11,075 research outputs found
Simultaneous core partitions: parameterizations and sums
Fix coprime . We re-prove, without Ehrhart reciprocity, a conjecture
of Armstrong (recently verified by Johnson) that the finitely many simultaneous
-cores have average size , and that the
subset of self-conjugate cores has the same average (first shown by
Chen--Huang--Wang). We similarly prove a recent conjecture of Fayers that the
average weighted by an inverse stabilizer---giving the "expected size of the
-core of a random -core"---is . We also prove
Fayers' conjecture that the analogous self-conjugate average is the same if
is odd, but instead if is even. In principle,
our explicit methods---or implicit variants thereof---extend to averages of
arbitrary powers.
The main new observation is that the stabilizers appearing in Fayers'
conjectures have simple formulas in Johnson's -coordinates parameterization
of -cores.
We also observe that the -coordinates extend to parameterize general
-cores. As an example application with , we count the number of
-cores for coprime , verifying a recent conjecture of
Amdeberhan and Leven.Comment: v4: updated references to match final EJC versio
The Generalized Legendre transform and its applications to inverse spectral problems
Let be a Riemannian manifold, an isometric
action on of an -torus and a bounded
-invariant smooth function. By -invariance the Schr\"odinger operator,
, restricts to a self-adjoint operator on
, being a weight of and a large
positive integer. Let be the asymptotic support of the
spectrum of this operator. We will show that extend to a function,
and that, modulo assumptions on and
one can recover from , i.e. prove that is spectrally determined. The
main ingredient in the proof of this result is the existence of a "generalized
Legendre transform" mapping the graph of onto the graph of .Comment: 23 page
Vertex Operator Superalgebras and Their Representations
After giving some definitions for vertex operator SUPERalgebras and their
modules, we construct an associative algebra corresponding to any vertex
operator superalgebra, such that the representations of the vertex operator
algebra are in one-to-one correspondence with those of the corresponding
associative algebra. A way is presented to decribe the fusion rules for the
vertex operator superalgebras via modules of the associative algebra. The above
are generalizations of Zhu's constructions for vertex operator algebras. Then
we deal in detail with vertex operator superalgebras corresponding to
Neveu-Schwarz algebras, super affine Kac-Moody algebras, and free fermions. We
use the machinery established above to find the rationality conditions,
classify the representations and compute the fusion rules. In the appendix, we
present explicit formulas for singular vectors and defining relations for the
integrable representations of super affine algebras. These formulas are not
only crucial for the theory of the corresponding vertex operator superalgebras
and their representations, but also of independent interest.Comment: 50 pages, to appear in Contemporary Mathematic
Pivotal estimation via square-root Lasso in nonparametric regression
We propose a self-tuning method that simultaneously
resolves three important practical problems in high-dimensional regression
analysis, namely it handles the unknown scale, heteroscedasticity and (drastic)
non-Gaussianity of the noise. In addition, our analysis allows for badly
behaved designs, for example, perfectly collinear regressors, and generates
sharp bounds even in extreme cases, such as the infinite variance case and the
noiseless case, in contrast to Lasso. We establish various nonasymptotic bounds
for including prediction norm rate and sparsity. Our
analysis is based on new impact factors that are tailored for bounding
prediction norm. In order to cover heteroscedastic non-Gaussian noise, we rely
on moderate deviation theory for self-normalized sums to achieve Gaussian-like
results under weak conditions. Moreover, we derive bounds on the performance of
ordinary least square (ols) applied to the model selected by accounting for possible misspecification of the selected model. Under
mild conditions, the rate of convergence of ols post
is as good as 's rate. As an application, we consider
the use of and ols post as
estimators of nuisance parameters in a generic semiparametric problem
(nonlinear moment condition or -problem), resulting in a construction of
-consistent and asymptotically normal estimators of the main
parameters.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1204 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Exit of Plasmodium Sporozoites from Oocysts Is an Active Process That Involves the Circumsporozoite Protein
Plasmodium sporozoites develop within oocysts residing in the mosquito midgut. Mature sporozoites exit the oocysts, enter the hemolymph, and invade the salivary glands. The circumsporozoite (CS) protein is the major surface protein of salivary gland and oocyst sporozoites. It is also found on the oocyst plasma membrane and on the inner surface of the oocyst capsule. CS protein contains a conserved motif of positively charged amino acids: region II-plus, which has been implicated in the initial stages of sporozoite invasion of hepatocytes. We investigated the function of region II-plus by generating mutant parasites in which the region had been substituted with alanines. Mutant parasites produced normal numbers of sporozoites in the oocysts, but the sporozoites were unable to exit the oocysts. In in vitro as well, there was a profound delay, upon trypsin treatment, in the release of mutant sporozoites from oocysts. We conclude that the exit of sporozoites from oocysts is an active process that involves the region II-plus of CS protein. In addition, the mutant sporozoites were not infective to young rats. These findings provide a new target for developing reagents that interfere with the transmission of malaria
- …