On the complexity of the Whitehead minimization problem

The Whitehead minimization problem consists in finding a minimum size element in the automorphic orbit of a word, a cyclic word or a finitely generated subgroup in a finite rank free group. We give the first fully polynomial algorithm to solve this problem, that is, an algorithm that is polynomial both in the length of the input word and in the rank of the free group. Earlier algorithms had an exponential dependency in the rank of the free group. It follows that the primitivity problem – to decide whether a word is an element of some basis of the free group – and the free factor problem can also be solved in polynomial time.

Euclidean geometric invariant of framed knots in manifolds

We present an invariant of a three dimensional manifold with a framed knot in it based on the Reidemeister torsion of an acyclic complex of Euclidean geometric origin. To show its nontriviality, we calculate the invariant for some framed (un)knots in lens spaces. An important feature of our work is that we are not using any nontrivial representation of the manifold fundamental group or knot group.

Mecanismes de reacció de sistemes catalítics amb èmfasi en carbens de Fischer i quantificació de l'aromaticitat en petites molècules inorgàniques cícliques i planes.

El tema principal del treball és sobre l'estudi teòric de l'estructura i reactivitat en carbens de Fischer de la forma (CO)5Cr=C(X)R (X= OH, NH2, OMe, NMe2 i R= CHCH2 i Ph). Particularment, el nostre interès va sorgir del tipus de reaccions de cicloadicció que donen lloc a la síntesi de productes naturals i fàrmacs de gran valor afegit. Hem estudiat els mecanismes de reacció dels casos més comuns de cicloanul•lació: la reacció de benzanul•lació de Dötz i ciclopentanulacions que es troben en competència amb el primer cas, derivats de l'inserció de acetilè i fenilacetilè. En l'últim pas de les reaccions que comporten la formació d'un sistema de més d'un anell, hi tenim una migració del complex metàl•lic de crom d'un extrem a un altre anomenat com rearranjament haptotròpic. Aleshores, hem investigat sobre els mecanismes de migració haptotròpica de Cr(CO)3 sobre hidrocarburs aromàtics policíclics analitzant l'efecte de la mida i la curvatura del sistema així com la complexació d'un segon fragment metàl•lic a la manera de coordinació als anells. D'una altre banda, vam estudiar l'aromaticitat en 54 cúmuls cíclics de molecules inorgàniques mitjançant metodologia desenvolupada al nostre grup de recerca i altres. Vam proposar la tècnica del scan-NICS com nova mesura quantitativa d'aromaticitat i reportar l'escassa correlació entre els distints índexs d'aromaticitat a la literatura. Finalment, com resultat de col•laboracions en estades de recerca, he desenvolupat propostes de mecanismes de reacció en sistemes catalítics de isonitrils i fosfinetà-amides i en dímers de gassos nobles.

Homotopy Batalin-Vilkovisky algebras

This paper provides an explicit cofibrant resolution of the operad encoding Batalin-Vilkovisky algebras. Thus it defines the notion of homotopy Batalin-Vilkovisky algebras with the required homotopy properties. To define this resolution we extend the theory of Koszul duality to operads and properads that are defined by quadratic and linear relations. The operad encoding Batalin-Vilkovisky algebras is shown to be Koszul in this sense. This allows us to prove a Poincaré-Birkhoff-Witt Theorem for such an operad and to give an explicit small quasi-free resolution for it. This particular resolution enables us to describe the deformation theory and homotopy theory of BV-algebras and of homotopy BV-algebras. We show that any topological conformal field theory carries a homotopy BV-algebra structure which lifts the BV-algebra structure on homology. The same result is proved for the singular chain complex of the double loop space of a topological space endowed with an action of the circle. We also prove the cyclic Deligne conjecture with this cofibrant resolution of the operad BV. We develop the general obstruction theory for algebras over the Koszul resolution of a properad and apply it to extend a conjecture of Lian-Zuckerman, showing that certain vertex algebras have an explicit homotopy BV-algebra structure.

Evaluación de un nuevo método de sutura para la cirugía endoscopica transluminal a través de orificios naturales en un modelo animal

Natural Orifice Transluminal Endoscopic Surgery (NOTES) is a novel, potentially less invasive alternative to laparoscopic surgery. However, the problems of transluminal access and closure represent significant obstacles to its successful introduction in humans. Objective: to evaluate the feasibility and safety of a novel device designed for transluminal access and closure in a survival porcine model. Subjects: Four adult female Yorkshire pigs were used in the study. Interventions: While under general anesthesia, the animals were prepared with multiple tap water enemas followed by instillation of an antibiotic suspension and povidone-iodine lavage. At a distance of 15 to 20 cm from the anus, the prototype device (LSI Solutions, Victor, NY, USA) deployed a circumscribing purse-string suture around the planned incision site and subsequently used a blade mechanism to create a 2.5-cm linear incision. The transcolonic incision was then closed by cinching and securing the purse-string suture with a titanium knot by use of a separate hand-activated suture-locking device. Main Outcome Measurements: The animals were monitored daily for signs of peritonitis and sepsis and were survived for 14 days. The peritoneal cavity was examined for peritonitis, and the colonic incision site was examined for wound dehiscence, pericolic abscess formation, and gross adhesions. Tissue samples from both incisional and random peritoneal sites were obtained for histologic examination. Results: Transcolonic incision and closure were successful in all 4 animals. The device performed in a rapid and reproducible fashion. All animals recovered without septic complications. At necropsy, there was no evidence of peritonitis, abscesses, or wound dehiscence. Salpingocolonic and colovesicular adhesions were noted in 3 of 4 animals. Histologic examination revealed microabscesses at the incision site in all animals. Conclusions: The prototype incision and closure device represents a promising solution to the problems of transluminal access for NOTES. The presence of incision-related adhesions and microabscesses signal the need for further refinement in aseptic technique. 

The generator problem for Z-stable C*-algebras

The generator problem was posed by Kadison in 1967, and it remains open until today. We provide a solution for the class of C*-algebras absorbing the Jiang-Su algebra Z tensorially. More precisely, we show that every unital, separable, Z-stable C*-algebra A is singly generated, which means that there exists an element x є A that is not contained in any proper sub-C*- algebra of A. To give applications of our result, we observe that Z can be embedded into the reduced group C*-algebra of a discrete group that contains a non-cyclic, free subgroup. It follows that certain tensor products with reduced group C*-algebras are singly generated. In particular, C*r (F ∞) ⨂ C*r (F ∞) is singly generated.

Electrochemically promoted carbon-halide bond cleavage in 4-nitrobenzyl halides

This manuscript reports the study of the carbon-halide bond cleavage in 4-nitrobenzyl halides, taking special attention to the iodide and fluoride derivatives. The electrochemical reduction mechanism has been disclosed for both compounds by terms of cyclic voltammetry and controlled potential electrolysis. In the case of 4-nitrobenzyl iodide, a first one electron irreversible wave leads to the corresponding 4-nitrobenzyl radical and iodide. However, in the case of 4-nitrobenzyl fluoride, a first one-electron reversible wave appears at –1.02 vs. SCE followed by one electron irreversible wave. In this second electron transfer process, the cleavage of the C-F bond is taking place, so the bond cleavage reaction occurs at the dianion level. To disclose and understand the electrochemical reduction mechanisms that allows to obtain important thermodynamic and kinetic data that would help in the understanding of C-X bond cleavage. This type of bond dissociation reactions are involved in the metabolism pathways of the human body.