Quaternionic R transform and non-hermitian random matrices

Using the Cayley-Dickson construction we rephrase and review the non-hermitian diagrammatic formalism [R. A. Janik, M. A. Nowak, G. Papp and I. Zahed, Nucl.Phys. B $\textbf{501}$, 603 (1997)], that generalizes the free probability calculus to asymptotically large non-hermitian random matrices. The main object in this generalization is a quaternionic extension of the R transform which is a generating function for planar (non-crossing) cumulants. We demonstrate that the quaternionic R transform generates all connected averages of all distinct powers of $X$ and its hermitian conjugate $X^\dagger$: \langle\langle \frac{1}{N} \mbox{Tr} X^{a} X^{\dagger b} X^c \ldots \rangle\rangle for $N\rightarrow \infty$. We show that the R transform for gaussian elliptic laws is given by a simple linear quaternionic map $\mathcal{R}(z+wj) = x + \sigma^2 \left(\mu e^{2i\phi} z + w j\right)$ where $(z,w)$ is the Cayley-Dickson pair of complex numbers forming a quaternion $q=(z,w)\equiv z+ wj$. This map has five real parameters $\Re e x$, $\Im m x$, $\phi$, $\sigma$ and $\mu$. We use the R transform to calculate the limiting eigenvalue densities of several products of gaussian random matrices.Comment: 27 pages, 16 figure

Commutative law for products of infinitely large isotropic random matrices

Ensembles of isotropic random matrices are defined by the invariance of the probability measure under the left (and right) multiplication by an arbitrary unitary matrix. We show that the multiplication of large isotropic random matrices is spectrally commutative and self-averaging in the limit of infinite matrix size $N \rightarrow \infty$. The notion of spectral commutativity means that the eigenvalue density of a product ABC... of such matrices is independent of the order of matrix multiplication, for example the matrix ABCD has the same eigenvalue density as ADCB. In turn, the notion of self-averaging means that the product of n independent but identically distributed random matrices, which we symbolically denote by AAA..., has the same eigenvalue density as the corresponding power A^n of a single matrix drawn from the underlying matrix ensemble. For example, the eigenvalue density of ABCCABC is the same as of A^2B^2C^3. We also discuss the singular behavior of the eigenvalue and singular value densities of isotropic matrices and their products for small eigenvalues $\lambda \rightarrow 0$. We show that the singularities at the origin of the eigenvalue density and of the singular value density are in one-to-one correspondence in the limit $N \rightarrow \infty$: the eigenvalue density of an isotropic random matrix has a power law singularity at the origin $\sim |\lambda|^{-s}$ with a power $s \in (0,2)$ when and only when the density of its singular values has a power law singularity $\sim \lambda^{-\sigma}$ with a power $\sigma = s/(4-s)$. These results are obtained analytically in the limit $N \rightarrow \infty$. We supplement these results with numerical simulations for large but finite N and discuss finite size effects for the most common ensembles of isotropic random matrices.Comment: 15 pages, 4 figure

Verification theorem and construction of epsilon-optimal controls for control of abstract evolution equations

We study several aspects of the dynamic programming approach to optimal control of abstract evolution equations, including a class of semilinear partial differential equations. We introduce and prove a verification theorem which provides a sufficient condition for optimality. Moreover we prove sub- and superoptimality principles of dynamic programming and give an explicit construction of $\epsilon$-optimal controls.optimal control of PDE; verification theorem; dynamic programming; $\epsilon$-optimal controls; Hamilton-Jacobi-Bellman equations

New spectral relations between products and powers of isotropic random matrices

We show that the limiting eigenvalue density of the product of n identically distributed random matrices from an isotropic unitary ensemble (IUE) is equal to the eigenvalue density of n-th power of a single matrix from this ensemble, in the limit when the size of the matrix tends to infinity. Using this observation one can derive the limiting density of the product of n independent identically distributed non-hermitian matrices with unitary invariant measures. In this paper we discuss two examples: the product of n Girko-Ginibre matrices and the product of n truncated unitary matrices. We also provide an evidence that the result holds also for isotropic orthogonal ensembles (IOE).Comment: 8 pages, 3 figures (in version 2 we added a figure and discussion on finite size effects for isotropic orthogonal ensemble

Fostering Resilience In Youth Through Positive Youth-Adult Relationships

Research examining adolescents has found that close, healthy relationships with supportive and caring adults can help foster resilience in youth. Resilience, as defined by Ann Masten (2001), is the “phenomena characterized as good outcomes in spite of serious threats to adaptation or development.” Promoting resilience in a school setting can help decrease at-risk behaviors such as truancy, substance use, and suicidal ideation/attempts. Kaleidoscope Connect is a program focusing on establishing social-emotional skills and resilience in youth and specifically targets youth-adult relationships. This research project examines 28 7th and 8th grade students who participated in the Kaleidoscope Connect program in 2016-2017 in a rural middle school in Western Montana. Two standardized self-report rating scales will be used to examine the levels of resilience (Resiliency Scales for Children and Adolescents) and problem behavior (Behavioral and Emotional Screening System) in these students. In addition, this study also examined data from the Student Support Card Surveys. One of these surveys asked students to report their number of anchors (relationships with adults) and their proximity to these individuals. Another survey specifically examined the level of support that they received from each of these anchors. The goal of this project is to examine the relationship between the presence of supportive adults in a student’s life and their risk level for problem behaviors. Due to the small sample size, descriptive analyses will be conducted to examine the results. Supportive and caring relationships with adults helps build resilience in children, with these relationships often occurring in the school setting. This research aims to highlight the importance of building positive youth-adult connections, which is critical due to an increase in at-risk youth and mental health needs during the pandemic

The Effects of Instructors Discussing Alcohol In The Classroom on Student Perceptions of Their Instructor

This thesis is comprised of two studies. Study I is a qualitative exploratory analysis which attempts to uncover students’ experiences with instructors discussing alcohol in the university classroom and how the appropriateness of this behavior is determined. How this mention of alcohol affects the students’ perception of the instructor, as well as the relationship between perceived appropriateness of the behavior and change in perception, are also investigated. Study I found that students have experienced their instructors mentioning alcohol in the classroom as normative student behavior, part of the formal curriculum, and their instructor disclosing personal alcohol use. Participants determined appropriateness based on perceived relevance to students or class. Study II sought to determine out how instructors discussing alcohol in the classroom affects the way students perceive them. Specifically, Study II examines how an instructor mentioning alcohol while advocating for safe drinking behavior, discussing their personal alcohol use behaviors, alcohol as part of the curriculum, student future alcohol use, and student past alcohol use affect rapport, homophily, and credibility. The data indicated that instructors advocating for student safe drinking had the most positive impact on all three measures, whereas instructors who discussed their personal drinking behaviors in the classroom had significantly lower scores on rapport, homophily, and credibility