Continuous cellobiose hydrolysis using self-immobilized β-glucosidase fromAspergillus phoenicis QM 329 in a fluidized-bed reactor

Aspergillus phoenicis QM 329 was grown in the shape of beads in shake flasks and in an air-lift fermentor. The production of β-glucosidase started when the carbon source, glucose, was consumed. The β-glucosidase activity was retained in the beads at a pH below 6.0. The influence of bead diameter on enzyme activity and the pH and temperature optima for cellobiose hydrolysis has been studied. The enzyme-containing beads were used in a fluidized-bed reactor for continuous cellobiose hydrolysis, and a productivity of 2.0 g/L-h at a substrate conversion of 76% was obtained. The self-immobilized β- glucosidase is a stable and reusable enzyme with a half-life of 700 h when operating at 50°C and pH 4.8.

     

Magnetic inelastic scattering of neutrons by a ferromagnetic crystal

This paper deals with a theoretical investigation of the energy spectrum of thermal neutrons inelastically scattered by a ferromagnetic crystal. The method of double-time temperature-dependent Green’s functions is used. Formulæ are derived for the line width and for an asymmetry parameter leading to a deviation from the Lorentzian form. The results are discussed and compar d with earlier work. Special attention is paid to the question of the deviation of the energy distribution from the purely Lorentzian form.     

Influence of fresnel losses and multiple-beam interference in the measurement of polarized light absorption

A theory is presented describing macroscopic optical effects observed in different applications of linear and circular dichroism spectroscopy. It is shown that the optical Fresnel losses and multiple-beam interference may influence this type of measurement.     

Strong correlation effects in the ionisation of CS2

The Hell photoelectron spectrum of CS₂ is presented. This spectrum is investigated by two different Green's function calculations. For the outer valence region the origin and assignment of satellite lines of significant intensity is clarified. Strong final state correlation effects are found for the inner valence region showing the so-called breakdown of single-particle picture of ionisation.     

Zig-Zag procedures for memory allocation and retrieval of dense equilateral arrays of dynamically varying order

A zig-zag method is introduced for efficient memory allocation and retrieval of dense equilateral arrays. These arrays are of dynamically varying order. No repacking of elements is required.     

Algorithms for confluent Vandermonde systems

Two compact algorithms are developed for solving systems of linear equationsV x=b andV ^T a=f, whereV=V( ₀, ₁, ..., _n ) is a confluent Vandermonde matrix of Hermite type. The solution is obtained by one forward and one backward vector recursion, starting with the right hand side. The total amount of storage is only 2n. The number of arithmetic operations needed isO(n ²) and compares favourably with other proposed methods.     

Perturbation theory for pseudo-inverses

A perturbation theory for pseudo-inverses is developed. The theory is based on a useful decomposition (theorem 2.1) ofB ⁺ -A ⁺ whereB andA arem ×n matrices. Sharp estimates of B ⁺ -A ⁺ are derived for unitary invariant norms whenA andB are of the same rank and B -A is small. Under similar conditions the perturbation of a linear systemAx=b is studied. Realistic bounds on the perturbation ofx=A ⁺ b andr=b=Ax are given. Finally it is seen thatA ⁺ andB ⁺ can be compared if and only ifR(A) andR(B) as well asR(A ^H ) andR(B ^H ) are in the acute case. Some theorems valid only in the acute case are also proved.This work was sponsored in part by The Swedish Institute of Applied Mathematics.     

Solving linear least squares problems by Gram-Schmidt orthogonalization

A general analysis of the condition of the linear least squares problem is given. The influence of rounding errors is studied in detail for a modified version of the Gram-Schmidt orthogonalization to obtain a factorizationA=QR of a givenm×n matrixA, whereR is upper triangular andQ ^T Q=I. Letx be the vector which minimizes ‖b?Ax‖₂ andr=b?Ax. It is shown that if inner-products are accumulated in double precision then the errors in the computedx andr are less than the errors resulting from some simultaneous initial perturbation δA, δb such that

$$\parallel \delta A\parallel _E /\parallel A\parallel _E \approx \parallel \delta b\parallel _2 /\parallel b\parallel _2 \approx 2 \cdot n^{3/2} machine units.$$     

Dynamic hashing

A new file organisation called dynamic hashing is presented. The organisation is based on normal hashing, but the allocated storage space can easily be increased and decreased without reorganising the file, according to the number of records actually stored in the file. The expected storage utilisation is analysed and is shown to be approximately 69% all the time. Algorithms for inserting and deleting a record are presented and analysed. Retrieval of a record is fast, requiring only one access to secondary storage. There are no overflow records. The proposed scheme necessitates maintenance of a relatively small index structured as a forest of binary trees or slightly modified binary tries. The expected size of the index is analysed and a compact representation of the index is suggested.     

Iterative refinement of linear least squares solutions I

An iterative procedure is developed for reducing the rounding errors in the computed least squares solution to an overdetermined system of equationsAx =b, whereA is anm ×n matrix (m n) of rankn. The method relies on computing accurate residuals to a certain augmented system of linear equations, by using double precision accumulation of inner products. To determine the corrections, two methods are given, based on a matrix decomposition ofA obtained either by orthogonal Householder transformations or by a modified Gram-Schmidt orthogonalization. It is shown that the rate of convergence in the iteration is independent of the right hand side,b, and depends linearly on the condition number, 2135;(A), of the rectangular matrixA. The limiting accuracy achieved will be approximately the same as that obtained by a double precision factorization.In a second part of this paper the case whenx is subject to linear constraints and/orA has rank less thann is covered. Here also ALGOL-programs embodying the derived algorithms will be given.This work was sponsored by the Swedish Natural Science Research Council.