Frost and flowers: Predicting the effects of climate change on spring frost damage using a large pollen count dataset

Background/Question/Methods 
Climate change is resulting in warmer average temperatures in much of North America, leading to earlier flowering dates for most tree genera. However, earlier flowering dates can increase the chance of frost damage which may in turn reduce or eliminate seed and fruit production. This has important implications for both forest regeneration and the organisms which depend on these resources.
Although specific examples, such as the “Easter frost” of 2007, show that frost damage can have extreme short term impacts on tree reproduction, we lack a general understanding of the frequency and importance of these events in non-agricultural contexts. Moreover, the magnitude by which climate change could increase these events is unknown.
In this study we address the following questions: 1) How good of a predictor are environmental variables (such as temperature) for pollen release date and duration across multiple genera of North American trees? 2) How frequently and to what extent has frost damage affected pollen production over the last decade? 3) Based on temporal and spatial variation in environmental factors and tree pollen phenology, what can we predict for future decades? 
To investigate these questions we make use of a large pollen count dataset, derived from asthma related data collection efforts. This dataset consists of 55 urban sites across North America, and contains six years of data. Pollen was collected daily and identified to the genus level. We also make use of nearby municipal weather stations for detailed daily weather data.
Results/Conclusions 
Certain genera, such as Acer, Alnus, Populus, and Quercus had relatively earlier pollen production times, putting them at higher risk of frost damage. Within each genus there was high variation in when the day of peak pollen counts occurred; for example Quercus peak days ranged from February 28th to June 7th. This variation was largely explained by environmental variables and site specific effects. 
We found many examples of days which had minimum temperatures sufficiently low to damage plant tissue but where pollen was still collected. For example, in the Quercus genus 223 out of 7417 pollen count observations fit these criteria, suggesting the importance of frost damage to this genus. Next, by comparing pollen count distributions between years, we show the extent to which frost damage has limited pollen production for multiple genera. Finally, we use hierarchical models to show how climate change may affect the impact of frost damage in the future.
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p-Adic valuation of weights in Abelian codes over /spl Zopf/(p/sup d/)

Counting polynomial techniques introduced by Wilson are used to provide analogs of a theorem of McEliece. McEliece's original theorem relates the greatest power of p dividing the Hamming weights of words in cyclic codes over GF (p) to the length of the smallest unity-product sequence of nonzeroes of the code. Calderbank, Li, and Poonen presented analogs for cyclic codes over /spl Zopf/(2/sup d/) using various weight functions (Hamming, Lee, and Euclidean weight as well as count of occurrences of a particular symbol). Some of these results were strengthened by Wilson, who also considered the alphabet /spl Zopf/(p/sup d/) for p an arbitrary prime. These previous results, new strengthened versions, and generalizations are proved here in a unified and comprehensive fashion for the larger class of Abelian codes over /spl Zopf/(p/sup d/) with p any prime. For Abelian codes over /spl Zopf//sub 4/, combinatorial methods for use with counting polynomials are developed. These show that the analogs of McEliece's theorem obtained by Wilson (for Hamming weight, Lee weight, and symbol counts) and the analog obtained here for Euclidean weight are sharp in the sense that they give the maximum power of 2 that divides the weights of all the codewords whose Fourier transforms have a specified support

p-Adic estimates of Hamming weights in Abelian codes over Galois rings

A generalization of McEliece's theorem on the p-adic valuation of Hamming weights of words in cyclic codes is proved in this paper by means of counting polynomial techniques introduced by Wilson along with a technique known as trace-averaging introduced here. The original theorem of McEliece concerned cyclic codes over prime fields. Delsarte and McEliece later extended this to Abelian codes over finite fields. Calderbank, Li, and Poonen extended McEliece's original theorem to cover cyclic codes over the rings /spl Zopf//sub 2//sup d/, Wilson strengthened their results and extended them to cyclic codes over /spl Zopf//sub p//sup d/, and Katz strengthened Wilson's results and extended them to Abelian codes over /spl Zopf//sub p//sup d/. It is natural to ask whether there is a single analogue of McEliece's theorem which correctly captures the behavior of codes over all finite fields and all rings of integers modulo prime powers. In this paper, this question is answered affirmatively: a single theorem for Abelian codes over Galois rings is presented. This theorem contains all previously mentioned results and more

Divisibility of Weil Sums of Binomials

Consider the Weil sum $W_{F,d}(u)=\sum_{x \in F} \psi(x^d+u x)$, where $F$ is a finite field of characteristic $p$, $\psi$ is the canonical additive character of $F$, $d$ is coprime to $|F^*|$, and $u \in F^*$. We say that $W_{F,d}(u)$ is three-valued when it assumes precisely three distinct values as $u$ runs through $F^*$: this is the minimum number of distinct values in the nondegenerate case, and three-valued $W_{F,d}$ are rare and desirable. When $W_{F,d}$ is three-valued, we give a lower bound on the $p$-adic valuation of the values. This enables us to prove the characteristic $3$ case of a 1976 conjecture of Helleseth: when $p=3$ and $[F:{\mathbb F}_3]$ is a power of $2$, we show that $W_{F,d}$ cannot be three-valued.Comment: 11 page

Proof of a Conjectured Three-Valued Family of Weil Sums of Binomials

We consider Weil sums of binomials of the form $W_{F,d}(a)=\sum_{x \in F} \psi(x^d-a x)$, where $F$ is a finite field, $\psi\colon F\to {\mathbb C}$ is the canonical additive character, $\gcd(d,|F^\times|)=1$, and $a \in F^\times$. If we fix $F$ and $d$ and examine the values of $W_{F,d}(a)$ as $a$ runs through $F^\times$, we always obtain at least three distinct values unless $d$ is degenerate (a power of the characteristic of $F$ modulo $|F^\times|$). Choices of $F$ and $d$ for which we obtain only three values are quite rare and desirable in a wide variety of applications. We show that if $F$ is a field of order $3^n$ with $n$ odd, and $d=3^r+2$ with $4 r \equiv 1 \pmod{n}$, then $W_{F,d}(a)$ assumes only the three values $0$ and $\pm 3^{(n+1)/2}$. This proves the 2001 conjecture of Dobbertin, Helleseth, Kumar, and Martinsen. The proof employs diverse methods involving trilinear forms, counting points on curves via multiplicative character sums, divisibility properties of Gauss sums, and graph theory.Comment: 19 page