This paper mainly studies the singularities of Gauss Map of pedal hypersurface in R n +1 . It contains the geometry of pedal hypersurfaces in R n +1 and their Gauss maps. The singularity of Gauss map of the pedal hypersurface using the rank of jacobian matrix of Gauss map is given and classified. The sets of singularities and its graphs under the Gauss map are plotted.

This paper mainly studies the Singularities of smooth mapping. The singularities of the families of Gauss maps corresponding to the family of mappings are studied and the shape of these families and their singularities using mathematica program are illustrated and plotted.

We present algorithms for computing the differential geometry properties of Frenet apparatus (t,n,b1,b2,κ1,κ2,κ3) of intersectioncurves of implicit–parametric–parametric and implicit–implicit–parametric hypersurfaces in R4, for transversal intersection. Some examples are given and plotted.

A brachiopod fauna of late Bathonian age recovered from the Kehailia Formation from Gebel Engabashi in northern Sinai consists of six species (two rhynchonellids and four terebratulids) referred to six genera, of which one genus and two species are new: Globirhynchia sphaerica (Cooper, 1989) new combination, Daghanirhynchia angulocostata Cooper, 1989, Ectyphoria sinaiensis new species, Cooperithyris circularis new genus and species, and new material: Avonothyris species A, and Ptyctothyris species A. The brachiopods described herein comprise a fauna located at the northern part of the Indo-African Faunal Realm within the Jurassic Ethiopian Province. They extend the The geographic distribution of those taxa that show great affinity with the Jurassic brachiopod fauna of Saudi Arabia described by cooper (1989). Differentiation of the endemic faunas that is so characteristic of many of these Ethiopian province faunas is becoming more well-defined

Thermolysis of N-arylnicotinamide oximes 1a–c (R = H, CH3 and Cl) under nitrogen gives rise to benzimidazoles 3a–c as the major products (60–62%), in addition to nicotinonitrile 4, arylamines 5a–c, nicotinic acid 6, phenols 7a–c and 8a–c, nicotinanilides 9a–c, 2-(pyridin-3-yl) benzoxazoles 10a and carbazoles 11a–c. In the presence of naphthalene, thermolysis of 1a gave - and -naphthols 12 and 13 beside the previous products. Also pyrolysis of 1a in boiling tetralin lead to the formation of 1-hydroxytetralin 14, -tetralone 15 and 1,1-bitetralyl 16 as the major products. The isolated products have been interpreted in the terms of a free radical mechanism involving the homolysis of N–O and/or C–N bonds.

Thermolysis of N-phenylbenzamide oximes I, II and III (R = Cl, NO2 and OCH3) under nitrogen gives rise to benzimidazoles as the major products (45–52%), in addition to benzonitrile, arylamines, benzoic acid, phenols, benzanilides, 2-phenyl benzoxazoles and carbazoles. In the presence of naphthalene, I gave - and -naphthols beside the previous products. Also heating of I under reflux boiling tetralin lead to the formation of 1-hydroxytetralin, -tetralone and 1,1-bitetralyl as the major products. The isolated products have been interpreted in the terms of a free radical mechanism involving the homolysis of N–O and/or C–N bonds.

Synthesis of seleniumcontaining amino acid analogues is described. These compounds were prepared in a concise and short synthetic route in good yields by nucleophilic substitution reaction of pyridineselenol and quinolineselenol derivatives with Nphthaloylglycyl chloride followed by hydrazinolysis. The newly synthesized compounds were screened against different strains of bacteria and fungi.

The compound of selenium containing heterocycles namely (I) 4,6-dimethyl-3-cyanopyridine-(2H)- selenone (C8H8N2Se) was used as model to investigate the structural and micro-structural features of some selected selenones. Visualization of molecular structure of selenium containing moiety was accurately investigated to clarify role of selenium ion in nucleation process of forming 3D-framenet of H-bonded of substituted selenones. Bond distances, torsions on angles of investigated compound were compared with others crystal data to confirm crystal structures of substituted selenones which is monoclinic crystal form with P21/n space group.

Selenium has long been recognized as a dietary antioxidant and is now known to be an essential component of the active sites of several enzymes, including glutathione peroxidase [2] and thioredoxin reductase, which catalyze reactions essential for the protection of cellular components against oxidative and radical damage [3]. Moreover, previous work in our laboratory describes the synthesis of pyrimidoselenolo- [2,3-b]quinoline [4] and pyrimidoselenolo[2,3-c]pyridazine derivatives [5] and indicates that certain compounds possess significant anti-inflammatory and analgesic activities with strong fungicidal effects. Furthermore, in Part 1 [6] and Part 2 [1] we published the synthesis of selenolo[2,3-b]pyridine, pyrido[3',2':4,5]selenolo- [3,2-d]pyrimidine, 2,4-dimethyl-7,8-dihydropyrrolo[1,2-a]pyrido[3',2':4,5]selenolo[3,2-d]pyrimidin-10(6H)-one, 7,9-dimethylpyrido[3',2':4,5]selenolo[3,2-e][1,2,4]triazolo[4,3-c]pyrimidine, 7,9-dimethylpyrido[3',2':4,5]selenolo[ 3,2-d]pyrimidine-2,4(1H,3H)-dithione, 7,9-dimethylpyrido[3',2':4,5]selenolo[2,3-e]tetrazolo[1,5-c]pyrimidine- 6(5H)thione, 9,11-dimethylpyrido[3',2':4,5]selenolo[2,3-e][1,2,4]triazolo[1,5-c]pyrimidine, 7,9-dimethylpyrido[ 3',2':4,5]selenolo[2,3-e]imidazo[1,2-c]pyrimidine, and 10,12-dimethylpyrido[3'',2'':4',5']selenolo- [3',2':4,5]pyrimido[1,6-a]pyrimidine derivatives. Prompted by these observations and in continuation of our work we report herein new classes of fused tri- and tetracyclic systems containing the selenolo[2,3-b]pyridine fragment in the hope that members of these series may find interesting biological applications.