Lam\'e polynomials, hyperelliptic reductions and Lam\'e band structure

The band structure of the Lam\'e equation, viewed as a one-dimensional Schr\"odinger equation with a periodic potential, is studied. At integer values of the degree parameter l, the dispersion relation is reduced to the l=1 dispersion relation, and a previously published l=2 dispersion relation is shown to be partially incorrect. The Hermite-Krichever Ansatz, which expresses Lam\'e equation solutions in terms of l=1 solutions, is the chief tool. It is based on a projection from a genus-l hyperelliptic curve, which parametrizes solutions, to an elliptic curve. A general formula for this covering is derived, and is used to reduce certain hyperelliptic integrals to elliptic ones. Degeneracies between band edges, which can occur if the Lam\'e equation parameters take complex values, are investigated. If the Lam\'e equation is viewed as a differential equation on an elliptic curve, a formula is conjectured for the number of points in elliptic moduli space (elliptic curve parameter space) at which degeneracies occur. Tables of spectral polynomials and Lam\'e polynomials, i.e., band edge solutions, are given. A table in the older literature is corrected.Comment: 38 pages, 1 figure; final revision

From transformation to chronification of migraine : pathophysiological and clinical aspects

Chronic migraine is a neurological disorder characterized by 15 or more headache days per month of which at least 8 days show typical migraine features. The process that describes the development from episodic migraine into chronic migraine is commonly referred to as migraine transformation or chronification. Ample studies have attempted to identify factors associated with migraine transformation from different perspectives. Understanding CM as a pathological brain state with trigeminovascular participation where biological changes occur, we have completed a comprehensive review on the clinical, epidemiological, genetic, molecular, structural, functional, physiological and preclinical evidence available

From transformation to chronification of migraine: Pathophysiological and clinical aspects

Chronic migraine is a neurological disorder characterized by 15 or more headache days per month of which at least 8 days show typical migraine features. The process that describes the development from episodic migraine into chronic migraine is commonly referred to as migraine transformation or chronification. Ample studies have attempted to identify factors associated with migraine transformation fr

Embryology and bony malformations of the craniovertebral junction

BACKGROUND: The embryology of the bony craniovertebral junction (CVJ) is reviewed with the purpose of explaining the genesis and unusual configurations of the numerous congenital malformations in this region. Functionally, the bony CVJ can be divided into a central pillar consisting of the basiocciput and dental pivot and a two-tiered ring revolving round the central pivot, comprising the foramen magnum rim and occipital condyles above and the atlantal ring below. Embryologically, the central pillar and the surrounding rings descend from different primordia, and accordingly, developmental anomalies at the CVJ can also be segregated into those affecting the central pillar and those affecting the surrounding rings, respectively. DISCUSSION: A logical classification of this seemingly unwieldy group of malformations is thus possible based on their ontogenetic lineage, morbid anatomy, and clinical relevance. Representative examples of the main constituents of this classification scheme are given, and their surgical treatments are selectively discussed

An 8.5-kDa ribonuclease from the extreme thermophilic archaebacterium Sulfolobus solfataricus

Protein p3, a ribonuclease we previously isolated from the archaebacterium Sulfolobus solfataricus [P. Fusi et al. (1993) Eur. J. Biochem. 211, 305-310], was subjected to complete amino acid sequencing. It consisted of 75 residues, with a calculated Mr of 8582, a pI of 10.1, and had some degree of monomethylation at Lys-4 and Lys-6. p2, a previously sequenced, 62-residue ribonuclease from the same organism, had an identical sequence for 57 consecutive residues starting from the N-terminus. p2 and p3 also showed a striking similarity to five other proteins previously isolated from Sulfolobus strains and identified as DNA-binding proteins. However, the C-terminus, 10 residue region of p3 did not show any similarity to these proteins; in contrast, it was significantly similar to stretches in three eubacterial ribonucleases from Bacillus strains. No difference between p2 and p3 has so far been detected as regards their catalytic properties. Available data suggest that these molecules have a narrow substrate specificity and probably play specific roles in RNA processing