149 research outputs found
Evolution equations of curvature tensors along the hyperbolic geometric flow
We consider the hyperbolic geometric flow introduced by Kong and Liu [KL]. When the Riemannian
metric evolve, then so does its curvature. Using the techniques and ideas of
S.Brendle [Br,BS], we derive evolution equations for the Levi-Civita connection
and the curvature tensors along the hyperbolic geometric flow. The method and
results are computed and written in global tensor form, different from the
local normal coordinate method in [DKL1]. In addition, we further show that any
solution to the hyperbolic geometric flow that develops a singularity in finite
time has unbounded Ricci curvature.Comment: 15 page
Static flow on complete noncompact manifolds I: short-time existence and asymptotic expansions at conformal infinity
In this paper, we study short-time existence of static flow on complete
noncompact asymptotically static manifolds from the point of view that the
stationary points of the evolution equations can be interpreted as static
solutions of the Einstein vacuum equations with negative cosmological constant.
For a static vacuum we also compute the asymptotic expansions of
and at conformal infinity.Comment: 25 page
Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t = -1
We prove that the hyperelliptic Torelli group is generated by Dehn twists about
separating curves that are preserved by the hyperelliptic involution. This verifies a
conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel
of the Burau representation evaluated at t = â1 and also the fundamental group of
the branch locus of the period mapping, and so we obtain analogous generating sets
for those. One application is that each component in Torelli space of the locus of
hyperelliptic curves becomes simply connected when curves of compact type are added
Quantum geometry and quantum algorithms
Motivated by algorithmic problems arising in quantum field theories whose
dynamical variables are geometric in nature, we provide a quantum algorithm
that efficiently approximates the colored Jones polynomial. The construction is
based on the complete solution of Chern-Simons topological quantum field theory
and its connection to Wess-Zumino-Witten conformal field theory. The colored
Jones polynomial is expressed as the expectation value of the evolution of the
q-deformed spin-network quantum automaton. A quantum circuit is constructed
capable of simulating the automaton and hence of computing such expectation
value. The latter is efficiently approximated using a standard sampling
procedure in quantum computation.Comment: Submitted to J. Phys. A: Math-Gen, for the special issue ``The
Quantum Universe'' in honor of G. C. Ghirard
Some new directions in infinite-combinatorial topology
We give a light introduction to selection principles in topology, a young
subfield of infinite-combinatorial topology. Emphasis is put on the modern
approach to the problems it deals with. Recent results are described, and open
problems are stated. Some results which do not appear elsewhere are also
included, with proofs.Comment: Small update
Blow-up solutions for linear perturbations of the Yamabe equation
For a smooth, compact Riemannian manifold (M,g) of dimension N \geg 3, we
are interested in the critical equation where \Delta_g is the Laplace--Beltrami
operator, S_g is the Scalar curvature of (M,g), , and
is a small parameter
Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres
We study existence and non-existence of constant scalar curvature metrics
conformal and arbitrarily close to homogeneous metrics on spheres, using
variational techniques. This describes all critical points of the
Hilbert-Einstein functional on such conformal classes, near homogeneous
metrics. Both bifurcation and local rigidity type phenomena are obtained for
1-parameter families of U(n+1), Sp(n+1) and Spin(9)-homogeneous metrics.Comment: LaTeX2e, 18 pages, 1 figure, revised version. To appear in Calc. Var.
and PDE
Understanding mixed sequence DNA recognition by novel designed compounds: the kinetic and thermodynamic behavior of azabenzimidazole diamidines
Sequence-specific recognition of DNA by small organic molecules offers a potentially effective approach for the external regulation of gene expression and is an important goal in cell biochemistry. Rational design of compounds from established modules can potentially yield compounds that bind strongly and selectively with specific DNA sequences. An initial approach is to start with common A·T bp recognition molecules and build in G·C recognition units. Here we report on the DNA interaction of a synthetic compound that specifically binds to a G·C bp in the minor groove of DNA by using an azabenzimidazole moiety. The detailed interactions were evaluated with biosensor-surface plasmon resonance (SPR), isothermal calorimetric (ITC), and mass spectrometry (ESI-MS) methods. The compound, DB2277, binds with single G·C bp containing sequences with subnanomolar potency and displays slow dissociation kinetics and high selectivity. A detailed thermodynamic and kinetic study at different experimental salt concentrations and temperatures shows that the binding free energy is salt concentration dependent but essentially temperature independent under our experimental conditions, and binding enthalpy is temperature dependent but salt concentration independent. The results show that in the proper compound structural context novel heterocyclic cations can be designed to strongly recognize complex DNA sequences
Asymptotic behavior of solutions to the -Yamabe equation near isolated singularities
-Yamabe equations are conformally invariant equations generalizing
the classical Yamabe equation. In an earlier work YanYan Li proved that an
admissible solution with an isolated singularity at to the
-Yamabe equation is asymptotically radially symmetric. In this work
we prove that an admissible solution with an isolated singularity at to the -Yamabe equation is asymptotic to a radial
solution to the same equation on . These results
generalize earlier pioneering work in this direction on the classical Yamabe
equation by Caffarelli, Gidas, and Spruck. In extending the work of Caffarelli
et al, we formulate and prove a general asymptotic approximation result for
solutions to certain ODEs which include the case for scalar curvature and
curvature cases. An alternative proof is also provided using
analysis of the linearized operators at the radial solutions, along the lines
of approach in a work by Korevaar, Mazzeo, Pacard, and Schoen.Comment: 55 page
The subseries number
Every conditionally convergent series of real numbers has a divergent subseries. How many subsets of the natural numbers are needed so that every conditionally convergent series diverges on the subseries corresponding to one of these sets? The answer to this question is defined to be the subseries number, a new cardinal characteristic of the continuum. This cardinal is bounded below by N1 and above by the cardinality of the continuum, but it is not provably equal to either. We define three natural variants of the subseries number, and compare them with each other, with their corresponding rearrangement numbers, and with several well-studied cardinal characteristics of the continuum. Many consistency results are obtained from these comparisons, and we obtain another by computing the value of the subseries number in the Laver model
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