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Large structures made of nowhere L q functions

Szymon Głąb, Pedro L. Kaufmann, Leonardo Pellegrini (2014)

Studia Mathematica

We say that a real-valued function f defined on a positive Borel measure space (X,μ) is nowhere q-integrable if, for each nonvoid open subset U of X, the restriction f | U is not in L q ( U ) . When (X,μ) has some natural properties, we show that certain sets of functions defined in X which are p-integrable for some p’s but nowhere q-integrable for some other q’s (0 < p,q < ∞) admit a variety of large linear and algebraic structures within them. The presented results answer a question of Bernal-González,...

Line graphs: their maximum nullities and zero forcing numbers

Shaun Fallat, Abolghasem Soltani (2016)

Czechoslovak Mathematical Journal

The maximum nullity over a collection of matrices associated with a graph has been attracting the attention of numerous researchers for at least three decades. Along these lines various zero forcing parameters have been devised and utilized for bounding the maximum nullity. The maximum nullity and zero forcing number, and their positive counterparts, for general families of line graphs associated with graphs possessing a variety of specific properties are analysed. Building upon earlier work, where...

Linear extensions of relations between vector spaces

Árpád Száz (2003)

Commentationes Mathematicae Universitatis Carolinae

Let X and Y be vector spaces over the same field K . Following the terminology of Richard Arens [Pacific J. Math. 11 (1961), 9–23], a relation F of X into Y is called linear if λ F ( x ) F ( λ x ) and F ( x ) + F ( y ) F ( x + y ) for all λ K { 0 } and x , y X . After improving and supplementing some former results on linear relations, we show that a relation Φ of a linearly independent subset E of X into Y can be extended to a linear relation F of X into Y if and only if there exists a linear subspace Z of Y such that Φ ( e ) Y | Z for all e E . Moreover, if E generates...

Linear forms and axioms of choice

Marianne Morillon (2009)

Commentationes Mathematicae Universitatis Carolinae

We work in set-theory without choice ZF. Given a commutative field 𝕂 , we consider the statement 𝐃 ( 𝕂 ) : “On every non null 𝕂 -vector space there exists a non-null linear form.” We investigate various statements which are equivalent to 𝐃 ( 𝕂 ) in ZF. Denoting by 2 the two-element field, we deduce that 𝐃 ( 2 ) implies the axiom of choice for pairs. We also deduce that 𝐃 ( ) implies the axiom of choice for linearly ordered sets isomorphic with .

Linear operators preserving maximal column ranks of nonbinary boolean matrices

Seok-Zun Song, Sung-Dae Yang, Sung-Min Hong, Young-Bae Jun, Seon-Jeong Kim (2000)

Discussiones Mathematicae - General Algebra and Applications

The maximal column rank of an m by n matrix is the maximal number of the columns of A which are linearly independent. We compare the maximal column rank with rank of matrices over a nonbinary Boolean algebra. We also characterize the linear operators which preserve the maximal column ranks of matrices over nonbinary Boolean algebra.

Linear operators that preserve Boolean rank of Boolean matrices

LeRoy B. Beasley, Seok-Zun Song (2013)

Czechoslovak Mathematical Journal

The Boolean rank of a nonzero m × n Boolean matrix A is the minimum number k such that there exist an m × k Boolean matrix B and a k × n Boolean matrix C such that A = B C . In the previous research L. B. Beasley and N. J. Pullman obtained that a linear operator preserves Boolean rank if and only if it preserves Boolean ranks 1 and 2 . In this paper we extend this characterizations of linear operators that preserve the Boolean ranks of Boolean matrices. That is, we obtain that a linear operator preserves Boolean rank...

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