The Hessian measures of a (semi-)convex function can be introduced as coefficients of a local Steiner formula. The investigation of Hessian measures is continued by the provision of a geometric characterization of the support of these measures. Then the Radon–Nikodym derivative and the absolute continuity of Hessian measures with respect to Lebesgue measure are explored. As special cases of the results, known results for surface area measures of convex bodies are recovered.

Large classes of integrable Hamiltonian systems can be expressed as systems over coadjoint orbits in a loop algebra defined over a semi-simple Lie algebra [gfr ]. These systems can then be integrated via the classical, symplectic Liouville–Arnold method. On the other hand, the existence of spectral curves as constants of motion allows one to integrate these systems in terms of flows of line bundles on the curves. This note links the symplectic geometry of the coadjoint orbits with the algebraic geometry of these curves for arbitrary semi-simple [gfr ], which then allows us to reconcile the two integration methods.

For normalised generalized matrix functions f and g, we say that f dominates g if f(A)[ges ]g(A) for every M-matrix A. We first demonstrate a finite set of test matrices for any such inequality. Then, using results from group representation theory, all comparisons among immanants in certain classes are determined. This work parallels ongoing research into gmf inequalities on positive semidefinite matrices, for which no finite set of test matrices is available. However, the inequalities for the two classes are quite different, and the test matrices permit more rapid progress in the M-matrix case. Just as in the positive semidefinite case, the gmf inequalities we prove may be used to verify previously unknown determinantal inequalities for M-matrices, such as the symmetrized Fischer inequalities recently proved in the positive semidefinite case.

It is proved that the following classes of finitely generated groups have $\Pi_1^1$-complete first-order theories: all finitely generated groups, the $n$-generated groups, and the strictly $n$-generated groups ($n\,{\geqslant}\,2$). Moreover, all those theories are distinct. Similar techniques show that quasi-finitely axiomatizable groups have a hyperarithmetical word problem, where a finitely generated group is quasi-finitely axiomatizable if it is the only finitely generated group satisfying an appropriate first-order sentence. The Turing degrees of word problems of quasi-finitely axiomatizable groups form a cofinal set in the Turing degrees of hyperarithmetical sets.

All groups and rings in this paper are finite, and p denotes a prime number.

R. Shepherd [28] and C. R. Leedham-Green and S. McKay [16] showed that any p-group of maximal class contains a subgroup of class at most 2 the index of which is bounded above by a function of p. These papers gave rise to a program for the classification of p-groups that used the notion of coclass proposed by C. R. Leedham-Green and M. F. Newman [21] in 1980. They made several conjectures. The strongest conjecture, Conjecture A, asserted that every p-group of coclass r contains a subgroup of class at most 2 the index of which is bounded above by a function depending only on p and r. These conjectures were proved in a long series of papers by C. R. Leedham-Green, S. McKay, S. Donkin, W. Plesken, A. Shalev and E. Zelmanov (cf. [3, 16–20, 27]). In a recent paper, A. Shalev [26] gave a proof of Conjecture A for all primes p with explicit bounds.

A classical space and a nonstandard space are used as models for white noise. The expectation of the product of certain generalized functionals is first proved to be finite. Then the covariance and the correlation between Donsker's delta functions are calculated and given meaning.

Let F be a non-lattice distribution function which lies in the domain of attraction of a non-normal stable distribution. Exact uniform convergence rates are obtained for the convergence of the normalised partial sums of random variables with distribution F. Second order regular variation conditions are assumed related to the domain of attraction conditions.

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.

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.

A necessary and sufficient condition for the almost sure existence of the exact packing measure on the boundary of a Galton–Watson tree is obtained with an explicit exact packing measure. In the case where the exact packing measure does not exist almost surely, a criterion which classifies functions $\ph$ according to whether the $\ph$ packing measure is zero or infinity is given.

Lower bounds are established for the number of zeros of the logarithmic derivative of a meromorphic function of small growth with few poles.

The main theorem shows that whenever certain amalgamated products act geometrically on a CAT(0) space, the space has non-locally connected boundary. One can now easily construct a wide variety of examples of one-ended CAT(0) groups with non-locally connected boundary. Applications of this theorem to right-angled Coxeter and Artin groups are considered. In particular, it is shown that the natural CAT(0) space on which a right-angled Artin group acts has locally connected boundary if and only if the group is ℤn for some n.

We characterise the profinite groups with polynomial subgroup growth. We deduce that the property PSG is extension-closed in the category of all groups, and subgroup-closed in the category of finitely generated profinite groups.

Ritt has shown that any complex polynomial p can be written as the composition of polynomials p1,…,pm, where each pj is prime in the sense that it cannot be written as a non-trivial composition of polynomials. The factors pj are not unique but the number m of them is, as is the set of the degrees of the pj. The paper extends Ritt's theory and, in particular, a third invariant of the decomposition is introduced.

A criterion is given for showing that certain one-relator groups are residually finite. This is applied to a one-relator group with torsion $G = \langle a_1, \ldots, a_r \mid W^n\rangle$ . It is shown that $G$ is residually finite provided that $W$ is outside the commutator subgroup and $n$ is sufficiently large. An important ingredient in the proof is a criterion which implies that a subgroup of a group is malnormal. A graded small-cancellation criterion is developed which detects whether a map $A \rightarrow B$ between graphs induces a $\pi_1$ -injection, and whether $\pi_1 A$ maps to a malnormal subgroup of $\pi_1 B$ .

A new coupling of one-dimensional random walks is described which tries to control the coupling by keeping the separation of the two random walks of constant sign. It turns out that among such monotone couplings there is an optimal one-step coupling which maximises the second moment of the difference (assuming this is finite), and this coupling is ‘fast’ in the sense that for a random walk with a unimodal step distribution the coupling time achieved by using the new coupling at each step is stochastically no larger than any other coupling. This is applied to the case of symmetric unimodal distributions.

To a finitely generated profinite group $G$, a formal Dirichlet series $P_G(s)\,{=}\,\sum_{n}{a_n}/{n^s}$ is associated, where $a_n \,{=}\,\sum_{|G:H|=n} \mu_G(H)$. It is proved that $G$ is prosoluble if and only if the sequence $\{a_n\}_{n \in \mathbb N}$ is multiplicative, that is, $a_{rs}\,{=}\,a_ra_s$ for any pair of coprime positive integers $r$ and $s$. This extends the analogous result on the probabilistic zeta function of finite groups.

Several new multiple-integral representations are proved for well-poised hypergeometric series and integrals. The results yield, in particular, transformations of the multiple integrals that cannot be achieved by evident changes of variable. All this generalizes some classical results of Whipple and Bailey in analysis and, on the other hand, certain analytic constructions with known connection to irrationality proofs for values of Riemann's zeta function at positive integers.

Let A be a Banach algebra, and let E be a Banach A-bimodule. A linear map S:A→E is intertwining if the bilinear map

formula here

is continuous, and a linear map D[ratio ]A→E is a derivation if δ1D=0, so that a derivation is an intertwining map. Derivations from A to E are not necessarily continuous.

The purpose of the present paper is to prove that the continuity of all intertwining maps from a Banach algebra A into each Banach A-bimodule follows from the fact that all derivations from A into each such bimodule are continuous; this resolves a question left open in [1, p. 36]. Indeed, we prove a somewhat stronger result involving left- (or right-) intertwining maps.