Examining Servant Leadership Behaviors in Higher Education: Explorations of a Compassion-Based Leadership Model through the Lens of University Leaders

A capstone submitted in partial fulfillment of the requirements for the degree of Doctor of Education in the Ernst and Sara Lane Volgenau College of Education at Morehead State University by Karol A. D. Johansen on March 31, 2023

Generating random density matrices

We study various methods to generate ensembles of random density matrices of a fixed size N, obtained by partial trace of pure states on composite systems. Structured ensembles of random pure states, invariant with respect to local unitary transformations are introduced. To analyze statistical properties of quantum entanglement in bi-partite systems we analyze the distribution of Schmidt coefficients of random pure states. Such a distribution is derived in the case of a superposition of k random maximally entangled states. For another ensemble, obtained by performing selective measurements in a maximally entangled basis on a multi--partite system, we show that this distribution is given by the Fuss-Catalan law and find the average entanglement entropy. A more general class of structured ensembles proposed, containing also the case of Bures, forms an extension of the standard ensemble of structureless random pure states, described asymptotically, as N \to \infty, by the Marchenko-Pastur distribution.Comment: 13 pages in latex with 8 figures include

Geometry of sets of quantum maps: a generic positive map acting on a high-dimensional system is not completely positive

We investigate the set a) of positive, trace preserving maps acting on density matrices of size N, and a sequence of its nested subsets: the sets of maps which are b) decomposable, c) completely positive, d) extended by identity impose positive partial transpose and e) are superpositive. Working with the Hilbert-Schmidt (Euclidean) measure we derive tight explicit two-sided bounds for the volumes of all five sets. A sample consequence is the fact that, as N increases, a generic positive map becomes not decomposable and, a fortiori, not completely positive. Due to the Jamiolkowski isomorphism, the results obtained for quantum maps are closely connected to similar relations between the volume of the set of quantum states and the volumes of its subsets (such as states with positive partial transpose or separable states) or supersets. Our approach depends on systematic use of duality to derive quantitative estimates, and on various tools of classical convexity, high-dimensional probability and geometry of Banach spaces, some of which are not standard.Comment: 34 pages in Latex including 3 figures in eps, ver 2: minor revision

Spectral density of generalized Wishart matrices and free multiplicative convolution

We investigate the level density for several ensembles of positive random matrices of a Wishart--like structure, $W=XX^{\dagger}$, where $X$ stands for a nonhermitian random matrix. In particular, making use of the Cauchy transform, we study free multiplicative powers of the Marchenko-Pastur (MP) distribution, ${\rm MP}^{\boxtimes s}$, which for an integer $s$ yield Fuss-Catalan distributions corresponding to a product of $s$ independent square random matrices, $X=X_1\cdots X_s$. New formulae for the level densities are derived for $s=3$ and $s=1/3$. Moreover, the level density corresponding to the generalized Bures distribution, given by the free convolution of arcsine and MP distributions is obtained. We also explain the reason of such a curious convolution. The technique proposed here allows for the derivation of the level densities for several other cases.Comment: 10 latex pages including 4 figures, Ver 4, minor improvements and references updat

CP^n, or, entanglement illustrated

We show that many topological and geometrical properties of complex projective space can be understood just by looking at a suitably constructed picture. The idea is to view CP^n as a set of flat tori parametrized by the positive octant of a round sphere. We pay particular attention to submanifolds of constant entanglement in CP^3 and give a few new results concerning them.Comment: 28 pages, 9 figure

Temperature dependence of the resonance and low energy spin excitations in superconducting FeTe0.6_{0.6}Se0.4_{0.4}

We use inelastic neutron scattering to study the temperature dependence of the low-energy spin excitations in single crystals of superconducting FeTe$_{0.6}$Se$_{0.4}$ ($T_c=14$ K). In the low-temperature superconducting state, the imaginary part of the dynamic susceptibility at the electron and hole Fermi surfaces nesting wave vector $Q=(0.5,0.5)$, $\chi^{\prime\prime}(Q,\omega)$, has a small spin gap, a two-dimensional neutron spin resonance above the spin gap, and increases linearly with increasing $\hbar\omega$ for energies above the resonance. While the intensity of the resonance decreases like an order parameter with increasing temperature and disappears at temperature slightly above $T_c$, the energy of the mode is weakly temperature dependent and vanishes concurrently above $T_c$. This suggests that in spite of its similarities with the resonance in electron-doped superconducting BaFe$_{2-x}$(Co,Ni)$_x$As$_2$, the mode in FeTe$_{0.6}$Se$_{0.4}$ is not directly associated with the superconducting electronic gap.Comment: 7 pages, 6 figure

Fast chromatin immunoprecipitation assay

Chromatin immunoprecipitation (ChIP) is a widely used method to explore in vivo interactions between proteins and DNA. The ChIP assay takes several days to complete, involves several tube transfers and uses either phenol–chlorophorm or spin columns to purify DNA. The traditional ChIP method becomes a challenge when handling multiple samples. We have developed an efficient and rapid Chelex resin-based ChIP procedure that dramatically reduces time of the assay and uses only a single tube to isolate PCR-ready DNA. This method greatly facilitates the probing of chromatin changes over many time points with several antibodies in one experiment