Completions of Alternating Sign Matrices

Brualdi, Richard A.; Kim, Hwa Kyung

doi:10.1007/s00373-014-1409-1

Cited by 7 publications

(10 citation statements)

References 4 publications

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“…We conjecture that α * n = α n (Q * n ). We remark that in [6] completions of certain (0, −1)-matrices to ASMs are considered where the matrices Q * n also occur, and there is a related conjecture. Let F 2m be the so-called 2m×2m diamond ASM, the ASM with the largest number of nonzeros [7].…”

Section: Asm-extensionsmentioning

confidence: 90%

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Alternating Sign Matrices: Extensions, König-Properties, and Primary Sum-Sequences

Brualdi

Dahl

2019

Graphs and Combinatorics

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This paper is concerned with properties of permutation matrices and alternating sign matrices (ASMs). An ASM is a square (0, ±1)-matrix such that, ignoring 0's, the 1's and −1's in each row and column alternate, beginning and ending with a 1. We study extensions of permutation matrices into ASMs by changing some zeros to +1 or −1. Furthermore, several properties concerning the term rank and line covering of ASMs are shown. An ASM A is determined by a sum-matrix Σ(A) whose entries are the sums of the entries of its leading submatrices (so determined by the entries of A). We show that those sums corresponding to the nonzero entries of a permutation matrix determine all the entries of the sum-matrix and investigate some of the properties of the resulting sequence of numbers. Finally, we investigate the lattice-properties of the set of ASMs (of order n), where the partial order comes from the Bruhat order for permutation matrices.

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Section: Asm-extensionsmentioning

confidence: 90%

“…We will also use the shorter notation π = (π 1 π 2 • • • π n ). Some recent developments concerning ASMs, related objects and generalizations may be found in [4], [5] and [6].…”

Section: Introductionmentioning

confidence: 99%

Alternating Sign Matrices: Extensions, König-Properties, and Primary Sum-Sequences

Brualdi

Dahl

2019

Graphs and Combinatorics

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“…Product (Figure 3e) is generated by multiplying the three map values for each grid cell. Union (Figure 3b) and intersect (Figure 3d) are achieved from the inclusion-exclusion principle (Brualdi, 1992). In our simple case with only three sets of independent distributions, we make the following calculation for union:…”

Section: Combining Distributionsmentioning

confidence: 99%

A Multivariate Approach for Mapping Lithospheric Domain Boundaries in East Antarctica

Stål

Reading

Halpin

et al. 2019

Geophysical Research Letters

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Beneath the ice of East Antarctica lies a continent that is likely to be as geologically complex as its neighbors in Gondwana. An improved model of the heterogeneous lithosphere is required to progress research on Antarctica's tectonic evolution and support interdisciplinary studies of cryosphere and solid Earth interaction. We make use of multiple data sets, which were updated following the field campaigns and compilations of the International Polar Year of 2007/2008. Seismic tomography results, gravity anomalies, and surface elevation are used in a novel method, which combines spatial multivariate data to map possible boundaries as projected likelihood functions. Six multivariate combinations are tested and compared with sparse geological observations in East Antarctica. The resulting lithospheric domain boundaries contribute to our understanding of the deep continental structure. New boundaries are suggested in the interior, and models agree with likely surface expressions of crustal tectonic boundaries exposed along the coast.

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“…We shall later use this connection to ASMs. We refer to [4] for ASM and completion problems (see also later), and to [2] for our recent study of ASMs and related matrix classes and polyhedra.…”

Section: For Example Ifmentioning

confidence: 99%

“…We now discuss the connection to ASMs, as given in Lemma 1.2. The following result from [4] concerns the completion of a matrix into an ASM, in the sense of replacing some zeros in the matrix by ones (and no other changes).…”

Section: Example 34 Considermentioning

confidence: 99%

The interval structure of (0,1)-matrices

Brualdi

Dahl

2019

Discrete Applied Mathematics

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Let A be an n × n (0, * )-matrix, so each entry is 0 or * . An A-interval matrix is a (0, 1)-matrix obtained from A by choosing some * 's so that in every interval of consecutive * 's, in a row or column of A, exactly one * is chosen and replaced with a 1, and every other * is replaced with a 0. We consider the existence questions for A-interval matrices, both in general, and for specific classes of such A defined by permutation matrices. Moreover, we discuss uniqueness and the number of A-permutation matrices, as well as properties of an associated graph.

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Completions of Alternating Sign Matrices

Cited by 7 publications

References 4 publications

Alternating Sign Matrices: Extensions, König-Properties, and Primary Sum-Sequences

Alternating Sign Matrices: Extensions, König-Properties, and Primary Sum-Sequences

A Multivariate Approach for Mapping Lithospheric Domain Boundaries in East Antarctica

The interval structure of (0,1)-matrices

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