A preferred vision for administering secondary schools : a reflective essay

There are many personal traits that have been found universally among effective leaders. Intelligence, self-reflection, honesty and not being afraid to seek assistance when confronted with difficult problems are examples of these traits. Effective leaders must have inherent people skills . They must have a genuine interest in and awareness of the needs of others. Emotional balance and remaining rational in times of conflict, crisis, or challenging circumstances are all positive personal attributes essential of effective leadership

Relativistic Wavepackets in Classically Chaotic Quantum Cosmological Billiards

Close to a spacelike singularity, pure gravity and supergravity in four to eleven spacetime dimensions admit a cosmological billiard description based on hyperbolic Kac-Moody groups. We investigate the quantum cosmological billiards of relativistic wavepackets towards the singularity, employing flat and hyperbolic space descriptions for the quantum billiards. We find that the strongly chaotic classical billiard motion of four-dimensional pure gravity corresponds to a spreading wavepacket subject to successive redshifts and tending to zero as the singularity is approached. We discuss the possible implications of these results in the context of singularity resolution and compare them with those of known semiclassical approaches. As an aside, we obtain exact solutions for the one-dimensional relativistic quantum billiards with moving walls.Comment: 18 pages, 10 figure

Integral equations PS-3 and moduli of pants

More than a hundred years ago H.Poincare and V.A.Steklov considered a problem for the Laplace equation with spectral parameter in the boundary conditions. Today similar problems for two adjacent domains with the spectral parameter in the conditions on the common boundary of the domains arises in a variety of situations: in justification and optimization of domain decomposition method, simple 2D models of oil extraction, (thermo)conductivity of composite materials. Singular 1D integral Poincare-Steklov equation with spectral parameter naturally emerges after reducing this 2D problem to the common boundary of the domains. We present a constructive representation for the eigenvalues and eigenfunctions of this integral equation in terms of moduli of explicitly constructed pants, one of the simplest Riemann surfaces with boundary. Essentially the solution of integral equation is reduced to the solution of three transcendent equations with three unknown numbers, moduli of pants. The discreet spectrum of the equation is related to certain surgery procedure ('grafting') invented by B.Maskit (1969), D.Hejhal (1975) and D.Sullivan- W.Thurston (1983).Comment: 27 pages, 13 figure

Supersymmetric quantum cosmological billiards

D=11 Supergravity near a space-like singularity admits a cosmological billiard description based on the hyperbolic Kac-Moody group E10. The quantization of this system via the supersymmetry constraint is shown to lead to wavefunctions involving automorphic (Maass wave) forms under the modular group W^+(E10)=PSL(2,O) with Dirichlet boundary conditions on the billiard domain. A general inequality for the Laplace eigenvalues of these automorphic forms implies that the wave function of the universe is generically complex and always tends to zero when approaching the initial singularity. We discuss possible implications of this result for the question of singularity resolution in quantum cosmology and comment on the differences with other approaches.Comment: 4 pages. v2: Added ref. Version to be published in PR

Analytic Continuation for Asymptotically AdS 3D Gravity

We have previously proposed that asymptotically AdS 3D wormholes and black holes can be analytically continued to the Euclidean signature. The analytic continuation procedure was described for non-rotating spacetimes, for which a plane t=0 of time symmetry exists. The resulting Euclidean manifolds turned out to be handlebodies whose boundary is the Schottky double of the geometry of the t=0 plane. In the present paper we generalize this analytic continuation map to the case of rotating wormholes. The Euclidean manifolds we obtain are quotients of the hyperbolic space by a certain quasi-Fuchsian group. The group is the Fenchel-Nielsen deformation of the group of the non-rotating spacetime. The angular velocity of an asymptotic region is shown to be related to the Fenchel-Nielsen twist. This solves the problem of classification of rotating black holes and wormholes in 2+1 dimensions: the spacetimes are parametrized by the moduli of the boundary of the corresponding Euclidean spaces. We also comment on the thermodynamics of the wormhole spacetimes.Comment: 28 pages, 14 figure

Numerical Study of Length Spectra and Low-lying Eigenvalue Spectra of Compact Hyperbolic 3-manifolds

In this paper, we numerically investigate the length spectra and the low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero eigenvalues have been successfully computed using the periodic orbit sum method, which are compared with various geometric quantities such as volume, diameter and length of the shortest periodic geodesic of the manifolds. The deviation of low-lying eigenvalue spectra of manifolds converging to a cusped hyperbolic manifold from the asymptotic distribution has been measured by $\zeta-$ function and spectral distance.Comment: 19 pages, 18 EPS figures and 2 GIF figures (fig.10) Description of cusped manifolds in section 2 is correcte

Eigenvalues of Laplacian with constant magnetic field on non-compact hyperbolic surfaces with finite area

We consider a magnetic Laplacian $-\Delta_A=(id+A)^\star (id+A)$ on a noncompact hyperbolic surface \mM with finite area. $A$ is a real one-form and the magnetic field $dA$ is constant in each cusp. When the harmonic component of $A$ satifies some quantified condition, the spectrum of $-\Delta_A$ is discrete. In this case we prove that the counting function of the eigenvalues of $-\Delta_{A}$ satisfies the classical Weyl formula, even when $dA=0.

Multiplicities of Periodic Orbit Lengths for Non-Arithmetic Models

Multiplicities of periodic orbit lengths for non-arithmetic Hecke triangle groups are discussed. It is demonstrated both numerically and analytically that at least for certain groups the mean multiplicity of periodic orbits with exactly the same length increases exponentially with the length. The main ingredient used is the construction of joint distribution of periodic orbits when group matrices are transformed by field isomorphisms. The method can be generalized to other groups for which traces of group matrices are integers of an algebraic field of finite degree

Casimir Effect in Hyperbolic Polygons

We derive a trace formula for the spectra of quantum mechanical systems in hyperbolic polygons which are the fundamental domains of discrete isometry groups acting in the two dimensional hyperboloid. Using this trace formula and the point splitting regularization method we calculate the Casimir energy for a scalar fields in such domains. The dependence of the vacuum energy on the number of vertexes is established.Comment: Latex, 1