盲数的概念、运算与性质

本文阐述了“盲数”产生的背景，给出了“盲数”的定义、运算与性质。“盲数”是用来表达和处理具有多种不确定性的较复杂信息的数学工具。     

Coning, Symmetry and Spherical Frameworks

In this paper, we combine separate works on (a) the transfer of infinitesimal rigidity results from an Euclidean space to the next higher dimension by coning (Whiteley in Topol. Struct. 8:53?C70, 1983), (b) the further transfer of these results to spherical space via associated rigidity matrices (Saliola and Whiteley in arXiv:0709.3354, 2007), and (c) the prediction of finite motions from symmetric infinitesimal motions at regular points of the symmetry-derived orbit rigidity matrix (Schulze and Whiteley in Discrete Comput. Geom. 46:561?C598, 2011). Each of these techniques is reworked and simplified to apply across several metrics, including the Minkowskian metric $\mathbb{M}^{d}$ and the hyperbolic metric ? ^d . This leads to a set of new results transferring infinitesimal and finite motions associated with corresponding symmetric frameworks among $\mathbb{E}^{d}$ , cones in $\mathbb{E}^{d+1}$ , $\mathbb{S}^{d}$ , $\mathbb{M}^{d}$ , and ? ^d . We also consider the further extensions associated with the other Cayley?CKlein geometries overlaid on the shared underlying projective geometry.     

Planarity, Symmetry and Counting Tilings

We axiomatize the geometrical properties of T-tetromino to what we call generalised k-T-tetromino. Using this set of axioms, we show that the number of tilings of a 2kn × 2km rectangular region is given by T(L _n,m ; 3, 3) if and only if the tile is a k-T-tetromino. This generalizes a result of Korn and Pak (Theor Comp Sci 319:3–27, 2004).     

On Locations and Distances

Regional science is about integrating different spatial disciplines, in particular spatial economics and economic geography. In this spirit a first study bears on the envelopment of production units inside an urban technological complex, comparing natural and logarithmic envelopments from the point of view of their respective representative efficiency. A second and related study treats the reverse problem: given Manhattan distances between production units, which are their respective locations?     

Games,Kernels, and Antitone Operations

In this note we want to point out the usefulness of antitone operations in the study of games on graphs. We demonstrate this by proving, among others, the generalizations from games to antitone operations of theorems stated by Fraenkel. This perspective makes also clear the general nature of some problems by separating what is general and what is particular.     

函数的对称性

作为启示学生数学概念的一种扩展性思考,介绍中心对称函数和轴对称函数概念.同时讨论对称函数的导函数的对称性和对称函数的原函数的对称性.     

Nonnegative Curvature, Symmetry and Fundamental Group

We prove a result on equivariant deformations of flat bundles, and as a corollary, we obtain two splitting in a finite cover theorems for isometric group actions on Riemannian manifolds with infinite fundamental groups, where the manifolds are either compact of Ric 0, or complete of sec 0. The result is used to construct Lie group actions that are isometric in some metric of scal > 0 and that are not isometric in any metric of Ric 0.     

Affine-Invariant Distances, Envelopes and Symmetry Sets

Affine invariant symmetry sets of planar curves are introduced and studied in this paper. Two different approaches are investigated. The first one is based on affine invariant distances, and defines the symmetry set as the closure of the locus of points on (at least) two affine normals and affine-equidistant from the corresponding points on the curve. The second approach is based on affine bitangent conics. In this case the symmetry set is defined as the closure of the locus of centers of conics with (at least) 3-point contact with the curve at two or more distinct points on the curve. This is equivalent to conic and curve having, at those points, the same affine tangent, or the same Euclidean tangent and curvature. Although the two analogous definitions for the classical Euclidean symmetry set are equivalent, this is not the case for the affine group. We present a number of properties of both affine symmetry sets, showing their similarities with and differences from the Euclidean case. We conclude the paper with a discussion of possible extensions to higher dimensions and other transformation groups, as well as to invariant Voronoi diagrams.     

The Inverse Gaussian Models: Analogues of Symmetry, Skewness and Kurtosis

The inverse Gaussian (IG) family is strikingly analogous to the Gaussian family in terms of having simple inference solutions, which use the familiar ², t and F distributions, for a variety of basic problems. Hence, the IG family, consisting of asymmetric distributions is widely used for modelling and analyzing nonnegative skew data. However, the process lacks measures of model appropriateness corresponding to and ₂, routinely employed in statistical analyses. We use known similarities between the two families to define a concept termed IG-symmetry, an analogue of the symmetry, and to develop IG-analogues ₁ and ₂ of and ₂, respectively. Interestingly, the asymptotic null distributions of the sample versions d ₁, d ₂ of ₁, ₂ are exactly the same as those of their normal counterparts and b ₂. Some applications are discussed, and the analogies between the two families, enhanced during this study are tabulated.     

Semistar Operations and Standard Closure Operations

Let R be a commutative ring. It is shown that there is an order isomorphism between a popular class of finite type closure operations on the ideals of R and the poset of semistar operations of finite type.     

The Joy and Pain of Skew Symmetry

In this paper, we review recent progress on two related issues. Firstly, the discretisation of partial differential equations of quantum mechanics in a semiclassical regime. Due to the presence of a small parameter, such equations exhibit high oscillations and multiscale behaviour, rendering them difficult to discretise. We describe a methodology, using symmetric Zassenhaus splittings in a free Lie algebra, which allows for their exceedingly fast and accurate numerics. The imperative of preserving the unitarity of the underlying flow takes us to the second theme of this paper, approximation of derivatives by skew-symmetric matrices. Here, we identify a gap in the elementary theory of finite-difference approximations: in the presence of Dirichlet boundary conditions, it is impossible to approximate the derivative to order higher than two on a uniform grid! This motivates the investigation of skew symmetry on non-uniform grids, an endeavour which, although still in its infancy, is already replete with interesting results. We conclude by discussing a number of generalisations and open problems.     

强收敛与弱收敛的微小区别

对于独立同分布随机变量序列{Xn,n≥1},证明了其满足弱大数律的一个必要条件是:对于任何r∈(0,1),E|X|r<∞都成立.另外我们还举例说明了这样的条件不是充分的.     

Operations on Hypermaps,and Outer Automorphisms

We present a group ℌ, isomorphic to PGL(2, ℤ), of operations on hypermaps, and a group ℌ, isomorphic to GL(2, ℤ), of operations on oriented hypermaps without boundary. These are induced by the group of automorphisms of a certain group G whose transitive permutation representations correspond to hypermaps.     

Operations on maps,and outer automorphisms

By representing maps on surfaces as transitive permutation representations of a certain group Γ, it is shown that there are exactly six invertible operations (such as duality) on maps; they are induced by the outer automorphisms of Γ, and form a group isomorphic to S₃. Various consequences are deduced, such as the result that each finite map has a finite reflexible cover which is invariant under all six operations.