Sparse Penalized Forward Selection for Support Vector Classification

We propose a new binary classification and variable selection technique especially designed for high-dimensional predictors. Among many predictors, typically, only a small fraction of them have significant impact on prediction. In such a situation, more interpretable models with better prediction accuracy can be obtained by variable selection along with classification. By adding an ?₁-type penalty to the loss function, common classification methods such as logistic regression or support vector machines (SVM) can perform variable selection. Existing penalized SVM methods all attempt to jointly solve all the parameters involved in the penalization problem altogether. When data dimension is very high, the joint optimization problem is very complex and involves a lot of memory allocation. In this article, we propose a new penalized forward search technique that can reduce high-dimensional optimization problems to one-dimensional optimization by iterating the selection steps. The new algorithm can be regarded as a forward selection version of the penalized SVM and its variants. The advantage of optimizing in one dimension is that the location of the optimum solution can be obtained with intelligent search by exploiting convexity and a piecewise linear or quadratic structure of the criterion function. In each step, the predictor that is most able to predict the outcome is chosen in the model. The search is then repeatedly used in an iterative fashion until convergence occurs. Comparison of our new classification rule with ?₁-SVM and other common methods show very promising performance, in that the proposed method leads to much leaner models without compromising misclassification rates, particularly for high-dimensional predictors.     

Complete Convergence of Martingale Arrays

We study complete convergence of martingale arrays under rather weak conditions. Our results considerably strengthen many of the results available in the literature. As a tool, we establish a martingale analogue of an inequality of Hoffman-Jørgensen which was earlier known only for independent random variables.     

Asymptotic Expansions of Posterior Distributions in Nonregular Cases

We study the asymptotic behaviour of the posterior distributions for a one-parameter family of discontinuous densities. It is shown that a suitably centered and normalized posterior converges almost surely to an exponential limit in the total variation norm. Further, asymptotic expansions for the density, distribution function, moments and quantiles of the posterior are also obtained. It is to be noted that, in view of the results of Ghosh et al. (1994, Statistical Decision Theory and Related Topics V, 183-199, Springer, New York) and Ghosal et al. (1995, Ann. Statist., 23, 2145-2152), the nonregular cases considered here are essentially the only ones for which the posterior distributions converge. The results obtained here are also supported by a simulation experiment.     

Self-dual connections,hyperbolic metrics and harmonic mappings on Riemann surfaces

The Sampson-Wolf model of Teichmüller space (using harmonic mappings) is shown to be exactly the same as the more recent Hitchin model (utilizing self-dual connections). Indeed, it is noted how the self-duality equations become the harmonicity equations. An interpretation of the modular group action in this model is mentioned.     

Emeritus Professor Shan-Fu Shen,Cornell University

A numerical analysis of blood flow in a rigid artery has been performed to observe the variations in flow pattern and haemodynamic parameters under the influence of multiple (double) stenoses. The interspacing distance, degree of stenotic severity and Reynolds number of flow have been varied. It is found that two stenoses interact when the interspacing distance is lower than a critical value that corresponds to the redevelopment length of fully developed flow after the proximal stenosis. In the case of non-interacting stenoses the haemodynamic parameters, like the peak centreline velocity (representing the peak systolic velocity ratio in medical term), the maximum wall shear stress, region of low shear stress and the irrecoverable pressure drop at the stenosis site, repeat themselves at each individual stenosis. However, when the stenoses interact hydrodynamically because of their proximity, the parameters behave differently, which can lead to a different haemodynamic profile in the arteries and pose difficulties in their pathological interpretation. This work evaluates the influence of double stenoses on the variations in the key haemodynamic features under different degree of stenoses and interspacing distance between the stenoses.     

Limit constructions over Riemann surfaces and their parameter spaces, and the commensurability group actions

To any compact hyperbolic Riemann surface X, we associate a new type of automorphism group — called its commensurability automorphism group, ComAut(X). The members of ComAut(X) arise from closed circuits, starting and ending at X, where the edges represent holomorphic covering maps amongst compact connected Riemann surfaces (and the vertices represent the covering surfaces). This group turns out to be the isotropy subgroup, at the point represented by X (in $ T_\infty $), for the action of the universal commensurability modular group on the universal direct limit of Teichmüller spaces, $ T_\infty $. Now, each point of $ T_\infty $ represents a complex structure on the universal hyperbolic solenoid. We notice that ComAut(X) acts by holomorphic automorphisms on that complex solenoid. Interestingly, this action turns out to be ergodic (with respect to the natural measure on the solenoid) if and only if the Fuchsian group uniformizing X is arithmetic. Furthermore, the action of the commensurability modular group, and of its isotropy subgroups, on some natural vector bundles over $ T_\infty $, are studied by us.     

Schiffer variation of complex structure and coordinates for Teichmüller spaces

Schiffer variation of complex structure on a Riemann surfaceX ₀ is achieved by punching out a parametric disc \(\bar D\) fromX ₀ and replacing it by another Jordan domain whose boundary curve is a holomorphic image of \(\partial \bar D\) . This change of structure depends on a complex parameter ε which determines the holomorphic mapping function around \(\partial \bar D\) . It is very natural to look for conditions under which these ε-parameters provide local coordinates for Teichmüller spaceT(X ₀), (or reduced Teichmüller spaceT ^#(X₀)). For compactX ₀ this problem was first solved by Patt [8] using a complicated analysis of periods and Ahlfors' [2] τ-coordinates. Using Gardiner's [6], [7] technique, (independently discovered by the present author), of interpreting Schiffer variation as a quasi conformal deformation of structure, we greatly simplify and generalize Patt's result. Theorems 1 and 2 below take care of all the finitedimensional Teichmüller spaces. In Theorem 3 we are able to analyse the situation for infinite dimensionalT(X ₀) also. Variational formulae for the dependence of classical moduli parameters on the ε's follow painlessly.     

Performing chemical reactions in virtual capillary of surface tension-confined microfluidic devices

In this paper we report a new method of fabrication of surface tensionconfined microfluidic devices on glass. We have also successfully carried out some well-known chemical reactions in these fluidic channels to demonstrate the usefulness of these wall-less microchannels. The confined flow path of liquid was achieved on the basis of extreme differences in hydrophobic and hydrophilic characters of the surface. The flow paths were fabricated by making parallel lines using permanent marker pen ink or other polymer on glass surfaces. Two mirror image patterned glass plates were then sandwiched one on top of the other, separated by a thin gap-created using a spacer. The aqueous liquid moves between the surfaces by capillary forces, confined to the hydrophilic areas without wetting the hydrophobic lines, achieving liquid confinement without physical side-walls. We have shown that the microfluidic devices designed in such a way can be very useful due to their simplicity and low fabrication cost. More importantly, we have also demonstrated that the minimum requirement of such a working device is a hydrophilic line surrounded by hydrophobic environment, two walls of which are constituted of air and the rest is made of a hydrophobic surface.     

Effect of interface states associated with transitional nanophase precipitates in the enhancement of red emission from SrAl12O19:Pr by Ti incorporation

SrAl₁₂O₁₉:Pr³⁺, Ti⁴⁺ phosphor suitable for field emission displays is prepared by the wet chemical gel-carbonate method and the mechanism of enhancement in red photoluminescence (PL) intensity with Ti⁴⁺ therein has been investigated. The PL spectra of Pr³⁺ show both ¹D₂-³H₄ and ³P₀-³H₆ emission in the red region with very weak intensity when excited at 355 nm. The emission intensity has increased by about 100 times at room temperature in the compositional range SrAl_12−xTi_xO_19+x/2:Pr³⁺, with 0.1≤x≤0.3 in comparison to Ti-free SrAl₁₂O₁₉:Pr³⁺. TEM investigations show the presence of exsolved nanophase of SrAl₈Ti₃O₁₉, the precipitation of which is preceded by the presence of defect centers at the interfacial regions between the semicoherent transient phase and the parent SrAl₁₂O₁₉ matrix. The presence of transitional nanophase and the associated defects modify the excitation-emission process by way of formation of electronic sub-levels at lower energy (3.5 eV) than the band gap of SrAl₁₂O₁₉ (∼7 eV) followed by non-resonance energy transfer to Pr³⁺ level, leading to magnetic-dipole related red emission with enhanced intensity. The PL intensity of Pr³⁺ decreases at high Ti⁴⁺ concentrations (x>0.3) due to higher extent of segregation of non-emissive SrAl₈Ti₃O₁₉:Pr³⁺ phase.     

Electron transport in sub-micron GaAs channels at 300 K

Transient velocity-field characteristics have been computed for GaAs channels having lengths of 0.1, 0.2, 0.5, 1, and 20 μm for electric fields between 1 and 50 kV/cm at 300 K. The results are compared with earlier calculations and the significant features of the computed results are discussed. It is found that the electron motion for all channel lengths and for all fields is significantly affected by collisions. The threshold field for negative differential mobility increases, and the magnitude of the differential mobility decreases with decrease in the length of the sample. The maximum steady-state velocity increases with decrease in the length and may be as high as 5.4×10⁷ cm/s for 0.1 μm samples.