Algebras associated to acyclic directed graphs

We construct and study a class of algebras associated to generalized layered graphs, i.e. directed graphs with a ranking function on their vertices. Each finite directed acyclic graph admits countably many structures of a generalized layered graph. We construct linear bases in such algebras and compute their Hilbert series. Our interest to generalized layered graphs and algebras associated to those graphs is motivated by their relations to factorizations of polynomials over noncommutative rings.Comment: 20 pages, Latex; an expanded and corrected version; to appear in "Advances of Applied Mathematics

Hilbert series of algebras associated to directed graphs

Few changes. We compute the Hilbert series of some algebras associated to directed graphs and related to factorizations of noncommutative polynomials.Comment: AMSLaTeX, 9 page

On a class of Koszul algebras associated to directed graphs

In math.QA/0506507 I. Gelfand and the authors introduced and studied a new class of algebras associated to directed graphs. In this paper we show that these algebras are Koszul for a large class of layered (i.e. ranked) graphs.Comment: 15 pages; AMSTE

Hilbert series of quadratic algebras associated with pseudo-roots of noncommutative polynomials

The quadratic algebras Q_n are associated with pseudo-roots of noncommutative polynomials. We compute the Hilbert series of the algebras Q_n and of the dual quadratic algebras Q_n^!Comment: Amstex, 24 page

On antipodes in pointed Hopf algebras

AbstractIf S is the antipode of a Hopf algebra H, the order of S is defined to be the smallest positive integer n such that Sn = I (in case such integers exist) or ∞ (if no such integers exist). Although in most familiar examples of Hopf algebras the antipode has order 1 or 2, examples are known of infinite dimensional Hopf algebras in which the antipode has infinite order or arbitrary even order [1, 4, 6] and also of finite dimensional Hopf algebras in which the antipode has arbitrary even order [3, 5]. Some sufficient conditions for the antipode to have order ⩽4 are known [2, 4], but the following questions remain open: Does the antipode of a finite dimensional Hopf algebra necessarily have finite order? If the antipode S of a Hopf algebra H has finite order is that order bounded by some function of dim H?In this paper, by constructing a certain basis for an arbitrary pointed coalgebra and studying the action of the antipode on the elements of such a basis for a pointed Hopf algebra, we obtain affirmative answers to the second question in case H is pointed and to the first question in case H is pointed over a field of prime characteristic.We use freely the definitions, notation, and results of [4]

Dissociation of Cerebral Blood Flow and Femoral Artery Blood Pressure Pulsatility After Cardiac Arrest and Resuscitation in a Rodent Model: Implications for Neurological Recovery.

Background Impaired neurological function affects 85% to 90% of cardiac arrest (CA) survivors. Pulsatile blood flow may play an important role in neurological recovery after CA. Cerebral blood flow (CBF) pulsatility immediately, during, and after CA and resuscitation has not been investigated. We characterized the effects of asphyxial CA on short-term (<2 hours after CA) CBF and femoral arterial blood pressure (ABP) pulsatility and studied their relationship to cerebrovascular resistance (CVR) and short-term neuroelectrical recovery. Methods and Results Male rats underwent asphyxial CA followed by cardiopulmonary resuscitation. A multimodal platform combining laser speckle imaging, ABP, and electroencephalography to monitor CBF, peripheral blood pressure, and brain electrophysiology, respectively, was used. CBF and ABP pulsatility and CVR were assessed during baseline, CA, and multiple time points after resuscitation. Neuroelectrical recovery, a surrogate for neurological outcome, was assessed using quantitative electroencephalography 90 minutes after resuscitation. We found that CBF pulsatility differs significantly from baseline at all experimental time points with sustained deficits during the 2 hours of postresuscitation monitoring, whereas ABP pulsatility was relatively unaffected. Alterations in CBF pulsatility were inversely correlated with changes in CVR, but ABP pulsatility had no association to CVR. Interestingly, despite small changes in ABP pulsatility, higher ABP pulsatility was associated with worse neuroelectrical recovery, whereas CBF pulsatility had no association. Conclusions Our results reveal, for the first time, that CBF pulsatility and CVR are significantly altered in the short-term postresuscitation period after CA. Nevertheless, higher ABP pulsatility appears to be inversely associated with neuroelectrical recovery, possibly caused by impaired cerebral autoregulation and/or more severe global cerebral ischemia