Triplets of mutually orthogonal circles associated with any ellipse

The author has shown (J. Geom. 94, 159–173, 2009) that, for any general point P on a given ellipse H, four concyclic notable points exist which determine a circle (denoted by Ω) orthogonal to Monge’s circle. Now, it is shown that a new set of notable concyclic points exists; such points determine a circle (denoted by Δ) orthogonal to both Monge’s circle and the circle Ω. Moreover, it is possible to introduce a new ellipse (denoted by H _Δ) concentric with the circle Δ, which is tangent to the ellipse H at P, shares the same circle Ω with the ellipse H and admits the circle Δ as its own Monge’s circle. Only elementary facts from trigonometry and analytic geometry are used.     

Director circles of conic sections

Given a conic section, the locus of a moving point in the plane of the conic section such that the two tangent lines drawn to the conic section from the moving point are all mutually perpendicular is a curve. In the case of an ellipse and hyperbola this curve is a circle referred to as the director circle. In the case of the parabola this curve coincides with the directrix of the parabola. The last section is devoted to the graphical illustrations of director circles for circles, parabolas, ellipses and hyperbolas using the built-in Maple V software of Scientific Work Place 3.0.     

New properties of the Busemann ellipsea

A combination of the Busemann ellipse, the inscribed unit circle and a circle of radius √2 about the same centre is considered. For supersonic two-dimensional potential gas flows, it is shown that the inclinations of the velocity vector in motion along an arbitrary characteristic, the characteristic itself and the characteristic of the other family have values equal to, respectively. the difference between the areas of the elliptical and circular (R = 1) sectors, the difference between the areas of the elliptical and circular (R = √2) sectors, and the area of the elliptical sector, apart from unimportant multiplicative and additive constants. The straight sides of the sectors in question are the semiminor antis of the ellipse and the radius vector of the velocity. The obvious analogy with one of Ke:pler's laws is pointed out. The existence of a point of intersection of the ellipse and the second circle illustrates a well-known result of Khristianovich concerning the points of inflexion of characteristics with a monotone velocity distribution. It is shown how the combination of the ellipse and the inscribed circle illustrates the simplification of the compatibility conditions and the Darboux equation for trans- and hypersonic flows.     

Pólya–Schur Inequality and the Green Energy of a Discrete Charge

Doklady Mathematics - The classical Pólya–Schur inequality for the logarithmic energy of a point charge distributed on a circle is generalized to the Green energy with respect to the...     

Two new sets of ellipse-related concyclic points

Two new circles (denoted by Γ _I and Γ _E ) are shown to be associated with any ellipse. Their analogies with two circles described by Barlotti are described. Two further new circles—denoted by Ω and Γ—are shown to be associated with any general point P of the ellipse. Tight relationships link the circles Ω and Γ with the circle K (previously introduced by the present author), as well as with Monge’s orthoptic circle, with Barlotti’s circles and with the circles Γ _I and Γ _E . In particular, the circle Ω is orthogonal to Monge’s circle. A new special point of the ellipse (the point T) is described. New properties of Fagnano’s point are described.     

Zeros of a family of hypergeometric para‐orthogonal polynomials on the unit circle

Para‐orthogonal polynomials derived from orthogonal polynomials on the unit circle are known to have all their zeros on the unit circle. In this note we study the zeros of a family of hypergeometric para‐orthogonal polynomials. As tools to study these polynomials, we obtain new results which can be considered as extensions of certain classical results associated with three term recurrence relations and differential equations satisfied by orthogonal polynomials on the real line. One of these results which might be considered as an extension of the classical Sturm comparison theorem, enables us to obtain monotonicity with respect to the parameters for the zeros of these para‐orthogonal polynomials. Finally, a monotonicity of the zeros of Meixner‐Pollaczek polynomials is proved.     

从圆内接三角形面积想到的

首先证明了正三角形的外接椭圆中面积最小的是一个圆.进而用初等方法证明了二维情形的F.John定理.     

Rigid sets in the plane

A compact set in the plane is rigid with respect to a norm if the norm isometries of the set act transitively on it. We show that if a norm has an infinite rigid set, then, up to linear transformation, the norm is Euclidean and the set is a circle. Our methods also yield a new characterisation of the ellipse.     

Intersection area of a stationary circle and a circle under a coupled axial rotation and translation

A circle is placed concentrically in a circle of equal or larger size. The circle is then rotated along a vertical axis, creating an ellipse, and translated along the horizontal axis. The intersection area of the circle and circle/ellipse is determined as function of the rotation angle and the relative size of the initial circles. This configuration corresponds to the closing of a ball valve used to control the flow of fluids through pipes.     

Stability theorem and extremum conditions for abnormal problems

We prove a generalized inverse function theorem in a neighborhood of a singular point of a mapping. As corollaries to this theorem, we obtain an inverse function theorem, an error bound theorem, and a tangent cone theorem that extend and strengthen the corresponding classical results in the irregular case. Using these corollaries, we establish necessary extremum conditions that are meaningful for abnormal problems.     

Reconstruction of Mobilities for Electrons and Holes in Semiconductors

From a simplified approximate semiconductor model, we develop a 1-D identification problem to recover the mobilities for electrons and holes in semiconductors based on the LBIC technique, and cast it as an optimization problem. Its solution is defined by the minimal point of some objective functional. On this argumentation, we derive the gradient operator of objective functional and the necessary condition for the solution of inverse problem. Our result provides a numerical approach to reconstruct the mobilities for electrons and holes in semiconductors.     

An Algorithm for Least-Squares Circle Fitting to Data with Specified Uncertainty Ellipses

The problem considered here is that of fitting a circle to aset of measured data points specified in terms of their cartesiancoordinates. It is assumed that the data adequately representsa circle and that associated with each data point there is anuncertainty ellipse describing the measurement error. A weightednonlinear least-squares problem is formulated in order to determineunbiased estimates of the centre and radius of the circle whichbests fits the given data. The resulting problem displays structurewhich is exploited when the Gauss-Newton algorithm is used toobtain a solution. In addition to estimates of the circle parametersthe algorithm produces error ellipses for the centre of thecircle and any point on its circumference.     

The inverse scattering problem by an elastic inclusion

In this work we consider the inverse elastic scattering problem by an inclusion in two dimensions. The elastic inclusion is placed in an isotropic homogeneous elastic medium. The inverse problem, using the third Betti’s formula (direct method), is equivalent to a system of four integral equations that are non linear with respect to the unknown boundary. Two equations are on the boundary and two on the unit circle where the far-field patterns of the scattered waves lie. We solve iteratively the system of integral equations by linearising only the far-field equations. Numerical results are presented that illustrate the feasibility of the proposed method.     

基于斜率的圆锥曲线新定义

为解决圆锥曲线对圆、椭圆、抛物线和双曲线统一的定义问题,可定义圆锥曲线是动点与二定点连线（或其中一连线为折线）斜率之积为定值的轨迹。此法不但较好地解决了圆锥曲线定义的不统一问题,而且数学推导也异常简单,有着明显的优点。此外,还论述了按此定义,用《几何画板》画各种圆锥曲线时,如何有效设置生成点的问题。     

Minimum-Area ellipse containing a finite set of points. I

From the geometric point of view, we consider the problem of construction of a minimum-area ellipse containing a given convex polygon. For an arbitrary triangle, we obtain an equation for the boundary of the minimum-area ellipse in explicit form. For a quadrangle, the problem of construction of a minimumarea ellipse is connected with the solution of a cubic equation. For an arbitrary polygon, we prove that if the boundary of the minimum-area ellipse has exactly three common points with the polygon, then this ellipse is the minimum-area ellipse for the triangle obtained. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 7, pp. 980–988, July, 1998.     

An extended version of Schur-Cohn-Fujiwara theorem in stability theory

This paper is concerned with root localization of a complex polynomial with respect to the unit circle in the more general case. The classical Schur-Cohn-Fujiwara theorem converts the inertia problem of a polynomial to that of an appropriate Hermitian matrix under the condition that the associated Bezout matrix is nonsingular. To complete it, we discuss an extended version of the Schur-Cohn-Fujiwara theorem to the singular case of that Bezout matrix. Our method is mainly based on a perturbation technique for a Bezout matrix. As an application of these results and methods, we further obtain an explicit formula for the number of roots of a polynomial located on the upper half part of the unit circle as well.     

The rigidity conjecture

A central question in dynamics is whether the topology of a system determines its geometry. This is known as rigidity. Under mild topological conditions rigidity holds for many classical cases, including: Kleinian groups, circle diffeomorphisms, unimodal interval maps, critical circle maps, and circle maps with a break point. More recent developments show that under similar topological conditions, rigidity does not hold for slightly more general systems. In this paper we state a conjecture which describes how topological classes are organized into rigidity classes.     

Spectrum of the Kerzman-Stein Operator for the Ellipse

The skew-hermitian part of the Cauchy operator, defined with respect to arclength measure on the boundary, is known as the Kerzman-Stein operator. For an ellipse, the eigenvalues of this operator are shown to have multiplicity two. For an ellipse with small eccentricity, we compute the leading coefficient in the asymptotic expansion of the eigenvalues.     

Complex velocity potential for an ellipse and a circle in translation and rotation

This paper investigates the analytical solutions of two-dimensional Neumann-type external boundary-value problem for an ellipse and a circle in complex analysis treatment. The transformation of harmonics between two planes is expanded in Laurent series. The circle theorem is extended so as to suit to the system of the ellipse and the circle. By means of Basset's process, the harmonic expressions of the system are found in the form of recurrence formulae which are suitable for numerical computations. The complex velocity potential of the system in translation and rotation is represented by the summation of the harmonic expressions.     

A new linear spectral transformation associated with derivatives of Dirac linear functionals

In this contribution, we analyze the regularity conditions of a perturbation on a quasi-definite linear functional by the addition of Dirac delta functionals supported on N points of the unit circle or on its complement. We also deal with a new example of linear spectral transformation. We introduce a perturbation of a quasi-definite linear functional by the addition of the first derivative of the Dirac linear functional when its support is a point on the unit circle or two points symmetric with respect to the unit circle. Necessary and sufficient conditions for the quasi-definiteness of the new linear functional are obtained. Outer relative asymptotics for the new sequence of monic orthogonal polynomials in terms of the original ones are obtained. Finally, we prove that this linear spectral transform can be decomposed as an iteration of Christoffel and Geronimus linear transformations.