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.     

Modeling the power of an optical parametric generator by discrete mode summation

An analytical expression for calculating the signal output power of an optical parametric generator (OPG) configuration was developed. The model is based on Heisenberg equations in space and radiation mode theory. A simple analytical expression can be obtained by assuming that all modes within the gain bandwidth of the nonlinear crystal have the same gain and the same interaction length. Another derivation considers the gain and interaction length of each individual mode. The model predictions are in excellent agreement with the measured signal power of OPGs based on 25- and 35-mm periodically poled LiNbO₃ crystal (with effective quadratic nonlinearity of ∼14.5 pm/V) and 47-mm periodically poled LiTaO₃ crystal (with effective quadratic nonlinearity of ∼9 pm/V). In addition, a comparison was made between the summation over discrete modes approach and the traditional approach of continuous integration over the beam parameters and pump frequency. We have found that the first approach, which is developed in this paper, predicts more accurately the output power of the OPG.     

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.     

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.     

Asymmetric banana-shaped liquid crystals with two different terminal alkoxy chains

Two series of asymmetric banana-shaped compounds have been synthesized and studied. In the 1,3-phenylene bis[4-(4'-alkoxybenzoyloxy)]benzoate series the lack of symmetry was derived solely from the difference in length of the two terminal alkoxy chains. In the 3,4'-biphenylene bis[4-(4'-alkoxybenzoyloxy)]benzoate series the asymmetric nature originates from the 3,4'-substitution of the central biphenyl group and from the difference in length of the two terminal chains. All the melting points of the asymmetrical compounds in the series with the central phenyl unit are lower than those of the symmetrical compounds. The liquid crystalline B₁ or B₂ phase was retained in all cases. In the series with the central biphenyl unit the compounds with the shortest chain attached to the para-position of the central biphenyl unit have the lowest melting points. A significant lowering of the melting points in comparison with the symmetrically substituted compounds, however, could not be achieved. All the compounds of both series show a layer spacing which is comparable to those of the symmetrically substituted parent compounds. The observed switching behaviour of both the symmetric and asymmetric compounds with a B₂ phase was antiferroelectric.     

A new six-coordinate organotin(IV) complex of OP[NC5H10]3: A comparison with an analogous five-coordinate complex by means of X-ray crystallography,Hirshfeld surface analysis and DFT calculations

A new six-coordinate organotin(IV)-phosphoric triamide complex of OP[NC₅H₁₀]₃ = OP was synthesized ([Cl₂Sn(CH₃)₂(OP)₂], 1) and characterized by X-ray crystallography and spectroscopic methods (FT-IR, UV–Vis, and ¹H/¹³C/³¹P-NMR). The crystal structures of 1 and the analogous previously reported five-coordinate complex [Cl₂Sn(CH₃)₂(OP)] (IZOVIE) were compared on a structural level and by computational means using Hirshfeld surface analysis, density functional theory calculations and the atom in molecule method. The investigation of intermolecular interactions in the crystal structures of the two complexes by the Hirshfeld surface method indicates that in the absence of normal hydrogen bonds, the chlorine-based interactions H?Cl/Cl?H (for 1 and IZOVIE) and Cl?Sn/Sn?Cl (for IZOVIE) play a determinant role in the molecular assemblies. However, the prominent contacts are of H?H type. From calculated electronic parameters such as bond order, Mulliken charge and electron delocalization energy, it was found that the Sn-OP contact has a lower strength in IZOVIE than in 1, suggesting more ionic character of the metal-oxygen contact in five-coordinate complex IZOVIE. Furthermore, we discuss the similarities and differences of the two complexes 1 and IZOVIE derived from the same ligand OP by density functional theory calculations to present an insight into the organotin(IV)-phosphoric triamide coordination chemistry affected by different geometries and coordination numbers.     

Probing the interaction of HTI-286 with tubulin using a stilbene analogue

HTI-286 is a synthetic analogue of the natural product hemiasterlin. HTI-286 is a potent antitumor agent that induces tubulin oligomerization. To investigate the binding stoichiometry and the binding site during this ligand-induced tubulin association, we synthesized an analogue of HTI-286 containing the chromophore stilbene. Using the distinct absorbance of the stilbene analogue, we determined the amounts of inhibitors bound to different tubulin oligomers by analytical ultracentrifugation. Herein we describe our findings based on these experiments. At the ratio of inhibitor to protein equal to or greater than 1, the stilbene analogue induces oligomerization of tubulin to a ring structure. The binding stoichiometry in the ring is one inhibitor per tubulin monomer (defined as an alpha/beta-heterodimer). At the ratio of inhibitor to protein less than 1, tubulin forms multiple intermediates, with the binding stoichiometry less than one inhibitor per tubulin monomer for all intermediates. The stable complex between the inhibitor and tubulin monomer was not detected under our experimental conditions. The binding site of the stilbene analogue does not overlap with the classic tubulin-binding agent, colchicine.