113 research outputs found
Macdonald polynomials in superspace: conjectural definition and positivity conjectures
We introduce a conjectural construction for an extension to superspace of the
Macdonald polynomials. The construction, which depends on certain orthogonality
and triangularity relations, is tested for high degrees. We conjecture a simple
form for the norm of the Macdonald polynomials in superspace, and a rather
non-trivial expression for their evaluation. We study the limiting cases q=0
and q=\infty, which lead to two families of Hall-Littlewood polynomials in
superspace. We also find that the Macdonald polynomials in superspace evaluated
at q=t=0 or q=t=\infty seem to generalize naturally the Schur functions. In
particular, their expansion coefficients in the corresponding Hall-Littlewood
bases appear to be polynomials in t with nonnegative integer coefficients. More
strikingly, we formulate a generalization of the Macdonald positivity
conjecture to superspace: the expansion coefficients of the Macdonald
superpolynomials expanded into a modified version of the Schur superpolynomial
basis (the q=t=0 family) are polynomials in q and t with nonnegative integer
coefficients.Comment: 18 page
Unitary designs and codes
A unitary design is a collection of unitary matrices that approximate the
entire unitary group, much like a spherical design approximates the entire unit
sphere. In this paper, we use irreducible representations of the unitary group
to find a general lower bound on the size of a unitary t-design in U(d), for
any d and t. We also introduce the notion of a unitary code - a subset of U(d)
in which the trace inner product of any pair of matrices is restricted to only
a small number of distinct values - and give an upper bound for the size of a
code of degree s in U(d) for any d and s. These bounds can be strengthened when
the particular inner product values that occur in the code or design are known.
Finally, we describe some constructions of designs: we give an upper bound on
the size of the smallest weighted unitary t-design in U(d), and we catalogue
some t-designs that arise from finite groups.Comment: 25 pages, no figure
The interaction of a gap with a free boundary in a two dimensional dimer system
Let be a fixed vertical lattice line of the unit triangular lattice in
the plane, and let \Cal H be the half plane to the left of . We
consider lozenge tilings of \Cal H that have a triangular gap of side-length
two and in which is a free boundary - i.e., tiles are allowed to
protrude out half-way across . We prove that the correlation function of
this gap near the free boundary has asymptotics ,
, where is the distance from the gap to the free boundary. This
parallels the electrostatic phenomenon by which the field of an electric charge
near a conductor can be obtained by the method of images.Comment: 34 pages, AmS-Te
UBC-Nepal expedition: The use of oral antioxidants does not alter cerebrovascular function at sea-level or high-altitude
Hypoxia is associated with an increased systemic and cerebral formation of free radicals and associated reactants that may be linked to impaired cerebral vascular function a neurological sequela. To what extent oral antioxidants prophylaxis impacts cerebrovascular function in humans throughout the course of acclimatization to the hypoxia of terrestrial high-altitude has not been examined. Thus, the purpose of the current study was to examine the influence of orally ingested antioxidants at clinically relevant doses (vitamin C, E, and alpha-lipoic acid) on cerebrovascular regulation at sea-level (344 m; n = 12; female n = 2 participants), and at high altitude (5050 m; n = 9; female n = 2), in a randomized, placebo-controlled, and double-blinded crossover design. Hypercapnic and hypoxic cerebrovascular reactivity tests of the internal carotid (ICA)] were conducted at sea-level, while global and regional cerebral blood flow [i.e. ICA and vertebral artery (VA)] were assessed after 10–12 days following arrival at 5050 m. At sea-level, acute administration of antioxidants did not alter cerebral hypoxic cerebrovascular reactivity (pre vs. post: 1.5 ± 0.7 vs. 1.2 ± 0.8 %∆CBF/-%∆SpO2; P = 0.96), or cerebral hypercapnic cerebrovascular reactivity (pre vs. post: 5.7 ± 2.0 vs. 5.8 ± 1.9 %∆CBF/∆mmHg; P = 0.33). Furthermore, global cerebral blood flow (P = 0.43), as well as cerebral vascular conductance (ICA P = 0.08; VA P = 0.32), were unaltered at 5050 m following antioxidant administration. In conclusion, these data show that an oral antioxidant cocktail known to attenuate systemic oxidative stress failed to alter cerebrovascular function at sea-level and cerebral blood flow during acclimatization to high-altitude
Functional central limit theorems for vicious walkers
We consider the diffusion scaling limit of the vicious walker model that is a
system of nonintersecting random walks. We prove a functional central limit
theorem for the model and derive two types of nonintersecting Brownian motions,
in which the nonintersecting condition is imposed in a finite time interval
for the first type and in an infinite time interval for
the second type, respectively. The limit process of the first type is a
temporally inhomogeneous diffusion, and that of the second type is a temporally
homogeneous diffusion that is identified with a Dyson's model of Brownian
motions studied in the random matrix theory. We show that these two types of
processes are related to each other by a multi-dimensional generalization of
Imhof's relation, whose original form relates the Brownian meander and the
three-dimensional Bessel process. We also study the vicious walkers with wall
restriction and prove a functional central limit theorem in the diffusion
scaling limit.Comment: AMS-LaTeX, 20 pages, 2 figures, v6: minor corrections made for
publicatio
Crystal Graphs and -Analogues of Weight Multiplicities for the Root System
We give an expression of the -analogues of the multiplicities of weights
in irreducible \sl_{n+1}-modules in terms of the geometry of the crystal
graph attached to the corresponding U_q(\sl_{n+1})-modules. As an
application, we describe multivariate polynomial analogues of the
multiplicities of the zero weight, refining Kostant's generalized exponents.Comment: LaTeX file with epic, eepic pictures, 17 pages, November 1994, to
appear in Lett. Math. Phy
Affine and toric hyperplane arrangements
We extend the Billera-Ehrenborg-Readdy map between the intersection lattice
and face lattice of a central hyperplane arrangement to affine and toric
hyperplane arrangements. For arrangements on the torus, we also generalize
Zaslavsky's fundamental results on the number of regions.Comment: 32 pages, 4 figure
Fluctuation properties of the TASEP with periodic initial configuration
We consider the joint distributions of particle positions for the continuous
time totally asymmetric simple exclusion process (TASEP). They are expressed as
Fredholm determinants with a kernel defining a signed determinantal point
process. We then consider certain periodic initial conditions and determine the
kernel in the scaling limit. This result has been announced first in a letter
by one of us and here we provide a self-contained derivation. Connections to
last passage directed percolation and random matrices are also briefly
discussed.Comment: 33 pages, 4 figure, LaTeX; We added several references to the general
framework and techniques use
Vicious walkers, friendly walkers and Young tableaux II: With a wall
We derive new results for the number of star and watermelon configurations of
vicious walkers in the presence of an impenetrable wall by showing that these
follow from standard results in the theory of Young tableaux, and combinatorial
descriptions of symmetric functions. For the problem of -friendly walkers,
we derive exact asymptotics for the number of stars and watermelons both in the
absence of a wall and in the presence of a wall.Comment: 35 pages, AmS-LaTeX; Definitions of n-friendly walkers clarified; the
statement of Theorem 4 and its proof were correcte
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