Computing the integer programming gap

We determine the maximal gap between the optimal values of an integer program and its linear programming relaxation, where the matrix and cost function are fixed but the right hand side is unspecified. Our formula involves irreducible decomposition of monomial ideals. The gap can be computed in polynomial time when the dimension is fixed. Partially supported by the National Science Foundation (DMS-0200729).     

Selective self-organization of guest molecules in self-assembled molecular boxes

This article describes the synthesis and binding properties of highly selective noncovalent molecular receptors 1(3).(DEB)6 and 3(3).(DEB)6 for different hydroxyl functionalized anthraquinones 2. These receptors are formed by the self-assembly of three calix[4]arene dimelamine derivative molecules (1 or 3) and six diethylbarbiturate (DEB) molecules to give 1(3).(DEB)6 or 3(3).(DEB)6. Encapsulation of 2 occurs in a highly organized manner; that is, a noncovalent hydrogen-bonded trimer of 2 is formed within the hydrogen-bonded receptors 1(3).(DEB)6 and 3(3).(DEB)6. Both receptors 1(3).(DEB)6 and 3(3).(DEB)6 change conformation from staggered to eclipsed upon complexation to afford a better fit for the 2(3) trimer. The receptor selectivity toward different anthraquinone derivatives 2 has been studied using 1H NMR spectroscopy, X-ray crystallography, UV spectroscopy, and isothermal microcalorimetry (ITC). The pi-pi stacking between the electron-deficient center ring of the anthraquinone derivatives 2a-c and 2e-g and the relatively electron-poor melamine units of the receptor is the driving force for the encapsulation of the guest molecules. The selectivity of the hydrogen-bonded host for the anthraquinone derivatives is the result of steric interactions between the guest molecules and the calix[4]arene aromatic rings of the host.     

Factors influencing the transport of short-chain alcohols through mesoporous gamma-alumina membranes

The pressure-driven transport of water, ethanol, and 1-propanol through supported gamma-alumina membranes with different pore diameters is reported. Water and alcohols had similar permeabilities when they were transported through gamma-alumina membranes with average pore diameters of 4.4 and 6.0 nm, and the permeability coefficient was found to be proportional to the square of pore size, in accordance with a viscous flow mechanism. For transport through membranes with an average diameter of 3.2 nm, the behavior of water was in accordance with the viscous flow mechanism, but the permeability of the membrane for ethanol and 1-propanol was much smaller than expected and could not be explained in terms of viscous flow. Although the low permeability of the membrane with 3.2 nm pores for ethanol and 1-propanol was partly due to the presence of small amounts of water in the alcohols, the permeability coefficients were still substantially smaller when water was absent. This intrinsic difference between water and alcohol may be due to differences in molecular size, chemisorption of alcohols on the oxide pore wall, which would lead to a reduction of the effective pore size, and/or a certain degree of translational ordering of the alcohol molecules inside the membrane pores, which leads to an effectively higher viscosity and, therefore, to a higher transport resistance.     

Temperature Controlled Sequential Gelation in Composite Microgel Suspensions

Depending on the volume fraction and interparticle interactions, colloidal suspensions can exhibit a variety of physical states, ranging from fluids, crystals, and glasses to gels. For microgel particles made of thermoresponsive polymers, both parameters can be tuned using environmental parameters such as temperature and ionic strength, making them excellent systems to experimentally study state transitions in colloidal suspensions. Using a simple two‐step synthesis it is shown that the properties of composite microgels, with a fluorescent latex core and a responsive microgel shell, can be finely tuned. With this system the transitions between glass, liquid, and gel states for suspensions composed of a single species are explored. Finally, a suspension of two species of microgels is demonstrated, with different transition temperatures, gels in a sequential manner. Upon increasing temperature a distinct core–sheath structure is formed with a primary gel composed of the species with lowest transition temperature, which acts as a scaffold for the aggregation of the second species.     

Generic and typical ranks of multi-way arrays

The concept of tensor rank was introduced in the 20s. In the 70s, when methods of Component Analysis on arrays with more than two indices became popular, tensor rank became a much studied topic. The generic rank may be seen as an upper bound to the number of factors that are needed to construct a random tensor. We explain in this paper how to obtain numerically in the complex field the generic rank of tensors of arbitrary dimensions, based on Terracini’s lemma, and compare it with the algebraic results already known in the real or complex fields. In particular, we examine the cases of symmetric tensors, tensors with symmetric matrix slices, complex tensors enjoying Hermitian symmetries, or merely tensors with free entries.     

Binding of small guest molecules to multivalent receptors

The complexation of phenol derivatives, aromatic carboxylic acids, and n-octylgalactopyranoside by hydrogen-bonded exo-receptors is described. The receptors are formed by self-assembly of differently functionalized calix[4]arene dimelamines with 5,5-diethyl barbiturate or butyl cyanurate. The multivalent complementary recognition site of the receptors is used very efficiently to complex multiple guests. A 1:6 binding mode was observed for phenol derivatives forming single hydrogen bonds with all six recognition sites of an ureido functionalized receptor assembly, while 1:3 complexation was observed for phenol derivatives which form two hydrogen bonds with two different ureido recognition sites of the same receptor. Aromatic carboxylic acids are complexed in a 1:6 ratio by receptors having six amino recognition sites. The complexation of n-octylgalactopyranoside by Gly-L-Ser functionalized receptors is also described, indicating that it is possible to use small peptidic fragments to complex biologically important molecules.     

Nonconventional morphologies in two-length scale block copolymer systems beyond the weak segregation theory

The order-disorder and order-order transitions (ODT and OOT) in the linear multiblock copolymers with two-length scale architecture A(fmN)(B(N2)A(N2))(n)B((1-f)mN) are studied under intermediate cooling below the ODT critical point where a nonconventional sequence of the OOTs was predicted previously [Smirnova et al., J. Chem. Phys. 124, 054907 (2006)] within the weak segregation theory (WST). To describe the ordered morphologies appearing in block copolymers (BCs) under cooling, we use the pseudospectral version of the self-consistent field theory (SCFT) with some modifications providing a good convergence speed and a high precision of the solution due to using the Ng iterations [J. Chem. Phys. 61, 2680 (1974)] and a reasonable choice of the predefined symmetries of the computation cell as well as initial guess for the iterations. The WST predicted sequence of the phase transitions is found to hold if the tails of the BCs under consideration are symmetric enough (mid R:0.5-fmid R:0.05, a large region of the face-centered cubic phase stability is found (up to our knowledge, first within the SCFT framework) inside of the body-centered cubic phase stability region. Occurrence of the two-dimensional and three-dimensional phases with the micelles formed, unlike the conventional diblock copolymers, by the longer (rather than shorter) tails, and its relationship to the BC architecture is first described in detail. The calculated spectra of the ordered phases show that nonmonotonous temperature dependence of the secondary peak scattering intensities accompanied by their vanishing and reappearance is rather a rule than exception.