Some GeometricaL Applications of Vectors

H. Lob

doi:10.1017/s0025557200205767

Some GeometricaL Applications of Vectors

The Mathematical Gazette ◽

10.1017/s0025557200205767 ◽

1936 ◽

Vol 20 (237) ◽

pp. 37-43

Author(s):

H. Lob

Keyword(s):

Unit Circle ◽

First Principles ◽

Unit Circles ◽

A Chain

The aim of the following notes is merely to indicate the straight-forward way in which vectors lend themselves to geometry. Some of the work is less direct than it might be, but this is partly due to the desire to keep to first principles as much as possible. The chain of theorems in § 6 has been suggested by a chain constructed many years ago by Mr. H. W. Richmond. In this, points are taken on a unit circle and the second unit circle through each pair of points is drawn ; the centres give rise to a chain of unit circles. 1.1. Relation between mutual distances of 4 points in a plane. Let O, A, B, C be the four given points.

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Unit Circles and Inverse Trigonometric Functions

Mathematics Teacher ◽

10.5951/mathteacher.108.2.0114 ◽

2014 ◽

Vol 108 (2) ◽

pp. 114-119

Author(s):

Azael Barrera

Keyword(s):

Unit Circle ◽

Trigonometric Functions ◽

Unit Circles ◽

Inverse Trigonometric Functions

A method to determine all the inverse trigonometric functions directly from the unit circle.

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Mn Adsorption on the GaAs(111)–(2×2)B Surface: First Principles Studies

Zeitschrift für Physikalische Chemie ◽

10.1515/zpch-2015-0705 ◽

2016 ◽

Vol 230 (5-7) ◽

Cited By ~ 5

Author(s):

Jonathan Guerrero-Sanchez ◽

J. Castro-Medina ◽

J. F. Rivas-Silva ◽

Noboru Takeuchi ◽

L. Morales de la Garza ◽

...

Keyword(s):

Density Functional Theory ◽

Magnetic Properties ◽

Magnetic Moment ◽

First Principles ◽

Density Functional ◽

Interstitial Atom ◽

Ferromagnetic Behavior ◽

Functional Theory ◽

Spin Polarized ◽

A Chain

AbstractMn adsorption on the GaAs(111)–(1×1)B surface electronic and magnetic properties are investigated using first principles total energy calculations within the periodic spin polarized density functional theory. Results show that one Mn atom adsorption on top of the surface drives to an interstitial Mn atom. The interstitial atom is bonded to three first monolayer As atoms forming a chain-like structure. This stable structure has a ferromagnetic behavior with a Mn magnetic moment of ∼ 3.98 μ

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Unit-circle-preserving mappings

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171204404098 ◽

2004 ◽

Vol 2004 (66) ◽

pp. 3577-3586 ◽

Cited By ~ 1

Author(s):

Soon-Mo Jung ◽

Byungbae Kim

Keyword(s):

Unit Circle ◽

Linear Isometry ◽

Unit Circles ◽

One To One

We prove that if a one-to-one mappingf:ℝn→ℝn(n≥2)preserves the unit circles, thenfis a linear isometry up to translation.

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First-principles design of nanomachines

Proceedings of the National Academy of Sciences ◽

10.1073/pnas.0901429106 ◽

2009 ◽

Vol 106 (17) ◽

pp. 6900-6903 ◽

Cited By ~ 16

Author(s):

Jayanth R. Banavar ◽

Marek Cieplak ◽

Trinh Xuan Hoang ◽

Amos Maritan

Keyword(s):

First Principles ◽

Reference Frames ◽

Hard Spheres ◽

Molecular Machines ◽

Thermal Fluctuations ◽

Chain Molecule ◽

Inherent Anisotropy ◽

Predictive Design ◽

Simple Chain ◽

A Chain

Learning from nature's amazing molecular machines, globular proteins, we present a framework for the predictive design of nanomachines. We show that the crucial ingredients for a chain molecule to behave as a machine are its inherent anisotropy and the coupling between the local Frenet coordinate reference frames of nearby monomers. We demonstrate that, even in the absence of heterogeneity, protein-like behavior is obtained for a simple chain molecule made up of just 30 hard spheres. This chain spontaneously switches between 2 distinct geometries, a single helix and a dual helix, merely because of thermal fluctuations.

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Intuition and Coherence in the Keystone Loop

Exemplars of Truth ◽

10.1093/oso/9780190884277.003.0006 ◽

2018 ◽

pp. 108-130

Author(s):

Keith Lehrer

Keyword(s):

First Principles ◽

Knowledge Claim ◽

The Other ◽

Wilfrid Sellars ◽

Thomas Reid ◽

Other Hand ◽

A Chain ◽

Knowledge Claims

Is knowledge and justification a matter of isolated intuition or coherence with a background system? One intuitionist, Thomas Reid, failed to acknowledge the controversy. He argued that knowledge was a matter of first principles, which drive intuition, but also claimed that the first principles depended on each other like links in a chain, as a coherentist might. Wilfrid Sellars, a famous coherentist, argued that all knowledge was explained by coherence with a background system, but, on the other hand, conceded that some knowledge claims were justified noninferentially, as an intuitionist might. This book suggests a resolution to the conflict in terms of a kind of knowledge requiring the knower be able to defend the target knowledge claim. The defense rests on exemplar representation of experience, yielding intuition, tied together in a keystone loop within a system to defend that representation, yielding coherence.

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On Pure Quasi-Quantum Quadratic Operators of 𝕄2(ℂ) II

Open Systems & Information Dynamics ◽

10.1142/s1230161215500249 ◽

2015 ◽

Vol 22 (04) ◽

pp. 1550024 ◽

Cited By ~ 1

Author(s):

Farrukh Mukhamedov

Keyword(s):

Unit Circle ◽

Early Paper ◽

Unit Sphere ◽

Quadratic Operator ◽

Completely Positive ◽

Potential Interest ◽

Unit Circles ◽

Weaker Conditions

In this paper we study quasi quantum quadratic operators (QQO) acting on the algebra of [Formula: see text] matrices [Formula: see text]. We consider two kinds of quasi QQO the corresponding quadratic operator maps from the unit circle into the sphere and from the sphere into the unit circle, respectively. In our early paper we have defined a q-purity of quasi QQO. This notion is equivalent to the invariance of the unit sphere in [Formula: see text]. But to check this condition, in general, is tricky. Therefore, it would be better to find weaker conditions to check the q-purity. One of the main results of this paper is to provide a criterion of q-purity of quasi QQO in terms of the unit circles. Moreover, we are able to classify all possible kinds of quadratic operators which can produce q-pure quasi QQO. We think that such result will allow one to check whether a given mapping is a pure channel or not. This finding suggests us to study such a class of nonpositive mappings. Correspondingly, the complement of this class will be of potential interest for physicist since this set contains all completely positive mappings.

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Observation of Atom Rows in Thorium Pyromellitate Using a Field Emission STEM

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s042482010007802x ◽

1977 ◽

Vol 35 ◽

pp. 80-81

Author(s):

H. Todokoro ◽

S. Nomura ◽

T. Komoda

Keyword(s):

Field Emission ◽

Amorphous Carbon ◽

Carbon Film ◽

Dark Field Image ◽

Dark Field ◽

Amorphous Carbon Film ◽

Atomic Size ◽

Wide Range ◽

A Chain

It is interesting to observe polymers at atomic size resolution. Some works have been reported for thorium pyromellitate by using a STEM (1), or a CTEM (2,3). The results showed that this polymer forms a chain in which thorium atoms are arranged. However, the distance between adjacent thorium atoms varies over a wide range (0.4-1.3nm) according to the different authors.The present authors have also observed thorium pyromellitate specimens by means of a field emission STEM, described in reference 4. The specimen was prepared by placing a drop of thorium pyromellitate in 10-3 CH3OH solution onto an amorphous carbon film about 2nm thick. The dark field image is shown in Fig. 1A. Thorium atoms are clearly observed as regular atom rows having a spacing of 0.85nm. This lattice gradually deteriorated by successive observations. The image changed to granular structures, as shown in Fig. 1B, which was taken after four scanning frames.

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Time-Resolved Cryo-Electron Microscopy of Oscillating Microtubules

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100181269 ◽

1990 ◽

Vol 48 (1) ◽

pp. 502-503

Author(s):

Eva-Maria Mandelkow ◽

Ron Milligan

Keyword(s):

Electron Microscopy ◽

Dynamic Instability ◽

Entire Solution ◽

Reaction Scheme ◽

Time Resolved ◽

X Ray ◽

X Ray Scattering ◽

A Chain ◽

Ray Scattering

Microtubules form part of the cytoskeleton of eukaryotic cells. They are hollow libers of about 25 nm diameter made up of 13 protofilaments, each of which consists of a chain of heterodimers of α-and β-tubulin. Microtubules can be assembled in vitro at 37°C in the presence of GTP which is hydrolyzed during the reaction, and they are disassembled at 4°C. In contrast to most other polymers microtubules show the behavior of “dynamic instability”, i.e. they can switch between phases of growth and phases of shrinkage, even at an overall steady state [1]. In certain conditions an entire solution can be synchronized, leading to autonomous oscillations in the degree of assembly which can be observed by X-ray scattering (Fig. 1), light scattering, or electron microscopy [2-5]. In addition such solutions are capable of generating spontaneous spatial patterns [6].In an earlier study we have analyzed the structure of microtubules and their cold-induced disassembly by cryo-EM [7]. One result was that disassembly takes place by loss of protofilament fragments (tubulin oligomers) which fray apart at the microtubule ends. We also looked at microtubule oscillations by time-resolved X-ray scattering and proposed a reaction scheme [4] which involves a cyclic interconversion of tubulin, microtubules, and oligomers (Fig. 2). The present study was undertaken to answer two questions: (a) What is the nature of the oscillations as seen by time-resolved cryo-EM? (b) Do microtubules disassemble by fraying protofilament fragments during oscillations at 37°C?

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