The class of the affine line is a zero divisor in the Grothendieck ring: an improvement

Lev A. Borisov has shown that the class of the affine line is a zero divisor in the Grothendieck ring of algebraic varieties over complex numbers. We improve the final formula by removing a factor.Comment: Comptes Rendus Math\'ematique, Elsevier Masson, 201

Equistarable graphs and counterexamples to three conjectures on equistable graphs

Equistable graphs are graphs admitting positive weights on vertices such that a subset of vertices is a maximal stable set if and only if it is of total weight $1$. In $1994$, Mahadev et al.~introduced a subclass of equistable graphs, called strongly equistable graphs, as graphs such that for every $c \le 1$ and every non-empty subset $T$ of vertices that is not a maximal stable set, there exist positive vertex weights such that every maximal stable set is of total weight $1$ and the total weight of $T$ does not equal $c$. Mahadev et al. conjectured that every equistable graph is strongly equistable. General partition graphs are the intersection graphs of set systems over a finite ground set $U$ such that every maximal stable set of the graph corresponds to a partition of $U$. In $2009$, Orlin proved that every general partition graph is equistable, and conjectured that the converse holds as well. Orlin's conjecture, if true, would imply the conjecture due to Mahadev, Peled, and Sun. An intermediate conjecture, one that would follow from Orlin's conjecture and would imply the conjecture by Mahadev, Peled, and Sun, was posed by Miklavi\v{c} and Milani\v{c} in $2011$, and states that every equistable graph has a clique intersecting all maximal stable sets. The above conjectures have been verified for several graph classes. We introduce the notion of equistarable graphs and based on it construct counterexamples to all three conjectures within the class of complements of line graphs of triangle-free graphs

Reduction to a single closed equation for 2 by 2 reaction-diffusion systems of Lotka-Volterra type

We consider general models of coupled reaction-diffusion systems for interacting variants of the same species. When the total population becomes large with intensive competition, we prove that the frequencies (i.e. proportions) of the variants can be approached by the solution of a simpler reaction-diffusion system, through a singular limit method and a relative compactness argument. As an example of application, we retrieve the classical bistable equation for Wolbachia's spread into an arthropod population from a system modeling interaction between infected and uninfected individuals

The Geography of Asset Trade and the Euro: Insiders and Outsiders

This paper analyzes the determinants of cross-border asset trade on cross-country data and a Swedish data set. We focus our analysis on the impact of the euro for the determinants of bond trade, equity and banking assets. With the help of a theoretical model, we attempt to disentangle the different effects that the euro may have had on asset holdings for both euro zone countries and countries outside of the euro zone such as Sweden. We find evidence that the euro has implied 1) a unilateral financial liberalization which makes it cheaper for all countries to buy euro zone assets. For bonds and equity holdings, this would translate into approximately 14% and 17% decrease in transaction costs. Using Swedish data, we find that this effect of the euro is larger for flows than for stocks. 2) a preferential financial liberalization which on top of the previous effect has decreased transaction costs inside the euro zone by approximately 17% and 10% for bonds and equity respectively. 3) a diversion effect due to the fact that lower transaction costs inside the euro zone have led euro countries to purchase less equity from outside the euro zone. Our empirical analysis also suggests that the elasticity of substitution between bonds inside the euro zone is higher than between bonds denominated in different currencies. We illustrate this effect for transaction costs generated by the difference in the legal system.Euro; Gravity Equation; International Asset Trade

Towards 1/N corrections to deep inelastic scattering from the gauge/gravity duality

$1/N^2$ corrections to deep inelastic scattering (DIS) of charged leptons from glueballs at strong coupling are investigated in the framework of the gauge/gravity duality. The structure functions $F_1$ and $F_2$ (and also $F_L$) are studied at subleading order in the $1/N^2$ expansion, in terms of $q^2$ and the Bjorken parameter $x$. The relevant type IIB supergravity one-loop diagrams (which correspond to DIS with two-hadron final states) are studied in detail, while $n$-loop diagrams (corresponding to DIS with $(n+1)$-hadron final states) are briefly discussed. The $1/N^{2n}$ and $\Lambda^2/q^2$ dependence of the structure functions is analyzed. Within this context two very different limits are considered: one is the large $N$ limit and the other one is when the virtual photon momentum transfer $q$ is much larger than the infrared confining scale $\Lambda$. These limits do not commute.Comment: 45 page

Graph towers, laminations and their invariant measures

In this paper we present a combinatorial machinery, consisting of a graph tower $\overleftarrow \Gamma$ and vector towers $\overleftarrow v$ on $\overleftarrow \Gamma$, which allows us to efficiently describe all invariant measures $\mu = \mu^{\overleftarrow v}$ on any given shift space over a finite alphabet. The new technology admits a number of direct applications, in particular concerning invariant measures on non-primitive substitution subshifts, minimal subshifts with many ergodic measures, or an efficient calculation of the measure of a given cylinder. It also applies to currents on a free group $F_N$, and in particular the set of projectively fixed currents under the action of a (possibly reducible) endomorphism $\varphi: F_N \to F_N$ is determined, when $\varphi$ is represented by a train track map.Comment: 52 pages, 3 figures. This is rather a new paper than a new version of the old one. The setting is much more general, and also closer to a symbolic dynamics spirit. Also, some of the work from the original paper has been removed and will be taken up in a forthcoming paper. Accepted in Journal of London Mathematical society. To appear 201