9,042 research outputs found
Solutions of special asymptotics to the Einstein constraint equations
We construct solutions with prescribed asymptotics to the Einstein constraint
equations using a cut-off technique. Moreover, we give various examples of
vacuum asymptotically flat manifolds whose center of mass and angular momentum
are ill-defined.Comment: 13 pages; the error in Lemma 3.5 fixed and typos corrected; to appear
in Class. Quantum Gra
Relative Periodic Solutions of the Complex Ginzburg-Landau Equation
A method of finding relative periodic orbits for differential equations with
continuous symmetries is described and its utility demonstrated by computing
relative periodic solutions for the one-dimensional complex Ginzburg-Landau
equation (CGLE) with periodic boundary conditions. A relative periodic solution
is a solution that is periodic in time, up to a transformation by an element of
the equation's symmetry group. With the method used, relative periodic
solutions are represented by a space-time Fourier series modified to include
the symmetry group element and are sought as solutions to a system of nonlinear
algebraic equations for the Fourier coefficients, group element, and time
period. The 77 relative periodic solutions found for the CGLE exhibit a wide
variety of temporal dynamics, with the sum of their positive Lyapunov exponents
varying from 5.19 to 60.35 and their unstable dimensions from 3 to 8.
Preliminary work indicates that weighted averages over the collection of
relative periodic solutions accurately approximate the value of several
functionals on typical trajectories.Comment: 32 pages, 12 figure
SERS biosensors based on cucurbituril-mediated nanoaggregates for wastewater-based epidemiology
Hierarchical self-assembly of nanoparticles (NPs) mediated by macrocyclic molecules, cucurbiturils (CBs), provides a facile method to fabricate surface-enhanced Raman spectroscopy (SERS) sensors for potential applications in biosensing and environmental monitoring. In contrast to conventional techniques for wastewater-based epidemiology (WBE), CB-NP SERS sensors offer great opportunities for on-site quantification of trace chemical and biological markers due to its high sensitivity, selectivity, reproducibility, multiplexing capability and tolerance against contamination. The working principles of the CB-Au NP nanocomposites including fabrication, sensing mechanisms and structure-property relationships are explained while the design guidelines and selected examples of CB-Au NP SERS sensors are discussed. The review concludes by highlighting recent advances in this area and exploring opportunities in the context of WBE
Specifying angular momentum and center of mass for vacuum initial data sets
We show that it is possible to perturb arbitrary vacuum asymptotically flat
spacetimes to new ones having exactly the same energy and linear momentum, but
with center of mass and angular momentum equal to any preassigned values
measured with respect to a fixed affine frame at infinity. This is in contrast
to the axisymmetric situation where a bound on the angular momentum by the mass
has been shown to hold for black hole solutions. Our construction involves
changing the solution at the linear level in a shell near infinity, and
perturbing to impose the vacuum constraint equations. The procedure involves
the perturbation correction of an approximate solution which is given
explicitly.Comment: (v2) a minor change in the introduction and a remark added after
Theorem 2.1; (v3) final version, appeared in Comm. Math. Phy
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