A variation on the unique product property of groups is described which seems natural from a geometric point of view. It is stronger than the unique product property, and hence implies, for example, that the group rings have no zero divisors. Some of its closure properties are described. It is shown that most surface groups satisfy this condition, and various other examples are given. It is explained how these ideas can give a more geometric interpretation of Promislow's example of a non-unique-product group.

Inverse Sturm–Liouville problems with eigenparameter-dependent boundary conditions are considered. Theorems analogous to those of both Hochstadt and Gelfand and Levitan are proved.

In particular, let ly = (1/r)(−(py′)′+qy), l˜y = (1/r˜)(−(p˜y′)′+q˜y),

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where det Δ = δ > 0, c ≠ 0, det [sum ] > 0, t ≠ 0 and (cs + dr − au − tb)2 < 4(cr − ta)(ds − ub). Denote by (l; α; Δ) the eigenvalue problem ly = λy with boundary conditions y(0)cosα+y′(0)sinα = 0 and (aλ+b)y(1) = (cλ+d)(py′)(1). Define (l˜; α; Δ) as above but with l replaced by l˜. Let wn denote the eigenfunction of (l; α; Δ) having eigenvalue λn and initial conditions wn(0) = sin α and pw′n(0) = −cos α and let γn = −awn(1)+cpw′n(1). Define w˜n and γ˜n similarly.

As sample results, it is proved that if (l; α; Δ) and (l˜; α; Δ) have the same spectrum, and (l; α; Σ) and (l˜; α; Σ) have the same spectrum or ∫10[mid ]wn[mid ]2rdt+([mid ]γn[mid ]2/δ) = ∫10[mid ]w˜n[mid ]2r˜dt+([mid ]γ˜n[mid ]2/δ) for all n, then q/r = q˜/r˜.

In the study of discrete groups it is important to find conditions for a group to be discrete. Given a discrete subgroup of Möbius transformations containing a parabolic element with fixed point ∞, a classical result, called Shimizu's lemma, gives a uniform bound on the radii of isometric circles of those elements of the group not fixing ∞. Recently Parker [8] has shown that if a discrete subgroup G of PU(1, n; C) contains a Heisenberg translation g, then any element of G not sharing a fixed point with g has an isometric sphere whose radius is bounded above by a function of the translation length of g at its centres. Parker's theorem is considered as a generalization of Shimizu's lemma. In [1] Basmajian and Miner have independently obtained qualitatively similar results for discrete subgroups of PU(1, 2; C) by using their stable basin theorem.

The purpose of this paper is twofold. First we improve the stable basin theorem, and second we show that under some conditions Parker's theorem yields the discreteness conditions of Basmajian and Miner for groups with a Heisenberg translation. The latter answers a question posed in [8].

The paper provides a new look at compensated compactness and paracommutators. It is shown that there is a one-to-one correspondence between compensated quantities and paracommutators, that the H1-regularity of compensated quantities is equivalent to bounded mean oscillation boundedness of paracommutators, that the weak convergence of compensated quantities is equivalent to vanishing mean oscillation compactness of paracommutators, that the Schatten–von Neumann Sp-property is a natural generalization of vanishing mean oscillation compactness, that the theory of paracommutators provides a ready-made tool for Sp-regularity of compensated quantities, and that the cut-off phenomenon of paracommutators exactly coincides with the vanishing moment properties of the compensated quantities. Some further examples are given.

It is shown that every irreducible admissible representation of Sp4(F), where F is a p-adic field, has at most one Fourier Jacobi model for given Fourier Jacobi data.

Let v be a measure on a separable metric space. For t, q ∈ R, the centred Hausdorff measures μh with the gauge function h(x, r) = rt(vB(x, r))q is studied. The dimension defined by these measures plays an important role in the study of multifractals. It is shown that if v is a doubling measure, then μh is equivalent to the usual spherical measure, and thus they define the same dimension. Moreover, it is shown that this is true even without the doubling condition, if q [ges ] 1 and t [ges ] 0 or if q [les ] 0. An example in R2 is also given to show the surprising fact that the above assertion is not necessarily true if 0 < q < 1. Another interesting question, which has been asked several times about the centred Hausdorff measure, is whether it is Borel regular. A positive answer is given, using the above equivalence for all gauge functions mentioned above.

In this paper we study the second-order parabolic equation

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in a domain [0,T]×ℝd ⊂ ℝd+1, where a = (aij)di,j=1 is matrix of bounded measurable coefficients, b = (bj)dj=1, and bˆ = (bˆj)dj=1 are measurable (in general, singular) vector fields, V is a measurable potential, T is a fixed positive number, and ∂tu = ∂u/∂t, and we employ the notation

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We introduce a new class of coefficients in the lower-order terms for which we prove the existence and the uniqueness of the weak fundamental solution, and for this we derive Gaussian upper and lower bounds.

In this paper we study the abstract quasi-linear evolution equation of second order

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in a general Banach space Z. It is well-known that the abstract quasi-linear theory due to Kato [10, 11] is widely applicable to quasi-linear partial differential equations of second order and that his theory is based on the theory of semigroups of class (C0). (For example, see the work of Hughes et al. [9] and Heard [8].) However, even in the special case where A(t, w, v) = A is independent of (t, w, v), it is found in [2] and [14] that there exist linear partial differential equations of second order for which Cauchy problems are not solvable by the theory of semigroups of class (C0) but fit into the mould of well-posed problems where the solution and its derivative depend continuously on the initial data if the initial condition is measured in the graph norm of a suitable power of A. (See also work by Krein and Khazan [13] and Fattorini [6, Chapter 8].) This kind of Cauchy problem has recently been studied extensively, using the theory of integrated semigroups or regularized semigroups. The theory of integrated semigroups was studied intensively by Arendt [1] and that of regularized semigroups was initiated by Da Prato [3] and renewed by Davies and Pang [4]. For the theory of regularized semigroups we refer the reader to [5] and [16].

The compression theorem is used to prove results for equivariant configuration spaces that are analogous to the well-known non-equivariant results of May, Milgram and Segal.

Let us consider the boundary value problem

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where Ω ⊂ ℝN is a bounded domain with smooth boundary (for example, such that certain Sobolev imbedding theorems hold). Let

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Then, if ϕ(s) = [mid ]s[mid ]p−1s, p > 1, problem (1) is fairly well understood and a great variety of existence results are available. These results are usually obtained using variational methods, monotone operator methods or fixed point and degree theory arguments in the Sobolev space W1,0p(Ω). If, on the other hand, we assume that ϕ is an odd nondecreasing function such that

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and

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then a Sobolev space setting for the problem is not appropriate and very general results are rather sparse. The first general existence results using the theory of monotone operators in Orlicz–Sobolev spaces were obtained in [5] and in [9, 10]. Other recent work that puts the problem into this framework is contained in [2] and [8].

It is in the spirit of these latter papers that we pursue the study of problem (1) and we assume that F[ratio ]Ω×ℝ→ℝ is a Carathéodory function that satisfies certain growth conditions to be specified later.

We note here that the problems to be studied, when formulated as operator equations, lead to the use of the topological degree for multivalued maps (cf. [4, 16]).

We shall see that a natural way of formulating the boundary value problem will be a variational inequality formulation of the problem in a suitable Orlicz–Sobolev space. In order to do this we shall have need of some facts about Orlicz–Sobolev spaces which we shall give now.

In this paper we continue our investigation in [5, 7, 8] on multipeak solutions to the problem

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where Δ = [sum ]Ni=1δ2/δx2i is the Laplace operator in ℝN, 2 < q < ∞ for N = 1, 2, 2 < q < 2N/(N−2) for N[ges ]3, and Q(x) is a bounded positive continuous function on ℝN satisfying the following conditions.

(Q1) Q has a strict local minimum at some point x0∈ℝN, that is, for some δ > 0

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for all 0 < [mid ]x−x0[mid ] < δ.

(Q2) There are constants C, θ > 0 such that

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for all [mid ]x−x0[mid ] [les ] δ, [mid ]y−y0[mid ] [les ] δ.

Our aim here is to show that corresponding to each strict local minimum point x0 of Q(x) in ℝN, and for each positive integer k, (1.1) has a positive solution with k-peaks concentrating near x0, provided ε is sufficiently small, that is, a solution with k-maximum points converging to x0, while vanishing as ε → 0 everywhere else in ℝN.

Let [ ] be the open unit disk in the complex plane [Copf ]. The Bergman space L2a([ ]) is the Hilbert space of analytic functions f in [ ] such that

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where dA is the normalized area measure on [ ]. If

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are two functions in L2a([ ]), then the inner product of f and g is given by

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We study multiplication operators on L2a([ ]) induced by analytic functions. Thus for φ ∈ H ∞([ ]), the space of bounded analytic functions in [ ], we define

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by

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It is easy to check that Mφ is a bounded linear operator on L2a([ ]) with

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The paper considers a mixing Smale space, the relations of stable and unstable equivalence on such a space and the C*-algebras which are constructed from them. In general, these associations are not functorial. However, it is shown that, if one restricts to the class of s-resolving, finite-to-one factor maps, then the construction of the stable C*-algebra is contravariant, while that of the unstable C*-algebra is covariant. The paper also discusses the constructions of these C*-algebras for Smale spaces which are not mixing.

Conditions for pointwise Fourier inversion using spherical Cesàro means of a given degree are established on rank 1 noncompact symmetric spaces.

It is shown that if a plane of PG(3,q), q even, meets an ovoid in a conic, then the ovoid must be an elliptic quadric. This is proved by using the generalized quadrangles T2([Cscr ]) ([Cscr ] a conic), W(q) and the isomorphism between them to show that every secant plane section of the ovoid must be a conic. The result then follows from a well-known theorem of Barlotti.

A linear operator between two Banach spaces X and Y is strictly-singular (or Kato) if it fails to be an isomorphism on any infinite dimensional subspace. A weaker notion for Banach lattices introduced in [8] is the following one: an operator T from a Banach lattice X to a Banach space Y is said to be disjointly strictly-singular if there is no disjoint sequence of non-null vectors (xn)n∈ℕ in X such that the restriction of T to the subspace [(xn)∞n=1] spanned by the vectors (xn)n∈ℕ is an isomorphism. Clearly every strictly-singular operator is disjointly strictly-singular but the converse is not true in general (consider for example the canonic inclusion Lq[0, 1][rarrhk ]Lp[0, 1] for 1[les ]p<q<∞). In the special case of considering Banach lattices X with a Schauder basis of disjoint vectors both concepts coincide. The notion of disjointly strictly-singular has turned out to be a useful tool in the study of lattice structure of function spaces (cf. [7–9]). In general the class of all disjointly strictly-singular operators is not an operator ideal since it fails to be stable with respect to the composition on the right.

The aim of this paper is to study when the inclusion operators between arbitrary rearrangement invariant function spaces E[0, 1] ≡ E on the probability space [0, 1] are disjointly strictly-singular operators.

The paper studies the heat kernel of the Schrödinger operator with magnetic fields and of uniformly elliptic operators with non-negative electric potentials in the reverse Hölder class which includes non-negative polynomials as typical examples. The main aim of the paper is to give a pointwise estimate of the heat kernel of the operators above which is affected by magnetic fields and non-negative degenerate electric potentials. A weighted smoothing estimate for the semigroup generated by the operators above is also given.

The tangent developables of space curves are classified locally and topologically. It turns out that only seven different tangent developables arise, except for very degenerate cases. An extension of the known classification due to Cleave, Shcherbak and Mond is given.

The paper considers finite subsets Λ ⊂ Zd which possess the extension property, namely that every collection {ck}k∈Λ−Λ of complex numbers which is positive definite with respect to Λ is the restriction of the Fourier coefficients of some positive measure on Td. All finite subsets of Z2 which possess the extension property are described.

Every convolution operator on a space of ultradifferentiable functions of Beurling or Roumieu type and on the corresponding space of ultradistributions is hypercyclic and chaotic when it is not a multiple of the identity. The operator of differentiation is hypercyclic on the space A−∞, but it need not be hypercyclic on radial weighted algebras of entire functions.