Michael W. Schroeder
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Michael W. Schroeder has a total of 3 co-authors. Here are the top co-authors by number of shared publications:
- Kathleen P. Kiernan
Department of Mathematics University of Wisconsin Madison 53706 WI
4 shared publications - Richard A. Brualdi
Department of Mathematics, University of Wisconsin, Madison, USA
4 shared publications - Seth A. Meyer
Department of Mathematics University of Wisconsin Madison 53706 WI
4 shared publications
Latest Publications
- Open Access
Patterns of Alternating Sign Matrices
We initiate a study of the zero-nonzero patterns of n by n alternating sign matrices. We characterize the row (column) sum vectors of these patterns and determine their minimum term rank. In the case of connected alternating sign matrices, we find the minimum number of nonzero entries and characterize the case of equality. We also study symmetric alternating sign matrices, in particular, those with only zeros on the main diagonal. These give rise to alternating signed graphs without loops, and we determine the maximum number of edges in such graphs. We also consider n by n alternating sign matrices whose patterns are maximal within the class of all n by n alternating sign matrices.
Open AccessPatterns of Alternating Sign Matrices
We initiate a study of the zero-nonzero patterns of n by n alternating sign matrices. We characterize the row (column) sum vectors of these patterns and determine their minimum term rank. In the case of connected alternating sign matrices, we find the minimum number of nonzero entries and characterize the case of equality. We also study symmetric alternating sign matrices, in particular, those with only zeros on the main diagonal. These give rise to alternating signed graphs without loops, and we determine the maximum number of edges in such graphs. We also consider n by n alternating sign matrices whose patterns are maximal within the class of all n by n alternating sign matrices.
- Open Access
On the t-Term Rank of a Matrix
For t a positive integer, the t-term rank of a (0,1)-matrix A is defined to be the largest number of 1s in A with at most one 1 in each column and at most t 1s in each row. Thus the 1-term rank is the ordinary term rank. We generalize some basic results for the term rank to the t-term rank, including a formula for the maximum term rank over a nonempty class of (0,1)-matrices with the the same row sum and column sum vectors. We also show the surprising result that in such a class there exists a matrix which realizes all of the maximum terms ranks between 1 and t.
Open AccessOn the t-Term Rank of a Matrix
For t a positive integer, the t-term rank of a (0,1)-matrix A is defined to be the largest number of 1s in A with at most one 1 in each column and at most t 1s in each row. Thus the 1-term rank is the ordinary term rank. We generalize some basic results for the term rank to the t-term rank, including a formula for the maximum term rank over a nonempty class of (0,1)-matrices with the the same row sum and column sum vectors. We also show the surprising result that in such a class there exists a matrix which realizes all of the maximum terms ranks between 1 and t.
- Open Access
- 1 January 2010
- Published by American Institute of Mathematical Sciences (AIMS) in Advances in Mathematics of Communications
- Vol. 4 (3) , 419-431
Combinatorial batch codes and transversal matroids
Combinatorial batch codes were defined by Paterson, Stinson, and Wei as purely combinatorial versions of the batch codes introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai. There are $n$ items and $m$ servers each of which stores a subset of the items. It is required that, for prescribed integers $k$ and $t$, any $k$ items can be retrieved by reading at most $t$ items from each server. Only the case $t=1$ is considered here. An optimal combinatorial batch code is one in which the total storage required is a minimum. We establish an important connection between combinatorial batch codes and transversal matroids, and exploit this connection to characterize optimal combinatorial batch codes if $n=m+1$ and $n=m+2$.- 1 January 2010
- Published by American Institute of Mathematical Sciences (AIMS) in Advances in Mathematics of Communications
- Vol. 4 (3) , 419-431
Open AccessCombinatorial batch codes and transversal matroids
Combinatorial batch codes were defined by Paterson, Stinson, and Wei as purely combinatorial versions of the batch codes introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai. There are $n$ items and $m$ servers each of which stores a subset of the items. It is required that, for prescribed integers $k$ and $t$, any $k$ items can be retrieved by reading at most $t$ items from each server. Only the case $t=1$ is considered here. An optimal combinatorial batch code is one in which the total storage required is a minimum. We establish an important connection between combinatorial batch codes and transversal matroids, and exploit this connection to characterize optimal combinatorial batch codes if $n=m+1$ and $n=m+2$. - Open Access
- 1 January 2010
- Published by American Institute of Mathematical Sciences (AIMS) in Advances in Mathematics of Communications
- Vol. 4 (4) , 597
Erratum
Erratum to "Combinatorial Batch Codes and Transversal Matroids" (Advances in Mathematics of Communication, Vol. 4, No. 3, 2010, 419--431.
- 1 January 2010
- Published by American Institute of Mathematical Sciences (AIMS) in Advances in Mathematics of Communications
- Vol. 4 (4) , 597
Open AccessErratum
Erratum to "Combinatorial Batch Codes and Transversal Matroids" (Advances in Mathematics of Communication, Vol. 4, No. 3, 2010, 419--431.