434 research outputs found
Mappings of least Dirichlet energy and their Hopf differentials
The paper is concerned with mappings between planar domains having least
Dirichlet energy. The existence and uniqueness (up to a conformal change of
variables in the domain) of the energy-minimal mappings is established within
the class of strong limits of homeomorphisms in the
Sobolev space , a result of considerable interest in the
mathematical models of Nonlinear Elasticity. The inner variation leads to the
Hopf differential and its trajectories.
For a pair of doubly connected domains, in which has finite conformal
modulus, we establish the following principle:
A mapping is energy-minimal if and only if
its Hopf-differential is analytic in and real along the boundary of .
In general, the energy-minimal mappings may not be injective, in which case
one observes the occurrence of cracks in . Nevertheless, cracks are
triggered only by the points in the boundary of where fails to be
convex. The general law of formation of cracks reads as follows:
Cracks propagate along vertical trajectories of the Hopf differential from
the boundary of toward the interior of where they eventually terminate
before making a crosscut.Comment: 51 pages, 4 figure
Pinwheel patterns and powder diffraction
Pinwheel patterns and their higher dimensional generalisations display
continuous circular or spherical symmetries in spite of being perfectly
ordered. The same symmetries show up in the corresponding diffraction images.
Interestingly, they also arise from amorphous systems, and also from regular
crystals when investigated by powder diffraction. We present first steps and
results towards a general frame to investigate such systems, with emphasis on
statistical properties that are helpful to understand and compare the
diffraction images. We concentrate on properties that are accessible via an
alternative substitution rule for the pinwheel tiling, based on two different
prototiles. Due to striking similarities, we compare our results with the toy
model for the powder diffraction of the square lattice.Comment: 7 pages, 4 figure
Pilot Study: Unique Response of Bone Tissue During an Investigation of Radio-Adaptive Effects in Mice
PURPOSE: We obtained bone tissue to evaluate the collateral effects of experiments designed to investigate molecular mechanisms of radio-adaptation in a mouse model. Radio-adaptation describes a process by which the prior exposure to low dose radiation can protect against the toxic effect of a subsequent high dose exposure. In the radio-adaptation experiments, C57Bl/6 mice were exposed to either a Sham or a priming Low Dose (5 cGy) of Cs-137 gamma rays before being exposed to either a Sham or High Dose (6 Gy) 24 hours later. ANALYSIS: Bone tissue were obtained from two experiments where mice were sacrificed at 3 days (n=3/group, 12 total) and at 14 days (n=6/group, 24 total) following high dose exposure. Tissues were analyzed to 1) evaluate a radio-adaptive response in bone tissue and 2) describe cellular and microstructural effects for two skeletal sites with different rates of bone turnover. One tibia and one lumbar vertebrae (LV2), collected at the 3-day time-point, were analyzed by bone histomorphometry and micro-CT to evaluate the cellular response and any evidence of microarchitectural impact. Likewise, tibia and LV2, collected at the 14-day time-point, were analyzed by micro-CT alone to evaluate resulting changes to bone structure and microarchitecture. The data were analyzed by 2-way ANOVA to evaluate the effects of the priming low dose radiation, of the high dose radiation, and of any interaction between the priming low and high doses of radiation. Bone histomorphometry was performed in the cancellous bone (aka trabecular bone) compartments of the proximal tibial metaphysis and of LV2. RESULTS: Cellular Response @ 3 Days The priming Low Dose radiation decreased osteoblast-covered bone perimeter in the proximal tibia and the total cell density in the bone marrow in the LV2. High Dose radiation, regardless of prior exposure to priming dose, dramatically reduced total cell density in bone marrow of both the long bone and vertebra. However, in the proximal tibia, High Dose radiation increased the osteoclast-covered bone perimeters, the density of adipocytes in bone marrow, and the area of bone marrow occupied by fat cells -- while in the LV2, adipocytes were rare and not stimulated by High Dose radiation. In an unexpected response, High Dose radiation dramatically increased (10-fold) osteoblast-covered bone perimeter in the LV2
Supersymmetric quantum cosmological billiards
D=11 Supergravity near a space-like singularity admits a cosmological
billiard description based on the hyperbolic Kac-Moody group E10. The
quantization of this system via the supersymmetry constraint is shown to lead
to wavefunctions involving automorphic (Maass wave) forms under the modular
group W^+(E10)=PSL(2,O) with Dirichlet boundary conditions on the billiard
domain. A general inequality for the Laplace eigenvalues of these automorphic
forms implies that the wave function of the universe is generically complex and
always tends to zero when approaching the initial singularity. We discuss
possible implications of this result for the question of singularity resolution
in quantum cosmology and comment on the differences with other approaches.Comment: 4 pages. v2: Added ref. Version to be published in PR
Random polynomials, random matrices, and -functions
We show that the Circular Orthogonal Ensemble of random matrices arises
naturally from a family of random polynomials. This sheds light on the
appearance of random matrix statistics in the zeros of the Riemann
zeta-function.Comment: Added background material. Final version. To appear in Nonlinearit
Evidence for the classical integrability of the complete AdS(4) x CP(3) superstring
We construct a zero-curvature Lax connection in a sub-sector of the
superstring theory on AdS(4) x CP(3) which is not described by the
OSp(6|4)/U(3) x SO(1,3) supercoset sigma-model. In this sub-sector worldsheet
fermions associated to eight broken supersymmetries of the type IIA background
are physical fields. As such, the prescription for the construction of the Lax
connection based on the Z_4-automorphism of the isometry superalgebra OSp(6|4)
does not do the job. So, to construct the Lax connection we have used an
alternative method which nevertheless relies on the isometry of the target
superspace and kappa-symmetry of the Green-Schwarz superstring.Comment: 1+26 pages; v2: minor typos corrected, acknowledgements adde
Doubly connected minimal surfaces and extremal harmonic mappings
The concept of a conformal deformation has two natural extensions:
quasiconformal and harmonic mappings. Both classes do not preserve the
conformal type of the domain, however they cannot change it in an arbitrary
way. Doubly connected domains are where one first observes nontrivial conformal
invariants. Herbert Groetzsch and Johannes C. C. Nitsche addressed this issue
for quasiconformal and harmonic mappings, respectively. Combining these
concepts we obtain sharp estimates for quasiconformal harmonic mappings between
doubly connected domains. We then apply our results to the Cauchy problem for
minimal surfaces, also known as the Bjorling problem. Specifically, we obtain a
sharp estimate of the modulus of a doubly connected minimal surface that
evolves from its inner boundary with a given initial slope.Comment: 35 pages, 2 figures. Minor edits, references adde
Sobolev Inequalities for Differential Forms and -cohomology
We study the relation between Sobolev inequalities for differential forms on
a Riemannian manifold and the -cohomology of that manifold.
The -cohomology of is defined to be the quotient of the space
of closed differential forms in modulo the exact forms which are
exterior differentials of forms in .Comment: This paper has appeared in the Journal of Geometric Analysis, (only
minor changes have been made since verion 1
Principal forms X^2 + nY^2 representing many integers
In 1966, Shanks and Schmid investigated the asymptotic behavior of the number
of positive integers less than or equal to x which are represented by the
quadratic form X^2+nY^2. Based on some numerical computations, they observed
that the constant occurring in the main term appears to be the largest for n=2.
In this paper, we prove that in fact this constant is unbounded as n runs
through positive integers with a fixed number of prime divisors.Comment: 10 pages, title has been changed, Sections 2 and 3 are new, to appear
in Abh. Math. Sem. Univ. Hambur
Non-Abelian adiabatic statistics and Hall viscosity in quantum Hall states and p_x+ip_y paired superfluids
Many trial wavefunctions for fractional quantum Hall states in a single
Landau level are given by functions called conformal blocks, taken from some
conformal field theory. Also, wavefunctions for certain paired states of
fermions in two dimensions, such as p_x+ip_y states, reduce to such a form at
long distances. Here we investigate the adiabatic transport of such
many-particle trial wavefunctions using methods from two-dimensional field
theory. One context for this is to calculate the statistics of widely-separated
quasiholes, which has been predicted to be non-Abelian in a variety of cases.
The Berry phase or matrix (holonomy) resulting from adiabatic transport around
a closed loop in parameter space is the same as the effect of analytic
continuation around the same loop with the particle coordinates held fixed
(monodromy), provided the trial functions are orthonormal and holomorphic in
the parameters so that the Berry vector potential (or connection) vanishes. We
show that this is the case (up to a simple area term) for paired states
(including the Moore-Read quantum Hall state), and present general conditions
for it to hold for other trial states (such as the Read-Rezayi series). We
argue that trial states based on a non-unitary conformal field theory do not
describe a gapped topological phase, at least in many cases. By considering
adiabatic variation of the aspect ratio of the torus, we calculate the Hall
viscosity, a non-dissipative viscosity coefficient analogous to Hall
conductivity, for paired states, Laughlin states, and more general quantum Hall
states. Hall viscosity is an invariant within a topological phase, and is
generally proportional to the "conformal spin density" in the ground state.Comment: 44 pages, RevTeX; v2 minor changes; v3 typos corrected, three small
addition
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