Geometric Permutations Induced by Line Transversals through a Fixed Point

Let ℒ be the space of line transversals to a finite family of pairwise disjoint compact convex sets in ℝ3. We prove that each connected component of ℒ can itself be represented as the space of transversals to some finite family of pairwise disjoint compact convex sets.

We show that a set of n disjoint unit spheres in ℝd admits at most two distinct geometric permutations, or line transversals, if n is large enough. This bound is optimal.

Contact determination in terms of edge-face intersection tests permits handling nonconvex polyhedra directly, without decomposing them into convex entities, which saves the decomposition time and avoids having to deal with fictitious features but requires checking all possible pairings. However, by considering only translations and departing from a noninterfering situation, the number of pairings to be checked decreases drastically. The set of critical pairings can be determined efficiently using the spherical face orientation graph (SFOG), a representation developed by the authors. An algorithm to exploit the SFOG in convex settings provides controlled evidence of the pruning potential of this approach: the number of critical pairings grows linearly with the complexity of the polyhedra, instead of quadratically as the total number of pairings does. Experiments with a similar algorithm on nonconvex settings confirm the expected potential of the approach: for workpieces with many concavities moving in close proximity, the contact determination procedure presented in this paper performs one order of magnitude faster than RAPID, at the expense of a much higher preprocessing time.

Let u be an eigenfunction of the Laplacian on a compact manifold with boundary, with Dirichlet or Neumann boundary conditions, and let −λ2 be the corresponding eigenvalue. We consider the problem of estimating max M u in terms of λ, for large λ, assuming ∫ M u 2 = 1. We prove that max M u ≤C M λ(n − 1)/2, which is optimal for some M. Our proof simplifies some of the arguments used before for such problems. We review the ‘wave equation method’ and discuss some special cases which may be handled by more direct methods. *This work was partially supported by NSF grant DMS-9306389 and by the Mathematical Sciences Research Institute, Berkeley.

Recycling lithium-ion battery materials is a crucial step to achieving a circular economy. A hydrometallurgical recycling process of spent lithium-ion battery materials involves leaching by acids and precipitation by bases. A large amount of Li2SO4 leachate solution is generated at the end of the recycling process. Such Li2SO4 solution can be separated by electrodialysis (ED) to form H2SO4 and LiOH solutions, which are reused for leaching and precipitation, respectively, making the recycling a closed-loop process. In this study, an ‘electrodialysis stack constructed with bipolar membranes’ (EDBM) is built with repetitive unit cells consisting of a bipolar membrane, anion exchange membrane (AEM), and cation exchange membrane (CEM), see Fig. 1(a). The effective area for each membrane is 121.8 cm2 (70mm×174mm). The gap between these membranes is 0.8mm, with a mesh inserted to allow flow mixing. All experiments were operated at constant current mode. The feed, acid, and base solutions were recirculated through their tanks at initial concentrations of 1.1, 0.1, and 0.1 mol/L. The ion concentrations of the solutions were measured with Inductively coupled plasma mass spectrometry periodically. The weight and the pH value of the solution tanks were also recorded.The trend of concentration variations of the present stack with up to 6 cell pairs is found to be consistent with that observed with a three-compartment configuration in our previous study [1]. The concentration of H2SO4 increases with time linearly, whereas that of LiOH levels off due to a significant electro-osmosis that drags water through the CEM into the concentrate channel. The present setup differs from the three-compartment design in the bipolar membrane, which splits water to produce H+ and OH- at the internal interface, eliminating the formation of bubbly flows in the unit cell. It is found that the voltage loss across the bipolar membrane ca. 1V accounts for a major part of the unit cell of about 1.27V.The effects of current density, number of cell pairs, and flow rates on the overall stack performance are investigated. Increasing the stack current density accelerates the separation process at the cost of higher voltage losses. It is found that increasing the number of cell pairs can improve the overall production efficiency and reduce energy consumption per cell pair. For instance, at 800 A/m2, the specific energy consumption for LiOH production is reduced to 3.40 kWh/kg with five cell pairs versus 5.83 kWh/kg with two cell pairs. It is also found that increasing the solution flow rate can result into the increase of LiOH solution concentration, ion recovery rate, and current efficiency (Fig. 1b), although the impact of flow rate is low.A small EDBM stack is constructed and tested for Li2SO4 separation in the present study. This work demonstrates that EDBM can be a simple and energy-saving alternative to the current chemical precipitation method to produce LiOH from Li2SO4. The present study provides new insights into optimal operation and design for Li2SO4 EDBM. The findings are helpful in determining how the stack can be scaled up for practical application. Furthermore, a 2D multiphysics model is currently under development to elucidate the coupled transport processes within the unit cell. The model will be validated with experimental data of the present setup. Figure 1

First Person is a series of interviews with the first authors of a selection of papers published in Journal of Cell Science, helping early-career researchers promote themselves alongside their papers. Maitreyi Rathod and Sushmita Chatterjee are co-first authors on ‘Mannose glycosylation is an integral step for NIS localization and function in human breast cancer cells’, published in JCS. Maitreyi conducted the research described in this article while a PhD student in Dr Abhijit De's lab at the ACTREC, Tata Memorial Centre, Navi Mumbai, India. Sushmita is now a a postdoctoral fellow in Prof. Dan Peer's lab at Tel Aviv University, Israel, investigating the use of RNAi in ovarian cancer and in order to target chemo-resistant ovarian cancer cells.

Electrodialysis has been used for treating toxic substances such as heavy metals and minimizing secondary environmental pollution problems effectively. However, electrodialysis depends on the operating parameters as well as fluid dynamics and electrical properties. This study provides design elements for the treatment of heavy metal-containing wastewater by electrodialysis. We found that the limiting current density (LCD) is proportional but not completely linear to the diluate concentration over a threshold value. In contrast, it is linear to the linear flow velocity for the whole range. As the number of cell pairs increases, because linear flow velocity and LCD increase, the removal efficiency of heavy metals also increases. Therefore, for highly concentrated wastewater, increasing the linear flow velocity, the applied voltage, and the number of cell pairs can effectively improve removal efficiency. It was found that the current efficiency is as low as 17% when the removal efficiency of heavy metals exceeds 95%. Thus, it is necessary to select an operating range that optimizes the operating and initial investment costs for the effective removal of heavy metals using electrodialysis.

Introduction of Julián F. Hillyer, Recipient of the Henry Baldwin Ward Medal For 2015

Polyspermy in the sunflower (Helianthus annuus L. cv. ‘Bismarckianus’) was studied. Flowers were pollinated with a mixture of small amounts of pollen from the pollinated florets' one-day circle (i.e., a whorl in the head comprising all the florets opening during one day) and large amounts of pollen from a single floret of Jerusalem artichoke, or girasole (Helianthus tuberosus L.). The artichoke pollen, but not the sunflower pollen, was labelled with P-32. In another experiment the labelled artichoke pollen was replaced by labelled pollen from another sunflower plant. The activity per pollen grain, as well as the activity per ovule, was determined by means of a Geiger-Müller counter. Main results: (1) When the sunflower's pollen tubes reach the ovule and double fertilization is effected by the plant's own sperm cells, the ovule also usually receives one or several pairs of sperm cells from the artichoke. (2) Each ovule in a one-day circle often receives the same number of sperm cells. (3) After pollination with labelled artichoke pollen from one floret, an activity corresponding to 1, 2, 4, 8 or 16 pollen grains per ovule was obtained. After pollination with active foreign sunflower pollen from one floret, the corresponding number was 2, 3, 5, 9 or 17. Besides the pair of sperm cells effecting double fertilization, the number of pairs received by an embryo sac thus corresponds to one of the members in the 2n series, with n representing the integer series from zero upwards. The highest n value found is 4. (4) When pollen from the same father is used, the number of sperm cells entering the embryo sac varies between the one-day circles. Thus, variation occurs among isogenic whorls of florets. An autoradiographic method was used to follow the localization after fertilization of the compounds containing the isotope. Literature on polyspermy in the Angiosperms is reviewed. The following general conclusions could be drawn: (1) Polyspermy is a widespread phenomenon among cross-fertilizing Angiosperms. (2) Polyspermy does not seem to occur among self-pollinating species. (3) When a large number of nuclei are involved, it is improbable that the number of pollen tubes reaching the embryo sac corresponds to the number of pairs of sperm nuclei entering the embryo sac. Hence, pollen tubes may sometimes empty their contents into other pollen tubes.

Ordered stabbing of pairwise disjoint convex sets in linear time

The distances between flats of a Poisson k-flat process in the d-dimensional Euclidean space with k < d/2 are discussed. Continuing an approach originally due to Rolf Schneider, the number of pairs of flats having distance less than a given threshold and midpoint in a fixed compact and convex set is considered. For a family of increasing convex subsets, the asymptotic variance is computed and a central limit theorem with an explicit rate of convergence is proven. Moreover, the asymptotic distribution of the m-th smallest distance between two flats is investigated and it is shown that the ordered distances form asymptotically after suitable rescaling an inhomogeneous Poisson point process on the positive real half-axis. A similar result with a homogeneous limiting process is derived for distances around a fixed, strictly positive value. Our proofs rely on recent findings based on the Wiener–Ito chaos decomposition and the Malliavin–Stein method.

The dynamics of Hermitian many-body quantum systems has long been a challenging subject due to the complexity induced by the particle-particle interactions. In contrast, this difficulty may be avoided in a well-designed non-Hermitian system. The exceptional point (EP) in a non-Hermitian system admits a peculiar dynamics: the final state being a particular eigenstate, coalescing state. In this work, we study the dynamic transition from a trivial insulating state to an $\ensuremath{\eta}$-pairing state in a composite non-Hermitian Hubbard system. The system consists of two subsystems, A and B, which are connected by unidirectional hoppings. We show that the dynamic transition from an insulating state to an $\ensuremath{\eta}$-pairing state occurs by the probability flow from A to B: the initial state is prepared as an insulating state of A, while B is left empty. The final state is an $\ensuremath{\eta}$-pairing state in B but empty in A. Analytical analyses and numerical simulations show that the speed of relaxation of the off-diagonal long-range order pair state depends on the order of the EP, which is determined by the number of pairs and the fidelity of the scheme is immune to the irregularity of the lattice.

DCG

We introduce a new measure for planar point sets S that captures a combinatorial distance that S is from being a convex set: The reflexivity rho(S) of S is given by the smallest number of reflex vertices in a simple polygonalization of S. We prove various combinatorial bounds and provide efficient algorithms to compute reflexivity, both exactly (in special cases) and approximately (in general). Our study considers also some closely related quantities, such as the convex cover number kappa_c(S) of a planar point set, which is the smallest number of convex chains that cover S, and the convex partition number kappa_p(S), which is given by the smallest number of convex chains with pairwise-disjoint convex hulls that cover S. We have proved that it is NP-complete to determine the convex cover or the convex partition number and have given logarithmic-approximation algorithms for determining each.

We consider a dynamic combinatorial optimization problem where at each time step, the decision maker selects a subset of cardinality K from N possible items and observes a feedback in the form of the index of one of the items in the said subset or none. Each of the N items is ascribed a certain value (reward), which is collected if the item is chosen. This problem is motivated by that of assortment selection in online retail, where items are products. Akin to that literature, it is assumed that the choice of the item given the subset is governed by a multinomial logit (MNL) choice model whose parameters are a priori unknown. The objective of the decision maker is to maximize the expected cumulative rewards over a finite horizon T or alternatively, minimize the regret relative to an oracle that knows the MNL choice model parameters. We formulate this problem as a multiarmed bandit problem that we refer to as the MNL-bandit problem. We present a Thompson sampling-based algorithm for this problem and show that it achieves near-optimal regret as well as attractive empirical performance. Funding: S. Agrawal is supported in part by the Division of Civil, Mechanical and Manufacturing Innovation [NSF Grant 1846792]. V. Goyal is supported in part by the Division of Civil, Mechanical and Manufacturing Innovation [NSF Grants 1351838 and 1636046]. A. Zeevi is supported in part by the Division of Computer and Network Systems [NSF Grant 0964170] and the United States-Israel Binational Science Foundation [Grant 2010466].