449 research outputs found
On the distribution of the nodal sets of random spherical harmonics
We study the length of the nodal set of eigenfunctions of the Laplacian on
the \spheredim-dimensional sphere. It is well known that the eigenspaces
corresponding to \eigval=n(n+\spheredim-1) are the spaces \eigspc of
spherical harmonics of degree , of dimension \eigspcdim. We use the
multiplicity of the eigenvalues to endow \eigspc with the Gaussian
probability measure and study the distribution of the \spheredim-dimensional
volume of the nodal sets of a randomly chosen function. The expected volume is
proportional to \sqrt{\eigval}. One of our main results is bounding the
variance of the volume to be O(\frac{\eigval}{\sqrt{\eigspcdim}}).
In addition to the volume of the nodal set, we study its Leray measure. For
every , the expected value of the Leray measure is .
We are able to determine that the asymptotic form of the variance is
\frac{const}{\eigspcdim}.Comment: 47 pages, accepted for publication in the Journal of Mathematical
Physics. Lemmas 2.5, 2.11 were proved for any dimension, some other,
suggested by the referee, modifications and corrections, were mad
Solution of a Generalized Stieltjes Problem
We present the exact solution for a set of nonlinear algebraic equations
. These
were encountered by us in a recent study of the low energy spectrum of the
Heisenberg ferromagnetic chain \cite{dhar}. These equations are low
(density) ``degenerations'' of more complicated transcendental equation of
Bethe's Ansatz for a ferromagnet, but are interesting in themselves. They
generalize, through a single parameter, the equations of Stieltjes,
, familiar from Random Matrix theory.
It is shown that the solutions of these set of equations is given by the
zeros of generalized associated Laguerre polynomials. These zeros are
interesting, since they provide one of the few known cases where the location
is along a nontrivial curve in the complex plane that is determined in this
work.
Using a ``Green's function'' and a saddle point technique we determine the
asymptotic distribution of zeros.Comment: 19 pages, 4 figure
Jacobi Crossover Ensembles of Random Matrices and Statistics of Transmission Eigenvalues
We study the transition in conductance properties of chaotic mesoscopic
cavities as time-reversal symmetry is broken. We consider the Brownian motion
model for transmission eigenvalues for both types of transitions, viz.,
orthogonal-unitary and symplectic-unitary crossovers depending on the presence
or absence of spin-rotation symmetry of the electron. In both cases the
crossover is governed by a Brownian motion parameter {\tau}, which measures the
extent of time-reversal symmetry breaking. It is shown that the results
obtained correspond to the Jacobi crossover ensembles of random matrices. We
derive the level density and the correlation functions of higher orders for the
transmission eigenvalues. We also obtain the exact expressions for the average
conductance, average shot-noise power and variance of conductance, as functions
of {\tau}, for arbitrary number of modes (channels) in the two leads connected
to the cavity. Moreover, we give the asymptotic result for the variance of
shot-noise power for both the crossovers, the exact results being too long. In
the {\tau} \rightarrow 0 and {\tau} \rightarrow \infty limits the known results
for the orthogonal (or symplectic) and unitary ensembles are reproduced. In the
weak time-reversal symmetry breaking regime our results are shown to be in
agreement with the semiclassical predictions.Comment: 24 pages, 5 figure
Parental Access to Children's Raw Genomic Data in Canada: Legal Rights and Professional Responsibility
Children with rare and common diseases now undergo whole genome sequencing (WGS) in clinical and research contexts. Parents sometimes request access to their child's raw genomic data, to pursue their own analyses or for onward sharing with health professionals and researchers. These requests raise legal, ethical, and practical issues for professionals and parents alike. The advent of widespread WGS in pediatrics occurs in a context where privacy and data protection law remains focused on giving individuals control-oriented rights with respect to their personal information. Acting in their child's stead and in their best interests, parents are generally the ones who will be exercising these informational rights on behalf of the child. In this paper, we map the contours of parental authority to access their child's raw genomic data. We consider three use cases: hospital-based researchers, healthcare professionals acting in a clinical-diagnostic capacity, and “pure” academic researchers at a public institution. Our research seeks to answer two principal questions: Do parents have a right of access to their child's raw WGS data? If so, what are the limits of this right? Primarily focused on the laws of Ontario, Canada's most populous province, with a secondary focus on Canada's three other most populous provinces (Quebec, British Columbia, and Alberta) and the European Union, our principal findings include (1) parents have a general right of access to information about their children, but that the access right is more capacious in the clinical context than in the research context; (2) the right of access extends to personal data in raw form; (3) a consideration of the best interests of the child may materially limit the legal rights of parents to access data about their child; (4) the ability to exercise rights of access are transferred from parents to children when they gain decision-making capacity in both the clinical and research contexts, but with more nuance in the former. With these findings in mind, we argue that professional guidelines, which are concerned with obligations to interpret and return results, may assist in furthering a child's best interests in the context of legal access rights. We conclude by crafting recommendations for healthcare professionals in the clinical and research contexts when faced with a parental request for a child's raw genomic data
Time evolution in the Morse potential using supersymmetry: dissociation of the NO molecule
We present an algebraic method for treating molecular vibrations in the Morse
potential perturbed by an external laser field. By the help of a complete and
normalizable basis we transform the Schr\"{o}dinger equation into a system of
coupled ordinary differential equations. We apply our method to calculate the
dissociation probability of the NO molecule excited by chirped laser pulses.
The dependence of the molecular dipole-moment on the interatomic separation is
determined by a quantum-chemical method, and the corresponding transition
dipole moments are given by approximate analytic expressions. These turn out to
be very small between neighboring stationary states around the vibrational
quantum number , therefore we propose to use additional pulses in order
to skip this trapping state, and to obtain a reasonable dissociation
probability.Comment: 4 pages, 3 figure
Preliminary interpretation of Titan plasma interaction as observed by the Cassini Plasma Spectrometer: Comparisons with Voyager 1
The Cassini Plasma Spectrometer (CAPS) instrument observed the plasma environment at Titan during the Cassini orbiter's TA encounter on October 26, 2004. Titan was in Saturn's magnetosphere during the Voyager 1 flyby and also during the TA encounter. CAPS measurements from this encounter are compared with measurements made by the Voyager 1 Plasma Science Instrument (PLS). The comparisons focus on the composition and nature of ambient and pickup ions. They lead to: A) the major ion components of Saturn's magnetosphere in the vicinity of Titan are H+, H-2(+) and O+/CH4+ ions; B) finite gyroradius effects are apparent in ambient O+ ions as the result of their absorption by Titan's extended atmosphere; C) the principal pickup ions are composed of H+, H-2(+), N+/CH2+, CH4+, and N-2(+); D) the pickup ions are in narrow energy ranges; and E) there is clear evidence of the slowing down of background ions due to pickup ion mass loading
Some comments on developments in exact solutions in statistical mechanics since 1944
Lars Onsager and Bruria Kaufman calculated the partition function of the
Ising model exactly in 1944 and 1949. Since then there have been many
developments in the exact solution of similar, but usually more complicated,
models. Here I shall mention a few, and show how some of the latest work seems
to be returning once again to the properties observed by Onsager and Kaufman.Comment: 28 pages, 5 figures, section on six-vertex model revise
Non-Maxwellian Proton Velocity Distributions in Nonradiative Shocks
The Balmer line profiles of nonradiative supernova remnant shocks provide the
means to measure the post-shock proton velocity distribution. While most
analyses assume a Maxwellian velocity distribution, this is unlikely to be
correct. In particular, neutral atoms that pass through the shock and become
ionized downstream form a nonthermal distribution similar to that of pickup
ions in the solar wind. We predict the H alpha line profiles from the
combination of pickup protons and the ordinary shocked protons, and we consider
the extent to which this distribution could affect the shock parameters derived
from H alpha profiles. The Maxwellian assumption could lead to an underestimate
of shock speed by up to about 15%. The isotropization of the pickup ion
population generates wave energy, and we find that for the most favorable
parameters this energy could significantly heat the thermal particles.
Sufficiently accurate profiles could constrain the strength and direction of
the magnetic field in the shocked plasma, and we discuss the distortions from a
Gaussian profile to be expected in Tycho's supernova remnant.Comment: 13 pages, 6 figure
Direct and inverse spectral transform for the relativistic Toda lattice and the connection with Laurent orthogonal polynomials
We introduce a spectral transform for the finite relativistic Toda lattice
(RTL) in generalized form. In the nonrelativistic case, Moser constructed a
spectral transform from the spectral theory of symmetric Jacobi matrices. Here
we use a non-symmetric generalized eigenvalue problem for a pair of bidiagonal
matrices (L,M) to define the spectral transform for the RTL. The inverse
spectral transform is described in terms of a terminating T-fraction. The
generalized eigenvalues are constants of motion and the auxiliary spectral data
have explicit time evolution. Using the connection with the theory of Laurent
orthogonal polynomials, we study the long-time behaviour of the RTL. As in the
case of the Toda lattice the matrix entries have asymptotic limits. We show
that L tends to an upper Hessenberg matrix with the generalized eigenvalues
sorted on the diagonal, while M tends to the identity matrix.Comment: 24 pages, 9 figure
Systematical Approach to the Exact Solution of the Dirac Equation for A Special Form of the Woods-Saxon Potential
Exact solution of the Dirac equation for a special form of the Woods-Saxon
potential is obtained for the s-states. The energy eigenvalues and
two-component spinor wave functions are derived by using a systematical method
which is called as Nikiforov-Uvarov. It is seen that the energy eigenvalues
strongly depend on the potential parameters. In addition, it is also shown that
the non-relativistic limit can be reached easily and directly.Comment: 10 pages, no figures, submitted for Publicatio
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