Matter Tensors and Coordinate Transformations

1996 ◽  
Author(s):  
Gerhard Oertel
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Coordinate Transformations ◽  
Ordered Arrays ◽  
The Subject ◽  
Three Dimensional Space ◽  
Three Components

Vectors, the subject of the previous two chapters, may be classified as members of a class of mathematical entities called tensors, insofar as they can be expressed in the form of ordered arrays, or matrices, and insofar as they further conform to conditions to be explored in the present chapter. Tensors can have various ranks, and vectors are tensors of the first rank, which in three-dimensional space have 31 or three components. Much of this, and later, chapters deals with tensors of the second rank which in the same space have 32 or nine components. Tensors of higher (nth) rank do exist and have 3n components, and so do, at least nominally, tensors of zero rank with a single, or 30, component, which makes them scalars. Tensors of the second rank for three dimensions are written as three-by-three matrices with each component marked by two subscripts, which may be either letters or numbers.

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.

Influence of Ball and Background Patterns on Perception of Visual Direction of a Moving Object

1979 ◽  
Vol 49 (2) ◽  
pp. 343-346 ◽  
Author(s):  
Marcella V. Ridenour
Keyword(s):  
Analysis Of Variance ◽  
Moving Objects ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Moving Object ◽  
Visual Direction ◽  
Visual Patterns ◽  
Prediction Time ◽  
The Subject ◽  
Three Dimensional Space

30 boys and 30 girls, 6 yr. old, participated in a study assessing the influence of the visual patterns of moving objects and their respective backgrounds on the prediction of objects' directionality. An apparatus was designed to permit modified spherical objects with interchangeable covers and backgrounds to move in three-dimensional space in three directions at selected speeds. The subject's task was to predict one of three possible directions of an object: the object either moved toward the subject's midline or toward a point 18 in. to the left or right of the midline. The movements of all objects started at the same place which was 19.5 ft. in front of the subject. Prediction time was recorded on 15 trials. Analysis of variance indicated that visual patterns of the moving object did not influence the prediction of the object's directionality. Visual patterns of the background behind the moving object did not influence the prediction of the object's directionality except during the conditions of a light nonpatterned moving object. It was concluded that visual patterns of the background and that of the moving object have a very limited influence on the prediction of direction.

scholarly journals Fashion Window: A Conceptual Analysis Of Department Store Holiday Windows

2021 ◽  
Author(s):  
McKenzie Bohn
Keyword(s):  
Research Design ◽  
Conceptual Analysis ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Gestalt Theory ◽  
Fashion Industry ◽  
Art Object ◽  
The Subject ◽  
Window Displays ◽  
Three Dimensional Space

Window displays in the fashion industry are unique sites of meaning that combine advertising and artwork in a three-dimensional space. The current body of research surrounding window displays approaches the subject from a marketer’s position and attempts to evaluate performance. This project shifts the focus to the artistic qualities of window displays as they are used by fashion retailers. The primary theoretical lens is gestalt theory, which has applications in both psychology and design. The specific windows examined are the Christmas windows at retailer Saks Fifth Avenue Toronto in December of 2018. An autoethnographic research design is employed, resulting in an exploratory empirical analysis that serves as an entry into an under-represented area of study: the fashion window as an art object. The key findings of the project are the application of gestalt theory to the design of the windows and the researcher’s observations to suggest an explanation of the public’s response to the displays.

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 On a set of quartic surfaces in space of four dimensions; and a certain involutory transformation

1926 ◽  
Vol 110 (756) ◽  
pp. 700-708
Keyword(s):  
Present Note ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Quartic Surface ◽  
Four Dimensions ◽  
Cubic Surfaces ◽  
Mutual Relations ◽  
Quartic Surfaces ◽  
Three Dimensional Space

There exists in space of four dimensions an interesting figure of 15 lines and 15 points, first considered by Stéphanos (‘Compt. Rendus,’ vol. 93, 1881), though suggested very clearly by Cremona’s discussion of cubic surfaces in three-dimensional space. In connection with the figure of 15 lines there arises a quartic surface, the intersection of two quadrics, which is of importance as giving rise by projection to the Cyclides, as Segre has shown in detail (‘Math. Ann.,’ vol. 24, 1884). The symmetry of the figure suggests, howrever, the consideration of 15 such quartic surfaces; and it is natural to inquire as to the mutual relations of these surfaces, in particular as to their intersections. In general, two surfaces in space of four dimensions meet in a finite number of points. It appears that in this case any two of these 15 surfaces have a curve in common; it is the purpose of the present note to determine the complete intersection of any two of these 15 surfaces. Similar results may be obtained for a system of cubic surfaces in three dimensions, corresponding to those here given for this system of quartic surfaces in four dimensions, since the surfaces have one point in common, which may be taken as the centre of a projection.

Perspective production in a savant autistic draughtsman

1995 ◽  
Vol 25 (3) ◽  
pp. 639-648 ◽  
Author(s):  
L. Mottron ◽  
S. Belleville
Keyword(s):  
Computational Analysis ◽  
Cognitive Deficit ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Theoretical Models ◽  
Linear Perspective ◽  
Vanishing Point ◽  
Vanishing Points ◽  
The Subject ◽  
Three Dimensional Space

SYNOPSISThis study examines perspective construction in an autistic patient (E.C.) with quasi-normal intelligence who exhibits exceptional ability when performing three-dimensional drawings of inanimate objects. Examination of E.C.'s spontaneous graphic productions showed that although his drawings approximate the ‘linear perspective’ system, the subject does not use vanishing points in his productions. Nevertheless, a formal computational analysis of E.C.'s accuracy in an experimental task showed that he was able to draw objects rotated in three-dimensional space more accurately than over-trained controls. This accuracy was not modified by suppressing graphic cues that permitted the construction of a vanishing point. E.C. was also able to detect a perspective incongruency between an object and a landscape at a level superior to that of control subjects. Since E.C. does not construct vanishing points in his drawings, it is proposed that his production of a precise realistic perspective is reached without the use of explicit or implicit perspective rules. ‘Special abilities’ in perspective are examined in relation to existing theoretical models of the cognitive deficit in autism and are compared to other special abilities in autism.

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.

On the Problem of “Reverse Perspective”: Definitions East and West

Leonardo ◽  
2010 ◽  
Vol 43 (5) ◽  
pp. 464-469 ◽  
Author(s):  
Clemena Antonova
Keyword(s):  
20Th Century ◽  
Dimensional Space ◽  
Three Dimensional ◽  
The West ◽  
Pictorial Space ◽  
History Of ◽  
The Subject ◽  
East And West ◽  
Three Dimensional Space

The author considers the history of the theory of “reverse perspective” in the 20th century. She identifies six distinct views on reverse perspective, some of which are mutually exclusive. The first four definitions have circulated in both Western and Russian scholarship, while two further views proposed by Russian authors are little known in the West. The most useful contribution of Russian theory to the subject is the suggestion of a pictorial space fundamentally different from the three-dimensional space frequently taken for granted by Western viewers.

THREE-D SINGLETONS AND 2-D C.F.T.

1992 ◽  
Vol 07 (10) ◽  
pp. 2193-2206 ◽  
Author(s):  
A.M. HARUN AR-RASHID ◽  
C. FRONSDAL ◽  
M. FLATO
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Alternative Formulation ◽  
Boundary Values ◽  
Witten Theory ◽  
Space Time Structure ◽  
Dimensional Space Time ◽  
Left And Right ◽  
Three Dimensional Space

Two-dimensional Wess-Zumino-Novikov-Witten theory is extended to three dimensions, where it becomes a scalar gauge theory of the singleton type. The three-dimensional formulation involves a scalar field valued in a compact group G, a Nakanishi-Lautrup field valued in Lie (G) and Faddeev-Popov ghosts. The physical sector, characterized by the vanishing of the Nakanishi-Lautrup field, coincides with the WZNW theory of the group G. Three-dimensional space-time structure involves a generalized metric, but only its boundary values are of consequence. An alternative formulation in terms of left and right movers (in three dimensions!) is also possible.

A notation for vectors and tensors

1955 ◽  
Vol 51 (3) ◽  
pp. 449-453
Author(s):  
F. C. Powell
Keyword(s):  
Scalar Product ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Usual Form ◽  
Three Dimensional Space ◽  
Scalar Moment

The vector notation commonly employed in elementary physics cannot be applied in its usual form to spaces of other than three dimensions. In plane dynamics, for instance, it cannot be used to represent the velocity (– ωx2, ωx1) at the point (x1, x2) due to a rotation ω about the origin, or the (scalar) moment about the origin of the force (F1, F2) acting at (x1, x2). In relativity physics the symbol ⋅ is often used to denote the scalar product of two vectors, it is true, and the tensor aαbβ – aβbα is sometimes denoted by a × b, but there exists no body of rules for the manipulation of these symbols that enables one to dispense with the suffix notation as in the case of vectors in three-dimensional space.

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