Barrett-Johnson inequalities for totally nonnegative matrices

Given a matrix $A$, let $A_{I,J}$ denote the submatrix of $A$ determined by rows $I$ and columns $J$. Fischer's Inequalities state that for each $n \times n$ Hermitian positive semidefinite matrix $A$, and each subset $I$ of $\{1,\dotsc,n\}$ and its complement $I^c$, we have $\det(A) \leq \det(A_{I,I})\det(A_{I^c,I^c})$. Barrett and Johnson (Linear Multilinear Algebra 34, 1993) extended these to state inequalities for sums of products of principal minors whose orders are given by nonincreasing integer sequences $(\lambda_1,\dotsc,\lambda_r)$, $(\mu_1,\dotsc,\mu_s)$ summing to $n$. Specifically, if $\lambda_1+\cdots+\lambda_i\leq \mu_1+\cdots+\mu_i$ for all $i$, then $$\lambda_1!\cdots\lambda_r! \sum_{(I_1,\dotsc,I_r)} \det(A_{I_1,I_1}) \cdots \det(A_{I_r,I_r}) ~\geq~ \mu_1!\cdots\mu_s! \sum_{(J_1,\dotsc,J_s)} \det(A_{J_1,J_1}) \cdots \det(A_{J_s,J_s}),$$ where sums are over sequences of disjoint subsets of $\{1,\dotsc,n\}$ satisfying $|I_k| = \lambda_k$, $|J_k| = \mu_k$. We show that these inequalities hold for totally nonnegative matrices as well

Polarization singularity anisotropy: determining monstardom

C points, that is isolated points of circular polarization in transverse fields of varying polarization, are classified morphologically into three distinct types, known as lemons, stars and monstars. These morphologies are interpreted here according to two natural parameters associated with the singularity, namely the anisotropy of the C point, and the polarization azimuth on the anisotropy axis. In addition to providing insight into singularity morphology, this observation applies to the densities of the various morphologies in isotropic random polarization speckle fields.Comment: 3 pages, 2 figures, Optics Letters styl

Singular optics and topological photonics

For centuries, singularities in wave fields have been a mainstay of interest in multiple areas of physics, ranging from plasma physics, fluid dynamics and atmospheric physics to optics and photonics [1]. In optics, energy and momentum flow around singularities in the interference field can twist to form vortices, which carry angular momentum [2–4]. Singular Optics is the branch of physics that encompasses studies of structured light with localized and extended singularities, such as optical vortices in scalar optical fields and polarization singularities in vector fields [2, 5, 6]. It has been pioneered by the publication of a seminal paper of Nye and Berry 'Dislocations in wave trains' in 1974 [7], and has recently celebrated its 40th anniversary

Roadmap on structured light

Structured light refers to the generation and application of custom light fields. As the tools and technology to create and detect structured light have evolved, steadily the applications have begun to emerge. This roadmap touches on the key fields within structured light from the perspective of experts in those areas, providing insight into the current state and the challenges their respective fields face. Collectively the roadmap outlines the venerable nature of structured light research and the exciting prospects for the future that are yet to be realized

Local Government-Supported Community Development: Community Priorities And Issues Of Autonomy

There is increasing support for local solutions to poverty through community-based organizations (CBOs). However, a dilemma remains: How can CBOs secure resources necessary for change and yet maintain autonomy in definition of development priorities and delivery strategies? The authors examine a community-development model used in central Florida that includes local government support in the formation and activities of a CBO, and they explore the threat to community autonomy associated with differences in development priorities between community-based and external forces. The results provide encouraging evidence that development models that include cooperation between local government and low-income communities do not necessarily produce sublimation of community priorities. © 1996 Sage Publications, Inc

The Impact Of Day Trips To Daytona Beach

Coastal areas and other tourist destinations are popular with \u27day trippers\u27, who have a substantial effect on local tourism economies and may help to stabilize seasonal demand fluctuations. However, day trippers can also accelerate resource degradation, burden municipalities with higher policing and maintenance costs, and create image problems. Daytona Beach is a destination for which day tripper visitation generates both positive and negative effects which local policy makers need to assess. Analyses of what influences beach area selection, frequency of visitation and party size, and annual per capita spending are used to shed light on public policy issues such as restricting beach driving and an over-dependence on events-based tourism