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The pentagram map, first proposed by R. Schwartz in 1992, is a well studied discrete integrable system on real planar polygons. It has been reframed in the language of difference operators using a known correspondence of certain degree 3 difference operators and polygons in P2. In this paper, we generalize the notion of a projective space by describing one-dimensional “subspaces” in free modules over stably finite rings. We then define polygons in such spaces and discuss how these projective spaces relate to known objects, such as Grassmannians. With these projective spaces, we generalize the correspondence of difference operators and polygons by proving that there is a one-to-one correspondence between properly bounded left (resp. right) difference operators, whose coefficients are from a stably finite ring R, and polygons in a certain left (resp. right) projective space over R. With this correspondence, we show that some known (and proposed) generalizations of the pentagram map can be understood as the usual pentagram map in a certain projective plane. Finally, we reframe these pentagram maps in the language of difference operators, showing that the pentagram map on pseudo-difference operators is a refactorization, and use this interpretation to find invariants.

Operator representations of deformed Lie type commutation relations, associated with group or semigroup actions of dynamical systems and iterated function systems are considered. In particular, it is shown that some multi-parameter deformed symmetric difference and multiplication operators satisfy these commutation relations. The operator representations are considered also in the context of twisted derivations. © Springer Nature Switzerland AG 2020.

A general class of multi-parametric families of unital associative complex algebras, defined by commutation relations associated with group or semigroup actions of dynamical systems and iterated function systems, is considered. A generalization of these commutation relations in three generators is also considered, modifying Lie algebra type commutation relations, typical for usual differential or difference operators, to relations satisfied by more general twisted difference operators associated with general twisting maps. General reordering and nested commutator formulas for arbitrary elements in these algebras are presented, and some special cases are considered, generalizing some well-known results in mathematics and physics. © Springer Nature Switzerland AG 2020.

We define, using difference operators, classes of functions defined on the set of points with integer coordinates, which are preserved under the formation of marginal functions. The duality between classes of functions with certain convexity properties and families of second-order difference operators plays an important role and is explained using notions from mathematical morphology., www.cb.uu.se/~kiselman

We shall study a new class of functions in discrete convexity which is defined using difference operators. The relation between this new class and the class of integrally convex functions is studied.

A crucial ingredient in the recent discovery by Ablowitz, Halburd, Herbst and Korhonen (2000, 2007) that a connection exists between discrete Painleve equations and (finite order) Nevanlinna theory is an estimate of the integrated average of log(+) vertical bar f(z + 1)/f(z)vertical bar on vertical bar z vertical bar = r. We obtained essentially the same estimate in our previous paper (2008) independent of Halburd et al. (2006). We continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences, difference quotients and logarithmic derivatives for finite order meromorphic functions. In addition to the potential applications of our new estimates in integrable systems, they are also of independent interest. In particular, our findings show that there are marked differences between the growth of meromorphic functions with Nevanlinna order less than and greater than one. We have established a

A crucial ingredient in the recent discovery by Ablowitz, Halburd, Herbst and Korhonen (2000, 2007) that a connection exists between discrete Painleve equations and (finite order) Nevanlinna theory is an estimate of the integrated average of log(+) vertical bar f(z + 1)/f(z)vertical bar on vertical bar z vertical bar = r. We obtained essentially the same estimate in our previous paper (2008) independent of Halburd et al. (2006). We continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences, difference quotients and logarithmic derivatives for finite order meromorphic functions. In addition to the potential applications of our new estimates in integrable systems, they are also of independent interest. In particular, our findings show that there are marked differences between the growth of meromorphic functions with Nevanlinna order less than and greater than one. We have established a

The solvability of elliptic functionally-differential equations with compressions of arguments in weight spaces were studied. Boundary value problems for elliptic differential-difference equations and nonlocal boundary value for nonlocal boundary-value problems for elliptic differential equations were also studied. Several theorems were used to prove the solvability of the differential equations. A positive solution to Kato's problem for a large class of operators including, differential-difference operators and functional-differential operators with dilations and compressions of arguments were also obtained.

A priori estimates was obtained for solutions of elliptic differential difference equations with degeneracy containing several different difference operators and constructed the Friedrichs extension of the corresponding sectorial operator. The Friedrichs extension of the differential difference operator LR was constructed and the properties of this extension were studied.