Temperature-Dependent and pH-Responsive Pillar[5]arene-Based Complexes and Hydrogen-Bond-Based Supramolecular Pentagonal Boxes in Water

Supramolecular systems in water are of paramount importance and those based on hydrogen bonds are both intriguing and scarce. Here, after studying the peculiar host–guest complexes formed between per-dimethylamino-pillar[5]arene ( 1 ) and the bis-sulfonates 2 a – c , we describe the formation of the first hydrogen-bond-based supramolecular pentagonal boxes (SPBs), which are stable in water. These pH-responsive SPBs are constructed from 1 as a body, benzene polycarboxylic acids 3 a , b as lid compounds, and 2 a – c as guests. We demonstrate that encapsulation of 2 a – c in pillar[5]arene 1 and in the highly stable water-soluble SPBs, that is, 1(3 a) ₂ and 1(3 b) ₂, is both temperature and pH dependent and, quite interestingly, depends, on the nature of the lid compounds used for capping the boxes even at high pH. We also highlight the difference in the ¹H NMR characteristics of 2 b and 2 c in the cavity of 1 and the SPBs.     

All-norm approximation algorithms

A major drawback in optimization problems and in particular in scheduling problems is that for every measure there may be a different optimal solution. In many cases the various measures are different ℓ_p norms. We address this problem by introducing the concept of an all-norm ρ-approximation algorithm, which supplies one solution that guarantees ρ-approximation to all ℓ_p norms simultaneously. Specifically, we consider the problem of scheduling in the restricted assignment model, where there are m machines and n jobs, each job is associated with a subset of the machines and should be assigned to one of them. Previous work considered approximation algorithms for each norm separately. Lenstra et al. [Math. Program. 46 (1990) 259–271] showed a 2-approximation algorithm for the problem with respect to the ℓ_∞ norm. For any fixed ℓ_p norm the previously known approximation algorithm has a performance of θ(p). We provide an all-norm 2-approximation polynomial algorithm for the restricted assignment problem. On the other hand, we show that for any given ℓ_p norm (p>1) there is no PTAS unless P=NP by showing an APX-hardness result. We also show for any given ℓ_p norm a FPTAS for any fixed number of machines.     

Path Selection in a Poisson field

A criterion for path selection for channels growing in a Poisson field is presented. We invoke a generalization of the principle of local symmetry. We then use this criterion to grow channels in a confined geometry. The channel trajectories reveal a self-similar shape as they reach steady state. Analyzing their paths, we identify a cause for branching that may result in a ramified structure in which the golden ratio appears.     

On parallel hashing and integer sorting

The problem of sorting n integers from a restricted range [1…m], where m is a superpolynomial in n, is considered. An o(n log n) randomized algorithm is given. Our algorithm takes O(n log log m) expected time and O(n) space. (Thus, for m = n^polylog(n) we have an O(n log log n) algorithm.) The algorithm is parallelizable. The resulting parallel algorithm achieves optimal speedup. Some features of the algorithm make us believe that it is relevant for practical applications. A result of independent interest is a parallel hashing technique. The expected construction time is logarithmic using an optimal number of processors, and searching for a value takes O(1) time in the worst case. This technique enables drastic reduction of space requirements for the price of using randomness. Applicability of the technique is demonstrated for the parallel sorting algorithm and for some parallel string matching algorithms. The parallel sorting algorithm is designed for a strong and nonstandard model of parallel computation. Efficient simulations of the strong model on a CRCW PRAM are introduced. One of the simulations even achieves optimal speedup. This is probably the first optimal speedup simulation of a certain kind.     

Physical limits of heat-bath algorithmic cooling

Simultaneous near-certain preparation of qubits (quantum bits) in their ground states is a key hurdle in quantum computing proposals as varied as liquid-state NMR and ion traps. "Closed-system" cooling mechanisms are of limited applicability due to the need for a continual supply of ancillas for fault tolerance, and to the high initial temperatures of some systems. "Open-system" mechanisms are therefore required. We describe a new, efficient initialization procedure for such open systems. With this procedure, an n-qubit device that is originally maximally mixed, but is in contact with a heat bath of bias epsilon>2(-n), can be almost perfectly initialized. This performance is optimal due to a newly discovered threshold effect: for bias epsilon<2(-n) no cooling procedure can, even in principle (running indefinitely without any decoherence), significantly initialize even a single qubit.     

Nonaqueous magnesium electrochemistry and its application in secondary batteries

A revolution in modern electronics has led to the miniaturization and evolution of many portable devices, such as cellular telephones and laptop computers, since the 1980s. This has led to an increasing demand for new and compatible energy storage technologies. Furthermore, a growing awareness of pollution issues has provided a strong impetus for the science and technology community to develop alternatives with ever-higher energy densities, with the ultimate goal of being able to propel electric vehicles. Magnesium's thermodynamic properties make this metal a natural candidate for utilization as an anode in high-energy-density, rechargeable battery systems. We report herein on the results of extensive studies on magnesium anodes and magnesium insertion electrodes in nonaqueous electrolyte solutions. Novel, rechargeable nonaqueous magnesium battery systems were developed based on the research. This work had two major challenges: one was to develop electrolyte solutions with especially high anodic stability in which magnesium anodes can function at a high level of cycling efficiency; the other was to develop a cathode that can reversibly intercalate Mg ions in these electrolyte systems. The new magnesium batteries consist of Mg metal anodes, an electrolyte with a general structure of Mg(AlX(3-n)R(n)R')(2) (R',R = alkyl groups, X = halide) in ethereal solutions (e.g., tetrahydrofuran, polyethers of the "glyme" family), and Chevrel phases of MgMo(3)S(4) stoichiometry as highly reversible cathodes. With their practical energy density expected to be >60 Wh/Kg, the battery systems can be cycled thousands of times with almost no capacity fading. The batteries are an environmentally friendly alternative to lead-acid and nickel-cadmium batteries and are composed of abundant, inexpensive, and nonpoisonous materials. The batteries are expected to provide superior results in large devices that require high-energy density, high cycle life, a high degree of safety, and low-cost components. Further developments in this field are in active progress.     

The density of 0's in recurrence double sequences

Let be a double sequence over a finite field satisfying a linear recurrence with constant coefficients, with at most finitely many nonzero elements on each row. Given a nonzero element g of , we show how to obtain an explicit formula for the number of g's in the first qⁿ rows of A. We also characterize the cases when the density of 0's is 1.