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Non-commutative geometry of finite groups
A finite set can be supplied with a group structure which can then be used to select (classes of) differential calculi on it via the notions of left-, right- and bicovariance. A correspondingExpand
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Differential calculi and linear connections
A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found thatExpand
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Discrete differential calculus graphs, topologies and gauge theory
Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a ‘‘reduction’’ of the ‘‘universalExpand
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Umbral Calculus, Discretization, and Quantum Mechanics on a Lattice
`Umbral calculus' deals with representations of the canonical commutation relations. We present a short exposition of it and discuss how this calculus can be used to discretize continuum models andExpand
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An algebraic scheme associated with the noncommutative KP hierarchy and some of its extensions
A well-known ansatz ('trace method') for soliton solutions turns the equations of the (non-commutative) KP hierarchy, and those of certain extensions, into families of algebraic sum identities. WeExpand
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Discrete Riemannian geometry
Within a framework of noncommutative geometry, we develop an analog of (pseudo-) Riemannian geometry on finite and discrete sets. On a finite set, there is a counterpart of the continuum metricExpand
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Discrete differential manifolds and dynamics on networks
A discrete differential manifold is a countable set together with an algebraic differential calculus on it. This structure has already been explored in previous work and provides a convenientExpand
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Differential calculus and gauge theory on finite sets
We develop differential calculus and gauge theory on a finite set G. An elegant formulation is obtained when G is supplied with a group structure and in particular for a cyclic group. Connes'Expand
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Matrix KP: tropical limit and Yang–Baxter maps
We study soliton solutions of matrix Kadomtsev–Petviashvili (KP) equations in a tropical limit, in which their support at fixed time is a planar graph and polarizations are attached to itsExpand
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A Noncommutative version of the nonlinear Schrodinger equation
We apply a (Moyal) deformation quantization to a bicomplex associated with the classical nonlinear Schrodinger equation. This induces a deformation of the latter equation to noncommutative space-timeExpand
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