Star-shaped Region Research Articles

On the Center of Surface Area of the Boundary of a Star-Shaped Region

Every line passing through the center of a circular disk bisects its area. Not every planar region has a point with this property. It turns out that if a compact connected region is star-shaped at its center of area, then it must be centrally symmetric. However, there exists a non-centrally symmetric star-shaped region whose boundary has a center of length, which is a point where every line passing through this point cut its length in half. In this article, we characterize such regions and provide a method to create them. Additionally, we provide new examples and a method to generate analogous objects in higher dimensions.

Blow-up Phenomena for a Reaction-diffusion Equation with Nonlocal Gradient Terms

In this paper, we investigate blow-up phenomena of the solution to a reaction-diffusion equation with nonlocal gradient absorption terms under Robin boundary condition on a bounded star-shaped region. Based on the method of auxiliary function and the technique of modified differential inequality, we establish some conditions on the nonlinearities for which the solution exists globally or blows up at finite time, when the sign of the constant $\sigma$ is either positive or negative. Moreover, upper and lower bounds for a blow-up time are derived under appropriate measure in higher dimensional spaces.

Star-Shaped Intense Uptake of ¹³¹I on Whole Body Scans Can Reflect Good Therapeutic Effects of Low-Dose Radioactive Iodine Treatment of 1.1 GBq.

BackgroundAfter initial radioactive iodine (RAI) treatment in differentiated thyroid cancer patients, we sometimes observe a star-shaped region of intense uptake of 131I on whole body scans (WBSs), called a ‘star artifact.’ We evaluated the clinical implications of star artifacts on the success rate of remnant ablation and long-term prognosis.MethodsTotal 636 patients who received 131I dose of 1.1 GBq for the initial RAI therapy and who did not show distant metastasis at the time of diagnosis were retrospectively evaluated. A negative second WBS was used for evaluating the ablation efficacy of the RAI therapy. Among them, 235 patients (36.9%) showed a star artifact on their first WBS.ResultsIn patients with first stimulated thyroglobulin (sTg) levels ≤2 ng/mL, patients with star artifacts had a higher rate of negative second WBS compared with those without star artifacts (77.8% vs. 63.9%, P=0.044), and showed significantly higher recurrence-free survival (P=0.043) during the median 8.0 years (range, 1.0 to 10.0) of follow-up. The 5- and 10-year recurrence rates (5YRR, 10YRR) were also significantly lower in patients with star artifacts compared with those without (0% vs. 4.9%, respectively, P=0.006 for 5YRR; 0% vs. 6.4%, respectively, P=0.005 for 10YRR). However, ablation success rate or recurrence-free survival was not different among patients whose first sTg levels >2 ng/mL regardless of star artifacts.ConclusionTherefore, star artifacts at initial RAI therapy imply a good ablation efficacy or a favorable long-term prognosis in patients with sTg levels ≤2 ng/mL.

Blow-up Phenomena for a Porous Medium Equation with Time-dependent Coefficients and Inner Absorption Term Under Nonlinear Boundary Flux

This paper deals with blow-up phenomena for an initial boundary value problem of a porous medium equation with time-dependent coefficients and inner absorption term in a bounded star-shaped region under nonlinear boundary flux. Using the auxiliary function method and modified differential inequality technique, we establish some conditions on time-dependent coefficient and nonlinear functions to guarantee that the solution $u(x,t)$ exists globally or blows up at some finite time $t^{\ast}$. Moreover, the upper and lower bounds for $t^{\ast}$ are derived in the higher dimensional space. Finally, some examples are presented to illustrate applications of our results.

Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux

This paper deals with blow-up phenomena for an initial boundary value problem of a nonlocal quasilinear parabolic equation with time-dependent coefficients in a bounded star-shaped region under nonlinear boundary flux. Using the auxiliary function method and modified differential inequality technique, we establish some conditions on time-dependent coefficients and nonlinearities to guarantee that the solution $u(x,t)$ exists globally or blows up at some finite time $t^{\ast}$. Moreover, upper and lower bounds of $t^{\ast}$ are obtained under suitable measure in high-dimensional spaces. Finally, some application examples are presented.

Blow-up analysis for a reaction–diffusion equation with weighted nonlocal inner absorptions under nonlinear boundary flux

A blow-up analysis for a nonlinear divergence form reaction–diffusion equation with weighted nonlocal inner absorption terms is considered under nonlinear boundary flux in a bounded star-shaped region. Based on the auxiliary function method and the modified differential inequality technique, we establish conditions on weight function and nonlinearities to guarantee the solution exists globally or blows up at finite time. Moreover, upper and lower bounds for the blow-up time are derived under appropriate measure in higher dimensional spaces. Three examples are presented to illustrate applications of our main results.

Blow-up analysis for a semilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux

A blow-up analysis for a nonlinear divergence form of parabolic equation with time-dependent coefficients is given under nonlinear boundary flux in a bounded star-shaped region. We establish some conditions on time-dependent coefficients and nonlinearities to guarantee existence of global solution or blow-up solution at some finite time t*. Moreover, an upper bound for t* is derived. Under somewhat more restrictive conditions, a lower bound for t* can be obtained. Finally, some application examples to verify the bounds of t* are presented.

Construction of control Lyapunov functions for damping stabilization of control affine systems

This paper considers the stabilization of nonlinear control affine systems that satisfy Jurdjevic–Quinn conditions. We first obtain a differential one-form associated to the system by taking the interior product of a non vanishing two-form with respect to the drift vector field. We then construct a homotopy operator on a star-shaped region centered at a desired equilibrium point that decomposes the system into an exact part and an anti-exact one. Integrating the exact one-form, we obtain a locally-defined dissipative potential that is used to generate the damping feedback controller. Applying the same decomposition approach on the entire control affine system under damping feedback, we compute a control Lyapunov function for the closed-loop system. Under Jurdjevic–Quinn conditions, it is shown that the obtained damping feedback is locally stabilizing the system to the desired equilibrium point provided that it is the maximal invariant set for the controlled dynamics. The technique is also applied to construct damping feedback controllers for the stabilization of periodic orbits. Examples are presented to illustrate the proposed method.

Ideal-specific elimination orders form a star-shaped region

This paper shows that for any given polynomial ideal I⊂K[x1,…,xn] the collection of Gröbner cones corresponding to I-specific elimination orders forms a star-shaped region which contrary to first intuition in general is not convex.Moreover we show that the corresponding region may contain Gröbner cones intersecting in the boundary of the Gröbner fan in the origin only. This implies that Gröbner walks aiming for the elimination of variables from a polynomial ideal can be terminated earlier than previously known. We provide a slightly improved stopping criterion for a known Gröbner walk algorithm for the elimination of variables.

Feedback Controller Design for the Four-Tank Process using Dissipative Hamiltonian Realization

This paper considers the problem of stabilizing the quadruple-tank process using an approximate dissipative Hamiltonian realization. The proposed approach consists in canceling by feedback the deviation of the system from a Hamiltonian system. First, we obtain a characteristic one-form for the system by taking the interior product of a non vanishing two-form with respect to the controlled vector field. We then construct a homotopy operator on a star-shaped region centered at a desired equilibrium point. The dynamics of the system is then decomposed into an exact part and an anti-exact one. The exact part is generated by a potential, hence stability of this part is guaranteed using the generating potential as a Lyapunov function. The stabilizing feedback controller is designed by canceling the anti-exact part of the characteristic one-form. Application of the resulting controller is illustrated by numerical simulations.

A Simple Algorithm for Complete Motion Planning of Translating Polyhedral Robots

We present a new sampling-based algorithm for complete motion planning. Our algorithm relies on computing a star-shaped roadmap of the free space. We partition the free space into star-shaped regions such that a single point, called the guard, can see every point in the star-shaped region. The resulting set of guards capture the intra-region connectivity—the connectivity between points belonging to the same star-shaped region. We capture the inter-region connectivity by computing connectors that link guards of adjacent regions. The guards and connectors are combined to obtain the star-shaped roadmap. We present an efficient algorithm to compute the roadmap in a deterministic manner without explicit computation of the free space. We show that the star-shaped roadmap captures the connectivity of the free space, thereby enabling us to perform complete motion planning. Our approach is relatively simple to implement. We apply our approach to perform motion planning of robots with translational and rotational degrees of freedom (dof). We highlight its performance in challenging scenarios with narrow passages or when there is no collision-free path for robots with low degrees of freedom.

Compactness of Hardy-Type Operators over Star-Shaped Regions in

AbstractWe study a compactness property of the operators between weighted Lebesgue spaces that average a function over certain domains involving a star-shaped region. The cases covered are (i) when the average is taken over a difference of two dilations of a star-shaped region in , and (ii) when the average is taken over all dilations of star-shaped regions in . These cases include, respectively, the average over annuli and the average over balls centered at origin.

A fast apparent horizon finder for three-dimensional Cartesian grids in numerical relativity**Appendix B on ‘multiprocessor and parallelization issues’ and appendix C on ‘searching for the critical parameter of a 1-parameter initial data sequence’ also appear in the preprint-archive version of this paper (gr-qc/0306056).

In 3 + 1 numerical simulations of dynamic black-hole spacetimes, it is useful to be able to find the apparent horizon(s) (AH) in each slice of a time evolution. A number of AH finders are available, but they often take many minutes to run, so they are too slow to be practically usable at each time step. Here I present a new AH finder, AHFINDERDIRECT, which is very fast and accurate: at typical resolutions it takes only a few seconds to find an AH to ∼10−5m accuracy on a GHz-class processor. I assume that an AH to be searched for is a Strahlkörper (‘star-shaped region’) with respect to some local origin, and so parametrize the AH shape by r = h(angle) for some single-valued function h:S2 → ℜ+. The AH equation then becomes a nonlinear elliptic PDE in h on S2, whose coefficients are algebraic functions of gij, Kij, and the Cartesian-coordinate spatial derivatives of gij. I discretize S2 using six angular patches (one each in the neighbourhood of the ±x, ± y, and ±z axes) to avoid coordinate singularities, and finite difference the AH equation in the angular coordinates using fourth-order finite differencing. I solve the resulting system of nonlinear algebraic equations (for h at the angular grid points) by Newton's method, using a ‘symbolic differentiation’ technique to compute the Jacobian matrix. AHFINDERDIRECT is implemented as a thorn in the CACTUS computational toolkit, and is freely available by anonymous CVS checkout.

Uniqueness and Complementary Energy in Nonlinear Elastostatics

Global uniqueness of the smooth stress and deformation to within the usual rigid-body translation and rotation is established in the null traction boundary value problem of nonlinear homogeneous elasticity on a n-dimensional star-shaped region. A complementary energy is postulated to be a function of the Biot stress and to be para-convex and rank-(n-1) convex, conditions analogous to quasi-convexity and rank-(n-2) of the stored energy function. Uniqueness follows immediately from an identity involving the complementary energy and the Piola-Kirchhoff stress. The interrelationship is discussed between the two conditions imposed on the complementary energy, and between these conditions and those known for uniqueness in the linear elastic traction boundary value problem.

Theory of Ordered Spaces¶II. The Local Differential Structure

In this paper we investigate the conditions under which the ordered spaces defined in [1] are locally diffeomorphic to ℝ N . In Sect.~1 we give an introduction and an overview of the results. In Sect. 2 we show that the axioms of [1] do not suffice to make light rays locally homeomorphic to ℝ. We introduce this structure via the new connectedness axiom 2.13, and work out some of its immediate consequences. In Sect. 3 we give the (somewhat involved) construction of timelike curves in a D-set, which are basic to everything that follows. They are used in Sect. 4 to prove (i) a nested interval theorem for ordered spaces; (ii) the contractibility of order intervals in D-sets; and (iii) that order intervals in D-sets are star-shaped. The notion of D-countability (meaning that a D-set has a countable base in the subspace topology) is introduced in Sect. 5. The Urysohn lemma shows that a D-countable ordered space is locally metrizable. If this space is also locally compact, then it has finite topological dimension N; these results are established in Sect. 6. The local differential structure now follows from known results: the embedding of such spaces in ℝ2 n +1, and the result that an open star-shaped region in ℝ n is diffeomorphic to ℝ n . In conclusion, we exhibit these inclusions in Fig. 3, and suggest the possibility that Wigner's position on the “Unreasonable effectiveness of mathematics in the natural sciences” may be open to reasonable doubt. The axioms of [1] are given in the Appendix.

EXPLICIT SOLUTIONS OF CLASSES OF EXTERIOR DIFFERENTIAL SYSTEMS

Explicit solutions of certain exterior differential systems of degrees one and two are constructed on a star-shaped region of a differentiable manifold. A closed differential ideal that is generated by 1-forms that are solutions of an exterior differential is constructed. This ideal does not satisfy the Frobenius theorem because there are no solutions that satisfy the necessary independence condition, although a modified version of the Frobenius theorem is exhibited in this case. Solutions of certain exterior differential systems whose associated curvature 2-forms have compact support in a star-shaped region are also constructed.

Angular homeostasis: III. The formalism of discrete orbits in ontogeny.

The formal properties of orbits in a plane are explored by elementary topology. The notions developed from first principles include: convex and polygonal orbits; convexity; orientation, winding number and interior; convex and star-shaped regions. It is shown that an orbit that is convex with respect to each of its interior points bounds a convex region. Also, an orbit that is convex with respect to a fixed point bounds a star-shaped region. Biological considerations that directed interest to these patterns are indicated, and the implications of the prospect of higher orders of star-shapedness mentioned.