208,126 research outputs found

    Jensen's functional equation on the symmetric group Sn\bold{S_n}

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    Two natural extensions of Jensen's functional equation on the real line are the equations f(xy)+f(xy1)=2f(x)f(xy)+f(xy^{-1}) = 2f(x) and f(xy)+f(y1x)=2f(x)f(xy)+f(y^{-1}x) = 2f(x), where ff is a map from a multiplicative group GG into an abelian additive group HH. In a series of papers \cite{Ng1}, \cite{Ng2}, \cite{Ng3}, C. T. Ng has solved these functional equations for the case where GG is a free group and the linear group GLn(R)GL_n(R), R=\z,\r, a quadratically closed field or a finite field. He has also mentioned, without detailed proof, in the above papers and in \cite{Ng4} that when GG is the symmetric group SnS_n the group of all solutions of these functional equations coincides with the group of all homomorphisms from (Sn,)(S_n,\cdot) to (H,+)(H,+). The aim of this paper is to give an elementary and direct proof of this fact.Comment: 8 pages, Abstract changed, the proof of Proposition 2.1 and Lemma 2.4 changed (minor), one reference added, final version, to be published in Aequationes Mathematicae (2011

    Hopf algebras of dimension pq, II

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    Let H be a Hopf algebra of dimension pq over an algebraically closed field of characteristic zero, where p, q are odd primes with p < q < 4p+12. We prove that H is semisimple and thus isomorphic to a group algebra, or the dual of a group algebra.Comment: 14pp, Late

    About quantum fluctuations and holographic principle in (4+n)-dimensional spacetime

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    In the article we present explicit expressions for quantum fluctuations of spacetime in the case of (4+n)(4+n)-dimensional spacetimes, and consider their holographic properties and some implications for clocks, black holes and computation. We also consider quantum fluctuations and their holographic properties in ADD model and estimate the typical size and mass of the clock to be used in precise measurements of spacetime fluctuations. Numerical estimations of phase incoherence of light from extra-galactic sources in ADD model are also presented.Comment: 5 page
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