SU(2) instantons with boundary jumps and spin tunneling in magnetic molecules

Coherent state path integrals are shown in general to contain instantons with jumps at the boundaries, i.e., boundary points lying outside classical phase space. Inclusion of these instantons is shown to resolve the "missing quench paradox" in the magnetic molecule Fe8, i.e., the fact that the tunneling between the ground Zeeman states of this molecule is quenched at only four magnetic field values, instead of the ten that would be expected from the topological Berry phase between interfering instantons. An approximate formula is found for the location of the four remaining quenches.     

A Prime Sensitive Hankel Determinant of Jacobi Symbol Enumerators

We show that the determinant of a Hankel matrix of odd dimension n whose entries are the enumerators of the Jacobi symbols which depend on the row and the column indices vanishes if and only if n is composite. If the dimension is a prime p, then the determinant evaluates to a polynomial of degree p − 1 which is the product of a power of p and the generating polynomial of the partial sums of Legendre symbols. The sign of the determinant is determined by the quadratic character of −1 modulo p. The proof of the evaluation makes use of elementary properties of Legendre symbols, quadratic Gauss sums, and orthogonality of trigonometric functions.     

A combinatorial interpretation of the inverse kostka matrix

The Kostka matrix K relates the. homogeneous and the Schur bases in the ring of symmetric functions where K _λ,μenumerates the number of column strict tableaux of shape λ and type μ. We make use of the Jacobi -Trudi identity to give a combinatorial interpretation for the inverse of the Kostka matrix in terms of certain types of signed rim hook tabloids. Using this interpretation, the matrix identity KK ^?1=Iis given a purely combinatorial proof. The generalized Jacobi-Trudi identity itself is also shown to admit a combinatorial proof via these rim hook tabloids. A further application of our combinatorial interpretation is a simple rule for the evaluation of a specialization of skew Schur functions that arises in the computation of plethysms.     

A multilinear operator for almost product evaluation of Hankel determinants

In a recent paper we have presented a method to evaluate certain Hankel determinants as almost products; i.e. as a sum of a small number of products. The technique to find the explicit form of the almost product relies on differential-convolution equations and trace calculations. In the trace calculations a number of intermediate nonlinear terms involving determinants occur, but only to cancel out in the end.In this paper, we introduce a class of multilinear operators γ acting on tuples of matrices as an alternative to the trace method. These operators do not produce extraneous nonlinear terms, and can be combined easily with differentiation.The paper is self contained. An example of an almost product evaluation using γ-operators is worked out in detail and tables of the γ-operator values on various forms of matrices are provided. We also present an explicit evaluation of a new class of Hankel determinants and conjectures.     

A parallel method for fast and practical high-order newton interpolation

We present parallel algorithms for the computation and evaluation of interpolating polynomials. The algorithms use parallel prefix techniques for the calculation of divided differences in the Newton representation of the interpolating polynomial. Forn+1 given input pairs, the proposed interpolation algorithm requires only 2 [log(n+1)]+2 parallel arithmetic steps and circuit sizeO(n ²), reducing the best known circuit size for parallel interpolation by a factor of logn. The algorithm for the computation of the divided differences is shown to be numerically stable and does not require equidistant points, precomputation, or the fast Fourier transform. We report on numerical experiments comparing this with other serial and parallel algorithms. The experiments indicate that the method can be very useful for very high-order interpolation, which is made possible for special sets of interpolation nodes.Supported in part by the National Science Foundation under Grant No. NSF DCR-8603722.Supported by the National Science Foundation under Grants No. US NSF MIP-8410110, US NSF DCR85-09970, and US NSF CCR-8717942 and AT&T under Grant AT&T AFFL67Sameh.     

Purification of Acetylcholinesterase by 9-Amino-1,2,3,4-tetrahydroacridine from Human Erythrocytes

The acetylcholinesterase enzyme was purified from human erythrocyte membranes using a simple and effective method in a single step. Tacrine (9-amino-1,2,3,4-tetrahydroacridine) is a well-known drug for the treatment of Alzheimer's disease, which inhibits cholinesterase. We have developed a tacrine ligand affinity resin that is easy to synthesize, inexpensive and selective for acetylcholinesterase. The affinity resin was synthesized by coupling tacrine as the ligand and l-tyrosine as the spacer arm to CNBr-activated Sepharose 4B. Acetylcholinesterase was purified with a yield of 23.5 %, a specific activity of 9.22 EU/mg proteins and 658-fold purification using the affinity resin in a single step. During purification, the enzyme activity was measured using acetylthiocholine iodide as a substrate and 5,5′-dithiobis-(2-nitrobenzoicacid) as the chromogenic agent. The molecular weight of the enzyme was determined as about 70 kDa monomer upon disulphide reduction by sodium dodecyl sulphate polyacrylamide gel electrophoresis. K _m, V _max, optimum pH and optimum temperature for acetylcholinesterase were found by means of graphics for acetylthiocholine iodide as the substrate. The optimum pH and optimum temperature of the acetylcholinesterase were determined to be 7.4 and 25–35 °C. The Michaelis–Menten constant (K _m) for the hydrolysis of acetylthiocholine iodide was found to be 0.25 mM, and the V _max was 0.090 μmol/mL/min. Maximum binding was achieved at 2 °C with pH 7.4 and an ionic strength of approximately 0.1 M. The capacity for the optimum condition was 0.07 mg protein/g gel for acetylcholinesterase.