Ramanujan’s partial theta series and parity in partitions

A partial theta series identity from Ramanujan’s lost notebook has a connection with some parity problems in partitions studied by Andrews in Ramanujan J., to appear  where 15 open problems are listed. In this paper, the partial theta series identity of Ramanujan is revisited and answers to Questions 9 and 10 of Andrews are provided.     

Empirical study of the scaling behavior of the amplitude–frequency distribution of the Hilbert–Huang transform and its application in sunspot time series analysis

Investigating long-range correlation by the Hurst exponent, H$H$, is crucial in the study of time series. Recently, empirical-mode-decomposition-based arbitrary-order Hilbert spectral analysis (EMD-HSA) has been proposed to numerically obtain without proof a scaling relationship, generated from the amplitude–frequency distribution, related to H$H$. We propose a formalism to empirically study EMD-HSA, to deduce its scaling exponent ξ(q)$\xi \left(q\right)$ from the perspective of EMD-based arbitrary-order Hilbert marginal spectrum (EMD-HMS), and to numerically compare the results with the expected H$H$. EMD-HSA and EMD-HMS experiments show that, by incompletely removing (quasi-)periodic trends, the sunspot series should have an H$H$ value around 0.12.     

Evolutionary snowdrift game incorporating costly punishment in structured populations

The role of punishment and the effects of a structured population in promoting cooperation are important issues. Within a recent model of snowdrift game (SG) incorporating a costly punishing strategy (P), we study the effects of a population connected through a square lattice. The punishers, who carry basically a cooperative (C) character, are willing to pay a cost α$\alpha$ so as to punish a non-cooperative (D) opponent by β$\beta$. Depending on α$\alpha$, β$\beta$, the cost-to-benefit ratio r$r$ in SG, and the initial conditions, the system evolves into different phases that could be homogeneous or inhomogeneous. The spatial structure imposes geometrical constraint on how one agent is affected by neighboring agents. Results of extensive numerical simulations, both for the steady state and the dynamics, are presented. Possible phases are identified and discussed, and isolated phases in the r–β$r\text{–}\beta$ space are identified as special local structures of strategies that are stable due to the lattice structure. In contrast to a well-mixed population where punishers are suppressed due to the cost of punishment, the altruistic punishing strategy can flourish and prevail for appropriate values of the parameters, implying an enhancement in cooperation by imposing punishments in a structured population. The system could evolve to a phase corresponding to the coexistence of C, D, and P strategies at some particular payoff parameters, and such a phase is absent in a well-mixed population. The pair approximation, a commonly used analytic approach, is extended from a two-strategy system to a three-strategy system. We show that the pair approximation can, at best, capture the numerical results only qualitatively. Due to the improper way of including spatial correlation imposed by the lattice structure, the approximation does not give the frequencies of C, D, and P accurately and fails to give the homogeneous AllD and AllP phases.     

Total synthesis of (-)-ulapualide A, a novel tris-oxazole macrolide from marine nudibranchs, based on some biosynthesis speculation

A new, second generation, total synthesis of ulapualide A (1), whose stereochemistry was recently determined from X-ray analysis of its complex with the protein actin, is described. The synthesis is designed and based on some speculation of the biosynthetic origin of the contiguous tris-oxazole unit in ulapualide A, alongside that of the related co-metabolites that contain only two oxazole rings, e.g. 6 and 7. The mono-oxazole carboxylic acid 67b and the mono-oxazole secondary 55b alcohol which, together, contain all of the 10 asymmetric centres in the natural metabolite, were first elaborated using a combination of contemporary asymmetric synthesis protocols. Esterification of 67b with 55b under Yamaguchi conditions gave the ester 77 which was then converted into the omega-amino acid 18a following simultaneous deprotection of the t-butyl ester and the N-Boc protecting groups. Macrolactamisation of 18a, using HATU, now gave the key intermediate macrolactam 17, containing two of the three oxazole rings in ulapualide A (1). A number of procedures were used to introduce the third oxazole ring in ulapualide A from 17, including: a) cyclodehydration to the oxazoline 78a followed by oxidation using nickel peroxide leading to 76; b) dehydration to the enamide 79, followed by conversion into the methoxyoxazoline 78b, via 80, and elimination of methanol from 78b using camphorsulfonic acid. The tris-oxazole macrolide 76 was next converted into the aldehyde 82b in four straightforward steps, which was then reacted with N-methylformamide, leading to the E-alkenylformamide 83. Removal of the TBDPS protection at C3 in 83 finally gave (-)-ulapualide A, whose 1H and 13C NMR spectroscopic data were indistinguishable from those obtained for naturally derived material. It is likely that the tris-oxazole unit in ulapualide A (1) is derived in nature from a cascade of cyclodehydrations from an acylated tris-serine precursor, e.g.9, followed by oxidation of the resulting tris-oxazoline intermediate, i.e.10. It is also plausible to speculate that the biosynthesis of metabolites related to ulapualide A, e.g. the bis-oxazole 6 and the imide 7, involve cyclisations of just two of the serine units in 9. These speculations were given some credence by carrying out pertinent interconversions involving the bis-oxazole amide 24, the enamide 25, the imide 26, the oxazoline 27 and the tris-oxazole 30 as model compounds. An alternative strategy to the tris-oxazole macrolide intermediate 76 was also examined, involving preliminary synthesis of the aldehyde 73, containing a shortened (C25-C34) side chain from 67b and 47b. A Wadsworth-Emmons olefination reaction between 73 and the phosphonate ester 74 led smoothly to the E-alkene 75, but we were not able to reduce selectively the conjugated enone group in 75 to 76 without simultaneous reduction of the oxazole alkene bond, using a variety of reagents and reaction conditions.     

Vector generalized linear and additive extreme value models

Over recent years parametric and nonparametric regression has slowly been adopted into extreme value data analysis. Its introduction has been characterized by piecemeal additions and embellishments, which has had a negative effect on software development and usage. The purpose of this article is to convey the classes of vector generalized linear and additive models (VGLMs and VGAMs) as offering significant advantages for extreme value data analysis, providing flexible smoothing within a unifying framework. In particular, VGLMs and VGAMs allow all parameters of extreme value distributions to be modelled as linear or smooth functions of covariates. We implement new auxiliary methodology by incorporating a quasi-Newton update for the working weight matrices within an iteratively reweighted least squares (IRLS) algorithm. A software implementation by the first author, called the vgam package for , is used to illustrate the potential of VGLMs and VGAMs.     

Main‐chain syndioregic nonlinear optical polymers. II. Extended pi conjugation and improved thermal properties

Recent results concerning the synthesis of new main‐chain syndioregic nonlinear optical polymers are presented. In particular, the synthesis of polymers with extended pi conjugation in the chromophore and chromophores with improved thermal stability are presented. The nonlinear optical coefficient of several of the polymers and the optical loss at 1.3 and 1.55 μm were measured and are discussed. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 2824–2839, 2000     

Multilevel Deep Feature Generation Framework for Automated Detection of Retinal Abnormalities Using OCT Images

Optical coherence tomography (OCT) images coupled with many learning techniques have been developed to diagnose retinal disorders. This work aims to develop a novel framework for extracting deep features from 18 pre-trained convolutional neural networks (CNN) and to attain high performance using OCT images. In this work, we have developed a new framework for automated detection of retinal disorders using transfer learning. This model consists of three phases: deep fused and multilevel feature extraction, using 18 pre-trained networks and tent maximal pooling, feature selection with ReliefF, and classification using the optimized classifier. The novelty of this proposed framework is the feature generation using widely used CNNs and to select the most suitable features for classification. The extracted features using our proposed intelligent feature extractor are fed to iterative ReliefF (IRF) to automatically select the best feature vector. The quadratic support vector machine (QSVM) is utilized as a classifier in this work. We have developed our model using two public OCT image datasets, and they are named database 1 (DB1) and database 2 (DB2). The proposed framework can attain 97.40% and 100% classification accuracies using the two OCT datasets, DB1 and DB2, respectively. These results illustrate the success of our model.