Quadratically constrained minimum cross-entropy analysis

Quadratically constrained minimum cross-entropy problem has recently been studied by Zhang and Brockett through an elaborately constructed dual. In this paper, we take a geometric programming approach to analyze this problem. Unlike Zhang and Brockett, we separate the probability constraint from general quadratic constraints and use two simple geometric inequalities to derive its dual problem. Furthermore, by using the dual perturbation method, we directly prove the strong duality theorem and derive a dual-to-primal conversion formula. As a by-product, the perturbation proof gives us insights to develop a computation procedure that avoids dual non-differentiability and allows us to use a general purpose optimizer to find an-optimal solution for the quadratically constrained minimum cross-entropy analysis.     

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.     

From dilute to dense self-avoiding walks on hypercubic lattices

A field-theoretic representation is presented to count the number of configurations of a single self-avoiding walk on a hypercubic lattice ind dimensions with periodic boundary conditions. We evaluate the connectivity constant as a function of the fractionf of sites occupied by the polymer chain. The meanfield approximation is exact in the limit of infinite dimensions, and corrections to it in powers ofd ^–1 can be systematically evaluated. The connectivity constant and the site entropy calculated throughout second order compare well with known results in two and three dimensions. We also find that the entropy per site develops a maximum atf1–(2d)^–1. Ford=2 (d=3), this maximum occurs atf~0.80 (f~0.86) and its value is about 50% (30%) higher than the entropy per site of a Hamiltonian walk (f=1).     

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。     

Dimerization of some substituted benzoic acids in aqueous solution from conductance measurements

Precision molar conductances of benzoic, o-toluic, 2,6-dimethylbenzoic, 2,3,6-trimethylbenzoic, and, o-fluorobenzoic acids have been determined in aqueous solution as a function of temperature and of concentration up to near saturation (<0.035 M). At the higher concentrations molar conductances are found to be less than anticipated for the simple dissociation of a 1-1 electrolyte. Although the deviations are only 1% or less they have been interpreted to show that these acids are dimerized in solution. The interpretation includes an assumption that the dimer ionizes to produce a triple ion. Increasing numbers of methyl groups lead to increasing dimerization. For those acids with two ortho groups the dimerization increases with increasing temperature while the other three show decreasing dimerization with increasing temperature. Temperature functions have been determined for the dimerization constants and from these functions standard changes in enthalpy, entropy, and heat capacity have been determined. Comparisons are made with dimerization studies in non-aqueous solvents. From these as well as the behavior of benzene in water it is concluded that a major factor driving the dimerization is hydrophobic interaction. To provide a limiting conductance of the triple ion needed in the dimerization calculations a conductance study was also made for o-Phenylbenzoic acid on the assumption that its anion provides an approximate model of the triple ion.     

Second Dissociation Constants of 4-[N-morpholino]butanesulfonic Acid and N-[2-hydroxyethyl]piperazine-N′-4-butanesulfonic Acid from 5 to 55°C

The values of the second dissociation constant, pK ₂, for the dissociation of the NH⁺ charge center of the zwitterionic buffer compounds 4-(N-morpholino)butanesulfonic acid (MOBS), and N-(2-hydroxyethyl)piperazine-N-4-butanesulfonic acid (HEPBS) have been determined from 5 to 55°C, including, 37°C at intervals of 5°C. The electromotive-force (emf) measurements have been made utilizing hydrogen electrodes and silver–silver chloride electrodes. The value of pK ₂ for MOBS was found to be 7.702 ± 0.0005, and 8.284 ± 0.0004 for HEPBS, at 25°C, respectively. The related thermodynamic quantities, G ^o, H ^o, S ^o, and C _p ^o for the dissociation processes of MOBS and HEPBS have been derived from the temperature coefficients of pK ₂. Both the MOBS and HEPBS buffer materials are useful as primary pH standards for the control of pH 7.3 to 8.6 in the region close to that of physiological fluids.     

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.     

Thermal Studies on Ionic Derivatives of Hafnium(IV) Involving Maltol Ligand

A number of ionic chelate complexes of maltol(A) and hafnium(IV) the type[(η₅−C₅H₅)₂HfL]⁺[MCl3]⁻ (B) [HL=maltol; M=Zn(II), Cd(II), Hg(II), Cu(II)]have been synthesized and characterized by spectral studies (IR, UV, ¹H NMR and ¹³C NMR). The stoichiometry of the complexes has been confirmed by conductance measurements. Thermogravimetric (TG) and differential thermal analytical (DTA) studies have been carried out for these complexes and from TG curves, the order, apparent activation energy and apparent activation entropy of the thermal decomposition reactions have been elucidated .The order in each case has been determined to be one and the degree of spontaneity and lability have been inferred from the apparent activation energy and entropy, respectively. Thermal parameters have been correlated with some structural aspects of the complexes concerned. From differential thermal analysis curves, the heat of reaction has been calculated. This revised version was published online in August 2006 with corrections to the Cover Date.     

Hydration heat capacities of hydrophobic solutes at 248–373 K

A new equation is suggested to define the temperature dependence of the Gibbs energy of hydration of hydrophobic substances: ΔG ⁰ = b ₀ + b ₁ T + b ₂lnT. According to this equation, the hydration heat capacity is in inverse proportion to temperature. Consistent values of hydration heat capacity of nonpolar solutes have been obtained for different temperatures using data on solubility and dissolution enthalpy. The contributions of the hydrocarbon radicals and OH group to the heat capacity of hydration of the compounds were found for the temperature range 248–373 K. The hydration heat capacity of the hydroxyl group has a weak dependence on temperature and increases by only 12 J/(mol·K) in the specified temperature interval. Changes in the hydration entropy of hydrophobic and OH groups are calculated for the temperature increasing from 248 K to 373 K.