Abstract If is an analytic family of pairwise eventually different functions then the following strong maximality condition fails: For any countable , no member which covered by finitely many from there such that for all are infinitely integers k f(k) = h(k) . However if V L exists a coanalytic satisfying this condition.

In this paper, we define a new coproduct on the space of decorated planar rooted forests to equip it with weighted infinitesimal unitary bialgebraic structure. We introduce concept $\Omega$-cocycle bialgebras weight $\lambda$ and then prove that $H_{\mathrm{RT}}(X,\Omega)$, together set grafting operations $\{ B^+_\omega \mid \omega\in \Omega\}$, is free bialgebra $X$, involving version Hochschild 1-cocycle condition. As an application, cocycle structure undecorated forests, which object...

We study smooth toroidal compactifications of Siegel varieties thoroughly from the viewpoints Hodge theory and K\"ahler-Einstein metric. observe that any cusp a space can be identified as set certain weight one polarized mixed structures. then infinity boundary divisors compactifications, obtain global volume form formula an arbitrary variety $\scr{A}_{g,\Gamma} (g>1)$ with compactification $\overline{\scr{A}}_{g,\Gamma}$ such...

Recent advances in stochastic PDEs, Hopf algebras of typed trees and integral equations have inspired the study algebraic structures with replicating operations. To understand their combinatorial nature, we first use rooted forests multiple decoration sets to construct free Hochschild 1-cocycle conditions. Applying universal property underlying operated method Gr\"obner-Shirshov bases, then objects category matching Rota-Baxter which is a generalization allow operators. Finally are equipped...

We equip a matrix algebra with weighted infinitesimal unitary bialgebraic structure, via construction of suitable coproduct. Furthermore, an Hopf algebra, under the view Aguiar, is constructed on algebra. By exploring relationship between bialgebras and pre-Lie algebras, we construct algebraic structure then new Lie also introduce associative Yang-Baxter equations (AYBEs) obtain solutions AYBEs bialgebras. give bijection equation weight $\lambda$ Rota-Baxter operators $-\lambda$ algebras. As...

Abstract This study presents an extension of a dry‐environment convection scheme (Tiedtke–Bechtold) to couple with boundary‐layer moist turbulence scheme. The deep and shallow convective updraught is modified develop in environment the large‐scale budget cloud condensate takes account influence compensation subsidence. An ambiguous layer introduced sub‐cloud transport mimic non‐local that ignored local Long‐term global simulation suggests improve low short‐wave radiative forcing. includes...

We introduce a new operator – modified λ-differential from usual differential operator. construct the free (modified) (k,∂)-algebra on (k,∂)-module, and apply it to erect universal enveloping algebra of Lie (k,∂)-algebra. The corresponding Poincaré-Birchoff-Witt theorems are obtained. Wronskian envelope (k,−λidk)-algebra is also constructed.

Unifying various generalizations of Reynolds operators, the relative cocycle weighted operators are studied. We give a characterization in context pre-Lie algebras. construct an explicit graded Lie algebra whose Maurer-Cartan elements given by operator, which makes it possible to cohomology for operators. This can be seen as certain with coefficients suitable representation. Finally, we introduce notion NS-pre-Lie algebras and show induce L-dendriform

The paper considers the possible cardinalities of maximal cofinitary groups and cofinalities permutation on set natural numbers. These two cardinals are compared with some other well-known cardinal invariants which related to covering properties in several forcing models. An application combinatorial group theory a construction partially ordered is presented.

Motivated by comatrix coalgebras, we introduce the concept of a Newtonian coalgebra. We construct an infinitesimal unitary bialgebra on matrix algebras, via construction suitable coproduct. As consequence, coalgebra is established. Furthermore, Hopf algebra, under view Aguiar, constructed algebras. By close relationship between pre-Lie algebras and bialgebras, erect algebra new Lie Finally, weighted non-commutative polynomial also given.

We introduce the notion of a matching Rota-Baxter algebra motivated by recent work on multiple pre-Lie algebras arising from study algebraic renormalization regularity structures~[10,18]. This is also related to iterated integrals with kernels and solutions associative polarized Yang-Baxter equation. Generalizing natural connection dendriform , we obtain algebras. As in classical case one operation, are which coincide aforementioned More general notions results tridendriform PostLie obtained.

As an algebraic meaning of the nonhomogenous associative Yang-Baxter equation, weighted infinitesimal bialgebras play important role in mathematics and mathematical physics. In this paper, we introduce concept Hopf modules show that any module carries a natural structure unitary over quasitriangular bialgebra. We decorate planar rooted forests new way, prove space forests, together with coproduct family grafting operations, is free $\Omega$-cocycle bialgebra weight zero on set. A...