Solving Medium-Density Subset Sum Problems in Expected Polynomial Time: An Enumeration Approach

The subset sum problem (SSP) can be briefly stated as: given a target integer $E$ and a set $A$ containing $n$ positive integer $a_j$, find a subset of $A$ summing to $E$. The \textit{density} $d$ of an SSP instance is defined by the ratio of $n$ to $m$, where $m$ is the logarithm of the largest integer within $A$. Based on the structural and statistical properties of subset sums, we present an improved enumeration scheme for SSP, and implement it as a complete and exact algorithm (EnumPlus). The algorithm always equivalently reduces an instance to be low-density, and then solve it by enumeration. Through this approach, we show the possibility to design a sole algorithm that can efficiently solve arbitrary density instance in a uniform way. Furthermore, our algorithm has considerable performance advantage over previous algorithms. Firstly, it extends the density scope, in which SSP can be solved in expected polynomial time. Specifically, It solves SSP in expected $O(n\log{n})$ time when density $d \geq c\cdot \sqrt{n}/\log{n}$, while the previously best density scope is $d \geq c\cdot n/(\log{n})^{2}$. In addition, the overall expected time and space requirement in the average case are proven to be $O(n^5\log n)$ and $O(n^5)$ respectively. Secondly, in the worst case, it slightly improves the previously best time complexity of exact algorithms for SSP. Specifically, the worst-case time complexity of our algorithm is proved to be $O((n-6)2^{n/2}+n)$, while the previously best result is $O(n2^{n/2})$.Comment: 11 pages, 1 figur

Characterization studies of 1-(4-cyano-2-oxo-1,2-dihydro-1-pyridyl)-3-(4-cyano-1,2-dihydro-1-pyridyl)propane formed from the reaction of hydroxide Ion with 1,3-Bis-(4-cyano pyridinium)propane

The aqueous alkaline reaction of 1,3-bis(4-cyanopyridinium)propane dibromide, a reactant constituted of two pyridinium rings linked by a three-methylene bridge, generates a novel compound,1 -(4-cyano-2-oxo-1,2-dihydro-1-pyridyl)-3-(4-cyano-1,2-dihydro-1-pyridyl)propane. The reaction pathway is attributed to the proximity of the OH- ion inserted between two pyridinium moieties, which occurs only in bis(pyridinium) derivatives connected by short methylene spacers, where charge-conformational effects are important.A reação em meio aquoso alcalino do dibrometo de 1,3-bis(4-cianopiridinium)propano, um composto constituído por dois anéis piridínicos conectados por uma ponte metilênica de três carbonos, gerou um novo composto, o 1-(4-ciano-2-oxo-1,2-diidro-1-piridil)-3-(4-ciano-1,2-diidro-1-piridil)propano. O resultado da reação é atribuído à proximidade do íon OH-, encapsulado entre os dois anéis piridínicos, fato este observado apenas em derivados bis-piridínicos conectados por pontes metilênicas de curta extensão, onde imperam efeitos de carga aliados à conformação.CNPqFAPESPFundação Araucári

Relationship between Structure, Entropy and Diffusivity in Water and Water-like Liquids

Anomalous behaviour of the excess entropy ($S_e$) and the associated scaling relationship with diffusivity are compared in liquids with very different underlying interactions but similar water-like anomalies: water (SPC/E and TIP3P models), tetrahedral ionic melts (SiO$_2$ and BeF$_2$) and a fluid with core-softened, two-scale ramp (2SRP) interactions. We demonstrate the presence of an excess entropy anomaly in the two water models. Using length and energy scales appropriate for onset of anomalous behaviour, the density range of the excess entropy anomaly is shown to be much narrower in water than in ionic melts or the 2SRP fluid. While the reduced diffusivities ($D^*$) conform to the excess entropy scaling relation, $D^* =A\exp (\alpha S_e)$ for all the systems (Y. Rosenfeld, Phys. Rev. A {\bf 1977}, {\it 15}, 2545), the exponential scaling parameter, $\alpha$, shows a small isochore-dependence in the case of water. Replacing $S_e$ by pair correlation-based approximants accentuates the isochore-dependence of the diffusivity scaling. Isochores with similar diffusivity scaling parameters are shown to have the temperature dependence of the corresponding entropic contribution. The relationship between diffusivity, excess entropy and pair correlation approximants to the excess entropy are very similar in all the tetrahedral liquids.Comment: 24 pages, 4 figures, to be published in Journal of Physical Chemistry

Influence of Molecular Dipole Orientations on Long-Range Exponential Interaction Forces at Hydrophobic Contacts in Aqueous Solutions

Strong and particularly long ranged (>100 nm) interaction forces between apposing hydrophobic lipid monolayers are now well understood in terms of a partial turnover of mobile lipid patches, giving rise to a correlated long-range electrostatic attraction. Here we describe similarly strong long-ranged attractive forces between self-assembled monolayers of carboranethiols, with dipole moments aligned either parallel or perpendicular to the surface, and hydrophobic lipid monolayers deposited on mica. We compare the interaction forces measured at very different length scales using atomic force microscope and surface forces apparatus measurements. Both systems gave a long-ranged exponential attraction with a decay length of 2.0 +/- 0.2 nm for dipole alignments perpendicular to the surface. The effect of dipole alignment parallel to the surface is larger than for perpendicular dipoles, likely due to greater lateral correlation of in-plane surface dipoles. The magnitudes and range of the measured interaction forces also depend on the surface area of the probe used: At extended surfaces, dipole alignment parallel to the surface leads to a stronger attraction due to electrostatic correlations of freely rotating surface dipoles and charge patches on the apposing surfaces. In contrast, perpendicular dipoles at extended surfaces, where molecular rotation cannot lead to large dipole correlations, do not depend on the scale of the probe used. Our results may be important to a range of scale-dependent interaction phenomena related to solvent/water structuring on dipolar and hydrophobic surfaces at interfaces

Salt-induced aggregation and fusion of dioctadecyldimethylammonium chloride and sodium dihexadecylphosphate vesicles.

Small dioctadecyldimethylammonium chloride (DODAC) vesicles prepared by sonication fuse upon addition of NaCl as detected by several methods (electron microscopy, trapped volume determinations, temperature-dependent phase transition curves, and osmometer behavior. In contrast, small sodium dihexadecyl phosphate (DHP) vesicles mainly aggregate upon NaCl addition as shown by electron microscopy and the lack of osmometer behavior. Scatter-derived absorbance changes of small and large DODAC or DHP vesicles as a function of time after salt addition were obtained for a range of NaCl or amphiphile concentration. These changes were interpreted in accordance with a phenomenological model based upon fundamental light-scattering laws and simple geometrical considerations. Short-range hydration repulsion between DODAC (or DHP) vesicles is possibly the main energy barrier for the fusion process

INTEGRATION OF THE NONLINEAR POISSON BOLTZMANN-EQUATION FOR CHARGED VESICLES IN ELECTROLYTIC SOLUTIONS

The Poisson-Boltzmann equation (PBE), with specific ion-surface interactions and a cell model, was used to calculate the electrostatic properties of aqueous solutions containing vesicles of ionic amphiphiles. Vesicles are assumed to be water- and ion-permeable hollow spheres and specific ion adsorption at the surfaces was calculated using a Volmer isotherm. We solved the PBE numerically for a range of amphiphile and salt concentrations (up to 0.1 M) and calculated co-ion and counterion distributions in the inside and outside of vesicles as well as the fields and electrical potentials. The calculations yield results that are consistent with measured values for vesicles of synthetic amphiphiles