Crystallography, geometry and physics in higher dimensions. V. Polar and mono-incommensurate point groups in the four-dimensional space %E4

1989 ◽  
Vol 45 (2) ◽  
pp. 187-193 ◽  
Author(s):  
R. Veysseyre ◽  
D. Weigel
Keyword(s):  
Euclidean Space ◽  
Dimensional Space ◽  
Physical Space ◽  
Higher Dimensions ◽  
Dimensional Euclidean Space ◽  
Acta Cryst ◽  
Magnetic Structures ◽  
Convenient Means ◽  
Incommensurate Structures ◽  
Point Groups

The crystallographic point groups of the four-dimensional Euclidean space {\bb E}4are a convenient means of studying some crystallized solids of physical space, for instance the groups of magnetic structures and the groups of mono-incommensurate structures, as is demonstrated by a simple example. The concept of polar crystallographic point groups defined here in {\bb E}4, and also in {\bb E}nenables the list and the WPV notation {geometric symbol of Weigel, Phan & Veysseyre [Acta Cryst.(1987), A43, 294-304]} of these special structures to be stated in a more precise way. This paper is especially concerned with the mono-incommensurate structures while a discussion on magnetic structures will be published later.

LP -bound for the Fourier Transform of Surface-Carried Measures Supported on Hypersurfaces with D∞ Type Singularities

2020 ◽  
pp. 350-359
Author(s):  
Nigina A. Soleeva
Keyword(s):  
Fourier Transform ◽  
Euclidean Space ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Convex Surfaces ◽  
The Fourier Transform

Estimate for Fourier transform of surface-carried measures supported on non-convex surfaces of three-dimensional Euclidean space is considered in this paper.The exact convergence exponent was found wherein the Fourier transform of measures is integrable in tree-dimensional space. This result gives an answer to the question posed by Erd¨osh and Salmhofer

The Osculatory Packing of a Three Dimensional Sphere

1973 ◽  
Vol 25 (2) ◽  
pp. 303-322 ◽  
Author(s):  
David W. Boyd
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Higher Dimensions ◽  
Dimensional Euclidean Space ◽  
Dimensional Sphere ◽  
Specific Knowledge ◽  
Precise Information ◽  
Practical Applications ◽  
Physical Problems ◽  
Three Dimensional Space

Packings by unequal spheres in three dimensional space have interested many authors. This is to some extent due to the practical applications of such investigations to engineering and physical problems (see, for example, [16; 17; 31]). There are a few general results known concerning complete packings by spheres in N-dimensional Euclidean space, due mainly to Larman [20; 21]. For osculatory packings, although there is a great deal of specific knowledge about the two-dimensional situation, the results for higher dimensions, such as [4], rely on general methods which do not give particularly precise information.

scholarly journals Explicit Solutions of the Wave Equation on Three Dimensional Space-Times: Two Examples with Dirichlet Boundary Conditions on a Disk

2006 ◽  
Vol 4 (4) ◽  
Author(s):  
Daniel Boykis ◽  
Patrick Moylan
Keyword(s):  
Wave Equation ◽  
Euclidean Space ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Flat Space ◽  
Dirichlet Boundary Conditions ◽  
Dimensional Euclidean Space ◽  
Curved Space

We study solutions of the wave equation with circular Dirichlet boundary conditions on a flat two-dimensional Euclidean space, and we also study the analogous problem on a certain curved space which is a Lorentzian variant of the 3-sphere. The curved space goes over into the usual flat space-time as the radius R of the curved space goes to infinity. We show, at least in some cases, that solutions of certain Dirichlet boundary value problems are obtained much more simply in the curved space than in the flat space. Since the flat space is the limit R → ∞ of the curved space, this gives an alternative method of obtaining solutions of a corresponding problem in Euclidean space.

On Some Applications of Graph Theory to Geometry

1967 ◽  
Vol 19 ◽  
pp. 968-971 ◽  
Author(s):  
P. Erdös
Keyword(s):  
Graph Theory ◽  
Euclidean Space ◽  
Dimensional Space ◽  
Dimensional Euclidean Space ◽  
Image Position

Let [Pn(k)] be the class of all subsets Pn(k) of the k-dimensional Euclidean space consisting of n distinct points and having diameter 1. Denote by dk(n, r) the maximum number of times a given distance r can occur among points of a set Pn(k).PutIn other words Dk(n) denotes the maximum number of times the same distance can occur between n suitably chosen points in k-dimensional space.

RELATIVISTIC STAR SOLUTIONS IN HIGHER DIMENSIONS

2004 ◽  
Vol 13 (02) ◽  
pp. 229-238 ◽  
Author(s):  
B. C. PAUL
Keyword(s):  
Dimensional Space ◽  
Compact Star ◽  
Compact Object ◽  
Higher Dimensions ◽  
Space Time ◽  
Matter Content ◽  
Dimensional Euclidean Space ◽  
Higher Dimensional Space Time ◽  
Higher Dimensional ◽  
Dimensional Space Time

A class of relativistic solutions of compact star which is in hydrostatic equilibrium is obtained in higher dimensions assuming a spherically symmetric space–time. The space–time geometry is assumed to be a (D-1)-spheroid immersed in a D-dimensional Euclidean space. It is noted that the parameter a which is the measure of spheroidicity of the space–time plays here an important role in determining the equation of state of the matter content in such a star. One obtains a realistic solution when the parameter a picks up values [Formula: see text] for D<9 and [Formula: see text] for space–time dimensions D≥9. It is also observed that the higher dimensional space–time accommodates a more massive compact object for a given size compared to that in the usual four dimensional space–times.

Universal Algebraic Varieties and Ideals in Physics: Field Theory on Algebraic Varieties

1997 ◽  
Vol 11 (21) ◽  
pp. 2533-2592
Author(s):  
Kazumoto Iguchi
Keyword(s):  
Field Theory ◽  
Euclidean Space ◽  
Vector Fields ◽  
High Energy Physics ◽  
Dimensional Space ◽  
High Energy ◽  
Dimensional Euclidean Space ◽  
Algebraic Varieties ◽  
Physical Systems ◽  
And Mathematics

A class of universal algebraic varieties in physics is discussed herein using the concepts of determinant ideals in algebraic geometry. It is shown that these algebraic varieties arise with very different physical contexts in many branches of physics and mathematics from high energy physics theory to chaos theory. In these physical systems the models are constructed by using the fields on usual manifolds such as vector fields in a Euclidean space and a Minkowskian space. But there is a universal mathematical aspect of linear algebra for linear vector spaces, where the linear independency and dependency are described using the Gramians of the vectors. These Gramians form a class of hypersurfaces in a higher-dimensional mathematical space: If there exist g vectors vi in an n-dimensional Euclidean space, the Gramian Gg is given as a g × g determinant Gg= Det [xij] with the inner products xij=(vi,vj), and exists in a g(g-1)/2-[g(g+1)/2-] dimensional space if the vectors are (not) normalized, xii=1(xii ≠ 1). It is also shown that the Gramians are invariant under automorphisms of the vectors. The mathematical structure of the Gramians is revealed to be equivalent to the concepts of determinant ideals Ig(v), each element of which is a g × g determinant constructed from components of an arbitrary N×N matrix with N>n and which have inclusion relation: R=I0(v)⊃ I1(v) ⊃···⊃ Ig(v) ⊃···, and Ig(v)=0 if g>n. In the various physical systems the ideals naturally emerge to give us dynamical flows on the hypersurfaces, and therefore, it is called the field theory on algebraic varieties. This viewpoint provides us a grand viewpoint in physics and mathematics.

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