Optimality for set functions with values in ordered vector spaces

Let (X, , ) be a finite atomless measure space,L a convex subfamily of , andY andZ locally convex Hausdorff topological vector spaces which are ordered by the conesC andD, respectively. LetF:LY beC-convex andG:LZ beD-convex set functions. Consider the following optimization problem (P): minimizeF(), subject to L andG() _D . The paper generalizes the Moreau-Rockafellar theorem with set functions. By applying this theorem, a Kuhn-Tucker type optimality condition and a Fritz John type optimality condition for problem (P) are established. The duality theorem for problem (P) is also studied.This work was partially supported by National Science Council, Taipei, Taiwan. This paper was written while the first author was visiting at the University of Iowa, 1987-88.The authors would like to express their gratitude to the two anonymous referees for their valuable comments. Also, they would like to thank Professor P. L. Yu for his encouragement and suggestions which improved the material presented here considerably.     

Upper Bound to the Expectation Value of the Squared Hamiltonian

An upper bound to the expectation value of the squared Hamiltonian H ² is derived which relies on replacing products of certain operators with products of the matrix representations of said operators to reduce the computational demands of H ². An example is given which shows the strength of the bound and an application with the Temple lower bound is shown.     

Discovery of High-Pressure Basic Granulite Terrain in North China Archaean Craton and Preliminary Study

High-pressure basic granulite terrain of 2.65 Ga from Jiouhuai'an in the junction of the Hebei and Shanxi provinces was discovered and reported recently. The three metamorphic mineral assemblages and their mineral reaction texture of the basic granulites have been identified, which represent three main metamorphic episodes of 1.4—1.5 Gpa, 800℃; 0.7—0.9 Gpa, 820℃ and 0.5—0.6 Gpa, 700℃. Their steep decompressional clockwise P-T path implies that a late Archaean continental collision and sole thrusting event perhaps occurred in North China Craton. This is important to study petrogenesis of high-pressure granulites, composition and texture of deep crust in North China Craton, and early Precambrian collision tectonism.     

THERMODYNAMIC STUDY OF ADSORPTION OF PHENOLIC COMPOUNDS FROM AQUEOUS SOLUTION BY A WATER-COMPATIBLE HYPERCROSSLINKED POLYMERIC ADSORBENT

Equilibrium data for the adsorption of phenolic compounds, i.e., phenol, p-cresol, p-chlorophenol and p-nitrophenol from aqueous solutions by a water-compatible hypercrosslinked polymeric adsorbent (NJ-8) within temperature range of 283-323 K were obtained and correlated with a Freundlich-type of isotherm equation, so that equilibrium constants KF and n were obtained. The capacities of equilibrium adsorption for all the four phenolic compounds on the NJ-8 from aqueous solutions are around 2 times as high as those of Amberlite XAD-4, which may be attributed to the unusual micropore structure and the partial polarity on the network. The values of the enthalpy (always negative) are indicative of an exothermic process, which manifests the adsorption of all the four phenolic compounds on the two polymeric adsorbents to be a process of physical adsorption. The negative values of free energy change show that the solute is more concentrated on the adsorbent than in the bulk solution. The absolute free energy values of adsorption for NJ-8 are always higher than those for Amberlite XAD-4, which indicates that phenolic compounds are preferentially adsorbed on NJ-8. The negative values of the adsorption entropy are consistent with the restricted mobilities of adsorbed molecules of phenolic compounds as compared with the molecules in solution. The adsorption entropy values of phenolic compounds for NJ-8 are lower than those for Amberlite XAD-4, which means the micropores of NJ-8 require more orderly arranged adsorbate.     

Global Existence of Solutions and Lower Bound Estimate of Blow-Up Time for the Keller-Segel Chemotaxis Model北大核心CSCD

A macroscopic nonlinear Keller-Segel model for chemotactic cell migration was considered, where the existence region of the model is a bounded convex one on Ω ⊂ RN(N≥2). The global existence of the solution on Ω ⊂ R3 was obtained by means of the energy estimate method. The lower bound of the blow-up time was proved for N = 3 and N = 2. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

Lower semicontinuous solutions of the Bellman equation for the minimum time problem

The minimum time problem associated with a nonlinear control system is considered, and the unicity of the lower semicontinuous solution of the corresponding Bellman equation is investigated. A main tool in our approach is the Kruzkov transformation that enables us to work on ⁿ –{0}, where {0} is the target set, instead of the unknown reachable set.This research was carried out while the first author was visiting the Department of Mathematics, University of Genova, Genova, Italy.     

The occupation time of Brownian motion in a ball

LetW be a Wiener process of dimensiond3, starting from 0, and letX(t) be the total time spent byW in the ball centered at 0 with radiust. We give an affirmative answer to a conjecture of Taylor and Tricot⁽¹⁶⁾ on the tail distribution ofX(t). Lévy's lower functions ofX(t) are characterized by an integral test.     

硝酸钆低水合物的制备及表征

本文用两种方法制得硝酸钆的低水合物Gd(NO_3)_3·5H_2O、Gd(NO_3)_3·3.5H_2O,其中Gd(NO_3)_3·3.5H_2O是文献中尚未报道的 ,给出了它们的M.P.和X-衍射特征值。     

Heterogeneous Catalytic Partial Oxidation of C3-C4 Alkanes with Nitrogen Monoxide

The possibility of making products of partial oxidation, alcohols and acids in particular, by the reaction of C₃-C₄ alkanes with nitrogen monoxide over oxide and zeolite catalysts is demonstrated. The most effective catalysts are CeO₂ and Fe-TsVK. The dependence of the rate and selectivity of the process on the reaction conditions has been established.     

Asymptotic Behavior of the t Expansion

The asymptotic behavior of a system's ground-state energy from the t expansion of Horn and Weinstein has been suggested to have the form E ₁(t)=E ₁+exp(–a _n t+b _n ). In the limit of very large t, this becomes E ₁(t)=E ₁+exp(–a ₁ t+b ₁). A simple analysis shows that the parameters are a ₁=E ₂–E ₁ and b ₁=ln[(E ₂–E ₁)|c ₂|²/|c ₁|²]. Functions are introduced which allow determination of a ₁, b ₁ and lower bounds to E ₁.