Rank and perimeter preserver of rank-1 matrices over max algebra

Seok-Zun Song; Kyung-Tae Kang

Displaying similar documents to “Rank and perimeter preserver of rank-1 matrices over max algebra”

Finite-rank perturbations of positive operators and isometries

Man-Duen Choi, Pei Yuan Wu (2006)

Studia Mathematica

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We completely characterize the ranks of A - B and $A^{1 / 2} - B^{1 / 2}$ for operators A and B on a Hilbert space satisfying A ≥ B ≥ 0. Namely, let l and m be nonnegative integers or infinity. Then l = rank(A - B) and $m = r a n k (A^{1 / 2} - B^{1 / 2})$ for some operators A and B with A ≥ B ≥ 0 on a Hilbert space of dimension n (1 ≤ n ≤ ∞) if and only if l = m = 0 or 0 < l ≤ m ≤ n. In particular, this answers in the negative the question posed by C. Benhida whether for positive operators A and B the finiteness of rank(A - B) implies that...

Rank α operators on the space C(T,X)

Dumitru Popa (2002)

Colloquium Mathematicae

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For 0 ≤ α < 1, an operator U ∈ L(X,Y) is called a rank α operator if $x ₙ \to^{τ_{α}} x$ implies Uxₙ → Ux in norm. We give some results on rank α operators, including an interpolation result and a characterization of rank α operators U: C(T,X) → Y in terms of their representing measures.

Chern rank of complex bundle

Bikram Banerjee (2019)

Commentationes Mathematicae Universitatis Carolinae

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Motivated by the work of A. C. Naolekar and A. S. Thakur (2014) we introduce notions of upper chern rank and even cup length of a finite connected CW-complex and prove that upper chern rank is a homotopy invariant. It turns out that determination of upper chern rank of a space $X$ sometimes helps to detect whether a generator of the top cohomology group can be realized as Euler class for some real (orientable) vector bundle over $X$ or not. For a closed connected $d$ -dimensional complex manifold...

Rational realization of the minimum ranks of nonnegative sign pattern matrices

Wei Fang, Wei Gao, Yubin Gao, Fei Gong, Guangming Jing, Zhongshan Li, Yan Ling Shao, Lihua Zhang (2016)

Czechoslovak Mathematical Journal

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A sign pattern matrix (or nonnegative sign pattern matrix) is a matrix whose entries are from the set ${+, -, 0}$ ( ${+, 0}$ , respectively). The minimum rank (or rational minimum rank) of a sign pattern matrix $𝒜$ is the minimum of the ranks of the matrices (rational matrices, respectively) whose entries have signs equal to the corresponding entries of $𝒜$ . Using a correspondence between sign patterns with minimum rank $r \geq 2$ and point-hyperplane configurations in $ℝ^{r - 1}$ and Steinitz’s theorem on the rational realizability...

On some finite 2-groups in which the derived group has two generators

Elliot Benjamin, Chip Snyder (2023)

Czechoslovak Mathematical Journal

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We show that any finite 2-group, whose abelianization has either 4-rank at most 2 or 8-rank 0 and whose commutator subgroup is generated by two elements, is metabelian. We also prove that the minimal order of any 2-group with nonabelian commutator subgroup of 2-rank 2 is $2^{12}$ .

Perimeter preserver of matrices over semifields

Seok-Zun Song, Kyung-Tae Kang, Young Bae Jun (2006)

Czechoslovak Mathematical Journal

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For a rank- $1$ matrix $A = {𝐚 𝐛}^{t}$ , 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 semifields. That is, a linear operator $T$ preserves the rank and perimeter of rank- $1$ matrices over semifields if and only if it has the form $T (A) = U A V$ , or $T (A) = U A^{t} V$ with some invertible matrices U and V.

Perimeter preservers of nonnegative integer matrices

Seok-Zun Song, Kyung-Tae Kang, Sucheol Yi (2004)

Commentationes Mathematicae Universitatis Carolinae

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We investigate the perimeter of nonnegative integer matrices. We also characterize the linear operators which preserve the rank and perimeter of nonnegative integer matrices. That is, a linear operator $T$ preserves the rank and perimeter of rank- $1$ matrices if and only if it has the form $T (A) = P (A \circ B) Q$ , or $T (A) = P (A^{t} \circ B) Q$ with appropriate permutation matrices $P$ and $Q$ and positive integer matrix $B$ , where $\circ$ denotes Hadamard product.

Opérateurs sur un espace $L^{1}$

Alain Costé, Françoise Lust-Piquard (1987)

Colloquium Mathematicae

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Infinite rank of elliptic curves over $ℚ^{a b}$

Bo-Hae Im, Michael Larsen (2013)

Acta Arithmetica

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If E is an elliptic curve defined over a quadratic field K, and the j-invariant of E is not 0 or 1728, then $E (ℚ^{a b})$ has infinite rank. If E is an elliptic curve in Legendre form, y² = x(x-1)(x-λ), where ℚ(λ) is a cubic field, then $E (K ℚ^{a b})$ has infinite rank. If λ ∈ K has a minimal polynomial P(x) of degree 4 and v² = P(u) is an elliptic curve of positive rank over ℚ, we prove that y² = x(x-1)(x-λ) has infinite rank over $K ℚ^{a b}$ .

$M_{p}$ -группы с ядром произвольного ранга

В.П. Шунков (1987)

Algebra i Logika

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Class groups of large ranks in biquadratic fields

Mahesh Kumar Ram (2024)

Czechoslovak Mathematical Journal

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For any integer $n > 1$ , we provide a parametric family of biquadratic fields with class groups having $n$ -rank at least 2. Moreover, in some cases, the $n$ -rank is bigger than 4.

On sets with rank one in simple homogeneous structures

Ove Ahlman, Vera Koponen (2015)

Fundamenta Mathematicae

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We study definable sets D of SU-rank 1 in $ℳ^{e q}$ , where ℳ is a countable homogeneous and simple structure in a language with finite relational vocabulary. Each such D can be seen as a ’canonically embedded structure’, which inherits all relations on D which are definable in $ℳ^{e q}$ , and has no other definable relations. Our results imply that if no relation symbol of the language of ℳ has arity higher than 2, then there is a close relationship between triviality of dependence and being a reduct...