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Rank and spectral multiplicity

100%
Ferenczi S., Kwiatkowski J.
Studia Mathematica
|
1992
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tom 102
|
nr 2
121-144
EN
For a dynamical system (X,T,μ), we investigate the connections between a metric invariant, the rank r(T), and a spectral invariant, the maximal multiplicity m(T). We build examples of systems for which the pair (m(T),r(T)) takes values (m,m) for any integer m ≥ 1 or (p-1, p) for any prime number p ≥ 3.
2
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Inverse limit of M -cocycles and applications

100%
Kwiatkowski J.
Fundamenta Mathematicae
|
1998
|
tom 157
|
nr 2-3
261-276
EN
For any m, 2 ≤ m < ∞, we construct an ergodic dynamical system having spectral multiplicity m and infinite rank. Given r > 1, 0 < b < 1 such that rb > 1 we construct a dynamical system (X, B, μ, T) with simple spectrum such that r(T) = r, F*(T) = b, and $#C(T)/wcl{T^n: n ∈ ℤ} = ∞$
3
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Convex independence and the structure of clone-free multipartite tournaments

88%
Parker D., Westhoff R., Wolf M.
Discussiones Mathematicae Graph Theory
|
2009
|
tom 29
|
nr 1
51-69
EN
We investigate the convex invariants associated with two-path convexity in clone-free multipartite tournaments. Specifically, we explore the relationship between the Helly number, Radon number and rank of such digraphs. The main result is a structural theorem that describes the arc relationships among certain vertices associated with vertices of a given convexly independent set. We use this to prove that the Helly number, Radon number, and rank coincide in any clone-free bipartite tournament. We then study the relationship between Helly independence and Radon independence in clone-free multipartite tournaments. We show that if the rank is at least 4 or the Helly number is at least 3, then the Helly number and the Radon number are equal.
4
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Rank and perimeter preserver of rank-1 matrices over max algebra

63%
Song S.-Z., Kang K.-T.
Discussiones Mathematicae - General Algebra and Applications
|
2003
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tom 23
|
nr 2
125-137
EN
For a rank-1 matrix $A = a ⊗ b^{t}$ over max algebra, we define the perimeter of A as the number of nonzero entries in both a and b. We characterize the linear operators which preserve the rank and perimeter of rank-1 matrices over max algebra. That is, a linear operator T preserves the rank and perimeter of rank-1 matrices if and only if it has the form T(A) = U ⊗ A ⊗ V, or $T(A) = U ⊗ A^{t} ⊗ V$ with some monomial matrices U and V.
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