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

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

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

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