| Title:
|
Equation with residuated functions (English) |
| Author:
|
Cuninghame-Green, R. A. |
| Author:
|
Zimmermann, K. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
42 |
| Issue:
|
4 |
| Year:
|
2001 |
| Pages:
|
729-740 |
| . |
| Category:
|
math |
| . |
| Summary:
|
The structure of solution-sets for the equation $F(x)=G(y)$ is discussed, where $F,G$ are given residuated functions mapping between partially-ordered sets. An algorithm is proposed which produces a solution in the event of finite termination: this solution is maximal relative to initial trial values of $x,y$. Properties are defined which are sufficient for finite termination. The particular case of max-based linear algebra is discussed, with application to the synchronisation problem for discrete-event systems; here, if data are rational, finite termination is assured. Numerical examples are given. For more general residuated real functions, lower semicontinuity is sufficient for convergence to a solution, if one exists. (English) |
| Keyword:
|
systems of nonlinear equations |
| Keyword:
|
residuation theory |
| Keyword:
|
max-algebras |
| MSC:
|
47H05 |
| MSC:
|
47J05 |
| MSC:
|
90C27 |
| MSC:
|
93C65 |
| idZBL:
|
Zbl 1068.93039 |
| idMR:
|
MR1883381 |
| . |
| Date available:
|
2009-01-08T19:18:13Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119288 |
| . |
| Reference:
|
[1] Baccelli F.L., Cohen G., Olsder G.J., Quadrat J.-P.: Synchronization and Linearity, An Algebra for Discrete Event Systems.Wiley, Chichester, 1992. Zbl 0824.93003, MR 1204266 |
| Reference:
|
[2] Blyth T.S., Janowitz M.: Residuation Theory.Pergamon, Oxford, 1972. Zbl 0301.06001, MR 0396359 |
| Reference:
|
[3] Cuninghame-Green R.A., Butkovic P.: The Equation $A øtimes x = B øtimes y$ over $(\{ - \infty \} \cup {\Bbb R}, \max,+)$.Theoretical Computer Science, Special Issue on $(\max,+)$ Algebra, to appear. MR 1957609 |
| Reference:
|
[4] Cuninghame-Green R.A., Cechlarova K.: Residuation in fuzzy algebra and some applications.Fuzzy Sets and Systems 71 227-239 (1995). Zbl 0845.04007, MR 1329610 |
| Reference:
|
[5] Cuninghame-Green R.A.: Minimax Algebra.Lecture Notes in Economics and Mathematical Systems No. 166, Springer-Verlag, Berlin, 1979. Zbl 0739.90073, MR 0580321 |
| Reference:
|
[6] Walkup E.A., Borriello G.: A General Linear Max-Plus Solution Technique.in Idempotency (ed. J. Gunawardena), Cambridge, 1998. Zbl 0898.68035 |
| Reference:
|
[7] Zimmermann U.: Linear and Combinatorial Optimization in Ordered Algebraic Structures.North Holland, Amsterdam, 1981. Zbl 0466.90045, MR 0609751 |
| . |
