No abstract

19750 school children, ages 6 to 15 years, were examined by the authors of this study, 1,220 (6.18%) had congenital abnormalities. In this group, 4.23% were boys and 1.88% were girls. Case histories revealed inbreeding amongst the parents (families) of children with congenital malformation to be 8.9% and 8.2% for the rest of the families in this study. There were 27 different congenital abnormalities identified, with prevalence rates of 0.05/1,000 to 15.85/1,000. The most prevalent abnormalities were umbilical hernia (15.85/1000), inguinal hernia (14.50/1,000), pectus carinatum and excavatum (7.68/1,000), undescended testes (9.00/1,000 boys), congenital nevus (3.54/1,000), retractile testis (4.45/1,000 boys), pilonidal sinus (2.63/1,000), pes planus (2.28/1,000), and hemangioma (1.16/1,000). Of the 19,750 children, 70 had multiple anomalies (3.75/1,000).

We note that all regular and semiregular polytopes in arbitrary dimensions can be obtained from the Coxeter-Dynkin diagrams. The vertices of a regular or semiregular polytope are the weights obtained as the orbit of the Coxeter-Weyl group acting on the highest weight representing a selected irreducible representation of the Lie group. This paper, in particular, deals with the determination of the vertices of the Platonic and Archimedean solids from the Coxeter diagrams A3, B3, and H3 in the context of the quaternionic representations of the root systems and the Coxeter-Weyl groups. We use Lie algebraic techniques in the derivation of vertices of the polyhedra and show that the polyhedra possessing the tetrahedral, octahedral, and icosahedral symmetries are related to the Coxeter-Weyl groups representing the symmetries of the diagrams of A3, B3, and H3, respectively. This technique leads to the determination of the vertices of all Platonic and Archimedean solids except two chiral polyhedra, snubcuboctahedron and snubicosidodecahedron.

Transmission probabilities of the scattering problem with a position dependent mass are studied. After sketching the basis of the theory, within the context of the Schrödinger equation for spatially varying effective mass, the simplest problem, namely, tranmission through a square well potential with a position dependent mass barrier is studied and its novel properties are obtained. The solutions presented here may be adventageous in the design of semiconductor devices.

Cayley-Dickson doubling procedure is used to construct the root systems of some celebrated Lie algebras in terms of the integer elements of the division algebras of real numbers, complex numbers, quaternions and octonions. Starting with the roots and weights of SU (2) expressed as the real numbers one can construct the root systems of the Lie algebras of SO(4), SP (2) ≈ SO(5), SO(8), SO(9), F4 and E8 in terms of the discrete elements of the division algebras. The roots themselves display the group structures besides the octonionic roots of E8 which form a closed octonion algebra. The automorphism group Aut(F4) of the Dynkin diagram of F4 of order 2304, the largest crystallographic group in 4-dimensional Euclidean space, is realized as the direct product of two binary octahedral group of quaternions preserving the quaternionic root system of F4 .The Weyl groups of many Lie algebras, such as, G2, SO(7), SO(8), SO(9), SU (3)XSU (3) and SP (3) × SU (2) have been constructed as the subgroups of Aut(F4) .We have also classified the other non-parabolic subgroups of Aut(F4) which are not Weyl groups. Two subgroups of orders192 with different conjugacy classes occur as maximal subgroups in the finite subgroups of the Lie group G2 of orders 12096 and 1344 and proves to be useful in their constructions. The triality of SO(8) manifesting itself as the cyclic symmetry of the quaternionic imaginary units e1, e2, e3 is used to show that SO(7) and SO(9) can be embedded triply symmetric way in SO(8) and F4 respectively. INTRODUCTIONThere are a few celebrated Lie algebras which seem to be playing important roles in understanding the underlying symmetries of the unified theory of all interactions. The most popular ones are the exceptional Lie groups G 2 , F 4 , E 6 , E 7 and E 8 and the related groups [1]. The groups Spin7 and G 2 are proposed as the holonomy groups for the compactification of the M-theory from 11 to 4 dimensional space-time [2]. It is also well known that two orthogonal groups SO(8) and SO(9) are the little groups of the massles particles of string theories in 10-dimensions and the M-theory in 11-dimensions respectively. The fact that SO(9) can be embedded in the exceptional Lie group F 4 in a triply symmetric way and the non-compact F 4(−25) can be embedded in the Lorenz group SO(25, 1) indicates the importance of the exceptional group F 4 [3]. The largest exceptional group E 8 which had been suggested as the unified theory of the electroweak and strong interactions with three generations of lepton-quark families [4] naturally occurred as the gauge symmetry of the E 8 × E 8 heterotic string theory [5]. It has many novel mathematical aspects [6] which has not been exploited in physics. It was known that a non-compact version of E 7 manifests itself as a global symmetry of the 11-dimensional supergravity [7]. Some of its maximal subgroups show themselves as local symmetries [8]. The E 6 has been suggested as a unified theory of electroweak and strong interactions [9].The Weyl groups of these groups are also impor...