CURVATURE-BASED INTERACTION POTENTIAL BETWEEN MICRO/NANO PLANAR CURVE AND PARTICLE LOCATED OUTSIDE THE CURVE

Based on the negative exponential pair-potential （I/R）n, the interaction potential between the micro/nano planar curve and the particle located outside the curve is studied. We verified that, whatever the value of exponent n may be the potential of particle/plane-curve is always of unified curvature form. Furthermore, we proved that the driving forces acted on the particle may be induced by the highly curved micro/nano curve, and the curvature and gradient of curvature are confirmed to be the essential factors forming the driving force. Through the idealized numerical experiments, the accuracy and reliability of the curvature-based potential are examined.

An efficient method for tracing planar implicit curves

This paper presents a method for tracing a planar implicit curve f（x, y）=0 on a rectangular region based on continuation scheme. First, according to the starting track-point and the starting track-direction of the curve, make a new fimction F（x, y）=0 where the same curve withf（x, y）=0 is defined. Then we trace the curve between the two domains where F（x, y）〉0 and F（x, y）〈0 alternately, according to the two rules presented in this paper. Equal step size or adaptive step size can be used, when we trace the curve. An irregular planar implicit curve （such as the curve with large curvatures at some points on the curve）, can be plotted if an adaptive step size is used. Moreover, this paper presents a scheme to search for the multiple points on the curve. Our method has the following advantages： （1） it can plot Co planar implicit curves; （2） it can plot the planar implicit curves with multiple points; （3） by the help of using the two rules, our method does not need to compute the tangent vector at the points on the curve, and directly searches for the direction of the tracing curve; （4） the tracing procedure costs only one of two evaluations of function f（x, y）=0 per moving step, while most existing similar methods cost more evaluations of the function.

Evaluation of spiral CT imaging in the diagnosis of the primary ureteral carcinoma

Objective: To explore the characteristics of the primary ureteral carcinoma (PUC) and discuss the value of spiral CT (SCT) in the diagnosis of PUC. Methods: The SCT findings of the primary ureteral carcinoma in 16 cases were analyzed and compared with the histopathological diagnosis and staging. Results: The transverse diameters of the lesions were 1.0–2.1 cm, and the longitudinal lengths were 1.5–15.2 cm. There were no statistically significant differences (P>0.1) in diam- eters and lengths among the low staging group (pT0–T2) and the high staging group (pT3–T4). The average CT value of the lesions was 43 HU on plain scanning, and 73 HU on CE scanning. The increment was 30 HU. The lesions were clearer on CE scanning. Curved planar reconstruction (CPR) could show the entire course of the urinary tract. Among 6 cases of pT3 stage, CT gave a correct diagnosis in 1 case. For 2 cases of pT4 stage, CT gave correct diagnoses in both cases. Conclusion: The carcinomatous lesions spread along the ureter. The longitudinal length of each lesion is longer than its transverse diameter. Tumor cannot be staged merely according to its diameter and length. CT is difficult to differentiate stage T0–T3, while for stage T4, CT diagnosis is accurate. Contrast enhancement CT scanning has the confirming and differentiating roles. CPR offers direct and easy observing images for clinical doctors.

The Location of Singular and Inflection Points for Planar Cubic B-Spline Curve

Using vectors between control points(a_i=P_(i+1)-P_i),parameters λ and μ(such that a_(i+1)=λ_(ai+μ_(a_i+2))are used to study the shape classification of planar parametric cubic B-spline curves. The regiosn of λμ space corresponding to different geometric features on the curves are investigated.These results are useful for curve design.

ON THE CONNECTION BETWEEN TWO PARTS OF HILBERT′S 16TH PROBLEM AND EQUIVARIANT BIFURCATIONP ROBLEM

This paper is a brief survey of our recent study on the connection between two parts of Hilbert′s 16th problem and equivariant bifurcation problem. We hope to understand the following questions: can we use the periodic solution family of ( m-1) degree planar Hamiltonian systems with Z q equivariant (or D q equivariant) symmetry to realize some schemes of ovals for planar algebraic curves? On the contrary, if an algebraic curve of degree m has maximal number of branches of ovals (it is called M -curve), can we make his all ovals become limit cycles of a planar polynomial system? What schemes of distribution of limit cycles can be realized by polynomial system.

On Singularity of Spline Space Over Morgan-Scott's Type Partition

Multivariate spline function is an important research object and tool in Computational Geometry. The singularity of multivariate spline spaces is a difficult problem that is ineritable in the research of the structure of multivariate spline spaces. The aim of this paper is to reveal the geometric significance of the singularity of bivariate spline space over Morgan-Scott type triangulation by using some new concepts proposed by the first author such as characteristic ratio, characteristic mapping of lines （or ponits）, and characteristic number of algebraic curve. With these concepts and the relevant results, a polished necessary and sufficient conditions for the singularity of spline space S u＋1^u （△MS^u） are geometrically given for any smoothness u by recursion. Moreover, the famous Pascal＇s theorem is generalized to algebraic plane curves of degree n≥3.