Evaluation of Euler Number of Complex Grassmann Manifold G(k, N) via Mathai-Quillen Formalism

In this paper, we provide a recipe for computing Euler number of Grassmann manifold G(k,N) by using Mathai-Quillen formalism (MQ formalism) and Atiyah-Jeffrey construction. Especially, we construct path-integral representation of Euler number of G(k,N). Our model corresponds to a finite dimensional toy-model of topological Yang-Mills theory which motivated Atiyah-Jeffrey construction. As a by-product, we construct free fermion realization of cohomology ring of G(k,N).Comment: 31 pages, Latex, final version accepted for publication in Journal of Geometry and Physic

A CASE OF BOW HUNTER'S STROKE PRESENTING SYNCOPE AS AN INITIAL SYMPTOM WHILE DRJVING

Bow Hunter's Stroke is a consequence of vertebrobasilar insufficiency as a result of mechanical occlusion or stenosis of the vertebral artery at the C1-C2 level by head rotation. It is rarely symptomatic in daily activities. We describe a case of Bow Hunter's Stroke(BHS) presenting syncope as an initial symptom while driving a car. A 59-year-old male patient suddenly lost consciousness while driving and his car dropped into a ravine 20 meters deep. On admission he was conscious, but suffering Central Cord Syndrome (CCS). We examined the cause of syncope. At the time of the accident, he turned his head to the rear in order to back his car and lost consciousness. Magnetic resonance angiography demonstrated the dominant vertebral artery (VA) in the left and the hypoplastic VA in the right. Cerebral angiography on turning the head 90 degrees to the right revealed the left VA occluded at the C1-C2 level. Therefore we diagnosed the patient with BHS. The vertigo symptom was intractable with conservative treatment, and we therefore performed C1-C2 posterior fusion. The post-operative course was uneventfu1 and he does not have the symptoms anymore. Syncope sometimes induces traffic accidents, but it is rare that BHS is detected on examination of common trauma. Therefore when vertigo, dizziness, or syncope is found in common trauma, BHS must be considered as a potential cause