The problem of completing a partially prescribed Toeplitz matrix to a positive definite Toeplitz matrix is considered. The main result is a characterization of the patterns of specified entries that guarantee the existence of a positive definite Toeplitz completion of a partial positive definite Toeplitz matrix. This characterization resolves affirmatively a conjecture of C. R. Johnson and L. Rodman.

Several results about the multiplicities of indecomposable injectives in the minimal injective resolution of a ring exist in the literature. Mostly these apply to universal enveloping algebras of finite dimensional solvable Lie algebras, and to Gorenstein noetherian PI local rings. These results are unified and extended to the much wider class of rings with Auslander dualizing complexes.

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 $f$ be a conformal map of the unit disk ${\bb D}$ onto the domain $G \subset \hat{\bb C} = {\bb C} \cup \{\infty\}$ . We shall always use the spherical metric in $\hat{\bb C}$ .

Carathéodory [3] introduced the concept of a prime end of $G$ in order to describe the boundary behaviour of $f$ in geometric terms; see for example [6, Chapter 9] or [12, Section 2.4]. There is a bijective map $\hat{f}$ of ${\bb T} = \partial {\bb D}$ onto the set of prime ends of $G$ .

A certain property of some type-definable subgroups of superstable groups with finite U-rank is closely related to the Mordell–Lang conjecture. This property is discussed in the context of algebraic groups.

It is proved that Thom spectra of generalized braid groups are the wedges of suspensions over the Eilenberg–MacLane spectrum for ℤ/2. The precise structure of the Thom spectra of the generalized braid groups of the types C and D is obtained. For the generalized braid groups of type C the natural pairing analogous to the pairing of the classical braids is studied. This pairing generates the multiplicative structure of the Thom spectrum such that the corresponding bordism theory has the coefficient ring isomorphic to the polynomial ring over ℤ/2 on one generator of dimension 1[ratio ]ℤ/2[s].

B. Hartley in [4] introduced the class of barely transitive groups. By definition, a group of permutations G on an infinite set X is called barely transitive if G itself is transitive on X while every orbit of every proper subgroup of G is finite. If G is locally finite and G′≠G then the theorem of B. Love, proved in [4], shows that G is a locally nilpotent p-group of Heineken–Mohamed type. However, it is not known if perfect barely transitive locally nilpotent p-groups exist. Obviously this is a more general question than the corresponding one about perfect minimal non FC-groups (see below for definition). In this work it will be shown that a barely transitive locally nilpotent p-group cannot be perfect if the stabilizer of a point is hypercentral and solvable.

Let $C_A/\Q$ be the curve $y^2 = x^5 + A$, and let $L(s, J_A)$ denote the $L$-series of its Jacobian. Under the assumption that the sign in the functional equation for $L(s, J_A)$ is $+1$, the critical value $L(1, J_A)$ is evaluated in terms of the value of a theta series for $\Q(\sqrt{5})$ depending on $A$ at a complex multiplication point coming from $\Q(\zeta_5)$.

The purpose of the paper is to introduce a new linearization method for local well-posedness of nonlinear evolution equations. This approach is based on an implicit function theorem of Nash–Moser type. The technique is illustrated by an application to a general class of fully nonlinear parabolic partial differential equations of arbitrary order on ℝn. The estimates required by the Nash–Moser technique are derived for the higher order Sobolev norms of the solutions of the linearized parabolic equation using semigroup theory and elliptic theory. In particular, a priori estimates, resolvent estimates and commutator estimates are involved. The general method based on a combination of Nash–Moser techniques with semigroup theory is applicable to other problems and has already been used to prove short-time solvability for some nonlinear Schrödinger type equation. This approach might be useful in other situations as well since it compensates for a loss of derivatives in the estimates of the solutions of the linearized equation.

The notion of a group of finitary automorphisms of an arbitrary module over an arbitrary ring is introduced, and it is shown how properties of such groups can be derived from the case where the ring is a division ring (that is, from the properties of finitary skew linear groups). The results are particularly strong if either the group is locally finite or the module is Noetherian.

We produce examples of Legendrian isotopy classes in $ST^*\mathbb{R}^n$ (the space of cooriented contact elements of $\mathbb{R}^n$, endowed with its standard contact structure) such that any element of those classes has a singular (non-immersed) wave front, that is, the projection to the base of the fibration $ST^*\mathbb{R}^n \rightarrow \mathbb{R}^n$, when restricted to such a Legendrian embedding, has local singularities. Furthermore, our examples are such that the underlying Legendrian homotopy classes contain some Legendrian embeddings whose wave fronts are immersed. These examples are motivated by a question of Arnold about the removability of singularities of wave fronts by means of Legendrian isotopy. The two key points, which are of independent interest, are as follows.

(1) A duality argument which allows the question to be translated into a problem about Legendrian submanifolds of the one-jets space of the sphere, and hence to use the technology of generating families.

(2) Several independent constructions of families of functions on compact manifolds such that the ‘critical sets’ of these families are connected. For example, we prove that given two compact, connected manifolds $M$ and $N$ of positive dimension, there exists

\[f~: N \times M \rightarrow \mathbb{R}, (x,y) \rightarrow f(x,y),\quad x\in N, \ y \in M\]

such that the equation $(\partial f/\partial y)(x,y)=0$ is regular and defines a smooth and connected submanifold of $M \times N$.

It is shown that the general plane section of a double structure on an integral curve C⊆P3 has a connected numerical character (that is, its Hilbert function is of decreasing type). The paper gives applications, particularly to the stability of the normal bundle of space curves.

A sufficient condition is given under which the sum, product and indeed any polynomial combination of a well-bounded operator and a commuting real scalar-type spectral operator is well-bounded. This generalizes a result of Gillespie for Hilbert space operators. It is shown in particular that if $X$ is a UMD space, then the sum of finitely many commuting real scalar-type spectral operators acting on $X$ is a well-bounded operator (a result which fails on general reflexive Banach spaces).

For a convex body $K$ in $\R^d$ and $1\le k\le d-1$, let $P_k(K)$ be the Minkowski sum (average) of all orthogonal projections of $K$ onto $k$-dimensional subspaces of $\R^d$. It is known that the operator $P_k$ is injective if $k\,{\geq}\, d/2$, $k\,{=}\,3$ for all $d$, and if $k=2$, $d\ne 14$.

It is shown that $P_{2k}(K)$ determines a convex body $K$ among all centrally symmetric convex bodies and $P_{2k+1}(K)$ determines a convex body $K$ among all bodies of constant width. Corresponding stability results are also given. Furthermore, it is shown that any convex body $K$ is determined by the two sets $P_k(K)$ and $P_{k'}(K)$ if $1<k<k'$. Concerning the range of $P_k$, $1\le k\le d-2$, it is shown that its closure (in the Hausdorff-metric) does not contain any polytopes other than singletons.

A fundamental aspect of the study of Diophantine equations is that of determining when an equation has a local solution. Artin once conjectured (see the preface to [1]) that if k is a complete, discretely valued field with finite residue class field, then every homogeneous form of degree d in greater than d2 variables whose coefficients are integers of k has a nontrivial zero. In this paper, we consider the case of this conjecture in which k is a p-adic field. Although a counterexample due to Terjanian [16] proved Artin's conjecture false in this situation, Ax and Kochen [2] have shown when [k:Qp] = n is finite, that given d, there exists a number p(d, n) such that Artin's conjecture is true provided that p is larger than p(d, n). Unfortunately, the methods of Ax and Kochen do not lead to explicit estimates for p(d, n). Cohen [5] found a method which determines the possible cardinalities of the residue class fields of all p-adic fields for which Artin's conjecture is false, and Brown [3] has used this to bound p(d, 1), but this bound is so large that one feels that it must be possible to do better. Hence, it is still an interesting problem to obtain estimates on the size of p(d, n).

The number of crepant valuations of an isolated canonical singularity $0\in X:(f\,{=}\,0)\,{\subset}\,\mathbb{C}^{n}$, which is assumed to be nondegenerate with respect to its Newton polyhedron, is calculated in terms of weightings and the Newton polyhedron of $f$.

There is a homotopy equivalence ϕ:M→M′ between closed smooth manifolds of an odd dimension such that ϕ*TM′, TM are stably isomorphic but not isomorphic to each other.

Until recently, the known symmetric nets were class-regular and therefore satisfied a certain geometric condition that defines the class of nets known as tactical symmetric nets. Thus the known symmetric nets were tactical. This paper gives a general method of construction for non-tactical symmetric nets, which are therefore not class-regular. It is not known whether there are tactical symmetric nets which are not class regular.

Criteria for selfadjointness of integral operators, including matrix operators, based on our earlier domination results are established. The range of applicability is elucidated by carefully chosen examples, not covered by previously known criteria.

In [17, 18, 19], we began to investigate the continuity properties of homomorphisms from (non-abelian) group algebras. Already in [19], we worked with general intertwining maps [3, 12]. These maps not only provide a unified approach to both homomorphisms and derivations, but also have some significance in their own right in connection with the cohomology comparison problem [4].

The present paper is a continuation of [17, 18, 19]; this time we focus on groups which are connected or factorizable in the sense of [26]. In [26], G. A. Willis showed that if G is a connected or factorizable, locally compact group, then every derivation from L1(G) into a Banach L1(G)-module is automatically continuous. For general intertwining maps from L1(G), this conclusion is false: if G is connected and, for some n∈ℕ, has an infinite number of inequivalent, n-dimensional, irreducible unitary representations, then there is a discontinuous homomorphism from L1(G into a Banach algebra by [18, Theorem 2.2] (provided that the continuum hypothesis is assumed). Hence, for an arbitrary intertwining map θ from L1(G), the best we can reasonably hope for is a result asserting the continuity of θ on a ‘large’, preferably dense subspace of L1(G). Even if the target space of θ is a Banach module (which implies that the continuity ideal [Iscr ](θ) of θ is closed), it is not a priori evident that θ is automatically continuous: the proofs of the automatic continuity theorems in [26] rely on the fact that we can always confine ourselves to restrictions to L1(G) of derivations from M(G) [25, Lemmas 3.1 and 3.4]. It is not clear if this strategy still works for an arbitrary intertwining map from L1(G) into a Banach LL1(G)-module.