PT-symmetric coupler with chi((2)) nonlinearity

We introduce the notion of a PT-symmetric dimer with a chi((2)) nonlinearity. Similarly to the Kerr case, we argue that such a nonlinearity should be accessible in a pair of optical waveguides with quadratic nonlinearity and gain and loss, respectively. An interesting feature of the problem is that because of the two harmonics, there exist in general two distinct gain and loss parameters, different values of which are considered herein. We find a number of traits that appear to be absent in the more standard cubic case. For instance, bifurcations of nonlinear modes from the linear solutions occur in two different ways depending on whether the first-or the second-harmonic amplitude is vanishing in the underlying linear eigenvector. Moreover, a host of interesting bifurcation phenomena appear to occur, including saddle-center and pitchfork bifurcations which our parametric variations elucidate. The existence and stability analysis of the stationary solutions is corroborated by numerical time-evolution simulations exploring the evolution of the different configurations, when unstable.

Asymmetric and Symmetric Triazenido Cyclopalladated Complexes: Synthesis, Structural Analysis and DFT Calculations

The reaction of [Pd{dmba}(l-N3)]2 (dmba = N,N-dimethylbenzylamine) with 1-(2-fluorophenyl)-3-(4- nitrophenyl)triazenido (L1 ) or 1,3-bis(4-nitrophenyl)triazenido (L2 ) anions, in methanol, and subsequent treatment with pyridine (py) allows the preparation of the corresponding cyclopalladated compounds [Pd(dmba)(L1 )(py)] (1) and [Pd(dmba)(L2 )(py)]py (2). The acentric mononuclear entities of (1) and (2) are connected by weak intermolecular non-classical CAHC hydrogen bonds, which results in 2-D arrangements by translation, along the [1 0 0] and [0 01] crystallographic directions, respectively.

Open-closed superstring amplitudes using vertex operators in AdS5xS5

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Open-closed superstring amplitudes using vertex operators in AdS(5) x S-5

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

Anderson-like Transition for a Class of Random Sparse Models in d >= 2 Dimensions

We show that the Kronecker sum of d >= 2 copies of a random one-dimensional sparse model displays a spectral transition of the type predicted by Anderson, from absolutely continuous around the center of the band to pure point around the boundaries. Possible applications to physics and open problems are discussed briefly.

Sharp estimates for eigenvalues of integral operators generated by dot product kernels on the sphere

We obtain explicit formulas for the eigenvalues of integral operators generated by continuous dot product kernels defined on the sphere via the usual gamma function. Using them, we present both, a procedure to describe sharp bounds for the eigenvalues and their asymptotic behavior near 0. We illustrate our results with examples, among them the integral operator generated by a Gaussian kernel. Finally, we sketch complex versions of our results to cover the cases when the sphere sits in a Hermitian space.

Differential forms on Carnot groups

The main goal of this thesis is to understand and link together some of the early works by Michel Rumin and Pierre Julg. The work is centered around the so-called Rumin complex, which is a construction in subRiemannian geometry. A Carnot manifold is a manifold endowed with a horizontal distribution. If further a metric is given, one gets a subRiemannian manifold. Such data arise in different contexts, such as: - formulation of the second principle of thermodynamics; - optimal control; - propagation of singularities for sums of squares of vector fields; - real hypersurfaces in complex manifolds; - ideal boundaries of rank one symmetric spaces; - asymptotic geometry of nilpotent groups; - modelization of human vision. Differential forms on a Carnot manifold have weights, which produces a filtered complex. In view of applications to nilpotent groups, Rumin has defined a substitute for the de Rham complex, adapted to this filtration. The presence of a filtered complex also suggests the use of the formal machinery of spectral sequences in the study of cohomology. The goal was indeed to understand the link between Rumin's operator and the differentials which appear in the various spectral sequences we have worked with: - the weight spectral sequence; - a special spectral sequence introduced by Julg and called by him Forman's spectral sequence; - Forman's spectral sequence (which turns out to be unrelated to the previous one). We will see that in general Rumin's operator depends on choices. However, in some special cases, it does not because it has an alternative interpretation as a differential in a natural spectral sequence. After defining Carnot groups and analysing their main properties, we will introduce the concept of weights of forms which will produce a splitting on the exterior differential operator d. We shall see how the Rumin complex arises from this splitting and proceed to carry out the complete computations in some key examples. From the third chapter onwards we will focus on Julg's paper, describing his new filtration and its relationship with the weight spectral sequence. We will study the connection between the spectral sequences and Rumin's complex in the n-dimensional Heisenberg group and the 7-dimensional quaternionic Heisenberg group and then generalize the result to Carnot groups using the weight filtration. Finally, we shall explain why Julg required the independence of choices in some special Rumin operators, introducing the Szego map and describing its main properties.

Spectral theory of differential operators on finite metric graphs and on bounded domains

Die vorliegende Arbeit widmet sich der Spektraltheorie von Differentialoperatoren auf metrischen Graphen und von indefiniten Differentialoperatoren auf beschränkten Gebieten. Sie besteht aus zwei Teilen. Im Ersten werden endliche, nicht notwendigerweise kompakte, metrische Graphen und die Hilberträume von quadratintegrierbaren Funktionen auf diesen betrachtet. Alle quasi-m-akkretiven Laplaceoperatoren auf solchen Graphen werden charakterisiert, und Abschätzungen an die negativen Eigenwerte selbstadjungierter Laplaceoperatoren werden hergeleitet. Weiterhin wird die Wohlgestelltheit eines gemischten Diffusions- und Transportproblems auf kompakten Graphen durch die Anwendung von Halbgruppenmethoden untersucht. Eine Verallgemeinerung des indefiniten Operators $-tfrac{d}{dx}sgn(x)tfrac{d}{dx}$ von Intervallen auf metrische Graphen wird eingeführt. Die Spektral- und Streutheorie der selbstadjungierten Realisierungen wird detailliert besprochen. Im zweiten Teil der Arbeit werden Operatoren untersucht, die mit indefiniten Formen der Art $langlegrad v, A(cdot)grad urangle$ mit $u,vin H_0^1(Omega)subset L^2(Omega)$ und $OmegasubsetR^d$ beschränkt, assoziiert sind. Das Eigenwertverhalten entspricht in Dimension $d=1$ einer verallgemeinerten Weylschen Asymptotik und für $dgeq 2$ werden Abschätzungen an die Eigenwerte bewiesen. Die Frage, wann indefinite Formmethoden für Dimensionen $dgeq 2$ anwendbar sind, bleibt offen und wird diskutiert.

Tree-level contribution to B[over ¯]→X_{d}γ using fragmentation functions

We evaluate the most important tree-level contributions connected with the b→uu ¯ ¯ dγ transition to the inclusive radiative decay B ¯ ¯ ¯ →X d γ using fragmentation functions. In this framework the singularities arising from collinear photon emission from the light quarks (u , u ¯ ¯ , and d ) can be absorbed into the (bare) quark-to-photon fragmentation function. We use as input the fragmentation function extracted by the ALEPH group from the two-jet cross section measured at the large electron positron (LEP) collider, where one of the jets is required to contain a photon. To get the quark-to-photon fragmentation function at the fragmentation scale μ F ∼m b , we use the evolution equation, which we solve numerically. We then calculate the (integrated) photon energy spectrum for b→uu ¯ ¯ dγ related to the operators P u 1,2 . For comparison, we also give the corresponding results when using nonzero (constituent) masses for the light quarks.

Pseudospectra in non-Hermitian quantum mechanics

We propose giving the mathematical concept of the pseudospectrum a central role in quantum mechanics with non-Hermitian operators. We relate pseudospectral properties to quasi-Hermiticity, similarity to self-adjoint operators, and basis properties of eigenfunctions. The abstract results are illustrated by unexpected wild properties of operators familiar from PT -symmetric quantum mechanics.

Asymptotically Toeplitz Hessenberg Operators and the Rieman mapping

In a recent work the authors have established a relation between the limits of the elements of the diagonals of the Hessenberg matrix D associated with a regular measure, whenever those limits exist, and the coe?cients of the Laurent series expansion of the Riemann mapping ?(z) of the support supp(?), when this is a Jordan arc or a connected nite union of Jordan arcs in the complex plane C. We extend here this result using asymptotic Toeplitz operator properties of the Hessenberg matriz.

SERBA: a B.I.E. program with linear elements for 2-D elastostatics analysis

The great developments that have occurred during the last few years in the finite element method and its applications has kept hidden other options for computation. The boundary integral element method now appears as a valid alternative and, in certain cases, has significant advantages. This method deals only with the boundary of the domain, while the F.E.M. analyses the whole domain. This has the following advantages: the dimensions of the problem to be studied are reduced by one, consequently simplifying the system of equations and preparation of input data. It is also possible to analyse infinite domains without discretization errors. These simplifications have the drawbacks of having to solve a full and non-symmetric matrix and some difficulties are incurred in the imposition of boundary conditions when complicated variations of the function over the boundary are assumed. In this paper a practical treatment of these problems, in particular boundary conditions imposition, has been carried out using the computer program shown below. Program SERBA solves general elastostatics problems in 2-dimensional continua using the boundary integral equation method. The boundary of the domain is discretized by line or elements over which the functions are assumed to vary linearly. Data (stresses and/or displacements) are introduced in the local co-ordinate system (element co-ordinates). Resulting stresses are obtained in local co-ordinates and displacements in a general system. The program has been written in Fortran ASCII and implemented on a 1108 Univac Computer. For 100 elements the core requirements are about 40 Kwords. Also available is a Fortran IV version (3 segments)implemented on a 21 MX Hewlett-Packard computer,using 15 Kwords.

Multiple four-stranded conformations of human telomere sequence d(CCCTAA) in solution

By detailed NMR analysis of a human telomere repeating unit, d(CCCTAA), we have found that three distinct tetramers, each of which consists of four symmetric single-strands, slowly exchange in a slightly acidic solution. Our new finding is a novel i-motif topology (T-form) where T4 is intercalated between C1 and C2 of the other duplex. The other two tetramers have a topology where C1 is intercalated between C2 and C3 of the other parallel duplex, resulting in the non-stacking T4 residues (R-form), and a topology where C1 is stacked between C3 and T4 of the other duplex (S-form). From the NMR denaturation profile, the R-form is the most stable of the three structures in the temperature range of 15–50°C, the S-form the second and the T-form the least stable. The thermodynamic parameters indicate that the T-form is the most enthalpically driven and entropically opposed, and its population is increased with decreasing temperature. The T-form structure determined by restrained molecular dynamics calculation suggests that inter-strand van der Waals contacts in the narrow grooves should contribute to the enthalpic stabilization of the T-form.

Geometry of Dirac Operators

Thesis (Master, Mathematics & Statistics) -- Queen's University, 2016-07-04 20:27:20.386