Linear maps preserving Drazin inverses of matrices over local rings

AbstractThe Brascamp–Lieb inequalities are a very general class of classical multilinear inequalities, well-known examples of which being Hölder’s inequality, Young’s convolution inequality, and the Loomis–Whitney inequality. Conventionally, a Brascamp–Lieb inequality is defined as a multilinear Lebesgue bound on the product of the pullbacks of a collection of functions $$f_j\in L^{q_j}(\mathbb {R}^{n_j})$$ f j ∈ L q j ( R n j ) , for $$j=1,\ldots ,m$$ j = 1 , … , m , under some corresponding linear maps $$B_j$$ B j . This regime is now fairly well understood (Bennett et al. in Geom Funct Anal 17(5):1343–1415, 2008), and moving forward there has been interest in nonlinear generalisations, where $$B_j$$ B j is now taken to belong to some suitable class of nonlinear maps. While there has been great recent progress on the question of local nonlinear Brascamp–Lieb inequalities (Bennett et al. in Duke Math J 169(17):3291–3338, 2020), there has been relatively little regarding global results; this paper represents some progress along this line of enquiry. We prove a global nonlinear Brascamp–Lieb inequality for ‘quasialgebraic’ maps, a class that encompasses polynomial and rational maps, as a consequence of the multilinear Kakeya-type inequalities of Zhang and Zorin-Kranich. We incorporate a natural affine-invariant weight that both compensates for local degeneracies and yields a constant with minimal dependence on the underlying maps. We then show that this inequality generalises Young’s convolution inequality on algebraic groups with suboptimal constant.

Download Full-text

Correction to: Classification of non-local rings with genus two zero-divisor graphs

Soft Computing ◽

10.1007/s00500-020-05533-z ◽

2021 ◽

Vol 25 (4) ◽

pp. 3355-3356

Author(s):

T. Asir ◽

K. Mano ◽

T. Tamizh Chelvam

Keyword(s):

Zero Divisor ◽

Local Rings ◽

Non Local ◽

Genus Two

Download Full-text

How Many Inflections are There in the Lyapunov Spectrum?

Communications in Mathematical Physics ◽

10.1007/s00220-021-04161-4 ◽

2021 ◽

Author(s):

O. Jenkinson ◽

M. Pollicott ◽

P. Vytnova

Keyword(s):

Piecewise Linear ◽

Lyapunov Spectrum ◽

Stat Phys ◽

Linear Map ◽

Piecewise Linear Map ◽

Expanding Map ◽

Lyapunov Spectra ◽

Linear Maps ◽

Inflection Points ◽

Piecewise Linear Maps

AbstractIommi and Kiwi (J Stat Phys 135:535–546, 2009) showed that the Lyapunov spectrum of an expanding map need not be concave, and posed various problems concerning the possible number of inflection points. In this paper we answer a conjecture in Iommi and Kiwi (2009) by proving that the Lyapunov spectrum of a two branch piecewise linear map has at most two points of inflection. We then answer a question in Iommi and Kiwi (2009) by proving that there exist finite branch piecewise linear maps whose Lyapunov spectra have arbitrarily many points of inflection. This approach is used to exhibit a countable branch piecewise linear map whose Lyapunov spectrum has infinitely many points of inflection.

Download Full-text