Pricing for Online Resource Allocation: Intervals and Paths

We present pricing mechanisms for several online resource allocation problems which obtain tight or nearly tight approximations to social welfare. In our settings, buyers arrive online and purchase bundles of items; buyers' values for the bundles are drawn from known distributions. This problem is closely related to the so-called prophet-inequality of Krengel and Sucheston and its extensions in recent literature. Motivated by applications to cloud economics, we consider two kinds of buyer preferences. In the first, items correspond to different units of time at which a resource is available; the items are arranged in a total order and buyers desire intervals of items. The second corresponds to bandwidth allocation over a tree network; the items are edges in the network and buyers desire paths. Because buyers' preferences have complementarities in the settings we consider, recent constant-factor approximations via item prices do not apply, and indeed strong negative results are known. We develop static, anonymous bundle pricing mechanisms. For the interval preferences setting, we show that static, anonymous bundle pricings achieve a sublogarithmic competitive ratio, which is optimal (within constant factors) over the class of all online allocation algorithms, truthful or not. For the path preferences setting, we obtain a nearly-tight logarithmic competitive ratio. Both of these results exhibit an exponential improvement over item pricings for these settings. Our results extend to settings where the seller has multiple copies of each item, with the competitive ratio decreasing linearly with supply. Such a gradual tradeoff between supply and the competitive ratio for welfare was previously known only for the single item prophet inequality

Experimental Nonnative Wood Addition Enhances Instream Habitat for Native Fishes and Investigating Dryland River Alterations

The rivers of the Colorado River Basin (CRB) have been degraded by human activities such flow regulation, water overallocation, and the introduction of invasive riparian vegetation (primarily tamarisk tamarix spp. and Russian olive Elaeagnus angustifolia). These stressors have resulted in widespread habitat loss and simplification, which is a major contributor to the endangerment of native fishes in the CRB. The objectives of this study were to 1) assess the effectiveness of enhancing native fish habitat by experimentally adding cut wood from nonnative Russian olive to the San Juan River, a highly degraded dryland river, and 2) determine the magnitude of channel narrowing and vegetation encroachment over time, which are indicators of riverine habitat alteration, for three dryland CRB tributaries. Addressing these objectives can provide valuable insights to assist managers in more effectively restoring habitats for endangered native fish species. Our results indicate that wood addition can lead to increased local abundances of native fishes, which is likely due to the increased food resources and higher habitat quality that wood addition facilitated. Additionally, we demonstrated that all three study rivers underwent substantial vegetation encroachment and channel narrowing, the latter of which coincided with a general decrease in river flows over time, with large decreases in flows corresponding to more pronounced channel narrowing. We therefore provide evidence supporting nonnative wood addition as an effective management action for enhancing native fish habitat while also underscoring the magnitude of river alteration in the region. Our findings emphasize that, when possible, preserving or restoring natural flows should be prioritized for the effective conservation of dryland rivers. In systems where managing flows is not feasible, managers may consider non-flow methods, like adding in-stream wood, to improve native fish habitats

The emergence of 4-cycles in polynomial maps over the extended integers

Let $f(x) \in \mathbb{Z}[x]$; for each integer $\alpha$ it is interesting to consider the number of iterates $n_{\alpha}$, if possible, needed to satisfy $f^{n_{\alpha}}(\alpha) = \alpha$. The sets $\{\alpha, f(\alpha), \ldots, f^{n_{\alpha} - 1}(\alpha), \alpha\}$ generated by the iterates of $f$ are called cycles. For $\mathbb{Z}[x]$ it is known that cycles of length 1 and 2 occur, and no others. While much is known for extensions to number fields, we concentrate on extending $\mathbb{Z}$ by adjoining reciprocals of primes. Let $\mathbb{Z}[1/p_1, \ldots, 1/p_n]$ denote $\mathbb{Z}$ extended by adding in the reciprocals of the $n$ primes $p_1, \ldots, p_n$ and all their products and powers with each other and the elements of $\mathbb{Z}$. Interestingly, cycles of length 4, called 4-cycles, emerge for polynomials in $\mathbb{Z}\left[1/p_1, \ldots, 1/p_n\right][x]$ under the appropriate conditions. The problem of finding criteria under which 4-cycles emerge is equivalent to determining how often a sum of four terms is zero, where the terms are $\pm 1$ times a product of elements from the list of $n$ primes. We investigate conditions on sets of primes under which 4-cycles emerge. We characterize when 4-cycles emerge if the set has one or two primes, and (assuming a generalization of the ABC conjecture) find conditions on sets of primes guaranteed not to cause 4-cycles to emerge.Comment: 14 pages, 1 figur

Synthesis, structures and cytotoxicity studies of p-sulfonatocalix[4]arene lanthanide complexes

A number of p-sulfonatocalix[4]arene complexes of the lanthanides (Tb, Gd, and Eu) have been prepared, some in the presence of tetraazamacrocycle 1,4,7,10-tetraazacyclododecane-1,4,7-triacetic acid (DO3A), and fully characterised. Crystal structure determinations reveal lanthanide coordination at the sulfonate group, bridging several calixarene units, giving coordination polymers. All complexes in this study have been determined to be relatively non-toxic using in vitro cell assays with CC₅₀ values in the range 30–170 μM