A nilpotent group without local functional equations for pro-isomorphic subgroups

The pro-isomorphic zeta function of a torsion-free finitely generated nilpotent group G enumerates finite index subgroups H such that H and G have isomorphic profinite completions. It admits an Euler product decomposition, indexed by the rational primes. We manufacture the first example of a torsion-free finitely generated nilpotent group G such that the local Euler factors of its pro-isomorphic zeta function do not satisfy functional equations. The group G has nilpotency class 4 and Hirsch length 25. It is obtained, via the Malcev correspondence, from a Z-Lie lattice L with a suitable algebraic automorphism group Aut(L).Comment: 16 page

On pro-isomorphic zeta functions of D∗D^*-groups of even Hirsch length

Pro-isomorphic zeta functions of finitely generated nilpotent groups form one of the group-theoretic generalisations of the Riemann zeta functions. They are Dirichlet generating functions enumerating finite-index subgroups whose profinite completion is isomorphic to that of the ambient group. We study pro-isomorphic zeta functions of $D^*$-groups; these form the building blocks of finitely generated class two nilpotent groups with centre of rank two, up to commensurability. These groups were classified by Grunewald and Segal, and can be indexed by primary polynomials whose companion matrices define commutator relations. We provide a key step towards the elucidation of the pro-isomorphic zeta functions of $D^*$-groups of even Hirsch length by describing the automorphism groups of the associated graded Lie rings. Utilizing this description of the automorphism groups, we calculate the local pro-isomorphic zeta functions of groups associated to the polynomials $x^2$ and $x^3$. In both cases, the local zeta functions are uniform in the prime~$p$ and satisfy functional equations.Comment: 29 page

Genetic Addiction Risk Score (GARS): Molecular Neurogenetic Evidence for Predisposition to Reward Deficiency Syndrome (RDS)

We have published extensively on the neuro- genetics of brain reward systems with reference to the genes related to dopaminergic function in particular. In 1996, we coined “ Reward Deficiency Syndrome ” (RDS), to portray behaviors found to have gene-based association with hypodopaminergic function. RDS as a useful concept has been embraced in many subsequent studies, to increase our understanding of Substance Use Disorder (SUD), addictions, and other obsessive, compulsive, and impulsive behaviors. Interestingly, albeit others, in one published study, we were able to describe lifetime RDS behaviors in a recovering addict (17 years sober) blindly by assessing resultant Genetic Addic- tion Risk Score (GARS ™ ) data only. We hypothesize that genetic testing at an early age may be an effective preventive strategy to reduce or eliminate pathological substance and behavioral seeking activity. Here, we consider a select number of genes, their polymorphisms, and associated risks for RDS whereby, utilizing GWAS, there is evidence for convergence to reward candidate genes. The evidence presented serves as a plausible brain-print providing relevant genetic information that will reinforce targeted therapies, to improve recovery and prevent relapse on an individualized basis. The primary driver of RDS is a hypodopaminergic trait (genes) as well as epige- netic states (methylation and deacetylation on chromatin structure). We now have entered a new era in addiction med- icine that embraces the neuroscience of addiction and RDS as a pathological condition in brain reward circuitry that calls for appropriate evidence-based therapy and early genetic diagno- sis and that requires further intensive investigation

Stationary and regenerative multivariate point processes

Imperial Users onl

Uniformity and Functional Equations for Local Zeta Functions of K\mathfrak{K}-Split Algebraic Groups

We study the local zeta functions of an algebraic group $\mathcal{G}$ defined over $\mathfrak{K}$ together with a faithful $\mathfrak{K}$-rational representation $\rho$ for a finite extension $\mathfrak{K}$ of $\mathbb{Q}$. These are given by integrals over $\mathfrak{p}$-adic points of $\mathcal{G}$ determined by $\rho$ for a prime $\mathfrak{p}$ of $\mathfrak{K}$. We prove that the local zeta functions are almost uniform for all $\mathfrak{K}$-split groups whose unipotent radical satisfies a certain lifting property. This property is automatically satisfied if $\mathcal{G}$ is reductive. We provide a further criterion in terms of invariants of $\mathcal{G}$ and $\rho$ which guarantees that the local zeta functions satisfy functional equations for almost all primes of $\mathfrak{K}$. We obtain these results by using a $\mathfrak{p}$-adic Bruhat decomposition of Iwahori and Matsumoto [IM] to express the zeta function as a weighted sum over the Weyl group $W$ associated to $\mathcal{G}$ of generating functions over lattice points of a polyhedral cone. The functional equation reflects an interplay between symmetries of the Weyl group and reciprocities of the combinatorial object. We construct families of groups with representations violating our second structural criterion whose local zeta functions do not satisfy functional equations. Our work generalizes results of Igusa [Igu] and du Sautoy and Lubotzky [dSL] and has implications for zeta functions of finitely generated torsion-free nilpotent groups.Comment: 22 page

Interaction of cocaine with some central dopominergic and serotonergic mechanisms

The present study ln male Wistar rats was designed to rate and analyze six specific cocaine-induced behaviors. These behavioral parameters have been defined by others as either dopaminergic (sniffing, grooming, and locomotor activity) or serotonergic (repetitive head movements, rearing, and Straub tail) in origin. Results were analyzed by analysis of variance in two ways : (i) as grouped dopaminergic or serotonergic scores, and (ii) as the net behavioral index (dopaminergic scores minus the serotonergic scores). The purpose of approaching the data in this way was to attempt to define the behavioral interactions of the two neurotransmitters. One conclusion that developed from this study was the indication that dopaminergic behaviors peak at lower doses of cocaine than do serotonergic behaviors. This relationship held true for all the individual parameters in addition to the dopaminergic and serotonergic totals. A dopaminergic blocker, haloperidol, significantly attenuated all responses elicited by cocaine. When the net behavioral index was analyzed, it was found that the response of the median dose of cocaine was significantly altered from a net dopaminergic score towards a net serotonergic score. In this sense, haloperidol was shown to have the capacity to attenuate dopaminergic-associated parameters to a greater extent than the serotonergic-associated parameters. Cyproheptadine, an antiserotonergic agent, did not significantly affect the net behavioral index; however, this compound did significantly increase the dopaminergic parameter of grooming at the high doses of cocaine and cyproheptadine. Also at this dose combination , gnawing was elicited -- a dopaminergic response seen under no other experimental conditions. Due to the antiserotonergic agent causing an increase in the dopaminergic parameters of grooming and gnawing, it is proposed that the serotonergic influence on these two dopaminergic behaviors is of an inhibitory type