A Dynamical Systems Construction of a Counterexample to the Finiteness Conjecture

Kozyakin, Victor

doi:10.1109/cdc.2005.1582511

Cited by 39 publications

(58 citation statements)

References 7 publications

(5 reference statements)

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“…We say in these cases that the set of matrices possess the finiteness property. It was conjectured that all sets of matrices have the finiteness property: this is the well known finiteness conjecture which has been disproved in [7], [8], and [12]. Nevertheless, we conjecture here that the sets of matrices with binary entries, and, in particular, those constructed in order to compute a capacity do always possess the finiteness property.…”

Section: Examplementioning

confidence: 63%

On the Complexity of Computing the Capacity of Codes That Avoid Forbidden Difference Patterns

Blondel

Jungers

Protasov

2006

IEEE Trans. Inform. Theory

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Abstract-Some questions related to the computation of the capacity of codes that avoid forbidden difference patterns are analysed. The maximal number of n-bit sequences whose pairwise differences do not contain some given forbidden difference patterns is known to increase exponentially with n; the coefficient of the exponent is the capacity of the forbidden patterns.In this paper, new inequalities for the capacity are given that allow for the approximation of the capacity with arbitrary high accuracy. The computational cost of the algorithm derived from these inequalities is fixed once the desired accuracy is given. Subsequently, a polynomial time algorithm is given for determining if the capacity of a set is positive while the same problem is shown to be NP-hard when the sets of forbidden patterns are defined over an extended set of symbols. Finally, the existence of extremal norms is proved for any set of matrices arising in the capacity computation. Based on this result, a second capacity approximating algorithm is proposed. The usefulness of this algorithm is illustrated by computing exactly the capacity of particular codes that were only known approximately.

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Section: Examplementioning

confidence: 63%

On the Complexity of Computing the Capacity of Codes That Avoid Forbidden Difference Patterns

Blondel

Jungers

Protasov

2006

IEEE Trans. Inform. Theory

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“…. , n a matrix A i be a finite convex combinations of matrices A and together with (14) this yields the required equality.…”

Section: Theorem 3 Implies Thatmentioning

confidence: 99%

“…The possibility of explicit calculation of the spectral characteristics of sets of matrices is conventionally associated with the validity of the finiteness conjecture according to which the limit in formulas (2) and (4) is attained at some finite value of n. For the generalized spectral radius this conjecture was set up by Lagarias and Wang [11] and subsequently disproved by Bousch and Mairesse [12]. Later on there appeared a few alternative counterexamples [13][14][15]. However, all these counterexamples were pure 'nonexistence' results which provided no constructive description of the sets of matrices for which the finiteness conjecture fails.…”

Section: Introductionmentioning

confidence: 99%

Hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non-negative matrices

Kozyakin

2016

Linear Algebra and its Applications

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Recently Blondel, Nesterov and Protasov proved [1,2] that the finiteness conjecture holds for the generalized and the lower spectral radii of the sets of non-negative matrices with independent row/column uncertainty. We show that this result can be obtained as a simple consequence of the so-called hourglass alternative used in [3], by the author and his companions, to analyze the minimax relations between the spectral radii of matrix products. Axiomatization of the statements that constitute the hourglass alternative makes it possible to define a new class of sets of positive matrices having the finiteness property, which includes the sets of non-negative matrices with independent row uncertainty. This class of matrices, supplemented by the zero and identity matrices, forms a semiring with the Minkowski operations of addition and multiplication of matrix sets, which gives means to construct new sets of non-negative matrices possessing the finiteness property for the generalized and the lower spectral radii.

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“…satisfies the following property: for any periodic switching law we have x(k) → 0, yet the switched system is not GUAS (see also Kozyakin (2005Kozyakin ( , 2007). This provides a counterexample to the finiteness conjecture asserting that there always exists an integer k and a matrix B ∈ Σ(k) such that ρ(A) = (ρ(B)) 1/k .…”

Section: Joint Spectral Radiusmentioning

confidence: 99%

Explicit construction of a Barabanov norm for a class of positive planar discrete-time linear switched systems

Teichner

Margaliot

2012

Automatica

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We consider the stability under arbitrary switching of a discrete-time linear switched system. A powerful approach for addressing this problem is based on studying the "most unstable" switching law (MUSL). If the solution of the switched system corresponding to the MUSL converges to the origin, then the switched system is stable for any switching law. The MUSL can be characterized using optimal control techniques. This variational approach leads to a Hamilton-Jacobi-Bellman equation describing the behavior of the switched system under the MUSL. The solution of this equation is sometimes referred to as a Barabanov norm of the switched system. Although the Barabanov norm was studied extensively, it seems that there are few examples where it was actually computed in closed-form. In this paper, we consider a special class of positive planar discretetime linear switched systems and provide a closed-form expression for a corresponding Barabanov norm and a MUSL. The unit circle in this norm is a parallelogram.

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A Dynamical Systems Construction of a Counterexample to the Finiteness Conjecture

Cited by 39 publications

References 7 publications

On the Complexity of Computing the Capacity of Codes That Avoid Forbidden Difference Patterns

On the Complexity of Computing the Capacity of Codes That Avoid Forbidden Difference Patterns

Hourglass alternative and the finiteness conjecture for the spectral characteristics of sets of non-negative matrices

Explicit construction of a Barabanov norm for a class of positive planar discrete-time linear switched systems

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