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Izv. Akad. Nauk SSSR Ser. Mat., 1991, Volume 55, Issue 1, Pages 93–109 (Mi izv1027)  

This article is cited in 14 scientific papers (total in 14 papers)

Endomorphisms of semimodules over semirings with an idempotent operation

P. I. Dudnikov, S. N. Samborskii


Abstract: For an arbitrary endomorphism $A$ of the free semimodule $K^n$ over an Abelian semiring $K$ with operations $\oplus$ and $\odot$ it is shown under the assumption that $\oplus$ is idempotent (and under certain other restrictions on $K$) that there exists a nontrivial “spectrum”, i.e., there exist a $\lambda\in K$ and a nontrivial subsemimodule $J$ such that $Af=\lambda\odot f$ for any $f\in J$. The same result is also obtained for endomorphism analogues of integral operators (in the sense of the theory of idempotent integration). In terms of this spectrum investigations are made of the asymptotic behavior of endomorphisms under iteration and of convergence of the “Neumann series” appearing in the solution of the equations $y=Ay\oplus f$. The simplest examples are connected with the semiring $\{K=R\cup \{-\infty\}, \oplus=\max, \odot=+\}$ and arise, for example, in dynamic programming problems.

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English version:
Mathematics of the USSR-Izvestiya, 1992, 38:1, 91–105

Bibliographic databases:

UDC: 512.55
MSC: Primary 16Y60; Secondary 90C39
Received: 11.11.1987

Citation: P. I. Dudnikov, S. N. Samborskii, “Endomorphisms of semimodules over semirings with an idempotent operation”, Izv. Akad. Nauk SSSR Ser. Mat., 55:1 (1991), 93–109; Math. USSR-Izv., 38:1 (1992), 91–105

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. A. Lesin, S. N. Samborskii, “Spectra of compact endomorphisms of a semimodule of continuous functions over an idempotent semiring”, Russian Math. Surveys, 48:3 (1993), 205–206  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. S. A. Lesin, S. N. Samborskii, “Spectra of Compact Endomorphisms of a Semimodule of Continuous Functions over an Idempotent Semiring”, Funct. Anal. Appl., 27:1 (1993), 64–66  mathnet  crossref  mathscinet  zmath  isi
    3. M. Gondran, M. Minoux, “Eigenvalues and eigen-functionals of diagonally dominant endomorphisms in Min-Max analysis”, Linear Algebra and its Applications, 282:1-3 (1998), 47  crossref
    4. G. L. Litvinov, V. P. Maslov, G. B. Shpiz, “Tensor products of idempotent semimodules. An algebraic approach”, Math. Notes, 65:4 (1999), 479–489  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. A. N. SOBOLEVSKIĬ, “AUBRY–MATHER THEORY AND IDEMPOTENT EIGENFUNCTIONS OF BELLMAN OPERATOR”, Commun. Contemp. Math, 01:04 (1999), 517  crossref
    6. P. Del Moral, M. Doisy, “Maslov Idempotent Probability Calculus. II”, Theory Probab Appl, 44:2 (2000), 319  mathnet  crossref  mathscinet  isi
    7. G. B. Shpiz, “Solution of algebraic equations in idempotent semifields”, Russian Math. Surveys, 55:5 (2000), 1003–1004  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    8. G. L. Litvinov, V. P. Maslov, G. B. Shpiz, “Idempotent Functional Analysis: An Algebraic Approach”, Math. Notes, 69:5 (2001), 696–729  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. Nicolas Baca�r, “Convergence of numerical methods and parameter dependence of min-plus eigenvalue problems, Frenkel-Kontorova models and homogenization of Hamilton–Jacobi equations”, M2AN, 35:6 (2001), 1185  crossref
    10. G. L. Litvinov, “The Maslov dequantization, idempotent and tropical mathematics: a brief introduction”, J. Math. Sci. (N. Y.), 140:3 (2007), 426–444  mathnet  crossref  mathscinet  zmath  elib  elib
    11. G. L. Litvinov, G. B. Shpiz, “Kernel theorems and nuclearity in idempotent mathematics. An algebraic approach”, J. Math. Sci. (N. Y.), 141:4 (2007), 1417–1428  mathnet  crossref  mathscinet  zmath  elib  elib
    12. V. D. Matveenko, “Optimal paths in oriented graphs and eigenvectors in $\max$-$\oplus$ systems”, Discrete Math. Appl., 19:4 (2009), 389–409  mathnet  crossref  crossref  mathscinet  elib
    13. Vechtomov E.M., “Stroenie polutel”, Vestn. Syktyvkarskogo gos. un-ta. Ser. 1: Matem. Mekh. Inform., 2009, no. 10, 3–42  elib
    14. DAVID CACHERA, THOMAS JENSEN, ARNAUD JOBIN, PASCAL SOTIN, “Long-run cost analysis by approximation of linear operators over dioids”, Math Struct Comp Sci, 2010, 1  crossref
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