scholarly journals Infinite Orders and Non-D-finite Property of 3-Dimensional Lattice Walks

10.37236/5408 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Daniel K. Du ◽  
Qing-Hu Hou ◽  
Rong-Hua Wang
Keyword(s):  
Generating Function ◽  
Infinite Groups ◽  
Dimensional Lattice ◽  
3 Dimensional ◽  
Lattice Walks

Recently, Bostan and his coauthors investigated lattice walks restricted to the non-negative octant $\mathbb{N}^3$. For the $35548$ non-trivial models with at most six steps, they found that many models associated to a group of order at least $200$ and conjectured these groups were in fact infinite groups. In this paper, we first confirm these conjectures and then consider the non-$D$-finite property of the generating function for some of these models.

scholarly journals Continued Classification of 3D Lattice Models in the Positive Octant

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Axel Bacher ◽  
Manuel Kauers ◽  
Rika Yatchak
Keyword(s):  
Asymptotic Behaviour ◽  
Generating Function ◽  
Asymptotic Form ◽  
Three Dimensional ◽  
Lattice Models ◽  
Dimensional Lattice ◽  
Lattice Walks ◽  
Small Set ◽  
International Audience

International audience We continue the investigations of lattice walks in the three-dimensional lattice restricted to the positive octant. We separate models which clearly have a D-finite generating function from models for which there is no reason to expect that their generating function is D-finite, and we isolate a small set of models whose nature remains unclear and requires further investigation. For these, we give some experimental results about their asymptotic behaviour, based on the inspection of a large number of initial terms. At least for some of them, the guessed asymptotic form seems to tip the balance towards non-D-finiteness.

scholarly journals On 3-Dimensional Lattice Walks Confined to the Positive Octant

2016 ◽  
Vol 20 (4) ◽  
pp. 661-704 ◽  
Author(s):  
Alin Bostan ◽  
Mireille Bousquet-Mélou ◽  
Manuel Kauers ◽  
Stephen Melczer
Keyword(s):  
Dimensional Lattice ◽  
3 Dimensional ◽  
Lattice Walks

Deviations from Ideal Decagonal Symmetry in Electron Diffraction Patterns of the “Decagonal” Phase in the Al-Co-Cu Alloy System

1991 ◽  
Vol 49 ◽  
pp. 914-915
Author(s):  
Atul S. Ramani ◽  
Earle R. Ryba ◽  
Paul R. Howell
Keyword(s):  
Electron Microscope ◽  
Alloy System ◽  
Nominal Composition ◽  
Dimensional Lattice ◽  
3 Dimensional ◽  
Decagonal Phase ◽  
Cu Alloy ◽  
Transmission Electron ◽  
Lattice Periodicity ◽  
Diffraction Patterns

The “decagonal” phase in the Al-Co-Cu system of nominal composition Al65CO15Cu20 first discovered by He et al. is especially suitable as a topic of investigation since it has been claimed that it is thermodynamically stable and is reported to be periodic in the dimension perpendicular to the plane of quasiperiodic 10-fold symmetry. It can thus be expected that it is an important link between fully periodic and fully quasiperiodic phases. In the present paper, we report important findings of our transmission electron microscope (TEM) study that concern deviations from ideal decagonal symmetry of selected area diffraction patterns (SADPs) obtained from several “decagonal” phase crystals and also observation of a lattice of main reflections on the 10-fold and 2-fold SADPs that implies complete 3-dimensional lattice periodicity and the fundamentally incommensurate nature of the “decagonal” phase. We also present diffraction evidence for a new transition phase that can be classified as being one-dimensionally quasiperiodic if the lattice of main reflections is ignored.

Finite-order Integration Weights Can be Dangerous

2007 ◽  
Vol 7 (3) ◽  
pp. 239-254 ◽  
Author(s):  
I.H. Sloan
Keyword(s):  
Function Space ◽  
Finite Order ◽  
Small Subset ◽  
Worst Case ◽  
Integration Rule ◽  
Dimensional Lattice ◽  
3 Dimensional ◽  
Sum Of Functions ◽  
Integration Rules

Abstract Finite-order weights have been introduced in recent years to describe the often occurring situation that multivariate integrands can be approximated by a sum of functions each depending only on a small subset of the variables. The aim of this paper is to demonstrate the danger of relying on this structure when designing lattice integration rules, if the true integrand has components lying outside the assumed finiteorder function space. It does this by proving, for weights of order two, the existence of 3-dimensional lattice integration rules for which the worst case error is of order O(N¯½), where N is the number of points, yet for which there exists a smooth 3- dimensional integrand for which the integration rule does not converge.

scholarly journals Statistics on Lattice Walks and q-Lassalle Numbers

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Lenny Tevlin
Keyword(s):  
Positive Integer ◽  
Generating Function ◽  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Integer Coefficients ◽  
Major Index ◽  
Lattice Walks ◽  
International Audience ◽  
The Difference ◽  
Positive Integers

International audience This paper contains two results. First, I propose a $q$-generalization of a certain sequence of positive integers, related to Catalan numbers, introduced by Zeilberger, see Lassalle (2010). These $q$-integers are palindromic polynomials in $q$ with positive integer coefficients. The positivity depends on the positivity of a certain difference of products of $q$-binomial coefficients.To this end, I introduce a new inversion/major statistics on lattice walks. The difference in $q$-binomial coefficients is then seen as a generating function of weighted walks that remain in the upper half-plan. Cet document contient deux résultats. Tout d’abord, je vous propose un $q$-generalization d’une certaine séquence de nombres entiers positifs, liés à nombres de Catalan, introduites par Zeilberger (Lassalle, 2010). Ces $q$-integers sont des polynômes palindromiques à $q$ à coefficients entiers positifs. La positivité dépend de la positivité d’une certaine différence de produits de $q$-coefficients binomial.Pour ce faire, je vous présente une nouvelle inversion/major index sur les chemins du réseau. La différence de $q$-binomial coefficients est alors considérée comme une fonction de génération de trajets pondérés qui restent dans le demi-plan supérieur.

Three-Dimensional Lattice Walks in the Upper Half-Space: 10795

2001 ◽  
Vol 108 (10) ◽  
pp. 980 ◽  
Author(s):  
Emeric Deutsch ◽  
Jim Brawner
Keyword(s):  
Half Space ◽  
Three Dimensional ◽  
Dimensional Lattice ◽  
Lattice Walks

Influence of Post-Deposition Annealing on Lattice Strain, Electrical Transport and Magnetic Properties in Epitaxial La0.8Ca0.2MnO3 CMR Films

1999 ◽  
Vol 574 ◽  
Author(s):  
T. K. Nath ◽  
R. A. Rao ◽  
D. Lavric ◽  
C. B. Eom
Keyword(s):  
Cell Volume ◽  
Annealing Time ◽  
Unit Cell Volume ◽  
Lattice Strain ◽  
Unit Cell ◽  
Clear Correlation ◽  
Dimensional Lattice ◽  
X Ray ◽  
3 Dimensional ◽  
Perovskite Unit Cell

AbstractThe effect of annealing on 3-dimensional lattice strain, crystallographic domain structure, magnetic and electrical properties of both 250 Å and 4000 Å thick epitaxial La0.8Ca0.2MnO3 (LCMO(x=0.2)) thin films grown on (001) LaAlO3 substrates have been studied. While short annealing time (∼2hrs. at 950 °C in oxygen of 1 atm. pressure) leads to anomalous increase of the peak temperature (Tp) and Curie temperature (Tc) above room temperature and that of the bulk material, longer annealing time (∼10 hrs.) restores the Tp and Tc to almost the same values as that of the as-grown films. Furthermore, as the annealing time is increased, the lattice strain relaxes with film's lattice parameter approaching the bulk value. In-plane and out-of-plane lattice parameters and strain states of the as-grown and annealed films were measured directly using normal and grazing incidence x-ray diffraction. A clear correlation is observed between Tp and perovskite unit cell volume for both the films. Tp is found to increase with the decrease of perovskite unit cell volume. This is attributed to the enhancement of overlap between Mn d orbitals and oxygen p orbitals leading to increased bandwidth and conductivity. Crystalline quality of the films as determined by the full width at half maximum (FWHM) of the x-ray rocking curves, improves with the annealing time. This work highlights the importance of controlling the 3-dimensional lattice strain for optimizing the properties of CMR films.

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