Rational Functions with a Unique Critical Point

Abstract We study existence and multiplicity of positive ground states for the scalar curvature equation $$\begin{aligned} \varDelta u+ K(|x|)\, u^{\frac{n+2}{n-2}}=0, \quad x\in {{\mathbb {R}}}^n\,, \quad n>2, \end{aligned}$$ Δ u + K ( | x | ) u n + 2 n - 2 = 0 , x ∈ R n , n > 2 , when the function $$K:{{\mathbb {R}}}^+\rightarrow {{\mathbb {R}}}^+$$ K : R + → R + is bounded above and below by two positive constants, i.e. $$0<\underline{K} \le K(r) \le \overline{K}$$ 0 < K ̲ ≤ K ( r ) ≤ K ¯ for every $$r > 0$$ r > 0 , it is decreasing in $$(0,{{{\mathcal {R}}}})$$ ( 0 , R ) and increasing in $$({{{\mathcal {R}}}},+\infty )$$ ( R , + ∞ ) for a certain $${{{\mathcal {R}}}}>0$$ R > 0 . We recall that in this case ground states have to be radial, so the problem is reduced to an ODE and, then, to a dynamical system via Fowler transformation. We provide a smallness non perturbative (i.e. computable) condition on the ratio $$\overline{K}/\underline{K}$$ K ¯ / K ̲ which guarantees the existence of a large number of ground states with fast decay, i.e. such that $$u(|x|) \sim |x|^{2-n}$$ u ( | x | ) ∼ | x | 2 - n as $$|x| \rightarrow +\infty $$ | x | → + ∞ , which are of bubble-tower type. We emphasize that if K(r) has a unique critical point and it is a maximum the radial ground state with fast decay, if it exists, is unique.

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Perturbations of weakly expanding critical orbits

Frontiers in Complex Dynamics ◽

10.23943/princeton/9780691159294.003.0010 ◽

2014 ◽

Author(s):

Genadi Levin

Keyword(s):

Critical Point ◽

Rational Function ◽

Critical Points ◽

Smooth Curve ◽

Rational Functions ◽

Rational Maps ◽

Dimensional Manifold ◽

Precise Form ◽

Special Cases ◽

Zero Number

This chapter studies perturbations of polynomials and rational functions with several (possibly, not all) summable critical points. It proves that there exists an r-dimensional manifold Δ‎ in an appropriate space containing f (a polynomial or a rational function which has r summable critical points) such that for every smooth curve in Δ‎ through f, the ratio between parameter and dynamical derivatives along forward iterates of at least one of these summable points tends to a non-zero number. In doing so the chapter establishes a precise form of this relation for rational maps with one critical point satisfying the summability condition (certain expansion rate assumption along the critical orbit). This result brings to a natural general form many previously known special cases studied over the years.

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Quadratic rational functions with a rational periodic critical point of period 3

Journal de Théorie des Nombres de Bordeaux ◽

10.5802/jtnb.1068 ◽

2019 ◽

Vol 31 (1) ◽

pp. 49-79

Author(s):

Solomon Vishkautsan

Keyword(s):

Critical Point ◽

Rational Functions

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

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100068382 ◽

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.

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

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100080377 ◽

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.

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The Effects of Drying Techniques on Clay-Rich Soil Texture

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100052341 ◽

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.

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Preservation of Plant Cell Surfaces For Scanning Electron Microscopy

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100072800 ◽

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.

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Scanning Electron Microscopy of Dried and Acid Treated Shells of Difflugia

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s042482010007268x ◽

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.

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