SIAM Digital Library
 
 
 

SIAM J. on Matrix Analysis and Applications

All Journal content published prior to 1997 is part of LOCUS.

Search Issue | RSS Feeds RSS
Previous Issue

1981

Volume 2, Issue 4, pp. 341-471


Fluctuation Results for Markov-Dependent Trials

R. B. Nain and Kanwar Sen

SIAM. J. on Algebraic and Discrete Methods 2, pp. 341-346 (6 pages)

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
Takacs (SIAM Rev., 21 (1979), pp. 222–228) obtained explicit results for the distributions of the number of changes in luck in $n$ Bernoulli trials with probability $p(0 < p < 1)$ for success. In this paper, the corresponding results have been obtained for Markov-dependent trials.

Weight Enumerators of Normalized Codes

Stephen M. Gagola, Jr.

SIAM. J. on Algebraic and Discrete Methods 2, pp. 347-380 (34 pages) | Cited 1 time

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
A linear code $C$ over a finite field is normalized if it contains the all ones vector. If a normalized code $C$ is also self-dual then its complete weight enumerator is invariant under the action of a linear group $G$, which is explicitly determined. The character of this representation is then used to calculate the Molien series for $G$.
Further restrictions on $C$ may lead to larger finite linear groups containing $G$. It is determined here that if the field is not $GF(2)$ or $GF(4)$ then there are only finitely many linear groups containing $G$ with the property that the only scalar matrices appearing are those already contained in $G$. In fact, if the characteristic is odd and $\tilde G$ is the unimodular subgroup of $G$, then the finite unimodular subgroups containing $\tilde G$ are contained in a unique, such maximal linear group. The classification of the finite simple groups is used for the proof of this last result.

Broadcasting in Trees with Multiple Originators

Arthur M. Farley and Andrzej Proskurowski

SIAM. J. on Algebraic and Discrete Methods 2, pp. 381-386 (6 pages) | Cited 3 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
Broadcasting is the information dissemination process in a communication network whereby all sites of the network become informed of a given message by calls made over lines of the network. We present an algorithm which, given a tree network and a time, determines a smallest set of subtrees covering sites of the network such that broadcast can be completed within the given time in each subtree. Information developed by the algorithm is sufficient to determine a satisfactory originator and calling scheme within each subtree.

The Bandwidth of Caterpillars with Hairs of Length 1 and 2

S. F. Assmann, G. W. Peck, M. M. Sysło, and J. Zak

SIAM. J. on Algebraic and Discrete Methods 2, pp. 387-393 (7 pages) | Cited 9 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
In this paper we show that the bandwidth of any caterpillar with hairs of length 1 and 2 is given by the maximum over all subcaterpillars of $\lceil (n - 1)/d \rceil$, where $n$ is the number of vertices and $d$ is the diameter of the subcaterpillar. We also give an $n\log n$ algorithm which produces a bandwidth labelling of such a caterpillar.

Covering Regions by Rectangles

Seth Chaiken, Daniel J. Kleitman, Michael Saks, and James Shearer

SIAM. J. on Algebraic and Discrete Methods 2, pp. 394-410 (17 pages) | Cited 11 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
A board$\mathcal{B}$ is a finite set of unit squares lying in the plane whose corners have integer coordinates. A rectangle of $\mathcal{B}$ is a rectangular subset of $\mathcal{B}$ and an antirectangle is a set of squares in $\mathcal{B}$ no two of which are in a common rectangle. We prove a conjecture of Chvátal that $\mathcal{B}$ if is convex in the horizontal and vertical directions, then the minimum number of rectangles whose union is $\mathcal{B}$ equals the maximum cardinality of an antirectangle. Our proof uses two analogous minimax theorems about covering the corners and covering the edges of the board.
We quote examples that illustrate the necessity of the hypotheses, and give some conjectures and open questions. The method of proof can give a polynomial running time algorithm for finding a minimum cover.

Minimean Location of Different Facilities on a Line Network

B. L. Hulme and P. J. Slater

SIAM. J. on Algebraic and Discrete Methods 2, pp. 411-415 (5 pages) | Cited 1 time

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
The $m$-mean median problem for a network $N$ is introduced in the context of locating $m$ different facilities on $m$ of the vertices in $V(N)$. If we let $d(v, w)$ denote the distance in $N$ between vertices $v$ and $w$, the problem is to select an $m$-set $S \subseteq V(N)$ with complement $\bar S = V(N) - S$ so as to minimize the sum $\sum_{u \in S} \sum_{v \in \bar S} d(u,v)$. That is, one seeks to partition $V(N)$ into two sets, a set $S$ of “facility vertices” and a set $\bar S$ of “customer vertices,” so as to minimize the average distance between a facility and a customer. Complete results are given for the special case of a line network.

Integer Rounding for Polymatroid and Branching Optimization Problems

S. Baum and L. E. Trotter, Jr.

SIAM. J. on Algebraic and Discrete Methods 2, pp. 416-425 (10 pages) | Cited 12 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
Where matrix $M\geqq 0$ and vector $w\geqq 0$ have rational entries, define $r^* (w) = \max \{ 1 \cdot y:yM\leqq w,y\geqq 0 \}$, $z^* (w) = \max \{ 1 \cdot y:yM\leqq w,y\geqq 0,y\,{\text{integral}} \}$. Integer round-down holds for $M$ if, for all integral $w\geqq 0$ , $\lfloor r^* (w) \rfloor = z^* (w)$. Similarly, when $\lceil r_* (w) \rceil = z_* (w)$ for all integral $w\geqq 0$, where $r_* (w) = \min \{ 1 \cdot y:yM\leqq w,y\geqq 0 \}$, $z_* (w) = \min \{ 1 \cdot y:yM\geqq w,y\geqq 0,y \,{\text{integral}} \}$, integer round-up holds for $M$. The integer round-down and round-up properties are shown to hold for certain matrices related to integral polymatroids and branchings in directed graphs.

Hypergeometric and Generalized Hypergeometric Group Testing

F. K. Hwang, Tien Tai Song, and Ding Zhu Du

SIAM. J. on Algebraic and Discrete Methods 2, pp. 426-428 (3 pages)

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
We consider two group testing problems involving a set of $n$ items. In the first problem extactly$d$ defectives, and in the second problem at most$d$ defectives, are distributed arbitrarily in the set. We show that any procedure for identifying all defectives in the first problem can be easily adapted to the second problem, with an increase of at most one in the maximum number of tests required. Some related problems are also described.

Acyclic Digraphs, Young Tableaux and Nilpotent Matrices

Emden R. Gansner

SIAM. J. on Algebraic and Discrete Methods 2, pp. 429-440 (12 pages)

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
A nilpotent matrix is associated with an acyclic digraph in such a way that the Jordan invariants of the matrix correspond to the maximum size of certain families of paths in the digraph. This allows one to associate an integer partition and a standard Young tableau with the digraph, extending the Robinson-Schensted map on permutations. The associated partition is characterized using matrices whose rows are paths in the digraph. This leads to a proof of a conjecture of Greene concerning the entries in the associated Young tableau. When the digraph is transitive, a second characterization is given for the partition. Most of the arguments used are algebraic in nature

Expected Number of Vertices of a Random Convex Polyhedron

D. G. Kelly and J. W. Tolle

SIAM. J. on Algebraic and Discrete Methods 2, pp. 441-451 (11 pages) | Cited 3 times

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
Given $m$ points on the unit sphere in $n$-space, the hyperplanes tangent to the sphere at the given points bound a convex polyhedron with $m$ facets. If the points are chosen independently at random from the uniform distribution on the sphere, the number $V_{mn} $ of the vertices of the polyhedron is a random variable. We obtain an integral expression for $EV_{mn}$ and asymptotic bounds of the form \[ \alpha ^n n^{(n - 6)/2} m\leqq EV_{mn} \leqq \beta ^n n^{(n - 5)/2} m. \]

Weight Enumerators of Self-Orthogonal Codes over $GF$(3)

C. L. Mallows and N. J. A. Sloane

SIAM. J. on Algebraic and Discrete Methods 2, pp. 452-460 (9 pages)

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
The Hamming and complete weight enumerators of maximally self-orthogonal codes over $GF$(3) of lengths $12m - 1$, $12m$ and $12m + 1$ are characterized. The results for length $12m + 1$ are believed to be new, while those for length $12m - 1$, and $12m$ have been considerably simplified.

Strong Connectivity in Directional Nearest-Neighbor Graphs

B. E. Flinchbaugh and L. K. Jones

SIAM. J. on Algebraic and Discrete Methods 2, pp. 461-463 (3 pages)

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
A Directional Nearest-Neighbor graph is defined on a finite set of points in the plane by drawing an arc from each point $X$ to its nearest neighbor in each of $r$ divisions of the plane relative to $X$. We prove that Directional Nearest-Neighbor graphs having $r = 4$ are strongly connected.

The Reliability of Standby Systems with a Faulty Switch

T. Downs and P. K. W. Chan

SIAM. J. on Algebraic and Discrete Methods 2, pp. 464-471 (8 pages)

Online Publication Date: July 31, 2006

Full Text: | Download PDF

Show Abstract
In this paper explicit expressions are derived for the reliability of standby systems with a faulty switch. Three modes of switch malfunction are included, and the expressions apply to hot, warm and cold standby. The expressions are derived by finding the exponential of the state transition matrix using constituent matrices.
Close

Your Recent History Recent History Help

Recently Viewed

  1. Radiative Decay of Bubble Oscillations in a Compressible Fluid
  2. Dynamics of Compressible Non-isothermal Fluids of Non-Newtonian Korteweg Type
  3. A Centennial of the Zaremba–Hopf–Oleinik Lemma
  4. Dynamics of Soliton-like Solutions for Slowly Varying, Generalized KdV Equations: Refraction versus ...
  5. A Singular Limit for Compressible Rotating Fluids
© SIAM By using SIAM Publications Online you agree to abide by the Terms and Conditions of Use. Banner art adapted from a figure by Hinke M. Osinga and Bernd Krauskopf (University of Bristol, UK).
close