In this paper, we demonstrate, first in literature known to us, that potential functions can be constructed in continuous dissipative chaotic systems and can be used to reveal their dynamical properties. To attain this aim, a Lorenz-like system is proposed and rigorously proved chaotic for exemplified analysis. We explicitly construct a potential function monotonically decreasing along the system's dynamics, revealing the structure of the chaotic strange attractor. The potential function can have different forms of construction. We also decompose the dynamical system to explain for the different origins of chaotic attractor and strange attractor. Consequently, reasons for the existence of both chaotic nonstrange attractors and nonchaotic strange attractors are clearly discussed within current decomposition framework.
We used randomly amplified polymorphic DNA (RAPD)-PCR to estimate genetic variation among isolates of Trichoderma associated with green mold on the cultivated mushroom Agaricus bisporus. Of 83 isolates examined, 66 were sampled during the recent green mold epidemic, while the remaining 17 isolates were collected just prior to the epidemic and date back to the 1950s.Trichoderma harzianum biotype 4 was identified by RAPD analysis as the cause of almost 90% of the epidemic-related episodes of green mold occurring in the major commercial mushroom-growing region in North America. Biotype 4 was more closely allied to T. harzianum biotype 2, the predominant pathogenic genotype in Europe, than to the less pathogenic biotype 1 and Trichoderma atroviride (formerly T. harzianum biotype 3). No variation in the RAPD patterns was observed among the isolates within biotype 2 or 4, suggesting that the two pathogenic biotypes were populations containing single clones. Considerable genetic variation, however, was noted among isolates of biotype 1 and T. atroviride from Europe. Biotype 4 was not represented by the preepidemic isolates of Trichoderma as determined by RAPD markers and PCR amplification of an arbitrary DNA sequence unique to the genomes of biotypes 2 and 4. Our findings suggest that the onset of the green mold epidemic in North America resulted from the recent introduction of a highly virulent genotype of the pathogen into cultivated mushrooms.
A conjecture by Mackey and Higson claims that there is close relationship between irreducible representations of a real reductive group and those of its Cartan motion group. The case of irreducible tempered unitary representations has been verified recently by Afgoustidis. We study the admissible representations of SL(2, R) by considering families of D-modules over its flag varieties. We make a conjecture which gives a geometric understanding of the Makcey-Higson bijection in the general case.
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