Critical invariant circles in asymmetric and multiharmonic generalized standard maps

Invariant circles play an important role as barriers to transport in the dynamics of area-preserving maps. KAM theory guarantees the persistence of some circles for near-integrable maps, but far from the integrable case all circles can be destroyed. A standard method for determining the existence or nonexistence of a circle, Greene’s residue criterion, requires the computation of long-period orbits, which can be difficult if the map has no reversing symmetry. We use de la Llave’s quasi-Newton, Fourier-based scheme to numerically compute the conjugacy of a Diophantine circle conjugate to rigid rotation, and the singularity of a norm of a derivative of the conjugacy to predict criticality. We study near-critical conjugacies for families of rotational invariant circles in generalizations of Chirikov’s standard map.A first goal is to obtain evidence to support the long-standing conjecture that when circles breakup they form cantori, as is known for twist maps by Aubry–Mather theory. The location of the largest gaps is compared to the maxima of the potential when anti-integrable theory applies. A second goal is to support the conjecture that locally most robust circles have noble rotation numbers, even when the map is not reversible. We show that relative robustness varies inversely with the discriminant for rotation numbers in quadratic algebraic fields. Finally, we observe that the rotation number of the globally most robust circle generically appears to be a piecewise-constant function in two-parameter families of maps.     

Seifert circles,braid index and the algebraic crossing number

We introduce new operations reducing the number of Seifert circles in link diagrams of a special type. The operations are similar to one described in [Mem. Amer. Math. Soc. 508 (1993)] and [Math. Proc. Cambridge Philos. Soc. 111 (2) (1992) 273]. We discuss a conjecture about the number of Seifert circles that can be canceled by applying the operation repeatedly. We translate the problem into one belonging to graph theory.     

Solving circle packing problems by global optimization: Numerical results and industrial applications

A (general) circle packing is an optimized arrangement of N arbitrary sized circles inside a container (e.g., a rectangle or a circle) such that no two circles overlap. In this paper, we present several circle packing problems, review their industrial applications, and some exact and heuristic strategies for their solution. We also present illustrative numerical results using ‘generic’ global optimization software packages. Our work highlights the relevance of global optimization in solving circle packing problems, and points towards the necessary advancements in both theory and numerical practice.     

Spiral hexagonal circle packings in the plane

We discuss an intriguing geometric algorithm which generates infinite spiral patterns of packed circles in the plane. Using Kleinian group and covering theory, we construct a complex parametrization of all such patterns and characterize those whose circles have mutually disjoint interiors. We prove that these coherent spirals, along with the regular hexagonal packing, give all possible hexagonal circle packings in the plane. Several examples are illustrated.The last two authors gratefully acknowledge support of the National Science Foundation and the Tennessee Science Alliance.     

Circles in torus-torus intersections

We present a procedure to compute all the circles in the intersection curve of two tori, based on the geometric properties of the circles embedded in a torus. By using the geometric constraints in computing the circles, our algorithm provides an efficient and robust solution.     

Topological relations between separating circles

The family of separating circles of two finite sets in the plane consists of all the circles that enclose the first set but exclude the second set. We prove some theoretical results on distances between families of circles, and properties about enclosure and intersection. Most of these results state that a property that involves one or more infinite families of circles can be verified by examining a finite subcollection of circles. As a result enclosure and intersection can be decided, and distances can be computed with simple geometric algorithms. Furthermore, the circles of the finite subcollections correspond to the vertices of a polytope in the parameter space of separating circles. A polytope of separating circle parameters is well-known computational geometry, but we prove some new properties and we introduce the concept of an elementary circular separation as a concise way to define such a polytope.     

The apollonian octets and an inversive form of Krause's theorem

Suppose we are given three disjoint circles in the Euclidean plane with the property that none of them contains the other two. Then there are eight distinct circles tangent to the given three, and R.M. Krause has shown that a certain alternating sum of the curvatures of these eight circles must vanish. We express this result in an inversively invariant way and determine the extent to which it generalizes to other configurations of three given circles.     

Willmore Surfaces of Revolution with Two Prescribed Boundary Circles

We consider the family of smooth embedded surfaces of revolution in ?³ having two concentric circles contained in two parallel planes of ?³ as boundary. Minimizing the Willmore functional within this class of surfaces we prove the existence of smooth axi-symmetric Willmore surfaces having these circles as boundary. When the radii of the circles tend to zero we prove convergence of these solutions to the round sphere.     

度量方程应用于Krause定理的推广

本文用距离几何的方法证明了主要定理,对曲率为Ｋ的ｎ维常曲率空间,其内任意ｎ＋１个ｎ－１维球Ｓｉ（ｉ＝１,２,…,ｎ＋１）,它们中的任一个都与其它球不变,则与Ｓｉ交角为βｉ（ｉ＝１,２,…,ｎ＋１）的ｎ－１维一般有２＾ｎ＋１个,当ｎ为偶数时,它们的测地线曲率之交错和为零;当ｎ为奇数时,此结论不成立,该定理包括非欧情形,而当ｎ＝２,βｉ＝１（ｉ＝１,２,…,ｎ＋１）时,就是ｉｉｌｋｅｒＪＢ在「１」中所     

作业成本法数学模型的创新

随着企业产品价格竞争愈演愈烈，成本的合理分配、计算日显重要，但在间接费用的分摊上，传统成本会计对成本信息反映失真的局限性日益显露，会计理论界和实务界开始寻求一种新的准确的成本计算方法，作业成本法应运而生。本将由作业成本法基本原理推导出其数学模型，并将对其数学模型进一步改进，建立比较数学模型。使作业成本法向实际应用更进一步。     

Zur Möbiusgeometrie der Berührkreisscharen

Making use of an earlier result of the theory of stripes in three-dimensional conformal spaceM ₃ we obtain a moving frame and derivational formulas for one-parameter families of tangent circles inM ₃. Avoiding invariant parameters we can set up a bijection of the tangent circles into the osculating circles that preserves the duple ratio. The following section deals with loxodromes on Dupin cyclides. Finally with the aid of a modified stereographic projection we show how to get the Frenet formulas of the euclidian theory of spacecurves from the derivational formulas for tangent circles inM ₃.     

Constant Mean Curvature Surfaces Foliated by Circles in Lorentz–Minkowski Space

We prove that a spacelike surface in L3 with nonzero constant mean curvature and foliated by pieces of circles in spacelike planes is a surface of revolution. When the planes containing the circles are timelike or null, examples of nonrotational constant mean curvature surfaces constructed by circles are presented. Finally, we prove that a nonzero constant mean curvature spacelike surface foliated by pieces of circles in parallel planes is a surface of revolution.     

On Miquel’s theorem and inversions in normed planes

Asplund and Grünbaum proved that Miquel’s six-circles theorem holds in strictly convex, smooth normed planes if the considered circles have equal radii. We extend this result in two directions. First we prove that Miquel’s theorem for circles of equal radii (more precisely, a generalized version of it) is true in every normed plane, without the assumptions of strict convexity and smoothness, and give also some properties of the circle configuration related to this theorem. Second we clarify the situation if the circles of the corresponding configuration do not necessarily have equal radii.     

Weyl–Titchmarsh theory for discrete symplectic systems with general linear dependence on spectral parameter

In this paper we develop the Weyl–Titchmarsh theory for discrete symplectic systems with general linear dependence on the spectral parameter. We generalize and complete several recent results concerning these systems, which have the spectral parameter only in the second equation. Our new theory includes characterizations of the Weyl discs and Weyl circles, their limiting behaviour, properties of square summable solutions including the analysis of the exact number of linearly independent square summable solutions and limit point/circle criteria. Some illustrative examples are also provided.     

Invariant circles for homogeneous polynomial vector fields on the 2-dimensional sphere

LexX be a homogeneous polynomial vector field of degreen≥3 on S² having finitely many invariant circles. Then, for such a vector fieldX we find upper bounds for the number of invariant circles, invariant great circles, invariant circles intersecting at a same point and parallel circles with the same director vector. We give examples of homogeneous polynomial vector fields of degree 3 on S² having finitely many invariant circles which are not great circles, which are limit cycles, but are not great circles and invariant great circles that are limit cycles. Moreover, for the casen=3 we determine the maximum number of parallel invariant circles with the same director vector. The authors are partially supported by a MCYT grant BFM2002-04236-C02-02 and by a CIRIT grant number 2001SGR 00173.     

Paare und Tripel halbkonzentrischer Kreise der euklidischen Ebene

In [1], semi-concentric circles have been defined and considered, also their special nets and pencils.—The isochordal curve and the isogonal curve of two circles, the isochordal lines and the isogonal points of three circles show many differences according to whether the two or three circles are semiconcentric or not. Results for semi-concentric circles are developed here, while results for other circles have been given in [2] and [5].     

A note on minimal sets of periods for cofrontier maps

Using an earlier fixed point result of the author and Nielsen-type arguments patterned on related results for torus self-maps by Alsedá et al. we describe minimal sets of periods for self-maps of cofrontiers that are inverse limits of circles. Special attention is given to the pseudocircle, ever present in dynamical systems. Our result extends a property of circle maps, proved by Efremova, and independently by Block et al. in the late 1970s. We explain why our natural generalization is rather unexpected and shows potential of Nielsen theory for new applications.     

Limit cycles of discontinuous piecewise differential systems formed by linear centers in R2 and separated by two circles

We show that discontinuous planar piecewise differential systems formed by linear centers and separated by two concentric circles can have at most three limit cycles. Usually is a difficult problem to provide the exact upper bound that a class of differential systems can exhibit. Here we also provide examples of such systems with zero, one, two, or three limit cycles.     

A result on strong unique continuation for the laplace operator

We prove that two linked circles in R³ evolving by curvature develop three dimensional fattening at finite time. The same result holds for a suitable class of planar closed linked curves.     

Two circles and their common tangents

Given two circles C ₁ and C ₂ in a plane such that neither one of the two circles is contained in the other, there are either four common tangents when the circles do not intersect at all or the circles have three common tangents when they touch each other externally or only two common tangents when the circles intersect exactly at two points. The article oulines analytical procedures for computing the equations of these common tangents. Using the built-in Maple of the software Scientific Work Place 3.0, the diagrams of the circles with their common tangents are incorporated in this article.