Related papers: Algebraic structures on valuations, their properti…

We study convex subsets of buildings, discuss some structural features and derive several characterizations of buildings.

We study G-graded Artinian algebras having Poincar\'e duality, considering in particular their Lefschetz properties. We also prove a correspondence between the toric setup and the G-graded one, provide an application to toric geometry, and…

We revisit a classical theme of (general or translation invariant) valuations on convex polyhedra. Our setting generalizes the classical one, in a ``dual'' direction to previously considered generalizations: while previous research was…

We discuss a selection of recent developments in arithmetic combinatorics having to do with ``approximate algebraic structure'' together with some of their applications.

Motivated by some recent developments in abstract theories of quadratic forms, we start to develop in this work an expansion of Linear Algebra to multivalued structures (a multialgebraic structure is essentially an algebraic structure but…

We are going to introduce a new algebraic, analytic structure that is a kind of generalization of the Hausdorff dimension and measure. We give many examples and study the basic properties and relations of such systems.

We introduce notions of concavity for functions on balanced polyhedral spaces, and we show that concave functions on such spaces satisfy several strong continuity properties.

We aim at studying collections of algebraic structures defined over a commutative ring and investigating the complexity of significant constructions carried out on these objects. The assignment of measures of size, via a multiplicity…

We define and study complex structures and generalizations on spaces consisting of geodesics or harmonic maps that are compatible with the symmetries of these spaces. The main results are about existence and uniqueness of such structures.

In this paper is discussed description of some algebraic structures in quantum theory by using formal recursive constructions with "complex Poincar\'e group" ISO(4,C).

The recent investigation of the gauge structure of extended geometry is generalised to situations when ancillary transformations appear in the commutator of two generalised diffeomorphisms. The relevant underlying algebraic structure turns…

The notion of a valuation on convex bodies is very classical. The notion of a valuation on a class of functions was recently introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on…

A general method to easily build global and relative operators for any number n of elementary systems if they are defined for 2 is presented. It is based on properties of the morphisms valued in the tensor products of algebras of the…

We study the notion of geometric structures for toposes: This generalizes the notion of (X,G) manifolds. We give some applications to algebraic geometry

In convex geometry, the constructions that assign to a convex body its difference body, projection body, or volume have the following properties: They are (1) invariant under volume-preserving linear changes of coordinates; (2) continuous;…

We study the Lie algebra structure of the Onsager algebra from the ideal theoretic point of view. A structure theorem of ideals in the Onsager algebra is obtained with the connection to the finite-dimensional representations. We also…

We study groups, exponential groups and ordered groups equipped with valuations. We investigate algebraic and topological features of such valued structures, and apply our findings in order to solve regular equations over groups using…

We give an explicit classification of translation-invariant, Lorentz-invariant continuous valuations on convex sets. We also classify the Lorentz-invariant even generalized valuations.

We develop the notion of deformations using a valuation ring as ring of coefficients. This permits to consider in particular the classical Gerstenhaber deformations of associative or Lie algebras as infinitesimal deformations and to solve…

The algebra of smooth translation-invariant valuations on convex bodies, introduced by S.Alesker in the early 2000s, was in part proved and in part conjectured to satisfy properties formally analogous to those of the cohomology ring of a…