Compact Ovoids in Quadrangles I: Geometric Constructions

This paper is about ovoids in infinite generalized quadrangles. Using the axiom of choice, Cameron showed that infinite quadrangles contain many ovoids. Therefore, we consider mainly closed ovoids in compact quadrangles. After deriving some basic properties of compact ovoids, we consider ovoids which arise from full imbeddings. This leads to restrictions for the topological parameters (m,m). For example, if there is a regular pair of lines or a full closed subquadrangle, then mm. The existence of full subquadrangles implies the nonexistence of ideal subquadrangles, so finite-dimensional quadrangles are either point-minimal or line-minimal. Another result is that (up to duality) such a quadrangle is spanned by the set of points on an ordinary quadrangle. This is useful for studying orbits of automorphism groups. Finally we prove general nonexistence results for ovoids in quadrangles with low-dimensional line pencils. As one consequence we show that the symplectic quadrangle over an algebraically closed field of characteristic 0 has no Zariski-closed ovoids or spreads.     

Geometric Algebra in Linear Algebra and Geometry

This article explores the use of geometric algebra in linear and multilinear algebra, and in affine, projective and conformal geometries. Our principal objective is to show how the rich algebraic tools of geometric algebra are fully compatible with and augment the more traditional tools of matrix algebra. The novel concept of an h-twistor makes possible a simple new proof of the striking relationship between conformal transformations in a pseudo-Euclidean space to isometries in a pseudo-Euclidean space of two higher dimensions. The utility of the h-twistor concept, which is a generalization of the idea of a Penrose twistor to a pseudo-Euclidean space of arbitrary signature, is amply demonstrated in a new treatment of the Schwarzian derivative.     

从积分几何的观点看几何不等式——纪念李国平院士吴新谋教授诞辰100周年

该文先介绍一些中国数学家在几何不等式方面的工作.作者用积分几何中著名的Poincarè公式及Blaschke公式估计一随机凸域包含另一域的包含测度, 得到了经典的等周不等式和Bonnesen -型不等式.还得到了一些诸如对称混合等周不等式、Minkowski -型和Bonnesen -型对称混合等似不等式在内的一些新的几何不等式.最后还研究了Gage -型等周不等式以及Ros -型等周不等式.     

利用辛几何构作两类Cartesian认证码

本文利用辛几何构作了两类 Cartesian认证码 ,计算了码的参数 .当编码规则按等概率分布选取时 ,计算出敌方成功的模仿攻击概率和成功的替换攻击概率     

Quasigroups, Geometry and Nonlinear Geometric Algebra

A survey of the methods of the theory of quasigroups and loops in algebra and geometry is presented in order to attract the attention of mathematicians and physicists to promising applications of this new branch of mathematics in applied sciences.     

Some Constructions of Cartesian Authentication Codes from Pseudo－symplectic Geometry

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Explicit Constructions of Universal R-Trees and Asymptotic Geometry of Hyperbolic Spaces

This paper presents explicit constructions of universal R-treesas certain spaces of functions, and also proves that a -universal R-tree can be isometricallyembedded at infinity into a complete simply connected manifoldof negative curvature, or into a non-abelian free group. Incontrast to asymptotic cone constructions, asymptotic spacesare built without using the axiom of choice. 2000 MathematicsSubject Classification L53C23, 20F67.     

Geometry of Diffeomorphism Groups, Complete integrability and Geometric statistics

We study the geometry of the space of densities Dens(M), which is the quotient space Diff(M)/Diff _μ (M) of the diffeomorphism group of a compact manifold M by the subgroup of volume-preserving diffeomorphisms, endowed with a right-invariant homogeneous Sobolev ${\dot{H}^1}$ -metric. We construct an explicit isometry from this space to (a subset of) an infinite-dimensional sphere and show that the associated Euler–Arnold equation is a completely integrable system in any space dimension whose smooth solutions break down in finite time. We also show that the ${\dot{H}^1}$ -metric induces the Fisher–Rao metric on the space of probability distributions and its Riemannian distance is the spherical version of the Hellinger distance.     

Prompting Teacher Geometric Reasoning through Coaching in a Dynamic Geometry Software Context

This study investigated the ways in which four middle grades teachers developed mathematical knowledge for teaching (MKT) geometry as they implemented dynamic geometry software in their classrooms with the assistance of a coach. Teachers developed various components of MKT by observing coaches teach, by dynamic discourse with students, which is discourse with respect to dynamic geometry software images, and by discussions with coaches. The dynamic geometry software environment proved productive as coaches guided teachers’ growth in explanations, examples, and definitions. The environment also helped teachers discover unnoticed abilities among their low achievers. Moreover, teachers developed confidence to teach with more expert uses of dynamic geometry software.     

The Geometry of Railway Geometric Overthrow Revisited Using Computer Algebra Methods

There are size standards for railway coaches and freight wagons in order to allow trains to pass from one railway line or even network to another. The maximum size and shape of the rolling stock is denoted gauge. On curves, vehicles sweep a larger path than on straight track. We shall focus on the geometric overthrow of the different kind of rolling stock (the geometric overthrow is that part of the vehicle element offset which is due to the track curve). We shall detail the influence of the kind of vehicle (two axles, classic bogies, shared bogies or Talgo vehicles) in the geometric overthrow. The computations can be easily carried out with a computer algebra system such as Maple and some curious results are obtained (for instance, a negative value (!) is obtained for Talgo coaches).     

From Multifractal Measures to Multifractal Wavelet Series

Given a positive locally finite Borel measure μ on R, a natural way to construct multifractal wavelet series is to set , where . Indeed, under suitable conditions, it is shown that the function F_μ inherits the multifractal properties of μ. The transposition of multifractal properties works with many classes of statistically selfsimilar multifractal measures, enlarging the class of processes which have self-similarity properties and controlled multifractal behaviors. Several perturbations of the wavelet coefficients and their impact on the multifractal nature of F_μ are studied. As an application, multifractal Gaussian processes associated with F_μ are created. We obtain results for the multifractal spectrum of the so-called W-cascades introduced by Arnéodo et al.     

A Geometric Build-Up Algorithm for Solving the Molecular Distance Geometry Problem with Sparse Distance Data

Nuclear magnetic resonance (NMR) structure modeling usually produces a sparse set of inter-atomic distances in protein. In order to calculate the three-dimensional structure of protein, current approaches need to estimate all other missing distances to build a full set of distances. However, the estimation step is costly and prone to introducing errors. In this report, we describe a geometric build-up algorithm for solving protein structure by using only a sparse set of inter-atomic distances. Such a sparse set of distances can be obtained by combining NMR data with our knowledge on certain bond lengths and bond angles. It can also include confident estimations on some missing distances. Our algorithm utilizes a simple geometric relationship between coordinates and distances. The coordinates for each atom are calculated by using the coordinates of previously determined atoms and their distances. We have implemented the algorithm and tested it on several proteins. Our results showed that our algorithm successfully determined the protein structures with sparse sets of distances. Therefore, our algorithm reduces the need of estimating the missing distances and promises a more efficient approach to NMR structure modeling.     

Self-Conformal Multifractal Measures

A self-conformal measure is a measure invariant under a set of conformal mappings. In this paper we describe the local structure of self-conformal measures. For such a measure we divide its support into sets of fixed local dimension and give a formula for the Hausdorff and packing dimensions of these sets. Moreover, we compute the generalized dimensions of the self-conformal measure.     

Constructions in Sasakian geometry

We first generalize the join construction described previously by the first two authors [4] for quasi-regular Sasakian-Einstein orbifolds to the general quasi-regular Sasakian case. This allows for the further construction of specific types of Sasakian structures that are preserved under the join operation, such as positive, negative, or null Sasakian structures, as well as Sasakian-Einstein structures. In particular, we show that there are families of Sasakian-Einstein structures on certain 7-manifolds homeomorphic to S ² × S ⁵. We next show how the join construction emerges as a special case of Lerman’s contact fibre bundle construction [32]. In particular, when both the base and the fiber of the contact fiber bundle are toric we show that the construction yields a new toric Sasakian manifold. Finally, we study toric Sasakian manifolds in dimension 5 and show that any simply-connected compact oriented 5-manifold with vanishing torsion admits regular toric Sasakian structures. This is accomplished by explicitly constructing circle bundles over the equivariant blow-ups of Hirzebruch surfaces. During the preparation of this work the first two authors were partially supported by NSF grants DMS-0203219 and DMS-0504367.     

Multifractal structure in Latin-American market indices

We study the multifractal nature of daily price and volatility returns of Latin-American stock markets employing the multifractal detrended fluctuation analysis. Comparing with the results obtained for a developed country (US) we conclude that the multifractality degree is higher for emerging markets. Moreover, we propose a stock market inefficiency ranking by considering the multifractality degree as a measure of inefficiency. Finally, we analyze the sources of multifractality quantifying the contributions of two factors, the long-range correlations of the time series and the broad fat-tail distributions. We find that the multifractal structure of Latin-American market indices can be mainly attributed to the latter.     

Constructions Using Conics

Conclusion  After recalling definitions and results about the constructibility of a geometric object, we have shown by more and more efficient methods how the works of Gauss, computer algebra systems (Maple), and dynamic geometry software (Cabri-Geometry, distributed by Texas Instruments) could be used together to construct regular polygons, using ruler, compass, and simple conics. In particular, we have given the list of small ₂-constructible polygons, and presented new ₂-constructions of the regular polygons with 19, 37, 73, and 97 sides. The ancient Greeks gave precedence to constructions using only ruler and compass, not because they did not know about the other curves (they invented a number of mechanical devices drawing some algebraic curves of degrees 2, 3, 4, and more), but for the neatness, perfection of reasoning, and the simplicity of the shapes involved (circle and straight line). Today’s tools such as Cabri-Geometry enlarge the notion of geometric simplicity by allowing the manipulation of algebraic expressions (the sequences defined by Gauss) and complex geometric objects (the conic sections). Some generalizations of the questions treated here may be considered: