Spectral Analysis of Symmetric and Anti-Symmetric Pairwise Kernels

This work investigates theoretical properties of symmetric and anti-symmetric kernels. First chapters give an overview of the theory of kernels used in supervised machine learning. Central focus is on the regularized least squares algorithm, which is motivated as a problem of function reconstruction through an abstract inverse problem. Brief review of reproducing kernel Hilbert spaces shows how kernels define an implicit hypothesis space with multiple equivalent characterizations and how this space may be modified by incorporating prior knowledge. Mathematical results of the abstract inverse problem, in particular spectral properties, pseudoinverse and regularization are recollected and then specialized to kernels. Symmetric and anti-symmetric kernels are applied in relation learning problems which incorporate prior knowledge that the relation is symmetric or anti-symmetric, respectively. Theoretical properties of these kernels are proved in a draft this thesis is based on and comprehensively referenced here. These proofs show that these kernels can be guaranteed to learn only symmetric or anti-symmetric relations, and they can learn any relations relative to the original kernel modified to learn only symmetric or anti-symmetric parts. Further results prove spectral properties of these kernels, central result being a simple inequality for the the trace of the estimator, also called the effective dimension. This quantity is used in learning bounds to guarantee smaller variance.

Spectral Analysis of Bounded Self-adjoint operators - A Linear Algebraic Approach

This thesis Entitled Spectral theory of bounded self-adjoint operators -A linear algebraic approach.The main results of the thesis can be classified as three different approaches to the spectral approximation problems. The truncation method and its perturbed versions are part of the classical linear algebraic approach to the subject. The usage of block Toeplitz-Laurent operators and the matrix valued symbols is considered as a particular example where the linear algebraic techniques are effective in simplifying problems in inverse spectral theory. The abstract approach to the spectral approximation problems via pre-conditioners and Korovkin-type theorems is an attempt to make the computations involved, well conditioned. However, in all these approaches, linear algebra comes as the central object. The objective of this study is to discuss the linear algebraic techniques in the spectral theory of bounded self-adjoint operators on a separable Hilbert space. The usage of truncation method in approximating the bounds of essential spectrum and the discrete spectral values outside these bounds is well known. The spectral gap prediction and related results was proved in the second chapter. The discrete versions of Borg-type theorems, proved in the third chapter, partly overlap with some known results in operator theory. The pure linear algebraic approach is the main novelty of the results proved here.

Raman and infrared spectral analysis of thallium niobyl phosphates: Tl2NbO2PO4, Tl3NaNb4O9(PO4)2 and TlNbOP2O7

Raman and infrared spectra of Tl2NbO2PO4, Tl3NaNb4O9(PO4)2 and TlNbOP2O7 are reported. The observed bands are assigned in terms of vibrations of NbO6 octahedra and PO4 tetrahedra in the first two compounds and in terms of NbO6 octahedra and P2O7 4− anion in the third compound. The NbO6 octahedra in all the title compounds are found to be corner-shared and distorted. The higher wavenumber values of the ν1 (NbO6) mode and other stretching modes indicate that the NbO6 octahedra in them are distorted in the order TlNbOP2O7 > Tl2NbO2PO4 > Tl3NaNb4O9(PO4)2. The splitting of the ν3 (PO4) mode indicates that PO4 tetrahedra is distorted more in Tl2NbO2PO4 than in Tl3NaNb4O9(PO4)2. The symmetry of P2O7 4− anion in TlNbOP2O7 is lowered. Bands indicate that the P–O–P bridge in the above compound has a bent P–O–P bridge configuration

Vibrational spectroscopy of a compound with a CS7 ring

The product of the Asinger reaction between elemental sulfur, n-butylamine and acetophenone is 8-(n-butylaminophenylmethyliden)-1,2,3,4,5,6,7-heptathiocane which contains a CS7 ring. A combination of infrared, Raman and inelastic neutron scattering spectroscopies with periodic density functional theory calculations is used to provide a complete assignment of the vibrational spectra of this unusual species. The similarity between the Raman spectra of the compound and that of elemental sulfur is particularly striking. Copyright (C) 2009 John Wiley & Sons, Ltd.

Spectral analysis of the elliptic sine-Gordon equation in the quarter plane

We study the elliptic sine-Gordon equation in the quarter plane using a spectral transform approach. We determine the Riemann-Hilbert problem associated with well-posed boundary value problems in this domain and use it to derive a formal representation of the solution. Our analysis is based on a generalization of the usual inverse scattering transform recently introduced by Fokas for studying linear elliptic problems.

Identification of lipophilic flavones and flavonols by comparative HPLC, TLC and UV spectral analysis

The identification of lipophilic flavones and flavonols using a combination of high performance liquid chromatography, thin layer chromatography and UV spectral analysis is discussed. Data are provided for the flavones, apigenin, luteolin and tricetin and twelve of their methyl ethers, 8-hydroxyluteolin, 6-hydroxyluteolin and scutellarein and fourteen of their methyl ethers, and some 6,8-dihydroxyapigenin and 6,8-dihydroxyluteolin derivatives. Data for some forty two flavonols with extra 6- and/or 8-hydroxylation, mostly 6-hydroxykaempferol and quercetagetin derivatives, are also presented. The remaining compounds analysed include fourteen 5-deoxyflavones, four 5-methoxyflavones and five 5-deoxyflavonols plus further 5-hydroxylated flavones and flavonols without B-ring oxidation or with 2-, 5- or 6-hydroxylation. Copyright © 2003 John Wiley & Sons, Ltd.

Spectral analysis of diffusions with jump boundary

In this paper we consider one-dimensional diffusions with constant coefficients in a finite interval with jump boundary and a certain deterministic jump distribution. We use coupling methods in order to identify the spectral gap in the case of a large drift and prove that there is a threshold drift above which the bottom of the spectrum no longer depends on the drift. As a corollary to our result we are able to answer two questions concerning elliptic eigenvalue problems with non-local boundary conditions formulated previously by Iddo Ben-Ari and Ross Pinsky.

Spectral analysis and the Aharonov-Bohm\~ effect on certain almost-Riemannian manifold

We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eigenfunctions. In particular in both cases we get a Weyl's law with leading term Elog E. We then study the drastic effect of Aharonov-Bohm magnetic potentials on the spectral properties. Other generalised Riemannian structures including conic and anti-conic type manifolds are also studied. In this case, the Aharonov-Bohm magnetic potential may affect the self-adjointness of the Laplace-Beltrami operator.

Spectral analysis and modeling of microcyclostratigraphy in late Paleozoic glaciogenic rhythmites, Parana Basin, Brazil

We investigate the depositional time scale of lithological couplets (fine sandstone/siltstone-siltstone/mudstone) from two distinctive outcrops of Permo-Carboniferous glacial rhythmites in the Itarare Group (Parana Basin, Brazil). Resolving the fundamental issue of time scale for these rhythmites is important in light of recent evidence for paleosecular variation measured in these sequences. Spectral analysis and tuning of high-resolution gray scale scans of sediment core microstratigraphy, which comprises pervasive laminations, reveal a comparable spectral content at both localities, with a frequency suite interpreted as that of short-term climate variability of Recent and modern times. This evidence for decadal- to centennial-scale deposition of these lithological couplets is discussed in light of the `varvic` character, i.e., annual time scale that was previously assumed for the rhythmites.

Self-adjoint extensions and spectral analysis in the Calogero problem

In this paper, we present a mathematically rigorous quantum-mechanical treatment of a one-dimensional motion of a particle in the Calogero potential alpha x(-2). Although the problem is quite old and well studied, we believe that our consideration based on a uniform approach to constructing a correct quantum-mechanical description for systems with singular potentials and/or boundaries, proposed in our previous works, adds some new points to its solution. To demonstrate that a consideration of the Calogero problem requires mathematical accuracy, we discuss some `paradoxes` inherent in the `naive` quantum-mechanical treatment. Using a self-adjoint extension method, we construct and study all possible self-adjoint operators (self-adjoint Hamiltonians) associated with a formal differential expression for the Calogero Hamiltonian. In particular, we discuss a spontaneous scale-symmetry breaking associated with self-adjoint extensions. A complete spectral analysis of all self-adjoint Hamiltonians is presented.

Self-adjoint extensions and spectral analysis in the generalized Kratzer problem

We present a mathematically rigorous quantum-mechanical treatment of a one-dimensional non-relativistic motion of a particle in the potential field V(x) = g(1)x(-1) + g(2)x(-2), x is an element of R(+) = [0, infinity). For g(2) > 0 and g(1) < 0, the potential is known as the Kratzer potential V(K)(x) and is usually used to describe molecular energy and structure, interactions between different molecules and interactions between non-bonded atoms. We construct all self-adjoint Schrodinger operators with the potential V(x) and represent rigorous solutions of the corresponding spectral problems. Solving the first part of the problem, we use a method of specifying self-adjoint extensions by (asymptotic) self-adjoint boundary conditions. Solving spectral problems, we follow Krein`s method of guiding functionals. This work is a continuation of our previous works devoted to the Coulomb, Calogero and Aharonov-Bohm potentials.

Molecular Ordering of Layer-by-Layer Polyelectrolyte Films Studied by Sum-Frequency Vibrational Spectroscopy

The molecular arrangement in organic thin films is crucial for their increasing technological applications. Here, we use vibrational spectroscopy by sum-frequency generation (SFG) to study the ordering of polyelectrolyte layers adsorbed on silica for all steps of layer-by-layer (LbL) self-assembly. In situ measurements during adsorption and rinsing showed that the adsorbed polymer has a disordered conformation and confirmed surface charge overcompensation upon polyelectrolyte adsorption by probing the interfacial electric field. In dry films, the polymer chains acquired a net orientational ordering, which was affected, however, by the adsorption of subsequent layers. Such a detailed characterization may allow the control of LbL film structure and functionality with unprecedented power.

Molecular Ordering in Layer-by-Layer Polyelectrolyte Films Studied by Sum-Frequency Vibrational Spectroscopy: The Effects of Drying Procedures

Sum-Frequency Vibrational Spectroscopy (SFVS) has been used to investigate the effect of nitrogen-flow drying on the molecular ordering of Layer-by-Layer (LbL) films of poly(allylamine hydrochloride) (PAH) alternated with poly(styrene sulfonate) (PSS). We find that films dried by spontaneous water evaporation are more ordered and homogeneous than films dried by nitrogen flow. The latter are quite inhomogeneous and may have regions with highly disordered polymer conformation. We propose that drying by spontaneous water evaporation reduces the effect of drag by the drying front, while during nitrogen-flow drying the fast evaporation of water ""freezes"" the disordered conformation of adsorbed polyelectrolyte molecules. These findings are important for many applications of LbL films, since device performance usually depends on film morphology and its molecular structure.