Backbone Decomposition of Multitype Superprocesses

In this paper, we provide a construction of the so-called backbone decomposition for multitype supercritical superprocesses. While backbone decompositions are fairly well known for both continuous-state branching processes and superprocesses in the one-type case, so far no such decompositions or even description of prolific genealogies have been given for the multitype cases. Here we focus on superprocesses, but by turning the movement off, we get the prolific backbone decomposition for multitype continuous-state branching processes as an easy consequence of our results.

Normal Forms and Cohomology Classes at Single Resonances

This chapter discusses the single-resonance non-degeneracy conditions and normal forms. It then formulates Theorem 3.3, which covers the forcing equivalence in the single-resonance regime. The classical partial averaging theory indicates that after a coordinate change, the system has the normal form away from punctures. In order to state the normal form, one needs an anisotropic norm adapted to the perturbative nature of the system. The chapter also uses the idea of Lochak to cover the action space with double resonances. A double resonance corresponds to a periodic orbit of the unperturbed system. Finally, the chapter looks at a lemma which is an easy consequence of the Dirichlet theorem.

DISCRETE METRIC SPACES: STRUCTURE, ENUMERATION, AND 0-1 LAWS

Fix an integer $r \ge 3$. We consider metric spaces on n points such that the distance between any two points lies in $\left\{ {1, \ldots ,r} \right\}$. Our main result describes their approximate structure for large n. As a consequence, we show that the number of these metric spaces is $\left\lceil {{{r + 1} \over 2}} \right\rceil ^{\left( {\matrix{ n \cr 2 \cr } } \right) + o\left( {n^2 } \right)} .$Related results in the continuous setting have recently been proved by Kozma, Meyerovitch, Peled, and Samotij [34]. When r is even, our structural characterization is more precise and implies that almost all such metric spaces have all distances at least $r/2$. As an easy consequence, when r is even, we improve the error term above from $o\left( {n^2 } \right)$ to $o\left( 1 \right)$, and also show a labeled first-order 0-1 law in the language ${\cal L}_r $, consisting of r binary relations, one for each element of $[r]$ . In particular, we show the almost sure theory T is the theory of the Fraïssé limit of the class of all finite simple complete edge-colored graphs with edge colors in $\left\{ {r/2, \ldots ,r} \right\}$.Our work can be viewed as an extension of a long line of research in extremal combinatorics to the colored setting, as well as an addition to the collection of known structures that admit logical 0-1 laws.

When are Multiples of Polygonal Numbers again Polygonal Numbers?

International audience Euler showed that there are infinitely many triangular numbers that are three times other triangular numbers. In general, it is an easy consequence of the Pell equation that for a given square-free m > 1, the relation ∆ = m∆' is satisfied by infinitely many pairs of triangular numbers ∆, ∆'. After recalling what is known about triangular numbers, we shall study this problem for higher polygonal numbers. Whereas there are always infinitely many triangular numbers which are fixed multiples of other triangular numbers, we give an example that this is false for higher polygonal numbers. However, as we will show, if there is one such solution, there are infinitely many. We will give conditions which conjecturally assure the existence of a solution. But due to the erratic behavior of the fundamental unit of Q(√ m), finding such a solution is exceedingly difficult. Finally, we also show in this paper that, given m > n > 1 with obvious exceptions, the system of simultaneous relations P = mP' , P = nP'' has only finitely many possibilities not just for triangular numbers, but for triplets P , P' , P'' of polygonal numbers, and give examples of such solutions.

An Extremal Graph Problem with a Transcendental Solution

We prove that the number of multigraphs with vertex set {1, . . .,n} such that every four vertices span at most nine edges isan2+o(n2)whereais transcendental (assuming Schanuel's conjecture from number theory). This is an easy consequence of the solution to a related problem about maximizing the product of the edge multiplicities in certain multigraphs, and appears to be the first explicit (somewhat natural) question in extremal graph theory whose solution is transcendental. These results may shed light on a question of Razborov, who asked whether there are conjectures or theorems in extremal combinatorics which cannot be proved by a certain class of finite methods that include Cauchy–Schwarz arguments.Our proof involves a novel application of Zykov symmetrization applied to multigraphs, a rather technical progressive induction, and a straightforward use of hypergraph containers.

Order and Spectrum Preserving Maps on Positive Operators

We describe the general form of surjective maps on the cone of all positive operators that preserve order and spectrum. The result is optimal as shown by counterexamples. As an easy consequence, we characterize surjective order and spectrum preserving maps on the set of all self-adjoint operators.

Existence of solutions of inverted variational inequalities

In this paper we introduce two new generalized variational inequalities and we give some existence results of the solutions for these variational inequalities involving operators belonging to a recently introduced class of operators. We show by examples, that our results fail outside of this class. Further, we establish a result that may be viewed as a generalization of Minty’s theorem, that is, we show that under some circumstances the set of solutions of these variational inequalities coincide. We also show, the condition that the operators, involved in these variational inequalities, belong to the above mentioned class, is essential in obtaining this result. As application, we show that Brouwer’s fixed point theorem is an easy consequence of our results.

A Spectral Erdős–Stone–Bollobás Theorem

Let r ≥ 3 and (c/rr)r log n ≥ 1. If G is a graph of order n and its largest eigenvalue μ(G) satisfies then G contains a complete r-partite subgraph with r − 1 parts of size ⌊(c/rr)r log n⌋ and one part of size greater than n1−cr−1.This result implies the Erdős–Stone–Bollobás theorem, the essential quantitative form of the Erdős–Stone theorem. Another easy consequence is that if F1, F2, . . . are r-chromatic graphs satisfying v(Fn) = o(log n), then

On the BP-homology of 2e × 2e

We study υ0- and υ1-divisibility properties of the [2e]-series associated to the universal 2-typical formal group law. This allows us to identify elements annihilating the toral class τ in BP*(2e × 2e). We conjecture that these elements form a minimal system of generators of the annihilator ideal of τ. This would provide a Landweber-type presentation for the BP-homology of 2e × 2e from which the relation hom:dimBP (Z2e × Z2e) = 2 would be an easy consequence.