Having a Heart for ALL Students

The student body in today’s general educational classrooms continues to increase in diversity, with many of the students being English language learners. This growth in the number of non-English speaking students brings challenges to educators. Students’ test scores are the measurement of success for educators, resulting in a decrease of caring about students as individuals. This article addresses the importance and responsibility for Christian educators to reach out to non-English speaking students

Observational Learning with Position Uncertainty

Observational learning is typically examined when agents have precise information about their position in the sequence of play. We present a model in which agents are uncertain about their positions. Agents are allowed to have arbitrary ex-ante beliefs about their positions: they may observe their position perfectly, imperfectly, or not at all. Agents sample the decisions of past individuals and receive a private signal about the state of the world. We show that social learning is robust to position uncertainty. Under any sampling rule satisfying a stationarity assumption, learning is complete if signal strength is unbounded. In cases with bounded signal strength, we show that agents achieve what we define as constrained efficient learning: individuals do at least as well as the most informed agent would do in isolation.social learning; information aggregation; herds; position uncertainty; observational learning

Invisibility and PT Symmetry: A Simple Geometrical Viewpoint

We give a simplified account of the properties of the transfer matrix for a complex one-dimensional potential, paying special attention to the particular instance of unidirectional invisibility. In appropriate variables, invisible potentials appear as performing null rotations, which lead to the helicity-gauge symmetry of massless particles. In hyperbolic geometry, this can be interpreted, via M\"{o}bius transformations, as parallel displacements, a geometric action that has no Euclidean analogy.Comment: 13 pages. No figure. Accepted for publication in Symmetr

Sliderule-like property of Wigner's little groups and cyclic S-matrices for multilayer optics

It is noted that two-by-two S-matrices in multilayer optics can be represented by the Sp(2) group whose algebraic property is the same as the group of Lorentz transformations applicable to two space-like and one time-like dimensions. It is noted also that Wigner's little groups have a sliderule-like property which allows us to perform multiplications by additions. It is shown that these two mathematical properties lead to a cyclic representation of the S-matrix for multilayer optics, as in the case of ABCD matrices for laser cavities. It is therefore possible to write the N-layer S-matrix as a multiplication of the N single-layer S-matrices resulting in the same mathematical expression with one of the parameters multiplied by N. In addition, it is noted, as in the case of lens optics, multilayer optics can serve as an analogue computer for the contraction of Wigner's little groups for internal space-time symmetries of relativistic particles.Comment: RevTex 13 pages, Secs. IV and V revised and expande