127 research outputs found
Topological calculation of the phase of the determinant of a non self-adjoint elliptic operator
We study the zeta-regularized determinant of a non self-adjoint elliptic
operator on a closed odd-dimensional manifold. We show that, if the spectrum of
the operator is symmetric with respect to the imaginary axis, then the
determinant is real and its sign is determined by the parity of the number of
the eigenvalues of the operator, which lie on the positive part of the
imaginary axis. It follows that, for many geometrically defined operators, the
phase of the determinant is a topological invariant. In numerous examples,
coming from geometry and physics, we calculate the phase of the determinants in
purely topological terms. Some of those examples were known in physical
literature, but no mathematically rigorous proofs and no general theory were
available until now.Comment: To appear in Communications of Mathematical Physic
Bounds on changes in Ritz values for a perturbed invariant subspace of a Hermitian matrix
The Rayleigh-Ritz method is widely used for eigenvalue approximation. Given a
matrix with columns that form an orthonormal basis for a subspace \X, and
a Hermitian matrix , the eigenvalues of are called Ritz values of
with respect to \X. If the subspace \X is -invariant then the Ritz
values are some of the eigenvalues of . If the -invariant subspace \X
is perturbed to give rise to another subspace \Y, then the vector of absolute
values of changes in Ritz values of represents the absolute eigenvalue
approximation error using \Y. We bound the error in terms of principal angles
between \X and \Y. We capitalize on ideas from a recent paper [DOI:
10.1137/060649070] by A. Knyazev and M. Argentati, where the vector of absolute
values of differences between Ritz values for subspaces \X and \Y was
weakly (sub-)majorized by a constant times the sine of the vector of principal
angles between \X and \Y, the constant being the spread of the spectrum of
. In that result no assumption was made on either subspace being
-invariant. It was conjectured there that if one of the trial subspaces is
-invariant then an analogous weak majorization bound should only involve
terms of the order of sine squared. Here we confirm this conjecture.
Specifically we prove that the absolute eigenvalue error is weakly majorized by
a constant times the sine squared of the vector of principal angles between the
subspaces \X and \Y, where the constant is proportional to the spread of
the spectrum of . For many practical cases we show that the proportionality
factor is simply one, and that this bound is sharp. For the general case we can
only prove the result with a slightly larger constant, which we believe is
artificial.Comment: 12 pages. Accepted to SIAM Journal on Matrix Analysis and
Applications (SIMAX
Entanglement of Collectively Interacting Harmonic Chains
We study the ground-state entanglement of one-dimensional harmonic chains
that are coupled to each other by a collective interaction as realized e.g. in
an anisotropic ion crystal. Due to the collective type of coupling, where each
chain interacts with every other one in the same way,the total system shows
critical behavior in the direction orthogonal to the chains while the isolated
harmonic chains can be gapped and non-critical. We derive lower and most
importantly upper bounds for the entanglement,quantified by the von Neumann
entropy, between a compact block of oscillators and its environment. For
sufficiently large size of the subsystems the bounds coincide and show that the
area law for entanglement is violated by a logarithmic correction.Comment: 5 pages, 1 figur
On one mechanism of light ablation of nanostructures
The mechanism of mechanical ablation of nanoparticles during the interaction with a high-power laser radiation pulse is proposed. A particle is polarized under a laser electric field, and electric forces acting on field-induced oppositesign charges cause rupture stresses. Upon reaching the stresses exceeding the maximum allowable values for a given material, a nanoparticle decays into two ones. This effect can be used for narrowing the size distribution of nanoparticles produced by the laser ablation method
Eigenvalues of rank one perturbations of unstructured matrices
Let be a fixed complex matrix and let be two vectors. The
eigenvalues of matrices form a system
of intersecting curves. The dependence of the intersections on the vectors
is studied
Periodic solutions for completely resonant nonlinear wave equations
We consider the nonlinear string equation with Dirichlet boundary conditions
, with odd and analytic,
, and we construct small amplitude periodic solutions with frequency
\o for a large Lebesgue measure set of \o close to 1. This extends previous
results where only a zero-measure set of frequencies could be treated (the ones
for which no small divisors appear). The proof is based on combining the
Lyapunov-Schmidt decomposition, which leads to two separate sets of equations
dealing with the resonant and nonresonant Fourier components, respectively the
Q and the P equations, with resummation techniques of divergent powers series,
allowing us to control the small divisors problem. The main difficulty with
respect the nonlinear wave equations , ,
is that not only the P equation but also the Q equation is infinite-dimensiona
Quantum state transformations and the Schubert calculus
Recent developments in mathematics have provided powerful tools for comparing
the eigenvalues of matrices related to each other via a moment map. In this
paper we survey some of the more concrete aspects of the approach with a
particular focus on applications to quantum information theory. After
discussing the connection between Horn's Problem and Nielsen's Theorem, we move
on to characterizing the eigenvalues of the partial trace of a matrix.Comment: 40 pages. Accepted for publication in Annals of Physic
Different Modes of Combustion Wave on a Lattice Burner
The stabilization of a planar premixed flame front on a lattice (porous) burner is considered. The developed model captures all the important features of the phenomenon, while also admitting qualitative analytical investigation. It has been rigorously mathematically proven that there exist two different stabilization regimes: one with flame front located nearby the surface of the burner, and another with the flame front located inside the lattice. These two regimes result in qualitatively different gas temperature profiles along the flow that is monotonic and non-monotonic, respectively. The boundary between the two regimes is described in terms of dependence of the lattice solid material temperature on flow Peclet number. With similar temperature profiles, such dependencies may be both monotonic and non-monotonic. The transition between the two types of dependencies is controlled by the Arrhenius number. Conclusions of the study are supported by numerical analysis. They also compare favorably with the available experimental data. The novelty of the present approach is a fundamentally rigorous analytical analysis of the problem. The proposed analytical model, based on δ-function approximation of the chemical source term, agrees well (within 7% relative error) with the model based on the distributed description of the chemical reaction zone. The obtained results are important from both a theoretical and practical point of view. They demonstrate the existence of the two qualitatively different operating regimes for lattice burners, thus impacting design solutions for such devices. The results will be of great interest to the broader academic community, particularly in research areas where similar wave structures may emerge
The Lagrange Equilibrium Points L_4 and L_5 in a Black Hole Binary System
We calculate the location and stability of the L_4 and L_5 Lagrange
equilibrium points in the circular restricted three-body problem as the binary
system evolves via gravitational radiation losses. Relative to the purely
Newtonian case, we find that the L_4 equilibrium point moves towards the
secondary mass and becomes slightly less stable, while the L_5 point moves away
from the secondary and gains in stability. We discuss a number of astrophysical
applications of these results, in particular as a mechanism for producing
electromagnetic counterparts to gravitational-wave signals.Comment: 10 pages, 4 figures, submitted to ApJ; comments welcom
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