2,226 research outputs found
Nonclassical spectral asymptotics and Dixmier traces: From circles to contact manifolds
We consider the spectral behavior and noncommutative geometry of commutators
, where is an operator of order with geometric origin and a
multiplication operator by a function. When is H\"{o}lder continuous, the
spectral asymptotics is governed by singularities. We study precise spectral
asymptotics through the computation of Dixmier traces; such computations have
only been considered in less singular settings. Even though a Weyl law fails
for these operators, and no pseudo-differential calculus is available,
variations of Connes' residue trace theorem and related integral formulas
continue to hold. On the circle, a large class of non-measurable Hankel
operators is obtained from H\"older continuous functions , displaying a wide
range of nonclassical spectral asymptotics beyond the Weyl law. The results
extend from Riemannian manifolds to contact manifolds and noncommutative tori.Comment: 40 page
Discontinuities without discontinuity: The Weakly-enforced Slip Method
Tectonic faults are commonly modelled as Volterra or Somigliana dislocations
in an elastic medium. Various solution methods exist for this problem. However,
the methods used in practice are often limiting, motivated by reasons of
computational efficiency rather than geophysical accuracy. A typical
geophysical application involves inverse problems for which many different
fault configurations need to be examined, each adding to the computational
load. In practice, this precludes conventional finite-element methods, which
suffer a large computational overhead on account of geometric changes. This
paper presents a new non-conforming finite-element method based on weak
imposition of the displacement discontinuity. The weak imposition of the
discontinuity enables the application of approximation spaces that are
independent of the dislocation geometry, thus enabling optimal reuse of
computational components. Such reuse of computational components renders
finite-element modeling a viable option for inverse problems in geophysical
applications. A detailed analysis of the approximation properties of the new
formulation is provided. The analysis is supported by numerical experiments in
2D and 3D.Comment: Submitted for publication in CMAM
Spin bit models from non-planar N=4 SYM
We study spin models underlying the non-planar dynamics of SYM
gauge theory. In particular, we derive the non-local spin chain Hamiltonian
generating dilatations in the gauge theory at leading order in
but exact in .
States in the spin chain are characterized by a spin-configuration and a
linking variable describing how sites in the chain are connected. Joining and
splitting of string/traces are mimicked by a twist operator acting on the
linking variable. The results are applied to a systematic study of non-planar
anomalous dimensions and operator mixing in SYM. Intriguingly, we
identify a sequence of SYM operators for which corrections to the one-loop
anomalous dimensions stop at the first non-planar order.Comment: 22 pages, 2 figures, some typos corrected in eqs. (2.6), (2.8),
(2.11), (2.14
Concordance and Mutation
We provide a framework for studying the interplay between concordance and
positive mutation and identify some of the basic structures relating the two.
The fundamental result in understanding knot concordance is the structure
theorem proved by Levine: for n>1 there is an isomorphism phi from the
concordance group C_n of knotted (2n-1)-spheres in S^{2n+1} to an algebraically
defined group G_{+-}; furthermore, G__{+-} is isomorphic to the infinite direct
sum Z^infty direct sum Z_2^infty direct sum Z_4^infty. It was a startling
consequence of the work of Casson and Gordon that in the classical case the
kernel of phi on C_1 is infinitely generated. Beyond this, little has been
discovered about the pair (C_1,phi).
In this paper we present a new approach to studying C_1 by introducing a
group, M, defined as the quotient of the set of knots by the equivalence
relation generated by concordance and positive mutation, with group operation
induced by connected sum. We prove there is a factorization of phi,
C_1-->M-->G_-. Our main result is that both maps have infinitely generated
kernels.
Among geometric constructions on classical knots, the most subtle is positive
mutation. Positive mutants are indistinguishable using classical abelian knot
invariants as well as by such modern invariants as the Jones, Homfly or
Kauffman polynomials. Distinguishing positive mutants up to concordance is a
far more difficult problem; only one example has been known until now. The
results in this paper provide, among other results, the first infinite families
of knots that are distinct from their positive mutants, even up to concordance.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol5/paper26.abs.htm
On stability of discretizations of the Helmholtz equation (extended version)
We review the stability properties of several discretizations of the
Helmholtz equation at large wavenumbers. For a model problem in a polygon, a
complete -explicit stability (including -explicit stability of the
continuous problem) and convergence theory for high order finite element
methods is developed. In particular, quasi-optimality is shown for a fixed
number of degrees of freedom per wavelength if the mesh size and the
approximation order are selected such that is sufficiently small and
, and, additionally, appropriate mesh refinement is used near
the vertices. We also review the stability properties of two classes of
numerical schemes that use piecewise solutions of the homogeneous Helmholtz
equation, namely, Least Squares methods and Discontinuous Galerkin (DG)
methods. The latter includes the Ultra Weak Variational Formulation
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