Applications of Danishefsky’s dienes in asymmetric Oxo-Diels-Alder reactions

Applications of Danishefsky’s dienes in catalytic asymmetric hetero-Diels-Alder (AHDA) reactions or specifically, their asymmetric Oxo-Diels-Alder (AOxo-DA) reactions with appropriate dienophiles are highlighted in detail, including the preparation of catalysts with discussion from a mechanistic points of view. Danishefsky’s dienes are effective and useful compounds for the synthesis of optically active six-membered rings such as dihydropyrones, dihydropyridones and dihydropyrans. Due to the broad range of the overall subject, we have limited ourselves, to the recent developments in the utility of Danishefsky’s dienes in the reaction with carbonyl compounds (aldehydes, ketones and 1,2-dicarbonyl compounds) in asymmetric Oxo-Diels-Alder (AOxo-DA) reactions.     

Some Identities Involving Square of Fibonacci Numbers and Lucas Numbers

By studying the properties of Chebyshev polynomials, some specific and meaningful identities for the calculation of square of Chebyshev polynomials, Fibonacci numbers and Lucas numbers are obtained.     

Extended large deviations

LetX ₁,X ₂,..., be a sequence ofi.i.d. random variables with a moment generating function finite in a neighborhood of 0. Further, for each integern1, letS _n denote the sum of the firstn terms in this sequence. We study the extended large deviation of such sums, meaning,P{S _n >n _n }, where _n is any sequence converging to infinity. We also derive functional extended large deviation theorems and then apply them to obtain functional versions of the Erdös-Rényi strong law of large numbers.Research partially supported by an NSF Grant.     

Chaotic Numbers of Complete Bipartite Graphs and Tripartite Graphs

In this paper, we study the chaotic numbers of complete bipartite graphs and complete tripartite graphs. For the complete bipartite graphs, we find closed-form formulas of the chaotic numbers and characterize all chaotic mappings. For the complete tripartite graphs, we develop an algorithm running in O(n ⁴ ₃) time to find the chaotic numbers, with n ₃ the number of vertices in the largest partite set.Research supported by NSC 90-2115-M-036-003.The author thanks the authors of Ref. 6, since his work was motivated by their work. Also, the author thanks the referees for helpful comments which made the paper more readable.     

Asymptotic enumeration theorems for the numbers of spanning trees and Eulerian trails in circulant digraphs and graphs

The asymptotic properties of the numbers of spanning trees and Eulerian trails in circulant digraphs and graphs are studied. Let be a directed circulant graph. Let and be the numbers of spanning trees and of Eulerian trails, respectively. Then

Ranking fuzzy numbers with integral value

Ranking fuzzy numbers is important in decision making. Since very often the alternatives are evaluated by fuzzy numbers in a vague environment, a comparison between these fuzzy numbers is indeed a comparison between alternatives. This paper proposes a method of ranking fuzzy numbers with integral value. The method, which is independent of the type of membership functions used and the normality of the functions, can rank more than two fuzzy numbers simultaneously. It is relatively simple in computation, especially in ranking triangular and trapezoidal fuzzy numbers. Further, an index of optimism is used to reflect the decision maker's optimistic attitude. Discussion on comparative advantages is included.     

缺陷形状对铝材料微喷射的影响

通过二维弹塑性流体力学欧拉程序MEPH,对包含不同形状不同角度缺陷铝材料的微射流现象进行数值模拟,并对其最大喷射速度以及喷射系数进行数值统计,得到不同形状、不同角度缺陷最大喷射速度及喷射系数的变化规律.并将计算结果与实验结果进行比较,基本吻合,能够反映试验规律.     

On the microdynamics of composite plates

Summary The subject of the contribution is an investigation of elastic composite plates with a microperiodic structure along the plate's midsurface. The computational models of the plates are largely based on the homogenization procedures leading to equations with constant coefficients. However, the homogenization theories neglect the microstructure length-scale effect on the plate's dynamic behaviour. The objective of this research is to formulate and investigate a refined theory of the composite plates under consideration, which would describe the microdynamic plate behaviour caused by the microstructure. The possible applications of the proposed theory are shown by an example.This research was partly supported by KBN, Warsaw, under grant 3 3310 92 03.     

The comparison of two approaches to numerical modelling of gas-particles turbulent flow and heat transfer in a pipe

The comparison of two theoretical approaches for the numerical investigation of turbulent gas–solid flows with heat transfer in a pipe are presented in this paper. The first approach is based on Eulerian–Eulerian modelling of investigated phenomena, the second one is formulated within the framework of the Eulerian–Lagrangian approach. The verification of numerical models under consideration. Their testing against available experimental data show good prognostic properties of the elaborated theoretical tool for research activities to study new physical fundamentals of turbulent gas-suspended particles flows in pipes and channels.     

Thermodynamic laws and consistent Eulerian formulation of finite elastoplasticity with thermal effects

Recently it has been demonstrated that, on the basis of the separation D=D^e+D^p arising from the split of the stress power and two consistency criteria for objective Eulerian rate formulations, it is possible to establish a consistent Eulerian rate formulation of finite elastoplasticity in terms of the Kirchhoff stress and the stretching, without involving additional deformation-like variables labelled “elastic” or “plastic”. It has further been demonstrated that this consistent formulation leads to a simple essential structure implied by the work postulate, namely, both the normality rule for plastic flow D^p and the convexity of the yield surface in Kirchhoff stress space. Here, we attempt to place such an Eulerian formulation on the thermodynamic grounds by extending it to a general case with thermal effects, where the consistency requirements are treated in a twofold sense. First, we propose a general constitutive formulation based on the foregoing separation as well as the two consistency criteria. This is accomplished by employing the corotational logarithmic rate and by incorporating an exactly integrable Eulerian rate equation for D^e for thermo-elastic behaviour. Then, we study the consistency of the formulation with thermodynamic laws. Towards this goal, simple forms of restrictions are derived, and consequences are discussed. It is shown that the proposed Eulerian formulation is free in the sense of thermodynamic consistency. Namely, a Helmholtz free energy function in explicit form may be found such that the restrictions from the thermodynamic laws can be fulfilled with positive internal dissipation for arbitrary forms of constitutive functions included in the constitutive formulation. In particular, that is the case for the foregoing essential constitutive structure in the purely mechanical case. These results eventually lead to a complete, explicit constitutive theory for coupled fields of deformation, stress and temperature in thermo-elastoplastic solids at finite deformations.