206 research outputs found
Far-from-constant mean curvature solutions of Einstein's constraint equations with positive Yamabe metrics
In this article we develop some new existence results for the Einstein
constraint equations using the Lichnerowicz-York conformal rescaling method.
The mean extrinsic curvature is taken to be an arbitrary smooth function
without restrictions on the size of its spatial derivatives, so that it can be
arbitrarily far from constant. The rescaled background metric belongs to the
positive Yamabe class, and the freely specifiable part of the data given by the
traceless-transverse part of the rescaled extrinsic curvature and the matter
fields are taken to be sufficiently small, with the matter energy density not
identically zero. Using topological fixed-point arguments and global barrier
constructions, we then establish existence of solutions to the constraints. Two
recent advances in the analysis of the Einstein constraint equations make this
result possible: A new type of topological fixed-point argument without
smallness conditions on spatial derivatives of the mean extrinsic curvature,
and a new construction of global super-solutions for the Hamiltonian constraint
that is similarly free of such conditions on the mean extrinsic curvature. For
clarity, we present our results only for strong solutions on closed manifolds.
However, our results also hold for weak solutions and for other cases such as
compact manifolds with boundary; these generalizations will appear elsewhere.
The existence results presented here for the Einstein constraints are
apparently the first such results that do not require smallness conditions on
spatial derivatives of the mean extrinsic curvature.Comment: 4 pages, no figures, accepted for publication in Physical Review
Letters. (Abstract shortenned and other minor changes reflecting v4 version
of arXiv:0712.0798
Convergence rates in expectation for Tikhonov-type regularization of Inverse Problems with Poisson data
In this paper we study a Tikhonov-type method for ill-posed nonlinear
operator equations \gdag = F(
ag) where \gdag is an integrable,
non-negative function. We assume that data are drawn from a Poisson process
with density t\gdag where may be interpreted as an exposure time. Such
problems occur in many photonic imaging applications including positron
emission tomography, confocal fluorescence microscopy, astronomic observations,
and phase retrieval problems in optics. Our approach uses a
Kullback-Leibler-type data fidelity functional and allows for general convex
penalty terms. We prove convergence rates of the expectation of the
reconstruction error under a variational source condition as both
for an a priori and for a Lepski{\u\i}-type parameter choice rule
On Bogovski\u{\i} and regularized Poincar\'e integral operators for de Rham complexes on Lipschitz domains
We study integral operators related to a regularized version of the classical
Poincar\'e path integral and the adjoint class generalizing Bogovski\u{\i}'s
integral operator, acting on differential forms in . We prove that these
operators are pseudodifferential operators of order -1. The Poincar\'e-type
operators map polynomials to polynomials and can have applications in finite
element analysis. For a domain starlike with respect to a ball, the special
support properties of the operators imply regularity for the de Rham complex
without boundary conditions (using Poincar\'e-type operators) and with full
Dirichlet boundary conditions (using Bogovski\u{\i}-type operators). For
bounded Lipschitz domains, the same regularity results hold, and in addition we
show that the cohomology spaces can always be represented by
functions.Comment: 23 page
Relativistic Elastostatics I: Bodies in Rigid Rotation
We consider elastic bodies in rigid rotation, both nonrelativistically and in
special relativity. Assuming a body to be in its natural state in the absence
of rotation, we prove the existence of solutions to the elastic field equations
for small angular velocity.Comment: 25 page
Strategies for enzymological studies and measurements of biological molecules with the cytolysin A nanopore
Pore-forming toxins are used in a variety of biotechnological applications. Typically, individual membrane proteins are reconstituted in artificial lipid bilayers where they form water-filled nanoscale apertures (nanopores). When a voltage is applied, the ionic current passing through a nanopore can be used for example to sequence biopolymers, identify molecules, or to study chemical or enzymatic reactions at the single-molecule level. Here we present strategies for studying individual enzymes and measuring molecules, also in highly complex biological samples such as blood
Математична модель контактного з’єднання метало-пластмасових циліндричних оболонок
We consider alpha scale spaces, a parameterized class (alpha is an element of (0, 1]) of scale space representations beyond the well-established Gaussian scale space, which are generated by the alpha-th power of the minus Laplace operator on a bounded domain using the Neumann boundary condition. The Neumann boundary condition ensures that there is no grey-value flux through the boundary. Thereby no artificial grey-values from outside the image affect the evolution proces, which is the case for the alpha scale spaces on an unbounded domain. Moreover, the connection between the a scale spaces which is not trivial in the unbounded domain case, becomes straightforward: The generator of the Gaussian semigroup extends to a compact, self-adjoint operator on the Hilbert space L-2(Omega) and therefore it has a complete countable set of eigen functions. Taking the alpha-th power of the Gaussian generator simply boils down to taking the alpha-th power of the corresponding eigenvalues. Consequently, all alpha scale spaces have exactly the same eigen-modes and can be implemented simultaneously as scale dependent Fourier series. The only difference between them is the (relative) contribution of each eigen-mode to the evolution proces. By introducing the notion of (non-dimensional) relative scale in each a scale space, we are able to compare the various alpha scale spaces. The case alpha = 0.5, where the generator equals the square root of the minus Laplace operator leads to Poisson scale space, which is at least as interesting as Gaussian scale space and can be extended to a (Clifford) analytic scale space
N-representability and stationarity in time-dependent density functional theory
To construct an N-representable time-dependent density-functional theory, a
generalization to the time domain of the Levy-Lieb (LL) constrained search
algorithm is required. That the action is only stationary in the Dirac-Frenkel
variational principle eliminates the possibility of basing the search on the
action itself. Instead, we use the norm of the partial functional derivative of
the action in the Hilbert space of the wave functions in place of the energy of
the LL search. The electron densities entering the formalism are
-representable, and the resulting universal action functional has a unique
stationary point in the density at that corresponding to the solution of the
Schr\"{o}dinger equation. The original Runge-Gross (RG) formulation is subsumed
within the new formalism. Concerns in the literature about the meaning of the
functional derivatives and the internal consistency of the RG formulation are
allayed by clarifying the nature of the functional derivatives entering the
formalism.Comment: 9 pages, 0 figures, Phys. Rev. A accepted. Introduction was expanded,
subsections reorganized, appendix and new references adde
Schrödinger operators with δ and δ′-potentials supported on hypersurfaces
Self-adjoint Schrödinger operators with δ and δ′-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman–Schwinger principle and a variant of Krein’s formula are shown. Furthermore, Schatten–von Neumann type estimates for the differences of the powers of the resolvents of the Schrödinger operators with δ and δ′-potentials, and the Schrödinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schrödinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity
Model validation for a noninvasive arterial stenosis detection problem
Copyright @ 2013 American Institute of Mathematical SciencesA current thrust in medical research is the development of a non-invasive method for detection, localization, and characterization of an arterial stenosis (a blockage or partial blockage in an artery). A method has been proposed to detect shear waves in the chest cavity which have been generated by disturbances in the blood flow resulting from a stenosis. In order to develop this methodology further, we use both one-dimensional pressure and shear wave experimental data from novel acoustic phantoms to validate corresponding viscoelastic mathematical models, which were developed in a concept paper [8] and refined herein. We estimate model parameters which give a good fit (in a sense to be precisely defined) to the experimental data, and use asymptotic error theory to provide confidence intervals for parameter estimates. Finally, since a robust error model is necessary for accurate parameter estimates and confidence analysis, we include a comparison of absolute and relative models for measurement error.The National Institute of Allergy and Infectious Diseases, the Air Force Office of Scientific Research, the Deopartment of Education and the Engineering and Physical Sciences Research Council (EPSRC)
Recruitment of the mitotic exit network to yeast centrosomes couples septin displacement to actomyosin constriction
The Mitotic Exit Network (MEN) promotes mitotic exit and cytokinesis but if and how MEN independently controls these two processes is unclear. Here, the authors report that MEN displaces septins from the cell division site to promote actomyosin ring constriction, independently of MEN control of mitotic exit
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