Search results for "Length"

Extremal length and Hölder continuity of conformal mappings

1986

Uncommon Suffix Tries

2011

Common assumptions on the source producing the words inserted in a suffix trie with $n$ leaves lead to a $\log n$ height and saturation level. We provide an example of a suffix trie whose height increases faster than a power of $n$ and another one whose saturation level is negligible with respect to $\log n$. Both are built from VLMC (Variable Length Markov Chain) probabilistic sources; they are easily extended to families of sources having the same properties. The first example corresponds to a ''logarithmic infinite comb'' and enjoys a non uniform polynomial mixing. The second one corresponds to a ''factorial infinite comb'' for which mixing is uniform and exponential.

Mahonian STAT on rearrangement class of words

2017

In 2000, Babson and Steingr\'{i}msson generalized the notion of permutation patterns to the so-called vincular patterns, and they showed that many Mahonian statistics can be expressed as sums of vincular pattern occurrence statistics. STAT is one of such Mahonian statistics discoverd by them. In 2016, Kitaev and the third author introduced a words analogue of STAT and proved a joint equidistribution result involving two sextuple statistics on the whole set of words with fixed length and alphabet. Moreover, their computer experiments hinted at a finer involution on $R(w)$, the rearrangement class of a given word $w$. We construct such an involution in this paper, which yields a comparable jo…

String attractors and combinatorics on words

2019

The notion of \emph{string attractor} has recently been introduced in [Prezza, 2017] and studied in [Kempa and Prezza, 2018] to provide a unifying framework for known dictionary-based compressors. A string attractor for a word $w=w[1]w[2]\cdots w[n]$ is a subset $\Gamma$ of the positions $\{1,\ldots,n\}$, such that all distinct factors of $w$ have an occurrence crossing at least one of the elements of $\Gamma$. While finding the smallest string attractor for a word is a NP-complete problem, it has been proved in [Kempa and Prezza, 2018] that dictionary compressors can be interpreted as algorithms approximating the smallest string attractor for a given word. In this paper we explore the noti…

Binary jumbled string matching for highly run-length compressible texts

2012

The Binary Jumbled String Matching problem is defined as: Given a string $s$ over $\{a,b\}$ of length $n$ and a query $(x,y)$, with $x,y$ non-negative integers, decide whether $s$ has a substring $t$ with exactly $x$ $a$'s and $y$ $b$'s. Previous solutions created an index of size O(n) in a pre-processing step, which was then used to answer queries in constant time. The fastest algorithms for construction of this index have running time $O(n^2/\log n)$ [Burcsi et al., FUN 2010; Moosa and Rahman, IPL 2010], or $O(n^2/\log^2 n)$ in the word-RAM model [Moosa and Rahman, JDA 2012]. We propose an index constructed directly from the run-length encoding of $s$. The construction time of our index i…

Carbon Monoxide in the Cold Debris of Supernova 1987A

2013

We report spectroscopic and imaging observations of rotational transitions of cold CO and SiO in the ejecta of SN1987A, the first such emission detected in a supernova remnant. In addition to line luminosities for the CO J=1-0, 2-1, 6-5, and 7-6 transitions, we present upper limits for all other transitions up to J=13-12, collectively measured from the Atacama Large Millimeter Array (ALMA), the Atacama Pathfinder EXperiment (APEX), and the Herschel Spectral and Photometric Imaging REceiver (SPIRE). Simple models show the lines are emitted from at least 0.01 solar masses of CO at a temperature > 14 K, confined within at most 35% of a spherical volume expanding at ~ 2000 km/s. Moreover, we…

Faraday patterns in bose-Einstein condensates.

2002

Temporal periodic modulation of the interatomic s-wave scattering length in Bose-Einstein condensates is shown to excite subharmonic patterns of atom density through a parametric resonance. The dominating wavelength of the spatial structures is shown to be primarily selected by the excitation frequency but also affected by the depth of the spatial modulation via a nonlinear resonance. These phenomena represent macroscopic quantum analogues of the Faraday waves excited in vertically shaken liquids.

Observation of Geometric Parametric Instability Induced by the Periodic Spatial Self-Imaging of Multimode Waves

2016

Spatio-temporal mode coupling in highly multimode physical systems permits new routes for exploring complex instabilities and forming coherent wave structures. We present here the first experimental demonstration of multiple geometric parametric instability sidebands, generated in the frequency domain through resonant space-time coupling, owing to the natural periodic spatial self-imaging of a multimode quasi-continuous-wave beam in a standard graded-index multimode fiber. The input beam was launched in the fiber by means of an amplified microchip laser emitting sub-nanosecond pulses at 1064 nm. The experimentally observed frequency spacing among sidebands agrees well with analytical predic…

Molecular equilibrium structures from experimental rotational constants and calculated vibration–rotation interaction constants

2002

A detailed study is carried out of the accuracy of molecular equilibrium geometries obtained from least-squares fits involving experimental rotational constants B(0) and sums of ab initio vibration-rotation interaction constants alpha(r)(B). The vibration-rotation interaction constants have been calculated for 18 single-configuration dominated molecules containing hydrogen and first-row atoms at various standard levels of ab initio theory. Comparisons with the experimental data and tests for the internal consistency of the calculations show that the equilibrium structures generated using Hartree-Fock vibration-rotation interaction constants have an accuracy similar to that obtained by a dir…

Fabrication of polarizing photonic crystal fibres and photonic crystal fibre tapers: Applications

2007

We report the fabrication of an anisotropic photonic crystal fibre with polarization properties and photonic crystal fibre tapers for supercontinuum generation. The anisotropy of the fibre was created by enlarging four airholes next to the silica core. Different polarization regimes as a function of the geometric parameters, including polarizing behaviour at 1.55 mum, were obtained. In the second part of the paper, we report the fabrication of photonic crystal fibre tapers. We present experimental results on supercontinuum generation in photonic crystal fibre tapers using quasi-continuous pump pulses of 7 ns duration at 532 nm and at 1064 nm.