Homothetic transformations and geometric loci: properties of triangles and quadrilaterals

In this paper, we use geometric transformations to find some interesting properties related with geometric loci. In particular, given a triangle or a cyclic quadrilateral, the locus generated by the centroid or by the orthocentre (for triangles) or by the anticentre (for cyclic quadrilaterals) when one vertex moves on the circumcircle of the figure are considered. These loci are studied in paragraphs 3 and 4. By means of the homothetic transformations some properties of triangles and quadrilaterals are found. The study of these properties can be used, with profit, in a classroom activity supported by Dynamic Geometry System.     

Cut loci and conjugate loci on Liouville surfaces

In the earlier paper (Itoh and Kiyohara, Manuscr Math 114:247?C264, 2004), we showed that the cut locus of a general point on two-dimensional ellipsoid is a segment of a curvature line and proved ??Jacobi??s last geometric statement?? on the singularities of the conjugate locus. In the present paper, we show that a wider class of Liouville surfaces possess such simple cut loci and conjugate loci. The results include the determination of cut loci and the set of poles on two-sheeted hyperboloids and elliptic paraboloids.     

Geometric relations between the zeros of polynomials

We consider the classical theorem of Grace, which gives a condition for a geometric relation between two arbitrary algebraic polynomials of the same degree. This theorem is one of the basic instruments in the geometry of polynomials. In some applications of the Grace theorem, one of the two polynomials is fixed. In this case, the condition in the Grace theorem may be changed. We explore this opportunity and introduce a new notion of locus of a polynomial. Using the loci of polynomials, we may improve some theorems in the geometry of polynomials. In general, the loci of a polynomial are not easy to describe. We prove some statements concerning the properties of a point set on the extended complex plane that is a locus of a polynomial.     

Bisoptic curves of hyperbolas

Given a hyperbola, we study its bisoptic curves, i.e. the geometric locus of points through which passes a pair of tangents making a fixed angle θ or 180° ? θ. This question has been addressed in a previous paper for parabolas and for ellipses, showing hyperbolas and spiric curves, respectively. Here the requested geometric locus can be empty. If not, it is a punctured spiric curve, and two cases occur: the curve can have either one loop or two loops. Finally, we reconstruct explicitly the spiric curve as the intersection of a plane with a self-intersecting torus.     

Balanced split sets and Hamilton�CJacobi equations

We study the singular set of solutions to Hamilton?CJacobi equations with a Hamiltonian independent of u. In a previous paper, we proved that the singular set is what we called a balanced split locus. In this paper, we find and classify all balanced split loci, identifying the cases where the only balanced split locus is the singular locus, and the cases where this does not hold. This clarifies the relationship between viscosity solutions and the classical approach of characteristics, providing equations for the singular set. Along the way, we prove more structure results about the singular sets.     

On the stability of a finite-difference scheme for nonlocal parabolic boundary-value problems

We deal with the stability analysis of difference schemes for a one-dimensional parabolic equation subject to integral conditions. It is based on the spectral structure of the transition matrix of a difference scheme. The stability domain is defined by using the hyperbola which is the locus of points where the transition matrix has trivial eigenvalues. The stability conditions obtained are much more general compared with those known in the literature. We analyze three separate cases of nonlocal integral conditions and solve an example illustrating the efficiency of the technique.     

A stochastic model for solitons

The soliton physics for the propagation of waves is represented by a stochastic model in which the particles of the wave can jump ahead according to some probability distribution. We demonstrate the presence of a steady state (stationary distribution) for the wavelength. It is shown that the stationary distribution is a convolution of geometric random variables. Approximations to the stationary distribution are investigated for a large number of particles. The model is rich and includes Gaussian cases as limit distribution for the wavelength (when suitably normalized). A sufficient Lindeberg‐like condition identifies a class of solitons with normal behavior. Our general model includes, among many other reasonable alternatives, an exponential aging soliton, of which the uniform soliton is one special subcase (with Gumbel's stationary distribution). With the proper interpretation, our model also includes the deterministic model proposed in Takahashi and Satsuma [A soliton cellular automaton, J Phys Soc Japan 59 (1990), 3514–3519]. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2004     

On the variety of rational space curves

We consider the locus of smooth rational curves of given degree in a given projective space, which are incident to a generic collection of linear spaces. When this locus is finite (resp. 1-dimensional) we give a recursive procedure to compute its degree (resp. geometric genus). The method is based on the elementary geometry of ruled surfaces.     

On Boundary Stability of Wave Equations with Variable Coefficients

In this paper, we consider the boundary stabilization of the wave equation with variable coefficients by Riemmannian geometry method subject to a different geometric condition which is motivated by the geometric multiplier identities. Several (multiplier) identities (inequalities) which have been built for constant wave equation by Kormornik and Zuazua are generalized to the variable coefficient case by some computational techniques in Riemmannian geometry, so that the precise estimates on the exponential decay rate are derived from those inequalitities. Also, the exponential decay for the solutions of semilinear wave equation with variable coefficients is obtained under natural growth and sign assumptions on the nonlinearity. Our method is rather general and can be adapted to other evolution systems with variable coefficients (e.g. elasticity plates) as well.     

Raising the order of geometric numerical integrators by composition and extrapolation

We analyse composition and polynomial extrapolation as procedures to raise the order of a geometric integrator for solving numerically differential equations. Methods up to order sixteen are constructed starting with basic symmetric schemes of order six and eight. If these are geometric integrators, then the new methods obtained by extrapolation preserve the geometric properties up to a higher order than the order of the method itself. We show that, for a number of problems, this is a very efficient procedure to obtain high accuracy. The relative performance of the different algorithms is examined on several numerical experiments. AMS subject classification 17B66, 34A50, 65L05     

Fuzzy正项几何规划的直接算法

将Fuzzy正项几何规划化的一变量有上、下界限制的Fuzzy正项几何规划,利用Fuzzy几何不等式,又将该变量有上、下界限制的Fuzzy正项几何规划化为单项Fuzzy正项几何规划,提出基于Fuzzy值集割平面法的两种直接算法,并通过一个数值例证实该方法的有效性。     

Complete Group Classification and Exact Solutions to the Generalized Short Pulse Equation

In this paper, the complete group classification is performed on the generalized short pulse equation, which includes a lot of important nonlinear wave equations as its special cases. In the sense of geometric symmetry, all of the vector fields of the equation are obtained in terms of the arbitrary functions. Then, the symmetry reductions and exact solutions to the equations are investigated. Especially, we develop the analytic power series method for constructing the exact power series solutions to the short pulse types of equations.     

On asymptotic theory for multivariate GARCH models

The paper investigates the asymptotic theory for a multivariate GARCH model in its general vector specification proposed by Bollerslev, Engle and Wooldridge (1988) [4], known as the VEC model. This model includes as important special cases the so-called BEKK model and many versions of factor GARCH models, which are often used in practice. We provide sufficient conditions for strict stationarity and geometric ergodicity. The strong consistency of the quasi-maximum likelihood estimator (QMLE) is proved under mild regularity conditions which allow the process to be integrated. In order to obtain asymptotic normality, the existence of sixth-order moments of the process is assumed.     

Improved Results on Geometric Hitting Set Problems

We consider the problem of computing minimum geometric hitting sets in which, given a set of geometric objects and a set of points, the goal is to compute the smallest subset of points that hit all geometric objects. The problem is known to be strongly NP-hard even for simple geometric objects like unit disks in the plane. Therefore, unless P = NP, it is not possible to get Fully Polynomial Time Approximation Algorithms (FPTAS) for such problems. We give the first PTAS for this problem when the geometric objects are half-spaces in ?³ and when they are an r-admissible set regions in the plane (this includes pseudo-disks as they are 2-admissible). Quite surprisingly, our algorithm is a very simple local-search algorithm which iterates over local improvements only.     

Correcting for biases in affected sib-pair linkage analysis caused by uncertainty in sibling relationship

When there is uncertainty in sibling relationship,the classical affected sib-pair(ASP) linkage tests may be severely biased.This can happen,for example,if some of the half sib-pairs are mixed with full sib-pairs.The genomic control method has been used in association analysis to adjust for population structures.We show that the same idea can be applied to ASP linkage analysis with uncertainty in sibling relationship.Assuming that,in addition to the candidate marker,null markers that are unlinked to the disease locus are also genotyped,we may use the information on these loci to estimate the proportion of half sib-pairs and to correct for the bias and variance distortion caused by the heterogeneity of sibling relationship.Unlike in association studies,the null loci are not required to be matched with the candidate marker in allele frequency for ASP linkage analysis.This makes our approach flexible in selecting null markers.In our simulations,using a number of 30 or more null loci can effectively remove the bias and variance distortion.It is also shown that,even the null loci are weakly linked to the disease locus,the proposed method can also provide satisfactory correction.     

Geometric loci and homothetic transformations

In this paper, we use a property of the centroids to study two general types of geometric loci: one in the plane and one in the space. Then from this result, we examine in detail the case of quadrilaterals and tetrahedra and, in particular, we determine some properties of the nine-point circle of a triangle. The study of those geometric loci is carried out by means of the homothetic transformations that facilitate and make more immediate the proofs of the properties.     

An outline of linear programming

Linear programming deals with the maximization or minimization of linear functions subject to linear inequality constraints. Definitions and practical examples are given in Section 2. Section 3 gives a geometric interpretation of the simplex algorithm. Section 4 develops the mathematical theory of duality and gives heuristic interpretations in terms of shadow prices. Section 5 studies those linear programs for which integer solutions are normally required and discusses those cases in which integer solutions arise naturally and those in which special techniques must be used.The preparation of this paper was supported in part by the Army Research Office under Contract No. DA-ARO-D-31-124-70-G42.     

Hybrid stress finite element formulation based on additional internal displacements – application of microscopic geometric perturbation and extension to linear dynamics

Based on the variational principle for hybrid stress finite element method, the assumed stress distribution can be determined through a variational process which includes a microscopic geometric perturbation. An interpretation of such geometric perturbation is discussed. For the creation of, what is called, hyper-bioelement and for the purpose of maintaining invariance, tensor notations of the physical quantities are represented by the use of covariant and contravariant basis vectors of the natural coordinate system. This process is illustrated by a simple 4-node plane element. Furthermore, based on the legitimate variational principle for linear dynamics and the presently proposed method to choose the assumed stress distribution, the finite element formulation different from the ordinary formula is advocated.