The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudomodular Hopf algebras. Pseudo-modular Hopf algebras are a class of Z_2-graded ribbon Hopf algebras that generalise the concept of a modular Hopf algebra. The quantum superalgebra Uq(osp(1|2n)) over C is considered with q a primitive Nth root of unity for all integers N &gt = 3. For such a q, a certain left ideal I of U_q(osp(1|2n)) is also a two-sided Hopf ideal, and the quotient algebra U^(N)_q(osp(1|2n)) = U_q(osp(1|2n))/I is a Z_2-graded ribbon Hopf algebra. For all n and all N &gt = 3, a finite collection of finite dimensional representations of U^(N)_q(osp(1|2n)) is defined. Each such representation of U^(N)_q(osp(1|2n)) is labelled by an integral dominant weight belonging to the truncated dominant Weyl chamber. Properties of these representations are considered: the quantum superdimension of each representation is calculated, each representation is shown to be self-dual, and more importantly, the decomposition of the tensor product of an arbitrary number of such representations is obtained for even N. It is proved that the quotient algebra U(N)^q_(osp(1|2n)), together with the set of finite dimensional representations discussed above, form a pseudo-modular Hopf algebra when N &gt = 6 is twice an odd number. Using this pseudo-modular Hopf algebra, we construct a topological invariant of 3-manifolds. This invariant is shown to be different to the topological invariants of 3-manifolds arising from quantum so(2n+1) at roots of unity.

It has been proven by J.C.Criscione that constitutive relations(mixed approach) based on a set of five strain invariants (Beta-1, Beta-2, Beta-3, Beta-4, Beta-5) are useful and stable for experimentally determining response terms for transversely isotropic material. On the other hand, Rivlin’s classical model is an unsuitable choice for determining response terms due to the co-alignment of the five invariants (I1, I2, I3, I4, I5). Despite this, however, a mixed (Lagrangian and Eulerian) approach causes unnecessary computational time and requires intricate calculation in the constitutive relation. Through changing the way to approach the derivation of a constitutive relation, we have verified that using an Eulerian approach causes shorter computational time and simpler calculation than using a mixed approach does. We applied this approach to a boundary value problem under specific deformation, i.e. finite extension and torsion to a fiber reinforced circular cylinder. The results under this deformation show that the computational time by Eulerian is less than half of the time by mixed. The main reason for the difference is that we have to determine two unit vectors on the cross fiber direction from the right Cauchy Green deformation tensor at every radius of the cylinder when we use a mixed approach. On the contrary, we directly use the left Cauchy Green deformation tensor in the constitutive relation by the Eulerian approach without defining the two cross fiber vectors. Moreover, the computational time by the Eulerian approach is not influenced by the degree of deformation even in the case of computational time by the Eulerian approach, possibly becoming the same as the computational time by the mixed approach. This is from the theoretical thought that the mixed approach is almost the same as the Eulerian approach under small deformation. This new constitutive relation by Eulerian approach will have more advantages with regard to saving computational time as the deformation gets more complicated. Therefore, since the Eulerain approach effectively shortens computational time, this may enhance the computational tools required to approach the problems with greater degrees of anisotropy and viscoelasticity.

Thesis (Ph. D.)--University of Texas at Austin, 2001. / Vita. Includes bibliographical references. Available also from UMI/Dissertation Abstracts International.

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Let G be a complex semisimple Lie Group with parabolic subgroup P, so that G/P is a generalized flag manifold. An algebraic construction of invariant differential operators between sections of homogeneous bundles over such spaces is given and it is shown how this leads to the classification of all such operators. As an example of a process which naturally generates such operators, the algebraic Penrose transform between generalized flag manifolds is given and computed for several cases, extending standard results in Twistor Theory to higher dimensions. It is then shown how to adapt the homogeneous construction to manifolds with a certain class of tangent bundle structure, including conformal manifolds. This leads to a natural definition of invariant differential operators on such manifolds, and an algebraic method for their construction. A curved analogue of the Penrose transform is given.

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The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudomodular Hopf algebras. Pseudo-modular Hopf algebras are a class of Z_2-graded ribbon Hopf algebras that generalise the concept of a modular Hopf algebra. The quantum superalgebra Uq(osp(1|2n)) over C is considered with q a primitive Nth root of unity for all integers N &gt = 3. For such a q, a certain left ideal I of U_q(osp(1|2n)) is also a two-sided Hopf ideal, and the quotient algebra U^(N)_q(osp(1|2n)) = U_q(osp(1|2n))/I is a Z_2-graded ribbon Hopf algebra. For all n and all N &gt = 3, a finite collection of finite dimensional representations of U^(N)_q(osp(1|2n)) is defined. Each such representation of U^(N)_q(osp(1|2n)) is labelled by an integral dominant weight belonging to the truncated dominant Weyl chamber. Properties of these representations are considered: the quantum superdimension of each representation is calculated, each representation is shown to be self-dual, and more importantly, the decomposition of the tensor product of an arbitrary number of such representations is obtained for even N. It is proved that the quotient algebra U(N)^q_(osp(1|2n)), together with the set of finite dimensional representations discussed above, form a pseudo-modular Hopf algebra when N &gt = 6 is twice an odd number. Using this pseudo-modular Hopf algebra, we construct a topological invariant of 3-manifolds. This invariant is shown to be different to the topological invariants of 3-manifolds arising from quantum so(2n+1) at roots of unity.

Thesis (Ph.D.)--Tufts University, 2004. / Adviser: Fulton B. Gonzalez. Submitted to the Dept. of Mathematics. Includes bibliographical references (leaves 72-73). Access restricted to members of the Tufts University community. Also available via the World Wide Web;

Thesis (Ph.D.) in Physics--University of Maine, 2007. / Includes vita. Includes bibliographical references (leaves 109-112).