Linking (n − 2)-dimensional panels inn-space I: (k − 1,k)-graphs and (k − 1,k)-frames

A (k – 1,k)-graph is a multi-graph satisfyinge (k – 1)v – k for every non-empty subset ofe edges onv vertices, with equality whene = |E(G)|. A (k – 1,k)-frame is a structure generalizing an (n – 2, 2)-framework inn-space, a structure consisting of a set of (n – 2)-dimensional bodies inn-space and a set of rigid bars each joining a pair of bodies using ball joints. We prove that a graph is the graph of a minimally rigid (with respect to edges) (k – 1,k)-frame if and only if it is a (k – 1,k)-graph. Rigidity here means infinitesimal rigidity or equivalently statical rigidity.     

Regularity and explicit representation of (0,1,...,m−2,m) interpolation on the zeros of (1−x 2)P n −2/(α,β) (x)

Necessary and sufficient conditions for the regularity andq-regularity of (0,1,...,m–2,m) interpolation on the zeros of (1–x ²)P _n ^–2/(,) (x) (,>–1) in a manageable form are established, whereP _n ^–2/(,) (x) stands for the (n–2)th Jacobi polynomial. Meanwhile, the explicit representation of the fundamental polynomials, when they exist, is given. Moreover, we show that under a mild assumption if the problem of (0,1,...,m–2,m) interpolation has an infinity of solutions then the general form of the solutions isf ₀(x)+C f(x) with an arbitrary constantC.This work is supported by the National Natural Science Foundation of China.     

Simple 3-designs and PSL(2, q) with q ≡ 1 (mod 4)

In this paper, we consider the action of (2, q) on the finite projective line for q ≡ 1 (mod 4) and construct several infinite families of simple 3-designs which admit PSL(2, q) as an automorphism group. Some of the designs are also minimal. We also indicate a general outline to obtain some other algebraic constructions of simple 3-designs.      

Some new 3-designs from PSL(2, q) with q ≡ 1 (mod4)

We determine the sizes of orbits from the action of subgroups of PSL(2,q) on projective line X = GF(q) ∪ {∞} with q a prime power and congruent to 1 modulo 4.As an example of its application,we construct some new families of simple 3-designs admitting PSL(2,q) as automorphism group.     

Propertyf −1(S) =g −1(S) for entire and meromorphicP-adic functions

LetW be an algebraically closed filed of characteristic zero, letK be an algebraically closed field of characteristic zero, complete for an ultrametric absolute value, and letA(K) (resp. ℳ(K)) be the set of entire (resp. meromorphic) functions inK. For everyn≥7, we show that the setS _n(b) of zeros of the polynomialx ⁿ−b (b≠0) is such that, iff, g ∈W[x] or iff, g ∈A(K), satisfyf ⁻¹(S _n(b))=g ⁻¹(S _n(b)), thenf ⁿ=g ⁿ. For everyn≥14, we show thatS _n(b) is such that iff, g ∈W({tx}) or iff, g ∈ ℳ(K) satisfyf ⁻¹(S _n(b))=g ⁻¹(S _n(b)), then eitherf ⁿ=g ⁿ, orfg is a constant. Analogous properties are true for complex entire and meromorphic functions withn≥8 andn≥15, respectively. For everyn≥9, we show that the setY _n(c) of zeros of the polynomial , (withc≠0 and 1) is an ursim ofn points forW[x], and forA(K). For everyn≥16, we show thatY _n(c) is an ursim ofn points forW(x), and for ℳ(K). We follow a method based on thep-adic Nevanlinna Theory and use certain improvement of a lemma obtained by Frank and Reinders.     

Embedding (p,p−1) graphs in their complements

A graphG is embeddable in its complement ifG is isomorphic with a subgraph of . A complete characterization is given of those (p,p−1) graphs which are embeddable in their complements. In particular, letG be a (p,p−1) graph wherep≧6 ifp is even andp≧9 ifp is odd; thenG is embeddable in if and only ifG is neither the starK _1,p−1 norK _1,n ∪C ₃ withn≧4.     

(S − 1,S) Perishable systems with stochastic leadtimes

This article discusses a one-to-one ordering perishable system, in which reorders are processed in the order of their arrival and the processing times are arbitrarily distributed, and as such, the leadtimes are not independent. The Markov renewal techniques are employed to obtain the various operating characteristics for the case of Poisson demand and exponential lifetimes. The problem of minimizing the steady state expected cost rate is also discussed, and in the special case of exponential processing times, the optimal stock level is derived explicitly.     

不动点集为P(1,n)×HP(1)的光滑对合

设(Mr,T)是一个在r维闭光滑流形上的不平凡光滑对合,可证对合的不动点集是P(1,n)×HP(1),n为奇数,则(Mr,T)协边于零.