On a length preserving curve flow

In this paper, we consider a new length preserving curve flow for closed convex curves in the plane. We show that the flow exists globally, the area of the region bounded by the evolving curve is increasing, and the evolving curve converges to the circle in C ^?? topology as t ?? ??.     

A non-local flow for Riemann surfaces

A non-local flow is defined for compact Riemann surfaces. Assuming the initial metric has positive Gauss curvature and is not conformal to the round sphere, the flow exists on some maximal time interval, and converges along a subsequence to a metric which admits a conformal Killing vector field. By a result of Tashiro (Trans Am Math Soc 117:251–275, 1965), the limiting metric must be conformal to the round sphere. Research supported in part by NSF Grant DMS-0500538.     

On a non-local perimeter-preserving curve evolution problem for convex plane curves

This paper deals with a non-local evolution problem for closed convex plane curves which preserves the perimeter of the evolving curve but enlarges the area it bounds and makes the evolving curve more and more circular during the evolution process. And the final shape of the evolving curve will be a circle in the C ^∞ metric as the time t goes to infinity. The first author is supported in part by the National Science Foundation of China (No.10671066) and the Shanghai Leading Academic Discipline Project (No. B407).     

Mappings preserving area 1 of triangles

This paper is concerned with the characterization of area-preserving mappings of real inner product spaces.     

A shape preserving area true approximation of histogram by rational splines

For a given histogram, we consider an application of a simple rational spline to a shape preserving area true approximation of the histogram. An algorithm for determination of the spline is as easy as one with a quadratic polynomial spline, while the latter does not always preserve the shape of the histogram. Some numerical examples are given at the end of the paper.     

State-dependent symplecticity and area preserving numerical methods

We introduce the definition of state-dependent symplecticity as a useful tool of investigation to discover nearby symplecticity in symmetric non-symplectic one-step methods applied to two-dimensional Hamiltonian systems. We first relate this property to Poisson systems and to the trapezoidal method, and then investigate Runge–Kutta and discrete gradient symmetric methods.     

The area within a curve

The area of a simple closed convex curve can be estimated in terms of the number of points of a square lattice that lie within the curve. We obtain the usual error bound without integration using a form of the Hardy—Littlewood—Ramanujan circle method, and also present simple estimates for the mean square error.     

A numerical scheme for regularized anisotropic curve shortening flow

Realistic interfacial energy densities are often non-convex, which results in backward parabolic behavior of the corresponding anisotropic curve shortening flow, thereby inducing phenomena such as the formation of corners and facets. Adding a term that is quadratic in the curvature to the interfacial energy yields a regularized evolution equation for the interface, which is fourth-order parabolic. Using a semi-implicit time discretization, we present a variational formulation of this equation, which allows the use of linear finite elements. The resulting linear system is shown to be uniquely solvable. We also present numerical examples.     

More Denjoy minimal sets for area preserving diffeomorphisms

For an area preserving, monotone twist diffeomorphism and an irrational number ω, we prove that if there is no invariant circle of angular rotation number ω, then there are uncountably many Denjoy minimal sets of angular rotation number ω. For each pair of positive integersn andR we prove that the space (with the vague topology) of Denjoy minimal sets of angular rotation number ω and intrinsic rotation number (ω+R)/n (mod. 1) contains a disk of dimensionn−1. Partially supported by NSF contract #MCS82-01604.     

A curve flow evolved by a fourth order parabolic equation

We study a fourth order curve flow, which is the gradient flow of a functional describing the shapes of human red blood cells. We prove that for any smooth closed initial curve in ℝ², the flow has a smooth solution for all time and the solution subconverges to a critical point of the functional.     

A singular non-local problem at resonance

We study a second order non-local problem using the coincidence degree theory. We show the existence of a solution whose derivative is singular at the right end-point of the interval, which is a new result for any resonant non-local problem. Under the non-local condition, we find a general way to ensure that the differential operator is a Fredholm operator of index zero.     

一种保持周长的平面曲线流

The purpose of this paper is to investigate a new type of evolution problem for closedconvex plane curves which will preserves the perimeter of the curve but expands the enclosedarea and the final limiting curve is a circle in the Hausdorff metric in the plane.     

Neutral curve behaviour in Taylor-Dean flow

Summary Numerical investigations reveal a class of neutral curve in infinite Taylor-Dean flow not previously observed. Based on the work of Hall, asymptotic expansions at the high-wavenumber end of the neutral curves are given.Previously, an interpretation had been given by Raney and Chang for the curves observed by Hughes and Reid. Using the above expansions and Rayleigh's criterion, an alternate explanation is proposed in this paper which takes into account the new phenomena.

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Résumé Des investigations numériques ont révélé un genre de courbe de stabilité neutre dans l'écoulement infini de Taylor-Dean non observé auparavant. Basé sur l'oeuvre de Hall, des développments asymptotiques à l'extrémité de ces courbes où le nombre d'ondes est élevé sont soumis. Auparavant, des courbes ont été observées par Hughes et Reid et une interprétation a été donné par Raney et Chang. Dans ce papier, on emploie les développements susmentionés et le criterium de Rayleigh pour proposer une autre explication qui fait état de nouveau phénomène.

     

A boosting method for maximization of the area under the ROC curve

We discuss receiver operating characteristic (ROC) curve and the area under the ROC curve (AUC) for binary classification problems in clinical fields. We propose a statistical method for combining multiple feature variables, based on a boosting algorithm for maximization of the AUC. In this iterative procedure, various simple classifiers that consist of the feature variables are combined flexibly into a single strong classifier. We consider a regularization to prevent overfitting to data in the algorithm using a penalty term for nonsmoothness. This regularization method not only improves the classification performance but also helps us to get a clearer understanding about how each feature variable is related to the binary outcome variable. We demonstrate the usefulness of score plots constructed componentwise by the boosting method. We describe two simulation studies and a real data analysis in order to illustrate the utility of our method.     

Mappings preserving the area equality of hyperbolic triangles are motions

It is shown that a mapping ${\varphi: \mathfrak{A}\rightarrow \mathfrak{B}}It is shown that a mapping j: \mathfrakA? \mathfrakB{\varphi: \mathfrak{A}\rightarrow \mathfrak{B}} between models \mathfrakA{\mathfrak{A}} and \mathfrakB{\mathfrak{B}} of elementary plane hyperbolic geometry, coordinatized by Euclidean ordered fields, that maps triangles having the same area and sharing a side into triangles that have the same property, must be a hyperbolic motion onto j(\mathfrakA){\varphi(\mathfrak{A})}. The relations that Tarski and Szmielew used as primitives for geometry, the equidistance relation ≡ and the betweenness relation B are shown to be positively existentially definable in terms of the quaternary relation Δ, with Δ(abcd) standing for “the triangles abc and abd have the same area.”     

Topological lower bounds on the distance between area preserving diffeomorphisms

Area preserving diffeomorphisms of the 2-disk which are Identity near the boundary form a group which can be equipped, using theL ²-norm on its Lie algebra, with a right invariant metric. In this paper we give a lower bound on the distance between diffeomorphisms which is invariant under area preserving changes of coordinates and which improves the lower bound induced by the Calabi invariant. In the case of renormalizable and infinitely renormalizable maps, our estimate can be improved and computed.