Derivation of the Rotation Matrix for an Axis-Angle Rotation Based on an Intuitive Interpretation of the Rotation Matrix

Author(s):  
Roshan Kumar Hota ◽  
Cheruvu Siva Kumar
Author(s):  
J. M. Cowley ◽  
Sumio Iijima

The imaging of detailed structures of crystal lattices with 3 to 4Å resolution, given the correct conditions of microscope defocus and crystal orientation and thickness, has been used by Iijima (this conference) for the study of new types of crystal structures and the defects in known structures associated with fluctuations of stoichiometry. The image intensities may be computed using n-beam dynamical diffraction theory involving several hundred beams (Fejes, this conference). However it is still important to have a suitable approximation to provide an immediate rough estimate of contrast and an evaluation of the intuitive interpretation in terms of an amplitude object.For crystals 100 to 150Å thick containing moderately heavy atoms the phase changes of the electron wave vary by about 10 radians suggesting that the “optimum defocus” theory of amplitude contrast for thin phase objects due to Scherzer and others can not apply, although it does predict the right defocus for optimum imaging.


2019 ◽  
Author(s):  
Shinichi Nakagawa ◽  
Malgorzata Lagisz ◽  
Rose E O'Dea ◽  
Joanna Rutkowska ◽  
Yefeng Yang ◽  
...  

‘Classic’ forest plots show the effect sizes from individual studies and the aggregate effect from a meta-analysis. However, in ecology and evolution meta-analyses routinely contain over 100 effect sizes, making the classic forest plot of limited use. We surveyed 102 meta-analyses in ecology and evolution, finding that only 11% use the classic forest plot. Instead, most used a ‘forest-like plot’, showing point estimates (with 95% confidence intervals; CIs) from a series of subgroups or categories in a meta-regression. We propose a modification of the forest-like plot, which we name the ‘orchard plot’. Orchard plots, in addition to showing overall mean effects and CIs from meta-analyses/regressions, also includes 95% prediction intervals (PIs), and the individual effect sizes scaled by their precision. The PI allows the user and reader to see the range in which an effect size from a future study may be expected to fall. The PI, therefore, provides an intuitive interpretation of any heterogeneity in the data. Supplementing the PI, the inclusion of underlying effect sizes also allows the user to see any influential or outlying effect sizes. We showcase the orchard plot with example datasets from ecology and evolution, using the R package, orchard, including several functions for visualizing meta-analytic data using forest-plot derivatives. We consider the orchard plot as a variant on the classic forest plot, cultivated to the needs of meta-analysts in ecology and evolution. Hopefully, the orchard plot will prove fruitful for visualizing large collections of heterogeneous effect sizes regardless of the field of study.


1993 ◽  
Vol 329 ◽  
Author(s):  
Frederick G. Anderson ◽  
H. Weidner ◽  
P. L. Summers ◽  
R. E. Peale ◽  
B. H. T. Chai

AbstractExpanding the crystal field in terms of operators that transform as the irreducible representations of the Td group leads to an intuitive interpretation of the crystal-field parameters. We apply this method to the crystal field experienced by Nd3+ dopants in the laser crystals YLiF4, YVO4, and KLiYF5.


1995 ◽  
Vol 32 (01) ◽  
pp. 1-10
Author(s):  
Ziad Taib

The functional differential equation y′(x) = ay(λx) + by(x) arises in many different situations. The purpose of this note is to show how it arises in some multitype branching process cell population models. We also show how its solution can be given an intuitive interpretation as the probability density function of an infinite sum of independent but not identically distributed random variables.


Sensors ◽  
2021 ◽  
Vol 21 (15) ◽  
pp. 5004
Author(s):  
Haohao Hu ◽  
Johannes Beck ◽  
Martin Lauer ◽  
Christoph Stiller

The fusion of motion data is key in the fields of robotic and automated driving. Most existing approaches are filter-based or pose-graph-based. By using filter-based approaches, parameters should be set very carefully and the motion data can usually only be fused in a time forward direction. Pose-graph-based approaches can fuse data in time forward and backward directions. However, pre-integration is needed by applying measurements from inertial measurement units. Additionally, both approaches only provide discrete fusion results. In this work, we address this problem and present a uniform B-spline-based continuous fusion approach, which can fuse motion measurements from an inertial measurement unit and pose data from other localization systems robustly, accurately and efficiently. In our continuous fusion approach, an axis-angle is applied as our rotation representation method and uniform B-spline as the back-end optimization base. Evaluation results performed on the real world data show that our approach provides accurate, robust and continuous fusion results, which again supports our continuous fusion concept.


2015 ◽  
Vol 45 (3) ◽  
pp. 577-599 ◽  
Author(s):  
Anas Abdallah ◽  
Jean-Philippe Boucher ◽  
Hélène Cossette

AbstractOne of the most critical problems in property/casualty insurance is to determine an appropriate reserve for incurred but unpaid losses. These provisions generally comprise most of the liabilities of a non-life insurance company. The global provisions are often determined under an assumption of independence between the lines of business. Recently, Shi and Frees (2011) proposed to put dependence between lines of business with a copula that captures dependence between two cells of two different runoff triangles. In this paper, we propose to generalize this model in two steps. First, by using an idea proposed by Barnett and Zehnwirth (1998), we will suppose a dependence between all the observations that belong to the same calendar year (CY) for each line of business. Thereafter, we will then suppose another dependence structure that links the CYs of different lines of business. This model is done by using hierarchical Archimedean copulas. We show that the model provides more flexibility than existing models, and offers a better, more realistic and more intuitive interpretation of the dependence between the lines of business. For illustration, the model is applied to a dataset from a major US property-casualty insurer, where a bootstrap method is proposed to estimate the distribution of the reserve.


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