A Note on Coverings of En by Convex Sets

1967 ◽  
Vol 10 (5) ◽  
pp. 669-673 ◽  
Author(s):  
J.H.H. Chalk
Keyword(s):  
Convex Sets ◽  
Independent Set ◽  
Linearly Independent ◽  
Image Position ◽  
Take All

Let (x1, x2, …, xn) denote the coordinates of a point of Euclidean n-space En. Let be a set of n+1 points of En with the property thatform a linearly independent set and define a lattice Λ of pointsby allowing u1, …, un to take all integer values.

scholarly journals On Schauder Bases for Spaces of Continuous Functions1)

1960 ◽  
Vol 3 (2) ◽  
pp. 173-184 ◽  
Author(s):  
H. W. Ellis ◽  
D. G. Kuehner
Keyword(s):  
Independent Set ◽  
Vector Spaces ◽  
Hamel Basis ◽  
Infinite Dimensional ◽  
Finite Dimensional Vector Space ◽  
Finite Dimensional ◽  
Linearly Independent ◽  
Schauder Bases ◽  
Image Position ◽  
Normed Vector

In a finite dimensional vector space V a set xi, i = 1, 2, …, n of vectors of V is said to be a basis, base, or coordinate system for V if the vectors xi are linearly independent and if each vector in V is a linear combination of the elements x1 with real coefficients. If a topology for V is defined in terms of a norm ||.|| then {xi} is a basis for V if and only if to each x ϵ V corresponds a unique set of constants ai such thatIn infinite dimensional normed vector spaces the above concepts of basis have different generalizations. The first or algebraic definition gives a Hamel basis which is a maximal linearly independent set [l, p. 2]. We shall be interested in the other or topological definition.

scholarly journals An elementary renewal theorem for random compact convex sets

1995 ◽  
Vol 27 (4) ◽  
pp. 931-942 ◽  
Author(s):  
Ilya S. Molchanov ◽  
Edward Omey ◽  
Eugene Kozarovitzky
Keyword(s):  
Support Function ◽  
Convex Sets ◽  
Compact Convex ◽  
Renewal Function ◽  
Renewal Theorem ◽  
Closed Sets ◽  
Random Closed Sets ◽  
Minkowski Sums ◽  
Image Position ◽  
Unit Vectors

A set-valued analog of the elementary renewal theorem for Minkowski sums of random closed sets is considered. The corresponding renewal function is defined as where are Minkowski (element-wise) sums of i.i.d. random compact convex sets. In this paper we determine the limit of H(tK)/t as t tends to infinity. For K containing the origin as an interior point, where hK(u) is the support function of K and is the set of all unit vectors u with EhA(u) > 0. Other set-valued generalizations of the renewal function are also suggested.

Kruskal's theorem for matroids

1968 ◽  
Vol 64 (1) ◽  
pp. 3-4 ◽  
Author(s):  
D. J. A. Welsh
Keyword(s):  
Vector Space ◽  
Finite Subset ◽  
Spanning Tree ◽  
General Theorem ◽  
Independent Set ◽  
Easy Method ◽  
Maximal Weight ◽  
Linearly Independent ◽  
Kruskal’S Theorem ◽  
Special Case

AbstractKruskal's theorem for obtaining a minimal (maximal) spanning tree of a graph is shown to be a special case of a more general theorem for matroid spaces in which each element of the matroid has an associated weight. Since any finite subset of a vector space can be regarded as a matroid space this theorem gives an easy method of selecting a linearly independent set of vectors of minimal (maximal) weight.

An Extremum Result

1962 ◽  
Vol 14 ◽  
pp. 597-601 ◽  
Author(s):  
J. Kiefer
Keyword(s):  
Probability Measure ◽  
Compact Space ◽  
Ad Hoc ◽  
Continuous Functions ◽  
Orthogonality Condition ◽  
Technical Detail ◽  
Real Numbers ◽  
Discrete Probability ◽  
Linearly Independent ◽  
Image Position

The main object of this paper is to prove the following:Theorem. Let f1, … ,fk be linearly independent continuous functions on a compact space. Then for 1 ≤ s ≤ k there exist real numbers aij, 1 ≤ i ≤ s, 1 ≤ j ≤ k, with {aij, 1 ≤ i, j ≤ s} n-singular, and a discrete probability measure ε*on, such that(a) the functions gi = Σj=1kaijfj 1 ≤ i ≤ s, are orthonormal (ε*) to the fj for s < j ≤ k;(b)The result in the case s = k was first proved in (2). The result when s < k, which because of the orthogonality condition of (a) is more general than that when s = k, was proved in (1) under a restriction which will be discussed in § 3. The present proof does not require this ad hoc restriction, and is more direct in approach than the method of (2) (although involving as much technical detail as the latter in the case when the latter applies).

26.—The Strong Limit-2 Case of Fourth-order Differential Equations

1974 ◽  
Vol 71 (4) ◽  
pp. 297-304
Author(s):  
V. Krishna Kumar
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Order Equation ◽  
Complex Numbers ◽  
Strong Limit ◽  
Fourth Order Equation ◽  
Linearly Independent ◽  
Image Position ◽  
Adjoint Operators ◽  
Bounded Below

SynopsisThe fourth-order equation considered isConditions are given on the coefficients r, p and q which ensure that this differential equation (*) is in the strong limit-2 case at ∞, i.e. is limit-2 at ∞. This implies that (*) has exactly two linearly independent solutions which are in the integrable-square space ℒ2(0, ∞) for all complex numbers λ with im [λ] ≠ 0. Additionally the conditions imply that self-adjoint operators generated by M[·] in ℒ2(0, ∞) are semi-bounded below. The results obtained are applied to the case when the coefficients r, p and q are powers of x ∈ [0, ∞).

An ordinal partition avoiding pentagrams

2000 ◽  
Vol 65 (3) ◽  
pp. 969-978 ◽  
Author(s):  
Jean A. Larson
Keyword(s):  
Independent Set ◽  
Order Type ◽  
Ramsey's Theorem ◽  
Ramsey’S Theorem ◽  
Image Position

AbstractSuppose that α = γ + δ where γ ≥ δ > 0. Then there is a graph which has no independent set of order type and has no pentagram (a pentagram is a set of five points with all pairs joined by edges). In the notation of Erdős and Rado. who generalized Ramsey's Theorem to this setting.

scholarly journals Inequalities relating different definitions of discrepancy

1974 ◽  
Vol 17 (1) ◽  
pp. 81-87 ◽  
Author(s):  
C. J. Smyth
Keyword(s):  
Convex Sets ◽  
Measurable Set ◽  
Image Position

LetP1, P2…, pN be N points in the unit s-dimensional closed square Q = [0, 1]s. For any measurable set S ⊆ Q, we define δ(S), the discrepancy of S, by , where V(S) is the s-dimensional volume of S, and n(S), is the number of indices i for which pi∈S. Let , where the supermum is taken over all s-balls B ∈ Q, and , the supermum in this case being taken over all convex sets C ∈ Q. Clearly Dc ≧ Dk. In this paper we establish Theorem , where φ1is a constant depending only on s.

The Schwarzian Derivative and Disconjugacy of nth order Linear Differential Equations

1969 ◽  
Vol 21 ◽  
pp. 235-249 ◽  
Author(s):  
Meira Lavie
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Schwarzian Derivative ◽  
Order Linear ◽  
Second Order Differential Equations ◽  
Number Of Zeros ◽  
Linearly Independent ◽  
Basic Relationship ◽  
Linear Differential ◽  
Image Position

In this paper we deal with the number of zeros of a solution of the nth order linear differential equation1.1where the functions pj(z) (j = 0, 1, …, n – 2) are assumed to be regular in a given domain D of the complex plane. The differential equation (1.1) is called disconjugate in D, if no (non-trivial) solution of (1.1) has more than (n – 1) zeros in D. (The zeros are counted by their multiplicity.)The ideas of this paper are related to those of Nehari (7; 9) on second order differential equations. In (7), he pointed out the following basic relationship. The function1.2where y1(z) and y2(z) are two linearly independent solutions of1.3is univalent in D, if and only if no solution of equation(1.3) has more than one zero in D, i.e., if and only if(1.3) is disconjugate in D.

Additive functions of intervals and Hausdorff measure

1946 ◽  
Vol 42 (1) ◽  
pp. 15-23 ◽  
Author(s):  
P. A. P. Moran
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Hausdorff Measure ◽  
Convex Sets ◽  
Infinite Sequence ◽  
Small Positive Number ◽  
Additive Functions ◽  
Image Position ◽  
Bounded Sets ◽  
Increasing Function

Consider bounded sets of points in a Euclidean space Rq of q dimensions. Let h(t) be a continuous increasing function, positive for t>0, and such that h(0) = 0. Then the Hausdroff measure h–mE of a set E in Rq, relative to the function h(t), is defined as follows. Let ε be a small positive number and suppose E is covered by a finite or enumerably infinite sequence of convex sets {Ui} (open or closed) of diameters di less than or equal to ε. Write h–mεE = greatest lower bound for any such sequence {Ui}. Then h–mεE is non-decreasing as ε tends to zero. We define

scholarly journals Three proofs of Minkowski's second inequality in the geometry of numbers

1965 ◽  
Vol 5 (4) ◽  
pp. 453-462 ◽  
Author(s):  
R. P. Bambah ◽  
Alan Woods ◽  
Hans Zassenhaus
Keyword(s):  
Convex Set ◽  
Geometry Of Numbers ◽  
Real Numbers ◽  
Open Convex ◽  
Positive Real ◽  
Original Proof ◽  
Linearly Independent ◽  
Image Position

Let K be a bounded, open convex set in euclidean n-space Rn, symmetric in the origin 0. Further let L be a lattice in Rn containing 0 and put extended over all positive real numbers ui for which uiK contains i linearly independent points of L. Denote the Jordan content of K by V(K) and the determinant of L by d(L). Minkowski's second inequality in the geometry of numbers states that Minkowski's original proof has been simplified by Weyl [6] and Cassels [7] and a different proof hasbeen given by Davenport [1].

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