Estimates of the first Dirichlet eigenvalue from exit time moment spectra

We compute the first Dirichlet eigenvalue of a geodesic ball in a rotationally symmetric model space in terms of the moment spectrum for the Brownian motion exit times from the ball. This expression implies an estimate as exact as you want for the first Dirichlet eigenvalue of a geodesic ball in these rotationally symmetric spaces, including the real space forms of constant curvature. As an application of the model space theory we prove lower and upper bounds for the first Dirichlet eigenvalues of extrinsic metric balls in submanifolds of ambient Riemannian spaces which have model space controlled curvatures. Moreover, from this general setting we thereby obtain new generalizations of the classical and celebrated results due to McKean and Cheung--Leung concerning the fundamental tones of Cartan-Hadamard manifolds and the fundamental tones of submanifolds with bounded mean curvature in hyperbolic spaces, respectively.Comment: 23 pages. arXiv admin note: text overlap with arXiv:1009.125

Small Dollar Loans, Big Problems: How States Protect Consumers From Abuses and How the Federal Government Can Help

Across America, drivers pass twice as many payday loan storefronts as Starbucks coffee shops.2 In twenty-nine states, there are more payday lender stores than McDonald’s restaurants.3 Numerous research studies warn of the dangers associated with payday loans, including significantly higher rates of bankruptcies, evictions, utility shut-offs, and involuntary bank account closures.4 Many states have recognized the dangers posed by payday and other types of small-dollar loans with predatory features, prompting them to adopt laws to combat the abusive nature of these loans. These laws, however, offer consumers varying degrees of protection. Historically, states have used their police powers to protect consumers from predatory lending. This Article discusses the extent to which each state’s current laws protect consumers from lending abuses associated with four common small-dollar loans: payday loans, auto-title loans, six-month installment loans, and one-year installment loans.5 Specifically, this Article highlights the findings from the 2010 Small Dollar Loan Products Scorecard (Scorecard), which updated the original 2008 Scorecard. 6 Both the 2008 and 2010 Scorecard grade state laws based on the maximum annual percentage rate (APR) they allow for the four typical small-dollar loan products listed above. Since the 2008 Scorecard, there has been significant state legislative activity across the country related to small-dollar loans. Only a handful of states, however, have enacted new measures that adequately protect consumers. This Article provides policy recommendations to guide ongoing reform efforts. The Article highlights three key points. First, states should continue their longstanding good fight on behalf of American families against abusive, small dollar lending, but they need help. Congress and the Consumer Financial Protection Bureau (CFPB), which President Obama established when he signed the Dodd-Frank Wall Street Reform and Consumer Protection Act into law on July 21, 2010, should join the battle.7 Second, the states and Congress should focus their reform efforts on enacting an across-the-board usury cap of 36% APR on all small-dollar loans. Third, the states, CFPB, and Congress should impose several restrictions on high-cost (over 36% APR), small-dollar lending to help curb its abusive nature. In this Article, Part II describes the methodology used by the 2010 Scorecard. Part III reports the major changes that have occurred in the two years since the Scorecard’s original 2008 publication. Finally, Part IV proposes several policy recommendations, at the state and federal level, with the focus in the latter category on opportunities for action by the newly created CFPB

The classification of complete stable area-stationary surfaces in the Heisenberg group H1\mathbb{H}^1

We prove that any $C^2$ complete, orientable, connected, stable area-stationary surface in the sub-Riemannian Heisenberg group $\mathbb{H}^1$ is either a Euclidean plane or congruent to the hyperbolic paraboloid $t=xy$.Comment: 32 pages, no figures, added reference missed in version

Parabolicity criteria and characterization results for submanifoldsof bounded mean curvature in model manifolds with weights

Let P be a submanifold properly immersed in a rotationally symmetric manifold having a pole and endowed with a weight e h. The aim of this paper is twofold. First, by assuming certain control on the h-mean curvature of P, we establish comparisons for the h-capacity of extrinsic balls in P, from which we deduce criteria ensuring the h-parabolicity or h-hyperbolicity of P. Second, we employ functions with geometric meaning to describe submanifolds of bounded h-mean curvature which are confined into some regions of the ambient manifold. As a consequence, we derive half-space and Bernstein-type theorems generalizing previous ones. Our results apply for some relevant h-minimal submanifolds appearing in the singularity theory of the mean curvature flow

Geometric analysis of Lorentzian distance function on spacelike hypersurfaces

Some analysis on the Lorentzian distance in a spacetime with controlled sectional (or Ricci) curvatures is done. In particular, we focus on the study of the restriction of such distance to a spacelike hypersurface satisfying the Omori-Yau maximum principle. As a consequence, and under appropriate hypotheses on the (sectional or Ricci) curvatures of the ambient spacetime, we obtain sharp estimates for the mean curvature of those hypersurfaces. Moreover, we also give a suficient condition for its hyperbolicity.Comment: Final version (January 2009). To appear in the Transactions of the American Mathematical Societ

Power and wavelength polarization bistability with very wide hysteresis cycles in a 1550nm-VCSEL subject to orthogonal optical injection

We have measured optical power and wavelength polarization bistability in a 1550nm-Vertical Cavity Surface-Emitting Laser (VCSEL) subject to orthogonally-polarized optical injection into the orthogonal polarization of the fundamental transverse mode. Optical bistability with very wide hysteresis cycles, up to four times wider than previously reported results has been measured for both the optical power and wavelength domain. We also report the experimental observation of three different shapes of polarization bistability, anticlockwise, clockwise and X-Shape bistability, all of them with wide hysteresis cycles. This rich variety of behaviour at the important wavelength of 1550 nm offers promise for the use of VCSELs for all-optical signal processing and optical switching/routing applications. ©2009 Optical Society of America

Ashes at Noon: Stories and poems about Quito, Ecuador

Ashes at Noon aims to pay tribute to the culture that raised me. I want to explore where I’m from, the narratives that made me who I am, and honor the environment of the Andes and all the ghosts that reside in it. My collection is anticolonial; it explores the pre-Columbian traditions we still abide to and the curuchupa—extremely Catholic, judgmental, prejudicial—and racist mindsets we’re taught to follow