From the Birkhoff-Gustavson normalization to the Bertrand-Darboux integrability condition

The Bertrand-Darboux integrability condition for a certain class of perturbed harmonic oscillators is studied from the viewpoint of the Birkhoff-Gustavson(BG)-normalization: By solving an inverse problem of the BG-normalization on computer algebra, it is shown that if the perturbed harmonic oscillators with a homogeneous-{\it cubic} polynomial potential and with a homogeneous-{\it quartic} polynomial potentials admit the same BG-normalization up to degree-4 then both oscillators satisfy the Bertrand-Darboux integrability condition.Comment: 23 pages, LaTeX (iop.sty), typos and Appendix adde

Cognitive and Socio-Emotional Deficits in Platelet-Derived Growth Factor Receptor-β Gene Knockout Mice

Platelet-derived growth factor (PDGF) is a potent mitogen. Extensive in vivo studies of PDGF and its receptor (PDGFR) genes have reported that PDGF plays an important role in embryogenesis and development of the central nervous system (CNS). Furthermore, PDGF and the β subunit of the PDGF receptor (PDGFR-β) have been reported to be associated with schizophrenia and autism. However, no study has reported on the effects of PDGF deletion on mice behavior. Here we generated novel mutant mice (PDGFR-β KO) in which PDGFR-β was conditionally deleted in CNS neurons using the Cre/loxP system. Mice without the Cre transgene but with floxed PDGFR-β were used as controls. Both groups of mice reached adulthood without any apparent anatomical defects. These mice were further examined by conducting several behavioral tests for spatial memory, social interaction, conditioning, prepulse inhibition, and forced swimming. The test results indicated that the PDGFR-β KO mice show deficits in all of these areas. Furthermore, an immunohistochemical study of the PDGFR-β KO mice brain indicated that the number of parvalbumin (calcium-binding protein)-positive (i.e., putatively γ-aminobutyric acid-ergic) neurons was low in the amygdala, hippocampus, and medial prefrontal cortex. Neurophysiological studies indicated that sensory-evoked gamma oscillation was low in the PDGFR-β KO mice, consistent with the observed reduction in the number of parvalbumin-positive neurons. These results suggest that PDGFR-β plays an important role in cognitive and socioemotional functions, and that deficits in this receptor may partly underlie the cognitive and socioemotional deficits observed in schizophrenic and autistic patients

GITAˉ¹: A symbolic computing program for an inverse problem of the Birkhoff-Gustavson normal form expansion

yesWe present briefly a symbolic computing program named GITAˉ¹ working on REDUCE 3.3 or later, which solves an inverse problem for the Birkhoff-Gustavs-n normal form (BGNF) expansionBelgorod State Universit

GITAˉ¹: A symbolic computing program for an inverse problem of the Birkhoff-Gustavson normal form expansion

We present briefly a symbolic computing program named GITAˉ¹ working on REDUCE 3.3 or later, which solves an inverse problem for the Birkhoff-Gustavs-n normal form (BGNF) expansionyesBelgorod State Universit

A comparison of algorithms for the normalization and quantization of polynomial hamiltonians

yesAlgorithms and programs for the normalization of polynomial Hamiltonians of classical mechanics by the Birkhoff-Gustavson and Deprit-Hori, as well as quasi-classical quantization procedures for normal forms, are compared. The algorithms and programs are represented in a universal pseudocode and implemented in the computer algebra systems REDUCE, MAPLE, and MATHEMATICA. Examples that illustrate the operation of these algorithms and programs for polynomial Hamiltonians of atomic systems in external electromagnetic fields are considere

A comparison of algorithms for the normalization and quantization of polynomial hamiltonians

Algorithms and programs for the normalization of polynomial Hamiltonians of classical mechanics by the Birkhoff-Gustavson and Deprit-Hori, as well as quasi-classical quantization procedures for normal forms, are compared. The algorithms and programs are represented in a universal pseudocode and implemented in the computer algebra systems REDUCE, MAPLE, and MATHEMATICA. Examples that illustrate the operation of these algorithms and programs for polynomial Hamiltonians of atomic systems in external electromagnetic fields are consideredye