29,239 research outputs found
Spectra of random linear combinations of matrices defined via representations and Coxeter generators of the symmetric group
We consider the asymptotic behavior as of the spectra of random
matrices of the form where for each the random variables are i.i.d.
standard Gaussian and the matrices are obtained by applying
an irreducible unitary representation of the symmetric group on
to the transposition that interchanges and
[thus, is both unitary and self-adjoint, with all eigenvalues
either +1 or -1]. Irreducible representations of the symmetric group on
are indexed by partitions of . A consequence of
the results we establish is that if
is the partition of corresponding to , is the corresponding conjugate partition of (i.e., the Young
diagram of is the transpose of the Young diagram of ),
for each , and
for each , then the spectral
measure of the resulting random matrix converges in distribution to a random
probability measure that is Gaussian with random mean and variance
, where is the constant and
is a standard Gaussian random variable.Comment: Published in at http://dx.doi.org/10.1214/08-AOP418 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Sign variation, the Grassmannian, and total positivity
The totally nonnegative Grassmannian is the set of k-dimensional subspaces V
of R^n whose nonzero Pluecker coordinates all have the same sign. Gantmakher
and Krein (1950) and Schoenberg and Whitney (1951) independently showed that V
is totally nonnegative iff every vector in V, when viewed as a sequence of n
numbers and ignoring any zeros, changes sign at most k-1 times. We generalize
this result from the totally nonnegative Grassmannian to the entire
Grassmannian, showing that if V is generic (i.e. has no zero Pluecker
coordinates), then the vectors in V change sign at most m times iff certain
sequences of Pluecker coordinates of V change sign at most m-k+1 times. We also
give an algorithm which, given a non-generic V whose vectors change sign at
most m times, perturbs V into a generic subspace whose vectors also change sign
at most m times. We deduce that among all V whose vectors change sign at most m
times, the generic subspaces are dense. These results generalize to oriented
matroids. As an application of our results, we characterize when a generalized
amplituhedron construction, in the sense of Arkani-Hamed and Trnka (2013), is
well defined. We also give two ways of obtaining the positroid cell of each V
in the totally nonnegative Grassmannian from the sign patterns of vectors in V.Comment: 28 pages. v2: We characterize when a generalized amplituhedron
construction is well defined, in new Section 4 (the previous Section 4 is now
Section 5); v3: Final version to appear in J. Combin. Theory Ser.
Discussion of "Statistical Inference: The Big Picture" by R. E. Kass
Discussion of "Statistical Inference: The Big Picture" by R. E. Kass
[arXiv:1106.2895]Comment: Published in at http://dx.doi.org/10.1214/11-STS337A the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Moment curves and cyclic symmetry for positive Grassmannians
We show that for each k and n, the cyclic shift map on the complex
Grassmannian Gr(k,n) has exactly fixed points. There is a unique
totally nonnegative fixed point, given by taking n equally spaced points on the
trigonometric moment curve (if k is odd) or the symmetric moment curve (if k is
even). We introduce a parameter q, and show that the fixed points of a
q-deformation of the cyclic shift map are precisely the critical points of the
mirror-symmetric superpotential on Gr(k,n). This follows from
results of Rietsch about the quantum cohomology ring of Gr(k,n). We survey many
other diverse contexts which feature moment curves and the cyclic shift map.Comment: 18 pages. v2: Minor change
A unified model for the spectrophotometric development of classical and recurrent novae: the role of asphericity of the ejecta
There is increasing evidence that the geometry, and not only the filling
factors, of nova ejecta is important in the interpretation of their spectral
and photometric developments. Ensembles of spectra and light curves have
provided general typographies. This Letter suggests how these can be
unified.The observed spread in the maximum magnitude - rate of decline (MMRD)
relation is argued to result from the range of opening angles and inclination
of the ejecta, and not only to their masses and velocities. The spectroscopic
classes can be similarly explained and linked to the behavior of the light
curves. The secondary maximum observed in some dust forming novae is a natural
consequence of the asphericity. Neither secondary ejections nor winds are
needed to explain the phenomenology. The spectrophotometric development of
classical novae can be understood within a single phenomenological model with
bipolar, although not jet-like, mass ejecta. High resolution spectropolarimetry
will be an essential analytical tool.Comment: 4 pages, no figures; accepted for A&A Letters (in press
The Galactic Center Isolated Nonthermal Filaments as Analogs of Cometary Plasma Tails
We propose a model for the origin of the isolated nonthermal filaments
observed at the Galactic center based on an analogy to cometary plasma tails.
We invoke the interaction between a large scale magnetized galactic wind and
embedded molecular clouds. As the advected wind magnetic field encounters a
dense molecular cloud, it is impeded and drapes around the cloud, ultimately
forming a current sheet in the wake. This draped field is further stretched by
the wind flow into a long, thin filament whose aspect ratio is determined by
the balance between the dynamical wind and amplified magnetic field pressures.
The key feature of this cometary model is that the filaments are dynamic
configurations, and not static structures. As such, they are local
amplifications of an otherwise weak field and not directly connected to any
static global field. The derived field strengths for the wind and wake are
consistent with observational estimates. Finally, the observed synchrotron
emission is naturally explained by the acceleration of electrons to high energy
by plasma and MHD turbulence generated in the cloud wake.Comment: Uses AAS aasms4.sty macros. ApJ (in press, vol. 521, 20 Aug
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