Stability of homomorphisms in the compact-open topology

Zlatoš, Pavol

doi:10.1007/s00012-010-0099-7

Cited by 5 publications

(3 citation statements)

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“…Skof [110] studied the stability of the Cauchy equation on an interval, Kominek [69] investigated the equation on an N -dimensional cube in the space R N , and Sikorska [100,101] dealt with such stability postulated for orthogonal vectors in a ball centered at the origin. A somewhat more abstract approach to the conditional stability is presented in [116,117].…”

mentioning

confidence: 99%

Recent developments of the conditional stability of the homomorphism equation

Brzdęk¹,

Fechner²,

Moslehian³

et al. 2015

Banach J. Math. Anal.

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The issue of Ulam's type stability of an equation is understood in the following way: when a mapping which satisfies the equation approximately (in some sense), it is "close" to a solution of it. In this expository paper, we present a survey and a discussion of selected recent results concerning such stability of the equations of homomorphisms, focussing especially on some conditional versions of them. Some historyThis is an expository paper, but because of the demands of the journal, we had to restrict very significantly the number of references. We apologize to the authors of all the papers that are connected with the subjects considered here, but had to be omitted.The property of additivity of a mapping is very important (not only in mathematics) and can be described by the following Cauchy functional equationwhere f is a mapping between semigroups endowed with binary operations denoted by + (we abuse the notation and we use the same symbol for operations in two different structures). Every mapping satisfying equation (1.1) is said to be

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mentioning

confidence: 99%

Recent developments of the conditional stability of the homomorphism equation

Brzdęk¹,

Fechner²,

Moslehian³

et al. 2015

Banach J. Math. Anal.

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“…A systematic and general approach to this topic in the realm of compact Hausdorff topological spaces, using nonstandard analysis was developed by Anderson [1]. The study of stability of the homomorphy property with respect to the compact-open topology was commenced by the second of the present authors [23], [24], [25]. The survey article by Boualem and Brouzet [4] reflects some recent development.…”

mentioning

confidence: 99%

A uniform stability principle for dual lattices

Vodička

Zlatoš

2019

JLA

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We prove a highly uniform stability or "almost-near" theorem for dual lattices of lattices L ⊆ R n . More precisely, we show that, for a vector x from the linear span of a lattice L ⊆ R n , subject to λ 1 (L) ≥ λ > 0, to be ε-close to some vector from the dual lattice L ′ of L, it is enough that the inner products u x are δ-close (with δ < 1/3) to some integers for all vectors u ∈ L satisfying u ≤ r, where r > 0 depends on n, λ, δ and ε, only. This generalizes an analogous result proved for integral lattices in [15]. The proof is nonconstructive, using the ultraproduct construction and a slight portion of nonstandard analysis.

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“…Similarly as Theorem D, also the last Theorem 3 admits a generalization to a stability result for continuous homomorphisms A → B between topological universal algebras, with A locally compact and B completely regular. This result is proved in Zlatoš [22] using the Arzèla-Ascoli theorem and the projective limit The compact subsets of Q are exactly the finite ones and each finite set F ⊆ Q generates a cyclic subgroup F of Q isomorphic to the free abelian group Z. Thus for any finite F ⊆ Q containing an element a = 0 there is even a homomorphism h : F → Z such that h(a) = 0 (and not only a nonzero partial homomorphism F → Z).…”

Section: Pavol Zlatošmentioning

confidence: 53%

10.4115/jla.v2i0.46

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We introduce the notion of stability of continuous homomorphisms of topological groups with respect to the compact-open topology on the space of all continuous functions between them and prove that the characters of any locally compact abelian group are stable in this sense. Using this notion we also generalize a global stability result on continuous homomorphisms between compact groups to a local version for continuous homomorphisms from any locally compact group to an arbitrary topological group and show the necessity of the assumptions of the theorem through some counterexamples. Finally, we separately discuss the special cases of metrizable and discrete groups, respectively. As an application of the latter we prove a purely algebraic result on extendability of finite partial functions to group homomorphisms.

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Stability of homomorphisms in the compact-open topology

Cited by 5 publications

References 10 publications

Recent developments of the conditional stability of the homomorphism equation

Recent developments of the conditional stability of the homomorphism equation

A uniform stability principle for dual lattices

10.4115/jla.v2i0.46

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