2,342 research outputs found
Abelian covers of surfaces and the homology of the level L mapping class group
We calculate the first homology group of the mapping class group with
coefficients in the first rational homology group of the universal abelian -cover of the surface. If the surface has one marked point, then the
answer is \Q^{\tau(L)}, where is the number of positive divisors of
. If the surface instead has one boundary component, then the answer is
\Q. We also perform the same calculation for the level subgroup of the
mapping class group. Set . If the surface has one
marked point, then the answer is \Q[H_L], the rational group ring of .
If the surface instead has one boundary component, then the answer is \Q.Comment: 32 pages, 10 figures; numerous corrections and simplifications; to
appear in J. Topol. Ana
Sufficient conditions for the existence of bound states in a central potential
We show how a large class of sufficient conditions for the existence of bound
states, in non-positive central potentials, can be constructed. These
sufficient conditions yield upper limits on the critical value,
, of the coupling constant (strength), , of the
potential, , for which a first -wave bound state appears.
These upper limits are significantly more stringent than hitherto known
results.Comment: 7 page
Necessary and sufficient conditions for existence of bound states in a central potential
We obtain, using the Birman-Schwinger method, a series of necessary
conditions for the existence of at least one bound state applicable to
arbitrary central potentials in the context of nonrelativistic quantum
mechanics. These conditions yield a monotonic series of lower limits on the
"critical" value of the strength of the potential (for which a first bound
state appears) which converges to the exact critical strength. We also obtain a
sufficient condition for the existence of bound states in a central monotonic
potential which yield an upper limit on the critical strength of the potential.Comment: 7 page
Upper and lower limits on the number of bound states in a central potential
In a recent paper new upper and lower limits were given, in the context of
the Schr\"{o}dinger or Klein-Gordon equations, for the number of S-wave
bound states possessed by a monotonically nondecreasing central potential
vanishing at infinity. In this paper these results are extended to the number
of bound states for the -th partial wave, and results are also
obtained for potentials that are not monotonic and even somewhere positive. New
results are also obtained for the case treated previously, including the
remarkably neat \textit{lower} limit with (valid in the Schr\"{o}dinger case, for a class of potentials
that includes the monotonically nondecreasing ones), entailing the following
\textit{lower} limit for the total number of bound states possessed by a
monotonically nondecreasing central potential vanishing at infinity: N\geq
\{\{(\sigma+1)/2\}\} {(\sigma+3)/2\} \}/2 (here the double braces denote of
course the integer part).Comment: 44 pages, 5 figure
Fisheye Consistency: Keeping Data in Synch in a Georeplicated World
Over the last thirty years, numerous consistency conditions for replicated
data have been proposed and implemented. Popular examples of such conditions
include linearizability (or atomicity), sequential consistency, causal
consistency, and eventual consistency. These consistency conditions are usually
defined independently from the computing entities (nodes) that manipulate the
replicated data; i.e., they do not take into account how computing entities
might be linked to one another, or geographically distributed. To address this
lack, as a first contribution, this paper introduces the notion of proximity
graph between computing nodes. If two nodes are connected in this graph, their
operations must satisfy a strong consistency condition, while the operations
invoked by other nodes are allowed to satisfy a weaker condition. The second
contribution is the use of such a graph to provide a generic approach to the
hybridization of data consistency conditions into the same system. We
illustrate this approach on sequential consistency and causal consistency, and
present a model in which all data operations are causally consistent, while
operations by neighboring processes in the proximity graph are sequentially
consistent. The third contribution of the paper is the design and the proof of
a distributed algorithm based on this proximity graph, which combines
sequential consistency and causal consistency (the resulting condition is
called fisheye consistency). In doing so the paper not only extends the domain
of consistency conditions, but provides a generic provably correct solution of
direct relevance to modern georeplicated systems
Abelian subgroups of Garside groups
In this paper, we show that for every abelian subgroup of a Garside
group, some conjugate consists of ultra summit elements and the
centralizer of is a finite index subgroup of the normalizer of .
Combining with the results on translation numbers in Garside groups, we obtain
an easy proof of the algebraic flat torus theorem for Garside groups and solve
several algorithmic problems concerning abelian subgroups of Garside groups.Comment: This article replaces our earlier preprint "Stable super summit sets
in Garside groups", arXiv:math.GT/060258
On the negative spectrum of two-dimensional Schr\"odinger operators with radial potentials
For a two-dimensional Schr\"odinger operator
with the radial potential , we study the behavior of
the number of its negative eigenvalues, as the coupling
parameter tends to infinity. We obtain the necessary and sufficient
conditions for the semi-classical growth and for
the validity of the Weyl asymptotic law.Comment: 13 page
Entanglement entropy of fermions in any dimension and the Widom conjecture
We show that entanglement entropy of free fermions scales faster then area
law, as opposed to the scaling for the harmonic lattice, for example.
We also suggest and provide evidence in support of an explicit formula for the
entanglement entropy of free fermions in any dimension , as the size of a subsystem
, where is the Fermi surface and
is the boundary of the region in real space. The expression for the constant
is based on a conjecture due to H. Widom. We
prove that a similar expression holds for the particle number fluctuations and
use it to prove a two sided estimates on the entropy .Comment: Final versio
The a priori Tan Theta Theorem for spectral subspaces
Let A be a self-adjoint operator on a separable Hilbert space H. Assume that
the spectrum of A consists of two disjoint components s_0 and s_1 such that the
set s_0 lies in a finite gap of the set s_1. Let V be a bounded self-adjoint
operator on H off-diagonal with respect to the partition spec(A)=s_0 \cup s_1.
It is known that if ||V||<\sqrt{2}d, where d=\dist(s_0,s_1), then the
perturbation V does not close the gaps between s_0 and s_1 and the spectrum of
the perturbed operator L=A+V consists of two isolated components s'_0 and s'_1
grown from s_0 and s_1, respectively. Furthermore, it is known that if V
satisfies the stronger bound ||V||< d then the following sharp norm estimate
holds: ||E_L(s'_0)-E_A(s_0)|| \leq sin(arctan(||V||/d)), where E_A(s_0) and
E_L(s'_0) are the spectral projections of A and L associated with the spectral
sets s_0 and s'_0, respectively. In the present work we prove that this
estimate remains valid and sharp also for d \leq ||V||< \sqrt{2}d, which
completely settles the issue.Comment: v3: some typos fixed; Examples adde
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