The symmetric groups Sn, alternating groups An, and (for n ≥ 3) dihedral groups Dn behave, by their very definition, as permutations on certain sets. The groups Sn and An both permute the set {1, 2, . . . , n} and Dn can be considered as a group of permutations of a regular n-gon, or even just of its n vertices, since rigid motions of the vertices determine where the rest of the n-gon goes. If ...

Glaucoma is the second leading cause for blindness worldwide. It is mainly caused by glaucomatous optic neuropathy (GON) characterized by retinal ganglion cell loss, which leads to visual field defect and blindness. Up to now, the main purpose of antiglaucomatous therapies has been to lower intraocular pressure (IOP) through surgeries and medications. However, it has been found that progressive...

occipital neuralgia is a condition manifested by chronic occipital headaches and is thought to be caused by irritation or trauma to the greater occipital nerve (gon). treatment for occipital neuralgia includes medications, nerve blocks, and pulsed radiofrequency ablation (prfa). landmark-guided gon blocks are the mainstay in both the diagnosis and treatment of occipital neuralgia. ultrasound is...

Let F={P1,...,Pm} be a family of sets. A partial order P(F, <) on F is naturally defined by the condition Pi< Pj iff Pi is contained in Pj. When the elements of F are disks (i.e. circles together with their interiors), P(F, <) is called a circle order; if the elements of F are n-polygons, P(F, <) is called an n-gon order. In this paper we study circle orders and n-gon orders. The crossing numbe...

This paper presents an approach to the generation of periodic signals through the application of Euclidean geometry in the time domain. In particular, the content of this paper is focused on waveforms derived from regular polygons. It is an attempt to show how the geometrical relations and proportions of regular polygons and star polygons with respect to their Schläfli symbol can be made audibl...

In the standard Coxeter presentation, the symmetric group Sn is generated by the ad jacent transpositions (1,2), (2,3), ... , (n1,n). For any given permutation, we consider all minimal-length factorizations thereof as a product of the generators. Any two transpositions (i,i+1) and (j,j+1) commute if the numbers i and j are not consecutive; thus, in any factorization, their order can be switche...

In wild-type Caenorhabditis elegans, the gonad is a complex epithelial tube that consists of long arms composed predominantly of germline tissue as well as somatic structures specialized for particular reproductive functions. In gon-1 mutants, the adult gonad is severely disorganized with essentially no arm extension and no recognizable somatic structure. The developmental defects in gon-1 muta...

Occipital stimulation in a small group of refractory chronic migraine and cluster headache patients has been suggested as a novel therapeutic approach with promising results. In an earlier study we have shown that a drug-induced block of the greater occipital nerve (GON) inhibits the nociceptive blink reflex (nBR). Now, we sought to examine the effects of low-frequency (3 Hz) short-time nocicep...

Occipital neuralgia is a rare pain syndrome characterized by periodic lancinating pain involving the occipital nerve complex. We present a unique case of entrapment of the greater occipital nerve (GON) within the semispinalis capitis, which was thought to be the cause of occipital neuralgia. A 66-year-old woman with refractory left occipital neuralgia revealed an abnormally low-loop of the left...

We consider the question of the existence of equiangular polygons with edge lengths in arithmetic progression, and show that they do not exist when the number of sides is a power of two and do exist if it is any other even number. A few results for small odd numbers are given. An Olympiad problem [1] asked the following. Prove that there exists a convex 1990-gon with the following two propertie...