The consistency with CH of some consequences of Martin’s axiom plus2^{\aleph _0 } > \aleph _1

We present here a (weak) axiom which implies some of the consequences of MA, but is consistent with GCH. We use the method of Jensen in his proof of consis (ZFC+GCH+SH). The third author wishes to thank the United States—Israel Binational Foundation for partially supporting this research by Grant 1110.     

Upper and lower bounds for graph-diameter problems with application to record allocation

This paper studies three graph problems with parameters n, the number of nodes, e, the number of edges, and k, the diameter of the graph. Given any two of these three parameters, the problem is to construct a directed graph which minimizes or maximizes the third. The first problem has its origin in a recent study of record allocation in a paged computer system. It is shown how to construct graphs that are optimal for all three problems in some cases and are asymptotically optimal for other cases. The solution of the second problem answers a question raised by Berge in “The Theory of Graphs and its Application,” 1962.     

The Budan-Fourier theorem for splines

The Budan-Fourier theorem for polynomials connects the number of zeros in an interval with the number of sign changes in the sequence of successive derivatives evaluated at the end-points. An extension is offered to splines with knots of arbitrary multiplicities, in which case the connection involves the number of zeros of the highest derivative. The theorem yields bounds on the number of zeros of splines and is a valuable tool in spline interpolation and approximation with boundary conditions.     

The 2-center problem in three dimensions

Let P be a set of n points in ${\mathbb{R}}^3$. The 2-center problem for P is to find two congruent balls of minimum radius whose union covers P. We present a randomized algorithm for computing a 2-center of P that runs in $O(\beta (r^{?})n^2{\log }^{4}n\log \log n)$ expected time; here $\beta (r)=1/{(1?r/r_0)}^3$, $r^{?}$ is the radius of the 2-center balls of P, and $r_0$ is the radius of the smallest enclosing ball of P. The algorithm is near quadratic as long as $r^{?}$ is not too close to $r_0$, which is equivalent to the condition that the centers of the two covering balls be not too close to each other. This improves an earlier slightly super-cubic algorithm of Agarwal, Efrat, and Sharir (2000) [2] (at the cost of making the algorithm performance depend on the center separation of the covering balls).     

Lossless compression of chemical fingerprints using integer entropy codes improves storage and retrieval

Many modern chemoinformatics systems for small molecules rely on large fingerprint vector representations, where the components of the vector record the presence or number of occurrences in the molecular graphs of particular combinatorial features, such as labeled paths or labeled trees. These large fingerprint vectors are often compressed to much shorter fingerprint vectors using a lossy compression scheme based on a simple modulo procedure. Here, we combine statistical models of fingerprints with integer entropy codes, such as Golomb and Elias codes, to encode the indices or the run lengths of the fingerprints. After reordering the fingerprint components by decreasing frequency order, the indices are monotone-increasing and the run lengths are quasi-monotone-increasing, and both exhibit power-law distribution trends. We take advantage of these statistical properties to derive new efficient, lossless, compression algorithms for monotone integer sequences: monotone value (MOV) coding and monotone length (MOL) coding. In contrast to lossy systems that use 1024 or more bits of storage per molecule, we can achieve lossless compression of long chemical fingerprints based on circular substructures in slightly over 300 bits per molecule, close to the Shannon entropy limit, using a MOL Elias Gamma code for run lengths. The improvement in storage comes at a modest computational cost. Furthermore, because the compression is lossless, uncompressed similarity (e.g., Tanimoto) between molecules can be computed exactly from their compressed representations, leading to significant improvements in retrival performance, as shown on six benchmark data sets of druglike molecules.     

Stereoselectivity in the synthesis of polyprenylphosphoryl β-d-ribofuranoses

Decaprenylphosphoryl β-d-arabinofuranose (DPA) is a key arabinose donor in mycobacteria. The ribo analog of DPA (DPR) has also been found in mycobacteria. It has recently been confirmed that DPA is formed via a two-step epimerization of DPR. The stereoselective synthesis of DPR as well as two shorter analogs of DPR is described.     

Self-sustained oscillations in pipe systems with multiple deep side branches: Prediction and reduction by detuning

Flow-induced pulsations are frequently observed in pipe networks. In the present work we focus on the case of flow-induced pulsations in a pipe system composed of six equally spaced deep closed side branches. These pulsations are self-sustained aeroacoustic oscillations driven by the instability of the flow along the closed branches. The prediction of pulsations in such complex systems has not yet been proved to be possible, indeed the methods proposed in the literature have only been applied to relatively simple geometries, mainly single or double side branch systems. We propose a prediction model of the self-sustained oscillations in multiple deep side branch systems. This has been established by means of an analytical model for the acoustic wave propagation in which a semi-analytical source model is included. Detuning of the acoustic resonator is often considered as a possible remedial measure to suppress pulsations. Although this countermeasure appears to be very effective for double side branch systems in cross configuration, its effectiveness has never been assessed for different geometries. The effectiveness of the length-detuning on the six side branch system appear to be limited and depends on the upstream and downstream acoustic boundary conditions of the main pipe.     

Scheduling programs with repetitive projects: A comparison of a simulated annealing,a genetic and a pair-wise swap algorithm

We address the problem of scheduling in programs involving the production of multiple units of the same product. Our study was motivated by a construction program for fast naval patrol boats. Other applications of this problem include procurement of multiple copies of aircraft, spacecraft, and weapon systems. In this problem we must decide how many units of the product to assign to each of a number of available crews (individuals, teams, subcontractors, etc.). These types of problems are characterized by two potentially conflicting considerations: 1) the need to complete each unit by its contractual due date, and 2) learning effects. Because of the first consideration, there is a tendency to use multiple crews for simultaneous production, so that meeting due dates is assured. However, the second consideration encourages assigning many units to a single crew so that learning effects are maximized. We study this scheduling problem with two different penalty cost structures and develop models for both versions. The models trade-off the penalty associated with late deliveries and the savings due to learning (and possibly incentive payments for early completion). We discuss different heuristic algorithms — simulated annealing, a genetic algorithm, and a pair-wise swap heuristic — as well as an exhaustive search to determine a baseline for comparisons. Our computational results show that the pair-wise swap algorithm is the most efficient solution procedure for these models.     

The de-Rham theorem and Shapiro lemma for Schwartz functions on Nash manifolds

In this paper we continue our work on Schwartz functions and generalized Schwartz functions on Nash (i.e. smooth semi-algebraic) manifolds. Our first goal is to prove analogs of the de-Rham theorem for de-Rham complexes with coefficients in Schwartz functions and generalized Schwartz functions. Using that we compute the cohomologies of the Lie algebra g of an algebraic group G with coefficients in the space of generalized Schwartz sections of G-equivariant bundle over a G-transitive variety M. We do it under some assumptions on topological properties of G and M. This computation for the classical case is known as the Shapiro lemma.