征入手,通过比较均值和方差是否一致来判定;MAR机制的检验利用Logit模型刻画缺失指示变量R的分布,通过估计参数的显著性来判定,NMAR机制则通过对数据的缺失模式和原因进行分析来识别.