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Sensor Location Problem for a Multigraph

Pilipchuk, L. A., Vishnevetskaya, T. S., Pesheva, Y. H. (2013)

Mathematica Balkanica New Series

MSC 2010: 05C50, 15A03, 15A06, 65K05, 90C08, 90C35We introduce sparse linear underdetermined systems with embedded network structure. Their structure is inherited from the non-homogeneous network ow programming problems with nodes of variable intensities. One of the new applications of the researched underdetermined systems is the sensor location problem (SLP) for a multigraph. That is the location of the minimum number of sensors in the nodes of the multigraph, in order to determine the arcs ow...

Several results on chordal bipartite graphs

Mihály Bakonyi, Aaron Bono (1997)

Czechoslovak Mathematical Journal

The question of generalizing results involving chordal graphs to similar concepts for chordal bipartite graphs is addressed. First, it is found that the removal of a bisimplicial edge from a chordal bipartite graph produces a chordal bipartite graph. As consequence, occurance of arithmetic zeros will not terminate perfect Gaussian elimination on sparse matrices having associated a chordal bipartite graph. Next, a property concerning minimal edge separators is presented. Finally, it is shown that,...

Simultaneous solution of linear equations and inequalities in max-algebra

Abdulhadi Aminu (2011)

Kybernetika

Let a ø p l u s b = max ( a , b ) and a ø t i m e s b = a + b for a , b ∈ ℝ . Max-algebra is an analogue of linear algebra developed on the pair of operations ( ø p l u s , ø t i m e s ) extended to matrices and vectors. The system of equations A ø t i m e s x = b and inequalities C ø t i m e s x ł e q d have each been studied in the literature. We consider a problem consisting of these two systems and present necessary and sufficient conditions for its solvability. We also develop a polynomial algorithm for solving max-linear program whose constraints are max-linear equations and inequalities.

Solutions of minus partial ordering equations over von Neumann regular rings

Yu Guan, Zhaojia Tong (2015)

Open Mathematics

In this paper, we mainly derive the general solutions of two systems of minus partial ordering equations over von Neumann regular rings. Meanwhile, some special cases are correspondingly presented. As applications, we give some necessary and sufficient conditions for the existence of solutions. It can be seen that some known results can be regarded as the special cases of this paper.

Solvability classes for core problems in matrix total least squares minimization

Iveta Hnětynková, Martin Plešinger, Jana Žáková (2019)

Applications of Mathematics

Linear matrix approximation problems A X ≈ B are often solved by the total least squares minimization (TLS). Unfortunately, the TLS solution may not exist in general. The so-called core problem theory brought an insight into this effect. Moreover, it simplified the solvability analysis if B is of column rank one by extracting a core problem having always a unique TLS solution. However, if the rank of B is larger, the core problem may stay unsolvable in the TLS sense, as shown for the first time by Hnětynková,...

Solving systems of two–sided (max, min)–linear equations

Martin Gavalec, Karel Zimmermann (2010)

Kybernetika

A finite iteration method for solving systems of (max, min)-linear equations is presented. The systems have variables on both sides of the equations. The algorithm has polynomial complexity and may be extended to wider classes of equations with a similar structure.

Sparse recovery with pre-Gaussian random matrices

Simon Foucart, Ming-Jun Lai (2010)

Studia Mathematica

For an m × N underdetermined system of linear equations with independent pre-Gaussian random coefficients satisfying simple moment conditions, it is proved that the s-sparse solutions of the system can be found by ℓ₁-minimization under the optimal condition m ≥ csln(eN/s). The main ingredient of the proof is a variation of a classical Restricted Isometry Property, where the inner norm becomes the ℓ₁-norm and the outer norm depends on probability distributions.

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