Tame group actions on central simple algebras

We study actions of linear algebraic groups on finite-dimensional central simple algebras. We describe the fixed algebra for a broad class of such actions.Comment: 19 pages, LaTeX; slightly revised; final version will appear in Journal of Algebr

Unipotent group actions on affine varieties

Algebraic actions of unipotent groups $U$ actions on affine $k-$varieties $X$ ($k$ an algebraically closed field of characteristic 0) for which the algebraic quotient $X//U$ has small dimension are considered$.$ In case $X$ is factorial, $O(X)^{\ast}=k^{\ast},$ and $X//U$ is one-dimensional, it is shown that $O(X)^{U}$=$k[f]$, and if some point in $X$ has trivial isotropy, then $X$ is $U$ equivariantly isomorphic to $U\times A^{1}(k).$ The main results are given distinct geometric and algebraic proofs. Links to the Abhyankar-Sathaye conjecture and a new equivalent formulation of the Sathaye conjecture are made.Comment: 10 pages. This submission comes out of an older submission ("A commuting derivations theorem on UFDs") and contains part of i

Hardy type derivations on fields of exponential logarithmic series

We consider the valued field \mathds{K}:=\mathbb{R}((\Gamma)) of formal series (with real coefficients and monomials in a totally ordered multiplicative group $\Gamma>$). We investigate how to endow \mathds{K} with a logarithm $l$, which satisfies some natural properties such as commuting with infinite products of monomials. In the article "Hardy type derivations on generalized series fields", we study derivations on \mathds{K}. Here, we investigate compatibility conditions between the logarithm and the derivation, i.e. when the logarithmic derivative is the derivative of the logarithm. We analyse sufficient conditions on a given derivation to construct a compatible logarithm via integration of logarithmic derivatives. In her monograph "Ordered exponential fields", the first author described the exponential closure \mathds{K}^{\rm{EL}} of (\mathds{K},l). Here we show how to extend such a log-compatible derivation on \mathds{K} to \mathds{K}^{\rm{EL}}.Comment: 25 page

Splitting fields and general differential Galois theory

An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on the search for prime differential ideals of special form in tensor products of differential rings. The main results demonstrating the work of the technique obtained are the theorem on the constructedness of the differential closure and the general theorem on the Galois correspondence for normal extensions..Comment: 33 pages, this version coincides with the published on

Adjunctive Atypical Antipsychotic Treatment for Major Depressive Disorder: A Meta-Analysis of Depression, Quality of Life, and Safety Outcomes

Atypical antipsychotic medications are widely prescribed for the adjunctive treatment of depression, yet their total risk-benefit profile is not well understood. We thus conducted a systematic review of the efficacy and safety profiles of atypical antipsychotic medications used for the adjunctive treatment of depression

Nasally-Administered Oxytocin Has Limited Effects on Owner-Directed Attachment Behavior in Pet Dogs (Canis lupus familiaris)

The present study explored the effects of intranasal oxytocin, a naturally occurring hormone, on the behavior of pet dogs during an attachment test. Each dog participated in two testing sessions. On one visit saline was administered nasally, and on another, oxytocin was administered nasally. For half of the dogs (n = 20), solutions were administered with a Mucosal Atomization Device (MAD) and for half of the dogs (n = 20), solutions were administered using a nasal spray bottle. Condition order was counterbalanced and a double-blind methodology was employed. Following a 30-min wait period after administration of solutions, dog-owner pairs participated in the Secure Base Test, a short attachment test consisting of three 2-min phases: (1) Baseline- the owner was present, dogs were able to freely explore the testing room (2) Alone- dogs were left alone in the testing room (3) Return- owners re-entered the room and were reunited with their dog. In each phase the dog was evaluated for contact seeking, exploration, and avoidance behaviors. Although, oxytocin administration was expected to increase owner-directed proximity and contact seeking behavior, this effect was not observed. In fact, in the baseline phase, dogs spent significantly more time seeking the proximity of their owners when they received saline than when they received OT (p \u3c 0.05). Sex differences were also assessed for the behavioral variables of interest in the Secure Base Test, and results indicated that OT did not affect dogs\u27 behavior in the alone phase, but when saline was administered, females spent significantly more time in contact with the door than males in the alone phase (p \u3c 0.05). Overall, the effects of nasally administered oxytocin on attachment related behavior appeared to be limited or inconsistent for this pet dog population

The theory of the exponential differential equations of semiabelian varieties

The complete first order theories of the exponential differential equations of semiabelian varieties are given. It is shown that these theories also arises from an amalgamation-with-predimension construction in the style of Hrushovski. The theory includes necessary and sufficient conditions for a system of equations to have a solution. The necessary condition generalizes Ax's differential fields version of Schanuel's conjecture to semiabelian varieties. There is a purely algebraic corollary, the "Weak CIT" for semiabelian varieties, which concerns the intersections of algebraic subgroups with algebraic varieties.Comment: 53 pages; v3: Substantial changes, including a completely new introductio

Rationality of quotients by linear actions of affine groups

Let G be the (special) affine group, semidirect product of SL_n and C^n. In this paper we study the representation theory of G and in particular the question of rationality for V/G where V is a generically free G-representation. We show that the answer to this question is positive if the dimension of V is sufficiently large and V is indecomposable. We have a more precise theorem if V is a two-step extension 0 -> S -> V -> Q -> 0 with S, Q completely reducible.Comment: 18 pages; dedicated to Fabrizio Catanese on the occasion of his 60th birthda

Is the function field of a reductive Lie algebra purely transcendental over the field of invariants for the adjoint action?

Let $k$ be a field of characteristic zero, let $G$ be a connected reductive algebraic group over $k$ and let $\mathfrak{g}$ be its Lie algebra. Let $k(G)$, respectively, $k(\mathfrak{g})$, be the field of $k$-rational functions on $G$, respectively, $\mathfrak{g}$. The conjugation action of $G$ on itself induces the adjoint action of $G$ on $\mathfrak{g}$. We investigate the question whether or not the field extensions $k(G)/k(G)^G$ and $k(\mathfrak{g})/k(\mathfrak{g})^G$ are purely transcendental. We show that the answer is the same for $k(G)/k(G)^G$ and $k(\mathfrak{g})/k(\mathfrak{g})^G$, and reduce the problem to the case where $G$ is simple. For simple groups we show that the answer is positive if $G$ is split of type ${\sf A}_{n}$ or ${\sf C}_n$, and negative for groups of other types, except possibly ${\sf G}_{2}$. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of $G$ on itself. The results and methods of this paper have played an important part in recent A. Premet's negative solution (arxiv:0907.2500) of the Gelfand--Kirillov conjecture for finite-dimensional simple Lie algebras of every type, other than ${\sf A}_n$, ${\sf C}_n$, and ${\sf G}_2$.Comment: Final version, 37 pages. To appear in Compositio Mathematica

Adjunctive Atypical Antipsychotic Treatment for Major Depressive Disorder: A Meta-Analysis of Depression, Quality of Life, and Safety Outcomes

Background: Atypical antipsychotic medications are widely prescribed for the adjunctive treatment of depression, yet their total risk–benefit profile is not well understood. We thus conducted a systematic review of the efficacy and safety profiles of atypical antipsychotic medications used for the adjunctive treatment of depression. Methods and Findings: We included randomized trials comparing adjunctive antipsychotic medication to placebo for treatment-resistant depression in adults. Our literature search (conducted in December 2011 and updated on December 14, 2012) identified 14 short-term trials of aripiprazole, olanzapine/fluoxetine combination (OFC), quetiapine, and risperidone. When possible, we supplemented published literature with data from manufacturers' clinical trial registries and US Food and Drug Administration New Drug Applications. Study duration ranged from 4 to 12 wk. All four drugs had statistically significant effects on remission, as follows: aripiprazole (odds ratio [OR], 2.01; 95% CI, 1.48–2.73), OFC (OR, 1.42; 95% CI, 1.01–2.0), quetiapine (OR, 1.79; 95% CI, 1.33–2.42), and risperidone (OR, 2.37; 95% CI, 1.31–4.30). The number needed to treat (NNT) was 19 for OFC and nine for each other drug. All drugs with the exception of OFC also had statistically significant effects on response rates, as follows: aripiprazole (OR, 2.07; 95% CI, 1.58–2.72; NNT, 7), OFC (OR, 1.30, 95% CI, 0.87–1.93), quetiapine (OR, 1.53, 95% CI, 1.17–2.0; NNT, 10), and risperidone (OR, 1.83, 95% CI, 1.16–2.88; NNT, 8). All four drugs showed statistically significant effects on clinician-rated depression severity measures (Hedges' g ranged from 0.26 to 0.48; mean difference of 2.69 points on the Montgomery–Asberg Depression Rating Scale across drugs). On measures of functioning and quality of life, these medications produced either no benefit or a very small benefit, except for risperidone, which had a small-to-moderate effect on quality of life (g = 0.49). Treatment was linked to several adverse events, including akathisia (aripiprazole), sedation (quetiapine, OFC, and aripiprazole), abnormal metabolic laboratory results (quetiapine and OFC), and weight gain (all four drugs, especially OFC). Shortcomings in study design and data reporting, as well as use of post hoc analyses, may have inflated the apparent benefits of treatment and reduced the apparent incidence of adverse events. Conclusions: Atypical antipsychotic medications for the adjunctive treatment of depression are efficacious in reducing observer-rated depressive symptoms, but clinicians should interpret these findings cautiously in light of (1) the small-to-moderate-sized benefits, (2) the lack of benefit with regards to quality of life or functional impairment, and (3) the abundant evidence of potential treatment-related harm