3,785 research outputs found
Bounds for extreme zeros of some classical orthogonal polynomials
We derive upper bounds for the smallest zero and lower bounds for the largest
zero of Laguerre, Jacobi and Gegenbauer polynomials. Our approach uses mixed
three term recurrence relations satisfied by polynomials corresponding to
different parameter(s) within the same classical family. We prove that
interlacing properties of the zeros impose restrictions on the possible
location of common zeros of the polynomials involved and deduce strict bounds
for the extreme zeros of polynomials belonging to each of these three classical
families. We show numerically that the bounds generated by our method improve
known lower (upper) bounds for the largest (smallest) zeros of polynomials in
these families, notably in the case of Jacobi and Gegenbauer polynomials
Understanding Equitable Assessment: How Preservice Teachers Make Meaning of DisAbility
Disproportionality of historically marginalized populations in special education continues to be a critical concern. The identification of students with disabilities is reliant on valid and reliable assessment that is free of bias. The extent to which this is possible given measurement constraints and an increasingly diverse student population is unclear. How teachers are trained to design, select, administer, score, and interpret assessment data related to the identification of students with disabilities is vastly under-researched considering the significant implications of assessment practices. In this study, six special education preservice teachers engaged in an assessment methods course during their second semester of an initial certification program. This study focuses on shifts in preservice teacher understanding and the associated learning experiences in the course. Findings from this study have the potential to inform general and special education teacher preparation coursework
Finite dimensional approximations to Wiener measure and path integral formulas on manifolds
Certain natural geometric approximation schemes are developed for Wiener
measure on a compact Riemannian manifold. These approximations closely mimic
the informal path integral formulas used in the physics literature for
representing the heat semi-group on Riemannian manifolds. The path space is
approximated by finite dimensional manifolds consisting of piecewise geodesic
paths adapted to partitions of . The finite dimensional manifolds of
piecewise geodesics carry both an and a type Riemannian
structures . It is proved that as the mesh of the partition tends to
,
where is the energy of the piecewise geodesic path , and
for and , is a ``normalization'' constant, is
the Riemannian volume form relative , and is Wiener measure on
paths on . Here and
where is the scalar curvature of . These results are also shown to
imply the well know integration by parts formula for the Wiener measure.Comment: 48 pages, latex2e using amsart and amssym
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