The equivalence of a smooth function to a Taylor polynomial in the neighborhood of a finite-type critical point

1969 ◽  
Vol 2 (4) ◽  
pp. 318-323 ◽  
Author(s):  
A. M. Samoilenko
Keyword(s):  
Critical Point ◽  
Finite Type ◽  
Smooth Function ◽  
Taylor Polynomial

scholarly journals On the Level Set of a Function with Degenerate Minimum Point

2015 ◽  
Vol 2015 ◽  
pp. 1-6
Author(s):  
Yasuhiko Kamiyama
Keyword(s):  
Critical Point ◽  
Level Set ◽  
Smooth Function ◽  
Closed Manifold ◽  
Minimum Point ◽  
Unique Point ◽  
Degenerate Minimum ◽  
Nondegenerate Critical Point ◽  
Smooth Closed Manifold

Forn≥2, letMbe ann-dimensional smooth closed manifold andf:M→Ra smooth function. We setminf(M)=mand assume thatmis attained by unique pointp∈Msuch thatpis a nondegenerate critical point. Then the Morse lemma tells us that ifais slightly bigger thanm,f-1(a)is diffeomorphic toSn-1. In this paper, we relax the condition onpfrom being nondegenerate to being an isolated critical point and obtain the same consequence. Some application to the topology of polygon spaces is also included.

scholarly journals Thermodynamics and signatures of criticality in a network of neurons

2015 ◽  
Vol 112 (37) ◽  
pp. 11508-11513 ◽  
Author(s):  
Gašper Tkačik ◽  
Thierry Mora ◽  
Olivier Marre ◽  
Dario Amodei ◽  
Stephanie E. Palmer ◽  
...  
Keyword(s):  
Neural Network ◽  
Critical Point ◽  
Thermodynamic Limit ◽  
Smooth Function ◽  
Functional Significance ◽  
Statistical Physics ◽  
Small Patch ◽  
Vertebrate Retina ◽  
The Individual ◽  
The Relationship

The activity of a neural network is defined by patterns of spiking and silence from the individual neurons. Because spikes are (relatively) sparse, patterns of activity with increasing numbers of spikes are less probable, but, with more spikes, the number of possible patterns increases. This tradeoff between probability and numerosity is mathematically equivalent to the relationship between entropy and energy in statistical physics. We construct this relationship for populations of up to N = 160 neurons in a small patch of the vertebrate retina, using a combination of direct and model-based analyses of experiments on the response of this network to naturalistic movies. We see signs of a thermodynamic limit, where the entropy per neuron approaches a smooth function of the energy per neuron as N increases. The form of this function corresponds to the distribution of activity being poised near an unusual kind of critical point. We suggest further tests of criticality, and give a brief discussion of its functional significance.

The Critical Point Drying Method Applied to Scanning Electron Microscopy of L-929 Cells

1970 ◽  
Vol 28 ◽  
pp. 278-279
Author(s):  
Charles TurnbiLL ◽  
Delbert E. Philpott
Keyword(s):  
Electron Microscopy ◽  
Carbon Dioxide ◽  
Surface Tension ◽  
Critical Point ◽  
Freeze Drying ◽  
Attractive Alternative ◽  
Crystal Formation ◽  
Critical Point Drying ◽  
Time Required ◽  
Scanning Electron

The advent of the scanning electron microscope (SCEM) has renewed interest in preparing specimens by avoiding the forces of surface tension. The present method of freeze drying by Boyde and Barger (1969) and Small and Marszalek (1969) does prevent surface tension but ice crystal formation and time required for pumping out the specimen to dryness has discouraged us. We believe an attractive alternative to freeze drying is the critical point method originated by Anderson (1951; for electron microscopy. He avoided surface tension effects during drying by first exchanging the specimen water with alcohol, amy L acetate and then with carbon dioxide. He then selected a specific temperature (36.5°C) and pressure (72 Atm.) at which carbon dioxide would pass from the liquid to the gaseous phase without the effect of surface tension This combination of temperature and, pressure is known as the "critical point" of the Liquid.

SEM Cold Stage Techniques for the Observation of Vegetative and Floral Apices of Oat (Avena sativa L.) Plants

1977 ◽  
Vol 35 ◽  
pp. 590-591
Author(s):  
B. K. Kirchoff ◽  
L.F. Allard ◽  
W.C. Bigelow
Keyword(s):  
Critical Point ◽  
Avena Sativa ◽  
Substantial Improvement ◽  
Fresh Tissue ◽  
Cold Stage ◽  
Critical Point Drying ◽  
Almost All ◽  
Cell Collapse ◽  
Conductive Paint ◽  
Carbon Based

In attempting to use the SEM to investigate the transition from the vegetative to the floral state in oat (Avena sativa L.) it was discovered that the procedures of fixation and critical point drying (CPD), and fresh tissue examination of the specimens gave unsatisfactory results. In most cases, by using these techniques, cells of the tissue were collapsed or otherwise visibly distorted. Figure 1 shows the results of fixation with 4.5% formaldehyde-gluteraldehyde followed by CPD. Almost all cellular detail has been obscured by the resulting shrinkage distortions. The larger cracks seen on the left of the picture may be due to dissection damage, rather than CPD. The results of observation of fresh tissue are seen in Fig. 2. Although there is a substantial improvement over CPD, some cell collapse still occurs.Due to these difficulties, it was decided to experiment with cold stage techniques. The specimens to be observed were dissected out and attached to the sample stub using a carbon based conductive paint in acetone.

The Effects of Drying Techniques on Clay-Rich Soil Texture

1974 ◽  
Vol 32 ◽  
pp. 466-467
Author(s):  
T. G. Naymik
Keyword(s):  
Critical Point ◽  
Liquid Nitrogen ◽  
Liquid Nitrogen Temperature ◽  
Drying Methods ◽  
Air Drying ◽  
Freeze Dried ◽  
Critical Point Drying ◽  
Set Up ◽  
The Relationship ◽  
Quartz Grains

Three techniques were incorporated for drying clay-rich specimens: air-drying, freeze-drying and critical point drying. In air-drying, the specimens were set out for several days to dry or were placed in an oven (80°F) for several hours. The freeze-dried specimens were frozen by immersion in liquid nitrogen or in isopentane at near liquid nitrogen temperature and then were immediately placed in the freeze-dry vacuum chamber. The critical point specimens were molded in agar immediately after sampling. When the agar had set up the dehydration series, water-alcohol-amyl acetate-CO2 was carried out. The objectives were to compare the fabric plasmas (clays and precipitates), fabricskeletons (quartz grains) and the relationship between them for each drying technique. The three drying methods are not only applicable to the study of treated soils, but can be incorporated into all SEM clay soil studies.

Preservation of Plant Cell Surfaces For Scanning Electron Microscopy

1973 ◽  
Vol 31 ◽  
pp. 472-473
Author(s):  
Linda M. Sicko ◽  
Thomas E. Jensen
Keyword(s):  
Electron Microscopy ◽  
Scanning Electron Microscopy ◽  
Critical Point ◽  
Filter Paper ◽  
Freeze Drying ◽  
Cell Surfaces ◽  
Air Drying ◽  
Popular Method ◽  
Critical Point Drying ◽  
Scanning Electron

The use of critical point drying is rapidly becoming a popular method of preparing biological samples for scanning electron microscopy. The procedure is rapid, and produces consistent results with a variety of samples. The preservation of surface details is much greater than that of air drying, and the procedure is less complicated than that of freeze drying. This paper will present results comparing conventional air-drying of plant specimens to critical point drying, both of fixed and unfixed material. The preservation of delicate structures which are easily damaged in processing and the use of filter paper as a vehicle for drying will be discussed.

Scanning Electron Microscopy of Dried and Acid Treated Shells of Difflugia

1973 ◽  
Vol 31 ◽  
pp. 448-449
Author(s):  
Barry S. Eckert ◽  
S. M. McGee-Russell
Keyword(s):  
Electron Microscopy ◽  
Scanning Electron Microscopy ◽  
Critical Point ◽  
Point Method ◽  
External Structure ◽  
Cell Locomotion ◽  
Single Opening ◽  
Scanning Electron

Difflugia lobostoma is a shelled amoeba. The shell is an external structure of considerable mass which presents the animal with special restrictions in cell locomotion which are met by the development of active pseudopodial lobopodia containing, apparently, an organized system of thick and thin microfilaments (Eckert and McGee-Russell, 1972). The shell is constructed of sand grains picked up from the environment, and cemented into place with a secretion. There is a single opening through which lobopods extend. The organization of the shell was studied by scanning electron microscopy (SEM).Intact shells or animals with shells were dried by the critical point method of Anderson (1966) or air dried, after primary fixation in glutaraldehyde.

Macromolecular structures of biological specimens are not obscured by controlled osmium impregnation

1983 ◽  
Vol 41 ◽  
pp. 606-607
Author(s):  
Klaus-Ruediger Peters ◽  
Samuel A. Green
Keyword(s):  
Thin Films ◽  
Electrical Conductivity ◽  
Critical Point ◽  
Metal Films ◽  
High Magnification ◽  
Osmium Impregnation ◽  
Instrumental Resolution ◽  
20 Nm ◽  
Continuous Metal

High magnification imaging of macromolecules on metal coated biological specimens is limited only by wet preparation procedures since recently obtained instrumental resolution allows visualization of topographic structures as smal l as 1-2 nm. Details of such dimensions may be visualized if continuous metal films with a thickness of 2 nm or less are applied. Such thin films give sufficient contrast in TEM as well as in SEM (SE-I image mode). The requisite increase in electrical conductivity for SEM of biological specimens is achieved through the use of ligand mediated wet osmiuum impregnation of the specimen before critical point (CP) drying. A commonly used ligand is thiocarbohvdrazide (TCH), first introduced to TEM for en block staining of lipids and glvcomacromolecules with osmium black. Now TCH is also used for SEM. However, after ligand mediated osinification nonspecific osmium black precipitates were often found obscuring surface details with large diffuse aggregates or with dense particular deposits, 2-20 nm in size. Thus, only low magnification work was considered possible after TCH appl ication.

SEM of non-coated hepatocellular cytoskeleton of perfused livers

1984 ◽  
Vol 42 ◽  
pp. 306-307
Author(s):  
S.W. French ◽  
N.C. Benson ◽  
C. Davis-Scibienski
Keyword(s):  
Ambient Temperature ◽  
Critical Point ◽  
Protease Inhibitors ◽  
Metal Deposition ◽  
Heat Damage ◽  
Detergent Extraction ◽  
Cytoskeletal Elements ◽  
Triton X ◽  
Excised Tissue

Previous SEM studies of liver cytoskeletal elements have encountered technical difficulties such as variable metal coating and heat damage which occurs during metal deposition. The majority of studies involving evaluation of the cell cytoskeleton have been limited to cells which could be isolated, maintained in culture as a monolayer and thus easily extracted. Detergent extraction of excised tissue by immersion has often been unsatisfactory beyond the depth of several cells. These disadvantages have been avoided in the present study. Whole C3H mouse livers were perfused in situ with 0.5% Triton X-100 in a modified Jahn's buffer including protease inhibitors. Perfusion was continued for 1 to 2 hours at ambient temperature. The liver was then perfused with a 2% buffered gluteraldehyde solution. Liver samples including spontaneous tumors were then maintained in buffered gluteraldehyde for 2 hours. Samples were processed for SEM and TEM using the modified thicarbohydrazide procedure of Malich and Wilson, cryofractured, and critical point dried (CPD). Some samples were mechanically fractured after CPD.

Rapid Critical Point Drying of Tissues in a Parr Bomb

1972 ◽  
Vol 30 ◽  
pp. 400-401
Author(s):  
C.A. Baechler ◽  
W. C. Pitchford ◽  
J. M. Riddle ◽  
C.B. Boyd ◽  
H. Kanagawa ◽  
...  
Keyword(s):  
Critical Point ◽  
Critical Pressure ◽  
Soft Tissues ◽  
Biological Tissues ◽  
Critical Temperatures ◽  
Air Drying ◽  
The Past ◽  
Critical Point Drying ◽  
Great Stress ◽  
Infiltration Techniques

Preservation of the topographic ultrastructure of soft biological tissues for examination by scanning electron microscopy has been accomplished in the past by using lengthy epoxy infiltration techniques, or dehydration in ethanol or acetone followed by air drying. Since the former technique requires several days of preparation and the latter technique subjects the tissues to great stress during the phase change encountered during air-drying, an alternate rapid, economical, and reliable method of surface structure preservation was developed. Turnbill and Philpott had used a fluorocarbon for the critical point drying of soft tissues and indicated the advantages of working with fluids having both moderately low critical pressures as well as low critical temperatures. Freon-116 (duPont) which has a critical temperature of 19. 7 C and a critical pressure of 432 psi was used in this study.

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