The results here generalise [2, Proposition 4.3] and [9, Theorem 5.11]. We shall prove the following.

THEOREM A. Let R be a Noetherian PI-ring. Let P be a non-idempotent prime ideal of R such that PRis projective. Then P is left localisable and RPis a prime principal left and right ideal ring.

We also have the following theorem.

THEOREM B. Let R be a Noetherian PI-ring. Let M be a non-idempotent maximal ideal of R such that MRis projective. Then M has the left AR-property and M contains a right regular element of R.

Almkvist proved that for a commutative ring A the characteristic polynomial of an endomorphism α : P → P of a finitely generated projective A-module determines (P, α) up to extensions. For a non-commutative ring A the generalized characteristic polynomial of an endomorphism of an endomorphism α : P → P of a finitely generated projective A-module is defined to be the Whitehead torsion [1 − xα] ∈ K1(A[[x]]), which is an equivalence class of formal power series with constant coefficient 1.

The paper gives an example of a non-commutative ring A and an endomorphism α : P → P for which the generalized characteristic polynomial does not determine (P, α) up to extensions. The phenomenon is traced back to the non-injectivity of the natural map [sum ]−1A[x] → A[[x]] where [sum ]−1A[x] is the Cohn localization of A[x] inverting the set [sum ] of matrices in A[x] sent to an invertible matrix by A[x] → A;x [map ] 0.

Let C be a smooth proper curve of genus 2 over an algebraically closed field k. Fix a Weierstrass point ∞in C(k) and identify C with its image in its Jacobian J under the Albanese embedding that uses ∞ as base point. For any integer N[ges ]1, we write JN for the group of points in J(k) of order dividing N and J*N for the subset of JN of points of order N. It follows from the Riemann–Roch theorem that C(k)∩J2 consists of the Weierstrass points of C and that C(k)∩J*3 and C(k)∩J* are empty (see [3]). The purpose of this paper is to study curves C with C(k)∩J*5 non-empty.

The most powerful geometric tools are those of differential geometry, but to apply such techniques to finitely generated groups seems hopeless at first glance since the natural metric on a finitely generated group is discrete. However Gromov recognized that a group can metrically resemble a manifold in such a way that geometric results about that manifold carry over to the group [18, 20]. This resemblance is formalized in the concept of a ‘quasi-isometry’. This paper contributes to an ongoing program to understand which groups are quasi-isometric to which simply connected, homogeneous, Riemannian manifolds [15, 18, 20] by proving that any group quasi-isometric to H2×R is a finite extension of a cocompact lattice in Isom(H2×R) or Isom(SL˜(2, R)).

Let [hfr ] be a reductive subalgebra of a semisimple Lie algebra [gfr ] and C[hfr ] ∈ U([hfr ]) be the Casimir element determined by the restriction of the Killing form on [gfr ] to [hfr ]. The paper studies eigenvalues of C[hfr ] on the isotropy representation [mfr ]≃[gfr ]/[hfr ]. Some general estimates connecting the eigenvalues and the Dynkin indices of [mfr ] are given. If [hfr ] is a symmetric subalgebra, it is shown that describing the maximal eigenvalue of C[hfr ] on exterior powers of [mfr ] is connected with possible dimensions of commutative Lie subalgebras in [mfr ], thereby extending a result of Kostant. In this situation, a formula is also given for the maximal eigenvalue of C[hfr ] on ∧ [mfr ]. More generally, a similar picture arises if [hfr ] = [gfr ]Θ, where Θ is an automorphism of finite order m and [mfr ] is replaced by the eigenspace of Θ corresponding to a primitive mth root of unity.

Let f be a continuous function on an open subset Ω of ℝ2 such that for every x ∈ Ω there exists a continuous map γ : [−1, 1] → Ω with γ(0) = x and f ∘ γ increasing on [−1, 1]. Then for every γ ∈ Ω there exists a continuous map γ : [0, 1) → Ω such that γ(0) = y, f ∘ γ is increasing on [0; 1), and for every compact subset K of Ω, max{t : γ(t) ∈ K} < 1. This result gives an answer to a question posed by M. Ortel. Furthermore, an example shows that this result is not valid in higher dimensions.

On page 212 of his lost notebook, Ramanujan defined a new class invariant λn and constructed a table of values for λn. The paper constructs a new class of series for 1/π associated with λn. The new method also yields a new proof of the Borweins' general series for 1/π belonging to Ramanujan's ‘theory of q2’.

We study, on the entire space ℝN(N [ges ] 1), the diffusive logistic equation

and its generalizations. Here p > 1 is a constant. Problem (1.1) plays an important role in understanding various population models and some other problems in applied mathematics. When λ = 1 and p = 2, it is also known as the Fisher equation and KPP equation, due to the pioneering works of Fisher [8] and Kolmogoroff, Petrovsky and Piscounoff [18].

The paper studies the periodic and anti-periodic eigenvalues of the one-dimensional p-Laplacian with a periodic potential. After a rotation number function ρ(λ) has been introduced, it is proved that for any non-negative integer n, the endpoints of the interval ρ−1(n/2) in ℝ yield the corresponding periodic or anti-periodic eigenvalues. However, as in the Dirichlet problem of the higher dimensional p-Laplacian, it remains open if these eigenvalues represent all periodic and anti-periodic eigenvalues. The result obtained is a partial generalization of the spectrum theory of the one-dimensional Schrödinger operators with periodic potentials.

It is proved that if X and Y are operator spaces such that every completely bounded operator from X into Y is completely compact and Z is a completely complemented subspace of X [oplus ] Y, then there exists a completely bounded automorphism τ: X [oplus ] Y → X [oplus ] Y with completely bounded inverse such that τZ = X0 [oplus ] Y0, where X0 and Y0 are completely complemented subspaces of X and Y, respectively. If X and Y are homogeneous, the existence is proved of such a τ under a weaker assumption that any operator from X to Y is strictly singular. An upper estimate is obtained for ∥τ∥cb∥τ−1∥cb if X and Y are separable homogeneous Hilbertian operator spaces. Also proved is the uniqueness of a ‘completely unconditional’ basis in X [oplus ] Y if X and Y satisfy certain conditions.

The paper studies the relation between the asymptotic values of the ratios area/length (F/L) and diameter/length (D/L) of a sequence of convex sets expanding over the whole hyperbolic plane. It is known that F/L goes to a value between 0 and 1 depending on the shape of the contour. In the paper, it is first of all seen that D/L has limit value between 0 and 1/2 in strong contrast with the euclidean situation in which the lower bound is 1/π (D/L = 1/π if and only if the convex set has constant width). Moreover, it is shown that, as the limit of D/L approaches 1/2, the possible limit values of F/L reduce. Examples of all possible limits F/L and D/L are given.

Let V and X be Hausdorff, locally convex, real, topological vector spaces with dim V > 1. It is shown that a map σ from an open, connected subset of V onto an open subset of X is homeomorphic and convexity-preserving if and only if σ is projective.

Let X be a locally compact metric absolute neighbourhood retract for metric spaces, U ⊂ X be an open subset and f: U → X be a continuous map. The aim of the paper is to study the fixed point index of the map that f induces in the hyperspace of X. For any compact isolated invariant set, K ⊂ U, this fixed point index produces, in a very natural way, a Conley-type (integer valued) index for K. This index is computed and it is shown that it only depends on what is called the attracting part of K. The index is used to obtain a characterization of isolating neighbourhoods of compact invariant sets with non-empty attracting part. This index also provides a characterization of compact isolated minimal sets that are attractors.

Let M1 and M2 be two simply connected closed manifolds of the same dimension. It is proved that

(1) if k is a coefficient field such that neither M1 nor M2 has the same cohomology as a sphere, then the sequence (bk)k[ges ]1 of Betti numbers of the free loop space on M1 #M2 is unbounded;

(2) if, moreover, the cohomology H*(M1;k) is not generated as algebra by only one element, then the sequence (bk)k[ges ]1 has an exponential growth.

Thanks to theorems of Gromoll and Meyer and of Gromov, this implies, in case 1, that there exist infinitely many closed geodesics on M1#M2 for each Riemannian metric, and, in case 2, that for a generic metric, the number of closed geodesics of length [les ]t grows exponentially with t.

Let ([sum ]A, T) be a topologically mixing subshift of finite type on an alphabet consisting of m symbols and let Φ:[sum ]A → Rd be a continuous function. Denote by σΦ(x) the ergodic limit limn→∞n−1 [sum ]n−1j=0 Φ(Tjx) when the limit exists. Possible ergodic limits are just mean values ∫ Φdμ for all T-invariant measures. For any possible ergodic limit α, the following variational formula is proved:

[formula here]

where hμ denotes the entropy of μ and htop denotes topological entropy. It is also proved that unless all points have the same ergodic limit, then the set of points whose ergodic limit does not exist has the same topological entropy as the whole space [sum ]A

It is shown that many hyperbolic monopoles can be distinguished from each other via their asymptotic values in contrast to the case of Euclidean monopoles.