Supersymmetry and Schr\"odinger-type operators with distributional matrix-valued potentials

Building on work on Miura's transformation by Kappeler, Perry, Shubin, and Topalov, we develop a detailed spectral theoretic treatment of Schr\"odinger operators with matrix-valued potentials, with special emphasis on distributional potential coefficients. Our principal method relies on a supersymmetric (factorization) formalism underlying Miura's transformation, which intimately connects the triple of operators $(D, H_1, H_2)$ of the form [D= (0 & A^*, A & 0) \text{in} L^2(\mathbb{R})^{2m} \text{and} H_1 = A^* A, H_2 = A A^* \text{in} L^2(\mathbb{R})^m.] Here $A= I_m (d/dx) + \phi$ in $L^2(\mathbb{R})^m$, with a matrix-valued coefficient $\phi = \phi^* \in L^1_{\text{loc}}(\mathbb{R})^{m \times m}$, $m \in \mathbb{N}$, thus explicitly permitting distributional potential coefficients $V_j$ in $H_j$, $j=1,2$, where [H_j = - I_m \frac{d^2}{dx^2} + V_j(x), \quad V_j(x) = \phi(x)^2 + (-1)^{j} \phi'(x), j=1,2.] Upon developing Weyl--Titchmarsh theory for these generalized Schr\"odinger operators $H_j$, with (possibly, distributional) matrix-valued potentials $V_j$, we provide some spectral theoretic applications, including a derivation of the corresponding spectral representations for $H_j$, $j=1,2$. Finally, we derive a local Borg--Marchenko uniqueness theorem for $H_j$, $j=1,2$, by employing the underlying supersymmetric structure and reducing it to the known local Borg--Marchenko uniqueness theorem for $D$.Comment: 36 page

Ariadne: An interface to support collaborative database browsing:Technical Report CSEG/3/1995

This paper outlines issues in the learning of information searching skills. We report on our observations of the learning of browsing skills and the subsequent iterative development and testing of the Ariadne system – intended to investigate and support the collaborative learning of search skills. A key part of this support is a mechanism for recording an interaction history and providing students with a visualisation of that history that they can reflect and comment upon

Inverse Spectral Theory for Sturm-Liouville Operators with Distributional Potentials

We discuss inverse spectral theory for singular differential operators on arbitrary intervals $(a,b) \subseteq \mathbb{R}$ associated with rather general differential expressions of the type $$\tau f = \frac{1}{r} \left(- \big(p[f' + s f]\big)' + s p[f' + s f] + qf\right),$$ where the coefficients $p$, $q$, $r$, $s$ are Lebesgue measurable on $(a,b)$ with $p^{-1}$, $q$, $r$, $s \in L^1_{\text{loc}}((a,b); dx)$ and real-valued with $p\not=0$ and $r>0$ a.e.\ on $(a,b)$. In particular, we explicitly permit certain distributional potential coefficients. The inverse spectral theory results derived in this paper include those implied by the spectral measure, by two-spectra and three-spectra, as well as local Borg-Marchenko-type inverse spectral results. The special cases of Schr\"odinger operators with distributional potentials and Sturm--Liouville operators in impedance form are isolated, in particular.Comment: 29 page

Inverse Spectral Problems for Schr\"odinger-Type Operators with Distributional Matrix-Valued Potentials

The principal purpose of this note is to provide a reconstruction procedure for distributional matrix-valued potential coefficients of Schr\"odinger-type operators on a half-line from the underlying Weyl-Titchmarsh function.Comment: 14 pages. arXiv admin note: text overlap with arXiv:1206.496

Improbable nationalists? Social democracy and national independence in Georgia 1918-1921

The Democratic Republic of Georgia (1918-1921) had the world’s first ever elected social-democratic government. However, despite attempts then and later to present it as an exemplar of democratic socialism, the main tasks its leaders faced, in the context of post-imperial revolutionary chaos, were those of nation-building and consolidation. This necessarily led the state into conflict with its neighbours and national minorities, and to adopt a series of measures and compromises which both undermined the socialist aspects of the regime and fostered nationalism and even chauvinism, against the intentions and ideology of its rulers. This in turn weakened the regime, facilitating its eventual overthrow by the Red Army

A Life Skills Toolkit: Curriculum Development for Sustainable Public Health Community Engagement

Introduction: Committee On Temporary Shelter (COTS) is a community organization that provides gateway housing opportunities to fourteen previously homeless veterans through its Canal Street program in Winooski, VT. Many of the residents struggle with physical fitness, poor nutrition, and mental illness, including PTSD. Research has shown that there is an increasing prevalence of overweight or obese veterans returning from service, and these individuals present a particular challenge to primary care physicians as their mental health issues are closely related to their level of fitness. It has been demonstrated that veterans often suffer from ingrained food insecurity, which negatively influences post-service eating behavior, and readjustment solutions are needed to ease reentry into civilian life. While literature recommendations exist outlining the important role of initiating easy-to-use exercise programs and the beneficial impact of exercise in a natural environment on veterans, there has been little research into more holistic approaches to improve the diminished quality of life impacting many individuals with PTSD. Recent literature shows decreased PTSD symptoms after a life skills intervention and that short-term nutritional education interventions have the capacity to favorably change eating behaviors in a low income population. Therefore, we decided that a comprehensive, yet personalized intervention was needed.https://scholarworks.uvm.edu/comphp_gallery/1211/thumbnail.jp

Holocene Vegetation, Climate, and Carbon History on Western Kodiak Island, Alaska

At Phalarope Pond, western Kodiak Island, a multidisciplinary study using pollen and spores, macrofossils, stable isotopes, and carbon accumulation provides the Holocene vegetation and climate history following the deglaciation that began over 16,000 cal years ago (yr BP) [years Before Present, as calibrated from 1950]. Following a cold and dry Younger Dryas, a warm and wet early Holocene was characterized by abundant ferns in a sedge tundra environment with maximum carbon accumulation, similar to high latitude peatlands globally. About 8,700 cal yr BP sedge and ferns declined and climate remained warm as drier conditions prevailed, limiting carbon sequestration. The abrupt shift in D/H (Deuterium/Hydrogen) isotopes of about 60 percent indicates a shift to cooler conditions or a more distal moisture source. Neoglaciation beginning about 3,700 cal yr BP is evident from increases in Artemisia, Empetrum and Betula, signifying cooler conditions, while Alnus declines, paralleling regional trends

A Unified Approach to Attractor Reconstruction

In the analysis of complex, nonlinear time series, scientists in a variety of disciplines have relied on a time delayed embedding of their data, i.e. attractor reconstruction. The process has focused primarily on heuristic and empirical arguments for selection of the key embedding parameters, delay and embedding dimension. This approach has left several long-standing, but common problems unresolved in which the standard approaches produce inferior results or give no guidance at all. We view the current reconstruction process as unnecessarily broken into separate problems. We propose an alternative approach that views the problem of choosing all embedding parameters as being one and the same problem addressable using a single statistical test formulated directly from the reconstruction theorems. This allows for varying time delays appropriate to the data and simultaneously helps decide on embedding dimension. A second new statistic, undersampling, acts as a check against overly long time delays and overly large embedding dimension. Our approach is more flexible than those currently used, but is more directly connected with the mathematical requirements of embedding. In addition, the statistics developed guide the user by allowing optimization and warning when embedding parameters are chosen beyond what the data can support. We demonstrate our approach on uni- and multivariate data, data possessing multiple time scales, and chaotic data. This unified approach resolves all the main issues in attractor reconstruction.Comment: 22 pages, revised version as submitted to CHAOS. Manuscript is currently under review. 4 Figures, 31 reference