Cauchy's functional equation and extensions: Goldie's equation and inequality, the Go{\l}\k{a}b-Schinzel equation and Beurling's equation

The Cauchy functional equation is not only the most important single functional equation, it is also central to regular variation. Classical Karamata regular variation involves a functional equation and inequality due to Goldie; we study this, and its counterpart in Beurling regular variation, together with the related Go{\l}\k{a}b-Schinzel equation.Comment: Companion paper to: Additivity, subadditivity and linearity: automatic continuity and quantifier weakenin

Additivity, subadditivity and linearity: automatic continuity and quantifier weakening

We study the interplay between additivity (as in the Cauchy functional equation), subadditivity and linearity. We obtain automatic continuity results in which additive or subadditive functions, under minimal regularity conditions, are continuous and so linear. We apply our results in the context of quantifier weakening in the theory of regular variation completing our programme of reducing the number of hard proofs there to zero.Comment: Companion paper to: Cauchy's functional equation and extensions: Goldie's equation and inequality, the Go{\l}\k{a}b-Schinzel equation and Beurling's equation Updated to refer to other developments and their publication detail

Plasmids from thermophillic bacilli

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A describing function for resonantly commutated H-bridge inverters

Abstract—The paper presents the derivation of a describing function to model the dynamic behavior of a metal oxide semiconductor field effect transistor-based, capacitively commutated H-bridge, including a comprehensive explanation of the various stages in the switching cycle. Expressions to model the resulting input current, are also given. The derived model allows the inverter to be accurately modeled within a control system simulation over a number of utility input voltage cycles, without resorting to computationally intensive switching-cycle level, time-domain SPICE simulations. Experimental measurements from a prototype H-bridge inverter employed in an induction heating application, are used to demonstrate a high degree of prediction accuracy over a large variation of load conditions is possible using the simplified model