14,464 research outputs found
Tropical linear algebra with the Lukasiewicz T-norm
The max-Lukasiewicz semiring is defined as the unit interval [0,1] equipped
with the arithmetics "a+b"=max(a,b) and "ab"=max(0,a+b-1). Linear algebra over
this semiring can be developed in the usual way. We observe that any problem of
the max-Lukasiewicz linear algebra can be equivalently formulated as a problem
of the tropical (max-plus) linear algebra. Based on this equivalence, we
develop a theory of the matrix powers and the eigenproblem over the
max-Lukasiewicz semiring.Comment: 27 page
Entropy algebras and Birkhoff factorization
We develop notions of Rota-Baxter structures and associated Birkhoff
factorizations, in the context of min-plus semirings and their thermodynamic
deformations, including deformations arising from quantum information measures
such as the von Neumann entropy. We consider examples related to Manin's
renormalization and computation program, to Markov random fields and to
counting functions and zeta functions of algebraic varieties.Comment: 28 pages, LaTe
Sparse solutions of linear Diophantine equations
We present structural results on solutions to the Diophantine system
,
with the smallest number of non-zero entries. Our tools are algebraic and
number theoretic in nature and include Siegel's Lemma, generating functions,
and commutative algebra. These results have some interesting consequences in
discrete optimization
Public Key Cryptography based on Semigroup Actions
A generalization of the original Diffie-Hellman key exchange in
found a new depth when Miller and Koblitz suggested that such a protocol could
be used with the group over an elliptic curve. In this paper, we propose a
further vast generalization where abelian semigroups act on finite sets. We
define a Diffie-Hellman key exchange in this setting and we illustrate how to
build interesting semigroup actions using finite (simple) semirings. The
practicality of the proposed extensions rely on the orbit sizes of the
semigroup actions and at this point it is an open question how to compute the
sizes of these orbits in general and also if there exists a square root attack
in general. In Section 2 a concrete practical semigroup action built from
simple semirings is presented. It will require further research to analyse this
system.Comment: 20 pages. To appear in Advances in Mathematics of Communication
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