Vibrational spectroscopy of bio-molecules: an algebraic approach

Porphyrins have received numerous considerable attentions during recent years because of their great biological importance. The most interesting areas of current research in molecular spectroscopy are the study of the vibrational ground and excited states of polyatomic molecules especially for the Metalloporphyrins. In this paper, we have calculated the fundamental and extrapolated vibrational energy levels of Magnesium, Nickel, Copper, Zinc, Metalloporphyrin molecules using U (2) algebraic model Hamiltonian. The results obtained by this method are in good agreement with the experimental data. This study gives a general approach for solving the vibrational spectra of Metalloporphyrin molecules.     

On the F-fundamental group scheme

Let X be a smooth projective variety defined over a perfect field k of positive characteristic, and let FX be the absolute Frobenius morphism of X. For any vector bundle E→X, and any polynomial g with non-negative integer coefficients, define the vector bundle using the powers of FX and the direct sum operation. We construct a neutral Tannakian category using the vector bundles with the property that there are two distinct polynomials f and g with non-negative integer coefficients such that . We also investigate the group scheme defined by this neutral Tannakian category.     

Morse theory for the space of Higgs <Emphasis Type="Italic">G</Emphasis>–bundles

Fix a C ^∞ principal G–bundle E⁰_G{E^0_G} on a compact connected Riemann surface X, where G is a connected complex reductive linear algebraic group. We consider the gradient flow of the Yang–Mills–Higgs functional on the cotangent bundle of the space of all smooth connections on E⁰_G{E^0_G}. We prove that this flow preserves the subset of Higgs G–bundles, and, furthermore, the flow emanating from any point of this subset has a limit. Given a Higgs G–bundle, we identify the limit point of the integral curve passing through it. These generalize the results of the second named author on Higgs vector bundles.     

Control of de-excitation to selected vibrational levels in the ground state of NaH molecule using two broadband ultrashort pulses

We show that the de-excitation to different vibrational levels of the ground state in NaH molecule can be controlled by using two delayed ultrashort pulses (4 fs Gaussian). A vibrational wave packet generated on the excited A¹Σ⁺ state by the first pulse is de-excited back to the ground state by a second pulse after a time delay. The cross-section for de-excitation of the wave packet to different vibrational levels of the ground electronic state can be controlled by controlling the delay time between the two pulses as well as by choosing a pulse duration much shorter than the vibrational period of the molecule, such that the de-excited wave packet remains localized in the Franck–Condon region of a particular vibrational level of the ground state. Hence, the de-excitation to a particular vibrational level can be enhanced by suppressing that in others. In spite of the large bandwidth of the pulse which includes nine vibrational levels of the upper state and five vibrational levels of the ground state, one can selectively de-excite the molecule to any one or two vibrational levels of the ground state by carefully choosing the delay time between the pulses and the pulse duration. We are designing the wave packet in the ground state by two short pulses and selectively distributing the population in one or two levels at various values of the delay time. In light molecules having small vibrational period, this selectivity in de-excitation to one or two vibrational levels in the ground state can be achieved only by using ultrashort (4 fs) pulses in the presence of which the localization of the wave packet in the Franck–Condon region of the vibrational levels are particularly possible. It has been shown that the de-excitation cross-section to a particular vibrational level oscillates with delay between the pulses which can be realized as a time-dependent quantum gate.     

An algebraic approach to the comparative study of the vibrational spectra of monofluoroacetylene (HCCF) and deuterated acetylene (HCCD)

Using the Lie algebraic method the vibrational energy levels of HCCF and HCCD are calculated for 102 and 110 vibrational bands, respectively, using the local Hamiltonian. A comparative study is made between the two. Better results are obtained than those published earlier from local mode analysis.     

Detonation re-initiation mechanism following the Mach reflection of a quenched detonation

This experimental study addresses the re-initiation mechanism of detonation waves following the Mach reflection of a shock–flame complex. The detonation diffraction around a cylinder is used to reproducibly generate the shock–flame complex of interest. The experiments are performed in methane–oxygen. We use a novel experimental technique of coupling a two-in-line-spark flash system with a double-frame camera in order to obtain microsecond time resolution permitting accurate schlieren velocimetry. The first series of experiments compares the non-reactive sequence of shock reflections with the reflection over a rough wall under identical conditions. It was found that the hot reaction products generated along the rough wall are entrained by the wall jet into a large vortex structure behind the Mach stem. The second series of experiments performed in more sensitive mixtures addressed the sequence of events leading to the detonation establishment along the Mach and transverse waves. Following ignition and jet entrainment, a detonation first appears along the Mach stem while the transverse wave remains non-reactive. The structure of the unburned tongue however indicates local instabilities and hot spot formation, leading to the rapid reaction of this gas. Numerical simulations are also reported, confirming the sequence of ignition events obtained experimentally.     

Approximate Hermitian–Einstein connections on principal bundles over a compact Riemann surface

Let $X$ be a compact connected Riemann surface and $G$ a connected reductive complex affine algebraic group. Given a holomorphic principal $G$ -bundle $E_G$ over $X$ , we construct a $C^\infty $ Hermitian structure on $E_G$ together with a $1$ -parameter family of $C^\infty $ automorphisms $\{F_t\}_{t\in \mathbb R }$ of the principal $G$ -bundle $E_G$ with the following property: Let $\nabla ^t$ be the connection on $E_G$ corresponding to the Hermitian structure and the new holomorphic structure on $E_G$ constructed using $F_t$ from the original holomorphic structure. As $t\rightarrow -\infty $ , the connection $\nabla ^t$ converges in $C^\infty $ Fréchet topology to the connection on $E_G$ given by the Hermitian–Einstein connection on the polystable principal bundle associated to $E_G$ . In particular, as $t\rightarrow -\infty $ , the curvature of $\nabla ^t$ converges in $C^\infty $ Fréchet topology to the curvature of the connection on $E_G$ given by the Hermitian–Einstein connection on the polystable principal bundle associated to $E_G$ . The family $\{F_t\}_{t\in \mathbb R }$ is constructed by generalizing the method of [6]. Given a holomorphic vector bundle $E$ on $X$ , in [6] a $1$ -parameter family of $C^\infty $ automorphisms of $E$ is constructed such that as $t\rightarrow -\infty $ , the curvature converges, in $C^0$ topology, to the curvature of the Hermitian–Einstein connection of the associated graded bundle.