Families of maps Singularities and its Gauss maps

Abstract: This paper mainly studies the Singularities of smooth mapping. The singularities of the families of Gauss maps corresponding to the family of mappings are studied and the shape of these families and their singularities using mathematica program are illustrated and plotted. By changing the control parameters we find some singular points for the family which can be classified according to the famous theorems in singularity theory. Using the Hessian matrix we can obtain the singular points and singular set. Geometrically these singularities can be plotted but the classification of them can't be a valuable for all points. Using the terminology of level set which tell us the type of singular points like folds (level sets is start line), cusp (level set is semicubical parabola). In general there is no existance of some famous types. where ∆ is the discriminant set and it is a plan. Remark 4 The family of contours is given from z = k (constant), and the family of zero level set corresponding to k = 0 are given through the figurer

Metamorphosis of plasma turbulence-shear flow dynamics through a transcritical bifurcation

The structural properties of an economical model for a confined plasma turbulence governor are investigated through bifurcation and stability analyses. A close relationship is demonstrated between the underlying bifurcation framework of the model and typical behavior associated with low- to high-confinement transitions such as shear flow stabilization of turbulence and oscillatory collective action. In particular, the analysis evinces two types of discontinuous transition that are qualitatively distinct. One involves classical hysteresis, governed by viscous dissipation. The other is intrinsically oscillatory and non-hysteretic, and thus provides a model for the so-called dithering transitions that are frequently observed. This metamorphosis, or transformation, of the system dynamics is an important late side-effect of symmetry-breaking, which manifests as an unusual non-symmetric transcritical bifurcation induced by a significant shear flow drive.Comment: 17 pages, revtex text, 9 figures comprised of 16 postscript files. Submitted to Phys. Rev.

A Novel Intracellular Isoform of Matrix Metalloproteinase-2 Induced by Oxidative Stress Activates Innate Immunity

Experimental and clinical evidence has pinpointed a critical role for matrix metalloproteinase-2 (MMP-2) in ischemic ventricular remodeling and systolic heart failure. Prior studies have demonstrated that transgenic expression of the full-length, 68 kDa, secreted form of MMP-2 induces severe systolic failure. These mice also had unexpected and severe mitochondrial structural abnormalities and dysfunction. We hypothesized that an additional intracellular isoform of MMP-2, which affects mitochondrial function is induced under conditions of systolic failure-associated oxidative stress.Western blots of cardiac mitochondria from the full length MMP-2 transgenics, ageing mice and a model of accelerated atherogenesis revealed a smaller 65 kDa MMP-2 isoform. Cultured cardiomyoblasts subjected to transient oxidative stress generated the 65 kDa MMP-2 isoform. The 65 kDa MMP-2 isoform was also induced by hypoxic culture of cardiomyoblasts. Genomic database analysis of the MMP-2 gene mapped transcriptional start sites and RNA transcripts induced by hypoxia or epigenetic modifiers within the first intron of the MMP-2 gene. Translation of these transcripts yields a 65 kDa N-terminal truncated isoform beginning at M(77), thereby deleting the signal sequence and inhibitory prodomain. Cellular trafficking studies demonstrated that the 65 kDa MMP-2 isoform is not secreted and is present in cytosolic and mitochondrial fractions, while the full length 68 kDa isoform was found only in the extracellular space. Expression of the 65 kDa MMP-2 isoform induced mitochondrial-nuclear stress signaling with activation of the pro-inflammatory NF-κB, NFAT and IRF transcriptional pathways. By microarray, the 65 kDa MMP-2 induces an innate immunity transcriptome, including viral stress response genes, innate immunity transcription factor IRF7, chemokines and pro-apoptosis genes.A novel N-terminal truncated intracellular isoform of MMP-2 is induced by oxidative stress. This isoform initiates a primary innate immune response that may contribute to progressive cardiac dysfunction in the setting of ischemia and systolic failure