scholarly journals Covering n-Permutations with (n+1)-Permutations

10.37236/2168 ◽  
2013 ◽  
Vol 20 (1) ◽  
Author(s):  
Taylor F Allison ◽  
Anant P Godbole ◽  
Kathryn M Hawley ◽  
Bill Kay
Keyword(s):  
Phase Transition ◽  
Upper Bounds ◽  
Probability Of Coverage

Let $S_n$ be the set of all permutations on $[n]:=\{1,2,\ldots,n\}$. We denote by $\kappa_n$ the smallest cardinality of a subset ${\cal A}$ of $S_{n+1}$ that "covers" $S_n$, in the sense that each $\pi\in S_n$ may be found as an order-isomorphic subsequence of some $\pi'$ in ${\cal A}$.  What are general upper bounds on $\kappa_n$?  If we randomly select $\nu_n$ elements of $S_{n+1}$, when does the probability that they cover $S_n$ transition from 0 to 1?  Can we provide a fine-magnification analysis that provides the "probability of coverage"  when $\nu_n$ is around the level given by the phase transition?   In this paper we answer these questions and raise others.

scholarly journals Tail Bounds for the Stable Marriage of Poisson and Lebesgue

2009 ◽  
Vol 61 (6) ◽  
pp. 1279-1299 ◽  
Author(s):  
Christopher Hoffman ◽  
Alexander E. Holroyd ◽  
Yuval Peres
Keyword(s):  
Phase Transition ◽  
Lower Bounds ◽  
Power Law ◽  
Critical Case ◽  
Voronoi Tessellation ◽  
Upper Bounds ◽  
Stable Marriage ◽  
Marriage Problem ◽  
Stable Allocation ◽  
Typical Site

Abstract Let 𝚵 be a discrete set in ℝd. Call the elements of 𝚵 centers. The well-known Voronoi tessellation partitions ℝd into polyhedral regions (of varying volumes) by allocating each site of ℝd to the closest center. Here we study allocations of ℝd to 𝚵 in which each center attempts to claima region of equal volume α.We focus on the case where 𝚵 arises from a Poisson process of unit intensity. In an earlier paper by the authors it was proved that there is a unique allocation which is stable in the sense of the Gale–Shapley marriage problem. We study the distance X from a typical site to its allocated center in the stable allocation.The model exhibits a phase transition in the appetite α. In the critical case α = 1 we prove a power law upper bound on X in dimension d = 1. (Power law lower bounds were proved earlier for all d). In the non-critical cases α < 1 and α > 1 we prove exponential upper bounds on X.

scholarly journals An Analysis of Phase Transition in NK Landscapes

2002 ◽  
Vol 17 ◽  
pp. 309-332 ◽  
Author(s):  
Y. Gao ◽  
J. Culberson
Keyword(s):  
Phase Transition ◽  
Probability Model ◽  
Fixed Ratio ◽  
Upper Bounds ◽  
Ratio Model ◽  
Problem Size ◽  
Threshold Phenomena ◽  
Uniform Probability ◽  
Random Instances ◽  
Nk Landscapes

In this paper, we analyze the decision version of the NK landscape model from the perspective of threshold phenomena and phase transitions under two random distributions, the uniform probability model and the fixed ratio model. For the uniform probability model, we prove that the phase transition is easy in the sense that there is a polynomial algorithm that can solve a random instance of the problem with the probability asymptotic to 1 as the problem size tends to infinity. For the fixed ratio model, we establish several upper bounds for the solubility threshold, and prove that random instances with parameters above these upper bounds can be solved polynomially. This, together with our empirical study for random instances generated below and in the phase transition region, suggests that the phase transition of the fixed ratio model is also easy.

Microstructure of laser-irradiated, conducting Kapton polyimide

1995 ◽  
Vol 53 ◽  
pp. 502-503
Author(s):  
D. L. Callahan ◽  
Z. Ball ◽  
H. M. Phillips ◽  
R. Sauerbrey
Keyword(s):  
Electron Microscopy ◽  
Phase Transition ◽  
Transmission Electron Microscopy ◽  
Cross Section ◽  
Film Surface ◽  
Cross Sectional ◽  
General Utility ◽  
Transmission Electron ◽  
Considerable Debate ◽  
Potential Applications

Ultraviolet laser-irradiation can be used to induce an insulator-to-conductor phase transition on the surface of Kapton polyimide. Such structures have potential applications as resistors or conductors for VLSI applications as well as general utility electrodes. Although the percolative nature of the phase transformation has been well-established, there has been little definitive work on the mechanism or extent of transformation. In particular, there has been considerable debate about whether or not the transition is primarily photothermal in nature, as we propose, or photochemical. In this study, cross-sectional optical microscopy and transmission electron microscopy are utilized to characterize the nature of microstructural changes associated with the laser-induced pyrolysis of polyimide.Laser-modified polyimide samples initially 12 μm thick were prepared in cross-section by standard ultramicrotomy. Resulting contraction in parallel to the film surface has led to distortions in apparent magnification. The scale bars shown are calibrated for the direction normal to the film surface only.

The structure of membranes and membrane proteins embedded in amorphous ice

1992 ◽  
Vol 50 (1) ◽  
pp. 432-433
Author(s):  
Uwe Lücken ◽  
Joachim Jäger
Keyword(s):  
Phase Transition ◽  
Transition Temperature ◽  
Membrane Proteins ◽  
Phase Transition Temperature ◽  
Lipid Vesicles ◽  
Freezing Stress ◽  
Amorphous Ice ◽  
Different Temperatures ◽  
Band Pass ◽  
Lipid Polymorphism

TEM imaging of frozen-hydrated lipid vesicles has been done by several groups Thermotrophic and lyotrophic polymorphism has been reported. By using image processing, computer simulation and tilt experiments, we tried to learn about the influence of freezing-stress and defocus artifacts on the lipid polymorphism and fine structure of the bilayer profile. We show integrated membrane proteins do modulate the bilayer structure and the morphology of the vesicles.Phase transitions of DMPC vesicles were visualized after freezing under equilibrium conditions at different temperatures in a controlled-environment vitrification system. Below the main phase transition temperature of 24°C (Fig. 1), vesicles show a facetted appearance due to the quasicrystalline areas. A gradual increase in temperature leads to melting processes with different morphology in the bilayer profile. Far above the phase transition temperature the bilayer profile is still present. In the band-pass-filtered images (Fig. 2) no significant change in the width of the bilayer profile is visible.

Electron Microscope study of commensurate-incommensurate phase transition in BaZnGeO4

1990 ◽  
Vol 48 (4) ◽  
pp. 172-173
Author(s):  
Naoki Yamamoto ◽  
Makoto Kikuchi ◽  
Tooru Atake ◽  
Akihiro Hamano ◽  
Yasutoshi Saito
Keyword(s):  
Phase Transition ◽  
Electron Microscope Study ◽  
Room Temperature ◽  
Hexagonal Lattice ◽  
Ferroelectric Materials ◽  
Temperature Phase ◽  
X Ray Diffraction ◽  
X Ray ◽  
Periodic Arrays ◽  
Modulation Wave Vector

BaZnGeO4 undergoes many phase transitions from I to V phase. The highest temperature phase I has a BaAl2O4 type structure with a hexagonal lattice. Recent X-ray diffraction study showed that the incommensurate (IC) lattice modulation appears along the c axis in the III and IV phases with a period of about 4c, and a commensurate (C) phase with a modulated period of 4c exists between the III and IV phases in the narrow temperature region (—58°C to —47°C on cooling), called the III' phase. The modulations in the IC phases are considered displacive type, but the detailed structures have not been studied. It is also not clear whether the modulation changes into periodic arrays of discommensurations (DC’s) near the III-III' and IV-V phase transition temperature as found in the ferroelectric materials such as Rb2ZnCl4.At room temperature (III phase) satellite reflections were seen around the fundamental reflections in a diffraction pattern (Fig.1) and they aligned along a certain direction deviated from the c* direction, which indicates that the modulation wave vector q tilts from the c* axis. The tilt angle is about 2 degree at room temperature and depends on temperature.

Dynamic low-dose electron diffraction and imaging of phase transitions in polymers

1993 ◽  
Vol 51 ◽  
pp. 890-891
Author(s):  
David C. Martin ◽  
Jun Liao
Keyword(s):  
Crystal Structure ◽  
Phase Transition ◽  
Electron Diffraction ◽  
Low Dose ◽  
Dark Field ◽  
Continuous Change ◽  
Selected Area Electron Diffraction ◽  
Resolution Electron ◽  
Chain Axis ◽  
High Resolution Electron Micrographs

By careful control of the electron beam it is possible to simultaneously induce and observe the phase transformation from monomer to polymer in certain solid-state polymcrizable diacetylenes. The continuous change in the crystal structure from DCHD diacetylene monomer (a=1.76 nm, b=1.36 nm, c=0.455 nm, γ=94 degrees, P2l/c) to polymer (a=1.74 nm, b=1.29 nm, c=0.49 nm, γ=108 degrees, P2l/c) occurs at a characteristic dose (10−4C/cm2) which is five orders of magnitude smaller than the critical end point dose (20 C/cm2). Previously we discussed the progress of this phase transition primarily as observed down the [001] zone (the chain axis direction). Here we report on the associated changes of the dark field (DF) images and selected area electron diffraction (SAED) patterns of the crystals as observed from the side (i.e., in the [hk0] zones).High resolution electron micrographs (HREM), DF images, and SAED patterns were obtained on a JEOL 4000 EX HREM operating at 400 kV.

Single Crystal Raman Spectra of the Phase Transition in KTCNQ

1982 ◽  
Vol 85 (1) ◽  
pp. 297-303 ◽  
Author(s):  
A. D. Bandrauk ◽  
K. D. Truong ◽  
S. Jandl
Keyword(s):  
Phase Transition ◽  
Single Crystal ◽  
Raman Spectra
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