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Green and Tao proved that the primes contains arbitrarily long arithmetic progressions. We show that, essentially the same proof leads to the following result: The primes in an short interval contains many arithmetic progressions of any…
We show that a class of algebras is closed under the taking of homomorphic images and direct products if and only if the class consists of all algebras that satisfy a set of (generally simultaneous) equations. For classes of regular…
Given an evolution algebra associated to a connected finite graph $\Gamma$, we exhibit a free action of the group of symmetries of $\Gamma$ on the set of automorphisms of the algebra. This allows us to explicitly describe this set and we…
We give a necessary and sufficient condition for an extension of valuation rings containing $\bf Q$ to be a filtered direct limit of smooth algebras.
A simple formal procedure makes the main properties of the lagrangian binomial extendable to functions depending to any kind of order of the time--derivatives of the lagrangian coordinates. Such a broadly formulated binomial can provide the…
In this work we study how to extend a partial action of a Hopf Algebra $A$ on an algebra $R$ to a partial action of a Hopf-Ore extension of $A$ on $R$. As consequence, we characterize all partial actions of rank one Hopf algebras (in…
We obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.
We formalize and study several competing notions of versality for an action of a linear algebraic group on an algebraic variety X. Our main result is that these notions of versality are equivalent to various statements concerning rational…
We give the basic definitions of group actions on (algebraic) stacks, and prove the existence of fixed points and quotients as (algebraic) stacks.
We prolonge the list of C*-algebras for which all extensions by any stable separable C*-algebra are semi-invertible. In particular, we handle certain amalgamations, both of C*-algebras and of groups. Concerning groups we consider both…
An algebraic deformation theory of coalgebra morphisms is constructed.
Finite-dimensional subalgebras of a Lie algebra of smooth vector fields on a circle, as well as piecewise-smooth global transformations of a circle on itself, are considered. A canonical forms of realizations of two- and three-dimensional…
An additive action on an algebraic variety is an effective action of the vector group with an open orbit. We describe projective surfaces with du Val singularities that admit an additive action with a finite number of orbits. In particular,…
We look at sequences of positive integers that can be realized as degree sequences of iterates of rational dominant maps of smooth projective varieties over arbitrary fields. New constraints on the degree growth of endomorphisms of the…
We interpret realizations of a graph on the sphere up to rotations as elements of a moduli space of curves of genus zero. We focus on those graphs that admit an assignment of edge lengths on the sphere resulting in a flexible object. Our…
We describe an approach to the question of finding real solutions to problems of enumerative geometry, in particular the question of whether a problem of enumerative geometry can have all of its solutions be real. We give some methods to…
We study the actions of a Lie group $G$ by birationally extendible automorphisms on a domain $D\subset C^n$. For a large class of such domains defined by polynomial inequalities, all automorphisms are of this type. In the cases 1) $G$ has…
Motivated by partition regularity problems of homogeneous quadratic equations, we prove multiple recurrence and convergence results for multiplicative measure preserving actions with iterates given by rational sequences involving…
Invariants withstand transformations and, therefore, represent the essence of objects or phenomena. In mathematics, transformations often constitute a group action. Since the 19th century, studying the structure of various types of…
We show that there exists a positive constant C such that the following holds: Given an infinite arithmetic progression A of real numbers and a sufficiently large integer n (depending on A), there needs at least Cn geometric progressions to…