Quantification of phenological patterns (e.g. migration, hibernation or reproduction) should involve statistical assessments of non‐uniform temporal patterns. Circular statistics (e.g. Rayleigh test or Hermans‐Rasson test) provide useful approaches for doing so based on the number of individuals that exhibit particular activities during a number of time intervals. This study used monthly reproductive activity as an example to illustrate problems in applying circular statistics to data when marginal totals characterize experimental designs (e.g. the number of reproductively active individuals per time interval depends on sampling effort or sampling success). We illustrate the nature of this problem by crafting four exemplar data sets and developing a bootstrapping simulation procedure to overcome complications that arise from the existence of marginal totals. In addition, we apply circular statistics and our bootstrapping simulation to empirical data on the reproductive phenology of six species of Neotropical bats from the Amazon. Because sampling effort or success can differ among time intervals, circular statistics can produce misleading results of two types: those suggesting uniform phenologies when empirical patterns are markedly modal, and those suggesting non‐uniform phenologies when empirical patterns are uniform. The bootstrapping simulation overcomes these limitations: the exemplar phenology in which the percentage of reproductively active individuals is modal is appropriately identified as non‐uniform based on the bootstrapping approach, and the exemplar phenology in which the percentage of reproductively active individuals is invariant is appropriately identified as uniform based on the bootstrapping approach. The reproductive phenology of each of the six empirical examples is non‐uniform based on the bootstrapping approach, and this is true for bats species with unimodal peaks or bimodal peaks. In addition to problems with marginal totals, a review of analyses of phenological patterns in ecology identified two other frequent issues in the application of circular statistics: sampling bias and pseudoreplication. Each of these issues and potential solutions are also discussed. By providing source code for the execution of the Rayleigh test and Hermans‐Rasson test, along with the code for the bootstrapping simulation, we offer a useful tool for assessing non‐random phenologies when marginal totals characterize experimental designs.
Sequences of high rank lattices with large systole containing a fixed genus surface group
In this paper we exhibit sequences of torsion-free lattices (both uniform and non-uniform) that have arbitrarily large systole, but all containing a thin surface subgroup of fixed genus.
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- Award ID(s):
- 1812397
- NSF-PAR ID:
- 10096665
- Date Published:
- Journal Name:
- New York journal of mathematics
- Volume:
- 25
- ISSN:
- 1076-9803
- Page Range / eLocation ID:
- 145-155
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract -
For an $r$-uniform hypergraph $H$, let $\nu^{(m)}(H)$ denote the maximum size of a set $M$ of edges in $H$ such that every two edges in $M$ intersect in less than $m$ vertices, and let $\tau^{(m)}(H)$ denote the minimum size of a collection $C$ of $m$-sets of vertices such that every edge in $H$ contains an element of $C$. The fractional analogues of these parameters are denoted by $\nu^{*(m)}(H)$ and $\tau^{*(m)}(H)$, respectively. Generalizing a famous conjecture of Tuza on covering triangles in a graph, Aharoni and Zerbib conjectured that for every $r$-uniform hypergraph $H$, $\tau^{(r-1)}(H)/\nu^{(r-1)}(H) \leq \lceil{\frac{r+1}{2}}\rceil$. In this paper we prove bounds on the ratio between the parameters $\tau^{(m)}$ and $\nu^{(m)}$, and their fractional analogues. Our main result is that, for every $r$-uniform hypergraph~$H$,\[ \tau^{*(r-1)}(H)/\nu^{(r-1)}(H) \le \begin{cases} \frac{3}{4}r - \frac{r}{4(r+1)} &\text{for }r\text{ even,}\\\frac{3}{4}r - \frac{r}{4(r+2)} &\text{for }r\text{ odd.} \\\end{cases} \]This improves the known bound of $r-1$.We also prove that, for every $r$-uniform hypergraph $H$, $\tau^{(m)}(H)/\nu^{*(m)}(H) \le \operatorname{ex}_m(r, m+1)$, where the Turán number $\operatorname{ex}_r(n, k)$ is the maximum number of edges in an $r$-uniform hypergraph on $n$ vertices that does not contain a copy of the complete $r$-uniform hypergraph on $k$ vertices. Finally, we prove further bounds in the special cases $(r,m)=(4,2)$ and $(r,m)=(4,3)$.more » « less
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A bstract Modular graph functions (MGFs) are SL(2 , ℤ)-invariant functions on the Poincaré upper half-plane associated with Feynman graphs of a conformal scalar field on a torus. The low-energy expansion of genus-one superstring amplitudes involves suitably regularized integrals of MGFs over the fundamental domain for SL(2 , ℤ). In earlier work, these integrals were evaluated for all MGFs up to two loops and for higher loops up to weight six. These results led to the conjectured uniform transcendentality of the genus-one four-graviton amplitude in Type II superstring theory. In this paper, we explicitly evaluate the integrals of several infinite families of three-loop MGFs and investigate their transcendental structure. Up to weight seven, the structure of the integral of each individual MGF is consistent with the uniform transcendentality of string amplitudes. Starting at weight eight, the transcendental weights obtained for the integrals of individual MGFs are no longer consistent with the uniform transcendentality of string amplitudes. However, in all the cases we examine, the violations of uniform transcendentality take on a special form given by the integrals of triple products of non-holomorphic Eisenstein series. If Type II superstring amplitudes do exhibit uniform transcendentality, then the special combinations of MGFs which enter the amplitudes must be such that these integrals of triple products of Eisenstein series precisely cancel one another. Whether this indeed is the case poses a novel challenge to the conjectured uniform transcendentality of genus-one string amplitudes.more » « less
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We present the general conditions to realize a fourth-order exceptional point of degeneracy (EPD) in two uniform (i.e., invariant along z) lossless and gainless coupled transmission lines (CTLs), namely, a degenerate band edge (DBE). Until now the DBE has been shown only in periodic structures. In contrast, the CTLs considered here are uniform and subdivided into four cases where the two TLs support combinations of forward propagation, backward propagation, and evanescent modes (when neglecting the mutual coupling). We demonstrate, for the first time, that a DBE is supported in uniform CTLs when there is proper coupling between: 1) propagating modes and evanescent modes, 2) forward and backward propagating modes, or 3) four evanescent modes (two in each direction). We also show that the loaded quality factor of uniform CTLs exhibiting a fourth-order EPD at k=0 is robust to series losses due to the fact that the degenerate modes do not advance in phase. We also provide a microstrip possible implementation of a uniform CTL exhibiting a DBE using periodic series capacitors with very subwavelength unit-cell length. Finally, we show an experimental verification of the existence DBE for a microstrip implementation of a CTL supporting coupled propagating and evanescent modes.more » « less
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