Product Integration Rules at Clenshaw-Curtis and Related Points: A Robust Implementation

Product integration rules generalizing the Fej?r, Clenshaw-Curtis,and Filippi quadrature rules respectively are derived for integralswith trigonometric and hyperbolic weight factors. The Chebyshevmoments of the weight functions are found to be given by well-conditionedexpressions, in terms of hypergeometric functions ₀F₁. An a priori error estimator is discussed which is shown bothto avoid wasteful invocation of the integration rule and toincrease significantly the robustness of the automatic quadratureprocedure. Then, specializing to extended Clenshaw-Curtis (ECC) rules,three types of a posteriori error estimates are considered andthe existence of a great risk of their failure is demonstratedby large scale validation tests. An empirical error estimator,superseding them for slowly varying integrands, is found toresult in a spectacular increase in the output reliability. Finally, enhancements in the control of the interval subdivisionstrategy aiming at increasing code robustness is discussed.Comparison with the code DQAWO of QUADPACK, with about a hundredthousand solved integrals, is illustrative of the increasedrobustness and error estimate reliability of our computer codeimplementation of the ECC rules.     

Boundary control of a hyperbolic system with time lags

This paper considers an optimal boundary control problem fora hyperbolic system in which constant time lags appear in thestate equation and in the boundary condition simultaneously.Making use of Lion's scheme, necessary and sufficient conditionsof optimality for the Neumann problem are derived.     

Time-optimal control of a parabolic system with time lags given in the integral form

In this paper the time-optimal control problem for a distributedparabolic system in which time lags appear in the integral formboth in the state equation and in the boundary condition ispresented. The particular properties of the optimal controlare discussed.     

稻壳SiO2/钒催化剂上苯乙酮液相氧化（英文）

Rice husk silica catalyst loaded with 10 wt% vanadium was synthesized from agricultural biomass via a sol-gel synthetic route at pH = 9. The catalyst was characterized by different physico-chemical methods. The FTIR spectra showed the formation of Si–O–V and V=O stretching bands. The presence of vanadium was confirmed by EDX elemental analysis. RH-10V possessed a high specific surface area of 276 m2/g and pore volume of 0.83 ml/g. The prepared catalyst possessed a narrow pore size distribution centered around 7.9 nm. The catalytic performance of RH-10V was tested in the oxidation of acetophenone at 70 ℃. RH-10V was found to be an active catalyst in the oxidation of acetophenone, producing 36.28% conversion efficiency. The products identified were benzoic acid, 2-hydroxyacetophenone, phenol, acetic acid, and 3-hydroxyacetophenone.     

Control of distributed time delay systems with Dirichlet boundary conditions

Various optimization problems associated with the optimal controlof distributed parameter systems with time lags appearing inthe boundary conditions have been studied recently by Wang (1975),Knowles (1978), Wong (1987), Kowalewski (1987a,b, 1988a,b,c,1990a,b,c,d, 1991, 1993a,b,c,d, 1995) and Kowalewski & Duda(1992). In this paper optimal boundary control problems fordistributed systems described by linear partial differentialequations of parabolic and hyperbolic type in which constanttime delays appear in the state equations are considered. Sufficientconditions for the existence of a unique solutions of such equationswith the Dirichlet boundary conditions are proved. The performancefunctional has the quadratic form. The time horizon T is fixed.Finally, we impose some constraints on the control. Making useof Lions' scheme (Lions 1971) necessary and sufficient conditionsof optimality for the Dirichlet problem with the quadratic performancefunctional and constrained control are derived. The flow chartof the algorithm, which can be used in the numerical solvingof certain optimization problems for distributed parameter systems,is also presented.     

Complex monodromy and the topology of real algebraic sets

A relation between the Euler characteristics of the Milnorfibres of a real analytic function is derived from a simple identity involvingcomplex monodromy and complex conjugation. A corollary is the result of Costeand Kurdyka that the Euler characteristic of the local link of an irreduciblealgebraic subset of a real algebraic set is generically constant modulo 4. Asimilar relation for iterated Milnor fibres of ordered sets of functions isused to define topological invariants of ordered collections of algebraicsubsets.     

Nucleophilic Substitution Reactions of Meta‐ and Para‐Substituted Benzylamines with Benzyl Bromide in Methanol Medium

The rates of reactions of para‐ and meta‐substituted benzylamines with benzyl bromide were measured using conductivity technique in methanol medium. The reaction followed a total second‐order path. The end product of the reaction is identified as dibenzylamine (X‐C₆H₄CH₂NHCH₂C₆H₅) (where X = 4‐OCH₃, 4‐CH₃, H, 4‐Cl, 4‐CF₃, 3‐CF₃, 4‐NO₂). Electron‐withdrawing groups such as chloro, trifluoromethyl, and nitro in the benzylamine moiety decrease the rate of the reaction, whereas the electron‐donating groups, such as methoxy and methyl, increase the rate compared to the unsubstituted compound. A mechanism involving formation of an S_N2‐type transition state between the amine nucleophiles and the benzyl bromide and its subsequent decomposition is proposed. Hammett's reaction constant ρ of the reaction decreases with an increase in temperature. Activation parameters were calculated and discussed.     

Minimum time problem for a parabolic system with time-varying lags

In this paper the time-optimal control problem for a distributedparabolic system in which time-varying lags appear both in thestate equation and in the boundary condition is presented. Theparticular properties of the optimal control are discussed.     

Feedback Control of a Distributed Parabolic System with Boundary Condition Involving a Time-Varying Lag

In this paper, a feedback control problem for a distributed-parametersystem with boundary condition involving a time-varying lagis considered. Neccessary and sufficient conditions for optimalityare derived. The optimal control is obtained in feedback form.Estimates for the solutions of parabolic systems with specifiedforms of feedback control are established.