scholarly journals On lattice points in n -dimensional star bodies I. Existence theorems

1946 ◽  
Vol 187 (1009) ◽  
pp. 151-187 ◽  
Keyword(s):  
Euclidean Space ◽  
Existence Theorems ◽  
Lattice Points ◽  
Infinite Group ◽  
Dimensional Euclidean Space ◽  
Star Body ◽  
Linear Transformations ◽  
Real Numbers ◽  
Negative Function ◽  
Star Bodies

Let F ( X ) = F ( x 1 ,..., x n ) be a continuous non-negative function of X satisfying F ( tX ) = | t | F ( X ) for all real numbers t . The set K in n -dimensional Euclidean space R n defined by F ( X )⩽ 1 is called a star body. The author studies the lattices Λ in R n which are of minimum determinant and have no point except (0, ..., 0) inside K . He investigates how many points of such lattices lie on, or near to, the boundary of K , and considers in detail the case when K admits an infinite group of linear transformations into itself.

scholarly journals An elementary proof of Minkowski's second inequality

1969 ◽  
Vol 10 (1-2) ◽  
pp. 177-181 ◽  
Author(s):  
I. Danicic
Keyword(s):  
Euclidean Space ◽  
Elementary Proof ◽  
Content Volume ◽  
Lattice Points ◽  
The Body ◽  
Dimensional Euclidean Space ◽  
Open Convex ◽  
Successive Minima ◽  
Linearly Independent ◽  
Independent Lattice

Let K be an open convex domain in n-dimensional Euclidean space, symmetric about the origin O, and of finite Jordan content (volume) V. With K are associated n positive constants λ1, λ2,…,λn, the ‘successive minima of K’ and n linearly independent lattice points (points with integer coordinates) P1, P2, …, Pn (not necessarily unique) such that all lattice points in the body λ,K are linearly dependent on P1, P2, …, Pj-1. The points P1,…, Pj lie in λK provided that λ > λj. For j = 1 this means that λ1K contains no lattice point other than the origin. Obviously

The anomaly of convex bodies

1953 ◽  
Vol 49 (1) ◽  
pp. 54-58 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Euclidean Space ◽  
Real Number ◽  
Convex Bodies ◽  
Dimensional Euclidean Space ◽  
Real Numbers

I write X for the point (x1, x2, …, xn) of n-dimensional Euclidean space Rn. The coordinates x1, x2, …, xn are real numbers. The origin (0, 0,…, 0) is denoted by O. If t is a real number, tX denotes the point (tx1, tx2, …, txn); in particular, − X is the point (−x1, −x2,…, −xn). Also X + Y denotes the point {x1 + y1, x2 + y2, …, xn + yn).

scholarly journals On the critical determinants of certain star bodies

2017 ◽  
Vol 25 (1) ◽  
pp. 5-11 ◽  
Author(s):  
Werner Georg Nowak
Keyword(s):  
Calculus Of Variations ◽  
Diophantine Approximation ◽  
Star Body ◽  
Geometry Of Numbers ◽  
Real Numbers ◽  
Critical Determinant ◽  
Simultaneous Diophantine Approximation ◽  
Star Bodies ◽  
Classic Paper

Abstract In a classic paper [14], W.G. Spohn established the to-date sharpest estimates from below for the simultaneous Diophantine approximation constants for three and more real numbers. As a by-result of his method which used Blichfeldt’s Theorem and the calculus of variations, he derived a bound for the critical determinant of the star body|x1|(|x1|3 + |x2|3 + |x3|3 ≤ 1.In this little note, after a brief exposition of the basics of the geometry of numbers and its significance for Diophantine approximation, this latter result is improved and extended to the star body|x1|(|x1|3 + |x22 + x32)3/2≤ 1.

scholarly journals On the Irreducibility of Convex Bodies

1959 ◽  
Vol 11 ◽  
pp. 256-261 ◽  
Author(s):  
A. C. Woods
Keyword(s):  
Euclidean Space ◽  
Convex Bodies ◽  
Dimensional Euclidean Space ◽  
Star Body ◽  
Closed Set ◽  
Absolute Value ◽  
Linearly Independent ◽  
The Absolute ◽  
Set Of Points ◽  
Rational Integral

We select a Cartesian co-ordinate system in ndimensional Euclidean space Rn with origin 0 and employ the usual pointvector notation.By a lattice Λ in Rn we mean the set of all rational integral combinations of n linearly independent points X1, X2, … , Xn of Rn. The points X1 X2, … , Xn are said to form a basis of Λ. Let {X1, X2, … , Xn) denote the determinant formed when the co-ordinates of Xi are taken in order as the ith row of the determinant for i = 1,2, … , n. The absolute value of this determinant is called the determinant d(Λ) of Λ. It is well known that d(Λ) is independent of the particular basis one takes for Λ.A star body in Rn is a closed set of points K such that if X ∈ K then every point of the form tX where — 1 < t < 1 is an inner point of K.

scholarly journals Decomposition of the n-Dimensional Lattice-Graph into Hamiltonian Lines

1961 ◽  
Vol 12 (3) ◽  
pp. 123-131 ◽  
Author(s):  
C. ST.J. A. Nash-Williams
Keyword(s):  
Euclidean Space ◽  
Unit Length ◽  
Lattice Points ◽  
Dimensional Euclidean Space ◽  
Dimensional Lattice ◽  
Edge Disjoint ◽  
Unordered Pair ◽  
Spanning Subgraph ◽  
Lattice Graph ◽  
Disjoint Sets

A graph G consists, for the purposes of this paper, of two disjoint sets V(G), E(G), whose elements are called vertices and edges respectively of G, together with a relationship whereby with each edge is associated an unordered pair of distinct vertices (called its end-vertices) which the edge is said to join, and whereby no two vertices are joined by more than one edge. An edge γ and vertex ξ are incident if ξ is an end-vertex of γ. A monomorphism [isomorphism] of a graph G into [onto] a graph H is a one-to-one function φ from V(G)∪E(G) into [onto] V(H)∪E(H) such that φ(V(G))⊂V(H), φ(E(G))⊂E(H) and an edge and vertex of G are incident in G if and only if their images under φ are incident in H. G and H are isomorphic (in symbols, G ≅ H) if there exists an isomorphism of G onto H. A subgraph of G is a graph H such that V(H) ⊂ V(G), E(H)⊂E(G) and an edge and vertex of H are incident in H if and only if they are incident in G; if V(H) = V(G), H is a spanning subgraph. A collection of graphs are edge-disjoint if no two of them have an edge in common. A decomposition of G is a set of edge-disjoint subgraphs of G which between them include all the edges and vertices of G. Ln is a graph whose vertices are the lattice points of n-dimensional Euclidean space, two vertices A and B being joined by an edge if and only if AB is of unit length (and therefore necessarily parallel to one of the co-ordinate axes). An endless Hamiltonian line of a graph G is a spanning subgraph of G which is isomorphic to L1. The object of this paper is to prove that Ln is decomposable into n endless Hamiltonian lines, a result previously established (1) for the case where n is a power of 2.

On the Symmetries of Spherical Harmonics

1954 ◽  
Vol 6 ◽  
pp. 135-157 ◽  
Author(s):  
Burnett Meyer
Keyword(s):  
Finite Group ◽  
Euclidean Space ◽  
Spherical Harmonics ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Linear Transformations ◽  
A Finite Group

Let be a finite group of transformations of three-dimensional Euclidean space, such that the distance between any two points is preserved by all transformations of the group. Such a group is a group of orthogonal linear transformations of three variables, or, geometrically speaking, a group of rotations and rotatory inversions. Thirty-two groups of this type are important in crystallography and are known as the crystallographic classes.

scholarly journals Unitary Representations of Some Linear Groups

1952 ◽  
Vol 4 ◽  
pp. 1-13 ◽  
Author(s):  
Seizô Itô
Keyword(s):  
Euclidean Space ◽  
Analogous Result ◽  
Unitary Representations ◽  
Dimensional Euclidean Space ◽  
Linear Groups ◽  
Complex Numbers ◽  
Two Dimensional ◽  
Linear Transformations ◽  
Straight Line ◽  
Irreducible Unitary Representations

Recently I. Gelfand and M. Neumark [2] have determined the types of irreducible unitary representations of the group G1 of linear transformations of the straight line. The analogous result is obtained for the group G2 of transformations z → az + b in the complex-number plane , where a and b run over all complex numbers with the exception of a = 0, which may be considered as the group of all sense-preserving similar transformations in the two-dimensional euclidean space E2.

scholarly journals Lattice points in a convex Set of given width

1976 ◽  
Vol 21 (4) ◽  
pp. 504-507 ◽  
Author(s):  
G. B. Elkington ◽  
J. Hammer
Keyword(s):  
Euclidean Space ◽  
Minimum Distance ◽  
Convex Set ◽  
Lattice Points ◽  
Dimensional Euclidean Space ◽  
Integral Lattice ◽  
Supporting Hyperplanes ◽  
Bounded Convex

Let S be a closed bounded convex set in d-dimensional Euclidean space Ed. The width w(S) of S is the minimum distance between supporting hyperplanes of S, and L(S) is the number of integral lattice points in the interior of S.

Lattice Octahedra

1960 ◽  
Vol 12 ◽  
pp. 297-302 ◽  
Author(s):  
L. J. Mordell
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Real Numbers ◽  
Linearly Independent ◽  
Image Position

Let Ai, A2, … , An be n linearly independent points in n-dimensional Euclidean space of a lattice Λ. The points ± A1, ±A2, . . , ±An define a closed n-dimensional octahedron (or “cross poly tope“) K with centre at the origin O. Our problem is to find a basis for the lattices Λ which have no points in K except ±A1, ±A2, … , ±An.Let the position of a point P in space be defined vectorially by1where the p are real numbers. We have the following results.When n = 2, it is well known that a basis is2When n = 3, Minkowski (1) proved that there are two types of lattices, with respective bases3When n = 4, there are six essentially different bases typified by A1, A2, A3 and one of4In all expressions of this kind, the signs are independent of each other and of any other signs. This result is a restatement of a result by Brunngraber (2) and a proof is given by Wolff (3).

Repeated vector products

2020 ◽  
Vol 104 (561) ◽  
pp. 460-468
Author(s):  
A. F. Beardon ◽  
N. Lord
Keyword(s):  
Euclidean Space ◽  
Binary Operation ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Vector Product ◽  
Real Numbers ◽  
Image Position

In [1] the second author observed that it is possible to have a binary operation * on a set X with the property that two different arrangements of brackets in a given combination x1 * … * xn of elements of X yield the same outcome for all choices of the xj. For example, for the operation of subtraction on the set of real numbers, we have $$\left[ {a - \left( {b - c} \right)} \right] - d = a - \left[ {b - \left( {c - d} \right)} \right]$$ for all real numbers a, b, c and d. The author then asked whether or not a similar example might hold for an n-fold vector product on three-dimensional Euclidean space3. We shall show here that no such example can exist; thus two different arrangements of brackets in a repeated vector product will, for some vectors, yield different answers.

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