Algebraic theory of vector-valued integration

We define a monad M on a category of measurable bornological sets, and we show how this monad gives rise to a theory of vector-valued integration that is related to the notion of Pettis integral. We show that an algebra X of this monad is a bornological locally convex vector space endowed with operations which associate vectors \int f dm in X to incoming maps f:T --> X and measures m on T. We prove that a Banach space is an M-algebra as soon as it has a Pettis integral for each incoming bounded weakly-measurable function. It follows that all separable Banach spaces, and all reflexive Banach spaces, are M-algebras.Comment: shortened, e.g. by citing references regarding basic lemmas; made changes to ordering of some lemmas and section

Idempotent convexity and algebras for the capacity monad and its submonads

Idempotent analogues of convexity are introduced. It is proved that the category of algebras for the capacity monad in the category of compacta is isomorphic to the category of $(\max,\min)$-idempotent biconvex compacta and their biaffine maps. It is also shown that the category of algebras for the monad of sup-measures ($(\max,\min)$-idempotent measures) is isomorphic to the category of $(\max,\min)$-idempotent convex compacta and their affine maps

Versal deformations of DqD_q-invariant 2-parameter families of planar vector fields

The paper deals with 2-parameter families of planar vector fields which are invariant under the group $D_q$ for q ≥ 3. The germs at z = 0 of such families are studied and versal families are found. We also give the phase portraits of the versal families

On a certain map of a triangle

The paper answers some questions asked by Sharkovski concerning the map F:(u,v) ↦ (u(4-u-v),uv) of the triangle Δ = {u,v ≥ 0: u+v ≤ 4}. We construct an absolutely continuous σ-finite invariant measure for F. We also prove the following strange phenomenon. The preimages of side I = Δ ∩ {v=0} form a dense subset $∪F^{-n}(I)$ of Δ and there is another dense set Λ consisting of points whose orbits approach the interval I but are not attracted by I