Continuous valuations on the space of Lipschitz functions on the sphere

We study real-valued valuations on the space of Lipschitz functions over the Euclidean unit sphere $S^{n?1}$. After introducing an appropriate notion of convergence, we show that continuous valuations are bounded on sets which are bounded with respect to the Lipschitz norm. This fact, in combination with measure theoretical arguments, will yield an integral representation for continuous and rotation invariant valuations on the space of Lipschitz functions over the 1-dimensional sphere.     

Dimension of the space of unitary equivariant translation invariant tensor valuations

Following the work of Semyon Alesker in the scalar valued case and of Thomas Wannerer in the vector valued case, the dimensions of the spaces of continuous translation invariant and unitary equivariant tensor valuations are computed. In addition, a basis in the vector valued case is presented.     

Translation invariant diffusions in the space of tempered distributions

In this paper we prove existence and pathwise uniqueness for a class of stochastic differential equations (with coefficients σ _ij , b _i and initial condition y in the space of tempered distributions) that may be viewed as a generalisation of Ito’s original equations with smooth coefficients. The solutions are characterized as the translates of a finite dimensional diffusion whose coefficients σ _ij $\tilde y$ , b _i $\tilde y$ are assumed to be locally Lipshitz.Here denotes convolution and $\tilde y$ is the distribution which on functions, is realised by the formula $\tilde y\left( r \right): = y\left( { - r} \right)$ . The expected value of the solution satisfies a non linear evolution equation which is related to the forward Kolmogorov equation associated with the above finite dimensional diffusion.     

Classification of invariant valuations on the quaternionic plane

We describe the orbit space of the action of the group Sp(2)Sp(1)$\mathrm{Sp}(2)\mathrm{Sp}(1)$ on the real Grassmann manifolds _Grk(H²)${\mathrm{Gr}}_k({\mathbb{H}}^2)$ in terms of certain quaternionic matrices of Moore rank not larger than 2. We then give a complete classification of valuations on the quaternionic plane H²${\mathbb{H}}^2$ which are invariant under the action of the group Sp(2)Sp(1)$\mathrm{Sp}(2)\mathrm{Sp}(1)$.     

Extensions of valuations to rational function fields over completions

Given a valued field $(K,v)$ and its completion $(\widehat{K},v)$, we study the set of all possible extensions of v to $\widehat{K}(X)$. We show that any such extension is closely connected with the underlying subextension $(K(X)|K,v)$. The connections between these extensions are studied via minimal pairs, key polynomials, pseudo-Cauchy sequences, and implicit constant fields. As a consequence, we obtain strong ramification theoretic properties of $(\widehat{K},v)$. We also give necessary and sufficient conditions for $(K(X),v)$ to be dense in $(\widehat{K}(X),v)$.     

Characterizations of Translation Generalized Quadrangles

If x is a regular point of the generalizedquadrangle of order (s,t), s 1 t, then x defines a dual net . If contains a line L of regularpoints and if for at least one point x on Lthe automorphism group of the dual net satisfies certain transitivityproperties, then is a translation generalized quadrangle. Thisresult has many applications. We give one example. Ifs=t 1, then is a dual affine plane. Let be a generalizedquadrangle of orders,s odd and s 1, which contains a lineL of regular points. If for at least one pointx on L the plane is Desarguesian, then is isomorphic to the classical generalizedquadrangleW(s).     

Translation planes admitting a pair of index three homology groups

It is shown that a translation plane of order which admits two homology groups of order must in fact admit symmetric homology groups of this order. It is further shown that a plane admitting such symmetric index 3 homology groups is, with a finite number of exceptions, a generalized André plane. A list of the possibly exceptional orders is determined. Received 20 March 2000.     

Chapter 3 Translation invariance in data envelopment analysis: A generalization

In this paper, we undertake a revision and a generalization of the results contained in the only paper so far published on the matter of translation invariance by allowing inputs and outputs to take not only zero but negative values. This broadens the field of application of the DEA methodology.     

Wavelet subspaces invariant under groups of translation operators

We study the action of translation operators on wavelet subspaces. This action gives rise to an equivalence relation on the set of all wavelets. We show by explicit construction that each of the associated equivalence classes is non-empty.     

Conformally invariant powers of the Dirac operator in Clifford analysis

The paper deals with conformally invariant higher‐order operators acting on spinor‐valued functions, such that their symbols are given by powers of the Dirac operator. A general classification result proves that these are unique, up to a constant multiple. A general construction for such an invariant operators on manifolds with a given conformal spin structure was described in (Conformally Invariant Powers of the Ambient Dirac Operator. ArXiv math.DG/0112033, preprint), generalizing the case of powers of the Laplace operator from (J. London Math. Soc. 1992; 46 :557–565). Although there is no hope to obtain explicit formulae for higher powers of the Laplace or Dirac operator on a general manifold, it is possible to write down an explicit formula on Einstein manifolds in case of the Laplace operator (see Laplacian Operators and Curvature on Conformally Einstein Manifolds. ArXiv: math/0506037, 2006). Here we shall treat the spinor case on the sphere. We shall compute the explicit form of such operators on the sphere, and we shall show that they coincide with operators studied in (J. Four. Anal. Appl. 2002; 8 (6):535–563). The methods used are coming from representation theory combined with traditional Clifford analysis techniques. Copyright © 2010 John Wiley & Sons, Ltd.     

复杂随机系统的生存性问题研究

为了刻画复杂随机系统的理性决策,提出了复杂随机系统的生存性及不变性的概念,给出并证明了复杂随机系统的生存性定理及不变性定理.并提出了均方相依锥,生存域与不变域的概念.得到了与文献中的一致的结论.     

Orthomorphisms and the construction of projective planes

We discuss a simple computational method for the construction of finite projective planes. The planes so constructed all possess a special group of automorphisms which we call the group of translations, but they are not always translation planes. Of the four planes of order 9, three admit the additive group of the field as a group of translations, and the present construction yields all three. The known planes of order 16 comprise four self-dual planes and eighteen other planes (nine dual pairs); of these, the method gives three of the four self-dual planes and six of the nine dual pairs, including the ``sporadic' (not translation) plane of Mathon.

     

Measures in the space of planes and convex bodies

The present paper gives two representations (so-called flag representations) for the measure of planes intersecting a convex body. These representations are obtained by means of a stochastic approximation of a convex body depending on a distribution P that are defined on the surface of the body. Also it is proved that the flag representations do not depend on the density of P.     

Non-existence of finite-time self-similar solutions of the Keller-Segel system in the scaling invariant class

Let us consider the problem whether there does exist a finite-time self-similar solution of the backward type to the semilinear Keller-Segel system. In the case of parabolic-elliptic type for n?3 we show that there is no such a solution with a finite mass in the scaling invariant class. On the other hand, in the case of parabolic-parabolic type for n?2, non-existence of finite-time self-similar solutions is proved in a larger class of a finite mass with some local bounds.     

The valuations of the near hexagons related to the Witt designs S(5,6,12) and S(5,8,24)

Valuations of dense near polygons were introduced in 16 . In the present paper, we classify all valuations of the near hexagons ??₁ and ??₂, which are related to the respective Witt designs S(5,6,12) and S(5,8,24). Using these classifications, we prove that if a dense near polygon S contains a hex H isomorphic to ??₁ or ??₂, then H is classical in S. We will use this result to determine all dense near octagons that contain a hex isomorphic to ??₁ or ??₂. As a by‐product, we obtain a purely geometrical proof for the nonexistence of regular near 2d‐gons, d ≥ 4, whose parameters s, t, t_i (0 ≤ i ≤ d) satisfy (s, t₂, t₃) = (2, 1, 11) or (2, 2, 14). The nonexistence of these regular near polygons can also be shown with the aid of eigenvalue techniques. © 2005 Wiley Periodicals, Inc. J Combin Designs 14: 214–228, 2006     

Conformally invariant powers of the Laplacian -- A complete nonexistence theorem

We show that on conformal manifolds of even dimension there is no conformally invariant natural differential operator between density bundles with leading part a power of the Laplacian for n/2$">. This shows that a large class of invariant operators on conformally flat manifolds do not generalise to arbitrarily curved manifolds and that the theorem of Graham, Jenne, Mason and Sparling, asserting the existence of curved version of for , is sharp.

     

An invariance principle for a finite dimensional stochastic approximation method in a Hilbert space

For the application of the classical Robbins-Monro procedure in a Hilbert space the statistician generally has to observe infinite dimensional vectors. A modified procedure is proposed, which works in appropriate finite dimensional subspaces of growing dimension. For this procedure an invariance principle is given together with some applications.