A Lower Bound for the Determinantal Complexity of a Hypersurface

We prove that the determinantal complexity of a hypersurface of degree \(d > 2\) is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the \(3 \times 3\) permanent is 7. We also prove that for \(n> 3\), there is no nonsingular hypersurface in \({\mathbb {P}}^n\) of degree d that has an expression as a determinant of a \(d \times d\) matrix of linear forms, while on the other hand for \(n \le 3\), a general determinantal expression is nonsingular. Finally, we answer a question of Ressayre by showing that the determinantal complexity of the unique (singular) cubic surface containing a single line is 5.     

Cluster Partition Function and Invariants of 3-Manifolds

The author reviews some recent developments in Chern-Simons theory on a hyperbolic 3-manifold M with complex gauge group G.The author focuses on the case of G =SL(N,C) and M being a knot complement:M =S3 \ k.The main result presented in this note is the cluster partition function,a computational tool that uses cluster algebra techniques to evaluate the Chern-Simons path integral for G =SL(N,C).He also reviews various applications and open questions regarding the cluster partition function and some of its relation with string theory.     

Beijing Lectures on the Grade Restriction Rule

The authors describe the relationships between categories of B-branes in different phases of the non-Abelian gauged linear sigma model.The relationship is described explicitly for the model proposed by Hori and Tong with non-Abelian gauge group that connects two non-birational Calabi-Yau varieties studied by Rφdland.A grade restriction rule for this model is derived using the hemisphere partition function and it is used to map B-type D-branes between the two Calabi-Yau varieties.     

Cleft extensions of a family of braided Hopf algebras

We introduce a family of braided Hopf algebras that (in characteristic zero) generalizes the rank 1 Hopf algebras introduced by Krop and Radford and we study its cleft extensions.     

Combined chemical and phase equilibrium for the hydration of ethylene to ethanol calculated by means of the Peng-Robinson-Stryjek-Vera equation of state and the Wong-Sandler mixing rules

Due to the economics of the ethylene market and the subsidized production of fermentation-based ethanol in some countries, use of the ethylene hydration process to make ethanol has been steadily declining. The economics of this process might improve by combining the reaction and separation in a reactive distillation column, whose conceptual design requires a study of the combined chemical and phase equilibrium (CPE) of the reacting system. In this work, the Peng-Robinson-Stryjek-Vera equation of state was combined with the UNIQUAC activity coefficient model through the Wong-Sandler (WS) mixing rules in order to correlate the available experimental data for the vapor-liquid equilibria (VLE) of the ethylene-water, ethylene-ethanol, and ethanol-water binary systems at 200 °C. The interaction energies of the UNIQUAC model and the binary interaction coefficient of the WS mixing rules were used as the fitting parameters. From the optimum values of these parameters, both the VLE and the combined CPE of the ethylene-water-ethanol ternary system were predicted at 200 °C and various pressures. At this temperature, the catalytic activity of a H-pentasil zeolite has already been reported to exhibit a maximum for ethylene hydration, and also the experimentally measured two-phase region of the ternary system is sufficiently wide. By means of the reactive flash method, the chemical equilibrium compositions of the liquid and vapor phases were determined for several pressures, and the equilibrium conversion and the vapor fraction were calculated as functions of the ethylene to water feed mole ratio. It turns out that the vapor-liquid mixed-phase hydration of ethylene achieves equilibrium conversions much higher than those computed for a vapor-phase reaction that would hypothetically occur at the same conditions of pressure and feed mole ratio. It was found that the reactive phase diagram of the ternary system exhibits a critical point at 200 °C and 155 atm.     

Hydrogen bonding interactions between α-, β-glucose,and methacrylic acid

Intermolecular interactions between α-, β-glucose, and methacrylic acid (MAA) have been investigated. Twenty-two possible conformations have been optimized at the DFT(B3LYP) level of theory with the 6-31G(d) basis set. The geometrical parameters for the most stable configurations of hydrogen bonding sites in the optimized systems have been determined. The binding energies ΔE _bind have been calculated at the MP2/6-311++G(d,p) level of approximation taking into account the basis set superposition error (BSSE) and the zero-point vibrational energies corrections. Results indicate that the most stabilized complexes form hydrogen bonds either through carboxylic and hemiacetal oxygen atoms acting as proton acceptors. Both, α- and β-anomers are studied in the pyranose six-membered ring. In all complexes, the nuclear quadrupole coupling constants (χ) for ¹⁷O nuclei were obtained about 10.0 MHz, while for the ²H atoms they vary from ≈200.0 to ≈350.0 kHz.     

Stereospecific ring expansion of chiral vinyl aziridines

In this report, it is demonstrated that chiral vinyl aziridines can be stereospecifically ring expanded. This synthetic approach allows controlled access to chiral 2,5-cis- or 2,5-trans-3-pyrroline products from starting materials with the appropriate aziridine geometry. Twenty three ring expansion examples, most of which feature a stereospecific cyclization, are presented.     

Limit cycles and invariant cylinders for a class of continuous and discontinuous vector field in dimention 2n

The subject of this paper concerns with the bifurcation of limit cycles and invariant cylinders from a global center of a linear differential system in dimension 2n perturbed inside a class of continuous and discontinuous piecewise linear differential systems. Our main results show that at most one limit cycle and at most one invariant cylinder can bifurcate using the expansion of the displacement function up to first order with respect to a small parameter. This upper bound is reached. For proving these results we use the averaging theory in a form where the differentiability of the system is not needed.     

Solving the vendor–buyer integrated inventory system with arithmetic–geometric inequality

In the past, economic order quantity (EOQ) and economic production quantity (EPQ) were treated independently from the viewpoints of the buyer or the vendor. In most cases, the optimal solution for one player was non-optimal to the other player. In today’s competitive markets, close cooperation between the vendor and the buyer is necessary to reduce the joint inventory cost and the response time of the vendor–buyer system. The successful experiences of National Semiconductor, Wal-Mart, and Procter and Gamble have demonstrated that integrating the supply chain has significantly influenced the company’s performance and market share (Simchi-Levi et al. (2000) [1]). Recently, Yang et al. (2007) [2] presented an inventory model to determine the economic lot size for both the vendor and buyer, and the number of deliveries in an integrated two stage supply chain. In this paper, we present an alternative approach to determine the global optimal inventory policy for the vendor–buyer integrated system using arithmetic–geometric inequality.