No description available.

In this thesis we present a slight generalisation of the Pearl complex or relative quantum homology to some non monotone Lagrangian submanifolds. First we develop the theory for the so called almost monotone Lagrangian submanifolds, We apply it to uniruling problems as well as estimates for the relative Gromov width. In the second part we develop the theory for toric fiber in toric Fano manifolds, recovering previous computaional results of Floer homology .

We show that the symplectic current obtained from the boundary term, which arises in the first variation of a local diffeomorphism invariant action, is covariantly conserved for any gravity theory described by that action.Therefore, a Poincar&eacute / invariant two-form can be constructed on the phase space, which is shown to be closed without reference to a specific theory.Finally, we show that one can obtain a charge expression for gravity theories in various dimensions, which plays the role of the Abbott-Deser-Tekin charge for spacetimes with nonconstant curvature backgrounds, by using the diffeomorphism invariance of the symplectic two-form. As an example, we calculate the conserved charges of some solutions of new massive gravity and compare the results with previous works.

When the finite-difference time-domain (FDTD) method is applied to light scattering computations, the far fields can be obtained by either a volume integration method, or a surface integration method. In the first study, we investigate the errors associated with the two near-to-far field transform methods. For a scatterer with a small refractive index, the surface approach is more accurate than its volume counterpart for computing the phase functions and extinction efficiencies; however, the volume integral approach is more accurate for computing other scattering matrix elements. If a large refractive index is involved, the results computed from the volume integration method become less accurate, whereas the surface method still retains the same order of accuracy as in the situation of a small refractive index. In my second study, a fourth order symplectic FDTD method is applied to the problem of light scattering by small particles. The total-field/ scattered-field (TF/SF) technique is generalized for providing the incident wave source conditions in the symplectic FDTD (SFDTD) scheme. Numerical examples demonstrate that the fourthorder symplectic FDTD scheme substantially improves the precision of the near field calculation. The major shortcoming of the fourth-order SFDTD scheme is that it requires more computer CPU time than the conventional second-order FDTD scheme if the same grid size is used. My third study is on multiple scattering theory. We develop a 3D Monte Carlo code for the solving vector radiative transfer equation, which is the equation governing the radiation field in a multiple scattering medium. The impulse-response relation for a plane-parallel scattering medium is studied using our 3D Monte Carlo code. For a collimated light beam source, the angular radiance distribution has a dark region as the detector moves away from the incident point. The dark region is gradually filled as multiple scattering increases. We have also studied the effects of the finite size of clouds. Extending the finite size of clouds to infinite layers leads to underestimating the reflected radiance in the multiple scattering region, especially for scattering angles around 90 degrees. The results have important applications in the field of remote sensing.

The rigid body has been one of the most noteworthy applications of Newtonian mechanics. Applying the principles of classical mechanics to the rigid body is by no means routine. The equations of motion, though discovered two hundred and fifty years ago by Euler, have remained quite elusive since their introduction. Understanding the rigid body has required the applications of concepts from integrable systems, algebraic geometry, Lie groups, representation theory, and symplectic geometry to name a few. Moreover, several important developments in these fields have in fact originated with the study of the rigid body and subsequently have grown into general theories with much wider applications.In this work, we study the stability of equilibria of non-degenerate orbits of the generalized rigid body. The energy-Casimir method introduced by V.I. Arnold in 1966 allows us to prove stability of certain non-degenerate equilibria of systems on Lie groups. Applied to the three dimensional rigid body, it recovers the classical Euler stability theorem [12]: rotations around the longest and shortest principal moments of inertia are stable equilibria. This method has not been applied to the analysis of rigid body dynamics beyond dimension n = 3. Furthermore, no conditions for the stability of equilibria are known at all beyond n = 4, in which case the conditions are not of the elegant longest/shortest type [10].Utilizing the rich geometric structures of the symmetry group G = SO(2n), we obtain stability results for generic equilibria of the even dimensional free rigid body. After obtaining a general expression for the generic equilibria, we apply the energy-Casimir method and find that indeed the classical longest/shortest conditions on the entries of the inertia matrix are suffcient to prove stability of generic equilibria for the generalized rigid body in even dimensions.

When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation $T$ of a real algebraic Lie group $G$. This requires defining an inner product on the Hilbert space $\mathbb{H}$ that carries the representation $T$. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of $T$, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet ($\mathfrak{F}_{\mathbb{H}_D}$, $\mathbb{H}_D$, $\mathfrak{F}^{\mathfrak{H}_D}$) of Bargmann spaces. Coherent state wave functions -- the elements of $\mathfrak{F}_{\mathbb{H}_D}$ and of $\mathfrak{F}^{\mathbb{H}_D}$ -- are used to define the inner product on $\mathbb{H}_D$ in a way that simplifies the calculation of the overlaps. This inner product and the group action $\Gamma$ of $G$ on $\mathfrak{F}^{\mathbb{H}_D}$ are used to formulate expressions for the matrix elements of $T$ with coefficients from the given subrepresentation. The process of finding an explicit expression for $\Gamma$ relies on matrix factorizations in the complexification of $G$ even though the representation $T$ does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and $\Gamma$ action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in $\mathbb{H}$. The scalar and vector coherent state methods will both be applied to Sp($n$) and Sp($n,\mathbb{R}$).

No description available.

No description available.

No description available.

When solving problems involving quantum mechanical systems, it is frequently desirable to find the matrix elements of a unitary representation $T$ of a real algebraic Lie group $G$. This requires defining an inner product on the Hilbert space $\mathbb{H}$ that carries the representation $T$. In the case where the representation is determined by a representation of a subgroup containing the lowest weight vector of $T$, this can be achieved through the coherent state construction. In both the scalar and vector coherent state methods, the process of finding the overlaps can be simplified by introducing the coherent state triplet ($\mathfrak{F}_{\mathbb{H}_D}$, $\mathbb{H}_D$, $\mathfrak{F}^{\mathfrak{H}_D}$) of Bargmann spaces. Coherent state wave functions -- the elements of $\mathfrak{F}_{\mathbb{H}_D}$ and of $\mathfrak{F}^{\mathbb{H}_D}$ -- are used to define the inner product on $\mathbb{H}_D$ in a way that simplifies the calculation of the overlaps. This inner product and the group action $\Gamma$ of $G$ on $\mathfrak{F}^{\mathbb{H}_D}$ are used to formulate expressions for the matrix elements of $T$ with coefficients from the given subrepresentation. The process of finding an explicit expression for $\Gamma$ relies on matrix factorizations in the complexification of $G$ even though the representation $T$ does not extend to the complexification. It will be shown that these factorizations are, in fact, justified, that the overlaps and $\Gamma$ action can be expressed in terms of the given subrepresentation, and that it is possible to find numerical values for the inner product in $\mathbb{H}$. The scalar and vector coherent state methods will both be applied to Sp($n$) and Sp($n,\mathbb{R}$).