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We introduce the notion of the slot length of a family of matrices over an arbitrary field ${\\mathbb {F}}$ . Using this definition it is shown that, if $n\\ge 5$ and A and B are $n\\times n$ complex matrices with A unicellular and the pair $\\{A,B\\}$ irreducible, the slot length s of $\\{A,B\\}$ satisfies $2\\le s\\le n-1$ , where both inequalities are sharp, for every n. It is conjectured that the slot length of any irreducible pair of $n\\times n$ matrices, where $n\\ge 5$ , is at most $n-1$ . The slot length of a family of rank-one complex matrices can be equal to n.

The main purpose of this article is to develop a theory that extends the domain of any local isometry to the whole space containing the domain, where a local isometry is an isometry between two proper subsets. In fact, the main purpose of this article consists of the following three detailed objectives: The first objective is to extend the bounded domain of any local isometry to the first-order generalized linear span. The second one is to extend the bounded domain of any local isometry to the second-order generalized linear span. The third objective of this article is to extend the bounded domain of any local isometry to the whole Hilbert space.

A binary modified de Bruijn sequence is an infinite and periodic binary sequence derived by removing a zero from the longest run of zeros in a binary de Bruijn sequence. The minimal polynomial of the modified sequence is its unique least-degree characteristic polynomial. Leveraging a recent characterization, we devise a novel general approach to determine the minimal polynomial. We translate the characterization into a problem of identifying a Hamiltonian cycle in a specially constructed graph. The graph is isomorphic to the modified de Bruijn–Good graph. Along the way, we demonstrate the usefulness of some computational tools from the cycle joining method in the modified setup.

Cyclic codes are a subclass of linear codes and have applications in consumer electronics, data storage systems, and communication systems as they have efficient encoding and decoding algorithms. In this paper, some monomials and trinomials over finite fields are employed to construct a number of families of cyclic codes. Lower bounds on the minimum weight of some families of the cyclic codes are developed. The minimum weights of other families of the codes constructed in this paper are determined. The dimensions of the codes are flexible. Many of the codes presented in this paper are optimal or almost optimal in the sense that they meet some bounds on linear codes. Open problems regarding cyclic codes from monomials and trinomials are also presented. [PUBLICATION ABSTRACT]

In this paper, a new family of binary sequences of period is proposed, where and . The presented family takes 7-valued correlation values , , , , , and . For and , it is proved that the proposed sequence family has linear spans , where l = 2, 3, 4, 5, 6, 7, and the distribution of linear span of sequences in is determined.

Summary The paper investigates the possibility of identifying localised damages for multi‐span and multi‐floor linear elastic frames using only natural frequencies measured in the undamaged and damaged configurations. Namely, frames of increasing complexity are studied by exploring one by one their significant substructures (i.e. multi‐span beams, floor by floor); the error function is defined and minimised on a database of finite element damaged models that only includes the natural frequencies of the local modes of the substructure, that is, the only modes significantly affected by the localised damage considered here. The performances and limits of the procedure are here discussed by means of numerical simulations on steel frames of increasing complexity; a particular attention is also devoted to the role of noise on the identification procedure. Copyright © 2015 John Wiley & Sons, Ltd.

In this paper we show that if a C*-algebra 𝒜 admits a certain 3 × 3 matrix decomposition, then every commutator in 𝒜 can be written as a linear combination of at most 84 projections in A. In certain C*-algebras, this is sufficient to allow us to show that every element is a linear combination of a fixed finite number of projections from the algebra.

In this paper, a recursion is derived to compute the linear span of the p-ary cascaded GMW sequences. It is the first time to determine the linear span of the p-ary cascaded GMW sequence without any restriction on the parameters completely. Whereas, the known result on the p-ary cascaded GMW sequence with the specific parameters in the literature could be viewed as a special case of the new result.

Not only the vectors of a vector space V are interesting, but also those subsets of the vector space that are themselves vector spaces when the opertaions of V are restricted to them. Such subsets are called vector subspaces or linear subspaces.

Examples of linear maps between normed spaces are constructed, including a one-to-one map from a countable-dimensional linear subspace of l2{l_2} onto l2{l_2}. We prove that the linear span of a countable-dimensional linearly independent subset of a normed linear space is, in many cases, countable dimensional.