Group generated by the Weierstrass points of a plane quartic

We describe the group generated by the Weierstrass points in the Jacobian of the curve This curve is the only curve of genus 3, apart from the fourth Fermat curve, possessing exactly twelve Weierstrass points.

     

The Manin-Mumford Conjecture: A Brief Survey

The Manin–Mumford conjecture asserts that if K is a fieldof characteristic zero, C a smooth proper geometrically irreduciblecurve over K, and J the Jacobian of C, then for any embeddingof C in J, the set C(K)J(K)_tors is finite. Although the conjecturewas proved by Raynaud in 1983, and several other proofs haveappeared since, a number of natural questions remain open, notablyconcerning bounds on the size of the intersection and the completedetermination of C(K)J(K)_tors for special families of curvesC. The first half of this survey paper presents the Manin–Mumfordconjecture and related general results, while the second describesrecent work mostly dealing with the above questions. 1991 MathematicsSubject Classification 11G10, 11G30, 11G35, 14G25, 14H25, 14H40.     

A technique for analytical calculation of observables in lattice gauge theories

It is shown that the partition function for a finite lattice factorizes into terms that can be associated with each vertex in the finite lattice. This factorization property forms the basis of a well-defined and efficient technique developed to calculate partition functions to high accuracy, on finite lattices for gauge theories. This technique, along with an expansion in finite lattices, provides a powerful means for calculating observables in lattice gauge theories. This is applied to SU(2) lattice gauge theory in four dimensions. The free energy, expectation value of a plaquette and specific heat are calculated. The results are very good both in the strong coupling and the weak coupling region and describe the crossover region quite well, agreeing all the way with the Monte Carlo data.     

Arbitrarily distortable Banach spaces of higher order

We study an ordinal rank on the class of Banach spaces with bases that quantifies the distortion of the norm of a given Banach space. The rank AD(?), introduced by P. Dodos, uses the transfinite Schreier families and has the property that AD(X) < ω₁ if and only if X is arbitrarily distortable. We prove several properties of this rank as well as some new results concerning higher order l₁ spreading models. We also compute this rank for several Banach spaces. In particular, it is shown that the class of Banach spaces \(\left( {X_0^{{\omega ^\xi }}} \right)\xi < {\omega _1}\), which each admit l₁ and c₀ spreading models hereditarily, and were introduced by S. A. Argyros, the first and third author, satisfy \(AD\left( {X_0^{{\omega ^\xi }}} \right) = {\omega ^\xi } + 1\). This answers some questions of Dodos.     

A dual method of constructing hereditarily indecomposable Banach spaces

A new method of defining hereditarily indecomposable Banach spaces is presented. This method provides a unified approach for constructing reflexive HI spaces and also HI spaces with no reflexive subspace. All the spaces presented here satisfy the property that the composition of any two strictly singular operators is a compact one. This yields the first known example of a Banach space with no reflexive subspace such that every operator has a non-trivial closed invariant subspace.     

Development of a validated HPLC method for the determination of B-complex vitamins in pharmaceuticals and biological fluids after solid phase extraction

An HPLC method was developed for the simultaneous determination of seven water-soluble vitamins, viz. thiamine, riboflavin, nicotinic acid, nicotinamide, pyridoxine, cyanocobalamin, and folic acid, in multivitamin pharmaceutical formulations and biological fluids (blood serum and urine). Separation was achieved at ambient temperature on a Phenomenex Luna C18 (150 x 4.6 mm) analytical column. Gradient elution was performed starting at a 99:1 A:B v/v composition, where A: 0.05 M CH3COONH4/CH3OH (99/1) and B: H2O/CH3OH (50/50), at a flow rate of 0.8 mL/min. After a 4-min isocratic elution the composition was changed to 100% of B in 18 min and elution continued isocratically for 8 min. Detection was performed with a photodiode array detector at 280 nm. Each vitamin was quantitatively determined at its maximum wavelength. Spectral comparison was used for peak identification in real samples. Detection limits were in the range of 1.6-3.4 ng, per 20-microL injection, while linearity held up to 25 ng/microL. Accuracy, intra-day repeatability (n = 6), and inter-day precision (n = 7) were found to be satisfactory. Theobromine (2 ng/microL) was used as internal standard. Sample preparation of biological fluids was performed by SPE on Supelclean LC-18 cartridges with methanol-water 85/15 v/v as eluent. Extraction recoveries from biological matrices ranged from 84.6% to 103.0%.     

Unstable Displacement of Non-aqueous Phase Liquids with Surfactant and Polymer

In this paper, we study two-phase multicomponent displacement of two immiscible fluids in both homogeneous and heterogeneous porous media. In many applications such as enhanced oil recovery, fluid mixing and spreading can be detrimental to the efficacy of the process. Here, we show that when an initially immobile phase is being displaced by a finite-size slug of solvents (surfactant and polymer), viscous fingering significantly enhances mixing and spreading of solvents. These effects are similar to those caused by medium heterogeneity and lead to poor displacement efficiency. We first quantify the displacement efficiency subject to different mobility ratios, Peclet numbers, and levels of medium heterogeneity. We observe a non-monotonic behavior in displacement efficiency as a function of mobility ratio, indicating that although stable frontal interface is desirable, miscible viscous fingering on the rear interface will eventually disintegrate the solvents slugs and reduce the displacement efficiency. Then, we show that miscible viscous fingering developing on the rear interface of the chemical slug could be greatly suppressed when viscosity contrast is gradually decreased using exponential or linear functions, leading to 10% increase in displacement efficiency while using the same amount of chemicals. To elucidate this low displacement efficiency, we study the evolution of mixing, spreading, and interfacial length and show that while higher viscosity ratios are quite effective in mobilizing the initially immobile phase in 1D displacements, they are in fact detrimental in 2D unstable displacements since they enhance mixing and spreading of solvents.