A soothing invisible hand: moderation potentials in optimal control

A moderation incentive is a continuously differentiable control-dependent cost term that is identically zero on the boundary of the admissible control region, and is subtracted from the `do or die' cost function to reward sub-maximal control utilization in optimal control systems. A moderation potential is a function on the cotangent bundle of the state space such that the solutions of Hamilton's equations satisfying appropriate boundary conditions are solutions of the synthesis problem - the control-parametrized Hamiltonian system central to Pontryagin's Maximum Principle. A multi-parameter family of moderation incentives for affinely controlled systems with quadratic control constraints possesses simple, readily calculated moderation potentials. One member of this family is a shifted version of the kinetic energy-style control cost term frequently used in geometric optimal control. The controls determined by this family approach those determined by a logarithmic penalty function as one of the parameters approaches zero, while the cost term itself is bounded.Comment: 26 pages, 6 figure

Relative Critical Points

Relative equilibria of Lagrangian and Hamiltonian systems with symmetry are critical points of appropriate scalar functions parametrized by the Lie algebra (or its dual) of the symmetry group. Setting aside the structures - symplectic, Poisson, or variational - generating dynamical systems from such functions highlights the common features of their construction and analysis, and supports the construction of analogous functions in non-Hamiltonian settings. If the symmetry group is nonabelian, the functions are invariant only with respect to the isotropy subgroup of the given parameter value. Replacing the parametrized family of functions with a single function on the product manifold and extending the action using the (co)adjoint action on the algebra or its dual yields a fully invariant function. An invariant map can be used to reverse the usual perspective: rather than selecting a parametrized family of functions and finding their critical points, conditions under which functions will be critical on specific orbits, typically distinguished by isotropy class, can be derived. This strategy is illustrated using several well-known mechanical systems - the Lagrange top, the double spherical pendulum, the free rigid body, and the Riemann ellipsoids - and generalizations of these systems

Geometric integration on spheres and some interesting applications

Geometric integration theory can be employed when numerically solving ODEs or PDEs with constraints. In this paper, we present several one-step algorithms of various orders for ODEs on a collection of spheres. To demonstrate the versatility of these algorithms, we present representative calculations for reduced free rigid body motion (a conservative ODE) and a discretization of micromagnetics (a dissipative PDE). We emphasize the role of isotropy in geometric integration and link numerical integration schemes to modern differential geometry through the use of partial connection forms; this theoretical framework generalizes moving frames and connections on principal bundles to manifolds with nonfree actions.Comment: This paper appeared in prin

Stability Properties of the Riemann Ellipsoids

We study the ellipticity and the ``Nekhoroshev stability'' (stability properties for finite, but very long, time scales) of the Riemann ellipsoids. We provide numerical evidence that the regions of ellipticity of the ellipsoids of types II and III are larger than those found by Chandrasekhar in the 60's and that all Riemann ellipsoids, except a finite number of codimension one subfamilies, are Nekhoroshev--stable. We base our analysis on a Hamiltonian formulation of the problem on a covering space, using recent results from Hamiltonian perturbation theory.Comment: 29 pages, 6 figure

An Exploration of Practicum Students\u27 Experiences of Meaning-Making Through Altruism

Finding meaning in one’s work as a counselor has been demonstrated as an important step in the development of altruism, an essential component of counselor effectiveness. Previous studies in counselor education-related research involving program outcomes focus on the core skills of counseling such as knowledge, skill building, self-appraisal and self-efficacy. Yet little investigation has concentrated on the internal rewards of the clinical experience, such as the meaning found in or the altruism development derived specifically from the practicum or internship. This dissertation research took a phenomenological approach to explore the meaning-making and altruism development of counselor education practicum students providing social and emotional support to adolescents identified as at-risk. Data were collected through semi-structured interviews to understand former practicum students’ conceptualization of meaning through work and thoughts related to personal altruism development. The themes illuminated through the study suggest that students found challenges in the experience, were able to collaborate toward successful outcomes, related the impact of the experience to working with at-risk adolescents, were able to describe the personal meaning derived from the practicum, and reflected on personal altruism development. Implications for counselor educators and supervisors and suggestions for future research are provided

Self-Efficacy Perceptions of Middle School Reading Teachers in Majority Minority Inclusive Classrooms on their Ability to Achieve Students Success.

According to the U.S. Department of Education National Center for Education Statistics (NCES, 2022), the number of students who require special accommodations in the classroom continues to increase. Because studies have shown that positive outcomes for students are directly linked to the self-efficacy of educators (Lotter et al., 2018; Neugebauer et al., 2019), educators who serve these populations must possess high levels of positive self-efficacy to handle the challenges associated with inclusive settings and specialized skills needed to achieve student success. The purpose of this study, which was guided and supported by the Social Cognitive Theory (Bandura, 1986), was to examine teacher self-efficacy perceptions of their ability to achieve student success in majority-minority inclusive classrooms and analyze the relationships between self-efficacy perceptions and age, gender, race, years of experience, level of education, and certification status of the participants. In this quantitative study, the research design was Correlational. The researcher investigated middle school reading teachers\u27 perceptions of their ability to achieve success. Data was collected via an online survey of the Teacher Self-Efficacy Scale (Bandura, 1999), a valid instrument, coded and analyzed via SPSS utilizing the Multiple Regression statistical model. Data results showed significant relationships between personal demographic independent variables, age, gender, and race, (p\u3c.001) and combined personal and social demographic predictor variables, age, gender, race, years of experience, level of education, and special education certification status, (p\u3c.001) and the dependent variable, teacher self-efficacy perceptions of their ability to achieve student success in majority-minority inclusive classrooms