Periodic distributions of dislocations

1964 ◽  
Vol 280 (1383) ◽  
pp. 528-544 ◽  
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Edge Length ◽  
Internal Stresses ◽  
Periodic Distribution ◽  
Stress And Deformation ◽  
The One ◽  
Distribution Of Dislocations ◽  
Distributed Dislocations ◽  
Three Dimensional Space

First, explicit expressions are obtained for the state of stress and deformation due to a periodic distribution of dislocations with respect to three-dimensional space and time. Further, equilibrium conditions for continuously distributed dislocations are derived from the law of energy conservation. The conditions are applied to determine several equilibrium states of periodic distributions. It was found that the distributions of edge and screw dislocations must have a phase difference of ½π when all the Burgers vectors are limited to the one direction. A sudden application of constant stress will cause the dislocations to move spontaneously to their new equilibrium positions. Also, an expression for dislocation velocity is established. In addition, expressions for internal stresses due to the periodic distribution of dislocations are used to find the stress field induced by a Frank network of dislocations. It was found that the normal stress acting on planes parallel to the network has a maximum value at a distance equal to one-half of the edge length of the hexagon of the net. The stress is propor­tional to the sum of the edge components of the three Burgers vectors at a node of the net­work, and decreases exponentially with distance from the network plane.

GEOMETRIC REGULARIZATION AND GAUGE INVARIANCE IN CHERN–SIMONS THEORIES

1992 ◽  
Vol 07 (02) ◽  
pp. 235-256 ◽  
Author(s):  
MANUEL ASOREY ◽  
FERNANDO FALCETO
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Gauge Condition ◽  
Simons Theory ◽  
Coupling Constant ◽  
Gauge Transformations ◽  
Yang Mills ◽  
The One ◽  
Chern Simons ◽  
Three Dimensional Space

Some perturbative aspects of Chern–Simons theories are analyzed in a geometric-regularization framework. In particular, we show that the independence from the gauge condition of the regularized theory, which insures its global meaning, does impose a new constraint on the parameters of the regularization. The condition turns out to be the one that arises in pure or topologically massive Yang–Mills theories in three-dimensional space–times. One-loop calculations show the existence of nonvanishing finite renormalizations of gauge fields and coupling constant which preserve the topological meaning of Chern–Simons theory. The existence of a (finite) gauge-field renormalization at one-loop level is compensated by the renormalization of gauge transformations in such a way that the one-loop effective action remains gauge-invariant with respect to renormalized gauge transformations. The independence of both renormalizations from the space–time volume indicates the topological nature of the theory.

On the Interpretation of Vowel “Quality”: The Dimension of Rounding

1989 ◽  
Vol 19 (1) ◽  
pp. 24-30 ◽  
Author(s):  
Leigh Lisker
Keyword(s):  
Vocal Tract ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Vowel Space ◽  
Tongue Position ◽  
Two Measures ◽  
The One ◽  
Jaw Position ◽  
Three Dimensional Space

The usual description of vowels in respect to their “phonetic quality” requires the linguist to locate them within a so-called “vowel space,” apparently articulatory in nature, and having three dimensions labeled high-low (or close-open), front-back, and unrounded-rounded. The first two are coordinates of tongue with associated jaw position, while the third specifies the posture of the lips. It is recognized that vowels can vary qualitatively in ways that this three-dimensional space does not account for. So, for example, vowels may differ in degree of nasalization, and they may be rhotacized or r-colored. Moreover, it is recognized that while this vowel space serves important functions within the community of linguists, both the two measures of tongue position and the one for the lips inadequately identify those aspects of vocal tract shapes that are primarily responsible for the distinctive phonetic qualities of vowels (Ladefoged 1971). With all this said, it remains true enough that almost any vowel pair of different qualities can be described as occupying different positions with the space. Someone hearing two vowels in sequence and detecting a quality difference will presumably also be able to diagnose the nature of the articulatory shift executed in going from one vowel to the other.

The Inversive Distance Between Two Circles

1967 ◽  
Vol 19 ◽  
pp. 1149-1152
Author(s):  
O. Bottema
Keyword(s):  
Hyperbolic Geometry ◽  
Euclidean Plane ◽  
Dimensional Space ◽  
Single Point ◽  
Three Dimensional ◽  
Complex Numbers ◽  
Inversive Geometry ◽  
The One ◽  
Three Dimensional Space

H. S. M. Coxeter (3) has recently studied the correspondence between two geometries the isomorphism of which was well known, but to which he was able to add some remarkable consequences. The two geometries are the inversive geometry of a plane E (the Euclidean plane completed with a single point at infinity or, what is the same thing, the plane of complex numbers to which ∞ is added) on the one hand, and the hyperbolic geometry of three-dimensional space S.

ON THE PHYSICAL PROPERTIES RELATED TO THE ALGEBRAIC STRUCTURE OF THE DIRAC EQUATION IN THREE-DIMENSIONAL SPACE–TIME

1998 ◽  
Vol 13 (09) ◽  
pp. 1523-1542
Author(s):  
C. A. LINHARES ◽  
JUAN A. MIGNACO
Keyword(s):  
Dirac Equation ◽  
Algebraic Structure ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Space Time ◽  
Four Dimensions ◽  
Dimensional Space Time ◽  
The One ◽  
Physical Consequences ◽  
Three Dimensional Space

We look for the physical consequences resulting from the SU(2) ⊗ SU(2) algebraic structure of the Dirac equation in three-dimensional space–time. We show how this is obtained from the general result we have proven relating the matrices of the Clifford–Dirac ring and the Lie algebra of unitary groups. It allows the introduction of a notion of chirality closely analogous to the one used in four dimensions. The irreducible representations for the Dirac matrices may be labelled with different chirality eigenvalues, and they are related through inversion of any single coordinate axis. We analyze the different discrete transformations for the space of solutions. Finally, we show that the spinor propagator is a direct sum of components with different chirality; the photon propagator receive separate contributions for both chiralities, and the result is that there is no generation of a topological mass at one-loop level. In the case of a charged particle in a constant "magnetic" field we have a good example where chirality plays a determinant role for the degeneracy of states.

Three-dimensional uncertain heat equation

2021 ◽  
pp. 2250012
Author(s):  
Tingqing Ye ◽  
Xiangfeng Yang
Keyword(s):  
Heat Source ◽  
Heat Equation ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Uncertainty Distribution ◽  
Change Rate ◽  
One Dimensional ◽  
Liu Process ◽  
The One ◽  
Three Dimensional Space

Heat equation is a partial differential equation describing the temperature change of an object with time. In the traditional heat equation, the strength of heat source is assumed to be certain. However, in practical application, the heat source is usually influenced by noise. To describe the noise, some researchers tried to employ a tool called Winner process. Unfortunately, it is unreasonable to apply Winner process in probability theory to modeling noise in heat equation because the change rate of temperature will tend to infinity. Thus, we employ Liu process in uncertainty theory to characterize the noise. By modeling the noise via Liu process, the one-dimensional uncertain heat equation was constructed. Since the real world is a three-dimensional space, the paper extends the one-dimensional uncertain heat equation to a three-dimensional uncertain heat equation. Later, the solution of the three-dimensional uncertain heat equation and the inverse uncertainty distribution of the solution are given. At last, a paradox of stochastic heat equation is introduced.

Diffraction contrast of dislocations in icosahedral quasicrystals

1990 ◽  
Vol 48 (4) ◽  
pp. 454-455
Author(s):  
K. Urban ◽  
Z. Zhang ◽  
M. Wollgarten ◽  
D. Gratias
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Complete Characterization ◽  
Extinction Condition ◽  
Burgers Vector ◽  
Diffraction Contrast ◽  
Lattice Geometry ◽  
Reference Space ◽  
Icosahedral Quasicrystals ◽  
The One

Recently dislocations have been observed by electron microscopy in the icosahedral quasicrystalline (IQ) phase of Al65Cu20Fe15. These dislocations exhibit diffraction contrast similar to that known for dislocations in conventional crystals. The contrast becomes extinct for certain diffraction vectors g. In the following the basis of electron diffraction contrast of dislocations in the IQ phase is described. Taking account of the six-dimensional nature of the Burgers vector a “strong” and a “weak” extinction condition are found.Dislocations in quasicrystals canot be described on the basis of simple shear or insertion of a lattice plane only. In order to achieve a complete characterization of these dislocations it is advantageous to make use of the one to one correspondence of the lattice geometry in our three-dimensional space (R3) and that in the six-dimensional reference space (R6) where full periodicity is recovered . Therefore the contrast extinction condition has to be written as gpbp + gobo = 0 (1). The diffraction vector g and the Burgers vector b decompose into two vectors gp, bp and go, bo in, respectively, the physical and the orthogonal three-dimensional sub-spaces of R6.

Extension of the Spatial PDI Model to Three Dimensions

1997 ◽  
Vol 84 (1) ◽  
pp. 176-178
Author(s):  
Frank O'Brien
Keyword(s):  
Population Density ◽  
Dimensional Space ◽  
Finite Lattice ◽  
Three Dimensional ◽  
Upper Bounds ◽  
Three Dimensions ◽  
Lower And Upper Bounds ◽  
Density Index ◽  
Three Dimensional Space

The author's population density index ( PDI) model is extended to three-dimensional distributions. A derived formula is presented that allows for the calculation of the lower and upper bounds of density in three-dimensional space for any finite lattice.

A Peptoid with Extended Shape in Water

2019 ◽  
Author(s):  
Jumpei Morimoto ◽  
Yasuhiro Fukuda ◽  
Takumu Watanabe ◽  
Daisuke Kuroda ◽  
Kouhei Tsumoto ◽  
...  
Keyword(s):  
Spectroscopic Analysis ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Biomolecular Recognition ◽  
Biological Application ◽  
New Class ◽  
Proteolytic Resistance ◽  
Pharmacokinetic Properties ◽  
Optically Pure ◽  
Three Dimensional Space

<div> <div> <div> <p>“Peptoids” was proposed, over decades ago, as a term describing analogs of peptides that exhibit better physicochemical and pharmacokinetic properties than peptides. Oligo-(N-substituted glycines) (oligo-NSG) was previously proposed as a peptoid due to its high proteolytic resistance and membrane permeability. However, oligo-NSG is conformationally flexible and is difficult to achieve a defined shape in water. This conformational flexibility is severely limiting biological application of oligo-NSG. Here, we propose oligo-(N-substituted alanines) (oligo-NSA) as a new peptoid that forms a defined shape in water. A synthetic method established in this study enabled the first isolation and conformational study of optically pure oligo-NSA. Computational simulations, crystallographic studies and spectroscopic analysis demonstrated the well-defined extended shape of oligo-NSA realized by backbone steric effects. The new class of peptoid achieves the constrained conformation without any assistance of N-substituents and serves as an ideal scaffold for displaying functional groups in well-defined three-dimensional space, which leads to effective biomolecular recognition. </p> </div> </div> </div>

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