342 research outputs found
Rank of Stably Dissipative Graphs
For the class of stably dissipative Lotka-Volterra systems we prove that the
rank of its defining matrix, which is the dimension of the associated invariant
foliation, is completely determined by the system's graph
Hardy-type Inequalities Via Auxiliary Sequences
We prove some Hardy-type inequalities via an approach that involves
constructing auxiliary sequences.Comment: 10 page
Convexity of the zeros of some orthogonal polynomials and related functions
We study convexity properties of the zeros of some special functions that
follow from the convexity theorem of Sturm. We prove results on the intervals
of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials,
as well as functions related to them, using transformations under which the
zeros remain unchanged. We give upper as well as lower bounds for the distance
between consecutive zeros in several cases
Optimal quasi-free approximation:reconstructing the spectrum from ground state energies
The sequence of ground state energy density at finite size, e_{L}, provides
much more information than usually believed. Having at disposal e_{L} for short
lattice sizes, we show how to re-construct an approximate quasi-particle
dispersion for any interacting model. The accuracy of this method relies on the
best possible quasi-free approximation to the model, consistent with the
observed values of the energy e_{L}. We also provide a simple criterion to
assess whether such a quasi-free approximation is valid. As a side effect, our
method is able to assess whether the nature of the quasi-particles is fermionic
or bosonic together with the effective boundary conditions of the model. When
applied to the spin-1/2 Heisenberg model, the method produces a band of Fermi
quasi-particles very close to the exact one of des Cloizeaux and Pearson. The
method is further tested on a spin-1/2 Heisenberg model with explicit
dimerization and on a spin-1 chain with single ion anisotropy. A connection
with the Riemann Hypothesis is also pointed out.Comment: 9 pages, 5 figures. One figure added showing convergence spee
Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time
Given a control region on a compact Riemannian manifold , we
consider the heat equation with a source term localized in . It is
known that any initial data in can be stirred to 0 in an arbitrarily
small time by applying a suitable control in , and,
as tends to 0, the norm of grows like times the norm of the
data. We investigate how depends on the geometry of . We prove
where is the largest distance of a point in from
. When is a segment of length controlled at one end, we prove
for some . Moreover, this bound implies where is the length of the longest generalized
geodesic in which does not intersect . The control transmutation
method used in proving this last result is of a broader interest.Comment: 26 pages, uses elsart.sty, typos and section 5.3 correcte
Grassmann-Gaussian integrals and generalized star products
In quantum scattering on networks there is a non-linear composition rule for
on-shell scattering matrices which serves as a replacement for the
multiplicative rule of transfer matrices valid in other physical contexts. In
this article, we show how this composition rule is obtained using Berezin
integration theory with Grassmann variables.Comment: 14 pages, 2 figures. In memory of Al.B. Zamolodichiko
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