scholarly journals A curve shortening flow rule for closed embedded plane curves with a prescribed rate of change in enclosed area

2016 ◽  
Vol 472 (2185) ◽  
pp. 20150629 ◽  
Author(s):  
Michael C. Dallaston ◽  
Scott W. McCue
Keyword(s):  
Flow Problem ◽  
Initial Conditions ◽  
Plane Curve ◽  
Rate Of Change ◽  
Plane Curves ◽  
Flow Rule ◽  
Closed Curves ◽  
Curve Shortening Flow ◽  
Standard Curve ◽  
Curve Shortening

Motivated by a problem from fluid mechanics, we consider a generalization of the standard curve shortening flow problem for a closed embedded plane curve such that the area enclosed by the curve is forced to decrease at a prescribed rate. Using formal asymptotic and numerical techniques, we derive possible extinction shapes as the curve contracts to a point, dependent on the rate of decreasing area; we find there is a wider class of extinction shapes than for standard curve shortening, for which initially simple closed curves are always asymptotically circular. We also provide numerical evidence that self-intersection is possible for non-convex initial conditions, distinguishing between pinch-off and coalescence of the curve interior.

scholarly journals Curve Flows with a Global Forcing Term

2021 ◽  
Author(s):  
Friederike Dittberner
Keyword(s):  
Finite Time ◽  
Exponential Convergence ◽  
Total Curvature ◽  
Closed Curves ◽  
Curve Shortening Flow ◽  
Smooth Curves ◽  
Forcing Term ◽  
Curve Flows ◽  
Curve Shortening ◽  
Area Preserving

AbstractWe consider embedded, smooth curves in the plane which are either closed or asymptotic to two lines. We study their behaviour under curve shortening flow with a global forcing term. We prove an analogue to Huisken’s distance comparison principle for curve shortening flow for initial curves whose local total curvature does not lie below $$-\pi $$ - π and show that this condition is sharp. With that, we can exclude singularities in finite time for bounded forcing terms. For immortal flows of closed curves whose forcing terms provide non-vanishing enclosed area and bounded length, we show convexity in finite time and smooth and exponential convergence to a circle. In particular, all of the above holds for the area preserving curve shortening flow.

scholarly journals Connected Numbers and the Embedded Topology of Plane Curves

2018 ◽  
Vol 61 (3) ◽  
pp. 650-658 ◽  
Author(s):  
Taketo Shirane
Keyword(s):  
Plane Curve ◽  
Smooth Curve ◽  
Plane Curves ◽  
Irreducible Curve ◽  
Galois Cover ◽  
Splitting Number

AbstractThe splitting number of a plane irreducible curve for a Galois cover is effective in distinguishing the embedded topology of plane curves. In this paper, we define the connected number of a plane curve (possibly reducible) for a Galois cover, which is similar to the splitting number. By using the connected number, we distinguish the embedded topology of Artal arrangements of degree b ≥ 4, where an Artal arrangement of degree b is a plane curve consisting of one smooth curve of degree b and three of its total inflectional tangents.

scholarly journals Plane Curves Which are Quantum Homogeneous Spaces

2021 ◽  
Author(s):  
Ken Brown ◽  
Angela Ankomaah Tabiri
Keyword(s):  
Hopf Algebras ◽  
Final Section ◽  
Homogeneous Spaces ◽  
Plane Curve ◽  
Plane Curves ◽  
Closed Field ◽  
Quantum Homogeneous Space ◽  
Central Elements ◽  
Pointed Hopf Algebras ◽  
Future Work

AbstractLet $\mathcal {C}$ C be a decomposable plane curve over an algebraically closed field k of characteristic 0. That is, $\mathcal {C}$ C is defined in k2 by an equation of the form g(x) = f(y), where g and f are polynomials of degree at least two. We use this data to construct three affine pointed Hopf algebras, A(x, a, g), A(y, b, f) and A(g, f), in the first two of which g [resp. f ] are skew primitive central elements, with the third being a factor of the tensor product of the first two. We conjecture that A(g, f) contains the coordinate ring $\mathcal {O}(\mathcal {C})$ O ( C ) of $\mathcal {C}$ C as a quantum homogeneous space, and prove this when each of g and f has degree at most five or is a power of the variable. We obtain many properties of these Hopf algebras, and show that, for small degrees, they are related to previously known algebras. For example, when g has degree three A(x, a, g) is a PBW deformation of the localisation at powers of a generator of the downup algebra A(− 1,− 1,0). The final section of the paper lists some questions for future work.

Kinetic Enzymatic Method for Determining Serum Creatinine

1975 ◽  
Vol 21 (10) ◽  
pp. 1422-1426 ◽  
Author(s):  
Gerald A Moss ◽  
Richard J L Bondar ◽  
Diane M Buzzelli
Keyword(s):  
Serum Creatinine ◽  
Accurate Determination ◽  
Test Subject ◽  
Rate Of Change ◽  
Enzymatic Method ◽  
Inhibitory Effects ◽  
Standard Curve ◽  
Analytical Recovery ◽  
Enzymatic Procedure

Abstract Creatinine amidohydrolase is used to measure serum creatinine in a totally enzymatic procedure. Creatine, produced by hydrolysis, is acted upon by creatine kinase, and then by pyruvate kinase and lactate dehydrogenase, to result in a change in absorbance at 340 nm. The amount of creatinine present is related to the rate of change in A340 and is determined from a standard curve. Absorbance and concentration are linearly related to 100 mg/liter and only 250 µl of serum is required. At 1.0 g/liter, heparin, oxalate, citrate, ethylenediaminetetraacetate, ascorbate, or glucose had no significant effect on the accurate determination of creatinine; higher concentrations (30 g/liter) had inhibitory effects on the test. Analytical recovery of creatinine added to either normal or abnormal sera averaged 102%. When results of this procedure and of the standard direct Jaffé test were compared, the latter were significantly higher. Unlike the Jaffé method, the present method of determining creatinine is rapid (about 10 min per test), subject to few or no interfering substances, and requires no serum deproteinization.

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