in this paper we study properties of symbols such that these belong to class of symbols sitting insidesm ρ,φ that we shall introduce as the following. so for because hypoelliptic pseudodifferential operatorsplays a key role in quantum mechanics we will investigate some properties of m−hypoelliptic pseudodifferential operators for which define base on this class of symbols. also we consider maxi...

A Rota–Baxter algebra, also known as a Baxter algebra, is an algebra with a linear operator satisfying a relation, called the Rota–Baxter relation, that generalizes the integration by parts formula. Most of the studies on Rota–Baxter algebras have been for commutative algebras. Free commutative Rota–Baxter algebras were constructed by Rota and Cartier in the 1970s. A later construction was obta...

In this paper, we study obstructions to the Dirichlet property by two approaches : density of non-positive points and functionals on adelic R-divisors. Applied to the algebraic dynamical systems, these results provide examples of nef adelic arithmetic R-Cartier divisor which does not have the Dirichlet property. We hope the obstructions obtained in the article will give ways toward criteria of ...

For some reason it seem convenient to define the notion of an effective Cartier divisor before anything else. Note that in Morphisms of Spaces, Section 13 we discussed the correspondence between closed subspaces and quasi-coherent sheaves of ideals. Moreover, in Properties of Spaces, Section 28, we discussed properties of quasi-coherent modules, in particular “locally generated by 1 element”. T...

Smooth surfaces have finitely generated canonical rings and projective canonical models. For normal surfaces, however, the graded ring of multicanonical sections is possibly nonnoetherian, such that the corresponding homogeneous spectrum is noncompact. I construct a canonical compactification by adding finitely many non-Q-Gorenstein points at infinity, provided that each Weil divisor is numeric...

We study isomorphism classes of symplectic dual pairs P ← S → P , where P is an integrable Poisson manifold, S is symplectic, and the two maps are complete, surjective Poisson submersions with connected and simply-connected fibres. For fixed P , these Morita self-equivalences of P form a group Pic(P ) under a natural “tensor product” operation. Variants of this construction are also studied, fo...

– For a base scheme X and an effective (Cartier) divisor D on X defined as a closed subscheme by an ideal I, there is code to compute an invertible sheaf corresponding to the class of D. Here, computing the explicit divisor map is essentially the same as computing the Riemann-Roch space if X is a variety. In fact, the Riemann-Roch space can be recovered during the computation of the associated ...

We introduce a notion of morphism of CohFT’s, on the basis of the analogy with A∞ morphisms, and discuss the relationship with morphisms of F -manifolds introduced by Manin and Hertling [5]. The structure maps of a morphism of CohFT’s have as input a cohomology class on the moduli space of scaled affine lines (complexified multiplihedron) studied in Ma’u-Woodward [9]. The main result is a compu...

We present several non-commutative extensions of the MacMahon Master Theorem, further extending the results of Cartier-Foata and Garoufalidis-LêZeilberger. The proofs are combinatorial and new even in the classical cases. We also give applications to the β-extension and Krattenthaler-Schlosser’s q-analogue. Introduction The MacMahon Master Theorem is one of the jewels in enumerative combinatori...

In this article we present a refinement of the base point free theorem for threefolds in positive characteristic. If $L$ is nef Cartier divisor numerical dimension at least one on projective Kawamata log terminal threefold $(X,\Delta)$ over perfect field $k$ characteristic $p \gg 0$ such that $L-(K_X+\Delta)$ big and nef, then show linear system $|mL|$ all sufficiently large integer $m>0$.