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.

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.

scholarly journals Improving the two-stage numerical integration in stability identification of oscillation with distributed delay

2019 ◽  
Vol 11 (1) ◽  
pp. 168781401881990
Author(s):  
Chigbogu Godwin Ozoegwu
Keyword(s):  
Numerical Integration ◽  
Distributed Delay ◽  
Original Method ◽  
Higher Order ◽  
Trapezoidal Rule ◽  
Point Of View ◽  
Engineering Systems ◽  
Two Stage ◽  
Integration Rule ◽  
Integration Rules

The vibration of the engineering systems with distributed delay is governed by delay integro-differential equations. Two-stage numerical integration approach was recently proposed for stability identification of such oscillators. This work improves the approach by handling the distributed delay—that is, the first-stage numerical integration—with tensor-based higher order numerical integration rules. The second-stage numerical integration of the arising methods remains the trapezoidal rule as in the original method. It is shown that local discretization error is of order [Formula: see text] irrespective of the order of the numerical integration rule used to handle the distributed delay. But [Formula: see text] is less weighted when higher order numerical integration rules are used to handle the distributed delay, suggesting higher accuracy. Results from theoretical error analyses, various numerical rate of convergence analyses, and stability computations were combined to conclude that—from application point of view—it is not necessary to increase the first-stage numerical integration rule beyond the first order (trapezoidal rule) though the best results are expected at the second order (Simpson’s 1/3 rule).

Impact of the Number of Response Categories on Linearity and Sensitivity of Self-Anchoring Scales

Methodology ◽  
2007 ◽  
Vol 3 (4) ◽  
pp. 160-169 ◽  
Author(s):  
Joeri Hofmans ◽  
Peter Theuns ◽  
Olivier Mairesse
Keyword(s):  
Rating Scale ◽  
Specific Pattern ◽  
Overt Response ◽  
Algebraic Models ◽  
Integration Rule ◽  
Functional Measurement ◽  
Main Effects ◽  
The Relationship ◽  
Integration Rules

Abstract. In this research, the relationship between the number of response categories and the linearity and sensitivity of self-anchoring scales is tested. According to the functional measurement paradigm, people integrate their impressions of stimuli using simple algebraic models. Then the integrated stimulus is transformed into an overt response on a rating scale. The combination of a particular algebraic integration rule along with a linear rating scale predicts specific patterns in a factorial plot. In two functional measurement experiments we manipulated the number of categories of different self-anchoring scales and attempted to replicate a specific pattern based on the integration rules found in previous research. For both experiments, the predicted pattern was observed for all scales. This indicates that the number of response categories does not impact on the linearity of the scale. Moreover, there is no relationship between the number of response categories and the sensitivity of the scale as measured by the F ratio for the main effects.

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.

scholarly journals THE DELAUNAY HIERARCHY

2002 ◽  
Vol 13 (02) ◽  
pp. 163-180 ◽  
Author(s):  
OLIVIER DEVILLERS
Keyword(s):  
Data Structure ◽  
Nearest Neighbor ◽  
Real Data ◽  
Small Sample ◽  
Small Subset ◽  
Point Location ◽  
Worst Case ◽  
Case Complexity ◽  
Worst Case Complexity ◽  
Set Of Points

We propose a new data structure to compute the Delaunay triangulation of a set of points in the plane. It combines good worst case complexity, fast behavior on real data, small memory occupation and the possibility of fully dynamic insertions and deletions. The location structure is organized into several levels. The lowest level just consists of the triangulation, then each level contains the triangulation of a small sample of the level below. Point location is done by walking in a triangulation to determine the nearest neighbor of the query at that level, then the walk restarts from the neighbor at the level below. Using a small subset (3%) to sample a level allows a small memory occupation; the walk and the use of the nearest neighbor to change levels quickly locate the query.

scholarly journals Two Heads Are Better Than One, but How Much?

2014 ◽  
Vol 61 (5) ◽  
pp. 356-367 ◽  
Author(s):  
Miguel A. Vadillo ◽  
Nerea Ortega-Castro ◽  
Itxaso Barberia ◽  
A. G. Baker
Keyword(s):  
Causal Learning ◽  
Present Series ◽  
Integration Rule ◽  
Causal Induction ◽  
Series Of Experiments ◽  
Causal Impact ◽  
The Individual ◽  
The Impact ◽  
Better Than ◽  
Integration Rules

Many theories of causal learning and causal induction differ in their assumptions about how people combine the causal impact of several causes presented in compound. Some theories propose that when several causes are present, their joint causal impact is equal to the linear sum of the individual impact of each cause. However, some recent theories propose that the causal impact of several causes needs to be combined by means of a noisy-OR integration rule. In other words, the probability of the effect given several causes would be equal to the sum of the probability of the effect given each cause in isolation minus the overlap between those probabilities. In the present series of experiments, participants were given information about the causal impact of several causes and then they were asked what compounds of those causes they would prefer to use if they wanted to produce the effect. The results of these experiments suggest that participants actually use a variety of strategies, including not only the linear and the noisy-OR integration rules, but also averaging the impact of several causes.

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