For a quiver $Q$ with underlying graph $\Gamma$, we take $ {\mathcal {M}}$ an associated toric Nakajima quiver variety. In this article, we give a direct relation between a specialization of the Tutte polynomial of $\Gamma$, the Kac polynomial of $Q$ and the Poincaré polynomial of $ {\mathcal {M}}$. We do this by giving a cell decomposition of $ {\mathcal {M}}$ indexed by spanning trees of $\Gamma$ and ‘geometrizing’ the deletion and contraction operators on graphs. These relations have been previously established in Hausel–Sturmfels [6] and Crawley-Boevey–Van den Bergh [3], however the methods here are more hands-on.

The administration of selective serotonin reuptake inhibitors (SSRIs) typically used as antidepressants increases alcohol consumption after an alcohol deprivation period in rats. However, the appearance of this effect after the treatment with selective noradrenaline reuptake inhibitors (SNRIs) has not been studied. In the present work we examined the effects of a 15-d treatment with the SNRI atomoxetine (1, 3 and 10 mg/kg, i.p.) in male rats trained to drink alcohol solutions in a 4-bottle choice test. The treatment with atomoxetine (10 mg/kg, i.p.) during an alcohol deprivation period increased alcohol consumption after relapse. This effect only lasted one week, disappearing thereafter. Treatment with atomoxetine did not cause a behavioral sensitized response to a challenge dose of amphetamine (1.5 mg/kg, i.p.), indicating the absence of a supersensitive dopaminergic transmission. This effect is markedly different from that of SSRI antidepressants that produced both long-lasting increases in alcohol consumption and behavioral sensitization. Clinical implications are discussed.

We study the relationship between the dual of the K-space defined by means of a polygon and the J-space generated by the dual N-tuple. The results complete the research started in [4]. Special attention is paid to the case when the N-tuple is formed by Banach lattices

In the analysis of stability in bifurcation problems it is often assumed that the (appropriate reduced) equations are in normal form. In the presence of symmetry, the truncated normal form is an equivariant polynomial map. Therefore, the determination of invariants and equivariants of the group of symmetries of the problem is an important step. In general, these are hard problems of invariant theory and, in most cases, they are tractable only through symbolic computer programs. Nevertheless, it is desirable to obtain some of the information about invariants and equivariants without actually computing them, for example, the number of linearly independent homogeneous invariants or equivariants of a certain degree. Generating functions for these dimensions are generally known as ‘Molien functions'.

We obtain formulae for the number of linearly independent homogeneous invariants or equivariants for Hopf bifurcation in terms of characters. We also show how to construct Molien functions for invariants and equivariants for Hopf bifurcation. Our results are then applied to the computation of the number of invariants and equivariants for Hopf bifurcation for several finite groups and the continuous group $\mathbb{O}(3)$.

We show that certain interpolation results for compact operators established by Cobos and co-workers cannot be extended to general closed operator ideals. We shall also characterize compactness of an embedding in terms of functions related to the classical K- and J-functionals of interpolation theory.

The paper establishes estimates for ideal measures of operators interpolated by the real method in terms of the measures of their restrictions to a sequence of spaces modelled on the intersection. It also shows that the estimates are optimal.

We establish an estimate for the measure of non-compactness of an interpolated operator acting from a J-space into a K-space. Our result refers to general Banach N-tuples. We also derive estimates for entropy numbers if some of the N-tuples reduce to a single Banach space.