16,555 research outputs found
Some Inequalities Related to the Seysen Measure of a Lattice
Given a lattice , a basis of together with its dual , the
orthogonality measure of was introduced
by M. Seysen in 1993. This measure is at the heart of the Seysen lattice
reduction algorithm and is linked with different geometrical properties of the
basis. In this paper, we explicit different expressions for this measure as
well as new inequalities.Comment: Typos correcte
Families of generalized Kloosterman sums
We construct p-adic relative cohomology for a family of toric exponential
sums which generalize the classical Kloosterman sums. Under natural hypotheses
such as quasi-homogeneity and nondegeneracy, this cohomology is acyclic except
in the top dimension. Our construction enables sufficiently sharp estimates for
the action of Frobenius on cohomology so that our earlier work may be applied
to the L-functions coming from linear algebra operations on these families to
deduce a number of basic properties.Comment: 36 pages, 4 figure
Far-off-resonant wave interaction in one-dimensional photonic crystals with quadratic nonlinearity
We extend a recently developed Hamiltonian formalism for nonlinear wave
interaction processes in spatially periodic dielectric structures to the
far-off-resonant regime, and investigate numerically the three-wave resonance
conditions in a one-dimensional optical medium with nonlinearity.
In particular, we demonstrate that the cascading of nonresonant wave
interaction processes generates an effective nonlinear response in
these systems. We obtain the corresponding coupling coefficients through
appropriate normal form transformations that formally lead to the Zakharov
equation for spatially periodic optical media.Comment: 14 pages, 4 figure
Mahler's work on the geometry of numbers
Mahler has written many papers on the geometry of numbers. Arguably, his most
influential achievements in this area are his compactness theorem for lattices,
his work on star bodies and their critical lattices, and his estimates for the
successive minima of reciprocal convex bodies and compound convex bodies. We
give a, by far not complete, overview of Mahler's work on these topics and
their impact.Comment: 17 pages. This paper will appear in "Mahler Selecta", a volume
dedicated to the work of Kurt Mahler and its impac
On the Proximity Factors of Lattice Reduction-Aided Decoding
Lattice reduction-aided decoding features reduced decoding complexity and
near-optimum performance in multi-input multi-output communications. In this
paper, a quantitative analysis of lattice reduction-aided decoding is
presented. To this aim, the proximity factors are defined to measure the
worst-case losses in distances relative to closest point search (in an infinite
lattice). Upper bounds on the proximity factors are derived, which are
functions of the dimension of the lattice alone. The study is then extended
to the dual-basis reduction. It is found that the bounds for dual basis
reduction may be smaller. Reasonably good bounds are derived in many cases. The
constant bounds on proximity factors not only imply the same diversity order in
fading channels, but also relate the error probabilities of (infinite) lattice
decoding and lattice reduction-aided decoding.Comment: remove redundant figure
Electronic structure of turbostratic graphene
We explore the rotational degree of freedom between graphene layers via the
simple prototype of the graphene twist bilayer, i.e., two layers rotated by
some angle . It is shown that, due to the weak interaction between
graphene layers, many features of this system can be understood by interference
conditions between the quantum states of the two layers, mathematically
expressed as Diophantine problems. Based on this general analysis we
demonstrate that while the Dirac cones from each layer are always effectively
degenerate, the Fermi velocity of the Dirac cones decreases as ; the form we derive for agrees with that found via a
continuum approximation in Phys. Rev. Lett., 99:256802, 2007. From tight
binding calculations for structures with we
find agreement with this formula for . In contrast, for
this formula breaks down and the Dirac bands become
strongly warped as the limit is approached. For an ideal system
of twisted layers the limit as is singular as for the Dirac point is fourfold degenerate, while at one has the
twofold degeneracy of the stacked bilayer. Interestingly, in this limit
the electronic properties are in an essential way determined \emph{globally},
in contrast to the 'nearsightedness' [W. Kohn. Phys. Rev. Lett., 76:3168,
1996.] of electronic structure generally found in condensed matter.Comment: Article as to be published in Phys. Rev B. Main changes: K-point
mapping tables fixed, several changes to presentation
Fourier-Space Crystallography as Group Cohomology
We reformulate Fourier-space crystallography in the language of cohomology of
groups. Once the problem is understood as a classification of linear functions
on the lattice, restricted by a particular group relation, and identified by
gauge transformation, the cohomological description becomes natural. We review
Fourier-space crystallography and group cohomology, quote the fact that
cohomology is dual to homology, and exhibit several results, previously
established for special cases or by intricate calculation, that fall
immediately out of the formalism. In particular, we prove that {\it two phase
functions are gauge equivalent if and only if they agree on all their
gauge-invariant integral linear combinations} and show how to find all these
linear combinations systematically.Comment: plain tex, 14 pages (replaced 5/8/01 to include archive preprint
number for reference 22
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