422 research outputs found
Discrete dynamical systems in group theory
In this expository paper we describe an unifying approach for many known
entropies in Mathematics. First we recall the notion of semigroup entropy h_S
in the category S of normed semigroups and contractive homomorphisms, recalling
also its properties. For a specific category X and a functor F from X to S, we
have the entropy h_F, defined by the composition of h_S with F, which
automatically satisfies the same properties proved for h_S. This general scheme
permits to obtain many of the known entropies as h_F, for appropriately chosen
categories X and functors F. In the last part we recall the definition and the
fundamental properties of the algebraic entropy for group endomorphisms, noting
how its deeper properties depend on the specific setting. Finally we discuss
the notion of growth for flows of groups, comparing it with the classical
notion of growth for finitely generated groups
Nonstandard analysis, deformation quantization and some logical aspects of (non)commutative algebraic geometry
This paper surveys results related to well-known works of B. Plotkin and V.
Remeslennikov on the edge of algebra, logic and geometry. We start from a brief
review of the paper and motivations. The first sections deal with model theory.
In Section 2.1 we describe the geometric equivalence, the elementary
equivalence, and the isotypicity of algebras. We look at these notions from the
positions of universal algebraic geometry and make emphasis on the cases of the
first order rigidity. In this setting Plotkin's problem on the structure of
automorphisms of (auto)endomorphisms of free objects, and auto-equivalence of
categories is pretty natural and important. Section 2.2 is dedicated to
particular cases of Plotkin's problem. Section 2.3 is devoted to Plotkin's
problem for automorphisms of the group of polynomial symplectomorphisms. This
setting has applications to mathematical physics through the use of model
theory (non-standard analysis) in the studying of homomorphisms between groups
of symplectomorphisms and automorphisms of the Weyl algebra. The last two
sections deal with algorithmic problems for noncommutative and commutative
algebraic geometry. Section 3.1 is devoted to the Gr\"obner basis in
non-commutative situation. Despite the existence of an algorithm for checking
equalities, the zero divisors and nilpotency problems are algorithmically
unsolvable. Section 3.2 is connected with the problem of embedding of algebraic
varieties; a sketch of the proof of its algorithmic undecidability over a field
of characteristic zero is given.Comment: In this review we partially used results of arXiv:1512.06533,
arXiv:math/0512273, arXiv:1812.01883 and arXiv:1606.01566 and put them in a
new contex
Markov semigroups, monoids, and groups
A group is Markov if it admits a prefix-closed regular language of unique
representatives with respect to some generating set, and strongly Markov if it
admits such a language of unique minimal-length representatives over every
generating set. This paper considers the natural generalizations of these
concepts to semigroups and monoids. Two distinct potential generalizations to
monoids are shown to be equivalent. Various interesting examples are presented,
including an example of a non-Markov monoid that nevertheless admits a regular
language of unique representatives over any generating set. It is shown that
all finitely generated commutative semigroups are strongly Markov, but that
finitely generated subsemigroups of virtually abelian or polycyclic groups need
not be. Potential connections with word-hyperbolic semigroups are investigated.
A study is made of the interaction of the classes of Markov and strongly Markov
semigroups with direct products, free products, and finite-index subsemigroups
and extensions. Several questions are posed.Comment: 40 pages; 3 figure
Rational Growth of a Class of Inverse Semigroups
AbstractWe give a sufficient condition for a finitely presented Rees quotient of a free inverse semigroup to have rational growth. Using related techniques we give a new proof that nonmonogenic free inverse semigroups have irrational growth. A new criterion for polynomial growth is proved and is used to show that polynomial growth implies rational growth. However we give an example of such a semigroup which has rational and exponential growth
Filtrations and Distortion in Infinite-Dimensional Algebras
A tame filtration of an algebra is defined by the growth of its terms, which
has to be majorated by an exponential function. A particular case is the degree
filtration used in the definition of the growth of finitely generated algebras.
The notion of tame filtration is useful in the study of possible distortion of
degrees of elements when one algebra is embedded as a subalgebra in another. A
geometric analogue is the distortion of the (Riemannian) metric of a (Lie)
subgroup when compared to the metric induced from the ambient (Lie) group. The
distortion of a subalgebra in an algebra also reflects the degree of complexity
of the membership problem for the elements of this algebra in this subalgebra.
One of our goals here is to investigate, mostly in the case of associative or
Lie algebras, if a tame filtration of an algebra can be induced from the degree
filtration of a larger algebra
Kolmogorov equations in infinite dimensions: Well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations
We develop a new method to uniquely solve a large class of heat equations,
so-called Kolmogorov equations in infinitely many variables. The equations are
analyzed in spaces of sequentially weakly continuous functions weighted by
proper (Lyapunov type) functions. This way for the first time the solutions are
constructed everywhere without exceptional sets for equations with possibly
nonlocally Lipschitz drifts. Apart from general analytic interest, the main
motivation is to apply this to uniquely solve martingale problems in the sense
of Stroock--Varadhan given by stochastic partial differential equations from
hydrodynamics, such as the stochastic Navier--Stokes equations. In this paper
this is done in the case of the stochastic generalized Burgers equation.
Uniqueness is shown in the sense of Markov flows.Comment: Published at http://dx.doi.org/10.1214/009117905000000666 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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