On Certain Functional Identities in EN

1971 ◽  
Vol 23 (2) ◽  
pp. 315-324 ◽  
Author(s):  
A. McD. Mercer
Keyword(s):  
Euclidean Space ◽  
Taylor Series ◽  
Dimensional Space ◽  
The Real ◽  
Higher Dimensional ◽  
Isolated Points ◽  
Functional Identities ◽  
Image Position ◽  
Special Case ◽  
Real Euclidean Space

1. If f is a real-valued function possessing a Taylor series convergent in (a — R, a + R), then it satisfies the following operational identity1.1in which D2 = d2/du2. Furthermore, when g is a solution of y″ + λ2y = 0 in (a – R, a + R), then g is such a function and (1.1) specializes to1.2In this note we generalize these results to the real Euclidean space EN, our conclusions being Theorems 1 and 2 below. Clearly, (1.2) is a special case of (1.1) but in higher-dimensional space it is of interest to allow g, now a solution of1.3to possess singularities at isolated points away from the origin. It is then necessary to consider not only a neighbourhood of the origin but annular regions also.

scholarly journals Space–times over normed division algebras, revisited

2020 ◽  
Vol 35 (10) ◽  
pp. 2050055
Author(s):  
R. Vilela Mendes
Keyword(s):  
Homogeneous Spaces ◽  
Dimensional Space ◽  
Internal Symmetry ◽  
Space Time ◽  
Mathematical Framework ◽  
The Real ◽  
Higher Dimensional ◽  
Dimensional Space Time ◽  
Symmetry Transformations ◽  
Extended Group

Normed division and Clifford algebras have been extensively used in the past as a mathematical framework to accommodate the structures of the Standard Model and grand unified theories. Less discussed has been the question of why such algebraic structures appear in Nature. One possibility could be an intrinsic complex, quaternionic or octonionic nature of the space–time manifold. Then, an obvious question is why space–time appears nevertheless to be simply parametrized by the real numbers. How the real slices of an higher-dimensional space–time manifold might be almost independent from each other is discussed here. This comes about as a result of the different nature of the representations of the real kinematical groups and those of the extended spaces. Some of the internal symmetry transformations might however appear as representations on homogeneous spaces of the extended group transformations that cannot be implemented on the elementary states.

The convergence of a sum of independent random variables

1963 ◽  
Vol 59 (2) ◽  
pp. 411-416
Author(s):  
G. De Barra ◽  
N. B. Slater
Keyword(s):  
Distribution Function ◽  
Euclidean Space ◽  
Rate Of Convergence ◽  
Radial Distribution Function ◽  
Random Variables ◽  
Covariance Matrices ◽  
Independent Random Variables ◽  
Second Moments ◽  
Image Position ◽  
Real Euclidean Space

Let Xν, ν= l, 2, …, n be n independent random variables in k-dimensional (real) Euclidean space Rk, which have, for each ν, finite fourth moments β4ii = l,…, k. In the case when the Xν are identically distributed, have zero means, and unit covariance matrices, Esseen(1) has discussed the rate of convergence of the distribution of the sumsIf denotes the projection of on the ith coordinate axis, Esseen proves that ifand ψ(a) denotes the corresponding normal (radial) distribution function of the same first and second moments as μn(a), thenwhere and C is a constant depending only on k. (C, without a subscript, will denote everywhere a constant depending only on k.)

A GENERAL ALGORITHM FOR THE GENERATION OF QUASILATTICES BY MEANS OF THE CUT AND PROJECTION METHOD USING THE SIMPLEX METHOD OF LINEAR PROGRAMMING AND THE MOORE-PENROSE GENERALIZED INVERSE

1989 ◽  
Vol 03 (05) ◽  
pp. 773-786
Author(s):  
J. L. ARAGÓN ◽  
G. VÁZQUEZ POLO ◽  
A. GÓMEZ
Keyword(s):  
Linear Programming ◽  
Euclidean Space ◽  
Projection Method ◽  
Simplex Method ◽  
Computational Algorithm ◽  
Generalized Inverse ◽  
Dimensional Space ◽  
General Algorithm ◽  
Higher Dimensional

A computational algorithm for the generation of quasiperiodic tiles based on the cut and projection method is presented. The algorithm is capable of projecting any type of lattice embedded in any euclidean space onto any subspace making it possible to generate quasiperiodic tiles with any desired symmetry. The simplex method of linear programming and the Moore-Penrose generalized inverse are used to construct the cut (strip) in the higher dimensional space which is to be projected.

scholarly journals MULTIDIMENSIONAL HYBRID BOUNDARY VALUE PROBLEM

2018 ◽  
Vol 58 (6) ◽  
pp. 402-413
Author(s):  
Marzena Szajewska ◽  
Agnieszka Maria Tereszkiewicz
Keyword(s):  
Euclidean Space ◽  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Special Functions ◽  
Boundary Value ◽  
Fundamental Region ◽  
The Real ◽  
Real Euclidean Space

The purpose of this paper is to discuss three types of boundary conditions for few families of special functions orthogonal on the fundamental region. Boundary value problems are considered on a simplex F in the real Euclidean space Rn of dimension n > 2.

Quantizations of Flag Manifolds and Conformal Space Time

1997 ◽  
Vol 09 (04) ◽  
pp. 453-465 ◽  
Author(s):  
R. Fioresi
Keyword(s):  
Euclidean Space ◽  
Minkowski Space ◽  
Conformal Group ◽  
Flag Manifolds ◽  
The Real ◽  
Rham Complex ◽  
De Rham ◽  
Big Cell ◽  
Real Complex ◽  
Real Euclidean Space

In this paper we work out the deformations of some flag manifolds and of complex Minkowski space viewed as an affine big cell inside G(2,4). All the deformations come in tandem with a coaction of the appropriate quantum group. In the case of the Minkowski space this allows us to define the quantum conformal group. We also give two involutions on the quantum complex Minkowski space, that respectively define the real Minkowski space and the real euclidean space. We also compute the quantum De Rham complex for both real (complex) Minkowski and 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.

scholarly journals COLLAPSING SHELLS OF RADIATION IN HIGHER DIMENSIONAL SPACE–TIME AND COSMIC CENSORSHIP CONJECTURE

2001 ◽  
Vol 16 (27) ◽  
pp. 4481-4488 ◽  
Author(s):  
S. G. GHOSH ◽  
R. V. SARAYKAR ◽  
A. BEESHAM
Keyword(s):  
Dimensional Space ◽  
Cosmic Censorship ◽  
Space Time ◽  
Naked Singularities ◽  
Higher Dimensional ◽  
Dimensional Space Time ◽  
Self Similar ◽  
Strong Curvature ◽  
Special Case ◽  
Cosmic Censorship Conjecture

Gravitational collapse of radiation shells in a non-self-similar higher dimensional spherically symmetric space–time is studied. Strong curvature naked singularities form a highly inhomogeneous collapse, violating the cosmic censorship conjecture. As a special case, self similar models can be constructed.

A Banach Space Whose Elements are Classes of Sets of Constant Width

1975 ◽  
Vol 18 (5) ◽  
pp. 679-689 ◽  
Author(s):  
J. E. Lewis
Keyword(s):  
Banach Space ◽  
Euclidean Space ◽  
Compact Subset ◽  
Compact Convex ◽  
Constant Width ◽  
The Real ◽  
Real Euclidean Space ◽  
Convex Subsets

Let K be a compact subset of the real Euclidean space En. We say that K has constant width if the distance between each pair of distinct parallel hyperplanes which support K is constant. The collection of all compact convex subsets of En which have constant width is denoted .

scholarly journals A generalization of Sonine's first finite integral

1963 ◽  
Vol 6 (2) ◽  
pp. 97-98 ◽  
Author(s):  
C. J. Tranter
Keyword(s):  
Bessel Function ◽  
The Real ◽  
Finite Integral ◽  
Image Position ◽  
Special Case

In this note I show thatwhere Jdenotes the Bessel function of the first kind of the orders and arguments indicated, n = 0, 1, 2, 3, … and the real parts of both μand v exceed — 1. This is a generalization of Sonine's first finite integral [1, p. 373] to which it reduces in the special case n = 0.

Celestial Tapestry

2020 ◽  
Author(s):  
Nicholas Mee
Keyword(s):  
Mathematics Curriculum ◽  
Dimensional Space ◽  
Golden Section ◽  
Single Point ◽  
Historical Context ◽  
Computer Art ◽  
Lightning Strike ◽  
Secret Message ◽  
Axiomatic Method ◽  
Higher Dimensional

Celestial Tapestry places mathematics within a vibrant cultural and historical context, highlighting links to the visual arts and design, and broader areas of artistic creativity. Threads are woven together telling of surprising influences that have passed between the arts and mathematics. The story involves many intriguing characters: Gaston Julia, who laid the foundations for fractals and computer art while recovering in hospital after suffering serious injury in the First World War; Charles Howard, Hinton who was imprisoned for bigamy but whose books had a huge influence on twentieth-century art; Michael Scott, the Scottish necromancer who was the dedicatee of Fibonacci’s Book of Calculation, the most important medieval book of mathematics; Richard of Wallingford, the pioneer clockmaker who suffered from leprosy and who never recovered from a lightning strike on his bedchamber; Alicia Stott Boole, the Victorian housewife who amazed mathematicians with her intuition for higher-dimensional space. The book includes more than 200 colour illustrations, puzzles to engage the reader, and many remarkable tales: the secret message in Hans Holbein’s The Ambassadors; the link between Viking runes, a Milanese banking dynasty, and modern sculpture; the connection between astrology, religion, and the Apocalypse; binary numbers and the I Ching. It also explains topics on the school mathematics curriculum: algorithms; arithmetic progressions; combinations and permutations; number sequences; the axiomatic method; geometrical proof; tessellations and polyhedra, as well as many essential topics for arts and humanities students: single-point perspective; fractals; computer art; the golden section; the higher-dimensional inspiration behind modern art.

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