Laurent series for inversion of linearly perturbed bounded linear operators on Banach space

In this paper we find necessary and sufficient conditions for the existence of a Laurent series expansion with a finite order pole at the origin for the inverse of a linearly perturbed bounded linear operator mapping one Banach space to another. In particular we show that the inversion defines linear projections that separate the Banach spaces into corresponding complementary subspaces. We present two pertinent applications.     

Writing representations over minimal fields

The chief aim of this paper is to describe a procedure which, given a d-dimensional absolutely irreducible matrix representation of a finite group over a finite field E, produces an equivalent representation such that all matrix entries lie in a subfield F of E which is as small as possible. The algorithm relies on a matrix version of Hilbert's Theorem 90, and is probabilistic with expected running time O(|E:F|d³) when |F| is bounded. Using similar methods we then describe an algorithm which takes as input a prime number and a power-conjugate presentation for a finite soluble group, and as output produces a full set of absolutely irreducible representations of the group over fields whose characteristic is the specified prime, each representation being written over its minimal field.     

Optimal Mathematical Models for Nonlinear Dynamical Systems

We propose a new method for the optimal causal representation of nonlinear systems. The proposed approach is based on the best constrained approximation of mappings in probability spaces by operators constructed from matrices of special form so that the approximant preserves the causality property. It is supposed that the observable input is contaminated with noise. The approximant minimises the mean-square difference between a desired output signal and the output signal of the approximating model. The method provides a numerically realisable mathematical model of the system. An analysis is given of the error associated with this representation.     

Eplt4 Proteinaceous Elicitor Produced in Pichia pastoris Has a Protective Effect Against Cercosporidium sofinum Infections of Soybean Leaves

A complementary DNA library was constructed from the mycelium of Trichoderma asperellum T4, and a highly expressed gene fragment named EplT4 was found. In order to find a more efficient and cost-effective way of obtaining EplT4, this study attempted to produce EplT4 using a Pichia pastoris expression system. The gene encoding EplT4, with an additional 6-His tag at the C-terminus, was cloned into the yeast vector pPIC9K and expressed in the P. pastoris strain GS115 to obtaining more protein for the further research. Transformants of P. pastoris were selected by PCR analysis, and the ability to secrete high levels of the EplT4 protein was determined. The optimal conditions for induction were assayed using the shake flask method and an enzyme-linked immunosorbent assay. The yield of purified EplT4 was approximately 20 mg/L by nickel affinity chromatography and gel-filtration chromatography. Western blot and matrix-assisted laser desorption/ionization time-of-flight mass spectrometer analysis revealed that the recombinant EplT4 was expressed in both its monomers and dimers. Soybean leaves treated with the EplT4 monomer demonstrated the induction of glucanase, chitinase III-A, cysteine proteinase inhibitor, and peroxidase genes. Early cellular events in plant defense response were also observed after incubation with EplT4. Soybean leaves protected by EplT4 against the pathogen Cercosporidium sofinum (Hara) indicated that EplT4 produced in P. pastoris was biologically active and would be potentially useful for improving food security.     

Sequential single-shot imaging of nanoscale dynamic interactions with a table-top soft x-ray laser

We demonstrate the first real-space recording of nanoscale dynamic interactions using single-shot soft x-ray (SXR) full-field laser microscopy. A sequence of real-space flash images acquired with a table-top SXR laser was used to capture the motion of a rapidly oscillating magnetic nanoprobe. Changes of 30 nm in the oscillation amplitude were detected when the nanoprobe was made to interact with stray fields from a magnetic sample. The table-top visualization of nanoscale dynamics in real space can significantly contribute to the understanding of nanoscale processes and can accelerate the development of new nanodevices.