713,005 research outputs found
Cayley graphs on the symmetric group generated by initial reversals have unit spectral gap
In a recent paper Gunnells, Scott and Walden have determined the complete
spectrum of the Schreier graph on the symmetric group corresponding to the
Young subgroup and generated by initial reversals. In
particular they find that the first nonzero eigenvalue, or spectral gap, of the
Laplacian is always 1, and report that "empirical evidence" suggests that this
also holds for the corresponding Cayley graph. We provide a simple proof of
this last assertion, based on the decomposition of the Laplacian of Cayley
graphs, into a direct sum of irreducible representation matrices of the
symmetric group.Comment: Shorter version. Published in the Electron. J. of Combinatoric
A classification of primitive permutation groups with finite stabilizers
We classify all infinite primitive permutation groups possessing a finite
point stabilizer, thus extending the seminal Aschbacher-O'Nan-Scott Theorem to
all primitive permutation groups with finite point stabilizers.Comment: Accepted in J. Algebra. Various changes, some due to the author, some
due to suggestions from readers and others due to the comments of anonymous
referee
Spectral synthesis for Banach Algebras II
This paper continues the study of spectral synthesis and the topologies tau-infinity and tau-r on the ideal space of a Banach algebra, concentrating particularly on the class of Haagerup tensor products of C*-algebras. For this class, it is shown that spectral synthesis is equivalent to the Hausdorffness of tau_infinity. Under a weak extra condition, spectral synthesis is shown to be equivalent to the Hausdorffness of tau_r
Tug-of-war and the infinity Laplacian
We prove that every bounded Lipschitz function F on a subset Y of a length
space X admits a tautest extension to X, i.e., a unique Lipschitz extension u
for which Lip_U u = Lip_{boundary of U} u for all open subsets U of X that do
not intersect Y.
This was previously known only for bounded domains R^n, in which case u is
infinity harmonic, that is, a viscosity solution to Delta_infty u = 0. We also
prove the first general uniqueness results for Delta_infty u = g on bounded
subsets of R^n (when g is uniformly continuous and bounded away from zero), and
analogous results for bounded length spaces.
The proofs rely on a new game-theoretic description of u. Let u^epsilon(x) be
the value of the following two-player zero-sum game, called tug-of-war: fix
x_0=x \in X minus Y. At the kth turn, the players toss a coin and the winner
chooses an x_k with d(x_k, x_{k-1})< epsilon. The game ends when x_k is in Y,
and player one's payoff is
F(x_k) - (epsilon^2/2) sum_{i=0}^{k-1} g(x_i)
We show that the u^\epsilon converge uniformly to u as epsilon tends to zero.
Even for bounded domains in R^n, the game theoretic description of
infinity-harmonic functions yields new intuition and estimates; for instance,
we prove power law bounds for infinity-harmonic functions in the unit disk with
boundary values supported in a delta-neighborhood of a Cantor set on the unit
circle.Comment: 44 pages, 4 figure
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