Alexander invariant for perturbations of Fredholm operators

The Alexander-type invariant for the bifurcation problems involving perturbations of linear Fredholm operators of nonnegative index is discussed and compared with another homotopy invariant (called the bifurcation index) detecting global bifurcations of solutions to multiparameter equations.     

Series representations for some mathematical constants

The authors apply a classical series identity involving the psi (or digamma) function with a view to deriving series representations for a number of known mathematical constants. Several closely-related consequences and results are also considered.     

Computing ODE symmetries as abnormal variational symmetries

We give a new computational method to obtain symmetries of ordinary differential equations. The proposed approach appears as an extension of a recent algorithm to compute variational symmetries of optimal control problems [P.D.F. Gouveia, D.F.M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control, Comput. Methods Appl. Math. 5 (4) (2005) 387-409], and is based on the resolution of a first order linear PDE that arises as a necessary and sufficient condition of invariance for abnormal optimal control problems. A computer algebra procedure is developed, which permits one to obtain ODE symmetries by the proposed method. Examples are given, and results compared with those obtained by previous available methods.     

Arithmetic of the Ramanujan-Göllnitz-Gordon continued fraction

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We extend the results of Chan and Huang [H.H. Chan, S.-S. Huang, On the Ramanujan-Göllnitz-Gordon continued fraction, Ramanujan J. 1 (1997) 75-90] and Vasuki, Srivatsa Kumar [K.R. Vasuki, B.R. Srivatsa Kumar, Certain identities for Ramanujan-Göllnitz-Gordon continued fraction, J. Comput. Appl. Math. 187 (2006) 87-95] to all odd primes p on the modular equations of the Ramanujan-Göllnitz-Gordon continued fraction v(τ) by computing the affine models of modular curves X(Γ) with Γ=Γ₁(8)∩Γ₀(16p). We then deduce the Kronecker congruence relations for these modular equations. Further, by showing that v(τ) is a modular unit over Z we give a new proof of the fact that the singular values of v(τ) are units at all imaginary quadratic arguments and obtain that they generate ray class fields modulo 8 over imaginary quadratic fields.

Video

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Density functional theory study of 13C NMR chemical shift of chlorinated compounds

The use of the standard density functional theory (DFT) leads to an overestimation of the paramagnetic contribution and underestimation of the shielding constants, especially for chlorinated carbon nuclei. For that reason, the predictions of chlorinated compounds often yield too high chemical shift values. In this study, the WC04 functional is shown to be capable of reducing the overestimation of the chemical shift of Cl‐bonded carbons in standard DFT functionals and to show a good performance in the prediction of ¹³C NMR chemical shifts of chlorinated organic compounds. The capability is attributed to the minimization of the contributions that intensively increase the chemical shift in the WC04. Extensive computations and analyses were performed to search for the optimal procedure for WC04. The B3LYP and mPW1PW91 standard functionals were also used to evaluate the performance. Through detailed comparisons between the basis set effects and the solvent effects on the results, the gas‐phase GIAO/WC04/6‐311+G(2d,p)//B3LYP/6‐31+G(d,p) was found to be specifically suitable for the prediction of ¹³C NMR chemical shifts of chlorides in both chlorinated and non‐chlorinated carbons. Further tests with eight molecules in the probe set sufficiently confirmed that WC04 was undoubtedly effective for accurately predicting ¹³C NMR chemical shifts of chlorinated organic compounds. Copyright © 2012 John Wiley & Sons, Ltd.