The critical points of the elastic energy among curves pinned at endpoints

<p style='text-indent:20px;'>In this paper we find curves minimizing the elastic energy among curves whose length is fixed and whose ends are pinned. Applying the shooting method, we can identify all critical points explicitly and determine which curve is the global minimizer. As a result we show that the critical points consist of wavelike elasticae and the minimizers do not have any loops or interior inflection points.</p>

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The elastic flow of curves on the sphere

Geometric Flows ◽

10.1515/geofl-2018-0001 ◽

2018 ◽

Vol 3 (1) ◽

pp. 1-13 ◽

Cited By ~ 2

Author(s):

Anna Dall’Acqua ◽

Tim Laux ◽

Lin ◽

Paola Pozzi ◽

Adrian Spener

Keyword(s):

Critical Points ◽

Elastic Energy ◽

Gradient Flow ◽

Closed Curves ◽

Long Time Existence ◽

Long Time ◽

Short Time ◽

Time Existence

Abstract We consider closed curves on the sphere moving by the L2-gradient flow of the elastic energy both with and without penalisation of the length and show short-time and long-time existence of the flow. Moreover, when the length is penalised, we prove sub-convergence to critical points.

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On the Points of Inflection of Bessel Functions of Positive Order, II

Canadian Journal of Mathematics ◽

10.4153/cjm-1991-037-4 ◽

1991 ◽

Vol 43 (3) ◽

pp. 628-651 ◽

Cited By ~ 4

Author(s):

R. Wong ◽

T. Lang

Keyword(s):

Bessel Function ◽

Critical Points ◽

Bessel Functions ◽

Positive Order ◽

Inflection Points ◽

Increasing Functions

Let jν, 1, jν,2, … denote the positive zeros of the Bessel function Jν(x), and similarly, let j'v,1, j'v,2, … denote the positive zeros of J'v(x), which are the positive critical points of Jv(x). It is well-known that when v is positive, both jν ,k. it and j'ν k are increasing functions of ν; see, e.g., [12, pp. 246 and 248]. Recently, Lorch and Szego [6] have attempted to show that the same is true for the positive zeros j″v,1, j″v,2, … of j″v(x), which are the positive inflection points of Jv(x).

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Nonlinear models in the height description of the Rhino sunflower cultivar

Ciência Rural ◽

10.1590/0103-8478cr20210213 ◽

2022 ◽

Vol 52 (3) ◽

Author(s):

Anderson Chuquel Mello ◽

Marcos Toebe ◽

Rafael Rodrigues de Souza ◽

João Antônio Paraginski ◽

Junior Carvalho Somavilla ◽

...

Keyword(s):

Growth Curve ◽

Critical Points ◽

Nonlinear Models ◽

Information Criterion ◽

Coefficient Of Determination ◽

Maximum Acceleration ◽

Inflection Points ◽

Best Fitting ◽

Microsoft Office

ABSTRACT: Sunflower produces achenes and oil of good quality, besides serving for production of silage, forage and biodiesel. Growth modeling allows knowing the growth pattern of the crop and optimizing the management. The research characterized the growth of the Rhino sunflower cultivar using the Logistic and Gompertz models and to make considerations regarding management based on critical points. The data used come from three uniformity trials with the Rhino confectionery sunflower cultivar carried out in the experimental area of the Federal University of Santa Maria - Campus Frederico Westphalen in the 2019/2020 agricultural harvest. In the first, second and third trials 14, 12 and 10 weekly height evaluations were performed on 10 plants, respectively. The data were adjusted for the thermal time accumulated. The parameters were estimated by ordinary least square’s method using the Gauss-Newton algorithm. The fitting quality of the models to the data was measured by the adjusted coefficient of determination, Akaike information criterion, Bayesian information criterion, and through intrinsic and parametric nonlinearity. The inflection points (IP), maximum acceleration (MAP), maximum deceleration (MDP) and asymptotic deceleration (ADP) were determined. Statistical analyses were performed with Microsoft Office Excel® and R software. The models satisfactorily described the height growth curve of sunflower, providing parameters with practical interpretations. The Logistics model has the best fitting quality, being the most suitable for characterizing the growth curve. The estimated critical points provide important information for crop management. Weeds must be controlled until the MAP. Covered fertilizer applications must be carried out between the MAP and IP range. ADP is an indicator of maturity, after reaching this point, the plants can be harvested for the production of silage without loss of volume and quality.

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On the Convergence of the Elastic Flow in the Hyperbolic Plane

Geometric Flows ◽

10.1515/geofl-2020-0002 ◽

2020 ◽

Vol 5 (1) ◽

pp. 40-77

Author(s):

Marius Müller ◽

Adrian Spener

Keyword(s):

Hyperbolic Space ◽

Elastic Energy ◽

Hyperbolic Plane ◽

Blow Up ◽

Type Inequality ◽

Gradient Flow ◽

Global Minimizer ◽

Infinite Time ◽

Closed Curves ◽

Convergence Results

AbstractWe examine the L2-gradient flow of Euler’s elastic energy for closed curves in hyperbolic space and prove convergence to the global minimizer for initial curves with elastic energy bounded by 16. We show the sharpness of this bound by constructing a class of curves whose lengths blow up in infinite time. The convergence results follow from a constrained sharp Reilly-type inequality.

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The relationship of the scleractinian corals to the rugose corals

Paleobiology ◽

10.1017/s0094837300025707 ◽

1980 ◽

Vol 6 (02) ◽

pp. 146-160 ◽

Cited By ~ 29

Author(s):

William A. Oliver

Keyword(s):

Critical Points ◽

Scleractinian Corals ◽

Convincing Evidence ◽

Early Triassic ◽

Rugose Corals ◽

History Of ◽

The Subject ◽

Relationship Of ◽

The Relationship ◽

Biologic Factors

The Mesozoic-Cenozoic coral Order Scleractinia has been suggested to have originated or evolved (1) by direct descent from the Paleozoic Order Rugosa or (2) by the development of a skeleton in members of one of the anemone groups that probably have existed throughout Phanerozoic time. In spite of much work on the subject, advocates of the direct descent hypothesis have failed to find convincing evidence of this relationship. Critical points are:(1) Rugosan septal insertion is serial; Scleractinian insertion is cyclic; no intermediate stages have been demonstrated. Apparent intermediates are Scleractinia having bilateral cyclic insertion or teratological Rugosa.(2) There is convincing evidence that the skeletons of many Rugosa were calcitic and none are known to be or to have been aragonitic. In contrast, the skeletons of all living Scleractinia are aragonitic and there is evidence that fossil Scleractinia were aragonitic also. The mineralogic difference is almost certainly due to intrinsic biologic factors.(3) No early Triassic corals of either group are known. This fact is not compelling (by itself) but is important in connection with points 1 and 2, because, given direct descent, both changes took place during this only stage in the history of the two groups in which there are no known corals.

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