1 Mirror operator ∇ψ * In this section, we discuss properties of distance generating functions and their subdif-ferentials. Let ψ be a proper, closed, convex function, and suppose that X is the effective domain of ψ (i.e. X = {x ∈ R n : ψ(x) < ∞}). The subdifferential of ψ at x ∈ X is ∂ψ(x) = {z ∈ E * : ψ(y) − ψ(x) − z, y − x ≥ 0 ∀y ∈ X }.

Definition 1. Let A be a ring. A graded A-algebra is an A-algebra R which is also a graded ring in such a way that if r ∈ Rd then ar ∈ Rd for all a ∈ A. That is, Rd is an A-submodule of R for all d ≥ 0. Equivalently a graded A-algebra is a morphism of graded rings A −→ R where we grade A by setting A0 = A,An = 0 for n > 0. A morphism of graded A-algebras is a morphism of A-algebras which preser...

The orientation morphism $Or(\cdot)(P)\colon \gamma\mapsto\dot{P}$ associates differential-polynomial flows $\dot{P}=Q(P)$ on spaces of bi-vectors $P$ finite-dimensional affine manifolds $N^d$ with (sums of) finite unoriented graphs $\gamma$ ordered sets edges and without multiple one-cycles. It is known that $d$-cocycles $\boldsymbol{\gamma}\in\ker d$ respect to the vertex-expanding differenti...

Intersection sheaves are usually defined for a proper flat surjective morphism of Noetherian schemes of relative dimension d and for d + 1 invertible sheaves on the ambient scheme. In this article, the construction is generalized to the equidimensional proper surjective morphisms over normal separated Noetherian schemes. Applications to the studies on family of effective algebraic cycles and on...

This paper examines the concept of simulation from a modelling viewpoint. How can one Mealy machine simulate the other one? We create formalism for simulation of Mealy machines. The injective s–morphism of the machine semigroups induces the simulation of machines [1]. We present the example of s–morphism such that it is not a homomorphism of semigroups. The story for the surjective s–morphisms ...

A word is cubefree if it contains no non-empty subword of the form xxx. A morphism h : Σ * → Σ * is k-uniform if h(a) has length k for all a ∈ Σ. A morphism is cubefree if it maps cubefree words to cubefree words. We show that for all k ≥ 0 there exists a k-uniform cubefree binary morphism.