G cubic transition between two circles with shape control

This paper describes a method for joining two circles with an S-shaped or with a broken back C-shaped transition curve, composed of at most two spiral segments. In highway and railway route design or car-like robot path planning, it is often desirable to have such a transition. It is shown that a single cubic curve can be used for blending or for a transition curve preserving G²$G^2$ continuity with local shape control parameter and more flexible constraints. Provision of the shape parameter and flexibility provide freedom to modify the shape in a stable manner which is an advantage over previous work by Meek, Walton, Sakai and Habib.     

Geometric interpolation by planar cubic <Emphasis Type="Italic">G</Emphasis><Superscript>1</Superscript> splines

In this paper, geometric interpolation by G ¹ cubic spline is studied. A wide class of sufficient conditions that admit a G ¹ cubic spline interpolant is determined. In particular, convex data as well as data with inflection points are included. The existence requirements are based upon geometric properties of data entirely, and can be easily verified in advance. The algorithm that carries out the verification is added. AMS subject classification (2000)  65D05, 65D07, 65D17     

On the k-sample Behrens-Fisher problem for high-dimensional data

For several decades, much attention has been paid to the two-sample Behrens-Fisher (BF) problem which tests the equality of the means or mean vectors of two normal populations with unequal variance/covariance structures. Little work, however, has been done for the k-sample BF problem for high dimensional data which tests the equality of the mean vectors of several high-dimensional normal populations with unequal covariance structures. In this paper we study this challenging problem via extending the famous Scheffe’s transformation method, which reduces the k-sample BF problem to a one-sample problem. The induced one-sample problem can be easily tested by the classical Hotelling’s T ² test when the size of the resulting sample is very large relative to its dimensionality. For high dimensional data, however, the dimensionality of the resulting sample is often very large, and even much larger than its sample size, which makes the classical Hotelling’s T ² test not powerful or not even well defined. To overcome this difficulty, we propose and study an L ²-norm based test. The asymptotic powers of the proposed L ²-norm based test and Hotelling’s T ² test are derived and theoretically compared. Methods for implementing the L ²-norm based test are described. Simulation studies are conducted to compare the L ²-norm based test and Hotelling’s T ² test when the latter can be well defined, and to compare the proposed implementation methods for the L ²-norm based test otherwise. The methodologies are motivated and illustrated by a real data example. The work was supported by the National University of Singapore Academic Research Grant (Grant No. R-155-000-085-112)     

On G 2 Continuous Spline Interpolation of Curves in ℝ d

In this paper the problem of G ² continuous interpolation of curves in ^d by polynomial splines of degree n is studied. The interpolation of the data points and two tangent directions at the boundary is considered. The case n = r + 2 = d, where r is the number of interior points interpolated by each segment of the spline curve, is studied in detail. It is shown that the problem is uniquely solvable asymptotically, e., when the data points are sampled regularly and sufficiently dense, and lie on a regular, convex parametric curve in ^d . In this case the optimal approximation order is also determined.     

The smallness problem for -algebras

Akemann showed that any von Neumann algebra with a weak^* separable dual space has a faithful normal representation on a separable Hilbert space. He posed the question: If a C^*-algebra has a weak^* separable state space, must it have a faithful representation on a separable Hilbert space? Wright solved this question negatively and showed that a unital C^*-algebra has the weak^* separable state space if and only if it has a unital completely positive map, into a type I factor on a separable Hilbert space, whose restriction to the self-adjoint part induces an order isomorphism. He called such a C^*-algebra almost separably representable. We say that a unital C^*-algebra is small if it has a unital complete isometry into a type I factor on a separable Hilbert space. In this paper we show that a unital C^*-algebra is small if and only if the state spaces of all n by n matrix algebras over the C^*-algebra are weak^*-separable. It is natural to ask whether almost separably representable algebras are small or not. We settle this question positively for simple C^*-algebras but the general question remains open.     

A New Construction of Central Relative (p a , p a , p a , 1)-Difference Sets

Semiregular relative difference sets (RDS) in a finite group E which avoid a central subgroup C are equivalent to orthogonal cocycles. For example, every abelian semiregular RDS must arise from a symmetric orthogonal cocycle, and vice versa. Here, we introduce a new construction for central (p ^a , p ^a , p ^a , 1)-RDS which derives from a novel type of orthogonal cocycle, an LP cocycle, defined in terms of a linearised permutation (LP) polynomial and multiplication in a finite presemifield. The construction yields many new non-abelian (p ^a , p ^a , p ^a , 1)-RDS. We show that the subset of the LP cocycles defined by the identity LP polynomial and multiplication in a commutative semifield determines the known abelian (p ^a , p ^a , p ^a , 1)-RDS, and give a second new construction using presemifields.We use this cohomological approach to identify equivalence classes of central (p ^a , p ^a , p ^a , 1)-RDS with elementary abelian C and E/C. We show that for p = 2, a 3 and p = 3, a 2, every central (p ^a , p ^a , p ^a , 1)-RDS is equivalent to one arising from an LP cocycle, and list them all by equivalence class. For p = 2, a = 4, we list the 32 distinct equivalence classes which arise from field multiplication. We prove that, for any p, there are at least a equivalence classes of central (p ^a , p ^a , p ^a , 1)-RDS, of which one is abelian and a – 1 are non-abelian.     

F*-Rings Are O*

O ^*-rings were introduced by Fuchs and recently characterized by Steinberg. A ring R is called O ^* if every partial order on R extends to a total order. We weaken the condition on the ordering of the ring by requiring that every partial order on R extends to an f-order. We call those rings F ^*-rings. We show that the two classes of rings coincide.     

Shape-Preserving C 2 Functional Interpolation via Parametric Cubics

The paper proposes a method for the construction of a shape preserving C ² function interpolating a given set of data. The constructed interpolant is a parametric cubic curve. The shape of the curve can be easily controlled via tension parameters which have an immediate geometric interpretation. The approximation order is investigated and numerical examples are presented.     

Classification of extensions of classifiable -algebras

For a certain class of extensions of C^*-algebras in which B and A belong to classifiable classes of C^*-algebras, we show that the functor which sends to its associated six term exact sequence in K-theory and the positive cones of K₀(B) and K₀(A) is a classification functor. We give two independent applications addressing the classification of a class of C^*-algebras arising from substitutional shift spaces on one hand and of graph algebras on the other. The former application leads to the answer of a question of Carlsen and the first named author concerning the completeness of stabilized Matsumoto algebras as an invariant of flow equivalence. The latter leads to the first classification result for nonsimple graph C^*-algebras.     

Tensioned Quasi-Interpolation Via Geometric Continuity

The paper proposes a method for the construction of C ² quasi-interpolating functions with tension properties. The constructed quasi-interpolant is a parametric cubic curve and its shape can be easily controlled via tension parameters which have an immediate geometric interpretation. Numerical examples are presented.     

An Asymptotic Expansion for the Distribution of Hotelling'sT-Statistic under Nonnormality

In this paper we obtain an asymptotic expansion for the distribution of Hotelling'sT²-statisticT²under nonnormality when the sample size is large. In the derivation we find an explicit Edgeworth expansion of the multivariatet-statistic. Our method is to use the Edgeworth expansion and to expand the characteristic function ofT².     

TheK-theory of AF embeddings of the rational rotation algebras

Our main result is the construction of an embedding ofC(T²) into an approximately finite-dimensionalC ^*-algebra which induces an injection onK ₀(C(T²)). The existence of this embedding implies that Cech cohomology cannot be extended to a stable, continuous homology theory forC ^*-algebras which admits a well-behaved Chern character. Homotopy properties ofC ^*-algebras are also considered. For example, we show that the second homotopy functor forC ^*-algebras is discontinuous. Similar embeddings are constructed for all the rational rotation algebras, with the consequence that none of the rational rotation algebras satisfies the homotopy property called semiprojectivity.     

Quantum Real Projective Space, Disc and Spheres

We define the C ^*-algebra of quantum real projective space R P _q ², classify its irreducible representations, and compute its K-theory. We also show that the q-disc of Klimek and Lesniewski can be obtained as a non-Galois Z ₂-quotient of the equator Podle quantum sphere. On the way, we provide the Cartesian coordinates for all Podle quantum spheres and determine an explicit form of isomorphisms between the C ^*-algebras of the equilateral spheres and the C ^*-algebra of the equator one.     

The Estimates for Sharp Maximal Functions of Multilinear Strongly Singular Integral Operators

In this paper, the author obtains that the multilinear operators of strongly singular integral operators and their dual operators are bounded from some L^p（R^n） to L^p（R^n） when the m-th order derivatives of A belong to L^p（R^n） for r large enough. By this result, the author gets the estimates for the Sharp maximal functions of the multilinear operators with the m-th order derivatives of A being Lipschitz functions. It follows that the multilinear operators are （L^p, L^p）-type operators for 1 〈 p 〈 ∞.     

Consecutive customer losses in oscillating GIX/M//n systems with state dependent services rates

We address the probability that k or more Consecutive Customer Losses take place during a busy period of a queue, the so-called k-CCL probability, for oscillating GI ^X /M//n systems with state dependent services rates, also denoted as GI ^X /M(m)−M(m)//n systems, in which the service rates oscillate between two forms according to the evolution of the number of customers in the system. We derive an efficient algorithm to compute k-CCL probabilities in these systems starting with an arbitrary number of customers in the system that involves solving a linear system of equations. The results derived are illustrated for specific sets of parameters.     

Hardy spaces estimates for a class of multilinear homogeneous operators

It is proved that a class of multilinear singular integral operators with homogeneous kernels are bounded operators from the product spaces to the Hardy spacesH ^r , (ℝ ⁿ ) and the weak Hardy spaceH ^r,∞ (ℝ ⁿ . As an application of this result, the L ^p ,(ℝ ⁿ ) boundedness of a class of commutator for the singular integral with homogeneous kernels is obtained. Project supparted in part by the National Natural Science Foundation of Chind (Grant No. 19131080) of China and Doctoral Programme Foundation of Institution of Higher Education (Grant No. 98002703) of China.     

Design and G connection of developable surfaces through Bézier geodesics

For potential application in shoemaking and garment manufacture industries, the G¹ connection of (1, k) developable surfaces with abutting geodesic is important. In this paper, we discuss the developable surface which contains a given 3D Bézier curve as geodesic and prove the corresponding conclusions in detail. Primarily we study G¹ connection of developable surfaces through abutting cubic Bézier geodesics and give some examples.     

Equilibrium probability calculations for a discrete-time bulk queue model

Many problems in management science and telecommunications can be solved by the analysis of aD ^X/D^m/1 queueing model. In this paper, we use the zeros, both inside and outside the unit circle, of the denominator of the generating function of the model to obtain an explicit closed-form solution for the equilibrium probabilities of the number of customers in the system. The moments of the number of customers in the queue or in the system are also studied. When there are infinitely many zeros outside the unit circle, we propose an approximation method using polynomials. This method yields correct values for a finite number of the probabilities, the number depending on the degree of the polynomial approximation.     

Relative K-homology and K-homology of an ideal

We provide a straightforward proof of one of the main results of the Baum-Douglas K-homology theory: If A is a separable nuclear C ^*-algebra and I an ideal of A, then the natural restriction map K ⁰(A, I) K ⁰(I) is an isomorphism of Abelian groups.     

Real <Emphasis Type="Italic">AW</Emphasis><Superscript>*</Superscript>-Algebras of type I

Let R be a real AW ^*-algebra, and suppose that its complexification M = R + iR is also a (complex) AW ^*-algebra. We prove that R is of type I if and only if so is M.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 79–81, 2004Original Russian Text Copyright © by Sh. A. Ayupov