Counting fixed points and rooted closed walks of the singular map x↦xxnx \mapsto x^{x^n} modulo powers of a prime

The "self-power" map $x \mapsto x^x$ modulo $m$ and its generalized form $x \mapsto x^{x^n}$ modulo $m$ are of considerable interest for both theoretical reasons and for potential applications to cryptography. In this paper, we use $p$-adic methods, primarily $p$-adic interpolation, Hensel's lemma, and lifting singular points modulo $p$, to count fixed points and rooted closed walks of equations related to these maps when $m$ is a prime power. In particular, we introduce a new technique for lifting singular solutions of several congruences in several unknowns using the left kernel of the Jacobian matrix.Comment: 18 pages. Version 2 shortens proofs, reduces redundancy, and introduces new technique for counting rooted closed walks. Version 3 updates title to agree with journal publicatio

Estimation from quantized Gaussian measurements: when and how to use dither

Subtractive dither is a powerful method for removing the signal dependence of quantization noise for coarsely quantized signals. However, estimation from dithered measurements often naively applies the sample mean or midrange, even when the total noise is not well described with a Gaussian or uniform distribution. We show that the generalized Gaussian distribution approximately describes subtractively dithered, quantized samples of a Gaussian signal. Furthermore, a generalized Gaussian fit leads to simple estimators based on order statistics that match the performance of more complicated maximum likelihood estimators requiring iterative solvers. The order statistics-based estimators outperform both the sample mean and midrange for nontrivial sums of Gaussian and uniform noise. Additional analysis of the generalized Gaussian approximation yields rules of thumb for determining when and how to apply dither to quantized measurements. Specifically, we find subtractive dither to be beneficial when the ratio between the Gaussian standard deviation and quantization interval length is roughly less than one-third. When that ratio is also greater than 0.822/K^0.930 for the number of measurements K > 20, estimators we present are more efficient than the midrange.https://arxiv.org/abs/1811.06856Accepted manuscrip

Pinch Keyboard: Natural Text Input for Immersive Virtual Environments

Text entry may be needed for system control tasks in immersive virtual environments, but no efficient and usable techniques exist. We present the pinch keyboard interaction technique, which simulates a standard QWERTY keyboard using Pinch Gloves™ and 6 DOF trackers. The system includes visual and auditory feedback and a simple method of calibration

Rationalizing Noneconomic Damages: A Health-Utilities Approach

Studdert et al examine why making compensation of noneconomic damages in personal-injury litigation more rational and predictable is socially valuable. Noneconomic-damages schedules as an alternative to caps are discussed, several potential approaches to construction of schedules are reviewed, and the use of a health-utilities approach as the most promising model is argued. An empirical analysis that combines health-utilities data created in a previous study with original empirical work is used to demonstrate how key steps in construction of a health-utilities-based schedule for noneconomic damages might proceed

USING RIGHTS OF FIRST REFUSAL FOR FARMLAND RETENTION

How can rights of first refusal protect prime agricultural land? This paper develops a theory for valuing rights of first refusal based on compensation for foreclosing future demand, information asymmetry, and advance purchase of market share. A procedure is developed for governments to use these rights to prevent conversion.Land Economics/Use,