On degrees of maps between Grassmannians

Let G_n,k denote the oriented grassmann manifold of orientedk-planes in ℝⁿ. It is shown that for any continuous mapf: G_n,k → G_n,k, dim G_n,k = dim G_m,l = l(m −l), the Brouwer’s degree is zero, providedl > 1,n ≠ m. Similar results for continuous mapsg: ℂG_m,l → ℂG_n,k,h: ℍG_m,l → ℍG_n,k, 1 ≤ l < k ≤ n/2, k(n — k) = l(m — l) are also obtained.     

A Class of Integrable and Nonintegrable Mappings and their Dynamics

We analyse a class of mappings which by construction do not belong to the QRT family. We show that some of the members of this class have invariants of high degree. A new linearisable mapping is also identified. A mapping which possesses confined singularities while having nonzero algebraic entropy is presented. Its dynamics are studied in detail and shown to be related intimately to the Fibonacci recurrence.      

Special functions as solutions to discrete Painlevé equations

We present results on special solutions of discrete Painlevé equations. These solutions exist only when one constraint among the parameters of the equation is satisfied and are obtained through the solutions of linear second-order (discrete) equations. These linear equations define the discrete analogues of special functions.     

Microstructure of polycarbonate seen by positrons as an in-situ probe

Positron Annihilation Lifetime Spectroscopy (PALS) is used to probe the micro-structural changes in the polymer polycarbonate in terms of the changes in free volume hole size and their content as a function of temperature. The measured spectra are best fitted to three lifetime component analysis. The average hole size in the amorphous regions is determined from the measured Ortho Positronium (O-Ps) lifetime ₃ by following the treatment of Nakanishi et al. On the other hand, information about defects in the crystalline regions is revealed by trapped positrons lifetime ₂. In this polymer the average hole radius varies from 2.71 Å to 2.77 Å. The present study indicates that this polymer has a glass transition temperature of 152°C. Further, we have calculated the trapping rates in the ordered and disordered regions of the polymer based on Goldanskii's kinetic equations and an attempt is made for the first time to estimate the activation energy in the amorphous and crystalline regions separately.     

Development and validation of a highly sensitive and robust LC‐ESI‐MS/MS method for simultaneous quantitation of simvastatin acid,amlodipine and valsartan in human plasma: application to a clinical pharmacokinetic study

A high‐throughput, simple, highly sensitive and specific LC‐MS/MS method has been developed for simultaneous estimation of simvastatin acid (SA), amlodipine (AD) and valsartan (VS) with 500 µL of human plasma using deuterated simvastatin acid as an internal standard (IS). The API‐4000 LC‐MS/MS was operated under the multiple reaction‐monitoring mode (MRM) using electrospray ionization. The assay procedure involved precipitation of SA, AD, VS and IS from plasma with acetonitrile. The total run time was 2.8 min and the elution of SA, AD, VS and IS occurred at 1.81, 1.12, 1.14 and 1.81 min, respectively; this was achieved with a mobile phase consisting of 0.02 m ammonium formate (pH 4.5):acetonitrile (20:80, v/v) at a flow rate of 0.50 mL/min on an X‐Terra C₁₈ column. A linear response function was established for the range of concentrations 0.5–50 ng/mL (r > 0.994) for VS and 0.2–50 ng/mL (r > 0.996) for SA and AD. The method validation parameters for all three analytes met the acceptance as per FDA guidelines. This novel method has been applied to human pharmacokinetic study. Copyright © 2009 John Wiley & Sons, Ltd.     

Reactive functionalization of poly(lactic acid), PLA: Effects of the reactive modifier,initiator and processing conditions on the final grafted maleic anhydride content and molecular weight of PLA

A response surface methodology (RSM) design was used to analyze the effects of maleic anhydride (MA) and 2,5-bis(tert-butylperoxy)-2,5-dimethyl hexane (Luperox or L101) content, and TSE screw speed on the degree of grafted MA (MA_g) and number average molecular weight (M_n) of maleated PLA (PLA-g-MA), which can be used as a reactive compatibilizer in production of PLA blends with various components. PLA-g-MA's FTIR peaks indicated that MA was grafted onto the PLA backbone and oligomeric MA was also present. A maximum of 0.52 wt% MA_g determined by titration was achieved at the expense of a 50% reduction of M_n and an increase in the polydispersity index to around 2.0. Generally, increasing L101 increased the degree of grafting and decreased M_n. L101 and MA_g had a large effect on the stability of PLA-g-MA's M_n during storage. Nominally, amounts of MA equal to 4.5 wt%, L101 between 0.45 and 0.65 wt%, and screw speed of 20 rpm provided the optimal conditions for grafting MA onto PLA.     

Fast head-related transfer function measurement via reciprocity

An efficient method for head-related transfer function (HRTF) measurement is presented. By applying the acoustical principle of reciprocity, one can swap the speaker and the microphone positions in the traditional (direct) HRTF measurement setup, that is, insert a microspeaker into the subject's ear and position several microphones around the subject, enabling simultaneous HRTF acquisition at all microphone positions. The setup used for reciprocal HRTF measurement is described, and the obtained HRTFs are compared with the analytical solution for a sound-hard sphere and with KEMAR manikin HRTF obtained by the direct method. The reciprocally measured sphere HRTF agrees well with the analytical solution. The reciprocally measured and the directly measured KEMAR HRTFs are not exactly identical but agree well in spectrum shape and feature positions. To evaluate if the observed differences are significant, an auditory localization model based on work by J. C. Middlebrooks [J. Acoust. Soc. Am. 92, 2607-2624 (1992)] was used to predict where a virtual sound source synthesized with the reciprocally measured HRTF would be localized if the directly measured HRTF were used for the localization. It was found that the predicted localization direction generally lies close to the measurement direction, indicating that the HRTFs obtained via the two methods are in good agreement.     

Synthesis of α-diazo-β-hydroxy esters using nanocrystalline MgO

Nanocrystalline magnesium oxide (NAP-MgO) has been found to be an effective heterogeneous, solid base catalyst for the direct aldol type reaction of ethyl diazoacetate and various aldehydes to afford the corresponding α-diazo-β-hydroxy esters in excellent yields with high selectivity under mild conditions.     

Orthogonal polynomial approach to discrete Lax pairs for initial boundary-value problems of the QD algorithm

Using orthogonal polynomial theory, we construct the Lax pair for the quotient-difference algorithm in the natural Rutishauser variables. We start by considering the family of orthogonal polynomials corresponding to a given linear form. Shifts on the linear form give rise to adjacent families. A compatible set of linear problems is made up from two relations connecting adjacent and original polynomials. Lax pairs for several initial boundary-value problems are derived and we recover the discrete-time Toda chain equations of Hirota and of Suris. This approach allows us to derive a Bäcklund transform that relates these two different discrete-time Toda systems. We also show that they yield the same bilinear equation up to a gauge transformation. The singularity confinement property is discussed as well.     

Schlesinger Transformations for Linearisable Equations

A simple axiomatic characterization of the noncommutative Itô algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. It is proved that every Itô algebra with a quotient identity has a faithful representation in a Minkowski space and is canonically decomposed into the orthogonal sum of quantum Brownian (Wiener) algebra and quantum Lévy (Poisson) algebra. In particular, every quantum thermal noise of a finite number of degrees of freedom is the orthogonal sum of a quantum Wiener noise and a quantum Poisson noise as it is stated by the Lévy–Khinchin Theorem in the classical case. Two basic examples of noncommutative Itô finite group algebras are considered.