scholarly journals Detecting (non)parallel evolution in multidimensional spaces: angles, correlations, and eigenanalysis

2021 ◽  
Author(s):  
Junya Watanabe
Keyword(s):  
Null Hypothesis ◽  
Dimensional Space ◽  
Parallel Evolution ◽  
Test Procedure ◽  
Rayleigh Test ◽  
Vector Correlation ◽  
Higher Dimensional ◽  
Evolutionary Trajectories ◽  
The Right ◽  
Biological Literature

Parallelism between evolutionary trajectories in a trait space is often seen as evidence for repeatability of phenotypic evolution, and angles between trajectories play a pivotal role in the analysis of parallelism. However, many biologists have been ignorant on properties of angles in multidimensional spaces, and unsound uses of angles are common in the biological literature. To remedy this situation, this study provides a brief overview on geometric and statistical aspects of angles in multidimensional spaces. Under the null hypothesis that trajectory vectors have no preferred directions, the angle between two independent vectors is concentrated around the right angle, with a more pronounced peak in a higher-dimensional space. This probability distribution is closely related to t- and beta distributions, which can be used for testing the null hypothesis concerning a pair of trajectories. A recently proposed method with eigenanalysis of a vector correlation matrix essentially boils down to the test of no correlation or concentration of multiple vectors, for which a simple test procedure is available in the statistical literature. Concentration of vectors can also be examined by tools of directional statistics such as the Rayleigh test. These frameworks provide biologists with baselines to make statistically justified inferences for (non)parallel evolution.

Celestial Tapestry

2020 ◽  
Author(s):  
Nicholas Mee
Keyword(s):  
Mathematics Curriculum ◽  
Dimensional Space ◽  
Golden Section ◽  
Single Point ◽  
Historical Context ◽  
Computer Art ◽  
Lightning Strike ◽  
Secret Message ◽  
Axiomatic Method ◽  
Higher Dimensional

Celestial Tapestry places mathematics within a vibrant cultural and historical context, highlighting links to the visual arts and design, and broader areas of artistic creativity. Threads are woven together telling of surprising influences that have passed between the arts and mathematics. The story involves many intriguing characters: Gaston Julia, who laid the foundations for fractals and computer art while recovering in hospital after suffering serious injury in the First World War; Charles Howard, Hinton who was imprisoned for bigamy but whose books had a huge influence on twentieth-century art; Michael Scott, the Scottish necromancer who was the dedicatee of Fibonacci’s Book of Calculation, the most important medieval book of mathematics; Richard of Wallingford, the pioneer clockmaker who suffered from leprosy and who never recovered from a lightning strike on his bedchamber; Alicia Stott Boole, the Victorian housewife who amazed mathematicians with her intuition for higher-dimensional space. The book includes more than 200 colour illustrations, puzzles to engage the reader, and many remarkable tales: the secret message in Hans Holbein’s The Ambassadors; the link between Viking runes, a Milanese banking dynasty, and modern sculpture; the connection between astrology, religion, and the Apocalypse; binary numbers and the I Ching. It also explains topics on the school mathematics curriculum: algorithms; arithmetic progressions; combinations and permutations; number sequences; the axiomatic method; geometrical proof; tessellations and polyhedra, as well as many essential topics for arts and humanities students: single-point perspective; fractals; computer art; the golden section; the higher-dimensional inspiration behind modern art.

Hypothesis testing with error correction models

2021 ◽  
pp. 1-9
Author(s):  
Patrick W. Kraft ◽  
Ellen M. Key ◽  
Matthew J. Lebo
Keyword(s):  
Hypothesis Testing ◽  
Error Correction ◽  
Null Hypothesis ◽  
Alternative Hypothesis ◽  
Correction Coefficient ◽  
Correction Model ◽  
Long Run ◽  
Independent Variable ◽  
The Right ◽  
General Error

Abstract Grant and Lebo (2016) and Keele et al. (2016) clarify the conditions under which the popular general error correction model (GECM) can be used and interpreted easily: In a bivariate GECM the data must be integrated in order to rely on the error correction coefficient, $\alpha _1^\ast$ , to test cointegration and measure the rate of error correction between a single exogenous x and a dependent variable, y. Here we demonstrate that even if the data are all integrated, the test on $\alpha _1^\ast$ is misunderstood when there is more than a single independent variable. The null hypothesis is that there is no cointegration between y and any x but the correct alternative hypothesis is that y is cointegrated with at least one—but not necessarily more than one—of the x's. A significant $\alpha _1^\ast$ can occur when some I(1) regressors are not cointegrated and the equation is not balanced. Thus, the correct limiting distributions of the right-hand-side long-run coefficients may be unknown. We use simulations to demonstrate the problem and then discuss implications for applied examples.

scholarly journals Aperiodic Crystals

2014 ◽  
Vol 70 (a1) ◽  
pp. C1-C1 ◽  
Author(s):  
Ted Janssen ◽  
Aloysio Janner
Keyword(s):  
Physical Properties ◽  
Dimensional Space ◽  
X Rays ◽  
New Techniques ◽  
New Family ◽  
Modulated Phases ◽  
Open Questions ◽  
Higher Dimensional ◽  
Lattice Periodicity ◽  
Aperiodic Crystals

2014 is the International Year of Crystallography. During at least fifty years after the discovery of diffraction of X-rays by crystals, it was believed that crystals have lattice periodicity, and crystals were defined by this property. Now it has become clear that there is a large class of compounds with interesting properties that should be called crystals as well, but are not lattice periodic. A method has been developed to describe and analyze these aperiodic crystals, using a higher-dimensional space. In this lecture the discovery of aperiodic crystals and the development of the formalism of the so-called superspace will be described. There are several classes of such materials. After the incommensurate modulated phases, incommensurate magnetic crystals, incommensurate composites and quasicrystals were discovered. They could all be studied using the same technique. Their main properties of these classes and the ways to characterize them will be discussed. The new family of aperiodic crystals has led also to new physical properties, to new techniques in crystallography and to interesting mathematical questions. Much has been done in the last fifty years by hundreds of crystallographers, crystal growers, physicists, chemists, mineralogists and mathematicians. Many new insights have been obtained. But there are still many questions, also of fundamental nature, to be answered. We end with a discussion of these open questions.

scholarly journals Cerebral hemispheres – cerebellum – kidney interaction in patients with acute cerebral ischemia

2021 ◽  
Vol 26 (1) ◽  
pp. 90-98
Author(s):  
O.M. Kononets ◽  
O.V. Tkachenko ◽  
O.O. Kamenetska
Keyword(s):  
Nervous System ◽  
Ischemic Stroke ◽  
Brain Magnetic Resonance Imaging ◽  
Environmental Parameters ◽  
The Body ◽  
Electrolyte Balance ◽  
Vector Correlation ◽  
Vertebrobasilar System ◽  
Acute Cerebral Ischemia ◽  
The Right

The nervous system, in particular the autonomic one, is well known to constantly regulate the internal functioning of the body, adapting it to changeable external and internal environmental parameters. In particular, there is a close multiple-vector correlation between the nervous system and the kidneys. The aim of this study was to specify the mechanisms, clinical and paraclinical characteristics of the concomitant lesions of the nervous system and the kidneys in patients with acute stroke. This paper presents the case report of 215 patients, aged 70 ± 8.44, who suffered from ischemic stroke. Among them, we examined 144 women and 71 men. The patients underwent a comprehensive examination, including a detailed clinical and neurological check-up (evaluating the patients’ condition severity with the National Institutes of Health Stroke Scale (NIHSS) and the Barthel index on admission and on the 21st day of the disease), laboratory analysis (electrolyte balance, nitrogen metabolism (on admission and on the 21st day of the disease) and instrumental examination (CT scan of the brain, the follow-up brain magnetic resonance imaging). The statistical methods were used to analyze the data. In the 1st day of the disease, all the surveyed patients with right hemispheric carotid stroke and the overwhelming majority of the patients with left hemispheric carotid stroke and ischemic stroke in the vertebrobasilar system had cerebral renal syndrome, represented by renal concentration-filtration dysfunction, accompanied by the reduced glomerular filtration rate. A reliable relationship was found between the renal concentration and filtration function and the right hemispheric ischemic focus in patients with ischemic stroke, the characteristics are to be specified.

scholarly journals Celestial Spheres

2018 ◽  
Vol 2 (2) ◽  
pp. 168-186
Author(s):  
Austin M. Freeman
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Good Evidence ◽  
Medieval Period ◽  
The Bible ◽  
Higher Dimensional ◽  
Spatial Dimensions ◽  
Modern Physics ◽  
The Subject ◽  
Three Dimensional Space

Angels probably have bodies. There is no good evidence (biblical, philosophical, or historical) to argue against their bodiliness; there is an abundance of evidence (biblical, philosophical, historical) that makes the case for angelic bodies. After surveying biblical texts alleged to demonstrate angelic incorporeality, the discussion moves to examine patristic, medieval, and some modern figures on the subject. In short, before the High Medieval period belief in angelic bodies was the norm, and afterwards it is the exception. A brief foray into modern physics and higher spatial dimensions (termed “hyperspace”), coupled with an analogical use of Edwin Abbott’s Flatland, serves to explain the way in which appealing to higher-dimensional angelic bodies matches the record of angelic activity in the Bible remarkably well. This position also cuts through a historical equivocation on the question of angelic embodiment. Angels do have bodies, but they are bodies very unlike our own. They do not have bodies in any three-dimensional space we can observe, but are nevertheless embodied beings.

scholarly journals Visualising higher-dimensional space-time and space-scale objects as projections to ℝ3

2017 ◽  
Vol 3 ◽  
pp. e123 ◽  
Author(s):  
Ken Arroyo Ohori ◽  
Hugo Ledoux ◽  
Jantien Stoter
Keyword(s):  
Dimensional Space ◽  
Space Time ◽  
Three Dimensions ◽  
Time And Space ◽  
3D Objects ◽  
Levels Of Detail ◽  
Higher Dimensional Space Time ◽  
Space Scale ◽  
Higher Dimensional ◽  
Dimensional Space Time

Objects of more than three dimensions can be used to model geographic phenomena that occur in space, time and scale. For instance, a single 4D object can be used to represent the changes in a 3D object’s shape across time or all its optimal representations at various levels of detail. In this paper, we look at how such higher-dimensional space-time and space-scale objects can be visualised as projections from ℝ4to ℝ3. We present three projections that we believe are particularly intuitive for this purpose: (i) a simple ‘long axis’ projection that puts 3D objects side by side; (ii) the well-known orthographic and perspective projections; and (iii) a projection to a 3-sphere (S3) followed by a stereographic projection to ℝ3, which results in an inwards-outwards fourth axis. Our focus is in using these projections from ℝ4to ℝ3, but they are formulated from ℝnto ℝn−1so as to be easily extensible and to incorporate other non-spatial characteristics. We present a prototype interactive visualiser that applies these projections from 4D to 3D in real-time using the programmable pipeline and compute shaders of the Metal graphics API.

On the structure of quasi-crystals in the higher-dimensional space

2013 ◽  
Vol 62 (2) ◽  
pp. 265-269 ◽  
Author(s):  
V. Ya. Shevchenko ◽  
G. V. Zhizhin ◽  
A. L. Mackay
Keyword(s):  
Dimensional Space ◽  
Higher Dimensional ◽  
Quasi Crystals

Test procedure and sample size determination for a proportion study using a double-sampling scheme with two fallible classifiers

2017 ◽  
Vol 28 (4) ◽  
pp. 1019-1043 ◽  
Author(s):  
Shi-Fang Qiu ◽  
Xiao-Song Zeng ◽  
Man-Lai Tang ◽  
Wai-Yin Poon
Keyword(s):  
Sample Size ◽  
Null Hypothesis ◽  
Test Procedure ◽  
Sample Size Determination ◽  
Double Sampling ◽  
Score Statistic ◽  
Wald Statistic ◽  
Test Procedures ◽  
Practical Applications ◽  
Logit Transformation

Double sampling is usually applied to collect necessary information for situations in which an infallible classifier is available for validating a subset of the sample that has already been classified by a fallible classifier. Inference procedures have previously been developed based on the partially validated data obtained by the double-sampling process. However, it could happen in practice that such infallible classifier or gold standard does not exist. In this article, we consider the case in which both classifiers are fallible and propose asymptotic and approximate unconditional test procedures based on six test statistics for a population proportion and five approximate sample size formulas based on the recommended test procedures under two models. Our results suggest that both asymptotic and approximate unconditional procedures based on the score statistic perform satisfactorily for small to large sample sizes and are highly recommended. When sample size is moderate or large, asymptotic procedures based on the Wald statistic with the variance being estimated under the null hypothesis, likelihood rate statistic, log- and logit-transformation statistics based on both models generally perform well and are hence recommended. The approximate unconditional procedures based on the log-transformation statistic under Model I, Wald statistic with the variance being estimated under the null hypothesis, log- and logit-transformation statistics under Model II are recommended when sample size is small. In general, sample size formulae based on the Wald statistic with the variance being estimated under the null hypothesis, likelihood rate statistic and score statistic are recommended in practical applications. The applicability of the proposed methods is illustrated by a real-data example.

Dynamical Variation of Weierstrass-Mandelbrot Function in Higher Dimensional Space

2013 ◽  
Vol 470 ◽  
pp. 767-771
Author(s):  
L. Zhang ◽  
Shu Tang Liu
Keyword(s):  
Hurst Exponent ◽  
Dimensional Space ◽  
Dynamical Behavior ◽  
Fractal Dimensions ◽  
Statistical Characteristics ◽  
3 Dimensional ◽  
Higher Dimensional ◽  
Complex Phenomena ◽  
Real Complex

Many real complex phenomena are related with Weierstrass-Mandelbrot function (WMF). Most researches focus on the systems as parameters fixed, such as calculations of its different fractal dimensions or the statistical characteristics of its generalized form and so on. Moreover, real systems always change according to different environments, so that to study the dynamical behavior of these systems as parameters change is important. However, there is few results about this aim. In this paper, we propose simulated results for the effects of parameters changeably on the graph of WMF in higher dimensional space. In addition, the relationships between the Hurst exponent of WMF and its parameters dynamically in 2-and 3-dimensional spaces are also given.

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