On an independence result in the theory of lawless sequences

The open data axiom LS3 for lawless sequences is actually an infinite list of axiom schemata: for each n we have LS3(n): A(a*,.... Un)AAi<jln CZi#Uj+%i ~ 3al... 2u,3a, y&Et41... Wn E htAi<jsn Bi*Bj+A(B19--*,/%I)); here oi, pi range over lawless sequences, and the ui range over finite sequences; ‘a E U ’ stands for ‘a has initial segment u’. In [D] it was shown that LS3(1) does not imply LS3(2) by using Cohen generic sequences. In [DL], this method was used to show that LS3(2) does not imply LS3(3). The aim of this note is to give simple proofs of these facts, by using the models described in [HM]. Our method also shows that LS3(3) does not imply LS3(4), but we have not been able to prove a similar independence result for larger n. For n L 4 a different approach seems necessary for showing LS3(n)f* 74LS3(n + 1). We observe here that the models described below all satisfy the axioms LSI (decidable equality) and LS2 (density), and that the models which show tha

Enumeration of graphs with a heavy-tailed degree sequence

In this paper, we asymptotically enumerate graphs with a given degree sequence d=(d_1,...,d_n) satisfying restrictions designed to permit heavy-tailed sequences in the sparse case (i.e. where the average degree is rather small). Our general result requires upper bounds on functions of M_k= \sum_{i=1}^n [d_i]_k for a few small integers k\ge 1. Note that M_1 is simply the total degree of the graphs. As special cases, we asymptotically enumerate graphs with (i) degree sequences satisfying M_2=o(M_1^{ 9/8}); (ii) degree sequences following a power law with parameter gamma>5/2; (iii) power-law degree sequences that mimic independent power-law "degrees" with parameter gamma>1+\sqrt{3}\approx 2.732; (iv) degree sequences following a certain "long-tailed" power law; (v) certain bi-valued sequences. A previous result on sparse graphs by McKay and the second author applies to a wide range of degree sequences but requires Delta =o(M_1^{1/3}), where Delta is the maximum degree. Our new result applies in some cases when Delta is only barely o(M_1^ {3/5}). Case (i) above generalises a result of Janson which requires M_2=O(M_1) (and hence M_1=O(n) and Delta=O(n^{1/2})). Cases (ii) and (iii) provide the first asymptotic enumeration results applicable to degree sequences of real-world networks following a power law, for which it has been empirically observed that 2<gamma<3.Comment: 34 page

Remarks on the degree growth of birational transformations

We look at sequences of positive integers that can be realized as degree sequences of iterates of rational dominant maps of smooth projective varieties over arbitrary fields. New constraints on the degree growth of endomorphisms of the affine space and examples of degree sequences are displayed. We also show that the set of all degree sequences of rational maps is countable; this generalizes a result of Bonifant and Fornaess.Comment: 12 page

Random series in Lp(X, Σ,μ) using unconditional basic sequences and lp stable sequences: A result on almost sure almost everywhere convergence

Here we study the almost sure almost everywhere convergence of random series of the form Σ∞ i=1αifi in the Lebesgue spaces L p(X, Σ,μ), where the ai's are centered random variables, and the fi's constitute an unconditional basic sequence or an lp stable sequence. We show that if one of these series converges in the norm topology almost surely, then it converges almost everywhere almost surely.Fil: Medina, Juan Miguel. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Cernuschi Frias, Bruno. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentin