Rate of Convergence for Cardy’s Formula

We show that crossing probabilities in 2D critical site percolation on the triangular lattice in a piecewise analytic Jordan domain converge with power law rate in the mesh size to their limit given by the Cardy–Smirnov formula. We use this result to obtain new upper and lower bounds of ${e^{O(\sqrt{{\rm log}\,{\rm log} R})}\, R^{-1/3}}$ for the probability that the cluster at the origin in the half-plane has diameter R, improving the previously known estimate of R ^?1/3+o(1).     

Synthesis and biological evaluation of 5-methyl-4-phenyl thiazole derivatives as anticancer agents

Abstract

New N-(5-methyl-4-phenylthiazol-2-yl)-2-(substituted thio)acetamides were synthesized and studied for their anticancer activity. The title compounds were procured by reacting 2-chloro-N-(5-methyl-4-phenylthiazol-2-yl)acetamide with some mercapto derivatives. The structural elucidation of the compounds was performed by ¹H-NMR, ¹³C-NMR and LC-MS/MS spectral data and elemental analyses. The synthesized compounds were investigated for their antitumor activities against A549 human lung adenocarcinoma cells and NIH/3T3 mouse embryoblast cell line for determining their selective cytotoxicity. 2-[(1-methyl-1H-tetrazol-5-yl)thio]-N-(5-methyl-4-phenylthiazol-2-yl)acetamide (4c) showed high selectivity, and whose IC₅₀ value was determined as 23.30?±?0.35?µM and >1000?µM against A549 human lung adenocarcinoma cells and NIH/3T3 mouse embryoblast cell lines, respectively. 2-[(1-Methyl-1H-imidazol-2-yl)thio]-N-(5-methyl-4-phenyl thiazol-2-yl)acetamide (4a) and 2-[(1-Methyl-1H-tetrazol-5-yl)thio]-N-(5-methyl-4-phenyl thiazol-2-yl)acetamide (4c) exhibited the highest apoptosis percentage among those tested, but not as high as the standard, cisplatin.     

Approximation algorithm for minimizing total latency in machine scheduling with deliveries

We study the problem of minimizing total latency in machine scheduling with deliveries, which is defined as follows. There is a set of n jobs to be processed by a single machine at a plant, where job J_i is associated with its processing time and a customer i located at location i to which the job is to be delivered. In addition, there is a single uncapacitated delivery vehicle available. All jobs (vehicle) are available for processing (delivery) at time 0. Our aim is to determine the sequence in which the jobs should be processed in the plant, the departure times of the vehicle from the plant, and the routing of the vehicle, so as to minimize the total latency (job delivery time). We present a 6e16.309691-approximation algorithm for the problem.     

A tournament approach to pattern avoiding matrices

We consider the following Turán-type problem: given a fixed tournament H, what is the least integer t = t(n,H) so that adding t edges to any n-vertex tournament, results in a digraph containing a copy of H. Similarly, what is the least integer t = t(T _n ,H) so that adding t edges to the n-vertex transitive tournament, results in a digraph containing a copy of H. Besides proving several results on these problems, our main contributions are the following:

• Pach and Tardos conjectured that if M is an acyclic 0/1 matrix, then any n × n matrix with n(log n) ^O(1) entries equal to 1 contains the pattern M. We show that this conjecture is equivalent to the assertion that t(T _n ,H) = n(log n) ^O(1) if and only if H belongs to a certain (natural) family of tournaments.
• We propose an approach for determining if t(n,H) = n(log n) ^O(1). This approach combines expansion in sparse graphs, together with certain structural characterizations of H-free tournaments. Our result opens the door for using structural graph theoretic tools in order to settle the Pach–Tardos conjecture.

     

Temperature Integration: An efficient procedure for calculation of free energy differences

We propose a method, Temperature Integration, which allows an efficient calculation of free energy differences between two systems of interest, with the same degrees of freedom, which may have rough energy landscapes. The method is based on calculating, for each single system, the difference between the values of lnZ$lnZ$ at two temperatures, using a Parallel Tempering procedure. If our two systems of interest have the same phase space volume, they have the same values of lnZ$lnZ$ at high-T$T$, and we can obtain the free energy difference between them, using the two single-system calculations described above. If the phase space volume of a system is known, our method can be used to calculate its absolute (versus relative) free energy as well. We apply our method and demonstrate its efficiency on a “toy model” of hard rods on a 1-dimensional ring.     

Avoider–Enforcer games played on edge disjoint hypergraphs

We analyze Avoider–Enforcer games played on edge disjoint hypergraphs, providing an analog of the classic and well known game Box, due to Chvátal and Erd?s. We consider both strict and monotone versions of Avoider–Enforcer games, and for each version we give a sufficient condition to win for each player. We also present applications of our results to several general Avoider–Enforcer games.     

Comparison of CP-PC-SAFT and SAFT-VR-Mie in Predicting Phase Equilibria of Binary Systems Comprising Gases and 1-Alkyl-3-methylimidazolium Ionic Liquids

This study compares performances of the Critical Point-based revision of Perturbed-Chain SAFT (CP-PC-SAFT) and the SAFT of Variable Range and Mie Potential (SAFT-VR-Mie) in predicting the available data on VLE, LLVE, critical loci and saturated phase densities of systems comprising CO, O₂, CH₄, H₂S, SO₂, propane, the refrigerants R22, R23, R114, R124, R125, R125, R134a, and R1234ze(E) and ionic liquids (ILs) with 1-alkyl-3-methylimidazolium ([C_nmim]⁺) cations and bis(trifluoromethanesulfonyl)imide ([NTf₂]⁻), tetrafluoroborate ([BF₄]⁻) and hexafluorophosphate ([PF₆]⁻) anions. Both models were implemented in the entirely predictive manner with k₁₂ = 0. The fundamental Global Phase Diagram considerations of the IL systems are discussed. It is demonstrated that despite a number of quantitative inaccuracies, both models are capable of reproducing the regularities characteristic for the considered systems, which makes them suitable for preliminary estimation of selectivity of the ILs in separating various gases.