Fractional Moment Estimates for Random Unitary Operators

We consider unitary analogs of $d-$dimensional Anderson models on $l^2(\Z^d)$ defined by the product $U_\omega=D_\omega S$ where $S$ is a deterministic unitary and $D_\omega$ is a diagonal matrix of i.i.d. random phases. The operator $S$ is an absolutely continuous band matrix which depends on parameters controlling the size of its off-diagonal elements. We adapt the method of Aizenman-Molchanov to get exponential estimates on fractional moments of the matrix elements of $U_\omega(U_\omega -z)^{-1}$, provided the distribution of phases is absolutely continuous and the parameters correspond to small off-diagonal elements of $S$. Such estimates imply almost sure localization for $U_\omega$

General Adiabatic Evolution with a Gap Condition

We consider the adiabatic regime of two parameters evolution semigroups generated by linear operators that are analytic in time and satisfy the following gap condition for all times: the spectrum of the generator consists in finitely many isolated eigenvalues of finite algebraic multiplicity, away from the rest of the spectrum. The restriction of the generator to the spectral subspace corresponding to the distinguished eigenvalues is not assumed to be diagonalizable. The presence of eigenilpotents in the spectral decomposition of the generator forbids the evolution to follow the instantaneous eigenprojectors of the generator in the adiabatic limit. Making use of superadiabatic renormalization, we construct a different set of time-dependent projectors, close to the instantaneous eigeprojectors of the generator in the adiabatic limit, and an approximation of the evolution semigroup which intertwines exactly between the values of these projectors at the initial and final times. Hence, the evolution semigroup follows the constructed set of projectors in the adiabatic regime, modulo error terms we control

Somatostatin Regulates Circadian Clock Function and Photic Processing

Daily and seasonal rhythms are programmed by neural circuits that use daily timing and duration of light to anticipate predictable environmental changes (i.e., day length, temperature, food, predation). Daily and annual changes in light modulate human health to produce both positive and negative effects, but neural mechanisms underlying light-driven changes in the brain remain poorly understood. In mammals, light is processed and encoded by the brain’s central clock, the suprachiasmatic nucleus (SCN). The SCN also encodes day length (i.e., photoperiod) to regulate annual fluctuations in mammalian physiology, but it’s not clear precisely how the SCN network achieves this. One signal that may contribute to SCN photoperiod encoding is the neuropeptide somatostatin (SST). In rodents, SST expression is modulated by photoperiod in hypothalamic regions regulated by the SCN, suggesting involvement of the central clock. The SCN expresses SST but its role in central clock function and photoperiodic encoding has not been examined. Here, using a range of genetic and imaging approaches, I demonstrate that SST signaling increases circadian robustness in a sexually dimorphic manner. First, I use cellular fate-mapping approaches to demonstrate that SCN SST is regulated by photoperiod in a manner that suggests de novo Sst transcription. Next, I use a battery of circadian behavioral assays to demonstrate that SST contributes to photoperiodic entrainment and circadian responses to light in a manner influenced by sex. However, lack of SST does not alter basic circadian properties, suggesting that SST signaling modulates specific circadian characteristics under particular conditions. Third, I demonstrate that SST regulates SCN neurochemistry via influence on neurons that mediate photic responses. Further, those same cells express a subtype of SST receptor capable of resetting molecular clock function. Last, I demonstrate that lack of SST enhances SCN photoperiodic encoding by modulating photic processing and network communication in a sex-dependent manner. Collectively, these results provide new insight into mechanisms that regulate seasonality and circadian clock function in mammals. The discovery of sexually divergent clock circuits may provide new insights relevant for understanding gender disparities in seasonal/circadian disease states

Smooth adiabatic evolutions with leaky power tails

Adiabatic evolutions with a gap condition have, under a range of circumstances, exponentially small tails that describe the leaking out of the spectral subspace. Adiabatic evolutions without a gap condition do not seem to have this feature in general. This is a known fact for eigenvalue crossing. We show that this is also the case for eigenvalues at the threshold of the continuous spectrum by considering the Friedrichs model.Comment: Final form, to appear in J. Phys. A; 11 pages, no figure

Localization for Random Unitary Operators

We consider unitary analogs of $1-$dimensional Anderson models on $l^2(\Z)$ defined by the product $U_\omega=D_\omega S$ where $S$ is a deterministic unitary and $D_\omega$ is a diagonal matrix of i.i.d. random phases. The operator $S$ is an absolutely continuous band matrix which depends on a parameter controlling the size of its off-diagonal elements. We prove that the spectrum of $U_\omega$ is pure point almost surely for all values of the parameter of $S$. We provide similar results for unitary operators defined on $l^2(\N)$ together with an application to orthogonal polynomials on the unit circle. We get almost sure localization for polynomials characterized by Verblunski coefficients of constant modulus and correlated random phases

Exponentially Accurate Semiclassical Tunneling Wave Functions in One Dimension

We study the time behavior of wave functions involved in tunneling through a smooth potential barrier in one dimension in the semiclassical limit. We determine the leading order component of the wave function that tunnels. It is exponentially small in $1/\hbar$. For a wide variety of incoming wave packets, the leading order tunneling component is Gaussian for sufficiently small $\hbar$. We prove this for both the large time asymptotics and for moderately large values of the time variable

Close to Uniform Prime Number Generation With Fewer Random Bits

In this paper, we analyze several variants of a simple method for generating prime numbers with fewer random bits. To generate a prime $p$ less than $x$, the basic idea is to fix a constant $q\propto x^{1-\varepsilon}$, pick a uniformly random $a<q$ coprime to $q$, and choose $p$ of the form $a+t\cdot q$, where only $t$ is updated if the primality test fails. We prove that variants of this approach provide prime generation algorithms requiring few random bits and whose output distribution is close to uniform, under less and less expensive assumptions: first a relatively strong conjecture by H.L. Montgomery, made precise by Friedlander and Granville; then the Extended Riemann Hypothesis; and finally fully unconditionally using the Barban-Davenport-Halberstam theorem. We argue that this approach has a number of desirable properties compared to previous algorithms.Comment: Full version of ICALP 2014 paper. Alternate version of IACR ePrint Report 2011/48

Is there a “Gestalt bias” in indulgence? Subjectively constructing food units into wholes (versus parts) increases desire to eat and actual consumption

In the present work we extend research into the unit bias effect and its extension—the portion size effect—by demonstrating the existence of a “Gestalt bias.” Drawing on the tenets of Gestalt psychology, we show that a unit bias effect can be observed for food portions that are composed of identical basic units, but which are subjectively grouped into, or perceived as a Gestalt—a larger whole. In three studies, we find that such subjectively constructed food wholes constitute a new (perceptual) unit that is perceived bigger than the units it is constructed from, thereby prompting increased eating and desire to eat

Correlated Markov Quantum Walks

We consider the discrete time unitary dynamics given by a quantum walk on $\Z^d$ performed by a particle with internal degree of freedom, called coin state, according to the following iterated rule: a unitary update of the coin state takes place, followed by a shift on the lattice, conditioned on the coin state of the particle. We study the large time behavior of the quantum mechanical probability distribution of the position observable in $\Z^d$ for random updates of the coin states of the following form. The random sequences of unitary updates are given by a site dependent function of a Markov chain in time, with the following properties: on each site, they share the same stationnary Markovian distribution and, for each fixed time, they form a deterministic periodic pattern on the lattice. We prove a Feynman-Kac formula to express the characteristic function of the averaged distribution over the randomness at time $n$ in terms of the nth power of an operator $M$. By analyzing the spectrum of $M$, we show that this distribution posesses a drift proportional to the time and its centered counterpart displays a diffusive behavior with a diffusion matrix we compute. Moderate and large deviations principles are also proven to hold for the averaged distribution and the limit of the suitably rescaled corresponding characteristic function is shown to satisfy a diffusion equation. An example of random updates for which the analysis of the distribution can be performed without averaging is worked out. The random distribution displays a deterministic drift proportional to time and its centered counterpart gives rise to a random diffusion matrix whose law we compute. We complete the picture by presenting an uncorrelated example.Comment: 37 pages. arXiv admin note: substantial text overlap with arXiv:1010.400

Exploring Principals’ Perceptions about Collective Efficacy in Urban K-8 New Jersey Schools Designated for the School Improvement Process

Leadership plays an important role in determining organizational culture and maintaining a commitment to the collective goal. Leaders not only play a critical role in developing and implementing policies to direct an organization toward a goal, but also serve as role models for those with whom they work. Research has shown that relationships between principals and teachers are critical to impact teaching practices, school climate, and students positively, yet these relationships can affect student learning and achievement. Research has also shown that students’ overall performance in school and any decrease in their overall achievement data are directly related to teacher self-efficacy (Hoy & Hoy, 2009). Prior to the designation of underperforming, a school’s data indicated academic deficits for 3 years or more. The public identifier as targeted and/or comprehensive may have further implications of reduced teacher and principal efficacy. The purpose of this qualitative narrative study was to explore principals’ perceptions about collective efficacy in urban K-8 schools in New Jersey designated for the school improvement process. Principals’ responses to the research question and sub-questions guided this study: 1. What are principals’ perceptions of community collective efficacy as a means towards school improvement? a. How do principals describe the value/possibility of community collective efficacy? b. How do principals describe the challenges of community collective efficacy? Collective efficacy in schools identified as in need of improvement is critical to enhancing student achievement. Significant findings revealed that collective efficacy contributes to improved student outcomes and can be fostered by effective leadership strategies. Teacher collective efficacy has also been shown to have a positive effect on the school environment for all parties involved in the educational process. Recommendations for districts included implementation of professional development, research-based and targeted interventions, and efficacious practices directly related to school improvement. Policy recommendations are relevant to teacher and leadership programs, including effectiveness and cultural programs. An avenue of future research is the investigation of whether increasing the DEI initiatives of K-12 schools leads to differences in student self-efficacy and teacher self-efficacy, which would, in turn, impact collective efficacy. An interesting concept is the idea that collective efficacy promotes inclusion. Recommendations for practice include an early warning indicator evaluative structure to support principals and schools at risk of being identified as in need of improvement. Keywords: efficacy, self-efficacy, comprehensive school, failing schools, principal perception, data-driven, school improvement, stigmatized, underperforming, school cultur