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The K-group of substitutional systems

  • El Kacimi, Aziz
  • Parthasarathy, Rajagopalan
In another article we associated a dynamical system to a non- properly ordered Bratteli diagram. In this article we describe how to compute the K-group K0 of the dynamical system in terms of the Bratteli diagram. In the case of properly ordered Bratteli diagrams this description coincides with what is already known, namely the so-called dimension group of the Bratteli diagram. The new group defined here is more relevant for non-properly ordered Bratteli diagrams. We use our main result to describe K0 of a substitutional system.

Stabilization in Hr ∞ (D)

  • Wick, Brett D.
It is shown that for H∞/R(D)functions f1 and f2 with inf/z ∈D (│f1(z)│+ │f2(z) │≥ ᵟ > 0 and f1 being positive on the real zeros of f2, then there exists H∞/R (D) functions g2 and g1, g1 -1 with norm controlled by a constant depending only on ᵟ g1f1 + g2f2 = 1 ∀ z ∈ D. These results are connected to the computation of the stable rank of the algebra H∞/R and to results in Control Theory.

A boundedness criterion for general maximal operators

  • Lerner, Andrei K.
  • Ombrosi, Sheldy
We consider maximal operators Mβ with respect to a basis β. In the case when Mβ satisfies a reversed weak type inequality, we obtain a boundedness criterion for Mβ on an arbitrary quasiBanach function space X. Being applied to specific β X this criterion yields new and short proofs of a number of well-known results. Our principal application is related to an open problem on the boundedness of the two-dimensional one-sided maximal function M+ Lp/w.

f-polynomials, h-polynomials and l2-Euler characteristics

  • Boros, Dan
We introduce a many-variable version of the f-polynomial and h-polynomial associated to a finite simplicial complex. In this context the h-polynomial is actually a rational function. We establish connections with the l2-Euler characteristic of right-angled buildings. When L is a triangulation of a sphere we obtain a new formula for the l2-Euler characteristic.

Homotopy classification of gerbes

  • Jardine, J. F.
Gerbes are locally connected presheaves of groupoids on a small Grothendieck site C. They are classified up to local weak equivalence by path components of a cocycle category taking values in the big 2-groupoid Iso(Gr(C)) consisting of all sheaves of groups on C, their isomorphisms and homotopies. If F is a full sub- presheaf of Iso(Gr(C)) then the set [*,BF] of morphisms in the homotopy category of simplicial presheaves classifies gerbes locally weakly equivalent to objects of F. Id St(пF)is the stack completion of the fundamental groupoid(пF)of F if L is a global section of St(пF) and if FL is the homotopy fibre over L of the canonical map BF --> B St(пF), then [*FL] is in bijective correspondence with Giraud's non-abelian cohomology object H2 (C, L) of equivalence classes of gerbes with band L.

Joining polynomial and exponential combinatorics for some entire maps

  • Garijo, Antoni
  • Jarque i Ribera, Xavier
  • Moreno Rocha, Mónica
We consider families of entire transcendental maps given by Fλ,m (z) = λzm exp (z) where m ≥ 2. All these maps have a superattracting fixed point at z = 0 and a free critical point at z = −m. In parameter planes we focus on the capture zones, i.e., we consider λ values for which the free critical point belongs to the basin of attraction of z = 0. We explain the connection between the dynamics near zero and the dynamics near infinity at the boundary of the immediate basin of attraction of the origin, thus, joining together exponential and polynomial behaviors in the same dynamical plane.

Algebraic webs invariant under endomorphisms

  • Dabija, Marius
  • Jonsson, Mattias
We classify noninvertible, holomorphic selfmaps of the projective plane that preserve an algebraic web. In doing so, we obtain interesting examples of critically finite maps.

On the sum product estimates and two variables expanders

  • Shen, Chun-Yen
Let Fp be the finite field of a prime order p. Let F: Fp x Fp --> Fp be a function defined by F(x, y) = x(f(x) + by), where b ∈ F*/p and f: Fp → Fp is any function. We prove that if A ⊂ Fp and │A│ < p1/2 then │A + A│+│F(A,A) │⪆│A│13/12. Taking f = 0 and b = 1, we get the well-known sum-product theorem by Bourgain, Katz and Tao, and Bourgain, Glibichuk and Konyagin, and also improve the previous known exponent from 14/13 to 13/12.

Some remarks about parametrizations of intrinsic regular surfaces in the Heisenberg group

  • Bigolin, Francesco
  • Vittone, Davide
We prove that, in general, H-regular surfaces in the Heisenberg group H1 are not bi-Lipschitz equivalent to the plane R2 en- dowed with the "parabolic" distance, which instead is the model space for C1 surfaces without characteristic points. In Heisenberg groups Hn, H-regular surfaces can be seen as intrinsic graphs: we show that such parametrizations do not belong to Sobolev classes of metric-space valued maps.

Index of an implicit differential equation

  • Challapa, L. S.
  • Ruas, M. A. S.

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