We give a lower estimate for the central value μ*n(e) of the nth convolution power μ*···*μ of a symmetric probability measure μ on a polycyclic group G of exponential growth whose support is finite and generates G. We also give a similar large time diagonal estimate for the fundamendal solution of the equation (∂/∂t + L)u = 0, where L is a left invariant sub-Laplacian on a unimodular amenable Lie group G of exponential growth.

Let G be a complex reductive group, and G^ its set of irreducible admissible representations. The Bruhat order on G^ is defined in a natural way. We prove that this Bruhat order is preserved by transfer. This gives new proofs of some results by the author on L-functions.

A pointwise estimate for the rate of approximation by polynomials , For 0 ≤ ƛ ≤ 1, integer r, and δn(x) = n-1 + φ(x), is achieved here. This formula bridges the gap between the classical estimate mentioned in most texts on approximation and obtained by Timan and others (ƛ = 0) and the recently developed estimate by Totik and first author (ƛ = 1 ). Furthermore, a matching converse result and estimates on derivatives of the approximating polynomials and their rate of approximation are derived. These results also cover the range between the classical pointwise results and the modern norm estimates for C[— 1,1].

Let u1 and u2 be subharmonic (plurisubharmonic) on overlapping sets K1 and K2 respectively. We seek to approximate u1 and u2 simultaneously by a single subharmonic (plurisubharmonic) function u.

We compute the loops of units in the integral alternative loop rings of six Moufang loops. Four of these are subloops of the loop of matrices of determinant one in Zorn's vector matrix algebra over a ring of integers while the remaining two are closely related to this interesting algebra. This paper thus serves, in part, to highlight a Moufang analogue of SL(2, Z) which the author suggests is worthy of further study.

Let k be a local field, with char(k) ≠ 2. A quadratic space V over k is a finite dimensional vector space together with a non-degenerate quadratic form Q: V → k.The special orthogonal group SO(V) consists of all linear maps T: V → V which satisfy:

Q(Tv) = Q(v) for all ν and det T = 1.

The concept of essential map and topological transversality due to A. Granas is extended to multi-valued maps in locally convex spaces and it is next applied to prove the solvability of boundary value problems for certain neutral functional differential equations. In order to achieve a required compactness property, the weak topology in a Sobolev space is considered. The topological tool established in the first part of the paper allows to avoid some obstacles which are encountered when trying to use standard degree-theoretical arguments.

Existence theorems and asymptotic properties will be obtained for boundary value problems of the form in an unbounded domain Ω⊆ RN(N ≥3) with smooth boundary, where Δ denotes the TV-dimensional Laplacian, τ — (N+ 2)/ (N — 2) is the critical Sobolev exponent, and is the completion of in the L2(Ω) norm of .

We investigate the spectrum for frames with block size four, and discuss several applications to the construction of other combinatorial designs.

Our main result is that a frame of type hu, having blocks of size four, exists if and only if u ≥ 5, h ≡ 0 mod 3 and h(u — 1) ≡ 0 mod 4, except possibly where

1. (i) h = 9 and u ∈ ﹛13,17,29,33,93,113,133,153,173,193﹜;

2. (ii) h ≡ 0 mod 12 and u ∈ ﹛8,12﹜,

3. h = 36 and u ∈ ﹛7,18,23,28,33,38,43,48﹜,

4. h = 24 or 120 and u ∈ ﹛7﹜,

5. h = 72 and u ∈ 2Z+ U ﹛n : n ≡ 3 mod4 and n ≤527﹜ U ﹛563﹜; or

6. (iii) h ≡ 6mod l2 and u ∈ (﹛17,29,33,563﹜ U ﹛n : n ≡ 3 or 11 mod 12 and n ≤ 527﹜ U ﹛n : n ≡ 7 mod 12 and n ≤ 259﹜), h = 18.

Additionally, we give a new recursive construction for resolvable group-divisible designs from frames: if there is a resolvable k-GDD of type gu, a k-frame of type ﹛mg)v where u ≥ m + 1, and a resolvable TD(k, mv) then there is a resolvable k-GDD of type (mg)uv. We use this to construct some new resolvable GDDs with group size three and block size four.

For graphs A, B and a positive integer r, the relation means that whenever Δ is an r-colouring of the vertices of A, then there is an embedding ϕ of B into A such that Δ ∘ ϕ is constant. A class of graphs has the Ramsey property if, for every , there is an such that . For a given finite graph G, let Forb(G) denote the class of all finite graphs which do not embed G. It is known that, if G is 2-connected, then Forb() has the Ramsey property, and Forb(G) has the Ramsey property if and only if Forb(G) also has the Ramsey property. In this paper we show that if neither G nor its complement is 2-connected, then either (i) G has a cut point adjacent to every other vertex, or (ii) G has a cut point adjacent to every other vertex except one. We show that Forb(G) has the Ramsey property if G is a path of length 2 or 3, but that Forb(G) does not have the Ramsey property if (i) holds and G is not the path of length 2.

Let 0 < wbe a smooth function on a complete Riemannian manifold Mn, and define L = — Δ — ▽ (log w) and Rw =Ric - w -1 Hess w.In this paper we show that if Rw ≥ —nK, (K ≥0), then the positive solutions of (L + ∂/∂t)u —0 satisfy a gradient estimate of the same form as that obtained by Li and Yau ([LY]) when Lis the Laplacian. This is used to obtain a parabolic Harnack inequality, which in turn, yields upper and lower Gaussian estimates for the heat kernel of L.The results obtained are applied to study the LPmapping properties of t→ e-tL μfor measures μ which are α-dimensional in a sense that generalises the local uniform α-dimensionality introduced by R. S. Strichartz ([St2], [St3]).

We present a series of conjectures for immanants, together with the supporting evidence we possess for them. The conjectures are loosely organized into three families. The first concerns inequalities involving the immanants of totally positive matrices (Le.,real matrices with nonnegative minors). This includes, for example, the conjecture that immanants of totally positive matrices are nonnegative. The second family involves the immanants of Jacobi-Trudi matrices. These conjectures were suggested by a previous conjecture of Goulden and Jackson (recently proved by C. Greene) that the immanants of Jacobi-Trudi matrices are polynomials with nonnegative coefficients. The third family involves geometric and combinatorial structures associated with total positivity and paths in acyclic digraphs.

Our main result is a Tauberian theorem for the general Euler-Borel summability method. Examples of the method include the discrete Borel, Euler, Meyer- Kônig, Taylor and Karamata methods. Every function/ analytic on the closed unit disk and satisfying some general conditions generates such a method, denoted by (£,ƒ). For instance the function ƒ(z) = exp(z — 1) generates the discrete Borel method. To each such function ƒ corresponds an even positive integer p = p(f).We show that if a sequence (sn)is summable (E,f)and as n→ ∞ m > n, (m— n)n-p(f)→0, then (sn)is convergent. If the Maclaurin coefficients of/ are nonnegative, then p(f) =2. In this case we may replace the condition . This generalizes the Tauberian theorems for Borel summability due to Hardy and Littlewood, and R. Schmidt. We have also found new examples of the method and proved that the exponent —p(f)in the Tauberian condition (*) is the best possible.