I exhibit a pair of non-symmetric operads that, while not themselves isomorphic, induce isomorphic monads. The existence of such a pair implies that if ‘algebraic theory’ is understood as meaning ‘monad’ then operads cannot be regarded as algebraic theories of a special kind.

Higher algebraic K-theoretic analogues of Hasse's norm theorem in class field theory are proved. The result gives a partial affirmative answer to a question posed by Bak and Rehmann.

An alternative proof is given of a result, originally due to Guido Mislin, giving necessary and sufficient conditions for the inclusion of a subgroup to induce an isomorphism in mod p cohomology.

Let $H$ be a not necessarily separable Hilbert space, and let $\mathcal{B}(H)$ denote the space of all bounded linear operators on $H$. It is proved that a commutative lattice $\mathcal{D}$ of self-adjoint projections in $H$ that contains $0$ and $I$ is spatially complete if and only if it is a closed subset of $\mathcal{B}(H)$ in the strong operator topology. Some related results are obtained concerning commutative lattice-ordered cones of self-adjoint operators that contain $\mathcal{D}$.

In this paper the authors develop a new approach to the problem of ‘propagation of smallness’ for harmonic functions in arbitrary domains, in ℝn (n[ges ]2). The main result of this paper is a certain logarithmic-convexity relation for the L2-norms of harmonic functions. As a consequence, new kinds of uniqueness results for harmonic functions are obtained. The method works also for analytic functions in [Copf ], with Lp-norms (p>0).

The notions of controlled truncations for operators in the Roe algebras $C^*(X)$ of a coarse space $(X, \cal{E})$ with uniformly locally finite coarse structure, and rank distributions on $(X, \cal{E})$ are introduced. It is shown that the controlled propagation operators in an ideal $I$ of $C^*(X)$ are exactly the controlled truncations of elements in $I$. It follows that the lattice of the ideals of $C^*(X)$ in which controlled propagation operators are dense is isomorphic to the lattice of all rank distributions on $(X, \cal{E})$. If $X$ is a discrete metric space with Yu's property A, then the ideal structure of the Roe algebra $C^*(X)$ is completely determined by the rank distributions on $X$.

We give a theoretical lower bound for the slope of a Siegel modular cusp form that is as least as good as Eichler's lower bound. In degrees $n=5,6$ and 7 we show that our new bound is strictly better. In the process we find the forms of smallest dyadic trace on the perfect core for ranks $n \le 8$. In degrees $n=5,6$ and 7 we settle the value of the generalized Hermite constant $\gamma_n'$ introduced by Bergé and Martinet and find all dual-critical pairs.

Tao has shown that hard summation (summing Fourier-type series by using terms in order of decreasing size of the Fourier coefficients) works for wavelets, but the present author has shown that it fails for classical Fourier series. This paper, which is intended for a general audience, exhibits the underlying ideas in the context of Haar and Walsh series, where many of the proofs simplify.

The following result is established.

THEOREM. Let G be a periodic, residually finite group with all subgroups sub-normal. Then G is nilpotent.

The well-known groups of Heineken and Mohamed [1] show that the hypothesis of residual finiteness cannot be omitted here, while examples in [5] show that a residually finite group with all subgroups subnormal need not be nilpotent. The proof of the Theorem will use the results of Möhres that a group with all subgroups subnormal is soluble [3] and that a periodic hypercentral group with all subgroups subnormal is nilpotent [4]. Borrowing an idea from [2], the plan is to construct certain subgroups H and K that intersect trivially, and to show that the subnormality of both leads to a contradiction.

The topological disc (De Paepe's) \[ P=\{(z^2,\bar{z}^2+\bar{z}^3):|z|\le 1\}\subset {\bb C}^2 \]

is shown here to have non-trivial polynomially convex hull. In fact, the authors show that this holds for all discs of the form $X=\{(z^2,f(\bar{z})):|z|\le r\}$ , where $f$ is holomorphic on $|z|\le r$ , and $f(z)=z^2+a_3z^3+\ldots$ , with all coefficients $a_n$ real, and at least one $a_{2n+1}\ne 0$ .

A complete characterization of the categorical quotients of $({\mathbb P}^1)^n$ by the diagonal action of $\SL(2, {\mathbb C})$ with respect to any polarization is given by M. Polito, in ‘$\SL(2, {\mathbb C})$-quotients de $({\mathbb P}^1)^n$’, C. R. Acad. Sci. Paris Sér. I 321 (1995) 1577–1582. In this paper, these categorical quotients are obtained by certain linear systems on ${\mathbb P}^{n-3}$?>, depending on the given polarization.

An inequality is established involving colengths of the tight closure of ideals of systems of parameters in local rings with some mild conditions. As an application, a proof is given of a result due to Goto and Nakamura (first conjectured by Watanabe and Yoshida), which states that the Hilbert–Samuel multiplicity of a parameter ideal is greater than or equal to the colength of the tight closure of the ideal. The result is also further refined.

Let $J$ be a Jacobi real symmetric matrix on $l_{2}$ with zero diagonal and non-diagonal entries of the form $\{1+p_{n}\}$. If $p_{n-1}\pm p_{n}=O(n^{-\alpha})$ with some $\alpha>2/3$, then the existence of bounded solutions of $Ju=\lambda u$ is proved for almost every $\lambda\in(-2,2)$ with the WKB-type asymptotic behavior.

Let $p{>}2$ be a prime, and let $k$ be a field of characteristic zero, linearly disjoint from the $p$th cyclotomic extension of $\Q$. Given a projective Galois representation $\varrho \colon \Gal(\kbar/k) {\To} \PGL_2(\F_p)$ with cyclotomic determinant, two twists $X_\varrho(p)$ and $ X'_\varrho(p)$ of a certain rational model of the modular curve $X(p)$ can be attached to it. The $k$-rational points of these twists classify the elliptic curves $E/k$ such that ${\rhobar}_{E, p}{=}\varrho$, where ${\rhobar}_{E, p}$ denotes the projective Galois representation associated with the $p$-torsion module $E[p]$. The octahedral ($p{=}3$) and icosahedral ($p{=}5$) genus-zero cases are discussed in further detail.

We consider the following strengthening of Hadwiger's Conjecture.

Let G be any graph of chromatic number k, S any subset of V(G) which takes all k colours in each proper k-colouring of G. Then there are k pairwise adjacent connected subgraphs of G, each of whose vertex sets has non-trivial intersection with S.

We show that the truth of this conjecture for all graphs of chromatic number [les ]k implies the truth of Hadwiger's Conjecture for all graphs of chromatic number [les ]k + 1. We also show that its truth implies the following statement (which is at first sight even stronger).

For any graph G of chromatic number k and any subset S of V(G), define χ(S; G) to be the least number of colours that can appear on S in any proper k-colouring of G, and h(S; G) to be the largest number of pairwise adjacent connected subgraphs of G each having non-trivial intersection with S. Then χ(S; G) [les ]h(S; G).

We define the number w(S; G) to be the largest cardinality of a subset T of S such that, however T is partitioned into pairs (possibly with one spare element), there are vertex-disjoint paths linking the elements in each pair, none passing through the spare element if it exists. We show that χ(S; G) [les ]([mid ]S[mid ]+w(S; G))/2 for any graph G and subset S of V(G).

Finally, we show that for any graph G, χ(S; G) [les ]h(S; G) whenever χ(S; G) [les ]3.

Sufficient conditions are given for the global existence of solutions to the Rayleigh equation.

We study the behaviour of non-negative solutions to the equation

formula here

This equation is an ‘explicitly’ soluble special case of a class of non-local evolution equations, of which the behaviour of solutions has been studied by Uchiyama [9]. In this paper, certain fine properties of solutions such as a local relaxation within their supports are obtained for the present special case. The asymptotic forms (for large time) of the solutions whose initial measures have compact supports are identified within the error of the order O(t−3/2). It is also shown that the comparison principle does not hold. With a simple change of variables, the equation ([midast ]) is transformed to the equation arising in Dyson's approach to the Wigner semi-circle law of eigenvalues of random matrices, and the present results have immediate consequences on the latter.

Let $s:\Q\,{\longrightarrow}\,\Q$ be the Dedekind sum, given by $s(h/k)=\sum_{\nu=1}^{k-1}({\nu/k}\,{-}\,{1/2})(\{{h\nu/k}\}\,{-}\,{1/2})$ when $\gcd(h,k)\,{=}\,1$. Then for every rational $\alpha\,{\ne}\,1/12$ there are infinitely many rational $x$ such that $s(x)\,{=}\,\alpha x$. Also, the fixed points of $s$ are dense in the real line.

A criterion of isomorphism for symmetry groups of set ideals is provided in terms of ideal quotients and cardinal invariants. Furthermore, set ideals with complete symmetry group are characterized. They form a wide class, comprising, for example, all uniform dense ideals and the ideals of ‘thin’ sets in separable metric spaces. If the symmetry group of a set ideal is not complete, then its outer automorphism group is shown to be cyclic of order 2.