A note on non-degenerate umbilics and the path formulation for bifurcation problems

2004 ◽  
Vol 134 (6) ◽  
pp. 1115-1126 ◽  
Author(s):  
Jacques-Elie Furter ◽  
Angela Maria Sitta
Keyword(s):  
Bifurcation Problems ◽  
Multiparameter Bifurcation ◽  
Two Parameters ◽  
Special Parameter

Path formulation can be used to classify and structure efficiently multiparameter bifurcation problems around fundamental singularities: the cores. The non-degenerate umbilic singularities are the generic cores for four situations in corank 2: the general or gradient problems and the Z2-equivariant (general or gradient) problems. Those categories determine an interesting ‘Russian doll’ type of structure in the universal unfoldings of the umbilic singularities.One advantage of our approach is that we can handle one, two or more parameters using the same framework (even considering some special parameter structure, for instance, some internal hierarchy). We classify the generic bifurcations that occur in those cases with one or two parameters.

scholarly journals NONDEGENERATE UMBILICS, THE PATH FORMULATION AND GRADIENT BIFURCATION PROBLEMS

2009 ◽  
Vol 19 (09) ◽  
pp. 2965-2977 ◽  
Author(s):  
JACQUES-ELIE FURTER ◽  
ANGELA MARIA SITTA
Keyword(s):  
Cylindrical Panel ◽  
Point Of View ◽  
Singular Behavior ◽  
Universal Unfolding ◽  
Buckling Modes ◽  
The Core ◽  
Bifurcation Problems ◽  
Multiparameter Bifurcation ◽  
Special Parameter

Parametrized contact-equivalence is a successful theory for the understanding and classification of the qualitative local behavior of bifurcation diagrams and their perturbations. Path formulation is an alternative point of view making explicit the singular behavior due to the core of the bifurcation germ (when the parameters vanish) from the effects of the way parameters enter. We show how to use path formulation to classify and structure efficiently multiparameter bifurcation problems in corank 2 problems. In particular, the nondegenerate umbilics singularities are the generic cores in four situations: the general or gradient problems, with or without ℤ2 symmetry where ℤ2 acts on the second component of ℝ2 via κ(x,y) = (x,-y). The universal unfolding of the umbilic singularities have an interesting "Russian doll" type of structure of miniversal unfoldings in all those categories. With the path formulation approach we can handle one, or more, parameter situations using the same framework. We can even consider some special parameter structure (for instance, some internal hierarchy of parameters). We classify the generic bifurcations with 1, 2 or 3 parameters that occur in those cases. Some results are known with one bifurcation parameter, but the others are new. We discuss some applications to the bifurcation of a loaded cylindrical panel. This problem has many natural parameters that provide concrete examples of our generic diagrams around the first interaction of the buckling modes.

Analysis of One Model of the Immune System

2003 ◽  
Vol 8 (2) ◽  
pp. 55-63
Author(s):  
A. Kavaliauskas
Keyword(s):  
Immune System ◽  
Tumor Cells ◽  
Stationary Point ◽  
Software Package ◽  
Medical Intervention ◽  
Research Results ◽  
Two Parameters ◽  
The Relationship ◽  
Special Parameter

The article deals with development of a tumor in the cylindrical space. A model with two parameters – lymphocytes and tumor cells – described in [1] is taken as a base of the research. Some generalization of the model is done. The research, results of which are described in [2, 3], was continued. The relationship between the surface and volumetric tumor cells of classical forms is obtained. It turns out, that even in general case the trivial stationary point is of a saddle type.The article particularly deals with the development of a cylindrical tumor. It is considered, that lymphocytes and the medical intervention (chemotherapy or irradiation) keeps down the cell fission of the tumor. Some special parameter is introduced to indicate the influence of the medical cure. In respect of that parameter the bifurcation values of the system are surveyed using the theory of [1, 4]. Some results of the survey are tested with the help of the software package MAPLE.

scholarly journals Origin Preserving Path Formulation for Multiparameter ℤ2-Equivariant Corank 2 Bifurcation Problems

2020 ◽  
Vol 30 (09) ◽  
pp. 2050140
Author(s):  
Jacques-Elie Furter
Keyword(s):  
Singularity Theory ◽  
Cylindrical Panel ◽  
Qualitative Behavior ◽  
Partial Classification ◽  
Equivariant Bifurcation ◽  
Bifurcation Problems ◽  
Multiparameter Bifurcation ◽  
Generic Bifurcation ◽  
Minimal Modification

A singularity theory, in the form of path formulation, is developed to analyze and organize the qualitative behavior of multiparameter [Formula: see text]-equivariant bifurcation problems of corank 2 and their deformations when the trivial solution is preserved as parameters vary. Path formulation allows for an efficient discussion of different parameter structures with a minimal modification of the algebra between cases. We give a partial classification of one-parameter problems. With a couple of parameter hierarchies, we show that the generic bifurcation problems are 2-determined and of topological codimension-0. We also show that the preservation of the trivial solutions is an important hypotheses for multiparameter bifurcation problems. We apply our results to the bifurcation of a cylindrical panel under axial compression.

Photo-electric Hβ and uvby photometry

1966 ◽  
Vol 24 ◽  
pp. 170-180
Author(s):  
D. L. Crawford
Keyword(s):  
Narrow Band ◽  
Line Strength ◽  
Interference Filter ◽  
B Stars ◽  
Relative Line ◽  
The Mean ◽  
F Stars ◽  
Two Parameters ◽  
Filter Photometry ◽  
The Continuum

Early in the 1950's Strömgren (1, 2, 3, 4, 5) introduced medium to narrow-band interference filter photometry at the McDonald Observatory. He used six interference filters to obtain two parameters of astrophysical interest. These parameters he calledlandc, for line and continuum hydrogen absorption. The first measured empirically the absorption line strength of Hβby means of a filter of half width 35Å centered on Hβand compared to the mean of two filters situated in the continuum near Hβ. The second index measured empirically the Balmer discontinuity by means of a filter situated below the Balmer discontinuity and two above it. He showed that these two indices could accurately predict the spectral type and luminosity of both B stars and A and F stars. He later derived (6) an indexmfrom the same filters. This index was a measure of the relative line blanketing near 4100Å compared to two filters above 4500Å. These three indices confirmed earlier work by many people, including Lindblad and Becker. References to this earlier work and to the systems discussed today can be found in Strömgren's article inBasic Astronomical Data(7).

Radiation Damage of Support Films

1981 ◽  
Vol 39 ◽  
pp. 28-29
Author(s):  
H.A. Cohen ◽  
W. Chiu
Keyword(s):  
Transfer Function ◽  
Low Temperatures ◽  
Carbon Support ◽  
Contrast Transfer Function ◽  
Electron Dose ◽  
Imaging Parameters ◽  
Negative Stain ◽  
Phase Corrections ◽  
Two Parameters ◽  
Medium Dose

The goal of imaging the finest detail possible in biological specimens leads to contradictory requirements for the choice of an electron dose. The dose should be as low as possible to minimize object damage, yet as high as possible to optimize image statistics. For specimens that are protected by low temperatures or for which the low resolution associated with negative stain is acceptable, the first condition may be partially relaxed, allowing the use of (for example) 6 to 10 e/Å2. However, this medium dose is marginal for obtaining the contrast transfer function (CTF) of the microscope, which is necessary to allow phase corrections to the image. We have explored two parameters that affect the CTF under medium dose conditions.Figure 1 displays the CTF for carbon (C, row 1) and triafol plus carbon (T+C, row 2). For any column, the images to which the CTF correspond were from a carbon covered hole (C) and the adjacent triafol plus carbon support film (T+C), both recorded on the same micrograph; therefore the imaging parameters of defocus, illumination angle, and electron statistics were identical.

Applications of Scanning Microscopy in Biomedical Research

1968 ◽  
Vol 26 ◽  
pp. 354-355
Author(s):  
T. L. Hayes
Keyword(s):  
Information Content ◽  
Biomedical Research ◽  
Biomedical Applications ◽  
Three Dimensional ◽  
Dental Enamel ◽  
Resolving Power ◽  
Whole Mount ◽  
Two Parameters ◽  
Content Information ◽  
Sectioned Tissue

Biomedical applications of the scanning electron microscope (SEM) have increased in number quite rapidly over the last several years. Studies have been made of cells, whole mount tissue, sectioned tissue, particles, human chromosomes, microorganisms, dental enamel and skeletal material. Many of the advantages of using this instrument for such investigations come from its ability to produce images that are high in information content. Information about the chemical make-up of the specimen, its electrical properties and its three dimensional architecture all may be represented in such images. Since the biological system is distinctive in its chemistry and often spatially scaled to the resolving power of the SEM, these images are particularly useful in biomedical research.In any form of microscopy there are two parameters that together determine the usefulness of the image. One parameter is the size of the volume being studied or resolving power of the instrument and the other is the amount of information about this volume that is displayed in the image. Both parameters are important in describing the performance of a microscope. The light microscope image, for example, is rich in information content (chemical, spatial, living specimen, etc.) but is very limited in resolving power.

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