Solution of a class of stochastic linear-convex control problems using deterministic equivalents

In this paper, we identify a new class of stochastic linearconvex optimal control problems, whose solution can be obtained by solving appropriate equivalent deterministic optimal control problems. The term linear-convex is meant to imply that the dynamics is linear and the cost function is convex in the state variables, linear in the control variables, and separable. Moreover, some of the coefficients in the dynamics are allowed to be random and the expectations of the control variables are allowed to be constrained. For any stochastic linear-convex problem, the equivalent deterministic problem is obtained. Furthermore, it is shown that the optimal feedback policy of the stochastic problem is affine in its current state, where the affine transformation depends explicitly on the optimal solution of the equivalent deterministic problem in a simple way. The result is illustrated by its application to a simple stochastic inventory control problem.This research was supported in part by NSERC Grant A4617, by SSHRC Grant 410-83-0888, and by an INRIA Post-Doctoral Fellowship.     

Estimation of entropy and other functionals of a multivariate density

For a multivariate density f with respect to Lebesgue measure , the estimation of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa8qaaeaacaWGkbGaaiikaiaadAgacaGGPaGaamOzaiaadsgacqaH% 8oqBaSqabeqaniabgUIiYdaaaa!4404!\[\int {J(f)fd\mu } \], and in particular % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa8qaaeaacaWGMbWaaWbaaSqabeaacaaIYaaaaOGaamizaiabeY7a% TbWcbeqab0Gaey4kIipaaaa!41E4!\[\int {f^2 d\mu } \] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% Waa8qaaeaacaWGMbGaciiBaiaac+gacaGGNbGaamOzaiaadsgacqaH% 8oqBaSqabeqaniabgUIiYdaaaa!44AC!\[\int {f\log fd\mu } \], is studied. These two particular functionals are important in a number of contexts. Asymptotic bias and variance terms are obtained for the estimators % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaybyaeqaleqabaGaey4jIKnaneaacaWGjbaaaOGaeyypa0Zaa8qa% aeaacaWGkbGaaiikamaawagabeWcbeqaaiabgEIizdqdbaGaamOzaa% aakiaacMcacaWGKbGaamOramaaBaaaleaacaWGobaabeaaaeqabeqd% cqGHRiI8aaaa!4994!\[\mathop I\limits^ \wedge = \int {J(\mathop f\limits^ \wedge )dF_N } \] and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaybyaeqaleqabaGaeSipIOdaneaacaWGjbaaaOGaeyypa0Zaa8qa% aeaacaWGkbGaaiikamaawagabeWcbeqaaiabgEIizdqdbaGaamOzaa% aakiaacMcadaGfGbqabSqabeaacqGHNis2a0qaaiaadAgaaaGccaWG% KbGaeqiVd0galeqabeqdcqGHRiI8aaaa!4C40!\[\mathop I\limits^ \sim = \int {J(\mathop f\limits^ \wedge )\mathop f\limits^ \wedge d\mu } \], where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaybyaeqaleqabaGaey4jIKnaneaacaWGMbaaaaaa!3E9C!\[{\mathop f\limits^ \wedge }\] is a kernel density estimate of f and F _n is the empirical distribution function based on the random sample X ₁ ,..., X _n from f. For the two functionalsmentioned above, a first order bias term for Î can be made zero by appropriate choices of non-unimodal kernels. Suggestions for the choice of bandwidth are given; for % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaybyaeqaleqabaGaey4jIKnaneaacaWGjbaaaOGaeyypa0Zaa8qa% aeaadaGfGbqabSqabeaacqGHNis2a0qaaiaadAgaaaGccaWGKbGaam% OramaaBaaaleaacaWGobaabeaaaeqabeqdcqGHRiI8aaaa!476C!\[\mathop I\limits^ \wedge = \int {\mathop f\limits^ \wedge dF_N } \], a study of optimal bandwidth is possible.This research was supported by an NSERC Grant and a UBC Killam Research Fellowship.     

配合物的生成热和配体的质子化热之间的直线焓关系——Ⅶ.Ni(Ⅱ)-2,2′-联吡啶-N-(间位取代苯基)氨基乙酸三元体系

有关Ni(Ⅱ)-N-(间位取代苯基)氨基乙酸二元体系的直线焓和直线熵关系,前文已有报道,本文报告Ni(Ⅱ)-2,2′-联吡啶-N-(间位取代苯基)氨基乙酸(Ni(Ⅱ)-biPy-m-RPhG:R=CH_3,H,CH_3O,Cl)三元体系的生成热研究,发现在此三元体系中亦存在良好的直线焓和直线熵关系。     

高聚物亚浓溶液区标度律的蒙特卡洛模拟

利用蒙特卡洛模拟方法对线形链和星形键在亚浓溶液区的标度律作了验证与讨论。我们的计算结果证明线形链渗透压Π与体积分数Φ在亚浓溶液区的标度规律是与德热纳的理论结果相一致的。同时还证明了星形链在亚浓溶液区具有与线形链相同的标度行为。     

Generalized classical theory of magnetism

We consider an Ising model with Kac potential ^dK(¦x¦) which may have arbitrary sign, and show, following Gates and Penrose, that the free energy in the classical limit0+ can be obtained from a variational principle. When the Fourier transform of the potential has its maximum atp=0 one recovers the usual mean-field theory of magnetism. When the maximum occurs forp ₀0, however, one obtains an oscillatory or helicoidal phase in which the magnetization near the critical point oscillates with period 2/¦p ₀¦. An example with a potential possessing parameter-dependent oscillations is shown to exhibit crossover phenomena and a multicritical Lifshitz point in the classical limit.     

Dynamical systems with relaxation time

We define mathematically a class of dynamical systems that exhibit relaxation corresponding to that observed in physical systems, and then show that this class is identical with the class ofK-mixing dynamical systems.     

Heat conductivity of three periodic Hard Disks via nonequilibrium molecular dynamics

We use a driving field, of the type first suggested by Evans, to generate a steady heat current in the simplest possible system, a two-dimensional periodic fluid of three hard disks. Hard-disk motion equations can be conveniently derived from repulsive constant-force or linear-force potentials by considering the infinitely repulsive limit of these potentials. We show that the isoenergetic and isokinetic forms of the nonequilibrium equations of motion generate steady-state heat conductivities differing by terms of order 1/N, whereN is the number of particles. The resulting conductivities appear to vary as the logarithm of the driving field strength. Even at low fields, the three-body periodic-system results lie well below Enskog's infinite-system prediction.     

Excess heat capacities and orientational order in systems containing hexadecane isomers of different molecular structure

Using the Picker flow microcalorimeter, excess heat capacities have been obtained at 25°C throughout the concentration range for 2,2-dimethylbutane,n-hexane, and cyclohexane each mixed with a series of hexadecane isomers of increasing degrees of orientational order, as determined by depolarized Rayleigh scattering. The isomers are 2,2,4,4,6,8,8-heptamethylnonane, 6-, 4-, and 2-methylpentadecane, andn-hexadecane. Thec _p ^E values are negative, increasing rapidly in magnitude with increase of orientational order, and are not predicted by the Prigogine—Flory theory which neglects order. Values ofc _p ^E are obtained at 10, 25, and 55°C for cyclohexane +6-, 4-, and 2-methylpentadecane which with other literature data lead to the temperature dependence of the thermodynamic excess functions for cyclohexane solutions of the five C₁₆ isomers. The excess enthalpy and entropy vary with the C₁₆ isomer and with temperature, but the corresponding variation of the excess free energy is small, indicating a high degree of enthalpy-entropy compensation. This is consistent with a rapid decrease with temperature of orientational order in the C₁₆ isomers.     

Entropy evaluation using the kinetic method: is it feasible?

The kinetic method is one of the most widely used experimental techniques for the measurement of thermochemical parameters by mass spectrometry. Recently it has been realized that it can also be used to determine reaction entropies, but the validity of this approach has not been established. This Perspective evaluates kinetic method plots in cases where there is a significant entropy difference between the competing fragmentation channels (i.e. between sample and reference compounds in the dissociating cluster ion). The concept underlying this study is to calculate mass spectra theoretically, based on known thermochemical parameters and as a function of experimental conditions. This can be done accurately using the RRKM-based MassKinetics software. The resulting mass spectra are then interpreted by the kinetic method, yielding DeltaH and DeltaS values. These values are, in turn, compared with the true values used to generate the calculated mass spectra. The results show that the reaction entropy difference between sample and reference has a very large influence on kinetic method plots. This should always be considered when studying energy-dependent mass spectra (using metastable ions or low- or high-energy collision-induced dissociation (CID)), even if only DeltaH is to be determined. Kinetic method plots are not strictly linear and this becomes a serious issue in the case of small molecules showing a large entropy effect. In such cases, results obtained at a low degree of excitation are more accurate. Energy and entropy effects can be evaluated in a relatively straightforward manner: first, the apparent Gibbs energy (DeltaG(app)) and effective temperature (T(eff)) are determined from kinetic method plots (intercept and slope, respectively), obtained from experiments using various degrees of excitation. Second, the resulting DeltaG(app) is plotted against T(eff), the slope yielding DeltaS while the intercept (extrapolation to zero temperature) yields DeltaH. This data evaluation yields more accurate results than alternative methods used in the literature. The resulting DeltaH values are fairly accurate, with errors, in most cases, <4 kJ mol(-1). On the other hand, DeltaS is systematically underestimated by 20-40%. Empirically scaling DeltaS values determined by the kinetic method by 1.35 results in a DeltaS value within 20% (or 10 J mol(-1) K(-1)) of the theoretical value.     

Thermodynamic characteristics of thermal dissociation of platinum tetrachloride

The pressure of thermal dissociation of platinum tetrachloride by the first step PtCl₄(s) = PtCl₃(s) + 0.5 Cl₂(g) was measured by the static method with a quartz membrane-gauge zero-pressure manometer. An approximating equation for the dissociation pressure vs. temperature was found. The enthalpy (52160±880 J mol⁻¹) and entropy (72.1±1.6 J mol⁻¹ K⁻¹) of dissociation were calculated. The heat of formation found for platinum tetrachloride (−246.3±1.3 kJ mol⁻¹) at 298.15 K agrees well with the value obtained by the calorimetric method (−245.6±1.9 kJ mol⁻¹).__________Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 10, pp. 2028–2031, October, 2004.