From Fibonacci Numbers to Central Limit Type Theorems

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers $\{F_n\}_{n=1}^{\infty}$. Lekkerkerker proved that the average number of summands for integers in $[F_n, F_{n+1})$ is $n/(\phi^2 + 1)$, with $\phi$ the golden mean. This has been generalized to the following: given nonnegative integers $c_1,c_2,...,c_L$ with $c_1,c_L>0$ and recursive sequence $\{H_n\}_{n=1}^{\infty}$ with $H_1=1$, $H_{n+1} =c_1H_n+c_2H_{n-1}+...+c_nH_1+1$ $(1\le n< L)$ and $H_{n+1}=c_1H_n+c_2H_{n-1}+...+c_LH_{n+1-L}$ $(n\geq L)$, every positive integer can be written uniquely as $\sum a_iH_i$ under natural constraints on the $a_i$'s, the mean and the variance of the numbers of summands for integers in $[H_{n}, H_{n+1})$ are of size $n$, and the distribution of the numbers of summands converges to a Gaussian as $n$ goes to the infinity. Previous approaches used number theory or ergodic theory. We convert the problem to a combinatorial one. In addition to re-deriving these results, our method generalizes to a multitude of other problems (in the sequel paper \cite{BM} we show how this perspective allows us to determine the distribution of gaps between summands in decompositions). For example, it is known that every integer can be written uniquely as a sum of the $\pm F_n$'s, such that every two terms of the same (opposite) sign differ in index by at least 4 (3). The presence of negative summands introduces complications and features not seen in previous problems. We prove that the distribution of the numbers of positive and negative summands converges to a bivariate normal with computable, negative correlation, namely $-(21-2\phi)/(29+2\phi) \approx -0.551058$.Comment: This is a companion paper to Kologlu, Kopp, Miller and Wang's On the number of summands in Zeckendorf decompositions. Version 2.0 (mostly correcting missing references to previous literature

Wilson Line Picture of Levin-Wen Partition Functions

Levin and Wen [Phys. Rev. B 71, 045110 (2005)] have recently given a lattice Hamiltonian description of doubled Chern-Simons theories. We relate the partition function of these theories to an expectation of Wilson loops that form a link in 2+1 dimensional spacetime known in the mathematical literature as Chain-Mail. This geometric construction gives physical interpretation of the Levin-Wen Hilbert space and Hamiltonian, its topological invariance, exactness under coarse-graining, and how two opposite chirality sectors of the doubled theory arise.Comment: Final published version; Appendix adde

Environmental Effects On Drosophila Brain Development And Learning

Brain development and behavior are sensitive to a variety of environmental influences including social interactions and physicochemical stressors. Sensory input in situ is a mosaic of both enrichment and stress, yet little is known about how multiple environmental factors interact to affect brain anatomical structures, circuits and cognitive function. In this study, we addressed these issues by testing the individual and combined effects of sub-adult thermal stress, larval density and early-adult living spatial enrichment on brain anatomy and olfactory associative learning in adult Drosophila melanogaster. In response to heat stress, the mushroom bodies (MBs) were the most volumetrically impaired among all of the brain structures, an effect highly correlated with reduced odor learning performance. However, MBs were not sensitive to either larval culture density or early-adult living conditions. Extreme larval crowding reduced the volume of the antennal lobes, optic lobes and central complex. Neither larval crowding nor early-adult spatial enrichment affected olfactory learning. These results illustrate that various brain structures react differently to environmental inputs, and that MB development and learning are highly sensitive to certain stressors (pre-adult hyperthermia) and resistant to others (larval crowding). © 2018. Published by The Company of Biologists Ltd

Efficiency Costs of Subsidy Rules for Crop Insurance

Participation in federal crop insurance programs has been encouraged through premium subsidies. The current subsidy depends on contract features as well as coverage levels. This type of subsidy rule causes farmers to choose contract designs and coverages that are not efficient for managing risk, in order to capture subsidy. Farmers are found to be as well off with a flat subsidy that is up to 25% less than the value of the current regressive proportional subsidy.crop insurance, futures, risk management, subsidy, Risk and Uncertainty,

Wavelength-independent coupler from fiber to an on-chip cavity, demonstrated over an 850nm span

A robust wide band (850 nm) fiber coupler to a whispering-gallery cavity with ultra-high quality factor is experimentally demonstrated. The device trades off ideality for broad-band, efficient input coupling. Output coupling efficiency can remain high enough for practical applications wherein pumping and power extraction must occur over very broad wavelength spans