Synthesis of 6-amino-1,2,3-triazolo[4,5-c] pyridin-4(5H)one (8-aza-3-deazaguanine) and 6-amino-1-(β-D-ribofuranosyl)-1,2,3-triazolo[4,5-c] pyridin-4(5H)one (8-Aza-3-deazaguanosine) via novel ring closure procedures

The total synthesis of 6-amino-1,2,3-triazolo[4,5-c]pyridin-4(5H)one (8-aza-3-deazaguanine, 3 ) and 6-amino-1-(β-D-ribofuranosyl)-1,2,3-triazolo[4,5-c]pyridin-4(5H)one (8-aza-3-deazaguano-sine, 22 ) has been described for the first time by a novel base-catalyzed ring closure of 4(5)-cyanomethyl-1,2,3-triazole-5(4)carboxamide (14) and methyl 5-cyanomethyl-1-(2,3,5-tri-O-ben-zoyl-β-D-ribofuranosyl)-1,2,3-triazole-4-carboxylate (17) , respectively. Under the catalysis of DBU, 2,4-dinitrophenylhydrazone of dimethyl 1,3-acetonedicarboxylate (7) was converted to methyl 5-methoxycarbonylmethyl-1-(2,4-dinitroanilino)-1,2,3-triazole-4-carboxylate (12) via dimethyl 2-diazo-3-iminoglutarate (8) . Catalytic reduction of 12 gave methyl 4(5)methoxycar-bonylmethyl-1,2,3-triazole-5(4)carboxylate (11) from which methyl 4(5)carbamoylmethyl-1,2,3-triazole-5(4)carboxylate (10) was obtained by ammonolysis. Dehydration of 10 provided methyl 4(5)cyanomethyl-1,2,3-triazole-5(4)carboxylate (13) which on amination gave 14 . The 1,2,3-triazole nucleosides 17, 18 and 19 were obtained from the stannic chloride-catalyzed condensation of the trimethylsilyl 13 and a fully acylated β-D-ribofuranose. The yield and ratio of the ribofuranosyl derivatives of 13 markedly depends on the ratio of stannic chloride used. The structures of the nucleosides 22 and 23 were established by a combination of NOE, ¹H-nmr and ¹³C-nmr spectroscopy.     

On the Numerical Solution to a Nonlinear Wave Equation Associated with the First Painlev′e Equation: an Operator-Splitting Approach

The main goal of this article is to discuss the numerical solution to a nonlinear wave equation associated with the first of the celebrated Painlevé transcendent ordinary differential equations. In order to solve numerically the above equation, whose solutions blow up in finite time, the authors advocate a numerical methodology based on the Strang’s symmetrized operator-splitting scheme. With this approach, one can decouple nonlinearity and differential operators, leading to the alternate solution at every time step of the equation as follows: (i) The first Painlevé ordinary differential equation, (ii) a linear wave equation with a constant coefficient. Assuming that the space dimension is two, the authors consider a fully discrete variant of the above scheme, where the space-time discretization of the linear wave equation sub-steps is achieved via a Galerkin/finite element space approximation combined with a second order accurate centered time discretization scheme. To handle the nonlinear sub-steps, a second order accurate centered explicit time discretization scheme with adaptively variable time step is used, in order to follow accurately the fast dynamic of the solution before it blows up. The results of numerical experiments are presented for different coefficients and boundary conditions. They show that the above methodology is robust and describes fairly accurately the evolution of a rather “violent” phenomenon.     

Pricing stock and bond derivatives with a multi-factor Gaussian model

The martingale approach to pricing contingent claims can be applied in a multiple state variable model. The idea is used to derive the prices of derivative securities (futures on stock and bond futures, options on stocks, bonds and futures) given a continuous time Gaussian multi-factor model of the returns of stocks and bonds. The bond market is similar to Langetieg's multi-factor model, which has closed-form solutions. This model is a generalization of Vasicek's model, where the term structure depends on state variables following correlated mean reverting processes. The stock market is affected by systematic and unsystematic risk.     

A QMR-based interior-point algorithm for solving linear programs

A new approach for the implementation of interior-point methods for solving linear programs is proposed. Its main feature is the iterative solution of the symmetric, but highly indefinite 2×2-block systems of linear equations that arise within the interior-point algorithm. These linear systems are solved by a symmetric variant of the quasi-minimal residual (QMR) algorithm, which is an iterative solver for general linear systems. The symmetric QMR algorithm can be combined with indefinite preconditioners, which is crucial for the efficient solution of highly indefinite linear systems, yet it still fully exploits the symmetry of the linear systems to be solved. To support the use of the symmetric QMR iteration, a novel stable reduction of the original unsymmetric 3×3-block systems to symmetric 2×2-block systems is introduced, and a measure for a low relative accuracy for the solution of these linear systems within the interior-point algorithm is proposed. Some indefinite preconditioners are discussed. Finally, we report results of a few preliminary numerical experiments to illustrate the features of the new approach.     

Partial permutation decoding for binary linear and $$Z_4$$-linear Hadamard codes

In this paper, s-\({\text {PD}}\)-sets of minimum size \(s+1\) for partial permutation decoding for the binary linear Hadamard code \(H_m\) of length \(2^m\), for all \(m\ge 4\) and \(2 \le s \le \lfloor {\frac{2^m}{1+m}}\rfloor -1\), are constructed. Moreover, recursive constructions to obtain s-\({\text {PD}}\)-sets of size \(l\ge s+1\) for \(H_{m+1}\) of length \(2^{m+1}\), from an s-\({\text {PD}}\)-set of the same size for \(H_m\), are also described. These results are generalized to find s-\({\text {PD}}\)-sets for the \({\mathbb {Z}}_4\)-linear Hadamard codes \(H_{\gamma , \delta }\) of length \(2^m\), \(m=\gamma +2\delta -1\), which are binary Hadamard codes (not necessarily linear) obtained as the Gray map image of quaternary linear codes of type \(2^\gamma 4^\delta \). Specifically, s-PD-sets of minimum size \(s+1\) for \(H_{\gamma , \delta }\), for all \(\delta \ge 3\) and \(2\le s \le \lfloor {\frac{2^{2\delta -2}}{\delta }}\rfloor -1\), are constructed and recursive constructions are described.     

Maximum likelihood estimation for stochastic volatility in mean models with heavy‐tailed distributions

In this article, we introduce a likelihood‐based estimation method for the stochastic volatility in mean (SVM) model with scale mixtures of normal (SMN) distributions. Our estimation method is based on the fact that the powerful hidden Markov model (HMM) machinery can be applied in order to evaluate an arbitrarily accurate approximation of the likelihood of an SVM model with SMN distributions. Likelihood‐based estimation of the parameters of stochastic volatility models, in general, and SVM models with SMN distributions, in particular, is usually regarded as challenging as the likelihood is a high‐dimensional multiple integral. However, the HMM approximation, which is very easy to implement, makes numerical maximum of the likelihood feasible and leads to simple formulae for forecast distributions, for computing appropriately defined residuals, and for decoding, that is, estimating the volatility of the process. Copyright © 2017 John Wiley & Sons, Ltd.     

Local Semicircle Law for Random Regular Graphs

We consider random d‐regular graphs on N vertices, with degree d at least (log N)⁴. We prove that the Green's function of the adjacency matrix and the Stieltjes transform of its empirical spectral measure are well approximated by Wigner's semicircle law, down to the optimal scale given by the typical eigenvalue spacing (up to a logarithmic correction). Aside from well‐known consequences for the local eigenvalue distribution, this result implies the complete (isotropic) delocalization of all eigenvectors and a probabilistic version of quantum unique ergodicity.© 2017 Wiley Periodicals, Inc.     

Trees in tournaments

Letf(n) be the smallest integer such that every tournament of orderf(n) contains every oriented tree of ordern. Sumner has just conjectures thatf(n)=2n–2, and F. K. Chung has shown thatf(n)(1+o(1))nlog₂ n. Here we show thatf(n)12n andf(n)(4+o(1))n.