$\psi:H^2(S_1,\QQ)\rightarrow H^2(S_2,\QQ)$ between any two K\"ahler $K3$

surfaces $S_1$ and $S_2$. We prove that the cohomology class of $\psi$ in $H^{2,2}(S_1\times S_2)$

is a polynomial in Chern classes of coherent analytic sheaves

over $S_1 \times S_2$. Consequently, the cohomology class of $\psi$ is algebraic

whenever $S_1$ and $S_2$ are algebraic.

]]>A for certain 2 g(C[[t]]). For these , which are equivalued, integral, and regular,

it is known that the affine Springer fiber, X, has a paving by affines resulting from

the intersection of Schubert cells with X. Our description of the elements of Xallow

us to understand these affine spaces and write down explicit dimension formulae. We

also explore some closure relations between the affine spaces and begin to describe the

moment map for the both the regular and extended torus action.

]]>It is natural to study the topology of these orbit closures since the study of the topology of Borel orbit closures in the flag variety (that is, Schubert varieties) has proved to be inter- esting, linking geometry and representation theory since the local intersection cohomology Betti numbers turned out to be the coefficients of Kazhdan-Lusztig polynomials.

We compute equivariant intersection cohomology with respect to a torus action because such actions often have convenient localization properties enabling us to use data from the moment graph (roughly speaking the collection of 0 and 1-dimensional orbits) to compute the equivariant (intersection) cohomology of the whole space, an approach commonly re- ferred to as GKM theory after Goresky, Kottowitz and MacPherson. Furthermore in the GKM setting we can recover ordinary intersection cohomology from the equivariant inter- section cohomology. Unfortunately the GKM theorems are not practical when computing intersection cohomology since for singular varieties we may not a priori know the local equivariant intersection cohomology at the torus fixed points. Braden and MacPherson address this problem, showing how to algorithmically apply GKM theory to compute the equivariant intersection cohomology for a large class of varieties that includes Schubert varieties.

Our setting is more complicated than that of Braden and MacPherson in that we must use some larger torus orbits than just the 0 and 1-dimensional orbits. Nonetheless we are able to extend the moment graph approach of Braden and MacPherson. We define a more general notion of moment graph and identify canonical sheaves on the generalized moment graph whose sections are the equivariant intersection cohomology of the Borel orbit closures of the wonderful compactification.

]]>wave solutions in a chain of periodically forced coupled nonlinear oscillators, Turing

instability in a HCV model and tumor dynamics.

Motivated by earlier studies of artificial perceptions of light called phosphenes, we

analyze traveling wave solutions in a chain of periodically forced coupled nonlinear

oscillators modeling this phenomenon. We examine the discrete model problem in its

co-traveling frame and systematically obtain the corresponding traveling waves in one

spatial dimension. Direct numerical simulations as well as linear stability analysis are

employed to reveal the parameter regions where the traveling waves are stable, and

these waves are, in turn, connected to the standing waves analyzed in earlier work.

We also consider a two-dimensional extension of the model and demonstrate the

robust evolution and stability of planar fronts and annihilation of radial ones. Finally,

we show that solutions that initially feature two symmetric fronts with bulged centers

evolve in qualitative agreement with experimental observations of phosphenes.

For hepatitis C virus (HCV) model, using the Routh-Hurwitz conditions, we prove

in most parameter regimes that there can be no Turing instability. The simulations

support this in all parameter regions of the model. We introduce a modified model

where Turing instability is observed.

For tumor dynamics model, we present the Fisher Kolomogorov equation (PDE)

and the effective particle methods (ODE) for single front solution and localized

solution with and without radiation. The predicted lifetimes of the patients from the

PDE and ODE are compared and show good quantitative agreement.

]]>Our methods are based on Conditional Random Fields, which allow us to capture temporal dependence in an individual’s physical activity type without requiring us to model the distribution of the observed features at each point in time. We develop three novel estimation strategies for Conditional Random Fields, evaluate their performance on classification tasks through simulation studies and demonstrate their use in applications with real physical activity data sets.

]]>The systematic analysis begins by considering the dark-bright (DB)-soliton interactions and multiple-dark-bright-soliton complexes in atomic two-component BECs. The interaction between two DB solitons in a homogeneous condensate and at the presence of the trap are both considered. Our analytical approximation relies in a Hamiltonian perturbation theory, which leads to an equation of motion of the centers of DB-soliton interacting pairs. Employing this equation, we demonstrate the existence of robust DB-soliton molecules, in the form of stationary two- and three-DB-soliton states. Also the equilibrium distance of the constituent solitons and the corresponding oscillation frequencies are found semianalytically, where the latter corresponds to the characteristic anomalous modes' eigenfrequencies that we numerically computed via a so called Bogoliubov-de Gennes (BdG) linearization analysis. Those studies are discussed in Chapter 2.

Then, we extend our studies to the dynamics of dark-bright (DB) solitons in binary BECs at finite temperature using a system of two-coupled dissipative GPs. We show that the effect of the bright soliton is to partially stabilize dark solitons against temperature-induced dissipation, thus providing longer lifetimes in Chapter 3.

Furthermore, the dark-dark (DD) solitons as a prototypical coherent structure that emerges in two-component BECs are studied and are connected to dark-bright (DB) solitons via SO(2) rotation. We obtained their beating frequency and their frequency of oscillation inside a parabolic trap. They are identified as exact periodic orbits in the Manakov limit of equal inter- and intra- species nonlinearity strengths with and without the trap and we showcase the persistence of such states upon weak deviations from this limit. Also we investigated in detail the effect of the deviation from the Manakov case by considering different from unity scattering length ratios in Chapter 4.

Next, we revisited Hamiltonian eigenvalue problems that typically arise in the linearization around a stationary state of a Hamiltonian nonlinear PDE. Also we presented a overview of the known facts for the eigenvalue counts of the corresponding unstable spectra. In particular, we focused on a straightforward plan to implement finite-dimensional techniques for locating this spectrum via the singular points of the meromorphic Krein Matrix and illustrated the value of the approach by considering realistic problems for recently observed experimentally multivortex and multisoliton solutions in atomic Bose-Einstein condensates in Chapter 5.

In the two dimensional scenario, we also examine the stability and dynamics of vortices under the effect of dissipation used as a simplified model for the inclusion of the effect of finite temperatures in atomic BECs, which enables an analytical prediction that can be compared directly to numerical results in Chapter 6.

In all the above studies, our analytical prediction from the equation of motion are in good agreement with the numerical results from the BdG analysis.

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