Perturbation analysis of a matrix differential equation x˙=ABx\dot x=ABx

Two complex matrix pairs $(A,B)$ and $(A',B')$ are contragrediently equivalent if there are nonsingular $S$ and $R$ such that $(A',B')=(S^{-1}AR,R^{-1}BS)$. M.I. Garc\'{\i}a-Planas and V.V. Sergeichuk (1999) constructed a miniversal deformation of a canonical pair $(A,B)$ for contragredient equivalence; that is, a simple normal form to which all matrix pairs $(A + \widetilde A, B+\widetilde B)$ close to $(A,B)$ can be reduced by contragredient equivalence transformations that smoothly depend on the entries of $\widetilde A$ and $\widetilde B$. Each perturbation $(\widetilde A,\widetilde B)$ of $(A,B)$ defines the first order induced perturbation $A\widetilde{B}+\widetilde{A}B$ of the matrix $AB$, which is the first order summand in the product $(A +\widetilde{A})(B+\widetilde{B}) = AB + A\widetilde{B}+\widetilde{A}B+ \widetilde A \widetilde B$. We find all canonical matrix pairs $(A,B)$, for which the first order induced perturbations $A\widetilde{B}+\widetilde{A}B$ are nonzero for all nonzero perturbations in the normal form of Garc\'{\i}a-Planas and Sergeichuk. This problem arises in the theory of matrix differential equations $\dot x=Cx$, whose product of two matrices: $C=AB$; using the substitution $x = Sy$, one can reduce $C$ by similarity transformations $S^{-1}CS$ and $(A,B)$ by contragredient equivalence transformations $(S^{-1}AR,R^{-1}BS)$

Simplest miniversal deformations of matrices, matrix pencils, and contragredient matrix pencils

V. I. Arnold [Russian Math. Surveys 26 (2) (1971) 29-43] constructed a simple normal form for a family of complex n-by-n matrices that smoothly depend on parameters with respect to similarity transformations that smoothly depend on the same parameters. We construct analogous normal forms for a family of real matrices and a family of matrix pencils that smoothly depend on parameters, simplifying their normal forms by D. M. Galin [Uspehi Mat. Nauk 27 (1) (1972) 241-242] and by A. Edelman, E. Elmroth, B. Kagstrom [Siam J. Matrix Anal. Appl. 18 (3) (1997) 653-692].Comment: 20 page

Detection and discrimination between ochratoxin producer and non-producer strains of Penicillium nordicum on a ham-based medium using an electronic nose

The aim of this work was to evaluate the potential use of qualitative volatile patterns produced by Penicillium nordicum to discriminate between ochratoxin A (OTA) producers and non-producer strains on a ham-based medium. Experiments were carried out on a 3% ham medium at two water activities (aw ; 0.995, 0.95) inoculated with P. nordicum spores and incubated at 25°C for up to 14days. Growing colonies were sampled after 1, 2, 3, 7 and 14days, placed in 30-ml vials, sealed and the head space analysed using a hybrid sensor electronic nose device. The effect of environmental conditions on growth and OTA production was evaluated based on the qualitative response. However, after 7days, it was possible to discriminate between strains grown at 0.995 aw, and after 14days, the OTA producer and non-producer strain and the controls could be discriminated at both aw levels. This study suggests that volatile patterns produced by P. nordicum strains may differ and be used to predict the presence of toxigenic contaminants in ham. This approach could be utilised in ham production as part of a quality assurance system for preventing OTA contaminatio

Rigid systems of second-order linear differential equations

We say that a system of differential equations d^2x(t)/dt^2=Adx(t)/dt+Bx(t)+Cu(t), in which A and B are m-by-m complex matrices and C is an m-by-n complex matrix, is rigid if it can be reduced by substitutions x(t)=Sy(t), u(t)=Udy(t)/dt+Vy(t)+Pv(t) with nonsingular S and P to each system obtained from it by a small enough perturbation of its matrices A,B,C. We prove that there exists a rigid system if and only if m<n(1+square_root{5})/2, and describe all rigid systems.Comment: 22 page