Monotonicity testing and shortest-path routing on the cube

We study the problem of monotonicity testing over the hypercube. As previously observed in several works, a positive answer to a natural question about routing properties of the hypercube network would imply the existence of efficient monotonicity testers. In particular, if any set of source-sink pairs on the directed hypercube (with all sources and all sinks distinct) can be connected with edge-disjoint paths, then monotonicity of functions $f:\{ 0,1\} ^n \to \mathcal{R}$ can be tested with O(n/∈) queries, for any totally ordered range $\mathcal{R}$ . More generally, if at least a µ(n) fraction of the pairs can always be connected with edge-disjoint paths then the query complexity is O(n/(∈µ(n))). We construct a family of instances of Ω(2 ⁿ ) pairs in n-dimensional hypercubes such that no more than roughly a $\frac{1} {{\sqrt n }}$ fraction of the pairs can be simultaneously connected with edge-disjoint paths. This answers an open question of Lehman and Ron [16], and suggests that the aforementioned appealing combinatorial approach for deriving query-complexity upper bounds from routing properties cannot yield, by itself, query-complexity bounds better than ≈ n ^3/2. Additionally, our construction can also be used to obtain a strong counterexample to Szymanski’s conjecture about routing on the hypercube. In particular, we show that for any δ > 0, the n-dimensional hypercube is not $n^{\tfrac{1} {2} - \delta }$ -realizable with shortest paths, while previously it was only known that hypercubes are not 1-realizable with shortest paths. We also prove a lower bound of Ω(n/∈) queries for one-sided non-adaptive testing of monotonicity over the n-dimensional hypercube, as well as additional bounds for specific classes of functions and testers.     

Spectral and temporal holograms with nonlinear optics

In this Letter we show how encoding techniques for computer-generated holograms may be used to arbitrarily shape a nonlinearly generated spectrum and consequently the temporal shape by modulating the quadratic nonlinear coefficient. We give examples of a modulation pattern and a simple setup that can generate high-order Hermite-Gauss and Airy functions through difference-frequency generation from a transform-limited Gaussian pulse, under practical fabrication considerations.     

Two new XP(O)[NHC(CH3)3]2 phosphoramidates,with X = (CH3)2N and [(CH3)3CNH]2P(O)(O)

In N,N′‐di‐tert‐butyl‐N′′,N′′‐dimethylphosphoric triamide, C₁₀H₂₆N₃OP, (I), and N,N′,N′′,N′′′‐tetra‐tert‐butoxybis(phosphonic diamide), C₁₆H₄₀N₄O₃P₂, (II), the extended structures are mediated by P(O)...(H—N)₂ interactions. The asymmetric unit of (I) consists of six independent molecules which aggregate through P(O)...(H—N)₂ hydrogen bonds, giving R₂¹(6) loops and forming two independent chains parallel to the a axis. Of the 12 independent tert‐butyl groups, five are disordered over two different positions with occupancies ranging from to . In the structure of (II), the asymmetric unit contains one molecule. P(O)...(H—N)₂ hydrogen bonds give S(6) and R₂²(8) rings, and the molecules form extended chains parallel to the c axis. The structures of (I) and (II), along with similar structures having (N)P(O)(NH)₂ and (NH)₂P(O)(O)P(O)(NH)₂ skeletons extracted from the Cambridge Structural Database, are used to compare hydrogen‐bond patterns in these families of phosphoramidates. The strengths of P(O)[...H—N]_x (x = 1, 2 or 3) hydrogen bonds are also analysed, using these compounds and previously reported structures with (N)₂P(O)(NH) and P(O)(NH)₃ fragments.     

Bond Fukui functions as descriptor of the electron density reorganization in π conjugated systems

The bond Fukui function is introduced and tested as a new reactivity index capable of predicting the evolution of bond breaking and formation processes during an organic reaction involving π conjugated systems. As an illustration, we examine many cases where substituted ethylenes and dienes may respond to different reagents to yield cycloaddition, Michael addition, and other reactions at double bonds.     

Relationship between solvation energy, chemical potential and hardness variations

Starting from a density functional theory (DFT) formalism describing the energy change from one ground state representing an isolated solute, to another one representing the same solute in the field of the solvent, it is possible to obtain a simple and useful expression for the solvation energy in terms of the variation of the electronic chemical potential and global hardness associated to the change from gas to solution phase. Since both properties may be obtained from an orbital theory within the approximate Self Consistent Reaction Field (SCRF) methodology, the proposed model is expected to be useful for the analysis of chemical reactivity in solution.     

Order statistics applications to queueing and scheduling problems

We prove several basic combinatorial identities and use them in two applications: the queue inference engine (QIE) and earliest due date rule (EDD) scheduling. Larson (1990) introduced the QIE. His objective was to deduce the behavior of a multiserver queueing system without observing the queue. With only a Poisson arrival assumption, he analyzed the performance during a busy period. Such a period starts once all servers are busy with the queue empty, and it ends as soon as a server becomes idle. We generalize the standard order statistics result for Poisson processes, and show how to sample a busy period in the M/M/c system. We derive simple expressions for the variance of the total waiting time in the M/M/c and M/D/1 queues given that n Poisson arrivals and departures occur during a busy period. We also perform a probabilistic analysis of the EDD for a one-machine scheduling problem with earliness and tardiness penalties. The schedule is without preemption and with no inserted idle time. The jobs are independent and each may have a different due date. For large n, we show that the variance of the total penalty costs of the EDD is linear in n. The mean of the total penalty costs of the EDD is known to be proportional to the square root of n (see Harel (1993)). This revised version was published online in June 2006 with corrections to the Cover Date.     

Liquid crystalline properties of salicylaldimine-based dimers: influence of terminal alkyl chain length and central part

The synthesis and thermotropic properties of four homologous series of salicylaldimine-based dimer liquid crystals are reported. Two 4-(4-alkoxy-2-hydroxybenzylideneamino)benzoyloxy groups are connected to a central part consisting of a 1,3-phenylene, 1,5-pentylene, 2,2-dimethyl-1,5-pentylene or 3,3-dimethyl-1,5-pentylene unit. The terminal alkoxy chains have been varied from 4 to 16 carbon atoms in length. All the compounds exhibit liquid crystalline phases whose behaviour depends on the nature of the central part and the length of the alkoxy terminal chains. All compounds of the series with the central phenyl part exhibit enantiotropic B-phases, and the sequence B₆-B₁-B₂ on increasing terminal chain length was observed. Replacement of the phenyl group with a pentyl central group partly suppresses the formation of B-phases. The longer homologues of this series show the B₁ phase, while the shorter exhibit an intercalated SmC_c mesophase. The introduction of methyl substituents to the pentyl spacer causes the melting points to fall dramatically and the formation of B-phases is totally suppressed. The compounds with the long tails show intercalated SmA_c phases and those with short tails show intercalated SmC_c phases.     

Locating two obnoxious facilities using the weighted maximin criterion

Given are a finite set of points P and a compact polygonal set S in R². The problem is to locate two new facilities in S, maximizing the minimum of all weighted distances between the points in P and the two new facilities, and the distance between the pair of new facilities. We present subquadratic algorithms.     

Propellanes. Part LXXXII. Preparation and Labelling of [20.3.3]Propellane-24,27-dione

The title diketone has been prepared by a synthetic sequence beginning with decane-1,10-dicarboxylic acid.     

Construction and characterization of a frequency-domain fluorescence lifetime imaging microscopy system

The construction of a homodyne frequency domain fluorescence lifetime imaging microscope is described. The system consists of (i) an intensity-modulated laser excitation source, (ii) an epifluorescence microscope, (iii) a gain-modulated microchannel plate (MCP) image intensifier, and (iv) a slow-scan CCD camera. The phase and modulation homogeneity of the MCP image intensifier were determined at frequencies of 40, 100, 160, and 240 MHz. The detected modulation depths were 65, 52, 32, and 23%, respectively, and were highly homogeneously distributed. The phasedistribution image revealed iris effects at frequencies of 160 and 240 MHz but was homogeneous at lower frequencies. Lifetime imaging of a solution of the fluorescent flavoprotein lipoamide dehydrogenase demonstrated (i) the accuracy of the determined lifetimes (< 60 ps), (ii) the time resolution of the instrument (< 50 ps), and (iii) the average precision for single pixel fluorescence lifetimes (50 ps is feasible). The imaging of tiny fluorescent microspheres revealed that even in a volume of 0.3 x 10^-15 L, the standard error in the lifetimes can be as low as 79 ps. The spatial resolution of the instrument is estimated to be < 400 nm in the object plane at a 100 x magnification.