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On sets of vectors of a finite vector space in which every subset of basis size is a basis

Simeon Ball (2012)

Journal of the European Mathematical Society

It is shown that the maximum size of a set S of vectors of a k -dimensional vector space over 𝔽 q , with the property that every subset of size k is a basis, is at most q + 1 , if k ≤ p , and at most q + k − p , if q ≥ k ≥ p + 1 ≥ 4 , where q = p k and p is prime. Moreover, for k ≤ p , the sets S of maximum size are classified, generalising Beniamino Segre’s “arc is a conic” theorem. These results have various implications. One such implication is that a k × ( p + 2 ) matrix, with k ≤ p and entries from 𝔽 p , has k columns which are linearly dependent. Another is...

On solution sets of information inequalities

Nihat Ay, Walter Wenzel (2012)

Kybernetika

We investigate solution sets of a special kind of linear inequality systems. In particular, we derive characterizations of these sets in terms of minimal solution sets. The studied inequalities emerge as information inequalities in the context of Bayesian networks. This allows to deduce structural properties of Bayesian networks, which is important within causal inference.

On some algebraic identities and the exterior product of double forms

Mohammed Larbi Labbi (2013)

Archivum Mathematicum

We use the exterior product of double forms to free from coordinates celebrated classical results of linear algebra about matrices and bilinear forms namely Cayley-Hamilton theorem, Laplace expansion of the determinant, Newton identities and Jacobi’s formula for the determinant. This coordinate free formalism is then used to easily generalize the previous results to higher multilinear forms namely to double forms. In particular, we show that the Cayley-Hamilton theorem once applied to the second...

On some free semigroups, generated by matrices

Piotr Słanina (2015)

Czechoslovak Mathematical Journal

Let A = 1 2 0 1 , B λ = 1 0 λ 1 . We call a complex number λ “semigroup free“ if the semigroup generated by A and B λ is free and “free” if the group generated by A and B λ is free. First families of semigroup free λ ’s were described by J. L. Brenner, A. Charnow (1978). In this paper we enlarge the set of known semigroup free λ ’s. To do it, we use a new version of “Ping-Pong Lemma” for semigroups embeddable in groups. At the end we present most of the known results related to semigroup free and free numbers in a common picture....

On some identities involving spherical means

Gianfranco Cimmino (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

For every positive definite quadratic form in n variables the reciprocal of the square root of the discriminant is equal to the arithmetic mean of the values assumed by the form on the n - 1 sphere centered at 0 and with radius 1 raised to the ( - n 2 )-th. power. Various consequences are deduced from this, in particular a simplification of some calculations from which one obtains the possibility of solving linear systems using spherical means rather than determinants.

On some properties of the Laplacian matrix revealed by the RCM algorithm

Francisco Pedroche, Miguel Rebollo, Carlos Carrascosa, Alberto Palomares (2016)

Czechoslovak Mathematical Journal

In this paper we present some theoretical results about the irreducibility of the Laplacian matrix ordered by the Reverse Cuthill-McKee (RCM) algorithm. We consider undirected graphs with no loops consisting of some connected components. RCM is a well-known scheme for numbering the nodes of a network in such a way that the corresponding adjacency matrix has a narrow bandwidth. Inspired by some properties of the eigenvectors of a Laplacian matrix, we derive some properties based on row sums of a...

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