Probability, Selfadjoint Operators, Unit Vectors and the Need for Complexness

2019 ◽  
pp. 175-187
Author(s):  
K. Kong Wan
Keyword(s):  
Selfadjoint Operators ◽  
Unit Vectors

Crystallochemical classifications of phyllosilicates based on the unified system of projection of chemical composition: I. The mica group

Clay Minerals ◽  
1990 ◽  
Vol 25 (1) ◽  
pp. 73-81 ◽  
Author(s):  
A. Wiewióra
Keyword(s):  
Chemical Composition ◽  
Divalent Cations ◽  
Layer Charge ◽  
Vector Representation ◽  
Similar Formula ◽  
Crystallochemical Formula ◽  
Trivalent Cations ◽  
Image Position ◽  
Composition I ◽  
Unit Vectors

AbstractA unified system of vector representation of chemical composition is proposed for the phyllosilicates based on projection of the composition, as given by crystallochemical formula, onto a field with orthogonal axes chosen for octahedral divalent cations, R2+, and Si (X, Y, respectively), and oblique axes for octahedral trivalent cations, R3+, and vacancies, □, (V, Z, respectively). Point coordinates for each set of axes were used to define the direction and length of the unit vectors for phyllosilicates belonging to different groups. Parallel to these fundamental directions the composition isolines were drawn in the projection fields. Applied to micas, this system enables control of the chemical composition by the general crystallochemical formula covering all varieties of Li-free dioctahedral and trioctahedral micas:where z (number of vacancies) = (y-x+ m)/2; m (layer charge) =1; u+y+z = 3. There is a similar formula for vacancy-free lithian micas:where w = m — x+y;m=1; u+y+w = 3, and for Li-free brittle micas:where z = (y — x+m)/2; m = 2; u+y+z = 3. Projection fields were used to classify micas.

Consistency conditions for a finite set of projections of a function

1979 ◽  
Vol 85 (1) ◽  
pp. 61-68 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Lebesgue Measure ◽  
Convex Subset ◽  
Compact Convex ◽  
Unit Vector ◽  
Compact Convex Subset ◽  
Finite Collection ◽  
Finite Set ◽  
Almost All ◽  
Dimensional Lebesgue Measure ◽  
Unit Vectors

Let H(μ, θ) be the hyperplane in Rn (n ≥ 2) that is perpendicular to the unit vector 6 and perpendicular distance μ from the origin; that is, H(μ, θ) = (x ∈ Rn: x. θ = μ). (Note that H(μ, θ) and H(−μ, −θ) are the same hyperplanes.) Let X be a proper compact convex subset of Rm. If f(x) ∈ L1(X) we will denote by F(μ, θ) the projection of f perpendicular to θ; that is, the integral of f(x) over H(μ, θ) with respect to (n − 1)-dimensional Lebesgue measure. By Fubini's Theorem, if f(x) ∈ L1(X), F(μ, θ) exists for almost all μ for every θ. Our aim in this paper is, given a finite collection of unit vectors θ1, …, θN, to characterize the F(μ, θi) that are the projections of some function f(x) with support in X for 1 ≤ i ≤ N.

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.

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