On the Geometry of Infinite-Dimensional Grassmannian Manifolds and Gauge Theory

This PhD Thesis is about certain infinite-dimensional Grassmannian manifolds that arise naturally in geometry, representation theory and mathematical physics. From the physics point of view one encounters these infinite-dimensional manifolds when trying to understand the second quantization of fermions. The many particle Hilbert space of the second quantized fermions is called the fermionic Fock space. A typical element of the fermionic Fock space can be thought to be a linear combination of the configurations m particles and n anti-particles . Geometrically the fermionic Fock space can be constructed as holomorphic sections of a certain (dual)determinant line bundle lying over the so called restricted Grassmannian manifold, which is a typical example of an infinite-dimensional Grassmannian manifold one encounters in QFT. The construction should be compared with its well-known finite-dimensional analogue, where one realizes an exterior power of a finite-dimensional vector space as the space of holomorphic sections of a determinant line bundle lying over a finite-dimensional Grassmannian manifold. The connection with infinite-dimensional representation theory stems from the fact that the restricted Grassmannian manifold is an infinite-dimensional homogeneous (Kähler) manifold, i.e. it is of the form G/H where G is a certain infinite-dimensional Lie group and H its subgroup. A central extension of G acts on the total space of the dual determinant line bundle and also on the space its holomorphic sections; thus G admits a (projective) representation on the fermionic Fock space. This construction also induces the so called basic representation for loop groups (of compact groups), which in turn are vitally important in string theory / conformal field theory. The Thesis consists of three chapters: the first chapter is an introduction to the backround material and the other two chapters are individually written research articles. The first article deals in a new way with the well-known question in Yang-Mills theory, when can one lift the action of the gauge transformation group on the space of connection one forms to the total space of the Fock bundle in a compatible way with the second quantized Dirac operator. In general there is an obstruction to this (called the Mickelsson-Faddeev anomaly) and various geometric interpretations for this anomaly, using such things as group extensions and bundle gerbes, have been given earlier. In this work we give a new geometric interpretation for the Faddeev-Mickelsson anomaly in terms of differentiable gerbes (certain sheaves of categories) and central extensions of Lie groupoids. The second research article deals with the question how to define a Dirac-like operator on the restricted Grassmannian manifold, which is an infinite-dimensional space and hence not in the landscape of standard Dirac operator theory. The construction relies heavily on infinite-dimensional representation theory and one of the most technically demanding challenges is to be able to introduce proper normal orderings for certain infinite sums of operators in such a way that all divergences will disappear and the infinite sum will make sense as a well-defined operator acting on a suitable Hilbert space of spinors. This research article was motivated by a more extensive ongoing project to construct twisted K-theory classes in Yang-Mills theory via a Dirac-like operator on the restricted Grassmannian manifold.

Nitrile Reversible ketenimine tautomerism in substituted alkylidene malononitriles and alkylidene cyanoacetates: A characteristic UV absorption band

Several alkylidene malononitriles (1b,1d,1e,2b and4b) and alkylidene cyanoacetates (1a,2a and4a) studied exhibit a long wavelength UV absorption band around 355 nm which shows a hyperchromic effect in the presence of ethanolic alkali. This band has been assigned to the ketenimine tautomer (5). Addition of water to1b,1e and2b gives the corresponding pyridine diols (7a,7b and8a) respectively. Similarly, addition of ethanol to1e and2b gave the corresponding ethoxypyridine derivatives (7c and8b). Mechanism of formation of these compounds is discussed. Structures, as well as mechanism of formation of1c,7c and10 obtained from1b,1e and2b respectively on standing at room temperature are also discussed.

Nitrileketenimine tautomerism in substituted alkylidene malononitriles and alkylidene cyanoacetates: A characteristic UV absorption band

Several alkylidene malononitriles (1b,1d,1e,2b and4b) and alkylidene cyanoacetates (1a,2a and4a) studied exhibit a long wavelength UV absorption band around 355 nm which shows a hyperchromic effect in the presence of ethanolic alkali. This band has been assigned to the ketenimine tautomer (5). Addition of water to1b,1e and2b gives the corresponding pyridine diols (7a,7b and8a) respectively. Similarly, addition of ethanol to1e and2b gave the corresponding ethoxypyridine derivatives (7c and8b). Mechanism of formation of these compounds is discussed. Structures, as well as mechanism of formation of1c,7c and10 obtained from1b,1e and2b respectively on standing at room temperature are also discussed.

Dual Drug Delivery Microcapsules via Layer-by-Layer Self-Assembly

The integration of hydrophobic and hydrophilic drugs in the polymer microcapsule offers the possibility of developing a new drug delivery system that combines the best features of these two distinct classes of material. Recently, we have reported the encapsulation of an uncharged water-insoluble drug in the polymer membrane. The hydrophobic drug is deposited using a layer-by-layer (LbL) technique, which is based on the sequential adsorption of oppositely charged polyelectrolytes onto a charged substrate. In this paper, we report the encapsulation of two different drugs, which are invariably different in structure and in their solubility in water. We have characterized these dual drug vehicular capsules by confocal laser scanning microscopy, atomic force microscopy, visible microscopy, and transmission electron microscopy. The growth of a thin film on a flat substrate by LbL was monitored by UV−vis spectra. The desorption kinetics of two drugs from the thin film was modeled by a second-order rate model.

Planar pursuit-evasion with variable speeds, part 1, extremal trajectory maps

Pursuit evasion in a plane is formulated with both players allowed to vary their speeds between fixed limits. A suitable choice of real-space coordinates confers open-loop optimality on the game. The solution in the small is described in terms of the individual players'' extremal trajectory maps (ETM). Each map is independent of role, adversary, and capture radius. An ETM depicts the actual real-space trajectories. A template method of generating constant control arcs is described. Examples of ETM for an aircraft flying at a constant altitude with fixed and varying speeds are presented.

Planar pursuit-evasion with variable speeds, part 2, barrier sections

An iterative method of constructing sections of the game surfaces from the players'' extremal trajectory maps is discussed. Barrier sections are presented for aircraft pursuit-evasion at constant altitude, with one aircraft flying at sustained speed and the other varying its speed.

(v, k, λ) Configurations and self-dual codes

The incidence matrix of a (v, k, λ) configuration is used to construct a (2v, v) and a (2v + 2, v + 1) self-dual code. If the incidence matrix is a circulant, the codes obtained are quasi-cyclic and extended quasi-cyclic, respectively. The weight distributions of some codes of this type are obtained.

Harmonic response of a shock mount employing dual-phase damping

In this note, the application of dual-phase damping to a simple shock mount experiencing a harmonic input is described. The damping ratio is a function of the relative displacement between the foundation and the mounted mass. The purpose of employing such a damping is to reduce the absolute transmissibility over the whole frequency range.

Normalized Two Stage SVQ for Minimum Complexity Wide-band LSF Quantization

We develop a two stage split vector quantization method with optimum bit allocation, for achieving minimum computational complexity. This also results in much lower memory requirement than the recently proposed switched split vector quantization method. To improve the rate-distortion performance further, a region specific normalization is introduced, which results in 1 bit/vector improvement over the typical two stage split vector quantizer, for wide-band LSF quantization.

Compact AM Signals for Maximum Clutter Rejection

The transmitted signal is assumed to consist of a close succession of rectangular pulses of equal width. A matched filter scheme is employed and a theory is developed for a computer-aided optimization of the envelope of monotone compact signals for maximum rejection of dense clutter of any given distribution in range. Specific results are presented and indeterminate cases are discussed.

A Simple Method of Obtaining the Band Strengths in the Electronic Spectra of Diatomic Molecules

Relative band strengths of diatomic molecules for which the product of Franck-Condon factor and r-centroid is approximately equal to 1 for (0,0) band can be determined by a simple method which will be in good agreement with the smoothed array of experimental values.

Third post-Newtonian angular momentum flux and the secular evolution of orbital elements for inspiralling compact binaries in quasi-elliptical orbits

The angular-momentum flux from an inspiralling binary system of compact objects moving in quasi-elliptical orbits is computed at the third post-Newtonian (3PN) order using the multipolar post-Minkowskian wave generation formalism. The 3PN angular-momentum flux involves the instantaneous, tail, and tail-of-tails contributions as for the 3PN energy flux, and in addition a contribution due to nonlinear memory. We average the angular-momentum flux over the binary's orbit using the 3PN quasi-Keplerian representation of elliptical orbits. The averaged angular-momentum flux provides the final input needed for gravitational-wave phasing of binaries moving in quasi-elliptical orbits. We obtain the evolution of orbital elements under 3PN gravitational radiation reaction in the quasi-elliptic case. For small eccentricities, we give simpler limiting expressions relevant for phasing up to order e(2). This work is important for the construction of templates for quasi-eccentric binaries, and for the comparison of post-Newtonian results with the numerical relativity simulations of the plunge and merger of eccentric binaries.

EIT as a tool to observe dual wavelength operation of a semiconductor laser

We propose a simplified technique for dual wavelength operation of an extended cavity semiconductor laser, and its characterization using electromagnetically induced transparency (EIT). In this laser cavity scheme light beam is made converging before it incidences on the cavity grating. The converging angle of the beam creates two longitudinal oscillating modes of resonating cavity. Frequency separation between the longitudinal modes are measured with the help of beat frequency generation in a photodiode and creating pair of EIT spectra in Rb vapor. The pair of EIT dips that are generated due to dual wavelength of this laser (that is used as control laser) can be used to estimate frequency difference between the generated wavelengths. Width of EIT spectra can be used to estimate line width of individual wavelength components.

Dual yolk-shell structure of carbon and silica-coated silicon for high-performance lithium-ion batteries

Silicon batteries have attracted much attention in recent years due to their high theoretical capacity, although a rapid capacity fade is normally observed, attributed mainly to volume expansion during lithiation. Here, we report for the first time successful synthesis of Si/void/SiO2/void/C nanostructures. The synthesis strategy only involves selective etching of SiO2 in Si/SiO2/C structures with hydrofluoric acid solution. Compared with reported results, such novel structures include a hard SiO2-coated layer, a conductive carbon-coated layer, and two internal void spaces. In the structures, the carbon can enhance conductivity, the SiO2 layer has mechanically strong qualities, and the two internal void spaces can confine and accommodate volume expansion of silicon during lithiation. Therefore, these specially designed dual yolk-shell structures exhibit a stable and high capacity of 956 mA h g−1 after 430 cycles with capacity retention of 83%, while the capacity of Si/C core-shell structures rapidly decreases in the first ten cycles under the same experimental conditions. The novel dual yolk-shell structures developed for Si can also be extended to other battery materials that undergo large volume changes.