Non-extendable 𝔽 q $$\mathbb{F}_{q}$$ -Quadratic Perfect Nonlinear Maps

An Algebraic Brascamp–Lieb Inequality

Journal of Geometric Analysis ◽

10.1007/s12220-021-00638-9 ◽

2021 ◽

Author(s):

Jennifer Duncan

Keyword(s):

General Class ◽

Recent Progress ◽

Algebraic Groups ◽

Hölder’S Inequality ◽

Duke Math ◽

Rational Maps ◽

Affine Invariant ◽

Linear Maps ◽

Nonlinear Maps ◽

Funct Anal

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.

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On the identification of nonlinear maps in a general interconnected system

Proceedings of the 1999 American Control Conference (Cat. No. 99CH36251) ◽

10.1109/acc.1999.782407 ◽

1999 ◽

Keyword(s):

Interconnected System ◽

Nonlinear Maps

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Thermodynamic processes expressed in terms of nonlinear maps

Quantum Studies Mathematics and Foundations ◽

10.1007/s40509-018-0165-y ◽

2018 ◽

Vol 6 (1) ◽

pp. 83-88

Author(s):

Yehuda Roth

Keyword(s):

Nonlinear Maps ◽

Thermodynamic Processes

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Stabilizing Feedback Controls for Nonlinear Hamiltonian Systems and Nonconservative Bilinear Systems in Elasticity

Journal of Dynamic Systems Measurement and Control ◽

10.1115/1.3149627 ◽

1982 ◽

Vol 104 (1) ◽

pp. 27-32 ◽

Cited By ~ 7

Author(s):

S. N. Singh

Keyword(s):

Asymptotic Stability ◽

Hamiltonian Systems ◽

Attitude Control ◽

Sufficient Conditions ◽

Bilinear Systems ◽

Feedback Controls ◽

Control Laws ◽

Nonlinear Maps ◽

The Stability ◽

Necessary And Sufficient

Using the invariance principle of LaSalle [1], sufficient conditions for the existence of linear and nonlinear control laws for local and global asymptotic stability of nonlinear Hamiltonian systems are derived. An instability theorem is also presented which identifies the control laws from the given class which cannot achieve asymptotic stability. Some of the stability results are based on certain results for the univalence of nonlinear maps. A similar approach for the stabilization of bilinear systems which include nonconservative systems in elasticity is used and a necessary and sufficient condition for stabilization is obtained. An application to attitude control of a gyrostat Satellite is presented.

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