20 research outputs found
A Small Contribution to Catalanā²s Equation
AbstractUsing recent results on linear forms in logarithms of algebraic numbers, we prove that any solution of the equation xp ā yq = Ļµ, where Ļµ = Ā± 1, p and q are odd primes, and p > q satisfies p < 3.42 Ā· 1028 and q < 5.6 Ā· 1019. We also combine our work with some results of Altonen and Inkeri to determine the six cases with q ā¤ 37 for which this equation may have solutions
States on pseudo effect algebras and integrals
We show that every state on an interval pseudo effect algebra satisfying
some kind of the Riesz Decomposition Properties (RDP) is an integral through a
regular Borel probability measure defined on the Borel -algebra of a
Choquet simplex . In particular, if satisfies the strongest type of
(RDP), the representing Borel probability measure can be uniquely chosen to
have its support in the set of the extreme points of $K.
The Lattice and Simplex Structure of States on Pseudo Effect Algebras
We study states, measures, and signed measures on pseudo effect algebras with
some kind of the Riesz Decomposition Property, (RDP). We show that the set of
all Jordan signed measures is always an Abelian Dedekind complete -group.
Therefore, the state space of the pseudo effect algebra with (RDP) is either
empty or a nonempty Choquet simplex or even a Bauer simplex. This will allow
represent states on pseudo effect algebras by standard integrals
Reversible maps and composites of involutions in groups of piecewise linear homeomorphisms of the real line
An element of a group is reversible if it is conjugate to its own inverse, and it is strongly reversible if it is conjugate to its inverse by an involution. A group element is strongly reversible if and only if it can be expressed as a composite of two involutions. In this paper the reversible maps, the strongly reversible maps, and those maps that can be expressed as a composite of involutions are determined in certain groups of piecewise linear homeomorphisms of the real line
Electron-hole asymmetry in two-terminal graphene devices
A theoretical model is proposed to describe asymmetric gate-voltage
dependence of conductance and noise in two-terminal ballistic graphene devices.
The model is analyzed independently within the self-consistent Hartree and
Thomas-Fermi approximations. Our results justify the prominent role of metal
contacts in recent experiments with suspended graphene flakes. The
contact-induced electrostatic potentials in graphene demonstrate a power-law
decay with the exponent varying from -1 to -0.5. Within our model we explain
electron-hole asymmetry and strong Fabri-Perot oscillations of the conductance
and noise at positive doping, which were observed in many experiments with
submicrometer samples. Limitations of the Thomas-Fermi approximation in a
vicinity of the Dirac point are discussed.Comment: 7 pages, 8 figure
Embedding finitely generated Abelian lattice-ordered groups : Higman's theorem and a realisation of \pi
Graham Higman proved that a finitely generated group can be embedded in a finitely presented group if and only if it has a recursively enumerable set of defining relations. The analogue for lattice-ordered groups is considered here. Clearly, the finitely generated lattice-ordered groups that can be embedded in finitely presented lattice-ordered groups must have recursively enumerable sets of defining relations. The converse direction is proved for a special class of lattice-ordered groups