Perturbations of the Spence–Abel equation and deformations of the dilogarithm function

Hartnick, Tobias; Ott, Andreas

doi:10.1007/s00208-016-1514-y

Cited by 3 publications

(5 citation statements)

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“…In recent years, the Ulam stability of various objects (for example functional equations and inequalities, differential, difference and integral equations, flows, groups, random dynamics, vector measures, and C*-algebras) has been studied by many researchers (for more information on this notion as well as its applications we refer the reader to papers [4,7,8,10,13,14,22,23,26,28,29,33,34,40,42,45,46] and books [12,39]).…”

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confidence: 99%

On perturbations of two general equations in several variables

Ciepliński

2022

Math. Ann.

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In this paper, we deal with perturbations of two general functional equations in several variables. Namely, we prove the generalized, in the spirit of Bourgin, Ulam stability of these equations in Banach spaces. In order to do this, we use the fixed point method. Moreover, as corollaries from our main results, we get several outcomes on approximate solutions of a few important classic equations. They include, among others, the functional equations which characterize multi-additive and multi-quadratic mappings. In consequence, the perturbation of homomorphisms of Banach spaces and quadratic mappings is also treated.

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confidence: 99%

On perturbations of two general equations in several variables

Ciepliński

2022

Math. Ann.

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“…There is a number of known bounded cohomology classes in higher degree, often emerging from explicit geometric constructions (e.g. [19,50,26,24,39,11,37,12,8,27,28,10,31]). On the other hand, a classical result due to Johnson [34] asserts that the bounded cohomology of an amenable group vanishes in all positive degrees.…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

“…Surjectivity of the comparison map was already studied by Dupont [19] and has since been established in many cases [19,50,26,24,11,37,12,27], while still almost nothing is known about injectivity. In fact, injectivity of the comparison map has so far only been proved in degree 2 by Burger and Monod [15], in degree 3 for certain groups of rank 1 by Burger and Monod [17], Bloch [6] and Pieters [45], and in degree 4 for SL 2 (R) by Hartnick and the author [28].…”

Section: Introduction and Statement Of The Resultsmentioning

confidence: 99%

Transgression in bounded cohomology and a conjecture of Monod

Ott

2019

J. Topol. Anal.

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We develop an algebro-analytic framework for the systematic study of the continuous bounded cohomology of Lie groups in large degree. As an application, we examine the continuous bounded cohomology of PSL(2, R) with trivial real coefficients in all degrees greater than two. We prove a vanishing result for strongly reducible classes, thus providing further evidence for a conjecture of Monod. On the cochain level, our method yields explicit formulas for cohomological primitives of arbitrary bounded cocycles. defined by L κ (α) = κ α.Corollary. The bounded Lefschetz map in (1) is zero in all positive degrees.Returning to Theorem 1, we note that in small degrees n = 3, 4, much stronger vanishing theorems apply: Burger and Monod [17] proved that H 3 cb (G; R) = 0, while Hartnick and the author [28] showed that H 4 cb (G; R) = 0. In large degree, on the other hand, our Theorem 1

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“…This would imply that bcd(H) coincides with the dimension of the associated symmetric space, thereby providing examples of groups of arbitrary bounded cohomological dimension. Plenty is known by now about surjectivity of the comparison map [19,25,7,34,23,28], while injectivity still remains mysterious in higher degrees. Indeed, injectivity has so far been established only in degree two for arbitrary H [14], and for some rank one groups in degree three [16,3,6].…”

Section: Introductionmentioning

confidence: 99%

“…In particular, the approach originally developed for free groups [24] and hyperbolic groups [21] has been extended to larger classes of groups including mapping class groups [2] and acylindrically hyperbolic groups [30,22]. Moreover, there has been some progress in constructing bounded cohomology classes in higher degree [36,28,5,23].…”

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confidence: 99%