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.

scholarly journals Chaotic Attractor Generation via Space Function Controls

2011 ◽  
Vol 2011 ◽  
pp. 1-11
Author(s):  
Hong Shi ◽  
Guangming Xie ◽  
Desheng Liu
Keyword(s):  
Linear Systems ◽  
Chaotic Attractor ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Underlying Mechanism ◽  
Two Dimensional ◽  
One Dimensional ◽  
Suitable Space ◽  
Space Function ◽  
Three Dimensional Space

The analysis of chaotic attractor generation is given, and the generation of novel chaotic attractor is introduced in this paper. The underlying mechanism involves two simple linear systems with one-dimensional, two-dimensional, or three-dimensional space functions. Moreover, it is demonstrated by simulation that various attractor patterns are generated conveniently by adjusting suitable space functions' parameters and the statistic behavior is also discussed.

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.

Cubic Hybrid Mechanism Based on S[T] Output Base and P/R Input Base

2012 ◽  
Vol 430-432 ◽  
pp. 1725-1728
Author(s):  
Jian Guo Luo ◽  
Mao Yan He
Keyword(s):  
Parallel Mechanism ◽  
Dimensional Space ◽  
Three Dimensional ◽  
One Dimensional ◽  
Hybrid Mechanism ◽  
Hybrid Manipulator ◽  
New Type ◽  
The Stability ◽  
Serial Mechanism ◽  
Three Dimensional Space

Based on the flexibility of single couple of serial mechanism and the stability of multi couples of parallel mechanism, a new type of S[T] output base of hybrid mechanism presented, component of sphere joint run through the tiger joint, this component still the output one with the capability of rotate in three dimensional space. Add serial branch including three translation couple P or/and rotation couple R to the new type of S[T] output base, put these members on one cubic frame, twenty seven configurations obtained with 3-DOF(degree of freedom) allow of three dimensional rotation, twenty seven configurations belong to three conditions obtained with 4-DOF allow of three dimensional rotation and one dimensional translation, nine configurations belong to three conditions obtained with 5-DOF allow of three dimensional rotation and two dimensional translation, one configuration obtained with 6-DOF allow of three dimensional rotation and three dimensional translation, all those sixty four configurations have no more than six translation couple or rotation couple, and the sum of two kind of couple is equal to six. Developing new type of hybrid manipulator based on the hybrid cubic mechanism constructed with S[T] output base and P/R input base will be possible in theory and useful.

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.

EFFECTIVE DIPOLE-RADIATION-FIELD THEORY I: ONE-DIMENSIONAL OSCILLATOR BEYOND STANDARD COUPLING

1994 ◽  
Vol 08 (17) ◽  
pp. 2307-2325 ◽  
Author(s):  
H. DEKKER
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Dipole Interaction ◽  
Quantum Mechanical ◽  
Renormalization Procedure ◽  
One Dimensional ◽  
Dipole Radiation ◽  
Consistent Treatment ◽  
Interaction Problem ◽  
Three Dimensional Space

A novel generalization is given of the standard dipole interaction between a charged particle and the electromagnetic field in the radiation gauge. The resulting nonlinear interaction problem is statistically linearized. The ensuing dynamics is solved exactly for a harmonically bound nonrelativistic electron in a finite region of three-dimensional space. The solution involves a generalized renormalization procedure and is free of runaway modes. The theory is particularly suited for a self-consistent treatment of the system's quantum mechanics. As a consequence of the generalized coupling an earlier noted ultraviolet quantum mechanical divergence is absent.

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.

scholarly journals A-topology for Minkowski space

1979 ◽  
Vol 21 (1) ◽  
pp. 53-64 ◽  
Author(s):  
Sribatsa Nanda
Keyword(s):  
Minkowski Space ◽  
Dimensional Space ◽  
Three Dimensional ◽  
One Dimensional ◽  
Group Of Homeomorphisms ◽  
Dimensional Space Time ◽  
The One ◽  
Induced Topology ◽  
Time Continuum ◽  
Inhomogeneous Lorentz Group

AbstractWe consider in this paper a topology (which we call the A-topology) on Minkowski space, the four-dimensional space–time continuum of special relativity and derive its group of homeomorphisms. We define the A-topology to be the finest topology on Minkowski space with respect to which the induced topology on time-like and light-like lines is one-dimensional Euclidean and the induced topology on space-like hyperplanes is three- dimensional Euclidean. It is then shown that the group of homeomorphisms of this topology is precisely the one generated by the inhomogeneous Lorentz group and the dilatations.

scholarly journals One-dimensional sensor learns to sense three-dimensional space

2020 ◽  
Vol 28 (13) ◽  
pp. 19374 ◽  
Author(s):  
Chen Zhu ◽  
Rex E. Gerald II ◽  
Yizheng Chen ◽  
Jie Huang
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
One Dimensional ◽  
Three Dimensional Space

Reflection from Curved Mirrors in a Plane

2019 ◽  
Vol 7 (1) ◽  
pp. 46-54 ◽  
Author(s):  
Л. Жихарев ◽  
L. Zhikharev
Keyword(s):  
Dimensional Space ◽  
Spherical Surface ◽  
Three Dimensional ◽  
Two Dimensional ◽  
Interesting Phenomenon ◽  
One Dimensional ◽  
Straight Line ◽  
The Family ◽  
Dimensional Object ◽  
Three Dimensional Space

Reflection from a certain mirror is one of the main types of transformations in geometry. On a plane a mirror represents a straight line. When reflecting, we obtain an object, each point of which is symmetric with respect to this straight line. In this paper have been considered examples of reflection from a circle – a general case of a straight line, if the latter is defined through a circle of infinite radius. While analyzing a simple reflection and generalization of this process to the cases of such curvature of the mirror, an interesting phenomenon was found – an increase in the reflection dimension by one, that is, under reflection of a one-dimensional object from the circle, a two-dimensional curve is obtained. Thus, under reflection of a point from the circle was obtained the family of Pascal's snails. The main cases, related to reflection from a circular mirror the simplest two-dimensional objects – a segment and a circle at their various arrangement, were also considered. In these examples, the reflections are two-dimensional objects – areas of bizarre shape, bounded by sections of curves – Pascal snails. The most interesting is the reflection of two-dimensional objects on a plane, because the reflection is too informative to fit in the appropriate space. To represent the models of obtained reflections, it was proposed to move into three-dimensional space, and also developed a general algorithm allowing obtain the object reflection from the curved mirror in the space of any dimension. Threedimensional models of the reflections obtained by this algorithm have been presented. This paper reveals the prospects for further research related to transition to three-dimensional space and reflection of objects from a spherical surface (possibility to obtain four-dimensional and five-dimensional reflections), as well as studies of reflections from geometric curves in the plane, and more complex surfaces in space.

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