1,530 research outputs found
Rational semimodules over the max-plus semiring and geometric approach of discrete event systems
We introduce rational semimodules over semirings whose addition is
idempotent, like the max-plus semiring, in order to extend the geometric
approach of linear control to discrete event systems. We say that a
subsemimodule of the free semimodule S^n over a semiring S is rational if it
has a generating family that is a rational subset of S^n, S^n being thought of
as a monoid under the entrywise product. We show that for various semirings of
max-plus type whose elements are integers, rational semimodules are stable
under the natural algebraic operations (union, product, direct and inverse
image, intersection, projection, etc). We show that the reachable and
observable spaces of max-plus linear dynamical systems are rational, and give
various examples.Comment: 24 pages, 9 postscript figures; example in section 4.3 expande
How to implement a modular form
AbstractWe present a model for Fourier expansions of arbitrary modular forms. This model takes precisions and symmetries of such Fourier expansions into account. The value of this approach is illustrated by studying a series of examples. An implementation of these ideas is provided by the author. We discuss the technical background of this implementation, and we explain how to implement arbitrary Fourier expansions and modular forms. The framework allows us to focus on the considerations of a mathematical nature during this procedure. We conclude with a list of currently available implementations and a discussion of possible computational research
Network Models
Networks can be combined in various ways, such as overlaying one on top of
another or setting two side by side. We introduce "network models" to encode
these ways of combining networks. Different network models describe different
kinds of networks. We show that each network model gives rise to an operad,
whose operations are ways of assembling a network of the given kind from
smaller parts. Such operads, and their algebras, can serve as tools for
designing networks. Technically, a network model is a lax symmetric monoidal
functor from the free symmetric monoidal category on some set to
, and the construction of the corresponding operad proceeds via a
symmetric monoidal version of the Grothendieck construction.Comment: 46 page
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