A general notion of completely monotone functionals on an ordered Banach algebra B into a proper H*-algebra A with an integral representation for such functionals is given. As an application of this result we have obtained a characterization for the generalized completely continuous monotone functions on weighted foundation semigroups. A generalized version of Bochner’s theorem on foundation se...

One can prove the following proposition (1) Let S, T be up-complete Scott top-lattices and M be a subset of SCMaps(S, T ). Then ⊔ SCMaps(S,T ) M is a continuous map from S into T . Let S be a non empty relational structure and let T be a non empty reflexive relational structure. One can check that every map from S into T which is constant is also monotone. Let S be a non empty relational struct...

It was recently shown that monotone gene action, i.e., order-preservation between allele content and corresponding genotypic values in the mapping from genotypes to phenotypes, is a prerequisite for achieving a predictable parent-offspring relationship across the whole allele frequency spectrum. Here we test the consequential prediction that the design principles underlying gene regulatory netw...

We extend the shuffle algebra perspective on scalar-valued non-commutative probability theory to operator-valued case. Given an space with B acting it (on left and right), we associate operators in operad of multilinear maps distribution free cumulants a random variable. These define representation PRO non-crossing partitions. Using concepts from higher category theory, specifically 2-monoidal ...

Discrete systems such as sets, monoids, groups are familiar categories. The internal structure of the latter two is defined by an algebraic operator. In this paper we concentrate on discrete systems that are characterized by unary operators; these include choice operators σ, encountered in economics and social theory, and closure operators φ, encountered in discrete geometry and data mining. Be...

We develop a quenched thermodynamic formalism for random dynamical systems generated by countably branched, piecewise-monotone mappings of the interval that satisfy covering condition. Given contracting potential $\varphi$ (in sense Liverani-Saussol-Vaienti), we prove there exists unique conformal measure $\nu_\varphi$ and equilibrium state $\mu_\varphi$. Further, quasi-compactness associated t...

We consider the Fermi–Ulam model, which can be described as a particle moving freely between two vertical rigid walls; left one being fixed, whereas right moves according to regular periodic function. The is elastically reflected when hitting walls. show that dynamics of model by an area-preserving monotone twist map. Thus, Aubry–Mather sets exist for every rotation number in interval. Conseque...

Boolean, free and monotone cumulants as well relations among them, have proven to be important in the study of non-commutative probability theory. Quite notably, Boolean were successfully used infinite divisibility via Bercovici--Pata bijection. On other hand, recent years concept infinitesimal has been developed, together with notion which can useful context combinatorial questions. In this ...