Principal bundles on compact complex manifolds with trivial tangent bundle

Let G be a connected complex Lie group and G ì G{Gamma subset G} a cocompact lattice. Let H be a complex Lie group. We prove that a holomorphic principal H-bundle E _H over G/Γ admits a holomorphic connection if and only if E _H is invariant. If G is simply connected, we show that a holomorphic principal H-bundle E _H over G/Γ admits a flat holomorphic connection if and only if E _H is homogeneous.     

Master-slave chaos synchronization via optimal fractional order PI λ D μ controller with bacterial foraging algorithm

Chaos synchronization in a master-slave configuration has been studied in this paper with a fractional order (FO) Proportional-Integral-Derivative (PID) controller using an intelligent Bacterial Foraging Optimization (BFO) algorithm. A?comparative study has been made to highlight the advantage of using a fractional order PI ^?? D ^?? controller over the conventional PID controller for chaos synchronization using two Lu systems as a representative example. Simulation results are presented to show the effectiveness of the proposed chaos synchronization technique over the existing methodologies.     

Chaos in a mapping describing elastoplastic oscillations

We study the local and global dynamical behavior of a two dimensional piecewise linear map which describes the asymptotic motions of a single degree of freedom, parametrically excited, elastoplastic oscillator after it has settled down to purely elastic oscillations. We give existence and stability conditions for periodic orbits and prove that chaos, in the form of a Smale horseshoe, exists at specific, but representative, parameter values. We interpret simulations of the elastoplastic oscillator itself in the light of these results.Partially supported by NSF grant number MSS-9016626.Partially supported by AFOSR 91-0329.     

One-parameter Lie groups admitted by time-dependent Schrödinger equation: Atoms,molecules and nucleons in harmonic oscillator field

Lie’s method of differential equation is used to obtain the one-parameter Lie groups admitted by the time-dependent Schrödinger equations for atoms, molecules and nucleons in harmonic oscillator field. This group for atoms and molecules is isomorphic to 10-parameter inhomogeneous orthogonal group in 4 dimensions, irrespective of the numbers of nuclei and electrons. For Z protons andN neutrons in a harmonic oscillator field, both isotropic and anisotropic, the r-parameter Lie groups are seraidirect products of an invariant subgroup and a factor group. In the case of isotropic oscillator field r is 1/2[3Z(3Z-1) +3N (3N -1)+2], while for the anisotropic oscillator field r is 1/2[3Z (Z+1)+3N(JV+1)+2].     

Bogomolov restriction theorem for Higgs bundles

Let (E,θ) be a stable Higgs bundle of rank r on a smooth complex projective surface X equipped with a polarization H. Let C⊂X be a smooth complete curve with [C]=n⋅H. If where , then we prove that the restriction of (E,θ) to C is a stable Higgs bundle. This is a Higgs bundle analog of Bogomolov's restriction theorem for stable vector bundles.     

Solutions to an integro-differential parabolic problem arising in the pricing of financial options in a Lévy market

We study an integro-differential parabolic problem modeling a process with jumps arising in financial mathematics. Under suitable conditions, we prove the existence of weak solutions to a more general integro-differential equation by using the Schaefer’s fixed point theorem and generalize the result to unbounded domains.     

A differential evolution-based regression framework for forecasting Bitcoin price

This research proposes a differential evolution-based regression framework for forecasting one day ahead price of Bitcoin. The maximal overlap discrete wavelet transformation first decomposes the original series into granular linear and nonlinear components. We then fit polynomial regression with interaction (PRI) and support vector regression (SVR) on linear and nonlinear components and obtain component-wise projections. The sum of these projections constitutes the final forecast. For accurate predictions, the PRI coefficients and tuning of the hyperparameters of SVR must be precisely estimated. Differential evolution, a metaheuristic optimization technique, helps to achieve these goals. We compare the forecast accuracy of the proposed regression framework with six advanced predictive modeling algorithms- multilayer perceptron neural network, random forest, adaptive neural fuzzy inference system, standalone SVR, multiple adaptive regression spline, and least absolute shrinkage and selection operator. Finally, we perform the numerical experimentation based on—(1) the daily closing prices of Bitcoin for January 10, 2013, to February 23, 2019, and (2) randomly generated surrogate time series through Monte Carlo analysis. The forecast accuracy of the proposed framework is higher than the other predictive modeling algorithms.

     

Harder–Narasimhan reduction for principal bundles over a compact Kähler manifold

Generalizing the Harder–Narasimhan filtration of a vector bundle it is shown that a principal G-bundle over a compact K?hler manifold admits a canonical reduction of its structure group to a parabolic subgroup of G. Here G is a complex connected reductive algebraic group; in the special case where , this reduction is the Harder–Narasimhan filtration of the vector bundle associated to by the standard representation of . The reduction of in question is determined by two conditions. If P denotes the parabolic subgroup, L its Levi factor and the canonical reduction, then the first condition says that the principal L-bundle obtained by extending the structure group of the P-bundle using the natural projection of P to L is semistable. Denoting by the Lie algebra of the unipotent radical of P, the second condition says that for any irreducible P-module V occurring in , the associated vector bundle is of positive degree; here is considered as a P-module using the adjoint action. The second condition has an equivalent reformulation which says that for any nontrivial character of P which can be expressed as a nonnegative integral combination of simple roots (with respect to any Borel subgroup contained in P), the line bundle associated to for is of positive degree. The equivalence of these two conditions is a consequence of a representation theoretic result proved here. Received: 10 November 1999 / Revised version: 31 October 2001 / Published online: 26 April 2002     

Spectroscopic studies on nanodispersions of CdS, HgS, their core-shells and composites prepared in micellar medium

CdS is a large band gap material compared to HgS. Both are interesting from academic and technological points of view. The nanodispersions (colloids) of CdS and HgS as well their core-shell products and composites (co-colloids) were prepared by varied modes of precursor addition in micellar solution of cationic surfactant cetyltrimethylammonium bromide (CTAB). The prepared dispersions were studied by spectroscopic and electron microscopic techniques.