On Small and Large Hyperplanes of DW(5, q)

We determine lower and upper bounds for the size of a hyperplane of the dual polar space DW(5, q). In some cases, we also determine all hyperplanes attaining these bounds.     

Valuations and hyperplanes of dual polar spaces

Valuations were introduced in De Bruyn and Vandecasteele (Valuations of near polygons,preprint, 2004) as a very important tool for classifying near polygons. In the present paper we study valuations of dual polar spaces. We will introduce the class of the SDPS-valuations and characterize these valuations. We will show that a valuation of a finite thick dual polar space is the extension of an SDPS-valuation if and only if no induced hex valuation is ovoidal or semi-classical. Each SDPS-valuation will also give rise to a geometric hyperplane of the dual polar space.     

Two classes of hyperplanes of dual polar spaces without subquadrangular quads

Let Π be one of the following polar spaces: (i) a nondegenerate polar space of rank n−1?2 which is embedded as a hyperplane in Q(2n,K); (ii) a nondegenerate polar space of rank n?2 which contains Q(2n,K) as a hyperplane. Let Δ and DQ(2n,K) denote the dual polar spaces associated with Π and Q(2n,K), respectively. We show that every locally singular hyperplane of DQ(2n,K) gives rise to a hyperplane of Δ without subquadrangular quads. Suppose Π is associated with a nonsingular quadric Q⁻(2n+?,K) of PG(2n+?,K), ?∈{−1,1}, described by a quadratic form of Witt-index , which becomes a quadratic form of Witt-index when regarded over a quadratic Galois extension of K. Then we show that the constructed hyperplanes of Δ arise from embedding.     

On the simple connectedness of hyperplane complements in dual polar spaces, II

Suppose Δ is a dual polar space of rank n and H is a hyperplane of Δ. Cardinali, De Bruyn and Pasini have already shown that if n≥4 and the line size is greater than or equal to 4 then the hyperplane complement Δ−H is simply connected. This paper is a follow-up, where we investigate the remaining cases. We prove that the hyperplane complements are simply connected in all cases except for three specific types of hyperplane occurring in the smallest case, when the rank and the line size are both 3.     

On embeddings of CAT(0) cube complexes into products of trees via colouring their hyperplanes

We prove that the contact graph of a 2-dimensional CAT(0) cube complex X of maximum degree Δ can be coloured with at most ?(Δ)=MΔ²⁶$?(\mathrm{\Delta })=M{\mathrm{\Delta }}^{26}$ colours, for a fixed constant M. This implies that X (and the associated median graph) isometrically embeds in the Cartesian product of at most ?(Δ)$?(\mathrm{\Delta })$ trees, and that the event structure whose domain is X admits a nice labelling with ?(Δ)$?(\mathrm{\Delta })$ labels. On the other hand, we present an example of a 5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes which cannot be embedded into a Cartesian product of a finite number of trees. This answers in the negative a question raised independently by F. Haglund, G. Niblo, M. Sageev, and the first author of this paper.     

On geometric SDPS-sets of elliptic dual polar spaces

Let n∈N?{0,1} and let K and K^′ be fields such that K^′ is a quadratic Galois extension of K. Let Q⁻(2n+1,K) be a nonsingular quadric of Witt index n in PG(2n+1,K) whose associated quadratic form defines a nonsingular quadric Q⁺(2n+1,K^′) of Witt index n+1 in PG(2n+1,K^′). For even n, we define a class of SDPS-sets of the dual polar space DQ⁻(2n+1,K) associated to Q⁻(2n+1,K), and call its members geometric SDPS-sets. We show that geometric SDPS-sets of DQ⁻(2n+1,K) are unique up to isomorphism and that they all arise from the spin embedding of DQ⁻(2n+1,K). We will use geometric SDPS-sets to describe the structure of the natural embedding of DQ⁻(2n+1,K) into one of the half-spin geometries for Q⁺(2n+1,K^′).     

Small Point Sets that Meet All Generators of W(2n+1,q)

Let W(2n+1,q), n1, be the symplectic polar space of finite order q and (projective) rank n. We investigate the smallest cardinality of a set of points that meets every generator of W(2n+1,q). For q even, we show that this cardinality is q ⁿ⁺¹+q ^{n–1, and we characterize all sets of this cardinality. For q odd, better bounds are known.     

关于mr(2,q)的一个新值

编码理论中关于寻找某一线性码的最大长度涉及到有限射影空间中关于t-blockingsets,(k,r-ares和caps集所含元素的个数的问题,本文研究了(k,r)-arc集的元素的个数,找到了使得(k,r)-arc集存在的最大k值,即mr(2,q)的一个新值,丰富了编码理论的相关内容。     

Homology representations arising from the half cube, II

In a previous work, we defined a family of subcomplexes of the n-dimensional half cube by removing the interiors of all half cube shaped faces of dimension at least k, and we proved that the reduced homology of such a subcomplex is concentrated in degree k−1. This homology module supports a natural action of the Coxeter group W(Dn) of type D. In this paper, we explicitly determine the characters (over C) of these homology representations, which turn out to be multiplicity free. Regarded as representations of the symmetric group Sn by restriction, the homology representations turn out to be direct sums of certain representations induced from parabolic subgroups. The latter representations of Sn agree (over C) with the representations of Sn on the (k−2)-nd homology of the complement of the k-equal real hyperplane arrangement.     

C~n中F(p,q,s)型空间上的Bergman型算子

设p0,s≥0,q+s-1,q+n-1.讨论了C~n中单位球上F(p,q,s)到本身或A(p,q,s)空间到L(p,q,s)空间上的Bergman型算子的有界性条件.     

F(p,q,s)空间的嵌入关系

为得到一类广泛的解析函数空间F(p,q,s)之间的嵌入关系,将每个空间F(p,q,s)对应为三维空间R(F)中一点(p,q,s),从而将空间之间的嵌入关系转化为三维空间中直线的单调性进行了研究,完善了已有的结果.     

On an integral-type operator from a weighted-type space to F(p,q,s) on the unit ball

We characterize the boundedness and compactness of an integral-type operator introduced by S. Stevi? from a weighted-type space to F(p,q,s) space on the unit ball of ?ⁿ.     

Flocks, Ovoids of Q(4,q)and Designs

We prove that an ovoid O of Q(4,q),q odd, is the Thas' ovoid associated with a semifield flock if and only if O represents, on the Klein quadric, a symplectic spread of PG(3,q), whose associated plane is a semifiled plane.     

On the (p,q)$(p,q)$-type strong law of large numbers for sequences of independent random variables

Li, Qi, and Rosalsky (Trans. Amer. Math. Soc., 368 (2016), no. 1, 539–561) introduced a refinement of the Marcinkiewicz–Zygmund strong law of large numbers (SLLN), the so-called $(p,q)$-type SLLN, where and $q>0$. They obtained sets of necessary and sufficient conditions for this new type SLLN for two cases: , $q>p$, and . Results for the case where and remain open problems. This paper gives a complete solution to these problems. We consider random variables taking values in a real separable Banach space $\mathbf{B}$, but the results are new even when $\mathbf{B}$ is the real line. Furthermore, the conditions for a sequence of random variables $\left\{X_n,n\ge 1\right\}$ satisfying the $(p,q)$-type SLLN are shown to provide an exact characterization of stable type p Banach spaces.     

$(p,q)$-Analogue of Mittag-Leffler Function with $(p,q)$-Laplace Transform

The aim of this paper is to define $(p, q)$-analogue of Mittag-Leffler Function, by using $(p, q)$-Gamma function. Some transformation formulae are also derived by using the $(p, q)$-derivative. The $(p, q)$-analogue for this function provides elegant generalization of $q$-analogue of Mittag-Leffler function in connection with $q$-calculus. Moreover, the $(p, q)$-Laplace Transform of the Mittag-Leffler function has been obtained. Some special cases have also been discussed.     

Trialities and 1-Systems of Q <Superscript>+</Superscript>(7,q)

The image of a 1-system of Q7q under a triality of the D₄-geometry, attached to Q7q, will be investigated. Attention will mainly be paid to the case of a locally hermitian, semiclassical 1-system of a Q6 q, embedded in Q7q. It is found that its image under a triality is always locally hermitian and semiclassical as well. Moreover, it is a proper 1-system of Q7q whenever the original 1-system of Q6q is not a spread of some generalized hexagon Hq on Q6,q. Finally, some results concerning isomorphisms will be obtained.AMS classification : 51A50, 51E12, 51E30communicated by: S. Ball     

Gleason问题在多复变F(p,q,s)空间上的可解性

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Sets of type (M,N) mod q with respect to hyperplanes in PG(r,q)

In this work we study the sets of type (M,N) mod q-with respect to hyperplanes in PG(r,q), where N-M is a coprime of q.