102,086 research outputs found
Improved bounds for the dimensions of planar distance sets
We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than 1, improving recent estimates of Keleti and Shmerkin, and of Liu in this regime. In particular, we prove that if dimH .A/ > 1, then the set of distances spanned by points of A has Hausdorff dimension at least 40=57 > 0:7 and there are many y 2 A such that the pinned distance set 1jx -yjW x 2 Aºhas Hausdorff dimension at least 29=42 and lower box-counting dimension at least 40=57. We use the approach and many results from the earlier work of Keleti and Shmerkin, but incorporate estimates from the recent work of Guth, Iosevich, Ou and Wang as additional input.Fil: Shmerkin, Pablo Sebastian. Consejo Nacional de Investigaciones CientÃficas y Técnicas; Argentina. Universidad Torcuato Di Tella. Departamento de Matemáticas y EstadÃstica; Argentina. University of British Columbia; Canad
Better Lower Bounds for Shortcut Sets and Additive Spanners via an Improved Alternation Product
We obtain improved lower bounds for additive spanners, additive emulators,
and diameter-reducing shortcut sets. Spanners and emulators are sparse graphs
that approximately preserve the distances of a given graph. A shortcut set is a
set of edges that when added to a directed graph, decreases its diameter. The
previous best known lower bounds for these three structures are given by Huang
and Pettie [SWAT 2018]. For -sized spanners, we improve the lower bound
on the additive stretch from to . For
-sized emulators, we improve the lower bound on the additive stretch from
to . For -sized shortcut sets, we
improve the lower bound on the graph diameter from to
. Our key technical contribution, which is the basis of all of
our bounds, is an improvement of a graph product known as an alternation
product.Comment: To appear in SODA 202
Improved Lower Bounds for Ginzburg-Landau Energies via Mass Displacement
We prove some improved estimates for the Ginzburg-Landau energy (with or
without magnetic field) in two dimensions, relating the asymptotic energy of an
arbitrary configuration to its vortices and their degrees, with possibly
unbounded numbers of vortices. The method is based on a localisation of the
``ball construction method" combined with a mass displacement idea which allows
to compensate for negative errors in the ball construction estimates by energy
``displaced" from close by.
Under good conditions, our main estimate allows to get a lower bound on the
energy which includes a finite order ``renormalized energy" of vortex
interaction, up to the best possible precision i.e. with only a error
per vortex, and is complemented by local compactness results on the vortices.
This is used crucially in a forthcoming paper relating minimizers of the
Ginzburg-Landau energy with the Abrikosov lattice. It can also serve to provide
lower bounds for weighted Ginzburg-Landau energies.Comment: 43 pages, to appear in "Analysis & PDE
Geometric lower bounds for generalized ranks
We revisit a geometric lower bound for Waring rank of polynomials (symmetric
rank of symmetric tensors) of Landsberg and Teitler and generalize it to a
lower bound for rank with respect to arbitrary varieties, improving the bound
given by the "non-Abelian" catalecticants recently introduced by Landsberg and
Ottaviani. This is applied to give lower bounds for ranks of multihomogeneous
polynomials (partially symmetric tensors); a special case is the simultaneous
Waring decomposition problem for a linear system of polynomials. We generalize
the classical Apolarity Lemma to multihomogeneous polynomials and give some
more general statements. Finally we revisit the lower bound of Ranestad and
Schreyer, and again generalize it to multihomogeneous polynomials and some more
general settings.Comment: 43 pages. v2: minor change
A Tight Lower Bound for Counting Hamiltonian Cycles via Matrix Rank
For even , the matchings connectivity matrix encodes which
pairs of perfect matchings on vertices form a single cycle. Cygan et al.
(STOC 2013) showed that the rank of over is
and used this to give an
time algorithm for counting Hamiltonian cycles modulo on graphs of
pathwidth . The same authors complemented their algorithm by an
essentially tight lower bound under the Strong Exponential Time Hypothesis
(SETH). This bound crucially relied on a large permutation submatrix within
, which enabled a "pattern propagation" commonly used in previous
related lower bounds, as initiated by Lokshtanov et al. (SODA 2011).
We present a new technique for a similar pattern propagation when only a
black-box lower bound on the asymptotic rank of is given; no
stronger structural insights such as the existence of large permutation
submatrices in are needed. Given appropriate rank bounds, our
technique yields lower bounds for counting Hamiltonian cycles (also modulo
fixed primes ) parameterized by pathwidth.
To apply this technique, we prove that the rank of over the
rationals is . We also show that the rank of
over is for any prime
and even for some primes.
As a consequence, we obtain that Hamiltonian cycles cannot be counted in time
for any unless SETH fails. This
bound is tight due to a time algorithm by Bodlaender et
al. (ICALP 2013). Under SETH, we also obtain that Hamiltonian cycles cannot be
counted modulo primes in time , indicating
that the modulus can affect the complexity in intricate ways.Comment: improved lower bounds modulo primes, improved figures, to appear in
SODA 201
Lower bounds for the spinless Salpeter equation
We obtain lower bounds on the ground state energy, in one and three
dimensions, for the spinless Salpeter equation (Schr\"odinger equation with a
relativistic kinetic energy operator) applicable to potentials for which the
attractive parts are in for some ( or 3). An extension
to confining potentials, which are not in , is also presented.Comment: 11 pages, 2 figures. Contribution to a special issue of Journal of
Nonlinear Mathematical Physics in honour of Francesco Calogero on the
occasion of his seventieth birthda
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