scholarly journals On Packings of Spheres in Hilbert Space

1955 ◽  
Vol 2 (3) ◽  
pp. 145-146 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Hilbert Space ◽  
Unit Sphere ◽  
Infinite Sequence ◽  
Real Hilbert Space ◽  
Real Numbers ◽  
Image Position

A point x in real Hilbert space is represented by an infinite sequence (x1, x2, x3, …) of real numbers such thatis convergent. The unit “sphere“ S consists of all points × for which ‖x‖ ≤ 1. The sphere of radius a and centre y is denoted by Sa(y) and consists of all points × for which ‖x−y‖ ≤ a.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

CHEBYSHEV SUBSETS OF A HILBERT SPACE SPHERE

2019 ◽  
Vol 107 (3) ◽  
pp. 289-301
Author(s):  
THEO BENDIT
Keyword(s):  
Hilbert Space ◽  
Unit Sphere ◽  
Stereographic Projection ◽  
Real Hilbert Space ◽  
Equivalent Formulation ◽  
Chebyshev Sets

The Chebyshev conjecture posits that Chebyshev subsets of a real Hilbert space $X$ are convex. Works by Asplund, Ficken and Klee have uncovered an equivalent formulation of the Chebyshev conjecture in terms of uniquely remotal subsets of $X$. In this tradition, we develop another equivalent formulation in terms of Chebyshev subsets of the unit sphere of $X\times \mathbb{R}$. We characterise such sets in terms of the image under stereographic projection. Such sets have superior structure to Chebyshev sets and uniquely remotal sets.

Quasifree representations of Clifford algebras

1993 ◽  
Vol 113 (3) ◽  
pp. 487-497
Author(s):  
P. L. Robinson
Keyword(s):  
Hilbert Space ◽  
Clifford Algebra ◽  
Clifford Algebras ◽  
Real Hilbert Space ◽  
Adjoint Operator ◽  
The State ◽  
Infinite Dimensional ◽  
Image Position

Let V be an infinite-dimensional real Hilbert space with associated C* Clifford algebra C[V]. To any state σ of the C* algebra C[V] there corresponds a skew-adjoint operator C of norm at most unity on V such thatwe refer to C as the covariance of the state σ.

Completeness properties in L2 of the eigenfunctions of two semi-linear differential operators

1980 ◽  
Vol 88 (3) ◽  
pp. 451-468 ◽  
Author(s):  
L. E. Fraenkel
Keyword(s):  
Hilbert Space ◽  
Product Space ◽  
Orthonormal Basis ◽  
Differential Operators ◽  
Real Hilbert Space ◽  
The Real ◽  
Linear Differential Operators ◽  
Linear Problems ◽  
Linear Differential ◽  
Image Position

This paper concerns the boundary-value problemsin which λ is a real parameter, u is to be a real-valued function in C2[0, 1], and problem (I) is that with the minus sign. (The differential operators are called semi-linear because the non-linearity is only in undifferentiated terms.) If we linearize the equations (for ‘ small’ solutions u) by neglecting , there result the eigenvalues λ = n2π2 (with n = 1,2,…) and corresponding normalized eigenfunctionsand it is well known ((2), p. 186) that the sequence {en} is complete in that it is an orthonormal basis for the real Hilbert space L2(0, 1). We shall be concerned with possible extensions of this result to the non-linear problems (I) and (II), for which non-trivial solutions (λ, u) bifurcate from the trivial solution (λ, 0) at the points {n2π2,0) in the product space × L2(0, 1). (Here denotes the real line.)

scholarly journals The Packing of Spheres in the Space lp

1958 ◽  
Vol 4 (1) ◽  
pp. 22-25 ◽  
Author(s):  
Jane A. C. Burlak ◽  
R. A. Rankin ◽  
A. P. Robertson
Keyword(s):  
Infinite Number ◽  
Unit Sphere ◽  
Complex Space ◽  
Infinite Sequence ◽  
Complex Numbers ◽  
The Real ◽  
The Space Lp ◽  
Fixed Radius ◽  
Image Position

A point x in the real or complex space lpis an infinite sequence,(x1, x2, x3,…) of real or complex numbers such that is convergent. Here p ≥ 1 and we writeThe unit sphere S consists of all points x ε lp for which ¶ x ¶ ≤ 1. The sphere of radius a≥ ≤ 0 and centre y is denoted by Sa(y) and consists of all points x ε lp such that ¶ x - y ¶ ≤ a. The sphere Sa(y) is contained in S if and only if ¶ y ¶≤1 - a, and the two spheres Sa(y) and Sa(z) do not overlap if and only if¶ y- z ¶≥ 2aThe statement that a finite or infinite number of spheres Sa (y) of fixed radius a can be packed in S means that each sphere Sa (y) is contained in S and that no two such spheres overlap.

scholarly journals Two convergence theorems to fixed point of a nonexpansive mapping on the unit sphere of a Hilbert space

Filomat ◽  
2012 ◽  
Vol 26 (5) ◽  
pp. 949-955 ◽  
Author(s):  
Yasunori Kimura ◽  
Kenzi Satô
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Nonexpansive Mapping ◽  
Unit Sphere ◽  
Real Hilbert Space ◽  
Projection Methods ◽  
Convergence Theorems ◽  
Iterative Schemes ◽  
Different Types

We consider iterative schemes converging to a fixed point of nonexpansive mapping defined on the unit sphere of a real Hilbert space by using two different types of projection methods

scholarly journals Existence and uniqueness of solutions for a nonlinear second order differential equation in Hilbert space

1990 ◽  
Vol 33 (1) ◽  
pp. 89-95 ◽  
Author(s):  
Dang Dinh Hai
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Order Differential Equation ◽  
Real Hilbert Space ◽  
Boundary Value ◽  
Uniqueness Of Solutions ◽  
Second Order Differential Equation ◽  
Image Position

This paper is concerned with the existence and uniqueness of solutions for the Picard boundary value problemin a real Hilbert space. Our theorems improve corresponding results of Mawhin for |k| large.

scholarly journals A Generalization of the Lax-Milgram Lemma

1980 ◽  
Vol 23 (2) ◽  
pp. 179-184 ◽  
Author(s):  
K. Inayatnoor ◽  
M. Aslam Noor
Keyword(s):  
Hilbert Space ◽  
Bilinear Form ◽  
Dual Space ◽  
Real Hilbert Space ◽  
Inner Product ◽  
Continuous Functional ◽  
Image Position

Let H be a real Hilbert space with its dual space H'. The norm and inner product in H are denoted by ||.|| and 〈.,.〉 respectively. We denote by 〈.,.〉, the pairing between H' and H.If a(u, v) is a bilinear form and F is a real-valued continuous functional on H, then we consider I[v], a functional defined by

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (3) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn} whose state space is the non-negative integers and whose transition probability matrix ║Pij║ possesses the representation where {Pr}, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z0 = k, a finite positive integer.

scholarly journals An estimate on the eigenvalues in bifurcation for gradient mappings

1997 ◽  
Vol 39 (2) ◽  
pp. 211-216 ◽  
Author(s):  
Raffaele Chiappinelli
Keyword(s):  
Hilbert Space ◽  
Eigenvalue Problem ◽  
Bifurcation Point ◽  
Nonlinear Operator ◽  
Real Hilbert Space ◽  
Image Position

Let H be a real Hilbert space and let A: H→H be a nonlinear operator such that A(0) = 0. We consider the eigenvalue problemRecall that λ0 ε ℝ is said to be a bifurcation point for (1.1) if every neighbourhood of (λ0, 0) in ℝ × H contains solutions of (1.1).

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