scholarly journals Geometrical Interpretation of Isomers

2021 ◽  
pp. 155-163
Author(s):  
Lemi Türker
Keyword(s):  
Present Article ◽  
Mathematical Analysis ◽  
Euclidean Geometry ◽  
Geometrical Interpretation ◽  
2D And 3D ◽  
Geometrical Relationships

The present article considers isomerism, which is one of the most important topics of chemistry. A model is proposed in 2D and 3D-Euclidean geometry starting from the very fundamental concepts and has established certain geometrical relationships between the mass of a molecule and its bonds and atoms. Some crucial angles are defined. Certain mathematical analysis have been presented as well.

scholarly journals “I have Learned Non-Euclidean Geometry Just from this Book” - Some facets of the correspondence between Friedrich Engel and David Hilbert

2021 ◽  
Vol 76 (3) ◽  
Author(s):  
Peter Ullrich
Keyword(s):  
Present Article ◽  
19Th Century ◽  
Euclidean Geometry ◽  
Academic Life ◽  
David Hilbert ◽  
Non Euclidean Geometry ◽  
The 19Th Century

AbstractFriedrich Engel and David Hilbert learned to know each other at Leipzig in 1885 and exchanged letters in particular during the next 15 years which contain interesting information on the academic life of mathematicians at the end of the 19th century. In the present article we will mainly discuss a statement by Hilbert himself on Moritz Pasch’s influence on his views of geometry, and on personnel politics concerning Hermann Minkowski and Eduard Study but also Engel himself.

Historically Speaking—: The Early Beginnings of Set Theory

1970 ◽  
Vol 63 (8) ◽  
pp. 690-692
Author(s):  
Phillip E. Johnson
Keyword(s):  
Present Article ◽  
Set Theory ◽  
Euclidean Geometry ◽  
Non Euclidean Geometry ◽  
The Creation

Georg Canttor created and largely developed the theory of sets in approximately the year. 1874-1897. In contrast to such developments as the calculus and non-Euclidean geometry, The creation of set theory was, according to all indications, Cantor's alone. AIso, set theory was not preceded by a long evoIutionary period such as is usually the case with big mathematical breakthroughs. The present article will concern itself primarily with the very earliest set-theoretic works of Cantor, namely, his first two papers in this area.1

BELL INEQUALITIES VIEWED IN TERMS OF EUCLIDEAN GEOMETRY AND GROUP THEORY

1995 ◽  
Vol 09 (07) ◽  
pp. 819-847
Author(s):  
ROBERT R. TUCCI
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Euclidean Geometry ◽  
Hidden Variables ◽  
Bell Inequalities ◽  
Geometrical Interpretation ◽  
Finite Group Theory

This paper presents a new, computer-implementable algorithm for determining Bell inequalities. The algorithm is very general: one can use it to derive the Bell inequalities for any of the experiments that are usually considered for this purpose, such as the Bohm-Bell experiment, the Clauser-Horne experiment, or the experiments with spin 1, 3/2, 2 and 5/2 particles proposed by Mermin, Schwarz and Garg. As an example, this paper applies the algorithm to the Clauser-Horne experiment. The algorithm, which is based on simple notions from n-dimensional Euclidean geometry, gives a simple geometrical interpretation to the Bell inequalities. The algorithm allows one to demonstrate that a set of Bell inequalities is complete. The algorithm uses special hidden variables that were first used by Wigner and Belinfante. This paper also shows how to use finite group theory to express a set of Bell inequalities in a form that makes its invariance under relabellings explicit.

scholarly journals INFORMATION AND MEMORY-BASED ANALYSIS FOR DECODING OF THE HUMAN LEARNING BETWEEN NORMAL AND VIRTUAL REALITY (VR) CONDITIONS

Fractals ◽  
2021 ◽  
Vol 29 (03) ◽  
pp. 2150163
Author(s):  
HAMIDREZA NAMAZI ◽  
MOHAMMAD HOSSEIN BABINI ◽  
KAMIL KUCA ◽  
ONDREJ KREJCAR
Keyword(s):  
Virtual Reality ◽  
Shannon Entropy ◽  
Mathematical Analysis ◽  
Strong Correlation ◽  
Hurst Exponent ◽  
Human Learning ◽  
Learning Ability ◽  
Eeg Signals ◽  
2D And 3D ◽  
Electroencephalogram Eeg

In this paper, we investigated the learning ability of students in normal versus virtual reality (VR) watching of videos by mathematical analysis of electroencephalogram (EEG) signals. We played six videos in the 2D and 3D modes for nine subjects and calculated the Shannon entropy of recorded EEG signals to investigate how much their embedded information changes between these modes. We also calculated the Hurst exponent of EEG signals to compare the changes in the memory of signals. The analysis results showed that watching the videos in a VR condition causes greater information and memory in EEG signals. A strong correlation was obtained between the increment of information and memory of EEG signals. These increments also have been verified based on the answers that subjects gave to the questions about the content of videos. Therefore, we can say that when subjects watch a video in a VR condition, more information is transferred to their brains that cause increments in their memory.

LA GÉOMÉTRIE DE L'ASTROLABE AU Xe SIÈCLE Geometry of the Astrolabe in the Tenth Century

2000 ◽  
Vol 10 (1) ◽  
pp. 7-77 ◽  
Author(s):  
Abgrall Philippe
Keyword(s):  
Present Article ◽  
Mathematical Analysis ◽  
Critical Edition ◽  
Eleventh Century ◽  
Stereographic Projection ◽  
Tenth Century ◽  
The Other ◽  
Conical Projection ◽  
Geometrical Transformation ◽  
History Of

Many studies on the astrolabe were written during the period from the ninth to the eleventh century, but very few of them related to projection, i.e., to the geometrical transformation underlying the design of the instrument. Among those that did, the treatise entitled The Art of the Astrolabe, written in the tenth century by Abū Sahl al-Qūhī, represents a particulary important phase in the history of geometry. This work recently appeared in a critical edition with translation and commentary by Roshdi Rashed. It contains the earliest known theory of the projection of the sphere, a theory developed in a commentary written by a contemporary mathematician, Ibn Sahl. Following R. Rashed, the present article offers here a thorough mathematical analysis of al-Qūhī's treatise and of the commentary by Ibn Sahl. It also presents, with commentary, an account of a contemporary treatise on the projection of the sphere, written by al-[Sdotu]āġānī. The latter work is concerned with the conical projection of a sphere on a plane, from a point on an axis of the sphere, other than its pole. The author consciously avoids the case of stereographic projection, but he studies all the other cases of conical projection which, if we employ the terms of al-Qūhī's theory, are compatible with the movement of the instrument (i.e. the rotation of the sphere around its axis). These three texts provide clear evidence of the emergence, during the second half of the tenth century, of a new field of study, that of projective geometry.

Density waves in disk galaxies

1967 ◽  
Vol 31 ◽  
pp. 313-317 ◽  
Author(s):  
C. C. Lin ◽  
F. H. Shu
Keyword(s):  
Mathematical Analysis ◽  
Spiral Structure ◽  
Stellar System ◽  
Collective Modes ◽  
Disk Galaxies ◽  
Spiral Pattern ◽  
Density Waves ◽  
The Galaxy ◽  
Detailed Mathematical Analysis

Density waves in the nature of those proposed by B. Lindblad are described by detailed mathematical analysis of collective modes in a disk-like stellar system. The treatment is centered around a hypothesis of quasi-stationary spiral structure. We examine (a) the mechanism for the maintenance of this spiral pattern, and (b) its consequences on the observable features of the galaxy.

Analysis of 2D and 3D defects associated with precipitation of Cu in silicon

1991 ◽  
Vol 49 ◽  
pp. 870-871
Author(s):  
P.M. Rice ◽  
MJ. Kim ◽  
R.W. Carpenter
Keyword(s):  
Colony Growth ◽  
Dark Field ◽  
Dark Side ◽  
Silicide Phase ◽  
Strain Fields ◽  
Near Surface ◽  
Unknown Composition ◽  
Secondary Precipitates ◽  
20 Nm ◽  
2D And 3D

Extrinsic gettering of Cu on near-surface dislocations in Si has been the topic of recent investigation. It was shown that the Cu precipitated hetergeneously on dislocations as Cu silicide along with voids, and also with a secondary planar precipitate of unknown composition. Here we report the results of investigations of the sense of the strain fields about the large (~100 nm) silicide precipitates, and further analysis of the small (~10-20 nm) planar precipitates.Numerous dark field images were analyzed in accordance with Ashby and Brown's criteria for determining the sense of the strain fields about precipitates. While the situation is complicated by the presence of dislocations and secondary precipitates, micrographs like those shown in Fig. 1(a) and 1(b) tend to show anomalously wide strain fields with the dark side on the side of negative g, indicating the strain fields about the silicide precipitates are vacancy in nature. This is in conflict with information reported on the η'' phase (the Cu silicide phase presumed to precipitate within the bulk) whose interstitial strain field is considered responsible for the interstitial Si atoms which cause the bounding dislocation to expand during star colony growth.

Computing cell tractions from particle displacements on silicone rubber substrata

1994 ◽  
Vol 52 ◽  
pp. 162-163
Author(s):  
Tim Oliver ◽  
Akira Ishihara ◽  
Ken Jacobsen ◽  
Micah Dembo
Keyword(s):  
Plane Stress ◽  
Linear Elasticity ◽  
Silicone Rubber ◽  
Mathematical Analysis ◽  
Elastic Recovery ◽  
Video Images ◽  
Cell Traction Forces ◽  
Latex Beads ◽  
Cell Traction ◽  
Better Than

In order to better understand the distribution of cell traction forces generated by rapidly locomoting cells, we have applied a mathematical analysis to our modified silicone rubber traction assay, based on the plane stress Green’s function of linear elasticity. To achieve this, we made crosslinked silicone rubber films into which we incorporated many more latex beads than previously possible (Figs. 1 and 6), using a modified airbrush. These films could be deformed by fish keratocytes, were virtually drift-free, and showed better than a 90% elastic recovery to micromanipulation (data not shown). Video images of cells locomoting on these films were recorded. From a pair of images representing the undisturbed and stressed states of the film, we recorded the cell’s outline and the associated displacements of bead centroids using Image-1 (Fig. 1). Next, using our own software, a mesh of quadrilaterals was plotted (Fig. 2) to represent the cell outline and to superimpose on the outline a traction density distribution. The net displacement of each bead in the film was calculated from centroid data and displayed with the mesh outline (Fig. 3).

Covalent organic frameworks: an ideal platform for designing ordered materials and advanced applications

2021 ◽  
Author(s):  
Ruoyang Liu ◽  
Ke Tian Tan ◽  
Yifan Gong ◽  
Yongzhi Chen ◽  
Zhuoer Li ◽  
...  
Keyword(s):  
Covalent Organic Frameworks ◽  
2D And 3D ◽  
Advanced Applications

Covalent organic frameworks offer a molecular platform for integrating organic units into periodically ordered yet extended 2D and 3D polymers to create topologically well-defined polygonal lattices and built-in discrete micropores and/or mesopores.

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