5,522 research outputs found
A fast and well-conditioned spectral method for singular integral equations
We develop a spectral method for solving univariate singular integral
equations over unions of intervals by utilizing Chebyshev and ultraspherical
polynomials to reformulate the equations as almost-banded infinite-dimensional
systems. This is accomplished by utilizing low rank approximations for sparse
representations of the bivariate kernels. The resulting system can be solved in
operations using an adaptive QR factorization, where is
the bandwidth and is the optimal number of unknowns needed to resolve the
true solution. The complexity is reduced to operations by
pre-caching the QR factorization when the same operator is used for multiple
right-hand sides. Stability is proved by showing that the resulting linear
operator can be diagonally preconditioned to be a compact perturbation of the
identity. Applications considered include the Faraday cage, and acoustic
scattering for the Helmholtz and gravity Helmholtz equations, including
spectrally accurate numerical evaluation of the far- and near-field solution.
The Julia software package SingularIntegralEquations.jl implements our method
with a convenient, user-friendly interface
The paradigm of the area law and the structure of transversal and longitudinal lightfront degrees of freedom
It is shown that an algebraically defined holographic projection of a QFT
onto the lightfront changes the local quantum properties in a very drastic way.
The expected ubiquitous vacuum polarization characteristic of QFT is confined
to the lightray (longitudinal) direction, whereas operators whose localization
is transversely separated are completely free of vacuum correlations. This
unexpected ''transverse return to QM'' combined with the rather universal
nature of the strongly longitudinal correlated vacuum correlations (which turn
out to be described by rather kinematical chiral theories) leads to a d-2
dimensional area structure of the d-1 dimensional lightfront theory. An
additive transcription in terms of an appropriately defined entropy related to
the vacuum restricted to the horizon is proposed and its model independent
universality aspects which permit its interpretation as a quantum candidate for
Bekenstein's area law are discussed. The transverse tensor product foliation
structure of lightfront degrees of freedom is essential for the simplifying
aspects of the algebraic lightcone holography. Key-words: Quantum field theory;
Mathematical physics, Quantum gravityComment: 16 pages latex, identical to version published in JPA: Math. Gen. 35
(2002) 9165-918
On Rigidity of 3d Asymptotic Symmetry Algebras
We study rigidity and stability of infinite dimensional algebras which are
not subject to the Hochschild-Serre factorization theorem. In particular, we
consider algebras appearing as asymptotic symmetries of three dimensional
spacetimes, the BMS3, u(1) Kac-Moody and Virasoro algebras. We construct and
classify the family of algebras which appear as deformations of BMS3, u(1)
Kac-Moody and their central extensions by direct computations and also by
cohomological analysis. The Virasoro algebra appears as a specific member in
this family of rigid algebras; for this case stabilization procedure is inverse
of the In\"on\"u-Wigner contraction relating Virasoro to BMS3 algebra. We
comment on the physical meaning of deformation and stabilization of these
algebras and relevance of the family of rigid algebras we obtainComment: 50 pages, one figure and two tables; v2: minor improvements,
references adde
Exact stabilization of entangled states in finite time by dissipative quantum circuits
Open quantum systems evolving according to discrete-time dynamics are
capable, unlike continuous-time counterparts, to converge to a stable
equilibrium in finite time with zero error. We consider dissipative quantum
circuits consisting of sequences of quantum channels subject to specified
quasi-locality constraints, and determine conditions under which stabilization
of a pure multipartite entangled state of interest may be exactly achieved in
finite time. Special emphasis is devoted to characterizing scenarios where
finite-time stabilization may be achieved robustly with respect to the order of
the applied quantum maps, as suitable for unsupervised control architectures.
We show that if a decomposition of the physical Hilbert space into virtual
subsystems is found, which is compatible with the locality constraint and
relative to which the target state factorizes, then robust stabilization may be
achieved by independently cooling each component. We further show that if the
same condition holds for a scalable class of pure states, a continuous-time
quasi-local Markov semigroup ensuring rapid mixing can be obtained. Somewhat
surprisingly, we find that the commutativity of the canonical parent
Hamiltonian one may associate to the target state does not directly relate to
its finite-time stabilizability properties, although in all cases where we can
guarantee robust stabilization, a (possibly non-canonical) commuting parent
Hamiltonian may be found. Beside graph states, quantum states amenable to
finite-time robust stabilization include a class of universal resource states
displaying two-dimensional symmetry-protected topological order, along with
tensor network states obtained by generalizing a construction due to Bravyi and
Vyalyi. Extensions to representative classes of mixed graph-product and thermal
states are also discussed.Comment: 20 + 9 pages, 9 figure
Advances in R-matrices and their applications (after Maulik-Okounkov, Kang-Kashiwara-Kim-Oh,...)
R-matrices are the solutions of the Yang-Baxter equation. At the origin of
the quantum group theory, they may be interpreted as intertwining operators.
Recent advances have been made independently in different directions.
Maulik-Okounkov have given a geometric approach to R-matrices with new tools in
symplectic geometry, the stable envelopes. Kang-Kashiwara-Kim-Oh proved a
conjecture on the categorification of cluster algebras by using R-matrices in a
crucial way. Eventually, a better understanding of the action of
transfer-matrices obtained from R-matrices led to the proof of several
conjectures about the corresponding quantum integrable systems.Comment: This is an English translation of the Bourbaki seminar 1129 (March
2017). The French version will appear in Ast\'erisqu
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