Resonances, scattering theory, and rigged Hilbert spaces

The problem of decaying states and resonances is examined within the framework of scattering theory in a rigged Hilbert space formalism. The stationary free,''in,'' and ''out'' eigenvectors of formal scattering theory, which have a rigorous setting in rigged Hilbert space, are considered to be analytic functions of the energy eigenvalue. The value of these analytic functions at any point of regularity, real or complex, is an eigenvector with eigenvalue equal to the position of the point. The poles of the eigenvector families give origin to other eigenvectors of the Hamiltonian: the singularities of the ''out'' eigenvector family are the same as those of the continued S matrix, so that resonances are seen as eigenvectors of the Hamiltonian with eigenvalue equal to their location in the complex energy plane. Cauchy theorem then provides for expansions in terms of ''complete'' sets of eigenvectors with complex eigenvalues of the Hamiltonian. Applying such expansions to the survival amplitude of a decaying state, one finds that resonances give discrete contributions with purely exponential time behavior; the background is of course present, but explicitly separated. The resolvent of the Hamiltonian, restricted to the nuclear space appearing in the rigged Hilbert space, can be continued across the absolutely continuous spectrum; the singularities of the continuation are the same as those of the ''out'' eigenvectors. The free, ''in'' and ''out'' eigenvectors with complex eigenvalues and those corresponding to resonances can be approximated by physical vectors in the Hilbert space, as plane waves can. The need for having some further physical information in addition to the specification of the total Hamiltonian is apparent in the proposed framework. The formalism is applied to the Lee–Friedrichs model and to the scattering of a spinless particle by a local central potential. Journal of Mathematical Physics is copyrighted by The American Institute of Physics.

Cynodon dactylon, Green Couch Grass, Bermuda Grass 'Grand Prix'

‘Grand Prix’ is a selection from a cross between ‘Wintergreen’ and ‘Couch 5’ (also designated C5). ‘Couch 5’ was a selection from an earlier series of crosses by the breeder between ‘Wintergreen’ and a number of Cynodon dactylon accessions, which were collected by the breeder from the Mornington Peninsula area of Victoria between 1986 and 1990. C5 was an experimental breeding line, and was not subsequently reserved as vegetative germplasm. Living material of C5 is no longer in existence. Following the crossing of ‘Couch 5’ and ‘Wintergreen’ in 1998, the resultant seed was germinated on moist blotting paper. Individual seedlings, a total of 150 in number, were planted into 150mm pots and these plants observed during 1998 and 1999. During the summer of 1999-2000, the majority of the seedling plants were culled on the basis of their shoot density, leaf texture, internode length, and colour. In the spring of 2000, the remaining 20 potted seedlings were planted individually into 4m2 plots at the Evergreen Turf farm at Pakenham (Victoria), and allowed to expand fully across these plots. The final selection of Seedling 12 (later designated DN12) in late 2002 was based on shoot density, leaf colour, turf quality, and reduced thatch accumulation as expressed in these plots. Propagation: the original plant has been multiplied through four (4) vegetative expansions prior to PBR application without showing any discernible off types. Breeder: David Nickson, Frankston, VIC. PBR Certificate Number 3133, Application Number 2005/291, granted 12 September 2006.

Cynodon dactylon, Green Couch Grass, Bermuda Grass 'Winter Gem'

‘Winter Gem’ is a selection from a cross between ‘Wintergreen’ and Couch 5 (also designated C5). Couch 5 was a selection from an earlier series of crosses by the breeder between ‘Wintergreen’ and a number of Cynodon dactylon accessions, which were collected by the breeder from the Mornington Peninsula area of Victoria between 1986 and 1990. C5 was an experimental breeding line, and was not subsequently reserved as vegetative germplasm. Living material of C5 is no longer in existence. Following the crossing of Couch 5 and ‘Wintergreen’ in 1998, the resultant seed was germinated on moist blotting paper. Individual seedlings, a total of 150 in number, were planted into 150mm pots and these plants observed during 1998 and 1999. During the summer of 1999-2000, the majority of the seedling plants were culled on the basis of their shoot density, leaf texture, internode length, and colour. In the spring of 2000, the remaining 20 potted seedlings were planted individually into 4m2 plots at the Evergreen Turf farm at Pakenham (Victoria), and allowed to expand fully across these plots. The final selection of Seedling 9 (later designated DN9) in late 2002 was based on shoot density, leaf texture, and retention of winter colour as expressed in these plots. Propagation: The original plant had been multiplied through four (4) vegetative expansions prior to PBR application without showing any discernible off types. Breeder: David Nickson, Frankston, VIC. PBR Certificate Number 3132, Application Number 2005/290, granted 11 September 2006.

Unsteady laminar compressible stagnation-point boundary-layer flow over a three-dimensional body: Higher order effects

The unsteady laminar compressible three-dimensional stagnation-point boundary-layer flow with variable properties has been studied when the velocity of the incident stream, mass transfer and wall temperature vary arbitrarily with time. The second-order unsteady boundary-layer equations for all the effects have been derived by using the method of matched asymptotic expansions. Both nodal and saddle point flows as well as cold and hot wall cases have been considered. The partial differential equations governing the flow have been solved numerically using an implicit finite-difference scheme. Computations have been carried out for an accelerating stream, a decelerating stream and a fluctuating stream. The results indicate that the unsteady free stream velocity distributions, the nature of the stagnation point, the mass transfer, the wall temperature and the variation of the density-viscosity product across the boundary significantly affect the skin friction and heat transfer. The variation of the wall temperature with time strongly affects the heat transfer whereas its effect is comparatively less on skin friction. Suction increases the skin friction and heat transfer but injection does the opposite. The skin friction in the x direction due to the combined effects of first- and second-order boundary layers is less than the skin-friction in the x direction due to the first-order boundary layers for all the parameters. The overall skin friction in the z direction and heat transfer are more or less than the first-order boundary layers depending upon the values of the various parameters.

Unsteady flow of a dilute suspension in a semi-infinite contracting or expanding pipe

In the present paper an exact similar solution of the Navier-Stokes equation for unsteady flow of a dilute suspension in a semi-infinite contracting or expanding circular pipe is presented. The effects of the Schmidt number (Sc), Reynolds number (|ε|), the volume fraction (α) and the relaxation time (τ) of the particulate phase on the flow characteristics are examined. The presence of the solid particles has been observed to influence the flow behaviour significantly. These solutions are valid down to the state of a completely collapsed pipe, since the nonlinearity is retained fully. The results may help understanding the flow near the heart and certain forced contractions or expansions of valved veins.

Motion of a viscous fluid with suspended particles in a curved tube

In this paper we have discussed the motion of a viscous fluid with suspended particles through a curved tube of small curvature ratio. The system is treated as two separate interacting continua. Solutions for axial and secondary velocities are obtained in the form of asymptotic expansions in powers of Dean Number. The streamline pattern for the particulate phase reveals many interesting features. The influence of the particulate continium on the fluid is described by the parameter τ which depends on the density ratio of the two continua. The concentration distribution of the particles in a given cross section is determined. It is noticed that the particles move closer to the wall for certain values of the concentration and the density ratio.

A conditionally linearized Monte Carlo filter in non-linear structural dynamics

State and parameter estimations of non-linear dynamical systems, based on incomplete and noisy measurements, are considered using Monte Carlo simulations. Given the measurements. the proposed method obtains the marginalized posterior distribution of an appropriately chosen (ideally small) subset of the state vector using a particle filter. Samples (particles) of the marginalized states are then used to construct a family of conditionally linearized system of equations and thus obtain the posterior distribution of the states using a bank of Kalman filters. Discrete process equations for the marginalized states are derived through truncated Ito-Taylor expansions. Increased analyticity and reduced dispersion of weights computed over a smaller sample space of marginalized states are the key features of the filter that help achieve smaller sample variance of the estimates. Numerical illustrations are provided for state/parameter estimations of a Duffing oscillator and a 3-DOF non-linear oscillator. Performance of the filter in parameter estimation is also assessed using measurements obtained through experiments on simple models in the laboratory. Despite an added computational cost, the results verify that the proposed filter generally produces estimates with lower sample variance over the standard sequential importance sampling (SIS) filter.

Molecular biology of progressive myoclonus epilepsy of Unverricht-Lundborg type (EPM1)

Progressive myoclonus epilepsy of Unverricht-Lundborg type (EPM1) is an autosomal recessively inherited disorder characterized by age of onset at 6-15 years, stimulus-sensitive myoclonus, tonic-clonic epileptic seizures and a progressive course. Mutations in the cystatin B (CSTB) gene underlie EPM1. The most common mutation underlying EPM1 is a dodecamer repeat expansion in the promoter region of CSTB. In addition, nine other mutations have been identified. CSTB, a cysteine protease inhibitor, is a ubiquitously expressed inhibitor of cathepsins, but its physiological function is unknown. The purpose of this study was to investigate CSTB gene expression and CSTB protein function in normal and pathological conditions. The basal CSTB promoter was mapped and characterized using different promoter-luciferase gene constructs. The binding activity of transcription factors to one ARE half, five Sp1 and four AP1 sites in the CSTB promoter was demonstrated. The CSTB promoter activity was clearly decreased using a CSTB promoter with "premutation" repeat expansions and in individuals with alike expansions. The expression of CSTB mRNA and protein was markedly reduced in patient cells. The endogenous CSTB protein localized to the nucleus, cytoplasm and lysosomes, and in differentiated cells merely to the cytoplasm. This suggests that the subcellular distribution of CSTB is dependent on the differentation status of the cells. The proteins representing patient missense mutations failed to associate with lysosomes, implying the importance of the lysosomal association for the proper physiological function of CSTB. Several alternatively spliced CSTB isoforms were identified. Of these CSTB2 was widely expressed with very low levels whereas the other alternatively spliced forms seemed to have limited tissue expression. In patients CSTB2 expression was reduced similarly to that of CSTB. The physiological relevance of CSTB alternative splicing remains unknown. The mouse Cstb transcript was shown to be present in all embryonic stages and adult tissues examined. The expression was highest at embryonic day 7 and in thymus, as well as in postnatal brain in the cortex, caudate putamen, thalamus, hippocampus, and in the Purkinje cell layer of the cerebellum. Our data implies that CSTB expression is tightly temporally and spatially regulated. The data presented in my thesis lay the basis for further understanding of the role of CSTB in health and disease.

Ranking of Prospective New Generation Location for a Power Network in a Deregulated System

In developing countries high rate of growth in demand of electric energy is felt, and so the addition of new generating units becomes necessary. In deregulated power systems private generating stations are encouraged to add new generations. Finding the appropriate location of new generator to be installed can be obtained by running repeated power flows, carrying system studies like analyzing the voltage profile, voltage stability, loss analysis etc. In this paper a new methodology is proposed which will mainly consider the existing network topology into account. A concept of T-index is introduced in this paper, which considers the electrical distances between generator and load nodes.This index is used for ranking significant new generation expansion locations and also indicates the amount of permissible generations that can be installed at these new locations. This concept facilitates for the medium and long term planning of power generation expansions within the available transmission corridors. Studies carried out on a sample 7-bus system, EHV equivalent 24-bus system and IEEE 39 bus system are presented for illustration purpose.

Evidence for reproductive philopatry in the bull shark Carcharhinus leucas

Reproductive philopatry in bull sharks Carcharhinus leucas was investigated by comparing mitochondrial (NADH dehydrogenase subunit 4, 797 base pairs and control region genes 837 base pairs) and nuclear (three microsatellite loci) DNA of juveniles sampled from 13 river systems across northern Australia. High mitochondrial and low microsatellite genetic diversity among juveniles sampled from different rivers (mitochondrial fST = 0.0767, P < 0.05; microsatellite FST = -0.0022, P > 0.05) supported female reproductive philopatry. Genetic structure was not further influenced by geographic distance (P > 0.05) or long-shore barriers to movement (P > 0.05). Additionally, results suggest that C. leucas in northern Australia has a long-term effective population size of 11 000-13 000 females and has undergone population bottlenecks and expansions that coincide with the timing of the last ice-ages.

Structure of shock waves at re-entry speeds

The unified structure of steady, one-dimensional shock waves in argon, in the absence of an external electric or magnetic field, is investigated. The analysis is based on a two-temperature, three-fluid continuum approach, using the Navier—Stokes equations as a model and including non-equilibrium collisional as well as radiative ionization phenomena. Quasi charge neutrality and zero velocity slip are assumed. The integral nature of the radiative terms is reduced to analytical forms through suitable spectral and directional approximations. The analysis is based on the method of matched asymptotic expansions. With respect to a suitably chosen small parameter, which is the ratio of atom-atom elastic collisional mean free-path to photon mean free-path, the following shock morphology emerges: within the radiation and electron thermal conduction dominated outer layer occurs an optically transparent discontinuity which consists of a chemically frozen heavy particle (atoms and ions) shock and a collisional ionization relaxation layer. Solutions are obtained for the first order with respect to the small parameter of the problem for two cases: (i) including electron thermal conduction and (ii) neglecting it in the analysis of the outer layer. It has been found that the influence of electron thermal conduction on the shock structure is substantial. Results for various free-stream conditions are presented in the form of tables and figures.

Conformal Field Theory Methods for Variants of Schramm-Loewner Evolutions

This thesis consists of an introduction, four research articles and an appendix. The thesis studies relations between two different approaches to continuum limit of models of two dimensional statistical mechanics at criticality. The approach of conformal field theory (CFT) could be thought of as the algebraic classification of some basic objects in these models. It has been succesfully used by physicists since 1980's. The other approach, Schramm-Loewner evolutions (SLEs), is a recently introduced set of mathematical methods to study random curves or interfaces occurring in the continuum limit of the models. The first and second included articles argue on basis of statistical mechanics what would be a plausible relation between SLEs and conformal field theory. The first article studies multiple SLEs, several random curves simultaneously in a domain. The proposed definition is compatible with a natural commutation requirement suggested by Dubédat. The curves of multiple SLE may form different topological configurations, ``pure geometries''. We conjecture a relation between the topological configurations and CFT concepts of conformal blocks and operator product expansions. Example applications of multiple SLEs include crossing probabilities for percolation and Ising model. The second article studies SLE variants that represent models with boundary conditions implemented by primary fields. The most well known of these, SLE(kappa, rho), is shown to be simple in terms of the Coulomb gas formalism of CFT. In the third article the space of local martingales for variants of SLE is shown to carry a representation of Virasoro algebra. Finding this structure is guided by the relation of SLEs and CFTs in general, but the result is established in a straightforward fashion. This article, too, emphasizes multiple SLEs and proposes a possible way of treating pure geometries in terms of Coulomb gas. The fourth article states results of applications of the Virasoro structure to the open questions of SLE reversibility and duality. Proofs of the stated results are provided in the appendix. The objective is an indirect computation of certain polynomial expected values. Provided that these expected values exist, in generic cases they are shown to possess the desired properties, thus giving support for both reversibility and duality.

Two-dimensional heat conduction with phase change in a semi-infinite mould

Analytical solution of a 2-dimensional problem of solidification of a superheated liquid in a semi-infinite mould has been studied in this paper. On the boundary, the prescribed temperature is such that the solidification starts simultaneously at all points of the boundary. Results are also given for the 2-dimensional ablation problem. The solution of the heat conduction equation has been obtained in terms of multiple Laplace integrals involving suitable unknown fictitious initial temperatures. These fictitious initial temperatures have interesting physical interpretations. By choosing suitable series expansions for fictitious initial temperatures and moving interface boundary, the unknown quantities can be determined. Solidification thickness has been calculated for short time and effect of parameters on the solidification thickness has been shown with the help of graphs.

Structure of shock waves in partially ionized argon

The paper presents a unified picture of the structure of steady one-dimensional shock waves in partially ionized argon in the absence of external electric and magnetic fields. The study is based on a two-temperature three-fluid continuum approach using the Navier-Stokes equations as a model and taking account of nonequilibrium ionization. The analysis of the governing equations is based on the method of matched asymptotic expansions and leads to three layers: (1) a broad thermal layer dominated by electron thermal conduction; (2) an atom-ion shock structured by heavy-particle collisional dissipative mechanisms; and (3) an ionization relaxation layer in which electron-atom inelastic collisions dominate.

Reed-Muller Canonical Forms in Multivalued Logic

Canonical forms for m-valued functions referred to as m-Reed-Muller canonical (m-RMC) forms that are a generalization of RMC forms of two-valued functions are proposed. m-RMC forms are based on the operations ?m (addition mod m) and .m (multiplication mod m) and do not, as in the cases of the generalizations proposed in the literature, require an m-valued function for m not a power of a prime, to be expressed by a canonical form for M-valued functions, where M > m is a power of a prime. Methods of obtaining the m-RMC forms from the truth vector or the sum of products representation of an m-valued function are discussed. Using a generalization of the Boolean difference to m-valued logic, series expansions for m-valued functions are derived.