The ‘isomorphism problem for integral group rings’ (IP) is the question whether for two finite groups G and H, the existence of an isomorphism of the integral group rings ℤG and ℤH implies that G and H are isomorphic. Though (IP) is not true in general [4], it is still an interesting question for which classes of finite groups (IP) has a positive solution. In this note, we want to show that (IP) holds for finite groups of Lie type associated to Chevalley groups of universal type.

Note that if U(ℤG) denotes the units of ℤG, and V(ℤG) stands for the units with augmentation 1, then U(ℤG)≅V(ℤG)×U(ℤ), and we can always assume that H[les ]V(ℤG).

Throughout this paper, X will denote a Banach space, S=S(X) and B=B(X) will be the unit sphere and the closed unit ball of X, respectively, and [Gscr ]=[Gscr ](X) will stand for the group of all surjective linear isometries on X. Unless explicitly stated otherwise, all Banach spaces will be assumed to be real. Nevertheless, by passing to real structures, the results remain true for complex spaces.

We show that every μ-constant family of isolated hypersurface singularities of type F(x, t) =f(x)+tg(x), where t is a parameter, is topologically trivial. The proof uses only the curve selection lemma, and hence, for an appropriately translated statement, also works over the reals and for some families of non-isolated singularities. Some applications to study the singularities at infinity of complex polynomials are given.

Let X be a real Banach space. A set K ⊆ X is called a total cone if it is closed under addition and non-negative scalar multiplication, does not contain both x and −x for any non-zero x∈X, and is such that K−K := {x − y : x, y ∈ K} is dense in X. Suppose that T is a bounded linear operator on X which leaves a closed total cone K invariant. We denote by σ(T) and r(T) the spectrum and spectral radius of T.

Krein and Rutman [5] showed that if T is compact, r(T) > 0 and K is normal (that is, inf{∥x+y∥ : x, y ∈ K, ∥x∥ = ∥y∥ = 1} > 0), then r(T) is an eigenvalue of T with an eigenvector in K. This result was later extended by Nussbaum [6] to any bounded operator T such that re(T) < r(T), where re(T) denotes the essential spectral radius of T, without the hypothesis of normality. The more general question of whether r(T) ∈ σ(T) for all bounded operators T was answered in the negative by Bonsall [1], who as well as giving counterexamples described a property of K called the bounded decomposition property, which is sufficient to guarantee that r(T) ∈ σ(T).

More recently, Toland [8] showed that if X is a separable Hilbert space and T is self-adjoint, then r(T) ∈ σ(T), without any extra hypotheses on K. In this paper we extend Toland's results to normal operators on Hilbert spaces, removing in passing the separability hypothesis.

Let ϕ[ratio ]G→G be an endomorphism of a finitely generated residually finite group. R. Hirshon asked if there exists n such that the restriction of ϕ to ϕn(G) is injective. We give an example to show that this is not always the case.

Let T=(T1, …, Tn) be a commuting n-tuple of bounded linear operators acting on some complex Banach space [Xscr ]. We show that if T has the single-valued extension property, then the local spectrum σT(x) coincides with the spectrum σ(T), for all vectors x∈[Xscr ], except on a set of the first Baire category.

A deterministic self-similar Cantor set F defines a natural probability space, the elements of which are neighbourhoods of various sizes of all points in F. Considering random variables on this probability space, some porosity and density parameters are explicitly calculated.

In the representation theory of finite groups, it is useful to know which ordinary irreducible representations remain irreducible modulo a prime p. For the symmetric groups [Sfr ]n, this amounts to determining which Specht modules are irreducible over a field of characteristic p. Throughout this note we work in characteristic 2, and in this case we classify the irreducible Specht modules, thereby verifying the conjecture in [3, p. 97].

Recall that a partition is 2-regular if all of its non-zero parts are distinct; otherwise the partition is 2-singular. The irreducible Specht modules Sλ with λ a 2-regular partition were classified in [2]. Let λ′ denote the partition conjugate to λ. If Sλ is irreducible, then Sλ′ is irreducible, since Sλ′ is isomorphic to the dual of Sλ tensored with the sign representation. It turns out that if neither λ nor λ′ is 2-regular, then Sλ is irreducible only if λ = (2, 2).

In order to state our theorem, we let l(k), for an integer k, be the least non-negative integer such that k<2l(k).

We establish the existence of self-homeomorphisms of Rn, n [ges ] 2, which are chaotic in the sense of Devaney, preserve volume and are spatially periodic. Moreover, we show that in the space of volume-preserving homeomorphisms of the n-torus with mean rotation zero, those with chaotic lifts to Rn are dense, with respect to the uniform topology. An application is given for fixed points of 2-dimensional torus homeomorphisms (Conley–Zehnder–Franks Theorem).

In 1938, I. J. Schoenberg asked for which positive numbers p is the function exp(−∥x∥p) positive definite, where the norm is taken from one of the spaces [lscr ]np, q>2. The solution of the problem was completed in 1991, by showing that for every p∈(0, 2], the function exp(−∥x∥p) is not positive definite for the [lscr ]nq norms with q>2 and n[ges ]3. We prove a similar result for a more general class of norms, which contains some Orlicz spaces and q-sums, and, in particular, present a simple proof of the answer to Schoenberg's original question. Some consequences concerning isometric embeddings in Lp spaces for 0<p[les ]2 are also discussed.

We solve the functional equations

formula here

for suitable functions f, g and h subject to x + y + z = 0. These equations essentially characterise the Weierstrass [pscr ]-function and its degenerations.

If two finitely generated, torsion-free, nilpotent groups of class two satisfy the two-arrow property (that is, they embed into each other with finite, relatively prime indices), then they necessarily belong to the same Mislin genus (that is, they have isomorphic localizations at every prime). Here we show that the other implication is false in general. We even provide counterexamples in the case where both groups have isomorphic localizations at every finite set of primes of bounded cardinality. The latter equivalence relation leads us to introduce the notion of n-genus for every positive integer n, which we show to be meaningful in various contexts. In particular, the two-arrow property is related to the n-genus in the context of topological spaces.

Let Γ be a lattice in a connected Lie group. We show that, besides a few exceptional cases, the deficiency of Γ is nonpositive.

We prove that a large and natural class of subhomogeneous C*-algebras with one-dimensional spectrum have the fragility property, that is, every morphism out of such a C*-algebra and into a C*-algebra of real rank zero can be approximated by a morphism with finite-dimensional range, provided that a K-theoretical obstruction vanishes.

Every ε-isometry between real normed spaces of the same finite dimension which maps the origin to the origin may be uniformly approximated to within 2ε by a linear isometry. Under a smoothness hypothesis, necessary and sufficient conditions are obtained for the same conclusion to hold for a given ε-isometry between infinite-dimensional Banach spaces.

Oscillation criteria for the first-order linear delay differential equation

formula here

are established. Our results improve the known results in the literature.

If two operator algebras A and B are strongly Morita equivalent (in the sense of [5]), then their C*- envelopes C*(A) and C*(B) are strongly Morita equivalent (in the usual C*-algebraic sense due to Rieffel). Moreover, if Y is an equivalence bimodule for a (strong) Morita equivalence of A and B, then the operation, Y[otimes ]hA−, of tensoring with Y, gives a bijection between the boundary representations of C*(A) for A and the boundary representations of C*(B) for B. Thus the ‘noncommutative Choquet boundaries’ of Morita equivalent A and B are the same. Other important objects associated with an operator algebra are also shown to be preserved by Morita equivalence, such as boundary ideals, the Shilov boundary ideal, Arveson's property of admissability, and the lattice of C*-algebras generated by an operator algebra.

Throughout this paper, we shall let Σ be a subset of [0, 1] having cardinality N. We shall consider Σ to be a set of slopes, and for any s∈Σ, we shall let es be the unit vector of slope s in ℝ2. Then, following [7], we define the maximal operator on ℝ2 associated with the set Σ by

formula here

The history of the bounds obtained on [Mfr ]0Σ is quite curious. The earliest study of related operators was carried out by Cordoba [2]. He obtained a bound of C√(1+logN) on the L2 operator norm of the Kakeya maximal operator over rectangles of length 1 and eccentricity N. This operator is analogous to [Mfr ]0Σ with

formula here

However, for arbitrary sets Σ, the best known result seems to be C(1+logN). This follows from Lemma 5.1 in [1], but a point of view which produces a proof appears already in [8]. However, in this paper, we prove the following.

Let Ω be an open subset of IRd, and let Tp for p∈[1, ∞) be consistent C0-semigroups given by kernels that satisfy an upper heat kernel estimate. Denoting their generators by Ap, we show that the spectrum σ(Ap) is independent of p∈[1, ∞). We also treat the case of weighted Lp-spaces for weights that satisfy a subexponential growth condition. An example shows that independence of the spectrum may fail for an exponential weight.

It is shown that every transcendental entire function f grows transcendentally in a sequence of cercles de remplissage. An example shows that if

formula here

then there may be no sequence of cercles de remplissage the union of which contains infinitely many zeros of f. It is also shown that every transcendental entire function f has a Hayman direction, that is, a direction θ such that, in every open sector containing θ, either f assumes all complex values infinitely often, or else every derivative of f assumes all complex values, except possibly zero, infinitely often.