Partial difference equations in m1⩾m2⩾⋯⩾mn⩾0 and their applications to combinatorics

Various discrete functions encountered in Combinatorics are solutions of Partial Difference Equations in the subset of Nⁿ given by m₁?m₂???m_n?0. Given a partial difference equation, it is described how to pass from the standard “easy” solution of an equation in Nⁿ to a solution of the same equation subject to certain “Dirichlet” or “Neumann” boundary conditions in the domain m₁?m₂???m_n?0 and related domains. Applications include a rather quick derivation of MacMahon's generating function for plane partitions, a generalization and q-analog of the Ballot problem, and a joint analog of the Ballot problem and Simon Newcomb's problem.     

Groups with T1 primitive ideal spaces

We give a simple necessary and sufficient condition for the group C¹-algebra of a connected locally compact group to have a T₁ primitive ideal space, i.e., to have the property that all primitive ideals are maximal. A companion result settles the same question almost entirely for almost connected groups. As a by-product of the method used, we show that every point in the primitive ideal space of the group C¹-algebra of a connected locally compact group is at least locally closed. Finally, we obtain an analog of these results for discrete finitely generated groups; in particular the primitive ideal space of the group C¹-algebra of a discrete finitely generated solvable group is T₁ if and only if the group is a finite extension of a nilpotent group.     

A note on Banach spaces containing c0 or C∞

It is shown that every separable Banach space X containing a subspace isomorphic to c₀ has a subspace Y with basis such that $\text{X}\text{Y}\text{~}\text{c}{}_0\text{C}{}_{\mathrm{\infty }}$ and the latter space has a shrinking basis and an unconditional FDD. Moreover, it is shown that X ⊕ C_∞ has a basis if X has the bounded approximation property.     

Einheiten in Z[D2m]

The unit of the group rings of the dihedral groups D_2m (m square free odd) over the rational integer ring are represented by matrices over algebraic number fields. Thereby their multiplicative structure is revealed.     

Triangular imbedding of Kn − K6

For n = 12s + 9 (s ≥ 4) we imbed K_n ? K₆ in an orientable surface of genus 12s² + 11s.     

Lower bounds on mt(r, s)

Useful lower bounds are obtained for m_t(r, s), s a prime power, r ? t ? 3, by relating (t ? 1)-flats of PG(r ? 1, s) to (t ? 1) and lower dimensional flats of its subgeometries. Explicit expressions in general are obtained for these bounds when t = 3. A general method is developed for deriving these bounds when t > 3.     

The genus of Kn − K2

The minimal genus of K_n ? K₂ is determined using the method of index two current graphs.     

The crossing number of C3 × Cn

The crossing number of the Cartesian product C₃ × C_n of a 3-cycle and an n-cycle is shown to be n.     

A note on contiguity and L1-norm

The interrelationship between the property of contiguity of two sequences of probability measures {P_n} and {Q_n} and the convergence to zero of ∥P_n ? Q_n∥ is extendended and clarified.     

K-theoretic amenability for SL2(Qp), and the action on the associated tree

It is proved that if a locally compact group G acts simplicially on a tree in such a way that the stabilizers of the vertices are amenable, then G is K-amenable. In particular, the canonical map from the full C¹-algebra onto the reduced C¹-algebra of G induces isomorphisms in K-theory. The main corollary of our result is that SL₂(Q_p) and some other groups over local fields are K-amenable.     

K2 and K3 of the circle

Let k be $\text{Z}\text{[}\text{1}\text{2}\text{]}$, $\text{Q}$ or $\text{R}$, and set $\text{A =}\text{k[x,y]}\text{(x}{}^2\text{+ y}{}^2\text{? 1)}$. We compute K₂(A) and K₃(A). Our method is to construct a map $\text{? : K}{}_1\text{(k[i])→K}{}_{\mathrm{1\; +\; 1}}\text{(A)}$ and compare this to a localization sequence.We give three applications. We show that ? accounts for the primitive elements in K₂(A), and compare our results to computations of Bloch [1] for group schemes. Secondly, we consider the problem of basepoint independence, and indicate the interplay of geometry upon the K-theory of affine schemes obtained by glueing points of Spec(A). Third, we can iterate the construction to compute the K-theory of the torus ring A ?_kA.     

A necessary condition for the local solvability of the operator Pm2(x, D) + P2m − 1(x, D)

It is shown that a necessary condition for the local solvability of the operator P(x, D) = P_m²(x, D) + P_{2m ? 1}(x, D), where P_m(x, D) is an mth-order homogeneous differential operator of principal type with real coefficients, is that along any null-bicharacteristic strip of P_m(x, ξ) the imaginary part of the sub-principal symbol cannot have an odd-order zero where its real part does not vanish.     

A note on the J1-convergence of a first-passage process

Let X₁, X₂, … be a sequence of independent and identically distributed random variables with mean zero such that the common distribution function belongs to the domain of attraction of a stable law G_α,β with 1<α<2 and β=1 or α=2. If S_n=X₁+…X_n and N(ξ)=min{k:S_k>ξ}, ξ>0, then it is shown that $\text{N(nt)}\text{B}{}^1\text{(n)}$, 0<t<1, converges weakly under the Skorohod J₁-topology to a stable subordinator of index $\text{1}\text{α}$, where B¹(n) depends on the norming constant for S_n.     

A note on hyperbolic transformations

We shall discuss the so‐called hyperbolic Householder and Givens transformations applied to complex matrices, including the case of zero hyperbolic energy of a transformed vector. For each case a numerically stable algorithm is available. Copyright © 2001 John Wiley & Sons, Ltd.     

Optimal packings of K4's into a Kn

In this paper we determine the maximum cardinality of a packing of K₄'s into K_n, that is, construct optimal constant weight codes with weight 4 and minimum distance 6.