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 $M$.

We give examples to show that unlike generalized free products of groups (g.f.p.) polygonal products of finitely generated (f.g.) nilpotent groups with cyclic amalgamations need not be residually finite (R) and polygonal products of finite p-groups with cyclic amalgamations need not be residually nilpotent. However, polygonal products f.g. abelian groups are R, and under certain conditions polygonal products of finite p-groups with cyclic amalgamations are R.

Let G be a graph and ζ(G) be the greatest integer n such that every set of n points in G lies on a cycle [8]. It is clear that ζ(G)≥2 for 2-connected planar graphs. Moreover, it is easy to construct arbitrarily large 2-connected planar graphs for which ζ=2. On the other hand, by a well-known theorem of Tutte [5], [6], if G is planar and 4-connected, it has a Hamiltonian cycle, i.e., ζ(G)=|V(G)| for all 4-connected (and hence for all 5-connected) planar graphs.

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.

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.

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.

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.

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.

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

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.

In the first part of this article we recall the definition and a few basic properties of convex functionals defined on a space of bounded measures. In the second part we show several results of approximation of the following type: Although a measure μ cannot be approximated in the sense of the norm by smooth functions, we can find an appropriate sequence of smooth functions which converge weakly to the measure μ, the corresponding value of the functional converging to the value of the functional at μ.

This article is part of a series on the existence theory of solution of variational problems of mechanics (perfect plasticity), which is based on a systematic utilization of the methods of convex analysis and the calculus of variations.

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.

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.

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 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 established sufficient conditions for the oscillation of all solutions of the delay differential equation

and of the neutral delay differential equation

where p, q, r and a are nonnegative constants and n is an odd natural number.

The main object of this paper is to show that the existence of a particular kind of isomorphism between the integral group rings of two finite groups implies that the groups themselves are isomorphic. The proof employs certain types of linear forms which are first discussed in general. These linear forms are in some way related to the bilinear forms used by Weidmann [3] in showing that groups with isomorphic character rings have the same character table, and a shorter and, in a sense, more natural proof of this result is included here as another application of these linear forms.

Unlike the (classical) Kolakoski sequence on the alphabet {1, 2}, its analogue on {1, 3} can be related to a primitive substitution rule. Using this connection, we prove that the corresponding bi-infinite fixed point is a regular generic model set and thus has a pure point diffraction spectrum. The Kolakoski-(3, 1) sequence is then obtained as a deformation, without losing the pure point diffraction property.