3,889 research outputs found
Well-rounded zeta-function of planar arithmetic lattices
We investigate the properties of the zeta-function of well-rounded
sublattices of a fixed arithmetic lattice in the plane. In particular, we show
that this function has abscissa of convergence at with a real pole of
order 2, improving upon a recent result of S. Kuehnlein. We use this result to
show that the number of well-rounded sublattices of a planar arithmetic lattice
of index less or equal is as . To obtain these
results, we produce a description of integral well-rounded sublattices of a
fixed planar integral well-rounded lattice and investigate convergence
properties of a zeta-function of similarity classes of such lattices, building
on some previous results of the author.Comment: 12 pages; to appear in PAM
On well-rounded sublattices of the hexagonal lattice
We produce an explicit parameterization of well-rounded sublattices of the
hexagonal lattice in the plane, splitting them into similarity classes. We use
this parameterization to study the number, the greatest minimal norm, and the
highest signal-to-noise ratio of well-rounded sublattices of the hexagonal
lattice of a fixed index. This investigation parallels earlier work by
Bernstein, Sloane, and Wright where similar questions were addressed on the
space of all sublattices of the hexagonal lattice. Our restriction is motivated
by the importance of well-rounded lattices for discrete optimization problems.
Finally, we also discuss the existence of a natural combinatorial structure on
the set of similarity classes of well-rounded sublattices of the hexagonal
lattice, induced by the action of a certain matrix monoid.Comment: 21 pages (minor correction to the proof of Lemma 2.1); to appear in
Discrete Mathematic
On similarity classes of well-rounded sublattices of
A lattice is called well-rounded if its minimal vectors span the
corresponding Euclidean space. In this paper we study the similarity classes of
well-rounded sublattices of . We relate the set of all such
similarity classes to a subset of primitive Pythagorean triples, and prove that
it has structure of a noncommutative infinitely generated monoid. We discuss
the structure of a given similarity class, and define a zeta function
corresponding to each similarity class. We relate it to Dedekind zeta of
, and investigate the growth of some related Dirichlet series,
which reflect on the distribution of well-rounded lattices. Finally, we
construct a sequence of similarity classes of well-rounded sublattices of
, which gives good circle packing density and converges to the
hexagonal lattice as fast as possible with respect to a natural metric we
define.Comment: 27 pages, 2 figures; added a lemma on Diophantine approximation by
quotients of Pythagorean triples; final version to be published in Journal of
Number Theor
On the difficulty of finding spines
We prove that the set of symplectic lattices in the Siegel space
whose systoles generate a subspace of dimension at least 3 in
does not contain any -equivariant
deformation retract of
On the number of perfect lattices
We show that the number of non-similar perfect -dimensional
lattices satisfies eventually the
inequalities for arbitrary
smallstrictly positive
Free and forced wave propagation in a Rayleigh-beam grid: flat bands, Dirac cones, and vibration localization vs isotropization
In-plane wave propagation in a periodic rectangular grid beam structure,
which includes rotational inertia (so-called 'Rayleigh beams'), is analyzed
both with a Floquet-Bloch exact formulation for free oscillations and with a
numerical treatment (developed with PML absorbing boundary conditions) for
forced vibrations (including Fourier representation and energy flux
evaluations), induced by a concentrated force or moment. A complex interplay is
observed between axial and flexural vibrations (not found in the common
idealization of out-of-plane motion), giving rise to several forms of vibration
localization: 'X-', 'cross-' and 'star-' shaped, and channel propagation. These
localizations are triggered by several factors, including rotational inertia
and slenderness of the beams and the type of forcing source (concentrated force
or moment). Although the considered grid of beams introduces an orthotropy in
the mechanical response, a surprising 'isotropization' of the vibration is
observed at special frequencies. Moreover, rotational inertia is shown to
'sharpen' degeneracies related to Dirac cones (which become more pronounced
when the aspect ratio of the grid is increased), while the slenderness can be
tuned to achieve a perfectly flat band in the dispersion diagram. The obtained
results can be exploited in the realization of metamaterials designed to
control wave propagation.Comment: 25 pages, 20 figure
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