Numerical solution of Navier–Stokes equations using multiquadric radial basis function networks

A numerical method based on radial basis function networks (RBFNs) for solving steady incompressible viscous flow problems (including Boussinesq materials) is presented in this paper. The method uses a ‘universal approximator’ based on neural network methodology to represent the solutions. The method is easy to implement and does not require any kind of ‘finite element‐type’ discretization of the domain and its boundary. Instead, two sets of random points distributed throughout the domain and on the boundary are required. The first set defines the centres of the RBFNs and the second defines the collocation points. The two sets of points can be different; however, experience shows that if the two sets are the same better results are obtained. In this work the two sets are identical and hence commonly referred to as the set of centres. Planar Poiseuille, driven cavity and natural convection flows are simulated to verify the method. The numerical solutions obtained using only relatively low densities of centres are in good agreement with analytical and benchmark solutions available in the literature. With uniformly distributed centres, the method achieves Reynolds number Re = 100 000 for the Poiseuille flow (assuming that laminar flow can be maintained) using the density of , Re = 400 for the driven cavity flow with a density of and Rayleigh number Ra = 1 000 000 for the natural convection flow with a density of . Copyright © 2001 John Wiley & Sons, Ltd.     

Reversing the Stereoselectivity of a Palladium‐Catalyzed O‐Glycosylation through an Inner‐Sphere or Outer‐Sphere Pathway

An efficient and concise method for the construction of various O‐glycosidic bonds by a palladium‐catalyzed reaction with a 3‐O‐picoloyl glucal has been developed. The stereochemistry of the anomeric center derives from either an inner‐sphere or outer‐sphere pathway. Harder nucleophiles, such as aliphatic alcohols and sodium phenoxides give β‐products, and α products result from using softer nucleophiles, such as phenol.     

C‐Selective and Diastereoselective Alkyl Addition to β,γ‐Alkynyl‐α‐imino Esters with Zinc(II)ate Complexes

Since umpolung α‐imino esters contain three electrophilic centers, regioselective alkyl addition with traditional organometallic reagents has been a serious problem in the practical synthesis of versatile chiral α‐amino acid derivatives. An unusual C‐alkyl addition to α‐imino esters using a Grignard reagent (RMgX)‐derived zinc(II)ate was developed. Zinc(II)ate complexes consist of a Lewis acidic [MgX]⁺ moiety, a nucleophilic [R₃Zn]^? moiety, and 2 [MgX₂]. Therefore, the ionically separated [R₃Zn]^? selectively attacks the imino carbon atom ,which is most strongly activated by chelation of [MgX]⁺. In particular, chiral β,γ‐alkynyl‐α‐imino esters can strongly promote highly regio‐ and diastereoselective C‐alkylation because of structural considerations, and the corresponding optically active α‐quaternary amino acid derivatives are obtained within 5 minutes in high to excellent yields.     

Adaptive Fourier decomposition‐based Dirac type time‐frequency distribution

The Dirac‐type time‐frequency distribution (TFD), regarded as ideal TFD, has long been desired. It, until the present time, cannot be implemented, due to the fact that there has been no appropriate representation of signals leading to such TFD. Instead, people have been developing other types of TFD, including the Wigner and the windowed Fourier transform types. This paper promotes a practical passage leading to a Dirac‐type TFD. Based on the proposed function decomposition method, viz., adaptive Fourier decomposition, we establish a rigorous and practical Dirac‐type TFD theory. We do follow the route of analytic signal representation of signals founded and developed by Garbo, Ville, Cohen, Boashash, Picinbono, and others. The difference, however, is that our treatment is theoretically throughout and rigorous. To well illustrate the new theory and the related TFD, we include several examples and experiments of which some are in comparison with the most commonly used TFDs. Copyright © 2016 John Wiley & Sons, Ltd.     

Two Novel Characterizations of Self-Decomposability on the Half-Line

Two novel characterizations of self-decomposable Bernstein functions are provided. The first one is purely analytic, stating that a function \(\varPsi \) is the Bernstein function of a self-decomposable probability law \(\pi \) on the positive half-axis if and only if alternating sums of \(\varPsi \) satisfy certain monotonicity conditions. The second characterization is of probabilistic nature, showing that \(\varPsi \) is a self-decomposable Bernstein function if and only if a related d-variate function \(C_{\psi ,d}\), \(\psi :=\exp (-\varPsi )\), is a d-variate copula for each \(d \ge 2\). A canonical stochastic construction is presented, in which \(\pi \) (respectively \(\varPsi \)) determines the probability law of an exchangeable sequence of random variables \(\{U_k\}_{k\in {\mathbb {N}}}\) such that \((U_1,\ldots ,U_d) \sim C_{\psi ,d}\) for each \(d \ge 2\). The random variables \(\{U_k\}_{k\in {\mathbb {N}}},\) are i.i.d. conditioned on an increasing Sato process whose law is characterized by \(\varPsi \). The probability law of \(\{U_k\}_{k \in {\mathbb {N}}}\) is studied in quite some detail.     

Practical and efficient synthesis of N-fused tricyclic indoles

A practical and efficient synthesis of methyl 6,7,8,9-tetrahydropyrido[1,2-a]indol-10-ylacetate derivatives 6 is reported. This synthetic approach featured the nucleophilic aromatic substitution of 2-piperidinemethanol derivatives 2 with aryl fluorides 1, and the intramolecular Heck coupling as key steps to afford the desired N-fused tricyclic indoles 6.