It is shown that, in characteristic zero, a finite subgroup of a general linear group is generated by pseudo-reflections if and only if its ring of coinvariants satisfies Poincaré duality.

We show that the Bogdanov-Takens system has at most one limit cycle. Similarly we show that the maximum number of limit cycles in the universal unfolding of the symmetric cusp of order 2 (resp. 3) is one (resp. 2). The proof uses the elementary technique of Liénard's equation, yielding a global result for all values of the parameters.

A lower bound on the number of points that can be placed in a square of side $\sigma$ such that no two points are within unit distance from each other is proven. The result is constructive, and the series of packings obtained contains many conjecturally optimal packings.

Let $M$ be an $m$ dimensional submanifold in the Euclidean space ${{\text{R}}^{n}}$ and $H$ be the mean curvature of $M$. We obtain some low geometric estimates of the total squaremean curvature $\int\limits_{M}{{{H}^{2}}d\sigma }$. The low bounds are geometric invariants involving the volume of $M$, the total scalar curvature of $M$, the Euler characteristic and the circumscribed ball of $M$.

What is a von Neumann algebra? What is a factor (i) of type I, (ii) of type II, (iii) of type III? What is a projection geometry? And finally, what is a continuous geometry?

The questions recall some of the most brilliant mathematical work of the past 30 years, work which was done by John von Neumann, partly in collaboration with F. J. Murray, and which grew out of von Neumann1 s analysis of linear operators in Hilbert space.

Let A be a local ring with maximal ideal m, let N(m ) be the order of the residue field A/m and let N be a subgroup of GLn (A) which is normalized by SLn(A). It follows from results of Klingenberg that N is normal in GLn(A) when n ≥ 3 or . Results of Lacroix show that this is also true when n = 2 and N(m) = 3, provided that N∩2(A) ≠ SL2(A)'.

The principal aim of this paper is to provide examples of non-normal subgroups of GL2(A) which are normalized by SL2(A). In the process we extend results of Lacroix and Levesque on GL2(A)-normalized subgroups of GL2(A), where 2 ∊ m and N(m) > 2.

It gives me quite extraordinary pleasure to have been asked to deliver the Jeffrey-Williams lecture of the Canadian Mathematical Society. The reasons are manifold. First of all Canada was my home for the most formative years of my life — from 16 to 23 — and was in fact the first country willing to take me on as an adopted son. I was of course born in Budapest, but in Europe the geographical accidents of birth are not taken seriously, rather I inherited my father's status and so managed to become stateless "by induction" so to speak.

We study the transfer of nondegeneracy between Lie triple systems and their standard Lie algebra envelopes as well as between Kantor pairs, their associated Lie triple systems, and their Lie algebra envelopes. We also show that simple Kantor pairs and Lie triple systems in characteristic 0 are nondegenerate.

Let $f$ be a modular form that is non-ordinary at $p$. Loeffler has recently constructed four two-variable $p$-adic $L$-functions associated with $f$. In the case where ${{a}_{p}}\,=\,0$, he showed that, as in the one-variable case, Pollack’s plus and minus splitting applies to these new objects. In this article, we show that such a splitting can be generalised to the case where ${{a}_{p}}\ne 0$ using Sprung’s logarithmic matrix.

It is known that the normalized standard generators of the free orthogonal quantum group $O_{N}^{+}$ converge in distribution to a free semicircular system as $N\,\to \,\infty$. In this note, we substantially improve this convergence result by proving that, in addition to distributional convergence, the operator normof any non-commutative polynomial in the normalized standard generators of $O_{N}^{+}$ converges as $N\,\to \,\infty$ to the operator norm of the corresponding non-commutative polynomial in a standard free semicircular system. Analogous strong convergence results are obtained for the generators of free unitary quantum groups. As applications of these results, we obtain a matrix-coefficient version of our strong convergence theorem, and we recover a well-known ${{\mathcal{L}}^{2}}\,-\,{{\mathcal{L}}^{\infty }}$ norm equivalence for noncommutative polynomials in free semicircular systems.

We consider the inviscid limit for the two-dimensional Navier–Stokes equations in the class of integrable and bounded vorticity fields. It is expected that the difference between the Navier–Stokes and Euler velocity fields vanishes in $L^2$ with an order proportional to the square root of the viscosity constant $\nu $. Here, we provide an order $ (\nu /|\log \nu | )^{\frac 12\exp (-Ct)}$ bound, which slightly improves upon earlier results by Chemin.

Let G(n;m) denote a graph of n vertices and m edges. Vertices of G will be denoted by x1, …, y1 …; edges will be denoted by (x, y) and triangles by (x, y, z). (G - x1-x2 - … - xk) will denote the graph G from which the vertices x1, …, xk and all edges incident to them have been omitted G - (xi, xj) denotes the graph G from which the edge (xi, xj) has been omitted.

Strong oscillation criteria will be obtained for the linear elliptic partial differential equation

(1)

in unbounded domains R of general type in n-dimensional Euclidean space En. It will be assumed throughout that B and each Aij are real-valued continuous functions in R, and that the matrix (Aij(x)) is symmetric and positive definite in R.

The purpose of this paper is two-fold. Sections 2 and 3 are motivated by an observation that certain theorems concerning "diminishing orbital diameters" (introduced in [1]) are true under weaker assumptions. Specifically, we investigate the relationship between that concept and alternate conditions such as "asymptotic regularity", and in the process we sharpen some metric space results established in [1;5]. Mention is made in these sections of examples which show that certain additional weakenings of our hypotheses cannot be made, but we include in detail only the one which seemed to us most intricate.

We construct an explicit continuous function F such that for each point x, every extended real number is a derived number of F at x and F has an infinite left and an infinite right derived number at x.

Necessary and sufficient conditions are found for the existence of two positive solutions of the semilinear elliptic equation Δu + q(|x|)u = f(x, u) in an exterior domain Ω⊂ℝn, n ≥ 1, where q, f are real-valued and locally Hölder continuous, and f(x, u) is nonincreasing in u for each fixed x∈Ω. An example is the singular stationary Klein-Gordon equation Δu — k2u = p(x)u-λ where k and λ are positive constants. In this case NASC are given for the existence of two positive solutions ui(x) in some exterior subdomain of Ω such that both |x|m exp[(-l)i-1k|x|]ui(x) are bounded and bounded away from zero in this subdomain, m = (n —1)/2, i = 1, 2.

The Sobolev inequality of order m asserts that if p ≧ 1, mp < n and 1/q = 1/p — m/n, then the Lq-norm of a smooth function with compact support in Rn is bounded by a constant times the sum of the Lp-norms of the partial derivatives of order m of that function. In this paper we show that that sum may be reduced to include only the completely mixed partial derivatives or order m, and in some circumstances even fewer partial derivatives.

For each finite group $G$ for which the integral group ring $\mathbb{Z}G$ has only trivial units, we give ring-theoretic conditions for a commutative ring $R$ under which the group ring $RG$ has nontrivial units. Several examples of rings satisfying the conditions and rings not satisfying the conditions are given. In addition, we extend a well-known result for fields by showing that if $R$ is a ring of finite characteristic and $RG$ has only trivial units, then $G$ has order at most 3.

We describe contracted semigroup algebras of Malcev nilpotent semigroups that are prime Noetherian maximal orders.

If A is a real $2n \times 2n$ positive definite matrix, then there exists a symplectic matrix M such that $M^TAM=\text {diag}(D, D),$ where D is a positive diagonal matrix with diagonal entries $d_1(A)\leqslant \cdots \leqslant d_n(A).$ We prove a maxmin principle for $d_k(A)$ akin to the classical Courant–Fisher–Weyl principle for Hermitian eigenvalues and use it to derive an analogue of the Weyl inequality $d_{i+j-1}(A+B)\geqslant d_i(A)+d_j(B).$