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Elaine Gorom-Alexander (Title: A One-Dimensional Symmetric Force-Based Blending Method for Atomistic-to-Continuum Coupling)
Dr Candidate: Madhumita Paul (Title: Spectral Theory of the Schrodinger Operator on Half Axis with Increasing Potential)
Jacob Raymond (Title: Shifts of Finite Type on Locally Finite Groups )
1 Title: A One-Dimensional Symmetric Force-Based Blending Method for Atomistic-to-Continuum Coupling
Abstract: Inspired by the blending method developed by [P. Seleson, S. Beneddine, and S. Prudhome, A Force-Based Coupling Scheme for Peridynamics and Classical Elasticity, (2013)] for the nonlocal-to-local coupling, we create a symmetric and consistent blended force-based Atomistic-to-Continuum (a/c) scheme for the atomistic chain in one-dimensional space.
The conditions for the well-posedness of the underlying model are established by analyzing an optimal blending size and blending type to ensure the $H^1$ semi-norm stability for the blended force-based operator. We present several numerical experiments to test and confirm the theoretical findings.
2 Title: Spectral Theory of the Schrodinger Operator on Half Axis with Increasing Potential
Abstract: By the spectrum corresponding to a problem for a Strum-Liouville operator or a Dirac system for the case of the half line is meant the complement of the set of point, in a neighborhood of which the spectral function ρ(dλ) is constant where ρ(dλ) = ρ(λ + ∆) − ρ(λ) for any interval ∆ ⊂ [0, ∞). By the point or discrete spectrum, it means the set of all points of discontinuity of the spectral function ρ(dλ) for the case of the half line.By the continuous spectrum, it means set of all points of continuity of ρ(dλ) which belong to the spectrum. The points of the discrete spectrum are called the eigenvalues and solutions of the problem corresponding to such points are called eigenfunction. The spectral measure (for the case of self-adjoint operators) has can be represented as absolute continuous spectrum, discrete spectrum and singular continuous spectrum. Spectral analysis of schr ̈odinger operator on half axis with increasing potential will be formulated with Dirichlet boundary condition at point 0.
3. Title: Shifts of Finite Type on Locally Finite Groups
Abstract: Let $G$ be a group and $\mathcal{A}$ be a finite alphabet. The set $\mathcal{A}^G$, which is the set of all ways to label elements of $G$ by symbols in $\mathcal{A}$, is acted upon by $G$ in a natural way by shifting the position of labels according to specific elements of the group. A closed subset of $\mathcal{A}^G$ for which this action by $G$ is closed is called a \textit{shift}, and shifts are the primary object of study in symbolic dynamics. A specific type of shift, called a \textit{shift of finite type} (SFT), are defined by a finite amount of information, and are generally easier to study than shifts in general. In the case that the group $G$ is locally finite (every finitely generated subgroup is finite), we show that many strong dynamical properties hold for every SFT. These properties include that every sofic shift is a SFT, every SFT is strongly irreducible, every SFT is entropy minimal, and every SFT has a unique measure of maximal entropy, among others. In addition, we show that for any of these properties, if a group $G$ is such that every SFT has this property, then $G$ must be locally finite.
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